LMFDB - Embedded newform 241.2.a.b.1.8

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:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 14 x^{10} + 44 x^{9} + 65 x^{8} - 219 x^{7} - 123 x^{6} + 444 x^{5} + 105 x^{4} + \cdots - 1 $ x^12 - 3x^11 - 14x^10 + 44x^9 + 65x^8 - 219x^7 - 123x^6 + 444x^5 + 105x^4 - 328x^3 - 45x^2 + 18*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$1.54879$ 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)

$f(q)$	$=$	$q+1.54879 q^{2} +2.81087 q^{3} +0.398765 q^{4} +0.334961 q^{5} +4.35346 q^{6} -4.24623 q^{7} -2.47998 q^{8} +4.90098 q^{9} +O(q^{10})$	\(q+1.54879 q^{2} +2.81087 q^{3} +0.398765 q^{4} +0.334961 q^{5} +4.35346 q^{6} -4.24623 q^{7} -2.47998 q^{8} +4.90098 q^{9} +0.518786 q^{10} +0.915418 q^{11} +1.12088 q^{12} +4.81392 q^{13} -6.57654 q^{14} +0.941532 q^{15} -4.63852 q^{16} -5.38915 q^{17} +7.59061 q^{18} -4.34799 q^{19} +0.133571 q^{20} -11.9356 q^{21} +1.41779 q^{22} +8.10534 q^{23} -6.97091 q^{24} -4.88780 q^{25} +7.45578 q^{26} +5.34340 q^{27} -1.69325 q^{28} -6.45221 q^{29} +1.45824 q^{30} +10.7804 q^{31} -2.22414 q^{32} +2.57312 q^{33} -8.34668 q^{34} -1.42232 q^{35} +1.95434 q^{36} +5.16908 q^{37} -6.73415 q^{38} +13.5313 q^{39} -0.830699 q^{40} -0.612344 q^{41} -18.4858 q^{42} -1.85213 q^{43} +0.365036 q^{44} +1.64164 q^{45} +12.5535 q^{46} -2.21116 q^{47} -13.0383 q^{48} +11.0305 q^{49} -7.57020 q^{50} -15.1482 q^{51} +1.91962 q^{52} +0.00846193 q^{53} +8.27584 q^{54} +0.306629 q^{55} +10.5306 q^{56} -12.2216 q^{57} -9.99315 q^{58} +8.85799 q^{59} +0.375450 q^{60} +3.78644 q^{61} +16.6967 q^{62} -20.8107 q^{63} +5.83230 q^{64} +1.61248 q^{65} +3.98523 q^{66} +4.67684 q^{67} -2.14900 q^{68} +22.7830 q^{69} -2.20289 q^{70} +1.48694 q^{71} -12.1544 q^{72} -10.3630 q^{73} +8.00585 q^{74} -13.7390 q^{75} -1.73383 q^{76} -3.88708 q^{77} +20.9572 q^{78} -17.6786 q^{79} -1.55372 q^{80} +0.316666 q^{81} -0.948396 q^{82} +7.45731 q^{83} -4.75950 q^{84} -1.80516 q^{85} -2.86857 q^{86} -18.1363 q^{87} -2.27022 q^{88} -0.520713 q^{89} +2.54256 q^{90} -20.4410 q^{91} +3.23212 q^{92} +30.3024 q^{93} -3.42463 q^{94} -1.45641 q^{95} -6.25177 q^{96} -6.33494 q^{97} +17.0840 q^{98} +4.48644 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 3 q^{2} + q^{3} + 13 q^{4} + 6 q^{5} - q^{6} + 3 q^{7} + 9 q^{8} + 15 q^{9}+O(q^{10}) $ 12 * q + 3 * q^2 + q^3 + 13 * q^4 + 6 * q^5 - q^6 + 3 * q^7 + 9 * q^8 + 15 * q^9	\( 12 q + 3 q^{2} + q^{3} + 13 q^{4} + 6 q^{5} - q^{6} + 3 q^{7} + 9 q^{8} + 15 q^{9} - 7 q^{10} + 22 q^{11} - 7 q^{12} - 5 q^{13} + 6 q^{14} + 13 q^{15} + 15 q^{16} - 4 q^{17} - q^{18} - 6 q^{19} + 10 q^{20} - 14 q^{21} - 12 q^{22} + 32 q^{23} - 15 q^{24} + 4 q^{25} + 8 q^{26} - 5 q^{27} - 11 q^{28} + 6 q^{29} - 19 q^{30} + 8 q^{31} + q^{32} - 24 q^{33} - 19 q^{34} + 15 q^{35} - 8 q^{36} - 8 q^{37} - 10 q^{38} + 31 q^{39} - 52 q^{40} - q^{41} - 49 q^{42} - 2 q^{43} + 42 q^{44} - 15 q^{45} - 25 q^{46} + 34 q^{47} - 49 q^{48} - 9 q^{49} - 27 q^{50} - 3 q^{51} - 41 q^{52} + 5 q^{53} - 40 q^{54} - 3 q^{55} + q^{56} - 22 q^{57} - 33 q^{58} + 26 q^{59} - 57 q^{60} - 26 q^{61} - 17 q^{62} - 4 q^{63} + 13 q^{64} - 25 q^{65} - 2 q^{66} + 6 q^{67} - 35 q^{68} - 2 q^{69} - 4 q^{70} + 94 q^{71} + 17 q^{72} - 22 q^{73} + 26 q^{74} - 20 q^{76} - 7 q^{77} + 54 q^{78} + 9 q^{79} + 19 q^{80} + 4 q^{81} + 15 q^{82} - 8 q^{83} + 2 q^{84} + 4 q^{85} + 9 q^{86} + 4 q^{87} + 6 q^{88} - 3 q^{89} + 11 q^{90} - 20 q^{91} + 36 q^{92} + 12 q^{93} + 48 q^{94} + 33 q^{95} - 23 q^{96} - 29 q^{97} + 28 q^{98} + 36 q^{99}+O(q^{100}) \) 12 * q + 3 * q^2 + q^3 + 13 * q^4 + 6 * q^5 - q^6 + 3 * q^7 + 9 * q^8 + 15 * q^9 - 7 * q^10 + 22 * q^11 - 7 * q^12 - 5 * q^13 + 6 * q^14 + 13 * q^15 + 15 * q^16 - 4 * q^17 - q^18 - 6 * q^19 + 10 * q^20 - 14 * q^21 - 12 * q^22 + 32 * q^23 - 15 * q^24 + 4 * q^25 + 8 * q^26 - 5 * q^27 - 11 * q^28 + 6 * q^29 - 19 * q^30 + 8 * q^31 + q^32 - 24 * q^33 - 19 * q^34 + 15 * q^35 - 8 * q^36 - 8 * q^37 - 10 * q^38 + 31 * q^39 - 52 * q^40 - q^41 - 49 * q^42 - 2 * q^43 + 42 * q^44 - 15 * q^45 - 25 * q^46 + 34 * q^47 - 49 * q^48 - 9 * q^49 - 27 * q^50 - 3 * q^51 - 41 * q^52 + 5 * q^53 - 40 * q^54 - 3 * q^55 + q^56 - 22 * q^57 - 33 * q^58 + 26 * q^59 - 57 * q^60 - 26 * q^61 - 17 * q^62 - 4 * q^63 + 13 * q^64 - 25 * q^65 - 2 * q^66 + 6 * q^67 - 35 * q^68 - 2 * q^69 - 4 * q^70 + 94 * q^71 + 17 * q^72 - 22 * q^73 + 26 * q^74 - 20 * q^76 - 7 * q^77 + 54 * q^78 + 9 * q^79 + 19 * q^80 + 4 * q^81 + 15 * q^82 - 8 * q^83 + 2 * q^84 + 4 * q^85 + 9 * q^86 + 4 * q^87 + 6 * q^88 - 3 * q^89 + 11 * q^90 - 20 * q^91 + 36 * q^92 + 12 * q^93 + 48 * q^94 + 33 * q^95 - 23 * q^96 - 29 * q^97 + 28 * q^98 + 36 * 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 $

$2$	1.54879	1.09516	0.547582	−	0.836752i	$-0.315549\pi$
$2$	1.54879	1.09516	0.547582	+	0.836752i	$0.315549\pi$
$3$	2.81087	1.62286	0.811428	−	0.584453i	$-0.198691\pi$
$3$	2.81087	1.62286	0.811428	+	0.584453i	$0.198691\pi$
$4$	0.398765	0.199382
$5$	0.334961	0.149799	0.0748996	−	0.997191i	$-0.476136\pi$
$5$	0.334961	0.149799	0.0748996	+	0.997191i	$0.476136\pi$
$6$	4.35346	1.77729
$7$	−4.24623	−1.60493	−0.802463	−	0.596702i	$-0.796477\pi$
$7$	−4.24623	−1.60493	−0.802463	+	0.596702i	$0.796477\pi$
$8$	−2.47998	−0.876807
$9$	4.90098	1.63366
$10$	0.518786	0.164055
$11$	0.915418	0.276009	0.138004	−	0.990432i	$-0.455931\pi$
$11$	0.915418	0.276009	0.138004	+	0.990432i	$0.455931\pi$
$12$	1.12088	0.323569
$13$	4.81392	1.33514	0.667571	−	0.744546i	$-0.267334\pi$
$13$	4.81392	1.33514	0.667571	+	0.744546i	$0.267334\pi$
$14$	−6.57654	−1.75766
$15$	0.941532	0.243103
$16$	−4.63852	−1.15963
$17$	−5.38915	−1.30706	−0.653530	−	0.756901i	$-0.726713\pi$
$17$	−5.38915	−1.30706	−0.653530	+	0.756901i	$0.726713\pi$
$18$	7.59061	1.78912
$19$	−4.34799	−0.997498	−0.498749	−	0.866747i	$-0.666207\pi$
$19$	−4.34799	−0.997498	−0.498749	+	0.866747i	$0.666207\pi$
$20$	0.133571	0.0298673
$21$	−11.9356	−2.60456
$22$	1.41779	0.302275
$23$	8.10534	1.69008	0.845040	−	0.534703i	$-0.179577\pi$
$23$	8.10534	1.69008	0.845040	+	0.534703i	$0.179577\pi$
$24$	−6.97091	−1.42293
$25$	−4.88780	−0.977560
$26$	7.45578	1.46220
$27$	5.34340	1.02834
$28$	−1.69325	−0.319994
$29$	−6.45221	−1.19815	−0.599073	−	0.800695i	$-0.704464\pi$
$29$	−6.45221	−1.19815	−0.599073	+	0.800695i	$0.704464\pi$
$30$	1.45824	0.266237
$31$	10.7804	1.93623	0.968113	−	0.250515i	$-0.0806000\pi$
$31$	10.7804	1.93623	0.968113	+	0.250515i	$0.0806000\pi$
$32$	−2.22414	−0.393176
$33$	2.57312	0.447922
$34$	−8.34668	−1.43144
$35$	−1.42232	−0.240417
$36$	1.95434	0.325723
$37$	5.16908	0.849792	0.424896	−	0.905242i	$-0.360311\pi$
$37$	5.16908	0.849792	0.424896	+	0.905242i	$0.360311\pi$
$38$	−6.73415	−1.09242
$39$	13.5313	2.16674
$40$	−0.830699	−0.131345
$41$	−0.612344	−0.0956321	−0.0478161	−	0.998856i	$-0.515226\pi$
$41$	−0.612344	−0.0956321	−0.0478161	+	0.998856i	$0.515226\pi$
$42$	−18.4858	−2.85242
$43$	−1.85213	−0.282447	−0.141223	−	0.989978i	$-0.545104\pi$
$43$	−1.85213	−0.282447	−0.141223	+	0.989978i	$0.545104\pi$
$44$	0.365036	0.0550313
$45$	1.64164	0.244721
$46$	12.5535	1.85091
$47$	−2.21116	−0.322531	−0.161265	−	0.986911i	$-0.551557\pi$
$47$	−2.21116	−0.322531	−0.161265	+	0.986911i	$0.551557\pi$
$48$	−13.0383	−1.88191
$49$	11.0305	1.57579
$50$	−7.57020	−1.07059
$51$	−15.1482	−2.12117
$52$	1.91962	0.266204
$53$	0.00846193	0.00116234	0.000581168	−	1.00000i	$-0.499815\pi$
$53$	0.00846193	0.00116234	0.000581168		1.00000i	$0.499815\pi$
$54$	8.27584	1.12620
$55$	0.306629	0.0413459
$56$	10.5306	1.40721
$57$	−12.2216	−1.61879
$58$	−9.99315	−1.31216
$59$	8.85799	1.15321	0.576606	−	0.817022i	$-0.304377\pi$
$59$	8.85799	1.15321	0.576606	+	0.817022i	$0.304377\pi$
$60$	0.375450	0.0484704
$61$	3.78644	0.484805	0.242402	−	0.970176i	$-0.422065\pi$
$61$	3.78644	0.484805	0.242402	+	0.970176i	$0.422065\pi$
$62$	16.6967	2.12048
$63$	−20.8107	−2.62190
$64$	5.83230	0.729037
$65$	1.61248	0.200003
$66$	3.98523	0.490548
$67$	4.67684	0.571367	0.285683	−	0.958324i	$-0.407779\pi$
$67$	4.67684	0.571367	0.285683	+	0.958324i	$0.407779\pi$
$68$	−2.14900	−0.260605
$69$	22.7830	2.74276
$70$	−2.20289	−0.263295
$71$	1.48694	0.176467	0.0882334	−	0.996100i	$-0.471878\pi$
$71$	1.48694	0.176467	0.0882334	+	0.996100i	$0.471878\pi$
$72$	−12.1544	−1.43240
$73$	−10.3630	−1.21290	−0.606449	−	0.795122i	$-0.707407\pi$
$73$	−10.3630	−1.21290	−0.606449	+	0.795122i	$0.707407\pi$
$74$	8.00585	0.930661
$75$	−13.7390	−1.58644
$76$	−1.73383	−0.198884
$77$	−3.88708	−0.442974
$78$	20.9572	2.37294
$79$	−17.6786	−1.98900	−0.994499	−	0.104743i	$-0.966598\pi$
$79$	−17.6786	−1.98900	−0.994499	+	0.104743i	$0.966598\pi$
$80$	−1.55372	−0.173712
$81$	0.316666	0.0351851
$82$	−0.948396	−0.104733
$83$	7.45731	0.818546	0.409273	−	0.912412i	$-0.365782\pi$
$83$	7.45731	0.818546	0.409273	+	0.912412i	$0.365782\pi$
$84$	−4.75950	−0.519304
$85$	−1.80516	−0.195797
$86$	−2.86857	−0.309325
$87$	−18.1363	−1.94442
$88$	−2.27022	−0.242006
$89$	−0.520713	−0.0551955	−0.0275978	−	0.999619i	$-0.508786\pi$
$89$	−0.520713	−0.0551955	−0.0275978	+	0.999619i	$0.508786\pi$
$90$	2.54256	0.268010
$91$	−20.4410	−2.14280
$92$	3.23212	0.336972
$93$	30.3024	3.14221
$94$	−3.42463	−0.353224
$95$	−1.45641	−0.149424
$96$	−6.25177	−0.638068
$97$	−6.33494	−0.643216	−0.321608	−	0.946873i	$-0.604223\pi$
$97$	−6.33494	−0.643216	−0.321608	+	0.946873i	$0.604223\pi$
$98$	17.0840	1.72574
$99$	4.48644	0.450904
$100$	−1.94908	−0.194908
$101$	7.20652	0.717076	0.358538	−	0.933515i	$-0.383275\pi$
$101$	7.20652	0.717076	0.358538	+	0.933515i	$0.383275\pi$
$102$	−23.4614	−2.32303
$103$	7.82738	0.771255	0.385627	−	0.922655i	$-0.373985\pi$
$103$	7.82738	0.771255	0.385627	+	0.922655i	$0.373985\pi$
$104$	−11.9385	−1.17066
$105$	−3.99797	−0.390161
$106$	0.0131058	0.00127295
$107$	3.28176	0.317260	0.158630	−	0.987338i	$-0.449292\pi$
$107$	3.28176	0.317260	0.158630	+	0.987338i	$0.449292\pi$
$108$	2.13076	0.205033
$109$	6.89529	0.660449	0.330224	−	0.943902i	$-0.392876\pi$
$109$	6.89529	0.660449	0.330224	+	0.943902i	$0.392876\pi$
$110$	0.474906	0.0452805
$111$	14.5296	1.37909
$112$	19.6962	1.86112
$113$	−14.4737	−1.36157	−0.680787	−	0.732481i	$-0.738362\pi$
$113$	−14.4737	−1.36157	−0.680787	+	0.732481i	$0.738362\pi$
$114$	−18.9288	−1.77284
$115$	2.71497	0.253173
$116$	−2.57292	−0.238889
$117$	23.5929	2.18117
$118$	13.7192	1.26296
$119$	22.8836	2.09773
$120$	−2.33499	−0.213154
$121$	−10.1620	−0.923819
$122$	5.86443	0.530940
$123$	−1.72122	−0.155197
$124$	4.29886	0.386049
$125$	−3.31203	−0.296237
$126$	−32.2315	−2.87141
$127$	6.96745	0.618261	0.309130	−	0.951020i	$-0.399962\pi$
$127$	6.96745	0.618261	0.309130	+	0.951020i	$0.399962\pi$
$128$	13.4813	1.19159
$129$	−5.20609	−0.458370
$130$	2.49740	0.219036
$131$	7.02108	0.613434	0.306717	−	0.951801i	$-0.400769\pi$
$131$	7.02108	0.613434	0.306717	+	0.951801i	$0.400769\pi$
$132$	1.02607	0.0893079
$133$	18.4626	1.60091
$134$	7.24346	0.625740
$135$	1.78983	0.154044
$136$	13.3650	1.14604
$137$	5.78101	0.493905	0.246953	−	0.969028i	$-0.420571\pi$
$137$	5.78101	0.493905	0.246953	+	0.969028i	$0.420571\pi$
$138$	35.2862	3.00376
$139$	2.34939	0.199272	0.0996362	−	0.995024i	$-0.468232\pi$
$139$	2.34939	0.199272	0.0996362	+	0.995024i	$0.468232\pi$
$140$	−0.567173	−0.0479349
$141$	−6.21527	−0.523421
$142$	2.30296	0.193260
$143$	4.40675	0.368511
$144$	−22.7333	−1.89444
$145$	−2.16124	−0.179481
$146$	−16.0502	−1.32832
$147$	31.0053	2.55727
$148$	2.06125	0.169434
$149$	4.44128	0.363844	0.181922	−	0.983313i	$-0.441768\pi$
$149$	4.44128	0.363844	0.181922	+	0.983313i	$0.441768\pi$
$150$	−21.2788	−1.73741
$151$	−5.33629	−0.434261	−0.217130	−	0.976143i	$-0.569670\pi$
$151$	−5.33629	−0.434261	−0.217130	+	0.976143i	$0.569670\pi$
$152$	10.7830	0.874613
$153$	−26.4121	−2.13529
$154$	−6.02028	−0.485128
$155$	3.61103	0.290045
$156$	5.39581	0.432010
$157$	−21.6127	−1.72488	−0.862442	−	0.506157i	$-0.831066\pi$
$157$	−21.6127	−1.72488	−0.862442	+	0.506157i	$0.831066\pi$
$158$	−27.3805	−2.17828
$159$	0.0237854	0.00188630
$160$	−0.745001	−0.0588975
$161$	−34.4172	−2.71245
$162$	0.490451	0.0385334
$163$	−22.0999	−1.73100	−0.865500	−	0.500909i	$-0.832999\pi$
$163$	−22.0999	−1.73100	−0.865500	+	0.500909i	$0.832999\pi$
$164$	−0.244181	−0.0190674
$165$	0.861895	0.0670984
$166$	11.5498	0.896442
$167$	−2.58299	−0.199877	−0.0999387	−	0.994994i	$-0.531865\pi$
$167$	−2.58299	−0.199877	−0.0999387	+	0.994994i	$0.531865\pi$
$168$	29.6001	2.28370
$169$	10.1739	0.782604
$170$	−2.79582	−0.214429
$171$	−21.3094	−1.62957
$172$	−0.738564	−0.0563149
$173$	9.53817	0.725174	0.362587	−	0.931950i	$-0.381894\pi$
$173$	9.53817	0.725174	0.362587	+	0.931950i	$0.381894\pi$
$174$	−28.0894	−2.12945
$175$	20.7547	1.56891
$176$	−4.24618	−0.320068
$177$	24.8986	1.87150
$178$	−0.806478	−0.0604481
$179$	11.6661	0.871963	0.435981	−	0.899956i	$-0.356401\pi$
$179$	11.6661	0.871963	0.435981	+	0.899956i	$0.356401\pi$
$180$	0.654628	0.0487931
$181$	−6.78106	−0.504032	−0.252016	−	0.967723i	$-0.581094\pi$
$181$	−6.78106	−0.504032	−0.252016	+	0.967723i	$0.581094\pi$
$182$	−31.6590	−2.34672
$183$	10.6432	0.786768
$184$	−20.1011	−1.48187
$185$	1.73144	0.127298
$186$	46.9322	3.44124
$187$	−4.93332	−0.360760
$188$	−0.881732	−0.0643069
$189$	−22.6893	−1.65041
$190$	−2.25568	−0.163644
$191$	9.30072	0.672976	0.336488	−	0.941688i	$-0.390761\pi$
$191$	9.30072	0.672976	0.336488	+	0.941688i	$0.390761\pi$
$192$	16.3938	1.18312
$193$	−1.27177	−0.0915440	−0.0457720	−	0.998952i	$-0.514575\pi$
$193$	−1.27177	−0.0915440	−0.0457720	+	0.998952i	$0.514575\pi$
$194$	−9.81153	−0.704427
$195$	4.53246	0.324576
$196$	4.39858	0.314184
$197$	9.51222	0.677718	0.338859	−	0.940837i	$-0.389959\pi$
$197$	9.51222	0.677718	0.338859	+	0.940837i	$0.389959\pi$
$198$	6.94858	0.493814
$199$	−18.4776	−1.30984	−0.654922	−	0.755696i	$-0.727299\pi$
$199$	−18.4776	−1.30984	−0.654922	+	0.755696i	$0.727299\pi$
$200$	12.1217	0.857132
$201$	13.1460	0.927246
$202$	11.1614	0.785315
$203$	27.3976	1.92293
$204$	−6.04056	−0.422924
$205$	−0.205112	−0.0143256
$206$	12.1230	0.844650
$207$	39.7241	2.76102
$208$	−22.3295	−1.54827
$209$	−3.98023	−0.275318
$210$	−6.19203	−0.427291
$211$	5.54459	0.381705	0.190853	−	0.981619i	$-0.438875\pi$
$211$	5.54459	0.381705	0.190853	+	0.981619i	$0.438875\pi$
$212$	0.00337432	0.000231749 0
$213$	4.17958	0.286380
$214$	5.08278	0.347451
$215$	−0.620391	−0.0423103
$216$	−13.2516	−0.901654
$217$	−45.7763	−3.10750
$218$	10.6794	0.723299
$219$	−29.1290	−1.96836
$220$	0.122273	0.00824365
$221$	−25.9429	−1.74511
$222$	22.5034	1.51033
$223$	−12.8984	−0.863744	−0.431872	−	0.901935i	$-0.642147\pi$
$223$	−12.8984	−0.863744	−0.431872	+	0.901935i	$0.642147\pi$
$224$	9.44422	0.631019
$225$	−23.9550	−1.59700
$226$	−22.4169	−1.49115
$227$	2.85469	0.189473	0.0947364	−	0.995502i	$-0.469799\pi$
$227$	2.85469	0.189473	0.0947364	+	0.995502i	$0.469799\pi$
$228$	−4.87356	−0.322759
$229$	−16.1705	−1.06857	−0.534287	−	0.845303i	$-0.679420\pi$
$229$	−16.1705	−1.06857	−0.534287	+	0.845303i	$0.679420\pi$
$230$	4.20494	0.277265
$231$	−10.9261	−0.718882
$232$	16.0014	1.05054
$233$	11.3792	0.745476	0.372738	−	0.927937i	$-0.378419\pi$
$233$	11.3792	0.745476	0.372738	+	0.927937i	$0.378419\pi$
$234$	36.5406	2.38874
$235$	−0.740653	−0.0483148
$236$	3.53226	0.229930
$237$	−49.6922	−3.22786
$238$	35.4420	2.29736
$239$	27.0358	1.74880	0.874400	−	0.485206i	$-0.161255\pi$
$239$	27.0358	1.74880	0.874400	+	0.485206i	$0.161255\pi$
$240$	−4.36731	−0.281909
$241$	1.00000	0.0644157
$241$	1.00000	0.0644157
$242$	−15.7389	−1.01173
$243$	−15.1401	−0.971238
$244$	1.50990	0.0966615
$245$	3.69479	0.236052
$246$	−2.66582	−0.169966
$247$	−20.9309	−1.33180
$248$	−26.7353	−1.69770
$249$	20.9615	1.32838
$250$	−5.12966	−0.324428
$251$	−16.5147	−1.04240	−0.521199	−	0.853435i	$-0.674515\pi$
$251$	−16.5147	−1.04240	−0.521199	+	0.853435i	$0.674515\pi$
$252$	−8.29858	−0.522761
$253$	7.41977	0.466477
$254$	10.7911	0.677097
$255$	−5.07405	−0.317750
$256$	9.21519	0.575949
$257$	−9.88138	−0.616383	−0.308192	−	0.951324i	$-0.599724\pi$
$257$	−9.88138	−0.616383	−0.308192	+	0.951324i	$0.599724\pi$
$258$	−8.06316	−0.501990
$259$	−21.9491	−1.36385
$260$	0.643000	0.0398771
$261$	−31.6222	−1.95736
$262$	10.8742	0.671811
$263$	−31.5776	−1.94716	−0.973579	−	0.228351i	$-0.926666\pi$
$263$	−31.5776	−1.94716	−0.973579	+	0.228351i	$0.926666\pi$
$264$	−6.38129	−0.392741
$265$	0.00283442	0.000174117 0
$266$	28.5948	1.75326
$267$	−1.46366	−0.0895744
$268$	1.86496	0.113921
$269$	4.21527	0.257010	0.128505	−	0.991709i	$-0.458982\pi$
$269$	4.21527	0.257010	0.128505	+	0.991709i	$0.458982\pi$
$270$	2.77209	0.168704
$271$	11.8060	0.717162	0.358581	−	0.933499i	$-0.383261\pi$
$271$	11.8060	0.717162	0.358581	+	0.933499i	$0.383261\pi$
$272$	24.9976	1.51570
$273$	−57.4571	−3.47746
$274$	8.95360	0.540907
$275$	−4.47438	−0.269815
$276$	9.08508	0.546857
$277$	8.01314	0.481463	0.240731	−	0.970592i	$-0.422613\pi$
$277$	8.01314	0.481463	0.240731	+	0.970592i	$0.422613\pi$
$278$	3.63872	0.218236
$279$	52.8348	3.16313
$280$	3.52734	0.210799
$281$	−21.6719	−1.29284	−0.646419	−	0.762982i	$-0.723734\pi$
$281$	−21.6719	−1.29284	−0.646419	+	0.762982i	$0.723734\pi$
$282$	−9.62618	−0.573231
$283$	2.15646	0.128188	0.0640941	−	0.997944i	$-0.479584\pi$
$283$	2.15646	0.128188	0.0640941	+	0.997944i	$0.479584\pi$
$284$	0.592938	0.0351844
$285$	−4.09377	−0.242494
$286$	6.82515	0.403580
$287$	2.60016	0.153482
$288$	−10.9005	−0.642316
$289$	12.0429	0.708406
$290$	−3.34732	−0.196561
$291$	−17.8067	−1.04385
$292$	−4.13240	−0.241831
$293$	26.9520	1.57455	0.787276	−	0.616601i	$-0.211491\pi$
$293$	26.9520	1.57455	0.787276	+	0.616601i	$0.211491\pi$
$294$	48.0208	2.80063
$295$	2.96708	0.172750
$296$	−12.8192	−0.745103
$297$	4.89145	0.283830
$298$	6.87863	0.398469
$299$	39.0185	2.25650
$300$	−5.47862	−0.316308
$301$	7.86457	0.453306
$302$	−8.26481	−0.475586
$303$	20.2566	1.16371
$304$	20.1682	1.15673
$305$	1.26831	0.0726234
$306$	−40.9069	−2.33849
$307$	16.6282	0.949020	0.474510	−	0.880250i	$-0.342625\pi$
$307$	16.6282	0.949020	0.474510	+	0.880250i	$0.342625\pi$
$308$	−1.55003	−0.0883211
$309$	22.0017	1.25163
$310$	5.59275	0.317647
$311$	−10.1432	−0.575167	−0.287584	−	0.957756i	$-0.592852\pi$
$311$	−10.1432	−0.575167	−0.287584	+	0.957756i	$0.592852\pi$
$312$	−33.5574	−1.89981
$313$	−26.7382	−1.51133	−0.755667	−	0.654956i	$-0.772687\pi$
$313$	−26.7382	−1.51133	−0.755667	+	0.654956i	$0.772687\pi$
$314$	−33.4737	−1.88903
$315$	−6.97078	−0.392759
$316$	−7.04961	−0.396571
$317$	−25.6438	−1.44030	−0.720151	−	0.693817i	$-0.755928\pi$
$317$	−25.6438	−1.44030	−0.720151	+	0.693817i	$0.755928\pi$
$318$	0.0368387	0.00206581
$319$	−5.90647	−0.330699
$320$	1.95359	0.109209
$321$	9.22460	0.514867
$322$	−53.3051	−2.97058
$323$	23.4320	1.30379
$324$	0.126275	0.00701529
$325$	−23.5295	−1.30518
$326$	−34.2282	−1.89573
$327$	19.3817	1.07181
$328$	1.51860	0.0838509
$329$	9.38910	0.517638
$330$	1.33490	0.0734837
$331$	21.9178	1.20471	0.602356	−	0.798228i	$-0.294229\pi$
$331$	21.9178	1.20471	0.602356	+	0.798228i	$0.294229\pi$
$332$	2.97371	0.163204
$333$	25.3336	1.38827
$334$	−4.00052	−0.218898
$335$	1.56656	0.0855903
$336$	55.3635	3.02033
$337$	5.64948	0.307747	0.153873	−	0.988091i	$-0.450825\pi$
$337$	5.64948	0.307747	0.153873	+	0.988091i	$0.450825\pi$
$338$	15.7572	0.857079
$339$	−40.6838	−2.20964
$340$	−0.719833	−0.0390384
$341$	9.86861	0.534415
$342$	−33.0039	−1.78465
$343$	−17.1144	−0.924093
$344$	4.59325	0.247651
$345$	7.63144	0.410863
$346$	14.7727	0.794184
$347$	19.7057	1.05786	0.528928	−	0.848667i	$-0.322594\pi$
$347$	19.7057	1.05786	0.528928	+	0.848667i	$0.322594\pi$
$348$	−7.23213	−0.387683
$349$	−31.5930	−1.69113	−0.845567	−	0.533869i	$-0.820738\pi$
$349$	−31.5930	−1.69113	−0.845567	+	0.533869i	$0.820738\pi$
$350$	32.1448	1.71821
$351$	25.7227	1.37298
$352$	−2.03602	−0.108520
$353$	12.7948	0.680999	0.340499	−	0.940245i	$-0.389404\pi$
$353$	12.7948	0.680999	0.340499	+	0.940245i	$0.389404\pi$
$354$	38.5629	2.04959
$355$	0.498066	0.0264346
$356$	−0.207642	−0.0110050
$357$	64.3227	3.40432
$358$	18.0683	0.954942
$359$	27.2866	1.44013	0.720066	−	0.693906i	$-0.244112\pi$
$359$	27.2866	1.44013	0.720066	+	0.693906i	$0.244112\pi$
$360$	−4.07124	−0.214573
$361$	−0.0949647	−0.00499814
$362$	−10.5025	−0.551998
$363$	−28.5641	−1.49923
$364$	−8.15117	−0.427237
$365$	−3.47121	−0.181691
$366$	16.4841	0.861639
$367$	25.9541	1.35479	0.677396	−	0.735619i	$-0.263109\pi$
$367$	25.9541	1.35479	0.677396	+	0.735619i	$0.263109\pi$
$368$	−37.5967	−1.95987
$369$	−3.00109	−0.156230
$370$	2.68165	0.139412
$371$	−0.0359313	−0.00186546
$372$	12.0835	0.626502
$373$	1.94963	0.100948	0.0504740	−	0.998725i	$-0.483927\pi$
$373$	1.94963	0.100948	0.0504740	+	0.998725i	$0.483927\pi$
$374$	−7.64070	−0.395091
$375$	−9.30968	−0.480750
$376$	5.48364	0.282797
$377$	−31.0604	−1.59969
$378$	−35.1411	−1.80747
$379$	22.0885	1.13461	0.567305	−	0.823508i	$-0.307986\pi$
$379$	22.0885	1.13461	0.567305	+	0.823508i	$0.307986\pi$
$380$	−0.580765	−0.0297926
$381$	19.5846	1.00335
$382$	14.4049	0.737019
$383$	−32.6589	−1.66879	−0.834396	−	0.551166i	$-0.814183\pi$
$383$	−32.6589	−1.66879	−0.834396	+	0.551166i	$0.814183\pi$
$384$	37.8942	1.93378
$385$	−1.30202	−0.0663571
$386$	−1.96971	−0.100256
$387$	−9.07724	−0.461422
$388$	−2.52615	−0.128246
$389$	−1.59569	−0.0809045	−0.0404522	−	0.999181i	$-0.512880\pi$
$389$	−1.59569	−0.0809045	−0.0404522	+	0.999181i	$0.512880\pi$
$390$	7.01985	0.355464
$391$	−43.6809	−2.20904
$392$	−27.3555	−1.38166
$393$	19.7353	0.995515
$394$	14.7325	0.742212
$395$	−5.92165	−0.297951
$396$	1.78904	0.0899024
$397$	−20.5064	−1.02919	−0.514593	−	0.857434i	$-0.672057\pi$
$397$	−20.5064	−1.02919	−0.514593	+	0.857434i	$0.672057\pi$
$398$	−28.6181	−1.43449
$399$	51.8959	2.59805
$400$	22.6721	1.13361
$401$	2.51432	0.125559	0.0627795	−	0.998027i	$-0.480004\pi$
$401$	2.51432	0.125559	0.0627795	+	0.998027i	$0.480004\pi$
$402$	20.3604	1.01549
$403$	51.8962	2.58514
$404$	2.87371	0.142972
$405$	0.106071	0.00527070
$406$	42.4333	2.10593
$407$	4.73187	0.234550
$408$	37.5672	1.85986
$409$	−3.99904	−0.197740	−0.0988698	−	0.995100i	$-0.531523\pi$
$409$	−3.99904	−0.197740	−0.0988698	+	0.995100i	$0.531523\pi$
$410$	−0.317676	−0.0156889
$411$	16.2497	0.801537
$412$	3.12128	0.153775
$413$	−37.6131	−1.85082
$414$	61.5245	3.02376
$415$	2.49791	0.122618
$416$	−10.7068	−0.524946
$417$	6.60382	0.323390
$418$	−6.16456	−0.301518
$419$	19.3567	0.945635	0.472818	−	0.881160i	$-0.343237\pi$
$419$	19.3567	0.945635	0.472818	+	0.881160i	$0.343237\pi$
$420$	−1.59425	−0.0777914
$421$	−4.81504	−0.234671	−0.117335	−	0.993092i	$-0.537435\pi$
$421$	−4.81504	−0.234671	−0.117335	+	0.993092i	$0.537435\pi$
$422$	8.58743	0.418029
$423$	−10.8368	−0.526905
$424$	−0.0209855	−0.00101914
$425$	26.3411	1.27773
$426$	6.47331	0.313633
$427$	−16.0781	−0.778075
$428$	1.30865	0.0632561
$429$	12.3868	0.598040
$430$	−0.960859	−0.0463367
$431$	26.8254	1.29213	0.646067	−	0.763281i	$-0.276413\pi$
$431$	26.8254	1.29213	0.646067	+	0.763281i	$0.276413\pi$
$432$	−24.7855	−1.19249
$433$	1.59224	0.0765181	0.0382591	−	0.999268i	$-0.487819\pi$
$433$	1.59224	0.0765181	0.0382591	+	0.999268i	$0.487819\pi$
$434$	−70.8981	−3.40322
$435$	−6.07496	−0.291272
$436$	2.74960	0.131682
$437$	−35.2419	−1.68585
$438$	−45.1149	−2.15567
$439$	−36.5165	−1.74284	−0.871418	−	0.490541i	$-0.836799\pi$
$439$	−36.5165	−1.74284	−0.871418	+	0.490541i	$0.836799\pi$
$440$	−0.760436	−0.0362524
$441$	54.0603	2.57430
$442$	−40.1803	−1.91118
$443$	−1.27451	−0.0605540	−0.0302770	−	0.999542i	$-0.509639\pi$
$443$	−1.27451	−0.0605540	−0.0302770	+	0.999542i	$0.509639\pi$
$444$	5.79390	0.274966
$445$	−0.174419	−0.00826825
$446$	−19.9770	−0.945940
$447$	12.4839	0.590466
$448$	−24.7653	−1.17005
$449$	31.9473	1.50768	0.753842	−	0.657055i	$-0.228198\pi$
$449$	31.9473	1.50768	0.753842	+	0.657055i	$0.228198\pi$
$450$	−37.1014	−1.74898
$451$	−0.560551	−0.0263953
$452$	−5.77162	−0.271474
$453$	−14.9996	−0.704742
$454$	4.42133	0.207504
$455$	−6.84696	−0.320990
$456$	30.3095	1.41937
$457$	12.8561	0.601384	0.300692	−	0.953721i	$-0.402782\pi$
$457$	12.8561	0.601384	0.300692	+	0.953721i	$0.402782\pi$
$458$	−25.0447	−1.17026
$459$	−28.7964	−1.34410
$460$	1.08264	0.0504782
$461$	35.6691	1.66127	0.830637	−	0.556814i	$-0.187976\pi$
$461$	35.6691	1.66127	0.830637	+	0.556814i	$0.187976\pi$
$462$	−16.9222	−0.787293
$463$	2.74466	0.127555	0.0637776	−	0.997964i	$-0.479685\pi$
$463$	2.74466	0.127555	0.0637776	+	0.997964i	$0.479685\pi$
$464$	29.9287	1.38940
$465$	10.1501	0.470701
$466$	17.6240	0.816418
$467$	−25.5934	−1.18432	−0.592160	−	0.805821i	$-0.701725\pi$
$467$	−25.5934	−1.18432	−0.592160	+	0.805821i	$0.701725\pi$
$468$	9.40804	0.434887
$469$	−19.8590	−0.917001
$470$	−1.14712	−0.0529126
$471$	−60.7505	−2.79924
$472$	−21.9677	−1.01114
$473$	−1.69547	−0.0779578
$474$	−76.9631	−3.53503
$475$	21.2521	0.975114
$476$	9.12517	0.418251
$477$	0.0414718	0.00189886
$478$	41.8729	1.91522
$479$	−0.610882	−0.0279119	−0.0139560	−	0.999903i	$-0.504442\pi$
$479$	−0.610882	−0.0279119	−0.0139560	+	0.999903i	$0.504442\pi$
$480$	−2.09410	−0.0955821
$481$	24.8836	1.13459
$482$	1.54879	0.0705457
$483$	−96.7421	−4.40192
$484$	−4.05225	−0.184193
$485$	−2.12196	−0.0963533
$486$	−23.4489	−1.06366
$487$	26.6270	1.20658	0.603291	−	0.797521i	$-0.293856\pi$
$487$	26.6270	1.20658	0.603291	+	0.797521i	$0.293856\pi$
$488$	−9.39032	−0.425080
$489$	−62.1200	−2.80916
$490$	5.72247	0.258515
$491$	−31.7856	−1.43447	−0.717233	−	0.696833i	$-0.754592\pi$
$491$	−31.7856	−1.43447	−0.717233	+	0.696833i	$0.754592\pi$
$492$	−0.686362	−0.0309436
$493$	34.7719	1.56605
$494$	−32.4177	−1.45854
$495$	1.50278	0.0675452
$496$	−50.0053	−2.24530
$497$	−6.31387	−0.283216
$498$	32.4651	1.45480
$499$	−17.2468	−0.772074	−0.386037	−	0.922483i	$-0.626156\pi$
$499$	−17.2468	−0.772074	−0.386037	+	0.922483i	$0.626156\pi$
$500$	−1.32072	−0.0590645
$501$	−7.26043	−0.324372
$502$	−25.5779	−1.14160
$503$	−23.6924	−1.05639	−0.528196	−	0.849122i	$-0.677132\pi$
$503$	−23.6924	−1.05639	−0.528196	+	0.849122i	$0.677132\pi$
$504$	51.6102	2.29890
$505$	2.41391	0.107417
$506$	11.4917	0.510868
$507$	28.5974	1.27005
$508$	2.77837	0.123270
$509$	20.3227	0.900787	0.450394	−	0.892830i	$-0.351284\pi$
$509$	20.3227	0.900787	0.450394	+	0.892830i	$0.351284\pi$
$510$	−7.85867	−0.347988
$511$	44.0037	1.94661
$512$	−12.6902	−0.560832
$513$	−23.2331	−1.02577
$514$	−15.3042	−0.675040
$515$	2.62187	0.115533
$516$	−2.07600	−0.0913910
$517$	−2.02413	−0.0890213
$518$	−33.9947	−1.49364
$519$	26.8106	1.17685
$520$	−3.99892	−0.175364
$521$	16.3912	0.718112	0.359056	−	0.933316i	$-0.383099\pi$
$521$	16.3912	0.718112	0.359056	+	0.933316i	$0.383099\pi$
$522$	−48.9762	−2.14363
$523$	0.258754	0.0113145	0.00565727	−	0.999984i	$-0.498199\pi$
$523$	0.258754	0.0113145	0.00565727	+	0.999984i	$0.498199\pi$
$524$	2.79976	0.122308
$525$	58.3389	2.54612
$526$	−48.9072	−2.13246
$527$	−58.0974	−2.53076
$528$	−11.9354	−0.519424
$529$	42.6965	1.85637
$530$	0.00438994	0.000190687 0
$531$	43.4128	1.88396
$532$	7.36223	0.319193
$533$	−2.94778	−0.127682
$534$	−2.26690	−0.0980985
$535$	1.09926	0.0475253
$536$	−11.5985	−0.500978
$537$	32.7918	1.41507
$538$	6.52859	0.281468
$539$	10.0975	0.434931
$540$	0.713723	0.0307137
$541$	−17.9084	−0.769941	−0.384970	−	0.922929i	$-0.625788\pi$
$541$	−17.9084	−0.769941	−0.384970	+	0.922929i	$0.625788\pi$
$542$	18.2850	0.785410
$543$	−19.0607	−0.817971
$544$	11.9862	0.513905
$545$	2.30965	0.0989347
$546$	−88.9892	−3.80839
$547$	12.2295	0.522897	0.261448	−	0.965217i	$-0.415800\pi$
$547$	12.2295	0.522897	0.261448	+	0.965217i	$0.415800\pi$
$548$	2.30526	0.0984760
$549$	18.5573	0.792006
$550$	−6.92989	−0.295492
$551$	28.0542	1.19515
$552$	−56.5016	−2.40487
$553$	75.0675	3.19219
$554$	12.4107	0.527280
$555$	4.86686	0.206587
$556$	0.936853	0.0397314
$557$	25.2541	1.07005	0.535024	−	0.844837i	$-0.320302\pi$
$557$	25.2541	1.07005	0.535024	+	0.844837i	$0.320302\pi$
$558$	81.8302	3.46415
$559$	−8.91600	−0.377107
$560$	6.59747	0.278794
$561$	−13.8669	−0.585461
$562$	−33.5654	−1.41587
$563$	−10.4115	−0.438794	−0.219397	−	0.975636i	$-0.570409\pi$
$563$	−10.4115	−0.438794	−0.219397	+	0.975636i	$0.570409\pi$
$564$	−2.47843	−0.104361
$565$	−4.84814	−0.203963
$566$	3.33991	0.140387
$567$	−1.34464	−0.0564695
$568$	−3.68758	−0.154727
$569$	−28.5127	−1.19532	−0.597658	−	0.801751i	$-0.703902\pi$
$569$	−28.5127	−1.19532	−0.597658	+	0.801751i	$0.703902\pi$
$570$	−6.34042	−0.265571
$571$	35.8278	1.49935	0.749674	−	0.661807i	$-0.230210\pi$
$571$	35.8278	1.49935	0.749674	+	0.661807i	$0.230210\pi$
$572$	1.75726	0.0734746
$573$	26.1431	1.09214
$574$	4.02711	0.168088
$575$	−39.6173	−1.65215
$576$	28.5840	1.19100
$577$	1.00615	0.0418864	0.0209432	−	0.999781i	$-0.493333\pi$
$577$	1.00615	0.0418864	0.0209432	+	0.999781i	$0.493333\pi$
$578$	18.6520	0.775820
$579$	−3.57478	−0.148563
$580$	−0.861827	−0.0357854
$581$	−31.6655	−1.31371
$582$	−27.5789	−1.14318
$583$	0.00774620	0.000320815 0
$584$	25.7001	1.06348
$585$	7.90272	0.326737
$586$	41.7431	1.72439
$587$	−7.69459	−0.317590	−0.158795	−	0.987312i	$-0.550761\pi$
$587$	−7.69459	−0.317590	−0.158795	+	0.987312i	$0.550761\pi$
$588$	12.3638	0.509875
$589$	−46.8733	−1.93138
$590$	4.59540	0.189190
$591$	26.7376	1.09984
$592$	−23.9769	−0.985443
$593$	−27.9165	−1.14639	−0.573197	−	0.819418i	$-0.694297\pi$
$593$	−27.9165	−1.14639	−0.573197	+	0.819418i	$0.694297\pi$
$594$	7.57585	0.310841
$595$	7.66511	0.314239
$596$	1.77103	0.0725441
$597$	−51.9382	−2.12569
$598$	60.4316	2.47123
$599$	43.2997	1.76918	0.884588	−	0.466372i	$-0.154439\pi$
$599$	43.2997	1.76918	0.884588	+	0.466372i	$0.154439\pi$
$600$	34.0724	1.39100
$601$	−25.4104	−1.03651	−0.518255	−	0.855226i	$-0.673418\pi$
$601$	−25.4104	−1.03651	−0.518255	+	0.855226i	$0.673418\pi$
$602$	12.1806	0.496444
$603$	22.9211	0.933419
$604$	−2.12792	−0.0865840
$605$	−3.40388	−0.138387
$606$	31.3733	1.27445
$607$	−2.58042	−0.104736	−0.0523679	−	0.998628i	$-0.516677\pi$
$607$	−2.58042	−0.104736	−0.0523679	+	0.998628i	$0.516677\pi$
$608$	9.67055	0.392192
$609$	77.0110	3.12064
$610$	1.96436	0.0795344
$611$	−10.6443	−0.430624
$612$	−10.5322	−0.425740
$613$	24.6593	0.995980	0.497990	−	0.867183i	$-0.334072\pi$
$613$	24.6593	0.995980	0.497990	+	0.867183i	$0.334072\pi$
$614$	25.7536	1.03933
$615$	−0.576542	−0.0232484
$616$	9.63989	0.388402
$617$	−17.2509	−0.694497	−0.347248	−	0.937773i	$-0.612884\pi$
$617$	−17.2509	−0.694497	−0.347248	+	0.937773i	$0.612884\pi$
$618$	34.0762	1.37074
$619$	−0.0379403	−0.00152495	−0.000762475	−	1.00000i	$-0.500243\pi$
$619$	−0.0379403	−0.00152495	−0.000762475		1.00000i	$0.500243\pi$
$620$	1.43995	0.0578299
$621$	43.3101	1.73797
$622$	−15.7097	−0.629902
$623$	2.21107	0.0885847
$624$	−62.7652	−2.51262
$625$	23.3296	0.933184
$626$	−41.4120	−1.65516
$627$	−11.1879	−0.446802
$628$	−8.61840	−0.343911
$629$	−27.8569	−1.11073
$630$	−10.7963	−0.430135
$631$	39.8274	1.58551	0.792753	−	0.609544i	$-0.208647\pi$
$631$	39.8274	1.58551	0.792753	+	0.609544i	$0.208647\pi$
$632$	43.8427	1.74397
$633$	15.5851	0.619452
$634$	−39.7171	−1.57737
$635$	2.33383	0.0926150
$636$	0.00948477	0.000376096 0
$637$	53.1000	2.10390
$638$	−9.14790	−0.362169
$639$	7.28744	0.288287
$640$	4.51572	0.178499
$641$	17.4965	0.691071	0.345536	−	0.938406i	$-0.387697\pi$
$641$	17.4965	0.691071	0.345536	+	0.938406i	$0.387697\pi$
$642$	14.2870	0.563863
$643$	19.8884	0.784323	0.392162	−	0.919896i	$-0.371727\pi$
$643$	19.8884	0.784323	0.392162	+	0.919896i	$0.371727\pi$
$644$	−13.7244	−0.540815
$645$	−1.74384	−0.0686636
$646$	36.2913	1.42786
$647$	−39.9620	−1.57107	−0.785535	−	0.618817i	$-0.787612\pi$
$647$	−39.9620	−1.57107	−0.785535	+	0.618817i	$0.787612\pi$
$648$	−0.785327	−0.0308506
$649$	8.10876	0.318297
$650$	−36.4424	−1.42939
$651$	−128.671	−5.04302
$652$	−8.81267	−0.345131
$653$	−45.0393	−1.76252	−0.881262	−	0.472629i	$-0.843305\pi$
$653$	−45.0393	−1.76252	−0.881262	+	0.472629i	$0.843305\pi$
$654$	30.0183	1.17381
$655$	2.35179	0.0918920
$656$	2.84037	0.110898
$657$	−50.7889	−1.98146
$658$	14.5418	0.566898
$659$	−28.6163	−1.11473	−0.557367	−	0.830266i	$-0.688188\pi$
$659$	−28.6163	−1.11473	−0.557367	+	0.830266i	$0.688188\pi$
$660$	0.343693	0.0133783
$661$	−21.0849	−0.820106	−0.410053	−	0.912062i	$-0.634490\pi$
$661$	−21.0849	−0.820106	−0.410053	+	0.912062i	$0.634490\pi$
$662$	33.9462	1.31936
$663$	−72.9222	−2.83206
$664$	−18.4940	−0.717707
$665$	6.18425	0.239815
$666$	39.2365	1.52038
$667$	−52.2974	−2.02496
$668$	−1.03000	−0.0398521
$669$	−36.2558	−1.40173
$670$	2.42628	0.0937354
$671$	3.46618	0.133810
$672$	26.5465	1.02405
$673$	11.9180	0.459406	0.229703	−	0.973261i	$-0.426224\pi$
$673$	11.9180	0.459406	0.229703	+	0.973261i	$0.426224\pi$
$674$	8.74988	0.337033
$675$	−26.1175	−1.00526
$676$	4.05698	0.156038
$677$	46.0077	1.76822	0.884111	−	0.467278i	$-0.154765\pi$
$677$	46.0077	1.76822	0.884111	+	0.467278i	$0.154765\pi$
$678$	−63.0108	−2.41992
$679$	26.8997	1.03231
$680$	4.47676	0.171676
$681$	8.02417	0.307487
$682$	15.2844	0.585272
$683$	41.5189	1.58868	0.794338	−	0.607477i	$-0.207818\pi$
$683$	41.5189	1.58868	0.794338	+	0.607477i	$0.207818\pi$
$684$	−8.49745	−0.324908
$685$	1.93642	0.0739866
$686$	−26.5068	−1.01203
$687$	−45.4531	−1.73414
$688$	8.59113	0.327534
$689$	0.0407351	0.00155188
$690$	11.8195	0.449962
$691$	28.5903	1.08763	0.543814	−	0.839206i	$-0.316980\pi$
$691$	28.5903	1.08763	0.543814	+	0.839206i	$0.316980\pi$
$692$	3.80349	0.144587
$693$	−19.0505	−0.723668
$694$	30.5200	1.15852
$695$	0.786954	0.0298509
$696$	44.9778	1.70488
$697$	3.30001	0.124997
$698$	−48.9311	−1.85207
$699$	31.9854	1.20980
$700$	8.27626	0.312813
$701$	19.2033	0.725298	0.362649	−	0.931926i	$-0.381872\pi$
$701$	19.2033	0.725298	0.362649	+	0.931926i	$0.381872\pi$
$702$	39.8392	1.50364
$703$	−22.4751	−0.847665
$704$	5.33899	0.201221
$705$	−2.08188	−0.0784080
$706$	19.8165	0.745805
$707$	−30.6006	−1.15085
$708$	9.92870	0.373144
$709$	4.05386	0.152246	0.0761229	−	0.997098i	$-0.475746\pi$
$709$	4.05386	0.152246	0.0761229	+	0.997098i	$0.475746\pi$
$710$	0.771402	0.0289502
$711$	−86.6425	−3.24935
$712$	1.29136	0.0483958
$713$	87.3792	3.27238
$714$	99.6227	3.72828
$715$	1.47609	0.0552027
$716$	4.65202	0.173854
$717$	75.9940	2.83805
$718$	42.2613	1.57718
$719$	13.0438	0.486452	0.243226	−	0.969970i	$-0.421794\pi$
$719$	13.0438	0.486452	0.243226	+	0.969970i	$0.421794\pi$
$720$	−7.61477	−0.283786
$721$	−33.2369	−1.23781
$722$	−0.147081	−0.00547378
$723$	2.81087	0.104537
$724$	−2.70405	−0.100495
$725$	31.5371	1.17126
$726$	−44.2399	−1.64190
$727$	−43.3367	−1.60727	−0.803635	−	0.595123i	$-0.797103\pi$
$727$	−43.3367	−1.60727	−0.803635	+	0.595123i	$0.797103\pi$
$728$	50.6935	1.87882
$729$	−43.5068	−1.61136
$730$	−5.37619	−0.198982
$731$	9.98139	0.369175
$732$	4.24413	0.156868
$733$	−6.99674	−0.258431	−0.129215	−	0.991617i	$-0.541246\pi$
$733$	−6.99674	−0.258431	−0.129215	+	0.991617i	$0.541246\pi$
$734$	40.1975	1.48372
$735$	10.3856	0.383078
$736$	−18.0274	−0.664499
$737$	4.28126	0.157702
$738$	−4.64807	−0.171098
$739$	−1.44307	−0.0530842	−0.0265421	−	0.999648i	$-0.508450\pi$
$739$	−1.44307	−0.0530842	−0.0265421	+	0.999648i	$0.508450\pi$
$740$	0.690438	0.0253810
$741$	−58.8340	−2.16132
$742$	−0.0556503	−0.00204299
$743$	−4.24266	−0.155648	−0.0778240	−	0.996967i	$-0.524797\pi$
$743$	−4.24266	−0.155648	−0.0778240	+	0.996967i	$0.524797\pi$
$744$	−75.1495	−2.75511
$745$	1.48766	0.0545036
$746$	3.01958	0.110555
$747$	36.5481	1.33723
$748$	−1.96723	−0.0719292
$749$	−13.9351	−0.509179
$750$	−14.4188	−0.526500
$751$	19.0193	0.694024	0.347012	−	0.937861i	$-0.387196\pi$
$751$	19.0193	0.694024	0.347012	+	0.937861i	$0.387196\pi$
$752$	10.2565	0.374016
$753$	−46.4206	−1.69166
$754$	−48.1063	−1.75193
$755$	−1.78745	−0.0650519
$756$	−9.04771	−0.329062
$757$	0.151878	0.00552010	0.00276005	−	0.999996i	$-0.499121\pi$
$757$	0.151878	0.00552010	0.00276005	+	0.999996i	$0.499121\pi$
$758$	34.2105	1.24258
$759$	20.8560	0.757025
$760$	3.61187	0.131016
$761$	−16.2184	−0.587916	−0.293958	−	0.955818i	$-0.594973\pi$
$761$	−16.2184	−0.587916	−0.293958	+	0.955818i	$0.594973\pi$
$762$	30.3325	1.09883
$763$	−29.2790	−1.05997
$764$	3.70880	0.134180
$765$	−8.84703	−0.319865
$766$	−50.5819	−1.82760
$767$	42.6417	1.53970
$768$	25.9027	0.934682
$769$	5.32816	0.192138	0.0960691	−	0.995375i	$-0.469373\pi$
$769$	5.32816	0.192138	0.0960691	+	0.995375i	$0.469373\pi$
$770$	−2.01656	−0.0726719
$771$	−27.7752	−1.00030
$772$	−0.507137	−0.0182523
$773$	−6.19612	−0.222859	−0.111430	−	0.993772i	$-0.535543\pi$
$773$	−6.19612	−0.222859	−0.111430	+	0.993772i	$0.535543\pi$
$774$	−14.0588	−0.505333
$775$	−52.6927	−1.89278
$776$	15.7106	0.563976
$777$	−61.6961	−2.21334
$778$	−2.47139	−0.0886036
$779$	2.66247	0.0953929
$780$	1.80739	0.0647148
$781$	1.36117	0.0487064
$782$	−67.6527	−2.41925
$783$	−34.4768	−1.23210
$784$	−51.1652	−1.82733
$785$	−7.23943	−0.258386
$786$	30.5660	1.09025
$787$	26.4924	0.944351	0.472176	−	0.881504i	$-0.343469\pi$
$787$	26.4924	0.944351	0.472176	+	0.881504i	$0.343469\pi$
$788$	3.79314	0.135125
$789$	−88.7604	−3.15996
$790$	−9.17142	−0.326304
$791$	61.4589	2.18523
$792$	−11.1263	−0.395356
$793$	18.2277	0.647283
$794$	−31.7602	−1.12713
$795$	0.00796718	0.000282567 0
$796$	−7.36823	−0.261160
$797$	41.8090	1.48095	0.740476	−	0.672083i	$-0.234600\pi$
$797$	41.8090	1.48095	0.740476	+	0.672083i	$0.234600\pi$
$798$	80.3761	2.84528
$799$	11.9163	0.421567
$800$	10.8712	0.384353
$801$	−2.55201	−0.0901707
$802$	3.89416	0.137508
$803$	−9.48648	−0.334770
$804$	5.24215	0.184877
$805$	−11.5284	−0.406323
$806$	80.3766	2.83115
$807$	11.8486	0.417090
$808$	−17.8721	−0.628737
$809$	1.66235	0.0584451	0.0292226	−	0.999573i	$-0.490697\pi$
$809$	1.66235	0.0584451	0.0292226	+	0.999573i	$0.490697\pi$
$810$	0.164282	0.00577228
$811$	−13.4156	−0.471085	−0.235542	−	0.971864i	$-0.575687\pi$
$811$	−13.4156	−0.471085	−0.235542	+	0.971864i	$0.575687\pi$
$812$	10.9252	0.383399
$813$	33.1851	1.16385
$814$	7.32869	0.256870
$815$	−7.40262	−0.259302
$816$	70.2651	2.45977
$817$	8.05304	0.281740
$818$	−6.19369	−0.216557
$819$	−100.181	−3.50061
$820$	−0.0817914	−0.00285628
$821$	23.1210	0.806930	0.403465	−	0.914995i	$-0.367806\pi$
$821$	23.1210	0.806930	0.403465	+	0.914995i	$0.367806\pi$
$822$	25.1674	0.877814
$823$	20.7084	0.721849	0.360924	−	0.932595i	$-0.382461\pi$
$823$	20.7084	0.721849	0.360924	+	0.932595i	$0.382461\pi$
$824$	−19.4118	−0.676241
$825$	−12.5769	−0.437871
$826$	−58.2550	−2.02695
$827$	−1.22288	−0.0425235	−0.0212618	−	0.999774i	$-0.506768\pi$
$827$	−1.22288	−0.0425235	−0.0212618	+	0.999774i	$0.506768\pi$
$828$	15.8406	0.550498
$829$	26.5008	0.920411	0.460205	−	0.887813i	$-0.347776\pi$
$829$	26.5008	0.920411	0.460205	+	0.887813i	$0.347776\pi$
$830$	3.86875	0.134286
$831$	22.5239	0.781344
$832$	28.0762	0.973368
$833$	−59.4450	−2.05965
$834$	10.2280	0.354165
$835$	−0.865201	−0.0299415
$836$	−1.58718	−0.0548936
$837$	57.6043	1.99110
$838$	29.9795	1.03562
$839$	27.5572	0.951380	0.475690	−	0.879613i	$-0.342198\pi$
$839$	27.5572	0.951380	0.475690	+	0.879613i	$0.342198\pi$
$840$	9.91489	0.342096
$841$	12.6310	0.435553
$842$	−7.45750	−0.257003
$843$	−60.9169	−2.09809
$844$	2.21099	0.0761053
$845$	3.40785	0.117234
$846$	−16.7840	−0.577047
$847$	43.1503	1.48266
$848$	−0.0392508	−0.00134788
$849$	6.06152	0.208031
$850$	40.7969	1.39932
$851$	41.8972	1.43622
$852$	1.66667	0.0570992
$853$	14.3366	0.490875	0.245437	−	0.969412i	$-0.421068\pi$
$853$	14.3366	0.490875	0.245437	+	0.969412i	$0.421068\pi$
$854$	−24.9017	−0.852119
$855$	−7.13783	−0.244109
$856$	−8.13872	−0.278176
$857$	33.5953	1.14759	0.573796	−	0.818998i	$-0.305470\pi$
$857$	33.5953	1.14759	0.573796	+	0.818998i	$0.305470\pi$
$858$	19.1846	0.654951
$859$	−31.1946	−1.06435	−0.532173	−	0.846635i	$-0.678625\pi$
$859$	−31.1946	−1.06435	−0.532173	+	0.846635i	$0.678625\pi$
$860$	−0.247390	−0.00843594
$861$	7.30870	0.249080
$862$	41.5470	1.41510
$863$	23.3830	0.795965	0.397983	−	0.917393i	$-0.369710\pi$
$863$	23.3830	0.795965	0.397983	+	0.917393i	$0.369710\pi$
$864$	−11.8845	−0.404318
$865$	3.19492	0.108631
$866$	2.46605	0.0837999
$867$	33.8510	1.14964
$868$	−18.2540	−0.619580
$869$	−16.1833	−0.548981
$870$	−9.40887	−0.318991
$871$	22.5139	0.762856
$872$	−17.1002	−0.579086
$873$	−31.0474	−1.05080
$874$	−54.5825	−1.84628
$875$	14.0637	0.475438
$876$	−11.6156	−0.392456
$877$	−35.6526	−1.20390	−0.601951	−	0.798533i	$-0.705610\pi$
$877$	−35.6526	−1.20390	−0.601951	+	0.798533i	$0.705610\pi$
$878$	−56.5565	−1.90869
$879$	75.7585	2.55527
$880$	−1.42231	−0.0479459
$881$	29.7222	1.00137	0.500683	−	0.865631i	$-0.333082\pi$
$881$	29.7222	1.00137	0.500683	+	0.865631i	$0.333082\pi$
$882$	83.7283	2.81928
$883$	−41.7428	−1.40476	−0.702379	−	0.711803i	$-0.747879\pi$
$883$	−41.7428	−1.40476	−0.702379	+	0.711803i	$0.747879\pi$
$884$	−10.3451	−0.347944
$885$	8.34008	0.280349
$886$	−1.97396	−0.0663165
$887$	−2.72462	−0.0914838	−0.0457419	−	0.998953i	$-0.514565\pi$
$887$	−2.72462	−0.0914838	−0.0457419	+	0.998953i	$0.514565\pi$
$888$	−36.0332	−1.20919
$889$	−29.5854	−0.992263
$890$	−0.270139	−0.00905508
$891$	0.289882	0.00971140
$892$	−5.14345	−0.172215
$893$	9.61410	0.321724
$894$	19.3349	0.646657
$895$	3.90768	0.130619
$896$	−57.2448	−1.91241
$897$	109.676	3.66197
$898$	49.4797	1.65116
$899$	−69.5577	−2.31988
$900$	−9.55242	−0.318414
$901$	−0.0456026	−0.00151924
$902$	−0.868178	−0.0289072
$903$	22.1063	0.735650
$904$	35.8946	1.19384
$905$	−2.27139	−0.0755037
$906$	−23.2313	−0.771808
$907$	−19.4428	−0.645587	−0.322794	−	0.946469i	$-0.604622\pi$
$907$	−19.4428	−0.645587	−0.322794	+	0.946469i	$0.604622\pi$
$908$	1.13835	0.0377775
$909$	35.3190	1.17146
$910$	−10.6045	−0.351537
$911$	35.4707	1.17520	0.587598	−	0.809153i	$-0.300074\pi$
$911$	35.4707	1.17520	0.587598	+	0.809153i	$0.300074\pi$
$912$	56.6902	1.87720
$913$	6.82655	0.225926
$914$	19.9115	0.658614
$915$	3.56506	0.117857
$916$	−6.44821	−0.213055
$917$	−29.8131	−0.984516
$918$	−44.5997	−1.47201
$919$	30.5669	1.00831	0.504154	−	0.863614i	$-0.331804\pi$
$919$	30.5669	1.00831	0.504154	+	0.863614i	$0.331804\pi$
$920$	−6.73310	−0.221984
$921$	46.7396	1.54012
$922$	55.2441	1.81937
$923$	7.15799	0.235608
$924$	−4.35693	−0.143332
$925$	−25.2654	−0.830723
$926$	4.25092	0.139694
$927$	38.3618	1.25997
$928$	14.3506	0.471082
$929$	−59.7630	−1.96076	−0.980380	−	0.197116i	$-0.936842\pi$
$929$	−59.7630	−1.96076	−0.980380	+	0.197116i	$0.936842\pi$
$930$	15.7205	0.515495
$931$	−47.9605	−1.57184
$932$	4.53762	0.148635
$933$	−28.5111	−0.933413
$934$	−39.6389	−1.29702
$935$	−1.65247	−0.0540416
$936$	−58.5101	−1.91246
$937$	29.4112	0.960824	0.480412	−	0.877043i	$-0.340487\pi$
$937$	29.4112	0.960824	0.480412	+	0.877043i	$0.340487\pi$
$938$	−30.7574	−1.00427
$939$	−75.1577	−2.45268
$940$	−0.295346	−0.00963313
$941$	45.0066	1.46717	0.733587	−	0.679595i	$-0.237845\pi$
$941$	45.0066	1.46717	0.733587	+	0.679595i	$0.237845\pi$
$942$	−94.0901	−3.06562
$943$	−4.96326	−0.161626
$944$	−41.0879	−1.33730
$945$	−7.60005	−0.247230
$946$	−2.62594	−0.0853765
$947$	−10.4456	−0.339437	−0.169719	−	0.985493i	$-0.554286\pi$
$947$	−10.4456	−0.339437	−0.169719	+	0.985493i	$0.554286\pi$
$948$	−19.8155	−0.643578
$949$	−49.8867	−1.61939
$950$	32.9152	1.06791
$951$	−72.0815	−2.33740
$952$	−56.7509	−1.83931
$953$	−48.6114	−1.57468	−0.787339	−	0.616521i	$-0.788542\pi$
$953$	−48.6114	−1.57468	−0.787339	+	0.616521i	$0.788542\pi$
$954$	0.0642313	0.00207956
$955$	3.11538	0.100811
$956$	10.7809	0.348680
$957$	−16.6023	−0.536676
$958$	−0.946131	−0.0305681
$959$	−24.5475	−0.792681
$960$	5.49129	0.177231
$961$	85.2180	2.74897
$962$	38.5395	1.24256
$963$	16.0839	0.518295
$964$	0.398765	0.0128434
$965$	−0.425994	−0.0137132
$966$	−149.834	−4.82082
$967$	−28.8087	−0.926426	−0.463213	−	0.886247i	$-0.653303\pi$
$967$	−28.8087	−0.926426	−0.463213	+	0.886247i	$0.653303\pi$
$968$	25.2016	0.810011
$969$	65.8642	2.11586
$970$	−3.28648	−0.105523
$971$	−4.01498	−0.128847	−0.0644234	−	0.997923i	$-0.520521\pi$
$971$	−4.01498	−0.128847	−0.0644234	+	0.997923i	$0.520521\pi$
$972$	−6.03734	−0.193648
$973$	−9.97605	−0.319817
$974$	41.2397	1.32140
$975$	−66.1383	−2.11812
$976$	−17.5635	−0.562193
$977$	4.44163	0.142100	0.0710501	−	0.997473i	$-0.477365\pi$
$977$	4.44163	0.142100	0.0710501	+	0.997473i	$0.477365\pi$
$978$	−96.2111	−3.07649
$979$	−0.476670	−0.0152344
$980$	1.47335	0.0470645
$981$	33.7937	1.07895
$982$	−49.2294	−1.57097
$983$	−16.5283	−0.527172	−0.263586	−	0.964636i	$-0.584905\pi$
$983$	−16.5283	−0.527172	−0.263586	+	0.964636i	$0.584905\pi$
$984$	4.26860	0.136078
$985$	3.18623	0.101522
$986$	53.8545	1.71508
$987$	26.3915	0.840051
$988$	−8.34651	−0.265538
$989$	−15.0121	−0.477358
$990$	2.32751	0.0739730
$991$	34.8094	1.10576	0.552878	−	0.833262i	$-0.313530\pi$
$991$	34.8094	1.10576	0.552878	+	0.833262i	$0.313530\pi$
$992$	−23.9772	−0.761278
$993$	61.6081	1.95507
$994$	−9.77890	−0.310168
$995$	−6.18929	−0.196214
$996$	8.35872	0.264856
$997$	−14.7239	−0.466312	−0.233156	−	0.972439i	$-0.574905\pi$
$997$	−14.7239	−0.466312	−0.233156	+	0.972439i	$0.574905\pi$
$998$	−26.7118	−0.845547
$999$	27.6205	0.873874

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	241.2.a.b.1.8	✓	12
3.2	odd	2		2169.2.a.h.1.5		12
4.3	odd	2		3856.2.a.n.1.2		12
5.4	even	2		6025.2.a.h.1.5		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
241.2.a.b.1.8	✓	12	1.1	even	1	trivial
2169.2.a.h.1.5		12	3.2	odd	2
3856.2.a.n.1.2		12	4.3	odd	2
6025.2.a.h.1.5		12	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)

\(f(q)\)	\(=\)	\(q+1.54879 q^{2} +2.81087 q^{3} +0.398765 q^{4} +0.334961 q^{5} +4.35346 q^{6} -4.24623 q^{7} -2.47998 q^{8} +4.90098 q^{9} +O(q^{10})\)	\(q+1.54879 q^{2} +2.81087 q^{3} +0.398765 q^{4} +0.334961 q^{5} +4.35346 q^{6} -4.24623 q^{7} -2.47998 q^{8} +4.90098 q^{9} +0.518786 q^{10} +0.915418 q^{11} +1.12088 q^{12} +4.81392 q^{13} -6.57654 q^{14} +0.941532 q^{15} -4.63852 q^{16} -5.38915 q^{17} +7.59061 q^{18} -4.34799 q^{19} +0.133571 q^{20} -11.9356 q^{21} +1.41779 q^{22} +8.10534 q^{23} -6.97091 q^{24} -4.88780 q^{25} +7.45578 q^{26} +5.34340 q^{27} -1.69325 q^{28} -6.45221 q^{29} +1.45824 q^{30} +10.7804 q^{31} -2.22414 q^{32} +2.57312 q^{33} -8.34668 q^{34} -1.42232 q^{35} +1.95434 q^{36} +5.16908 q^{37} -6.73415 q^{38} +13.5313 q^{39} -0.830699 q^{40} -0.612344 q^{41} -18.4858 q^{42} -1.85213 q^{43} +0.365036 q^{44} +1.64164 q^{45} +12.5535 q^{46} -2.21116 q^{47} -13.0383 q^{48} +11.0305 q^{49} -7.57020 q^{50} -15.1482 q^{51} +1.91962 q^{52} +0.00846193 q^{53} +8.27584 q^{54} +0.306629 q^{55} +10.5306 q^{56} -12.2216 q^{57} -9.99315 q^{58} +8.85799 q^{59} +0.375450 q^{60} +3.78644 q^{61} +16.6967 q^{62} -20.8107 q^{63} +5.83230 q^{64} +1.61248 q^{65} +3.98523 q^{66} +4.67684 q^{67} -2.14900 q^{68} +22.7830 q^{69} -2.20289 q^{70} +1.48694 q^{71} -12.1544 q^{72} -10.3630 q^{73} +8.00585 q^{74} -13.7390 q^{75} -1.73383 q^{76} -3.88708 q^{77} +20.9572 q^{78} -17.6786 q^{79} -1.55372 q^{80} +0.316666 q^{81} -0.948396 q^{82} +7.45731 q^{83} -4.75950 q^{84} -1.80516 q^{85} -2.86857 q^{86} -18.1363 q^{87} -2.27022 q^{88} -0.520713 q^{89} +2.54256 q^{90} -20.4410 q^{91} +3.23212 q^{92} +30.3024 q^{93} -3.42463 q^{94} -1.45641 q^{95} -6.25177 q^{96} -6.33494 q^{97} +17.0840 q^{98} +4.48644 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 3 q^{2} + q^{3} + 13 q^{4} + 6 q^{5} - q^{6} + 3 q^{7} + 9 q^{8} + 15 q^{9}+O(q^{10}) \) 12 * q + 3 * q^2 + q^3 + 13 * q^4 + 6 * q^5 - q^6 + 3 * q^7 + 9 * q^8 + 15 * q^9	\( 12 q + 3 q^{2} + q^{3} + 13 q^{4} + 6 q^{5} - q^{6} + 3 q^{7} + 9 q^{8} + 15 q^{9} - 7 q^{10} + 22 q^{11} - 7 q^{12} - 5 q^{13} + 6 q^{14} + 13 q^{15} + 15 q^{16} - 4 q^{17} - q^{18} - 6 q^{19} + 10 q^{20} - 14 q^{21} - 12 q^{22} + 32 q^{23} - 15 q^{24} + 4 q^{25} + 8 q^{26} - 5 q^{27} - 11 q^{28} + 6 q^{29} - 19 q^{30} + 8 q^{31} + q^{32} - 24 q^{33} - 19 q^{34} + 15 q^{35} - 8 q^{36} - 8 q^{37} - 10 q^{38} + 31 q^{39} - 52 q^{40} - q^{41} - 49 q^{42} - 2 q^{43} + 42 q^{44} - 15 q^{45} - 25 q^{46} + 34 q^{47} - 49 q^{48} - 9 q^{49} - 27 q^{50} - 3 q^{51} - 41 q^{52} + 5 q^{53} - 40 q^{54} - 3 q^{55} + q^{56} - 22 q^{57} - 33 q^{58} + 26 q^{59} - 57 q^{60} - 26 q^{61} - 17 q^{62} - 4 q^{63} + 13 q^{64} - 25 q^{65} - 2 q^{66} + 6 q^{67} - 35 q^{68} - 2 q^{69} - 4 q^{70} + 94 q^{71} + 17 q^{72} - 22 q^{73} + 26 q^{74} - 20 q^{76} - 7 q^{77} + 54 q^{78} + 9 q^{79} + 19 q^{80} + 4 q^{81} + 15 q^{82} - 8 q^{83} + 2 q^{84} + 4 q^{85} + 9 q^{86} + 4 q^{87} + 6 q^{88} - 3 q^{89} + 11 q^{90} - 20 q^{91} + 36 q^{92} + 12 q^{93} + 48 q^{94} + 33 q^{95} - 23 q^{96} - 29 q^{97} + 28 q^{98} + 36 q^{99}+O(q^{100}) \) 12 * q + 3 * q^2 + q^3 + 13 * q^4 + 6 * q^5 - q^6 + 3 * q^7 + 9 * q^8 + 15 * q^9 - 7 * q^10 + 22 * q^11 - 7 * q^12 - 5 * q^13 + 6 * q^14 + 13 * q^15 + 15 * q^16 - 4 * q^17 - q^18 - 6 * q^19 + 10 * q^20 - 14 * q^21 - 12 * q^22 + 32 * q^23 - 15 * q^24 + 4 * q^25 + 8 * q^26 - 5 * q^27 - 11 * q^28 + 6 * q^29 - 19 * q^30 + 8 * q^31 + q^32 - 24 * q^33 - 19 * q^34 + 15 * q^35 - 8 * q^36 - 8 * q^37 - 10 * q^38 + 31 * q^39 - 52 * q^40 - q^41 - 49 * q^42 - 2 * q^43 + 42 * q^44 - 15 * q^45 - 25 * q^46 + 34 * q^47 - 49 * q^48 - 9 * q^49 - 27 * q^50 - 3 * q^51 - 41 * q^52 + 5 * q^53 - 40 * q^54 - 3 * q^55 + q^56 - 22 * q^57 - 33 * q^58 + 26 * q^59 - 57 * q^60 - 26 * q^61 - 17 * q^62 - 4 * q^63 + 13 * q^64 - 25 * q^65 - 2 * q^66 + 6 * q^67 - 35 * q^68 - 2 * q^69 - 4 * q^70 + 94 * q^71 + 17 * q^72 - 22 * q^73 + 26 * q^74 - 20 * q^76 - 7 * q^77 + 54 * q^78 + 9 * q^79 + 19 * q^80 + 4 * q^81 + 15 * q^82 - 8 * q^83 + 2 * q^84 + 4 * q^85 + 9 * q^86 + 4 * q^87 + 6 * q^88 - 3 * q^89 + 11 * q^90 - 20 * q^91 + 36 * q^92 + 12 * q^93 + 48 * q^94 + 33 * q^95 - 23 * q^96 - 29 * q^97 + 28 * q^98 + 36 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more