LMFDB - Embedded newform 289.4.b.f.288.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [289,4,Mod(288,289)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(289, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("289.288");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 289 = 17^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	289.b (of order $2$, degree $1$, not 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:	$17.0515519917$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			288.19
Character	$\chi$	$=$	289.288
Dual form			289.4.b.f.288.20

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.73276 q^{2} -3.65511i q^{3} +5.93351 q^{4} -14.5154i q^{5} -13.6437i q^{6} -12.9694i q^{7} -7.71372 q^{8} +13.6402 q^{9} +O(q^{10})$	\(q+3.73276 q^{2} -3.65511i q^{3} +5.93351 q^{4} -14.5154i q^{5} -13.6437i q^{6} -12.9694i q^{7} -7.71372 q^{8} +13.6402 q^{9} -54.1823i q^{10} +20.2490i q^{11} -21.6876i q^{12} -90.7856 q^{13} -48.4118i q^{14} -53.0552 q^{15} -76.2615 q^{16} +50.9155 q^{18} +127.359 q^{19} -86.1270i q^{20} -47.4047 q^{21} +75.5848i q^{22} +69.5078i q^{23} +28.1945i q^{24} -85.6954 q^{25} -338.881 q^{26} -148.544i q^{27} -76.9542i q^{28} -43.9568i q^{29} -198.042 q^{30} -218.817i q^{31} -222.956 q^{32} +74.0124 q^{33} -188.256 q^{35} +80.9340 q^{36} -41.5125i q^{37} +475.399 q^{38} +331.832i q^{39} +111.967i q^{40} -440.141i q^{41} -176.951 q^{42} +310.790 q^{43} +120.148i q^{44} -197.992i q^{45} +259.456i q^{46} +84.3763 q^{47} +278.744i q^{48} +174.794 q^{49} -319.880 q^{50} -538.677 q^{52} -47.6186 q^{53} -554.480i q^{54} +293.922 q^{55} +100.043i q^{56} -465.510i q^{57} -164.080i q^{58} +1.73428 q^{59} -314.804 q^{60} -159.400i q^{61} -816.792i q^{62} -176.905i q^{63} -222.151 q^{64} +1317.78i q^{65} +276.271 q^{66} +141.756 q^{67} +254.059 q^{69} -702.714 q^{70} +447.302i q^{71} -105.216 q^{72} +757.081i q^{73} -154.956i q^{74} +313.226i q^{75} +755.684 q^{76} +262.618 q^{77} +1238.65i q^{78} -529.801i q^{79} +1106.96i q^{80} -174.662 q^{81} -1642.94i q^{82} +762.097 q^{83} -281.276 q^{84} +1160.11 q^{86} -160.667 q^{87} -156.195i q^{88} -397.434 q^{89} -739.056i q^{90} +1177.44i q^{91} +412.425i q^{92} -799.801 q^{93} +314.957 q^{94} -1848.66i q^{95} +814.930i q^{96} -427.919i q^{97} +652.464 q^{98} +276.200i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 96 q^{4} + 102 q^{8} - 216 q^{9}+O(q^{10}) $ 24 * q + 96 * q^4 + 102 * q^8 - 216 * q^9	$ 24 q + 96 q^{4} + 102 q^{8} - 216 q^{9} - 144 q^{13} + 276 q^{15} + 384 q^{16} + 378 q^{18} + 132 q^{19} + 492 q^{21} - 888 q^{25} - 1056 q^{26} - 3780 q^{30} + 2706 q^{32} + 1932 q^{33} - 132 q^{35} + 1326 q^{36} - 120 q^{38} - 1608 q^{42} - 972 q^{43} - 1776 q^{47} + 1140 q^{49} + 870 q^{50} + 450 q^{52} - 2052 q^{53} + 1944 q^{55} + 1584 q^{59} - 2916 q^{60} - 3714 q^{64} + 1188 q^{66} + 1248 q^{67} - 3012 q^{69} + 3300 q^{70} + 2910 q^{72} - 1350 q^{76} + 2016 q^{77} + 5040 q^{81} - 1344 q^{83} - 1554 q^{84} + 5556 q^{86} - 1452 q^{87} - 1812 q^{89} - 264 q^{93} - 1470 q^{94} + 3822 q^{98}+O(q^{100}) $ 24 * q + 96 * q^4 + 102 * q^8 - 216 * q^9 - 144 * q^13 + 276 * q^15 + 384 * q^16 + 378 * q^18 + 132 * q^19 + 492 * q^21 - 888 * q^25 - 1056 * q^26 - 3780 * q^30 + 2706 * q^32 + 1932 * q^33 - 132 * q^35 + 1326 * q^36 - 120 * q^38 - 1608 * q^42 - 972 * q^43 - 1776 * q^47 + 1140 * q^49 + 870 * q^50 + 450 * q^52 - 2052 * q^53 + 1944 * q^55 + 1584 * q^59 - 2916 * q^60 - 3714 * q^64 + 1188 * q^66 + 1248 * q^67 - 3012 * q^69 + 3300 * q^70 + 2910 * q^72 - 1350 * q^76 + 2016 * q^77 + 5040 * q^81 - 1344 * q^83 - 1554 * q^84 + 5556 * q^86 - 1452 * q^87 - 1812 * q^89 - 264 * q^93 - 1470 * q^94 + 3822 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/289\mathbb{Z}\right)^\times$.

$n$	$3$
$\chi(n)$	$-1$

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 $

$2$	3.73276	1.31973	0.659865	−	0.751384i	$-0.270613\pi$
$2$	3.73276	1.31973	0.659865	+	0.751384i	$0.270613\pi$
$3$	− 3.65511i	− 0.703426i	−0.936108	−	0.351713i	$-0.885599\pi$
$3$	− 3.65511i	− 0.703426i	0.936108	−	0.351713i	$-0.114401\pi$
$4$	5.93351	0.741689
$5$	− 14.5154i	− 1.29829i	−0.760664	−	0.649146i	$-0.775126\pi$
$5$	− 14.5154i	− 1.29829i	0.760664	−	0.649146i	$-0.224874\pi$
$6$	− 13.6437i	− 0.928333i
$7$	− 12.9694i	− 0.700284i	−0.936697	−	0.350142i	$-0.886133\pi$
$7$	− 12.9694i	− 0.700284i	0.936697	−	0.350142i	$-0.113867\pi$
$8$	−7.71372	−0.340901
$9$	13.6402	0.505191
$10$	− 54.1823i	− 1.71340i
$11$	20.2490i	0.555028i	0.960722	+	0.277514i	$0.0895105\pi$
$11$	20.2490i	0.555028i	−0.960722	+	0.277514i	$0.910489\pi$
$12$	− 21.6876i	− 0.521723i
$13$	−90.7856	−1.93688	−0.968438	−	0.249253i	$-0.919815\pi$
$13$	−90.7856	−1.93688	−0.968438	+	0.249253i	$0.919815\pi$
$14$	− 48.4118i	− 0.924186i
$15$	−53.0552	−0.913253
$16$	−76.2615	−1.19159
$17$	0	0
$17$	0	0
$18$	50.9155	0.666716
$19$	127.359	1.53779	0.768897	−	0.639373i	$-0.220806\pi$
$19$	127.359	1.53779	0.768897	+	0.639373i	$0.220806\pi$
$20$	− 86.1270i	− 0.962929i
$21$	−47.4047	−0.492598
$22$	75.5848i	0.732488i
$23$	69.5078i	0.630147i	0.949067	+	0.315073i	$0.102029\pi$
$23$	69.5078i	0.630147i	−0.949067	+	0.315073i	$0.897971\pi$
$24$	28.1945i	0.239799i
$25$	−85.6954	−0.685563
$26$	−338.881	−2.55616
$27$	− 148.544i	− 1.05879i
$28$	− 76.9542i	− 0.519392i
$29$	− 43.9568i	− 0.281468i	−0.990047	−	0.140734i	$-0.955054\pi$
$29$	− 43.9568i	− 0.281468i	0.990047	−	0.140734i	$-0.0449462\pi$
$30$	−198.042	−1.20525
$31$	− 218.817i	− 1.26776i	−0.773430	−	0.633882i	$-0.781460\pi$
$31$	− 218.817i	− 1.26776i	0.773430	−	0.633882i	$-0.218540\pi$
$32$	−222.956	−1.23167
$33$	74.0124	0.390422
$34$	0	0
$35$	−188.256	−0.909173
$36$	80.9340	0.374695
$37$	− 41.5125i	− 0.184449i	−0.995738	−	0.0922245i	$-0.970602\pi$
$37$	− 41.5125i	− 0.184449i	0.995738	−	0.0922245i	$-0.0293977\pi$
$38$	475.399	2.02947
$39$	331.832i	1.36245i
$40$	111.967i	0.442590i
$41$	− 440.141i	− 1.67655i	−0.545249	−	0.838274i	$-0.683565\pi$
$41$	− 440.141i	− 1.67655i	0.545249	−	0.838274i	$-0.316435\pi$
$42$	−176.951	−0.650097
$43$	310.790	1.10221	0.551106	−	0.834435i	$-0.314206\pi$
$43$	310.790	1.10221	0.551106	+	0.834435i	$0.314206\pi$
$44$	120.148i	0.411658i
$45$	− 197.992i	− 0.655886i
$46$	259.456i	0.831624i
$47$	84.3763	0.261863	0.130931	−	0.991391i	$-0.458203\pi$
$47$	84.3763	0.261863	0.130931	+	0.991391i	$0.458203\pi$
$48$	278.744i	0.838194i
$49$	174.794	0.509603
$50$	−319.880	−0.904759
$51$	0	0
$52$	−538.677	−1.43656
$53$	−47.6186	−0.123414	−0.0617068	−	0.998094i	$-0.519654\pi$
$53$	−47.6186	−0.123414	−0.0617068	+	0.998094i	$0.519654\pi$
$54$	− 554.480i	− 1.39732i
$55$	293.922	0.720589
$56$	100.043i	0.238728i
$57$	− 465.510i	− 1.08172i
$58$	− 164.080i	− 0.371462i
$59$	1.73428	0.00382685	0.00191342	−	0.999998i	$-0.499391\pi$
$59$	1.73428	0.00382685	0.00191342	+	0.999998i	$0.499391\pi$
$60$	−314.804	−0.677350
$61$	− 159.400i	− 0.334575i	−0.985908	−	0.167287i	$-0.946499\pi$
$61$	− 159.400i	− 0.334575i	0.985908	−	0.167287i	$-0.0535007\pi$
$62$	− 816.792i	− 1.67311i
$63$	− 176.905i	− 0.353777i
$64$	−222.151	−0.433888
$65$	1317.78i	2.51463i
$66$	276.271	0.515251
$67$	141.756	0.258482	0.129241	−	0.991613i	$-0.458746\pi$
$67$	141.756	0.258482	0.129241	+	0.991613i	$0.458746\pi$
$68$	0	0
$69$	254.059	0.443262
$70$	−702.714	−1.19986
$71$	447.302i	0.747675i	0.927494	+	0.373838i	$0.121958\pi$
$71$	447.302i	0.747675i	−0.927494	+	0.373838i	$0.878042\pi$
$72$	−105.216	−0.172220
$73$	757.081i	1.21383i	0.794766	+	0.606916i	$0.207593\pi$
$73$	757.081i	1.21383i	−0.794766	+	0.606916i	$0.792407\pi$
$74$	− 154.956i	− 0.243423i
$75$	313.226i	0.482243i
$76$	755.684	1.14056
$77$	262.618	0.388677
$78$	1238.65i	1.79807i
$79$	− 529.801i	− 0.754523i	−0.926107	−	0.377261i	$-0.876866\pi$
$79$	− 529.801i	− 0.754523i	0.926107	−	0.377261i	$-0.123134\pi$
$80$	1106.96i	1.54703i
$81$	−174.662	−0.239591
$82$	− 1642.94i	− 2.21259i
$83$	762.097	1.00784	0.503922	−	0.863749i	$-0.331890\pi$
$83$	762.097	1.00784	0.503922	+	0.863749i	$0.331890\pi$
$84$	−281.276	−0.365354
$85$	0	0
$86$	1160.11	1.45462
$87$	−160.667	−0.197992
$88$	− 156.195i	− 0.189210i
$89$	−397.434	−0.473348	−0.236674	−	0.971589i	$-0.576057\pi$
$89$	−397.434	−0.473348	−0.236674	+	0.971589i	$0.576057\pi$
$90$	− 739.056i	− 0.865593i
$91$	1177.44i	1.35636i
$92$	412.425i	0.467373i
$93$	−799.801	−0.891779
$94$	314.957	0.345588
$95$	− 1848.66i	− 1.99651i
$96$	814.930i	0.866391i
$97$	− 427.919i	− 0.447924i	−0.974598	−	0.223962i	$-0.928101\pi$
$97$	− 427.919i	− 0.447924i	0.974598	−	0.223962i	$-0.0718991\pi$
$98$	652.464	0.672538
$99$	276.200i	0.280395i
$100$	−508.474	−0.508474
$101$	1591.35	1.56778	0.783889	−	0.620901i	$-0.213233\pi$
$101$	1591.35	1.56778	0.783889	+	0.620901i	$0.213233\pi$
$102$	0	0
$103$	−1094.59	−1.04712	−0.523558	−	0.851990i	$-0.675396\pi$
$103$	−1094.59	−1.04712	−0.523558	+	0.851990i	$0.675396\pi$
$104$	700.295	0.660284
$105$	688.096i	0.639536i
$106$	−177.749	−0.162873
$107$	− 1321.45i	− 1.19392i	−0.802271	−	0.596960i	$-0.796375\pi$
$107$	− 1321.45i	− 1.19392i	0.802271	−	0.596960i	$-0.203625\pi$
$108$	− 881.389i	− 0.785293i
$109$	1324.25i	1.16367i	0.813307	+	0.581835i	$0.197665\pi$
$109$	1324.25i	1.16367i	−0.813307	+	0.581835i	$0.802335\pi$
$110$	1097.14	0.950983
$111$	−151.733	−0.129746
$112$	989.069i	0.834449i
$113$	1621.26i	1.34969i	0.737958	+	0.674847i	$0.235790\pi$
$113$	1621.26i	1.34969i	−0.737958	+	0.674847i	$0.764210\pi$
$114$	− 1737.64i	− 1.42758i
$115$	1008.93	0.818115
$116$	− 260.818i	− 0.208762i
$117$	−1238.33	−0.978493
$118$	6.47365	0.00505041
$119$	0	0
$120$	409.253	0.311329
$121$	920.977	0.691944
$122$	− 595.001i	− 0.441548i
$123$	−1608.76	−1.17933
$124$	− 1298.35i	− 0.940287i
$125$	− 570.520i	− 0.408231i
$126$	− 660.345i	− 0.466890i
$127$	−423.287	−0.295753	−0.147876	−	0.989006i	$-0.547244\pi$
$127$	−423.287	−0.295753	−0.147876	+	0.989006i	$0.547244\pi$
$128$	954.415	0.659056
$129$	− 1135.97i	− 0.775325i
$130$	4918.98i	3.31864i
$131$	− 792.977i	− 0.528876i	−0.964403	−	0.264438i	$-0.914814\pi$
$131$	− 792.977i	− 0.528876i	0.964403	−	0.264438i	$-0.0851865\pi$
$132$	439.153	0.289571
$133$	− 1651.77i	− 1.07689i
$134$	529.143	0.341127
$135$	−2156.17	−1.37462
$136$	0	0
$137$	809.512	0.504827	0.252414	−	0.967619i	$-0.418776\pi$
$137$	809.512	0.504827	0.252414	+	0.967619i	$0.418776\pi$
$138$	948.341	0.584986
$139$	2842.04i	1.73424i	0.498102	+	0.867119i	$0.334031\pi$
$139$	2842.04i	1.73424i	−0.498102	+	0.867119i	$0.665969\pi$
$140$	−1117.02	−0.674323
$141$	− 308.405i	− 0.184201i
$142$	1669.67i	0.986730i
$143$	− 1838.32i	− 1.07502i
$144$	−1040.22	−0.601979
$145$	−638.048	−0.365428
$146$	2826.00i	1.60193i
$147$	− 638.891i	− 0.358468i
$148$	− 246.315i	− 0.136804i
$149$	−1010.84	−0.555782	−0.277891	−	0.960613i	$-0.589635\pi$
$149$	−1010.84	−0.555782	−0.277891	+	0.960613i	$0.589635\pi$
$150$	1169.20i	0.636431i
$151$	−1340.29	−0.722327	−0.361163	−	0.932503i	$-0.617620\pi$
$151$	−1340.29	−0.722327	−0.361163	+	0.932503i	$0.617620\pi$
$152$	−982.409	−0.524236
$153$	0	0
$154$	980.292	0.512949
$155$	−3176.21	−1.64593
$156$	1968.93i	1.01051i
$157$	−947.372	−0.481583	−0.240792	−	0.970577i	$-0.577407\pi$
$157$	−947.372	−0.481583	−0.240792	+	0.970577i	$0.577407\pi$
$158$	− 1977.62i	− 0.995766i
$159$	174.051i	0.0868123i
$160$	3236.29i	1.59907i
$161$	901.477	0.441282
$162$	−651.970	−0.316195
$163$	− 966.144i	− 0.464259i	−0.972685	−	0.232129i	$-0.925431\pi$
$163$	− 966.144i	− 0.464259i	0.972685	−	0.232129i	$-0.0745693\pi$
$164$	− 2611.58i	− 1.24348i
$165$	− 1074.32i	− 0.506881i
$166$	2844.73	1.33008
$167$	1193.13i	0.552857i	0.961034	+	0.276429i	$0.0891510\pi$
$167$	1193.13i	0.552857i	−0.961034	+	0.276429i	$0.910849\pi$
$168$	365.667	0.167927
$169$	6045.03	2.75149
$170$	0	0
$171$	1737.19	0.776880
$172$	1844.08	0.817498
$173$	− 8.50924i	− 0.00373957i	−0.999998	−	0.00186978i	$-0.999405\pi$
$173$	− 8.50924i	− 0.00373957i	0.999998	−	0.00186978i	$-0.000595171\pi$
$174$	−599.731	−0.261296
$175$	1111.42i	0.480089i
$176$	− 1544.22i	− 0.661364i
$177$	− 6.33899i	− 0.00269191i
$178$	−1483.53	−0.624691
$179$	909.300	0.379689	0.189844	−	0.981814i	$-0.439202\pi$
$179$	909.300	0.379689	0.189844	+	0.981814i	$0.439202\pi$
$180$	− 1174.79i	− 0.486463i
$181$	− 2448.71i	− 1.00559i	−0.864407	−	0.502793i	$-0.832306\pi$
$181$	− 2448.71i	− 1.00559i	0.864407	−	0.502793i	$-0.167694\pi$
$182$	4395.10i	1.79003i
$183$	−582.624	−0.235349
$184$	− 536.164i	− 0.214818i
$185$	−602.568	−0.239469
$186$	−2985.47	−1.17691
$187$	0	0
$188$	500.648	0.194221
$189$	−1926.54	−0.741454
$190$	− 6900.59i	− 2.63485i
$191$	845.418	0.320274	0.160137	−	0.987095i	$-0.448806\pi$
$191$	845.418	0.320274	0.160137	+	0.987095i	$0.448806\pi$
$192$	811.986i	0.305208i
$193$	7.70682i	0.00287435i	0.999999	+	0.00143717i	$0.000457467\pi$
$193$	7.70682i	0.00287435i	−0.999999	+	0.00143717i	$0.999543\pi$
$194$	− 1597.32i	− 0.591139i
$195$	4816.65	1.76886
$196$	1037.14	0.377967
$197$	5024.07i	1.81701i	0.417879	+	0.908503i	$0.362773\pi$
$197$	5024.07i	1.81701i	−0.417879	+	0.908503i	$0.637227\pi$
$198$	1030.99i	0.370046i
$199$	1210.87i	0.431339i	0.976466	+	0.215669i	$0.0691934\pi$
$199$	1210.87i	0.431339i	−0.976466	+	0.215669i	$0.930807\pi$
$200$	661.030	0.233709
$201$	− 518.135i	− 0.181823i
$202$	5940.15	2.06905
$203$	−570.095	−0.197107
$204$	0	0
$205$	−6388.80	−2.17665
$206$	−4085.84	−1.38191
$207$	948.098i	0.318345i
$208$	6923.45	2.30796
$209$	2578.89i	0.853519i
$210$	2568.50i	0.844016i
$211$	3047.09i	0.994172i	0.867701	+	0.497086i	$0.165597\pi$
$211$	3047.09i	0.994172i	−0.867701	+	0.497086i	$0.834403\pi$
$212$	−282.545	−0.0915344
$213$	1634.94	0.525934
$214$	− 4932.66i	− 1.57565i
$215$	− 4511.23i	− 1.43099i
$216$	1145.83i	0.360943i
$217$	−2837.93	−0.887795
$218$	4943.10i	1.53573i
$219$	2767.22	0.853841
$220$	1743.99	0.534453
$221$	0	0
$222$	−566.382	−0.171230
$223$	2883.35	0.865846	0.432923	−	0.901431i	$-0.357482\pi$
$223$	2883.35	0.865846	0.432923	+	0.901431i	$0.357482\pi$
$224$	2891.62i	0.862520i
$225$	−1168.90	−0.346340
$226$	6051.78i	1.78123i
$227$	− 5402.18i	− 1.57954i	−0.613403	−	0.789770i	$-0.710200\pi$
$227$	− 5402.18i	− 1.57954i	0.613403	−	0.789770i	$-0.289800\pi$
$228$	− 2762.11i	− 0.802303i
$229$	1524.25	0.439849	0.219925	−	0.975517i	$-0.429419\pi$
$229$	1524.25	0.439849	0.219925	+	0.975517i	$0.429419\pi$
$230$	3766.10	1.07969
$231$	− 959.899i	− 0.273406i
$232$	339.070i	0.0959528i
$233$	− 1685.79i	− 0.473989i	−0.971511	−	0.236995i	$-0.923838\pi$
$233$	− 1685.79i	− 0.473989i	0.971511	−	0.236995i	$-0.0761624\pi$
$234$	−4622.39	−1.29135
$235$	− 1224.75i	− 0.339975i
$236$	10.2904	0.00283833
$237$	−1936.48	−0.530751
$238$	0	0
$239$	−4887.91	−1.32290	−0.661449	−	0.749990i	$-0.730058\pi$
$239$	−4887.91	−1.32290	−0.661449	+	0.749990i	$0.730058\pi$
$240$	4046.07	1.08822
$241$	− 3416.56i	− 0.913195i	−0.889673	−	0.456597i	$-0.849068\pi$
$241$	− 3416.56i	− 0.913195i	0.889673	−	0.456597i	$-0.150932\pi$
$242$	3437.79	0.913179
$243$	− 3372.29i	− 0.890257i
$244$	− 945.800i	− 0.248150i
$245$	− 2537.19i	− 0.661614i
$246$	−6005.13	−1.55640
$247$	−11562.3	−2.97852
$248$	1687.89i	0.432183i
$249$	− 2785.55i	− 0.708944i
$250$	− 2129.62i	− 0.538755i
$251$	2357.42	0.592825	0.296413	−	0.955060i	$-0.404210\pi$
$251$	2357.42	0.592825	0.296413	+	0.955060i	$0.404210\pi$
$252$	− 1049.67i	− 0.262392i
$253$	−1407.47	−0.349749
$254$	−1580.03	−0.390314
$255$	0	0
$256$	5339.81	1.30366
$257$	−2033.64	−0.493598	−0.246799	−	0.969067i	$-0.579379\pi$
$257$	−2033.64	−0.493598	−0.246799	+	0.969067i	$0.579379\pi$
$258$	− 4240.32i	− 1.02322i
$259$	−538.393	−0.129167
$260$	7819.09i	1.86507i
$261$	− 599.578i	− 0.142195i
$262$	− 2960.00i	− 0.697974i
$263$	−5377.07	−1.26070	−0.630351	−	0.776310i	$-0.717089\pi$
$263$	−5377.07	−1.26070	−0.630351	+	0.776310i	$0.717089\pi$
$264$	−570.911	−0.133095
$265$	691.200i	0.160227i
$266$	− 6165.66i	− 1.42121i
$267$	1452.67i	0.332965i
$268$	841.113	0.191713
$269$	3012.15i	0.682728i	0.939931	+	0.341364i	$0.110889\pi$
$269$	3012.15i	0.682728i	−0.939931	+	0.341364i	$0.889111\pi$
$270$	−8048.48	−1.81413
$271$	−1340.23	−0.300417	−0.150209	−	0.988654i	$-0.547995\pi$
$271$	−1340.23	−0.300417	−0.150209	+	0.988654i	$0.547995\pi$
$272$	0	0
$273$	4303.67	0.954102
$274$	3021.72	0.666236
$275$	− 1735.25i	− 0.380507i
$276$	1507.46	0.328762
$277$	− 4917.83i	− 1.06673i	−0.845885	−	0.533365i	$-0.820927\pi$
$277$	− 4917.83i	− 1.06673i	0.845885	−	0.533365i	$-0.179073\pi$
$278$	10608.7i	2.28873i
$279$	− 2984.70i	− 0.640463i
$280$	1452.15	0.309938
$281$	697.896	0.148160	0.0740801	−	0.997252i	$-0.476398\pi$
$281$	697.896	0.148160	0.0740801	+	0.997252i	$0.476398\pi$
$282$	− 1151.20i	− 0.243096i
$283$	5379.29i	1.12991i	0.825121	+	0.564957i	$0.191107\pi$
$283$	5379.29i	1.12991i	−0.825121	+	0.564957i	$0.808893\pi$
$284$	2654.07i	0.554542i
$285$	−6757.04	−1.40439
$286$	− 6862.01i	− 1.41874i
$287$	−5708.38	−1.17406
$288$	−3041.16	−0.622230
$289$	0	0
$290$	−2381.68	−0.482266
$291$	−1564.09	−0.315081
$292$	4492.15i	0.900285i
$293$	2876.89	0.573616	0.286808	−	0.957988i	$-0.407406\pi$
$293$	2876.89	0.573616	0.286808	+	0.957988i	$0.407406\pi$
$294$	− 2384.83i	− 0.473081i
$295$	− 25.1737i	− 0.00496837i
$296$	320.216i	0.0628789i
$297$	3007.88	0.587659
$298$	−3773.24	−0.733482
$299$	− 6310.31i	− 1.22052i
$300$	1858.53i	0.357674i
$301$	− 4030.78i	− 0.771861i
$302$	−5002.99	−0.953276
$303$	− 5816.58i	− 1.10282i
$304$	−9712.57	−1.83241
$305$	−2313.74	−0.434376
$306$	0	0
$307$	−3765.61	−0.700048	−0.350024	−	0.936741i	$-0.613827\pi$
$307$	−3765.61	−0.700048	−0.350024	+	0.936741i	$0.613827\pi$
$308$	1558.25	0.288277
$309$	4000.84i	0.736570i
$310$	−11856.0	−2.17218
$311$	1043.69i	0.190297i	0.995463	+	0.0951483i	$0.0303325\pi$
$311$	1043.69i	0.190297i	−0.995463	+	0.0951483i	$0.969667\pi$
$312$	− 2559.66i	− 0.464461i
$313$	− 801.837i	− 0.144800i	−0.997376	−	0.0724002i	$-0.976934\pi$
$313$	− 801.837i	− 0.144800i	0.997376	−	0.0724002i	$-0.0230659\pi$
$314$	−3536.31	−0.635560
$315$	−2567.84	−0.459306
$316$	− 3143.58i	− 0.559621i
$317$	806.122i	0.142827i	0.997447	+	0.0714137i	$0.0227511\pi$
$317$	806.122i	0.142827i	−0.997447	+	0.0714137i	$0.977249\pi$
$318$	649.692i	0.114569i
$319$	890.082	0.156223
$320$	3224.60i	0.563314i
$321$	−4830.05	−0.839836
$322$	3365.00	0.582373
$323$	0	0
$324$	−1036.36	−0.177702
$325$	7779.91	1.32785
$326$	− 3606.38i	− 0.612697i
$327$	4840.27	0.818556
$328$	3395.12i	0.571538i
$329$	− 1094.31i	− 0.183378i
$330$	− 4010.17i	− 0.668947i
$331$	−2607.01	−0.432913	−0.216457	−	0.976292i	$-0.569450\pi$
$331$	−2607.01	−0.432913	−0.216457	+	0.976292i	$0.569450\pi$
$332$	4521.91	0.747506
$333$	− 566.237i	− 0.0931820i
$334$	4453.67i	0.729623i
$335$	− 2057.64i	− 0.335585i
$336$	3615.16	0.586973
$337$	− 1067.31i	− 0.172523i	−0.996273	−	0.0862614i	$-0.972508\pi$
$337$	− 1067.31i	− 0.172523i	0.996273	−	0.0862614i	$-0.0274920\pi$
$338$	22564.6	3.63123
$339$	5925.89	0.949410
$340$	0	0
$341$	4430.83	0.703645
$342$	6484.53	1.02527
$343$	− 6715.49i	− 1.05715i
$344$	−2397.35	−0.375745
$345$	− 3687.75i	− 0.575484i
$346$	− 31.7630i	− 0.00493522i
$347$	11380.2i	1.76058i	0.474439	+	0.880288i	$0.342651\pi$
$347$	11380.2i	1.76058i	−0.474439	+	0.880288i	$0.657349\pi$
$348$	−953.319	−0.146848
$349$	2447.03	0.375320	0.187660	−	0.982234i	$-0.439910\pi$
$349$	2447.03	0.375320	0.187660	+	0.982234i	$0.439910\pi$
$350$	4148.67i	0.633588i
$351$	13485.7i	2.05075i
$352$	− 4514.65i	− 0.683613i
$353$	5229.05	0.788426	0.394213	−	0.919019i	$-0.371017\pi$
$353$	5229.05	0.788426	0.394213	+	0.919019i	$0.371017\pi$
$354$	− 23.6619i	− 0.00355259i
$355$	6492.74	0.970701
$356$	−2358.18	−0.351077
$357$	0	0
$358$	3394.20	0.501087
$359$	−9108.24	−1.33904	−0.669519	−	0.742795i	$-0.733500\pi$
$359$	−9108.24	−1.33904	−0.669519	+	0.742795i	$0.733500\pi$
$360$	1527.25i	0.223592i
$361$	9361.22	1.36481
$362$	− 9140.44i	− 1.32710i
$363$	− 3366.27i	− 0.486731i
$364$	6986.34i	1.00600i
$365$	10989.3	1.57591
$366$	−2174.80	−0.310597
$367$	1155.57i	0.164360i	0.996617	+	0.0821801i	$0.0261883\pi$
$367$	1155.57i	0.164360i	−0.996617	+	0.0821801i	$0.973812\pi$
$368$	− 5300.77i	− 0.750875i
$369$	− 6003.59i	− 0.846977i
$370$	−2249.24	−0.316034
$371$	617.586i	0.0864245i
$372$	−4745.62	−0.661422
$373$	−9949.97	−1.38121	−0.690603	−	0.723234i	$-0.742655\pi$
$373$	−9949.97	−1.38121	−0.690603	+	0.723234i	$0.742655\pi$
$374$	0	0
$375$	−2085.31	−0.287160
$376$	−650.855	−0.0892694
$377$	3990.64i	0.545169i
$378$	−7191.30	−0.978520
$379$	1443.78i	0.195678i	0.995202	+	0.0978388i	$0.0311930\pi$
$379$	1443.78i	0.195678i	−0.995202	+	0.0978388i	$0.968807\pi$
$380$	− 10969.0i	− 1.48079i
$381$	1547.16i	0.208040i
$382$	3155.74	0.422675
$383$	8249.59	1.10061	0.550306	−	0.834963i	$-0.314511\pi$
$383$	8249.59	1.10061	0.550306	+	0.834963i	$0.314511\pi$
$384$	− 3488.49i	− 0.463598i
$385$	− 3812.00i	− 0.504617i
$386$	28.7677i	0.00379336i
$387$	4239.23	0.556827
$388$	− 2539.06i	− 0.332220i
$389$	9396.96	1.22479	0.612397	−	0.790551i	$-0.290206\pi$
$389$	9396.96	1.22479	0.612397	+	0.790551i	$0.290206\pi$
$390$	17979.4	2.33442
$391$	0	0
$392$	−1348.31	−0.173724
$393$	−2898.42	−0.372025
$394$	18753.6i	2.39796i
$395$	−7690.25	−0.979591
$396$	1638.83i	0.207966i
$397$	11970.7i	1.51333i	0.653800	+	0.756667i	$0.273174\pi$
$397$	11970.7i	1.51333i	−0.653800	+	0.756667i	$0.726826\pi$
$398$	4519.90i	0.569251i
$399$	−6037.40	−0.757514
$400$	6535.26	0.816908
$401$	10710.9i	1.33386i	0.745122	+	0.666928i	$0.232391\pi$
$401$	10710.9i	1.33386i	−0.745122	+	0.666928i	$0.767609\pi$
$402$	− 1934.08i	− 0.239958i
$403$	19865.4i	2.45550i
$404$	9442.31	1.16280
$405$	2535.27i	0.311059i
$406$	−2128.03	−0.260129
$407$	840.587	0.102374
$408$	0	0
$409$	−1220.58	−0.147565	−0.0737824	−	0.997274i	$-0.523507\pi$
$409$	−1220.58	−0.147565	−0.0737824	+	0.997274i	$0.523507\pi$
$410$	−23847.9	−2.87259
$411$	− 2958.86i	− 0.355109i
$412$	−6494.75	−0.776634
$413$	− 22.4926i	− 0.00267988i
$414$	3539.02i	0.420129i
$415$	− 11062.1i	− 1.30848i
$416$	20241.2	2.38560
$417$	10388.0	1.21991
$418$	9626.37i	1.12641i
$419$	− 10940.2i	− 1.27557i	−0.770215	−	0.637784i	$-0.779851\pi$
$419$	− 10940.2i	− 1.27557i	0.770215	−	0.637784i	$-0.220149\pi$
$420$	4082.82i	0.474337i
$421$	−2304.87	−0.266823	−0.133411	−	0.991061i	$-0.542593\pi$
$421$	−2304.87	−0.266823	−0.133411	+	0.991061i	$0.542593\pi$
$422$	11374.1i	1.31204i
$423$	1150.91	0.132291
$424$	367.316	0.0420718
$425$	0	0
$426$	6102.83	0.694092
$427$	−2067.32	−0.234297
$428$	− 7840.84i	− 0.885518i
$429$	−6719.26	−0.756198
$430$	− 16839.4i	− 1.88852i
$431$	− 5741.54i	− 0.641671i	−0.947135	−	0.320835i	$-0.896036\pi$
$431$	− 5741.54i	− 0.641671i	0.947135	−	0.320835i	$-0.103964\pi$
$432$	11328.2i	1.26164i
$433$	−4553.54	−0.505379	−0.252690	−	0.967547i	$-0.581315\pi$
$433$	−4553.54	−0.505379	−0.252690	+	0.967547i	$0.581315\pi$
$434$	−10593.3	−1.17165
$435$	2332.14i	0.257051i
$436$	7857.44i	0.863080i
$437$	8852.42i	0.969036i
$438$	10329.4	1.12684
$439$	10786.4i	1.17268i	0.810065	+	0.586340i	$0.199432\pi$
$439$	10786.4i	1.17268i	−0.810065	+	0.586340i	$0.800568\pi$
$440$	−2267.23	−0.245650
$441$	2384.22	0.257447
$442$	0	0
$443$	13133.5	1.40856	0.704280	−	0.709922i	$-0.251270\pi$
$443$	13133.5	1.40856	0.704280	+	0.709922i	$0.251270\pi$
$444$	−900.307	−0.0962313
$445$	5768.90i	0.614544i
$446$	10762.9	1.14268
$447$	3694.74i	0.390952i
$448$	2881.17i	0.303845i
$449$	714.661i	0.0751157i	0.999294	+	0.0375578i	$0.0119578\pi$
$449$	714.661i	0.0751157i	−0.999294	+	0.0375578i	$0.988042\pi$
$450$	−4363.22	−0.457076
$451$	8912.42	0.930531
$452$	9619.77i	1.00105i
$453$	4898.91i	0.508104i
$454$	− 20165.1i	− 2.08457i
$455$	17090.9	1.76096
$456$	3590.81i	0.368761i
$457$	976.357	0.0999388	0.0499694	−	0.998751i	$-0.484088\pi$
$457$	976.357	0.0999388	0.0499694	+	0.998751i	$0.484088\pi$
$458$	5689.67	0.580482
$459$	0	0
$460$	5986.50	0.606787
$461$	17713.0	1.78953	0.894767	−	0.446533i	$-0.147341\pi$
$461$	17713.0	1.78953	0.894767	+	0.446533i	$0.147341\pi$
$462$	− 3583.08i	− 0.360822i
$463$	−13597.7	−1.36488	−0.682441	−	0.730941i	$-0.739082\pi$
$463$	−13597.7	−1.36488	−0.682441	+	0.730941i	$0.739082\pi$
$464$	3352.21i	0.335393i
$465$	11609.4i	1.15779i
$466$	− 6292.63i	− 0.625538i
$467$	−9115.78	−0.903272	−0.451636	−	0.892202i	$-0.649159\pi$
$467$	−9115.78	−0.903272	−0.451636	+	0.892202i	$0.649159\pi$
$468$	−7347.64	−0.725737
$469$	− 1838.50i	− 0.181011i
$470$	− 4571.71i	− 0.448675i
$471$	3462.75i	0.338758i
$472$	−13.3777	−0.00130458
$473$	6293.20i	0.611758i
$474$	−7228.43	−0.700448
$475$	−10914.0	−1.05425
$476$	0	0
$477$	−649.525	−0.0623474
$478$	−18245.4	−1.74587
$479$	11483.2i	1.09537i	0.836686	+	0.547683i	$0.184490\pi$
$479$	11483.2i	1.09537i	−0.836686	+	0.547683i	$0.815510\pi$
$480$	11829.0	1.12483
$481$	3768.74i	0.357255i
$482$	− 12753.2i	− 1.20517i
$483$	− 3295.00i	− 0.310409i
$484$	5464.63	0.513207
$485$	−6211.40	−0.581536
$486$	− 12588.0i	− 1.17490i
$487$	4117.57i	0.383131i	0.981480	+	0.191565i	$0.0613564\pi$
$487$	4117.57i	0.383131i	−0.981480	+	0.191565i	$0.938644\pi$
$488$	1229.57i	0.114057i
$489$	−3531.36	−0.326572
$490$	− 9470.74i	− 0.873152i
$491$	11870.1	1.09102	0.545509	−	0.838105i	$-0.316336\pi$
$491$	11870.1	1.09102	0.545509	+	0.838105i	$0.316336\pi$
$492$	−9545.61	−0.874694
$493$	0	0
$494$	−43159.4	−3.93084
$495$	4009.14	0.364035
$496$	16687.3i	1.51065i
$497$	5801.25	0.523585
$498$	− 10397.8i	− 0.935615i
$499$	− 16770.4i	− 1.50450i	−0.658875	−	0.752252i	$-0.728967\pi$
$499$	− 16770.4i	− 1.50450i	0.658875	−	0.752252i	$-0.271033\pi$
$500$	− 3385.19i	− 0.302780i
$501$	4361.02	0.388895
$502$	8799.69	0.782369
$503$	14451.7i	1.28105i	0.767938	+	0.640525i	$0.221283\pi$
$503$	14451.7i	1.28105i	−0.767938	+	0.640525i	$0.778717\pi$
$504$	1364.60i	0.120603i
$505$	− 23099.1i	− 2.03544i
$506$	−5253.73	−0.461575
$507$	− 22095.2i	− 1.93547i
$508$	−2511.57	−0.219357
$509$	22279.2	1.94010	0.970049	−	0.242910i	$-0.0781019\pi$
$509$	22279.2	1.94010	0.970049	+	0.242910i	$0.0781019\pi$
$510$	0	0
$511$	9818.92	0.850026
$512$	12296.9	1.06143
$513$	− 18918.4i	− 1.62820i
$514$	−7591.08	−0.651417
$515$	15888.3i	1.35946i
$516$	− 6740.31i	− 0.575049i
$517$	1708.54i	0.145341i
$518$	−2009.69	−0.170465
$519$	−31.1022	−0.00263051
$520$	− 10165.0i	− 0.857242i
$521$	16004.3i	1.34580i	0.739735	+	0.672898i	$0.234951\pi$
$521$	16004.3i	1.34580i	−0.739735	+	0.672898i	$0.765049\pi$
$522$	− 2238.08i	− 0.187659i
$523$	−22158.6	−1.85264	−0.926319	−	0.376739i	$-0.877045\pi$
$523$	−22158.6	−1.85264	−0.926319	+	0.376739i	$0.877045\pi$
$524$	− 4705.14i	− 0.392261i
$525$	4062.37	0.337707
$526$	−20071.3	−1.66379
$527$	0	0
$528$	−5644.30	−0.465221
$529$	7335.66	0.602915
$530$	2580.09i	0.211456i
$531$	23.6559	0.00193329
$532$	− 9800.79i	− 0.798718i
$533$	39958.5i	3.24727i
$534$	5422.46i	0.439424i
$535$	−19181.3	−1.55006
$536$	−1093.47	−0.0881169
$537$	− 3323.59i	− 0.267083i
$538$	11243.6i	0.901017i
$539$	3539.40i	0.282844i
$540$	−12793.7	−1.01954
$541$	3099.41i	0.246310i	0.992387	+	0.123155i	$0.0393013\pi$
$541$	3099.41i	0.246310i	−0.992387	+	0.123155i	$0.960699\pi$
$542$	−5002.75	−0.396470
$543$	−8950.30	−0.707355
$544$	0	0
$545$	19221.9	1.51078
$546$	16064.6	1.25916
$547$	− 21795.9i	− 1.70370i	−0.523783	−	0.851852i	$-0.675480\pi$
$547$	− 21795.9i	− 1.70370i	0.523783	−	0.851852i	$-0.324520\pi$
$548$	4803.25	0.374424
$549$	− 2174.24i	− 0.169024i
$550$	− 6477.27i	− 0.502167i
$551$	− 5598.28i	− 0.432839i
$552$	−1959.74	−0.151109
$553$	−6871.22	−0.528380
$554$	− 18357.1i	− 1.40780i
$555$	2202.45i	0.168449i
$556$	16863.3i	1.28626i
$557$	12689.7	0.965315	0.482658	−	0.875809i	$-0.339672\pi$
$557$	12689.7	0.965315	0.482658	+	0.875809i	$0.339672\pi$
$558$	− 11141.2i	− 0.845239i
$559$	−28215.3	−2.13485
$560$	14356.7	1.08336
$561$	0	0
$562$	2605.08	0.195532
$563$	10849.0	0.812129	0.406065	−	0.913844i	$-0.366901\pi$
$563$	10849.0	0.812129	0.406065	+	0.913844i	$0.366901\pi$
$564$	− 1829.92i	− 0.136620i
$565$	23533.2	1.75230
$566$	20079.6i	1.49118i
$567$	2265.26i	0.167781i
$568$	− 3450.36i	− 0.254884i
$569$	5582.88	0.411329	0.205665	−	0.978623i	$-0.434064\pi$
$569$	5582.88	0.411329	0.205665	+	0.978623i	$0.434064\pi$
$570$	−25222.4	−1.85342
$571$	− 20899.4i	− 1.53172i	−0.643008	−	0.765859i	$-0.722314\pi$
$571$	− 20899.4i	− 1.53172i	0.643008	−	0.765859i	$-0.277686\pi$
$572$	− 10907.7i	− 0.797331i
$573$	− 3090.10i	− 0.225289i
$574$	−21308.0	−1.54944
$575$	− 5956.50i	− 0.432006i
$576$	−3030.17	−0.219196
$577$	−7924.36	−0.571742	−0.285871	−	0.958268i	$-0.592283\pi$
$577$	−7924.36	−0.571742	−0.285871	+	0.958268i	$0.592283\pi$
$578$	0	0
$579$	28.1693	0.00202189
$580$	−3785.86	−0.271034
$581$	− 9883.97i	− 0.705776i
$582$	−5838.38	−0.415823
$583$	− 964.230i	− 0.0684980i
$584$	− 5839.91i	− 0.413797i
$585$	17974.8i	1.27037i
$586$	10738.7	0.757019
$587$	−1139.92	−0.0801529	−0.0400764	−	0.999197i	$-0.512760\pi$
$587$	−1139.92	−0.0801529	−0.0400764	+	0.999197i	$0.512760\pi$
$588$	− 3790.86i	− 0.265872i
$589$	− 27868.2i	− 1.94956i
$590$	− 93.9673i	− 0.00655690i
$591$	18363.5	1.27813
$592$	3165.81i	0.219787i
$593$	−2645.81	−0.183222	−0.0916109	−	0.995795i	$-0.529202\pi$
$593$	−2645.81	−0.183222	−0.0916109	+	0.995795i	$0.529202\pi$
$594$	11227.7	0.775552
$595$	0	0
$596$	−5997.84	−0.412217
$597$	4425.87	0.303415
$598$	− 23554.9i	− 1.61075i
$599$	−21822.7	−1.48856	−0.744282	−	0.667865i	$-0.767208\pi$
$599$	−21822.7	−1.48856	−0.744282	+	0.667865i	$0.767208\pi$
$600$	− 2416.14i	− 0.164397i
$601$	− 10909.3i	− 0.740434i	−0.928945	−	0.370217i	$-0.879283\pi$
$601$	− 10909.3i	− 0.740434i	0.928945	−	0.370217i	$-0.120717\pi$
$602$	− 15045.9i	− 1.01865i
$603$	1933.58	0.130583
$604$	−7952.63	−0.535741
$605$	− 13368.3i	− 0.898345i
$606$	− 21711.9i	− 1.45542i
$607$	15308.1i	1.02362i	0.859100	+	0.511808i	$0.171024\pi$
$607$	15308.1i	1.02362i	−0.859100	+	0.511808i	$0.828976\pi$
$608$	−28395.4	−1.89406
$609$	2083.76i	0.138651i
$610$	−8636.65	−0.573259
$611$	−7660.16	−0.507196
$612$	0	0
$613$	24096.7	1.58769	0.793847	−	0.608118i	$-0.208075\pi$
$613$	24096.7	1.58769	0.793847	+	0.608118i	$0.208075\pi$
$614$	−14056.1	−0.923875
$615$	23351.8i	1.53111i
$616$	−2025.76	−0.132501
$617$	7030.91i	0.458758i	0.973337	+	0.229379i	$0.0736696\pi$
$617$	7030.91i	0.458758i	−0.973337	+	0.229379i	$0.926330\pi$
$618$	14934.2i	0.972073i
$619$	− 19270.2i	− 1.25127i	−0.780116	−	0.625635i	$-0.784840\pi$
$619$	− 19270.2i	− 1.25127i	0.780116	−	0.625635i	$-0.215160\pi$
$620$	−18846.0	−1.22077
$621$	10325.0	0.667194
$622$	3895.85i	0.251140i
$623$	5154.50i	0.331478i
$624$	− 25306.0i	− 1.62348i
$625$	−18993.2	−1.21557
$626$	− 2993.07i	− 0.191097i
$627$	9426.12	0.600388
$628$	−5621.24	−0.357185
$629$	0	0
$630$	−9585.14	−0.606160
$631$	1474.23	0.0930079	0.0465040	−	0.998918i	$-0.485192\pi$
$631$	1474.23	0.0930079	0.0465040	+	0.998918i	$0.485192\pi$
$632$	4086.74i	0.257218i
$633$	11137.5	0.699327
$634$	3009.06i	0.188494i
$635$	6144.15i	0.383974i
$636$	1032.73i	0.0643877i
$637$	−15868.8	−0.987038
$638$	3322.46	0.206172
$639$	6101.27i	0.377719i
$640$	− 13853.7i	− 0.855648i
$641$	18747.7i	1.15521i	0.816317	+	0.577604i	$0.196012\pi$
$641$	18747.7i	1.15521i	−0.816317	+	0.577604i	$0.803988\pi$
$642$	−18029.4	−1.10836
$643$	− 2324.44i	− 0.142561i	−0.997456	−	0.0712807i	$-0.977291\pi$
$643$	− 2324.44i	− 0.142561i	0.997456	−	0.0712807i	$-0.0227086\pi$
$644$	5348.92	0.327294
$645$	−16489.1	−1.00660
$646$	0	0
$647$	−11883.3	−0.722071	−0.361036	−	0.932552i	$-0.617577\pi$
$647$	−11883.3	−0.722071	−0.361036	+	0.932552i	$0.617577\pi$
$648$	1347.29	0.0816768
$649$	35.1175i	0.00212401i
$650$	29040.5	1.75241
$651$	10373.0i	0.624498i
$652$	− 5732.62i	− 0.344336i
$653$	873.094i	0.0523228i	0.999658	+	0.0261614i	$0.00832839\pi$
$653$	873.094i	0.0523228i	−0.999658	+	0.0261614i	$0.991672\pi$
$654$	18067.6	1.08027
$655$	−11510.3	−0.686636
$656$	33565.8i	1.99775i
$657$	10326.7i	0.613217i
$658$	− 4084.81i	− 0.242010i
$659$	−965.182	−0.0570534	−0.0285267	−	0.999593i	$-0.509082\pi$
$659$	−965.182	−0.0570534	−0.0285267	+	0.999593i	$0.509082\pi$
$660$	− 6374.47i	− 0.375948i
$661$	−10564.2	−0.621631	−0.310815	−	0.950470i	$-0.600602\pi$
$661$	−10564.2	−0.621631	−0.310815	+	0.950470i	$0.600602\pi$
$662$	−9731.35	−0.571329
$663$	0	0
$664$	−5878.60	−0.343575
$665$	−23976.0	−1.39812
$666$	− 2113.63i	− 0.122975i
$667$	3055.34	0.177366
$668$	7079.45i	0.410048i
$669$	− 10539.0i	− 0.609059i
$670$	− 7680.69i	− 0.442882i
$671$	3227.69	0.185698
$672$	10569.2	0.606719
$673$	24755.2i	1.41789i	0.705262	+	0.708947i	$0.250829\pi$
$673$	24755.2i	1.41789i	−0.705262	+	0.708947i	$0.749171\pi$
$674$	− 3984.02i	− 0.227684i
$675$	12729.6i	0.725868i
$676$	35868.2	2.04075
$677$	34038.8i	1.93238i	0.257840	+	0.966188i	$0.416989\pi$
$677$	34038.8i	1.93238i	−0.257840	+	0.966188i	$0.583011\pi$
$678$	22119.9	1.25297
$679$	−5549.87	−0.313674
$680$	0	0
$681$	−19745.6	−1.11109
$682$	16539.2	0.928622
$683$	− 23062.1i	− 1.29202i	−0.763330	−	0.646008i	$-0.776437\pi$
$683$	− 23062.1i	− 1.29202i	0.763330	−	0.646008i	$-0.223563\pi$
$684$	10307.6	0.576203
$685$	− 11750.4i	− 0.655413i
$686$	− 25067.3i	− 1.39515i
$687$	− 5571.31i	− 0.309401i
$688$	−23701.4	−1.31338
$689$	4323.08	0.239037
$690$	− 13765.5i	− 0.759484i
$691$	− 14331.9i	− 0.789018i	−0.918892	−	0.394509i	$-0.870915\pi$
$691$	− 14331.9i	− 0.789018i	0.918892	−	0.394509i	$-0.129085\pi$
$692$	− 50.4896i	− 0.00277360i
$693$	3582.16	0.196356
$694$	42479.5i	2.32349i
$695$	41253.3	2.25155
$696$	1239.34	0.0674958
$697$	0	0
$698$	9134.19	0.495321
$699$	−6161.73	−0.333417
$700$	6594.62i	0.356076i
$701$	28211.9	1.52004	0.760020	−	0.649900i	$-0.225189\pi$
$701$	28211.9	1.52004	0.760020	+	0.649900i	$0.225189\pi$
$702$	50338.9i	2.70644i
$703$	− 5286.97i	− 0.283644i
$704$	− 4498.34i	− 0.240820i
$705$	−4476.61	−0.239147
$706$	19518.8	1.04051
$707$	− 20639.0i	− 1.09789i
$708$	− 37.6124i	− 0.00199656i
$709$	− 5397.14i	− 0.285887i	−0.989731	−	0.142943i	$-0.954343\pi$
$709$	− 5397.14i	− 0.285887i	0.989731	−	0.142943i	$-0.0456567\pi$
$710$	24235.8	1.28106
$711$	− 7226.57i	− 0.381178i
$712$	3065.70	0.161365
$713$	15209.5	0.798878
$714$	0	0
$715$	−26683.9	−1.39569
$716$	5395.34	0.281611
$717$	17865.9i	0.930562i
$718$	−33998.9	−1.76717
$719$	− 17858.0i	− 0.926276i	−0.886286	−	0.463138i	$-0.846723\pi$
$719$	− 17858.0i	− 0.926276i	0.886286	−	0.463138i	$-0.153277\pi$
$720$	15099.2i	0.781545i
$721$	14196.2i	0.733279i
$722$	34943.2	1.80118
$723$	−12487.9	−0.642366
$724$	− 14529.4i	− 0.745831i
$725$	3766.89i	0.192964i
$726$	− 12565.5i	− 0.642354i
$727$	27634.4	1.40977	0.704885	−	0.709322i	$-0.250999\pi$
$727$	27634.4	1.40977	0.704885	+	0.709322i	$0.250999\pi$
$728$	− 9082.43i	− 0.462386i
$729$	−17042.0	−0.865821
$730$	41020.4	2.07977
$731$	0	0
$732$	−3457.00	−0.174555
$733$	−16773.4	−0.845209	−0.422605	−	0.906314i	$-0.638884\pi$
$733$	−16773.4	−0.845209	−0.422605	+	0.906314i	$0.638884\pi$
$734$	4313.46i	0.216911i
$735$	−9273.72	−0.465396
$736$	− 15497.2i	− 0.776134i
$737$	2870.43i	0.143465i
$738$	− 22410.0i	− 1.11778i
$739$	26618.2	1.32499	0.662495	−	0.749067i	$-0.269498\pi$
$739$	26618.2	1.32499	0.662495	+	0.749067i	$0.269498\pi$
$740$	−3575.34	−0.177611
$741$	42261.6i	2.09517i
$742$	2305.30i	0.114057i
$743$	3231.55i	0.159561i	0.996812	+	0.0797806i	$0.0254220\pi$
$743$	3231.55i	0.159561i	−0.996812	+	0.0797806i	$0.974578\pi$
$744$	6169.44	0.304009
$745$	14672.7i	0.721567i
$746$	−37140.9	−1.82282
$747$	10395.1	0.509154
$748$	0	0
$749$	−17138.5	−0.836083
$750$	−7783.98	−0.378974
$751$	− 6094.29i	− 0.296117i	−0.988979	−	0.148058i	$-0.952698\pi$
$751$	− 6094.29i	− 0.296117i	0.988979	−	0.148058i	$-0.0473023\pi$
$752$	−6434.67	−0.312032
$753$	− 8616.63i	− 0.417009i
$754$	14896.1i	0.719476i
$755$	19454.8i	0.937791i
$756$	−11431.1	−0.549928
$757$	4282.74	0.205626	0.102813	−	0.994701i	$-0.467216\pi$
$757$	4282.74	0.205626	0.102813	+	0.994701i	$0.467216\pi$
$758$	5389.27i	0.258242i
$759$	5144.44i	0.246023i
$760$	14260.0i	0.680612i
$761$	−35191.9	−1.67636	−0.838178	−	0.545397i	$-0.816379\pi$
$761$	−35191.9	−1.67636	−0.838178	+	0.545397i	$0.816379\pi$
$762$	5775.18i	0.274557i
$763$	17174.7	0.814899
$764$	5016.29	0.237543
$765$	0	0
$766$	30793.8	1.45251
$767$	−157.448	−0.00741213
$768$	− 19517.6i	− 0.917032i
$769$	23142.5	1.08523	0.542614	−	0.839982i	$-0.317434\pi$
$769$	23142.5	1.08523	0.542614	+	0.839982i	$0.317434\pi$
$770$	− 14229.3i	− 0.665958i
$771$	7433.17i	0.347210i
$772$	45.7285i	0.00213187i
$773$	4345.19	0.202181	0.101090	−	0.994877i	$-0.467767\pi$
$773$	4345.19	0.202181	0.101090	+	0.994877i	$0.467767\pi$
$774$	15824.0	0.734862
$775$	18751.6i	0.869133i
$776$	3300.85i	0.152698i
$777$	1967.89i	0.0908592i
$778$	35076.6	1.61640
$779$	− 56055.7i	− 2.57818i
$780$	28579.6	1.31194
$781$	−9057.42	−0.414981
$782$	0	0
$783$	−6529.53	−0.298016
$784$	−13330.0	−0.607236
$785$	13751.4i	0.625236i
$786$	−10819.1	−0.490973
$787$	15543.2i	0.704010i	0.935998	+	0.352005i	$0.114500\pi$
$787$	15543.2i	0.704010i	−0.935998	+	0.352005i	$0.885500\pi$
$788$	29810.3i	1.34765i
$789$	19653.8i	0.886811i
$790$	−28705.9	−1.29280
$791$	21026.8	0.945169
$792$	− 2130.53i	− 0.0955872i
$793$	14471.2i	0.648030i
$794$	44683.9i	1.99719i
$795$	2526.41	0.112708
$796$	7184.72i	0.319919i
$797$	−15559.4	−0.691519	−0.345759	−	0.938323i	$-0.612379\pi$
$797$	−15559.4	−0.691519	−0.345759	+	0.938323i	$0.612379\pi$
$798$	−22536.2	−0.999714
$799$	0	0
$800$	19106.3	0.844389
$801$	−5421.07	−0.239131
$802$	39981.2i	1.76033i
$803$	−15330.2	−0.673711
$804$	− 3074.36i	− 0.134856i
$805$	− 13085.3i	− 0.572913i
$806$	74153.0i	3.24060i
$807$	11009.7	0.480249
$808$	−12275.3	−0.534458
$809$	− 9433.22i	− 0.409956i	−0.978767	−	0.204978i	$-0.934288\pi$
$809$	− 9433.22i	− 0.409956i	0.978767	−	0.204978i	$-0.0657123\pi$
$810$	9463.57i	0.410514i
$811$	− 12596.0i	− 0.545381i	−0.962102	−	0.272691i	$-0.912086\pi$
$811$	− 12596.0i	− 0.545381i	0.962102	−	0.272691i	$-0.0879135\pi$
$812$	−3382.66	−0.146192
$813$	4898.68i	0.211321i
$814$	3137.71	0.135107
$815$	−14023.9	−0.602744
$816$	0	0
$817$	39581.8	1.69497
$818$	−4556.15	−0.194746
$819$	16060.4i	0.685223i
$820$	−37908.0	−1.61440
$821$	− 24332.2i	− 1.03435i	−0.855880	−	0.517175i	$-0.826984\pi$
$821$	− 24332.2i	− 1.03435i	0.855880	−	0.517175i	$-0.173016\pi$
$822$	− 11044.7i	− 0.468648i
$823$	− 40516.8i	− 1.71607i	−0.513591	−	0.858035i	$-0.671685\pi$
$823$	− 40516.8i	− 1.71607i	0.513591	−	0.858035i	$-0.328315\pi$
$824$	8443.35	0.356964
$825$	−6342.53	−0.267659
$826$	− 83.9596i	− 0.00353672i
$827$	− 3544.77i	− 0.149049i	−0.997219	−	0.0745247i	$-0.976256\pi$
$827$	− 3544.77i	− 0.149049i	0.997219	−	0.0745247i	$-0.0237440\pi$
$828$	5625.55i	0.236113i
$829$	−25265.7	−1.05852	−0.529260	−	0.848460i	$-0.677530\pi$
$829$	−25265.7	−1.05852	−0.529260	+	0.848460i	$0.677530\pi$
$830$	− 41292.2i	− 1.72684i
$831$	−17975.2	−0.750366
$832$	20168.1	0.840388
$833$	0	0
$834$	38775.9	1.60995
$835$	17318.7	0.717771
$836$	15301.9i	0.633045i
$837$	−32504.0	−1.34230
$838$	− 40837.1i	− 1.68341i
$839$	− 32421.0i	− 1.33408i	−0.745021	−	0.667042i	$-0.767560\pi$
$839$	− 32421.0i	− 1.33408i	0.745021	−	0.667042i	$-0.232440\pi$
$840$	− 5307.78i	− 0.218019i
$841$	22456.8	0.920776
$842$	−8603.51	−0.352134
$843$	− 2550.89i	− 0.104220i
$844$	18079.9i	0.737366i
$845$	− 87745.7i	− 3.57224i
$846$	4296.06	0.174588
$847$	− 11944.6i	− 0.484557i
$848$	3631.47	0.147058
$849$	19661.9	0.794811
$850$	0	0
$851$	2885.44	0.116230
$852$	9700.91	0.390080
$853$	− 31041.4i	− 1.24600i	−0.782222	−	0.623000i	$-0.785913\pi$
$853$	− 31041.4i	− 1.24600i	0.782222	−	0.623000i	$-0.214087\pi$
$854$	−7716.83	−0.309209
$855$	− 25216.0i	− 1.00862i
$856$	10193.3i	0.407009i
$857$	− 6852.25i	− 0.273125i	−0.990631	−	0.136563i	$-0.956394\pi$
$857$	− 6852.25i	− 0.273125i	0.990631	−	0.136563i	$-0.0436055\pi$
$858$	−25081.4	−0.997978
$859$	−31049.1	−1.23327	−0.616636	−	0.787248i	$-0.711505\pi$
$859$	−31049.1	−1.23327	−0.616636	+	0.787248i	$0.711505\pi$
$860$	− 26767.4i	− 1.06135i
$861$	20864.8i	0.825864i
$862$	− 21431.8i	− 0.846833i
$863$	31789.3	1.25391	0.626953	−	0.779057i	$-0.284302\pi$
$863$	31789.3	1.25391	0.626953	+	0.779057i	$0.284302\pi$
$864$	33118.9i	1.30408i
$865$	−123.515	−0.00485505
$866$	−16997.3	−0.666964
$867$	0	0
$868$	−16838.9	−0.658467
$869$	10728.0	0.418781
$870$	8705.31i	0.339239i
$871$	−12869.4	−0.500648
$872$	− 10214.9i	− 0.396697i
$873$	− 5836.89i	− 0.226287i
$874$	33044.0i	1.27887i
$875$	−7399.32	−0.285877
$876$	16419.3	0.633284
$877$	− 34579.2i	− 1.33142i	−0.746210	−	0.665711i	$-0.768128\pi$
$877$	− 34579.2i	− 1.33142i	0.746210	−	0.665711i	$-0.231872\pi$
$878$	40263.0i	1.54762i
$879$	− 10515.3i	− 0.403497i
$880$	−22414.9	−0.858644
$881$	14088.5i	0.538765i	0.963033	+	0.269383i	$0.0868197\pi$
$881$	14088.5i	0.538765i	−0.963033	+	0.269383i	$0.913180\pi$
$882$	8899.71	0.339761
$883$	4779.77	0.182165	0.0910827	−	0.995843i	$-0.470967\pi$
$883$	4779.77	0.182165	0.0910827	+	0.995843i	$0.470967\pi$
$884$	0	0
$885$	−92.0126	−0.00349488
$886$	49024.3	1.85892
$887$	14254.5i	0.539594i	0.962917	+	0.269797i	$0.0869567\pi$
$887$	14254.5i	0.539594i	−0.962917	+	0.269797i	$0.913043\pi$
$888$	1170.42	0.0442307
$889$	5489.79i	0.207111i
$890$	21533.9i	0.811032i
$891$	− 3536.73i	− 0.132980i
$892$	17108.4	0.642188
$893$	10746.1	0.402691
$894$	13791.6i	0.515951i
$895$	− 13198.8i	− 0.492947i
$896$	− 12378.2i	− 0.461526i
$897$	−23064.9	−0.858544
$898$	2667.66i	0.0991325i
$899$	−9618.49	−0.356835
$900$	−6935.67	−0.256877
$901$	0	0
$902$	33268.0	1.22805
$903$	−14732.9	−0.542947
$904$	− 12506.0i	− 0.460113i
$905$	−35543.8	−1.30554
$906$	18286.5i	0.670560i
$907$	− 36585.5i	− 1.33936i	−0.742650	−	0.669680i	$-0.766431\pi$
$907$	− 36585.5i	− 1.33936i	0.742650	−	0.669680i	$-0.233569\pi$
$908$	− 32053.9i	− 1.17153i
$909$	21706.3	0.792028
$910$	63796.3	2.32399
$911$	− 19721.3i	− 0.717230i	−0.933486	−	0.358615i	$-0.883249\pi$
$911$	− 19721.3i	− 0.717230i	0.933486	−	0.358615i	$-0.116751\pi$
$912$	35500.5i	1.28897i
$913$	15431.7i	0.559382i
$914$	3644.51	0.131892
$915$	8456.99i	0.305551i
$916$	9044.16	0.326231
$917$	−10284.5	−0.370363
$918$	0	0
$919$	−28189.0	−1.01183	−0.505913	−	0.862585i	$-0.668844\pi$
$919$	−28189.0	−1.01183	−0.505913	+	0.862585i	$0.668844\pi$
$920$	−7782.60	−0.278897
$921$	13763.7i	0.492432i
$922$	66118.3	2.36170
$923$	− 40608.5i	− 1.44815i
$924$	− 5695.57i	− 0.202782i
$925$	3557.43i	0.126451i
$926$	−50757.1	−1.80128
$927$	−14930.4	−0.528994
$928$	9800.45i	0.346676i
$929$	30866.1i	1.09008i	0.838410	+	0.545040i	$0.183485\pi$
$929$	30866.1i	1.09008i	−0.838410	+	0.545040i	$0.816515\pi$
$930$	43335.1i	1.52797i
$931$	22261.5	0.783664
$932$	− 10002.6i	− 0.351552i
$933$	3814.80	0.133860
$934$	−34027.0	−1.19208
$935$	0	0
$936$	9552.13	0.333570
$937$	−13990.7	−0.487785	−0.243893	−	0.969802i	$-0.578424\pi$
$937$	−13990.7	−0.487785	−0.243893	+	0.969802i	$0.578424\pi$
$938$	− 6862.68i	− 0.238885i
$939$	−2930.80	−0.101856
$940$	− 7267.08i	− 0.252155i
$941$	4495.60i	0.155741i	0.996963	+	0.0778706i	$0.0248121\pi$
$941$	4495.60i	0.155741i	−0.996963	+	0.0778706i	$0.975188\pi$
$942$	12925.6i	0.447070i
$943$	30593.2	1.05647
$944$	−132.259	−0.00456002
$945$	27964.3i	0.962624i
$946$	23491.0i	0.807356i
$947$	29959.5i	1.02804i	0.857779	+	0.514019i	$0.171844\pi$
$947$	29959.5i	1.02804i	−0.857779	+	0.514019i	$0.828156\pi$
$948$	−11490.1	−0.393652
$949$	− 68732.1i	− 2.35104i
$950$	−40739.5	−1.39133
$951$	2946.47	0.100469
$952$	0	0
$953$	−44526.8	−1.51350	−0.756750	−	0.653704i	$-0.773214\pi$
$953$	−44526.8	−1.51350	−0.756750	+	0.653704i	$0.773214\pi$
$954$	−2424.52	−0.0822818
$955$	− 12271.5i	− 0.415809i
$956$	−29002.5	−0.981178
$957$	− 3253.35i	− 0.109891i
$958$	42864.0i	1.44559i
$959$	− 10498.9i	− 0.353522i
$960$	11786.3	0.396250
$961$	−18089.9	−0.607227
$962$	14067.8i	0.471480i
$963$	− 18024.8i	− 0.603158i
$964$	− 20272.2i	− 0.677306i
$965$	111.867	0.00373174
$966$	− 12299.4i	− 0.409656i
$967$	−31696.1	−1.05406	−0.527031	−	0.849846i	$-0.676695\pi$
$967$	−31696.1	−1.05406	−0.527031	+	0.849846i	$0.676695\pi$
$968$	−7104.16	−0.235885
$969$	0	0
$970$	−23185.7	−0.767471
$971$	47645.1	1.57467	0.787335	−	0.616525i	$-0.211460\pi$
$971$	47645.1	1.57467	0.787335	+	0.616525i	$0.211460\pi$
$972$	− 20009.5i	− 0.660293i
$973$	36859.7	1.21446
$974$	15369.9i	0.505629i
$975$	− 28436.4i	− 0.934046i
$976$	12156.1i	0.398675i
$977$	−47682.5	−1.56141	−0.780706	−	0.624898i	$-0.785140\pi$
$977$	−47682.5	−1.56141	−0.780706	+	0.624898i	$0.785140\pi$
$978$	−13181.7	−0.430987
$979$	− 8047.66i	− 0.262721i
$980$	− 15054.5i	− 0.490711i
$981$	18063.0i	0.587875i
$982$	44308.2	1.43985
$983$	42569.9i	1.38125i	0.723212	+	0.690626i	$0.242665\pi$
$983$	42569.9i	1.38125i	−0.723212	+	0.690626i	$0.757335\pi$
$984$	12409.6	0.402035
$985$	72926.1	2.35900
$986$	0	0
$987$	−3999.84	−0.128993
$988$	−68605.2	−2.20913
$989$	21602.4i	0.694555i
$990$	14965.2	0.480428
$991$	36264.9i	1.16245i	0.813742	+	0.581227i	$0.197427\pi$
$991$	36264.9i	1.16245i	−0.813742	+	0.581227i	$0.802573\pi$
$992$	48786.7i	1.56147i
$993$	9528.92i	0.304523i
$994$	21654.7	0.690991
$995$	17576.2	0.560004
$996$	− 16528.1i	− 0.525816i
$997$	− 37027.1i	− 1.17619i	−0.808792	−	0.588094i	$-0.799878\pi$
$997$	− 37027.1i	− 1.17619i	0.808792	−	0.588094i	$-0.200122\pi$
$998$	− 62600.0i	− 1.98554i
$999$	−6166.44	−0.195293

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	289.4.b.f.288.19		24
17.4	even	4		289.4.a.i.1.3	yes	12
17.13	even	4		289.4.a.h.1.3	✓	12
17.16	even	2	inner	289.4.b.f.288.20		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
289.4.a.h.1.3	✓	12	17.13	even	4
289.4.a.i.1.3	yes	12	17.4	even	4
289.4.b.f.288.19		24	1.1	even	1	trivial
289.4.b.f.288.20		24	17.16	even	2	inner

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:	\( N \)	\(=\)	\( 289 = 17^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	289.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.73276 q^{2} -3.65511i q^{3} +5.93351 q^{4} -14.5154i q^{5} -13.6437i q^{6} -12.9694i q^{7} -7.71372 q^{8} +13.6402 q^{9} +O(q^{10})\)	\(q+3.73276 q^{2} -3.65511i q^{3} +5.93351 q^{4} -14.5154i q^{5} -13.6437i q^{6} -12.9694i q^{7} -7.71372 q^{8} +13.6402 q^{9} -54.1823i q^{10} +20.2490i q^{11} -21.6876i q^{12} -90.7856 q^{13} -48.4118i q^{14} -53.0552 q^{15} -76.2615 q^{16} +50.9155 q^{18} +127.359 q^{19} -86.1270i q^{20} -47.4047 q^{21} +75.5848i q^{22} +69.5078i q^{23} +28.1945i q^{24} -85.6954 q^{25} -338.881 q^{26} -148.544i q^{27} -76.9542i q^{28} -43.9568i q^{29} -198.042 q^{30} -218.817i q^{31} -222.956 q^{32} +74.0124 q^{33} -188.256 q^{35} +80.9340 q^{36} -41.5125i q^{37} +475.399 q^{38} +331.832i q^{39} +111.967i q^{40} -440.141i q^{41} -176.951 q^{42} +310.790 q^{43} +120.148i q^{44} -197.992i q^{45} +259.456i q^{46} +84.3763 q^{47} +278.744i q^{48} +174.794 q^{49} -319.880 q^{50} -538.677 q^{52} -47.6186 q^{53} -554.480i q^{54} +293.922 q^{55} +100.043i q^{56} -465.510i q^{57} -164.080i q^{58} +1.73428 q^{59} -314.804 q^{60} -159.400i q^{61} -816.792i q^{62} -176.905i q^{63} -222.151 q^{64} +1317.78i q^{65} +276.271 q^{66} +141.756 q^{67} +254.059 q^{69} -702.714 q^{70} +447.302i q^{71} -105.216 q^{72} +757.081i q^{73} -154.956i q^{74} +313.226i q^{75} +755.684 q^{76} +262.618 q^{77} +1238.65i q^{78} -529.801i q^{79} +1106.96i q^{80} -174.662 q^{81} -1642.94i q^{82} +762.097 q^{83} -281.276 q^{84} +1160.11 q^{86} -160.667 q^{87} -156.195i q^{88} -397.434 q^{89} -739.056i q^{90} +1177.44i q^{91} +412.425i q^{92} -799.801 q^{93} +314.957 q^{94} -1848.66i q^{95} +814.930i q^{96} -427.919i q^{97} +652.464 q^{98} +276.200i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 96 q^{4} + 102 q^{8} - 216 q^{9}+O(q^{10}) \) 24 * q + 96 * q^4 + 102 * q^8 - 216 * q^9	\( 24 q + 96 q^{4} + 102 q^{8} - 216 q^{9} - 144 q^{13} + 276 q^{15} + 384 q^{16} + 378 q^{18} + 132 q^{19} + 492 q^{21} - 888 q^{25} - 1056 q^{26} - 3780 q^{30} + 2706 q^{32} + 1932 q^{33} - 132 q^{35} + 1326 q^{36} - 120 q^{38} - 1608 q^{42} - 972 q^{43} - 1776 q^{47} + 1140 q^{49} + 870 q^{50} + 450 q^{52} - 2052 q^{53} + 1944 q^{55} + 1584 q^{59} - 2916 q^{60} - 3714 q^{64} + 1188 q^{66} + 1248 q^{67} - 3012 q^{69} + 3300 q^{70} + 2910 q^{72} - 1350 q^{76} + 2016 q^{77} + 5040 q^{81} - 1344 q^{83} - 1554 q^{84} + 5556 q^{86} - 1452 q^{87} - 1812 q^{89} - 264 q^{93} - 1470 q^{94} + 3822 q^{98}+O(q^{100}) \) 24 * q + 96 * q^4 + 102 * q^8 - 216 * q^9 - 144 * q^13 + 276 * q^15 + 384 * q^16 + 378 * q^18 + 132 * q^19 + 492 * q^21 - 888 * q^25 - 1056 * q^26 - 3780 * q^30 + 2706 * q^32 + 1932 * q^33 - 132 * q^35 + 1326 * q^36 - 120 * q^38 - 1608 * q^42 - 972 * q^43 - 1776 * q^47 + 1140 * q^49 + 870 * q^50 + 450 * q^52 - 2052 * q^53 + 1944 * q^55 + 1584 * q^59 - 2916 * q^60 - 3714 * q^64 + 1188 * q^66 + 1248 * q^67 - 3012 * q^69 + 3300 * q^70 + 2910 * q^72 - 1350 * q^76 + 2016 * q^77 + 5040 * q^81 - 1344 * q^83 - 1554 * q^84 + 5556 * q^86 - 1452 * q^87 - 1812 * q^89 - 264 * q^93 - 1470 * q^94 + 3822 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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