LMFDB - Embedded newform 882.3.c.c.685.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [882,3,Mod(685,882)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(882, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("882.685");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 882 = 2 \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	882.c (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:	$24.0327593166$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 2x^{2} + 4 $ x^4 + 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 126)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			685.3
Root			$0.707107 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	882.685
Dual form			882.3.c.c.685.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{2} +2.00000 q^{4} -4.89898i q^{5} +2.82843 q^{8} +O(q^{10})$	$q+1.41421 q^{2} +2.00000 q^{4} -4.89898i q^{5} +2.82843 q^{8} -6.92820i q^{10} -16.9706 q^{11} +1.73205i q^{13} +4.00000 q^{16} -4.89898i q^{17} -29.4449i q^{19} -9.79796i q^{20} -24.0000 q^{22} -8.48528 q^{23} +1.00000 q^{25} +2.44949i q^{26} -33.9411 q^{29} -12.1244i q^{31} +5.65685 q^{32} -6.92820i q^{34} -47.0000 q^{37} -41.6413i q^{38} -13.8564i q^{40} -68.5857i q^{41} +31.0000 q^{43} -33.9411 q^{44} -12.0000 q^{46} +83.2827i q^{47} +1.41421 q^{50} +3.46410i q^{52} -76.3675 q^{53} +83.1384i q^{55} -48.0000 q^{58} -83.2827i q^{59} +83.1384i q^{61} -17.1464i q^{62} +8.00000 q^{64} +8.48528 q^{65} -31.0000 q^{67} -9.79796i q^{68} +59.3970 q^{71} -81.4064i q^{73} -66.4680 q^{74} -58.8897i q^{76} +41.0000 q^{79} -19.5959i q^{80} -96.9948i q^{82} +4.89898i q^{83} -24.0000 q^{85} +43.8406 q^{86} -48.0000 q^{88} -58.7878i q^{89} -16.9706 q^{92} +117.779i q^{94} -144.250 q^{95} -41.5692i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{4}+O(q^{10}) $ 4 * q + 8 * q^4	$ 4 q + 8 q^{4} + 16 q^{16} - 96 q^{22} + 4 q^{25} - 188 q^{37} + 124 q^{43} - 48 q^{46} - 192 q^{58} + 32 q^{64} - 124 q^{67} + 164 q^{79} - 96 q^{85} - 192 q^{88}+O(q^{100}) $ 4 * q + 8 * q^4 + 16 * q^16 - 96 * q^22 + 4 * q^25 - 188 * q^37 + 124 * q^43 - 48 * q^46 - 192 * q^58 + 32 * q^64 - 124 * q^67 + 164 * q^79 - 96 * q^85 - 192 * q^88

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/882\mathbb{Z}\right)^\times$.

$n$	$199$	$785$
$\chi(n)$	$-1$	$1$

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 $

$2$	1.41421	0.707107
$2$	1.41421	0.707107
$3$	0	0
$3$	0	0
$4$	2.00000	0.500000
$5$	− 4.89898i	− 0.979796i	−0.871780	−	0.489898i	$-0.837034\pi$
$5$	− 4.89898i	− 0.979796i	0.871780	−	0.489898i	$-0.162966\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	2.82843	0.353553
$9$	0	0
$10$	− 6.92820i	− 0.692820i
$11$	−16.9706	−1.54278	−0.771389	−	0.636364i	$-0.780438\pi$
$11$	−16.9706	−1.54278	−0.771389	+	0.636364i	$0.780438\pi$
$12$	0	0
$13$	1.73205i	0.133235i	0.997779	+	0.0666173i	$0.0212207\pi$
$13$	1.73205i	0.133235i	−0.997779	+	0.0666173i	$0.978779\pi$
$14$	0	0
$15$	0	0
$16$	4.00000	0.250000
$17$	− 4.89898i	− 0.288175i	−0.989565	−	0.144088i	$-0.953975\pi$
$17$	− 4.89898i	− 0.288175i	0.989565	−	0.144088i	$-0.0460247\pi$
$18$	0	0
$19$	− 29.4449i	− 1.54973i	−0.632127	−	0.774865i	$-0.717818\pi$
$19$	− 29.4449i	− 1.54973i	0.632127	−	0.774865i	$-0.282182\pi$
$20$	− 9.79796i	− 0.489898i
$21$	0	0
$22$	−24.0000	−1.09091
$23$	−8.48528	−0.368925	−0.184463	−	0.982840i	$-0.559054\pi$
$23$	−8.48528	−0.368925	−0.184463	+	0.982840i	$0.559054\pi$
$24$	0	0
$25$	1.00000	0.0400000
$26$	2.44949i	0.0942111i
$27$	0	0
$28$	0	0
$29$	−33.9411	−1.17038	−0.585192	−	0.810895i	$-0.698981\pi$
$29$	−33.9411	−1.17038	−0.585192	+	0.810895i	$0.698981\pi$
$30$	0	0
$31$	− 12.1244i	− 0.391108i	−0.980693	−	0.195554i	$-0.937349\pi$
$31$	− 12.1244i	− 0.391108i	0.980693	−	0.195554i	$-0.0626505\pi$
$32$	5.65685	0.176777
$33$	0	0
$34$	− 6.92820i	− 0.203771i
$35$	0	0
$36$	0	0
$37$	−47.0000	−1.27027	−0.635135	−	0.772401i	$-0.719056\pi$
$37$	−47.0000	−1.27027	−0.635135	+	0.772401i	$0.719056\pi$
$38$	− 41.6413i	− 1.09582i
$39$	0	0
$40$	− 13.8564i	− 0.346410i
$41$	− 68.5857i	− 1.67282i	−0.548103	−	0.836411i	$-0.684650\pi$
$41$	− 68.5857i	− 1.67282i	0.548103	−	0.836411i	$-0.315350\pi$
$42$	0	0
$43$	31.0000	0.720930	0.360465	−	0.932773i	$-0.382618\pi$
$43$	31.0000	0.720930	0.360465	+	0.932773i	$0.382618\pi$
$44$	−33.9411	−0.771389
$45$	0	0
$46$	−12.0000	−0.260870
$47$	83.2827i	1.77197i	0.463713	+	0.885986i	$0.346517\pi$
$47$	83.2827i	1.77197i	−0.463713	+	0.885986i	$0.653483\pi$
$48$	0	0
$49$	0	0
$50$	1.41421	0.0282843
$51$	0	0
$52$	3.46410i	0.0666173i
$53$	−76.3675	−1.44090	−0.720448	−	0.693509i	$-0.756064\pi$
$53$	−76.3675	−1.44090	−0.720448	+	0.693509i	$0.756064\pi$
$54$	0	0
$55$	83.1384i	1.51161i
$56$	0	0
$57$	0	0
$58$	−48.0000	−0.827586
$59$	− 83.2827i	− 1.41157i	−0.708426	−	0.705785i	$-0.750594\pi$
$59$	− 83.2827i	− 1.41157i	0.708426	−	0.705785i	$-0.249406\pi$
$60$	0	0
$61$	83.1384i	1.36293i	0.731853	+	0.681463i	$0.238656\pi$
$61$	83.1384i	1.36293i	−0.731853	+	0.681463i	$0.761344\pi$
$62$	− 17.1464i	− 0.276555i
$63$	0	0
$64$	8.00000	0.125000
$65$	8.48528	0.130543
$66$	0	0
$67$	−31.0000	−0.462687	−0.231343	−	0.972872i	$-0.574312\pi$
$67$	−31.0000	−0.462687	−0.231343	+	0.972872i	$0.574312\pi$
$68$	− 9.79796i	− 0.144088i
$69$	0	0
$70$	0	0
$71$	59.3970	0.836577	0.418289	−	0.908314i	$-0.362630\pi$
$71$	59.3970	0.836577	0.418289	+	0.908314i	$0.362630\pi$
$72$	0	0
$73$	− 81.4064i	− 1.11516i	−0.830125	−	0.557578i	$-0.811731\pi$
$73$	− 81.4064i	− 1.11516i	0.830125	−	0.557578i	$-0.188269\pi$
$74$	−66.4680	−0.898217
$75$	0	0
$76$	− 58.8897i	− 0.774865i
$77$	0	0
$78$	0	0
$79$	41.0000	0.518987	0.259494	−	0.965745i	$-0.416444\pi$
$79$	41.0000	0.518987	0.259494	+	0.965745i	$0.416444\pi$
$80$	− 19.5959i	− 0.244949i
$81$	0	0
$82$	− 96.9948i	− 1.18286i
$83$	4.89898i	0.0590238i	0.999564	+	0.0295119i	$0.00939530\pi$
$83$	4.89898i	0.0590238i	−0.999564	+	0.0295119i	$0.990605\pi$
$84$	0	0
$85$	−24.0000	−0.282353
$86$	43.8406	0.509775
$87$	0	0
$88$	−48.0000	−0.545455
$89$	− 58.7878i	− 0.660537i	−0.943887	−	0.330268i	$-0.892861\pi$
$89$	− 58.7878i	− 0.660537i	0.943887	−	0.330268i	$-0.107139\pi$
$90$	0	0
$91$	0	0
$92$	−16.9706	−0.184463
$93$	0	0
$94$	117.779i	1.25297i
$95$	−144.250	−1.51842
$96$	0	0
$97$	− 41.5692i	− 0.428549i	−0.976774	−	0.214274i	$-0.931261\pi$
$97$	− 41.5692i	− 0.428549i	0.976774	−	0.214274i	$-0.0687387\pi$
$98$	0	0
$99$	0	0
$100$	2.00000	0.0200000
$101$	− 176.363i	− 1.74617i	−0.487567	−	0.873085i	$-0.662116\pi$
$101$	− 176.363i	− 1.74617i	0.487567	−	0.873085i	$-0.337884\pi$
$102$	0	0
$103$	29.4449i	0.285872i	0.989732	+	0.142936i	$0.0456544\pi$
$103$	29.4449i	0.285872i	−0.989732	+	0.142936i	$0.954346\pi$
$104$	4.89898i	0.0471056i
$105$	0	0
$106$	−108.000	−1.01887
$107$	−144.250	−1.34813	−0.674064	−	0.738673i	$-0.735453\pi$
$107$	−144.250	−1.34813	−0.674064	+	0.738673i	$0.735453\pi$
$108$	0	0
$109$	169.000	1.55046	0.775229	−	0.631680i	$-0.217634\pi$
$109$	169.000	1.55046	0.775229	+	0.631680i	$0.217634\pi$
$110$	117.576i	1.06887i
$111$	0	0
$112$	0	0
$113$	−59.3970	−0.525637	−0.262818	−	0.964845i	$-0.584652\pi$
$113$	−59.3970	−0.525637	−0.262818	+	0.964845i	$0.584652\pi$
$114$	0	0
$115$	41.5692i	0.361471i
$116$	−67.8823	−0.585192
$117$	0	0
$118$	− 117.779i	− 0.998131i
$119$	0	0
$120$	0	0
$121$	167.000	1.38017
$122$	117.576i	0.963734i
$123$	0	0
$124$	− 24.2487i	− 0.195554i
$125$	− 127.373i	− 1.01899i
$126$	0	0
$127$	209.000	1.64567	0.822835	−	0.568281i	$-0.192391\pi$
$127$	209.000	1.64567	0.822835	+	0.568281i	$0.192391\pi$
$128$	11.3137	0.0883883
$129$	0	0
$130$	12.0000	0.0923077
$131$	58.7878i	0.448761i	0.974502	+	0.224381i	$0.0720359\pi$
$131$	58.7878i	0.448761i	−0.974502	+	0.224381i	$0.927964\pi$
$132$	0	0
$133$	0	0
$134$	−43.8406	−0.327169
$135$	0	0
$136$	− 13.8564i	− 0.101885i
$137$	152.735	1.11485	0.557427	−	0.830226i	$-0.311789\pi$
$137$	152.735	1.11485	0.557427	+	0.830226i	$0.311789\pi$
$138$	0	0
$139$	195.722i	1.40807i	0.710165	+	0.704035i	$0.248620\pi$
$139$	195.722i	1.40807i	−0.710165	+	0.704035i	$0.751380\pi$
$140$	0	0
$141$	0	0
$142$	84.0000	0.591549
$143$	− 29.3939i	− 0.205552i
$144$	0	0
$145$	166.277i	1.14674i
$146$	− 115.126i	− 0.788534i
$147$	0	0
$148$	−94.0000	−0.635135
$149$	50.9117	0.341689	0.170845	−	0.985298i	$-0.445350\pi$
$149$	50.9117	0.341689	0.170845	+	0.985298i	$0.445350\pi$
$150$	0	0
$151$	10.0000	0.0662252	0.0331126	−	0.999452i	$-0.489458\pi$
$151$	10.0000	0.0662252	0.0331126	+	0.999452i	$0.489458\pi$
$152$	− 83.2827i	− 0.547912i
$153$	0	0
$154$	0	0
$155$	−59.3970	−0.383206
$156$	0	0
$157$	41.5692i	0.264772i	0.991198	+	0.132386i	$0.0422639\pi$
$157$	41.5692i	0.264772i	−0.991198	+	0.132386i	$0.957736\pi$
$158$	57.9828	0.366979
$159$	0	0
$160$	− 27.7128i	− 0.173205i
$161$	0	0
$162$	0	0
$163$	86.0000	0.527607	0.263804	−	0.964576i	$-0.415023\pi$
$163$	86.0000	0.527607	0.263804	+	0.964576i	$0.415023\pi$
$164$	− 137.171i	− 0.836411i
$165$	0	0
$166$	6.92820i	0.0417362i
$167$	− 181.262i	− 1.08540i	−0.839926	−	0.542701i	$-0.817402\pi$
$167$	− 181.262i	− 1.08540i	0.839926	−	0.542701i	$-0.182598\pi$
$168$	0	0
$169$	166.000	0.982249
$170$	−33.9411	−0.199654
$171$	0	0
$172$	62.0000	0.360465
$173$	44.0908i	0.254860i	0.991848	+	0.127430i	$0.0406728\pi$
$173$	44.0908i	0.254860i	−0.991848	+	0.127430i	$0.959327\pi$
$174$	0	0
$175$	0	0
$176$	−67.8823	−0.385695
$177$	0	0
$178$	− 83.1384i	− 0.467070i
$179$	−8.48528	−0.0474038	−0.0237019	−	0.999719i	$-0.507545\pi$
$179$	−8.48528	−0.0474038	−0.0237019	+	0.999719i	$0.507545\pi$
$180$	0	0
$181$	− 43.3013i	− 0.239234i	−0.992820	−	0.119617i	$-0.961833\pi$
$181$	− 43.3013i	− 0.239234i	0.992820	−	0.119617i	$-0.0381666\pi$
$182$	0	0
$183$	0	0
$184$	−24.0000	−0.130435
$185$	230.252i	1.24461i
$186$	0	0
$187$	83.1384i	0.444591i
$188$	166.565i	0.885986i
$189$	0	0
$190$	−204.000	−1.07368
$191$	76.3675	0.399830	0.199915	−	0.979813i	$-0.435933\pi$
$191$	76.3675	0.399830	0.199915	+	0.979813i	$0.435933\pi$
$192$	0	0
$193$	−287.000	−1.48705	−0.743523	−	0.668710i	$-0.766847\pi$
$193$	−287.000	−1.48705	−0.743523	+	0.668710i	$0.766847\pi$
$194$	− 58.7878i	− 0.303030i
$195$	0	0
$196$	0	0
$197$	127.279	0.646087	0.323044	−	0.946384i	$-0.395294\pi$
$197$	127.279	0.646087	0.323044	+	0.946384i	$0.395294\pi$
$198$	0	0
$199$	207.846i	1.04445i	0.852807	+	0.522226i	$0.174898\pi$
$199$	207.846i	1.04445i	−0.852807	+	0.522226i	$0.825102\pi$
$200$	2.82843	0.0141421
$201$	0	0
$202$	− 249.415i	− 1.23473i
$203$	0	0
$204$	0	0
$205$	−336.000	−1.63902
$206$	41.6413i	0.202142i
$207$	0	0
$208$	6.92820i	0.0333087i
$209$	499.696i	2.39089i
$210$	0	0
$211$	82.0000	0.388626	0.194313	−	0.980940i	$-0.437752\pi$
$211$	82.0000	0.388626	0.194313	+	0.980940i	$0.437752\pi$
$212$	−152.735	−0.720448
$213$	0	0
$214$	−204.000	−0.953271
$215$	− 151.868i	− 0.706364i
$216$	0	0
$217$	0	0
$218$	239.002	1.09634
$219$	0	0
$220$	166.277i	0.755804i
$221$	8.48528	0.0383949
$222$	0	0
$223$	41.5692i	0.186409i	0.995647	+	0.0932045i	$0.0297110\pi$
$223$	41.5692i	0.186409i	−0.995647	+	0.0932045i	$0.970289\pi$
$224$	0	0
$225$	0	0
$226$	−84.0000	−0.371681
$227$	382.120i	1.68335i	0.539985	+	0.841675i	$0.318430\pi$
$227$	382.120i	1.68335i	−0.539985	+	0.841675i	$0.681570\pi$
$228$	0	0
$229$	81.4064i	0.355486i	0.984077	+	0.177743i	$0.0568796\pi$
$229$	81.4064i	0.355486i	−0.984077	+	0.177743i	$0.943120\pi$
$230$	58.7878i	0.255599i
$231$	0	0
$232$	−96.0000	−0.413793
$233$	229.103	0.983273	0.491636	−	0.870801i	$-0.336399\pi$
$233$	229.103	0.983273	0.491636	+	0.870801i	$0.336399\pi$
$234$	0	0
$235$	408.000	1.73617
$236$	− 166.565i	− 0.705785i
$237$	0	0
$238$	0	0
$239$	−67.8823	−0.284026	−0.142013	−	0.989865i	$-0.545358\pi$
$239$	−67.8823	−0.284026	−0.142013	+	0.989865i	$0.545358\pi$
$240$	0	0
$241$	− 457.261i	− 1.89735i	−0.316252	−	0.948675i	$-0.602425\pi$
$241$	− 457.261i	− 1.89735i	0.316252	−	0.948675i	$-0.397575\pi$
$242$	236.174	0.975924
$243$	0	0
$244$	166.277i	0.681463i
$245$	0	0
$246$	0	0
$247$	51.0000	0.206478
$248$	− 34.2929i	− 0.138278i
$249$	0	0
$250$	− 180.133i	− 0.720533i
$251$	347.828i	1.38577i	0.721050	+	0.692884i	$0.243660\pi$
$251$	347.828i	1.38577i	−0.721050	+	0.692884i	$0.756340\pi$
$252$	0	0
$253$	144.000	0.569170
$254$	295.571	1.16366
$255$	0	0
$256$	16.0000	0.0625000
$257$	− 161.666i	− 0.629052i	−0.949249	−	0.314526i	$-0.898154\pi$
$257$	− 161.666i	− 0.629052i	0.949249	−	0.314526i	$-0.101846\pi$
$258$	0	0
$259$	0	0
$260$	16.9706	0.0652714
$261$	0	0
$262$	83.1384i	0.317322i
$263$	254.558	0.967903	0.483951	−	0.875095i	$-0.339201\pi$
$263$	254.558	0.967903	0.483951	+	0.875095i	$0.339201\pi$
$264$	0	0
$265$	374.123i	1.41178i
$266$	0	0
$267$	0	0
$268$	−62.0000	−0.231343
$269$	19.5959i	0.0728473i	0.999336	+	0.0364236i	$0.0115966\pi$
$269$	19.5959i	0.0728473i	−0.999336	+	0.0364236i	$0.988403\pi$
$270$	0	0
$271$	− 41.5692i	− 0.153392i	−0.997055	−	0.0766960i	$-0.975563\pi$
$271$	− 41.5692i	− 0.153392i	0.997055	−	0.0766960i	$-0.0244371\pi$
$272$	− 19.5959i	− 0.0720438i
$273$	0	0
$274$	216.000	0.788321
$275$	−16.9706	−0.0617111
$276$	0	0
$277$	−337.000	−1.21661	−0.608303	−	0.793705i	$-0.708150\pi$
$277$	−337.000	−1.21661	−0.608303	+	0.793705i	$0.708150\pi$
$278$	276.792i	0.995656i
$279$	0	0
$280$	0	0
$281$	246.073	0.875705	0.437853	−	0.899047i	$-0.355739\pi$
$281$	246.073	0.875705	0.437853	+	0.899047i	$0.355739\pi$
$282$	0	0
$283$	− 195.722i	− 0.691596i	−0.938309	−	0.345798i	$-0.887608\pi$
$283$	− 195.722i	− 0.691596i	0.938309	−	0.345798i	$-0.112392\pi$
$284$	118.794	0.418289
$285$	0	0
$286$	− 41.5692i	− 0.145347i
$287$	0	0
$288$	0	0
$289$	265.000	0.916955
$290$	235.151i	0.810866i
$291$	0	0
$292$	− 162.813i	− 0.557578i
$293$	− 97.9796i	− 0.334401i	−0.985923	−	0.167201i	$-0.946527\pi$
$293$	− 97.9796i	− 0.334401i	0.985923	−	0.167201i	$-0.0534728\pi$
$294$	0	0
$295$	−408.000	−1.38305
$296$	−132.936	−0.449108
$297$	0	0
$298$	72.0000	0.241611
$299$	− 14.6969i	− 0.0491536i
$300$	0	0
$301$	0	0
$302$	14.1421	0.0468283
$303$	0	0
$304$	− 117.779i	− 0.387432i
$305$	407.294	1.33539
$306$	0	0
$307$	71.0141i	0.231316i	0.993289	+	0.115658i	$0.0368977\pi$
$307$	71.0141i	0.231316i	−0.993289	+	0.115658i	$0.963102\pi$
$308$	0	0
$309$	0	0
$310$	−84.0000	−0.270968
$311$	215.555i	0.693103i	0.938031	+	0.346552i	$0.112647\pi$
$311$	215.555i	0.693103i	−0.938031	+	0.346552i	$0.887353\pi$
$312$	0	0
$313$	− 292.717i	− 0.935197i	−0.883941	−	0.467598i	$-0.845119\pi$
$313$	− 292.717i	− 0.935197i	0.883941	−	0.467598i	$-0.154881\pi$
$314$	58.7878i	0.187222i
$315$	0	0
$316$	82.0000	0.259494
$317$	−237.588	−0.749489	−0.374744	−	0.927128i	$-0.622269\pi$
$317$	−237.588	−0.749489	−0.374744	+	0.927128i	$0.622269\pi$
$318$	0	0
$319$	576.000	1.80564
$320$	− 39.1918i	− 0.122474i
$321$	0	0
$322$	0	0
$323$	−144.250	−0.446594
$324$	0	0
$325$	1.73205i	0.00532939i
$326$	121.622	0.373075
$327$	0	0
$328$	− 193.990i	− 0.591432i
$329$	0	0
$330$	0	0
$331$	−185.000	−0.558912	−0.279456	−	0.960158i	$-0.590154\pi$
$331$	−185.000	−0.558912	−0.279456	+	0.960158i	$0.590154\pi$
$332$	9.79796i	0.0295119i
$333$	0	0
$334$	− 256.344i	− 0.767496i
$335$	151.868i	0.453338i
$336$	0	0
$337$	−359.000	−1.06528	−0.532641	−	0.846341i	$-0.678800\pi$
$337$	−359.000	−1.06528	−0.532641	+	0.846341i	$0.678800\pi$
$338$	234.759	0.694555
$339$	0	0
$340$	−48.0000	−0.141176
$341$	205.757i	0.603393i
$342$	0	0
$343$	0	0
$344$	87.6812	0.254887
$345$	0	0
$346$	62.3538i	0.180213i
$347$	466.690	1.34493	0.672465	−	0.740129i	$-0.265236\pi$
$347$	466.690	1.34493	0.672465	+	0.740129i	$0.265236\pi$
$348$	0	0
$349$	− 581.969i	− 1.66753i	−0.552117	−	0.833767i	$-0.686180\pi$
$349$	− 581.969i	− 1.66753i	0.552117	−	0.833767i	$-0.313820\pi$
$350$	0	0
$351$	0	0
$352$	−96.0000	−0.272727
$353$	289.040i	0.818810i	0.912353	+	0.409405i	$0.134264\pi$
$353$	289.040i	0.818810i	−0.912353	+	0.409405i	$0.865736\pi$
$354$	0	0
$355$	− 290.985i	− 0.819675i
$356$	− 117.576i	− 0.330268i
$357$	0	0
$358$	−12.0000	−0.0335196
$359$	339.411	0.945435	0.472718	−	0.881214i	$-0.343273\pi$
$359$	339.411	0.945435	0.472718	+	0.881214i	$0.343273\pi$
$360$	0	0
$361$	−506.000	−1.40166
$362$	− 61.2372i	− 0.169164i
$363$	0	0
$364$	0	0
$365$	−398.808	−1.09263
$366$	0	0
$367$	154.153i	0.420034i	0.977698	+	0.210017i	$0.0673520\pi$
$367$	154.153i	0.420034i	−0.977698	+	0.210017i	$0.932648\pi$
$368$	−33.9411	−0.0922313
$369$	0	0
$370$	325.626i	0.880069i
$371$	0	0
$372$	0	0
$373$	−289.000	−0.774799	−0.387399	−	0.921912i	$-0.626627\pi$
$373$	−289.000	−0.774799	−0.387399	+	0.921912i	$0.626627\pi$
$374$	117.576i	0.314373i
$375$	0	0
$376$	235.559i	0.626486i
$377$	− 58.7878i	− 0.155936i
$378$	0	0
$379$	7.00000	0.0184697	0.00923483	−	0.999957i	$-0.497060\pi$
$379$	7.00000	0.0184697	0.00923483	+	0.999957i	$0.497060\pi$
$380$	−288.500	−0.759209
$381$	0	0
$382$	108.000	0.282723
$383$	− 494.797i	− 1.29190i	−0.763381	−	0.645949i	$-0.776462\pi$
$383$	− 494.797i	− 1.29190i	0.763381	−	0.645949i	$-0.223538\pi$
$384$	0	0
$385$	0	0
$386$	−405.879	−1.05150
$387$	0	0
$388$	− 83.1384i	− 0.214274i
$389$	−229.103	−0.588953	−0.294476	−	0.955659i	$-0.595145\pi$
$389$	−229.103	−0.588953	−0.294476	+	0.955659i	$0.595145\pi$
$390$	0	0
$391$	41.5692i	0.106315i
$392$	0	0
$393$	0	0
$394$	180.000	0.456853
$395$	− 200.858i	− 0.508502i
$396$	0	0
$397$	− 81.4064i	− 0.205054i	−0.994730	−	0.102527i	$-0.967307\pi$
$397$	− 81.4064i	− 0.205054i	0.994730	−	0.102527i	$-0.0326928\pi$
$398$	293.939i	0.738540i
$399$	0	0
$400$	4.00000	0.0100000
$401$	−93.3381	−0.232763	−0.116382	−	0.993205i	$-0.537130\pi$
$401$	−93.3381	−0.232763	−0.116382	+	0.993205i	$0.537130\pi$
$402$	0	0
$403$	21.0000	0.0521092
$404$	− 352.727i	− 0.873085i
$405$	0	0
$406$	0	0
$407$	797.616	1.95975
$408$	0	0
$409$	− 417.424i	− 1.02060i	−0.859997	−	0.510299i	$-0.829535\pi$
$409$	− 417.424i	− 1.02060i	0.859997	−	0.510299i	$-0.170465\pi$
$410$	−475.176	−1.15897
$411$	0	0
$412$	58.8897i	0.142936i
$413$	0	0
$414$	0	0
$415$	24.0000	0.0578313
$416$	9.79796i	0.0235528i
$417$	0	0
$418$	706.677i	1.69061i
$419$	− 19.5959i	− 0.0467683i	−0.999727	−	0.0233842i	$-0.992556\pi$
$419$	− 19.5959i	− 0.0467683i	0.999727	−	0.0233842i	$-0.00744408\pi$
$420$	0	0
$421$	407.000	0.966746	0.483373	−	0.875415i	$-0.339412\pi$
$421$	407.000	0.966746	0.483373	+	0.875415i	$0.339412\pi$
$422$	115.966	0.274800
$423$	0	0
$424$	−216.000	−0.509434
$425$	− 4.89898i	− 0.0115270i
$426$	0	0
$427$	0	0
$428$	−288.500	−0.674064
$429$	0	0
$430$	− 214.774i	− 0.499475i
$431$	−161.220	−0.374061	−0.187031	−	0.982354i	$-0.559886\pi$
$431$	−161.220	−0.374061	−0.187031	+	0.982354i	$0.559886\pi$
$432$	0	0
$433$	168.009i	0.388011i	0.981000	+	0.194006i	$0.0621480\pi$
$433$	168.009i	0.388011i	−0.981000	+	0.194006i	$0.937852\pi$
$434$	0	0
$435$	0	0
$436$	338.000	0.775229
$437$	249.848i	0.571734i
$438$	0	0
$439$	540.400i	1.23098i	0.788145	+	0.615490i	$0.211042\pi$
$439$	540.400i	1.23098i	−0.788145	+	0.615490i	$0.788958\pi$
$440$	235.151i	0.534434i
$441$	0	0
$442$	12.0000	0.0271493
$443$	127.279	0.287312	0.143656	−	0.989628i	$-0.454114\pi$
$443$	127.279	0.287312	0.143656	+	0.989628i	$0.454114\pi$
$444$	0	0
$445$	−288.000	−0.647191
$446$	58.7878i	0.131811i
$447$	0	0
$448$	0	0
$449$	110.309	0.245676	0.122838	−	0.992427i	$-0.460800\pi$
$449$	110.309	0.245676	0.122838	+	0.992427i	$0.460800\pi$
$450$	0	0
$451$	1163.94i	2.58079i
$452$	−118.794	−0.262818
$453$	0	0
$454$	540.400i	1.19031i
$455$	0	0
$456$	0	0
$457$	25.0000	0.0547046	0.0273523	−	0.999626i	$-0.491292\pi$
$457$	25.0000	0.0547046	0.0273523	+	0.999626i	$0.491292\pi$
$458$	115.126i	0.251367i
$459$	0	0
$460$	83.1384i	0.180736i
$461$	78.3837i	0.170030i	0.996380	+	0.0850148i	$0.0270938\pi$
$461$	78.3837i	0.170030i	−0.996380	+	0.0850148i	$0.972906\pi$
$462$	0	0
$463$	521.000	1.12527	0.562635	−	0.826705i	$-0.309788\pi$
$463$	521.000	1.12527	0.562635	+	0.826705i	$0.309788\pi$
$464$	−135.765	−0.292596
$465$	0	0
$466$	324.000	0.695279
$467$	220.454i	0.472064i	0.971745	+	0.236032i	$0.0758471\pi$
$467$	220.454i	0.472064i	−0.971745	+	0.236032i	$0.924153\pi$
$468$	0	0
$469$	0	0
$470$	576.999	1.22766
$471$	0	0
$472$	− 235.559i	− 0.499065i
$473$	−526.087	−1.11224
$474$	0	0
$475$	− 29.4449i	− 0.0619892i
$476$	0	0
$477$	0	0
$478$	−96.0000	−0.200837
$479$	− 876.917i	− 1.83073i	−0.402631	−	0.915363i	$-0.631904\pi$
$479$	− 876.917i	− 1.83073i	0.402631	−	0.915363i	$-0.368096\pi$
$480$	0	0
$481$	− 81.4064i	− 0.169244i
$482$	− 646.665i	− 1.34163i
$483$	0	0
$484$	334.000	0.690083
$485$	−203.647	−0.419890
$486$	0	0
$487$	127.000	0.260780	0.130390	−	0.991463i	$-0.458377\pi$
$487$	127.000	0.260780	0.130390	+	0.991463i	$0.458377\pi$
$488$	235.151i	0.481867i
$489$	0	0
$490$	0	0
$491$	−627.911	−1.27884	−0.639420	−	0.768857i	$-0.720826\pi$
$491$	−627.911	−1.27884	−0.639420	+	0.768857i	$0.720826\pi$
$492$	0	0
$493$	166.277i	0.337276i
$494$	72.1249	0.146002
$495$	0	0
$496$	− 48.4974i	− 0.0977771i
$497$	0	0
$498$	0	0
$499$	−233.000	−0.466934	−0.233467	−	0.972365i	$-0.575007\pi$
$499$	−233.000	−0.466934	−0.233467	+	0.972365i	$0.575007\pi$
$500$	− 254.747i	− 0.509494i
$501$	0	0
$502$	491.902i	0.979885i
$503$	− 538.888i	− 1.07135i	−0.844425	−	0.535674i	$-0.820058\pi$
$503$	− 538.888i	− 1.07135i	0.844425	−	0.535674i	$-0.179942\pi$
$504$	0	0
$505$	−864.000	−1.71089
$506$	203.647	0.402464
$507$	0	0
$508$	418.000	0.822835
$509$	− 318.434i	− 0.625606i	−0.949818	−	0.312803i	$-0.898732\pi$
$509$	− 318.434i	− 0.625606i	0.949818	−	0.312803i	$-0.101268\pi$
$510$	0	0
$511$	0	0
$512$	22.6274	0.0441942
$513$	0	0
$514$	− 228.631i	− 0.444807i
$515$	144.250	0.280097
$516$	0	0
$517$	− 1413.35i	− 2.73376i
$518$	0	0
$519$	0	0
$520$	24.0000	0.0461538
$521$	− 568.282i	− 1.09075i	−0.838192	−	0.545376i	$-0.816387\pi$
$521$	− 568.282i	− 1.09075i	0.838192	−	0.545376i	$-0.183613\pi$
$522$	0	0
$523$	− 528.275i	− 1.01009i	−0.863094	−	0.505043i	$-0.831476\pi$
$523$	− 528.275i	− 1.01009i	0.863094	−	0.505043i	$-0.168524\pi$
$524$	117.576i	0.224381i
$525$	0	0
$526$	360.000	0.684411
$527$	−59.3970	−0.112708
$528$	0	0
$529$	−457.000	−0.863894
$530$	529.090i	0.998283i
$531$	0	0
$532$	0	0
$533$	118.794	0.222878
$534$	0	0
$535$	706.677i	1.32089i
$536$	−87.6812	−0.163584
$537$	0	0
$538$	27.7128i	0.0515108i
$539$	0	0
$540$	0	0
$541$	335.000	0.619224	0.309612	−	0.950863i	$-0.399801\pi$
$541$	335.000	0.619224	0.309612	+	0.950863i	$0.399801\pi$
$542$	− 58.7878i	− 0.108464i
$543$	0	0
$544$	− 27.7128i	− 0.0509427i
$545$	− 827.928i	− 1.51913i
$546$	0	0
$547$	−658.000	−1.20293	−0.601463	−	0.798901i	$-0.705415\pi$
$547$	−658.000	−1.20293	−0.601463	+	0.798901i	$0.705415\pi$
$548$	305.470	0.557427
$549$	0	0
$550$	−24.0000	−0.0436364
$551$	999.392i	1.81378i
$552$	0	0
$553$	0	0
$554$	−476.590	−0.860271
$555$	0	0
$556$	391.443i	0.704035i
$557$	−271.529	−0.487485	−0.243742	−	0.969840i	$-0.578375\pi$
$557$	−271.529	−0.487485	−0.243742	+	0.969840i	$0.578375\pi$
$558$	0	0
$559$	53.6936i	0.0960529i
$560$	0	0
$561$	0	0
$562$	348.000	0.619217
$563$	− 14.6969i	− 0.0261047i	−0.999915	−	0.0130523i	$-0.995845\pi$
$563$	− 14.6969i	− 0.0261047i	0.999915	−	0.0130523i	$-0.00415481\pi$
$564$	0	0
$565$	290.985i	0.515017i
$566$	− 276.792i	− 0.489032i
$567$	0	0
$568$	168.000	0.295775
$569$	−848.528	−1.49126	−0.745631	−	0.666359i	$-0.767852\pi$
$569$	−848.528	−1.49126	−0.745631	+	0.666359i	$0.767852\pi$
$570$	0	0
$571$	449.000	0.786340	0.393170	−	0.919466i	$-0.371378\pi$
$571$	449.000	0.786340	0.393170	+	0.919466i	$0.371378\pi$
$572$	− 58.7878i	− 0.102776i
$573$	0	0
$574$	0	0
$575$	−8.48528	−0.0147570
$576$	0	0
$577$	− 292.717i	− 0.507308i	−0.967295	−	0.253654i	$-0.918368\pi$
$577$	− 292.717i	− 0.507308i	0.967295	−	0.253654i	$-0.0816324\pi$
$578$	374.767	0.648385
$579$	0	0
$580$	332.554i	0.573369i
$581$	0	0
$582$	0	0
$583$	1296.00	2.22298
$584$	− 230.252i	− 0.394267i
$585$	0	0
$586$	− 138.564i	− 0.236457i
$587$	529.090i	0.901345i	0.892689	+	0.450673i	$0.148816\pi$
$587$	529.090i	0.901345i	−0.892689	+	0.450673i	$0.851184\pi$
$588$	0	0
$589$	−357.000	−0.606112
$590$	−576.999	−0.977965
$591$	0	0
$592$	−188.000	−0.317568
$593$	1048.38i	1.76793i	0.467554	+	0.883964i	$0.345135\pi$
$593$	1048.38i	1.76793i	−0.467554	+	0.883964i	$0.654865\pi$
$594$	0	0
$595$	0	0
$596$	101.823	0.170845
$597$	0	0
$598$	− 20.7846i	− 0.0347569i
$599$	−644.881	−1.07660	−0.538298	−	0.842754i	$-0.680933\pi$
$599$	−644.881	−1.07660	−0.538298	+	0.842754i	$0.680933\pi$
$600$	0	0
$601$	− 458.993i	− 0.763716i	−0.924221	−	0.381858i	$-0.875284\pi$
$601$	− 458.993i	− 0.763716i	0.924221	−	0.381858i	$-0.124716\pi$
$602$	0	0
$603$	0	0
$604$	20.0000	0.0331126
$605$	− 818.130i	− 1.35228i
$606$	0	0
$607$	− 1051.35i	− 1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$607$	− 1051.35i	− 1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$608$	− 166.565i	− 0.273956i
$609$	0	0
$610$	576.000	0.944262
$611$	−144.250	−0.236088
$612$	0	0
$613$	290.000	0.473083	0.236542	−	0.971621i	$-0.423986\pi$
$613$	290.000	0.473083	0.236542	+	0.971621i	$0.423986\pi$
$614$	100.429i	0.163565i
$615$	0	0
$616$	0	0
$617$	−729.734	−1.18271	−0.591357	−	0.806410i	$-0.701407\pi$
$617$	−729.734	−1.18271	−0.591357	+	0.806410i	$0.701407\pi$
$618$	0	0
$619$	− 819.260i	− 1.32352i	−0.749715	−	0.661761i	$-0.769809\pi$
$619$	− 819.260i	− 1.32352i	0.749715	−	0.661761i	$-0.230191\pi$
$620$	−118.794	−0.191603
$621$	0	0
$622$	304.841i	0.490098i
$623$	0	0
$624$	0	0
$625$	−599.000	−0.958400
$626$	− 413.964i	− 0.661284i
$627$	0	0
$628$	83.1384i	0.132386i
$629$	230.252i	0.366060i
$630$	0	0
$631$	−58.0000	−0.0919176	−0.0459588	−	0.998943i	$-0.514634\pi$
$631$	−58.0000	−0.0919176	−0.0459588	+	0.998943i	$0.514634\pi$
$632$	115.966	0.183490
$633$	0	0
$634$	−336.000	−0.529968
$635$	− 1023.89i	− 1.61242i
$636$	0	0
$637$	0	0
$638$	814.587	1.27678
$639$	0	0
$640$	− 55.4256i	− 0.0866025i
$641$	−958.837	−1.49585	−0.747923	−	0.663786i	$-0.768949\pi$
$641$	−958.837	−1.49585	−0.747923	+	0.663786i	$0.768949\pi$
$642$	0	0
$643$	760.370i	1.18254i	0.806475	+	0.591268i	$0.201372\pi$
$643$	760.370i	1.18254i	−0.806475	+	0.591268i	$0.798628\pi$
$644$	0	0
$645$	0	0
$646$	−204.000	−0.315789
$647$	352.727i	0.545172i	0.962131	+	0.272586i	$0.0878790\pi$
$647$	352.727i	0.545172i	−0.962131	+	0.272586i	$0.912121\pi$
$648$	0	0
$649$	1413.35i	2.17774i
$650$	2.44949i	0.00376845i
$651$	0	0
$652$	172.000	0.263804
$653$	441.235	0.675704	0.337852	−	0.941199i	$-0.390300\pi$
$653$	441.235	0.675704	0.337852	+	0.941199i	$0.390300\pi$
$654$	0	0
$655$	288.000	0.439695
$656$	− 274.343i	− 0.418206i
$657$	0	0
$658$	0	0
$659$	−161.220	−0.244644	−0.122322	−	0.992490i	$-0.539034\pi$
$659$	−161.220	−0.244644	−0.122322	+	0.992490i	$0.539034\pi$
$660$	0	0
$661$	− 833.116i	− 1.26039i	−0.776438	−	0.630194i	$-0.782975\pi$
$661$	− 833.116i	− 1.26039i	0.776438	−	0.630194i	$-0.217025\pi$
$662$	−261.630	−0.395211
$663$	0	0
$664$	13.8564i	0.0208681i
$665$	0	0
$666$	0	0
$667$	288.000	0.431784
$668$	− 362.524i	− 0.542701i
$669$	0	0
$670$	214.774i	0.320559i
$671$	− 1410.91i	− 2.10269i
$672$	0	0
$673$	−263.000	−0.390788	−0.195394	−	0.980725i	$-0.562598\pi$
$673$	−263.000	−0.390788	−0.195394	+	0.980725i	$0.562598\pi$
$674$	−507.703	−0.753268
$675$	0	0
$676$	332.000	0.491124
$677$	499.696i	0.738103i	0.929409	+	0.369052i	$0.120317\pi$
$677$	499.696i	0.738103i	−0.929409	+	0.369052i	$0.879683\pi$
$678$	0	0
$679$	0	0
$680$	−67.8823	−0.0998268
$681$	0	0
$682$	290.985i	0.426664i
$683$	−958.837	−1.40386	−0.701930	−	0.712246i	$-0.747678\pi$
$683$	−958.837	−1.40386	−0.701930	+	0.712246i	$0.747678\pi$
$684$	0	0
$685$	− 748.246i	− 1.09233i
$686$	0	0
$687$	0	0
$688$	124.000	0.180233
$689$	− 132.272i	− 0.191977i
$690$	0	0
$691$	− 1234.95i	− 1.78720i	−0.448868	−	0.893598i	$-0.648173\pi$
$691$	− 1234.95i	− 1.78720i	0.448868	−	0.893598i	$-0.351827\pi$
$692$	88.1816i	0.127430i
$693$	0	0
$694$	660.000	0.951009
$695$	958.837	1.37962
$696$	0	0
$697$	−336.000	−0.482066
$698$	− 823.029i	− 1.17912i
$699$	0	0
$700$	0	0
$701$	975.807	1.39202	0.696011	−	0.718031i	$-0.254956\pi$
$701$	975.807	1.39202	0.696011	+	0.718031i	$0.254956\pi$
$702$	0	0
$703$	1383.91i	1.96858i
$704$	−135.765	−0.192847
$705$	0	0
$706$	408.764i	0.578986i
$707$	0	0
$708$	0	0
$709$	−1106.00	−1.55994	−0.779972	−	0.625815i	$-0.784767\pi$
$709$	−1106.00	−1.55994	−0.779972	+	0.625815i	$0.784767\pi$
$710$	− 411.514i	− 0.579598i
$711$	0	0
$712$	− 166.277i	− 0.233535i
$713$	102.879i	0.144290i
$714$	0	0
$715$	−144.000	−0.201399
$716$	−16.9706	−0.0237019
$717$	0	0
$718$	480.000	0.668524
$719$	− 685.857i	− 0.953904i	−0.878929	−	0.476952i	$-0.841742\pi$
$719$	− 685.857i	− 0.953904i	0.878929	−	0.476952i	$-0.158258\pi$
$720$	0	0
$721$	0	0
$722$	−715.592	−0.991125
$723$	0	0
$724$	− 86.6025i	− 0.119617i
$725$	−33.9411	−0.0468153
$726$	0	0
$727$	427.817i	0.588468i	0.955733	+	0.294234i	$0.0950646\pi$
$727$	427.817i	0.588468i	−0.955733	+	0.294234i	$0.904935\pi$
$728$	0	0
$729$	0	0
$730$	−564.000	−0.772603
$731$	− 151.868i	− 0.207754i
$732$	0	0
$733$	− 39.8372i	− 0.0543481i	−0.999631	−	0.0271741i	$-0.991349\pi$
$733$	− 39.8372i	− 0.0543481i	0.999631	−	0.0271741i	$-0.00865084\pi$
$734$	218.005i	0.297009i
$735$	0	0
$736$	−48.0000	−0.0652174
$737$	526.087	0.713823
$738$	0	0
$739$	−487.000	−0.658999	−0.329499	−	0.944156i	$-0.606880\pi$
$739$	−487.000	−0.658999	−0.329499	+	0.944156i	$0.606880\pi$
$740$	460.504i	0.622303i
$741$	0	0
$742$	0	0
$743$	−509.117	−0.685218	−0.342609	−	0.939478i	$-0.611311\pi$
$743$	−509.117	−0.685218	−0.342609	+	0.939478i	$0.611311\pi$
$744$	0	0
$745$	− 249.415i	− 0.334786i
$746$	−408.708	−0.547866
$747$	0	0
$748$	166.277i	0.222295i
$749$	0	0
$750$	0	0
$751$	−545.000	−0.725699	−0.362850	−	0.931848i	$-0.618196\pi$
$751$	−545.000	−0.725699	−0.362850	+	0.931848i	$0.618196\pi$
$752$	333.131i	0.442993i
$753$	0	0
$754$	− 83.1384i	− 0.110263i
$755$	− 48.9898i	− 0.0648871i
$756$	0	0
$757$	−770.000	−1.01717	−0.508587	−	0.861011i	$-0.669832\pi$
$757$	−770.000	−1.01717	−0.508587	+	0.861011i	$0.669832\pi$
$758$	9.89949	0.0130600
$759$	0	0
$760$	−408.000	−0.536842
$761$	− 171.464i	− 0.225314i	−0.993634	−	0.112657i	$-0.964064\pi$
$761$	− 171.464i	− 0.225314i	0.993634	−	0.112657i	$-0.0359362\pi$
$762$	0	0
$763$	0	0
$764$	152.735	0.199915
$765$	0	0
$766$	− 699.749i	− 0.913510i
$767$	144.250	0.188070
$768$	0	0
$769$	− 704.945i	− 0.916703i	−0.888771	−	0.458352i	$-0.848440\pi$
$769$	− 704.945i	− 0.916703i	0.888771	−	0.458352i	$-0.151560\pi$
$770$	0	0
$771$	0	0
$772$	−574.000	−0.743523
$773$	921.008i	1.19147i	0.803180	+	0.595736i	$0.203140\pi$
$773$	921.008i	1.19147i	−0.803180	+	0.595736i	$0.796860\pi$
$774$	0	0
$775$	− 12.1244i	− 0.0156443i
$776$	− 117.576i	− 0.151515i
$777$	0	0
$778$	−324.000	−0.416452
$779$	−2019.50	−2.59242
$780$	0	0
$781$	−1008.00	−1.29065
$782$	58.7878i	0.0751762i
$783$	0	0
$784$	0	0
$785$	203.647	0.259423
$786$	0	0
$787$	457.261i	0.581018i	0.956872	+	0.290509i	$0.0938247\pi$
$787$	457.261i	0.581018i	−0.956872	+	0.290509i	$0.906175\pi$
$788$	254.558	0.323044
$789$	0	0
$790$	− 284.056i	− 0.359565i
$791$	0	0
$792$	0	0
$793$	−144.000	−0.181589
$794$	− 115.126i	− 0.144995i
$795$	0	0
$796$	415.692i	0.522226i
$797$	14.6969i	0.0184403i	0.999957	+	0.00922016i	$0.00293491\pi$
$797$	14.6969i	0.0184403i	−0.999957	+	0.00922016i	$0.997065\pi$
$798$	0	0
$799$	408.000	0.510638
$800$	5.65685	0.00707107
$801$	0	0
$802$	−132.000	−0.164589
$803$	1381.51i	1.72044i
$804$	0	0
$805$	0	0
$806$	29.6985	0.0368468
$807$	0	0
$808$	− 498.831i	− 0.617365i
$809$	941.866	1.16424	0.582118	−	0.813105i	$-0.302224\pi$
$809$	941.866	1.16424	0.582118	+	0.813105i	$0.302224\pi$
$810$	0	0
$811$	498.831i	0.615081i	0.951535	+	0.307540i	$0.0995059\pi$
$811$	498.831i	0.615081i	−0.951535	+	0.307540i	$0.900494\pi$
$812$	0	0
$813$	0	0
$814$	1128.00	1.38575
$815$	− 421.312i	− 0.516948i
$816$	0	0
$817$	− 912.791i	− 1.11725i
$818$	− 590.327i	− 0.721671i
$819$	0	0
$820$	−672.000	−0.819512
$821$	602.455	0.733806	0.366903	−	0.930259i	$-0.380418\pi$
$821$	602.455	0.733806	0.366903	+	0.930259i	$0.380418\pi$
$822$	0	0
$823$	38.0000	0.0461725	0.0230863	−	0.999733i	$-0.492651\pi$
$823$	38.0000	0.0461725	0.0230863	+	0.999733i	$0.492651\pi$
$824$	83.2827i	0.101071i
$825$	0	0
$826$	0	0
$827$	−687.308	−0.831086	−0.415543	−	0.909574i	$-0.636408\pi$
$827$	−687.308	−0.831086	−0.415543	+	0.909574i	$0.636408\pi$
$828$	0	0
$829$	− 833.116i	− 1.00497i	−0.864587	−	0.502483i	$-0.832420\pi$
$829$	− 833.116i	− 1.00497i	0.864587	−	0.502483i	$-0.167580\pi$
$830$	33.9411	0.0408929
$831$	0	0
$832$	13.8564i	0.0166543i
$833$	0	0
$834$	0	0
$835$	−888.000	−1.06347
$836$	999.392i	1.19544i
$837$	0	0
$838$	− 27.7128i	− 0.0330702i
$839$	244.949i	0.291953i	0.989288	+	0.145977i	$0.0466325\pi$
$839$	244.949i	0.291953i	−0.989288	+	0.145977i	$0.953368\pi$
$840$	0	0
$841$	311.000	0.369798
$842$	575.585	0.683593
$843$	0	0
$844$	164.000	0.194313
$845$	− 813.231i	− 0.962403i
$846$	0	0
$847$	0	0
$848$	−305.470	−0.360224
$849$	0	0
$850$	− 6.92820i	− 0.00815083i
$851$	398.808	0.468635
$852$	0	0
$853$	1245.34i	1.45996i	0.683469	+	0.729979i	$0.260470\pi$
$853$	1245.34i	1.45996i	−0.683469	+	0.729979i	$0.739530\pi$
$854$	0	0
$855$	0	0
$856$	−408.000	−0.476636
$857$	− 1224.74i	− 1.42911i	−0.699581	−	0.714554i	$-0.746630\pi$
$857$	− 1224.74i	− 1.42911i	0.699581	−	0.714554i	$-0.253370\pi$
$858$	0	0
$859$	− 249.415i	− 0.290355i	−0.989406	−	0.145178i	$-0.953625\pi$
$859$	− 249.415i	− 0.290355i	0.989406	−	0.145178i	$-0.0463754\pi$
$860$	− 303.737i	− 0.353182i
$861$	0	0
$862$	−228.000	−0.264501
$863$	−661.852	−0.766920	−0.383460	−	0.923557i	$-0.625268\pi$
$863$	−661.852	−0.766920	−0.383460	+	0.923557i	$0.625268\pi$
$864$	0	0
$865$	216.000	0.249711
$866$	237.601i	0.274365i
$867$	0	0
$868$	0	0
$869$	−695.793	−0.800682
$870$	0	0
$871$	− 53.6936i	− 0.0616459i
$872$	478.004	0.548170
$873$	0	0
$874$	353.338i	0.404277i
$875$	0	0
$876$	0	0
$877$	−574.000	−0.654504	−0.327252	−	0.944937i	$-0.606123\pi$
$877$	−574.000	−0.654504	−0.327252	+	0.944937i	$0.606123\pi$
$878$	764.241i	0.870434i
$879$	0	0
$880$	332.554i	0.377902i
$881$	161.666i	0.183503i	0.995782	+	0.0917516i	$0.0292466\pi$
$881$	161.666i	0.183503i	−0.995782	+	0.0917516i	$0.970753\pi$
$882$	0	0
$883$	1735.00	1.96489	0.982446	−	0.186546i	$-0.0597294\pi$
$883$	1735.00	1.96489	0.982446	+	0.186546i	$0.0597294\pi$
$884$	16.9706	0.0191975
$885$	0	0
$886$	180.000	0.203160
$887$	− 195.959i	− 0.220924i	−0.993880	−	0.110462i	$-0.964767\pi$
$887$	− 195.959i	− 0.220924i	0.993880	−	0.110462i	$-0.0352330\pi$
$888$	0	0
$889$	0	0
$890$	−407.294	−0.457633
$891$	0	0
$892$	83.1384i	0.0932045i
$893$	2452.25	2.74608
$894$	0	0
$895$	41.5692i	0.0464461i
$896$	0	0
$897$	0	0
$898$	156.000	0.173719
$899$	411.514i	0.457747i
$900$	0	0
$901$	374.123i	0.415231i
$902$	1646.06i	1.82490i
$903$	0	0
$904$	−168.000	−0.185841
$905$	−212.132	−0.234400
$906$	0	0
$907$	751.000	0.828004	0.414002	−	0.910276i	$-0.364131\pi$
$907$	751.000	0.828004	0.414002	+	0.910276i	$0.364131\pi$
$908$	764.241i	0.841675i
$909$	0	0
$910$	0	0
$911$	1247.34	1.36919	0.684597	−	0.728921i	$-0.259978\pi$
$911$	1247.34	1.36919	0.684597	+	0.728921i	$0.259978\pi$
$912$	0	0
$913$	− 83.1384i	− 0.0910607i
$914$	35.3553	0.0386820
$915$	0	0
$916$	162.813i	0.177743i
$917$	0	0
$918$	0	0
$919$	1015.00	1.10446	0.552231	−	0.833691i	$-0.313777\pi$
$919$	1015.00	1.10446	0.552231	+	0.833691i	$0.313777\pi$
$920$	117.576i	0.127799i
$921$	0	0
$922$	110.851i	0.120229i
$923$	102.879i	0.111461i
$924$	0	0
$925$	−47.0000	−0.0508108
$926$	736.805	0.795686
$927$	0	0
$928$	−192.000	−0.206897
$929$	− 1092.47i	− 1.17597i	−0.808873	−	0.587983i	$-0.799922\pi$
$929$	− 1092.47i	− 1.17597i	0.808873	−	0.587983i	$-0.200078\pi$
$930$	0	0
$931$	0	0
$932$	458.205	0.491636
$933$	0	0
$934$	311.769i	0.333800i
$935$	407.294	0.435608
$936$	0	0
$937$	− 1747.64i	− 1.86514i	−0.360985	−	0.932572i	$-0.617559\pi$
$937$	− 1747.64i	− 1.86514i	0.360985	−	0.932572i	$-0.382441\pi$
$938$	0	0
$939$	0	0
$940$	816.000	0.868085
$941$	1239.44i	1.31715i	0.752513	+	0.658577i	$0.228841\pi$
$941$	1239.44i	1.31715i	−0.752513	+	0.658577i	$0.771159\pi$
$942$	0	0
$943$	581.969i	0.617146i
$944$	− 333.131i	− 0.352893i
$945$	0	0
$946$	−744.000	−0.786469
$947$	1204.91	1.27234	0.636172	−	0.771547i	$-0.280517\pi$
$947$	1204.91	1.27234	0.636172	+	0.771547i	$0.280517\pi$
$948$	0	0
$949$	141.000	0.148577
$950$	− 41.6413i	− 0.0438330i
$951$	0	0
$952$	0	0
$953$	1026.72	1.07735	0.538677	−	0.842512i	$-0.318924\pi$
$953$	1026.72	1.07735	0.538677	+	0.842512i	$0.318924\pi$
$954$	0	0
$955$	− 374.123i	− 0.391752i
$956$	−135.765	−0.142013
$957$	0	0
$958$	− 1240.15i	− 1.29452i
$959$	0	0
$960$	0	0
$961$	814.000	0.847034
$962$	− 115.126i	− 0.119674i
$963$	0	0
$964$	− 914.523i	− 0.948675i
$965$	1406.01i	1.45700i
$966$	0	0
$967$	−895.000	−0.925543	−0.462771	−	0.886478i	$-0.653145\pi$
$967$	−895.000	−0.925543	−0.462771	+	0.886478i	$0.653145\pi$
$968$	472.347	0.487962
$969$	0	0
$970$	−288.000	−0.296907
$971$	210.656i	0.216948i	0.994099	+	0.108474i	$0.0345964\pi$
$971$	210.656i	0.216948i	−0.994099	+	0.108474i	$0.965404\pi$
$972$	0	0
$973$	0	0
$974$	179.605	0.184400
$975$	0	0
$976$	332.554i	0.340731i
$977$	−1255.82	−1.28539	−0.642693	−	0.766124i	$-0.722183\pi$
$977$	−1255.82	−1.28539	−0.642693	+	0.766124i	$0.722183\pi$
$978$	0	0
$979$	997.661i	1.01906i
$980$	0	0
$981$	0	0
$982$	−888.000	−0.904277
$983$	1180.65i	1.20107i	0.799598	+	0.600536i	$0.205046\pi$
$983$	1180.65i	1.20107i	−0.799598	+	0.600536i	$0.794954\pi$
$984$	0	0
$985$	− 623.538i	− 0.633034i
$986$	235.151i	0.238490i
$987$	0	0
$988$	102.000	0.103239
$989$	−263.044	−0.265969
$990$	0	0
$991$	655.000	0.660949	0.330474	−	0.943815i	$-0.392791\pi$
$991$	655.000	0.660949	0.330474	+	0.943815i	$0.392791\pi$
$992$	− 68.5857i	− 0.0691388i
$993$	0	0
$994$	0	0
$995$	1018.23	1.02335
$996$	0	0
$997$	458.993i	0.460375i	0.973146	+	0.230187i	$0.0739339\pi$
$997$	458.993i	0.460375i	−0.973146	+	0.230187i	$0.926066\pi$
$998$	−329.512	−0.330172
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.3.c.c.685.3		4
3.2	odd	2	inner	882.3.c.c.685.2		4
7.2	even	3		882.3.n.c.325.1		4
7.3	odd	6		882.3.n.c.19.1		4
7.4	even	3		126.3.n.b.19.1	✓	4
7.5	odd	6		126.3.n.b.73.1	yes	4
7.6	odd	2	inner	882.3.c.c.685.4		4
21.2	odd	6		882.3.n.c.325.2		4
21.5	even	6		126.3.n.b.73.2	yes	4
21.11	odd	6		126.3.n.b.19.2	yes	4
21.17	even	6		882.3.n.c.19.2		4
21.20	even	2	inner	882.3.c.c.685.1		4
28.11	odd	6		1008.3.cg.i.145.2		4
28.19	even	6		1008.3.cg.i.577.2		4
84.11	even	6		1008.3.cg.i.145.1		4
84.47	odd	6		1008.3.cg.i.577.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.3.n.b.19.1	✓	4	7.4	even	3
126.3.n.b.19.2	yes	4	21.11	odd	6
126.3.n.b.73.1	yes	4	7.5	odd	6
126.3.n.b.73.2	yes	4	21.5	even	6
882.3.c.c.685.1		4	21.20	even	2	inner
882.3.c.c.685.2		4	3.2	odd	2	inner
882.3.c.c.685.3		4	1.1	even	1	trivial
882.3.c.c.685.4		4	7.6	odd	2	inner
882.3.n.c.19.1		4	7.3	odd	6
882.3.n.c.19.2		4	21.17	even	6
882.3.n.c.325.1		4	7.2	even	3
882.3.n.c.325.2		4	21.2	odd	6
1008.3.cg.i.145.1		4	84.11	even	6
1008.3.cg.i.145.2		4	28.11	odd	6
1008.3.cg.i.577.1		4	84.47	odd	6
1008.3.cg.i.577.2		4	28.19	even	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	882.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} +2.00000 q^{4} -4.89898i q^{5} +2.82843 q^{8} +O(q^{10})\)	\(q+1.41421 q^{2} +2.00000 q^{4} -4.89898i q^{5} +2.82843 q^{8} -6.92820i q^{10} -16.9706 q^{11} +1.73205i q^{13} +4.00000 q^{16} -4.89898i q^{17} -29.4449i q^{19} -9.79796i q^{20} -24.0000 q^{22} -8.48528 q^{23} +1.00000 q^{25} +2.44949i q^{26} -33.9411 q^{29} -12.1244i q^{31} +5.65685 q^{32} -6.92820i q^{34} -47.0000 q^{37} -41.6413i q^{38} -13.8564i q^{40} -68.5857i q^{41} +31.0000 q^{43} -33.9411 q^{44} -12.0000 q^{46} +83.2827i q^{47} +1.41421 q^{50} +3.46410i q^{52} -76.3675 q^{53} +83.1384i q^{55} -48.0000 q^{58} -83.2827i q^{59} +83.1384i q^{61} -17.1464i q^{62} +8.00000 q^{64} +8.48528 q^{65} -31.0000 q^{67} -9.79796i q^{68} +59.3970 q^{71} -81.4064i q^{73} -66.4680 q^{74} -58.8897i q^{76} +41.0000 q^{79} -19.5959i q^{80} -96.9948i q^{82} +4.89898i q^{83} -24.0000 q^{85} +43.8406 q^{86} -48.0000 q^{88} -58.7878i q^{89} -16.9706 q^{92} +117.779i q^{94} -144.250 q^{95} -41.5692i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{4}+O(q^{10}) \) 4 * q + 8 * q^4	\( 4 q + 8 q^{4} + 16 q^{16} - 96 q^{22} + 4 q^{25} - 188 q^{37} + 124 q^{43} - 48 q^{46} - 192 q^{58} + 32 q^{64} - 124 q^{67} + 164 q^{79} - 96 q^{85} - 192 q^{88}+O(q^{100}) \) 4 * q + 8 * q^4 + 16 * q^16 - 96 * q^22 + 4 * q^25 - 188 * q^37 + 124 * q^43 - 48 * q^46 - 192 * q^58 + 32 * q^64 - 124 * q^67 + 164 * q^79 - 96 * q^85 - 192 * q^88

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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