LMFDB - Embedded newform 1629.4.a.e.1.9

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1629,4,Mod(1,1629)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1629.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1629, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1629 = 3^{2} \cdot 181 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1629.a (trivial)

Newform invariants

comment:select newform

sage:traces = [23] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$96.1141113994$
Analytic rank:	$1$
Dimension:	$23$
Twist minimal:	no (minimal twist has level 543)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Character	$\chi$	$=$	1629.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.91929 q^{2} -4.31632 q^{4} +17.8997 q^{5} -9.33325 q^{7} +23.6386 q^{8} -34.3548 q^{10} +56.1277 q^{11} -52.8683 q^{13} +17.9132 q^{14} -10.8389 q^{16} +45.4222 q^{17} -26.3643 q^{19} -77.2608 q^{20} -107.725 q^{22} -173.853 q^{23} +195.399 q^{25} +101.470 q^{26} +40.2853 q^{28} +133.466 q^{29} -187.129 q^{31} -168.306 q^{32} -87.1784 q^{34} -167.062 q^{35} -114.278 q^{37} +50.6008 q^{38} +423.124 q^{40} +49.4326 q^{41} -379.975 q^{43} -242.265 q^{44} +333.674 q^{46} -366.733 q^{47} -255.890 q^{49} -375.028 q^{50} +228.196 q^{52} +313.361 q^{53} +1004.67 q^{55} -220.625 q^{56} -256.160 q^{58} -185.879 q^{59} +314.519 q^{61} +359.154 q^{62} +409.739 q^{64} -946.326 q^{65} -830.782 q^{67} -196.056 q^{68} +320.642 q^{70} -657.110 q^{71} +552.051 q^{73} +219.333 q^{74} +113.797 q^{76} -523.853 q^{77} -124.685 q^{79} -194.013 q^{80} -94.8756 q^{82} +973.588 q^{83} +813.043 q^{85} +729.284 q^{86} +1326.78 q^{88} -291.060 q^{89} +493.433 q^{91} +750.403 q^{92} +703.868 q^{94} -471.913 q^{95} -862.897 q^{97} +491.129 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 23 q - 7 q^{2} + 79 q^{4} - 29 q^{5} - 6 q^{7} - 84 q^{8} + 41 q^{10} - 45 q^{11} + 166 q^{13} + 23 q^{14} + 195 q^{16} - 369 q^{17} - 115 q^{19} - 162 q^{20} - 130 q^{22} - 221 q^{23} + 712 q^{25} - 227 q^{26}+ \cdots - 10181 q^{98}+O(q^{100}) $ 23 * q - 7 * q^2 + 79 * q^4 - 29 * q^5 - 6 * q^7 - 84 * q^8 + 41 * q^10 - 45 * q^11 + 166 * q^13 + 23 * q^14 + 195 * q^16 - 369 * q^17 - 115 * q^19 - 162 * q^20 - 130 * q^22 - 221 * q^23 + 712 * q^25 - 227 * q^26 - 265 * q^28 - 599 * q^29 - 8 * q^31 - 1084 * q^32 + 409 * q^34 - 210 * q^35 + 400 * q^37 - 507 * q^38 + 806 * q^40 - 1329 * q^41 - 656 * q^43 - 344 * q^44 + 146 * q^46 - 1079 * q^47 + 2033 * q^49 - 3481 * q^50 + 2191 * q^52 - 1218 * q^53 + 1470 * q^55 - 3224 * q^56 + 3488 * q^58 - 2075 * q^59 + 249 * q^61 - 4385 * q^62 + 2556 * q^64 - 4512 * q^65 + 1020 * q^67 - 6214 * q^68 + 264 * q^70 - 571 * q^71 + 3247 * q^73 - 3078 * q^74 + 2386 * q^76 - 3290 * q^77 - 1627 * q^79 - 7008 * q^80 + 3929 * q^82 - 4312 * q^83 + 2298 * q^85 - 2775 * q^86 + 1476 * q^88 - 5539 * q^89 - 1488 * q^91 - 5387 * q^92 + 1645 * q^94 - 1796 * q^95 + 4202 * q^97 - 10181 * q^98

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.91929	−0.678572	−0.339286	−	0.940683i	$-0.610185\pi$
$2$	−1.91929	−0.678572	−0.339286	+	0.940683i	$0.610185\pi$
$3$	0	0
$3$	0	0
$4$	−4.31632	−0.539540
$5$	17.8997	1.60100	0.800499	−	0.599334i	$-0.204568\pi$
$5$	17.8997	1.60100	0.800499	+	0.599334i	$0.204568\pi$
$6$	0	0
$7$	−9.33325	−0.503948	−0.251974	−	0.967734i	$-0.581080\pi$
$7$	−9.33325	−0.503948	−0.251974	+	0.967734i	$0.581080\pi$
$8$	23.6386	1.04469
$9$	0	0
$10$	−34.3548	−1.08639
$11$	56.1277	1.53847	0.769233	−	0.638968i	$-0.220639\pi$
$11$	56.1277	1.53847	0.769233	+	0.638968i	$0.220639\pi$
$12$	0	0
$13$	−52.8683	−1.12792	−0.563962	−	0.825800i	$-0.690724\pi$
$13$	−52.8683	−1.12792	−0.563962	+	0.825800i	$0.690724\pi$
$14$	17.9132	0.341965
$15$	0	0
$16$	−10.8389	−0.169357
$17$	45.4222	0.648029	0.324015	−	0.946052i	$-0.394967\pi$
$17$	45.4222	0.648029	0.324015	+	0.946052i	$0.394967\pi$
$18$	0	0
$19$	−26.3643	−0.318336	−0.159168	−	0.987251i	$-0.550881\pi$
$19$	−26.3643	−0.318336	−0.159168	+	0.987251i	$0.550881\pi$
$20$	−77.2608	−0.863802
$21$	0	0
$22$	−107.725	−1.04396
$23$	−173.853	−1.57612	−0.788060	−	0.615598i	$-0.788914\pi$
$23$	−173.853	−1.57612	−0.788060	+	0.615598i	$0.788914\pi$
$24$	0	0
$25$	195.399	1.56319
$26$	101.470	0.765379
$27$	0	0
$28$	40.2853	0.271900
$29$	133.466	0.854621	0.427311	−	0.904105i	$-0.359461\pi$
$29$	133.466	0.854621	0.427311	+	0.904105i	$0.359461\pi$
$30$	0	0
$31$	−187.129	−1.08417	−0.542085	−	0.840323i	$-0.682365\pi$
$31$	−187.129	−1.08417	−0.542085	+	0.840323i	$0.682365\pi$
$32$	−168.306	−0.929768
$33$	0	0
$34$	−87.1784	−0.439735
$35$	−167.062	−0.806820
$36$	0	0
$37$	−114.278	−0.507761	−0.253881	−	0.967236i	$-0.581707\pi$
$37$	−114.278	−0.507761	−0.253881	+	0.967236i	$0.581707\pi$
$38$	50.6008	0.216014
$39$	0	0
$40$	423.124	1.67254
$41$	49.4326	0.188294	0.0941472	−	0.995558i	$-0.469988\pi$
$41$	49.4326	0.188294	0.0941472	+	0.995558i	$0.469988\pi$
$42$	0	0
$43$	−379.975	−1.34757	−0.673787	−	0.738925i	$-0.735334\pi$
$43$	−379.975	−1.34757	−0.673787	+	0.738925i	$0.735334\pi$
$44$	−242.265	−0.830063
$45$	0	0
$46$	333.674	1.06951
$47$	−366.733	−1.13816	−0.569080	−	0.822282i	$-0.692701\pi$
$47$	−366.733	−1.13816	−0.569080	+	0.822282i	$0.692701\pi$
$48$	0	0
$49$	−255.890	−0.746036
$50$	−375.028	−1.06074
$51$	0	0
$52$	228.196	0.608560
$53$	313.361	0.812140	0.406070	−	0.913842i	$-0.366899\pi$
$53$	313.361	0.812140	0.406070	+	0.913842i	$0.366899\pi$
$54$	0	0
$55$	1004.67	2.46308
$56$	−220.625	−0.526469
$57$	0	0
$58$	−256.160	−0.579922
$59$	−185.879	−0.410160	−0.205080	−	0.978745i	$-0.565745\pi$
$59$	−185.879	−0.410160	−0.205080	+	0.978745i	$0.565745\pi$
$60$	0	0
$61$	314.519	0.660164	0.330082	−	0.943952i	$-0.392924\pi$
$61$	314.519	0.660164	0.330082	+	0.943952i	$0.392924\pi$
$62$	359.154	0.735688
$63$	0	0
$64$	409.739	0.800272
$65$	−946.326	−1.80581
$66$	0	0
$67$	−830.782	−1.51487	−0.757435	−	0.652911i	$-0.773548\pi$
$67$	−830.782	−1.51487	−0.757435	+	0.652911i	$0.773548\pi$
$68$	−196.056	−0.349637
$69$	0	0
$70$	320.642	0.547486
$71$	−657.110	−1.09837	−0.549187	−	0.835699i	$-0.685063\pi$
$71$	−657.110	−1.09837	−0.549187	+	0.835699i	$0.685063\pi$
$72$	0	0
$73$	552.051	0.885105	0.442552	−	0.896743i	$-0.354073\pi$
$73$	552.051	0.885105	0.442552	+	0.896743i	$0.354073\pi$
$74$	219.333	0.344553
$75$	0	0
$76$	113.797	0.171755
$77$	−523.853	−0.775307
$78$	0	0
$79$	−124.685	−0.177571	−0.0887855	−	0.996051i	$-0.528299\pi$
$79$	−124.685	−0.177571	−0.0887855	+	0.996051i	$0.528299\pi$
$80$	−194.013	−0.271141
$81$	0	0
$82$	−94.8756	−0.127771
$83$	973.588	1.28753	0.643766	−	0.765223i	$-0.277371\pi$
$83$	973.588	1.28753	0.643766	+	0.765223i	$0.277371\pi$
$84$	0	0
$85$	813.043	1.03749
$86$	729.284	0.914427
$87$	0	0
$88$	1326.78	1.60722
$89$	−291.060	−0.346655	−0.173327	−	0.984864i	$-0.555452\pi$
$89$	−291.060	−0.346655	−0.173327	+	0.984864i	$0.555452\pi$
$90$	0	0
$91$	493.433	0.568416
$92$	750.403	0.850379
$93$	0	0
$94$	703.868	0.772324
$95$	−471.913	−0.509656
$96$	0	0
$97$	−862.897	−0.903236	−0.451618	−	0.892211i	$-0.649153\pi$
$97$	−862.897	−0.903236	−0.451618	+	0.892211i	$0.649153\pi$
$98$	491.129	0.506240
$99$	0	0
$100$	−843.405	−0.843405
$101$	297.950	0.293536	0.146768	−	0.989171i	$-0.453113\pi$
$101$	297.950	0.293536	0.146768	+	0.989171i	$0.453113\pi$
$102$	0	0
$103$	676.140	0.646816	0.323408	−	0.946260i	$-0.395171\pi$
$103$	676.140	0.646816	0.323408	+	0.946260i	$0.395171\pi$
$104$	−1249.73	−1.17833
$105$	0	0
$106$	−601.431	−0.551096
$107$	1704.19	1.53972	0.769861	−	0.638212i	$-0.220326\pi$
$107$	1704.19	1.53972	0.769861	+	0.638212i	$0.220326\pi$
$108$	0	0
$109$	−1857.97	−1.63268	−0.816338	−	0.577575i	$-0.803999\pi$
$109$	−1857.97	−1.63268	−0.816338	+	0.577575i	$0.803999\pi$
$110$	−1928.25	−1.67138
$111$	0	0
$112$	101.162	0.0853473
$113$	609.016	0.507003	0.253502	−	0.967335i	$-0.418418\pi$
$113$	609.016	0.507003	0.253502	+	0.967335i	$0.418418\pi$
$114$	0	0
$115$	−3111.91	−2.52337
$116$	−576.082	−0.461102
$117$	0	0
$118$	356.757	0.278323
$119$	−423.936	−0.326573
$120$	0	0
$121$	1819.31	1.36688
$122$	−603.653	−0.447969
$123$	0	0
$124$	807.706	0.584953
$125$	1260.12	0.901671
$126$	0	0
$127$	−475.597	−0.332302	−0.166151	−	0.986100i	$-0.553134\pi$
$127$	−475.597	−0.332302	−0.166151	+	0.986100i	$0.553134\pi$
$128$	560.038	0.386725
$129$	0	0
$130$	1816.28	1.22537
$131$	−2459.31	−1.64023	−0.820117	−	0.572195i	$-0.806092\pi$
$131$	−2459.31	−1.64023	−0.820117	+	0.572195i	$0.806092\pi$
$132$	0	0
$133$	246.065	0.160425
$134$	1594.51	1.02795
$135$	0	0
$136$	1073.72	0.676989
$137$	1367.08	0.852539	0.426269	−	0.904596i	$-0.359828\pi$
$137$	1367.08	0.852539	0.426269	+	0.904596i	$0.359828\pi$
$138$	0	0
$139$	−922.129	−0.562690	−0.281345	−	0.959607i	$-0.590781\pi$
$139$	−922.129	−0.562690	−0.281345	+	0.959607i	$0.590781\pi$
$140$	721.094	0.435311
$141$	0	0
$142$	1261.19	0.745326
$143$	−2967.37	−1.73527
$144$	0	0
$145$	2389.00	1.36825
$146$	−1059.55	−0.600607
$147$	0	0
$148$	493.260	0.273957
$149$	1403.31	0.771565	0.385783	−	0.922590i	$-0.373931\pi$
$149$	1403.31	0.771565	0.385783	+	0.922590i	$0.373931\pi$
$150$	0	0
$151$	−2957.18	−1.59372	−0.796860	−	0.604164i	$-0.793507\pi$
$151$	−2957.18	−1.59372	−0.796860	+	0.604164i	$0.793507\pi$
$152$	−623.216	−0.332562
$153$	0	0
$154$	1005.43	0.526102
$155$	−3349.55	−1.73575
$156$	0	0
$157$	736.658	0.374469	0.187235	−	0.982315i	$-0.440048\pi$
$157$	736.658	0.374469	0.187235	+	0.982315i	$0.440048\pi$
$158$	239.306	0.120495
$159$	0	0
$160$	−3012.63	−1.48856
$161$	1622.61	0.794283
$162$	0	0
$163$	2207.52	1.06077	0.530387	−	0.847755i	$-0.322046\pi$
$163$	2207.52	1.06077	0.530387	+	0.847755i	$0.322046\pi$
$164$	−213.367	−0.101592
$165$	0	0
$166$	−1868.60	−0.873683
$167$	2640.85	1.22368	0.611841	−	0.790981i	$-0.290429\pi$
$167$	2640.85	1.22368	0.611841	+	0.790981i	$0.290429\pi$
$168$	0	0
$169$	598.055	0.272214
$170$	−1560.47	−0.704014
$171$	0	0
$172$	1640.09	0.727070
$173$	2061.26	0.905865	0.452933	−	0.891545i	$-0.350378\pi$
$173$	2061.26	0.905865	0.452933	+	0.891545i	$0.350378\pi$
$174$	0	0
$175$	−1823.71	−0.787769
$176$	−608.360	−0.260551
$177$	0	0
$178$	558.629	0.235230
$179$	466.392	0.194747	0.0973737	−	0.995248i	$-0.468956\pi$
$179$	466.392	0.194747	0.0973737	+	0.995248i	$0.468956\pi$
$180$	0	0
$181$	−181.000	−0.0743294
$181$	−181.000	−0.0743294
$182$	−947.042	−0.385711
$183$	0	0
$184$	−4109.63	−1.64656
$185$	−2045.54	−0.812925
$186$	0	0
$187$	2549.44	0.996970
$188$	1582.94	0.614083
$189$	0	0
$190$	905.740	0.345838
$191$	1137.33	0.430860	0.215430	−	0.976519i	$-0.430885\pi$
$191$	1137.33	0.430860	0.215430	+	0.976519i	$0.430885\pi$
$192$	0	0
$193$	−330.564	−0.123288	−0.0616438	−	0.998098i	$-0.519634\pi$
$193$	−330.564	−0.123288	−0.0616438	+	0.998098i	$0.519634\pi$
$194$	1656.15	0.612911
$195$	0	0
$196$	1104.50	0.402516
$197$	−3983.77	−1.44077	−0.720385	−	0.693575i	$-0.756035\pi$
$197$	−3983.77	−1.44077	−0.720385	+	0.693575i	$0.756035\pi$
$198$	0	0
$199$	126.519	0.0450688	0.0225344	−	0.999746i	$-0.492826\pi$
$199$	126.519	0.0450688	0.0225344	+	0.999746i	$0.492826\pi$
$200$	4618.97	1.63305
$201$	0	0
$202$	−571.853	−0.199185
$203$	−1245.67	−0.430685
$204$	0	0
$205$	884.828	0.301459
$206$	−1297.71	−0.438911
$207$	0	0
$208$	573.033	0.191022
$209$	−1479.77	−0.489750
$210$	0	0
$211$	−1067.29	−0.348223	−0.174112	−	0.984726i	$-0.555705\pi$
$211$	−1067.29	−0.348223	−0.174112	+	0.984726i	$0.555705\pi$
$212$	−1352.56	−0.438182
$213$	0	0
$214$	−3270.84	−1.04481
$215$	−6801.44	−2.15746
$216$	0	0
$217$	1746.52	0.546366
$218$	3566.00	1.10789
$219$	0	0
$220$	−4336.47	−1.32893
$221$	−2401.39	−0.730928
$222$	0	0
$223$	−725.792	−0.217949	−0.108975	−	0.994045i	$-0.534757\pi$
$223$	−725.792	−0.217949	−0.108975	+	0.994045i	$0.534757\pi$
$224$	1570.84	0.468555
$225$	0	0
$226$	−1168.88	−0.344038
$227$	−2043.12	−0.597385	−0.298693	−	0.954349i	$-0.596551\pi$
$227$	−2043.12	−0.597385	−0.298693	+	0.954349i	$0.596551\pi$
$228$	0	0
$229$	−1829.51	−0.527935	−0.263968	−	0.964532i	$-0.585031\pi$
$229$	−1829.51	−0.527935	−0.263968	+	0.964532i	$0.585031\pi$
$230$	5972.66	1.71229
$231$	0	0
$232$	3154.95	0.892813
$233$	−2548.04	−0.716428	−0.358214	−	0.933639i	$-0.616614\pi$
$233$	−2548.04	−0.716428	−0.358214	+	0.933639i	$0.616614\pi$
$234$	0	0
$235$	−6564.41	−1.82219
$236$	802.315	0.221298
$237$	0	0
$238$	813.658	0.221603
$239$	−1554.00	−0.420585	−0.210293	−	0.977638i	$-0.567442\pi$
$239$	−1554.00	−0.420585	−0.210293	+	0.977638i	$0.567442\pi$
$240$	0	0
$241$	−4707.35	−1.25820	−0.629102	−	0.777323i	$-0.716577\pi$
$241$	−4707.35	−1.25820	−0.629102	+	0.777323i	$0.716577\pi$
$242$	−3491.79	−0.927525
$243$	0	0
$244$	−1357.56	−0.356184
$245$	−4580.36	−1.19440
$246$	0	0
$247$	1393.84	0.359059
$248$	−4423.46	−1.13262
$249$	0	0
$250$	−2418.55	−0.611849
$251$	−3157.56	−0.794038	−0.397019	−	0.917810i	$-0.629955\pi$
$251$	−3157.56	−0.794038	−0.397019	+	0.917810i	$0.629955\pi$
$252$	0	0
$253$	−9757.94	−2.42481
$254$	912.809	0.225491
$255$	0	0
$256$	−4352.79	−1.06269
$257$	−7952.97	−1.93032	−0.965160	−	0.261659i	$-0.915730\pi$
$257$	−7952.97	−1.93032	−0.965160	+	0.261659i	$0.915730\pi$
$258$	0	0
$259$	1066.58	0.255885
$260$	4084.64	0.974303
$261$	0	0
$262$	4720.13	1.11302
$263$	3955.26	0.927344	0.463672	−	0.886007i	$-0.346532\pi$
$263$	3955.26	0.927344	0.463672	+	0.886007i	$0.346532\pi$
$264$	0	0
$265$	5609.06	1.30023
$266$	−472.270	−0.108860
$267$	0	0
$268$	3585.92	0.817332
$269$	−3982.59	−0.902686	−0.451343	−	0.892351i	$-0.649055\pi$
$269$	−3982.59	−0.902686	−0.451343	+	0.892351i	$0.649055\pi$
$270$	0	0
$271$	−2930.71	−0.656931	−0.328465	−	0.944516i	$-0.606531\pi$
$271$	−2930.71	−0.656931	−0.328465	+	0.944516i	$0.606531\pi$
$272$	−492.325	−0.109748
$273$	0	0
$274$	−2623.83	−0.578509
$275$	10967.3	2.40492
$276$	0	0
$277$	6179.33	1.34036	0.670180	−	0.742199i	$-0.266217\pi$
$277$	6179.33	1.34036	0.670180	+	0.742199i	$0.266217\pi$
$278$	1769.83	0.381826
$279$	0	0
$280$	−3949.12	−0.842876
$281$	−4110.88	−0.872720	−0.436360	−	0.899772i	$-0.643733\pi$
$281$	−4110.88	−0.872720	−0.436360	+	0.899772i	$0.643733\pi$
$282$	0	0
$283$	−1097.57	−0.230543	−0.115271	−	0.993334i	$-0.536774\pi$
$283$	−1097.57	−0.230543	−0.115271	+	0.993334i	$0.536774\pi$
$284$	2836.29	0.592616
$285$	0	0
$286$	5695.26	1.17751
$287$	−461.367	−0.0948907
$288$	0	0
$289$	−2849.83	−0.580058
$290$	−4585.19	−0.928454
$291$	0	0
$292$	−2382.83	−0.477549
$293$	−2388.52	−0.476241	−0.238121	−	0.971236i	$-0.576531\pi$
$293$	−2388.52	−0.476241	−0.238121	+	0.971236i	$0.576531\pi$
$294$	0	0
$295$	−3327.19	−0.656665
$296$	−2701.37	−0.530453
$297$	0	0
$298$	−2693.35	−0.523563
$299$	9191.29	1.77775
$300$	0	0
$301$	3546.41	0.679108
$302$	5675.69	1.08145
$303$	0	0
$304$	285.759	0.0539126
$305$	5629.79	1.05692
$306$	0	0
$307$	2096.37	0.389728	0.194864	−	0.980830i	$-0.437574\pi$
$307$	2096.37	0.389728	0.194864	+	0.980830i	$0.437574\pi$
$308$	2261.12	0.418309
$309$	0	0
$310$	6428.76	1.17784
$311$	−6079.47	−1.10847	−0.554237	−	0.832359i	$-0.686990\pi$
$311$	−6079.47	−1.10847	−0.554237	+	0.832359i	$0.686990\pi$
$312$	0	0
$313$	−10614.5	−1.91683	−0.958416	−	0.285374i	$-0.907882\pi$
$313$	−10614.5	−1.91683	−0.958416	+	0.285374i	$0.907882\pi$
$314$	−1413.86	−0.254105
$315$	0	0
$316$	538.178	0.0958066
$317$	8429.82	1.49358	0.746791	−	0.665058i	$-0.231593\pi$
$317$	8429.82	1.49358	0.746791	+	0.665058i	$0.231593\pi$
$318$	0	0
$319$	7491.13	1.31481
$320$	7334.21	1.28123
$321$	0	0
$322$	−3114.26	−0.538978
$323$	−1197.52	−0.206291
$324$	0	0
$325$	−10330.4	−1.76316
$326$	−4236.88	−0.719812
$327$	0	0
$328$	1168.52	0.196709
$329$	3422.81	0.573574
$330$	0	0
$331$	10309.5	1.71197	0.855985	−	0.517000i	$-0.172951\pi$
$331$	10309.5	1.71197	0.855985	+	0.517000i	$0.172951\pi$
$332$	−4202.31	−0.694674
$333$	0	0
$334$	−5068.56	−0.830357
$335$	−14870.8	−2.42530
$336$	0	0
$337$	−3675.84	−0.594172	−0.297086	−	0.954851i	$-0.596015\pi$
$337$	−3675.84	−0.594172	−0.297086	+	0.954851i	$0.596015\pi$
$338$	−1147.84	−0.184717
$339$	0	0
$340$	−3509.35	−0.559769
$341$	−10503.1	−1.66796
$342$	0	0
$343$	5589.59	0.879912
$344$	−8982.09	−1.40780
$345$	0	0
$346$	−3956.16	−0.614695
$347$	−4456.24	−0.689405	−0.344703	−	0.938712i	$-0.612020\pi$
$347$	−4456.24	−0.689405	−0.344703	+	0.938712i	$0.612020\pi$
$348$	0	0
$349$	1369.42	0.210038	0.105019	−	0.994470i	$-0.466510\pi$
$349$	1369.42	0.210038	0.105019	+	0.994470i	$0.466510\pi$
$350$	3500.23	0.534558
$351$	0	0
$352$	−9446.62	−1.43042
$353$	−3692.28	−0.556715	−0.278357	−	0.960478i	$-0.589790\pi$
$353$	−3692.28	−0.556715	−0.278357	+	0.960478i	$0.589790\pi$
$354$	0	0
$355$	−11762.1	−1.75849
$356$	1256.31	0.187034
$357$	0	0
$358$	−895.143	−0.132150
$359$	−9194.46	−1.35171	−0.675856	−	0.737033i	$-0.736226\pi$
$359$	−9194.46	−1.35171	−0.675856	+	0.737033i	$0.736226\pi$
$360$	0	0
$361$	−6163.92	−0.898662
$362$	347.392	0.0504379
$363$	0	0
$364$	−2129.81	−0.306683
$365$	9881.54	1.41705
$366$	0	0
$367$	448.057	0.0637286	0.0318643	−	0.999492i	$-0.489856\pi$
$367$	448.057	0.0637286	0.0318643	+	0.999492i	$0.489856\pi$
$368$	1884.37	0.266928
$369$	0	0
$370$	3925.99	0.551628
$371$	−2924.68	−0.409276
$372$	0	0
$373$	8424.64	1.16947	0.584734	−	0.811225i	$-0.301199\pi$
$373$	8424.64	1.16947	0.584734	+	0.811225i	$0.301199\pi$
$374$	−4893.12	−0.676517
$375$	0	0
$376$	−8669.06	−1.18902
$377$	−7056.12	−0.963949
$378$	0	0
$379$	−12814.9	−1.73683	−0.868413	−	0.495842i	$-0.834859\pi$
$379$	−12814.9	−1.73683	−0.868413	+	0.495842i	$0.834859\pi$
$380$	2036.93	0.274979
$381$	0	0
$382$	−2182.87	−0.292369
$383$	13873.8	1.85096	0.925480	−	0.378797i	$-0.123662\pi$
$383$	13873.8	1.85096	0.925480	+	0.378797i	$0.123662\pi$
$384$	0	0
$385$	−9376.82	−1.24126
$386$	634.449	0.0836596
$387$	0	0
$388$	3724.54	0.487332
$389$	−7520.78	−0.980253	−0.490127	−	0.871651i	$-0.663049\pi$
$389$	−7520.78	−0.980253	−0.490127	+	0.871651i	$0.663049\pi$
$390$	0	0
$391$	−7896.76	−1.02137
$392$	−6048.89	−0.779376
$393$	0	0
$394$	7646.01	0.977666
$395$	−2231.82	−0.284291
$396$	0	0
$397$	−6446.95	−0.815020	−0.407510	−	0.913201i	$-0.633603\pi$
$397$	−6446.95	−0.815020	−0.407510	+	0.913201i	$0.633603\pi$
$398$	−242.827	−0.0305824
$399$	0	0
$400$	−2117.91	−0.264738
$401$	13100.6	1.63145	0.815727	−	0.578437i	$-0.196337\pi$
$401$	13100.6	1.63145	0.815727	+	0.578437i	$0.196337\pi$
$402$	0	0
$403$	9893.17	1.22286
$404$	−1286.05	−0.158374
$405$	0	0
$406$	2390.81	0.292251
$407$	−6414.15	−0.781173
$408$	0	0
$409$	13140.5	1.58864	0.794321	−	0.607499i	$-0.207827\pi$
$409$	13140.5	1.58864	0.794321	+	0.607499i	$0.207827\pi$
$410$	−1698.24	−0.204562
$411$	0	0
$412$	−2918.43	−0.348983
$413$	1734.86	0.206699
$414$	0	0
$415$	17426.9	2.06134
$416$	8898.05	1.04871
$417$	0	0
$418$	2840.11	0.332330
$419$	4021.66	0.468904	0.234452	−	0.972128i	$-0.424670\pi$
$419$	4021.66	0.468904	0.234452	+	0.972128i	$0.424670\pi$
$420$	0	0
$421$	−2939.00	−0.340233	−0.170117	−	0.985424i	$-0.554414\pi$
$421$	−2939.00	−0.340233	−0.170117	+	0.985424i	$0.554414\pi$
$422$	2048.44	0.236295
$423$	0	0
$424$	7407.41	0.848434
$425$	8875.45	1.01299
$426$	0	0
$427$	−2935.48	−0.332688
$428$	−7355.82	−0.830741
$429$	0	0
$430$	13054.0	1.46399
$431$	9679.41	1.08177	0.540883	−	0.841098i	$-0.318090\pi$
$431$	9679.41	1.08177	0.540883	+	0.841098i	$0.318090\pi$
$432$	0	0
$433$	−75.8532	−0.00841864	−0.00420932	−	0.999991i	$-0.501340\pi$
$433$	−75.8532	−0.00841864	−0.00420932	+	0.999991i	$0.501340\pi$
$434$	−3352.08	−0.370749
$435$	0	0
$436$	8019.60	0.880893
$437$	4583.50	0.501736
$438$	0	0
$439$	−8154.32	−0.886525	−0.443263	−	0.896392i	$-0.646179\pi$
$439$	−8154.32	−0.886525	−0.443263	+	0.896392i	$0.646179\pi$
$440$	23749.0	2.57315
$441$	0	0
$442$	4608.97	0.495988
$443$	−3120.41	−0.334662	−0.167331	−	0.985901i	$-0.553515\pi$
$443$	−3120.41	−0.334662	−0.167331	+	0.985901i	$0.553515\pi$
$444$	0	0
$445$	−5209.88	−0.554994
$446$	1393.01	0.147894
$447$	0	0
$448$	−3824.20	−0.403296
$449$	−17954.3	−1.88712	−0.943561	−	0.331200i	$-0.892547\pi$
$449$	−17954.3	−1.88712	−0.943561	+	0.331200i	$0.892547\pi$
$450$	0	0
$451$	2774.53	0.289685
$452$	−2628.70	−0.273548
$453$	0	0
$454$	3921.34	0.405369
$455$	8832.30	0.910032
$456$	0	0
$457$	4171.14	0.426953	0.213477	−	0.976948i	$-0.431521\pi$
$457$	4171.14	0.426953	0.213477	+	0.976948i	$0.431521\pi$
$458$	3511.36	0.358242
$459$	0	0
$460$	13432.0	1.36146
$461$	4166.77	0.420967	0.210483	−	0.977597i	$-0.432496\pi$
$461$	4166.77	0.420967	0.210483	+	0.977597i	$0.432496\pi$
$462$	0	0
$463$	−14769.2	−1.48247	−0.741233	−	0.671248i	$-0.765758\pi$
$463$	−14769.2	−1.48247	−0.741233	+	0.671248i	$0.765758\pi$
$464$	−1446.62	−0.144736
$465$	0	0
$466$	4890.44	0.486148
$467$	−9126.12	−0.904296	−0.452148	−	0.891943i	$-0.649342\pi$
$467$	−9126.12	−0.904296	−0.452148	+	0.891943i	$0.649342\pi$
$468$	0	0
$469$	7753.90	0.763416
$470$	12599.0	1.23649
$471$	0	0
$472$	−4393.93	−0.428490
$473$	−21327.1	−2.07320
$474$	0	0
$475$	−5151.57	−0.497621
$476$	1829.84	0.176199
$477$	0	0
$478$	2982.58	0.285398
$479$	1469.89	0.140211	0.0701054	−	0.997540i	$-0.477666\pi$
$479$	1469.89	0.140211	0.0701054	+	0.997540i	$0.477666\pi$
$480$	0	0
$481$	6041.67	0.572717
$482$	9034.78	0.853782
$483$	0	0
$484$	−7852.73	−0.737484
$485$	−15445.6	−1.44608
$486$	0	0
$487$	12909.7	1.20123	0.600613	−	0.799540i	$-0.294923\pi$
$487$	12909.7	1.20123	0.600613	+	0.799540i	$0.294923\pi$
$488$	7434.78	0.689666
$489$	0	0
$490$	8791.05	0.810488
$491$	−18447.9	−1.69560	−0.847801	−	0.530314i	$-0.822074\pi$
$491$	−18447.9	−1.69560	−0.847801	+	0.530314i	$0.822074\pi$
$492$	0	0
$493$	6062.32	0.553819
$494$	−2675.18	−0.243648
$495$	0	0
$496$	2028.26	0.183612
$497$	6132.97	0.553524
$498$	0	0
$499$	−10980.3	−0.985058	−0.492529	−	0.870296i	$-0.663928\pi$
$499$	−10980.3	−0.985058	−0.492529	+	0.870296i	$0.663928\pi$
$500$	−5439.10	−0.486487
$501$	0	0
$502$	6060.29	0.538812
$503$	4028.20	0.357075	0.178537	−	0.983933i	$-0.442863\pi$
$503$	4028.20	0.357075	0.178537	+	0.983933i	$0.442863\pi$
$504$	0	0
$505$	5333.21	0.469950
$506$	18728.3	1.64541
$507$	0	0
$508$	2052.83	0.179290
$509$	2731.61	0.237871	0.118936	−	0.992902i	$-0.462052\pi$
$509$	2731.61	0.237871	0.118936	+	0.992902i	$0.462052\pi$
$510$	0	0
$511$	−5152.43	−0.446047
$512$	3873.97	0.334389
$513$	0	0
$514$	15264.1	1.30986
$515$	12102.7	1.03555
$516$	0	0
$517$	−20583.9	−1.75102
$518$	−2047.09	−0.173637
$519$	0	0
$520$	−22369.8	−1.88650
$521$	−16986.4	−1.42838	−0.714192	−	0.699950i	$-0.753205\pi$
$521$	−16986.4	−1.42838	−0.714192	+	0.699950i	$0.753205\pi$
$522$	0	0
$523$	21093.9	1.76362	0.881810	−	0.471605i	$-0.156325\pi$
$523$	21093.9	1.76362	0.881810	+	0.471605i	$0.156325\pi$
$524$	10615.2	0.884972
$525$	0	0
$526$	−7591.29	−0.629270
$527$	−8499.79	−0.702574
$528$	0	0
$529$	18057.7	1.48416
$530$	−10765.4	−0.882303
$531$	0	0
$532$	−1062.09	−0.0865557
$533$	−2613.42	−0.212382
$534$	0	0
$535$	30504.5	2.46509
$536$	−19638.5	−1.58257
$537$	0	0
$538$	7643.75	0.612538
$539$	−14362.5	−1.14775
$540$	0	0
$541$	−10959.2	−0.870928	−0.435464	−	0.900206i	$-0.643416\pi$
$541$	−10959.2	−0.870928	−0.435464	+	0.900206i	$0.643416\pi$
$542$	5624.90	0.445775
$543$	0	0
$544$	−7644.82	−0.602516
$545$	−33257.2	−2.61391
$546$	0	0
$547$	23945.6	1.87174	0.935870	−	0.352346i	$-0.114616\pi$
$547$	23945.6	1.87174	0.935870	+	0.352346i	$0.114616\pi$
$548$	−5900.77	−0.459979
$549$	0	0
$550$	−21049.5	−1.63191
$551$	−3518.74	−0.272057
$552$	0	0
$553$	1163.71	0.0894866
$554$	−11859.9	−0.909531
$555$	0	0
$556$	3980.20	0.303594
$557$	−11376.7	−0.865437	−0.432718	−	0.901529i	$-0.642446\pi$
$557$	−11376.7	−0.865437	−0.432718	+	0.901529i	$0.642446\pi$
$558$	0	0
$559$	20088.6	1.51996
$560$	1810.77	0.136641
$561$	0	0
$562$	7889.97	0.592204
$563$	19299.8	1.44474	0.722370	−	0.691507i	$-0.243053\pi$
$563$	19299.8	1.44474	0.722370	+	0.691507i	$0.243053\pi$
$564$	0	0
$565$	10901.2	0.811711
$566$	2106.55	0.156440
$567$	0	0
$568$	−15533.2	−1.14746
$569$	−10855.8	−0.799820	−0.399910	−	0.916554i	$-0.630959\pi$
$569$	−10855.8	−0.799820	−0.399910	+	0.916554i	$0.630959\pi$
$570$	0	0
$571$	−12876.9	−0.943748	−0.471874	−	0.881666i	$-0.656422\pi$
$571$	−12876.9	−0.943748	−0.471874	+	0.881666i	$0.656422\pi$
$572$	12808.1	0.936249
$573$	0	0
$574$	885.498	0.0643902
$575$	−33970.7	−2.46378
$576$	0	0
$577$	16662.5	1.20220	0.601100	−	0.799174i	$-0.294729\pi$
$577$	16662.5	1.20220	0.601100	+	0.799174i	$0.294729\pi$
$578$	5469.65	0.393612
$579$	0	0
$580$	−10311.7	−0.738223
$581$	−9086.74	−0.648849
$582$	0	0
$583$	17588.2	1.24945
$584$	13049.7	0.924659
$585$	0	0
$586$	4584.27	0.323164
$587$	9224.90	0.648641	0.324320	−	0.945947i	$-0.394864\pi$
$587$	9224.90	0.648641	0.324320	+	0.945947i	$0.394864\pi$
$588$	0	0
$589$	4933.52	0.345131
$590$	6385.84	0.445595
$591$	0	0
$592$	1238.64	0.0859931
$593$	4407.67	0.305230	0.152615	−	0.988286i	$-0.451231\pi$
$593$	4407.67	0.305230	0.152615	+	0.988286i	$0.451231\pi$
$594$	0	0
$595$	−7588.34	−0.522843
$596$	−6057.11	−0.416290
$597$	0	0
$598$	−17640.8	−1.20633
$599$	25472.0	1.73749	0.868745	−	0.495259i	$-0.164927\pi$
$599$	25472.0	1.73749	0.868745	+	0.495259i	$0.164927\pi$
$600$	0	0
$601$	4524.23	0.307067	0.153533	−	0.988143i	$-0.450935\pi$
$601$	4524.23	0.307067	0.153533	+	0.988143i	$0.450935\pi$
$602$	−6806.59	−0.460824
$603$	0	0
$604$	12764.1	0.859875
$605$	32565.2	2.18837
$606$	0	0
$607$	−7020.38	−0.469437	−0.234719	−	0.972063i	$-0.575417\pi$
$607$	−7020.38	−0.469437	−0.234719	+	0.972063i	$0.575417\pi$
$608$	4437.27	0.295979
$609$	0	0
$610$	−10805.2	−0.717197
$611$	19388.6	1.28376
$612$	0	0
$613$	−2803.90	−0.184745	−0.0923723	−	0.995725i	$-0.529445\pi$
$613$	−2803.90	−0.184745	−0.0923723	+	0.995725i	$0.529445\pi$
$614$	−4023.55	−0.264458
$615$	0	0
$616$	−12383.2	−0.809955
$617$	8131.95	0.530599	0.265300	−	0.964166i	$-0.414529\pi$
$617$	8131.95	0.530599	0.265300	+	0.964166i	$0.414529\pi$
$618$	0	0
$619$	−21886.2	−1.42113	−0.710567	−	0.703630i	$-0.751561\pi$
$619$	−21886.2	−1.42113	−0.710567	+	0.703630i	$0.751561\pi$
$620$	14457.7	0.936508
$621$	0	0
$622$	11668.3	0.752179
$623$	2716.53	0.174696
$624$	0	0
$625$	−1869.06	−0.119620
$626$	20372.4	1.30071
$627$	0	0
$628$	−3179.65	−0.202041
$629$	−5190.75	−0.329044
$630$	0	0
$631$	4452.70	0.280918	0.140459	−	0.990087i	$-0.455142\pi$
$631$	4452.70	0.280918	0.140459	+	0.990087i	$0.455142\pi$
$632$	−2947.37	−0.185507
$633$	0	0
$634$	−16179.3	−1.01350
$635$	−8513.04	−0.532015
$636$	0	0
$637$	13528.5	0.841473
$638$	−14377.7	−0.892191
$639$	0	0
$640$	10024.5	0.619146
$641$	15513.1	0.955899	0.477949	−	0.878387i	$-0.341380\pi$
$641$	15513.1	0.955899	0.477949	+	0.878387i	$0.341380\pi$
$642$	0	0
$643$	−10772.9	−0.660717	−0.330358	−	0.943856i	$-0.607170\pi$
$643$	−10772.9	−0.660717	−0.330358	+	0.943856i	$0.607170\pi$
$644$	−7003.70	−0.428547
$645$	0	0
$646$	2298.40	0.139983
$647$	−2770.41	−0.168340	−0.0841701	−	0.996451i	$-0.526824\pi$
$647$	−2770.41	−0.168340	−0.0841701	+	0.996451i	$0.526824\pi$
$648$	0	0
$649$	−10433.0	−0.631017
$650$	19827.1	1.19643
$651$	0	0
$652$	−9528.35	−0.572330
$653$	31205.7	1.87010	0.935049	−	0.354518i	$-0.115355\pi$
$653$	31205.7	1.87010	0.935049	+	0.354518i	$0.115355\pi$
$654$	0	0
$655$	−44020.9	−2.62601
$656$	−535.793	−0.0318891
$657$	0	0
$658$	−6569.38	−0.389211
$659$	6071.71	0.358908	0.179454	−	0.983766i	$-0.442567\pi$
$659$	6071.71	0.358908	0.179454	+	0.983766i	$0.442567\pi$
$660$	0	0
$661$	20654.8	1.21540	0.607700	−	0.794167i	$-0.292092\pi$
$661$	20654.8	1.21540	0.607700	+	0.794167i	$0.292092\pi$
$662$	−19787.0	−1.16170
$663$	0	0
$664$	23014.3	1.34507
$665$	4404.49	0.256840
$666$	0	0
$667$	−23203.4	−1.34699
$668$	−11398.7	−0.660225
$669$	0	0
$670$	28541.3	1.64574
$671$	17653.2	1.01564
$672$	0	0
$673$	−10494.7	−0.601101	−0.300551	−	0.953766i	$-0.597170\pi$
$673$	−10494.7	−0.601101	−0.300551	+	0.953766i	$0.597170\pi$
$674$	7055.01	0.403188
$675$	0	0
$676$	−2581.40	−0.146870
$677$	−12244.8	−0.695133	−0.347567	−	0.937655i	$-0.612992\pi$
$677$	−12244.8	−0.695133	−0.347567	+	0.937655i	$0.612992\pi$
$678$	0	0
$679$	8053.64	0.455184
$680$	19219.2	1.08386
$681$	0	0
$682$	20158.5	1.13183
$683$	−14313.6	−0.801897	−0.400949	−	0.916101i	$-0.631319\pi$
$683$	−14313.6	−0.801897	−0.400949	+	0.916101i	$0.631319\pi$
$684$	0	0
$685$	24470.4	1.36491
$686$	−10728.1	−0.597084
$687$	0	0
$688$	4118.50	0.228222
$689$	−16566.8	−0.916033
$690$	0	0
$691$	−15583.7	−0.857936	−0.428968	−	0.903320i	$-0.641123\pi$
$691$	−15583.7	−0.857936	−0.428968	+	0.903320i	$0.641123\pi$
$692$	−8897.05	−0.488750
$693$	0	0
$694$	8552.83	0.467811
$695$	−16505.8	−0.900866
$696$	0	0
$697$	2245.34	0.122020
$698$	−2628.31	−0.142526
$699$	0	0
$700$	7871.71	0.425032
$701$	28768.4	1.55003	0.775014	−	0.631944i	$-0.217743\pi$
$701$	28768.4	1.55003	0.775014	+	0.631944i	$0.217743\pi$
$702$	0	0
$703$	3012.86	0.161639
$704$	22997.7	1.23119
$705$	0	0
$706$	7086.57	0.377771
$707$	−2780.84	−0.147927
$708$	0	0
$709$	9566.70	0.506749	0.253374	−	0.967368i	$-0.418460\pi$
$709$	9566.70	0.506749	0.253374	+	0.967368i	$0.418460\pi$
$710$	22574.8	1.19327
$711$	0	0
$712$	−6880.25	−0.362146
$713$	32532.8	1.70878
$714$	0	0
$715$	−53115.1	−2.77817
$716$	−2013.10	−0.105074
$717$	0	0
$718$	17646.8	0.917235
$719$	19674.1	1.02047	0.510237	−	0.860034i	$-0.329558\pi$
$719$	19674.1	1.02047	0.510237	+	0.860034i	$0.329558\pi$
$720$	0	0
$721$	−6310.58	−0.325962
$722$	11830.4	0.609807
$723$	0	0
$724$	781.253	0.0401037
$725$	26079.2	1.33594
$726$	0	0
$727$	−34245.6	−1.74704	−0.873520	−	0.486789i	$-0.838168\pi$
$727$	−34245.6	−1.74704	−0.873520	+	0.486789i	$0.838168\pi$
$728$	11664.1	0.593818
$729$	0	0
$730$	−18965.6	−0.961571
$731$	−17259.3	−0.873267
$732$	0	0
$733$	30906.4	1.55737	0.778685	−	0.627415i	$-0.215887\pi$
$733$	30906.4	1.55737	0.778685	+	0.627415i	$0.215887\pi$
$734$	−859.953	−0.0432445
$735$	0	0
$736$	29260.4	1.46543
$737$	−46629.9	−2.33057
$738$	0	0
$739$	−25421.1	−1.26540	−0.632699	−	0.774398i	$-0.718053\pi$
$739$	−25421.1	−1.26540	−0.632699	+	0.774398i	$0.718053\pi$
$740$	8829.20	0.438605
$741$	0	0
$742$	5613.31	0.277724
$743$	33013.8	1.63009	0.815046	−	0.579396i	$-0.196712\pi$
$743$	33013.8	1.63009	0.815046	+	0.579396i	$0.196712\pi$
$744$	0	0
$745$	25118.7	1.23527
$746$	−16169.4	−0.793569
$747$	0	0
$748$	−11004.2	−0.537905
$749$	−15905.6	−0.775940
$750$	0	0
$751$	−9966.57	−0.484268	−0.242134	−	0.970243i	$-0.577847\pi$
$751$	−9966.57	−0.484268	−0.242134	+	0.970243i	$0.577847\pi$
$752$	3974.97	0.192756
$753$	0	0
$754$	13542.8	0.654109
$755$	−52932.6	−2.55154
$756$	0	0
$757$	23753.5	1.14047	0.570236	−	0.821481i	$-0.306852\pi$
$757$	23753.5	1.14047	0.570236	+	0.821481i	$0.306852\pi$
$758$	24595.5	1.17856
$759$	0	0
$760$	−11155.4	−0.532432
$761$	−10526.6	−0.501431	−0.250716	−	0.968061i	$-0.580666\pi$
$761$	−10526.6	−0.501431	−0.250716	+	0.968061i	$0.580666\pi$
$762$	0	0
$763$	17340.9	0.822784
$764$	−4909.07	−0.232466
$765$	0	0
$766$	−26627.9	−1.25601
$767$	9827.13	0.462630
$768$	0	0
$769$	2950.58	0.138362	0.0691811	−	0.997604i	$-0.477961\pi$
$769$	2950.58	0.138362	0.0691811	+	0.997604i	$0.477961\pi$
$770$	17996.9	0.842288
$771$	0	0
$772$	1426.82	0.0665186
$773$	10231.2	0.476056	0.238028	−	0.971258i	$-0.423499\pi$
$773$	10231.2	0.476056	0.238028	+	0.971258i	$0.423499\pi$
$774$	0	0
$775$	−36564.8	−1.69477
$776$	−20397.7	−0.943601
$777$	0	0
$778$	14434.6	0.665173
$779$	−1303.26	−0.0599410
$780$	0	0
$781$	−36882.0	−1.68981
$782$	15156.2	0.693075
$783$	0	0
$784$	2773.56	0.126347
$785$	13186.0	0.599525
$786$	0	0
$787$	−27741.6	−1.25652	−0.628261	−	0.778003i	$-0.716233\pi$
$787$	−27741.6	−1.25652	−0.628261	+	0.778003i	$0.716233\pi$
$788$	17195.2	0.777352
$789$	0	0
$790$	4283.51	0.192912
$791$	−5684.10	−0.255503
$792$	0	0
$793$	−16628.1	−0.744615
$794$	12373.6	0.553050
$795$	0	0
$796$	−546.096	−0.0243164
$797$	34080.0	1.51465	0.757326	−	0.653037i	$-0.226506\pi$
$797$	34080.0	1.51465	0.757326	+	0.653037i	$0.226506\pi$
$798$	0	0
$799$	−16657.8	−0.737561
$800$	−32886.8	−1.45341
$801$	0	0
$802$	−25143.9	−1.10706
$803$	30985.3	1.36170
$804$	0	0
$805$	29044.2	1.27165
$806$	−18987.9	−0.829801
$807$	0	0
$808$	7043.12	0.306653
$809$	26201.8	1.13870	0.569348	−	0.822096i	$-0.307196\pi$
$809$	26201.8	1.13870	0.569348	+	0.822096i	$0.307196\pi$
$810$	0	0
$811$	13358.8	0.578412	0.289206	−	0.957267i	$-0.406609\pi$
$811$	13358.8	0.578412	0.289206	+	0.957267i	$0.406609\pi$
$812$	5376.72	0.232372
$813$	0	0
$814$	12310.6	0.530083
$815$	39513.9	1.69830
$816$	0	0
$817$	10017.8	0.428982
$818$	−25220.4	−1.07801
$819$	0	0
$820$	−3819.20	−0.162649
$821$	35994.5	1.53010	0.765052	−	0.643968i	$-0.222713\pi$
$821$	35994.5	1.53010	0.765052	+	0.643968i	$0.222713\pi$
$822$	0	0
$823$	−38743.7	−1.64097	−0.820486	−	0.571667i	$-0.806297\pi$
$823$	−38743.7	−1.64097	−0.820486	+	0.571667i	$0.806297\pi$
$824$	15983.0	0.675721
$825$	0	0
$826$	−3329.70	−0.140260
$827$	−34159.4	−1.43632	−0.718162	−	0.695876i	$-0.755016\pi$
$827$	−34159.4	−1.43632	−0.718162	+	0.695876i	$0.755016\pi$
$828$	0	0
$829$	35296.0	1.47875	0.739374	−	0.673295i	$-0.235122\pi$
$829$	35296.0	1.47875	0.739374	+	0.673295i	$0.235122\pi$
$830$	−33447.4	−1.39876
$831$	0	0
$832$	−21662.2	−0.902647
$833$	−11623.1	−0.483453
$834$	0	0
$835$	47270.4	1.95911
$836$	6387.15	0.264239
$837$	0	0
$838$	−7718.74	−0.318185
$839$	39727.0	1.63472	0.817358	−	0.576129i	$-0.195438\pi$
$839$	39727.0	1.63472	0.817358	+	0.576129i	$0.195438\pi$
$840$	0	0
$841$	−6575.82	−0.269622
$842$	5640.80	0.230873
$843$	0	0
$844$	4606.75	0.187880
$845$	10705.0	0.435815
$846$	0	0
$847$	−16980.1	−0.688835
$848$	−3396.48	−0.137542
$849$	0	0
$850$	−17034.6	−0.687390
$851$	19867.5	0.800293
$852$	0	0
$853$	−7068.86	−0.283743	−0.141872	−	0.989885i	$-0.545312\pi$
$853$	−7068.86	−0.283743	−0.141872	+	0.989885i	$0.545312\pi$
$854$	5634.05	0.225753
$855$	0	0
$856$	40284.7	1.60853
$857$	7467.01	0.297629	0.148815	−	0.988865i	$-0.452454\pi$
$857$	7467.01	0.297629	0.148815	+	0.988865i	$0.452454\pi$
$858$	0	0
$859$	−38191.8	−1.51698	−0.758490	−	0.651685i	$-0.774063\pi$
$859$	−38191.8	−1.51698	−0.758490	+	0.651685i	$0.774063\pi$
$860$	29357.2	1.16404
$861$	0	0
$862$	−18577.6	−0.734056
$863$	41733.3	1.64614	0.823069	−	0.567941i	$-0.192260\pi$
$863$	41733.3	1.64614	0.823069	+	0.567941i	$0.192260\pi$
$864$	0	0
$865$	36895.9	1.45029
$866$	145.584	0.00571266
$867$	0	0
$868$	−7538.53	−0.294786
$869$	−6998.26	−0.273187
$870$	0	0
$871$	43922.0	1.70866
$872$	−43919.9	−1.70564
$873$	0	0
$874$	−8797.09	−0.340464
$875$	−11761.1	−0.454396
$876$	0	0
$877$	16676.0	0.642087	0.321043	−	0.947065i	$-0.395966\pi$
$877$	16676.0	0.642087	0.321043	+	0.947065i	$0.395966\pi$
$878$	15650.5	0.601571
$879$	0	0
$880$	−10889.5	−0.417141
$881$	19171.1	0.733135	0.366567	−	0.930391i	$-0.380533\pi$
$881$	19171.1	0.733135	0.366567	+	0.930391i	$0.380533\pi$
$882$	0	0
$883$	7041.22	0.268353	0.134177	−	0.990957i	$-0.457161\pi$
$883$	7041.22	0.268353	0.134177	+	0.990957i	$0.457161\pi$
$884$	10365.2	0.394365
$885$	0	0
$886$	5988.99	0.227093
$887$	−41940.1	−1.58761	−0.793805	−	0.608173i	$-0.791903\pi$
$887$	−41940.1	−1.58761	−0.793805	+	0.608173i	$0.791903\pi$
$888$	0	0
$889$	4438.86	0.167463
$890$	9999.29	0.376603
$891$	0	0
$892$	3132.75	0.117592
$893$	9668.67	0.362318
$894$	0	0
$895$	8348.28	0.311790
$896$	−5226.98	−0.194890
$897$	0	0
$898$	34459.6	1.28055
$899$	−24975.3	−0.926555
$900$	0	0
$901$	14233.5	0.526290
$902$	−5325.14	−0.196572
$903$	0	0
$904$	14396.3	0.529661
$905$	−3239.85	−0.119001
$906$	0	0
$907$	−45038.2	−1.64881	−0.824404	−	0.566001i	$-0.808490\pi$
$907$	−45038.2	−1.64881	−0.824404	+	0.566001i	$0.808490\pi$
$908$	8818.74	0.322313
$909$	0	0
$910$	−16951.8	−0.617523
$911$	48746.1	1.77281	0.886405	−	0.462910i	$-0.153195\pi$
$911$	48746.1	1.77281	0.886405	+	0.462910i	$0.153195\pi$
$912$	0	0
$913$	54645.2	1.98082
$914$	−8005.63	−0.289719
$915$	0	0
$916$	7896.73	0.284842
$917$	22953.3	0.826593
$918$	0	0
$919$	−37028.2	−1.32910	−0.664552	−	0.747242i	$-0.731378\pi$
$919$	−37028.2	−1.32910	−0.664552	+	0.747242i	$0.731378\pi$
$920$	−73561.2	−2.63613
$921$	0	0
$922$	−7997.24	−0.285656
$923$	34740.3	1.23888
$924$	0	0
$925$	−22329.8	−0.793729
$926$	28346.3	1.00596
$927$	0	0
$928$	−22463.1	−0.794599
$929$	−38766.8	−1.36910	−0.684552	−	0.728964i	$-0.740002\pi$
$929$	−38766.8	−1.36910	−0.684552	+	0.728964i	$0.740002\pi$
$930$	0	0
$931$	6746.38	0.237490
$932$	10998.2	0.386541
$933$	0	0
$934$	17515.7	0.613631
$935$	45634.2	1.59615
$936$	0	0
$937$	46941.3	1.63661	0.818306	−	0.574782i	$-0.194913\pi$
$937$	46941.3	1.63661	0.818306	+	0.574782i	$0.194913\pi$
$938$	−14882.0	−0.518033
$939$	0	0
$940$	28334.1	0.983145
$941$	−39286.7	−1.36101	−0.680505	−	0.732743i	$-0.738240\pi$
$941$	−39286.7	−1.36101	−0.680505	+	0.732743i	$0.738240\pi$
$942$	0	0
$943$	−8593.98	−0.296775
$944$	2014.72	0.0694636
$945$	0	0
$946$	40933.0	1.40681
$947$	−10859.1	−0.372622	−0.186311	−	0.982491i	$-0.559653\pi$
$947$	−10859.1	−0.372622	−0.186311	+	0.982491i	$0.559653\pi$
$948$	0	0
$949$	−29186.0	−0.998331
$950$	9887.36	0.337672
$951$	0	0
$952$	−10021.3	−0.341167
$953$	−37223.0	−1.26524	−0.632618	−	0.774464i	$-0.718020\pi$
$953$	−37223.0	−1.26524	−0.632618	+	0.774464i	$0.718020\pi$
$954$	0	0
$955$	20357.8	0.689805
$956$	6707.56	0.226922
$957$	0	0
$958$	−2821.15	−0.0951432
$959$	−12759.3	−0.429635
$960$	0	0
$961$	5226.11	0.175426
$962$	−11595.7	−0.388630
$963$	0	0
$964$	20318.4	0.678851
$965$	−5916.99	−0.197383
$966$	0	0
$967$	−8408.61	−0.279630	−0.139815	−	0.990178i	$-0.544651\pi$
$967$	−8408.61	−0.279630	−0.139815	+	0.990178i	$0.544651\pi$
$968$	43006.0	1.42796
$969$	0	0
$970$	29644.6	0.981269
$971$	20660.0	0.682813	0.341406	−	0.939916i	$-0.389097\pi$
$971$	20660.0	0.682813	0.341406	+	0.939916i	$0.389097\pi$
$972$	0	0
$973$	8606.46	0.283567
$974$	−24777.6	−0.815118
$975$	0	0
$976$	−3409.03	−0.111804
$977$	−50316.1	−1.64765	−0.823825	−	0.566844i	$-0.808164\pi$
$977$	−50316.1	−1.64765	−0.823825	+	0.566844i	$0.808164\pi$
$978$	0	0
$979$	−16336.5	−0.533317
$980$	19770.3	0.644427
$981$	0	0
$982$	35406.8	1.15059
$983$	−1034.36	−0.0335614	−0.0167807	−	0.999859i	$-0.505342\pi$
$983$	−1034.36	−0.0335614	−0.0167807	+	0.999859i	$0.505342\pi$
$984$	0	0
$985$	−71308.2	−2.30667
$986$	−11635.4	−0.375807
$987$	0	0
$988$	−6016.24	−0.193727
$989$	66059.7	2.12394
$990$	0	0
$991$	34133.6	1.09414	0.547068	−	0.837088i	$-0.315744\pi$
$991$	34133.6	1.09414	0.547068	+	0.837088i	$0.315744\pi$
$992$	31494.9	1.00803
$993$	0	0
$994$	−11771.0	−0.375606
$995$	2264.65	0.0721551
$996$	0	0
$997$	12536.8	0.398241	0.199120	−	0.979975i	$-0.436192\pi$
$997$	12536.8	0.398241	0.199120	+	0.979975i	$0.436192\pi$
$998$	21074.3	0.668433
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1629.4.a.e.1.9		23
3.2	odd	2		543.4.a.c.1.15	✓	23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
543.4.a.c.1.15	✓	23	3.2	odd	2
1629.4.a.e.1.9		23	1.1	even	1	trivial

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1629 = 3^{2} \cdot 181 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1629.a (trivial)

\(f(q)\)	\(=\)	\(q-1.91929 q^{2} -4.31632 q^{4} +17.8997 q^{5} -9.33325 q^{7} +23.6386 q^{8} -34.3548 q^{10} +56.1277 q^{11} -52.8683 q^{13} +17.9132 q^{14} -10.8389 q^{16} +45.4222 q^{17} -26.3643 q^{19} -77.2608 q^{20} -107.725 q^{22} -173.853 q^{23} +195.399 q^{25} +101.470 q^{26} +40.2853 q^{28} +133.466 q^{29} -187.129 q^{31} -168.306 q^{32} -87.1784 q^{34} -167.062 q^{35} -114.278 q^{37} +50.6008 q^{38} +423.124 q^{40} +49.4326 q^{41} -379.975 q^{43} -242.265 q^{44} +333.674 q^{46} -366.733 q^{47} -255.890 q^{49} -375.028 q^{50} +228.196 q^{52} +313.361 q^{53} +1004.67 q^{55} -220.625 q^{56} -256.160 q^{58} -185.879 q^{59} +314.519 q^{61} +359.154 q^{62} +409.739 q^{64} -946.326 q^{65} -830.782 q^{67} -196.056 q^{68} +320.642 q^{70} -657.110 q^{71} +552.051 q^{73} +219.333 q^{74} +113.797 q^{76} -523.853 q^{77} -124.685 q^{79} -194.013 q^{80} -94.8756 q^{82} +973.588 q^{83} +813.043 q^{85} +729.284 q^{86} +1326.78 q^{88} -291.060 q^{89} +493.433 q^{91} +750.403 q^{92} +703.868 q^{94} -471.913 q^{95} -862.897 q^{97} +491.129 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q - 7 q^{2} + 79 q^{4} - 29 q^{5} - 6 q^{7} - 84 q^{8} + 41 q^{10} - 45 q^{11} + 166 q^{13} + 23 q^{14} + 195 q^{16} - 369 q^{17} - 115 q^{19} - 162 q^{20} - 130 q^{22} - 221 q^{23} + 712 q^{25} - 227 q^{26}+ \cdots - 10181 q^{98}+O(q^{100}) \) 23 * q - 7 * q^2 + 79 * q^4 - 29 * q^5 - 6 * q^7 - 84 * q^8 + 41 * q^10 - 45 * q^11 + 166 * q^13 + 23 * q^14 + 195 * q^16 - 369 * q^17 - 115 * q^19 - 162 * q^20 - 130 * q^22 - 221 * q^23 + 712 * q^25 - 227 * q^26 - 265 * q^28 - 599 * q^29 - 8 * q^31 - 1084 * q^32 + 409 * q^34 - 210 * q^35 + 400 * q^37 - 507 * q^38 + 806 * q^40 - 1329 * q^41 - 656 * q^43 - 344 * q^44 + 146 * q^46 - 1079 * q^47 + 2033 * q^49 - 3481 * q^50 + 2191 * q^52 - 1218 * q^53 + 1470 * q^55 - 3224 * q^56 + 3488 * q^58 - 2075 * q^59 + 249 * q^61 - 4385 * q^62 + 2556 * q^64 - 4512 * q^65 + 1020 * q^67 - 6214 * q^68 + 264 * q^70 - 571 * q^71 + 3247 * q^73 - 3078 * q^74 + 2386 * q^76 - 3290 * q^77 - 1627 * q^79 - 7008 * q^80 + 3929 * q^82 - 4312 * q^83 + 2298 * q^85 - 2775 * q^86 + 1476 * q^88 - 5539 * q^89 - 1488 * q^91 - 5387 * q^92 + 1645 * q^94 - 1796 * q^95 + 4202 * q^97 - 10181 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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