LMFDB - Embedded newform 269.2.a.c.1.9

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [269,2,Mod(1,269)] 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("269.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(269, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 269 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	269.a (trivial)

Newform invariants

comment:select newform

sage:traces = [16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	yes
Analytic conductor:	$2.14797581437$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{15} - 28 x^{14} + 27 x^{13} + 314 x^{12} - 283 x^{11} - 1803 x^{10} + 1435 x^{9} + \cdots + 172 $ x^16 - x^15 - 28x^14 + 27x^13 + 314x^12 - 283x^11 - 1803x^10 + 1435x^9 + 5637x^8 - 3547x^7 - 9470x^6 + 3701x^5 + 7860x^4 - 1001x^3 - 2363x^2 - 43x + 172
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$0.282877$ of defining polynomial
Character	$\chi$	$=$	269.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.282877 q^{2} -1.37910 q^{3} -1.91998 q^{4} +1.97313 q^{5} -0.390116 q^{6} +2.44119 q^{7} -1.10887 q^{8} -1.09807 q^{9} +0.558153 q^{10} +3.43591 q^{11} +2.64785 q^{12} +4.69311 q^{13} +0.690556 q^{14} -2.72116 q^{15} +3.52629 q^{16} +1.90195 q^{17} -0.310618 q^{18} -1.23155 q^{19} -3.78838 q^{20} -3.36666 q^{21} +0.971939 q^{22} +6.50000 q^{23} +1.52925 q^{24} -1.10675 q^{25} +1.32757 q^{26} +5.65167 q^{27} -4.68704 q^{28} -5.54253 q^{29} -0.769751 q^{30} +4.88221 q^{31} +3.21525 q^{32} -4.73848 q^{33} +0.538017 q^{34} +4.81680 q^{35} +2.10827 q^{36} -10.2236 q^{37} -0.348377 q^{38} -6.47229 q^{39} -2.18795 q^{40} -5.60633 q^{41} -0.952349 q^{42} -2.85084 q^{43} -6.59688 q^{44} -2.16664 q^{45} +1.83870 q^{46} -1.80999 q^{47} -4.86312 q^{48} -1.04058 q^{49} -0.313073 q^{50} -2.62299 q^{51} -9.01068 q^{52} +2.04207 q^{53} +1.59872 q^{54} +6.77951 q^{55} -2.70697 q^{56} +1.69844 q^{57} -1.56785 q^{58} -1.02290 q^{59} +5.22457 q^{60} +12.6403 q^{61} +1.38106 q^{62} -2.68060 q^{63} -6.14306 q^{64} +9.26012 q^{65} -1.34041 q^{66} +1.65225 q^{67} -3.65171 q^{68} -8.96418 q^{69} +1.36256 q^{70} -4.59265 q^{71} +1.21762 q^{72} +5.17859 q^{73} -2.89202 q^{74} +1.52632 q^{75} +2.36455 q^{76} +8.38772 q^{77} -1.83086 q^{78} +13.5494 q^{79} +6.95784 q^{80} -4.50003 q^{81} -1.58590 q^{82} -13.9606 q^{83} +6.46392 q^{84} +3.75280 q^{85} -0.806435 q^{86} +7.64373 q^{87} -3.80998 q^{88} -9.64059 q^{89} -0.612891 q^{90} +11.4568 q^{91} -12.4799 q^{92} -6.73308 q^{93} -0.512003 q^{94} -2.43001 q^{95} -4.43416 q^{96} -16.2488 q^{97} -0.294355 q^{98} -3.77287 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + q^{2} + 5 q^{3} + 25 q^{4} - q^{5} - 2 q^{6} + 11 q^{7} + 21 q^{9} + 7 q^{10} + 16 q^{11} - 4 q^{12} - q^{13} - 9 q^{14} - 5 q^{15} + 39 q^{16} - 2 q^{17} - 8 q^{18} + 35 q^{19} - 16 q^{20} - 3 q^{22}+ \cdots + 90 q^{99}+O(q^{100}) $ 16 * q + q^2 + 5 * q^3 + 25 * q^4 - q^5 - 2 * q^6 + 11 * q^7 + 21 * q^9 + 7 * q^10 + 16 * q^11 - 4 * q^12 - q^13 - 9 * q^14 - 5 * q^15 + 39 * q^16 - 2 * q^17 - 8 * q^18 + 35 * q^19 - 16 * q^20 - 3 * q^22 + q^23 - 15 * q^24 + 13 * q^25 - 14 * q^26 + 11 * q^27 + 11 * q^28 + 2 * q^29 - 13 * q^30 + 13 * q^31 - 29 * q^32 - 8 * q^33 - 10 * q^34 + 9 * q^35 + 11 * q^36 + 4 * q^37 - 32 * q^38 + 11 * q^39 + 6 * q^40 - 2 * q^41 - 45 * q^42 + 51 * q^43 + 8 * q^44 - 35 * q^45 + 20 * q^46 - 9 * q^47 - 40 * q^48 + 27 * q^49 - 58 * q^50 + 40 * q^51 - 38 * q^52 - 29 * q^53 - 49 * q^54 - 8 * q^55 - 65 * q^56 - 16 * q^57 - 26 * q^58 - 13 * q^59 - 95 * q^60 + 14 * q^61 - 47 * q^62 - 14 * q^63 + 32 * q^64 - 29 * q^65 - 82 * q^66 + 25 * q^67 - 45 * q^68 - 12 * q^69 - 44 * q^70 - 9 * q^71 - 30 * q^72 + 4 * q^73 + 8 * q^74 + 22 * q^75 + 58 * q^76 - 10 * q^77 - 17 * q^78 + 51 * q^79 - 7 * q^80 - 21 * q^82 - 2 * q^83 - 18 * q^84 + 4 * q^85 + 10 * q^86 + q^87 - 18 * q^88 - 35 * q^89 + 16 * q^90 + 65 * q^91 + 69 * q^92 - 36 * q^93 + 24 * q^94 + 18 * q^95 - 28 * q^96 - 14 * q^97 + 24 * q^98 + 90 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.282877	0.200024	0.100012	−	0.994986i	$-0.468112\pi$
$2$	0.282877	0.200024	0.100012	+	0.994986i	$0.468112\pi$
$3$	−1.37910	−0.796227	−0.398113	−	0.917336i	$-0.630335\pi$
$3$	−1.37910	−0.796227	−0.398113	+	0.917336i	$0.630335\pi$
$4$	−1.91998	−0.959990
$5$	1.97313	0.882412	0.441206	−	0.897406i	$-0.354551\pi$
$5$	1.97313	0.882412	0.441206	+	0.897406i	$0.354551\pi$
$6$	−0.390116	−0.159264
$7$	2.44119	0.922684	0.461342	−	0.887222i	$-0.347368\pi$
$7$	2.44119	0.922684	0.461342	+	0.887222i	$0.347368\pi$
$8$	−1.10887	−0.392045
$9$	−1.09807	−0.366023
$10$	0.558153	0.176503
$11$	3.43591	1.03597	0.517983	−	0.855391i	$-0.326683\pi$
$11$	3.43591	1.03597	0.517983	+	0.855391i	$0.326683\pi$
$12$	2.64785	0.764370
$13$	4.69311	1.30163	0.650817	−	0.759235i	$-0.274427\pi$
$13$	4.69311	1.30163	0.650817	+	0.759235i	$0.274427\pi$
$14$	0.690556	0.184559
$15$	−2.72116	−0.702600
$16$	3.52629	0.881572
$17$	1.90195	0.461291	0.230645	−	0.973038i	$-0.425916\pi$
$17$	1.90195	0.461291	0.230645	+	0.973038i	$0.425916\pi$
$18$	−0.310618	−0.0732134
$19$	−1.23155	−0.282537	−0.141269	−	0.989971i	$-0.545118\pi$
$19$	−1.23155	−0.282537	−0.141269	+	0.989971i	$0.545118\pi$
$20$	−3.78838	−0.847107
$21$	−3.36666	−0.734666
$22$	0.971939	0.207218
$23$	6.50000	1.35534	0.677672	−	0.735365i	$-0.262989\pi$
$23$	6.50000	1.35534	0.677672	+	0.735365i	$0.262989\pi$
$24$	1.52925	0.312157
$25$	−1.10675	−0.221349
$26$	1.32757	0.260358
$27$	5.65167	1.08766
$28$	−4.68704	−0.885768
$29$	−5.54253	−1.02922	−0.514611	−	0.857424i	$-0.672064\pi$
$29$	−5.54253	−1.02922	−0.514611	+	0.857424i	$0.672064\pi$
$30$	−0.769751	−0.140537
$31$	4.88221	0.876871	0.438436	−	0.898763i	$-0.355533\pi$
$31$	4.88221	0.876871	0.438436	+	0.898763i	$0.355533\pi$
$32$	3.21525	0.568380
$33$	−4.73848	−0.824864
$34$	0.538017	0.0922692
$35$	4.81680	0.814187
$36$	2.10827	0.351379
$37$	−10.2236	−1.68075	−0.840377	−	0.542003i	$-0.817666\pi$
$37$	−10.2236	−1.68075	−0.840377	+	0.542003i	$0.817666\pi$
$38$	−0.348377	−0.0565142
$39$	−6.47229	−1.03640
$40$	−2.18795	−0.345945
$41$	−5.60633	−0.875561	−0.437780	−	0.899082i	$-0.644235\pi$
$41$	−5.60633	−0.875561	−0.437780	+	0.899082i	$0.644235\pi$
$42$	−0.952349	−0.146951
$43$	−2.85084	−0.434749	−0.217374	−	0.976088i	$-0.569749\pi$
$43$	−2.85084	−0.434749	−0.217374	+	0.976088i	$0.569749\pi$
$44$	−6.59688	−0.994518
$45$	−2.16664	−0.322983
$46$	1.83870	0.271101
$47$	−1.80999	−0.264014	−0.132007	−	0.991249i	$-0.542142\pi$
$47$	−1.80999	−0.264014	−0.132007	+	0.991249i	$0.542142\pi$
$48$	−4.86312	−0.701931
$49$	−1.04058	−0.148654
$50$	−0.313073	−0.0442752
$51$	−2.62299	−0.367292
$52$	−9.01068	−1.24956
$53$	2.04207	0.280500	0.140250	−	0.990116i	$-0.455209\pi$
$53$	2.04207	0.280500	0.140250	+	0.990116i	$0.455209\pi$
$54$	1.59872	0.217559
$55$	6.77951	0.914149
$56$	−2.70697	−0.361734
$57$	1.69844	0.224964
$58$	−1.56785	−0.205869
$59$	−1.02290	−0.133170	−0.0665850	−	0.997781i	$-0.521210\pi$
$59$	−1.02290	−0.133170	−0.0665850	+	0.997781i	$0.521210\pi$
$60$	5.22457	0.674489
$61$	12.6403	1.61843	0.809213	−	0.587515i	$-0.199894\pi$
$61$	12.6403	1.61843	0.809213	+	0.587515i	$0.199894\pi$
$62$	1.38106	0.175395
$63$	−2.68060	−0.337724
$64$	−6.14306	−0.767882
$65$	9.26012	1.14858
$66$	−1.34041	−0.164992
$67$	1.65225	0.201854	0.100927	−	0.994894i	$-0.467819\pi$
$67$	1.65225	0.201854	0.100927	+	0.994894i	$0.467819\pi$
$68$	−3.65171	−0.442835
$69$	−8.96418	−1.07916
$70$	1.36256	0.162857
$71$	−4.59265	−0.545047	−0.272523	−	0.962149i	$-0.587858\pi$
$71$	−4.59265	−0.545047	−0.272523	+	0.962149i	$0.587858\pi$
$72$	1.21762	0.143498
$73$	5.17859	0.606108	0.303054	−	0.952973i	$-0.401994\pi$
$73$	5.17859	0.606108	0.303054	+	0.952973i	$0.401994\pi$
$74$	−2.89202	−0.336191
$75$	1.52632	0.176244
$76$	2.36455	0.271233
$77$	8.38772	0.955870
$78$	−1.83086	−0.207304
$79$	13.5494	1.52443	0.762213	−	0.647326i	$-0.224113\pi$
$79$	13.5494	1.52443	0.762213	+	0.647326i	$0.224113\pi$
$80$	6.95784	0.777910
$81$	−4.50003	−0.500004
$82$	−1.58590	−0.175133
$83$	−13.9606	−1.53237	−0.766187	−	0.642618i	$-0.777848\pi$
$83$	−13.9606	−1.53237	−0.766187	+	0.642618i	$0.777848\pi$
$84$	6.46392	0.705272
$85$	3.75280	0.407049
$86$	−0.806435	−0.0869601
$87$	7.64373	0.819494
$88$	−3.80998	−0.406145
$89$	−9.64059	−1.02190	−0.510950	−	0.859610i	$-0.670706\pi$
$89$	−9.64059	−1.02190	−0.510950	+	0.859610i	$0.670706\pi$
$90$	−0.612891	−0.0646044
$91$	11.4568	1.20100
$92$	−12.4799	−1.30112
$93$	−6.73308	−0.698188
$94$	−0.512003	−0.0528090
$95$	−2.43001	−0.249314
$96$	−4.43416	−0.452560
$97$	−16.2488	−1.64982	−0.824908	−	0.565267i	$-0.808773\pi$
$97$	−16.2488	−1.64982	−0.824908	+	0.565267i	$0.808773\pi$
$98$	−0.294355	−0.0297344
$99$	−3.77287	−0.379188
$100$	2.12493	0.212493
$101$	15.4748	1.53980	0.769900	−	0.638165i	$-0.220306\pi$
$101$	15.4748	1.53980	0.769900	+	0.638165i	$0.220306\pi$
$102$	−0.741982	−0.0734672
$103$	11.6936	1.15220	0.576101	−	0.817379i	$-0.304574\pi$
$103$	11.6936	1.15220	0.576101	+	0.817379i	$0.304574\pi$
$104$	−5.20405	−0.510299
$105$	−6.64287	−0.648278
$106$	0.577654	0.0561067
$107$	0.0867365	0.00838514	0.00419257	−	0.999991i	$-0.498665\pi$
$107$	0.0867365	0.00838514	0.00419257	+	0.999991i	$0.498665\pi$
$108$	−10.8511	−1.04415
$109$	3.18524	0.305091	0.152545	−	0.988296i	$-0.451253\pi$
$109$	3.18524	0.305091	0.152545	+	0.988296i	$0.451253\pi$
$110$	1.91776	0.182852
$111$	14.0994	1.33826
$112$	8.60835	0.813413
$113$	−2.15866	−0.203070	−0.101535	−	0.994832i	$-0.532375\pi$
$113$	−2.15866	−0.203070	−0.101535	+	0.994832i	$0.532375\pi$
$114$	0.480448	0.0449981
$115$	12.8254	1.19597
$116$	10.6416	0.988043
$117$	−5.15336	−0.476428
$118$	−0.289354	−0.0266372
$119$	4.64303	0.425626
$120$	3.01741	0.275451
$121$	0.805487	0.0732261
$122$	3.57565	0.323724
$123$	7.73171	0.697145
$124$	−9.37375	−0.841788
$125$	−12.0494	−1.07773
$126$	−0.758279	−0.0675528
$127$	3.13665	0.278333	0.139166	−	0.990269i	$-0.455558\pi$
$127$	3.13665	0.278333	0.139166	+	0.990269i	$0.455558\pi$
$128$	−8.16822	−0.721975
$129$	3.93160	0.346158
$130$	2.61947	0.229743
$131$	−19.1069	−1.66938	−0.834689	−	0.550721i	$-0.814353\pi$
$131$	−19.1069	−1.66938	−0.834689	+	0.550721i	$0.814353\pi$
$132$	9.09779	0.791861
$133$	−3.00645	−0.260693
$134$	0.467383	0.0403757
$135$	11.1515	0.959768
$136$	−2.10902	−0.180847
$137$	−7.77105	−0.663925	−0.331963	−	0.943293i	$-0.607711\pi$
$137$	−7.77105	−0.663925	−0.331963	+	0.943293i	$0.607711\pi$
$138$	−2.53576	−0.215858
$139$	7.50884	0.636891	0.318446	−	0.947941i	$-0.396839\pi$
$139$	7.50884	0.636891	0.318446	+	0.947941i	$0.396839\pi$
$140$	−9.24816	−0.781612
$141$	2.49616	0.210215
$142$	−1.29915	−0.109022
$143$	16.1251	1.34845
$144$	−3.87211	−0.322676
$145$	−10.9361	−0.908198
$146$	1.46490	0.121236
$147$	1.43507	0.118362
$148$	19.6292	1.61351
$149$	−21.5548	−1.76584	−0.882921	−	0.469522i	$-0.844426\pi$
$149$	−21.5548	−1.76584	−0.882921	+	0.469522i	$0.844426\pi$
$150$	0.431760	0.0352531
$151$	−8.16873	−0.664762	−0.332381	−	0.943145i	$-0.607852\pi$
$151$	−8.16873	−0.664762	−0.332381	+	0.943145i	$0.607852\pi$
$152$	1.36563	0.110767
$153$	−2.08848	−0.168843
$154$	2.37269	0.191197
$155$	9.63325	0.773761
$156$	12.4267	0.994930
$157$	19.5185	1.55774	0.778872	−	0.627183i	$-0.215792\pi$
$157$	19.5185	1.55774	0.778872	+	0.627183i	$0.215792\pi$
$158$	3.83281	0.304922
$159$	−2.81623	−0.223342
$160$	6.34411	0.501546
$161$	15.8677	1.25055
$162$	−1.27295	−0.100013
$163$	5.85613	0.458687	0.229344	−	0.973345i	$-0.426342\pi$
$163$	5.85613	0.458687	0.229344	+	0.973345i	$0.426342\pi$
$164$	10.7640	0.840530
$165$	−9.34965	−0.727870
$166$	−3.94912	−0.306511
$167$	−16.1245	−1.24775	−0.623876	−	0.781523i	$-0.714443\pi$
$167$	−16.1245	−1.24775	−0.623876	+	0.781523i	$0.714443\pi$
$168$	3.73319	0.288022
$169$	9.02526	0.694251
$170$	1.06158	0.0814194
$171$	1.35233	0.103415
$172$	5.47355	0.417354
$173$	−13.5232	−1.02815	−0.514074	−	0.857746i	$-0.671864\pi$
$173$	−13.5232	−1.02815	−0.514074	+	0.857746i	$0.671864\pi$
$174$	2.16223	0.163918
$175$	−2.70178	−0.204236
$176$	12.1160	0.913279
$177$	1.41068	0.106034
$178$	−2.72710	−0.204405
$179$	22.1712	1.65715	0.828577	−	0.559875i	$-0.189151\pi$
$179$	22.1712	1.65715	0.828577	+	0.559875i	$0.189151\pi$
$180$	4.15990	0.310061
$181$	3.61117	0.268416	0.134208	−	0.990953i	$-0.457151\pi$
$181$	3.61117	0.268416	0.134208	+	0.990953i	$0.457151\pi$
$182$	3.24085	0.240228
$183$	−17.4323	−1.28863
$184$	−7.20766	−0.531356
$185$	−20.1726	−1.48312
$186$	−1.90463	−0.139654
$187$	6.53494	0.477882
$188$	3.47514	0.253451
$189$	13.7968	1.00357
$190$	−0.687394	−0.0498688
$191$	−7.41632	−0.536626	−0.268313	−	0.963332i	$-0.586466\pi$
$191$	−7.41632	−0.536626	−0.268313	+	0.963332i	$0.586466\pi$
$192$	8.47192	0.611408
$193$	−23.0099	−1.65629	−0.828145	−	0.560514i	$-0.810604\pi$
$193$	−23.0099	−1.65629	−0.828145	+	0.560514i	$0.810604\pi$
$194$	−4.59640	−0.330003
$195$	−12.7707	−0.914528
$196$	1.99789	0.142707
$197$	0.810204	0.0577247	0.0288623	−	0.999583i	$-0.490812\pi$
$197$	0.810204	0.0577247	0.0288623	+	0.999583i	$0.490812\pi$
$198$	−1.06726	−0.0758466
$199$	16.0676	1.13900	0.569500	−	0.821991i	$-0.307137\pi$
$199$	16.0676	1.13900	0.569500	+	0.821991i	$0.307137\pi$
$200$	1.22724	0.0867789
$201$	−2.27863	−0.160722
$202$	4.37746	0.307997
$203$	−13.5304	−0.949647
$204$	5.03609	0.352597
$205$	−11.0620	−0.772605
$206$	3.30784	0.230468
$207$	−7.13745	−0.496087
$208$	16.5493	1.14748
$209$	−4.23150	−0.292699
$210$	−1.87911	−0.129671
$211$	−9.98098	−0.687119	−0.343559	−	0.939131i	$-0.611633\pi$
$211$	−9.98098	−0.687119	−0.343559	+	0.939131i	$0.611633\pi$
$212$	−3.92074	−0.269278
$213$	6.33374	0.433981
$214$	0.0245357	0.00167723
$215$	−5.62508	−0.383627
$216$	−6.26697	−0.426413
$217$	11.9184	0.809075
$218$	0.901030	0.0610255
$219$	−7.14182	−0.482599
$220$	−13.0165	−0.877574
$221$	8.92606	0.600432
$222$	3.98840	0.267684
$223$	−19.6088	−1.31310	−0.656550	−	0.754283i	$-0.727985\pi$
$223$	−19.6088	−1.31310	−0.656550	+	0.754283i	$0.727985\pi$
$224$	7.84903	0.524436
$225$	1.21529	0.0810191
$226$	−0.610634	−0.0406188
$227$	−11.4505	−0.759998	−0.379999	−	0.924987i	$-0.624076\pi$
$227$	−11.4505	−0.759998	−0.379999	+	0.924987i	$0.624076\pi$
$228$	−3.26097	−0.215963
$229$	22.6433	1.49631	0.748156	−	0.663523i	$-0.230939\pi$
$229$	22.6433	1.49631	0.748156	+	0.663523i	$0.230939\pi$
$230$	3.62799	0.239223
$231$	−11.5675	−0.761089
$232$	6.14595	0.403501
$233$	−15.8879	−1.04085	−0.520427	−	0.853906i	$-0.674227\pi$
$233$	−15.8879	−1.04085	−0.520427	+	0.853906i	$0.674227\pi$
$234$	−1.45776	−0.0952971
$235$	−3.57134	−0.232969
$236$	1.96395	0.127842
$237$	−18.6860	−1.21379
$238$	1.31340	0.0851353
$239$	7.35894	0.476010	0.238005	−	0.971264i	$-0.423507\pi$
$239$	7.35894	0.476010	0.238005	+	0.971264i	$0.423507\pi$
$240$	−9.59558	−0.619392
$241$	−13.1161	−0.844882	−0.422441	−	0.906390i	$-0.638827\pi$
$241$	−13.1161	−0.844882	−0.422441	+	0.906390i	$0.638827\pi$
$242$	0.227853	0.0146470
$243$	−10.7490	−0.689548
$244$	−24.2692	−1.55367
$245$	−2.05320	−0.131174
$246$	2.18712	0.139446
$247$	−5.77980	−0.367760
$248$	−5.41374	−0.343773
$249$	19.2531	1.22012
$250$	−3.40850	−0.215572
$251$	−0.274032	−0.0172967	−0.00864837	−	0.999963i	$-0.502753\pi$
$251$	−0.274032	−0.0172967	−0.00864837	+	0.999963i	$0.502753\pi$
$252$	5.14670	0.324212
$253$	22.3334	1.40409
$254$	0.887284	0.0556732
$255$	−5.17551	−0.324103
$256$	9.97552	0.623470
$257$	−5.65777	−0.352922	−0.176461	−	0.984308i	$-0.556465\pi$
$257$	−5.65777	−0.352922	−0.176461	+	0.984308i	$0.556465\pi$
$258$	1.11216	0.0692399
$259$	−24.9578	−1.55080
$260$	−17.7793	−1.10262
$261$	6.08609	0.376719
$262$	−5.40490	−0.333916
$263$	18.5754	1.14541	0.572703	−	0.819763i	$-0.305895\pi$
$263$	18.5754	1.14541	0.572703	+	0.819763i	$0.305895\pi$
$264$	5.25436	0.323384
$265$	4.02928	0.247517
$266$	−0.850455	−0.0521447
$267$	13.2954	0.813665
$268$	−3.17229	−0.193778
$269$	1.00000	0.0609711
$269$	1.00000	0.0609711
$270$	3.15449	0.191976
$271$	−5.19643	−0.315661	−0.157830	−	0.987466i	$-0.550450\pi$
$271$	−5.19643	−0.315661	−0.157830	+	0.987466i	$0.550450\pi$
$272$	6.70683	0.406661
$273$	−15.8001	−0.956266
$274$	−2.19825	−0.132801
$275$	−3.80269	−0.229311
$276$	17.2111	1.03598
$277$	6.13328	0.368513	0.184257	−	0.982878i	$-0.441012\pi$
$277$	6.13328	0.368513	0.184257	+	0.982878i	$0.441012\pi$
$278$	2.12407	0.127393
$279$	−5.36101	−0.320955
$280$	−5.34120	−0.319198
$281$	19.4792	1.16203	0.581017	−	0.813891i	$-0.302655\pi$
$281$	19.4792	1.16203	0.581017	+	0.813891i	$0.302655\pi$
$282$	0.706105	0.0420480
$283$	1.43510	0.0853076	0.0426538	−	0.999090i	$-0.486419\pi$
$283$	1.43510	0.0853076	0.0426538	+	0.999090i	$0.486419\pi$
$284$	8.81780	0.523240
$285$	3.35124	0.198511
$286$	4.56141	0.269722
$287$	−13.6861	−0.807866
$288$	−3.53056	−0.208040
$289$	−13.3826	−0.787211
$290$	−3.09358	−0.181661
$291$	22.4088	1.31363
$292$	−9.94279	−0.581858
$293$	−29.8473	−1.74370	−0.871848	−	0.489776i	$-0.837078\pi$
$293$	−29.8473	−1.74370	−0.871848	+	0.489776i	$0.837078\pi$
$294$	0.405947	0.0236753
$295$	−2.01832	−0.117511
$296$	11.3367	0.658931
$297$	19.4186	1.12678
$298$	−6.09736	−0.353210
$299$	30.5052	1.76416
$300$	−2.93051	−0.169193
$301$	−6.95944	−0.401136
$302$	−2.31074	−0.132968
$303$	−21.3414	−1.22603
$304$	−4.34280	−0.249077
$305$	24.9410	1.42812
$306$	−0.590781	−0.0337727
$307$	−14.6428	−0.835710	−0.417855	−	0.908514i	$-0.637218\pi$
$307$	−14.6428	−0.835710	−0.417855	+	0.908514i	$0.637218\pi$
$308$	−16.1043	−0.917626
$309$	−16.1267	−0.917413
$310$	2.72502	0.154771
$311$	−14.9926	−0.850155	−0.425077	−	0.905157i	$-0.639753\pi$
$311$	−14.9926	−0.850155	−0.425077	+	0.905157i	$0.639753\pi$
$312$	7.17693	0.406314
$313$	−14.3883	−0.813272	−0.406636	−	0.913590i	$-0.633298\pi$
$313$	−14.3883	−0.813272	−0.406636	+	0.913590i	$0.633298\pi$
$314$	5.52132	0.311586
$315$	−5.28918	−0.298012
$316$	−26.0146	−1.46344
$317$	18.4565	1.03662	0.518309	−	0.855193i	$-0.326562\pi$
$317$	18.4565	1.03662	0.518309	+	0.855193i	$0.326562\pi$
$318$	−0.796646	−0.0446737
$319$	−19.0436	−1.06624
$320$	−12.1211	−0.677589
$321$	−0.119619	−0.00667647
$322$	4.48861	0.250141
$323$	−2.34235	−0.130332
$324$	8.63998	0.479999
$325$	−5.19408	−0.288116
$326$	1.65656	0.0917484
$327$	−4.39278	−0.242922
$328$	6.21669	0.343259
$329$	−4.41853	−0.243601
$330$	−2.64480	−0.145591
$331$	29.1435	1.60187	0.800935	−	0.598752i	$-0.204336\pi$
$331$	29.1435	1.60187	0.800935	+	0.598752i	$0.204336\pi$
$332$	26.8041	1.47106
$333$	11.2263	0.615195
$334$	−4.56125	−0.249580
$335$	3.26011	0.178119
$336$	−11.8718	−0.647661
$337$	−13.3401	−0.726682	−0.363341	−	0.931656i	$-0.618364\pi$
$337$	−13.3401	−0.726682	−0.363341	+	0.931656i	$0.618364\pi$
$338$	2.55303	0.138867
$339$	2.97702	0.161689
$340$	−7.20531	−0.390763
$341$	16.7748	0.908409
$342$	0.382542	0.0206855
$343$	−19.6286	−1.05984
$344$	3.16121	0.170441
$345$	−17.6875	−0.952264
$346$	−3.82539	−0.205654
$347$	15.4841	0.831232	0.415616	−	0.909540i	$-0.363566\pi$
$347$	15.4841	0.831232	0.415616	+	0.909540i	$0.363566\pi$
$348$	−14.6758	−0.786706
$349$	4.95774	0.265382	0.132691	−	0.991157i	$-0.457638\pi$
$349$	4.95774	0.265382	0.132691	+	0.991157i	$0.457638\pi$
$350$	−0.764271	−0.0408520
$351$	26.5239	1.41574
$352$	11.0473	0.588823
$353$	−19.7221	−1.04970	−0.524851	−	0.851194i	$-0.675879\pi$
$353$	−19.7221	−1.04970	−0.524851	+	0.851194i	$0.675879\pi$
$354$	0.399050	0.0212092
$355$	−9.06190	−0.480956
$356$	18.5098	0.981015
$357$	−6.40322	−0.338895
$358$	6.27172	0.331471
$359$	10.0142	0.528527	0.264263	−	0.964451i	$-0.414871\pi$
$359$	10.0142	0.528527	0.264263	+	0.964451i	$0.414871\pi$
$360$	2.40252	0.126624
$361$	−17.4833	−0.920173
$362$	1.02152	0.0536897
$363$	−1.11085	−0.0583046
$364$	−21.9968	−1.15295
$365$	10.2180	0.534837
$366$	−4.93119	−0.257758
$367$	−1.28284	−0.0669637	−0.0334819	−	0.999439i	$-0.510660\pi$
$367$	−1.28284	−0.0669637	−0.0334819	+	0.999439i	$0.510660\pi$
$368$	22.9209	1.19483
$369$	6.15614	0.320476
$370$	−5.70634	−0.296659
$371$	4.98509	0.258813
$372$	12.9274	0.670254
$373$	33.4573	1.73235	0.866176	−	0.499739i	$-0.166571\pi$
$373$	33.4573	1.73235	0.866176	+	0.499739i	$0.166571\pi$
$374$	1.84858	0.0955878
$375$	16.6174	0.858120
$376$	2.00704	0.103505
$377$	−26.0117	−1.33967
$378$	3.90279	0.200738
$379$	32.2690	1.65755	0.828775	−	0.559583i	$-0.189039\pi$
$379$	32.2690	1.65755	0.828775	+	0.559583i	$0.189039\pi$
$380$	4.66558	0.239339
$381$	−4.32577	−0.221616
$382$	−2.09790	−0.107338
$383$	−13.8572	−0.708071	−0.354036	−	0.935232i	$-0.615191\pi$
$383$	−13.8572	−0.708071	−0.354036	+	0.935232i	$0.615191\pi$
$384$	11.2648	0.574856
$385$	16.5501	0.843471
$386$	−6.50897	−0.331298
$387$	3.13042	0.159128
$388$	31.1974	1.58381
$389$	29.3155	1.48636	0.743178	−	0.669094i	$-0.233317\pi$
$389$	29.3155	1.48636	0.743178	+	0.669094i	$0.233317\pi$
$390$	−3.61253	−0.182927
$391$	12.3627	0.625208
$392$	1.15387	0.0582791
$393$	26.3504	1.32920
$394$	0.229188	0.0115463
$395$	26.7348	1.34517
$396$	7.24384	0.364017
$397$	−4.33222	−0.217428	−0.108714	−	0.994073i	$-0.534673\pi$
$397$	−4.33222	−0.217428	−0.108714	+	0.994073i	$0.534673\pi$
$398$	4.54514	0.227827
$399$	4.14621	0.207570
$400$	−3.90271	−0.195135
$401$	11.9708	0.597794	0.298897	−	0.954285i	$-0.403381\pi$
$401$	11.9708	0.597794	0.298897	+	0.954285i	$0.403381\pi$
$402$	−0.644570	−0.0321482
$403$	22.9127	1.14137
$404$	−29.7113	−1.47819
$405$	−8.87916	−0.441209
$406$	−3.82743	−0.189952
$407$	−35.1275	−1.74120
$408$	2.90856	0.143995
$409$	10.5969	0.523985	0.261992	−	0.965070i	$-0.415620\pi$
$409$	10.5969	0.523985	0.261992	+	0.965070i	$0.415620\pi$
$410$	−3.12919	−0.154540
$411$	10.7171	0.528635
$412$	−22.4514	−1.10610
$413$	−2.49709	−0.122874
$414$	−2.01902	−0.0992293
$415$	−27.5461	−1.35218
$416$	15.0895	0.739823
$417$	−10.3555	−0.507110
$418$	−1.19699	−0.0585468
$419$	0.320387	0.0156519	0.00782595	−	0.999969i	$-0.497509\pi$
$419$	0.320387	0.0156519	0.00782595	+	0.999969i	$0.497509\pi$
$420$	12.7542	0.622340
$421$	24.0501	1.17213	0.586064	−	0.810265i	$-0.300677\pi$
$421$	24.0501	1.17213	0.586064	+	0.810265i	$0.300677\pi$
$422$	−2.82338	−0.137440
$423$	1.98749	0.0966352
$424$	−2.26439	−0.109969
$425$	−2.10498	−0.102106
$426$	1.79167	0.0868065
$427$	30.8574	1.49330
$428$	−0.166532	−0.00804965
$429$	−22.2382	−1.07367
$430$	−1.59120	−0.0767346
$431$	−22.7364	−1.09518	−0.547588	−	0.836748i	$-0.684454\pi$
$431$	−22.7364	−1.09518	−0.547588	+	0.836748i	$0.684454\pi$
$432$	19.9294	0.958854
$433$	−26.8292	−1.28933	−0.644663	−	0.764467i	$-0.723002\pi$
$433$	−26.8292	−1.28933	−0.644663	+	0.764467i	$0.723002\pi$
$434$	3.37144	0.161834
$435$	15.0821	0.723131
$436$	−6.11561	−0.292884
$437$	−8.00508	−0.382935
$438$	−2.02025	−0.0965314
$439$	26.3083	1.25562	0.627812	−	0.778365i	$-0.283951\pi$
$439$	26.3083	1.25562	0.627812	+	0.778365i	$0.283951\pi$
$440$	−7.51760	−0.358387
$441$	1.14263	0.0544109
$442$	2.52497	0.120101
$443$	6.31703	0.300131	0.150066	−	0.988676i	$-0.452051\pi$
$443$	6.31703	0.300131	0.150066	+	0.988676i	$0.452051\pi$
$444$	−27.0707	−1.28472
$445$	−19.0222	−0.901737
$446$	−5.54686	−0.262651
$447$	29.7264	1.40601
$448$	−14.9964	−0.708513
$449$	23.1873	1.09428	0.547138	−	0.837042i	$-0.315717\pi$
$449$	23.1873	1.09428	0.547138	+	0.837042i	$0.315717\pi$
$450$	0.343776	0.0162057
$451$	−19.2628	−0.907052
$452$	4.14459	0.194945
$453$	11.2655	0.529301
$454$	−3.23908	−0.152018
$455$	22.6057	1.05977
$456$	−1.88335	−0.0881958
$457$	28.8658	1.35029	0.675144	−	0.737686i	$-0.264082\pi$
$457$	28.8658	1.35029	0.675144	+	0.737686i	$0.264082\pi$
$458$	6.40526	0.299298
$459$	10.7492	0.501729
$460$	−24.6244	−1.14812
$461$	−9.40857	−0.438201	−0.219100	−	0.975702i	$-0.570312\pi$
$461$	−9.40857	−0.438201	−0.219100	+	0.975702i	$0.570312\pi$
$462$	−3.27219	−0.152236
$463$	38.4998	1.78924	0.894618	−	0.446831i	$-0.147447\pi$
$463$	38.4998	1.78924	0.894618	+	0.446831i	$0.147447\pi$
$464$	−19.5446	−0.907333
$465$	−13.2853	−0.616089
$466$	−4.49432	−0.208196
$467$	29.2898	1.35537	0.677686	−	0.735352i	$-0.262983\pi$
$467$	29.2898	1.35537	0.677686	+	0.735352i	$0.262983\pi$
$468$	9.89435	0.457367
$469$	4.03346	0.186248
$470$	−1.01025	−0.0465993
$471$	−26.9180	−1.24032
$472$	1.13426	0.0522087
$473$	−9.79522	−0.450385
$474$	−5.28584	−0.242787
$475$	1.36302	0.0625394
$476$	−8.91453	−0.408597
$477$	−2.24234	−0.102670
$478$	2.08167	0.0952134
$479$	−14.2259	−0.650000	−0.325000	−	0.945714i	$-0.605364\pi$
$479$	−14.2259	−0.650000	−0.325000	+	0.945714i	$0.605364\pi$
$480$	−8.74919	−0.399344
$481$	−47.9806	−2.18773
$482$	−3.71023	−0.168997
$483$	−21.8833	−0.995724
$484$	−1.54652	−0.0702964
$485$	−32.0610	−1.45582
$486$	−3.04064	−0.137926
$487$	−34.8431	−1.57889	−0.789446	−	0.613820i	$-0.789632\pi$
$487$	−34.8431	−1.57889	−0.789446	+	0.613820i	$0.789632\pi$
$488$	−14.0165	−0.634496
$489$	−8.07622	−0.365219
$490$	−0.580802	−0.0262380
$491$	−29.5777	−1.33482	−0.667411	−	0.744689i	$-0.732598\pi$
$491$	−29.5777	−1.33482	−0.667411	+	0.744689i	$0.732598\pi$
$492$	−14.8447	−0.669252
$493$	−10.5416	−0.474771
$494$	−1.63497	−0.0735608
$495$	−7.44438	−0.334600
$496$	17.2161	0.773025
$497$	−11.2115	−0.502906
$498$	5.44626	0.244053
$499$	−22.3770	−1.00173	−0.500866	−	0.865525i	$-0.666985\pi$
$499$	−22.3770	−1.00173	−0.500866	+	0.865525i	$0.666985\pi$
$500$	23.1347	1.03461
$501$	22.2374	0.993494
$502$	−0.0775172	−0.00345976
$503$	1.82300	0.0812836	0.0406418	−	0.999174i	$-0.487060\pi$
$503$	1.82300	0.0812836	0.0406418	+	0.999174i	$0.487060\pi$
$504$	2.97244	0.132403
$505$	30.5338	1.35874
$506$	6.31760	0.280852
$507$	−12.4468	−0.552781
$508$	−6.02231	−0.267197
$509$	−2.76277	−0.122458	−0.0612288	−	0.998124i	$-0.519502\pi$
$509$	−2.76277	−0.122458	−0.0612288	+	0.998124i	$0.519502\pi$
$510$	−1.46403	−0.0648283
$511$	12.6419	0.559246
$512$	19.1583	0.846684
$513$	−6.96032	−0.307305
$514$	−1.60045	−0.0705928
$515$	23.0730	1.01672
$516$	−7.54860	−0.332309
$517$	−6.21895	−0.273509
$518$	−7.05998	−0.310198
$519$	18.6499	0.818639
$520$	−10.2683	−0.450294
$521$	14.7532	0.646352	0.323176	−	0.946339i	$-0.395249\pi$
$521$	14.7532	0.646352	0.323176	+	0.946339i	$0.395249\pi$
$522$	1.72161	0.0753529
$523$	3.00571	0.131431	0.0657154	−	0.997838i	$-0.479067\pi$
$523$	3.00571	0.131431	0.0657154	+	0.997838i	$0.479067\pi$
$524$	36.6849	1.60259
$525$	3.72604	0.162618
$526$	5.25454	0.229109
$527$	9.28573	0.404493
$528$	−16.7093	−0.727177
$529$	19.2500	0.836956
$530$	1.13979	0.0495093
$531$	1.12321	0.0487434
$532$	5.77233	0.250262
$533$	−26.3111	−1.13966
$534$	3.76095	0.162752
$535$	0.171143	0.00739914
$536$	−1.83213	−0.0791360
$537$	−30.5764	−1.31947
$538$	0.282877	0.0121957
$539$	−3.57534	−0.154001
$540$	−21.4106	−0.921368
$541$	22.5866	0.971076	0.485538	−	0.874216i	$-0.338624\pi$
$541$	22.5866	0.971076	0.485538	+	0.874216i	$0.338624\pi$
$542$	−1.46995	−0.0631397
$543$	−4.98018	−0.213720
$544$	6.11524	0.262189
$545$	6.28491	0.269216
$546$	−4.46948	−0.191276
$547$	6.83246	0.292135	0.146067	−	0.989275i	$-0.453338\pi$
$547$	6.83246	0.292135	0.146067	+	0.989275i	$0.453338\pi$
$548$	14.9203	0.637362
$549$	−13.8799	−0.592382
$550$	−1.07569	−0.0458676
$551$	6.82591	0.290793
$552$	9.94011	0.423079
$553$	33.0767	1.40656
$554$	1.73496	0.0737114
$555$	27.8201	1.18090
$556$	−14.4168	−0.611409
$557$	22.0266	0.933298	0.466649	−	0.884443i	$-0.345461\pi$
$557$	22.0266	0.933298	0.466649	+	0.884443i	$0.345461\pi$
$558$	−1.51650	−0.0641987
$559$	−13.3793	−0.565883
$560$	16.9854	0.717765
$561$	−9.01236	−0.380502
$562$	5.51022	0.232435
$563$	−36.1440	−1.52329	−0.761645	−	0.647995i	$-0.775608\pi$
$563$	−36.1440	−1.52329	−0.761645	+	0.647995i	$0.775608\pi$
$564$	−4.79258	−0.201804
$565$	−4.25932	−0.179191
$566$	0.405955	0.0170636
$567$	−10.9854	−0.461345
$568$	5.09265	0.213683
$569$	−4.48775	−0.188136	−0.0940682	−	0.995566i	$-0.529987\pi$
$569$	−4.48775	−0.188136	−0.0940682	+	0.995566i	$0.529987\pi$
$570$	0.947988	0.0397068
$571$	−15.3385	−0.641896	−0.320948	−	0.947097i	$-0.604001\pi$
$571$	−15.3385	−0.641896	−0.320948	+	0.947097i	$0.604001\pi$
$572$	−30.9599	−1.29450
$573$	10.2279	0.427276
$574$	−3.87148	−0.161593
$575$	−7.19386	−0.300005
$576$	6.74551	0.281063
$577$	25.5418	1.06332	0.531659	−	0.846958i	$-0.321569\pi$
$577$	25.5418	1.06332	0.531659	+	0.846958i	$0.321569\pi$
$578$	−3.78562	−0.157461
$579$	31.7331	1.31878
$580$	20.9972	0.871861
$581$	−34.0805	−1.41390
$582$	6.33892	0.262757
$583$	7.01638	0.290589
$584$	−5.74239	−0.237622
$585$	−10.1683	−0.420406
$586$	−8.44309	−0.348781
$587$	6.30137	0.260085	0.130043	−	0.991508i	$-0.458489\pi$
$587$	6.30137	0.260085	0.130043	+	0.991508i	$0.458489\pi$
$588$	−2.75530	−0.113627
$589$	−6.01269	−0.247749
$590$	−0.570934	−0.0235050
$591$	−1.11736	−0.0459619
$592$	−36.0514	−1.48171
$593$	−11.9031	−0.488801	−0.244400	−	0.969674i	$-0.578591\pi$
$593$	−11.9031	−0.488801	−0.244400	+	0.969674i	$0.578591\pi$
$594$	5.49307	0.225384
$595$	9.16131	0.375577
$596$	41.3849	1.69519
$597$	−22.1589	−0.906902
$598$	8.62920	0.352874
$599$	−44.5084	−1.81856	−0.909282	−	0.416180i	$-0.863368\pi$
$599$	−44.5084	−1.81856	−0.909282	+	0.416180i	$0.863368\pi$
$600$	−1.69249	−0.0690957
$601$	18.3848	0.749930	0.374965	−	0.927039i	$-0.377655\pi$
$601$	18.3848	0.749930	0.374965	+	0.927039i	$0.377655\pi$
$602$	−1.96866	−0.0802367
$603$	−1.81429	−0.0738834
$604$	15.6838	0.638165
$605$	1.58933	0.0646156
$606$	−6.03697	−0.245235
$607$	−21.1238	−0.857389	−0.428694	−	0.903450i	$-0.641026\pi$
$607$	−21.1238	−0.857389	−0.428694	+	0.903450i	$0.641026\pi$
$608$	−3.95974	−0.160589
$609$	18.6598	0.756134
$610$	7.05523	0.285658
$611$	−8.49446	−0.343649
$612$	4.00983	0.162088
$613$	2.99053	0.120786	0.0603931	−	0.998175i	$-0.480765\pi$
$613$	2.99053	0.120786	0.0603931	+	0.998175i	$0.480765\pi$
$614$	−4.14211	−0.167162
$615$	15.2557	0.615169
$616$	−9.30090	−0.374744
$617$	−10.4767	−0.421775	−0.210888	−	0.977510i	$-0.567635\pi$
$617$	−10.4767	−0.421775	−0.210888	+	0.977510i	$0.567635\pi$
$618$	−4.56185	−0.183505
$619$	−9.34147	−0.375465	−0.187733	−	0.982220i	$-0.560114\pi$
$619$	−9.34147	−0.375465	−0.187733	+	0.982220i	$0.560114\pi$
$620$	−18.4957	−0.742804
$621$	36.7358	1.47416
$622$	−4.24107	−0.170051
$623$	−23.5345	−0.942892
$624$	−22.8232	−0.913657
$625$	−18.2414	−0.729655
$626$	−4.07010	−0.162674
$627$	5.83568	0.233055
$628$	−37.4751	−1.49542
$629$	−19.4448	−0.775316
$630$	−1.49618	−0.0596094
$631$	−21.8463	−0.869688	−0.434844	−	0.900506i	$-0.643196\pi$
$631$	−21.8463	−0.869688	−0.434844	+	0.900506i	$0.643196\pi$
$632$	−15.0245	−0.597644
$633$	13.7648	0.547102
$634$	5.22090	0.207348
$635$	6.18902	0.245604
$636$	5.40711	0.214406
$637$	−4.88355	−0.193493
$638$	−5.38700	−0.213273
$639$	5.04305	0.199500
$640$	−16.1170	−0.637079
$641$	39.9747	1.57891	0.789453	−	0.613811i	$-0.210364\pi$
$641$	39.9747	1.57891	0.789453	+	0.613811i	$0.210364\pi$
$642$	−0.0338373	−0.00133545
$643$	−15.1256	−0.596495	−0.298247	−	0.954489i	$-0.596402\pi$
$643$	−15.1256	−0.596495	−0.298247	+	0.954489i	$0.596402\pi$
$644$	−30.4658	−1.20052
$645$	7.75757	0.305454
$646$	−0.662596	−0.0260695
$647$	43.0209	1.69133	0.845664	−	0.533716i	$-0.179205\pi$
$647$	43.0209	1.69133	0.845664	+	0.533716i	$0.179205\pi$
$648$	4.98995	0.196024
$649$	−3.51459	−0.137960
$650$	−1.46928	−0.0576301
$651$	−16.4368	−0.644207
$652$	−11.2437	−0.440336
$653$	33.2486	1.30112	0.650559	−	0.759455i	$-0.274534\pi$
$653$	33.2486	1.30112	0.650559	+	0.759455i	$0.274534\pi$
$654$	−1.24262	−0.0485901
$655$	−37.7005	−1.47308
$656$	−19.7695	−0.771870
$657$	−5.68645	−0.221850
$658$	−1.24990	−0.0487261
$659$	42.2827	1.64710	0.823549	−	0.567245i	$-0.191991\pi$
$659$	42.2827	1.64710	0.823549	+	0.567245i	$0.191991\pi$
$660$	17.9512	0.698748
$661$	−16.5817	−0.644955	−0.322477	−	0.946577i	$-0.604516\pi$
$661$	−16.5817	−0.644955	−0.322477	+	0.946577i	$0.604516\pi$
$662$	8.24400	0.320412
$663$	−12.3100	−0.478080
$664$	15.4805	0.600759
$665$	−5.93213	−0.230038
$666$	3.17564	0.123054
$667$	−36.0264	−1.39495
$668$	30.9588	1.19783
$669$	27.0425	1.04552
$670$	0.922208	0.0356280
$671$	43.4310	1.67663
$672$	−10.8246	−0.417569
$673$	12.5066	0.482096	0.241048	−	0.970513i	$-0.422509\pi$
$673$	12.5066	0.482096	0.241048	+	0.970513i	$0.422509\pi$
$674$	−3.77361	−0.145354
$675$	−6.25497	−0.240754
$676$	−17.3283	−0.666474
$677$	41.3779	1.59028	0.795141	−	0.606424i	$-0.207397\pi$
$677$	41.3779	1.59028	0.795141	+	0.606424i	$0.207397\pi$
$678$	0.842129	0.0323418
$679$	−39.6664	−1.52226
$680$	−4.16137	−0.159581
$681$	15.7915	0.605130
$682$	4.74521	0.181703
$683$	−12.2490	−0.468694	−0.234347	−	0.972153i	$-0.575295\pi$
$683$	−12.2490	−0.468694	−0.234347	+	0.972153i	$0.575295\pi$
$684$	−2.59645	−0.0992776
$685$	−15.3333	−0.585856
$686$	−5.55247	−0.211994
$687$	−31.2275	−1.19140
$688$	−10.0529	−0.383262
$689$	9.58367	0.365109
$690$	−5.00338	−0.190476
$691$	−29.4843	−1.12164	−0.560819	−	0.827939i	$-0.689514\pi$
$691$	−29.4843	−1.12164	−0.560819	+	0.827939i	$0.689514\pi$
$692$	25.9643	0.987013
$693$	−9.21030	−0.349871
$694$	4.38010	0.166266
$695$	14.8159	0.562000
$696$	−8.47591	−0.321278
$697$	−10.6630	−0.403888
$698$	1.40243	0.0530827
$699$	21.9111	0.828755
$700$	5.18737	0.196064
$701$	−16.4542	−0.621465	−0.310733	−	0.950497i	$-0.600574\pi$
$701$	−16.4542	−0.621465	−0.310733	+	0.950497i	$0.600574\pi$
$702$	7.50298	0.283182
$703$	12.5909	0.474875
$704$	−21.1070	−0.795500
$705$	4.92526	0.185496
$706$	−5.57892	−0.209966
$707$	37.7770	1.42075
$708$	−2.70849	−0.101791
$709$	−46.2347	−1.73638	−0.868190	−	0.496232i	$-0.834717\pi$
$709$	−46.2347	−1.73638	−0.868190	+	0.496232i	$0.834717\pi$
$710$	−2.56340	−0.0962027
$711$	−14.8782	−0.557976
$712$	10.6902	0.400631
$713$	31.7344	1.18846
$714$	−1.81132	−0.0677870
$715$	31.8170	1.18989
$716$	−42.5683	−1.59085
$717$	−10.1487	−0.379012
$718$	2.83277	0.105718
$719$	−23.8436	−0.889218	−0.444609	−	0.895725i	$-0.646657\pi$
$719$	−23.8436	−0.889218	−0.444609	+	0.895725i	$0.646657\pi$
$720$	−7.64019	−0.284733
$721$	28.5462	1.06312
$722$	−4.94561	−0.184057
$723$	18.0885	0.672717
$724$	−6.93338	−0.257677
$725$	6.13418	0.227818
$726$	−0.314234	−0.0116623
$727$	38.1659	1.41549	0.707747	−	0.706466i	$-0.249712\pi$
$727$	38.1659	1.41549	0.707747	+	0.706466i	$0.249712\pi$
$728$	−12.7041	−0.470845
$729$	28.3241	1.04904
$730$	2.89045	0.106980
$731$	−5.42215	−0.200546
$732$	33.4697	1.23708
$733$	−50.9077	−1.88032	−0.940159	−	0.340737i	$-0.889323\pi$
$733$	−50.9077	−1.88032	−0.940159	+	0.340737i	$0.889323\pi$
$734$	−0.362885	−0.0133943
$735$	2.83158	0.104444
$736$	20.8991	0.770351
$737$	5.67698	0.209114
$738$	1.74143	0.0641028
$739$	48.7104	1.79184	0.895920	−	0.444216i	$-0.146518\pi$
$739$	48.7104	1.79184	0.895920	+	0.444216i	$0.146518\pi$
$740$	38.7309	1.42378
$741$	7.97095	0.292820
$742$	1.41017	0.0517688
$743$	0.675629	0.0247864	0.0123932	−	0.999923i	$-0.496055\pi$
$743$	0.675629	0.0247864	0.0123932	+	0.999923i	$0.496055\pi$
$744$	7.46612	0.273721
$745$	−42.5306	−1.55820
$746$	9.46428	0.346512
$747$	15.3297	0.560885
$748$	−12.5470	−0.458762
$749$	0.211741	0.00773683
$750$	4.70068	0.171644
$751$	20.4428	0.745967	0.372983	−	0.927838i	$-0.378335\pi$
$751$	20.4428	0.745967	0.372983	+	0.927838i	$0.378335\pi$
$752$	−6.38253	−0.232747
$753$	0.377919	0.0137721
$754$	−7.35810	−0.267966
$755$	−16.1180	−0.586594
$756$	−26.4896	−0.963418
$757$	−21.5003	−0.781441	−0.390720	−	0.920509i	$-0.627774\pi$
$757$	−21.5003	−0.781441	−0.390720	+	0.920509i	$0.627774\pi$
$758$	9.12815	0.331549
$759$	−30.8001	−1.11797
$760$	2.69457	0.0977423
$761$	29.7165	1.07722	0.538612	−	0.842554i	$-0.318949\pi$
$761$	29.7165	1.07722	0.538612	+	0.842554i	$0.318949\pi$
$762$	−1.22366	−0.0443285
$763$	7.77579	0.281503
$764$	14.2392	0.515156
$765$	−4.12084	−0.148989
$766$	−3.91988	−0.141631
$767$	−4.80057	−0.173339
$768$	−13.7573	−0.496423
$769$	1.14904	0.0414355	0.0207178	−	0.999785i	$-0.493405\pi$
$769$	1.14904	0.0414355	0.0207178	+	0.999785i	$0.493405\pi$
$770$	4.68163	0.168714
$771$	7.80266	0.281006
$772$	44.1786	1.59002
$773$	3.89004	0.139915	0.0699576	−	0.997550i	$-0.477714\pi$
$773$	3.89004	0.139915	0.0699576	+	0.997550i	$0.477714\pi$
$774$	0.885522	0.0318294
$775$	−5.40337	−0.194095
$776$	18.0178	0.646802
$777$	34.4195	1.23479
$778$	8.29268	0.297307
$779$	6.90448	0.247379
$780$	24.5195	0.877938
$781$	−15.7799	−0.564650
$782$	3.49711	0.125056
$783$	−31.3245	−1.11945
$784$	−3.66938	−0.131049
$785$	38.5126	1.37457
$786$	7.45392	0.265872
$787$	19.9603	0.711508	0.355754	−	0.934580i	$-0.384224\pi$
$787$	19.9603	0.711508	0.355754	+	0.934580i	$0.384224\pi$
$788$	−1.55558	−0.0554151
$789$	−25.6174	−0.912003
$790$	7.56264	0.269067
$791$	−5.26970	−0.187369
$792$	4.18363	0.148659
$793$	59.3223	2.10660
$794$	−1.22548	−0.0434907
$795$	−5.55680	−0.197079
$796$	−30.8494	−1.09343
$797$	−42.9686	−1.52203	−0.761014	−	0.648736i	$-0.775298\pi$
$797$	−42.9686	−1.52203	−0.761014	+	0.648736i	$0.775298\pi$
$798$	1.17287	0.0415190
$799$	−3.44251	−0.121787
$800$	−3.55846	−0.125811
$801$	10.5860	0.374040
$802$	3.38626	0.119573
$803$	17.7932	0.627908
$804$	4.37492	0.154291
$805$	31.3092	1.10350
$806$	6.48148	0.228300
$807$	−1.37910	−0.0485468
$808$	−17.1595	−0.603671
$809$	28.0766	0.987119	0.493560	−	0.869712i	$-0.335695\pi$
$809$	28.0766	0.987119	0.493560	+	0.869712i	$0.335695\pi$
$810$	−2.51171	−0.0882524
$811$	−1.23855	−0.0434915	−0.0217458	−	0.999764i	$-0.506922\pi$
$811$	−1.23855	−0.0434915	−0.0217458	+	0.999764i	$0.506922\pi$
$812$	25.9781	0.911652
$813$	7.16642	0.251337
$814$	−9.93673	−0.348282
$815$	11.5549	0.404751
$816$	−9.24942	−0.323794
$817$	3.51095	0.122833
$818$	2.99762	0.104809
$819$	−12.5803	−0.439593
$820$	21.2389	0.741694
$821$	9.91227	0.345941	0.172970	−	0.984927i	$-0.444664\pi$
$821$	9.91227	0.345941	0.172970	+	0.984927i	$0.444664\pi$
$822$	3.03161	0.105740
$823$	4.82339	0.168133	0.0840663	−	0.996460i	$-0.473209\pi$
$823$	4.82339	0.168133	0.0840663	+	0.996460i	$0.473209\pi$
$824$	−12.9667	−0.451715
$825$	5.24430	0.182583
$826$	−0.706369	−0.0245777
$827$	24.6328	0.856566	0.428283	−	0.903645i	$-0.359119\pi$
$827$	24.6328	0.856566	0.428283	+	0.903645i	$0.359119\pi$
$828$	13.7038	0.476239
$829$	14.6300	0.508122	0.254061	−	0.967188i	$-0.418234\pi$
$829$	14.6300	0.508122	0.254061	+	0.967188i	$0.418234\pi$
$830$	−7.79215	−0.270469
$831$	−8.45844	−0.293420
$832$	−28.8300	−0.999502
$833$	−1.97913	−0.0685728
$834$	−2.92932	−0.101434
$835$	−31.8158	−1.10103
$836$	8.12440	0.280988
$837$	27.5926	0.953741
$838$	0.0906298	0.00313076
$839$	−42.5008	−1.46729	−0.733645	−	0.679533i	$-0.762182\pi$
$839$	−42.5008	−1.46729	−0.733645	+	0.679533i	$0.762182\pi$
$840$	7.36608	0.254154
$841$	1.71964	0.0592979
$842$	6.80320	0.234454
$843$	−26.8639	−0.925243
$844$	19.1633	0.659627
$845$	17.8080	0.612615
$846$	0.562215	0.0193293
$847$	1.96635	0.0675646
$848$	7.20094	0.247281
$849$	−1.97915	−0.0679242
$850$	−0.595449	−0.0204237
$851$	−66.4535	−2.27800
$852$	−12.1607	−0.416617
$853$	27.8087	0.952153	0.476077	−	0.879404i	$-0.342058\pi$
$853$	27.8087	0.952153	0.476077	+	0.879404i	$0.342058\pi$
$854$	8.72884	0.298695
$855$	2.66832	0.0912548
$856$	−0.0961796	−0.00328735
$857$	−25.9123	−0.885149	−0.442574	−	0.896732i	$-0.645935\pi$
$857$	−25.9123	−0.885149	−0.442574	+	0.896732i	$0.645935\pi$
$858$	−6.29067	−0.214760
$859$	53.1905	1.81484	0.907418	−	0.420229i	$-0.138050\pi$
$859$	53.1905	1.81484	0.907418	+	0.420229i	$0.138050\pi$
$860$	10.8000	0.368278
$861$	18.8746	0.643244
$862$	−6.43160	−0.219061
$863$	−27.3604	−0.931360	−0.465680	−	0.884953i	$-0.654190\pi$
$863$	−27.3604	−0.931360	−0.465680	+	0.884953i	$0.654190\pi$
$864$	18.1715	0.618207
$865$	−26.6830	−0.907251
$866$	−7.58934	−0.257896
$867$	18.4560	0.626798
$868$	−22.8831	−0.776704
$869$	46.5545	1.57925
$870$	4.26637	0.144643
$871$	7.75419	0.262741
$872$	−3.53202	−0.119609
$873$	17.8423	0.603871
$874$	−2.26445	−0.0765961
$875$	−29.4150	−0.994407
$876$	13.7122	0.463291
$877$	−58.1170	−1.96247	−0.981235	−	0.192813i	$-0.938239\pi$
$877$	−58.1170	−1.96247	−0.981235	+	0.192813i	$0.938239\pi$
$878$	7.44199	0.251155
$879$	41.1625	1.38838
$880$	23.9065	0.805888
$881$	−46.7912	−1.57644	−0.788218	−	0.615396i	$-0.788996\pi$
$881$	−46.7912	−1.57644	−0.788218	+	0.615396i	$0.788996\pi$
$882$	0.323223	0.0108835
$883$	25.8997	0.871593	0.435797	−	0.900045i	$-0.356467\pi$
$883$	25.8997	0.871593	0.435797	+	0.900045i	$0.356467\pi$
$884$	−17.1379	−0.576409
$885$	2.78347	0.0935653
$886$	1.78694	0.0600334
$887$	47.8720	1.60738	0.803692	−	0.595046i	$-0.202866\pi$
$887$	47.8720	1.60738	0.803692	+	0.595046i	$0.202866\pi$
$888$	−15.6345	−0.524658
$889$	7.65716	0.256813
$890$	−5.38092	−0.180369
$891$	−15.4617	−0.517987
$892$	37.6484	1.26056
$893$	2.22909	0.0745937
$894$	8.40890	0.281236
$895$	43.7468	1.46229
$896$	−19.9402	−0.666155
$897$	−42.0699	−1.40467
$898$	6.55915	0.218882
$899$	−27.0598	−0.902495
$900$	−2.33333	−0.0777775
$901$	3.88392	0.129392
$902$	−5.44900	−0.181432
$903$	9.59780	0.319395
$904$	2.39367	0.0796124
$905$	7.12532	0.236854
$906$	3.18675	0.105873
$907$	−55.0189	−1.82687	−0.913437	−	0.406980i	$-0.866582\pi$
$907$	−55.0189	−1.82687	−0.913437	+	0.406980i	$0.866582\pi$
$908$	21.9848	0.729591
$909$	−16.9924	−0.563603
$910$	6.39463	0.211980
$911$	−46.9311	−1.55490	−0.777448	−	0.628947i	$-0.783486\pi$
$911$	−46.9311	−1.55490	−0.777448	+	0.628947i	$0.783486\pi$
$912$	5.98918	0.198322
$913$	−47.9674	−1.58749
$914$	8.16547	0.270090
$915$	−34.3963	−1.13711
$916$	−43.4747	−1.43645
$917$	−46.6436	−1.54031
$918$	3.04069	0.100358
$919$	−10.0327	−0.330948	−0.165474	−	0.986214i	$-0.552915\pi$
$919$	−10.0327	−0.330948	−0.165474	+	0.986214i	$0.552915\pi$
$920$	−14.2217	−0.468874
$921$	20.1940	0.665415
$922$	−2.66146	−0.0876506
$923$	−21.5538	−0.709452
$924$	22.2095	0.730638
$925$	11.3150	0.372034
$926$	10.8907	0.357890
$927$	−12.8404	−0.421733
$928$	−17.8206	−0.584990
$929$	−24.5291	−0.804773	−0.402387	−	0.915470i	$-0.631819\pi$
$929$	−24.5291	−0.804773	−0.402387	+	0.915470i	$0.631819\pi$
$930$	−3.75809	−0.123233
$931$	1.28153	0.0420003
$932$	30.5045	0.999209
$933$	20.6764	0.676916
$934$	8.28540	0.271107
$935$	12.8943	0.421689
$936$	5.71441	0.186781
$937$	8.89765	0.290673	0.145337	−	0.989382i	$-0.453573\pi$
$937$	8.89765	0.290673	0.145337	+	0.989382i	$0.453573\pi$
$938$	1.14097	0.0372540
$939$	19.8429	0.647549
$940$	6.85691	0.223648
$941$	−36.1992	−1.18006	−0.590030	−	0.807381i	$-0.700884\pi$
$941$	−36.1992	−1.18006	−0.590030	+	0.807381i	$0.700884\pi$
$942$	−7.61448	−0.248093
$943$	−36.4411	−1.18669
$944$	−3.60704	−0.117399
$945$	27.2229	0.885562
$946$	−2.77084	−0.0900877
$947$	15.3584	0.499081	0.249541	−	0.968364i	$-0.419720\pi$
$947$	15.3584	0.499081	0.249541	+	0.968364i	$0.419720\pi$
$948$	35.8768	1.16523
$949$	24.3037	0.788931
$950$	0.385565	0.0125094
$951$	−25.4534	−0.825383
$952$	−5.14852	−0.166864
$953$	−5.78646	−0.187442	−0.0937209	−	0.995599i	$-0.529876\pi$
$953$	−5.78646	−0.187442	−0.0937209	+	0.995599i	$0.529876\pi$
$954$	−0.634305	−0.0205364
$955$	−14.6334	−0.473525
$956$	−14.1290	−0.456965
$957$	26.2632	0.848968
$958$	−4.02418	−0.130015
$959$	−18.9706	−0.612593
$960$	16.7162	0.539514
$961$	−7.16401	−0.231097
$962$	−13.5726	−0.437597
$963$	−0.0952428	−0.00306916
$964$	25.1826	0.811079
$965$	−45.4016	−1.46153
$966$	−6.19027	−0.199169
$967$	−27.8199	−0.894629	−0.447315	−	0.894377i	$-0.647620\pi$
$967$	−27.8199	−0.894629	−0.447315	+	0.894377i	$0.647620\pi$
$968$	−0.893181	−0.0287079
$969$	3.23035	0.103774
$970$	−9.06931	−0.291198
$971$	−26.2776	−0.843290	−0.421645	−	0.906761i	$-0.638547\pi$
$971$	−26.2776	−0.843290	−0.421645	+	0.906761i	$0.638547\pi$
$972$	20.6379	0.661959
$973$	18.3305	0.587649
$974$	−9.85630	−0.315816
$975$	7.16319	0.229406
$976$	44.5734	1.42676
$977$	−38.8447	−1.24275	−0.621377	−	0.783512i	$-0.713426\pi$
$977$	−38.8447	−1.24275	−0.621377	+	0.783512i	$0.713426\pi$
$978$	−2.28457	−0.0730525
$979$	−33.1242	−1.05865
$980$	3.94210	0.125926
$981$	−3.49762	−0.111670
$982$	−8.36684	−0.266996
$983$	15.9632	0.509147	0.254574	−	0.967053i	$-0.418065\pi$
$983$	15.9632	0.509147	0.254574	+	0.967053i	$0.418065\pi$
$984$	−8.57347	−0.273312
$985$	1.59864	0.0509369
$986$	−2.98198	−0.0949655
$987$	6.09361	0.193962
$988$	11.0971	0.353046
$989$	−18.5304	−0.589234
$990$	−2.10584	−0.0669280
$991$	14.6111	0.464138	0.232069	−	0.972699i	$-0.425450\pi$
$991$	14.6111	0.464138	0.232069	+	0.972699i	$0.425450\pi$
$992$	15.6975	0.498396
$993$	−40.1919	−1.27545
$994$	−3.17148	−0.100593
$995$	31.7035	1.00507
$996$	−36.9656	−1.17130
$997$	44.1438	1.39805	0.699024	−	0.715098i	$-0.253618\pi$
$997$	44.1438	1.39805	0.699024	+	0.715098i	$0.253618\pi$
$998$	−6.32992	−0.200370
$999$	−57.7805	−1.82809

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	269.2.a.c.1.9	✓	16
3.2	odd	2		2421.2.a.i.1.8		16
4.3	odd	2		4304.2.a.l.1.12		16
5.4	even	2		6725.2.a.i.1.8		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
269.2.a.c.1.9	✓	16	1.1	even	1	trivial
2421.2.a.i.1.8		16	3.2	odd	2
4304.2.a.l.1.12		16	4.3	odd	2
6725.2.a.i.1.8		16	5.4	even	2

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:	\( N \)	\(=\)	\( 269 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	269.a (trivial)

\(f(q)\)	\(=\)	\(q+0.282877 q^{2} -1.37910 q^{3} -1.91998 q^{4} +1.97313 q^{5} -0.390116 q^{6} +2.44119 q^{7} -1.10887 q^{8} -1.09807 q^{9} +0.558153 q^{10} +3.43591 q^{11} +2.64785 q^{12} +4.69311 q^{13} +0.690556 q^{14} -2.72116 q^{15} +3.52629 q^{16} +1.90195 q^{17} -0.310618 q^{18} -1.23155 q^{19} -3.78838 q^{20} -3.36666 q^{21} +0.971939 q^{22} +6.50000 q^{23} +1.52925 q^{24} -1.10675 q^{25} +1.32757 q^{26} +5.65167 q^{27} -4.68704 q^{28} -5.54253 q^{29} -0.769751 q^{30} +4.88221 q^{31} +3.21525 q^{32} -4.73848 q^{33} +0.538017 q^{34} +4.81680 q^{35} +2.10827 q^{36} -10.2236 q^{37} -0.348377 q^{38} -6.47229 q^{39} -2.18795 q^{40} -5.60633 q^{41} -0.952349 q^{42} -2.85084 q^{43} -6.59688 q^{44} -2.16664 q^{45} +1.83870 q^{46} -1.80999 q^{47} -4.86312 q^{48} -1.04058 q^{49} -0.313073 q^{50} -2.62299 q^{51} -9.01068 q^{52} +2.04207 q^{53} +1.59872 q^{54} +6.77951 q^{55} -2.70697 q^{56} +1.69844 q^{57} -1.56785 q^{58} -1.02290 q^{59} +5.22457 q^{60} +12.6403 q^{61} +1.38106 q^{62} -2.68060 q^{63} -6.14306 q^{64} +9.26012 q^{65} -1.34041 q^{66} +1.65225 q^{67} -3.65171 q^{68} -8.96418 q^{69} +1.36256 q^{70} -4.59265 q^{71} +1.21762 q^{72} +5.17859 q^{73} -2.89202 q^{74} +1.52632 q^{75} +2.36455 q^{76} +8.38772 q^{77} -1.83086 q^{78} +13.5494 q^{79} +6.95784 q^{80} -4.50003 q^{81} -1.58590 q^{82} -13.9606 q^{83} +6.46392 q^{84} +3.75280 q^{85} -0.806435 q^{86} +7.64373 q^{87} -3.80998 q^{88} -9.64059 q^{89} -0.612891 q^{90} +11.4568 q^{91} -12.4799 q^{92} -6.73308 q^{93} -0.512003 q^{94} -2.43001 q^{95} -4.43416 q^{96} -16.2488 q^{97} -0.294355 q^{98} -3.77287 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + q^{2} + 5 q^{3} + 25 q^{4} - q^{5} - 2 q^{6} + 11 q^{7} + 21 q^{9} + 7 q^{10} + 16 q^{11} - 4 q^{12} - q^{13} - 9 q^{14} - 5 q^{15} + 39 q^{16} - 2 q^{17} - 8 q^{18} + 35 q^{19} - 16 q^{20} - 3 q^{22}+ \cdots + 90 q^{99}+O(q^{100}) \) 16 * q + q^2 + 5 * q^3 + 25 * q^4 - q^5 - 2 * q^6 + 11 * q^7 + 21 * q^9 + 7 * q^10 + 16 * q^11 - 4 * q^12 - q^13 - 9 * q^14 - 5 * q^15 + 39 * q^16 - 2 * q^17 - 8 * q^18 + 35 * q^19 - 16 * q^20 - 3 * q^22 + q^23 - 15 * q^24 + 13 * q^25 - 14 * q^26 + 11 * q^27 + 11 * q^28 + 2 * q^29 - 13 * q^30 + 13 * q^31 - 29 * q^32 - 8 * q^33 - 10 * q^34 + 9 * q^35 + 11 * q^36 + 4 * q^37 - 32 * q^38 + 11 * q^39 + 6 * q^40 - 2 * q^41 - 45 * q^42 + 51 * q^43 + 8 * q^44 - 35 * q^45 + 20 * q^46 - 9 * q^47 - 40 * q^48 + 27 * q^49 - 58 * q^50 + 40 * q^51 - 38 * q^52 - 29 * q^53 - 49 * q^54 - 8 * q^55 - 65 * q^56 - 16 * q^57 - 26 * q^58 - 13 * q^59 - 95 * q^60 + 14 * q^61 - 47 * q^62 - 14 * q^63 + 32 * q^64 - 29 * q^65 - 82 * q^66 + 25 * q^67 - 45 * q^68 - 12 * q^69 - 44 * q^70 - 9 * q^71 - 30 * q^72 + 4 * q^73 + 8 * q^74 + 22 * q^75 + 58 * q^76 - 10 * q^77 - 17 * q^78 + 51 * q^79 - 7 * q^80 - 21 * q^82 - 2 * q^83 - 18 * q^84 + 4 * q^85 + 10 * q^86 + q^87 - 18 * q^88 - 35 * q^89 + 16 * q^90 + 65 * q^91 + 69 * q^92 - 36 * q^93 + 24 * q^94 + 18 * q^95 - 28 * q^96 - 14 * q^97 + 24 * q^98 + 90 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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