LMFDB - Embedded newform 289.4.b.d.288.4

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:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 43x^{6} + 505x^{4} + 1528x^{2} + 1296 $ x^8 + 43x^6 + 505x^4 + 1528*x^2 + 1296
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			288.4
Root			$-1.22501i$ of defining polynomial
Character	$\chi$	$=$	289.288
Dual form			289.4.b.d.288.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.22501 q^{2} +0.534684i q^{3} -6.49935 q^{4} +20.9528i q^{5} -0.654993i q^{6} +15.0235i q^{7} +17.7618 q^{8} +26.7141 q^{9} +O(q^{10})$	\(q-1.22501 q^{2} +0.534684i q^{3} -6.49935 q^{4} +20.9528i q^{5} -0.654993i q^{6} +15.0235i q^{7} +17.7618 q^{8} +26.7141 q^{9} -25.6674i q^{10} +45.4168i q^{11} -3.47510i q^{12} +3.14456 q^{13} -18.4039i q^{14} -11.2031 q^{15} +30.2364 q^{16} -32.7250 q^{18} +63.2180 q^{19} -136.180i q^{20} -8.03282 q^{21} -55.6360i q^{22} +114.551i q^{23} +9.49698i q^{24} -314.020 q^{25} -3.85211 q^{26} +28.7201i q^{27} -97.6429i q^{28} +96.6197i q^{29} +13.7239 q^{30} -194.578i q^{31} -179.135 q^{32} -24.2837 q^{33} -314.784 q^{35} -173.624 q^{36} -73.6682i q^{37} -77.4426 q^{38} +1.68135i q^{39} +372.161i q^{40} +341.047i q^{41} +9.84027 q^{42} +281.677 q^{43} -295.180i q^{44} +559.736i q^{45} -140.326i q^{46} +36.2294 q^{47} +16.1669i q^{48} +117.295 q^{49} +384.678 q^{50} -20.4376 q^{52} -191.274 q^{53} -35.1824i q^{54} -951.610 q^{55} +266.845i q^{56} +33.8017i q^{57} -118.360i q^{58} -104.171 q^{59} +72.8131 q^{60} -517.547i q^{61} +238.359i q^{62} +401.339i q^{63} -22.4497 q^{64} +65.8874i q^{65} +29.7477 q^{66} -560.893 q^{67} -61.2488 q^{69} +385.613 q^{70} -333.262i q^{71} +474.492 q^{72} -378.895i q^{73} +90.2443i q^{74} -167.902i q^{75} -410.876 q^{76} -682.319 q^{77} -2.05967i q^{78} -877.994i q^{79} +633.538i q^{80} +705.925 q^{81} -417.786i q^{82} -1195.27 q^{83} +52.2081 q^{84} -345.057 q^{86} -51.6611 q^{87} +806.687i q^{88} +783.884 q^{89} -685.682i q^{90} +47.2422i q^{91} -744.509i q^{92} +104.038 q^{93} -44.3813 q^{94} +1324.59i q^{95} -95.7804i q^{96} -1605.30i q^{97} -143.688 q^{98} +1213.27i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{2} + 22 q^{4} - 120 q^{8} - 12 q^{9}+O(q^{10}) $ 8 * q - 2 * q^2 + 22 * q^4 - 120 * q^8 - 12 * q^9	$ 8 q - 2 q^{2} + 22 q^{4} - 120 q^{8} - 12 q^{9} - 44 q^{13} - 108 q^{15} + 126 q^{16} - 668 q^{18} - 44 q^{19} - 704 q^{21} - 756 q^{25} + 896 q^{26} + 626 q^{30} - 662 q^{32} + 188 q^{33} - 484 q^{35} - 282 q^{36} - 1048 q^{38} - 2910 q^{42} + 228 q^{43} + 20 q^{47} - 2012 q^{49} + 1610 q^{50} - 3074 q^{52} - 100 q^{53} - 2632 q^{55} - 1992 q^{59} + 434 q^{60} - 300 q^{64} + 2180 q^{66} - 1736 q^{67} - 2256 q^{69} - 2104 q^{70} - 78 q^{72} + 1746 q^{76} + 1788 q^{77} + 2160 q^{81} - 1700 q^{83} + 886 q^{84} + 4822 q^{86} + 768 q^{87} + 1568 q^{89} + 3100 q^{93} - 2238 q^{94} - 3754 q^{98}+O(q^{100}) $ 8 * q - 2 * q^2 + 22 * q^4 - 120 * q^8 - 12 * q^9 - 44 * q^13 - 108 * q^15 + 126 * q^16 - 668 * q^18 - 44 * q^19 - 704 * q^21 - 756 * q^25 + 896 * q^26 + 626 * q^30 - 662 * q^32 + 188 * q^33 - 484 * q^35 - 282 * q^36 - 1048 * q^38 - 2910 * q^42 + 228 * q^43 + 20 * q^47 - 2012 * q^49 + 1610 * q^50 - 3074 * q^52 - 100 * q^53 - 2632 * q^55 - 1992 * q^59 + 434 * q^60 - 300 * q^64 + 2180 * q^66 - 1736 * q^67 - 2256 * q^69 - 2104 * q^70 - 78 * q^72 + 1746 * q^76 + 1788 * q^77 + 2160 * q^81 - 1700 * q^83 + 886 * q^84 + 4822 * q^86 + 768 * q^87 + 1568 * q^89 + 3100 * q^93 - 2238 * q^94 - 3754 * 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 $

$2$	−1.22501	−0.433106	−0.216553	−	0.976271i	$-0.569481\pi$
$2$	−1.22501	−0.433106	−0.216553	+	0.976271i	$0.569481\pi$
$3$	0.534684i	0.102900i	0.998676	+	0.0514500i	$0.0163843\pi$
$3$	0.534684i	0.102900i	−0.998676	+	0.0514500i	$0.983616\pi$
$4$	−6.49935	−0.812419
$5$	20.9528i	1.87408i	0.349227	+	0.937038i	$0.386444\pi$
$5$	20.9528i	1.87408i	−0.349227	+	0.937038i	$0.613556\pi$
$6$	− 0.654993i	− 0.0445666i
$7$	15.0235i	0.811191i	0.914053	+	0.405596i	$0.132936\pi$
$7$	15.0235i	0.811191i	−0.914053	+	0.405596i	$0.867064\pi$
$8$	17.7618	0.784970
$9$	26.7141	0.989412
$10$	− 25.6674i	− 0.811674i
$11$	45.4168i	1.24488i	0.782667	+	0.622440i	$0.213859\pi$
$11$	45.4168i	1.24488i	−0.782667	+	0.622440i	$0.786141\pi$
$12$	− 3.47510i	− 0.0835979i
$13$	3.14456	0.0670880	0.0335440	−	0.999437i	$-0.489321\pi$
$13$	3.14456	0.0670880	0.0335440	+	0.999437i	$0.489321\pi$
$14$	− 18.4039i	− 0.351332i
$15$	−11.2031	−0.192843
$16$	30.2364	0.472444
$17$	0	0
$17$	0	0
$18$	−32.7250	−0.428520
$19$	63.2180	0.763326	0.381663	−	0.924302i	$-0.375351\pi$
$19$	63.2180	0.763326	0.381663	+	0.924302i	$0.375351\pi$
$20$	− 136.180i	− 1.52254i
$21$	−8.03282	−0.0834716
$22$	− 55.6360i	− 0.539166i
$23$	114.551i	1.03850i	0.854621	+	0.519252i	$0.173789\pi$
$23$	114.551i	1.03850i	−0.854621	+	0.519252i	$0.826211\pi$
$24$	9.49698i	0.0807734i
$25$	−314.020	−2.51216
$26$	−3.85211	−0.0290562
$27$	28.7201i	0.204711i
$28$	− 97.6429i	− 0.659027i
$29$	96.6197i	0.618684i	0.950951	+	0.309342i	$0.100109\pi$
$29$	96.6197i	0.618684i	−0.950951	+	0.309342i	$0.899891\pi$
$30$	13.7239	0.0835213
$31$	− 194.578i	− 1.12733i	−0.826004	−	0.563664i	$-0.809391\pi$
$31$	− 194.578i	− 1.12733i	0.826004	−	0.563664i	$-0.190609\pi$
$32$	−179.135	−0.989588
$33$	−24.2837	−0.128098
$34$	0	0
$35$	−314.784	−1.52023
$36$	−173.624	−0.803817
$37$	− 73.6682i	− 0.327324i	−0.986516	−	0.163662i	$-0.947669\pi$
$37$	− 73.6682i	− 0.327324i	0.986516	−	0.163662i	$-0.0523306\pi$
$38$	−77.4426	−0.330601
$39$	1.68135i	0.00690336i
$40$	372.161i	1.47109i
$41$	341.047i	1.29909i	0.760324	+	0.649545i	$0.225040\pi$
$41$	341.047i	1.29909i	−0.760324	+	0.649545i	$0.774960\pi$
$42$	9.84027	0.0361521
$43$	281.677	0.998962	0.499481	−	0.866325i	$-0.333524\pi$
$43$	281.677	0.998962	0.499481	+	0.866325i	$0.333524\pi$
$44$	− 295.180i	− 1.01137i
$45$	559.736i	1.85423i
$46$	− 140.326i	− 0.449783i
$47$	36.2294	0.112438	0.0562191	−	0.998418i	$-0.482095\pi$
$47$	36.2294	0.112438	0.0562191	+	0.998418i	$0.482095\pi$
$48$	16.1669i	0.0486145i
$49$	117.295	0.341968
$50$	384.678	1.08803
$51$	0	0
$52$	−20.4376	−0.0545036
$53$	−191.274	−0.495727	−0.247863	−	0.968795i	$-0.579728\pi$
$53$	−191.274	−0.495727	−0.247863	+	0.968795i	$0.579728\pi$
$54$	− 35.1824i	− 0.0886614i
$55$	−951.610	−2.33300
$56$	266.845i	0.636761i
$57$	33.8017i	0.0785463i
$58$	− 118.360i	− 0.267956i
$59$	−104.171	−0.229864	−0.114932	−	0.993373i	$-0.536665\pi$
$59$	−104.171	−0.229864	−0.114932	+	0.993373i	$0.536665\pi$
$60$	72.8131	0.156669
$61$	− 517.547i	− 1.08631i	−0.839631	−	0.543157i	$-0.817229\pi$
$61$	− 517.547i	− 1.08631i	0.839631	−	0.543157i	$-0.182771\pi$
$62$	238.359i	0.488253i
$63$	401.339i	0.802602i
$64$	−22.4497	−0.0438470
$65$	65.8874i	0.125728i
$66$	29.7477	0.0554802
$67$	−560.893	−1.02275	−0.511373	−	0.859359i	$-0.670863\pi$
$67$	−560.893	−1.02275	−0.511373	+	0.859359i	$0.670863\pi$
$68$	0	0
$69$	−61.2488	−0.106862
$70$	385.613	0.658423
$71$	− 333.262i	− 0.557056i	−0.960428	−	0.278528i	$-0.910154\pi$
$71$	− 333.262i	− 0.557056i	0.960428	−	0.278528i	$-0.0898465\pi$
$72$	474.492	0.776658
$73$	− 378.895i	− 0.607484i	−0.952754	−	0.303742i	$-0.901764\pi$
$73$	− 378.895i	− 0.607484i	0.952754	−	0.303742i	$-0.0982360\pi$
$74$	90.2443i	0.141766i
$75$	− 167.902i	− 0.258502i
$76$	−410.876	−0.620141
$77$	−682.319	−1.00984
$78$	− 2.05967i	− 0.00298989i
$79$	− 877.994i	− 1.25041i	−0.780462	−	0.625203i	$-0.785016\pi$
$79$	− 877.994i	− 1.25041i	0.780462	−	0.625203i	$-0.214984\pi$
$80$	633.538i	0.885396i
$81$	705.925	0.968347
$82$	− 417.786i	− 0.562643i
$83$	−1195.27	−1.58069	−0.790346	−	0.612661i	$-0.790099\pi$
$83$	−1195.27	−1.58069	−0.790346	+	0.612661i	$0.790099\pi$
$84$	52.2081	0.0678139
$85$	0	0
$86$	−345.057	−0.432656
$87$	−51.6611	−0.0636626
$88$	806.687i	0.977194i
$89$	783.884	0.933613	0.466806	−	0.884360i	$-0.345405\pi$
$89$	783.884	0.933613	0.466806	+	0.884360i	$0.345405\pi$
$90$	− 685.682i	− 0.803080i
$91$	47.2422i	0.0544212i
$92$	− 744.509i	− 0.843701i
$93$	104.038	0.116002
$94$	−44.3813	−0.0486977
$95$	1324.59i	1.43053i
$96$	− 95.7804i	− 0.101829i
$97$	− 1605.30i	− 1.68034i	−0.542320	−	0.840172i	$-0.682454\pi$
$97$	− 1605.30i	− 1.68034i	0.542320	−	0.840172i	$-0.317546\pi$
$98$	−143.688	−0.148109
$99$	1213.27i	1.23170i
$100$	2040.93	2.04093
$101$	118.682	0.116923	0.0584617	−	0.998290i	$-0.481380\pi$
$101$	118.682	0.116923	0.0584617	+	0.998290i	$0.481380\pi$
$102$	0	0
$103$	1399.63	1.33893	0.669466	−	0.742843i	$-0.266523\pi$
$103$	1399.63	1.33893	0.669466	+	0.742843i	$0.266523\pi$
$104$	55.8532	0.0526621
$105$	− 168.310i	− 0.156432i
$106$	234.312	0.214702
$107$	631.693i	0.570730i	0.958419	+	0.285365i	$0.0921148\pi$
$107$	631.693i	0.570730i	−0.958419	+	0.285365i	$0.907885\pi$
$108$	− 186.662i	− 0.166311i
$109$	1231.00i	1.08173i	0.841109	+	0.540866i	$0.181904\pi$
$109$	1231.00i	1.08173i	−0.841109	+	0.540866i	$0.818096\pi$
$110$	1165.73	1.01044
$111$	39.3892	0.0336816
$112$	454.256i	0.383242i
$113$	− 343.826i	− 0.286234i	−0.989706	−	0.143117i	$-0.954288\pi$
$113$	− 343.826i	− 0.286234i	0.989706	−	0.143117i	$-0.0457125\pi$
$114$	− 41.4073i	− 0.0340189i
$115$	−2400.17	−1.94624
$116$	− 627.966i	− 0.502631i
$117$	84.0041	0.0663776
$118$	127.611	0.0995555
$119$	0	0
$120$	−198.988	−0.151376
$121$	−731.689	−0.549729
$122$	634.000i	0.470489i
$123$	−182.353	−0.133676
$124$	1264.63i	0.915862i
$125$	− 3960.51i	− 2.83391i
$126$	− 491.644i	− 0.347612i
$127$	2346.48	1.63950	0.819749	−	0.572723i	$-0.194113\pi$
$127$	2346.48	1.63950	0.819749	+	0.572723i	$0.194113\pi$
$128$	1460.58	1.00858
$129$	150.608i	0.102793i
$130$	− 80.7126i	− 0.0544536i
$131$	− 399.454i	− 0.266416i	−0.991088	−	0.133208i	$-0.957472\pi$
$131$	− 399.454i	− 0.266416i	0.991088	−	0.133208i	$-0.0425278\pi$
$132$	157.828	0.104069
$133$	949.754i	0.619204i
$134$	687.099	0.442958
$135$	−601.767	−0.383643
$136$	0	0
$137$	1431.60	0.892773	0.446387	−	0.894840i	$-0.352711\pi$
$137$	1431.60	0.892773	0.446387	+	0.894840i	$0.352711\pi$
$138$	75.0303	0.0462826
$139$	811.640i	0.495269i	0.968854	+	0.247635i	$0.0796532\pi$
$139$	811.640i	0.495269i	−0.968854	+	0.247635i	$0.920347\pi$
$140$	2045.89	1.23507
$141$	19.3713i	0.0115699i
$142$	408.250i	0.241264i
$143$	142.816i	0.0835166i
$144$	807.739	0.467441
$145$	−2024.46	−1.15946
$146$	464.150i	0.263105i
$147$	62.7159i	0.0351886i
$148$	478.796i	0.265924i
$149$	1760.04	0.967704	0.483852	−	0.875150i	$-0.339237\pi$
$149$	1760.04	0.967704	0.483852	+	0.875150i	$0.339237\pi$
$150$	205.681i	0.111959i
$151$	−1669.08	−0.899523	−0.449761	−	0.893149i	$-0.648491\pi$
$151$	−1669.08	−0.899523	−0.449761	+	0.893149i	$0.648491\pi$
$152$	1122.87	0.599188
$153$	0	0
$154$	835.847	0.437367
$155$	4076.95	2.11270
$156$	− 10.9277i	− 0.00560842i
$157$	1167.18	0.593321	0.296661	−	0.954983i	$-0.404127\pi$
$157$	1167.18	0.593321	0.296661	+	0.954983i	$0.404127\pi$
$158$	1075.55i	0.541558i
$159$	− 102.271i	− 0.0510103i
$160$	− 3753.37i	− 1.85456i
$161$	−1720.96	−0.842426
$162$	−864.764	−0.419397
$163$	2173.56i	1.04446i	0.852805	+	0.522229i	$0.174899\pi$
$163$	2173.56i	1.04446i	−0.852805	+	0.522229i	$0.825101\pi$
$164$	− 2216.59i	− 1.05540i
$165$	− 508.811i	− 0.240066i
$166$	1464.21	0.684607
$167$	− 1132.85i	− 0.524924i	−0.964942	−	0.262462i	$-0.915466\pi$
$167$	− 1132.85i	− 0.524924i	0.964942	−	0.262462i	$-0.0845345\pi$
$168$	−142.678	−0.0655227
$169$	−2187.11	−0.995499
$170$	0	0
$171$	1688.81	0.755244
$172$	−1830.72	−0.811575
$173$	− 46.3472i	− 0.0203683i	−0.999948	−	0.0101841i	$-0.996758\pi$
$173$	− 46.3472i	− 0.0203683i	0.999948	−	0.0101841i	$-0.00324177\pi$
$174$	63.2853	0.0275727
$175$	− 4717.68i	− 2.03785i
$176$	1373.24i	0.588136i
$177$	− 55.6988i	− 0.0236530i
$178$	−960.265	−0.404353
$179$	643.960	0.268893	0.134446	−	0.990921i	$-0.457074\pi$
$179$	643.960	0.268893	0.134446	+	0.990921i	$0.457074\pi$
$180$	− 3637.92i	− 1.50641i
$181$	371.082i	0.152388i	0.997093	+	0.0761942i	$0.0242769\pi$
$181$	371.082i	0.152388i	−0.997093	+	0.0761942i	$0.975723\pi$
$182$	− 57.8722i	− 0.0235702i
$183$	276.724	0.111782
$184$	2034.64i	0.815194i
$185$	1543.56	0.613430
$186$	−127.447	−0.0502412
$187$	0	0
$188$	−235.468	−0.0913470
$189$	−431.476	−0.166059
$190$	− 1622.64i	− 0.619572i
$191$	−1874.95	−0.710298	−0.355149	−	0.934810i	$-0.615570\pi$
$191$	−1874.95	−0.710298	−0.355149	+	0.934810i	$0.615570\pi$
$192$	− 12.0035i	− 0.00451186i
$193$	− 819.870i	− 0.305780i	−0.988243	−	0.152890i	$-0.951142\pi$
$193$	− 819.870i	− 0.305780i	0.988243	−	0.152890i	$-0.0488580\pi$
$194$	1966.51i	0.727767i
$195$	−35.2289	−0.0129374
$196$	−762.343	−0.277822
$197$	1067.82i	0.386189i	0.981180	+	0.193094i	$0.0618523\pi$
$197$	1067.82i	0.386189i	−0.981180	+	0.193094i	$0.938148\pi$
$198$	− 1486.27i	− 0.533457i
$199$	2378.01i	0.847099i	0.905873	+	0.423550i	$0.139216\pi$
$199$	2378.01i	0.847099i	−0.905873	+	0.423550i	$0.860784\pi$
$200$	−5577.58	−1.97197
$201$	− 299.901i	− 0.105241i
$202$	−145.386	−0.0506402
$203$	−1451.56	−0.501871
$204$	0	0
$205$	−7145.90	−2.43459
$206$	−1714.56	−0.579899
$207$	3060.14i	1.02751i
$208$	95.0802	0.0316953
$209$	2871.16i	0.950250i
$210$	206.181i	0.0677518i
$211$	3536.41i	1.15382i	0.816807	+	0.576910i	$0.195742\pi$
$211$	3536.41i	1.15382i	−0.816807	+	0.576910i	$0.804258\pi$
$212$	1243.16	0.402738
$213$	178.190	0.0573211
$214$	− 773.830i	− 0.247187i
$215$	5901.93i	1.87213i
$216$	510.122i	0.160692i
$217$	2923.23	0.914479
$218$	− 1507.99i	− 0.468505i
$219$	202.589	0.0625101
$220$	6184.85	1.89538
$221$	0	0
$222$	−48.2522	−0.0145877
$223$	−4920.87	−1.47769	−0.738847	−	0.673873i	$-0.764629\pi$
$223$	−4920.87	−1.47769	−0.738847	+	0.673873i	$0.764629\pi$
$224$	− 2691.22i	− 0.802745i
$225$	−8388.78	−2.48556
$226$	421.190i	0.123970i
$227$	5114.37i	1.49538i	0.664045	+	0.747692i	$0.268838\pi$
$227$	5114.37i	1.49538i	−0.664045	+	0.747692i	$0.731162\pi$
$228$	− 219.689i	− 0.0638125i
$229$	−1829.79	−0.528016	−0.264008	−	0.964521i	$-0.585045\pi$
$229$	−1829.79	−0.528016	−0.264008	+	0.964521i	$0.585045\pi$
$230$	2940.23	0.842927
$231$	− 364.825i	− 0.103912i
$232$	1716.14i	0.485648i
$233$	− 5252.53i	− 1.47684i	−0.674338	−	0.738422i	$-0.735571\pi$
$233$	− 5252.53i	− 1.47684i	0.674338	−	0.738422i	$-0.264429\pi$
$234$	−102.906	−0.0287486
$235$	759.108i	0.210718i
$236$	677.047	0.186746
$237$	469.450	0.128667
$238$	0	0
$239$	355.927	0.0963306	0.0481653	−	0.998839i	$-0.484663\pi$
$239$	355.927	0.0963306	0.0481653	+	0.998839i	$0.484663\pi$
$240$	−338.743	−0.0911073
$241$	2229.06i	0.595794i	0.954598	+	0.297897i	$0.0962853\pi$
$241$	2229.06i	0.595794i	−0.954598	+	0.297897i	$0.903715\pi$
$242$	896.325	0.238091
$243$	1152.89i	0.304353i
$244$	3363.72i	0.882542i
$245$	2457.66i	0.640875i
$246$	223.384	0.0578960
$247$	198.793	0.0512100
$248$	− 3456.05i	− 0.884918i
$249$	− 639.089i	− 0.162653i
$250$	4851.66i	1.22738i
$251$	1388.08	0.349063	0.174531	−	0.984652i	$-0.444159\pi$
$251$	1388.08	0.349063	0.174531	+	0.984652i	$0.444159\pi$
$252$	− 2608.44i	− 0.652049i
$253$	−5202.56	−1.29281
$254$	−2874.46	−0.710077
$255$	0	0
$256$	−1609.62	−0.392975
$257$	7401.42	1.79645	0.898225	−	0.439535i	$-0.144857\pi$
$257$	7401.42	1.79645	0.898225	+	0.439535i	$0.144857\pi$
$258$	− 184.497i	− 0.0445204i
$259$	1106.75	0.265522
$260$	− 428.225i	− 0.102144i
$261$	2581.11i	0.612133i
$262$	489.335i	0.115386i
$263$	−6738.57	−1.57992	−0.789959	−	0.613160i	$-0.789898\pi$
$263$	−6738.57	−1.57992	−0.789959	+	0.613160i	$0.789898\pi$
$264$	−431.323	−0.100553
$265$	− 4007.73i	− 0.929030i
$266$	− 1163.46i	− 0.268181i
$267$	419.130i	0.0960688i
$268$	3645.44	0.830899
$269$	− 3474.39i	− 0.787500i	−0.919218	−	0.393750i	$-0.871178\pi$
$269$	− 3474.39i	− 0.787500i	0.919218	−	0.393750i	$-0.128822\pi$
$270$	737.170	0.166158
$271$	−1524.81	−0.341791	−0.170896	−	0.985289i	$-0.554666\pi$
$271$	−1524.81	−0.341791	−0.170896	+	0.985289i	$0.554666\pi$
$272$	0	0
$273$	−25.2597	−0.00559994
$274$	−1753.72	−0.386665
$275$	− 14261.8i	− 3.12734i
$276$	398.077	0.0868168
$277$	− 5379.31i	− 1.16683i	−0.812175	−	0.583414i	$-0.801717\pi$
$277$	− 5379.31i	− 1.16683i	0.812175	−	0.583414i	$-0.198283\pi$
$278$	− 994.267i	− 0.214504i
$279$	− 5197.97i	− 1.11539i
$280$	−5591.14	−1.19334
$281$	5595.31	1.18786	0.593929	−	0.804517i	$-0.297576\pi$
$281$	5595.31	1.18786	0.593929	+	0.804517i	$0.297576\pi$
$282$	− 23.7300i	− 0.00501100i
$283$	3604.25i	0.757069i	0.925587	+	0.378534i	$0.123572\pi$
$283$	3604.25i	0.757069i	−0.925587	+	0.378534i	$0.876428\pi$
$284$	2165.99i	0.452563i
$285$	−708.240	−0.147202
$286$	− 174.951i	− 0.0361715i
$287$	−5123.72	−1.05381
$288$	−4785.42	−0.979110
$289$	0	0
$290$	2479.98	0.502170
$291$	858.328	0.172907
$292$	2462.57i	0.493531i
$293$	3857.83	0.769204	0.384602	−	0.923083i	$-0.374339\pi$
$293$	3857.83	0.769204	0.384602	+	0.923083i	$0.374339\pi$
$294$	− 76.8275i	− 0.0152404i
$295$	− 2182.69i	− 0.430783i
$296$	− 1308.48i	− 0.256939i
$297$	−1304.38	−0.254840
$298$	−2156.06	−0.419119
$299$	360.213i	0.0696712i
$300$	1091.25i	0.210012i
$301$	4231.77i	0.810349i
$302$	2044.64	0.389589
$303$	63.4572i	0.0120314i
$304$	1911.48	0.360629
$305$	10844.1	2.03584
$306$	0	0
$307$	−4799.00	−0.892161	−0.446080	−	0.894993i	$-0.647180\pi$
$307$	−4799.00	−0.892161	−0.446080	+	0.894993i	$0.647180\pi$
$308$	4434.63	0.820411
$309$	748.362i	0.137776i
$310$	−4994.30	−0.915023
$311$	2633.07i	0.480090i	0.970762	+	0.240045i	$0.0771622\pi$
$311$	2633.07i	0.480090i	−0.970762	+	0.240045i	$0.922838\pi$
$312$	29.8638i	0.00541893i
$313$	3669.10i	0.662587i	0.943528	+	0.331293i	$0.107485\pi$
$313$	3669.10i	0.662587i	−0.943528	+	0.331293i	$0.892515\pi$
$314$	−1429.81	−0.256971
$315$	−8409.18	−1.50414
$316$	5706.39i	1.01585i
$317$	− 3963.01i	− 0.702161i	−0.936345	−	0.351080i	$-0.885814\pi$
$317$	− 3963.01i	− 0.702161i	0.936345	−	0.351080i	$-0.114186\pi$
$318$	125.283i	0.0220929i
$319$	−4388.16	−0.770188
$320$	− 470.384i	− 0.0821727i
$321$	−337.756	−0.0587281
$322$	2108.19	0.364860
$323$	0	0
$324$	−4588.05	−0.786703
$325$	−987.456	−0.168536
$326$	− 2662.64i	− 0.452361i
$327$	−658.199	−0.111310
$328$	6057.63i	1.01975i
$329$	544.291i	0.0912090i
$330$	623.298i	0.103974i
$331$	11619.0	1.92942	0.964710	−	0.263316i	$-0.0848163\pi$
$331$	11619.0	1.92942	0.964710	+	0.263316i	$0.0848163\pi$
$332$	7768.45	1.28418
$333$	− 1967.98i	− 0.323858i
$334$	1387.75i	0.227348i
$335$	− 11752.3i	− 1.91671i
$336$	−242.883	−0.0394357
$337$	1616.09i	0.261229i	0.991433	+	0.130615i	$0.0416951\pi$
$337$	1616.09i	0.261229i	−0.991433	+	0.130615i	$0.958305\pi$
$338$	2679.23	0.431157
$339$	183.838	0.0294535
$340$	0	0
$341$	8837.09	1.40339
$342$	−2068.81	−0.327101
$343$	6915.23i	1.08859i
$344$	5003.10	0.784155
$345$	− 1283.33i	− 0.200268i
$346$	56.7757i	0.00882162i
$347$	6435.64i	0.995629i	0.867284	+	0.497814i	$0.165864\pi$
$347$	6435.64i	0.995629i	−0.867284	+	0.497814i	$0.834136\pi$
$348$	335.763	0.0517207
$349$	5582.36	0.856210	0.428105	−	0.903729i	$-0.359181\pi$
$349$	5582.36	0.856210	0.428105	+	0.903729i	$0.359181\pi$
$350$	5779.20i	0.882603i
$351$	90.3120i	0.0137336i
$352$	− 8135.73i	− 1.23192i
$353$	−7725.05	−1.16477	−0.582384	−	0.812914i	$-0.697880\pi$
$353$	−7725.05	−1.16477	−0.582384	+	0.812914i	$0.697880\pi$
$354$	68.2316i	0.0102443i
$355$	6982.78	1.04397
$356$	−5094.74	−0.758485
$357$	0	0
$358$	−788.856	−0.116459
$359$	−10767.6	−1.58299	−0.791494	−	0.611178i	$-0.790696\pi$
$359$	−10767.6	−1.58299	−0.791494	+	0.611178i	$0.790696\pi$
$360$	9941.94i	1.45552i
$361$	−2862.49	−0.417333
$362$	− 454.579i	− 0.0660004i
$363$	− 391.222i	− 0.0565671i
$364$	− 307.044i	− 0.0442128i
$365$	7938.92	1.13847
$366$	−338.990	−0.0484134
$367$	− 3600.73i	− 0.512143i	−0.966658	−	0.256071i	$-0.917572\pi$
$367$	− 3600.73i	− 0.512143i	0.966658	−	0.256071i	$-0.0824282\pi$
$368$	3463.62i	0.490635i
$369$	9110.78i	1.28533i
$370$	−1890.87	−0.265680
$371$	− 2873.60i	− 0.402129i
$372$	−676.177	−0.0942423
$373$	−7071.35	−0.981610	−0.490805	−	0.871269i	$-0.663297\pi$
$373$	−7071.35	−0.981610	−0.490805	+	0.871269i	$0.663297\pi$
$374$	0	0
$375$	2117.62	0.291609
$376$	643.501	0.0882607
$377$	303.827i	0.0415063i
$378$	528.562	0.0719214
$379$	4374.71i	0.592912i	0.955046	+	0.296456i	$0.0958049\pi$
$379$	4374.71i	0.592912i	−0.955046	+	0.296456i	$0.904195\pi$
$380$	− 8609.00i	− 1.16219i
$381$	1254.62i	0.168704i
$382$	2296.83	0.307634
$383$	−6532.97	−0.871591	−0.435795	−	0.900046i	$-0.643533\pi$
$383$	−6532.97	−0.871591	−0.435795	+	0.900046i	$0.643533\pi$
$384$	780.948i	0.103783i
$385$	− 14296.5i	− 1.89251i
$386$	1004.35i	0.132435i
$387$	7524.75	0.988384
$388$	10433.4i	1.36514i
$389$	2513.01	0.327544	0.163772	−	0.986498i	$-0.447634\pi$
$389$	2513.01	0.327544	0.163772	+	0.986498i	$0.447634\pi$
$390$	43.1558	0.00560328
$391$	0	0
$392$	2083.38	0.268435
$393$	213.582	0.0274142
$394$	− 1308.09i	− 0.167261i
$395$	18396.4	2.34336
$396$	− 7885.47i	− 1.00066i
$397$	− 1967.83i	− 0.248772i	−0.992234	−	0.124386i	$-0.960304\pi$
$397$	− 1967.83i	− 0.248772i	0.992234	−	0.124386i	$-0.0396961\pi$
$398$	− 2913.09i	− 0.366884i
$399$	−507.818	−0.0637161
$400$	−9494.85	−1.18686
$401$	12269.2i	1.52792i	0.645266	+	0.763958i	$0.276747\pi$
$401$	12269.2i	1.52792i	−0.645266	+	0.763958i	$0.723253\pi$
$402$	367.381i	0.0455804i
$403$	− 611.861i	− 0.0756302i
$404$	−771.354	−0.0949908
$405$	14791.1i	1.81476i
$406$	1778.18	0.217363
$407$	3345.78	0.407479
$408$	0	0
$409$	11894.5	1.43801	0.719004	−	0.695006i	$-0.244598\pi$
$409$	11894.5	1.43801	0.719004	+	0.695006i	$0.244598\pi$
$410$	8753.80	1.05444
$411$	765.454i	0.0918664i
$412$	−9096.71	−1.08777
$413$	− 1565.02i	− 0.186464i
$414$	− 3748.70i	− 0.445020i
$415$	− 25044.2i	− 2.96234i
$416$	−563.299	−0.0663895
$417$	−433.971	−0.0509632
$418$	− 3517.20i	− 0.411559i
$419$	− 137.704i	− 0.0160555i	−0.999968	−	0.00802776i	$-0.997445\pi$
$419$	− 137.704i	− 0.0160555i	0.999968	−	0.00802776i	$-0.00255534\pi$
$420$	1093.91i	0.127089i
$421$	6240.45	0.722425	0.361213	−	0.932483i	$-0.382363\pi$
$421$	6240.45	0.722425	0.361213	+	0.932483i	$0.382363\pi$
$422$	− 4332.13i	− 0.499727i
$423$	967.836	0.111248
$424$	−3397.38	−0.389130
$425$	0	0
$426$	−218.285	−0.0248261
$427$	7775.36	0.881209
$428$	− 4105.60i	− 0.463672i
$429$	−76.3614	−0.00859386
$430$	− 7229.91i	− 0.810831i
$431$	− 10990.9i	− 1.22834i	−0.789173	−	0.614171i	$-0.789491\pi$
$431$	− 10990.9i	− 1.22834i	0.789173	−	0.614171i	$-0.210509\pi$
$432$	868.392i	0.0967142i
$433$	−1643.15	−0.182367	−0.0911835	−	0.995834i	$-0.529065\pi$
$433$	−1643.15	−0.182367	−0.0911835	+	0.995834i	$0.529065\pi$
$434$	−3580.98	−0.396066
$435$	− 1082.44i	− 0.119309i
$436$	− 8000.73i	− 0.878820i
$437$	7241.70i	0.792717i
$438$	−248.174	−0.0270735
$439$	− 13532.5i	− 1.47124i	−0.677397	−	0.735618i	$-0.736892\pi$
$439$	− 13532.5i	− 1.47124i	0.677397	−	0.735618i	$-0.263108\pi$
$440$	−16902.4	−1.83134
$441$	3133.44	0.338347
$442$	0	0
$443$	−7579.72	−0.812920	−0.406460	−	0.913669i	$-0.633237\pi$
$443$	−7579.72	−0.812920	−0.406460	+	0.913669i	$0.633237\pi$
$444$	−256.005	−0.0273636
$445$	16424.6i	1.74966i
$446$	6028.11	0.639998
$447$	941.064i	0.0995768i
$448$	− 337.272i	− 0.0355683i
$449$	− 8617.72i	− 0.905781i	−0.891566	−	0.452890i	$-0.850393\pi$
$449$	− 8617.72i	− 0.905781i	0.891566	−	0.452890i	$-0.149607\pi$
$450$	10276.3	1.07651
$451$	−15489.3	−1.61721
$452$	2234.64i	0.232542i
$453$	− 892.432i	− 0.0925609i
$454$	− 6265.15i	− 0.647660i
$455$	−989.857	−0.101990
$456$	600.380i	0.0616565i
$457$	12066.5	1.23512	0.617558	−	0.786525i	$-0.288122\pi$
$457$	12066.5	1.23512	0.617558	+	0.786525i	$0.288122\pi$
$458$	2241.50	0.228687
$459$	0	0
$460$	15599.6	1.58116
$461$	8906.49	0.899819	0.449910	−	0.893074i	$-0.351456\pi$
$461$	8906.49	0.899819	0.449910	+	0.893074i	$0.351456\pi$
$462$	446.914i	0.0450050i
$463$	−3044.63	−0.305607	−0.152803	−	0.988257i	$-0.548830\pi$
$463$	−3044.63	−0.305607	−0.152803	+	0.988257i	$0.548830\pi$
$464$	2921.43i	0.292293i
$465$	2179.88i	0.217397i
$466$	6434.40i	0.639631i
$467$	2838.99	0.281312	0.140656	−	0.990058i	$-0.455079\pi$
$467$	2838.99	0.281312	0.140656	+	0.990058i	$0.455079\pi$
$468$	−545.972	−0.0539265
$469$	− 8426.57i	− 0.829643i
$470$	− 929.914i	− 0.0912632i
$471$	624.075i	0.0610528i
$472$	−1850.28	−0.180436
$473$	12792.9i	1.24359i
$474$	−575.080	−0.0557264
$475$	−19851.7	−1.91760
$476$	0	0
$477$	−5109.72	−0.490478
$478$	−436.014	−0.0417214
$479$	− 502.893i	− 0.0479703i	−0.999712	−	0.0239852i	$-0.992365\pi$
$479$	− 502.893i	− 0.0479703i	0.999712	−	0.0239852i	$-0.00763544\pi$
$480$	2006.87	0.190835
$481$	− 231.654i	− 0.0219595i
$482$	− 2730.62i	− 0.258042i
$483$	− 920.170i	− 0.0866856i
$484$	4755.50	0.446610
$485$	33635.5	3.14909
$486$	− 1412.30i	− 0.131817i
$487$	6031.80i	0.561246i	0.959818	+	0.280623i	$0.0905411\pi$
$487$	6031.80i	0.561246i	−0.959818	+	0.280623i	$0.909459\pi$
$488$	− 9192.59i	− 0.852724i
$489$	−1162.17	−0.107475
$490$	− 3010.66i	− 0.277567i
$491$	16555.8	1.52169	0.760846	−	0.648932i	$-0.224784\pi$
$491$	16555.8	1.52169	0.760846	+	0.648932i	$0.224784\pi$
$492$	1185.17	0.108601
$493$	0	0
$494$	−243.523	−0.0221794
$495$	−25421.4	−2.30830
$496$	− 5883.32i	− 0.532599i
$497$	5006.76	0.451879
$498$	782.890i	0.0704461i
$499$	5163.14i	0.463194i	0.972812	+	0.231597i	$0.0743951\pi$
$499$	5163.14i	0.463194i	−0.972812	+	0.231597i	$0.925605\pi$
$500$	25740.7i	2.30232i
$501$	605.716	0.0540147
$502$	−1700.41	−0.151181
$503$	10736.0i	0.951683i	0.879531	+	0.475842i	$0.157856\pi$
$503$	10736.0i	0.951683i	−0.879531	+	0.475842i	$0.842144\pi$
$504$	7128.52i	0.630019i
$505$	2486.71i	0.219123i
$506$	6373.18	0.559926
$507$	− 1169.41i	− 0.102437i
$508$	−15250.6	−1.33196
$509$	18206.2	1.58542	0.792709	−	0.609600i	$-0.208670\pi$
$509$	18206.2	1.58542	0.792709	+	0.609600i	$0.208670\pi$
$510$	0	0
$511$	5692.32	0.492786
$512$	−9712.82	−0.838379
$513$	1815.63i	0.156261i
$514$	−9066.81	−0.778054
$515$	29326.2i	2.50926i
$516$	− 978.856i	− 0.0835111i
$517$	1645.42i	0.139972i
$518$	−1355.78	−0.114999
$519$	24.7811	0.00209589
$520$	1170.28i	0.0986927i
$521$	2147.80i	0.180608i	0.995914	+	0.0903039i	$0.0287838\pi$
$521$	2147.80i	0.180608i	−0.995914	+	0.0903039i	$0.971216\pi$
$522$	− 3161.88i	− 0.265119i
$523$	−21222.1	−1.77434	−0.887169	−	0.461444i	$-0.847331\pi$
$523$	−21222.1	−1.77434	−0.887169	+	0.461444i	$0.847331\pi$
$524$	2596.19i	0.216441i
$525$	2522.47	0.209694
$526$	8254.82	0.684272
$527$	0	0
$528$	−734.251	−0.0605192
$529$	−955.000	−0.0784910
$530$	4909.51i	0.402368i
$531$	−2782.85	−0.227430
$532$	− 6172.78i	− 0.503053i
$533$	1072.44i	0.0871533i
$534$	− 513.439i	− 0.0416080i
$535$	−13235.7	−1.06959
$536$	−9962.50	−0.802825
$537$	344.315i	0.0276691i
$538$	4256.16i	0.341071i
$539$	5327.17i	0.425710i
$540$	3911.09	0.311679
$541$	− 5123.54i	− 0.407169i	−0.979057	−	0.203584i	$-0.934741\pi$
$541$	− 5123.54i	− 0.407169i	0.979057	−	0.203584i	$-0.0652591\pi$
$542$	1867.90	0.148032
$543$	−198.412	−0.0156808
$544$	0	0
$545$	−25793.0	−2.02725
$546$	30.9433	0.00242537
$547$	− 17905.9i	− 1.39964i	−0.714319	−	0.699820i	$-0.753264\pi$
$547$	− 17905.9i	− 1.39964i	0.714319	−	0.699820i	$-0.246736\pi$
$548$	−9304.48	−0.725306
$549$	− 13825.8i	− 1.07481i
$550$	17470.8i	1.35447i
$551$	6108.10i	0.472258i
$552$	−1087.89	−0.0838835
$553$	13190.5	1.01432
$554$	6589.70i	0.505360i
$555$	825.315i	0.0631220i
$556$	− 5275.14i	− 0.402366i
$557$	−416.095	−0.0316526	−0.0158263	−	0.999875i	$-0.505038\pi$
$557$	−416.095	−0.0316526	−0.0158263	+	0.999875i	$0.505038\pi$
$558$	6367.56i	0.483083i
$559$	885.750	0.0670183
$560$	−9517.94	−0.718226
$561$	0	0
$562$	−6854.31	−0.514469
$563$	23092.3	1.72864	0.864321	−	0.502940i	$-0.167748\pi$
$563$	23092.3	1.72864	0.864321	+	0.502940i	$0.167748\pi$
$564$	− 125.901i	− 0.00939961i
$565$	7204.12	0.536424
$566$	− 4415.24i	− 0.327891i
$567$	10605.4i	0.785515i
$568$	− 5919.35i	− 0.437272i
$569$	−11328.2	−0.834630	−0.417315	−	0.908762i	$-0.637029\pi$
$569$	−11328.2	−0.834630	−0.417315	+	0.908762i	$0.637029\pi$
$570$	867.600	0.0637540
$571$	− 15325.7i	− 1.12323i	−0.827400	−	0.561613i	$-0.810181\pi$
$571$	− 15325.7i	− 1.12323i	0.827400	−	0.561613i	$-0.189819\pi$
$572$	− 928.211i	− 0.0678505i
$573$	− 1002.51i	− 0.0730896i
$574$	6276.60	0.456412
$575$	− 35971.4i	− 2.60889i
$576$	−599.723	−0.0433828
$577$	9112.72	0.657482	0.328741	−	0.944420i	$-0.393376\pi$
$577$	9112.72	0.657482	0.328741	+	0.944420i	$0.393376\pi$
$578$	0	0
$579$	438.372	0.0314648
$580$	13157.6	0.941968
$581$	− 17957.0i	− 1.28224i
$582$	−1051.46	−0.0748873
$583$	− 8687.06i	− 0.617121i
$584$	− 6729.87i	− 0.476856i
$585$	1760.12i	0.124397i
$586$	−4725.87	−0.333147
$587$	−13653.4	−0.960025	−0.480013	−	0.877262i	$-0.659368\pi$
$587$	−13653.4	−0.960025	−0.480013	+	0.877262i	$0.659368\pi$
$588$	− 407.613i	− 0.0285879i
$589$	− 12300.8i	− 0.860519i
$590$	2673.81i	0.186575i
$591$	−570.947	−0.0397388
$592$	− 2227.46i	− 0.154642i
$593$	−21660.3	−1.49997	−0.749986	−	0.661453i	$-0.769940\pi$
$593$	−21660.3	−1.49997	−0.749986	+	0.661453i	$0.769940\pi$
$594$	1597.87	0.110373
$595$	0	0
$596$	−11439.1	−0.786181
$597$	−1271.49	−0.0871665
$598$	− 441.265i	− 0.0301750i
$599$	4638.62	0.316408	0.158204	−	0.987406i	$-0.449430\pi$
$599$	4638.62	0.316408	0.158204	+	0.987406i	$0.449430\pi$
$600$	− 2982.24i	− 0.202916i
$601$	− 18.6165i	− 0.00126353i	−1.00000		0.000631766i	$-0.999799\pi$
$601$	− 18.6165i	− 0.00126353i	1.00000		0.000631766i	$-0.000201097\pi$
$602$	− 5183.96i	− 0.350967i
$603$	−14983.8	−1.01192
$604$	10847.9	0.730789
$605$	− 15330.9i	− 1.03023i
$606$	− 77.7357i	− 0.00521088i
$607$	21481.4i	1.43641i	0.695831	+	0.718206i	$0.255036\pi$
$607$	21481.4i	1.43641i	−0.695831	+	0.718206i	$0.744964\pi$
$608$	−11324.5	−0.755379
$609$	− 776.129i	− 0.0516426i
$610$	−13284.1	−0.881733
$611$	113.925	0.00754326
$612$	0	0
$613$	−4978.56	−0.328030	−0.164015	−	0.986458i	$-0.552445\pi$
$613$	−4978.56	−0.328030	−0.164015	+	0.986458i	$0.552445\pi$
$614$	5878.82	0.386400
$615$	− 3820.80i	− 0.250520i
$616$	−12119.2	−0.792691
$617$	320.723i	0.0209268i	0.999945	+	0.0104634i	$0.00333066\pi$
$617$	320.723i	0.0209268i	−0.999945	+	0.0104634i	$0.996669\pi$
$618$	− 916.750i	− 0.0596717i
$619$	− 5305.17i	− 0.344480i	−0.985055	−	0.172240i	$-0.944900\pi$
$619$	− 5305.17i	− 0.344480i	0.985055	−	0.172240i	$-0.0551004\pi$
$620$	−26497.5	−1.71640
$621$	−3289.92	−0.212593
$622$	− 3225.54i	− 0.207930i
$623$	11776.7i	0.757339i
$624$	50.8379i	0.00326145i
$625$	43731.2	2.79880
$626$	− 4494.68i	− 0.286970i
$627$	−1535.16	−0.0977808
$628$	−7585.94	−0.482025
$629$	0	0
$630$	10301.3	0.651451
$631$	−19724.5	−1.24440	−0.622202	−	0.782857i	$-0.713762\pi$
$631$	−19724.5	−1.24440	−0.622202	+	0.782857i	$0.713762\pi$
$632$	− 15594.8i	− 0.981531i
$633$	−1890.86	−0.118728
$634$	4854.73i	0.304110i
$635$	49165.3i	3.07255i
$636$	664.697i	0.0414417i
$637$	368.842	0.0229420
$638$	5375.54	0.333573
$639$	− 8902.81i	− 0.551158i
$640$	30603.2i	1.89015i
$641$	8160.70i	0.502852i	0.967876	+	0.251426i	$0.0808996\pi$
$641$	8160.70i	0.502852i	−0.967876	+	0.251426i	$0.919100\pi$
$642$	413.755	0.0254355
$643$	31644.9i	1.94083i	0.241449	+	0.970413i	$0.422377\pi$
$643$	31644.9i	1.94083i	−0.241449	+	0.970413i	$0.577623\pi$
$644$	11185.1	0.684403
$645$	−3155.67	−0.192642
$646$	0	0
$647$	23434.3	1.42395	0.711976	−	0.702204i	$-0.247801\pi$
$647$	23434.3	1.42395	0.711976	+	0.702204i	$0.247801\pi$
$648$	12538.5	0.760123
$649$	− 4731.14i	− 0.286153i
$650$	1209.64	0.0729940
$651$	1563.01i	0.0940999i
$652$	− 14126.8i	− 0.848538i
$653$	7603.42i	0.455658i	0.973701	+	0.227829i	$0.0731628\pi$
$653$	7603.42i	0.455658i	−0.973701	+	0.227829i	$0.926837\pi$
$654$	806.300	0.0482092
$655$	8369.69	0.499283
$656$	10312.0i	0.613747i
$657$	− 10121.8i	− 0.601051i
$658$	− 666.762i	− 0.0395032i
$659$	−11917.6	−0.704468	−0.352234	−	0.935912i	$-0.614578\pi$
$659$	−11917.6	−0.704468	−0.352234	+	0.935912i	$0.614578\pi$
$660$	3306.94i	0.195034i
$661$	7319.30	0.430693	0.215346	−	0.976538i	$-0.430912\pi$
$661$	7319.30	0.430693	0.215346	+	0.976538i	$0.430912\pi$
$662$	−14233.4	−0.835643
$663$	0	0
$664$	−21230.1	−1.24080
$665$	−19900.0	−1.16044
$666$	2410.80i	0.140265i
$667$	−11067.9	−0.642506
$668$	7362.77i	0.426458i
$669$	− 2631.11i	− 0.152055i
$670$	14396.7i	0.830137i
$671$	23505.4	1.35233
$672$	1438.96	0.0826025
$673$	− 20860.7i	− 1.19483i	−0.801933	−	0.597414i	$-0.796195\pi$
$673$	− 20860.7i	− 1.19483i	0.801933	−	0.597414i	$-0.203805\pi$
$674$	− 1979.73i	− 0.113140i
$675$	− 9018.69i	− 0.514266i
$676$	14214.8	0.808763
$677$	− 9840.97i	− 0.558670i	−0.960194	−	0.279335i	$-0.909886\pi$
$677$	− 9840.97i	− 0.558670i	0.960194	−	0.279335i	$-0.0901139\pi$
$678$	−225.204	−0.0127565
$679$	24117.2	1.36308
$680$	0	0
$681$	−2734.57	−0.153875
$682$	−10825.5	−0.607816
$683$	16967.2i	0.950560i	0.879835	+	0.475280i	$0.157653\pi$
$683$	16967.2i	0.950560i	−0.879835	+	0.475280i	$0.842347\pi$
$684$	−10976.2	−0.613574
$685$	29996.1i	1.67312i
$686$	− 8471.22i	− 0.471476i
$687$	− 978.358i	− 0.0543329i
$688$	8516.90	0.471953
$689$	−601.473	−0.0332573
$690$	1572.10i	0.0867372i
$691$	24669.3i	1.35813i	0.734079	+	0.679064i	$0.237614\pi$
$691$	24669.3i	1.35813i	−0.734079	+	0.679064i	$0.762386\pi$
$692$	301.227i	0.0165476i
$693$	−18227.5	−0.999144
$694$	− 7883.72i	− 0.431213i
$695$	−17006.1	−0.928172
$696$	−917.595	−0.0499732
$697$	0	0
$698$	−6838.45	−0.370830
$699$	2808.45	0.151967
$700$	30661.8i	1.65558i
$701$	−28908.3	−1.55756	−0.778780	−	0.627297i	$-0.784161\pi$
$701$	−28908.3	−1.55756	−0.778780	+	0.627297i	$0.784161\pi$
$702$	− 110.633i	− 0.00594812i
$703$	− 4657.16i	− 0.249855i
$704$	− 1019.59i	− 0.0545843i
$705$	−405.883	−0.0216829
$706$	9463.26	0.504468
$707$	1783.01i	0.0948473i
$708$	362.006i	0.0192162i
$709$	− 7775.90i	− 0.411890i	−0.978564	−	0.205945i	$-0.933973\pi$
$709$	− 7775.90i	− 0.411890i	0.978564	−	0.205945i	$-0.0660268\pi$
$710$	−8553.98	−0.452148
$711$	− 23454.8i	− 1.23717i
$712$	13923.2	0.732858
$713$	22289.1	1.17073
$714$	0	0
$715$	−2992.40	−0.156516
$716$	−4185.32	−0.218454
$717$	190.309i	0.00991242i
$718$	13190.4	0.685601
$719$	− 30789.3i	− 1.59700i	−0.601992	−	0.798502i	$-0.705626\pi$
$719$	− 30789.3i	− 1.59700i	0.601992	−	0.798502i	$-0.294374\pi$
$720$	16924.4i	0.876021i
$721$	21027.4i	1.08613i
$722$	3506.57	0.180750
$723$	−1191.84	−0.0613073
$724$	− 2411.79i	− 0.123803i
$725$	− 30340.6i	− 1.55423i
$726$	479.251i	0.0244996i
$727$	−25161.8	−1.28363	−0.641815	−	0.766860i	$-0.721818\pi$
$727$	−25161.8	−1.28363	−0.641815	+	0.766860i	$0.721818\pi$
$728$	839.109i	0.0427190i
$729$	18443.5	0.937029
$730$	−9725.25	−0.493079
$731$	0	0
$732$	−1798.53	−0.0908136
$733$	−2453.96	−0.123655	−0.0618274	−	0.998087i	$-0.519693\pi$
$733$	−2453.96	−0.123655	−0.0618274	+	0.998087i	$0.519693\pi$
$734$	4410.92i	0.221812i
$735$	−1314.07	−0.0659460
$736$	− 20520.1i	− 1.02769i
$737$	− 25474.0i	− 1.27320i
$738$	− 11160.8i	− 0.556686i
$739$	28203.6	1.40391	0.701953	−	0.712224i	$-0.252312\pi$
$739$	28203.6	1.40391	0.701953	+	0.712224i	$0.252312\pi$
$740$	−10032.1	−0.498362
$741$	106.291i	0.00526951i
$742$	3520.19i	0.174165i
$743$	− 25185.5i	− 1.24356i	−0.783191	−	0.621781i	$-0.786410\pi$
$743$	− 25185.5i	− 1.24356i	0.783191	−	0.621781i	$-0.213590\pi$
$744$	1847.90	0.0910581
$745$	36877.7i	1.81355i
$746$	8662.47	0.425141
$747$	−31930.4	−1.56395
$748$	0	0
$749$	−9490.23	−0.462971
$750$	−2594.11	−0.126298
$751$	− 16814.8i	− 0.817020i	−0.912754	−	0.408510i	$-0.866048\pi$
$751$	− 16814.8i	− 0.817020i	0.912754	−	0.408510i	$-0.133952\pi$
$752$	1095.45	0.0531208
$753$	742.184i	0.0359186i
$754$	− 372.190i	− 0.0179766i
$755$	− 34972.0i	− 1.68577i
$756$	2804.31	0.134910
$757$	−31469.3	−1.51093	−0.755464	−	0.655191i	$-0.772588\pi$
$757$	−31469.3	−1.51093	−0.755464	+	0.655191i	$0.772588\pi$
$758$	− 5359.06i	− 0.256794i
$759$	− 2781.73i	− 0.133031i
$760$	23527.2i	1.12292i
$761$	16621.7	0.791771	0.395885	−	0.918300i	$-0.370438\pi$
$761$	16621.7	0.791771	0.395885	+	0.918300i	$0.370438\pi$
$762$	− 1536.93i	− 0.0730669i
$763$	−18494.0	−0.877492
$764$	12186.0	0.577059
$765$	0	0
$766$	8002.95	0.377491
$767$	−327.573	−0.0154211
$768$	− 860.641i	− 0.0404371i
$769$	35810.5	1.67927	0.839635	−	0.543151i	$-0.182769\pi$
$769$	35810.5	1.67927	0.839635	+	0.543151i	$0.182769\pi$
$770$	17513.3i	0.819658i
$771$	3957.42i	0.184855i
$772$	5328.63i	0.248422i
$773$	−29648.3	−1.37953	−0.689765	−	0.724033i	$-0.742286\pi$
$773$	−29648.3	−1.37953	−0.689765	+	0.724033i	$0.742286\pi$
$774$	−9217.89	−0.428075
$775$	61101.3i	2.83203i
$776$	− 28513.1i	− 1.31902i
$777$	591.763i	0.0273223i
$778$	−3078.45	−0.141861
$779$	21560.3i	0.991629i
$780$	228.965	0.0105106
$781$	15135.7	0.693468
$782$	0	0
$783$	−2774.93	−0.126651
$784$	3546.58	0.161561
$785$	24455.8i	1.11193i
$786$	−261.640	−0.0118733
$787$	− 12772.1i	− 0.578496i	−0.957254	−	0.289248i	$-0.906595\pi$
$787$	− 12772.1i	− 0.578496i	0.957254	−	0.289248i	$-0.0934051\pi$
$788$	− 6940.15i	− 0.313747i
$789$	− 3603.01i	− 0.162574i
$790$	−22535.8	−1.01492
$791$	5165.46	0.232190
$792$	21549.9i	0.966847i
$793$	− 1627.46i	− 0.0728786i
$794$	2410.61i	0.107745i
$795$	2142.87	0.0955972
$796$	− 15455.5i	− 0.688200i
$797$	12572.8	0.558785	0.279392	−	0.960177i	$-0.409867\pi$
$797$	12572.8	0.558785	0.279392	+	0.960177i	$0.409867\pi$
$798$	622.082	0.0275958
$799$	0	0
$800$	56251.9	2.48601
$801$	20940.8	0.923727
$802$	− 15029.9i	− 0.661750i
$803$	17208.2	0.756245
$804$	1949.16i	0.0854995i
$805$	− 36058.9i	− 1.57877i
$806$	749.535i	0.0327559i
$807$	1857.70	0.0810338
$808$	2108.00	0.0917813
$809$	− 14249.3i	− 0.619259i	−0.950857	−	0.309629i	$-0.899795\pi$
$809$	− 14249.3i	− 0.619259i	0.950857	−	0.309629i	$-0.100205\pi$
$810$	− 18119.2i	− 0.785982i
$811$	25534.0i	1.10557i	0.833323	+	0.552786i	$0.186435\pi$
$811$	25534.0i	1.10557i	−0.833323	+	0.552786i	$0.813565\pi$
$812$	9434.23	0.407730
$813$	− 815.290i	− 0.0351703i
$814$	−4098.61	−0.176482
$815$	−45542.3	−1.95739
$816$	0	0
$817$	17807.1	0.762533
$818$	−14570.9	−0.622810
$819$	1262.03i	0.0538450i
$820$	46443.7	1.97791
$821$	− 34196.6i	− 1.45368i	−0.686809	−	0.726838i	$-0.740989\pi$
$821$	− 34196.6i	− 1.45368i	0.686809	−	0.726838i	$-0.259011\pi$
$822$	− 937.689i	− 0.0397879i
$823$	− 35235.8i	− 1.49240i	−0.665723	−	0.746199i	$-0.731877\pi$
$823$	− 35235.8i	− 1.49240i	0.665723	−	0.746199i	$-0.268123\pi$
$824$	24860.1	1.05102
$825$	7625.57	0.321804
$826$	1917.16i	0.0807586i
$827$	6761.12i	0.284289i	0.989846	+	0.142145i	$0.0453998\pi$
$827$	6761.12i	0.284289i	−0.989846	+	0.142145i	$0.954600\pi$
$828$	− 19888.9i	− 0.834767i
$829$	26249.2	1.09973	0.549863	−	0.835255i	$-0.314680\pi$
$829$	26249.2	1.09973	0.549863	+	0.835255i	$0.314680\pi$
$830$	30679.3i	1.28301i
$831$	2876.23	0.120067
$832$	−70.5944	−0.00294161
$833$	0	0
$834$	531.619	0.0220725
$835$	23736.3	0.983748
$836$	− 18660.7i	− 0.772001i
$837$	5588.28	0.230776
$838$	168.688i	0.00695374i
$839$	8884.87i	0.365602i	0.983150	+	0.182801i	$0.0585163\pi$
$839$	8884.87i	0.365602i	−0.983150	+	0.182801i	$0.941484\pi$
$840$	− 2989.50i	− 0.122795i
$841$	15053.6	0.617230
$842$	−7644.61	−0.312887
$843$	2991.73i	0.122231i
$844$	− 22984.3i	− 0.937386i
$845$	− 45826.1i	− 1.86564i
$846$	−1185.61	−0.0481821
$847$	− 10992.5i	− 0.445935i
$848$	−5783.44	−0.234203
$849$	−1927.14	−0.0779024
$850$	0	0
$851$	8438.79	0.339927
$852$	−1158.12	−0.0465687
$853$	39637.2i	1.59104i	0.605931	+	0.795518i	$0.292801\pi$
$853$	39637.2i	1.59104i	−0.605931	+	0.795518i	$0.707199\pi$
$854$	−9524.89	−0.381657
$855$	35385.4i	1.41538i
$856$	11220.0i	0.448006i
$857$	18348.8i	0.731367i	0.930739	+	0.365684i	$0.119165\pi$
$857$	18348.8i	0.731367i	−0.930739	+	0.365684i	$0.880835\pi$
$858$	93.5435	0.00372205
$859$	−3064.71	−0.121730	−0.0608652	−	0.998146i	$-0.519386\pi$
$859$	−3064.71	−0.121730	−0.0608652	+	0.998146i	$0.519386\pi$
$860$	− 38358.7i	− 1.52095i
$861$	− 2739.57i	− 0.108437i
$862$	13464.0i	0.532002i
$863$	−19334.7	−0.762643	−0.381321	−	0.924443i	$-0.624531\pi$
$863$	−19334.7	−0.762643	−0.381321	+	0.924443i	$0.624531\pi$
$864$	− 5144.76i	− 0.202579i
$865$	971.103	0.0381717
$866$	2012.88	0.0789843
$867$	0	0
$868$	−18999.1	−0.742940
$869$	39875.7	1.55661
$870$	1326.00i	0.0516733i
$871$	−1763.76	−0.0686140
$872$	21864.9i	0.849127i
$873$	− 42884.1i	− 1.66255i
$874$	− 8871.15i	− 0.343331i
$875$	59500.6	2.29884
$876$	−1316.70	−0.0507844
$877$	20294.5i	0.781409i	0.920516	+	0.390704i	$0.127769\pi$
$877$	20294.5i	0.781409i	−0.920516	+	0.390704i	$0.872231\pi$
$878$	16577.5i	0.637201i
$879$	2062.72i	0.0791511i
$880$	−28773.3	−1.10221
$881$	− 30744.7i	− 1.17573i	−0.808961	−	0.587863i	$-0.799969\pi$
$881$	− 30744.7i	− 1.17573i	0.808961	−	0.587863i	$-0.200031\pi$
$882$	−3838.49	−0.146540
$883$	−9156.84	−0.348983	−0.174492	−	0.984659i	$-0.555828\pi$
$883$	−9156.84	−0.348983	−0.174492	+	0.984659i	$0.555828\pi$
$884$	0	0
$885$	1167.05	0.0443275
$886$	9285.23	0.352081
$887$	14045.4i	0.531677i	0.964018	+	0.265838i	$0.0856488\pi$
$887$	14045.4i	0.531677i	−0.964018	+	0.265838i	$0.914351\pi$
$888$	699.625	0.0264391
$889$	35252.3i	1.32995i
$890$	− 20120.3i	− 0.757789i
$891$	32060.9i	1.20548i
$892$	31982.5	1.20051
$893$	2290.35	0.0858271
$894$	− 1152.81i	− 0.0431273i
$895$	13492.8i	0.503926i
$896$	21943.0i	0.818150i
$897$	−192.600	−0.00716917
$898$	10556.8i	0.392299i
$899$	18800.0	0.697459
$900$	54521.6	2.01932
$901$	0	0
$902$	18974.5	0.700424
$903$	−2262.66	−0.0833850
$904$	− 6106.98i	− 0.224685i
$905$	−7775.21	−0.285588
$906$	1093.24i	0.0400887i
$907$	− 28078.7i	− 1.02794i	−0.857810	−	0.513968i	$-0.828175\pi$
$907$	− 28078.7i	− 1.02794i	0.857810	−	0.513968i	$-0.171825\pi$
$908$	− 33240.1i	− 1.21488i
$909$	3170.47	0.115685
$910$	1212.58	0.0441723
$911$	41942.5i	1.52538i	0.646767	+	0.762688i	$0.276121\pi$
$911$	41942.5i	1.52538i	−0.646767	+	0.762688i	$0.723879\pi$
$912$	1022.04i	0.0371087i
$913$	− 54285.2i	− 1.96777i
$914$	−14781.6	−0.534937
$915$	5798.16i	0.209488i
$916$	11892.4	0.428970
$917$	6001.19	0.216114
$918$	0	0
$919$	18355.9	0.658873	0.329437	−	0.944178i	$-0.393141\pi$
$919$	18355.9	0.658873	0.329437	+	0.944178i	$0.393141\pi$
$920$	−42631.5	−1.52774
$921$	− 2565.95i	− 0.0918034i
$922$	−10910.5	−0.389717
$923$	− 1047.96i	− 0.0373718i
$924$	2371.13i	0.0844203i
$925$	23133.3i	0.822291i
$926$	3729.70	0.132360
$927$	37389.9	1.32475
$928$	− 17307.9i	− 0.612242i
$929$	31074.3i	1.09743i	0.836008	+	0.548717i	$0.184883\pi$
$929$	31074.3i	1.09743i	−0.836008	+	0.548717i	$0.815117\pi$
$930$	− 2670.37i	− 0.0941559i
$931$	7415.16	0.261033
$932$	34138.1i	1.19982i
$933$	−1407.86	−0.0494013
$934$	−3477.79	−0.121838
$935$	0	0
$936$	1492.07	0.0521045
$937$	−12245.2	−0.426930	−0.213465	−	0.976951i	$-0.568475\pi$
$937$	−12245.2	−0.426930	−0.213465	+	0.976951i	$0.568475\pi$
$938$	10322.6i	0.359324i
$939$	−1961.81	−0.0681802
$940$	− 4933.71i	− 0.171191i
$941$	11112.2i	0.384960i	0.981301	+	0.192480i	$0.0616530\pi$
$941$	11112.2i	0.384960i	−0.981301	+	0.192480i	$0.938347\pi$
$942$	− 764.497i	− 0.0264423i
$943$	−39067.4	−1.34911
$944$	−3149.77	−0.108598
$945$	− 9040.63i	− 0.311208i
$946$	− 15671.4i	− 0.538606i
$947$	40553.8i	1.39157i	0.718248	+	0.695787i	$0.244944\pi$
$947$	40553.8i	1.39157i	−0.718248	+	0.695787i	$0.755056\pi$
$948$	−3051.12	−0.104531
$949$	− 1191.46i	− 0.0407549i
$950$	24318.6	0.830524
$951$	2118.96	0.0722524
$952$	0	0
$953$	−10668.5	−0.362631	−0.181316	−	0.983425i	$-0.558036\pi$
$953$	−10668.5	−0.362631	−0.181316	+	0.983425i	$0.558036\pi$
$954$	6259.45	0.212429
$955$	− 39285.5i	− 1.33115i
$956$	−2313.30	−0.0782608
$957$	− 2346.28i	− 0.0792524i
$958$	616.049i	0.0207762i
$959$	21507.6i	0.724210i
$960$	251.507	0.00845557
$961$	−8069.41	−0.270867
$962$	283.778i	0.00951080i
$963$	16875.1i	0.564687i
$964$	− 14487.5i	− 0.484035i
$965$	17178.6	0.573055
$966$	1127.22i	0.0375441i
$967$	−12007.5	−0.399311	−0.199656	−	0.979866i	$-0.563982\pi$
$967$	−12007.5	−0.399311	−0.199656	+	0.979866i	$0.563982\pi$
$968$	−12996.1	−0.431520
$969$	0	0
$970$	−41203.8	−1.36389
$971$	−21024.0	−0.694842	−0.347421	−	0.937709i	$-0.612943\pi$
$971$	−21024.0	−0.694842	−0.347421	+	0.937709i	$0.612943\pi$
$972$	− 7493.03i	− 0.247263i
$973$	−12193.7	−0.401758
$974$	− 7389.01i	− 0.243079i
$975$	− 527.977i	− 0.0173424i
$976$	− 15648.8i	− 0.513222i
$977$	−21887.9	−0.716740	−0.358370	−	0.933580i	$-0.616667\pi$
$977$	−21887.9	−0.716740	−0.358370	+	0.933580i	$0.616667\pi$
$978$	1423.67	0.0465480
$979$	35601.5i	1.16224i
$980$	− 15973.2i	− 0.520659i
$981$	32885.2i	1.07028i
$982$	−20281.0	−0.659054
$983$	38731.8i	1.25672i	0.777924	+	0.628358i	$0.216273\pi$
$983$	38731.8i	1.25672i	−0.777924	+	0.628358i	$0.783727\pi$
$984$	−3238.92	−0.104932
$985$	−22373.9	−0.723747
$986$	0	0
$987$	−291.024	−0.00938541
$988$	−1292.02	−0.0416040
$989$	32266.5i	1.03743i
$990$	31141.5	0.999739
$991$	− 16776.8i	− 0.537771i	−0.963172	−	0.268886i	$-0.913345\pi$
$991$	− 16776.8i	− 0.537771i	0.963172	−	0.268886i	$-0.0866554\pi$
$992$	34855.6i	1.11559i
$993$	6212.49i	0.198537i
$994$	−6133.33	−0.195712
$995$	−49826.0	−1.58753
$996$	4153.67i	0.132143i
$997$	− 51595.5i	− 1.63896i	−0.573105	−	0.819482i	$-0.694261\pi$
$997$	− 51595.5i	− 1.63896i	0.573105	−	0.819482i	$-0.305739\pi$
$998$	− 6324.90i	− 0.200612i
$999$	2115.76	0.0670066

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	289.4.b.d.288.4		8
17.4	even	4		289.4.a.d.1.3	yes	4
17.13	even	4		289.4.a.c.1.3	✓	4
17.16	even	2	inner	289.4.b.d.288.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
289.4.a.c.1.3	✓	4	17.13	even	4
289.4.a.d.1.3	yes	4	17.4	even	4
289.4.b.d.288.3		8	17.16	even	2	inner
289.4.b.d.288.4		8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.22501 q^{2} +0.534684i q^{3} -6.49935 q^{4} +20.9528i q^{5} -0.654993i q^{6} +15.0235i q^{7} +17.7618 q^{8} +26.7141 q^{9} +O(q^{10})\)	\(q-1.22501 q^{2} +0.534684i q^{3} -6.49935 q^{4} +20.9528i q^{5} -0.654993i q^{6} +15.0235i q^{7} +17.7618 q^{8} +26.7141 q^{9} -25.6674i q^{10} +45.4168i q^{11} -3.47510i q^{12} +3.14456 q^{13} -18.4039i q^{14} -11.2031 q^{15} +30.2364 q^{16} -32.7250 q^{18} +63.2180 q^{19} -136.180i q^{20} -8.03282 q^{21} -55.6360i q^{22} +114.551i q^{23} +9.49698i q^{24} -314.020 q^{25} -3.85211 q^{26} +28.7201i q^{27} -97.6429i q^{28} +96.6197i q^{29} +13.7239 q^{30} -194.578i q^{31} -179.135 q^{32} -24.2837 q^{33} -314.784 q^{35} -173.624 q^{36} -73.6682i q^{37} -77.4426 q^{38} +1.68135i q^{39} +372.161i q^{40} +341.047i q^{41} +9.84027 q^{42} +281.677 q^{43} -295.180i q^{44} +559.736i q^{45} -140.326i q^{46} +36.2294 q^{47} +16.1669i q^{48} +117.295 q^{49} +384.678 q^{50} -20.4376 q^{52} -191.274 q^{53} -35.1824i q^{54} -951.610 q^{55} +266.845i q^{56} +33.8017i q^{57} -118.360i q^{58} -104.171 q^{59} +72.8131 q^{60} -517.547i q^{61} +238.359i q^{62} +401.339i q^{63} -22.4497 q^{64} +65.8874i q^{65} +29.7477 q^{66} -560.893 q^{67} -61.2488 q^{69} +385.613 q^{70} -333.262i q^{71} +474.492 q^{72} -378.895i q^{73} +90.2443i q^{74} -167.902i q^{75} -410.876 q^{76} -682.319 q^{77} -2.05967i q^{78} -877.994i q^{79} +633.538i q^{80} +705.925 q^{81} -417.786i q^{82} -1195.27 q^{83} +52.2081 q^{84} -345.057 q^{86} -51.6611 q^{87} +806.687i q^{88} +783.884 q^{89} -685.682i q^{90} +47.2422i q^{91} -744.509i q^{92} +104.038 q^{93} -44.3813 q^{94} +1324.59i q^{95} -95.7804i q^{96} -1605.30i q^{97} -143.688 q^{98} +1213.27i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{2} + 22 q^{4} - 120 q^{8} - 12 q^{9}+O(q^{10}) \) 8 * q - 2 * q^2 + 22 * q^4 - 120 * q^8 - 12 * q^9	\( 8 q - 2 q^{2} + 22 q^{4} - 120 q^{8} - 12 q^{9} - 44 q^{13} - 108 q^{15} + 126 q^{16} - 668 q^{18} - 44 q^{19} - 704 q^{21} - 756 q^{25} + 896 q^{26} + 626 q^{30} - 662 q^{32} + 188 q^{33} - 484 q^{35} - 282 q^{36} - 1048 q^{38} - 2910 q^{42} + 228 q^{43} + 20 q^{47} - 2012 q^{49} + 1610 q^{50} - 3074 q^{52} - 100 q^{53} - 2632 q^{55} - 1992 q^{59} + 434 q^{60} - 300 q^{64} + 2180 q^{66} - 1736 q^{67} - 2256 q^{69} - 2104 q^{70} - 78 q^{72} + 1746 q^{76} + 1788 q^{77} + 2160 q^{81} - 1700 q^{83} + 886 q^{84} + 4822 q^{86} + 768 q^{87} + 1568 q^{89} + 3100 q^{93} - 2238 q^{94} - 3754 q^{98}+O(q^{100}) \) 8 * q - 2 * q^2 + 22 * q^4 - 120 * q^8 - 12 * q^9 - 44 * q^13 - 108 * q^15 + 126 * q^16 - 668 * q^18 - 44 * q^19 - 704 * q^21 - 756 * q^25 + 896 * q^26 + 626 * q^30 - 662 * q^32 + 188 * q^33 - 484 * q^35 - 282 * q^36 - 1048 * q^38 - 2910 * q^42 + 228 * q^43 + 20 * q^47 - 2012 * q^49 + 1610 * q^50 - 3074 * q^52 - 100 * q^53 - 2632 * q^55 - 1992 * q^59 + 434 * q^60 - 300 * q^64 + 2180 * q^66 - 1736 * q^67 - 2256 * q^69 - 2104 * q^70 - 78 * q^72 + 1746 * q^76 + 1788 * q^77 + 2160 * q^81 - 1700 * q^83 + 886 * q^84 + 4822 * q^86 + 768 * q^87 + 1568 * q^89 + 3100 * q^93 - 2238 * q^94 - 3754 * q^98

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