LMFDB - Embedded newform 8034.2.a.r.1.8

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:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 4x^{8} - 9x^{7} + 45x^{6} + 7x^{5} - 123x^{4} + 37x^{3} + 87x^{2} - 54x + 8 $ x^9 - 4x^8 - 9x^7 + 45x^6 + 7x^5 - 123x^4 + 37x^3 + 87x^2 - 54x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-1.44298$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.44298 q^{5} -1.00000 q^{6} +0.367530 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.44298 q^{5} -1.00000 q^{6} +0.367530 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.44298 q^{10} +0.547866 q^{11} -1.00000 q^{12} +1.00000 q^{13} +0.367530 q^{14} -1.44298 q^{15} +1.00000 q^{16} -3.55497 q^{17} +1.00000 q^{18} -4.08520 q^{19} +1.44298 q^{20} -0.367530 q^{21} +0.547866 q^{22} -6.44178 q^{23} -1.00000 q^{24} -2.91782 q^{25} +1.00000 q^{26} -1.00000 q^{27} +0.367530 q^{28} +0.496780 q^{29} -1.44298 q^{30} +1.35014 q^{31} +1.00000 q^{32} -0.547866 q^{33} -3.55497 q^{34} +0.530337 q^{35} +1.00000 q^{36} -1.55757 q^{37} -4.08520 q^{38} -1.00000 q^{39} +1.44298 q^{40} -0.751095 q^{41} -0.367530 q^{42} +10.7331 q^{43} +0.547866 q^{44} +1.44298 q^{45} -6.44178 q^{46} -7.51682 q^{47} -1.00000 q^{48} -6.86492 q^{49} -2.91782 q^{50} +3.55497 q^{51} +1.00000 q^{52} +1.23606 q^{53} -1.00000 q^{54} +0.790559 q^{55} +0.367530 q^{56} +4.08520 q^{57} +0.496780 q^{58} +3.24791 q^{59} -1.44298 q^{60} -12.4908 q^{61} +1.35014 q^{62} +0.367530 q^{63} +1.00000 q^{64} +1.44298 q^{65} -0.547866 q^{66} -7.76227 q^{67} -3.55497 q^{68} +6.44178 q^{69} +0.530337 q^{70} -10.7114 q^{71} +1.00000 q^{72} -12.6599 q^{73} -1.55757 q^{74} +2.91782 q^{75} -4.08520 q^{76} +0.201357 q^{77} -1.00000 q^{78} -15.8651 q^{79} +1.44298 q^{80} +1.00000 q^{81} -0.751095 q^{82} +8.69957 q^{83} -0.367530 q^{84} -5.12974 q^{85} +10.7331 q^{86} -0.496780 q^{87} +0.547866 q^{88} -3.90904 q^{89} +1.44298 q^{90} +0.367530 q^{91} -6.44178 q^{92} -1.35014 q^{93} -7.51682 q^{94} -5.89486 q^{95} -1.00000 q^{96} +0.761776 q^{97} -6.86492 q^{98} +0.547866 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 9 q^{2} - 9 q^{3} + 9 q^{4} - 4 q^{5} - 9 q^{6} - 4 q^{7} + 9 q^{8} + 9 q^{9}+O(q^{10}) $ 9 * q + 9 * q^2 - 9 * q^3 + 9 * q^4 - 4 * q^5 - 9 * q^6 - 4 * q^7 + 9 * q^8 + 9 * q^9	\( 9 q + 9 q^{2} - 9 q^{3} + 9 q^{4} - 4 q^{5} - 9 q^{6} - 4 q^{7} + 9 q^{8} + 9 q^{9} - 4 q^{10} - 5 q^{11} - 9 q^{12} + 9 q^{13} - 4 q^{14} + 4 q^{15} + 9 q^{16} - 6 q^{17} + 9 q^{18} - 4 q^{19} - 4 q^{20} + 4 q^{21} - 5 q^{22} - 6 q^{23} - 9 q^{24} - 11 q^{25} + 9 q^{26} - 9 q^{27} - 4 q^{28} - 19 q^{29} + 4 q^{30} - 6 q^{31} + 9 q^{32} + 5 q^{33} - 6 q^{34} + 10 q^{35} + 9 q^{36} - 13 q^{37} - 4 q^{38} - 9 q^{39} - 4 q^{40} - 18 q^{41} + 4 q^{42} - 20 q^{43} - 5 q^{44} - 4 q^{45} - 6 q^{46} + 14 q^{47} - 9 q^{48} - 3 q^{49} - 11 q^{50} + 6 q^{51} + 9 q^{52} - 3 q^{53} - 9 q^{54} - 4 q^{55} - 4 q^{56} + 4 q^{57} - 19 q^{58} - 9 q^{59} + 4 q^{60} - 24 q^{61} - 6 q^{62} - 4 q^{63} + 9 q^{64} - 4 q^{65} + 5 q^{66} - 4 q^{67} - 6 q^{68} + 6 q^{69} + 10 q^{70} - 9 q^{71} + 9 q^{72} - 24 q^{73} - 13 q^{74} + 11 q^{75} - 4 q^{76} + 3 q^{77} - 9 q^{78} - 15 q^{79} - 4 q^{80} + 9 q^{81} - 18 q^{82} + 20 q^{83} + 4 q^{84} - 31 q^{85} - 20 q^{86} + 19 q^{87} - 5 q^{88} + 3 q^{89} - 4 q^{90} - 4 q^{91} - 6 q^{92} + 6 q^{93} + 14 q^{94} - 4 q^{95} - 9 q^{96} - 19 q^{97} - 3 q^{98} - 5 q^{99}+O(q^{100}) \) 9 * q + 9 * q^2 - 9 * q^3 + 9 * q^4 - 4 * q^5 - 9 * q^6 - 4 * q^7 + 9 * q^8 + 9 * q^9 - 4 * q^10 - 5 * q^11 - 9 * q^12 + 9 * q^13 - 4 * q^14 + 4 * q^15 + 9 * q^16 - 6 * q^17 + 9 * q^18 - 4 * q^19 - 4 * q^20 + 4 * q^21 - 5 * q^22 - 6 * q^23 - 9 * q^24 - 11 * q^25 + 9 * q^26 - 9 * q^27 - 4 * q^28 - 19 * q^29 + 4 * q^30 - 6 * q^31 + 9 * q^32 + 5 * q^33 - 6 * q^34 + 10 * q^35 + 9 * q^36 - 13 * q^37 - 4 * q^38 - 9 * q^39 - 4 * q^40 - 18 * q^41 + 4 * q^42 - 20 * q^43 - 5 * q^44 - 4 * q^45 - 6 * q^46 + 14 * q^47 - 9 * q^48 - 3 * q^49 - 11 * q^50 + 6 * q^51 + 9 * q^52 - 3 * q^53 - 9 * q^54 - 4 * q^55 - 4 * q^56 + 4 * q^57 - 19 * q^58 - 9 * q^59 + 4 * q^60 - 24 * q^61 - 6 * q^62 - 4 * q^63 + 9 * q^64 - 4 * q^65 + 5 * q^66 - 4 * q^67 - 6 * q^68 + 6 * q^69 + 10 * q^70 - 9 * q^71 + 9 * q^72 - 24 * q^73 - 13 * q^74 + 11 * q^75 - 4 * q^76 + 3 * q^77 - 9 * q^78 - 15 * q^79 - 4 * q^80 + 9 * q^81 - 18 * q^82 + 20 * q^83 + 4 * q^84 - 31 * q^85 - 20 * q^86 + 19 * q^87 - 5 * q^88 + 3 * q^89 - 4 * q^90 - 4 * q^91 - 6 * q^92 + 6 * q^93 + 14 * q^94 - 4 * q^95 - 9 * q^96 - 19 * q^97 - 3 * q^98 - 5 * 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
$5$	1.44298	0.645319	0.322660	−	0.946515i	$-0.395423\pi$
$5$	1.44298	0.645319	0.322660	+	0.946515i	$0.395423\pi$
$6$	−1.00000	−0.408248
$7$	0.367530	0.138913	0.0694566	−	0.997585i	$-0.477873\pi$
$7$	0.367530	0.138913	0.0694566	+	0.997585i	$0.477873\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	1.44298	0.456310
$11$	0.547866	0.165188	0.0825939	−	0.996583i	$-0.473680\pi$
$11$	0.547866	0.165188	0.0825939	+	0.996583i	$0.473680\pi$
$12$	−1.00000	−0.288675
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0.367530	0.0982265
$15$	−1.44298	−0.372575
$16$	1.00000	0.250000
$17$	−3.55497	−0.862206	−0.431103	−	0.902303i	$-0.641876\pi$
$17$	−3.55497	−0.862206	−0.431103	+	0.902303i	$0.641876\pi$
$18$	1.00000	0.235702
$19$	−4.08520	−0.937210	−0.468605	−	0.883408i	$-0.655243\pi$
$19$	−4.08520	−0.937210	−0.468605	+	0.883408i	$0.655243\pi$
$20$	1.44298	0.322660
$21$	−0.367530	−0.0802016
$22$	0.547866	0.116805
$23$	−6.44178	−1.34320	−0.671602	−	0.740912i	$-0.734394\pi$
$23$	−6.44178	−1.34320	−0.671602	+	0.740912i	$0.734394\pi$
$24$	−1.00000	−0.204124
$25$	−2.91782	−0.583563
$26$	1.00000	0.196116
$27$	−1.00000	−0.192450
$28$	0.367530	0.0694566
$29$	0.496780	0.0922497	0.0461248	−	0.998936i	$-0.485313\pi$
$29$	0.496780	0.0922497	0.0461248	+	0.998936i	$0.485313\pi$
$30$	−1.44298	−0.263450
$31$	1.35014	0.242493	0.121246	−	0.992622i	$-0.461311\pi$
$31$	1.35014	0.242493	0.121246	+	0.992622i	$0.461311\pi$
$32$	1.00000	0.176777
$33$	−0.547866	−0.0953713
$34$	−3.55497	−0.609672
$35$	0.530337	0.0896433
$36$	1.00000	0.166667
$37$	−1.55757	−0.256063	−0.128031	−	0.991770i	$-0.540866\pi$
$37$	−1.55757	−0.256063	−0.128031	+	0.991770i	$0.540866\pi$
$38$	−4.08520	−0.662707
$39$	−1.00000	−0.160128
$40$	1.44298	0.228155
$41$	−0.751095	−0.117301	−0.0586507	−	0.998279i	$-0.518680\pi$
$41$	−0.751095	−0.117301	−0.0586507	+	0.998279i	$0.518680\pi$
$42$	−0.367530	−0.0567111
$43$	10.7331	1.63678	0.818392	−	0.574660i	$-0.194866\pi$
$43$	10.7331	1.63678	0.818392	+	0.574660i	$0.194866\pi$
$44$	0.547866	0.0825939
$45$	1.44298	0.215106
$46$	−6.44178	−0.949789
$47$	−7.51682	−1.09644	−0.548220	−	0.836334i	$-0.684694\pi$
$47$	−7.51682	−1.09644	−0.548220	+	0.836334i	$0.684694\pi$
$48$	−1.00000	−0.144338
$49$	−6.86492	−0.980703
$50$	−2.91782	−0.412641
$51$	3.55497	0.497795
$52$	1.00000	0.138675
$53$	1.23606	0.169786	0.0848928	−	0.996390i	$-0.472945\pi$
$53$	1.23606	0.169786	0.0848928	+	0.996390i	$0.472945\pi$
$54$	−1.00000	−0.136083
$55$	0.790559	0.106599
$56$	0.367530	0.0491132
$57$	4.08520	0.541098
$58$	0.496780	0.0652304
$59$	3.24791	0.422841	0.211421	−	0.977395i	$-0.432191\pi$
$59$	3.24791	0.422841	0.211421	+	0.977395i	$0.432191\pi$
$60$	−1.44298	−0.186288
$61$	−12.4908	−1.59928	−0.799639	−	0.600481i	$-0.794976\pi$
$61$	−12.4908	−1.59928	−0.799639	+	0.600481i	$0.794976\pi$
$62$	1.35014	0.171468
$63$	0.367530	0.0463044
$64$	1.00000	0.125000
$65$	1.44298	0.178979
$66$	−0.547866	−0.0674377
$67$	−7.76227	−0.948312	−0.474156	−	0.880441i	$-0.657247\pi$
$67$	−7.76227	−0.948312	−0.474156	+	0.880441i	$0.657247\pi$
$68$	−3.55497	−0.431103
$69$	6.44178	0.775499
$70$	0.530337	0.0633874
$71$	−10.7114	−1.27121	−0.635607	−	0.772012i	$-0.719250\pi$
$71$	−10.7114	−1.27121	−0.635607	+	0.772012i	$0.719250\pi$
$72$	1.00000	0.117851
$73$	−12.6599	−1.48173	−0.740864	−	0.671656i	$-0.765583\pi$
$73$	−12.6599	−1.48173	−0.740864	+	0.671656i	$0.765583\pi$
$74$	−1.55757	−0.181064
$75$	2.91782	0.336920
$76$	−4.08520	−0.468605
$77$	0.201357	0.0229468
$78$	−1.00000	−0.113228
$79$	−15.8651	−1.78497	−0.892484	−	0.451080i	$-0.851039\pi$
$79$	−15.8651	−1.78497	−0.892484	+	0.451080i	$0.851039\pi$
$80$	1.44298	0.161330
$81$	1.00000	0.111111
$82$	−0.751095	−0.0829446
$83$	8.69957	0.954901	0.477451	−	0.878659i	$-0.341561\pi$
$83$	8.69957	0.954901	0.477451	+	0.878659i	$0.341561\pi$
$84$	−0.367530	−0.0401008
$85$	−5.12974	−0.556398
$86$	10.7331	1.15738
$87$	−0.496780	−0.0532604
$88$	0.547866	0.0584027
$89$	−3.90904	−0.414357	−0.207179	−	0.978303i	$-0.566428\pi$
$89$	−3.90904	−0.414357	−0.207179	+	0.978303i	$0.566428\pi$
$90$	1.44298	0.152103
$91$	0.367530	0.0385276
$92$	−6.44178	−0.671602
$93$	−1.35014	−0.140003
$94$	−7.51682	−0.775301
$95$	−5.89486	−0.604800
$96$	−1.00000	−0.102062
$97$	0.761776	0.0773466	0.0386733	−	0.999252i	$-0.487687\pi$
$97$	0.761776	0.0773466	0.0386733	+	0.999252i	$0.487687\pi$
$98$	−6.86492	−0.693462
$99$	0.547866	0.0550626
$100$	−2.91782	−0.291782
$101$	4.45108	0.442899	0.221449	−	0.975172i	$-0.428921\pi$
$101$	4.45108	0.442899	0.221449	+	0.975172i	$0.428921\pi$
$102$	3.55497	0.351994
$103$	−1.00000	−0.0985329
$103$	−1.00000	−0.0985329
$104$	1.00000	0.0980581
$105$	−0.530337	−0.0517556
$106$	1.23606	0.120057
$107$	11.8666	1.14719	0.573596	−	0.819138i	$-0.305548\pi$
$107$	11.8666	1.14719	0.573596	+	0.819138i	$0.305548\pi$
$108$	−1.00000	−0.0962250
$109$	17.5556	1.68152	0.840762	−	0.541405i	$-0.182108\pi$
$109$	17.5556	1.68152	0.840762	+	0.541405i	$0.182108\pi$
$110$	0.790559	0.0753768
$111$	1.55757	0.147838
$112$	0.367530	0.0347283
$113$	−18.7990	−1.76846	−0.884230	−	0.467052i	$-0.845316\pi$
$113$	−18.7990	−1.76846	−0.884230	+	0.467052i	$0.845316\pi$
$114$	4.08520	0.382614
$115$	−9.29535	−0.866796
$116$	0.496780	0.0461248
$117$	1.00000	0.0924500
$118$	3.24791	0.298994
$119$	−1.30656	−0.119772
$120$	−1.44298	−0.131725
$121$	−10.6998	−0.972713
$122$	−12.4908	−1.13086
$123$	0.751095	0.0677240
$124$	1.35014	0.121246
$125$	−11.4252	−1.02190
$126$	0.367530	0.0327422
$127$	2.24137	0.198890	0.0994449	−	0.995043i	$-0.468293\pi$
$127$	2.24137	0.198890	0.0994449	+	0.995043i	$0.468293\pi$
$128$	1.00000	0.0883883
$129$	−10.7331	−0.944998
$130$	1.44298	0.126558
$131$	9.91155	0.865976	0.432988	−	0.901400i	$-0.357459\pi$
$131$	9.91155	0.865976	0.432988	+	0.901400i	$0.357459\pi$
$132$	−0.547866	−0.0476856
$133$	−1.50143	−0.130191
$134$	−7.76227	−0.670558
$135$	−1.44298	−0.124192
$136$	−3.55497	−0.304836
$137$	−12.7428	−1.08869	−0.544346	−	0.838861i	$-0.683222\pi$
$137$	−12.7428	−1.08869	−0.544346	+	0.838861i	$0.683222\pi$
$138$	6.44178	0.548361
$139$	8.50060	0.721011	0.360506	−	0.932757i	$-0.382604\pi$
$139$	8.50060	0.721011	0.360506	+	0.932757i	$0.382604\pi$
$140$	0.530337	0.0448217
$141$	7.51682	0.633030
$142$	−10.7114	−0.898885
$143$	0.547866	0.0458149
$144$	1.00000	0.0833333
$145$	0.716842	0.0595305
$146$	−12.6599	−1.04774
$147$	6.86492	0.566209
$148$	−1.55757	−0.128031
$149$	9.60182	0.786612	0.393306	−	0.919408i	$-0.371331\pi$
$149$	9.60182	0.786612	0.393306	+	0.919408i	$0.371331\pi$
$150$	2.91782	0.238239
$151$	13.1514	1.07024	0.535122	−	0.844775i	$-0.320266\pi$
$151$	13.1514	1.07024	0.535122	+	0.844775i	$0.320266\pi$
$152$	−4.08520	−0.331354
$153$	−3.55497	−0.287402
$154$	0.201357	0.0162258
$155$	1.94823	0.156485
$156$	−1.00000	−0.0800641
$157$	7.00388	0.558971	0.279485	−	0.960150i	$-0.409836\pi$
$157$	7.00388	0.558971	0.279485	+	0.960150i	$0.409836\pi$
$158$	−15.8651	−1.26216
$159$	−1.23606	−0.0980258
$160$	1.44298	0.114077
$161$	−2.36755	−0.186589
$162$	1.00000	0.0785674
$163$	3.96067	0.310224	0.155112	−	0.987897i	$-0.450426\pi$
$163$	3.96067	0.310224	0.155112	+	0.987897i	$0.450426\pi$
$164$	−0.751095	−0.0586507
$165$	−0.790559	−0.0615449
$166$	8.69957	0.675217
$167$	19.6742	1.52243	0.761217	−	0.648497i	$-0.224602\pi$
$167$	19.6742	1.52243	0.761217	+	0.648497i	$0.224602\pi$
$168$	−0.367530	−0.0283555
$169$	1.00000	0.0769231
$170$	−5.12974	−0.393433
$171$	−4.08520	−0.312403
$172$	10.7331	0.818392
$173$	11.0291	0.838528	0.419264	−	0.907864i	$-0.362288\pi$
$173$	11.0291	0.838528	0.419264	+	0.907864i	$0.362288\pi$
$174$	−0.496780	−0.0376608
$175$	−1.07238	−0.0810646
$176$	0.547866	0.0412970
$177$	−3.24791	−0.244128
$178$	−3.90904	−0.292995
$179$	18.9259	1.41459	0.707294	−	0.706920i	$-0.249916\pi$
$179$	18.9259	1.41459	0.707294	+	0.706920i	$0.249916\pi$
$180$	1.44298	0.107553
$181$	8.01462	0.595722	0.297861	−	0.954609i	$-0.403727\pi$
$181$	8.01462	0.595722	0.297861	+	0.954609i	$0.403727\pi$
$182$	0.367530	0.0272431
$183$	12.4908	0.923344
$184$	−6.44178	−0.474894
$185$	−2.24754	−0.165242
$186$	−1.35014	−0.0989973
$187$	−1.94765	−0.142426
$188$	−7.51682	−0.548220
$189$	−0.367530	−0.0267339
$190$	−5.89486	−0.427658
$191$	−9.99549	−0.723249	−0.361624	−	0.932324i	$-0.617778\pi$
$191$	−9.99549	−0.723249	−0.361624	+	0.932324i	$0.617778\pi$
$192$	−1.00000	−0.0721688
$193$	20.0860	1.44582	0.722912	−	0.690940i	$-0.242803\pi$
$193$	20.0860	1.44582	0.722912	+	0.690940i	$0.242803\pi$
$194$	0.761776	0.0546923
$195$	−1.44298	−0.103334
$196$	−6.86492	−0.490352
$197$	−19.3711	−1.38014	−0.690068	−	0.723744i	$-0.742419\pi$
$197$	−19.3711	−1.38014	−0.690068	+	0.723744i	$0.742419\pi$
$198$	0.547866	0.0389352
$199$	−11.5330	−0.817556	−0.408778	−	0.912634i	$-0.634045\pi$
$199$	−11.5330	−0.817556	−0.408778	+	0.912634i	$0.634045\pi$
$200$	−2.91782	−0.206321
$201$	7.76227	0.547508
$202$	4.45108	0.313177
$203$	0.182581	0.0128147
$204$	3.55497	0.248898
$205$	−1.08381	−0.0756968
$206$	−1.00000	−0.0696733
$207$	−6.44178	−0.447735
$208$	1.00000	0.0693375
$209$	−2.23814	−0.154816
$210$	−0.530337	−0.0365967
$211$	−25.4564	−1.75249	−0.876245	−	0.481866i	$-0.839959\pi$
$211$	−25.4564	−1.75249	−0.876245	+	0.481866i	$0.839959\pi$
$212$	1.23606	0.0848928
$213$	10.7114	0.733936
$214$	11.8666	0.811187
$215$	15.4876	1.05625
$216$	−1.00000	−0.0680414
$217$	0.496218	0.0336855
$218$	17.5556	1.18902
$219$	12.6599	0.855476
$220$	0.790559	0.0532994
$221$	−3.55497	−0.239133
$222$	1.55757	0.104537
$223$	17.1504	1.14848	0.574239	−	0.818688i	$-0.305298\pi$
$223$	17.1504	1.14848	0.574239	+	0.818688i	$0.305298\pi$
$224$	0.367530	0.0245566
$225$	−2.91782	−0.194521
$226$	−18.7990	−1.25049
$227$	−17.7510	−1.17818	−0.589088	−	0.808069i	$-0.700513\pi$
$227$	−17.7510	−1.17818	−0.589088	+	0.808069i	$0.700513\pi$
$228$	4.08520	0.270549
$229$	−6.19924	−0.409657	−0.204829	−	0.978798i	$-0.565664\pi$
$229$	−6.19924	−0.409657	−0.204829	+	0.978798i	$0.565664\pi$
$230$	−9.29535	−0.612917
$231$	−0.201357	−0.0132483
$232$	0.496780	0.0326152
$233$	−19.5650	−1.28174	−0.640872	−	0.767648i	$-0.721427\pi$
$233$	−19.5650	−1.28174	−0.640872	+	0.767648i	$0.721427\pi$
$234$	1.00000	0.0653720
$235$	−10.8466	−0.707554
$236$	3.24791	0.211421
$237$	15.8651	1.03055
$238$	−1.30656	−0.0846915
$239$	−11.5577	−0.747607	−0.373804	−	0.927508i	$-0.621947\pi$
$239$	−11.5577	−0.747607	−0.373804	+	0.927508i	$0.621947\pi$
$240$	−1.44298	−0.0931438
$241$	10.8027	0.695865	0.347932	−	0.937520i	$-0.386884\pi$
$241$	10.8027	0.695865	0.347932	+	0.937520i	$0.386884\pi$
$242$	−10.6998	−0.687812
$243$	−1.00000	−0.0641500
$244$	−12.4908	−0.799639
$245$	−9.90593	−0.632867
$246$	0.751095	0.0478881
$247$	−4.08520	−0.259935
$248$	1.35014	0.0857342
$249$	−8.69957	−0.551313
$250$	−11.4252	−0.722595
$251$	11.7031	0.738694	0.369347	−	0.929292i	$-0.379581\pi$
$251$	11.7031	0.738694	0.369347	+	0.929292i	$0.379581\pi$
$252$	0.367530	0.0231522
$253$	−3.52923	−0.221881
$254$	2.24137	0.140636
$255$	5.12974	0.321237
$256$	1.00000	0.0625000
$257$	−2.51465	−0.156859	−0.0784296	−	0.996920i	$-0.524991\pi$
$257$	−2.51465	−0.156859	−0.0784296	+	0.996920i	$0.524991\pi$
$258$	−10.7331	−0.668214
$259$	−0.572453	−0.0355705
$260$	1.44298	0.0894897
$261$	0.496780	0.0307499
$262$	9.91155	0.612338
$263$	16.2335	1.00100	0.500500	−	0.865736i	$-0.333149\pi$
$263$	16.2335	1.00100	0.500500	+	0.865736i	$0.333149\pi$
$264$	−0.547866	−0.0337188
$265$	1.78360	0.109566
$266$	−1.50143	−0.0920588
$267$	3.90904	0.239229
$268$	−7.76227	−0.474156
$269$	13.1976	0.804675	0.402337	−	0.915491i	$-0.368198\pi$
$269$	13.1976	0.804675	0.402337	+	0.915491i	$0.368198\pi$
$270$	−1.44298	−0.0878168
$271$	−26.4860	−1.60891	−0.804454	−	0.594015i	$-0.797542\pi$
$271$	−26.4860	−1.60891	−0.804454	+	0.594015i	$0.797542\pi$
$272$	−3.55497	−0.215552
$273$	−0.367530	−0.0222439
$274$	−12.7428	−0.769822
$275$	−1.59857	−0.0963975
$276$	6.44178	0.387750
$277$	0.916739	0.0550815	0.0275408	−	0.999621i	$-0.491232\pi$
$277$	0.916739	0.0550815	0.0275408	+	0.999621i	$0.491232\pi$
$278$	8.50060	0.509832
$279$	1.35014	0.0808310
$280$	0.530337	0.0316937
$281$	0.265437	0.0158346	0.00791731	−	0.999969i	$-0.497480\pi$
$281$	0.265437	0.0158346	0.00791731	+	0.999969i	$0.497480\pi$
$282$	7.51682	0.447620
$283$	−0.758999	−0.0451178	−0.0225589	−	0.999746i	$-0.507181\pi$
$283$	−0.758999	−0.0451178	−0.0225589	+	0.999746i	$0.507181\pi$
$284$	−10.7114	−0.635607
$285$	5.89486	0.349181
$286$	0.547866	0.0323960
$287$	−0.276050	−0.0162947
$288$	1.00000	0.0589256
$289$	−4.36220	−0.256600
$290$	0.716842	0.0420944
$291$	−0.761776	−0.0446561
$292$	−12.6599	−0.740864
$293$	4.69443	0.274251	0.137126	−	0.990554i	$-0.456214\pi$
$293$	4.69443	0.274251	0.137126	+	0.990554i	$0.456214\pi$
$294$	6.86492	0.400370
$295$	4.68666	0.272868
$296$	−1.55757	−0.0905319
$297$	−0.547866	−0.0317904
$298$	9.60182	0.556219
$299$	−6.44178	−0.372538
$300$	2.91782	0.168460
$301$	3.94474	0.227371
$302$	13.1514	0.756776
$303$	−4.45108	−0.255708
$304$	−4.08520	−0.234302
$305$	−18.0239	−1.03205
$306$	−3.55497	−0.203224
$307$	11.2493	0.642034	0.321017	−	0.947074i	$-0.395975\pi$
$307$	11.2493	0.642034	0.321017	+	0.947074i	$0.395975\pi$
$308$	0.201357	0.0114734
$309$	1.00000	0.0568880
$310$	1.94823	0.110652
$311$	27.2339	1.54429	0.772146	−	0.635445i	$-0.219183\pi$
$311$	27.2339	1.54429	0.772146	+	0.635445i	$0.219183\pi$
$312$	−1.00000	−0.0566139
$313$	−5.84803	−0.330550	−0.165275	−	0.986247i	$-0.552851\pi$
$313$	−5.84803	−0.330550	−0.165275	+	0.986247i	$0.552851\pi$
$314$	7.00388	0.395252
$315$	0.530337	0.0298811
$316$	−15.8651	−0.892484
$317$	−7.97888	−0.448139	−0.224069	−	0.974573i	$-0.571934\pi$
$317$	−7.97888	−0.448139	−0.224069	+	0.974573i	$0.571934\pi$
$318$	−1.23606	−0.0693147
$319$	0.272169	0.0152385
$320$	1.44298	0.0806649
$321$	−11.8666	−0.662332
$322$	−2.36755	−0.131938
$323$	14.5228	0.808068
$324$	1.00000	0.0555556
$325$	−2.91782	−0.161851
$326$	3.96067	0.219361
$327$	−17.5556	−0.970828
$328$	−0.751095	−0.0414723
$329$	−2.76265	−0.152310
$330$	−0.790559	−0.0435188
$331$	−28.0019	−1.53912	−0.769561	−	0.638574i	$-0.779525\pi$
$331$	−28.0019	−1.53912	−0.769561	+	0.638574i	$0.779525\pi$
$332$	8.69957	0.477451
$333$	−1.55757	−0.0853543
$334$	19.6742	1.07652
$335$	−11.2008	−0.611964
$336$	−0.367530	−0.0200504
$337$	−9.31312	−0.507318	−0.253659	−	0.967294i	$-0.581634\pi$
$337$	−9.31312	−0.507318	−0.253659	+	0.967294i	$0.581634\pi$
$338$	1.00000	0.0543928
$339$	18.7990	1.02102
$340$	−5.12974	−0.278199
$341$	0.739698	0.0400569
$342$	−4.08520	−0.220902
$343$	−5.09577	−0.275146
$344$	10.7331	0.578691
$345$	9.29535	0.500445
$346$	11.0291	0.592929
$347$	−9.47773	−0.508791	−0.254396	−	0.967100i	$-0.581877\pi$
$347$	−9.47773	−0.508791	−0.254396	+	0.967100i	$0.581877\pi$
$348$	−0.496780	−0.0266302
$349$	−17.8767	−0.956918	−0.478459	−	0.878110i	$-0.658804\pi$
$349$	−17.8767	−0.956918	−0.478459	+	0.878110i	$0.658804\pi$
$350$	−1.07238	−0.0573213
$351$	−1.00000	−0.0533761
$352$	0.547866	0.0292014
$353$	−30.8518	−1.64208	−0.821039	−	0.570872i	$-0.806605\pi$
$353$	−30.8518	−1.64208	−0.821039	+	0.570872i	$0.806605\pi$
$354$	−3.24791	−0.172624
$355$	−15.4564	−0.820339
$356$	−3.90904	−0.207179
$357$	1.30656	0.0691503
$358$	18.9259	1.00026
$359$	−5.25381	−0.277285	−0.138643	−	0.990342i	$-0.544274\pi$
$359$	−5.25381	−0.277285	−0.138643	+	0.990342i	$0.544274\pi$
$360$	1.44298	0.0760516
$361$	−2.31111	−0.121638
$362$	8.01462	0.421239
$363$	10.6998	0.561596
$364$	0.367530	0.0192638
$365$	−18.2679	−0.956187
$366$	12.4908	0.652903
$367$	−8.29246	−0.432863	−0.216432	−	0.976298i	$-0.569442\pi$
$367$	−8.29246	−0.432863	−0.216432	+	0.976298i	$0.569442\pi$
$368$	−6.44178	−0.335801
$369$	−0.751095	−0.0391005
$370$	−2.24754	−0.116844
$371$	0.454288	0.0235855
$372$	−1.35014	−0.0700017
$373$	−19.8937	−1.03006	−0.515028	−	0.857173i	$-0.672219\pi$
$373$	−19.8937	−1.03006	−0.515028	+	0.857173i	$0.672219\pi$
$374$	−1.94765	−0.100710
$375$	11.4252	0.589996
$376$	−7.51682	−0.387650
$377$	0.496780	0.0255855
$378$	−0.367530	−0.0189037
$379$	15.8718	0.815277	0.407639	−	0.913143i	$-0.366352\pi$
$379$	15.8718	0.815277	0.407639	+	0.913143i	$0.366352\pi$
$380$	−5.89486	−0.302400
$381$	−2.24137	−0.114829
$382$	−9.99549	−0.511414
$383$	−17.5544	−0.896988	−0.448494	−	0.893786i	$-0.648039\pi$
$383$	−17.5544	−0.896988	−0.448494	+	0.893786i	$0.648039\pi$
$384$	−1.00000	−0.0510310
$385$	0.290554	0.0148080
$386$	20.0860	1.02235
$387$	10.7331	0.545595
$388$	0.761776	0.0386733
$389$	−28.0548	−1.42243	−0.711217	−	0.702972i	$-0.751856\pi$
$389$	−28.0548	−1.42243	−0.711217	+	0.702972i	$0.751856\pi$
$390$	−1.44298	−0.0730680
$391$	22.9003	1.15812
$392$	−6.86492	−0.346731
$393$	−9.91155	−0.499971
$394$	−19.3711	−0.975904
$395$	−22.8930	−1.15187
$396$	0.547866	0.0275313
$397$	−31.9046	−1.60125	−0.800624	−	0.599168i	$-0.795498\pi$
$397$	−31.9046	−1.60125	−0.800624	+	0.599168i	$0.795498\pi$
$398$	−11.5330	−0.578099
$399$	1.50143	0.0751657
$400$	−2.91782	−0.145891
$401$	−19.1941	−0.958508	−0.479254	−	0.877676i	$-0.659093\pi$
$401$	−19.1941	−0.958508	−0.479254	+	0.877676i	$0.659093\pi$
$402$	7.76227	0.387147
$403$	1.35014	0.0672554
$404$	4.45108	0.221449
$405$	1.44298	0.0717021
$406$	0.182581	0.00906136
$407$	−0.853340	−0.0422985
$408$	3.55497	0.175997
$409$	−25.2693	−1.24949	−0.624743	−	0.780831i	$-0.714796\pi$
$409$	−25.2693	−1.24949	−0.624743	+	0.780831i	$0.714796\pi$
$410$	−1.08381	−0.0535257
$411$	12.7428	0.628557
$412$	−1.00000	−0.0492665
$413$	1.19370	0.0587382
$414$	−6.44178	−0.316596
$415$	12.5533	0.616216
$416$	1.00000	0.0490290
$417$	−8.50060	−0.416276
$418$	−2.23814	−0.109471
$419$	−20.3789	−0.995575	−0.497788	−	0.867299i	$-0.665854\pi$
$419$	−20.3789	−0.995575	−0.497788	+	0.867299i	$0.665854\pi$
$420$	−0.530337	−0.0258778
$421$	32.7240	1.59487	0.797436	−	0.603404i	$-0.206189\pi$
$421$	32.7240	1.59487	0.797436	+	0.603404i	$0.206189\pi$
$422$	−25.4564	−1.23920
$423$	−7.51682	−0.365480
$424$	1.23606	0.0600283
$425$	10.3727	0.503152
$426$	10.7114	0.518971
$427$	−4.59073	−0.222161
$428$	11.8666	0.573596
$429$	−0.547866	−0.0264512
$430$	15.4876	0.746880
$431$	−10.4093	−0.501398	−0.250699	−	0.968065i	$-0.580660\pi$
$431$	−10.4093	−0.501398	−0.250699	+	0.968065i	$0.580660\pi$
$432$	−1.00000	−0.0481125
$433$	−8.79314	−0.422571	−0.211286	−	0.977424i	$-0.567765\pi$
$433$	−8.79314	−0.422571	−0.211286	+	0.977424i	$0.567765\pi$
$434$	0.496218	0.0238192
$435$	−0.716842	−0.0343699
$436$	17.5556	0.840762
$437$	26.3160	1.25886
$438$	12.6599	0.604913
$439$	30.7549	1.46785	0.733926	−	0.679229i	$-0.237686\pi$
$439$	30.7549	1.46785	0.733926	+	0.679229i	$0.237686\pi$
$440$	0.790559	0.0376884
$441$	−6.86492	−0.326901
$442$	−3.55497	−0.169093
$443$	−21.1042	−1.00269	−0.501346	−	0.865247i	$-0.667162\pi$
$443$	−21.1042	−1.00269	−0.501346	+	0.865247i	$0.667162\pi$
$444$	1.55757	0.0739190
$445$	−5.64065	−0.267393
$446$	17.1504	0.812097
$447$	−9.60182	−0.454151
$448$	0.367530	0.0173641
$449$	17.3977	0.821049	0.410524	−	0.911850i	$-0.365346\pi$
$449$	17.3977	0.821049	0.410524	+	0.911850i	$0.365346\pi$
$450$	−2.91782	−0.137547
$451$	−0.411500	−0.0193768
$452$	−18.7990	−0.884230
$453$	−13.1514	−0.617905
$454$	−17.7510	−0.833097
$455$	0.530337	0.0248626
$456$	4.08520	0.191307
$457$	−11.1130	−0.519846	−0.259923	−	0.965629i	$-0.583697\pi$
$457$	−11.1130	−0.519846	−0.259923	+	0.965629i	$0.583697\pi$
$458$	−6.19924	−0.289672
$459$	3.55497	0.165932
$460$	−9.29535	−0.433398
$461$	1.40726	0.0655424	0.0327712	−	0.999463i	$-0.489567\pi$
$461$	1.40726	0.0655424	0.0327712	+	0.999463i	$0.489567\pi$
$462$	−0.201357	−0.00936798
$463$	24.9641	1.16018	0.580091	−	0.814552i	$-0.303017\pi$
$463$	24.9641	1.16018	0.580091	+	0.814552i	$0.303017\pi$
$464$	0.496780	0.0230624
$465$	−1.94823	−0.0903469
$466$	−19.5650	−0.906330
$467$	39.0266	1.80594	0.902969	−	0.429706i	$-0.141383\pi$
$467$	39.0266	1.80594	0.902969	+	0.429706i	$0.141383\pi$
$468$	1.00000	0.0462250
$469$	−2.85286	−0.131733
$470$	−10.8466	−0.500316
$471$	−7.00388	−0.322722
$472$	3.24791	0.149497
$473$	5.88031	0.270377
$474$	15.8651	0.728710
$475$	11.9199	0.546921
$476$	−1.30656	−0.0598859
$477$	1.23606	0.0565952
$478$	−11.5577	−0.528638
$479$	−6.49329	−0.296686	−0.148343	−	0.988936i	$-0.547394\pi$
$479$	−6.49329	−0.296686	−0.148343	+	0.988936i	$0.547394\pi$
$480$	−1.44298	−0.0658626
$481$	−1.55757	−0.0710191
$482$	10.8027	0.492051
$483$	2.36755	0.107727
$484$	−10.6998	−0.486356
$485$	1.09923	0.0499133
$486$	−1.00000	−0.0453609
$487$	7.81244	0.354015	0.177008	−	0.984209i	$-0.443358\pi$
$487$	7.81244	0.354015	0.177008	+	0.984209i	$0.443358\pi$
$488$	−12.4908	−0.565430
$489$	−3.96067	−0.179108
$490$	−9.90593	−0.447504
$491$	17.4032	0.785395	0.392697	−	0.919668i	$-0.371542\pi$
$491$	17.4032	0.785395	0.392697	+	0.919668i	$0.371542\pi$
$492$	0.751095	0.0338620
$493$	−1.76604	−0.0795382
$494$	−4.08520	−0.183802
$495$	0.790559	0.0355330
$496$	1.35014	0.0606232
$497$	−3.93678	−0.176589
$498$	−8.69957	−0.389837
$499$	−6.53910	−0.292730	−0.146365	−	0.989231i	$-0.546757\pi$
$499$	−6.53910	−0.292730	−0.146365	+	0.989231i	$0.546757\pi$
$500$	−11.4252	−0.510952
$501$	−19.6742	−0.878978
$502$	11.7031	0.522335
$503$	−5.06136	−0.225675	−0.112837	−	0.993613i	$-0.535994\pi$
$503$	−5.06136	−0.225675	−0.112837	+	0.993613i	$0.535994\pi$
$504$	0.367530	0.0163711
$505$	6.42280	0.285811
$506$	−3.52923	−0.156894
$507$	−1.00000	−0.0444116
$508$	2.24137	0.0994449
$509$	−14.1467	−0.627043	−0.313522	−	0.949581i	$-0.601509\pi$
$509$	−14.1467	−0.627043	−0.313522	+	0.949581i	$0.601509\pi$
$510$	5.12974	0.227149
$511$	−4.65288	−0.205831
$512$	1.00000	0.0441942
$513$	4.08520	0.180366
$514$	−2.51465	−0.110916
$515$	−1.44298	−0.0635852
$516$	−10.7331	−0.472499
$517$	−4.11821	−0.181119
$518$	−0.572453	−0.0251522
$519$	−11.0291	−0.484124
$520$	1.44298	0.0632788
$521$	12.1702	0.533188	0.266594	−	0.963809i	$-0.414102\pi$
$521$	12.1702	0.533188	0.266594	+	0.963809i	$0.414102\pi$
$522$	0.496780	0.0217435
$523$	7.32120	0.320134	0.160067	−	0.987106i	$-0.448829\pi$
$523$	7.32120	0.320134	0.160067	+	0.987106i	$0.448829\pi$
$524$	9.91155	0.432988
$525$	1.07238	0.0468027
$526$	16.2335	0.707814
$527$	−4.79972	−0.209079
$528$	−0.547866	−0.0238428
$529$	18.4965	0.804198
$530$	1.78360	0.0774748
$531$	3.24791	0.140947
$532$	−1.50143	−0.0650954
$533$	−0.751095	−0.0325335
$534$	3.90904	0.169161
$535$	17.1233	0.740305
$536$	−7.76227	−0.335279
$537$	−18.9259	−0.816713
$538$	13.1976	0.568991
$539$	−3.76106	−0.162000
$540$	−1.44298	−0.0620959
$541$	17.0796	0.734308	0.367154	−	0.930160i	$-0.380332\pi$
$541$	17.0796	0.734308	0.367154	+	0.930160i	$0.380332\pi$
$542$	−26.4860	−1.13767
$543$	−8.01462	−0.343940
$544$	−3.55497	−0.152418
$545$	25.3324	1.08512
$546$	−0.367530	−0.0157288
$547$	−11.1203	−0.475470	−0.237735	−	0.971330i	$-0.576405\pi$
$547$	−11.1203	−0.475470	−0.237735	+	0.971330i	$0.576405\pi$
$548$	−12.7428	−0.544346
$549$	−12.4908	−0.533093
$550$	−1.59857	−0.0681634
$551$	−2.02945	−0.0864573
$552$	6.44178	0.274180
$553$	−5.83091	−0.247956
$554$	0.916739	0.0389485
$555$	2.24754	0.0954027
$556$	8.50060	0.360506
$557$	−24.0397	−1.01859	−0.509297	−	0.860591i	$-0.670094\pi$
$557$	−24.0397	−1.01859	−0.509297	+	0.860591i	$0.670094\pi$
$558$	1.35014	0.0571561
$559$	10.7331	0.453962
$560$	0.530337	0.0224108
$561$	1.94765	0.0822297
$562$	0.265437	0.0111968
$563$	31.6791	1.33512	0.667558	−	0.744558i	$-0.267340\pi$
$563$	31.6791	1.33512	0.667558	+	0.744558i	$0.267340\pi$
$564$	7.51682	0.316515
$565$	−27.1265	−1.14122
$566$	−0.758999	−0.0319031
$567$	0.367530	0.0154348
$568$	−10.7114	−0.449442
$569$	39.0007	1.63500	0.817498	−	0.575932i	$-0.195361\pi$
$569$	39.0007	1.63500	0.817498	+	0.575932i	$0.195361\pi$
$570$	5.89486	0.246908
$571$	−29.0311	−1.21491	−0.607457	−	0.794352i	$-0.707810\pi$
$571$	−29.0311	−1.21491	−0.607457	+	0.794352i	$0.707810\pi$
$572$	0.547866	0.0229074
$573$	9.99549	0.417568
$574$	−0.276050	−0.0115221
$575$	18.7959	0.783845
$576$	1.00000	0.0416667
$577$	36.8361	1.53351	0.766754	−	0.641941i	$-0.221871\pi$
$577$	36.8361	1.53351	0.766754	+	0.641941i	$0.221871\pi$
$578$	−4.36220	−0.181444
$579$	−20.0860	−0.834747
$580$	0.716842	0.0297652
$581$	3.19735	0.132648
$582$	−0.761776	−0.0315766
$583$	0.677194	0.0280465
$584$	−12.6599	−0.523870
$585$	1.44298	0.0596598
$586$	4.69443	0.193925
$587$	−9.61426	−0.396823	−0.198411	−	0.980119i	$-0.563578\pi$
$587$	−9.61426	−0.396823	−0.198411	+	0.980119i	$0.563578\pi$
$588$	6.86492	0.283105
$589$	−5.51561	−0.227267
$590$	4.68666	0.192947
$591$	19.3711	0.796822
$592$	−1.55757	−0.0640157
$593$	24.6024	1.01030	0.505150	−	0.863032i	$-0.331437\pi$
$593$	24.6024	1.01030	0.505150	+	0.863032i	$0.331437\pi$
$594$	−0.547866	−0.0224792
$595$	−1.88533	−0.0772911
$596$	9.60182	0.393306
$597$	11.5330	0.472016
$598$	−6.44178	−0.263424
$599$	−16.3558	−0.668278	−0.334139	−	0.942524i	$-0.608446\pi$
$599$	−16.3558	−0.668278	−0.334139	+	0.942524i	$0.608446\pi$
$600$	2.91782	0.119119
$601$	−43.4714	−1.77324	−0.886618	−	0.462502i	$-0.846952\pi$
$601$	−43.4714	−1.77324	−0.886618	+	0.462502i	$0.846952\pi$
$602$	3.94474	0.160776
$603$	−7.76227	−0.316104
$604$	13.1514	0.535122
$605$	−15.4396	−0.627710
$606$	−4.45108	−0.180813
$607$	18.5048	0.751086	0.375543	−	0.926805i	$-0.377456\pi$
$607$	18.5048	0.751086	0.375543	+	0.926805i	$0.377456\pi$
$608$	−4.08520	−0.165677
$609$	−0.182581	−0.00739857
$610$	−18.0239	−0.729766
$611$	−7.51682	−0.304098
$612$	−3.55497	−0.143701
$613$	−4.31437	−0.174256	−0.0871278	−	0.996197i	$-0.527769\pi$
$613$	−4.31437	−0.174256	−0.0871278	+	0.996197i	$0.527769\pi$
$614$	11.2493	0.453986
$615$	1.08381	0.0437036
$616$	0.201357	0.00811291
$617$	15.3995	0.619960	0.309980	−	0.950743i	$-0.399678\pi$
$617$	15.3995	0.619960	0.309980	+	0.950743i	$0.399678\pi$
$618$	1.00000	0.0402259
$619$	−26.5851	−1.06854	−0.534272	−	0.845313i	$-0.679414\pi$
$619$	−26.5851	−1.06854	−0.534272	+	0.845313i	$0.679414\pi$
$620$	1.94823	0.0782427
$621$	6.44178	0.258500
$622$	27.2339	1.09198
$623$	−1.43669	−0.0575597
$624$	−1.00000	−0.0400320
$625$	−1.89727	−0.0758909
$626$	−5.84803	−0.233734
$627$	2.23814	0.0893829
$628$	7.00388	0.279485
$629$	5.53711	0.220779
$630$	0.530337	0.0211291
$631$	−27.2053	−1.08303	−0.541514	−	0.840692i	$-0.682149\pi$
$631$	−27.2053	−1.08303	−0.541514	+	0.840692i	$0.682149\pi$
$632$	−15.8651	−0.631081
$633$	25.4564	1.01180
$634$	−7.97888	−0.316882
$635$	3.23425	0.128347
$636$	−1.23606	−0.0490129
$637$	−6.86492	−0.271998
$638$	0.272169	0.0107753
$639$	−10.7114	−0.423738
$640$	1.44298	0.0570387
$641$	−9.72247	−0.384014	−0.192007	−	0.981394i	$-0.561500\pi$
$641$	−9.72247	−0.384014	−0.192007	+	0.981394i	$0.561500\pi$
$642$	−11.8666	−0.468339
$643$	11.8052	0.465551	0.232776	−	0.972530i	$-0.425219\pi$
$643$	11.8052	0.465551	0.232776	+	0.972530i	$0.425219\pi$
$644$	−2.36755	−0.0932944
$645$	−15.4876	−0.609825
$646$	14.5228	0.571391
$647$	20.3152	0.798674	0.399337	−	0.916804i	$-0.369240\pi$
$647$	20.3152	0.798674	0.399337	+	0.916804i	$0.369240\pi$
$648$	1.00000	0.0392837
$649$	1.77942	0.0698483
$650$	−2.91782	−0.114446
$651$	−0.496218	−0.0194483
$652$	3.96067	0.155112
$653$	−6.05275	−0.236862	−0.118431	−	0.992962i	$-0.537786\pi$
$653$	−6.05275	−0.236862	−0.118431	+	0.992962i	$0.537786\pi$
$654$	−17.5556	−0.686479
$655$	14.3021	0.558831
$656$	−0.751095	−0.0293253
$657$	−12.6599	−0.493909
$658$	−2.76265	−0.107699
$659$	48.6361	1.89460	0.947298	−	0.320355i	$-0.103802\pi$
$659$	48.6361	1.89460	0.947298	+	0.320355i	$0.103802\pi$
$660$	−0.790559	−0.0307724
$661$	−25.4492	−0.989859	−0.494929	−	0.868933i	$-0.664806\pi$
$661$	−25.4492	−0.989859	−0.494929	+	0.868933i	$0.664806\pi$
$662$	−28.0019	−1.08832
$663$	3.55497	0.138064
$664$	8.69957	0.337609
$665$	−2.16654	−0.0840146
$666$	−1.55757	−0.0603546
$667$	−3.20015	−0.123910
$668$	19.6742	0.761217
$669$	−17.1504	−0.663074
$670$	−11.2008	−0.432724
$671$	−6.84327	−0.264181
$672$	−0.367530	−0.0141778
$673$	11.5630	0.445721	0.222860	−	0.974850i	$-0.428461\pi$
$673$	11.5630	0.445721	0.222860	+	0.974850i	$0.428461\pi$
$674$	−9.31312	−0.358728
$675$	2.91782	0.112307
$676$	1.00000	0.0384615
$677$	29.0327	1.11582	0.557910	−	0.829902i	$-0.311604\pi$
$677$	29.0327	1.11582	0.557910	+	0.829902i	$0.311604\pi$
$678$	18.7990	0.721971
$679$	0.279975	0.0107445
$680$	−5.12974	−0.196717
$681$	17.7510	0.680220
$682$	0.739698	0.0283245
$683$	−5.69397	−0.217874	−0.108937	−	0.994049i	$-0.534745\pi$
$683$	−5.69397	−0.217874	−0.108937	+	0.994049i	$0.534745\pi$
$684$	−4.08520	−0.156202
$685$	−18.3876	−0.702554
$686$	−5.09577	−0.194557
$687$	6.19924	0.236516
$688$	10.7331	0.409196
$689$	1.23606	0.0470901
$690$	9.29535	0.353868
$691$	−29.9649	−1.13992	−0.569960	−	0.821673i	$-0.693041\pi$
$691$	−29.9649	−1.13992	−0.569960	+	0.821673i	$0.693041\pi$
$692$	11.0291	0.419264
$693$	0.201357	0.00764892
$694$	−9.47773	−0.359770
$695$	12.2662	0.465282
$696$	−0.496780	−0.0188304
$697$	2.67012	0.101138
$698$	−17.8767	−0.676643
$699$	19.5650	0.740015
$700$	−1.07238	−0.0405323
$701$	17.1846	0.649052	0.324526	−	0.945877i	$-0.394795\pi$
$701$	17.1846	0.649052	0.324526	+	0.945877i	$0.394795\pi$
$702$	−1.00000	−0.0377426
$703$	6.36299	0.239985
$704$	0.547866	0.0206485
$705$	10.8466	0.408507
$706$	−30.8518	−1.16112
$707$	1.63590	0.0615245
$708$	−3.24791	−0.122064
$709$	−2.26108	−0.0849167	−0.0424584	−	0.999098i	$-0.513519\pi$
$709$	−2.26108	−0.0849167	−0.0424584	+	0.999098i	$0.513519\pi$
$710$	−15.4564	−0.580067
$711$	−15.8651	−0.594989
$712$	−3.90904	−0.146497
$713$	−8.69733	−0.325718
$714$	1.30656	0.0488966
$715$	0.790559	0.0295652
$716$	18.9259	0.707294
$717$	11.5577	0.431631
$718$	−5.25381	−0.196070
$719$	−27.1574	−1.01280	−0.506399	−	0.862299i	$-0.669024\pi$
$719$	−27.1574	−1.01280	−0.506399	+	0.862299i	$0.669024\pi$
$720$	1.44298	0.0537766
$721$	−0.367530	−0.0136875
$722$	−2.31111	−0.0860108
$723$	−10.8027	−0.401758
$724$	8.01462	0.297861
$725$	−1.44951	−0.0538335
$726$	10.6998	0.397108
$727$	−21.8260	−0.809480	−0.404740	−	0.914432i	$-0.632638\pi$
$727$	−21.8260	−0.809480	−0.404740	+	0.914432i	$0.632638\pi$
$728$	0.367530	0.0136216
$729$	1.00000	0.0370370
$730$	−18.2679	−0.676126
$731$	−38.1559	−1.41125
$732$	12.4908	0.461672
$733$	28.4759	1.05178	0.525891	−	0.850552i	$-0.323732\pi$
$733$	28.4759	1.05178	0.525891	+	0.850552i	$0.323732\pi$
$734$	−8.29246	−0.306080
$735$	9.90593	0.365386
$736$	−6.44178	−0.237447
$737$	−4.25268	−0.156650
$738$	−0.751095	−0.0276482
$739$	−22.0525	−0.811214	−0.405607	−	0.914048i	$-0.632940\pi$
$739$	−22.0525	−0.811214	−0.405607	+	0.914048i	$0.632940\pi$
$740$	−2.24754	−0.0826212
$741$	4.08520	0.150074
$742$	0.454288	0.0166774
$743$	50.5530	1.85461	0.927305	−	0.374306i	$-0.122119\pi$
$743$	50.5530	1.85461	0.927305	+	0.374306i	$0.122119\pi$
$744$	−1.35014	−0.0494987
$745$	13.8552	0.507616
$746$	−19.8937	−0.728360
$747$	8.69957	0.318300
$748$	−1.94765	−0.0712130
$749$	4.36135	0.159360
$750$	11.4252	0.417190
$751$	−3.70047	−0.135032	−0.0675161	−	0.997718i	$-0.521507\pi$
$751$	−3.70047	−0.135032	−0.0675161	+	0.997718i	$0.521507\pi$
$752$	−7.51682	−0.274110
$753$	−11.7031	−0.426485
$754$	0.496780	0.0180916
$755$	18.9771	0.690648
$756$	−0.367530	−0.0133669
$757$	22.6460	0.823082	0.411541	−	0.911391i	$-0.364991\pi$
$757$	22.6460	0.823082	0.411541	+	0.911391i	$0.364991\pi$
$758$	15.8718	0.576488
$759$	3.52923	0.128103
$760$	−5.89486	−0.213829
$761$	47.7971	1.73265	0.866323	−	0.499485i	$-0.166477\pi$
$761$	47.7971	1.73265	0.866323	+	0.499485i	$0.166477\pi$
$762$	−2.24137	−0.0811964
$763$	6.45221	0.233586
$764$	−9.99549	−0.361624
$765$	−5.12974	−0.185466
$766$	−17.5544	−0.634266
$767$	3.24791	0.117275
$768$	−1.00000	−0.0360844
$769$	34.1465	1.23135	0.615677	−	0.787998i	$-0.288882\pi$
$769$	34.1465	1.23135	0.615677	+	0.787998i	$0.288882\pi$
$770$	0.290554	0.0104708
$771$	2.51465	0.0905627
$772$	20.0860	0.722912
$773$	5.38187	0.193572	0.0967861	−	0.995305i	$-0.469144\pi$
$773$	5.38187	0.193572	0.0967861	+	0.995305i	$0.469144\pi$
$774$	10.7331	0.385794
$775$	−3.93947	−0.141510
$776$	0.761776	0.0273462
$777$	0.572453	0.0205367
$778$	−28.0548	−1.00581
$779$	3.06838	0.109936
$780$	−1.44298	−0.0516669
$781$	−5.86844	−0.209989
$782$	22.9003	0.818914
$783$	−0.496780	−0.0177535
$784$	−6.86492	−0.245176
$785$	10.1064	0.360715
$786$	−9.91155	−0.353533
$787$	27.6134	0.984310	0.492155	−	0.870508i	$-0.336209\pi$
$787$	27.6134	0.984310	0.492155	+	0.870508i	$0.336209\pi$
$788$	−19.3711	−0.690068
$789$	−16.2335	−0.577928
$790$	−22.8930	−0.814498
$791$	−6.90919	−0.245662
$792$	0.547866	0.0194676
$793$	−12.4908	−0.443560
$794$	−31.9046	−1.13225
$795$	−1.78360	−0.0632579
$796$	−11.5330	−0.408778
$797$	−8.58448	−0.304078	−0.152039	−	0.988375i	$-0.548584\pi$
$797$	−8.58448	−0.304078	−0.152039	+	0.988375i	$0.548584\pi$
$798$	1.50143	0.0531502
$799$	26.7220	0.945358
$800$	−2.91782	−0.103160
$801$	−3.90904	−0.138119
$802$	−19.1941	−0.677767
$803$	−6.93592	−0.244763
$804$	7.76227	0.273754
$805$	−3.41632	−0.120409
$806$	1.35014	0.0475568
$807$	−13.1976	−0.464579
$808$	4.45108	0.156588
$809$	34.8384	1.22485	0.612426	−	0.790528i	$-0.290194\pi$
$809$	34.8384	1.22485	0.612426	+	0.790528i	$0.290194\pi$
$810$	1.44298	0.0507011
$811$	−16.2717	−0.571375	−0.285688	−	0.958323i	$-0.592222\pi$
$811$	−16.2717	−0.571375	−0.285688	+	0.958323i	$0.592222\pi$
$812$	0.182581	0.00640735
$813$	26.4860	0.928903
$814$	−0.853340	−0.0299096
$815$	5.71516	0.200193
$816$	3.55497	0.124449
$817$	−43.8469	−1.53401
$818$	−25.2693	−0.883520
$819$	0.367530	0.0128425
$820$	−1.08381	−0.0378484
$821$	−0.468662	−0.0163564	−0.00817821	−	0.999967i	$-0.502603\pi$
$821$	−0.468662	−0.0163564	−0.00817821	+	0.999967i	$0.502603\pi$
$822$	12.7428	0.444457
$823$	−2.14677	−0.0748317	−0.0374158	−	0.999300i	$-0.511913\pi$
$823$	−2.14677	−0.0748317	−0.0374158	+	0.999300i	$0.511913\pi$
$824$	−1.00000	−0.0348367
$825$	1.59857	0.0556551
$826$	1.19370	0.0415342
$827$	−2.31041	−0.0803407	−0.0401704	−	0.999193i	$-0.512790\pi$
$827$	−2.31041	−0.0803407	−0.0401704	+	0.999193i	$0.512790\pi$
$828$	−6.44178	−0.223867
$829$	−13.9682	−0.485134	−0.242567	−	0.970135i	$-0.577989\pi$
$829$	−13.9682	−0.485134	−0.242567	+	0.970135i	$0.577989\pi$
$830$	12.5533	0.435731
$831$	−0.916739	−0.0318013
$832$	1.00000	0.0346688
$833$	24.4046	0.845568
$834$	−8.50060	−0.294352
$835$	28.3894	0.982456
$836$	−2.23814	−0.0774078
$837$	−1.35014	−0.0466678
$838$	−20.3789	−0.703978
$839$	−17.4439	−0.602229	−0.301115	−	0.953588i	$-0.597359\pi$
$839$	−17.4439	−0.602229	−0.301115	+	0.953588i	$0.597359\pi$
$840$	−0.530337	−0.0182984
$841$	−28.7532	−0.991490
$842$	32.7240	1.12774
$843$	−0.265437	−0.00914213
$844$	−25.4564	−0.876245
$845$	1.44298	0.0496399
$846$	−7.51682	−0.258434
$847$	−3.93251	−0.135123
$848$	1.23606	0.0424464
$849$	0.758999	0.0260488
$850$	10.3727	0.355782
$851$	10.0335	0.343945
$852$	10.7114	0.366968
$853$	−9.14256	−0.313035	−0.156518	−	0.987675i	$-0.550027\pi$
$853$	−9.14256	−0.313035	−0.156518	+	0.987675i	$0.550027\pi$
$854$	−4.59073	−0.157091
$855$	−5.89486	−0.201600
$856$	11.8666	0.405594
$857$	47.4325	1.62026	0.810131	−	0.586249i	$-0.199396\pi$
$857$	47.4325	1.62026	0.810131	+	0.586249i	$0.199396\pi$
$858$	−0.547866	−0.0187038
$859$	34.7702	1.18634	0.593171	−	0.805076i	$-0.297876\pi$
$859$	34.7702	1.18634	0.593171	+	0.805076i	$0.297876\pi$
$860$	15.4876	0.528124
$861$	0.276050	0.00940775
$862$	−10.4093	−0.354542
$863$	10.4815	0.356794	0.178397	−	0.983959i	$-0.442909\pi$
$863$	10.4815	0.356794	0.178397	+	0.983959i	$0.442909\pi$
$864$	−1.00000	−0.0340207
$865$	15.9148	0.541118
$866$	−8.79314	−0.298803
$867$	4.36220	0.148148
$868$	0.496218	0.0168427
$869$	−8.69197	−0.294855
$870$	−0.716842	−0.0243032
$871$	−7.76227	−0.263014
$872$	17.5556	0.594508
$873$	0.761776	0.0257822
$874$	26.3160	0.890152
$875$	−4.19911	−0.141956
$876$	12.6599	0.427738
$877$	−33.6797	−1.13728	−0.568642	−	0.822585i	$-0.692531\pi$
$877$	−33.6797	−1.13728	−0.568642	+	0.822585i	$0.692531\pi$
$878$	30.7549	1.03793
$879$	−4.69443	−0.158339
$880$	0.790559	0.0266497
$881$	−41.6950	−1.40474	−0.702371	−	0.711811i	$-0.747875\pi$
$881$	−41.6950	−1.40474	−0.702371	+	0.711811i	$0.747875\pi$
$882$	−6.86492	−0.231154
$883$	50.7231	1.70697	0.853484	−	0.521119i	$-0.174485\pi$
$883$	50.7231	1.70697	0.853484	+	0.521119i	$0.174485\pi$
$884$	−3.55497	−0.119567
$885$	−4.68666	−0.157540
$886$	−21.1042	−0.709011
$887$	−5.84804	−0.196358	−0.0981790	−	0.995169i	$-0.531302\pi$
$887$	−5.84804	−0.196358	−0.0981790	+	0.995169i	$0.531302\pi$
$888$	1.55757	0.0522686
$889$	0.823772	0.0276284
$890$	−5.64065	−0.189075
$891$	0.547866	0.0183542
$892$	17.1504	0.574239
$893$	30.7077	1.02760
$894$	−9.60182	−0.321133
$895$	27.3096	0.912860
$896$	0.367530	0.0122783
$897$	6.44178	0.215085
$898$	17.3977	0.580569
$899$	0.670724	0.0223699
$900$	−2.91782	−0.0972605
$901$	−4.39415	−0.146390
$902$	−0.411500	−0.0137014
$903$	−3.94474	−0.131273
$904$	−18.7990	−0.625245
$905$	11.5649	0.384431
$906$	−13.1514	−0.436925
$907$	12.8499	0.426673	0.213336	−	0.976979i	$-0.431567\pi$
$907$	12.8499	0.426673	0.213336	+	0.976979i	$0.431567\pi$
$908$	−17.7510	−0.589088
$909$	4.45108	0.147633
$910$	0.530337	0.0175805
$911$	−7.46535	−0.247338	−0.123669	−	0.992324i	$-0.539466\pi$
$911$	−7.46535	−0.247338	−0.123669	+	0.992324i	$0.539466\pi$
$912$	4.08520	0.135275
$913$	4.76620	0.157738
$914$	−11.1130	−0.367587
$915$	18.0239	0.595852
$916$	−6.19924	−0.204829
$917$	3.64279	0.120295
$918$	3.55497	0.117331
$919$	47.9151	1.58057	0.790286	−	0.612738i	$-0.209932\pi$
$919$	47.9151	1.58057	0.790286	+	0.612738i	$0.209932\pi$
$920$	−9.29535	−0.306458
$921$	−11.2493	−0.370678
$922$	1.40726	0.0463455
$923$	−10.7114	−0.352572
$924$	−0.201357	−0.00662416
$925$	4.54470	0.149429
$926$	24.9641	0.820372
$927$	−1.00000	−0.0328443
$928$	0.496780	0.0163076
$929$	58.8597	1.93113	0.965563	−	0.260170i	$-0.0837785\pi$
$929$	58.8597	1.93113	0.965563	+	0.260170i	$0.0837785\pi$
$930$	−1.94823	−0.0638849
$931$	28.0446	0.919125
$932$	−19.5650	−0.640872
$933$	−27.2339	−0.891597
$934$	39.0266	1.27699
$935$	−2.81041	−0.0919102
$936$	1.00000	0.0326860
$937$	27.9848	0.914224	0.457112	−	0.889409i	$-0.348884\pi$
$937$	27.9848	0.914224	0.457112	+	0.889409i	$0.348884\pi$
$938$	−2.85286	−0.0931493
$939$	5.84803	0.190843
$940$	−10.8466	−0.353777
$941$	30.5569	0.996126	0.498063	−	0.867141i	$-0.334045\pi$
$941$	30.5569	0.996126	0.498063	+	0.867141i	$0.334045\pi$
$942$	−7.00388	−0.228199
$943$	4.83839	0.157560
$944$	3.24791	0.105710
$945$	−0.530337	−0.0172519
$946$	5.88031	0.191185
$947$	28.3802	0.922234	0.461117	−	0.887339i	$-0.347449\pi$
$947$	28.3802	0.922234	0.461117	+	0.887339i	$0.347449\pi$
$948$	15.8651	0.515276
$949$	−12.6599	−0.410957
$950$	11.9199	0.386732
$951$	7.97888	0.258733
$952$	−1.30656	−0.0423457
$953$	−50.2727	−1.62849	−0.814247	−	0.580519i	$-0.802850\pi$
$953$	−50.2727	−1.62849	−0.814247	+	0.580519i	$0.802850\pi$
$954$	1.23606	0.0400189
$955$	−14.4233	−0.466726
$956$	−11.5577	−0.373804
$957$	−0.272169	−0.00879797
$958$	−6.49329	−0.209789
$959$	−4.68337	−0.151234
$960$	−1.44298	−0.0465719
$961$	−29.1771	−0.941197
$962$	−1.55757	−0.0502181
$963$	11.8666	0.382397
$964$	10.8027	0.347932
$965$	28.9837	0.933018
$966$	2.36755	0.0761746
$967$	−25.3126	−0.813998	−0.406999	−	0.913429i	$-0.633425\pi$
$967$	−25.3126	−0.813998	−0.406999	+	0.913429i	$0.633425\pi$
$968$	−10.6998	−0.343906
$969$	−14.5228	−0.466538
$970$	1.09923	0.0352940
$971$	4.02520	0.129175	0.0645874	−	0.997912i	$-0.479427\pi$
$971$	4.02520	0.129175	0.0645874	+	0.997912i	$0.479427\pi$
$972$	−1.00000	−0.0320750
$973$	3.12422	0.100158
$974$	7.81244	0.250327
$975$	2.91782	0.0934449
$976$	−12.4908	−0.399820
$977$	−0.313974	−0.0100449	−0.00502246	−	0.999987i	$-0.501599\pi$
$977$	−0.313974	−0.0100449	−0.00502246	+	0.999987i	$0.501599\pi$
$978$	−3.96067	−0.126648
$979$	−2.14163	−0.0684468
$980$	−9.90593	−0.316433
$981$	17.5556	0.560508
$982$	17.4032	0.555358
$983$	−59.2236	−1.88894	−0.944470	−	0.328599i	$-0.893424\pi$
$983$	−59.2236	−1.88894	−0.944470	+	0.328599i	$0.893424\pi$
$984$	0.751095	0.0239440
$985$	−27.9521	−0.890628
$986$	−1.76604	−0.0562420
$987$	2.76265	0.0879363
$988$	−4.08520	−0.129968
$989$	−69.1404	−2.19854
$990$	0.790559	0.0251256
$991$	−12.3984	−0.393849	−0.196924	−	0.980419i	$-0.563095\pi$
$991$	−12.3984	−0.393849	−0.196924	+	0.980419i	$0.563095\pi$
$992$	1.35014	0.0428671
$993$	28.0019	0.888612
$994$	−3.93678	−0.124867
$995$	−16.6419	−0.527584
$996$	−8.69957	−0.275656
$997$	−57.3758	−1.81711	−0.908555	−	0.417766i	$-0.862813\pi$
$997$	−57.3758	−1.81711	−0.908555	+	0.417766i	$0.862813\pi$
$998$	−6.53910	−0.206992
$999$	1.55757	0.0492793

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8034.2.a.r.1.8	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8034.2.a.r.1.8	✓	9	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.44298 q^{5} -1.00000 q^{6} +0.367530 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.44298 q^{5} -1.00000 q^{6} +0.367530 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.44298 q^{10} +0.547866 q^{11} -1.00000 q^{12} +1.00000 q^{13} +0.367530 q^{14} -1.44298 q^{15} +1.00000 q^{16} -3.55497 q^{17} +1.00000 q^{18} -4.08520 q^{19} +1.44298 q^{20} -0.367530 q^{21} +0.547866 q^{22} -6.44178 q^{23} -1.00000 q^{24} -2.91782 q^{25} +1.00000 q^{26} -1.00000 q^{27} +0.367530 q^{28} +0.496780 q^{29} -1.44298 q^{30} +1.35014 q^{31} +1.00000 q^{32} -0.547866 q^{33} -3.55497 q^{34} +0.530337 q^{35} +1.00000 q^{36} -1.55757 q^{37} -4.08520 q^{38} -1.00000 q^{39} +1.44298 q^{40} -0.751095 q^{41} -0.367530 q^{42} +10.7331 q^{43} +0.547866 q^{44} +1.44298 q^{45} -6.44178 q^{46} -7.51682 q^{47} -1.00000 q^{48} -6.86492 q^{49} -2.91782 q^{50} +3.55497 q^{51} +1.00000 q^{52} +1.23606 q^{53} -1.00000 q^{54} +0.790559 q^{55} +0.367530 q^{56} +4.08520 q^{57} +0.496780 q^{58} +3.24791 q^{59} -1.44298 q^{60} -12.4908 q^{61} +1.35014 q^{62} +0.367530 q^{63} +1.00000 q^{64} +1.44298 q^{65} -0.547866 q^{66} -7.76227 q^{67} -3.55497 q^{68} +6.44178 q^{69} +0.530337 q^{70} -10.7114 q^{71} +1.00000 q^{72} -12.6599 q^{73} -1.55757 q^{74} +2.91782 q^{75} -4.08520 q^{76} +0.201357 q^{77} -1.00000 q^{78} -15.8651 q^{79} +1.44298 q^{80} +1.00000 q^{81} -0.751095 q^{82} +8.69957 q^{83} -0.367530 q^{84} -5.12974 q^{85} +10.7331 q^{86} -0.496780 q^{87} +0.547866 q^{88} -3.90904 q^{89} +1.44298 q^{90} +0.367530 q^{91} -6.44178 q^{92} -1.35014 q^{93} -7.51682 q^{94} -5.89486 q^{95} -1.00000 q^{96} +0.761776 q^{97} -6.86492 q^{98} +0.547866 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{2} - 9 q^{3} + 9 q^{4} - 4 q^{5} - 9 q^{6} - 4 q^{7} + 9 q^{8} + 9 q^{9}+O(q^{10}) \) 9 * q + 9 * q^2 - 9 * q^3 + 9 * q^4 - 4 * q^5 - 9 * q^6 - 4 * q^7 + 9 * q^8 + 9 * q^9	\( 9 q + 9 q^{2} - 9 q^{3} + 9 q^{4} - 4 q^{5} - 9 q^{6} - 4 q^{7} + 9 q^{8} + 9 q^{9} - 4 q^{10} - 5 q^{11} - 9 q^{12} + 9 q^{13} - 4 q^{14} + 4 q^{15} + 9 q^{16} - 6 q^{17} + 9 q^{18} - 4 q^{19} - 4 q^{20} + 4 q^{21} - 5 q^{22} - 6 q^{23} - 9 q^{24} - 11 q^{25} + 9 q^{26} - 9 q^{27} - 4 q^{28} - 19 q^{29} + 4 q^{30} - 6 q^{31} + 9 q^{32} + 5 q^{33} - 6 q^{34} + 10 q^{35} + 9 q^{36} - 13 q^{37} - 4 q^{38} - 9 q^{39} - 4 q^{40} - 18 q^{41} + 4 q^{42} - 20 q^{43} - 5 q^{44} - 4 q^{45} - 6 q^{46} + 14 q^{47} - 9 q^{48} - 3 q^{49} - 11 q^{50} + 6 q^{51} + 9 q^{52} - 3 q^{53} - 9 q^{54} - 4 q^{55} - 4 q^{56} + 4 q^{57} - 19 q^{58} - 9 q^{59} + 4 q^{60} - 24 q^{61} - 6 q^{62} - 4 q^{63} + 9 q^{64} - 4 q^{65} + 5 q^{66} - 4 q^{67} - 6 q^{68} + 6 q^{69} + 10 q^{70} - 9 q^{71} + 9 q^{72} - 24 q^{73} - 13 q^{74} + 11 q^{75} - 4 q^{76} + 3 q^{77} - 9 q^{78} - 15 q^{79} - 4 q^{80} + 9 q^{81} - 18 q^{82} + 20 q^{83} + 4 q^{84} - 31 q^{85} - 20 q^{86} + 19 q^{87} - 5 q^{88} + 3 q^{89} - 4 q^{90} - 4 q^{91} - 6 q^{92} + 6 q^{93} + 14 q^{94} - 4 q^{95} - 9 q^{96} - 19 q^{97} - 3 q^{98} - 5 q^{99}+O(q^{100}) \) 9 * q + 9 * q^2 - 9 * q^3 + 9 * q^4 - 4 * q^5 - 9 * q^6 - 4 * q^7 + 9 * q^8 + 9 * q^9 - 4 * q^10 - 5 * q^11 - 9 * q^12 + 9 * q^13 - 4 * q^14 + 4 * q^15 + 9 * q^16 - 6 * q^17 + 9 * q^18 - 4 * q^19 - 4 * q^20 + 4 * q^21 - 5 * q^22 - 6 * q^23 - 9 * q^24 - 11 * q^25 + 9 * q^26 - 9 * q^27 - 4 * q^28 - 19 * q^29 + 4 * q^30 - 6 * q^31 + 9 * q^32 + 5 * q^33 - 6 * q^34 + 10 * q^35 + 9 * q^36 - 13 * q^37 - 4 * q^38 - 9 * q^39 - 4 * q^40 - 18 * q^41 + 4 * q^42 - 20 * q^43 - 5 * q^44 - 4 * q^45 - 6 * q^46 + 14 * q^47 - 9 * q^48 - 3 * q^49 - 11 * q^50 + 6 * q^51 + 9 * q^52 - 3 * q^53 - 9 * q^54 - 4 * q^55 - 4 * q^56 + 4 * q^57 - 19 * q^58 - 9 * q^59 + 4 * q^60 - 24 * q^61 - 6 * q^62 - 4 * q^63 + 9 * q^64 - 4 * q^65 + 5 * q^66 - 4 * q^67 - 6 * q^68 + 6 * q^69 + 10 * q^70 - 9 * q^71 + 9 * q^72 - 24 * q^73 - 13 * q^74 + 11 * q^75 - 4 * q^76 + 3 * q^77 - 9 * q^78 - 15 * q^79 - 4 * q^80 + 9 * q^81 - 18 * q^82 + 20 * q^83 + 4 * q^84 - 31 * q^85 - 20 * q^86 + 19 * q^87 - 5 * q^88 + 3 * q^89 - 4 * q^90 - 4 * q^91 - 6 * q^92 + 6 * q^93 + 14 * q^94 - 4 * q^95 - 9 * q^96 - 19 * q^97 - 3 * q^98 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more