LMFDB - Embedded newform 4013.2.a.b.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4013,2,Mod(1,4013)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4013, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4013.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4013 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4013.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$32.0439663311$
Analytic rank:	$1$
Dimension:	$157$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Character	$\chi$	$=$	4013.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.78090 q^{2} -3.27207 q^{3} +5.73341 q^{4} +3.80445 q^{5} +9.09929 q^{6} +2.37189 q^{7} -10.3822 q^{8} +7.70642 q^{9} +O(q^{10})$	\(q-2.78090 q^{2} -3.27207 q^{3} +5.73341 q^{4} +3.80445 q^{5} +9.09929 q^{6} +2.37189 q^{7} -10.3822 q^{8} +7.70642 q^{9} -10.5798 q^{10} -4.68102 q^{11} -18.7601 q^{12} -3.85571 q^{13} -6.59599 q^{14} -12.4484 q^{15} +17.4052 q^{16} +3.20172 q^{17} -21.4308 q^{18} -0.257912 q^{19} +21.8125 q^{20} -7.76098 q^{21} +13.0175 q^{22} -0.119984 q^{23} +33.9714 q^{24} +9.47384 q^{25} +10.7223 q^{26} -15.3997 q^{27} +13.5990 q^{28} +2.90338 q^{29} +34.6178 q^{30} -3.59389 q^{31} -27.6376 q^{32} +15.3166 q^{33} -8.90366 q^{34} +9.02373 q^{35} +44.1840 q^{36} -7.11438 q^{37} +0.717228 q^{38} +12.6161 q^{39} -39.4987 q^{40} +10.2829 q^{41} +21.5825 q^{42} -8.48379 q^{43} -26.8382 q^{44} +29.3187 q^{45} +0.333664 q^{46} +10.2517 q^{47} -56.9509 q^{48} -1.37414 q^{49} -26.3458 q^{50} -10.4762 q^{51} -22.1064 q^{52} -5.15252 q^{53} +42.8251 q^{54} -17.8087 q^{55} -24.6255 q^{56} +0.843905 q^{57} -8.07400 q^{58} +10.8839 q^{59} -71.3718 q^{60} -4.84265 q^{61} +9.99425 q^{62} +18.2788 q^{63} +42.0470 q^{64} -14.6688 q^{65} -42.5940 q^{66} -3.85425 q^{67} +18.3568 q^{68} +0.392597 q^{69} -25.0941 q^{70} -6.61352 q^{71} -80.0099 q^{72} -1.72512 q^{73} +19.7844 q^{74} -30.9990 q^{75} -1.47872 q^{76} -11.1029 q^{77} -35.0842 q^{78} -12.1359 q^{79} +66.2171 q^{80} +27.2696 q^{81} -28.5956 q^{82} -11.9047 q^{83} -44.4969 q^{84} +12.1808 q^{85} +23.5926 q^{86} -9.50004 q^{87} +48.5995 q^{88} +14.4576 q^{89} -81.5323 q^{90} -9.14532 q^{91} -0.687919 q^{92} +11.7594 q^{93} -28.5090 q^{94} -0.981213 q^{95} +90.4319 q^{96} -1.89343 q^{97} +3.82134 q^{98} -36.0739 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 157 q - 15 q^{2} - 51 q^{3} + 137 q^{4} - 13 q^{5} - 15 q^{6} - 49 q^{7} - 42 q^{8} + 144 q^{9}+O(q^{10}) $ 157 * q - 15 * q^2 - 51 * q^3 + 137 * q^4 - 13 * q^5 - 15 * q^6 - 49 * q^7 - 42 * q^8 + 144 * q^9	\( 157 q - 15 q^{2} - 51 q^{3} + 137 q^{4} - 13 q^{5} - 15 q^{6} - 49 q^{7} - 42 q^{8} + 144 q^{9} - 61 q^{10} - 27 q^{11} - 93 q^{12} - 97 q^{13} - 12 q^{14} - 36 q^{15} + 105 q^{16} - 45 q^{17} - 68 q^{18} - 128 q^{19} - 30 q^{20} - 26 q^{21} - 68 q^{22} - 41 q^{23} - 40 q^{24} + 102 q^{25} - 5 q^{26} - 189 q^{27} - 115 q^{28} - 26 q^{29} - 12 q^{30} - 88 q^{31} - 89 q^{32} - 52 q^{33} - 61 q^{34} - 87 q^{35} + 110 q^{36} - 62 q^{37} - 37 q^{38} - 20 q^{39} - 161 q^{40} - 34 q^{41} - 53 q^{42} - 254 q^{43} - 19 q^{44} - 46 q^{45} - 52 q^{46} - 76 q^{47} - 162 q^{48} + 96 q^{49} - 54 q^{50} - 76 q^{51} - 259 q^{52} - 48 q^{53} - 12 q^{54} - 194 q^{55} - 10 q^{56} - 30 q^{57} - 52 q^{58} - 64 q^{59} - 31 q^{60} - 107 q^{61} - 51 q^{62} - 106 q^{63} + 54 q^{64} - 17 q^{65} - 13 q^{66} - 193 q^{67} - 118 q^{68} - 55 q^{69} - 86 q^{70} - 11 q^{71} - 172 q^{72} - 173 q^{73} - 11 q^{74} - 209 q^{75} - 213 q^{76} - 84 q^{77} - 30 q^{78} - 111 q^{79} - 6 q^{80} + 157 q^{81} - 117 q^{82} - 154 q^{83} - 6 q^{84} - 91 q^{85} + 28 q^{86} - 165 q^{87} - 165 q^{88} - 32 q^{89} - 103 q^{90} - 200 q^{91} - 86 q^{92} - 39 q^{93} - 118 q^{94} - 22 q^{95} - 28 q^{96} - 151 q^{97} - 38 q^{98} - 91 q^{99}+O(q^{100}) \) 157 * q - 15 * q^2 - 51 * q^3 + 137 * q^4 - 13 * q^5 - 15 * q^6 - 49 * q^7 - 42 * q^8 + 144 * q^9 - 61 * q^10 - 27 * q^11 - 93 * q^12 - 97 * q^13 - 12 * q^14 - 36 * q^15 + 105 * q^16 - 45 * q^17 - 68 * q^18 - 128 * q^19 - 30 * q^20 - 26 * q^21 - 68 * q^22 - 41 * q^23 - 40 * q^24 + 102 * q^25 - 5 * q^26 - 189 * q^27 - 115 * q^28 - 26 * q^29 - 12 * q^30 - 88 * q^31 - 89 * q^32 - 52 * q^33 - 61 * q^34 - 87 * q^35 + 110 * q^36 - 62 * q^37 - 37 * q^38 - 20 * q^39 - 161 * q^40 - 34 * q^41 - 53 * q^42 - 254 * q^43 - 19 * q^44 - 46 * q^45 - 52 * q^46 - 76 * q^47 - 162 * q^48 + 96 * q^49 - 54 * q^50 - 76 * q^51 - 259 * q^52 - 48 * q^53 - 12 * q^54 - 194 * q^55 - 10 * q^56 - 30 * q^57 - 52 * q^58 - 64 * q^59 - 31 * q^60 - 107 * q^61 - 51 * q^62 - 106 * q^63 + 54 * q^64 - 17 * q^65 - 13 * q^66 - 193 * q^67 - 118 * q^68 - 55 * q^69 - 86 * q^70 - 11 * q^71 - 172 * q^72 - 173 * q^73 - 11 * q^74 - 209 * q^75 - 213 * q^76 - 84 * q^77 - 30 * q^78 - 111 * q^79 - 6 * q^80 + 157 * q^81 - 117 * q^82 - 154 * q^83 - 6 * q^84 - 91 * q^85 + 28 * q^86 - 165 * q^87 - 165 * q^88 - 32 * q^89 - 103 * q^90 - 200 * q^91 - 86 * q^92 - 39 * q^93 - 118 * q^94 - 22 * q^95 - 28 * q^96 - 151 * q^97 - 38 * q^98 - 91 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	−2.78090	−1.96639	−0.983197	−	0.182548i	$-0.941565\pi$
$2$	−2.78090	−1.96639	−0.983197	+	0.182548i	$0.941565\pi$
$3$	−3.27207	−1.88913	−0.944564	−	0.328327i	$-0.893515\pi$
$3$	−3.27207	−1.88913	−0.944564	+	0.328327i	$0.893515\pi$
$4$	5.73341	2.86670
$5$	3.80445	1.70140	0.850701	−	0.525650i	$-0.176178\pi$
$5$	3.80445	1.70140	0.850701	+	0.525650i	$0.176178\pi$
$6$	9.09929	3.71477
$7$	2.37189	0.896490	0.448245	−	0.893911i	$-0.352049\pi$
$7$	2.37189	0.896490	0.448245	+	0.893911i	$0.352049\pi$
$8$	−10.3822	−3.67068
$9$	7.70642	2.56881
$10$	−10.5798	−3.34563
$11$	−4.68102	−1.41138	−0.705691	−	0.708520i	$-0.749363\pi$
$11$	−4.68102	−1.41138	−0.705691	+	0.708520i	$0.749363\pi$
$12$	−18.7601	−5.41557
$13$	−3.85571	−1.06938	−0.534691	−	0.845048i	$-0.679572\pi$
$13$	−3.85571	−1.06938	−0.534691	+	0.845048i	$0.679572\pi$
$14$	−6.59599	−1.76285
$15$	−12.4484	−3.21417
$16$	17.4052	4.35129
$17$	3.20172	0.776531	0.388265	−	0.921548i	$-0.373074\pi$
$17$	3.20172	0.776531	0.388265	+	0.921548i	$0.373074\pi$
$18$	−21.4308	−5.05128
$19$	−0.257912	−0.0591691	−0.0295845	−	0.999562i	$-0.509418\pi$
$19$	−0.257912	−0.0591691	−0.0295845	+	0.999562i	$0.509418\pi$
$20$	21.8125	4.87742
$21$	−7.76098	−1.69358
$22$	13.0175	2.77533
$23$	−0.119984	−0.0250185	−0.0125092	−	0.999922i	$-0.503982\pi$
$23$	−0.119984	−0.0250185	−0.0125092	+	0.999922i	$0.503982\pi$
$24$	33.9714	6.93438
$25$	9.47384	1.89477
$26$	10.7223	2.10282
$27$	−15.3997	−2.96367
$28$	13.5990	2.56997
$29$	2.90338	0.539144	0.269572	−	0.962980i	$-0.413118\pi$
$29$	2.90338	0.539144	0.269572	+	0.962980i	$0.413118\pi$
$30$	34.6178	6.32032
$31$	−3.59389	−0.645482	−0.322741	−	0.946487i	$-0.604604\pi$
$31$	−3.59389	−0.645482	−0.322741	+	0.946487i	$0.604604\pi$
$32$	−27.6376	−4.88568
$33$	15.3166	2.66628
$34$	−8.90366	−1.52697
$35$	9.02373	1.52529
$36$	44.1840	7.36401
$37$	−7.11438	−1.16960	−0.584799	−	0.811179i	$-0.698826\pi$
$37$	−7.11438	−1.16960	−0.584799	+	0.811179i	$0.698826\pi$
$38$	0.717228	0.116350
$39$	12.6161	2.02020
$40$	−39.4987	−6.24530
$41$	10.2829	1.60591	0.802956	−	0.596038i	$-0.203259\pi$
$41$	10.2829	1.60591	0.802956	+	0.596038i	$0.203259\pi$
$42$	21.5825	3.33025
$43$	−8.48379	−1.29377	−0.646883	−	0.762589i	$-0.723928\pi$
$43$	−8.48379	−1.29377	−0.646883	+	0.762589i	$0.723928\pi$
$44$	−26.8382	−4.04602
$45$	29.3187	4.37057
$46$	0.333664	0.0491961
$47$	10.2517	1.49536	0.747682	−	0.664057i	$-0.231167\pi$
$47$	10.2517	1.49536	0.747682	+	0.664057i	$0.231167\pi$
$48$	−56.9509	−8.22015
$49$	−1.37414	−0.196306
$50$	−26.3458	−3.72586
$51$	−10.4762	−1.46697
$52$	−22.1064	−3.06560
$53$	−5.15252	−0.707754	−0.353877	−	0.935292i	$-0.615137\pi$
$53$	−5.15252	−0.707754	−0.353877	+	0.935292i	$0.615137\pi$
$54$	42.8251	5.82775
$55$	−17.8087	−2.40133
$56$	−24.6255	−3.29073
$57$	0.843905	0.111778
$58$	−8.07400	−1.06017
$59$	10.8839	1.41696	0.708480	−	0.705731i	$-0.249381\pi$
$59$	10.8839	1.41696	0.708480	+	0.705731i	$0.249381\pi$
$60$	−71.3718	−9.21406
$61$	−4.84265	−0.620037	−0.310019	−	0.950730i	$-0.600335\pi$
$61$	−4.84265	−0.620037	−0.310019	+	0.950730i	$0.600335\pi$
$62$	9.99425	1.26927
$63$	18.2788	2.30291
$64$	42.0470	5.25587
$65$	−14.6688	−1.81945
$66$	−42.5940	−5.24296
$67$	−3.85425	−0.470872	−0.235436	−	0.971890i	$-0.575652\pi$
$67$	−3.85425	−0.470872	−0.235436	+	0.971890i	$0.575652\pi$
$68$	18.3568	2.22608
$69$	0.392597	0.0472631
$70$	−25.0941	−2.99932
$71$	−6.61352	−0.784880	−0.392440	−	0.919778i	$-0.628369\pi$
$71$	−6.61352	−0.784880	−0.392440	+	0.919778i	$0.628369\pi$
$72$	−80.0099	−9.42925
$73$	−1.72512	−0.201911	−0.100955	−	0.994891i	$-0.532190\pi$
$73$	−1.72512	−0.201911	−0.100955	+	0.994891i	$0.532190\pi$
$74$	19.7844	2.29989
$75$	−30.9990	−3.57946
$76$	−1.47872	−0.169620
$77$	−11.1029	−1.26529
$78$	−35.0842	−3.97251
$79$	−12.1359	−1.36540	−0.682699	−	0.730699i	$-0.739194\pi$
$79$	−12.1359	−1.36540	−0.682699	+	0.730699i	$0.739194\pi$
$80$	66.2171	7.40329
$81$	27.2696	3.02996
$82$	−28.5956	−3.15786
$83$	−11.9047	−1.30671	−0.653356	−	0.757051i	$-0.726640\pi$
$83$	−11.9047	−1.30671	−0.653356	+	0.757051i	$0.726640\pi$
$84$	−44.4969	−4.85501
$85$	12.1808	1.32119
$86$	23.5926	2.54405
$87$	−9.50004	−1.01851
$88$	48.5995	5.18073
$89$	14.4576	1.53250	0.766251	−	0.642542i	$-0.222120\pi$
$89$	14.4576	1.53250	0.766251	+	0.642542i	$0.222120\pi$
$90$	−81.5323	−8.59426
$91$	−9.14532	−0.958690
$92$	−0.687919	−0.0717205
$93$	11.7594	1.21940
$94$	−28.5090	−2.94048
$95$	−0.981213	−0.100670
$96$	90.4319	9.22967
$97$	−1.89343	−0.192248	−0.0961241	−	0.995369i	$-0.530645\pi$
$97$	−1.89343	−0.192248	−0.0961241	+	0.995369i	$0.530645\pi$
$98$	3.82134	0.386014
$99$	−36.0739	−3.62557
$100$	54.3174	5.43174
$101$	−7.80505	−0.776632	−0.388316	−	0.921526i	$-0.626943\pi$
$101$	−7.80505	−0.776632	−0.388316	+	0.921526i	$0.626943\pi$
$102$	29.1334	2.88463
$103$	−5.49805	−0.541739	−0.270870	−	0.962616i	$-0.587311\pi$
$103$	−5.49805	−0.541739	−0.270870	+	0.962616i	$0.587311\pi$
$104$	40.0309	3.92535
$105$	−29.5263	−2.88147
$106$	14.3287	1.39172
$107$	3.16675	0.306141	0.153071	−	0.988215i	$-0.451084\pi$
$107$	3.16675	0.306141	0.153071	+	0.988215i	$0.451084\pi$
$108$	−88.2928	−8.49598
$109$	−11.8156	−1.13173	−0.565866	−	0.824497i	$-0.691458\pi$
$109$	−11.8156	−1.13173	−0.565866	+	0.824497i	$0.691458\pi$
$110$	49.5243	4.72196
$111$	23.2787	2.20952
$112$	41.2831	3.90089
$113$	3.26832	0.307458	0.153729	−	0.988113i	$-0.450872\pi$
$113$	3.26832	0.307458	0.153729	+	0.988113i	$0.450872\pi$
$114$	−2.34682	−0.219799
$115$	−0.456474	−0.0425664
$116$	16.6462	1.54557
$117$	−29.7137	−2.74703
$118$	−30.2670	−2.78630
$119$	7.59412	0.696152
$120$	129.242	11.7982
$121$	10.9120	0.991999
$122$	13.4669	1.21924
$123$	−33.6462	−3.03377
$124$	−20.6052	−1.85041
$125$	17.0205	1.52236
$126$	−50.8314	−4.52843
$127$	−4.04447	−0.358889	−0.179444	−	0.983768i	$-0.557430\pi$
$127$	−4.04447	−0.358889	−0.179444	+	0.983768i	$0.557430\pi$
$128$	−61.6534	−5.44944
$129$	27.7595	2.44409
$130$	40.7926	3.57775
$131$	−8.39808	−0.733744	−0.366872	−	0.930271i	$-0.619571\pi$
$131$	−8.39808	−0.733744	−0.366872	+	0.930271i	$0.619571\pi$
$132$	87.8165	7.64344
$133$	−0.611739	−0.0530445
$134$	10.7183	0.925920
$135$	−58.5874	−5.04240
$136$	−33.2410	−2.85039
$137$	−5.70042	−0.487020	−0.243510	−	0.969898i	$-0.578299\pi$
$137$	−5.70042	−0.487020	−0.243510	+	0.969898i	$0.578299\pi$
$138$	−1.09177	−0.0929378
$139$	19.9098	1.68873	0.844364	−	0.535769i	$-0.179978\pi$
$139$	19.9098	1.68873	0.844364	+	0.535769i	$0.179978\pi$
$140$	51.7368	4.37256
$141$	−33.5442	−2.82494
$142$	18.3915	1.54338
$143$	18.0487	1.50931
$144$	134.131	11.1776
$145$	11.0458	0.917300
$146$	4.79740	0.397036
$147$	4.49627	0.370846
$148$	−40.7897	−3.35289
$149$	−18.6982	−1.53182	−0.765910	−	0.642948i	$-0.777711\pi$
$149$	−18.6982	−1.53182	−0.765910	+	0.642948i	$0.777711\pi$
$150$	86.2052	7.03862
$151$	2.20879	0.179748	0.0898742	−	0.995953i	$-0.471353\pi$
$151$	2.20879	0.179748	0.0898742	+	0.995953i	$0.471353\pi$
$152$	2.67770	0.217191
$153$	24.6738	1.99476
$154$	30.8760	2.48806
$155$	−13.6728	−1.09822
$156$	72.3335	5.79131
$157$	15.5360	1.23990	0.619952	−	0.784639i	$-0.287152\pi$
$157$	15.5360	1.23990	0.619952	+	0.784639i	$0.287152\pi$
$158$	33.7488	2.68491
$159$	16.8594	1.33704
$160$	−105.146	−8.31250
$161$	−0.284590	−0.0224288
$162$	−75.8341	−5.95809
$163$	−11.8145	−0.925386	−0.462693	−	0.886519i	$-0.653117\pi$
$163$	−11.8145	−0.925386	−0.462693	+	0.886519i	$0.653117\pi$
$164$	58.9558	4.60367
$165$	58.2713	4.53642
$166$	33.1058	2.56951
$167$	4.74632	0.367282	0.183641	−	0.982993i	$-0.441212\pi$
$167$	4.74632	0.367282	0.183641	+	0.982993i	$0.441212\pi$
$168$	80.5764	6.21660
$169$	1.86649	0.143576
$170$	−33.8735	−2.59798
$171$	−1.98758	−0.151994
$172$	−48.6411	−3.70885
$173$	−10.8413	−0.824250	−0.412125	−	0.911127i	$-0.635213\pi$
$173$	−10.8413	−0.824250	−0.412125	+	0.911127i	$0.635213\pi$
$174$	26.4187	2.00279
$175$	22.4709	1.69864
$176$	−81.4740	−6.14133
$177$	−35.6127	−2.67682
$178$	−40.2051	−3.01350
$179$	16.7175	1.24952	0.624762	−	0.780816i	$-0.285196\pi$
$179$	16.7175	1.24952	0.624762	+	0.780816i	$0.285196\pi$
$180$	168.096	12.5291
$181$	21.6019	1.60565	0.802827	−	0.596212i	$-0.203328\pi$
$181$	21.6019	1.60565	0.802827	+	0.596212i	$0.203328\pi$
$182$	25.4322	1.88516
$183$	15.8455	1.17133
$184$	1.24571	0.0918347
$185$	−27.0663	−1.98995
$186$	−32.7019	−2.39782
$187$	−14.9873	−1.09598
$188$	58.7772	4.28677
$189$	−36.5264	−2.65690
$190$	2.72866	0.197958
$191$	−11.2835	−0.816444	−0.408222	−	0.912883i	$-0.633851\pi$
$191$	−11.2835	−0.816444	−0.408222	+	0.912883i	$0.633851\pi$
$192$	−137.580	−9.92902
$193$	20.4034	1.46867	0.734336	−	0.678787i	$-0.237494\pi$
$193$	20.4034	1.46867	0.734336	+	0.678787i	$0.237494\pi$
$194$	5.26543	0.378036
$195$	47.9974	3.43717
$196$	−7.87850	−0.562750
$197$	−15.0401	−1.07156	−0.535782	−	0.844356i	$-0.679983\pi$
$197$	−15.0401	−1.07156	−0.535782	+	0.844356i	$0.679983\pi$
$198$	100.318	7.12929
$199$	−21.0105	−1.48939	−0.744696	−	0.667404i	$-0.767405\pi$
$199$	−21.0105	−1.48939	−0.744696	+	0.667404i	$0.767405\pi$
$200$	−98.3597	−6.95508
$201$	12.6114	0.889537
$202$	21.7051	1.52716
$203$	6.88649	0.483337
$204$	−60.0646	−4.20536
$205$	39.1206	2.73230
$206$	15.2895	1.06527
$207$	−0.924649	−0.0642675
$208$	−67.1093	−4.65319
$209$	1.20729	0.0835102
$210$	82.1096	5.66610
$211$	−17.5576	−1.20871	−0.604357	−	0.796714i	$-0.706570\pi$
$211$	−17.5576	−1.20871	−0.604357	+	0.796714i	$0.706570\pi$
$212$	−29.5415	−2.02892
$213$	21.6399	1.48274
$214$	−8.80642	−0.601995
$215$	−32.2762	−2.20122
$216$	159.883	10.8787
$217$	−8.52431	−0.578668
$218$	32.8581	2.22543
$219$	5.64472	0.381435
$220$	−102.105	−6.88390
$221$	−12.3449	−0.830408
$222$	−64.7358	−4.34478
$223$	−0.215086	−0.0144032	−0.00720162	−	0.999974i	$-0.502292\pi$
$223$	−0.215086	−0.0144032	−0.00720162	+	0.999974i	$0.502292\pi$
$224$	−65.5532	−4.37996
$225$	73.0093	4.86729
$226$	−9.08888	−0.604583
$227$	−4.58591	−0.304377	−0.152189	−	0.988351i	$-0.548632\pi$
$227$	−4.58591	−0.304377	−0.152189	+	0.988351i	$0.548632\pi$
$228$	4.83845	0.320434
$229$	−21.6947	−1.43363	−0.716814	−	0.697264i	$-0.754400\pi$
$229$	−21.6947	−1.43363	−0.716814	+	0.697264i	$0.754400\pi$
$230$	1.26941	0.0837024
$231$	36.3293	2.39030
$232$	−30.1436	−1.97902
$233$	2.67271	0.175095	0.0875476	−	0.996160i	$-0.472097\pi$
$233$	2.67271	0.175095	0.0875476	+	0.996160i	$0.472097\pi$
$234$	82.6308	5.40175
$235$	39.0021	2.54422
$236$	62.4017	4.06200
$237$	39.7096	2.57941
$238$	−21.1185	−1.36891
$239$	1.73128	0.111987	0.0559934	−	0.998431i	$-0.482167\pi$
$239$	1.73128	0.111987	0.0559934	+	0.998431i	$0.482167\pi$
$240$	−216.667	−13.9858
$241$	6.67211	0.429789	0.214894	−	0.976637i	$-0.431059\pi$
$241$	6.67211	0.429789	0.214894	+	0.976637i	$0.431059\pi$
$242$	−30.3452	−1.95066
$243$	−43.0288	−2.76030
$244$	−27.7649	−1.77746
$245$	−5.22784	−0.333994
$246$	93.5667	5.96559
$247$	0.994433	0.0632743
$248$	37.3126	2.36936
$249$	38.9530	2.46855
$250$	−47.3323	−2.99356
$251$	27.8718	1.75925	0.879627	−	0.475663i	$-0.157792\pi$
$251$	27.8718	1.75925	0.879627	+	0.475663i	$0.157792\pi$
$252$	104.800	6.60176
$253$	0.561649	0.0353106
$254$	11.2473	0.705716
$255$	−39.8563	−2.49590
$256$	87.3579	5.45987
$257$	−25.6734	−1.60146	−0.800732	−	0.599023i	$-0.795556\pi$
$257$	−25.6734	−1.60146	−0.800732	+	0.599023i	$0.795556\pi$
$258$	−77.1965	−4.80604
$259$	−16.8745	−1.04853
$260$	−84.1025	−5.21582
$261$	22.3746	1.38495
$262$	23.3542	1.44283
$263$	0.841132	0.0518664	0.0259332	−	0.999664i	$-0.491744\pi$
$263$	0.841132	0.0518664	0.0259332	+	0.999664i	$0.491744\pi$
$264$	−159.021	−9.78706
$265$	−19.6025	−1.20417
$266$	1.70118	0.104306
$267$	−47.3062	−2.89509
$268$	−22.0980	−1.34985
$269$	−27.0225	−1.64759	−0.823796	−	0.566886i	$-0.808148\pi$
$269$	−27.0225	−1.64759	−0.823796	+	0.566886i	$0.808148\pi$
$270$	162.926	9.91535
$271$	−28.4964	−1.73103	−0.865517	−	0.500879i	$-0.833010\pi$
$271$	−28.4964	−1.73103	−0.865517	+	0.500879i	$0.833010\pi$
$272$	55.7265	3.37891
$273$	29.9241	1.81109
$274$	15.8523	0.957673
$275$	−44.3473	−2.67424
$276$	2.25092	0.135489
$277$	−12.1308	−0.728870	−0.364435	−	0.931229i	$-0.618738\pi$
$277$	−12.1308	−0.728870	−0.364435	+	0.931229i	$0.618738\pi$
$278$	−55.3672	−3.32071
$279$	−27.6960	−1.65812
$280$	−93.6866	−5.59885
$281$	−5.70855	−0.340543	−0.170272	−	0.985397i	$-0.554465\pi$
$281$	−5.70855	−0.340543	−0.170272	+	0.985397i	$0.554465\pi$
$282$	93.2832	5.55494
$283$	10.4513	0.621266	0.310633	−	0.950530i	$-0.399459\pi$
$283$	10.4513	0.621266	0.310633	+	0.950530i	$0.399459\pi$
$284$	−37.9180	−2.25002
$285$	3.21059	0.190179
$286$	−50.1916	−2.96789
$287$	24.3898	1.43968
$288$	−212.987	−12.5504
$289$	−6.74900	−0.397000
$290$	−30.7171	−1.80377
$291$	6.19541	0.363182
$292$	−9.89084	−0.578818
$293$	12.1992	0.712682	0.356341	−	0.934356i	$-0.384024\pi$
$293$	12.1992	0.712682	0.356341	+	0.934356i	$0.384024\pi$
$294$	−12.5037	−0.729230
$295$	41.4071	2.41082
$296$	73.8632	4.29321
$297$	72.0864	4.18288
$298$	51.9979	3.01216
$299$	0.462625	0.0267543
$300$	−177.730	−10.2612
$301$	−20.1226	−1.15985
$302$	−6.14242	−0.353456
$303$	25.5387	1.46716
$304$	−4.48900	−0.257462
$305$	−18.4236	−1.05493
$306$	−68.6153	−3.92248
$307$	−11.6847	−0.666882	−0.333441	−	0.942771i	$-0.608210\pi$
$307$	−11.6847	−0.666882	−0.333441	+	0.942771i	$0.608210\pi$
$308$	−63.6573	−3.62721
$309$	17.9900	1.02342
$310$	38.0226	2.15954
$311$	16.4233	0.931281	0.465641	−	0.884974i	$-0.345824\pi$
$311$	16.4233	0.931281	0.465641	+	0.884974i	$0.345824\pi$
$312$	−130.984	−7.41549
$313$	26.2163	1.48183	0.740917	−	0.671597i	$-0.234391\pi$
$313$	26.2163	1.48183	0.740917	+	0.671597i	$0.234391\pi$
$314$	−43.2040	−2.43814
$315$	69.5407	3.91817
$316$	−69.5803	−3.91420
$317$	8.16811	0.458767	0.229383	−	0.973336i	$-0.426329\pi$
$317$	8.16811	0.458767	0.229383	+	0.973336i	$0.426329\pi$
$318$	−46.8843	−2.62914
$319$	−13.5908	−0.760938
$320$	159.966	8.94235
$321$	−10.3618	−0.578340
$322$	0.791415	0.0441038
$323$	−0.825762	−0.0459466
$324$	156.348	8.68599
$325$	−36.5284	−2.02623
$326$	32.8550	1.81967
$327$	38.6615	2.13799
$328$	−106.759	−5.89478
$329$	24.3159	1.34058
$330$	−162.047	−8.92038
$331$	7.86922	0.432531	0.216266	−	0.976335i	$-0.430612\pi$
$331$	7.86922	0.432531	0.216266	+	0.976335i	$0.430612\pi$
$332$	−68.2546	−3.74596
$333$	−54.8264	−3.00447
$334$	−13.1991	−0.722220
$335$	−14.6633	−0.801142
$336$	−135.081	−7.36928
$337$	−9.19381	−0.500819	−0.250409	−	0.968140i	$-0.580565\pi$
$337$	−9.19381	−0.500819	−0.250409	+	0.968140i	$0.580565\pi$
$338$	−5.19052	−0.282327
$339$	−10.6942	−0.580827
$340$	69.8374	3.78746
$341$	16.8231	0.911022
$342$	5.52725	0.298880
$343$	−19.8625	−1.07248
$344$	88.0808	4.74900
$345$	1.49361	0.0804135
$346$	30.1486	1.62080
$347$	−30.9554	−1.66177	−0.830887	−	0.556441i	$-0.812166\pi$
$347$	−30.9554	−1.66177	−0.830887	+	0.556441i	$0.812166\pi$
$348$	−54.4676	−2.91977
$349$	32.0111	1.71352	0.856759	−	0.515718i	$-0.172475\pi$
$349$	32.0111	1.71352	0.856759	+	0.515718i	$0.172475\pi$
$350$	−62.4893	−3.34020
$351$	59.3768	3.16930
$352$	129.372	6.89556
$353$	10.4110	0.554120	0.277060	−	0.960853i	$-0.410640\pi$
$353$	10.4110	0.554120	0.277060	+	0.960853i	$0.410640\pi$
$354$	99.0355	5.26368
$355$	−25.1608	−1.33540
$356$	82.8913	4.39323
$357$	−24.8485	−1.31512
$358$	−46.4897	−2.45705
$359$	−21.6949	−1.14501	−0.572506	−	0.819901i	$-0.694029\pi$
$359$	−21.6949	−1.14501	−0.572506	+	0.819901i	$0.694029\pi$
$360$	−304.394	−16.0429
$361$	−18.9335	−0.996499
$362$	−60.0726	−3.15735
$363$	−35.7048	−1.87401
$364$	−52.4338	−2.74828
$365$	−6.56315	−0.343531
$366$	−44.0646	−2.30330
$367$	25.3595	1.32376	0.661879	−	0.749611i	$-0.269759\pi$
$367$	25.3595	1.32376	0.661879	+	0.749611i	$0.269759\pi$
$368$	−2.08835	−0.108863
$369$	79.2439	4.12527
$370$	75.2687	3.91303
$371$	−12.2212	−0.634494
$372$	67.4217	3.49565
$373$	13.5487	0.701524	0.350762	−	0.936465i	$-0.385923\pi$
$373$	13.5487	0.701524	0.350762	+	0.936465i	$0.385923\pi$
$374$	41.6783	2.15513
$375$	−55.6921	−2.87593
$376$	−106.436	−5.48900
$377$	−11.1946	−0.576550
$378$	101.576	5.22452
$379$	7.16923	0.368259	0.184129	−	0.982902i	$-0.441053\pi$
$379$	7.16923	0.368259	0.184129	+	0.982902i	$0.441053\pi$
$380$	−5.62570	−0.288592
$381$	13.2338	0.677987
$382$	31.3783	1.60545
$383$	20.1968	1.03201	0.516005	−	0.856585i	$-0.327418\pi$
$383$	20.1968	1.03201	0.516005	+	0.856585i	$0.327418\pi$
$384$	201.734	10.2947
$385$	−42.2403	−2.15277
$386$	−56.7399	−2.88799
$387$	−65.3796	−3.32343
$388$	−10.8558	−0.551119
$389$	10.1438	0.514310	0.257155	−	0.966370i	$-0.417215\pi$
$389$	10.1438	0.514310	0.257155	+	0.966370i	$0.417215\pi$
$390$	−133.476	−6.75883
$391$	−0.384156	−0.0194276
$392$	14.2666	0.720574
$393$	27.4791	1.38614
$394$	41.8251	2.10712
$395$	−46.1705	−2.32309
$396$	−206.827	−10.3934
$397$	−3.53064	−0.177198	−0.0885988	−	0.996067i	$-0.528239\pi$
$397$	−3.53064	−0.177198	−0.0885988	+	0.996067i	$0.528239\pi$
$398$	58.4280	2.92873
$399$	2.00165	0.100208
$400$	164.894	8.24468
$401$	13.0477	0.651569	0.325785	−	0.945444i	$-0.394372\pi$
$401$	13.0477	0.651569	0.325785	+	0.945444i	$0.394372\pi$
$402$	−35.0710	−1.74918
$403$	13.8570	0.690266
$404$	−44.7496	−2.22637
$405$	103.746	5.15517
$406$	−19.1506	−0.950431
$407$	33.3026	1.65075
$408$	108.767	5.38476
$409$	7.28115	0.360030	0.180015	−	0.983664i	$-0.442385\pi$
$409$	7.28115	0.360030	0.180015	+	0.983664i	$0.442385\pi$
$410$	−108.790	−5.37278
$411$	18.6522	0.920043
$412$	−31.5226	−1.55301
$413$	25.8153	1.27029
$414$	2.57136	0.126375
$415$	−45.2909	−2.22324
$416$	106.562	5.22465
$417$	−65.1462	−3.19023
$418$	−3.35736	−0.164214
$419$	18.0548	0.882037	0.441018	−	0.897498i	$-0.354617\pi$
$419$	18.0548	0.882037	0.441018	+	0.897498i	$0.354617\pi$
$420$	−169.286	−8.26032
$421$	29.1081	1.41864	0.709322	−	0.704885i	$-0.249001\pi$
$421$	29.1081	1.41864	0.709322	+	0.704885i	$0.249001\pi$
$422$	48.8259	2.37681
$423$	79.0039	3.84130
$424$	53.4947	2.59793
$425$	30.3326	1.47135
$426$	−60.1783	−2.91565
$427$	−11.4862	−0.555857
$428$	18.1563	0.877617
$429$	−59.0564	−2.85127
$430$	89.7568	4.32846
$431$	24.3356	1.17220	0.586102	−	0.810237i	$-0.300662\pi$
$431$	24.3356	1.17220	0.586102	+	0.810237i	$0.300662\pi$
$432$	−268.034	−12.8958
$433$	−3.56497	−0.171322	−0.0856608	−	0.996324i	$-0.527300\pi$
$433$	−3.56497	−0.171322	−0.0856608	+	0.996324i	$0.527300\pi$
$434$	23.7053	1.13789
$435$	−36.1424	−1.73290
$436$	−67.7439	−3.24434
$437$	0.0309454	0.00148032
$438$	−15.6974	−0.750051
$439$	6.32302	0.301781	0.150891	−	0.988550i	$-0.451786\pi$
$439$	6.32302	0.301781	0.150891	+	0.988550i	$0.451786\pi$
$440$	184.894	8.81450
$441$	−10.5897	−0.504271
$442$	34.3299	1.63291
$443$	−6.07037	−0.288412	−0.144206	−	0.989548i	$-0.546063\pi$
$443$	−6.07037	−0.288412	−0.144206	+	0.989548i	$0.546063\pi$
$444$	133.466	6.33404
$445$	55.0032	2.60740
$446$	0.598134	0.0283225
$447$	61.1819	2.89380
$448$	99.7308	4.71184
$449$	15.0231	0.708986	0.354493	−	0.935059i	$-0.384653\pi$
$449$	15.0231	0.708986	0.354493	+	0.935059i	$0.384653\pi$
$450$	−203.032	−9.57101
$451$	−48.1343	−2.26656
$452$	18.7386	0.881391
$453$	−7.22729	−0.339568
$454$	12.7530	0.598526
$455$	−34.7929	−1.63112
$456$	−8.76163	−0.410301
$457$	−2.44117	−0.114193	−0.0570966	−	0.998369i	$-0.518184\pi$
$457$	−2.44117	−0.114193	−0.0570966	+	0.998369i	$0.518184\pi$
$458$	60.3309	2.81908
$459$	−49.3055	−2.30139
$460$	−2.61715	−0.122025
$461$	−21.2044	−0.987588	−0.493794	−	0.869579i	$-0.664390\pi$
$461$	−21.2044	−0.987588	−0.493794	+	0.869579i	$0.664390\pi$
$462$	−101.028	−4.70026
$463$	−21.5904	−1.00339	−0.501695	−	0.865044i	$-0.667290\pi$
$463$	−21.5904	−1.00339	−0.501695	+	0.865044i	$0.667290\pi$
$464$	50.5338	2.34597
$465$	44.7382	2.07469
$466$	−7.43255	−0.344306
$467$	−20.5370	−0.950340	−0.475170	−	0.879894i	$-0.657613\pi$
$467$	−20.5370	−0.950340	−0.475170	+	0.879894i	$0.657613\pi$
$468$	−170.361	−7.87493
$469$	−9.14186	−0.422132
$470$	−108.461	−5.00293
$471$	−50.8347	−2.34234
$472$	−112.999	−5.20120
$473$	39.7128	1.82600
$474$	−110.428	−5.07214
$475$	−2.44342	−0.112112
$476$	43.5402	1.99566
$477$	−39.7075	−1.81808
$478$	−4.81451	−0.220210
$479$	−1.23264	−0.0563208	−0.0281604	−	0.999603i	$-0.508965\pi$
$479$	−1.23264	−0.0563208	−0.0281604	+	0.999603i	$0.508965\pi$
$480$	344.044	15.7034
$481$	27.4310	1.25075
$482$	−18.5545	−0.845134
$483$	0.931196	0.0423709
$484$	62.5629	2.84377
$485$	−7.20344	−0.327092
$486$	119.659	5.42784
$487$	−15.6331	−0.708403	−0.354201	−	0.935169i	$-0.615247\pi$
$487$	−15.6331	−0.708403	−0.354201	+	0.935169i	$0.615247\pi$
$488$	50.2775	2.27596
$489$	38.6579	1.74817
$490$	14.5381	0.656765
$491$	39.6796	1.79071	0.895357	−	0.445348i	$-0.146920\pi$
$491$	39.6796	1.79071	0.895357	+	0.445348i	$0.146920\pi$
$492$	−192.907	−8.69693
$493$	9.29580	0.418662
$494$	−2.76542	−0.124422
$495$	−137.241	−6.16854
$496$	−62.5523	−2.80868
$497$	−15.6865	−0.703637
$498$	−108.324	−4.85414
$499$	−29.6217	−1.32605	−0.663024	−	0.748598i	$-0.730727\pi$
$499$	−29.6217	−1.32605	−0.663024	+	0.748598i	$0.730727\pi$
$500$	97.5854	4.36415
$501$	−15.5303	−0.693842
$502$	−77.5088	−3.45939
$503$	15.8623	0.707266	0.353633	−	0.935384i	$-0.384946\pi$
$503$	15.8623	0.707266	0.353633	+	0.935384i	$0.384946\pi$
$504$	−189.775	−8.45323
$505$	−29.6939	−1.32136
$506$	−1.56189	−0.0694345
$507$	−6.10728	−0.271234
$508$	−23.1886	−1.02883
$509$	−19.6121	−0.869293	−0.434647	−	0.900601i	$-0.643127\pi$
$509$	−19.6121	−0.869293	−0.434647	+	0.900601i	$0.643127\pi$
$510$	110.836	4.90792
$511$	−4.09180	−0.181011
$512$	−119.627	−5.28682
$513$	3.97177	0.175358
$514$	71.3952	3.14911
$515$	−20.9171	−0.921716
$516$	159.157	7.00648
$517$	−47.9885	−2.11053
$518$	46.9264	2.06183
$519$	35.4735	1.55711
$520$	152.296	6.67860
$521$	−42.3042	−1.85338	−0.926690	−	0.375827i	$-0.877359\pi$
$521$	−42.3042	−1.85338	−0.926690	+	0.375827i	$0.877359\pi$
$522$	−62.2216	−2.72337
$523$	−11.1933	−0.489447	−0.244723	−	0.969593i	$-0.578697\pi$
$523$	−11.1933	−0.489447	−0.244723	+	0.969593i	$0.578697\pi$
$524$	−48.1496	−2.10343
$525$	−73.5262	−3.20895
$526$	−2.33910	−0.101990
$527$	−11.5066	−0.501237
$528$	266.588	11.6018
$529$	−22.9856	−0.999374
$530$	54.5127	2.36788
$531$	83.8756	3.63989
$532$	−3.50735	−0.152063
$533$	−39.6477	−1.71733
$534$	131.554	5.69289
$535$	12.0477	0.520870
$536$	40.0158	1.72842
$537$	−54.7007	−2.36051
$538$	75.1469	3.23981
$539$	6.43238	0.277062
$540$	−335.906	−14.4551
$541$	−1.64172	−0.0705831	−0.0352916	−	0.999377i	$-0.511236\pi$
$541$	−1.64172	−0.0705831	−0.0352916	+	0.999377i	$0.511236\pi$
$542$	79.2457	3.40390
$543$	−70.6827	−3.03329
$544$	−88.4877	−3.79388
$545$	−44.9520	−1.92553
$546$	−83.2159	−3.56131
$547$	−7.18688	−0.307289	−0.153644	−	0.988126i	$-0.549101\pi$
$547$	−7.18688	−0.307289	−0.153644	+	0.988126i	$0.549101\pi$
$548$	−32.6829	−1.39614
$549$	−37.3194	−1.59275
$550$	123.325	5.25861
$551$	−0.748816	−0.0319006
$552$	−4.07603	−0.173487
$553$	−28.7851	−1.22407
$554$	33.7346	1.43325
$555$	88.5627	3.75928
$556$	114.151	4.84109
$557$	20.2174	0.856639	0.428319	−	0.903627i	$-0.359106\pi$
$557$	20.2174	0.856639	0.428319	+	0.903627i	$0.359106\pi$
$558$	77.0199	3.26051
$559$	32.7110	1.38353
$560$	157.060	6.63698
$561$	49.0395	2.07045
$562$	15.8749	0.669642
$563$	8.95897	0.377576	0.188788	−	0.982018i	$-0.439544\pi$
$563$	8.95897	0.377576	0.188788	+	0.982018i	$0.439544\pi$
$564$	−192.323	−8.09826
$565$	12.4342	0.523109
$566$	−29.0640	−1.22165
$567$	64.6805	2.71633
$568$	68.6631	2.88104
$569$	−8.98239	−0.376561	−0.188281	−	0.982115i	$-0.560291\pi$
$569$	−8.98239	−0.376561	−0.188281	+	0.982115i	$0.560291\pi$
$570$	−8.92834	−0.373967
$571$	−12.1345	−0.507813	−0.253907	−	0.967229i	$-0.581716\pi$
$571$	−12.1345	−0.507813	−0.253907	+	0.967229i	$0.581716\pi$
$572$	103.480	4.32673
$573$	36.9203	1.54237
$574$	−67.8256	−2.83099
$575$	−1.13671	−0.0474041
$576$	324.032	13.5013
$577$	−33.1798	−1.38129	−0.690647	−	0.723192i	$-0.742674\pi$
$577$	−33.1798	−1.38129	−0.690647	+	0.723192i	$0.742674\pi$
$578$	18.7683	0.780658
$579$	−66.7614	−2.77451
$580$	63.3298	2.62963
$581$	−28.2367	−1.17145
$582$	−17.2288	−0.714158
$583$	24.1191	0.998911
$584$	17.9107	0.741148
$585$	−113.044	−4.67380
$586$	−33.9246	−1.40141
$587$	−25.9537	−1.07123	−0.535613	−	0.844464i	$-0.679919\pi$
$587$	−25.9537	−1.07123	−0.535613	+	0.844464i	$0.679919\pi$
$588$	25.7790	1.06311
$589$	0.926908	0.0381926
$590$	−115.149	−4.74061
$591$	49.2123	2.02432
$592$	−123.827	−5.08926
$593$	−34.7233	−1.42591	−0.712957	−	0.701207i	$-0.752645\pi$
$593$	−34.7233	−1.42591	−0.712957	+	0.701207i	$0.752645\pi$
$594$	−200.465	−8.22518
$595$	28.8915	1.18443
$596$	−107.205	−4.39127
$597$	68.7476	2.81365
$598$	−1.28651	−0.0526094
$599$	12.3809	0.505869	0.252935	−	0.967483i	$-0.418604\pi$
$599$	12.3809	0.505869	0.252935	+	0.967483i	$0.418604\pi$
$600$	321.839	13.1390
$601$	11.2320	0.458163	0.229082	−	0.973407i	$-0.426428\pi$
$601$	11.2320	0.458163	0.229082	+	0.973407i	$0.426428\pi$
$602$	55.9590	2.28072
$603$	−29.7025	−1.20958
$604$	12.6639	0.515286
$605$	41.5141	1.68779
$606$	−71.0205	−2.88501
$607$	3.79688	0.154111	0.0770553	−	0.997027i	$-0.475448\pi$
$607$	3.79688	0.154111	0.0770553	+	0.997027i	$0.475448\pi$
$608$	7.12806	0.289081
$609$	−22.5330	−0.913085
$610$	51.2342	2.07441
$611$	−39.5276	−1.59911
$612$	141.465	5.71838
$613$	38.8427	1.56884	0.784420	−	0.620230i	$-0.212961\pi$
$613$	38.8427	1.56884	0.784420	+	0.620230i	$0.212961\pi$
$614$	32.4940	1.31135
$615$	−128.005	−5.16167
$616$	115.273	4.64447
$617$	−33.0943	−1.33233	−0.666163	−	0.745806i	$-0.732065\pi$
$617$	−33.0943	−1.33233	−0.666163	+	0.745806i	$0.732065\pi$
$618$	−50.0284	−2.01244
$619$	9.92150	0.398779	0.199389	−	0.979920i	$-0.436104\pi$
$619$	9.92150	0.398779	0.199389	+	0.979920i	$0.436104\pi$
$620$	−78.3916	−3.14828
$621$	1.84772	0.0741466
$622$	−45.6716	−1.83127
$623$	34.2918	1.37387
$624$	219.586	8.79047
$625$	17.3844	0.695375
$626$	−72.9050	−2.91387
$627$	−3.95034	−0.157761
$628$	89.0740	3.55444
$629$	−22.7783	−0.908228
$630$	−193.386	−7.70467
$631$	12.9603	0.515942	0.257971	−	0.966153i	$-0.416946\pi$
$631$	12.9603	0.515942	0.257971	+	0.966153i	$0.416946\pi$
$632$	125.998	5.01194
$633$	57.4495	2.28341
$634$	−22.7147	−0.902116
$635$	−15.3870	−0.610614
$636$	96.6618	3.83289
$637$	5.29828	0.209925
$638$	37.7946	1.49630
$639$	−50.9665	−2.01620
$640$	−234.557	−9.27168
$641$	−39.5580	−1.56245	−0.781223	−	0.624252i	$-0.785404\pi$
$641$	−39.5580	−1.56245	−0.781223	+	0.624252i	$0.785404\pi$
$642$	28.8152	1.13725
$643$	13.7982	0.544146	0.272073	−	0.962277i	$-0.412291\pi$
$643$	13.7982	0.544146	0.272073	+	0.962277i	$0.412291\pi$
$644$	−1.63167	−0.0642967
$645$	105.610	4.15838
$646$	2.29636	0.0903491
$647$	−13.1931	−0.518676	−0.259338	−	0.965787i	$-0.583504\pi$
$647$	−13.1931	−0.518676	−0.259338	+	0.965787i	$0.583504\pi$
$648$	−283.120	−11.1220
$649$	−50.9477	−1.99987
$650$	101.582	3.98436
$651$	27.8921	1.09318
$652$	−67.7376	−2.65281
$653$	−6.64266	−0.259947	−0.129974	−	0.991517i	$-0.541489\pi$
$653$	−6.64266	−0.259947	−0.129974	+	0.991517i	$0.541489\pi$
$654$	−107.514	−4.20413
$655$	−31.9501	−1.24839
$656$	178.975	6.98779
$657$	−13.2945	−0.518669
$658$	−67.6201	−2.63611
$659$	26.3721	1.02731	0.513656	−	0.857996i	$-0.328291\pi$
$659$	26.3721	1.02731	0.513656	+	0.857996i	$0.328291\pi$
$660$	334.093	13.0046
$661$	3.29903	0.128317	0.0641586	−	0.997940i	$-0.479564\pi$
$661$	3.29903	0.128317	0.0641586	+	0.997940i	$0.479564\pi$
$662$	−21.8835	−0.850527
$663$	40.3933	1.56875
$664$	123.598	4.79652
$665$	−2.32733	−0.0902500
$666$	152.467	5.90797
$667$	−0.348360	−0.0134885
$668$	27.2126	1.05289
$669$	0.703777	0.0272096
$670$	40.7772	1.57536
$671$	22.6685	0.875109
$672$	214.495	8.27431
$673$	−5.60010	−0.215868	−0.107934	−	0.994158i	$-0.534424\pi$
$673$	−5.60010	−0.215868	−0.107934	+	0.994158i	$0.534424\pi$
$674$	25.5671	0.984807
$675$	−145.894	−5.61547
$676$	10.7014	0.411591
$677$	26.5204	1.01926	0.509631	−	0.860393i	$-0.329782\pi$
$677$	26.5204	1.01926	0.509631	+	0.860393i	$0.329782\pi$
$678$	29.7394	1.14213
$679$	−4.49100	−0.172349
$680$	−126.464	−4.84966
$681$	15.0054	0.575008
$682$	−46.7834	−1.79143
$683$	38.9845	1.49170	0.745850	−	0.666114i	$-0.232043\pi$
$683$	38.9845	1.49170	0.745850	+	0.666114i	$0.232043\pi$
$684$	−11.3956	−0.435721
$685$	−21.6870	−0.828616
$686$	55.2357	2.10891
$687$	70.9866	2.70831
$688$	−147.662	−5.62955
$689$	19.8666	0.756858
$690$	−4.15359	−0.158125
$691$	−39.2340	−1.49253	−0.746266	−	0.665648i	$-0.768155\pi$
$691$	−39.2340	−1.49253	−0.746266	+	0.665648i	$0.768155\pi$
$692$	−62.1577	−2.36288
$693$	−85.5634	−3.25028
$694$	86.0839	3.26770
$695$	75.7459	2.87321
$696$	98.6317	3.73863
$697$	32.9228	1.24704
$698$	−89.0198	−3.36945
$699$	−8.74529	−0.330777
$700$	128.835	4.86950
$701$	−40.8295	−1.54211	−0.771055	−	0.636768i	$-0.780271\pi$
$701$	−40.8295	−1.54211	−0.771055	+	0.636768i	$0.780271\pi$
$702$	−165.121	−6.23209
$703$	1.83488	0.0692040
$704$	−196.823	−7.41804
$705$	−127.617	−4.80635
$706$	−28.9519	−1.08962
$707$	−18.5127	−0.696243
$708$	−204.182	−7.67365
$709$	11.2817	0.423694	0.211847	−	0.977303i	$-0.432052\pi$
$709$	11.2817	0.423694	0.211847	+	0.977303i	$0.432052\pi$
$710$	69.9697	2.62592
$711$	−93.5245	−3.50744
$712$	−150.102	−5.62532
$713$	0.431210	0.0161490
$714$	69.1012	2.58605
$715$	68.6652	2.56793
$716$	95.8482	3.58201
$717$	−5.66485	−0.211558
$718$	60.3313	2.25154
$719$	−34.0470	−1.26974	−0.634870	−	0.772619i	$-0.718946\pi$
$719$	−34.0470	−1.26974	−0.634870	+	0.772619i	$0.718946\pi$
$720$	510.296	19.0176
$721$	−13.0408	−0.485664
$722$	52.6521	1.95951
$723$	−21.8316	−0.811926
$724$	123.852	4.60294
$725$	27.5061	1.02155
$726$	99.2914	3.68505
$727$	−11.9412	−0.442875	−0.221437	−	0.975175i	$-0.571075\pi$
$727$	−11.9412	−0.442875	−0.221437	+	0.975175i	$0.571075\pi$
$728$	94.9489	3.51904
$729$	58.9844	2.18461
$730$	18.2515	0.675517
$731$	−27.1627	−1.00465
$732$	90.8485	3.35786
$733$	−16.0363	−0.592313	−0.296157	−	0.955139i	$-0.595705\pi$
$733$	−16.0363	−0.592313	−0.296157	+	0.955139i	$0.595705\pi$
$734$	−70.5224	−2.60303
$735$	17.1058	0.630958
$736$	3.31607	0.122232
$737$	18.0419	0.664580
$738$	−220.370	−8.11192
$739$	−17.6003	−0.647437	−0.323718	−	0.946153i	$-0.604933\pi$
$739$	−17.6003	−0.647437	−0.323718	+	0.946153i	$0.604933\pi$
$740$	−155.182	−5.70461
$741$	−3.25385	−0.119533
$742$	33.9860	1.24767
$743$	−34.8498	−1.27852	−0.639258	−	0.768992i	$-0.720758\pi$
$743$	−34.8498	−1.27852	−0.639258	+	0.768992i	$0.720758\pi$
$744$	−122.089	−4.47602
$745$	−71.1365	−2.60624
$746$	−37.6775	−1.37947
$747$	−91.7427	−3.35669
$748$	−85.9285	−3.14186
$749$	7.51119	0.274453
$750$	154.874	5.65521
$751$	−11.4495	−0.417800	−0.208900	−	0.977937i	$-0.566988\pi$
$751$	−11.4495	−0.417800	−0.208900	+	0.977937i	$0.566988\pi$
$752$	178.433	6.50677
$753$	−91.1985	−3.32346
$754$	31.1310	1.13372
$755$	8.40322	0.305824
$756$	−209.421	−7.61656
$757$	−28.2068	−1.02519	−0.512597	−	0.858629i	$-0.671317\pi$
$757$	−28.2068	−1.02519	−0.512597	+	0.858629i	$0.671317\pi$
$758$	−19.9369	−0.724141
$759$	−1.83775	−0.0667062
$760$	10.1872	0.369528
$761$	4.25018	0.154069	0.0770344	−	0.997028i	$-0.475455\pi$
$761$	4.25018	0.154069	0.0770344	+	0.997028i	$0.475455\pi$
$762$	−36.8018	−1.33319
$763$	−28.0254	−1.01459
$764$	−64.6928	−2.34050
$765$	93.8701	3.39388
$766$	−56.1654	−2.02934
$767$	−41.9650	−1.51527
$768$	−285.841	−10.3144
$769$	−9.99829	−0.360547	−0.180274	−	0.983616i	$-0.557698\pi$
$769$	−9.99829	−0.360547	−0.180274	+	0.983616i	$0.557698\pi$
$770$	117.466	4.23319
$771$	84.0051	3.02537
$772$	116.981	4.21025
$773$	44.4981	1.60048	0.800242	−	0.599677i	$-0.204704\pi$
$773$	44.4981	1.60048	0.800242	+	0.599677i	$0.204704\pi$
$774$	181.814	6.53518
$775$	−34.0479	−1.22304
$776$	19.6580	0.705681
$777$	55.2146	1.98081
$778$	−28.2089	−1.01134
$779$	−2.65207	−0.0950203
$780$	275.189	9.85335
$781$	30.9580	1.10777
$782$	1.06830	0.0382023
$783$	−44.7111	−1.59785
$784$	−23.9171	−0.854183
$785$	59.1058	2.10958
$786$	−76.4166	−2.72569
$787$	−25.2737	−0.900910	−0.450455	−	0.892799i	$-0.648738\pi$
$787$	−25.2737	−0.900910	−0.450455	+	0.892799i	$0.648738\pi$
$788$	−86.2312	−3.07186
$789$	−2.75224	−0.0979823
$790$	128.396	4.56811
$791$	7.75210	0.275633
$792$	374.528	13.3083
$793$	18.6718	0.663056
$794$	9.81835	0.348440
$795$	64.1407	2.27484
$796$	−120.462	−4.26965
$797$	−18.9930	−0.672765	−0.336383	−	0.941725i	$-0.609204\pi$
$797$	−18.9930	−0.672765	−0.336383	+	0.941725i	$0.609204\pi$
$798$	−5.56639	−0.197048
$799$	32.8231	1.16120
$800$	−261.834	−9.25722
$801$	111.416	3.93670
$802$	−36.2843	−1.28124
$803$	8.07535	0.284973
$804$	72.3061	2.55004
$805$	−1.08271	−0.0381604
$806$	−38.5349	−1.35734
$807$	88.4195	3.11251
$808$	81.0340	2.85076
$809$	26.7432	0.940240	0.470120	−	0.882602i	$-0.344211\pi$
$809$	26.7432	0.940240	0.470120	+	0.882602i	$0.344211\pi$
$810$	−288.507	−10.1371
$811$	−7.53549	−0.264607	−0.132303	−	0.991209i	$-0.542237\pi$
$811$	−7.53549	−0.264607	−0.132303	+	0.991209i	$0.542237\pi$
$812$	39.4831	1.38558
$813$	93.2422	3.27015
$814$	−92.6112	−3.24602
$815$	−44.9478	−1.57445
$816$	−182.341	−6.38320
$817$	2.18807	0.0765509
$818$	−20.2482	−0.707960
$819$	−70.4776	−2.46269
$820$	224.294	7.83270
$821$	10.0074	0.349260	0.174630	−	0.984634i	$-0.444127\pi$
$821$	10.0074	0.349260	0.174630	+	0.984634i	$0.444127\pi$
$822$	−51.8698	−1.80917
$823$	44.5787	1.55392	0.776958	−	0.629552i	$-0.216761\pi$
$823$	44.5787	1.55392	0.776958	+	0.629552i	$0.216761\pi$
$824$	57.0821	1.98855
$825$	145.107	5.05198
$826$	−71.7899	−2.49789
$827$	0.269507	0.00937166	0.00468583	−	0.999989i	$-0.498508\pi$
$827$	0.269507	0.00937166	0.00468583	+	0.999989i	$0.498508\pi$
$828$	−5.30139	−0.184236
$829$	−44.4503	−1.54382	−0.771911	−	0.635731i	$-0.780699\pi$
$829$	−44.4503	−1.54382	−0.771911	+	0.635731i	$0.780699\pi$
$830$	125.949	4.37177
$831$	39.6928	1.37693
$832$	−162.121	−5.62053
$833$	−4.39961	−0.152437
$834$	181.165	6.27324
$835$	18.0571	0.624893
$836$	6.92190	0.239399
$837$	55.3449	1.91300
$838$	−50.2087	−1.73443
$839$	−38.1015	−1.31541	−0.657705	−	0.753276i	$-0.728472\pi$
$839$	−38.1015	−1.31541	−0.657705	+	0.753276i	$0.728472\pi$
$840$	306.549	10.5769
$841$	−20.5704	−0.709324
$842$	−80.9469	−2.78961
$843$	18.6787	0.643330
$844$	−100.665	−3.46502
$845$	7.10097	0.244281
$846$	−219.702	−7.55351
$847$	25.8820	0.889318
$848$	−89.6805	−3.07964
$849$	−34.1974	−1.17365
$850$	−84.3518	−2.89324
$851$	0.853614	0.0292615
$852$	124.070	4.25058
$853$	0.551051	0.0188676	0.00943382	−	0.999956i	$-0.496997\pi$
$853$	0.551051	0.0188676	0.00943382	+	0.999956i	$0.496997\pi$
$854$	31.9420	1.09303
$855$	−7.56164	−0.258603
$856$	−32.8780	−1.12375
$857$	0.0288594	0.000985818 0	0.000492909	−	1.00000i	$-0.499843\pi$
$857$	0.0288594	0.000985818 0	0.000492909		1.00000i	$0.499843\pi$
$858$	164.230	5.60672
$859$	−10.6732	−0.364165	−0.182082	−	0.983283i	$-0.558284\pi$
$859$	−10.6732	−0.364165	−0.182082	+	0.983283i	$0.558284\pi$
$860$	−185.052	−6.31024
$861$	−79.8050	−2.71975
$862$	−67.6749	−2.30502
$863$	−14.6511	−0.498730	−0.249365	−	0.968410i	$-0.580222\pi$
$863$	−14.6511	−0.498730	−0.249365	+	0.968410i	$0.580222\pi$
$864$	425.610	14.4796
$865$	−41.2452	−1.40238
$866$	9.91383	0.336886
$867$	22.0832	0.749983
$868$	−48.8734	−1.65887
$869$	56.8086	1.92710
$870$	100.508	3.40756
$871$	14.8609	0.503542
$872$	122.673	4.15422
$873$	−14.5915	−0.493848
$874$	−0.0860561	−0.00291089
$875$	40.3707	1.36478
$876$	32.3635	1.09346
$877$	−30.6064	−1.03351	−0.516753	−	0.856135i	$-0.672859\pi$
$877$	−30.6064	−1.03351	−0.516753	+	0.856135i	$0.672859\pi$
$878$	−17.5837	−0.593421
$879$	−39.9164	−1.34635
$880$	−309.964	−10.4489
$881$	−14.9270	−0.502902	−0.251451	−	0.967870i	$-0.580908\pi$
$881$	−14.9270	−0.502902	−0.251451	+	0.967870i	$0.580908\pi$
$882$	29.4489	0.991595
$883$	−19.4227	−0.653627	−0.326813	−	0.945089i	$-0.605975\pi$
$883$	−19.4227	−0.653627	−0.326813	+	0.945089i	$0.605975\pi$
$884$	−70.7783	−2.38053
$885$	−135.487	−4.55434
$886$	16.8811	0.567132
$887$	14.4044	0.483652	0.241826	−	0.970320i	$-0.422254\pi$
$887$	14.4044	0.483652	0.241826	+	0.970320i	$0.422254\pi$
$888$	−241.685	−8.11043
$889$	−9.59303	−0.321740
$890$	−152.958	−5.12718
$891$	−127.650	−4.27643
$892$	−1.23318	−0.0412899
$893$	−2.64404	−0.0884793
$894$	−170.141	−5.69036
$895$	63.6008	2.12594
$896$	−146.235	−4.88537
$897$	−1.51374	−0.0505422
$898$	−41.7779	−1.39415
$899$	−10.4344	−0.348007
$900$	418.592	13.9531
$901$	−16.4969	−0.549593
$902$	133.857	4.45694
$903$	65.8425	2.19110
$904$	−33.9325	−1.12858
$905$	82.1832	2.73186
$906$	20.0984	0.667724
$907$	−51.3436	−1.70484	−0.852418	−	0.522861i	$-0.824865\pi$
$907$	−51.3436	−1.70484	−0.852418	+	0.522861i	$0.824865\pi$
$908$	−26.2929	−0.872560
$909$	−60.1490	−1.99502
$910$	96.7556	3.20742
$911$	−2.50431	−0.0829715	−0.0414857	−	0.999139i	$-0.513209\pi$
$911$	−2.50431	−0.0829715	−0.0414857	+	0.999139i	$0.513209\pi$
$912$	14.6883	0.486378
$913$	55.7263	1.84427
$914$	6.78865	0.224549
$915$	60.2832	1.99290
$916$	−124.385	−4.10979
$917$	−19.9193	−0.657794
$918$	137.114	4.52543
$919$	4.47057	0.147471	0.0737353	−	0.997278i	$-0.476508\pi$
$919$	4.47057	0.147471	0.0737353	+	0.997278i	$0.476508\pi$
$920$	4.73923	0.156248
$921$	38.2332	1.25983
$922$	58.9674	1.94199
$923$	25.4998	0.839336
$924$	208.291	6.85227
$925$	−67.4005	−2.21611
$926$	60.0407	1.97306
$927$	−42.3703	−1.39162
$928$	−80.2422	−2.63408
$929$	−26.3764	−0.865382	−0.432691	−	0.901542i	$-0.642436\pi$
$929$	−26.3764	−0.865382	−0.432691	+	0.901542i	$0.642436\pi$
$930$	−124.413	−4.07965
$931$	0.354407	0.0116152
$932$	15.3237	0.501946
$933$	−53.7382	−1.75931
$934$	57.1114	1.86874
$935$	−57.0185	−1.86471
$936$	308.495	10.0835
$937$	28.5502	0.932694	0.466347	−	0.884602i	$-0.345570\pi$
$937$	28.5502	0.932694	0.466347	+	0.884602i	$0.345570\pi$
$938$	25.4226	0.830078
$939$	−85.7815	−2.79937
$940$	223.615	7.29351
$941$	−7.80803	−0.254535	−0.127267	−	0.991868i	$-0.540621\pi$
$941$	−7.80803	−0.254535	−0.127267	+	0.991868i	$0.540621\pi$
$942$	141.366	4.60596
$943$	−1.23378	−0.0401774
$944$	189.436	6.16560
$945$	−138.963	−4.52046
$946$	−110.437	−3.59063
$947$	47.1421	1.53191	0.765957	−	0.642892i	$-0.222266\pi$
$947$	47.1421	1.53191	0.765957	+	0.642892i	$0.222266\pi$
$948$	227.671	7.39442
$949$	6.65158	0.215919
$950$	6.79490	0.220456
$951$	−26.7266	−0.866669
$952$	−78.8440	−2.55535
$953$	−19.8208	−0.642058	−0.321029	−	0.947069i	$-0.604029\pi$
$953$	−19.8208	−0.642058	−0.321029	+	0.947069i	$0.604029\pi$
$954$	110.423	3.57506
$955$	−42.9274	−1.38910
$956$	9.92611	0.321033
$957$	44.4699	1.43751
$958$	3.42785	0.110749
$959$	−13.5208	−0.436609
$960$	−523.418	−16.8932
$961$	−18.0839	−0.583353
$962$	−76.2829	−2.45946
$963$	24.4043	0.786418
$964$	38.2540	1.23208
$965$	77.6238	2.49880
$966$	−2.58956	−0.0833178
$967$	27.7979	0.893920	0.446960	−	0.894554i	$-0.352507\pi$
$967$	27.7979	0.893920	0.446960	+	0.894554i	$0.352507\pi$
$968$	−113.291	−3.64131
$969$	2.70195	0.0867990
$970$	20.0321	0.643191
$971$	47.4551	1.52291	0.761453	−	0.648221i	$-0.224487\pi$
$971$	47.4551	1.52291	0.761453	+	0.648221i	$0.224487\pi$
$972$	−246.702	−7.91297
$973$	47.2239	1.51393
$974$	43.4741	1.39300
$975$	119.523	3.82781
$976$	−84.2870	−2.69796
$977$	−13.9723	−0.447013	−0.223506	−	0.974702i	$-0.571750\pi$
$977$	−13.9723	−0.447013	−0.223506	+	0.974702i	$0.571750\pi$
$978$	−107.504	−3.43759
$979$	−67.6763	−2.16295
$980$	−29.9734	−0.957464
$981$	−91.0562	−2.90720
$982$	−110.345	−3.52125
$983$	29.1742	0.930514	0.465257	−	0.885176i	$-0.345962\pi$
$983$	29.1742	0.930514	0.465257	+	0.885176i	$0.345962\pi$
$984$	349.323	11.1360
$985$	−57.2194	−1.82316
$986$	−25.8507	−0.823254
$987$	−79.5633	−2.53253
$988$	5.70149	0.181389
$989$	1.01792	0.0323680
$990$	381.655	12.1298
$991$	21.3876	0.679400	0.339700	−	0.940534i	$-0.389674\pi$
$991$	21.3876	0.679400	0.339700	+	0.940534i	$0.389674\pi$
$992$	99.3264	3.15362
$993$	−25.7486	−0.817107
$994$	43.6227	1.38363
$995$	−79.9332	−2.53405
$996$	223.334	7.07660
$997$	22.6506	0.717353	0.358676	−	0.933462i	$-0.383228\pi$
$997$	22.6506	0.717353	0.358676	+	0.933462i	$0.383228\pi$
$998$	82.3749	2.60753
$999$	109.559	3.46631

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4013.2.a.b.1.3	✓	157

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4013.2.a.b.1.3	✓	157	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4013 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4013.a (trivial)

\(f(q)\)	\(=\)	\(q-2.78090 q^{2} -3.27207 q^{3} +5.73341 q^{4} +3.80445 q^{5} +9.09929 q^{6} +2.37189 q^{7} -10.3822 q^{8} +7.70642 q^{9} +O(q^{10})\)	\(q-2.78090 q^{2} -3.27207 q^{3} +5.73341 q^{4} +3.80445 q^{5} +9.09929 q^{6} +2.37189 q^{7} -10.3822 q^{8} +7.70642 q^{9} -10.5798 q^{10} -4.68102 q^{11} -18.7601 q^{12} -3.85571 q^{13} -6.59599 q^{14} -12.4484 q^{15} +17.4052 q^{16} +3.20172 q^{17} -21.4308 q^{18} -0.257912 q^{19} +21.8125 q^{20} -7.76098 q^{21} +13.0175 q^{22} -0.119984 q^{23} +33.9714 q^{24} +9.47384 q^{25} +10.7223 q^{26} -15.3997 q^{27} +13.5990 q^{28} +2.90338 q^{29} +34.6178 q^{30} -3.59389 q^{31} -27.6376 q^{32} +15.3166 q^{33} -8.90366 q^{34} +9.02373 q^{35} +44.1840 q^{36} -7.11438 q^{37} +0.717228 q^{38} +12.6161 q^{39} -39.4987 q^{40} +10.2829 q^{41} +21.5825 q^{42} -8.48379 q^{43} -26.8382 q^{44} +29.3187 q^{45} +0.333664 q^{46} +10.2517 q^{47} -56.9509 q^{48} -1.37414 q^{49} -26.3458 q^{50} -10.4762 q^{51} -22.1064 q^{52} -5.15252 q^{53} +42.8251 q^{54} -17.8087 q^{55} -24.6255 q^{56} +0.843905 q^{57} -8.07400 q^{58} +10.8839 q^{59} -71.3718 q^{60} -4.84265 q^{61} +9.99425 q^{62} +18.2788 q^{63} +42.0470 q^{64} -14.6688 q^{65} -42.5940 q^{66} -3.85425 q^{67} +18.3568 q^{68} +0.392597 q^{69} -25.0941 q^{70} -6.61352 q^{71} -80.0099 q^{72} -1.72512 q^{73} +19.7844 q^{74} -30.9990 q^{75} -1.47872 q^{76} -11.1029 q^{77} -35.0842 q^{78} -12.1359 q^{79} +66.2171 q^{80} +27.2696 q^{81} -28.5956 q^{82} -11.9047 q^{83} -44.4969 q^{84} +12.1808 q^{85} +23.5926 q^{86} -9.50004 q^{87} +48.5995 q^{88} +14.4576 q^{89} -81.5323 q^{90} -9.14532 q^{91} -0.687919 q^{92} +11.7594 q^{93} -28.5090 q^{94} -0.981213 q^{95} +90.4319 q^{96} -1.89343 q^{97} +3.82134 q^{98} -36.0739 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 157 q - 15 q^{2} - 51 q^{3} + 137 q^{4} - 13 q^{5} - 15 q^{6} - 49 q^{7} - 42 q^{8} + 144 q^{9}+O(q^{10}) \) 157 * q - 15 * q^2 - 51 * q^3 + 137 * q^4 - 13 * q^5 - 15 * q^6 - 49 * q^7 - 42 * q^8 + 144 * q^9	\( 157 q - 15 q^{2} - 51 q^{3} + 137 q^{4} - 13 q^{5} - 15 q^{6} - 49 q^{7} - 42 q^{8} + 144 q^{9} - 61 q^{10} - 27 q^{11} - 93 q^{12} - 97 q^{13} - 12 q^{14} - 36 q^{15} + 105 q^{16} - 45 q^{17} - 68 q^{18} - 128 q^{19} - 30 q^{20} - 26 q^{21} - 68 q^{22} - 41 q^{23} - 40 q^{24} + 102 q^{25} - 5 q^{26} - 189 q^{27} - 115 q^{28} - 26 q^{29} - 12 q^{30} - 88 q^{31} - 89 q^{32} - 52 q^{33} - 61 q^{34} - 87 q^{35} + 110 q^{36} - 62 q^{37} - 37 q^{38} - 20 q^{39} - 161 q^{40} - 34 q^{41} - 53 q^{42} - 254 q^{43} - 19 q^{44} - 46 q^{45} - 52 q^{46} - 76 q^{47} - 162 q^{48} + 96 q^{49} - 54 q^{50} - 76 q^{51} - 259 q^{52} - 48 q^{53} - 12 q^{54} - 194 q^{55} - 10 q^{56} - 30 q^{57} - 52 q^{58} - 64 q^{59} - 31 q^{60} - 107 q^{61} - 51 q^{62} - 106 q^{63} + 54 q^{64} - 17 q^{65} - 13 q^{66} - 193 q^{67} - 118 q^{68} - 55 q^{69} - 86 q^{70} - 11 q^{71} - 172 q^{72} - 173 q^{73} - 11 q^{74} - 209 q^{75} - 213 q^{76} - 84 q^{77} - 30 q^{78} - 111 q^{79} - 6 q^{80} + 157 q^{81} - 117 q^{82} - 154 q^{83} - 6 q^{84} - 91 q^{85} + 28 q^{86} - 165 q^{87} - 165 q^{88} - 32 q^{89} - 103 q^{90} - 200 q^{91} - 86 q^{92} - 39 q^{93} - 118 q^{94} - 22 q^{95} - 28 q^{96} - 151 q^{97} - 38 q^{98} - 91 q^{99}+O(q^{100}) \) 157 * q - 15 * q^2 - 51 * q^3 + 137 * q^4 - 13 * q^5 - 15 * q^6 - 49 * q^7 - 42 * q^8 + 144 * q^9 - 61 * q^10 - 27 * q^11 - 93 * q^12 - 97 * q^13 - 12 * q^14 - 36 * q^15 + 105 * q^16 - 45 * q^17 - 68 * q^18 - 128 * q^19 - 30 * q^20 - 26 * q^21 - 68 * q^22 - 41 * q^23 - 40 * q^24 + 102 * q^25 - 5 * q^26 - 189 * q^27 - 115 * q^28 - 26 * q^29 - 12 * q^30 - 88 * q^31 - 89 * q^32 - 52 * q^33 - 61 * q^34 - 87 * q^35 + 110 * q^36 - 62 * q^37 - 37 * q^38 - 20 * q^39 - 161 * q^40 - 34 * q^41 - 53 * q^42 - 254 * q^43 - 19 * q^44 - 46 * q^45 - 52 * q^46 - 76 * q^47 - 162 * q^48 + 96 * q^49 - 54 * q^50 - 76 * q^51 - 259 * q^52 - 48 * q^53 - 12 * q^54 - 194 * q^55 - 10 * q^56 - 30 * q^57 - 52 * q^58 - 64 * q^59 - 31 * q^60 - 107 * q^61 - 51 * q^62 - 106 * q^63 + 54 * q^64 - 17 * q^65 - 13 * q^66 - 193 * q^67 - 118 * q^68 - 55 * q^69 - 86 * q^70 - 11 * q^71 - 172 * q^72 - 173 * q^73 - 11 * q^74 - 209 * q^75 - 213 * q^76 - 84 * q^77 - 30 * q^78 - 111 * q^79 - 6 * q^80 + 157 * q^81 - 117 * q^82 - 154 * q^83 - 6 * q^84 - 91 * q^85 + 28 * q^86 - 165 * q^87 - 165 * q^88 - 32 * q^89 - 103 * q^90 - 200 * q^91 - 86 * q^92 - 39 * q^93 - 118 * q^94 - 22 * q^95 - 28 * q^96 - 151 * q^97 - 38 * q^98 - 91 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more