LMFDB - Embedded newform 4011.2.a.l.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4011,2,Mod(1,4011)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4011, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4011.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4011 = 3 \cdot 7 \cdot 191 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4011.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:	$32.0279962507$
Analytic rank:	$0$
Dimension:	$28$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Character	$\chi$	$=$	4011.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.70611 q^{2} -1.00000 q^{3} +0.910828 q^{4} +3.20896 q^{5} +1.70611 q^{6} +1.00000 q^{7} +1.85825 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.70611 q^{2} -1.00000 q^{3} +0.910828 q^{4} +3.20896 q^{5} +1.70611 q^{6} +1.00000 q^{7} +1.85825 q^{8} +1.00000 q^{9} -5.47486 q^{10} +3.93073 q^{11} -0.910828 q^{12} +1.80679 q^{13} -1.70611 q^{14} -3.20896 q^{15} -4.99205 q^{16} +5.44043 q^{17} -1.70611 q^{18} -4.39873 q^{19} +2.92281 q^{20} -1.00000 q^{21} -6.70628 q^{22} +7.22623 q^{23} -1.85825 q^{24} +5.29743 q^{25} -3.08259 q^{26} -1.00000 q^{27} +0.910828 q^{28} -2.63405 q^{29} +5.47486 q^{30} -1.92672 q^{31} +4.80050 q^{32} -3.93073 q^{33} -9.28200 q^{34} +3.20896 q^{35} +0.910828 q^{36} +9.41420 q^{37} +7.50474 q^{38} -1.80679 q^{39} +5.96306 q^{40} +7.72760 q^{41} +1.70611 q^{42} +5.41740 q^{43} +3.58022 q^{44} +3.20896 q^{45} -12.3288 q^{46} +2.97412 q^{47} +4.99205 q^{48} +1.00000 q^{49} -9.03803 q^{50} -5.44043 q^{51} +1.64568 q^{52} +3.19507 q^{53} +1.70611 q^{54} +12.6136 q^{55} +1.85825 q^{56} +4.39873 q^{57} +4.49400 q^{58} -0.792180 q^{59} -2.92281 q^{60} +0.304948 q^{61} +3.28721 q^{62} +1.00000 q^{63} +1.79389 q^{64} +5.79792 q^{65} +6.70628 q^{66} -15.0535 q^{67} +4.95530 q^{68} -7.22623 q^{69} -5.47486 q^{70} -6.17969 q^{71} +1.85825 q^{72} +9.66203 q^{73} -16.0617 q^{74} -5.29743 q^{75} -4.00649 q^{76} +3.93073 q^{77} +3.08259 q^{78} -5.54886 q^{79} -16.0193 q^{80} +1.00000 q^{81} -13.1842 q^{82} -1.55518 q^{83} -0.910828 q^{84} +17.4581 q^{85} -9.24270 q^{86} +2.63405 q^{87} +7.30430 q^{88} -7.33957 q^{89} -5.47486 q^{90} +1.80679 q^{91} +6.58185 q^{92} +1.92672 q^{93} -5.07419 q^{94} -14.1154 q^{95} -4.80050 q^{96} -1.82774 q^{97} -1.70611 q^{98} +3.93073 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q - 6 q^{2} - 28 q^{3} + 34 q^{4} + 8 q^{5} + 6 q^{6} + 28 q^{7} - 15 q^{8} + 28 q^{9}+O(q^{10}) $ 28 * q - 6 * q^2 - 28 * q^3 + 34 * q^4 + 8 * q^5 + 6 * q^6 + 28 * q^7 - 15 * q^8 + 28 * q^9	\( 28 q - 6 q^{2} - 28 q^{3} + 34 q^{4} + 8 q^{5} + 6 q^{6} + 28 q^{7} - 15 q^{8} + 28 q^{9} + 4 q^{10} - 8 q^{11} - 34 q^{12} + 26 q^{13} - 6 q^{14} - 8 q^{15} + 62 q^{16} + 9 q^{17} - 6 q^{18} + 25 q^{19} + 20 q^{20} - 28 q^{21} + 3 q^{22} - 30 q^{23} + 15 q^{24} + 42 q^{25} + 25 q^{26} - 28 q^{27} + 34 q^{28} - 5 q^{29} - 4 q^{30} + 18 q^{31} - 26 q^{32} + 8 q^{33} + 30 q^{34} + 8 q^{35} + 34 q^{36} + 36 q^{37} - 2 q^{38} - 26 q^{39} + 28 q^{40} + 21 q^{41} + 6 q^{42} + 8 q^{43} - 20 q^{44} + 8 q^{45} + 24 q^{46} + 6 q^{47} - 62 q^{48} + 28 q^{49} - 48 q^{50} - 9 q^{51} + 54 q^{52} - 12 q^{53} + 6 q^{54} + 15 q^{55} - 15 q^{56} - 25 q^{57} + 19 q^{58} + 33 q^{59} - 20 q^{60} + 48 q^{61} + 28 q^{63} + 75 q^{64} + 21 q^{65} - 3 q^{66} + 27 q^{67} + 19 q^{68} + 30 q^{69} + 4 q^{70} - 45 q^{71} - 15 q^{72} + 61 q^{73} - 31 q^{74} - 42 q^{75} + 63 q^{76} - 8 q^{77} - 25 q^{78} + 35 q^{79} + 84 q^{80} + 28 q^{81} + 11 q^{82} + 43 q^{83} - 34 q^{84} + 43 q^{85} - q^{86} + 5 q^{87} - 27 q^{88} + 25 q^{89} + 4 q^{90} + 26 q^{91} - 102 q^{92} - 18 q^{93} + 55 q^{94} - 43 q^{95} + 26 q^{96} + 40 q^{97} - 6 q^{98} - 8 q^{99}+O(q^{100}) \) 28 * q - 6 * q^2 - 28 * q^3 + 34 * q^4 + 8 * q^5 + 6 * q^6 + 28 * q^7 - 15 * q^8 + 28 * q^9 + 4 * q^10 - 8 * q^11 - 34 * q^12 + 26 * q^13 - 6 * q^14 - 8 * q^15 + 62 * q^16 + 9 * q^17 - 6 * q^18 + 25 * q^19 + 20 * q^20 - 28 * q^21 + 3 * q^22 - 30 * q^23 + 15 * q^24 + 42 * q^25 + 25 * q^26 - 28 * q^27 + 34 * q^28 - 5 * q^29 - 4 * q^30 + 18 * q^31 - 26 * q^32 + 8 * q^33 + 30 * q^34 + 8 * q^35 + 34 * q^36 + 36 * q^37 - 2 * q^38 - 26 * q^39 + 28 * q^40 + 21 * q^41 + 6 * q^42 + 8 * q^43 - 20 * q^44 + 8 * q^45 + 24 * q^46 + 6 * q^47 - 62 * q^48 + 28 * q^49 - 48 * q^50 - 9 * q^51 + 54 * q^52 - 12 * q^53 + 6 * q^54 + 15 * q^55 - 15 * q^56 - 25 * q^57 + 19 * q^58 + 33 * q^59 - 20 * q^60 + 48 * q^61 + 28 * q^63 + 75 * q^64 + 21 * q^65 - 3 * q^66 + 27 * q^67 + 19 * q^68 + 30 * q^69 + 4 * q^70 - 45 * q^71 - 15 * q^72 + 61 * q^73 - 31 * q^74 - 42 * q^75 + 63 * q^76 - 8 * q^77 - 25 * q^78 + 35 * q^79 + 84 * q^80 + 28 * q^81 + 11 * q^82 + 43 * q^83 - 34 * q^84 + 43 * q^85 - q^86 + 5 * q^87 - 27 * q^88 + 25 * q^89 + 4 * q^90 + 26 * q^91 - 102 * q^92 - 18 * q^93 + 55 * q^94 - 43 * q^95 + 26 * q^96 + 40 * q^97 - 6 * q^98 - 8 * 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 $

$2$	−1.70611	−1.20641	−0.603203	−	0.797588i	$-0.706109\pi$
$2$	−1.70611	−1.20641	−0.603203	+	0.797588i	$0.706109\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0.910828	0.455414
$5$	3.20896	1.43509	0.717545	−	0.696512i	$-0.245266\pi$
$5$	3.20896	1.43509	0.717545	+	0.696512i	$0.245266\pi$
$6$	1.70611	0.696519
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.85825	0.656991
$9$	1.00000	0.333333
$10$	−5.47486	−1.73130
$11$	3.93073	1.18516	0.592580	−	0.805511i	$-0.298109\pi$
$11$	3.93073	1.18516	0.592580	+	0.805511i	$0.298109\pi$
$12$	−0.910828	−0.262933
$13$	1.80679	0.501114	0.250557	−	0.968102i	$-0.419386\pi$
$13$	1.80679	0.501114	0.250557	+	0.968102i	$0.419386\pi$
$14$	−1.70611	−0.455978
$15$	−3.20896	−0.828550
$16$	−4.99205	−1.24801
$17$	5.44043	1.31950	0.659749	−	0.751486i	$-0.270662\pi$
$17$	5.44043	1.31950	0.659749	+	0.751486i	$0.270662\pi$
$18$	−1.70611	−0.402135
$19$	−4.39873	−1.00914	−0.504569	−	0.863371i	$-0.668349\pi$
$19$	−4.39873	−1.00914	−0.504569	+	0.863371i	$0.668349\pi$
$20$	2.92281	0.653561
$21$	−1.00000	−0.218218
$22$	−6.70628	−1.42978
$23$	7.22623	1.50677	0.753386	−	0.657578i	$-0.228419\pi$
$23$	7.22623	1.50677	0.753386	+	0.657578i	$0.228419\pi$
$24$	−1.85825	−0.379314
$25$	5.29743	1.05949
$26$	−3.08259	−0.604546
$27$	−1.00000	−0.192450
$28$	0.910828	0.172130
$29$	−2.63405	−0.489132	−0.244566	−	0.969633i	$-0.578645\pi$
$29$	−2.63405	−0.489132	−0.244566	+	0.969633i	$0.578645\pi$
$30$	5.47486	0.999567
$31$	−1.92672	−0.346050	−0.173025	−	0.984917i	$-0.555354\pi$
$31$	−1.92672	−0.346050	−0.173025	+	0.984917i	$0.555354\pi$
$32$	4.80050	0.848617
$33$	−3.93073	−0.684253
$34$	−9.28200	−1.59185
$35$	3.20896	0.542413
$36$	0.910828	0.151805
$37$	9.41420	1.54768	0.773842	−	0.633379i	$-0.218332\pi$
$37$	9.41420	1.54768	0.773842	+	0.633379i	$0.218332\pi$
$38$	7.50474	1.21743
$39$	−1.80679	−0.289318
$40$	5.96306	0.942842
$41$	7.72760	1.20685	0.603424	−	0.797421i	$-0.293803\pi$
$41$	7.72760	1.20685	0.603424	+	0.797421i	$0.293803\pi$
$42$	1.70611	0.263259
$43$	5.41740	0.826146	0.413073	−	0.910698i	$-0.364456\pi$
$43$	5.41740	0.826146	0.413073	+	0.910698i	$0.364456\pi$
$44$	3.58022	0.539739
$45$	3.20896	0.478364
$46$	−12.3288	−1.81778
$47$	2.97412	0.433820	0.216910	−	0.976192i	$-0.430402\pi$
$47$	2.97412	0.433820	0.216910	+	0.976192i	$0.430402\pi$
$48$	4.99205	0.720540
$49$	1.00000	0.142857
$50$	−9.03803	−1.27817
$51$	−5.44043	−0.761813
$52$	1.64568	0.228214
$53$	3.19507	0.438876	0.219438	−	0.975626i	$-0.429578\pi$
$53$	3.19507	0.438876	0.219438	+	0.975626i	$0.429578\pi$
$54$	1.70611	0.232173
$55$	12.6136	1.70081
$56$	1.85825	0.248319
$57$	4.39873	0.582626
$58$	4.49400	0.590091
$59$	−0.792180	−0.103133	−0.0515665	−	0.998670i	$-0.516421\pi$
$59$	−0.792180	−0.103133	−0.0515665	+	0.998670i	$0.516421\pi$
$60$	−2.92281	−0.377333
$61$	0.304948	0.0390446	0.0195223	−	0.999809i	$-0.493785\pi$
$61$	0.304948	0.0390446	0.0195223	+	0.999809i	$0.493785\pi$
$62$	3.28721	0.417476
$63$	1.00000	0.125988
$64$	1.79389	0.224236
$65$	5.79792	0.719144
$66$	6.70628	0.825487
$67$	−15.0535	−1.83907	−0.919536	−	0.393005i	$-0.871436\pi$
$67$	−15.0535	−1.83907	−0.919536	+	0.393005i	$0.871436\pi$
$68$	4.95530	0.600918
$69$	−7.22623	−0.869936
$70$	−5.47486	−0.654370
$71$	−6.17969	−0.733395	−0.366697	−	0.930340i	$-0.619512\pi$
$71$	−6.17969	−0.733395	−0.366697	+	0.930340i	$0.619512\pi$
$72$	1.85825	0.218997
$73$	9.66203	1.13086	0.565428	−	0.824798i	$-0.308711\pi$
$73$	9.66203	1.13086	0.565428	+	0.824798i	$0.308711\pi$
$74$	−16.0617	−1.86713
$75$	−5.29743	−0.611695
$76$	−4.00649	−0.459576
$77$	3.93073	0.447949
$78$	3.08259	0.349035
$79$	−5.54886	−0.624295	−0.312148	−	0.950034i	$-0.601048\pi$
$79$	−5.54886	−0.624295	−0.312148	+	0.950034i	$0.601048\pi$
$80$	−16.0193	−1.79101
$81$	1.00000	0.111111
$82$	−13.1842	−1.45595
$83$	−1.55518	−0.170703	−0.0853517	−	0.996351i	$-0.527201\pi$
$83$	−1.55518	−0.170703	−0.0853517	+	0.996351i	$0.527201\pi$
$84$	−0.910828	−0.0993795
$85$	17.4581	1.89360
$86$	−9.24270	−0.996666
$87$	2.63405	0.282400
$88$	7.30430	0.778641
$89$	−7.33957	−0.777993	−0.388997	−	0.921239i	$-0.627178\pi$
$89$	−7.33957	−0.777993	−0.388997	+	0.921239i	$0.627178\pi$
$90$	−5.47486	−0.577101
$91$	1.80679	0.189403
$92$	6.58185	0.686205
$93$	1.92672	0.199792
$94$	−5.07419	−0.523363
$95$	−14.1154	−1.44821
$96$	−4.80050	−0.489949
$97$	−1.82774	−0.185579	−0.0927894	−	0.995686i	$-0.529578\pi$
$97$	−1.82774	−0.185579	−0.0927894	+	0.995686i	$0.529578\pi$
$98$	−1.70611	−0.172344
$99$	3.93073	0.395054
$100$	4.82505	0.482505
$101$	13.9788	1.39094	0.695470	−	0.718555i	$-0.255196\pi$
$101$	13.9788	1.39094	0.695470	+	0.718555i	$0.255196\pi$
$102$	9.28200	0.919055
$103$	2.43892	0.240314	0.120157	−	0.992755i	$-0.461660\pi$
$103$	2.43892	0.240314	0.120157	+	0.992755i	$0.461660\pi$
$104$	3.35747	0.329227
$105$	−3.20896	−0.313163
$106$	−5.45115	−0.529463
$107$	−0.857412	−0.0828891	−0.0414446	−	0.999141i	$-0.513196\pi$
$107$	−0.857412	−0.0828891	−0.0414446	+	0.999141i	$0.513196\pi$
$108$	−0.910828	−0.0876445
$109$	2.96663	0.284151	0.142076	−	0.989856i	$-0.454622\pi$
$109$	2.96663	0.284151	0.142076	+	0.989856i	$0.454622\pi$
$110$	−21.5202	−2.05187
$111$	−9.41420	−0.893556
$112$	−4.99205	−0.471704
$113$	−6.50624	−0.612056	−0.306028	−	0.952023i	$-0.599000\pi$
$113$	−6.50624	−0.612056	−0.306028	+	0.952023i	$0.599000\pi$
$114$	−7.50474	−0.702884
$115$	23.1887	2.16236
$116$	−2.39917	−0.222757
$117$	1.80679	0.167038
$118$	1.35155	0.124420
$119$	5.44043	0.498724
$120$	−5.96306	−0.544350
$121$	4.45067	0.404606
$122$	−0.520277	−0.0471036
$123$	−7.72760	−0.696774
$124$	−1.75491	−0.157596
$125$	0.954443	0.0853680
$126$	−1.70611	−0.151993
$127$	−13.9503	−1.23789	−0.618946	−	0.785434i	$-0.712440\pi$
$127$	−13.9503	−1.23789	−0.618946	+	0.785434i	$0.712440\pi$
$128$	−12.6616	−1.11914
$129$	−5.41740	−0.476975
$130$	−9.89192	−0.867579
$131$	−3.99333	−0.348899	−0.174449	−	0.984666i	$-0.555815\pi$
$131$	−3.99333	−0.348899	−0.174449	+	0.984666i	$0.555815\pi$
$132$	−3.58022	−0.311618
$133$	−4.39873	−0.381419
$134$	25.6829	2.21867
$135$	−3.20896	−0.276183
$136$	10.1097	0.866899
$137$	5.32682	0.455101	0.227550	−	0.973766i	$-0.426928\pi$
$137$	5.32682	0.455101	0.227550	+	0.973766i	$0.426928\pi$
$138$	12.3288	1.04949
$139$	2.29228	0.194428	0.0972142	−	0.995263i	$-0.469007\pi$
$139$	2.29228	0.194428	0.0972142	+	0.995263i	$0.469007\pi$
$140$	2.92281	0.247023
$141$	−2.97412	−0.250466
$142$	10.5433	0.884771
$143$	7.10202	0.593900
$144$	−4.99205	−0.416004
$145$	−8.45258	−0.701948
$146$	−16.4845	−1.36427
$147$	−1.00000	−0.0824786
$148$	8.57472	0.704837
$149$	−21.6580	−1.77429	−0.887147	−	0.461488i	$-0.847316\pi$
$149$	−21.6580	−1.77429	−0.887147	+	0.461488i	$0.847316\pi$
$150$	9.03803	0.737952
$151$	4.08106	0.332112	0.166056	−	0.986116i	$-0.446897\pi$
$151$	4.08106	0.332112	0.166056	+	0.986116i	$0.446897\pi$
$152$	−8.17396	−0.662995
$153$	5.44043	0.439833
$154$	−6.70628	−0.540408
$155$	−6.18278	−0.496613
$156$	−1.64568	−0.131760
$157$	−5.22506	−0.417005	−0.208503	−	0.978022i	$-0.566859\pi$
$157$	−5.22506	−0.417005	−0.208503	+	0.978022i	$0.566859\pi$
$158$	9.46699	0.753153
$159$	−3.19507	−0.253385
$160$	15.4046	1.21784
$161$	7.22623	0.569506
$162$	−1.70611	−0.134045
$163$	3.98186	0.311884	0.155942	−	0.987766i	$-0.450159\pi$
$163$	3.98186	0.311884	0.155942	+	0.987766i	$0.450159\pi$
$164$	7.03851	0.549616
$165$	−12.6136	−0.981965
$166$	2.65332	0.205938
$167$	−16.5291	−1.27906	−0.639530	−	0.768766i	$-0.720871\pi$
$167$	−16.5291	−1.27906	−0.639530	+	0.768766i	$0.720871\pi$
$168$	−1.85825	−0.143367
$169$	−9.73551	−0.748885
$170$	−29.7856	−2.28445
$171$	−4.39873	−0.336380
$172$	4.93432	0.376238
$173$	−6.84872	−0.520699	−0.260349	−	0.965514i	$-0.583838\pi$
$173$	−6.84872	−0.520699	−0.260349	+	0.965514i	$0.583838\pi$
$174$	−4.49400	−0.340689
$175$	5.29743	0.400448
$176$	−19.6224	−1.47910
$177$	0.792180	0.0595439
$178$	12.5222	0.938575
$179$	−8.39954	−0.627811	−0.313905	−	0.949454i	$-0.601637\pi$
$179$	−8.39954	−0.627811	−0.313905	+	0.949454i	$0.601637\pi$
$180$	2.92281	0.217854
$181$	−26.2922	−1.95428	−0.977141	−	0.212591i	$-0.931810\pi$
$181$	−26.2922	−1.95428	−0.977141	+	0.212591i	$0.931810\pi$
$182$	−3.08259	−0.228497
$183$	−0.304948	−0.0225424
$184$	13.4282	0.989937
$185$	30.2098	2.22107
$186$	−3.28721	−0.241030
$187$	21.3849	1.56382
$188$	2.70891	0.197568
$189$	−1.00000	−0.0727393
$190$	24.0824	1.74712
$191$	−1.00000	−0.0723575
$191$	−1.00000	−0.0723575
$192$	−1.79389	−0.129463
$193$	−4.43744	−0.319414	−0.159707	−	0.987164i	$-0.551055\pi$
$193$	−4.43744	−0.319414	−0.159707	+	0.987164i	$0.551055\pi$
$194$	3.11833	0.223883
$195$	−5.79792	−0.415198
$196$	0.910828	0.0650592
$197$	0.822598	0.0586077	0.0293039	−	0.999571i	$-0.490671\pi$
$197$	0.822598	0.0586077	0.0293039	+	0.999571i	$0.490671\pi$
$198$	−6.70628	−0.476595
$199$	19.2383	1.36377	0.681883	−	0.731461i	$-0.261161\pi$
$199$	19.2383	1.36377	0.681883	+	0.731461i	$0.261161\pi$
$200$	9.84396	0.696073
$201$	15.0535	1.06179
$202$	−23.8494	−1.67804
$203$	−2.63405	−0.184874
$204$	−4.95530	−0.346940
$205$	24.7976	1.73194
$206$	−4.16108	−0.289916
$207$	7.22623	0.502258
$208$	−9.01959	−0.625396
$209$	−17.2903	−1.19599
$210$	5.47486	0.377801
$211$	26.7920	1.84444	0.922220	−	0.386665i	$-0.126373\pi$
$211$	26.7920	1.84444	0.922220	+	0.386665i	$0.126373\pi$
$212$	2.91016	0.199870
$213$	6.17969	0.423426
$214$	1.46284	0.0999979
$215$	17.3842	1.18559
$216$	−1.85825	−0.126438
$217$	−1.92672	−0.130795
$218$	−5.06141	−0.342802
$219$	−9.66203	−0.652900
$220$	11.4888	0.774575
$221$	9.82972	0.661219
$222$	16.0617	1.07799
$223$	−5.12252	−0.343030	−0.171515	−	0.985182i	$-0.554866\pi$
$223$	−5.12252	−0.343030	−0.171515	+	0.985182i	$0.554866\pi$
$224$	4.80050	0.320747
$225$	5.29743	0.353162
$226$	11.1004	0.738388
$227$	4.50097	0.298740	0.149370	−	0.988781i	$-0.452275\pi$
$227$	4.50097	0.298740	0.149370	+	0.988781i	$0.452275\pi$
$228$	4.00649	0.265336
$229$	−7.61185	−0.503005	−0.251502	−	0.967857i	$-0.580925\pi$
$229$	−7.61185	−0.503005	−0.251502	+	0.967857i	$0.580925\pi$
$230$	−39.5626	−2.60868
$231$	−3.93073	−0.258623
$232$	−4.89474	−0.321355
$233$	−19.3040	−1.26465	−0.632323	−	0.774705i	$-0.717898\pi$
$233$	−19.3040	−1.26465	−0.632323	+	0.774705i	$0.717898\pi$
$234$	−3.08259	−0.201515
$235$	9.54384	0.622572
$236$	−0.721540	−0.0469682
$237$	5.54886	0.360437
$238$	−9.28200	−0.601663
$239$	−8.56173	−0.553812	−0.276906	−	0.960897i	$-0.589309\pi$
$239$	−8.56173	−0.553812	−0.276906	+	0.960897i	$0.589309\pi$
$240$	16.0193	1.03404
$241$	−27.1502	−1.74890	−0.874450	−	0.485116i	$-0.838778\pi$
$241$	−27.1502	−1.74890	−0.874450	+	0.485116i	$0.838778\pi$
$242$	−7.59336	−0.488119
$243$	−1.00000	−0.0641500
$244$	0.277755	0.0177815
$245$	3.20896	0.205013
$246$	13.1842	0.840592
$247$	−7.94759	−0.505693
$248$	−3.58034	−0.227352
$249$	1.55518	0.0985557
$250$	−1.62839	−0.102988
$251$	−24.1749	−1.52590	−0.762952	−	0.646455i	$-0.776251\pi$
$251$	−24.1749	−1.52590	−0.762952	+	0.646455i	$0.776251\pi$
$252$	0.910828	0.0573768
$253$	28.4044	1.78577
$254$	23.8009	1.49340
$255$	−17.4581	−1.09327
$256$	18.0143	1.12590
$257$	−17.5612	−1.09544	−0.547718	−	0.836663i	$-0.684503\pi$
$257$	−17.5612	−1.09544	−0.547718	+	0.836663i	$0.684503\pi$
$258$	9.24270	0.575426
$259$	9.41420	0.584970
$260$	5.28091	0.327508
$261$	−2.63405	−0.163044
$262$	6.81308	0.420913
$263$	−2.60071	−0.160367	−0.0801834	−	0.996780i	$-0.525551\pi$
$263$	−2.60071	−0.160367	−0.0801834	+	0.996780i	$0.525551\pi$
$264$	−7.30430	−0.449548
$265$	10.2528	0.629827
$266$	7.50474	0.460145
$267$	7.33957	0.449175
$268$	−13.7111	−0.837540
$269$	29.5694	1.80288	0.901439	−	0.432905i	$-0.142512\pi$
$269$	29.5694	1.80288	0.901439	+	0.432905i	$0.142512\pi$
$270$	5.47486	0.333189
$271$	−1.40916	−0.0856005	−0.0428003	−	0.999084i	$-0.513628\pi$
$271$	−1.40916	−0.0856005	−0.0428003	+	0.999084i	$0.513628\pi$
$272$	−27.1589	−1.64675
$273$	−1.80679	−0.109352
$274$	−9.08816	−0.549036
$275$	20.8228	1.25566
$276$	−6.58185	−0.396181
$277$	7.42225	0.445960	0.222980	−	0.974823i	$-0.428422\pi$
$277$	7.42225	0.445960	0.222980	+	0.974823i	$0.428422\pi$
$278$	−3.91089	−0.234559
$279$	−1.92672	−0.115350
$280$	5.96306	0.356361
$281$	22.2116	1.32504	0.662518	−	0.749046i	$-0.269488\pi$
$281$	22.2116	1.32504	0.662518	+	0.749046i	$0.269488\pi$
$282$	5.07419	0.302164
$283$	−2.15591	−0.128155	−0.0640777	−	0.997945i	$-0.520411\pi$
$283$	−2.15591	−0.128155	−0.0640777	+	0.997945i	$0.520411\pi$
$284$	−5.62864	−0.333998
$285$	14.1154	0.836122
$286$	−12.1169	−0.716485
$287$	7.72760	0.456146
$288$	4.80050	0.282872
$289$	12.5983	0.741077
$290$	14.4211	0.846834
$291$	1.82774	0.107144
$292$	8.80045	0.515008
$293$	4.32175	0.252479	0.126240	−	0.992000i	$-0.459709\pi$
$293$	4.32175	0.252479	0.126240	+	0.992000i	$0.459709\pi$
$294$	1.70611	0.0995026
$295$	−2.54207	−0.148005
$296$	17.4940	1.01682
$297$	−3.93073	−0.228084
$298$	36.9510	2.14052
$299$	13.0563	0.755064
$300$	−4.82505	−0.278574
$301$	5.41740	0.312254
$302$	−6.96275	−0.400662
$303$	−13.9788	−0.803060
$304$	21.9587	1.25942
$305$	0.978567	0.0560326
$306$	−9.28200	−0.530617
$307$	15.2215	0.868735	0.434368	−	0.900736i	$-0.356972\pi$
$307$	15.2215	0.868735	0.434368	+	0.900736i	$0.356972\pi$
$308$	3.58022	0.204002
$309$	−2.43892	−0.138746
$310$	10.5485	0.599117
$311$	23.4432	1.32934	0.664671	−	0.747136i	$-0.268572\pi$
$311$	23.4432	1.32934	0.664671	+	0.747136i	$0.268572\pi$
$312$	−3.35747	−0.190080
$313$	1.04720	0.0591912	0.0295956	−	0.999562i	$-0.490578\pi$
$313$	1.04720	0.0591912	0.0295956	+	0.999562i	$0.490578\pi$
$314$	8.91455	0.503077
$315$	3.20896	0.180804
$316$	−5.05405	−0.284313
$317$	6.89749	0.387402	0.193701	−	0.981061i	$-0.437951\pi$
$317$	6.89749	0.387402	0.193701	+	0.981061i	$0.437951\pi$
$318$	5.45115	0.305685
$319$	−10.3538	−0.579700
$320$	5.75651	0.321799
$321$	0.857412	0.0478561
$322$	−12.3288	−0.687056
$323$	−23.9310	−1.33156
$324$	0.910828	0.0506016
$325$	9.57135	0.530923
$326$	−6.79352	−0.376258
$327$	−2.96663	−0.164055
$328$	14.3598	0.792889
$329$	2.97412	0.163969
$330$	21.5202	1.18465
$331$	8.88105	0.488147	0.244073	−	0.969757i	$-0.421516\pi$
$331$	8.88105	0.488147	0.244073	+	0.969757i	$0.421516\pi$
$332$	−1.41650	−0.0777407
$333$	9.41420	0.515895
$334$	28.2006	1.54307
$335$	−48.3060	−2.63924
$336$	4.99205	0.272339
$337$	−16.1445	−0.879446	−0.439723	−	0.898134i	$-0.644923\pi$
$337$	−16.1445	−0.879446	−0.439723	+	0.898134i	$0.644923\pi$
$338$	16.6099	0.903459
$339$	6.50624	0.353371
$340$	15.9014	0.862372
$341$	−7.57344	−0.410125
$342$	7.50474	0.405810
$343$	1.00000	0.0539949
$344$	10.0669	0.542770
$345$	−23.1887	−1.24844
$346$	11.6847	0.628174
$347$	−1.37477	−0.0738014	−0.0369007	−	0.999319i	$-0.511749\pi$
$347$	−1.37477	−0.0738014	−0.0369007	+	0.999319i	$0.511749\pi$
$348$	2.39917	0.128609
$349$	21.9812	1.17663	0.588313	−	0.808633i	$-0.299792\pi$
$349$	21.9812	1.17663	0.588313	+	0.808633i	$0.299792\pi$
$350$	−9.03803	−0.483103
$351$	−1.80679	−0.0964394
$352$	18.8695	1.00575
$353$	29.9433	1.59372	0.796860	−	0.604164i	$-0.206493\pi$
$353$	29.9433	1.59372	0.796860	+	0.604164i	$0.206493\pi$
$354$	−1.35155	−0.0718341
$355$	−19.8304	−1.05249
$356$	−6.68509	−0.354309
$357$	−5.44043	−0.287938
$358$	14.3306	0.757394
$359$	−33.0426	−1.74392	−0.871962	−	0.489574i	$-0.837152\pi$
$359$	−33.0426	−1.74392	−0.871962	+	0.489574i	$0.837152\pi$
$360$	5.96306	0.314281
$361$	0.348855	0.0183608
$362$	44.8575	2.35766
$363$	−4.45067	−0.233600
$364$	1.64568	0.0862569
$365$	31.0051	1.62288
$366$	0.520277	0.0271953
$367$	−0.212691	−0.0111024	−0.00555118	−	0.999985i	$-0.501767\pi$
$367$	−0.212691	−0.0111024	−0.00555118	+	0.999985i	$0.501767\pi$
$368$	−36.0737	−1.88047
$369$	7.72760	0.402283
$370$	−51.5414	−2.67951
$371$	3.19507	0.165880
$372$	1.75491	0.0909881
$373$	−0.316532	−0.0163894	−0.00819470	−	0.999966i	$-0.502608\pi$
$373$	−0.316532	−0.0163894	−0.00819470	+	0.999966i	$0.502608\pi$
$374$	−36.4851	−1.88660
$375$	−0.954443	−0.0492872
$376$	5.52667	0.285016
$377$	−4.75919	−0.245111
$378$	1.70611	0.0877531
$379$	−0.0824038	−0.00423280	−0.00211640	−	0.999998i	$-0.500674\pi$
$379$	−0.0824038	−0.00423280	−0.00211640	+	0.999998i	$0.500674\pi$
$380$	−12.8567	−0.659533
$381$	13.9503	0.714697
$382$	1.70611	0.0872924
$383$	0.780066	0.0398595	0.0199298	−	0.999801i	$-0.493656\pi$
$383$	0.780066	0.0398595	0.0199298	+	0.999801i	$0.493656\pi$
$384$	12.6616	0.646134
$385$	12.6136	0.642847
$386$	7.57078	0.385343
$387$	5.41740	0.275382
$388$	−1.66476	−0.0845152
$389$	13.8214	0.700774	0.350387	−	0.936605i	$-0.386050\pi$
$389$	13.8214	0.700774	0.350387	+	0.936605i	$0.386050\pi$
$390$	9.89192	0.500897
$391$	39.3138	1.98818
$392$	1.85825	0.0938559
$393$	3.99333	0.201437
$394$	−1.40345	−0.0707047
$395$	−17.8061	−0.895920
$396$	3.58022	0.179913
$397$	−14.7117	−0.738359	−0.369179	−	0.929358i	$-0.620361\pi$
$397$	−14.7117	−0.738359	−0.369179	+	0.929358i	$0.620361\pi$
$398$	−32.8228	−1.64526
$399$	4.39873	0.220212
$400$	−26.4450	−1.32225
$401$	−8.25988	−0.412479	−0.206239	−	0.978502i	$-0.566123\pi$
$401$	−8.25988	−0.412479	−0.206239	+	0.978502i	$0.566123\pi$
$402$	−25.6829	−1.28095
$403$	−3.48119	−0.173410
$404$	12.7323	0.633454
$405$	3.20896	0.159455
$406$	4.49400	0.223033
$407$	37.0047	1.83425
$408$	−10.1097	−0.500505
$409$	29.6754	1.46735	0.733677	−	0.679498i	$-0.237803\pi$
$409$	29.6754	1.46735	0.733677	+	0.679498i	$0.237803\pi$
$410$	−42.3075	−2.08942
$411$	−5.32682	−0.262752
$412$	2.22144	0.109443
$413$	−0.792180	−0.0389806
$414$	−12.3288	−0.605926
$415$	−4.99052	−0.244975
$416$	8.67351	0.425254
$417$	−2.29228	−0.112253
$418$	29.4992	1.44285
$419$	25.7654	1.25872	0.629362	−	0.777112i	$-0.283316\pi$
$419$	25.7654	1.25872	0.629362	+	0.777112i	$0.283316\pi$
$420$	−2.92281	−0.142619
$421$	12.8555	0.626537	0.313268	−	0.949665i	$-0.398576\pi$
$421$	12.8555	0.626537	0.313268	+	0.949665i	$0.398576\pi$
$422$	−45.7103	−2.22514
$423$	2.97412	0.144607
$424$	5.93724	0.288338
$425$	28.8203	1.39799
$426$	−10.5433	−0.510823
$427$	0.304948	0.0147575
$428$	−0.780955	−0.0377489
$429$	−7.10202	−0.342889
$430$	−29.6595	−1.43031
$431$	−10.5768	−0.509464	−0.254732	−	0.967012i	$-0.581987\pi$
$431$	−10.5768	−0.509464	−0.254732	+	0.967012i	$0.581987\pi$
$432$	4.99205	0.240180
$433$	15.0281	0.722205	0.361102	−	0.932526i	$-0.382400\pi$
$433$	15.0281	0.722205	0.361102	+	0.932526i	$0.382400\pi$
$434$	3.28721	0.157791
$435$	8.45258	0.405270
$436$	2.70209	0.129407
$437$	−31.7862	−1.52054
$438$	16.4845	0.787662
$439$	−17.3614	−0.828616	−0.414308	−	0.910137i	$-0.635976\pi$
$439$	−17.3614	−0.828616	−0.414308	+	0.910137i	$0.635976\pi$
$440$	23.4392	1.11742
$441$	1.00000	0.0476190
$442$	−16.7706	−0.797698
$443$	24.5590	1.16683	0.583416	−	0.812173i	$-0.301716\pi$
$443$	24.5590	1.16683	0.583416	+	0.812173i	$0.301716\pi$
$444$	−8.57472	−0.406938
$445$	−23.5524	−1.11649
$446$	8.73962	0.413833
$447$	21.6580	1.02439
$448$	1.79389	0.0847531
$449$	27.0570	1.27690	0.638449	−	0.769664i	$-0.279576\pi$
$449$	27.0570	1.27690	0.638449	+	0.769664i	$0.279576\pi$
$450$	−9.03803	−0.426057
$451$	30.3751	1.43031
$452$	−5.92607	−0.278739
$453$	−4.08106	−0.191745
$454$	−7.67918	−0.360402
$455$	5.79792	0.271811
$456$	8.17396	0.382781
$457$	11.9615	0.559537	0.279769	−	0.960067i	$-0.409742\pi$
$457$	11.9615	0.559537	0.279769	+	0.960067i	$0.409742\pi$
$458$	12.9867	0.606828
$459$	−5.44043	−0.253938
$460$	21.1209	0.984767
$461$	12.3868	0.576910	0.288455	−	0.957494i	$-0.406858\pi$
$461$	12.3868	0.576910	0.288455	+	0.957494i	$0.406858\pi$
$462$	6.70628	0.312005
$463$	−27.1828	−1.26329	−0.631645	−	0.775257i	$-0.717620\pi$
$463$	−27.1828	−1.26329	−0.631645	+	0.775257i	$0.717620\pi$
$464$	13.1493	0.610442
$465$	6.18278	0.286720
$466$	32.9348	1.52568
$467$	−33.2236	−1.53741	−0.768703	−	0.639606i	$-0.779097\pi$
$467$	−33.2236	−1.53741	−0.768703	+	0.639606i	$0.779097\pi$
$468$	1.64568	0.0760714
$469$	−15.0535	−0.695104
$470$	−16.2829	−0.751074
$471$	5.22506	0.240758
$472$	−1.47207	−0.0677575
$473$	21.2944	0.979115
$474$	−9.46699	−0.434833
$475$	−23.3020	−1.06917
$476$	4.95530	0.227126
$477$	3.19507	0.146292
$478$	14.6073	0.668122
$479$	38.1854	1.74474	0.872368	−	0.488849i	$-0.162583\pi$
$479$	38.1854	1.74474	0.872368	+	0.488849i	$0.162583\pi$
$480$	−15.4046	−0.703122
$481$	17.0095	0.775566
$482$	46.3214	2.10988
$483$	−7.22623	−0.328805
$484$	4.05380	0.184263
$485$	−5.86514	−0.266322
$486$	1.70611	0.0773909
$487$	−8.87994	−0.402388	−0.201194	−	0.979551i	$-0.564482\pi$
$487$	−8.87994	−0.402388	−0.201194	+	0.979551i	$0.564482\pi$
$488$	0.566671	0.0256520
$489$	−3.98186	−0.180066
$490$	−5.47486	−0.247329
$491$	−9.29509	−0.419482	−0.209741	−	0.977757i	$-0.567262\pi$
$491$	−9.29509	−0.419482	−0.209741	+	0.977757i	$0.567262\pi$
$492$	−7.03851	−0.317321
$493$	−14.3304	−0.645409
$494$	13.5595	0.610071
$495$	12.6136	0.566938
$496$	9.61830	0.431874
$497$	−6.17969	−0.277197
$498$	−2.65332	−0.118898
$499$	−1.31114	−0.0586946	−0.0293473	−	0.999569i	$-0.509343\pi$
$499$	−1.31114	−0.0586946	−0.0293473	+	0.999569i	$0.509343\pi$
$500$	0.869334	0.0388778
$501$	16.5291	0.738466
$502$	41.2451	1.84086
$503$	−38.7918	−1.72964	−0.864821	−	0.502081i	$-0.832568\pi$
$503$	−38.7918	−1.72964	−0.864821	+	0.502081i	$0.832568\pi$
$504$	1.85825	0.0827731
$505$	44.8574	1.99613
$506$	−48.4611	−2.15436
$507$	9.73551	0.432369
$508$	−12.7064	−0.563753
$509$	0.468496	0.0207657	0.0103829	−	0.999946i	$-0.496695\pi$
$509$	0.468496	0.0207657	0.0103829	+	0.999946i	$0.496695\pi$
$510$	29.7856	1.31893
$511$	9.66203	0.427423
$512$	−5.41138	−0.239151
$513$	4.39873	0.194209
$514$	29.9614	1.32154
$515$	7.82641	0.344873
$516$	−4.93432	−0.217221
$517$	11.6905	0.514147
$518$	−16.0617	−0.705711
$519$	6.84872	0.300626
$520$	10.7740	0.472471
$521$	20.1685	0.883596	0.441798	−	0.897115i	$-0.354341\pi$
$521$	20.1685	0.883596	0.441798	+	0.897115i	$0.354341\pi$
$522$	4.49400	0.196697
$523$	−5.19874	−0.227325	−0.113663	−	0.993519i	$-0.536258\pi$
$523$	−5.19874	−0.227325	−0.113663	+	0.993519i	$0.536258\pi$
$524$	−3.63724	−0.158893
$525$	−5.29743	−0.231199
$526$	4.43712	0.193467
$527$	−10.4822	−0.456612
$528$	19.6224	0.853956
$529$	29.2184	1.27036
$530$	−17.4925	−0.759827
$531$	−0.792180	−0.0343777
$532$	−4.00649	−0.173703
$533$	13.9622	0.604768
$534$	−12.5222	−0.541887
$535$	−2.75140	−0.118953
$536$	−27.9731	−1.20825
$537$	8.39954	0.362467
$538$	−50.4488	−2.17500
$539$	3.93073	0.169309
$540$	−2.92281	−0.125778
$541$	4.08058	0.175438	0.0877189	−	0.996145i	$-0.472042\pi$
$541$	4.08058	0.175438	0.0877189	+	0.996145i	$0.472042\pi$
$542$	2.40419	0.103269
$543$	26.2922	1.12831
$544$	26.1168	1.11975
$545$	9.51979	0.407783
$546$	3.08259	0.131923
$547$	−14.5474	−0.622001	−0.311000	−	0.950410i	$-0.600664\pi$
$547$	−14.5474	−0.622001	−0.311000	+	0.950410i	$0.600664\pi$
$548$	4.85181	0.207259
$549$	0.304948	0.0130149
$550$	−35.5261	−1.51484
$551$	11.5865	0.493602
$552$	−13.4282	−0.571540
$553$	−5.54886	−0.235961
$554$	−12.6632	−0.538008
$555$	−30.2098	−1.28233
$556$	2.08787	0.0885454
$557$	−23.1018	−0.978854	−0.489427	−	0.872044i	$-0.662794\pi$
$557$	−23.1018	−0.978854	−0.489427	+	0.872044i	$0.662794\pi$
$558$	3.28721	0.139159
$559$	9.78811	0.413993
$560$	−16.0193	−0.676938
$561$	−21.3849	−0.902871
$562$	−37.8956	−1.59853
$563$	−11.8331	−0.498708	−0.249354	−	0.968412i	$-0.580218\pi$
$563$	−11.8331	−0.498708	−0.249354	+	0.968412i	$0.580218\pi$
$564$	−2.70891	−0.114066
$565$	−20.8783	−0.878356
$566$	3.67822	0.154607
$567$	1.00000	0.0419961
$568$	−11.4834	−0.481834
$569$	−33.6095	−1.40898	−0.704492	−	0.709712i	$-0.748825\pi$
$569$	−33.6095	−1.40898	−0.704492	+	0.709712i	$0.748825\pi$
$570$	−24.0824	−1.00870
$571$	38.8077	1.62405	0.812026	−	0.583621i	$-0.198365\pi$
$571$	38.8077	1.62405	0.812026	+	0.583621i	$0.198365\pi$
$572$	6.46872	0.270471
$573$	1.00000	0.0417756
$574$	−13.1842	−0.550297
$575$	38.2804	1.59640
$576$	1.79389	0.0747452
$577$	15.4937	0.645009	0.322505	−	0.946568i	$-0.395475\pi$
$577$	15.4937	0.645009	0.322505	+	0.946568i	$0.395475\pi$
$578$	−21.4942	−0.894039
$579$	4.43744	0.184414
$580$	−7.69885	−0.319677
$581$	−1.55518	−0.0645198
$582$	−3.11833	−0.129259
$583$	12.5590	0.520139
$584$	17.9545	0.742962
$585$	5.79792	0.239715
$586$	−7.37340	−0.304592
$587$	18.9088	0.780448	0.390224	−	0.920720i	$-0.372398\pi$
$587$	18.9088	0.780448	0.390224	+	0.920720i	$0.372398\pi$
$588$	−0.910828	−0.0375619
$589$	8.47515	0.349212
$590$	4.33707	0.178554
$591$	−0.822598	−0.0338372
$592$	−46.9961	−1.93153
$593$	14.1788	0.582252	0.291126	−	0.956685i	$-0.405970\pi$
$593$	14.1788	0.582252	0.291126	+	0.956685i	$0.405970\pi$
$594$	6.70628	0.275162
$595$	17.4581	0.715714
$596$	−19.7267	−0.808038
$597$	−19.2383	−0.787371
$598$	−22.2755	−0.910914
$599$	−26.9763	−1.10222	−0.551110	−	0.834432i	$-0.685796\pi$
$599$	−26.9763	−1.10222	−0.551110	+	0.834432i	$0.685796\pi$
$600$	−9.84396	−0.401878
$601$	−0.954648	−0.0389409	−0.0194704	−	0.999810i	$-0.506198\pi$
$601$	−0.954648	−0.0389409	−0.0194704	+	0.999810i	$0.506198\pi$
$602$	−9.24270	−0.376705
$603$	−15.0535	−0.613024
$604$	3.71714	0.151248
$605$	14.2820	0.580647
$606$	23.8494	0.968816
$607$	−18.8223	−0.763975	−0.381988	−	0.924167i	$-0.624760\pi$
$607$	−18.8223	−0.763975	−0.381988	+	0.924167i	$0.624760\pi$
$608$	−21.1161	−0.856372
$609$	2.63405	0.106737
$610$	−1.66955	−0.0675980
$611$	5.37362	0.217393
$612$	4.95530	0.200306
$613$	46.3481	1.87198	0.935990	−	0.352026i	$-0.114507\pi$
$613$	46.3481	1.87198	0.935990	+	0.352026i	$0.114507\pi$
$614$	−25.9696	−1.04805
$615$	−24.7976	−0.999934
$616$	7.30430	0.294298
$617$	−47.6249	−1.91731	−0.958654	−	0.284576i	$-0.908147\pi$
$617$	−47.6249	−1.91731	−0.958654	+	0.284576i	$0.908147\pi$
$618$	4.16108	0.167383
$619$	−18.5321	−0.744866	−0.372433	−	0.928059i	$-0.621476\pi$
$619$	−18.5321	−0.744866	−0.372433	+	0.928059i	$0.621476\pi$
$620$	−5.63145	−0.226165
$621$	−7.22623	−0.289979
$622$	−39.9968	−1.60372
$623$	−7.33957	−0.294054
$624$	9.01959	0.361073
$625$	−23.4244	−0.936975
$626$	−1.78664	−0.0714085
$627$	17.2903	0.690506
$628$	−4.75913	−0.189910
$629$	51.2173	2.04217
$630$	−5.47486	−0.218123
$631$	41.4846	1.65148	0.825738	−	0.564055i	$-0.190759\pi$
$631$	41.4846	1.65148	0.825738	+	0.564055i	$0.190759\pi$
$632$	−10.3112	−0.410156
$633$	−26.7920	−1.06489
$634$	−11.7679	−0.467363
$635$	−44.7661	−1.77649
$636$	−2.91016	−0.115395
$637$	1.80679	0.0715877
$638$	17.6647	0.699353
$639$	−6.17969	−0.244465
$640$	−40.6305	−1.60606
$641$	−30.1593	−1.19122	−0.595611	−	0.803273i	$-0.703090\pi$
$641$	−30.1593	−1.19122	−0.595611	+	0.803273i	$0.703090\pi$
$642$	−1.46284	−0.0577338
$643$	−0.166191	−0.00655394	−0.00327697	−	0.999995i	$-0.501043\pi$
$643$	−0.166191	−0.00655394	−0.00327697	+	0.999995i	$0.501043\pi$
$644$	6.58185	0.259361
$645$	−17.3842	−0.684503
$646$	40.8291	1.60640
$647$	−37.1284	−1.45967	−0.729834	−	0.683624i	$-0.760403\pi$
$647$	−37.1284	−1.45967	−0.729834	+	0.683624i	$0.760403\pi$
$648$	1.85825	0.0729990
$649$	−3.11385	−0.122229
$650$	−16.3298	−0.640508
$651$	1.92672	0.0755143
$652$	3.62679	0.142036
$653$	21.8820	0.856308	0.428154	−	0.903706i	$-0.359164\pi$
$653$	21.8820	0.856308	0.428154	+	0.903706i	$0.359164\pi$
$654$	5.06141	0.197917
$655$	−12.8144	−0.500701
$656$	−38.5765	−1.50616
$657$	9.66203	0.376952
$658$	−5.07419	−0.197813
$659$	−32.2102	−1.25473	−0.627367	−	0.778724i	$-0.715867\pi$
$659$	−32.2102	−1.25473	−0.627367	+	0.778724i	$0.715867\pi$
$660$	−11.4888	−0.447201
$661$	−37.7797	−1.46946	−0.734730	−	0.678360i	$-0.762691\pi$
$661$	−37.7797	−1.46946	−0.734730	+	0.678360i	$0.762691\pi$
$662$	−15.1521	−0.588903
$663$	−9.82972	−0.381755
$664$	−2.88992	−0.112151
$665$	−14.1154	−0.547370
$666$	−16.0617	−0.622378
$667$	−19.0343	−0.737010
$668$	−15.0552	−0.582502
$669$	5.12252	0.198048
$670$	82.4155	3.18399
$671$	1.19867	0.0462741
$672$	−4.80050	−0.185183
$673$	10.7687	0.415102	0.207551	−	0.978224i	$-0.433451\pi$
$673$	10.7687	0.415102	0.207551	+	0.978224i	$0.433451\pi$
$674$	27.5443	1.06097
$675$	−5.29743	−0.203898
$676$	−8.86737	−0.341053
$677$	−5.33415	−0.205008	−0.102504	−	0.994733i	$-0.532685\pi$
$677$	−5.33415	−0.205008	−0.102504	+	0.994733i	$0.532685\pi$
$678$	−11.1004	−0.426308
$679$	−1.82774	−0.0701422
$680$	32.4416	1.24408
$681$	−4.50097	−0.172478
$682$	12.9212	0.494777
$683$	1.62884	0.0623260	0.0311630	−	0.999514i	$-0.490079\pi$
$683$	1.62884	0.0623260	0.0311630	+	0.999514i	$0.490079\pi$
$684$	−4.00649	−0.153192
$685$	17.0935	0.653111
$686$	−1.70611	−0.0651398
$687$	7.61185	0.290410
$688$	−27.0439	−1.03104
$689$	5.77282	0.219927
$690$	39.5626	1.50612
$691$	−11.2585	−0.428295	−0.214147	−	0.976801i	$-0.568697\pi$
$691$	−11.2585	−0.428295	−0.214147	+	0.976801i	$0.568697\pi$
$692$	−6.23801	−0.237134
$693$	3.93073	0.149316
$694$	2.34551	0.0890344
$695$	7.35583	0.279022
$696$	4.89474	0.185535
$697$	42.0415	1.59243
$698$	−37.5025	−1.41949
$699$	19.3040	0.730143
$700$	4.82505	0.182370
$701$	6.50146	0.245557	0.122778	−	0.992434i	$-0.460820\pi$
$701$	6.50146	0.245557	0.122778	+	0.992434i	$0.460820\pi$
$702$	3.08259	0.116345
$703$	−41.4105	−1.56183
$704$	7.05129	0.265755
$705$	−9.54384	−0.359442
$706$	−51.0867	−1.92267
$707$	13.9788	0.525726
$708$	0.721540	0.0271171
$709$	35.1037	1.31835	0.659174	−	0.751991i	$-0.270906\pi$
$709$	35.1037	1.31835	0.659174	+	0.751991i	$0.270906\pi$
$710$	33.8329	1.26973
$711$	−5.54886	−0.208098
$712$	−13.6388	−0.511135
$713$	−13.9229	−0.521418
$714$	9.28200	0.347370
$715$	22.7901	0.852301
$716$	−7.65054	−0.285914
$717$	8.56173	0.319744
$718$	56.3745	2.10388
$719$	−42.4511	−1.58316	−0.791579	−	0.611067i	$-0.790741\pi$
$719$	−42.4511	−1.58316	−0.791579	+	0.611067i	$0.790741\pi$
$720$	−16.0193	−0.597004
$721$	2.43892	0.0908303
$722$	−0.595187	−0.0221506
$723$	27.1502	1.00973
$724$	−23.9477	−0.890008
$725$	−13.9537	−0.518228
$726$	7.59336	0.281816
$727$	−12.7938	−0.474497	−0.237248	−	0.971449i	$-0.576246\pi$
$727$	−12.7938	−0.474497	−0.237248	+	0.971449i	$0.576246\pi$
$728$	3.35747	0.124436
$729$	1.00000	0.0370370
$730$	−52.8982	−1.95785
$731$	29.4730	1.09010
$732$	−0.277755	−0.0102661
$733$	−15.1048	−0.557908	−0.278954	−	0.960305i	$-0.589988\pi$
$733$	−15.1048	−0.557908	−0.278954	+	0.960305i	$0.589988\pi$
$734$	0.362875	0.0133940
$735$	−3.20896	−0.118364
$736$	34.6895	1.27867
$737$	−59.1711	−2.17960
$738$	−13.1842	−0.485316
$739$	−13.5179	−0.497262	−0.248631	−	0.968598i	$-0.579981\pi$
$739$	−13.5179	−0.497262	−0.248631	+	0.968598i	$0.579981\pi$
$740$	27.5159	1.01151
$741$	7.94759	0.291962
$742$	−5.45115	−0.200118
$743$	9.89506	0.363014	0.181507	−	0.983390i	$-0.441902\pi$
$743$	9.89506	0.363014	0.181507	+	0.983390i	$0.441902\pi$
$744$	3.58034	0.131262
$745$	−69.4997	−2.54627
$746$	0.540040	0.0197723
$747$	−1.55518	−0.0569011
$748$	19.4780	0.712185
$749$	−0.857412	−0.0313291
$750$	1.62839	0.0594604
$751$	54.6563	1.99444	0.997219	−	0.0745324i	$-0.0237464\pi$
$751$	54.6563	1.99444	0.997219	+	0.0745324i	$0.0237464\pi$
$752$	−14.8470	−0.541413
$753$	24.1749	0.880981
$754$	8.11972	0.295703
$755$	13.0960	0.476611
$756$	−0.910828	−0.0331265
$757$	−11.3589	−0.412848	−0.206424	−	0.978463i	$-0.566183\pi$
$757$	−11.3589	−0.412848	−0.206424	+	0.978463i	$0.566183\pi$
$758$	0.140590	0.00510647
$759$	−28.4044	−1.03101
$760$	−26.2299	−0.951459
$761$	−41.9899	−1.52213	−0.761066	−	0.648674i	$-0.775324\pi$
$761$	−41.9899	−1.52213	−0.761066	+	0.648674i	$0.775324\pi$
$762$	−23.8009	−0.862214
$763$	2.96663	0.107399
$764$	−0.910828	−0.0329526
$765$	17.4581	0.631200
$766$	−1.33088	−0.0480867
$767$	−1.43130	−0.0516814
$768$	−18.0143	−0.650037
$769$	−14.3841	−0.518703	−0.259351	−	0.965783i	$-0.583509\pi$
$769$	−14.3841	−0.518703	−0.259351	+	0.965783i	$0.583509\pi$
$770$	−21.5202	−0.775534
$771$	17.5612	0.632451
$772$	−4.04175	−0.145466
$773$	20.9323	0.752881	0.376441	−	0.926441i	$-0.377148\pi$
$773$	20.9323	0.752881	0.376441	+	0.926441i	$0.377148\pi$
$774$	−9.24270	−0.332222
$775$	−10.2067	−0.366635
$776$	−3.39640	−0.121924
$777$	−9.41420	−0.337732
$778$	−23.5810	−0.845418
$779$	−33.9916	−1.21788
$780$	−5.28091	−0.189087
$781$	−24.2907	−0.869191
$782$	−67.0739	−2.39856
$783$	2.63405	0.0941334
$784$	−4.99205	−0.178287
$785$	−16.7670	−0.598440
$786$	−6.81308	−0.243014
$787$	31.5464	1.12451	0.562254	−	0.826965i	$-0.309934\pi$
$787$	31.5464	1.12451	0.562254	+	0.826965i	$0.309934\pi$
$788$	0.749246	0.0266908
$789$	2.60071	0.0925878
$790$	30.3792	1.08084
$791$	−6.50624	−0.231335
$792$	7.30430	0.259547
$793$	0.550978	0.0195658
$794$	25.0998	0.890760
$795$	−10.2528	−0.363631
$796$	17.5228	0.621079
$797$	46.4530	1.64545	0.822725	−	0.568440i	$-0.192453\pi$
$797$	46.4530	1.64545	0.822725	+	0.568440i	$0.192453\pi$
$798$	−7.50474	−0.265665
$799$	16.1805	0.572425
$800$	25.4303	0.899098
$801$	−7.33957	−0.259331
$802$	14.0923	0.497617
$803$	37.9789	1.34025
$804$	13.7111	0.483554
$805$	23.1887	0.817294
$806$	5.93931	0.209203
$807$	−29.5694	−1.04089
$808$	25.9761	0.913836
$809$	31.2144	1.09744	0.548719	−	0.836007i	$-0.315116\pi$
$809$	31.2144	1.09744	0.548719	+	0.836007i	$0.315116\pi$
$810$	−5.47486	−0.192367
$811$	−1.97135	−0.0692235	−0.0346117	−	0.999401i	$-0.511019\pi$
$811$	−1.97135	−0.0692235	−0.0346117	+	0.999401i	$0.511019\pi$
$812$	−2.39917	−0.0841944
$813$	1.40916	0.0494215
$814$	−63.1343	−2.21286
$815$	12.7776	0.447581
$816$	27.1589	0.950752
$817$	−23.8297	−0.833695
$818$	−50.6296	−1.77022
$819$	1.80679	0.0631344
$820$	22.5863	0.788748
$821$	28.1243	0.981544	0.490772	−	0.871288i	$-0.336715\pi$
$821$	28.1243	0.981544	0.490772	+	0.871288i	$0.336715\pi$
$822$	9.08816	0.316986
$823$	6.94493	0.242085	0.121042	−	0.992647i	$-0.461376\pi$
$823$	6.94493	0.242085	0.121042	+	0.992647i	$0.461376\pi$
$824$	4.53214	0.157884
$825$	−20.8228	−0.724957
$826$	1.35155	0.0470264
$827$	−38.2034	−1.32846	−0.664231	−	0.747527i	$-0.731241\pi$
$827$	−38.2034	−1.32846	−0.664231	+	0.747527i	$0.731241\pi$
$828$	6.58185	0.228735
$829$	−2.42712	−0.0842974	−0.0421487	−	0.999111i	$-0.513420\pi$
$829$	−2.42712	−0.0842974	−0.0421487	+	0.999111i	$0.513420\pi$
$830$	8.51440	0.295539
$831$	−7.42225	−0.257475
$832$	3.24118	0.112368
$833$	5.44043	0.188500
$834$	3.91089	0.135423
$835$	−53.0413	−1.83557
$836$	−15.7484	−0.544672
$837$	1.92672	0.0665973
$838$	−43.9588	−1.51853
$839$	−22.8629	−0.789314	−0.394657	−	0.918829i	$-0.629137\pi$
$839$	−22.8629	−0.789314	−0.394657	+	0.918829i	$0.629137\pi$
$840$	−5.96306	−0.205745
$841$	−22.0618	−0.760750
$842$	−21.9329	−0.755858
$843$	−22.2116	−0.765010
$844$	24.4030	0.839984
$845$	−31.2409	−1.07472
$846$	−5.07419	−0.174454
$847$	4.45067	0.152927
$848$	−15.9499	−0.547723
$849$	2.15591	0.0739905
$850$	−49.1708	−1.68654
$851$	68.0291	2.33201
$852$	5.62864	0.192834
$853$	−43.5618	−1.49153	−0.745764	−	0.666210i	$-0.767915\pi$
$853$	−43.5618	−1.49153	−0.745764	+	0.666210i	$0.767915\pi$
$854$	−0.520277	−0.0178035
$855$	−14.1154	−0.482735
$856$	−1.59329	−0.0544574
$857$	51.6378	1.76391	0.881956	−	0.471331i	$-0.156226\pi$
$857$	51.6378	1.76391	0.881956	+	0.471331i	$0.156226\pi$
$858$	12.1169	0.413663
$859$	−41.0150	−1.39941	−0.699707	−	0.714430i	$-0.746686\pi$
$859$	−41.0150	−1.39941	−0.699707	+	0.714430i	$0.746686\pi$
$860$	15.8340	0.539936
$861$	−7.72760	−0.263356
$862$	18.0452	0.614620
$863$	39.0004	1.32759	0.663794	−	0.747915i	$-0.268945\pi$
$863$	39.0004	1.32759	0.663794	+	0.747915i	$0.268945\pi$
$864$	−4.80050	−0.163316
$865$	−21.9773	−0.747250
$866$	−25.6397	−0.871272
$867$	−12.5983	−0.427861
$868$	−1.75491	−0.0595657
$869$	−21.8111	−0.739890
$870$	−14.4211	−0.488920
$871$	−27.1985	−0.921585
$872$	5.51274	0.186685
$873$	−1.82774	−0.0618596
$874$	54.2310	1.83439
$875$	0.954443	0.0322661
$876$	−8.80045	−0.297340
$877$	35.8744	1.21139	0.605696	−	0.795696i	$-0.292895\pi$
$877$	35.8744	1.21139	0.605696	+	0.795696i	$0.292895\pi$
$878$	29.6206	0.999647
$879$	−4.32175	−0.145769
$880$	−62.9676	−2.12264
$881$	11.3755	0.383249	0.191624	−	0.981468i	$-0.438624\pi$
$881$	11.3755	0.383249	0.191624	+	0.981468i	$0.438624\pi$
$882$	−1.70611	−0.0574479
$883$	7.13105	0.239979	0.119989	−	0.992775i	$-0.461714\pi$
$883$	7.13105	0.239979	0.119989	+	0.992775i	$0.461714\pi$
$884$	8.95319	0.301128
$885$	2.54207	0.0854509
$886$	−41.9004	−1.40767
$887$	0.756378	0.0253967	0.0126983	−	0.999919i	$-0.495958\pi$
$887$	0.756378	0.0253967	0.0126983	+	0.999919i	$0.495958\pi$
$888$	−17.4940	−0.587059
$889$	−13.9503	−0.467879
$890$	40.1831	1.34694
$891$	3.93073	0.131685
$892$	−4.66574	−0.156221
$893$	−13.0824	−0.437785
$894$	−36.9510	−1.23583
$895$	−26.9538	−0.900966
$896$	−12.6616	−0.422994
$897$	−13.0563	−0.435937
$898$	−46.1623	−1.54046
$899$	5.07510	0.169264
$900$	4.82505	0.160835
$901$	17.3825	0.579097
$902$	−51.8235	−1.72553
$903$	−5.41740	−0.180280
$904$	−12.0902	−0.402116
$905$	−84.3706	−2.80457
$906$	6.96275	0.231322
$907$	−25.8551	−0.858505	−0.429253	−	0.903184i	$-0.641223\pi$
$907$	−25.8551	−0.858505	−0.429253	+	0.903184i	$0.641223\pi$
$908$	4.09961	0.136050
$909$	13.9788	0.463647
$910$	−9.89192	−0.327914
$911$	33.3009	1.10331	0.551654	−	0.834073i	$-0.313997\pi$
$911$	33.3009	1.10331	0.551654	+	0.834073i	$0.313997\pi$
$912$	−21.9587	−0.727125
$913$	−6.11301	−0.202311
$914$	−20.4078	−0.675029
$915$	−0.978567	−0.0323504
$916$	−6.93308	−0.229076
$917$	−3.99333	−0.131871
$918$	9.28200	0.306352
$919$	−54.2262	−1.78876	−0.894379	−	0.447311i	$-0.852382\pi$
$919$	−54.2262	−1.78876	−0.894379	+	0.447311i	$0.852382\pi$
$920$	43.0904	1.42065
$921$	−15.2215	−0.501565
$922$	−21.1333	−0.695987
$923$	−11.1654	−0.367514
$924$	−3.58022	−0.117781
$925$	49.8710	1.63975
$926$	46.3770	1.52404
$927$	2.43892	0.0801048
$928$	−12.6448	−0.415086
$929$	9.27730	0.304378	0.152189	−	0.988351i	$-0.451368\pi$
$929$	9.27730	0.304378	0.152189	+	0.988351i	$0.451368\pi$
$930$	−10.5485	−0.345900
$931$	−4.39873	−0.144163
$932$	−17.5826	−0.575937
$933$	−23.4432	−0.767496
$934$	56.6833	1.85473
$935$	68.6233	2.24422
$936$	3.35747	0.109742
$937$	52.8757	1.72738	0.863688	−	0.504028i	$-0.168149\pi$
$937$	52.8757	1.72738	0.863688	+	0.504028i	$0.168149\pi$
$938$	25.6829	0.838577
$939$	−1.04720	−0.0341740
$940$	8.69280	0.283528
$941$	43.3714	1.41387	0.706933	−	0.707281i	$-0.250078\pi$
$941$	43.3714	1.41387	0.706933	+	0.707281i	$0.250078\pi$
$942$	−8.91455	−0.290452
$943$	55.8414	1.81845
$944$	3.95460	0.128711
$945$	−3.20896	−0.104388
$946$	−36.3306	−1.18121
$947$	16.6006	0.539447	0.269724	−	0.962938i	$-0.413068\pi$
$947$	16.6006	0.539447	0.269724	+	0.962938i	$0.413068\pi$
$948$	5.05405	0.164148
$949$	17.4573	0.566687
$950$	39.7559	1.28985
$951$	−6.89749	−0.223666
$952$	10.1097	0.327657
$953$	10.8465	0.351352	0.175676	−	0.984448i	$-0.443789\pi$
$953$	10.8465	0.351352	0.175676	+	0.984448i	$0.443789\pi$
$954$	−5.45115	−0.176488
$955$	−3.20896	−0.103840
$956$	−7.79827	−0.252214
$957$	10.3538	0.334690
$958$	−65.1487	−2.10486
$959$	5.32682	0.172012
$960$	−5.75651	−0.185790
$961$	−27.2877	−0.880249
$962$	−29.0201	−0.935647
$963$	−0.857412	−0.0276297
$964$	−24.7292	−0.796474
$965$	−14.2396	−0.458388
$966$	12.3288	0.396672
$967$	−12.8538	−0.413351	−0.206675	−	0.978410i	$-0.566264\pi$
$967$	−12.8538	−0.413351	−0.206675	+	0.978410i	$0.566264\pi$
$968$	8.27047	0.265823
$969$	23.9310	0.768775
$970$	10.0066	0.321293
$971$	−50.4774	−1.61990	−0.809949	−	0.586501i	$-0.800505\pi$
$971$	−50.4774	−1.61990	−0.809949	+	0.586501i	$0.800505\pi$
$972$	−0.910828	−0.0292148
$973$	2.29228	0.0734870
$974$	15.1502	0.485443
$975$	−9.57135	−0.306529
$976$	−1.52232	−0.0487281
$977$	47.8738	1.53162	0.765809	−	0.643068i	$-0.222339\pi$
$977$	47.8738	1.53162	0.765809	+	0.643068i	$0.222339\pi$
$978$	6.79352	0.217233
$979$	−28.8499	−0.922047
$980$	2.92281	0.0933658
$981$	2.96663	0.0947171
$982$	15.8585	0.506065
$983$	−0.401995	−0.0128217	−0.00641083	−	0.999979i	$-0.502041\pi$
$983$	−0.401995	−0.0128217	−0.00641083	+	0.999979i	$0.502041\pi$
$984$	−14.3598	−0.457774
$985$	2.63969	0.0841074
$986$	24.4493	0.778624
$987$	−2.97412	−0.0946673
$988$	−7.23889	−0.230300
$989$	39.1474	1.24481
$990$	−21.5202	−0.683957
$991$	50.0038	1.58842	0.794212	−	0.607641i	$-0.207884\pi$
$991$	50.0038	1.58842	0.794212	+	0.607641i	$0.207884\pi$
$992$	−9.24925	−0.293664
$993$	−8.88105	−0.281832
$994$	10.5433	0.334412
$995$	61.7350	1.95713
$996$	1.41650	0.0448836
$997$	−39.2717	−1.24375	−0.621873	−	0.783118i	$-0.713628\pi$
$997$	−39.2717	−1.24375	−0.621873	+	0.783118i	$0.713628\pi$
$998$	2.23695	0.0708095
$999$	−9.41420	−0.297852

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4011.2.a.l.1.9	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4011.2.a.l.1.9	✓	28	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:	\( N \)	\(=\)	\( 4011 = 3 \cdot 7 \cdot 191 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4011.a (trivial)

\(f(q)\)	\(=\)	\(q-1.70611 q^{2} -1.00000 q^{3} +0.910828 q^{4} +3.20896 q^{5} +1.70611 q^{6} +1.00000 q^{7} +1.85825 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.70611 q^{2} -1.00000 q^{3} +0.910828 q^{4} +3.20896 q^{5} +1.70611 q^{6} +1.00000 q^{7} +1.85825 q^{8} +1.00000 q^{9} -5.47486 q^{10} +3.93073 q^{11} -0.910828 q^{12} +1.80679 q^{13} -1.70611 q^{14} -3.20896 q^{15} -4.99205 q^{16} +5.44043 q^{17} -1.70611 q^{18} -4.39873 q^{19} +2.92281 q^{20} -1.00000 q^{21} -6.70628 q^{22} +7.22623 q^{23} -1.85825 q^{24} +5.29743 q^{25} -3.08259 q^{26} -1.00000 q^{27} +0.910828 q^{28} -2.63405 q^{29} +5.47486 q^{30} -1.92672 q^{31} +4.80050 q^{32} -3.93073 q^{33} -9.28200 q^{34} +3.20896 q^{35} +0.910828 q^{36} +9.41420 q^{37} +7.50474 q^{38} -1.80679 q^{39} +5.96306 q^{40} +7.72760 q^{41} +1.70611 q^{42} +5.41740 q^{43} +3.58022 q^{44} +3.20896 q^{45} -12.3288 q^{46} +2.97412 q^{47} +4.99205 q^{48} +1.00000 q^{49} -9.03803 q^{50} -5.44043 q^{51} +1.64568 q^{52} +3.19507 q^{53} +1.70611 q^{54} +12.6136 q^{55} +1.85825 q^{56} +4.39873 q^{57} +4.49400 q^{58} -0.792180 q^{59} -2.92281 q^{60} +0.304948 q^{61} +3.28721 q^{62} +1.00000 q^{63} +1.79389 q^{64} +5.79792 q^{65} +6.70628 q^{66} -15.0535 q^{67} +4.95530 q^{68} -7.22623 q^{69} -5.47486 q^{70} -6.17969 q^{71} +1.85825 q^{72} +9.66203 q^{73} -16.0617 q^{74} -5.29743 q^{75} -4.00649 q^{76} +3.93073 q^{77} +3.08259 q^{78} -5.54886 q^{79} -16.0193 q^{80} +1.00000 q^{81} -13.1842 q^{82} -1.55518 q^{83} -0.910828 q^{84} +17.4581 q^{85} -9.24270 q^{86} +2.63405 q^{87} +7.30430 q^{88} -7.33957 q^{89} -5.47486 q^{90} +1.80679 q^{91} +6.58185 q^{92} +1.92672 q^{93} -5.07419 q^{94} -14.1154 q^{95} -4.80050 q^{96} -1.82774 q^{97} -1.70611 q^{98} +3.93073 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 6 q^{2} - 28 q^{3} + 34 q^{4} + 8 q^{5} + 6 q^{6} + 28 q^{7} - 15 q^{8} + 28 q^{9}+O(q^{10}) \) 28 * q - 6 * q^2 - 28 * q^3 + 34 * q^4 + 8 * q^5 + 6 * q^6 + 28 * q^7 - 15 * q^8 + 28 * q^9	\( 28 q - 6 q^{2} - 28 q^{3} + 34 q^{4} + 8 q^{5} + 6 q^{6} + 28 q^{7} - 15 q^{8} + 28 q^{9} + 4 q^{10} - 8 q^{11} - 34 q^{12} + 26 q^{13} - 6 q^{14} - 8 q^{15} + 62 q^{16} + 9 q^{17} - 6 q^{18} + 25 q^{19} + 20 q^{20} - 28 q^{21} + 3 q^{22} - 30 q^{23} + 15 q^{24} + 42 q^{25} + 25 q^{26} - 28 q^{27} + 34 q^{28} - 5 q^{29} - 4 q^{30} + 18 q^{31} - 26 q^{32} + 8 q^{33} + 30 q^{34} + 8 q^{35} + 34 q^{36} + 36 q^{37} - 2 q^{38} - 26 q^{39} + 28 q^{40} + 21 q^{41} + 6 q^{42} + 8 q^{43} - 20 q^{44} + 8 q^{45} + 24 q^{46} + 6 q^{47} - 62 q^{48} + 28 q^{49} - 48 q^{50} - 9 q^{51} + 54 q^{52} - 12 q^{53} + 6 q^{54} + 15 q^{55} - 15 q^{56} - 25 q^{57} + 19 q^{58} + 33 q^{59} - 20 q^{60} + 48 q^{61} + 28 q^{63} + 75 q^{64} + 21 q^{65} - 3 q^{66} + 27 q^{67} + 19 q^{68} + 30 q^{69} + 4 q^{70} - 45 q^{71} - 15 q^{72} + 61 q^{73} - 31 q^{74} - 42 q^{75} + 63 q^{76} - 8 q^{77} - 25 q^{78} + 35 q^{79} + 84 q^{80} + 28 q^{81} + 11 q^{82} + 43 q^{83} - 34 q^{84} + 43 q^{85} - q^{86} + 5 q^{87} - 27 q^{88} + 25 q^{89} + 4 q^{90} + 26 q^{91} - 102 q^{92} - 18 q^{93} + 55 q^{94} - 43 q^{95} + 26 q^{96} + 40 q^{97} - 6 q^{98} - 8 q^{99}+O(q^{100}) \) 28 * q - 6 * q^2 - 28 * q^3 + 34 * q^4 + 8 * q^5 + 6 * q^6 + 28 * q^7 - 15 * q^8 + 28 * q^9 + 4 * q^10 - 8 * q^11 - 34 * q^12 + 26 * q^13 - 6 * q^14 - 8 * q^15 + 62 * q^16 + 9 * q^17 - 6 * q^18 + 25 * q^19 + 20 * q^20 - 28 * q^21 + 3 * q^22 - 30 * q^23 + 15 * q^24 + 42 * q^25 + 25 * q^26 - 28 * q^27 + 34 * q^28 - 5 * q^29 - 4 * q^30 + 18 * q^31 - 26 * q^32 + 8 * q^33 + 30 * q^34 + 8 * q^35 + 34 * q^36 + 36 * q^37 - 2 * q^38 - 26 * q^39 + 28 * q^40 + 21 * q^41 + 6 * q^42 + 8 * q^43 - 20 * q^44 + 8 * q^45 + 24 * q^46 + 6 * q^47 - 62 * q^48 + 28 * q^49 - 48 * q^50 - 9 * q^51 + 54 * q^52 - 12 * q^53 + 6 * q^54 + 15 * q^55 - 15 * q^56 - 25 * q^57 + 19 * q^58 + 33 * q^59 - 20 * q^60 + 48 * q^61 + 28 * q^63 + 75 * q^64 + 21 * q^65 - 3 * q^66 + 27 * q^67 + 19 * q^68 + 30 * q^69 + 4 * q^70 - 45 * q^71 - 15 * q^72 + 61 * q^73 - 31 * q^74 - 42 * q^75 + 63 * q^76 - 8 * q^77 - 25 * q^78 + 35 * q^79 + 84 * q^80 + 28 * q^81 + 11 * q^82 + 43 * q^83 - 34 * q^84 + 43 * q^85 - q^86 + 5 * q^87 - 27 * q^88 + 25 * q^89 + 4 * q^90 + 26 * q^91 - 102 * q^92 - 18 * q^93 + 55 * q^94 - 43 * q^95 + 26 * q^96 + 40 * q^97 - 6 * q^98 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more