LMFDB - Embedded newform 464.3.d.b.175.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [464,3,Mod(175,464)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(464, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 0]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("464.175");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 464 = 2^{4} \cdot 29 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	464.d (of order $2$, degree $1$, 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:	$12.6430842663$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 69 x^{18} + 1795 x^{16} + 24222 x^{14} + 189561 x^{12} + 892623 x^{10} + 2508433 x^{8} + \cdots + 21609 $ x^20 + 69x^18 + 1795x^16 + 24222x^14 + 189561x^12 + 892623x^10 + 2508433x^8 + 3989370x^6 + 3148495x^4 + 862449*x^2 + 21609
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{36}\cdot 7^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			175.16
Root			$0.166656i$ of defining polynomial
Character	$\chi$	$=$	464.175
Dual form			464.3.d.b.175.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.92832i q^{3} +7.62576 q^{5} -5.54000i q^{7} -6.43170 q^{9} +9.38498i q^{11} +23.1937 q^{13} +29.9564i q^{15} +3.39182 q^{17} -37.5237i q^{19} +21.7629 q^{21} +8.17550i q^{23} +33.1521 q^{25} +10.0891i q^{27} +5.38516 q^{29} +4.41568i q^{31} -36.8672 q^{33} -42.2467i q^{35} -0.0159584 q^{37} +91.1123i q^{39} -75.8415 q^{41} +16.0497i q^{43} -49.0466 q^{45} +35.9472i q^{47} +18.3084 q^{49} +13.3242i q^{51} -71.0080 q^{53} +71.5676i q^{55} +147.405 q^{57} +64.7708i q^{59} +41.0583 q^{61} +35.6316i q^{63} +176.870 q^{65} +65.6441i q^{67} -32.1160 q^{69} -32.6479i q^{71} -122.284 q^{73} +130.232i q^{75} +51.9928 q^{77} -85.2166i q^{79} -97.5185 q^{81} -80.5050i q^{83} +25.8652 q^{85} +21.1547i q^{87} -19.5031 q^{89} -128.493i q^{91} -17.3462 q^{93} -286.146i q^{95} +174.513 q^{97} -60.3614i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 8 q^{5} - 68 q^{9} + 16 q^{13} + 40 q^{17} - 48 q^{21} + 188 q^{25} - 120 q^{33} - 80 q^{37} - 72 q^{41} + 72 q^{45} - 28 q^{49} + 96 q^{53} + 104 q^{57} - 96 q^{61} - 80 q^{65} + 352 q^{69} - 312 q^{73}+ \cdots + 56 q^{97}+O(q^{100}) $ 20 * q - 8 * q^5 - 68 * q^9 + 16 * q^13 + 40 * q^17 - 48 * q^21 + 188 * q^25 - 120 * q^33 - 80 * q^37 - 72 * q^41 + 72 * q^45 - 28 * q^49 + 96 * q^53 + 104 * q^57 - 96 * q^61 - 80 * q^65 + 352 * q^69 - 312 * q^73 - 192 * q^77 + 60 * q^81 + 336 * q^85 - 152 * q^89 - 472 * q^93 + 56 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/464\mathbb{Z}\right)^\times$.

$n$	$117$	$175$	$321$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	3.92832i	1.30944i	0.755871	+	0.654720i	$0.227213\pi$
$3$	3.92832i	1.30944i	−0.755871	+	0.654720i	$0.772787\pi$
$4$	0	0
$5$	7.62576	1.52515	0.762576	−	0.646899i	$-0.223935\pi$
$5$	7.62576	1.52515	0.762576	+	0.646899i	$0.223935\pi$
$6$	0	0
$7$	− 5.54000i	− 0.791428i	−0.918374	−	0.395714i	$-0.870497\pi$
$7$	− 5.54000i	− 0.791428i	0.918374	−	0.395714i	$-0.129503\pi$
$8$	0	0
$9$	−6.43170	−0.714633
$10$	0	0
$11$	9.38498i	0.853180i	0.904445	+	0.426590i	$0.140285\pi$
$11$	9.38498i	0.853180i	−0.904445	+	0.426590i	$0.859715\pi$
$12$	0	0
$13$	23.1937	1.78413	0.892066	−	0.451905i	$-0.149255\pi$
$13$	23.1937	1.78413	0.892066	+	0.451905i	$0.149255\pi$
$14$	0	0
$15$	29.9564i	1.99709i
$16$	0	0
$17$	3.39182	0.199519	0.0997595	−	0.995012i	$-0.468193\pi$
$17$	3.39182	0.199519	0.0997595	+	0.995012i	$0.468193\pi$
$18$	0	0
$19$	− 37.5237i	− 1.97493i	−0.157840	−	0.987465i	$-0.550453\pi$
$19$	− 37.5237i	− 1.97493i	0.157840	−	0.987465i	$-0.449547\pi$
$20$	0	0
$21$	21.7629	1.03633
$22$	0	0
$23$	8.17550i	0.355456i	0.984080	+	0.177728i	$0.0568748\pi$
$23$	8.17550i	0.355456i	−0.984080	+	0.177728i	$0.943125\pi$
$24$	0	0
$25$	33.1521	1.32609
$26$	0	0
$27$	10.0891i	0.373670i
$28$	0	0
$29$	5.38516	0.185695
$29$	5.38516	0.185695
$30$	0	0
$31$	4.41568i	0.142441i	0.997461	+	0.0712206i	$0.0226894\pi$
$31$	4.41568i	0.142441i	−0.997461	+	0.0712206i	$0.977311\pi$
$32$	0	0
$33$	−36.8672	−1.11719
$34$	0	0
$35$	− 42.2467i	− 1.20705i
$36$	0	0
$37$	−0.0159584	−0.000431309 0	−0.000215654	−	1.00000i	$-0.500069\pi$
$37$	−0.0159584	−0.000431309 0	−0.000215654		1.00000i	$0.500069\pi$
$38$	0	0
$39$	91.1123i	2.33621i
$40$	0	0
$41$	−75.8415	−1.84979	−0.924896	−	0.380220i	$-0.875848\pi$
$41$	−75.8415	−1.84979	−0.924896	+	0.380220i	$0.875848\pi$
$42$	0	0
$43$	16.0497i	0.373248i	0.982431	+	0.186624i	$0.0597546\pi$
$43$	16.0497i	0.373248i	−0.982431	+	0.186624i	$0.940245\pi$
$44$	0	0
$45$	−49.0466	−1.08992
$46$	0	0
$47$	35.9472i	0.764834i	0.923990	+	0.382417i	$0.124908\pi$
$47$	35.9472i	0.764834i	−0.923990	+	0.382417i	$0.875092\pi$
$48$	0	0
$49$	18.3084	0.373641
$50$	0	0
$51$	13.3242i	0.261258i
$52$	0	0
$53$	−71.0080	−1.33977	−0.669887	−	0.742463i	$-0.733658\pi$
$53$	−71.0080	−1.33977	−0.669887	+	0.742463i	$0.733658\pi$
$54$	0	0
$55$	71.5676i	1.30123i
$56$	0	0
$57$	147.405	2.58605
$58$	0	0
$59$	64.7708i	1.09781i	0.835885	+	0.548905i	$0.184955\pi$
$59$	64.7708i	1.09781i	−0.835885	+	0.548905i	$0.815045\pi$
$60$	0	0
$61$	41.0583	0.673087	0.336544	−	0.941668i	$-0.390742\pi$
$61$	41.0583	0.673087	0.336544	+	0.941668i	$0.390742\pi$
$62$	0	0
$63$	35.6316i	0.565581i
$64$	0	0
$65$	176.870	2.72107
$66$	0	0
$67$	65.6441i	0.979763i	0.871789	+	0.489882i	$0.162960\pi$
$67$	65.6441i	0.979763i	−0.871789	+	0.489882i	$0.837040\pi$
$68$	0	0
$69$	−32.1160	−0.465449
$70$	0	0
$71$	− 32.6479i	− 0.459830i	−0.973211	−	0.229915i	$-0.926155\pi$
$71$	− 32.6479i	− 0.459830i	0.973211	−	0.229915i	$-0.0738448\pi$
$72$	0	0
$73$	−122.284	−1.67513	−0.837565	−	0.546338i	$-0.816021\pi$
$73$	−122.284	−1.67513	−0.837565	+	0.546338i	$0.816021\pi$
$74$	0	0
$75$	130.232i	1.73643i
$76$	0	0
$77$	51.9928	0.675231
$78$	0	0
$79$	− 85.2166i	− 1.07869i	−0.842085	−	0.539345i	$-0.818672\pi$
$79$	− 85.2166i	− 1.07869i	0.842085	−	0.539345i	$-0.181328\pi$
$80$	0	0
$81$	−97.5185	−1.20393
$82$	0	0
$83$	− 80.5050i	− 0.969940i	−0.874531	−	0.484970i	$-0.838831\pi$
$83$	− 80.5050i	− 0.969940i	0.874531	−	0.484970i	$-0.161169\pi$
$84$	0	0
$85$	25.8652	0.304297
$86$	0	0
$87$	21.1547i	0.243157i
$88$	0	0
$89$	−19.5031	−0.219136	−0.109568	−	0.993979i	$-0.534947\pi$
$89$	−19.5031	−0.219136	−0.109568	+	0.993979i	$0.534947\pi$
$90$	0	0
$91$	− 128.493i	− 1.41201i
$92$	0	0
$93$	−17.3462	−0.186518
$94$	0	0
$95$	− 286.146i	− 3.01207i
$96$	0	0
$97$	174.513	1.79910	0.899551	−	0.436816i	$-0.143894\pi$
$97$	174.513	1.79910	0.899551	+	0.436816i	$0.143894\pi$
$98$	0	0
$99$	− 60.3614i	− 0.609711i
$100$	0	0
$101$	−145.157	−1.43720	−0.718600	−	0.695424i	$-0.755217\pi$
$101$	−145.157	−1.43720	−0.718600	+	0.695424i	$0.755217\pi$
$102$	0	0
$103$	85.0272i	0.825507i	0.910843	+	0.412753i	$0.135433\pi$
$103$	85.0272i	0.825507i	−0.910843	+	0.412753i	$0.864567\pi$
$104$	0	0
$105$	165.958	1.58056
$106$	0	0
$107$	− 135.791i	− 1.26907i	−0.772893	−	0.634537i	$-0.781191\pi$
$107$	− 135.791i	− 1.26907i	0.772893	−	0.634537i	$-0.218809\pi$
$108$	0	0
$109$	−8.73480	−0.0801358	−0.0400679	−	0.999197i	$-0.512757\pi$
$109$	−8.73480	−0.0801358	−0.0400679	+	0.999197i	$0.512757\pi$
$110$	0	0
$111$	− 0.0626898i	0 0.000564773i
$112$	0	0
$113$	−16.7233	−0.147994	−0.0739970	−	0.997258i	$-0.523576\pi$
$113$	−16.7233	−0.147994	−0.0739970	+	0.997258i	$0.523576\pi$
$114$	0	0
$115$	62.3443i	0.542125i
$116$	0	0
$117$	−149.175	−1.27500
$118$	0	0
$119$	− 18.7907i	− 0.157905i
$120$	0	0
$121$	32.9221	0.272083
$122$	0	0
$123$	− 297.930i	− 2.42219i
$124$	0	0
$125$	62.1662	0.497330
$126$	0	0
$127$	74.2716i	0.584815i	0.956294	+	0.292408i	$0.0944564\pi$
$127$	74.2716i	0.584815i	−0.956294	+	0.292408i	$0.905544\pi$
$128$	0	0
$129$	−63.0482	−0.488746
$130$	0	0
$131$	156.350i	1.19351i	0.802422	+	0.596756i	$0.203544\pi$
$131$	156.350i	1.19351i	−0.802422	+	0.596756i	$0.796456\pi$
$132$	0	0
$133$	−207.881	−1.56301
$134$	0	0
$135$	76.9370i	0.569904i
$136$	0	0
$137$	99.1270	0.723554	0.361777	−	0.932265i	$-0.382170\pi$
$137$	99.1270	0.723554	0.361777	+	0.932265i	$0.382170\pi$
$138$	0	0
$139$	− 213.058i	− 1.53279i	−0.642368	−	0.766396i	$-0.722048\pi$
$139$	− 213.058i	− 1.53279i	0.642368	−	0.766396i	$-0.277952\pi$
$140$	0	0
$141$	−141.212	−1.00150
$142$	0	0
$143$	217.673i	1.52219i
$144$	0	0
$145$	41.0659	0.283213
$146$	0	0
$147$	71.9214i	0.489261i
$148$	0	0
$149$	−30.4204	−0.204164	−0.102082	−	0.994776i	$-0.532550\pi$
$149$	−30.4204	−0.204164	−0.102082	+	0.994776i	$0.532550\pi$
$150$	0	0
$151$	− 263.348i	− 1.74402i	−0.489484	−	0.872012i	$-0.662815\pi$
$151$	− 263.348i	− 1.74402i	0.489484	−	0.872012i	$-0.337185\pi$
$152$	0	0
$153$	−21.8152	−0.142583
$154$	0	0
$155$	33.6729i	0.217244i
$156$	0	0
$157$	−15.9827	−0.101801	−0.0509004	−	0.998704i	$-0.516209\pi$
$157$	−15.9827	−0.101801	−0.0509004	+	0.998704i	$0.516209\pi$
$158$	0	0
$159$	− 278.942i	− 1.75435i
$160$	0	0
$161$	45.2922	0.281318
$162$	0	0
$163$	− 100.548i	− 0.616861i	−0.951247	−	0.308430i	$-0.900196\pi$
$163$	− 100.548i	− 0.616861i	0.951247	−	0.308430i	$-0.0998037\pi$
$164$	0	0
$165$	−281.140	−1.70388
$166$	0	0
$167$	− 110.099i	− 0.659274i	−0.944108	−	0.329637i	$-0.893074\pi$
$167$	− 110.099i	− 0.659274i	0.944108	−	0.329637i	$-0.106926\pi$
$168$	0	0
$169$	368.948	2.18313
$170$	0	0
$171$	241.341i	1.41135i
$172$	0	0
$173$	237.942	1.37539	0.687695	−	0.726000i	$-0.258623\pi$
$173$	237.942	1.37539	0.687695	+	0.726000i	$0.258623\pi$
$174$	0	0
$175$	− 183.663i	− 1.04950i
$176$	0	0
$177$	−254.440	−1.43752
$178$	0	0
$179$	58.3673i	0.326074i	0.986620	+	0.163037i	$0.0521291\pi$
$179$	58.3673i	0.326074i	−0.986620	+	0.163037i	$0.947871\pi$
$180$	0	0
$181$	−68.2620	−0.377138	−0.188569	−	0.982060i	$-0.560385\pi$
$181$	−68.2620	−0.377138	−0.188569	+	0.982060i	$0.560385\pi$
$182$	0	0
$183$	161.290i	0.881367i
$184$	0	0
$185$	−0.121695	−0.000657811 0
$186$	0	0
$187$	31.8322i	0.170226i
$188$	0	0
$189$	55.8936	0.295733
$190$	0	0
$191$	− 98.0523i	− 0.513363i	−0.966496	−	0.256681i	$-0.917371\pi$
$191$	− 98.0523i	− 0.513363i	0.966496	−	0.256681i	$-0.0826291\pi$
$192$	0	0
$193$	−256.470	−1.32886	−0.664431	−	0.747350i	$-0.731326\pi$
$193$	−256.470	−1.32886	−0.664431	+	0.747350i	$0.731326\pi$
$194$	0	0
$195$	694.800i	3.56308i
$196$	0	0
$197$	−308.149	−1.56421	−0.782104	−	0.623148i	$-0.785853\pi$
$197$	−308.149	−1.56421	−0.782104	+	0.623148i	$0.785853\pi$
$198$	0	0
$199$	99.3770i	0.499382i	0.968326	+	0.249691i	$0.0803290\pi$
$199$	99.3770i	0.499382i	−0.968326	+	0.249691i	$0.919671\pi$
$200$	0	0
$201$	−257.871	−1.28294
$202$	0	0
$203$	− 29.8338i	− 0.146965i
$204$	0	0
$205$	−578.349	−2.82121
$206$	0	0
$207$	− 52.5824i	− 0.254021i
$208$	0	0
$209$	352.159	1.68497
$210$	0	0
$211$	− 171.658i	− 0.813547i	−0.913529	−	0.406774i	$-0.866654\pi$
$211$	− 171.658i	− 0.813547i	0.913529	−	0.406774i	$-0.133346\pi$
$212$	0	0
$213$	128.252	0.602120
$214$	0	0
$215$	122.391i	0.569260i
$216$	0	0
$217$	24.4629	0.112732
$218$	0	0
$219$	− 480.373i	− 2.19348i
$220$	0	0
$221$	78.6690	0.355968
$222$	0	0
$223$	− 334.207i	− 1.49869i	−0.662182	−	0.749343i	$-0.730369\pi$
$223$	− 334.207i	− 1.49869i	0.662182	−	0.749343i	$-0.269631\pi$
$224$	0	0
$225$	−213.225	−0.947665
$226$	0	0
$227$	77.1942i	0.340062i	0.985439	+	0.170031i	$0.0543868\pi$
$227$	77.1942i	0.340062i	−0.985439	+	0.170031i	$0.945613\pi$
$228$	0	0
$229$	89.0065	0.388675	0.194337	−	0.980935i	$-0.437744\pi$
$229$	89.0065	0.388675	0.194337	+	0.980935i	$0.437744\pi$
$230$	0	0
$231$	204.244i	0.884174i
$232$	0	0
$233$	361.682	1.55228	0.776142	−	0.630558i	$-0.217174\pi$
$233$	361.682	1.55228	0.776142	+	0.630558i	$0.217174\pi$
$234$	0	0
$235$	274.124i	1.16649i
$236$	0	0
$237$	334.758	1.41248
$238$	0	0
$239$	− 51.3625i	− 0.214906i	−0.994210	−	0.107453i	$-0.965730\pi$
$239$	− 51.3625i	− 0.214906i	0.994210	−	0.107453i	$-0.0342695\pi$
$240$	0	0
$241$	−208.917	−0.866874	−0.433437	−	0.901184i	$-0.642699\pi$
$241$	−208.917	−0.866874	−0.433437	+	0.901184i	$0.642699\pi$
$242$	0	0
$243$	− 292.282i	− 1.20281i
$244$	0	0
$245$	139.616	0.569860
$246$	0	0
$247$	− 870.313i	− 3.52353i
$248$	0	0
$249$	316.249	1.27008
$250$	0	0
$251$	120.972i	0.481960i	0.970530	+	0.240980i	$0.0774689\pi$
$251$	120.972i	0.481960i	−0.970530	+	0.240980i	$0.922531\pi$
$252$	0	0
$253$	−76.7269	−0.303268
$254$	0	0
$255$	101.607i	0.398458i
$256$	0	0
$257$	−33.6370	−0.130883	−0.0654416	−	0.997856i	$-0.520846\pi$
$257$	−33.6370	−0.130883	−0.0654416	+	0.997856i	$0.520846\pi$
$258$	0	0
$259$	0.0884097i	0 0.000341350i
$260$	0	0
$261$	−34.6358	−0.132704
$262$	0	0
$263$	355.007i	1.34984i	0.737893	+	0.674918i	$0.235821\pi$
$263$	355.007i	1.34984i	−0.737893	+	0.674918i	$0.764179\pi$
$264$	0	0
$265$	−541.490	−2.04336
$266$	0	0
$267$	− 76.6145i	− 0.286946i
$268$	0	0
$269$	−93.6950	−0.348308	−0.174154	−	0.984718i	$-0.555719\pi$
$269$	−93.6950	−0.348308	−0.174154	+	0.984718i	$0.555719\pi$
$270$	0	0
$271$	87.2682i	0.322023i	0.986953	+	0.161011i	$0.0514756\pi$
$271$	87.2682i	0.322023i	−0.986953	+	0.161011i	$0.948524\pi$
$272$	0	0
$273$	504.762	1.84895
$274$	0	0
$275$	311.132i	1.13139i
$276$	0	0
$277$	368.254	1.32944	0.664719	−	0.747093i	$-0.268551\pi$
$277$	368.254	1.32944	0.664719	+	0.747093i	$0.268551\pi$
$278$	0	0
$279$	− 28.4003i	− 0.101793i
$280$	0	0
$281$	275.626	0.980876	0.490438	−	0.871476i	$-0.336837\pi$
$281$	275.626	0.980876	0.490438	+	0.871476i	$0.336837\pi$
$282$	0	0
$283$	− 382.779i	− 1.35257i	−0.736638	−	0.676287i	$-0.763588\pi$
$283$	− 382.779i	− 1.35257i	0.736638	−	0.676287i	$-0.236412\pi$
$284$	0	0
$285$	1124.07	3.94412
$286$	0	0
$287$	420.162i	1.46398i
$288$	0	0
$289$	−277.496	−0.960192
$290$	0	0
$291$	685.542i	2.35582i
$292$	0	0
$293$	−465.001	−1.58703	−0.793517	−	0.608548i	$-0.791752\pi$
$293$	−465.001	−1.58703	−0.793517	+	0.608548i	$0.791752\pi$
$294$	0	0
$295$	493.926i	1.67433i
$296$	0	0
$297$	−94.6861	−0.318808
$298$	0	0
$299$	189.620i	0.634181i
$300$	0	0
$301$	88.9151	0.295399
$302$	0	0
$303$	− 570.224i	− 1.88193i
$304$	0	0
$305$	313.101	1.02656
$306$	0	0
$307$	− 322.409i	− 1.05019i	−0.851042	−	0.525097i	$-0.824029\pi$
$307$	− 322.409i	− 1.05019i	0.851042	−	0.525097i	$-0.175971\pi$
$308$	0	0
$309$	−334.014	−1.08095
$310$	0	0
$311$	540.440i	1.73775i	0.495032	+	0.868875i	$0.335156\pi$
$311$	540.440i	1.73775i	−0.495032	+	0.868875i	$0.664844\pi$
$312$	0	0
$313$	−285.630	−0.912555	−0.456277	−	0.889838i	$-0.650817\pi$
$313$	−285.630	−0.912555	−0.456277	+	0.889838i	$0.650817\pi$
$314$	0	0
$315$	271.718i	0.862596i
$316$	0	0
$317$	193.635	0.610837	0.305419	−	0.952218i	$-0.401204\pi$
$317$	193.635	0.610837	0.305419	+	0.952218i	$0.401204\pi$
$318$	0	0
$319$	50.5397i	0.158432i
$320$	0	0
$321$	533.430	1.66178
$322$	0	0
$323$	− 127.274i	− 0.394036i
$324$	0	0
$325$	768.921	2.36591
$326$	0	0
$327$	− 34.3131i	− 0.104933i
$328$	0	0
$329$	199.147	0.605311
$330$	0	0
$331$	441.209i	1.33296i	0.745524	+	0.666479i	$0.232199\pi$
$331$	441.209i	1.33296i	−0.745524	+	0.666479i	$0.767801\pi$
$332$	0	0
$333$	0.102640	0.000308228 0
$334$	0	0
$335$	500.586i	1.49429i
$336$	0	0
$337$	302.671	0.898133	0.449066	−	0.893498i	$-0.351757\pi$
$337$	302.671	0.898133	0.449066	+	0.893498i	$0.351757\pi$
$338$	0	0
$339$	− 65.6946i	− 0.193789i
$340$	0	0
$341$	−41.4411	−0.121528
$342$	0	0
$343$	− 372.889i	− 1.08714i
$344$	0	0
$345$	−244.909	−0.709880
$346$	0	0
$347$	− 363.008i	− 1.04613i	−0.852292	−	0.523066i	$-0.824788\pi$
$347$	− 363.008i	− 1.04613i	0.852292	−	0.523066i	$-0.175212\pi$
$348$	0	0
$349$	12.1746	0.0348841	0.0174421	−	0.999848i	$-0.494448\pi$
$349$	12.1746	0.0348841	0.0174421	+	0.999848i	$0.494448\pi$
$350$	0	0
$351$	234.004i	0.666677i
$352$	0	0
$353$	70.0382	0.198409	0.0992043	−	0.995067i	$-0.468370\pi$
$353$	70.0382	0.198409	0.0992043	+	0.995067i	$0.468370\pi$
$354$	0	0
$355$	− 248.965i	− 0.701310i
$356$	0	0
$357$	73.8159	0.206767
$358$	0	0
$359$	362.358i	1.00935i	0.863308	+	0.504677i	$0.168388\pi$
$359$	362.358i	1.00935i	−0.863308	+	0.504677i	$0.831612\pi$
$360$	0	0
$361$	−1047.02	−2.90035
$362$	0	0
$363$	129.328i	0.356277i
$364$	0	0
$365$	−932.511	−2.55483
$366$	0	0
$367$	228.998i	0.623973i	0.950086	+	0.311987i	$0.100994\pi$
$367$	228.998i	0.623973i	−0.950086	+	0.311987i	$0.899006\pi$
$368$	0	0
$369$	487.790	1.32192
$370$	0	0
$371$	393.384i	1.06034i
$372$	0	0
$373$	−233.561	−0.626169	−0.313085	−	0.949725i	$-0.601362\pi$
$373$	−233.561	−0.626169	−0.313085	+	0.949725i	$0.601362\pi$
$374$	0	0
$375$	244.209i	0.651224i
$376$	0	0
$377$	124.902	0.331305
$378$	0	0
$379$	414.460i	1.09356i	0.837276	+	0.546781i	$0.184147\pi$
$379$	414.460i	1.09356i	−0.837276	+	0.546781i	$0.815853\pi$
$380$	0	0
$381$	−291.762	−0.765781
$382$	0	0
$383$	361.668i	0.944303i	0.881517	+	0.472152i	$0.156523\pi$
$383$	361.668i	0.944303i	−0.881517	+	0.472152i	$0.843477\pi$
$384$	0	0
$385$	396.484	1.02983
$386$	0	0
$387$	− 103.227i	− 0.266736i
$388$	0	0
$389$	−467.835	−1.20266	−0.601330	−	0.799001i	$-0.705362\pi$
$389$	−467.835	−1.20266	−0.601330	+	0.799001i	$0.705362\pi$
$390$	0	0
$391$	27.7298i	0.0709203i
$392$	0	0
$393$	−614.194	−1.56283
$394$	0	0
$395$	− 649.841i	− 1.64517i
$396$	0	0
$397$	64.9294	0.163550	0.0817751	−	0.996651i	$-0.473941\pi$
$397$	64.9294	0.163550	0.0817751	+	0.996651i	$0.473941\pi$
$398$	0	0
$399$	− 816.623i	− 2.04667i
$400$	0	0
$401$	−250.312	−0.624221	−0.312110	−	0.950046i	$-0.601036\pi$
$401$	−250.312	−0.624221	−0.312110	+	0.950046i	$0.601036\pi$
$402$	0	0
$403$	102.416i	0.254134i
$404$	0	0
$405$	−743.652	−1.83618
$406$	0	0
$407$	− 0.149770i	0 0.000367984i
$408$	0	0
$409$	−275.819	−0.674375	−0.337187	−	0.941438i	$-0.609476\pi$
$409$	−275.819	−0.674375	−0.337187	+	0.941438i	$0.609476\pi$
$410$	0	0
$411$	389.402i	0.947451i
$412$	0	0
$413$	358.830	0.868837
$414$	0	0
$415$	− 613.911i	− 1.47930i
$416$	0	0
$417$	836.961	2.00710
$418$	0	0
$419$	106.224i	0.253517i	0.991934	+	0.126758i	$0.0404573\pi$
$419$	106.224i	0.253517i	−0.991934	+	0.126758i	$0.959543\pi$
$420$	0	0
$421$	427.248	1.01484	0.507420	−	0.861699i	$-0.330599\pi$
$421$	427.248	1.01484	0.507420	+	0.861699i	$0.330599\pi$
$422$	0	0
$423$	− 231.202i	− 0.546576i
$424$	0	0
$425$	112.446	0.264579
$426$	0	0
$427$	− 227.463i	− 0.532700i
$428$	0	0
$429$	−855.088	−1.99321
$430$	0	0
$431$	682.610i	1.58378i	0.610662	+	0.791891i	$0.290903\pi$
$431$	682.610i	1.58378i	−0.610662	+	0.791891i	$0.709097\pi$
$432$	0	0
$433$	31.9951	0.0738917	0.0369459	−	0.999317i	$-0.488237\pi$
$433$	31.9951	0.0738917	0.0369459	+	0.999317i	$0.488237\pi$
$434$	0	0
$435$	161.320i	0.370851i
$436$	0	0
$437$	306.775	0.702001
$438$	0	0
$439$	832.197i	1.89566i	0.318769	+	0.947832i	$0.396731\pi$
$439$	832.197i	1.89566i	−0.318769	+	0.947832i	$0.603269\pi$
$440$	0	0
$441$	−117.754	−0.267017
$442$	0	0
$443$	− 608.511i	− 1.37361i	−0.726840	−	0.686807i	$-0.759012\pi$
$443$	− 608.511i	− 1.37361i	0.726840	−	0.686807i	$-0.240988\pi$
$444$	0	0
$445$	−148.726	−0.334216
$446$	0	0
$447$	− 119.501i	− 0.267340i
$448$	0	0
$449$	−384.818	−0.857055	−0.428528	−	0.903529i	$-0.640968\pi$
$449$	−384.818	−0.857055	−0.428528	+	0.903529i	$0.640968\pi$
$450$	0	0
$451$	− 711.771i	− 1.57821i
$452$	0	0
$453$	1034.51	2.28370
$454$	0	0
$455$	− 979.857i	− 2.15353i
$456$	0	0
$457$	−330.157	−0.722444	−0.361222	−	0.932480i	$-0.617640\pi$
$457$	−330.157	−0.722444	−0.361222	+	0.932480i	$0.617640\pi$
$458$	0	0
$459$	34.2204i	0.0745544i
$460$	0	0
$461$	−873.905	−1.89567	−0.947836	−	0.318758i	$-0.896734\pi$
$461$	−873.905	−1.89567	−0.947836	+	0.318758i	$0.896734\pi$
$462$	0	0
$463$	269.148i	0.581313i	0.956828	+	0.290656i	$0.0938736\pi$
$463$	269.148i	0.581313i	−0.956828	+	0.290656i	$0.906126\pi$
$464$	0	0
$465$	−132.278	−0.284469
$466$	0	0
$467$	− 719.153i	− 1.53994i	−0.638079	−	0.769971i	$-0.720271\pi$
$467$	− 719.153i	− 1.53994i	0.638079	−	0.769971i	$-0.279729\pi$
$468$	0	0
$469$	363.668	0.775412
$470$	0	0
$471$	− 62.7852i	− 0.133302i
$472$	0	0
$473$	−150.626	−0.318448
$474$	0	0
$475$	− 1243.99i	− 2.61893i
$476$	0	0
$477$	456.702	0.957447
$478$	0	0
$479$	96.7800i	0.202046i	0.994884	+	0.101023i	$0.0322116\pi$
$479$	96.7800i	0.202046i	−0.994884	+	0.101023i	$0.967788\pi$
$480$	0	0
$481$	−0.370135	−0.000769512 0
$482$	0	0
$483$	177.922i	0.368369i
$484$	0	0
$485$	1330.79	2.74390
$486$	0	0
$487$	279.484i	0.573889i	0.957947	+	0.286945i	$0.0926395\pi$
$487$	279.484i	0.573889i	−0.957947	+	0.286945i	$0.907360\pi$
$488$	0	0
$489$	394.986	0.807742
$490$	0	0
$491$	− 394.911i	− 0.804299i	−0.915574	−	0.402149i	$-0.868263\pi$
$491$	− 394.911i	− 0.804299i	0.915574	−	0.402149i	$-0.131737\pi$
$492$	0	0
$493$	18.2655	0.0370497
$494$	0	0
$495$	− 460.301i	− 0.929902i
$496$	0	0
$497$	−180.869	−0.363922
$498$	0	0
$499$	86.1332i	0.172612i	0.996269	+	0.0863059i	$0.0275062\pi$
$499$	86.1332i	0.172612i	−0.996269	+	0.0863059i	$0.972494\pi$
$500$	0	0
$501$	432.503	0.863279
$502$	0	0
$503$	− 63.2534i	− 0.125752i	−0.998021	−	0.0628761i	$-0.979973\pi$
$503$	− 63.2534i	− 0.125752i	0.998021	−	0.0628761i	$-0.0200273\pi$
$504$	0	0
$505$	−1106.93	−2.19195
$506$	0	0
$507$	1449.35i	2.85867i
$508$	0	0
$509$	473.051	0.929373	0.464687	−	0.885475i	$-0.346167\pi$
$509$	473.051	0.929373	0.464687	+	0.885475i	$0.346167\pi$
$510$	0	0
$511$	677.456i	1.32574i
$512$	0	0
$513$	378.580	0.737973
$514$	0	0
$515$	648.397i	1.25902i
$516$	0	0
$517$	−337.364	−0.652541
$518$	0	0
$519$	934.714i	1.80099i
$520$	0	0
$521$	365.912	0.702326	0.351163	−	0.936314i	$-0.385786\pi$
$521$	365.912	0.702326	0.351163	+	0.936314i	$0.385786\pi$
$522$	0	0
$523$	− 118.474i	− 0.226528i	−0.993565	−	0.113264i	$-0.963869\pi$
$523$	− 118.474i	− 0.226528i	0.993565	−	0.113264i	$-0.0361305\pi$
$524$	0	0
$525$	721.486	1.37426
$526$	0	0
$527$	14.9772i	0.0284197i
$528$	0	0
$529$	462.161	0.873651
$530$	0	0
$531$	− 416.586i	− 0.784531i
$532$	0	0
$533$	−1759.05	−3.30027
$534$	0	0
$535$	− 1035.51i	− 1.93553i
$536$	0	0
$537$	−229.286	−0.426975
$538$	0	0
$539$	171.824i	0.318784i
$540$	0	0
$541$	270.289	0.499610	0.249805	−	0.968296i	$-0.419633\pi$
$541$	270.289	0.499610	0.249805	+	0.968296i	$0.419633\pi$
$542$	0	0
$543$	− 268.155i	− 0.493839i
$544$	0	0
$545$	−66.6094	−0.122219
$546$	0	0
$547$	− 675.879i	− 1.23561i	−0.786331	−	0.617805i	$-0.788022\pi$
$547$	− 675.879i	− 1.23561i	0.786331	−	0.617805i	$-0.211978\pi$
$548$	0	0
$549$	−264.075	−0.481010
$550$	0	0
$551$	− 202.071i	− 0.366735i
$552$	0	0
$553$	−472.099	−0.853706
$554$	0	0
$555$	− 0.478057i	0 0.000861364i
$556$	0	0
$557$	654.060	1.17426	0.587128	−	0.809494i	$-0.300259\pi$
$557$	654.060	1.17426	0.587128	+	0.809494i	$0.300259\pi$
$558$	0	0
$559$	372.251i	0.665924i
$560$	0	0
$561$	−125.047	−0.222900
$562$	0	0
$563$	− 394.468i	− 0.700654i	−0.936628	−	0.350327i	$-0.886071\pi$
$563$	− 394.468i	− 0.700654i	0.936628	−	0.350327i	$-0.113929\pi$
$564$	0	0
$565$	−127.528	−0.225713
$566$	0	0
$567$	540.252i	0.952826i
$568$	0	0
$569$	5.31708	0.00934460	0.00467230	−	0.999989i	$-0.498513\pi$
$569$	5.31708	0.00934460	0.00467230	+	0.999989i	$0.498513\pi$
$570$	0	0
$571$	− 801.525i	− 1.40372i	−0.712314	−	0.701861i	$-0.752353\pi$
$571$	− 801.525i	− 1.40372i	0.712314	−	0.701861i	$-0.247647\pi$
$572$	0	0
$573$	385.181	0.672218
$574$	0	0
$575$	271.035i	0.471366i
$576$	0	0
$577$	−332.817	−0.576805	−0.288403	−	0.957509i	$-0.593124\pi$
$577$	−332.817	−0.576805	−0.288403	+	0.957509i	$0.593124\pi$
$578$	0	0
$579$	− 1007.50i	− 1.74007i
$580$	0	0
$581$	−445.998	−0.767638
$582$	0	0
$583$	− 666.409i	− 1.14307i
$584$	0	0
$585$	−1137.57	−1.94457
$586$	0	0
$587$	− 529.513i	− 0.902066i	−0.892507	−	0.451033i	$-0.851056\pi$
$587$	− 529.513i	− 0.902066i	0.892507	−	0.451033i	$-0.148944\pi$
$588$	0	0
$589$	165.692	0.281311
$590$	0	0
$591$	− 1210.51i	− 2.04824i
$592$	0	0
$593$	939.990	1.58514	0.792571	−	0.609779i	$-0.208742\pi$
$593$	939.990	1.58514	0.792571	+	0.609779i	$0.208742\pi$
$594$	0	0
$595$	− 143.293i	− 0.240829i
$596$	0	0
$597$	−390.385	−0.653911
$598$	0	0
$599$	− 683.114i	− 1.14042i	−0.821498	−	0.570212i	$-0.806861\pi$
$599$	− 683.114i	− 1.14042i	0.821498	−	0.570212i	$-0.193139\pi$
$600$	0	0
$601$	−554.853	−0.923216	−0.461608	−	0.887084i	$-0.652727\pi$
$601$	−554.853	−0.923216	−0.461608	+	0.887084i	$0.652727\pi$
$602$	0	0
$603$	− 422.204i	− 0.700172i
$604$	0	0
$605$	251.056	0.414968
$606$	0	0
$607$	− 102.168i	− 0.168317i	−0.996452	−	0.0841585i	$-0.973180\pi$
$607$	− 102.168i	− 0.168317i	0.996452	−	0.0841585i	$-0.0268202\pi$
$608$	0	0
$609$	117.197	0.192441
$610$	0	0
$611$	833.749i	1.36456i
$612$	0	0
$613$	−728.098	−1.18776	−0.593881	−	0.804553i	$-0.702405\pi$
$613$	−728.098	−1.18776	−0.593881	+	0.804553i	$0.702405\pi$
$614$	0	0
$615$	− 2271.94i	− 3.69421i
$616$	0	0
$617$	37.0181	0.0599969	0.0299984	−	0.999550i	$-0.490450\pi$
$617$	37.0181	0.0599969	0.0299984	+	0.999550i	$0.490450\pi$
$618$	0	0
$619$	− 56.3733i	− 0.0910716i	−0.998963	−	0.0455358i	$-0.985500\pi$
$619$	− 56.3733i	− 0.0910716i	0.998963	−	0.0455358i	$-0.0144995\pi$
$620$	0	0
$621$	−82.4834	−0.132824
$622$	0	0
$623$	108.047i	0.173430i
$624$	0	0
$625$	−354.739	−0.567583
$626$	0	0
$627$	1383.39i	2.20637i
$628$	0	0
$629$	−0.0541282	−8.60543e−5 0
$630$	0	0
$631$	1011.93i	1.60369i	0.597532	+	0.801845i	$0.296148\pi$
$631$	1011.93i	1.60369i	−0.597532	+	0.801845i	$0.703852\pi$
$632$	0	0
$633$	674.329	1.06529
$634$	0	0
$635$	566.377i	0.891932i
$636$	0	0
$637$	424.641	0.666626
$638$	0	0
$639$	209.982i	0.328610i
$640$	0	0
$641$	−45.5733	−0.0710972	−0.0355486	−	0.999368i	$-0.511318\pi$
$641$	−45.5733	−0.0710972	−0.0355486	+	0.999368i	$0.511318\pi$
$642$	0	0
$643$	− 624.626i	− 0.971424i	−0.874119	−	0.485712i	$-0.838560\pi$
$643$	− 624.626i	− 0.971424i	0.874119	−	0.485712i	$-0.161440\pi$
$644$	0	0
$645$	−480.790	−0.745411
$646$	0	0
$647$	− 186.160i	− 0.287727i	−0.989598	−	0.143864i	$-0.954047\pi$
$647$	− 186.160i	− 0.287727i	0.989598	−	0.143864i	$-0.0459527\pi$
$648$	0	0
$649$	−607.873	−0.936630
$650$	0	0
$651$	96.0979i	0.147616i
$652$	0	0
$653$	676.168	1.03548	0.517740	−	0.855538i	$-0.326773\pi$
$653$	676.168	1.03548	0.517740	+	0.855538i	$0.326773\pi$
$654$	0	0
$655$	1192.29i	1.82029i
$656$	0	0
$657$	786.497	1.19710
$658$	0	0
$659$	− 678.525i	− 1.02963i	−0.857302	−	0.514814i	$-0.827861\pi$
$659$	− 678.525i	− 1.02963i	0.857302	−	0.514814i	$-0.172139\pi$
$660$	0	0
$661$	125.237	0.189467	0.0947333	−	0.995503i	$-0.469800\pi$
$661$	125.237	0.189467	0.0947333	+	0.995503i	$0.469800\pi$
$662$	0	0
$663$	309.037i	0.466119i
$664$	0	0
$665$	−1585.25	−2.38383
$666$	0	0
$667$	44.0264i	0.0660066i
$668$	0	0
$669$	1312.87	1.96244
$670$	0	0
$671$	385.332i	0.574265i
$672$	0	0
$673$	−1042.29	−1.54872	−0.774362	−	0.632743i	$-0.781929\pi$
$673$	−1042.29	−1.54872	−0.774362	+	0.632743i	$0.781929\pi$
$674$	0	0
$675$	334.475i	0.495519i
$676$	0	0
$677$	−648.606	−0.958059	−0.479029	−	0.877799i	$-0.659011\pi$
$677$	−648.606	−0.958059	−0.479029	+	0.877799i	$0.659011\pi$
$678$	0	0
$679$	− 966.801i	− 1.42386i
$680$	0	0
$681$	−303.243	−0.445291
$682$	0	0
$683$	967.574i	1.41665i	0.705885	+	0.708327i	$0.250550\pi$
$683$	967.574i	1.41665i	−0.705885	+	0.708327i	$0.749450\pi$
$684$	0	0
$685$	755.918	1.10353
$686$	0	0
$687$	349.646i	0.508946i
$688$	0	0
$689$	−1646.94	−2.39033
$690$	0	0
$691$	1049.17i	1.51834i	0.650891	+	0.759171i	$0.274395\pi$
$691$	1049.17i	1.51834i	−0.650891	+	0.759171i	$0.725605\pi$
$692$	0	0
$693$	−334.402	−0.482543
$694$	0	0
$695$	− 1624.73i	− 2.33774i
$696$	0	0
$697$	−257.241	−0.369069
$698$	0	0
$699$	1420.80i	2.03262i
$700$	0	0
$701$	47.3109	0.0674906	0.0337453	−	0.999430i	$-0.489256\pi$
$701$	47.3109	0.0674906	0.0337453	+	0.999430i	$0.489256\pi$
$702$	0	0
$703$	0.598819i	0 0.000851805i
$704$	0	0
$705$	−1076.85	−1.52744
$706$	0	0
$707$	804.170i	1.13744i
$708$	0	0
$709$	847.156	1.19486	0.597430	−	0.801921i	$-0.296188\pi$
$709$	847.156	1.19486	0.597430	+	0.801921i	$0.296188\pi$
$710$	0	0
$711$	548.087i	0.770868i
$712$	0	0
$713$	−36.1004	−0.0506317
$714$	0	0
$715$	1659.92i	2.32156i
$716$	0	0
$717$	201.768	0.281406
$718$	0	0
$719$	− 1041.20i	− 1.44812i	−0.689739	−	0.724058i	$-0.742275\pi$
$719$	− 1041.20i	− 1.44812i	0.689739	−	0.724058i	$-0.257725\pi$
$720$	0	0
$721$	471.050	0.653329
$722$	0	0
$723$	− 820.691i	− 1.13512i
$724$	0	0
$725$	178.530	0.246248
$726$	0	0
$727$	176.419i	0.242667i	0.992612	+	0.121334i	$0.0387171\pi$
$727$	176.419i	0.242667i	−0.992612	+	0.121334i	$0.961283\pi$
$728$	0	0
$729$	270.511	0.371071
$730$	0	0
$731$	54.4376i	0.0744701i
$732$	0	0
$733$	−199.309	−0.271908	−0.135954	−	0.990715i	$-0.543410\pi$
$733$	−199.309	−0.271908	−0.135954	+	0.990715i	$0.543410\pi$
$734$	0	0
$735$	548.455i	0.746197i
$736$	0	0
$737$	−616.069	−0.835915
$738$	0	0
$739$	1262.49i	1.70838i	0.519963	+	0.854189i	$0.325946\pi$
$739$	1262.49i	1.70838i	−0.519963	+	0.854189i	$0.674054\pi$
$740$	0	0
$741$	3418.87	4.61386
$742$	0	0
$743$	773.056i	1.04045i	0.854028	+	0.520226i	$0.174152\pi$
$743$	773.056i	1.04045i	−0.854028	+	0.520226i	$0.825848\pi$
$744$	0	0
$745$	−231.979	−0.311381
$746$	0	0
$747$	517.784i	0.693151i
$748$	0	0
$749$	−752.281	−1.00438
$750$	0	0
$751$	− 1466.80i	− 1.95313i	−0.215223	−	0.976565i	$-0.569048\pi$
$751$	− 1466.80i	− 1.95313i	0.215223	−	0.976565i	$-0.430952\pi$
$752$	0	0
$753$	−475.217	−0.631098
$754$	0	0
$755$	− 2008.23i	− 2.65990i
$756$	0	0
$757$	271.705	0.358923	0.179462	−	0.983765i	$-0.442564\pi$
$757$	271.705	0.358923	0.179462	+	0.983765i	$0.442564\pi$
$758$	0	0
$759$	− 301.408i	− 0.397112i
$760$	0	0
$761$	1211.23	1.59163	0.795817	−	0.605537i	$-0.207042\pi$
$761$	1211.23	1.59163	0.795817	+	0.605537i	$0.207042\pi$
$762$	0	0
$763$	48.3908i	0.0634217i
$764$	0	0
$765$	−166.357	−0.217461
$766$	0	0
$767$	1502.27i	1.95864i
$768$	0	0
$769$	1036.42	1.34775	0.673873	−	0.738847i	$-0.264629\pi$
$769$	1036.42	1.34775	0.673873	+	0.738847i	$0.264629\pi$
$770$	0	0
$771$	− 132.137i	− 0.171384i
$772$	0	0
$773$	−1053.73	−1.36317	−0.681583	−	0.731741i	$-0.738708\pi$
$773$	−1053.73	−1.36317	−0.681583	+	0.731741i	$0.738708\pi$
$774$	0	0
$775$	146.389i	0.188889i
$776$	0	0
$777$	−0.347301	−0.000446977 0
$778$	0	0
$779$	2845.85i	3.65321i
$780$	0	0
$781$	306.400	0.392318
$782$	0	0
$783$	54.3315i	0.0693889i
$784$	0	0
$785$	−121.880	−0.155261
$786$	0	0
$787$	701.506i	0.891368i	0.895190	+	0.445684i	$0.147039\pi$
$787$	701.506i	0.891368i	−0.895190	+	0.445684i	$0.852961\pi$
$788$	0	0
$789$	−1394.58	−1.76753
$790$	0	0
$791$	92.6471i	0.117127i
$792$	0	0
$793$	952.295	1.20088
$794$	0	0
$795$	− 2127.15i	− 2.67566i
$796$	0	0
$797$	1406.70	1.76500	0.882498	−	0.470316i	$-0.155860\pi$
$797$	1406.70	1.76500	0.882498	+	0.470316i	$0.155860\pi$
$798$	0	0
$799$	121.926i	0.152599i
$800$	0	0
$801$	125.438	0.156602
$802$	0	0
$803$	− 1147.64i	− 1.42919i
$804$	0	0
$805$	345.388	0.429053
$806$	0	0
$807$	− 368.064i	− 0.456089i
$808$	0	0
$809$	432.584	0.534714	0.267357	−	0.963598i	$-0.413850\pi$
$809$	432.584	0.534714	0.267357	+	0.963598i	$0.413850\pi$
$810$	0	0
$811$	908.477i	1.12019i	0.828427	+	0.560097i	$0.189236\pi$
$811$	908.477i	1.12019i	−0.828427	+	0.560097i	$0.810764\pi$
$812$	0	0
$813$	−342.817	−0.421670
$814$	0	0
$815$	− 766.757i	− 0.940806i
$816$	0	0
$817$	602.242	0.737138
$818$	0	0
$819$	826.429i	1.00907i
$820$	0	0
$821$	139.911	0.170416	0.0852078	−	0.996363i	$-0.472845\pi$
$821$	139.911	0.170416	0.0852078	+	0.996363i	$0.472845\pi$
$822$	0	0
$823$	− 444.474i	− 0.540066i	−0.962851	−	0.270033i	$-0.912965\pi$
$823$	− 444.474i	− 0.540066i	0.962851	−	0.270033i	$-0.0870346\pi$
$824$	0	0
$825$	−1222.23	−1.48149
$826$	0	0
$827$	431.195i	0.521396i	0.965420	+	0.260698i	$0.0839527\pi$
$827$	431.195i	0.521396i	−0.965420	+	0.260698i	$0.916047\pi$
$828$	0	0
$829$	−83.4125	−0.100618	−0.0503091	−	0.998734i	$-0.516021\pi$
$829$	−83.4125	−0.100618	−0.0503091	+	0.998734i	$0.516021\pi$
$830$	0	0
$831$	1446.62i	1.74082i
$832$	0	0
$833$	62.0990	0.0745486
$834$	0	0
$835$	− 839.586i	− 1.00549i
$836$	0	0
$837$	−44.5502	−0.0532261
$838$	0	0
$839$	835.598i	0.995945i	0.867193	+	0.497973i	$0.165922\pi$
$839$	835.598i	0.995945i	−0.867193	+	0.497973i	$0.834078\pi$
$840$	0	0
$841$	29.0000	0.0344828
$842$	0	0
$843$	1082.75i	1.28440i
$844$	0	0
$845$	2813.51	3.32960
$846$	0	0
$847$	− 182.388i	− 0.215334i
$848$	0	0
$849$	1503.68	1.77112
$850$	0	0
$851$	− 0.130468i	0 0.000153312i
$852$	0	0
$853$	58.9242	0.0690788	0.0345394	−	0.999403i	$-0.489004\pi$
$853$	58.9242	0.0690788	0.0345394	+	0.999403i	$0.489004\pi$
$854$	0	0
$855$	1840.41i	2.15252i
$856$	0	0
$857$	712.595	0.831500	0.415750	−	0.909479i	$-0.363519\pi$
$857$	712.595	0.831500	0.415750	+	0.909479i	$0.363519\pi$
$858$	0	0
$859$	− 729.876i	− 0.849681i	−0.905268	−	0.424841i	$-0.860330\pi$
$859$	− 729.876i	− 0.849681i	0.905268	−	0.424841i	$-0.139670\pi$
$860$	0	0
$861$	−1650.53	−1.91699
$862$	0	0
$863$	− 736.184i	− 0.853052i	−0.904475	−	0.426526i	$-0.859737\pi$
$863$	− 736.184i	− 0.853052i	0.904475	−	0.426526i	$-0.140263\pi$
$864$	0	0
$865$	1814.49	2.09768
$866$	0	0
$867$	− 1090.09i	− 1.25731i
$868$	0	0
$869$	799.756	0.920318
$870$	0	0
$871$	1522.53i	1.74803i
$872$	0	0
$873$	−1122.41	−1.28570
$874$	0	0
$875$	− 344.401i	− 0.393601i
$876$	0	0
$877$	−402.913	−0.459422	−0.229711	−	0.973259i	$-0.573778\pi$
$877$	−402.913	−0.459422	−0.229711	+	0.973259i	$0.573778\pi$
$878$	0	0
$879$	− 1826.67i	− 2.07813i
$880$	0	0
$881$	1086.82	1.23362	0.616810	−	0.787112i	$-0.288425\pi$
$881$	1086.82	1.23362	0.616810	+	0.787112i	$0.288425\pi$
$882$	0	0
$883$	− 1470.42i	− 1.66526i	−0.553833	−	0.832628i	$-0.686835\pi$
$883$	− 1470.42i	− 1.66526i	0.553833	−	0.832628i	$-0.313165\pi$
$884$	0	0
$885$	−1940.30	−2.19243
$886$	0	0
$887$	76.3569i	0.0860844i	0.999073	+	0.0430422i	$0.0137050\pi$
$887$	76.3569i	0.0860844i	−0.999073	+	0.0430422i	$0.986295\pi$
$888$	0	0
$889$	411.464	0.462839
$890$	0	0
$891$	− 915.210i	− 1.02717i
$892$	0	0
$893$	1348.87	1.51049
$894$	0	0
$895$	445.095i	0.497313i
$896$	0	0
$897$	−744.889	−0.830422
$898$	0	0
$899$	23.7792i	0.0264507i
$900$	0	0
$901$	−240.847	−0.267310
$902$	0	0
$903$	349.287i	0.386807i
$904$	0	0
$905$	−520.549	−0.575192
$906$	0	0
$907$	842.164i	0.928516i	0.885700	+	0.464258i	$0.153679\pi$
$907$	842.164i	0.928516i	−0.885700	+	0.464258i	$0.846321\pi$
$908$	0	0
$909$	933.607	1.02707
$910$	0	0
$911$	− 793.529i	− 0.871052i	−0.900176	−	0.435526i	$-0.856562\pi$
$911$	− 793.529i	− 0.871052i	0.900176	−	0.435526i	$-0.143438\pi$
$912$	0	0
$913$	755.538	0.827534
$914$	0	0
$915$	1229.96i	1.34422i
$916$	0	0
$917$	866.180	0.944580
$918$	0	0
$919$	36.0028i	0.0391761i	0.999808	+	0.0195880i	$0.00623546\pi$
$919$	36.0028i	0.0391761i	−0.999808	+	0.0195880i	$0.993765\pi$
$920$	0	0
$921$	1266.53	1.37517
$922$	0	0
$923$	− 757.227i	− 0.820397i
$924$	0	0
$925$	−0.529056	−0.000571953 0
$926$	0	0
$927$	− 546.869i	− 0.589935i
$928$	0	0
$929$	−522.582	−0.562521	−0.281260	−	0.959631i	$-0.590752\pi$
$929$	−522.582	−0.562521	−0.281260	+	0.959631i	$0.590752\pi$
$930$	0	0
$931$	− 686.999i	− 0.737915i
$932$	0	0
$933$	−2123.02	−2.27548
$934$	0	0
$935$	242.745i	0.259620i
$936$	0	0
$937$	328.762	0.350867	0.175434	−	0.984491i	$-0.443867\pi$
$937$	328.762	0.350867	0.175434	+	0.984491i	$0.443867\pi$
$938$	0	0
$939$	− 1122.04i	− 1.19494i
$940$	0	0
$941$	−1532.08	−1.62814	−0.814070	−	0.580766i	$-0.802753\pi$
$941$	−1532.08	−1.62814	−0.814070	+	0.580766i	$0.802753\pi$
$942$	0	0
$943$	− 620.042i	− 0.657521i
$944$	0	0
$945$	426.231	0.451038
$946$	0	0
$947$	− 886.618i	− 0.936239i	−0.883665	−	0.468119i	$-0.844932\pi$
$947$	− 886.618i	− 0.936239i	0.883665	−	0.468119i	$-0.155068\pi$
$948$	0	0
$949$	−2836.23	−2.98865
$950$	0	0
$951$	760.662i	0.799855i
$952$	0	0
$953$	1248.28	1.30984	0.654920	−	0.755699i	$-0.272702\pi$
$953$	1248.28	1.30984	0.654920	+	0.755699i	$0.272702\pi$
$954$	0	0
$955$	− 747.722i	− 0.782955i
$956$	0	0
$957$	−198.536	−0.207457
$958$	0	0
$959$	− 549.163i	− 0.572641i
$960$	0	0
$961$	941.502	0.979710
$962$	0	0
$963$	873.367i	0.906923i
$964$	0	0
$965$	−1955.78	−2.02672
$966$	0	0
$967$	1048.21i	1.08398i	0.840386	+	0.541989i	$0.182329\pi$
$967$	1048.21i	1.08398i	−0.840386	+	0.541989i	$0.817671\pi$
$968$	0	0
$969$	499.971	0.515966
$970$	0	0
$971$	1023.67i	1.05424i	0.849790	+	0.527122i	$0.176729\pi$
$971$	1023.67i	1.05424i	−0.849790	+	0.527122i	$0.823271\pi$
$972$	0	0
$973$	−1180.34	−1.21310
$974$	0	0
$975$	3020.57i	3.09802i
$976$	0	0
$977$	543.045	0.555829	0.277914	−	0.960606i	$-0.410357\pi$
$977$	543.045	0.555829	0.277914	+	0.960606i	$0.410357\pi$
$978$	0	0
$979$	− 183.036i	− 0.186963i
$980$	0	0
$981$	56.1796	0.0572677
$982$	0	0
$983$	− 1592.68i	− 1.62022i	−0.586275	−	0.810112i	$-0.699406\pi$
$983$	− 1592.68i	− 1.62022i	0.586275	−	0.810112i	$-0.300594\pi$
$984$	0	0
$985$	−2349.87	−2.38565
$986$	0	0
$987$	782.314i	0.792618i
$988$	0	0
$989$	−131.214	−0.132673
$990$	0	0
$991$	− 836.707i	− 0.844306i	−0.906525	−	0.422153i	$-0.861274\pi$
$991$	− 836.707i	− 0.844306i	0.906525	−	0.422153i	$-0.138726\pi$
$992$	0	0
$993$	−1733.21	−1.74543
$994$	0	0
$995$	757.825i	0.761633i
$996$	0	0
$997$	−76.2615	−0.0764910	−0.0382455	−	0.999268i	$-0.512177\pi$
$997$	−76.2615	−0.0764910	−0.0382455	+	0.999268i	$0.512177\pi$
$998$	0	0
$999$	− 0.161006i	0 0.000161167i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	464.3.d.b.175.16	yes	20
4.3	odd	2	inner	464.3.d.b.175.5	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
464.3.d.b.175.5	✓	20	4.3	odd	2	inner
464.3.d.b.175.16	yes	20	1.1	even	1	trivial

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 464 = 2^{4} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	464.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.92832i q^{3} +7.62576 q^{5} -5.54000i q^{7} -6.43170 q^{9} +9.38498i q^{11} +23.1937 q^{13} +29.9564i q^{15} +3.39182 q^{17} -37.5237i q^{19} +21.7629 q^{21} +8.17550i q^{23} +33.1521 q^{25} +10.0891i q^{27} +5.38516 q^{29} +4.41568i q^{31} -36.8672 q^{33} -42.2467i q^{35} -0.0159584 q^{37} +91.1123i q^{39} -75.8415 q^{41} +16.0497i q^{43} -49.0466 q^{45} +35.9472i q^{47} +18.3084 q^{49} +13.3242i q^{51} -71.0080 q^{53} +71.5676i q^{55} +147.405 q^{57} +64.7708i q^{59} +41.0583 q^{61} +35.6316i q^{63} +176.870 q^{65} +65.6441i q^{67} -32.1160 q^{69} -32.6479i q^{71} -122.284 q^{73} +130.232i q^{75} +51.9928 q^{77} -85.2166i q^{79} -97.5185 q^{81} -80.5050i q^{83} +25.8652 q^{85} +21.1547i q^{87} -19.5031 q^{89} -128.493i q^{91} -17.3462 q^{93} -286.146i q^{95} +174.513 q^{97} -60.3614i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 8 q^{5} - 68 q^{9} + 16 q^{13} + 40 q^{17} - 48 q^{21} + 188 q^{25} - 120 q^{33} - 80 q^{37} - 72 q^{41} + 72 q^{45} - 28 q^{49} + 96 q^{53} + 104 q^{57} - 96 q^{61} - 80 q^{65} + 352 q^{69} - 312 q^{73}+ \cdots + 56 q^{97}+O(q^{100}) \) 20 * q - 8 * q^5 - 68 * q^9 + 16 * q^13 + 40 * q^17 - 48 * q^21 + 188 * q^25 - 120 * q^33 - 80 * q^37 - 72 * q^41 + 72 * q^45 - 28 * q^49 + 96 * q^53 + 104 * q^57 - 96 * q^61 - 80 * q^65 + 352 * q^69 - 312 * q^73 - 192 * q^77 + 60 * q^81 + 336 * q^85 - 152 * q^89 - 472 * q^93 + 56 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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