LMFDB - Embedded newform 882.4.a.z.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [882,4,Mod(1,882)]

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, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("882.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 882 = 2 \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	882.a (trivial)

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:	yes
Analytic conductor:	$52.0396846251$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{1345}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 336 $ x^2 - x - 336
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 42)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-17.8371$ of defining polynomial
Character	$\chi$	$=$	882.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{2} +4.00000 q^{4} +20.8371 q^{5} -8.00000 q^{8} +O(q^{10})$	$q-2.00000 q^{2} +4.00000 q^{4} +20.8371 q^{5} -8.00000 q^{8} -41.6742 q^{10} -15.1629 q^{11} -2.16288 q^{13} +16.0000 q^{16} -119.348 q^{17} +33.5114 q^{19} +83.3485 q^{20} +30.3258 q^{22} -0.651517 q^{23} +309.186 q^{25} +4.32576 q^{26} +163.208 q^{29} +223.326 q^{31} -32.0000 q^{32} +238.697 q^{34} +168.534 q^{37} -67.0227 q^{38} -166.697 q^{40} -323.023 q^{41} +221.557 q^{43} -60.6515 q^{44} +1.30303 q^{46} +508.045 q^{47} -618.371 q^{50} -8.65152 q^{52} +176.511 q^{53} -315.951 q^{55} -326.417 q^{58} +454.928 q^{59} -38.6515 q^{61} -446.652 q^{62} +64.0000 q^{64} -45.0682 q^{65} +141.792 q^{67} -477.394 q^{68} -602.742 q^{71} +1102.30 q^{73} -337.068 q^{74} +134.045 q^{76} -116.303 q^{79} +333.394 q^{80} +646.045 q^{82} -568.928 q^{83} -2486.88 q^{85} -443.114 q^{86} +121.303 q^{88} -383.159 q^{89} -2.60607 q^{92} -1016.09 q^{94} +698.280 q^{95} -334.701 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{2} + 8 q^{4} + 5 q^{5} - 16 q^{8}+O(q^{10}) $ 2 * q - 4 * q^2 + 8 * q^4 + 5 * q^5 - 16 * q^8	$ 2 q - 4 q^{2} + 8 q^{4} + 5 q^{5} - 16 q^{8} - 10 q^{10} - 67 q^{11} - 41 q^{13} + 32 q^{16} - 92 q^{17} - 43 q^{19} + 20 q^{20} + 134 q^{22} - 148 q^{23} + 435 q^{25} + 82 q^{26} - 77 q^{29} + 520 q^{31} - 64 q^{32} + 184 q^{34} + 7 q^{37} + 86 q^{38} - 40 q^{40} - 426 q^{41} - 107 q^{43} - 268 q^{44} + 296 q^{46} + 576 q^{47} - 870 q^{50} - 164 q^{52} + 243 q^{53} + 505 q^{55} + 154 q^{58} - 7 q^{59} - 224 q^{61} - 1040 q^{62} + 128 q^{64} + 570 q^{65} + 687 q^{67} - 368 q^{68} - 472 q^{71} + 921 q^{73} - 14 q^{74} - 172 q^{76} - 526 q^{79} + 80 q^{80} + 852 q^{82} - 221 q^{83} - 2920 q^{85} + 214 q^{86} + 536 q^{88} + 774 q^{89} - 592 q^{92} - 1152 q^{94} + 1910 q^{95} - 1953 q^{97}+O(q^{100}) $ 2 * q - 4 * q^2 + 8 * q^4 + 5 * q^5 - 16 * q^8 - 10 * q^10 - 67 * q^11 - 41 * q^13 + 32 * q^16 - 92 * q^17 - 43 * q^19 + 20 * q^20 + 134 * q^22 - 148 * q^23 + 435 * q^25 + 82 * q^26 - 77 * q^29 + 520 * q^31 - 64 * q^32 + 184 * q^34 + 7 * q^37 + 86 * q^38 - 40 * q^40 - 426 * q^41 - 107 * q^43 - 268 * q^44 + 296 * q^46 + 576 * q^47 - 870 * q^50 - 164 * q^52 + 243 * q^53 + 505 * q^55 + 154 * q^58 - 7 * q^59 - 224 * q^61 - 1040 * q^62 + 128 * q^64 + 570 * q^65 + 687 * q^67 - 368 * q^68 - 472 * q^71 + 921 * q^73 - 14 * q^74 - 172 * q^76 - 526 * q^79 + 80 * q^80 + 852 * q^82 - 221 * q^83 - 2920 * q^85 + 214 * q^86 + 536 * q^88 + 774 * q^89 - 592 * q^92 - 1152 * q^94 + 1910 * q^95 - 1953 * q^97

Display 100 coefficients 10 coefficients

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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	20.8371	1.86373	0.931864	−	0.362807i	$-0.118182\pi$
$5$	20.8371	1.86373	0.931864	+	0.362807i	$0.118182\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−8.00000	−0.353553
$9$	0	0
$10$	−41.6742	−1.31786
$11$	−15.1629	−0.415616	−0.207808	−	0.978170i	$-0.566633\pi$
$11$	−15.1629	−0.415616	−0.207808	+	0.978170i	$0.566633\pi$
$12$	0	0
$13$	−2.16288	−0.0461442	−0.0230721	−	0.999734i	$-0.507345\pi$
$13$	−2.16288	−0.0461442	−0.0230721	+	0.999734i	$0.507345\pi$
$14$	0	0
$15$	0	0
$16$	16.0000	0.250000
$17$	−119.348	−1.70272	−0.851361	−	0.524581i	$-0.824222\pi$
$17$	−119.348	−1.70272	−0.851361	+	0.524581i	$0.824222\pi$
$18$	0	0
$19$	33.5114	0.404633	0.202317	−	0.979320i	$-0.435153\pi$
$19$	33.5114	0.404633	0.202317	+	0.979320i	$0.435153\pi$
$20$	83.3485	0.931864
$21$	0	0
$22$	30.3258	0.293885
$23$	−0.651517	−0.00590655	−0.00295327	−	0.999996i	$-0.500940\pi$
$23$	−0.651517	−0.00590655	−0.00295327	+	0.999996i	$0.500940\pi$
$24$	0	0
$25$	309.186	2.47348
$26$	4.32576	0.0326289
$27$	0	0
$28$	0	0
$29$	163.208	1.04507	0.522535	−	0.852618i	$-0.324986\pi$
$29$	163.208	1.04507	0.522535	+	0.852618i	$0.324986\pi$
$30$	0	0
$31$	223.326	1.29389	0.646943	−	0.762538i	$-0.276047\pi$
$31$	223.326	1.29389	0.646943	+	0.762538i	$0.276047\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	238.697	1.20401
$35$	0	0
$36$	0	0
$37$	168.534	0.748833	0.374417	−	0.927261i	$-0.377843\pi$
$37$	168.534	0.748833	0.374417	+	0.927261i	$0.377843\pi$
$38$	−67.0227	−0.286119
$39$	0	0
$40$	−166.697	−0.658928
$41$	−323.023	−1.23043	−0.615216	−	0.788359i	$-0.710931\pi$
$41$	−323.023	−1.23043	−0.615216	+	0.788359i	$0.710931\pi$
$42$	0	0
$43$	221.557	0.785746	0.392873	−	0.919593i	$-0.371481\pi$
$43$	221.557	0.785746	0.392873	+	0.919593i	$0.371481\pi$
$44$	−60.6515	−0.207808
$45$	0	0
$46$	1.30303	0.00417656
$47$	508.045	1.57672	0.788362	−	0.615211i	$-0.210929\pi$
$47$	508.045	1.57672	0.788362	+	0.615211i	$0.210929\pi$
$48$	0	0
$49$	0	0
$50$	−618.371	−1.74902
$51$	0	0
$52$	−8.65152	−0.0230721
$53$	176.511	0.457466	0.228733	−	0.973489i	$-0.426542\pi$
$53$	176.511	0.457466	0.228733	+	0.973489i	$0.426542\pi$
$54$	0	0
$55$	−315.951	−0.774596
$56$	0	0
$57$	0	0
$58$	−326.417	−0.738976
$59$	454.928	1.00384	0.501920	−	0.864914i	$-0.332627\pi$
$59$	454.928	1.00384	0.501920	+	0.864914i	$0.332627\pi$
$60$	0	0
$61$	−38.6515	−0.0811282	−0.0405641	−	0.999177i	$-0.512915\pi$
$61$	−38.6515	−0.0811282	−0.0405641	+	0.999177i	$0.512915\pi$
$62$	−446.652	−0.914916
$63$	0	0
$64$	64.0000	0.125000
$65$	−45.0682	−0.0860003
$66$	0	0
$67$	141.792	0.258546	0.129273	−	0.991609i	$-0.458736\pi$
$67$	141.792	0.258546	0.129273	+	0.991609i	$0.458736\pi$
$68$	−477.394	−0.851361
$69$	0	0
$70$	0	0
$71$	−602.742	−1.00750	−0.503749	−	0.863850i	$-0.668046\pi$
$71$	−602.742	−1.00750	−0.503749	+	0.863850i	$0.668046\pi$
$72$	0	0
$73$	1102.30	1.76732	0.883660	−	0.468129i	$-0.155072\pi$
$73$	1102.30	1.76732	0.883660	+	0.468129i	$0.155072\pi$
$74$	−337.068	−0.529505
$75$	0	0
$76$	134.045	0.202317
$77$	0	0
$78$	0	0
$79$	−116.303	−0.165634	−0.0828172	−	0.996565i	$-0.526392\pi$
$79$	−116.303	−0.165634	−0.0828172	+	0.996565i	$0.526392\pi$
$80$	333.394	0.465932
$81$	0	0
$82$	646.045	0.870046
$83$	−568.928	−0.752385	−0.376193	−	0.926542i	$-0.622767\pi$
$83$	−568.928	−0.752385	−0.376193	+	0.926542i	$0.622767\pi$
$84$	0	0
$85$	−2486.88	−3.17341
$86$	−443.114	−0.555607
$87$	0	0
$88$	121.303	0.146943
$89$	−383.159	−0.456346	−0.228173	−	0.973621i	$-0.573275\pi$
$89$	−383.159	−0.456346	−0.228173	+	0.973621i	$0.573275\pi$
$90$	0	0
$91$	0	0
$92$	−2.60607	−0.00295327
$93$	0	0
$94$	−1016.09	−1.11491
$95$	698.280	0.754127
$96$	0	0
$97$	−334.701	−0.350348	−0.175174	−	0.984538i	$-0.556049\pi$
$97$	−334.701	−0.350348	−0.175174	+	0.984538i	$0.556049\pi$
$98$	0	0
$99$	0	0
$100$	1236.74	1.23674
$101$	−14.7424	−0.0145240	−0.00726201	−	0.999974i	$-0.502312\pi$
$101$	−14.7424	−0.0145240	−0.00726201	+	0.999974i	$0.502312\pi$
$102$	0	0
$103$	841.420	0.804928	0.402464	−	0.915436i	$-0.368154\pi$
$103$	841.420	0.804928	0.402464	+	0.915436i	$0.368154\pi$
$104$	17.3030	0.0163144
$105$	0	0
$106$	−353.023	−0.323477
$107$	715.670	0.646603	0.323301	−	0.946296i	$-0.395207\pi$
$107$	715.670	0.646603	0.323301	+	0.946296i	$0.395207\pi$
$108$	0	0
$109$	600.019	0.527260	0.263630	−	0.964624i	$-0.415080\pi$
$109$	600.019	0.527260	0.263630	+	0.964624i	$0.415080\pi$
$110$	631.901	0.547722
$111$	0	0
$112$	0	0
$113$	−622.644	−0.518349	−0.259174	−	0.965831i	$-0.583450\pi$
$113$	−622.644	−0.518349	−0.259174	+	0.965831i	$0.583450\pi$
$114$	0	0
$115$	−13.5757	−0.0110082
$116$	652.833	0.522535
$117$	0	0
$118$	−909.856	−0.709822
$119$	0	0
$120$	0	0
$121$	−1101.09	−0.827263
$122$	77.3030	0.0573663
$123$	0	0
$124$	893.303	0.646943
$125$	3837.90	2.74618
$126$	0	0
$127$	−180.076	−0.125820	−0.0629100	−	0.998019i	$-0.520038\pi$
$127$	−180.076	−0.125820	−0.0629100	+	0.998019i	$0.520038\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	90.1363	0.0608114
$131$	217.860	0.145302	0.0726508	−	0.997357i	$-0.476854\pi$
$131$	217.860	0.145302	0.0726508	+	0.997357i	$0.476854\pi$
$132$	0	0
$133$	0	0
$134$	−283.583	−0.182820
$135$	0	0
$136$	954.788	0.602003
$137$	2601.86	1.62257	0.811283	−	0.584654i	$-0.198770\pi$
$137$	2601.86	1.62257	0.811283	+	0.584654i	$0.198770\pi$
$138$	0	0
$139$	2651.55	1.61800	0.808998	−	0.587811i	$-0.200010\pi$
$139$	2651.55	1.61800	0.808998	+	0.587811i	$0.200010\pi$
$140$	0	0
$141$	0	0
$142$	1205.48	0.712409
$143$	32.7955	0.0191783
$144$	0	0
$145$	3400.79	1.94773
$146$	−2204.60	−1.24968
$147$	0	0
$148$	674.136	0.374417
$149$	−581.023	−0.319458	−0.159729	−	0.987161i	$-0.551062\pi$
$149$	−581.023	−0.319458	−0.159729	+	0.987161i	$0.551062\pi$
$150$	0	0
$151$	−615.390	−0.331654	−0.165827	−	0.986155i	$-0.553029\pi$
$151$	−615.390	−0.331654	−0.165827	+	0.986155i	$0.553029\pi$
$152$	−268.091	−0.143059
$153$	0	0
$154$	0	0
$155$	4653.47	2.41145
$156$	0	0
$157$	306.932	0.156024	0.0780122	−	0.996952i	$-0.475143\pi$
$157$	306.932	0.156024	0.0780122	+	0.996952i	$0.475143\pi$
$158$	232.606	0.117121
$159$	0	0
$160$	−666.788	−0.329464
$161$	0	0
$162$	0	0
$163$	3514.50	1.68882	0.844408	−	0.535701i	$-0.179953\pi$
$163$	3514.50	1.68882	0.844408	+	0.535701i	$0.179953\pi$
$164$	−1292.09	−0.615216
$165$	0	0
$166$	1137.86	0.532017
$167$	1123.30	0.520502	0.260251	−	0.965541i	$-0.416195\pi$
$167$	1123.30	0.520502	0.260251	+	0.965541i	$0.416195\pi$
$168$	0	0
$169$	−2192.32	−0.997871
$170$	4973.76	2.24394
$171$	0	0
$172$	886.227	0.392873
$173$	1530.60	0.672655	0.336327	−	0.941745i	$-0.390815\pi$
$173$	1530.60	0.672655	0.336327	+	0.941745i	$0.390815\pi$
$174$	0	0
$175$	0	0
$176$	−242.606	−0.103904
$177$	0	0
$178$	766.318	0.322685
$179$	−3413.43	−1.42532	−0.712659	−	0.701511i	$-0.752509\pi$
$179$	−3413.43	−1.42532	−0.712659	+	0.701511i	$0.752509\pi$
$180$	0	0
$181$	−1286.71	−0.528399	−0.264200	−	0.964468i	$-0.585108\pi$
$181$	−1286.71	−0.528399	−0.264200	+	0.964468i	$0.585108\pi$
$182$	0	0
$183$	0	0
$184$	5.21213	0.00208828
$185$	3511.77	1.39562
$186$	0	0
$187$	1809.67	0.707679
$188$	2032.18	0.788362
$189$	0	0
$190$	−1396.56	−0.533248
$191$	−1055.30	−0.399783	−0.199891	−	0.979818i	$-0.564059\pi$
$191$	−1055.30	−0.399783	−0.199891	+	0.979818i	$0.564059\pi$
$192$	0	0
$193$	−4770.84	−1.77934	−0.889670	−	0.456604i	$-0.849066\pi$
$193$	−4770.84	−1.77934	−0.889670	+	0.456604i	$0.849066\pi$
$194$	669.402	0.247733
$195$	0	0
$196$	0	0
$197$	−1622.31	−0.586725	−0.293363	−	0.956001i	$-0.594774\pi$
$197$	−1622.31	−0.586725	−0.293363	+	0.956001i	$0.594774\pi$
$198$	0	0
$199$	3550.14	1.26464	0.632318	−	0.774709i	$-0.282104\pi$
$199$	3550.14	1.26464	0.632318	+	0.774709i	$0.282104\pi$
$200$	−2473.48	−0.874509
$201$	0	0
$202$	29.4848	0.0102700
$203$	0	0
$204$	0	0
$205$	−6730.86	−2.29319
$206$	−1682.84	−0.569170
$207$	0	0
$208$	−34.6061	−0.0115361
$209$	−508.129	−0.168172
$210$	0	0
$211$	4653.39	1.51826	0.759129	−	0.650941i	$-0.225625\pi$
$211$	4653.39	1.51826	0.759129	+	0.650941i	$0.225625\pi$
$212$	706.045	0.228733
$213$	0	0
$214$	−1431.34	−0.457217
$215$	4616.61	1.46442
$216$	0	0
$217$	0	0
$218$	−1200.04	−0.372829
$219$	0	0
$220$	−1263.80	−0.387298
$221$	258.136	0.0785707
$222$	0	0
$223$	4649.53	1.39621	0.698107	−	0.715993i	$-0.254026\pi$
$223$	4649.53	1.39621	0.698107	+	0.715993i	$0.254026\pi$
$224$	0	0
$225$	0	0
$226$	1245.29	0.366528
$227$	4151.72	1.21392	0.606958	−	0.794734i	$-0.292389\pi$
$227$	4151.72	1.21392	0.606958	+	0.794734i	$0.292389\pi$
$228$	0	0
$229$	−4263.63	−1.23034	−0.615172	−	0.788393i	$-0.710913\pi$
$229$	−4263.63	−1.23034	−0.615172	+	0.788393i	$0.710913\pi$
$230$	27.1515	0.00778398
$231$	0	0
$232$	−1305.67	−0.369488
$233$	−3049.90	−0.857535	−0.428768	−	0.903415i	$-0.641052\pi$
$233$	−3049.90	−0.857535	−0.428768	+	0.903415i	$0.641052\pi$
$234$	0	0
$235$	10586.2	2.93859
$236$	1819.71	0.501920
$237$	0	0
$238$	0	0
$239$	−3987.20	−1.07912	−0.539562	−	0.841946i	$-0.681410\pi$
$239$	−3987.20	−1.07912	−0.539562	+	0.841946i	$0.681410\pi$
$240$	0	0
$241$	624.648	0.166959	0.0834795	−	0.996509i	$-0.473397\pi$
$241$	624.648	0.166959	0.0834795	+	0.996509i	$0.473397\pi$
$242$	2202.17	0.584963
$243$	0	0
$244$	−154.606	−0.0405641
$245$	0	0
$246$	0	0
$247$	−72.4810	−0.0186715
$248$	−1786.61	−0.457458
$249$	0	0
$250$	−7675.80	−1.94184
$251$	−1328.78	−0.334152	−0.167076	−	0.985944i	$-0.553432\pi$
$251$	−1328.78	−0.334152	−0.167076	+	0.985944i	$0.553432\pi$
$252$	0	0
$253$	9.87887	0.00245486
$254$	360.152	0.0889682
$255$	0	0
$256$	256.000	0.0625000
$257$	−3226.18	−0.783049	−0.391525	−	0.920168i	$-0.628052\pi$
$257$	−3226.18	−0.783049	−0.391525	+	0.920168i	$0.628052\pi$
$258$	0	0
$259$	0	0
$260$	−180.273	−0.0430001
$261$	0	0
$262$	−435.720	−0.102744
$263$	−3250.61	−0.762135	−0.381067	−	0.924547i	$-0.624443\pi$
$263$	−3250.61	−0.762135	−0.381067	+	0.924547i	$0.624443\pi$
$264$	0	0
$265$	3677.99	0.852593
$266$	0	0
$267$	0	0
$268$	567.167	0.129273
$269$	2826.04	0.640546	0.320273	−	0.947325i	$-0.396225\pi$
$269$	2826.04	0.640546	0.320273	+	0.947325i	$0.396225\pi$
$270$	0	0
$271$	2396.77	0.537245	0.268622	−	0.963246i	$-0.413432\pi$
$271$	2396.77	0.537245	0.268622	+	0.963246i	$0.413432\pi$
$272$	−1909.58	−0.425680
$273$	0	0
$274$	−5203.71	−1.14733
$275$	−4688.14	−1.02802
$276$	0	0
$277$	1820.47	0.394878	0.197439	−	0.980315i	$-0.436738\pi$
$277$	1820.47	0.394878	0.197439	+	0.980315i	$0.436738\pi$
$278$	−5303.10	−1.14410
$279$	0	0
$280$	0	0
$281$	−3083.81	−0.654679	−0.327339	−	0.944907i	$-0.606152\pi$
$281$	−3083.81	−0.654679	−0.327339	+	0.944907i	$0.606152\pi$
$282$	0	0
$283$	−2554.77	−0.536626	−0.268313	−	0.963332i	$-0.586466\pi$
$283$	−2554.77	−0.536626	−0.268313	+	0.963332i	$0.586466\pi$
$284$	−2410.97	−0.503749
$285$	0	0
$286$	−65.5910	−0.0135611
$287$	0	0
$288$	0	0
$289$	9331.06	1.89926
$290$	−6801.58	−1.37725
$291$	0	0
$292$	4409.20	0.883660
$293$	−1846.47	−0.368163	−0.184081	−	0.982911i	$-0.558931\pi$
$293$	−1846.47	−0.368163	−0.184081	+	0.982911i	$0.558931\pi$
$294$	0	0
$295$	9479.39	1.87089
$296$	−1348.27	−0.264753
$297$	0	0
$298$	1162.05	0.225891
$299$	1.40915	0.000272553 0
$300$	0	0
$301$	0	0
$302$	1230.78	0.234515
$303$	0	0
$304$	536.182	0.101158
$305$	−805.386	−0.151201
$306$	0	0
$307$	−7041.50	−1.30905	−0.654527	−	0.756039i	$-0.727132\pi$
$307$	−7041.50	−1.30905	−0.654527	+	0.756039i	$0.727132\pi$
$308$	0	0
$309$	0	0
$310$	−9306.93	−1.70516
$311$	−2685.99	−0.489738	−0.244869	−	0.969556i	$-0.578745\pi$
$311$	−2685.99	−0.489738	−0.244869	+	0.969556i	$0.578745\pi$
$312$	0	0
$313$	−2219.19	−0.400754	−0.200377	−	0.979719i	$-0.564217\pi$
$313$	−2219.19	−0.400754	−0.200377	+	0.979719i	$0.564217\pi$
$314$	−613.864	−0.110326
$315$	0	0
$316$	−465.212	−0.0828172
$317$	−2221.26	−0.393560	−0.196780	−	0.980448i	$-0.563048\pi$
$317$	−2221.26	−0.393560	−0.196780	+	0.980448i	$0.563048\pi$
$318$	0	0
$319$	−2474.71	−0.434348
$320$	1333.58	0.232966
$321$	0	0
$322$	0	0
$323$	−3999.53	−0.688978
$324$	0	0
$325$	−668.731	−0.114137
$326$	−7029.00	−1.19417
$327$	0	0
$328$	2584.18	0.435023
$329$	0	0
$330$	0	0
$331$	4154.06	0.689812	0.344906	−	0.938637i	$-0.387911\pi$
$331$	4154.06	0.689812	0.344906	+	0.938637i	$0.387911\pi$
$332$	−2275.71	−0.376193
$333$	0	0
$334$	−2246.61	−0.368050
$335$	2954.53	0.481860
$336$	0	0
$337$	−254.167	−0.0410841	−0.0205420	−	0.999789i	$-0.506539\pi$
$337$	−254.167	−0.0410841	−0.0205420	+	0.999789i	$0.506539\pi$
$338$	4384.64	0.705601
$339$	0	0
$340$	−9947.52	−1.58671
$341$	−3386.26	−0.537761
$342$	0	0
$343$	0	0
$344$	−1772.45	−0.277803
$345$	0	0
$346$	−3061.20	−0.475639
$347$	6224.64	0.962986	0.481493	−	0.876450i	$-0.340095\pi$
$347$	6224.64	0.962986	0.481493	+	0.876450i	$0.340095\pi$
$348$	0	0
$349$	−9732.21	−1.49270	−0.746352	−	0.665552i	$-0.768196\pi$
$349$	−9732.21	−1.49270	−0.746352	+	0.665552i	$0.768196\pi$
$350$	0	0
$351$	0	0
$352$	485.212	0.0734713
$353$	1425.61	0.214951	0.107476	−	0.994208i	$-0.465723\pi$
$353$	1425.61	0.214951	0.107476	+	0.994208i	$0.465723\pi$
$354$	0	0
$355$	−12559.4	−1.87770
$356$	−1532.64	−0.228173
$357$	0	0
$358$	6826.86	1.00785
$359$	5766.49	0.847754	0.423877	−	0.905720i	$-0.360669\pi$
$359$	5766.49	0.847754	0.423877	+	0.905720i	$0.360669\pi$
$360$	0	0
$361$	−5735.99	−0.836272
$362$	2573.42	0.373635
$363$	0	0
$364$	0	0
$365$	22968.7	3.29381
$366$	0	0
$367$	−11545.3	−1.64213	−0.821065	−	0.570834i	$-0.806620\pi$
$367$	−11545.3	−1.64213	−0.821065	+	0.570834i	$0.806620\pi$
$368$	−10.4243	−0.00147664
$369$	0	0
$370$	−7023.53	−0.986854
$371$	0	0
$372$	0	0
$373$	−6479.57	−0.899463	−0.449731	−	0.893164i	$-0.648480\pi$
$373$	−6479.57	−0.899463	−0.449731	+	0.893164i	$0.648480\pi$
$374$	−3619.33	−0.500404
$375$	0	0
$376$	−4064.36	−0.557456
$377$	−353.000	−0.0482239
$378$	0	0
$379$	611.996	0.0829449	0.0414725	−	0.999140i	$-0.486795\pi$
$379$	611.996	0.0829449	0.0414725	+	0.999140i	$0.486795\pi$
$380$	2793.12	0.377063
$381$	0	0
$382$	2110.59	0.282689
$383$	4360.81	0.581794	0.290897	−	0.956754i	$-0.406046\pi$
$383$	4360.81	0.581794	0.290897	+	0.956754i	$0.406046\pi$
$384$	0	0
$385$	0	0
$386$	9541.68	1.25818
$387$	0	0
$388$	−1338.80	−0.175174
$389$	13146.9	1.71356	0.856781	−	0.515681i	$-0.172461\pi$
$389$	13146.9	1.71356	0.856781	+	0.515681i	$0.172461\pi$
$390$	0	0
$391$	77.7575	0.0100572
$392$	0	0
$393$	0	0
$394$	3244.62	0.414877
$395$	−2423.42	−0.308697
$396$	0	0
$397$	8478.04	1.07179	0.535895	−	0.844285i	$-0.319974\pi$
$397$	8478.04	1.07179	0.535895	+	0.844285i	$0.319974\pi$
$398$	−7100.27	−0.894232
$399$	0	0
$400$	4946.97	0.618371
$401$	−2803.00	−0.349065	−0.174533	−	0.984651i	$-0.555841\pi$
$401$	−2803.00	−0.349065	−0.174533	+	0.984651i	$0.555841\pi$
$402$	0	0
$403$	−483.027	−0.0597054
$404$	−58.9697	−0.00726201
$405$	0	0
$406$	0	0
$407$	−2555.46	−0.311227
$408$	0	0
$409$	6385.39	0.771973	0.385987	−	0.922504i	$-0.373861\pi$
$409$	6385.39	0.771973	0.385987	+	0.922504i	$0.373861\pi$
$410$	13461.7	1.62153
$411$	0	0
$412$	3365.68	0.402464
$413$	0	0
$414$	0	0
$415$	−11854.8	−1.40224
$416$	69.2121	0.00815722
$417$	0	0
$418$	1016.26	0.118916
$419$	4831.66	0.563346	0.281673	−	0.959510i	$-0.409111\pi$
$419$	4831.66	0.563346	0.281673	+	0.959510i	$0.409111\pi$
$420$	0	0
$421$	7475.37	0.865385	0.432693	−	0.901542i	$-0.357564\pi$
$421$	7475.37	0.865385	0.432693	+	0.901542i	$0.357564\pi$
$422$	−9306.77	−1.07357
$423$	0	0
$424$	−1412.09	−0.161739
$425$	−36900.8	−4.21165
$426$	0	0
$427$	0	0
$428$	2862.68	0.323301
$429$	0	0
$430$	−9233.21	−1.03550
$431$	−6991.93	−0.781414	−0.390707	−	0.920515i	$-0.627769\pi$
$431$	−6991.93	−0.781414	−0.390707	+	0.920515i	$0.627769\pi$
$432$	0	0
$433$	7699.26	0.854510	0.427255	−	0.904131i	$-0.359481\pi$
$433$	7699.26	0.854510	0.427255	+	0.904131i	$0.359481\pi$
$434$	0	0
$435$	0	0
$436$	2400.08	0.263630
$437$	−21.8332	−0.00238999
$438$	0	0
$439$	−9412.32	−1.02329	−0.511646	−	0.859196i	$-0.670964\pi$
$439$	−9412.32	−1.02329	−0.511646	+	0.859196i	$0.670964\pi$
$440$	2527.61	0.273861
$441$	0	0
$442$	−516.273	−0.0555579
$443$	6258.18	0.671185	0.335593	−	0.942007i	$-0.391063\pi$
$443$	6258.18	0.671185	0.335593	+	0.942007i	$0.391063\pi$
$444$	0	0
$445$	−7983.93	−0.850505
$446$	−9299.07	−0.987273
$447$	0	0
$448$	0	0
$449$	11633.8	1.22279	0.611396	−	0.791325i	$-0.290608\pi$
$449$	11633.8	1.22279	0.611396	+	0.791325i	$0.290608\pi$
$450$	0	0
$451$	4897.95	0.511387
$452$	−2490.58	−0.259174
$453$	0	0
$454$	−8303.43	−0.858369
$455$	0	0
$456$	0	0
$457$	−13104.6	−1.34138	−0.670688	−	0.741740i	$-0.734001\pi$
$457$	−13104.6	−1.34138	−0.670688	+	0.741740i	$0.734001\pi$
$458$	8527.26	0.869985
$459$	0	0
$460$	−54.3029	−0.00550410
$461$	−2594.63	−0.262134	−0.131067	−	0.991373i	$-0.541840\pi$
$461$	−2594.63	−0.262134	−0.131067	+	0.991373i	$0.541840\pi$
$462$	0	0
$463$	−14136.2	−1.41893	−0.709465	−	0.704741i	$-0.751063\pi$
$463$	−14136.2	−1.41893	−0.709465	+	0.704741i	$0.751063\pi$
$464$	2611.33	0.261267
$465$	0	0
$466$	6099.80	0.606369
$467$	15590.2	1.54482	0.772409	−	0.635125i	$-0.219052\pi$
$467$	15590.2	1.54482	0.772409	+	0.635125i	$0.219052\pi$
$468$	0	0
$469$	0	0
$470$	−21172.4	−2.07789
$471$	0	0
$472$	−3639.42	−0.354911
$473$	−3359.44	−0.326569
$474$	0	0
$475$	10361.2	1.00085
$476$	0	0
$477$	0	0
$478$	7974.41	0.763056
$479$	−8453.51	−0.806369	−0.403184	−	0.915119i	$-0.632097\pi$
$479$	−8453.51	−0.806369	−0.403184	+	0.915119i	$0.632097\pi$
$480$	0	0
$481$	−364.519	−0.0345543
$482$	−1249.30	−0.118058
$483$	0	0
$484$	−4404.35	−0.413632
$485$	−6974.20	−0.652953
$486$	0	0
$487$	−4011.07	−0.373221	−0.186611	−	0.982434i	$-0.559750\pi$
$487$	−4011.07	−0.373221	−0.186611	+	0.982434i	$0.559750\pi$
$488$	309.212	0.0286831
$489$	0	0
$490$	0	0
$491$	−13927.9	−1.28016	−0.640079	−	0.768309i	$-0.721098\pi$
$491$	−13927.9	−1.28016	−0.640079	+	0.768309i	$0.721098\pi$
$492$	0	0
$493$	−19478.7	−1.77946
$494$	144.962	0.0132027
$495$	0	0
$496$	3573.21	0.323472
$497$	0	0
$498$	0	0
$499$	−3947.55	−0.354141	−0.177071	−	0.984198i	$-0.556662\pi$
$499$	−3947.55	−0.354141	−0.177071	+	0.984198i	$0.556662\pi$
$500$	15351.6	1.37309
$501$	0	0
$502$	2657.57	0.236281
$503$	−13725.3	−1.21666	−0.608331	−	0.793684i	$-0.708161\pi$
$503$	−13725.3	−1.21666	−0.608331	+	0.793684i	$0.708161\pi$
$504$	0	0
$505$	−307.190	−0.0270688
$506$	−19.7577	−0.00173585
$507$	0	0
$508$	−720.303	−0.0629100
$509$	7830.10	0.681853	0.340926	−	0.940090i	$-0.389259\pi$
$509$	7830.10	0.681853	0.340926	+	0.940090i	$0.389259\pi$
$510$	0	0
$511$	0	0
$512$	−512.000	−0.0441942
$513$	0	0
$514$	6452.36	0.553700
$515$	17532.8	1.50017
$516$	0	0
$517$	−7703.43	−0.655312
$518$	0	0
$519$	0	0
$520$	360.545	0.0304057
$521$	−5907.39	−0.496751	−0.248376	−	0.968664i	$-0.579897\pi$
$521$	−5907.39	−0.496751	−0.248376	+	0.968664i	$0.579897\pi$
$522$	0	0
$523$	−7908.06	−0.661176	−0.330588	−	0.943775i	$-0.607247\pi$
$523$	−7908.06	−0.661176	−0.330588	+	0.943775i	$0.607247\pi$
$524$	871.439	0.0726508
$525$	0	0
$526$	6501.23	0.538911
$527$	−26653.6	−2.20313
$528$	0	0
$529$	−12166.6	−0.999965
$530$	−7355.98	−0.602874
$531$	0	0
$532$	0	0
$533$	698.659	0.0567773
$534$	0	0
$535$	14912.5	1.20509
$536$	−1134.33	−0.0914100
$537$	0	0
$538$	−5652.08	−0.452934
$539$	0	0
$540$	0	0
$541$	−3941.04	−0.313195	−0.156598	−	0.987662i	$-0.550053\pi$
$541$	−3941.04	−0.313195	−0.156598	+	0.987662i	$0.550053\pi$
$542$	−4793.54	−0.379889
$543$	0	0
$544$	3819.15	0.301001
$545$	12502.7	0.982670
$546$	0	0
$547$	−1828.71	−0.142943	−0.0714717	−	0.997443i	$-0.522770\pi$
$547$	−1828.71	−0.142943	−0.0714717	+	0.997443i	$0.522770\pi$
$548$	10407.4	0.811283
$549$	0	0
$550$	9376.29	0.726920
$551$	5469.33	0.422870
$552$	0	0
$553$	0	0
$554$	−3640.93	−0.279221
$555$	0	0
$556$	10606.2	0.808998
$557$	−22532.0	−1.71402	−0.857011	−	0.515298i	$-0.827681\pi$
$557$	−22532.0	−1.71402	−0.857011	+	0.515298i	$0.827681\pi$
$558$	0	0
$559$	−479.201	−0.0362577
$560$	0	0
$561$	0	0
$562$	6167.62	0.462928
$563$	23355.7	1.74836	0.874179	−	0.485604i	$-0.161400\pi$
$563$	23355.7	1.74836	0.874179	+	0.485604i	$0.161400\pi$
$564$	0	0
$565$	−12974.1	−0.966062
$566$	5109.54	0.379452
$567$	0	0
$568$	4821.94	0.356204
$569$	20887.6	1.53894	0.769468	−	0.638686i	$-0.220522\pi$
$569$	20887.6	1.53894	0.769468	+	0.638686i	$0.220522\pi$
$570$	0	0
$571$	23745.3	1.74029	0.870147	−	0.492792i	$-0.164024\pi$
$571$	23745.3	1.74029	0.870147	+	0.492792i	$0.164024\pi$
$572$	131.182	0.00958914
$573$	0	0
$574$	0	0
$575$	−201.440	−0.0146098
$576$	0	0
$577$	−2454.39	−0.177084	−0.0885422	−	0.996072i	$-0.528221\pi$
$577$	−2454.39	−0.177084	−0.0885422	+	0.996072i	$0.528221\pi$
$578$	−18662.1	−1.34298
$579$	0	0
$580$	13603.2	0.973863
$581$	0	0
$582$	0	0
$583$	−2676.42	−0.190130
$584$	−8818.39	−0.624842
$585$	0	0
$586$	3692.93	0.260330
$587$	18567.5	1.30556	0.652780	−	0.757547i	$-0.273603\pi$
$587$	18567.5	1.30556	0.652780	+	0.757547i	$0.273603\pi$
$588$	0	0
$589$	7483.95	0.523550
$590$	−18958.8	−1.32292
$591$	0	0
$592$	2696.55	0.187208
$593$	17112.9	1.18507	0.592533	−	0.805546i	$-0.298128\pi$
$593$	17112.9	1.18507	0.592533	+	0.805546i	$0.298128\pi$
$594$	0	0
$595$	0	0
$596$	−2324.09	−0.159729
$597$	0	0
$598$	−2.81830	−0.000192724 0
$599$	−23264.8	−1.58694	−0.793469	−	0.608611i	$-0.791727\pi$
$599$	−23264.8	−1.58694	−0.793469	+	0.608611i	$0.791727\pi$
$600$	0	0
$601$	−25322.3	−1.71867	−0.859334	−	0.511416i	$-0.829121\pi$
$601$	−25322.3	−1.71867	−0.859334	+	0.511416i	$0.829121\pi$
$602$	0	0
$603$	0	0
$604$	−2461.56	−0.165827
$605$	−22943.5	−1.54179
$606$	0	0
$607$	21734.4	1.45333	0.726665	−	0.686992i	$-0.241069\pi$
$607$	21734.4	1.45333	0.726665	+	0.686992i	$0.241069\pi$
$608$	−1072.36	−0.0715297
$609$	0	0
$610$	1610.77	0.106915
$611$	−1098.84	−0.0727567
$612$	0	0
$613$	−13572.4	−0.894262	−0.447131	−	0.894468i	$-0.647554\pi$
$613$	−13572.4	−0.894262	−0.447131	+	0.894468i	$0.647554\pi$
$614$	14083.0	0.925641
$615$	0	0
$616$	0	0
$617$	8497.12	0.554427	0.277213	−	0.960808i	$-0.410589\pi$
$617$	8497.12	0.554427	0.277213	+	0.960808i	$0.410589\pi$
$618$	0	0
$619$	22982.9	1.49235	0.746173	−	0.665752i	$-0.231889\pi$
$619$	22982.9	1.49235	0.746173	+	0.665752i	$0.231889\pi$
$620$	18613.9	1.20573
$621$	0	0
$622$	5371.98	0.346297
$623$	0	0
$624$	0	0
$625$	41322.5	2.64464
$626$	4438.38	0.283376
$627$	0	0
$628$	1227.73	0.0780122
$629$	−20114.3	−1.27505
$630$	0	0
$631$	−15717.9	−0.991635	−0.495817	−	0.868427i	$-0.665131\pi$
$631$	−15717.9	−0.991635	−0.495817	+	0.868427i	$0.665131\pi$
$632$	930.424	0.0585606
$633$	0	0
$634$	4442.52	0.278289
$635$	−3752.26	−0.234494
$636$	0	0
$637$	0	0
$638$	4949.42	0.307131
$639$	0	0
$640$	−2667.15	−0.164732
$641$	−29107.4	−1.79356	−0.896780	−	0.442478i	$-0.854100\pi$
$641$	−29107.4	−1.79356	−0.896780	+	0.442478i	$0.854100\pi$
$642$	0	0
$643$	3112.26	0.190880	0.0954398	−	0.995435i	$-0.469574\pi$
$643$	3112.26	0.190880	0.0954398	+	0.995435i	$0.469574\pi$
$644$	0	0
$645$	0	0
$646$	7999.06	0.487181
$647$	−7857.59	−0.477456	−0.238728	−	0.971087i	$-0.576730\pi$
$647$	−7857.59	−0.477456	−0.238728	+	0.971087i	$0.576730\pi$
$648$	0	0
$649$	−6898.02	−0.417213
$650$	1337.46	0.0807071
$651$	0	0
$652$	14058.0	0.844408
$653$	19522.0	1.16992	0.584958	−	0.811063i	$-0.301111\pi$
$653$	19522.0	1.16992	0.584958	+	0.811063i	$0.301111\pi$
$654$	0	0
$655$	4539.57	0.270803
$656$	−5168.36	−0.307608
$657$	0	0
$658$	0	0
$659$	−664.061	−0.0392536	−0.0196268	−	0.999807i	$-0.506248\pi$
$659$	−664.061	−0.0392536	−0.0196268	+	0.999807i	$0.506248\pi$
$660$	0	0
$661$	−15921.6	−0.936883	−0.468442	−	0.883494i	$-0.655184\pi$
$661$	−15921.6	−0.936883	−0.468442	+	0.883494i	$0.655184\pi$
$662$	−8308.11	−0.487770
$663$	0	0
$664$	4551.42	0.266008
$665$	0	0
$666$	0	0
$667$	−106.333	−0.00617276
$668$	4493.21	0.260251
$669$	0	0
$670$	−5909.06	−0.340727
$671$	586.068	0.0337182
$672$	0	0
$673$	24631.0	1.41078	0.705391	−	0.708819i	$-0.250771\pi$
$673$	24631.0	1.41078	0.705391	+	0.708819i	$0.250771\pi$
$674$	508.333	0.0290508
$675$	0	0
$676$	−8769.29	−0.498935
$677$	17092.8	0.970353	0.485177	−	0.874416i	$-0.338755\pi$
$677$	17092.8	0.970353	0.485177	+	0.874416i	$0.338755\pi$
$678$	0	0
$679$	0	0
$680$	19895.0	1.12197
$681$	0	0
$682$	6772.52	0.380254
$683$	19163.6	1.07361	0.536804	−	0.843707i	$-0.319632\pi$
$683$	19163.6	1.07361	0.536804	+	0.843707i	$0.319632\pi$
$684$	0	0
$685$	54215.2	3.02402
$686$	0	0
$687$	0	0
$688$	3544.91	0.196437
$689$	−381.773	−0.0211094
$690$	0	0
$691$	−8095.87	−0.445704	−0.222852	−	0.974852i	$-0.571537\pi$
$691$	−8095.87	−0.445704	−0.222852	+	0.974852i	$0.571537\pi$
$692$	6122.39	0.336327
$693$	0	0
$694$	−12449.3	−0.680934
$695$	55250.7	3.01551
$696$	0	0
$697$	38552.3	2.09508
$698$	19464.4	1.05550
$699$	0	0
$700$	0	0
$701$	−12354.7	−0.665664	−0.332832	−	0.942986i	$-0.608004\pi$
$701$	−12354.7	−0.665664	−0.332832	+	0.942986i	$0.608004\pi$
$702$	0	0
$703$	5647.81	0.303003
$704$	−970.424	−0.0519520
$705$	0	0
$706$	−2851.23	−0.151993
$707$	0	0
$708$	0	0
$709$	3828.82	0.202813	0.101406	−	0.994845i	$-0.467666\pi$
$709$	3828.82	0.202813	0.101406	+	0.994845i	$0.467666\pi$
$710$	25118.8	1.32774
$711$	0	0
$712$	3065.27	0.161343
$713$	−145.500	−0.00764241
$714$	0	0
$715$	683.363	0.0357431
$716$	−13653.7	−0.712659
$717$	0	0
$718$	−11533.0	−0.599453
$719$	1223.00	0.0634356	0.0317178	−	0.999497i	$-0.489902\pi$
$719$	1223.00	0.0634356	0.0317178	+	0.999497i	$0.489902\pi$
$720$	0	0
$721$	0	0
$722$	11472.0	0.591333
$723$	0	0
$724$	−5146.83	−0.264200
$725$	50461.7	2.58496
$726$	0	0
$727$	−6368.21	−0.324875	−0.162437	−	0.986719i	$-0.551936\pi$
$727$	−6368.21	−0.324875	−0.162437	+	0.986719i	$0.551936\pi$
$728$	0	0
$729$	0	0
$730$	−45937.5	−2.32907
$731$	−26442.5	−1.33791
$732$	0	0
$733$	25154.0	1.26751	0.633753	−	0.773535i	$-0.281513\pi$
$733$	25154.0	1.26751	0.633753	+	0.773535i	$0.281513\pi$
$734$	23090.7	1.16116
$735$	0	0
$736$	20.8485	0.00104414
$737$	−2149.97	−0.107456
$738$	0	0
$739$	−10739.1	−0.534566	−0.267283	−	0.963618i	$-0.586126\pi$
$739$	−10739.1	−0.534566	−0.267283	+	0.963618i	$0.586126\pi$
$740$	14047.1	0.697811
$741$	0	0
$742$	0	0
$743$	−28166.3	−1.39074	−0.695370	−	0.718652i	$-0.744760\pi$
$743$	−28166.3	−1.39074	−0.695370	+	0.718652i	$0.744760\pi$
$744$	0	0
$745$	−12106.8	−0.595383
$746$	12959.1	0.636016
$747$	0	0
$748$	7238.67	0.353839
$749$	0	0
$750$	0	0
$751$	28657.0	1.39242	0.696211	−	0.717837i	$-0.254868\pi$
$751$	28657.0	1.39242	0.696211	+	0.717837i	$0.254868\pi$
$752$	8128.73	0.394181
$753$	0	0
$754$	706.000	0.0340995
$755$	−12823.0	−0.618113
$756$	0	0
$757$	−23604.1	−1.13330	−0.566648	−	0.823960i	$-0.691760\pi$
$757$	−23604.1	−1.13330	−0.566648	+	0.823960i	$0.691760\pi$
$758$	−1223.99	−0.0586509
$759$	0	0
$760$	−5586.24	−0.266624
$761$	4630.97	0.220595	0.110297	−	0.993899i	$-0.464820\pi$
$761$	4630.97	0.220595	0.110297	+	0.993899i	$0.464820\pi$
$762$	0	0
$763$	0	0
$764$	−4221.18	−0.199891
$765$	0	0
$766$	−8721.62	−0.411390
$767$	−983.954	−0.0463214
$768$	0	0
$769$	−33276.8	−1.56046	−0.780228	−	0.625495i	$-0.784897\pi$
$769$	−33276.8	−1.56046	−0.780228	+	0.625495i	$0.784897\pi$
$770$	0	0
$771$	0	0
$772$	−19083.4	−0.889670
$773$	−22938.8	−1.06734	−0.533668	−	0.845694i	$-0.679187\pi$
$773$	−22938.8	−1.06734	−0.533668	+	0.845694i	$0.679187\pi$
$774$	0	0
$775$	69049.1	3.20041
$776$	2677.61	0.123867
$777$	0	0
$778$	−26293.8	−1.21167
$779$	−10824.9	−0.497873
$780$	0	0
$781$	9139.31	0.418733
$782$	−155.515	−0.00711152
$783$	0	0
$784$	0	0
$785$	6395.58	0.290787
$786$	0	0
$787$	13514.5	0.612120	0.306060	−	0.952012i	$-0.400989\pi$
$787$	13514.5	0.612120	0.306060	+	0.952012i	$0.400989\pi$
$788$	−6489.24	−0.293363
$789$	0	0
$790$	4846.84	0.218282
$791$	0	0
$792$	0	0
$793$	83.5986	0.00374360
$794$	−16956.1	−0.757870
$795$	0	0
$796$	14200.5	0.632318
$797$	10473.4	0.465480	0.232740	−	0.972539i	$-0.425231\pi$
$797$	10473.4	0.465480	0.232740	+	0.972539i	$0.425231\pi$
$798$	0	0
$799$	−60634.5	−2.68472
$800$	−9893.94	−0.437254
$801$	0	0
$802$	5606.00	0.246826
$803$	−16714.0	−0.734527
$804$	0	0
$805$	0	0
$806$	966.053	0.0422181
$807$	0	0
$808$	117.939	0.00513501
$809$	−23568.0	−1.02423	−0.512117	−	0.858916i	$-0.671139\pi$
$809$	−23568.0	−1.02423	−0.512117	+	0.858916i	$0.671139\pi$
$810$	0	0
$811$	6704.22	0.290280	0.145140	−	0.989411i	$-0.453637\pi$
$811$	6704.22	0.290280	0.145140	+	0.989411i	$0.453637\pi$
$812$	0	0
$813$	0	0
$814$	5110.92	0.220071
$815$	73232.1	3.14749
$816$	0	0
$817$	7424.67	0.317939
$818$	−12770.8	−0.545867
$819$	0	0
$820$	−26923.5	−1.14659
$821$	−24539.4	−1.04316	−0.521579	−	0.853203i	$-0.674657\pi$
$821$	−24539.4	−1.04316	−0.521579	+	0.853203i	$0.674657\pi$
$822$	0	0
$823$	−31117.0	−1.31795	−0.658973	−	0.752167i	$-0.729009\pi$
$823$	−31117.0	−1.31795	−0.658973	+	0.752167i	$0.729009\pi$
$824$	−6731.36	−0.284585
$825$	0	0
$826$	0	0
$827$	−31244.9	−1.31377	−0.656887	−	0.753989i	$-0.728127\pi$
$827$	−31244.9	−1.31377	−0.656887	+	0.753989i	$0.728127\pi$
$828$	0	0
$829$	−4231.50	−0.177281	−0.0886405	−	0.996064i	$-0.528252\pi$
$829$	−4231.50	−0.177281	−0.0886405	+	0.996064i	$0.528252\pi$
$830$	23709.6	0.991535
$831$	0	0
$832$	−138.424	−0.00576803
$833$	0	0
$834$	0	0
$835$	23406.4	0.970074
$836$	−2032.51	−0.0840861
$837$	0	0
$838$	−9663.32	−0.398346
$839$	−38670.4	−1.59124	−0.795621	−	0.605795i	$-0.792855\pi$
$839$	−38670.4	−1.59124	−0.795621	+	0.605795i	$0.792855\pi$
$840$	0	0
$841$	2247.96	0.0921710
$842$	−14950.7	−0.611920
$843$	0	0
$844$	18613.5	0.759129
$845$	−45681.7	−1.85976
$846$	0	0
$847$	0	0
$848$	2824.18	0.114367
$849$	0	0
$850$	73801.7	2.97809
$851$	−109.803	−0.00442302
$852$	0	0
$853$	19944.4	0.800565	0.400282	−	0.916392i	$-0.368912\pi$
$853$	19944.4	0.800565	0.400282	+	0.916392i	$0.368912\pi$
$854$	0	0
$855$	0	0
$856$	−5725.36	−0.228609
$857$	13882.2	0.553334	0.276667	−	0.960966i	$-0.410770\pi$
$857$	13882.2	0.553334	0.276667	+	0.960966i	$0.410770\pi$
$858$	0	0
$859$	4157.16	0.165123	0.0825614	−	0.996586i	$-0.473690\pi$
$859$	4157.16	0.165123	0.0825614	+	0.996586i	$0.473690\pi$
$860$	18466.4	0.732209
$861$	0	0
$862$	13983.9	0.552543
$863$	16237.9	0.640493	0.320246	−	0.947334i	$-0.396234\pi$
$863$	16237.9	0.640493	0.320246	+	0.947334i	$0.396234\pi$
$864$	0	0
$865$	31893.3	1.25365
$866$	−15398.5	−0.604230
$867$	0	0
$868$	0	0
$869$	1763.49	0.0688403
$870$	0	0
$871$	−306.678	−0.0119304
$872$	−4800.15	−0.186415
$873$	0	0
$874$	43.6664	0.00168998
$875$	0	0
$876$	0	0
$877$	−16489.4	−0.634900	−0.317450	−	0.948275i	$-0.602827\pi$
$877$	−16489.4	−0.634900	−0.317450	+	0.948275i	$0.602827\pi$
$878$	18824.6	0.723577
$879$	0	0
$880$	−5055.21	−0.193649
$881$	45411.7	1.73662	0.868309	−	0.496023i	$-0.165207\pi$
$881$	45411.7	1.73662	0.868309	+	0.496023i	$0.165207\pi$
$882$	0	0
$883$	−2206.85	−0.0841070	−0.0420535	−	0.999115i	$-0.513390\pi$
$883$	−2206.85	−0.0841070	−0.0420535	+	0.999115i	$0.513390\pi$
$884$	1032.55	0.0392854
$885$	0	0
$886$	−12516.4	−0.474600
$887$	28146.2	1.06545	0.532727	−	0.846287i	$-0.321167\pi$
$887$	28146.2	1.06545	0.532727	+	0.846287i	$0.321167\pi$
$888$	0	0
$889$	0	0
$890$	15967.9	0.601398
$891$	0	0
$892$	18598.1	0.698107
$893$	17025.3	0.637995
$894$	0	0
$895$	−71126.1	−2.65641
$896$	0	0
$897$	0	0
$898$	−23267.6	−0.864644
$899$	36448.6	1.35220
$900$	0	0
$901$	−21066.4	−0.778937
$902$	−9795.91	−0.361605
$903$	0	0
$904$	4981.15	0.183264
$905$	−26811.3	−0.984793
$906$	0	0
$907$	−5042.25	−0.184592	−0.0922960	−	0.995732i	$-0.529421\pi$
$907$	−5042.25	−0.184592	−0.0922960	+	0.995732i	$0.529421\pi$
$908$	16606.9	0.606958
$909$	0	0
$910$	0	0
$911$	−29647.3	−1.07822	−0.539110	−	0.842235i	$-0.681239\pi$
$911$	−29647.3	−1.07822	−0.539110	+	0.842235i	$0.681239\pi$
$912$	0	0
$913$	8626.59	0.312704
$914$	26209.3	0.948496
$915$	0	0
$916$	−17054.5	−0.615172
$917$	0	0
$918$	0	0
$919$	−11891.3	−0.426830	−0.213415	−	0.976962i	$-0.568459\pi$
$919$	−11891.3	−0.426830	−0.213415	+	0.976962i	$0.568459\pi$
$920$	108.606	0.00389199
$921$	0	0
$922$	5189.26	0.185357
$923$	1303.66	0.0464902
$924$	0	0
$925$	52108.3	1.85223
$926$	28272.4	1.00333
$927$	0	0
$928$	−5222.67	−0.184744
$929$	−39188.5	−1.38400	−0.691999	−	0.721898i	$-0.743270\pi$
$929$	−39188.5	−1.38400	−0.691999	+	0.721898i	$0.743270\pi$
$930$	0	0
$931$	0	0
$932$	−12199.6	−0.428768
$933$	0	0
$934$	−31180.5	−1.09235
$935$	37708.2	1.31892
$936$	0	0
$937$	−9716.23	−0.338757	−0.169379	−	0.985551i	$-0.554176\pi$
$937$	−9716.23	−0.338757	−0.169379	+	0.985551i	$0.554176\pi$
$938$	0	0
$939$	0	0
$940$	42344.8	1.46929
$941$	−6995.87	−0.242358	−0.121179	−	0.992631i	$-0.538667\pi$
$941$	−6995.87	−0.242358	−0.121179	+	0.992631i	$0.538667\pi$
$942$	0	0
$943$	210.455	0.00726760
$944$	7278.85	0.250960
$945$	0	0
$946$	6718.88	0.230919
$947$	−14979.2	−0.514002	−0.257001	−	0.966411i	$-0.582734\pi$
$947$	−14979.2	−0.514002	−0.257001	+	0.966411i	$0.582734\pi$
$948$	0	0
$949$	−2384.14	−0.0815516
$950$	−20722.5	−0.707711
$951$	0	0
$952$	0	0
$953$	−29393.3	−0.999100	−0.499550	−	0.866285i	$-0.666501\pi$
$953$	−29393.3	−0.999100	−0.499550	+	0.866285i	$0.666501\pi$
$954$	0	0
$955$	−21989.3	−0.745087
$956$	−15948.8	−0.539562
$957$	0	0
$958$	16907.0	0.570189
$959$	0	0
$960$	0	0
$961$	20083.4	0.674143
$962$	729.038	0.0244336
$963$	0	0
$964$	2498.59	0.0834795
$965$	−99410.6	−3.31621
$966$	0	0
$967$	−7133.95	−0.237241	−0.118621	−	0.992940i	$-0.537847\pi$
$967$	−7133.95	−0.237241	−0.118621	+	0.992940i	$0.537847\pi$
$968$	8808.70	0.292482
$969$	0	0
$970$	13948.4	0.461707
$971$	−9688.13	−0.320192	−0.160096	−	0.987101i	$-0.551180\pi$
$971$	−9688.13	−0.320192	−0.160096	+	0.987101i	$0.551180\pi$
$972$	0	0
$973$	0	0
$974$	8022.14	0.263907
$975$	0	0
$976$	−618.424	−0.0202820
$977$	−21305.7	−0.697676	−0.348838	−	0.937183i	$-0.613424\pi$
$977$	−21305.7	−0.697676	−0.348838	+	0.937183i	$0.613424\pi$
$978$	0	0
$979$	5809.79	0.189665
$980$	0	0
$981$	0	0
$982$	27855.8	0.905208
$983$	−37280.8	−1.20964	−0.604818	−	0.796364i	$-0.706754\pi$
$983$	−37280.8	−1.20964	−0.604818	+	0.796364i	$0.706754\pi$
$984$	0	0
$985$	−33804.3	−1.09350
$986$	38957.3	1.25827
$987$	0	0
$988$	−289.924	−0.00933574
$989$	−144.348	−0.00464105
$990$	0	0
$991$	−51397.1	−1.64751	−0.823755	−	0.566946i	$-0.808125\pi$
$991$	−51397.1	−1.64751	−0.823755	+	0.566946i	$0.808125\pi$
$992$	−7146.42	−0.228729
$993$	0	0
$994$	0	0
$995$	73974.6	2.35694
$996$	0	0
$997$	34373.8	1.09191	0.545953	−	0.837816i	$-0.316168\pi$
$997$	34373.8	1.09191	0.545953	+	0.837816i	$0.316168\pi$
$998$	7895.10	0.250416
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.4.a.z.1.2		2
3.2	odd	2		294.4.a.m.1.1		2
7.2	even	3		882.4.g.bf.361.1		4
7.3	odd	6		126.4.g.g.37.2		4
7.4	even	3		882.4.g.bf.667.1		4
7.5	odd	6		126.4.g.g.109.2		4
7.6	odd	2		882.4.a.v.1.1		2
12.11	even	2		2352.4.a.ca.1.1		2
21.2	odd	6		294.4.e.l.67.2		4
21.5	even	6		42.4.e.c.25.1	✓	4
21.11	odd	6		294.4.e.l.79.2		4
21.17	even	6		42.4.e.c.37.1	yes	4
21.20	even	2		294.4.a.n.1.2		2
84.47	odd	6		336.4.q.j.193.1		4
84.59	odd	6		336.4.q.j.289.1		4
84.83	odd	2		2352.4.a.bq.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.4.e.c.25.1	✓	4	21.5	even	6
42.4.e.c.37.1	yes	4	21.17	even	6
126.4.g.g.37.2		4	7.3	odd	6
126.4.g.g.109.2		4	7.5	odd	6
294.4.a.m.1.1		2	3.2	odd	2
294.4.a.n.1.2		2	21.20	even	2
294.4.e.l.67.2		4	21.2	odd	6
294.4.e.l.79.2		4	21.11	odd	6
336.4.q.j.193.1		4	84.47	odd	6
336.4.q.j.289.1		4	84.59	odd	6
882.4.a.v.1.1		2	7.6	odd	2
882.4.a.z.1.2		2	1.1	even	1	trivial
882.4.g.bf.361.1		4	7.2	even	3
882.4.g.bf.667.1		4	7.4	even	3
2352.4.a.bq.1.2		2	84.83	odd	2
2352.4.a.ca.1.1		2	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	882.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +4.00000 q^{4} +20.8371 q^{5} -8.00000 q^{8} +O(q^{10})\)	\(q-2.00000 q^{2} +4.00000 q^{4} +20.8371 q^{5} -8.00000 q^{8} -41.6742 q^{10} -15.1629 q^{11} -2.16288 q^{13} +16.0000 q^{16} -119.348 q^{17} +33.5114 q^{19} +83.3485 q^{20} +30.3258 q^{22} -0.651517 q^{23} +309.186 q^{25} +4.32576 q^{26} +163.208 q^{29} +223.326 q^{31} -32.0000 q^{32} +238.697 q^{34} +168.534 q^{37} -67.0227 q^{38} -166.697 q^{40} -323.023 q^{41} +221.557 q^{43} -60.6515 q^{44} +1.30303 q^{46} +508.045 q^{47} -618.371 q^{50} -8.65152 q^{52} +176.511 q^{53} -315.951 q^{55} -326.417 q^{58} +454.928 q^{59} -38.6515 q^{61} -446.652 q^{62} +64.0000 q^{64} -45.0682 q^{65} +141.792 q^{67} -477.394 q^{68} -602.742 q^{71} +1102.30 q^{73} -337.068 q^{74} +134.045 q^{76} -116.303 q^{79} +333.394 q^{80} +646.045 q^{82} -568.928 q^{83} -2486.88 q^{85} -443.114 q^{86} +121.303 q^{88} -383.159 q^{89} -2.60607 q^{92} -1016.09 q^{94} +698.280 q^{95} -334.701 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{2} + 8 q^{4} + 5 q^{5} - 16 q^{8}+O(q^{10}) \) 2 * q - 4 * q^2 + 8 * q^4 + 5 * q^5 - 16 * q^8	\( 2 q - 4 q^{2} + 8 q^{4} + 5 q^{5} - 16 q^{8} - 10 q^{10} - 67 q^{11} - 41 q^{13} + 32 q^{16} - 92 q^{17} - 43 q^{19} + 20 q^{20} + 134 q^{22} - 148 q^{23} + 435 q^{25} + 82 q^{26} - 77 q^{29} + 520 q^{31} - 64 q^{32} + 184 q^{34} + 7 q^{37} + 86 q^{38} - 40 q^{40} - 426 q^{41} - 107 q^{43} - 268 q^{44} + 296 q^{46} + 576 q^{47} - 870 q^{50} - 164 q^{52} + 243 q^{53} + 505 q^{55} + 154 q^{58} - 7 q^{59} - 224 q^{61} - 1040 q^{62} + 128 q^{64} + 570 q^{65} + 687 q^{67} - 368 q^{68} - 472 q^{71} + 921 q^{73} - 14 q^{74} - 172 q^{76} - 526 q^{79} + 80 q^{80} + 852 q^{82} - 221 q^{83} - 2920 q^{85} + 214 q^{86} + 536 q^{88} + 774 q^{89} - 592 q^{92} - 1152 q^{94} + 1910 q^{95} - 1953 q^{97}+O(q^{100}) \) 2 * q - 4 * q^2 + 8 * q^4 + 5 * q^5 - 16 * q^8 - 10 * q^10 - 67 * q^11 - 41 * q^13 + 32 * q^16 - 92 * q^17 - 43 * q^19 + 20 * q^20 + 134 * q^22 - 148 * q^23 + 435 * q^25 + 82 * q^26 - 77 * q^29 + 520 * q^31 - 64 * q^32 + 184 * q^34 + 7 * q^37 + 86 * q^38 - 40 * q^40 - 426 * q^41 - 107 * q^43 - 268 * q^44 + 296 * q^46 + 576 * q^47 - 870 * q^50 - 164 * q^52 + 243 * q^53 + 505 * q^55 + 154 * q^58 - 7 * q^59 - 224 * q^61 - 1040 * q^62 + 128 * q^64 + 570 * q^65 + 687 * q^67 - 368 * q^68 - 472 * q^71 + 921 * q^73 - 14 * q^74 - 172 * q^76 - 526 * q^79 + 80 * q^80 + 852 * q^82 - 221 * q^83 - 2920 * q^85 + 214 * q^86 + 536 * q^88 + 774 * q^89 - 592 * q^92 - 1152 * q^94 + 1910 * q^95 - 1953 * q^97

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