LMFDB - Embedded newform 2151.4.a.e.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2151,4,Mod(1,2151)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2151, 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("2151.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2151 = 3^{2} \cdot 239 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2151.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:	$126.913108422$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	no (minimal twist has level 717)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Character	$\chi$	$=$	2151.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.0917574 q^{2} -7.99158 q^{4} -10.9396 q^{5} -5.88604 q^{7} -1.46735 q^{8} +O(q^{10})$	\(q+0.0917574 q^{2} -7.99158 q^{4} -10.9396 q^{5} -5.88604 q^{7} -1.46735 q^{8} -1.00379 q^{10} -46.1672 q^{11} +8.83154 q^{13} -0.540088 q^{14} +63.7980 q^{16} +83.0165 q^{17} -69.4159 q^{19} +87.4246 q^{20} -4.23618 q^{22} -122.537 q^{23} -5.32530 q^{25} +0.810359 q^{26} +47.0388 q^{28} -5.43359 q^{29} -191.955 q^{31} +17.5927 q^{32} +7.61738 q^{34} +64.3909 q^{35} -193.654 q^{37} -6.36943 q^{38} +16.0522 q^{40} -485.027 q^{41} -217.229 q^{43} +368.949 q^{44} -11.2436 q^{46} -414.922 q^{47} -308.355 q^{49} -0.488636 q^{50} -70.5780 q^{52} -669.101 q^{53} +505.050 q^{55} +8.63686 q^{56} -0.498572 q^{58} -141.816 q^{59} +524.288 q^{61} -17.6133 q^{62} -508.770 q^{64} -96.6134 q^{65} -85.4839 q^{67} -663.433 q^{68} +5.90834 q^{70} -223.940 q^{71} -667.699 q^{73} -17.7692 q^{74} +554.743 q^{76} +271.742 q^{77} +837.497 q^{79} -697.924 q^{80} -44.5049 q^{82} -378.698 q^{83} -908.167 q^{85} -19.9324 q^{86} +67.7433 q^{88} +1282.00 q^{89} -51.9828 q^{91} +979.261 q^{92} -38.0721 q^{94} +759.382 q^{95} +1689.96 q^{97} -28.2938 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 3 q^{2} + 151 q^{4} + 14 q^{5} + 72 q^{7} - 57 q^{8}+O(q^{10}) $ 32 * q - 3 * q^2 + 151 * q^4 + 14 * q^5 + 72 * q^7 - 57 * q^8	$ 32 q - 3 q^{2} + 151 q^{4} + 14 q^{5} + 72 q^{7} - 57 q^{8} + 32 q^{10} - 154 q^{11} + 100 q^{13} - 42 q^{14} + 719 q^{16} - 32 q^{17} + 202 q^{19} + 132 q^{20} + 265 q^{22} - 552 q^{23} + 1086 q^{25} + 280 q^{26} + 390 q^{28} + 154 q^{29} + 560 q^{31} - 444 q^{32} + 156 q^{34} - 394 q^{35} + 914 q^{37} - 111 q^{38} + 257 q^{40} + 914 q^{41} + 1722 q^{43} - 1243 q^{44} + 584 q^{46} - 380 q^{47} + 2446 q^{49} + 454 q^{50} + 1552 q^{52} - 370 q^{53} + 918 q^{55} + 499 q^{56} + 2446 q^{58} - 492 q^{59} + 668 q^{61} - 578 q^{62} + 6475 q^{64} - 736 q^{65} + 4548 q^{67} - 5253 q^{68} + 7793 q^{70} - 258 q^{71} + 3096 q^{73} - 449 q^{74} + 6814 q^{76} - 3804 q^{77} + 2864 q^{79} + 1052 q^{80} + 14145 q^{82} - 2364 q^{83} + 3088 q^{85} - 2811 q^{86} + 8329 q^{88} + 4172 q^{89} + 7350 q^{91} - 13644 q^{92} + 6122 q^{94} - 3336 q^{95} + 6370 q^{97} - 1572 q^{98}+O(q^{100}) $ 32 * q - 3 * q^2 + 151 * q^4 + 14 * q^5 + 72 * q^7 - 57 * q^8 + 32 * q^10 - 154 * q^11 + 100 * q^13 - 42 * q^14 + 719 * q^16 - 32 * q^17 + 202 * q^19 + 132 * q^20 + 265 * q^22 - 552 * q^23 + 1086 * q^25 + 280 * q^26 + 390 * q^28 + 154 * q^29 + 560 * q^31 - 444 * q^32 + 156 * q^34 - 394 * q^35 + 914 * q^37 - 111 * q^38 + 257 * q^40 + 914 * q^41 + 1722 * q^43 - 1243 * q^44 + 584 * q^46 - 380 * q^47 + 2446 * q^49 + 454 * q^50 + 1552 * q^52 - 370 * q^53 + 918 * q^55 + 499 * q^56 + 2446 * q^58 - 492 * q^59 + 668 * q^61 - 578 * q^62 + 6475 * q^64 - 736 * q^65 + 4548 * q^67 - 5253 * q^68 + 7793 * q^70 - 258 * q^71 + 3096 * q^73 - 449 * q^74 + 6814 * q^76 - 3804 * q^77 + 2864 * q^79 + 1052 * q^80 + 14145 * q^82 - 2364 * q^83 + 3088 * q^85 - 2811 * q^86 + 8329 * q^88 + 4172 * q^89 + 7350 * q^91 - 13644 * q^92 + 6122 * q^94 - 3336 * q^95 + 6370 * q^97 - 1572 * q^98

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 $

$2$	0.0917574	0.0324412	0.0162206	−	0.999868i	$-0.494837\pi$
$2$	0.0917574	0.0324412	0.0162206	+	0.999868i	$0.494837\pi$
$3$	0	0
$3$	0	0
$4$	−7.99158	−0.998948
$5$	−10.9396	−0.978467	−0.489233	−	0.872153i	$-0.662723\pi$
$5$	−10.9396	−0.978467	−0.489233	+	0.872153i	$0.662723\pi$
$6$	0	0
$7$	−5.88604	−0.317816	−0.158908	−	0.987293i	$-0.550797\pi$
$7$	−5.88604	−0.317816	−0.158908	+	0.987293i	$0.550797\pi$
$8$	−1.46735	−0.0648482
$9$	0	0
$10$	−1.00379	−0.0317426
$11$	−46.1672	−1.26545	−0.632724	−	0.774377i	$-0.718063\pi$
$11$	−46.1672	−1.26545	−0.632724	+	0.774377i	$0.718063\pi$
$12$	0	0
$13$	8.83154	0.188418	0.0942088	−	0.995552i	$-0.469968\pi$
$13$	8.83154	0.188418	0.0942088	+	0.995552i	$0.469968\pi$
$14$	−0.540088	−0.0103103
$15$	0	0
$16$	63.7980	0.996844
$17$	83.0165	1.18438	0.592190	−	0.805798i	$-0.298263\pi$
$17$	83.0165	1.18438	0.592190	+	0.805798i	$0.298263\pi$
$18$	0	0
$19$	−69.4159	−0.838163	−0.419082	−	0.907949i	$-0.637648\pi$
$19$	−69.4159	−0.838163	−0.419082	+	0.907949i	$0.637648\pi$
$20$	87.4246	0.977437
$21$	0	0
$22$	−4.23618	−0.0410526
$23$	−122.537	−1.11090	−0.555449	−	0.831551i	$-0.687454\pi$
$23$	−122.537	−1.11090	−0.555449	+	0.831551i	$0.687454\pi$
$24$	0	0
$25$	−5.32530	−0.0426024
$26$	0.810359	0.00611248
$27$	0	0
$28$	47.0388	0.317482
$29$	−5.43359	−0.0347929	−0.0173964	−	0.999849i	$-0.505538\pi$
$29$	−5.43359	−0.0347929	−0.0173964	+	0.999849i	$0.505538\pi$
$30$	0	0
$31$	−191.955	−1.11213	−0.556066	−	0.831138i	$-0.687690\pi$
$31$	−191.955	−1.11213	−0.556066	+	0.831138i	$0.687690\pi$
$32$	17.5927	0.0971869
$33$	0	0
$34$	7.61738	0.0384227
$35$	64.3909	0.310973
$36$	0	0
$37$	−193.654	−0.860445	−0.430222	−	0.902723i	$-0.641565\pi$
$37$	−193.654	−0.860445	−0.430222	+	0.902723i	$0.641565\pi$
$38$	−6.36943	−0.0271910
$39$	0	0
$40$	16.0522	0.0634518
$41$	−485.027	−1.84753	−0.923763	−	0.382966i	$-0.874903\pi$
$41$	−485.027	−1.84753	−0.923763	+	0.382966i	$0.874903\pi$
$42$	0	0
$43$	−217.229	−0.770397	−0.385199	−	0.922834i	$-0.625867\pi$
$43$	−217.229	−0.770397	−0.385199	+	0.922834i	$0.625867\pi$
$44$	368.949	1.26412
$45$	0	0
$46$	−11.2436	−0.0360388
$47$	−414.922	−1.28771	−0.643857	−	0.765146i	$-0.722667\pi$
$47$	−414.922	−1.28771	−0.643857	+	0.765146i	$0.722667\pi$
$48$	0	0
$49$	−308.355	−0.898993
$50$	−0.488636	−0.00138207
$51$	0	0
$52$	−70.5780	−0.188219
$53$	−669.101	−1.73411	−0.867057	−	0.498208i	$-0.833992\pi$
$53$	−669.101	−1.73411	−0.867057	+	0.498208i	$0.833992\pi$
$54$	0	0
$55$	505.050	1.23820
$56$	8.63686	0.0206098
$57$	0	0
$58$	−0.498572	−0.00112872
$59$	−141.816	−0.312930	−0.156465	−	0.987684i	$-0.550010\pi$
$59$	−141.816	−0.312930	−0.156465	+	0.987684i	$0.550010\pi$
$60$	0	0
$61$	524.288	1.10046	0.550231	−	0.835012i	$-0.314540\pi$
$61$	524.288	1.10046	0.550231	+	0.835012i	$0.314540\pi$
$62$	−17.6133	−0.0360789
$63$	0	0
$64$	−508.770	−0.993691
$65$	−96.6134	−0.184360
$66$	0	0
$67$	−85.4839	−0.155873	−0.0779367	−	0.996958i	$-0.524833\pi$
$67$	−85.4839	−0.155873	−0.0779367	+	0.996958i	$0.524833\pi$
$68$	−663.433	−1.18313
$69$	0	0
$70$	5.90834	0.0100883
$71$	−223.940	−0.374321	−0.187161	−	0.982329i	$-0.559928\pi$
$71$	−223.940	−0.374321	−0.187161	+	0.982329i	$0.559928\pi$
$72$	0	0
$73$	−667.699	−1.07052	−0.535262	−	0.844686i	$-0.679787\pi$
$73$	−667.699	−1.07052	−0.535262	+	0.844686i	$0.679787\pi$
$74$	−17.7692	−0.0279138
$75$	0	0
$76$	554.743	0.837281
$77$	271.742	0.402180
$78$	0	0
$79$	837.497	1.19273	0.596366	−	0.802713i	$-0.296611\pi$
$79$	837.497	1.19273	0.596366	+	0.802713i	$0.296611\pi$
$80$	−697.924	−0.975379
$81$	0	0
$82$	−44.5049	−0.0599358
$83$	−378.698	−0.500813	−0.250406	−	0.968141i	$-0.580564\pi$
$83$	−378.698	−0.500813	−0.250406	+	0.968141i	$0.580564\pi$
$84$	0	0
$85$	−908.167	−1.15888
$86$	−19.9324	−0.0249926
$87$	0	0
$88$	67.7433	0.0820620
$89$	1282.00	1.52687	0.763434	−	0.645886i	$-0.223512\pi$
$89$	1282.00	1.52687	0.763434	+	0.645886i	$0.223512\pi$
$90$	0	0
$91$	−51.9828	−0.0598822
$92$	979.261	1.10973
$93$	0	0
$94$	−38.0721	−0.0417749
$95$	759.382	0.820115
$96$	0	0
$97$	1689.96	1.76896	0.884479	−	0.466579i	$-0.154514\pi$
$97$	1689.96	1.76896	0.884479	+	0.466579i	$0.154514\pi$
$98$	−28.2938	−0.0291644
$99$	0	0
$100$	42.5576	0.0425576
$101$	−1538.68	−1.51588	−0.757942	−	0.652322i	$-0.773795\pi$
$101$	−1538.68	−1.51588	−0.757942	+	0.652322i	$0.773795\pi$
$102$	0	0
$103$	−479.190	−0.458407	−0.229204	−	0.973378i	$-0.573612\pi$
$103$	−479.190	−0.458407	−0.229204	+	0.973378i	$0.573612\pi$
$104$	−12.9589	−0.0122185
$105$	0	0
$106$	−61.3950	−0.0562567
$107$	−1976.10	−1.78539	−0.892695	−	0.450660i	$-0.851189\pi$
$107$	−1976.10	−1.78539	−0.892695	+	0.450660i	$0.851189\pi$
$108$	0	0
$109$	1927.34	1.69363	0.846816	−	0.531885i	$-0.178516\pi$
$109$	1927.34	1.69363	0.846816	+	0.531885i	$0.178516\pi$
$110$	46.3421	0.0401686
$111$	0	0
$112$	−375.518	−0.316813
$113$	−1579.83	−1.31520	−0.657602	−	0.753365i	$-0.728429\pi$
$113$	−1579.83	−1.31520	−0.657602	+	0.753365i	$0.728429\pi$
$114$	0	0
$115$	1340.50	1.08698
$116$	43.4230	0.0347562
$117$	0	0
$118$	−13.0127	−0.0101518
$119$	−488.639	−0.376415
$120$	0	0
$121$	800.409	0.601359
$122$	48.1073	0.0357003
$123$	0	0
$124$	1534.02	1.11096
$125$	1425.71	1.02015
$126$	0	0
$127$	474.836	0.331771	0.165885	−	0.986145i	$-0.446952\pi$
$127$	474.836	0.331771	0.165885	+	0.986145i	$0.446952\pi$
$128$	−187.425	−0.129423
$129$	0	0
$130$	−8.86500	−0.00598086
$131$	1859.88	1.24045	0.620225	−	0.784424i	$-0.287041\pi$
$131$	1859.88	1.24045	0.620225	+	0.784424i	$0.287041\pi$
$132$	0	0
$133$	408.585	0.266382
$134$	−7.84378	−0.00505671
$135$	0	0
$136$	−121.814	−0.0768049
$137$	181.238	0.113023	0.0565117	−	0.998402i	$-0.482002\pi$
$137$	181.238	0.113023	0.0565117	+	0.998402i	$0.482002\pi$
$138$	0	0
$139$	−431.368	−0.263224	−0.131612	−	0.991301i	$-0.542015\pi$
$139$	−431.368	−0.263224	−0.131612	+	0.991301i	$0.542015\pi$
$140$	−514.585	−0.310645
$141$	0	0
$142$	−20.5482	−0.0121434
$143$	−407.727	−0.238433
$144$	0	0
$145$	59.4413	0.0340437
$146$	−61.2664	−0.0347290
$147$	0	0
$148$	1547.60	0.859539
$149$	−890.121	−0.489406	−0.244703	−	0.969598i	$-0.578691\pi$
$149$	−890.121	−0.489406	−0.244703	+	0.969598i	$0.578691\pi$
$150$	0	0
$151$	1567.92	0.845004	0.422502	−	0.906362i	$-0.361152\pi$
$151$	1567.92	0.845004	0.422502	+	0.906362i	$0.361152\pi$
$152$	101.857	0.0543534
$153$	0	0
$154$	24.9343	0.0130472
$155$	2099.91	1.08818
$156$	0	0
$157$	1961.08	0.996885	0.498443	−	0.866923i	$-0.333905\pi$
$157$	1961.08	0.996885	0.498443	+	0.866923i	$0.333905\pi$
$158$	76.8466	0.0386936
$159$	0	0
$160$	−192.457	−0.0950942
$161$	721.256	0.353061
$162$	0	0
$163$	7.99117	0.00383998	0.00191999	−	0.999998i	$-0.499389\pi$
$163$	7.99117	0.00383998	0.00191999	+	0.999998i	$0.499389\pi$
$164$	3876.13	1.84558
$165$	0	0
$166$	−34.7483	−0.0162469
$167$	−3362.61	−1.55812	−0.779062	−	0.626946i	$-0.784304\pi$
$167$	−3362.61	−1.55812	−0.779062	+	0.626946i	$0.784304\pi$
$168$	0	0
$169$	−2119.00	−0.964499
$170$	−83.3311	−0.0375953
$171$	0	0
$172$	1736.00	0.769586
$173$	−2874.38	−1.26321	−0.631605	−	0.775290i	$-0.717604\pi$
$173$	−2874.38	−1.26321	−0.631605	+	0.775290i	$0.717604\pi$
$174$	0	0
$175$	31.3449	0.0135397
$176$	−2945.37	−1.26145
$177$	0	0
$178$	117.633	0.0495334
$179$	356.073	0.148682	0.0743411	−	0.997233i	$-0.476315\pi$
$179$	356.073	0.148682	0.0743411	+	0.997233i	$0.476315\pi$
$180$	0	0
$181$	−3672.09	−1.50798	−0.753989	−	0.656887i	$-0.771873\pi$
$181$	−3672.09	−1.50798	−0.753989	+	0.656887i	$0.771873\pi$
$182$	−4.76981	−0.00194265
$183$	0	0
$184$	179.804	0.0720397
$185$	2118.49	0.841917
$186$	0	0
$187$	−3832.64	−1.49877
$188$	3315.88	1.28636
$189$	0	0
$190$	69.6789	0.0266055
$191$	−3684.77	−1.39592	−0.697959	−	0.716137i	$-0.745908\pi$
$191$	−3684.77	−1.39592	−0.697959	+	0.716137i	$0.745908\pi$
$192$	0	0
$193$	327.254	0.122053	0.0610266	−	0.998136i	$-0.480563\pi$
$193$	327.254	0.122053	0.0610266	+	0.998136i	$0.480563\pi$
$194$	155.066	0.0573871
$195$	0	0
$196$	2464.24	0.898047
$197$	3779.46	1.36688	0.683440	−	0.730007i	$-0.260483\pi$
$197$	3779.46	1.36688	0.683440	+	0.730007i	$0.260483\pi$
$198$	0	0
$199$	−2895.86	−1.03157	−0.515784	−	0.856719i	$-0.672499\pi$
$199$	−2895.86	−1.03157	−0.515784	+	0.856719i	$0.672499\pi$
$200$	7.81407	0.00276269
$201$	0	0
$202$	−141.185	−0.0491770
$203$	31.9823	0.0110577
$204$	0	0
$205$	5306.00	1.80774
$206$	−43.9692	−0.0148713
$207$	0	0
$208$	563.435	0.187823
$209$	3204.74	1.06065
$210$	0	0
$211$	2384.16	0.777879	0.388940	−	0.921263i	$-0.372841\pi$
$211$	2384.16	0.777879	0.388940	+	0.921263i	$0.372841\pi$
$212$	5347.17	1.73229
$213$	0	0
$214$	−181.322	−0.0579201
$215$	2376.39	0.753808
$216$	0	0
$217$	1129.85	0.353454
$218$	176.848	0.0549434
$219$	0	0
$220$	−4036.15	−1.23690
$221$	733.164	0.223158
$222$	0	0
$223$	4018.25	1.20665	0.603323	−	0.797497i	$-0.293843\pi$
$223$	4018.25	1.20665	0.603323	+	0.797497i	$0.293843\pi$
$224$	−103.551	−0.0308876
$225$	0	0
$226$	−144.961	−0.0426668
$227$	−4377.83	−1.28003	−0.640014	−	0.768363i	$-0.721072\pi$
$227$	−4377.83	−1.28003	−0.640014	+	0.768363i	$0.721072\pi$
$228$	0	0
$229$	−577.065	−0.166522	−0.0832609	−	0.996528i	$-0.526533\pi$
$229$	−577.065	−0.166522	−0.0832609	+	0.996528i	$0.526533\pi$
$230$	123.001	0.0352628
$231$	0	0
$232$	7.97296	0.00225625
$233$	2917.42	0.820286	0.410143	−	0.912021i	$-0.365479\pi$
$233$	2917.42	0.820286	0.410143	+	0.912021i	$0.365479\pi$
$234$	0	0
$235$	4539.07	1.25998
$236$	1133.33	0.312600
$237$	0	0
$238$	−44.8362	−0.0122113
$239$	239.000	0.0646846
$239$	239.000	0.0646846
$240$	0	0
$241$	−3432.39	−0.917425	−0.458712	−	0.888585i	$-0.651689\pi$
$241$	−3432.39	−0.917425	−0.458712	+	0.888585i	$0.651689\pi$
$242$	73.4435	0.0195088
$243$	0	0
$244$	−4189.89	−1.09930
$245$	3373.27	0.879635
$246$	0	0
$247$	−613.049	−0.157925
$248$	281.664	0.0721197
$249$	0	0
$250$	130.819	0.0330949
$251$	12.5634	0.00315933	0.00157967	−	0.999999i	$-0.499497\pi$
$251$	12.5634	0.00315933	0.00157967	+	0.999999i	$0.499497\pi$
$252$	0	0
$253$	5657.17	1.40578
$254$	43.5697	0.0107630
$255$	0	0
$256$	4052.96	0.989492
$257$	4258.47	1.03360	0.516802	−	0.856105i	$-0.327122\pi$
$257$	4258.47	1.03360	0.516802	+	0.856105i	$0.327122\pi$
$258$	0	0
$259$	1139.85	0.273463
$260$	772.094	0.184166
$261$	0	0
$262$	170.658	0.0402416
$263$	−3230.82	−0.757494	−0.378747	−	0.925500i	$-0.623645\pi$
$263$	−3230.82	−0.757494	−0.378747	+	0.925500i	$0.623645\pi$
$264$	0	0
$265$	7319.69	1.69677
$266$	37.4907	0.00864174
$267$	0	0
$268$	683.151	0.155709
$269$	−713.612	−0.161746	−0.0808730	−	0.996724i	$-0.525771\pi$
$269$	−713.612	−0.161746	−0.0808730	+	0.996724i	$0.525771\pi$
$270$	0	0
$271$	6567.39	1.47210	0.736052	−	0.676925i	$-0.236688\pi$
$271$	6567.39	1.47210	0.736052	+	0.676925i	$0.236688\pi$
$272$	5296.29	1.18064
$273$	0	0
$274$	16.6299	0.00366661
$275$	245.854	0.0539112
$276$	0	0
$277$	1439.95	0.312339	0.156170	−	0.987730i	$-0.450085\pi$
$277$	1439.95	0.312339	0.156170	+	0.987730i	$0.450085\pi$
$278$	−39.5812	−0.00853929
$279$	0	0
$280$	−94.4837	−0.0201660
$281$	2801.39	0.594723	0.297361	−	0.954765i	$-0.403893\pi$
$281$	2801.39	0.594723	0.297361	+	0.954765i	$0.403893\pi$
$282$	0	0
$283$	−5226.20	−1.09776	−0.548878	−	0.835902i	$-0.684945\pi$
$283$	−5226.20	−1.09776	−0.548878	+	0.835902i	$0.684945\pi$
$284$	1789.64	0.373927
$285$	0	0
$286$	−37.4120	−0.00773503
$287$	2854.89	0.587174
$288$	0	0
$289$	1978.75	0.402757
$290$	5.45418	0.00110442
$291$	0	0
$292$	5335.97	1.06940
$293$	743.651	0.148275	0.0741374	−	0.997248i	$-0.476380\pi$
$293$	743.651	0.148275	0.0741374	+	0.997248i	$0.476380\pi$
$294$	0	0
$295$	1551.41	0.306191
$296$	284.157	0.0557983
$297$	0	0
$298$	−81.6752	−0.0158769
$299$	−1082.19	−0.209313
$300$	0	0
$301$	1278.62	0.244845
$302$	143.868	0.0274129
$303$	0	0
$304$	−4428.60	−0.835518
$305$	−5735.50	−1.07677
$306$	0	0
$307$	−3755.96	−0.698254	−0.349127	−	0.937075i	$-0.613522\pi$
$307$	−3755.96	−0.698254	−0.349127	+	0.937075i	$0.613522\pi$
$308$	−2171.65	−0.401757
$309$	0	0
$310$	192.682	0.0353020
$311$	151.743	0.0276674	0.0138337	−	0.999904i	$-0.495596\pi$
$311$	151.743	0.0276674	0.0138337	+	0.999904i	$0.495596\pi$
$312$	0	0
$313$	8974.23	1.62062	0.810309	−	0.586003i	$-0.199299\pi$
$313$	8974.23	1.62062	0.810309	+	0.586003i	$0.199299\pi$
$314$	179.943	0.0323401
$315$	0	0
$316$	−6692.92	−1.19148
$317$	5155.94	0.913522	0.456761	−	0.889590i	$-0.349010\pi$
$317$	5155.94	0.913522	0.456761	+	0.889590i	$0.349010\pi$
$318$	0	0
$319$	250.854	0.0440286
$320$	5565.73	0.972294
$321$	0	0
$322$	66.1806	0.0114537
$323$	−5762.67	−0.992704
$324$	0	0
$325$	−47.0306	−0.00802704
$326$	0.733249	0.000124573 0
$327$	0	0
$328$	711.703	0.119809
$329$	2442.24	0.409256
$330$	0	0
$331$	3006.91	0.499319	0.249659	−	0.968334i	$-0.419681\pi$
$331$	3006.91	0.499319	0.249659	+	0.968334i	$0.419681\pi$
$332$	3026.39	0.500286
$333$	0	0
$334$	−308.545	−0.0505474
$335$	935.159	0.152517
$336$	0	0
$337$	−3490.91	−0.564280	−0.282140	−	0.959373i	$-0.591044\pi$
$337$	−3490.91	−0.564280	−0.282140	+	0.959373i	$0.591044\pi$
$338$	−194.434	−0.0312895
$339$	0	0
$340$	7257.69	1.15766
$341$	8862.01	1.40735
$342$	0	0
$343$	3833.90	0.603531
$344$	318.750	0.0499588
$345$	0	0
$346$	−263.746	−0.0409800
$347$	−12265.8	−1.89759	−0.948796	−	0.315889i	$-0.897697\pi$
$347$	−12265.8	−1.89759	−0.948796	+	0.315889i	$0.897697\pi$
$348$	0	0
$349$	−11664.1	−1.78901	−0.894506	−	0.447057i	$-0.852472\pi$
$349$	−11664.1	−1.78901	−0.894506	+	0.447057i	$0.852472\pi$
$350$	2.87613	0.000439245 0
$351$	0	0
$352$	−812.206	−0.122985
$353$	12733.0	1.91985	0.959926	−	0.280253i	$-0.0904185\pi$
$353$	12733.0	1.91985	0.959926	+	0.280253i	$0.0904185\pi$
$354$	0	0
$355$	2449.81	0.366261
$356$	−10245.2	−1.52526
$357$	0	0
$358$	32.6723	0.00482342
$359$	−6203.32	−0.911974	−0.455987	−	0.889986i	$-0.650714\pi$
$359$	−6203.32	−0.911974	−0.455987	+	0.889986i	$0.650714\pi$
$360$	0	0
$361$	−2040.43	−0.297482
$362$	−336.941	−0.0489206
$363$	0	0
$364$	415.425	0.0598191
$365$	7304.36	1.04747
$366$	0	0
$367$	−4415.91	−0.628089	−0.314044	−	0.949408i	$-0.601684\pi$
$367$	−4415.91	−0.628089	−0.314044	+	0.949408i	$0.601684\pi$
$368$	−7817.59	−1.10739
$369$	0	0
$370$	194.387	0.0273127
$371$	3938.35	0.551130
$372$	0	0
$373$	13780.9	1.91300	0.956501	−	0.291729i	$-0.0942306\pi$
$373$	13780.9	1.91300	0.956501	+	0.291729i	$0.0942306\pi$
$374$	−351.673	−0.0486219
$375$	0	0
$376$	608.834	0.0835058
$377$	−47.9870	−0.00655558
$378$	0	0
$379$	−10004.4	−1.35591	−0.677955	−	0.735103i	$-0.737134\pi$
$379$	−10004.4	−1.35591	−0.677955	+	0.735103i	$0.737134\pi$
$380$	−6068.66	−0.819252
$381$	0	0
$382$	−338.105	−0.0452852
$383$	−9598.18	−1.28053	−0.640266	−	0.768153i	$-0.721176\pi$
$383$	−9598.18	−1.28053	−0.640266	+	0.768153i	$0.721176\pi$
$384$	0	0
$385$	−2972.75	−0.393520
$386$	30.0280	0.00395955
$387$	0	0
$388$	−13505.4	−1.76710
$389$	4645.87	0.605540	0.302770	−	0.953064i	$-0.402089\pi$
$389$	4645.87	0.605540	0.302770	+	0.953064i	$0.402089\pi$
$390$	0	0
$391$	−10172.6	−1.31573
$392$	452.463	0.0582980
$393$	0	0
$394$	346.793	0.0443431
$395$	−9161.88	−1.16705
$396$	0	0
$397$	−11136.4	−1.40786	−0.703929	−	0.710270i	$-0.748573\pi$
$397$	−11136.4	−1.40786	−0.703929	+	0.710270i	$0.748573\pi$
$398$	−265.716	−0.0334652
$399$	0	0
$400$	−339.744	−0.0424680
$401$	−11788.1	−1.46800	−0.734000	−	0.679149i	$-0.762349\pi$
$401$	−11788.1	−1.46800	−0.734000	+	0.679149i	$0.762349\pi$
$402$	0	0
$403$	−1695.26	−0.209545
$404$	12296.5	1.51429
$405$	0	0
$406$	2.93462	0.000358726 0
$407$	8940.44	1.08885
$408$	0	0
$409$	8821.85	1.06653	0.533267	−	0.845947i	$-0.320964\pi$
$409$	8821.85	1.06653	0.533267	+	0.845947i	$0.320964\pi$
$410$	486.865	0.0586452
$411$	0	0
$412$	3829.48	0.457925
$413$	834.733	0.0994541
$414$	0	0
$415$	4142.80	0.490029
$416$	155.371	0.0183117
$417$	0	0
$418$	294.059	0.0344088
$419$	7989.72	0.931559	0.465780	−	0.884901i	$-0.345774\pi$
$419$	7989.72	0.931559	0.465780	+	0.884901i	$0.345774\pi$
$420$	0	0
$421$	−5077.89	−0.587841	−0.293920	−	0.955830i	$-0.594960\pi$
$421$	−5077.89	−0.587841	−0.293920	+	0.955830i	$0.594960\pi$
$422$	218.765	0.0252353
$423$	0	0
$424$	981.803	0.112454
$425$	−442.088	−0.0504575
$426$	0	0
$427$	−3085.98	−0.349745
$428$	15792.2	1.78351
$429$	0	0
$430$	218.052	0.0244544
$431$	−10833.8	−1.21077	−0.605387	−	0.795931i	$-0.706982\pi$
$431$	−10833.8	−1.21077	−0.605387	+	0.795931i	$0.706982\pi$
$432$	0	0
$433$	−5270.55	−0.584958	−0.292479	−	0.956272i	$-0.594480\pi$
$433$	−5270.55	−0.584958	−0.292479	+	0.956272i	$0.594480\pi$
$434$	103.672	0.0114664
$435$	0	0
$436$	−15402.5	−1.69185
$437$	8505.99	0.931114
$438$	0	0
$439$	18030.1	1.96021	0.980105	−	0.198482i	$-0.0636010\pi$
$439$	18030.1	1.96021	0.980105	+	0.198482i	$0.0636010\pi$
$440$	−741.084	−0.0802950
$441$	0	0
$442$	67.2732	0.00723950
$443$	−8049.40	−0.863293	−0.431646	−	0.902043i	$-0.642067\pi$
$443$	−8049.40	−0.863293	−0.431646	+	0.902043i	$0.642067\pi$
$444$	0	0
$445$	−14024.5	−1.49399
$446$	368.704	0.0391450
$447$	0	0
$448$	2994.64	0.315811
$449$	−8424.10	−0.885430	−0.442715	−	0.896662i	$-0.645985\pi$
$449$	−8424.10	−0.885430	−0.442715	+	0.896662i	$0.645985\pi$
$450$	0	0
$451$	22392.3	2.33795
$452$	12625.4	1.31382
$453$	0	0
$454$	−401.698	−0.0415256
$455$	568.670	0.0585927
$456$	0	0
$457$	1535.02	0.157123	0.0785613	−	0.996909i	$-0.474967\pi$
$457$	1535.02	0.157123	0.0785613	+	0.996909i	$0.474967\pi$
$458$	−52.9500	−0.00540216
$459$	0	0
$460$	−10712.7	−1.08583
$461$	10941.3	1.10539	0.552696	−	0.833383i	$-0.313599\pi$
$461$	10941.3	1.10539	0.552696	+	0.833383i	$0.313599\pi$
$462$	0	0
$463$	−1019.69	−0.102352	−0.0511759	−	0.998690i	$-0.516297\pi$
$463$	−1019.69	−0.102352	−0.0511759	+	0.998690i	$0.516297\pi$
$464$	−346.652	−0.0346830
$465$	0	0
$466$	267.695	0.0266110
$467$	3388.41	0.335753	0.167877	−	0.985808i	$-0.446309\pi$
$467$	3388.41	0.335753	0.167877	+	0.985808i	$0.446309\pi$
$468$	0	0
$469$	503.162	0.0495391
$470$	416.494	0.0408754
$471$	0	0
$472$	208.093	0.0202929
$473$	10028.8	0.974898
$474$	0	0
$475$	369.661	0.0357078
$476$	3904.99	0.376019
$477$	0	0
$478$	21.9300	0.00209844
$479$	−1981.51	−0.189013	−0.0945066	−	0.995524i	$-0.530127\pi$
$479$	−1981.51	−0.189013	−0.0945066	+	0.995524i	$0.530127\pi$
$480$	0	0
$481$	−1710.26	−0.162123
$482$	−314.947	−0.0297623
$483$	0	0
$484$	−6396.54	−0.600727
$485$	−18487.4	−1.73087
$486$	0	0
$487$	16738.2	1.55745	0.778725	−	0.627365i	$-0.215867\pi$
$487$	16738.2	1.55745	0.778725	+	0.627365i	$0.215867\pi$
$488$	−769.312	−0.0713630
$489$	0	0
$490$	309.523	0.0285364
$491$	−623.025	−0.0572642	−0.0286321	−	0.999590i	$-0.509115\pi$
$491$	−623.025	−0.0572642	−0.0286321	+	0.999590i	$0.509115\pi$
$492$	0	0
$493$	−451.078	−0.0412080
$494$	−56.2518	−0.00512326
$495$	0	0
$496$	−12246.3	−1.10862
$497$	1318.12	0.118965
$498$	0	0
$499$	−78.2604	−0.00702088	−0.00351044	−	0.999994i	$-0.501117\pi$
$499$	−78.2604	−0.00702088	−0.00351044	+	0.999994i	$0.501117\pi$
$500$	−11393.6	−1.01908
$501$	0	0
$502$	1.15278	0.000102492 0
$503$	−1517.02	−0.134474	−0.0672370	−	0.997737i	$-0.521418\pi$
$503$	−1517.02	−0.134474	−0.0672370	+	0.997737i	$0.521418\pi$
$504$	0	0
$505$	16832.5	1.48324
$506$	519.088	0.0456053
$507$	0	0
$508$	−3794.69	−0.331422
$509$	−17484.0	−1.52252	−0.761260	−	0.648446i	$-0.775419\pi$
$509$	−17484.0	−1.52252	−0.761260	+	0.648446i	$0.775419\pi$
$510$	0	0
$511$	3930.10	0.340230
$512$	1871.29	0.161524
$513$	0	0
$514$	390.746	0.0335313
$515$	5242.14	0.448537
$516$	0	0
$517$	19155.8	1.62953
$518$	104.590	0.00887146
$519$	0	0
$520$	141.765	0.0119554
$521$	−2366.92	−0.199034	−0.0995169	−	0.995036i	$-0.531730\pi$
$521$	−2366.92	−0.199034	−0.0995169	+	0.995036i	$0.531730\pi$
$522$	0	0
$523$	6208.75	0.519101	0.259550	−	0.965730i	$-0.416426\pi$
$523$	6208.75	0.519101	0.259550	+	0.965730i	$0.416426\pi$
$524$	−14863.4	−1.23914
$525$	0	0
$526$	−296.452	−0.0245740
$527$	−15935.4	−1.31719
$528$	0	0
$529$	2848.23	0.234095
$530$	671.636	0.0550453
$531$	0	0
$532$	−3265.24	−0.266102
$533$	−4283.54	−0.348106
$534$	0	0
$535$	21617.7	1.74695
$536$	125.434	0.0101081
$537$	0	0
$538$	−65.4792	−0.00524723
$539$	14235.9	1.13763
$540$	0	0
$541$	3659.77	0.290842	0.145421	−	0.989370i	$-0.453546\pi$
$541$	3659.77	0.290842	0.145421	+	0.989370i	$0.453546\pi$
$542$	602.607	0.0477568
$543$	0	0
$544$	1460.49	0.115106
$545$	−21084.3	−1.65716
$546$	0	0
$547$	−7266.86	−0.568023	−0.284011	−	0.958821i	$-0.591665\pi$
$547$	−7266.86	−0.568023	−0.284011	+	0.958821i	$0.591665\pi$
$548$	−1448.38	−0.112904
$549$	0	0
$550$	22.5590	0.00174894
$551$	377.178	0.0291621
$552$	0	0
$553$	−4929.54	−0.379069
$554$	132.126	0.0101326
$555$	0	0
$556$	3447.31	0.262947
$557$	21606.1	1.64359	0.821795	−	0.569783i	$-0.192973\pi$
$557$	21606.1	1.64359	0.821795	+	0.569783i	$0.192973\pi$
$558$	0	0
$559$	−1918.46	−0.145156
$560$	4108.01	0.309991
$561$	0	0
$562$	257.049	0.0192935
$563$	15880.6	1.18879	0.594395	−	0.804174i	$-0.297392\pi$
$563$	15880.6	1.18879	0.594395	+	0.804174i	$0.297392\pi$
$564$	0	0
$565$	17282.7	1.28688
$566$	−479.542	−0.0356125
$567$	0	0
$568$	328.598	0.0242740
$569$	15562.9	1.14662	0.573312	−	0.819337i	$-0.305658\pi$
$569$	15562.9	1.14662	0.573312	+	0.819337i	$0.305658\pi$
$570$	0	0
$571$	−8272.59	−0.606300	−0.303150	−	0.952943i	$-0.598038\pi$
$571$	−8272.59	−0.606300	−0.303150	+	0.952943i	$0.598038\pi$
$572$	3258.39	0.238182
$573$	0	0
$574$	261.957	0.0190486
$575$	652.545	0.0473270
$576$	0	0
$577$	25537.0	1.84249	0.921246	−	0.388981i	$-0.127173\pi$
$577$	25537.0	1.84249	0.921246	+	0.388981i	$0.127173\pi$
$578$	181.565	0.0130659
$579$	0	0
$580$	−475.030	−0.0340078
$581$	2229.03	0.159166
$582$	0	0
$583$	30890.5	2.19443
$584$	979.746	0.0694215
$585$	0	0
$586$	68.2355	0.00481021
$587$	−16283.5	−1.14496	−0.572480	−	0.819919i	$-0.694018\pi$
$587$	−16283.5	−1.14496	−0.572480	+	0.819919i	$0.694018\pi$
$588$	0	0
$589$	13324.7	0.932149
$590$	142.353	0.00993320
$591$	0	0
$592$	−12354.7	−0.857729
$593$	2745.07	0.190096	0.0950478	−	0.995473i	$-0.469700\pi$
$593$	2745.07	0.190096	0.0950478	+	0.995473i	$0.469700\pi$
$594$	0	0
$595$	5345.51	0.368310
$596$	7113.47	0.488891
$597$	0	0
$598$	−99.2987	−0.00679034
$599$	−2932.30	−0.200018	−0.100009	−	0.994987i	$-0.531887\pi$
$599$	−2932.30	−0.200018	−0.100009	+	0.994987i	$0.531887\pi$
$600$	0	0
$601$	−26220.8	−1.77965	−0.889825	−	0.456303i	$-0.849173\pi$
$601$	−26220.8	−1.77965	−0.889825	+	0.456303i	$0.849173\pi$
$602$	117.323	0.00794304
$603$	0	0
$604$	−12530.2	−0.844114
$605$	−8756.15	−0.588410
$606$	0	0
$607$	−28195.3	−1.88536	−0.942678	−	0.333704i	$-0.891701\pi$
$607$	−28195.3	−1.88536	−0.942678	+	0.333704i	$0.891701\pi$
$608$	−1221.21	−0.0814585
$609$	0	0
$610$	−526.275	−0.0349315
$611$	−3664.40	−0.242628
$612$	0	0
$613$	−9056.10	−0.596693	−0.298346	−	0.954458i	$-0.596435\pi$
$613$	−9056.10	−0.596693	−0.298346	+	0.954458i	$0.596435\pi$
$614$	−344.637	−0.0226522
$615$	0	0
$616$	−398.740	−0.0260806
$617$	11283.6	0.736238	0.368119	−	0.929779i	$-0.380002\pi$
$617$	11283.6	0.736238	0.368119	+	0.929779i	$0.380002\pi$
$618$	0	0
$619$	21397.8	1.38942	0.694711	−	0.719289i	$-0.255532\pi$
$619$	21397.8	1.38942	0.694711	+	0.719289i	$0.255532\pi$
$620$	−16781.6	−1.08704
$621$	0	0
$622$	13.9236	0.000897564 0
$623$	−7545.88	−0.485263
$624$	0	0
$625$	−14931.0	−0.955583
$626$	823.452	0.0525747
$627$	0	0
$628$	−15672.1	−0.995836
$629$	−16076.4	−1.01909
$630$	0	0
$631$	21656.0	1.36626	0.683132	−	0.730295i	$-0.260618\pi$
$631$	21656.0	1.36626	0.683132	+	0.730295i	$0.260618\pi$
$632$	−1228.90	−0.0773464
$633$	0	0
$634$	473.096	0.0296357
$635$	−5194.51	−0.324627
$636$	0	0
$637$	−2723.24	−0.169386
$638$	23.0177	0.00142834
$639$	0	0
$640$	2050.35	0.126637
$641$	−8942.03	−0.550997	−0.275499	−	0.961301i	$-0.588843\pi$
$641$	−8942.03	−0.550997	−0.275499	+	0.961301i	$0.588843\pi$
$642$	0	0
$643$	10528.6	0.645733	0.322866	−	0.946445i	$-0.395354\pi$
$643$	10528.6	0.645733	0.322866	+	0.946445i	$0.395354\pi$
$644$	−5763.97	−0.352690
$645$	0	0
$646$	−528.768	−0.0322045
$647$	−2415.12	−0.146752	−0.0733758	−	0.997304i	$-0.523377\pi$
$647$	−2415.12	−0.146752	−0.0733758	+	0.997304i	$0.523377\pi$
$648$	0	0
$649$	6547.24	0.395996
$650$	−4.31541	−0.000260407 0
$651$	0	0
$652$	−63.8621	−0.00383594
$653$	−2275.27	−0.136352	−0.0681762	−	0.997673i	$-0.521718\pi$
$653$	−2275.27	−0.136352	−0.0681762	+	0.997673i	$0.521718\pi$
$654$	0	0
$655$	−20346.4	−1.21374
$656$	−30943.8	−1.84169
$657$	0	0
$658$	224.094	0.0132767
$659$	32118.7	1.89858	0.949291	−	0.314400i	$-0.101803\pi$
$659$	32118.7	1.89858	0.949291	+	0.314400i	$0.101803\pi$
$660$	0	0
$661$	−8026.10	−0.472283	−0.236142	−	0.971719i	$-0.575883\pi$
$661$	−8026.10	−0.472283	−0.236142	+	0.971719i	$0.575883\pi$
$662$	275.906	0.0161985
$663$	0	0
$664$	555.681	0.0324768
$665$	−4469.75	−0.260646
$666$	0	0
$667$	665.814	0.0386513
$668$	26872.6	1.55648
$669$	0	0
$670$	85.8078	0.00494783
$671$	−24204.9	−1.39258
$672$	0	0
$673$	18088.6	1.03605	0.518027	−	0.855364i	$-0.326667\pi$
$673$	18088.6	1.03605	0.518027	+	0.855364i	$0.326667\pi$
$674$	−320.317	−0.0183059
$675$	0	0
$676$	16934.2	0.963484
$677$	19661.7	1.11619	0.558096	−	0.829776i	$-0.311532\pi$
$677$	19661.7	1.11619	0.558096	+	0.829776i	$0.311532\pi$
$678$	0	0
$679$	−9947.15	−0.562204
$680$	1332.60	0.0751511
$681$	0	0
$682$	813.156	0.0456559
$683$	18816.8	1.05418	0.527089	−	0.849810i	$-0.323284\pi$
$683$	18816.8	1.05418	0.527089	+	0.849810i	$0.323284\pi$
$684$	0	0
$685$	−1982.67	−0.110590
$686$	351.789	0.0195792
$687$	0	0
$688$	−13858.8	−0.767966
$689$	−5909.19	−0.326738
$690$	0	0
$691$	−755.273	−0.0415802	−0.0207901	−	0.999784i	$-0.506618\pi$
$691$	−755.273	−0.0415802	−0.0207901	+	0.999784i	$0.506618\pi$
$692$	22970.9	1.26188
$693$	0	0
$694$	−1125.48	−0.0615601
$695$	4718.99	0.257556
$696$	0	0
$697$	−40265.3	−2.18817
$698$	−1070.27	−0.0580376
$699$	0	0
$700$	−250.496	−0.0135255
$701$	1660.96	0.0894914	0.0447457	−	0.998998i	$-0.485752\pi$
$701$	1660.96	0.0894914	0.0447457	+	0.998998i	$0.485752\pi$
$702$	0	0
$703$	13442.6	0.721193
$704$	23488.5	1.25746
$705$	0	0
$706$	1168.35	0.0622822
$707$	9056.73	0.481773
$708$	0	0
$709$	−4218.84	−0.223472	−0.111736	−	0.993738i	$-0.535641\pi$
$709$	−4218.84	−0.223472	−0.111736	+	0.993738i	$0.535641\pi$
$710$	224.789	0.0118819
$711$	0	0
$712$	−1881.13	−0.0990146
$713$	23521.5	1.23547
$714$	0	0
$715$	4460.37	0.233298
$716$	−2845.58	−0.148526
$717$	0	0
$718$	−569.201	−0.0295855
$719$	−37015.4	−1.91995	−0.959974	−	0.280090i	$-0.909636\pi$
$719$	−37015.4	−1.91995	−0.959974	+	0.280090i	$0.909636\pi$
$720$	0	0
$721$	2820.53	0.145689
$722$	−187.225	−0.00965066
$723$	0	0
$724$	29345.8	1.50639
$725$	28.9355	0.00148226
$726$	0	0
$727$	2835.79	0.144668	0.0723341	−	0.997380i	$-0.476955\pi$
$727$	2835.79	0.144668	0.0723341	+	0.997380i	$0.476955\pi$
$728$	76.2768	0.00388325
$729$	0	0
$730$	670.229	0.0339812
$731$	−18033.6	−0.912443
$732$	0	0
$733$	−16718.4	−0.842442	−0.421221	−	0.906958i	$-0.638398\pi$
$733$	−16718.4	−0.842442	−0.421221	+	0.906958i	$0.638398\pi$
$734$	−405.192	−0.0203759
$735$	0	0
$736$	−2155.75	−0.107965
$737$	3946.55	0.197250
$738$	0	0
$739$	30479.9	1.51721	0.758607	−	0.651548i	$-0.225880\pi$
$739$	30479.9	1.51721	0.758607	+	0.651548i	$0.225880\pi$
$740$	−16930.1	−0.841031
$741$	0	0
$742$	361.373	0.0178793
$743$	5538.13	0.273451	0.136726	−	0.990609i	$-0.456342\pi$
$743$	5538.13	0.273451	0.136726	+	0.990609i	$0.456342\pi$
$744$	0	0
$745$	9737.56	0.478868
$746$	1264.50	0.0620600
$747$	0	0
$748$	30628.9	1.49719
$749$	11631.4	0.567426
$750$	0	0
$751$	−11119.8	−0.540302	−0.270151	−	0.962818i	$-0.587074\pi$
$751$	−11119.8	−0.540302	−0.270151	+	0.962818i	$0.587074\pi$
$752$	−26471.2	−1.28365
$753$	0	0
$754$	−4.40316	−0.000212671 0
$755$	−17152.4	−0.826808
$756$	0	0
$757$	609.384	0.0292582	0.0146291	−	0.999893i	$-0.495343\pi$
$757$	609.384	0.0292582	0.0146291	+	0.999893i	$0.495343\pi$
$758$	−917.976	−0.0439873
$759$	0	0
$760$	−1114.28	−0.0531830
$761$	−29044.3	−1.38352	−0.691758	−	0.722130i	$-0.743163\pi$
$761$	−29044.3	−1.38352	−0.691758	+	0.722130i	$0.743163\pi$
$762$	0	0
$763$	−11344.4	−0.538264
$764$	29447.1	1.39445
$765$	0	0
$766$	−880.704	−0.0415420
$767$	−1252.45	−0.0589614
$768$	0	0
$769$	11129.1	0.521881	0.260941	−	0.965355i	$-0.415967\pi$
$769$	11129.1	0.521881	0.260941	+	0.965355i	$0.415967\pi$
$770$	−272.772	−0.0127662
$771$	0	0
$772$	−2615.28	−0.121925
$773$	−41963.2	−1.95254	−0.976268	−	0.216566i	$-0.930514\pi$
$773$	−41963.2	−1.95254	−0.976268	+	0.216566i	$0.930514\pi$
$774$	0	0
$775$	1022.22	0.0473795
$776$	−2479.75	−0.114714
$777$	0	0
$778$	426.293	0.0196444
$779$	33668.6	1.54853
$780$	0	0
$781$	10338.7	0.473684
$782$	−933.409	−0.0426837
$783$	0	0
$784$	−19672.4	−0.896155
$785$	−21453.4	−0.975419
$786$	0	0
$787$	−41037.3	−1.85873	−0.929367	−	0.369158i	$-0.879646\pi$
$787$	−41037.3	−1.85873	−0.929367	+	0.369158i	$0.879646\pi$
$788$	−30203.8	−1.36544
$789$	0	0
$790$	−840.670	−0.0378604
$791$	9298.95	0.417993
$792$	0	0
$793$	4630.27	0.207346
$794$	−1021.85	−0.0456726
$795$	0	0
$796$	23142.5	1.03048
$797$	−20880.3	−0.928002	−0.464001	−	0.885835i	$-0.653587\pi$
$797$	−20880.3	−0.928002	−0.464001	+	0.885835i	$0.653587\pi$
$798$	0	0
$799$	−34445.3	−1.52514
$800$	−93.6865	−0.00414040
$801$	0	0
$802$	−1081.64	−0.0476236
$803$	30825.8	1.35469
$804$	0	0
$805$	−7890.24	−0.345459
$806$	−155.552	−0.00679789
$807$	0	0
$808$	2257.78	0.0983023
$809$	21424.1	0.931065	0.465533	−	0.885031i	$-0.345863\pi$
$809$	21424.1	0.931065	0.465533	+	0.885031i	$0.345863\pi$
$810$	0	0
$811$	26388.1	1.14255	0.571277	−	0.820757i	$-0.306448\pi$
$811$	26388.1	1.14255	0.571277	+	0.820757i	$0.306448\pi$
$812$	−255.589	−0.0110461
$813$	0	0
$814$	820.352	0.0353235
$815$	−87.4201	−0.00375729
$816$	0	0
$817$	15079.1	0.645719
$818$	809.470	0.0345996
$819$	0	0
$820$	−42403.3	−1.80584
$821$	35930.3	1.52738	0.763688	−	0.645586i	$-0.223387\pi$
$821$	35930.3	1.52738	0.763688	+	0.645586i	$0.223387\pi$
$822$	0	0
$823$	37021.4	1.56803	0.784013	−	0.620745i	$-0.213170\pi$
$823$	37021.4	1.56803	0.784013	+	0.620745i	$0.213170\pi$
$824$	703.137	0.0297269
$825$	0	0
$826$	76.5930	0.00322641
$827$	−12548.8	−0.527649	−0.263824	−	0.964571i	$-0.584984\pi$
$827$	−12548.8	−0.527649	−0.263824	+	0.964571i	$0.584984\pi$
$828$	0	0
$829$	4417.33	0.185067	0.0925334	−	0.995710i	$-0.470504\pi$
$829$	4417.33	0.185067	0.0925334	+	0.995710i	$0.470504\pi$
$830$	380.133	0.0158971
$831$	0	0
$832$	−4493.22	−0.187229
$833$	−25598.5	−1.06475
$834$	0	0
$835$	36785.6	1.52457
$836$	−25610.9	−1.05954
$837$	0	0
$838$	733.116	0.0302209
$839$	13736.1	0.565223	0.282612	−	0.959234i	$-0.408799\pi$
$839$	13736.1	0.565223	0.282612	+	0.959234i	$0.408799\pi$
$840$	0	0
$841$	−24359.5	−0.998789
$842$	−465.934	−0.0190702
$843$	0	0
$844$	−19053.2	−0.777061
$845$	23181.0	0.943730
$846$	0	0
$847$	−4711.24	−0.191122
$848$	−42687.3	−1.72864
$849$	0	0
$850$	−40.5649	−0.00163690
$851$	23729.7	0.955866
$852$	0	0
$853$	−13102.8	−0.525945	−0.262973	−	0.964803i	$-0.584703\pi$
$853$	−13102.8	−0.525945	−0.262973	+	0.964803i	$0.584703\pi$
$854$	−283.162	−0.0113461
$855$	0	0
$856$	2899.62	0.115779
$857$	−22997.1	−0.916648	−0.458324	−	0.888785i	$-0.651550\pi$
$857$	−22997.1	−0.916648	−0.458324	+	0.888785i	$0.651550\pi$
$858$	0	0
$859$	−37450.7	−1.48754	−0.743772	−	0.668433i	$-0.766965\pi$
$859$	−37450.7	−1.48754	−0.743772	+	0.668433i	$0.766965\pi$
$860$	−18991.1	−0.753015
$861$	0	0
$862$	−994.078	−0.0392789
$863$	14201.2	0.560157	0.280078	−	0.959977i	$-0.409640\pi$
$863$	14201.2	0.560157	0.280078	+	0.959977i	$0.409640\pi$
$864$	0	0
$865$	31444.6	1.23601
$866$	−483.612	−0.0189767
$867$	0	0
$868$	−9029.32	−0.353082
$869$	−38664.9	−1.50934
$870$	0	0
$871$	−754.954	−0.0293693
$872$	−2828.08	−0.109829
$873$	0	0
$874$	780.488	0.0302064
$875$	−8391.76	−0.324221
$876$	0	0
$877$	−18220.1	−0.701538	−0.350769	−	0.936462i	$-0.614080\pi$
$877$	−18220.1	−0.701538	−0.350769	+	0.936462i	$0.614080\pi$
$878$	1654.40	0.0635914
$879$	0	0
$880$	32221.2	1.23429
$881$	−47525.5	−1.81745	−0.908725	−	0.417395i	$-0.862943\pi$
$881$	−47525.5	−1.81745	−0.908725	+	0.417395i	$0.862943\pi$
$882$	0	0
$883$	7183.31	0.273768	0.136884	−	0.990587i	$-0.456291\pi$
$883$	7183.31	0.273768	0.136884	+	0.990587i	$0.456291\pi$
$884$	−5859.14	−0.222923
$885$	0	0
$886$	−738.593	−0.0280062
$887$	6802.54	0.257505	0.128753	−	0.991677i	$-0.458903\pi$
$887$	6802.54	0.257505	0.128753	+	0.991677i	$0.458903\pi$
$888$	0	0
$889$	−2794.90	−0.105442
$890$	−1286.85	−0.0484667
$891$	0	0
$892$	−32112.2	−1.20538
$893$	28802.2	1.07931
$894$	0	0
$895$	−3895.29	−0.145481
$896$	1103.19	0.0411329
$897$	0	0
$898$	−772.974	−0.0287244
$899$	1043.00	0.0386943
$900$	0	0
$901$	−55546.4	−2.05385
$902$	2054.66	0.0758457
$903$	0	0
$904$	2318.16	0.0852886
$905$	40171.1	1.47551
$906$	0	0
$907$	−7613.89	−0.278738	−0.139369	−	0.990241i	$-0.544507\pi$
$907$	−7613.89	−0.278738	−0.139369	+	0.990241i	$0.544507\pi$
$908$	34985.8	1.27868
$909$	0	0
$910$	52.1797	0.00190082
$911$	8102.65	0.294679	0.147340	−	0.989086i	$-0.452929\pi$
$911$	8102.65	0.294679	0.147340	+	0.989086i	$0.452929\pi$
$912$	0	0
$913$	17483.4	0.633753
$914$	140.849	0.00509724
$915$	0	0
$916$	4611.66	0.166347
$917$	−10947.4	−0.394235
$918$	0	0
$919$	−53645.0	−1.92556	−0.962778	−	0.270293i	$-0.912879\pi$
$919$	−53645.0	−1.92556	−0.962778	+	0.270293i	$0.912879\pi$
$920$	−1966.98	−0.0704885
$921$	0	0
$922$	1003.94	0.0358602
$923$	−1977.74	−0.0705287
$924$	0	0
$925$	1031.26	0.0366570
$926$	−93.5639	−0.00332041
$927$	0	0
$928$	−95.5916	−0.00338141
$929$	−4954.07	−0.174960	−0.0874799	−	0.996166i	$-0.527881\pi$
$929$	−4954.07	−0.174960	−0.0874799	+	0.996166i	$0.527881\pi$
$930$	0	0
$931$	21404.7	0.753503
$932$	−23314.8	−0.819423
$933$	0	0
$934$	310.911	0.0108922
$935$	41927.5	1.46650
$936$	0	0
$937$	−45370.3	−1.58184	−0.790919	−	0.611920i	$-0.790397\pi$
$937$	−45370.3	−1.58184	−0.790919	+	0.611920i	$0.790397\pi$
$938$	46.1688	0.00160711
$939$	0	0
$940$	−36274.4	−1.25866
$941$	31644.6	1.09626	0.548131	−	0.836392i	$-0.315339\pi$
$941$	31644.6	1.09626	0.548131	+	0.836392i	$0.315339\pi$
$942$	0	0
$943$	59433.6	2.05241
$944$	−9047.56	−0.311942
$945$	0	0
$946$	920.221	0.0316268
$947$	−2810.23	−0.0964311	−0.0482155	−	0.998837i	$-0.515353\pi$
$947$	−2810.23	−0.0964311	−0.0482155	+	0.998837i	$0.515353\pi$
$948$	0	0
$949$	−5896.81	−0.201706
$950$	33.9191	0.00115840
$951$	0	0
$952$	717.002	0.0244098
$953$	−22457.5	−0.763347	−0.381673	−	0.924297i	$-0.624652\pi$
$953$	−22457.5	−0.763347	−0.381673	+	0.924297i	$0.624652\pi$
$954$	0	0
$955$	40309.9	1.36586
$956$	−1909.99	−0.0646165
$957$	0	0
$958$	−181.818	−0.00613181
$959$	−1066.77	−0.0359207
$960$	0	0
$961$	7055.64	0.236838
$962$	−156.929	−0.00525945
$963$	0	0
$964$	27430.2	0.916459
$965$	−3580.03	−0.119425
$966$	0	0
$967$	−31827.2	−1.05842	−0.529211	−	0.848490i	$-0.677512\pi$
$967$	−31827.2	−1.05842	−0.529211	+	0.848490i	$0.677512\pi$
$968$	−1174.48	−0.0389971
$969$	0	0
$970$	−1696.36	−0.0561513
$971$	22057.7	0.729007	0.364504	−	0.931202i	$-0.381239\pi$
$971$	22057.7	0.729007	0.364504	+	0.931202i	$0.381239\pi$
$972$	0	0
$973$	2539.05	0.0836569
$974$	1535.85	0.0505255
$975$	0	0
$976$	33448.5	1.09699
$977$	26811.5	0.877968	0.438984	−	0.898495i	$-0.355338\pi$
$977$	26811.5	0.877968	0.438984	+	0.898495i	$0.355338\pi$
$978$	0	0
$979$	−59186.1	−1.93217
$980$	−26957.8	−0.878709
$981$	0	0
$982$	−57.1672	−0.00185772
$983$	−768.877	−0.0249475	−0.0124737	−	0.999922i	$-0.503971\pi$
$983$	−768.877	−0.0249475	−0.0124737	+	0.999922i	$0.503971\pi$
$984$	0	0
$985$	−41345.7	−1.33745
$986$	−41.3898	−0.00133683
$987$	0	0
$988$	4899.23	0.157758
$989$	26618.5	0.855833
$990$	0	0
$991$	−24379.6	−0.781475	−0.390738	−	0.920502i	$-0.627780\pi$
$991$	−24379.6	−0.781475	−0.390738	+	0.920502i	$0.627780\pi$
$992$	−3377.01	−0.108085
$993$	0	0
$994$	120.947	0.00385937
$995$	31679.5	1.00935
$996$	0	0
$997$	47746.6	1.51670	0.758350	−	0.651848i	$-0.226006\pi$
$997$	47746.6	1.51670	0.758350	+	0.651848i	$0.226006\pi$
$998$	−7.18097	−0.000227765 0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.e.1.16		32
3.2	odd	2		717.4.a.c.1.17	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
717.4.a.c.1.17	✓	32	3.2	odd	2
2151.4.a.e.1.16		32	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2151 = 3^{2} \cdot 239 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2151.a (trivial)

\(f(q)\)	\(=\)	\(q+0.0917574 q^{2} -7.99158 q^{4} -10.9396 q^{5} -5.88604 q^{7} -1.46735 q^{8} +O(q^{10})\)	\(q+0.0917574 q^{2} -7.99158 q^{4} -10.9396 q^{5} -5.88604 q^{7} -1.46735 q^{8} -1.00379 q^{10} -46.1672 q^{11} +8.83154 q^{13} -0.540088 q^{14} +63.7980 q^{16} +83.0165 q^{17} -69.4159 q^{19} +87.4246 q^{20} -4.23618 q^{22} -122.537 q^{23} -5.32530 q^{25} +0.810359 q^{26} +47.0388 q^{28} -5.43359 q^{29} -191.955 q^{31} +17.5927 q^{32} +7.61738 q^{34} +64.3909 q^{35} -193.654 q^{37} -6.36943 q^{38} +16.0522 q^{40} -485.027 q^{41} -217.229 q^{43} +368.949 q^{44} -11.2436 q^{46} -414.922 q^{47} -308.355 q^{49} -0.488636 q^{50} -70.5780 q^{52} -669.101 q^{53} +505.050 q^{55} +8.63686 q^{56} -0.498572 q^{58} -141.816 q^{59} +524.288 q^{61} -17.6133 q^{62} -508.770 q^{64} -96.6134 q^{65} -85.4839 q^{67} -663.433 q^{68} +5.90834 q^{70} -223.940 q^{71} -667.699 q^{73} -17.7692 q^{74} +554.743 q^{76} +271.742 q^{77} +837.497 q^{79} -697.924 q^{80} -44.5049 q^{82} -378.698 q^{83} -908.167 q^{85} -19.9324 q^{86} +67.7433 q^{88} +1282.00 q^{89} -51.9828 q^{91} +979.261 q^{92} -38.0721 q^{94} +759.382 q^{95} +1689.96 q^{97} -28.2938 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 3 q^{2} + 151 q^{4} + 14 q^{5} + 72 q^{7} - 57 q^{8}+O(q^{10}) \) 32 * q - 3 * q^2 + 151 * q^4 + 14 * q^5 + 72 * q^7 - 57 * q^8	\( 32 q - 3 q^{2} + 151 q^{4} + 14 q^{5} + 72 q^{7} - 57 q^{8} + 32 q^{10} - 154 q^{11} + 100 q^{13} - 42 q^{14} + 719 q^{16} - 32 q^{17} + 202 q^{19} + 132 q^{20} + 265 q^{22} - 552 q^{23} + 1086 q^{25} + 280 q^{26} + 390 q^{28} + 154 q^{29} + 560 q^{31} - 444 q^{32} + 156 q^{34} - 394 q^{35} + 914 q^{37} - 111 q^{38} + 257 q^{40} + 914 q^{41} + 1722 q^{43} - 1243 q^{44} + 584 q^{46} - 380 q^{47} + 2446 q^{49} + 454 q^{50} + 1552 q^{52} - 370 q^{53} + 918 q^{55} + 499 q^{56} + 2446 q^{58} - 492 q^{59} + 668 q^{61} - 578 q^{62} + 6475 q^{64} - 736 q^{65} + 4548 q^{67} - 5253 q^{68} + 7793 q^{70} - 258 q^{71} + 3096 q^{73} - 449 q^{74} + 6814 q^{76} - 3804 q^{77} + 2864 q^{79} + 1052 q^{80} + 14145 q^{82} - 2364 q^{83} + 3088 q^{85} - 2811 q^{86} + 8329 q^{88} + 4172 q^{89} + 7350 q^{91} - 13644 q^{92} + 6122 q^{94} - 3336 q^{95} + 6370 q^{97} - 1572 q^{98}+O(q^{100}) \) 32 * q - 3 * q^2 + 151 * q^4 + 14 * q^5 + 72 * q^7 - 57 * q^8 + 32 * q^10 - 154 * q^11 + 100 * q^13 - 42 * q^14 + 719 * q^16 - 32 * q^17 + 202 * q^19 + 132 * q^20 + 265 * q^22 - 552 * q^23 + 1086 * q^25 + 280 * q^26 + 390 * q^28 + 154 * q^29 + 560 * q^31 - 444 * q^32 + 156 * q^34 - 394 * q^35 + 914 * q^37 - 111 * q^38 + 257 * q^40 + 914 * q^41 + 1722 * q^43 - 1243 * q^44 + 584 * q^46 - 380 * q^47 + 2446 * q^49 + 454 * q^50 + 1552 * q^52 - 370 * q^53 + 918 * q^55 + 499 * q^56 + 2446 * q^58 - 492 * q^59 + 668 * q^61 - 578 * q^62 + 6475 * q^64 - 736 * q^65 + 4548 * q^67 - 5253 * q^68 + 7793 * q^70 - 258 * q^71 + 3096 * q^73 - 449 * q^74 + 6814 * q^76 - 3804 * q^77 + 2864 * q^79 + 1052 * q^80 + 14145 * q^82 - 2364 * q^83 + 3088 * q^85 - 2811 * q^86 + 8329 * q^88 + 4172 * q^89 + 7350 * q^91 - 13644 * q^92 + 6122 * q^94 - 3336 * q^95 + 6370 * q^97 - 1572 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more