LMFDB - Embedded newform 8001.2.a.v.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8001,2,Mod(1,8001)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8001, 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("8001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8001 = 3^{2} \cdot 7 \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8001.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:	$63.8883066572$
Analytic rank:	$0$
Dimension:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - 4 x^{18} - 22 x^{17} + 101 x^{16} + 178 x^{15} - 1035 x^{14} - 583 x^{13} + 5572 x^{12} + \cdots + 210 $ x^19 - 4x^18 - 22x^17 + 101x^16 + 178x^15 - 1035x^14 - 583x^13 + 5572x^12 + 21x^11 - 17032x^10 + 4985x^9 + 29792x^8 - 13249x^7 - 28600x^6 + 14000x^5 + 13725x^4 - 5723x^3 - 2913x^2 + 608x + 210
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 2667)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Root			$-1.48888$ of defining polynomial
Character	$\chi$	$=$	8001.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.48888 q^{2} +0.216778 q^{4} +2.91159 q^{5} +1.00000 q^{7} -2.65501 q^{8} +O(q^{10})$	\(q+1.48888 q^{2} +0.216778 q^{4} +2.91159 q^{5} +1.00000 q^{7} -2.65501 q^{8} +4.33502 q^{10} -4.14635 q^{11} +3.36668 q^{13} +1.48888 q^{14} -4.38656 q^{16} -4.79573 q^{17} +0.147823 q^{19} +0.631168 q^{20} -6.17343 q^{22} +5.23645 q^{23} +3.47735 q^{25} +5.01260 q^{26} +0.216778 q^{28} +1.23094 q^{29} +6.14440 q^{31} -1.22106 q^{32} -7.14029 q^{34} +2.91159 q^{35} +7.83605 q^{37} +0.220092 q^{38} -7.73031 q^{40} -0.0490986 q^{41} -8.18235 q^{43} -0.898836 q^{44} +7.79646 q^{46} +3.35398 q^{47} +1.00000 q^{49} +5.17738 q^{50} +0.729822 q^{52} -4.90402 q^{53} -12.0725 q^{55} -2.65501 q^{56} +1.83273 q^{58} +12.0362 q^{59} +10.8592 q^{61} +9.14831 q^{62} +6.95510 q^{64} +9.80239 q^{65} -1.83840 q^{67} -1.03961 q^{68} +4.33502 q^{70} +9.72828 q^{71} +7.33222 q^{73} +11.6670 q^{74} +0.0320448 q^{76} -4.14635 q^{77} +6.96474 q^{79} -12.7719 q^{80} -0.0731022 q^{82} +13.2685 q^{83} -13.9632 q^{85} -12.1826 q^{86} +11.0086 q^{88} +11.1337 q^{89} +3.36668 q^{91} +1.13515 q^{92} +4.99369 q^{94} +0.430400 q^{95} -0.471210 q^{97} +1.48888 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q - 4 q^{2} + 22 q^{4} - 5 q^{5} + 19 q^{7} - 9 q^{8}+O(q^{10}) $ 19 * q - 4 * q^2 + 22 * q^4 - 5 * q^5 + 19 * q^7 - 9 * q^8	$ 19 q - 4 q^{2} + 22 q^{4} - 5 q^{5} + 19 q^{7} - 9 q^{8} + 9 q^{11} + 24 q^{13} - 4 q^{14} + 20 q^{16} - 17 q^{17} + 23 q^{19} - 5 q^{20} - 3 q^{22} + 17 q^{23} + 38 q^{25} - 28 q^{26} + 22 q^{28} - 2 q^{29} + 16 q^{31} - 17 q^{32} + 29 q^{34} - 5 q^{35} + 56 q^{37} - 2 q^{38} - 13 q^{40} + 7 q^{41} + 19 q^{43} + 29 q^{44} + 10 q^{46} - 25 q^{47} + 19 q^{49} + 9 q^{50} + 16 q^{52} - 18 q^{53} + 10 q^{55} - 9 q^{56} + 31 q^{58} - 11 q^{59} + 26 q^{61} - 26 q^{62} + 45 q^{64} - 27 q^{65} + 24 q^{67} - 14 q^{68} + 32 q^{71} + 51 q^{73} + 12 q^{76} + 9 q^{77} + 30 q^{79} + 30 q^{80} - 52 q^{82} - q^{83} + 44 q^{85} + 24 q^{86} - 30 q^{88} - 5 q^{89} + 24 q^{91} + 88 q^{92} + 7 q^{94} + 24 q^{95} + 5 q^{97} - 4 q^{98}+O(q^{100}) $ 19 * q - 4 * q^2 + 22 * q^4 - 5 * q^5 + 19 * q^7 - 9 * q^8 + 9 * q^11 + 24 * q^13 - 4 * q^14 + 20 * q^16 - 17 * q^17 + 23 * q^19 - 5 * q^20 - 3 * q^22 + 17 * q^23 + 38 * q^25 - 28 * q^26 + 22 * q^28 - 2 * q^29 + 16 * q^31 - 17 * q^32 + 29 * q^34 - 5 * q^35 + 56 * q^37 - 2 * q^38 - 13 * q^40 + 7 * q^41 + 19 * q^43 + 29 * q^44 + 10 * q^46 - 25 * q^47 + 19 * q^49 + 9 * q^50 + 16 * q^52 - 18 * q^53 + 10 * q^55 - 9 * q^56 + 31 * q^58 - 11 * q^59 + 26 * q^61 - 26 * q^62 + 45 * q^64 - 27 * q^65 + 24 * q^67 - 14 * q^68 + 32 * q^71 + 51 * q^73 + 12 * q^76 + 9 * q^77 + 30 * q^79 + 30 * q^80 - 52 * q^82 - q^83 + 44 * q^85 + 24 * q^86 - 30 * q^88 - 5 * q^89 + 24 * q^91 + 88 * q^92 + 7 * q^94 + 24 * q^95 + 5 * q^97 - 4 * 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 $

$2$	1.48888	1.05280	0.526400	−	0.850237i	$-0.323541\pi$
$2$	1.48888	1.05280	0.526400	+	0.850237i	$0.323541\pi$
$3$	0	0
$3$	0	0
$4$	0.216778	0.108389
$5$	2.91159	1.30210	0.651051	−	0.759034i	$-0.274328\pi$
$5$	2.91159	1.30210	0.651051	+	0.759034i	$0.274328\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−2.65501	−0.938689
$9$	0	0
$10$	4.33502	1.37085
$11$	−4.14635	−1.25017	−0.625085	−	0.780556i	$-0.714936\pi$
$11$	−4.14635	−1.25017	−0.625085	+	0.780556i	$0.714936\pi$
$12$	0	0
$13$	3.36668	0.933749	0.466874	−	0.884324i	$-0.345380\pi$
$13$	3.36668	0.933749	0.466874	+	0.884324i	$0.345380\pi$
$14$	1.48888	0.397921
$15$	0	0
$16$	−4.38656	−1.09664
$17$	−4.79573	−1.16314	−0.581568	−	0.813498i	$-0.697560\pi$
$17$	−4.79573	−1.16314	−0.581568	+	0.813498i	$0.697560\pi$
$18$	0	0
$19$	0.147823	0.0339130	0.0169565	−	0.999856i	$-0.494602\pi$
$19$	0.147823	0.0339130	0.0169565	+	0.999856i	$0.494602\pi$
$20$	0.631168	0.141134
$21$	0	0
$22$	−6.17343	−1.31618
$23$	5.23645	1.09187	0.545937	−	0.837826i	$-0.316174\pi$
$23$	5.23645	1.09187	0.545937	+	0.837826i	$0.316174\pi$
$24$	0	0
$25$	3.47735	0.695471
$26$	5.01260	0.983051
$27$	0	0
$28$	0.216778	0.0409672
$29$	1.23094	0.228580	0.114290	−	0.993447i	$-0.463541\pi$
$29$	1.23094	0.228580	0.114290	+	0.993447i	$0.463541\pi$
$30$	0	0
$31$	6.14440	1.10357	0.551784	−	0.833987i	$-0.313947\pi$
$31$	6.14440	1.10357	0.551784	+	0.833987i	$0.313947\pi$
$32$	−1.22106	−0.215855
$33$	0	0
$34$	−7.14029	−1.22455
$35$	2.91159	0.492148
$36$	0	0
$37$	7.83605	1.28824	0.644120	−	0.764925i	$-0.277224\pi$
$37$	7.83605	1.28824	0.644120	+	0.764925i	$0.277224\pi$
$38$	0.220092	0.0357036
$39$	0	0
$40$	−7.73031	−1.22227
$41$	−0.0490986	−0.00766792	−0.00383396	−	0.999993i	$-0.501220\pi$
$41$	−0.0490986	−0.00766792	−0.00383396	+	0.999993i	$0.501220\pi$
$42$	0	0
$43$	−8.18235	−1.24780	−0.623899	−	0.781505i	$-0.714452\pi$
$43$	−8.18235	−1.24780	−0.623899	+	0.781505i	$0.714452\pi$
$44$	−0.898836	−0.135505
$45$	0	0
$46$	7.79646	1.14953
$47$	3.35398	0.489229	0.244614	−	0.969620i	$-0.421339\pi$
$47$	3.35398	0.489229	0.244614	+	0.969620i	$0.421339\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	5.17738	0.732192
$51$	0	0
$52$	0.729822	0.101208
$53$	−4.90402	−0.673619	−0.336809	−	0.941573i	$-0.609348\pi$
$53$	−4.90402	−0.673619	−0.336809	+	0.941573i	$0.609348\pi$
$54$	0	0
$55$	−12.0725	−1.62785
$56$	−2.65501	−0.354791
$57$	0	0
$58$	1.83273	0.240649
$59$	12.0362	1.56698	0.783492	−	0.621402i	$-0.213436\pi$
$59$	12.0362	1.56698	0.783492	+	0.621402i	$0.213436\pi$
$60$	0	0
$61$	10.8592	1.39037	0.695186	−	0.718830i	$-0.255322\pi$
$61$	10.8592	1.39037	0.695186	+	0.718830i	$0.255322\pi$
$62$	9.14831	1.16184
$63$	0	0
$64$	6.95510	0.869388
$65$	9.80239	1.21584
$66$	0	0
$67$	−1.83840	−0.224596	−0.112298	−	0.993675i	$-0.535821\pi$
$67$	−1.83840	−0.224596	−0.112298	+	0.993675i	$0.535821\pi$
$68$	−1.03961	−0.126071
$69$	0	0
$70$	4.33502	0.518134
$71$	9.72828	1.15453	0.577267	−	0.816555i	$-0.304119\pi$
$71$	9.72828	1.15453	0.577267	+	0.816555i	$0.304119\pi$
$72$	0	0
$73$	7.33222	0.858171	0.429086	−	0.903264i	$-0.358836\pi$
$73$	7.33222	0.858171	0.429086	+	0.903264i	$0.358836\pi$
$74$	11.6670	1.35626
$75$	0	0
$76$	0.0320448	0.00367579
$77$	−4.14635	−0.472520
$78$	0	0
$79$	6.96474	0.783595	0.391797	−	0.920052i	$-0.371853\pi$
$79$	6.96474	0.783595	0.391797	+	0.920052i	$0.371853\pi$
$80$	−12.7719	−1.42794
$81$	0	0
$82$	−0.0731022	−0.00807279
$83$	13.2685	1.45641	0.728204	−	0.685360i	$-0.240355\pi$
$83$	13.2685	1.45641	0.728204	+	0.685360i	$0.240355\pi$
$84$	0	0
$85$	−13.9632	−1.51452
$86$	−12.1826	−1.31368
$87$	0	0
$88$	11.0086	1.17352
$89$	11.1337	1.18017	0.590086	−	0.807340i	$-0.299094\pi$
$89$	11.1337	1.18017	0.590086	+	0.807340i	$0.299094\pi$
$90$	0	0
$91$	3.36668	0.352924
$92$	1.13515	0.118347
$93$	0	0
$94$	4.99369	0.515060
$95$	0.430400	0.0441581
$96$	0	0
$97$	−0.471210	−0.0478441	−0.0239220	−	0.999714i	$-0.507615\pi$
$97$	−0.471210	−0.0478441	−0.0239220	+	0.999714i	$0.507615\pi$
$98$	1.48888	0.150400
$99$	0	0
$100$	0.753813	0.0753813
$101$	5.16598	0.514034	0.257017	−	0.966407i	$-0.417260\pi$
$101$	5.16598	0.514034	0.257017	+	0.966407i	$0.417260\pi$
$102$	0	0
$103$	−2.38399	−0.234901	−0.117451	−	0.993079i	$-0.537472\pi$
$103$	−2.38399	−0.234901	−0.117451	+	0.993079i	$0.537472\pi$
$104$	−8.93857	−0.876499
$105$	0	0
$106$	−7.30152	−0.709186
$107$	−16.9004	−1.63382	−0.816909	−	0.576766i	$-0.804314\pi$
$107$	−16.9004	−1.63382	−0.816909	+	0.576766i	$0.804314\pi$
$108$	0	0
$109$	11.0708	1.06039	0.530193	−	0.847877i	$-0.322119\pi$
$109$	11.0708	1.06039	0.530193	+	0.847877i	$0.322119\pi$
$110$	−17.9745	−1.71380
$111$	0	0
$112$	−4.38656	−0.414491
$113$	−14.9881	−1.40996	−0.704982	−	0.709225i	$-0.749045\pi$
$113$	−14.9881	−1.40996	−0.704982	+	0.709225i	$0.749045\pi$
$114$	0	0
$115$	15.2464	1.42173
$116$	0.266840	0.0247755
$117$	0	0
$118$	17.9206	1.64972
$119$	−4.79573	−0.439624
$120$	0	0
$121$	6.19220	0.562927
$122$	16.1680	1.46378
$123$	0	0
$124$	1.33197	0.119615
$125$	−4.43332	−0.396528
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	12.7975	1.13115
$129$	0	0
$130$	14.5946	1.28003
$131$	5.79631	0.506426	0.253213	−	0.967411i	$-0.418513\pi$
$131$	5.79631	0.506426	0.253213	+	0.967411i	$0.418513\pi$
$132$	0	0
$133$	0.147823	0.0128179
$134$	−2.73716	−0.236455
$135$	0	0
$136$	12.7327	1.09182
$137$	−22.1491	−1.89233	−0.946164	−	0.323688i	$-0.895077\pi$
$137$	−22.1491	−1.89233	−0.946164	+	0.323688i	$0.895077\pi$
$138$	0	0
$139$	16.1472	1.36959	0.684794	−	0.728737i	$-0.259892\pi$
$139$	16.1472	1.36959	0.684794	+	0.728737i	$0.259892\pi$
$140$	0.631168	0.0533434
$141$	0	0
$142$	14.4843	1.21549
$143$	−13.9594	−1.16735
$144$	0	0
$145$	3.58399	0.297634
$146$	10.9168	0.903483
$147$	0	0
$148$	1.69868	0.139631
$149$	−11.4146	−0.935123	−0.467562	−	0.883961i	$-0.654867\pi$
$149$	−11.4146	−0.935123	−0.467562	+	0.883961i	$0.654867\pi$
$150$	0	0
$151$	21.7197	1.76752	0.883761	−	0.467939i	$-0.155003\pi$
$151$	21.7197	1.76752	0.883761	+	0.467939i	$0.155003\pi$
$152$	−0.392472	−0.0318337
$153$	0	0
$154$	−6.17343	−0.497469
$155$	17.8900	1.43696
$156$	0	0
$157$	−13.6447	−1.08897	−0.544484	−	0.838771i	$-0.683275\pi$
$157$	−13.6447	−1.08897	−0.544484	+	0.838771i	$0.683275\pi$
$158$	10.3697	0.824969
$159$	0	0
$160$	−3.55523	−0.281066
$161$	5.23645	0.412690
$162$	0	0
$163$	−0.934160	−0.0731690	−0.0365845	−	0.999331i	$-0.511648\pi$
$163$	−0.934160	−0.0731690	−0.0365845	+	0.999331i	$0.511648\pi$
$164$	−0.0106435	−0.000831118 0
$165$	0	0
$166$	19.7553	1.53331
$167$	−14.1430	−1.09442	−0.547209	−	0.836996i	$-0.684310\pi$
$167$	−14.1430	−1.09442	−0.547209	+	0.836996i	$0.684310\pi$
$168$	0	0
$169$	−1.66547	−0.128113
$170$	−20.7896	−1.59449
$171$	0	0
$172$	−1.77375	−0.135247
$173$	4.53586	0.344855	0.172428	−	0.985022i	$-0.444839\pi$
$173$	4.53586	0.344855	0.172428	+	0.985022i	$0.444839\pi$
$174$	0	0
$175$	3.47735	0.262863
$176$	18.1882	1.37099
$177$	0	0
$178$	16.5768	1.24249
$179$	0.416214	0.0311093	0.0155546	−	0.999879i	$-0.495049\pi$
$179$	0.416214	0.0311093	0.0155546	+	0.999879i	$0.495049\pi$
$180$	0	0
$181$	24.2415	1.80186	0.900928	−	0.433969i	$-0.142887\pi$
$181$	24.2415	1.80186	0.900928	+	0.433969i	$0.142887\pi$
$182$	5.01260	0.371558
$183$	0	0
$184$	−13.9028	−1.02493
$185$	22.8154	1.67742
$186$	0	0
$187$	19.8848	1.45412
$188$	0.727069	0.0530270
$189$	0	0
$190$	0.640816	0.0464897
$191$	26.6373	1.92741	0.963704	−	0.266975i	$-0.0860240\pi$
$191$	26.6373	1.92741	0.963704	+	0.266975i	$0.0860240\pi$
$192$	0	0
$193$	−19.0498	−1.37123	−0.685616	−	0.727964i	$-0.740467\pi$
$193$	−19.0498	−1.37123	−0.685616	+	0.727964i	$0.740467\pi$
$194$	−0.701577	−0.0503703
$195$	0	0
$196$	0.216778	0.0154841
$197$	5.83599	0.415797	0.207898	−	0.978150i	$-0.433338\pi$
$197$	5.83599	0.415797	0.207898	+	0.978150i	$0.433338\pi$
$198$	0	0
$199$	22.8300	1.61837	0.809187	−	0.587551i	$-0.199908\pi$
$199$	22.8300	1.61837	0.809187	+	0.587551i	$0.199908\pi$
$200$	−9.23241	−0.652830
$201$	0	0
$202$	7.69154	0.541175
$203$	1.23094	0.0863950
$204$	0	0
$205$	−0.142955	−0.00998442
$206$	−3.54948	−0.247304
$207$	0	0
$208$	−14.7681	−1.02399
$209$	−0.612926	−0.0423970
$210$	0	0
$211$	−25.8008	−1.77620	−0.888099	−	0.459652i	$-0.847974\pi$
$211$	−25.8008	−1.77620	−0.888099	+	0.459652i	$0.847974\pi$
$212$	−1.06308	−0.0730128
$213$	0	0
$214$	−25.1627	−1.72009
$215$	−23.8237	−1.62476
$216$	0	0
$217$	6.14440	0.417109
$218$	16.4831	1.11638
$219$	0	0
$220$	−2.61704	−0.176441
$221$	−16.1457	−1.08608
$222$	0	0
$223$	−8.47086	−0.567251	−0.283625	−	0.958935i	$-0.591537\pi$
$223$	−8.47086	−0.567251	−0.283625	+	0.958935i	$0.591537\pi$
$224$	−1.22106	−0.0815857
$225$	0	0
$226$	−22.3156	−1.48441
$227$	8.01720	0.532120	0.266060	−	0.963956i	$-0.414278\pi$
$227$	8.01720	0.532120	0.266060	+	0.963956i	$0.414278\pi$
$228$	0	0
$229$	20.2786	1.34005	0.670023	−	0.742340i	$-0.266284\pi$
$229$	20.2786	1.34005	0.670023	+	0.742340i	$0.266284\pi$
$230$	22.7001	1.49680
$231$	0	0
$232$	−3.26816	−0.214565
$233$	6.73232	0.441049	0.220524	−	0.975381i	$-0.429223\pi$
$233$	6.73232	0.441049	0.220524	+	0.975381i	$0.429223\pi$
$234$	0	0
$235$	9.76542	0.637026
$236$	2.60919	0.169844
$237$	0	0
$238$	−7.14029	−0.462836
$239$	−13.7010	−0.886247	−0.443123	−	0.896461i	$-0.646130\pi$
$239$	−13.7010	−0.886247	−0.443123	+	0.896461i	$0.646130\pi$
$240$	0	0
$241$	6.48260	0.417581	0.208791	−	0.977960i	$-0.433047\pi$
$241$	6.48260	0.417581	0.208791	+	0.977960i	$0.433047\pi$
$242$	9.21947	0.592650
$243$	0	0
$244$	2.35402	0.150701
$245$	2.91159	0.186015
$246$	0	0
$247$	0.497673	0.0316662
$248$	−16.3135	−1.03591
$249$	0	0
$250$	−6.60071	−0.417465
$251$	−21.9241	−1.38384	−0.691920	−	0.721975i	$-0.743235\pi$
$251$	−21.9241	−1.38384	−0.691920	+	0.721975i	$0.743235\pi$
$252$	0	0
$253$	−21.7121	−1.36503
$254$	1.48888	0.0934209
$255$	0	0
$256$	5.14376	0.321485
$257$	−25.1781	−1.57056	−0.785282	−	0.619138i	$-0.787482\pi$
$257$	−25.1781	−1.57056	−0.785282	+	0.619138i	$0.787482\pi$
$258$	0	0
$259$	7.83605	0.486909
$260$	2.12494	0.131783
$261$	0	0
$262$	8.63004	0.533166
$263$	−2.57791	−0.158961	−0.0794804	−	0.996836i	$-0.525326\pi$
$263$	−2.57791	−0.158961	−0.0794804	+	0.996836i	$0.525326\pi$
$264$	0	0
$265$	−14.2785	−0.877121
$266$	0.220092	0.0134947
$267$	0	0
$268$	−0.398523	−0.0243437
$269$	−15.9217	−0.970760	−0.485380	−	0.874303i	$-0.661319\pi$
$269$	−15.9217	−0.970760	−0.485380	+	0.874303i	$0.661319\pi$
$270$	0	0
$271$	2.87200	0.174462	0.0872308	−	0.996188i	$-0.472198\pi$
$271$	2.87200	0.174462	0.0872308	+	0.996188i	$0.472198\pi$
$272$	21.0368	1.27554
$273$	0	0
$274$	−32.9775	−1.99224
$275$	−14.4183	−0.869457
$276$	0	0
$277$	−5.15941	−0.309999	−0.154999	−	0.987915i	$-0.549538\pi$
$277$	−5.15941	−0.309999	−0.154999	+	0.987915i	$0.549538\pi$
$278$	24.0413	1.44190
$279$	0	0
$280$	−7.73031	−0.461974
$281$	−8.81624	−0.525933	−0.262966	−	0.964805i	$-0.584701\pi$
$281$	−8.81624	−0.525933	−0.262966	+	0.964805i	$0.584701\pi$
$282$	0	0
$283$	−8.37512	−0.497849	−0.248925	−	0.968523i	$-0.580077\pi$
$283$	−8.37512	−0.497849	−0.248925	+	0.968523i	$0.580077\pi$
$284$	2.10888	0.125139
$285$	0	0
$286$	−20.7840	−1.22898
$287$	−0.0490986	−0.00289820
$288$	0	0
$289$	5.99902	0.352884
$290$	5.33615	0.313349
$291$	0	0
$292$	1.58946	0.0930163
$293$	14.4763	0.845717	0.422858	−	0.906196i	$-0.361027\pi$
$293$	14.4763	0.845717	0.422858	+	0.906196i	$0.361027\pi$
$294$	0	0
$295$	35.0446	2.04037
$296$	−20.8048	−1.20926
$297$	0	0
$298$	−16.9951	−0.984498
$299$	17.6294	1.01954
$300$	0	0
$301$	−8.18235	−0.471623
$302$	32.3381	1.86085
$303$	0	0
$304$	−0.648436	−0.0371903
$305$	31.6174	1.81041
$306$	0	0
$307$	3.23887	0.184852	0.0924260	−	0.995720i	$-0.470538\pi$
$307$	3.23887	0.184852	0.0924260	+	0.995720i	$0.470538\pi$
$308$	−0.898836	−0.0512160
$309$	0	0
$310$	26.6361	1.51283
$311$	16.0805	0.911840	0.455920	−	0.890021i	$-0.349310\pi$
$311$	16.0805	0.911840	0.455920	+	0.890021i	$0.349310\pi$
$312$	0	0
$313$	−21.3203	−1.20510	−0.602548	−	0.798083i	$-0.705848\pi$
$313$	−21.3203	−1.20510	−0.602548	+	0.798083i	$0.705848\pi$
$314$	−20.3154	−1.14647
$315$	0	0
$316$	1.50980	0.0849330
$317$	−4.98067	−0.279742	−0.139871	−	0.990170i	$-0.544669\pi$
$317$	−4.98067	−0.279742	−0.139871	+	0.990170i	$0.544669\pi$
$318$	0	0
$319$	−5.10390	−0.285764
$320$	20.2504	1.13203
$321$	0	0
$322$	7.79646	0.434480
$323$	−0.708920	−0.0394454
$324$	0	0
$325$	11.7071	0.649395
$326$	−1.39086	−0.0770324
$327$	0	0
$328$	0.130357	0.00719779
$329$	3.35398	0.184911
$330$	0	0
$331$	−4.19417	−0.230532	−0.115266	−	0.993335i	$-0.536772\pi$
$331$	−4.19417	−0.230532	−0.115266	+	0.993335i	$0.536772\pi$
$332$	2.87632	0.157859
$333$	0	0
$334$	−21.0573	−1.15220
$335$	−5.35265	−0.292447
$336$	0	0
$337$	10.8184	0.589314	0.294657	−	0.955603i	$-0.404795\pi$
$337$	10.8184	0.589314	0.294657	+	0.955603i	$0.404795\pi$
$338$	−2.47970	−0.134878
$339$	0	0
$340$	−3.02691	−0.164157
$341$	−25.4768	−1.37965
$342$	0	0
$343$	1.00000	0.0539949
$344$	21.7243	1.17129
$345$	0	0
$346$	6.75338	0.363064
$347$	−17.1092	−0.918471	−0.459236	−	0.888315i	$-0.651877\pi$
$347$	−17.1092	−0.918471	−0.459236	+	0.888315i	$0.651877\pi$
$348$	0	0
$349$	9.72106	0.520356	0.260178	−	0.965561i	$-0.416219\pi$
$349$	9.72106	0.520356	0.260178	+	0.965561i	$0.416219\pi$
$350$	5.17738	0.276742
$351$	0	0
$352$	5.06295	0.269856
$353$	−12.6247	−0.671946	−0.335973	−	0.941872i	$-0.609065\pi$
$353$	−12.6247	−0.671946	−0.335973	+	0.941872i	$0.609065\pi$
$354$	0	0
$355$	28.3247	1.50332
$356$	2.41354	0.127918
$357$	0	0
$358$	0.619694	0.0327519
$359$	9.48768	0.500741	0.250370	−	0.968150i	$-0.419448\pi$
$359$	9.48768	0.500741	0.250370	+	0.968150i	$0.419448\pi$
$360$	0	0
$361$	−18.9781	−0.998850
$362$	36.0928	1.89699
$363$	0	0
$364$	0.729822	0.0382530
$365$	21.3484	1.11743
$366$	0	0
$367$	9.78894	0.510978	0.255489	−	0.966812i	$-0.417763\pi$
$367$	9.78894	0.510978	0.255489	+	0.966812i	$0.417763\pi$
$368$	−22.9700	−1.19739
$369$	0	0
$370$	33.9695	1.76599
$371$	−4.90402	−0.254604
$372$	0	0
$373$	−29.8399	−1.54505	−0.772525	−	0.634984i	$-0.781006\pi$
$373$	−29.8399	−1.54505	−0.772525	+	0.634984i	$0.781006\pi$
$374$	29.6061	1.53090
$375$	0	0
$376$	−8.90486	−0.459233
$377$	4.14418	0.213436
$378$	0	0
$379$	−19.1294	−0.982610	−0.491305	−	0.870988i	$-0.663480\pi$
$379$	−19.1294	−0.982610	−0.491305	+	0.870988i	$0.663480\pi$
$380$	0.0933013	0.00478625
$381$	0	0
$382$	39.6599	2.02918
$383$	−9.49745	−0.485297	−0.242648	−	0.970114i	$-0.578016\pi$
$383$	−9.49745	−0.485297	−0.242648	+	0.970114i	$0.578016\pi$
$384$	0	0
$385$	−12.0725	−0.615270
$386$	−28.3629	−1.44363
$387$	0	0
$388$	−0.102148	−0.00518577
$389$	−27.4189	−1.39020	−0.695098	−	0.718915i	$-0.744639\pi$
$389$	−27.4189	−1.39020	−0.695098	+	0.718915i	$0.744639\pi$
$390$	0	0
$391$	−25.1126	−1.27000
$392$	−2.65501	−0.134098
$393$	0	0
$394$	8.68911	0.437751
$395$	20.2785	1.02032
$396$	0	0
$397$	13.5805	0.681585	0.340792	−	0.940139i	$-0.389305\pi$
$397$	13.5805	0.681585	0.340792	+	0.940139i	$0.389305\pi$
$398$	33.9912	1.70383
$399$	0	0
$400$	−15.2536	−0.762681
$401$	−12.6146	−0.629941	−0.314970	−	0.949101i	$-0.601995\pi$
$401$	−12.6146	−0.629941	−0.314970	+	0.949101i	$0.601995\pi$
$402$	0	0
$403$	20.6862	1.03045
$404$	1.11987	0.0557156
$405$	0	0
$406$	1.83273	0.0909567
$407$	−32.4910	−1.61052
$408$	0	0
$409$	−0.129840	−0.00642020	−0.00321010	−	0.999995i	$-0.501022\pi$
$409$	−0.129840	−0.00642020	−0.00321010	+	0.999995i	$0.501022\pi$
$410$	−0.212844	−0.0105116
$411$	0	0
$412$	−0.516795	−0.0254607
$413$	12.0362	0.592264
$414$	0	0
$415$	38.6324	1.89639
$416$	−4.11093	−0.201555
$417$	0	0
$418$	−0.912576	−0.0446356
$419$	−14.5425	−0.710449	−0.355224	−	0.934781i	$-0.615596\pi$
$419$	−14.5425	−0.710449	−0.355224	+	0.934781i	$0.615596\pi$
$420$	0	0
$421$	12.9591	0.631586	0.315793	−	0.948828i	$-0.397730\pi$
$421$	12.9591	0.631586	0.315793	+	0.948828i	$0.397730\pi$
$422$	−38.4144	−1.86998
$423$	0	0
$424$	13.0202	0.632318
$425$	−16.6764	−0.808926
$426$	0	0
$427$	10.8592	0.525511
$428$	−3.66362	−0.177088
$429$	0	0
$430$	−35.4707	−1.71055
$431$	26.1264	1.25846	0.629232	−	0.777218i	$-0.283370\pi$
$431$	26.1264	1.25846	0.629232	+	0.777218i	$0.283370\pi$
$432$	0	0
$433$	27.6129	1.32699	0.663496	−	0.748180i	$-0.269072\pi$
$433$	27.6129	1.32699	0.663496	+	0.748180i	$0.269072\pi$
$434$	9.14831	0.439133
$435$	0	0
$436$	2.39990	0.114934
$437$	0.774068	0.0370287
$438$	0	0
$439$	4.55135	0.217224	0.108612	−	0.994084i	$-0.465359\pi$
$439$	4.55135	0.217224	0.108612	+	0.994084i	$0.465359\pi$
$440$	32.0525	1.52804
$441$	0	0
$442$	−24.0391	−1.14342
$443$	−2.30191	−0.109367	−0.0546836	−	0.998504i	$-0.517415\pi$
$443$	−2.30191	−0.109367	−0.0546836	+	0.998504i	$0.517415\pi$
$444$	0	0
$445$	32.4168	1.53670
$446$	−12.6121	−0.597202
$447$	0	0
$448$	6.95510	0.328598
$449$	25.9591	1.22509	0.612544	−	0.790437i	$-0.290146\pi$
$449$	25.9591	1.22509	0.612544	+	0.790437i	$0.290146\pi$
$450$	0	0
$451$	0.203580	0.00958621
$452$	−3.24910	−0.152825
$453$	0	0
$454$	11.9367	0.560216
$455$	9.80239	0.459543
$456$	0	0
$457$	−11.9556	−0.559258	−0.279629	−	0.960108i	$-0.590211\pi$
$457$	−11.9556	−0.559258	−0.279629	+	0.960108i	$0.590211\pi$
$458$	30.1925	1.41080
$459$	0	0
$460$	3.30508	0.154100
$461$	7.55639	0.351936	0.175968	−	0.984396i	$-0.443694\pi$
$461$	7.55639	0.351936	0.175968	+	0.984396i	$0.443694\pi$
$462$	0	0
$463$	−1.92100	−0.0892762	−0.0446381	−	0.999003i	$-0.514213\pi$
$463$	−1.92100	−0.0892762	−0.0446381	+	0.999003i	$0.514213\pi$
$464$	−5.39959	−0.250670
$465$	0	0
$466$	10.0236	0.464336
$467$	1.55517	0.0719648	0.0359824	−	0.999352i	$-0.488544\pi$
$467$	1.55517	0.0719648	0.0359824	+	0.999352i	$0.488544\pi$
$468$	0	0
$469$	−1.83840	−0.0848892
$470$	14.5396	0.670661
$471$	0	0
$472$	−31.9564	−1.47091
$473$	33.9269	1.55996
$474$	0	0
$475$	0.514033	0.0235855
$476$	−1.03961	−0.0476504
$477$	0	0
$478$	−20.3993	−0.933041
$479$	−3.79672	−0.173477	−0.0867384	−	0.996231i	$-0.527644\pi$
$479$	−3.79672	−0.173477	−0.0867384	+	0.996231i	$0.527644\pi$
$480$	0	0
$481$	26.3815	1.20289
$482$	9.65185	0.439630
$483$	0	0
$484$	1.34233	0.0610151
$485$	−1.37197	−0.0622979
$486$	0	0
$487$	42.8719	1.94271	0.971357	−	0.237626i	$-0.0763694\pi$
$487$	42.8719	1.94271	0.971357	+	0.237626i	$0.0763694\pi$
$488$	−28.8312	−1.30513
$489$	0	0
$490$	4.33502	0.195836
$491$	22.3751	1.00977	0.504887	−	0.863185i	$-0.331534\pi$
$491$	22.3751	1.00977	0.504887	+	0.863185i	$0.331534\pi$
$492$	0	0
$493$	−5.90325	−0.265869
$494$	0.740978	0.0333382
$495$	0	0
$496$	−26.9528	−1.21022
$497$	9.72828	0.436373
$498$	0	0
$499$	−5.93137	−0.265525	−0.132762	−	0.991148i	$-0.542385\pi$
$499$	−5.93137	−0.265525	−0.132762	+	0.991148i	$0.542385\pi$
$500$	−0.961046	−0.0429793
$501$	0	0
$502$	−32.6425	−1.45691
$503$	−7.56282	−0.337210	−0.168605	−	0.985684i	$-0.553926\pi$
$503$	−7.56282	−0.337210	−0.168605	+	0.985684i	$0.553926\pi$
$504$	0	0
$505$	15.0412	0.669325
$506$	−32.3268	−1.43710
$507$	0	0
$508$	0.216778	0.00961796
$509$	3.13987	0.139172	0.0695861	−	0.997576i	$-0.477832\pi$
$509$	3.13987	0.139172	0.0695861	+	0.997576i	$0.477832\pi$
$510$	0	0
$511$	7.33222	0.324358
$512$	−17.9365	−0.792688
$513$	0	0
$514$	−37.4872	−1.65349
$515$	−6.94119	−0.305865
$516$	0	0
$517$	−13.9068	−0.611619
$518$	11.6670	0.512618
$519$	0	0
$520$	−26.0255	−1.14129
$521$	−17.8111	−0.780321	−0.390160	−	0.920747i	$-0.627580\pi$
$521$	−17.8111	−0.780321	−0.390160	+	0.920747i	$0.627580\pi$
$522$	0	0
$523$	33.8549	1.48037	0.740185	−	0.672403i	$-0.234738\pi$
$523$	33.8549	1.48037	0.740185	+	0.672403i	$0.234738\pi$
$524$	1.25651	0.0548910
$525$	0	0
$526$	−3.83821	−0.167354
$527$	−29.4669	−1.28360
$528$	0	0
$529$	4.42036	0.192189
$530$	−21.2590	−0.923433
$531$	0	0
$532$	0.0320448	0.00138932
$533$	−0.165299	−0.00715991
$534$	0	0
$535$	−49.2069	−2.12740
$536$	4.88096	0.210825
$537$	0	0
$538$	−23.7055	−1.02202
$539$	−4.14635	−0.178596
$540$	0	0
$541$	3.35902	0.144415	0.0722077	−	0.997390i	$-0.476996\pi$
$541$	3.35902	0.144415	0.0722077	+	0.997390i	$0.476996\pi$
$542$	4.27608	0.183673
$543$	0	0
$544$	5.85589	0.251069
$545$	32.2335	1.38073
$546$	0	0
$547$	−10.8941	−0.465798	−0.232899	−	0.972501i	$-0.574821\pi$
$547$	−10.8941	−0.465798	−0.232899	+	0.972501i	$0.574821\pi$
$548$	−4.80144	−0.205107
$549$	0	0
$550$	−21.4672	−0.915365
$551$	0.181961	0.00775181
$552$	0	0
$553$	6.96474	0.296171
$554$	−7.68177	−0.326367
$555$	0	0
$556$	3.50035	0.148448
$557$	24.0627	1.01957	0.509784	−	0.860302i	$-0.329725\pi$
$557$	24.0627	1.01957	0.509784	+	0.860302i	$0.329725\pi$
$558$	0	0
$559$	−27.5474	−1.16513
$560$	−12.7719	−0.539710
$561$	0	0
$562$	−13.1264	−0.553702
$563$	27.1050	1.14234	0.571169	−	0.820832i	$-0.306490\pi$
$563$	27.1050	1.14234	0.571169	+	0.820832i	$0.306490\pi$
$564$	0	0
$565$	−43.6393	−1.83592
$566$	−12.4696	−0.524136
$567$	0	0
$568$	−25.8287	−1.08375
$569$	42.7501	1.79218	0.896089	−	0.443875i	$-0.146397\pi$
$569$	42.7501	1.79218	0.896089	+	0.443875i	$0.146397\pi$
$570$	0	0
$571$	18.7978	0.786665	0.393333	−	0.919396i	$-0.371322\pi$
$571$	18.7978	0.786665	0.393333	+	0.919396i	$0.371322\pi$
$572$	−3.02609	−0.126527
$573$	0	0
$574$	−0.0731022	−0.00305123
$575$	18.2090	0.759366
$576$	0	0
$577$	−1.84444	−0.0767849	−0.0383925	−	0.999263i	$-0.512224\pi$
$577$	−1.84444	−0.0767849	−0.0383925	+	0.999263i	$0.512224\pi$
$578$	8.93185	0.371516
$579$	0	0
$580$	0.776930	0.0322603
$581$	13.2685	0.550470
$582$	0	0
$583$	20.3338	0.842139
$584$	−19.4671	−0.805556
$585$	0	0
$586$	21.5536	0.890371
$587$	−0.0278962	−0.00115140	−0.000575700	−	1.00000i	$-0.500183\pi$
$587$	−0.0278962	−0.00115140	−0.000575700		1.00000i	$0.500183\pi$
$588$	0	0
$589$	0.908285	0.0374252
$590$	52.1773	2.14811
$591$	0	0
$592$	−34.3733	−1.41274
$593$	4.26821	0.175274	0.0876371	−	0.996152i	$-0.472068\pi$
$593$	4.26821	0.175274	0.0876371	+	0.996152i	$0.472068\pi$
$594$	0	0
$595$	−13.9632	−0.572435
$596$	−2.47444	−0.101357
$597$	0	0
$598$	26.2482	1.07337
$599$	28.3223	1.15722	0.578610	−	0.815605i	$-0.303596\pi$
$599$	28.3223	1.15722	0.578610	+	0.815605i	$0.303596\pi$
$600$	0	0
$601$	−5.52773	−0.225481	−0.112740	−	0.993624i	$-0.535963\pi$
$601$	−5.52773	−0.225481	−0.112740	+	0.993624i	$0.535963\pi$
$602$	−12.1826	−0.496525
$603$	0	0
$604$	4.70834	0.191580
$605$	18.0291	0.732989
$606$	0	0
$607$	−0.370522	−0.0150390	−0.00751951	−	0.999972i	$-0.502394\pi$
$607$	−0.370522	−0.0150390	−0.00751951	+	0.999972i	$0.502394\pi$
$608$	−0.180501	−0.00732029
$609$	0	0
$610$	47.0747	1.90600
$611$	11.2918	0.456817
$612$	0	0
$613$	15.5785	0.629211	0.314605	−	0.949223i	$-0.398128\pi$
$613$	15.5785	0.629211	0.314605	+	0.949223i	$0.398128\pi$
$614$	4.82230	0.194612
$615$	0	0
$616$	11.0086	0.443549
$617$	3.80748	0.153283	0.0766417	−	0.997059i	$-0.475580\pi$
$617$	3.80748	0.153283	0.0766417	+	0.997059i	$0.475580\pi$
$618$	0	0
$619$	−2.32509	−0.0934531	−0.0467265	−	0.998908i	$-0.514879\pi$
$619$	−2.32509	−0.0934531	−0.0467265	+	0.998908i	$0.514879\pi$
$620$	3.87815	0.155750
$621$	0	0
$622$	23.9420	0.959986
$623$	11.1337	0.446063
$624$	0	0
$625$	−30.2948	−1.21179
$626$	−31.7435	−1.26872
$627$	0	0
$628$	−2.95788	−0.118032
$629$	−37.5796	−1.49840
$630$	0	0
$631$	−36.8091	−1.46535	−0.732673	−	0.680581i	$-0.761728\pi$
$631$	−36.8091	−1.46535	−0.732673	+	0.680581i	$0.761728\pi$
$632$	−18.4915	−0.735551
$633$	0	0
$634$	−7.41564	−0.294513
$635$	2.91159	0.115543
$636$	0	0
$637$	3.36668	0.133393
$638$	−7.59912	−0.300852
$639$	0	0
$640$	37.2610	1.47287
$641$	44.7785	1.76864	0.884322	−	0.466878i	$-0.154621\pi$
$641$	44.7785	1.76864	0.884322	+	0.466878i	$0.154621\pi$
$642$	0	0
$643$	−13.7232	−0.541191	−0.270595	−	0.962693i	$-0.587221\pi$
$643$	−13.7232	−0.541191	−0.270595	+	0.962693i	$0.587221\pi$
$644$	1.13515	0.0447310
$645$	0	0
$646$	−1.05550	−0.0415281
$647$	−38.4468	−1.51150	−0.755750	−	0.654861i	$-0.772727\pi$
$647$	−38.4468	−1.51150	−0.755750	+	0.654861i	$0.772727\pi$
$648$	0	0
$649$	−49.9064	−1.95900
$650$	17.4306	0.683683
$651$	0	0
$652$	−0.202505	−0.00793072
$653$	8.68324	0.339801	0.169901	−	0.985461i	$-0.445655\pi$
$653$	8.68324	0.339801	0.169901	+	0.985461i	$0.445655\pi$
$654$	0	0
$655$	16.8765	0.659419
$656$	0.215374	0.00840895
$657$	0	0
$658$	4.99369	0.194674
$659$	19.4390	0.757237	0.378619	−	0.925553i	$-0.376399\pi$
$659$	19.4390	0.757237	0.378619	+	0.925553i	$0.376399\pi$
$660$	0	0
$661$	15.8557	0.616715	0.308358	−	0.951270i	$-0.400221\pi$
$661$	15.8557	0.616715	0.308358	+	0.951270i	$0.400221\pi$
$662$	−6.24464	−0.242705
$663$	0	0
$664$	−35.2281	−1.36711
$665$	0.430400	0.0166902
$666$	0	0
$667$	6.44575	0.249580
$668$	−3.06589	−0.118623
$669$	0	0
$670$	−7.96948	−0.307888
$671$	−45.0258	−1.73820
$672$	0	0
$673$	−46.4164	−1.78922	−0.894611	−	0.446846i	$-0.852547\pi$
$673$	−46.4164	−1.78922	−0.894611	+	0.446846i	$0.852547\pi$
$674$	16.1073	0.620430
$675$	0	0
$676$	−0.361038	−0.0138861
$677$	−29.2402	−1.12379	−0.561896	−	0.827208i	$-0.689928\pi$
$677$	−29.2402	−1.12379	−0.561896	+	0.827208i	$0.689928\pi$
$678$	0	0
$679$	−0.471210	−0.0180834
$680$	37.0725	1.42166
$681$	0	0
$682$	−37.9321	−1.45249
$683$	−47.5983	−1.82130	−0.910649	−	0.413182i	$-0.864417\pi$
$683$	−47.5983	−1.82130	−0.910649	+	0.413182i	$0.864417\pi$
$684$	0	0
$685$	−64.4892	−2.46400
$686$	1.48888	0.0568459
$687$	0	0
$688$	35.8924	1.36839
$689$	−16.5103	−0.628991
$690$	0	0
$691$	42.7666	1.62692	0.813459	−	0.581622i	$-0.197582\pi$
$691$	42.7666	1.62692	0.813459	+	0.581622i	$0.197582\pi$
$692$	0.983274	0.0373785
$693$	0	0
$694$	−25.4737	−0.966967
$695$	47.0140	1.78334
$696$	0	0
$697$	0.235464	0.00891883
$698$	14.4735	0.547831
$699$	0	0
$700$	0.753813	0.0284915
$701$	−35.1099	−1.32608	−0.663042	−	0.748582i	$-0.730735\pi$
$701$	−35.1099	−1.32608	−0.663042	+	0.748582i	$0.730735\pi$
$702$	0	0
$703$	1.15835	0.0436880
$704$	−28.8383	−1.08688
$705$	0	0
$706$	−18.7967	−0.707425
$707$	5.16598	0.194287
$708$	0	0
$709$	12.5880	0.472754	0.236377	−	0.971661i	$-0.424040\pi$
$709$	12.5880	0.472754	0.236377	+	0.971661i	$0.424040\pi$
$710$	42.1723	1.58270
$711$	0	0
$712$	−29.5602	−1.10781
$713$	32.1748	1.20496
$714$	0	0
$715$	−40.6441	−1.52000
$716$	0.0902259	0.00337190
$717$	0	0
$718$	14.1261	0.527180
$719$	0.862145	0.0321526	0.0160763	−	0.999871i	$-0.494883\pi$
$719$	0.862145	0.0321526	0.0160763	+	0.999871i	$0.494883\pi$
$720$	0	0
$721$	−2.38399	−0.0887843
$722$	−28.2563	−1.05159
$723$	0	0
$724$	5.25502	0.195301
$725$	4.28041	0.158970
$726$	0	0
$727$	33.1990	1.23128	0.615641	−	0.788027i	$-0.288897\pi$
$727$	33.1990	1.23128	0.615641	+	0.788027i	$0.288897\pi$
$728$	−8.93857	−0.331286
$729$	0	0
$730$	31.7853	1.17643
$731$	39.2404	1.45136
$732$	0	0
$733$	−40.4452	−1.49388	−0.746939	−	0.664893i	$-0.768477\pi$
$733$	−40.4452	−1.49388	−0.746939	+	0.664893i	$0.768477\pi$
$734$	14.5746	0.537958
$735$	0	0
$736$	−6.39403	−0.235687
$737$	7.62263	0.280783
$738$	0	0
$739$	5.49746	0.202227	0.101114	−	0.994875i	$-0.467759\pi$
$739$	5.49746	0.202227	0.101114	+	0.994875i	$0.467759\pi$
$740$	4.94587	0.181814
$741$	0	0
$742$	−7.30152	−0.268047
$743$	−40.3053	−1.47866	−0.739329	−	0.673345i	$-0.764857\pi$
$743$	−40.3053	−1.47866	−0.739329	+	0.673345i	$0.764857\pi$
$744$	0	0
$745$	−33.2347	−1.21763
$746$	−44.4281	−1.62663
$747$	0	0
$748$	4.31058	0.157610
$749$	−16.9004	−0.617525
$750$	0	0
$751$	5.92678	0.216271	0.108136	−	0.994136i	$-0.465512\pi$
$751$	5.92678	0.216271	0.108136	+	0.994136i	$0.465512\pi$
$752$	−14.7125	−0.536508
$753$	0	0
$754$	6.17020	0.224706
$755$	63.2388	2.30149
$756$	0	0
$757$	12.4772	0.453493	0.226747	−	0.973954i	$-0.427191\pi$
$757$	12.4772	0.453493	0.226747	+	0.973954i	$0.427191\pi$
$758$	−28.4814	−1.03449
$759$	0	0
$760$	−1.14272	−0.0414507
$761$	31.7213	1.14989	0.574947	−	0.818190i	$-0.305022\pi$
$761$	31.7213	1.14989	0.574947	+	0.818190i	$0.305022\pi$
$762$	0	0
$763$	11.0708	0.400789
$764$	5.77438	0.208910
$765$	0	0
$766$	−14.1406	−0.510921
$767$	40.5221	1.46317
$768$	0	0
$769$	22.6399	0.816415	0.408207	−	0.912889i	$-0.366154\pi$
$769$	22.6399	0.816415	0.408207	+	0.912889i	$0.366154\pi$
$770$	−17.9745	−0.647756
$771$	0	0
$772$	−4.12957	−0.148626
$773$	35.4318	1.27439	0.637197	−	0.770701i	$-0.280094\pi$
$773$	35.4318	1.27439	0.637197	+	0.770701i	$0.280094\pi$
$774$	0	0
$775$	21.3663	0.767499
$776$	1.25107	0.0449107
$777$	0	0
$778$	−40.8236	−1.46360
$779$	−0.00725792	−0.000260042 0
$780$	0	0
$781$	−40.3368	−1.44336
$782$	−37.3897	−1.33705
$783$	0	0
$784$	−4.38656	−0.156663
$785$	−39.7279	−1.41795
$786$	0	0
$787$	−29.1953	−1.04070	−0.520349	−	0.853954i	$-0.674198\pi$
$787$	−29.1953	−1.04070	−0.520349	+	0.853954i	$0.674198\pi$
$788$	1.26511	0.0450678
$789$	0	0
$790$	30.1923	1.07419
$791$	−14.9881	−0.532917
$792$	0	0
$793$	36.5593	1.29826
$794$	20.2198	0.717573
$795$	0	0
$796$	4.94904	0.175414
$797$	−23.7870	−0.842579	−0.421290	−	0.906926i	$-0.638422\pi$
$797$	−23.7870	−0.842579	−0.421290	+	0.906926i	$0.638422\pi$
$798$	0	0
$799$	−16.0848	−0.569039
$800$	−4.24607	−0.150121
$801$	0	0
$802$	−18.7816	−0.663202
$803$	−30.4019	−1.07286
$804$	0	0
$805$	15.2464	0.537364
$806$	30.7994	1.08486
$807$	0	0
$808$	−13.7157	−0.482518
$809$	41.1655	1.44730	0.723652	−	0.690165i	$-0.242462\pi$
$809$	41.1655	1.44730	0.723652	+	0.690165i	$0.242462\pi$
$810$	0	0
$811$	−16.1626	−0.567544	−0.283772	−	0.958892i	$-0.591586\pi$
$811$	−16.1626	−0.567544	−0.283772	+	0.958892i	$0.591586\pi$
$812$	0.266840	0.00936427
$813$	0	0
$814$	−48.3754	−1.69556
$815$	−2.71989	−0.0952736
$816$	0	0
$817$	−1.20954	−0.0423165
$818$	−0.193318	−0.00675919
$819$	0	0
$820$	−0.0309895	−0.00108220
$821$	−18.2548	−0.637097	−0.318548	−	0.947907i	$-0.603195\pi$
$821$	−18.2548	−0.637097	−0.318548	+	0.947907i	$0.603195\pi$
$822$	0	0
$823$	42.2479	1.47267	0.736334	−	0.676618i	$-0.236555\pi$
$823$	42.2479	1.47267	0.736334	+	0.676618i	$0.236555\pi$
$824$	6.32951	0.220499
$825$	0	0
$826$	17.9206	0.623536
$827$	46.8151	1.62792	0.813961	−	0.580920i	$-0.197307\pi$
$827$	46.8151	1.62792	0.813961	+	0.580920i	$0.197307\pi$
$828$	0	0
$829$	−47.1289	−1.63685	−0.818426	−	0.574611i	$-0.805153\pi$
$829$	−47.1289	−1.63685	−0.818426	+	0.574611i	$0.805153\pi$
$830$	57.5193	1.99652
$831$	0	0
$832$	23.4156	0.811790
$833$	−4.79573	−0.166162
$834$	0	0
$835$	−41.1786	−1.42504
$836$	−0.132869	−0.00459536
$837$	0	0
$838$	−21.6521	−0.747961
$839$	−42.7882	−1.47721	−0.738606	−	0.674137i	$-0.764516\pi$
$839$	−42.7882	−1.47721	−0.738606	+	0.674137i	$0.764516\pi$
$840$	0	0
$841$	−27.4848	−0.947751
$842$	19.2945	0.664934
$843$	0	0
$844$	−5.59304	−0.192520
$845$	−4.84917	−0.166817
$846$	0	0
$847$	6.19220	0.212766
$848$	21.5118	0.738718
$849$	0	0
$850$	−24.8293	−0.851638
$851$	41.0331	1.40660
$852$	0	0
$853$	−26.0928	−0.893402	−0.446701	−	0.894683i	$-0.647401\pi$
$853$	−26.0928	−0.893402	−0.446701	+	0.894683i	$0.647401\pi$
$854$	16.1680	0.553259
$855$	0	0
$856$	44.8706	1.53365
$857$	34.3896	1.17473	0.587363	−	0.809324i	$-0.300166\pi$
$857$	34.3896	1.17473	0.587363	+	0.809324i	$0.300166\pi$
$858$	0	0
$859$	1.52930	0.0521790	0.0260895	−	0.999660i	$-0.491695\pi$
$859$	1.52930	0.0521790	0.0260895	+	0.999660i	$0.491695\pi$
$860$	−5.16444	−0.176106
$861$	0	0
$862$	38.8992	1.32491
$863$	37.4603	1.27516	0.637582	−	0.770383i	$-0.279935\pi$
$863$	37.4603	1.27516	0.637582	+	0.770383i	$0.279935\pi$
$864$	0	0
$865$	13.2066	0.449037
$866$	41.1125	1.39706
$867$	0	0
$868$	1.33197	0.0452100
$869$	−28.8782	−0.979627
$870$	0	0
$871$	−6.18929	−0.209716
$872$	−29.3930	−0.995373
$873$	0	0
$874$	1.15250	0.0389838
$875$	−4.43332	−0.149874
$876$	0	0
$877$	−16.6275	−0.561470	−0.280735	−	0.959785i	$-0.590578\pi$
$877$	−16.6275	−0.561470	−0.280735	+	0.959785i	$0.590578\pi$
$878$	6.77643	0.228694
$879$	0	0
$880$	52.9566	1.78517
$881$	−27.3896	−0.922780	−0.461390	−	0.887197i	$-0.652649\pi$
$881$	−27.3896	−0.922780	−0.461390	+	0.887197i	$0.652649\pi$
$882$	0	0
$883$	−10.6137	−0.357179	−0.178589	−	0.983924i	$-0.557153\pi$
$883$	−10.6137	−0.357179	−0.178589	+	0.983924i	$0.557153\pi$
$884$	−3.50003	−0.117719
$885$	0	0
$886$	−3.42728	−0.115142
$887$	−21.4033	−0.718652	−0.359326	−	0.933212i	$-0.616993\pi$
$887$	−21.4033	−0.718652	−0.359326	+	0.933212i	$0.616993\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	48.2649	1.61784
$891$	0	0
$892$	−1.83629	−0.0614837
$893$	0.495796	0.0165912
$894$	0	0
$895$	1.21184	0.0405075
$896$	12.7975	0.427534
$897$	0	0
$898$	38.6502	1.28977
$899$	7.56339	0.252253
$900$	0	0
$901$	23.5183	0.783510
$902$	0.303107	0.0100924
$903$	0	0
$904$	39.7937	1.32352
$905$	70.5812	2.34620
$906$	0	0
$907$	49.9246	1.65772	0.828859	−	0.559458i	$-0.188991\pi$
$907$	49.9246	1.65772	0.828859	+	0.559458i	$0.188991\pi$
$908$	1.73795	0.0576759
$909$	0	0
$910$	14.5946	0.483807
$911$	32.6343	1.08122	0.540611	−	0.841273i	$-0.318193\pi$
$911$	32.6343	1.08122	0.540611	+	0.841273i	$0.318193\pi$
$912$	0	0
$913$	−55.0158	−1.82076
$914$	−17.8005	−0.588787
$915$	0	0
$916$	4.39595	0.145246
$917$	5.79631	0.191411
$918$	0	0
$919$	−38.9648	−1.28533	−0.642666	−	0.766147i	$-0.722172\pi$
$919$	−38.9648	−1.28533	−0.642666	+	0.766147i	$0.722172\pi$
$920$	−40.4793	−1.33456
$921$	0	0
$922$	11.2506	0.370519
$923$	32.7520	1.07804
$924$	0	0
$925$	27.2487	0.895933
$926$	−2.86014	−0.0939901
$927$	0	0
$928$	−1.50305	−0.0493402
$929$	−33.5112	−1.09947	−0.549734	−	0.835340i	$-0.685271\pi$
$929$	−33.5112	−1.09947	−0.549734	+	0.835340i	$0.685271\pi$
$930$	0	0
$931$	0.147823	0.00484471
$932$	1.45942	0.0478048
$933$	0	0
$934$	2.31547	0.0757646
$935$	57.8963	1.89341
$936$	0	0
$937$	33.7304	1.10193	0.550963	−	0.834530i	$-0.314261\pi$
$937$	33.7304	1.10193	0.550963	+	0.834530i	$0.314261\pi$
$938$	−2.73716	−0.0893714
$939$	0	0
$940$	2.11693	0.0690465
$941$	−25.1685	−0.820469	−0.410235	−	0.911980i	$-0.634553\pi$
$941$	−25.1685	−0.820469	−0.410235	+	0.911980i	$0.634553\pi$
$942$	0	0
$943$	−0.257102	−0.00837240
$944$	−52.7977	−1.71842
$945$	0	0
$946$	50.5132	1.64233
$947$	3.93697	0.127934	0.0639672	−	0.997952i	$-0.479625\pi$
$947$	3.93697	0.127934	0.0639672	+	0.997952i	$0.479625\pi$
$948$	0	0
$949$	24.6852	0.801316
$950$	0.765336	0.0248308
$951$	0	0
$952$	12.7327	0.412670
$953$	−53.3391	−1.72782	−0.863911	−	0.503645i	$-0.831992\pi$
$953$	−53.3391	−1.72782	−0.863911	+	0.503645i	$0.831992\pi$
$954$	0	0
$955$	77.5569	2.50968
$956$	−2.97008	−0.0960593
$957$	0	0
$958$	−5.65288	−0.182636
$959$	−22.1491	−0.715233
$960$	0	0
$961$	6.75371	0.217862
$962$	39.2790	1.26641
$963$	0	0
$964$	1.40529	0.0452612
$965$	−55.4651	−1.78548
$966$	0	0
$967$	−34.1465	−1.09808	−0.549038	−	0.835797i	$-0.685006\pi$
$967$	−34.1465	−1.09808	−0.549038	+	0.835797i	$0.685006\pi$
$968$	−16.4404	−0.528413
$969$	0	0
$970$	−2.04270	−0.0655873
$971$	30.5079	0.979045	0.489522	−	0.871991i	$-0.337171\pi$
$971$	30.5079	0.979045	0.489522	+	0.871991i	$0.337171\pi$
$972$	0	0
$973$	16.1472	0.517655
$974$	63.8314	2.04529
$975$	0	0
$976$	−47.6344	−1.52474
$977$	16.3516	0.523135	0.261568	−	0.965185i	$-0.415761\pi$
$977$	16.3516	0.523135	0.261568	+	0.965185i	$0.415761\pi$
$978$	0	0
$979$	−46.1643	−1.47542
$980$	0.631168	0.0201619
$981$	0	0
$982$	33.3140	1.06309
$983$	−41.7219	−1.33072	−0.665360	−	0.746522i	$-0.731722\pi$
$983$	−41.7219	−1.33072	−0.665360	+	0.746522i	$0.731722\pi$
$984$	0	0
$985$	16.9920	0.541410
$986$	−8.78926	−0.279907
$987$	0	0
$988$	0.107885	0.00343226
$989$	−42.8464	−1.36244
$990$	0	0
$991$	−46.3370	−1.47194	−0.735972	−	0.677012i	$-0.763275\pi$
$991$	−46.3370	−1.47194	−0.735972	+	0.677012i	$0.763275\pi$
$992$	−7.50270	−0.238211
$993$	0	0
$994$	14.4843	0.459414
$995$	66.4715	2.10729
$996$	0	0
$997$	45.1748	1.43070	0.715350	−	0.698766i	$-0.246267\pi$
$997$	45.1748	1.43070	0.715350	+	0.698766i	$0.246267\pi$
$998$	−8.83113	−0.279545
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.v.1.15		19
3.2	odd	2		2667.2.a.q.1.5	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.q.1.5	✓	19	3.2	odd	2
8001.2.a.v.1.15		19	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:	\( N \)	\(=\)	\( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8001.a (trivial)

\(f(q)\)	\(=\)	\(q+1.48888 q^{2} +0.216778 q^{4} +2.91159 q^{5} +1.00000 q^{7} -2.65501 q^{8} +O(q^{10})\)	\(q+1.48888 q^{2} +0.216778 q^{4} +2.91159 q^{5} +1.00000 q^{7} -2.65501 q^{8} +4.33502 q^{10} -4.14635 q^{11} +3.36668 q^{13} +1.48888 q^{14} -4.38656 q^{16} -4.79573 q^{17} +0.147823 q^{19} +0.631168 q^{20} -6.17343 q^{22} +5.23645 q^{23} +3.47735 q^{25} +5.01260 q^{26} +0.216778 q^{28} +1.23094 q^{29} +6.14440 q^{31} -1.22106 q^{32} -7.14029 q^{34} +2.91159 q^{35} +7.83605 q^{37} +0.220092 q^{38} -7.73031 q^{40} -0.0490986 q^{41} -8.18235 q^{43} -0.898836 q^{44} +7.79646 q^{46} +3.35398 q^{47} +1.00000 q^{49} +5.17738 q^{50} +0.729822 q^{52} -4.90402 q^{53} -12.0725 q^{55} -2.65501 q^{56} +1.83273 q^{58} +12.0362 q^{59} +10.8592 q^{61} +9.14831 q^{62} +6.95510 q^{64} +9.80239 q^{65} -1.83840 q^{67} -1.03961 q^{68} +4.33502 q^{70} +9.72828 q^{71} +7.33222 q^{73} +11.6670 q^{74} +0.0320448 q^{76} -4.14635 q^{77} +6.96474 q^{79} -12.7719 q^{80} -0.0731022 q^{82} +13.2685 q^{83} -13.9632 q^{85} -12.1826 q^{86} +11.0086 q^{88} +11.1337 q^{89} +3.36668 q^{91} +1.13515 q^{92} +4.99369 q^{94} +0.430400 q^{95} -0.471210 q^{97} +1.48888 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 4 q^{2} + 22 q^{4} - 5 q^{5} + 19 q^{7} - 9 q^{8}+O(q^{10}) \) 19 * q - 4 * q^2 + 22 * q^4 - 5 * q^5 + 19 * q^7 - 9 * q^8	\( 19 q - 4 q^{2} + 22 q^{4} - 5 q^{5} + 19 q^{7} - 9 q^{8} + 9 q^{11} + 24 q^{13} - 4 q^{14} + 20 q^{16} - 17 q^{17} + 23 q^{19} - 5 q^{20} - 3 q^{22} + 17 q^{23} + 38 q^{25} - 28 q^{26} + 22 q^{28} - 2 q^{29} + 16 q^{31} - 17 q^{32} + 29 q^{34} - 5 q^{35} + 56 q^{37} - 2 q^{38} - 13 q^{40} + 7 q^{41} + 19 q^{43} + 29 q^{44} + 10 q^{46} - 25 q^{47} + 19 q^{49} + 9 q^{50} + 16 q^{52} - 18 q^{53} + 10 q^{55} - 9 q^{56} + 31 q^{58} - 11 q^{59} + 26 q^{61} - 26 q^{62} + 45 q^{64} - 27 q^{65} + 24 q^{67} - 14 q^{68} + 32 q^{71} + 51 q^{73} + 12 q^{76} + 9 q^{77} + 30 q^{79} + 30 q^{80} - 52 q^{82} - q^{83} + 44 q^{85} + 24 q^{86} - 30 q^{88} - 5 q^{89} + 24 q^{91} + 88 q^{92} + 7 q^{94} + 24 q^{95} + 5 q^{97} - 4 q^{98}+O(q^{100}) \) 19 * q - 4 * q^2 + 22 * q^4 - 5 * q^5 + 19 * q^7 - 9 * q^8 + 9 * q^11 + 24 * q^13 - 4 * q^14 + 20 * q^16 - 17 * q^17 + 23 * q^19 - 5 * q^20 - 3 * q^22 + 17 * q^23 + 38 * q^25 - 28 * q^26 + 22 * q^28 - 2 * q^29 + 16 * q^31 - 17 * q^32 + 29 * q^34 - 5 * q^35 + 56 * q^37 - 2 * q^38 - 13 * q^40 + 7 * q^41 + 19 * q^43 + 29 * q^44 + 10 * q^46 - 25 * q^47 + 19 * q^49 + 9 * q^50 + 16 * q^52 - 18 * q^53 + 10 * q^55 - 9 * q^56 + 31 * q^58 - 11 * q^59 + 26 * q^61 - 26 * q^62 + 45 * q^64 - 27 * q^65 + 24 * q^67 - 14 * q^68 + 32 * q^71 + 51 * q^73 + 12 * q^76 + 9 * q^77 + 30 * q^79 + 30 * q^80 - 52 * q^82 - q^83 + 44 * q^85 + 24 * q^86 - 30 * q^88 - 5 * q^89 + 24 * q^91 + 88 * q^92 + 7 * q^94 + 24 * q^95 + 5 * q^97 - 4 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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