LMFDB - Embedded newform 882.3.c.a.685.4

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:	4.0.2048.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 4x^{2} + 2 $ x^4 + 4*x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 98)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			685.4
Root			$0.765367i$ of defining polynomial
Character	$\chi$	$=$	882.685
Dual form			882.3.c.a.685.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{2} +2.00000 q^{4} +8.15640i q^{5} +2.82843 q^{8} +O(q^{10})$	$q+1.41421 q^{2} +2.00000 q^{4} +8.15640i q^{5} +2.82843 q^{8} +11.5349i q^{10} -17.6569 q^{11} -1.21371i q^{13} +4.00000 q^{16} +16.6298i q^{17} -0.371418i q^{19} +16.3128i q^{20} -24.9706 q^{22} -3.02944 q^{23} -41.5269 q^{25} -1.71644i q^{26} -2.38478 q^{29} -1.26810i q^{31} +5.65685 q^{32} +23.5181i q^{34} -27.7574 q^{37} -0.525265i q^{38} +23.0698i q^{40} -28.7988i q^{41} -55.3137 q^{43} -35.3137 q^{44} -4.28427 q^{46} +58.3855i q^{47} -58.7279 q^{50} -2.42742i q^{52} +1.37258 q^{53} -144.016i q^{55} -3.37258 q^{58} -98.7735i q^{59} +82.2975i q^{61} -1.79337i q^{62} +8.00000 q^{64} +9.89949 q^{65} +97.9411 q^{67} +33.2597i q^{68} -64.5685 q^{71} +91.9940i q^{73} -39.2548 q^{74} -0.742837i q^{76} -19.7157 q^{79} +32.6256i q^{80} -40.7276i q^{82} -43.4495i q^{83} -135.640 q^{85} -78.2254 q^{86} -49.9411 q^{88} +7.86191i q^{89} -6.05887 q^{92} +82.5695i q^{94} +3.02944 q^{95} +47.2126i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{4}+O(q^{10}) $ 4 * q + 8 * q^4	$ 4 q + 8 q^{4} - 48 q^{11} + 16 q^{16} - 32 q^{22} - 80 q^{23} - 36 q^{25} + 64 q^{29} - 128 q^{37} - 176 q^{43} - 96 q^{44} + 96 q^{46} - 184 q^{50} + 96 q^{53} - 104 q^{58} + 32 q^{64} + 256 q^{67} - 32 q^{71} + 24 q^{74} - 192 q^{79} - 288 q^{85} - 64 q^{86} - 64 q^{88} - 160 q^{92} + 80 q^{95}+O(q^{100}) $ 4 * q + 8 * q^4 - 48 * q^11 + 16 * q^16 - 32 * q^22 - 80 * q^23 - 36 * q^25 + 64 * q^29 - 128 * q^37 - 176 * q^43 - 96 * q^44 + 96 * q^46 - 184 * q^50 + 96 * q^53 - 104 * q^58 + 32 * q^64 + 256 * q^67 - 32 * q^71 + 24 * q^74 - 192 * q^79 - 288 * q^85 - 64 * q^86 - 64 * q^88 - 160 * q^92 + 80 * q^95

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
$5$	8.15640i	1.63128i	0.578559	+	0.815640i	$0.303615\pi$
$5$	8.15640i	1.63128i	−0.578559	+	0.815640i	$0.696385\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	2.82843	0.353553
$9$	0	0
$10$	11.5349i	1.15349i
$11$	−17.6569	−1.60517	−0.802584	−	0.596539i	$-0.796542\pi$
$11$	−17.6569	−1.60517	−0.802584	+	0.596539i	$0.796542\pi$
$12$	0	0
$13$	− 1.21371i	− 0.0933622i	−0.998910	−	0.0466811i	$-0.985136\pi$
$13$	− 1.21371i	− 0.0933622i	0.998910	−	0.0466811i	$-0.0148645\pi$
$14$	0	0
$15$	0	0
$16$	4.00000	0.250000
$17$	16.6298i	0.978225i	0.872221	+	0.489113i	$0.162679\pi$
$17$	16.6298i	0.978225i	−0.872221	+	0.489113i	$0.837321\pi$
$18$	0	0
$19$	− 0.371418i	− 0.0195483i	−0.999952	−	0.00977417i	$-0.996889\pi$
$19$	− 0.371418i	− 0.0195483i	0.999952	−	0.00977417i	$-0.00311126\pi$
$20$	16.3128i	0.815640i
$21$	0	0
$22$	−24.9706	−1.13503
$23$	−3.02944	−0.131715	−0.0658573	−	0.997829i	$-0.520978\pi$
$23$	−3.02944	−0.131715	−0.0658573	+	0.997829i	$0.520978\pi$
$24$	0	0
$25$	−41.5269	−1.66108
$26$	− 1.71644i	− 0.0660170i
$27$	0	0
$28$	0	0
$29$	−2.38478	−0.0822337	−0.0411168	−	0.999154i	$-0.513092\pi$
$29$	−2.38478	−0.0822337	−0.0411168	+	0.999154i	$0.513092\pi$
$30$	0	0
$31$	− 1.26810i	− 0.0409065i	−0.999791	−	0.0204532i	$-0.993489\pi$
$31$	− 1.26810i	− 0.0409065i	0.999791	−	0.0204532i	$-0.00651092\pi$
$32$	5.65685	0.176777
$33$	0	0
$34$	23.5181i	0.691710i
$35$	0	0
$36$	0	0
$37$	−27.7574	−0.750199	−0.375099	−	0.926985i	$-0.622391\pi$
$37$	−27.7574	−0.750199	−0.375099	+	0.926985i	$0.622391\pi$
$38$	− 0.525265i	− 0.0138228i
$39$	0	0
$40$	23.0698i	0.576745i
$41$	− 28.7988i	− 0.702409i	−0.936299	−	0.351205i	$-0.885772\pi$
$41$	− 28.7988i	− 0.702409i	0.936299	−	0.351205i	$-0.114228\pi$
$42$	0	0
$43$	−55.3137	−1.28637	−0.643183	−	0.765713i	$-0.722386\pi$
$43$	−55.3137	−1.28637	−0.643183	+	0.765713i	$0.722386\pi$
$44$	−35.3137	−0.802584
$45$	0	0
$46$	−4.28427	−0.0931363
$47$	58.3855i	1.24224i	0.783714	+	0.621122i	$0.213323\pi$
$47$	58.3855i	1.24224i	−0.783714	+	0.621122i	$0.786677\pi$
$48$	0	0
$49$	0	0
$50$	−58.7279	−1.17456
$51$	0	0
$52$	− 2.42742i	− 0.0466811i
$53$	1.37258	0.0258978	0.0129489	−	0.999916i	$-0.495878\pi$
$53$	1.37258	0.0258978	0.0129489	+	0.999916i	$0.495878\pi$
$54$	0	0
$55$	− 144.016i	− 2.61848i
$56$	0	0
$57$	0	0
$58$	−3.37258	−0.0581480
$59$	− 98.7735i	− 1.67413i	−0.547105	−	0.837064i	$-0.684270\pi$
$59$	− 98.7735i	− 1.67413i	0.547105	−	0.837064i	$-0.315730\pi$
$60$	0	0
$61$	82.2975i	1.34914i	0.738211	+	0.674570i	$0.235671\pi$
$61$	82.2975i	1.34914i	−0.738211	+	0.674570i	$0.764329\pi$
$62$	− 1.79337i	− 0.0289253i
$63$	0	0
$64$	8.00000	0.125000
$65$	9.89949	0.152300
$66$	0	0
$67$	97.9411	1.46181	0.730904	−	0.682480i	$-0.239099\pi$
$67$	97.9411	1.46181	0.730904	+	0.682480i	$0.239099\pi$
$68$	33.2597i	0.489113i
$69$	0	0
$70$	0	0
$71$	−64.5685	−0.909416	−0.454708	−	0.890641i	$-0.650256\pi$
$71$	−64.5685	−0.909416	−0.454708	+	0.890641i	$0.650256\pi$
$72$	0	0
$73$	91.9940i	1.26019i	0.776517	+	0.630096i	$0.216984\pi$
$73$	91.9940i	1.26019i	−0.776517	+	0.630096i	$0.783016\pi$
$74$	−39.2548	−0.530471
$75$	0	0
$76$	− 0.742837i	− 0.00977417i
$77$	0	0
$78$	0	0
$79$	−19.7157	−0.249566	−0.124783	−	0.992184i	$-0.539823\pi$
$79$	−19.7157	−0.249566	−0.124783	+	0.992184i	$0.539823\pi$
$80$	32.6256i	0.407820i
$81$	0	0
$82$	− 40.7276i	− 0.496678i
$83$	− 43.4495i	− 0.523488i	−0.965137	−	0.261744i	$-0.915702\pi$
$83$	− 43.4495i	− 0.523488i	0.965137	−	0.261744i	$-0.0842977\pi$
$84$	0	0
$85$	−135.640	−1.59576
$86$	−78.2254	−0.909598
$87$	0	0
$88$	−49.9411	−0.567513
$89$	7.86191i	0.0883360i	0.999024	+	0.0441680i	$0.0140637\pi$
$89$	7.86191i	0.0883360i	−0.999024	+	0.0441680i	$0.985936\pi$
$90$	0	0
$91$	0	0
$92$	−6.05887	−0.0658573
$93$	0	0
$94$	82.5695i	0.878399i
$95$	3.02944	0.0318888
$96$	0	0
$97$	47.2126i	0.486728i	0.969935	+	0.243364i	$0.0782510\pi$
$97$	47.2126i	0.486728i	−0.969935	+	0.243364i	$0.921749\pi$
$98$	0	0
$99$	0	0
$100$	−83.0538	−0.830538
$101$	196.532i	1.94586i	0.231092	+	0.972932i	$0.425770\pi$
$101$	196.532i	1.94586i	−0.231092	+	0.972932i	$0.574230\pi$
$102$	0	0
$103$	119.090i	1.15621i	0.815963	+	0.578105i	$0.196207\pi$
$103$	119.090i	1.15621i	−0.815963	+	0.578105i	$0.803793\pi$
$104$	− 3.43289i	− 0.0330085i
$105$	0	0
$106$	1.94113	0.0183125
$107$	147.480	1.37832	0.689160	−	0.724609i	$-0.257980\pi$
$107$	147.480	1.37832	0.689160	+	0.724609i	$0.257980\pi$
$108$	0	0
$109$	−130.385	−1.19619	−0.598095	−	0.801425i	$-0.704076\pi$
$109$	−130.385	−1.19619	−0.598095	+	0.801425i	$0.704076\pi$
$110$	− 203.670i	− 1.85155i
$111$	0	0
$112$	0	0
$113$	1.94113	0.0171781	0.00858905	−	0.999963i	$-0.497266\pi$
$113$	1.94113	0.0171781	0.00858905	+	0.999963i	$0.497266\pi$
$114$	0	0
$115$	− 24.7093i	− 0.214864i
$116$	−4.76955	−0.0411168
$117$	0	0
$118$	− 139.687i	− 1.18379i
$119$	0	0
$120$	0	0
$121$	190.765	1.57657
$122$	116.386i	0.953986i
$123$	0	0
$124$	− 2.53620i	− 0.0204532i
$125$	− 134.800i	− 1.07840i
$126$	0	0
$127$	22.3431	0.175930	0.0879651	−	0.996124i	$-0.471964\pi$
$127$	22.3431	0.175930	0.0879651	+	0.996124i	$0.471964\pi$
$128$	11.3137	0.0883883
$129$	0	0
$130$	14.0000	0.107692
$131$	36.3662i	0.277605i	0.990320	+	0.138802i	$0.0443252\pi$
$131$	36.3662i	0.277605i	−0.990320	+	0.138802i	$0.955675\pi$
$132$	0	0
$133$	0	0
$134$	138.510	1.03365
$135$	0	0
$136$	47.0363i	0.345855i
$137$	75.9239	0.554189	0.277094	−	0.960843i	$-0.410629\pi$
$137$	75.9239	0.554189	0.277094	+	0.960843i	$0.410629\pi$
$138$	0	0
$139$	109.751i	0.789578i	0.918772	+	0.394789i	$0.129182\pi$
$139$	109.751i	0.789578i	−0.918772	+	0.394789i	$0.870818\pi$
$140$	0	0
$141$	0	0
$142$	−91.3137	−0.643054
$143$	21.4303i	0.149862i
$144$	0	0
$145$	− 19.4512i	− 0.134146i
$146$	130.099i	0.891090i
$147$	0	0
$148$	−55.5147	−0.375099
$149$	184.000	1.23490	0.617450	−	0.786610i	$-0.288166\pi$
$149$	184.000	1.23490	0.617450	+	0.786610i	$0.288166\pi$
$150$	0	0
$151$	216.284	1.43235	0.716173	−	0.697923i	$-0.245892\pi$
$151$	216.284	1.43235	0.716173	+	0.697923i	$0.245892\pi$
$152$	− 1.05053i	− 0.00691138i
$153$	0	0
$154$	0	0
$155$	10.3431	0.0667300
$156$	0	0
$157$	182.570i	1.16287i	0.813594	+	0.581433i	$0.197508\pi$
$157$	182.570i	1.16287i	−0.813594	+	0.581433i	$0.802492\pi$
$158$	−27.8823	−0.176470
$159$	0	0
$160$	46.1396i	0.288372i
$161$	0	0
$162$	0	0
$163$	−159.078	−0.975940	−0.487970	−	0.872860i	$-0.662262\pi$
$163$	−159.078	−0.975940	−0.487970	+	0.872860i	$0.662262\pi$
$164$	− 57.5976i	− 0.351205i
$165$	0	0
$166$	− 61.4469i	− 0.370162i
$167$	− 74.3455i	− 0.445183i	−0.974912	−	0.222591i	$-0.928548\pi$
$167$	− 74.3455i	− 0.445183i	0.974912	−	0.222591i	$-0.0714516\pi$
$168$	0	0
$169$	167.527	0.991284
$170$	−191.823	−1.12837
$171$	0	0
$172$	−110.627	−0.643183
$173$	188.648i	1.09045i	0.838290	+	0.545225i	$0.183556\pi$
$173$	188.648i	1.09045i	−0.838290	+	0.545225i	$0.816444\pi$
$174$	0	0
$175$	0	0
$176$	−70.6274	−0.401292
$177$	0	0
$178$	11.1184i	0.0624630i
$179$	209.872	1.17247	0.586235	−	0.810141i	$-0.300610\pi$
$179$	209.872	1.17247	0.586235	+	0.810141i	$0.300610\pi$
$180$	0	0
$181$	− 315.609i	− 1.74369i	−0.489778	−	0.871847i	$-0.662922\pi$
$181$	− 315.609i	− 1.74369i	0.489778	−	0.871847i	$-0.337078\pi$
$182$	0	0
$183$	0	0
$184$	−8.56854	−0.0465682
$185$	− 226.400i	− 1.22378i
$186$	0	0
$187$	− 293.631i	− 1.57022i
$188$	116.771i	0.621122i
$189$	0	0
$190$	4.28427	0.0225488
$191$	−326.794	−1.71096	−0.855482	−	0.517833i	$-0.826739\pi$
$191$	−326.794	−1.71096	−0.855482	+	0.517833i	$0.826739\pi$
$192$	0	0
$193$	108.451	0.561921	0.280961	−	0.959719i	$-0.409347\pi$
$193$	108.451	0.561921	0.280961	+	0.959719i	$0.409347\pi$
$194$	66.7688i	0.344169i
$195$	0	0
$196$	0	0
$197$	−40.5097	−0.205633	−0.102816	−	0.994700i	$-0.532785\pi$
$197$	−40.5097	−0.205633	−0.102816	+	0.994700i	$0.532785\pi$
$198$	0	0
$199$	− 245.390i	− 1.23311i	−0.787310	−	0.616557i	$-0.788527\pi$
$199$	− 245.390i	− 1.23311i	0.787310	−	0.616557i	$-0.211473\pi$
$200$	−117.456	−0.587279
$201$	0	0
$202$	277.939i	1.37593i
$203$	0	0
$204$	0	0
$205$	234.894	1.14583
$206$	168.418i	0.817563i
$207$	0	0
$208$	− 4.85483i	− 0.0233405i
$209$	6.55808i	0.0313784i
$210$	0	0
$211$	−36.2843	−0.171963	−0.0859817	−	0.996297i	$-0.527403\pi$
$211$	−36.2843	−0.171963	−0.0859817	+	0.996297i	$0.527403\pi$
$212$	2.74517	0.0129489
$213$	0	0
$214$	208.569	0.974619
$215$	− 451.161i	− 2.09842i
$216$	0	0
$217$	0	0
$218$	−184.392	−0.845834
$219$	0	0
$220$	− 288.033i	− 1.30924i
$221$	20.1838	0.0913293
$222$	0	0
$223$	88.9100i	0.398700i	0.979928	+	0.199350i	$0.0638830\pi$
$223$	88.9100i	0.398700i	−0.979928	+	0.199350i	$0.936117\pi$
$224$	0	0
$225$	0	0
$226$	2.74517	0.0121468
$227$	− 320.287i	− 1.41096i	−0.708732	−	0.705478i	$-0.750732\pi$
$227$	− 320.287i	− 1.41096i	0.708732	−	0.705478i	$-0.249268\pi$
$228$	0	0
$229$	2.90376i	0.0126802i	0.999980	+	0.00634008i	$0.00201812\pi$
$229$	2.90376i	0.0126802i	−0.999980	+	0.00634008i	$0.997982\pi$
$230$	− 34.9442i	− 0.151931i
$231$	0	0
$232$	−6.74517	−0.0290740
$233$	16.6102	0.0712883	0.0356441	−	0.999365i	$-0.488652\pi$
$233$	16.6102	0.0712883	0.0356441	+	0.999365i	$0.488652\pi$
$234$	0	0
$235$	−476.215	−2.02645
$236$	− 197.547i	− 0.837064i
$237$	0	0
$238$	0	0
$239$	397.470	1.66305	0.831527	−	0.555484i	$-0.187467\pi$
$239$	397.470	1.66305	0.831527	+	0.555484i	$0.187467\pi$
$240$	0	0
$241$	− 302.018i	− 1.25319i	−0.779347	−	0.626593i	$-0.784449\pi$
$241$	− 302.018i	− 1.25319i	0.779347	−	0.626593i	$-0.215551\pi$
$242$	269.782	1.11480
$243$	0	0
$244$	164.595i	0.674570i
$245$	0	0
$246$	0	0
$247$	−0.450793	−0.00182507
$248$	− 3.58673i	− 0.0144626i
$249$	0	0
$250$	− 190.636i	− 0.762545i
$251$	− 17.9974i	− 0.0717027i	−0.999357	−	0.0358514i	$-0.988586\pi$
$251$	− 17.9974i	− 0.0717027i	0.999357	−	0.0358514i	$-0.0114143\pi$
$252$	0	0
$253$	53.4903	0.211424
$254$	31.5980	0.124401
$255$	0	0
$256$	16.0000	0.0625000
$257$	445.668i	1.73412i	0.498206	+	0.867059i	$0.333992\pi$
$257$	445.668i	1.73412i	−0.498206	+	0.867059i	$0.666008\pi$
$258$	0	0
$259$	0	0
$260$	19.7990	0.0761500
$261$	0	0
$262$	51.4296i	0.196296i
$263$	33.5391	0.127525	0.0637626	−	0.997965i	$-0.479690\pi$
$263$	33.5391	0.127525	0.0637626	+	0.997965i	$0.479690\pi$
$264$	0	0
$265$	11.1953i	0.0422466i
$266$	0	0
$267$	0	0
$268$	195.882	0.730904
$269$	− 131.716i	− 0.489651i	−0.969567	−	0.244826i	$-0.921269\pi$
$269$	− 131.716i	− 0.489651i	0.969567	−	0.244826i	$-0.0787307\pi$
$270$	0	0
$271$	454.132i	1.67576i	0.545851	+	0.837882i	$0.316207\pi$
$271$	454.132i	1.67576i	−0.545851	+	0.837882i	$0.683793\pi$
$272$	66.5193i	0.244556i
$273$	0	0
$274$	107.373	0.391871
$275$	733.235	2.66631
$276$	0	0
$277$	53.7645	0.194096	0.0970478	−	0.995280i	$-0.469060\pi$
$277$	53.7645	0.194096	0.0970478	+	0.995280i	$0.469060\pi$
$278$	155.212i	0.558316i
$279$	0	0
$280$	0	0
$281$	162.218	0.577289	0.288645	−	0.957436i	$-0.406795\pi$
$281$	162.218	0.577289	0.288645	+	0.957436i	$0.406795\pi$
$282$	0	0
$283$	− 225.037i	− 0.795182i	−0.917563	−	0.397591i	$-0.869846\pi$
$283$	− 225.037i	− 0.795182i	0.917563	−	0.397591i	$-0.130154\pi$
$284$	−129.137	−0.454708
$285$	0	0
$286$	30.3070i	0.105968i
$287$	0	0
$288$	0	0
$289$	12.4487	0.0430751
$290$	− 27.5081i	− 0.0948557i
$291$	0	0
$292$	183.988i	0.630096i
$293$	− 92.6226i	− 0.316118i	−0.987430	−	0.158059i	$-0.949476\pi$
$293$	− 92.6226i	− 0.316118i	0.987430	−	0.158059i	$-0.0505236\pi$
$294$	0	0
$295$	805.637	2.73097
$296$	−78.5097	−0.265235
$297$	0	0
$298$	260.215	0.873206
$299$	3.67685i	0.0122972i
$300$	0	0
$301$	0	0
$302$	305.872	1.01282
$303$	0	0
$304$	− 1.48567i	− 0.00488708i
$305$	−671.252	−2.20083
$306$	0	0
$307$	430.230i	1.40140i	0.713457	+	0.700700i	$0.247129\pi$
$307$	430.230i	1.40140i	−0.713457	+	0.700700i	$0.752871\pi$
$308$	0	0
$309$	0	0
$310$	14.6274	0.0471852
$311$	− 59.0008i	− 0.189713i	−0.995491	−	0.0948567i	$-0.969761\pi$
$311$	− 59.0008i	− 0.189713i	0.995491	−	0.0948567i	$-0.0302393\pi$
$312$	0	0
$313$	− 313.725i	− 1.00232i	−0.865356	−	0.501158i	$-0.832907\pi$
$313$	− 313.725i	− 1.00232i	0.865356	−	0.501158i	$-0.167093\pi$
$314$	258.193i	0.822270i
$315$	0	0
$316$	−39.4315	−0.124783
$317$	−470.548	−1.48438	−0.742190	−	0.670190i	$-0.766213\pi$
$317$	−470.548	−1.48438	−0.742190	+	0.670190i	$0.766213\pi$
$318$	0	0
$319$	42.1076	0.131999
$320$	65.2512i	0.203910i
$321$	0	0
$322$	0	0
$323$	6.17662	0.0191227
$324$	0	0
$325$	50.4016i	0.155082i
$326$	−224.971	−0.690094
$327$	0	0
$328$	− 81.4552i	− 0.248339i
$329$	0	0
$330$	0	0
$331$	308.666	0.932526	0.466263	−	0.884646i	$-0.345600\pi$
$331$	308.666	0.932526	0.466263	+	0.884646i	$0.345600\pi$
$332$	− 86.8991i	− 0.261744i
$333$	0	0
$334$	− 105.140i	− 0.314792i
$335$	798.847i	2.38462i
$336$	0	0
$337$	189.865	0.563398	0.281699	−	0.959503i	$-0.409102\pi$
$337$	189.865	0.563398	0.281699	+	0.959503i	$0.409102\pi$
$338$	236.919	0.700943
$339$	0	0
$340$	−271.279	−0.797880
$341$	22.3907i	0.0656618i
$342$	0	0
$343$	0	0
$344$	−156.451	−0.454799
$345$	0	0
$346$	266.788i	0.771064i
$347$	−288.950	−0.832710	−0.416355	−	0.909202i	$-0.636693\pi$
$347$	−288.950	−0.832710	−0.416355	+	0.909202i	$0.636693\pi$
$348$	0	0
$349$	339.050i	0.971489i	0.874101	+	0.485745i	$0.161452\pi$
$349$	339.050i	0.971489i	−0.874101	+	0.485745i	$0.838548\pi$
$350$	0	0
$351$	0	0
$352$	−99.8823	−0.283756
$353$	498.029i	1.41085i	0.708787	+	0.705423i	$0.249243\pi$
$353$	498.029i	1.41085i	−0.708787	+	0.705423i	$0.750757\pi$
$354$	0	0
$355$	− 526.647i	− 1.48351i
$356$	15.7238i	0.0441680i
$357$	0	0
$358$	296.804	0.829062
$359$	117.941	0.328527	0.164263	−	0.986417i	$-0.447475\pi$
$359$	117.941	0.328527	0.164263	+	0.986417i	$0.447475\pi$
$360$	0	0
$361$	360.862	0.999618
$362$	− 446.338i	− 1.23298i
$363$	0	0
$364$	0	0
$365$	−750.340	−2.05573
$366$	0	0
$367$	− 175.066i	− 0.477020i	−0.971140	−	0.238510i	$-0.923341\pi$
$367$	− 175.066i	− 0.477020i	0.971140	−	0.238510i	$-0.0766589\pi$
$368$	−12.1177	−0.0329287
$369$	0	0
$370$	− 320.178i	− 0.865347i
$371$	0	0
$372$	0	0
$373$	−526.156	−1.41061	−0.705304	−	0.708905i	$-0.749189\pi$
$373$	−526.156	−1.41061	−0.705304	+	0.708905i	$0.749189\pi$
$374$	− 415.256i	− 1.11031i
$375$	0	0
$376$	165.139i	0.439199i
$377$	2.89442i	0.00767751i
$378$	0	0
$379$	−383.245	−1.01120	−0.505600	−	0.862768i	$-0.668729\pi$
$379$	−383.245	−1.01120	−0.505600	+	0.862768i	$0.668729\pi$
$380$	6.05887	0.0159444
$381$	0	0
$382$	−462.156	−1.20983
$383$	0.127451i	0 0.000332769i	1.00000		0.000166385i	$5.29619e-5\pi$
$383$	0.127451i	0 0.000332769i	−1.00000		0.000166385i	$0.999947\pi$
$384$	0	0
$385$	0	0
$386$	153.373	0.397338
$387$	0	0
$388$	94.4253i	0.243364i
$389$	372.909	0.958634	0.479317	−	0.877642i	$-0.340884\pi$
$389$	372.909	0.958634	0.479317	+	0.877642i	$0.340884\pi$
$390$	0	0
$391$	− 50.3790i	− 0.128847i
$392$	0	0
$393$	0	0
$394$	−57.2893	−0.145404
$395$	− 160.809i	− 0.407112i
$396$	0	0
$397$	72.5164i	0.182661i	0.995821	+	0.0913305i	$0.0291120\pi$
$397$	72.5164i	0.182661i	−0.995821	+	0.0913305i	$0.970888\pi$
$398$	− 347.034i	− 0.871944i
$399$	0	0
$400$	−166.108	−0.415269
$401$	12.3116	0.0307023	0.0153511	−	0.999882i	$-0.495113\pi$
$401$	12.3116	0.0307023	0.0153511	+	0.999882i	$0.495113\pi$
$402$	0	0
$403$	−1.53911	−0.00381912
$404$	393.064i	0.972932i
$405$	0	0
$406$	0	0
$407$	490.108	1.20420
$408$	0	0
$409$	− 185.279i	− 0.453004i	−0.974011	−	0.226502i	$-0.927271\pi$
$409$	− 185.279i	− 0.453004i	0.974011	−	0.226502i	$-0.0727290\pi$
$410$	332.191	0.810222
$411$	0	0
$412$	238.179i	0.578105i
$413$	0	0
$414$	0	0
$415$	354.392	0.853956
$416$	− 6.86577i	− 0.0165043i
$417$	0	0
$418$	9.27452i	0.0221879i
$419$	− 491.676i	− 1.17345i	−0.809785	−	0.586726i	$-0.800417\pi$
$419$	− 491.676i	− 1.17345i	0.809785	−	0.586726i	$-0.199583\pi$
$420$	0	0
$421$	−471.373	−1.11965	−0.559825	−	0.828611i	$-0.689132\pi$
$421$	−471.373	−1.11965	−0.559825	+	0.828611i	$0.689132\pi$
$422$	−51.3137	−0.121596
$423$	0	0
$424$	3.88225	0.00915625
$425$	− 690.586i	− 1.62491i
$426$	0	0
$427$	0	0
$428$	294.960	0.689160
$429$	0	0
$430$	− 638.038i	− 1.48381i
$431$	457.255	1.06092	0.530458	−	0.847711i	$-0.322020\pi$
$431$	457.255	1.06092	0.530458	+	0.847711i	$0.322020\pi$
$432$	0	0
$433$	248.941i	0.574921i	0.957792	+	0.287461i	$0.0928110\pi$
$433$	248.941i	0.574921i	−0.957792	+	0.287461i	$0.907189\pi$
$434$	0	0
$435$	0	0
$436$	−260.770	−0.598095
$437$	1.12519i	0.00257480i
$438$	0	0
$439$	430.001i	0.979501i	0.871863	+	0.489751i	$0.162912\pi$
$439$	430.001i	0.979501i	−0.871863	+	0.489751i	$0.837088\pi$
$440$	− 407.340i	− 0.925773i
$441$	0	0
$442$	28.5442	0.0645795
$443$	−380.353	−0.858585	−0.429293	−	0.903165i	$-0.641237\pi$
$443$	−380.353	−0.858585	−0.429293	+	0.903165i	$0.641237\pi$
$444$	0	0
$445$	−64.1249	−0.144101
$446$	125.738i	0.281923i
$447$	0	0
$448$	0	0
$449$	−326.627	−0.727455	−0.363728	−	0.931505i	$-0.618496\pi$
$449$	−326.627	−0.727455	−0.363728	+	0.931505i	$0.618496\pi$
$450$	0	0
$451$	508.496i	1.12749i
$452$	3.88225	0.00858905
$453$	0	0
$454$	− 452.954i	− 0.997697i
$455$	0	0
$456$	0	0
$457$	−621.117	−1.35912	−0.679559	−	0.733621i	$-0.737829\pi$
$457$	−621.117	−1.35912	−0.679559	+	0.733621i	$0.737829\pi$
$458$	4.10653i	0.00896622i
$459$	0	0
$460$	− 49.4186i	− 0.107432i
$461$	170.939i	0.370801i	0.982663	+	0.185401i	$0.0593583\pi$
$461$	170.939i	0.370801i	−0.982663	+	0.185401i	$0.940642\pi$
$462$	0	0
$463$	491.813	1.06223	0.531116	−	0.847299i	$-0.321773\pi$
$463$	491.813	1.06223	0.531116	+	0.847299i	$0.321773\pi$
$464$	−9.53911	−0.0205584
$465$	0	0
$466$	23.4903	0.0504084
$467$	− 300.342i	− 0.643132i	−0.946887	−	0.321566i	$-0.895791\pi$
$467$	− 300.342i	− 0.643132i	0.946887	−	0.321566i	$-0.104209\pi$
$468$	0	0
$469$	0	0
$470$	−673.470	−1.43292
$471$	0	0
$472$	− 279.374i	− 0.591893i
$473$	976.666	2.06483
$474$	0	0
$475$	15.4239i	0.0324713i
$476$	0	0
$477$	0	0
$478$	562.108	1.17596
$479$	161.823i	0.337834i	0.985630	+	0.168917i	$0.0540270\pi$
$479$	161.823i	0.337834i	−0.985630	+	0.168917i	$0.945973\pi$
$480$	0	0
$481$	33.6893i	0.0700402i
$482$	− 427.118i	− 0.886136i
$483$	0	0
$484$	381.529	0.788283
$485$	−385.085	−0.793990
$486$	0	0
$487$	770.392	1.58191	0.790957	−	0.611872i	$-0.209583\pi$
$487$	770.392	1.58191	0.790957	+	0.611872i	$0.209583\pi$
$488$	232.773i	0.476993i
$489$	0	0
$490$	0	0
$491$	363.578	0.740484	0.370242	−	0.928935i	$-0.379275\pi$
$491$	363.578	0.740484	0.370242	+	0.928935i	$0.379275\pi$
$492$	0	0
$493$	− 39.6584i	− 0.0804431i
$494$	−0.637518	−0.00129052
$495$	0	0
$496$	− 5.07241i	− 0.0102266i
$497$	0	0
$498$	0	0
$499$	224.402	0.449703	0.224852	−	0.974393i	$-0.427810\pi$
$499$	224.402	0.449703	0.224852	+	0.974393i	$0.427810\pi$
$500$	− 269.600i	− 0.539201i
$501$	0	0
$502$	− 25.4521i	− 0.0507015i
$503$	790.766i	1.57210i	0.618163	+	0.786050i	$0.287877\pi$
$503$	790.766i	1.57210i	−0.618163	+	0.786050i	$0.712123\pi$
$504$	0	0
$505$	−1603.00	−3.17425
$506$	75.6468	0.149500
$507$	0	0
$508$	44.6863	0.0879651
$509$	409.968i	0.805438i	0.915324	+	0.402719i	$0.131935\pi$
$509$	409.968i	0.805438i	−0.915324	+	0.402719i	$0.868065\pi$
$510$	0	0
$511$	0	0
$512$	22.6274	0.0441942
$513$	0	0
$514$	630.270i	1.22621i
$515$	−971.342	−1.88610
$516$	0	0
$517$	− 1030.90i	− 1.99401i
$518$	0	0
$519$	0	0
$520$	28.0000	0.0538462
$521$	420.351i	0.806816i	0.915020	+	0.403408i	$0.132174\pi$
$521$	420.351i	0.806816i	−0.915020	+	0.403408i	$0.867826\pi$
$522$	0	0
$523$	− 319.672i	− 0.611227i	−0.952156	−	0.305613i	$-0.901138\pi$
$523$	− 319.672i	− 0.611227i	0.952156	−	0.305613i	$-0.0988615\pi$
$524$	72.7324i	0.138802i
$525$	0	0
$526$	47.4315	0.0901739
$527$	21.0883	0.0400158
$528$	0	0
$529$	−519.823	−0.982651
$530$	15.8326i	0.0298728i
$531$	0	0
$532$	0	0
$533$	−34.9533	−0.0655785
$534$	0	0
$535$	1202.91i	2.24843i
$536$	277.019	0.516827
$537$	0	0
$538$	− 186.275i	− 0.346236i
$539$	0	0
$540$	0	0
$541$	−361.608	−0.668407	−0.334203	−	0.942501i	$-0.608467\pi$
$541$	−361.608	−0.668407	−0.334203	+	0.942501i	$0.608467\pi$
$542$	642.240i	1.18494i
$543$	0	0
$544$	94.0725i	0.172927i
$545$	− 1063.47i	− 1.95132i
$546$	0	0
$547$	93.4701	0.170878	0.0854389	−	0.996343i	$-0.472771\pi$
$547$	93.4701	0.170878	0.0854389	+	0.996343i	$0.472771\pi$
$548$	151.848	0.277094
$549$	0	0
$550$	1036.95	1.88536
$551$	0.885750i	0.00160753i
$552$	0	0
$553$	0	0
$554$	76.0345	0.137246
$555$	0	0
$556$	219.503i	0.394789i
$557$	−628.627	−1.12860	−0.564298	−	0.825572i	$-0.690853\pi$
$557$	−628.627	−1.12860	−0.564298	+	0.825572i	$0.690853\pi$
$558$	0	0
$559$	67.1347i	0.120098i
$560$	0	0
$561$	0	0
$562$	229.411	0.408205
$563$	− 282.424i	− 0.501642i	−0.968034	−	0.250821i	$-0.919300\pi$
$563$	− 282.424i	− 0.501642i	0.968034	−	0.250821i	$-0.0807005\pi$
$564$	0	0
$565$	15.8326i	0.0280223i
$566$	− 318.250i	− 0.562279i
$567$	0	0
$568$	−182.627	−0.321527
$569$	−625.664	−1.09959	−0.549793	−	0.835301i	$-0.685293\pi$
$569$	−625.664	−1.09959	−0.549793	+	0.835301i	$0.685293\pi$
$570$	0	0
$571$	−661.401	−1.15832	−0.579160	−	0.815214i	$-0.696619\pi$
$571$	−661.401	−1.15832	−0.579160	+	0.815214i	$0.696619\pi$
$572$	42.8605i	0.0749310i
$573$	0	0
$574$	0	0
$575$	125.803	0.218788
$576$	0	0
$577$	334.988i	0.580569i	0.956940	+	0.290285i	$0.0937499\pi$
$577$	334.988i	0.580569i	−0.956940	+	0.290285i	$0.906250\pi$
$578$	17.6051	0.0304587
$579$	0	0
$580$	− 38.9024i	− 0.0670731i
$581$	0	0
$582$	0	0
$583$	−24.2355	−0.0415703
$584$	260.198i	0.445545i
$585$	0	0
$586$	− 130.988i	− 0.223529i
$587$	1083.88i	1.84648i	0.384227	+	0.923239i	$0.374468\pi$
$587$	1083.88i	1.84648i	−0.384227	+	0.923239i	$0.625532\pi$
$588$	0	0
$589$	−0.470996	−0.000799654 0
$590$	1139.34	1.93109
$591$	0	0
$592$	−111.029	−0.187550
$593$	− 145.827i	− 0.245914i	−0.992412	−	0.122957i	$-0.960762\pi$
$593$	− 145.827i	− 0.245914i	0.992412	−	0.122957i	$-0.0392377\pi$
$594$	0	0
$595$	0	0
$596$	368.000	0.617450
$597$	0	0
$598$	5.19986i	0.00869541i
$599$	154.246	0.257505	0.128753	−	0.991677i	$-0.458903\pi$
$599$	154.246	0.257505	0.128753	+	0.991677i	$0.458903\pi$
$600$	0	0
$601$	− 361.658i	− 0.601760i	−0.953662	−	0.300880i	$-0.902720\pi$
$601$	− 361.658i	− 0.601760i	0.953662	−	0.300880i	$-0.0972804\pi$
$602$	0	0
$603$	0	0
$604$	432.569	0.716173
$605$	1555.95i	2.57182i
$606$	0	0
$607$	− 361.943i	− 0.596282i	−0.954522	−	0.298141i	$-0.903633\pi$
$607$	− 361.943i	− 0.596282i	0.954522	−	0.298141i	$-0.0963666\pi$
$608$	− 2.10106i	− 0.00345569i
$609$	0	0
$610$	−949.294	−1.55622
$611$	70.8629	0.115979
$612$	0	0
$613$	−513.360	−0.837454	−0.418727	−	0.908112i	$-0.637524\pi$
$613$	−513.360	−0.837454	−0.418727	+	0.908112i	$0.637524\pi$
$614$	608.436i	0.990939i
$615$	0	0
$616$	0	0
$617$	−613.767	−0.994761	−0.497380	−	0.867533i	$-0.665705\pi$
$617$	−613.767	−0.994761	−0.497380	+	0.867533i	$0.665705\pi$
$618$	0	0
$619$	− 454.594i	− 0.734400i	−0.930142	−	0.367200i	$-0.880316\pi$
$619$	− 454.594i	− 0.734400i	0.930142	−	0.367200i	$-0.119684\pi$
$620$	20.6863	0.0333650
$621$	0	0
$622$	− 83.4398i	− 0.134148i
$623$	0	0
$624$	0	0
$625$	61.3116	0.0980986
$626$	− 443.674i	− 0.708745i
$627$	0	0
$628$	365.140i	0.581433i
$629$	− 461.600i	− 0.733864i
$630$	0	0
$631$	40.7351	0.0645564	0.0322782	−	0.999479i	$-0.489724\pi$
$631$	40.7351	0.0645564	0.0322782	+	0.999479i	$0.489724\pi$
$632$	−55.7645	−0.0882350
$633$	0	0
$634$	−665.456	−1.04961
$635$	182.240i	0.286992i
$636$	0	0
$637$	0	0
$638$	59.5492	0.0933373
$639$	0	0
$640$	92.2792i	0.144186i
$641$	−153.650	−0.239703	−0.119852	−	0.992792i	$-0.538242\pi$
$641$	−153.650	−0.239703	−0.119852	+	0.992792i	$0.538242\pi$
$642$	0	0
$643$	− 899.887i	− 1.39951i	−0.714382	−	0.699756i	$-0.753292\pi$
$643$	− 899.887i	− 1.39951i	0.714382	−	0.699756i	$-0.246708\pi$
$644$	0	0
$645$	0	0
$646$	8.73506	0.0135218
$647$	709.014i	1.09585i	0.836528	+	0.547924i	$0.184582\pi$
$647$	709.014i	1.09585i	−0.836528	+	0.547924i	$0.815418\pi$
$648$	0	0
$649$	1744.03i	2.68726i
$650$	71.2786i	0.109659i
$651$	0	0
$652$	−318.156	−0.487970
$653$	848.222	1.29896	0.649481	−	0.760378i	$-0.274986\pi$
$653$	848.222	1.29896	0.649481	+	0.760378i	$0.274986\pi$
$654$	0	0
$655$	−296.617	−0.452851
$656$	− 115.195i	− 0.175602i
$657$	0	0
$658$	0	0
$659$	−437.803	−0.664345	−0.332172	−	0.943219i	$-0.607782\pi$
$659$	−437.803	−0.664345	−0.332172	+	0.943219i	$0.607782\pi$
$660$	0	0
$661$	333.242i	0.504149i	0.967708	+	0.252074i	$0.0811128\pi$
$661$	333.242i	0.504149i	−0.967708	+	0.252074i	$0.918887\pi$
$662$	436.520	0.659395
$663$	0	0
$664$	− 122.894i	− 0.185081i
$665$	0	0
$666$	0	0
$667$	7.22453	0.0108314
$668$	− 148.691i	− 0.222591i
$669$	0	0
$670$	1129.74i	1.68618i
$671$	− 1453.12i	− 2.16560i
$672$	0	0
$673$	366.253	0.544209	0.272105	−	0.962268i	$-0.412280\pi$
$673$	366.253	0.544209	0.272105	+	0.962268i	$0.412280\pi$
$674$	268.510	0.398382
$675$	0	0
$676$	335.054	0.495642
$677$	− 942.533i	− 1.39222i	−0.717935	−	0.696110i	$-0.754912\pi$
$677$	− 942.533i	− 1.39222i	0.717935	−	0.696110i	$-0.245088\pi$
$678$	0	0
$679$	0	0
$680$	−383.647	−0.564186
$681$	0	0
$682$	31.6652i	0.0464299i
$683$	215.500	0.315520	0.157760	−	0.987477i	$-0.449573\pi$
$683$	215.500	0.315520	0.157760	+	0.987477i	$0.449573\pi$
$684$	0	0
$685$	619.266i	0.904038i
$686$	0	0
$687$	0	0
$688$	−221.255	−0.321591
$689$	− 1.66592i	− 0.00241787i
$690$	0	0
$691$	− 916.732i	− 1.32668i	−0.748320	−	0.663338i	$-0.769139\pi$
$691$	− 916.732i	− 1.32668i	0.748320	−	0.663338i	$-0.230861\pi$
$692$	377.296i	0.545225i
$693$	0	0
$694$	−408.638	−0.588815
$695$	−895.176	−1.28802
$696$	0	0
$697$	478.919	0.687115
$698$	479.489i	0.686947i
$699$	0	0
$700$	0	0
$701$	−541.894	−0.773029	−0.386515	−	0.922283i	$-0.626321\pi$
$701$	−541.894	−0.773029	−0.386515	+	0.922283i	$0.626321\pi$
$702$	0	0
$703$	10.3096i	0.0146651i
$704$	−141.255	−0.200646
$705$	0	0
$706$	704.319i	0.997619i
$707$	0	0
$708$	0	0
$709$	−401.497	−0.566287	−0.283143	−	0.959078i	$-0.591377\pi$
$709$	−401.497	−0.566287	−0.283143	+	0.959078i	$0.591377\pi$
$710$	− 744.791i	− 1.04900i
$711$	0	0
$712$	22.2368i	0.0312315i
$713$	3.84163i	0.00538799i
$714$	0	0
$715$	−174.794	−0.244467
$716$	419.744	0.586235
$717$	0	0
$718$	166.794	0.232304
$719$	306.327i	0.426046i	0.977047	+	0.213023i	$0.0683309\pi$
$719$	306.327i	0.426046i	−0.977047	+	0.213023i	$0.931669\pi$
$720$	0	0
$721$	0	0
$722$	510.336	0.706837
$723$	0	0
$724$	− 631.217i	− 0.871847i
$725$	99.0324	0.136596
$726$	0	0
$727$	520.704i	0.716237i	0.933676	+	0.358119i	$0.116582\pi$
$727$	520.704i	0.716237i	−0.933676	+	0.358119i	$0.883418\pi$
$728$	0	0
$729$	0	0
$730$	−1061.14	−1.45362
$731$	− 919.858i	− 1.25836i
$732$	0	0
$733$	− 359.775i	− 0.490825i	−0.969419	−	0.245412i	$-0.921077\pi$
$733$	− 359.775i	− 0.490825i	0.969419	−	0.245412i	$-0.0789234\pi$
$734$	− 247.581i	− 0.337304i
$735$	0	0
$736$	−17.1371	−0.0232841
$737$	−1729.33	−2.34645
$738$	0	0
$739$	−857.588	−1.16047	−0.580235	−	0.814449i	$-0.697039\pi$
$739$	−857.588	−1.16047	−0.580235	+	0.814449i	$0.697039\pi$
$740$	− 452.800i	− 0.611892i
$741$	0	0
$742$	0	0
$743$	863.038	1.16156	0.580779	−	0.814061i	$-0.302748\pi$
$743$	863.038	1.16156	0.580779	+	0.814061i	$0.302748\pi$
$744$	0	0
$745$	1500.78i	2.01447i
$746$	−744.098	−0.997450
$747$	0	0
$748$	− 587.261i	− 0.785108i
$749$	0	0
$750$	0	0
$751$	−1486.04	−1.97875	−0.989373	−	0.145398i	$-0.953554\pi$
$751$	−1486.04	−1.97875	−0.989373	+	0.145398i	$0.953554\pi$
$752$	233.542i	0.310561i
$753$	0	0
$754$	4.09333i	0.00542882i
$755$	1764.10i	2.33656i
$756$	0	0
$757$	480.693	0.634998	0.317499	−	0.948259i	$-0.397157\pi$
$757$	480.693	0.634998	0.317499	+	0.948259i	$0.397157\pi$
$758$	−541.990	−0.715026
$759$	0	0
$760$	8.56854	0.0112744
$761$	1222.02i	1.60581i	0.596107	+	0.802905i	$0.296713\pi$
$761$	1222.02i	1.60581i	−0.596107	+	0.802905i	$0.703287\pi$
$762$	0	0
$763$	0	0
$764$	−653.588	−0.855482
$765$	0	0
$766$	0.180242i	0 0.000235303i
$767$	−119.882	−0.156300
$768$	0	0
$769$	− 290.136i	− 0.377289i	−0.982045	−	0.188645i	$-0.939591\pi$
$769$	− 290.136i	− 0.377289i	0.982045	−	0.188645i	$-0.0604094\pi$
$770$	0	0
$771$	0	0
$772$	216.902	0.280961
$773$	− 1131.59i	− 1.46390i	−0.681359	−	0.731949i	$-0.738611\pi$
$773$	− 1131.59i	− 1.46390i	0.681359	−	0.731949i	$-0.261389\pi$
$774$	0	0
$775$	52.6603i	0.0679488i
$776$	133.538i	0.172084i
$777$	0	0
$778$	527.373	0.677857
$779$	−10.6964	−0.0137309
$780$	0	0
$781$	1140.08	1.45977
$782$	− 71.2467i	− 0.0911083i
$783$	0	0
$784$	0	0
$785$	−1489.11	−1.89696
$786$	0	0
$787$	− 667.719i	− 0.848435i	−0.905560	−	0.424218i	$-0.860549\pi$
$787$	− 667.719i	− 0.848435i	0.905560	−	0.424218i	$-0.139451\pi$
$788$	−81.0193	−0.102816
$789$	0	0
$790$	− 227.419i	− 0.287872i
$791$	0	0
$792$	0	0
$793$	99.8852	0.125959
$794$	102.554i	0.129161i
$795$	0	0
$796$	− 490.780i	− 0.616557i
$797$	779.700i	0.978294i	0.872201	+	0.489147i	$0.162692\pi$
$797$	779.700i	0.978294i	−0.872201	+	0.489147i	$0.837308\pi$
$798$	0	0
$799$	−970.940	−1.21519
$800$	−234.912	−0.293640
$801$	0	0
$802$	17.4113	0.0217098
$803$	− 1624.32i	− 2.02282i
$804$	0	0
$805$	0	0
$806$	−2.17662	−0.00270053
$807$	0	0
$808$	555.877i	0.687967i
$809$	−1379.82	−1.70559	−0.852796	−	0.522245i	$-0.825095\pi$
$809$	−1379.82	−1.70559	−0.852796	+	0.522245i	$0.825095\pi$
$810$	0	0
$811$	736.976i	0.908725i	0.890817	+	0.454363i	$0.150133\pi$
$811$	736.976i	0.908725i	−0.890817	+	0.454363i	$0.849867\pi$
$812$	0	0
$813$	0	0
$814$	693.117	0.851495
$815$	− 1297.51i	− 1.59203i
$816$	0	0
$817$	20.5445i	0.0251463i
$818$	− 262.024i	− 0.320322i
$819$	0	0
$820$	469.789	0.572913
$821$	−4.15642	−0.00506263	−0.00253132	−	0.999997i	$-0.500806\pi$
$821$	−4.15642	−0.00506263	−0.00253132	+	0.999997i	$0.500806\pi$
$822$	0	0
$823$	1228.08	1.49220	0.746098	−	0.665836i	$-0.231925\pi$
$823$	1228.08	1.49220	0.746098	+	0.665836i	$0.231925\pi$
$824$	336.836i	0.408782i
$825$	0	0
$826$	0	0
$827$	−21.2548	−0.0257011	−0.0128506	−	0.999917i	$-0.504091\pi$
$827$	−21.2548	−0.0257011	−0.0128506	+	0.999917i	$0.504091\pi$
$828$	0	0
$829$	− 1223.92i	− 1.47639i	−0.674590	−	0.738193i	$-0.735679\pi$
$829$	− 1223.92i	− 1.47639i	0.674590	−	0.738193i	$-0.264321\pi$
$830$	501.186	0.603838
$831$	0	0
$832$	− 9.70967i	− 0.0116703i
$833$	0	0
$834$	0	0
$835$	606.392	0.726218
$836$	13.1162i	0.0156892i
$837$	0	0
$838$	− 695.335i	− 0.829756i
$839$	− 73.5125i	− 0.0876192i	−0.999040	−	0.0438096i	$-0.986050\pi$
$839$	− 73.5125i	− 0.0876192i	0.999040	−	0.0438096i	$-0.0139495\pi$
$840$	0	0
$841$	−835.313	−0.993238
$842$	−666.621	−0.791712
$843$	0	0
$844$	−72.5685	−0.0859817
$845$	1366.42i	1.61706i
$846$	0	0
$847$	0	0
$848$	5.49033	0.00647445
$849$	0	0
$850$	− 976.635i	− 1.14898i
$851$	84.0892	0.0988122
$852$	0	0
$853$	− 704.025i	− 0.825352i	−0.910878	−	0.412676i	$-0.864594\pi$
$853$	− 704.025i	− 0.825352i	0.910878	−	0.412676i	$-0.135406\pi$
$854$	0	0
$855$	0	0
$856$	417.137	0.487310
$857$	161.299i	0.188213i	0.995562	+	0.0941067i	$0.0299995\pi$
$857$	161.299i	0.188213i	−0.995562	+	0.0941067i	$0.970001\pi$
$858$	0	0
$859$	902.603i	1.05076i	0.850868	+	0.525380i	$0.176077\pi$
$859$	902.603i	1.05076i	−0.850868	+	0.525380i	$0.823923\pi$
$860$	− 902.322i	− 1.04921i
$861$	0	0
$862$	646.656	0.750181
$863$	802.705	0.930133	0.465066	−	0.885276i	$-0.346030\pi$
$863$	802.705	0.930133	0.465066	+	0.885276i	$0.346030\pi$
$864$	0	0
$865$	−1538.69	−1.77883
$866$	352.056i	0.406531i
$867$	0	0
$868$	0	0
$869$	348.118	0.400596
$870$	0	0
$871$	− 118.872i	− 0.136478i
$872$	−368.784	−0.422917
$873$	0	0
$874$	1.59126i	0.00182066i
$875$	0	0
$876$	0	0
$877$	1462.85	1.66802	0.834008	−	0.551753i	$-0.186041\pi$
$877$	1462.85	1.66802	0.834008	+	0.551753i	$0.186041\pi$
$878$	608.113i	0.692612i
$879$	0	0
$880$	− 576.066i	− 0.654620i
$881$	− 851.416i	− 0.966420i	−0.875505	−	0.483210i	$-0.839471\pi$
$881$	− 851.416i	− 0.966420i	0.875505	−	0.483210i	$-0.160529\pi$
$882$	0	0
$883$	−306.303	−0.346889	−0.173444	−	0.984844i	$-0.555490\pi$
$883$	−306.303	−0.346889	−0.173444	+	0.984844i	$0.555490\pi$
$884$	40.3675	0.0456646
$885$	0	0
$886$	−537.901	−0.607111
$887$	− 560.594i	− 0.632011i	−0.948757	−	0.316005i	$-0.897658\pi$
$887$	− 560.594i	− 0.632011i	0.948757	−	0.316005i	$-0.102342\pi$
$888$	0	0
$889$	0	0
$890$	−90.6863	−0.101895
$891$	0	0
$892$	177.820i	0.199350i
$893$	21.6854	0.0242838
$894$	0	0
$895$	1711.80i	1.91263i
$896$	0	0
$897$	0	0
$898$	−461.921	−0.514389
$899$	3.02414i	0.00336389i
$900$	0	0
$901$	22.8258i	0.0253339i
$902$	719.122i	0.797252i
$903$	0	0
$904$	5.49033	0.00607338
$905$	2574.23	2.84445
$906$	0	0
$907$	1.70563	0.00188052	0.000940258	−	1.00000i	$-0.499701\pi$
$907$	1.70563	0.00188052	0.000940258		1.00000i	$0.499701\pi$
$908$	− 640.574i	− 0.705478i
$909$	0	0
$910$	0	0
$911$	629.539	0.691042	0.345521	−	0.938411i	$-0.387702\pi$
$911$	629.539	0.691042	0.345521	+	0.938411i	$0.387702\pi$
$912$	0	0
$913$	767.182i	0.840287i
$914$	−878.392	−0.961041
$915$	0	0
$916$	5.80751i	0.00634008i
$917$	0	0
$918$	0	0
$919$	−562.940	−0.612557	−0.306279	−	0.951942i	$-0.599084\pi$
$919$	−562.940	−0.612557	−0.306279	+	0.951942i	$0.599084\pi$
$920$	− 69.8885i	− 0.0759657i
$921$	0	0
$922$	241.745i	0.262196i
$923$	78.3674i	0.0849051i
$924$	0	0
$925$	1152.68	1.24614
$926$	695.529	0.751111
$927$	0	0
$928$	−13.4903	−0.0145370
$929$	184.016i	0.198080i	0.995083	+	0.0990398i	$0.0315771\pi$
$929$	184.016i	0.198080i	−0.995083	+	0.0990398i	$0.968423\pi$
$930$	0	0
$931$	0	0
$932$	33.2203	0.0356441
$933$	0	0
$934$	− 424.748i	− 0.454763i
$935$	2394.97	2.56146
$936$	0	0
$937$	200.616i	0.214104i	0.994253	+	0.107052i	$0.0341412\pi$
$937$	200.616i	0.214104i	−0.994253	+	0.107052i	$0.965859\pi$
$938$	0	0
$939$	0	0
$940$	−952.431	−1.01322
$941$	715.285i	0.760133i	0.924959	+	0.380066i	$0.124099\pi$
$941$	715.285i	0.760133i	−0.924959	+	0.380066i	$0.875901\pi$
$942$	0	0
$943$	87.2441i	0.0925176i
$944$	− 395.094i	− 0.418532i
$945$	0	0
$946$	1381.21	1.46006
$947$	−865.588	−0.914032	−0.457016	−	0.889459i	$-0.651082\pi$
$947$	−865.588	−0.914032	−0.457016	+	0.889459i	$0.651082\pi$
$948$	0	0
$949$	111.654	0.117654
$950$	21.8126i	0.0229607i
$951$	0	0
$952$	0	0
$953$	1499.57	1.57352	0.786762	−	0.617257i	$-0.211756\pi$
$953$	1499.57	1.57352	0.786762	+	0.617257i	$0.211756\pi$
$954$	0	0
$955$	− 2665.46i	− 2.79106i
$956$	794.940	0.831527
$957$	0	0
$958$	228.852i	0.238885i
$959$	0	0
$960$	0	0
$961$	959.392	0.998327
$962$	47.6439i	0.0495259i
$963$	0	0
$964$	− 604.035i	− 0.626593i
$965$	884.568i	0.916651i
$966$	0	0
$967$	606.969	0.627682	0.313841	−	0.949475i	$-0.398384\pi$
$967$	606.969	0.627682	0.313841	+	0.949475i	$0.398384\pi$
$968$	539.563	0.557400
$969$	0	0
$970$	−544.593	−0.561436
$971$	406.226i	0.418358i	0.977877	+	0.209179i	$0.0670791\pi$
$971$	406.226i	0.418358i	−0.977877	+	0.209179i	$0.932921\pi$
$972$	0	0
$973$	0	0
$974$	1089.50	1.11858
$975$	0	0
$976$	329.190i	0.337285i
$977$	−283.591	−0.290267	−0.145133	−	0.989412i	$-0.546361\pi$
$977$	−283.591	−0.290267	−0.145133	+	0.989412i	$0.546361\pi$
$978$	0	0
$979$	− 138.817i	− 0.141794i
$980$	0	0
$981$	0	0
$982$	514.177	0.523601
$983$	956.266i	0.972803i	0.873735	+	0.486402i	$0.161691\pi$
$983$	956.266i	0.972803i	−0.873735	+	0.486402i	$0.838309\pi$
$984$	0	0
$985$	− 330.413i	− 0.335445i
$986$	− 56.0855i	− 0.0568818i
$987$	0	0
$988$	−0.901587	−0.000912537 0
$989$	167.569	0.169433
$990$	0	0
$991$	−945.568	−0.954155	−0.477078	−	0.878861i	$-0.658304\pi$
$991$	−945.568	−0.954155	−0.477078	+	0.878861i	$0.658304\pi$
$992$	− 7.17346i	− 0.00723131i
$993$	0	0
$994$	0	0
$995$	2001.50	2.01156
$996$	0	0
$997$	940.358i	0.943187i	0.881816	+	0.471594i	$0.156321\pi$
$997$	940.358i	0.943187i	−0.881816	+	0.471594i	$0.843679\pi$
$998$	317.352	0.317988
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.3.c.a.685.4		4
3.2	odd	2		98.3.b.a.97.2	yes	4
7.2	even	3		882.3.n.j.325.2		8
7.3	odd	6		882.3.n.j.19.2		8
7.4	even	3		882.3.n.j.19.1		8
7.5	odd	6		882.3.n.j.325.1		8
7.6	odd	2	inner	882.3.c.a.685.3		4
12.11	even	2		784.3.c.b.97.2		4
21.2	odd	6		98.3.d.b.31.3		8
21.5	even	6		98.3.d.b.31.4		8
21.11	odd	6		98.3.d.b.19.4		8
21.17	even	6		98.3.d.b.19.3		8
21.20	even	2		98.3.b.a.97.1	✓	4
84.11	even	6		784.3.s.j.705.2		8
84.23	even	6		784.3.s.j.129.3		8
84.47	odd	6		784.3.s.j.129.2		8
84.59	odd	6		784.3.s.j.705.3		8
84.83	odd	2		784.3.c.b.97.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
98.3.b.a.97.1	✓	4	21.20	even	2
98.3.b.a.97.2	yes	4	3.2	odd	2
98.3.d.b.19.3		8	21.17	even	6
98.3.d.b.19.4		8	21.11	odd	6
98.3.d.b.31.3		8	21.2	odd	6
98.3.d.b.31.4		8	21.5	even	6
784.3.c.b.97.2		4	12.11	even	2
784.3.c.b.97.3		4	84.83	odd	2
784.3.s.j.129.2		8	84.47	odd	6
784.3.s.j.129.3		8	84.23	even	6
784.3.s.j.705.2		8	84.11	even	6
784.3.s.j.705.3		8	84.59	odd	6
882.3.c.a.685.3		4	7.6	odd	2	inner
882.3.c.a.685.4		4	1.1	even	1	trivial
882.3.n.j.19.1		8	7.4	even	3
882.3.n.j.19.2		8	7.3	odd	6
882.3.n.j.325.1		8	7.5	odd	6
882.3.n.j.325.2		8	7.2	even	3

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)

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} +2.00000 q^{4} +8.15640i q^{5} +2.82843 q^{8} +O(q^{10})\)	\(q+1.41421 q^{2} +2.00000 q^{4} +8.15640i q^{5} +2.82843 q^{8} +11.5349i q^{10} -17.6569 q^{11} -1.21371i q^{13} +4.00000 q^{16} +16.6298i q^{17} -0.371418i q^{19} +16.3128i q^{20} -24.9706 q^{22} -3.02944 q^{23} -41.5269 q^{25} -1.71644i q^{26} -2.38478 q^{29} -1.26810i q^{31} +5.65685 q^{32} +23.5181i q^{34} -27.7574 q^{37} -0.525265i q^{38} +23.0698i q^{40} -28.7988i q^{41} -55.3137 q^{43} -35.3137 q^{44} -4.28427 q^{46} +58.3855i q^{47} -58.7279 q^{50} -2.42742i q^{52} +1.37258 q^{53} -144.016i q^{55} -3.37258 q^{58} -98.7735i q^{59} +82.2975i q^{61} -1.79337i q^{62} +8.00000 q^{64} +9.89949 q^{65} +97.9411 q^{67} +33.2597i q^{68} -64.5685 q^{71} +91.9940i q^{73} -39.2548 q^{74} -0.742837i q^{76} -19.7157 q^{79} +32.6256i q^{80} -40.7276i q^{82} -43.4495i q^{83} -135.640 q^{85} -78.2254 q^{86} -49.9411 q^{88} +7.86191i q^{89} -6.05887 q^{92} +82.5695i q^{94} +3.02944 q^{95} +47.2126i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{4}+O(q^{10}) \) 4 * q + 8 * q^4	\( 4 q + 8 q^{4} - 48 q^{11} + 16 q^{16} - 32 q^{22} - 80 q^{23} - 36 q^{25} + 64 q^{29} - 128 q^{37} - 176 q^{43} - 96 q^{44} + 96 q^{46} - 184 q^{50} + 96 q^{53} - 104 q^{58} + 32 q^{64} + 256 q^{67} - 32 q^{71} + 24 q^{74} - 192 q^{79} - 288 q^{85} - 64 q^{86} - 64 q^{88} - 160 q^{92} + 80 q^{95}+O(q^{100}) \) 4 * q + 8 * q^4 - 48 * q^11 + 16 * q^16 - 32 * q^22 - 80 * q^23 - 36 * q^25 + 64 * q^29 - 128 * q^37 - 176 * q^43 - 96 * q^44 + 96 * q^46 - 184 * q^50 + 96 * q^53 - 104 * q^58 + 32 * q^64 + 256 * q^67 - 32 * q^71 + 24 * q^74 - 192 * q^79 - 288 * q^85 - 64 * q^86 - 64 * q^88 - 160 * q^92 + 80 * q^95

Introduction

L-functions

Modular forms

Varieties

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Database

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