LMFDB - Embedded newform 289.4.b.e.288.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [289,4,Mod(288,289)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("289.288"); S:= CuspForms(chi, 4); N := Newforms(S);

sage: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")

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:traces = [12,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	no
Analytic conductor:	$17.0515519917$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} - 34 x^{10} + 124 x^{9} + 671 x^{8} - 1984 x^{7} - 5452 x^{6} + 8264 x^{5} + \cdots + 300356 $ x^12 - 4x^11 - 34x^10 + 124x^9 + 671x^8 - 1984x^7 - 5452x^6 + 8264x^5 + 24252x^4 + 25328x^3 - 31008x^2 - 174688*x + 300356
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 17)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			288.8
Root			$-0.705468 + 1.84776i$ of defining polynomial
Character	$\chi$	$=$	289.288
Dual form			289.4.b.e.288.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.70547 q^{2} +4.44379i q^{3} -5.09138 q^{4} -6.80983i q^{5} +7.57875i q^{6} -13.8760i q^{7} -22.3269 q^{8} +7.25269 q^{9} -11.6140i q^{10} +30.8361i q^{11} -22.6250i q^{12} +66.0130 q^{13} -23.6651i q^{14} +30.2615 q^{15} +2.65318 q^{16} +12.3692 q^{18} +79.7150 q^{19} +34.6714i q^{20} +61.6622 q^{21} +52.5899i q^{22} +28.8986i q^{23} -99.2163i q^{24} +78.6262 q^{25} +112.583 q^{26} +152.212i q^{27} +70.6481i q^{28} -266.201i q^{29} +51.6100 q^{30} +35.7906i q^{31} +183.140 q^{32} -137.029 q^{33} -94.4935 q^{35} -36.9262 q^{36} +357.875i q^{37} +135.951 q^{38} +293.348i q^{39} +152.043i q^{40} +32.7452i q^{41} +105.163 q^{42} +516.157 q^{43} -156.998i q^{44} -49.3896i q^{45} +49.2857i q^{46} -210.602 q^{47} +11.7902i q^{48} +150.456 q^{49} +134.094 q^{50} -336.097 q^{52} -87.2324 q^{53} +259.593i q^{54} +209.989 q^{55} +309.809i q^{56} +354.237i q^{57} -453.997i q^{58} +310.728 q^{59} -154.073 q^{60} -365.247i q^{61} +61.0397i q^{62} -100.639i q^{63} +291.115 q^{64} -449.537i q^{65} -233.699 q^{66} -660.131 q^{67} -128.420 q^{69} -161.156 q^{70} -398.030i q^{71} -161.930 q^{72} -643.443i q^{73} +610.344i q^{74} +349.399i q^{75} -405.859 q^{76} +427.882 q^{77} +500.296i q^{78} -384.511i q^{79} -18.0677i q^{80} -480.576 q^{81} +55.8460i q^{82} +153.488 q^{83} -313.946 q^{84} +880.289 q^{86} +1182.94 q^{87} -688.475i q^{88} -599.053 q^{89} -84.2324i q^{90} -915.998i q^{91} -147.134i q^{92} -159.046 q^{93} -359.175 q^{94} -542.846i q^{95} +813.838i q^{96} -44.5693i q^{97} +256.598 q^{98} +223.645i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 8 q^{2} + 16 q^{4} + 96 q^{8} + 36 q^{9} - 8 q^{13} + 192 q^{15} - 184 q^{16} + 352 q^{19} - 256 q^{21} + 492 q^{25} + 784 q^{26} + 744 q^{30} - 24 q^{32} - 1400 q^{33} - 632 q^{35} + 856 q^{36}+ \cdots + 3464 q^{98}+O(q^{100}) $ 12 * q + 8 * q^2 + 16 * q^4 + 96 * q^8 + 36 * q^9 - 8 * q^13 + 192 * q^15 - 184 * q^16 + 352 * q^19 - 256 * q^21 + 492 * q^25 + 784 * q^26 + 744 * q^30 - 24 * q^32 - 1400 * q^33 - 632 * q^35 + 856 * q^36 - 624 * q^38 + 1664 * q^42 + 1200 * q^43 - 1512 * q^47 + 1052 * q^49 - 2856 * q^50 + 792 * q^52 + 2504 * q^53 - 1424 * q^55 + 3408 * q^59 + 2808 * q^60 + 272 * q^64 - 272 * q^66 - 1080 * q^67 - 344 * q^69 - 2600 * q^70 + 248 * q^72 - 896 * q^76 - 848 * q^77 - 2404 * q^81 + 2960 * q^83 + 4768 * q^84 - 1200 * q^86 + 160 * q^87 - 2144 * q^89 - 3800 * q^93 - 5984 * q^94 + 3464 * 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 $

$2$	1.70547	0.602974	0.301487	−	0.953470i	$-0.402517\pi$
$2$	1.70547	0.602974	0.301487	+	0.953470i	$0.402517\pi$
$3$	4.44379i	0.855209i	0.903966	+	0.427604i	$0.140642\pi$
$3$	4.44379i	0.855209i	−0.903966	+	0.427604i	$0.859358\pi$
$4$	−5.09138	−0.636422
$5$	− 6.80983i	− 0.609090i	−0.952498	−	0.304545i	$-0.901496\pi$
$5$	− 6.80983i	− 0.609090i	0.952498	−	0.304545i	$-0.0985044\pi$
$6$	7.57875i	0.515668i
$7$	− 13.8760i	− 0.749235i	−0.927179	−	0.374618i	$-0.877774\pi$
$7$	− 13.8760i	− 0.749235i	0.927179	−	0.374618i	$-0.122226\pi$
$8$	−22.3269	−0.986720
$9$	7.25269	0.268618
$10$	− 11.6140i	− 0.367265i
$11$	30.8361i	0.845221i	0.906311	+	0.422610i	$0.138886\pi$
$11$	30.8361i	0.845221i	−0.906311	+	0.422610i	$0.861114\pi$
$12$	− 22.6250i	− 0.544274i
$13$	66.0130	1.40836	0.704181	−	0.710021i	$-0.251314\pi$
$13$	66.0130	1.40836	0.704181	+	0.710021i	$0.251314\pi$
$14$	− 23.6651i	− 0.451769i
$15$	30.2615	0.520899
$16$	2.65318	0.0414559
$17$	0	0
$17$	0	0
$18$	12.3692	0.161970
$19$	79.7150	0.962520	0.481260	−	0.876578i	$-0.340179\pi$
$19$	79.7150	0.962520	0.481260	+	0.876578i	$0.340179\pi$
$20$	34.6714i	0.387639i
$21$	61.6622	0.640752
$22$	52.5899i	0.509646i
$23$	28.8986i	0.261990i	0.991383	+	0.130995i	$0.0418173\pi$
$23$	28.8986i	0.261990i	−0.991383	+	0.130995i	$0.958183\pi$
$24$	− 99.2163i	− 0.843851i
$25$	78.6262	0.629009
$26$	112.583	0.849205
$27$	152.212i	1.08493i
$28$	70.6481i	0.476830i
$29$	− 266.201i	− 1.70456i	−0.523084	−	0.852281i	$-0.675219\pi$
$29$	− 266.201i	− 1.70456i	0.523084	−	0.852281i	$-0.324781\pi$
$30$	51.6100	0.314089
$31$	35.7906i	0.207361i	0.994611	+	0.103680i	$0.0330619\pi$
$31$	35.7906i	0.207361i	−0.994611	+	0.103680i	$0.966938\pi$
$32$	183.140	1.01172
$33$	−137.029	−0.722840
$34$	0	0
$35$	−94.4935	−0.456352
$36$	−36.9262	−0.170955
$37$	357.875i	1.59011i	0.606535	+	0.795057i	$0.292559\pi$
$37$	357.875i	1.59011i	−0.606535	+	0.795057i	$0.707441\pi$
$38$	135.951	0.580374
$39$	293.348i	1.20444i
$40$	152.043i	0.601001i
$41$	32.7452i	0.124730i	0.998053	+	0.0623652i	$0.0198644\pi$
$41$	32.7452i	0.124730i	−0.998053	+	0.0623652i	$0.980136\pi$
$42$	105.163	0.386357
$43$	516.157	1.83054	0.915270	−	0.402841i	$-0.131977\pi$
$43$	516.157	1.83054	0.915270	+	0.402841i	$0.131977\pi$
$44$	− 156.998i	− 0.537917i
$45$	− 49.3896i	− 0.163613i
$46$	49.2857i	0.157973i
$47$	−210.602	−0.653606	−0.326803	−	0.945093i	$-0.605971\pi$
$47$	−210.602	−0.653606	−0.326803	+	0.945093i	$0.605971\pi$
$48$	11.7902i	0.0354534i
$49$	150.456	0.438647
$50$	134.094	0.379276
$51$	0	0
$52$	−336.097	−0.896313
$53$	−87.2324	−0.226081	−0.113040	−	0.993590i	$-0.536059\pi$
$53$	−87.2324	−0.226081	−0.113040	+	0.993590i	$0.536059\pi$
$54$	259.593i	0.654186i
$55$	209.989	0.514815
$56$	309.809i	0.739285i
$57$	354.237i	0.823155i
$58$	− 453.997i	− 1.02781i
$59$	310.728	0.685651	0.342825	−	0.939399i	$-0.388616\pi$
$59$	310.728	0.685651	0.342825	+	0.939399i	$0.388616\pi$
$60$	−154.073	−0.331512
$61$	− 365.247i	− 0.766641i	−0.923615	−	0.383321i	$-0.874780\pi$
$61$	− 365.247i	− 0.766641i	0.923615	−	0.383321i	$-0.125220\pi$
$62$	61.0397i	0.125033i
$63$	− 100.639i	− 0.201258i
$64$	291.115	0.568583
$65$	− 449.537i	− 0.857819i
$66$	−233.699	−0.435854
$67$	−660.131	−1.20370	−0.601850	−	0.798609i	$-0.705569\pi$
$67$	−660.131	−1.20370	−0.601850	+	0.798609i	$0.705569\pi$
$68$	0	0
$69$	−128.420	−0.224056
$70$	−161.156	−0.275168
$71$	− 398.030i	− 0.665316i	−0.943047	−	0.332658i	$-0.892054\pi$
$71$	− 398.030i	− 0.665316i	0.943047	−	0.332658i	$-0.107946\pi$
$72$	−161.930	−0.265051
$73$	− 643.443i	− 1.03163i	−0.856699	−	0.515817i	$-0.827489\pi$
$73$	− 643.443i	− 1.03163i	0.856699	−	0.515817i	$-0.172511\pi$
$74$	610.344i	0.958797i
$75$	349.399i	0.537934i
$76$	−405.859	−0.612569
$77$	427.882	0.633269
$78$	500.296i	0.726248i
$79$	− 384.511i	− 0.547605i	−0.961786	−	0.273803i	$-0.911718\pi$
$79$	− 384.511i	− 0.547605i	0.961786	−	0.273803i	$-0.0882815\pi$
$80$	− 18.0677i	− 0.0252504i
$81$	−480.576	−0.659226
$82$	55.8460i	0.0752092i
$83$	153.488	0.202983	0.101491	−	0.994836i	$-0.467639\pi$
$83$	153.488	0.202983	0.101491	+	0.994836i	$0.467639\pi$
$84$	−313.946	−0.407789
$85$	0	0
$86$	880.289	1.10377
$87$	1182.94	1.45776
$88$	− 688.475i	− 0.833996i
$89$	−599.053	−0.713477	−0.356738	−	0.934204i	$-0.616111\pi$
$89$	−599.053	−0.713477	−0.356738	+	0.934204i	$0.616111\pi$
$90$	− 84.2324i	− 0.0986542i
$91$	− 915.998i	− 1.05519i
$92$	− 147.134i	− 0.166737i
$93$	−159.046	−0.177336
$94$	−359.175	−0.394107
$95$	− 542.846i	− 0.586261i
$96$	813.838i	0.865229i
$97$	− 44.5693i	− 0.0466529i	−0.999728	−	0.0233265i	$-0.992574\pi$
$97$	− 44.5693i	− 0.0466529i	0.999728	−	0.0233265i	$-0.00742571\pi$
$98$	256.598	0.264492
$99$	223.645i	0.227042i
$100$	−400.316	−0.400316
$101$	1478.33	1.45643	0.728215	−	0.685348i	$-0.240350\pi$
$101$	1478.33	1.45643	0.728215	+	0.685348i	$0.240350\pi$
$102$	0	0
$103$	618.133	0.591325	0.295662	−	0.955293i	$-0.404460\pi$
$103$	618.133	0.591325	0.295662	+	0.955293i	$0.404460\pi$
$104$	−1473.87	−1.38966
$105$	− 419.909i	− 0.390276i
$106$	−148.772	−0.136321
$107$	− 138.979i	− 0.125567i	−0.998027	−	0.0627833i	$-0.980002\pi$
$107$	− 138.979i	− 0.125567i	0.998027	−	0.0627833i	$-0.0199977\pi$
$108$	− 774.969i	− 0.690476i
$109$	823.466i	0.723612i	0.932253	+	0.361806i	$0.117840\pi$
$109$	823.466i	0.723612i	−0.932253	+	0.361806i	$0.882160\pi$
$110$	358.129	0.310420
$111$	−1590.32	−1.35988
$112$	− 36.8156i	− 0.0310602i
$113$	− 602.474i	− 0.501558i	−0.968044	−	0.250779i	$-0.919313\pi$
$113$	− 602.474i	− 0.501558i	0.968044	−	0.250779i	$-0.0806867\pi$
$114$	604.140i	0.496341i
$115$	196.795	0.159576
$116$	1355.33i	1.08482i
$117$	478.772	0.378312
$118$	529.937	0.413430
$119$	0	0
$120$	−675.646	−0.513982
$121$	380.136	0.285602
$122$	− 622.917i	− 0.462265i
$123$	−145.513	−0.106671
$124$	− 182.223i	− 0.131969i
$125$	− 1386.66i	− 0.992213i
$126$	− 171.636i	− 0.121354i
$127$	−1506.98	−1.05294	−0.526468	−	0.850195i	$-0.676484\pi$
$127$	−1506.98	−1.05294	−0.526468	+	0.850195i	$0.676484\pi$
$128$	−968.636	−0.668876
$129$	2293.70i	1.56549i
$130$	− 766.671i	− 0.517243i
$131$	2783.58i	1.85651i	0.371946	+	0.928254i	$0.378691\pi$
$131$	2783.58i	1.85651i	−0.371946	+	0.928254i	$0.621309\pi$
$132$	697.667	0.460032
$133$	− 1106.13i	− 0.721154i
$134$	−1125.83	−0.725799
$135$	1036.54	0.660822
$136$	0	0
$137$	−629.783	−0.392744	−0.196372	−	0.980529i	$-0.562916\pi$
$137$	−629.783	−0.392744	−0.196372	+	0.980529i	$0.562916\pi$
$138$	−219.015	−0.135100
$139$	2654.10i	1.61955i	0.586740	+	0.809775i	$0.300411\pi$
$139$	2654.10i	1.61955i	−0.586740	+	0.809775i	$0.699589\pi$
$140$	481.102	0.290432
$141$	− 935.872i	− 0.558969i
$142$	− 678.827i	− 0.401168i
$143$	2035.58i	1.19038i
$144$	19.2427	0.0111358
$145$	−1812.78	−1.03823
$146$	− 1097.37i	− 0.622048i
$147$	668.595i	0.375134i
$148$	− 1822.08i	− 1.01198i
$149$	−2216.30	−1.21857	−0.609283	−	0.792953i	$-0.708543\pi$
$149$	−2216.30	−1.21857	−0.609283	+	0.792953i	$0.708543\pi$
$150$	595.888i	0.324360i
$151$	−2133.15	−1.14963	−0.574813	−	0.818285i	$-0.694925\pi$
$151$	−2133.15	−1.14963	−0.574813	+	0.818285i	$0.694925\pi$
$152$	−1779.79	−0.949737
$153$	0	0
$154$	729.740	0.381845
$155$	243.728	0.126301
$156$	− 1493.55i	− 0.766534i
$157$	−345.107	−0.175430	−0.0877152	−	0.996146i	$-0.527957\pi$
$157$	−345.107	−0.175430	−0.0877152	+	0.996146i	$0.527957\pi$
$158$	− 655.770i	− 0.330192i
$159$	− 387.643i	− 0.193346i
$160$	− 1247.16i	− 0.616227i
$161$	400.998	0.196292
$162$	−819.606	−0.397496
$163$	− 678.526i	− 0.326050i	−0.986622	−	0.163025i	$-0.947875\pi$
$163$	− 678.526i	− 0.326050i	0.986622	−	0.163025i	$-0.0521252\pi$
$164$	− 166.718i	− 0.0793812i
$165$	933.146i	0.440275i
$166$	261.770	0.122393
$167$	799.790i	0.370596i	0.982682	+	0.185298i	$0.0593251\pi$
$167$	799.790i	0.370596i	−0.982682	+	0.185298i	$0.940675\pi$
$168$	−1376.73	−0.632243
$169$	2160.71	0.983483
$170$	0	0
$171$	578.148	0.258550
$172$	−2627.95	−1.16500
$173$	− 1885.48i	− 0.828615i	−0.910137	−	0.414307i	$-0.864024\pi$
$173$	− 1885.48i	− 0.828615i	0.910137	−	0.414307i	$-0.135976\pi$
$174$	2017.47	0.878989
$175$	− 1091.02i	− 0.471276i
$176$	81.8136i	0.0350394i
$177$	1380.81i	0.586374i
$178$	−1021.66	−0.430208
$179$	3588.86	1.49857	0.749286	−	0.662247i	$-0.230397\pi$
$179$	3588.86	1.49857	0.749286	+	0.662247i	$0.230397\pi$
$180$	251.461i	0.104127i
$181$	− 461.347i	− 0.189457i	−0.995503	−	0.0947284i	$-0.969802\pi$
$181$	− 461.347i	− 0.189457i	0.995503	−	0.0947284i	$-0.0301983\pi$
$182$	− 1562.20i	− 0.636255i
$183$	1623.08	0.655638
$184$	− 645.218i	− 0.258511i
$185$	2437.07	0.968522
$186$	−271.248	−0.106929
$187$	0	0
$188$	1072.26	0.415969
$189$	2112.10	0.812870
$190$	− 925.806i	− 0.353500i
$191$	−4519.91	−1.71230	−0.856149	−	0.516728i	$-0.827150\pi$
$191$	−4519.91	−1.71230	−0.856149	+	0.516728i	$0.827150\pi$
$192$	1293.65i	0.486257i
$193$	2928.92i	1.09238i	0.837663	+	0.546188i	$0.183922\pi$
$193$	2928.92i	1.09238i	−0.837663	+	0.546188i	$0.816078\pi$
$194$	− 76.0116i	− 0.0281305i
$195$	1997.65	0.733614
$196$	−766.027	−0.279165
$197$	− 1926.13i	− 0.696606i	−0.937382	−	0.348303i	$-0.886758\pi$
$197$	− 1926.13i	− 0.696606i	0.937382	−	0.348303i	$-0.113242\pi$
$198$	381.419i	0.136900i
$199$	3086.61i	1.09952i	0.835323	+	0.549759i	$0.185280\pi$
$199$	3086.61i	1.09952i	−0.835323	+	0.549759i	$0.814720\pi$
$200$	−1755.48	−0.620656
$201$	− 2933.49i	− 1.02941i
$202$	2521.25	0.878190
$203$	−3693.81	−1.27712
$204$	0	0
$205$	222.990	0.0759721
$206$	1054.21	0.356553
$207$	209.593i	0.0703754i
$208$	175.144	0.0583849
$209$	2458.10i	0.813541i
$210$	− 716.142i	− 0.235326i
$211$	− 3602.40i	− 1.17535i	−0.809097	−	0.587676i	$-0.800043\pi$
$211$	− 3602.40i	− 1.17535i	0.809097	−	0.587676i	$-0.199957\pi$
$212$	444.133	0.143883
$213$	1768.76	0.568984
$214$	− 237.024i	− 0.0757133i
$215$	− 3514.94i	− 1.11496i
$216$	− 3398.42i	− 1.07053i
$217$	496.631	0.155362
$218$	1404.39i	0.436319i
$219$	2859.33	0.882262
$220$	−1069.13	−0.327640
$221$	0	0
$222$	−2712.24	−0.819972
$223$	1577.43	0.473688	0.236844	−	0.971548i	$-0.423887\pi$
$223$	1577.43	0.473688	0.236844	+	0.971548i	$0.423887\pi$
$224$	− 2541.26i	− 0.758014i
$225$	570.252	0.168963
$226$	− 1027.50i	− 0.302426i
$227$	4292.90i	1.25520i	0.778537	+	0.627599i	$0.215962\pi$
$227$	4292.90i	1.25520i	−0.778537	+	0.627599i	$0.784038\pi$
$228$	− 1803.55i	− 0.523874i
$229$	−580.893	−0.167627	−0.0838133	−	0.996481i	$-0.526710\pi$
$229$	−580.893	−0.167627	−0.0838133	+	0.996481i	$0.526710\pi$
$230$	335.627	0.0962200
$231$	1901.42i	0.541577i
$232$	5943.45i	1.68193i
$233$	− 671.778i	− 0.188883i	−0.995530	−	0.0944413i	$-0.969894\pi$
$233$	− 671.778i	− 0.188883i	0.995530	−	0.0944413i	$-0.0301065\pi$
$234$	816.530	0.228112
$235$	1434.16i	0.398105i
$236$	−1582.04	−0.436363
$237$	1708.69	0.468317
$238$	0	0
$239$	−273.635	−0.0740584	−0.0370292	−	0.999314i	$-0.511789\pi$
$239$	−273.635	−0.0740584	−0.0370292	+	0.999314i	$0.511789\pi$
$240$	80.2891	0.0215943
$241$	6486.56i	1.73376i	0.498518	+	0.866879i	$0.333878\pi$
$241$	6486.56i	1.73376i	−0.498518	+	0.866879i	$0.666122\pi$
$242$	648.310	0.172211
$243$	1974.14i	0.521158i
$244$	1859.61i	0.487908i
$245$	− 1024.58i	− 0.267175i
$246$	−248.168	−0.0643195
$247$	5262.22	1.35558
$248$	− 799.093i	− 0.204607i
$249$	682.071i	0.173592i
$250$	− 2364.90i	− 0.598279i
$251$	−2527.58	−0.635615	−0.317808	−	0.948155i	$-0.602947\pi$
$251$	−2527.58	−0.635615	−0.317808	+	0.948155i	$0.602947\pi$
$252$	512.389i	0.128085i
$253$	−891.120	−0.221440
$254$	−2570.11	−0.634893
$255$	0	0
$256$	−3980.89	−0.971898
$257$	−1702.44	−0.413211	−0.206605	−	0.978424i	$-0.566242\pi$
$257$	−1702.44	−0.413211	−0.206605	+	0.978424i	$0.566242\pi$
$258$	3911.82i	0.943952i
$259$	4965.88	1.19137
$260$	2288.76i	0.545935i
$261$	− 1930.67i	− 0.457877i
$262$	4747.31i	1.11943i
$263$	4274.40	1.00217	0.501085	−	0.865398i	$-0.332934\pi$
$263$	4274.40	1.00217	0.501085	+	0.865398i	$0.332934\pi$
$264$	3059.44	0.713241
$265$	594.038i	0.137704i
$266$	− 1886.46i	− 0.434837i
$267$	− 2662.07i	− 0.610172i
$268$	3360.98	0.766061
$269$	− 7996.34i	− 1.81244i	−0.422810	−	0.906218i	$-0.638956\pi$
$269$	− 7996.34i	− 1.81244i	0.422810	−	0.906218i	$-0.361044\pi$
$270$	1767.78	0.398458
$271$	474.268	0.106309	0.0531545	−	0.998586i	$-0.483072\pi$
$271$	474.268	0.106309	0.0531545	+	0.998586i	$0.483072\pi$
$272$	0	0
$273$	4070.51	0.902411
$274$	−1074.07	−0.236815
$275$	2424.52i	0.531652i
$276$	653.833	0.142595
$277$	2826.25i	0.613043i	0.951864	+	0.306521i	$0.0991651\pi$
$277$	2826.25i	0.613043i	−0.951864	+	0.306521i	$0.900835\pi$
$278$	4526.48i	0.976547i
$279$	259.578i	0.0557008i
$280$	2109.75	0.450291
$281$	2498.15	0.530346	0.265173	−	0.964201i	$-0.414571\pi$
$281$	2498.15	0.530346	0.265173	+	0.964201i	$0.414571\pi$
$282$	− 1596.10i	− 0.337044i
$283$	2417.07i	0.507704i	0.967243	+	0.253852i	$0.0816976\pi$
$283$	2417.07i	0.507704i	−0.967243	+	0.253852i	$0.918302\pi$
$284$	2026.52i	0.423422i
$285$	2412.29	0.501375
$286$	3471.62i	0.717766i
$287$	454.374	0.0934524
$288$	1328.26	0.271766
$289$	0	0
$290$	−3091.65	−0.626027
$291$	198.057	0.0398980
$292$	3276.01i	0.656555i
$293$	6652.17	1.32636	0.663181	−	0.748459i	$-0.269206\pi$
$293$	6652.17	1.32636	0.663181	+	0.748459i	$0.269206\pi$
$294$	1140.27i	0.226196i
$295$	− 2116.01i	− 0.417623i
$296$	− 7990.24i	− 1.56900i
$297$	−4693.62	−0.917008
$298$	−3779.83	−0.734764
$299$	1907.68i	0.368977i
$300$	− 1778.92i	− 0.342353i
$301$	− 7162.21i	− 1.37150i
$302$	−3638.02	−0.693194
$303$	6569.40i	1.24555i
$304$	211.498	0.0399021
$305$	−2487.27	−0.466954
$306$	0	0
$307$	−2511.18	−0.466842	−0.233421	−	0.972376i	$-0.574992\pi$
$307$	−2511.18	−0.466842	−0.233421	+	0.972376i	$0.574992\pi$
$308$	−2178.51	−0.403027
$309$	2746.85i	0.505706i
$310$	415.670	0.0761563
$311$	− 3585.10i	− 0.653674i	−0.945081	−	0.326837i	$-0.894017\pi$
$311$	− 3585.10i	− 0.653674i	0.945081	−	0.326837i	$-0.105983\pi$
$312$	− 6549.56i	− 1.18845i
$313$	− 2612.32i	− 0.471747i	−0.971784	−	0.235874i	$-0.924205\pi$
$313$	− 2612.32i	− 0.471747i	0.971784	−	0.235874i	$-0.0757952\pi$
$314$	−588.570	−0.105780
$315$	−685.332	−0.122584
$316$	1957.69i	0.348508i
$317$	− 3903.80i	− 0.691669i	−0.938296	−	0.345834i	$-0.887596\pi$
$317$	− 3903.80i	− 0.691669i	0.938296	−	0.345834i	$-0.112404\pi$
$318$	− 661.112i	− 0.116583i
$319$	8208.60	1.44073
$320$	− 1982.44i	− 0.346318i
$321$	617.594	0.107386
$322$	683.890	0.118359
$323$	0	0
$324$	2446.79	0.419546
$325$	5190.35	0.885873
$326$	− 1157.20i	− 0.196600i
$327$	−3659.31	−0.618839
$328$	− 731.101i	− 0.123074i
$329$	2922.32i	0.489704i
$330$	1591.45i	0.265474i
$331$	6236.03	1.03554	0.517769	−	0.855520i	$-0.326763\pi$
$331$	6236.03	1.03554	0.517769	+	0.855520i	$0.326763\pi$
$332$	−781.468	−0.129183
$333$	2595.55i	0.427134i
$334$	1364.02i	0.223460i
$335$	4495.38i	0.733161i
$336$	163.601	0.0265630
$337$	− 3842.08i	− 0.621043i	−0.950566	−	0.310521i	$-0.899496\pi$
$337$	− 3842.08i	− 0.621043i	0.950566	−	0.310521i	$-0.100504\pi$
$338$	3685.02	0.593014
$339$	2677.27	0.428936
$340$	0	0
$341$	−1103.64	−0.175265
$342$	986.014	0.155899
$343$	− 6847.21i	− 1.07788i
$344$	−11524.2	−1.80623
$345$	874.516i	0.136471i
$346$	− 3215.62i	− 0.499633i
$347$	8301.86i	1.28434i	0.766561	+	0.642172i	$0.221966\pi$
$347$	8301.86i	1.28434i	−0.766561	+	0.642172i	$0.778034\pi$
$348$	−6022.81	−0.927748
$349$	−4929.17	−0.756024	−0.378012	−	0.925801i	$-0.623392\pi$
$349$	−4929.17	−0.756024	−0.378012	+	0.925801i	$0.623392\pi$
$350$	− 1860.70i	− 0.284167i
$351$	10048.0i	1.52798i
$352$	5647.33i	0.855124i
$353$	−9607.54	−1.44861	−0.724303	−	0.689482i	$-0.757838\pi$
$353$	−9607.54	−1.44861	−0.724303	+	0.689482i	$0.757838\pi$
$354$	2354.93i	0.353568i
$355$	−2710.52	−0.405237
$356$	3050.00	0.454073
$357$	0	0
$358$	6120.69	0.903600
$359$	1962.76	0.288553	0.144277	−	0.989537i	$-0.453915\pi$
$359$	1962.76	0.288553	0.144277	+	0.989537i	$0.453915\pi$
$360$	1102.72i	0.161440i
$361$	−504.522	−0.0735561
$362$	− 786.813i	− 0.114238i
$363$	1689.25i	0.244249i
$364$	4663.69i	0.671549i
$365$	−4381.74	−0.628358
$366$	2768.12	0.395333
$367$	3458.29i	0.491883i	0.969285	+	0.245941i	$0.0790971\pi$
$367$	3458.29i	0.491883i	−0.969285	+	0.245941i	$0.920903\pi$
$368$	76.6732i	0.0108610i
$369$	237.491i	0.0335049i
$370$	4156.34	0.583994
$371$	1210.44i	0.169388i
$372$	809.763	0.112861
$373$	2661.22	0.369418	0.184709	−	0.982793i	$-0.440866\pi$
$373$	2661.22	0.369418	0.184709	+	0.982793i	$0.440866\pi$
$374$	0	0
$375$	6162.03	0.848549
$376$	4702.10	0.644926
$377$	− 17572.7i	− 2.40064i
$378$	3602.11	0.490140
$379$	− 5583.03i	− 0.756677i	−0.925667	−	0.378339i	$-0.876495\pi$
$379$	− 5583.03i	− 0.756677i	0.925667	−	0.378339i	$-0.123505\pi$
$380$	2763.83i	0.373110i
$381$	− 6696.71i	− 0.900480i
$382$	−7708.56	−1.03247
$383$	−7832.13	−1.04492	−0.522459	−	0.852665i	$-0.674985\pi$
$383$	−7832.13	−1.04492	−0.522459	+	0.852665i	$0.674985\pi$
$384$	− 4304.42i	− 0.572029i
$385$	− 2913.81i	− 0.385718i
$386$	4995.18i	0.658674i
$387$	3743.53	0.491717
$388$	226.919i	0.0296910i
$389$	−7432.95	−0.968807	−0.484403	−	0.874845i	$-0.660963\pi$
$389$	−7432.95	−0.968807	−0.484403	+	0.874845i	$0.660963\pi$
$390$	3406.93	0.442350
$391$	0	0
$392$	−3359.22	−0.432821
$393$	−12369.7	−1.58770
$394$	− 3284.96i	− 0.420035i
$395$	−2618.45	−0.333541
$396$	− 1138.66i	− 0.144494i
$397$	− 9540.26i	− 1.20608i	−0.797713	−	0.603038i	$-0.793957\pi$
$397$	− 9540.26i	− 1.20608i	0.797713	−	0.603038i	$-0.206043\pi$
$398$	5264.12i	0.662981i
$399$	4915.40	0.616737
$400$	208.609	0.0260761
$401$	− 8663.30i	− 1.07886i	−0.842029	−	0.539432i	$-0.818639\pi$
$401$	− 8663.30i	− 1.07886i	0.842029	−	0.539432i	$-0.181361\pi$
$402$	− 5002.97i	− 0.620710i
$403$	2362.64i	0.292039i
$404$	−7526.75	−0.926905
$405$	3272.64i	0.401528i
$406$	−6299.68	−0.770069
$407$	−11035.4	−1.34400
$408$	0	0
$409$	5279.17	0.638235	0.319117	−	0.947715i	$-0.396614\pi$
$409$	5279.17	0.638235	0.319117	+	0.947715i	$0.396614\pi$
$410$	380.302	0.0458092
$411$	− 2798.62i	− 0.335878i
$412$	−3147.15	−0.376332
$413$	− 4311.68i	− 0.513714i
$414$	357.454i	0.0424346i
$415$	− 1045.23i	− 0.123635i
$416$	12089.6	1.42486
$417$	−11794.3	−1.38505
$418$	4192.21i	0.490544i
$419$	− 9408.24i	− 1.09695i	−0.836166	−	0.548476i	$-0.815208\pi$
$419$	− 9408.24i	− 1.09695i	0.836166	−	0.548476i	$-0.184792\pi$
$420$	2137.92i	0.248380i
$421$	−8191.97	−0.948343	−0.474171	−	0.880433i	$-0.657252\pi$
$421$	−8191.97	−0.948343	−0.474171	+	0.880433i	$0.657252\pi$
$422$	− 6143.77i	− 0.708706i
$423$	−1527.43	−0.175570
$424$	1947.63	0.223079
$425$	0	0
$426$	3016.57	0.343083
$427$	−5068.18	−0.574395
$428$	707.595i	0.0799133i
$429$	−9045.70	−1.01802
$430$	− 5994.62i	− 0.672294i
$431$	− 1950.29i	− 0.217964i	−0.994044	−	0.108982i	$-0.965241\pi$
$431$	− 1950.29i	− 0.217964i	0.994044	−	0.108982i	$-0.0347590\pi$
$432$	403.845i	0.0449769i
$433$	−11152.3	−1.23775	−0.618876	−	0.785489i	$-0.712412\pi$
$433$	−11152.3	−1.23775	−0.618876	+	0.785489i	$0.712412\pi$
$434$	846.988	0.0936791
$435$	− 8055.64i	− 0.887905i
$436$	− 4192.58i	− 0.460523i
$437$	2303.65i	0.252171i
$438$	4876.49	0.531981
$439$	470.158i	0.0511149i	0.999673	+	0.0255574i	$0.00813607\pi$
$439$	470.158i	0.0511149i	−0.999673	+	0.0255574i	$0.991864\pi$
$440$	−4688.40	−0.507979
$441$	1091.21	0.117829
$442$	0	0
$443$	−13132.9	−1.40849	−0.704246	−	0.709956i	$-0.748715\pi$
$443$	−13132.9	−1.40849	−0.704246	+	0.709956i	$0.748715\pi$
$444$	8096.93	0.865457
$445$	4079.45i	0.434572i
$446$	2690.25	0.285621
$447$	− 9848.78i	− 1.04213i
$448$	− 4039.51i	− 0.426002i
$449$	10433.0i	1.09658i	0.836288	+	0.548291i	$0.184721\pi$
$449$	10433.0i	1.09658i	−0.836288	+	0.548291i	$0.815279\pi$
$450$	972.546	0.101881
$451$	−1009.73	−0.105425
$452$	3067.43i	0.319203i
$453$	− 9479.28i	− 0.983169i
$454$	7321.40i	0.756851i
$455$	−6237.79	−0.642708
$456$	− 7909.02i	− 0.812224i
$457$	253.385	0.0259362	0.0129681	−	0.999916i	$-0.495872\pi$
$457$	253.385	0.0259362	0.0129681	+	0.999916i	$0.495872\pi$
$458$	−990.694	−0.101074
$459$	0	0
$460$	−1001.96	−0.101558
$461$	−15560.2	−1.57204	−0.786019	−	0.618202i	$-0.787861\pi$
$461$	−15560.2	−1.57204	−0.786019	+	0.618202i	$0.787861\pi$
$462$	3242.81i	0.326557i
$463$	10727.9	1.07682	0.538409	−	0.842684i	$-0.319026\pi$
$463$	10727.9	1.07682	0.538409	+	0.842684i	$0.319026\pi$
$464$	− 706.279i	− 0.0706641i
$465$	1083.08i	0.108014i
$466$	− 1145.70i	− 0.113891i
$467$	1960.48	0.194262	0.0971308	−	0.995272i	$-0.469033\pi$
$467$	1960.48	0.194262	0.0971308	+	0.995272i	$0.469033\pi$
$468$	−2437.61	−0.240766
$469$	9160.00i	0.901854i
$470$	2445.92i	0.240047i
$471$	− 1533.59i	− 0.150030i
$472$	−6937.61	−0.676545
$473$	15916.3i	1.54721i
$474$	2914.11	0.282383
$475$	6267.68	0.605434
$476$	0	0
$477$	−632.670	−0.0607295
$478$	−466.675	−0.0446553
$479$	− 4294.19i	− 0.409617i	−0.978802	−	0.204808i	$-0.934343\pi$
$479$	− 4294.19i	− 0.409617i	0.978802	−	0.204808i	$-0.0656571\pi$
$480$	5542.10	0.527002
$481$	23624.4i	2.23946i
$482$	11062.6i	1.04541i
$483$	1781.95i	0.167871i
$484$	−1935.42	−0.181764
$485$	−303.510	−0.0284158
$486$	3366.84i	0.314244i
$487$	18013.5i	1.67612i	0.545578	+	0.838060i	$0.316310\pi$
$487$	18013.5i	1.67612i	−0.545578	+	0.838060i	$0.683690\pi$
$488$	8154.85i	0.756460i
$489$	3015.23	0.278841
$490$	− 1747.39i	− 0.161100i
$491$	−13733.2	−1.26226	−0.631129	−	0.775678i	$-0.717408\pi$
$491$	−13733.2	−1.26226	−0.631129	+	0.775678i	$0.717408\pi$
$492$	740.862	0.0678875
$493$	0	0
$494$	8974.55	0.817377
$495$	1522.98	0.138289
$496$	94.9587i	0.00859632i
$497$	−5523.07	−0.498478
$498$	1163.25i	0.104672i
$499$	− 7747.71i	− 0.695060i	−0.937669	−	0.347530i	$-0.887020\pi$
$499$	− 7747.71i	− 0.695060i	0.937669	−	0.347530i	$-0.112980\pi$
$500$	7060.01i	0.631467i
$501$	−3554.10	−0.316937
$502$	−4310.71	−0.383259
$503$	− 19203.7i	− 1.70229i	−0.524934	−	0.851143i	$-0.675910\pi$
$503$	− 19203.7i	− 1.70229i	0.524934	−	0.851143i	$-0.324090\pi$
$504$	2246.95i	0.198586i
$505$	− 10067.2i	− 0.887098i
$506$	−1519.78	−0.133522
$507$	9601.76i	0.841083i
$508$	7672.61	0.670112
$509$	151.325	0.0131775	0.00658876	−	0.999978i	$-0.497903\pi$
$509$	151.325	0.0131775	0.00658876	+	0.999978i	$0.497903\pi$
$510$	0	0
$511$	−8928.43	−0.772936
$512$	959.802	0.0828470
$513$	12133.6i	1.04427i
$514$	−2903.45	−0.249155
$515$	− 4209.38i	− 0.360170i
$516$	− 11678.1i	− 0.996315i
$517$	− 6494.14i	− 0.552441i
$518$	8469.15	0.718365
$519$	8378.68	0.708638
$520$	10036.8i	0.846427i
$521$	8548.62i	0.718852i	0.933174	+	0.359426i	$0.117027\pi$
$521$	8548.62i	0.718852i	−0.933174	+	0.359426i	$0.882973\pi$
$522$	− 3292.70i	− 0.276088i
$523$	17874.0	1.49441	0.747205	−	0.664593i	$-0.231395\pi$
$523$	17874.0	1.49441	0.747205	+	0.664593i	$0.231395\pi$
$524$	− 14172.3i	− 1.18152i
$525$	4848.26	0.403039
$526$	7289.85	0.604283
$527$	0	0
$528$	−363.563	−0.0299660
$529$	11331.9	0.931361
$530$	1013.11i	0.0830317i
$531$	2253.62	0.184178
$532$	5631.71i	0.458958i
$533$	2161.61i	0.175666i
$534$	− 4540.07i	− 0.367918i
$535$	−946.425	−0.0764813
$536$	14738.7	1.18771
$537$	15948.2i	1.28159i
$538$	− 13637.5i	− 1.09285i
$539$	4639.47i	0.370753i
$540$	−5277.41	−0.420562
$541$	2122.24i	0.168655i	0.996438	+	0.0843273i	$0.0268741\pi$
$541$	2122.24i	0.168655i	−0.996438	+	0.0843273i	$0.973126\pi$
$542$	808.849	0.0641015
$543$	2050.13	0.162025
$544$	0	0
$545$	5607.67	0.440745
$546$	6942.12	0.544130
$547$	284.066i	0.0222044i	0.999938	+	0.0111022i	$0.00353401\pi$
$547$	284.066i	0.0222044i	−0.999938	+	0.0111022i	$0.996466\pi$
$548$	3206.46	0.249951
$549$	− 2649.03i	− 0.205934i
$550$	4134.95i	0.320572i
$551$	− 21220.2i	− 1.64067i
$552$	2867.21	0.221081
$553$	−5335.48	−0.410285
$554$	4820.08i	0.369649i
$555$	10829.8i	0.828289i
$556$	− 13513.0i	− 1.03072i
$557$	9915.36	0.754268	0.377134	−	0.926159i	$-0.376910\pi$
$557$	9915.36	0.754268	0.377134	+	0.926159i	$0.376910\pi$
$558$	442.702i	0.0335862i
$559$	34073.1	2.57806
$560$	−250.708	−0.0189185
$561$	0	0
$562$	4260.52	0.319785
$563$	5048.52	0.377922	0.188961	−	0.981985i	$-0.439488\pi$
$563$	5048.52	0.377922	0.188961	+	0.981985i	$0.439488\pi$
$564$	4764.88i	0.355741i
$565$	−4102.75	−0.305494
$566$	4122.24i	0.306132i
$567$	6668.48i	0.493915i
$568$	8886.78i	0.656481i
$569$	−10964.5	−0.807828	−0.403914	−	0.914797i	$-0.632350\pi$
$569$	−10964.5	−0.807828	−0.403914	+	0.914797i	$0.632350\pi$
$570$	4114.09	0.302316
$571$	− 6757.57i	− 0.495263i	−0.968854	−	0.247632i	$-0.920348\pi$
$571$	− 6757.57i	− 0.495263i	0.968854	−	0.247632i	$-0.0796523\pi$
$572$	− 10363.9i	− 0.757582i
$573$	− 20085.5i	− 1.46437i
$574$	774.920	0.0563494
$575$	2272.19i	0.164794i
$576$	2111.36	0.152732
$577$	−18050.9	−1.30237	−0.651187	−	0.758917i	$-0.725729\pi$
$577$	−18050.9	−1.30237	−0.651187	+	0.758917i	$0.725729\pi$
$578$	0	0
$579$	−13015.5	−0.934209
$580$	9229.57	0.660754
$581$	− 2129.81i	− 0.152082i
$582$	337.780	0.0240574
$583$	− 2689.90i	− 0.191088i
$584$	14366.1i	1.01793i
$585$	− 3260.36i	− 0.230426i
$586$	11345.1	0.799762
$587$	7447.06	0.523634	0.261817	−	0.965118i	$-0.415678\pi$
$587$	7447.06	0.523634	0.261817	+	0.965118i	$0.415678\pi$
$588$	− 3404.07i	− 0.238744i
$589$	2853.04i	0.199589i
$590$	− 3608.78i	− 0.251816i
$591$	8559.34	0.595743
$592$	949.505i	0.0659196i
$593$	1229.32	0.0851298	0.0425649	−	0.999094i	$-0.486447\pi$
$593$	1229.32	0.0851298	0.0425649	+	0.999094i	$0.486447\pi$
$594$	−8004.82	−0.552932
$595$	0	0
$596$	11284.0	0.775523
$597$	−13716.3	−0.940317
$598$	3253.49i	0.222484i
$599$	10655.6	0.726840	0.363420	−	0.931625i	$-0.381609\pi$
$599$	10655.6	0.726840	0.363420	+	0.931625i	$0.381609\pi$
$600$	− 7800.99i	− 0.530790i
$601$	− 2134.49i	− 0.144871i	−0.997373	−	0.0724356i	$-0.976923\pi$
$601$	− 2134.49i	− 0.144871i	0.997373	−	0.0724356i	$-0.0230772\pi$
$602$	− 12214.9i	− 0.826982i
$603$	−4787.73	−0.323336
$604$	10860.7	0.731647
$605$	− 2588.67i	− 0.173957i
$606$	11203.9i	0.751036i
$607$	− 12364.9i	− 0.826812i	−0.910547	−	0.413406i	$-0.864339\pi$
$607$	− 12364.9i	− 0.826812i	0.910547	−	0.413406i	$-0.135661\pi$
$608$	14599.0	0.973797
$609$	− 16414.5i	− 1.09220i
$610$	−4241.96	−0.281561
$611$	−13902.5	−0.920513
$612$	0	0
$613$	4888.87	0.322120	0.161060	−	0.986945i	$-0.448509\pi$
$613$	4888.87	0.322120	0.161060	+	0.986945i	$0.448509\pi$
$614$	−4282.73	−0.281493
$615$	990.920i	0.0649720i
$616$	−9553.30	−0.624859
$617$	24750.0i	1.61491i	0.589932	+	0.807453i	$0.299155\pi$
$617$	24750.0i	1.61491i	−0.589932	+	0.807453i	$0.700845\pi$
$618$	4684.67i	0.304927i
$619$	− 10794.8i	− 0.700937i	−0.936574	−	0.350469i	$-0.886022\pi$
$619$	− 10794.8i	− 0.700937i	0.936574	−	0.350469i	$-0.113978\pi$
$620$	−1240.91	−0.0803809
$621$	−4398.72	−0.284242
$622$	− 6114.28i	− 0.394148i
$623$	8312.47i	0.534562i
$624$	778.304i	0.0499313i
$625$	385.346	0.0246621
$626$	− 4455.22i	− 0.284451i
$627$	−10923.3	−0.695748
$628$	1757.07	0.111648
$629$	0	0
$630$	−1168.81	−0.0739152
$631$	7579.90	0.478211	0.239106	−	0.970994i	$-0.423146\pi$
$631$	7579.90	0.478211	0.239106	+	0.970994i	$0.423146\pi$
$632$	8584.94i	0.540333i
$633$	16008.3	1.00517
$634$	− 6657.80i	− 0.417058i
$635$	10262.3i	0.641333i
$636$	1973.64i	0.123050i
$637$	9932.03	0.617773
$638$	13999.5	0.868723
$639$	− 2886.79i	− 0.178716i
$640$	6596.25i	0.407406i
$641$	− 3181.53i	− 0.196042i	−0.995184	−	0.0980208i	$-0.968749\pi$
$641$	− 3181.53i	− 0.196042i	0.995184	−	0.0980208i	$-0.0312512\pi$
$642$	1053.29	0.0647507
$643$	10552.8i	0.647219i	0.946191	+	0.323609i	$0.104896\pi$
$643$	10552.8i	0.647219i	−0.946191	+	0.323609i	$0.895104\pi$
$644$	−2041.63	−0.124925
$645$	15619.7	0.953526
$646$	0	0
$647$	−10952.0	−0.665483	−0.332742	−	0.943018i	$-0.607974\pi$
$647$	−10952.0	−0.665483	−0.332742	+	0.943018i	$0.607974\pi$
$648$	10729.8	0.650471
$649$	9581.64i	0.579526i
$650$	8851.97	0.534158
$651$	2206.93i	0.132867i
$652$	3454.63i	0.207506i
$653$	6526.76i	0.391136i	0.980690	+	0.195568i	$0.0626550\pi$
$653$	6526.76i	0.391136i	−0.980690	+	0.195568i	$0.937345\pi$
$654$	−6240.84	−0.373144
$655$	18955.7	1.13078
$656$	86.8789i	0.00517081i
$657$	− 4666.69i	− 0.277116i
$658$	4983.92i	0.295279i
$659$	−25717.9	−1.52022	−0.760111	−	0.649793i	$-0.774855\pi$
$659$	−25717.9	−1.52022	−0.760111	+	0.649793i	$0.774855\pi$
$660$	− 4751.00i	− 0.280201i
$661$	−11808.9	−0.694878	−0.347439	−	0.937703i	$-0.612949\pi$
$661$	−11808.9	−0.694878	−0.347439	+	0.937703i	$0.612949\pi$
$662$	10635.3	0.624403
$663$	0	0
$664$	−3426.93	−0.200287
$665$	−7532.54	−0.439247
$666$	4426.64i	0.257550i
$667$	7692.85	0.446579
$668$	− 4072.03i	− 0.235856i
$669$	7009.76i	0.405102i
$670$	7666.73i	0.442077i
$671$	11262.8	0.647981
$672$	11292.8	0.648260
$673$	6282.85i	0.359860i	0.983679	+	0.179930i	$0.0575872\pi$
$673$	6282.85i	0.359860i	−0.983679	+	0.179930i	$0.942413\pi$
$674$	− 6552.54i	− 0.374472i
$675$	11967.8i	0.682433i
$676$	−11001.0	−0.625910
$677$	− 18502.8i	− 1.05040i	−0.850979	−	0.525200i	$-0.823991\pi$
$677$	− 18502.8i	− 1.05040i	0.850979	−	0.525200i	$-0.176009\pi$
$678$	4566.00	0.258638
$679$	−618.446	−0.0349540
$680$	0	0
$681$	−19076.8	−1.07346
$682$	−1882.22	−0.105680
$683$	137.518i	0.00770424i	0.999993	+	0.00385212i	$0.00122617\pi$
$683$	137.518i	0.00770424i	−0.999993	+	0.00385212i	$0.998774\pi$
$684$	−2943.57	−0.164547
$685$	4288.71i	0.239217i
$686$	− 11677.7i	− 0.649936i
$687$	− 2581.37i	− 0.143356i
$688$	1369.46	0.0758867
$689$	−5758.47	−0.318404
$690$	1491.46i	0.0822882i
$691$	− 303.235i	− 0.0166940i	−0.999965	−	0.00834702i	$-0.997343\pi$
$691$	− 303.235i	− 0.0166940i	0.999965	−	0.00834702i	$-0.00265697\pi$
$692$	9599.69i	0.527349i
$693$	3103.30	0.170108
$694$	14158.6i	0.774426i
$695$	18074.0	0.986452
$696$	−26411.5	−1.43840
$697$	0	0
$698$	−8406.54	−0.455863
$699$	2985.24	0.161534
$700$	5554.79i	0.299931i
$701$	−23434.3	−1.26263	−0.631313	−	0.775528i	$-0.717484\pi$
$701$	−23434.3	−1.26263	−0.631313	+	0.775528i	$0.717484\pi$
$702$	17136.5i	0.921331i
$703$	28528.0i	1.53052i
$704$	8976.83i	0.480578i
$705$	−6373.13	−0.340463
$706$	−16385.4	−0.873472
$707$	− 20513.4i	− 1.09121i
$708$	− 7030.24i	− 0.373182i
$709$	52.1860i	0.00276430i	0.999999	+	0.00138215i	$0.000439952\pi$
$709$	52.1860i	0.00276430i	−0.999999	+	0.00138215i	$0.999560\pi$
$710$	−4622.70	−0.244348
$711$	− 2788.74i	− 0.147097i
$712$	13375.0	0.704002
$713$	−1034.30	−0.0543265
$714$	0	0
$715$	13862.0	0.725046
$716$	−18272.3	−0.953725
$717$	− 1215.98i	− 0.0633354i
$718$	3347.43	0.173990
$719$	2244.80i	0.116435i	0.998304	+	0.0582175i	$0.0185417\pi$
$719$	2244.80i	0.116435i	−0.998304	+	0.0582175i	$0.981458\pi$
$720$	− 131.039i	− 0.00678271i
$721$	− 8577.23i	− 0.443041i
$722$	−860.445	−0.0443524
$723$	−28824.9	−1.48272
$724$	2348.89i	0.120575i
$725$	− 20930.4i	− 1.07219i
$726$	2880.96i	0.147276i
$727$	−2971.90	−0.151612	−0.0758059	−	0.997123i	$-0.524153\pi$
$727$	−2971.90	−0.151612	−0.0758059	+	0.997123i	$0.524153\pi$
$728$	20451.4i	1.04118i
$729$	−21748.2	−1.10492
$730$	−7472.91	−0.378883
$731$	0	0
$732$	−8263.73	−0.417263
$733$	−31704.8	−1.59760	−0.798802	−	0.601594i	$-0.794532\pi$
$733$	−31704.8	−1.59760	−0.798802	+	0.601594i	$0.794532\pi$
$734$	5898.00i	0.296593i
$735$	4553.02	0.228491
$736$	5292.50i	0.265060i
$737$	− 20355.9i	− 1.01739i
$738$	405.034i	0.0202026i
$739$	36985.6	1.84105	0.920526	−	0.390681i	$-0.127760\pi$
$739$	36985.6	1.84105	0.920526	+	0.390681i	$0.127760\pi$
$740$	−12408.0	−0.616389
$741$	23384.2i	1.15930i
$742$	2064.36i	0.102136i
$743$	− 10601.9i	− 0.523481i	−0.965138	−	0.261741i	$-0.915704\pi$
$743$	− 10601.9i	− 0.523481i	0.965138	−	0.261741i	$-0.0842965\pi$
$744$	3551.01	0.174981
$745$	15092.6i	0.742217i
$746$	4538.63	0.222750
$747$	1113.21	0.0545248
$748$	0	0
$749$	−1928.48	−0.0940789
$750$	10509.1	0.511653
$751$	− 29161.5i	− 1.41694i	−0.705743	−	0.708468i	$-0.749387\pi$
$751$	− 29161.5i	− 1.41694i	0.705743	−	0.708468i	$-0.250613\pi$
$752$	−558.765	−0.0270958
$753$	− 11232.0i	− 0.543583i
$754$	− 29969.7i	− 1.44752i
$755$	14526.4i	0.700225i
$756$	−10753.5	−0.517329
$757$	34447.9	1.65394	0.826969	−	0.562248i	$-0.190063\pi$
$757$	34447.9	1.65394	0.826969	+	0.562248i	$0.190063\pi$
$758$	− 9521.67i	− 0.456257i
$759$	− 3959.96i	− 0.189377i
$760$	12120.1i	0.578476i
$761$	−14414.5	−0.686632	−0.343316	−	0.939220i	$-0.611550\pi$
$761$	−14414.5	−0.686632	−0.343316	+	0.939220i	$0.611550\pi$
$762$	− 11421.0i	− 0.542966i
$763$	11426.4	0.542156
$764$	23012.6	1.08975
$765$	0	0
$766$	−13357.5	−0.630058
$767$	20512.1	0.965644
$768$	− 17690.3i	− 0.831175i
$769$	−7049.33	−0.330566	−0.165283	−	0.986246i	$-0.552854\pi$
$769$	−7049.33	−0.330566	−0.165283	+	0.986246i	$0.552854\pi$
$770$	− 4969.40i	− 0.232578i
$771$	− 7565.28i	− 0.353381i
$772$	− 14912.3i	− 0.695212i
$773$	−26768.3	−1.24552	−0.622760	−	0.782413i	$-0.713989\pi$
$773$	−26768.3	−1.24552	−0.622760	+	0.782413i	$0.713989\pi$
$774$	6384.47	0.296492
$775$	2814.08i	0.130432i
$776$	995.097i	0.0460334i
$777$	22067.3i	1.01887i
$778$	−12676.7	−0.584165
$779$	2610.29i	0.120055i
$780$	−10170.8	−0.466889
$781$	12273.7	0.562339
$782$	0	0
$783$	40519.0	1.84934
$784$	399.186	0.0181845
$785$	2350.12i	0.106853i
$786$	−21096.1	−0.957343
$787$	− 26085.5i	− 1.18151i	−0.806851	−	0.590755i	$-0.798830\pi$
$787$	− 26085.5i	− 1.18151i	0.806851	−	0.590755i	$-0.201170\pi$
$788$	9806.68i	0.443336i
$789$	18994.6i	0.857065i
$790$	−4465.69	−0.201116
$791$	−8359.95	−0.375785
$792$	− 4993.30i	− 0.224027i
$793$	− 24111.1i	− 1.07971i
$794$	− 16270.6i	− 0.727232i
$795$	−2639.78	−0.117765
$796$	− 15715.1i	− 0.699758i
$797$	−18108.6	−0.804818	−0.402409	−	0.915460i	$-0.631827\pi$
$797$	−18108.6	−0.804818	−0.402409	+	0.915460i	$0.631827\pi$
$798$	8383.06	0.371876
$799$	0	0
$800$	14399.6	0.636379
$801$	−4344.75	−0.191653
$802$	− 14775.0i	− 0.650527i
$803$	19841.2	0.871958
$804$	14935.5i	0.655142i
$805$	− 2730.73i	− 0.119560i
$806$	4029.41i	0.176092i
$807$	35534.1	1.55001
$808$	−33006.6	−1.43709
$809$	− 7304.62i	− 0.317450i	−0.987323	−	0.158725i	$-0.949262\pi$
$809$	− 7304.62i	− 0.317450i	0.987323	−	0.158725i	$-0.0507383\pi$
$810$	5581.38i	0.242111i
$811$	19511.7i	0.844817i	0.906405	+	0.422409i	$0.138815\pi$
$811$	19511.7i	0.844817i	−0.906405	+	0.422409i	$0.861185\pi$
$812$	18806.6	0.812786
$813$	2107.55i	0.0909163i
$814$	−18820.6	−0.810395
$815$	−4620.65	−0.198594
$816$	0	0
$817$	41145.5	1.76193
$818$	9003.45	0.384839
$819$	− 6643.45i	− 0.283444i
$820$	−1135.32	−0.0483503
$821$	− 29364.3i	− 1.24826i	−0.781320	−	0.624131i	$-0.785453\pi$
$821$	− 29364.3i	− 1.24826i	0.781320	−	0.624131i	$-0.214547\pi$
$822$	− 4772.96i	− 0.202526i
$823$	28112.4i	1.19069i	0.803470	+	0.595345i	$0.202985\pi$
$823$	28112.4i	1.19069i	−0.803470	+	0.595345i	$0.797015\pi$
$824$	−13801.0	−0.583472
$825$	−10774.1	−0.454673
$826$	− 7353.43i	− 0.309756i
$827$	39360.5i	1.65502i	0.561453	+	0.827509i	$0.310243\pi$
$827$	39360.5i	1.65502i	−0.561453	+	0.827509i	$0.689757\pi$
$828$	− 1067.12i	− 0.0447885i
$829$	15525.3	0.650443	0.325221	−	0.945638i	$-0.394561\pi$
$829$	15525.3	0.650443	0.325221	+	0.945638i	$0.394561\pi$
$830$	− 1782.61i	− 0.0745485i
$831$	−12559.3	−0.524280
$832$	19217.3	0.800771
$833$	0	0
$834$	−20114.7	−0.835151
$835$	5446.44	0.225727
$836$	− 12515.1i	− 0.517756i
$837$	−5447.75	−0.224972
$838$	− 16045.5i	− 0.661433i
$839$	− 30007.9i	− 1.23479i	−0.786654	−	0.617394i	$-0.788189\pi$
$839$	− 30007.9i	− 1.23479i	0.786654	−	0.617394i	$-0.211811\pi$
$840$	9375.29i	0.385093i
$841$	−46474.0	−1.90553
$842$	−13971.1	−0.571826
$843$	11101.3i	0.453556i
$844$	18341.2i	0.748020i
$845$	− 14714.1i	− 0.599029i
$846$	−2604.99	−0.105864
$847$	− 5274.78i	− 0.213983i
$848$	−231.443	−0.00937239
$849$	−10741.0	−0.434193
$850$	0	0
$851$	−10342.1	−0.416595
$852$	−9005.44	−0.362114
$853$	− 4351.43i	− 0.174666i	−0.996179	−	0.0873331i	$-0.972166\pi$
$853$	− 4351.43i	− 0.174666i	0.996179	−	0.0873331i	$-0.0278344\pi$
$854$	−8643.62	−0.346345
$855$	− 3937.09i	− 0.157480i
$856$	3102.98i	0.123899i
$857$	− 25861.5i	− 1.03082i	−0.856944	−	0.515410i	$-0.827640\pi$
$857$	− 25861.5i	− 1.03082i	0.856944	−	0.515410i	$-0.172360\pi$
$858$	−15427.2	−0.613840
$859$	−19084.0	−0.758017	−0.379009	−	0.925393i	$-0.623735\pi$
$859$	−19084.0	−0.758017	−0.379009	+	0.925393i	$0.623735\pi$
$860$	17895.9i	0.709588i
$861$	2019.14i	0.0799213i
$862$	− 3326.16i	− 0.131426i
$863$	24790.2	0.977831	0.488916	−	0.872331i	$-0.337393\pi$
$863$	24790.2	0.977831	0.488916	+	0.872331i	$0.337393\pi$
$864$	27876.1i	1.09765i
$865$	−12839.8	−0.504701
$866$	−19019.9	−0.746332
$867$	0	0
$868$	−2528.54	−0.0988757
$869$	11856.8	0.462847
$870$	− 13738.6i	− 0.535383i
$871$	−43577.2	−1.69524
$872$	− 18385.5i	− 0.714003i
$873$	− 323.248i	− 0.0125318i
$874$	3928.81i	0.152053i
$875$	−19241.3	−0.743401
$876$	−14557.9	−0.561491
$877$	14394.0i	0.554219i	0.960838	+	0.277109i	$0.0893764\pi$
$877$	14394.0i	0.554219i	−0.960838	+	0.277109i	$0.910624\pi$
$878$	801.839i	0.0308209i
$879$	29560.9i	1.13432i
$880$	557.137	0.0213421
$881$	− 15328.6i	− 0.586190i	−0.956083	−	0.293095i	$-0.905315\pi$
$881$	− 15328.6i	− 0.586190i	0.956083	−	0.293095i	$-0.0946853\pi$
$882$	1861.02	0.0710475
$883$	8385.90	0.319602	0.159801	−	0.987149i	$-0.448915\pi$
$883$	8385.90	0.319602	0.159801	+	0.987149i	$0.448915\pi$
$884$	0	0
$885$	9403.11	0.357155
$886$	−22397.7	−0.849283
$887$	9215.90i	0.348861i	0.984669	+	0.174430i	$0.0558084\pi$
$887$	9215.90i	0.348861i	−0.984669	+	0.174430i	$0.944192\pi$
$888$	35507.0	1.34182
$889$	20910.9i	0.788897i
$890$	6957.37i	0.262035i
$891$	− 14819.1i	− 0.557191i
$892$	−8031.28	−0.301465
$893$	−16788.1	−0.629108
$894$	− 16796.8i	− 0.628376i
$895$	− 24439.6i	− 0.912765i
$896$	13440.8i	0.501146i
$897$	−8477.36	−0.315553
$898$	17793.2i	0.661211i
$899$	9527.49	0.353459
$900$	−2903.37	−0.107532
$901$	0	0
$902$	−1722.07	−0.0635684
$903$	31827.4	1.17292
$904$	13451.4i	0.494897i
$905$	−3141.70	−0.115396
$906$	− 16166.6i	− 0.592825i
$907$	7468.98i	0.273433i	0.990610	+	0.136716i	$0.0436549\pi$
$907$	7468.98i	0.273433i	−0.990610	+	0.136716i	$0.956345\pi$
$908$	− 21856.8i	− 0.798836i
$909$	10721.9	0.391224
$910$	−10638.4	−0.387536
$911$	30698.3i	1.11644i	0.829692	+	0.558222i	$0.188516\pi$
$911$	30698.3i	1.11644i	−0.829692	+	0.558222i	$0.811484\pi$
$912$	939.854i	0.0341246i
$913$	4732.98i	0.171565i
$914$	432.140	0.0156389
$915$	− 11052.9i	− 0.399343i
$916$	2957.55	0.106681
$917$	38625.1	1.39096
$918$	0	0
$919$	−24550.8	−0.881237	−0.440619	−	0.897694i	$-0.645241\pi$
$919$	−24550.8	−0.881237	−0.440619	+	0.897694i	$0.645241\pi$
$920$	−4393.82	−0.157457
$921$	− 11159.2i	− 0.399247i
$922$	−26537.4	−0.947898
$923$	− 26275.1i	− 0.937006i
$924$	− 9680.86i	− 0.344672i
$925$	28138.3i	1.00020i
$926$	18296.0	0.649293
$927$	4483.13	0.158841
$928$	− 48752.1i	− 1.72453i
$929$	− 41655.0i	− 1.47110i	−0.677468	−	0.735552i	$-0.736923\pi$
$929$	− 41655.0i	− 1.47110i	0.677468	−	0.735552i	$-0.263077\pi$
$930$	1847.15i	0.0651296i
$931$	11993.6	0.422206
$932$	3420.28i	0.120209i
$933$	15931.5	0.559027
$934$	3343.54	0.117135
$935$	0	0
$936$	−10689.5	−0.373288
$937$	−83.7368	−0.00291949	−0.00145975	−	0.999999i	$-0.500465\pi$
$937$	−83.7368	−0.00291949	−0.00145975	+	0.999999i	$0.500465\pi$
$938$	15622.1i	0.543795i
$939$	11608.6	0.403442
$940$	− 7301.88i	− 0.253363i
$941$	− 49294.1i	− 1.70770i	−0.520523	−	0.853848i	$-0.674263\pi$
$941$	− 49294.1i	− 1.70770i	0.520523	−	0.853848i	$-0.325737\pi$
$942$	− 2615.48i	− 0.0904639i
$943$	−946.293	−0.0326782
$944$	824.418	0.0284243
$945$	− 14383.0i	− 0.495111i
$946$	27144.7i	0.932927i
$947$	9445.27i	0.324108i	0.986782	+	0.162054i	$0.0518118\pi$
$947$	9445.27i	0.324108i	−0.986782	+	0.162054i	$0.948188\pi$
$948$	−8699.57	−0.298047
$949$	− 42475.5i	− 1.45291i
$950$	10689.3	0.365061
$951$	17347.7	0.591521
$952$	0	0
$953$	41693.4	1.41719	0.708595	−	0.705616i	$-0.249329\pi$
$953$	41693.4	1.41719	0.708595	+	0.705616i	$0.249329\pi$
$954$	−1079.00	−0.0366183
$955$	30779.8i	1.04294i
$956$	1393.18	0.0471324
$957$	36477.3i	1.23213i
$958$	− 7323.60i	− 0.246988i
$959$	8738.88i	0.294258i
$960$	8809.56	0.296174
$961$	28510.0	0.957002
$962$	40290.6i	1.35033i
$963$	− 1007.97i	− 0.0337295i
$964$	− 33025.5i	− 1.10340i
$965$	19945.5	0.665355
$966$	3039.06i	0.101222i
$967$	−32380.0	−1.07681	−0.538403	−	0.842688i	$-0.680972\pi$
$967$	−32380.0	−1.07681	−0.538403	+	0.842688i	$0.680972\pi$
$968$	−8487.28	−0.281809
$969$	0	0
$970$	−517.626	−0.0171340
$971$	−5401.25	−0.178511	−0.0892556	−	0.996009i	$-0.528449\pi$
$971$	−5401.25	−0.178511	−0.0892556	+	0.996009i	$0.528449\pi$
$972$	− 10051.1i	− 0.331676i
$973$	36828.3	1.21342
$974$	30721.5i	1.01066i
$975$	23064.8i	0.757606i
$976$	− 969.066i	− 0.0317818i
$977$	25086.2	0.821473	0.410737	−	0.911754i	$-0.365272\pi$
$977$	25086.2	0.821473	0.410737	+	0.911754i	$0.365272\pi$
$978$	5142.37	0.168134
$979$	− 18472.4i	− 0.603045i
$980$	5216.52i	0.170036i
$981$	5972.35i	0.194375i
$982$	−23421.4	−0.761108
$983$	59631.6i	1.93485i	0.253166	+	0.967423i	$0.418528\pi$
$983$	59631.6i	1.93485i	−0.253166	+	0.967423i	$0.581472\pi$
$984$	3248.86	0.105254
$985$	−13116.7	−0.424296
$986$	0	0
$987$	−12986.2	−0.418799
$988$	−26792.0	−0.862719
$989$	14916.2i	0.479584i
$990$	2597.40	0.0833846
$991$	40063.8i	1.28423i	0.766609	+	0.642114i	$0.221942\pi$
$991$	40063.8i	1.28423i	−0.766609	+	0.642114i	$0.778058\pi$
$992$	6554.70i	0.209790i
$993$	27711.6i	0.885601i
$994$	−9419.43	−0.300569
$995$	21019.3	0.669706
$996$	− 3472.68i	− 0.110478i
$997$	− 10818.4i	− 0.343652i	−0.985127	−	0.171826i	$-0.945033\pi$
$997$	− 10818.4i	− 0.343652i	0.985127	−	0.171826i	$-0.0549667\pi$
$998$	− 13213.5i	− 0.419103i
$999$	−54472.8	−1.72517

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	289.4.b.e.288.8		12
17.4	even	4		289.4.a.g.1.6		12
17.7	odd	16		17.4.d.a.2.2	✓	12
17.12	odd	16		17.4.d.a.9.2	yes	12
17.13	even	4		289.4.a.g.1.5		12
17.16	even	2	inner	289.4.b.e.288.7		12
51.29	even	16		153.4.l.a.145.2		12
51.41	even	16		153.4.l.a.19.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.4.d.a.2.2	✓	12	17.7	odd	16
17.4.d.a.9.2	yes	12	17.12	odd	16
153.4.l.a.19.2		12	51.41	even	16
153.4.l.a.145.2		12	51.29	even	16
289.4.a.g.1.5		12	17.13	even	4
289.4.a.g.1.6		12	17.4	even	4
289.4.b.e.288.7		12	17.16	even	2	inner
289.4.b.e.288.8		12	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.70547 q^{2} +4.44379i q^{3} -5.09138 q^{4} -6.80983i q^{5} +7.57875i q^{6} -13.8760i q^{7} -22.3269 q^{8} +7.25269 q^{9} -11.6140i q^{10} +30.8361i q^{11} -22.6250i q^{12} +66.0130 q^{13} -23.6651i q^{14} +30.2615 q^{15} +2.65318 q^{16} +12.3692 q^{18} +79.7150 q^{19} +34.6714i q^{20} +61.6622 q^{21} +52.5899i q^{22} +28.8986i q^{23} -99.2163i q^{24} +78.6262 q^{25} +112.583 q^{26} +152.212i q^{27} +70.6481i q^{28} -266.201i q^{29} +51.6100 q^{30} +35.7906i q^{31} +183.140 q^{32} -137.029 q^{33} -94.4935 q^{35} -36.9262 q^{36} +357.875i q^{37} +135.951 q^{38} +293.348i q^{39} +152.043i q^{40} +32.7452i q^{41} +105.163 q^{42} +516.157 q^{43} -156.998i q^{44} -49.3896i q^{45} +49.2857i q^{46} -210.602 q^{47} +11.7902i q^{48} +150.456 q^{49} +134.094 q^{50} -336.097 q^{52} -87.2324 q^{53} +259.593i q^{54} +209.989 q^{55} +309.809i q^{56} +354.237i q^{57} -453.997i q^{58} +310.728 q^{59} -154.073 q^{60} -365.247i q^{61} +61.0397i q^{62} -100.639i q^{63} +291.115 q^{64} -449.537i q^{65} -233.699 q^{66} -660.131 q^{67} -128.420 q^{69} -161.156 q^{70} -398.030i q^{71} -161.930 q^{72} -643.443i q^{73} +610.344i q^{74} +349.399i q^{75} -405.859 q^{76} +427.882 q^{77} +500.296i q^{78} -384.511i q^{79} -18.0677i q^{80} -480.576 q^{81} +55.8460i q^{82} +153.488 q^{83} -313.946 q^{84} +880.289 q^{86} +1182.94 q^{87} -688.475i q^{88} -599.053 q^{89} -84.2324i q^{90} -915.998i q^{91} -147.134i q^{92} -159.046 q^{93} -359.175 q^{94} -542.846i q^{95} +813.838i q^{96} -44.5693i q^{97} +256.598 q^{98} +223.645i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 8 q^{2} + 16 q^{4} + 96 q^{8} + 36 q^{9} - 8 q^{13} + 192 q^{15} - 184 q^{16} + 352 q^{19} - 256 q^{21} + 492 q^{25} + 784 q^{26} + 744 q^{30} - 24 q^{32} - 1400 q^{33} - 632 q^{35} + 856 q^{36}+ \cdots + 3464 q^{98}+O(q^{100}) \) 12 * q + 8 * q^2 + 16 * q^4 + 96 * q^8 + 36 * q^9 - 8 * q^13 + 192 * q^15 - 184 * q^16 + 352 * q^19 - 256 * q^21 + 492 * q^25 + 784 * q^26 + 744 * q^30 - 24 * q^32 - 1400 * q^33 - 632 * q^35 + 856 * q^36 - 624 * q^38 + 1664 * q^42 + 1200 * q^43 - 1512 * q^47 + 1052 * q^49 - 2856 * q^50 + 792 * q^52 + 2504 * q^53 - 1424 * q^55 + 3408 * q^59 + 2808 * q^60 + 272 * q^64 - 272 * q^66 - 1080 * q^67 - 344 * q^69 - 2600 * q^70 + 248 * q^72 - 896 * q^76 - 848 * q^77 - 2404 * q^81 + 2960 * q^83 + 4768 * q^84 - 1200 * q^86 + 160 * q^87 - 2144 * q^89 - 3800 * q^93 - 5984 * q^94 + 3464 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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