LMFDB - Embedded newform 144.2.k.b.109.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [144,2,Mod(37,144)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(144, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("144.37"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$144 = 2^{4} \cdot 3^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	144.k (of order $4$ , degree $2$ , minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$1.14984578911$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	8.0.18939904.2
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{8} - 4x^{7} + 14x^{6} - 28x^{5} + 43x^{4} - 44x^{3} + 30x^{2} - 12x + 2$ x^8 - 4x^7 + 14x^6 - 28x^5 + 43x^4 - 44x^3 + 30x^2 - 12*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$2^{2}$
Twist minimal:	no (minimal twist has level 48)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			109.1
Root			$0.500000 + 0.0297061i$ of defining polynomial
Character	$\chi$	$=$	144.109
Dual form			144.2.k.b.37.1

$q$ -expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.874559 - 1.11137i) q^{2} +(-0.470294 + 1.94392i) q^{4} +(0.334904 - 0.334904i) q^{5} -4.55765i q^{7} +(2.57172 - 1.17740i) q^{8} +(-0.665096 - 0.0793096i) q^{10} +(2.47363 - 2.47363i) q^{11} +(-0.0594122 - 0.0594122i) q^{13} +(-5.06524 + 3.98593i) q^{14} +(-3.55765 - 1.82843i) q^{16} -3.61706 q^{17} +(2.55765 + 2.55765i) q^{19} +(0.493523 + 0.808530i) q^{20} +(-4.91245 - 0.585786i) q^{22} -2.82843i q^{23} +4.77568i q^{25} +(-0.0140696 + 0.117988i) q^{26} +(8.85970 + 2.14343i) q^{28} +(5.16333 + 5.16333i) q^{29} -0.557647 q^{31} +(1.07931 + 5.55294i) q^{32} +(3.16333 + 4.01990i) q^{34} +(-1.52637 - 1.52637i) q^{35} +(4.38607 - 4.38607i) q^{37} +(0.605684 - 5.07931i) q^{38} +(0.466962 - 1.25559i) q^{40} +9.27391i q^{41} +(-1.61040 + 1.61040i) q^{43} +(3.64520 + 5.97186i) q^{44} +(-3.14343 + 2.47363i) q^{46} -2.82843 q^{47} -13.7721 q^{49} +(5.30755 - 4.17661i) q^{50} +(0.143434 - 0.0875513i) q^{52} +(0.493523 - 0.493523i) q^{53} -1.65685i q^{55} +(-5.36618 - 11.7210i) q^{56} +(1.22274 - 10.2540i) q^{58} +(-4.00000 + 4.00000i) q^{59} +(2.72922 + 2.72922i) q^{61} +(0.487695 + 0.619753i) q^{62} +(5.22746 - 6.05588i) q^{64} -0.0397948 q^{65} +(3.77568 + 3.77568i) q^{67} +(1.70108 - 7.03127i) q^{68} +(-0.361465 + 3.03127i) q^{70} -9.11529i q^{71} -0.541560i q^{73} +(-8.71044 - 1.03868i) q^{74} +(-6.17471 + 3.76901i) q^{76} +(-11.2739 - 11.2739i) q^{77} -10.9937 q^{79} +(-1.80382 + 0.579123i) q^{80} +(10.3068 - 8.11058i) q^{82} +(10.6417 + 10.6417i) q^{83} +(-1.21137 + 1.21137i) q^{85} +(3.19813 + 0.381362i) q^{86} +(3.44902 - 9.27391i) q^{88} -14.6533i q^{89} +(-0.270780 + 0.270780i) q^{91} +(5.49824 + 1.33019i) q^{92} +(2.47363 + 3.14343i) q^{94} +1.71313 q^{95} +4.31724 q^{97} +(12.0446 + 15.3060i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$8 q - 4 q^{4} + 12 q^{8} - 8 q^{10} + 8 q^{11} - 12 q^{14} - 8 q^{19} - 16 q^{20} - 20 q^{26} + 8 q^{28} + 16 q^{29} + 24 q^{31} - 24 q^{35} - 16 q^{37} + 8 q^{38} + 16 q^{40} - 8 q^{43} + 40 q^{44} - 8 q^{46}+ \cdots + 40 q^{98}+O(q^{100})$ 8 * q - 4 * q^4 + 12 * q^8 - 8 * q^10 + 8 * q^11 - 12 * q^14 - 8 * q^19 - 16 * q^20 - 20 * q^26 + 8 * q^28 + 16 * q^29 + 24 * q^31 - 24 * q^35 - 16 * q^37 + 8 * q^38 + 16 * q^40 - 8 * q^43 + 40 * q^44 - 8 * q^46 - 8 * q^49 + 36 * q^50 - 16 * q^52 - 16 * q^53 - 16 * q^58 - 32 * q^59 + 16 * q^61 + 12 * q^62 + 8 * q^64 + 16 * q^65 - 16 * q^67 - 32 * q^68 + 32 * q^70 - 52 * q^74 + 8 * q^76 - 16 * q^77 - 24 * q^79 - 8 * q^80 + 40 * q^82 + 40 * q^83 - 16 * q^85 + 16 * q^86 + 32 * q^88 - 8 * q^91 + 16 * q^92 + 8 * q^94 + 48 * q^95 + 40 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/144\mathbb{Z}\right)^\times$ .

$n$	$37$	$65$	$127$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$ -expansion $a_n$ , the Satake parameters $\alpha_p$ , and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$ .

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$\alpha_n$			$\theta_n$
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$\alpha_p$			$\theta_p$

$2$	−0.874559	−	1.11137i	−0.618406	−	0.785858i
$2$	−0.874559	−	1.11137i	−0.618406	−	0.785858i
$3$		0			0
$3$		0			0
$4$	−0.470294	+	1.94392i	−0.235147	+	0.971960i
$5$	0.334904	−	0.334904i	0.149774	−	0.149774i	−0.628243	−	0.778017i	$-0.716226\pi$
$5$	0.334904	−	0.334904i	0.149774	−	0.149774i	0.778017	+	0.628243i	$0.216226\pi$
$6$		0			0
$7$		−	4.55765i		−	1.72263i	−0.508072	−	0.861314i	$-0.669642\pi$
$7$		−	4.55765i		−	1.72263i	0.508072	−	0.861314i	$-0.330358\pi$
$8$	2.57172	−	1.17740i	0.909239	−	0.416274i
$9$		0			0
$10$	−0.665096	−	0.0793096i	−0.210322	−	0.0250799i
$11$	2.47363	−	2.47363i	0.745826	−	0.745826i	−0.227866	−	0.973692i	$-0.573175\pi$
$11$	2.47363	−	2.47363i	0.745826	−	0.745826i	0.973692	+	0.227866i	$0.0731749\pi$
$12$		0			0
$13$	−0.0594122	−	0.0594122i	−0.0164780	−	0.0164780i	0.698820	−	0.715298i	$-0.253709\pi$
$13$	−0.0594122	−	0.0594122i	−0.0164780	−	0.0164780i	−0.715298	+	0.698820i	$0.753709\pi$
$14$	−5.06524	+	3.98593i	−1.35374	+	1.06528i
$15$		0			0
$16$	−3.55765	−	1.82843i	−0.889412	−	0.457107i
$17$	−3.61706			−0.877266			−0.438633	−	0.898666i	$-0.644537\pi$
$17$	−3.61706			−0.877266			−0.438633	+	0.898666i	$0.644537\pi$
$18$		0			0
$19$	2.55765	+	2.55765i	0.586765	+	0.586765i	0.936754	−	0.349989i	$-0.113815\pi$
$19$	2.55765	+	2.55765i	0.586765	+	0.586765i	−0.349989	+	0.936754i	$0.613815\pi$
$20$	0.493523	+	0.808530i	0.110355	+	0.180793i
$21$		0			0
$22$	−4.91245	−	0.585786i	−1.04734	−	0.124890i
$23$		−	2.82843i		−	0.589768i	−0.955533	−	0.294884i	$-0.904719\pi$
$23$		−	2.82843i		−	0.589768i	0.955533	−	0.294884i	$-0.0952810\pi$
$24$		0			0
$25$			4.77568i			0.955136i
$26$	−0.0140696	+	0.117988i	−0.00275927	+	0.0231394i
$27$		0			0
$28$	8.85970	+	2.14343i	1.67433	+	0.405071i
$29$	5.16333	+	5.16333i	0.958807	+	0.958807i	0.999184	−	0.0403780i	$-0.0128562\pi$
$29$	5.16333	+	5.16333i	0.958807	+	0.958807i	−0.0403780	+	0.999184i	$0.512856\pi$
$30$		0			0
$31$	−0.557647			−0.100156			−0.0500782	−	0.998745i	$-0.515947\pi$
$31$	−0.557647			−0.100156			−0.0500782	+	0.998745i	$0.515947\pi$
$32$	1.07931	+	5.55294i	0.190797	+	0.981630i
$33$		0			0
$34$	3.16333	+	4.01990i	0.542507	+	0.689407i
$35$	−1.52637	−	1.52637i	−0.258004	−	0.258004i
$36$		0			0
$37$	4.38607	−	4.38607i	0.721066	−	0.721066i	−0.247756	−	0.968822i	$-0.579693\pi$
$37$	4.38607	−	4.38607i	0.721066	−	0.721066i	0.968822	+	0.247756i	$0.0796932\pi$
$38$	0.605684	−	5.07931i	0.0982549	−	0.823973i
$39$		0			0
$40$	0.466962	−	1.25559i	0.0738332	−	0.198527i
$41$			9.27391i			1.44834i	0.689620	+	0.724171i	$0.257777\pi$
$41$			9.27391i			1.44834i	−0.689620	+	0.724171i	$0.742223\pi$
$42$		0			0
$43$	−1.61040	+	1.61040i	−0.245583	+	0.245583i	−0.819155	−	0.573572i	$-0.805557\pi$
$43$	−1.61040	+	1.61040i	−0.245583	+	0.245583i	0.573572	+	0.819155i	$0.305557\pi$
$44$	3.64520	+	5.97186i	0.549534	+	0.900292i
$45$		0			0
$46$	−3.14343	+	2.47363i	−0.463474	+	0.364716i
$47$	−2.82843			−0.412568			−0.206284	−	0.978492i	$-0.566137\pi$
$47$	−2.82843			−0.412568			−0.206284	+	0.978492i	$0.566137\pi$
$48$		0			0
$49$	−13.7721			−1.96745
$50$	5.30755	−	4.17661i	0.750601	−	0.590662i
$51$		0			0
$52$	0.143434	−	0.0875513i	0.0198907	−	0.0121412i
$53$	0.493523	−	0.493523i	0.0677906	−	0.0677906i	−0.672399	−	0.740189i	$-0.734736\pi$
$53$	0.493523	−	0.493523i	0.0677906	−	0.0677906i	0.740189	+	0.672399i	$0.234736\pi$
$54$		0			0
$55$		−	1.65685i		−	0.223410i
$56$	−5.36618	−	11.7210i	−0.717086	−	1.56628i
$57$		0			0
$58$	1.22274	−	10.2540i	0.160554	−	1.34642i
$59$	−4.00000	+	4.00000i	−0.520756	+	0.520756i	−0.917800	−	0.397044i	$-0.870036\pi$
$59$	−4.00000	+	4.00000i	−0.520756	+	0.520756i	0.397044	+	0.917800i	$0.370036\pi$
$60$		0			0
$61$	2.72922	+	2.72922i	0.349441	+	0.349441i	0.859901	−	0.510460i	$-0.170525\pi$
$61$	2.72922	+	2.72922i	0.349441	+	0.349441i	−0.510460	+	0.859901i	$0.670525\pi$
$62$	0.487695	+	0.619753i	0.0619374	+	0.0787088i
$63$		0			0
$64$	5.22746	−	6.05588i	0.653432	−	0.756985i
$65$	−0.0397948			−0.00493593
$66$		0			0
$67$	3.77568	+	3.77568i	0.461273	+	0.461273i	0.899072	−	0.437800i	$-0.144242\pi$
$67$	3.77568	+	3.77568i	0.461273	+	0.461273i	−0.437800	+	0.899072i	$0.644242\pi$
$68$	1.70108	−	7.03127i	0.206286	−	0.852667i
$69$		0			0
$70$	−0.361465	+	3.03127i	−0.0432033	+	0.362306i
$71$		−	9.11529i		−	1.08179i	−0.841091	−	0.540893i	$-0.818086\pi$
$71$		−	9.11529i		−	1.08179i	0.841091	−	0.540893i	$-0.181914\pi$
$72$		0			0
$73$		−	0.541560i		−	0.0633848i	−0.999498	−	0.0316924i	$-0.989910\pi$
$73$		−	0.541560i		−	0.0633848i	0.999498	−	0.0316924i	$-0.0100897\pi$
$74$	−8.71044	−	1.03868i	−1.01257	−	0.120744i
$75$		0			0
$76$	−6.17471	+	3.76901i	−0.708287	+	0.432336i
$77$	−11.2739	−	11.2739i	−1.28478	−	1.28478i
$78$		0			0
$79$	−10.9937			−1.23689			−0.618445	−	0.785828i	$-0.712237\pi$
$79$	−10.9937			−1.23689			−0.618445	+	0.785828i	$0.712237\pi$
$80$	−1.80382	+	0.579123i	−0.201673	+	0.0647479i
$81$		0			0
$82$	10.3068	−	8.11058i	1.13819	−	0.895664i
$83$	10.6417	+	10.6417i	1.16807	+	1.16807i	0.982660	+	0.185415i	$0.0593628\pi$
$83$	10.6417	+	10.6417i	1.16807	+	1.16807i	0.185415	+	0.982660i	$0.440637\pi$
$84$		0			0
$85$	−1.21137	+	1.21137i	−0.131391	+	0.131391i
$86$	3.19813	+	0.381362i	0.344864	+	0.0411234i
$87$		0			0
$88$	3.44902	−	9.27391i	0.367666	−	0.988603i
$89$		−	14.6533i		−	1.55325i	−0.629964	−	0.776625i	$-0.716930\pi$
$89$		−	14.6533i		−	1.55325i	0.629964	−	0.776625i	$-0.283070\pi$
$90$		0			0
$91$	−0.270780	+	0.270780i	−0.0283854	+	0.0283854i
$92$	5.49824	+	1.33019i	0.573231	+	0.138682i
$93$		0			0
$94$	2.47363	+	3.14343i	0.255135	+	0.324220i
$95$	1.71313			0.175764
$96$		0			0
$97$	4.31724			0.438349			0.219175	−	0.975686i	$-0.429664\pi$
$97$	4.31724			0.438349			0.219175	+	0.975686i	$0.429664\pi$
$98$	12.0446	+	15.3060i	1.21668	+	1.54614i
$99$		0			0
$100$	−9.28354	−	2.24597i	−0.928354	−	0.224597i
$101$	0.453728	−	0.453728i	0.0451477	−	0.0451477i	−0.684173	−	0.729320i	$-0.739836\pi$
$101$	0.453728	−	0.453728i	0.0451477	−	0.0451477i	0.729320	+	0.684173i	$0.239836\pi$
$102$		0			0
$103$			1.33686i			0.131724i	0.997829	+	0.0658622i	$0.0209798\pi$
$103$			1.33686i			0.131724i	−0.997829	+	0.0658622i	$0.979020\pi$
$104$	−0.222743	−	0.0828394i	−0.0218418	−	0.00812307i
$105$		0			0
$106$	−0.980103	−	0.116873i	−0.0951960	−	0.0113517i
$107$	−6.06255	+	6.06255i	−0.586088	+	0.586088i	−0.936570	−	0.350481i	$-0.886018\pi$
$107$	−6.06255	+	6.06255i	−0.586088	+	0.586088i	0.350481	+	0.936570i	$0.386018\pi$
$108$		0			0
$109$	5.71627	+	5.71627i	0.547519	+	0.547519i	0.925722	−	0.378203i	$-0.123458\pi$
$109$	5.71627	+	5.71627i	0.547519	+	0.547519i	−0.378203	+	0.925722i	$0.623458\pi$
$110$	−1.84138	+	1.44902i	−0.175569	+	0.138158i
$111$		0			0
$112$	−8.33333	+	16.2145i	−0.787425	+	1.53213i
$113$	9.55136			0.898516			0.449258	−	0.893402i	$-0.351688\pi$
$113$	9.55136			0.898516			0.449258	+	0.893402i	$0.351688\pi$
$114$		0			0
$115$	−0.947252	−	0.947252i	−0.0883317	−	0.0883317i
$116$	−12.4654	+	7.60882i	−1.15738	+	0.706461i
$117$		0			0
$118$	7.94372	+	0.947252i	0.731279	+	0.0872016i
$119$			16.4853i			1.51120i
$120$		0			0
$121$		−	1.23765i		−	0.112514i
$122$	0.646314	−	5.42004i	0.0585146	−	0.490707i
$123$		0			0
$124$	0.262258	−	1.08402i	0.0235515	−	0.0973480i
$125$	3.27391	+	3.27391i	0.292828	+	0.292828i
$126$		0			0
$127$	5.09921			0.452481			0.226241	−	0.974071i	$-0.427356\pi$
$127$	5.09921			0.452481			0.226241	+	0.974071i	$0.427356\pi$
$128$	−11.3021	−	0.513421i	−0.998970	−	0.0453804i
$129$		0			0
$130$	0.0348029	+	0.0442268i	0.00305241	+	0.00387894i
$131$	−2.11882	−	2.11882i	−0.185123	−	0.185123i	0.608461	−	0.793584i	$-0.291787\pi$
$131$	−2.11882	−	2.11882i	−0.185123	−	0.185123i	−0.793584	+	0.608461i	$0.791787\pi$
$132$		0			0
$133$	11.6569	−	11.6569i	1.01078	−	1.01078i
$134$	0.894129	−	7.49824i	0.0772410	−	0.647749i
$135$		0			0
$136$	−9.30205	+	4.25873i	−0.797644	+	0.365183i
$137$			3.37941i			0.288723i	0.989525	+	0.144361i	$0.0461127\pi$
$137$			3.37941i			0.288723i	−0.989525	+	0.144361i	$0.953887\pi$
$138$		0			0
$139$	5.88118	−	5.88118i	0.498835	−	0.498835i	−0.412240	−	0.911075i	$-0.635254\pi$
$139$	5.88118	−	5.88118i	0.498835	−	0.498835i	0.911075	+	0.412240i	$0.135254\pi$
$140$	3.68499	−	2.24930i	0.311439	−	0.190101i
$141$		0			0
$142$	−10.1305	+	7.97186i	−0.850131	+	0.668984i
$143$	−0.293927			−0.0245794
$144$		0			0
$145$	3.45844			0.287208
$146$	−0.601874	+	0.473626i	−0.0498115	+	0.0391975i
$147$		0			0
$148$	6.46343	+	10.5889i	0.531291	+	0.870404i
$149$	9.99176	−	9.99176i	0.818557	−	0.818557i	−0.167342	−	0.985899i	$-0.553518\pi$
$149$	9.99176	−	9.99176i	0.818557	−	0.818557i	0.985899	+	0.167342i	$0.0535185\pi$
$150$		0			0
$151$			9.97685i			0.811905i	0.913894	+	0.405952i	$0.133060\pi$
$151$			9.97685i			0.811905i	−0.913894	+	0.405952i	$0.866940\pi$
$152$	9.58892	+	3.56617i	0.777764	+	0.289255i
$153$		0			0
$154$	−2.66981	+	22.3892i	−0.215139	+	1.80417i
$155$	−0.186758	+	0.186758i	−0.0150008	+	0.0150008i
$156$		0			0
$157$	−16.1618	−	16.1618i	−1.28985	−	1.28985i	−0.934877	−	0.354971i	$-0.884491\pi$
$157$	−16.1618	−	16.1618i	−1.28985	−	1.28985i	−0.354971	−	0.934877i	$-0.615509\pi$
$158$	9.61465	+	12.2181i	0.764900	+	0.972020i
$159$		0			0
$160$	2.22117	+	1.49824i	0.175599	+	0.118446i
$161$	−12.8910			−1.01595
$162$		0			0
$163$	−7.50490	−	7.50490i	−0.587829	−	0.587829i	0.349214	−	0.937043i	$-0.386449\pi$
$163$	−7.50490	−	7.50490i	−0.587829	−	0.587829i	−0.937043	+	0.349214i	$0.886449\pi$
$164$	−18.0277	−	4.36147i	−1.40773	−	0.340573i
$165$		0			0
$166$	2.52008	−	21.1336i	0.195596	−	1.64029i
$167$			5.83822i			0.451775i	0.974153	+	0.225888i	$0.0725282\pi$
$167$			5.83822i			0.451775i	−0.974153	+	0.225888i	$0.927472\pi$
$168$		0			0
$169$		−	12.9929i		−	0.999457i
$170$	2.40569	+	0.286867i	0.184508	+	0.0220017i
$171$		0			0
$172$	−2.37312	−	3.88784i	−0.180949	−	0.296445i
$173$	3.62530	+	3.62530i	0.275627	+	0.275627i	0.831360	−	0.555734i	$-0.187563\pi$
$173$	3.62530	+	3.62530i	0.275627	+	0.275627i	−0.555734	+	0.831360i	$0.687563\pi$
$174$		0			0
$175$	21.7659			1.64534
$176$	−13.3231	+	4.27744i	−1.00427	+	0.322424i
$177$		0			0
$178$	−16.2853	+	12.8152i	−1.22063	+	0.960539i
$179$	−9.28334	−	9.28334i	−0.693869	−	0.693869i	0.269212	−	0.963081i	$-0.413237\pi$
$179$	−9.28334	−	9.28334i	−0.693869	−	0.693869i	−0.963081	+	0.269212i	$0.913237\pi$
$180$		0			0
$181$	−10.8316	+	10.8316i	−0.805104	+	0.805104i	−0.983888	−	0.178785i	$-0.942783\pi$
$181$	−10.8316	+	10.8316i	−0.805104	+	0.805104i	0.178785	+	0.983888i	$0.442783\pi$
$182$	0.537750	+	0.0641242i	0.0398607	+	0.00475320i
$183$		0			0
$184$	−3.33019	−	7.27391i	−0.245505	−	0.536240i
$185$		−	2.93783i		−	0.215993i
$186$		0			0
$187$	−8.94725	+	8.94725i	−0.654288	+	0.654288i
$188$	1.33019	−	5.49824i	0.0970142	−	0.401000i
$189$		0			0
$190$	−1.49824	−	1.90393i	−0.108693	−	0.138125i
$191$	8.63001			0.624446			0.312223	−	0.950009i	$-0.398926\pi$
$191$	8.63001			0.624446			0.312223	+	0.950009i	$0.398926\pi$
$192$		0			0
$193$	11.4514			0.824288			0.412144	−	0.911119i	$-0.364780\pi$
$193$	11.4514			0.824288			0.412144	+	0.911119i	$0.364780\pi$
$194$	−3.77568	−	4.79806i	−0.271078	−	0.344480i
$195$		0			0
$196$	6.47696	−	26.7720i	0.462640	−	1.91228i
$197$	−7.48999	+	7.48999i	−0.533640	+	0.533640i	−0.921654	−	0.388014i	$-0.873161\pi$
$197$	−7.48999	+	7.48999i	−0.533640	+	0.533640i	0.388014	+	0.921654i	$0.373161\pi$
$198$		0			0
$199$		−	3.68000i		−	0.260868i	−0.991457	−	0.130434i	$-0.958363\pi$
$199$		−	3.68000i		−	0.260868i	0.991457	−	0.130434i	$-0.0416371\pi$
$200$	5.62289	+	12.2817i	0.397598	+	0.868447i
$201$		0			0
$202$	−0.901073	−	0.107449i	−0.0633993	−	0.00756007i
$203$	23.5326	−	23.5326i	1.65167	−	1.65167i
$204$		0			0
$205$	3.10587	+	3.10587i	0.216923	+	0.216923i
$206$	1.48574	−	1.16916i	0.103517	−	0.0814592i
$207$		0			0
$208$	0.102737	+	0.319999i	0.00712351	+	0.0221879i
$209$	12.6533			0.875249
$210$		0			0
$211$	10.1188	+	10.1188i	0.696609	+	0.696609i	0.963677	−	0.267069i	$-0.0860551\pi$
$211$	10.1188	+	10.1188i	0.696609	+	0.696609i	−0.267069	+	0.963677i	$0.586055\pi$
$212$	0.727268	+	1.19147i	0.0499490	+	0.0818305i
$213$		0			0
$214$	12.0398	+	1.43569i	0.823023	+	0.0981417i
$215$			1.07866i			0.0735637i
$216$		0			0
$217$			2.54156i			0.172532i
$218$	1.35369	−	11.3521i	0.0916832	−	0.768862i
$219$		0			0
$220$	3.22079	+	0.779208i	0.217146	+	0.0525342i
$221$	0.214897	+	0.214897i	0.0144556	+	0.0144556i
$222$		0			0
$223$	−4.86156			−0.325554			−0.162777	−	0.986663i	$-0.552045\pi$
$223$	−4.86156			−0.325554			−0.162777	+	0.986663i	$0.552045\pi$
$224$	25.3083	−	4.91911i	1.69098	−	0.328672i
$225$		0			0
$226$	−8.35322	−	10.6151i	−0.555648	−	0.706106i
$227$	−10.6417	−	10.6417i	−0.706312	−	0.706312i	0.259445	−	0.965758i	$-0.416460\pi$
$227$	−10.6417	−	10.6417i	−0.706312	−	0.706312i	−0.965758	+	0.259445i	$0.916460\pi$
$228$		0			0
$229$	−20.1712	+	20.1712i	−1.33295	+	1.33295i	−0.430229	+	0.902720i	$0.641567\pi$
$229$	−20.1712	+	20.1712i	−1.33295	+	1.33295i	−0.902720	+	0.430229i	$0.858433\pi$
$230$	−0.224321	+	1.88118i	−0.0147913	+	0.124041i
$231$		0			0
$232$	19.3579	+	7.19932i	1.27091	+	0.472658i
$233$		−	13.5702i		−	0.889014i	−0.895775	−	0.444507i	$-0.853379\pi$
$233$		−	13.5702i		−	0.889014i	0.895775	−	0.444507i	$-0.146621\pi$
$234$		0			0
$235$	−0.947252	+	0.947252i	−0.0617919	+	0.0617919i
$236$	−5.89450	−	9.65685i	−0.383699	−	0.628608i
$237$		0			0
$238$	18.3213	−	14.4173i	1.18759	−	0.934538i
$239$	−29.3629			−1.89933			−0.949665	−	0.313267i	$-0.898576\pi$
$239$	−29.3629			−1.89933			−0.949665	+	0.313267i	$0.898576\pi$
$240$		0			0
$241$	24.0063			1.54638			0.773190	−	0.634175i	$-0.218660\pi$
$241$	24.0063			1.54638			0.773190	+	0.634175i	$0.218660\pi$
$242$	−1.37549	+	1.08240i	−0.0884197	+	0.0695791i
$243$		0			0
$244$	−6.58892	+	4.02185i	−0.421812	+	0.257473i
$245$	−4.61235	+	4.61235i	−0.294672	+	0.294672i
$246$		0			0
$247$		−	0.303911i		−	0.0193374i
$248$	−1.43411	+	0.656574i	−0.0910661	+	0.0416925i
$249$		0			0
$250$	0.775305	−	6.50176i	0.0490346	−	0.411208i
$251$	−15.7570	+	15.7570i	−0.994571	+	0.994571i	−0.999985	−	0.00541463i	$-0.998276\pi$
$251$	−15.7570	+	15.7570i	−0.994571	+	0.994571i	0.00541463	+	0.999985i	$0.498276\pi$
$252$		0			0
$253$	−6.99647	−	6.99647i	−0.439864	−	0.439864i
$254$	−4.45956	−	5.66711i	−0.279817	−	0.355586i
$255$		0			0
$256$	9.31371	+	13.0098i	0.582107	+	0.813112i
$257$	−8.66038			−0.540220			−0.270110	−	0.962829i	$-0.587060\pi$
$257$	−8.66038			−0.540220			−0.270110	+	0.962829i	$0.587060\pi$
$258$		0			0
$259$	−19.9902	−	19.9902i	−1.24213	−	1.24213i
$260$	0.0187152	−	0.0773578i	0.00116067	−	0.00479753i
$261$		0			0
$262$	−0.501765	+	4.20784i	−0.0309991	+	0.259961i
$263$		−	13.3208i		−	0.821394i	−0.911772	−	0.410697i	$-0.865285\pi$
$263$		−	13.3208i		−	0.821394i	0.911772	−	0.410697i	$-0.134715\pi$
$264$		0			0
$265$		−	0.330566i		−	0.0203065i
$266$	−23.1497	−	2.76049i	−1.41940	−	0.169257i
$267$		0			0
$268$	−9.11529	+	5.56394i	−0.556805	+	0.339872i
$269$	11.6714	+	11.6714i	0.711616	+	0.711616i	0.966873	−	0.255257i	$-0.0821602\pi$
$269$	11.6714	+	11.6714i	0.711616	+	0.711616i	−0.255257	+	0.966873i	$0.582160\pi$
$270$		0			0
$271$	−21.9769			−1.33500			−0.667499	−	0.744610i	$-0.732635\pi$
$271$	−21.9769			−1.33500			−0.667499	+	0.744610i	$0.732635\pi$
$272$	12.8682	+	6.61353i	0.780251	+	0.401004i
$273$		0			0
$274$	3.75578	−	2.95549i	0.226895	−	0.178548i
$275$	11.8132	+	11.8132i	0.712365	+	0.712365i
$276$		0			0
$277$	−10.9504	+	10.9504i	−0.657945	+	0.657945i	−0.954893	−	0.296949i	$-0.904031\pi$
$277$	−10.9504	+	10.9504i	−0.657945	+	0.657945i	0.296949	+	0.954893i	$0.404031\pi$
$278$	−11.6796	−	1.39274i	−0.700496	−	0.0835309i
$279$		0			0
$280$	−5.72256	−	2.12825i	−0.341988	−	0.127187i
$281$			22.8910i			1.36556i	0.730624	+	0.682780i	$0.239229\pi$
$281$			22.8910i			1.36556i	−0.730624	+	0.682780i	$0.760771\pi$
$282$		0			0
$283$	4.48528	−	4.48528i	0.266622	−	0.266622i	−0.561115	−	0.827738i	$-0.689628\pi$
$283$	4.48528	−	4.48528i	0.266622	−	0.266622i	0.827738	+	0.561115i	$0.189628\pi$
$284$	17.7194	+	4.28687i	1.05145	+	0.254379i
$285$		0			0
$286$	0.257057	+	0.326662i	0.0152001	+	0.0193159i
$287$	42.2672			2.49496
$288$		0			0
$289$	−3.91688			−0.230405
$290$	−3.02461	−	3.84361i	−0.177611	−	0.225705i
$291$		0			0
$292$	1.05275	+	0.254692i	0.0616074	+	0.0149047i
$293$	−21.6221	+	21.6221i	−1.26318	+	1.26318i	−0.313636	+	0.949543i	$0.601547\pi$
$293$	−21.6221	+	21.6221i	−1.26318	+	1.26318i	−0.949543	+	0.313636i	$0.898453\pi$
$294$		0			0
$295$			2.67923i			0.155991i
$296$	6.11557	−	16.4439i	0.355461	−	0.955783i
$297$		0			0
$298$	−19.8429	−	2.36618i	−1.14947	−	0.137069i
$299$	−0.168043	+	0.168043i	−0.00971818	+	0.00971818i
$300$		0			0
$301$	7.33962	+	7.33962i	0.423048	+	0.423048i
$302$	11.0880	−	8.72534i	0.638042	−	0.502087i
$303$		0			0
$304$	−4.42274	−	13.7757i	−0.253661	−	0.790089i
$305$	1.82805			0.104674
$306$		0			0
$307$	−12.1118	−	12.1118i	−0.691255	−	0.691255i	0.271253	−	0.962508i	$-0.412562\pi$
$307$	−12.1118	−	12.1118i	−0.691255	−	0.691255i	−0.962508	+	0.271253i	$0.912562\pi$
$308$	27.2176	−	16.6135i	1.55087	−	0.946644i
$309$		0			0
$310$	0.370889	+	0.0442268i	0.0210651	+	0.00251191i
$311$			26.8651i			1.52338i	0.647943	+	0.761689i	$0.275630\pi$
$311$			26.8651i			1.52338i	−0.647943	+	0.761689i	$0.724370\pi$
$312$		0			0
$313$			19.6890i			1.11289i	0.830885	+	0.556445i	$0.187835\pi$
$313$			19.6890i			1.11289i	−0.830885	+	0.556445i	$0.812165\pi$
$314$	−3.82731	+	32.0961i	−0.215988	+	1.81129i
$315$		0			0
$316$	5.17027	−	21.3709i	0.290851	−	1.20221i
$317$	−21.3447	−	21.3447i	−1.19884	−	1.19884i	−0.974515	−	0.224323i	$-0.927983\pi$
$317$	−21.3447	−	21.3447i	−1.19884	−	1.19884i	−0.224323	−	0.974515i	$-0.572017\pi$
$318$		0			0
$319$	25.5443			1.43021
$320$	−0.277444	−	3.77883i	−0.0155096	−	0.211243i
$321$		0			0
$322$	11.2739	+	14.3267i	0.628271	+	0.798394i
$323$	−9.25116	−	9.25116i	−0.514748	−	0.514748i
$324$		0			0
$325$	0.283734	−	0.283734i	0.0157387	−	0.0157387i
$326$	−1.77726	+	14.9042i	−0.0984331	+	0.825468i
$327$		0			0
$328$	10.9191	+	23.8499i	0.602907	+	1.31689i
$329$			12.8910i			0.710702i
$330$		0			0
$331$	14.6926	−	14.6926i	0.807576	−	0.807576i	−0.176690	−	0.984266i	$-0.556539\pi$
$331$	14.6926	−	14.6926i	0.807576	−	0.807576i	0.984266	+	0.176690i	$0.0565391\pi$
$332$	−25.6913	+	15.6818i	−1.40999	+	0.860653i
$333$		0			0
$334$	6.48844	−	5.10587i	0.355032	−	0.279381i
$335$	2.52898			0.138173
$336$		0			0
$337$	−23.0098			−1.25342			−0.626712	−	0.779251i	$-0.715600\pi$
$337$	−23.0098			−1.25342			−0.626712	+	0.779251i	$0.715600\pi$
$338$	−14.4400	+	11.3631i	−0.785432	+	0.618071i
$339$		0			0
$340$	−1.78510	−	2.92450i	−0.0968108	−	0.158603i
$341$	−1.37941	+	1.37941i	−0.0746993	+	0.0746993i
$342$		0			0
$343$			30.8651i			1.66656i
$344$	−2.24540	+	6.03756i	−0.121064	+	0.325524i
$345$		0			0
$346$	0.858518	−	7.19960i	0.0461542	−	0.387053i
$347$	10.9026	−	10.9026i	0.585284	−	0.585284i	−0.351067	−	0.936350i	$-0.614181\pi$
$347$	10.9026	−	10.9026i	0.585284	−	0.585284i	0.936350	+	0.351067i	$0.114181\pi$
$348$		0			0
$349$	20.0563	+	20.0563i	1.07359	+	1.07359i	0.997068	+	0.0765186i	$0.0243805\pi$
$349$	20.0563	+	20.0563i	1.07359	+	1.07359i	0.0765186	+	0.997068i	$0.475620\pi$
$350$	−19.0355	−	24.1900i	−1.01749	−	1.29301i
$351$		0			0
$352$	16.4057	+	11.0661i	0.874426	+	0.589824i
$353$	12.2117			0.649965			0.324983	−	0.945720i	$-0.394642\pi$
$353$	12.2117			0.649965			0.324983	+	0.945720i	$0.394642\pi$
$354$		0			0
$355$	−3.05275	−	3.05275i	−0.162023	−	0.162023i
$356$	28.4849	+	6.89137i	1.50970	+	0.365242i
$357$		0			0
$358$	−2.19841	+	18.4361i	−0.116190	+	0.974376i
$359$		−	33.4780i		−	1.76690i	−0.468522	−	0.883452i	$-0.655214\pi$
$359$		−	33.4780i		−	1.76690i	0.468522	−	0.883452i	$-0.344786\pi$
$360$		0			0
$361$		−	5.91688i		−	0.311415i
$362$	21.5107	+	2.56505i	1.13058	+	0.134816i
$363$		0			0
$364$	−0.399028	−	0.653720i	−0.0209148	−	0.0342643i
$365$	−0.181370	−	0.181370i	−0.00949337	−	0.00949337i
$366$		0			0
$367$	−0.702379			−0.0366639			−0.0183319	−	0.999832i	$-0.505836\pi$
$367$	−0.702379			−0.0366639			−0.0183319	+	0.999832i	$0.505836\pi$
$368$	−5.17157	+	10.0625i	−0.269587	+	0.524546i
$369$		0			0
$370$	−3.26502	+	2.56930i	−0.169740	+	0.133572i
$371$	−2.24930	−	2.24930i	−0.116778	−	0.116778i
$372$		0			0
$373$	18.9598	−	18.9598i	0.981702	−	0.981702i	−0.0181339	−	0.999836i	$-0.505773\pi$
$373$	18.9598	−	18.9598i	0.981702	−	0.981702i	0.999836	+	0.0181339i	$0.00577250\pi$
$374$	17.7686	+	2.11882i	0.918793	+	0.109562i
$375$		0			0
$376$	−7.27391	+	3.33019i	−0.375123	+	0.171742i
$377$		−	0.613530i		−	0.0315984i
$378$		0			0
$379$	−1.77844	+	1.77844i	−0.0913523	+	0.0913523i	−0.751306	−	0.659954i	$-0.770576\pi$
$379$	−1.77844	+	1.77844i	−0.0913523	+	0.0913523i	0.659954	+	0.751306i	$0.270576\pi$
$380$	−0.805676	+	3.33019i	−0.0413303	+	0.170835i
$381$		0			0
$382$	−7.54745	−	9.59115i	−0.386161	−	0.490726i
$383$	−25.4880			−1.30238			−0.651188	−	0.758916i	$-0.725729\pi$
$383$	−25.4880			−1.30238			−0.651188	+	0.758916i	$0.725729\pi$
$384$		0			0
$385$	−7.55136			−0.384853
$386$	−10.0149	−	12.7267i	−0.509745	−	0.647774i
$387$		0			0
$388$	−2.03037	+	8.39236i	−0.103076	+	0.426058i
$389$	11.7049	−	11.7049i	0.593462	−	0.593462i	−0.345103	−	0.938565i	$-0.612156\pi$
$389$	11.7049	−	11.7049i	0.593462	−	0.593462i	0.938565	+	0.345103i	$0.112156\pi$
$390$		0			0
$391$			10.2306i			0.517383i
$392$	−35.4181	+	16.2153i	−1.78888	+	0.818998i
$393$		0			0
$394$	14.8746	+	1.77373i	0.749372	+	0.0893591i
$395$	−3.68184	+	3.68184i	−0.185253	+	0.185253i
$396$		0			0
$397$	−9.04646	−	9.04646i	−0.454029	−	0.454029i	0.442661	−	0.896689i	$-0.354035\pi$
$397$	−9.04646	−	9.04646i	−0.454029	−	0.454029i	−0.896689	+	0.442661i	$0.854035\pi$
$398$	−4.08985	+	3.21838i	−0.205006	+	0.161323i
$399$		0			0
$400$	8.73198	−	16.9902i	0.436599	−	0.849509i
$401$	18.0853			0.903137			0.451568	−	0.892237i	$-0.350865\pi$
$401$	18.0853			0.903137			0.451568	+	0.892237i	$0.350865\pi$
$402$		0			0
$403$	0.0331311	+	0.0331311i	0.00165038	+	0.00165038i
$404$	0.668626	+	1.09540i	0.0332654	+	0.0544980i
$405$		0			0
$406$	−46.7342	−	5.57283i	−2.31938	−	0.276575i
$407$		−	21.6990i		−	1.07558i
$408$		0			0
$409$		−	25.2271i		−	1.24740i	−0.781665	−	0.623699i	$-0.785629\pi$
$409$		−	25.2271i		−	1.24740i	0.781665	−	0.623699i	$-0.214371\pi$
$410$	0.735510	−	6.16804i	0.0363243	−	0.304618i
$411$		0			0
$412$	−2.59874	−	0.628715i	−0.128031	−	0.0309746i
$413$	18.2306	+	18.2306i	0.897069	+	0.897069i
$414$		0			0
$415$	7.12787			0.349894
$416$	0.265788	−	0.394036i	0.0130313	−	0.0193192i
$417$		0			0
$418$	−11.0661	−	14.0625i	−0.541259	−	0.687822i
$419$	−7.25283	−	7.25283i	−0.354324	−	0.354324i	0.507392	−	0.861716i	$-0.330610\pi$
$419$	−7.25283	−	7.25283i	−0.354324	−	0.354324i	−0.861716	+	0.507392i	$0.830610\pi$
$420$		0			0
$421$	2.39550	−	2.39550i	0.116749	−	0.116749i	−0.646318	−	0.763068i	$-0.723692\pi$
$421$	2.39550	−	2.39550i	0.116749	−	0.116749i	0.763068	+	0.646318i	$0.223692\pi$
$422$	2.39627	−	20.0953i	0.116648	−	0.978223i
$423$		0			0
$424$	0.688127	−	1.85028i	0.0334184	−	0.0898574i
$425$		−	17.2739i		−	0.837908i
$426$		0			0
$427$	12.4388	−	12.4388i	0.601957	−	0.601957i
$428$	−8.93392	−	14.6363i	−0.431838	−	0.707471i
$429$		0			0
$430$	1.19879	−	0.943348i	0.0578107	−	0.0454923i
$431$	4.42454			0.213123			0.106561	−	0.994306i	$-0.466016\pi$
$431$	4.42454			0.213123			0.106561	+	0.994306i	$0.466016\pi$
$432$		0			0
$433$	7.31371			0.351474			0.175737	−	0.984437i	$-0.443769\pi$
$433$	7.31371			0.351474			0.175737	+	0.984437i	$0.443769\pi$
$434$	2.82462	−	2.22274i	0.135586	−	0.106695i
$435$		0			0
$436$	−13.8003	+	8.42364i	−0.660914	+	0.403419i
$437$	7.23412	−	7.23412i	0.346055	−	0.346055i
$438$		0			0
$439$		−	29.6533i		−	1.41527i	−0.706576	−	0.707637i	$-0.749761\pi$
$439$		−	29.6533i		−	1.41527i	0.706576	−	0.707637i	$-0.250239\pi$
$440$	−1.95078	−	4.26096i	−0.0929999	−	0.203133i
$441$		0			0
$442$	0.0508905	−	0.426771i	0.00242061	−	0.0202994i
$443$	−10.3056	+	10.3056i	−0.489633	+	0.489633i	−0.908190	−	0.418557i	$-0.862536\pi$
$443$	−10.3056	+	10.3056i	−0.489633	+	0.489633i	0.418557	+	0.908190i	$0.362536\pi$
$444$		0			0
$445$	−4.90746	−	4.90746i	−0.232636	−	0.232636i
$446$	4.25172	+	5.40300i	0.201325	+	0.255839i
$447$		0			0
$448$	−27.6006	−	23.8249i	−1.30400	−	1.12562i
$449$	6.48844			0.306208			0.153104	−	0.988210i	$-0.451073\pi$
$449$	6.48844			0.306208			0.153104	+	0.988210i	$0.451073\pi$
$450$		0			0
$451$	22.9402	+	22.9402i	1.08021	+	1.08021i
$452$	−4.49195	+	18.5671i	−0.211283	+	0.873322i
$453$		0			0
$454$	−2.52008	+	21.1336i	−0.118273	+	0.991850i
$455$			0.181370i			0.00850278i
$456$		0			0
$457$		−	9.00353i		−	0.421167i	−0.977576	−	0.210584i	$-0.932464\pi$
$457$		−	9.00353i		−	0.421167i	0.977576	−	0.210584i	$-0.0675364\pi$
$458$	40.0586	+	4.77679i	1.87181	+	0.223205i
$459$		0			0
$460$	2.28687	−	1.39589i	0.106626	−	0.0650839i
$461$	−14.6218	−	14.6218i	−0.681004	−	0.681004i	0.279223	−	0.960226i	$-0.409923\pi$
$461$	−14.6218	−	14.6218i	−0.681004	−	0.681004i	−0.960226	+	0.279223i	$0.909923\pi$
$462$		0			0
$463$	−18.6435			−0.866437			−0.433219	−	0.901289i	$-0.642622\pi$
$463$	−18.6435			−0.866437			−0.433219	+	0.901289i	$0.642622\pi$
$464$	−8.92854	−	27.8101i	−0.414497	−	1.29105i
$465$		0			0
$466$	−15.0815	+	11.8679i	−0.698639	+	0.549772i
$467$	23.5138	+	23.5138i	1.08809	+	1.08809i	0.995725	+	0.0923633i	$0.0294421\pi$
$467$	23.5138	+	23.5138i	1.08809	+	1.08809i	0.0923633	+	0.995725i	$0.470558\pi$
$468$		0			0
$469$	17.2082	−	17.2082i	0.794601	−	0.794601i
$470$	1.88118	+	0.224321i	0.0867722	+	0.0103472i
$471$		0			0
$472$	−5.57726	+	14.9965i	−0.256714	+	0.690268i
$473$			7.96703i			0.366325i
$474$		0			0
$475$	−12.2145	+	12.2145i	−0.560440	+	0.560440i
$476$	−32.0461	−	7.75293i	−1.46883	−	0.355355i
$477$		0			0
$478$	25.6796	+	32.6331i	1.17456	+	1.49260i
$479$	1.08864			0.0497412			0.0248706	−	0.999691i	$-0.492083\pi$
$479$	1.08864			0.0497412			0.0248706	+	0.999691i	$0.492083\pi$
$480$		0			0
$481$	−0.521173			−0.0237634
$482$	−20.9949	−	26.6799i	−0.956291	−	1.21524i
$483$		0			0
$484$	2.40589	+	0.582059i	0.109359	+	0.0264572i
$485$	1.44586	−	1.44586i	0.0656531	−	0.0656531i
$486$		0			0
$487$		−	35.3298i		−	1.60095i	−0.599369	−	0.800473i	$-0.704582\pi$
$487$		−	35.3298i		−	1.60095i	0.599369	−	0.800473i	$-0.295418\pi$
$488$	10.2322	+	3.80540i	0.463188	+	0.172262i
$489$		0			0
$490$	9.15980	+	1.09226i	0.413798	+	0.0493434i
$491$	12.8910	−	12.8910i	0.581761	−	0.581761i	−0.353626	−	0.935387i	$-0.615051\pi$
$491$	12.8910	−	12.8910i	0.581761	−	0.581761i	0.935387	+	0.353626i	$0.115051\pi$
$492$		0			0
$493$	−18.6761	−	18.6761i	−0.841128	−	0.841128i
$494$	−0.337758	+	0.265788i	−0.0151964	+	0.0119584i
$495$		0			0
$496$	1.98391	+	1.01962i	0.0890803	+	0.0457822i
$497$	−41.5443			−1.86352
$498$		0			0
$499$	14.3798	+	14.3798i	0.643728	+	0.643728i	0.951470	−	0.307742i	$-0.0995734\pi$
$499$	14.3798	+	14.3798i	0.643728	+	0.643728i	−0.307742	+	0.951470i	$0.599573\pi$
$500$	−7.90393	+	4.82452i	−0.353474	+	0.215759i
$501$		0			0
$502$	31.2922	+	3.73145i	1.39664	+	0.166543i
$503$			30.2969i			1.35087i	0.737420	+	0.675435i	$0.236044\pi$
$503$			30.2969i			1.35087i	−0.737420	+	0.675435i	$0.763956\pi$
$504$		0			0
$505$		−	0.303911i		−	0.0135239i
$506$	−1.65685	+	13.8945i	−0.0736562	+	0.617686i
$507$		0			0
$508$	−2.39813	+	9.91245i	−0.106400	+	0.439794i
$509$	−10.5825	−	10.5825i	−0.469063	−	0.469063i	0.432548	−	0.901611i	$-0.357615\pi$
$509$	−10.5825	−	10.5825i	−0.469063	−	0.469063i	−0.901611	+	0.432548i	$0.857615\pi$
$510$		0			0
$511$	−2.46824			−0.109188
$512$	6.31333	−	21.7288i	0.279013	−	0.960287i
$513$		0			0
$514$	7.57401	+	9.62491i	0.334075	+	0.424536i
$515$	0.447718	+	0.447718i	0.0197288	+	0.0197288i
$516$		0			0
$517$	−6.99647	+	6.99647i	−0.307704	+	0.307704i
$518$	−4.73393	+	39.6991i	−0.207997	+	1.74428i
$519$		0			0
$520$	−0.102341	+	0.0468544i	−0.00448794	+	0.00205470i
$521$		−	24.9049i		−	1.09110i	−0.838078	−	0.545551i	$-0.816320\pi$
$521$		−	24.9049i		−	1.09110i	0.838078	−	0.545551i	$-0.183680\pi$
$522$		0			0
$523$	−12.9008	+	12.9008i	−0.564112	+	0.564112i	−0.930473	−	0.366361i	$-0.880604\pi$
$523$	−12.9008	+	12.9008i	−0.564112	+	0.564112i	0.366361	+	0.930473i	$0.380604\pi$
$524$	5.11529	−	3.12235i	0.223463	−	0.136401i
$525$		0			0
$526$	−14.8043	+	11.6498i	−0.645499	+	0.507955i
$527$	2.01704			0.0878638
$528$		0			0
$529$	15.0000			0.652174
$530$	−0.367381	+	0.289099i	−0.0159580	+	0.0125577i
$531$		0			0
$532$	17.1778	+	28.1421i	0.744754	+	1.22012i
$533$	0.550984	−	0.550984i	0.0238657	−	0.0238657i
$534$		0			0
$535$			4.06074i			0.175561i
$536$	14.1555	+	5.26449i	0.611423	+	0.227391i
$537$		0			0
$538$	2.76393	−	23.1785i	0.119161	−	0.999297i
$539$	−34.0671	+	34.0671i	−1.46738	+	1.46738i
$540$		0			0
$541$	18.2767	+	18.2767i	0.785776	+	0.785776i	0.980799	−	0.195023i	$-0.0624782\pi$
$541$	18.2767	+	18.2767i	0.785776	+	0.785776i	−0.195023	+	0.980799i	$0.562478\pi$
$542$	19.2200	+	24.4245i	0.825572	+	1.04912i
$543$		0			0
$544$	−3.90393	−	20.0853i	−0.167379	−	0.861150i
$545$	3.82880			0.164008
$546$		0			0
$547$	13.7355	+	13.7355i	0.587287	+	0.587287i	0.936896	−	0.349609i	$-0.113685\pi$
$547$	13.7355	+	13.7355i	0.587287	+	0.587287i	−0.349609	+	0.936896i	$0.613685\pi$
$548$	−6.56930	−	1.58932i	−0.280627	−	0.0678922i
$549$		0			0
$550$	2.79753	−	23.4603i	0.119287	−	1.00035i
$551$			26.4120i			1.12519i
$552$		0			0
$553$			50.1055i			2.13070i
$554$	21.7467	+	2.59319i	0.923929	+	0.110174i
$555$		0			0
$556$	8.66665	+	14.1984i	0.367548	+	0.602147i
$557$	27.5525	+	27.5525i	1.16744	+	1.16744i	0.982808	+	0.184631i	$0.0591089\pi$
$557$	27.5525	+	27.5525i	1.16744	+	1.16744i	0.184631	+	0.982808i	$0.440891\pi$
$558$		0			0
$559$	0.191354			0.00809342
$560$	2.63944	+	8.22117i	0.111537	+	0.347408i
$561$		0			0
$562$	25.4404	−	20.0195i	1.07314	−	0.844472i
$563$	19.8928	+	19.8928i	0.838383	+	0.838383i	0.988646	−	0.150263i	$-0.0480121\pi$
$563$	19.8928	+	19.8928i	0.838383	+	0.838383i	−0.150263	+	0.988646i	$0.548012\pi$
$564$		0			0
$565$	3.19879	−	3.19879i	0.134574	−	0.134574i
$566$	−8.90746	−	1.06217i	−0.374408	−	0.0446464i
$567$		0			0
$568$	−10.7324	−	23.4420i	−0.450320	−	0.983603i
$569$		−	13.4849i		−	0.565317i	−0.959221	−	0.282658i	$-0.908784\pi$
$569$		−	13.4849i		−	0.565317i	0.959221	−	0.282658i	$-0.0912163\pi$
$570$		0			0
$571$	−14.8284	+	14.8284i	−0.620550	+	0.620550i	−0.945672	−	0.325122i	$-0.894595\pi$
$571$	−14.8284	+	14.8284i	−0.620550	+	0.620550i	0.325122	+	0.945672i	$0.394595\pi$
$572$	0.138232	−	0.571371i	0.00577977	−	0.0238902i
$573$		0			0
$574$	−36.9652	−	46.9746i	−1.54290	−	1.96068i
$575$	13.5077			0.563308
$576$		0			0
$577$	−11.6176			−0.483648			−0.241824	−	0.970320i	$-0.577746\pi$
$577$	−11.6176			−0.483648			−0.241824	+	0.970320i	$0.577746\pi$
$578$	3.42554	+	4.35311i	0.142484	+	0.181066i
$579$		0			0
$580$	−1.62648	+	6.72293i	−0.0675360	+	0.279154i
$581$	48.5010	−	48.5010i	2.01216	−	2.01216i
$582$		0			0
$583$		−	2.44158i		−	0.101120i
$584$	−0.637633	−	1.39274i	−0.0263854	−	0.0576319i
$585$		0			0
$586$	42.9401	+	5.12040i	1.77384	+	0.211522i
$587$	17.0268	−	17.0268i	0.702773	−	0.702773i	−0.262232	−	0.965005i	$-0.584459\pi$
$587$	17.0268	−	17.0268i	0.702773	−	0.702773i	0.965005	+	0.262232i	$0.0844585\pi$
$588$		0			0
$589$	−1.42627	−	1.42627i	−0.0587682	−	0.0587682i
$590$	2.97762	−	2.34315i	0.122587	−	0.0964658i
$591$		0			0
$592$	−23.6237	+	7.58449i	−0.970929	+	0.311721i
$593$	−41.5372			−1.70573			−0.852865	−	0.522132i	$-0.825137\pi$
$593$	−41.5372			−1.70573			−0.852865	+	0.522132i	$0.825137\pi$
$594$		0			0
$595$	5.52099	+	5.52099i	0.226338	+	0.226338i
$596$	14.7241	+	24.1222i	0.603123	+	0.988085i
$597$		0			0
$598$	0.333722	+	0.0397948i	0.0136469	+	0.00162733i
$599$			6.43160i			0.262788i	0.991330	+	0.131394i	$0.0419453\pi$
$599$			6.43160i			0.262788i	−0.991330	+	0.131394i	$0.958055\pi$
$600$		0			0
$601$			3.45844i			0.141073i	0.997509	+	0.0705364i	$0.0224711\pi$
$601$			3.45844i			0.141073i	−0.997509	+	0.0705364i	$0.977529\pi$
$602$	1.73812	−	14.5760i	0.0708403	−	0.594072i
$603$		0			0
$604$	−19.3942	−	4.69205i	−0.789139	−	0.190917i
$605$	−0.414494	−	0.414494i	−0.0168516	−	0.0168516i
$606$		0			0
$607$	−30.1019			−1.22180			−0.610900	−	0.791708i	$-0.709192\pi$
$607$	−30.1019			−1.22180			−0.610900	+	0.791708i	$0.709192\pi$
$608$	−11.4420	+	16.9629i	−0.464033	+	0.687938i
$609$		0			0
$610$	−1.59874	−	2.03165i	−0.0647311	−	0.0822590i
$611$	0.168043	+	0.168043i	0.00679829	+	0.00679829i
$612$		0			0
$613$	2.50490	−	2.50490i	0.101172	−	0.101172i	−0.654709	−	0.755881i	$-0.727209\pi$
$613$	2.50490	−	2.50490i	0.101172	−	0.101172i	0.755881	+	0.654709i	$0.227209\pi$
$614$	−2.86822	+	24.0531i	−0.115752	+	0.970705i
$615$		0			0
$616$	−42.2672	−	15.7194i	−1.70300	−	0.633353i
$617$			22.9098i			0.922315i	0.887318	+	0.461157i	$0.152566\pi$
$617$			22.9098i			0.922315i	−0.887318	+	0.461157i	$0.847434\pi$
$618$		0			0
$619$	−28.6104	+	28.6104i	−1.14995	+	1.14995i	−0.163386	+	0.986562i	$0.552242\pi$
$619$	−28.6104	+	28.6104i	−1.14995	+	1.14995i	−0.986562	+	0.163386i	$0.947758\pi$
$620$	−0.275212	−	0.450874i	−0.0110528	−	0.0181076i
$621$		0			0
$622$	29.8571	−	23.4951i	1.19716	−	0.942067i
$623$	−66.7847			−2.67567
$624$		0			0
$625$	−21.6855			−0.867420
$626$	21.8818	−	17.2192i	0.874574	−	0.688218i
$627$		0			0
$628$	39.0179	−	23.8164i	1.55698	−	0.950377i
$629$	−15.8647	+	15.8647i	−0.632567	+	0.632567i
$630$		0			0
$631$		−	11.1851i		−	0.445270i	−0.974902	−	0.222635i	$-0.928534\pi$
$631$		−	11.1851i		−	0.445270i	0.974902	−	0.222635i	$-0.0714659\pi$
$632$	−28.2727	+	12.9440i	−1.12463	+	0.514885i
$633$		0			0
$634$	−5.05470	+	42.3891i	−0.200748	+	1.68349i
$635$	1.70774	−	1.70774i	0.0677698	−	0.0677698i
$636$		0			0
$637$	0.818234	+	0.818234i	0.0324196	+	0.0324196i
$638$	−22.3400	−	28.3892i	−0.884449	−	1.12394i
$639$		0			0
$640$	−3.95705	+	3.61316i	−0.156416	+	0.142823i
$641$	6.69312			0.264362			0.132181	−	0.991226i	$-0.457802\pi$
$641$	6.69312			0.264362			0.132181	+	0.991226i	$0.457802\pi$
$642$		0			0
$643$	−17.9410	−	17.9410i	−0.707522	−	0.707522i	0.258491	−	0.966014i	$-0.416775\pi$
$643$	−17.9410	−	17.9410i	−0.707522	−	0.707522i	−0.966014	+	0.258491i	$0.916775\pi$
$644$	6.06255	−	25.0590i	0.238898	−	0.987464i
$645$		0			0
$646$	−2.19079	+	18.3722i	−0.0861957	+	0.722843i
$647$			6.72999i			0.264583i	0.991211	+	0.132292i	$0.0422335\pi$
$647$			6.72999i			0.264583i	−0.991211	+	0.132292i	$0.957766\pi$
$648$		0			0
$649$			19.7890i			0.776786i
$650$	−0.563475	−	0.0671918i	−0.0221013	−	0.00263548i
$651$		0			0
$652$	18.1184	−	11.0594i	0.709572	−	0.433120i
$653$	−26.1731	−	26.1731i	−1.02423	−	1.02423i	−0.999699	−	0.0245347i	$-0.992190\pi$
$653$	−26.1731	−	26.1731i	−1.02423	−	1.02423i	−0.0245347	−	0.999699i	$-0.507810\pi$
$654$		0			0
$655$	−1.41921			−0.0554529
$656$	16.9567	−	32.9933i	0.662047	−	1.28817i
$657$		0			0
$658$	14.3267	−	11.2739i	0.558511	−	0.439503i
$659$	−13.9741	−	13.9741i	−0.544353	−	0.544353i	0.380449	−	0.924802i	$-0.375770\pi$
$659$	−13.9741	−	13.9741i	−0.544353	−	0.544353i	−0.924802	+	0.380449i	$0.875770\pi$
$660$		0			0
$661$	11.9241	−	11.9241i	0.463794	−	0.463794i	−0.436103	−	0.899897i	$-0.643642\pi$
$661$	11.9241	−	11.9241i	0.463794	−	0.463794i	0.899897	+	0.436103i	$0.143642\pi$
$662$	−29.1784	−	3.47939i	−1.13405	−	0.135230i
$663$		0			0
$664$	39.8969	+	14.8379i	1.54830	+	0.575820i
$665$		−	7.80785i		−	0.302776i
$666$		0			0
$667$	14.6041	−	14.6041i	0.565473	−	0.565473i
$668$	−11.3490	−	2.74568i	−0.439108	−	0.106234i
$669$		0			0
$670$	−2.21174	−	2.81064i	−0.0854470	−	0.108584i
$671$	13.5021			0.521244
$672$		0			0
$673$	−37.3066			−1.43807			−0.719033	−	0.694976i	$-0.755415\pi$
$673$	−37.3066			−1.43807			−0.719033	+	0.694976i	$0.755415\pi$
$674$	20.1234	+	25.5724i	0.775125	+	0.985013i
$675$		0			0
$676$	25.2572	+	6.11050i	0.971432	+	0.235019i
$677$	−0.447461	+	0.447461i	−0.0171973	+	0.0171973i	−0.715653	−	0.698456i	$-0.753871\pi$
$677$	−0.447461	+	0.447461i	−0.0171973	+	0.0171973i	0.698456	+	0.715653i	$0.253871\pi$
$678$		0			0
$679$		−	19.6764i		−	0.755113i
$680$	−1.68903	+	4.54156i	−0.0647713	+	0.174161i
$681$		0			0
$682$	2.73941	+	0.326662i	0.104898	+	0.0125085i
$683$	4.27521	−	4.27521i	0.163586	−	0.163586i	−0.620567	−	0.784153i	$-0.713098\pi$
$683$	4.27521	−	4.27521i	0.163586	−	0.163586i	0.784153	+	0.620567i	$0.213098\pi$
$684$		0			0
$685$	1.13178	+	1.13178i	0.0432430	+	0.0432430i
$686$	34.3026	−	26.9933i	1.30968	−	1.03061i
$687$		0			0
$688$	8.67371	−	2.78473i	0.330682	−	0.106167i
$689$	−0.0586426			−0.00223410
$690$		0			0
$691$	20.0786	+	20.0786i	0.763827	+	0.763827i	0.977012	−	0.213185i	$-0.0683836\pi$
$691$	20.0786	+	20.0786i	0.763827	+	0.763827i	−0.213185	+	0.977012i	$0.568384\pi$
$692$	−8.75225	+	5.34234i	−0.332711	+	0.203085i
$693$		0			0
$694$	−21.6519	−	2.58188i	−0.821893	−	0.0980069i
$695$		−	3.93926i		−	0.149425i
$696$		0			0
$697$		−	33.5443i		−	1.27058i
$698$	4.74958	−	39.8303i	0.179774	−	1.50760i
$699$		0			0
$700$	−10.2364	+	42.3111i	−0.386898	+	1.59921i
$701$	10.4467	+	10.4467i	0.394565	+	0.394565i	0.876311	−	0.481746i	$-0.159997\pi$
$701$	10.4467	+	10.4467i	0.394565	+	0.394565i	−0.481746	+	0.876311i	$0.659997\pi$
$702$		0			0
$703$	22.4361			0.846192
$704$	−2.04922	−	27.9108i	−0.0772328	−	1.05193i
$705$		0			0
$706$	−10.6799	−	13.5718i	−0.401943	−	0.510781i
$707$	−2.06793	−	2.06793i	−0.0777727	−	0.0777727i
$708$		0			0
$709$	16.0916	−	16.0916i	0.604332	−	0.604332i	−0.337127	−	0.941459i	$-0.609455\pi$
$709$	16.0916	−	16.0916i	0.604332	−	0.604332i	0.941459	+	0.337127i	$0.109455\pi$
$710$	−0.722930	+	6.06255i	−0.0271311	+	0.227523i
$711$		0			0
$712$	−17.2528	−	37.6842i	−0.646577	−	1.41228i
$713$			1.57726i			0.0590690i
$714$		0			0
$715$	−0.0984373	+	0.0984373i	−0.00368135	+	0.00368135i
$716$	22.4120	−	13.6802i	0.837574	−	0.511252i
$717$		0			0
$718$	−37.2065	+	29.2785i	−1.38854	+	1.09266i
$719$	30.9957			1.15594			0.577972	−	0.816057i	$-0.303844\pi$
$719$	30.9957			1.15594			0.577972	+	0.816057i	$0.303844\pi$
$720$		0			0
$721$	6.09292			0.226912
$722$	−6.57585	+	5.17466i	−0.244728	+	0.192581i
$723$		0			0
$724$	−15.9617	−	26.1497i	−0.593211	−	0.971846i
$725$	−24.6584	+	24.6584i	−0.915790	+	0.915790i
$726$		0			0
$727$			41.1117i			1.52475i	0.647135	+	0.762375i	$0.275967\pi$
$727$			41.1117i			1.52475i	−0.647135	+	0.762375i	$0.724033\pi$
$728$	−0.377553	+	1.01519i	−0.0139930	+	0.0376253i
$729$		0			0
$730$	−0.0429509	+	0.360189i	−0.00158968	+	0.0133312i
$731$	5.82490	−	5.82490i	0.215442	−	0.215442i
$732$		0			0
$733$	0.146061	+	0.146061i	0.00539490	+	0.00539490i	0.709799	−	0.704404i	$-0.248786\pi$
$733$	0.146061	+	0.146061i	0.00539490	+	0.00539490i	−0.704404	+	0.709799i	$0.748786\pi$
$734$	0.614272	+	0.780604i	0.0226732	+	0.0288126i
$735$		0			0
$736$	15.7061	−	3.05275i	0.578934	−	0.112526i
$737$	18.6792			0.688058
$738$		0			0
$739$	−1.50766	−	1.50766i	−0.0554601	−	0.0554601i	0.678833	−	0.734293i	$-0.262486\pi$
$739$	−1.50766	−	1.50766i	−0.0554601	−	0.0554601i	−0.734293	+	0.678833i	$0.762486\pi$
$740$	5.71090	+	1.38164i	0.209937	+	0.0507902i
$741$		0			0
$742$	−0.532664	+	4.46696i	−0.0195547	+	0.163987i
$743$		−	40.5175i		−	1.48644i	−0.669046	−	0.743221i	$-0.733297\pi$
$743$		−	40.5175i		−	1.48644i	0.669046	−	0.743221i	$-0.266703\pi$
$744$		0			0
$745$		−	6.69256i		−	0.245196i
$746$	−37.6529	−	4.48993i	−1.37857	−	0.164388i
$747$		0			0
$748$	−13.1849	−	21.6006i	−0.482088	−	0.789795i
$749$	27.6309	+	27.6309i	1.00961	+	1.00961i
$750$		0			0
$751$	−12.5843			−0.459208			−0.229604	−	0.973284i	$-0.573743\pi$
$751$	−12.5843			−0.459208			−0.229604	+	0.973284i	$0.573743\pi$
$752$	10.0625	+	5.17157i	0.366943	+	0.188588i
$753$		0			0
$754$	−0.681859	+	0.536568i	−0.0248319	+	0.0195406i
$755$	3.34129	+	3.34129i	0.121602	+	0.121602i
$756$		0			0
$757$	−7.49900	+	7.49900i	−0.272556	+	0.272556i	−0.830128	−	0.557572i	$-0.811733\pi$
$757$	−7.49900	+	7.49900i	−0.272556	+	0.272556i	0.557572	+	0.830128i	$0.311733\pi$
$758$	3.53186	+	0.421157i	0.128283	+	0.0152971i
$759$		0			0
$760$	4.40569	−	2.01704i	0.159811	−	0.0731659i
$761$			42.8182i			1.55216i	0.630635	+	0.776079i	$0.282794\pi$
$761$			42.8182i			1.55216i	−0.630635	+	0.776079i	$0.717206\pi$
$762$		0			0
$763$	26.0527	−	26.0527i	0.943172	−	0.943172i
$764$	−4.05864	+	16.7761i	−0.146837	+	0.606936i
$765$		0			0
$766$	22.2908	+	28.3267i	0.805398	+	1.02348i
$767$	0.475298			0.0171620
$768$		0			0
$769$	12.7455			0.459614			0.229807	−	0.973236i	$-0.426190\pi$
$769$	12.7455			0.459614			0.229807	+	0.973236i	$0.426190\pi$
$770$	6.60411	+	8.39236i	0.237995	+	0.302440i
$771$		0			0
$772$	−5.38551	+	22.2606i	−0.193829	+	0.801175i
$773$	22.8765	−	22.8765i	0.822809	−	0.822809i	−0.163701	−	0.986510i	$-0.552343\pi$
$773$	22.8765	−	22.8765i	0.822809	−	0.822809i	0.986510	+	0.163701i	$0.0523432\pi$
$774$		0			0
$775$		−	2.66314i		−	0.0956630i
$776$	11.1027	−	5.08312i	0.398564	−	0.182473i
$777$		0			0
$778$	−23.2451	−	2.77187i	−0.833377	−	0.0993763i
$779$	−23.7194	+	23.7194i	−0.849836	+	0.849836i
$780$		0			0
$781$	−22.5478	−	22.5478i	−0.806825	−	0.806825i
$782$	11.3700	−	8.94725i	0.406590	−	0.319953i
$783$		0			0
$784$	48.9964	+	25.1814i	1.74987	+	0.899335i
$785$	−10.8253			−0.386370
$786$		0			0
$787$	5.20470	+	5.20470i	0.185528	+	0.185528i	0.793759	−	0.608232i	$-0.208121\pi$
$787$	5.20470	+	5.20470i	0.185528	+	0.185528i	−0.608232	+	0.793759i	$0.708121\pi$
$788$	−11.0374	−	18.0824i	−0.393193	−	0.644160i
$789$		0			0
$790$	7.31187	+	0.871906i	0.260145	+	0.0310210i
$791$		−	43.5317i		−	1.54781i
$792$		0			0
$793$		−	0.324298i		−	0.0115162i
$794$	−2.14232	+	17.9656i	−0.0760280	+	0.637577i
$795$		0			0
$796$	7.15363	+	1.73068i	0.253554	+	0.0613424i
$797$	−17.0149	−	17.0149i	−0.602698	−	0.602698i	0.338330	−	0.941028i	$-0.390138\pi$
$797$	−17.0149	−	17.0149i	−0.602698	−	0.602698i	−0.941028	+	0.338330i	$0.890138\pi$
$798$		0			0
$799$	10.2306			0.361932
$800$	−26.5190	+	5.15444i	−0.937589	+	0.182237i
$801$		0			0
$802$	−15.8167	−	20.0995i	−0.558506	−	0.709738i
$803$	−1.33962	−	1.33962i	−0.0472740	−	0.0472740i
$804$		0			0
$805$	−4.31724	+	4.31724i	−0.152163	+	0.152163i
$806$	0.00784586	−	0.0657960i	0.000276359	−	0.00231756i
$807$		0			0
$808$	0.632641	−	1.70108i	0.0222562	−	0.0598438i
$809$			7.83586i			0.275494i	0.990467	+	0.137747i	$0.0439861\pi$
$809$			7.83586i			0.275494i	−0.990467	+	0.137747i	$0.956014\pi$
$810$		0			0
$811$	32.3396	−	32.3396i	1.13560	−	1.13560i	0.146366	−	0.989230i	$-0.453242\pi$
$811$	32.3396	−	32.3396i	1.13560	−	1.13560i	0.989230	−	0.146366i	$-0.0467578\pi$
$812$	34.6783	+	56.8128i	1.21697	+	1.99374i
$813$		0			0
$814$	−24.1157	+	18.9771i	−0.845254	+	0.665146i
$815$	−5.02684			−0.176083
$816$		0			0
$817$	−8.23765			−0.288199
$818$	−28.0366	+	22.0625i	−0.980278	+	0.771399i
$819$		0			0
$820$	−7.49824	+	4.57689i	−0.261850	+	0.159832i
$821$	−19.3541	+	19.3541i	−0.675464	+	0.675464i	−0.958970	−	0.283507i	$-0.908502\pi$
$821$	−19.3541	+	19.3541i	−0.675464	+	0.675464i	0.283507	+	0.958970i	$0.408502\pi$
$822$		0			0
$823$		−	28.8560i		−	1.00586i	−0.864328	−	0.502929i	$-0.832256\pi$
$823$		−	28.8560i		−	1.00586i	0.864328	−	0.502929i	$-0.167744\pi$
$824$	1.57401	+	3.43801i	0.0548334	+	0.119769i
$825$		0			0
$826$	4.31724	−	36.2047i	0.150216	−	1.25972i
$827$	10.1984	−	10.1984i	0.354634	−	0.354634i	−0.507197	−	0.861830i	$-0.669318\pi$
$827$	10.1984	−	10.1984i	0.354634	−	0.354634i	0.861830	+	0.507197i	$0.169318\pi$
$828$		0			0
$829$	15.3794	+	15.3794i	0.534148	+	0.534148i	0.921804	−	0.387656i	$-0.126715\pi$
$829$	15.3794	+	15.3794i	0.534148	+	0.534148i	−0.387656	+	0.921804i	$0.626715\pi$
$830$	−6.23375	−	7.92172i	−0.216376	−	0.274967i
$831$		0			0
$832$	−0.670368	+	0.0492187i	−0.0232408	+	0.00170635i
$833$	49.8147			1.72598
$834$		0			0
$835$	1.95524	+	1.95524i	0.0676640	+	0.0676640i
$836$	−5.95078	+	24.5970i	−0.205812	+	0.850707i
$837$		0			0
$838$	−1.71756	+	14.4036i	−0.0593323	+	0.497565i
$839$		−	44.4557i		−	1.53478i	−0.641181	−	0.767390i	$-0.721555\pi$
$839$		−	44.4557i		−	1.53478i	0.641181	−	0.767390i	$-0.278445\pi$
$840$		0			0
$841$			24.3200i			0.838620i
$842$	−4.75729	−	0.567285i	−0.163947	−	0.0195499i
$843$		0			0
$844$	−24.4290	+	14.9114i	−0.840881	+	0.513270i
$845$	−4.35139	−	4.35139i	−0.149692	−	0.149692i
$846$		0			0
$847$	−5.64077			−0.193819
$848$	−2.65815	+	0.853410i	−0.0912813	+	0.0293062i
$849$		0			0
$850$	−19.1977	+	15.1071i	−0.658477	+	0.518168i
$851$	−12.4057	−	12.4057i	−0.425262	−	0.425262i
$852$		0			0
$853$	−11.7131	+	11.7131i	−0.401049	+	0.401049i	−0.878603	−	0.477553i	$-0.841524\pi$
$853$	−11.7131	+	11.7131i	−0.401049	+	0.401049i	0.477553	+	0.878603i	$0.341524\pi$
$854$	−24.7026	−	2.94567i	−0.845307	−	0.100799i
$855$		0			0
$856$	−8.45311	+	22.7292i	−0.288921	+	0.776868i
$857$		−	19.0888i		−	0.652062i	−0.945359	−	0.326031i	$-0.894289\pi$
$857$		−	19.0888i		−	0.652062i	0.945359	−	0.326031i	$-0.105711\pi$
$858$		0			0
$859$	−38.1323	+	38.1323i	−1.30106	+	1.30106i	−0.373379	+	0.927679i	$0.621801\pi$
$859$	−38.1323	+	38.1323i	−1.30106	+	1.30106i	−0.927679	+	0.373379i	$0.878199\pi$
$860$	−2.09682	−	0.507285i	−0.0715010	−	0.0172983i
$861$		0			0
$862$	−3.86952	−	4.91731i	−0.131796	−	0.167484i
$863$	−3.64533			−0.124089			−0.0620443	−	0.998073i	$-0.519762\pi$
$863$	−3.64533			−0.124089			−0.0620443	+	0.998073i	$0.519762\pi$
$864$		0			0
$865$	2.42826			0.0825632
$866$	−6.39627	−	8.12825i	−0.217354	−	0.276209i
$867$		0			0
$868$	−4.94059	−	1.19528i	−0.167694	−	0.0405704i
$869$	−27.1943	+	27.1943i	−0.922504	+	0.922504i
$870$		0			0
$871$		−	0.448643i		−	0.0152017i
$872$	21.4310	+	7.97028i	0.725744	+	0.269908i
$873$		0			0
$874$	−14.3665	−	1.71313i	−0.485953	−	0.0579476i
$875$	14.9213	−	14.9213i	0.504433	−	0.504433i
$876$		0			0
$877$	40.0563	+	40.0563i	1.35260	+	1.35260i	0.882738	+	0.469866i	$0.155698\pi$
$877$	40.0563	+	40.0563i	1.35260	+	1.35260i	0.469866	+	0.882738i	$0.344302\pi$
$878$	−32.9558	+	25.9336i	−1.11221	+	0.875215i
$879$		0			0
$880$	−3.02944	+	5.89450i	−0.102122	+	0.198704i
$881$	20.0118			0.674214			0.337107	−	0.941466i	$-0.390552\pi$
$881$	20.0118			0.674214			0.337107	+	0.941466i	$0.390552\pi$
$882$		0			0
$883$	−10.6273	−	10.6273i	−0.357636	−	0.357636i	0.505305	−	0.862941i	$-0.331380\pi$
$883$	−10.6273	−	10.6273i	−0.357636	−	0.357636i	−0.862941	+	0.505305i	$0.831380\pi$
$884$	−0.518808	+	0.316678i	−0.0174494	+	0.0106510i
$885$		0			0
$886$	20.4662	+	2.44049i	0.687574	+	0.0819900i
$887$			26.1180i			0.876958i	0.898742	+	0.438479i	$0.144483\pi$
$887$			26.1180i			0.876958i	−0.898742	+	0.438479i	$0.855517\pi$
$888$		0			0
$889$		−	23.2404i		−	0.779458i
$890$	−1.16215	+	9.74587i	−0.0389553	+	0.326682i
$891$		0			0
$892$	2.28636	−	9.45048i	0.0765530	−	0.316425i
$893$	−7.23412	−	7.23412i	−0.242081	−	0.242081i
$894$		0			0
$895$	−6.21805			−0.207847
$896$	−2.33999	+	51.5108i	−0.0781736	+	1.72085i
$897$		0			0
$898$	−5.67452	−	7.21107i	−0.189361	−	0.240636i
$899$	−2.87932	−	2.87932i	−0.0960306	−	0.0960306i
$900$		0			0
$901$	−1.78510	+	1.78510i	−0.0594704	+	0.0594704i
$902$	5.43253	−	45.5576i	0.180884	−	1.51690i
$903$		0			0
$904$	24.5634	−	11.2458i	0.816966	−	0.374029i
$905$			7.25507i			0.241167i
$906$		0			0
$907$	36.2378	−	36.2378i	1.20326	−	1.20326i	0.230087	−	0.973170i	$-0.426099\pi$
$907$	36.2378	−	36.2378i	1.20326	−	1.20326i	0.973170	−	0.230087i	$-0.0739010\pi$
$908$	25.6913	−	15.6818i	0.852594	−	0.520420i
$909$		0			0
$910$	0.201570	−	0.158619i	0.00668198	−	0.00525817i
$911$	21.0535			0.697533			0.348767	−	0.937210i	$-0.386601\pi$
$911$	21.0535			0.697533			0.348767	+	0.937210i	$0.386601\pi$
$912$		0			0
$913$	52.6470			1.74236
$914$	−10.0063	+	7.87412i	−0.330978	+	0.260453i
$915$		0			0
$916$	−29.7248	−	48.6975i	−0.982134	−	1.60901i
$917$	−9.65685	+	9.65685i	−0.318897	+	0.318897i
$918$		0			0
$919$			17.8839i			0.589937i	0.955507	+	0.294968i	$0.0953091\pi$
$919$			17.8839i			0.589937i	−0.955507	+	0.294968i	$0.904691\pi$
$920$	−3.55136	−	1.32077i	−0.117085	−	0.0435444i
$921$		0			0
$922$	−3.46262	+	29.0378i	−0.114035	+	0.956310i
$923$	−0.541560	+	0.541560i	−0.0178257	+	0.0178257i
$924$		0			0
$925$	20.9465	+	20.9465i	0.688716	+	0.688716i
$926$	16.3048	+	20.7199i	0.535810	+	0.680897i
$927$		0			0
$928$	−23.0988	+	34.2445i	−0.758256	+	1.12413i
$929$	−10.2774			−0.337192			−0.168596	−	0.985685i	$-0.553923\pi$
$929$	−10.2774			−0.337192			−0.168596	+	0.985685i	$0.553923\pi$
$930$		0			0
$931$	−35.2243	−	35.2243i	−1.15443	−	1.15443i
$932$	26.3794	+	6.38198i	0.864086	+	0.209049i
$933$		0			0
$934$	5.56837	−	46.6968i	0.182203	−	1.52796i
$935$			5.99294i			0.195990i
$936$		0			0
$937$		−	13.5780i		−	0.443574i	−0.975095	−	0.221787i	$-0.928811\pi$
$937$		−	13.5780i		−	0.443574i	0.975095	−	0.221787i	$-0.0711890\pi$
$938$	−34.1743	−	4.07513i	−1.11583	−	0.133058i
$939$		0			0
$940$	−1.39589	−	2.28687i	−0.0455290	−	0.0745894i
$941$	3.95902	+	3.95902i	0.129060	+	0.129060i	0.768686	−	0.639626i	$-0.220911\pi$
$941$	3.95902	+	3.95902i	0.129060	+	0.129060i	−0.639626	+	0.768686i	$0.720911\pi$
$942$		0			0
$943$	26.2306			0.854186
$944$	21.5443	−	6.91688i	0.701207	−	0.225125i
$945$		0			0
$946$	8.85433	−	6.96764i	0.287879	−	0.226537i
$947$	33.1708	+	33.1708i	1.07791	+	1.07791i	0.996697	+	0.0812084i	$0.0258779\pi$
$947$	33.1708	+	33.1708i	1.07791	+	1.07791i	0.0812084	+	0.996697i	$0.474122\pi$
$948$		0			0
$949$	−0.0321752	+	0.0321752i	−0.00104445	+	0.00104445i
$950$	24.2572	+	2.89255i	0.787006	+	0.0938468i
$951$		0			0
$952$	19.4098	+	42.3955i	0.629075	+	1.37405i
$953$			5.59115i			0.181115i	0.995891	+	0.0905576i	$0.0288649\pi$
$953$			5.59115i			0.181115i	−0.995891	+	0.0905576i	$0.971135\pi$
$954$		0			0
$955$	2.89023	−	2.89023i	0.0935255	−	0.0935255i
$956$	13.8092	−	57.0792i	0.446622	−	1.84607i
$957$		0			0
$958$	−0.952079	−	1.20988i	−0.0307603	−	0.0390896i
$959$	15.4022			0.497362
$960$		0			0
$961$	−30.6890			−0.989969
$962$	0.455796	+	0.579216i	0.0146955	+	0.0186747i
$963$		0			0
$964$	−11.2900	+	46.6663i	−0.363626	+	1.50302i
$965$	3.83511	−	3.83511i	0.123457	−	0.123457i
$966$		0			0
$967$			30.7561i			0.989048i	0.869164	+	0.494524i	$0.164658\pi$
$967$			30.7561i			0.989048i	−0.869164	+	0.494524i	$0.835342\pi$
$968$	−1.45721	−	3.18288i	−0.0468365	−	0.102302i
$969$		0			0
$970$	−2.87138	−	0.342398i	−0.0921944	−	0.0109937i
$971$	−8.03756	+	8.03756i	−0.257938	+	0.257938i	−0.824215	−	0.566277i	$-0.808383\pi$
$971$	−8.03756	+	8.03756i	−0.257938	+	0.257938i	0.566277	+	0.824215i	$0.308383\pi$
$972$		0			0
$973$	−26.8043	−	26.8043i	−0.859307	−	0.859307i
$974$	−39.2645	+	30.8980i	−1.25812	+	0.990035i
$975$		0			0
$976$	−4.71942	−	14.6998i	−0.151065	−	0.470529i
$977$	−22.8323			−0.730471			−0.365235	−	0.930915i	$-0.619012\pi$
$977$	−22.8323			−0.730471			−0.365235	+	0.930915i	$0.619012\pi$
$978$		0			0
$979$	−36.2468	−	36.2468i	−1.15845	−	1.15845i
$980$	−6.79687	−	11.1352i	−0.217118	−	0.355701i
$981$		0			0
$982$	−25.6006	−	3.05275i	−0.816947	−	0.0974171i
$983$			46.3557i			1.47852i	0.673422	+	0.739258i	$0.264824\pi$
$983$			46.3557i			1.47852i	−0.673422	+	0.739258i	$0.735176\pi$
$984$		0			0
$985$			5.01686i			0.159850i
$986$	−4.42274	+	37.0894i	−0.140849	+	1.18117i
$987$		0			0
$988$	0.590778	+	0.142927i	0.0187952	+	0.00454713i
$989$	4.55489	+	4.55489i	0.144837	+	0.144837i
$990$		0			0
$991$	3.43683			0.109175			0.0545873	−	0.998509i	$-0.482616\pi$
$991$	3.43683			0.109175			0.0545873	+	0.998509i	$0.482616\pi$
$992$	−0.601874	−	3.09658i	−0.0191095	−	0.0983165i
$993$		0			0
$994$	36.3329	+	46.1712i	1.15241	+	1.46446i
$995$	−1.23245	−	1.23245i	−0.0390712	−	0.0390712i
$996$		0			0
$997$	−21.9430	+	21.9430i	−0.694940	+	0.694940i	−0.963315	−	0.268374i	$-0.913514\pi$
$997$	−21.9430	+	21.9430i	−0.694940	+	0.694940i	0.268374	+	0.963315i	$0.413514\pi$
$998$	3.40532	−	28.5573i	0.107793	−	0.903964i
$999$		0			0

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	144.2.k.b.109.1		8
3.2	odd	2		48.2.j.a.13.4	✓	8
4.3	odd	2		576.2.k.b.145.3		8
8.3	odd	2		1152.2.k.f.289.2		8
8.5	even	2		1152.2.k.c.289.2		8
12.11	even	2		192.2.j.a.145.3		8
16.3	odd	4		1152.2.k.f.865.2		8
16.5	even	4	inner	144.2.k.b.37.1		8
16.11	odd	4		576.2.k.b.433.3		8
16.13	even	4		1152.2.k.c.865.2		8
24.5	odd	2		384.2.j.b.289.4		8
24.11	even	2		384.2.j.a.289.2		8
32.5	even	8		9216.2.a.y.1.3		4
32.11	odd	8		9216.2.a.bn.1.2		4
32.21	even	8		9216.2.a.bo.1.2		4
32.27	odd	8		9216.2.a.x.1.3		4
48.5	odd	4		48.2.j.a.37.4	yes	8
48.11	even	4		192.2.j.a.49.3		8
48.29	odd	4		384.2.j.b.97.4		8
48.35	even	4		384.2.j.a.97.2		8
96.5	odd	8		3072.2.a.t.1.2		4
96.11	even	8		3072.2.a.o.1.3		4
96.29	odd	8		3072.2.d.f.1537.2		8
96.35	even	8		3072.2.d.i.1537.6		8
96.53	odd	8		3072.2.a.i.1.3		4
96.59	even	8		3072.2.a.n.1.2		4
96.77	odd	8		3072.2.d.f.1537.7		8
96.83	even	8		3072.2.d.i.1537.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.2.j.a.13.4	✓	8	3.2	odd	2
48.2.j.a.37.4	yes	8	48.5	odd	4
144.2.k.b.37.1		8	16.5	even	4	inner
144.2.k.b.109.1		8	1.1	even	1	trivial
192.2.j.a.49.3		8	48.11	even	4
192.2.j.a.145.3		8	12.11	even	2
384.2.j.a.97.2		8	48.35	even	4
384.2.j.a.289.2		8	24.11	even	2
384.2.j.b.97.4		8	48.29	odd	4
384.2.j.b.289.4		8	24.5	odd	2
576.2.k.b.145.3		8	4.3	odd	2
576.2.k.b.433.3		8	16.11	odd	4
1152.2.k.c.289.2		8	8.5	even	2
1152.2.k.c.865.2		8	16.13	even	4
1152.2.k.f.289.2		8	8.3	odd	2
1152.2.k.f.865.2		8	16.3	odd	4
3072.2.a.i.1.3		4	96.53	odd	8
3072.2.a.n.1.2		4	96.59	even	8
3072.2.a.o.1.3		4	96.11	even	8
3072.2.a.t.1.2		4	96.5	odd	8
3072.2.d.f.1537.2		8	96.29	odd	8
3072.2.d.f.1537.7		8	96.77	odd	8
3072.2.d.i.1537.3		8	96.83	even	8
3072.2.d.i.1537.6		8	96.35	even	8
9216.2.a.x.1.3		4	32.27	odd	8
9216.2.a.y.1.3		4	32.5	even	8
9216.2.a.bn.1.2		4	32.11	odd	8
9216.2.a.bo.1.2		4	32.21	even	8

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	144.2.k.b.109.1

Level	$144$
Weight	$2$
Character	144.109
Analytic conductor	$1.150$
Analytic rank	$0$
Dimension	$8$
Inner twists	$2$