LMFDB - Embedded newform 6038.2.a.b.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6038,2,Mod(1,6038)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6038, 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("6038.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6038 = 2 \cdot 3019 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6038.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:	$48.2136727404$
Analytic rank:	$1$
Dimension:	$54$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Character	$\chi$	$=$	6038.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -2.02410 q^{3} +1.00000 q^{4} -4.35432 q^{5} -2.02410 q^{6} -1.05145 q^{7} +1.00000 q^{8} +1.09700 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.02410 q^{3} +1.00000 q^{4} -4.35432 q^{5} -2.02410 q^{6} -1.05145 q^{7} +1.00000 q^{8} +1.09700 q^{9} -4.35432 q^{10} +2.88356 q^{11} -2.02410 q^{12} +0.0392106 q^{13} -1.05145 q^{14} +8.81359 q^{15} +1.00000 q^{16} -1.17954 q^{17} +1.09700 q^{18} -1.02879 q^{19} -4.35432 q^{20} +2.12825 q^{21} +2.88356 q^{22} -5.84820 q^{23} -2.02410 q^{24} +13.9601 q^{25} +0.0392106 q^{26} +3.85188 q^{27} -1.05145 q^{28} +0.295209 q^{29} +8.81359 q^{30} -2.53642 q^{31} +1.00000 q^{32} -5.83663 q^{33} -1.17954 q^{34} +4.57836 q^{35} +1.09700 q^{36} +4.45208 q^{37} -1.02879 q^{38} -0.0793663 q^{39} -4.35432 q^{40} +8.75905 q^{41} +2.12825 q^{42} -1.03046 q^{43} +2.88356 q^{44} -4.77667 q^{45} -5.84820 q^{46} +2.95692 q^{47} -2.02410 q^{48} -5.89445 q^{49} +13.9601 q^{50} +2.38752 q^{51} +0.0392106 q^{52} +7.65978 q^{53} +3.85188 q^{54} -12.5560 q^{55} -1.05145 q^{56} +2.08237 q^{57} +0.295209 q^{58} +6.95279 q^{59} +8.81359 q^{60} -0.523332 q^{61} -2.53642 q^{62} -1.15344 q^{63} +1.00000 q^{64} -0.170735 q^{65} -5.83663 q^{66} -0.647740 q^{67} -1.17954 q^{68} +11.8374 q^{69} +4.57836 q^{70} +7.87283 q^{71} +1.09700 q^{72} -1.23174 q^{73} +4.45208 q^{74} -28.2567 q^{75} -1.02879 q^{76} -3.03193 q^{77} -0.0793663 q^{78} +13.7496 q^{79} -4.35432 q^{80} -11.0876 q^{81} +8.75905 q^{82} -9.54977 q^{83} +2.12825 q^{84} +5.13611 q^{85} -1.03046 q^{86} -0.597533 q^{87} +2.88356 q^{88} +7.56611 q^{89} -4.77667 q^{90} -0.0412281 q^{91} -5.84820 q^{92} +5.13398 q^{93} +2.95692 q^{94} +4.47967 q^{95} -2.02410 q^{96} -12.1742 q^{97} -5.89445 q^{98} +3.16326 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 54 q + 54 q^{2} - 21 q^{3} + 54 q^{4} - 14 q^{5} - 21 q^{6} - 44 q^{7} + 54 q^{8} + 39 q^{9}+O(q^{10}) $ 54 * q + 54 * q^2 - 21 * q^3 + 54 * q^4 - 14 * q^5 - 21 * q^6 - 44 * q^7 + 54 * q^8 + 39 * q^9	\( 54 q + 54 q^{2} - 21 q^{3} + 54 q^{4} - 14 q^{5} - 21 q^{6} - 44 q^{7} + 54 q^{8} + 39 q^{9} - 14 q^{10} - 31 q^{11} - 21 q^{12} - 34 q^{13} - 44 q^{14} - 22 q^{15} + 54 q^{16} - 40 q^{17} + 39 q^{18} - 44 q^{19} - 14 q^{20} - 3 q^{21} - 31 q^{22} - 33 q^{23} - 21 q^{24} + 14 q^{25} - 34 q^{26} - 66 q^{27} - 44 q^{28} - 22 q^{30} - 65 q^{31} + 54 q^{32} - 43 q^{33} - 40 q^{34} - 46 q^{35} + 39 q^{36} - 58 q^{37} - 44 q^{38} - 36 q^{39} - 14 q^{40} - 49 q^{41} - 3 q^{42} - 47 q^{43} - 31 q^{44} - 45 q^{45} - 33 q^{46} - 66 q^{47} - 21 q^{48} + 16 q^{49} + 14 q^{50} - 33 q^{51} - 34 q^{52} - 16 q^{53} - 66 q^{54} - 50 q^{55} - 44 q^{56} - 33 q^{57} - 70 q^{59} - 22 q^{60} - 40 q^{61} - 65 q^{62} - 117 q^{63} + 54 q^{64} - 33 q^{65} - 43 q^{66} - 82 q^{67} - 40 q^{68} - q^{69} - 46 q^{70} - 60 q^{71} + 39 q^{72} - 92 q^{73} - 58 q^{74} - 68 q^{75} - 44 q^{76} + 13 q^{77} - 36 q^{78} - 57 q^{79} - 14 q^{80} + 26 q^{81} - 49 q^{82} - 77 q^{83} - 3 q^{84} - 24 q^{85} - 47 q^{86} - 61 q^{87} - 31 q^{88} - 54 q^{89} - 45 q^{90} - 46 q^{91} - 33 q^{92} - 24 q^{93} - 66 q^{94} - 66 q^{95} - 21 q^{96} - 137 q^{97} + 16 q^{98} - 71 q^{99}+O(q^{100}) \) 54 * q + 54 * q^2 - 21 * q^3 + 54 * q^4 - 14 * q^5 - 21 * q^6 - 44 * q^7 + 54 * q^8 + 39 * q^9 - 14 * q^10 - 31 * q^11 - 21 * q^12 - 34 * q^13 - 44 * q^14 - 22 * q^15 + 54 * q^16 - 40 * q^17 + 39 * q^18 - 44 * q^19 - 14 * q^20 - 3 * q^21 - 31 * q^22 - 33 * q^23 - 21 * q^24 + 14 * q^25 - 34 * q^26 - 66 * q^27 - 44 * q^28 - 22 * q^30 - 65 * q^31 + 54 * q^32 - 43 * q^33 - 40 * q^34 - 46 * q^35 + 39 * q^36 - 58 * q^37 - 44 * q^38 - 36 * q^39 - 14 * q^40 - 49 * q^41 - 3 * q^42 - 47 * q^43 - 31 * q^44 - 45 * q^45 - 33 * q^46 - 66 * q^47 - 21 * q^48 + 16 * q^49 + 14 * q^50 - 33 * q^51 - 34 * q^52 - 16 * q^53 - 66 * q^54 - 50 * q^55 - 44 * q^56 - 33 * q^57 - 70 * q^59 - 22 * q^60 - 40 * q^61 - 65 * q^62 - 117 * q^63 + 54 * q^64 - 33 * q^65 - 43 * q^66 - 82 * q^67 - 40 * q^68 - q^69 - 46 * q^70 - 60 * q^71 + 39 * q^72 - 92 * q^73 - 58 * q^74 - 68 * q^75 - 44 * q^76 + 13 * q^77 - 36 * q^78 - 57 * q^79 - 14 * q^80 + 26 * q^81 - 49 * q^82 - 77 * q^83 - 3 * q^84 - 24 * q^85 - 47 * q^86 - 61 * q^87 - 31 * q^88 - 54 * q^89 - 45 * q^90 - 46 * q^91 - 33 * q^92 - 24 * q^93 - 66 * q^94 - 66 * q^95 - 21 * q^96 - 137 * q^97 + 16 * q^98 - 71 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−2.02410	−1.16862	−0.584308	−	0.811532i	$-0.698634\pi$
$3$	−2.02410	−1.16862	−0.584308	+	0.811532i	$0.698634\pi$
$4$	1.00000	0.500000
$5$	−4.35432	−1.94731	−0.973655	−	0.228025i	$-0.926773\pi$
$5$	−4.35432	−1.94731	−0.973655	+	0.228025i	$0.926773\pi$
$6$	−2.02410	−0.826337
$7$	−1.05145	−0.397412	−0.198706	−	0.980059i	$-0.563674\pi$
$7$	−1.05145	−0.397412	−0.198706	+	0.980059i	$0.563674\pi$
$8$	1.00000	0.353553
$9$	1.09700	0.365665
$10$	−4.35432	−1.37696
$11$	2.88356	0.869427	0.434714	−	0.900569i	$-0.356850\pi$
$11$	2.88356	0.869427	0.434714	+	0.900569i	$0.356850\pi$
$12$	−2.02410	−0.584308
$13$	0.0392106	0.0108751	0.00543753	−	0.999985i	$-0.498269\pi$
$13$	0.0392106	0.0108751	0.00543753	+	0.999985i	$0.498269\pi$
$14$	−1.05145	−0.281013
$15$	8.81359	2.27566
$16$	1.00000	0.250000
$17$	−1.17954	−0.286082	−0.143041	−	0.989717i	$-0.545688\pi$
$17$	−1.17954	−0.286082	−0.143041	+	0.989717i	$0.545688\pi$
$18$	1.09700	0.258565
$19$	−1.02879	−0.236020	−0.118010	−	0.993012i	$-0.537652\pi$
$19$	−1.02879	−0.236020	−0.118010	+	0.993012i	$0.537652\pi$
$20$	−4.35432	−0.973655
$21$	2.12825	0.464422
$22$	2.88356	0.614778
$23$	−5.84820	−1.21943	−0.609717	−	0.792620i	$-0.708717\pi$
$23$	−5.84820	−1.21943	−0.609717	+	0.792620i	$0.708717\pi$
$24$	−2.02410	−0.413168
$25$	13.9601	2.79202
$26$	0.0392106	0.00768983
$27$	3.85188	0.741294
$28$	−1.05145	−0.198706
$29$	0.295209	0.0548189	0.0274095	−	0.999624i	$-0.491274\pi$
$29$	0.295209	0.0548189	0.0274095	+	0.999624i	$0.491274\pi$
$30$	8.81359	1.60913
$31$	−2.53642	−0.455555	−0.227777	−	0.973713i	$-0.573146\pi$
$31$	−2.53642	−0.455555	−0.227777	+	0.973713i	$0.573146\pi$
$32$	1.00000	0.176777
$33$	−5.83663	−1.01603
$34$	−1.17954	−0.202290
$35$	4.57836	0.773885
$36$	1.09700	0.182833
$37$	4.45208	0.731918	0.365959	−	0.930631i	$-0.380741\pi$
$37$	4.45208	0.731918	0.365959	+	0.930631i	$0.380741\pi$
$38$	−1.02879	−0.166891
$39$	−0.0793663	−0.0127088
$40$	−4.35432	−0.688478
$41$	8.75905	1.36793	0.683967	−	0.729513i	$-0.260253\pi$
$41$	8.75905	1.36793	0.683967	+	0.729513i	$0.260253\pi$
$42$	2.12825	0.328396
$43$	−1.03046	−0.157143	−0.0785716	−	0.996908i	$-0.525036\pi$
$43$	−1.03046	−0.157143	−0.0785716	+	0.996908i	$0.525036\pi$
$44$	2.88356	0.434714
$45$	−4.77667	−0.712064
$46$	−5.84820	−0.862269
$47$	2.95692	0.431312	0.215656	−	0.976469i	$-0.430811\pi$
$47$	2.95692	0.431312	0.215656	+	0.976469i	$0.430811\pi$
$48$	−2.02410	−0.292154
$49$	−5.89445	−0.842064
$50$	13.9601	1.97426
$51$	2.38752	0.334320
$52$	0.0392106	0.00543753
$53$	7.65978	1.05215	0.526076	−	0.850437i	$-0.323663\pi$
$53$	7.65978	1.05215	0.526076	+	0.850437i	$0.323663\pi$
$54$	3.85188	0.524174
$55$	−12.5560	−1.69304
$56$	−1.05145	−0.140506
$57$	2.08237	0.275817
$58$	0.295209	0.0387628
$59$	6.95279	0.905176	0.452588	−	0.891720i	$-0.350501\pi$
$59$	6.95279	0.905176	0.452588	+	0.891720i	$0.350501\pi$
$60$	8.81359	1.13783
$61$	−0.523332	−0.0670058	−0.0335029	−	0.999439i	$-0.510666\pi$
$61$	−0.523332	−0.0670058	−0.0335029	+	0.999439i	$0.510666\pi$
$62$	−2.53642	−0.322126
$63$	−1.15344	−0.145320
$64$	1.00000	0.125000
$65$	−0.170735	−0.0211771
$66$	−5.83663	−0.718440
$67$	−0.647740	−0.0791341	−0.0395670	−	0.999217i	$-0.512598\pi$
$67$	−0.647740	−0.0791341	−0.0395670	+	0.999217i	$0.512598\pi$
$68$	−1.17954	−0.143041
$69$	11.8374	1.42505
$70$	4.57836	0.547219
$71$	7.87283	0.934333	0.467167	−	0.884169i	$-0.345275\pi$
$71$	7.87283	0.934333	0.467167	+	0.884169i	$0.345275\pi$
$72$	1.09700	0.129282
$73$	−1.23174	−0.144164	−0.0720822	−	0.997399i	$-0.522964\pi$
$73$	−1.23174	−0.144164	−0.0720822	+	0.997399i	$0.522964\pi$
$74$	4.45208	0.517544
$75$	−28.2567	−3.26280
$76$	−1.02879	−0.118010
$77$	−3.03193	−0.345521
$78$	−0.0793663	−0.00898647
$79$	13.7496	1.54695	0.773474	−	0.633829i	$-0.218518\pi$
$79$	13.7496	1.54695	0.773474	+	0.633829i	$0.218518\pi$
$80$	−4.35432	−0.486828
$81$	−11.0876	−1.23195
$82$	8.75905	0.967275
$83$	−9.54977	−1.04822	−0.524112	−	0.851649i	$-0.675603\pi$
$83$	−9.54977	−1.04822	−0.524112	+	0.851649i	$0.675603\pi$
$84$	2.12825	0.232211
$85$	5.13611	0.557090
$86$	−1.03046	−0.111117
$87$	−0.597533	−0.0640623
$88$	2.88356	0.307389
$89$	7.56611	0.802006	0.401003	−	0.916077i	$-0.368662\pi$
$89$	7.56611	0.802006	0.401003	+	0.916077i	$0.368662\pi$
$90$	−4.77667	−0.503505
$91$	−0.0412281	−0.00432188
$92$	−5.84820	−0.609717
$93$	5.13398	0.532369
$94$	2.95692	0.304984
$95$	4.47967	0.459604
$96$	−2.02410	−0.206584
$97$	−12.1742	−1.23610	−0.618052	−	0.786137i	$-0.712078\pi$
$97$	−12.1742	−1.23610	−0.618052	+	0.786137i	$0.712078\pi$
$98$	−5.89445	−0.595429
$99$	3.16326	0.317920
$100$	13.9601	1.39601
$101$	−6.37539	−0.634375	−0.317188	−	0.948363i	$-0.602738\pi$
$101$	−6.37539	−0.634375	−0.317188	+	0.948363i	$0.602738\pi$
$102$	2.38752	0.236400
$103$	5.02466	0.495094	0.247547	−	0.968876i	$-0.420375\pi$
$103$	5.02466	0.495094	0.247547	+	0.968876i	$0.420375\pi$
$104$	0.0392106	0.00384492
$105$	−9.26708	−0.904375
$106$	7.65978	0.743984
$107$	−14.7877	−1.42958	−0.714792	−	0.699337i	$-0.753479\pi$
$107$	−14.7877	−1.42958	−0.714792	+	0.699337i	$0.753479\pi$
$108$	3.85188	0.370647
$109$	1.90268	0.182243	0.0911216	−	0.995840i	$-0.470955\pi$
$109$	1.90268	0.182243	0.0911216	+	0.995840i	$0.470955\pi$
$110$	−12.5560	−1.19716
$111$	−9.01148	−0.855332
$112$	−1.05145	−0.0993530
$113$	−3.40628	−0.320436	−0.160218	−	0.987082i	$-0.551220\pi$
$113$	−3.40628	−0.320436	−0.160218	+	0.987082i	$0.551220\pi$
$114$	2.08237	0.195032
$115$	25.4649	2.37461
$116$	0.295209	0.0274095
$117$	0.0430139	0.00397663
$118$	6.95279	0.640056
$119$	1.24024	0.113692
$120$	8.81359	0.804567
$121$	−2.68506	−0.244096
$122$	−0.523332	−0.0473802
$123$	−17.7292	−1.59859
$124$	−2.53642	−0.227777
$125$	−39.0151	−3.48962
$126$	−1.15344	−0.102757
$127$	10.9274	0.969654	0.484827	−	0.874610i	$-0.338883\pi$
$127$	10.9274	0.969654	0.484827	+	0.874610i	$0.338883\pi$
$128$	1.00000	0.0883883
$129$	2.08575	0.183640
$130$	−0.170735	−0.0149745
$131$	2.89435	0.252881	0.126440	−	0.991974i	$-0.459645\pi$
$131$	2.89435	0.252881	0.126440	+	0.991974i	$0.459645\pi$
$132$	−5.83663	−0.508014
$133$	1.08172	0.0937973
$134$	−0.647740	−0.0559562
$135$	−16.7723	−1.44353
$136$	−1.17954	−0.101145
$137$	−3.12412	−0.266911	−0.133456	−	0.991055i	$-0.542607\pi$
$137$	−3.12412	−0.266911	−0.133456	+	0.991055i	$0.542607\pi$
$138$	11.8374	1.00766
$139$	−17.4903	−1.48351	−0.741754	−	0.670672i	$-0.766006\pi$
$139$	−17.4903	−1.48351	−0.741754	+	0.670672i	$0.766006\pi$
$140$	4.57836	0.386942
$141$	−5.98512	−0.504038
$142$	7.87283	0.660674
$143$	0.113066	0.00945508
$144$	1.09700	0.0914164
$145$	−1.28543	−0.106749
$146$	−1.23174	−0.101940
$147$	11.9310	0.984050
$148$	4.45208	0.365959
$149$	12.5734	1.03006	0.515028	−	0.857174i	$-0.327782\pi$
$149$	12.5734	1.03006	0.515028	+	0.857174i	$0.327782\pi$
$150$	−28.2567	−2.30715
$151$	−18.6934	−1.52124	−0.760622	−	0.649194i	$-0.775106\pi$
$151$	−18.6934	−1.52124	−0.760622	+	0.649194i	$0.775106\pi$
$152$	−1.02879	−0.0834457
$153$	−1.29396	−0.104610
$154$	−3.03193	−0.244320
$155$	11.0444	0.887107
$156$	−0.0793663	−0.00635439
$157$	−24.6973	−1.97106	−0.985531	−	0.169495i	$-0.945786\pi$
$157$	−24.6973	−1.97106	−0.985531	+	0.169495i	$0.945786\pi$
$158$	13.7496	1.09386
$159$	−15.5042	−1.22956
$160$	−4.35432	−0.344239
$161$	6.14911	0.484617
$162$	−11.0876	−0.871123
$163$	14.5423	1.13904	0.569519	−	0.821978i	$-0.307129\pi$
$163$	14.5423	1.13904	0.569519	+	0.821978i	$0.307129\pi$
$164$	8.75905	0.683967
$165$	25.4146	1.97852
$166$	−9.54977	−0.741206
$167$	−11.7540	−0.909555	−0.454777	−	0.890605i	$-0.650281\pi$
$167$	−11.7540	−0.909555	−0.454777	+	0.890605i	$0.650281\pi$
$168$	2.12825	0.164198
$169$	−12.9985	−0.999882
$170$	5.13611	0.393922
$171$	−1.12858	−0.0863044
$172$	−1.03046	−0.0785716
$173$	16.0781	1.22239	0.611196	−	0.791479i	$-0.290689\pi$
$173$	16.0781	1.22239	0.611196	+	0.791479i	$0.290689\pi$
$174$	−0.597533	−0.0452989
$175$	−14.6784	−1.10958
$176$	2.88356	0.217357
$177$	−14.0732	−1.05780
$178$	7.56611	0.567104
$179$	−12.7661	−0.954184	−0.477092	−	0.878853i	$-0.658309\pi$
$179$	−12.7661	−0.954184	−0.477092	+	0.878853i	$0.658309\pi$
$180$	−4.77667	−0.356032
$181$	−17.9019	−1.33063	−0.665317	−	0.746561i	$-0.731704\pi$
$181$	−17.9019	−1.33063	−0.665317	+	0.746561i	$0.731704\pi$
$182$	−0.0412281	−0.00305603
$183$	1.05928	0.0783041
$184$	−5.84820	−0.431135
$185$	−19.3858	−1.42527
$186$	5.13398	0.376442
$187$	−3.40129	−0.248727
$188$	2.95692	0.215656
$189$	−4.05007	−0.294599
$190$	4.47967	0.324989
$191$	7.71806	0.558459	0.279230	−	0.960224i	$-0.409921\pi$
$191$	7.71806	0.558459	0.279230	+	0.960224i	$0.409921\pi$
$192$	−2.02410	−0.146077
$193$	0.0722925	0.00520373	0.00260187	−	0.999997i	$-0.499172\pi$
$193$	0.0722925	0.00520373	0.00260187	+	0.999997i	$0.499172\pi$
$194$	−12.1742	−0.874058
$195$	0.345586	0.0247479
$196$	−5.89445	−0.421032
$197$	−7.95957	−0.567096	−0.283548	−	0.958958i	$-0.591511\pi$
$197$	−7.95957	−0.567096	−0.283548	+	0.958958i	$0.591511\pi$
$198$	3.16326	0.224803
$199$	−2.18216	−0.154689	−0.0773444	−	0.997004i	$-0.524644\pi$
$199$	−2.18216	−0.154689	−0.0773444	+	0.997004i	$0.524644\pi$
$200$	13.9601	0.987128
$201$	1.31109	0.0924774
$202$	−6.37539	−0.448571
$203$	−0.310398	−0.0217857
$204$	2.38752	0.167160
$205$	−38.1397	−2.66379
$206$	5.02466	0.350085
$207$	−6.41545	−0.445905
$208$	0.0392106	0.00271877
$209$	−2.96658	−0.205202
$210$	−9.26708	−0.639490
$211$	17.1188	1.17850	0.589252	−	0.807949i	$-0.299422\pi$
$211$	17.1188	1.17850	0.589252	+	0.807949i	$0.299422\pi$
$212$	7.65978	0.526076
$213$	−15.9354	−1.09188
$214$	−14.7877	−1.01087
$215$	4.48694	0.306007
$216$	3.85188	0.262087
$217$	2.66693	0.181043
$218$	1.90268	0.128865
$219$	2.49317	0.168473
$220$	−12.5560	−0.846522
$221$	−0.0462507	−0.00311116
$222$	−9.01148	−0.604811
$223$	−2.79015	−0.186842	−0.0934210	−	0.995627i	$-0.529780\pi$
$223$	−2.79015	−0.186842	−0.0934210	+	0.995627i	$0.529780\pi$
$224$	−1.05145	−0.0702532
$225$	15.3142	1.02094
$226$	−3.40628	−0.226583
$227$	−7.26292	−0.482057	−0.241028	−	0.970518i	$-0.577485\pi$
$227$	−7.26292	−0.482057	−0.241028	+	0.970518i	$0.577485\pi$
$228$	2.08237	0.137909
$229$	27.1878	1.79662	0.898311	−	0.439361i	$-0.144795\pi$
$229$	27.1878	1.79662	0.898311	+	0.439361i	$0.144795\pi$
$230$	25.4649	1.67911
$231$	6.13695	0.403782
$232$	0.295209	0.0193814
$233$	0.154827	0.0101431	0.00507153	−	0.999987i	$-0.498386\pi$
$233$	0.154827	0.0101431	0.00507153	+	0.999987i	$0.498386\pi$
$234$	0.0430139	0.00281191
$235$	−12.8754	−0.839898
$236$	6.95279	0.452588
$237$	−27.8306	−1.80779
$238$	1.24024	0.0803926
$239$	16.1247	1.04302	0.521510	−	0.853245i	$-0.325369\pi$
$239$	16.1247	1.04302	0.521510	+	0.853245i	$0.325369\pi$
$240$	8.81359	0.568915
$241$	−18.3279	−1.18060	−0.590302	−	0.807183i	$-0.700991\pi$
$241$	−18.3279	−1.18060	−0.590302	+	0.807183i	$0.700991\pi$
$242$	−2.68506	−0.172602
$243$	10.8868	0.698388
$244$	−0.523332	−0.0335029
$245$	25.6663	1.63976
$246$	−17.7292	−1.13037
$247$	−0.0403394	−0.00256673
$248$	−2.53642	−0.161063
$249$	19.3297	1.22497
$250$	−39.0151	−2.46753
$251$	4.90449	0.309569	0.154784	−	0.987948i	$-0.450532\pi$
$251$	4.90449	0.309569	0.154784	+	0.987948i	$0.450532\pi$
$252$	−1.15344	−0.0726599
$253$	−16.8636	−1.06021
$254$	10.9274	0.685649
$255$	−10.3960	−0.651025
$256$	1.00000	0.0625000
$257$	−21.4002	−1.33491	−0.667455	−	0.744650i	$-0.732616\pi$
$257$	−21.4002	−1.33491	−0.667455	+	0.744650i	$0.732616\pi$
$258$	2.08575	0.129853
$259$	−4.68116	−0.290873
$260$	−0.170735	−0.0105886
$261$	0.323843	0.0200454
$262$	2.89435	0.178814
$263$	−11.4654	−0.706985	−0.353493	−	0.935437i	$-0.615006\pi$
$263$	−11.4654	−0.706985	−0.353493	+	0.935437i	$0.615006\pi$
$264$	−5.83663	−0.359220
$265$	−33.3531	−2.04887
$266$	1.08172	0.0663247
$267$	−15.3146	−0.937237
$268$	−0.647740	−0.0395670
$269$	17.1024	1.04275	0.521376	−	0.853327i	$-0.325419\pi$
$269$	17.1024	1.04275	0.521376	+	0.853327i	$0.325419\pi$
$270$	−16.7723	−1.02073
$271$	−31.1834	−1.89426	−0.947129	−	0.320852i	$-0.896031\pi$
$271$	−31.1834	−1.89426	−0.947129	+	0.320852i	$0.896031\pi$
$272$	−1.17954	−0.0715204
$273$	0.0834500	0.00505062
$274$	−3.12412	−0.188735
$275$	40.2548	2.42746
$276$	11.8374	0.712525
$277$	−21.0331	−1.26376	−0.631879	−	0.775067i	$-0.717716\pi$
$277$	−21.0331	−1.26376	−0.631879	+	0.775067i	$0.717716\pi$
$278$	−17.4903	−1.04900
$279$	−2.78245	−0.166581
$280$	4.57836	0.273610
$281$	16.5693	0.988444	0.494222	−	0.869336i	$-0.335453\pi$
$281$	16.5693	0.988444	0.494222	+	0.869336i	$0.335453\pi$
$282$	−5.98512	−0.356409
$283$	−32.6615	−1.94153	−0.970763	−	0.240042i	$-0.922839\pi$
$283$	−32.6615	−1.94153	−0.970763	+	0.240042i	$0.922839\pi$
$284$	7.87283	0.467167
$285$	−9.06732	−0.537102
$286$	0.113066	0.00668575
$287$	−9.20973	−0.543633
$288$	1.09700	0.0646411
$289$	−15.6087	−0.918157
$290$	−1.28543	−0.0754833
$291$	24.6419	1.44453
$292$	−1.23174	−0.0720822
$293$	−16.5401	−0.966282	−0.483141	−	0.875543i	$-0.660504\pi$
$293$	−16.5401	−0.966282	−0.483141	+	0.875543i	$0.660504\pi$
$294$	11.9310	0.695828
$295$	−30.2747	−1.76266
$296$	4.45208	0.258772
$297$	11.1071	0.644501
$298$	12.5734	0.728359
$299$	−0.229311	−0.0132614
$300$	−28.2567	−1.63140
$301$	1.08348	0.0624506
$302$	−18.6934	−1.07568
$303$	12.9045	0.741342
$304$	−1.02879	−0.0590050
$305$	2.27875	0.130481
$306$	−1.29396	−0.0739706
$307$	16.6884	0.952460	0.476230	−	0.879321i	$-0.342003\pi$
$307$	16.6884	0.952460	0.476230	+	0.879321i	$0.342003\pi$
$308$	−3.03193	−0.172760
$309$	−10.1704	−0.578576
$310$	11.0444	0.627279
$311$	−21.8681	−1.24003	−0.620013	−	0.784592i	$-0.712873\pi$
$311$	−21.8681	−1.24003	−0.620013	+	0.784592i	$0.712873\pi$
$312$	−0.0793663	−0.00449323
$313$	−0.572303	−0.0323485	−0.0161742	−	0.999869i	$-0.505149\pi$
$313$	−0.572303	−0.0323485	−0.0161742	+	0.999869i	$0.505149\pi$
$314$	−24.6973	−1.39375
$315$	5.02245	0.282983
$316$	13.7496	0.773474
$317$	12.3661	0.694552	0.347276	−	0.937763i	$-0.387107\pi$
$317$	12.3661	0.694552	0.347276	+	0.937763i	$0.387107\pi$
$318$	−15.5042	−0.869432
$319$	0.851254	0.0476611
$320$	−4.35432	−0.243414
$321$	29.9319	1.67064
$322$	6.14911	0.342676
$323$	1.21350	0.0675210
$324$	−11.0876	−0.615977
$325$	0.547384	0.0303634
$326$	14.5423	0.805422
$327$	−3.85121	−0.212973
$328$	8.75905	0.483637
$329$	−3.10907	−0.171409
$330$	25.4146	1.39903
$331$	9.02390	0.495999	0.247999	−	0.968760i	$-0.420227\pi$
$331$	9.02390	0.495999	0.247999	+	0.968760i	$0.420227\pi$
$332$	−9.54977	−0.524112
$333$	4.88392	0.267637
$334$	−11.7540	−0.643152
$335$	2.82047	0.154099
$336$	2.12825	0.116106
$337$	−17.4716	−0.951739	−0.475869	−	0.879516i	$-0.657867\pi$
$337$	−17.4716	−0.951739	−0.475869	+	0.879516i	$0.657867\pi$
$338$	−12.9985	−0.707023
$339$	6.89467	0.374467
$340$	5.13611	0.278545
$341$	−7.31394	−0.396072
$342$	−1.12858	−0.0610264
$343$	13.5579	0.732058
$344$	−1.03046	−0.0555585
$345$	−51.5436	−2.77501
$346$	16.0781	0.864362
$347$	−22.2187	−1.19276	−0.596381	−	0.802701i	$-0.703395\pi$
$347$	−22.2187	−1.19276	−0.596381	+	0.802701i	$0.703395\pi$
$348$	−0.597533	−0.0320312
$349$	12.1111	0.648294	0.324147	−	0.946007i	$-0.394923\pi$
$349$	12.1111	0.648294	0.324147	+	0.946007i	$0.394923\pi$
$350$	−14.6784	−0.784593
$351$	0.151034	0.00806162
$352$	2.88356	0.153694
$353$	9.41829	0.501285	0.250642	−	0.968080i	$-0.419358\pi$
$353$	9.41829	0.501285	0.250642	+	0.968080i	$0.419358\pi$
$354$	−14.0732	−0.747981
$355$	−34.2808	−1.81944
$356$	7.56611	0.401003
$357$	−2.51037	−0.132863
$358$	−12.7661	−0.674710
$359$	14.5675	0.768843	0.384421	−	0.923158i	$-0.374401\pi$
$359$	14.5675	0.768843	0.384421	+	0.923158i	$0.374401\pi$
$360$	−4.77667	−0.251753
$361$	−17.9416	−0.944294
$362$	−17.9019	−0.940901
$363$	5.43484	0.285255
$364$	−0.0412281	−0.00216094
$365$	5.36339	0.280733
$366$	1.05928	0.0553693
$367$	3.82888	0.199866	0.0999330	−	0.994994i	$-0.468137\pi$
$367$	3.82888	0.199866	0.0999330	+	0.994994i	$0.468137\pi$
$368$	−5.84820	−0.304858
$369$	9.60864	0.500206
$370$	−19.3858	−1.00782
$371$	−8.05391	−0.418138
$372$	5.13398	0.266185
$373$	34.6896	1.79616	0.898079	−	0.439835i	$-0.144963\pi$
$373$	34.6896	1.79616	0.898079	+	0.439835i	$0.144963\pi$
$374$	−3.40129	−0.175877
$375$	78.9706	4.07802
$376$	2.95692	0.152492
$377$	0.0115753	0.000596159 0
$378$	−4.05007	−0.208313
$379$	−29.6533	−1.52319	−0.761593	−	0.648056i	$-0.775582\pi$
$379$	−29.6533	−1.52319	−0.761593	+	0.648056i	$0.775582\pi$
$380$	4.47967	0.229802
$381$	−22.1183	−1.13315
$382$	7.71806	0.394890
$383$	36.9674	1.88895	0.944473	−	0.328588i	$-0.106573\pi$
$383$	36.9674	1.88895	0.944473	+	0.328588i	$0.106573\pi$
$384$	−2.02410	−0.103292
$385$	13.2020	0.672837
$386$	0.0722925	0.00367959
$387$	−1.13041	−0.0574619
$388$	−12.1742	−0.618052
$389$	28.1325	1.42638	0.713188	−	0.700973i	$-0.247251\pi$
$389$	28.1325	1.42638	0.713188	+	0.700973i	$0.247251\pi$
$390$	0.345586	0.0174994
$391$	6.89821	0.348857
$392$	−5.89445	−0.297714
$393$	−5.85847	−0.295520
$394$	−7.95957	−0.400997
$395$	−59.8700	−3.01239
$396$	3.16326	0.158960
$397$	32.5647	1.63438	0.817188	−	0.576371i	$-0.195532\pi$
$397$	32.5647	1.63438	0.817188	+	0.576371i	$0.195532\pi$
$398$	−2.18216	−0.109382
$399$	−2.18952	−0.109613
$400$	13.9601	0.698005
$401$	−17.5796	−0.877883	−0.438942	−	0.898516i	$-0.644646\pi$
$401$	−17.5796	−0.877883	−0.438942	+	0.898516i	$0.644646\pi$
$402$	1.31109	0.0653914
$403$	−0.0994546	−0.00495419
$404$	−6.37539	−0.317188
$405$	48.2789	2.39900
$406$	−0.310398	−0.0154048
$407$	12.8379	0.636350
$408$	2.38752	0.118200
$409$	−17.0636	−0.843740	−0.421870	−	0.906656i	$-0.638626\pi$
$409$	−17.0636	−0.843740	−0.421870	+	0.906656i	$0.638626\pi$
$410$	−38.1397	−1.88358
$411$	6.32354	0.311917
$412$	5.02466	0.247547
$413$	−7.31054	−0.359728
$414$	−6.41545	−0.315302
$415$	41.5828	2.04122
$416$	0.0392106	0.00192246
$417$	35.4022	1.73365
$418$	−2.96658	−0.145100
$419$	−17.3220	−0.846237	−0.423118	−	0.906074i	$-0.639065\pi$
$419$	−17.3220	−0.846237	−0.423118	+	0.906074i	$0.639065\pi$
$420$	−9.26708	−0.452187
$421$	19.5775	0.954148	0.477074	−	0.878863i	$-0.341697\pi$
$421$	19.5775	0.954148	0.477074	+	0.878863i	$0.341697\pi$
$422$	17.1188	0.833329
$423$	3.24374	0.157716
$424$	7.65978	0.371992
$425$	−16.4666	−0.798745
$426$	−15.9354	−0.772074
$427$	0.550259	0.0266289
$428$	−14.7877	−0.714792
$429$	−0.228858	−0.0110494
$430$	4.48694	0.216379
$431$	−6.70802	−0.323114	−0.161557	−	0.986863i	$-0.551652\pi$
$431$	−6.70802	−0.323114	−0.161557	+	0.986863i	$0.551652\pi$
$432$	3.85188	0.185324
$433$	2.34769	0.112823	0.0564114	−	0.998408i	$-0.482034\pi$
$433$	2.34769	0.112823	0.0564114	+	0.998408i	$0.482034\pi$
$434$	2.66693	0.128017
$435$	2.60185	0.124749
$436$	1.90268	0.0911216
$437$	6.01655	0.287811
$438$	2.49317	0.119128
$439$	−4.05818	−0.193686	−0.0968432	−	0.995300i	$-0.530875\pi$
$439$	−4.05818	−0.193686	−0.0968432	+	0.995300i	$0.530875\pi$
$440$	−12.5560	−0.598582
$441$	−6.46619	−0.307914
$442$	−0.0462507	−0.00219992
$443$	8.79717	0.417966	0.208983	−	0.977919i	$-0.432985\pi$
$443$	8.79717	0.417966	0.208983	+	0.977919i	$0.432985\pi$
$444$	−9.01148	−0.427666
$445$	−32.9452	−1.56175
$446$	−2.79015	−0.132117
$447$	−25.4499	−1.20374
$448$	−1.05145	−0.0496765
$449$	−3.03760	−0.143353	−0.0716765	−	0.997428i	$-0.522835\pi$
$449$	−3.03760	−0.143353	−0.0716765	+	0.997428i	$0.522835\pi$
$450$	15.3142	0.721917
$451$	25.2573	1.18932
$452$	−3.40628	−0.160218
$453$	37.8373	1.77775
$454$	−7.26292	−0.340866
$455$	0.179520	0.00841605
$456$	2.08237	0.0975161
$457$	1.59766	0.0747354	0.0373677	−	0.999302i	$-0.488103\pi$
$457$	1.59766	0.0747354	0.0373677	+	0.999302i	$0.488103\pi$
$458$	27.1878	1.27040
$459$	−4.54346	−0.212071
$460$	25.4649	1.18731
$461$	10.2915	0.479321	0.239660	−	0.970857i	$-0.422964\pi$
$461$	10.2915	0.479321	0.239660	+	0.970857i	$0.422964\pi$
$462$	6.13695	0.285517
$463$	−4.34510	−0.201934	−0.100967	−	0.994890i	$-0.532194\pi$
$463$	−4.34510	−0.201934	−0.100967	+	0.994890i	$0.532194\pi$
$464$	0.295209	0.0137047
$465$	−22.3550	−1.03669
$466$	0.154827	0.00717222
$467$	3.50020	0.161970	0.0809850	−	0.996715i	$-0.474193\pi$
$467$	3.50020	0.161970	0.0809850	+	0.996715i	$0.474193\pi$
$468$	0.0430139	0.00198832
$469$	0.681069	0.0314488
$470$	−12.8754	−0.593898
$471$	49.9900	2.30342
$472$	6.95279	0.320028
$473$	−2.97139	−0.136625
$474$	−27.8306	−1.27830
$475$	−14.3620	−0.658973
$476$	1.24024	0.0568462
$477$	8.40275	0.384736
$478$	16.1247	0.737526
$479$	5.70181	0.260523	0.130261	−	0.991480i	$-0.458418\pi$
$479$	5.70181	0.260523	0.130261	+	0.991480i	$0.458418\pi$
$480$	8.81359	0.402284
$481$	0.174569	0.00795966
$482$	−18.3279	−0.834812
$483$	−12.4464	−0.566332
$484$	−2.68506	−0.122048
$485$	53.0104	2.40708
$486$	10.8868	0.493835
$487$	−20.5713	−0.932175	−0.466087	−	0.884739i	$-0.654337\pi$
$487$	−20.5713	−0.932175	−0.466087	+	0.884739i	$0.654337\pi$
$488$	−0.523332	−0.0236901
$489$	−29.4351	−1.33110
$490$	25.6663	1.15948
$491$	−16.7873	−0.757601	−0.378801	−	0.925478i	$-0.623663\pi$
$491$	−16.7873	−0.757601	−0.378801	+	0.925478i	$0.623663\pi$
$492$	−17.7292	−0.799295
$493$	−0.348212	−0.0156827
$494$	−0.0403394	−0.00181495
$495$	−13.7738	−0.619088
$496$	−2.53642	−0.113889
$497$	−8.27792	−0.371315
$498$	19.3297	0.866186
$499$	11.0362	0.494048	0.247024	−	0.969009i	$-0.420547\pi$
$499$	11.0362	0.494048	0.247024	+	0.969009i	$0.420547\pi$
$500$	−39.0151	−1.74481
$501$	23.7914	1.06292
$502$	4.90449	0.218898
$503$	−14.3329	−0.639072	−0.319536	−	0.947574i	$-0.603527\pi$
$503$	−14.3329	−0.639072	−0.319536	+	0.947574i	$0.603527\pi$
$504$	−1.15344	−0.0513783
$505$	27.7605	1.23533
$506$	−16.8636	−0.749680
$507$	26.3102	1.16848
$508$	10.9274	0.484827
$509$	15.5431	0.688937	0.344468	−	0.938798i	$-0.388059\pi$
$509$	15.5431	0.688937	0.344468	+	0.938798i	$0.388059\pi$
$510$	−10.3960	−0.460344
$511$	1.29512	0.0572927
$512$	1.00000	0.0441942
$513$	−3.96276	−0.174960
$514$	−21.4002	−0.943924
$515$	−21.8790	−0.964102
$516$	2.08575	0.0918201
$517$	8.52648	0.374994
$518$	−4.68116	−0.205678
$519$	−32.5437	−1.42851
$520$	−0.170735	−0.00748724
$521$	−5.38846	−0.236073	−0.118036	−	0.993009i	$-0.537660\pi$
$521$	−5.38846	−0.236073	−0.118036	+	0.993009i	$0.537660\pi$
$522$	0.323843	0.0141742
$523$	17.0224	0.744339	0.372170	−	0.928165i	$-0.378614\pi$
$523$	17.0224	0.744339	0.372170	+	0.928165i	$0.378614\pi$
$524$	2.89435	0.126440
$525$	29.7106	1.29668
$526$	−11.4654	−0.499914
$527$	2.99182	0.130326
$528$	−5.83663	−0.254007
$529$	11.2014	0.487017
$530$	−33.3531	−1.44877
$531$	7.62719	0.330992
$532$	1.08172	0.0468986
$533$	0.343447	0.0148764
$534$	−15.3146	−0.662727
$535$	64.3905	2.78384
$536$	−0.647740	−0.0279781
$537$	25.8399	1.11508
$538$	17.1024	0.737337
$539$	−16.9970	−0.732113
$540$	−16.7723	−0.721765
$541$	−9.11760	−0.391996	−0.195998	−	0.980604i	$-0.562795\pi$
$541$	−9.11760	−0.391996	−0.195998	+	0.980604i	$0.562795\pi$
$542$	−31.1834	−1.33944
$543$	36.2352	1.55500
$544$	−1.17954	−0.0505726
$545$	−8.28486	−0.354884
$546$	0.0834500	0.00357133
$547$	−6.53847	−0.279565	−0.139782	−	0.990182i	$-0.544640\pi$
$547$	−6.53847	−0.279565	−0.139782	+	0.990182i	$0.544640\pi$
$548$	−3.12412	−0.133456
$549$	−0.574093	−0.0245017
$550$	40.2548	1.71647
$551$	−0.303707	−0.0129384
$552$	11.8374	0.503831
$553$	−14.4570	−0.614776
$554$	−21.0331	−0.893612
$555$	39.2389	1.66560
$556$	−17.4903	−0.741754
$557$	37.2581	1.57868	0.789338	−	0.613959i	$-0.210424\pi$
$557$	37.2581	1.57868	0.789338	+	0.613959i	$0.210424\pi$
$558$	−2.78245	−0.117790
$559$	−0.0404048	−0.00170894
$560$	4.57836	0.193471
$561$	6.88457	0.290667
$562$	16.5693	0.698935
$563$	−10.0771	−0.424700	−0.212350	−	0.977194i	$-0.568112\pi$
$563$	−10.0771	−0.424700	−0.212350	+	0.977194i	$0.568112\pi$
$564$	−5.98512	−0.252019
$565$	14.8320	0.623989
$566$	−32.6615	−1.37287
$567$	11.6581	0.489594
$568$	7.87283	0.330337
$569$	26.2711	1.10134	0.550670	−	0.834723i	$-0.314372\pi$
$569$	26.2711	1.10134	0.550670	+	0.834723i	$0.314372\pi$
$570$	−9.06732	−0.379788
$571$	30.2592	1.26631	0.633155	−	0.774025i	$-0.281760\pi$
$571$	30.2592	1.26631	0.633155	+	0.774025i	$0.281760\pi$
$572$	0.113066	0.00472754
$573$	−15.6222	−0.652625
$574$	−9.20973	−0.384407
$575$	−81.6413	−3.40468
$576$	1.09700	0.0457082
$577$	−32.7840	−1.36482	−0.682408	−	0.730972i	$-0.739067\pi$
$577$	−32.7840	−1.36482	−0.682408	+	0.730972i	$0.739067\pi$
$578$	−15.6087	−0.649235
$579$	−0.146328	−0.00608117
$580$	−1.28543	−0.0533747
$581$	10.0411	0.416577
$582$	24.6419	1.02144
$583$	22.0875	0.914770
$584$	−1.23174	−0.0509698
$585$	−0.187296	−0.00774374
$586$	−16.5401	−0.683264
$587$	9.98585	0.412160	0.206080	−	0.978535i	$-0.433929\pi$
$587$	9.98585	0.412160	0.206080	+	0.978535i	$0.433929\pi$
$588$	11.9310	0.492025
$589$	2.60944	0.107520
$590$	−30.2747	−1.24639
$591$	16.1110	0.662717
$592$	4.45208	0.182980
$593$	−24.5275	−1.00722	−0.503611	−	0.863930i	$-0.667996\pi$
$593$	−24.5275	−1.00722	−0.503611	+	0.863930i	$0.667996\pi$
$594$	11.1071	0.455731
$595$	−5.40039	−0.221394
$596$	12.5734	0.515028
$597$	4.41691	0.180772
$598$	−0.229311	−0.00937723
$599$	3.03187	0.123879	0.0619394	−	0.998080i	$-0.480271\pi$
$599$	3.03187	0.123879	0.0619394	+	0.998080i	$0.480271\pi$
$600$	−28.2567	−1.15357
$601$	−5.45235	−0.222406	−0.111203	−	0.993798i	$-0.535470\pi$
$601$	−5.45235	−0.222406	−0.111203	+	0.993798i	$0.535470\pi$
$602$	1.08348	0.0441593
$603$	−0.710569	−0.0289366
$604$	−18.6934	−0.760622
$605$	11.6916	0.475331
$606$	12.9045	0.524208
$607$	−0.164581	−0.00668014	−0.00334007	−	0.999994i	$-0.501063\pi$
$607$	−0.164581	−0.00668014	−0.00334007	+	0.999994i	$0.501063\pi$
$608$	−1.02879	−0.0417229
$609$	0.628279	0.0254591
$610$	2.27875	0.0922640
$611$	0.115943	0.00469054
$612$	−1.29396	−0.0523051
$613$	−37.2926	−1.50624	−0.753118	−	0.657886i	$-0.771451\pi$
$613$	−37.2926	−1.50624	−0.753118	+	0.657886i	$0.771451\pi$
$614$	16.6884	0.673491
$615$	77.1987	3.11295
$616$	−3.03193	−0.122160
$617$	14.7703	0.594628	0.297314	−	0.954780i	$-0.403909\pi$
$617$	14.7703	0.594628	0.297314	+	0.954780i	$0.403909\pi$
$618$	−10.1704	−0.409115
$619$	−0.0673733	−0.00270796	−0.00135398	−	0.999999i	$-0.500431\pi$
$619$	−0.0673733	−0.00270796	−0.00135398	+	0.999999i	$0.500431\pi$
$620$	11.0444	0.443553
$621$	−22.5265	−0.903958
$622$	−21.8681	−0.876831
$623$	−7.95541	−0.318727
$624$	−0.0793663	−0.00317720
$625$	100.084	4.00335
$626$	−0.572303	−0.0228738
$627$	6.00466	0.239803
$628$	−24.6973	−0.985531
$629$	−5.25143	−0.209388
$630$	5.02245	0.200099
$631$	9.13293	0.363576	0.181788	−	0.983338i	$-0.441812\pi$
$631$	9.13293	0.363576	0.181788	+	0.983338i	$0.441812\pi$
$632$	13.7496	0.546928
$633$	−34.6502	−1.37722
$634$	12.3661	0.491122
$635$	−47.5816	−1.88822
$636$	−15.5042	−0.614781
$637$	−0.231125	−0.00915750
$638$	0.851254	0.0337015
$639$	8.63647	0.341653
$640$	−4.35432	−0.172120
$641$	−5.31710	−0.210013	−0.105006	−	0.994472i	$-0.533486\pi$
$641$	−5.31710	−0.210013	−0.105006	+	0.994472i	$0.533486\pi$
$642$	29.9319	1.18132
$643$	17.8583	0.704263	0.352131	−	0.935951i	$-0.385457\pi$
$643$	17.8583	0.704263	0.352131	+	0.935951i	$0.385457\pi$
$644$	6.14911	0.242309
$645$	−9.08203	−0.357605
$646$	1.21350	0.0477446
$647$	−26.9720	−1.06038	−0.530189	−	0.847880i	$-0.677879\pi$
$647$	−26.9720	−1.06038	−0.530189	+	0.847880i	$0.677879\pi$
$648$	−11.0876	−0.435562
$649$	20.0488	0.786985
$650$	0.547384	0.0214701
$651$	−5.39814	−0.211570
$652$	14.5423	0.569519
$653$	18.6789	0.730963	0.365481	−	0.930819i	$-0.380904\pi$
$653$	18.6789	0.730963	0.365481	+	0.930819i	$0.380904\pi$
$654$	−3.85121	−0.150594
$655$	−12.6029	−0.492437
$656$	8.75905	0.341983
$657$	−1.35122	−0.0527159
$658$	−3.10907	−0.121204
$659$	13.8484	0.539456	0.269728	−	0.962937i	$-0.413066\pi$
$659$	13.8484	0.539456	0.269728	+	0.962937i	$0.413066\pi$
$660$	25.4146	0.989260
$661$	−20.1514	−0.783800	−0.391900	−	0.920008i	$-0.628182\pi$
$661$	−20.1514	−0.783800	−0.391900	+	0.920008i	$0.628182\pi$
$662$	9.02390	0.350724
$663$	0.0936161	0.00363575
$664$	−9.54977	−0.370603
$665$	−4.71017	−0.182652
$666$	4.88392	0.189248
$667$	−1.72644	−0.0668480
$668$	−11.7540	−0.454777
$669$	5.64755	0.218347
$670$	2.82047	0.108964
$671$	−1.50906	−0.0582566
$672$	2.12825	0.0820991
$673$	−6.40679	−0.246964	−0.123482	−	0.992347i	$-0.539406\pi$
$673$	−6.40679	−0.246964	−0.123482	+	0.992347i	$0.539406\pi$
$674$	−17.4716	−0.672981
$675$	53.7726	2.06971
$676$	−12.9985	−0.499941
$677$	−25.7663	−0.990278	−0.495139	−	0.868814i	$-0.664883\pi$
$677$	−25.7663	−0.990278	−0.495139	+	0.868814i	$0.664883\pi$
$678$	6.89467	0.264788
$679$	12.8006	0.491243
$680$	5.13611	0.196961
$681$	14.7009	0.563340
$682$	−7.31394	−0.280065
$683$	−29.7027	−1.13654	−0.568271	−	0.822841i	$-0.692388\pi$
$683$	−29.7027	−1.13654	−0.568271	+	0.822841i	$0.692388\pi$
$684$	−1.12858	−0.0431522
$685$	13.6034	0.519759
$686$	13.5579	0.517643
$687$	−55.0309	−2.09956
$688$	−1.03046	−0.0392858
$689$	0.300345	0.0114422
$690$	−51.5436	−1.96223
$691$	−35.4171	−1.34733	−0.673664	−	0.739038i	$-0.735281\pi$
$691$	−35.4171	−1.34733	−0.673664	+	0.739038i	$0.735281\pi$
$692$	16.0781	0.611196
$693$	−3.32602	−0.126345
$694$	−22.2187	−0.843410
$695$	76.1584	2.88885
$696$	−0.597533	−0.0226494
$697$	−10.3317	−0.391341
$698$	12.1111	0.458413
$699$	−0.313386	−0.0118533
$700$	−14.6784	−0.554791
$701$	12.8430	0.485072	0.242536	−	0.970142i	$-0.422021\pi$
$701$	12.8430	0.485072	0.242536	+	0.970142i	$0.422021\pi$
$702$	0.151034	0.00570043
$703$	−4.58025	−0.172747
$704$	2.88356	0.108678
$705$	26.0611	0.981519
$706$	9.41829	0.354462
$707$	6.70343	0.252108
$708$	−14.0732	−0.528902
$709$	25.7888	0.968518	0.484259	−	0.874925i	$-0.339089\pi$
$709$	25.7888	0.968518	0.484259	+	0.874925i	$0.339089\pi$
$710$	−34.2808	−1.28654
$711$	15.0832	0.565665
$712$	7.56611	0.283552
$713$	14.8335	0.555519
$714$	−2.51037	−0.0939481
$715$	−0.492327	−0.0184120
$716$	−12.7661	−0.477092
$717$	−32.6381	−1.21889
$718$	14.5675	0.543654
$719$	22.8911	0.853695	0.426848	−	0.904324i	$-0.359624\pi$
$719$	22.8911	0.853695	0.426848	+	0.904324i	$0.359624\pi$
$720$	−4.77667	−0.178016
$721$	−5.28319	−0.196756
$722$	−17.9416	−0.667717
$723$	37.0975	1.37967
$724$	−17.9019	−0.665317
$725$	4.12114	0.153055
$726$	5.43484	0.201706
$727$	−36.8145	−1.36538	−0.682688	−	0.730710i	$-0.739189\pi$
$727$	−36.8145	−1.36538	−0.682688	+	0.730710i	$0.739189\pi$
$728$	−0.0412281	−0.00152802
$729$	11.2268	0.415806
$730$	5.36339	0.198508
$731$	1.21547	0.0449558
$732$	1.05928	0.0391520
$733$	3.95259	0.145992	0.0729961	−	0.997332i	$-0.476744\pi$
$733$	3.95259	0.145992	0.0729961	+	0.997332i	$0.476744\pi$
$734$	3.82888	0.141327
$735$	−51.9512	−1.91625
$736$	−5.84820	−0.215567
$737$	−1.86780	−0.0688013
$738$	9.60864	0.353699
$739$	1.50784	0.0554669	0.0277334	−	0.999615i	$-0.491171\pi$
$739$	1.50784	0.0554669	0.0277334	+	0.999615i	$0.491171\pi$
$740$	−19.3858	−0.712636
$741$	0.0816511	0.00299953
$742$	−8.05391	−0.295668
$743$	−27.2248	−0.998781	−0.499390	−	0.866377i	$-0.666443\pi$
$743$	−27.2248	−0.998781	−0.499390	+	0.866377i	$0.666443\pi$
$744$	5.13398	0.188221
$745$	−54.7487	−2.00584
$746$	34.6896	1.27008
$747$	−10.4761	−0.383299
$748$	−3.40129	−0.124364
$749$	15.5486	0.568134
$750$	78.9706	2.88360
$751$	−42.9106	−1.56583	−0.782916	−	0.622128i	$-0.786268\pi$
$751$	−42.9106	−1.56583	−0.782916	+	0.622128i	$0.786268\pi$
$752$	2.95692	0.107828
$753$	−9.92720	−0.361767
$754$	0.0115753	0.000421548 0
$755$	81.3969	2.96234
$756$	−4.05007	−0.147300
$757$	8.45866	0.307435	0.153718	−	0.988115i	$-0.450875\pi$
$757$	8.45866	0.307435	0.153718	+	0.988115i	$0.450875\pi$
$758$	−29.6533	−1.07706
$759$	34.1338	1.23898
$760$	4.47967	0.162495
$761$	10.5867	0.383766	0.191883	−	0.981418i	$-0.438541\pi$
$761$	10.5867	0.383766	0.191883	+	0.981418i	$0.438541\pi$
$762$	−22.1183	−0.801261
$763$	−2.00057	−0.0724257
$764$	7.71806	0.279230
$765$	5.63430	0.203708
$766$	36.9674	1.33569
$767$	0.272623	0.00984385
$768$	−2.02410	−0.0730386
$769$	−33.8508	−1.22069	−0.610345	−	0.792136i	$-0.708969\pi$
$769$	−33.8508	−1.22069	−0.610345	+	0.792136i	$0.708969\pi$
$770$	13.2020	0.475767
$771$	43.3163	1.56000
$772$	0.0722925	0.00260187
$773$	12.7005	0.456806	0.228403	−	0.973567i	$-0.426650\pi$
$773$	12.7005	0.456806	0.228403	+	0.973567i	$0.426650\pi$
$774$	−1.13041	−0.0406317
$775$	−35.4087	−1.27192
$776$	−12.1742	−0.437029
$777$	9.47516	0.339919
$778$	28.1325	1.00860
$779$	−9.01120	−0.322860
$780$	0.345586	0.0123740
$781$	22.7018	0.812335
$782$	6.89821	0.246679
$783$	1.13711	0.0406369
$784$	−5.89445	−0.210516
$785$	107.540	3.83827
$786$	−5.85847	−0.208965
$787$	10.2902	0.366807	0.183404	−	0.983038i	$-0.441289\pi$
$787$	10.2902	0.366807	0.183404	+	0.983038i	$0.441289\pi$
$788$	−7.95957	−0.283548
$789$	23.2071	0.826195
$790$	−59.8700	−2.13008
$791$	3.58155	0.127345
$792$	3.16326	0.112402
$793$	−0.0205201	−0.000728692 0
$794$	32.5647	1.15568
$795$	67.5102	2.39434
$796$	−2.18216	−0.0773444
$797$	11.9387	0.422890	0.211445	−	0.977390i	$-0.432183\pi$
$797$	11.9387	0.422890	0.211445	+	0.977390i	$0.432183\pi$
$798$	−2.18952	−0.0775081
$799$	−3.48783	−0.123390
$800$	13.9601	0.493564
$801$	8.29999	0.293266
$802$	−17.5796	−0.620757
$803$	−3.55180	−0.125340
$804$	1.31109	0.0462387
$805$	−26.7752	−0.943701
$806$	−0.0994546	−0.00350314
$807$	−34.6170	−1.21858
$808$	−6.37539	−0.224286
$809$	33.2388	1.16861	0.584307	−	0.811532i	$-0.301366\pi$
$809$	33.2388	1.16861	0.584307	+	0.811532i	$0.301366\pi$
$810$	48.2789	1.69635
$811$	−10.9839	−0.385696	−0.192848	−	0.981229i	$-0.561772\pi$
$811$	−10.9839	−0.385696	−0.192848	+	0.981229i	$0.561772\pi$
$812$	−0.310398	−0.0108929
$813$	63.1185	2.21366
$814$	12.8379	0.449967
$815$	−63.3217	−2.21806
$816$	2.38752	0.0835800
$817$	1.06012	0.0370890
$818$	−17.0636	−0.596615
$819$	−0.0452271	−0.00158036
$820$	−38.1397	−1.33190
$821$	−24.8138	−0.866009	−0.433004	−	0.901392i	$-0.642547\pi$
$821$	−24.8138	−0.866009	−0.433004	+	0.901392i	$0.642547\pi$
$822$	6.32354	0.220559
$823$	−32.4989	−1.13284	−0.566420	−	0.824117i	$-0.691672\pi$
$823$	−32.4989	−1.13284	−0.566420	+	0.824117i	$0.691672\pi$
$824$	5.02466	0.175042
$825$	−81.4799	−2.83677
$826$	−7.31054	−0.254366
$827$	−50.7407	−1.76443	−0.882214	−	0.470850i	$-0.843947\pi$
$827$	−50.7407	−1.76443	−0.882214	+	0.470850i	$0.843947\pi$
$828$	−6.41545	−0.222952
$829$	55.2834	1.92007	0.960036	−	0.279878i	$-0.0902940\pi$
$829$	55.2834	1.92007	0.960036	+	0.279878i	$0.0902940\pi$
$830$	41.5828	1.44336
$831$	42.5732	1.47685
$832$	0.0392106	0.00135938
$833$	6.95276	0.240899
$834$	35.4022	1.22588
$835$	51.1808	1.77119
$836$	−2.96658	−0.102601
$837$	−9.76999	−0.337700
$838$	−17.3220	−0.598380
$839$	−2.59106	−0.0894532	−0.0447266	−	0.998999i	$-0.514242\pi$
$839$	−2.59106	−0.0894532	−0.0447266	+	0.998999i	$0.514242\pi$
$840$	−9.26708	−0.319745
$841$	−28.9129	−0.996995
$842$	19.5775	0.674685
$843$	−33.5381	−1.15511
$844$	17.1188	0.589252
$845$	56.5994	1.94708
$846$	3.24374	0.111522
$847$	2.82321	0.0970068
$848$	7.65978	0.263038
$849$	66.1103	2.26890
$850$	−16.4666	−0.564798
$851$	−26.0367	−0.892525
$852$	−15.9354	−0.545939
$853$	9.68917	0.331751	0.165875	−	0.986147i	$-0.446955\pi$
$853$	9.68917	0.331751	0.165875	+	0.986147i	$0.446955\pi$
$854$	0.550259	0.0188295
$855$	4.91418	0.168061
$856$	−14.7877	−0.505434
$857$	29.0457	0.992181	0.496090	−	0.868271i	$-0.334768\pi$
$857$	29.0457	0.992181	0.496090	+	0.868271i	$0.334768\pi$
$858$	−0.228858	−0.00781308
$859$	−32.6952	−1.11555	−0.557773	−	0.829993i	$-0.688344\pi$
$859$	−32.6952	−1.11555	−0.557773	+	0.829993i	$0.688344\pi$
$860$	4.48694	0.153003
$861$	18.6414	0.635299
$862$	−6.70802	−0.228476
$863$	−3.76621	−0.128203	−0.0641016	−	0.997943i	$-0.520418\pi$
$863$	−3.76621	−0.128203	−0.0641016	+	0.997943i	$0.520418\pi$
$864$	3.85188	0.131044
$865$	−70.0090	−2.38038
$866$	2.34769	0.0797778
$867$	31.5936	1.07297
$868$	2.66693	0.0905215
$869$	39.6478	1.34496
$870$	2.60185	0.0882110
$871$	−0.0253983	−0.000860588 0
$872$	1.90268	0.0644327
$873$	−13.3551	−0.452001
$874$	6.01655	0.203513
$875$	41.0226	1.38682
$876$	2.49317	0.0842365
$877$	39.5048	1.33398	0.666991	−	0.745066i	$-0.267582\pi$
$877$	39.5048	1.33398	0.666991	+	0.745066i	$0.267582\pi$
$878$	−4.05818	−0.136957
$879$	33.4788	1.12921
$880$	−12.5560	−0.423261
$881$	23.4987	0.791692	0.395846	−	0.918317i	$-0.370451\pi$
$881$	23.4987	0.791692	0.395846	+	0.918317i	$0.370451\pi$
$882$	−6.46619	−0.217728
$883$	−40.4409	−1.36095	−0.680473	−	0.732773i	$-0.738226\pi$
$883$	−40.4409	−1.36095	−0.680473	+	0.732773i	$0.738226\pi$
$884$	−0.0462507	−0.00155558
$885$	61.2791	2.05987
$886$	8.79717	0.295547
$887$	−29.8610	−1.00263	−0.501316	−	0.865264i	$-0.667151\pi$
$887$	−29.8610	−1.00263	−0.501316	+	0.865264i	$0.667151\pi$
$888$	−9.01148	−0.302406
$889$	−11.4897	−0.385352
$890$	−32.9452	−1.10433
$891$	−31.9718	−1.07109
$892$	−2.79015	−0.0934210
$893$	−3.04205	−0.101798
$894$	−25.4499	−0.851173
$895$	55.5877	1.85809
$896$	−1.05145	−0.0351266
$897$	0.464150	0.0154975
$898$	−3.03760	−0.101366
$899$	−0.748774	−0.0249730
$900$	15.3142	0.510472
$901$	−9.03506	−0.301001
$902$	25.2573	0.840975
$903$	−2.19307	−0.0729809
$904$	−3.40628	−0.113291
$905$	77.9504	2.59116
$906$	37.8373	1.25706
$907$	0.612637	0.0203423	0.0101711	−	0.999948i	$-0.496762\pi$
$907$	0.612637	0.0203423	0.0101711	+	0.999948i	$0.496762\pi$
$908$	−7.26292	−0.241028
$909$	−6.99378	−0.231969
$910$	0.179520	0.00595104
$911$	−14.5007	−0.480430	−0.240215	−	0.970720i	$-0.577218\pi$
$911$	−14.5007	−0.480430	−0.240215	+	0.970720i	$0.577218\pi$
$912$	2.08237	0.0689543
$913$	−27.5374	−0.911354
$914$	1.59766	0.0528459
$915$	−4.61243	−0.152482
$916$	27.1878	0.898311
$917$	−3.04328	−0.100498
$918$	−4.54346	−0.149957
$919$	11.5521	0.381070	0.190535	−	0.981680i	$-0.438978\pi$
$919$	11.5521	0.381070	0.190535	+	0.981680i	$0.438978\pi$
$920$	25.4649	0.839553
$921$	−33.7791	−1.11306
$922$	10.2915	0.338931
$923$	0.308699	0.0101609
$924$	6.13695	0.201891
$925$	62.1515	2.04353
$926$	−4.34510	−0.142789
$927$	5.51203	0.181039
$928$	0.295209	0.00969071
$929$	22.8970	0.751225	0.375612	−	0.926777i	$-0.377432\pi$
$929$	22.8970	0.751225	0.375612	+	0.926777i	$0.377432\pi$
$930$	−22.3550	−0.733049
$931$	6.06413	0.198744
$932$	0.154827	0.00507153
$933$	44.2633	1.44912
$934$	3.50020	0.114530
$935$	14.8103	0.484349
$936$	0.0430139	0.00140595
$937$	−22.1870	−0.724816	−0.362408	−	0.932020i	$-0.618045\pi$
$937$	−22.1870	−0.724816	−0.362408	+	0.932020i	$0.618045\pi$
$938$	0.681069	0.0222377
$939$	1.15840	0.0378030
$940$	−12.8754	−0.419949
$941$	−3.32319	−0.108333	−0.0541665	−	0.998532i	$-0.517250\pi$
$941$	−3.32319	−0.108333	−0.0541665	+	0.998532i	$0.517250\pi$
$942$	49.9900	1.62876
$943$	−51.2246	−1.66810
$944$	6.95279	0.226294
$945$	17.6353	0.573676
$946$	−2.97139	−0.0966082
$947$	−24.0372	−0.781103	−0.390551	−	0.920581i	$-0.627716\pi$
$947$	−24.0372	−0.781103	−0.390551	+	0.920581i	$0.627716\pi$
$948$	−27.8306	−0.903894
$949$	−0.0482973	−0.00156780
$950$	−14.3620	−0.465964
$951$	−25.0304	−0.811665
$952$	1.24024	0.0401963
$953$	−19.0487	−0.617049	−0.308524	−	0.951216i	$-0.599835\pi$
$953$	−19.0487	−0.617049	−0.308524	+	0.951216i	$0.599835\pi$
$954$	8.40275	0.272049
$955$	−33.6069	−1.08749
$956$	16.1247	0.521510
$957$	−1.72303	−0.0556975
$958$	5.70181	0.184217
$959$	3.28486	0.106074
$960$	8.81359	0.284457
$961$	−24.5666	−0.792470
$962$	0.174569	0.00562833
$963$	−16.2221	−0.522750
$964$	−18.3279	−0.590302
$965$	−0.314785	−0.0101333
$966$	−12.4464	−0.400457
$967$	−35.6378	−1.14603	−0.573017	−	0.819543i	$-0.694227\pi$
$967$	−35.6378	−1.14603	−0.573017	+	0.819543i	$0.694227\pi$
$968$	−2.68506	−0.0863010
$969$	−2.45625	−0.0789062
$970$	53.0104	1.70206
$971$	−45.3624	−1.45575	−0.727874	−	0.685711i	$-0.759491\pi$
$971$	−45.3624	−1.45575	−0.727874	+	0.685711i	$0.759491\pi$
$972$	10.8868	0.349194
$973$	18.3902	0.589564
$974$	−20.5713	−0.659147
$975$	−1.10796	−0.0354832
$976$	−0.523332	−0.0167514
$977$	2.17868	0.0697021	0.0348510	−	0.999393i	$-0.488904\pi$
$977$	2.17868	0.0697021	0.0348510	+	0.999393i	$0.488904\pi$
$978$	−29.4351	−0.941230
$979$	21.8173	0.697285
$980$	25.6663	0.819880
$981$	2.08723	0.0666401
$982$	−16.7873	−0.535705
$983$	−48.1135	−1.53458	−0.767291	−	0.641299i	$-0.778396\pi$
$983$	−48.1135	−1.53458	−0.767291	+	0.641299i	$0.778396\pi$
$984$	−17.7292	−0.565187
$985$	34.6585	1.10431
$986$	−0.348212	−0.0110893
$987$	6.29308	0.200311
$988$	−0.0403394	−0.00128337
$989$	6.02631	0.191626
$990$	−13.7738	−0.437761
$991$	−31.3341	−0.995362	−0.497681	−	0.867360i	$-0.665815\pi$
$991$	−31.3341	−0.995362	−0.497681	+	0.867360i	$0.665815\pi$
$992$	−2.53642	−0.0805315
$993$	−18.2653	−0.579632
$994$	−8.27792	−0.262560
$995$	9.50180	0.301227
$996$	19.3297	0.612486
$997$	43.9673	1.39246	0.696230	−	0.717819i	$-0.254860\pi$
$997$	43.9673	1.39246	0.696230	+	0.717819i	$0.254860\pi$
$998$	11.0362	0.349344
$999$	17.1489	0.542567

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6038.2.a.b.1.14	✓	54

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6038.2.a.b.1.14	✓	54	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:	\( N \)	\(=\)	\( 6038 = 2 \cdot 3019 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6038.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.02410 q^{3} +1.00000 q^{4} -4.35432 q^{5} -2.02410 q^{6} -1.05145 q^{7} +1.00000 q^{8} +1.09700 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.02410 q^{3} +1.00000 q^{4} -4.35432 q^{5} -2.02410 q^{6} -1.05145 q^{7} +1.00000 q^{8} +1.09700 q^{9} -4.35432 q^{10} +2.88356 q^{11} -2.02410 q^{12} +0.0392106 q^{13} -1.05145 q^{14} +8.81359 q^{15} +1.00000 q^{16} -1.17954 q^{17} +1.09700 q^{18} -1.02879 q^{19} -4.35432 q^{20} +2.12825 q^{21} +2.88356 q^{22} -5.84820 q^{23} -2.02410 q^{24} +13.9601 q^{25} +0.0392106 q^{26} +3.85188 q^{27} -1.05145 q^{28} +0.295209 q^{29} +8.81359 q^{30} -2.53642 q^{31} +1.00000 q^{32} -5.83663 q^{33} -1.17954 q^{34} +4.57836 q^{35} +1.09700 q^{36} +4.45208 q^{37} -1.02879 q^{38} -0.0793663 q^{39} -4.35432 q^{40} +8.75905 q^{41} +2.12825 q^{42} -1.03046 q^{43} +2.88356 q^{44} -4.77667 q^{45} -5.84820 q^{46} +2.95692 q^{47} -2.02410 q^{48} -5.89445 q^{49} +13.9601 q^{50} +2.38752 q^{51} +0.0392106 q^{52} +7.65978 q^{53} +3.85188 q^{54} -12.5560 q^{55} -1.05145 q^{56} +2.08237 q^{57} +0.295209 q^{58} +6.95279 q^{59} +8.81359 q^{60} -0.523332 q^{61} -2.53642 q^{62} -1.15344 q^{63} +1.00000 q^{64} -0.170735 q^{65} -5.83663 q^{66} -0.647740 q^{67} -1.17954 q^{68} +11.8374 q^{69} +4.57836 q^{70} +7.87283 q^{71} +1.09700 q^{72} -1.23174 q^{73} +4.45208 q^{74} -28.2567 q^{75} -1.02879 q^{76} -3.03193 q^{77} -0.0793663 q^{78} +13.7496 q^{79} -4.35432 q^{80} -11.0876 q^{81} +8.75905 q^{82} -9.54977 q^{83} +2.12825 q^{84} +5.13611 q^{85} -1.03046 q^{86} -0.597533 q^{87} +2.88356 q^{88} +7.56611 q^{89} -4.77667 q^{90} -0.0412281 q^{91} -5.84820 q^{92} +5.13398 q^{93} +2.95692 q^{94} +4.47967 q^{95} -2.02410 q^{96} -12.1742 q^{97} -5.89445 q^{98} +3.16326 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q + 54 q^{2} - 21 q^{3} + 54 q^{4} - 14 q^{5} - 21 q^{6} - 44 q^{7} + 54 q^{8} + 39 q^{9}+O(q^{10}) \) 54 * q + 54 * q^2 - 21 * q^3 + 54 * q^4 - 14 * q^5 - 21 * q^6 - 44 * q^7 + 54 * q^8 + 39 * q^9	\( 54 q + 54 q^{2} - 21 q^{3} + 54 q^{4} - 14 q^{5} - 21 q^{6} - 44 q^{7} + 54 q^{8} + 39 q^{9} - 14 q^{10} - 31 q^{11} - 21 q^{12} - 34 q^{13} - 44 q^{14} - 22 q^{15} + 54 q^{16} - 40 q^{17} + 39 q^{18} - 44 q^{19} - 14 q^{20} - 3 q^{21} - 31 q^{22} - 33 q^{23} - 21 q^{24} + 14 q^{25} - 34 q^{26} - 66 q^{27} - 44 q^{28} - 22 q^{30} - 65 q^{31} + 54 q^{32} - 43 q^{33} - 40 q^{34} - 46 q^{35} + 39 q^{36} - 58 q^{37} - 44 q^{38} - 36 q^{39} - 14 q^{40} - 49 q^{41} - 3 q^{42} - 47 q^{43} - 31 q^{44} - 45 q^{45} - 33 q^{46} - 66 q^{47} - 21 q^{48} + 16 q^{49} + 14 q^{50} - 33 q^{51} - 34 q^{52} - 16 q^{53} - 66 q^{54} - 50 q^{55} - 44 q^{56} - 33 q^{57} - 70 q^{59} - 22 q^{60} - 40 q^{61} - 65 q^{62} - 117 q^{63} + 54 q^{64} - 33 q^{65} - 43 q^{66} - 82 q^{67} - 40 q^{68} - q^{69} - 46 q^{70} - 60 q^{71} + 39 q^{72} - 92 q^{73} - 58 q^{74} - 68 q^{75} - 44 q^{76} + 13 q^{77} - 36 q^{78} - 57 q^{79} - 14 q^{80} + 26 q^{81} - 49 q^{82} - 77 q^{83} - 3 q^{84} - 24 q^{85} - 47 q^{86} - 61 q^{87} - 31 q^{88} - 54 q^{89} - 45 q^{90} - 46 q^{91} - 33 q^{92} - 24 q^{93} - 66 q^{94} - 66 q^{95} - 21 q^{96} - 137 q^{97} + 16 q^{98} - 71 q^{99}+O(q^{100}) \) 54 * q + 54 * q^2 - 21 * q^3 + 54 * q^4 - 14 * q^5 - 21 * q^6 - 44 * q^7 + 54 * q^8 + 39 * q^9 - 14 * q^10 - 31 * q^11 - 21 * q^12 - 34 * q^13 - 44 * q^14 - 22 * q^15 + 54 * q^16 - 40 * q^17 + 39 * q^18 - 44 * q^19 - 14 * q^20 - 3 * q^21 - 31 * q^22 - 33 * q^23 - 21 * q^24 + 14 * q^25 - 34 * q^26 - 66 * q^27 - 44 * q^28 - 22 * q^30 - 65 * q^31 + 54 * q^32 - 43 * q^33 - 40 * q^34 - 46 * q^35 + 39 * q^36 - 58 * q^37 - 44 * q^38 - 36 * q^39 - 14 * q^40 - 49 * q^41 - 3 * q^42 - 47 * q^43 - 31 * q^44 - 45 * q^45 - 33 * q^46 - 66 * q^47 - 21 * q^48 + 16 * q^49 + 14 * q^50 - 33 * q^51 - 34 * q^52 - 16 * q^53 - 66 * q^54 - 50 * q^55 - 44 * q^56 - 33 * q^57 - 70 * q^59 - 22 * q^60 - 40 * q^61 - 65 * q^62 - 117 * q^63 + 54 * q^64 - 33 * q^65 - 43 * q^66 - 82 * q^67 - 40 * q^68 - q^69 - 46 * q^70 - 60 * q^71 + 39 * q^72 - 92 * q^73 - 58 * q^74 - 68 * q^75 - 44 * q^76 + 13 * q^77 - 36 * q^78 - 57 * q^79 - 14 * q^80 + 26 * q^81 - 49 * q^82 - 77 * q^83 - 3 * q^84 - 24 * q^85 - 47 * q^86 - 61 * q^87 - 31 * q^88 - 54 * q^89 - 45 * q^90 - 46 * q^91 - 33 * q^92 - 24 * q^93 - 66 * q^94 - 66 * q^95 - 21 * q^96 - 137 * q^97 + 16 * q^98 - 71 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more