LMFDB - Embedded newform 882.5.b.e.197.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [882,5,Mod(197,882)]

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([1, 0]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("882.197");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 882 = 2 \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	882.b (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:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 13x^{4} - 660x^{3} + 702x^{2} - 5832x + 157464 $ x^6 - 2x^5 + 13x^4 - 660x^3 + 702x^2 - 5832*x + 157464
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2}\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 126)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			197.4
Root			$-4.52284 + 5.79171i$ of defining polynomial
Character	$\chi$	$=$	882.197
Dual form			882.5.b.e.197.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.82843i q^{2} -8.00000 q^{4} -32.9258i q^{5} -22.6274i q^{8} +O(q^{10})$	$q+2.82843i q^{2} -8.00000 q^{4} -32.9258i q^{5} -22.6274i q^{8} +93.1283 q^{10} -12.7494i q^{11} -31.2973 q^{13} +64.0000 q^{16} +96.6298i q^{17} -487.811 q^{19} +263.407i q^{20} +36.0607 q^{22} +873.855i q^{23} -459.111 q^{25} -88.5220i q^{26} +1119.99i q^{29} +938.692 q^{31} +181.019i q^{32} -273.310 q^{34} +1487.06 q^{37} -1379.74i q^{38} -745.027 q^{40} -2076.81i q^{41} -2878.65 q^{43} +101.995i q^{44} -2471.63 q^{46} +1116.18i q^{47} -1298.56i q^{50} +250.378 q^{52} +344.990i q^{53} -419.784 q^{55} -3167.82 q^{58} -1919.85i q^{59} +6518.97 q^{61} +2655.02i q^{62} -512.000 q^{64} +1030.49i q^{65} +5029.07 q^{67} -773.039i q^{68} -1627.82i q^{71} -3581.38 q^{73} +4206.04i q^{74} +3902.49 q^{76} +4868.56 q^{79} -2107.25i q^{80} +5874.11 q^{82} -11175.2i q^{83} +3181.62 q^{85} -8142.07i q^{86} -288.486 q^{88} -1115.44i q^{89} -6990.84i q^{92} -3157.02 q^{94} +16061.6i q^{95} +11134.8 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 48 q^{4}+O(q^{10}) $ 6 * q - 48 * q^4	$ 6 q - 48 q^{4} - 104 q^{10} - 86 q^{13} + 384 q^{16} - 174 q^{19} + 472 q^{22} - 2170 q^{25} - 1682 q^{31} - 1744 q^{34} + 6018 q^{37} + 832 q^{40} - 2130 q^{43} - 656 q^{46} + 688 q^{52} + 12680 q^{55} + 7040 q^{58} + 20632 q^{61} - 3072 q^{64} - 74 q^{67} + 2942 q^{73} + 1392 q^{76} + 10526 q^{79} + 25464 q^{82} - 33056 q^{85} - 3776 q^{88} + 11640 q^{94} + 46436 q^{97}+O(q^{100}) $ 6 * q - 48 * q^4 - 104 * q^10 - 86 * q^13 + 384 * q^16 - 174 * q^19 + 472 * q^22 - 2170 * q^25 - 1682 * q^31 - 1744 * q^34 + 6018 * q^37 + 832 * q^40 - 2130 * q^43 - 656 * q^46 + 688 * q^52 + 12680 * q^55 + 7040 * q^58 + 20632 * q^61 - 3072 * q^64 - 74 * q^67 + 2942 * q^73 + 1392 * q^76 + 10526 * q^79 + 25464 * q^82 - 33056 * q^85 - 3776 * q^88 + 11640 * q^94 + 46436 * q^97

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 $

$2$	2.82843i	0.707107i
$2$	2.82843i	0.707107i
$3$	0	0
$3$	0	0
$4$	−8.00000	−0.500000
$5$	− 32.9258i	− 1.31703i	−0.752566	−	0.658517i	$-0.771184\pi$
$5$	− 32.9258i	− 1.31703i	0.752566	−	0.658517i	$-0.228816\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	− 22.6274i	− 0.353553i
$9$	0	0
$10$	93.1283	0.931283
$11$	− 12.7494i	− 0.105367i	−0.998611	−	0.0526834i	$-0.983223\pi$
$11$	− 12.7494i	− 0.105367i	0.998611	−	0.0526834i	$-0.0167774\pi$
$12$	0	0
$13$	−31.2973	−0.185191	−0.0925954	−	0.995704i	$-0.529516\pi$
$13$	−31.2973	−0.185191	−0.0925954	+	0.995704i	$0.529516\pi$
$14$	0	0
$15$	0	0
$16$	64.0000	0.250000
$17$	96.6298i	0.334359i	0.985926	+	0.167180i	$0.0534660\pi$
$17$	96.6298i	0.334359i	−0.985926	+	0.167180i	$0.946534\pi$
$18$	0	0
$19$	−487.811	−1.35128	−0.675638	−	0.737234i	$-0.736132\pi$
$19$	−487.811	−1.35128	−0.675638	+	0.737234i	$0.736132\pi$
$20$	263.407i	0.658517i
$21$	0	0
$22$	36.0607	0.0745056
$23$	873.855i	1.65190i	0.563744	+	0.825950i	$0.309361\pi$
$23$	873.855i	1.65190i	−0.563744	+	0.825950i	$0.690639\pi$
$24$	0	0
$25$	−459.111	−0.734578
$26$	− 88.5220i	− 0.130950i
$27$	0	0
$28$	0	0
$29$	1119.99i	1.33174i	0.746069	+	0.665869i	$0.231939\pi$
$29$	1119.99i	1.33174i	−0.746069	+	0.665869i	$0.768061\pi$
$30$	0	0
$31$	938.692	0.976787	0.488393	−	0.872624i	$-0.337583\pi$
$31$	938.692	0.976787	0.488393	+	0.872624i	$0.337583\pi$
$32$	181.019i	0.176777i
$33$	0	0
$34$	−273.310	−0.236428
$35$	0	0
$36$	0	0
$37$	1487.06	1.08624	0.543119	−	0.839656i	$-0.317243\pi$
$37$	1487.06	1.08624	0.543119	+	0.839656i	$0.317243\pi$
$38$	− 1379.74i	− 0.955496i
$39$	0	0
$40$	−745.027	−0.465642
$41$	− 2076.81i	− 1.23546i	−0.786390	−	0.617731i	$-0.788052\pi$
$41$	− 2076.81i	− 1.23546i	0.786390	−	0.617731i	$-0.211948\pi$
$42$	0	0
$43$	−2878.65	−1.55687	−0.778436	−	0.627725i	$-0.783986\pi$
$43$	−2878.65	−1.55687	−0.778436	+	0.627725i	$0.783986\pi$
$44$	101.995i	0.0526834i
$45$	0	0
$46$	−2471.63	−1.16807
$47$	1116.18i	0.505286i	0.967560	+	0.252643i	$0.0812998\pi$
$47$	1116.18i	0.505286i	−0.967560	+	0.252643i	$0.918700\pi$
$48$	0	0
$49$	0	0
$50$	− 1298.56i	− 0.519425i
$51$	0	0
$52$	250.378	0.0925954
$53$	344.990i	0.122816i	0.998113	+	0.0614079i	$0.0195591\pi$
$53$	344.990i	0.122816i	−0.998113	+	0.0614079i	$0.980441\pi$
$54$	0	0
$55$	−419.784	−0.138772
$56$	0	0
$57$	0	0
$58$	−3167.82	−0.941681
$59$	− 1919.85i	− 0.551524i	−0.961226	−	0.275762i	$-0.911070\pi$
$59$	− 1919.85i	− 0.551524i	0.961226	−	0.275762i	$-0.0889301\pi$
$60$	0	0
$61$	6518.97	1.75194	0.875970	−	0.482366i	$-0.160222\pi$
$61$	6518.97	1.75194	0.875970	+	0.482366i	$0.160222\pi$
$62$	2655.02i	0.690692i
$63$	0	0
$64$	−512.000	−0.125000
$65$	1030.49i	0.243903i
$66$	0	0
$67$	5029.07	1.12031	0.560155	−	0.828388i	$-0.310742\pi$
$67$	5029.07	1.12031	0.560155	+	0.828388i	$0.310742\pi$
$68$	− 773.039i	− 0.167180i
$69$	0	0
$70$	0	0
$71$	− 1627.82i	− 0.322916i	−0.986880	−	0.161458i	$-0.948380\pi$
$71$	− 1627.82i	− 0.322916i	0.986880	−	0.161458i	$-0.0516196\pi$
$72$	0	0
$73$	−3581.38	−0.672054	−0.336027	−	0.941852i	$-0.609083\pi$
$73$	−3581.38	−0.672054	−0.336027	+	0.941852i	$0.609083\pi$
$74$	4206.04i	0.768087i
$75$	0	0
$76$	3902.49	0.675638
$77$	0	0
$78$	0	0
$79$	4868.56	0.780093	0.390047	−	0.920795i	$-0.372459\pi$
$79$	4868.56	0.780093	0.390047	+	0.920795i	$0.372459\pi$
$80$	− 2107.25i	− 0.329258i
$81$	0	0
$82$	5874.11	0.873603
$83$	− 11175.2i	− 1.62219i	−0.584917	−	0.811094i	$-0.698873\pi$
$83$	− 11175.2i	− 1.62219i	0.584917	−	0.811094i	$-0.301127\pi$
$84$	0	0
$85$	3181.62	0.440362
$86$	− 8142.07i	− 1.10087i
$87$	0	0
$88$	−288.486	−0.0372528
$89$	− 1115.44i	− 0.140820i	−0.997518	−	0.0704101i	$-0.977569\pi$
$89$	− 1115.44i	− 0.140820i	0.997518	−	0.0704101i	$-0.0224308\pi$
$90$	0	0
$91$	0	0
$92$	− 6990.84i	− 0.825950i
$93$	0	0
$94$	−3157.02	−0.357291
$95$	16061.6i	1.77968i
$96$	0	0
$97$	11134.8	1.18342	0.591712	−	0.806149i	$-0.298452\pi$
$97$	11134.8	1.18342	0.591712	+	0.806149i	$0.298452\pi$
$98$	0	0
$99$	0	0
$100$	3672.89	0.367289
$101$	15768.4i	1.54577i	0.634544	+	0.772886i	$0.281188\pi$
$101$	15768.4i	1.54577i	−0.634544	+	0.772886i	$0.718812\pi$
$102$	0	0
$103$	11222.5	1.05782	0.528912	−	0.848676i	$-0.322600\pi$
$103$	11222.5	1.05782	0.528912	+	0.848676i	$0.322600\pi$
$104$	708.176i	0.0654749i
$105$	0	0
$106$	−975.778	−0.0868439
$107$	− 6706.10i	− 0.585737i	−0.956153	−	0.292869i	$-0.905390\pi$
$107$	− 6706.10i	− 0.585737i	0.956153	−	0.292869i	$-0.0946098\pi$
$108$	0	0
$109$	5807.79	0.488830	0.244415	−	0.969671i	$-0.421404\pi$
$109$	5807.79	0.488830	0.244415	+	0.969671i	$0.421404\pi$
$110$	− 1187.33i	− 0.0981264i
$111$	0	0
$112$	0	0
$113$	− 24819.8i	− 1.94375i	−0.235493	−	0.971876i	$-0.575671\pi$
$113$	− 24819.8i	− 1.94375i	0.235493	−	0.971876i	$-0.424329\pi$
$114$	0	0
$115$	28772.4	2.17561
$116$	− 8959.94i	− 0.665869i
$117$	0	0
$118$	5430.17	0.389986
$119$	0	0
$120$	0	0
$121$	14478.5	0.988898
$122$	18438.4i	1.23881i
$123$	0	0
$124$	−7509.54	−0.488393
$125$	− 5462.03i	− 0.349570i
$126$	0	0
$127$	−2259.69	−0.140101	−0.0700506	−	0.997543i	$-0.522316\pi$
$127$	−2259.69	−0.140101	−0.0700506	+	0.997543i	$0.522316\pi$
$128$	− 1448.15i	− 0.0883883i
$129$	0	0
$130$	−2914.66	−0.172465
$131$	6613.44i	0.385376i	0.981260	+	0.192688i	$0.0617205\pi$
$131$	6613.44i	0.385376i	−0.981260	+	0.192688i	$0.938279\pi$
$132$	0	0
$133$	0	0
$134$	14224.4i	0.792178i
$135$	0	0
$136$	2186.48	0.118214
$137$	− 1461.52i	− 0.0778689i	−0.999242	−	0.0389345i	$-0.987604\pi$
$137$	− 1461.52i	− 0.0778689i	0.999242	−	0.0389345i	$-0.0123964\pi$
$138$	0	0
$139$	2116.42	0.109540	0.0547699	−	0.998499i	$-0.482557\pi$
$139$	2116.42	0.109540	0.0547699	+	0.998499i	$0.482557\pi$
$140$	0	0
$141$	0	0
$142$	4604.16	0.228336
$143$	399.021i	0.0195130i
$144$	0	0
$145$	36876.7	1.75394
$146$	− 10129.7i	− 0.475214i
$147$	0	0
$148$	−11896.5	−0.543119
$149$	31964.1i	1.43976i	0.694099	+	0.719879i	$0.255803\pi$
$149$	31964.1i	1.43976i	−0.694099	+	0.719879i	$0.744197\pi$
$150$	0	0
$151$	−29744.3	−1.30452	−0.652258	−	0.757997i	$-0.726178\pi$
$151$	−29744.3	−1.30452	−0.652258	+	0.757997i	$0.726178\pi$
$152$	11037.9i	0.477748i
$153$	0	0
$154$	0	0
$155$	− 30907.2i	− 1.28646i
$156$	0	0
$157$	29356.1	1.19097	0.595484	−	0.803368i	$-0.296961\pi$
$157$	29356.1	1.19097	0.595484	+	0.803368i	$0.296961\pi$
$158$	13770.4i	0.551609i
$159$	0	0
$160$	5960.21	0.232821
$161$	0	0
$162$	0	0
$163$	−5838.52	−0.219750	−0.109875	−	0.993945i	$-0.535045\pi$
$163$	−5838.52	−0.219750	−0.109875	+	0.993945i	$0.535045\pi$
$164$	16614.5i	0.617731i
$165$	0	0
$166$	31608.4	1.14706
$167$	40057.9i	1.43633i	0.695871	+	0.718167i	$0.255019\pi$
$167$	40057.9i	1.43633i	−0.695871	+	0.718167i	$0.744981\pi$
$168$	0	0
$169$	−27581.5	−0.965704
$170$	8998.98i	0.311383i
$171$	0	0
$172$	23029.2	0.778436
$173$	12911.3i	0.431399i	0.976460	+	0.215700i	$0.0692032\pi$
$173$	12911.3i	0.431399i	−0.976460	+	0.215700i	$0.930797\pi$
$174$	0	0
$175$	0	0
$176$	− 815.960i	− 0.0263417i
$177$	0	0
$178$	3154.93	0.0995750
$179$	14180.9i	0.442587i	0.975207	+	0.221294i	$0.0710279\pi$
$179$	14180.9i	0.442587i	−0.975207	+	0.221294i	$0.928972\pi$
$180$	0	0
$181$	5402.52	0.164907	0.0824536	−	0.996595i	$-0.473724\pi$
$181$	5402.52	0.164907	0.0824536	+	0.996595i	$0.473724\pi$
$182$	0	0
$183$	0	0
$184$	19773.1	0.584035
$185$	− 48962.7i	− 1.43061i
$186$	0	0
$187$	1231.97	0.0352304
$188$	− 8929.41i	− 0.252643i
$189$	0	0
$190$	−45429.0	−1.25842
$191$	− 16386.5i	− 0.449180i	−0.974453	−	0.224590i	$-0.927896\pi$
$191$	− 16386.5i	− 0.449180i	0.974453	−	0.224590i	$-0.0721043\pi$
$192$	0	0
$193$	70750.2	1.89938	0.949692	−	0.313185i	$-0.101396\pi$
$193$	70750.2	1.89938	0.949692	+	0.313185i	$0.101396\pi$
$194$	31494.1i	0.836808i
$195$	0	0
$196$	0	0
$197$	52991.9i	1.36546i	0.730673	+	0.682728i	$0.239206\pi$
$197$	52991.9i	1.36546i	−0.730673	+	0.682728i	$0.760794\pi$
$198$	0	0
$199$	45943.6	1.16016	0.580081	−	0.814559i	$-0.303021\pi$
$199$	45943.6	1.16016	0.580081	+	0.814559i	$0.303021\pi$
$200$	10388.5i	0.259713i
$201$	0	0
$202$	−44599.9	−1.09303
$203$	0	0
$204$	0	0
$205$	−68380.7	−1.62714
$206$	31741.9i	0.747995i
$207$	0	0
$208$	−2003.02	−0.0462977
$209$	6219.28i	0.142380i
$210$	0	0
$211$	−88468.3	−1.98711	−0.993557	−	0.113337i	$-0.963846\pi$
$211$	−88468.3	−1.98711	−0.993557	+	0.113337i	$0.963846\pi$
$212$	− 2759.92i	− 0.0614079i
$213$	0	0
$214$	18967.7	0.414179
$215$	94782.1i	2.05045i
$216$	0	0
$217$	0	0
$218$	16426.9i	0.345655i
$219$	0	0
$220$	3358.27	0.0693858
$221$	− 3024.25i	− 0.0619203i
$222$	0	0
$223$	47557.8	0.956339	0.478170	−	0.878268i	$-0.341300\pi$
$223$	47557.8	0.956339	0.478170	+	0.878268i	$0.341300\pi$
$224$	0	0
$225$	0	0
$226$	70200.9	1.37444
$227$	− 16107.1i	− 0.312583i	−0.987711	−	0.156291i	$-0.950046\pi$
$227$	− 16107.1i	− 0.312583i	0.987711	−	0.156291i	$-0.0499539\pi$
$228$	0	0
$229$	72943.5	1.39096	0.695482	−	0.718544i	$-0.255191\pi$
$229$	72943.5	1.39096	0.695482	+	0.718544i	$0.255191\pi$
$230$	81380.7i	1.53839i
$231$	0	0
$232$	25342.5	0.470841
$233$	94681.0i	1.74402i	0.489489	+	0.872009i	$0.337183\pi$
$233$	94681.0i	1.74402i	−0.489489	+	0.872009i	$0.662817\pi$
$234$	0	0
$235$	36751.1	0.665479
$236$	15358.8i	0.275762i
$237$	0	0
$238$	0	0
$239$	− 105335.i	− 1.84407i	−0.387104	−	0.922036i	$-0.626524\pi$
$239$	− 105335.i	− 1.84407i	0.387104	−	0.922036i	$-0.373476\pi$
$240$	0	0
$241$	−81614.3	−1.40518	−0.702590	−	0.711594i	$-0.747973\pi$
$241$	−81614.3	−1.40518	−0.702590	+	0.711594i	$0.747973\pi$
$242$	40951.2i	0.699256i
$243$	0	0
$244$	−52151.8	−0.875970
$245$	0	0
$246$	0	0
$247$	15267.1	0.250244
$248$	− 21240.2i	− 0.345346i
$249$	0	0
$250$	15449.0	0.247183
$251$	− 42230.8i	− 0.670319i	−0.942161	−	0.335160i	$-0.891210\pi$
$251$	− 42230.8i	− 0.670319i	0.942161	−	0.335160i	$-0.108790\pi$
$252$	0	0
$253$	11141.1	0.174055
$254$	− 6391.38i	− 0.0990666i
$255$	0	0
$256$	4096.00	0.0625000
$257$	1501.76i	0.0227371i	0.999935	+	0.0113686i	$0.00361880\pi$
$257$	1501.76i	0.0227371i	−0.999935	+	0.0113686i	$0.996381\pi$
$258$	0	0
$259$	0	0
$260$	− 8243.91i	− 0.121951i
$261$	0	0
$262$	−18705.6	−0.272502
$263$	− 62313.0i	− 0.900881i	−0.892807	−	0.450440i	$-0.851267\pi$
$263$	− 62313.0i	− 0.900881i	0.892807	−	0.450440i	$-0.148733\pi$
$264$	0	0
$265$	11359.1	0.161753
$266$	0	0
$267$	0	0
$268$	−40232.5	−0.560155
$269$	84769.9i	1.17149i	0.810497	+	0.585743i	$0.199197\pi$
$269$	84769.9i	1.17149i	−0.810497	+	0.585743i	$0.800803\pi$
$270$	0	0
$271$	−3360.71	−0.0457607	−0.0228803	−	0.999738i	$-0.507284\pi$
$271$	−3360.71	−0.0457607	−0.0228803	+	0.999738i	$0.507284\pi$
$272$	6184.31i	0.0835898i
$273$	0	0
$274$	4133.81	0.0550616
$275$	5853.38i	0.0774001i
$276$	0	0
$277$	13936.4	0.181632	0.0908160	−	0.995868i	$-0.471052\pi$
$277$	13936.4	0.181632	0.0908160	+	0.995868i	$0.471052\pi$
$278$	5986.13i	0.0774563i
$279$	0	0
$280$	0	0
$281$	− 26527.4i	− 0.335956i	−0.985791	−	0.167978i	$-0.946276\pi$
$281$	− 26527.4i	− 0.335956i	0.985791	−	0.167978i	$-0.0537237\pi$
$282$	0	0
$283$	131777.	1.64538	0.822690	−	0.568491i	$-0.192472\pi$
$283$	131777.	1.64538	0.822690	+	0.568491i	$0.192472\pi$
$284$	13022.5i	0.161458i
$285$	0	0
$286$	−1128.60	−0.0137978
$287$	0	0
$288$	0	0
$289$	74183.7	0.888204
$290$	104303.i	1.24023i
$291$	0	0
$292$	28651.0	0.336027
$293$	49721.0i	0.579169i	0.957153	+	0.289584i	$0.0935171\pi$
$293$	49721.0i	0.579169i	−0.957153	+	0.289584i	$0.906483\pi$
$294$	0	0
$295$	−63212.8	−0.726375
$296$	− 33648.3i	− 0.384043i
$297$	0	0
$298$	−90408.1	−1.01806
$299$	− 27349.3i	− 0.305917i
$300$	0	0
$301$	0	0
$302$	− 84129.5i	− 0.922432i
$303$	0	0
$304$	−31219.9	−0.337819
$305$	− 214643.i	− 2.30736i
$306$	0	0
$307$	−103928.	−1.10269	−0.551347	−	0.834276i	$-0.685886\pi$
$307$	−103928.	−1.10269	−0.551347	+	0.834276i	$0.685886\pi$
$308$	0	0
$309$	0	0
$310$	87418.8	0.909665
$311$	34305.1i	0.354681i	0.984150	+	0.177340i	$0.0567493\pi$
$311$	34305.1i	0.354681i	−0.984150	+	0.177340i	$0.943251\pi$
$312$	0	0
$313$	3891.57	0.0397225	0.0198612	−	0.999803i	$-0.493678\pi$
$313$	3891.57	0.0397225	0.0198612	+	0.999803i	$0.493678\pi$
$314$	83031.7i	0.842141i
$315$	0	0
$316$	−38948.5	−0.390047
$317$	50090.7i	0.498470i	0.968443	+	0.249235i	$0.0801791\pi$
$317$	50090.7i	0.498470i	−0.968443	+	0.249235i	$0.919821\pi$
$318$	0	0
$319$	14279.2	0.140321
$320$	16858.0i	0.164629i
$321$	0	0
$322$	0	0
$323$	− 47137.1i	− 0.451812i
$324$	0	0
$325$	14368.9	0.136037
$326$	− 16513.8i	− 0.155386i
$327$	0	0
$328$	−46992.9	−0.436801
$329$	0	0
$330$	0	0
$331$	−86506.7	−0.789576	−0.394788	−	0.918772i	$-0.629182\pi$
$331$	−86506.7	−0.789576	−0.394788	+	0.918772i	$0.629182\pi$
$332$	89402.0i	0.811094i
$333$	0	0
$334$	−113301.	−1.01564
$335$	− 165586.i	− 1.47549i
$336$	0	0
$337$	154655.	1.36177	0.680886	−	0.732390i	$-0.261595\pi$
$337$	154655.	1.36177	0.680886	+	0.732390i	$0.261595\pi$
$338$	− 78012.2i	− 0.682856i
$339$	0	0
$340$	−25453.0	−0.220181
$341$	− 11967.7i	− 0.102921i
$342$	0	0
$343$	0	0
$344$	65136.5i	0.550437i
$345$	0	0
$346$	−36518.8	−0.305045
$347$	139873.i	1.16165i	0.814029	+	0.580824i	$0.197270\pi$
$347$	139873.i	1.16165i	−0.814029	+	0.580824i	$0.802730\pi$
$348$	0	0
$349$	216650.	1.77872	0.889360	−	0.457207i	$-0.151150\pi$
$349$	216650.	1.77872	0.889360	+	0.457207i	$0.151150\pi$
$350$	0	0
$351$	0	0
$352$	2307.88	0.0186264
$353$	− 115966.i	− 0.930642i	−0.885142	−	0.465321i	$-0.845939\pi$
$353$	− 115966.i	− 0.930642i	0.885142	−	0.465321i	$-0.154061\pi$
$354$	0	0
$355$	−53597.3	−0.425291
$356$	8923.50i	0.0704101i
$357$	0	0
$358$	−40109.7	−0.312956
$359$	17569.5i	0.136323i	0.997674	+	0.0681617i	$0.0217134\pi$
$359$	17569.5i	0.136323i	−0.997674	+	0.0681617i	$0.978287\pi$
$360$	0	0
$361$	107638.	0.825947
$362$	15280.6i	0.116607i
$363$	0	0
$364$	0	0
$365$	117920.i	0.885118i
$366$	0	0
$367$	91800.3	0.681572	0.340786	−	0.940141i	$-0.389307\pi$
$367$	91800.3	0.681572	0.340786	+	0.940141i	$0.389307\pi$
$368$	55926.7i	0.412975i
$369$	0	0
$370$	138487.	1.01160
$371$	0	0
$372$	0	0
$373$	129948.	0.934012	0.467006	−	0.884254i	$-0.345333\pi$
$373$	129948.	0.934012	0.467006	+	0.884254i	$0.345333\pi$
$374$	3484.54i	0.0249116i
$375$	0	0
$376$	25256.2	0.178646
$377$	− 35052.7i	− 0.246626i
$378$	0	0
$379$	119943.	0.835019	0.417509	−	0.908673i	$-0.362903\pi$
$379$	119943.	0.835019	0.417509	+	0.908673i	$0.362903\pi$
$380$	− 128493.i	− 0.889838i
$381$	0	0
$382$	46348.1	0.317618
$383$	− 57138.2i	− 0.389519i	−0.980851	−	0.194760i	$-0.937607\pi$
$383$	− 57138.2i	− 0.389519i	0.980851	−	0.194760i	$-0.0623927\pi$
$384$	0	0
$385$	0	0
$386$	200112.i	1.34307i
$387$	0	0
$388$	−89078.7	−0.591712
$389$	201870.i	1.33405i	0.745034	+	0.667026i	$0.232433\pi$
$389$	201870.i	1.33405i	−0.745034	+	0.667026i	$0.767567\pi$
$390$	0	0
$391$	−84440.4	−0.552328
$392$	0	0
$393$	0	0
$394$	−149884.	−0.965523
$395$	− 160302.i	− 1.02741i
$396$	0	0
$397$	169811.	1.07742	0.538708	−	0.842492i	$-0.318912\pi$
$397$	169811.	1.07742	0.538708	+	0.842492i	$0.318912\pi$
$398$	129948.i	0.820359i
$399$	0	0
$400$	−29383.1	−0.183644
$401$	171845.i	1.06868i	0.845270	+	0.534339i	$0.179439\pi$
$401$	171845.i	1.06868i	−0.845270	+	0.534339i	$0.820561\pi$
$402$	0	0
$403$	−29378.5	−0.180892
$404$	− 126147.i	− 0.772886i
$405$	0	0
$406$	0	0
$407$	− 18959.1i	− 0.114453i
$408$	0	0
$409$	130985.	0.783023	0.391511	−	0.920173i	$-0.371952\pi$
$409$	130985.	0.783023	0.391511	+	0.920173i	$0.371952\pi$
$410$	− 193410.i	− 1.15056i
$411$	0	0
$412$	−89779.7	−0.528912
$413$	0	0
$414$	0	0
$415$	−367954.	−2.13648
$416$	− 5665.41i	− 0.0327374i
$417$	0	0
$418$	−17590.8	−0.100678
$419$	99074.5i	0.564331i	0.959366	+	0.282165i	$0.0910527\pi$
$419$	99074.5i	0.564331i	−0.959366	+	0.282165i	$0.908947\pi$
$420$	0	0
$421$	68914.0	0.388815	0.194408	−	0.980921i	$-0.437722\pi$
$421$	68914.0	0.388815	0.194408	+	0.980921i	$0.437722\pi$
$422$	− 250226.i	− 1.40510i
$423$	0	0
$424$	7806.23	0.0434220
$425$	− 44363.8i	− 0.245613i
$426$	0	0
$427$	0	0
$428$	53648.8i	0.292869i
$429$	0	0
$430$	−268084.	−1.44989
$431$	3716.93i	0.0200092i	0.999950	+	0.0100046i	$0.00318462\pi$
$431$	3716.93i	0.0200092i	−0.999950	+	0.0100046i	$0.996815\pi$
$432$	0	0
$433$	−178185.	−0.950376	−0.475188	−	0.879884i	$-0.657620\pi$
$433$	−178185.	−0.950376	−0.475188	+	0.879884i	$0.657620\pi$
$434$	0	0
$435$	0	0
$436$	−46462.3	−0.244415
$437$	− 426276.i	− 2.23217i
$438$	0	0
$439$	−235586.	−1.22242	−0.611209	−	0.791469i	$-0.709317\pi$
$439$	−235586.	−1.22242	−0.611209	+	0.791469i	$0.709317\pi$
$440$	9498.63i	0.0490632i
$441$	0	0
$442$	8553.87	0.0437843
$443$	163368.i	0.832454i	0.909261	+	0.416227i	$0.136648\pi$
$443$	163368.i	0.832454i	−0.909261	+	0.416227i	$0.863352\pi$
$444$	0	0
$445$	−36726.7	−0.185465
$446$	134514.i	0.676234i
$447$	0	0
$448$	0	0
$449$	− 108998.i	− 0.540664i	−0.962767	−	0.270332i	$-0.912866\pi$
$449$	− 108998.i	− 0.540664i	0.962767	−	0.270332i	$-0.0871335\pi$
$450$	0	0
$451$	−26478.0	−0.130177
$452$	198558.i	0.971876i
$453$	0	0
$454$	45557.7	0.221029
$455$	0	0
$456$	0	0
$457$	−228791.	−1.09549	−0.547743	−	0.836647i	$-0.684513\pi$
$457$	−228791.	−1.09549	−0.547743	+	0.836647i	$0.684513\pi$
$458$	206315.i	0.983560i
$459$	0	0
$460$	−230179.	−1.08780
$461$	− 334485.i	− 1.57389i	−0.617022	−	0.786946i	$-0.711661\pi$
$461$	− 334485.i	− 1.57389i	0.617022	−	0.786946i	$-0.288339\pi$
$462$	0	0
$463$	−69250.7	−0.323044	−0.161522	−	0.986869i	$-0.551640\pi$
$463$	−69250.7	−0.323044	−0.161522	+	0.986869i	$0.551640\pi$
$464$	71679.5i	0.332935i
$465$	0	0
$466$	−267798.	−1.23321
$467$	− 80112.0i	− 0.367336i	−0.982988	−	0.183668i	$-0.941203\pi$
$467$	− 80112.0i	− 0.367336i	0.982988	−	0.183668i	$-0.0587972\pi$
$468$	0	0
$469$	0	0
$470$	103948.i	0.470564i
$471$	0	0
$472$	−43441.3	−0.194993
$473$	36701.1i	0.164043i
$474$	0	0
$475$	223959.	0.992618
$476$	0	0
$477$	0	0
$478$	297933.	1.30396
$479$	− 242654.i	− 1.05759i	−0.848751	−	0.528793i	$-0.822645\pi$
$479$	− 242654.i	− 1.05759i	0.848751	−	0.528793i	$-0.177355\pi$
$480$	0	0
$481$	−46540.9	−0.201161
$482$	− 230840.i	− 0.993613i
$483$	0	0
$484$	−115828.	−0.494449
$485$	− 366624.i	− 1.55861i
$486$	0	0
$487$	−308414.	−1.30040	−0.650199	−	0.759764i	$-0.725315\pi$
$487$	−308414.	−1.30040	−0.650199	+	0.759764i	$0.725315\pi$
$488$	− 147507.i	− 0.619404i
$489$	0	0
$490$	0	0
$491$	180044.i	0.746818i	0.927667	+	0.373409i	$0.121811\pi$
$491$	180044.i	0.746818i	−0.927667	+	0.373409i	$0.878189\pi$
$492$	0	0
$493$	−108225.	−0.445279
$494$	43182.0i	0.176949i
$495$	0	0
$496$	60076.3	0.244197
$497$	0	0
$498$	0	0
$499$	271024.	1.08844	0.544222	−	0.838941i	$-0.316825\pi$
$499$	271024.	1.08844	0.544222	+	0.838941i	$0.316825\pi$
$500$	43696.2i	0.174785i
$501$	0	0
$502$	119447.	0.473987
$503$	− 108563.i	− 0.429087i	−0.976714	−	0.214544i	$-0.931174\pi$
$503$	− 108563.i	− 0.429087i	0.976714	−	0.214544i	$-0.0688264\pi$
$504$	0	0
$505$	519189.	2.03584
$506$	31511.8i	0.123076i
$507$	0	0
$508$	18077.6	0.0700506
$509$	− 307599.i	− 1.18727i	−0.804734	−	0.593636i	$-0.797692\pi$
$509$	− 307599.i	− 1.18727i	0.804734	−	0.593636i	$-0.202308\pi$
$510$	0	0
$511$	0	0
$512$	11585.2i	0.0441942i
$513$	0	0
$514$	−4247.63	−0.0160776
$515$	− 369509.i	− 1.39319i
$516$	0	0
$517$	14230.6	0.0532404
$518$	0	0
$519$	0	0
$520$	23317.3	0.0862326
$521$	− 176446.i	− 0.650034i	−0.945708	−	0.325017i	$-0.894630\pi$
$521$	− 176446.i	− 0.650034i	0.945708	−	0.325017i	$-0.105370\pi$
$522$	0	0
$523$	−270558.	−0.989139	−0.494569	−	0.869138i	$-0.664674\pi$
$523$	−270558.	−0.989139	−0.494569	+	0.869138i	$0.664674\pi$
$524$	− 52907.5i	− 0.192688i
$525$	0	0
$526$	176248.	0.637019
$527$	90705.6i	0.326598i
$528$	0	0
$529$	−483781.	−1.72877
$530$	32128.3i	0.114376i
$531$	0	0
$532$	0	0
$533$	64998.5i	0.228796i
$534$	0	0
$535$	−220804.	−0.771436
$536$	− 113795.i	− 0.396089i
$537$	0	0
$538$	−239766.	−0.828366
$539$	0	0
$540$	0	0
$541$	90959.2	0.310779	0.155390	−	0.987853i	$-0.450337\pi$
$541$	90959.2	0.310779	0.155390	+	0.987853i	$0.450337\pi$
$542$	− 9505.52i	− 0.0323577i
$543$	0	0
$544$	−17491.9	−0.0591069
$545$	− 191226.i	− 0.643806i
$546$	0	0
$547$	464521.	1.55250	0.776248	−	0.630427i	$-0.217120\pi$
$547$	464521.	1.55250	0.776248	+	0.630427i	$0.217120\pi$
$548$	11692.2i	0.0389345i
$549$	0	0
$550$	−16555.9	−0.0547301
$551$	− 546344.i	− 1.79955i
$552$	0	0
$553$	0	0
$554$	39418.2i	0.128433i
$555$	0	0
$556$	−16931.3	−0.0547699
$557$	− 91882.2i	− 0.296156i	−0.988976	−	0.148078i	$-0.952691\pi$
$557$	− 91882.2i	− 0.296156i	0.988976	−	0.148078i	$-0.0473087\pi$
$558$	0	0
$559$	90094.0	0.288318
$560$	0	0
$561$	0	0
$562$	75030.8	0.237556
$563$	584759.i	1.84484i	0.386183	+	0.922422i	$0.373793\pi$
$563$	584759.i	1.84484i	−0.386183	+	0.922422i	$0.626207\pi$
$564$	0	0
$565$	−817212.	−2.55999
$566$	372721.i	1.16346i
$567$	0	0
$568$	−36833.3	−0.114168
$569$	− 24357.2i	− 0.0752319i	−0.999292	−	0.0376159i	$-0.988024\pi$
$569$	− 24357.2i	− 0.0752319i	0.999292	−	0.0376159i	$-0.0119764\pi$
$570$	0	0
$571$	222870.	0.683566	0.341783	−	0.939779i	$-0.388969\pi$
$571$	222870.	0.683566	0.341783	+	0.939779i	$0.388969\pi$
$572$	− 3192.17i	− 0.00975649i
$573$	0	0
$574$	0	0
$575$	− 401197.i	− 1.21345i
$576$	0	0
$577$	−90484.1	−0.271782	−0.135891	−	0.990724i	$-0.543390\pi$
$577$	−90484.1	−0.271782	−0.135891	+	0.990724i	$0.543390\pi$
$578$	209823.i	0.628055i
$579$	0	0
$580$	−295013.	−0.876972
$581$	0	0
$582$	0	0
$583$	4398.41	0.0129407
$584$	81037.3i	0.237607i
$585$	0	0
$586$	−140632.	−0.409534
$587$	− 255906.i	− 0.742685i	−0.928496	−	0.371343i	$-0.878898\pi$
$587$	− 255906.i	− 0.742685i	0.928496	−	0.371343i	$-0.121102\pi$
$588$	0	0
$589$	−457904.	−1.31991
$590$	− 178793.i	− 0.513625i
$591$	0	0
$592$	95171.9	0.271560
$593$	− 443523.i	− 1.26127i	−0.776081	−	0.630633i	$-0.782795\pi$
$593$	− 443523.i	− 1.26127i	0.776081	−	0.630633i	$-0.217205\pi$
$594$	0	0
$595$	0	0
$596$	− 255713.i	− 0.719879i
$597$	0	0
$598$	77355.4	0.216316
$599$	− 386117.i	− 1.07613i	−0.842903	−	0.538066i	$-0.819155\pi$
$599$	− 386117.i	− 1.07613i	0.842903	−	0.538066i	$-0.180845\pi$
$600$	0	0
$601$	192808.	0.533798	0.266899	−	0.963725i	$-0.414001\pi$
$601$	192808.	0.533798	0.266899	+	0.963725i	$0.414001\pi$
$602$	0	0
$603$	0	0
$604$	237954.	0.652258
$605$	− 476715.i	− 1.30241i
$606$	0	0
$607$	608590.	1.65176	0.825881	−	0.563845i	$-0.190678\pi$
$607$	608590.	1.65176	0.825881	+	0.563845i	$0.190678\pi$
$608$	− 88303.2i	− 0.238874i
$609$	0	0
$610$	607101.	1.63155
$611$	− 34933.3i	− 0.0935744i
$612$	0	0
$613$	6432.34	0.0171178	0.00855890	−	0.999963i	$-0.497276\pi$
$613$	6432.34	0.0171178	0.00855890	+	0.999963i	$0.497276\pi$
$614$	− 293952.i	− 0.779722i
$615$	0	0
$616$	0	0
$617$	463579.i	1.21774i	0.793271	+	0.608869i	$0.208376\pi$
$617$	463579.i	1.21774i	−0.793271	+	0.608869i	$0.791624\pi$
$618$	0	0
$619$	−534613.	−1.39527	−0.697635	−	0.716453i	$-0.745764\pi$
$619$	−534613.	−1.39527	−0.697635	+	0.716453i	$0.745764\pi$
$620$	247258.i	0.643230i
$621$	0	0
$622$	−97029.4	−0.250797
$623$	0	0
$624$	0	0
$625$	−466786.	−1.19497
$626$	11007.0i	0.0280880i
$627$	0	0
$628$	−234849.	−0.595484
$629$	143694.i	0.363194i
$630$	0	0
$631$	439949.	1.10495	0.552477	−	0.833528i	$-0.313683\pi$
$631$	439949.	1.10495	0.552477	+	0.833528i	$0.313683\pi$
$632$	− 110163.i	− 0.275805i
$633$	0	0
$634$	−141678.	−0.352471
$635$	74402.3i	0.184518i
$636$	0	0
$637$	0	0
$638$	40387.7i	0.0992219i
$639$	0	0
$640$	−47681.7	−0.116410
$641$	70616.3i	0.171866i	0.996301	+	0.0859328i	$0.0273870\pi$
$641$	70616.3i	0.171866i	−0.996301	+	0.0859328i	$0.972613\pi$
$642$	0	0
$643$	360434.	0.871773	0.435887	−	0.900002i	$-0.356435\pi$
$643$	360434.	0.871773	0.435887	+	0.900002i	$0.356435\pi$
$644$	0	0
$645$	0	0
$646$	133324.	0.319479
$647$	134034.i	0.320189i	0.987102	+	0.160094i	$0.0511799\pi$
$647$	134034.i	0.320189i	−0.987102	+	0.160094i	$0.948820\pi$
$648$	0	0
$649$	−24477.0	−0.0581123
$650$	40641.5i	0.0961928i
$651$	0	0
$652$	46708.2	0.109875
$653$	− 390337.i	− 0.915405i	−0.889105	−	0.457702i	$-0.848673\pi$
$653$	− 390337.i	− 0.915405i	0.889105	−	0.457702i	$-0.151327\pi$
$654$	0	0
$655$	217753.	0.507553
$656$	− 132916.i	− 0.308865i
$657$	0	0
$658$	0	0
$659$	− 64213.9i	− 0.147863i	−0.997263	−	0.0739313i	$-0.976445\pi$
$659$	− 64213.9i	− 0.147863i	0.997263	−	0.0739313i	$-0.0235546\pi$
$660$	0	0
$661$	142798.	0.326827	0.163413	−	0.986558i	$-0.447750\pi$
$661$	142798.	0.326827	0.163413	+	0.986558i	$0.447750\pi$
$662$	− 244678.i	− 0.558314i
$663$	0	0
$664$	−252867.	−0.573530
$665$	0	0
$666$	0	0
$667$	−978710.	−2.19990
$668$	− 320463.i	− 0.718167i
$669$	0	0
$670$	468349.	1.04333
$671$	− 83112.8i	− 0.184596i
$672$	0	0
$673$	−388106.	−0.856879	−0.428440	−	0.903570i	$-0.640937\pi$
$673$	−388106.	−0.856879	−0.428440	+	0.903570i	$0.640937\pi$
$674$	437430.i	0.962918i
$675$	0	0
$676$	220652.	0.482852
$677$	867232.i	1.89216i	0.323933	+	0.946080i	$0.394995\pi$
$677$	867232.i	1.89216i	−0.323933	+	0.946080i	$0.605005\pi$
$678$	0	0
$679$	0	0
$680$	− 71991.8i	− 0.155692i
$681$	0	0
$682$	33849.9	0.0727760
$683$	154067.i	0.330269i	0.986271	+	0.165134i	$0.0528058\pi$
$683$	154067.i	0.330269i	−0.986271	+	0.165134i	$0.947194\pi$
$684$	0	0
$685$	−48121.8	−0.102556
$686$	0	0
$687$	0	0
$688$	−184234.	−0.389218
$689$	− 10797.2i	− 0.0227444i
$690$	0	0
$691$	−287694.	−0.602525	−0.301262	−	0.953541i	$-0.597408\pi$
$691$	−287694.	−0.602525	−0.301262	+	0.953541i	$0.597408\pi$
$692$	− 103291.i	− 0.215700i
$693$	0	0
$694$	−395620.	−0.821409
$695$	− 69684.9i	− 0.144268i
$696$	0	0
$697$	200682.	0.413088
$698$	612779.i	1.25775i
$699$	0	0
$700$	0	0
$701$	519721.i	1.05763i	0.848737	+	0.528816i	$0.177364\pi$
$701$	519721.i	1.05763i	−0.848737	+	0.528816i	$0.822636\pi$
$702$	0	0
$703$	−725404.	−1.46781
$704$	6527.68i	0.0131708i
$705$	0	0
$706$	328002.	0.658063
$707$	0	0
$708$	0	0
$709$	604688.	1.20293	0.601463	−	0.798901i	$-0.294585\pi$
$709$	604688.	1.20293	0.601463	+	0.798901i	$0.294585\pi$
$710$	− 151596.i	− 0.300726i
$711$	0	0
$712$	−25239.5	−0.0497875
$713$	820280.i	1.61355i
$714$	0	0
$715$	13138.1	0.0256992
$716$	− 113447.i	− 0.221294i
$717$	0	0
$718$	−49694.1	−0.0963952
$719$	965322.i	1.86730i	0.358184	+	0.933651i	$0.383396\pi$
$719$	965322.i	1.86730i	−0.358184	+	0.933651i	$0.616604\pi$
$720$	0	0
$721$	0	0
$722$	304447.i	0.584033i
$723$	0	0
$724$	−43220.2	−0.0824536
$725$	− 514201.i	− 0.978266i
$726$	0	0
$727$	−843561.	−1.59606	−0.798028	−	0.602621i	$-0.794123\pi$
$727$	−843561.	−1.59606	−0.798028	+	0.602621i	$0.794123\pi$
$728$	0	0
$729$	0	0
$730$	−333528.	−0.625873
$731$	− 278164.i	− 0.520554i
$732$	0	0
$733$	−17344.9	−0.0322823	−0.0161411	−	0.999870i	$-0.505138\pi$
$733$	−17344.9	−0.0322823	−0.0161411	+	0.999870i	$0.505138\pi$
$734$	259650.i	0.481944i
$735$	0	0
$736$	−158185.	−0.292017
$737$	− 64117.5i	− 0.118043i
$738$	0	0
$739$	106206.	0.194473	0.0972364	−	0.995261i	$-0.469000\pi$
$739$	106206.	0.194473	0.0972364	+	0.995261i	$0.469000\pi$
$740$	391702.i	0.715306i
$741$	0	0
$742$	0	0
$743$	513818.i	0.930747i	0.885114	+	0.465374i	$0.154080\pi$
$743$	513818.i	0.930747i	−0.885114	+	0.465374i	$0.845920\pi$
$744$	0	0
$745$	1.05244e6	1.89621
$746$	367549.i	0.660446i
$747$	0	0
$748$	−9855.76	−0.0176152
$749$	0	0
$750$	0	0
$751$	561.101	0.000994858 0	0.000497429	−	1.00000i	$-0.499842\pi$
$751$	561.101	0.000994858 0	0.000497429		1.00000i	$0.499842\pi$
$752$	71435.3i	0.126321i
$753$	0	0
$754$	99144.0	0.174391
$755$	979355.i	1.71809i
$756$	0	0
$757$	−484297.	−0.845124	−0.422562	−	0.906334i	$-0.638869\pi$
$757$	−484297.	−0.845124	−0.422562	+	0.906334i	$0.638869\pi$
$758$	339250.i	0.590448i
$759$	0	0
$760$	363432.	0.629211
$761$	− 831927.i	− 1.43653i	−0.695768	−	0.718267i	$-0.744936\pi$
$761$	− 831927.i	− 1.43653i	0.695768	−	0.718267i	$-0.255064\pi$
$762$	0	0
$763$	0	0
$764$	131092.i	0.224590i
$765$	0	0
$766$	161611.	0.275432
$767$	60086.2i	0.102137i
$768$	0	0
$769$	−313466.	−0.530076	−0.265038	−	0.964238i	$-0.585384\pi$
$769$	−313466.	−0.530076	−0.265038	+	0.964238i	$0.585384\pi$
$770$	0	0
$771$	0	0
$772$	−566001.	−0.949692
$773$	346553.i	0.579976i	0.957030	+	0.289988i	$0.0936514\pi$
$773$	346553.i	0.579976i	−0.957030	+	0.289988i	$0.906349\pi$
$774$	0	0
$775$	−430964.	−0.717526
$776$	− 251953.i	− 0.418404i
$777$	0	0
$778$	−570975.	−0.943317
$779$	1.01309e6i	1.66945i
$780$	0	0
$781$	−20753.7	−0.0340246
$782$	− 238834.i	− 0.390555i
$783$	0	0
$784$	0	0
$785$	− 966576.i	− 1.56854i
$786$	0	0
$787$	592027.	0.955855	0.477927	−	0.878399i	$-0.341388\pi$
$787$	592027.	0.955855	0.477927	+	0.878399i	$0.341388\pi$
$788$	− 423936.i	− 0.682728i
$789$	0	0
$790$	453401.	0.726488
$791$	0	0
$792$	0	0
$793$	−204026.	−0.324443
$794$	480297.i	0.761849i
$795$	0	0
$796$	−367549.	−0.580081
$797$	− 583565.i	− 0.918698i	−0.888256	−	0.459349i	$-0.848083\pi$
$797$	− 583565.i	− 0.918698i	0.888256	−	0.459349i	$-0.151917\pi$
$798$	0	0
$799$	−107856.	−0.168947
$800$	− 83108.0i	− 0.129856i
$801$	0	0
$802$	−486050.	−0.755670
$803$	45660.4i	0.0708122i
$804$	0	0
$805$	0	0
$806$	− 83094.9i	− 0.127910i
$807$	0	0
$808$	356799.	0.546513
$809$	510353.i	0.779782i	0.920861	+	0.389891i	$0.127487\pi$
$809$	510353.i	0.779782i	−0.920861	+	0.389891i	$0.872513\pi$
$810$	0	0
$811$	−364449.	−0.554109	−0.277055	−	0.960854i	$-0.589358\pi$
$811$	−364449.	−0.554109	−0.277055	+	0.960854i	$0.589358\pi$
$812$	0	0
$813$	0	0
$814$	53624.4	0.0809308
$815$	192238.i	0.289418i
$816$	0	0
$817$	1.40424e6	2.10376
$818$	370481.i	0.553681i
$819$	0	0
$820$	547046.	0.813572
$821$	243877.i	0.361814i	0.983500	+	0.180907i	$0.0579032\pi$
$821$	243877.i	0.361814i	−0.983500	+	0.180907i	$0.942097\pi$
$822$	0	0
$823$	−182693.	−0.269725	−0.134863	−	0.990864i	$-0.543059\pi$
$823$	−182693.	−0.269725	−0.134863	+	0.990864i	$0.543059\pi$
$824$	− 253935.i	− 0.373998i
$825$	0	0
$826$	0	0
$827$	464044.i	0.678497i	0.940697	+	0.339249i	$0.110173\pi$
$827$	464044.i	0.678497i	−0.940697	+	0.339249i	$0.889827\pi$
$828$	0	0
$829$	794796.	1.15650	0.578251	−	0.815859i	$-0.303735\pi$
$829$	794796.	1.15650	0.578251	+	0.815859i	$0.303735\pi$
$830$	− 1.04073e6i	− 1.51072i
$831$	0	0
$832$	16024.2	0.0231489
$833$	0	0
$834$	0	0
$835$	1.31894e6	1.89170
$836$	− 49754.3i	− 0.0711898i
$837$	0	0
$838$	−280225.	−0.399042
$839$	510979.i	0.725903i	0.931808	+	0.362952i	$0.118231\pi$
$839$	510979.i	0.725903i	−0.931808	+	0.362952i	$0.881769\pi$
$840$	0	0
$841$	−547101.	−0.773527
$842$	194918.i	0.274934i
$843$	0	0
$844$	707746.	0.993557
$845$	908144.i	1.27187i
$846$	0	0
$847$	0	0
$848$	22079.3i	0.0307040i
$849$	0	0
$850$	125480.	0.173675
$851$	1.29947e6i	1.79436i
$852$	0	0
$853$	−364523.	−0.500988	−0.250494	−	0.968118i	$-0.580593\pi$
$853$	−364523.	−0.500988	−0.250494	+	0.968118i	$0.580593\pi$
$854$	0	0
$855$	0	0
$856$	−151742.	−0.207089
$857$	820187.i	1.11674i	0.829593	+	0.558369i	$0.188573\pi$
$857$	820187.i	1.11674i	−0.829593	+	0.558369i	$0.811427\pi$
$858$	0	0
$859$	−405456.	−0.549486	−0.274743	−	0.961518i	$-0.588593\pi$
$859$	−405456.	−0.549486	−0.274743	+	0.961518i	$0.588593\pi$
$860$	− 758257.i	− 1.02523i
$861$	0	0
$862$	−10513.1	−0.0141487
$863$	389932.i	0.523561i	0.965127	+	0.261781i	$0.0843097\pi$
$863$	389932.i	0.523561i	−0.965127	+	0.261781i	$0.915690\pi$
$864$	0	0
$865$	425117.	0.568167
$866$	− 503983.i	− 0.672017i
$867$	0	0
$868$	0	0
$869$	− 62071.2i	− 0.0821959i
$870$	0	0
$871$	−157396.	−0.207471
$872$	− 131415.i	− 0.172828i
$873$	0	0
$874$	1.20569e6	1.57838
$875$	0	0
$876$	0	0
$877$	587488.	0.763835	0.381918	−	0.924196i	$-0.375264\pi$
$877$	587488.	0.763835	0.381918	+	0.924196i	$0.375264\pi$
$878$	− 666337.i	− 0.864380i
$879$	0	0
$880$	−26866.2	−0.0346929
$881$	790614.i	1.01862i	0.860583	+	0.509310i	$0.170099\pi$
$881$	790614.i	1.01862i	−0.860583	+	0.509310i	$0.829901\pi$
$882$	0	0
$883$	511516.	0.656051	0.328026	−	0.944669i	$-0.393617\pi$
$883$	511516.	0.656051	0.328026	+	0.944669i	$0.393617\pi$
$884$	24194.0i	0.0309601i
$885$	0	0
$886$	−462075.	−0.588634
$887$	1.12332e6i	1.42776i	0.700269	+	0.713879i	$0.253063\pi$
$887$	1.12332e6i	1.42776i	−0.700269	+	0.713879i	$0.746937\pi$
$888$	0	0
$889$	0	0
$890$	− 103879.i	− 0.131144i
$891$	0	0
$892$	−380462.	−0.478170
$893$	− 544483.i	− 0.682781i
$894$	0	0
$895$	466919.	0.582902
$896$	0	0
$897$	0	0
$898$	308294.	0.382307
$899$	1.05133e6i	1.30082i
$900$	0	0
$901$	−33336.3	−0.0410646
$902$	− 74891.2i	− 0.0920487i
$903$	0	0
$904$	−561607.	−0.687220
$905$	− 177883.i	− 0.217188i
$906$	0	0
$907$	735492.	0.894053	0.447026	−	0.894521i	$-0.352483\pi$
$907$	735492.	0.894053	0.447026	+	0.894521i	$0.352483\pi$
$908$	128857.i	0.156291i
$909$	0	0
$910$	0	0
$911$	− 945510.i	− 1.13928i	−0.821895	−	0.569638i	$-0.807083\pi$
$911$	− 945510.i	− 1.13928i	0.821895	−	0.569638i	$-0.192917\pi$
$912$	0	0
$913$	−142477.	−0.170925
$914$	− 647119.i	− 0.774625i
$915$	0	0
$916$	−583548.	−0.695482
$917$	0	0
$918$	0	0
$919$	1.42714e6	1.68980	0.844899	−	0.534926i	$-0.179661\pi$
$919$	1.42714e6	1.68980	0.844899	+	0.534926i	$0.179661\pi$
$920$	− 651045.i	− 0.769193i
$921$	0	0
$922$	946067.	1.11291
$923$	50946.2i	0.0598010i
$924$	0	0
$925$	−682726.	−0.797927
$926$	− 195870.i	− 0.228427i
$927$	0	0
$928$	−202740.	−0.235420
$929$	− 1.23609e6i	− 1.43225i	−0.697973	−	0.716124i	$-0.745914\pi$
$929$	− 1.23609e6i	− 1.43225i	0.697973	−	0.716124i	$-0.254086\pi$
$930$	0	0
$931$	0	0
$932$	− 757448.i	− 0.872009i
$933$	0	0
$934$	226591.	0.259746
$935$	− 40563.7i	− 0.0463996i
$936$	0	0
$937$	−443156.	−0.504752	−0.252376	−	0.967629i	$-0.581212\pi$
$937$	−443156.	−0.504752	−0.252376	+	0.967629i	$0.581212\pi$
$938$	0	0
$939$	0	0
$940$	−294008.	−0.332739
$941$	598692.i	0.676120i	0.941124	+	0.338060i	$0.109771\pi$
$941$	598692.i	0.676120i	−0.941124	+	0.338060i	$0.890229\pi$
$942$	0	0
$943$	1.81483e6	2.04086
$944$	− 122871.i	− 0.137881i
$945$	0	0
$946$	−103806.	−0.115996
$947$	− 303712.i	− 0.338658i	−0.985560	−	0.169329i	$-0.945840\pi$
$947$	− 303712.i	− 0.338658i	0.985560	−	0.169329i	$-0.0541601\pi$
$948$	0	0
$949$	112087.	0.124458
$950$	633453.i	0.701887i
$951$	0	0
$952$	0	0
$953$	− 604699.i	− 0.665815i	−0.942960	−	0.332907i	$-0.891970\pi$
$953$	− 604699.i	− 0.665815i	0.942960	−	0.332907i	$-0.108030\pi$
$954$	0	0
$955$	−539541.	−0.591586
$956$	842682.i	0.922036i
$957$	0	0
$958$	686328.	0.747826
$959$	0	0
$960$	0	0
$961$	−42378.5	−0.0458879
$962$	− 131638.i	− 0.142243i
$963$	0	0
$964$	652914.	0.702590
$965$	− 2.32951e6i	− 2.50155i
$966$	0	0
$967$	−868964.	−0.929285	−0.464643	−	0.885498i	$-0.653817\pi$
$967$	−868964.	−0.929285	−0.464643	+	0.885498i	$0.653817\pi$
$968$	− 327610.i	− 0.349628i
$969$	0	0
$970$	1.03697e6	1.10210
$971$	1.07743e6i	1.14274i	0.820691	+	0.571372i	$0.193589\pi$
$971$	1.07743e6i	1.14274i	−0.820691	+	0.571372i	$0.806411\pi$
$972$	0	0
$973$	0	0
$974$	− 872327.i	− 0.919521i
$975$	0	0
$976$	417214.	0.437985
$977$	458986.i	0.480851i	0.970668	+	0.240425i	$0.0772869\pi$
$977$	458986.i	0.480851i	−0.970668	+	0.240425i	$0.922713\pi$
$978$	0	0
$979$	−14221.1	−0.0148378
$980$	0	0
$981$	0	0
$982$	−509240.	−0.528080
$983$	58950.5i	0.0610071i	0.999535	+	0.0305036i	$0.00971109\pi$
$983$	58950.5i	0.0610071i	−0.999535	+	0.0305036i	$0.990289\pi$
$984$	0	0
$985$	1.74480e6	1.79835
$986$	− 306105.i	− 0.314860i
$987$	0	0
$988$	−122137.	−0.125122
$989$	− 2.51553e6i	− 2.57179i
$990$	0	0
$991$	−1.80219e6	−1.83508	−0.917538	−	0.397647i	$-0.869827\pi$
$991$	−1.80219e6	−1.83508	−0.917538	+	0.397647i	$0.869827\pi$
$992$	169921.i	0.172673i
$993$	0	0
$994$	0	0
$995$	− 1.51273e6i	− 1.52797i
$996$	0	0
$997$	−869382.	−0.874622	−0.437311	−	0.899310i	$-0.644069\pi$
$997$	−869382.	−0.874622	−0.437311	+	0.899310i	$0.644069\pi$
$998$	766571.i	0.769646i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.5.b.e.197.4		6
3.2	odd	2	inner	882.5.b.e.197.3		6
7.3	odd	6		126.5.s.b.107.1	yes	12
7.5	odd	6		126.5.s.b.53.6	yes	12
7.6	odd	2		882.5.b.h.197.6		6
21.5	even	6		126.5.s.b.53.1	✓	12
21.17	even	6		126.5.s.b.107.6	yes	12
21.20	even	2		882.5.b.h.197.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.5.s.b.53.1	✓	12	21.5	even	6
126.5.s.b.53.6	yes	12	7.5	odd	6
126.5.s.b.107.1	yes	12	7.3	odd	6
126.5.s.b.107.6	yes	12	21.17	even	6
882.5.b.e.197.3		6	3.2	odd	2	inner
882.5.b.e.197.4		6	1.1	even	1	trivial
882.5.b.h.197.1		6	21.20	even	2
882.5.b.h.197.6		6	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	882.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.82843i q^{2} -8.00000 q^{4} -32.9258i q^{5} -22.6274i q^{8} +O(q^{10})\)	\(q+2.82843i q^{2} -8.00000 q^{4} -32.9258i q^{5} -22.6274i q^{8} +93.1283 q^{10} -12.7494i q^{11} -31.2973 q^{13} +64.0000 q^{16} +96.6298i q^{17} -487.811 q^{19} +263.407i q^{20} +36.0607 q^{22} +873.855i q^{23} -459.111 q^{25} -88.5220i q^{26} +1119.99i q^{29} +938.692 q^{31} +181.019i q^{32} -273.310 q^{34} +1487.06 q^{37} -1379.74i q^{38} -745.027 q^{40} -2076.81i q^{41} -2878.65 q^{43} +101.995i q^{44} -2471.63 q^{46} +1116.18i q^{47} -1298.56i q^{50} +250.378 q^{52} +344.990i q^{53} -419.784 q^{55} -3167.82 q^{58} -1919.85i q^{59} +6518.97 q^{61} +2655.02i q^{62} -512.000 q^{64} +1030.49i q^{65} +5029.07 q^{67} -773.039i q^{68} -1627.82i q^{71} -3581.38 q^{73} +4206.04i q^{74} +3902.49 q^{76} +4868.56 q^{79} -2107.25i q^{80} +5874.11 q^{82} -11175.2i q^{83} +3181.62 q^{85} -8142.07i q^{86} -288.486 q^{88} -1115.44i q^{89} -6990.84i q^{92} -3157.02 q^{94} +16061.6i q^{95} +11134.8 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 48 q^{4}+O(q^{10}) \) 6 * q - 48 * q^4	\( 6 q - 48 q^{4} - 104 q^{10} - 86 q^{13} + 384 q^{16} - 174 q^{19} + 472 q^{22} - 2170 q^{25} - 1682 q^{31} - 1744 q^{34} + 6018 q^{37} + 832 q^{40} - 2130 q^{43} - 656 q^{46} + 688 q^{52} + 12680 q^{55} + 7040 q^{58} + 20632 q^{61} - 3072 q^{64} - 74 q^{67} + 2942 q^{73} + 1392 q^{76} + 10526 q^{79} + 25464 q^{82} - 33056 q^{85} - 3776 q^{88} + 11640 q^{94} + 46436 q^{97}+O(q^{100}) \) 6 * q - 48 * q^4 - 104 * q^10 - 86 * q^13 + 384 * q^16 - 174 * q^19 + 472 * q^22 - 2170 * q^25 - 1682 * q^31 - 1744 * q^34 + 6018 * q^37 + 832 * q^40 - 2130 * q^43 - 656 * q^46 + 688 * q^52 + 12680 * q^55 + 7040 * q^58 + 20632 * q^61 - 3072 * q^64 - 74 * q^67 + 2942 * q^73 + 1392 * q^76 + 10526 * q^79 + 25464 * q^82 - 33056 * q^85 - 3776 * q^88 + 11640 * q^94 + 46436 * q^97

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