LMFDB - Embedded newform 882.5.c.b.685.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [882,5,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, 5, names="a")

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

chi := DirichletCharacter("882.685");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 882 = 2 \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 5 $
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:	$91.1723074400$
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^{4} $
Twist minimal:	no (minimal twist has level 14)
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			882.5.c.b.685.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.82843 q^{2} +8.00000 q^{4} -5.79050i q^{5} +22.6274 q^{8} +O(q^{10})$	$q+2.82843 q^{2} +8.00000 q^{4} -5.79050i q^{5} +22.6274 q^{8} -16.3780i q^{10} -10.0294 q^{11} +190.220i q^{13} +64.0000 q^{16} +421.771i q^{17} -432.248i q^{19} -46.3240i q^{20} -28.3675 q^{22} -921.823 q^{23} +591.470 q^{25} +538.022i q^{26} -877.882 q^{29} -724.200i q^{31} +181.019 q^{32} +1192.95i q^{34} -540.675 q^{37} -1222.58i q^{38} -131.024i q^{40} +894.280i q^{41} -1246.82 q^{43} -80.2355 q^{44} -2607.31 q^{46} -1750.73i q^{47} +1672.93 q^{50} +1521.76i q^{52} -812.203 q^{53} +58.0754i q^{55} -2483.03 q^{58} +2854.51i q^{59} +5834.20i q^{61} -2048.35i q^{62} +512.000 q^{64} +1101.47 q^{65} -2203.79 q^{67} +3374.17i q^{68} -3408.24 q^{71} +9395.66i q^{73} -1529.26 q^{74} -3457.98i q^{76} +2352.23 q^{79} -370.592i q^{80} +2529.41i q^{82} -3750.16i q^{83} +2442.26 q^{85} -3526.54 q^{86} -226.940 q^{88} +6363.96i q^{89} -7374.58 q^{92} -4951.81i q^{94} -2502.93 q^{95} +6370.47i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 32 q^{4}+O(q^{10}) $ 4 * q + 32 * q^4	$ 4 q + 32 q^{4} - 108 q^{11} + 256 q^{16} + 192 q^{22} - 972 q^{23} + 1144 q^{25} - 3240 q^{29} + 892 q^{37} + 2344 q^{43} - 864 q^{44} - 7680 q^{46} + 3456 q^{50} + 5508 q^{53} - 768 q^{58} + 2048 q^{64} - 6048 q^{65} + 10124 q^{67} - 18792 q^{71} - 8640 q^{74} - 1588 q^{79} + 10380 q^{85} - 20736 q^{86} + 1536 q^{88} - 7776 q^{92} + 17820 q^{95}+O(q^{100}) $ 4 * q + 32 * q^4 - 108 * q^11 + 256 * q^16 + 192 * q^22 - 972 * q^23 + 1144 * q^25 - 3240 * q^29 + 892 * q^37 + 2344 * q^43 - 864 * q^44 - 7680 * q^46 + 3456 * q^50 + 5508 * q^53 - 768 * q^58 + 2048 * q^64 - 6048 * q^65 + 10124 * q^67 - 18792 * q^71 - 8640 * q^74 - 1588 * q^79 + 10380 * q^85 - 20736 * q^86 + 1536 * q^88 - 7776 * q^92 + 17820 * 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 $

$2$	2.82843	0.707107
$2$	2.82843	0.707107
$3$	0	0
$3$	0	0
$4$	8.00000	0.500000
$5$	− 5.79050i	− 0.231620i	−0.993271	−	0.115810i	$-0.963054\pi$
$5$	− 5.79050i	− 0.231620i	0.993271	−	0.115810i	$-0.0369464\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	22.6274	0.353553
$9$	0	0
$10$	− 16.3780i	− 0.163780i
$11$	−10.0294	−0.0828879	−0.0414440	−	0.999141i	$-0.513196\pi$
$11$	−10.0294	−0.0828879	−0.0414440	+	0.999141i	$0.513196\pi$
$12$	0	0
$13$	190.220i	1.12556i	0.826607	+	0.562780i	$0.190268\pi$
$13$	190.220i	1.12556i	−0.826607	+	0.562780i	$0.809732\pi$
$14$	0	0
$15$	0	0
$16$	64.0000	0.250000
$17$	421.771i	1.45942i	0.683759	+	0.729708i	$0.260344\pi$
$17$	421.771i	1.45942i	−0.683759	+	0.729708i	$0.739656\pi$
$18$	0	0
$19$	− 432.248i	− 1.19736i	−0.800987	−	0.598681i	$-0.795692\pi$
$19$	− 432.248i	− 1.19736i	0.800987	−	0.598681i	$-0.204308\pi$
$20$	− 46.3240i	− 0.115810i
$21$	0	0
$22$	−28.3675	−0.0586106
$23$	−921.823	−1.74258	−0.871288	−	0.490772i	$-0.836715\pi$
$23$	−921.823	−1.74258	−0.871288	+	0.490772i	$0.836715\pi$
$24$	0	0
$25$	591.470	0.946352
$26$	538.022i	0.795891i
$27$	0	0
$28$	0	0
$29$	−877.882	−1.04386	−0.521928	−	0.852990i	$-0.674787\pi$
$29$	−877.882	−1.04386	−0.521928	+	0.852990i	$0.674787\pi$
$30$	0	0
$31$	− 724.200i	− 0.753590i	−0.926297	−	0.376795i	$-0.877026\pi$
$31$	− 724.200i	− 0.753590i	0.926297	−	0.376795i	$-0.122974\pi$
$32$	181.019	0.176777
$33$	0	0
$34$	1192.95i	1.03196i
$35$	0	0
$36$	0	0
$37$	−540.675	−0.394942	−0.197471	−	0.980309i	$-0.563273\pi$
$37$	−540.675	−0.394942	−0.197471	+	0.980309i	$0.563273\pi$
$38$	− 1222.58i	− 0.846663i
$39$	0	0
$40$	− 131.024i	− 0.0818900i
$41$	894.280i	0.531993i	0.963974	+	0.265996i	$0.0857009\pi$
$41$	894.280i	0.531993i	−0.963974	+	0.265996i	$0.914299\pi$
$42$	0	0
$43$	−1246.82	−0.674322	−0.337161	−	0.941447i	$-0.609467\pi$
$43$	−1246.82	−0.674322	−0.337161	+	0.941447i	$0.609467\pi$
$44$	−80.2355	−0.0414440
$45$	0	0
$46$	−2607.31	−1.23219
$47$	− 1750.73i	− 0.792543i	−0.918133	−	0.396271i	$-0.870304\pi$
$47$	− 1750.73i	− 0.792543i	0.918133	−	0.396271i	$-0.129696\pi$
$48$	0	0
$49$	0	0
$50$	1672.93	0.669172
$51$	0	0
$52$	1521.76i	0.562780i
$53$	−812.203	−0.289143	−0.144571	−	0.989494i	$-0.546180\pi$
$53$	−812.203	−0.289143	−0.144571	+	0.989494i	$0.546180\pi$
$54$	0	0
$55$	58.0754i	0.0191985i
$56$	0	0
$57$	0	0
$58$	−2483.03	−0.738117
$59$	2854.51i	0.820025i	0.912080	+	0.410012i	$0.134476\pi$
$59$	2854.51i	0.820025i	−0.912080	+	0.410012i	$0.865524\pi$
$60$	0	0
$61$	5834.20i	1.56791i	0.620816	+	0.783956i	$0.286801\pi$
$61$	5834.20i	1.56791i	−0.620816	+	0.783956i	$0.713199\pi$
$62$	− 2048.35i	− 0.532868i
$63$	0	0
$64$	512.000	0.125000
$65$	1101.47	0.260702
$66$	0	0
$67$	−2203.79	−0.490930	−0.245465	−	0.969405i	$-0.578941\pi$
$67$	−2203.79	−0.490930	−0.245465	+	0.969405i	$0.578941\pi$
$68$	3374.17i	0.729708i
$69$	0	0
$70$	0	0
$71$	−3408.24	−0.676103	−0.338052	−	0.941128i	$-0.609768\pi$
$71$	−3408.24	−0.676103	−0.338052	+	0.941128i	$0.609768\pi$
$72$	0	0
$73$	9395.66i	1.76312i	0.472074	+	0.881559i	$0.343506\pi$
$73$	9395.66i	1.76312i	−0.472074	+	0.881559i	$0.656494\pi$
$74$	−1529.26	−0.279266
$75$	0	0
$76$	− 3457.98i	− 0.598681i
$77$	0	0
$78$	0	0
$79$	2352.23	0.376900	0.188450	−	0.982083i	$-0.439654\pi$
$79$	2352.23	0.376900	0.188450	+	0.982083i	$0.439654\pi$
$80$	− 370.592i	− 0.0579050i
$81$	0	0
$82$	2529.41i	0.376176i
$83$	− 3750.16i	− 0.544370i	−0.962245	−	0.272185i	$-0.912254\pi$
$83$	− 3750.16i	− 0.544370i	0.962245	−	0.272185i	$-0.0877462\pi$
$84$	0	0
$85$	2442.26	0.338030
$86$	−3526.54	−0.476817
$87$	0	0
$88$	−226.940	−0.0293053
$89$	6363.96i	0.803429i	0.915765	+	0.401714i	$0.131586\pi$
$89$	6363.96i	0.803429i	−0.915765	+	0.401714i	$0.868414\pi$
$90$	0	0
$91$	0	0
$92$	−7374.58	−0.871288
$93$	0	0
$94$	− 4951.81i	− 0.560413i
$95$	−2502.93	−0.277333
$96$	0	0
$97$	6370.47i	0.677062i	0.940955	+	0.338531i	$0.109930\pi$
$97$	6370.47i	0.677062i	−0.940955	+	0.338531i	$0.890070\pi$
$98$	0	0
$99$	0	0
$100$	4731.76	0.473176
$101$	9110.83i	0.893131i	0.894751	+	0.446566i	$0.147353\pi$
$101$	9110.83i	0.893131i	−0.894751	+	0.446566i	$0.852647\pi$
$102$	0	0
$103$	− 16111.8i	− 1.51869i	−0.650686	−	0.759347i	$-0.725518\pi$
$103$	− 16111.8i	− 1.51869i	0.650686	−	0.759347i	$-0.274482\pi$
$104$	4304.18i	0.397946i
$105$	0	0
$106$	−2297.26	−0.204455
$107$	−20551.4	−1.79504	−0.897518	−	0.440977i	$-0.854632\pi$
$107$	−20551.4	−1.79504	−0.897518	+	0.440977i	$0.854632\pi$
$108$	0	0
$109$	−14257.7	−1.20005	−0.600023	−	0.799983i	$-0.704842\pi$
$109$	−14257.7	−1.20005	−0.600023	+	0.799983i	$0.704842\pi$
$110$	164.262i	0.0135754i
$111$	0	0
$112$	0	0
$113$	10304.7	0.807010	0.403505	−	0.914978i	$-0.367792\pi$
$113$	10304.7	0.807010	0.403505	+	0.914978i	$0.367792\pi$
$114$	0	0
$115$	5337.81i	0.403615i
$116$	−7023.06	−0.521928
$117$	0	0
$118$	8073.76i	0.579845i
$119$	0	0
$120$	0	0
$121$	−14540.4	−0.993130
$122$	16501.6i	1.10868i
$123$	0	0
$124$	− 5793.60i	− 0.376795i
$125$	− 7043.97i	− 0.450814i
$126$	0	0
$127$	−2116.70	−0.131236	−0.0656179	−	0.997845i	$-0.520902\pi$
$127$	−2116.70	−0.131236	−0.0656179	+	0.997845i	$0.520902\pi$
$128$	1448.15	0.0883883
$129$	0	0
$130$	3115.42	0.184344
$131$	6057.15i	0.352960i	0.984304	+	0.176480i	$0.0564711\pi$
$131$	6057.15i	0.352960i	−0.984304	+	0.176480i	$0.943529\pi$
$132$	0	0
$133$	0	0
$134$	−6233.25	−0.347140
$135$	0	0
$136$	9543.59i	0.515981i
$137$	−2860.25	−0.152392	−0.0761962	−	0.997093i	$-0.524278\pi$
$137$	−2860.25	−0.152392	−0.0761962	+	0.997093i	$0.524278\pi$
$138$	0	0
$139$	16966.2i	0.878124i	0.898457	+	0.439062i	$0.144689\pi$
$139$	16966.2i	0.878124i	−0.898457	+	0.439062i	$0.855311\pi$
$140$	0	0
$141$	0	0
$142$	−9639.95	−0.478077
$143$	− 1907.80i	− 0.0932953i
$144$	0	0
$145$	5083.38i	0.241778i
$146$	26574.9i	1.24671i
$147$	0	0
$148$	−4325.40	−0.197471
$149$	14910.1	0.671596	0.335798	−	0.941934i	$-0.390994\pi$
$149$	14910.1	0.671596	0.335798	+	0.941934i	$0.390994\pi$
$150$	0	0
$151$	−2161.45	−0.0947961	−0.0473981	−	0.998876i	$-0.515093\pi$
$151$	−2161.45	−0.0947961	−0.0473981	+	0.998876i	$0.515093\pi$
$152$	− 9780.65i	− 0.423332i
$153$	0	0
$154$	0	0
$155$	−4193.48	−0.174546
$156$	0	0
$157$	41553.7i	1.68582i	0.538056	+	0.842909i	$0.319159\pi$
$157$	41553.7i	1.68582i	−0.538056	+	0.842909i	$0.680841\pi$
$158$	6653.11	0.266508
$159$	0	0
$160$	− 1048.19i	− 0.0409450i
$161$	0	0
$162$	0	0
$163$	−11765.1	−0.442814	−0.221407	−	0.975182i	$-0.571065\pi$
$163$	−11765.1	−0.442814	−0.221407	+	0.975182i	$0.571065\pi$
$164$	7154.24i	0.265996i
$165$	0	0
$166$	− 10607.1i	− 0.384927i
$167$	13600.3i	0.487660i	0.969818	+	0.243830i	$0.0784038\pi$
$167$	13600.3i	0.487660i	−0.969818	+	0.243830i	$0.921596\pi$
$168$	0	0
$169$	−7622.52	−0.266886
$170$	6907.77	0.239023
$171$	0	0
$172$	−9974.57	−0.337161
$173$	− 45153.1i	− 1.50867i	−0.656487	−	0.754337i	$-0.727958\pi$
$173$	− 45153.1i	− 1.50867i	0.656487	−	0.754337i	$-0.272042\pi$
$174$	0	0
$175$	0	0
$176$	−641.884	−0.0207220
$177$	0	0
$178$	18000.0i	0.568110i
$179$	−19852.4	−0.619593	−0.309797	−	0.950803i	$-0.600261\pi$
$179$	−19852.4	−0.619593	−0.309797	+	0.950803i	$0.600261\pi$
$180$	0	0
$181$	− 21398.5i	− 0.653170i	−0.945168	−	0.326585i	$-0.894102\pi$
$181$	− 21398.5i	− 0.653170i	0.945168	−	0.326585i	$-0.105898\pi$
$182$	0	0
$183$	0	0
$184$	−20858.5	−0.616094
$185$	3130.78i	0.0914764i
$186$	0	0
$187$	− 4230.13i	− 0.120968i
$188$	− 14005.8i	− 0.396271i
$189$	0	0
$190$	−7079.36	−0.196104
$191$	−11625.9	−0.318683	−0.159341	−	0.987224i	$-0.550937\pi$
$191$	−11625.9	−0.318683	−0.159341	+	0.987224i	$0.550937\pi$
$192$	0	0
$193$	−52546.9	−1.41069	−0.705347	−	0.708862i	$-0.749209\pi$
$193$	−52546.9	−1.41069	−0.705347	+	0.708862i	$0.749209\pi$
$194$	18018.4i	0.478755i
$195$	0	0
$196$	0	0
$197$	54811.7	1.41235	0.706173	−	0.708039i	$-0.250420\pi$
$197$	54811.7	1.41235	0.706173	+	0.708039i	$0.250420\pi$
$198$	0	0
$199$	− 5972.87i	− 0.150826i	−0.997152	−	0.0754131i	$-0.975972\pi$
$199$	− 5972.87i	− 0.150826i	0.997152	−	0.0754131i	$-0.0240276\pi$
$200$	13383.4	0.334586
$201$	0	0
$202$	25769.3i	0.631539i
$203$	0	0
$204$	0	0
$205$	5178.33	0.123220
$206$	− 45571.1i	− 1.07388i
$207$	0	0
$208$	12174.1i	0.281390i
$209$	4335.20i	0.0992469i
$210$	0	0
$211$	3042.82	0.0683457	0.0341729	−	0.999416i	$-0.489120\pi$
$211$	3042.82	0.0683457	0.0341729	+	0.999416i	$0.489120\pi$
$212$	−6497.62	−0.144571
$213$	0	0
$214$	−58128.1	−1.26928
$215$	7219.71i	0.156186i
$216$	0	0
$217$	0	0
$218$	−40327.0	−0.848560
$219$	0	0
$220$	464.604i	0.00959925i
$221$	−80229.2	−1.64266
$222$	0	0
$223$	42104.7i	0.846683i	0.905970	+	0.423341i	$0.139143\pi$
$223$	42104.7i	0.846683i	−0.905970	+	0.423341i	$0.860857\pi$
$224$	0	0
$225$	0	0
$226$	29146.1	0.570642
$227$	92189.5i	1.78908i	0.446989	+	0.894540i	$0.352496\pi$
$227$	92189.5i	1.78908i	−0.446989	+	0.894540i	$0.647504\pi$
$228$	0	0
$229$	− 23963.1i	− 0.456953i	−0.973549	−	0.228477i	$-0.926626\pi$
$229$	− 23963.1i	− 0.456953i	0.973549	−	0.228477i	$-0.0733745\pi$
$230$	15097.6i	0.285399i
$231$	0	0
$232$	−19864.2	−0.369059
$233$	35348.1	0.651110	0.325555	−	0.945523i	$-0.394449\pi$
$233$	35348.1	0.651110	0.325555	+	0.945523i	$0.394449\pi$
$234$	0	0
$235$	−10137.6	−0.183569
$236$	22836.0i	0.410012i
$237$	0	0
$238$	0	0
$239$	−14393.5	−0.251983	−0.125992	−	0.992031i	$-0.540211\pi$
$239$	−14393.5	−0.251983	−0.125992	+	0.992031i	$0.540211\pi$
$240$	0	0
$241$	− 7499.94i	− 0.129129i	−0.997914	−	0.0645645i	$-0.979434\pi$
$241$	− 7499.94i	− 0.129129i	0.997914	−	0.0645645i	$-0.0205658\pi$
$242$	−41126.5	−0.702249
$243$	0	0
$244$	46673.6i	0.783956i
$245$	0	0
$246$	0	0
$247$	82222.1	1.34770
$248$	− 16386.8i	− 0.266434i
$249$	0	0
$250$	− 19923.4i	− 0.318774i
$251$	45414.9i	0.720860i	0.932786	+	0.360430i	$0.117370\pi$
$251$	45414.9i	0.720860i	−0.932786	+	0.360430i	$0.882630\pi$
$252$	0	0
$253$	9245.36	0.144438
$254$	−5986.94	−0.0927977
$255$	0	0
$256$	4096.00	0.0625000
$257$	− 51727.9i	− 0.783175i	−0.920141	−	0.391588i	$-0.871926\pi$
$257$	− 51727.9i	− 0.783175i	0.920141	−	0.391588i	$-0.128074\pi$
$258$	0	0
$259$	0	0
$260$	8811.73	0.130351
$261$	0	0
$262$	17132.2i	0.249580i
$263$	89833.9	1.29876	0.649380	−	0.760464i	$-0.275029\pi$
$263$	89833.9	1.29876	0.649380	+	0.760464i	$0.275029\pi$
$264$	0	0
$265$	4703.06i	0.0669713i
$266$	0	0
$267$	0	0
$268$	−17630.3	−0.245465
$269$	− 23763.8i	− 0.328406i	−0.986427	−	0.164203i	$-0.947495\pi$
$269$	− 23763.8i	− 0.328406i	0.986427	−	0.164203i	$-0.0525052\pi$
$270$	0	0
$271$	− 110819.i	− 1.50895i	−0.656331	−	0.754473i	$-0.727892\pi$
$271$	− 110819.i	− 1.50895i	0.656331	−	0.754473i	$-0.272108\pi$
$272$	26993.4i	0.364854i
$273$	0	0
$274$	−8090.01	−0.107758
$275$	−5932.11	−0.0784412
$276$	0	0
$277$	−7122.38	−0.0928252	−0.0464126	−	0.998922i	$-0.514779\pi$
$277$	−7122.38	−0.0928252	−0.0464126	+	0.998922i	$0.514779\pi$
$278$	47987.7i	0.620927i
$279$	0	0
$280$	0	0
$281$	−90209.2	−1.14245	−0.571226	−	0.820792i	$-0.693532\pi$
$281$	−90209.2	−1.14245	−0.571226	+	0.820792i	$0.693532\pi$
$282$	0	0
$283$	72449.3i	0.904610i	0.891863	+	0.452305i	$0.149398\pi$
$283$	72449.3i	0.904610i	−0.891863	+	0.452305i	$0.850602\pi$
$284$	−27265.9	−0.338052
$285$	0	0
$286$	− 5396.06i	− 0.0659698i
$287$	0	0
$288$	0	0
$289$	−94369.9	−1.12989
$290$	14378.0i	0.170963i
$291$	0	0
$292$	75165.2i	0.881559i
$293$	− 89167.8i	− 1.03866i	−0.854574	−	0.519329i	$-0.826182\pi$
$293$	− 89167.8i	− 1.03866i	0.854574	−	0.519329i	$-0.173818\pi$
$294$	0	0
$295$	16529.0	0.189934
$296$	−12234.1	−0.139633
$297$	0	0
$298$	42172.2	0.474890
$299$	− 175349.i	− 1.96137i
$300$	0	0
$301$	0	0
$302$	−6113.49	−0.0670310
$303$	0	0
$304$	− 27663.9i	− 0.299341i
$305$	33782.9	0.363160
$306$	0	0
$307$	113110.i	1.20012i	0.799954	+	0.600061i	$0.204857\pi$
$307$	113110.i	1.20012i	−0.799954	+	0.600061i	$0.795143\pi$
$308$	0	0
$309$	0	0
$310$	−11860.9	−0.123423
$311$	− 65344.2i	− 0.675595i	−0.941219	−	0.337797i	$-0.890318\pi$
$311$	− 65344.2i	− 0.675595i	0.941219	−	0.337797i	$-0.109682\pi$
$312$	0	0
$313$	1958.86i	0.0199947i	0.999950	+	0.00999734i	$0.00318231\pi$
$313$	1958.86i	0.0199947i	−0.999950	+	0.00999734i	$0.996818\pi$
$314$	117532.i	1.19205i
$315$	0	0
$316$	18817.8	0.188450
$317$	86151.2	0.857320	0.428660	−	0.903466i	$-0.358986\pi$
$317$	86151.2	0.857320	0.428660	+	0.903466i	$0.358986\pi$
$318$	0	0
$319$	8804.66	0.0865230
$320$	− 2964.74i	− 0.0289525i
$321$	0	0
$322$	0	0
$323$	182310.	1.74745
$324$	0	0
$325$	112509.i	1.06518i
$326$	−33276.8	−0.313117
$327$	0	0
$328$	20235.2i	0.188088i
$329$	0	0
$330$	0	0
$331$	−90622.3	−0.827140	−0.413570	−	0.910472i	$-0.635718\pi$
$331$	−90622.3	−0.827140	−0.413570	+	0.910472i	$0.635718\pi$
$332$	− 30001.3i	− 0.272185i
$333$	0	0
$334$	38467.6i	0.344827i
$335$	12761.0i	0.113709i
$336$	0	0
$337$	−26197.5	−0.230675	−0.115338	−	0.993326i	$-0.536795\pi$
$337$	−26197.5	−0.230675	−0.115338	+	0.993326i	$0.536795\pi$
$338$	−21559.7	−0.188717
$339$	0	0
$340$	19538.1	0.169015
$341$	7263.32i	0.0624635i
$342$	0	0
$343$	0	0
$344$	−28212.3	−0.238409
$345$	0	0
$346$	− 127712.i	− 1.06679i
$347$	160661.	1.33429	0.667146	−	0.744927i	$-0.267516\pi$
$347$	160661.	1.33429	0.667146	+	0.744927i	$0.267516\pi$
$348$	0	0
$349$	− 18120.2i	− 0.148769i	−0.997230	−	0.0743845i	$-0.976301\pi$
$349$	− 18120.2i	− 0.148769i	0.997230	−	0.0743845i	$-0.0236992\pi$
$350$	0	0
$351$	0	0
$352$	−1815.52	−0.0146527
$353$	− 13229.5i	− 0.106168i	−0.998590	−	0.0530839i	$-0.983095\pi$
$353$	− 13229.5i	− 0.106168i	0.998590	−	0.0530839i	$-0.0169051\pi$
$354$	0	0
$355$	19735.4i	0.156599i
$356$	50911.7i	0.401714i
$357$	0	0
$358$	−56151.0	−0.438119
$359$	107735.	0.835924	0.417962	−	0.908464i	$-0.362744\pi$
$359$	107735.	0.835924	0.417962	+	0.908464i	$0.362744\pi$
$360$	0	0
$361$	−56517.2	−0.433677
$362$	− 60524.1i	− 0.461861i
$363$	0	0
$364$	0	0
$365$	54405.5	0.408373
$366$	0	0
$367$	− 70427.2i	− 0.522887i	−0.965219	−	0.261444i	$-0.915801\pi$
$367$	− 70427.2i	− 0.522887i	0.965219	−	0.261444i	$-0.0841986\pi$
$368$	−58996.6	−0.435644
$369$	0	0
$370$	8855.18i	0.0646836i
$371$	0	0
$372$	0	0
$373$	191053.	1.37321	0.686605	−	0.727030i	$-0.259100\pi$
$373$	191053.	1.37321	0.686605	+	0.727030i	$0.259100\pi$
$374$	− 11964.6i	− 0.0855372i
$375$	0	0
$376$	− 39614.4i	− 0.280206i
$377$	− 166990.i	− 1.17492i
$378$	0	0
$379$	74979.5	0.521992	0.260996	−	0.965340i	$-0.415949\pi$
$379$	74979.5	0.521992	0.260996	+	0.965340i	$0.415949\pi$
$380$	−20023.4	−0.138667
$381$	0	0
$382$	−32882.9	−0.225343
$383$	128637.i	0.876936i	0.898747	+	0.438468i	$0.144479\pi$
$383$	128637.i	0.876936i	−0.898747	+	0.438468i	$0.855521\pi$
$384$	0	0
$385$	0	0
$386$	−148625.	−0.997511
$387$	0	0
$388$	50963.8i	0.338531i
$389$	−26308.0	−0.173856	−0.0869278	−	0.996215i	$-0.527705\pi$
$389$	−26308.0	−0.173856	−0.0869278	+	0.996215i	$0.527705\pi$
$390$	0	0
$391$	− 388798.i	− 2.54314i
$392$	0	0
$393$	0	0
$394$	155031.	0.998679
$395$	− 13620.6i	− 0.0872975i
$396$	0	0
$397$	− 88026.3i	− 0.558511i	−0.960217	−	0.279255i	$-0.909912\pi$
$397$	− 88026.3i	− 0.558511i	0.960217	−	0.279255i	$-0.0900876\pi$
$398$	− 16893.8i	− 0.106650i
$399$	0	0
$400$	37854.1	0.236588
$401$	−264600.	−1.64551	−0.822755	−	0.568397i	$-0.807564\pi$
$401$	−264600.	−1.64551	−0.822755	+	0.568397i	$0.807564\pi$
$402$	0	0
$403$	137757.	0.848211
$404$	72886.6i	0.446566i
$405$	0	0
$406$	0	0
$407$	5422.67	0.0327359
$408$	0	0
$409$	− 140171.i	− 0.837937i	−0.908001	−	0.418968i	$-0.862392\pi$
$409$	− 140171.i	− 0.837937i	0.908001	−	0.418968i	$-0.137608\pi$
$410$	14646.5	0.0871298
$411$	0	0
$412$	− 128895.i	− 0.759347i
$413$	0	0
$414$	0	0
$415$	−21715.3	−0.126087
$416$	34433.4i	0.198973i
$417$	0	0
$418$	12261.8i	0.0701781i
$419$	− 319409.i	− 1.81936i	−0.415306	−	0.909682i	$-0.636325\pi$
$419$	− 319409.i	− 1.81936i	0.415306	−	0.909682i	$-0.363675\pi$
$420$	0	0
$421$	315726.	1.78134	0.890670	−	0.454651i	$-0.150236\pi$
$421$	315726.	1.78134	0.890670	+	0.454651i	$0.150236\pi$
$422$	8606.40	0.0483277
$423$	0	0
$424$	−18378.0	−0.102227
$425$	249465.i	1.38112i
$426$	0	0
$427$	0	0
$428$	−164411.	−0.897518
$429$	0	0
$430$	20420.4i	0.110440i
$431$	−168531.	−0.907245	−0.453623	−	0.891194i	$-0.649869\pi$
$431$	−168531.	−0.907245	−0.453623	+	0.891194i	$0.649869\pi$
$432$	0	0
$433$	17104.3i	0.0912284i	0.998959	+	0.0456142i	$0.0145245\pi$
$433$	17104.3i	0.0912284i	−0.998959	+	0.0456142i	$0.985476\pi$
$434$	0	0
$435$	0	0
$436$	−114062.	−0.600023
$437$	398456.i	2.08649i
$438$	0	0
$439$	221981.i	1.15183i	0.817511	+	0.575913i	$0.195353\pi$
$439$	221981.i	1.15183i	−0.817511	+	0.575913i	$0.804647\pi$
$440$	1314.10i	0.00678769i
$441$	0	0
$442$	−226922.	−1.16154
$443$	−34230.7	−0.174425	−0.0872124	−	0.996190i	$-0.527796\pi$
$443$	−34230.7	−0.174425	−0.0872124	+	0.996190i	$0.527796\pi$
$444$	0	0
$445$	36850.5	0.186090
$446$	119090.i	0.598695i
$447$	0	0
$448$	0	0
$449$	187206.	0.928598	0.464299	−	0.885679i	$-0.346306\pi$
$449$	187206.	0.928598	0.464299	+	0.885679i	$0.346306\pi$
$450$	0	0
$451$	− 8969.13i	− 0.0440958i
$452$	82437.6	0.403505
$453$	0	0
$454$	260751.i	1.26507i
$455$	0	0
$456$	0	0
$457$	147845.	0.707904	0.353952	−	0.935264i	$-0.384838\pi$
$457$	147845.	0.707904	0.353952	+	0.935264i	$0.384838\pi$
$458$	− 67777.9i	− 0.323115i
$459$	0	0
$460$	42702.5i	0.201808i
$461$	73979.5i	0.348105i	0.984736	+	0.174052i	$0.0556862\pi$
$461$	73979.5i	0.348105i	−0.984736	+	0.174052i	$0.944314\pi$
$462$	0	0
$463$	−66987.3	−0.312486	−0.156243	−	0.987719i	$-0.549938\pi$
$463$	−66987.3	−0.312486	−0.156243	+	0.987719i	$0.549938\pi$
$464$	−56184.5	−0.260964
$465$	0	0
$466$	99979.6	0.460405
$467$	− 232034.i	− 1.06394i	−0.846763	−	0.531970i	$-0.821452\pi$
$467$	− 232034.i	− 1.06394i	0.846763	−	0.531970i	$-0.178548\pi$
$468$	0	0
$469$	0	0
$470$	−28673.4	−0.129803
$471$	0	0
$472$	64590.1i	0.289922i
$473$	12504.9	0.0558931
$474$	0	0
$475$	− 255662.i	− 1.13313i
$476$	0	0
$477$	0	0
$478$	−40711.1	−0.178179
$479$	− 220049.i	− 0.959065i	−0.877524	−	0.479532i	$-0.840806\pi$
$479$	− 220049.i	− 0.959065i	0.877524	−	0.479532i	$-0.159194\pi$
$480$	0	0
$481$	− 102847.i	− 0.444531i
$482$	− 21213.0i	− 0.0913079i
$483$	0	0
$484$	−116323.	−0.496565
$485$	36888.2	0.156821
$486$	0	0
$487$	139656.	0.588847	0.294423	−	0.955675i	$-0.404872\pi$
$487$	139656.	0.588847	0.294423	+	0.955675i	$0.404872\pi$
$488$	132013.i	0.554341i
$489$	0	0
$490$	0	0
$491$	284872.	1.18164	0.590821	−	0.806802i	$-0.298804\pi$
$491$	284872.	1.18164	0.590821	+	0.806802i	$0.298804\pi$
$492$	0	0
$493$	− 370265.i	− 1.52342i
$494$	232559.	0.952970
$495$	0	0
$496$	− 46348.8i	− 0.188397i
$497$	0	0
$498$	0	0
$499$	−313573.	−1.25933	−0.629663	−	0.776869i	$-0.716807\pi$
$499$	−313573.	−1.25933	−0.629663	+	0.776869i	$0.716807\pi$
$500$	− 56351.7i	− 0.225407i
$501$	0	0
$502$	128453.i	0.509725i
$503$	192865.i	0.762285i	0.924516	+	0.381143i	$0.124469\pi$
$503$	192865.i	0.762285i	−0.924516	+	0.381143i	$0.875531\pi$
$504$	0	0
$505$	52756.2	0.206867
$506$	26149.8	0.102133
$507$	0	0
$508$	−16933.6	−0.0656179
$509$	− 173256.i	− 0.668734i	−0.942443	−	0.334367i	$-0.891477\pi$
$509$	− 173256.i	− 0.668734i	0.942443	−	0.334367i	$-0.108523\pi$
$510$	0	0
$511$	0	0
$512$	11585.2	0.0441942
$513$	0	0
$514$	− 146309.i	− 0.553789i
$515$	−93295.5	−0.351760
$516$	0	0
$517$	17558.8i	0.0656922i
$518$	0	0
$519$	0	0
$520$	24923.3	0.0921721
$521$	− 168311.i	− 0.620064i	−0.950726	−	0.310032i	$-0.899660\pi$
$521$	− 168311.i	− 0.620064i	0.950726	−	0.310032i	$-0.100340\pi$
$522$	0	0
$523$	452722.i	1.65512i	0.561380	+	0.827558i	$0.310271\pi$
$523$	452722.i	1.65512i	−0.561380	+	0.827558i	$0.689729\pi$
$524$	48457.2i	0.176480i
$525$	0	0
$526$	254089.	0.918362
$527$	305447.	1.09980
$528$	0	0
$529$	569916.	2.03657
$530$	13302.3i	0.0473558i
$531$	0	0
$532$	0	0
$533$	−170110.	−0.598790
$534$	0	0
$535$	119003.i	0.415766i
$536$	−49866.0	−0.173570
$537$	0	0
$538$	− 67214.1i	− 0.232218i
$539$	0	0
$540$	0	0
$541$	−14472.4	−0.0494477	−0.0247238	−	0.999694i	$-0.507871\pi$
$541$	−14472.4	−0.0494477	−0.0247238	+	0.999694i	$0.507871\pi$
$542$	− 313442.i	− 1.06699i
$543$	0	0
$544$	76348.7i	0.257991i
$545$	82559.4i	0.277954i
$546$	0	0
$547$	−370524.	−1.23835	−0.619173	−	0.785255i	$-0.712532\pi$
$547$	−370524.	−1.23835	−0.619173	+	0.785255i	$0.712532\pi$
$548$	−22882.0	−0.0761962
$549$	0	0
$550$	−16778.5	−0.0554663
$551$	379463.i	1.24987i
$552$	0	0
$553$	0	0
$554$	−20145.1	−0.0656373
$555$	0	0
$556$	135730.i	0.439062i
$557$	437281.	1.40945	0.704727	−	0.709479i	$-0.251070\pi$
$557$	437281.	1.40945	0.704727	+	0.709479i	$0.251070\pi$
$558$	0	0
$559$	− 237170.i	− 0.758990i
$560$	0	0
$561$	0	0
$562$	−255150.	−0.807836
$563$	− 4062.71i	− 0.0128174i	−0.999979	−	0.00640869i	$-0.997960\pi$
$563$	− 4062.71i	− 0.0128174i	0.999979	−	0.00640869i	$-0.00203996\pi$
$564$	0	0
$565$	− 59669.4i	− 0.186919i
$566$	204918.i	0.639656i
$567$	0	0
$568$	−77119.6	−0.239039
$569$	−433364.	−1.33853	−0.669266	−	0.743023i	$-0.733391\pi$
$569$	−433364.	−1.33853	−0.669266	+	0.743023i	$0.733391\pi$
$570$	0	0
$571$	47636.6	0.146106	0.0730530	−	0.997328i	$-0.476726\pi$
$571$	47636.6	0.146106	0.0730530	+	0.997328i	$0.476726\pi$
$572$	− 15262.4i	− 0.0466477i
$573$	0	0
$574$	0	0
$575$	−545230.	−1.64909
$576$	0	0
$577$	− 24131.0i	− 0.0724809i	−0.999343	−	0.0362404i	$-0.988462\pi$
$577$	− 24131.0i	− 0.0724809i	0.999343	−	0.0362404i	$-0.0115382\pi$
$578$	−266918.	−0.798956
$579$	0	0
$580$	40667.0i	0.120889i
$581$	0	0
$582$	0	0
$583$	8145.93	0.0239665
$584$	212599.i	0.623356i
$585$	0	0
$586$	− 252205.i	− 0.734442i
$587$	− 680260.i	− 1.97423i	−0.160003	−	0.987117i	$-0.551150\pi$
$587$	− 680260.i	− 1.97423i	0.160003	−	0.987117i	$-0.448850\pi$
$588$	0	0
$589$	−313034.	−0.902320
$590$	46751.1	0.134304
$591$	0	0
$592$	−34603.2	−0.0987354
$593$	− 352622.i	− 1.00277i	−0.865225	−	0.501383i	$-0.832825\pi$
$593$	− 352622.i	− 1.00277i	0.865225	−	0.501383i	$-0.167175\pi$
$594$	0	0
$595$	0	0
$596$	119281.	0.335798
$597$	0	0
$598$	− 495961.i	− 1.38690i
$599$	382596.	1.06632	0.533159	−	0.846015i	$-0.321005\pi$
$599$	382596.	1.06632	0.533159	+	0.846015i	$0.321005\pi$
$600$	0	0
$601$	8474.87i	0.0234630i	0.999931	+	0.0117315i	$0.00373434\pi$
$601$	8474.87i	0.0234630i	−0.999931	+	0.0117315i	$0.996266\pi$
$602$	0	0
$603$	0	0
$604$	−17291.6	−0.0473981
$605$	84196.2i	0.230029i
$606$	0	0
$607$	− 222052.i	− 0.602668i	−0.953519	−	0.301334i	$-0.902568\pi$
$607$	− 222052.i	− 0.602668i	0.953519	−	0.301334i	$-0.0974319\pi$
$608$	− 78245.2i	− 0.211666i
$609$	0	0
$610$	95552.6	0.256793
$611$	333023.	0.892055
$612$	0	0
$613$	13140.4	0.0349694	0.0174847	−	0.999847i	$-0.494434\pi$
$613$	13140.4	0.0349694	0.0174847	+	0.999847i	$0.494434\pi$
$614$	319924.i	0.848615i
$615$	0	0
$616$	0	0
$617$	235797.	0.619394	0.309697	−	0.950835i	$-0.399772\pi$
$617$	235797.	0.619394	0.309697	+	0.950835i	$0.399772\pi$
$618$	0	0
$619$	− 617960.i	− 1.61279i	−0.591374	−	0.806397i	$-0.701414\pi$
$619$	− 617960.i	− 1.61279i	0.591374	−	0.806397i	$-0.298586\pi$
$620$	−33547.8	−0.0872732
$621$	0	0
$622$	− 184821.i	− 0.477718i
$623$	0	0
$624$	0	0
$625$	328881.	0.841935
$626$	5540.49i	0.0141384i
$627$	0	0
$628$	332430.i	0.842909i
$629$	− 228041.i	− 0.576384i
$630$	0	0
$631$	453628.	1.13931	0.569653	−	0.821885i	$-0.307077\pi$
$631$	453628.	1.13931	0.569653	+	0.821885i	$0.307077\pi$
$632$	53224.9	0.133254
$633$	0	0
$634$	243672.	0.606217
$635$	12256.8i	0.0303968i
$636$	0	0
$637$	0	0
$638$	24903.4	0.0611810
$639$	0	0
$640$	− 8385.54i	− 0.0204725i
$641$	144938.	0.352750	0.176375	−	0.984323i	$-0.443563\pi$
$641$	144938.	0.352750	0.176375	+	0.984323i	$0.443563\pi$
$642$	0	0
$643$	238775.i	0.577520i	0.957401	+	0.288760i	$0.0932430\pi$
$643$	238775.i	0.577520i	−0.957401	+	0.288760i	$0.906757\pi$
$644$	0	0
$645$	0	0
$646$	515650.	1.23563
$647$	33276.6i	0.0794933i	0.999210	+	0.0397466i	$0.0126551\pi$
$647$	33276.6i	0.0794933i	−0.999210	+	0.0397466i	$0.987345\pi$
$648$	0	0
$649$	− 28629.1i	− 0.0679701i
$650$	318224.i	0.753193i
$651$	0	0
$652$	−94121.0	−0.221407
$653$	−77564.4	−0.181901	−0.0909507	−	0.995855i	$-0.528991\pi$
$653$	−77564.4	−0.181901	−0.0909507	+	0.995855i	$0.528991\pi$
$654$	0	0
$655$	35073.9	0.0817526
$656$	57233.9i	0.132998i
$657$	0	0
$658$	0	0
$659$	762599.	1.75600	0.878001	−	0.478658i	$-0.158877\pi$
$659$	762599.	1.75600	0.878001	+	0.478658i	$0.158877\pi$
$660$	0	0
$661$	499668.i	1.14361i	0.820389	+	0.571805i	$0.193757\pi$
$661$	499668.i	1.14361i	−0.820389	+	0.571805i	$0.806243\pi$
$662$	−256318.	−0.584876
$663$	0	0
$664$	− 84856.5i	− 0.192464i
$665$	0	0
$666$	0	0
$667$	809252.	1.81900
$668$	108803.i	0.243830i
$669$	0	0
$670$	36093.6i	0.0804046i
$671$	− 58513.8i	− 0.129961i
$672$	0	0
$673$	601790.	1.32866	0.664332	−	0.747438i	$-0.268716\pi$
$673$	601790.	1.32866	0.664332	+	0.747438i	$0.268716\pi$
$674$	−74097.8	−0.163112
$675$	0	0
$676$	−60980.2	−0.133443
$677$	161904.i	0.353248i	0.984278	+	0.176624i	$0.0565177\pi$
$677$	161904.i	0.353248i	−0.984278	+	0.176624i	$0.943482\pi$
$678$	0	0
$679$	0	0
$680$	55262.1	0.119512
$681$	0	0
$682$	20543.8i	0.0441683i
$683$	13434.3	0.0287987	0.0143994	−	0.999896i	$-0.495416\pi$
$683$	13434.3	0.0287987	0.0143994	+	0.999896i	$0.495416\pi$
$684$	0	0
$685$	16562.3i	0.0352971i
$686$	0	0
$687$	0	0
$688$	−79796.5	−0.168580
$689$	− 154497.i	− 0.325448i
$690$	0	0
$691$	− 246175.i	− 0.515571i	−0.966202	−	0.257785i	$-0.917007\pi$
$691$	− 246175.i	− 0.515571i	0.966202	−	0.257785i	$-0.0829927\pi$
$692$	− 361225.i	− 0.754337i
$693$	0	0
$694$	454417.	0.943487
$695$	98242.9	0.203391
$696$	0	0
$697$	−377182.	−0.776399
$698$	− 51251.7i	− 0.105196i
$699$	0	0
$700$	0	0
$701$	−750473.	−1.52721	−0.763605	−	0.645683i	$-0.776573\pi$
$701$	−750473.	−1.52721	−0.763605	+	0.645683i	$0.776573\pi$
$702$	0	0
$703$	233706.i	0.472889i
$704$	−5135.07	−0.0103610
$705$	0	0
$706$	− 37418.6i	− 0.0750720i
$707$	0	0
$708$	0	0
$709$	−595336.	−1.18432	−0.592161	−	0.805820i	$-0.701725\pi$
$709$	−595336.	−1.18432	−0.592161	+	0.805820i	$0.701725\pi$
$710$	55820.1i	0.110732i
$711$	0	0
$712$	144000.i	0.284055i
$713$	667584.i	1.31319i
$714$	0	0
$715$	−11047.1	−0.0216091
$716$	−158819.	−0.309797
$717$	0	0
$718$	304720.	0.591088
$719$	26728.2i	0.0517026i	0.999666	+	0.0258513i	$0.00822964\pi$
$719$	26728.2i	0.0517026i	−0.999666	+	0.0258513i	$0.991770\pi$
$720$	0	0
$721$	0	0
$722$	−159855.	−0.306656
$723$	0	0
$724$	− 171188.i	− 0.326585i
$725$	−519241.	−0.987855
$726$	0	0
$727$	− 514869.i	− 0.974154i	−0.873359	−	0.487077i	$-0.838063\pi$
$727$	− 514869.i	− 0.974154i	0.873359	−	0.487077i	$-0.161937\pi$
$728$	0	0
$729$	0	0
$730$	153882.	0.288764
$731$	− 525873.i	− 0.984116i
$732$	0	0
$733$	241789.i	0.450016i	0.974357	+	0.225008i	$0.0722409\pi$
$733$	241789.i	0.450016i	−0.974357	+	0.225008i	$0.927759\pi$
$734$	− 199198.i	− 0.369737i
$735$	0	0
$736$	−166868.	−0.308047
$737$	22102.7	0.0406922
$738$	0	0
$739$	−671522.	−1.22962	−0.614811	−	0.788675i	$-0.710768\pi$
$739$	−671522.	−1.22962	−0.614811	+	0.788675i	$0.710768\pi$
$740$	25046.2i	0.0457382i
$741$	0	0
$742$	0	0
$743$	134542.	0.243714	0.121857	−	0.992548i	$-0.461115\pi$
$743$	134542.	0.243714	0.121857	+	0.992548i	$0.461115\pi$
$744$	0	0
$745$	− 86337.0i	− 0.155555i
$746$	540381.	0.971006
$747$	0	0
$748$	− 33841.0i	− 0.0604840i
$749$	0	0
$750$	0	0
$751$	446561.	0.791774	0.395887	−	0.918299i	$-0.370437\pi$
$751$	446561.	0.791774	0.395887	+	0.918299i	$0.370437\pi$
$752$	− 112047.i	− 0.198136i
$753$	0	0
$754$	− 472320.i	− 0.830795i
$755$	12515.9i	0.0219567i
$756$	0	0
$757$	−939360.	−1.63923	−0.819616	−	0.572913i	$-0.805813\pi$
$757$	−939360.	−1.63923	−0.819616	+	0.572913i	$0.805813\pi$
$758$	212074.	0.369104
$759$	0	0
$760$	−56634.9	−0.0980520
$761$	− 991448.i	− 1.71199i	−0.516986	−	0.855994i	$-0.672946\pi$
$761$	− 991448.i	− 1.71199i	0.516986	−	0.855994i	$-0.327054\pi$
$762$	0	0
$763$	0	0
$764$	−93007.0	−0.159341
$765$	0	0
$766$	363840.i	0.620087i
$767$	−542983.	−0.922987
$768$	0	0
$769$	180922.i	0.305942i	0.988231	+	0.152971i	$0.0488841\pi$
$769$	180922.i	0.305942i	−0.988231	+	0.152971i	$0.951116\pi$
$770$	0	0
$771$	0	0
$772$	−420375.	−0.705347
$773$	− 19270.4i	− 0.0322502i	−0.999870	−	0.0161251i	$-0.994867\pi$
$773$	− 19270.4i	− 0.0322502i	0.999870	−	0.0161251i	$-0.00513300\pi$
$774$	0	0
$775$	− 428342.i	− 0.713161i
$776$	144147.i	0.239377i
$777$	0	0
$778$	−74410.3	−0.122935
$779$	386551.	0.636988
$780$	0	0
$781$	34182.7	0.0560408
$782$	− 1.09969e6i	− 1.79827i
$783$	0	0
$784$	0	0
$785$	240617.	0.390469
$786$	0	0
$787$	− 278163.i	− 0.449107i	−0.974462	−	0.224553i	$-0.927908\pi$
$787$	− 278163.i	− 0.449107i	0.974462	−	0.224553i	$-0.0720923\pi$
$788$	438494.	0.706173
$789$	0	0
$790$	− 38524.8i	− 0.0617286i
$791$	0	0
$792$	0	0
$793$	−1.10978e6	−1.76478
$794$	− 248976.i	− 0.394927i
$795$	0	0
$796$	− 47783.0i	− 0.0754131i
$797$	− 60214.6i	− 0.0947950i	−0.998876	−	0.0473975i	$-0.984907\pi$
$797$	− 60214.6i	− 0.0947950i	0.998876	−	0.0473975i	$-0.0150927\pi$
$798$	0	0
$799$	738406.	1.15665
$800$	107068.	0.167293
$801$	0	0
$802$	−748400.	−1.16355
$803$	− 94233.1i	− 0.146141i
$804$	0	0
$805$	0	0
$806$	389636.	0.599775
$807$	0	0
$808$	206155.i	0.315770i
$809$	−70473.9	−0.107679	−0.0538395	−	0.998550i	$-0.517146\pi$
$809$	−70473.9	−0.107679	−0.0538395	+	0.998550i	$0.517146\pi$
$810$	0	0
$811$	1.08434e6i	1.64863i	0.566133	+	0.824314i	$0.308439\pi$
$811$	1.08434e6i	1.64863i	−0.566133	+	0.824314i	$0.691561\pi$
$812$	0	0
$813$	0	0
$814$	15337.6	0.0231478
$815$	68125.9i	0.102565i
$816$	0	0
$817$	538936.i	0.807408i
$818$	− 396463.i	− 0.592511i
$819$	0	0
$820$	41426.6	0.0616101
$821$	647690.	0.960906	0.480453	−	0.877020i	$-0.340472\pi$
$821$	647690.	0.960906	0.480453	+	0.877020i	$0.340472\pi$
$822$	0	0
$823$	1.07902e6	1.59306	0.796529	−	0.604600i	$-0.206667\pi$
$823$	1.07902e6	1.59306	0.796529	+	0.604600i	$0.206667\pi$
$824$	− 364569.i	− 0.536940i
$825$	0	0
$826$	0	0
$827$	−9026.16	−0.0131975	−0.00659875	−	0.999978i	$-0.502100\pi$
$827$	−9026.16	−0.0131975	−0.00659875	+	0.999978i	$0.502100\pi$
$828$	0	0
$829$	756024.i	1.10009i	0.835136	+	0.550043i	$0.185389\pi$
$829$	756024.i	1.10009i	−0.835136	+	0.550043i	$0.814611\pi$
$830$	−61420.2	−0.0891569
$831$	0	0
$832$	97392.5i	0.140695i
$833$	0	0
$834$	0	0
$835$	78752.7	0.112952
$836$	34681.6i	0.0496234i
$837$	0	0
$838$	− 903426.i	− 1.28648i
$839$	654438.i	0.929704i	0.885388	+	0.464852i	$0.153893\pi$
$839$	654438.i	0.929704i	−0.885388	+	0.464852i	$0.846107\pi$
$840$	0	0
$841$	63396.2	0.0896337
$842$	893009.	1.25960
$843$	0	0
$844$	24342.6	0.0341729
$845$	44138.2i	0.0618160i
$846$	0	0
$847$	0	0
$848$	−51981.0	−0.0722857
$849$	0	0
$850$	705594.i	0.976600i
$851$	498407.	0.688216
$852$	0	0
$853$	123790.i	0.170133i	0.996375	+	0.0850664i	$0.0271102\pi$
$853$	123790.i	0.170133i	−0.996375	+	0.0850664i	$0.972890\pi$
$854$	0	0
$855$	0	0
$856$	−465025.	−0.634641
$857$	− 173232.i	− 0.235866i	−0.993022	−	0.117933i	$-0.962373\pi$
$857$	− 173232.i	− 0.235866i	0.993022	−	0.117933i	$-0.0376268\pi$
$858$	0	0
$859$	267312.i	0.362270i	0.983458	+	0.181135i	$0.0579770\pi$
$859$	267312.i	0.362270i	−0.983458	+	0.181135i	$0.942023\pi$
$860$	57757.7i	0.0780932i
$861$	0	0
$862$	−476677.	−0.641519
$863$	−1.23904e6	−1.66366	−0.831830	−	0.555030i	$-0.812707\pi$
$863$	−1.23904e6	−1.66366	−0.831830	+	0.555030i	$0.812707\pi$
$864$	0	0
$865$	−261459.	−0.349439
$866$	48378.4i	0.0645083i
$867$	0	0
$868$	0	0
$869$	−23591.6	−0.0312404
$870$	0	0
$871$	− 419204.i	− 0.552572i
$872$	−322616.	−0.424280
$873$	0	0
$874$	1.12700e6i	1.47537i
$875$	0	0
$876$	0	0
$877$	697922.	0.907419	0.453710	−	0.891150i	$-0.350100\pi$
$877$	697922.	0.907419	0.453710	+	0.891150i	$0.350100\pi$
$878$	627857.i	0.814464i
$879$	0	0
$880$	3716.83i	0.00479962i
$881$	1.44660e6i	1.86378i	0.362735	+	0.931892i	$0.381843\pi$
$881$	1.44660e6i	1.86378i	−0.362735	+	0.931892i	$0.618157\pi$
$882$	0	0
$883$	−539049.	−0.691364	−0.345682	−	0.938352i	$-0.612352\pi$
$883$	−539049.	−0.691364	−0.345682	+	0.938352i	$0.612352\pi$
$884$	−641833.	−0.821330
$885$	0	0
$886$	−96819.1	−0.123337
$887$	983274.i	1.24976i	0.780720	+	0.624881i	$0.214853\pi$
$887$	983274.i	1.24976i	−0.780720	+	0.624881i	$0.785147\pi$
$888$	0	0
$889$	0	0
$890$	104229.	0.131586
$891$	0	0
$892$	336838.i	0.423341i
$893$	−756748.	−0.948961
$894$	0	0
$895$	114955.i	0.143510i
$896$	0	0
$897$	0	0
$898$	529499.	0.656618
$899$	635762.i	0.786639i
$900$	0	0
$901$	− 342564.i	− 0.421980i
$902$	− 25368.5i	− 0.0311804i
$903$	0	0
$904$	233169.	0.285321
$905$	−123908.	−0.151287
$906$	0	0
$907$	293081.	0.356266	0.178133	−	0.984006i	$-0.442994\pi$
$907$	293081.	0.356266	0.178133	+	0.984006i	$0.442994\pi$
$908$	737516.i	0.894540i
$909$	0	0
$910$	0	0
$911$	−1.39546e6	−1.68143	−0.840716	−	0.541477i	$-0.817865\pi$
$911$	−1.39546e6	−1.68143	−0.840716	+	0.541477i	$0.817865\pi$
$912$	0	0
$913$	37612.0i	0.0451217i
$914$	418169.	0.500564
$915$	0	0
$916$	− 191705.i	− 0.228477i
$917$	0	0
$918$	0	0
$919$	−860318.	−1.01866	−0.509328	−	0.860572i	$-0.670106\pi$
$919$	−860318.	−1.01866	−0.509328	+	0.860572i	$0.670106\pi$
$920$	120781.i	0.142700i
$921$	0	0
$922$	209246.i	0.246147i
$923$	− 648314.i	− 0.760995i
$924$	0	0
$925$	−319793.	−0.373754
$926$	−189469.	−0.220961
$927$	0	0
$928$	−158914.	−0.184529
$929$	1.08037e6i	1.25182i	0.779894	+	0.625911i	$0.215273\pi$
$929$	1.08037e6i	1.25182i	−0.779894	+	0.625911i	$0.784727\pi$
$930$	0	0
$931$	0	0
$932$	282785.	0.325555
$933$	0	0
$934$	− 656290.i	− 0.752319i
$935$	−24494.5	−0.0280186
$936$	0	0
$937$	659462.i	0.751122i	0.926798	+	0.375561i	$0.122550\pi$
$937$	659462.i	0.751122i	−0.926798	+	0.375561i	$0.877450\pi$
$938$	0	0
$939$	0	0
$940$	−81100.7	−0.0917844
$941$	890694.i	1.00589i	0.864319	+	0.502944i	$0.167750\pi$
$941$	890694.i	1.00589i	−0.864319	+	0.502944i	$0.832250\pi$
$942$	0	0
$943$	− 824368.i	− 0.927038i
$944$	182688.i	0.205006i
$945$	0	0
$946$	35369.2	0.0395224
$947$	−1.52254e6	−1.69773	−0.848863	−	0.528613i	$-0.822712\pi$
$947$	−1.52254e6	−1.69773	−0.848863	+	0.528613i	$0.822712\pi$
$948$	0	0
$949$	−1.78724e6	−1.98450
$950$	− 723121.i	− 0.801242i
$951$	0	0
$952$	0	0
$953$	464150.	0.511061	0.255530	−	0.966801i	$-0.417750\pi$
$953$	464150.	0.511061	0.255530	+	0.966801i	$0.417750\pi$
$954$	0	0
$955$	67319.6i	0.0738133i
$956$	−115148.	−0.125992
$957$	0	0
$958$	− 622392.i	− 0.678161i
$959$	0	0
$960$	0	0
$961$	399056.	0.432103
$962$	− 290895.i	− 0.314331i
$963$	0	0
$964$	− 59999.5i	− 0.0645645i
$965$	304273.i	0.326745i
$966$	0	0
$967$	61596.4	0.0658723	0.0329361	−	0.999457i	$-0.489514\pi$
$967$	61596.4	0.0658723	0.0329361	+	0.999457i	$0.489514\pi$
$968$	−329012.	−0.351124
$969$	0	0
$970$	104336.	0.110889
$971$	772298.i	0.819118i	0.912284	+	0.409559i	$0.134317\pi$
$971$	772298.i	0.819118i	−0.912284	+	0.409559i	$0.865683\pi$
$972$	0	0
$973$	0	0
$974$	395007.	0.416378
$975$	0	0
$976$	373389.i	0.391978i
$977$	−1.00334e6	−1.05114	−0.525570	−	0.850750i	$-0.676148\pi$
$977$	−1.00334e6	−1.05114	−0.525570	+	0.850750i	$0.676148\pi$
$978$	0	0
$979$	− 63826.9i	− 0.0665945i
$980$	0	0
$981$	0	0
$982$	805739.	0.835548
$983$	− 523081.i	− 0.541330i	−0.962674	−	0.270665i	$-0.912756\pi$
$983$	− 523081.i	− 0.541330i	0.962674	−	0.270665i	$-0.0872435\pi$
$984$	0	0
$985$	− 317387.i	− 0.327127i
$986$	− 1.04727e6i	− 1.07722i
$987$	0	0
$988$	657776.	0.673852
$989$	1.14935e6	1.17506
$990$	0	0
$991$	−1.67549e6	−1.70606	−0.853029	−	0.521863i	$-0.825237\pi$
$991$	−1.67549e6	−1.70606	−0.853029	+	0.521863i	$0.825237\pi$
$992$	− 131094.i	− 0.133217i
$993$	0	0
$994$	0	0
$995$	−34585.9	−0.0349344
$996$	0	0
$997$	656503.i	0.660460i	0.943901	+	0.330230i	$0.107126\pi$
$997$	656503.i	0.660460i	−0.943901	+	0.330230i	$0.892874\pi$
$998$	−886919.	−0.890477
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.5.c.b.685.3		4
3.2	odd	2		98.5.b.b.97.1		4
7.2	even	3		126.5.n.a.73.1		4
7.3	odd	6		126.5.n.a.19.1		4
7.6	odd	2	inner	882.5.c.b.685.4		4
12.11	even	2		784.5.c.b.97.3		4
21.2	odd	6		14.5.d.a.3.2	✓	4
21.5	even	6		98.5.d.a.31.2		4
21.11	odd	6		98.5.d.a.19.2		4
21.17	even	6		14.5.d.a.5.2	yes	4
21.20	even	2		98.5.b.b.97.2		4
84.23	even	6		112.5.s.b.17.1		4
84.59	odd	6		112.5.s.b.33.1		4
84.83	odd	2		784.5.c.b.97.2		4
105.2	even	12		350.5.i.a.199.1		8
105.17	odd	12		350.5.i.a.299.4		8
105.23	even	12		350.5.i.a.199.4		8
105.38	odd	12		350.5.i.a.299.1		8
105.44	odd	6		350.5.k.a.101.1		4
105.59	even	6		350.5.k.a.201.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.5.d.a.3.2	✓	4	21.2	odd	6
14.5.d.a.5.2	yes	4	21.17	even	6
98.5.b.b.97.1		4	3.2	odd	2
98.5.b.b.97.2		4	21.20	even	2
98.5.d.a.19.2		4	21.11	odd	6
98.5.d.a.31.2		4	21.5	even	6
112.5.s.b.17.1		4	84.23	even	6
112.5.s.b.33.1		4	84.59	odd	6
126.5.n.a.19.1		4	7.3	odd	6
126.5.n.a.73.1		4	7.2	even	3
350.5.i.a.199.1		8	105.2	even	12
350.5.i.a.199.4		8	105.23	even	12
350.5.i.a.299.1		8	105.38	odd	12
350.5.i.a.299.4		8	105.17	odd	12
350.5.k.a.101.1		4	105.44	odd	6
350.5.k.a.201.1		4	105.59	even	6
784.5.c.b.97.2		4	84.83	odd	2
784.5.c.b.97.3		4	12.11	even	2
882.5.c.b.685.3		4	1.1	even	1	trivial
882.5.c.b.685.4		4	7.6	odd	2	inner

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:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	882.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.82843 q^{2} +8.00000 q^{4} -5.79050i q^{5} +22.6274 q^{8} +O(q^{10})\)	\(q+2.82843 q^{2} +8.00000 q^{4} -5.79050i q^{5} +22.6274 q^{8} -16.3780i q^{10} -10.0294 q^{11} +190.220i q^{13} +64.0000 q^{16} +421.771i q^{17} -432.248i q^{19} -46.3240i q^{20} -28.3675 q^{22} -921.823 q^{23} +591.470 q^{25} +538.022i q^{26} -877.882 q^{29} -724.200i q^{31} +181.019 q^{32} +1192.95i q^{34} -540.675 q^{37} -1222.58i q^{38} -131.024i q^{40} +894.280i q^{41} -1246.82 q^{43} -80.2355 q^{44} -2607.31 q^{46} -1750.73i q^{47} +1672.93 q^{50} +1521.76i q^{52} -812.203 q^{53} +58.0754i q^{55} -2483.03 q^{58} +2854.51i q^{59} +5834.20i q^{61} -2048.35i q^{62} +512.000 q^{64} +1101.47 q^{65} -2203.79 q^{67} +3374.17i q^{68} -3408.24 q^{71} +9395.66i q^{73} -1529.26 q^{74} -3457.98i q^{76} +2352.23 q^{79} -370.592i q^{80} +2529.41i q^{82} -3750.16i q^{83} +2442.26 q^{85} -3526.54 q^{86} -226.940 q^{88} +6363.96i q^{89} -7374.58 q^{92} -4951.81i q^{94} -2502.93 q^{95} +6370.47i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 32 q^{4}+O(q^{10}) \) 4 * q + 32 * q^4	\( 4 q + 32 q^{4} - 108 q^{11} + 256 q^{16} + 192 q^{22} - 972 q^{23} + 1144 q^{25} - 3240 q^{29} + 892 q^{37} + 2344 q^{43} - 864 q^{44} - 7680 q^{46} + 3456 q^{50} + 5508 q^{53} - 768 q^{58} + 2048 q^{64} - 6048 q^{65} + 10124 q^{67} - 18792 q^{71} - 8640 q^{74} - 1588 q^{79} + 10380 q^{85} - 20736 q^{86} + 1536 q^{88} - 7776 q^{92} + 17820 q^{95}+O(q^{100}) \) 4 * q + 32 * q^4 - 108 * q^11 + 256 * q^16 + 192 * q^22 - 972 * q^23 + 1144 * q^25 - 3240 * q^29 + 892 * q^37 + 2344 * q^43 - 864 * q^44 - 7680 * q^46 + 3456 * q^50 + 5508 * q^53 - 768 * q^58 + 2048 * q^64 - 6048 * q^65 + 10124 * q^67 - 18792 * q^71 - 8640 * q^74 - 1588 * q^79 + 10380 * q^85 - 20736 * q^86 + 1536 * q^88 - 7776 * q^92 + 17820 * q^95

Introduction

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