LMFDB - Embedded newform 8010.2.a.bo.1.1

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:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 21x^{6} - 22x^{5} + 59x^{4} + 103x^{3} + 27x^{2} - 18x - 6 $ x^8 - x^7 - 21x^6 - 22x^5 + 59x^4 + 103x^3 + 27x^2 - 18x - 6
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.95321$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -4.45998 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -4.45998 q^{7} -1.00000 q^{8} -1.00000 q^{10} +0.479969 q^{11} +5.00211 q^{13} +4.45998 q^{14} +1.00000 q^{16} +6.43486 q^{17} +5.04993 q^{19} +1.00000 q^{20} -0.479969 q^{22} +5.25410 q^{23} +1.00000 q^{25} -5.00211 q^{26} -4.45998 q^{28} -8.95888 q^{29} +7.90719 q^{31} -1.00000 q^{32} -6.43486 q^{34} -4.45998 q^{35} -6.18146 q^{37} -5.04993 q^{38} -1.00000 q^{40} +10.6396 q^{41} +11.4332 q^{43} +0.479969 q^{44} -5.25410 q^{46} +0.180329 q^{47} +12.8914 q^{49} -1.00000 q^{50} +5.00211 q^{52} -12.3383 q^{53} +0.479969 q^{55} +4.45998 q^{56} +8.95888 q^{58} +7.58962 q^{59} -10.9811 q^{61} -7.90719 q^{62} +1.00000 q^{64} +5.00211 q^{65} -8.29989 q^{67} +6.43486 q^{68} +4.45998 q^{70} -4.33739 q^{71} -8.00975 q^{73} +6.18146 q^{74} +5.04993 q^{76} -2.14065 q^{77} -10.4975 q^{79} +1.00000 q^{80} -10.6396 q^{82} +7.69871 q^{83} +6.43486 q^{85} -11.4332 q^{86} -0.479969 q^{88} -1.00000 q^{89} -22.3093 q^{91} +5.25410 q^{92} -0.180329 q^{94} +5.04993 q^{95} -12.4923 q^{97} -12.8914 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} + 8 q^{4} + 8 q^{5} - 5 q^{7} - 8 q^{8}+O(q^{10}) $ 8 * q - 8 * q^2 + 8 * q^4 + 8 * q^5 - 5 * q^7 - 8 * q^8	$ 8 q - 8 q^{2} + 8 q^{4} + 8 q^{5} - 5 q^{7} - 8 q^{8} - 8 q^{10} + 3 q^{11} - 2 q^{13} + 5 q^{14} + 8 q^{16} + 9 q^{17} + 5 q^{19} + 8 q^{20} - 3 q^{22} + 8 q^{23} + 8 q^{25} + 2 q^{26} - 5 q^{28} + 10 q^{29} + 2 q^{31} - 8 q^{32} - 9 q^{34} - 5 q^{35} - 16 q^{37} - 5 q^{38} - 8 q^{40} + 19 q^{41} - 4 q^{43} + 3 q^{44} - 8 q^{46} + 17 q^{47} + 11 q^{49} - 8 q^{50} - 2 q^{52} + 16 q^{53} + 3 q^{55} + 5 q^{56} - 10 q^{58} + 30 q^{59} - 15 q^{61} - 2 q^{62} + 8 q^{64} - 2 q^{65} - 3 q^{67} + 9 q^{68} + 5 q^{70} + 32 q^{71} - 20 q^{73} + 16 q^{74} + 5 q^{76} - 2 q^{77} - q^{79} + 8 q^{80} - 19 q^{82} + 13 q^{83} + 9 q^{85} + 4 q^{86} - 3 q^{88} - 8 q^{89} - 8 q^{91} + 8 q^{92} - 17 q^{94} + 5 q^{95} - 40 q^{97} - 11 q^{98}+O(q^{100}) $ 8 * q - 8 * q^2 + 8 * q^4 + 8 * q^5 - 5 * q^7 - 8 * q^8 - 8 * q^10 + 3 * q^11 - 2 * q^13 + 5 * q^14 + 8 * q^16 + 9 * q^17 + 5 * q^19 + 8 * q^20 - 3 * q^22 + 8 * q^23 + 8 * q^25 + 2 * q^26 - 5 * q^28 + 10 * q^29 + 2 * q^31 - 8 * q^32 - 9 * q^34 - 5 * q^35 - 16 * q^37 - 5 * q^38 - 8 * q^40 + 19 * q^41 - 4 * q^43 + 3 * q^44 - 8 * q^46 + 17 * q^47 + 11 * q^49 - 8 * q^50 - 2 * q^52 + 16 * q^53 + 3 * q^55 + 5 * q^56 - 10 * q^58 + 30 * q^59 - 15 * q^61 - 2 * q^62 + 8 * q^64 - 2 * q^65 - 3 * q^67 + 9 * q^68 + 5 * q^70 + 32 * q^71 - 20 * q^73 + 16 * q^74 + 5 * q^76 - 2 * q^77 - q^79 + 8 * q^80 - 19 * q^82 + 13 * q^83 + 9 * q^85 + 4 * q^86 - 3 * q^88 - 8 * q^89 - 8 * q^91 + 8 * q^92 - 17 * q^94 + 5 * q^95 - 40 * q^97 - 11 * 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
$7$	−4.45998	−1.68571	−0.842856	−	0.538139i	$-0.819128\pi$
$7$	−4.45998	−1.68571	−0.842856	+	0.538139i	$0.819128\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−1.00000	−0.316228
$11$	0.479969	0.144716	0.0723581	−	0.997379i	$-0.476948\pi$
$11$	0.479969	0.144716	0.0723581	+	0.997379i	$0.476948\pi$
$12$	0	0
$13$	5.00211	1.38733	0.693667	−	0.720296i	$-0.255994\pi$
$13$	5.00211	1.38733	0.693667	+	0.720296i	$0.255994\pi$
$14$	4.45998	1.19198
$15$	0	0
$16$	1.00000	0.250000
$17$	6.43486	1.56068	0.780341	−	0.625354i	$-0.215045\pi$
$17$	6.43486	1.56068	0.780341	+	0.625354i	$0.215045\pi$
$18$	0	0
$19$	5.04993	1.15853	0.579266	−	0.815138i	$-0.303339\pi$
$19$	5.04993	1.15853	0.579266	+	0.815138i	$0.303339\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	−0.479969	−0.102330
$23$	5.25410	1.09555	0.547777	−	0.836624i	$-0.315474\pi$
$23$	5.25410	1.09555	0.547777	+	0.836624i	$0.315474\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−5.00211	−0.980994
$27$	0	0
$28$	−4.45998	−0.842856
$29$	−8.95888	−1.66362	−0.831811	−	0.555059i	$-0.812696\pi$
$29$	−8.95888	−1.66362	−0.831811	+	0.555059i	$0.812696\pi$
$30$	0	0
$31$	7.90719	1.42017	0.710087	−	0.704114i	$-0.248656\pi$
$31$	7.90719	1.42017	0.710087	+	0.704114i	$0.248656\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−6.43486	−1.10357
$35$	−4.45998	−0.753874
$36$	0	0
$37$	−6.18146	−1.01622	−0.508112	−	0.861291i	$-0.669657\pi$
$37$	−6.18146	−1.01622	−0.508112	+	0.861291i	$0.669657\pi$
$38$	−5.04993	−0.819207
$39$	0	0
$40$	−1.00000	−0.158114
$41$	10.6396	1.66162	0.830809	−	0.556557i	$-0.187878\pi$
$41$	10.6396	1.66162	0.830809	+	0.556557i	$0.187878\pi$
$42$	0	0
$43$	11.4332	1.74354	0.871772	−	0.489913i	$-0.162971\pi$
$43$	11.4332	1.74354	0.871772	+	0.489913i	$0.162971\pi$
$44$	0.479969	0.0723581
$45$	0	0
$46$	−5.25410	−0.774674
$47$	0.180329	0.0263037	0.0131518	−	0.999914i	$-0.495814\pi$
$47$	0.180329	0.0263037	0.0131518	+	0.999914i	$0.495814\pi$
$48$	0	0
$49$	12.8914	1.84163
$50$	−1.00000	−0.141421
$51$	0	0
$52$	5.00211	0.693667
$53$	−12.3383	−1.69479	−0.847396	−	0.530962i	$-0.821831\pi$
$53$	−12.3383	−1.69479	−0.847396	+	0.530962i	$0.821831\pi$
$54$	0	0
$55$	0.479969	0.0647190
$56$	4.45998	0.595989
$57$	0	0
$58$	8.95888	1.17636
$59$	7.58962	0.988085	0.494042	−	0.869438i	$-0.335519\pi$
$59$	7.58962	0.988085	0.494042	+	0.869438i	$0.335519\pi$
$60$	0	0
$61$	−10.9811	−1.40599	−0.702993	−	0.711197i	$-0.748153\pi$
$61$	−10.9811	−1.40599	−0.702993	+	0.711197i	$0.748153\pi$
$62$	−7.90719	−1.00421
$63$	0	0
$64$	1.00000	0.125000
$65$	5.00211	0.620435
$66$	0	0
$67$	−8.29989	−1.01399	−0.506996	−	0.861948i	$-0.669244\pi$
$67$	−8.29989	−1.01399	−0.506996	+	0.861948i	$0.669244\pi$
$68$	6.43486	0.780341
$69$	0	0
$70$	4.45998	0.533069
$71$	−4.33739	−0.514754	−0.257377	−	0.966311i	$-0.582858\pi$
$71$	−4.33739	−0.514754	−0.257377	+	0.966311i	$0.582858\pi$
$72$	0	0
$73$	−8.00975	−0.937470	−0.468735	−	0.883339i	$-0.655290\pi$
$73$	−8.00975	−0.937470	−0.468735	+	0.883339i	$0.655290\pi$
$74$	6.18146	0.718580
$75$	0	0
$76$	5.04993	0.579266
$77$	−2.14065	−0.243950
$78$	0	0
$79$	−10.4975	−1.18106	−0.590530	−	0.807016i	$-0.701081\pi$
$79$	−10.4975	−1.18106	−0.590530	+	0.807016i	$0.701081\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	−10.6396	−1.17494
$83$	7.69871	0.845044	0.422522	−	0.906353i	$-0.361145\pi$
$83$	7.69871	0.845044	0.422522	+	0.906353i	$0.361145\pi$
$84$	0	0
$85$	6.43486	0.697958
$86$	−11.4332	−1.23287
$87$	0	0
$88$	−0.479969	−0.0511649
$89$	−1.00000	−0.106000
$89$	−1.00000	−0.106000
$90$	0	0
$91$	−22.3093	−2.33865
$92$	5.25410	0.547777
$93$	0	0
$94$	−0.180329	−0.0185995
$95$	5.04993	0.518112
$96$	0	0
$97$	−12.4923	−1.26840	−0.634201	−	0.773168i	$-0.718671\pi$
$97$	−12.4923	−1.26840	−0.634201	+	0.773168i	$0.718671\pi$
$98$	−12.8914	−1.30223
$99$	0	0
$100$	1.00000	0.100000
$101$	15.8857	1.58069	0.790344	−	0.612663i	$-0.209902\pi$
$101$	15.8857	1.58069	0.790344	+	0.612663i	$0.209902\pi$
$102$	0	0
$103$	−5.07106	−0.499666	−0.249833	−	0.968289i	$-0.580376\pi$
$103$	−5.07106	−0.499666	−0.249833	+	0.968289i	$0.580376\pi$
$104$	−5.00211	−0.490497
$105$	0	0
$106$	12.3383	1.19840
$107$	7.08632	0.685060	0.342530	−	0.939507i	$-0.388716\pi$
$107$	7.08632	0.685060	0.342530	+	0.939507i	$0.388716\pi$
$108$	0	0
$109$	8.78679	0.841621	0.420811	−	0.907148i	$-0.361746\pi$
$109$	8.78679	0.841621	0.420811	+	0.907148i	$0.361746\pi$
$110$	−0.479969	−0.0457633
$111$	0	0
$112$	−4.45998	−0.421428
$113$	12.1207	1.14022	0.570109	−	0.821569i	$-0.306901\pi$
$113$	12.1207	1.14022	0.570109	+	0.821569i	$0.306901\pi$
$114$	0	0
$115$	5.25410	0.489947
$116$	−8.95888	−0.831811
$117$	0	0
$118$	−7.58962	−0.698681
$119$	−28.6993	−2.63086
$120$	0	0
$121$	−10.7696	−0.979057
$122$	10.9811	0.994182
$123$	0	0
$124$	7.90719	0.710087
$125$	1.00000	0.0894427
$126$	0	0
$127$	−13.6747	−1.21343	−0.606715	−	0.794920i	$-0.707513\pi$
$127$	−13.6747	−1.21343	−0.606715	+	0.794920i	$0.707513\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−5.00211	−0.438714
$131$	1.28471	0.112245	0.0561227	−	0.998424i	$-0.482126\pi$
$131$	1.28471	0.112245	0.0561227	+	0.998424i	$0.482126\pi$
$132$	0	0
$133$	−22.5226	−1.95295
$134$	8.29989	0.717001
$135$	0	0
$136$	−6.43486	−0.551785
$137$	0.667885	0.0570613	0.0285307	−	0.999593i	$-0.490917\pi$
$137$	0.667885	0.0570613	0.0285307	+	0.999593i	$0.490917\pi$
$138$	0	0
$139$	0.268139	0.0227433	0.0113716	−	0.999935i	$-0.496380\pi$
$139$	0.268139	0.0227433	0.0113716	+	0.999935i	$0.496380\pi$
$140$	−4.45998	−0.376937
$141$	0	0
$142$	4.33739	0.363986
$143$	2.40086	0.200770
$144$	0	0
$145$	−8.95888	−0.743994
$146$	8.00975	0.662891
$147$	0	0
$148$	−6.18146	−0.508112
$149$	−5.62211	−0.460581	−0.230291	−	0.973122i	$-0.573968\pi$
$149$	−5.62211	−0.460581	−0.230291	+	0.973122i	$0.573968\pi$
$150$	0	0
$151$	16.3873	1.33358	0.666791	−	0.745245i	$-0.267667\pi$
$151$	16.3873	1.33358	0.666791	+	0.745245i	$0.267667\pi$
$152$	−5.04993	−0.409603
$153$	0	0
$154$	2.14065	0.172499
$155$	7.90719	0.635121
$156$	0	0
$157$	−13.8126	−1.10236	−0.551181	−	0.834386i	$-0.685823\pi$
$157$	−13.8126	−1.10236	−0.551181	+	0.834386i	$0.685823\pi$
$158$	10.4975	0.835136
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	−23.4331	−1.84679
$162$	0	0
$163$	−11.8212	−0.925908	−0.462954	−	0.886382i	$-0.653211\pi$
$163$	−11.8212	−0.925908	−0.462954	+	0.886382i	$0.653211\pi$
$164$	10.6396	0.830809
$165$	0	0
$166$	−7.69871	−0.597536
$167$	−15.5973	−1.20696	−0.603479	−	0.797379i	$-0.706219\pi$
$167$	−15.5973	−1.20696	−0.603479	+	0.797379i	$0.706219\pi$
$168$	0	0
$169$	12.0211	0.924698
$170$	−6.43486	−0.493531
$171$	0	0
$172$	11.4332	0.871772
$173$	−1.33332	−0.101370	−0.0506852	−	0.998715i	$-0.516141\pi$
$173$	−1.33332	−0.101370	−0.0506852	+	0.998715i	$0.516141\pi$
$174$	0	0
$175$	−4.45998	−0.337143
$176$	0.479969	0.0361790
$177$	0	0
$178$	1.00000	0.0749532
$179$	1.97238	0.147423	0.0737113	−	0.997280i	$-0.476516\pi$
$179$	1.97238	0.147423	0.0737113	+	0.997280i	$0.476516\pi$
$180$	0	0
$181$	13.1334	0.976197	0.488098	−	0.872789i	$-0.337691\pi$
$181$	13.1334	0.976197	0.488098	+	0.872789i	$0.337691\pi$
$182$	22.3093	1.65367
$183$	0	0
$184$	−5.25410	−0.387337
$185$	−6.18146	−0.454470
$186$	0	0
$187$	3.08853	0.225856
$188$	0.180329	0.0131518
$189$	0	0
$190$	−5.04993	−0.366360
$191$	22.6304	1.63748	0.818738	−	0.574167i	$-0.194674\pi$
$191$	22.6304	1.63748	0.818738	+	0.574167i	$0.194674\pi$
$192$	0	0
$193$	19.4004	1.39647	0.698237	−	0.715867i	$-0.253968\pi$
$193$	19.4004	1.39647	0.698237	+	0.715867i	$0.253968\pi$
$194$	12.4923	0.896896
$195$	0	0
$196$	12.8914	0.920814
$197$	17.2500	1.22901	0.614505	−	0.788913i	$-0.289356\pi$
$197$	17.2500	1.22901	0.614505	+	0.788913i	$0.289356\pi$
$198$	0	0
$199$	3.37668	0.239366	0.119683	−	0.992812i	$-0.461812\pi$
$199$	3.37668	0.239366	0.119683	+	0.992812i	$0.461812\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	−15.8857	−1.11772
$203$	39.9564	2.80439
$204$	0	0
$205$	10.6396	0.743099
$206$	5.07106	0.353317
$207$	0	0
$208$	5.00211	0.346834
$209$	2.42381	0.167658
$210$	0	0
$211$	14.9198	1.02712	0.513562	−	0.858053i	$-0.328326\pi$
$211$	14.9198	1.02712	0.513562	+	0.858053i	$0.328326\pi$
$212$	−12.3383	−0.847396
$213$	0	0
$214$	−7.08632	−0.484411
$215$	11.4332	0.779736
$216$	0	0
$217$	−35.2659	−2.39401
$218$	−8.78679	−0.595116
$219$	0	0
$220$	0.479969	0.0323595
$221$	32.1878	2.16519
$222$	0	0
$223$	4.08935	0.273843	0.136921	−	0.990582i	$-0.456279\pi$
$223$	4.08935	0.273843	0.136921	+	0.990582i	$0.456279\pi$
$224$	4.45998	0.297995
$225$	0	0
$226$	−12.1207	−0.806255
$227$	26.1264	1.73407	0.867034	−	0.498250i	$-0.166024\pi$
$227$	26.1264	1.73407	0.867034	+	0.498250i	$0.166024\pi$
$228$	0	0
$229$	11.2417	0.742872	0.371436	−	0.928459i	$-0.378866\pi$
$229$	11.2417	0.742872	0.371436	+	0.928459i	$0.378866\pi$
$230$	−5.25410	−0.346445
$231$	0	0
$232$	8.95888	0.588179
$233$	25.0391	1.64036	0.820182	−	0.572103i	$-0.193872\pi$
$233$	25.0391	1.64036	0.820182	+	0.572103i	$0.193872\pi$
$234$	0	0
$235$	0.180329	0.0117634
$236$	7.58962	0.494042
$237$	0	0
$238$	28.6993	1.86030
$239$	−12.2839	−0.794578	−0.397289	−	0.917694i	$-0.630049\pi$
$239$	−12.2839	−0.794578	−0.397289	+	0.917694i	$0.630049\pi$
$240$	0	0
$241$	−4.29702	−0.276795	−0.138398	−	0.990377i	$-0.544195\pi$
$241$	−4.29702	−0.276795	−0.138398	+	0.990377i	$0.544195\pi$
$242$	10.7696	0.692298
$243$	0	0
$244$	−10.9811	−0.702993
$245$	12.8914	0.823601
$246$	0	0
$247$	25.2603	1.60727
$248$	−7.90719	−0.502107
$249$	0	0
$250$	−1.00000	−0.0632456
$251$	14.8391	0.936637	0.468319	−	0.883560i	$-0.344860\pi$
$251$	14.8391	0.936637	0.468319	+	0.883560i	$0.344860\pi$
$252$	0	0
$253$	2.52180	0.158544
$254$	13.6747	0.858024
$255$	0	0
$256$	1.00000	0.0625000
$257$	−6.82802	−0.425920	−0.212960	−	0.977061i	$-0.568310\pi$
$257$	−6.82802	−0.425920	−0.212960	+	0.977061i	$0.568310\pi$
$258$	0	0
$259$	27.5691	1.71306
$260$	5.00211	0.310217
$261$	0	0
$262$	−1.28471	−0.0793695
$263$	2.32502	0.143367	0.0716834	−	0.997427i	$-0.477163\pi$
$263$	2.32502	0.143367	0.0716834	+	0.997427i	$0.477163\pi$
$264$	0	0
$265$	−12.3383	−0.757934
$266$	22.5226	1.38095
$267$	0	0
$268$	−8.29989	−0.506996
$269$	22.3485	1.36261	0.681305	−	0.732000i	$-0.261413\pi$
$269$	22.3485	1.36261	0.681305	+	0.732000i	$0.261413\pi$
$270$	0	0
$271$	23.7542	1.44296	0.721481	−	0.692434i	$-0.243462\pi$
$271$	23.7542	1.44296	0.721481	+	0.692434i	$0.243462\pi$
$272$	6.43486	0.390171
$273$	0	0
$274$	−0.667885	−0.0403484
$275$	0.479969	0.0289432
$276$	0	0
$277$	−28.5293	−1.71416	−0.857081	−	0.515181i	$-0.827725\pi$
$277$	−28.5293	−1.71416	−0.857081	+	0.515181i	$0.827725\pi$
$278$	−0.268139	−0.0160819
$279$	0	0
$280$	4.45998	0.266535
$281$	−14.7735	−0.881315	−0.440658	−	0.897675i	$-0.645255\pi$
$281$	−14.7735	−0.881315	−0.440658	+	0.897675i	$0.645255\pi$
$282$	0	0
$283$	20.2313	1.20263	0.601314	−	0.799013i	$-0.294644\pi$
$283$	20.2313	1.20263	0.601314	+	0.799013i	$0.294644\pi$
$284$	−4.33739	−0.257377
$285$	0	0
$286$	−2.40086	−0.141966
$287$	−47.4521	−2.80101
$288$	0	0
$289$	24.4074	1.43573
$290$	8.95888	0.526083
$291$	0	0
$292$	−8.00975	−0.468735
$293$	−22.4882	−1.31378	−0.656888	−	0.753988i	$-0.728127\pi$
$293$	−22.4882	−1.31378	−0.656888	+	0.753988i	$0.728127\pi$
$294$	0	0
$295$	7.58962	0.441885
$296$	6.18146	0.359290
$297$	0	0
$298$	5.62211	0.325680
$299$	26.2815	1.51990
$300$	0	0
$301$	−50.9917	−2.93911
$302$	−16.3873	−0.942985
$303$	0	0
$304$	5.04993	0.289633
$305$	−10.9811	−0.628776
$306$	0	0
$307$	−1.14339	−0.0652567	−0.0326284	−	0.999468i	$-0.510388\pi$
$307$	−1.14339	−0.0652567	−0.0326284	+	0.999468i	$0.510388\pi$
$308$	−2.14065	−0.121975
$309$	0	0
$310$	−7.90719	−0.449098
$311$	9.87775	0.560116	0.280058	−	0.959983i	$-0.409646\pi$
$311$	9.87775	0.560116	0.280058	+	0.959983i	$0.409646\pi$
$312$	0	0
$313$	33.9942	1.92146	0.960732	−	0.277477i	$-0.0894984\pi$
$313$	33.9942	1.92146	0.960732	+	0.277477i	$0.0894984\pi$
$314$	13.8126	0.779488
$315$	0	0
$316$	−10.4975	−0.590530
$317$	4.69492	0.263693	0.131847	−	0.991270i	$-0.457909\pi$
$317$	4.69492	0.263693	0.131847	+	0.991270i	$0.457909\pi$
$318$	0	0
$319$	−4.29999	−0.240753
$320$	1.00000	0.0559017
$321$	0	0
$322$	23.4331	1.30588
$323$	32.4956	1.80810
$324$	0	0
$325$	5.00211	0.277467
$326$	11.8212	0.654716
$327$	0	0
$328$	−10.6396	−0.587471
$329$	−0.804262	−0.0443404
$330$	0	0
$331$	10.2480	0.563284	0.281642	−	0.959520i	$-0.409121\pi$
$331$	10.2480	0.563284	0.281642	+	0.959520i	$0.409121\pi$
$332$	7.69871	0.422522
$333$	0	0
$334$	15.5973	0.853448
$335$	−8.29989	−0.453471
$336$	0	0
$337$	0.883821	0.0481448	0.0240724	−	0.999710i	$-0.492337\pi$
$337$	0.883821	0.0481448	0.0240724	+	0.999710i	$0.492337\pi$
$338$	−12.0211	−0.653860
$339$	0	0
$340$	6.43486	0.348979
$341$	3.79521	0.205522
$342$	0	0
$343$	−26.2755	−1.41874
$344$	−11.4332	−0.616436
$345$	0	0
$346$	1.33332	0.0716797
$347$	−0.454293	−0.0243877	−0.0121939	−	0.999926i	$-0.503882\pi$
$347$	−0.454293	−0.0243877	−0.0121939	+	0.999926i	$0.503882\pi$
$348$	0	0
$349$	−35.1964	−1.88402	−0.942011	−	0.335583i	$-0.891067\pi$
$349$	−35.1964	−1.88402	−0.942011	+	0.335583i	$0.891067\pi$
$350$	4.45998	0.238396
$351$	0	0
$352$	−0.479969	−0.0255825
$353$	6.45262	0.343438	0.171719	−	0.985146i	$-0.445068\pi$
$353$	6.45262	0.343438	0.171719	+	0.985146i	$0.445068\pi$
$354$	0	0
$355$	−4.33739	−0.230205
$356$	−1.00000	−0.0529999
$357$	0	0
$358$	−1.97238	−0.104244
$359$	7.88378	0.416090	0.208045	−	0.978119i	$-0.433290\pi$
$359$	7.88378	0.416090	0.208045	+	0.978119i	$0.433290\pi$
$360$	0	0
$361$	6.50177	0.342199
$362$	−13.1334	−0.690275
$363$	0	0
$364$	−22.3093	−1.16932
$365$	−8.00975	−0.419249
$366$	0	0
$367$	−28.1658	−1.47024	−0.735121	−	0.677936i	$-0.762875\pi$
$367$	−28.1658	−1.47024	−0.735121	+	0.677936i	$0.762875\pi$
$368$	5.25410	0.273889
$369$	0	0
$370$	6.18146	0.321359
$371$	55.0284	2.85693
$372$	0	0
$373$	−36.7764	−1.90421	−0.952106	−	0.305769i	$-0.901086\pi$
$373$	−36.7764	−1.90421	−0.952106	+	0.305769i	$0.901086\pi$
$374$	−3.08853	−0.159704
$375$	0	0
$376$	−0.180329	−0.00929975
$377$	−44.8133	−2.30800
$378$	0	0
$379$	11.2814	0.579488	0.289744	−	0.957104i	$-0.406430\pi$
$379$	11.2814	0.579488	0.289744	+	0.957104i	$0.406430\pi$
$380$	5.04993	0.259056
$381$	0	0
$382$	−22.6304	−1.15787
$383$	−2.27807	−0.116404	−0.0582019	−	0.998305i	$-0.518537\pi$
$383$	−2.27807	−0.116404	−0.0582019	+	0.998305i	$0.518537\pi$
$384$	0	0
$385$	−2.14065	−0.109098
$386$	−19.4004	−0.987456
$387$	0	0
$388$	−12.4923	−0.634201
$389$	−6.78573	−0.344050	−0.172025	−	0.985093i	$-0.555031\pi$
$389$	−6.78573	−0.344050	−0.172025	+	0.985093i	$0.555031\pi$
$390$	0	0
$391$	33.8094	1.70981
$392$	−12.8914	−0.651114
$393$	0	0
$394$	−17.2500	−0.869042
$395$	−10.4975	−0.528186
$396$	0	0
$397$	−21.9651	−1.10240	−0.551199	−	0.834374i	$-0.685830\pi$
$397$	−21.9651	−1.10240	−0.551199	+	0.834374i	$0.685830\pi$
$398$	−3.37668	−0.169258
$399$	0	0
$400$	1.00000	0.0500000
$401$	8.67138	0.433028	0.216514	−	0.976280i	$-0.430531\pi$
$401$	8.67138	0.433028	0.216514	+	0.976280i	$0.430531\pi$
$402$	0	0
$403$	39.5526	1.97026
$404$	15.8857	0.790344
$405$	0	0
$406$	−39.9564	−1.98300
$407$	−2.96691	−0.147064
$408$	0	0
$409$	−7.88421	−0.389849	−0.194924	−	0.980818i	$-0.562446\pi$
$409$	−7.88421	−0.389849	−0.194924	+	0.980818i	$0.562446\pi$
$410$	−10.6396	−0.525450
$411$	0	0
$412$	−5.07106	−0.249833
$413$	−33.8495	−1.66563
$414$	0	0
$415$	7.69871	0.377915
$416$	−5.00211	−0.245248
$417$	0	0
$418$	−2.42381	−0.118552
$419$	23.9437	1.16973	0.584864	−	0.811131i	$-0.301148\pi$
$419$	23.9437	1.16973	0.584864	+	0.811131i	$0.301148\pi$
$420$	0	0
$421$	10.0014	0.487439	0.243720	−	0.969846i	$-0.421632\pi$
$421$	10.0014	0.487439	0.243720	+	0.969846i	$0.421632\pi$
$422$	−14.9198	−0.726286
$423$	0	0
$424$	12.3383	0.599199
$425$	6.43486	0.312137
$426$	0	0
$427$	48.9754	2.37009
$428$	7.08632	0.342530
$429$	0	0
$430$	−11.4332	−0.551357
$431$	4.68528	0.225682	0.112841	−	0.993613i	$-0.464005\pi$
$431$	4.68528	0.225682	0.112841	+	0.993613i	$0.464005\pi$
$432$	0	0
$433$	−9.86694	−0.474175	−0.237087	−	0.971488i	$-0.576193\pi$
$433$	−9.86694	−0.474175	−0.237087	+	0.971488i	$0.576193\pi$
$434$	35.2659	1.69282
$435$	0	0
$436$	8.78679	0.420811
$437$	26.5328	1.26924
$438$	0	0
$439$	−30.3016	−1.44622	−0.723109	−	0.690734i	$-0.757288\pi$
$439$	−30.3016	−1.44622	−0.723109	+	0.690734i	$0.757288\pi$
$440$	−0.479969	−0.0228816
$441$	0	0
$442$	−32.1878	−1.53102
$443$	−36.2974	−1.72454	−0.862272	−	0.506446i	$-0.830959\pi$
$443$	−36.2974	−1.72454	−0.862272	+	0.506446i	$0.830959\pi$
$444$	0	0
$445$	−1.00000	−0.0474045
$446$	−4.08935	−0.193636
$447$	0	0
$448$	−4.45998	−0.210714
$449$	21.4542	1.01249	0.506243	−	0.862391i	$-0.331034\pi$
$449$	21.4542	1.01249	0.506243	+	0.862391i	$0.331034\pi$
$450$	0	0
$451$	5.10666	0.240463
$452$	12.1207	0.570109
$453$	0	0
$454$	−26.1264	−1.22617
$455$	−22.3093	−1.04588
$456$	0	0
$457$	−25.7369	−1.20392	−0.601962	−	0.798525i	$-0.705614\pi$
$457$	−25.7369	−1.20392	−0.601962	+	0.798525i	$0.705614\pi$
$458$	−11.2417	−0.525290
$459$	0	0
$460$	5.25410	0.244973
$461$	15.3643	0.715585	0.357793	−	0.933801i	$-0.383529\pi$
$461$	15.3643	0.715585	0.357793	+	0.933801i	$0.383529\pi$
$462$	0	0
$463$	−7.53505	−0.350184	−0.175092	−	0.984552i	$-0.556022\pi$
$463$	−7.53505	−0.350184	−0.175092	+	0.984552i	$0.556022\pi$
$464$	−8.95888	−0.415905
$465$	0	0
$466$	−25.0391	−1.15991
$467$	−18.9685	−0.877759	−0.438880	−	0.898546i	$-0.644625\pi$
$467$	−18.9685	−0.877759	−0.438880	+	0.898546i	$0.644625\pi$
$468$	0	0
$469$	37.0173	1.70930
$470$	−0.180329	−0.00831795
$471$	0	0
$472$	−7.58962	−0.349341
$473$	5.48757	0.252319
$474$	0	0
$475$	5.04993	0.231707
$476$	−28.6993	−1.31543
$477$	0	0
$478$	12.2839	0.561851
$479$	25.8726	1.18215	0.591075	−	0.806616i	$-0.298704\pi$
$479$	25.8726	1.18215	0.591075	+	0.806616i	$0.298704\pi$
$480$	0	0
$481$	−30.9203	−1.40984
$482$	4.29702	0.195724
$483$	0	0
$484$	−10.7696	−0.489529
$485$	−12.4923	−0.567247
$486$	0	0
$487$	7.91132	0.358496	0.179248	−	0.983804i	$-0.442634\pi$
$487$	7.91132	0.358496	0.179248	+	0.983804i	$0.442634\pi$
$488$	10.9811	0.497091
$489$	0	0
$490$	−12.8914	−0.582374
$491$	−15.9387	−0.719305	−0.359653	−	0.933086i	$-0.617105\pi$
$491$	−15.9387	−0.719305	−0.359653	+	0.933086i	$0.617105\pi$
$492$	0	0
$493$	−57.6491	−2.59639
$494$	−25.2603	−1.13651
$495$	0	0
$496$	7.90719	0.355044
$497$	19.3447	0.867727
$498$	0	0
$499$	25.8367	1.15661	0.578304	−	0.815821i	$-0.303715\pi$
$499$	25.8367	1.15661	0.578304	+	0.815821i	$0.303715\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	−14.8391	−0.662303
$503$	−39.6058	−1.76594	−0.882968	−	0.469432i	$-0.844459\pi$
$503$	−39.6058	−1.76594	−0.882968	+	0.469432i	$0.844459\pi$
$504$	0	0
$505$	15.8857	0.706905
$506$	−2.52180	−0.112108
$507$	0	0
$508$	−13.6747	−0.606715
$509$	19.8128	0.878188	0.439094	−	0.898441i	$-0.355300\pi$
$509$	19.8128	0.878188	0.439094	+	0.898441i	$0.355300\pi$
$510$	0	0
$511$	35.7233	1.58030
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	6.82802	0.301171
$515$	−5.07106	−0.223458
$516$	0	0
$517$	0.0865523	0.00380657
$518$	−27.5691	−1.21132
$519$	0	0
$520$	−5.00211	−0.219357
$521$	−21.3309	−0.934522	−0.467261	−	0.884119i	$-0.654759\pi$
$521$	−21.3309	−0.934522	−0.467261	+	0.884119i	$0.654759\pi$
$522$	0	0
$523$	12.5506	0.548801	0.274400	−	0.961616i	$-0.411521\pi$
$523$	12.5506	0.548801	0.274400	+	0.961616i	$0.411521\pi$
$524$	1.28471	0.0561227
$525$	0	0
$526$	−2.32502	−0.101376
$527$	50.8817	2.21644
$528$	0	0
$529$	4.60552	0.200240
$530$	12.3383	0.535940
$531$	0	0
$532$	−22.5226	−0.976477
$533$	53.2202	2.30522
$534$	0	0
$535$	7.08632	0.306368
$536$	8.29989	0.358500
$537$	0	0
$538$	−22.3485	−0.963511
$539$	6.18747	0.266513
$540$	0	0
$541$	−31.7148	−1.36352	−0.681762	−	0.731574i	$-0.738786\pi$
$541$	−31.7148	−1.36352	−0.681762	+	0.731574i	$0.738786\pi$
$542$	−23.7542	−1.02033
$543$	0	0
$544$	−6.43486	−0.275892
$545$	8.78679	0.376385
$546$	0	0
$547$	41.7434	1.78482	0.892410	−	0.451226i	$-0.149013\pi$
$547$	41.7434	1.78482	0.892410	+	0.451226i	$0.149013\pi$
$548$	0.667885	0.0285307
$549$	0	0
$550$	−0.479969	−0.0204660
$551$	−45.2417	−1.92736
$552$	0	0
$553$	46.8186	1.99093
$554$	28.5293	1.21210
$555$	0	0
$556$	0.268139	0.0113716
$557$	13.1524	0.557287	0.278643	−	0.960395i	$-0.410115\pi$
$557$	13.1524	0.557287	0.278643	+	0.960395i	$0.410115\pi$
$558$	0	0
$559$	57.1900	2.41888
$560$	−4.45998	−0.188468
$561$	0	0
$562$	14.7735	0.623184
$563$	8.92230	0.376030	0.188015	−	0.982166i	$-0.439795\pi$
$563$	8.92230	0.376030	0.188015	+	0.982166i	$0.439795\pi$
$564$	0	0
$565$	12.1207	0.509921
$566$	−20.2313	−0.850387
$567$	0	0
$568$	4.33739	0.181993
$569$	32.4952	1.36227	0.681136	−	0.732157i	$-0.261486\pi$
$569$	32.4952	1.36227	0.681136	+	0.732157i	$0.261486\pi$
$570$	0	0
$571$	35.8296	1.49942	0.749711	−	0.661765i	$-0.230192\pi$
$571$	35.8296	1.49942	0.749711	+	0.661765i	$0.230192\pi$
$572$	2.40086	0.100385
$573$	0	0
$574$	47.4521	1.98061
$575$	5.25410	0.219111
$576$	0	0
$577$	−0.239868	−0.00998583	−0.00499292	−	0.999988i	$-0.501589\pi$
$577$	−0.239868	−0.00998583	−0.00499292	+	0.999988i	$0.501589\pi$
$578$	−24.4074	−1.01521
$579$	0	0
$580$	−8.95888	−0.371997
$581$	−34.3361	−1.42450
$582$	0	0
$583$	−5.92199	−0.245264
$584$	8.00975	0.331446
$585$	0	0
$586$	22.4882	0.928980
$587$	−37.9842	−1.56778	−0.783888	−	0.620903i	$-0.786766\pi$
$587$	−37.9842	−1.56778	−0.783888	+	0.620903i	$0.786766\pi$
$588$	0	0
$589$	39.9308	1.64532
$590$	−7.58962	−0.312460
$591$	0	0
$592$	−6.18146	−0.254056
$593$	−15.7498	−0.646769	−0.323384	−	0.946268i	$-0.604821\pi$
$593$	−15.7498	−0.646769	−0.323384	+	0.946268i	$0.604821\pi$
$594$	0	0
$595$	−28.6993	−1.17656
$596$	−5.62211	−0.230291
$597$	0	0
$598$	−26.2815	−1.07473
$599$	−17.1835	−0.702099	−0.351049	−	0.936357i	$-0.614175\pi$
$599$	−17.1835	−0.702099	−0.351049	+	0.936357i	$0.614175\pi$
$600$	0	0
$601$	−27.2620	−1.11204	−0.556020	−	0.831169i	$-0.687672\pi$
$601$	−27.2620	−1.11204	−0.556020	+	0.831169i	$0.687672\pi$
$602$	50.9917	2.07827
$603$	0	0
$604$	16.3873	0.666791
$605$	−10.7696	−0.437848
$606$	0	0
$607$	28.6121	1.16133	0.580665	−	0.814143i	$-0.302793\pi$
$607$	28.6121	1.16133	0.580665	+	0.814143i	$0.302793\pi$
$608$	−5.04993	−0.204802
$609$	0	0
$610$	10.9811	0.444612
$611$	0.902024	0.0364920
$612$	0	0
$613$	21.2549	0.858477	0.429238	−	0.903191i	$-0.358782\pi$
$613$	21.2549	0.858477	0.429238	+	0.903191i	$0.358782\pi$
$614$	1.14339	0.0461435
$615$	0	0
$616$	2.14065	0.0862493
$617$	−19.5840	−0.788422	−0.394211	−	0.919020i	$-0.628982\pi$
$617$	−19.5840	−0.788422	−0.394211	+	0.919020i	$0.628982\pi$
$618$	0	0
$619$	−23.0661	−0.927103	−0.463552	−	0.886070i	$-0.653425\pi$
$619$	−23.0661	−0.927103	−0.463552	+	0.886070i	$0.653425\pi$
$620$	7.90719	0.317561
$621$	0	0
$622$	−9.87775	−0.396062
$623$	4.45998	0.178685
$624$	0	0
$625$	1.00000	0.0400000
$626$	−33.9942	−1.35868
$627$	0	0
$628$	−13.8126	−0.551181
$629$	−39.7768	−1.58600
$630$	0	0
$631$	30.0890	1.19782	0.598911	−	0.800815i	$-0.295600\pi$
$631$	30.0890	1.19782	0.598911	+	0.800815i	$0.295600\pi$
$632$	10.4975	0.417568
$633$	0	0
$634$	−4.69492	−0.186459
$635$	−13.6747	−0.542662
$636$	0	0
$637$	64.4841	2.55495
$638$	4.29999	0.170238
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	−28.1522	−1.11194	−0.555972	−	0.831201i	$-0.687654\pi$
$641$	−28.1522	−1.11194	−0.555972	+	0.831201i	$0.687654\pi$
$642$	0	0
$643$	−9.97528	−0.393387	−0.196693	−	0.980465i	$-0.563020\pi$
$643$	−9.97528	−0.393387	−0.196693	+	0.980465i	$0.563020\pi$
$644$	−23.4331	−0.923395
$645$	0	0
$646$	−32.4956	−1.27852
$647$	−11.1058	−0.436613	−0.218307	−	0.975880i	$-0.570053\pi$
$647$	−11.1058	−0.436613	−0.218307	+	0.975880i	$0.570053\pi$
$648$	0	0
$649$	3.64279	0.142992
$650$	−5.00211	−0.196199
$651$	0	0
$652$	−11.8212	−0.462954
$653$	20.1152	0.787170	0.393585	−	0.919288i	$-0.371235\pi$
$653$	20.1152	0.787170	0.393585	+	0.919288i	$0.371235\pi$
$654$	0	0
$655$	1.28471	0.0501977
$656$	10.6396	0.415405
$657$	0	0
$658$	0.804262	0.0313534
$659$	23.7129	0.923723	0.461861	−	0.886952i	$-0.347182\pi$
$659$	23.7129	0.923723	0.461861	+	0.886952i	$0.347182\pi$
$660$	0	0
$661$	−14.9215	−0.580381	−0.290190	−	0.956969i	$-0.593719\pi$
$661$	−14.9215	−0.580381	−0.290190	+	0.956969i	$0.593719\pi$
$662$	−10.2480	−0.398302
$663$	0	0
$664$	−7.69871	−0.298768
$665$	−22.5226	−0.873387
$666$	0	0
$667$	−47.0708	−1.82259
$668$	−15.5973	−0.603479
$669$	0	0
$670$	8.29989	0.320653
$671$	−5.27059	−0.203469
$672$	0	0
$673$	−21.1446	−0.815063	−0.407531	−	0.913191i	$-0.633610\pi$
$673$	−21.1446	−0.815063	−0.407531	+	0.913191i	$0.633610\pi$
$674$	−0.883821	−0.0340435
$675$	0	0
$676$	12.0211	0.462349
$677$	−11.1083	−0.426925	−0.213462	−	0.976951i	$-0.568474\pi$
$677$	−11.1083	−0.426925	−0.213462	+	0.976951i	$0.568474\pi$
$678$	0	0
$679$	55.7154	2.13816
$680$	−6.43486	−0.246766
$681$	0	0
$682$	−3.79521	−0.145326
$683$	8.57039	0.327937	0.163969	−	0.986466i	$-0.447570\pi$
$683$	8.57039	0.327937	0.163969	+	0.986466i	$0.447570\pi$
$684$	0	0
$685$	0.667885	0.0255186
$686$	26.2755	1.00320
$687$	0	0
$688$	11.4332	0.435886
$689$	−61.7173	−2.35124
$690$	0	0
$691$	−9.30405	−0.353943	−0.176971	−	0.984216i	$-0.556630\pi$
$691$	−9.30405	−0.353943	−0.176971	+	0.984216i	$0.556630\pi$
$692$	−1.33332	−0.0506852
$693$	0	0
$694$	0.454293	0.0172447
$695$	0.268139	0.0101711
$696$	0	0
$697$	68.4640	2.59326
$698$	35.1964	1.33220
$699$	0	0
$700$	−4.45998	−0.168571
$701$	16.8849	0.637733	0.318867	−	0.947800i	$-0.396698\pi$
$701$	16.8849	0.637733	0.318867	+	0.947800i	$0.396698\pi$
$702$	0	0
$703$	−31.2159	−1.17733
$704$	0.479969	0.0180895
$705$	0	0
$706$	−6.45262	−0.242847
$707$	−70.8499	−2.66459
$708$	0	0
$709$	8.42459	0.316392	0.158196	−	0.987408i	$-0.449432\pi$
$709$	8.42459	0.316392	0.158196	+	0.987408i	$0.449432\pi$
$710$	4.33739	0.162780
$711$	0	0
$712$	1.00000	0.0374766
$713$	41.5452	1.55588
$714$	0	0
$715$	2.40086	0.0897870
$716$	1.97238	0.0737113
$717$	0	0
$718$	−7.88378	−0.294220
$719$	16.0847	0.599858	0.299929	−	0.953962i	$-0.403037\pi$
$719$	16.0847	0.599858	0.299929	+	0.953962i	$0.403037\pi$
$720$	0	0
$721$	22.6168	0.842294
$722$	−6.50177	−0.241971
$723$	0	0
$724$	13.1334	0.488098
$725$	−8.95888	−0.332724
$726$	0	0
$727$	6.26371	0.232308	0.116154	−	0.993231i	$-0.462943\pi$
$727$	6.26371	0.232308	0.116154	+	0.993231i	$0.462943\pi$
$728$	22.3093	0.826837
$729$	0	0
$730$	8.00975	0.296454
$731$	73.5709	2.72112
$732$	0	0
$733$	−41.6191	−1.53724	−0.768619	−	0.639707i	$-0.779056\pi$
$733$	−41.6191	−1.53724	−0.768619	+	0.639707i	$0.779056\pi$
$734$	28.1658	1.03962
$735$	0	0
$736$	−5.25410	−0.193669
$737$	−3.98369	−0.146741
$738$	0	0
$739$	−34.9026	−1.28391	−0.641956	−	0.766741i	$-0.721877\pi$
$739$	−34.9026	−1.28391	−0.641956	+	0.766741i	$0.721877\pi$
$740$	−6.18146	−0.227235
$741$	0	0
$742$	−55.0284	−2.02016
$743$	33.2880	1.22122	0.610609	−	0.791932i	$-0.290925\pi$
$743$	33.2880	1.22122	0.610609	+	0.791932i	$0.290925\pi$
$744$	0	0
$745$	−5.62211	−0.205978
$746$	36.7764	1.34648
$747$	0	0
$748$	3.08853	0.112928
$749$	−31.6048	−1.15481
$750$	0	0
$751$	−6.43989	−0.234995	−0.117497	−	0.993073i	$-0.537487\pi$
$751$	−6.43989	−0.234995	−0.117497	+	0.993073i	$0.537487\pi$
$752$	0.180329	0.00657592
$753$	0	0
$754$	44.8133	1.63200
$755$	16.3873	0.596396
$756$	0	0
$757$	36.3510	1.32120	0.660600	−	0.750738i	$-0.270302\pi$
$757$	36.3510	1.32120	0.660600	+	0.750738i	$0.270302\pi$
$758$	−11.2814	−0.409760
$759$	0	0
$760$	−5.04993	−0.183180
$761$	−25.5108	−0.924766	−0.462383	−	0.886680i	$-0.653005\pi$
$761$	−25.5108	−0.924766	−0.462383	+	0.886680i	$0.653005\pi$
$762$	0	0
$763$	−39.1889	−1.41873
$764$	22.6304	0.818738
$765$	0	0
$766$	2.27807	0.0823099
$767$	37.9641	1.37080
$768$	0	0
$769$	47.6531	1.71841	0.859207	−	0.511629i	$-0.170958\pi$
$769$	47.6531	1.71841	0.859207	+	0.511629i	$0.170958\pi$
$770$	2.14065	0.0771437
$771$	0	0
$772$	19.4004	0.698237
$773$	51.8530	1.86502	0.932511	−	0.361141i	$-0.117613\pi$
$773$	51.8530	1.86502	0.932511	+	0.361141i	$0.117613\pi$
$774$	0	0
$775$	7.90719	0.284035
$776$	12.4923	0.448448
$777$	0	0
$778$	6.78573	0.243280
$779$	53.7290	1.92504
$780$	0	0
$781$	−2.08182	−0.0744932
$782$	−33.8094	−1.20902
$783$	0	0
$784$	12.8914	0.460407
$785$	−13.8126	−0.492992
$786$	0	0
$787$	10.1572	0.362066	0.181033	−	0.983477i	$-0.442056\pi$
$787$	10.1572	0.362066	0.181033	+	0.983477i	$0.442056\pi$
$788$	17.2500	0.614505
$789$	0	0
$790$	10.4975	0.373484
$791$	−54.0579	−1.92208
$792$	0	0
$793$	−54.9286	−1.95057
$794$	21.9651	0.779513
$795$	0	0
$796$	3.37668	0.119683
$797$	30.5167	1.08096	0.540479	−	0.841357i	$-0.318243\pi$
$797$	30.5167	1.08096	0.540479	+	0.841357i	$0.318243\pi$
$798$	0	0
$799$	1.16039	0.0410517
$800$	−1.00000	−0.0353553
$801$	0	0
$802$	−8.67138	−0.306197
$803$	−3.84443	−0.135667
$804$	0	0
$805$	−23.4331	−0.825910
$806$	−39.5526	−1.39318
$807$	0	0
$808$	−15.8857	−0.558858
$809$	−20.7349	−0.728999	−0.364499	−	0.931204i	$-0.618760\pi$
$809$	−20.7349	−0.728999	−0.364499	+	0.931204i	$0.618760\pi$
$810$	0	0
$811$	22.9458	0.805736	0.402868	−	0.915258i	$-0.368013\pi$
$811$	22.9458	0.805736	0.402868	+	0.915258i	$0.368013\pi$
$812$	39.9564	1.40219
$813$	0	0
$814$	2.96691	0.103990
$815$	−11.8212	−0.414079
$816$	0	0
$817$	57.7367	2.01995
$818$	7.88421	0.275665
$819$	0	0
$820$	10.6396	0.371549
$821$	−19.2153	−0.670618	−0.335309	−	0.942108i	$-0.608841\pi$
$821$	−19.2153	−0.670618	−0.335309	+	0.942108i	$0.608841\pi$
$822$	0	0
$823$	−9.48348	−0.330573	−0.165287	−	0.986246i	$-0.552855\pi$
$823$	−9.48348	−0.330573	−0.165287	+	0.986246i	$0.552855\pi$
$824$	5.07106	0.176659
$825$	0	0
$826$	33.8495	1.17778
$827$	6.60277	0.229601	0.114800	−	0.993389i	$-0.463377\pi$
$827$	6.60277	0.229601	0.114800	+	0.993389i	$0.463377\pi$
$828$	0	0
$829$	−29.3896	−1.02074	−0.510371	−	0.859955i	$-0.670492\pi$
$829$	−29.3896	−1.02074	−0.510371	+	0.859955i	$0.670492\pi$
$830$	−7.69871	−0.267226
$831$	0	0
$832$	5.00211	0.173417
$833$	82.9543	2.87420
$834$	0	0
$835$	−15.5973	−0.539768
$836$	2.42381	0.0838292
$837$	0	0
$838$	−23.9437	−0.827123
$839$	20.8688	0.720469	0.360235	−	0.932862i	$-0.382697\pi$
$839$	20.8688	0.720469	0.360235	+	0.932862i	$0.382697\pi$
$840$	0	0
$841$	51.2615	1.76764
$842$	−10.0014	−0.344672
$843$	0	0
$844$	14.9198	0.513562
$845$	12.0211	0.413537
$846$	0	0
$847$	48.0323	1.65041
$848$	−12.3383	−0.423698
$849$	0	0
$850$	−6.43486	−0.220714
$851$	−32.4780	−1.11333
$852$	0	0
$853$	36.3082	1.24317	0.621585	−	0.783347i	$-0.286489\pi$
$853$	36.3082	1.24317	0.621585	+	0.783347i	$0.286489\pi$
$854$	−48.9754	−1.67590
$855$	0	0
$856$	−7.08632	−0.242205
$857$	−54.0279	−1.84556	−0.922779	−	0.385330i	$-0.874088\pi$
$857$	−54.0279	−1.84556	−0.922779	+	0.385330i	$0.874088\pi$
$858$	0	0
$859$	26.6279	0.908531	0.454265	−	0.890866i	$-0.349902\pi$
$859$	26.6279	0.908531	0.454265	+	0.890866i	$0.349902\pi$
$860$	11.4332	0.389868
$861$	0	0
$862$	−4.68528	−0.159581
$863$	33.7336	1.14830	0.574152	−	0.818749i	$-0.305332\pi$
$863$	33.7336	1.14830	0.574152	+	0.818749i	$0.305332\pi$
$864$	0	0
$865$	−1.33332	−0.0453342
$866$	9.86694	0.335292
$867$	0	0
$868$	−35.2659	−1.19700
$869$	−5.03848	−0.170919
$870$	0	0
$871$	−41.5169	−1.40675
$872$	−8.78679	−0.297558
$873$	0	0
$874$	−26.5328	−0.897485
$875$	−4.45998	−0.150775
$876$	0	0
$877$	−4.00605	−0.135275	−0.0676373	−	0.997710i	$-0.521546\pi$
$877$	−4.00605	−0.135275	−0.0676373	+	0.997710i	$0.521546\pi$
$878$	30.3016	1.02263
$879$	0	0
$880$	0.479969	0.0161798
$881$	−33.3263	−1.12279	−0.561396	−	0.827547i	$-0.689736\pi$
$881$	−33.3263	−1.12279	−0.561396	+	0.827547i	$0.689736\pi$
$882$	0	0
$883$	−30.7719	−1.03556	−0.517779	−	0.855514i	$-0.673241\pi$
$883$	−30.7719	−1.03556	−0.517779	+	0.855514i	$0.673241\pi$
$884$	32.1878	1.08259
$885$	0	0
$886$	36.2974	1.21944
$887$	21.0231	0.705886	0.352943	−	0.935645i	$-0.385181\pi$
$887$	21.0231	0.705886	0.352943	+	0.935645i	$0.385181\pi$
$888$	0	0
$889$	60.9886	2.04549
$890$	1.00000	0.0335201
$891$	0	0
$892$	4.08935	0.136921
$893$	0.910648	0.0304737
$894$	0	0
$895$	1.97238	0.0659294
$896$	4.45998	0.148997
$897$	0	0
$898$	−21.4542	−0.715936
$899$	−70.8396	−2.36263
$900$	0	0
$901$	−79.3950	−2.64503
$902$	−5.10666	−0.170033
$903$	0	0
$904$	−12.1207	−0.403128
$905$	13.1334	0.436568
$906$	0	0
$907$	27.1081	0.900110	0.450055	−	0.893001i	$-0.351404\pi$
$907$	27.1081	0.900110	0.450055	+	0.893001i	$0.351404\pi$
$908$	26.1264	0.867034
$909$	0	0
$910$	22.3093	0.739545
$911$	47.8291	1.58465	0.792324	−	0.610100i	$-0.208871\pi$
$911$	47.8291	1.58465	0.792324	+	0.610100i	$0.208871\pi$
$912$	0	0
$913$	3.69515	0.122291
$914$	25.7369	0.851303
$915$	0	0
$916$	11.2417	0.371436
$917$	−5.72977	−0.189214
$918$	0	0
$919$	−18.8743	−0.622607	−0.311304	−	0.950310i	$-0.600766\pi$
$919$	−18.8743	−0.622607	−0.311304	+	0.950310i	$0.600766\pi$
$920$	−5.25410	−0.173222
$921$	0	0
$922$	−15.3643	−0.505995
$923$	−21.6961	−0.714136
$924$	0	0
$925$	−6.18146	−0.203245
$926$	7.53505	0.247617
$927$	0	0
$928$	8.95888	0.294090
$929$	−50.1184	−1.64433	−0.822166	−	0.569247i	$-0.807235\pi$
$929$	−50.1184	−1.64433	−0.822166	+	0.569247i	$0.807235\pi$
$930$	0	0
$931$	65.1006	2.13359
$932$	25.0391	0.820182
$933$	0	0
$934$	18.9685	0.620670
$935$	3.08853	0.101006
$936$	0	0
$937$	−5.29410	−0.172951	−0.0864753	−	0.996254i	$-0.527560\pi$
$937$	−5.29410	−0.172951	−0.0864753	+	0.996254i	$0.527560\pi$
$938$	−37.0173	−1.20866
$939$	0	0
$940$	0.180329	0.00588168
$941$	−0.846632	−0.0275994	−0.0137997	−	0.999905i	$-0.504393\pi$
$941$	−0.846632	−0.0275994	−0.0137997	+	0.999905i	$0.504393\pi$
$942$	0	0
$943$	55.9012	1.82039
$944$	7.58962	0.247021
$945$	0	0
$946$	−5.48757	−0.178416
$947$	29.8290	0.969313	0.484657	−	0.874705i	$-0.338945\pi$
$947$	29.8290	0.969313	0.484657	+	0.874705i	$0.338945\pi$
$948$	0	0
$949$	−40.0656	−1.30058
$950$	−5.04993	−0.163841
$951$	0	0
$952$	28.6993	0.930150
$953$	8.28712	0.268446	0.134223	−	0.990951i	$-0.457146\pi$
$953$	8.28712	0.268446	0.134223	+	0.990951i	$0.457146\pi$
$954$	0	0
$955$	22.6304	0.732302
$956$	−12.2839	−0.397289
$957$	0	0
$958$	−25.8726	−0.835907
$959$	−2.97875	−0.0961890
$960$	0	0
$961$	31.5237	1.01689
$962$	30.9203	0.996910
$963$	0	0
$964$	−4.29702	−0.138398
$965$	19.4004	0.624522
$966$	0	0
$967$	−20.3567	−0.654626	−0.327313	−	0.944916i	$-0.606143\pi$
$967$	−20.3567	−0.654626	−0.327313	+	0.944916i	$0.606143\pi$
$968$	10.7696	0.346149
$969$	0	0
$970$	12.4923	0.401104
$971$	−20.0857	−0.644579	−0.322290	−	0.946641i	$-0.604452\pi$
$971$	−20.0857	−0.644579	−0.322290	+	0.946641i	$0.604452\pi$
$972$	0	0
$973$	−1.19590	−0.0383387
$974$	−7.91132	−0.253495
$975$	0	0
$976$	−10.9811	−0.351496
$977$	17.3256	0.554296	0.277148	−	0.960827i	$-0.410611\pi$
$977$	17.3256	0.554296	0.277148	+	0.960827i	$0.410611\pi$
$978$	0	0
$979$	−0.479969	−0.0153399
$980$	12.8914	0.411800
$981$	0	0
$982$	15.9387	0.508625
$983$	9.82708	0.313435	0.156718	−	0.987643i	$-0.449909\pi$
$983$	9.82708	0.313435	0.156718	+	0.987643i	$0.449909\pi$
$984$	0	0
$985$	17.2500	0.549630
$986$	57.6491	1.83592
$987$	0	0
$988$	25.2603	0.803636
$989$	60.0710	1.91015
$990$	0	0
$991$	19.3994	0.616243	0.308122	−	0.951347i	$-0.400300\pi$
$991$	19.3994	0.616243	0.308122	+	0.951347i	$0.400300\pi$
$992$	−7.90719	−0.251054
$993$	0	0
$994$	−19.3447	−0.613576
$995$	3.37668	0.107048
$996$	0	0
$997$	9.41168	0.298071	0.149035	−	0.988832i	$-0.452383\pi$
$997$	9.41168	0.298071	0.149035	+	0.988832i	$0.452383\pi$
$998$	−25.8367	−0.817845
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8010.2.a.bo.1.1	✓	8
3.2	odd	2		8010.2.a.bp.1.1	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8010.2.a.bo.1.1	✓	8	1.1	even	1	trivial
8010.2.a.bp.1.1	yes	8	3.2	odd	2

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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -4.45998 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -4.45998 q^{7} -1.00000 q^{8} -1.00000 q^{10} +0.479969 q^{11} +5.00211 q^{13} +4.45998 q^{14} +1.00000 q^{16} +6.43486 q^{17} +5.04993 q^{19} +1.00000 q^{20} -0.479969 q^{22} +5.25410 q^{23} +1.00000 q^{25} -5.00211 q^{26} -4.45998 q^{28} -8.95888 q^{29} +7.90719 q^{31} -1.00000 q^{32} -6.43486 q^{34} -4.45998 q^{35} -6.18146 q^{37} -5.04993 q^{38} -1.00000 q^{40} +10.6396 q^{41} +11.4332 q^{43} +0.479969 q^{44} -5.25410 q^{46} +0.180329 q^{47} +12.8914 q^{49} -1.00000 q^{50} +5.00211 q^{52} -12.3383 q^{53} +0.479969 q^{55} +4.45998 q^{56} +8.95888 q^{58} +7.58962 q^{59} -10.9811 q^{61} -7.90719 q^{62} +1.00000 q^{64} +5.00211 q^{65} -8.29989 q^{67} +6.43486 q^{68} +4.45998 q^{70} -4.33739 q^{71} -8.00975 q^{73} +6.18146 q^{74} +5.04993 q^{76} -2.14065 q^{77} -10.4975 q^{79} +1.00000 q^{80} -10.6396 q^{82} +7.69871 q^{83} +6.43486 q^{85} -11.4332 q^{86} -0.479969 q^{88} -1.00000 q^{89} -22.3093 q^{91} +5.25410 q^{92} -0.180329 q^{94} +5.04993 q^{95} -12.4923 q^{97} -12.8914 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} + 8 q^{4} + 8 q^{5} - 5 q^{7} - 8 q^{8}+O(q^{10}) \) 8 * q - 8 * q^2 + 8 * q^4 + 8 * q^5 - 5 * q^7 - 8 * q^8	\( 8 q - 8 q^{2} + 8 q^{4} + 8 q^{5} - 5 q^{7} - 8 q^{8} - 8 q^{10} + 3 q^{11} - 2 q^{13} + 5 q^{14} + 8 q^{16} + 9 q^{17} + 5 q^{19} + 8 q^{20} - 3 q^{22} + 8 q^{23} + 8 q^{25} + 2 q^{26} - 5 q^{28} + 10 q^{29} + 2 q^{31} - 8 q^{32} - 9 q^{34} - 5 q^{35} - 16 q^{37} - 5 q^{38} - 8 q^{40} + 19 q^{41} - 4 q^{43} + 3 q^{44} - 8 q^{46} + 17 q^{47} + 11 q^{49} - 8 q^{50} - 2 q^{52} + 16 q^{53} + 3 q^{55} + 5 q^{56} - 10 q^{58} + 30 q^{59} - 15 q^{61} - 2 q^{62} + 8 q^{64} - 2 q^{65} - 3 q^{67} + 9 q^{68} + 5 q^{70} + 32 q^{71} - 20 q^{73} + 16 q^{74} + 5 q^{76} - 2 q^{77} - q^{79} + 8 q^{80} - 19 q^{82} + 13 q^{83} + 9 q^{85} + 4 q^{86} - 3 q^{88} - 8 q^{89} - 8 q^{91} + 8 q^{92} - 17 q^{94} + 5 q^{95} - 40 q^{97} - 11 q^{98}+O(q^{100}) \) 8 * q - 8 * q^2 + 8 * q^4 + 8 * q^5 - 5 * q^7 - 8 * q^8 - 8 * q^10 + 3 * q^11 - 2 * q^13 + 5 * q^14 + 8 * q^16 + 9 * q^17 + 5 * q^19 + 8 * q^20 - 3 * q^22 + 8 * q^23 + 8 * q^25 + 2 * q^26 - 5 * q^28 + 10 * q^29 + 2 * q^31 - 8 * q^32 - 9 * q^34 - 5 * q^35 - 16 * q^37 - 5 * q^38 - 8 * q^40 + 19 * q^41 - 4 * q^43 + 3 * q^44 - 8 * q^46 + 17 * q^47 + 11 * q^49 - 8 * q^50 - 2 * q^52 + 16 * q^53 + 3 * q^55 + 5 * q^56 - 10 * q^58 + 30 * q^59 - 15 * q^61 - 2 * q^62 + 8 * q^64 - 2 * q^65 - 3 * q^67 + 9 * q^68 + 5 * q^70 + 32 * q^71 - 20 * q^73 + 16 * q^74 + 5 * q^76 - 2 * q^77 - q^79 + 8 * q^80 - 19 * q^82 + 13 * q^83 + 9 * q^85 + 4 * q^86 - 3 * q^88 - 8 * q^89 - 8 * q^91 + 8 * q^92 - 17 * q^94 + 5 * q^95 - 40 * q^97 - 11 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more