LMFDB - Embedded newform 8010.2.a.bn.1.5

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			1.5
Root			$0.327888$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.20963 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.20963 q^{7} +1.00000 q^{8} +1.00000 q^{10} -6.18365 q^{11} +3.16169 q^{13} +1.20963 q^{14} +1.00000 q^{16} +7.07027 q^{17} +7.52787 q^{19} +1.00000 q^{20} -6.18365 q^{22} -0.264276 q^{23} +1.00000 q^{25} +3.16169 q^{26} +1.20963 q^{28} -7.83309 q^{29} +1.25757 q^{31} +1.00000 q^{32} +7.07027 q^{34} +1.20963 q^{35} +6.93568 q^{37} +7.52787 q^{38} +1.00000 q^{40} +12.3086 q^{41} -2.10929 q^{43} -6.18365 q^{44} -0.264276 q^{46} -2.63046 q^{47} -5.53679 q^{49} +1.00000 q^{50} +3.16169 q^{52} -0.462435 q^{53} -6.18365 q^{55} +1.20963 q^{56} -7.83309 q^{58} -5.46914 q^{59} -4.77845 q^{61} +1.25757 q^{62} +1.00000 q^{64} +3.16169 q^{65} +2.60784 q^{67} +7.07027 q^{68} +1.20963 q^{70} -7.98808 q^{71} -8.54418 q^{73} +6.93568 q^{74} +7.52787 q^{76} -7.47993 q^{77} -9.15202 q^{79} +1.00000 q^{80} +12.3086 q^{82} +14.6185 q^{83} +7.07027 q^{85} -2.10929 q^{86} -6.18365 q^{88} -1.00000 q^{89} +3.82448 q^{91} -0.264276 q^{92} -2.63046 q^{94} +7.52787 q^{95} +1.47145 q^{97} -5.53679 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	1.20963	0.457198	0.228599	−	0.973521i	$-0.426586\pi$
$7$	1.20963	0.457198	0.228599	+	0.973521i	$0.426586\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	1.00000	0.316228
$11$	−6.18365	−1.86444	−0.932220	−	0.361891i	$-0.882131\pi$
$11$	−6.18365	−1.86444	−0.932220	+	0.361891i	$0.882131\pi$
$12$	0	0
$13$	3.16169	0.876895	0.438448	−	0.898757i	$-0.355528\pi$
$13$	3.16169	0.876895	0.438448	+	0.898757i	$0.355528\pi$
$14$	1.20963	0.323287
$15$	0	0
$16$	1.00000	0.250000
$17$	7.07027	1.71479	0.857396	−	0.514657i	$-0.172081\pi$
$17$	7.07027	1.71479	0.857396	+	0.514657i	$0.172081\pi$
$18$	0	0
$19$	7.52787	1.72701	0.863506	−	0.504338i	$-0.168263\pi$
$19$	7.52787	1.72701	0.863506	+	0.504338i	$0.168263\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	−6.18365	−1.31836
$23$	−0.264276	−0.0551055	−0.0275527	−	0.999620i	$-0.508771\pi$
$23$	−0.264276	−0.0551055	−0.0275527	+	0.999620i	$0.508771\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	3.16169	0.620059
$27$	0	0
$28$	1.20963	0.228599
$29$	−7.83309	−1.45457	−0.727285	−	0.686336i	$-0.759218\pi$
$29$	−7.83309	−1.45457	−0.727285	+	0.686336i	$0.759218\pi$
$30$	0	0
$31$	1.25757	0.225866	0.112933	−	0.993603i	$-0.463975\pi$
$31$	1.25757	0.225866	0.112933	+	0.993603i	$0.463975\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	7.07027	1.21254
$35$	1.20963	0.204465
$36$	0	0
$37$	6.93568	1.14022	0.570109	−	0.821569i	$-0.306901\pi$
$37$	6.93568	1.14022	0.570109	+	0.821569i	$0.306901\pi$
$38$	7.52787	1.22118
$39$	0	0
$40$	1.00000	0.158114
$41$	12.3086	1.92228	0.961138	−	0.276070i	$-0.0890321\pi$
$41$	12.3086	1.92228	0.961138	+	0.276070i	$0.0890321\pi$
$42$	0	0
$43$	−2.10929	−0.321664	−0.160832	−	0.986982i	$-0.551418\pi$
$43$	−2.10929	−0.321664	−0.160832	+	0.986982i	$0.551418\pi$
$44$	−6.18365	−0.932220
$45$	0	0
$46$	−0.264276	−0.0389654
$47$	−2.63046	−0.383692	−0.191846	−	0.981425i	$-0.561447\pi$
$47$	−2.63046	−0.383692	−0.191846	+	0.981425i	$0.561447\pi$
$48$	0	0
$49$	−5.53679	−0.790970
$50$	1.00000	0.141421
$51$	0	0
$52$	3.16169	0.438448
$53$	−0.462435	−0.0635203	−0.0317601	−	0.999496i	$-0.510111\pi$
$53$	−0.462435	−0.0635203	−0.0317601	+	0.999496i	$0.510111\pi$
$54$	0	0
$55$	−6.18365	−0.833803
$56$	1.20963	0.161644
$57$	0	0
$58$	−7.83309	−1.02854
$59$	−5.46914	−0.712021	−0.356011	−	0.934482i	$-0.615863\pi$
$59$	−5.46914	−0.712021	−0.356011	+	0.934482i	$0.615863\pi$
$60$	0	0
$61$	−4.77845	−0.611818	−0.305909	−	0.952061i	$-0.598960\pi$
$61$	−4.77845	−0.611818	−0.305909	+	0.952061i	$0.598960\pi$
$62$	1.25757	0.159712
$63$	0	0
$64$	1.00000	0.125000
$65$	3.16169	0.392160
$66$	0	0
$67$	2.60784	0.318598	0.159299	−	0.987230i	$-0.449077\pi$
$67$	2.60784	0.318598	0.159299	+	0.987230i	$0.449077\pi$
$68$	7.07027	0.857396
$69$	0	0
$70$	1.20963	0.144579
$71$	−7.98808	−0.948011	−0.474005	−	0.880522i	$-0.657192\pi$
$71$	−7.98808	−0.948011	−0.474005	+	0.880522i	$0.657192\pi$
$72$	0	0
$73$	−8.54418	−1.00002	−0.500010	−	0.866019i	$-0.666670\pi$
$73$	−8.54418	−1.00002	−0.500010	+	0.866019i	$0.666670\pi$
$74$	6.93568	0.806256
$75$	0	0
$76$	7.52787	0.863506
$77$	−7.47993	−0.852418
$78$	0	0
$79$	−9.15202	−1.02968	−0.514841	−	0.857286i	$-0.672149\pi$
$79$	−9.15202	−1.02968	−0.514841	+	0.857286i	$0.672149\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	12.3086	1.35925
$83$	14.6185	1.60459	0.802296	−	0.596926i	$-0.203611\pi$
$83$	14.6185	1.60459	0.802296	+	0.596926i	$0.203611\pi$
$84$	0	0
$85$	7.07027	0.766879
$86$	−2.10929	−0.227451
$87$	0	0
$88$	−6.18365	−0.659179
$89$	−1.00000	−0.106000
$89$	−1.00000	−0.106000
$90$	0	0
$91$	3.82448	0.400914
$92$	−0.264276	−0.0275527
$93$	0	0
$94$	−2.63046	−0.271311
$95$	7.52787	0.772343
$96$	0	0
$97$	1.47145	0.149403	0.0747014	−	0.997206i	$-0.476200\pi$
$97$	1.47145	0.149403	0.0747014	+	0.997206i	$0.476200\pi$
$98$	−5.53679	−0.559301
$99$	0	0
$100$	1.00000	0.100000
$101$	−10.5215	−1.04693	−0.523466	−	0.852046i	$-0.675361\pi$
$101$	−10.5215	−1.04693	−0.523466	+	0.852046i	$0.675361\pi$
$102$	0	0
$103$	12.4636	1.22807	0.614035	−	0.789279i	$-0.289545\pi$
$103$	12.4636	1.22807	0.614035	+	0.789279i	$0.289545\pi$
$104$	3.16169	0.310029
$105$	0	0
$106$	−0.462435	−0.0449156
$107$	−7.83875	−0.757800	−0.378900	−	0.925438i	$-0.623698\pi$
$107$	−7.83875	−0.757800	−0.378900	+	0.925438i	$0.623698\pi$
$108$	0	0
$109$	11.5688	1.10809	0.554046	−	0.832486i	$-0.313083\pi$
$109$	11.5688	1.10809	0.554046	+	0.832486i	$0.313083\pi$
$110$	−6.18365	−0.589588
$111$	0	0
$112$	1.20963	0.114299
$113$	19.8364	1.86606	0.933028	−	0.359805i	$-0.117157\pi$
$113$	19.8364	1.86606	0.933028	+	0.359805i	$0.117157\pi$
$114$	0	0
$115$	−0.264276	−0.0246439
$116$	−7.83309	−0.727285
$117$	0	0
$118$	−5.46914	−0.503475
$119$	8.55242	0.783999
$120$	0	0
$121$	27.2375	2.47614
$122$	−4.77845	−0.432620
$123$	0	0
$124$	1.25757	0.112933
$125$	1.00000	0.0894427
$126$	0	0
$127$	18.1282	1.60862	0.804311	−	0.594209i	$-0.202535\pi$
$127$	18.1282	1.60862	0.804311	+	0.594209i	$0.202535\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	3.16169	0.277299
$131$	9.80263	0.856460	0.428230	−	0.903670i	$-0.359137\pi$
$131$	9.80263	0.856460	0.428230	+	0.903670i	$0.359137\pi$
$132$	0	0
$133$	9.10595	0.789586
$134$	2.60784	0.225283
$135$	0	0
$136$	7.07027	0.606271
$137$	−11.4691	−0.979875	−0.489937	−	0.871758i	$-0.662980\pi$
$137$	−11.4691	−0.979875	−0.489937	+	0.871758i	$0.662980\pi$
$138$	0	0
$139$	−10.7866	−0.914904	−0.457452	−	0.889234i	$-0.651238\pi$
$139$	−10.7866	−0.914904	−0.457452	+	0.889234i	$0.651238\pi$
$140$	1.20963	0.102232
$141$	0	0
$142$	−7.98808	−0.670345
$143$	−19.5508	−1.63492
$144$	0	0
$145$	−7.83309	−0.650503
$146$	−8.54418	−0.707121
$147$	0	0
$148$	6.93568	0.570109
$149$	6.16840	0.505335	0.252667	−	0.967553i	$-0.418692\pi$
$149$	6.16840	0.505335	0.252667	+	0.967553i	$0.418692\pi$
$150$	0	0
$151$	−3.20561	−0.260869	−0.130434	−	0.991457i	$-0.541637\pi$
$151$	−3.20561	−0.260869	−0.130434	+	0.991457i	$0.541637\pi$
$152$	7.52787	0.610591
$153$	0	0
$154$	−7.47993	−0.602750
$155$	1.25757	0.101010
$156$	0	0
$157$	18.9970	1.51613	0.758063	−	0.652181i	$-0.226146\pi$
$157$	18.9970	1.51613	0.758063	+	0.652181i	$0.226146\pi$
$158$	−9.15202	−0.728095
$159$	0	0
$160$	1.00000	0.0790569
$161$	−0.319677	−0.0251941
$162$	0	0
$163$	23.4230	1.83463	0.917317	−	0.398157i	$-0.130350\pi$
$163$	23.4230	1.83463	0.917317	+	0.398157i	$0.130350\pi$
$164$	12.3086	0.961138
$165$	0	0
$166$	14.6185	1.13462
$167$	0.988527	0.0764945	0.0382473	−	0.999268i	$-0.487823\pi$
$167$	0.988527	0.0764945	0.0382473	+	0.999268i	$0.487823\pi$
$168$	0	0
$169$	−3.00371	−0.231054
$170$	7.07027	0.542265
$171$	0	0
$172$	−2.10929	−0.160832
$173$	12.8951	0.980397	0.490198	−	0.871611i	$-0.336924\pi$
$173$	12.8951	0.980397	0.490198	+	0.871611i	$0.336924\pi$
$174$	0	0
$175$	1.20963	0.0914395
$176$	−6.18365	−0.466110
$177$	0	0
$178$	−1.00000	−0.0749532
$179$	−19.8285	−1.48205	−0.741027	−	0.671475i	$-0.765661\pi$
$179$	−19.8285	−1.48205	−0.741027	+	0.671475i	$0.765661\pi$
$180$	0	0
$181$	20.1561	1.49819	0.749095	−	0.662463i	$-0.230489\pi$
$181$	20.1561	1.49819	0.749095	+	0.662463i	$0.230489\pi$
$182$	3.82448	0.283489
$183$	0	0
$184$	−0.264276	−0.0194827
$185$	6.93568	0.509921
$186$	0	0
$187$	−43.7201	−3.19713
$188$	−2.63046	−0.191846
$189$	0	0
$190$	7.52787	0.546129
$191$	8.85282	0.640568	0.320284	−	0.947322i	$-0.396222\pi$
$191$	8.85282	0.640568	0.320284	+	0.947322i	$0.396222\pi$
$192$	0	0
$193$	4.25214	0.306076	0.153038	−	0.988220i	$-0.451094\pi$
$193$	4.25214	0.306076	0.153038	+	0.988220i	$0.451094\pi$
$194$	1.47145	0.105644
$195$	0	0
$196$	−5.53679	−0.395485
$197$	14.8409	1.05737	0.528685	−	0.848818i	$-0.322685\pi$
$197$	14.8409	1.05737	0.528685	+	0.848818i	$0.322685\pi$
$198$	0	0
$199$	−22.6702	−1.60705	−0.803524	−	0.595273i	$-0.797044\pi$
$199$	−22.6702	−1.60705	−0.803524	+	0.595273i	$0.797044\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	−10.5215	−0.740293
$203$	−9.47515	−0.665025
$204$	0	0
$205$	12.3086	0.859668
$206$	12.4636	0.868377
$207$	0	0
$208$	3.16169	0.219224
$209$	−46.5497	−3.21991
$210$	0	0
$211$	10.5320	0.725050	0.362525	−	0.931974i	$-0.381915\pi$
$211$	10.5320	0.725050	0.362525	+	0.931974i	$0.381915\pi$
$212$	−0.462435	−0.0317601
$213$	0	0
$214$	−7.83875	−0.535846
$215$	−2.10929	−0.143852
$216$	0	0
$217$	1.52120	0.103265
$218$	11.5688	0.783539
$219$	0	0
$220$	−6.18365	−0.416902
$221$	22.3540	1.50369
$222$	0	0
$223$	8.47585	0.567585	0.283792	−	0.958886i	$-0.408407\pi$
$223$	8.47585	0.567585	0.283792	+	0.958886i	$0.408407\pi$
$224$	1.20963	0.0808219
$225$	0	0
$226$	19.8364	1.31950
$227$	−10.5272	−0.698714	−0.349357	−	0.936990i	$-0.613600\pi$
$227$	−10.5272	−0.698714	−0.349357	+	0.936990i	$0.613600\pi$
$228$	0	0
$229$	−3.35717	−0.221848	−0.110924	−	0.993829i	$-0.535381\pi$
$229$	−3.35717	−0.221848	−0.110924	+	0.993829i	$0.535381\pi$
$230$	−0.264276	−0.0174259
$231$	0	0
$232$	−7.83309	−0.514268
$233$	10.1710	0.666323	0.333162	−	0.942870i	$-0.391884\pi$
$233$	10.1710	0.666323	0.333162	+	0.942870i	$0.391884\pi$
$234$	0	0
$235$	−2.63046	−0.171592
$236$	−5.46914	−0.356011
$237$	0	0
$238$	8.55242	0.554371
$239$	14.6929	0.950406	0.475203	−	0.879876i	$-0.342375\pi$
$239$	14.6929	0.950406	0.475203	+	0.879876i	$0.342375\pi$
$240$	0	0
$241$	−0.476590	−0.0306999	−0.0153499	−	0.999882i	$-0.504886\pi$
$241$	−0.476590	−0.0306999	−0.0153499	+	0.999882i	$0.504886\pi$
$242$	27.2375	1.75089
$243$	0	0
$244$	−4.77845	−0.305909
$245$	−5.53679	−0.353733
$246$	0	0
$247$	23.8008	1.51441
$248$	1.25757	0.0798558
$249$	0	0
$250$	1.00000	0.0632456
$251$	−30.2949	−1.91219	−0.956097	−	0.293050i	$-0.905330\pi$
$251$	−30.2949	−1.91219	−0.956097	+	0.293050i	$0.905330\pi$
$252$	0	0
$253$	1.63419	0.102741
$254$	18.1282	1.13747
$255$	0	0
$256$	1.00000	0.0625000
$257$	1.44993	0.0904440	0.0452220	−	0.998977i	$-0.485600\pi$
$257$	1.44993	0.0904440	0.0452220	+	0.998977i	$0.485600\pi$
$258$	0	0
$259$	8.38961	0.521305
$260$	3.16169	0.196080
$261$	0	0
$262$	9.80263	0.605609
$263$	11.8126	0.728398	0.364199	−	0.931321i	$-0.381343\pi$
$263$	11.8126	0.728398	0.364199	+	0.931321i	$0.381343\pi$
$264$	0	0
$265$	−0.462435	−0.0284071
$266$	9.10595	0.558321
$267$	0	0
$268$	2.60784	0.159299
$269$	−15.1337	−0.922717	−0.461358	−	0.887214i	$-0.652638\pi$
$269$	−15.1337	−0.922717	−0.461358	+	0.887214i	$0.652638\pi$
$270$	0	0
$271$	13.6075	0.826594	0.413297	−	0.910596i	$-0.364377\pi$
$271$	13.6075	0.826594	0.413297	+	0.910596i	$0.364377\pi$
$272$	7.07027	0.428698
$273$	0	0
$274$	−11.4691	−0.692876
$275$	−6.18365	−0.372888
$276$	0	0
$277$	30.8839	1.85564	0.927818	−	0.373034i	$-0.121682\pi$
$277$	30.8839	1.85564	0.927818	+	0.373034i	$0.121682\pi$
$278$	−10.7866	−0.646935
$279$	0	0
$280$	1.20963	0.0722893
$281$	−26.1034	−1.55720	−0.778599	−	0.627522i	$-0.784069\pi$
$281$	−26.1034	−1.55720	−0.778599	+	0.627522i	$0.784069\pi$
$282$	0	0
$283$	−7.81142	−0.464341	−0.232170	−	0.972675i	$-0.574583\pi$
$283$	−7.81142	−0.464341	−0.232170	+	0.972675i	$0.574583\pi$
$284$	−7.98808	−0.474005
$285$	0	0
$286$	−19.5508	−1.15606
$287$	14.8888	0.878859
$288$	0	0
$289$	32.9887	1.94051
$290$	−7.83309	−0.459975
$291$	0	0
$292$	−8.54418	−0.500010
$293$	−8.96391	−0.523677	−0.261839	−	0.965112i	$-0.584329\pi$
$293$	−8.96391	−0.523677	−0.261839	+	0.965112i	$0.584329\pi$
$294$	0	0
$295$	−5.46914	−0.318426
$296$	6.93568	0.403128
$297$	0	0
$298$	6.16840	0.357326
$299$	−0.835561	−0.0483217
$300$	0	0
$301$	−2.55146	−0.147064
$302$	−3.20561	−0.184462
$303$	0	0
$304$	7.52787	0.431753
$305$	−4.77845	−0.273613
$306$	0	0
$307$	−12.3822	−0.706688	−0.353344	−	0.935493i	$-0.614956\pi$
$307$	−12.3822	−0.706688	−0.353344	+	0.935493i	$0.614956\pi$
$308$	−7.47993	−0.426209
$309$	0	0
$310$	1.25757	0.0714252
$311$	5.84011	0.331162	0.165581	−	0.986196i	$-0.447050\pi$
$311$	5.84011	0.331162	0.165581	+	0.986196i	$0.447050\pi$
$312$	0	0
$313$	−1.23272	−0.0696774	−0.0348387	−	0.999393i	$-0.511092\pi$
$313$	−1.23272	−0.0696774	−0.0348387	+	0.999393i	$0.511092\pi$
$314$	18.9970	1.07206
$315$	0	0
$316$	−9.15202	−0.514841
$317$	1.63700	0.0919434	0.0459717	−	0.998943i	$-0.485362\pi$
$317$	1.63700	0.0919434	0.0459717	+	0.998943i	$0.485362\pi$
$318$	0	0
$319$	48.4371	2.71196
$320$	1.00000	0.0559017
$321$	0	0
$322$	−0.319677	−0.0178149
$323$	53.2241	2.96147
$324$	0	0
$325$	3.16169	0.175379
$326$	23.4230	1.29728
$327$	0	0
$328$	12.3086	0.679627
$329$	−3.18188	−0.175423
$330$	0	0
$331$	−26.6907	−1.46705	−0.733526	−	0.679661i	$-0.762127\pi$
$331$	−26.6907	−1.46705	−0.733526	+	0.679661i	$0.762127\pi$
$332$	14.6185	0.802296
$333$	0	0
$334$	0.988527	0.0540898
$335$	2.60784	0.142481
$336$	0	0
$337$	34.0877	1.85687	0.928437	−	0.371490i	$-0.121153\pi$
$337$	34.0877	1.85687	0.928437	+	0.371490i	$0.121153\pi$
$338$	−3.00371	−0.163380
$339$	0	0
$340$	7.07027	0.383439
$341$	−7.77637	−0.421114
$342$	0	0
$343$	−15.1649	−0.818827
$344$	−2.10929	−0.113725
$345$	0	0
$346$	12.8951	0.693245
$347$	8.76185	0.470360	0.235180	−	0.971952i	$-0.424432\pi$
$347$	8.76185	0.470360	0.235180	+	0.971952i	$0.424432\pi$
$348$	0	0
$349$	30.2008	1.61661	0.808305	−	0.588765i	$-0.200385\pi$
$349$	30.2008	1.61661	0.808305	+	0.588765i	$0.200385\pi$
$350$	1.20963	0.0646575
$351$	0	0
$352$	−6.18365	−0.329590
$353$	−18.8254	−1.00197	−0.500986	−	0.865455i	$-0.667029\pi$
$353$	−18.8254	−1.00197	−0.500986	+	0.865455i	$0.667029\pi$
$354$	0	0
$355$	−7.98808	−0.423963
$356$	−1.00000	−0.0529999
$357$	0	0
$358$	−19.8285	−1.04797
$359$	−13.4597	−0.710378	−0.355189	−	0.934795i	$-0.615583\pi$
$359$	−13.4597	−0.710378	−0.355189	+	0.934795i	$0.615583\pi$
$360$	0	0
$361$	37.6689	1.98257
$362$	20.1561	1.05938
$363$	0	0
$364$	3.82448	0.200457
$365$	−8.54418	−0.447223
$366$	0	0
$367$	−8.28334	−0.432387	−0.216193	−	0.976351i	$-0.569364\pi$
$367$	−8.28334	−0.432387	−0.216193	+	0.976351i	$0.569364\pi$
$368$	−0.264276	−0.0137764
$369$	0	0
$370$	6.93568	0.360569
$371$	−0.559375	−0.0290413
$372$	0	0
$373$	−35.3587	−1.83080	−0.915401	−	0.402542i	$-0.868127\pi$
$373$	−35.3587	−1.83080	−0.915401	+	0.402542i	$0.868127\pi$
$374$	−43.7201	−2.26071
$375$	0	0
$376$	−2.63046	−0.135656
$377$	−24.7658	−1.27551
$378$	0	0
$379$	−12.3268	−0.633185	−0.316592	−	0.948562i	$-0.602539\pi$
$379$	−12.3268	−0.633185	−0.316592	+	0.948562i	$0.602539\pi$
$380$	7.52787	0.386172
$381$	0	0
$382$	8.85282	0.452950
$383$	24.8502	1.26978	0.634892	−	0.772601i	$-0.281045\pi$
$383$	24.8502	1.26978	0.634892	+	0.772601i	$0.281045\pi$
$384$	0	0
$385$	−7.47993	−0.381213
$386$	4.25214	0.216428
$387$	0	0
$388$	1.47145	0.0747014
$389$	−18.4175	−0.933803	−0.466902	−	0.884309i	$-0.654630\pi$
$389$	−18.4175	−0.933803	−0.466902	+	0.884309i	$0.654630\pi$
$390$	0	0
$391$	−1.86851	−0.0944944
$392$	−5.53679	−0.279650
$393$	0	0
$394$	14.8409	0.747674
$395$	−9.15202	−0.460488
$396$	0	0
$397$	−21.4986	−1.07898	−0.539492	−	0.841991i	$-0.681384\pi$
$397$	−21.4986	−1.07898	−0.539492	+	0.841991i	$0.681384\pi$
$398$	−22.6702	−1.13635
$399$	0	0
$400$	1.00000	0.0500000
$401$	−14.0476	−0.701505	−0.350752	−	0.936468i	$-0.614074\pi$
$401$	−14.0476	−0.701505	−0.350752	+	0.936468i	$0.614074\pi$
$402$	0	0
$403$	3.97605	0.198061
$404$	−10.5215	−0.523466
$405$	0	0
$406$	−9.47515	−0.470244
$407$	−42.8878	−2.12587
$408$	0	0
$409$	−34.8061	−1.72105	−0.860526	−	0.509406i	$-0.829865\pi$
$409$	−34.8061	−1.72105	−0.860526	+	0.509406i	$0.829865\pi$
$410$	12.3086	0.607877
$411$	0	0
$412$	12.4636	0.614035
$413$	−6.61564	−0.325534
$414$	0	0
$415$	14.6185	0.717596
$416$	3.16169	0.155015
$417$	0	0
$418$	−46.5497	−2.27682
$419$	−22.1464	−1.08192	−0.540961	−	0.841047i	$-0.681940\pi$
$419$	−22.1464	−1.08192	−0.540961	+	0.841047i	$0.681940\pi$
$420$	0	0
$421$	−15.1840	−0.740023	−0.370011	−	0.929027i	$-0.620646\pi$
$421$	−15.1840	−0.740023	−0.370011	+	0.929027i	$0.620646\pi$
$422$	10.5320	0.512688
$423$	0	0
$424$	−0.462435	−0.0224578
$425$	7.07027	0.342959
$426$	0	0
$427$	−5.78016	−0.279722
$428$	−7.83875	−0.378900
$429$	0	0
$430$	−2.10929	−0.101719
$431$	−1.31376	−0.0632818	−0.0316409	−	0.999499i	$-0.510073\pi$
$431$	−1.31376	−0.0632818	−0.0316409	+	0.999499i	$0.510073\pi$
$432$	0	0
$433$	13.5237	0.649907	0.324954	−	0.945730i	$-0.394651\pi$
$433$	13.5237	0.649907	0.324954	+	0.945730i	$0.394651\pi$
$434$	1.52120	0.0730197
$435$	0	0
$436$	11.5688	0.554046
$437$	−1.98944	−0.0951678
$438$	0	0
$439$	−24.2822	−1.15893	−0.579463	−	0.814999i	$-0.696738\pi$
$439$	−24.2822	−1.15893	−0.579463	+	0.814999i	$0.696738\pi$
$440$	−6.18365	−0.294794
$441$	0	0
$442$	22.3540	1.06327
$443$	−12.1897	−0.579149	−0.289574	−	0.957155i	$-0.593514\pi$
$443$	−12.1897	−0.579149	−0.289574	+	0.957155i	$0.593514\pi$
$444$	0	0
$445$	−1.00000	−0.0474045
$446$	8.47585	0.401343
$447$	0	0
$448$	1.20963	0.0571497
$449$	17.8944	0.844490	0.422245	−	0.906482i	$-0.361242\pi$
$449$	17.8944	0.844490	0.422245	+	0.906482i	$0.361242\pi$
$450$	0	0
$451$	−76.1119	−3.58397
$452$	19.8364	0.933028
$453$	0	0
$454$	−10.5272	−0.494066
$455$	3.82448	0.179294
$456$	0	0
$457$	23.8353	1.11497	0.557485	−	0.830187i	$-0.311766\pi$
$457$	23.8353	1.11497	0.557485	+	0.830187i	$0.311766\pi$
$458$	−3.35717	−0.156870
$459$	0	0
$460$	−0.264276	−0.0123220
$461$	−20.0223	−0.932529	−0.466265	−	0.884645i	$-0.654401\pi$
$461$	−20.0223	−0.932529	−0.466265	+	0.884645i	$0.654401\pi$
$462$	0	0
$463$	−37.9390	−1.76317	−0.881587	−	0.472022i	$-0.843524\pi$
$463$	−37.9390	−1.76317	−0.881587	+	0.472022i	$0.843524\pi$
$464$	−7.83309	−0.363642
$465$	0	0
$466$	10.1710	0.471162
$467$	3.96070	0.183279	0.0916397	−	0.995792i	$-0.470789\pi$
$467$	3.96070	0.183279	0.0916397	+	0.995792i	$0.470789\pi$
$468$	0	0
$469$	3.15452	0.145662
$470$	−2.63046	−0.121334
$471$	0	0
$472$	−5.46914	−0.251738
$473$	13.0431	0.599723
$474$	0	0
$475$	7.52787	0.345403
$476$	8.55242	0.392000
$477$	0	0
$478$	14.6929	0.672039
$479$	34.3271	1.56844	0.784222	−	0.620480i	$-0.213062\pi$
$479$	34.3271	1.56844	0.784222	+	0.620480i	$0.213062\pi$
$480$	0	0
$481$	21.9285	0.999853
$482$	−0.476590	−0.0217081
$483$	0	0
$484$	27.2375	1.23807
$485$	1.47145	0.0668150
$486$	0	0
$487$	−28.7272	−1.30175	−0.650876	−	0.759184i	$-0.725598\pi$
$487$	−28.7272	−1.30175	−0.650876	+	0.759184i	$0.725598\pi$
$488$	−4.77845	−0.216310
$489$	0	0
$490$	−5.53679	−0.250127
$491$	−22.5983	−1.01985	−0.509923	−	0.860220i	$-0.670326\pi$
$491$	−22.5983	−1.01985	−0.509923	+	0.860220i	$0.670326\pi$
$492$	0	0
$493$	−55.3821	−2.49428
$494$	23.8008	1.07085
$495$	0	0
$496$	1.25757	0.0564666
$497$	−9.66263	−0.433428
$498$	0	0
$499$	17.4254	0.780066	0.390033	−	0.920801i	$-0.372464\pi$
$499$	17.4254	0.780066	0.390033	+	0.920801i	$0.372464\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	−30.2949	−1.35213
$503$	3.58639	0.159909	0.0799546	−	0.996799i	$-0.474522\pi$
$503$	3.58639	0.159909	0.0799546	+	0.996799i	$0.474522\pi$
$504$	0	0
$505$	−10.5215	−0.468202
$506$	1.63419	0.0726487
$507$	0	0
$508$	18.1282	0.804311
$509$	21.3424	0.945986	0.472993	−	0.881066i	$-0.343174\pi$
$509$	21.3424	0.945986	0.472993	+	0.881066i	$0.343174\pi$
$510$	0	0
$511$	−10.3353	−0.457207
$512$	1.00000	0.0441942
$513$	0	0
$514$	1.44993	0.0639535
$515$	12.4636	0.549210
$516$	0	0
$517$	16.2658	0.715370
$518$	8.38961	0.368618
$519$	0	0
$520$	3.16169	0.138649
$521$	3.02727	0.132627	0.0663136	−	0.997799i	$-0.478876\pi$
$521$	3.02727	0.132627	0.0663136	+	0.997799i	$0.478876\pi$
$522$	0	0
$523$	−25.3020	−1.10638	−0.553189	−	0.833056i	$-0.686589\pi$
$523$	−25.3020	−1.10638	−0.553189	+	0.833056i	$0.686589\pi$
$524$	9.80263	0.428230
$525$	0	0
$526$	11.8126	0.515055
$527$	8.89136	0.387314
$528$	0	0
$529$	−22.9302	−0.996963
$530$	−0.462435	−0.0200869
$531$	0	0
$532$	9.10595	0.394793
$533$	38.9159	1.68563
$534$	0	0
$535$	−7.83875	−0.338899
$536$	2.60784	0.112641
$537$	0	0
$538$	−15.1337	−0.652459
$539$	34.2376	1.47472
$540$	0	0
$541$	−21.9178	−0.942320	−0.471160	−	0.882048i	$-0.656165\pi$
$541$	−21.9178	−0.942320	−0.471160	+	0.882048i	$0.656165\pi$
$542$	13.6075	0.584490
$543$	0	0
$544$	7.07027	0.303135
$545$	11.5688	0.495554
$546$	0	0
$547$	40.2624	1.72149	0.860747	−	0.509033i	$-0.169997\pi$
$547$	40.2624	1.72149	0.860747	+	0.509033i	$0.169997\pi$
$548$	−11.4691	−0.489937
$549$	0	0
$550$	−6.18365	−0.263672
$551$	−58.9665	−2.51206
$552$	0	0
$553$	−11.0706	−0.470768
$554$	30.8839	1.31213
$555$	0	0
$556$	−10.7866	−0.457452
$557$	6.91358	0.292938	0.146469	−	0.989215i	$-0.453209\pi$
$557$	6.91358	0.292938	0.146469	+	0.989215i	$0.453209\pi$
$558$	0	0
$559$	−6.66893	−0.282066
$560$	1.20963	0.0511162
$561$	0	0
$562$	−26.1034	−1.10111
$563$	−27.2108	−1.14680	−0.573400	−	0.819276i	$-0.694376\pi$
$563$	−27.2108	−1.14680	−0.573400	+	0.819276i	$0.694376\pi$
$564$	0	0
$565$	19.8364	0.834525
$566$	−7.81142	−0.328339
$567$	0	0
$568$	−7.98808	−0.335172
$569$	−10.7063	−0.448831	−0.224415	−	0.974494i	$-0.572047\pi$
$569$	−10.7063	−0.448831	−0.224415	+	0.974494i	$0.572047\pi$
$570$	0	0
$571$	−10.6222	−0.444525	−0.222263	−	0.974987i	$-0.571344\pi$
$571$	−10.6222	−0.444525	−0.222263	+	0.974987i	$0.571344\pi$
$572$	−19.5508	−0.817460
$573$	0	0
$574$	14.8888	0.621447
$575$	−0.264276	−0.0110211
$576$	0	0
$577$	9.73558	0.405298	0.202649	−	0.979251i	$-0.435045\pi$
$577$	9.73558	0.405298	0.202649	+	0.979251i	$0.435045\pi$
$578$	32.9887	1.37215
$579$	0	0
$580$	−7.83309	−0.325252
$581$	17.6830	0.733616
$582$	0	0
$583$	2.85953	0.118430
$584$	−8.54418	−0.353561
$585$	0	0
$586$	−8.96391	−0.370296
$587$	0.457090	0.0188661	0.00943307	−	0.999956i	$-0.496997\pi$
$587$	0.457090	0.0188661	0.00943307	+	0.999956i	$0.496997\pi$
$588$	0	0
$589$	9.46683	0.390074
$590$	−5.46914	−0.225161
$591$	0	0
$592$	6.93568	0.285055
$593$	−0.151772	−0.00623254	−0.00311627	−	0.999995i	$-0.500992\pi$
$593$	−0.151772	−0.00623254	−0.00311627	+	0.999995i	$0.500992\pi$
$594$	0	0
$595$	8.55242	0.350615
$596$	6.16840	0.252667
$597$	0	0
$598$	−0.835561	−0.0341686
$599$	33.0532	1.35052	0.675259	−	0.737580i	$-0.264032\pi$
$599$	33.0532	1.35052	0.675259	+	0.737580i	$0.264032\pi$
$600$	0	0
$601$	31.3191	1.27753	0.638766	−	0.769401i	$-0.279445\pi$
$601$	31.3191	1.27753	0.638766	+	0.769401i	$0.279445\pi$
$602$	−2.55146	−0.103990
$603$	0	0
$604$	−3.20561	−0.130434
$605$	27.2375	1.10736
$606$	0	0
$607$	−20.1827	−0.819190	−0.409595	−	0.912267i	$-0.634330\pi$
$607$	−20.1827	−0.819190	−0.409595	+	0.912267i	$0.634330\pi$
$608$	7.52787	0.305296
$609$	0	0
$610$	−4.77845	−0.193474
$611$	−8.31670	−0.336458
$612$	0	0
$613$	−40.2224	−1.62457	−0.812283	−	0.583264i	$-0.801775\pi$
$613$	−40.2224	−1.62457	−0.812283	+	0.583264i	$0.801775\pi$
$614$	−12.3822	−0.499704
$615$	0	0
$616$	−7.47993	−0.301375
$617$	−13.6966	−0.551406	−0.275703	−	0.961243i	$-0.588911\pi$
$617$	−13.6966	−0.551406	−0.275703	+	0.961243i	$0.588911\pi$
$618$	0	0
$619$	31.6824	1.27342	0.636712	−	0.771102i	$-0.280294\pi$
$619$	31.6824	1.27342	0.636712	+	0.771102i	$0.280294\pi$
$620$	1.25757	0.0505052
$621$	0	0
$622$	5.84011	0.234167
$623$	−1.20963	−0.0484628
$624$	0	0
$625$	1.00000	0.0400000
$626$	−1.23272	−0.0492693
$627$	0	0
$628$	18.9970	0.758063
$629$	49.0371	1.95524
$630$	0	0
$631$	−18.4268	−0.733559	−0.366779	−	0.930308i	$-0.619540\pi$
$631$	−18.4268	−0.733559	−0.366779	+	0.930308i	$0.619540\pi$
$632$	−9.15202	−0.364048
$633$	0	0
$634$	1.63700	0.0650138
$635$	18.1282	0.719398
$636$	0	0
$637$	−17.5056	−0.693598
$638$	48.4371	1.91764
$639$	0	0
$640$	1.00000	0.0395285
$641$	38.8928	1.53617	0.768087	−	0.640345i	$-0.221209\pi$
$641$	38.8928	1.53617	0.768087	+	0.640345i	$0.221209\pi$
$642$	0	0
$643$	−0.293987	−0.0115937	−0.00579686	−	0.999983i	$-0.501845\pi$
$643$	−0.293987	−0.0115937	−0.00579686	+	0.999983i	$0.501845\pi$
$644$	−0.319677	−0.0125970
$645$	0	0
$646$	53.2241	2.09407
$647$	0.398872	0.0156813	0.00784065	−	0.999969i	$-0.497504\pi$
$647$	0.398872	0.0156813	0.00784065	+	0.999969i	$0.497504\pi$
$648$	0	0
$649$	33.8193	1.32752
$650$	3.16169	0.124012
$651$	0	0
$652$	23.4230	0.917317
$653$	−34.7771	−1.36093	−0.680467	−	0.732778i	$-0.738223\pi$
$653$	−34.7771	−1.36093	−0.680467	+	0.732778i	$0.738223\pi$
$654$	0	0
$655$	9.80263	0.383021
$656$	12.3086	0.480569
$657$	0	0
$658$	−3.18188	−0.124043
$659$	−5.42513	−0.211333	−0.105667	−	0.994402i	$-0.533698\pi$
$659$	−5.42513	−0.211333	−0.105667	+	0.994402i	$0.533698\pi$
$660$	0	0
$661$	−11.3870	−0.442903	−0.221452	−	0.975171i	$-0.571080\pi$
$661$	−11.3870	−0.442903	−0.221452	+	0.975171i	$0.571080\pi$
$662$	−26.6907	−1.03736
$663$	0	0
$664$	14.6185	0.567309
$665$	9.10595	0.353114
$666$	0	0
$667$	2.07010	0.0801547
$668$	0.988527	0.0382473
$669$	0	0
$670$	2.60784	0.100750
$671$	29.5483	1.14070
$672$	0	0
$673$	0.248918	0.00959510	0.00479755	−	0.999988i	$-0.498473\pi$
$673$	0.248918	0.00959510	0.00479755	+	0.999988i	$0.498473\pi$
$674$	34.0877	1.31301
$675$	0	0
$676$	−3.00371	−0.115527
$677$	39.3279	1.51149	0.755747	−	0.654864i	$-0.227274\pi$
$677$	39.3279	1.51149	0.755747	+	0.654864i	$0.227274\pi$
$678$	0	0
$679$	1.77991	0.0683066
$680$	7.07027	0.271133
$681$	0	0
$682$	−7.77637	−0.297773
$683$	37.9132	1.45071	0.725355	−	0.688375i	$-0.241676\pi$
$683$	37.9132	1.45071	0.725355	+	0.688375i	$0.241676\pi$
$684$	0	0
$685$	−11.4691	−0.438213
$686$	−15.1649	−0.578998
$687$	0	0
$688$	−2.10929	−0.0804160
$689$	−1.46208	−0.0557006
$690$	0	0
$691$	17.8350	0.678476	0.339238	−	0.940701i	$-0.389831\pi$
$691$	17.8350	0.678476	0.339238	+	0.940701i	$0.389831\pi$
$692$	12.8951	0.490198
$693$	0	0
$694$	8.76185	0.332595
$695$	−10.7866	−0.409158
$696$	0	0
$697$	87.0249	3.29630
$698$	30.2008	1.14312
$699$	0	0
$700$	1.20963	0.0457198
$701$	39.0714	1.47571	0.737854	−	0.674960i	$-0.235839\pi$
$701$	39.0714	1.47571	0.737854	+	0.674960i	$0.235839\pi$
$702$	0	0
$703$	52.2109	1.96917
$704$	−6.18365	−0.233055
$705$	0	0
$706$	−18.8254	−0.708502
$707$	−12.7272	−0.478655
$708$	0	0
$709$	2.85649	0.107278	0.0536388	−	0.998560i	$-0.482918\pi$
$709$	2.85649	0.107278	0.0536388	+	0.998560i	$0.482918\pi$
$710$	−7.98808	−0.299787
$711$	0	0
$712$	−1.00000	−0.0374766
$713$	−0.332346	−0.0124465
$714$	0	0
$715$	−19.5508	−0.731158
$716$	−19.8285	−0.741027
$717$	0	0
$718$	−13.4597	−0.502313
$719$	−46.6929	−1.74135	−0.870675	−	0.491858i	$-0.836318\pi$
$719$	−46.6929	−1.74135	−0.870675	+	0.491858i	$0.836318\pi$
$720$	0	0
$721$	15.0763	0.561471
$722$	37.6689	1.40189
$723$	0	0
$724$	20.1561	0.749095
$725$	−7.83309	−0.290914
$726$	0	0
$727$	17.8993	0.663850	0.331925	−	0.943306i	$-0.392302\pi$
$727$	17.8993	0.663850	0.331925	+	0.943306i	$0.392302\pi$
$728$	3.82448	0.141745
$729$	0	0
$730$	−8.54418	−0.316234
$731$	−14.9133	−0.551587
$732$	0	0
$733$	−32.2416	−1.19087	−0.595434	−	0.803404i	$-0.703020\pi$
$733$	−32.2416	−1.19087	−0.595434	+	0.803404i	$0.703020\pi$
$734$	−8.28334	−0.305744
$735$	0	0
$736$	−0.264276	−0.00974136
$737$	−16.1260	−0.594007
$738$	0	0
$739$	−18.9033	−0.695370	−0.347685	−	0.937611i	$-0.613032\pi$
$739$	−18.9033	−0.695370	−0.347685	+	0.937611i	$0.613032\pi$
$740$	6.93568	0.254961
$741$	0	0
$742$	−0.559375	−0.0205353
$743$	−41.0115	−1.50457	−0.752284	−	0.658839i	$-0.771048\pi$
$743$	−41.0115	−1.50457	−0.752284	+	0.658839i	$0.771048\pi$
$744$	0	0
$745$	6.16840	0.225993
$746$	−35.3587	−1.29457
$747$	0	0
$748$	−43.7201	−1.59856
$749$	−9.48199	−0.346464
$750$	0	0
$751$	−54.3085	−1.98175	−0.990873	−	0.134799i	$-0.956961\pi$
$751$	−54.3085	−1.98175	−0.990873	+	0.134799i	$0.956961\pi$
$752$	−2.63046	−0.0959229
$753$	0	0
$754$	−24.7658	−0.901918
$755$	−3.20561	−0.116664
$756$	0	0
$757$	12.1111	0.440184	0.220092	−	0.975479i	$-0.429364\pi$
$757$	12.1111	0.440184	0.220092	+	0.975479i	$0.429364\pi$
$758$	−12.3268	−0.447729
$759$	0	0
$760$	7.52787	0.273065
$761$	4.67371	0.169422	0.0847109	−	0.996406i	$-0.473003\pi$
$761$	4.67371	0.169422	0.0847109	+	0.996406i	$0.473003\pi$
$762$	0	0
$763$	13.9940	0.506617
$764$	8.85282	0.320284
$765$	0	0
$766$	24.8502	0.897873
$767$	−17.2917	−0.624368
$768$	0	0
$769$	17.4124	0.627908	0.313954	−	0.949438i	$-0.398346\pi$
$769$	17.4124	0.627908	0.313954	+	0.949438i	$0.398346\pi$
$770$	−7.47993	−0.269558
$771$	0	0
$772$	4.25214	0.153038
$773$	−40.7657	−1.46624	−0.733120	−	0.680099i	$-0.761937\pi$
$773$	−40.7657	−1.46624	−0.733120	+	0.680099i	$0.761937\pi$
$774$	0	0
$775$	1.25757	0.0451733
$776$	1.47145	0.0528219
$777$	0	0
$778$	−18.4175	−0.660299
$779$	92.6573	3.31979
$780$	0	0
$781$	49.3955	1.76751
$782$	−1.86851	−0.0668177
$783$	0	0
$784$	−5.53679	−0.197743
$785$	18.9970	0.678032
$786$	0	0
$787$	−6.98519	−0.248995	−0.124498	−	0.992220i	$-0.539732\pi$
$787$	−6.98519	−0.248995	−0.124498	+	0.992220i	$0.539732\pi$
$788$	14.8409	0.528685
$789$	0	0
$790$	−9.15202	−0.325614
$791$	23.9948	0.853156
$792$	0	0
$793$	−15.1080	−0.536500
$794$	−21.4986	−0.762957
$795$	0	0
$796$	−22.6702	−0.803524
$797$	−35.0490	−1.24150	−0.620750	−	0.784009i	$-0.713172\pi$
$797$	−35.0490	−1.24150	−0.620750	+	0.784009i	$0.713172\pi$
$798$	0	0
$799$	−18.5981	−0.657952
$800$	1.00000	0.0353553
$801$	0	0
$802$	−14.0476	−0.496039
$803$	52.8342	1.86448
$804$	0	0
$805$	−0.319677	−0.0112671
$806$	3.97605	0.140050
$807$	0	0
$808$	−10.5215	−0.370147
$809$	−31.4052	−1.10415	−0.552074	−	0.833795i	$-0.686163\pi$
$809$	−31.4052	−1.10415	−0.552074	+	0.833795i	$0.686163\pi$
$810$	0	0
$811$	24.5786	0.863071	0.431535	−	0.902096i	$-0.357972\pi$
$811$	24.5786	0.863071	0.431535	+	0.902096i	$0.357972\pi$
$812$	−9.47515	−0.332513
$813$	0	0
$814$	−42.8878	−1.50322
$815$	23.4230	0.820473
$816$	0	0
$817$	−15.8785	−0.555518
$818$	−34.8061	−1.21697
$819$	0	0
$820$	12.3086	0.429834
$821$	−29.5683	−1.03194	−0.515970	−	0.856607i	$-0.672568\pi$
$821$	−29.5683	−1.03194	−0.515970	+	0.856607i	$0.672568\pi$
$822$	0	0
$823$	−20.2185	−0.704774	−0.352387	−	0.935854i	$-0.614630\pi$
$823$	−20.2185	−0.704774	−0.352387	+	0.935854i	$0.614630\pi$
$824$	12.4636	0.434188
$825$	0	0
$826$	−6.61564	−0.230188
$827$	11.9257	0.414696	0.207348	−	0.978267i	$-0.433517\pi$
$827$	11.9257	0.414696	0.207348	+	0.978267i	$0.433517\pi$
$828$	0	0
$829$	−13.1521	−0.456791	−0.228395	−	0.973568i	$-0.573348\pi$
$829$	−13.1521	−0.456791	−0.228395	+	0.973568i	$0.573348\pi$
$830$	14.6185	0.507417
$831$	0	0
$832$	3.16169	0.109612
$833$	−39.1466	−1.35635
$834$	0	0
$835$	0.988527	0.0342094
$836$	−46.5497	−1.60996
$837$	0	0
$838$	−22.1464	−0.765035
$839$	20.2487	0.699062	0.349531	−	0.936925i	$-0.386341\pi$
$839$	20.2487	0.699062	0.349531	+	0.936925i	$0.386341\pi$
$840$	0	0
$841$	32.3574	1.11577
$842$	−15.1840	−0.523275
$843$	0	0
$844$	10.5320	0.362525
$845$	−3.00371	−0.103331
$846$	0	0
$847$	32.9473	1.13208
$848$	−0.462435	−0.0158801
$849$	0	0
$850$	7.07027	0.242508
$851$	−1.83294	−0.0628323
$852$	0	0
$853$	−3.03231	−0.103824	−0.0519121	−	0.998652i	$-0.516532\pi$
$853$	−3.03231	−0.103824	−0.0519121	+	0.998652i	$0.516532\pi$
$854$	−5.78016	−0.197793
$855$	0	0
$856$	−7.83875	−0.267923
$857$	40.7851	1.39319	0.696597	−	0.717463i	$-0.254697\pi$
$857$	40.7851	1.39319	0.696597	+	0.717463i	$0.254697\pi$
$858$	0	0
$859$	−44.9989	−1.53534	−0.767671	−	0.640844i	$-0.778584\pi$
$859$	−44.9989	−1.53534	−0.767671	+	0.640844i	$0.778584\pi$
$860$	−2.10929	−0.0719262
$861$	0	0
$862$	−1.31376	−0.0447470
$863$	−14.7164	−0.500953	−0.250477	−	0.968123i	$-0.580587\pi$
$863$	−14.7164	−0.500953	−0.250477	+	0.968123i	$0.580587\pi$
$864$	0	0
$865$	12.8951	0.438447
$866$	13.5237	0.459554
$867$	0	0
$868$	1.52120	0.0516327
$869$	56.5929	1.91978
$870$	0	0
$871$	8.24518	0.279377
$872$	11.5688	0.391770
$873$	0	0
$874$	−1.98944	−0.0672938
$875$	1.20963	0.0408930
$876$	0	0
$877$	−26.0266	−0.878854	−0.439427	−	0.898278i	$-0.644819\pi$
$877$	−26.0266	−0.878854	−0.439427	+	0.898278i	$0.644819\pi$
$878$	−24.2822	−0.819484
$879$	0	0
$880$	−6.18365	−0.208451
$881$	6.77315	0.228193	0.114097	−	0.993470i	$-0.463603\pi$
$881$	6.77315	0.228193	0.114097	+	0.993470i	$0.463603\pi$
$882$	0	0
$883$	37.7055	1.26889	0.634445	−	0.772968i	$-0.281229\pi$
$883$	37.7055	1.26889	0.634445	+	0.772968i	$0.281229\pi$
$884$	22.3540	0.751847
$885$	0	0
$886$	−12.1897	−0.409520
$887$	−33.9967	−1.14150	−0.570749	−	0.821125i	$-0.693347\pi$
$887$	−33.9967	−1.14150	−0.570749	+	0.821125i	$0.693347\pi$
$888$	0	0
$889$	21.9285	0.735458
$890$	−1.00000	−0.0335201
$891$	0	0
$892$	8.47585	0.283792
$893$	−19.8018	−0.662640
$894$	0	0
$895$	−19.8285	−0.662795
$896$	1.20963	0.0404109
$897$	0	0
$898$	17.8944	0.597144
$899$	−9.85067	−0.328538
$900$	0	0
$901$	−3.26954	−0.108924
$902$	−76.1119	−2.53425
$903$	0	0
$904$	19.8364	0.659750
$905$	20.1561	0.670011
$906$	0	0
$907$	−46.6713	−1.54970	−0.774848	−	0.632148i	$-0.782173\pi$
$907$	−46.6713	−1.54970	−0.774848	+	0.632148i	$0.782173\pi$
$908$	−10.5272	−0.349357
$909$	0	0
$910$	3.82448	0.126780
$911$	52.5575	1.74131	0.870654	−	0.491895i	$-0.163696\pi$
$911$	52.5575	1.74131	0.870654	+	0.491895i	$0.163696\pi$
$912$	0	0
$913$	−90.3959	−2.99167
$914$	23.8353	0.788403
$915$	0	0
$916$	−3.35717	−0.110924
$917$	11.8576	0.391571
$918$	0	0
$919$	35.9283	1.18516	0.592582	−	0.805510i	$-0.298108\pi$
$919$	35.9283	1.18516	0.592582	+	0.805510i	$0.298108\pi$
$920$	−0.264276	−0.00871294
$921$	0	0
$922$	−20.0223	−0.659398
$923$	−25.2558	−0.831306
$924$	0	0
$925$	6.93568	0.228044
$926$	−37.9390	−1.24675
$927$	0	0
$928$	−7.83309	−0.257134
$929$	36.2409	1.18902	0.594512	−	0.804087i	$-0.297345\pi$
$929$	36.2409	1.18902	0.594512	+	0.804087i	$0.297345\pi$
$930$	0	0
$931$	−41.6803	−1.36602
$932$	10.1710	0.333162
$933$	0	0
$934$	3.96070	0.129598
$935$	−43.7201	−1.42980
$936$	0	0
$937$	9.27714	0.303071	0.151535	−	0.988452i	$-0.451578\pi$
$937$	9.27714	0.303071	0.151535	+	0.988452i	$0.451578\pi$
$938$	3.15452	0.102999
$939$	0	0
$940$	−2.63046	−0.0857961
$941$	50.4630	1.64505	0.822524	−	0.568731i	$-0.192565\pi$
$941$	50.4630	1.64505	0.822524	+	0.568731i	$0.192565\pi$
$942$	0	0
$943$	−3.25286	−0.105928
$944$	−5.46914	−0.178005
$945$	0	0
$946$	13.0431	0.424068
$947$	−12.9703	−0.421478	−0.210739	−	0.977542i	$-0.567587\pi$
$947$	−12.9703	−0.421478	−0.210739	+	0.977542i	$0.567587\pi$
$948$	0	0
$949$	−27.0141	−0.876913
$950$	7.52787	0.244236
$951$	0	0
$952$	8.55242	0.277186
$953$	−13.1405	−0.425664	−0.212832	−	0.977089i	$-0.568269\pi$
$953$	−13.1405	−0.425664	−0.212832	+	0.977089i	$0.568269\pi$
$954$	0	0
$955$	8.85282	0.286471
$956$	14.6929	0.475203
$957$	0	0
$958$	34.3271	1.10906
$959$	−13.8734	−0.447996
$960$	0	0
$961$	−29.4185	−0.948984
$962$	21.9285	0.707003
$963$	0	0
$964$	−0.476590	−0.0153499
$965$	4.25214	0.136881
$966$	0	0
$967$	34.0057	1.09355	0.546775	−	0.837279i	$-0.315855\pi$
$967$	34.0057	1.09355	0.546775	+	0.837279i	$0.315855\pi$
$968$	27.2375	0.875447
$969$	0	0
$970$	1.47145	0.0472453
$971$	−13.3191	−0.427429	−0.213715	−	0.976896i	$-0.568556\pi$
$971$	−13.3191	−0.427429	−0.213715	+	0.976896i	$0.568556\pi$
$972$	0	0
$973$	−13.0478	−0.418292
$974$	−28.7272	−0.920478
$975$	0	0
$976$	−4.77845	−0.152954
$977$	−38.6755	−1.23734	−0.618670	−	0.785651i	$-0.712328\pi$
$977$	−38.6755	−1.23734	−0.618670	+	0.785651i	$0.712328\pi$
$978$	0	0
$979$	6.18365	0.197630
$980$	−5.53679	−0.176866
$981$	0	0
$982$	−22.5983	−0.721140
$983$	−29.8787	−0.952983	−0.476491	−	0.879179i	$-0.658092\pi$
$983$	−29.8787	−0.952983	−0.476491	+	0.879179i	$0.658092\pi$
$984$	0	0
$985$	14.8409	0.472870
$986$	−55.3821	−1.76373
$987$	0	0
$988$	23.8008	0.757205
$989$	0.557436	0.0177254
$990$	0	0
$991$	20.6525	0.656047	0.328024	−	0.944669i	$-0.393617\pi$
$991$	20.6525	0.656047	0.328024	+	0.944669i	$0.393617\pi$
$992$	1.25757	0.0399279
$993$	0	0
$994$	−9.66263	−0.306480
$995$	−22.6702	−0.718694
$996$	0	0
$997$	26.3348	0.834030	0.417015	−	0.908900i	$-0.363076\pi$
$997$	26.3348	0.834030	0.417015	+	0.908900i	$0.363076\pi$
$998$	17.4254	0.551590
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8010.2.a.bn.1.5		7
3.2	odd	2		2670.2.a.t.1.5	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2670.2.a.t.1.5	✓	7	3.2	odd	2
8010.2.a.bn.1.5		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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.20963 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.20963 q^{7} +1.00000 q^{8} +1.00000 q^{10} -6.18365 q^{11} +3.16169 q^{13} +1.20963 q^{14} +1.00000 q^{16} +7.07027 q^{17} +7.52787 q^{19} +1.00000 q^{20} -6.18365 q^{22} -0.264276 q^{23} +1.00000 q^{25} +3.16169 q^{26} +1.20963 q^{28} -7.83309 q^{29} +1.25757 q^{31} +1.00000 q^{32} +7.07027 q^{34} +1.20963 q^{35} +6.93568 q^{37} +7.52787 q^{38} +1.00000 q^{40} +12.3086 q^{41} -2.10929 q^{43} -6.18365 q^{44} -0.264276 q^{46} -2.63046 q^{47} -5.53679 q^{49} +1.00000 q^{50} +3.16169 q^{52} -0.462435 q^{53} -6.18365 q^{55} +1.20963 q^{56} -7.83309 q^{58} -5.46914 q^{59} -4.77845 q^{61} +1.25757 q^{62} +1.00000 q^{64} +3.16169 q^{65} +2.60784 q^{67} +7.07027 q^{68} +1.20963 q^{70} -7.98808 q^{71} -8.54418 q^{73} +6.93568 q^{74} +7.52787 q^{76} -7.47993 q^{77} -9.15202 q^{79} +1.00000 q^{80} +12.3086 q^{82} +14.6185 q^{83} +7.07027 q^{85} -2.10929 q^{86} -6.18365 q^{88} -1.00000 q^{89} +3.82448 q^{91} -0.264276 q^{92} -2.63046 q^{94} +7.52787 q^{95} +1.47145 q^{97} -5.53679 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 7 q^{2} + 7 q^{4} + 7 q^{5} + q^{7} + 7 q^{8}+O(q^{10}) \) 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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