LMFDB - Embedded newform 289.4.b.b.288.1

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:	$6$
Coefficient field:	6.0.27793984.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{3} + 49x^{2} - 14x + 2 $ x^6 - 2x^3 + 49x^2 - 14*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 17)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			288.1
Root			$0.143705 - 0.143705i$ of defining polynomial
Character	$\chi$	$=$	289.288
Dual form			289.4.b.b.288.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-4.67129 q^{2} -7.62999i q^{3} +13.8209 q^{4} -11.9174i q^{5} +35.6419i q^{6} -26.1222i q^{7} -27.1912 q^{8} -31.2167 q^{9} +55.6696i q^{10} +3.24412i q^{11} -105.453i q^{12} -20.0515 q^{13} +122.024i q^{14} -90.9296 q^{15} +16.4506 q^{16} +145.822 q^{18} -57.3466 q^{19} -164.709i q^{20} -199.312 q^{21} -15.1542i q^{22} -77.0438i q^{23} +207.469i q^{24} -17.0243 q^{25} +93.6662 q^{26} +32.1732i q^{27} -361.033i q^{28} -286.162i q^{29} +424.758 q^{30} -8.54816i q^{31} +140.684 q^{32} +24.7526 q^{33} -311.309 q^{35} -431.443 q^{36} +357.982i q^{37} +267.882 q^{38} +152.992i q^{39} +324.049i q^{40} -194.467i q^{41} +931.044 q^{42} +74.2619 q^{43} +44.8367i q^{44} +372.021i q^{45} +359.894i q^{46} +23.6130 q^{47} -125.518i q^{48} -339.369 q^{49} +79.5255 q^{50} -277.130 q^{52} -104.330 q^{53} -150.290i q^{54} +38.6614 q^{55} +710.295i q^{56} +437.553i q^{57} +1336.75i q^{58} -249.363 q^{59} -1256.73 q^{60} +370.384i q^{61} +39.9309i q^{62} +815.448i q^{63} -788.781 q^{64} +238.961i q^{65} -115.626 q^{66} +939.650 q^{67} -587.843 q^{69} +1454.21 q^{70} -520.197i q^{71} +848.820 q^{72} +348.741i q^{73} -1672.24i q^{74} +129.895i q^{75} -792.583 q^{76} +84.7434 q^{77} -714.672i q^{78} +953.827i q^{79} -196.049i q^{80} -597.369 q^{81} +908.412i q^{82} +1414.28 q^{83} -2754.68 q^{84} -346.899 q^{86} -2183.41 q^{87} -88.2115i q^{88} -486.132 q^{89} -1737.82i q^{90} +523.788i q^{91} -1064.82i q^{92} -65.2223 q^{93} -110.303 q^{94} +683.422i q^{95} -1073.42i q^{96} -685.281i q^{97} +1585.29 q^{98} -101.271i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{2} + 50 q^{4} + 78 q^{8} - 118 q^{9} + 60 q^{13} - 216 q^{15} + 274 q^{16} - 206 q^{18} - 160 q^{19} - 384 q^{21} + 446 q^{25} - 52 q^{26} + 800 q^{30} + 142 q^{32} - 664 q^{33} - 664 q^{35}+ \cdots + 5714 q^{98}+O(q^{100}) $ 6 * q - 2 * q^2 + 50 * q^4 + 78 * q^8 - 118 * q^9 + 60 * q^13 - 216 * q^15 + 274 * q^16 - 206 * q^18 - 160 * q^19 - 384 * q^21 + 446 * q^25 - 52 * q^26 + 800 * q^30 + 142 * q^32 - 664 * q^33 - 664 * q^35 - 2626 * q^36 + 1448 * q^38 + 2256 * q^42 - 1112 * q^43 + 1280 * q^47 + 538 * q^49 + 1094 * q^50 - 1548 * q^52 - 604 * q^53 + 152 * q^55 - 1272 * q^59 - 2656 * q^60 - 1838 * q^64 - 4936 * q^66 + 2016 * q^67 + 1152 * q^69 + 3008 * q^70 - 1854 * q^72 + 1816 * q^76 + 1008 * q^77 - 1010 * q^81 + 4792 * q^83 - 4080 * q^84 - 2528 * q^86 - 2856 * q^87 - 340 * q^89 - 1264 * q^93 + 4032 * q^94 + 5714 * q^98

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 $

$2$	−4.67129	−1.65155	−0.825775	−	0.564000i	$-0.809262\pi$
$2$	−4.67129	−1.65155	−0.825775	+	0.564000i	$0.809262\pi$
$3$	− 7.62999i	− 1.46839i	−0.678938	−	0.734196i	$-0.737559\pi$
$3$	− 7.62999i	− 1.46839i	0.678938	−	0.734196i	$-0.262441\pi$
$4$	13.8209	1.72762
$5$	− 11.9174i	− 1.06592i	−0.846139	−	0.532962i	$-0.821079\pi$
$5$	− 11.9174i	− 1.06592i	0.846139	−	0.532962i	$-0.178921\pi$
$6$	35.6419i	2.42512i
$7$	− 26.1222i	− 1.41047i	−0.708975	−	0.705233i	$-0.750842\pi$
$7$	− 26.1222i	− 1.41047i	0.708975	−	0.705233i	$-0.249158\pi$
$8$	−27.1912	−1.20169
$9$	−31.2167	−1.15617
$10$	55.6696i	1.76043i
$11$	3.24412i	0.0889216i	0.999011	+	0.0444608i	$0.0141570\pi$
$11$	3.24412i	0.0889216i	−0.999011	+	0.0444608i	$0.985843\pi$
$12$	− 105.453i	− 2.53682i
$13$	−20.0515	−0.427790	−0.213895	−	0.976857i	$-0.568615\pi$
$13$	−20.0515	−0.427790	−0.213895	+	0.976857i	$0.568615\pi$
$14$	122.024i	2.32945i
$15$	−90.9296	−1.56519
$16$	16.4506	0.257041
$17$	0	0
$17$	0	0
$18$	145.822	1.90948
$19$	−57.3466	−0.692432	−0.346216	−	0.938155i	$-0.612534\pi$
$19$	−57.3466	−0.692432	−0.346216	+	0.938155i	$0.612534\pi$
$20$	− 164.709i	− 1.84151i
$21$	−199.312	−2.07112
$22$	− 15.1542i	− 0.146858i
$23$	− 77.0438i	− 0.698467i	−0.937036	−	0.349233i	$-0.886442\pi$
$23$	− 77.0438i	− 0.698467i	0.937036	−	0.349233i	$-0.113558\pi$
$24$	207.469i	1.76456i
$25$	−17.0243	−0.136195
$26$	93.6662	0.706517
$27$	32.1732i	0.229323i
$28$	− 361.033i	− 2.43674i
$29$	− 286.162i	− 1.83238i	−0.400744	−	0.916190i	$-0.631248\pi$
$29$	− 286.162i	− 1.83238i	0.400744	−	0.916190i	$-0.368752\pi$
$30$	424.758	2.58500
$31$	− 8.54816i	− 0.0495256i	−0.999693	−	0.0247628i	$-0.992117\pi$
$31$	− 8.54816i	− 0.0495256i	0.999693	−	0.0247628i	$-0.00788305\pi$
$32$	140.684	0.777178
$33$	24.7526	0.130572
$34$	0	0
$35$	−311.309	−1.50345
$36$	−431.443	−1.99742
$37$	357.982i	1.59059i	0.606221	+	0.795296i	$0.292685\pi$
$37$	357.982i	1.59059i	−0.606221	+	0.795296i	$0.707315\pi$
$38$	267.882	1.14359
$39$	152.992i	0.628164i
$40$	324.049i	1.28091i
$41$	− 194.467i	− 0.740748i	−0.928883	−	0.370374i	$-0.879230\pi$
$41$	− 194.467i	− 0.740748i	0.928883	−	0.370374i	$-0.120770\pi$
$42$	931.044	3.42055
$43$	74.2619	0.263368	0.131684	−	0.991292i	$-0.457962\pi$
$43$	74.2619	0.263368	0.131684	+	0.991292i	$0.457962\pi$
$44$	44.8367i	0.153622i
$45$	372.021i	1.23239i
$46$	359.894i	1.15355i
$47$	23.6130	0.0732831	0.0366416	−	0.999328i	$-0.488334\pi$
$47$	23.6130	0.0732831	0.0366416	+	0.999328i	$0.488334\pi$
$48$	− 125.518i	− 0.377437i
$49$	−339.369	−0.989415
$50$	79.5255	0.224932
$51$	0	0
$52$	−277.130	−0.739058
$53$	−104.330	−0.270393	−0.135197	−	0.990819i	$-0.543167\pi$
$53$	−104.330	−0.270393	−0.135197	+	0.990819i	$0.543167\pi$
$54$	− 150.290i	− 0.378739i
$55$	38.6614	0.0947837
$56$	710.295i	1.69495i
$57$	437.553i	1.01676i
$58$	1336.75i	3.02627i
$59$	−249.363	−0.550243	−0.275122	−	0.961409i	$-0.588718\pi$
$59$	−249.363	−0.550243	−0.275122	+	0.961409i	$0.588718\pi$
$60$	−1256.73	−2.70405
$61$	370.384i	0.777424i	0.921359	+	0.388712i	$0.127080\pi$
$61$	370.384i	0.777424i	−0.921359	+	0.388712i	$0.872920\pi$
$62$	39.9309i	0.0817940i
$63$	815.448i	1.63074i
$64$	−788.781	−1.54059
$65$	238.961i	0.455992i
$66$	−115.626	−0.215646
$67$	939.650	1.71338	0.856691	−	0.515830i	$-0.172517\pi$
$67$	939.650	1.71338	0.856691	+	0.515830i	$0.172517\pi$
$68$	0	0
$69$	−587.843	−1.02562
$70$	1454.21	2.48302
$71$	− 520.197i	− 0.869522i	−0.900546	−	0.434761i	$-0.856833\pi$
$71$	− 520.197i	− 0.869522i	0.900546	−	0.434761i	$-0.143167\pi$
$72$	848.820	1.38937
$73$	348.741i	0.559137i	0.960126	+	0.279568i	$0.0901914\pi$
$73$	348.741i	0.559137i	−0.960126	+	0.279568i	$0.909809\pi$
$74$	− 1672.24i	− 2.62694i
$75$	129.895i	0.199987i
$76$	−792.583	−1.19626
$77$	84.7434	0.125421
$78$	− 714.672i	− 1.03744i
$79$	953.827i	1.35840i	0.733951	+	0.679202i	$0.237674\pi$
$79$	953.827i	1.35840i	−0.733951	+	0.679202i	$0.762326\pi$
$80$	− 196.049i	− 0.273986i
$81$	−597.369	−0.819437
$82$	908.412i	1.22338i
$83$	1414.28	1.87033	0.935166	−	0.354211i	$-0.115250\pi$
$83$	1414.28	1.87033	0.935166	+	0.354211i	$0.115250\pi$
$84$	−2754.68	−3.57809
$85$	0	0
$86$	−346.899	−0.434966
$87$	−2183.41	−2.69065
$88$	− 88.2115i	− 0.106857i
$89$	−486.132	−0.578987	−0.289493	−	0.957180i	$-0.593487\pi$
$89$	−486.132	−0.578987	−0.289493	+	0.957180i	$0.593487\pi$
$90$	− 1737.82i	− 2.03536i
$91$	523.788i	0.603384i
$92$	− 1064.82i	− 1.20668i
$93$	−65.2223	−0.0727230
$94$	−110.303	−0.121031
$95$	683.422i	0.738080i
$96$	− 1073.42i	− 1.14120i
$97$	− 685.281i	− 0.717317i	−0.933469	−	0.358659i	$-0.883234\pi$
$97$	− 685.281i	− 0.717317i	0.933469	−	0.358659i	$-0.116766\pi$
$98$	1585.29	1.63407
$99$	− 101.271i	− 0.102809i
$100$	−235.292	−0.235292
$101$	864.755	0.851944	0.425972	−	0.904736i	$-0.359932\pi$
$101$	864.755	0.851944	0.425972	+	0.904736i	$0.359932\pi$
$102$	0	0
$103$	1880.91	1.79933	0.899665	−	0.436580i	$-0.143810\pi$
$103$	1880.91	1.79933	0.899665	+	0.436580i	$0.143810\pi$
$104$	545.224	0.514073
$105$	2375.28i	2.20765i
$106$	487.355	0.446567
$107$	− 32.8149i	− 0.0296480i	−0.999890	−	0.0148240i	$-0.995281\pi$
$107$	− 32.8149i	− 0.0296480i	0.999890	−	0.0148240i	$-0.00471880\pi$
$108$	444.663i	0.396183i
$109$	− 528.727i	− 0.464613i	−0.972643	−	0.232307i	$-0.925373\pi$
$109$	− 528.727i	− 0.464613i	0.972643	−	0.232307i	$-0.0746273\pi$
$110$	−180.599	−0.156540
$111$	2731.40	2.33561
$112$	− 429.727i	− 0.362548i
$113$	− 414.691i	− 0.345229i	−0.984989	−	0.172614i	$-0.944779\pi$
$113$	− 414.691i	− 0.345229i	0.984989	−	0.172614i	$-0.0552215\pi$
$114$	− 2043.94i	− 1.67923i
$115$	−918.161	−0.744513
$116$	− 3955.03i	− 3.16565i
$117$	625.940	0.494600
$118$	1164.85	0.908754
$119$	0	0
$120$	2472.49	1.88088
$121$	1320.48	0.992093
$122$	− 1730.17i	− 1.28395i
$123$	−1483.78	−1.08771
$124$	− 118.143i	− 0.0855613i
$125$	− 1286.79i	− 0.920751i
$126$	− 3809.19i	− 2.69325i
$127$	−596.093	−0.416494	−0.208247	−	0.978076i	$-0.566776\pi$
$127$	−596.093	−0.416494	−0.208247	+	0.978076i	$0.566776\pi$
$128$	2559.15	1.76718
$129$	− 566.617i	− 0.386728i
$130$	− 1116.26i	− 0.753094i
$131$	121.819i	0.0812472i	0.999175	+	0.0406236i	$0.0129344\pi$
$131$	121.819i	0.0812472i	−0.999175	+	0.0406236i	$0.987066\pi$
$132$	342.103	0.225578
$133$	1498.02i	0.976652i
$134$	−4389.38	−2.82973
$135$	383.420	0.244441
$136$	0	0
$137$	−897.365	−0.559614	−0.279807	−	0.960056i	$-0.590270\pi$
$137$	−897.365	−0.559614	−0.279807	+	0.960056i	$0.590270\pi$
$138$	2745.98	1.69387
$139$	− 2113.61i	− 1.28974i	−0.764292	−	0.644871i	$-0.776911\pi$
$139$	− 2113.61i	− 1.28974i	0.764292	−	0.644871i	$-0.223089\pi$
$140$	−4302.57	−2.59738
$141$	− 180.167i	− 0.107608i
$142$	2429.99i	1.43606i
$143$	− 65.0493i	− 0.0380398i
$144$	−513.534	−0.297184
$145$	−3410.31	−1.95318
$146$	− 1629.07i	− 0.923442i
$147$	2589.38i	1.45285i
$148$	4947.65i	2.74793i
$149$	2580.76	1.41895	0.709476	−	0.704729i	$-0.248932\pi$
$149$	2580.76	1.41895	0.709476	+	0.704729i	$0.248932\pi$
$150$	− 606.778i	− 0.330288i
$151$	−1342.77	−0.723662	−0.361831	−	0.932244i	$-0.617848\pi$
$151$	−1342.77	−0.723662	−0.361831	+	0.932244i	$0.617848\pi$
$152$	1559.32	0.832091
$153$	0	0
$154$	−395.861	−0.207139
$155$	−101.872	−0.0527906
$156$	2114.50i	1.08523i
$157$	−2495.82	−1.26871	−0.634357	−	0.773041i	$-0.718735\pi$
$157$	−2495.82	−1.26871	−0.634357	+	0.773041i	$0.718735\pi$
$158$	− 4455.60i	− 2.24347i
$159$	796.036i	0.397043i
$160$	− 1676.59i	− 0.828413i
$161$	−2012.55	−0.985164
$162$	2790.48	1.35334
$163$	− 1961.58i	− 0.942595i	−0.881974	−	0.471297i	$-0.843786\pi$
$163$	− 1961.58i	− 0.942595i	0.881974	−	0.471297i	$-0.156214\pi$
$164$	− 2687.72i	− 1.27973i
$165$	− 294.986i	− 0.139180i
$166$	−6606.51	−3.08894
$167$	2179.24i	1.00979i	0.863182	+	0.504894i	$0.168468\pi$
$167$	2179.24i	1.00979i	−0.863182	+	0.504894i	$0.831532\pi$
$168$	5419.54	2.48885
$169$	−1794.94	−0.816995
$170$	0	0
$171$	1790.17	0.800571
$172$	1026.37	0.454999
$173$	3111.45i	1.36739i	0.729766	+	0.683697i	$0.239629\pi$
$173$	3111.45i	1.36739i	−0.729766	+	0.683697i	$0.760371\pi$
$174$	10199.4	4.44374
$175$	444.713i	0.192098i
$176$	53.3677i	0.0228565i
$177$	1902.64i	0.807972i
$178$	2270.86	0.956226
$179$	−810.106	−0.338269	−0.169135	−	0.985593i	$-0.554097\pi$
$179$	−810.106	−0.338269	−0.169135	+	0.985593i	$0.554097\pi$
$180$	5141.68i	2.12910i
$181$	3356.23i	1.37827i	0.724634	+	0.689134i	$0.242009\pi$
$181$	3356.23i	1.37827i	−0.724634	+	0.689134i	$0.757991\pi$
$182$	− 2446.77i	− 0.996519i
$183$	2826.03	1.14156
$184$	2094.92i	0.839343i
$185$	4266.22	1.69545
$186$	304.672	0.120106
$187$	0	0
$188$	326.353	0.126605
$189$	840.434	0.323453
$190$	− 3192.46i	− 1.21898i
$191$	1338.41	0.507038	0.253519	−	0.967330i	$-0.418412\pi$
$191$	1338.41	0.507038	0.253519	+	0.967330i	$0.418412\pi$
$192$	6018.39i	2.26219i
$193$	227.465i	0.0848358i	0.999100	+	0.0424179i	$0.0135061\pi$
$193$	227.465i	0.0848358i	−0.999100	+	0.0424179i	$0.986494\pi$
$194$	3201.15i	1.18468i
$195$	1823.27	0.669575
$196$	−4690.40	−1.70933
$197$	− 815.549i	− 0.294952i	−0.989066	−	0.147476i	$-0.952885\pi$
$197$	− 815.549i	− 0.294952i	0.989066	−	0.147476i	$-0.0471148\pi$
$198$	473.064i	0.169794i
$199$	1866.90i	0.665030i	0.943098	+	0.332515i	$0.107897\pi$
$199$	1866.90i	0.665030i	−0.943098	+	0.332515i	$0.892103\pi$
$200$	462.912	0.163664
$201$	− 7169.52i	− 2.51591i
$202$	−4039.52	−1.40703
$203$	−7475.19	−2.58451
$204$	0	0
$205$	−2317.54	−0.789581
$206$	−8786.25	−2.97168
$207$	2405.05i	0.807549i
$208$	−329.859	−0.109960
$209$	− 186.039i	− 0.0615721i
$210$	− 11095.6i	− 3.64605i
$211$	1102.88i	0.359836i	0.983682	+	0.179918i	$0.0575833\pi$
$211$	1102.88i	0.359836i	−0.983682	+	0.179918i	$0.942417\pi$
$212$	−1441.94	−0.467135
$213$	−3969.10	−1.27680
$214$	153.288i	0.0489651i
$215$	− 885.008i	− 0.280731i
$216$	− 874.828i	− 0.275576i
$217$	−223.297	−0.0698542
$218$	2469.84i	0.767332i
$219$	2660.89	0.821032
$220$	534.337	0.163750
$221$	0	0
$222$	−12759.2	−3.85738
$223$	568.848	0.170820	0.0854100	−	0.996346i	$-0.472780\pi$
$223$	568.848	0.170820	0.0854100	+	0.996346i	$0.472780\pi$
$224$	− 3674.98i	− 1.09618i
$225$	531.442	0.157464
$226$	1937.14i	0.570163i
$227$	− 2106.99i	− 0.616061i	−0.951377	−	0.308030i	$-0.900330\pi$
$227$	− 2106.99i	− 0.616061i	0.951377	−	0.308030i	$-0.0996698\pi$
$228$	6047.39i	1.75657i
$229$	−4336.30	−1.25131	−0.625656	−	0.780099i	$-0.715169\pi$
$229$	−4336.30	−1.25131	−0.625656	+	0.780099i	$0.715169\pi$
$230$	4289.00	1.22960
$231$	− 646.591i	− 0.184167i
$232$	7781.11i	2.20196i
$233$	− 4517.39i	− 1.27014i	−0.772453	−	0.635072i	$-0.780970\pi$
$233$	− 4517.39i	− 1.27014i	0.772453	−	0.635072i	$-0.219030\pi$
$234$	−2923.95	−0.816856
$235$	− 281.405i	− 0.0781143i
$236$	−3446.43	−0.950609
$237$	7277.69	1.99467
$238$	0	0
$239$	5300.88	1.43467	0.717333	−	0.696731i	$-0.245363\pi$
$239$	5300.88	1.43467	0.717333	+	0.696731i	$0.245363\pi$
$240$	−1495.85	−0.402319
$241$	− 1368.82i	− 0.365864i	−0.983126	−	0.182932i	$-0.941441\pi$
$241$	− 1368.82i	− 0.365864i	0.983126	−	0.182932i	$-0.0585588\pi$
$242$	−6168.32	−1.63849
$243$	5426.60i	1.43258i
$244$	5119.05i	1.34309i
$245$	4044.40i	1.05464i
$246$	6931.17	1.79640
$247$	1149.88	0.296216
$248$	232.435i	0.0595146i
$249$	− 10790.9i	− 2.74638i
$250$	6010.96i	1.52067i
$251$	−5547.63	−1.39507	−0.697536	−	0.716549i	$-0.745720\pi$
$251$	−5547.63	−1.39507	−0.697536	+	0.716549i	$0.745720\pi$
$252$	11270.3i	2.81730i
$253$	249.939	0.0621088
$254$	2784.52	0.687860
$255$	0	0
$256$	−5644.28	−1.37800
$257$	193.949	0.0470748	0.0235374	−	0.999723i	$-0.492507\pi$
$257$	193.949	0.0470748	0.0235374	+	0.999723i	$0.492507\pi$
$258$	2646.83i	0.638700i
$259$	9351.29	2.24348
$260$	3302.67i	0.787780i
$261$	8933.04i	2.11855i
$262$	− 569.052i	− 0.134184i
$263$	1345.63	0.315494	0.157747	−	0.987480i	$-0.449577\pi$
$263$	1345.63	0.315494	0.157747	+	0.987480i	$0.449577\pi$
$264$	−673.052	−0.156907
$265$	1243.34i	0.288218i
$266$	− 6997.67i	− 1.61299i
$267$	3709.18i	0.850180i
$268$	12986.8	2.96007
$269$	− 3083.04i	− 0.698797i	−0.936974	−	0.349398i	$-0.886386\pi$
$269$	− 3083.04i	− 0.698797i	0.936974	−	0.349398i	$-0.113614\pi$
$270$	−1791.07	−0.403707
$271$	−422.163	−0.0946294	−0.0473147	−	0.998880i	$-0.515066\pi$
$271$	−422.163	−0.0946294	−0.0473147	+	0.998880i	$0.515066\pi$
$272$	0	0
$273$	3996.50	0.886004
$274$	4191.85	0.924230
$275$	− 55.2288i	− 0.0121106i
$276$	−8124.54	−1.77188
$277$	− 8260.00i	− 1.79168i	−0.444377	−	0.895840i	$-0.646575\pi$
$277$	− 8260.00i	− 1.79168i	0.444377	−	0.895840i	$-0.353425\pi$
$278$	9873.28i	2.13007i
$279$	266.845i	0.0572602i
$280$	8464.86	1.80669
$281$	−3321.91	−0.705226	−0.352613	−	0.935769i	$-0.614707\pi$
$281$	−3321.91	−0.705226	−0.352613	+	0.935769i	$0.614707\pi$
$282$	841.611i	0.177721i
$283$	7954.43i	1.67082i	0.549629	+	0.835409i	$0.314769\pi$
$283$	7954.43i	1.67082i	−0.549629	+	0.835409i	$0.685231\pi$
$284$	− 7189.61i	− 1.50220i
$285$	5214.50	1.08379
$286$	303.864i	0.0628246i
$287$	−5079.91	−1.04480
$288$	−4391.69	−0.898552
$289$	0	0
$290$	15930.5	3.22577
$291$	−5228.69	−1.05330
$292$	4819.92i	0.965974i
$293$	−1171.99	−0.233681	−0.116841	−	0.993151i	$-0.537277\pi$
$293$	−1171.99	−0.233681	−0.116841	+	0.993151i	$0.537277\pi$
$294$	− 12095.8i	− 2.39945i
$295$	2971.76i	0.586517i
$296$	− 9733.98i	− 1.91141i
$297$	−104.374	−0.0203918
$298$	−12055.5	−2.34347
$299$	1544.84i	0.298798i
$300$	1795.27i	0.345500i
$301$	− 1939.88i	− 0.371472i
$302$	6272.46	1.19516
$303$	− 6598.07i	− 1.25099i
$304$	−943.387	−0.177983
$305$	4414.01	0.828675
$306$	0	0
$307$	865.763	0.160950	0.0804751	−	0.996757i	$-0.474356\pi$
$307$	865.763	0.160950	0.0804751	+	0.996757i	$0.474356\pi$
$308$	1171.23	0.216679
$309$	− 14351.3i	− 2.64212i
$310$	475.872	0.0871862
$311$	6994.83i	1.27537i	0.770297	+	0.637685i	$0.220108\pi$
$311$	6994.83i	1.27537i	−0.770297	+	0.637685i	$0.779892\pi$
$312$	− 4160.05i	− 0.754861i
$313$	− 3442.33i	− 0.621635i	−0.950470	−	0.310818i	$-0.899397\pi$
$313$	− 3442.33i	− 0.621635i	0.950470	−	0.310818i	$-0.100603\pi$
$314$	11658.7	2.09534
$315$	9718.02	1.73825
$316$	13182.8i	2.34680i
$317$	− 2066.15i	− 0.366078i	−0.983106	−	0.183039i	$-0.941407\pi$
$317$	− 2066.15i	− 0.366078i	0.983106	−	0.183039i	$-0.0585935\pi$
$318$	− 3718.52i	− 0.655736i
$319$	928.344	0.162938
$320$	9400.22i	1.64215i
$321$	−250.377	−0.0435349
$322$	9401.21	1.62705
$323$	0	0
$324$	−8256.20	−1.41567
$325$	341.362	0.0582627
$326$	9163.11i	1.55674i
$327$	−4034.18	−0.682234
$328$	5287.80i	0.890152i
$329$	− 616.823i	− 0.103363i
$330$	1377.96i	0.229862i
$331$	−9027.44	−1.49907	−0.749536	−	0.661964i	$-0.769723\pi$
$331$	−9027.44	−1.49907	−0.749536	+	0.661964i	$0.769723\pi$
$332$	19546.7	3.23121
$333$	− 11175.0i	− 1.83900i
$334$	− 10179.8i	− 1.66771i
$335$	− 11198.2i	− 1.82633i
$336$	−3278.81	−0.532362
$337$	204.309i	0.0330250i	0.999864	+	0.0165125i	$0.00525633\pi$
$337$	204.309i	0.0330250i	−0.999864	+	0.0165125i	$0.994744\pi$
$338$	8384.67	1.34931
$339$	−3164.09	−0.506931
$340$	0	0
$341$	27.7312	0.00440390
$342$	−8362.39	−1.32218
$343$	− 94.8397i	− 0.0149296i
$344$	−2019.27	−0.316488
$345$	7005.56i	1.09324i
$346$	− 14534.5i	− 2.25832i
$347$	− 143.063i	− 0.0221326i	−0.999939	−	0.0110663i	$-0.996477\pi$
$347$	− 143.063i	− 0.0221326i	0.999939	−	0.0110663i	$-0.00352259\pi$
$348$	−30176.8	−4.64841
$349$	3998.42	0.613268	0.306634	−	0.951828i	$-0.400797\pi$
$349$	3998.42	0.613268	0.306634	+	0.951828i	$0.400797\pi$
$350$	− 2077.38i	− 0.317259i
$351$	− 645.119i	− 0.0981024i
$352$	456.396i	0.0691079i
$353$	−5809.57	−0.875956	−0.437978	−	0.898986i	$-0.644305\pi$
$353$	−5809.57	−0.875956	−0.437978	+	0.898986i	$0.644305\pi$
$354$	− 8887.77i	− 1.33441i
$355$	−6199.39	−0.926844
$356$	−6718.79	−1.00027
$357$	0	0
$358$	3784.24	0.558668
$359$	4895.37	0.719687	0.359844	−	0.933013i	$-0.382830\pi$
$359$	4895.37	0.719687	0.359844	+	0.933013i	$0.382830\pi$
$360$	− 10115.7i	− 1.48096i
$361$	−3570.37	−0.520538
$362$	− 15677.9i	− 2.27628i
$363$	− 10075.2i	− 1.45678i
$364$	7239.24i	1.04242i
$365$	4156.08	0.595998
$366$	−13201.2	−1.88535
$367$	− 528.151i	− 0.0751206i	−0.999294	−	0.0375603i	$-0.988041\pi$
$367$	− 528.151i	− 0.0751206i	0.999294	−	0.0375603i	$-0.0119586\pi$
$368$	− 1267.42i	− 0.179535i
$369$	6070.62i	0.856433i
$370$	−19928.7	−2.80012
$371$	2725.33i	0.381380i
$372$	−901.433	−0.125637
$373$	−10113.5	−1.40390	−0.701950	−	0.712226i	$-0.747687\pi$
$373$	−10113.5	−1.40390	−0.701950	+	0.712226i	$0.747687\pi$
$374$	0	0
$375$	−9818.18	−1.35202
$376$	−642.066	−0.0880639
$377$	5737.98i	0.783875i
$378$	−3925.91	−0.534198
$379$	729.385i	0.0988548i	0.998778	+	0.0494274i	$0.0157396\pi$
$379$	729.385i	0.0988548i	−0.998778	+	0.0494274i	$0.984260\pi$
$380$	9445.52i	1.27512i
$381$	4548.18i	0.611576i
$382$	−6252.12	−0.837399
$383$	1608.08	0.214540	0.107270	−	0.994230i	$-0.465789\pi$
$383$	1608.08	0.214540	0.107270	+	0.994230i	$0.465789\pi$
$384$	− 19526.3i	− 2.59491i
$385$	− 1009.92i	− 0.133689i
$386$	− 1062.56i	− 0.140111i
$387$	−2318.21	−0.304499
$388$	− 9471.22i	− 1.23925i
$389$	−9824.09	−1.28047	−0.640233	−	0.768181i	$-0.721162\pi$
$389$	−9824.09	−1.28047	−0.640233	+	0.768181i	$0.721162\pi$
$390$	−8517.02	−1.10584
$391$	0	0
$392$	9227.87	1.18897
$393$	929.478	0.119303
$394$	3809.66i	0.487127i
$395$	11367.1	1.44796
$396$	− 1399.65i	− 0.177614i
$397$	− 2876.88i	− 0.363694i	−0.983327	−	0.181847i	$-0.941792\pi$
$397$	− 2876.88i	− 0.363694i	0.983327	−	0.181847i	$-0.0582076\pi$
$398$	− 8720.82i	− 1.09833i
$399$	11429.9	1.43411
$400$	−280.061	−0.0350076
$401$	6515.91i	0.811444i	0.913996	+	0.405722i	$0.132980\pi$
$401$	6515.91i	0.811444i	−0.913996	+	0.405722i	$0.867020\pi$
$402$	33490.9i	4.15516i
$403$	171.403i	0.0211866i
$404$	11951.7	1.47183
$405$	7119.09i	0.873457i
$406$	34918.8	4.26845
$407$	−1161.34	−0.141438
$408$	0	0
$409$	−8870.10	−1.07237	−0.536183	−	0.844101i	$-0.680134\pi$
$409$	−8870.10	−1.07237	−0.536183	+	0.844101i	$0.680134\pi$
$410$	10825.9	1.30403
$411$	6846.89i	0.821732i
$412$	25995.9	3.10855
$413$	6513.92i	0.776099i
$414$	− 11234.7i	− 1.33371i
$415$	− 16854.5i	− 1.99363i
$416$	−2820.93	−0.332469
$417$	−16126.8	−1.89385
$418$	869.041i	0.101689i
$419$	1009.53i	0.117706i	0.998267	+	0.0588531i	$0.0187443\pi$
$419$	1009.53i	0.117706i	−0.998267	+	0.0588531i	$0.981256\pi$
$420$	32828.6i	3.81398i
$421$	−3253.60	−0.376652	−0.188326	−	0.982107i	$-0.560306\pi$
$421$	−3253.60	−0.376652	−0.188326	+	0.982107i	$0.560306\pi$
$422$	− 5151.87i	− 0.594287i
$423$	−737.119	−0.0847280
$424$	2836.86	0.324930
$425$	0	0
$426$	18540.8	2.10870
$427$	9675.25	1.09653
$428$	− 453.532i	− 0.0512203i
$429$	−496.325	−0.0558573
$430$	4134.13i	0.463640i
$431$	2352.51i	0.262915i	0.991322	+	0.131457i	$0.0419656\pi$
$431$	2352.51i	0.262915i	−0.991322	+	0.131457i	$0.958034\pi$
$432$	529.269i	0.0589455i
$433$	5860.51	0.650434	0.325217	−	0.945639i	$-0.394563\pi$
$433$	5860.51	0.650434	0.325217	+	0.945639i	$0.394563\pi$
$434$	1043.08	0.115368
$435$	26020.6i	2.86803i
$436$	− 7307.50i	− 0.802674i
$437$	4418.20i	0.483641i
$438$	−12429.8	−1.35597
$439$	2894.17i	0.314650i	0.987547	+	0.157325i	$0.0502870\pi$
$439$	2894.17i	0.314650i	−0.987547	+	0.157325i	$0.949713\pi$
$440$	−1051.25	−0.113901
$441$	10594.0	1.14394
$442$	0	0
$443$	−8256.85	−0.885541	−0.442771	−	0.896635i	$-0.646004\pi$
$443$	−8256.85	−0.885541	−0.442771	+	0.896635i	$0.646004\pi$
$444$	37750.5	4.03504
$445$	5793.42i	0.617156i
$446$	−2657.25	−0.282118
$447$	− 19691.1i	− 2.08358i
$448$	20604.7i	2.17295i
$449$	− 15487.1i	− 1.62779i	−0.581009	−	0.813897i	$-0.697342\pi$
$449$	− 15487.1i	− 1.62779i	0.581009	−	0.813897i	$-0.302658\pi$
$450$	−2482.52	−0.260060
$451$	630.874	0.0658685
$452$	− 5731.42i	− 0.596423i
$453$	10245.3i	1.06262i
$454$	9842.35i	1.01745i
$455$	6242.19	0.643162
$456$	− 11897.6i	− 1.22184i
$457$	16055.6	1.64343	0.821716	−	0.569897i	$-0.193017\pi$
$457$	16055.6	1.64343	0.821716	+	0.569897i	$0.193017\pi$
$458$	20256.1	2.06661
$459$	0	0
$460$	−12689.8	−1.28623
$461$	−14064.0	−1.42088	−0.710440	−	0.703758i	$-0.751504\pi$
$461$	−14064.0	−1.42088	−0.710440	+	0.703758i	$0.751504\pi$
$462$	3020.41i	0.304161i
$463$	−8071.30	−0.810162	−0.405081	−	0.914281i	$-0.632757\pi$
$463$	−8071.30	−0.810162	−0.405081	+	0.914281i	$0.632757\pi$
$464$	− 4707.55i	− 0.470997i
$465$	777.280i	0.0775172i
$466$	21102.0i	2.09771i
$467$	−8582.41	−0.850421	−0.425211	−	0.905094i	$-0.639800\pi$
$467$	−8582.41	−0.850421	−0.425211	+	0.905094i	$0.639800\pi$
$468$	8651.07	0.854479
$469$	− 24545.7i	− 2.41667i
$470$	1314.53i	0.129010i
$471$	19043.1i	1.86297i
$472$	6780.49	0.661224
$473$	240.914i	0.0234191i
$474$	−33996.2	−3.29429
$475$	976.286	0.0943054
$476$	0	0
$477$	3256.84	0.312621
$478$	−24761.9	−2.36942
$479$	− 6320.96i	− 0.602948i	−0.953474	−	0.301474i	$-0.902521\pi$
$479$	− 6320.96i	− 0.602948i	0.953474	−	0.301474i	$-0.0974787\pi$
$480$	−12792.4	−1.21643
$481$	− 7178.07i	− 0.680441i
$482$	6394.13i	0.604242i
$483$	15355.8i	1.44661i
$484$	18250.2	1.71396
$485$	−8166.77	−0.764606
$486$	− 25349.2i	− 2.36597i
$487$	− 7336.47i	− 0.682643i	−0.939947	−	0.341321i	$-0.889126\pi$
$487$	− 7336.47i	− 0.682643i	0.939947	−	0.341321i	$-0.110874\pi$
$488$	− 10071.2i	− 0.934225i
$489$	−14966.8	−1.38410
$490$	− 18892.6i	− 1.74179i
$491$	−6672.53	−0.613294	−0.306647	−	0.951823i	$-0.599207\pi$
$491$	−6672.53	−0.613294	−0.306647	+	0.951823i	$0.599207\pi$
$492$	−20507.2	−1.87914
$493$	0	0
$494$	−5371.43	−0.489215
$495$	−1206.88	−0.109586
$496$	− 140.623i	− 0.0127301i
$497$	−13588.7	−1.22643
$498$	50407.6i	4.53578i
$499$	− 17920.9i	− 1.60772i	−0.594821	−	0.803858i	$-0.702777\pi$
$499$	− 17920.9i	− 1.60772i	0.594821	−	0.803858i	$-0.297223\pi$
$500$	− 17784.6i	− 1.59070i
$501$	16627.6	1.48276
$502$	25914.6	2.30403
$503$	11325.3i	1.00392i	0.864891	+	0.501959i	$0.167387\pi$
$503$	11325.3i	1.00392i	−0.864891	+	0.501959i	$0.832613\pi$
$504$	− 22173.0i	− 1.95965i
$505$	− 10305.6i	− 0.908108i
$506$	−1167.54	−0.102576
$507$	13695.4i	1.19967i
$508$	−8238.56	−0.719541
$509$	−8313.78	−0.723972	−0.361986	−	0.932184i	$-0.617901\pi$
$509$	−8313.78	−0.723972	−0.361986	+	0.932184i	$0.617901\pi$
$510$	0	0
$511$	9109.87	0.788644
$512$	5892.85	0.508651
$513$	− 1845.02i	− 0.158791i
$514$	−905.993	−0.0777464
$515$	− 22415.5i	− 1.91795i
$516$	− 7831.17i	− 0.668117i
$517$	76.6033i	0.00651646i
$518$	−43682.6	−3.70521
$519$	23740.3	2.00787
$520$	− 6497.65i	− 0.547963i
$521$	− 5121.64i	− 0.430677i	−0.976539	−	0.215339i	$-0.930914\pi$
$521$	− 5121.64i	− 0.430677i	0.976539	−	0.215339i	$-0.0690856\pi$
$522$	− 41728.8i	− 3.49889i
$523$	13378.5	1.11855	0.559275	−	0.828982i	$-0.311080\pi$
$523$	13378.5	1.11855	0.559275	+	0.828982i	$0.311080\pi$
$524$	1683.65i	0.140364i
$525$	3393.15	0.282075
$526$	−6285.81	−0.521054
$527$	0	0
$528$	407.195	0.0335623
$529$	6231.26	0.512144
$530$	− 5808.01i	− 0.476007i
$531$	7784.29	0.636176
$532$	20704.0i	1.68728i
$533$	3899.35i	0.316885i
$534$	− 17326.6i	− 1.40411i
$535$	−391.068	−0.0316025
$536$	−25550.2	−2.05896
$537$	6181.10i	0.496712i
$538$	14401.8i	1.15410i
$539$	− 1100.95i	− 0.0879804i
$540$	5299.23	0.422301
$541$	9906.81i	0.787296i	0.919261	+	0.393648i	$0.128787\pi$
$541$	9906.81i	0.787296i	−0.919261	+	0.393648i	$0.871213\pi$
$542$	1972.04	0.156285
$543$	25608.0	2.02384
$544$	0	0
$545$	−6301.05	−0.495243
$546$	−18668.8	−1.46328
$547$	16399.6i	1.28189i	0.767585	+	0.640947i	$0.221458\pi$
$547$	16399.6i	1.28189i	−0.767585	+	0.640947i	$0.778542\pi$
$548$	−12402.4	−0.966798
$549$	− 11562.2i	− 0.898836i
$550$	257.990i	0.0200013i
$551$	16410.4i	1.26880i
$552$	15984.2	1.23248
$553$	24916.1	1.91598
$554$	38584.8i	2.95905i
$555$	− 32551.2i	− 2.48959i
$556$	− 29212.0i	− 2.22818i
$557$	22044.3	1.67692	0.838461	−	0.544962i	$-0.183456\pi$
$557$	22044.3	1.67692	0.838461	+	0.544962i	$0.183456\pi$
$558$	− 1246.51i	− 0.0945681i
$559$	−1489.06	−0.112666
$560$	−5121.22	−0.386448
$561$	0	0
$562$	15517.6	1.16472
$563$	−12048.8	−0.901947	−0.450973	−	0.892537i	$-0.648923\pi$
$563$	−12048.8	−0.901947	−0.450973	+	0.892537i	$0.648923\pi$
$564$	− 2490.07i	− 0.185906i
$565$	−4942.04	−0.367988
$566$	− 37157.4i	− 2.75944i
$567$	15604.6i	1.15579i
$568$	14144.8i	1.04490i
$569$	23785.4	1.75243	0.876217	−	0.481916i	$-0.160059\pi$
$569$	23785.4	1.75243	0.876217	+	0.481916i	$0.160059\pi$
$570$	−24358.4	−1.78993
$571$	10878.3i	0.797271i	0.917110	+	0.398635i	$0.130516\pi$
$571$	10878.3i	0.797271i	−0.917110	+	0.398635i	$0.869484\pi$
$572$	− 899.041i	− 0.0657182i
$573$	− 10212.1i	− 0.744531i
$574$	23729.7	1.72554
$575$	1311.62i	0.0951274i
$576$	24623.1	1.78119
$577$	6315.86	0.455689	0.227845	−	0.973698i	$-0.426832\pi$
$577$	6315.86	0.455689	0.227845	+	0.973698i	$0.426832\pi$
$578$	0	0
$579$	1735.56	0.124572
$580$	−47133.7	−3.37434
$581$	− 36944.1i	− 2.63804i
$582$	24424.7	1.73958
$583$	− 338.459i	− 0.0240438i
$584$	− 9482.69i	− 0.671912i
$585$	− 7459.58i	− 0.527206i
$586$	5474.72	0.385936
$587$	−18192.1	−1.27916	−0.639581	−	0.768724i	$-0.720892\pi$
$587$	−18192.1	−1.27916	−0.639581	+	0.768724i	$0.720892\pi$
$588$	35787.7i	2.50996i
$589$	490.207i	0.0342931i
$590$	− 13882.0i	− 0.968663i
$591$	−6222.63	−0.433104
$592$	5889.03i	0.408848i
$593$	9828.72	0.680636	0.340318	−	0.940310i	$-0.389465\pi$
$593$	9828.72	0.680636	0.340318	+	0.940310i	$0.389465\pi$
$594$	487.559	0.0336781
$595$	0	0
$596$	35668.5	2.45140
$597$	14244.4	0.976524
$598$	− 7216.40i	− 0.493479i
$599$	4662.57	0.318043	0.159021	−	0.987275i	$-0.449166\pi$
$599$	4662.57	0.318043	0.159021	+	0.987275i	$0.449166\pi$
$600$	− 3532.01i	− 0.240323i
$601$	− 21658.6i	− 1.47000i	−0.678066	−	0.735001i	$-0.737182\pi$
$601$	− 21658.6i	− 1.47000i	0.678066	−	0.735001i	$-0.262818\pi$
$602$	9061.76i	0.613504i
$603$	−29332.8	−1.98097
$604$	−18558.3	−1.25021
$605$	− 15736.6i	− 1.05750i
$606$	30821.5i	2.06607i
$607$	25764.7i	1.72283i	0.507902	+	0.861415i	$0.330421\pi$
$607$	25764.7i	1.72283i	−0.507902	+	0.861415i	$0.669579\pi$
$608$	−8067.76	−0.538143
$609$	57035.6i	3.79507i
$610$	−20619.1	−1.36860
$611$	−473.475	−0.0313498
$612$	0	0
$613$	16018.1	1.05541	0.527705	−	0.849428i	$-0.323053\pi$
$613$	16018.1	1.05541	0.527705	+	0.849428i	$0.323053\pi$
$614$	−4044.23	−0.265817
$615$	17682.8i	1.15941i
$616$	−2304.28	−0.150718
$617$	22250.3i	1.45180i	0.687798	+	0.725902i	$0.258578\pi$
$617$	22250.3i	1.45180i	−0.687798	+	0.725902i	$0.741422\pi$
$618$	67038.9i	4.36360i
$619$	3765.95i	0.244534i	0.992497	+	0.122267i	$0.0390164\pi$
$619$	3765.95i	0.244534i	−0.992497	+	0.122267i	$0.960984\pi$
$620$	−1407.96	−0.0912018
$621$	2478.74	0.160175
$622$	− 32674.9i	− 2.10634i
$623$	12698.8i	0.816642i
$624$	2516.82i	0.161464i
$625$	−17463.2	−1.11765
$626$	16080.1i	1.02666i
$627$	−1419.47	−0.0904120
$628$	−34494.5	−2.19185
$629$	0	0
$630$	−45395.7	−2.87080
$631$	20806.5	1.31267	0.656334	−	0.754470i	$-0.272106\pi$
$631$	20806.5	1.31267	0.656334	+	0.754470i	$0.272106\pi$
$632$	− 25935.7i	− 1.63239i
$633$	8414.96	0.528380
$634$	9651.60i	0.604596i
$635$	7103.88i	0.443951i
$636$	11002.0i	0.685937i
$637$	6804.85	0.423262
$638$	−4336.56	−0.269100
$639$	16238.8i	1.00532i
$640$	− 30498.4i	− 1.88368i
$641$	2439.58i	0.150324i	0.997171	+	0.0751620i	$0.0239474\pi$
$641$	2439.58i	0.150324i	−0.997171	+	0.0751620i	$0.976053\pi$
$642$	1169.58	0.0719000
$643$	− 19320.1i	− 1.18493i	−0.805595	−	0.592466i	$-0.798154\pi$
$643$	− 19320.1i	− 1.18493i	0.805595	−	0.592466i	$-0.201846\pi$
$644$	−27815.4	−1.70199
$645$	−6752.60	−0.412222
$646$	0	0
$647$	−14067.1	−0.854766	−0.427383	−	0.904071i	$-0.640564\pi$
$647$	−14067.1	−0.854766	−0.427383	+	0.904071i	$0.640564\pi$
$648$	16243.2	0.984712
$649$	− 808.963i	− 0.0489285i
$650$	−1594.60	−0.0962238
$651$	1703.75i	0.102573i
$652$	− 27110.9i	− 1.62844i
$653$	− 15893.7i	− 0.952478i	−0.879316	−	0.476239i	$-0.842000\pi$
$653$	− 15893.7i	− 0.952478i	0.879316	−	0.476239i	$-0.158000\pi$
$654$	18844.8	1.12674
$655$	1451.77	0.0866033
$656$	− 3199.11i	− 0.190403i
$657$	− 10886.5i	− 0.646459i
$658$	2881.36i	0.170710i
$659$	−9653.54	−0.570635	−0.285318	−	0.958433i	$-0.592099\pi$
$659$	−9653.54	−0.570635	−0.285318	+	0.958433i	$0.592099\pi$
$660$	− 4076.98i	− 0.240449i
$661$	−5389.06	−0.317111	−0.158555	−	0.987350i	$-0.550684\pi$
$661$	−5389.06	−0.317111	−0.158555	+	0.987350i	$0.550684\pi$
$662$	42169.7	2.47579
$663$	0	0
$664$	−38456.0	−2.24757
$665$	17852.5	1.04104
$666$	52201.7i	3.03720i
$667$	−22047.0	−1.27986
$668$	30119.1i	1.74453i
$669$	− 4340.30i	− 0.250831i
$670$	52309.9i	3.01628i
$671$	−1201.57	−0.0691297
$672$	−28040.1	−1.60963
$673$	3032.18i	0.173673i	0.996223	+	0.0868366i	$0.0276758\pi$
$673$	3032.18i	0.173673i	−0.996223	+	0.0868366i	$0.972324\pi$
$674$	− 954.386i	− 0.0545424i
$675$	− 547.726i	− 0.0312326i
$676$	−24807.7	−1.41145
$677$	− 22029.2i	− 1.25059i	−0.780388	−	0.625295i	$-0.784979\pi$
$677$	− 22029.2i	− 1.25059i	0.780388	−	0.625295i	$-0.215021\pi$
$678$	14780.4	0.837222
$679$	−17901.1	−1.01175
$680$	0	0
$681$	−16076.3	−0.904618
$682$	−129.540	−0.00727326
$683$	− 9040.72i	− 0.506491i	−0.967402	−	0.253246i	$-0.918502\pi$
$683$	− 9040.72i	− 0.506491i	0.967402	−	0.253246i	$-0.0814980\pi$
$684$	24741.8	1.38308
$685$	10694.3i	0.596506i
$686$	443.023i	0.0246570i
$687$	33085.9i	1.83742i
$688$	1221.65	0.0676964
$689$	2091.97	0.115672
$690$	− 32725.0i	− 1.80553i
$691$	22863.5i	1.25871i	0.777118	+	0.629355i	$0.216681\pi$
$691$	22863.5i	1.25871i	−0.777118	+	0.629355i	$0.783319\pi$
$692$	43003.1i	2.36233i
$693$	−2645.41	−0.145008
$694$	668.288i	0.0365531i
$695$	−25188.7	−1.37477
$696$	59369.7	3.23334
$697$	0	0
$698$	−18677.8	−1.01284
$699$	−34467.6	−1.86507
$700$	6146.34i	0.331871i
$701$	1753.00	0.0944507	0.0472253	−	0.998884i	$-0.484962\pi$
$701$	1753.00	0.0944507	0.0472253	+	0.998884i	$0.484962\pi$
$702$	3013.54i	0.162021i
$703$	− 20529.1i	− 1.10138i
$704$	− 2558.90i	− 0.136992i
$705$	−2147.12	−0.114702
$706$	27138.2	1.44668
$707$	− 22589.3i	− 1.20164i
$708$	26296.2i	1.39587i
$709$	11547.0i	0.611645i	0.952089	+	0.305823i	$0.0989315\pi$
$709$	11547.0i	0.611645i	−0.952089	+	0.305823i	$0.901069\pi$
$710$	28959.2	1.53073
$711$	− 29775.3i	− 1.57055i
$712$	13218.5	0.695765
$713$	−658.582	−0.0345920
$714$	0	0
$715$	−775.218	−0.0405476
$716$	−11196.4	−0.584399
$717$	− 40445.6i	− 2.10665i
$718$	−22867.7	−1.18860
$719$	− 10289.8i	− 0.533720i	−0.963735	−	0.266860i	$-0.914014\pi$
$719$	− 10289.8i	− 0.533720i	0.963735	−	0.266860i	$-0.0859861\pi$
$720$	6119.99i	0.316776i
$721$	− 49133.4i	− 2.53790i
$722$	16678.2	0.859695
$723$	−10444.0	−0.537231
$724$	46386.2i	2.38112i
$725$	4871.72i	0.249560i
$726$	47064.2i	2.40595i
$727$	2950.10	0.150499	0.0752497	−	0.997165i	$-0.476025\pi$
$727$	2950.10	0.150499	0.0752497	+	0.997165i	$0.476025\pi$
$728$	− 14242.5i	− 0.725083i
$729$	25275.9	1.28415
$730$	−19414.2	−0.984320
$731$	0	0
$732$	39058.3	1.97218
$733$	−24348.2	−1.22691	−0.613453	−	0.789731i	$-0.710220\pi$
$733$	−24348.2	−1.22691	−0.613453	+	0.789731i	$0.710220\pi$
$734$	2467.15i	0.124065i
$735$	30858.7	1.54863
$736$	− 10838.8i	− 0.542833i
$737$	3048.33i	0.152357i
$738$	− 28357.6i	− 1.41444i
$739$	29233.5	1.45517	0.727585	−	0.686017i	$-0.240642\pi$
$739$	29233.5	1.45517	0.727585	+	0.686017i	$0.240642\pi$
$740$	58963.1	2.92909
$741$	− 8773.59i	− 0.434961i
$742$	− 12730.8i	− 0.629868i
$743$	15340.6i	0.757457i	0.925508	+	0.378729i	$0.123639\pi$
$743$	15340.6i	0.757457i	−0.925508	+	0.378729i	$0.876361\pi$
$744$	1773.47	0.0873908
$745$	− 30755.9i	− 1.51250i
$746$	47242.9	2.31861
$747$	−44149.2	−2.16243
$748$	0	0
$749$	−857.197	−0.0418175
$750$	45863.5	2.23293
$751$	39862.6i	1.93689i	0.249223	+	0.968446i	$0.419825\pi$
$751$	39862.6i	1.93689i	−0.249223	+	0.968446i	$0.580175\pi$
$752$	388.448	0.0188368
$753$	42328.3i	2.04851i
$754$	− 26803.7i	− 1.29461i
$755$	16002.3i	0.771369i
$756$	11615.6	0.558802
$757$	−26375.1	−1.26634	−0.633169	−	0.774013i	$-0.718246\pi$
$757$	−26375.1	−1.26634	−0.633169	+	0.774013i	$0.718246\pi$
$758$	− 3407.17i	− 0.163264i
$759$	− 1907.03i	− 0.0912000i
$760$	− 18583.1i	− 0.886946i
$761$	7848.63	0.373867	0.186933	−	0.982373i	$-0.440145\pi$
$761$	7848.63	0.373867	0.186933	+	0.982373i	$0.440145\pi$
$762$	− 21245.9i	− 1.01005i
$763$	−13811.5	−0.655322
$764$	18498.1	0.875967
$765$	0	0
$766$	−7511.78	−0.354323
$767$	5000.10	0.235389
$768$	43065.8i	2.02344i
$769$	31818.9	1.49209	0.746046	−	0.665895i	$-0.231950\pi$
$769$	31818.9	1.49209	0.746046	+	0.665895i	$0.231950\pi$
$770$	4717.63i	0.220794i
$771$	− 1479.83i	− 0.0691242i
$772$	3143.78i	0.146564i
$773$	29559.8	1.37541	0.687706	−	0.725990i	$-0.258618\pi$
$773$	29559.8	1.37541	0.687706	+	0.725990i	$0.258618\pi$
$774$	10829.0	0.502896
$775$	145.527i	0.00674512i
$776$	18633.6i	0.861996i
$777$	− 71350.2i	− 3.29430i
$778$	45891.2	2.11475
$779$	11152.0i	0.512917i
$780$	25199.3	1.15677
$781$	1687.58	0.0773193
$782$	0	0
$783$	9206.75	0.420208
$784$	−5582.84	−0.254320
$785$	29743.6i	1.35235i
$786$	−4341.86	−0.197034
$787$	− 28038.7i	− 1.26998i	−0.772521	−	0.634989i	$-0.781005\pi$
$787$	− 28038.7i	− 1.26998i	0.772521	−	0.634989i	$-0.218995\pi$
$788$	− 11271.6i	− 0.509563i
$789$	− 10267.1i	− 0.463269i
$790$	−53099.2	−2.39137
$791$	−10832.6	−0.486934
$792$	2753.67i	0.123545i
$793$	− 7426.75i	− 0.332574i
$794$	13438.7i	0.600659i
$795$	9486.68	0.423217
$796$	25802.3i	1.14892i
$797$	−5320.45	−0.236462	−0.118231	−	0.992986i	$-0.537722\pi$
$797$	−5320.45	−0.236462	−0.118231	+	0.992986i	$0.537722\pi$
$798$	−53392.2	−2.36850
$799$	0	0
$800$	−2395.05	−0.105847
$801$	15175.4	0.669409
$802$	− 30437.7i	− 1.34014i
$803$	−1131.35	−0.0497194
$804$	− 99089.4i	− 4.34653i
$805$	23984.4i	1.05011i
$806$	− 800.673i	− 0.0349907i
$807$	−23523.6	−1.02611
$808$	−23513.8	−1.02378
$809$	− 30934.0i	− 1.34435i	−0.740390	−	0.672177i	$-0.765359\pi$
$809$	− 30934.0i	− 1.34435i	0.740390	−	0.672177i	$-0.234641\pi$
$810$	− 33255.3i	− 1.44256i
$811$	− 40364.5i	− 1.74771i	−0.486189	−	0.873854i	$-0.661613\pi$
$811$	− 40364.5i	− 1.74771i	0.486189	−	0.873854i	$-0.338387\pi$
$812$	−103314.	−4.46504
$813$	3221.10i	0.138953i
$814$	5424.94	0.233592
$815$	−23376.9	−1.00473
$816$	0	0
$817$	−4258.66	−0.182364
$818$	41434.8	1.77107
$819$	− 16350.9i	− 0.697616i
$820$	−32030.6	−1.36409
$821$	19799.7i	0.841672i	0.907137	+	0.420836i	$0.138263\pi$
$821$	19799.7i	0.841672i	−0.907137	+	0.420836i	$0.861737\pi$
$822$	− 31983.8i	− 1.35713i
$823$	− 18756.4i	− 0.794419i	−0.917728	−	0.397210i	$-0.869979\pi$
$823$	− 18756.4i	− 0.794419i	0.917728	−	0.397210i	$-0.130021\pi$
$824$	−51144.1	−2.16225
$825$	−421.395	−0.0177832
$826$	− 30428.4i	− 1.28177i
$827$	− 20958.0i	− 0.881234i	−0.897695	−	0.440617i	$-0.854760\pi$
$827$	− 20958.0i	− 0.881234i	0.897695	−	0.440617i	$-0.145240\pi$
$828$	33240.0i	1.39513i
$829$	−31320.3	−1.31218	−0.656091	−	0.754682i	$-0.727791\pi$
$829$	−31320.3	−1.31218	−0.656091	+	0.754682i	$0.727791\pi$
$830$	78732.4i	3.29258i
$831$	−63023.7	−2.63089
$832$	15816.2	0.659049
$833$	0	0
$834$	75333.0	3.12778
$835$	25970.8	1.07636
$836$	− 2571.23i	− 0.106373i
$837$	275.021	0.0113574
$838$	− 4715.82i	− 0.194398i
$839$	30290.6i	1.24642i	0.782053	+	0.623212i	$0.214173\pi$
$839$	30290.6i	1.24642i	−0.782053	+	0.623212i	$0.785827\pi$
$840$	− 64586.8i	− 2.65292i
$841$	−57499.9	−2.35762
$842$	15198.5	0.622060
$843$	25346.1i	1.03555i
$844$	15242.8i	0.621659i
$845$	21391.0i	0.870855i
$846$	3443.29	0.139933
$847$	− 34493.7i	− 1.39931i
$848$	−1716.29	−0.0695021
$849$	60692.2	2.45342
$850$	0	0
$851$	27580.3	1.11098
$852$	−54856.6	−2.20582
$853$	− 21111.8i	− 0.847425i	−0.905797	−	0.423712i	$-0.860727\pi$
$853$	− 21111.8i	− 0.847425i	0.905797	−	0.423712i	$-0.139273\pi$
$854$	−45195.9	−1.81097
$855$	− 21334.2i	− 0.853348i
$856$	892.277i	0.0356278i
$857$	39983.0i	1.59369i	0.604184	+	0.796845i	$0.293499\pi$
$857$	39983.0i	1.59369i	−0.604184	+	0.796845i	$0.706501\pi$
$858$	2318.48	0.0922512
$859$	39503.3	1.56907	0.784537	−	0.620082i	$-0.212901\pi$
$859$	39503.3	1.56907	0.784537	+	0.620082i	$0.212901\pi$
$860$	− 12231.6i	− 0.484995i
$861$	38759.6i	1.53418i
$862$	− 10989.2i	− 0.434217i
$863$	26019.9	1.02634	0.513168	−	0.858288i	$-0.328472\pi$
$863$	26019.9	1.02634	0.513168	+	0.858288i	$0.328472\pi$
$864$	4526.26i	0.178225i
$865$	37080.4	1.45754
$866$	−27376.1	−1.07422
$867$	0	0
$868$	−3086.17	−0.120681
$869$	−3094.33	−0.120791
$870$	− 121550.i	− 4.73669i
$871$	−18841.4	−0.732968
$872$	14376.7i	0.558323i
$873$	21392.2i	0.829343i
$874$	− 20638.7i	− 0.798757i
$875$	−33613.8	−1.29869
$876$	36775.9	1.41843
$877$	− 15038.3i	− 0.579027i	−0.957174	−	0.289514i	$-0.906506\pi$
$877$	− 15038.3i	− 0.579027i	0.957174	−	0.289514i	$-0.0934935\pi$
$878$	− 13519.5i	− 0.519660i
$879$	8942.30i	0.343136i
$880$	636.004	0.0243633
$881$	− 18334.3i	− 0.701133i	−0.936538	−	0.350567i	$-0.885989\pi$
$881$	− 18334.3i	− 0.701133i	0.936538	−	0.350567i	$-0.114011\pi$
$882$	−49487.5	−1.88927
$883$	26659.5	1.01604	0.508020	−	0.861345i	$-0.330378\pi$
$883$	26659.5	1.01604	0.508020	+	0.861345i	$0.330378\pi$
$884$	0	0
$885$	22674.5	0.861237
$886$	38570.1	1.46251
$887$	11473.7i	0.434327i	0.976135	+	0.217163i	$0.0696804\pi$
$887$	11473.7i	0.434327i	−0.976135	+	0.217163i	$0.930320\pi$
$888$	−74270.1	−2.80669
$889$	15571.3i	0.587450i
$890$	− 27062.7i	− 1.01926i
$891$	− 1937.94i	− 0.0728656i
$892$	7862.01	0.295111
$893$	−1354.12	−0.0507436
$894$	91983.0i	3.44113i
$895$	9654.36i	0.360569i
$896$	− 66850.7i	− 2.49255i
$897$	11787.1	0.438752
$898$	72344.6i	2.68838i
$899$	−2446.16	−0.0907498
$900$	7345.03	0.272038
$901$	0	0
$902$	−2946.99	−0.108785
$903$	−14801.3	−0.545466
$904$	11276.0i	0.414860i
$905$	39997.5	1.46913
$906$	− 47858.8i	− 1.75497i
$907$	20361.6i	0.745421i	0.927948	+	0.372710i	$0.121571\pi$
$907$	20361.6i	0.745421i	−0.927948	+	0.372710i	$0.878429\pi$
$908$	− 29120.5i	− 1.06432i
$909$	−26994.8	−0.984995
$910$	−29159.1	−1.06221
$911$	19261.9i	0.700523i	0.936652	+	0.350262i	$0.113907\pi$
$911$	19261.9i	0.700523i	−0.936652	+	0.350262i	$0.886093\pi$
$912$	7198.03i	0.261349i
$913$	4588.09i	0.166313i
$914$	−75000.3	−2.71421
$915$	− 33678.9i	− 1.21682i
$916$	−59931.7	−2.16179
$917$	3182.18	0.114596
$918$	0	0
$919$	−21191.8	−0.760666	−0.380333	−	0.924850i	$-0.624191\pi$
$919$	−21191.8	−0.760666	−0.380333	+	0.924850i	$0.624191\pi$
$920$	24965.9	0.894677
$921$	− 6605.76i	− 0.236338i
$922$	65697.0	2.34665
$923$	10430.7i	0.371973i
$924$	− 8936.49i	− 0.318170i
$925$	− 6094.40i	− 0.216630i
$926$	37703.4	1.33802
$927$	−58715.6	−2.08034
$928$	− 40258.5i	− 1.42409i
$929$	− 40815.0i	− 1.44144i	−0.693227	−	0.720719i	$-0.743812\pi$
$929$	− 40815.0i	− 1.44144i	0.693227	−	0.720719i	$-0.256188\pi$
$930$	− 3630.90i	− 0.128024i
$931$	19461.7	0.685102
$932$	− 62434.5i	− 2.19432i
$933$	53370.4	1.87274
$934$	40090.9	1.40451
$935$	0	0
$936$	−17020.1	−0.594358
$937$	38439.1	1.34018	0.670092	−	0.742278i	$-0.266255\pi$
$937$	38439.1	1.34018	0.670092	+	0.742278i	$0.266255\pi$
$938$	114660.i	3.99124i
$939$	−26264.9	−0.912804
$940$	− 3889.28i	− 0.134951i
$941$	2244.08i	0.0777415i	0.999244	+	0.0388708i	$0.0123761\pi$
$941$	2244.08i	0.0777415i	−0.999244	+	0.0388708i	$0.987624\pi$
$942$	− 88955.6i	− 3.07678i
$943$	−14982.5	−0.517388
$944$	−4102.18	−0.141435
$945$	− 10015.8i	− 0.344776i
$946$	− 1125.38i	− 0.0386778i
$947$	− 42289.0i	− 1.45112i	−0.688160	−	0.725559i	$-0.741581\pi$
$947$	− 42289.0i	− 1.45112i	0.688160	−	0.725559i	$-0.258419\pi$
$948$	100584.	3.44602
$949$	− 6992.76i	− 0.239193i
$950$	−4560.51	−0.155750
$951$	−15764.7	−0.537546
$952$	0	0
$953$	37426.2	1.27214	0.636072	−	0.771629i	$-0.280558\pi$
$953$	37426.2	1.27214	0.636072	+	0.771629i	$0.280558\pi$
$954$	−15213.6	−0.516309
$955$	− 15950.4i	− 0.540464i
$956$	73263.0	2.47855
$957$	− 7083.25i	− 0.239257i
$958$	29527.0i	0.995799i
$959$	23441.2i	0.789317i
$960$	71723.5	2.41132
$961$	29717.9	0.997547
$962$	33530.8i	1.12378i
$963$	1024.37i	0.0342782i
$964$	− 18918.3i	− 0.632072i
$965$	2710.80	0.0904286
$966$	− 71731.1i	− 2.38914i
$967$	−1088.56	−0.0362003	−0.0181001	−	0.999836i	$-0.505762\pi$
$967$	−1088.56	−0.0362003	−0.0181001	+	0.999836i	$0.505762\pi$
$968$	−35905.4	−1.19219
$969$	0	0
$970$	38149.3	1.26278
$971$	−39506.5	−1.30569	−0.652845	−	0.757491i	$-0.726425\pi$
$971$	−39506.5	−1.30569	−0.652845	+	0.757491i	$0.726425\pi$
$972$	75000.6i	2.47494i
$973$	−55212.1	−1.81914
$974$	34270.8i	1.12742i
$975$	− 2604.59i	− 0.0855525i
$976$	6093.05i	0.199830i
$977$	43326.8	1.41878	0.709389	−	0.704817i	$-0.248971\pi$
$977$	43326.8	1.41878	0.709389	+	0.704817i	$0.248971\pi$
$978$	69914.4	2.28591
$979$	− 1577.07i	− 0.0514845i
$980$	55897.3i	1.82202i
$981$	16505.1i	0.537174i
$982$	31169.3	1.01288
$983$	10664.1i	0.346014i	0.984921	+	0.173007i	$0.0553483\pi$
$983$	10664.1i	0.346014i	−0.984921	+	0.173007i	$0.944652\pi$
$984$	40345.9	1.30709
$985$	−9719.22	−0.314396
$986$	0	0
$987$	−4706.35	−0.151778
$988$	15892.4	0.511747
$989$	− 5721.42i	− 0.183954i
$990$	5637.69	0.180987
$991$	15461.4i	0.495609i	0.968810	+	0.247804i	$0.0797090\pi$
$991$	15461.4i	0.495609i	−0.968810	+	0.247804i	$0.920291\pi$
$992$	− 1202.59i	− 0.0384902i
$993$	68879.2i	2.20122i
$994$	63476.7	2.02551
$995$	22248.6	0.708871
$996$	− 149141.i	− 4.74469i
$997$	− 46061.1i	− 1.46316i	−0.681756	−	0.731579i	$-0.738784\pi$
$997$	− 46061.1i	− 1.46316i	0.681756	−	0.731579i	$-0.261216\pi$
$998$	83713.7i	2.65522i
$999$	−11517.4	−0.364760

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	289.4.b.b.288.1		6
17.4	even	4		17.4.a.b.1.3	✓	3
17.13	even	4		289.4.a.b.1.3		3
17.16	even	2	inner	289.4.b.b.288.2		6
51.38	odd	4		153.4.a.g.1.1		3
68.55	odd	4		272.4.a.h.1.3		3
85.4	even	4		425.4.a.g.1.1		3
85.38	odd	4		425.4.b.f.324.2		6
85.72	odd	4		425.4.b.f.324.5		6
119.55	odd	4		833.4.a.d.1.3		3
136.21	even	4		1088.4.a.v.1.3		3
136.123	odd	4		1088.4.a.x.1.1		3
187.21	odd	4		2057.4.a.e.1.1		3
204.191	even	4		2448.4.a.bi.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.4.a.b.1.3	✓	3	17.4	even	4
153.4.a.g.1.1		3	51.38	odd	4
272.4.a.h.1.3		3	68.55	odd	4
289.4.a.b.1.3		3	17.13	even	4
289.4.b.b.288.1		6	1.1	even	1	trivial
289.4.b.b.288.2		6	17.16	even	2	inner
425.4.a.g.1.1		3	85.4	even	4
425.4.b.f.324.2		6	85.38	odd	4
425.4.b.f.324.5		6	85.72	odd	4
833.4.a.d.1.3		3	119.55	odd	4
1088.4.a.v.1.3		3	136.21	even	4
1088.4.a.x.1.1		3	136.123	odd	4
2057.4.a.e.1.1		3	187.21	odd	4
2448.4.a.bi.1.3		3	204.191	even	4

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)

\(f(q)\)	\(=\)	\(q-4.67129 q^{2} -7.62999i q^{3} +13.8209 q^{4} -11.9174i q^{5} +35.6419i q^{6} -26.1222i q^{7} -27.1912 q^{8} -31.2167 q^{9} +55.6696i q^{10} +3.24412i q^{11} -105.453i q^{12} -20.0515 q^{13} +122.024i q^{14} -90.9296 q^{15} +16.4506 q^{16} +145.822 q^{18} -57.3466 q^{19} -164.709i q^{20} -199.312 q^{21} -15.1542i q^{22} -77.0438i q^{23} +207.469i q^{24} -17.0243 q^{25} +93.6662 q^{26} +32.1732i q^{27} -361.033i q^{28} -286.162i q^{29} +424.758 q^{30} -8.54816i q^{31} +140.684 q^{32} +24.7526 q^{33} -311.309 q^{35} -431.443 q^{36} +357.982i q^{37} +267.882 q^{38} +152.992i q^{39} +324.049i q^{40} -194.467i q^{41} +931.044 q^{42} +74.2619 q^{43} +44.8367i q^{44} +372.021i q^{45} +359.894i q^{46} +23.6130 q^{47} -125.518i q^{48} -339.369 q^{49} +79.5255 q^{50} -277.130 q^{52} -104.330 q^{53} -150.290i q^{54} +38.6614 q^{55} +710.295i q^{56} +437.553i q^{57} +1336.75i q^{58} -249.363 q^{59} -1256.73 q^{60} +370.384i q^{61} +39.9309i q^{62} +815.448i q^{63} -788.781 q^{64} +238.961i q^{65} -115.626 q^{66} +939.650 q^{67} -587.843 q^{69} +1454.21 q^{70} -520.197i q^{71} +848.820 q^{72} +348.741i q^{73} -1672.24i q^{74} +129.895i q^{75} -792.583 q^{76} +84.7434 q^{77} -714.672i q^{78} +953.827i q^{79} -196.049i q^{80} -597.369 q^{81} +908.412i q^{82} +1414.28 q^{83} -2754.68 q^{84} -346.899 q^{86} -2183.41 q^{87} -88.2115i q^{88} -486.132 q^{89} -1737.82i q^{90} +523.788i q^{91} -1064.82i q^{92} -65.2223 q^{93} -110.303 q^{94} +683.422i q^{95} -1073.42i q^{96} -685.281i q^{97} +1585.29 q^{98} -101.271i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{2} + 50 q^{4} + 78 q^{8} - 118 q^{9} + 60 q^{13} - 216 q^{15} + 274 q^{16} - 206 q^{18} - 160 q^{19} - 384 q^{21} + 446 q^{25} - 52 q^{26} + 800 q^{30} + 142 q^{32} - 664 q^{33} - 664 q^{35}+ \cdots + 5714 q^{98}+O(q^{100}) \) 6 * q - 2 * q^2 + 50 * q^4 + 78 * q^8 - 118 * q^9 + 60 * q^13 - 216 * q^15 + 274 * q^16 - 206 * q^18 - 160 * q^19 - 384 * q^21 + 446 * q^25 - 52 * q^26 + 800 * q^30 + 142 * q^32 - 664 * q^33 - 664 * q^35 - 2626 * q^36 + 1448 * q^38 + 2256 * q^42 - 1112 * q^43 + 1280 * q^47 + 538 * q^49 + 1094 * q^50 - 1548 * q^52 - 604 * q^53 + 152 * q^55 - 1272 * q^59 - 2656 * q^60 - 1838 * q^64 - 4936 * q^66 + 2016 * q^67 + 1152 * q^69 + 3008 * q^70 - 1854 * q^72 + 1816 * q^76 + 1008 * q^77 - 1010 * q^81 + 4792 * q^83 - 4080 * q^84 - 2528 * q^86 - 2856 * q^87 - 340 * q^89 - 1264 * q^93 + 4032 * q^94 + 5714 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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