LMFDB - Embedded newform 216.3.e.c.161.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [216,3,Mod(161,216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(216, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("216.161");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 216 = 2^{3} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	216.e (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:	$5.88557371018$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.4
Root			$-0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	216.161
Dual form			216.3.e.c.161.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.82843i q^{5} -11.4853 q^{7} +O(q^{10})$	$q+5.82843i q^{5} -11.4853 q^{7} -14.6569i q^{11} -18.9706 q^{13} -5.31371i q^{17} -8.97056 q^{19} +16.2843i q^{23} -8.97056 q^{25} +42.0000i q^{29} -9.48528 q^{31} -66.9411i q^{35} -52.9706 q^{37} +62.9117i q^{41} +2.97056 q^{43} -75.9411i q^{47} +82.9117 q^{49} +2.91674i q^{53} +85.4264 q^{55} +61.1960i q^{59} +119.882 q^{61} -110.569i q^{65} -41.0294 q^{67} -10.2843i q^{71} -20.0883 q^{73} +168.338i q^{77} -40.0589 q^{79} +108.598i q^{83} +30.9706 q^{85} -106.971i q^{89} +217.882 q^{91} -52.2843i q^{95} -56.8823 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{7}+O(q^{10}) $ 4 * q - 12 * q^7	$ 4 q - 12 q^{7} - 8 q^{13} + 32 q^{19} + 32 q^{25} - 4 q^{31} - 144 q^{37} - 56 q^{43} + 128 q^{49} + 172 q^{55} + 208 q^{61} - 232 q^{67} - 284 q^{73} - 296 q^{79} + 56 q^{85} + 600 q^{91} + 44 q^{97}+O(q^{100}) $ 4 * q - 12 * q^7 - 8 * q^13 + 32 * q^19 + 32 * q^25 - 4 * q^31 - 144 * q^37 - 56 * q^43 + 128 * q^49 + 172 * q^55 + 208 * q^61 - 232 * q^67 - 284 * q^73 - 296 * q^79 + 56 * q^85 + 600 * q^91 + 44 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/216\mathbb{Z}\right)^\times$.

$n$	$55$	$109$	$137$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	5.82843i	1.16569i	0.812585	+	0.582843i	$0.198060\pi$
$5$	5.82843i	1.16569i	−0.812585	+	0.582843i	$0.801940\pi$
$6$	0	0
$7$	−11.4853	−1.64075	−0.820377	−	0.571823i	$-0.806237\pi$
$7$	−11.4853	−1.64075	−0.820377	+	0.571823i	$0.806237\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 14.6569i	− 1.33244i	−0.745755	−	0.666221i	$-0.767911\pi$
$11$	− 14.6569i	− 1.33244i	0.745755	−	0.666221i	$-0.232089\pi$
$12$	0	0
$13$	−18.9706	−1.45927	−0.729637	−	0.683835i	$-0.760311\pi$
$13$	−18.9706	−1.45927	−0.729637	+	0.683835i	$0.760311\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 5.31371i	− 0.312571i	−0.987712	−	0.156286i	$-0.950048\pi$
$17$	− 5.31371i	− 0.312571i	0.987712	−	0.156286i	$-0.0499520\pi$
$18$	0	0
$19$	−8.97056	−0.472135	−0.236067	−	0.971737i	$-0.575859\pi$
$19$	−8.97056	−0.472135	−0.236067	+	0.971737i	$0.575859\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	16.2843i	0.708012i	0.935243	+	0.354006i	$0.115181\pi$
$23$	16.2843i	0.708012i	−0.935243	+	0.354006i	$0.884819\pi$
$24$	0	0
$25$	−8.97056	−0.358823
$26$	0	0
$27$	0	0
$28$	0	0
$29$	42.0000i	1.44828i	0.689655	+	0.724138i	$0.257762\pi$
$29$	42.0000i	1.44828i	−0.689655	+	0.724138i	$0.742238\pi$
$30$	0	0
$31$	−9.48528	−0.305977	−0.152988	−	0.988228i	$-0.548890\pi$
$31$	−9.48528	−0.305977	−0.152988	+	0.988228i	$0.548890\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 66.9411i	− 1.91260i
$36$	0	0
$37$	−52.9706	−1.43164	−0.715818	−	0.698286i	$-0.753946\pi$
$37$	−52.9706	−1.43164	−0.715818	+	0.698286i	$0.753946\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	62.9117i	1.53443i	0.641389	+	0.767216i	$0.278358\pi$
$41$	62.9117i	1.53443i	−0.641389	+	0.767216i	$0.721642\pi$
$42$	0	0
$43$	2.97056	0.0690829	0.0345414	−	0.999403i	$-0.489003\pi$
$43$	2.97056	0.0690829	0.0345414	+	0.999403i	$0.489003\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 75.9411i	− 1.61577i	−0.589341	−	0.807884i	$-0.700613\pi$
$47$	− 75.9411i	− 1.61577i	0.589341	−	0.807884i	$-0.299387\pi$
$48$	0	0
$49$	82.9117	1.69208
$50$	0	0
$51$	0	0
$52$	0	0
$53$	2.91674i	0.0550328i	0.999621	+	0.0275164i	$0.00875985\pi$
$53$	2.91674i	0.0550328i	−0.999621	+	0.0275164i	$0.991240\pi$
$54$	0	0
$55$	85.4264	1.55321
$56$	0	0
$57$	0	0
$58$	0	0
$59$	61.1960i	1.03722i	0.855011	+	0.518610i	$0.173550\pi$
$59$	61.1960i	1.03722i	−0.855011	+	0.518610i	$0.826450\pi$
$60$	0	0
$61$	119.882	1.96528	0.982641	−	0.185515i	$-0.0593954\pi$
$61$	119.882	1.96528	0.982641	+	0.185515i	$0.0593954\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 110.569i	− 1.70105i
$66$	0	0
$67$	−41.0294	−0.612380	−0.306190	−	0.951970i	$-0.599054\pi$
$67$	−41.0294	−0.612380	−0.306190	+	0.951970i	$0.599054\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 10.2843i	− 0.144849i	−0.997374	−	0.0724244i	$-0.976926\pi$
$71$	− 10.2843i	− 0.144849i	0.997374	−	0.0724244i	$-0.0230736\pi$
$72$	0	0
$73$	−20.0883	−0.275182	−0.137591	−	0.990489i	$-0.543936\pi$
$73$	−20.0883	−0.275182	−0.137591	+	0.990489i	$0.543936\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	168.338i	2.18621i
$78$	0	0
$79$	−40.0589	−0.507074	−0.253537	−	0.967326i	$-0.581594\pi$
$79$	−40.0589	−0.507074	−0.253537	+	0.967326i	$0.581594\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	108.598i	1.30841i	0.756318	+	0.654205i	$0.226997\pi$
$83$	108.598i	1.30841i	−0.756318	+	0.654205i	$0.773003\pi$
$84$	0	0
$85$	30.9706	0.364360
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 106.971i	− 1.20192i	−0.799280	−	0.600958i	$-0.794786\pi$
$89$	− 106.971i	− 1.20192i	0.799280	−	0.600958i	$-0.205214\pi$
$90$	0	0
$91$	217.882	2.39431
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 52.2843i	− 0.550361i
$96$	0	0
$97$	−56.8823	−0.586415	−0.293207	−	0.956049i	$-0.594723\pi$
$97$	−56.8823	−0.586415	−0.293207	+	0.956049i	$0.594723\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1.54416i	0.0152887i	0.999971	+	0.00764434i	$0.00243329\pi$
$101$	1.54416i	0.0152887i	−0.999971	+	0.00764434i	$0.997567\pi$
$102$	0	0
$103$	67.9411	0.659623	0.329811	−	0.944047i	$-0.393015\pi$
$103$	67.9411	0.659623	0.329811	+	0.944047i	$0.393015\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 64.8823i	− 0.606376i	−0.952931	−	0.303188i	$-0.901949\pi$
$107$	− 64.8823i	− 0.606376i	0.952931	−	0.303188i	$-0.0980510\pi$
$108$	0	0
$109$	−51.0294	−0.468160	−0.234080	−	0.972217i	$-0.575208\pi$
$109$	−51.0294	−0.468160	−0.234080	+	0.972217i	$0.575208\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 208.971i	− 1.84930i	−0.380822	−	0.924649i	$-0.624359\pi$
$113$	− 208.971i	− 1.84930i	0.380822	−	0.924649i	$-0.375641\pi$
$114$	0	0
$115$	−94.9117	−0.825319
$116$	0	0
$117$	0	0
$118$	0	0
$119$	61.0294i	0.512852i
$120$	0	0
$121$	−93.8234	−0.775400
$122$	0	0
$123$	0	0
$124$	0	0
$125$	93.4264i	0.747411i
$126$	0	0
$127$	−200.161	−1.57607	−0.788037	−	0.615628i	$-0.788903\pi$
$127$	−200.161	−1.57607	−0.788037	+	0.615628i	$0.788903\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	117.686i	0.898369i	0.893439	+	0.449184i	$0.148285\pi$
$131$	117.686i	0.898369i	−0.893439	+	0.449184i	$0.851715\pi$
$132$	0	0
$133$	103.029	0.774657
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 19.2061i	− 0.140190i	−0.997540	−	0.0700951i	$-0.977670\pi$
$137$	− 19.2061i	− 0.140190i	0.997540	−	0.0700951i	$-0.0223303\pi$
$138$	0	0
$139$	−175.882	−1.26534	−0.632670	−	0.774422i	$-0.718041\pi$
$139$	−175.882	−1.26534	−0.632670	+	0.774422i	$0.718041\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	278.049i	1.94440i
$144$	0	0
$145$	−244.794	−1.68823
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 121.534i	− 0.815665i	−0.913057	−	0.407832i	$-0.866285\pi$
$149$	− 121.534i	− 0.815665i	0.913057	−	0.407832i	$-0.133715\pi$
$150$	0	0
$151$	−126.397	−0.837066	−0.418533	−	0.908202i	$-0.637456\pi$
$151$	−126.397	−0.837066	−0.418533	+	0.908202i	$0.637456\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 55.2843i	− 0.356673i
$156$	0	0
$157$	−25.8234	−0.164480	−0.0822401	−	0.996613i	$-0.526207\pi$
$157$	−25.8234	−0.164480	−0.0822401	+	0.996613i	$0.526207\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 187.029i	− 1.16167i
$162$	0	0
$163$	0.176624	0.00108358	0.000541790	−	1.00000i	$-0.499828\pi$
$163$	0.176624	0.00108358	0.000541790		1.00000i	$0.499828\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	128.059i	0.766820i	0.923578	+	0.383410i	$0.125250\pi$
$167$	128.059i	0.766820i	−0.923578	+	0.383410i	$0.874750\pi$
$168$	0	0
$169$	190.882	1.12948
$170$	0	0
$171$	0	0
$172$	0	0
$173$	7.54416i	0.0436078i	0.999762	+	0.0218039i	$0.00694095\pi$
$173$	7.54416i	0.0436078i	−0.999762	+	0.0218039i	$0.993059\pi$
$174$	0	0
$175$	103.029	0.588740
$176$	0	0
$177$	0	0
$178$	0	0
$179$	100.882i	0.563588i	0.959475	+	0.281794i	$0.0909295\pi$
$179$	100.882i	0.563588i	−0.959475	+	0.281794i	$0.909071\pi$
$180$	0	0
$181$	−129.941	−0.717907	−0.358953	−	0.933355i	$-0.616866\pi$
$181$	−129.941	−0.717907	−0.358953	+	0.933355i	$0.616866\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 308.735i	− 1.66884i
$186$	0	0
$187$	−77.8823	−0.416483
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 17.1472i	− 0.0897758i	−0.998992	−	0.0448879i	$-0.985707\pi$
$191$	− 17.1472i	− 0.0897758i	0.998992	−	0.0448879i	$-0.0142931\pi$
$192$	0	0
$193$	83.8528	0.434471	0.217235	−	0.976119i	$-0.430296\pi$
$193$	83.8528	0.434471	0.217235	+	0.976119i	$0.430296\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	306.161i	1.55412i	0.629427	+	0.777060i	$0.283290\pi$
$197$	306.161i	1.55412i	−0.629427	+	0.777060i	$0.716710\pi$
$198$	0	0
$199$	254.103	1.27690	0.638449	−	0.769664i	$-0.279576\pi$
$199$	254.103	1.27690	0.638449	+	0.769664i	$0.279576\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 482.382i	− 2.37627i
$204$	0	0
$205$	−366.676	−1.78866
$206$	0	0
$207$	0	0
$208$	0	0
$209$	131.480i	0.629092i
$210$	0	0
$211$	−243.029	−1.15180	−0.575899	−	0.817521i	$-0.695348\pi$
$211$	−243.029	−1.15180	−0.575899	+	0.817521i	$0.695348\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	17.3137i	0.0805289i
$216$	0	0
$217$	108.941	0.502033
$218$	0	0
$219$	0	0
$220$	0	0
$221$	100.804i	0.456127i
$222$	0	0
$223$	61.7645	0.276971	0.138485	−	0.990364i	$-0.455777\pi$
$223$	61.7645	0.276971	0.138485	+	0.990364i	$0.455777\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	123.941i	0.545996i	0.962015	+	0.272998i	$0.0880153\pi$
$227$	123.941i	0.545996i	−0.962015	+	0.272998i	$0.911985\pi$
$228$	0	0
$229$	269.529	1.17698	0.588491	−	0.808504i	$-0.299722\pi$
$229$	269.529	1.17698	0.588491	+	0.808504i	$0.299722\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	54.1665i	0.232474i	0.993221	+	0.116237i	$0.0370833\pi$
$233$	54.1665i	0.232474i	−0.993221	+	0.116237i	$0.962917\pi$
$234$	0	0
$235$	442.617	1.88348
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 336.167i	− 1.40655i	−0.710916	−	0.703277i	$-0.751719\pi$
$239$	− 336.167i	− 1.40655i	0.710916	−	0.703277i	$-0.248281\pi$
$240$	0	0
$241$	267.588	1.11032	0.555162	−	0.831743i	$-0.312656\pi$
$241$	267.588	1.11032	0.555162	+	0.831743i	$0.312656\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	483.245i	1.97243i
$246$	0	0
$247$	170.177	0.688974
$248$	0	0
$249$	0	0
$250$	0	0
$251$	272.039i	1.08382i	0.840437	+	0.541910i	$0.182299\pi$
$251$	272.039i	1.08382i	−0.840437	+	0.541910i	$0.817701\pi$
$252$	0	0
$253$	238.676	0.943384
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 182.902i	− 0.711679i	−0.934547	−	0.355840i	$-0.884195\pi$
$257$	− 182.902i	− 0.711679i	0.934547	−	0.355840i	$-0.115805\pi$
$258$	0	0
$259$	608.382	2.34896
$260$	0	0
$261$	0	0
$262$	0	0
$263$	170.225i	0.647245i	0.946186	+	0.323622i	$0.104901\pi$
$263$	170.225i	0.647245i	−0.946186	+	0.323622i	$0.895099\pi$
$264$	0	0
$265$	−17.0000	−0.0641509
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 76.9605i	− 0.286098i	−0.989716	−	0.143049i	$-0.954309\pi$
$269$	− 76.9605i	− 0.286098i	0.989716	−	0.143049i	$-0.0456907\pi$
$270$	0	0
$271$	−361.603	−1.33433	−0.667164	−	0.744911i	$-0.732492\pi$
$271$	−361.603	−1.33433	−0.667164	+	0.744911i	$0.732492\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	131.480i	0.478110i
$276$	0	0
$277$	−169.823	−0.613081	−0.306540	−	0.951858i	$-0.599171\pi$
$277$	−169.823	−0.613081	−0.306540	+	0.951858i	$0.599171\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 464.725i	− 1.65383i	−0.562330	−	0.826913i	$-0.690095\pi$
$281$	− 464.725i	− 1.65383i	0.562330	−	0.826913i	$-0.309905\pi$
$282$	0	0
$283$	−69.3238	−0.244960	−0.122480	−	0.992471i	$-0.539085\pi$
$283$	−69.3238	−0.244960	−0.122480	+	0.992471i	$0.539085\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 722.558i	− 2.51763i
$288$	0	0
$289$	260.765	0.902299
$290$	0	0
$291$	0	0
$292$	0	0
$293$	39.9411i	0.136318i	0.997674	+	0.0681589i	$0.0217125\pi$
$293$	39.9411i	0.136318i	−0.997674	+	0.0681589i	$0.978288\pi$
$294$	0	0
$295$	−356.676	−1.20907
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 308.922i	− 1.03318i
$300$	0	0
$301$	−34.1177	−0.113348
$302$	0	0
$303$	0	0
$304$	0	0
$305$	698.725i	2.29090i
$306$	0	0
$307$	−495.235	−1.61314	−0.806571	−	0.591137i	$-0.798679\pi$
$307$	−495.235	−1.61314	−0.806571	+	0.591137i	$0.798679\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 148.617i	− 0.477869i	−0.971036	−	0.238935i	$-0.923202\pi$
$311$	− 148.617i	− 0.477869i	0.971036	−	0.238935i	$-0.0767982\pi$
$312$	0	0
$313$	239.558	0.765362	0.382681	−	0.923880i	$-0.375001\pi$
$313$	239.558	0.765362	0.382681	+	0.923880i	$0.375001\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 159.250i	− 0.502365i	−0.967940	−	0.251183i	$-0.919181\pi$
$317$	− 159.250i	− 0.502365i	0.967940	−	0.251183i	$-0.0808195\pi$
$318$	0	0
$319$	615.588	1.92974
$320$	0	0
$321$	0	0
$322$	0	0
$323$	47.6670i	0.147576i
$324$	0	0
$325$	170.177	0.523620
$326$	0	0
$327$	0	0
$328$	0	0
$329$	872.205i	2.65108i
$330$	0	0
$331$	−94.4996	−0.285497	−0.142749	−	0.989759i	$-0.545594\pi$
$331$	−94.4996	−0.285497	−0.142749	+	0.989759i	$0.545594\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 239.137i	− 0.713842i
$336$	0	0
$337$	−435.235	−1.29150	−0.645749	−	0.763550i	$-0.723455\pi$
$337$	−435.235	−1.29150	−0.645749	+	0.763550i	$0.723455\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	139.024i	0.407696i
$342$	0	0
$343$	−389.485	−1.13553
$344$	0	0
$345$	0	0
$346$	0	0
$347$	3.49957i	0.0100852i	0.999987	+	0.00504260i	$0.00160512\pi$
$347$	3.49957i	0.0100852i	−0.999987	+	0.00504260i	$0.998395\pi$
$348$	0	0
$349$	26.9117	0.0771109	0.0385554	−	0.999256i	$-0.487724\pi$
$349$	26.9117	0.0771109	0.0385554	+	0.999256i	$0.487724\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	291.245i	0.825056i	0.910945	+	0.412528i	$0.135354\pi$
$353$	291.245i	0.825056i	−0.910945	+	0.412528i	$0.864646\pi$
$354$	0	0
$355$	59.9411	0.168848
$356$	0	0
$357$	0	0
$358$	0	0
$359$	51.9411i	0.144683i	0.997380	+	0.0723414i	$0.0230471\pi$
$359$	51.9411i	0.144683i	−0.997380	+	0.0723414i	$0.976953\pi$
$360$	0	0
$361$	−280.529	−0.777089
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 117.083i	− 0.320776i
$366$	0	0
$367$	222.750	0.606949	0.303474	−	0.952840i	$-0.401853\pi$
$367$	222.750	0.606949	0.303474	+	0.952840i	$0.401853\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 33.4996i	− 0.0902953i
$372$	0	0
$373$	−498.382	−1.33614	−0.668072	−	0.744097i	$-0.732880\pi$
$373$	−498.382	−1.33614	−0.668072	+	0.744097i	$0.732880\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 796.764i	− 2.11343i
$378$	0	0
$379$	−160.000	−0.422164	−0.211082	−	0.977468i	$-0.567699\pi$
$379$	−160.000	−0.422164	−0.211082	+	0.977468i	$0.567699\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	296.205i	0.773382i	0.922209	+	0.386691i	$0.126382\pi$
$383$	296.205i	0.773382i	−0.922209	+	0.386691i	$0.873618\pi$
$384$	0	0
$385$	−981.146	−2.54843
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 108.848i	− 0.279814i	−0.990165	−	0.139907i	$-0.955320\pi$
$389$	− 108.848i	− 0.279814i	0.990165	−	0.139907i	$-0.0446804\pi$
$390$	0	0
$391$	86.5299	0.221304
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 233.480i	− 0.591089i
$396$	0	0
$397$	83.4416	0.210180	0.105090	−	0.994463i	$-0.466487\pi$
$397$	83.4416	0.210180	0.105090	+	0.994463i	$0.466487\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	199.529i	0.497579i	0.968558	+	0.248789i	$0.0800327\pi$
$401$	199.529i	0.497579i	−0.968558	+	0.248789i	$0.919967\pi$
$402$	0	0
$403$	179.941	0.446504
$404$	0	0
$405$	0	0
$406$	0	0
$407$	776.382i	1.90757i
$408$	0	0
$409$	−299.235	−0.731627	−0.365814	−	0.930688i	$-0.619209\pi$
$409$	−299.235	−0.731627	−0.365814	+	0.930688i	$0.619209\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 702.853i	− 1.70182i
$414$	0	0
$415$	−632.955	−1.52519
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 21.7645i	− 0.0519439i	−0.999663	−	0.0259720i	$-0.991732\pi$
$419$	− 21.7645i	− 0.0519439i	0.999663	−	0.0259720i	$-0.00826806\pi$
$420$	0	0
$421$	25.8234	0.0613382	0.0306691	−	0.999530i	$-0.490236\pi$
$421$	25.8234	0.0613382	0.0306691	+	0.999530i	$0.490236\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	47.6670i	0.112158i
$426$	0	0
$427$	−1376.88	−3.22455
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 312.843i	− 0.725853i	−0.931818	−	0.362927i	$-0.881778\pi$
$431$	− 312.843i	− 0.725853i	0.931818	−	0.362927i	$-0.118222\pi$
$432$	0	0
$433$	672.823	1.55386	0.776932	−	0.629584i	$-0.216775\pi$
$433$	672.823	1.55386	0.776932	+	0.629584i	$0.216775\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 146.079i	− 0.334277i
$438$	0	0
$439$	58.1026	0.132352	0.0661761	−	0.997808i	$-0.478920\pi$
$439$	58.1026	0.132352	0.0661761	+	0.997808i	$0.478920\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 634.431i	− 1.43212i	−0.698037	−	0.716062i	$-0.745943\pi$
$443$	− 634.431i	− 1.43212i	0.698037	−	0.716062i	$-0.254057\pi$
$444$	0	0
$445$	623.470	1.40106
$446$	0	0
$447$	0	0
$448$	0	0
$449$	44.7553i	0.0996777i	0.998757	+	0.0498388i	$0.0158708\pi$
$449$	44.7553i	0.0996777i	−0.998757	+	0.0498388i	$0.984129\pi$
$450$	0	0
$451$	922.087	2.04454
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1269.91i	2.79101i
$456$	0	0
$457$	−765.705	−1.67550	−0.837751	−	0.546052i	$-0.816130\pi$
$457$	−765.705	−1.67550	−0.837751	+	0.546052i	$0.816130\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 254.584i	− 0.552242i	−0.961123	−	0.276121i	$-0.910951\pi$
$461$	− 254.584i	− 0.552242i	0.961123	−	0.276121i	$-0.0890491\pi$
$462$	0	0
$463$	822.279	1.77598	0.887991	−	0.459862i	$-0.152101\pi$
$463$	822.279	1.77598	0.887991	+	0.459862i	$0.152101\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	505.971i	1.08345i	0.840556	+	0.541724i	$0.182228\pi$
$467$	505.971i	1.08345i	−0.840556	+	0.541724i	$0.817772\pi$
$468$	0	0
$469$	471.235	1.00476
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 43.5391i	− 0.0920488i
$474$	0	0
$475$	80.4710	0.169413
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 710.902i	− 1.48414i	−0.670324	−	0.742068i	$-0.733845\pi$
$479$	− 710.902i	− 1.48414i	0.670324	−	0.742068i	$-0.266155\pi$
$480$	0	0
$481$	1004.88	2.08915
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 331.534i	− 0.683575i
$486$	0	0
$487$	−861.882	−1.76978	−0.884889	−	0.465801i	$-0.845766\pi$
$487$	−861.882	−1.76978	−0.884889	+	0.465801i	$0.845766\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	516.235i	1.05139i	0.850672	+	0.525697i	$0.176195\pi$
$491$	516.235i	1.05139i	−0.850672	+	0.525697i	$0.823805\pi$
$492$	0	0
$493$	223.176	0.452689
$494$	0	0
$495$	0	0
$496$	0	0
$497$	118.118i	0.237661i
$498$	0	0
$499$	−193.383	−0.387540	−0.193770	−	0.981047i	$-0.562072\pi$
$499$	−193.383	−0.387540	−0.193770	+	0.981047i	$0.562072\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	627.265i	1.24705i	0.781804	+	0.623524i	$0.214300\pi$
$503$	627.265i	1.24705i	−0.781804	+	0.623524i	$0.785700\pi$
$504$	0	0
$505$	−9.00000	−0.0178218
$506$	0	0
$507$	0	0
$508$	0	0
$509$	524.543i	1.03054i	0.857029	+	0.515268i	$0.172308\pi$
$509$	524.543i	1.03054i	−0.857029	+	0.515268i	$0.827692\pi$
$510$	0	0
$511$	230.720	0.451507
$512$	0	0
$513$	0	0
$514$	0	0
$515$	395.990i	0.768912i
$516$	0	0
$517$	−1113.06	−2.15292
$518$	0	0
$519$	0	0
$520$	0	0
$521$	477.744i	0.916976i	0.888701	+	0.458488i	$0.151609\pi$
$521$	477.744i	0.916976i	−0.888701	+	0.458488i	$0.848391\pi$
$522$	0	0
$523$	481.411	0.920480	0.460240	−	0.887794i	$-0.347763\pi$
$523$	481.411	0.920480	0.460240	+	0.887794i	$0.347763\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	50.4020i	0.0956395i
$528$	0	0
$529$	263.823	0.498719
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 1193.47i	− 2.23916i
$534$	0	0
$535$	378.161	0.706844
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 1215.22i	− 2.25459i
$540$	0	0
$541$	−227.294	−0.420136	−0.210068	−	0.977687i	$-0.567369\pi$
$541$	−227.294	−0.420136	−0.210068	+	0.977687i	$0.567369\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 297.421i	− 0.545727i
$546$	0	0
$547$	665.765	1.21712	0.608560	−	0.793508i	$-0.291748\pi$
$547$	665.765	1.21712	0.608560	+	0.793508i	$0.291748\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 376.764i	− 0.683782i
$552$	0	0
$553$	460.087	0.831985
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 827.465i	− 1.48557i	−0.669527	−	0.742787i	$-0.733503\pi$
$557$	− 827.465i	− 1.48557i	0.669527	−	0.742787i	$-0.266497\pi$
$558$	0	0
$559$	−56.3532	−0.100811
$560$	0	0
$561$	0	0
$562$	0	0
$563$	883.108i	1.56857i	0.620398	+	0.784287i	$0.286971\pi$
$563$	883.108i	1.56857i	−0.620398	+	0.784287i	$0.713029\pi$
$564$	0	0
$565$	1217.97	2.15570
$566$	0	0
$567$	0	0
$568$	0	0
$569$	994.607i	1.74799i	0.485933	+	0.873996i	$0.338480\pi$
$569$	994.607i	1.74799i	−0.485933	+	0.873996i	$0.661520\pi$
$570$	0	0
$571$	992.705	1.73854	0.869269	−	0.494340i	$-0.164590\pi$
$571$	992.705	1.73854	0.869269	+	0.494340i	$0.164590\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 146.079i	− 0.254051i
$576$	0	0
$577$	251.588	0.436028	0.218014	−	0.975946i	$-0.430042\pi$
$577$	251.588	0.436028	0.218014	+	0.975946i	$0.430042\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 1247.28i	− 2.14678i
$582$	0	0
$583$	42.7502	0.0733280
$584$	0	0
$585$	0	0
$586$	0	0
$587$	793.254i	1.35137i	0.737191	+	0.675685i	$0.236152\pi$
$587$	793.254i	1.35137i	−0.737191	+	0.675685i	$0.763848\pi$
$588$	0	0
$589$	85.0883	0.144462
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 117.245i	− 0.197715i	−0.995102	−	0.0988573i	$-0.968481\pi$
$593$	− 117.245i	− 0.197715i	0.995102	−	0.0988573i	$-0.0315187\pi$
$594$	0	0
$595$	−355.706	−0.597825
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 875.470i	− 1.46155i	−0.682617	−	0.730776i	$-0.739158\pi$
$599$	− 875.470i	− 1.46155i	0.682617	−	0.730776i	$-0.260842\pi$
$600$	0	0
$601$	428.029	0.712195	0.356098	−	0.934449i	$-0.384107\pi$
$601$	428.029	0.712195	0.356098	+	0.934449i	$0.384107\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 546.843i	− 0.903872i
$606$	0	0
$607$	314.000	0.517298	0.258649	−	0.965971i	$-0.416723\pi$
$607$	314.000	0.517298	0.258649	+	0.965971i	$0.416723\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1440.65i	2.35785i
$612$	0	0
$613$	−482.118	−0.786489	−0.393244	−	0.919434i	$-0.628647\pi$
$613$	−482.118	−0.786489	−0.393244	+	0.919434i	$0.628647\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	884.382i	1.43336i	0.697403	+	0.716679i	$0.254339\pi$
$617$	884.382i	1.43336i	−0.697403	+	0.716679i	$0.745661\pi$
$618$	0	0
$619$	171.117	0.276441	0.138220	−	0.990401i	$-0.455862\pi$
$619$	171.117	0.276441	0.138220	+	0.990401i	$0.455862\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1228.59i	1.97205i
$624$	0	0
$625$	−768.793	−1.23007
$626$	0	0
$627$	0	0
$628$	0	0
$629$	281.470i	0.447488i
$630$	0	0
$631$	128.632	0.203855	0.101927	−	0.994792i	$-0.467499\pi$
$631$	128.632	0.203855	0.101927	+	0.994792i	$0.467499\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 1166.63i	− 1.83721i
$636$	0	0
$637$	−1572.88	−2.46920
$638$	0	0
$639$	0	0
$640$	0	0
$641$	12.8730i	0.0200827i	0.999950	+	0.0100414i	$0.00319632\pi$
$641$	12.8730i	0.0200827i	−0.999950	+	0.0100414i	$0.996804\pi$
$642$	0	0
$643$	−701.823	−1.09148	−0.545741	−	0.837954i	$-0.683752\pi$
$643$	−701.823	−1.09148	−0.545741	+	0.837954i	$0.683752\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	271.216i	0.419190i	0.977788	+	0.209595i	$0.0672146\pi$
$647$	271.216i	0.419190i	−0.977788	+	0.209595i	$0.932785\pi$
$648$	0	0
$649$	896.940	1.38203
$650$	0	0
$651$	0	0
$652$	0	0
$653$	392.897i	0.601679i	0.953675	+	0.300840i	$0.0972669\pi$
$653$	392.897i	0.601679i	−0.953675	+	0.300840i	$0.902733\pi$
$654$	0	0
$655$	−685.926	−1.04722
$656$	0	0
$657$	0	0
$658$	0	0
$659$	288.765i	0.438186i	0.975704	+	0.219093i	$0.0703098\pi$
$659$	288.765i	0.438186i	−0.975704	+	0.219093i	$0.929690\pi$
$660$	0	0
$661$	−119.029	−0.180075	−0.0900374	−	0.995938i	$-0.528699\pi$
$661$	−119.029	−0.180075	−0.0900374	+	0.995938i	$0.528699\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	600.500i	0.903007i
$666$	0	0
$667$	−683.939	−1.02540
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 1757.10i	− 2.61862i
$672$	0	0
$673$	−372.176	−0.553010	−0.276505	−	0.961012i	$-0.589176\pi$
$673$	−372.176	−0.553010	−0.276505	+	0.961012i	$0.589176\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	382.431i	0.564890i	0.959283	+	0.282445i	$0.0911455\pi$
$677$	382.431i	0.564890i	−0.959283	+	0.282445i	$0.908855\pi$
$678$	0	0
$679$	653.309	0.962163
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 920.745i	− 1.34809i	−0.738690	−	0.674045i	$-0.764555\pi$
$683$	− 920.745i	− 1.34809i	0.738690	−	0.674045i	$-0.235445\pi$
$684$	0	0
$685$	111.941	0.163418
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 55.3322i	− 0.0803080i
$690$	0	0
$691$	−806.940	−1.16779	−0.583893	−	0.811831i	$-0.698471\pi$
$691$	−806.940	−1.16779	−0.583893	+	0.811831i	$0.698471\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 1025.12i	− 1.47499i
$696$	0	0
$697$	334.294	0.479619
$698$	0	0
$699$	0	0
$700$	0	0
$701$	1139.95i	1.62618i	0.582136	+	0.813092i	$0.302217\pi$
$701$	1139.95i	1.62618i	−0.582136	+	0.813092i	$0.697783\pi$
$702$	0	0
$703$	475.176	0.675926
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 17.7351i	− 0.0250850i
$708$	0	0
$709$	324.824	0.458144	0.229072	−	0.973409i	$-0.426431\pi$
$709$	324.824	0.458144	0.229072	+	0.973409i	$0.426431\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 154.461i	− 0.216635i
$714$	0	0
$715$	−1620.59	−2.26656
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 251.334i	− 0.349560i	−0.984607	−	0.174780i	$-0.944079\pi$
$719$	− 251.334i	− 0.349560i	0.984607	−	0.174780i	$-0.0559215\pi$
$720$	0	0
$721$	−780.323	−1.08228
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 376.764i	− 0.519674i
$726$	0	0
$727$	−960.484	−1.32116	−0.660581	−	0.750755i	$-0.729690\pi$
$727$	−960.484	−1.32116	−0.660581	+	0.750755i	$0.729690\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 15.7847i	− 0.0215933i
$732$	0	0
$733$	6.79394	0.00926868	0.00463434	−	0.999989i	$-0.498525\pi$
$733$	6.79394	0.00926868	0.00463434	+	0.999989i	$0.498525\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	601.362i	0.815960i
$738$	0	0
$739$	1045.65	1.41495	0.707474	−	0.706739i	$-0.249835\pi$
$739$	1045.65	1.41495	0.707474	+	0.706739i	$0.249835\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	634.991i	0.854631i	0.904103	+	0.427315i	$0.140541\pi$
$743$	634.991i	0.854631i	−0.904103	+	0.427315i	$0.859459\pi$
$744$	0	0
$745$	708.352	0.950809
$746$	0	0
$747$	0	0
$748$	0	0
$749$	745.191i	0.994914i
$750$	0	0
$751$	−1072.07	−1.42753	−0.713763	−	0.700387i	$-0.753011\pi$
$751$	−1072.07	−1.42753	−0.713763	+	0.700387i	$0.753011\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 736.696i	− 0.975756i
$756$	0	0
$757$	−382.999	−0.505943	−0.252972	−	0.967474i	$-0.581408\pi$
$757$	−382.999	−0.505943	−0.252972	+	0.967474i	$0.581408\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 168.666i	− 0.221637i	−0.993841	−	0.110819i	$-0.964653\pi$
$761$	− 168.666i	− 0.221637i	0.993841	−	0.110819i	$-0.0353473\pi$
$762$	0	0
$763$	586.087	0.768136
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 1160.92i	− 1.51359i
$768$	0	0
$769$	−429.029	−0.557905	−0.278952	−	0.960305i	$-0.589987\pi$
$769$	−429.029	−0.557905	−0.278952	+	0.960305i	$0.589987\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 529.882i	− 0.685488i	−0.939429	−	0.342744i	$-0.888644\pi$
$773$	− 529.882i	− 0.685488i	0.939429	−	0.342744i	$-0.111356\pi$
$774$	0	0
$775$	85.0883	0.109791
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 564.353i	− 0.724459i
$780$	0	0
$781$	−150.735	−0.193003
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 150.510i	− 0.191732i
$786$	0	0
$787$	−335.411	−0.426190	−0.213095	−	0.977032i	$-0.568354\pi$
$787$	−335.411	−0.426190	−0.213095	+	0.977032i	$0.568354\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	2400.09i	3.03424i
$792$	0	0
$793$	−2274.23	−2.86789
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 1119.73i	− 1.40493i	−0.711718	−	0.702465i	$-0.752083\pi$
$797$	− 1119.73i	− 1.40493i	0.711718	−	0.702465i	$-0.247917\pi$
$798$	0	0
$799$	−403.529	−0.505043
$800$	0	0
$801$	0	0
$802$	0	0
$803$	294.431i	0.366664i
$804$	0	0
$805$	1090.09	1.35415
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1047.77i	1.29515i	0.762003	+	0.647574i	$0.224216\pi$
$809$	1047.77i	1.29515i	−0.762003	+	0.647574i	$0.775784\pi$
$810$	0	0
$811$	336.735	0.415210	0.207605	−	0.978213i	$-0.433433\pi$
$811$	336.735	0.415210	0.207605	+	0.978213i	$0.433433\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1.02944i	0.00126311i
$816$	0	0
$817$	−26.6476	−0.0326164
$818$	0	0
$819$	0	0
$820$	0	0
$821$	698.235i	0.850470i	0.905083	+	0.425235i	$0.139808\pi$
$821$	698.235i	0.850470i	−0.905083	+	0.425235i	$0.860192\pi$
$822$	0	0
$823$	285.014	0.346311	0.173156	−	0.984894i	$-0.444604\pi$
$823$	285.014	0.346311	0.173156	+	0.984894i	$0.444604\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 458.902i	− 0.554899i	−0.960740	−	0.277450i	$-0.910511\pi$
$827$	− 458.902i	− 0.554899i	0.960740	−	0.277450i	$-0.0894891\pi$
$828$	0	0
$829$	−654.471	−0.789470	−0.394735	−	0.918795i	$-0.629164\pi$
$829$	−654.471	−0.789470	−0.394735	+	0.918795i	$0.629164\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 440.569i	− 0.528894i
$834$	0	0
$835$	−746.382	−0.893870
$836$	0	0
$837$	0	0
$838$	0	0
$839$	909.558i	1.08410i	0.840347	+	0.542049i	$0.182351\pi$
$839$	909.558i	1.08410i	−0.840347	+	0.542049i	$0.817649\pi$
$840$	0	0
$841$	−923.000	−1.09750
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1112.54i	1.31662i
$846$	0	0
$847$	1077.59	1.27224
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 862.587i	− 1.01362i
$852$	0	0
$853$	−308.940	−0.362181	−0.181090	−	0.983466i	$-0.557963\pi$
$853$	−308.940	−0.362181	−0.181090	+	0.983466i	$0.557963\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1008.47i	1.17674i	0.808591	+	0.588372i	$0.200231\pi$
$857$	1008.47i	1.17674i	−0.808591	+	0.588372i	$0.799769\pi$
$858$	0	0
$859$	1537.65	1.79004	0.895020	−	0.446025i	$-0.147161\pi$
$859$	1537.65	1.79004	0.895020	+	0.446025i	$0.147161\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 442.627i	− 0.512894i	−0.966558	−	0.256447i	$-0.917448\pi$
$863$	− 442.627i	− 0.512894i	0.966558	−	0.256447i	$-0.0825519\pi$
$864$	0	0
$865$	−43.9706	−0.0508330
$866$	0	0
$867$	0	0
$868$	0	0
$869$	587.137i	0.675647i
$870$	0	0
$871$	778.352	0.893630
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 1073.03i	− 1.22632i
$876$	0	0
$877$	1527.59	1.74183	0.870917	−	0.491431i	$-0.163526\pi$
$877$	1527.59	1.74183	0.870917	+	0.491431i	$0.163526\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 608.079i	− 0.690215i	−0.938563	−	0.345107i	$-0.887843\pi$
$881$	− 608.079i	− 0.690215i	0.938563	−	0.345107i	$-0.112157\pi$
$882$	0	0
$883$	1056.26	1.19622	0.598111	−	0.801413i	$-0.295918\pi$
$883$	1056.26	1.19622	0.598111	+	0.801413i	$0.295918\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 626.049i	− 0.705805i	−0.935660	−	0.352902i	$-0.885195\pi$
$887$	− 626.049i	− 0.705805i	0.935660	−	0.352902i	$-0.114805\pi$
$888$	0	0
$889$	2298.91	2.58595
$890$	0	0
$891$	0	0
$892$	0	0
$893$	681.235i	0.762861i
$894$	0	0
$895$	−587.985	−0.656966
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 398.382i	− 0.443139i
$900$	0	0
$901$	15.4987	0.0172017
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 757.352i	− 0.836853i
$906$	0	0
$907$	1247.68	1.37561	0.687803	−	0.725897i	$-0.258575\pi$
$907$	1247.68	1.37561	0.687803	+	0.725897i	$0.258575\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 279.214i	− 0.306492i	−0.988188	−	0.153246i	$-0.951027\pi$
$911$	− 279.214i	− 0.306492i	0.988188	−	0.153246i	$-0.0489727\pi$
$912$	0	0
$913$	1591.70	1.74338
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 1351.66i	− 1.47400i
$918$	0	0
$919$	−857.219	−0.932774	−0.466387	−	0.884581i	$-0.654445\pi$
$919$	−857.219	−0.932774	−0.466387	+	0.884581i	$0.654445\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	195.098i	0.211374i
$924$	0	0
$925$	475.176	0.513704
$926$	0	0
$927$	0	0
$928$	0	0
$929$	452.569i	0.487157i	0.969881	+	0.243578i	$0.0783213\pi$
$929$	452.569i	0.487157i	−0.969881	+	0.243578i	$0.921679\pi$
$930$	0	0
$931$	−743.765	−0.798888
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 453.931i	− 0.485488i
$936$	0	0
$937$	−375.353	−0.400590	−0.200295	−	0.979736i	$-0.564190\pi$
$937$	−375.353	−0.400590	−0.200295	+	0.979736i	$0.564190\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 830.897i	− 0.882993i	−0.897263	−	0.441497i	$-0.854448\pi$
$941$	− 830.897i	− 0.882993i	0.897263	−	0.441497i	$-0.145552\pi$
$942$	0	0
$943$	−1024.47	−1.08640
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 116.823i	− 0.123362i	−0.998096	−	0.0616808i	$-0.980354\pi$
$947$	− 116.823i	− 0.123362i	0.998096	−	0.0616808i	$-0.0196461\pi$
$948$	0	0
$949$	381.087	0.401566
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 762.177i	− 0.799766i	−0.916566	−	0.399883i	$-0.869051\pi$
$953$	− 762.177i	− 0.799766i	0.916566	−	0.399883i	$-0.130949\pi$
$954$	0	0
$955$	99.9411	0.104650
$956$	0	0
$957$	0	0
$958$	0	0
$959$	220.587i	0.230018i
$960$	0	0
$961$	−871.029	−0.906378
$962$	0	0
$963$	0	0
$964$	0	0
$965$	488.730i	0.506456i
$966$	0	0
$967$	372.072	0.384770	0.192385	−	0.981320i	$-0.438378\pi$
$967$	372.072	0.384770	0.192385	+	0.981320i	$0.438378\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1442.09i	1.48516i	0.669760	+	0.742578i	$0.266397\pi$
$971$	1442.09i	1.48516i	−0.669760	+	0.742578i	$0.733603\pi$
$972$	0	0
$973$	2020.06	2.07611
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1493.09i	1.52824i	0.645076	+	0.764118i	$0.276825\pi$
$977$	1493.09i	1.52824i	−0.645076	+	0.764118i	$0.723175\pi$
$978$	0	0
$979$	−1567.85	−1.60148
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1263.59i	1.28544i	0.766101	+	0.642720i	$0.222194\pi$
$983$	1263.59i	1.28544i	−0.766101	+	0.642720i	$0.777806\pi$
$984$	0	0
$985$	−1784.44	−1.81161
$986$	0	0
$987$	0	0
$988$	0	0
$989$	48.3734i	0.0489115i
$990$	0	0
$991$	543.132	0.548065	0.274032	−	0.961720i	$-0.411642\pi$
$991$	543.132	0.548065	0.274032	+	0.961720i	$0.411642\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	1481.02i	1.48846i
$996$	0	0
$997$	1236.21	1.23992	0.619962	−	0.784631i	$-0.287148\pi$
$997$	1236.21	1.23992	0.619962	+	0.784631i	$0.287148\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	216.3.e.c.161.4	yes	4
3.2	odd	2	inner	216.3.e.c.161.1	✓	4
4.3	odd	2		432.3.e.h.161.4		4
8.3	odd	2		1728.3.e.s.1025.1		4
8.5	even	2		1728.3.e.p.1025.1		4
9.2	odd	6		648.3.m.f.377.4		8
9.4	even	3		648.3.m.f.593.4		8
9.5	odd	6		648.3.m.f.593.1		8
9.7	even	3		648.3.m.f.377.1		8
12.11	even	2		432.3.e.h.161.1		4
24.5	odd	2		1728.3.e.p.1025.4		4
24.11	even	2		1728.3.e.s.1025.4		4
36.7	odd	6		1296.3.q.l.1025.1		8
36.11	even	6		1296.3.q.l.1025.4		8
36.23	even	6		1296.3.q.l.593.1		8
36.31	odd	6		1296.3.q.l.593.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.3.e.c.161.1	✓	4	3.2	odd	2	inner
216.3.e.c.161.4	yes	4	1.1	even	1	trivial
432.3.e.h.161.1		4	12.11	even	2
432.3.e.h.161.4		4	4.3	odd	2
648.3.m.f.377.1		8	9.7	even	3
648.3.m.f.377.4		8	9.2	odd	6
648.3.m.f.593.1		8	9.5	odd	6
648.3.m.f.593.4		8	9.4	even	3
1296.3.q.l.593.1		8	36.23	even	6
1296.3.q.l.593.4		8	36.31	odd	6
1296.3.q.l.1025.1		8	36.7	odd	6
1296.3.q.l.1025.4		8	36.11	even	6
1728.3.e.p.1025.1		4	8.5	even	2
1728.3.e.p.1025.4		4	24.5	odd	2
1728.3.e.s.1025.1		4	8.3	odd	2
1728.3.e.s.1025.4		4	24.11	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:	\( N \)	\(=\)	\( 216 = 2^{3} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	216.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+5.82843i q^{5} -11.4853 q^{7} +O(q^{10})\)	\(q+5.82843i q^{5} -11.4853 q^{7} -14.6569i q^{11} -18.9706 q^{13} -5.31371i q^{17} -8.97056 q^{19} +16.2843i q^{23} -8.97056 q^{25} +42.0000i q^{29} -9.48528 q^{31} -66.9411i q^{35} -52.9706 q^{37} +62.9117i q^{41} +2.97056 q^{43} -75.9411i q^{47} +82.9117 q^{49} +2.91674i q^{53} +85.4264 q^{55} +61.1960i q^{59} +119.882 q^{61} -110.569i q^{65} -41.0294 q^{67} -10.2843i q^{71} -20.0883 q^{73} +168.338i q^{77} -40.0589 q^{79} +108.598i q^{83} +30.9706 q^{85} -106.971i q^{89} +217.882 q^{91} -52.2843i q^{95} -56.8823 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{7}+O(q^{10}) \) 4 * q - 12 * q^7	\( 4 q - 12 q^{7} - 8 q^{13} + 32 q^{19} + 32 q^{25} - 4 q^{31} - 144 q^{37} - 56 q^{43} + 128 q^{49} + 172 q^{55} + 208 q^{61} - 232 q^{67} - 284 q^{73} - 296 q^{79} + 56 q^{85} + 600 q^{91} + 44 q^{97}+O(q^{100}) \) 4 * q - 12 * q^7 - 8 * q^13 + 32 * q^19 + 32 * q^25 - 4 * q^31 - 144 * q^37 - 56 * q^43 + 128 * q^49 + 172 * q^55 + 208 * q^61 - 232 * q^67 - 284 * q^73 - 296 * q^79 + 56 * q^85 + 600 * q^91 + 44 * q^97

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