LMFDB - Embedded newform 441.4.c.b.440.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,4,Mod(440,441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

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

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

chi := DirichletCharacter("441.440");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 441 = 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	441.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:	$26.0198423125$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			440.20
Character	$\chi$	$=$	441.440
Dual form			441.4.c.b.440.19

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.641037i q^{2} +7.58907 q^{4} +12.0729 q^{5} +9.99318i q^{8} +O(q^{10})$	$q+0.641037i q^{2} +7.58907 q^{4} +12.0729 q^{5} +9.99318i q^{8} +7.73920i q^{10} -43.9393i q^{11} -66.9783i q^{13} +54.3066 q^{16} +39.4310 q^{17} -57.9144i q^{19} +91.6223 q^{20} +28.1667 q^{22} -46.1519i q^{23} +20.7555 q^{25} +42.9356 q^{26} +201.623i q^{29} -185.521i q^{31} +114.758i q^{32} +25.2767i q^{34} -410.805 q^{37} +37.1253 q^{38} +120.647i q^{40} +408.886 q^{41} +129.530 q^{43} -333.458i q^{44} +29.5851 q^{46} -514.681 q^{47} +13.3051i q^{50} -508.303i q^{52} +437.584i q^{53} -530.475i q^{55} -129.248 q^{58} +607.906 q^{59} +24.5192i q^{61} +118.926 q^{62} +360.888 q^{64} -808.624i q^{65} +441.499 q^{67} +299.245 q^{68} +1150.89i q^{71} +525.077i q^{73} -263.341i q^{74} -439.517i q^{76} -151.850 q^{79} +655.639 q^{80} +262.111i q^{82} +1302.05 q^{83} +476.047 q^{85} +83.0338i q^{86} +439.093 q^{88} +583.190 q^{89} -350.250i q^{92} -329.929i q^{94} -699.196i q^{95} -1091.84i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 96 q^{4}+O(q^{10}) $ 24 * q - 96 * q^4	$ 24 q - 96 q^{4} + 144 q^{16} + 624 q^{22} + 312 q^{25} - 864 q^{37} + 1248 q^{43} - 3888 q^{46} - 7440 q^{58} - 3360 q^{64} - 2688 q^{67} + 480 q^{79} + 13248 q^{85} - 7248 q^{88}+O(q^{100}) $ 24 * q - 96 * q^4 + 144 * q^16 + 624 * q^22 + 312 * q^25 - 864 * q^37 + 1248 * q^43 - 3888 * q^46 - 7440 * q^58 - 3360 * q^64 - 2688 * q^67 + 480 * q^79 + 13248 * q^85 - 7248 * q^88

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/441\mathbb{Z}\right)^\times$.

$n$	$199$	$344$
$\chi(n)$	$-1$	$-1$

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 $

$2$	0.641037i	0.226641i	0.993558	+	0.113320i	$0.0361487\pi$
$2$	0.641037i	0.226641i	−0.993558	+	0.113320i	$0.963851\pi$
$3$	0	0
$3$	0	0
$4$	7.58907	0.948634
$5$	12.0729	1.07984	0.539918	−	0.841718i	$-0.318455\pi$
$5$	12.0729	1.07984	0.539918	+	0.841718i	$0.318455\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	9.99318i	0.441640i
$9$	0	0
$10$	7.73920i	0.244735i
$11$	− 43.9393i	− 1.20438i	−0.798353	−	0.602190i	$-0.794295\pi$
$11$	− 43.9393i	− 1.20438i	0.798353	−	0.602190i	$-0.205705\pi$
$12$	0	0
$13$	− 66.9783i	− 1.42896i	−0.699657	−	0.714479i	$-0.746664\pi$
$13$	− 66.9783i	− 1.42896i	0.699657	−	0.714479i	$-0.253336\pi$
$14$	0	0
$15$	0	0
$16$	54.3066	0.848540
$17$	39.4310	0.562554	0.281277	−	0.959627i	$-0.409242\pi$
$17$	39.4310	0.562554	0.281277	+	0.959627i	$0.409242\pi$
$18$	0	0
$19$	− 57.9144i	− 0.699288i	−0.936883	−	0.349644i	$-0.886302\pi$
$19$	− 57.9144i	− 0.699288i	0.936883	−	0.349644i	$-0.113698\pi$
$20$	91.6223	1.02437
$21$	0	0
$22$	28.1667	0.272962
$23$	− 46.1519i	− 0.418406i	−0.977872	−	0.209203i	$-0.932913\pi$
$23$	− 46.1519i	− 0.418406i	0.977872	−	0.209203i	$-0.0670869\pi$
$24$	0	0
$25$	20.7555	0.166044
$26$	42.9356	0.323860
$27$	0	0
$28$	0	0
$29$	201.623i	1.29105i	0.763738	+	0.645526i	$0.223362\pi$
$29$	201.623i	1.29105i	−0.763738	+	0.645526i	$0.776638\pi$
$30$	0	0
$31$	− 185.521i	− 1.07486i	−0.843309	−	0.537428i	$-0.819396\pi$
$31$	− 185.521i	− 1.07486i	0.843309	−	0.537428i	$-0.180604\pi$
$32$	114.758i	0.633954i
$33$	0	0
$34$	25.2767i	0.127498i
$35$	0	0
$36$	0	0
$37$	−410.805	−1.82529	−0.912647	−	0.408748i	$-0.865966\pi$
$37$	−410.805	−1.82529	−0.912647	+	0.408748i	$0.865966\pi$
$38$	37.1253	0.158487
$39$	0	0
$40$	120.647i	0.476899i
$41$	408.886	1.55749	0.778746	−	0.627339i	$-0.215856\pi$
$41$	408.886	1.55749	0.778746	+	0.627339i	$0.215856\pi$
$42$	0	0
$43$	129.530	0.459377	0.229688	−	0.973264i	$-0.426229\pi$
$43$	129.530	0.459377	0.229688	+	0.973264i	$0.426229\pi$
$44$	− 333.458i	− 1.14252i
$45$	0	0
$46$	29.5851	0.0948279
$47$	−514.681	−1.59732	−0.798658	−	0.601785i	$-0.794456\pi$
$47$	−514.681	−1.59732	−0.798658	+	0.601785i	$0.794456\pi$
$48$	0	0
$49$	0	0
$50$	13.3051i	0.0376324i
$51$	0	0
$52$	− 508.303i	− 1.35556i
$53$	437.584i	1.13409i	0.823687	+	0.567045i	$0.191913\pi$
$53$	437.584i	1.13409i	−0.823687	+	0.567045i	$0.808087\pi$
$54$	0	0
$55$	− 530.475i	− 1.30053i
$56$	0	0
$57$	0	0
$58$	−129.248	−0.292605
$59$	607.906	1.34140	0.670701	−	0.741728i	$-0.265994\pi$
$59$	607.906	1.34140	0.670701	+	0.741728i	$0.265994\pi$
$60$	0	0
$61$	24.5192i	0.0514649i	0.999669	+	0.0257324i	$0.00819179\pi$
$61$	24.5192i	0.0514649i	−0.999669	+	0.0257324i	$0.991808\pi$
$62$	118.926	0.243607
$63$	0	0
$64$	360.888	0.704860
$65$	− 808.624i	− 1.54304i
$66$	0	0
$67$	441.499	0.805041	0.402520	−	0.915411i	$-0.368134\pi$
$67$	441.499	0.805041	0.402520	+	0.915411i	$0.368134\pi$
$68$	299.245	0.533658
$69$	0	0
$70$	0	0
$71$	1150.89i	1.92373i	0.273518	+	0.961867i	$0.411813\pi$
$71$	1150.89i	1.92373i	−0.273518	+	0.961867i	$0.588187\pi$
$72$	0	0
$73$	525.077i	0.841858i	0.907094	+	0.420929i	$0.138296\pi$
$73$	525.077i	0.841858i	−0.907094	+	0.420929i	$0.861704\pi$
$74$	− 263.341i	− 0.413686i
$75$	0	0
$76$	− 439.517i	− 0.663369i
$77$	0	0
$78$	0	0
$79$	−151.850	−0.216259	−0.108130	−	0.994137i	$-0.534486\pi$
$79$	−151.850	−0.216259	−0.108130	+	0.994137i	$0.534486\pi$
$80$	655.639	0.916284
$81$	0	0
$82$	262.111i	0.352992i
$83$	1302.05	1.72191	0.860956	−	0.508680i	$-0.169866\pi$
$83$	1302.05	1.72191	0.860956	+	0.508680i	$0.169866\pi$
$84$	0	0
$85$	476.047	0.607466
$86$	83.0338i	0.104114i
$87$	0	0
$88$	439.093	0.531903
$89$	583.190	0.694584	0.347292	−	0.937757i	$-0.387101\pi$
$89$	583.190	0.694584	0.347292	+	0.937757i	$0.387101\pi$
$90$	0	0
$91$	0	0
$92$	− 350.250i	− 0.396914i
$93$	0	0
$94$	− 329.929i	− 0.362017i
$95$	− 699.196i	− 0.755116i
$96$	0	0
$97$	− 1091.84i	− 1.14288i	−0.820643	−	0.571441i	$-0.806384\pi$
$97$	− 1091.84i	− 1.14288i	0.820643	−	0.571441i	$-0.193616\pi$
$98$	0	0
$99$	0	0
$100$	157.515	0.157515
$101$	156.773	0.154451	0.0772254	−	0.997014i	$-0.475394\pi$
$101$	156.773	0.154451	0.0772254	+	0.997014i	$0.475394\pi$
$102$	0	0
$103$	− 241.078i	− 0.230622i	−0.993329	−	0.115311i	$-0.963213\pi$
$103$	− 241.078i	− 0.230622i	0.993329	−	0.115311i	$-0.0367865\pi$
$104$	669.326	0.631085
$105$	0	0
$106$	−280.507	−0.257031
$107$	− 12.8374i	− 0.0115985i	−0.999983	−	0.00579926i	$-0.998154\pi$
$107$	− 12.8374i	− 0.0115985i	0.999983	−	0.00579926i	$-0.00184597\pi$
$108$	0	0
$109$	−2066.40	−1.81583	−0.907915	−	0.419154i	$-0.862327\pi$
$109$	−2066.40	−1.81583	−0.907915	+	0.419154i	$0.862327\pi$
$110$	340.055	0.294754
$111$	0	0
$112$	0	0
$113$	87.2159i	0.0726069i	0.999341	+	0.0363035i	$0.0115583\pi$
$113$	87.2159i	0.0726069i	−0.999341	+	0.0363035i	$0.988442\pi$
$114$	0	0
$115$	− 557.188i	− 0.451809i
$116$	1530.13i	1.22474i
$117$	0	0
$118$	389.691i	0.304016i
$119$	0	0
$120$	0	0
$121$	−599.659	−0.450533
$122$	−15.7177	−0.0116640
$123$	0	0
$124$	− 1407.93i	− 1.01965i
$125$	−1258.54	−0.900535
$126$	0	0
$127$	1933.38	1.35086	0.675432	−	0.737423i	$-0.263957\pi$
$127$	1933.38	1.35086	0.675432	+	0.737423i	$0.263957\pi$
$128$	1149.41i	0.793704i
$129$	0	0
$130$	518.358	0.349716
$131$	−1354.08	−0.903105	−0.451553	−	0.892244i	$-0.649130\pi$
$131$	−1354.08	−0.903105	−0.451553	+	0.892244i	$0.649130\pi$
$132$	0	0
$133$	0	0
$134$	283.018i	0.182455i
$135$	0	0
$136$	394.041i	0.248446i
$137$	62.2722i	0.0388341i	0.999811	+	0.0194171i	$0.00618103\pi$
$137$	62.2722i	0.0388341i	−0.999811	+	0.0194171i	$0.993819\pi$
$138$	0	0
$139$	2355.95i	1.43762i	0.695208	+	0.718808i	$0.255312\pi$
$139$	2355.95i	1.43762i	−0.695208	+	0.718808i	$0.744688\pi$
$140$	0	0
$141$	0	0
$142$	−737.761	−0.435997
$143$	−2942.98	−1.72101
$144$	0	0
$145$	2434.18i	1.39412i
$146$	−336.594	−0.190799
$147$	0	0
$148$	−3117.63	−1.73154
$149$	− 358.256i	− 0.196976i	−0.995138	−	0.0984881i	$-0.968599\pi$
$149$	− 358.256i	− 0.196976i	0.995138	−	0.0984881i	$-0.0314006\pi$
$150$	0	0
$151$	460.319	0.248081	0.124040	−	0.992277i	$-0.460415\pi$
$151$	460.319	0.248081	0.124040	+	0.992277i	$0.460415\pi$
$152$	578.749	0.308834
$153$	0	0
$154$	0	0
$155$	− 2239.78i	− 1.16067i
$156$	0	0
$157$	753.605i	0.383084i	0.981484	+	0.191542i	$0.0613489\pi$
$157$	753.605i	0.383084i	−0.981484	+	0.191542i	$0.938651\pi$
$158$	− 97.3416i	− 0.0490132i
$159$	0	0
$160$	1385.46i	0.684566i
$161$	0	0
$162$	0	0
$163$	2413.72	1.15986	0.579929	−	0.814667i	$-0.303080\pi$
$163$	2413.72	1.15986	0.579929	+	0.814667i	$0.303080\pi$
$164$	3103.06	1.47749
$165$	0	0
$166$	834.663i	0.390256i
$167$	−2912.06	−1.34935	−0.674676	−	0.738114i	$-0.735717\pi$
$167$	−2912.06	−1.34935	−0.674676	+	0.738114i	$0.735717\pi$
$168$	0	0
$169$	−2289.10	−1.04192
$170$	305.164i	0.137677i
$171$	0	0
$172$	983.016	0.435780
$173$	−1739.54	−0.764478	−0.382239	−	0.924063i	$-0.624847\pi$
$173$	−1739.54	−0.764478	−0.382239	+	0.924063i	$0.624847\pi$
$174$	0	0
$175$	0	0
$176$	− 2386.19i	− 1.02197i
$177$	0	0
$178$	373.846i	0.157421i
$179$	− 2636.53i	− 1.10091i	−0.834864	−	0.550456i	$-0.814454\pi$
$179$	− 2636.53i	− 1.10091i	0.834864	−	0.550456i	$-0.185546\pi$
$180$	0	0
$181$	− 208.975i	− 0.0858175i	−0.999079	−	0.0429087i	$-0.986338\pi$
$181$	− 208.975i	− 0.0858175i	0.999079	−	0.0429087i	$-0.0136625\pi$
$182$	0	0
$183$	0	0
$184$	461.204	0.184785
$185$	−4959.62	−1.97102
$186$	0	0
$187$	− 1732.57i	− 0.677529i
$188$	−3905.95	−1.51527
$189$	0	0
$190$	448.211	0.171140
$191$	770.313i	0.291822i	0.989298	+	0.145911i	$0.0466112\pi$
$191$	770.313i	0.291822i	−0.989298	+	0.145911i	$0.953389\pi$
$192$	0	0
$193$	860.837	0.321059	0.160529	−	0.987031i	$-0.448680\pi$
$193$	860.837	0.321059	0.160529	+	0.987031i	$0.448680\pi$
$194$	699.911	0.259024
$195$	0	0
$196$	0	0
$197$	1034.13i	0.374003i	0.982360	+	0.187001i	$0.0598769\pi$
$197$	1034.13i	0.374003i	−0.982360	+	0.187001i	$0.940123\pi$
$198$	0	0
$199$	5161.09i	1.83849i	0.393684	+	0.919246i	$0.371200\pi$
$199$	5161.09i	1.83849i	−0.393684	+	0.919246i	$0.628800\pi$
$200$	207.414i	0.0733318i
$201$	0	0
$202$	100.498i	0.0350049i
$203$	0	0
$204$	0	0
$205$	4936.44	1.68184
$206$	154.540	0.0522685
$207$	0	0
$208$	− 3637.36i	− 1.21253i
$209$	−2544.72	−0.842209
$210$	0	0
$211$	136.191	0.0444349	0.0222174	−	0.999753i	$-0.492927\pi$
$211$	136.191	0.0444349	0.0222174	+	0.999753i	$0.492927\pi$
$212$	3320.85i	1.07584i
$213$	0	0
$214$	8.22927	0.00262870
$215$	1563.81	0.496051
$216$	0	0
$217$	0	0
$218$	− 1324.64i	− 0.411541i
$219$	0	0
$220$	− 4025.82i	− 1.23373i
$221$	− 2641.02i	− 0.803866i
$222$	0	0
$223$	2544.16i	0.763989i	0.924165	+	0.381994i	$0.124763\pi$
$223$	2544.16i	0.763989i	−0.924165	+	0.381994i	$0.875237\pi$
$224$	0	0
$225$	0	0
$226$	−55.9086	−0.0164557
$227$	−4438.19	−1.29768	−0.648840	−	0.760925i	$-0.724746\pi$
$227$	−4438.19	−1.29768	−0.648840	+	0.760925i	$0.724746\pi$
$228$	0	0
$229$	3108.73i	0.897077i	0.893763	+	0.448539i	$0.148055\pi$
$229$	3108.73i	0.897077i	−0.893763	+	0.448539i	$0.851945\pi$
$230$	357.179	0.102398
$231$	0	0
$232$	−2014.86	−0.570181
$233$	4588.02i	1.29000i	0.764181	+	0.645002i	$0.223143\pi$
$233$	4588.02i	1.29000i	−0.764181	+	0.645002i	$0.776857\pi$
$234$	0	0
$235$	−6213.70	−1.72484
$236$	4613.45	1.27250
$237$	0	0
$238$	0	0
$239$	− 738.555i	− 0.199888i	−0.994993	−	0.0999439i	$-0.968134\pi$
$239$	− 738.555i	− 0.199888i	0.994993	−	0.0999439i	$-0.0318663\pi$
$240$	0	0
$241$	249.739i	0.0667515i	0.999443	+	0.0333757i	$0.0106258\pi$
$241$	249.739i	0.0667515i	−0.999443	+	0.0333757i	$0.989374\pi$
$242$	− 384.404i	− 0.102109i
$243$	0	0
$244$	186.078i	0.0488213i
$245$	0	0
$246$	0	0
$247$	−3879.01	−0.999253
$248$	1853.94	0.474700
$249$	0	0
$250$	− 806.768i	− 0.204098i
$251$	4098.68	1.03070	0.515351	−	0.856979i	$-0.327662\pi$
$251$	4098.68	1.03070	0.515351	+	0.856979i	$0.327662\pi$
$252$	0	0
$253$	−2027.88	−0.503920
$254$	1239.37i	0.306161i
$255$	0	0
$256$	2150.30	0.524974
$257$	−2675.05	−0.649280	−0.324640	−	0.945838i	$-0.605243\pi$
$257$	−2675.05	−0.649280	−0.324640	+	0.945838i	$0.605243\pi$
$258$	0	0
$259$	0	0
$260$	− 6136.71i	− 1.46378i
$261$	0	0
$262$	− 868.018i	− 0.204681i
$263$	− 1884.79i	− 0.441904i	−0.975285	−	0.220952i	$-0.929084\pi$
$263$	− 1884.79i	− 0.441904i	0.975285	−	0.220952i	$-0.0709165\pi$
$264$	0	0
$265$	5282.91i	1.22463i
$266$	0	0
$267$	0	0
$268$	3350.57	0.763689
$269$	−1207.22	−0.273626	−0.136813	−	0.990597i	$-0.543686\pi$
$269$	−1207.22	−0.273626	−0.136813	+	0.990597i	$0.543686\pi$
$270$	0	0
$271$	− 4547.57i	− 1.01935i	−0.860366	−	0.509677i	$-0.829765\pi$
$271$	− 4547.57i	− 1.01935i	0.860366	−	0.509677i	$-0.170235\pi$
$272$	2141.36	0.477350
$273$	0	0
$274$	−39.9188	−0.00880140
$275$	− 911.983i	− 0.199981i
$276$	0	0
$277$	−4272.73	−0.926799	−0.463400	−	0.886149i	$-0.653371\pi$
$277$	−4272.73	−0.926799	−0.463400	+	0.886149i	$0.653371\pi$
$278$	−1510.25	−0.325823
$279$	0	0
$280$	0	0
$281$	8218.49i	1.74475i	0.488840	+	0.872374i	$0.337420\pi$
$281$	8218.49i	1.74475i	−0.488840	+	0.872374i	$0.662580\pi$
$282$	0	0
$283$	4817.89i	1.01199i	0.862536	+	0.505996i	$0.168875\pi$
$283$	4817.89i	1.01199i	−0.862536	+	0.505996i	$0.831125\pi$
$284$	8734.16i	1.82492i
$285$	0	0
$286$	− 1886.56i	− 0.390051i
$287$	0	0
$288$	0	0
$289$	−3358.20	−0.683533
$290$	−1560.40	−0.315966
$291$	0	0
$292$	3984.85i	0.798615i
$293$	−6857.68	−1.36734	−0.683669	−	0.729792i	$-0.739617\pi$
$293$	−6857.68	−1.36734	−0.683669	+	0.729792i	$0.739617\pi$
$294$	0	0
$295$	7339.21	1.44849
$296$	− 4105.25i	− 0.806123i
$297$	0	0
$298$	229.655	0.0446428
$299$	−3091.18	−0.597884
$300$	0	0
$301$	0	0
$302$	295.081i	0.0562253i
$303$	0	0
$304$	− 3145.13i	− 0.593374i
$305$	296.018i	0.0555736i
$306$	0	0
$307$	− 7537.04i	− 1.40118i	−0.713565	−	0.700589i	$-0.752921\pi$
$307$	− 7537.04i	− 1.40118i	0.713565	−	0.700589i	$-0.247079\pi$
$308$	0	0
$309$	0	0
$310$	1435.78	0.263055
$311$	5587.52	1.01878	0.509388	−	0.860537i	$-0.329872\pi$
$311$	5587.52	1.01878	0.509388	+	0.860537i	$0.329872\pi$
$312$	0	0
$313$	4221.53i	0.762348i	0.924503	+	0.381174i	$0.124480\pi$
$313$	4221.53i	0.762348i	−0.924503	+	0.381174i	$0.875520\pi$
$314$	−483.089	−0.0868226
$315$	0	0
$316$	−1152.40	−0.205151
$317$	− 8265.70i	− 1.46450i	−0.681034	−	0.732252i	$-0.738469\pi$
$317$	− 8265.70i	− 1.46450i	0.681034	−	0.732252i	$-0.261531\pi$
$318$	0	0
$319$	8859.18	1.55492
$320$	4356.98	0.761133
$321$	0	0
$322$	0	0
$323$	− 2283.62i	− 0.393387i
$324$	0	0
$325$	− 1390.17i	− 0.237270i
$326$	1547.28i	0.262871i
$327$	0	0
$328$	4086.07i	0.687851i
$329$	0	0
$330$	0	0
$331$	−7309.44	−1.21379	−0.606893	−	0.794783i	$-0.707584\pi$
$331$	−7309.44	−1.21379	−0.606893	+	0.794783i	$0.707584\pi$
$332$	9881.36	1.63346
$333$	0	0
$334$	− 1866.74i	− 0.305818i
$335$	5330.19	0.869311
$336$	0	0
$337$	127.665	0.0206361	0.0103180	−	0.999947i	$-0.496716\pi$
$337$	127.665	0.0206361	0.0103180	+	0.999947i	$0.496716\pi$
$338$	− 1467.40i	− 0.236141i
$339$	0	0
$340$	3612.76	0.576263
$341$	−8151.66	−1.29454
$342$	0	0
$343$	0	0
$344$	1294.42i	0.202879i
$345$	0	0
$346$	− 1115.11i	− 0.173262i
$347$	− 5170.57i	− 0.799916i	−0.916533	−	0.399958i	$-0.869025\pi$
$347$	− 5170.57i	− 0.799916i	0.916533	−	0.399958i	$-0.130975\pi$
$348$	0	0
$349$	− 11271.1i	− 1.72874i	−0.502855	−	0.864370i	$-0.667717\pi$
$349$	− 11271.1i	− 1.72874i	0.502855	−	0.864370i	$-0.332283\pi$
$350$	0	0
$351$	0	0
$352$	5042.38	0.763522
$353$	1657.92	0.249978	0.124989	−	0.992158i	$-0.460110\pi$
$353$	1657.92	0.249978	0.124989	+	0.992158i	$0.460110\pi$
$354$	0	0
$355$	13894.6i	2.07732i
$356$	4425.87	0.658906
$357$	0	0
$358$	1690.11	0.249512
$359$	285.803i	0.0420170i	0.999779	+	0.0210085i	$0.00668770\pi$
$359$	285.803i	0.0420170i	−0.999779	+	0.0210085i	$0.993312\pi$
$360$	0	0
$361$	3504.92	0.510996
$362$	133.961	0.0194498
$363$	0	0
$364$	0	0
$365$	6339.22i	0.909068i
$366$	0	0
$367$	4406.07i	0.626690i	0.949639	+	0.313345i	$0.101450\pi$
$367$	4406.07i	0.626690i	−0.949639	+	0.313345i	$0.898550\pi$
$368$	− 2506.35i	− 0.355034i
$369$	0	0
$370$	− 3179.30i	− 0.446713i
$371$	0	0
$372$	0	0
$373$	−12545.1	−1.74144	−0.870722	−	0.491776i	$-0.836348\pi$
$373$	−12545.1	−1.74144	−0.870722	+	0.491776i	$0.836348\pi$
$374$	1110.64	0.153556
$375$	0	0
$376$	− 5143.29i	− 0.705439i
$377$	13504.4	1.84486
$378$	0	0
$379$	4695.85	0.636437	0.318219	−	0.948017i	$-0.396915\pi$
$379$	4695.85	0.636437	0.318219	+	0.948017i	$0.396915\pi$
$380$	− 5306.25i	− 0.716329i
$381$	0	0
$382$	−493.800	−0.0661387
$383$	7190.02	0.959250	0.479625	−	0.877474i	$-0.340773\pi$
$383$	7190.02	0.959250	0.479625	+	0.877474i	$0.340773\pi$
$384$	0	0
$385$	0	0
$386$	551.828i	0.0727651i
$387$	0	0
$388$	− 8286.06i	− 1.08418i
$389$	9187.81i	1.19753i	0.800923	+	0.598767i	$0.204342\pi$
$389$	9187.81i	1.19753i	−0.800923	+	0.598767i	$0.795658\pi$
$390$	0	0
$391$	− 1819.81i	− 0.235376i
$392$	0	0
$393$	0	0
$394$	−662.914	−0.0847643
$395$	−1833.28	−0.233524
$396$	0	0
$397$	− 11951.8i	− 1.51094i	−0.655182	−	0.755471i	$-0.727408\pi$
$397$	− 11951.8i	− 1.51094i	0.655182	−	0.755471i	$-0.272592\pi$
$398$	−3308.45	−0.416677
$399$	0	0
$400$	1127.16	0.140895
$401$	− 6844.67i	− 0.852386i	−0.904632	−	0.426193i	$-0.859854\pi$
$401$	− 6844.67i	− 0.852386i	0.904632	−	0.426193i	$-0.140146\pi$
$402$	0	0
$403$	−12425.9	−1.53592
$404$	1189.76	0.146517
$405$	0	0
$406$	0	0
$407$	18050.5i	2.19835i
$408$	0	0
$409$	− 5166.93i	− 0.624666i	−0.949973	−	0.312333i	$-0.898890\pi$
$409$	− 5166.93i	− 0.624666i	0.949973	−	0.312333i	$-0.101110\pi$
$410$	3164.45i	0.381173i
$411$	0	0
$412$	− 1829.56i	− 0.218776i
$413$	0	0
$414$	0	0
$415$	15719.6	1.85938
$416$	7686.29	0.905893
$417$	0	0
$418$	− 1631.26i	− 0.190879i
$419$	13900.4	1.62071	0.810356	−	0.585938i	$-0.199274\pi$
$419$	13900.4	1.62071	0.810356	+	0.585938i	$0.199274\pi$
$420$	0	0
$421$	7800.31	0.903003	0.451501	−	0.892270i	$-0.350889\pi$
$421$	7800.31	0.903003	0.451501	+	0.892270i	$0.350889\pi$
$422$	87.3033i	0.0100708i
$423$	0	0
$424$	−4372.85	−0.500859
$425$	818.411	0.0934089
$426$	0	0
$427$	0	0
$428$	− 97.4241i	− 0.0110027i
$429$	0	0
$430$	1002.46i	0.112426i
$431$	− 2085.65i	− 0.233091i	−0.993185	−	0.116546i	$-0.962818\pi$
$431$	− 2085.65i	− 0.233091i	0.993185	−	0.116546i	$-0.0371821\pi$
$432$	0	0
$433$	1186.95i	0.131735i	0.997828	+	0.0658674i	$0.0209814\pi$
$433$	1186.95i	0.131735i	−0.997828	+	0.0658674i	$0.979019\pi$
$434$	0	0
$435$	0	0
$436$	−15682.1	−1.72256
$437$	−2672.86	−0.292586
$438$	0	0
$439$	− 8692.50i	− 0.945035i	−0.881321	−	0.472518i	$-0.843345\pi$
$439$	− 8692.50i	− 0.945035i	0.881321	−	0.472518i	$-0.156655\pi$
$440$	5301.13	0.574367
$441$	0	0
$442$	1692.99	0.182189
$443$	6758.69i	0.724864i	0.932010	+	0.362432i	$0.118054\pi$
$443$	6758.69i	0.724864i	−0.932010	+	0.362432i	$0.881946\pi$
$444$	0	0
$445$	7040.81	0.750037
$446$	−1630.90	−0.173151
$447$	0	0
$448$	0	0
$449$	− 6721.89i	− 0.706516i	−0.935526	−	0.353258i	$-0.885074\pi$
$449$	− 6721.89i	− 0.706516i	0.935526	−	0.353258i	$-0.114926\pi$
$450$	0	0
$451$	− 17966.1i	− 1.87581i
$452$	661.888i	0.0688774i
$453$	0	0
$454$	− 2845.05i	− 0.294107i
$455$	0	0
$456$	0	0
$457$	−3301.86	−0.337975	−0.168988	−	0.985618i	$-0.554050\pi$
$457$	−3301.86	−0.337975	−0.168988	+	0.985618i	$0.554050\pi$
$458$	−1992.81	−0.203314
$459$	0	0
$460$	− 4228.54i	− 0.428602i
$461$	7241.19	0.731574	0.365787	−	0.930699i	$-0.380800\pi$
$461$	7241.19	0.731574	0.365787	+	0.930699i	$0.380800\pi$
$462$	0	0
$463$	−7585.65	−0.761415	−0.380707	−	0.924696i	$-0.624319\pi$
$463$	−7585.65	−0.761415	−0.380707	+	0.924696i	$0.624319\pi$
$464$	10949.5i	1.09551i
$465$	0	0
$466$	−2941.09	−0.292368
$467$	−2048.56	−0.202990	−0.101495	−	0.994836i	$-0.532363\pi$
$467$	−2048.56	−0.202990	−0.101495	+	0.994836i	$0.532363\pi$
$468$	0	0
$469$	0	0
$470$	− 3983.21i	− 0.390919i
$471$	0	0
$472$	6074.92i	0.592417i
$473$	− 5691.47i	− 0.553264i
$474$	0	0
$475$	− 1202.04i	− 0.116113i
$476$	0	0
$477$	0	0
$478$	473.442	0.0453027
$479$	3297.04	0.314500	0.157250	−	0.987559i	$-0.449737\pi$
$479$	3297.04	0.314500	0.157250	+	0.987559i	$0.449737\pi$
$480$	0	0
$481$	27515.0i	2.60827i
$482$	−160.092	−0.0151286
$483$	0	0
$484$	−4550.85	−0.427390
$485$	− 13181.7i	− 1.23413i
$486$	0	0
$487$	−1179.71	−0.109769	−0.0548846	−	0.998493i	$-0.517479\pi$
$487$	−1179.71	−0.109769	−0.0548846	+	0.998493i	$0.517479\pi$
$488$	−245.024	−0.0227290
$489$	0	0
$490$	0	0
$491$	7573.80i	0.696132i	0.937470	+	0.348066i	$0.113162\pi$
$491$	7573.80i	0.696132i	−0.937470	+	0.348066i	$0.886838\pi$
$492$	0	0
$493$	7950.21i	0.726287i
$494$	− 2486.59i	− 0.226472i
$495$	0	0
$496$	− 10075.0i	− 0.912059i
$497$	0	0
$498$	0	0
$499$	−1819.53	−0.163233	−0.0816167	−	0.996664i	$-0.526008\pi$
$499$	−1819.53	−0.163233	−0.0816167	+	0.996664i	$0.526008\pi$
$500$	−9551.12	−0.854278
$501$	0	0
$502$	2627.41i	0.233599i
$503$	−15635.9	−1.38602	−0.693012	−	0.720926i	$-0.743717\pi$
$503$	−15635.9	−1.38602	−0.693012	+	0.720926i	$0.743717\pi$
$504$	0	0
$505$	1892.71	0.166781
$506$	− 1299.95i	− 0.114209i
$507$	0	0
$508$	14672.6	1.28147
$509$	−3293.79	−0.286827	−0.143413	−	0.989663i	$-0.545808\pi$
$509$	−3293.79	−0.286827	−0.143413	+	0.989663i	$0.545808\pi$
$510$	0	0
$511$	0	0
$512$	10573.7i	0.912685i
$513$	0	0
$514$	− 1714.81i	− 0.147153i
$515$	− 2910.51i	− 0.249034i
$516$	0	0
$517$	22614.7i	1.92378i
$518$	0	0
$519$	0	0
$520$	8080.73	0.681468
$521$	3298.04	0.277332	0.138666	−	0.990339i	$-0.455719\pi$
$521$	3298.04	0.277332	0.138666	+	0.990339i	$0.455719\pi$
$522$	0	0
$523$	− 17826.4i	− 1.49043i	−0.666824	−	0.745215i	$-0.732347\pi$
$523$	− 17826.4i	− 1.49043i	0.666824	−	0.745215i	$-0.267653\pi$
$524$	−10276.2	−0.856716
$525$	0	0
$526$	1208.22	0.100154
$527$	− 7315.28i	− 0.604665i
$528$	0	0
$529$	10037.0	0.824937
$530$	−3386.54	−0.277551
$531$	0	0
$532$	0	0
$533$	− 27386.5i	− 2.22559i
$534$	0	0
$535$	− 154.985i	− 0.0125245i
$536$	4411.98i	0.355538i
$537$	0	0
$538$	− 773.871i	− 0.0620148i
$539$	0	0
$540$	0	0
$541$	10418.2	0.827939	0.413969	−	0.910291i	$-0.364142\pi$
$541$	10418.2	0.827939	0.413969	+	0.910291i	$0.364142\pi$
$542$	2915.16	0.231027
$543$	0	0
$544$	4525.02i	0.356633i
$545$	−24947.5	−1.96080
$546$	0	0
$547$	−996.607	−0.0779010	−0.0389505	−	0.999241i	$-0.512401\pi$
$547$	−996.607	−0.0779010	−0.0389505	+	0.999241i	$0.512401\pi$
$548$	472.588i	0.0368394i
$549$	0	0
$550$	584.615	0.0453238
$551$	11676.9	0.902818
$552$	0	0
$553$	0	0
$554$	− 2738.98i	− 0.210051i
$555$	0	0
$556$	17879.4i	1.36377i
$557$	11137.4i	0.847229i	0.905843	+	0.423614i	$0.139239\pi$
$557$	11137.4i	0.847229i	−0.905843	+	0.423614i	$0.860761\pi$
$558$	0	0
$559$	− 8675.73i	− 0.656430i
$560$	0	0
$561$	0	0
$562$	−5268.36	−0.395431
$563$	−4191.49	−0.313766	−0.156883	−	0.987617i	$-0.550145\pi$
$563$	−4191.49	−0.313766	−0.156883	+	0.987617i	$0.550145\pi$
$564$	0	0
$565$	1052.95i	0.0784035i
$566$	−3088.45	−0.229359
$567$	0	0
$568$	−11501.0	−0.849598
$569$	2997.85i	0.220873i	0.993883	+	0.110436i	$0.0352248\pi$
$569$	2997.85i	0.220873i	−0.993883	+	0.110436i	$0.964775\pi$
$570$	0	0
$571$	1762.22	0.129154	0.0645768	−	0.997913i	$-0.479430\pi$
$571$	1762.22	0.129154	0.0645768	+	0.997913i	$0.479430\pi$
$572$	−22334.5	−1.63261
$573$	0	0
$574$	0	0
$575$	− 957.907i	− 0.0694739i
$576$	0	0
$577$	15400.1i	1.11111i	0.831478	+	0.555557i	$0.187495\pi$
$577$	15400.1i	1.11111i	−0.831478	+	0.555557i	$0.812505\pi$
$578$	− 2152.73i	− 0.154917i
$579$	0	0
$580$	18473.2i	1.32251i
$581$	0	0
$582$	0	0
$583$	19227.1	1.36587
$584$	−5247.19	−0.371798
$585$	0	0
$586$	− 4396.03i	− 0.309895i
$587$	−2642.97	−0.185838	−0.0929190	−	0.995674i	$-0.529620\pi$
$587$	−2642.97	−0.185838	−0.0929190	+	0.995674i	$0.529620\pi$
$588$	0	0
$589$	−10744.3	−0.751635
$590$	4704.71i	0.328288i
$591$	0	0
$592$	−22309.4	−1.54884
$593$	−1201.90	−0.0832312	−0.0416156	−	0.999134i	$-0.513250\pi$
$593$	−1201.90	−0.0832312	−0.0416156	+	0.999134i	$0.513250\pi$
$594$	0	0
$595$	0	0
$596$	− 2718.83i	− 0.186858i
$597$	0	0
$598$	− 1981.56i	− 0.135505i
$599$	− 9853.11i	− 0.672099i	−0.941844	−	0.336049i	$-0.890909\pi$
$599$	− 9853.11i	− 0.672099i	0.941844	−	0.336049i	$-0.109091\pi$
$600$	0	0
$601$	7573.82i	0.514047i	0.966405	+	0.257024i	$0.0827418\pi$
$601$	7573.82i	0.514047i	−0.966405	+	0.257024i	$0.917258\pi$
$602$	0	0
$603$	0	0
$604$	3493.39	0.235338
$605$	−7239.64	−0.486501
$606$	0	0
$607$	19797.0i	1.32378i	0.749599	+	0.661892i	$0.230246\pi$
$607$	19797.0i	1.32378i	−0.749599	+	0.661892i	$0.769754\pi$
$608$	6646.14	0.443317
$609$	0	0
$610$	−189.759	−0.0125952
$611$	34472.4i	2.28250i
$612$	0	0
$613$	−10527.7	−0.693656	−0.346828	−	0.937929i	$-0.612741\pi$
$613$	−10527.7	−0.693656	−0.346828	+	0.937929i	$0.612741\pi$
$614$	4831.52	0.317564
$615$	0	0
$616$	0	0
$617$	− 23133.1i	− 1.50940i	−0.656067	−	0.754702i	$-0.727781\pi$
$617$	− 23133.1i	− 1.50940i	0.656067	−	0.754702i	$-0.272219\pi$
$618$	0	0
$619$	− 16344.4i	− 1.06129i	−0.847595	−	0.530643i	$-0.821951\pi$
$619$	− 16344.4i	− 1.06129i	0.847595	−	0.530643i	$-0.178049\pi$
$620$	− 16997.9i	− 1.10105i
$621$	0	0
$622$	3581.81i	0.230896i
$623$	0	0
$624$	0	0
$625$	−17788.6	−1.13847
$626$	−2706.16	−0.172779
$627$	0	0
$628$	5719.17i	0.363407i
$629$	−16198.4	−1.02683
$630$	0	0
$631$	−11124.2	−0.701819	−0.350909	−	0.936409i	$-0.614127\pi$
$631$	−11124.2	−0.701819	−0.350909	+	0.936409i	$0.614127\pi$
$632$	− 1517.47i	− 0.0955088i
$633$	0	0
$634$	5298.62	0.331917
$635$	23341.5	1.45871
$636$	0	0
$637$	0	0
$638$	5679.07i	0.352408i
$639$	0	0
$640$	13876.7i	0.857070i
$641$	3582.68i	0.220760i	0.993889	+	0.110380i	$0.0352068\pi$
$641$	3582.68i	0.220760i	−0.993889	+	0.110380i	$0.964793\pi$
$642$	0	0
$643$	14145.6i	0.867570i	0.901016	+	0.433785i	$0.142822\pi$
$643$	14145.6i	0.867570i	−0.901016	+	0.433785i	$0.857178\pi$
$644$	0	0
$645$	0	0
$646$	1463.89	0.0891577
$647$	−23914.6	−1.45314	−0.726568	−	0.687095i	$-0.758886\pi$
$647$	−23914.6	−1.45314	−0.726568	+	0.687095i	$0.758886\pi$
$648$	0	0
$649$	− 26711.0i	− 1.61556i
$650$	891.152	0.0537751
$651$	0	0
$652$	18317.9	1.10028
$653$	25410.7i	1.52281i	0.648276	+	0.761406i	$0.275490\pi$
$653$	25410.7i	1.52281i	−0.648276	+	0.761406i	$0.724510\pi$
$654$	0	0
$655$	−16347.7	−0.975205
$656$	22205.2	1.32160
$657$	0	0
$658$	0	0
$659$	− 14977.5i	− 0.885342i	−0.896684	−	0.442671i	$-0.854031\pi$
$659$	− 14977.5i	− 0.885342i	0.896684	−	0.442671i	$-0.145969\pi$
$660$	0	0
$661$	− 102.280i	− 0.00601852i	−0.999995	−	0.00300926i	$-0.999042\pi$
$661$	− 102.280i	− 0.00601852i	0.999995	−	0.00300926i	$-0.000957878\pi$
$662$	− 4685.63i	− 0.275094i
$663$	0	0
$664$	13011.6i	0.760465i
$665$	0	0
$666$	0	0
$667$	9305.30	0.540184
$668$	−22099.8	−1.28004
$669$	0	0
$670$	3416.85i	0.197022i
$671$	1077.35	0.0619833
$672$	0	0
$673$	7778.89	0.445549	0.222774	−	0.974870i	$-0.428489\pi$
$673$	7778.89	0.445549	0.222774	+	0.974870i	$0.428489\pi$
$674$	81.8380i	0.00467698i
$675$	0	0
$676$	−17372.1	−0.988400
$677$	−20416.1	−1.15902	−0.579510	−	0.814965i	$-0.696756\pi$
$677$	−20416.1	−1.15902	−0.579510	+	0.814965i	$0.696756\pi$
$678$	0	0
$679$	0	0
$680$	4757.22i	0.268281i
$681$	0	0
$682$	− 5225.52i	− 0.293395i
$683$	26805.4i	1.50173i	0.660458	+	0.750863i	$0.270362\pi$
$683$	26805.4i	1.50173i	−0.660458	+	0.750863i	$0.729638\pi$
$684$	0	0
$685$	751.808i	0.0419345i
$686$	0	0
$687$	0	0
$688$	7034.35	0.389800
$689$	29308.6	1.62056
$690$	0	0
$691$	− 32711.2i	− 1.80086i	−0.435003	−	0.900429i	$-0.643253\pi$
$691$	− 32711.2i	− 1.80086i	0.435003	−	0.900429i	$-0.356747\pi$
$692$	−13201.5	−0.725210
$693$	0	0
$694$	3314.53	0.181294
$695$	28443.2i	1.55239i
$696$	0	0
$697$	16122.8	0.876174
$698$	7225.22	0.391803
$699$	0	0
$700$	0	0
$701$	− 503.993i	− 0.0271548i	−0.999908	−	0.0135774i	$-0.995678\pi$
$701$	− 503.993i	− 0.0271548i	0.999908	−	0.0135774i	$-0.00432196\pi$
$702$	0	0
$703$	23791.5i	1.27641i
$704$	− 15857.2i	− 0.848920i
$705$	0	0
$706$	1062.79i	0.0566553i
$707$	0	0
$708$	0	0
$709$	13787.5	0.730327	0.365163	−	0.930943i	$-0.381013\pi$
$709$	13787.5	0.730327	0.365163	+	0.930943i	$0.381013\pi$
$710$	−8906.94	−0.470805
$711$	0	0
$712$	5827.92i	0.306756i
$713$	−8562.15	−0.449726
$714$	0	0
$715$	−35530.4	−1.85841
$716$	− 20008.8i	− 1.04436i
$717$	0	0
$718$	−183.210	−0.00952276
$719$	32223.1	1.67138	0.835688	−	0.549205i	$-0.185069\pi$
$719$	32223.1	1.67138	0.835688	+	0.549205i	$0.185069\pi$
$720$	0	0
$721$	0	0
$722$	2246.78i	0.115813i
$723$	0	0
$724$	− 1585.92i	− 0.0814094i
$725$	4184.80i	0.214372i
$726$	0	0
$727$	− 33807.4i	− 1.72469i	−0.506325	−	0.862343i	$-0.668996\pi$
$727$	− 33807.4i	− 1.72469i	0.506325	−	0.862343i	$-0.331004\pi$
$728$	0	0
$729$	0	0
$730$	−4063.67	−0.206032
$731$	5107.51	0.258424
$732$	0	0
$733$	16442.8i	0.828551i	0.910152	+	0.414275i	$0.135965\pi$
$733$	16442.8i	0.828551i	−0.910152	+	0.414275i	$0.864035\pi$
$734$	−2824.46	−0.142034
$735$	0	0
$736$	5296.30	0.265250
$737$	− 19399.2i	− 0.969575i
$738$	0	0
$739$	−20205.4	−1.00578	−0.502888	−	0.864351i	$-0.667729\pi$
$739$	−20205.4	−1.00578	−0.502888	+	0.864351i	$0.667729\pi$
$740$	−37638.9	−1.86977
$741$	0	0
$742$	0	0
$743$	− 14573.9i	− 0.719601i	−0.933029	−	0.359800i	$-0.882845\pi$
$743$	− 14573.9i	− 0.719601i	0.933029	−	0.359800i	$-0.117155\pi$
$744$	0	0
$745$	− 4325.19i	− 0.212702i
$746$	− 8041.85i	− 0.394682i
$747$	0	0
$748$	− 13148.6i	− 0.642727i
$749$	0	0
$750$	0	0
$751$	2689.06	0.130659	0.0653297	−	0.997864i	$-0.479190\pi$
$751$	2689.06	0.130659	0.0653297	+	0.997864i	$0.479190\pi$
$752$	−27950.5	−1.35539
$753$	0	0
$754$	8656.82i	0.418120i
$755$	5557.39	0.267886
$756$	0	0
$757$	11192.8	0.537395	0.268697	−	0.963225i	$-0.413407\pi$
$757$	11192.8	0.537395	0.268697	+	0.963225i	$0.413407\pi$
$758$	3010.22i	0.144243i
$759$	0	0
$760$	6987.19	0.333490
$761$	−29063.4	−1.38442	−0.692212	−	0.721694i	$-0.743364\pi$
$761$	−29063.4	−1.38442	−0.692212	+	0.721694i	$0.743364\pi$
$762$	0	0
$763$	0	0
$764$	5845.96i	0.276832i
$765$	0	0
$766$	4609.07i	0.217405i
$767$	− 40716.6i	− 1.91681i
$768$	0	0
$769$	40870.3i	1.91654i	0.285861	+	0.958271i	$0.407720\pi$
$769$	40870.3i	1.91654i	−0.285861	+	0.958271i	$0.592280\pi$
$770$	0	0
$771$	0	0
$772$	6532.95	0.304567
$773$	40291.6	1.87476	0.937378	−	0.348313i	$-0.113245\pi$
$773$	40291.6	1.87476	0.937378	+	0.348313i	$0.113245\pi$
$774$	0	0
$775$	− 3850.59i	− 0.178474i
$776$	10911.0	0.504743
$777$	0	0
$778$	−5889.73	−0.271410
$779$	− 23680.4i	− 1.08914i
$780$	0	0
$781$	50569.1	2.31691
$782$	1166.57	0.0533458
$783$	0	0
$784$	0	0
$785$	9098.22i	0.413668i
$786$	0	0
$787$	5001.02i	0.226515i	0.993566	+	0.113257i	$0.0361284\pi$
$787$	5001.02i	0.226515i	−0.993566	+	0.113257i	$0.963872\pi$
$788$	7848.07i	0.354792i
$789$	0	0
$790$	− 1175.20i	− 0.0529262i
$791$	0	0
$792$	0	0
$793$	1642.25	0.0735411
$794$	7661.56	0.342441
$795$	0	0
$796$	39167.9i	1.74406i
$797$	−8762.59	−0.389444	−0.194722	−	0.980858i	$-0.562380\pi$
$797$	−8762.59	−0.389444	−0.194722	+	0.980858i	$0.562380\pi$
$798$	0	0
$799$	−20294.4	−0.898577
$800$	2381.86i	0.105264i
$801$	0	0
$802$	4387.69	0.193185
$803$	23071.5	1.01392
$804$	0	0
$805$	0	0
$806$	− 7965.46i	− 0.348103i
$807$	0	0
$808$	1566.66i	0.0682116i
$809$	38643.8i	1.67941i	0.543043	+	0.839705i	$0.317272\pi$
$809$	38643.8i	1.67941i	−0.543043	+	0.839705i	$0.682728\pi$
$810$	0	0
$811$	− 3577.67i	− 0.154906i	−0.996996	−	0.0774530i	$-0.975321\pi$
$811$	− 3577.67i	− 0.154906i	0.996996	−	0.0774530i	$-0.0246788\pi$
$812$	0	0
$813$	0	0
$814$	−11571.0	−0.498236
$815$	29140.6	1.25246
$816$	0	0
$817$	− 7501.68i	− 0.321237i
$818$	3312.20	0.141575
$819$	0	0
$820$	37463.0	1.59545
$821$	− 21009.0i	− 0.893081i	−0.894763	−	0.446541i	$-0.852656\pi$
$821$	− 21009.0i	− 0.893081i	0.894763	−	0.446541i	$-0.147344\pi$
$822$	0	0
$823$	31309.2	1.32609	0.663044	−	0.748580i	$-0.269264\pi$
$823$	31309.2	1.32609	0.663044	+	0.748580i	$0.269264\pi$
$824$	2409.13	0.101852
$825$	0	0
$826$	0	0
$827$	27813.3i	1.16949i	0.811219	+	0.584743i	$0.198805\pi$
$827$	27813.3i	1.16949i	−0.811219	+	0.584743i	$0.801195\pi$
$828$	0	0
$829$	− 35061.5i	− 1.46892i	−0.678652	−	0.734460i	$-0.737435\pi$
$829$	− 35061.5i	− 1.46892i	0.678652	−	0.734460i	$-0.262565\pi$
$830$	10076.8i	0.421412i
$831$	0	0
$832$	− 24171.7i	− 1.00722i
$833$	0	0
$834$	0	0
$835$	−35157.1	−1.45708
$836$	−19312.0	−0.798948
$837$	0	0
$838$	8910.66i	0.367320i
$839$	4808.56	0.197866	0.0989332	−	0.995094i	$-0.468457\pi$
$839$	4808.56	0.197866	0.0989332	+	0.995094i	$0.468457\pi$
$840$	0	0
$841$	−16263.0	−0.666816
$842$	5000.29i	0.204657i
$843$	0	0
$844$	1033.56	0.0421524
$845$	−27636.1	−1.12510
$846$	0	0
$847$	0	0
$848$	23763.7i	0.962320i
$849$	0	0
$850$	524.632i	0.0211703i
$851$	18959.4i	0.763714i
$852$	0	0
$853$	− 9486.81i	− 0.380800i	−0.981707	−	0.190400i	$-0.939022\pi$
$853$	− 9486.81i	− 0.380800i	0.981707	−	0.190400i	$-0.0609784\pi$
$854$	0	0
$855$	0	0
$856$	128.287	0.00512237
$857$	−22524.7	−0.897818	−0.448909	−	0.893577i	$-0.648187\pi$
$857$	−22524.7	−0.897818	−0.448909	+	0.893577i	$0.648187\pi$
$858$	0	0
$859$	− 27644.3i	− 1.09804i	−0.835811	−	0.549018i	$-0.815002\pi$
$859$	− 27644.3i	− 1.09804i	0.835811	−	0.549018i	$-0.184998\pi$
$860$	11867.9	0.470571
$861$	0	0
$862$	1336.98	0.0528280
$863$	− 20158.0i	− 0.795116i	−0.917577	−	0.397558i	$-0.869858\pi$
$863$	− 20158.0i	− 0.795116i	0.917577	−	0.397558i	$-0.130142\pi$
$864$	0	0
$865$	−21001.3	−0.825510
$866$	−760.879	−0.0298565
$867$	0	0
$868$	0	0
$869$	6672.18i	0.260458i
$870$	0	0
$871$	− 29570.9i	− 1.15037i
$872$	− 20649.9i	− 0.801944i
$873$	0	0
$874$	− 1713.40i	− 0.0663120i
$875$	0	0
$876$	0	0
$877$	−30973.1	−1.19258	−0.596288	−	0.802771i	$-0.703358\pi$
$877$	−30973.1	−1.19258	−0.596288	+	0.802771i	$0.703358\pi$
$878$	5572.22	0.214184
$879$	0	0
$880$	− 28808.3i	− 1.10355i
$881$	10973.0	0.419626	0.209813	−	0.977741i	$-0.432714\pi$
$881$	10973.0	0.419626	0.209813	+	0.977741i	$0.432714\pi$
$882$	0	0
$883$	42025.5	1.60167	0.800833	−	0.598888i	$-0.204390\pi$
$883$	42025.5	1.60167	0.800833	+	0.598888i	$0.204390\pi$
$884$	− 20042.9i	− 0.762574i
$885$	0	0
$886$	−4332.57	−0.164284
$887$	4681.12	0.177200	0.0886002	−	0.996067i	$-0.471761\pi$
$887$	4681.12	0.177200	0.0886002	+	0.996067i	$0.471761\pi$
$888$	0	0
$889$	0	0
$890$	4513.42i	0.169989i
$891$	0	0
$892$	19307.8i	0.724746i
$893$	29807.4i	1.11698i
$894$	0	0
$895$	− 31830.6i	− 1.18880i
$896$	0	0
$897$	0	0
$898$	4308.98	0.160125
$899$	37405.4	1.38770
$900$	0	0
$901$	17254.3i	0.637986i
$902$	11517.0	0.425136
$903$	0	0
$904$	−871.564	−0.0320661
$905$	− 2522.94i	− 0.0926687i
$906$	0	0
$907$	−48406.5	−1.77212	−0.886060	−	0.463571i	$-0.846568\pi$
$907$	−48406.5	−1.77212	−0.886060	+	0.463571i	$0.846568\pi$
$908$	−33681.8	−1.23102
$909$	0	0
$910$	0	0
$911$	− 20920.5i	− 0.760842i	−0.924813	−	0.380421i	$-0.875779\pi$
$911$	− 20920.5i	− 0.760842i	0.924813	−	0.380421i	$-0.124221\pi$
$912$	0	0
$913$	− 57211.1i	− 2.07384i
$914$	− 2116.62i	− 0.0765990i
$915$	0	0
$916$	23592.4i	0.850998i
$917$	0	0
$918$	0	0
$919$	33471.4	1.20143	0.600717	−	0.799462i	$-0.294882\pi$
$919$	33471.4	1.20143	0.600717	+	0.799462i	$0.294882\pi$
$920$	5568.08	0.199537
$921$	0	0
$922$	4641.87i	0.165805i
$923$	77084.5	2.74893
$924$	0	0
$925$	−8526.48	−0.303080
$926$	− 4862.68i	− 0.172568i
$927$	0	0
$928$	−23137.9	−0.818468
$929$	9370.97	0.330949	0.165475	−	0.986214i	$-0.447084\pi$
$929$	9370.97	0.330949	0.165475	+	0.986214i	$0.447084\pi$
$930$	0	0
$931$	0	0
$932$	34818.8i	1.22374i
$933$	0	0
$934$	− 1313.20i	− 0.0460057i
$935$	− 20917.2i	− 0.731620i
$936$	0	0
$937$	− 29702.0i	− 1.03556i	−0.855513	−	0.517781i	$-0.826758\pi$
$937$	− 29702.0i	− 1.03556i	0.855513	−	0.517781i	$-0.173242\pi$
$938$	0	0
$939$	0	0
$940$	−47156.2	−1.63624
$941$	−4903.33	−0.169866	−0.0849331	−	0.996387i	$-0.527068\pi$
$941$	−4903.33	−0.169866	−0.0849331	+	0.996387i	$0.527068\pi$
$942$	0	0
$943$	− 18870.8i	− 0.651664i
$944$	33013.3	1.13823
$945$	0	0
$946$	3648.44	0.125392
$947$	21004.5i	0.720755i	0.932807	+	0.360378i	$0.117352\pi$
$947$	21004.5i	0.720755i	−0.932807	+	0.360378i	$0.882648\pi$
$948$	0	0
$949$	35168.8	1.20298
$950$	770.555	0.0263159
$951$	0	0
$952$	0	0
$953$	− 4598.25i	− 0.156298i	−0.996942	−	0.0781489i	$-0.975099\pi$
$953$	− 4598.25i	− 0.156298i	0.996942	−	0.0781489i	$-0.0249010\pi$
$954$	0	0
$955$	9299.94i	0.315119i
$956$	− 5604.95i	− 0.189620i
$957$	0	0
$958$	2113.52i	0.0712785i
$959$	0	0
$960$	0	0
$961$	−4627.06	−0.155317
$962$	−17638.2	−0.591140
$963$	0	0
$964$	1895.29i	0.0633227i
$965$	10392.8	0.346691
$966$	0	0
$967$	52408.6	1.74286	0.871431	−	0.490518i	$-0.163192\pi$
$967$	52408.6	1.74286	0.871431	+	0.490518i	$0.163192\pi$
$968$	− 5992.50i	− 0.198973i
$969$	0	0
$970$	8449.97	0.279703
$971$	48712.4	1.60994	0.804971	−	0.593314i	$-0.202181\pi$
$971$	48712.4	1.60994	0.804971	+	0.593314i	$0.202181\pi$
$972$	0	0
$973$	0	0
$974$	− 756.236i	− 0.0248782i
$975$	0	0
$976$	1331.55i	0.0436700i
$977$	11655.7i	0.381676i	0.981622	+	0.190838i	$0.0611206\pi$
$977$	11655.7i	0.381676i	−0.981622	+	0.190838i	$0.938879\pi$
$978$	0	0
$979$	− 25624.9i	− 0.836544i
$980$	0	0
$981$	0	0
$982$	−4855.09	−0.157772
$983$	40457.2	1.31270	0.656350	−	0.754457i	$-0.272100\pi$
$983$	40457.2	1.31270	0.656350	+	0.754457i	$0.272100\pi$
$984$	0	0
$985$	12484.9i	0.403861i
$986$	−5096.38	−0.164606
$987$	0	0
$988$	−29438.1	−0.947925
$989$	− 5978.07i	− 0.192206i
$990$	0	0
$991$	9660.88	0.309675	0.154838	−	0.987940i	$-0.450515\pi$
$991$	9660.88	0.309675	0.154838	+	0.987940i	$0.450515\pi$
$992$	21290.0	0.681410
$993$	0	0
$994$	0	0
$995$	62309.4i	1.98527i
$996$	0	0
$997$	− 4997.01i	− 0.158733i	−0.996846	−	0.0793665i	$-0.974710\pi$
$997$	− 4997.01i	− 0.158733i	0.996846	−	0.0793665i	$-0.0252897\pi$
$998$	− 1166.39i	− 0.0369954i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.4.c.b.440.20	yes	24
3.2	odd	2	inner	441.4.c.b.440.5	✓	24
7.2	even	3		441.4.p.d.80.13		48
7.3	odd	6		441.4.p.d.215.12		48
7.4	even	3		441.4.p.d.215.11		48
7.5	odd	6		441.4.p.d.80.14		48
7.6	odd	2	inner	441.4.c.b.440.6	yes	24
21.2	odd	6		441.4.p.d.80.12		48
21.5	even	6		441.4.p.d.80.11		48
21.11	odd	6		441.4.p.d.215.14		48
21.17	even	6		441.4.p.d.215.13		48
21.20	even	2	inner	441.4.c.b.440.19	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.4.c.b.440.5	✓	24	3.2	odd	2	inner
441.4.c.b.440.6	yes	24	7.6	odd	2	inner
441.4.c.b.440.19	yes	24	21.20	even	2	inner
441.4.c.b.440.20	yes	24	1.1	even	1	trivial
441.4.p.d.80.11		48	21.5	even	6
441.4.p.d.80.12		48	21.2	odd	6
441.4.p.d.80.13		48	7.2	even	3
441.4.p.d.80.14		48	7.5	odd	6
441.4.p.d.215.11		48	7.4	even	3
441.4.p.d.215.12		48	7.3	odd	6
441.4.p.d.215.13		48	21.17	even	6
441.4.p.d.215.14		48	21.11	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	441.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.641037i q^{2} +7.58907 q^{4} +12.0729 q^{5} +9.99318i q^{8} +O(q^{10})\)	\(q+0.641037i q^{2} +7.58907 q^{4} +12.0729 q^{5} +9.99318i q^{8} +7.73920i q^{10} -43.9393i q^{11} -66.9783i q^{13} +54.3066 q^{16} +39.4310 q^{17} -57.9144i q^{19} +91.6223 q^{20} +28.1667 q^{22} -46.1519i q^{23} +20.7555 q^{25} +42.9356 q^{26} +201.623i q^{29} -185.521i q^{31} +114.758i q^{32} +25.2767i q^{34} -410.805 q^{37} +37.1253 q^{38} +120.647i q^{40} +408.886 q^{41} +129.530 q^{43} -333.458i q^{44} +29.5851 q^{46} -514.681 q^{47} +13.3051i q^{50} -508.303i q^{52} +437.584i q^{53} -530.475i q^{55} -129.248 q^{58} +607.906 q^{59} +24.5192i q^{61} +118.926 q^{62} +360.888 q^{64} -808.624i q^{65} +441.499 q^{67} +299.245 q^{68} +1150.89i q^{71} +525.077i q^{73} -263.341i q^{74} -439.517i q^{76} -151.850 q^{79} +655.639 q^{80} +262.111i q^{82} +1302.05 q^{83} +476.047 q^{85} +83.0338i q^{86} +439.093 q^{88} +583.190 q^{89} -350.250i q^{92} -329.929i q^{94} -699.196i q^{95} -1091.84i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 96 q^{4}+O(q^{10}) \) 24 * q - 96 * q^4	\( 24 q - 96 q^{4} + 144 q^{16} + 624 q^{22} + 312 q^{25} - 864 q^{37} + 1248 q^{43} - 3888 q^{46} - 7440 q^{58} - 3360 q^{64} - 2688 q^{67} + 480 q^{79} + 13248 q^{85} - 7248 q^{88}+O(q^{100}) \) 24 * q - 96 * q^4 + 144 * q^16 + 624 * q^22 + 312 * q^25 - 864 * q^37 + 1248 * q^43 - 3888 * q^46 - 7440 * q^58 - 3360 * q^64 - 2688 * q^67 + 480 * q^79 + 13248 * q^85 - 7248 * q^88

Introduction

L-functions

Modular forms

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Database

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