LMFDB - Embedded newform 8034.2.a.bd.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8034,2,Mod(1,8034)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8034, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8034.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8034 = 2 \cdot 3 \cdot 13 \cdot 103 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8034.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:	$64.1518129839$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 5 x^{15} - 36 x^{14} + 196 x^{13} + 498 x^{12} - 3101 x^{11} - 3150 x^{10} + 25368 x^{9} + \cdots - 66432 $ x^16 - 5x^15 - 36x^14 + 196x^13 + 498x^12 - 3101x^11 - 3150x^10 + 25368x^9 + 6763x^8 - 113788x^7 + 19731x^6 + 270913x^5 - 122680x^4 - 296326x^3 + 185524x^2 + 94528*x - 66432
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-2.07316$ of defining polynomial
Character	$\chi$	$=$	8034.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.07316 q^{5} +1.00000 q^{6} -0.689093 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.07316 q^{5} +1.00000 q^{6} -0.689093 q^{7} +1.00000 q^{8} +1.00000 q^{9} -2.07316 q^{10} -5.68444 q^{11} +1.00000 q^{12} -1.00000 q^{13} -0.689093 q^{14} -2.07316 q^{15} +1.00000 q^{16} +4.26526 q^{17} +1.00000 q^{18} +1.18518 q^{19} -2.07316 q^{20} -0.689093 q^{21} -5.68444 q^{22} -6.71682 q^{23} +1.00000 q^{24} -0.702017 q^{25} -1.00000 q^{26} +1.00000 q^{27} -0.689093 q^{28} +7.38369 q^{29} -2.07316 q^{30} -0.861742 q^{31} +1.00000 q^{32} -5.68444 q^{33} +4.26526 q^{34} +1.42860 q^{35} +1.00000 q^{36} +7.06113 q^{37} +1.18518 q^{38} -1.00000 q^{39} -2.07316 q^{40} +4.45306 q^{41} -0.689093 q^{42} +6.54240 q^{43} -5.68444 q^{44} -2.07316 q^{45} -6.71682 q^{46} +13.4396 q^{47} +1.00000 q^{48} -6.52515 q^{49} -0.702017 q^{50} +4.26526 q^{51} -1.00000 q^{52} -11.5160 q^{53} +1.00000 q^{54} +11.7847 q^{55} -0.689093 q^{56} +1.18518 q^{57} +7.38369 q^{58} +12.3728 q^{59} -2.07316 q^{60} -11.8569 q^{61} -0.861742 q^{62} -0.689093 q^{63} +1.00000 q^{64} +2.07316 q^{65} -5.68444 q^{66} -11.1954 q^{67} +4.26526 q^{68} -6.71682 q^{69} +1.42860 q^{70} -8.51079 q^{71} +1.00000 q^{72} +15.2639 q^{73} +7.06113 q^{74} -0.702017 q^{75} +1.18518 q^{76} +3.91710 q^{77} -1.00000 q^{78} +3.53462 q^{79} -2.07316 q^{80} +1.00000 q^{81} +4.45306 q^{82} +17.6723 q^{83} -0.689093 q^{84} -8.84255 q^{85} +6.54240 q^{86} +7.38369 q^{87} -5.68444 q^{88} +9.33143 q^{89} -2.07316 q^{90} +0.689093 q^{91} -6.71682 q^{92} -0.861742 q^{93} +13.4396 q^{94} -2.45706 q^{95} +1.00000 q^{96} +0.658392 q^{97} -6.52515 q^{98} -5.68444 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{2} + 16 q^{3} + 16 q^{4} + 5 q^{5} + 16 q^{6} + 4 q^{7} + 16 q^{8} + 16 q^{9}+O(q^{10}) $ 16 * q + 16 * q^2 + 16 * q^3 + 16 * q^4 + 5 * q^5 + 16 * q^6 + 4 * q^7 + 16 * q^8 + 16 * q^9	\( 16 q + 16 q^{2} + 16 q^{3} + 16 q^{4} + 5 q^{5} + 16 q^{6} + 4 q^{7} + 16 q^{8} + 16 q^{9} + 5 q^{10} + 18 q^{11} + 16 q^{12} - 16 q^{13} + 4 q^{14} + 5 q^{15} + 16 q^{16} + 17 q^{17} + 16 q^{18} + 8 q^{19} + 5 q^{20} + 4 q^{21} + 18 q^{22} + 9 q^{23} + 16 q^{24} + 17 q^{25} - 16 q^{26} + 16 q^{27} + 4 q^{28} + 14 q^{29} + 5 q^{30} + 12 q^{31} + 16 q^{32} + 18 q^{33} + 17 q^{34} + 16 q^{35} + 16 q^{36} + 31 q^{37} + 8 q^{38} - 16 q^{39} + 5 q^{40} + 29 q^{41} + 4 q^{42} + 30 q^{43} + 18 q^{44} + 5 q^{45} + 9 q^{46} - q^{47} + 16 q^{48} + 36 q^{49} + 17 q^{50} + 17 q^{51} - 16 q^{52} + 12 q^{53} + 16 q^{54} + 30 q^{55} + 4 q^{56} + 8 q^{57} + 14 q^{58} + 38 q^{59} + 5 q^{60} + 12 q^{62} + 4 q^{63} + 16 q^{64} - 5 q^{65} + 18 q^{66} + 28 q^{67} + 17 q^{68} + 9 q^{69} + 16 q^{70} + 32 q^{71} + 16 q^{72} + 20 q^{73} + 31 q^{74} + 17 q^{75} + 8 q^{76} + 26 q^{77} - 16 q^{78} + 13 q^{79} + 5 q^{80} + 16 q^{81} + 29 q^{82} + 39 q^{83} + 4 q^{84} + 31 q^{85} + 30 q^{86} + 14 q^{87} + 18 q^{88} + 9 q^{89} + 5 q^{90} - 4 q^{91} + 9 q^{92} + 12 q^{93} - q^{94} - 20 q^{95} + 16 q^{96} + 35 q^{97} + 36 q^{98} + 18 q^{99}+O(q^{100}) \) 16 * q + 16 * q^2 + 16 * q^3 + 16 * q^4 + 5 * q^5 + 16 * q^6 + 4 * q^7 + 16 * q^8 + 16 * q^9 + 5 * q^10 + 18 * q^11 + 16 * q^12 - 16 * q^13 + 4 * q^14 + 5 * q^15 + 16 * q^16 + 17 * q^17 + 16 * q^18 + 8 * q^19 + 5 * q^20 + 4 * q^21 + 18 * q^22 + 9 * q^23 + 16 * q^24 + 17 * q^25 - 16 * q^26 + 16 * q^27 + 4 * q^28 + 14 * q^29 + 5 * q^30 + 12 * q^31 + 16 * q^32 + 18 * q^33 + 17 * q^34 + 16 * q^35 + 16 * q^36 + 31 * q^37 + 8 * q^38 - 16 * q^39 + 5 * q^40 + 29 * q^41 + 4 * q^42 + 30 * q^43 + 18 * q^44 + 5 * q^45 + 9 * q^46 - q^47 + 16 * q^48 + 36 * q^49 + 17 * q^50 + 17 * q^51 - 16 * q^52 + 12 * q^53 + 16 * q^54 + 30 * q^55 + 4 * q^56 + 8 * q^57 + 14 * q^58 + 38 * q^59 + 5 * q^60 + 12 * q^62 + 4 * q^63 + 16 * q^64 - 5 * q^65 + 18 * q^66 + 28 * q^67 + 17 * q^68 + 9 * q^69 + 16 * q^70 + 32 * q^71 + 16 * q^72 + 20 * q^73 + 31 * q^74 + 17 * q^75 + 8 * q^76 + 26 * q^77 - 16 * q^78 + 13 * q^79 + 5 * q^80 + 16 * q^81 + 29 * q^82 + 39 * q^83 + 4 * q^84 + 31 * q^85 + 30 * q^86 + 14 * q^87 + 18 * q^88 + 9 * q^89 + 5 * q^90 - 4 * q^91 + 9 * q^92 + 12 * q^93 - q^94 - 20 * q^95 + 16 * q^96 + 35 * q^97 + 36 * q^98 + 18 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	−2.07316	−0.927144	−0.463572	−	0.886059i	$-0.653433\pi$
$5$	−2.07316	−0.927144	−0.463572	+	0.886059i	$0.653433\pi$
$6$	1.00000	0.408248
$7$	−0.689093	−0.260453	−0.130226	−	0.991484i	$-0.541570\pi$
$7$	−0.689093	−0.260453	−0.130226	+	0.991484i	$0.541570\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−2.07316	−0.655590
$11$	−5.68444	−1.71392	−0.856961	−	0.515382i	$-0.827650\pi$
$11$	−5.68444	−1.71392	−0.856961	+	0.515382i	$0.827650\pi$
$12$	1.00000	0.288675
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−0.689093	−0.184168
$15$	−2.07316	−0.535287
$16$	1.00000	0.250000
$17$	4.26526	1.03448	0.517238	−	0.855841i	$-0.326960\pi$
$17$	4.26526	1.03448	0.517238	+	0.855841i	$0.326960\pi$
$18$	1.00000	0.235702
$19$	1.18518	0.271899	0.135949	−	0.990716i	$-0.456592\pi$
$19$	1.18518	0.271899	0.135949	+	0.990716i	$0.456592\pi$
$20$	−2.07316	−0.463572
$21$	−0.689093	−0.150372
$22$	−5.68444	−1.21193
$23$	−6.71682	−1.40055	−0.700277	−	0.713871i	$-0.746940\pi$
$23$	−6.71682	−1.40055	−0.700277	+	0.713871i	$0.746940\pi$
$24$	1.00000	0.204124
$25$	−0.702017	−0.140403
$26$	−1.00000	−0.196116
$27$	1.00000	0.192450
$28$	−0.689093	−0.130226
$29$	7.38369	1.37112	0.685558	−	0.728018i	$-0.259558\pi$
$29$	7.38369	1.37112	0.685558	+	0.728018i	$0.259558\pi$
$30$	−2.07316	−0.378505
$31$	−0.861742	−0.154773	−0.0773867	−	0.997001i	$-0.524658\pi$
$31$	−0.861742	−0.154773	−0.0773867	+	0.997001i	$0.524658\pi$
$32$	1.00000	0.176777
$33$	−5.68444	−0.989533
$34$	4.26526	0.731485
$35$	1.42860	0.241477
$36$	1.00000	0.166667
$37$	7.06113	1.16084	0.580421	−	0.814316i	$-0.302888\pi$
$37$	7.06113	1.16084	0.580421	+	0.814316i	$0.302888\pi$
$38$	1.18518	0.192261
$39$	−1.00000	−0.160128
$40$	−2.07316	−0.327795
$41$	4.45306	0.695451	0.347726	−	0.937596i	$-0.386954\pi$
$41$	4.45306	0.695451	0.347726	+	0.937596i	$0.386954\pi$
$42$	−0.689093	−0.106329
$43$	6.54240	0.997707	0.498853	−	0.866686i	$-0.333755\pi$
$43$	6.54240	0.997707	0.498853	+	0.866686i	$0.333755\pi$
$44$	−5.68444	−0.856961
$45$	−2.07316	−0.309048
$46$	−6.71682	−0.990341
$47$	13.4396	1.96037	0.980185	−	0.198083i	$-0.0634715\pi$
$47$	13.4396	1.96037	0.980185	+	0.198083i	$0.0634715\pi$
$48$	1.00000	0.144338
$49$	−6.52515	−0.932164
$50$	−0.702017	−0.0992802
$51$	4.26526	0.597255
$52$	−1.00000	−0.138675
$53$	−11.5160	−1.58185	−0.790924	−	0.611914i	$-0.790400\pi$
$53$	−11.5160	−1.58185	−0.790924	+	0.611914i	$0.790400\pi$
$54$	1.00000	0.136083
$55$	11.7847	1.58905
$56$	−0.689093	−0.0920839
$57$	1.18518	0.156981
$58$	7.38369	0.969526
$59$	12.3728	1.61080	0.805398	−	0.592734i	$-0.201951\pi$
$59$	12.3728	1.61080	0.805398	+	0.592734i	$0.201951\pi$
$60$	−2.07316	−0.267644
$61$	−11.8569	−1.51812	−0.759061	−	0.651020i	$-0.774342\pi$
$61$	−11.8569	−1.51812	−0.759061	+	0.651020i	$0.774342\pi$
$62$	−0.861742	−0.109441
$63$	−0.689093	−0.0868175
$64$	1.00000	0.125000
$65$	2.07316	0.257144
$66$	−5.68444	−0.699706
$67$	−11.1954	−1.36773	−0.683865	−	0.729608i	$-0.739702\pi$
$67$	−11.1954	−1.36773	−0.683865	+	0.729608i	$0.739702\pi$
$68$	4.26526	0.517238
$69$	−6.71682	−0.808610
$70$	1.42860	0.170750
$71$	−8.51079	−1.01005	−0.505023	−	0.863106i	$-0.668516\pi$
$71$	−8.51079	−1.01005	−0.505023	+	0.863106i	$0.668516\pi$
$72$	1.00000	0.117851
$73$	15.2639	1.78650	0.893252	−	0.449557i	$-0.148418\pi$
$73$	15.2639	1.78650	0.893252	+	0.449557i	$0.148418\pi$
$74$	7.06113	0.820840
$75$	−0.702017	−0.0810620
$76$	1.18518	0.135949
$77$	3.91710	0.446395
$78$	−1.00000	−0.113228
$79$	3.53462	0.397675	0.198838	−	0.980032i	$-0.436283\pi$
$79$	3.53462	0.397675	0.198838	+	0.980032i	$0.436283\pi$
$80$	−2.07316	−0.231786
$81$	1.00000	0.111111
$82$	4.45306	0.491758
$83$	17.6723	1.93979	0.969895	−	0.243525i	$-0.0783039\pi$
$83$	17.6723	1.93979	0.969895	+	0.243525i	$0.0783039\pi$
$84$	−0.689093	−0.0751862
$85$	−8.84255	−0.959109
$86$	6.54240	0.705485
$87$	7.38369	0.791615
$88$	−5.68444	−0.605963
$89$	9.33143	0.989130	0.494565	−	0.869141i	$-0.335327\pi$
$89$	9.33143	0.989130	0.494565	+	0.869141i	$0.335327\pi$
$90$	−2.07316	−0.218530
$91$	0.689093	0.0722365
$92$	−6.71682	−0.700277
$93$	−0.861742	−0.0893585
$94$	13.4396	1.38619
$95$	−2.45706	−0.252089
$96$	1.00000	0.102062
$97$	0.658392	0.0668496	0.0334248	−	0.999441i	$-0.489359\pi$
$97$	0.658392	0.0668496	0.0334248	+	0.999441i	$0.489359\pi$
$98$	−6.52515	−0.659140
$99$	−5.68444	−0.571307
$100$	−0.702017	−0.0702017
$101$	6.82535	0.679147	0.339574	−	0.940579i	$-0.389717\pi$
$101$	6.82535	0.679147	0.339574	+	0.940579i	$0.389717\pi$
$102$	4.26526	0.422323
$103$	1.00000	0.0985329
$103$	1.00000	0.0985329
$104$	−1.00000	−0.0980581
$105$	1.42860	0.139417
$106$	−11.5160	−1.11854
$107$	2.46030	0.237846	0.118923	−	0.992903i	$-0.462056\pi$
$107$	2.46030	0.237846	0.118923	+	0.992903i	$0.462056\pi$
$108$	1.00000	0.0962250
$109$	11.7260	1.12314	0.561572	−	0.827428i	$-0.310197\pi$
$109$	11.7260	1.12314	0.561572	+	0.827428i	$0.310197\pi$
$110$	11.7847	1.12363
$111$	7.06113	0.670213
$112$	−0.689093	−0.0651131
$113$	6.59652	0.620548	0.310274	−	0.950647i	$-0.399579\pi$
$113$	6.59652	0.620548	0.310274	+	0.950647i	$0.399579\pi$
$114$	1.18518	0.111002
$115$	13.9250	1.29852
$116$	7.38369	0.685558
$117$	−1.00000	−0.0924500
$118$	12.3728	1.13901
$119$	−2.93916	−0.269432
$120$	−2.07316	−0.189253
$121$	21.3128	1.93753
$122$	−11.8569	−1.07347
$123$	4.45306	0.401519
$124$	−0.861742	−0.0773867
$125$	11.8212	1.05732
$126$	−0.689093	−0.0613893
$127$	−10.3423	−0.917732	−0.458866	−	0.888506i	$-0.651744\pi$
$127$	−10.3423	−0.917732	−0.458866	+	0.888506i	$0.651744\pi$
$128$	1.00000	0.0883883
$129$	6.54240	0.576026
$130$	2.07316	0.181828
$131$	14.9451	1.30576	0.652880	−	0.757461i	$-0.273560\pi$
$131$	14.9451	1.30576	0.652880	+	0.757461i	$0.273560\pi$
$132$	−5.68444	−0.494767
$133$	−0.816698	−0.0708167
$134$	−11.1954	−0.967131
$135$	−2.07316	−0.178429
$136$	4.26526	0.365743
$137$	−13.2838	−1.13491	−0.567455	−	0.823404i	$-0.692072\pi$
$137$	−13.2838	−1.13491	−0.567455	+	0.823404i	$0.692072\pi$
$138$	−6.71682	−0.571774
$139$	−11.0521	−0.937423	−0.468712	−	0.883351i	$-0.655282\pi$
$139$	−11.0521	−0.937423	−0.468712	+	0.883351i	$0.655282\pi$
$140$	1.42860	0.120739
$141$	13.4396	1.13182
$142$	−8.51079	−0.714210
$143$	5.68444	0.475356
$144$	1.00000	0.0833333
$145$	−15.3076	−1.27122
$146$	15.2639	1.26325
$147$	−6.52515	−0.538185
$148$	7.06113	0.580421
$149$	16.1808	1.32558	0.662791	−	0.748804i	$-0.269372\pi$
$149$	16.1808	1.32558	0.662791	+	0.748804i	$0.269372\pi$
$150$	−0.702017	−0.0573195
$151$	−7.86045	−0.639674	−0.319837	−	0.947473i	$-0.603628\pi$
$151$	−7.86045	−0.639674	−0.319837	+	0.947473i	$0.603628\pi$
$152$	1.18518	0.0961306
$153$	4.26526	0.344826
$154$	3.91710	0.315649
$155$	1.78653	0.143497
$156$	−1.00000	−0.0800641
$157$	3.44324	0.274800	0.137400	−	0.990516i	$-0.456125\pi$
$157$	3.44324	0.274800	0.137400	+	0.990516i	$0.456125\pi$
$158$	3.53462	0.281199
$159$	−11.5160	−0.913281
$160$	−2.07316	−0.163898
$161$	4.62851	0.364778
$162$	1.00000	0.0785674
$163$	−17.5028	−1.37093	−0.685463	−	0.728107i	$-0.740400\pi$
$163$	−17.5028	−1.37093	−0.685463	+	0.728107i	$0.740400\pi$
$164$	4.45306	0.347726
$165$	11.7847	0.917440
$166$	17.6723	1.37164
$167$	−5.68878	−0.440211	−0.220106	−	0.975476i	$-0.570640\pi$
$167$	−5.68878	−0.440211	−0.220106	+	0.975476i	$0.570640\pi$
$168$	−0.689093	−0.0531647
$169$	1.00000	0.0769231
$170$	−8.84255	−0.678193
$171$	1.18518	0.0906328
$172$	6.54240	0.498853
$173$	12.5224	0.952058	0.476029	−	0.879430i	$-0.342076\pi$
$173$	12.5224	0.952058	0.476029	+	0.879430i	$0.342076\pi$
$174$	7.38369	0.559756
$175$	0.483755	0.0365684
$176$	−5.68444	−0.428480
$177$	12.3728	0.929994
$178$	9.33143	0.699420
$179$	20.7165	1.54843	0.774214	−	0.632924i	$-0.218146\pi$
$179$	20.7165	1.54843	0.774214	+	0.632924i	$0.218146\pi$
$180$	−2.07316	−0.154524
$181$	15.3155	1.13839	0.569195	−	0.822202i	$-0.307255\pi$
$181$	15.3155	1.13839	0.569195	+	0.822202i	$0.307255\pi$
$182$	0.689093	0.0510789
$183$	−11.8569	−0.876488
$184$	−6.71682	−0.495171
$185$	−14.6388	−1.07627
$186$	−0.861742	−0.0631860
$187$	−24.2456	−1.77301
$188$	13.4396	0.980185
$189$	−0.689093	−0.0501241
$190$	−2.45706	−0.178254
$191$	−9.93969	−0.719211	−0.359605	−	0.933104i	$-0.617089\pi$
$191$	−9.93969	−0.719211	−0.359605	+	0.933104i	$0.617089\pi$
$192$	1.00000	0.0721688
$193$	8.98429	0.646703	0.323352	−	0.946279i	$-0.395190\pi$
$193$	8.98429	0.646703	0.323352	+	0.946279i	$0.395190\pi$
$194$	0.658392	0.0472698
$195$	2.07316	0.148462
$196$	−6.52515	−0.466082
$197$	−8.10619	−0.577542	−0.288771	−	0.957398i	$-0.593247\pi$
$197$	−8.10619	−0.577542	−0.288771	+	0.957398i	$0.593247\pi$
$198$	−5.68444	−0.403975
$199$	25.6181	1.81602	0.908011	−	0.418947i	$-0.137601\pi$
$199$	25.6181	1.81602	0.908011	+	0.418947i	$0.137601\pi$
$200$	−0.702017	−0.0496401
$201$	−11.1954	−0.789660
$202$	6.82535	0.480230
$203$	−5.08805	−0.357111
$204$	4.26526	0.298628
$205$	−9.23190	−0.644784
$206$	1.00000	0.0696733
$207$	−6.71682	−0.466851
$208$	−1.00000	−0.0693375
$209$	−6.73707	−0.466013
$210$	1.42860	0.0985826
$211$	−18.2873	−1.25895	−0.629475	−	0.777021i	$-0.716730\pi$
$211$	−18.2873	−1.25895	−0.629475	+	0.777021i	$0.716730\pi$
$212$	−11.5160	−0.790924
$213$	−8.51079	−0.583150
$214$	2.46030	0.168183
$215$	−13.5634	−0.925018
$216$	1.00000	0.0680414
$217$	0.593820	0.0403111
$218$	11.7260	0.794183
$219$	15.2639	1.03144
$220$	11.7847	0.794526
$221$	−4.26526	−0.286912
$222$	7.06113	0.473912
$223$	14.2669	0.955383	0.477691	−	0.878528i	$-0.341474\pi$
$223$	14.2669	0.955383	0.477691	+	0.878528i	$0.341474\pi$
$224$	−0.689093	−0.0460419
$225$	−0.702017	−0.0468011
$226$	6.59652	0.438794
$227$	12.5881	0.835501	0.417750	−	0.908562i	$-0.362819\pi$
$227$	12.5881	0.835501	0.417750	+	0.908562i	$0.362819\pi$
$228$	1.18518	0.0784903
$229$	−14.9209	−0.986000	−0.493000	−	0.870029i	$-0.664100\pi$
$229$	−14.9209	−0.986000	−0.493000	+	0.870029i	$0.664100\pi$
$230$	13.9250	0.918189
$231$	3.91710	0.257726
$232$	7.38369	0.484763
$233$	26.7892	1.75502	0.877511	−	0.479557i	$-0.159203\pi$
$233$	26.7892	1.75502	0.877511	+	0.479557i	$0.159203\pi$
$234$	−1.00000	−0.0653720
$235$	−27.8625	−1.81755
$236$	12.3728	0.805398
$237$	3.53462	0.229598
$238$	−2.93916	−0.190517
$239$	−18.1853	−1.17631	−0.588153	−	0.808749i	$-0.700145\pi$
$239$	−18.1853	−1.17631	−0.588153	+	0.808749i	$0.700145\pi$
$240$	−2.07316	−0.133822
$241$	20.8017	1.33995	0.669977	−	0.742382i	$-0.266304\pi$
$241$	20.8017	1.33995	0.669977	+	0.742382i	$0.266304\pi$
$242$	21.3128	1.37004
$243$	1.00000	0.0641500
$244$	−11.8569	−0.759061
$245$	13.5277	0.864251
$246$	4.45306	0.283917
$247$	−1.18518	−0.0754111
$248$	−0.861742	−0.0547207
$249$	17.6723	1.11994
$250$	11.8212	0.747637
$251$	−12.6097	−0.795915	−0.397957	−	0.917404i	$-0.630281\pi$
$251$	−12.6097	−0.795915	−0.397957	+	0.917404i	$0.630281\pi$
$252$	−0.689093	−0.0434088
$253$	38.1813	2.40044
$254$	−10.3423	−0.648934
$255$	−8.84255	−0.553742
$256$	1.00000	0.0625000
$257$	−10.2870	−0.641686	−0.320843	−	0.947132i	$-0.603966\pi$
$257$	−10.2870	−0.641686	−0.320843	+	0.947132i	$0.603966\pi$
$258$	6.54240	0.407312
$259$	−4.86577	−0.302344
$260$	2.07316	0.128572
$261$	7.38369	0.457039
$262$	14.9451	0.923312
$263$	22.5027	1.38758	0.693789	−	0.720178i	$-0.255940\pi$
$263$	22.5027	1.38758	0.693789	+	0.720178i	$0.255940\pi$
$264$	−5.68444	−0.349853
$265$	23.8746	1.46660
$266$	−0.816698	−0.0500749
$267$	9.33143	0.571074
$268$	−11.1954	−0.683865
$269$	−9.26209	−0.564719	−0.282360	−	0.959309i	$-0.591117\pi$
$269$	−9.26209	−0.564719	−0.282360	+	0.959309i	$0.591117\pi$
$270$	−2.07316	−0.126168
$271$	−28.4705	−1.72946	−0.864729	−	0.502239i	$-0.832510\pi$
$271$	−28.4705	−1.72946	−0.864729	+	0.502239i	$0.832510\pi$
$272$	4.26526	0.258619
$273$	0.689093	0.0417058
$274$	−13.2838	−0.802503
$275$	3.99057	0.240641
$276$	−6.71682	−0.404305
$277$	−27.7241	−1.66578	−0.832890	−	0.553439i	$-0.813315\pi$
$277$	−27.7241	−1.66578	−0.832890	+	0.553439i	$0.813315\pi$
$278$	−11.0521	−0.662858
$279$	−0.861742	−0.0515911
$280$	1.42860	0.0853750
$281$	−14.8944	−0.888525	−0.444262	−	0.895897i	$-0.646534\pi$
$281$	−14.8944	−0.888525	−0.444262	+	0.895897i	$0.646534\pi$
$282$	13.4396	0.800318
$283$	−13.0060	−0.773126	−0.386563	−	0.922263i	$-0.626338\pi$
$283$	−13.0060	−0.773126	−0.386563	+	0.922263i	$0.626338\pi$
$284$	−8.51079	−0.505023
$285$	−2.45706	−0.145544
$286$	5.68444	0.336128
$287$	−3.06857	−0.181132
$288$	1.00000	0.0589256
$289$	1.19241	0.0701418
$290$	−15.3076	−0.898891
$291$	0.658392	0.0385956
$292$	15.2639	0.893252
$293$	12.4176	0.725443	0.362721	−	0.931898i	$-0.381848\pi$
$293$	12.4176	0.725443	0.362721	+	0.931898i	$0.381848\pi$
$294$	−6.52515	−0.380555
$295$	−25.6507	−1.49344
$296$	7.06113	0.410420
$297$	−5.68444	−0.329844
$298$	16.1808	0.937329
$299$	6.71682	0.388444
$300$	−0.702017	−0.0405310
$301$	−4.50832	−0.259855
$302$	−7.86045	−0.452318
$303$	6.82535	0.392106
$304$	1.18518	0.0679746
$305$	24.5812	1.40752
$306$	4.26526	0.243828
$307$	2.81469	0.160643	0.0803213	−	0.996769i	$-0.474405\pi$
$307$	2.81469	0.160643	0.0803213	+	0.996769i	$0.474405\pi$
$308$	3.91710	0.223198
$309$	1.00000	0.0568880
$310$	1.78653	0.101468
$311$	2.78907	0.158154	0.0790768	−	0.996869i	$-0.474803\pi$
$311$	2.78907	0.158154	0.0790768	+	0.996869i	$0.474803\pi$
$312$	−1.00000	−0.0566139
$313$	26.0513	1.47251	0.736254	−	0.676705i	$-0.236593\pi$
$313$	26.0513	1.47251	0.736254	+	0.676705i	$0.236593\pi$
$314$	3.44324	0.194313
$315$	1.42860	0.0804924
$316$	3.53462	0.198838
$317$	22.8382	1.28272	0.641362	−	0.767239i	$-0.278370\pi$
$317$	22.8382	1.28272	0.641362	+	0.767239i	$0.278370\pi$
$318$	−11.5160	−0.645787
$319$	−41.9721	−2.34999
$320$	−2.07316	−0.115893
$321$	2.46030	0.137321
$322$	4.62851	0.257937
$323$	5.05509	0.281273
$324$	1.00000	0.0555556
$325$	0.702017	0.0389409
$326$	−17.5028	−0.969391
$327$	11.7260	0.648447
$328$	4.45306	0.245879
$329$	−9.26115	−0.510584
$330$	11.7847	0.648728
$331$	15.3109	0.841565	0.420782	−	0.907162i	$-0.361756\pi$
$331$	15.3109	0.841565	0.420782	+	0.907162i	$0.361756\pi$
$332$	17.6723	0.969895
$333$	7.06113	0.386948
$334$	−5.68878	−0.311276
$335$	23.2097	1.26808
$336$	−0.689093	−0.0375931
$337$	−13.9339	−0.759028	−0.379514	−	0.925186i	$-0.623909\pi$
$337$	−13.9339	−0.759028	−0.379514	+	0.925186i	$0.623909\pi$
$338$	1.00000	0.0543928
$339$	6.59652	0.358274
$340$	−8.84255	−0.479555
$341$	4.89852	0.265270
$342$	1.18518	0.0640871
$343$	9.32008	0.503237
$344$	6.54240	0.352743
$345$	13.9250	0.749698
$346$	12.5224	0.673207
$347$	−3.49399	−0.187567	−0.0937835	−	0.995593i	$-0.529896\pi$
$347$	−3.49399	−0.187567	−0.0937835	+	0.995593i	$0.529896\pi$
$348$	7.38369	0.395807
$349$	7.89884	0.422815	0.211408	−	0.977398i	$-0.432195\pi$
$349$	7.89884	0.422815	0.211408	+	0.977398i	$0.432195\pi$
$350$	0.483755	0.0258578
$351$	−1.00000	−0.0533761
$352$	−5.68444	−0.302981
$353$	18.7688	0.998961	0.499481	−	0.866325i	$-0.333524\pi$
$353$	18.7688	0.998961	0.499481	+	0.866325i	$0.333524\pi$
$354$	12.3728	0.657605
$355$	17.6442	0.936458
$356$	9.33143	0.494565
$357$	−2.93916	−0.155557
$358$	20.7165	1.09490
$359$	−13.1161	−0.692240	−0.346120	−	0.938190i	$-0.612501\pi$
$359$	−13.1161	−0.692240	−0.346120	+	0.938190i	$0.612501\pi$
$360$	−2.07316	−0.109265
$361$	−17.5954	−0.926071
$362$	15.3155	0.804964
$363$	21.3128	1.11863
$364$	0.689093	0.0361183
$365$	−31.6445	−1.65635
$366$	−11.8569	−0.619771
$367$	−10.9930	−0.573829	−0.286914	−	0.957956i	$-0.592630\pi$
$367$	−10.9930	−0.573829	−0.286914	+	0.957956i	$0.592630\pi$
$368$	−6.71682	−0.350138
$369$	4.45306	0.231817
$370$	−14.6388	−0.761037
$371$	7.93561	0.411997
$372$	−0.861742	−0.0446792
$373$	−23.7264	−1.22851	−0.614254	−	0.789108i	$-0.710543\pi$
$373$	−23.7264	−1.22851	−0.614254	+	0.789108i	$0.710543\pi$
$374$	−24.2456	−1.25371
$375$	11.8212	0.610443
$376$	13.4396	0.693096
$377$	−7.38369	−0.380279
$378$	−0.689093	−0.0354431
$379$	10.2569	0.526864	0.263432	−	0.964678i	$-0.415146\pi$
$379$	10.2569	0.526864	0.263432	+	0.964678i	$0.415146\pi$
$380$	−2.45706	−0.126045
$381$	−10.3423	−0.529853
$382$	−9.93969	−0.508559
$383$	−11.4208	−0.583578	−0.291789	−	0.956483i	$-0.594250\pi$
$383$	−11.4208	−0.583578	−0.291789	+	0.956483i	$0.594250\pi$
$384$	1.00000	0.0510310
$385$	−8.12077	−0.413873
$386$	8.98429	0.457288
$387$	6.54240	0.332569
$388$	0.658392	0.0334248
$389$	−8.69549	−0.440879	−0.220440	−	0.975401i	$-0.570749\pi$
$389$	−8.69549	−0.440879	−0.220440	+	0.975401i	$0.570749\pi$
$390$	2.07316	0.104978
$391$	−28.6490	−1.44884
$392$	−6.52515	−0.329570
$393$	14.9451	0.753881
$394$	−8.10619	−0.408384
$395$	−7.32782	−0.368702
$396$	−5.68444	−0.285654
$397$	29.3393	1.47250	0.736250	−	0.676710i	$-0.236595\pi$
$397$	29.3393	1.47250	0.736250	+	0.676710i	$0.236595\pi$
$398$	25.6181	1.28412
$399$	−0.816698	−0.0408860
$400$	−0.702017	−0.0351009
$401$	29.1653	1.45645	0.728223	−	0.685340i	$-0.240346\pi$
$401$	29.1653	1.45645	0.728223	+	0.685340i	$0.240346\pi$
$402$	−11.1954	−0.558374
$403$	0.861742	0.0429264
$404$	6.82535	0.339574
$405$	−2.07316	−0.103016
$406$	−5.08805	−0.252516
$407$	−40.1385	−1.98959
$408$	4.26526	0.211162
$409$	−38.6595	−1.91159	−0.955794	−	0.294036i	$-0.905001\pi$
$409$	−38.6595	−1.91159	−0.955794	+	0.294036i	$0.905001\pi$
$410$	−9.23190	−0.455931
$411$	−13.2838	−0.655241
$412$	1.00000	0.0492665
$413$	−8.52598	−0.419536
$414$	−6.71682	−0.330114
$415$	−36.6375	−1.79846
$416$	−1.00000	−0.0490290
$417$	−11.0521	−0.541222
$418$	−6.73707	−0.329521
$419$	−19.3560	−0.945602	−0.472801	−	0.881169i	$-0.656757\pi$
$419$	−19.3560	−0.945602	−0.472801	+	0.881169i	$0.656757\pi$
$420$	1.42860	0.0697084
$421$	−1.82366	−0.0888799	−0.0444400	−	0.999012i	$-0.514150\pi$
$421$	−1.82366	−0.0888799	−0.0444400	+	0.999012i	$0.514150\pi$
$422$	−18.2873	−0.890212
$423$	13.4396	0.653457
$424$	−11.5160	−0.559268
$425$	−2.99428	−0.145244
$426$	−8.51079	−0.412349
$427$	8.17051	0.395399
$428$	2.46030	0.118923
$429$	5.68444	0.274447
$430$	−13.5634	−0.654087
$431$	33.2779	1.60294	0.801471	−	0.598034i	$-0.204051\pi$
$431$	33.2779	1.60294	0.801471	+	0.598034i	$0.204051\pi$
$432$	1.00000	0.0481125
$433$	6.91887	0.332500	0.166250	−	0.986084i	$-0.446834\pi$
$433$	6.91887	0.332500	0.166250	+	0.986084i	$0.446834\pi$
$434$	0.593820	0.0285043
$435$	−15.3076	−0.733941
$436$	11.7260	0.561572
$437$	−7.96063	−0.380808
$438$	15.2639	0.729337
$439$	40.7948	1.94703	0.973515	−	0.228625i	$-0.0734230\pi$
$439$	40.7948	1.94703	0.973515	+	0.228625i	$0.0734230\pi$
$440$	11.7847	0.561815
$441$	−6.52515	−0.310721
$442$	−4.26526	−0.202878
$443$	−9.06802	−0.430834	−0.215417	−	0.976522i	$-0.569111\pi$
$443$	−9.06802	−0.430834	−0.215417	+	0.976522i	$0.569111\pi$
$444$	7.06113	0.335106
$445$	−19.3455	−0.917066
$446$	14.2669	0.675558
$447$	16.1808	0.765326
$448$	−0.689093	−0.0325566
$449$	−17.2449	−0.813836	−0.406918	−	0.913465i	$-0.633396\pi$
$449$	−17.2449	−0.813836	−0.406918	+	0.913465i	$0.633396\pi$
$450$	−0.702017	−0.0330934
$451$	−25.3131	−1.19195
$452$	6.59652	0.310274
$453$	−7.86045	−0.369316
$454$	12.5881	0.590788
$455$	−1.42860	−0.0669737
$456$	1.18518	0.0555011
$457$	6.59265	0.308391	0.154196	−	0.988040i	$-0.450721\pi$
$457$	6.59265	0.308391	0.154196	+	0.988040i	$0.450721\pi$
$458$	−14.9209	−0.697208
$459$	4.26526	0.199085
$460$	13.9250	0.649258
$461$	−17.4764	−0.813959	−0.406980	−	0.913437i	$-0.633418\pi$
$461$	−17.4764	−0.813959	−0.406980	+	0.913437i	$0.633418\pi$
$462$	3.91710	0.182240
$463$	39.3751	1.82992	0.914959	−	0.403547i	$-0.132223\pi$
$463$	39.3751	1.82992	0.914959	+	0.403547i	$0.132223\pi$
$464$	7.38369	0.342779
$465$	1.78653	0.0828482
$466$	26.7892	1.24099
$467$	13.6099	0.629790	0.314895	−	0.949126i	$-0.398031\pi$
$467$	13.6099	0.629790	0.314895	+	0.949126i	$0.398031\pi$
$468$	−1.00000	−0.0462250
$469$	7.71464	0.356229
$470$	−27.8625	−1.28520
$471$	3.44324	0.158656
$472$	12.3728	0.569503
$473$	−37.1898	−1.70999
$474$	3.53462	0.162350
$475$	−0.832015	−0.0381755
$476$	−2.93916	−0.134716
$477$	−11.5160	−0.527283
$478$	−18.1853	−0.831774
$479$	−38.0883	−1.74030	−0.870150	−	0.492788i	$-0.835978\pi$
$479$	−38.0883	−1.74030	−0.870150	+	0.492788i	$0.835978\pi$
$480$	−2.07316	−0.0946263
$481$	−7.06113	−0.321960
$482$	20.8017	0.947490
$483$	4.62851	0.210605
$484$	21.3128	0.968764
$485$	−1.36495	−0.0619792
$486$	1.00000	0.0453609
$487$	−11.5090	−0.521525	−0.260762	−	0.965403i	$-0.583974\pi$
$487$	−11.5090	−0.521525	−0.260762	+	0.965403i	$0.583974\pi$
$488$	−11.8569	−0.536737
$489$	−17.5028	−0.791505
$490$	13.5277	0.611118
$491$	38.8918	1.75516	0.877581	−	0.479428i	$-0.159156\pi$
$491$	38.8918	1.75516	0.877581	+	0.479428i	$0.159156\pi$
$492$	4.45306	0.200759
$493$	31.4933	1.41839
$494$	−1.18518	−0.0533237
$495$	11.7847	0.529684
$496$	−0.861742	−0.0386934
$497$	5.86472	0.263069
$498$	17.6723	0.791916
$499$	27.5230	1.23210	0.616048	−	0.787708i	$-0.288733\pi$
$499$	27.5230	1.23210	0.616048	+	0.787708i	$0.288733\pi$
$500$	11.8212	0.528659
$501$	−5.68878	−0.254156
$502$	−12.6097	−0.562797
$503$	−37.2434	−1.66060	−0.830299	−	0.557317i	$-0.811831\pi$
$503$	−37.2434	−1.66060	−0.830299	+	0.557317i	$0.811831\pi$
$504$	−0.689093	−0.0306946
$505$	−14.1500	−0.629668
$506$	38.1813	1.69737
$507$	1.00000	0.0444116
$508$	−10.3423	−0.458866
$509$	6.64666	0.294608	0.147304	−	0.989091i	$-0.452940\pi$
$509$	6.64666	0.294608	0.147304	+	0.989091i	$0.452940\pi$
$510$	−8.84255	−0.391555
$511$	−10.5182	−0.465299
$512$	1.00000	0.0441942
$513$	1.18518	0.0523269
$514$	−10.2870	−0.453740
$515$	−2.07316	−0.0913542
$516$	6.54240	0.288013
$517$	−76.3967	−3.35992
$518$	−4.86577	−0.213790
$519$	12.5224	0.549671
$520$	2.07316	0.0909140
$521$	−3.40087	−0.148995	−0.0744975	−	0.997221i	$-0.523735\pi$
$521$	−3.40087	−0.148995	−0.0744975	+	0.997221i	$0.523735\pi$
$522$	7.38369	0.323175
$523$	−12.2756	−0.536774	−0.268387	−	0.963311i	$-0.586491\pi$
$523$	−12.2756	−0.536774	−0.268387	+	0.963311i	$0.586491\pi$
$524$	14.9451	0.652880
$525$	0.483755	0.0211128
$526$	22.5027	0.981166
$527$	−3.67555	−0.160109
$528$	−5.68444	−0.247383
$529$	22.1157	0.961551
$530$	23.8746	1.03704
$531$	12.3728	0.536932
$532$	−0.816698	−0.0354083
$533$	−4.45306	−0.192883
$534$	9.33143	0.403811
$535$	−5.10059	−0.220518
$536$	−11.1954	−0.483566
$537$	20.7165	0.893985
$538$	−9.26209	−0.399317
$539$	37.0918	1.59766
$540$	−2.07316	−0.0892145
$541$	8.88394	0.381950	0.190975	−	0.981595i	$-0.438835\pi$
$541$	8.88394	0.381950	0.190975	+	0.981595i	$0.438835\pi$
$542$	−28.4705	−1.22291
$543$	15.3155	0.657250
$544$	4.26526	0.182871
$545$	−24.3098	−1.04132
$546$	0.689093	0.0294904
$547$	−33.8910	−1.44907	−0.724537	−	0.689236i	$-0.757946\pi$
$547$	−33.8910	−1.44907	−0.724537	+	0.689236i	$0.757946\pi$
$548$	−13.2838	−0.567455
$549$	−11.8569	−0.506041
$550$	3.99057	0.170159
$551$	8.75099	0.372805
$552$	−6.71682	−0.285887
$553$	−2.43568	−0.103576
$554$	−27.7241	−1.17788
$555$	−14.6388	−0.621384
$556$	−11.0521	−0.468712
$557$	20.1782	0.854976	0.427488	−	0.904021i	$-0.359399\pi$
$557$	20.1782	0.854976	0.427488	+	0.904021i	$0.359399\pi$
$558$	−0.861742	−0.0364804
$559$	−6.54240	−0.276714
$560$	1.42860	0.0603693
$561$	−24.2456	−1.02365
$562$	−14.8944	−0.628282
$563$	−10.7090	−0.451330	−0.225665	−	0.974205i	$-0.572456\pi$
$563$	−10.7090	−0.451330	−0.225665	+	0.974205i	$0.572456\pi$
$564$	13.4396	0.565910
$565$	−13.6756	−0.575338
$566$	−13.0060	−0.546683
$567$	−0.689093	−0.0289392
$568$	−8.51079	−0.357105
$569$	−11.3295	−0.474959	−0.237480	−	0.971393i	$-0.576321\pi$
$569$	−11.3295	−0.474959	−0.237480	+	0.971393i	$0.576321\pi$
$570$	−2.45706	−0.102915
$571$	−17.5000	−0.732354	−0.366177	−	0.930545i	$-0.619333\pi$
$571$	−17.5000	−0.732354	−0.366177	+	0.930545i	$0.619333\pi$
$572$	5.68444	0.237678
$573$	−9.93969	−0.415237
$574$	−3.06857	−0.128080
$575$	4.71532	0.196643
$576$	1.00000	0.0416667
$577$	−22.5401	−0.938355	−0.469177	−	0.883104i	$-0.655449\pi$
$577$	−22.5401	−0.938355	−0.469177	+	0.883104i	$0.655449\pi$
$578$	1.19241	0.0495977
$579$	8.98429	0.373374
$580$	−15.3076	−0.635612
$581$	−12.1779	−0.505223
$582$	0.658392	0.0272912
$583$	65.4622	2.71117
$584$	15.2639	0.631624
$585$	2.07316	0.0857145
$586$	12.4176	0.512965
$587$	31.5369	1.30167	0.650834	−	0.759220i	$-0.274419\pi$
$587$	31.5369	1.30167	0.650834	+	0.759220i	$0.274419\pi$
$588$	−6.52515	−0.269093
$589$	−1.02132	−0.0420827
$590$	−25.6507	−1.05602
$591$	−8.10619	−0.333444
$592$	7.06113	0.290211
$593$	−3.89375	−0.159897	−0.0799487	−	0.996799i	$-0.525476\pi$
$593$	−3.89375	−0.159897	−0.0799487	+	0.996799i	$0.525476\pi$
$594$	−5.68444	−0.233235
$595$	6.09334	0.249802
$596$	16.1808	0.662791
$597$	25.6181	1.04848
$598$	6.71682	0.274671
$599$	19.5441	0.798549	0.399275	−	0.916831i	$-0.369262\pi$
$599$	19.5441	0.798549	0.399275	+	0.916831i	$0.369262\pi$
$600$	−0.702017	−0.0286597
$601$	−11.2559	−0.459137	−0.229569	−	0.973292i	$-0.573732\pi$
$601$	−11.2559	−0.459137	−0.229569	+	0.973292i	$0.573732\pi$
$602$	−4.50832	−0.183745
$603$	−11.1954	−0.455910
$604$	−7.86045	−0.319837
$605$	−44.1848	−1.79637
$606$	6.82535	0.277261
$607$	−20.9102	−0.848719	−0.424359	−	0.905494i	$-0.639501\pi$
$607$	−20.9102	−0.848719	−0.424359	+	0.905494i	$0.639501\pi$
$608$	1.18518	0.0480653
$609$	−5.08805	−0.206178
$610$	24.5812	0.995266
$611$	−13.4396	−0.543709
$612$	4.26526	0.172413
$613$	20.6897	0.835648	0.417824	−	0.908528i	$-0.362793\pi$
$613$	20.6897	0.835648	0.417824	+	0.908528i	$0.362793\pi$
$614$	2.81469	0.113591
$615$	−9.23190	−0.372266
$616$	3.91710	0.157825
$617$	3.13495	0.126208	0.0631042	−	0.998007i	$-0.479900\pi$
$617$	3.13495	0.126208	0.0631042	+	0.998007i	$0.479900\pi$
$618$	1.00000	0.0402259
$619$	0.820917	0.0329955	0.0164977	−	0.999864i	$-0.494748\pi$
$619$	0.820917	0.0329955	0.0164977	+	0.999864i	$0.494748\pi$
$620$	1.78653	0.0717487
$621$	−6.71682	−0.269537
$622$	2.78907	0.111831
$623$	−6.43022	−0.257621
$624$	−1.00000	−0.0400320
$625$	−20.9971	−0.839883
$626$	26.0513	1.04122
$627$	−6.73707	−0.269053
$628$	3.44324	0.137400
$629$	30.1175	1.20086
$630$	1.42860	0.0569167
$631$	−9.37858	−0.373355	−0.186678	−	0.982421i	$-0.559772\pi$
$631$	−9.37858	−0.373355	−0.186678	+	0.982421i	$0.559772\pi$
$632$	3.53462	0.140599
$633$	−18.2873	−0.726855
$634$	22.8382	0.907022
$635$	21.4412	0.850870
$636$	−11.5160	−0.456640
$637$	6.52515	0.258536
$638$	−41.9721	−1.66169
$639$	−8.51079	−0.336682
$640$	−2.07316	−0.0819488
$641$	−10.1375	−0.400408	−0.200204	−	0.979754i	$-0.564160\pi$
$641$	−10.1375	−0.400408	−0.200204	+	0.979754i	$0.564160\pi$
$642$	2.46030	0.0971004
$643$	−25.2869	−0.997220	−0.498610	−	0.866826i	$-0.666156\pi$
$643$	−25.2869	−0.997220	−0.498610	+	0.866826i	$0.666156\pi$
$644$	4.62851	0.182389
$645$	−13.5634	−0.534059
$646$	5.05509	0.198890
$647$	31.9150	1.25471	0.627353	−	0.778735i	$-0.284138\pi$
$647$	31.9150	1.25471	0.627353	+	0.778735i	$0.284138\pi$
$648$	1.00000	0.0392837
$649$	−70.3322	−2.76078
$650$	0.702017	0.0275354
$651$	0.593820	0.0232736
$652$	−17.5028	−0.685463
$653$	23.4266	0.916752	0.458376	−	0.888758i	$-0.348431\pi$
$653$	23.4266	0.916752	0.458376	+	0.888758i	$0.348431\pi$
$654$	11.7260	0.458522
$655$	−30.9836	−1.21063
$656$	4.45306	0.173863
$657$	15.2639	0.595501
$658$	−9.26115	−0.361037
$659$	−9.29248	−0.361983	−0.180992	−	0.983485i	$-0.557931\pi$
$659$	−9.29248	−0.361983	−0.180992	+	0.983485i	$0.557931\pi$
$660$	11.7847	0.458720
$661$	−4.81807	−0.187401	−0.0937006	−	0.995600i	$-0.529870\pi$
$661$	−4.81807	−0.187401	−0.0937006	+	0.995600i	$0.529870\pi$
$662$	15.3109	0.595076
$663$	−4.26526	−0.165649
$664$	17.6723	0.685819
$665$	1.69314	0.0656573
$666$	7.06113	0.273613
$667$	−49.5949	−1.92032
$668$	−5.68878	−0.220106
$669$	14.2669	0.551591
$670$	23.2097	0.896670
$671$	67.3998	2.60194
$672$	−0.689093	−0.0265823
$673$	37.1855	1.43340	0.716699	−	0.697383i	$-0.245652\pi$
$673$	37.1855	1.43340	0.716699	+	0.697383i	$0.245652\pi$
$674$	−13.9339	−0.536714
$675$	−0.702017	−0.0270207
$676$	1.00000	0.0384615
$677$	−22.5424	−0.866375	−0.433188	−	0.901304i	$-0.642611\pi$
$677$	−22.5424	−0.866375	−0.433188	+	0.901304i	$0.642611\pi$
$678$	6.59652	0.253338
$679$	−0.453693	−0.0174111
$680$	−8.84255	−0.339096
$681$	12.5881	0.482376
$682$	4.89852	0.187574
$683$	−15.6051	−0.597112	−0.298556	−	0.954392i	$-0.596505\pi$
$683$	−15.6051	−0.597112	−0.298556	+	0.954392i	$0.596505\pi$
$684$	1.18518	0.0453164
$685$	27.5394	1.05223
$686$	9.32008	0.355842
$687$	−14.9209	−0.569268
$688$	6.54240	0.249427
$689$	11.5160	0.438726
$690$	13.9250	0.530117
$691$	50.8893	1.93592	0.967960	−	0.251104i	$-0.0807936\pi$
$691$	50.8893	1.93592	0.967960	+	0.251104i	$0.0807936\pi$
$692$	12.5224	0.476029
$693$	3.91710	0.148798
$694$	−3.49399	−0.132630
$695$	22.9127	0.869127
$696$	7.38369	0.279878
$697$	18.9934	0.719428
$698$	7.89884	0.298975
$699$	26.7892	1.01326
$700$	0.483755	0.0182842
$701$	−28.0683	−1.06013	−0.530063	−	0.847958i	$-0.677832\pi$
$701$	−28.0683	−1.06013	−0.530063	+	0.847958i	$0.677832\pi$
$702$	−1.00000	−0.0377426
$703$	8.36870	0.315631
$704$	−5.68444	−0.214240
$705$	−27.8625	−1.04936
$706$	18.7688	0.706372
$707$	−4.70330	−0.176886
$708$	12.3728	0.464997
$709$	28.9558	1.08746	0.543729	−	0.839261i	$-0.317012\pi$
$709$	28.9558	1.08746	0.543729	+	0.839261i	$0.317012\pi$
$710$	17.6442	0.662175
$711$	3.53462	0.132558
$712$	9.33143	0.349710
$713$	5.78817	0.216769
$714$	−2.93916	−0.109995
$715$	−11.7847	−0.440724
$716$	20.7165	0.774214
$717$	−18.1853	−0.679141
$718$	−13.1161	−0.489488
$719$	23.1870	0.864730	0.432365	−	0.901699i	$-0.357679\pi$
$719$	23.1870	0.864730	0.432365	+	0.901699i	$0.357679\pi$
$720$	−2.07316	−0.0772620
$721$	−0.689093	−0.0256632
$722$	−17.5954	−0.654831
$723$	20.8017	0.773622
$724$	15.3155	0.569195
$725$	−5.18348	−0.192510
$726$	21.3128	0.790992
$727$	8.31723	0.308469	0.154235	−	0.988034i	$-0.450709\pi$
$727$	8.31723	0.308469	0.154235	+	0.988034i	$0.450709\pi$
$728$	0.689093	0.0255395
$729$	1.00000	0.0370370
$730$	−31.6445	−1.17121
$731$	27.9050	1.03210
$732$	−11.8569	−0.438244
$733$	6.75335	0.249441	0.124720	−	0.992192i	$-0.460197\pi$
$733$	6.75335	0.249441	0.124720	+	0.992192i	$0.460197\pi$
$734$	−10.9930	−0.405758
$735$	13.5277	0.498976
$736$	−6.71682	−0.247585
$737$	63.6393	2.34418
$738$	4.45306	0.163919
$739$	8.26466	0.304020	0.152010	−	0.988379i	$-0.451425\pi$
$739$	8.26466	0.304020	0.152010	+	0.988379i	$0.451425\pi$
$740$	−14.6388	−0.538134
$741$	−1.18518	−0.0435386
$742$	7.93561	0.291326
$743$	−44.9745	−1.64995	−0.824977	−	0.565166i	$-0.808812\pi$
$743$	−44.9745	−1.64995	−0.824977	+	0.565166i	$0.808812\pi$
$744$	−0.861742	−0.0315930
$745$	−33.5454	−1.22901
$746$	−23.7264	−0.868687
$747$	17.6723	0.646596
$748$	−24.2456	−0.886506
$749$	−1.69538	−0.0619477
$750$	11.8212	0.431648
$751$	38.1761	1.39306	0.696532	−	0.717526i	$-0.254725\pi$
$751$	38.1761	1.39306	0.696532	+	0.717526i	$0.254725\pi$
$752$	13.4396	0.490093
$753$	−12.6097	−0.459522
$754$	−7.38369	−0.268898
$755$	16.2960	0.593070
$756$	−0.689093	−0.0250621
$757$	−45.6248	−1.65826	−0.829131	−	0.559055i	$-0.811164\pi$
$757$	−45.6248	−1.65826	−0.829131	+	0.559055i	$0.811164\pi$
$758$	10.2569	0.372549
$759$	38.1813	1.38589
$760$	−2.45706	−0.0891270
$761$	7.05451	0.255726	0.127863	−	0.991792i	$-0.459188\pi$
$761$	7.05451	0.255726	0.127863	+	0.991792i	$0.459188\pi$
$762$	−10.3423	−0.374662
$763$	−8.08028	−0.292526
$764$	−9.93969	−0.359605
$765$	−8.84255	−0.319703
$766$	−11.4208	−0.412652
$767$	−12.3728	−0.446755
$768$	1.00000	0.0360844
$769$	29.4584	1.06230	0.531149	−	0.847278i	$-0.321760\pi$
$769$	29.4584	1.06230	0.531149	+	0.847278i	$0.321760\pi$
$770$	−8.12077	−0.292652
$771$	−10.2870	−0.370477
$772$	8.98429	0.323352
$773$	6.80735	0.244843	0.122422	−	0.992478i	$-0.460934\pi$
$773$	6.80735	0.244843	0.122422	+	0.992478i	$0.460934\pi$
$774$	6.54240	0.235162
$775$	0.604958	0.0217307
$776$	0.658392	0.0236349
$777$	−4.86577	−0.174559
$778$	−8.69549	−0.311749
$779$	5.27767	0.189092
$780$	2.07316	0.0742310
$781$	48.3790	1.73114
$782$	−28.6490	−1.02448
$783$	7.38369	0.263872
$784$	−6.52515	−0.233041
$785$	−7.13837	−0.254779
$786$	14.9451	0.533075
$787$	−11.9853	−0.427228	−0.213614	−	0.976918i	$-0.568524\pi$
$787$	−11.9853	−0.427228	−0.213614	+	0.976918i	$0.568524\pi$
$788$	−8.10619	−0.288771
$789$	22.5027	0.801119
$790$	−7.32782	−0.260712
$791$	−4.54561	−0.161623
$792$	−5.68444	−0.201988
$793$	11.8569	0.421051
$794$	29.3393	1.04121
$795$	23.8746	0.846743
$796$	25.6181	0.908011
$797$	−11.0110	−0.390031	−0.195015	−	0.980800i	$-0.562476\pi$
$797$	−11.0110	−0.390031	−0.195015	+	0.980800i	$0.562476\pi$
$798$	−0.816698	−0.0289108
$799$	57.3234	2.02796
$800$	−0.702017	−0.0248201
$801$	9.33143	0.329710
$802$	29.1653	1.02986
$803$	−86.7666	−3.06193
$804$	−11.1954	−0.394830
$805$	−9.59563	−0.338202
$806$	0.861742	0.0303536
$807$	−9.26209	−0.326041
$808$	6.82535	0.240115
$809$	13.7580	0.483704	0.241852	−	0.970313i	$-0.422245\pi$
$809$	13.7580	0.483704	0.241852	+	0.970313i	$0.422245\pi$
$810$	−2.07316	−0.0728433
$811$	−25.5093	−0.895753	−0.447877	−	0.894095i	$-0.647820\pi$
$811$	−25.5093	−0.895753	−0.447877	+	0.894095i	$0.647820\pi$
$812$	−5.08805	−0.178555
$813$	−28.4705	−0.998503
$814$	−40.1385	−1.40686
$815$	36.2861	1.27105
$816$	4.26526	0.149314
$817$	7.75391	0.271275
$818$	−38.6595	−1.35170
$819$	0.689093	0.0240788
$820$	−9.23190	−0.322392
$821$	−21.1540	−0.738279	−0.369139	−	0.929374i	$-0.620347\pi$
$821$	−21.1540	−0.738279	−0.369139	+	0.929374i	$0.620347\pi$
$822$	−13.2838	−0.463325
$823$	3.00048	0.104590	0.0522951	−	0.998632i	$-0.483346\pi$
$823$	3.00048	0.104590	0.0522951	+	0.998632i	$0.483346\pi$
$824$	1.00000	0.0348367
$825$	3.99057	0.138934
$826$	−8.52598	−0.296657
$827$	23.5895	0.820288	0.410144	−	0.912021i	$-0.365478\pi$
$827$	23.5895	0.820288	0.410144	+	0.912021i	$0.365478\pi$
$828$	−6.71682	−0.233426
$829$	10.8688	0.377490	0.188745	−	0.982026i	$-0.439558\pi$
$829$	10.8688	0.377490	0.188745	+	0.982026i	$0.439558\pi$
$830$	−36.6375	−1.27171
$831$	−27.7241	−0.961738
$832$	−1.00000	−0.0346688
$833$	−27.8314	−0.964302
$834$	−11.0521	−0.382701
$835$	11.7937	0.408139
$836$	−6.73707	−0.233006
$837$	−0.861742	−0.0297862
$838$	−19.3560	−0.668642
$839$	5.47786	0.189117	0.0945584	−	0.995519i	$-0.469856\pi$
$839$	5.47786	0.189117	0.0945584	+	0.995519i	$0.469856\pi$
$840$	1.42860	0.0492913
$841$	25.5189	0.879962
$842$	−1.82366	−0.0628476
$843$	−14.8944	−0.512990
$844$	−18.2873	−0.629475
$845$	−2.07316	−0.0713188
$846$	13.4396	0.462064
$847$	−14.6865	−0.504634
$848$	−11.5160	−0.395462
$849$	−13.0060	−0.446365
$850$	−2.99428	−0.102703
$851$	−47.4283	−1.62582
$852$	−8.51079	−0.291575
$853$	−19.5763	−0.670279	−0.335140	−	0.942169i	$-0.608783\pi$
$853$	−19.5763	−0.670279	−0.335140	+	0.942169i	$0.608783\pi$
$854$	8.17051	0.279589
$855$	−2.45706	−0.0840297
$856$	2.46030	0.0840914
$857$	14.7550	0.504022	0.252011	−	0.967724i	$-0.418908\pi$
$857$	14.7550	0.504022	0.252011	+	0.967724i	$0.418908\pi$
$858$	5.68444	0.194063
$859$	−20.7850	−0.709176	−0.354588	−	0.935023i	$-0.615379\pi$
$859$	−20.7850	−0.709176	−0.354588	+	0.935023i	$0.615379\pi$
$860$	−13.5634	−0.462509
$861$	−3.06857	−0.104577
$862$	33.2779	1.13345
$863$	−23.3414	−0.794550	−0.397275	−	0.917700i	$-0.630044\pi$
$863$	−23.3414	−0.794550	−0.397275	+	0.917700i	$0.630044\pi$
$864$	1.00000	0.0340207
$865$	−25.9608	−0.882695
$866$	6.91887	0.235113
$867$	1.19241	0.0404964
$868$	0.593820	0.0201556
$869$	−20.0923	−0.681584
$870$	−15.3076	−0.518975
$871$	11.1954	0.379340
$872$	11.7260	0.397091
$873$	0.658392	0.0222832
$874$	−7.96063	−0.269272
$875$	−8.14589	−0.275381
$876$	15.2639	0.515719
$877$	17.8422	0.602487	0.301243	−	0.953547i	$-0.402598\pi$
$877$	17.8422	0.602487	0.301243	+	0.953547i	$0.402598\pi$
$878$	40.7948	1.37676
$879$	12.4176	0.418835
$880$	11.7847	0.397263
$881$	−41.1070	−1.38493	−0.692465	−	0.721451i	$-0.743475\pi$
$881$	−41.1070	−1.38493	−0.692465	+	0.721451i	$0.743475\pi$
$882$	−6.52515	−0.219713
$883$	5.14880	0.173271	0.0866355	−	0.996240i	$-0.472388\pi$
$883$	5.14880	0.173271	0.0866355	+	0.996240i	$0.472388\pi$
$884$	−4.26526	−0.143456
$885$	−25.6507	−0.862239
$886$	−9.06802	−0.304646
$887$	45.3150	1.52153	0.760764	−	0.649028i	$-0.224824\pi$
$887$	45.3150	1.52153	0.760764	+	0.649028i	$0.224824\pi$
$888$	7.06113	0.236956
$889$	7.12681	0.239026
$890$	−19.3455	−0.648464
$891$	−5.68444	−0.190436
$892$	14.2669	0.477691
$893$	15.9283	0.533022
$894$	16.1808	0.541167
$895$	−42.9487	−1.43562
$896$	−0.689093	−0.0230210
$897$	6.71682	0.224268
$898$	−17.2449	−0.575469
$899$	−6.36284	−0.212212
$900$	−0.702017	−0.0234006
$901$	−49.1188	−1.63639
$902$	−25.3131	−0.842835
$903$	−4.50832	−0.150027
$904$	6.59652	0.219397
$905$	−31.7514	−1.05545
$906$	−7.86045	−0.261146
$907$	36.5079	1.21222	0.606112	−	0.795379i	$-0.292728\pi$
$907$	36.5079	1.21222	0.606112	+	0.795379i	$0.292728\pi$
$908$	12.5881	0.417750
$909$	6.82535	0.226382
$910$	−1.42860	−0.0473576
$911$	−29.8396	−0.988629	−0.494315	−	0.869283i	$-0.664581\pi$
$911$	−29.8396	−0.988629	−0.494315	+	0.869283i	$0.664581\pi$
$912$	1.18518	0.0392452
$913$	−100.457	−3.32465
$914$	6.59265	0.218065
$915$	24.5812	0.812631
$916$	−14.9209	−0.493000
$917$	−10.2986	−0.340089
$918$	4.26526	0.140774
$919$	−53.9824	−1.78072	−0.890358	−	0.455262i	$-0.849546\pi$
$919$	−53.9824	−1.78072	−0.890358	+	0.455262i	$0.849546\pi$
$920$	13.9250	0.459095
$921$	2.81469	0.0927471
$922$	−17.4764	−0.575556
$923$	8.51079	0.280136
$924$	3.91710	0.128863
$925$	−4.95704	−0.162986
$926$	39.3751	1.29395
$927$	1.00000	0.0328443
$928$	7.38369	0.242382
$929$	−5.66487	−0.185858	−0.0929292	−	0.995673i	$-0.529623\pi$
$929$	−5.66487	−0.185858	−0.0929292	+	0.995673i	$0.529623\pi$
$930$	1.78653	0.0585825
$931$	−7.73347	−0.253454
$932$	26.7892	0.877511
$933$	2.78907	0.0913100
$934$	13.6099	0.445329
$935$	50.2649	1.64384
$936$	−1.00000	−0.0326860
$937$	−19.8912	−0.649816	−0.324908	−	0.945746i	$-0.605333\pi$
$937$	−19.8912	−0.649816	−0.324908	+	0.945746i	$0.605333\pi$
$938$	7.71464	0.251892
$939$	26.0513	0.850153
$940$	−27.8625	−0.908773
$941$	−2.83346	−0.0923682	−0.0461841	−	0.998933i	$-0.514706\pi$
$941$	−2.83346	−0.0923682	−0.0461841	+	0.998933i	$0.514706\pi$
$942$	3.44324	0.112187
$943$	−29.9104	−0.974017
$944$	12.3728	0.402699
$945$	1.42860	0.0464723
$946$	−37.1898	−1.20915
$947$	29.9877	0.974470	0.487235	−	0.873271i	$-0.338006\pi$
$947$	29.9877	0.974470	0.487235	+	0.873271i	$0.338006\pi$
$948$	3.53462	0.114799
$949$	−15.2639	−0.495487
$950$	−0.832015	−0.0269941
$951$	22.8382	0.740581
$952$	−2.93916	−0.0952586
$953$	10.5614	0.342117	0.171059	−	0.985261i	$-0.445281\pi$
$953$	10.5614	0.342117	0.171059	+	0.985261i	$0.445281\pi$
$954$	−11.5160	−0.372845
$955$	20.6065	0.666812
$956$	−18.1853	−0.588153
$957$	−41.9721	−1.35677
$958$	−38.0883	−1.23058
$959$	9.15376	0.295590
$960$	−2.07316	−0.0669109
$961$	−30.2574	−0.976045
$962$	−7.06113	−0.227660
$963$	2.46030	0.0792821
$964$	20.8017	0.669977
$965$	−18.6258	−0.599587
$966$	4.62851	0.148920
$967$	6.94037	0.223187	0.111594	−	0.993754i	$-0.464404\pi$
$967$	6.94037	0.223187	0.111594	+	0.993754i	$0.464404\pi$
$968$	21.3128	0.685019
$969$	5.05509	0.162393
$970$	−1.36495	−0.0438259
$971$	−44.9595	−1.44282	−0.721409	−	0.692509i	$-0.756505\pi$
$971$	−44.9595	−1.44282	−0.721409	+	0.692509i	$0.756505\pi$
$972$	1.00000	0.0320750
$973$	7.61589	0.244154
$974$	−11.5090	−0.368774
$975$	0.702017	0.0224825
$976$	−11.8569	−0.379530
$977$	17.8412	0.570792	0.285396	−	0.958410i	$-0.407875\pi$
$977$	17.8412	0.570792	0.285396	+	0.958410i	$0.407875\pi$
$978$	−17.5028	−0.559678
$979$	−53.0439	−1.69529
$980$	13.5277	0.432125
$981$	11.7260	0.374381
$982$	38.8918	1.24109
$983$	−12.1084	−0.386199	−0.193099	−	0.981179i	$-0.561854\pi$
$983$	−12.1084	−0.386199	−0.193099	+	0.981179i	$0.561854\pi$
$984$	4.45306	0.141958
$985$	16.8054	0.535465
$986$	31.4933	1.00295
$987$	−9.26115	−0.294786
$988$	−1.18518	−0.0377055
$989$	−43.9441	−1.39734
$990$	11.7847	0.374543
$991$	−11.0206	−0.350080	−0.175040	−	0.984561i	$-0.556005\pi$
$991$	−11.0206	−0.350080	−0.175040	+	0.984561i	$0.556005\pi$
$992$	−0.861742	−0.0273603
$993$	15.3109	0.485878
$994$	5.86472	0.186018
$995$	−53.1104	−1.68371
$996$	17.6723	0.559969
$997$	−0.0290697	−0.000920647 0	−0.000460324	−	1.00000i	$-0.500147\pi$
$997$	−0.0290697	−0.000920647 0	−0.000460324		1.00000i	$0.500147\pi$
$998$	27.5230	0.871224
$999$	7.06113	0.223404

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8034.2.a.bd.1.5	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8034.2.a.bd.1.5	✓	16	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:	\( N \)	\(=\)	\( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8034.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.07316 q^{5} +1.00000 q^{6} -0.689093 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.07316 q^{5} +1.00000 q^{6} -0.689093 q^{7} +1.00000 q^{8} +1.00000 q^{9} -2.07316 q^{10} -5.68444 q^{11} +1.00000 q^{12} -1.00000 q^{13} -0.689093 q^{14} -2.07316 q^{15} +1.00000 q^{16} +4.26526 q^{17} +1.00000 q^{18} +1.18518 q^{19} -2.07316 q^{20} -0.689093 q^{21} -5.68444 q^{22} -6.71682 q^{23} +1.00000 q^{24} -0.702017 q^{25} -1.00000 q^{26} +1.00000 q^{27} -0.689093 q^{28} +7.38369 q^{29} -2.07316 q^{30} -0.861742 q^{31} +1.00000 q^{32} -5.68444 q^{33} +4.26526 q^{34} +1.42860 q^{35} +1.00000 q^{36} +7.06113 q^{37} +1.18518 q^{38} -1.00000 q^{39} -2.07316 q^{40} +4.45306 q^{41} -0.689093 q^{42} +6.54240 q^{43} -5.68444 q^{44} -2.07316 q^{45} -6.71682 q^{46} +13.4396 q^{47} +1.00000 q^{48} -6.52515 q^{49} -0.702017 q^{50} +4.26526 q^{51} -1.00000 q^{52} -11.5160 q^{53} +1.00000 q^{54} +11.7847 q^{55} -0.689093 q^{56} +1.18518 q^{57} +7.38369 q^{58} +12.3728 q^{59} -2.07316 q^{60} -11.8569 q^{61} -0.861742 q^{62} -0.689093 q^{63} +1.00000 q^{64} +2.07316 q^{65} -5.68444 q^{66} -11.1954 q^{67} +4.26526 q^{68} -6.71682 q^{69} +1.42860 q^{70} -8.51079 q^{71} +1.00000 q^{72} +15.2639 q^{73} +7.06113 q^{74} -0.702017 q^{75} +1.18518 q^{76} +3.91710 q^{77} -1.00000 q^{78} +3.53462 q^{79} -2.07316 q^{80} +1.00000 q^{81} +4.45306 q^{82} +17.6723 q^{83} -0.689093 q^{84} -8.84255 q^{85} +6.54240 q^{86} +7.38369 q^{87} -5.68444 q^{88} +9.33143 q^{89} -2.07316 q^{90} +0.689093 q^{91} -6.71682 q^{92} -0.861742 q^{93} +13.4396 q^{94} -2.45706 q^{95} +1.00000 q^{96} +0.658392 q^{97} -6.52515 q^{98} -5.68444 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{2} + 16 q^{3} + 16 q^{4} + 5 q^{5} + 16 q^{6} + 4 q^{7} + 16 q^{8} + 16 q^{9}+O(q^{10}) \) 16 * q + 16 * q^2 + 16 * q^3 + 16 * q^4 + 5 * q^5 + 16 * q^6 + 4 * q^7 + 16 * q^8 + 16 * q^9	\( 16 q + 16 q^{2} + 16 q^{3} + 16 q^{4} + 5 q^{5} + 16 q^{6} + 4 q^{7} + 16 q^{8} + 16 q^{9} + 5 q^{10} + 18 q^{11} + 16 q^{12} - 16 q^{13} + 4 q^{14} + 5 q^{15} + 16 q^{16} + 17 q^{17} + 16 q^{18} + 8 q^{19} + 5 q^{20} + 4 q^{21} + 18 q^{22} + 9 q^{23} + 16 q^{24} + 17 q^{25} - 16 q^{26} + 16 q^{27} + 4 q^{28} + 14 q^{29} + 5 q^{30} + 12 q^{31} + 16 q^{32} + 18 q^{33} + 17 q^{34} + 16 q^{35} + 16 q^{36} + 31 q^{37} + 8 q^{38} - 16 q^{39} + 5 q^{40} + 29 q^{41} + 4 q^{42} + 30 q^{43} + 18 q^{44} + 5 q^{45} + 9 q^{46} - q^{47} + 16 q^{48} + 36 q^{49} + 17 q^{50} + 17 q^{51} - 16 q^{52} + 12 q^{53} + 16 q^{54} + 30 q^{55} + 4 q^{56} + 8 q^{57} + 14 q^{58} + 38 q^{59} + 5 q^{60} + 12 q^{62} + 4 q^{63} + 16 q^{64} - 5 q^{65} + 18 q^{66} + 28 q^{67} + 17 q^{68} + 9 q^{69} + 16 q^{70} + 32 q^{71} + 16 q^{72} + 20 q^{73} + 31 q^{74} + 17 q^{75} + 8 q^{76} + 26 q^{77} - 16 q^{78} + 13 q^{79} + 5 q^{80} + 16 q^{81} + 29 q^{82} + 39 q^{83} + 4 q^{84} + 31 q^{85} + 30 q^{86} + 14 q^{87} + 18 q^{88} + 9 q^{89} + 5 q^{90} - 4 q^{91} + 9 q^{92} + 12 q^{93} - q^{94} - 20 q^{95} + 16 q^{96} + 35 q^{97} + 36 q^{98} + 18 q^{99}+O(q^{100}) \) 16 * q + 16 * q^2 + 16 * q^3 + 16 * q^4 + 5 * q^5 + 16 * q^6 + 4 * q^7 + 16 * q^8 + 16 * q^9 + 5 * q^10 + 18 * q^11 + 16 * q^12 - 16 * q^13 + 4 * q^14 + 5 * q^15 + 16 * q^16 + 17 * q^17 + 16 * q^18 + 8 * q^19 + 5 * q^20 + 4 * q^21 + 18 * q^22 + 9 * q^23 + 16 * q^24 + 17 * q^25 - 16 * q^26 + 16 * q^27 + 4 * q^28 + 14 * q^29 + 5 * q^30 + 12 * q^31 + 16 * q^32 + 18 * q^33 + 17 * q^34 + 16 * q^35 + 16 * q^36 + 31 * q^37 + 8 * q^38 - 16 * q^39 + 5 * q^40 + 29 * q^41 + 4 * q^42 + 30 * q^43 + 18 * q^44 + 5 * q^45 + 9 * q^46 - q^47 + 16 * q^48 + 36 * q^49 + 17 * q^50 + 17 * q^51 - 16 * q^52 + 12 * q^53 + 16 * q^54 + 30 * q^55 + 4 * q^56 + 8 * q^57 + 14 * q^58 + 38 * q^59 + 5 * q^60 + 12 * q^62 + 4 * q^63 + 16 * q^64 - 5 * q^65 + 18 * q^66 + 28 * q^67 + 17 * q^68 + 9 * q^69 + 16 * q^70 + 32 * q^71 + 16 * q^72 + 20 * q^73 + 31 * q^74 + 17 * q^75 + 8 * q^76 + 26 * q^77 - 16 * q^78 + 13 * q^79 + 5 * q^80 + 16 * q^81 + 29 * q^82 + 39 * q^83 + 4 * q^84 + 31 * q^85 + 30 * q^86 + 14 * q^87 + 18 * q^88 + 9 * q^89 + 5 * q^90 - 4 * q^91 + 9 * q^92 + 12 * q^93 - q^94 - 20 * q^95 + 16 * q^96 + 35 * q^97 + 36 * q^98 + 18 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more