LMFDB - Embedded newform 8010.2.a.bn.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8010,2,Mod(1,8010)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8010, 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("8010.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8010 = 2 \cdot 3^{2} \cdot 5 \cdot 89 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8010.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:	$63.9601720190$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 24x^{5} + 34x^{4} + 111x^{3} - 127x^{2} - 20x + 16 $ x^7 - 3x^6 - 24x^5 + 34x^4 + 111x^3 - 127x^2 - 20x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 2670)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-3.29882$ of defining polynomial
Character	$\chi$	$=$	8010.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -3.64705 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -3.64705 q^{7} +1.00000 q^{8} +1.00000 q^{10} +3.31719 q^{11} +5.17045 q^{13} -3.64705 q^{14} +1.00000 q^{16} +1.97320 q^{17} +5.28045 q^{19} +1.00000 q^{20} +3.31719 q^{22} +5.77849 q^{23} +1.00000 q^{25} +5.17045 q^{26} -3.64705 q^{28} +1.32256 q^{29} -10.4645 q^{31} +1.00000 q^{32} +1.97320 q^{34} -3.64705 q^{35} -10.2715 q^{37} +5.28045 q^{38} +1.00000 q^{40} -3.47543 q^{41} +0.262874 q^{43} +3.31719 q^{44} +5.77849 q^{46} +7.66848 q^{47} +6.30098 q^{49} +1.00000 q^{50} +5.17045 q^{52} +4.24666 q^{53} +3.31719 q^{55} -3.64705 q^{56} +1.32256 q^{58} -6.43940 q^{59} +3.19112 q^{61} -10.4645 q^{62} +1.00000 q^{64} +5.17045 q^{65} +2.21986 q^{67} +1.97320 q^{68} -3.64705 q^{70} +4.83817 q^{71} +7.45234 q^{73} -10.2715 q^{74} +5.28045 q^{76} -12.0980 q^{77} +7.23248 q^{79} +1.00000 q^{80} -3.47543 q^{82} -8.50666 q^{83} +1.97320 q^{85} +0.262874 q^{86} +3.31719 q^{88} -1.00000 q^{89} -18.8569 q^{91} +5.77849 q^{92} +7.66848 q^{94} +5.28045 q^{95} +13.5570 q^{97} +6.30098 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 7 q^{2} + 7 q^{4} + 7 q^{5} + q^{7} + 7 q^{8}+O(q^{10}) $ 7 * q + 7 * q^2 + 7 * q^4 + 7 * q^5 + q^7 + 7 * q^8	$ 7 q + 7 q^{2} + 7 q^{4} + 7 q^{5} + q^{7} + 7 q^{8} + 7 q^{10} - q^{11} + q^{14} + 7 q^{16} + 9 q^{17} + 9 q^{19} + 7 q^{20} - q^{22} + 8 q^{23} + 7 q^{25} + q^{28} + 4 q^{29} + 16 q^{31} + 7 q^{32} + 9 q^{34} + q^{35} + 2 q^{37} + 9 q^{38} + 7 q^{40} + q^{41} - 10 q^{43} - q^{44} + 8 q^{46} + 13 q^{47} + 26 q^{49} + 7 q^{50} + 24 q^{53} - q^{55} + q^{56} + 4 q^{58} + 6 q^{59} + 23 q^{61} + 16 q^{62} + 7 q^{64} + 5 q^{67} + 9 q^{68} + q^{70} + 8 q^{71} - 2 q^{73} + 2 q^{74} + 9 q^{76} + 6 q^{77} + 7 q^{79} + 7 q^{80} + q^{82} + 7 q^{83} + 9 q^{85} - 10 q^{86} - q^{88} - 7 q^{89} + 48 q^{91} + 8 q^{92} + 13 q^{94} + 9 q^{95} + 30 q^{97} + 26 q^{98}+O(q^{100}) $ 7 * q + 7 * q^2 + 7 * q^4 + 7 * q^5 + q^7 + 7 * q^8 + 7 * q^10 - q^11 + q^14 + 7 * q^16 + 9 * q^17 + 9 * q^19 + 7 * q^20 - q^22 + 8 * q^23 + 7 * q^25 + q^28 + 4 * q^29 + 16 * q^31 + 7 * q^32 + 9 * q^34 + q^35 + 2 * q^37 + 9 * q^38 + 7 * q^40 + q^41 - 10 * q^43 - q^44 + 8 * q^46 + 13 * q^47 + 26 * q^49 + 7 * q^50 + 24 * q^53 - q^55 + q^56 + 4 * q^58 + 6 * q^59 + 23 * q^61 + 16 * q^62 + 7 * q^64 + 5 * q^67 + 9 * q^68 + q^70 + 8 * q^71 - 2 * q^73 + 2 * q^74 + 9 * q^76 + 6 * q^77 + 7 * q^79 + 7 * q^80 + q^82 + 7 * q^83 + 9 * q^85 - 10 * q^86 - q^88 - 7 * q^89 + 48 * q^91 + 8 * q^92 + 13 * q^94 + 9 * q^95 + 30 * q^97 + 26 * q^98

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
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−3.64705	−1.37846	−0.689228	−	0.724545i	$-0.742050\pi$
$7$	−3.64705	−1.37846	−0.689228	+	0.724545i	$0.742050\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	1.00000	0.316228
$11$	3.31719	1.00017	0.500085	−	0.865976i	$-0.333302\pi$
$11$	3.31719	1.00017	0.500085	+	0.865976i	$0.333302\pi$
$12$	0	0
$13$	5.17045	1.43402	0.717012	−	0.697061i	$-0.245509\pi$
$13$	5.17045	1.43402	0.717012	+	0.697061i	$0.245509\pi$
$14$	−3.64705	−0.974715
$15$	0	0
$16$	1.00000	0.250000
$17$	1.97320	0.478570	0.239285	−	0.970949i	$-0.423087\pi$
$17$	1.97320	0.478570	0.239285	+	0.970949i	$0.423087\pi$
$18$	0	0
$19$	5.28045	1.21142	0.605709	−	0.795686i	$-0.292889\pi$
$19$	5.28045	1.21142	0.605709	+	0.795686i	$0.292889\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	3.31719	0.707227
$23$	5.77849	1.20490	0.602449	−	0.798157i	$-0.294192\pi$
$23$	5.77849	1.20490	0.602449	+	0.798157i	$0.294192\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	5.17045	1.01401
$27$	0	0
$28$	−3.64705	−0.689228
$29$	1.32256	0.245593	0.122796	−	0.992432i	$-0.460814\pi$
$29$	1.32256	0.245593	0.122796	+	0.992432i	$0.460814\pi$
$30$	0	0
$31$	−10.4645	−1.87949	−0.939744	−	0.341878i	$-0.888937\pi$
$31$	−10.4645	−1.87949	−0.939744	+	0.341878i	$0.888937\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	1.97320	0.338400
$35$	−3.64705	−0.616464
$36$	0	0
$37$	−10.2715	−1.68862	−0.844312	−	0.535852i	$-0.819990\pi$
$37$	−10.2715	−1.68862	−0.844312	+	0.535852i	$0.819990\pi$
$38$	5.28045	0.856602
$39$	0	0
$40$	1.00000	0.158114
$41$	−3.47543	−0.542771	−0.271385	−	0.962471i	$-0.587482\pi$
$41$	−3.47543	−0.542771	−0.271385	+	0.962471i	$0.587482\pi$
$42$	0	0
$43$	0.262874	0.0400879	0.0200440	−	0.999799i	$-0.493619\pi$
$43$	0.262874	0.0400879	0.0200440	+	0.999799i	$0.493619\pi$
$44$	3.31719	0.500085
$45$	0	0
$46$	5.77849	0.851992
$47$	7.66848	1.11856	0.559282	−	0.828978i	$-0.311077\pi$
$47$	7.66848	1.11856	0.559282	+	0.828978i	$0.311077\pi$
$48$	0	0
$49$	6.30098	0.900140
$50$	1.00000	0.141421
$51$	0	0
$52$	5.17045	0.717012
$53$	4.24666	0.583324	0.291662	−	0.956521i	$-0.405792\pi$
$53$	4.24666	0.583324	0.291662	+	0.956521i	$0.405792\pi$
$54$	0	0
$55$	3.31719	0.447289
$56$	−3.64705	−0.487358
$57$	0	0
$58$	1.32256	0.173660
$59$	−6.43940	−0.838338	−0.419169	−	0.907908i	$-0.637679\pi$
$59$	−6.43940	−0.838338	−0.419169	+	0.907908i	$0.637679\pi$
$60$	0	0
$61$	3.19112	0.408581	0.204291	−	0.978910i	$-0.434511\pi$
$61$	3.19112	0.408581	0.204291	+	0.978910i	$0.434511\pi$
$62$	−10.4645	−1.32900
$63$	0	0
$64$	1.00000	0.125000
$65$	5.17045	0.641315
$66$	0	0
$67$	2.21986	0.271199	0.135600	−	0.990764i	$-0.456704\pi$
$67$	2.21986	0.271199	0.135600	+	0.990764i	$0.456704\pi$
$68$	1.97320	0.239285
$69$	0	0
$70$	−3.64705	−0.435906
$71$	4.83817	0.574185	0.287093	−	0.957903i	$-0.407311\pi$
$71$	4.83817	0.574185	0.287093	+	0.957903i	$0.407311\pi$
$72$	0	0
$73$	7.45234	0.872231	0.436115	−	0.899891i	$-0.356354\pi$
$73$	7.45234	0.872231	0.436115	+	0.899891i	$0.356354\pi$
$74$	−10.2715	−1.19404
$75$	0	0
$76$	5.28045	0.605709
$77$	−12.0980	−1.37869
$78$	0	0
$79$	7.23248	0.813718	0.406859	−	0.913491i	$-0.366624\pi$
$79$	7.23248	0.813718	0.406859	+	0.913491i	$0.366624\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	−3.47543	−0.383797
$83$	−8.50666	−0.933727	−0.466863	−	0.884329i	$-0.654616\pi$
$83$	−8.50666	−0.933727	−0.466863	+	0.884329i	$0.654616\pi$
$84$	0	0
$85$	1.97320	0.214023
$86$	0.262874	0.0283464
$87$	0	0
$88$	3.31719	0.353613
$89$	−1.00000	−0.106000
$89$	−1.00000	−0.106000
$90$	0	0
$91$	−18.8569	−1.97674
$92$	5.77849	0.602449
$93$	0	0
$94$	7.66848	0.790944
$95$	5.28045	0.541763
$96$	0	0
$97$	13.5570	1.37650	0.688251	−	0.725473i	$-0.258379\pi$
$97$	13.5570	1.37650	0.688251	+	0.725473i	$0.258379\pi$
$98$	6.30098	0.636495
$99$	0	0
$100$	1.00000	0.100000
$101$	−15.8727	−1.57939	−0.789697	−	0.613497i	$-0.789762\pi$
$101$	−15.8727	−1.57939	−0.789697	+	0.613497i	$0.789762\pi$
$102$	0	0
$103$	−6.99104	−0.688848	−0.344424	−	0.938814i	$-0.611926\pi$
$103$	−6.99104	−0.688848	−0.344424	+	0.938814i	$0.611926\pi$
$104$	5.17045	0.507004
$105$	0	0
$106$	4.24666	0.412472
$107$	−0.922600	−0.0891911	−0.0445955	−	0.999005i	$-0.514200\pi$
$107$	−0.922600	−0.0891911	−0.0445955	+	0.999005i	$0.514200\pi$
$108$	0	0
$109$	8.45593	0.809931	0.404966	−	0.914332i	$-0.367284\pi$
$109$	8.45593	0.809931	0.404966	+	0.914332i	$0.367284\pi$
$110$	3.31719	0.316281
$111$	0	0
$112$	−3.64705	−0.344614
$113$	1.80502	0.169802	0.0849011	−	0.996389i	$-0.472943\pi$
$113$	1.80502	0.169802	0.0849011	+	0.996389i	$0.472943\pi$
$114$	0	0
$115$	5.77849	0.538847
$116$	1.32256	0.122796
$117$	0	0
$118$	−6.43940	−0.592795
$119$	−7.19634	−0.659688
$120$	0	0
$121$	0.00373415	0.000339468 0
$122$	3.19112	0.288911
$123$	0	0
$124$	−10.4645	−0.939744
$125$	1.00000	0.0894427
$126$	0	0
$127$	−18.1701	−1.61234	−0.806168	−	0.591686i	$-0.798462\pi$
$127$	−18.1701	−1.61234	−0.806168	+	0.591686i	$0.798462\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	5.17045	0.453478
$131$	−3.80680	−0.332601	−0.166301	−	0.986075i	$-0.553182\pi$
$131$	−3.80680	−0.332601	−0.166301	+	0.986075i	$0.553182\pi$
$132$	0	0
$133$	−19.2581	−1.66989
$134$	2.21986	0.191767
$135$	0	0
$136$	1.97320	0.169200
$137$	−12.4394	−1.06277	−0.531385	−	0.847131i	$-0.678328\pi$
$137$	−12.4394	−1.06277	−0.531385	+	0.847131i	$0.678328\pi$
$138$	0	0
$139$	17.9285	1.52067	0.760337	−	0.649529i	$-0.225034\pi$
$139$	17.9285	1.52067	0.760337	+	0.649529i	$0.225034\pi$
$140$	−3.64705	−0.308232
$141$	0	0
$142$	4.83817	0.406010
$143$	17.1513	1.43427
$144$	0	0
$145$	1.32256	0.109832
$146$	7.45234	0.616760
$147$	0	0
$148$	−10.2715	−0.844312
$149$	13.8565	1.13517	0.567585	−	0.823315i	$-0.307878\pi$
$149$	13.8565	1.13517	0.567585	+	0.823315i	$0.307878\pi$
$150$	0	0
$151$	17.8048	1.44894	0.724468	−	0.689308i	$-0.242085\pi$
$151$	17.8048	1.44894	0.724468	+	0.689308i	$0.242085\pi$
$152$	5.28045	0.428301
$153$	0	0
$154$	−12.0980	−0.974881
$155$	−10.4645	−0.840533
$156$	0	0
$157$	17.7199	1.41420	0.707099	−	0.707115i	$-0.250004\pi$
$157$	17.7199	1.41420	0.707099	+	0.707115i	$0.250004\pi$
$158$	7.23248	0.575385
$159$	0	0
$160$	1.00000	0.0790569
$161$	−21.0744	−1.66090
$162$	0	0
$163$	−0.0734716	−0.00575474	−0.00287737	−	0.999996i	$-0.500916\pi$
$163$	−0.0734716	−0.00575474	−0.00287737	+	0.999996i	$0.500916\pi$
$164$	−3.47543	−0.271385
$165$	0	0
$166$	−8.50666	−0.660245
$167$	7.17887	0.555518	0.277759	−	0.960651i	$-0.410408\pi$
$167$	7.17887	0.555518	0.277759	+	0.960651i	$0.410408\pi$
$168$	0	0
$169$	13.7335	1.05643
$170$	1.97320	0.151337
$171$	0	0
$172$	0.262874	0.0200440
$173$	9.83136	0.747464	0.373732	−	0.927537i	$-0.378078\pi$
$173$	9.83136	0.747464	0.373732	+	0.927537i	$0.378078\pi$
$174$	0	0
$175$	−3.64705	−0.275691
$176$	3.31719	0.250042
$177$	0	0
$178$	−1.00000	−0.0749532
$179$	−5.12462	−0.383032	−0.191516	−	0.981489i	$-0.561340\pi$
$179$	−5.12462	−0.383032	−0.191516	+	0.981489i	$0.561340\pi$
$180$	0	0
$181$	13.1767	0.979418	0.489709	−	0.871886i	$-0.337103\pi$
$181$	13.1767	0.979418	0.489709	+	0.871886i	$0.337103\pi$
$182$	−18.8569	−1.39777
$183$	0	0
$184$	5.77849	0.425996
$185$	−10.2715	−0.755175
$186$	0	0
$187$	6.54546	0.478651
$188$	7.66848	0.559282
$189$	0	0
$190$	5.28045	0.383084
$191$	14.8940	1.07769	0.538847	−	0.842404i	$-0.318860\pi$
$191$	14.8940	1.07769	0.538847	+	0.842404i	$0.318860\pi$
$192$	0	0
$193$	2.80109	0.201627	0.100814	−	0.994905i	$-0.467855\pi$
$193$	2.80109	0.201627	0.100814	+	0.994905i	$0.467855\pi$
$194$	13.5570	0.973334
$195$	0	0
$196$	6.30098	0.450070
$197$	8.05584	0.573955	0.286978	−	0.957937i	$-0.407349\pi$
$197$	8.05584	0.573955	0.286978	+	0.957937i	$0.407349\pi$
$198$	0	0
$199$	2.91824	0.206869	0.103434	−	0.994636i	$-0.467017\pi$
$199$	2.91824	0.206869	0.103434	+	0.994636i	$0.467017\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	−15.8727	−1.11680
$203$	−4.82344	−0.338539
$204$	0	0
$205$	−3.47543	−0.242734
$206$	−6.99104	−0.487089
$207$	0	0
$208$	5.17045	0.358506
$209$	17.5163	1.21162
$210$	0	0
$211$	6.25294	0.430470	0.215235	−	0.976562i	$-0.430948\pi$
$211$	6.25294	0.430470	0.215235	+	0.976562i	$0.430948\pi$
$212$	4.24666	0.291662
$213$	0	0
$214$	−0.922600	−0.0630676
$215$	0.262874	0.0179279
$216$	0	0
$217$	38.1647	2.59079
$218$	8.45593	0.572708
$219$	0	0
$220$	3.31719	0.223645
$221$	10.2023	0.686281
$222$	0	0
$223$	15.1255	1.01288	0.506438	−	0.862276i	$-0.330962\pi$
$223$	15.1255	1.01288	0.506438	+	0.862276i	$0.330962\pi$
$224$	−3.64705	−0.243679
$225$	0	0
$226$	1.80502	0.120068
$227$	−18.1179	−1.20253	−0.601263	−	0.799051i	$-0.705336\pi$
$227$	−18.1179	−1.20253	−0.601263	+	0.799051i	$0.705336\pi$
$228$	0	0
$229$	17.6936	1.16923	0.584614	−	0.811311i	$-0.301246\pi$
$229$	17.6936	1.16923	0.584614	+	0.811311i	$0.301246\pi$
$230$	5.77849	0.381022
$231$	0	0
$232$	1.32256	0.0868302
$233$	15.8674	1.03950	0.519752	−	0.854317i	$-0.326024\pi$
$233$	15.8674	1.03950	0.519752	+	0.854317i	$0.326024\pi$
$234$	0	0
$235$	7.66848	0.500237
$236$	−6.43940	−0.419169
$237$	0	0
$238$	−7.19634	−0.466470
$239$	−5.85824	−0.378938	−0.189469	−	0.981887i	$-0.560677\pi$
$239$	−5.85824	−0.378938	−0.189469	+	0.981887i	$0.560677\pi$
$240$	0	0
$241$	20.8972	1.34611	0.673055	−	0.739592i	$-0.264982\pi$
$241$	20.8972	1.34611	0.673055	+	0.739592i	$0.264982\pi$
$242$	0.00373415	0.000240040 0
$243$	0	0
$244$	3.19112	0.204291
$245$	6.30098	0.402555
$246$	0	0
$247$	27.3023	1.73720
$248$	−10.4645	−0.664500
$249$	0	0
$250$	1.00000	0.0632456
$251$	8.59941	0.542790	0.271395	−	0.962468i	$-0.412515\pi$
$251$	8.59941	0.542790	0.271395	+	0.962468i	$0.412515\pi$
$252$	0	0
$253$	19.1683	1.20510
$254$	−18.1701	−1.14009
$255$	0	0
$256$	1.00000	0.0625000
$257$	20.1377	1.25615	0.628076	−	0.778152i	$-0.283843\pi$
$257$	20.1377	1.25615	0.628076	+	0.778152i	$0.283843\pi$
$258$	0	0
$259$	37.4607	2.32769
$260$	5.17045	0.320658
$261$	0	0
$262$	−3.80680	−0.235185
$263$	−19.3840	−1.19527	−0.597636	−	0.801768i	$-0.703893\pi$
$263$	−19.3840	−1.19527	−0.597636	+	0.801768i	$0.703893\pi$
$264$	0	0
$265$	4.24666	0.260870
$266$	−19.2581	−1.18079
$267$	0	0
$268$	2.21986	0.135600
$269$	31.4358	1.91667	0.958336	−	0.285644i	$-0.0922074\pi$
$269$	31.4358	1.91667	0.958336	+	0.285644i	$0.0922074\pi$
$270$	0	0
$271$	−1.48617	−0.0902784	−0.0451392	−	0.998981i	$-0.514373\pi$
$271$	−1.48617	−0.0902784	−0.0451392	+	0.998981i	$0.514373\pi$
$272$	1.97320	0.119643
$273$	0	0
$274$	−12.4394	−0.751492
$275$	3.31719	0.200034
$276$	0	0
$277$	−13.0295	−0.782868	−0.391434	−	0.920206i	$-0.628021\pi$
$277$	−13.0295	−0.782868	−0.391434	+	0.920206i	$0.628021\pi$
$278$	17.9285	1.07528
$279$	0	0
$280$	−3.64705	−0.217953
$281$	−26.4224	−1.57623	−0.788115	−	0.615527i	$-0.788943\pi$
$281$	−26.4224	−1.57623	−0.788115	+	0.615527i	$0.788943\pi$
$282$	0	0
$283$	1.51396	0.0899956	0.0449978	−	0.998987i	$-0.485672\pi$
$283$	1.51396	0.0899956	0.0449978	+	0.998987i	$0.485672\pi$
$284$	4.83817	0.287093
$285$	0	0
$286$	17.1513	1.01418
$287$	12.6751	0.748185
$288$	0	0
$289$	−13.1065	−0.770971
$290$	1.32256	0.0776633
$291$	0	0
$292$	7.45234	0.436115
$293$	−22.9170	−1.33882	−0.669411	−	0.742892i	$-0.733454\pi$
$293$	−22.9170	−1.33882	−0.669411	+	0.742892i	$0.733454\pi$
$294$	0	0
$295$	−6.43940	−0.374916
$296$	−10.2715	−0.597018
$297$	0	0
$298$	13.8565	0.802686
$299$	29.8774	1.72785
$300$	0	0
$301$	−0.958715	−0.0552594
$302$	17.8048	1.02455
$303$	0	0
$304$	5.28045	0.302855
$305$	3.19112	0.182723
$306$	0	0
$307$	−17.4930	−0.998376	−0.499188	−	0.866494i	$-0.666368\pi$
$307$	−17.4930	−0.998376	−0.499188	+	0.866494i	$0.666368\pi$
$308$	−12.0980	−0.689345
$309$	0	0
$310$	−10.4645	−0.594346
$311$	−20.7523	−1.17675	−0.588376	−	0.808587i	$-0.700233\pi$
$311$	−20.7523	−1.17675	−0.588376	+	0.808587i	$0.700233\pi$
$312$	0	0
$313$	−26.1280	−1.47684	−0.738421	−	0.674340i	$-0.764428\pi$
$313$	−26.1280	−1.47684	−0.738421	+	0.674340i	$0.764428\pi$
$314$	17.7199	0.999989
$315$	0	0
$316$	7.23248	0.406859
$317$	−33.6785	−1.89157	−0.945787	−	0.324788i	$-0.894707\pi$
$317$	−33.6785	−1.89157	−0.945787	+	0.324788i	$0.894707\pi$
$318$	0	0
$319$	4.38717	0.245635
$320$	1.00000	0.0559017
$321$	0	0
$322$	−21.0744	−1.17443
$323$	10.4194	0.579749
$324$	0	0
$325$	5.17045	0.286805
$326$	−0.0734716	−0.00406921
$327$	0	0
$328$	−3.47543	−0.191898
$329$	−27.9673	−1.54189
$330$	0	0
$331$	−11.7065	−0.643449	−0.321724	−	0.946833i	$-0.604262\pi$
$331$	−11.7065	−0.643449	−0.321724	+	0.946833i	$0.604262\pi$
$332$	−8.50666	−0.466863
$333$	0	0
$334$	7.17887	0.392811
$335$	2.21986	0.121284
$336$	0	0
$337$	11.9327	0.650018	0.325009	−	0.945711i	$-0.394633\pi$
$337$	11.9327	0.650018	0.325009	+	0.945711i	$0.394633\pi$
$338$	13.7335	0.747006
$339$	0	0
$340$	1.97320	0.107012
$341$	−34.7129	−1.87981
$342$	0	0
$343$	2.54937	0.137653
$344$	0.262874	0.0141732
$345$	0	0
$346$	9.83136	0.528537
$347$	−35.7909	−1.92136	−0.960679	−	0.277663i	$-0.910440\pi$
$347$	−35.7909	−1.92136	−0.960679	+	0.277663i	$0.910440\pi$
$348$	0	0
$349$	26.7581	1.43233	0.716165	−	0.697931i	$-0.245896\pi$
$349$	26.7581	1.43233	0.716165	+	0.697931i	$0.245896\pi$
$350$	−3.64705	−0.194943
$351$	0	0
$352$	3.31719	0.176807
$353$	−8.82551	−0.469734	−0.234867	−	0.972027i	$-0.575466\pi$
$353$	−8.82551	−0.469734	−0.234867	+	0.972027i	$0.575466\pi$
$354$	0	0
$355$	4.83817	0.256783
$356$	−1.00000	−0.0529999
$357$	0	0
$358$	−5.12462	−0.270845
$359$	4.69895	0.248001	0.124001	−	0.992282i	$-0.460428\pi$
$359$	4.69895	0.248001	0.124001	+	0.992282i	$0.460428\pi$
$360$	0	0
$361$	8.88317	0.467535
$362$	13.1767	0.692553
$363$	0	0
$364$	−18.8569	−0.988369
$365$	7.45234	0.390073
$366$	0	0
$367$	−15.8388	−0.826778	−0.413389	−	0.910554i	$-0.635655\pi$
$367$	−15.8388	−0.826778	−0.413389	+	0.910554i	$0.635655\pi$
$368$	5.77849	0.301224
$369$	0	0
$370$	−10.2715	−0.533990
$371$	−15.4878	−0.804086
$372$	0	0
$373$	−1.40360	−0.0726755	−0.0363378	−	0.999340i	$-0.511569\pi$
$373$	−1.40360	−0.0726755	−0.0363378	+	0.999340i	$0.511569\pi$
$374$	6.54546	0.338458
$375$	0	0
$376$	7.66848	0.395472
$377$	6.83822	0.352186
$378$	0	0
$379$	−24.1508	−1.24054	−0.620272	−	0.784387i	$-0.712978\pi$
$379$	−24.1508	−1.24054	−0.620272	+	0.784387i	$0.712978\pi$
$380$	5.28045	0.270881
$381$	0	0
$382$	14.8940	0.762044
$383$	−4.93861	−0.252351	−0.126176	−	0.992008i	$-0.540270\pi$
$383$	−4.93861	−0.252351	−0.126176	+	0.992008i	$0.540270\pi$
$384$	0	0
$385$	−12.0980	−0.616569
$386$	2.80109	0.142572
$387$	0	0
$388$	13.5570	0.688251
$389$	13.0166	0.659970	0.329985	−	0.943986i	$-0.392956\pi$
$389$	13.0166	0.659970	0.329985	+	0.943986i	$0.392956\pi$
$390$	0	0
$391$	11.4021	0.576628
$392$	6.30098	0.318247
$393$	0	0
$394$	8.05584	0.405848
$395$	7.23248	0.363906
$396$	0	0
$397$	7.07387	0.355027	0.177514	−	0.984118i	$-0.443195\pi$
$397$	7.07387	0.355027	0.177514	+	0.984118i	$0.443195\pi$
$398$	2.91824	0.146278
$399$	0	0
$400$	1.00000	0.0500000
$401$	25.7088	1.28384	0.641918	−	0.766773i	$-0.278139\pi$
$401$	25.7088	1.28384	0.641918	+	0.766773i	$0.278139\pi$
$402$	0	0
$403$	−54.1064	−2.69523
$404$	−15.8727	−0.789697
$405$	0	0
$406$	−4.82344	−0.239383
$407$	−34.0725	−1.68891
$408$	0	0
$409$	22.2724	1.10130	0.550650	−	0.834736i	$-0.314380\pi$
$409$	22.2724	1.10130	0.550650	+	0.834736i	$0.314380\pi$
$410$	−3.47543	−0.171639
$411$	0	0
$412$	−6.99104	−0.344424
$413$	23.4848	1.15561
$414$	0	0
$415$	−8.50666	−0.417575
$416$	5.17045	0.253502
$417$	0	0
$418$	17.5163	0.856748
$419$	3.22620	0.157610	0.0788052	−	0.996890i	$-0.474889\pi$
$419$	3.22620	0.157610	0.0788052	+	0.996890i	$0.474889\pi$
$420$	0	0
$421$	25.6092	1.24812	0.624058	−	0.781378i	$-0.285483\pi$
$421$	25.6092	1.24812	0.624058	+	0.781378i	$0.285483\pi$
$422$	6.25294	0.304388
$423$	0	0
$424$	4.24666	0.206236
$425$	1.97320	0.0957141
$426$	0	0
$427$	−11.6382	−0.563211
$428$	−0.922600	−0.0445955
$429$	0	0
$430$	0.262874	0.0126769
$431$	−4.11340	−0.198136	−0.0990678	−	0.995081i	$-0.531586\pi$
$431$	−4.11340	−0.198136	−0.0990678	+	0.995081i	$0.531586\pi$
$432$	0	0
$433$	−35.5734	−1.70955	−0.854774	−	0.519000i	$-0.826305\pi$
$433$	−35.5734	−1.70955	−0.854774	+	0.519000i	$0.826305\pi$
$434$	38.1647	1.83197
$435$	0	0
$436$	8.45593	0.404966
$437$	30.5130	1.45964
$438$	0	0
$439$	−34.0959	−1.62731	−0.813654	−	0.581349i	$-0.802525\pi$
$439$	−34.0959	−1.62731	−0.813654	+	0.581349i	$0.802525\pi$
$440$	3.31719	0.158141
$441$	0	0
$442$	10.2023	0.485274
$443$	−18.2063	−0.865007	−0.432504	−	0.901632i	$-0.642370\pi$
$443$	−18.2063	−0.865007	−0.432504	+	0.901632i	$0.642370\pi$
$444$	0	0
$445$	−1.00000	−0.0474045
$446$	15.1255	0.716211
$447$	0	0
$448$	−3.64705	−0.172307
$449$	3.52934	0.166560	0.0832798	−	0.996526i	$-0.473460\pi$
$449$	3.52934	0.166560	0.0832798	+	0.996526i	$0.473460\pi$
$450$	0	0
$451$	−11.5287	−0.542863
$452$	1.80502	0.0849011
$453$	0	0
$454$	−18.1179	−0.850314
$455$	−18.8569	−0.884025
$456$	0	0
$457$	9.11268	0.426273	0.213137	−	0.977022i	$-0.431632\pi$
$457$	9.11268	0.426273	0.213137	+	0.977022i	$0.431632\pi$
$458$	17.6936	0.826769
$459$	0	0
$460$	5.77849	0.269423
$461$	5.95122	0.277176	0.138588	−	0.990350i	$-0.455744\pi$
$461$	5.95122	0.277176	0.138588	+	0.990350i	$0.455744\pi$
$462$	0	0
$463$	5.31935	0.247211	0.123606	−	0.992331i	$-0.460554\pi$
$463$	5.31935	0.247211	0.123606	+	0.992331i	$0.460554\pi$
$464$	1.32256	0.0613982
$465$	0	0
$466$	15.8674	0.735041
$467$	−20.5248	−0.949775	−0.474888	−	0.880046i	$-0.657511\pi$
$467$	−20.5248	−0.949775	−0.474888	+	0.880046i	$0.657511\pi$
$468$	0	0
$469$	−8.09594	−0.373836
$470$	7.66848	0.353721
$471$	0	0
$472$	−6.43940	−0.296397
$473$	0.872003	0.0400947
$474$	0	0
$475$	5.28045	0.242284
$476$	−7.19634	−0.329844
$477$	0	0
$478$	−5.85824	−0.267950
$479$	−33.9066	−1.54923	−0.774616	−	0.632431i	$-0.782057\pi$
$479$	−33.9066	−1.54923	−0.774616	+	0.632431i	$0.782057\pi$
$480$	0	0
$481$	−53.1082	−2.42153
$482$	20.8972	0.951844
$483$	0	0
$484$	0.00373415	0.000169734 0
$485$	13.5570	0.615591
$486$	0	0
$487$	33.2364	1.50609	0.753043	−	0.657971i	$-0.228585\pi$
$487$	33.2364	1.50609	0.753043	+	0.657971i	$0.228585\pi$
$488$	3.19112	0.144455
$489$	0	0
$490$	6.30098	0.284649
$491$	39.9859	1.80454	0.902269	−	0.431173i	$-0.141900\pi$
$491$	39.9859	1.80454	0.902269	+	0.431173i	$0.141900\pi$
$492$	0	0
$493$	2.60967	0.117533
$494$	27.3023	1.22839
$495$	0	0
$496$	−10.4645	−0.469872
$497$	−17.6451	−0.791489
$498$	0	0
$499$	5.04411	0.225805	0.112903	−	0.993606i	$-0.463985\pi$
$499$	5.04411	0.225805	0.112903	+	0.993606i	$0.463985\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	8.59941	0.383811
$503$	15.5393	0.692861	0.346431	−	0.938076i	$-0.387394\pi$
$503$	15.5393	0.692861	0.346431	+	0.938076i	$0.387394\pi$
$504$	0	0
$505$	−15.8727	−0.706327
$506$	19.1683	0.852136
$507$	0	0
$508$	−18.1701	−0.806168
$509$	−2.82763	−0.125332	−0.0626661	−	0.998035i	$-0.519960\pi$
$509$	−2.82763	−0.125332	−0.0626661	+	0.998035i	$0.519960\pi$
$510$	0	0
$511$	−27.1791	−1.20233
$512$	1.00000	0.0441942
$513$	0	0
$514$	20.1377	0.888234
$515$	−6.99104	−0.308062
$516$	0	0
$517$	25.4378	1.11875
$518$	37.4607	1.64593
$519$	0	0
$520$	5.17045	0.226739
$521$	44.7613	1.96103	0.980514	−	0.196449i	$-0.0629412\pi$
$521$	44.7613	1.96103	0.980514	+	0.196449i	$0.0629412\pi$
$522$	0	0
$523$	−30.2236	−1.32158	−0.660792	−	0.750569i	$-0.729780\pi$
$523$	−30.2236	−1.32158	−0.660792	+	0.750569i	$0.729780\pi$
$524$	−3.80680	−0.166301
$525$	0	0
$526$	−19.3840	−0.845184
$527$	−20.6486	−0.899467
$528$	0	0
$529$	10.3909	0.451779
$530$	4.24666	0.184463
$531$	0	0
$532$	−19.2581	−0.834943
$533$	−17.9695	−0.778347
$534$	0	0
$535$	−0.922600	−0.0398875
$536$	2.21986	0.0958833
$537$	0	0
$538$	31.4358	1.35529
$539$	20.9015	0.900292
$540$	0	0
$541$	41.8184	1.79792	0.898958	−	0.438036i	$-0.144326\pi$
$541$	41.8184	1.79792	0.898958	+	0.438036i	$0.144326\pi$
$542$	−1.48617	−0.0638365
$543$	0	0
$544$	1.97320	0.0846001
$545$	8.45593	0.362212
$546$	0	0
$547$	1.02155	0.0436783	0.0218392	−	0.999761i	$-0.493048\pi$
$547$	1.02155	0.0436783	0.0218392	+	0.999761i	$0.493048\pi$
$548$	−12.4394	−0.531385
$549$	0	0
$550$	3.31719	0.141445
$551$	6.98370	0.297516
$552$	0	0
$553$	−26.3772	−1.12167
$554$	−13.0295	−0.553571
$555$	0	0
$556$	17.9285	0.760337
$557$	21.5881	0.914716	0.457358	−	0.889283i	$-0.348796\pi$
$557$	21.5881	0.914716	0.457358	+	0.889283i	$0.348796\pi$
$558$	0	0
$559$	1.35918	0.0574871
$560$	−3.64705	−0.154116
$561$	0	0
$562$	−26.4224	−1.11456
$563$	2.30168	0.0970042	0.0485021	−	0.998823i	$-0.484555\pi$
$563$	2.30168	0.0970042	0.0485021	+	0.998823i	$0.484555\pi$
$564$	0	0
$565$	1.80502	0.0759379
$566$	1.51396	0.0636365
$567$	0	0
$568$	4.83817	0.203005
$569$	−6.03242	−0.252892	−0.126446	−	0.991973i	$-0.540357\pi$
$569$	−6.03242	−0.252892	−0.126446	+	0.991973i	$0.540357\pi$
$570$	0	0
$571$	21.0639	0.881498	0.440749	−	0.897630i	$-0.354713\pi$
$571$	21.0639	0.881498	0.440749	+	0.897630i	$0.354713\pi$
$572$	17.1513	0.717134
$573$	0	0
$574$	12.6751	0.529047
$575$	5.77849	0.240980
$576$	0	0
$577$	−26.3543	−1.09714	−0.548572	−	0.836103i	$-0.684828\pi$
$577$	−26.3543	−1.09714	−0.548572	+	0.836103i	$0.684828\pi$
$578$	−13.1065	−0.545158
$579$	0	0
$580$	1.32256	0.0549162
$581$	31.0242	1.28710
$582$	0	0
$583$	14.0870	0.583423
$584$	7.45234	0.308380
$585$	0	0
$586$	−22.9170	−0.946691
$587$	2.47549	0.102175	0.0510873	−	0.998694i	$-0.483731\pi$
$587$	2.47549	0.102175	0.0510873	+	0.998694i	$0.483731\pi$
$588$	0	0
$589$	−55.2576	−2.27685
$590$	−6.43940	−0.265106
$591$	0	0
$592$	−10.2715	−0.422156
$593$	−23.5656	−0.967724	−0.483862	−	0.875144i	$-0.660766\pi$
$593$	−23.5656	−0.967724	−0.483862	+	0.875144i	$0.660766\pi$
$594$	0	0
$595$	−7.19634	−0.295021
$596$	13.8565	0.567585
$597$	0	0
$598$	29.8774	1.22178
$599$	−0.499637	−0.0204146	−0.0102073	−	0.999948i	$-0.503249\pi$
$599$	−0.499637	−0.0204146	−0.0102073	+	0.999948i	$0.503249\pi$
$600$	0	0
$601$	30.8622	1.25889	0.629447	−	0.777043i	$-0.283282\pi$
$601$	30.8622	1.25889	0.629447	+	0.777043i	$0.283282\pi$
$602$	−0.958715	−0.0390743
$603$	0	0
$604$	17.8048	0.724468
$605$	0.00373415	0.000151815 0
$606$	0	0
$607$	41.2491	1.67425	0.837124	−	0.547013i	$-0.184235\pi$
$607$	41.2491	1.67425	0.837124	+	0.547013i	$0.184235\pi$
$608$	5.28045	0.214151
$609$	0	0
$610$	3.19112	0.129205
$611$	39.6495	1.60405
$612$	0	0
$613$	−32.6385	−1.31826	−0.659128	−	0.752030i	$-0.729075\pi$
$613$	−32.6385	−1.31826	−0.659128	+	0.752030i	$0.729075\pi$
$614$	−17.4930	−0.705959
$615$	0	0
$616$	−12.0980	−0.487440
$617$	−32.4153	−1.30499	−0.652495	−	0.757793i	$-0.726278\pi$
$617$	−32.4153	−1.30499	−0.652495	+	0.757793i	$0.726278\pi$
$618$	0	0
$619$	−28.1198	−1.13023	−0.565116	−	0.825012i	$-0.691168\pi$
$619$	−28.1198	−1.13023	−0.565116	+	0.825012i	$0.691168\pi$
$620$	−10.4645	−0.420266
$621$	0	0
$622$	−20.7523	−0.832090
$623$	3.64705	0.146116
$624$	0	0
$625$	1.00000	0.0400000
$626$	−26.1280	−1.04429
$627$	0	0
$628$	17.7199	0.707099
$629$	−20.2677	−0.808125
$630$	0	0
$631$	−22.7633	−0.906194	−0.453097	−	0.891461i	$-0.649681\pi$
$631$	−22.7633	−0.906194	−0.453097	+	0.891461i	$0.649681\pi$
$632$	7.23248	0.287693
$633$	0	0
$634$	−33.6785	−1.33754
$635$	−18.1701	−0.721059
$636$	0	0
$637$	32.5789	1.29082
$638$	4.38717	0.173690
$639$	0	0
$640$	1.00000	0.0395285
$641$	−38.0419	−1.50256	−0.751281	−	0.659982i	$-0.770564\pi$
$641$	−38.0419	−1.50256	−0.751281	+	0.659982i	$0.770564\pi$
$642$	0	0
$643$	27.8509	1.09833	0.549166	−	0.835713i	$-0.314945\pi$
$643$	27.8509	1.09833	0.549166	+	0.835713i	$0.314945\pi$
$644$	−21.0744	−0.830449
$645$	0	0
$646$	10.4194	0.409944
$647$	−26.1100	−1.02649	−0.513244	−	0.858243i	$-0.671557\pi$
$647$	−26.1100	−1.02649	−0.513244	+	0.858243i	$0.671557\pi$
$648$	0	0
$649$	−21.3607	−0.838481
$650$	5.17045	0.202802
$651$	0	0
$652$	−0.0734716	−0.00287737
$653$	35.1409	1.37517	0.687584	−	0.726104i	$-0.258671\pi$
$653$	35.1409	1.37517	0.687584	+	0.726104i	$0.258671\pi$
$654$	0	0
$655$	−3.80680	−0.148744
$656$	−3.47543	−0.135693
$657$	0	0
$658$	−27.9673	−1.09028
$659$	−3.02960	−0.118017	−0.0590083	−	0.998257i	$-0.518794\pi$
$659$	−3.02960	−0.118017	−0.0590083	+	0.998257i	$0.518794\pi$
$660$	0	0
$661$	−43.8067	−1.70388	−0.851941	−	0.523638i	$-0.824574\pi$
$661$	−43.8067	−1.70388	−0.851941	+	0.523638i	$0.824574\pi$
$662$	−11.7065	−0.454987
$663$	0	0
$664$	−8.50666	−0.330122
$665$	−19.2581	−0.746796
$666$	0	0
$667$	7.64239	0.295914
$668$	7.17887	0.277759
$669$	0	0
$670$	2.21986	0.0857607
$671$	10.5855	0.408651
$672$	0	0
$673$	−36.1293	−1.39268	−0.696341	−	0.717711i	$-0.745190\pi$
$673$	−36.1293	−1.39268	−0.696341	+	0.717711i	$0.745190\pi$
$674$	11.9327	0.459632
$675$	0	0
$676$	13.7335	0.528213
$677$	−31.9720	−1.22878	−0.614391	−	0.789002i	$-0.710598\pi$
$677$	−31.9720	−1.22878	−0.614391	+	0.789002i	$0.710598\pi$
$678$	0	0
$679$	−49.4430	−1.89745
$680$	1.97320	0.0756686
$681$	0	0
$682$	−34.7129	−1.32922
$683$	−31.3713	−1.20039	−0.600194	−	0.799855i	$-0.704910\pi$
$683$	−31.3713	−1.20039	−0.600194	+	0.799855i	$0.704910\pi$
$684$	0	0
$685$	−12.4394	−0.475285
$686$	2.54937	0.0973354
$687$	0	0
$688$	0.262874	0.0100220
$689$	21.9572	0.836501
$690$	0	0
$691$	−29.7786	−1.13283	−0.566416	−	0.824119i	$-0.691671\pi$
$691$	−29.7786	−1.13283	−0.566416	+	0.824119i	$0.691671\pi$
$692$	9.83136	0.373732
$693$	0	0
$694$	−35.7909	−1.35860
$695$	17.9285	0.680066
$696$	0	0
$697$	−6.85770	−0.259754
$698$	26.7581	1.01281
$699$	0	0
$700$	−3.64705	−0.137846
$701$	34.9766	1.32105	0.660523	−	0.750806i	$-0.270335\pi$
$701$	34.9766	1.32105	0.660523	+	0.750806i	$0.270335\pi$
$702$	0	0
$703$	−54.2381	−2.04563
$704$	3.31719	0.125021
$705$	0	0
$706$	−8.82551	−0.332152
$707$	57.8886	2.17713
$708$	0	0
$709$	0.336739	0.0126465	0.00632325	−	0.999980i	$-0.497987\pi$
$709$	0.336739	0.0126465	0.00632325	+	0.999980i	$0.497987\pi$
$710$	4.83817	0.181573
$711$	0	0
$712$	−1.00000	−0.0374766
$713$	−60.4693	−2.26459
$714$	0	0
$715$	17.1513	0.641424
$716$	−5.12462	−0.191516
$717$	0	0
$718$	4.69895	0.175363
$719$	8.41957	0.313997	0.156998	−	0.987599i	$-0.449818\pi$
$719$	8.41957	0.313997	0.156998	+	0.987599i	$0.449818\pi$
$720$	0	0
$721$	25.4967	0.949546
$722$	8.88317	0.330597
$723$	0	0
$724$	13.1767	0.489709
$725$	1.32256	0.0491186
$726$	0	0
$727$	−44.5902	−1.65376	−0.826880	−	0.562379i	$-0.809886\pi$
$727$	−44.5902	−1.65376	−0.826880	+	0.562379i	$0.809886\pi$
$728$	−18.8569	−0.698883
$729$	0	0
$730$	7.45234	0.275824
$731$	0.518702	0.0191849
$732$	0	0
$733$	14.1992	0.524460	0.262230	−	0.965005i	$-0.415542\pi$
$733$	14.1992	0.524460	0.262230	+	0.965005i	$0.415542\pi$
$734$	−15.8388	−0.584621
$735$	0	0
$736$	5.77849	0.212998
$737$	7.36369	0.271245
$738$	0	0
$739$	5.05209	0.185844	0.0929221	−	0.995673i	$-0.470379\pi$
$739$	5.05209	0.185844	0.0929221	+	0.995673i	$0.470379\pi$
$740$	−10.2715	−0.377588
$741$	0	0
$742$	−15.4878	−0.568575
$743$	−36.3476	−1.33346	−0.666732	−	0.745298i	$-0.732307\pi$
$743$	−36.3476	−1.33346	−0.666732	+	0.745298i	$0.732307\pi$
$744$	0	0
$745$	13.8565	0.507663
$746$	−1.40360	−0.0513893
$747$	0	0
$748$	6.54546	0.239326
$749$	3.36477	0.122946
$750$	0	0
$751$	−3.96324	−0.144621	−0.0723104	−	0.997382i	$-0.523037\pi$
$751$	−3.96324	−0.144621	−0.0723104	+	0.997382i	$0.523037\pi$
$752$	7.66848	0.279641
$753$	0	0
$754$	6.83822	0.249033
$755$	17.8048	0.647984
$756$	0	0
$757$	37.9573	1.37958	0.689791	−	0.724009i	$-0.257702\pi$
$757$	37.9573	1.37958	0.689791	+	0.724009i	$0.257702\pi$
$758$	−24.1508	−0.877197
$759$	0	0
$760$	5.28045	0.191542
$761$	4.62457	0.167641	0.0838203	−	0.996481i	$-0.473288\pi$
$761$	4.62457	0.167641	0.0838203	+	0.996481i	$0.473288\pi$
$762$	0	0
$763$	−30.8392	−1.11645
$764$	14.8940	0.538847
$765$	0	0
$766$	−4.93861	−0.178439
$767$	−33.2946	−1.20220
$768$	0	0
$769$	21.8987	0.789686	0.394843	−	0.918749i	$-0.370799\pi$
$769$	21.8987	0.789686	0.394843	+	0.918749i	$0.370799\pi$
$770$	−12.0980	−0.435980
$771$	0	0
$772$	2.80109	0.100814
$773$	−7.38708	−0.265695	−0.132847	−	0.991136i	$-0.542412\pi$
$773$	−7.38708	−0.265695	−0.132847	+	0.991136i	$0.542412\pi$
$774$	0	0
$775$	−10.4645	−0.375898
$776$	13.5570	0.486667
$777$	0	0
$778$	13.0166	0.466669
$779$	−18.3518	−0.657523
$780$	0	0
$781$	16.0491	0.574283
$782$	11.4021	0.407738
$783$	0	0
$784$	6.30098	0.225035
$785$	17.7199	0.632449
$786$	0	0
$787$	14.1732	0.505219	0.252610	−	0.967568i	$-0.418711\pi$
$787$	14.1732	0.505219	0.252610	+	0.967568i	$0.418711\pi$
$788$	8.05584	0.286978
$789$	0	0
$790$	7.23248	0.257320
$791$	−6.58301	−0.234065
$792$	0	0
$793$	16.4995	0.585915
$794$	7.07387	0.251042
$795$	0	0
$796$	2.91824	0.103434
$797$	−1.44394	−0.0511470	−0.0255735	−	0.999673i	$-0.508141\pi$
$797$	−1.44394	−0.0511470	−0.0255735	+	0.999673i	$0.508141\pi$
$798$	0	0
$799$	15.1314	0.535311
$800$	1.00000	0.0353553
$801$	0	0
$802$	25.7088	0.907810
$803$	24.7208	0.872379
$804$	0	0
$805$	−21.0744	−0.742776
$806$	−54.1064	−1.90582
$807$	0	0
$808$	−15.8727	−0.558400
$809$	−27.0894	−0.952412	−0.476206	−	0.879334i	$-0.657988\pi$
$809$	−27.0894	−0.952412	−0.476206	+	0.879334i	$0.657988\pi$
$810$	0	0
$811$	−4.91901	−0.172730	−0.0863649	−	0.996264i	$-0.527525\pi$
$811$	−4.91901	−0.172730	−0.0863649	+	0.996264i	$0.527525\pi$
$812$	−4.82344	−0.169269
$813$	0	0
$814$	−34.0725	−1.19424
$815$	−0.0734716	−0.00257360
$816$	0	0
$817$	1.38809	0.0485633
$818$	22.2724	0.778736
$819$	0	0
$820$	−3.47543	−0.121367
$821$	51.0293	1.78093	0.890467	−	0.455048i	$-0.150378\pi$
$821$	51.0293	1.78093	0.890467	+	0.455048i	$0.150378\pi$
$822$	0	0
$823$	−38.3477	−1.33672	−0.668358	−	0.743840i	$-0.733003\pi$
$823$	−38.3477	−1.33672	−0.668358	+	0.743840i	$0.733003\pi$
$824$	−6.99104	−0.243545
$825$	0	0
$826$	23.4848	0.817141
$827$	4.83305	0.168062	0.0840308	−	0.996463i	$-0.473221\pi$
$827$	4.83305	0.168062	0.0840308	+	0.996463i	$0.473221\pi$
$828$	0	0
$829$	−13.4499	−0.467133	−0.233566	−	0.972341i	$-0.575040\pi$
$829$	−13.4499	−0.467133	−0.233566	+	0.972341i	$0.575040\pi$
$830$	−8.50666	−0.295270
$831$	0	0
$832$	5.17045	0.179253
$833$	12.4331	0.430780
$834$	0	0
$835$	7.17887	0.248435
$836$	17.5163	0.605812
$837$	0	0
$838$	3.22620	0.111447
$839$	9.94062	0.343188	0.171594	−	0.985168i	$-0.445108\pi$
$839$	9.94062	0.343188	0.171594	+	0.985168i	$0.445108\pi$
$840$	0	0
$841$	−27.2508	−0.939684
$842$	25.6092	0.882552
$843$	0	0
$844$	6.25294	0.215235
$845$	13.7335	0.472448
$846$	0	0
$847$	−0.0136186	−0.000467942 0
$848$	4.24666	0.145831
$849$	0	0
$850$	1.97320	0.0676801
$851$	−59.3537	−2.03462
$852$	0	0
$853$	1.34983	0.0462173	0.0231087	−	0.999733i	$-0.492644\pi$
$853$	1.34983	0.0462173	0.0231087	+	0.999733i	$0.492644\pi$
$854$	−11.6382	−0.398250
$855$	0	0
$856$	−0.922600	−0.0315338
$857$	−8.85996	−0.302650	−0.151325	−	0.988484i	$-0.548354\pi$
$857$	−8.85996	−0.302650	−0.151325	+	0.988484i	$0.548354\pi$
$858$	0	0
$859$	32.5580	1.11086	0.555432	−	0.831562i	$-0.312553\pi$
$859$	32.5580	1.11086	0.555432	+	0.831562i	$0.312553\pi$
$860$	0.262874	0.00896393
$861$	0	0
$862$	−4.11340	−0.140103
$863$	−1.14806	−0.0390804	−0.0195402	−	0.999809i	$-0.506220\pi$
$863$	−1.14806	−0.0390804	−0.0195402	+	0.999809i	$0.506220\pi$
$864$	0	0
$865$	9.83136	0.334276
$866$	−35.5734	−1.20883
$867$	0	0
$868$	38.1647	1.29540
$869$	23.9915	0.813856
$870$	0	0
$871$	11.4777	0.388906
$872$	8.45593	0.286354
$873$	0	0
$874$	30.5130	1.03212
$875$	−3.64705	−0.123293
$876$	0	0
$877$	22.9787	0.775936	0.387968	−	0.921673i	$-0.373177\pi$
$877$	22.9787	0.775936	0.387968	+	0.921673i	$0.373177\pi$
$878$	−34.0959	−1.15068
$879$	0	0
$880$	3.31719	0.111822
$881$	−33.3006	−1.12193	−0.560963	−	0.827841i	$-0.689569\pi$
$881$	−33.3006	−1.12193	−0.560963	+	0.827841i	$0.689569\pi$
$882$	0	0
$883$	−25.8555	−0.870108	−0.435054	−	0.900404i	$-0.643271\pi$
$883$	−25.8555	−0.870108	−0.435054	+	0.900404i	$0.643271\pi$
$884$	10.2023	0.343141
$885$	0	0
$886$	−18.2063	−0.611653
$887$	−20.4749	−0.687480	−0.343740	−	0.939065i	$-0.611694\pi$
$887$	−20.4749	−0.687480	−0.343740	+	0.939065i	$0.611694\pi$
$888$	0	0
$889$	66.2673	2.22253
$890$	−1.00000	−0.0335201
$891$	0	0
$892$	15.1255	0.506438
$893$	40.4931	1.35505
$894$	0	0
$895$	−5.12462	−0.171297
$896$	−3.64705	−0.121839
$897$	0	0
$898$	3.52934	0.117775
$899$	−13.8400	−0.461589
$900$	0	0
$901$	8.37950	0.279162
$902$	−11.5287	−0.383862
$903$	0	0
$904$	1.80502	0.0600342
$905$	13.1767	0.438009
$906$	0	0
$907$	−17.7741	−0.590179	−0.295090	−	0.955470i	$-0.595350\pi$
$907$	−17.7741	−0.590179	−0.295090	+	0.955470i	$0.595350\pi$
$908$	−18.1179	−0.601263
$909$	0	0
$910$	−18.8569	−0.625100
$911$	−43.1417	−1.42935	−0.714674	−	0.699457i	$-0.753425\pi$
$911$	−43.1417	−1.42935	−0.714674	+	0.699457i	$0.753425\pi$
$912$	0	0
$913$	−28.2182	−0.933885
$914$	9.11268	0.301421
$915$	0	0
$916$	17.6936	0.584614
$917$	13.8836	0.458476
$918$	0	0
$919$	−35.9117	−1.18462	−0.592310	−	0.805710i	$-0.701784\pi$
$919$	−35.9117	−1.18462	−0.592310	+	0.805710i	$0.701784\pi$
$920$	5.77849	0.190511
$921$	0	0
$922$	5.95122	0.195993
$923$	25.0155	0.823396
$924$	0	0
$925$	−10.2715	−0.337725
$926$	5.31935	0.174805
$927$	0	0
$928$	1.32256	0.0434151
$929$	−23.7927	−0.780613	−0.390306	−	0.920685i	$-0.627631\pi$
$929$	−23.7927	−0.780613	−0.390306	+	0.920685i	$0.627631\pi$
$930$	0	0
$931$	33.2720	1.09045
$932$	15.8674	0.519752
$933$	0	0
$934$	−20.5248	−0.671592
$935$	6.54546	0.214059
$936$	0	0
$937$	16.9236	0.552869	0.276434	−	0.961033i	$-0.410847\pi$
$937$	16.9236	0.552869	0.276434	+	0.961033i	$0.410847\pi$
$938$	−8.09594	−0.264342
$939$	0	0
$940$	7.66848	0.250118
$941$	12.7509	0.415666	0.207833	−	0.978164i	$-0.433359\pi$
$941$	12.7509	0.415666	0.207833	+	0.978164i	$0.433359\pi$
$942$	0	0
$943$	−20.0827	−0.653983
$944$	−6.43940	−0.209585
$945$	0	0
$946$	0.872003	0.0283513
$947$	29.5868	0.961442	0.480721	−	0.876873i	$-0.340375\pi$
$947$	29.5868	0.961442	0.480721	+	0.876873i	$0.340375\pi$
$948$	0	0
$949$	38.5320	1.25080
$950$	5.28045	0.171320
$951$	0	0
$952$	−7.19634	−0.233235
$953$	−36.9503	−1.19694	−0.598468	−	0.801147i	$-0.704224\pi$
$953$	−36.9503	−1.19694	−0.598468	+	0.801147i	$0.704224\pi$
$954$	0	0
$955$	14.8940	0.481959
$956$	−5.85824	−0.189469
$957$	0	0
$958$	−33.9066	−1.09547
$959$	45.3671	1.46498
$960$	0	0
$961$	78.5068	2.53248
$962$	−53.1082	−1.71228
$963$	0	0
$964$	20.8972	0.673055
$965$	2.80109	0.0901704
$966$	0	0
$967$	26.3349	0.846874	0.423437	−	0.905926i	$-0.360824\pi$
$967$	26.3349	0.846874	0.423437	+	0.905926i	$0.360824\pi$
$968$	0.00373415	0.000120020 0
$969$	0	0
$970$	13.5570	0.435288
$971$	−12.8622	−0.412766	−0.206383	−	0.978471i	$-0.566169\pi$
$971$	−12.8622	−0.412766	−0.206383	+	0.978471i	$0.566169\pi$
$972$	0	0
$973$	−65.3861	−2.09618
$974$	33.2364	1.06496
$975$	0	0
$976$	3.19112	0.102145
$977$	−6.35960	−0.203462	−0.101731	−	0.994812i	$-0.532438\pi$
$977$	−6.35960	−0.203462	−0.101731	+	0.994812i	$0.532438\pi$
$978$	0	0
$979$	−3.31719	−0.106018
$980$	6.30098	0.201277
$981$	0	0
$982$	39.9859	1.27600
$983$	−23.7659	−0.758013	−0.379007	−	0.925394i	$-0.623734\pi$
$983$	−23.7659	−0.758013	−0.379007	+	0.925394i	$0.623734\pi$
$984$	0	0
$985$	8.05584	0.256681
$986$	2.60967	0.0831087
$987$	0	0
$988$	27.3023	0.868602
$989$	1.51902	0.0483019
$990$	0	0
$991$	14.0985	0.447854	0.223927	−	0.974606i	$-0.428112\pi$
$991$	14.0985	0.447854	0.223927	+	0.974606i	$0.428112\pi$
$992$	−10.4645	−0.332250
$993$	0	0
$994$	−17.6451	−0.559667
$995$	2.91824	0.0925145
$996$	0	0
$997$	−34.3061	−1.08649	−0.543243	−	0.839576i	$-0.682804\pi$
$997$	−34.3061	−1.08649	−0.543243	+	0.839576i	$0.682804\pi$
$998$	5.04411	0.159668
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8010.2.a.bn.1.2		7
3.2	odd	2		2670.2.a.t.1.2	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2670.2.a.t.1.2	✓	7	3.2	odd	2
8010.2.a.bn.1.2		7	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:	\( N \)	\(=\)	\( 8010 = 2 \cdot 3^{2} \cdot 5 \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8010.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -3.64705 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -3.64705 q^{7} +1.00000 q^{8} +1.00000 q^{10} +3.31719 q^{11} +5.17045 q^{13} -3.64705 q^{14} +1.00000 q^{16} +1.97320 q^{17} +5.28045 q^{19} +1.00000 q^{20} +3.31719 q^{22} +5.77849 q^{23} +1.00000 q^{25} +5.17045 q^{26} -3.64705 q^{28} +1.32256 q^{29} -10.4645 q^{31} +1.00000 q^{32} +1.97320 q^{34} -3.64705 q^{35} -10.2715 q^{37} +5.28045 q^{38} +1.00000 q^{40} -3.47543 q^{41} +0.262874 q^{43} +3.31719 q^{44} +5.77849 q^{46} +7.66848 q^{47} +6.30098 q^{49} +1.00000 q^{50} +5.17045 q^{52} +4.24666 q^{53} +3.31719 q^{55} -3.64705 q^{56} +1.32256 q^{58} -6.43940 q^{59} +3.19112 q^{61} -10.4645 q^{62} +1.00000 q^{64} +5.17045 q^{65} +2.21986 q^{67} +1.97320 q^{68} -3.64705 q^{70} +4.83817 q^{71} +7.45234 q^{73} -10.2715 q^{74} +5.28045 q^{76} -12.0980 q^{77} +7.23248 q^{79} +1.00000 q^{80} -3.47543 q^{82} -8.50666 q^{83} +1.97320 q^{85} +0.262874 q^{86} +3.31719 q^{88} -1.00000 q^{89} -18.8569 q^{91} +5.77849 q^{92} +7.66848 q^{94} +5.28045 q^{95} +13.5570 q^{97} +6.30098 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 7 q^{2} + 7 q^{4} + 7 q^{5} + q^{7} + 7 q^{8}+O(q^{10}) \) 7 * q + 7 * q^2 + 7 * q^4 + 7 * q^5 + q^7 + 7 * q^8	\( 7 q + 7 q^{2} + 7 q^{4} + 7 q^{5} + q^{7} + 7 q^{8} + 7 q^{10} - q^{11} + q^{14} + 7 q^{16} + 9 q^{17} + 9 q^{19} + 7 q^{20} - q^{22} + 8 q^{23} + 7 q^{25} + q^{28} + 4 q^{29} + 16 q^{31} + 7 q^{32} + 9 q^{34} + q^{35} + 2 q^{37} + 9 q^{38} + 7 q^{40} + q^{41} - 10 q^{43} - q^{44} + 8 q^{46} + 13 q^{47} + 26 q^{49} + 7 q^{50} + 24 q^{53} - q^{55} + q^{56} + 4 q^{58} + 6 q^{59} + 23 q^{61} + 16 q^{62} + 7 q^{64} + 5 q^{67} + 9 q^{68} + q^{70} + 8 q^{71} - 2 q^{73} + 2 q^{74} + 9 q^{76} + 6 q^{77} + 7 q^{79} + 7 q^{80} + q^{82} + 7 q^{83} + 9 q^{85} - 10 q^{86} - q^{88} - 7 q^{89} + 48 q^{91} + 8 q^{92} + 13 q^{94} + 9 q^{95} + 30 q^{97} + 26 q^{98}+O(q^{100}) \) 7 * q + 7 * q^2 + 7 * q^4 + 7 * q^5 + q^7 + 7 * q^8 + 7 * q^10 - q^11 + q^14 + 7 * q^16 + 9 * q^17 + 9 * q^19 + 7 * q^20 - q^22 + 8 * q^23 + 7 * q^25 + q^28 + 4 * q^29 + 16 * q^31 + 7 * q^32 + 9 * q^34 + q^35 + 2 * q^37 + 9 * q^38 + 7 * q^40 + q^41 - 10 * q^43 - q^44 + 8 * q^46 + 13 * q^47 + 26 * q^49 + 7 * q^50 + 24 * q^53 - q^55 + q^56 + 4 * q^58 + 6 * q^59 + 23 * q^61 + 16 * q^62 + 7 * q^64 + 5 * q^67 + 9 * q^68 + q^70 + 8 * q^71 - 2 * q^73 + 2 * q^74 + 9 * q^76 + 6 * q^77 + 7 * q^79 + 7 * q^80 + q^82 + 7 * q^83 + 9 * q^85 - 10 * q^86 - q^88 - 7 * q^89 + 48 * q^91 + 8 * q^92 + 13 * q^94 + 9 * q^95 + 30 * q^97 + 26 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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