LMFDB - Embedded newform 336.3.m.c.127.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [336,3,Mod(127,336)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(336, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 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("336.127");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 336 = 2^{4} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	336.m (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:	$9.15533688251$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - x^{2} - 2x + 4 $ x^4 - x^3 - x^2 - 2*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.3
Root			$1.39564 + 0.228425i$ of defining polynomial
Character	$\chi$	$=$	336.127
Dual form			336.3.m.c.127.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205i q^{3} -3.58258 q^{5} -2.64575i q^{7} -3.00000 q^{9} +O(q^{10})$	$q+1.73205i q^{3} -3.58258 q^{5} -2.64575i q^{7} -3.00000 q^{9} -0.913701i q^{11} -1.16515 q^{13} -6.20520i q^{15} -26.7477 q^{17} -17.5112i q^{19} +4.58258 q^{21} -27.1805i q^{23} -12.1652 q^{25} -5.19615i q^{27} -2.00000 q^{29} -45.6054i q^{31} +1.58258 q^{33} +9.47860i q^{35} +47.4955 q^{37} -2.01810i q^{39} -42.5735 q^{41} -14.6192i q^{43} +10.7477 q^{45} +8.37420i q^{47} -7.00000 q^{49} -46.3284i q^{51} -41.8258 q^{53} +3.27340i q^{55} +30.3303 q^{57} +27.0296i q^{59} -11.0091 q^{61} +7.93725i q^{63} +4.17424 q^{65} +71.8722i q^{67} +47.0780 q^{69} +55.6561i q^{71} -95.4955 q^{73} -21.0707i q^{75} -2.41742 q^{77} -63.7998i q^{79} +9.00000 q^{81} +32.5118i q^{83} +95.8258 q^{85} -3.46410i q^{87} -120.904 q^{89} +3.08270i q^{91} +78.9909 q^{93} +62.7352i q^{95} +107.495 q^{97} +2.74110i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5} - 12 q^{9}+O(q^{10}) $ 4 * q + 4 * q^5 - 12 * q^9	$ 4 q + 4 q^{5} - 12 q^{9} + 32 q^{13} - 52 q^{17} - 12 q^{25} - 8 q^{29} - 12 q^{33} + 80 q^{37} + 68 q^{41} - 12 q^{45} - 28 q^{49} + 16 q^{53} + 48 q^{57} - 264 q^{61} + 200 q^{65} + 60 q^{69} - 272 q^{73} - 28 q^{77} + 36 q^{81} + 200 q^{85} - 172 q^{89} + 96 q^{93} + 320 q^{97}+O(q^{100}) $ 4 * q + 4 * q^5 - 12 * q^9 + 32 * q^13 - 52 * q^17 - 12 * q^25 - 8 * q^29 - 12 * q^33 + 80 * q^37 + 68 * q^41 - 12 * q^45 - 28 * q^49 + 16 * q^53 + 48 * q^57 - 264 * q^61 + 200 * q^65 + 60 * q^69 - 272 * q^73 - 28 * q^77 + 36 * q^81 + 200 * q^85 - 172 * q^89 + 96 * q^93 + 320 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/336\mathbb{Z}\right)^\times$.

$n$	$85$	$113$	$127$	$241$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$3$	1.73205i	0.577350i
$3$	1.73205i	0.577350i
$4$	0	0
$5$	−3.58258	−0.716515	−0.358258	−	0.933623i	$-0.616629\pi$
$5$	−3.58258	−0.716515	−0.358258	+	0.933623i	$0.616629\pi$
$6$	0	0
$7$	− 2.64575i	− 0.377964i
$7$	− 2.64575i	− 0.377964i
$8$	0	0
$9$	−3.00000	−0.333333
$10$	0	0
$11$	− 0.913701i	− 0.0830637i	−0.999137	−	0.0415318i	$-0.986776\pi$
$11$	− 0.913701i	− 0.0830637i	0.999137	−	0.0415318i	$-0.0132238\pi$
$12$	0	0
$13$	−1.16515	−0.0896270	−0.0448135	−	0.998995i	$-0.514269\pi$
$13$	−1.16515	−0.0896270	−0.0448135	+	0.998995i	$0.514269\pi$
$14$	0	0
$15$	− 6.20520i	− 0.413680i
$16$	0	0
$17$	−26.7477	−1.57340	−0.786698	−	0.617338i	$-0.788211\pi$
$17$	−26.7477	−1.57340	−0.786698	+	0.617338i	$0.788211\pi$
$18$	0	0
$19$	− 17.5112i	− 0.921643i	−0.887493	−	0.460821i	$-0.847555\pi$
$19$	− 17.5112i	− 0.921643i	0.887493	−	0.460821i	$-0.152445\pi$
$20$	0	0
$21$	4.58258	0.218218
$22$	0	0
$23$	− 27.1805i	− 1.18176i	−0.806759	−	0.590881i	$-0.798780\pi$
$23$	− 27.1805i	− 1.18176i	0.806759	−	0.590881i	$-0.201220\pi$
$24$	0	0
$25$	−12.1652	−0.486606
$26$	0	0
$27$	− 5.19615i	− 0.192450i
$28$	0	0
$29$	−2.00000	−0.0689655	−0.0344828	−	0.999405i	$-0.510978\pi$
$29$	−2.00000	−0.0689655	−0.0344828	+	0.999405i	$0.510978\pi$
$30$	0	0
$31$	− 45.6054i	− 1.47114i	−0.677447	−	0.735571i	$-0.736914\pi$
$31$	− 45.6054i	− 1.47114i	0.677447	−	0.735571i	$-0.263086\pi$
$32$	0	0
$33$	1.58258	0.0479568
$34$	0	0
$35$	9.47860i	0.270817i
$36$	0	0
$37$	47.4955	1.28366	0.641830	−	0.766847i	$-0.278175\pi$
$37$	47.4955	1.28366	0.641830	+	0.766847i	$0.278175\pi$
$38$	0	0
$39$	− 2.01810i	− 0.0517462i
$40$	0	0
$41$	−42.5735	−1.03838	−0.519189	−	0.854660i	$-0.673766\pi$
$41$	−42.5735	−1.03838	−0.519189	+	0.854660i	$0.673766\pi$
$42$	0	0
$43$	− 14.6192i	− 0.339982i	−0.985446	−	0.169991i	$-0.945626\pi$
$43$	− 14.6192i	− 0.339982i	0.985446	−	0.169991i	$-0.0543738\pi$
$44$	0	0
$45$	10.7477	0.238838
$46$	0	0
$47$	8.37420i	0.178175i	0.996024	+	0.0890873i	$0.0283950\pi$
$47$	8.37420i	0.178175i	−0.996024	+	0.0890873i	$0.971605\pi$
$48$	0	0
$49$	−7.00000	−0.142857
$50$	0	0
$51$	− 46.3284i	− 0.908400i
$52$	0	0
$53$	−41.8258	−0.789165	−0.394583	−	0.918860i	$-0.629111\pi$
$53$	−41.8258	−0.789165	−0.394583	+	0.918860i	$0.629111\pi$
$54$	0	0
$55$	3.27340i	0.0595164i
$56$	0	0
$57$	30.3303	0.532111
$58$	0	0
$59$	27.0296i	0.458129i	0.973411	+	0.229065i	$0.0735667\pi$
$59$	27.0296i	0.458129i	−0.973411	+	0.229065i	$0.926433\pi$
$60$	0	0
$61$	−11.0091	−0.180477	−0.0902385	−	0.995920i	$-0.528763\pi$
$61$	−11.0091	−0.180477	−0.0902385	+	0.995920i	$0.528763\pi$
$62$	0	0
$63$	7.93725i	0.125988i
$64$	0	0
$65$	4.17424	0.0642191
$66$	0	0
$67$	71.8722i	1.07272i	0.843989	+	0.536360i	$0.180201\pi$
$67$	71.8722i	1.07272i	−0.843989	+	0.536360i	$0.819799\pi$
$68$	0	0
$69$	47.0780	0.682290
$70$	0	0
$71$	55.6561i	0.783889i	0.919989	+	0.391945i	$0.128197\pi$
$71$	55.6561i	0.783889i	−0.919989	+	0.391945i	$0.871803\pi$
$72$	0	0
$73$	−95.4955	−1.30816	−0.654078	−	0.756427i	$-0.726943\pi$
$73$	−95.4955	−1.30816	−0.654078	+	0.756427i	$0.726943\pi$
$74$	0	0
$75$	− 21.0707i	− 0.280942i
$76$	0	0
$77$	−2.41742	−0.0313951
$78$	0	0
$79$	− 63.7998i	− 0.807593i	−0.914849	−	0.403796i	$-0.867690\pi$
$79$	− 63.7998i	− 0.807593i	0.914849	−	0.403796i	$-0.132310\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	32.5118i	0.391709i	0.980633	+	0.195854i	$0.0627480\pi$
$83$	32.5118i	0.391709i	−0.980633	+	0.195854i	$0.937252\pi$
$84$	0	0
$85$	95.8258	1.12736
$86$	0	0
$87$	− 3.46410i	− 0.0398173i
$88$	0	0
$89$	−120.904	−1.35847	−0.679235	−	0.733921i	$-0.737688\pi$
$89$	−120.904	−1.35847	−0.679235	+	0.733921i	$0.737688\pi$
$90$	0	0
$91$	3.08270i	0.0338758i
$92$	0	0
$93$	78.9909	0.849365
$94$	0	0
$95$	62.7352i	0.660371i
$96$	0	0
$97$	107.495	1.10820	0.554100	−	0.832450i	$-0.313062\pi$
$97$	107.495	1.10820	0.554100	+	0.832450i	$0.313062\pi$
$98$	0	0
$99$	2.74110i	0.0276879i
$100$	0	0
$101$	−104.417	−1.03384	−0.516918	−	0.856035i	$-0.672921\pi$
$101$	−104.417	−1.03384	−0.516918	+	0.856035i	$0.672921\pi$
$102$	0	0
$103$	119.305i	1.15830i	0.815220	+	0.579151i	$0.196616\pi$
$103$	119.305i	1.15830i	−0.815220	+	0.579151i	$0.803384\pi$
$104$	0	0
$105$	−16.4174	−0.156356
$106$	0	0
$107$	67.0019i	0.626186i	0.949722	+	0.313093i	$0.101365\pi$
$107$	67.0019i	0.626186i	−0.949722	+	0.313093i	$0.898635\pi$
$108$	0	0
$109$	161.303	1.47984	0.739922	−	0.672693i	$-0.234862\pi$
$109$	161.303	1.47984	0.739922	+	0.672693i	$0.234862\pi$
$110$	0	0
$111$	82.2645i	0.741122i
$112$	0	0
$113$	32.6788	0.289193	0.144596	−	0.989491i	$-0.453812\pi$
$113$	32.6788	0.289193	0.144596	+	0.989491i	$0.453812\pi$
$114$	0	0
$115$	97.3762i	0.846750i
$116$	0	0
$117$	3.49545	0.0298757
$118$	0	0
$119$	70.7678i	0.594688i
$120$	0	0
$121$	120.165	0.993100
$122$	0	0
$123$	− 73.7394i	− 0.599508i
$124$	0	0
$125$	133.147	1.06518
$126$	0	0
$127$	− 56.4902i	− 0.444805i	−0.974955	−	0.222402i	$-0.928610\pi$
$127$	− 56.4902i	− 0.444805i	0.974955	−	0.222402i	$-0.0713899\pi$
$128$	0	0
$129$	25.3212	0.196288
$130$	0	0
$131$	140.471i	1.07230i	0.844123	+	0.536149i	$0.180122\pi$
$131$	140.471i	1.07230i	−0.844123	+	0.536149i	$0.819878\pi$
$132$	0	0
$133$	−46.3303	−0.348348
$134$	0	0
$135$	18.6156i	0.137893i
$136$	0	0
$137$	−228.156	−1.66537	−0.832686	−	0.553745i	$-0.813198\pi$
$137$	−228.156	−1.66537	−0.832686	+	0.553745i	$0.813198\pi$
$138$	0	0
$139$	− 176.256i	− 1.26803i	−0.773320	−	0.634015i	$-0.781406\pi$
$139$	− 176.256i	− 1.26803i	0.773320	−	0.634015i	$-0.218594\pi$
$140$	0	0
$141$	−14.5045	−0.102869
$142$	0	0
$143$	1.06460i	0.00744475i
$144$	0	0
$145$	7.16515	0.0494148
$146$	0	0
$147$	− 12.1244i	− 0.0824786i
$148$	0	0
$149$	274.835	1.84453	0.922265	−	0.386559i	$-0.126337\pi$
$149$	274.835	1.84453	0.922265	+	0.386559i	$0.126337\pi$
$150$	0	0
$151$	175.875i	1.16473i	0.812926	+	0.582367i	$0.197873\pi$
$151$	175.875i	1.16473i	−0.812926	+	0.582367i	$0.802127\pi$
$152$	0	0
$153$	80.2432	0.524465
$154$	0	0
$155$	163.385i	1.05410i
$156$	0	0
$157$	55.9818	0.356572	0.178286	−	0.983979i	$-0.442945\pi$
$157$	55.9818	0.356572	0.178286	+	0.983979i	$0.442945\pi$
$158$	0	0
$159$	− 72.4443i	− 0.455625i
$160$	0	0
$161$	−71.9129	−0.446664
$162$	0	0
$163$	− 237.386i	− 1.45636i	−0.685387	−	0.728179i	$-0.740367\pi$
$163$	− 237.386i	− 1.45636i	0.685387	−	0.728179i	$-0.259633\pi$
$164$	0	0
$165$	−5.66970	−0.0343618
$166$	0	0
$167$	− 237.386i	− 1.42147i	−0.703457	−	0.710737i	$-0.748361\pi$
$167$	− 237.386i	− 1.42147i	0.703457	−	0.710737i	$-0.251639\pi$
$168$	0	0
$169$	−167.642	−0.991967
$170$	0	0
$171$	52.5336i	0.307214i
$172$	0	0
$173$	−173.078	−1.00045	−0.500226	−	0.865895i	$-0.666750\pi$
$173$	−173.078	−1.00045	−0.500226	+	0.865895i	$0.666750\pi$
$174$	0	0
$175$	32.1860i	0.183920i
$176$	0	0
$177$	−46.8167	−0.264501
$178$	0	0
$179$	− 136.284i	− 0.761363i	−0.924706	−	0.380681i	$-0.875689\pi$
$179$	− 136.284i	− 0.761363i	0.924706	−	0.380681i	$-0.124311\pi$
$180$	0	0
$181$	80.5045	0.444776	0.222388	−	0.974958i	$-0.428615\pi$
$181$	80.5045	0.444776	0.222388	+	0.974958i	$0.428615\pi$
$182$	0	0
$183$	− 19.0683i	− 0.104198i
$184$	0	0
$185$	−170.156	−0.919762
$186$	0	0
$187$	24.4394i	0.130692i
$188$	0	0
$189$	−13.7477	−0.0727393
$190$	0	0
$191$	− 131.183i	− 0.686823i	−0.939185	−	0.343411i	$-0.888417\pi$
$191$	− 131.183i	− 0.686823i	0.939185	−	0.343411i	$-0.111583\pi$
$192$	0	0
$193$	−171.495	−0.888577	−0.444289	−	0.895884i	$-0.646544\pi$
$193$	−171.495	−0.888577	−0.444289	+	0.895884i	$0.646544\pi$
$194$	0	0
$195$	7.23000i	0.0370769i
$196$	0	0
$197$	57.1652	0.290178	0.145089	−	0.989419i	$-0.453653\pi$
$197$	57.1652	0.290178	0.145089	+	0.989419i	$0.453653\pi$
$198$	0	0
$199$	− 6.70601i	− 0.0336985i	−0.999858	−	0.0168493i	$-0.994636\pi$
$199$	− 6.70601i	− 0.0336985i	0.999858	−	0.0168493i	$-0.00536354\pi$
$200$	0	0
$201$	−124.486	−0.619335
$202$	0	0
$203$	5.29150i	0.0260665i
$204$	0	0
$205$	152.523	0.744013
$206$	0	0
$207$	81.5415i	0.393920i
$208$	0	0
$209$	−16.0000	−0.0765550
$210$	0	0
$211$	− 178.767i	− 0.847236i	−0.905841	−	0.423618i	$-0.860760\pi$
$211$	− 178.767i	− 0.847236i	0.905841	−	0.423618i	$-0.139240\pi$
$212$	0	0
$213$	−96.3992	−0.452579
$214$	0	0
$215$	52.3744i	0.243602i
$216$	0	0
$217$	−120.661	−0.556040
$218$	0	0
$219$	− 165.403i	− 0.755265i
$220$	0	0
$221$	31.1652	0.141019
$222$	0	0
$223$	114.506i	0.513480i	0.966480	+	0.256740i	$0.0826484\pi$
$223$	114.506i	0.513480i	−0.966480	+	0.256740i	$0.917352\pi$
$224$	0	0
$225$	36.4955	0.162202
$226$	0	0
$227$	307.813i	1.35600i	0.735061	+	0.678001i	$0.237154\pi$
$227$	307.813i	1.35600i	−0.735061	+	0.678001i	$0.762846\pi$
$228$	0	0
$229$	−356.156	−1.55527	−0.777633	−	0.628718i	$-0.783580\pi$
$229$	−356.156	−1.55527	−0.777633	+	0.628718i	$0.783580\pi$
$230$	0	0
$231$	− 4.18710i	− 0.0181260i
$232$	0	0
$233$	213.514	0.916368	0.458184	−	0.888857i	$-0.348500\pi$
$233$	213.514	0.916368	0.458184	+	0.888857i	$0.348500\pi$
$234$	0	0
$235$	− 30.0012i	− 0.127665i
$236$	0	0
$237$	110.505	0.466264
$238$	0	0
$239$	− 402.988i	− 1.68614i	−0.537801	−	0.843072i	$-0.680745\pi$
$239$	− 402.988i	− 1.68614i	0.537801	−	0.843072i	$-0.319255\pi$
$240$	0	0
$241$	−344.468	−1.42933	−0.714664	−	0.699468i	$-0.753421\pi$
$241$	−344.468	−1.42933	−0.714664	+	0.699468i	$0.753421\pi$
$242$	0	0
$243$	15.5885i	0.0641500i
$244$	0	0
$245$	25.0780	0.102359
$246$	0	0
$247$	20.4032i	0.0826041i
$248$	0	0
$249$	−56.3121	−0.226153
$250$	0	0
$251$	− 491.601i	− 1.95857i	−0.202491	−	0.979284i	$-0.564904\pi$
$251$	− 491.601i	− 1.95857i	0.202491	−	0.979284i	$-0.435096\pi$
$252$	0	0
$253$	−24.8348	−0.0981615
$254$	0	0
$255$	165.975i	0.650883i
$256$	0	0
$257$	335.372	1.30495	0.652475	−	0.757811i	$-0.273731\pi$
$257$	335.372	1.30495	0.652475	+	0.757811i	$0.273731\pi$
$258$	0	0
$259$	− 125.661i	− 0.485178i
$260$	0	0
$261$	6.00000	0.0229885
$262$	0	0
$263$	− 484.601i	− 1.84259i	−0.388865	−	0.921295i	$-0.627133\pi$
$263$	− 484.601i	− 1.84259i	0.388865	−	0.921295i	$-0.372867\pi$
$264$	0	0
$265$	149.844	0.565449
$266$	0	0
$267$	− 209.412i	− 0.784313i
$268$	0	0
$269$	−27.9311	−0.103833	−0.0519165	−	0.998651i	$-0.516533\pi$
$269$	−27.9311	−0.103833	−0.0519165	+	0.998651i	$0.516533\pi$
$270$	0	0
$271$	− 16.5262i	− 0.0609823i	−0.999535	−	0.0304912i	$-0.990293\pi$
$271$	− 16.5262i	− 0.0609823i	0.999535	−	0.0304912i	$-0.00970714\pi$
$272$	0	0
$273$	−5.33939	−0.0195582
$274$	0	0
$275$	11.1153i	0.0404193i
$276$	0	0
$277$	−359.459	−1.29769	−0.648843	−	0.760922i	$-0.724747\pi$
$277$	−359.459	−1.29769	−0.648843	+	0.760922i	$0.724747\pi$
$278$	0	0
$279$	136.816i	0.490381i
$280$	0	0
$281$	259.147	0.922231	0.461116	−	0.887340i	$-0.347449\pi$
$281$	259.147	0.922231	0.461116	+	0.887340i	$0.347449\pi$
$282$	0	0
$283$	177.241i	0.626294i	0.949705	+	0.313147i	$0.101383\pi$
$283$	177.241i	0.626294i	−0.949705	+	0.313147i	$0.898617\pi$
$284$	0	0
$285$	−108.661	−0.381265
$286$	0	0
$287$	112.639i	0.392470i
$288$	0	0
$289$	426.441	1.47557
$290$	0	0
$291$	186.188i	0.639820i
$292$	0	0
$293$	−296.069	−1.01047	−0.505237	−	0.862981i	$-0.668595\pi$
$293$	−296.069	−1.01047	−0.505237	+	0.862981i	$0.668595\pi$
$294$	0	0
$295$	− 96.8356i	− 0.328256i
$296$	0	0
$297$	−4.74773	−0.0159856
$298$	0	0
$299$	31.6694i	0.105918i
$300$	0	0
$301$	−38.6788	−0.128501
$302$	0	0
$303$	− 180.856i	− 0.596885i
$304$	0	0
$305$	39.4409	0.129314
$306$	0	0
$307$	170.854i	0.556527i	0.960505	+	0.278263i	$0.0897588\pi$
$307$	170.854i	0.556527i	−0.960505	+	0.278263i	$0.910241\pi$
$308$	0	0
$309$	−206.642	−0.668746
$310$	0	0
$311$	− 419.967i	− 1.35038i	−0.737645	−	0.675188i	$-0.764062\pi$
$311$	− 419.967i	− 1.35038i	0.737645	−	0.675188i	$-0.235938\pi$
$312$	0	0
$313$	−238.624	−0.762378	−0.381189	−	0.924497i	$-0.624485\pi$
$313$	−238.624	−0.762378	−0.381189	+	0.924497i	$0.624485\pi$
$314$	0	0
$315$	− 28.4358i	− 0.0902724i
$316$	0	0
$317$	585.441	1.84682	0.923408	−	0.383819i	$-0.125391\pi$
$317$	585.441	1.84682	0.923408	+	0.383819i	$0.125391\pi$
$318$	0	0
$319$	1.82740i	0.00572853i
$320$	0	0
$321$	−116.051	−0.361529
$322$	0	0
$323$	468.385i	1.45011i
$324$	0	0
$325$	14.1742	0.0436131
$326$	0	0
$327$	279.385i	0.854389i
$328$	0	0
$329$	22.1561	0.0673436
$330$	0	0
$331$	− 71.4112i	− 0.215744i	−0.994165	−	0.107872i	$-0.965596\pi$
$331$	− 71.4112i	− 0.215744i	0.994165	−	0.107872i	$-0.0344037\pi$
$332$	0	0
$333$	−142.486	−0.427887
$334$	0	0
$335$	− 257.488i	− 0.768620i
$336$	0	0
$337$	416.955	1.23725	0.618627	−	0.785685i	$-0.287689\pi$
$337$	416.955	1.23725	0.618627	+	0.785685i	$0.287689\pi$
$338$	0	0
$339$	56.6013i	0.166966i
$340$	0	0
$341$	−41.6697	−0.122199
$342$	0	0
$343$	18.5203i	0.0539949i
$344$	0	0
$345$	−168.661	−0.488871
$346$	0	0
$347$	546.812i	1.57583i	0.615785	+	0.787914i	$0.288839\pi$
$347$	546.812i	1.57583i	−0.615785	+	0.787914i	$0.711161\pi$
$348$	0	0
$349$	−97.6151	−0.279700	−0.139850	−	0.990173i	$-0.544662\pi$
$349$	−97.6151	−0.279700	−0.139850	+	0.990173i	$0.544662\pi$
$350$	0	0
$351$	6.05430i	0.0172487i
$352$	0	0
$353$	315.858	0.894783	0.447391	−	0.894338i	$-0.352353\pi$
$353$	315.858	0.894783	0.447391	+	0.894338i	$0.352353\pi$
$354$	0	0
$355$	− 199.392i	− 0.561668i
$356$	0	0
$357$	−122.573	−0.343343
$358$	0	0
$359$	31.2797i	0.0871301i	0.999051	+	0.0435650i	$0.0138716\pi$
$359$	31.2797i	0.0871301i	−0.999051	+	0.0435650i	$0.986128\pi$
$360$	0	0
$361$	54.3576	0.150575
$362$	0	0
$363$	208.132i	0.573367i
$364$	0	0
$365$	342.120	0.937314
$366$	0	0
$367$	501.437i	1.36631i	0.730271	+	0.683157i	$0.239394\pi$
$367$	501.437i	1.36631i	−0.730271	+	0.683157i	$0.760606\pi$
$368$	0	0
$369$	127.720	0.346126
$370$	0	0
$371$	110.661i	0.298276i
$372$	0	0
$373$	−257.267	−0.689723	−0.344861	−	0.938654i	$-0.612074\pi$
$373$	−257.267	−0.689723	−0.344861	+	0.938654i	$0.612074\pi$
$374$	0	0
$375$	230.617i	0.614980i
$376$	0	0
$377$	2.33030	0.00618117
$378$	0	0
$379$	169.169i	0.446356i	0.974778	+	0.223178i	$0.0716431\pi$
$379$	169.169i	0.446356i	−0.974778	+	0.223178i	$0.928357\pi$
$380$	0	0
$381$	97.8439	0.256808
$382$	0	0
$383$	436.858i	1.14062i	0.821429	+	0.570311i	$0.193177\pi$
$383$	436.858i	1.14062i	−0.821429	+	0.570311i	$0.806823\pi$
$384$	0	0
$385$	8.66061	0.0224951
$386$	0	0
$387$	43.8576i	0.113327i
$388$	0	0
$389$	−215.982	−0.555223	−0.277612	−	0.960693i	$-0.589543\pi$
$389$	−215.982	−0.555223	−0.277612	+	0.960693i	$0.589543\pi$
$390$	0	0
$391$	727.017i	1.85938i
$392$	0	0
$393$	−243.303	−0.619092
$394$	0	0
$395$	228.568i	0.578652i
$396$	0	0
$397$	256.606	0.646363	0.323181	−	0.946337i	$-0.395248\pi$
$397$	256.606	0.646363	0.323181	+	0.946337i	$0.395248\pi$
$398$	0	0
$399$	− 80.2464i	− 0.201119i
$400$	0	0
$401$	557.441	1.39013	0.695063	−	0.718948i	$-0.255376\pi$
$401$	557.441	1.39013	0.695063	+	0.718948i	$0.255376\pi$
$402$	0	0
$403$	53.1372i	0.131854i
$404$	0	0
$405$	−32.2432	−0.0796128
$406$	0	0
$407$	− 43.3966i	− 0.106626i
$408$	0	0
$409$	16.1561	0.0395014	0.0197507	−	0.999805i	$-0.493713\pi$
$409$	16.1561	0.0395014	0.0197507	+	0.999805i	$0.493713\pi$
$410$	0	0
$411$	− 395.178i	− 0.961503i
$412$	0	0
$413$	71.5136	0.173157
$414$	0	0
$415$	− 116.476i	− 0.280665i
$416$	0	0
$417$	305.285	0.732098
$418$	0	0
$419$	− 582.652i	− 1.39058i	−0.718730	−	0.695289i	$-0.755276\pi$
$419$	− 582.652i	− 1.39058i	0.718730	−	0.695289i	$-0.244724\pi$
$420$	0	0
$421$	−662.762	−1.57426	−0.787128	−	0.616789i	$-0.788433\pi$
$421$	−662.762	−1.57426	−0.787128	+	0.616789i	$0.788433\pi$
$422$	0	0
$423$	− 25.1226i	− 0.0593915i
$424$	0	0
$425$	325.390	0.765624
$426$	0	0
$427$	29.1273i	0.0682139i
$428$	0	0
$429$	−1.84394	−0.00429823
$430$	0	0
$431$	− 286.336i	− 0.664354i	−0.943217	−	0.332177i	$-0.892217\pi$
$431$	− 286.336i	− 0.664354i	0.943217	−	0.332177i	$-0.107783\pi$
$432$	0	0
$433$	554.900	1.28152	0.640762	−	0.767739i	$-0.278618\pi$
$433$	554.900	1.28152	0.640762	+	0.767739i	$0.278618\pi$
$434$	0	0
$435$	12.4104i	0.0285297i
$436$	0	0
$437$	−475.964	−1.08916
$438$	0	0
$439$	364.843i	0.831078i	0.909575	+	0.415539i	$0.136407\pi$
$439$	364.843i	0.831078i	−0.909575	+	0.415539i	$0.863593\pi$
$440$	0	0
$441$	21.0000	0.0476190
$442$	0	0
$443$	445.241i	1.00506i	0.864560	+	0.502529i	$0.167597\pi$
$443$	445.241i	1.00506i	−0.864560	+	0.502529i	$0.832403\pi$
$444$	0	0
$445$	433.147	0.973364
$446$	0	0
$447$	476.028i	1.06494i
$448$	0	0
$449$	256.955	0.572282	0.286141	−	0.958188i	$-0.407627\pi$
$449$	256.955	0.572282	0.286141	+	0.958188i	$0.407627\pi$
$450$	0	0
$451$	38.8994i	0.0862515i
$452$	0	0
$453$	−304.624	−0.672460
$454$	0	0
$455$	− 11.0440i	− 0.0242725i
$456$	0	0
$457$	−643.982	−1.40915	−0.704575	−	0.709629i	$-0.748862\pi$
$457$	−643.982	−1.40915	−0.704575	+	0.709629i	$0.748862\pi$
$458$	0	0
$459$	138.985i	0.302800i
$460$	0	0
$461$	30.8856	0.0669970	0.0334985	−	0.999439i	$-0.489335\pi$
$461$	30.8856	0.0669970	0.0334985	+	0.999439i	$0.489335\pi$
$462$	0	0
$463$	− 435.794i	− 0.941239i	−0.882336	−	0.470619i	$-0.844030\pi$
$463$	− 435.794i	− 0.941239i	0.882336	−	0.470619i	$-0.155970\pi$
$464$	0	0
$465$	−282.991	−0.608583
$466$	0	0
$467$	22.0084i	0.0471272i	0.999722	+	0.0235636i	$0.00750123\pi$
$467$	22.0084i	0.0471272i	−0.999722	+	0.0235636i	$0.992499\pi$
$468$	0	0
$469$	190.156	0.405450
$470$	0	0
$471$	96.9634i	0.205867i
$472$	0	0
$473$	−13.3576	−0.0282401
$474$	0	0
$475$	213.027i	0.448477i
$476$	0	0
$477$	125.477	0.263055
$478$	0	0
$479$	− 662.677i	− 1.38346i	−0.722157	−	0.691729i	$-0.756849\pi$
$479$	− 662.677i	− 1.38346i	0.722157	−	0.691729i	$-0.243151\pi$
$480$	0	0
$481$	−55.3394	−0.115051
$482$	0	0
$483$	− 124.557i	− 0.257881i
$484$	0	0
$485$	−385.111	−0.794042
$486$	0	0
$487$	24.5024i	0.0503129i	0.999684	+	0.0251565i	$0.00800840\pi$
$487$	24.5024i	0.0503129i	−0.999684	+	0.0251565i	$0.991992\pi$
$488$	0	0
$489$	411.165	0.840829
$490$	0	0
$491$	684.136i	1.39335i	0.717386	+	0.696676i	$0.245339\pi$
$491$	684.136i	1.39335i	−0.717386	+	0.696676i	$0.754661\pi$
$492$	0	0
$493$	53.4955	0.108510
$494$	0	0
$495$	− 9.82020i	− 0.0198388i
$496$	0	0
$497$	147.252	0.296282
$498$	0	0
$499$	− 818.768i	− 1.64082i	−0.571777	−	0.820409i	$-0.693746\pi$
$499$	− 818.768i	− 1.64082i	0.571777	−	0.820409i	$-0.306254\pi$
$500$	0	0
$501$	411.165	0.820689
$502$	0	0
$503$	− 693.298i	− 1.37833i	−0.724606	−	0.689163i	$-0.757978\pi$
$503$	− 693.298i	− 1.37833i	0.724606	−	0.689163i	$-0.242022\pi$
$504$	0	0
$505$	374.083	0.740759
$506$	0	0
$507$	− 290.365i	− 0.572712i
$508$	0	0
$509$	−87.5462	−0.171996	−0.0859982	−	0.996295i	$-0.527408\pi$
$509$	−87.5462	−0.171996	−0.0859982	+	0.996295i	$0.527408\pi$
$510$	0	0
$511$	252.657i	0.494437i
$512$	0	0
$513$	−90.9909	−0.177370
$514$	0	0
$515$	− 427.419i	− 0.829941i
$516$	0	0
$517$	7.65151	0.0147998
$518$	0	0
$519$	− 299.780i	− 0.577611i
$520$	0	0
$521$	190.573	0.365784	0.182892	−	0.983133i	$-0.441454\pi$
$521$	190.573	0.365784	0.182892	+	0.983133i	$0.441454\pi$
$522$	0	0
$523$	− 767.379i	− 1.46726i	−0.679547	−	0.733632i	$-0.737824\pi$
$523$	− 767.379i	− 1.46726i	0.679547	−	0.733632i	$-0.262176\pi$
$524$	0	0
$525$	−55.7477	−0.106186
$526$	0	0
$527$	1219.84i	2.31469i
$528$	0	0
$529$	−209.780	−0.396560
$530$	0	0
$531$	− 81.0888i	− 0.152710i
$532$	0	0
$533$	49.6046	0.0930667
$534$	0	0
$535$	− 240.040i	− 0.448672i
$536$	0	0
$537$	236.051	0.439573
$538$	0	0
$539$	6.39590i	0.0118662i
$540$	0	0
$541$	−491.495	−0.908494	−0.454247	−	0.890876i	$-0.650092\pi$
$541$	−491.495	−0.908494	−0.454247	+	0.890876i	$0.650092\pi$
$542$	0	0
$543$	139.438i	0.256792i
$544$	0	0
$545$	−577.880	−1.06033
$546$	0	0
$547$	1043.66i	1.90798i	0.299839	+	0.953990i	$0.403067\pi$
$547$	1043.66i	1.90798i	−0.299839	+	0.953990i	$0.596933\pi$
$548$	0	0
$549$	33.0273	0.0601590
$550$	0	0
$551$	35.0224i	0.0635616i
$552$	0	0
$553$	−168.798	−0.305241
$554$	0	0
$555$	− 294.719i	− 0.531025i
$556$	0	0
$557$	149.826	0.268987	0.134493	−	0.990914i	$-0.457059\pi$
$557$	149.826	0.268987	0.134493	+	0.990914i	$0.457059\pi$
$558$	0	0
$559$	17.0336i	0.0304715i
$560$	0	0
$561$	−42.3303	−0.0754551
$562$	0	0
$563$	− 863.309i	− 1.53341i	−0.642000	−	0.766704i	$-0.721895\pi$
$563$	− 863.309i	− 1.53341i	0.642000	−	0.766704i	$-0.278105\pi$
$564$	0	0
$565$	−117.074	−0.207211
$566$	0	0
$567$	− 23.8118i	− 0.0419961i
$568$	0	0
$569$	−765.056	−1.34456	−0.672281	−	0.740296i	$-0.734685\pi$
$569$	−765.056	−1.34456	−0.672281	+	0.740296i	$0.734685\pi$
$570$	0	0
$571$	124.850i	0.218652i	0.994006	+	0.109326i	$0.0348692\pi$
$571$	124.850i	0.218652i	−0.994006	+	0.109326i	$0.965131\pi$
$572$	0	0
$573$	227.216	0.396537
$574$	0	0
$575$	330.655i	0.575052i
$576$	0	0
$577$	930.900	1.61334	0.806672	−	0.590999i	$-0.201266\pi$
$577$	930.900	1.61334	0.806672	+	0.590999i	$0.201266\pi$
$578$	0	0
$579$	− 297.039i	− 0.513020i
$580$	0	0
$581$	86.0182	0.148052
$582$	0	0
$583$	38.2162i	0.0655510i
$584$	0	0
$585$	−12.5227	−0.0214064
$586$	0	0
$587$	− 333.681i	− 0.568452i	−0.958757	−	0.284226i	$-0.908263\pi$
$587$	− 333.681i	− 0.568452i	0.958757	−	0.284226i	$-0.0917366\pi$
$588$	0	0
$589$	−798.606	−1.35587
$590$	0	0
$591$	99.0129i	0.167535i
$592$	0	0
$593$	244.977	0.413114	0.206557	−	0.978435i	$-0.433774\pi$
$593$	244.977	0.413114	0.206557	+	0.978435i	$0.433774\pi$
$594$	0	0
$595$	− 253.531i	− 0.426103i
$596$	0	0
$597$	11.6151	0.0194559
$598$	0	0
$599$	− 597.598i	− 0.997660i	−0.866700	−	0.498830i	$-0.833763\pi$
$599$	− 597.598i	− 0.997660i	0.866700	−	0.498830i	$-0.166237\pi$
$600$	0	0
$601$	−236.955	−0.394267	−0.197134	−	0.980377i	$-0.563163\pi$
$601$	−236.955	−0.394267	−0.197134	+	0.980377i	$0.563163\pi$
$602$	0	0
$603$	− 215.617i	− 0.357573i
$604$	0	0
$605$	−430.501	−0.711571
$606$	0	0
$607$	− 579.236i	− 0.954261i	−0.878833	−	0.477130i	$-0.841677\pi$
$607$	− 579.236i	− 0.954261i	0.878833	−	0.477130i	$-0.158323\pi$
$608$	0	0
$609$	−9.16515	−0.0150495
$610$	0	0
$611$	− 9.75721i	− 0.0159693i
$612$	0	0
$613$	10.0000	0.0163132	0.00815661	−	0.999967i	$-0.497404\pi$
$613$	10.0000	0.0163132	0.00815661	+	0.999967i	$0.497404\pi$
$614$	0	0
$615$	264.177i	0.429556i
$616$	0	0
$617$	−1160.32	−1.88059	−0.940294	−	0.340364i	$-0.889450\pi$
$617$	−1160.32	−1.88059	−0.940294	+	0.340364i	$0.889450\pi$
$618$	0	0
$619$	− 516.279i	− 0.834053i	−0.908894	−	0.417027i	$-0.863072\pi$
$619$	− 516.279i	− 0.834053i	0.908894	−	0.417027i	$-0.136928\pi$
$620$	0	0
$621$	−141.234	−0.227430
$622$	0	0
$623$	319.881i	0.513453i
$624$	0	0
$625$	−172.880	−0.276608
$626$	0	0
$627$	− 27.7128i	− 0.0441991i
$628$	0	0
$629$	−1270.40	−2.01971
$630$	0	0
$631$	− 1212.67i	− 1.92183i	−0.276845	−	0.960915i	$-0.589289\pi$
$631$	− 1212.67i	− 1.92183i	0.276845	−	0.960915i	$-0.410711\pi$
$632$	0	0
$633$	309.633	0.489152
$634$	0	0
$635$	202.381i	0.318709i
$636$	0	0
$637$	8.15606	0.0128039
$638$	0	0
$639$	− 166.968i	− 0.261296i
$640$	0	0
$641$	−40.1197	−0.0625892	−0.0312946	−	0.999510i	$-0.509963\pi$
$641$	−40.1197	−0.0625892	−0.0312946	+	0.999510i	$0.509963\pi$
$642$	0	0
$643$	198.852i	0.309256i	0.987973	+	0.154628i	$0.0494179\pi$
$643$	198.852i	0.309256i	−0.987973	+	0.154628i	$0.950582\pi$
$644$	0	0
$645$	−90.7152	−0.140644
$646$	0	0
$647$	− 211.484i	− 0.326869i	−0.986554	−	0.163435i	$-0.947743\pi$
$647$	− 211.484i	− 0.326869i	0.986554	−	0.163435i	$-0.0522573\pi$
$648$	0	0
$649$	24.6970	0.0380539
$650$	0	0
$651$	− 208.990i	− 0.321030i
$652$	0	0
$653$	−994.000	−1.52221	−0.761103	−	0.648632i	$-0.775342\pi$
$653$	−994.000	−1.52221	−0.761103	+	0.648632i	$0.775342\pi$
$654$	0	0
$655$	− 503.248i	− 0.768318i
$656$	0	0
$657$	286.486	0.436052
$658$	0	0
$659$	900.356i	1.36625i	0.730303	+	0.683123i	$0.239379\pi$
$659$	900.356i	1.36625i	−0.730303	+	0.683123i	$0.760621\pi$
$660$	0	0
$661$	683.945	1.03471	0.517357	−	0.855770i	$-0.326916\pi$
$661$	683.945	1.03471	0.517357	+	0.855770i	$0.326916\pi$
$662$	0	0
$663$	53.9796i	0.0814172i
$664$	0	0
$665$	165.982	0.249597
$666$	0	0
$667$	54.3610i	0.0815008i
$668$	0	0
$669$	−198.330	−0.296458
$670$	0	0
$671$	10.0590i	0.0149911i
$672$	0	0
$673$	122.211	0.181591	0.0907954	−	0.995870i	$-0.471059\pi$
$673$	122.211	0.181591	0.0907954	+	0.995870i	$0.471059\pi$
$674$	0	0
$675$	63.2120i	0.0936474i
$676$	0	0
$677$	191.931	0.283502	0.141751	−	0.989902i	$-0.454727\pi$
$677$	191.931	0.283502	0.141751	+	0.989902i	$0.454727\pi$
$678$	0	0
$679$	− 284.406i	− 0.418860i
$680$	0	0
$681$	−533.147	−0.782888
$682$	0	0
$683$	303.340i	0.444129i	0.975032	+	0.222065i	$0.0712796\pi$
$683$	303.340i	0.444129i	−0.975032	+	0.222065i	$0.928720\pi$
$684$	0	0
$685$	817.386	1.19326
$686$	0	0
$687$	− 616.880i	− 0.897934i
$688$	0	0
$689$	48.7333	0.0707305
$690$	0	0
$691$	− 329.725i	− 0.477170i	−0.971122	−	0.238585i	$-0.923316\pi$
$691$	− 329.725i	− 0.477170i	0.971122	−	0.238585i	$-0.0766836\pi$
$692$	0	0
$693$	7.25227	0.0104650
$694$	0	0
$695$	631.451i	0.908563i
$696$	0	0
$697$	1138.74	1.63378
$698$	0	0
$699$	369.816i	0.529065i
$700$	0	0
$701$	712.918	1.01700	0.508501	−	0.861061i	$-0.330200\pi$
$701$	712.918	1.01700	0.508501	+	0.861061i	$0.330200\pi$
$702$	0	0
$703$	− 831.703i	− 1.18308i
$704$	0	0
$705$	51.9636	0.0737073
$706$	0	0
$707$	276.263i	0.390753i
$708$	0	0
$709$	−975.248	−1.37553	−0.687763	−	0.725935i	$-0.741407\pi$
$709$	−975.248	−1.37553	−0.687763	+	0.725935i	$0.741407\pi$
$710$	0	0
$711$	191.399i	0.269198i
$712$	0	0
$713$	−1239.58	−1.73854
$714$	0	0
$715$	− 3.81401i	− 0.00533428i
$716$	0	0
$717$	697.996	0.973495
$718$	0	0
$719$	30.2864i	0.0421230i	0.999778	+	0.0210615i	$0.00670457\pi$
$719$	30.2864i	0.0421230i	−0.999778	+	0.0210615i	$0.993295\pi$
$720$	0	0
$721$	315.652	0.437797
$722$	0	0
$723$	− 596.636i	− 0.825223i
$724$	0	0
$725$	24.3303	0.0335590
$726$	0	0
$727$	1292.40i	1.77771i	0.458186	+	0.888856i	$0.348499\pi$
$727$	1292.40i	1.77771i	−0.458186	+	0.888856i	$0.651501\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	0	0
$731$	391.031i	0.534926i
$732$	0	0
$733$	1053.37	1.43706	0.718532	−	0.695494i	$-0.244814\pi$
$733$	1053.37	1.43706	0.718532	+	0.695494i	$0.244814\pi$
$734$	0	0
$735$	43.4364i	0.0590972i
$736$	0	0
$737$	65.6697	0.0891041
$738$	0	0
$739$	− 167.341i	− 0.226443i	−0.993570	−	0.113222i	$-0.963883\pi$
$739$	− 167.341i	− 0.226443i	0.993570	−	0.113222i	$-0.0361170\pi$
$740$	0	0
$741$	−35.3394	−0.0476915
$742$	0	0
$743$	− 618.001i	− 0.831765i	−0.909418	−	0.415883i	$-0.863473\pi$
$743$	− 618.001i	− 0.831765i	0.909418	−	0.415883i	$-0.136527\pi$
$744$	0	0
$745$	−984.617	−1.32163
$746$	0	0
$747$	− 97.5355i	− 0.130570i
$748$	0	0
$749$	177.270	0.236676
$750$	0	0
$751$	1215.11i	1.61798i	0.587820	+	0.808992i	$0.299986\pi$
$751$	1215.11i	1.61798i	−0.587820	+	0.808992i	$0.700014\pi$
$752$	0	0
$753$	851.477	1.13078
$754$	0	0
$755$	− 630.085i	− 0.834550i
$756$	0	0
$757$	1011.80	1.33659	0.668296	−	0.743896i	$-0.267024\pi$
$757$	1011.80	1.33659	0.668296	+	0.743896i	$0.267024\pi$
$758$	0	0
$759$	− 43.0152i	− 0.0566735i
$760$	0	0
$761$	209.042	0.274693	0.137347	−	0.990523i	$-0.456143\pi$
$761$	209.042	0.274693	0.137347	+	0.990523i	$0.456143\pi$
$762$	0	0
$763$	− 426.768i	− 0.559329i
$764$	0	0
$765$	−287.477	−0.375787
$766$	0	0
$767$	− 31.4936i	− 0.0410607i
$768$	0	0
$769$	7.00909	0.00911455	0.00455728	−	0.999990i	$-0.498549\pi$
$769$	7.00909	0.00911455	0.00455728	+	0.999990i	$0.498549\pi$
$770$	0	0
$771$	580.881i	0.753413i
$772$	0	0
$773$	4.59167	0.00594006	0.00297003	−	0.999996i	$-0.499055\pi$
$773$	4.59167	0.00594006	0.00297003	+	0.999996i	$0.499055\pi$
$774$	0	0
$775$	554.797i	0.715867i
$776$	0	0
$777$	217.652	0.280118
$778$	0	0
$779$	745.513i	0.957013i
$780$	0	0
$781$	50.8530	0.0651127
$782$	0	0
$783$	10.3923i	0.0132724i
$784$	0	0
$785$	−200.559	−0.255489
$786$	0	0
$787$	732.642i	0.930930i	0.885066	+	0.465465i	$0.154113\pi$
$787$	732.642i	0.930930i	−0.885066	+	0.465465i	$0.845887\pi$
$788$	0	0
$789$	839.354	1.06382
$790$	0	0
$791$	− 86.4599i	− 0.109305i
$792$	0	0
$793$	12.8273	0.0161756
$794$	0	0
$795$	259.537i	0.326462i
$796$	0	0
$797$	78.8856	0.0989782	0.0494891	−	0.998775i	$-0.484241\pi$
$797$	78.8856	0.0989782	0.0494891	+	0.998775i	$0.484241\pi$
$798$	0	0
$799$	− 223.991i	− 0.280339i
$800$	0	0
$801$	362.711	0.452823
$802$	0	0
$803$	87.2542i	0.108660i
$804$	0	0
$805$	257.633	0.320041
$806$	0	0
$807$	− 48.3780i	− 0.0599480i
$808$	0	0
$809$	−1022.83	−1.26431	−0.632155	−	0.774842i	$-0.717830\pi$
$809$	−1022.83	−1.26431	−0.632155	+	0.774842i	$0.717830\pi$
$810$	0	0
$811$	− 73.3182i	− 0.0904047i	−0.998978	−	0.0452024i	$-0.985607\pi$
$811$	− 73.3182i	− 0.0904047i	0.998978	−	0.0452024i	$-0.0143933\pi$
$812$	0	0
$813$	28.6242	0.0352082
$814$	0	0
$815$	850.454i	1.04350i
$816$	0	0
$817$	−256.000	−0.313341
$818$	0	0
$819$	− 9.24810i	− 0.0112919i
$820$	0	0
$821$	336.395	0.409739	0.204869	−	0.978789i	$-0.434323\pi$
$821$	336.395	0.409739	0.204869	+	0.978789i	$0.434323\pi$
$822$	0	0
$823$	333.937i	0.405755i	0.979204	+	0.202878i	$0.0650294\pi$
$823$	333.937i	0.405755i	−0.979204	+	0.202878i	$0.934971\pi$
$824$	0	0
$825$	−19.2523	−0.0233361
$826$	0	0
$827$	1652.01i	1.99759i	0.0490993	+	0.998794i	$0.484365\pi$
$827$	1652.01i	1.99759i	−0.0490993	+	0.998794i	$0.515635\pi$
$828$	0	0
$829$	1011.07	1.21963	0.609816	−	0.792543i	$-0.291244\pi$
$829$	1011.07	1.21963	0.609816	+	0.792543i	$0.291244\pi$
$830$	0	0
$831$	− 622.601i	− 0.749219i
$832$	0	0
$833$	187.234	0.224771
$834$	0	0
$835$	850.454i	1.01851i
$836$	0	0
$837$	−236.973	−0.283122
$838$	0	0
$839$	− 337.718i	− 0.402524i	−0.979537	−	0.201262i	$-0.935496\pi$
$839$	− 337.718i	− 0.402524i	0.979537	−	0.201262i	$-0.0645042\pi$
$840$	0	0
$841$	−837.000	−0.995244
$842$	0	0
$843$	448.856i	0.532450i
$844$	0	0
$845$	600.592	0.710759
$846$	0	0
$847$	− 317.927i	− 0.375357i
$848$	0	0
$849$	−306.991	−0.361591
$850$	0	0
$851$	− 1290.95i	− 1.51698i
$852$	0	0
$853$	−1285.51	−1.50704	−0.753521	−	0.657424i	$-0.771646\pi$
$853$	−1285.51	−1.50704	−0.753521	+	0.657424i	$0.771646\pi$
$854$	0	0
$855$	− 188.206i	− 0.220124i
$856$	0	0
$857$	1169.04	1.36411	0.682055	−	0.731301i	$-0.261087\pi$
$857$	1169.04	1.36411	0.682055	+	0.731301i	$0.261087\pi$
$858$	0	0
$859$	− 1283.62i	− 1.49432i	−0.664642	−	0.747162i	$-0.731416\pi$
$859$	− 1283.62i	− 1.49432i	0.664642	−	0.747162i	$-0.268584\pi$
$860$	0	0
$861$	−195.096	−0.226593
$862$	0	0
$863$	1019.52i	1.18137i	0.806904	+	0.590683i	$0.201142\pi$
$863$	1019.52i	1.18137i	−0.806904	+	0.590683i	$0.798858\pi$
$864$	0	0
$865$	620.065	0.716838
$866$	0	0
$867$	738.617i	0.851923i
$868$	0	0
$869$	−58.2939	−0.0670816
$870$	0	0
$871$	− 83.7420i	− 0.0961447i
$872$	0	0
$873$	−322.486	−0.369400
$874$	0	0
$875$	− 352.274i	− 0.402599i
$876$	0	0
$877$	301.194	0.343437	0.171718	−	0.985146i	$-0.445068\pi$
$877$	301.194	0.343437	0.171718	+	0.985146i	$0.445068\pi$
$878$	0	0
$879$	− 512.806i	− 0.583398i
$880$	0	0
$881$	−1530.67	−1.73743	−0.868715	−	0.495313i	$-0.835053\pi$
$881$	−1530.67	−1.73743	−0.868715	+	0.495313i	$0.835053\pi$
$882$	0	0
$883$	319.301i	0.361609i	0.983519	+	0.180805i	$0.0578702\pi$
$883$	319.301i	0.361609i	−0.983519	+	0.180805i	$0.942130\pi$
$884$	0	0
$885$	167.724	0.189519
$886$	0	0
$887$	1315.17i	1.48271i	0.671110	+	0.741357i	$0.265818\pi$
$887$	1315.17i	1.48271i	−0.671110	+	0.741357i	$0.734182\pi$
$888$	0	0
$889$	−149.459	−0.168120
$890$	0	0
$891$	− 8.22330i	− 0.00922930i
$892$	0	0
$893$	146.642	0.164213
$894$	0	0
$895$	488.248i	0.545528i
$896$	0	0
$897$	−54.8530	−0.0611517
$898$	0	0
$899$	91.2108i	0.101458i
$900$	0	0
$901$	1118.74	1.24167
$902$	0	0
$903$	− 66.9936i	− 0.0741901i
$904$	0	0
$905$	−288.414	−0.318689
$906$	0	0
$907$	1585.93i	1.74854i	0.485441	+	0.874270i	$0.338659\pi$
$907$	1585.93i	1.74854i	−0.485441	+	0.874270i	$0.661341\pi$
$908$	0	0
$909$	313.252	0.344612
$910$	0	0
$911$	− 350.279i	− 0.384499i	−0.981346	−	0.192250i	$-0.938422\pi$
$911$	− 350.279i	− 0.384499i	0.981346	−	0.192250i	$-0.0615783\pi$
$912$	0	0
$913$	29.7061	0.0325368
$914$	0	0
$915$	68.3136i	0.0746597i
$916$	0	0
$917$	371.652	0.405291
$918$	0	0
$919$	− 431.552i	− 0.469588i	−0.972045	−	0.234794i	$-0.924558\pi$
$919$	− 431.552i	− 0.469588i	0.972045	−	0.234794i	$-0.0754416\pi$
$920$	0	0
$921$	−295.927	−0.321311
$922$	0	0
$923$	− 64.8478i	− 0.0702577i
$924$	0	0
$925$	−577.789	−0.624637
$926$	0	0
$927$	− 357.915i	− 0.386101i
$928$	0	0
$929$	620.802	0.668248	0.334124	−	0.942529i	$-0.391560\pi$
$929$	620.802	0.668248	0.334124	+	0.942529i	$0.391560\pi$
$930$	0	0
$931$	122.578i	0.131663i
$932$	0	0
$933$	727.405	0.779640
$934$	0	0
$935$	− 87.5560i	− 0.0936428i
$936$	0	0
$937$	−298.936	−0.319036	−0.159518	−	0.987195i	$-0.550994\pi$
$937$	−298.936	−0.319036	−0.159518	+	0.987195i	$0.550994\pi$
$938$	0	0
$939$	− 413.309i	− 0.440159i
$940$	0	0
$941$	786.951	0.836292	0.418146	−	0.908380i	$-0.362680\pi$
$941$	786.951	0.836292	0.418146	+	0.908380i	$0.362680\pi$
$942$	0	0
$943$	1157.17i	1.22711i
$944$	0	0
$945$	49.2523	0.0521188
$946$	0	0
$947$	629.775i	0.665021i	0.943100	+	0.332511i	$0.107896\pi$
$947$	629.775i	0.665021i	−0.943100	+	0.332511i	$0.892104\pi$
$948$	0	0
$949$	111.267	0.117246
$950$	0	0
$951$	1014.01i	1.06626i
$952$	0	0
$953$	465.855	0.488830	0.244415	−	0.969671i	$-0.421404\pi$
$953$	465.855	0.488830	0.244415	+	0.969671i	$0.421404\pi$
$954$	0	0
$955$	469.974i	0.492119i
$956$	0	0
$957$	−3.16515	−0.00330737
$958$	0	0
$959$	603.644i	0.629452i
$960$	0	0
$961$	−1118.85	−1.16426
$962$	0	0
$963$	− 201.006i	− 0.208729i
$964$	0	0
$965$	614.395	0.636679
$966$	0	0
$967$	− 362.538i	− 0.374910i	−0.982273	−	0.187455i	$-0.939976\pi$
$967$	− 362.538i	− 0.374910i	0.982273	−	0.187455i	$-0.0600239\pi$
$968$	0	0
$969$	−811.267	−0.837220
$970$	0	0
$971$	− 1530.12i	− 1.57582i	−0.615792	−	0.787908i	$-0.711164\pi$
$971$	− 1530.12i	− 1.57582i	0.615792	−	0.787908i	$-0.288836\pi$
$972$	0	0
$973$	−466.330	−0.479271
$974$	0	0
$975$	24.5505i	0.0251800i
$976$	0	0
$977$	−841.441	−0.861250	−0.430625	−	0.902531i	$-0.641707\pi$
$977$	−841.441	−0.861250	−0.430625	+	0.902531i	$0.641707\pi$
$978$	0	0
$979$	110.470i	0.112839i
$980$	0	0
$981$	−483.909	−0.493281
$982$	0	0
$983$	− 1611.63i	− 1.63951i	−0.572717	−	0.819753i	$-0.694111\pi$
$983$	− 1611.63i	− 1.63951i	0.572717	−	0.819753i	$-0.305889\pi$
$984$	0	0
$985$	−204.798	−0.207917
$986$	0	0
$987$	38.3754i	0.0388809i
$988$	0	0
$989$	−397.358	−0.401777
$990$	0	0
$991$	− 368.180i	− 0.371523i	−0.982595	−	0.185762i	$-0.940525\pi$
$991$	− 368.180i	− 0.371523i	0.982595	−	0.185762i	$-0.0594753\pi$
$992$	0	0
$993$	123.688	0.124560
$994$	0	0
$995$	24.0248i	0.0241455i
$996$	0	0
$997$	924.642	0.927425	0.463712	−	0.885986i	$-0.346517\pi$
$997$	924.642	0.927425	0.463712	+	0.885986i	$0.346517\pi$
$998$	0	0
$999$	− 246.794i	− 0.247041i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.3.m.c.127.3	yes	4
3.2	odd	2		1008.3.m.b.127.3		4
4.3	odd	2	inner	336.3.m.c.127.1	✓	4
7.6	odd	2		2352.3.m.f.1471.2		4
8.3	odd	2		1344.3.m.a.127.4		4
8.5	even	2		1344.3.m.a.127.2		4
12.11	even	2		1008.3.m.b.127.4		4
28.27	even	2		2352.3.m.f.1471.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.3.m.c.127.1	✓	4	4.3	odd	2	inner
336.3.m.c.127.3	yes	4	1.1	even	1	trivial
1008.3.m.b.127.3		4	3.2	odd	2
1008.3.m.b.127.4		4	12.11	even	2
1344.3.m.a.127.2		4	8.5	even	2
1344.3.m.a.127.4		4	8.3	odd	2
2352.3.m.f.1471.2		4	7.6	odd	2
2352.3.m.f.1471.4		4	28.27	even	2

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:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	336.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.73205i q^{3} -3.58258 q^{5} -2.64575i q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q+1.73205i q^{3} -3.58258 q^{5} -2.64575i q^{7} -3.00000 q^{9} -0.913701i q^{11} -1.16515 q^{13} -6.20520i q^{15} -26.7477 q^{17} -17.5112i q^{19} +4.58258 q^{21} -27.1805i q^{23} -12.1652 q^{25} -5.19615i q^{27} -2.00000 q^{29} -45.6054i q^{31} +1.58258 q^{33} +9.47860i q^{35} +47.4955 q^{37} -2.01810i q^{39} -42.5735 q^{41} -14.6192i q^{43} +10.7477 q^{45} +8.37420i q^{47} -7.00000 q^{49} -46.3284i q^{51} -41.8258 q^{53} +3.27340i q^{55} +30.3303 q^{57} +27.0296i q^{59} -11.0091 q^{61} +7.93725i q^{63} +4.17424 q^{65} +71.8722i q^{67} +47.0780 q^{69} +55.6561i q^{71} -95.4955 q^{73} -21.0707i q^{75} -2.41742 q^{77} -63.7998i q^{79} +9.00000 q^{81} +32.5118i q^{83} +95.8258 q^{85} -3.46410i q^{87} -120.904 q^{89} +3.08270i q^{91} +78.9909 q^{93} +62.7352i q^{95} +107.495 q^{97} +2.74110i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5} - 12 q^{9}+O(q^{10}) \) 4 * q + 4 * q^5 - 12 * q^9	\( 4 q + 4 q^{5} - 12 q^{9} + 32 q^{13} - 52 q^{17} - 12 q^{25} - 8 q^{29} - 12 q^{33} + 80 q^{37} + 68 q^{41} - 12 q^{45} - 28 q^{49} + 16 q^{53} + 48 q^{57} - 264 q^{61} + 200 q^{65} + 60 q^{69} - 272 q^{73} - 28 q^{77} + 36 q^{81} + 200 q^{85} - 172 q^{89} + 96 q^{93} + 320 q^{97}+O(q^{100}) \) 4 * q + 4 * q^5 - 12 * q^9 + 32 * q^13 - 52 * q^17 - 12 * q^25 - 8 * q^29 - 12 * q^33 + 80 * q^37 + 68 * q^41 - 12 * q^45 - 28 * q^49 + 16 * q^53 + 48 * q^57 - 264 * q^61 + 200 * q^65 + 60 * q^69 - 272 * q^73 - 28 * q^77 + 36 * q^81 + 200 * q^85 - 172 * q^89 + 96 * q^93 + 320 * q^97

Introduction

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