LMFDB - Embedded newform 3276.2.cf.c.2773.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3276,2,Mod(1765,3276)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3276, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 5]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3276.1765");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$3276 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 13$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	3276.cf (of order $6$ , degree $2$ , minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$26.1589917022$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{16} + 38x^{14} + 587x^{12} + 4762x^{10} + 21849x^{8} + 56552x^{6} + 76456x^{4} + 42624x^{2} + 2704$ x^16 + 38x^14 + 587x^12 + 4762x^10 + 21849x^8 + 56552x^6 + 76456x^4 + 42624*x^2 + 2704
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$2^{2}$
Twist minimal:	no (minimal twist has level 364)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			2773.5
Root			$-0.268953i$ of defining polynomial
Character	$\chi$	$=$	3276.2773
Dual form			3276.2.cf.c.1765.4

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.35585i q^{5} +(0.866025 - 0.500000i) q^{7} +(-2.42763 - 1.40159i) q^{11} +(0.303042 - 3.59279i) q^{13} +(2.90934 + 5.03912i) q^{17} +(2.75793 - 1.59229i) q^{19} +(-3.77674 + 6.54150i) q^{23} +3.16167 q^{25} +(3.60830 - 6.24977i) q^{29} +8.04407i q^{31} +(0.677925 + 1.17420i) q^{35} +(-6.02532 - 3.47872i) q^{37} +(-4.38164 - 2.52974i) q^{41} +(3.13427 + 5.42872i) q^{43} -2.64315i q^{47} +(0.500000 - 0.866025i) q^{49} +11.0887 q^{53} +(1.90035 - 3.29150i) q^{55} +(5.45213 - 3.14779i) q^{59} +(-6.40404 - 11.0921i) q^{61} +(4.87129 + 0.410880i) q^{65} +(4.29803 + 2.48147i) q^{67} +(14.2993 - 8.25572i) q^{71} +15.4351i q^{73} -2.80318 q^{77} +14.7661 q^{79} +9.79310i q^{83} +(-6.83230 + 3.94463i) q^{85} +(7.24707 + 4.18410i) q^{89} +(-1.53395 - 3.26297i) q^{91} +(2.15891 + 3.73934i) q^{95} +(-10.5685 + 6.10170i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$16 q - 6 q^{11} + 10 q^{13} - 2 q^{17} - 44 q^{25} + 22 q^{29} + 6 q^{35} + 12 q^{37} - 36 q^{41} + 6 q^{43} + 8 q^{49} - 8 q^{53} + 2 q^{55} + 18 q^{59} + 4 q^{61} + 30 q^{65} + 24 q^{67} - 36 q^{71} + 24 q^{77}+ \cdots - 42 q^{97}+O(q^{100})$ 16 * q - 6 * q^11 + 10 * q^13 - 2 * q^17 - 44 * q^25 + 22 * q^29 + 6 * q^35 + 12 * q^37 - 36 * q^41 + 6 * q^43 + 8 * q^49 - 8 * q^53 + 2 * q^55 + 18 * q^59 + 4 * q^61 + 30 * q^65 + 24 * q^67 - 36 * q^71 + 24 * q^77 + 8 * q^79 + 42 * q^85 + 36 * q^89 - 2 * q^91 + 30 * q^95 - 42 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3276\mathbb{Z}\right)^\times$ .

$n$	$1639$	$2017$	$2341$	$2549$
$\chi(n)$	$1$	$e\left(\frac{1}{6}\right)$	$1$	$1$

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			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$			1.35585i			0.606355i	0.952934	+	0.303177i	$0.0980475\pi$
$5$			1.35585i			0.606355i	−0.952934	+	0.303177i	$0.901953\pi$
$6$		0			0
$7$	0.866025	−	0.500000i	0.327327	−	0.188982i
$7$	0.866025	−	0.500000i	0.327327	−	0.188982i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.42763	−	1.40159i	−0.731958	−	0.422596i	0.0871803	−	0.996193i	$-0.472214\pi$
$11$	−2.42763	−	1.40159i	−0.731958	−	0.422596i	−0.819138	+	0.573597i	$0.805548\pi$
$12$		0			0
$13$	0.303042	−	3.59279i	0.0840488	−	0.996462i
$13$	0.303042	−	3.59279i	0.0840488	−	0.996462i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.90934	+	5.03912i	0.705618	+	1.22217i	0.966468	+	0.256787i	$0.0826640\pi$
$17$	2.90934	+	5.03912i	0.705618	+	1.22217i	−0.260850	+	0.965379i	$0.584003\pi$
$18$		0			0
$19$	2.75793	−	1.59229i	0.632713	−	0.365297i	−0.149089	−	0.988824i	$-0.547634\pi$
$19$	2.75793	−	1.59229i	0.632713	−	0.365297i	0.781802	+	0.623527i	$0.214301\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.77674	+	6.54150i	−0.787504	+	1.36400i	0.139988	+	0.990153i	$0.455294\pi$
$23$	−3.77674	+	6.54150i	−0.787504	+	1.36400i	−0.927492	+	0.373843i	$0.878040\pi$
$24$		0			0
$25$	3.16167			0.632334
$26$		0			0
$27$		0			0
$28$		0			0
$29$	3.60830	−	6.24977i	0.670045	−	1.16055i	−0.307845	−	0.951436i	$-0.599608\pi$
$29$	3.60830	−	6.24977i	0.670045	−	1.16055i	0.977891	−	0.209116i	$-0.0670587\pi$
$30$		0			0
$31$			8.04407i			1.44476i	0.691498	+	0.722379i	$0.256951\pi$
$31$			8.04407i			1.44476i	−0.691498	+	0.722379i	$0.743049\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	0.677925	+	1.17420i	0.114590	+	0.198476i
$36$		0			0
$37$	−6.02532	−	3.47872i	−0.990557	−	0.571898i	−0.0851160	−	0.996371i	$-0.527126\pi$
$37$	−6.02532	−	3.47872i	−0.990557	−	0.571898i	−0.905441	+	0.424473i	$0.860459\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−4.38164	−	2.52974i	−0.684298	−	0.395079i	0.117175	−	0.993111i	$-0.462616\pi$
$41$	−4.38164	−	2.52974i	−0.684298	−	0.395079i	−0.801472	+	0.598032i	$0.795950\pi$
$42$		0			0
$43$	3.13427	+	5.42872i	0.477972	+	0.827872i	0.999681	−	0.0252518i	$-0.00803875\pi$
$43$	3.13427	+	5.42872i	0.477972	+	0.827872i	−0.521709	+	0.853123i	$0.674705\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	2.64315i		−	0.385542i	−0.981244	−	0.192771i	$-0.938252\pi$
$47$		−	2.64315i		−	0.385542i	0.981244	−	0.192771i	$-0.0617475\pi$
$48$		0			0
$49$	0.500000	−	0.866025i	0.0714286	−	0.123718i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	11.0887			1.52315			0.761574	−	0.648079i	$-0.224427\pi$
$53$	11.0887			1.52315			0.761574	+	0.648079i	$0.224427\pi$
$54$		0			0
$55$	1.90035	−	3.29150i	0.256243	−	0.443826i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	5.45213	−	3.14779i	0.709807	−	0.409807i	−0.101183	−	0.994868i	$-0.532263\pi$
$59$	5.45213	−	3.14779i	0.709807	−	0.409807i	0.810989	+	0.585061i	$0.198929\pi$
$60$		0			0
$61$	−6.40404	−	11.0921i	−0.819953	−	1.42020i	−0.905716	−	0.423885i	$-0.860666\pi$
$61$	−6.40404	−	11.0921i	−0.819953	−	1.42020i	0.0857630	−	0.996316i	$-0.472667\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	4.87129	+	0.410880i	0.604209	+	0.0509634i
$66$		0			0
$67$	4.29803	+	2.48147i	0.525088	+	0.303160i	0.739014	−	0.673690i	$-0.235292\pi$
$67$	4.29803	+	2.48147i	0.525088	+	0.303160i	−0.213926	+	0.976850i	$0.568625\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	14.2993	−	8.25572i	1.69702	−	0.979773i	0.748453	−	0.663188i	$-0.230797\pi$
$71$	14.2993	−	8.25572i	1.69702	−	0.979773i	0.948564	−	0.316586i	$-0.102536\pi$
$72$		0			0
$73$			15.4351i			1.80654i	0.429071	+	0.903271i	$0.358841\pi$
$73$			15.4351i			1.80654i	−0.429071	+	0.903271i	$0.641159\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−2.80318			−0.319452
$78$		0			0
$79$	14.7661			1.66131			0.830656	−	0.556786i	$-0.187966\pi$
$79$	14.7661			1.66131			0.830656	+	0.556786i	$0.187966\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			9.79310i			1.07493i	0.843285	+	0.537466i	$0.180618\pi$
$83$			9.79310i			1.07493i	−0.843285	+	0.537466i	$0.819382\pi$
$84$		0			0
$85$	−6.83230	+	3.94463i	−0.741067	+	0.427855i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	7.24707	+	4.18410i	0.768188	+	0.443513i	0.832228	−	0.554434i	$-0.187065\pi$
$89$	7.24707	+	4.18410i	0.768188	+	0.443513i	−0.0640399	+	0.997947i	$0.520398\pi$
$90$		0			0
$91$	−1.53395	−	3.26297i	−0.160802	−	0.342052i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	2.15891	+	3.73934i	0.221499	+	0.383648i
$96$		0			0
$97$	−10.5685	+	6.10170i	−1.07306	+	0.619534i	−0.929017	−	0.370037i	$-0.879345\pi$
$97$	−10.5685	+	6.10170i	−1.07306	+	0.619534i	−0.144047	+	0.989571i	$0.546012\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	1.21143	−	2.09825i	0.120541	−	0.208784i	−0.799440	−	0.600746i	$-0.794870\pi$
$101$	1.21143	−	2.09825i	0.120541	−	0.208784i	0.919981	+	0.391962i	$0.128204\pi$
$102$		0			0
$103$	14.4909			1.42783			0.713917	−	0.700230i	$-0.246919\pi$
$103$	14.4909			1.42783			0.713917	+	0.700230i	$0.246919\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	3.24370	−	5.61826i	0.313581	−	0.543138i	−0.665554	−	0.746350i	$-0.731805\pi$
$107$	3.24370	−	5.61826i	0.313581	−	0.543138i	0.979135	+	0.203212i	$0.0651381\pi$
$108$		0			0
$109$		−	2.93644i		−	0.281260i	−0.990062	−	0.140630i	$-0.955087\pi$
$109$		−	2.93644i		−	0.281260i	0.990062	−	0.140630i	$-0.0449129\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−1.66909	−	2.89095i	−0.157015	−	0.271957i	0.776776	−	0.629777i	$-0.216854\pi$
$113$	−1.66909	−	2.89095i	−0.157015	−	0.271957i	−0.933791	+	0.357819i	$0.883520\pi$
$114$		0			0
$115$	−8.86929	−	5.12069i	−0.827066	−	0.477507i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	5.03912	+	2.90934i	0.461936	+	0.266699i
$120$		0			0
$121$	−1.57108	−	2.72119i	−0.142825	−	0.247381i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			11.0660i			0.989773i
$126$		0			0
$127$	−6.27512	+	10.8688i	−0.556827	+	0.964452i	0.440932	+	0.897540i	$0.354648\pi$
$127$	−6.27512	+	10.8688i	−0.556827	+	0.964452i	−0.997759	+	0.0669115i	$0.978685\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−1.26001			−0.110087			−0.0550437	−	0.998484i	$-0.517530\pi$
$131$	−1.26001			−0.110087			−0.0550437	+	0.998484i	$0.517530\pi$
$132$		0			0
$133$	1.59229	−	2.75793i	0.138069	−	0.239143i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	7.20794	−	4.16151i	0.615816	−	0.355542i	−0.159422	−	0.987211i	$-0.550963\pi$
$137$	7.20794	−	4.16151i	0.615816	−	0.355542i	0.775238	+	0.631669i	$0.217630\pi$
$138$		0			0
$139$	2.89438	+	5.01322i	0.245498	+	0.425216i	0.962272	−	0.272091i	$-0.0877151\pi$
$139$	2.89438	+	5.01322i	0.245498	+	0.425216i	−0.716773	+	0.697306i	$0.754382\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−5.77131	+	8.29723i	−0.482621	+	0.693849i
$144$		0			0
$145$	8.47375	+	4.89232i	0.703707	+	0.406285i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−12.3009	+	7.10193i	−1.00773	+	0.581812i	−0.910526	−	0.413452i	$-0.864323\pi$
$149$	−12.3009	+	7.10193i	−1.00773	+	0.581812i	−0.0972028	+	0.995265i	$0.530990\pi$
$150$		0			0
$151$			11.1509i			0.907451i	0.891142	+	0.453725i	$0.149905\pi$
$151$			11.1509i			0.907451i	−0.891142	+	0.453725i	$0.850095\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−10.9066			−0.876036
$156$		0			0
$157$	4.04749			0.323025			0.161513	−	0.986871i	$-0.448363\pi$
$157$	4.04749			0.323025			0.161513	+	0.986871i	$0.448363\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			7.55347i			0.595297i
$162$		0			0
$163$	−4.56675	+	2.63661i	−0.357695	+	0.206515i	−0.668069	−	0.744099i	$-0.732879\pi$
$163$	−4.56675	+	2.63661i	−0.357695	+	0.206515i	0.310374	+	0.950615i	$0.399546\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	4.63307	+	2.67491i	0.358518	+	0.206990i	0.668430	−	0.743775i	$-0.266966\pi$
$167$	4.63307	+	2.67491i	0.358518	+	0.206990i	−0.309912	+	0.950765i	$0.600300\pi$
$168$		0			0
$169$	−12.8163	−	2.17754i	−0.985872	−	0.167503i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	4.87155	+	8.43777i	0.370377	+	0.641511i	0.989623	−	0.143685i	$-0.0458952\pi$
$173$	4.87155	+	8.43777i	0.370377	+	0.641511i	−0.619247	+	0.785197i	$0.712562\pi$
$174$		0			0
$175$	2.73809	−	1.58083i	0.206980	−	0.119500i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−5.70263	+	9.87724i	−0.426234	+	0.738259i	−0.996535	−	0.0831767i	$-0.973493\pi$
$179$	−5.70263	+	9.87724i	−0.426234	+	0.738259i	0.570301	+	0.821436i	$0.306827\pi$
$180$		0			0
$181$	25.4727			1.89337			0.946685	−	0.322160i	$-0.104409\pi$
$181$	25.4727			1.89337			0.946685	+	0.322160i	$0.104409\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	4.71663	−	8.16943i	0.346773	−	0.600629i
$186$		0			0
$187$		−	16.3108i		−	1.19277i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	8.18905	+	14.1838i	0.592539	+	1.02631i	0.993889	+	0.110382i	$0.0352075\pi$
$191$	8.18905	+	14.1838i	0.592539	+	1.02631i	−0.401351	+	0.915924i	$0.631459\pi$
$192$		0			0
$193$	8.28132	+	4.78122i	0.596103	+	0.344160i	0.767507	−	0.641041i	$-0.221497\pi$
$193$	8.28132	+	4.78122i	0.596103	+	0.344160i	−0.171404	+	0.985201i	$0.554830\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−2.60716	−	1.50525i	−0.185753	−	0.107244i	0.404240	−	0.914653i	$-0.367536\pi$
$197$	−2.60716	−	1.50525i	−0.185753	−	0.107244i	−0.589993	+	0.807409i	$0.700869\pi$
$198$		0			0
$199$	−2.30985	−	4.00077i	−0.163741	−	0.283607i	0.772467	−	0.635055i	$-0.219023\pi$
$199$	−2.30985	−	4.00077i	−0.163741	−	0.283607i	−0.936207	+	0.351448i	$0.885689\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	7.21661i		−	0.506507i
$204$		0			0
$205$	3.42995	−	5.94085i	0.239558	−	0.414927i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−8.92697			−0.617492
$210$		0			0
$211$	−1.16932	+	2.02533i	−0.0804995	+	0.139429i	−0.903465	−	0.428663i	$-0.858985\pi$
$211$	−1.16932	+	2.02533i	−0.0804995	+	0.139429i	0.822965	+	0.568092i	$0.192318\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−7.36053	+	4.24960i	−0.501984	+	0.289821i
$216$		0			0
$217$	4.02203	+	6.96637i	0.273033	+	0.472908i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	18.9862	−	8.92559i	1.27715	−	0.600400i
$222$		0			0
$223$	−4.21429	−	2.43312i	−0.282210	−	0.162934i	0.352214	−	0.935920i	$-0.385429\pi$
$223$	−4.21429	−	2.43312i	−0.282210	−	0.162934i	−0.634423	+	0.772986i	$0.718762\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	24.3177	−	14.0398i	1.61402	−	0.931855i	0.625594	−	0.780149i	$-0.284857\pi$
$227$	24.3177	−	14.0398i	1.61402	−	0.931855i	0.988426	−	0.151706i	$-0.0484766\pi$
$228$		0			0
$229$		−	0.179280i		−	0.0118471i	−0.999982	−	0.00592357i	$-0.998114\pi$
$229$		−	0.179280i		−	0.0118471i	0.999982	−	0.00592357i	$-0.00188554\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−15.1384			−0.991747			−0.495873	−	0.868395i	$-0.665152\pi$
$233$	−15.1384			−0.991747			−0.495873	+	0.868395i	$0.665152\pi$
$234$		0			0
$235$	3.58371			0.233775
$236$		0			0
$237$		0			0
$238$		0			0
$239$			12.8703i			0.832512i	0.909247	+	0.416256i	$0.136658\pi$
$239$			12.8703i			0.832512i	−0.909247	+	0.416256i	$0.863342\pi$
$240$		0			0
$241$	8.51452	−	4.91586i	0.548469	−	0.316659i	−0.200035	−	0.979789i	$-0.564106\pi$
$241$	8.51452	−	4.91586i	0.548469	−	0.316659i	0.748504	+	0.663130i	$0.230772\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	1.17420	+	0.677925i	0.0750170	+	0.0433111i
$246$		0			0
$247$	−4.88501	−	10.3912i	−0.310825	−	0.661177i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	8.66099	+	15.0013i	0.546677	+	0.946872i	0.998499	+	0.0547641i	$0.0174407\pi$
$251$	8.66099	+	15.0013i	0.546677	+	0.946872i	−0.451823	+	0.892108i	$0.649226\pi$
$252$		0			0
$253$	18.3370	−	10.5869i	1.15284	−	0.665592i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	5.35302	−	9.27170i	0.333912	−	0.578353i	−0.649363	−	0.760478i	$-0.724964\pi$
$257$	5.35302	−	9.27170i	0.333912	−	0.578353i	0.983275	+	0.182126i	$0.0582978\pi$
$258$		0			0
$259$	−6.95744			−0.432314
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−7.03484	+	12.1847i	−0.433787	+	0.751341i	−0.997196	−	0.0748372i	$-0.976156\pi$
$263$	−7.03484	+	12.1847i	−0.433787	+	0.751341i	0.563409	+	0.826178i	$0.309490\pi$
$264$		0			0
$265$			15.0346i			0.923567i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−0.272011	−	0.471138i	−0.0165848	−	0.0287258i	0.857614	−	0.514294i	$-0.171946\pi$
$269$	−0.272011	−	0.471138i	−0.0165848	−	0.0287258i	−0.874199	+	0.485568i	$0.838613\pi$
$270$		0			0
$271$	−0.641910	−	0.370607i	−0.0389933	−	0.0225128i	0.480377	−	0.877062i	$-0.340500\pi$
$271$	−0.641910	−	0.370607i	−0.0389933	−	0.0225128i	−0.519370	+	0.854550i	$0.673833\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−7.67536	−	4.43137i	−0.462842	−	0.267222i
$276$		0			0
$277$	−7.06675	−	12.2400i	−0.424600	−	0.735429i	0.571783	−	0.820405i	$-0.306252\pi$
$277$	−7.06675	−	12.2400i	−0.424600	−	0.735429i	−0.996383	+	0.0849763i	$0.972919\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	25.8051i		−	1.53940i	−0.638404	−	0.769702i	$-0.720405\pi$
$281$		−	25.8051i		−	1.53940i	0.638404	−	0.769702i	$-0.279595\pi$
$282$		0			0
$283$	13.6409	−	23.6267i	0.810865	−	1.40446i	−0.101394	−	0.994846i	$-0.532330\pi$
$283$	13.6409	−	23.6267i	0.810865	−	1.40446i	0.912259	−	0.409613i	$-0.134336\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−5.05949			−0.298652
$288$		0			0
$289$	−8.42851	+	14.5986i	−0.495795	+	0.858742i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	1.50010	−	0.866083i	0.0876367	−	0.0505971i	−0.455541	−	0.890215i	$-0.650554\pi$
$293$	1.50010	−	0.866083i	0.0876367	−	0.0505971i	0.543178	+	0.839618i	$0.317221\pi$
$294$		0			0
$295$	4.26793	+	7.39227i	0.248489	+	0.430395i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	22.3577	+	15.5514i	1.29298	+	0.899360i
$300$		0			0
$301$	5.42872	+	3.13427i	0.312906	+	0.180656i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	15.0393	−	8.68292i	0.861145	−	0.497182i
$306$		0			0
$307$			21.4365i			1.22345i	0.791072	+	0.611723i	$0.209523\pi$
$307$			21.4365i			1.22345i	−0.791072	+	0.611723i	$0.790477\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	27.2724			1.54647			0.773237	−	0.634117i	$-0.218636\pi$
$311$	27.2724			1.54647			0.773237	+	0.634117i	$0.218636\pi$
$312$		0			0
$313$	7.28221			0.411615			0.205807	−	0.978593i	$-0.434018\pi$
$313$	7.28221			0.411615			0.205807	+	0.978593i	$0.434018\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			2.54836i			0.143130i	0.997436	+	0.0715651i	$0.0227994\pi$
$317$			2.54836i			0.143130i	−0.997436	+	0.0715651i	$0.977201\pi$
$318$		0			0
$319$	−17.5192	+	10.1147i	−0.980890	+	0.566317i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	16.0475	+	9.26503i	0.892907	+	0.515520i
$324$		0			0
$325$	0.958120	−	11.3592i	0.0531469	−	0.630096i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−1.32157	−	2.28903i	−0.0728607	−	0.126198i
$330$		0			0
$331$	0.169450	−	0.0978320i	0.00931381	−	0.00537733i	−0.495336	−	0.868702i	$-0.664955\pi$
$331$	0.169450	−	0.0978320i	0.00931381	−	0.00537733i	0.504650	+	0.863324i	$0.331622\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−3.36450	+	5.82749i	−0.183822	+	0.318390i
$336$		0			0
$337$	2.00900			0.109437			0.0547185	−	0.998502i	$-0.482574\pi$
$337$	2.00900			0.109437			0.0547185	+	0.998502i	$0.482574\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	11.2745	−	19.5280i	0.610549	−	1.05750i
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	2.27450	+	3.93956i	0.122102	+	0.211486i	0.920596	−	0.390516i	$-0.127703\pi$
$347$	2.27450	+	3.93956i	0.122102	+	0.211486i	−0.798495	+	0.602002i	$0.794370\pi$
$348$		0			0
$349$	4.65700	+	2.68872i	0.249284	+	0.143924i	0.619436	−	0.785047i	$-0.287361\pi$
$349$	4.65700	+	2.68872i	0.249284	+	0.143924i	−0.370153	+	0.928971i	$0.620695\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−7.10223	−	4.10047i	−0.378013	−	0.218246i	0.298940	−	0.954272i	$-0.403367\pi$
$353$	−7.10223	−	4.10047i	−0.378013	−	0.218246i	−0.676954	+	0.736026i	$0.736700\pi$
$354$		0			0
$355$	11.1935	+	19.3877i	0.594090	+	1.02899i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	31.7664i		−	1.67657i	−0.545233	−	0.838284i	$-0.683559\pi$
$359$		−	31.7664i		−	1.67657i	0.545233	−	0.838284i	$-0.316441\pi$
$360$		0			0
$361$	−4.42921	+	7.67162i	−0.233117	+	0.403770i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−20.9277			−1.09541
$366$		0			0
$367$	−8.66870	+	15.0146i	−0.452502	+	0.783757i	−0.998541	−	0.0540031i	$-0.982802\pi$
$367$	−8.66870	+	15.0146i	−0.452502	+	0.783757i	0.546038	+	0.837760i	$0.316135\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	9.60308	−	5.54434i	0.498567	−	0.287848i
$372$		0			0
$373$	−8.04814	−	13.9398i	−0.416717	−	0.721775i	0.578890	−	0.815406i	$-0.303486\pi$
$373$	−8.04814	−	13.9398i	−0.416717	−	0.721775i	−0.995607	+	0.0936306i	$0.970153\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−21.3607	−	14.8578i	−1.10013	−	0.765218i
$378$		0			0
$379$	−15.2907	−	8.82811i	−0.785432	−	0.453470i	0.0529197	−	0.998599i	$-0.483147\pi$
$379$	−15.2907	−	8.82811i	−0.785432	−	0.453470i	−0.838352	+	0.545129i	$0.816481\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−7.12262	+	4.11225i	−0.363949	+	0.210126i	−0.670812	−	0.741628i	$-0.734054\pi$
$383$	−7.12262	+	4.11225i	−0.363949	+	0.210126i	0.306863	+	0.951754i	$0.400721\pi$
$384$		0			0
$385$		−	3.80070i		−	0.193702i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−15.1688			−0.769090			−0.384545	−	0.923106i	$-0.625642\pi$
$389$	−15.1688			−0.769090			−0.384545	+	0.923106i	$0.625642\pi$
$390$		0			0
$391$	−43.9512			−2.22271
$392$		0			0
$393$		0			0
$394$		0			0
$395$			20.0206i			1.00734i
$396$		0			0
$397$	−24.5139	+	14.1531i	−1.23032	+	0.710325i	−0.967096	−	0.254410i	$-0.918119\pi$
$397$	−24.5139	+	14.1531i	−1.23032	+	0.710325i	−0.263223	+	0.964735i	$0.584785\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−14.8257	−	8.55962i	−0.740360	−	0.427447i	0.0818403	−	0.996645i	$-0.473920\pi$
$401$	−14.8257	−	8.55962i	−0.740360	−	0.427447i	−0.822200	+	0.569198i	$0.807254\pi$
$402$		0			0
$403$	28.9007	+	2.43769i	1.43965	+	0.121430i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	9.75149	+	16.8901i	0.483364	+	0.837210i
$408$		0			0
$409$	9.82072	−	5.66999i	0.485603	−	0.280363i	−0.237145	−	0.971474i	$-0.576212\pi$
$409$	9.82072	−	5.66999i	0.485603	−	0.280363i	0.722749	+	0.691111i	$0.242878\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	3.14779	−	5.45213i	0.154893	−	0.268282i
$414$		0			0
$415$	−13.2780			−0.651790
$416$		0			0
$417$		0			0
$418$		0			0
$419$	0.0694276	−	0.120252i	0.00339176	−	0.00587470i	−0.864325	−	0.502934i	$-0.832254\pi$
$419$	0.0694276	−	0.120252i	0.00339176	−	0.00587470i	0.867716	+	0.497060i	$0.165587\pi$
$420$		0			0
$421$		−	24.3258i		−	1.18557i	−0.805362	−	0.592783i	$-0.798029\pi$
$421$		−	24.3258i		−	1.18557i	0.805362	−	0.592783i	$-0.201971\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	9.19837	+	15.9320i	0.446186	+	0.772818i
$426$		0			0
$427$	−11.0921	−	6.40404i	−0.536785	−	0.309913i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−24.9349	−	14.3962i	−1.20107	−	0.693440i	−0.240280	−	0.970704i	$-0.577239\pi$
$431$	−24.9349	−	14.3962i	−1.20107	−	0.693440i	−0.960794	+	0.277263i	$0.910573\pi$
$432$		0			0
$433$	−12.6367	−	21.8873i	−0.607279	−	1.05184i	−0.991687	−	0.128675i	$-0.958928\pi$
$433$	−12.6367	−	21.8873i	−0.607279	−	1.05184i	0.384408	−	0.923163i	$-0.374406\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			24.0547i			1.15069i
$438$		0			0
$439$	3.01512	−	5.22234i	0.143904	−	0.249248i	−0.785060	−	0.619420i	$-0.787368\pi$
$439$	3.01512	−	5.22234i	0.143904	−	0.249248i	0.928963	+	0.370172i	$0.120701\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−23.4988			−1.11646			−0.558230	−	0.829686i	$-0.688519\pi$
$443$	−23.4988			−1.11646			−0.558230	+	0.829686i	$0.688519\pi$
$444$		0			0
$445$	−5.67301	+	9.82594i	−0.268927	+	0.465794i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−20.2543	+	11.6938i	−0.955859	+	0.551865i	−0.894896	−	0.446275i	$-0.852750\pi$
$449$	−20.2543	+	11.6938i	−0.955859	+	0.551865i	−0.0609627	+	0.998140i	$0.519417\pi$
$450$		0			0
$451$	7.09133	+	12.2826i	0.333918	+	0.578363i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	4.42410	−	2.07981i	0.207405	−	0.0975031i
$456$		0			0
$457$	−22.6616	−	13.0837i	−1.06007	−	0.612030i	−0.134616	−	0.990898i	$-0.542980\pi$
$457$	−22.6616	−	13.0837i	−1.06007	−	0.612030i	−0.925451	+	0.378868i	$0.876313\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	21.8524	−	12.6165i	1.01777	−	0.587609i	0.104312	−	0.994545i	$-0.466736\pi$
$461$	21.8524	−	12.6165i	1.01777	−	0.587609i	0.913457	+	0.406936i	$0.133403\pi$
$462$		0			0
$463$		−	18.1600i		−	0.843968i	−0.906603	−	0.421984i	$-0.861334\pi$
$463$		−	18.1600i		−	0.843968i	0.906603	−	0.421984i	$-0.138666\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	12.6328			0.584577			0.292289	−	0.956330i	$-0.405583\pi$
$467$	12.6328			0.584577			0.292289	+	0.956330i	$0.405583\pi$
$468$		0			0
$469$	4.96294			0.229167
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	17.5719i		−	0.807956i
$474$		0			0
$475$	8.71966	−	5.03430i	0.400086	−	0.230990i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−10.6585	−	6.15368i	−0.486999	−	0.281169i	0.236330	−	0.971673i	$-0.424055\pi$
$479$	−10.6585	−	6.15368i	−0.486999	−	0.281169i	−0.723329	+	0.690504i	$0.757389\pi$
$480$		0			0
$481$	−14.3243	+	20.5935i	−0.653130	+	0.938984i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−8.27299	−	14.3292i	−0.375657	−	0.650658i
$486$		0			0
$487$	13.5393	−	7.81693i	0.613525	−	0.354219i	−0.160819	−	0.986984i	$-0.551413\pi$
$487$	13.5393	−	7.81693i	0.613525	−	0.354219i	0.774344	+	0.632765i	$0.218080\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−5.10543	+	8.84286i	−0.230405	+	0.399073i	−0.957927	−	0.287011i	$-0.907338\pi$
$491$	−5.10543	+	8.84286i	−0.230405	+	0.399073i	0.727523	+	0.686084i	$0.240672\pi$
$492$		0			0
$493$	41.9911			1.89119
$494$		0			0
$495$		0			0
$496$		0			0
$497$	8.25572	−	14.2993i	0.370319	−	0.641412i
$498$		0			0
$499$		−	0.681003i		−	0.0304859i	−0.999884	−	0.0152430i	$-0.995148\pi$
$499$		−	0.681003i		−	0.0304859i	0.999884	−	0.0152430i	$-0.00485217\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−17.6701	−	30.6056i	−0.787873	−	1.36464i	−0.927268	−	0.374399i	$-0.877849\pi$
$503$	−17.6701	−	30.6056i	−0.787873	−	1.36464i	0.139395	−	0.990237i	$-0.455484\pi$
$504$		0			0
$505$	2.84492	+	1.64251i	0.126597	+	0.0730909i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−14.5717	−	8.41300i	−0.645881	−	0.372900i	0.140995	−	0.990010i	$-0.454970\pi$
$509$	−14.5717	−	8.41300i	−0.645881	−	0.372900i	−0.786876	+	0.617111i	$0.788303\pi$
$510$		0			0
$511$	7.71755	+	13.3672i	0.341404	+	0.591330i
$512$		0			0
$513$		0			0
$514$		0			0
$515$			19.6475i			0.865774i
$516$		0			0
$517$	−3.70461	+	6.41658i	−0.162929	+	0.282201i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−4.76619			−0.208811			−0.104405	−	0.994535i	$-0.533294\pi$
$521$	−4.76619			−0.208811			−0.104405	+	0.994535i	$0.533294\pi$
$522$		0			0
$523$	0.319080	−	0.552662i	0.0139524	−	0.0241662i	−0.858965	−	0.512034i	$-0.828892\pi$
$523$	0.319080	−	0.552662i	0.0139524	−	0.0241662i	0.872917	+	0.487868i	$0.162225\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−40.5351	+	23.4029i	−1.76573	+	1.01945i
$528$		0			0
$529$	−17.0275	−	29.4924i	−0.740325	−	1.28228i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−10.4167	+	14.9757i	−0.451196	+	0.648670i
$534$		0			0
$535$	7.61752	+	4.39798i	0.329334	+	0.190141i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−2.42763	+	1.40159i	−0.104565	+	0.0603708i
$540$		0			0
$541$			22.2050i			0.954668i	0.878722	+	0.477334i	$0.158397\pi$
$541$			22.2050i			0.954668i	−0.878722	+	0.477334i	$0.841603\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	3.98138			0.170544
$546$		0			0
$547$	16.2608			0.695263			0.347631	−	0.937631i	$-0.386986\pi$
$547$	16.2608			0.695263			0.347631	+	0.937631i	$0.386986\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	22.9819i		−	0.979062i
$552$		0			0
$553$	12.7878	−	7.38303i	0.543792	−	0.313958i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	18.5915	+	10.7338i	0.787747	+	0.454806i	0.839169	−	0.543871i	$-0.183042\pi$
$557$	18.5915	+	10.7338i	0.787747	+	0.454806i	−0.0514219	+	0.998677i	$0.516375\pi$
$558$		0			0
$559$	20.4541	−	9.61566i	0.865115	−	0.406699i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	5.59103	+	9.68395i	0.235634	+	0.408130i	0.959457	−	0.281856i	$-0.0909500\pi$
$563$	5.59103	+	9.68395i	0.235634	+	0.408130i	−0.723823	+	0.689986i	$0.757617\pi$
$564$		0			0
$565$	3.91969	−	2.26304i	0.164903	−	0.0952066i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	6.58165	−	11.3998i	0.275917	−	0.477903i	−0.694449	−	0.719542i	$-0.744352\pi$
$569$	6.58165	−	11.3998i	0.275917	−	0.477903i	0.970366	+	0.241639i	$0.0776851\pi$
$570$		0			0
$571$	−17.4199			−0.729001			−0.364501	−	0.931203i	$-0.618760\pi$
$571$	−17.4199			−0.729001			−0.364501	+	0.931203i	$0.618760\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−11.9408	+	20.6821i	−0.497965	+	0.862501i
$576$		0			0
$577$		−	8.47023i		−	0.352620i	−0.984335	−	0.176310i	$-0.943584\pi$
$577$		−	8.47023i		−	0.352620i	0.984335	−	0.176310i	$-0.0564161\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	4.89655	+	8.48107i	0.203143	+	0.351854i
$582$		0			0
$583$	−26.9192	−	15.5418i	−1.11488	−	0.643676i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−18.2459	−	10.5342i	−0.753087	−	0.434795i	0.0737213	−	0.997279i	$-0.476512\pi$
$587$	−18.2459	−	10.5342i	−0.753087	−	0.434795i	−0.826808	+	0.562484i	$0.809846\pi$
$588$		0			0
$589$	12.8085	+	22.1850i	0.527765	+	0.914116i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	13.2761i		−	0.545185i	−0.962130	−	0.272593i	$-0.912119\pi$
$593$		−	13.2761i		−	0.545185i	0.962130	−	0.272593i	$-0.0878811\pi$
$594$		0			0
$595$	−3.94463	+	6.83230i	−0.161714	+	0.280097i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−5.39084			−0.220264			−0.110132	−	0.993917i	$-0.535127\pi$
$599$	−5.39084			−0.220264			−0.110132	+	0.993917i	$0.535127\pi$
$600$		0			0
$601$	12.2204	−	21.1663i	0.498480	−	0.863392i	−0.501519	−	0.865147i	$-0.667225\pi$
$601$	12.2204	−	21.1663i	0.498480	−	0.863392i	0.999998	+	0.00175467i	$0.000558528\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	3.68953	−	2.13015i	0.150001	−	0.0866029i
$606$		0			0
$607$	12.9291	+	22.3939i	0.524776	+	0.908939i	0.999584	+	0.0288492i	$0.00918427\pi$
$607$	12.9291	+	22.3939i	0.524776	+	0.908939i	−0.474808	+	0.880090i	$0.657482\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−9.49628	−	0.800985i	−0.384178	−	0.0324044i
$612$		0			0
$613$	12.2706	+	7.08441i	0.495603	+	0.286136i	0.726896	−	0.686748i	$-0.240962\pi$
$613$	12.2706	+	7.08441i	0.495603	+	0.286136i	−0.231293	+	0.972884i	$0.574296\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−39.8763	+	23.0226i	−1.60536	+	0.926855i	−0.614970	+	0.788550i	$0.710832\pi$
$617$	−39.8763	+	23.0226i	−1.60536	+	0.926855i	−0.990390	+	0.138305i	$0.955835\pi$
$618$		0			0
$619$		−	12.0284i		−	0.483464i	−0.970343	−	0.241732i	$-0.922285\pi$
$619$		−	12.0284i		−	0.483464i	0.970343	−	0.241732i	$-0.0777155\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	8.36820			0.335265
$624$		0			0
$625$	0.804500			0.0321800
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	40.4831i		−	1.61417i
$630$		0			0
$631$	21.0476	−	12.1518i	0.837890	−	0.483756i	−0.0186564	−	0.999826i	$-0.505939\pi$
$631$	21.0476	−	12.1518i	0.837890	−	0.483756i	0.856546	+	0.516070i	$0.172606\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−14.7365	−	8.50812i	−0.584800	−	0.337634i
$636$		0			0
$637$	−2.95993	−	2.05884i	−0.117277	−	0.0815742i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	8.88276	+	15.3854i	0.350848	+	0.607687i	0.986398	−	0.164373i	$-0.0525600\pi$
$641$	8.88276	+	15.3854i	0.350848	+	0.607687i	−0.635550	+	0.772060i	$0.719227\pi$
$642$		0			0
$643$	−37.3742	+	21.5780i	−1.47389	+	0.850953i	−0.999568	−	0.0293964i	$-0.990641\pi$
$643$	−37.3742	+	21.5780i	−1.47389	+	0.850953i	−0.474326	+	0.880349i	$0.657308\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	5.39974	−	9.35262i	0.212286	−	0.367689i	−0.740144	−	0.672449i	$-0.765243\pi$
$647$	5.39974	−	9.35262i	0.212286	−	0.367689i	0.952429	+	0.304759i	$0.0985760\pi$
$648$		0			0
$649$	−17.6477			−0.692731
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−1.53280	+	2.65489i	−0.0599831	+	0.103894i	−0.894458	−	0.447153i	$-0.852438\pi$
$653$	−1.53280	+	2.65489i	−0.0599831	+	0.103894i	0.834475	+	0.551047i	$0.185771\pi$
$654$		0			0
$655$		−	1.70838i		−	0.0667520i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−1.82436	−	3.15988i	−0.0710670	−	0.123092i	0.828302	−	0.560282i	$-0.189307\pi$
$659$	−1.82436	−	3.15988i	−0.0710670	−	0.123092i	−0.899369	+	0.437190i	$0.855974\pi$
$660$		0			0
$661$	−23.5777	−	13.6126i	−0.917068	−	0.529469i	−0.0343693	−	0.999409i	$-0.510942\pi$
$661$	−23.5777	−	13.6126i	−0.917068	−	0.529469i	−0.882698	+	0.469940i	$0.844276\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	3.73934	+	2.15891i	0.145005	+	0.0837189i
$666$		0			0
$667$	27.2552	+	47.2074i	1.05533	+	1.82788i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			35.9034i			1.38604i
$672$		0			0
$673$	14.1485	−	24.5059i	0.545384	−	0.944633i	−0.453199	−	0.891410i	$-0.649717\pi$
$673$	14.1485	−	24.5059i	0.545384	−	0.944633i	0.998583	−	0.0532233i	$-0.0169495\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	28.8361			1.10826			0.554131	−	0.832429i	$-0.313051\pi$
$677$	28.8361			1.10826			0.554131	+	0.832429i	$0.313051\pi$
$678$		0			0
$679$	−6.10170	+	10.5685i	−0.234162	+	0.405580i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−35.4243	+	20.4522i	−1.35547	+	0.782583i	−0.989010	−	0.147849i	$-0.952765\pi$
$683$	−35.4243	+	20.4522i	−1.35547	+	0.782583i	−0.366464	+	0.930432i	$0.619432\pi$
$684$		0			0
$685$	5.64238	+	9.77290i	0.215584	+	0.373403i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	3.36034	−	39.8393i	0.128019	−	1.51776i
$690$		0			0
$691$	−36.5934	−	21.1272i	−1.39208	−	0.803717i	−0.398534	−	0.917154i	$-0.630481\pi$
$691$	−36.5934	−	21.1272i	−1.39208	−	0.803717i	−0.993545	+	0.113436i	$0.963814\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−6.79717	+	3.92435i	−0.257831	+	0.148859i
$696$		0			0
$697$		−	29.4395i		−	1.11510i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	30.3734			1.14719			0.573594	−	0.819140i	$-0.305549\pi$
$701$	30.3734			1.14719			0.573594	+	0.819140i	$0.305549\pi$
$702$		0			0
$703$	−22.1566			−0.835650
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	2.42285i		−	0.0911208i
$708$		0			0
$709$	−20.2907	+	11.7148i	−0.762033	+	0.439960i	−0.830025	−	0.557726i	$-0.811674\pi$
$709$	−20.2907	+	11.7148i	−0.762033	+	0.439960i	0.0679919	+	0.997686i	$0.478341\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−52.6203	−	30.3803i	−1.97064	−	1.13775i
$714$		0			0
$715$	−11.2498	−	7.82503i	−0.420719	−	0.292639i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	3.51491	+	6.08800i	0.131084	+	0.227044i	0.924095	−	0.382164i	$-0.124821\pi$
$719$	3.51491	+	6.08800i	0.131084	+	0.227044i	−0.793011	+	0.609208i	$0.791488\pi$
$720$		0			0
$721$	12.5495	−	7.24547i	0.467369	−	0.269835i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	11.4083	−	19.7597i	0.423692	−	0.733857i
$726$		0			0
$727$	−20.6810			−0.767015			−0.383507	−	0.923538i	$-0.625284\pi$
$727$	−20.6810			−0.767015			−0.383507	+	0.923538i	$0.625284\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−18.2373	+	31.5880i	−0.674531	+	1.16832i
$732$		0			0
$733$		−	39.0035i		−	1.44063i	−0.693648	−	0.720314i	$-0.743998\pi$
$733$		−	39.0035i		−	1.44063i	0.693648	−	0.720314i	$-0.256002\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−6.95602	−	12.0482i	−0.256228	−	0.443800i
$738$		0			0
$739$	24.8910	+	14.3708i	0.915629	+	0.528639i	0.882238	−	0.470803i	$-0.156036\pi$
$739$	24.8910	+	14.3708i	0.915629	+	0.528639i	0.0333913	+	0.999442i	$0.489369\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	5.48059	+	3.16422i	0.201063	+	0.116084i	0.597151	−	0.802129i	$-0.296299\pi$
$743$	5.48059	+	3.16422i	0.201063	+	0.116084i	−0.396088	+	0.918213i	$0.629633\pi$
$744$		0			0
$745$	−9.62915	−	16.6782i	−0.352785	−	0.611041i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	6.48741i		−	0.237045i
$750$		0			0
$751$	−15.7203	+	27.2283i	−0.573641	+	0.993575i	0.422547	+	0.906341i	$0.361136\pi$
$751$	−15.7203	+	27.2283i	−0.573641	+	0.993575i	−0.996188	+	0.0872339i	$0.972197\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−15.1190			−0.550237
$756$		0			0
$757$	−12.7212	+	22.0337i	−0.462358	+	0.800828i	−0.999078	−	0.0429328i	$-0.986330\pi$
$757$	−12.7212	+	22.0337i	−0.462358	+	0.800828i	0.536720	+	0.843760i	$0.319663\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	17.8492	−	10.3053i	0.647034	−	0.373565i	−0.140285	−	0.990111i	$-0.544802\pi$
$761$	17.8492	−	10.3053i	0.647034	−	0.373565i	0.787319	+	0.616546i	$0.211469\pi$
$762$		0			0
$763$	−1.46822	−	2.54304i	−0.0531532	−	0.0920641i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−9.65713	−	20.5423i	−0.348699	−	0.741739i
$768$		0			0
$769$	7.95173	+	4.59093i	0.286747	+	0.165553i	0.636474	−	0.771298i	$-0.280392\pi$
$769$	7.95173	+	4.59093i	0.286747	+	0.165553i	−0.349727	+	0.936852i	$0.613726\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	7.35736	−	4.24777i	0.264626	−	0.152782i	−0.361817	−	0.932249i	$-0.617843\pi$
$773$	7.35736	−	4.24777i	0.264626	−	0.152782i	0.626443	+	0.779467i	$0.284510\pi$
$774$		0			0
$775$			25.4327i			0.913569i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−16.1124			−0.577285
$780$		0			0
$781$	−46.2846			−1.65619
$782$		0			0
$783$		0			0
$784$		0			0
$785$			5.48780i			0.195868i
$786$		0			0
$787$	1.70464	−	0.984173i	0.0607638	−	0.0350820i	−0.469310	−	0.883033i	$-0.655497\pi$
$787$	1.70464	−	0.984173i	0.0607638	−	0.0350820i	0.530074	+	0.847951i	$0.322164\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−2.89095	−	1.66909i	−0.102790	−	0.0593460i
$792$		0			0
$793$	−41.7924	+	19.6470i	−1.48409	+	0.697686i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	19.2449	+	33.3332i	0.681690	+	1.18072i	0.974465	+	0.224540i	$0.0720879\pi$
$797$	19.2449	+	33.3332i	0.681690	+	1.18072i	−0.292775	+	0.956181i	$0.594579\pi$
$798$		0			0
$799$	13.3191	−	7.68981i	0.471197	−	0.272046i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	21.6337	−	37.4707i	0.763437	−	1.32231i
$804$		0			0
$805$	−10.2414			−0.360961
$806$		0			0
$807$		0			0
$808$		0			0
$809$	0.899878	−	1.55863i	0.0316380	−	0.0547987i	−0.849773	−	0.527149i	$-0.823261\pi$
$809$	0.899878	−	1.55863i	0.0316380	−	0.0547987i	0.881411	+	0.472350i	$0.156594\pi$
$810$		0			0
$811$			49.9235i			1.75305i	0.481357	+	0.876525i	$0.340144\pi$
$811$			49.9235i			1.75305i	−0.481357	+	0.876525i	$0.659856\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−3.57485	−	6.19183i	−0.125222	−	0.216890i
$816$		0			0
$817$	17.2882	+	9.98135i	0.604838	+	0.349203i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−3.65310	−	2.10912i	−0.127494	−	0.0736087i	0.434897	−	0.900480i	$-0.356785\pi$
$821$	−3.65310	−	2.10912i	−0.127494	−	0.0736087i	−0.562391	+	0.826872i	$0.690118\pi$
$822$		0			0
$823$	7.24912	+	12.5558i	0.252688	+	0.437669i	0.964265	−	0.264939i	$-0.0853519\pi$
$823$	7.24912	+	12.5558i	0.252688	+	0.437669i	−0.711577	+	0.702608i	$0.752019\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	26.2618i		−	0.913213i	−0.889669	−	0.456607i	$-0.849065\pi$
$827$		−	26.2618i		−	0.913213i	0.889669	−	0.456607i	$-0.150935\pi$
$828$		0			0
$829$	−27.5790	+	47.7682i	−0.957858	+	1.65906i	−0.230170	+	0.973150i	$0.573928\pi$
$829$	−27.5790	+	47.7682i	−0.957858	+	1.65906i	−0.727688	+	0.685909i	$0.759405\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	5.81868			0.201605
$834$		0			0
$835$	−3.62677	+	6.28175i	−0.125510	+	0.217389i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	6.07974	−	3.51014i	0.209896	−	0.121184i	−0.391367	−	0.920235i	$-0.627998\pi$
$839$	6.07974	−	3.51014i	0.209896	−	0.121184i	0.601263	+	0.799051i	$0.294664\pi$
$840$		0			0
$841$	−11.5397	−	19.9874i	−0.397922	−	0.689221i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	2.95241	−	17.3770i	0.101566	−	0.597788i
$846$		0			0
$847$	−2.72119	−	1.57108i	−0.0935012	−	0.0539829i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	45.5121	−	26.2764i	1.56013	−	0.900744i
$852$		0			0
$853$		−	57.6508i		−	1.97393i	−0.160945	−	0.986963i	$-0.551454\pi$
$853$		−	57.6508i		−	1.97393i	0.160945	−	0.986963i	$-0.448546\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	2.50654			0.0856217			0.0428109	−	0.999083i	$-0.486369\pi$
$857$	2.50654			0.0856217			0.0428109	+	0.999083i	$0.486369\pi$
$858$		0			0
$859$	17.1305			0.584486			0.292243	−	0.956344i	$-0.405598\pi$
$859$	17.1305			0.584486			0.292243	+	0.956344i	$0.405598\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			55.6429i			1.89411i	0.321079	+	0.947053i	$0.395955\pi$
$863$			55.6429i			1.89411i	−0.321079	+	0.947053i	$0.604045\pi$
$864$		0			0
$865$	−11.4403	+	6.60509i	−0.388984	+	0.224580i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−35.8465	−	20.6960i	−1.21601	−	0.702064i
$870$		0			0
$871$	10.2179	−	14.6899i	0.346220	−	0.497750i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	5.53300	+	9.58344i	0.187050	+	0.323979i
$876$		0			0
$877$	34.7751	−	20.0774i	1.17427	−	0.677967i	0.219590	−	0.975592i	$-0.429528\pi$
$877$	34.7751	−	20.0774i	1.17427	−	0.677967i	0.954683	+	0.297626i	$0.0961947\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−25.3837	+	43.9658i	−0.855197	+	1.48124i	0.0212650	+	0.999774i	$0.493231\pi$
$881$	−25.3837	+	43.9658i	−0.855197	+	1.48124i	−0.876462	+	0.481471i	$0.840103\pi$
$882$		0			0
$883$	−46.3312			−1.55917			−0.779585	−	0.626297i	$-0.784570\pi$
$883$	−46.3312			−1.55917			−0.779585	+	0.626297i	$0.784570\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−5.12752	+	8.88113i	−0.172165	+	0.298199i	−0.939177	−	0.343435i	$-0.888410\pi$
$887$	−5.12752	+	8.88113i	−0.172165	+	0.298199i	0.767011	+	0.641634i	$0.221743\pi$
$888$		0			0
$889$			12.5502i			0.420921i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−4.20866	−	7.28961i	−0.140837	−	0.243938i
$894$		0			0
$895$	−13.3921	−	7.73191i	−0.447647	−	0.258449i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	50.2736	+	29.0255i	1.67672	+	0.968053i
$900$		0			0
$901$	32.2607	+	55.8772i	1.07476	+	1.86154i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			34.5372i			1.14805i
$906$		0			0
$907$	26.1472	−	45.2883i	0.868205	−	1.50377i	0.00437495	−	0.999990i	$-0.498607\pi$
$907$	26.1472	−	45.2883i	0.868205	−	1.50377i	0.863830	−	0.503784i	$-0.168059\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−50.9172			−1.68696			−0.843481	−	0.537159i	$-0.819498\pi$
$911$	−50.9172			−1.68696			−0.843481	+	0.537159i	$0.819498\pi$
$912$		0			0
$913$	13.7259	−	23.7740i	0.454262	−	0.786805i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−1.09120	+	0.630004i	−0.0360346	+	0.0208046i
$918$		0			0
$919$	−11.4323	−	19.8013i	−0.377117	−	0.653186i	0.613524	−	0.789676i	$-0.289751\pi$
$919$	−11.4323	−	19.8013i	−0.377117	−	0.653186i	−0.990641	+	0.136490i	$0.956418\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−25.3278	−	53.8763i	−0.833674	−	1.77336i
$924$		0			0
$925$	−19.0501	−	10.9986i	−0.626363	−	0.361631i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−28.8090	+	16.6329i	−0.945193	+	0.545708i	−0.891584	−	0.452854i	$-0.850406\pi$
$929$	−28.8090	+	16.6329i	−0.945193	+	0.545708i	−0.0536089	+	0.998562i	$0.517072\pi$
$930$		0			0
$931$		−	3.18458i		−	0.104371i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	22.1150			0.723239
$936$		0			0
$937$	−51.2179			−1.67322			−0.836608	−	0.547802i	$-0.815465\pi$
$937$	−51.2179			−1.67322			−0.836608	+	0.547802i	$0.815465\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	10.2918i		−	0.335502i	−0.985829	−	0.167751i	$-0.946350\pi$
$941$		−	10.2918i		−	0.335502i	0.985829	−	0.167751i	$-0.0536504\pi$
$942$		0			0
$943$	33.0966	−	19.1083i	1.07777	−	0.622253i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−30.7341	−	17.7444i	−0.998725	−	0.576614i	−0.0908539	−	0.995864i	$-0.528960\pi$
$947$	−30.7341	−	17.7444i	−0.998725	−	0.576614i	−0.907871	+	0.419250i	$0.862293\pi$
$948$		0			0
$949$	55.4551	+	4.67749i	1.80015	+	0.151838i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−16.6676	−	28.8691i	−0.539916	−	0.935163i	−0.998908	−	0.0467220i	$-0.985123\pi$
$953$	−16.6676	−	28.8691i	−0.539916	−	0.935163i	0.458992	−	0.888441i	$-0.348211\pi$
$954$		0			0
$955$	−19.2312	+	11.1031i	−0.622306	+	0.359289i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	4.16151	−	7.20794i	0.134382	−	0.232757i
$960$		0			0
$961$	−33.7070			−1.08732
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−6.48263	+	11.2282i	−0.208683	+	0.361450i
$966$		0			0
$967$			29.7131i			0.955509i	0.878494	+	0.477754i	$0.158549\pi$
$967$			29.7131i			0.955509i	−0.878494	+	0.477754i	$0.841451\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−4.41851	−	7.65308i	−0.141797	−	0.245599i	0.786377	−	0.617747i	$-0.211955\pi$
$971$	−4.41851	−	7.65308i	−0.141797	−	0.245599i	−0.928173	+	0.372148i	$0.878621\pi$
$972$		0			0
$973$	5.01322	+	2.89438i	0.160716	+	0.0927896i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	12.8437	+	7.41533i	0.410907	+	0.237237i	0.691180	−	0.722683i	$-0.257091\pi$
$977$	12.8437	+	7.41533i	0.410907	+	0.237237i	−0.280272	+	0.959921i	$0.590425\pi$
$978$		0			0
$979$	−11.7288	−	20.3149i	−0.374854	−	0.649266i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	20.3390i		−	0.648714i	−0.945935	−	0.324357i	$-0.894852\pi$
$983$		−	20.3390i		−	0.648714i	0.945935	−	0.324357i	$-0.105148\pi$
$984$		0			0
$985$	2.04089	−	3.53493i	0.0650282	−	0.112632i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−47.3493			−1.50562
$990$		0			0
$991$	25.4453	−	44.0726i	0.808297	−	1.40001i	−0.105745	−	0.994393i	$-0.533723\pi$
$991$	25.4453	−	44.0726i	0.808297	−	1.40001i	0.914042	−	0.405619i	$-0.132944\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	5.42445	−	3.13181i	0.171967	−	0.0992849i
$996$		0			0
$997$	3.69697	+	6.40334i	0.117084	+	0.202796i	0.918611	−	0.395163i	$-0.129312\pi$
$997$	3.69697	+	6.40334i	0.117084	+	0.202796i	−0.801527	+	0.597959i	$0.795979\pi$
$998$		0			0
$999$		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	3276.2.cf.c.2773.5		16
3.2	odd	2		364.2.u.a.225.5	✓	16
12.11	even	2		1456.2.cc.f.225.4		16
13.10	even	6	inner	3276.2.cf.c.1765.4		16
21.2	odd	6		2548.2.bb.d.1733.5		16
21.5	even	6		2548.2.bb.c.1733.4		16
21.11	odd	6		2548.2.bq.e.1941.4		16
21.17	even	6		2548.2.bq.c.1941.5		16
21.20	even	2		2548.2.u.c.589.4		16
39.17	odd	6		4732.2.g.k.337.8		16
39.20	even	12		4732.2.a.t.1.4		8
39.23	odd	6		364.2.u.a.309.5	yes	16
39.32	even	12		4732.2.a.s.1.4		8
39.35	odd	6		4732.2.g.k.337.7		16
156.23	even	6		1456.2.cc.f.673.4		16
273.23	odd	6		2548.2.bq.e.361.4		16
273.62	even	6		2548.2.u.c.1765.4		16
273.101	even	6		2548.2.bb.c.569.4		16
273.179	odd	6		2548.2.bb.d.569.5		16
273.257	even	6		2548.2.bq.c.361.5		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
364.2.u.a.225.5	✓	16	3.2	odd	2
364.2.u.a.309.5	yes	16	39.23	odd	6
1456.2.cc.f.225.4		16	12.11	even	2
1456.2.cc.f.673.4		16	156.23	even	6
2548.2.u.c.589.4		16	21.20	even	2
2548.2.u.c.1765.4		16	273.62	even	6
2548.2.bb.c.569.4		16	273.101	even	6
2548.2.bb.c.1733.4		16	21.5	even	6
2548.2.bb.d.569.5		16	273.179	odd	6
2548.2.bb.d.1733.5		16	21.2	odd	6
2548.2.bq.c.361.5		16	273.257	even	6
2548.2.bq.c.1941.5		16	21.17	even	6
2548.2.bq.e.361.4		16	273.23	odd	6
2548.2.bq.e.1941.4		16	21.11	odd	6
3276.2.cf.c.1765.4		16	13.10	even	6	inner
3276.2.cf.c.2773.5		16	1.1	even	1	trivial
4732.2.a.s.1.4		8	39.32	even	12
4732.2.a.t.1.4		8	39.20	even	12
4732.2.g.k.337.7		16	39.35	odd	6
4732.2.g.k.337.8		16	39.17	odd	6

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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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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	3276.2.cf.c.2773.5

Level	$3276$
Weight	$2$
Character	3276.2773
Analytic conductor	$26.159$
Analytic rank	$0$
Dimension	$16$
Inner twists	$2$