LMFDB - Embedded newform 4334.2.a.a.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4334,2,Mod(1,4334)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4334, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4334.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4334 = 2 \cdot 11 \cdot 197 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4334.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:	$34.6071642360$
Analytic rank:	$1$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - x^{14} - 23 x^{13} + 19 x^{12} + 194 x^{11} - 124 x^{10} - 761 x^{9} + 353 x^{8} + 1417 x^{7} - 465 x^{6} - 1128 x^{5} + 288 x^{4} + 316 x^{3} - 79 x^{2} - 20 x + 4 $ x^15 - x^14 - 23x^13 + 19x^12 + 194x^11 - 124x^10 - 761x^9 + 353x^8 + 1417x^7 - 465x^6 - 1128x^5 + 288x^4 + 316x^3 - 79x^2 - 20*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Root			$-1.77498$ of defining polynomial
Character	$\chi$	$=$	4334.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.77498 q^{3} +1.00000 q^{4} -0.781935 q^{5} -1.77498 q^{6} +2.50677 q^{7} -1.00000 q^{8} +0.150553 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.77498 q^{3} +1.00000 q^{4} -0.781935 q^{5} -1.77498 q^{6} +2.50677 q^{7} -1.00000 q^{8} +0.150553 q^{9} +0.781935 q^{10} +1.00000 q^{11} +1.77498 q^{12} -1.62749 q^{13} -2.50677 q^{14} -1.38792 q^{15} +1.00000 q^{16} +0.818716 q^{17} -0.150553 q^{18} -2.63893 q^{19} -0.781935 q^{20} +4.44947 q^{21} -1.00000 q^{22} -0.372117 q^{23} -1.77498 q^{24} -4.38858 q^{25} +1.62749 q^{26} -5.05771 q^{27} +2.50677 q^{28} -9.27335 q^{29} +1.38792 q^{30} -1.78637 q^{31} -1.00000 q^{32} +1.77498 q^{33} -0.818716 q^{34} -1.96014 q^{35} +0.150553 q^{36} -1.76213 q^{37} +2.63893 q^{38} -2.88876 q^{39} +0.781935 q^{40} +1.44923 q^{41} -4.44947 q^{42} -6.20306 q^{43} +1.00000 q^{44} -0.117723 q^{45} +0.372117 q^{46} -3.71682 q^{47} +1.77498 q^{48} -0.716080 q^{49} +4.38858 q^{50} +1.45320 q^{51} -1.62749 q^{52} -4.17393 q^{53} +5.05771 q^{54} -0.781935 q^{55} -2.50677 q^{56} -4.68405 q^{57} +9.27335 q^{58} +1.81440 q^{59} -1.38792 q^{60} -9.44494 q^{61} +1.78637 q^{62} +0.377403 q^{63} +1.00000 q^{64} +1.27259 q^{65} -1.77498 q^{66} -5.26558 q^{67} +0.818716 q^{68} -0.660501 q^{69} +1.96014 q^{70} +15.4660 q^{71} -0.150553 q^{72} +15.1608 q^{73} +1.76213 q^{74} -7.78964 q^{75} -2.63893 q^{76} +2.50677 q^{77} +2.88876 q^{78} -14.1141 q^{79} -0.781935 q^{80} -9.42899 q^{81} -1.44923 q^{82} +12.5488 q^{83} +4.44947 q^{84} -0.640183 q^{85} +6.20306 q^{86} -16.4600 q^{87} -1.00000 q^{88} -9.46795 q^{89} +0.117723 q^{90} -4.07975 q^{91} -0.372117 q^{92} -3.17078 q^{93} +3.71682 q^{94} +2.06348 q^{95} -1.77498 q^{96} -1.58308 q^{97} +0.716080 q^{98} +0.150553 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - 15 q^{2} - q^{3} + 15 q^{4} - 7 q^{5} + q^{6} + q^{7} - 15 q^{8} + 2 q^{9}+O(q^{10}) $ 15 * q - 15 * q^2 - q^3 + 15 * q^4 - 7 * q^5 + q^6 + q^7 - 15 * q^8 + 2 * q^9	\( 15 q - 15 q^{2} - q^{3} + 15 q^{4} - 7 q^{5} + q^{6} + q^{7} - 15 q^{8} + 2 q^{9} + 7 q^{10} + 15 q^{11} - q^{12} - q^{13} - q^{14} - 6 q^{15} + 15 q^{16} - 6 q^{17} - 2 q^{18} - 14 q^{19} - 7 q^{20} - 3 q^{21} - 15 q^{22} + 2 q^{23} + q^{24} - 10 q^{25} + q^{26} - 7 q^{27} + q^{28} + 8 q^{29} + 6 q^{30} - 33 q^{31} - 15 q^{32} - q^{33} + 6 q^{34} - 8 q^{35} + 2 q^{36} - 9 q^{37} + 14 q^{38} - 9 q^{39} + 7 q^{40} - 10 q^{41} + 3 q^{42} - 6 q^{43} + 15 q^{44} - 20 q^{45} - 2 q^{46} - q^{47} - q^{48} - 30 q^{49} + 10 q^{50} + 12 q^{51} - q^{52} + 6 q^{53} + 7 q^{54} - 7 q^{55} - q^{56} - 24 q^{57} - 8 q^{58} - 15 q^{59} - 6 q^{60} - 25 q^{61} + 33 q^{62} + 12 q^{63} + 15 q^{64} + 31 q^{65} + q^{66} - 13 q^{67} - 6 q^{68} - 43 q^{69} + 8 q^{70} - 4 q^{71} - 2 q^{72} - 4 q^{73} + 9 q^{74} - 5 q^{75} - 14 q^{76} + q^{77} + 9 q^{78} - 20 q^{79} - 7 q^{80} + 11 q^{81} + 10 q^{82} + q^{83} - 3 q^{84} - q^{85} + 6 q^{86} + 22 q^{87} - 15 q^{88} - 41 q^{89} + 20 q^{90} - 31 q^{91} + 2 q^{92} + 14 q^{93} + q^{94} + 41 q^{95} + q^{96} - 57 q^{97} + 30 q^{98} + 2 q^{99}+O(q^{100}) \) 15 * q - 15 * q^2 - q^3 + 15 * q^4 - 7 * q^5 + q^6 + q^7 - 15 * q^8 + 2 * q^9 + 7 * q^10 + 15 * q^11 - q^12 - q^13 - q^14 - 6 * q^15 + 15 * q^16 - 6 * q^17 - 2 * q^18 - 14 * q^19 - 7 * q^20 - 3 * q^21 - 15 * q^22 + 2 * q^23 + q^24 - 10 * q^25 + q^26 - 7 * q^27 + q^28 + 8 * q^29 + 6 * q^30 - 33 * q^31 - 15 * q^32 - q^33 + 6 * q^34 - 8 * q^35 + 2 * q^36 - 9 * q^37 + 14 * q^38 - 9 * q^39 + 7 * q^40 - 10 * q^41 + 3 * q^42 - 6 * q^43 + 15 * q^44 - 20 * q^45 - 2 * q^46 - q^47 - q^48 - 30 * q^49 + 10 * q^50 + 12 * q^51 - q^52 + 6 * q^53 + 7 * q^54 - 7 * q^55 - q^56 - 24 * q^57 - 8 * q^58 - 15 * q^59 - 6 * q^60 - 25 * q^61 + 33 * q^62 + 12 * q^63 + 15 * q^64 + 31 * q^65 + q^66 - 13 * q^67 - 6 * q^68 - 43 * q^69 + 8 * q^70 - 4 * q^71 - 2 * q^72 - 4 * q^73 + 9 * q^74 - 5 * q^75 - 14 * q^76 + q^77 + 9 * q^78 - 20 * q^79 - 7 * q^80 + 11 * q^81 + 10 * q^82 + q^83 - 3 * q^84 - q^85 + 6 * q^86 + 22 * q^87 - 15 * q^88 - 41 * q^89 + 20 * q^90 - 31 * q^91 + 2 * q^92 + 14 * q^93 + q^94 + 41 * q^95 + q^96 - 57 * q^97 + 30 * q^98 + 2 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.77498	1.02479	0.512393	−	0.858751i	$-0.328759\pi$
$3$	1.77498	1.02479	0.512393	+	0.858751i	$0.328759\pi$
$4$	1.00000	0.500000
$5$	−0.781935	−0.349692	−0.174846	−	0.984596i	$-0.555943\pi$
$5$	−0.781935	−0.349692	−0.174846	+	0.984596i	$0.555943\pi$
$6$	−1.77498	−0.724632
$7$	2.50677	0.947472	0.473736	−	0.880667i	$-0.342905\pi$
$7$	2.50677	0.947472	0.473736	+	0.880667i	$0.342905\pi$
$8$	−1.00000	−0.353553
$9$	0.150553	0.0501844
$10$	0.781935	0.247270
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	1.77498	0.512393
$13$	−1.62749	−0.451385	−0.225692	−	0.974199i	$-0.572464\pi$
$13$	−1.62749	−0.451385	−0.225692	+	0.974199i	$0.572464\pi$
$14$	−2.50677	−0.669964
$15$	−1.38792	−0.358359
$16$	1.00000	0.250000
$17$	0.818716	0.198568	0.0992839	−	0.995059i	$-0.468345\pi$
$17$	0.818716	0.198568	0.0992839	+	0.995059i	$0.468345\pi$
$18$	−0.150553	−0.0354857
$19$	−2.63893	−0.605413	−0.302706	−	0.953084i	$-0.597890\pi$
$19$	−2.63893	−0.605413	−0.302706	+	0.953084i	$0.597890\pi$
$20$	−0.781935	−0.174846
$21$	4.44947	0.970955
$22$	−1.00000	−0.213201
$23$	−0.372117	−0.0775918	−0.0387959	−	0.999247i	$-0.512352\pi$
$23$	−0.372117	−0.0775918	−0.0387959	+	0.999247i	$0.512352\pi$
$24$	−1.77498	−0.362316
$25$	−4.38858	−0.877715
$26$	1.62749	0.319177
$27$	−5.05771	−0.973357
$28$	2.50677	0.473736
$29$	−9.27335	−1.72202	−0.861009	−	0.508590i	$-0.830167\pi$
$29$	−9.27335	−1.72202	−0.861009	+	0.508590i	$0.830167\pi$
$30$	1.38792	0.253398
$31$	−1.78637	−0.320842	−0.160421	−	0.987049i	$-0.551285\pi$
$31$	−1.78637	−0.320842	−0.160421	+	0.987049i	$0.551285\pi$
$32$	−1.00000	−0.176777
$33$	1.77498	0.308984
$34$	−0.818716	−0.140409
$35$	−1.96014	−0.331323
$36$	0.150553	0.0250922
$37$	−1.76213	−0.289692	−0.144846	−	0.989454i	$-0.546269\pi$
$37$	−1.76213	−0.289692	−0.144846	+	0.989454i	$0.546269\pi$
$38$	2.63893	0.428092
$39$	−2.88876	−0.462572
$40$	0.781935	0.123635
$41$	1.44923	0.226332	0.113166	−	0.993576i	$-0.463901\pi$
$41$	1.44923	0.226332	0.113166	+	0.993576i	$0.463901\pi$
$42$	−4.44947	−0.686569
$43$	−6.20306	−0.945958	−0.472979	−	0.881074i	$-0.656821\pi$
$43$	−6.20306	−0.945958	−0.472979	+	0.881074i	$0.656821\pi$
$44$	1.00000	0.150756
$45$	−0.117723	−0.0175491
$46$	0.372117	0.0548657
$47$	−3.71682	−0.542154	−0.271077	−	0.962558i	$-0.587380\pi$
$47$	−3.71682	−0.542154	−0.271077	+	0.962558i	$0.587380\pi$
$48$	1.77498	0.256196
$49$	−0.716080	−0.102297
$50$	4.38858	0.620639
$51$	1.45320	0.203489
$52$	−1.62749	−0.225692
$53$	−4.17393	−0.573334	−0.286667	−	0.958030i	$-0.592547\pi$
$53$	−4.17393	−0.573334	−0.286667	+	0.958030i	$0.592547\pi$
$54$	5.05771	0.688267
$55$	−0.781935	−0.105436
$56$	−2.50677	−0.334982
$57$	−4.68405	−0.620418
$58$	9.27335	1.21765
$59$	1.81440	0.236214	0.118107	−	0.993001i	$-0.462317\pi$
$59$	1.81440	0.236214	0.118107	+	0.993001i	$0.462317\pi$
$60$	−1.38792	−0.179180
$61$	−9.44494	−1.20930	−0.604650	−	0.796491i	$-0.706687\pi$
$61$	−9.44494	−1.20930	−0.604650	+	0.796491i	$0.706687\pi$
$62$	1.78637	0.226870
$63$	0.377403	0.0475483
$64$	1.00000	0.125000
$65$	1.27259	0.157846
$66$	−1.77498	−0.218485
$67$	−5.26558	−0.643293	−0.321646	−	0.946860i	$-0.604236\pi$
$67$	−5.26558	−0.643293	−0.321646	+	0.946860i	$0.604236\pi$
$68$	0.818716	0.0992839
$69$	−0.660501	−0.0795150
$70$	1.96014	0.234281
$71$	15.4660	1.83548	0.917740	−	0.397182i	$-0.130012\pi$
$71$	15.4660	1.83548	0.917740	+	0.397182i	$0.130012\pi$
$72$	−0.150553	−0.0177429
$73$	15.1608	1.77444	0.887218	−	0.461351i	$-0.152635\pi$
$73$	15.1608	1.77444	0.887218	+	0.461351i	$0.152635\pi$
$74$	1.76213	0.204843
$75$	−7.78964	−0.899470
$76$	−2.63893	−0.302706
$77$	2.50677	0.285674
$78$	2.88876	0.327088
$79$	−14.1141	−1.58796	−0.793978	−	0.607946i	$-0.791994\pi$
$79$	−14.1141	−1.58796	−0.793978	+	0.607946i	$0.791994\pi$
$80$	−0.781935	−0.0874230
$81$	−9.42899	−1.04767
$82$	−1.44923	−0.160041
$83$	12.5488	1.37741	0.688707	−	0.725040i	$-0.258179\pi$
$83$	12.5488	1.37741	0.688707	+	0.725040i	$0.258179\pi$
$84$	4.44947	0.485477
$85$	−0.640183	−0.0694376
$86$	6.20306	0.668893
$87$	−16.4600	−1.76470
$88$	−1.00000	−0.106600
$89$	−9.46795	−1.00360	−0.501800	−	0.864983i	$-0.667329\pi$
$89$	−9.46795	−1.00360	−0.501800	+	0.864983i	$0.667329\pi$
$90$	0.117723	0.0124091
$91$	−4.07975	−0.427674
$92$	−0.372117	−0.0387959
$93$	−3.17078	−0.328794
$94$	3.71682	0.383361
$95$	2.06348	0.211708
$96$	−1.77498	−0.181158
$97$	−1.58308	−0.160738	−0.0803688	−	0.996765i	$-0.525610\pi$
$97$	−1.58308	−0.160738	−0.0803688	+	0.996765i	$0.525610\pi$
$98$	0.716080	0.0723350
$99$	0.150553	0.0151312
$100$	−4.38858	−0.438858
$101$	14.9101	1.48361	0.741806	−	0.670614i	$-0.233969\pi$
$101$	14.9101	1.48361	0.741806	+	0.670614i	$0.233969\pi$
$102$	−1.45320	−0.143889
$103$	5.87321	0.578704	0.289352	−	0.957223i	$-0.406560\pi$
$103$	5.87321	0.578704	0.289352	+	0.957223i	$0.406560\pi$
$104$	1.62749	0.159589
$105$	−3.47920	−0.339535
$106$	4.17393	0.405408
$107$	−16.9770	−1.64122	−0.820612	−	0.571486i	$-0.806367\pi$
$107$	−16.9770	−1.64122	−0.820612	+	0.571486i	$0.806367\pi$
$108$	−5.05771	−0.486678
$109$	−3.16635	−0.303282	−0.151641	−	0.988436i	$-0.548456\pi$
$109$	−3.16635	−0.303282	−0.151641	+	0.988436i	$0.548456\pi$
$110$	0.781935	0.0745546
$111$	−3.12774	−0.296872
$112$	2.50677	0.236868
$113$	−0.907314	−0.0853529	−0.0426765	−	0.999089i	$-0.513588\pi$
$113$	−0.907314	−0.0853529	−0.0426765	+	0.999089i	$0.513588\pi$
$114$	4.68405	0.438702
$115$	0.290972	0.0271333
$116$	−9.27335	−0.861009
$117$	−0.245024	−0.0226525
$118$	−1.81440	−0.167029
$119$	2.05234	0.188137
$120$	1.38792	0.126699
$121$	1.00000	0.0909091
$122$	9.44494	0.855105
$123$	2.57236	0.231942
$124$	−1.78637	−0.160421
$125$	7.34126	0.656622
$126$	−0.377403	−0.0336217
$127$	−11.2218	−0.995778	−0.497889	−	0.867241i	$-0.665891\pi$
$127$	−11.2218	−0.995778	−0.497889	+	0.867241i	$0.665891\pi$
$128$	−1.00000	−0.0883883
$129$	−11.0103	−0.969404
$130$	−1.27259	−0.111614
$131$	19.0329	1.66291	0.831455	−	0.555591i	$-0.187508\pi$
$131$	19.0329	1.66291	0.831455	+	0.555591i	$0.187508\pi$
$132$	1.77498	0.154492
$133$	−6.61521	−0.573612
$134$	5.26558	0.454877
$135$	3.95480	0.340375
$136$	−0.818716	−0.0702043
$137$	−17.5646	−1.50065	−0.750324	−	0.661070i	$-0.770103\pi$
$137$	−17.5646	−1.50065	−0.750324	+	0.661070i	$0.770103\pi$
$138$	0.660501	0.0562256
$139$	2.44108	0.207049	0.103525	−	0.994627i	$-0.466988\pi$
$139$	2.44108	0.207049	0.103525	+	0.994627i	$0.466988\pi$
$140$	−1.96014	−0.165662
$141$	−6.59728	−0.555591
$142$	−15.4660	−1.29788
$143$	−1.62749	−0.136098
$144$	0.150553	0.0125461
$145$	7.25116	0.602176
$146$	−15.1608	−1.25472
$147$	−1.27103	−0.104833
$148$	−1.76213	−0.144846
$149$	11.3385	0.928887	0.464444	−	0.885603i	$-0.346254\pi$
$149$	11.3385	0.928887	0.464444	+	0.885603i	$0.346254\pi$
$150$	7.78964	0.636021
$151$	−1.88511	−0.153408	−0.0767040	−	0.997054i	$-0.524440\pi$
$151$	−1.88511	−0.153408	−0.0767040	+	0.997054i	$0.524440\pi$
$152$	2.63893	0.214046
$153$	0.123260	0.00996500
$154$	−2.50677	−0.202002
$155$	1.39683	0.112196
$156$	−2.88876	−0.231286
$157$	15.0113	1.19803	0.599016	−	0.800737i	$-0.295559\pi$
$157$	15.0113	1.19803	0.599016	+	0.800737i	$0.295559\pi$
$158$	14.1141	1.12286
$159$	−7.40865	−0.587544
$160$	0.781935	0.0618174
$161$	−0.932815	−0.0735161
$162$	9.42899	0.740812
$163$	−10.5044	−0.822765	−0.411382	−	0.911463i	$-0.634954\pi$
$163$	−10.5044	−0.822765	−0.411382	+	0.911463i	$0.634954\pi$
$164$	1.44923	0.113166
$165$	−1.38792	−0.108049
$166$	−12.5488	−0.973978
$167$	1.67807	0.129853	0.0649266	−	0.997890i	$-0.479319\pi$
$167$	1.67807	0.129853	0.0649266	+	0.997890i	$0.479319\pi$
$168$	−4.44947	−0.343284
$169$	−10.3513	−0.796252
$170$	0.640183	0.0490998
$171$	−0.397300	−0.0303823
$172$	−6.20306	−0.472979
$173$	6.87018	0.522330	0.261165	−	0.965294i	$-0.415893\pi$
$173$	6.87018	0.522330	0.261165	+	0.965294i	$0.415893\pi$
$174$	16.4600	1.24783
$175$	−11.0012	−0.831611
$176$	1.00000	0.0753778
$177$	3.22052	0.242069
$178$	9.46795	0.709653
$179$	−15.2669	−1.14110	−0.570552	−	0.821261i	$-0.693271\pi$
$179$	−15.2669	−1.14110	−0.570552	+	0.821261i	$0.693271\pi$
$180$	−0.117723	−0.00877454
$181$	25.0359	1.86091	0.930453	−	0.366410i	$-0.119413\pi$
$181$	25.0359	1.86091	0.930453	+	0.366410i	$0.119413\pi$
$182$	4.07975	0.302411
$183$	−16.7646	−1.23927
$184$	0.372117	0.0274329
$185$	1.37787	0.101303
$186$	3.17078	0.232493
$187$	0.818716	0.0598705
$188$	−3.71682	−0.271077
$189$	−12.6785	−0.922228
$190$	−2.06348	−0.149700
$191$	−1.20417	−0.0871308	−0.0435654	−	0.999051i	$-0.513872\pi$
$191$	−1.20417	−0.0871308	−0.0435654	+	0.999051i	$0.513872\pi$
$192$	1.77498	0.128098
$193$	−14.9340	−1.07497	−0.537485	−	0.843273i	$-0.680626\pi$
$193$	−14.9340	−1.07497	−0.537485	+	0.843273i	$0.680626\pi$
$194$	1.58308	0.113659
$195$	2.25883	0.161758
$196$	−0.716080	−0.0511485
$197$	1.00000	0.0712470
$197$	1.00000	0.0712470
$198$	−0.150553	−0.0106993
$199$	−15.2442	−1.08064	−0.540318	−	0.841461i	$-0.681696\pi$
$199$	−15.2442	−1.08064	−0.540318	+	0.841461i	$0.681696\pi$
$200$	4.38858	0.310319
$201$	−9.34629	−0.659237
$202$	−14.9101	−1.04907
$203$	−23.2462	−1.63156
$204$	1.45320	0.101745
$205$	−1.13321	−0.0791467
$206$	−5.87321	−0.409206
$207$	−0.0560234	−0.00389390
$208$	−1.62749	−0.112846
$209$	−2.63893	−0.182539
$210$	3.47920	0.240088
$211$	1.69924	0.116981	0.0584904	−	0.998288i	$-0.481371\pi$
$211$	1.69924	0.116981	0.0584904	+	0.998288i	$0.481371\pi$
$212$	−4.17393	−0.286667
$213$	27.4519	1.88097
$214$	16.9770	1.16052
$215$	4.85039	0.330794
$216$	5.05771	0.344134
$217$	−4.47804	−0.303989
$218$	3.16635	0.214452
$219$	26.9101	1.81842
$220$	−0.781935	−0.0527181
$221$	−1.33245	−0.0896305
$222$	3.12774	0.209920
$223$	−17.4207	−1.16658	−0.583288	−	0.812265i	$-0.698234\pi$
$223$	−17.4207	−1.16658	−0.583288	+	0.812265i	$0.698234\pi$
$224$	−2.50677	−0.167491
$225$	−0.660714	−0.0440476
$226$	0.907314	0.0603536
$227$	17.0235	1.12989	0.564945	−	0.825129i	$-0.308897\pi$
$227$	17.0235	1.12989	0.564945	+	0.825129i	$0.308897\pi$
$228$	−4.68405	−0.310209
$229$	24.1710	1.59727	0.798634	−	0.601818i	$-0.205557\pi$
$229$	24.1710	1.59727	0.798634	+	0.601818i	$0.205557\pi$
$230$	−0.290972	−0.0191861
$231$	4.44947	0.292754
$232$	9.27335	0.608825
$233$	−8.79647	−0.576276	−0.288138	−	0.957589i	$-0.593036\pi$
$233$	−8.79647	−0.576276	−0.288138	+	0.957589i	$0.593036\pi$
$234$	0.245024	0.0160177
$235$	2.90631	0.189587
$236$	1.81440	0.118107
$237$	−25.0522	−1.62731
$238$	−2.05234	−0.133033
$239$	9.49252	0.614020	0.307010	−	0.951706i	$-0.400671\pi$
$239$	9.49252	0.614020	0.307010	+	0.951706i	$0.400671\pi$
$240$	−1.38792	−0.0895898
$241$	−4.32087	−0.278332	−0.139166	−	0.990269i	$-0.544442\pi$
$241$	−4.32087	−0.278332	−0.139166	+	0.990269i	$0.544442\pi$
$242$	−1.00000	−0.0642824
$243$	−1.56314	−0.100275
$244$	−9.44494	−0.604650
$245$	0.559928	0.0357725
$246$	−2.57236	−0.164008
$247$	4.29484	0.273274
$248$	1.78637	0.113435
$249$	22.2739	1.41155
$250$	−7.34126	−0.464302
$251$	−18.6422	−1.17669	−0.588343	−	0.808612i	$-0.700219\pi$
$251$	−18.6422	−1.17669	−0.588343	+	0.808612i	$0.700219\pi$
$252$	0.377403	0.0237741
$253$	−0.372117	−0.0233948
$254$	11.2218	0.704121
$255$	−1.13631	−0.0711586
$256$	1.00000	0.0625000
$257$	12.1480	0.757774	0.378887	−	0.925443i	$-0.376307\pi$
$257$	12.1480	0.757774	0.378887	+	0.925443i	$0.376307\pi$
$258$	11.0103	0.685472
$259$	−4.41725	−0.274475
$260$	1.27259	0.0789228
$261$	−1.39613	−0.0864184
$262$	−19.0329	−1.17586
$263$	13.4102	0.826910	0.413455	−	0.910525i	$-0.364322\pi$
$263$	13.4102	0.826910	0.413455	+	0.910525i	$0.364322\pi$
$264$	−1.77498	−0.109242
$265$	3.26375	0.200490
$266$	6.61521	0.405605
$267$	−16.8054	−1.02848
$268$	−5.26558	−0.321646
$269$	−19.4803	−1.18774	−0.593868	−	0.804563i	$-0.702400\pi$
$269$	−19.4803	−1.18774	−0.593868	+	0.804563i	$0.702400\pi$
$270$	−3.95480	−0.240682
$271$	−15.5878	−0.946893	−0.473446	−	0.880823i	$-0.656990\pi$
$271$	−15.5878	−0.946893	−0.473446	+	0.880823i	$0.656990\pi$
$272$	0.818716	0.0496420
$273$	−7.24148	−0.438274
$274$	17.5646	1.06112
$275$	−4.38858	−0.264641
$276$	−0.660501	−0.0397575
$277$	−27.0289	−1.62401	−0.812004	−	0.583651i	$-0.801623\pi$
$277$	−27.0289	−1.62401	−0.812004	+	0.583651i	$0.801623\pi$
$278$	−2.44108	−0.146406
$279$	−0.268944	−0.0161013
$280$	1.96014	0.117141
$281$	3.39112	0.202297	0.101149	−	0.994871i	$-0.467748\pi$
$281$	3.39112	0.202297	0.101149	+	0.994871i	$0.467748\pi$
$282$	6.59728	0.392862
$283$	3.95128	0.234879	0.117440	−	0.993080i	$-0.462531\pi$
$283$	3.95128	0.234879	0.117440	+	0.993080i	$0.462531\pi$
$284$	15.4660	0.917740
$285$	3.66263	0.216955
$286$	1.62749	0.0962355
$287$	3.63291	0.214444
$288$	−0.150553	−0.00887143
$289$	−16.3297	−0.960571
$290$	−7.25116	−0.425803
$291$	−2.80994	−0.164722
$292$	15.1608	0.887218
$293$	7.09396	0.414434	0.207217	−	0.978295i	$-0.433559\pi$
$293$	7.09396	0.414434	0.207217	+	0.978295i	$0.433559\pi$
$294$	1.27103	0.0741278
$295$	−1.41874	−0.0826022
$296$	1.76213	0.102422
$297$	−5.05771	−0.293478
$298$	−11.3385	−0.656822
$299$	0.605618	0.0350238
$300$	−7.78964	−0.449735
$301$	−15.5497	−0.896269
$302$	1.88511	0.108476
$303$	26.4652	1.52038
$304$	−2.63893	−0.151353
$305$	7.38533	0.422883
$306$	−0.123260	−0.00704632
$307$	10.7934	0.616013	0.308006	−	0.951384i	$-0.400338\pi$
$307$	10.7934	0.616013	0.308006	+	0.951384i	$0.400338\pi$
$308$	2.50677	0.142837
$309$	10.4248	0.593048
$310$	−1.39683	−0.0793346
$311$	−32.2969	−1.83139	−0.915693	−	0.401878i	$-0.868358\pi$
$311$	−32.2969	−1.83139	−0.915693	+	0.401878i	$0.868358\pi$
$312$	2.88876	0.163544
$313$	16.1264	0.911518	0.455759	−	0.890103i	$-0.349368\pi$
$313$	16.1264	0.911518	0.455759	+	0.890103i	$0.349368\pi$
$314$	−15.0113	−0.847136
$315$	−0.295104	−0.0166273
$316$	−14.1141	−0.793978
$317$	31.5608	1.77263	0.886317	−	0.463080i	$-0.153256\pi$
$317$	31.5608	1.77263	0.886317	+	0.463080i	$0.153256\pi$
$318$	7.40865	0.415456
$319$	−9.27335	−0.519208
$320$	−0.781935	−0.0437115
$321$	−30.1337	−1.68190
$322$	0.932815	0.0519837
$323$	−2.16054	−0.120216
$324$	−9.42899	−0.523833
$325$	7.14237	0.396187
$326$	10.5044	0.581782
$327$	−5.62021	−0.310798
$328$	−1.44923	−0.0800206
$329$	−9.31723	−0.513675
$330$	1.38792	0.0764024
$331$	−16.6835	−0.917008	−0.458504	−	0.888692i	$-0.651614\pi$
$331$	−16.6835	−0.917008	−0.458504	+	0.888692i	$0.651614\pi$
$332$	12.5488	0.688707
$333$	−0.265294	−0.0145380
$334$	−1.67807	−0.0918200
$335$	4.11734	0.224954
$336$	4.44947	0.242739
$337$	19.5671	1.06589	0.532945	−	0.846150i	$-0.321085\pi$
$337$	19.5671	1.06589	0.532945	+	0.846150i	$0.321085\pi$
$338$	10.3513	0.563035
$339$	−1.61046	−0.0874684
$340$	−0.640183	−0.0347188
$341$	−1.78637	−0.0967376
$342$	0.397300	0.0214835
$343$	−19.3425	−1.04440
$344$	6.20306	0.334447
$345$	0.516469	0.0278057
$346$	−6.87018	−0.369343
$347$	−8.13041	−0.436463	−0.218232	−	0.975897i	$-0.570029\pi$
$347$	−8.13041	−0.436463	−0.218232	+	0.975897i	$0.570029\pi$
$348$	−16.4600	−0.882349
$349$	29.7361	1.59174	0.795868	−	0.605470i	$-0.207015\pi$
$349$	29.7361	1.59174	0.795868	+	0.605470i	$0.207015\pi$
$350$	11.0012	0.588038
$351$	8.23138	0.439358
$352$	−1.00000	−0.0533002
$353$	−6.94727	−0.369766	−0.184883	−	0.982761i	$-0.559191\pi$
$353$	−6.94727	−0.369766	−0.184883	+	0.982761i	$0.559191\pi$
$354$	−3.22052	−0.171168
$355$	−12.0934	−0.641853
$356$	−9.46795	−0.501800
$357$	3.64286	0.192800
$358$	15.2669	0.806883
$359$	16.6568	0.879113	0.439557	−	0.898215i	$-0.355136\pi$
$359$	16.6568	0.879113	0.439557	+	0.898215i	$0.355136\pi$
$360$	0.117723	0.00620453
$361$	−12.0360	−0.633475
$362$	−25.0359	−1.31586
$363$	1.77498	0.0931623
$364$	−4.07975	−0.213837
$365$	−11.8548	−0.620506
$366$	16.7646	0.876299
$367$	−29.9141	−1.56150	−0.780752	−	0.624841i	$-0.785164\pi$
$367$	−29.9141	−1.56150	−0.780752	+	0.624841i	$0.785164\pi$
$368$	−0.372117	−0.0193980
$369$	0.218187	0.0113584
$370$	−1.37787	−0.0716320
$371$	−10.4631	−0.543218
$372$	−3.17078	−0.164397
$373$	0.196826	0.0101913	0.00509563	−	0.999987i	$-0.498378\pi$
$373$	0.196826	0.0101913	0.00509563	+	0.999987i	$0.498378\pi$
$374$	−0.818716	−0.0423348
$375$	13.0306	0.672897
$376$	3.71682	0.191680
$377$	15.0923	0.777293
$378$	12.6785	0.652114
$379$	36.7410	1.88726	0.943630	−	0.331003i	$-0.107387\pi$
$379$	36.7410	1.88726	0.943630	+	0.331003i	$0.107387\pi$
$380$	2.06348	0.105854
$381$	−19.9186	−1.02046
$382$	1.20417	0.0616108
$383$	−6.97054	−0.356178	−0.178089	−	0.984014i	$-0.556991\pi$
$383$	−6.97054	−0.356178	−0.178089	+	0.984014i	$0.556991\pi$
$384$	−1.77498	−0.0905791
$385$	−1.96014	−0.0998978
$386$	14.9340	0.760119
$387$	−0.933890	−0.0474723
$388$	−1.58308	−0.0803688
$389$	−9.98330	−0.506173	−0.253087	−	0.967444i	$-0.581446\pi$
$389$	−9.98330	−0.506173	−0.253087	+	0.967444i	$0.581446\pi$
$390$	−2.25883	−0.114380
$391$	−0.304659	−0.0154072
$392$	0.716080	0.0361675
$393$	33.7830	1.70413
$394$	−1.00000	−0.0503793
$395$	11.0363	0.555296
$396$	0.150553	0.00756558
$397$	−16.4016	−0.823171	−0.411586	−	0.911371i	$-0.635025\pi$
$397$	−16.4016	−0.823171	−0.411586	+	0.911371i	$0.635025\pi$
$398$	15.2442	0.764125
$399$	−11.7419	−0.587829
$400$	−4.38858	−0.219429
$401$	16.9990	0.848887	0.424444	−	0.905454i	$-0.360470\pi$
$401$	16.9990	0.848887	0.424444	+	0.905454i	$0.360470\pi$
$402$	9.34629	0.466151
$403$	2.90731	0.144823
$404$	14.9101	0.741806
$405$	7.37286	0.366360
$406$	23.2462	1.15369
$407$	−1.76213	−0.0873454
$408$	−1.45320	−0.0719444
$409$	−7.86890	−0.389092	−0.194546	−	0.980893i	$-0.562323\pi$
$409$	−7.86890	−0.389092	−0.194546	+	0.980893i	$0.562323\pi$
$410$	1.13321	0.0559651
$411$	−31.1769	−1.53784
$412$	5.87321	0.289352
$413$	4.54828	0.223806
$414$	0.0560234	0.00275340
$415$	−9.81237	−0.481670
$416$	1.62749	0.0797943
$417$	4.33286	0.212181
$418$	2.63893	0.129074
$419$	12.8174	0.626173	0.313086	−	0.949725i	$-0.398637\pi$
$419$	12.8174	0.626173	0.313086	+	0.949725i	$0.398637\pi$
$420$	−3.47920	−0.169768
$421$	−6.50870	−0.317214	−0.158607	−	0.987342i	$-0.550700\pi$
$421$	−6.50870	−0.317214	−0.158607	+	0.987342i	$0.550700\pi$
$422$	−1.69924	−0.0827180
$423$	−0.559579	−0.0272076
$424$	4.17393	0.202704
$425$	−3.59300	−0.174286
$426$	−27.4519	−1.33005
$427$	−23.6763	−1.14578
$428$	−16.9770	−0.820612
$429$	−2.88876	−0.139471
$430$	−4.85039	−0.233907
$431$	1.23118	0.0593040	0.0296520	−	0.999560i	$-0.490560\pi$
$431$	1.23118	0.0593040	0.0296520	+	0.999560i	$0.490560\pi$
$432$	−5.05771	−0.243339
$433$	−2.24503	−0.107889	−0.0539447	−	0.998544i	$-0.517179\pi$
$433$	−2.24503	−0.107889	−0.0539447	+	0.998544i	$0.517179\pi$
$434$	4.47804	0.214953
$435$	12.8707	0.617101
$436$	−3.16635	−0.151641
$437$	0.981993	0.0469751
$438$	−26.9101	−1.28581
$439$	14.6966	0.701432	0.350716	−	0.936482i	$-0.385938\pi$
$439$	14.6966	0.701432	0.350716	+	0.936482i	$0.385938\pi$
$440$	0.781935	0.0372773
$441$	−0.107808	−0.00513371
$442$	1.33245	0.0633783
$443$	−17.1435	−0.814512	−0.407256	−	0.913314i	$-0.633514\pi$
$443$	−17.1435	−0.814512	−0.407256	+	0.913314i	$0.633514\pi$
$444$	−3.12774	−0.148436
$445$	7.40332	0.350951
$446$	17.4207	0.824894
$447$	20.1256	0.951910
$448$	2.50677	0.118434
$449$	30.0085	1.41619	0.708093	−	0.706119i	$-0.249556\pi$
$449$	30.0085	1.41619	0.708093	+	0.706119i	$0.249556\pi$
$450$	0.660714	0.0311464
$451$	1.44923	0.0682418
$452$	−0.907314	−0.0426765
$453$	−3.34603	−0.157210
$454$	−17.0235	−0.798953
$455$	3.19010	0.149554
$456$	4.68405	0.219351
$457$	23.8644	1.11633	0.558165	−	0.829730i	$-0.311506\pi$
$457$	23.8644	1.11633	0.558165	+	0.829730i	$0.311506\pi$
$458$	−24.1710	−1.12944
$459$	−4.14083	−0.193277
$460$	0.290972	0.0135666
$461$	26.3947	1.22933	0.614663	−	0.788790i	$-0.289292\pi$
$461$	26.3947	1.22933	0.614663	+	0.788790i	$0.289292\pi$
$462$	−4.44947	−0.207008
$463$	−29.1966	−1.35688	−0.678440	−	0.734656i	$-0.737343\pi$
$463$	−29.1966	−1.35688	−0.678440	+	0.734656i	$0.737343\pi$
$464$	−9.27335	−0.430504
$465$	2.47934	0.114977
$466$	8.79647	0.407489
$467$	11.7938	0.545753	0.272876	−	0.962049i	$-0.412025\pi$
$467$	11.7938	0.545753	0.272876	+	0.962049i	$0.412025\pi$
$468$	−0.245024	−0.0113262
$469$	−13.1996	−0.609502
$470$	−2.90631	−0.134058
$471$	26.6447	1.22772
$472$	−1.81440	−0.0835143
$473$	−6.20306	−0.285217
$474$	25.0522	1.15069
$475$	11.5812	0.531380
$476$	2.05234	0.0940687
$477$	−0.628399	−0.0287724
$478$	−9.49252	−0.434178
$479$	−15.2391	−0.696294	−0.348147	−	0.937440i	$-0.613189\pi$
$479$	−15.2391	−0.696294	−0.348147	+	0.937440i	$0.613189\pi$
$480$	1.38792	0.0633496
$481$	2.86784	0.130762
$482$	4.32087	0.196810
$483$	−1.65573	−0.0753382
$484$	1.00000	0.0454545
$485$	1.23787	0.0562087
$486$	1.56314	0.0709055
$487$	−37.6125	−1.70439	−0.852193	−	0.523228i	$-0.824728\pi$
$487$	−37.6125	−1.70439	−0.852193	+	0.523228i	$0.824728\pi$
$488$	9.44494	0.427552
$489$	−18.6450	−0.843157
$490$	−0.559928	−0.0252950
$491$	−10.2636	−0.463191	−0.231596	−	0.972812i	$-0.574395\pi$
$491$	−10.2636	−0.463191	−0.231596	+	0.972812i	$0.574395\pi$
$492$	2.57236	0.115971
$493$	−7.59224	−0.341937
$494$	−4.29484	−0.193234
$495$	−0.117723	−0.00529125
$496$	−1.78637	−0.0802106
$497$	38.7698	1.73907
$498$	−22.2739	−0.998118
$499$	15.7335	0.704328	0.352164	−	0.935938i	$-0.385446\pi$
$499$	15.7335	0.704328	0.352164	+	0.935938i	$0.385446\pi$
$500$	7.34126	0.328311
$501$	2.97854	0.133072
$502$	18.6422	0.832042
$503$	14.4975	0.646413	0.323207	−	0.946328i	$-0.395239\pi$
$503$	14.4975	0.646413	0.323207	+	0.946328i	$0.395239\pi$
$504$	−0.377403	−0.0168109
$505$	−11.6587	−0.518807
$506$	0.372117	0.0165426
$507$	−18.3733	−0.815987
$508$	−11.2218	−0.497889
$509$	17.8700	0.792075	0.396038	−	0.918234i	$-0.370385\pi$
$509$	17.8700	0.792075	0.396038	+	0.918234i	$0.370385\pi$
$510$	1.13631	0.0503167
$511$	38.0047	1.68123
$512$	−1.00000	−0.0441942
$513$	13.3470	0.589283
$514$	−12.1480	−0.535827
$515$	−4.59247	−0.202368
$516$	−11.0103	−0.484702
$517$	−3.71682	−0.163466
$518$	4.41725	0.194083
$519$	12.1944	0.535276
$520$	−1.27259	−0.0558069
$521$	0.778605	0.0341113	0.0170557	−	0.999855i	$-0.494571\pi$
$521$	0.778605	0.0341113	0.0170557	+	0.999855i	$0.494571\pi$
$522$	1.39613	0.0611070
$523$	27.8500	1.21780	0.608898	−	0.793249i	$-0.291612\pi$
$523$	27.8500	1.21780	0.608898	+	0.793249i	$0.291612\pi$
$524$	19.0329	0.831455
$525$	−19.5269	−0.852222
$526$	−13.4102	−0.584714
$527$	−1.46253	−0.0637090
$528$	1.77498	0.0772461
$529$	−22.8615	−0.993980
$530$	−3.26375	−0.141768
$531$	0.273163	0.0118543
$532$	−6.61521	−0.286806
$533$	−2.35862	−0.102163
$534$	16.8054	0.727242
$535$	13.2749	0.573923
$536$	5.26558	0.227438
$537$	−27.0985	−1.16939
$538$	19.4803	0.839856
$539$	−0.716080	−0.0308437
$540$	3.95480	0.170188
$541$	−23.9261	−1.02866	−0.514332	−	0.857591i	$-0.671960\pi$
$541$	−23.9261	−1.02866	−0.514332	+	0.857591i	$0.671960\pi$
$542$	15.5878	0.669554
$543$	44.4383	1.90703
$544$	−0.818716	−0.0351022
$545$	2.47588	0.106055
$546$	7.24148	0.309907
$547$	−20.7243	−0.886109	−0.443054	−	0.896495i	$-0.646105\pi$
$547$	−20.7243	−0.886109	−0.443054	+	0.896495i	$0.646105\pi$
$548$	−17.5646	−0.750324
$549$	−1.42197	−0.0606880
$550$	4.38858	0.187130
$551$	24.4718	1.04253
$552$	0.660501	0.0281128
$553$	−35.3808	−1.50454
$554$	27.0289	1.14835
$555$	2.44569	0.103814
$556$	2.44108	0.103525
$557$	−43.5313	−1.84448	−0.922240	−	0.386619i	$-0.873643\pi$
$557$	−43.5313	−1.84448	−0.922240	+	0.386619i	$0.873643\pi$
$558$	0.268944	0.0113853
$559$	10.0954	0.426991
$560$	−1.96014	−0.0828308
$561$	1.45320	0.0613544
$562$	−3.39112	−0.143046
$563$	17.4341	0.734760	0.367380	−	0.930071i	$-0.380255\pi$
$563$	17.4341	0.734760	0.367380	+	0.930071i	$0.380255\pi$
$564$	−6.59728	−0.277796
$565$	0.709461	0.0298472
$566$	−3.95128	−0.166085
$567$	−23.6364	−0.992634
$568$	−15.4660	−0.648940
$569$	−16.5164	−0.692403	−0.346202	−	0.938160i	$-0.612529\pi$
$569$	−16.5164	−0.692403	−0.346202	+	0.938160i	$0.612529\pi$
$570$	−3.66263	−0.153411
$571$	−12.6596	−0.529787	−0.264893	−	0.964278i	$-0.585337\pi$
$571$	−12.6596	−0.529787	−0.264893	+	0.964278i	$0.585337\pi$
$572$	−1.62749	−0.0680488
$573$	−2.13738	−0.0892903
$574$	−3.63291	−0.151635
$575$	1.63307	0.0681036
$576$	0.150553	0.00627305
$577$	−35.4381	−1.47531	−0.737653	−	0.675180i	$-0.764066\pi$
$577$	−35.4381	−1.47531	−0.737653	+	0.675180i	$0.764066\pi$
$578$	16.3297	0.679226
$579$	−26.5075	−1.10161
$580$	7.25116	0.301088
$581$	31.4571	1.30506
$582$	2.80994	0.116476
$583$	−4.17393	−0.172867
$584$	−15.1608	−0.627358
$585$	0.191593	0.00792138
$586$	−7.09396	−0.293049
$587$	6.83618	0.282159	0.141080	−	0.989998i	$-0.454943\pi$
$587$	6.83618	0.282159	0.141080	+	0.989998i	$0.454943\pi$
$588$	−1.27103	−0.0524163
$589$	4.71412	0.194242
$590$	1.41874	0.0584086
$591$	1.77498	0.0730129
$592$	−1.76213	−0.0724229
$593$	3.16742	0.130070	0.0650352	−	0.997883i	$-0.479284\pi$
$593$	3.16742	0.130070	0.0650352	+	0.997883i	$0.479284\pi$
$594$	5.05771	0.207520
$595$	−1.60479	−0.0657902
$596$	11.3385	0.464444
$597$	−27.0582	−1.10742
$598$	−0.605618	−0.0247655
$599$	4.38361	0.179109	0.0895547	−	0.995982i	$-0.471456\pi$
$599$	4.38361	0.179109	0.0895547	+	0.995982i	$0.471456\pi$
$600$	7.78964	0.318011
$601$	34.6226	1.41228	0.706142	−	0.708070i	$-0.250434\pi$
$601$	34.6226	1.41228	0.706142	+	0.708070i	$0.250434\pi$
$602$	15.5497	0.633758
$603$	−0.792749	−0.0322832
$604$	−1.88511	−0.0767040
$605$	−0.781935	−0.0317902
$606$	−26.4652	−1.07507
$607$	13.4099	0.544290	0.272145	−	0.962256i	$-0.412267\pi$
$607$	13.4099	0.544290	0.272145	+	0.962256i	$0.412267\pi$
$608$	2.63893	0.107023
$609$	−41.2615	−1.67200
$610$	−7.38533	−0.299023
$611$	6.04909	0.244720
$612$	0.123260	0.00498250
$613$	−21.9274	−0.885640	−0.442820	−	0.896610i	$-0.646022\pi$
$613$	−21.9274	−0.885640	−0.442820	+	0.896610i	$0.646022\pi$
$614$	−10.7934	−0.435587
$615$	−2.01142	−0.0811083
$616$	−2.50677	−0.101001
$617$	−23.5090	−0.946437	−0.473219	−	0.880945i	$-0.656908\pi$
$617$	−23.5090	−0.946437	−0.473219	+	0.880945i	$0.656908\pi$
$618$	−10.4248	−0.419348
$619$	−39.8837	−1.60306	−0.801530	−	0.597955i	$-0.795980\pi$
$619$	−39.8837	−1.60306	−0.801530	+	0.597955i	$0.795980\pi$
$620$	1.39683	0.0560980
$621$	1.88206	0.0755245
$622$	32.2969	1.29499
$623$	−23.7340	−0.950884
$624$	−2.88876	−0.115643
$625$	16.2025	0.648100
$626$	−16.1264	−0.644541
$627$	−4.68405	−0.187063
$628$	15.0113	0.599016
$629$	−1.44268	−0.0575235
$630$	0.295104	0.0117572
$631$	−32.2398	−1.28345	−0.641723	−	0.766937i	$-0.721780\pi$
$631$	−32.2398	−1.28345	−0.641723	+	0.766937i	$0.721780\pi$
$632$	14.1141	0.561428
$633$	3.01613	0.119880
$634$	−31.5608	−1.25344
$635$	8.77476	0.348216
$636$	−7.40865	−0.293772
$637$	1.16541	0.0461753
$638$	9.27335	0.367135
$639$	2.32846	0.0921124
$640$	0.781935	0.0309087
$641$	−47.3531	−1.87033	−0.935167	−	0.354207i	$-0.884751\pi$
$641$	−47.3531	−1.87033	−0.935167	+	0.354207i	$0.884751\pi$
$642$	30.1337	1.18928
$643$	−42.0430	−1.65801	−0.829006	−	0.559240i	$-0.811093\pi$
$643$	−42.0430	−1.65801	−0.829006	+	0.559240i	$0.811093\pi$
$644$	−0.932815	−0.0367580
$645$	8.60935	0.338993
$646$	2.16054	0.0850052
$647$	−35.8141	−1.40800	−0.703998	−	0.710202i	$-0.748604\pi$
$647$	−35.8141	−1.40800	−0.703998	+	0.710202i	$0.748604\pi$
$648$	9.42899	0.370406
$649$	1.81440	0.0712213
$650$	−7.14237	−0.280147
$651$	−7.94843	−0.311523
$652$	−10.5044	−0.411382
$653$	−45.4473	−1.77849	−0.889246	−	0.457429i	$-0.848770\pi$
$653$	−45.4473	−1.77849	−0.889246	+	0.457429i	$0.848770\pi$
$654$	5.62021	0.219768
$655$	−14.8825	−0.581507
$656$	1.44923	0.0565831
$657$	2.28250	0.0890489
$658$	9.31723	0.363223
$659$	−14.6584	−0.571012	−0.285506	−	0.958377i	$-0.592162\pi$
$659$	−14.6584	−0.571012	−0.285506	+	0.958377i	$0.592162\pi$
$660$	−1.38792	−0.0540247
$661$	−34.1543	−1.32845	−0.664224	−	0.747534i	$-0.731238\pi$
$661$	−34.1543	−1.32845	−0.664224	+	0.747534i	$0.731238\pi$
$662$	16.6835	0.648422
$663$	−2.36508	−0.0918520
$664$	−12.5488	−0.486989
$665$	5.17267	0.200587
$666$	0.265294	0.0102799
$667$	3.45077	0.133615
$668$	1.67807	0.0649266
$669$	−30.9214	−1.19549
$670$	−4.11734	−0.159067
$671$	−9.44494	−0.364618
$672$	−4.44947	−0.171642
$673$	38.1298	1.46980	0.734899	−	0.678177i	$-0.237230\pi$
$673$	38.1298	1.46980	0.734899	+	0.678177i	$0.237230\pi$
$674$	−19.5671	−0.753698
$675$	22.1962	0.854330
$676$	−10.3513	−0.398126
$677$	22.3458	0.858817	0.429408	−	0.903110i	$-0.358722\pi$
$677$	22.3458	0.858817	0.429408	+	0.903110i	$0.358722\pi$
$678$	1.61046	0.0618495
$679$	−3.96843	−0.152294
$680$	0.640183	0.0245499
$681$	30.2164	1.15789
$682$	1.78637	0.0684038
$683$	41.9925	1.60680	0.803400	−	0.595440i	$-0.203022\pi$
$683$	41.9925	1.60680	0.803400	+	0.595440i	$0.203022\pi$
$684$	−0.397300	−0.0151911
$685$	13.7344	0.524765
$686$	19.3425	0.738499
$687$	42.9031	1.63686
$688$	−6.20306	−0.236490
$689$	6.79304	0.258794
$690$	−0.516469	−0.0196616
$691$	0.267171	0.0101636	0.00508182	−	0.999987i	$-0.498382\pi$
$691$	0.267171	0.0101636	0.00508182	+	0.999987i	$0.498382\pi$
$692$	6.87018	0.261165
$693$	0.377403	0.0143363
$694$	8.13041	0.308626
$695$	−1.90876	−0.0724035
$696$	16.4600	0.623915
$697$	1.18651	0.0449423
$698$	−29.7361	−1.12553
$699$	−15.6136	−0.590559
$700$	−11.0012	−0.415805
$701$	9.31232	0.351722	0.175861	−	0.984415i	$-0.443729\pi$
$701$	9.31232	0.351722	0.175861	+	0.984415i	$0.443729\pi$
$702$	−8.23138	−0.310673
$703$	4.65013	0.175383
$704$	1.00000	0.0376889
$705$	5.15864	0.194286
$706$	6.94727	0.261464
$707$	37.3763	1.40568
$708$	3.22052	0.121034
$709$	37.3501	1.40271	0.701356	−	0.712811i	$-0.252578\pi$
$709$	37.3501	1.40271	0.701356	+	0.712811i	$0.252578\pi$
$710$	12.0934	0.453858
$711$	−2.12492	−0.0796906
$712$	9.46795	0.354826
$713$	0.664741	0.0248947
$714$	−3.64286	−0.136331
$715$	1.27259	0.0475923
$716$	−15.2669	−0.570552
$717$	16.8490	0.629239
$718$	−16.6568	−0.621627
$719$	−37.4487	−1.39660	−0.698301	−	0.715804i	$-0.746060\pi$
$719$	−37.4487	−1.39660	−0.698301	+	0.715804i	$0.746060\pi$
$720$	−0.117723	−0.00438727
$721$	14.7228	0.548306
$722$	12.0360	0.447935
$723$	−7.66946	−0.285230
$724$	25.0359	0.930453
$725$	40.6968	1.51144
$726$	−1.77498	−0.0658757
$727$	20.0366	0.743115	0.371558	−	0.928410i	$-0.378824\pi$
$727$	20.0366	0.743115	0.371558	+	0.928410i	$0.378824\pi$
$728$	4.07975	0.151206
$729$	25.5124	0.944905
$730$	11.8548	0.438764
$731$	−5.07855	−0.187837
$732$	−16.7646	−0.619637
$733$	−12.2045	−0.450783	−0.225392	−	0.974268i	$-0.572366\pi$
$733$	−12.2045	−0.450783	−0.225392	+	0.974268i	$0.572366\pi$
$734$	29.9141	1.10415
$735$	0.993861	0.0366591
$736$	0.372117	0.0137164
$737$	−5.26558	−0.193960
$738$	−0.218187	−0.00803157
$739$	39.4330	1.45057	0.725284	−	0.688450i	$-0.241709\pi$
$739$	39.4330	1.45057	0.725284	+	0.688450i	$0.241709\pi$
$740$	1.37787	0.0506515
$741$	7.62326	0.280047
$742$	10.4631	0.384113
$743$	−26.2310	−0.962321	−0.481160	−	0.876633i	$-0.659785\pi$
$743$	−26.2310	−0.962321	−0.481160	+	0.876633i	$0.659785\pi$
$744$	3.17078	0.116246
$745$	−8.86598	−0.324824
$746$	−0.196826	−0.00720632
$747$	1.88927	0.0691246
$748$	0.818716	0.0299352
$749$	−42.5574	−1.55501
$750$	−13.0306	−0.475810
$751$	20.7380	0.756741	0.378370	−	0.925654i	$-0.376485\pi$
$751$	20.7380	0.756741	0.378370	+	0.925654i	$0.376485\pi$
$752$	−3.71682	−0.135538
$753$	−33.0895	−1.20585
$754$	−15.0923	−0.549629
$755$	1.47403	0.0536456
$756$	−12.6785	−0.461114
$757$	−49.6783	−1.80559	−0.902795	−	0.430072i	$-0.858488\pi$
$757$	−49.6783	−1.80559	−0.902795	+	0.430072i	$0.858488\pi$
$758$	−36.7410	−1.33449
$759$	−0.660501	−0.0239747
$760$	−2.06348	−0.0748501
$761$	−38.6052	−1.39944	−0.699719	−	0.714418i	$-0.746692\pi$
$761$	−38.6052	−1.39944	−0.699719	+	0.714418i	$0.746692\pi$
$762$	19.9186	0.721573
$763$	−7.93733	−0.287351
$764$	−1.20417	−0.0435654
$765$	−0.0963815	−0.00348468
$766$	6.97054	0.251856
$767$	−2.95291	−0.106623
$768$	1.77498	0.0640491
$769$	−20.8456	−0.751713	−0.375856	−	0.926678i	$-0.622651\pi$
$769$	−20.8456	−0.751713	−0.375856	+	0.926678i	$0.622651\pi$
$770$	1.96014	0.0706384
$771$	21.5625	0.776556
$772$	−14.9340	−0.537485
$773$	26.9810	0.970439	0.485219	−	0.874392i	$-0.338740\pi$
$773$	26.9810	0.970439	0.485219	+	0.874392i	$0.338740\pi$
$774$	0.933890	0.0335680
$775$	7.83964	0.281608
$776$	1.58308	0.0568293
$777$	−7.84054	−0.281278
$778$	9.98330	0.357919
$779$	−3.82443	−0.137025
$780$	2.25883	0.0808789
$781$	15.4660	0.553418
$782$	0.304659	0.0108946
$783$	46.9019	1.67614
$784$	−0.716080	−0.0255743
$785$	−11.7379	−0.418942
$786$	−33.7830	−1.20500
$787$	−28.2984	−1.00873	−0.504365	−	0.863491i	$-0.668273\pi$
$787$	−28.2984	−1.00873	−0.504365	+	0.863491i	$0.668273\pi$
$788$	1.00000	0.0356235
$789$	23.8029	0.847405
$790$	−11.0363	−0.392653
$791$	−2.27443	−0.0808695
$792$	−0.150553	−0.00534967
$793$	15.3716	0.545860
$794$	16.4016	0.582070
$795$	5.79308	0.205460
$796$	−15.2442	−0.540318
$797$	54.8757	1.94380	0.971899	−	0.235397i	$-0.0756390\pi$
$797$	54.8757	1.94380	0.971899	+	0.235397i	$0.0756390\pi$
$798$	11.7419	0.415658
$799$	−3.04302	−0.107654
$800$	4.38858	0.155160
$801$	−1.42543	−0.0503651
$802$	−16.9990	−0.600254
$803$	15.1608	0.535012
$804$	−9.34629	−0.329618
$805$	0.729401	0.0257080
$806$	−2.90731	−0.102406
$807$	−34.5771	−1.21717
$808$	−14.9101	−0.524536
$809$	10.4403	0.367061	0.183531	−	0.983014i	$-0.441247\pi$
$809$	10.4403	0.367061	0.183531	+	0.983014i	$0.441247\pi$
$810$	−7.37286	−0.259056
$811$	7.76663	0.272723	0.136362	−	0.990659i	$-0.456459\pi$
$811$	7.76663	0.272723	0.136362	+	0.990659i	$0.456459\pi$
$812$	−23.2462	−0.815782
$813$	−27.6681	−0.970361
$814$	1.76213	0.0617625
$815$	8.21372	0.287714
$816$	1.45320	0.0508723
$817$	16.3695	0.572695
$818$	7.86890	0.275130
$819$	−0.614219	−0.0214626
$820$	−1.13321	−0.0395733
$821$	22.4742	0.784354	0.392177	−	0.919890i	$-0.371722\pi$
$821$	22.4742	0.784354	0.392177	+	0.919890i	$0.371722\pi$
$822$	31.1769	1.08742
$823$	30.8250	1.07449	0.537246	−	0.843426i	$-0.319465\pi$
$823$	30.8250	1.07449	0.537246	+	0.843426i	$0.319465\pi$
$824$	−5.87321	−0.204603
$825$	−7.78964	−0.271200
$826$	−4.54828	−0.158255
$827$	2.59362	0.0901890	0.0450945	−	0.998983i	$-0.485641\pi$
$827$	2.59362	0.0901890	0.0450945	+	0.998983i	$0.485641\pi$
$828$	−0.0560234	−0.00194695
$829$	7.86562	0.273184	0.136592	−	0.990627i	$-0.456385\pi$
$829$	7.86562	0.273184	0.136592	+	0.990627i	$0.456385\pi$
$830$	9.81237	0.340592
$831$	−47.9757	−1.66426
$832$	−1.62749	−0.0564231
$833$	−0.586266	−0.0203129
$834$	−4.33286	−0.150035
$835$	−1.31214	−0.0454086
$836$	−2.63893	−0.0912694
$837$	9.03497	0.312294
$838$	−12.8174	−0.442771
$839$	23.9608	0.827220	0.413610	−	0.910454i	$-0.364268\pi$
$839$	23.9608	0.827220	0.413610	+	0.910454i	$0.364268\pi$
$840$	3.47920	0.120044
$841$	56.9950	1.96535
$842$	6.50870	0.224305
$843$	6.01918	0.207311
$844$	1.69924	0.0584904
$845$	8.09403	0.278443
$846$	0.559579	0.0192387
$847$	2.50677	0.0861338
$848$	−4.17393	−0.143334
$849$	7.01345	0.240701
$850$	3.59300	0.123239
$851$	0.655718	0.0224777
$852$	27.4519	0.940486
$853$	−11.5589	−0.395770	−0.197885	−	0.980225i	$-0.563407\pi$
$853$	−11.5589	−0.395770	−0.197885	+	0.980225i	$0.563407\pi$
$854$	23.6763	0.810188
$855$	0.310663	0.0106244
$856$	16.9770	0.580260
$857$	16.9502	0.579009	0.289504	−	0.957177i	$-0.406510\pi$
$857$	16.9502	0.579009	0.289504	+	0.957177i	$0.406510\pi$
$858$	2.88876	0.0986208
$859$	17.0536	0.581860	0.290930	−	0.956744i	$-0.406035\pi$
$859$	17.0536	0.581860	0.290930	+	0.956744i	$0.406035\pi$
$860$	4.85039	0.165397
$861$	6.44833	0.219759
$862$	−1.23118	−0.0419343
$863$	33.8655	1.15279	0.576397	−	0.817170i	$-0.304458\pi$
$863$	33.8655	1.15279	0.576397	+	0.817170i	$0.304458\pi$
$864$	5.05771	0.172067
$865$	−5.37203	−0.182655
$866$	2.24503	0.0762893
$867$	−28.9849	−0.984379
$868$	−4.47804	−0.151995
$869$	−14.1141	−0.478787
$870$	−12.8707	−0.436356
$871$	8.56968	0.290372
$872$	3.16635	0.107226
$873$	−0.238338	−0.00806652
$874$	−0.981993	−0.0332164
$875$	18.4029	0.622131
$876$	26.9101	0.909208
$877$	19.9918	0.675074	0.337537	−	0.941312i	$-0.390406\pi$
$877$	19.9918	0.675074	0.337537	+	0.941312i	$0.390406\pi$
$878$	−14.6966	−0.495987
$879$	12.5916	0.424705
$880$	−0.781935	−0.0263590
$881$	−41.1816	−1.38744	−0.693721	−	0.720244i	$-0.744030\pi$
$881$	−41.1816	−1.38744	−0.693721	+	0.720244i	$0.744030\pi$
$882$	0.107808	0.00363008
$883$	−34.1202	−1.14824	−0.574118	−	0.818773i	$-0.694655\pi$
$883$	−34.1202	−1.14824	−0.574118	+	0.818773i	$0.694655\pi$
$884$	−1.33245	−0.0448152
$885$	−2.51823	−0.0846495
$886$	17.1435	0.575947
$887$	−27.7652	−0.932264	−0.466132	−	0.884715i	$-0.654353\pi$
$887$	−27.7652	−0.932264	−0.466132	+	0.884715i	$0.654353\pi$
$888$	3.12774	0.104960
$889$	−28.1306	−0.943471
$890$	−7.40332	−0.248160
$891$	−9.42899	−0.315883
$892$	−17.4207	−0.583288
$893$	9.80844	0.328227
$894$	−20.1256	−0.673102
$895$	11.9378	0.399035
$896$	−2.50677	−0.0837455
$897$	1.07496	0.0358918
$898$	−30.0085	−1.00139
$899$	16.5657	0.552496
$900$	−0.660714	−0.0220238
$901$	−3.41727	−0.113846
$902$	−1.44923	−0.0482542
$903$	−27.6004	−0.918483
$904$	0.907314	0.0301768
$905$	−19.5765	−0.650744
$906$	3.34603	0.111164
$907$	18.7838	0.623706	0.311853	−	0.950130i	$-0.399050\pi$
$907$	18.7838	0.623706	0.311853	+	0.950130i	$0.399050\pi$
$908$	17.0235	0.564945
$909$	2.24476	0.0744541
$910$	−3.19010	−0.105751
$911$	−17.0719	−0.565617	−0.282808	−	0.959176i	$-0.591266\pi$
$911$	−17.0719	−0.565617	−0.282808	+	0.959176i	$0.591266\pi$
$912$	−4.68405	−0.155105
$913$	12.5488	0.415306
$914$	−23.8644	−0.789365
$915$	13.1088	0.433364
$916$	24.1710	0.798634
$917$	47.7112	1.57556
$918$	4.14083	0.136668
$919$	57.6392	1.90134	0.950671	−	0.310201i	$-0.100396\pi$
$919$	57.6392	1.90134	0.950671	+	0.310201i	$0.100396\pi$
$920$	−0.290972	−0.00959305
$921$	19.1581	0.631281
$922$	−26.3947	−0.869264
$923$	−25.1708	−0.828507
$924$	4.44947	0.146377
$925$	7.73323	0.254267
$926$	29.1966	0.959458
$927$	0.884230	0.0290419
$928$	9.27335	0.304413
$929$	−6.15552	−0.201956	−0.100978	−	0.994889i	$-0.532197\pi$
$929$	−6.15552	−0.201956	−0.100978	+	0.994889i	$0.532197\pi$
$930$	−2.47934	−0.0813009
$931$	1.88969	0.0619320
$932$	−8.79647	−0.288138
$933$	−57.3263	−1.87678
$934$	−11.7938	−0.385905
$935$	−0.640183	−0.0209362
$936$	0.245024	0.00800885
$937$	−35.2150	−1.15042	−0.575212	−	0.818004i	$-0.695080\pi$
$937$	−35.2150	−1.15042	−0.575212	+	0.818004i	$0.695080\pi$
$938$	13.1996	0.430983
$939$	28.6240	0.934110
$940$	2.90631	0.0947934
$941$	34.0646	1.11047	0.555237	−	0.831692i	$-0.312628\pi$
$941$	34.0646	1.11047	0.555237	+	0.831692i	$0.312628\pi$
$942$	−26.6447	−0.868132
$943$	−0.539285	−0.0175616
$944$	1.81440	0.0590535
$945$	9.91380	0.322496
$946$	6.20306	0.201679
$947$	24.8820	0.808556	0.404278	−	0.914636i	$-0.367523\pi$
$947$	24.8820	0.808556	0.404278	+	0.914636i	$0.367523\pi$
$948$	−25.0522	−0.813657
$949$	−24.6740	−0.800953
$950$	−11.5812	−0.375743
$951$	56.0198	1.81657
$952$	−2.05234	−0.0665166
$953$	28.6111	0.926806	0.463403	−	0.886148i	$-0.346628\pi$
$953$	28.6111	0.926806	0.463403	+	0.886148i	$0.346628\pi$
$954$	0.628399	0.0203452
$955$	0.941584	0.0304689
$956$	9.49252	0.307010
$957$	−16.4600	−0.532076
$958$	15.2391	0.492355
$959$	−44.0306	−1.42182
$960$	−1.38792	−0.0447949
$961$	−27.8089	−0.897060
$962$	−2.86784	−0.0924630
$963$	−2.55593	−0.0823638
$964$	−4.32087	−0.139166
$965$	11.6774	0.375909
$966$	1.65573	0.0532721
$967$	52.0373	1.67341	0.836703	−	0.547656i	$-0.184480\pi$
$967$	52.0373	1.67341	0.836703	+	0.547656i	$0.184480\pi$
$968$	−1.00000	−0.0321412
$969$	−3.83491	−0.123195
$970$	−1.23787	−0.0397455
$971$	28.2403	0.906274	0.453137	−	0.891441i	$-0.350305\pi$
$971$	28.2403	0.906274	0.453137	+	0.891441i	$0.350305\pi$
$972$	−1.56314	−0.0501377
$973$	6.11923	0.196173
$974$	37.6125	1.20518
$975$	12.6776	0.406007
$976$	−9.44494	−0.302325
$977$	39.4882	1.26334	0.631669	−	0.775238i	$-0.282370\pi$
$977$	39.4882	1.26334	0.631669	+	0.775238i	$0.282370\pi$
$978$	18.6450	0.596202
$979$	−9.46795	−0.302597
$980$	0.559928	0.0178862
$981$	−0.476704	−0.0152200
$982$	10.2636	0.327526
$983$	11.0518	0.352499	0.176250	−	0.984346i	$-0.443603\pi$
$983$	11.0518	0.352499	0.176250	+	0.984346i	$0.443603\pi$
$984$	−2.57236	−0.0820039
$985$	−0.781935	−0.0249145
$986$	7.59224	0.241786
$987$	−16.5379	−0.526407
$988$	4.29484	0.136637
$989$	2.30827	0.0733986
$990$	0.117723	0.00374148
$991$	54.0951	1.71839	0.859193	−	0.511652i	$-0.170966\pi$
$991$	54.0951	1.71839	0.859193	+	0.511652i	$0.170966\pi$
$992$	1.78637	0.0567174
$993$	−29.6129	−0.939736
$994$	−38.7698	−1.22970
$995$	11.9200	0.377890
$996$	22.2739	0.705776
$997$	46.1210	1.46067	0.730333	−	0.683091i	$-0.239365\pi$
$997$	46.1210	1.46067	0.730333	+	0.683091i	$0.239365\pi$
$998$	−15.7335	−0.498035
$999$	8.91232	0.281973

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4334.2.a.a.1.13	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4334.2.a.a.1.13	✓	15	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4334 = 2 \cdot 11 \cdot 197 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4334.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.77498 q^{3} +1.00000 q^{4} -0.781935 q^{5} -1.77498 q^{6} +2.50677 q^{7} -1.00000 q^{8} +0.150553 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.77498 q^{3} +1.00000 q^{4} -0.781935 q^{5} -1.77498 q^{6} +2.50677 q^{7} -1.00000 q^{8} +0.150553 q^{9} +0.781935 q^{10} +1.00000 q^{11} +1.77498 q^{12} -1.62749 q^{13} -2.50677 q^{14} -1.38792 q^{15} +1.00000 q^{16} +0.818716 q^{17} -0.150553 q^{18} -2.63893 q^{19} -0.781935 q^{20} +4.44947 q^{21} -1.00000 q^{22} -0.372117 q^{23} -1.77498 q^{24} -4.38858 q^{25} +1.62749 q^{26} -5.05771 q^{27} +2.50677 q^{28} -9.27335 q^{29} +1.38792 q^{30} -1.78637 q^{31} -1.00000 q^{32} +1.77498 q^{33} -0.818716 q^{34} -1.96014 q^{35} +0.150553 q^{36} -1.76213 q^{37} +2.63893 q^{38} -2.88876 q^{39} +0.781935 q^{40} +1.44923 q^{41} -4.44947 q^{42} -6.20306 q^{43} +1.00000 q^{44} -0.117723 q^{45} +0.372117 q^{46} -3.71682 q^{47} +1.77498 q^{48} -0.716080 q^{49} +4.38858 q^{50} +1.45320 q^{51} -1.62749 q^{52} -4.17393 q^{53} +5.05771 q^{54} -0.781935 q^{55} -2.50677 q^{56} -4.68405 q^{57} +9.27335 q^{58} +1.81440 q^{59} -1.38792 q^{60} -9.44494 q^{61} +1.78637 q^{62} +0.377403 q^{63} +1.00000 q^{64} +1.27259 q^{65} -1.77498 q^{66} -5.26558 q^{67} +0.818716 q^{68} -0.660501 q^{69} +1.96014 q^{70} +15.4660 q^{71} -0.150553 q^{72} +15.1608 q^{73} +1.76213 q^{74} -7.78964 q^{75} -2.63893 q^{76} +2.50677 q^{77} +2.88876 q^{78} -14.1141 q^{79} -0.781935 q^{80} -9.42899 q^{81} -1.44923 q^{82} +12.5488 q^{83} +4.44947 q^{84} -0.640183 q^{85} +6.20306 q^{86} -16.4600 q^{87} -1.00000 q^{88} -9.46795 q^{89} +0.117723 q^{90} -4.07975 q^{91} -0.372117 q^{92} -3.17078 q^{93} +3.71682 q^{94} +2.06348 q^{95} -1.77498 q^{96} -1.58308 q^{97} +0.716080 q^{98} +0.150553 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - 15 q^{2} - q^{3} + 15 q^{4} - 7 q^{5} + q^{6} + q^{7} - 15 q^{8} + 2 q^{9}+O(q^{10}) \) 15 * q - 15 * q^2 - q^3 + 15 * q^4 - 7 * q^5 + q^6 + q^7 - 15 * q^8 + 2 * q^9	\( 15 q - 15 q^{2} - q^{3} + 15 q^{4} - 7 q^{5} + q^{6} + q^{7} - 15 q^{8} + 2 q^{9} + 7 q^{10} + 15 q^{11} - q^{12} - q^{13} - q^{14} - 6 q^{15} + 15 q^{16} - 6 q^{17} - 2 q^{18} - 14 q^{19} - 7 q^{20} - 3 q^{21} - 15 q^{22} + 2 q^{23} + q^{24} - 10 q^{25} + q^{26} - 7 q^{27} + q^{28} + 8 q^{29} + 6 q^{30} - 33 q^{31} - 15 q^{32} - q^{33} + 6 q^{34} - 8 q^{35} + 2 q^{36} - 9 q^{37} + 14 q^{38} - 9 q^{39} + 7 q^{40} - 10 q^{41} + 3 q^{42} - 6 q^{43} + 15 q^{44} - 20 q^{45} - 2 q^{46} - q^{47} - q^{48} - 30 q^{49} + 10 q^{50} + 12 q^{51} - q^{52} + 6 q^{53} + 7 q^{54} - 7 q^{55} - q^{56} - 24 q^{57} - 8 q^{58} - 15 q^{59} - 6 q^{60} - 25 q^{61} + 33 q^{62} + 12 q^{63} + 15 q^{64} + 31 q^{65} + q^{66} - 13 q^{67} - 6 q^{68} - 43 q^{69} + 8 q^{70} - 4 q^{71} - 2 q^{72} - 4 q^{73} + 9 q^{74} - 5 q^{75} - 14 q^{76} + q^{77} + 9 q^{78} - 20 q^{79} - 7 q^{80} + 11 q^{81} + 10 q^{82} + q^{83} - 3 q^{84} - q^{85} + 6 q^{86} + 22 q^{87} - 15 q^{88} - 41 q^{89} + 20 q^{90} - 31 q^{91} + 2 q^{92} + 14 q^{93} + q^{94} + 41 q^{95} + q^{96} - 57 q^{97} + 30 q^{98} + 2 q^{99}+O(q^{100}) \) 15 * q - 15 * q^2 - q^3 + 15 * q^4 - 7 * q^5 + q^6 + q^7 - 15 * q^8 + 2 * q^9 + 7 * q^10 + 15 * q^11 - q^12 - q^13 - q^14 - 6 * q^15 + 15 * q^16 - 6 * q^17 - 2 * q^18 - 14 * q^19 - 7 * q^20 - 3 * q^21 - 15 * q^22 + 2 * q^23 + q^24 - 10 * q^25 + q^26 - 7 * q^27 + q^28 + 8 * q^29 + 6 * q^30 - 33 * q^31 - 15 * q^32 - q^33 + 6 * q^34 - 8 * q^35 + 2 * q^36 - 9 * q^37 + 14 * q^38 - 9 * q^39 + 7 * q^40 - 10 * q^41 + 3 * q^42 - 6 * q^43 + 15 * q^44 - 20 * q^45 - 2 * q^46 - q^47 - q^48 - 30 * q^49 + 10 * q^50 + 12 * q^51 - q^52 + 6 * q^53 + 7 * q^54 - 7 * q^55 - q^56 - 24 * q^57 - 8 * q^58 - 15 * q^59 - 6 * q^60 - 25 * q^61 + 33 * q^62 + 12 * q^63 + 15 * q^64 + 31 * q^65 + q^66 - 13 * q^67 - 6 * q^68 - 43 * q^69 + 8 * q^70 - 4 * q^71 - 2 * q^72 - 4 * q^73 + 9 * q^74 - 5 * q^75 - 14 * q^76 + q^77 + 9 * q^78 - 20 * q^79 - 7 * q^80 + 11 * q^81 + 10 * q^82 + q^83 - 3 * q^84 - q^85 + 6 * q^86 + 22 * q^87 - 15 * q^88 - 41 * q^89 + 20 * q^90 - 31 * q^91 + 2 * q^92 + 14 * q^93 + q^94 + 41 * q^95 + q^96 - 57 * q^97 + 30 * q^98 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more