LMFDB - Embedded newform 241.2.a.a.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [241,2,Mod(1,241)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(241, 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("241.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 241 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	241.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:	$1.92439468871$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	7.7.31056073.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 3x^{5} + 11x^{4} + x^{3} - 9x^{2} + 1 $ x^7 - 3x^6 - 3x^5 + 11x^4 + x^3 - 9x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.27758$ of defining polynomial
Character	$\chi$	$=$	241.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.277577 q^{2} -0.494846 q^{3} -1.92295 q^{4} -1.23324 q^{5} -0.137358 q^{6} +1.36627 q^{7} -1.08892 q^{8} -2.75513 q^{9} +O(q^{10})$	\(q+0.277577 q^{2} -0.494846 q^{3} -1.92295 q^{4} -1.23324 q^{5} -0.137358 q^{6} +1.36627 q^{7} -1.08892 q^{8} -2.75513 q^{9} -0.342320 q^{10} -4.69806 q^{11} +0.951564 q^{12} -0.0431968 q^{13} +0.379244 q^{14} +0.610264 q^{15} +3.54364 q^{16} -7.31430 q^{17} -0.764761 q^{18} -0.697489 q^{19} +2.37146 q^{20} -0.676090 q^{21} -1.30407 q^{22} +1.41195 q^{23} +0.538848 q^{24} -3.47911 q^{25} -0.0119904 q^{26} +2.84790 q^{27} -2.62726 q^{28} +8.30334 q^{29} +0.169395 q^{30} +3.39655 q^{31} +3.16148 q^{32} +2.32481 q^{33} -2.03028 q^{34} -1.68494 q^{35} +5.29798 q^{36} +7.15948 q^{37} -0.193607 q^{38} +0.0213757 q^{39} +1.34290 q^{40} +5.45541 q^{41} -0.187667 q^{42} -11.7568 q^{43} +9.03414 q^{44} +3.39774 q^{45} +0.391925 q^{46} -5.24836 q^{47} -1.75356 q^{48} -5.13332 q^{49} -0.965723 q^{50} +3.61945 q^{51} +0.0830653 q^{52} -8.57769 q^{53} +0.790512 q^{54} +5.79385 q^{55} -1.48776 q^{56} +0.345149 q^{57} +2.30482 q^{58} -12.9925 q^{59} -1.17351 q^{60} +10.1636 q^{61} +0.942804 q^{62} -3.76423 q^{63} -6.20973 q^{64} +0.0532721 q^{65} +0.645315 q^{66} +10.1259 q^{67} +14.0650 q^{68} -0.698697 q^{69} -0.467700 q^{70} +1.86703 q^{71} +3.00012 q^{72} +6.47826 q^{73} +1.98731 q^{74} +1.72162 q^{75} +1.34124 q^{76} -6.41880 q^{77} +0.00593342 q^{78} -12.9436 q^{79} -4.37017 q^{80} +6.85611 q^{81} +1.51430 q^{82} -2.32915 q^{83} +1.30009 q^{84} +9.02030 q^{85} -3.26342 q^{86} -4.10887 q^{87} +5.11582 q^{88} -14.5180 q^{89} +0.943135 q^{90} -0.0590183 q^{91} -2.71511 q^{92} -1.68077 q^{93} -1.45682 q^{94} +0.860173 q^{95} -1.56444 q^{96} -2.23725 q^{97} -1.42489 q^{98} +12.9438 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 4 q^{2} - 3 q^{3} + 2 q^{4} - 8 q^{5} - 5 q^{6} - 7 q^{7} - 6 q^{8} - 2 q^{9}+O(q^{10}) $ 7 * q - 4 * q^2 - 3 * q^3 + 2 * q^4 - 8 * q^5 - 5 * q^6 - 7 * q^7 - 6 * q^8 - 2 * q^9	\( 7 q - 4 q^{2} - 3 q^{3} + 2 q^{4} - 8 q^{5} - 5 q^{6} - 7 q^{7} - 6 q^{8} - 2 q^{9} + 3 q^{10} - 18 q^{11} + q^{12} - q^{13} - 6 q^{14} - 11 q^{15} + 4 q^{16} - 2 q^{17} + 8 q^{18} - 6 q^{19} - 8 q^{20} - 2 q^{21} + 10 q^{22} - 22 q^{23} - 3 q^{24} + 5 q^{25} + 8 q^{26} + 3 q^{27} + 9 q^{28} - 16 q^{29} + 29 q^{30} - 18 q^{31} - 6 q^{32} + 4 q^{33} + 11 q^{34} + 7 q^{35} - 7 q^{36} + 8 q^{37} + 16 q^{38} - 9 q^{39} + 14 q^{40} - 15 q^{41} + 19 q^{42} + 14 q^{43} - 4 q^{44} + 3 q^{45} + 11 q^{46} - 10 q^{47} + 31 q^{48} + 6 q^{49} - 4 q^{50} + 13 q^{51} + 27 q^{52} + 15 q^{53} + 16 q^{54} + 29 q^{55} + 13 q^{56} + 14 q^{57} + 17 q^{58} - 18 q^{59} + 15 q^{60} + 4 q^{61} + 13 q^{62} - 16 q^{63} + 2 q^{64} - 7 q^{65} + 16 q^{66} + 18 q^{67} - 15 q^{68} + 26 q^{69} + 8 q^{70} - 50 q^{71} + 30 q^{72} + 10 q^{74} + 16 q^{75} - 20 q^{76} + 17 q^{77} - 32 q^{78} - 15 q^{79} - 11 q^{80} - 9 q^{81} + 45 q^{82} - 24 q^{83} + 6 q^{84} - 2 q^{85} - 23 q^{86} + 12 q^{87} + 8 q^{88} - 13 q^{89} - 39 q^{90} - 12 q^{91} - 10 q^{92} + 14 q^{93} - 32 q^{94} - 41 q^{95} - 15 q^{96} + q^{97} + 9 q^{98} - 20 q^{99}+O(q^{100}) \) 7 * q - 4 * q^2 - 3 * q^3 + 2 * q^4 - 8 * q^5 - 5 * q^6 - 7 * q^7 - 6 * q^8 - 2 * q^9 + 3 * q^10 - 18 * q^11 + q^12 - q^13 - 6 * q^14 - 11 * q^15 + 4 * q^16 - 2 * q^17 + 8 * q^18 - 6 * q^19 - 8 * q^20 - 2 * q^21 + 10 * q^22 - 22 * q^23 - 3 * q^24 + 5 * q^25 + 8 * q^26 + 3 * q^27 + 9 * q^28 - 16 * q^29 + 29 * q^30 - 18 * q^31 - 6 * q^32 + 4 * q^33 + 11 * q^34 + 7 * q^35 - 7 * q^36 + 8 * q^37 + 16 * q^38 - 9 * q^39 + 14 * q^40 - 15 * q^41 + 19 * q^42 + 14 * q^43 - 4 * q^44 + 3 * q^45 + 11 * q^46 - 10 * q^47 + 31 * q^48 + 6 * q^49 - 4 * q^50 + 13 * q^51 + 27 * q^52 + 15 * q^53 + 16 * q^54 + 29 * q^55 + 13 * q^56 + 14 * q^57 + 17 * q^58 - 18 * q^59 + 15 * q^60 + 4 * q^61 + 13 * q^62 - 16 * q^63 + 2 * q^64 - 7 * q^65 + 16 * q^66 + 18 * q^67 - 15 * q^68 + 26 * q^69 + 8 * q^70 - 50 * q^71 + 30 * q^72 + 10 * q^74 + 16 * q^75 - 20 * q^76 + 17 * q^77 - 32 * q^78 - 15 * q^79 - 11 * q^80 - 9 * q^81 + 45 * q^82 - 24 * q^83 + 6 * q^84 - 2 * q^85 - 23 * q^86 + 12 * q^87 + 8 * q^88 - 13 * q^89 - 39 * q^90 - 12 * q^91 - 10 * q^92 + 14 * q^93 - 32 * q^94 - 41 * q^95 - 15 * q^96 + q^97 + 9 * q^98 - 20 * 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 $

$2$	0.277577	0.196277	0.0981383	−	0.995173i	$-0.468711\pi$
$2$	0.277577	0.196277	0.0981383	+	0.995173i	$0.468711\pi$
$3$	−0.494846	−0.285699	−0.142850	−	0.989744i	$-0.545627\pi$
$3$	−0.494846	−0.285699	−0.142850	+	0.989744i	$0.545627\pi$
$4$	−1.92295	−0.961475
$5$	−1.23324	−0.551523	−0.275761	−	0.961226i	$-0.588930\pi$
$5$	−1.23324	−0.551523	−0.275761	+	0.961226i	$0.588930\pi$
$6$	−0.137358	−0.0560761
$7$	1.36627	0.516400	0.258200	−	0.966092i	$-0.416871\pi$
$7$	1.36627	0.516400	0.258200	+	0.966092i	$0.416871\pi$
$8$	−1.08892	−0.384992
$9$	−2.75513	−0.918376
$10$	−0.342320	−0.108251
$11$	−4.69806	−1.41652	−0.708259	−	0.705952i	$-0.750519\pi$
$11$	−4.69806	−1.41652	−0.708259	+	0.705952i	$0.750519\pi$
$12$	0.951564	0.274693
$13$	−0.0431968	−0.0119806	−0.00599032	−	0.999982i	$-0.501907\pi$
$13$	−0.0431968	−0.0119806	−0.00599032	+	0.999982i	$0.501907\pi$
$14$	0.379244	0.101357
$15$	0.610264	0.157570
$16$	3.54364	0.885911
$17$	−7.31430	−1.77398	−0.886989	−	0.461790i	$-0.847207\pi$
$17$	−7.31430	−1.77398	−0.886989	+	0.461790i	$0.847207\pi$
$18$	−0.764761	−0.180256
$19$	−0.697489	−0.160015	−0.0800075	−	0.996794i	$-0.525494\pi$
$19$	−0.697489	−0.160015	−0.0800075	+	0.996794i	$0.525494\pi$
$20$	2.37146	0.530275
$21$	−0.676090	−0.147535
$22$	−1.30407	−0.278030
$23$	1.41195	0.294412	0.147206	−	0.989106i	$-0.452972\pi$
$23$	1.41195	0.294412	0.147206	+	0.989106i	$0.452972\pi$
$24$	0.538848	0.109992
$25$	−3.47911	−0.695823
$26$	−0.0119904	−0.00235152
$27$	2.84790	0.548078
$28$	−2.62726	−0.496506
$29$	8.30334	1.54189	0.770946	−	0.636901i	$-0.219784\pi$
$29$	8.30334	1.54189	0.770946	+	0.636901i	$0.219784\pi$
$30$	0.169395	0.0309272
$31$	3.39655	0.610038	0.305019	−	0.952346i	$-0.401337\pi$
$31$	3.39655	0.610038	0.305019	+	0.952346i	$0.401337\pi$
$32$	3.16148	0.558875
$33$	2.32481	0.404698
$34$	−2.03028	−0.348191
$35$	−1.68494	−0.284806
$36$	5.29798	0.882996
$37$	7.15948	1.17701	0.588506	−	0.808493i	$-0.299716\pi$
$37$	7.15948	1.17701	0.588506	+	0.808493i	$0.299716\pi$
$38$	−0.193607	−0.0314072
$39$	0.0213757	0.00342286
$40$	1.34290	0.212332
$41$	5.45541	0.851992	0.425996	−	0.904725i	$-0.359924\pi$
$41$	5.45541	0.851992	0.425996	+	0.904725i	$0.359924\pi$
$42$	−0.187667	−0.0289577
$43$	−11.7568	−1.79290	−0.896448	−	0.443149i	$-0.853861\pi$
$43$	−11.7568	−1.79290	−0.896448	+	0.443149i	$0.853861\pi$
$44$	9.03414	1.36195
$45$	3.39774	0.506505
$46$	0.391925	0.0577862
$47$	−5.24836	−0.765551	−0.382776	−	0.923841i	$-0.625032\pi$
$47$	−5.24836	−0.765551	−0.382776	+	0.923841i	$0.625032\pi$
$48$	−1.75356	−0.253104
$49$	−5.13332	−0.733331
$50$	−0.965723	−0.136574
$51$	3.61945	0.506824
$52$	0.0830653	0.0115191
$53$	−8.57769	−1.17824	−0.589118	−	0.808047i	$-0.700525\pi$
$53$	−8.57769	−1.17824	−0.589118	+	0.808047i	$0.700525\pi$
$54$	0.790512	0.107575
$55$	5.79385	0.781242
$56$	−1.48776	−0.198810
$57$	0.345149	0.0457162
$58$	2.30482	0.302637
$59$	−12.9925	−1.69148	−0.845738	−	0.533598i	$-0.820840\pi$
$59$	−12.9925	−1.69148	−0.845738	+	0.533598i	$0.820840\pi$
$60$	−1.17351	−0.151499
$61$	10.1636	1.30132	0.650658	−	0.759371i	$-0.274493\pi$
$61$	10.1636	1.30132	0.650658	+	0.759371i	$0.274493\pi$
$62$	0.942804	0.119736
$63$	−3.76423	−0.474249
$64$	−6.20973	−0.776216
$65$	0.0532721	0.00660759
$66$	0.645315	0.0794328
$67$	10.1259	1.23707	0.618536	−	0.785757i	$-0.287726\pi$
$67$	10.1259	1.23707	0.618536	+	0.785757i	$0.287726\pi$
$68$	14.0650	1.70564
$69$	−0.698697	−0.0841132
$70$	−0.467700	−0.0559008
$71$	1.86703	0.221576	0.110788	−	0.993844i	$-0.464663\pi$
$71$	1.86703	0.221576	0.110788	+	0.993844i	$0.464663\pi$
$72$	3.00012	0.353567
$73$	6.47826	0.758223	0.379111	−	0.925351i	$-0.376230\pi$
$73$	6.47826	0.758223	0.379111	+	0.925351i	$0.376230\pi$
$74$	1.98731	0.231020
$75$	1.72162	0.198796
$76$	1.34124	0.153850
$77$	−6.41880	−0.731490
$78$	0.00593342	0.000671827 0
$79$	−12.9436	−1.45627	−0.728136	−	0.685432i	$-0.759613\pi$
$79$	−12.9436	−1.45627	−0.728136	+	0.685432i	$0.759613\pi$
$80$	−4.37017	−0.488600
$81$	6.85611	0.761790
$82$	1.51430	0.167226
$83$	−2.32915	−0.255657	−0.127829	−	0.991796i	$-0.540801\pi$
$83$	−2.32915	−0.255657	−0.127829	+	0.991796i	$0.540801\pi$
$84$	1.30009	0.141851
$85$	9.02030	0.978389
$86$	−3.26342	−0.351904
$87$	−4.10887	−0.440517
$88$	5.11582	0.545348
$89$	−14.5180	−1.53890	−0.769450	−	0.638707i	$-0.779470\pi$
$89$	−14.5180	−1.53890	−0.769450	+	0.638707i	$0.779470\pi$
$90$	0.943135	0.0994151
$91$	−0.0590183	−0.00618680
$92$	−2.71511	−0.283070
$93$	−1.68077	−0.174287
$94$	−1.45682	−0.150260
$95$	0.860173	0.0882519
$96$	−1.56444	−0.159670
$97$	−2.23725	−0.227158	−0.113579	−	0.993529i	$-0.536231\pi$
$97$	−2.23725	−0.227158	−0.113579	+	0.993529i	$0.536231\pi$
$98$	−1.42489	−0.143936
$99$	12.9438	1.30090
$100$	6.69017	0.669017
$101$	−4.01607	−0.399614	−0.199807	−	0.979835i	$-0.564031\pi$
$101$	−4.01607	−0.399614	−0.199807	+	0.979835i	$0.564031\pi$
$102$	1.00468	0.0994778
$103$	−5.25187	−0.517482	−0.258741	−	0.965947i	$-0.583308\pi$
$103$	−5.25187	−0.517482	−0.258741	+	0.965947i	$0.583308\pi$
$104$	0.0470379	0.00461245
$105$	0.833783	0.0813689
$106$	−2.38097	−0.231260
$107$	5.60385	0.541745	0.270873	−	0.962615i	$-0.412688\pi$
$107$	5.60385	0.541745	0.270873	+	0.962615i	$0.412688\pi$
$108$	−5.47637	−0.526964
$109$	−4.77557	−0.457416	−0.228708	−	0.973495i	$-0.573450\pi$
$109$	−4.77557	−0.457416	−0.228708	+	0.973495i	$0.573450\pi$
$110$	1.60824	0.153340
$111$	−3.54284	−0.336271
$112$	4.84155	0.457484
$113$	−12.6186	−1.18706	−0.593531	−	0.804811i	$-0.702267\pi$
$113$	−12.6186	−1.18706	−0.593531	+	0.804811i	$0.702267\pi$
$114$	0.0958056	0.00897301
$115$	−1.74127	−0.162375
$116$	−15.9669	−1.48249
$117$	0.119013	0.0110027
$118$	−3.60642	−0.331997
$119$	−9.99327	−0.916082
$120$	−0.664530	−0.0606630
$121$	11.0718	1.00653
$122$	2.82118	0.255418
$123$	−2.69959	−0.243413
$124$	−6.53139	−0.586536
$125$	10.4568	0.935285
$126$	−1.04487	−0.0930840
$127$	0.427651	0.0379479	0.0189740	−	0.999820i	$-0.493960\pi$
$127$	0.427651	0.0379479	0.0189740	+	0.999820i	$0.493960\pi$
$128$	−8.04663	−0.711229
$129$	5.81780	0.512229
$130$	0.0147871	0.00129692
$131$	−13.3633	−1.16756	−0.583780	−	0.811912i	$-0.698427\pi$
$131$	−13.3633	−1.16756	−0.583780	+	0.811912i	$0.698427\pi$
$132$	−4.47050	−0.389107
$133$	−0.952955	−0.0826317
$134$	2.81071	0.242808
$135$	−3.51215	−0.302278
$136$	7.96470	0.682967
$137$	11.2702	0.962874	0.481437	−	0.876481i	$-0.340115\pi$
$137$	11.2702	0.962874	0.481437	+	0.876481i	$0.340115\pi$
$138$	−0.193942	−0.0165095
$139$	−0.927184	−0.0786427	−0.0393214	−	0.999227i	$-0.512520\pi$
$139$	−0.927184	−0.0786427	−0.0393214	+	0.999227i	$0.512520\pi$
$140$	3.24005	0.273834
$141$	2.59713	0.218717
$142$	0.518245	0.0434902
$143$	0.202941	0.0169708
$144$	−9.76319	−0.813599
$145$	−10.2400	−0.850388
$146$	1.79822	0.148821
$147$	2.54020	0.209512
$148$	−13.7673	−1.13167
$149$	12.8757	1.05481	0.527407	−	0.849612i	$-0.323164\pi$
$149$	12.8757	1.05481	0.527407	+	0.849612i	$0.323164\pi$
$150$	0.477884	0.0390190
$151$	−21.2436	−1.72878	−0.864390	−	0.502823i	$-0.832295\pi$
$151$	−21.2436	−1.72878	−0.864390	+	0.502823i	$0.832295\pi$
$152$	0.759511	0.0616045
$153$	20.1518	1.62918
$154$	−1.78171	−0.143574
$155$	−4.18876	−0.336450
$156$	−0.0411045	−0.00329099
$157$	17.1798	1.37110	0.685550	−	0.728026i	$-0.259562\pi$
$157$	17.1798	1.37110	0.685550	+	0.728026i	$0.259562\pi$
$158$	−3.59286	−0.285832
$159$	4.24463	0.336621
$160$	−3.89887	−0.308232
$161$	1.92910	0.152034
$162$	1.90310	0.149522
$163$	11.1081	0.870055	0.435028	−	0.900417i	$-0.356739\pi$
$163$	11.1081	0.870055	0.435028	+	0.900417i	$0.356739\pi$
$164$	−10.4905	−0.819170
$165$	−2.86706	−0.223200
$166$	−0.646519	−0.0501796
$167$	−22.3791	−1.73174	−0.865872	−	0.500266i	$-0.833236\pi$
$167$	−22.3791	−1.73174	−0.865872	+	0.500266i	$0.833236\pi$
$168$	0.736209	0.0567998
$169$	−12.9981	−0.999856
$170$	2.50383	0.192035
$171$	1.92167	0.146954
$172$	22.6078	1.72383
$173$	4.16100	0.316355	0.158178	−	0.987411i	$-0.449438\pi$
$173$	4.16100	0.316355	0.158178	+	0.987411i	$0.449438\pi$
$174$	−1.14053	−0.0864633
$175$	−4.75339	−0.359323
$176$	−16.6482	−1.25491
$177$	6.42927	0.483254
$178$	−4.02985	−0.302050
$179$	−5.79009	−0.432772	−0.216386	−	0.976308i	$-0.569427\pi$
$179$	−5.79009	−0.432772	−0.216386	+	0.976308i	$0.569427\pi$
$180$	−6.53369	−0.486992
$181$	17.4917	1.30015	0.650076	−	0.759869i	$-0.274737\pi$
$181$	17.4917	1.30015	0.650076	+	0.759869i	$0.274737\pi$
$182$	−0.0163821	−0.00121432
$183$	−5.02941	−0.371785
$184$	−1.53750	−0.113346
$185$	−8.82937	−0.649148
$186$	−0.466542	−0.0342085
$187$	34.3630	2.51287
$188$	10.0923	0.736059
$189$	3.89099	0.283028
$190$	0.238764	0.0173218
$191$	−21.6074	−1.56346	−0.781728	−	0.623620i	$-0.785661\pi$
$191$	−21.6074	−1.56346	−0.781728	+	0.623620i	$0.785661\pi$
$192$	3.07286	0.221764
$193$	2.30886	0.166195	0.0830977	−	0.996541i	$-0.473519\pi$
$193$	2.30886	0.166195	0.0830977	+	0.996541i	$0.473519\pi$
$194$	−0.621008	−0.0445858
$195$	−0.0263615	−0.00188778
$196$	9.87112	0.705080
$197$	1.91876	0.136706	0.0683531	−	0.997661i	$-0.478226\pi$
$197$	1.91876	0.136706	0.0683531	+	0.997661i	$0.478226\pi$
$198$	3.59289	0.255336
$199$	21.2430	1.50588	0.752938	−	0.658091i	$-0.228636\pi$
$199$	21.2430	1.50588	0.752938	+	0.658091i	$0.228636\pi$
$200$	3.78848	0.267886
$201$	−5.01074	−0.353430
$202$	−1.11477	−0.0784349
$203$	11.3446	0.796232
$204$	−6.96002	−0.487299
$205$	−6.72784	−0.469893
$206$	−1.45780	−0.101570
$207$	−3.89010	−0.270381
$208$	−0.153074	−0.0106138
$209$	3.27685	0.226664
$210$	0.231439	0.0159708
$211$	−19.0961	−1.31463	−0.657314	−	0.753616i	$-0.728308\pi$
$211$	−19.0961	−1.31463	−0.657314	+	0.753616i	$0.728308\pi$
$212$	16.4945	1.13285
$213$	−0.923892	−0.0633040
$214$	1.55550	0.106332
$215$	14.4990	0.988823
$216$	−3.10114	−0.211006
$217$	4.64058	0.315023
$218$	−1.32559	−0.0897802
$219$	−3.20574	−0.216624
$220$	−11.1413	−0.751145
$221$	0.315954	0.0212534
$222$	−0.983410	−0.0660022
$223$	8.41781	0.563698	0.281849	−	0.959459i	$-0.409052\pi$
$223$	8.41781	0.563698	0.281849	+	0.959459i	$0.409052\pi$
$224$	4.31942	0.288603
$225$	9.58540	0.639027
$226$	−3.50265	−0.232993
$227$	1.82854	0.121364	0.0606821	−	0.998157i	$-0.480672\pi$
$227$	1.82854	0.121364	0.0606821	+	0.998157i	$0.480672\pi$
$228$	−0.663705	−0.0439550
$229$	2.52038	0.166551	0.0832756	−	0.996527i	$-0.473462\pi$
$229$	2.52038	0.166551	0.0832756	+	0.996527i	$0.473462\pi$
$230$	−0.483338	−0.0318704
$231$	3.17631	0.208986
$232$	−9.04169	−0.593616
$233$	16.3812	1.07317	0.536585	−	0.843847i	$-0.319714\pi$
$233$	16.3812	1.07317	0.536585	+	0.843847i	$0.319714\pi$
$234$	0.0330352	0.00215958
$235$	6.47249	0.422219
$236$	24.9839	1.62631
$237$	6.40510	0.416056
$238$	−2.77390	−0.179805
$239$	−10.7430	−0.694909	−0.347455	−	0.937697i	$-0.612954\pi$
$239$	−10.7430	−0.694909	−0.347455	+	0.937697i	$0.612954\pi$
$240$	2.16256	0.139593
$241$	−1.00000	−0.0644157
$241$	−1.00000	−0.0644157
$242$	3.07327	0.197557
$243$	−11.9364	−0.765721
$244$	−19.5441	−1.25118
$245$	6.33063	0.404449
$246$	−0.749343	−0.0477764
$247$	0.0301293	0.00191708
$248$	−3.69857	−0.234860
$249$	1.15257	0.0730411
$250$	2.90257	0.183575
$251$	23.8819	1.50741	0.753706	−	0.657212i	$-0.228264\pi$
$251$	23.8819	1.50741	0.753706	+	0.657212i	$0.228264\pi$
$252$	7.23844	0.455979
$253$	−6.63342	−0.417040
$254$	0.118706	0.00744829
$255$	−4.46366	−0.279525
$256$	10.1859	0.636619
$257$	5.25407	0.327740	0.163870	−	0.986482i	$-0.447602\pi$
$257$	5.25407	0.327740	0.163870	+	0.986482i	$0.447602\pi$
$258$	1.61489	0.100539
$259$	9.78175	0.607808
$260$	−0.102440	−0.00635304
$261$	−22.8768	−1.41604
$262$	−3.70936	−0.229165
$263$	−1.10964	−0.0684235	−0.0342118	−	0.999415i	$-0.510892\pi$
$263$	−1.10964	−0.0684235	−0.0342118	+	0.999415i	$0.510892\pi$
$264$	−2.53154	−0.155806
$265$	10.5784	0.649824
$266$	−0.264519	−0.0162187
$267$	7.18414	0.439662
$268$	−19.4715	−1.18941
$269$	−2.84634	−0.173544	−0.0867722	−	0.996228i	$-0.527655\pi$
$269$	−2.84634	−0.173544	−0.0867722	+	0.996228i	$0.527655\pi$
$270$	−0.974892	−0.0593301
$271$	13.1029	0.795946	0.397973	−	0.917397i	$-0.369714\pi$
$271$	13.1029	0.795946	0.397973	+	0.917397i	$0.369714\pi$
$272$	−25.9193	−1.57159
$273$	0.0292049	0.00176756
$274$	3.12834	0.188990
$275$	16.3451	0.985646
$276$	1.34356	0.0808728
$277$	−23.4634	−1.40978	−0.704888	−	0.709318i	$-0.749003\pi$
$277$	−23.4634	−1.40978	−0.704888	+	0.709318i	$0.749003\pi$
$278$	−0.257365	−0.0154357
$279$	−9.35792	−0.560244
$280$	1.83476	0.109648
$281$	8.91165	0.531625	0.265812	−	0.964025i	$-0.414360\pi$
$281$	8.91165	0.531625	0.265812	+	0.964025i	$0.414360\pi$
$282$	0.720903	0.0429291
$283$	−15.8040	−0.939453	−0.469726	−	0.882812i	$-0.655647\pi$
$283$	−15.8040	−0.939453	−0.469726	+	0.882812i	$0.655647\pi$
$284$	−3.59021	−0.213040
$285$	−0.425653	−0.0252135
$286$	0.0563318	0.00333097
$287$	7.45354	0.439968
$288$	−8.71027	−0.513258
$289$	36.4990	2.14700
$290$	−2.84240	−0.166911
$291$	1.10709	0.0648988
$292$	−12.4574	−0.729013
$293$	32.9425	1.92452	0.962261	−	0.272127i	$-0.0877272\pi$
$293$	32.9425	1.92452	0.962261	+	0.272127i	$0.0877272\pi$
$294$	0.705102	0.0411224
$295$	16.0229	0.932888
$296$	−7.79611	−0.453140
$297$	−13.3796	−0.776363
$298$	3.57399	0.207036
$299$	−0.0609917	−0.00352724
$300$	−3.31060	−0.191138
$301$	−16.0629	−0.925851
$302$	−5.89674	−0.339319
$303$	1.98733	0.114169
$304$	−2.47165	−0.141759
$305$	−12.5342	−0.717705
$306$	5.59369	0.319770
$307$	20.2343	1.15483	0.577416	−	0.816450i	$-0.304061\pi$
$307$	20.2343	1.15483	0.577416	+	0.816450i	$0.304061\pi$
$308$	12.3430	0.703309
$309$	2.59886	0.147844
$310$	−1.16271	−0.0660372
$311$	2.27289	0.128884	0.0644420	−	0.997921i	$-0.479473\pi$
$311$	2.27289	0.128884	0.0644420	+	0.997921i	$0.479473\pi$
$312$	−0.0232765	−0.00131777
$313$	13.1142	0.741257	0.370629	−	0.928781i	$-0.379142\pi$
$313$	13.1142	0.741257	0.370629	+	0.928781i	$0.379142\pi$
$314$	4.76873	0.269115
$315$	4.64221	0.261559
$316$	24.8900	1.40017
$317$	−7.86525	−0.441757	−0.220878	−	0.975301i	$-0.570892\pi$
$317$	−7.86525	−0.441757	−0.220878	+	0.975301i	$0.570892\pi$
$318$	1.17821	0.0660709
$319$	−39.0096	−2.18412
$320$	7.65810	0.428101
$321$	−2.77304	−0.154776
$322$	0.535473	0.0298407
$323$	5.10164	0.283863
$324$	−13.1840	−0.732443
$325$	0.150287	0.00833640
$326$	3.08336	0.170772
$327$	2.36317	0.130683
$328$	−5.94051	−0.328010
$329$	−7.17065	−0.395331
$330$	−0.795830	−0.0438090
$331$	−19.3752	−1.06496	−0.532480	−	0.846443i	$-0.678740\pi$
$331$	−19.3752	−1.06496	−0.532480	+	0.846443i	$0.678740\pi$
$332$	4.47884	0.245808
$333$	−19.7253	−1.08094
$334$	−6.21192	−0.339901
$335$	−12.4876	−0.682273
$336$	−2.39582	−0.130703
$337$	−26.2459	−1.42970	−0.714852	−	0.699276i	$-0.753506\pi$
$337$	−26.2459	−1.42970	−0.714852	+	0.699276i	$0.753506\pi$
$338$	−3.60798	−0.196249
$339$	6.24428	0.339143
$340$	−17.3456	−0.940697
$341$	−15.9572	−0.864130
$342$	0.533412	0.0288436
$343$	−16.5773	−0.895092
$344$	12.8022	0.690250
$345$	0.861662	0.0463903
$346$	1.15500	0.0620931
$347$	−3.65189	−0.196044	−0.0980219	−	0.995184i	$-0.531252\pi$
$347$	−3.65189	−0.196044	−0.0980219	+	0.995184i	$0.531252\pi$
$348$	7.90116	0.423547
$349$	−0.818652	−0.0438214	−0.0219107	−	0.999760i	$-0.506975\pi$
$349$	−0.818652	−0.0438214	−0.0219107	+	0.999760i	$0.506975\pi$
$350$	−1.31943	−0.0705267
$351$	−0.123020	−0.00656633
$352$	−14.8528	−0.791658
$353$	−28.5367	−1.51886	−0.759428	−	0.650591i	$-0.774521\pi$
$353$	−28.5367	−1.51886	−0.759428	+	0.650591i	$0.774521\pi$
$354$	1.78462	0.0948514
$355$	−2.30250	−0.122204
$356$	27.9173	1.47961
$357$	4.94513	0.261724
$358$	−1.60720	−0.0849430
$359$	−15.8623	−0.837180	−0.418590	−	0.908175i	$-0.637476\pi$
$359$	−15.8623	−0.837180	−0.418590	+	0.908175i	$0.637476\pi$
$360$	−3.69987	−0.195000
$361$	−18.5135	−0.974395
$362$	4.85531	0.255189
$363$	−5.47882	−0.287563
$364$	0.113489	0.00594845
$365$	−7.98926	−0.418177
$366$	−1.39605	−0.0729727
$367$	13.9937	0.730463	0.365232	−	0.930917i	$-0.380990\pi$
$367$	13.9937	0.730463	0.365232	+	0.930917i	$0.380990\pi$
$368$	5.00344	0.260822
$369$	−15.0304	−0.782449
$370$	−2.45083	−0.127413
$371$	−11.7194	−0.608441
$372$	3.23203	0.167573
$373$	12.2336	0.633434	0.316717	−	0.948520i	$-0.397419\pi$
$373$	12.2336	0.633434	0.316717	+	0.948520i	$0.397419\pi$
$374$	9.53839	0.493218
$375$	−5.17450	−0.267210
$376$	5.71505	0.294731
$377$	−0.358678	−0.0184728
$378$	1.08005	0.0555517
$379$	0.284829	0.0146307	0.00731535	−	0.999973i	$-0.497671\pi$
$379$	0.284829	0.0146307	0.00731535	+	0.999973i	$0.497671\pi$
$380$	−1.65407	−0.0848520
$381$	−0.211621	−0.0108417
$382$	−5.99772	−0.306870
$383$	29.8268	1.52408	0.762038	−	0.647532i	$-0.224199\pi$
$383$	29.8268	1.52408	0.762038	+	0.647532i	$0.224199\pi$
$384$	3.98184	0.203197
$385$	7.91593	0.403433
$386$	0.640887	0.0326203
$387$	32.3915	1.64655
$388$	4.30211	0.218407
$389$	−8.88544	−0.450510	−0.225255	−	0.974300i	$-0.572321\pi$
$389$	−8.88544	−0.450510	−0.225255	+	0.974300i	$0.572321\pi$
$390$	−0.00731734	−0.000370528 0
$391$	−10.3274	−0.522280
$392$	5.58978	0.282327
$393$	6.61279	0.333571
$394$	0.532605	0.0268322
$395$	15.9626	0.803167
$396$	−24.8902	−1.25078
$397$	−29.5831	−1.48473	−0.742367	−	0.669993i	$-0.766297\pi$
$397$	−29.5831	−1.48473	−0.742367	+	0.669993i	$0.766297\pi$
$398$	5.89657	0.295569
$399$	0.471566	0.0236078
$400$	−12.3287	−0.616437
$401$	6.30306	0.314760	0.157380	−	0.987538i	$-0.449695\pi$
$401$	6.30306	0.314760	0.157380	+	0.987538i	$0.449695\pi$
$402$	−1.39087	−0.0693701
$403$	−0.146720	−0.00730864
$404$	7.72270	0.384219
$405$	−8.45525	−0.420145
$406$	3.14899	0.156282
$407$	−33.6357	−1.66726
$408$	−3.94130	−0.195123
$409$	−7.63163	−0.377360	−0.188680	−	0.982039i	$-0.560421\pi$
$409$	−7.63163	−0.377360	−0.188680	+	0.982039i	$0.560421\pi$
$410$	−1.86750	−0.0922290
$411$	−5.57699	−0.275092
$412$	10.0991	0.497546
$413$	−17.7512	−0.873478
$414$	−1.07980	−0.0530694
$415$	2.87240	0.141001
$416$	−0.136566	−0.00669568
$417$	0.458813	0.0224682
$418$	0.909578	0.0444889
$419$	2.34760	0.114688	0.0573439	−	0.998354i	$-0.481737\pi$
$419$	2.34760	0.114688	0.0573439	+	0.998354i	$0.481737\pi$
$420$	−1.60332	−0.0782342
$421$	8.61693	0.419963	0.209982	−	0.977705i	$-0.432660\pi$
$421$	8.61693	0.419963	0.209982	+	0.977705i	$0.432660\pi$
$422$	−5.30064	−0.258031
$423$	14.4599	0.703064
$424$	9.34043	0.453611
$425$	25.4473	1.23437
$426$	−0.256451	−0.0124251
$427$	13.8862	0.671999
$428$	−10.7759	−0.520875
$429$	−0.100425	−0.00484854
$430$	4.02459	0.194083
$431$	−2.73295	−0.131641	−0.0658207	−	0.997831i	$-0.520967\pi$
$431$	−2.73295	−0.131641	−0.0658207	+	0.997831i	$0.520967\pi$
$432$	10.0919	0.485549
$433$	14.8768	0.714934	0.357467	−	0.933926i	$-0.383640\pi$
$433$	14.8768	0.714934	0.357467	+	0.933926i	$0.383640\pi$
$434$	1.28812	0.0618317
$435$	5.06723	0.242955
$436$	9.18318	0.439795
$437$	−0.984819	−0.0471103
$438$	−0.889839	−0.0425182
$439$	−19.8115	−0.945552	−0.472776	−	0.881183i	$-0.656748\pi$
$439$	−19.8115	−0.945552	−0.472776	+	0.881183i	$0.656748\pi$
$440$	−6.30904	−0.300772
$441$	14.1430	0.673474
$442$	0.0877017	0.00417154
$443$	−39.5818	−1.88059	−0.940293	−	0.340366i	$-0.889449\pi$
$443$	−39.5818	−1.88059	−0.940293	+	0.340366i	$0.889449\pi$
$444$	6.81270	0.323316
$445$	17.9041	0.848738
$446$	2.33659	0.110641
$447$	−6.37146	−0.301360
$448$	−8.48414	−0.400838
$449$	−22.9205	−1.08169	−0.540843	−	0.841124i	$-0.681895\pi$
$449$	−22.9205	−1.08169	−0.540843	+	0.841124i	$0.681895\pi$
$450$	2.66069	0.125426
$451$	−25.6299	−1.20686
$452$	24.2650	1.14133
$453$	10.5123	0.493911
$454$	0.507560	0.0238210
$455$	0.0727838	0.00341216
$456$	−0.375841	−0.0176003
$457$	−9.66494	−0.452107	−0.226053	−	0.974115i	$-0.572582\pi$
$457$	−9.66494	−0.452107	−0.226053	+	0.974115i	$0.572582\pi$
$458$	0.699599	0.0326901
$459$	−20.8304	−0.972279
$460$	3.34839	0.156119
$461$	6.28046	0.292510	0.146255	−	0.989247i	$-0.453278\pi$
$461$	6.28046	0.292510	0.146255	+	0.989247i	$0.453278\pi$
$462$	0.881672	0.0410191
$463$	−7.64337	−0.355218	−0.177609	−	0.984101i	$-0.556836\pi$
$463$	−7.64337	−0.355218	−0.177609	+	0.984101i	$0.556836\pi$
$464$	29.4241	1.36598
$465$	2.07279	0.0961234
$466$	4.54705	0.210638
$467$	25.8943	1.19824	0.599122	−	0.800658i	$-0.295516\pi$
$467$	25.8943	1.19824	0.599122	+	0.800658i	$0.295516\pi$
$468$	−0.228856	−0.0105789
$469$	13.8346	0.638823
$470$	1.79662	0.0828717
$471$	−8.50136	−0.391722
$472$	14.1478	0.651205
$473$	55.2342	2.53967
$474$	1.77791	0.0816621
$475$	2.42664	0.111342
$476$	19.2166	0.880790
$477$	23.6326	1.08206
$478$	−2.98202	−0.136394
$479$	12.6131	0.576309	0.288155	−	0.957584i	$-0.406958\pi$
$479$	12.6131	0.576309	0.288155	+	0.957584i	$0.406958\pi$
$480$	1.92934	0.0880618
$481$	−0.309267	−0.0141013
$482$	−0.277577	−0.0126433
$483$	−0.954605	−0.0434360
$484$	−21.2905	−0.967749
$485$	2.75907	0.125283
$486$	−3.31328	−0.150293
$487$	31.7807	1.44012	0.720061	−	0.693910i	$-0.244114\pi$
$487$	31.7807	1.44012	0.720061	+	0.693910i	$0.244114\pi$
$488$	−11.0674	−0.500996
$489$	−5.49680	−0.248574
$490$	1.75724	0.0793839
$491$	−10.2151	−0.461001	−0.230500	−	0.973072i	$-0.574036\pi$
$491$	−10.2151	−0.461001	−0.230500	+	0.973072i	$0.574036\pi$
$492$	5.19117	0.234036
$493$	−60.7331	−2.73528
$494$	0.00836321	0.000376278 0
$495$	−15.9628	−0.717474
$496$	12.0361	0.540439
$497$	2.55086	0.114422
$498$	0.319927	0.0143363
$499$	−9.35390	−0.418738	−0.209369	−	0.977837i	$-0.567141\pi$
$499$	−9.35390	−0.418738	−0.209369	+	0.977837i	$0.567141\pi$
$500$	−20.1079	−0.899253
$501$	11.0742	0.494758
$502$	6.62907	0.295870
$503$	−13.7659	−0.613792	−0.306896	−	0.951743i	$-0.599290\pi$
$503$	−13.7659	−0.613792	−0.306896	+	0.951743i	$0.599290\pi$
$504$	4.09896	0.182582
$505$	4.95278	0.220396
$506$	−1.84129	−0.0818552
$507$	6.43207	0.285658
$508$	−0.822353	−0.0364860
$509$	22.6367	1.00335	0.501677	−	0.865055i	$-0.332717\pi$
$509$	22.6367	1.00335	0.501677	+	0.865055i	$0.332717\pi$
$510$	−1.23901	−0.0548642
$511$	8.85102	0.391546
$512$	18.9206	0.836182
$513$	−1.98638	−0.0877008
$514$	1.45841	0.0643277
$515$	6.47682	0.285403
$516$	−11.1873	−0.492496
$517$	24.6571	1.08442
$518$	2.71519	0.119299
$519$	−2.05905	−0.0903824
$520$	−0.0580091	−0.00254387
$521$	−10.8738	−0.476389	−0.238194	−	0.971218i	$-0.576555\pi$
$521$	−10.8738	−0.476389	−0.238194	+	0.971218i	$0.576555\pi$
$522$	−6.35007	−0.277935
$523$	−3.43232	−0.150085	−0.0750424	−	0.997180i	$-0.523909\pi$
$523$	−3.43232	−0.150085	−0.0750424	+	0.997180i	$0.523909\pi$
$524$	25.6970	1.12258
$525$	2.35219	0.102658
$526$	−0.308011	−0.0134299
$527$	−24.8434	−1.08219
$528$	8.23831	0.358526
$529$	−21.0064	−0.913322
$530$	2.93631	0.127545
$531$	35.7959	1.55341
$532$	1.83249	0.0794483
$533$	−0.235656	−0.0102074
$534$	1.99415	0.0862955
$535$	−6.91091	−0.298785
$536$	−11.0263	−0.476263
$537$	2.86520	0.123643
$538$	−0.790079	−0.0340627
$539$	24.1167	1.03878
$540$	6.75369	0.290633
$541$	−1.00758	−0.0433193	−0.0216597	−	0.999765i	$-0.506895\pi$
$541$	−1.00758	−0.0433193	−0.0216597	+	0.999765i	$0.506895\pi$
$542$	3.63707	0.156226
$543$	−8.65571	−0.371452
$544$	−23.1240	−0.991433
$545$	5.88943	0.252275
$546$	0.00810662	0.000346931 0
$547$	10.6391	0.454894	0.227447	−	0.973791i	$-0.426962\pi$
$547$	10.6391	0.454894	0.227447	+	0.973791i	$0.426962\pi$
$548$	−21.6720	−0.925780
$549$	−28.0020	−1.19510
$550$	4.53702	0.193459
$551$	−5.79149	−0.246726
$552$	0.760826	0.0323829
$553$	−17.6844	−0.752019
$554$	−6.51289	−0.276706
$555$	4.36917	0.185461
$556$	1.78293	0.0756130
$557$	18.9092	0.801207	0.400603	−	0.916252i	$-0.368801\pi$
$557$	18.9092	0.801207	0.400603	+	0.916252i	$0.368801\pi$
$558$	−2.59755	−0.109963
$559$	0.507856	0.0214800
$560$	−5.97081	−0.252313
$561$	−17.0044	−0.717926
$562$	2.47367	0.104346
$563$	28.1768	1.18751	0.593756	−	0.804645i	$-0.297644\pi$
$563$	28.1768	1.18751	0.593756	+	0.804645i	$0.297644\pi$
$564$	−4.99415	−0.210291
$565$	15.5618	0.654692
$566$	−4.38684	−0.184393
$567$	9.36727	0.393388
$568$	−2.03305	−0.0853049
$569$	−0.499818	−0.0209534	−0.0104767	−	0.999945i	$-0.503335\pi$
$569$	−0.499818	−0.0209534	−0.0104767	+	0.999945i	$0.503335\pi$
$570$	−0.118151	−0.00494882
$571$	−28.2303	−1.18140	−0.590702	−	0.806890i	$-0.701149\pi$
$571$	−28.2303	−1.18140	−0.590702	+	0.806890i	$0.701149\pi$
$572$	−0.390246	−0.0163170
$573$	10.6923	0.446678
$574$	2.06893	0.0863555
$575$	−4.91233	−0.204858
$576$	17.1086	0.712858
$577$	−36.3728	−1.51422	−0.757111	−	0.653287i	$-0.773390\pi$
$577$	−36.3728	−1.51422	−0.757111	+	0.653287i	$0.773390\pi$
$578$	10.1313	0.421406
$579$	−1.14253	−0.0474819
$580$	19.6911	0.817627
$581$	−3.18224	−0.132021
$582$	0.307303	0.0127381
$583$	40.2985	1.66899
$584$	−7.05432	−0.291910
$585$	−0.146771	−0.00606825
$586$	9.14409	0.377739
$587$	16.1223	0.665437	0.332718	−	0.943026i	$-0.392034\pi$
$587$	16.1223	0.665437	0.332718	+	0.943026i	$0.392034\pi$
$588$	−4.88468	−0.201441
$589$	−2.36905	−0.0976152
$590$	4.44758	0.183104
$591$	−0.949491	−0.0390569
$592$	25.3706	1.04273
$593$	24.8487	1.02041	0.510207	−	0.860052i	$-0.329569\pi$
$593$	24.8487	1.02041	0.510207	+	0.860052i	$0.329569\pi$
$594$	−3.71387	−0.152382
$595$	12.3241	0.505240
$596$	−24.7593	−1.01418
$597$	−10.5120	−0.430228
$598$	−0.0169299	−0.000692315 0
$599$	−1.50155	−0.0613517	−0.0306759	−	0.999529i	$-0.509766\pi$
$599$	−1.50155	−0.0613517	−0.0306759	+	0.999529i	$0.509766\pi$
$600$	−1.87471	−0.0765349
$601$	−24.0828	−0.982356	−0.491178	−	0.871059i	$-0.663434\pi$
$601$	−24.0828	−0.982356	−0.491178	+	0.871059i	$0.663434\pi$
$602$	−4.45870	−0.181723
$603$	−27.8981	−1.13610
$604$	40.8504	1.66218
$605$	−13.6542	−0.555121
$606$	0.551638	0.0224088
$607$	16.0760	0.652506	0.326253	−	0.945283i	$-0.394214\pi$
$607$	16.0760	0.652506	0.326253	+	0.945283i	$0.394214\pi$
$608$	−2.20510	−0.0894284
$609$	−5.61381	−0.227483
$610$	−3.47920	−0.140869
$611$	0.226712	0.00917179
$612$	−38.7510	−1.56642
$613$	43.5233	1.75789	0.878945	−	0.476923i	$-0.158248\pi$
$613$	43.5233	1.75789	0.878945	+	0.476923i	$0.158248\pi$
$614$	5.61658	0.226667
$615$	3.32924	0.134248
$616$	6.98957	0.281618
$617$	−3.82544	−0.154006	−0.0770032	−	0.997031i	$-0.524535\pi$
$617$	−3.82544	−0.154006	−0.0770032	+	0.997031i	$0.524535\pi$
$618$	0.721385	0.0290183
$619$	8.69881	0.349635	0.174817	−	0.984601i	$-0.444066\pi$
$619$	8.69881	0.349635	0.174817	+	0.984601i	$0.444066\pi$
$620$	8.05479	0.323488
$621$	4.02109	0.161361
$622$	0.630903	0.0252969
$623$	−19.8354	−0.794687
$624$	0.0757480	0.00303235
$625$	4.49981	0.179992
$626$	3.64020	0.145492
$627$	−1.62153	−0.0647578
$628$	−33.0360	−1.31828
$629$	−52.3666	−2.08799
$630$	1.28857	0.0513379
$631$	−28.0264	−1.11571	−0.557856	−	0.829938i	$-0.688376\pi$
$631$	−28.0264	−1.11571	−0.557856	+	0.829938i	$0.688376\pi$
$632$	14.0946	0.560653
$633$	9.44961	0.375588
$634$	−2.18321	−0.0867065
$635$	−0.527398	−0.0209291
$636$	−8.16222	−0.323653
$637$	0.221743	0.00878578
$638$	−10.8282	−0.428692
$639$	−5.14391	−0.203490
$640$	9.92345	0.392259
$641$	22.7621	0.899048	0.449524	−	0.893268i	$-0.351594\pi$
$641$	22.7621	0.899048	0.449524	+	0.893268i	$0.351594\pi$
$642$	−0.769733	−0.0303789
$643$	35.9559	1.41796	0.708981	−	0.705228i	$-0.249155\pi$
$643$	35.9559	1.41796	0.708981	+	0.705228i	$0.249155\pi$
$644$	−3.70956	−0.146177
$645$	−7.17476	−0.282506
$646$	1.41610	0.0557157
$647$	−1.94619	−0.0765127	−0.0382564	−	0.999268i	$-0.512180\pi$
$647$	−1.94619	−0.0765127	−0.0382564	+	0.999268i	$0.512180\pi$
$648$	−7.46577	−0.293283
$649$	61.0395	2.39601
$650$	0.0417161	0.00163624
$651$	−2.29637	−0.0900019
$652$	−21.3604	−0.836537
$653$	24.3392	0.952467	0.476233	−	0.879319i	$-0.342002\pi$
$653$	24.3392	0.952467	0.476233	+	0.879319i	$0.342002\pi$
$654$	0.655961	0.0256501
$655$	16.4802	0.643936
$656$	19.3320	0.754789
$657$	−17.8484	−0.696334
$658$	−1.99041	−0.0775942
$659$	−6.16907	−0.240313	−0.120156	−	0.992755i	$-0.538340\pi$
$659$	−6.16907	−0.240313	−0.120156	+	0.992755i	$0.538340\pi$
$660$	5.51321	0.214602
$661$	−48.1704	−1.87361	−0.936807	−	0.349848i	$-0.886233\pi$
$661$	−48.1704	−1.87361	−0.936807	+	0.349848i	$0.886233\pi$
$662$	−5.37812	−0.209027
$663$	−0.156349	−0.00607208
$664$	2.53626	0.0984260
$665$	1.17522	0.0455732
$666$	−5.47529	−0.212163
$667$	11.7239	0.453951
$668$	43.0338	1.66503
$669$	−4.16551	−0.161048
$670$	−3.46628	−0.133914
$671$	−47.7492	−1.84334
$672$	−2.13744	−0.0824537
$673$	−19.2071	−0.740381	−0.370190	−	0.928956i	$-0.620708\pi$
$673$	−19.2071	−0.740381	−0.370190	+	0.928956i	$0.620708\pi$
$674$	−7.28525	−0.280617
$675$	−9.90817	−0.381366
$676$	24.9948	0.961337
$677$	−14.9094	−0.573014	−0.286507	−	0.958078i	$-0.592494\pi$
$677$	−14.9094	−0.573014	−0.286507	+	0.958078i	$0.592494\pi$
$678$	1.73327	0.0665658
$679$	−3.05667	−0.117304
$680$	−9.82240	−0.376672
$681$	−0.904843	−0.0346737
$682$	−4.42935	−0.169609
$683$	−22.8682	−0.875029	−0.437515	−	0.899211i	$-0.644141\pi$
$683$	−22.8682	−0.875029	−0.437515	+	0.899211i	$0.644141\pi$
$684$	−3.69528	−0.141293
$685$	−13.8988	−0.531047
$686$	−4.60149	−0.175686
$687$	−1.24720	−0.0475835
$688$	−41.6619	−1.58835
$689$	0.370529	0.0141160
$690$	0.239178	0.00910534
$691$	−35.5210	−1.35128	−0.675641	−	0.737231i	$-0.736133\pi$
$691$	−35.5210	−1.35128	−0.675641	+	0.737231i	$0.736133\pi$
$692$	−8.00140	−0.304168
$693$	17.6846	0.671783
$694$	−1.01368	−0.0384788
$695$	1.14344	0.0433732
$696$	4.47424	0.169596
$697$	−39.9025	−1.51142
$698$	−0.227239	−0.00860112
$699$	−8.10617	−0.306604
$700$	9.14054	0.345480
$701$	9.22303	0.348349	0.174175	−	0.984715i	$-0.444274\pi$
$701$	9.22303	0.348349	0.174175	+	0.984715i	$0.444274\pi$
$702$	−0.0341476	−0.00128882
$703$	−4.99366	−0.188339
$704$	29.1737	1.09952
$705$	−3.20288	−0.120628
$706$	−7.92114	−0.298116
$707$	−5.48701	−0.206360
$708$	−12.3632	−0.464636
$709$	28.7763	1.08072	0.540358	−	0.841435i	$-0.318289\pi$
$709$	28.7763	1.08072	0.540358	+	0.841435i	$0.318289\pi$
$710$	−0.639122	−0.0239858
$711$	35.6614	1.33741
$712$	15.8089	0.592464
$713$	4.79575	0.179602
$714$	1.37265	0.0513703
$715$	−0.250276	−0.00935978
$716$	11.1341	0.416100
$717$	5.31614	0.198535
$718$	−4.40301	−0.164319
$719$	−24.1878	−0.902055	−0.451027	−	0.892510i	$-0.648942\pi$
$719$	−24.1878	−0.902055	−0.451027	+	0.892510i	$0.648942\pi$
$720$	12.0404	0.448718
$721$	−7.17544	−0.267227
$722$	−5.13893	−0.191251
$723$	0.494846	0.0184035
$724$	−33.6358	−1.25006
$725$	−28.8883	−1.07288
$726$	−1.52080	−0.0564420
$727$	9.73994	0.361235	0.180617	−	0.983553i	$-0.442190\pi$
$727$	9.73994	0.361235	0.180617	+	0.983553i	$0.442190\pi$
$728$	0.0642663	0.00238187
$729$	−14.6617	−0.543024
$730$	−2.21764	−0.0820784
$731$	85.9928	3.18056
$732$	9.67131	0.357462
$733$	−4.38060	−0.161801	−0.0809006	−	0.996722i	$-0.525780\pi$
$733$	−4.38060	−0.161801	−0.0809006	+	0.996722i	$0.525780\pi$
$734$	3.88432	0.143373
$735$	−3.13268	−0.115551
$736$	4.46384	0.164539
$737$	−47.5719	−1.75234
$738$	−4.17208	−0.153577
$739$	−6.21520	−0.228630	−0.114315	−	0.993445i	$-0.536467\pi$
$739$	−6.21520	−0.228630	−0.114315	+	0.993445i	$0.536467\pi$
$740$	16.9784	0.624140
$741$	−0.0149094	−0.000547709 0
$742$	−3.25304	−0.119423
$743$	26.7097	0.979885	0.489942	−	0.871755i	$-0.337018\pi$
$743$	26.7097	0.979885	0.489942	+	0.871755i	$0.337018\pi$
$744$	1.83022	0.0670992
$745$	−15.8788	−0.581754
$746$	3.39578	0.124328
$747$	6.41710	0.234790
$748$	−66.0784	−2.41607
$749$	7.65635	0.279757
$750$	−1.43632	−0.0524471
$751$	46.9924	1.71478	0.857388	−	0.514670i	$-0.172085\pi$
$751$	46.9924	1.71478	0.857388	+	0.514670i	$0.172085\pi$
$752$	−18.5983	−0.678210
$753$	−11.8178	−0.430666
$754$	−0.0995608	−0.00362579
$755$	26.1985	0.953461
$756$	−7.48217	−0.272124
$757$	−35.2425	−1.28091	−0.640456	−	0.767995i	$-0.721255\pi$
$757$	−35.2425	−1.28091	−0.640456	+	0.767995i	$0.721255\pi$
$758$	0.0790621	0.00287166
$759$	3.28252	0.119148
$760$	−0.936661	−0.0339763
$761$	−47.1109	−1.70777	−0.853885	−	0.520461i	$-0.825760\pi$
$761$	−47.1109	−1.70777	−0.853885	+	0.520461i	$0.825760\pi$
$762$	−0.0587413	−0.00212797
$763$	−6.52469	−0.236210
$764$	41.5499	1.50322
$765$	−24.8521	−0.898529
$766$	8.27923	0.299141
$767$	0.561234	0.0202650
$768$	−5.04045	−0.181881
$769$	−53.8827	−1.94306	−0.971530	−	0.236915i	$-0.923864\pi$
$769$	−53.8827	−1.94306	−0.971530	+	0.236915i	$0.923864\pi$
$770$	2.19728	0.0791845
$771$	−2.59995	−0.0936350
$772$	−4.43983	−0.159793
$773$	−7.33241	−0.263728	−0.131864	−	0.991268i	$-0.542096\pi$
$773$	−7.33241	−0.263728	−0.131864	+	0.991268i	$0.542096\pi$
$774$	8.99114	0.323180
$775$	−11.8170	−0.424478
$776$	2.43618	0.0874539
$777$	−4.84045	−0.173650
$778$	−2.46639	−0.0884245
$779$	−3.80509	−0.136332
$780$	0.0506918	0.00181506
$781$	−8.77143	−0.313866
$782$	−2.86666	−0.102511
$783$	23.6471	0.845078
$784$	−18.1906	−0.649666
$785$	−21.1869	−0.756193
$786$	1.83556	0.0654722
$787$	−0.274880	−0.00979841	−0.00489921	−	0.999988i	$-0.501559\pi$
$787$	−0.274880	−0.00979841	−0.00489921	+	0.999988i	$0.501559\pi$
$788$	−3.68969	−0.131440
$789$	0.549102	0.0195485
$790$	4.43086	0.157643
$791$	−17.2404	−0.612998
$792$	−14.0947	−0.500835
$793$	−0.439035	−0.0155906
$794$	−8.21160	−0.291419
$795$	−5.23466	−0.185654
$796$	−40.8493	−1.44786
$797$	−7.42858	−0.263134	−0.131567	−	0.991307i	$-0.542001\pi$
$797$	−7.42858	−0.263134	−0.131567	+	0.991307i	$0.542001\pi$
$798$	0.130896	0.00463366
$799$	38.3881	1.35807
$800$	−10.9991	−0.388878
$801$	39.9988	1.41329
$802$	1.74958	0.0617800
$803$	−30.4353	−1.07404
$804$	9.63541	0.339815
$805$	−2.37904	−0.0838502
$806$	−0.0407261	−0.00143452
$807$	1.40850	0.0495815
$808$	4.37318	0.153848
$809$	42.6491	1.49946	0.749730	−	0.661743i	$-0.230183\pi$
$809$	42.6491	1.49946	0.749730	+	0.661743i	$0.230183\pi$
$810$	−2.34698	−0.0824646
$811$	25.6139	0.899425	0.449713	−	0.893173i	$-0.351526\pi$
$811$	25.6139	0.899425	0.449713	+	0.893173i	$0.351526\pi$
$812$	−21.8150	−0.765558
$813$	−6.48392	−0.227401
$814$	−9.33649	−0.327244
$815$	−13.6990	−0.479855
$816$	12.8260	0.449001
$817$	8.20024	0.286890
$818$	−2.11837	−0.0740669
$819$	0.162603	0.00568181
$820$	12.9373	0.451790
$821$	21.8573	0.762826	0.381413	−	0.924405i	$-0.375438\pi$
$821$	21.8573	0.762826	0.381413	+	0.924405i	$0.375438\pi$
$822$	−1.54804	−0.0539942
$823$	1.58270	0.0551693	0.0275847	−	0.999619i	$-0.491218\pi$
$823$	1.58270	0.0551693	0.0275847	+	0.999619i	$0.491218\pi$
$824$	5.71887	0.199226
$825$	−8.08830	−0.281598
$826$	−4.92732	−0.171443
$827$	36.2198	1.25949	0.629743	−	0.776803i	$-0.283160\pi$
$827$	36.2198	1.25949	0.629743	+	0.776803i	$0.283160\pi$
$828$	7.48047	0.259964
$829$	30.5957	1.06263	0.531316	−	0.847174i	$-0.321698\pi$
$829$	30.5957	1.06263	0.531316	+	0.847174i	$0.321698\pi$
$830$	0.797314	0.0276752
$831$	11.6107	0.402772
$832$	0.268241	0.00929957
$833$	37.5466	1.30091
$834$	0.127356	0.00440998
$835$	27.5988	0.955096
$836$	−6.30122	−0.217932
$837$	9.67303	0.334349
$838$	0.651640	0.0225105
$839$	−9.85906	−0.340373	−0.170186	−	0.985412i	$-0.554437\pi$
$839$	−9.85906	−0.340373	−0.170186	+	0.985412i	$0.554437\pi$
$840$	−0.907924	−0.0313264
$841$	39.9455	1.37743
$842$	2.39186	0.0824290
$843$	−4.40989	−0.151885
$844$	36.7208	1.26398
$845$	16.0298	0.551443
$846$	4.01374	0.137995
$847$	15.1270	0.519769
$848$	−30.3963	−1.04381
$849$	7.82056	0.268401
$850$	7.06358	0.242279
$851$	10.1088	0.346526
$852$	1.77660	0.0608653
$853$	20.6701	0.707730	0.353865	−	0.935296i	$-0.384867\pi$
$853$	20.6701	0.707730	0.353865	+	0.935296i	$0.384867\pi$
$854$	3.85448	0.131898
$855$	−2.36989	−0.0810484
$856$	−6.10216	−0.208567
$857$	36.0858	1.23267	0.616334	−	0.787485i	$-0.288617\pi$
$857$	36.0858	1.23267	0.616334	+	0.787485i	$0.288617\pi$
$858$	−0.0278756	−0.000951656 0
$859$	40.7566	1.39060	0.695299	−	0.718720i	$-0.255272\pi$
$859$	40.7566	1.39060	0.695299	+	0.718720i	$0.255272\pi$
$860$	−27.8808	−0.950729
$861$	−3.68835	−0.125699
$862$	−0.758604	−0.0258381
$863$	21.5125	0.732295	0.366148	−	0.930557i	$-0.380677\pi$
$863$	21.5125	0.732295	0.366148	+	0.930557i	$0.380677\pi$
$864$	9.00357	0.306308
$865$	−5.13152	−0.174477
$866$	4.12947	0.140325
$867$	−18.0614	−0.613396
$868$	−8.92362	−0.302887
$869$	60.8100	2.06284
$870$	1.40655	0.0476864
$871$	−0.437405	−0.0148209
$872$	5.20022	0.176102
$873$	6.16390	0.208616
$874$	−0.273363	−0.00924665
$875$	14.2868	0.482981
$876$	6.16448	0.208278
$877$	−47.8912	−1.61717	−0.808586	−	0.588378i	$-0.799767\pi$
$877$	−47.8912	−1.61717	−0.808586	+	0.588378i	$0.799767\pi$
$878$	−5.49922	−0.185590
$879$	−16.3015	−0.549835
$880$	20.5313	0.692111
$881$	−18.2689	−0.615496	−0.307748	−	0.951468i	$-0.599575\pi$
$881$	−18.2689	−0.615496	−0.307748	+	0.951468i	$0.599575\pi$
$882$	3.92576	0.132187
$883$	−37.0354	−1.24634	−0.623170	−	0.782087i	$-0.714155\pi$
$883$	−37.0354	−1.24634	−0.623170	+	0.782087i	$0.714155\pi$
$884$	−0.607565	−0.0204346
$885$	−7.92885	−0.266525
$886$	−10.9870	−0.369115
$887$	−13.7619	−0.462080	−0.231040	−	0.972944i	$-0.574213\pi$
$887$	−13.7619	−0.462080	−0.231040	+	0.972944i	$0.574213\pi$
$888$	3.85787	0.129462
$889$	0.584285	0.0195963
$890$	4.96978	0.166587
$891$	−32.2104	−1.07909
$892$	−16.1870	−0.541982
$893$	3.66067	0.122500
$894$	−1.76857	−0.0591499
$895$	7.14059	0.238684
$896$	−10.9938	−0.367278
$897$	0.0301815	0.00100773
$898$	−6.36221	−0.212310
$899$	28.2027	0.940613
$900$	−18.4323	−0.614409
$901$	62.7398	2.09017
$902$	−7.11426	−0.236879
$903$	7.94866	0.264515
$904$	13.7407	0.457009
$905$	−21.5716	−0.717063
$906$	2.91797	0.0969432
$907$	−27.1842	−0.902637	−0.451319	−	0.892363i	$-0.649046\pi$
$907$	−27.1842	−0.902637	−0.451319	+	0.892363i	$0.649046\pi$
$908$	−3.51619	−0.116689
$909$	11.0648	0.366996
$910$	0.0202031	0.000669727 0
$911$	29.4894	0.977027	0.488513	−	0.872556i	$-0.337539\pi$
$911$	29.4894	0.977027	0.488513	+	0.872556i	$0.337539\pi$
$912$	1.22309	0.0405004
$913$	10.9425	0.362143
$914$	−2.68277	−0.0887380
$915$	6.20248	0.205048
$916$	−4.84656	−0.160135
$917$	−18.2579	−0.602928
$918$	−5.78204	−0.190836
$919$	−44.2354	−1.45919	−0.729596	−	0.683878i	$-0.760292\pi$
$919$	−44.2354	−1.45919	−0.729596	+	0.683878i	$0.760292\pi$
$920$	1.89611	0.0625129
$921$	−10.0128	−0.329935
$922$	1.74331	0.0574130
$923$	−0.0806498	−0.00265462
$924$	−6.10789	−0.200935
$925$	−24.9086	−0.818991
$926$	−2.12163	−0.0697210
$927$	14.4696	0.475243
$928$	26.2508	0.861726
$929$	17.7425	0.582112	0.291056	−	0.956706i	$-0.405993\pi$
$929$	17.7425	0.582112	0.291056	+	0.956706i	$0.405993\pi$
$930$	0.575360	0.0188668
$931$	3.58043	0.117344
$932$	−31.5003	−1.03183
$933$	−1.12473	−0.0368220
$934$	7.18766	0.235187
$935$	−42.3779	−1.38591
$936$	−0.129596	−0.00423596
$937$	46.1680	1.50824	0.754121	−	0.656735i	$-0.228063\pi$
$937$	46.1680	1.50824	0.754121	+	0.656735i	$0.228063\pi$
$938$	3.84017	0.125386
$939$	−6.48949	−0.211777
$940$	−12.4463	−0.405953
$941$	−22.0383	−0.718428	−0.359214	−	0.933255i	$-0.616955\pi$
$941$	−22.0383	−0.718428	−0.359214	+	0.933255i	$0.616955\pi$
$942$	−2.35978	−0.0768859
$943$	7.70276	0.250836
$944$	−46.0407	−1.49850
$945$	−4.79853	−0.156096
$946$	15.3317	0.498478
$947$	−34.5940	−1.12416	−0.562078	−	0.827085i	$-0.689998\pi$
$947$	−34.5940	−1.12416	−0.562078	+	0.827085i	$0.689998\pi$
$948$	−12.3167	−0.400028
$949$	−0.279840	−0.00908399
$950$	0.673581	0.0218539
$951$	3.89209	0.126210
$952$	10.8819	0.352684
$953$	−20.9799	−0.679607	−0.339803	−	0.940496i	$-0.610361\pi$
$953$	−20.9799	−0.679607	−0.339803	+	0.940496i	$0.610361\pi$
$954$	6.55988	0.212384
$955$	26.6471	0.862281
$956$	20.6583	0.668138
$957$	19.3037	0.624001
$958$	3.50112	0.113116
$959$	15.3980	0.497228
$960$	−3.78958	−0.122308
$961$	−19.4635	−0.627854
$962$	−0.0858453	−0.00276777
$963$	−15.4393	−0.497526
$964$	1.92295	0.0619341
$965$	−2.84738	−0.0916605
$966$	−0.264976	−0.00852548
$967$	−0.0716187	−0.00230310	−0.00115155	−	0.999999i	$-0.500367\pi$
$967$	−0.0716187	−0.00230310	−0.00115155	+	0.999999i	$0.500367\pi$
$968$	−12.0563	−0.387504
$969$	−2.52453	−0.0810995
$970$	0.765853	0.0245901
$971$	−8.18205	−0.262575	−0.131287	−	0.991344i	$-0.541911\pi$
$971$	−8.18205	−0.262575	−0.131287	+	0.991344i	$0.541911\pi$
$972$	22.9531	0.736222
$973$	−1.26678	−0.0406111
$974$	8.82161	0.282663
$975$	−0.0743687	−0.00238170
$976$	36.0162	1.15285
$977$	22.6886	0.725872	0.362936	−	0.931814i	$-0.381774\pi$
$977$	22.6886	0.725872	0.362936	+	0.931814i	$0.381774\pi$
$978$	−1.52579	−0.0487893
$979$	68.2062	2.17988
$980$	−12.1735	−0.388868
$981$	13.1573	0.420080
$982$	−2.83548	−0.0904837
$983$	−18.8463	−0.601103	−0.300551	−	0.953766i	$-0.597171\pi$
$983$	−18.8463	−0.601103	−0.300551	+	0.953766i	$0.597171\pi$
$984$	2.93964	0.0937122
$985$	−2.36630	−0.0753966
$986$	−16.8581	−0.536872
$987$	3.54836	0.112946
$988$	−0.0579372	−0.00184323
$989$	−16.6000	−0.527850
$990$	−4.43090	−0.140823
$991$	−16.8310	−0.534655	−0.267328	−	0.963606i	$-0.586141\pi$
$991$	−16.8310	−0.534655	−0.267328	+	0.963606i	$0.586141\pi$
$992$	10.7381	0.340935
$993$	9.58775	0.304258
$994$	0.708060	0.0224583
$995$	−26.1978	−0.830525
$996$	−2.21633	−0.0702272
$997$	60.1343	1.90447	0.952236	−	0.305363i	$-0.0987779\pi$
$997$	60.1343	1.90447	0.952236	+	0.305363i	$0.0987779\pi$
$998$	−2.59643	−0.0821885
$999$	20.3895	0.645095

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	241.2.a.a.1.5	✓	7
3.2	odd	2		2169.2.a.e.1.3		7
4.3	odd	2		3856.2.a.j.1.4		7
5.4	even	2		6025.2.a.f.1.3		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
241.2.a.a.1.5	✓	7	1.1	even	1	trivial
2169.2.a.e.1.3		7	3.2	odd	2
3856.2.a.j.1.4		7	4.3	odd	2
6025.2.a.f.1.3		7	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 241 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	241.a (trivial)

\(f(q)\)	\(=\)	\(q+0.277577 q^{2} -0.494846 q^{3} -1.92295 q^{4} -1.23324 q^{5} -0.137358 q^{6} +1.36627 q^{7} -1.08892 q^{8} -2.75513 q^{9} +O(q^{10})\)	\(q+0.277577 q^{2} -0.494846 q^{3} -1.92295 q^{4} -1.23324 q^{5} -0.137358 q^{6} +1.36627 q^{7} -1.08892 q^{8} -2.75513 q^{9} -0.342320 q^{10} -4.69806 q^{11} +0.951564 q^{12} -0.0431968 q^{13} +0.379244 q^{14} +0.610264 q^{15} +3.54364 q^{16} -7.31430 q^{17} -0.764761 q^{18} -0.697489 q^{19} +2.37146 q^{20} -0.676090 q^{21} -1.30407 q^{22} +1.41195 q^{23} +0.538848 q^{24} -3.47911 q^{25} -0.0119904 q^{26} +2.84790 q^{27} -2.62726 q^{28} +8.30334 q^{29} +0.169395 q^{30} +3.39655 q^{31} +3.16148 q^{32} +2.32481 q^{33} -2.03028 q^{34} -1.68494 q^{35} +5.29798 q^{36} +7.15948 q^{37} -0.193607 q^{38} +0.0213757 q^{39} +1.34290 q^{40} +5.45541 q^{41} -0.187667 q^{42} -11.7568 q^{43} +9.03414 q^{44} +3.39774 q^{45} +0.391925 q^{46} -5.24836 q^{47} -1.75356 q^{48} -5.13332 q^{49} -0.965723 q^{50} +3.61945 q^{51} +0.0830653 q^{52} -8.57769 q^{53} +0.790512 q^{54} +5.79385 q^{55} -1.48776 q^{56} +0.345149 q^{57} +2.30482 q^{58} -12.9925 q^{59} -1.17351 q^{60} +10.1636 q^{61} +0.942804 q^{62} -3.76423 q^{63} -6.20973 q^{64} +0.0532721 q^{65} +0.645315 q^{66} +10.1259 q^{67} +14.0650 q^{68} -0.698697 q^{69} -0.467700 q^{70} +1.86703 q^{71} +3.00012 q^{72} +6.47826 q^{73} +1.98731 q^{74} +1.72162 q^{75} +1.34124 q^{76} -6.41880 q^{77} +0.00593342 q^{78} -12.9436 q^{79} -4.37017 q^{80} +6.85611 q^{81} +1.51430 q^{82} -2.32915 q^{83} +1.30009 q^{84} +9.02030 q^{85} -3.26342 q^{86} -4.10887 q^{87} +5.11582 q^{88} -14.5180 q^{89} +0.943135 q^{90} -0.0590183 q^{91} -2.71511 q^{92} -1.68077 q^{93} -1.45682 q^{94} +0.860173 q^{95} -1.56444 q^{96} -2.23725 q^{97} -1.42489 q^{98} +12.9438 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 4 q^{2} - 3 q^{3} + 2 q^{4} - 8 q^{5} - 5 q^{6} - 7 q^{7} - 6 q^{8} - 2 q^{9}+O(q^{10}) \) 7 * q - 4 * q^2 - 3 * q^3 + 2 * q^4 - 8 * q^5 - 5 * q^6 - 7 * q^7 - 6 * q^8 - 2 * q^9	\( 7 q - 4 q^{2} - 3 q^{3} + 2 q^{4} - 8 q^{5} - 5 q^{6} - 7 q^{7} - 6 q^{8} - 2 q^{9} + 3 q^{10} - 18 q^{11} + q^{12} - q^{13} - 6 q^{14} - 11 q^{15} + 4 q^{16} - 2 q^{17} + 8 q^{18} - 6 q^{19} - 8 q^{20} - 2 q^{21} + 10 q^{22} - 22 q^{23} - 3 q^{24} + 5 q^{25} + 8 q^{26} + 3 q^{27} + 9 q^{28} - 16 q^{29} + 29 q^{30} - 18 q^{31} - 6 q^{32} + 4 q^{33} + 11 q^{34} + 7 q^{35} - 7 q^{36} + 8 q^{37} + 16 q^{38} - 9 q^{39} + 14 q^{40} - 15 q^{41} + 19 q^{42} + 14 q^{43} - 4 q^{44} + 3 q^{45} + 11 q^{46} - 10 q^{47} + 31 q^{48} + 6 q^{49} - 4 q^{50} + 13 q^{51} + 27 q^{52} + 15 q^{53} + 16 q^{54} + 29 q^{55} + 13 q^{56} + 14 q^{57} + 17 q^{58} - 18 q^{59} + 15 q^{60} + 4 q^{61} + 13 q^{62} - 16 q^{63} + 2 q^{64} - 7 q^{65} + 16 q^{66} + 18 q^{67} - 15 q^{68} + 26 q^{69} + 8 q^{70} - 50 q^{71} + 30 q^{72} + 10 q^{74} + 16 q^{75} - 20 q^{76} + 17 q^{77} - 32 q^{78} - 15 q^{79} - 11 q^{80} - 9 q^{81} + 45 q^{82} - 24 q^{83} + 6 q^{84} - 2 q^{85} - 23 q^{86} + 12 q^{87} + 8 q^{88} - 13 q^{89} - 39 q^{90} - 12 q^{91} - 10 q^{92} + 14 q^{93} - 32 q^{94} - 41 q^{95} - 15 q^{96} + q^{97} + 9 q^{98} - 20 q^{99}+O(q^{100}) \) 7 * q - 4 * q^2 - 3 * q^3 + 2 * q^4 - 8 * q^5 - 5 * q^6 - 7 * q^7 - 6 * q^8 - 2 * q^9 + 3 * q^10 - 18 * q^11 + q^12 - q^13 - 6 * q^14 - 11 * q^15 + 4 * q^16 - 2 * q^17 + 8 * q^18 - 6 * q^19 - 8 * q^20 - 2 * q^21 + 10 * q^22 - 22 * q^23 - 3 * q^24 + 5 * q^25 + 8 * q^26 + 3 * q^27 + 9 * q^28 - 16 * q^29 + 29 * q^30 - 18 * q^31 - 6 * q^32 + 4 * q^33 + 11 * q^34 + 7 * q^35 - 7 * q^36 + 8 * q^37 + 16 * q^38 - 9 * q^39 + 14 * q^40 - 15 * q^41 + 19 * q^42 + 14 * q^43 - 4 * q^44 + 3 * q^45 + 11 * q^46 - 10 * q^47 + 31 * q^48 + 6 * q^49 - 4 * q^50 + 13 * q^51 + 27 * q^52 + 15 * q^53 + 16 * q^54 + 29 * q^55 + 13 * q^56 + 14 * q^57 + 17 * q^58 - 18 * q^59 + 15 * q^60 + 4 * q^61 + 13 * q^62 - 16 * q^63 + 2 * q^64 - 7 * q^65 + 16 * q^66 + 18 * q^67 - 15 * q^68 + 26 * q^69 + 8 * q^70 - 50 * q^71 + 30 * q^72 + 10 * q^74 + 16 * q^75 - 20 * q^76 + 17 * q^77 - 32 * q^78 - 15 * q^79 - 11 * q^80 - 9 * q^81 + 45 * q^82 - 24 * q^83 + 6 * q^84 - 2 * q^85 - 23 * q^86 + 12 * q^87 + 8 * q^88 - 13 * q^89 - 39 * q^90 - 12 * q^91 - 10 * q^92 + 14 * q^93 - 32 * q^94 - 41 * q^95 - 15 * q^96 + q^97 + 9 * q^98 - 20 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more