LMFDB - Embedded newform 6045.2.a.n.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6045,2,Mod(1,6045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6045, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6045.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6045 = 3 \cdot 5 \cdot 13 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6045.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:	$48.2695680219$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	6045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -2.00000 q^{4} -1.00000 q^{5} -4.12311 q^{7} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{3} -2.00000 q^{4} -1.00000 q^{5} -4.12311 q^{7} +1.00000 q^{9} -0.438447 q^{11} +2.00000 q^{12} -1.00000 q^{13} +1.00000 q^{15} +4.00000 q^{16} +0.438447 q^{17} -2.00000 q^{19} +2.00000 q^{20} +4.12311 q^{21} -0.438447 q^{23} +1.00000 q^{25} -1.00000 q^{27} +8.24621 q^{28} +1.43845 q^{29} -1.00000 q^{31} +0.438447 q^{33} +4.12311 q^{35} -2.00000 q^{36} +2.68466 q^{37} +1.00000 q^{39} -5.00000 q^{41} -0.561553 q^{43} +0.876894 q^{44} -1.00000 q^{45} +2.87689 q^{47} -4.00000 q^{48} +10.0000 q^{49} -0.438447 q^{51} +2.00000 q^{52} +5.56155 q^{53} +0.438447 q^{55} +2.00000 q^{57} +7.68466 q^{59} -2.00000 q^{60} -2.43845 q^{61} -4.12311 q^{63} -8.00000 q^{64} +1.00000 q^{65} +10.5616 q^{67} -0.876894 q^{68} +0.438447 q^{69} -0.684658 q^{71} +13.3693 q^{73} -1.00000 q^{75} +4.00000 q^{76} +1.80776 q^{77} +15.8078 q^{79} -4.00000 q^{80} +1.00000 q^{81} +10.8078 q^{83} -8.24621 q^{84} -0.438447 q^{85} -1.43845 q^{87} -5.80776 q^{89} +4.12311 q^{91} +0.876894 q^{92} +1.00000 q^{93} +2.00000 q^{95} +4.12311 q^{97} -0.438447 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 4 q^{4} - 2 q^{5} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - 4 * q^4 - 2 * q^5 + 2 * q^9	$ 2 q - 2 q^{3} - 4 q^{4} - 2 q^{5} + 2 q^{9} - 5 q^{11} + 4 q^{12} - 2 q^{13} + 2 q^{15} + 8 q^{16} + 5 q^{17} - 4 q^{19} + 4 q^{20} - 5 q^{23} + 2 q^{25} - 2 q^{27} + 7 q^{29} - 2 q^{31} + 5 q^{33} - 4 q^{36} - 7 q^{37} + 2 q^{39} - 10 q^{41} + 3 q^{43} + 10 q^{44} - 2 q^{45} + 14 q^{47} - 8 q^{48} + 20 q^{49} - 5 q^{51} + 4 q^{52} + 7 q^{53} + 5 q^{55} + 4 q^{57} + 3 q^{59} - 4 q^{60} - 9 q^{61} - 16 q^{64} + 2 q^{65} + 17 q^{67} - 10 q^{68} + 5 q^{69} + 11 q^{71} + 2 q^{73} - 2 q^{75} + 8 q^{76} - 17 q^{77} + 11 q^{79} - 8 q^{80} + 2 q^{81} + q^{83} - 5 q^{85} - 7 q^{87} + 9 q^{89} + 10 q^{92} + 2 q^{93} + 4 q^{95} - 5 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 4 * q^4 - 2 * q^5 + 2 * q^9 - 5 * q^11 + 4 * q^12 - 2 * q^13 + 2 * q^15 + 8 * q^16 + 5 * q^17 - 4 * q^19 + 4 * q^20 - 5 * q^23 + 2 * q^25 - 2 * q^27 + 7 * q^29 - 2 * q^31 + 5 * q^33 - 4 * q^36 - 7 * q^37 + 2 * q^39 - 10 * q^41 + 3 * q^43 + 10 * q^44 - 2 * q^45 + 14 * q^47 - 8 * q^48 + 20 * q^49 - 5 * q^51 + 4 * q^52 + 7 * q^53 + 5 * q^55 + 4 * q^57 + 3 * q^59 - 4 * q^60 - 9 * q^61 - 16 * q^64 + 2 * q^65 + 17 * q^67 - 10 * q^68 + 5 * q^69 + 11 * q^71 + 2 * q^73 - 2 * q^75 + 8 * q^76 - 17 * q^77 + 11 * q^79 - 8 * q^80 + 2 * q^81 + q^83 - 5 * q^85 - 7 * q^87 + 9 * q^89 + 10 * q^92 + 2 * q^93 + 4 * q^95 - 5 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0		−	1.00000i	$-0.5\pi$
$2$	0	0			1.00000i	$0.5\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−2.00000	−1.00000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−4.12311	−1.55839	−0.779194	−	0.626783i	$-0.784371\pi$
$7$	−4.12311	−1.55839	−0.779194	+	0.626783i	$0.784371\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.438447	−0.132197	−0.0660984	−	0.997813i	$-0.521055\pi$
$11$	−0.438447	−0.132197	−0.0660984	+	0.997813i	$0.521055\pi$
$12$	2.00000	0.577350
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	1.00000	0.258199
$16$	4.00000	1.00000
$17$	0.438447	0.106339	0.0531695	−	0.998586i	$-0.483068\pi$
$17$	0.438447	0.106339	0.0531695	+	0.998586i	$0.483068\pi$
$18$	0	0
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	2.00000	0.447214
$21$	4.12311	0.899735
$22$	0	0
$23$	−0.438447	−0.0914226	−0.0457113	−	0.998955i	$-0.514555\pi$
$23$	−0.438447	−0.0914226	−0.0457113	+	0.998955i	$0.514555\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	8.24621	1.55839
$29$	1.43845	0.267113	0.133556	−	0.991041i	$-0.457360\pi$
$29$	1.43845	0.267113	0.133556	+	0.991041i	$0.457360\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	0	0
$33$	0.438447	0.0763239
$34$	0	0
$35$	4.12311	0.696932
$36$	−2.00000	−0.333333
$37$	2.68466	0.441355	0.220678	−	0.975347i	$-0.429173\pi$
$37$	2.68466	0.441355	0.220678	+	0.975347i	$0.429173\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	−5.00000	−0.780869	−0.390434	−	0.920631i	$-0.627675\pi$
$41$	−5.00000	−0.780869	−0.390434	+	0.920631i	$0.627675\pi$
$42$	0	0
$43$	−0.561553	−0.0856360	−0.0428180	−	0.999083i	$-0.513634\pi$
$43$	−0.561553	−0.0856360	−0.0428180	+	0.999083i	$0.513634\pi$
$44$	0.876894	0.132197
$45$	−1.00000	−0.149071
$46$	0	0
$47$	2.87689	0.419638	0.209819	−	0.977740i	$-0.432712\pi$
$47$	2.87689	0.419638	0.209819	+	0.977740i	$0.432712\pi$
$48$	−4.00000	−0.577350
$49$	10.0000	1.42857
$50$	0	0
$51$	−0.438447	−0.0613949
$52$	2.00000	0.277350
$53$	5.56155	0.763938	0.381969	−	0.924175i	$-0.375246\pi$
$53$	5.56155	0.763938	0.381969	+	0.924175i	$0.375246\pi$
$54$	0	0
$55$	0.438447	0.0591202
$56$	0	0
$57$	2.00000	0.264906
$58$	0	0
$59$	7.68466	1.00046	0.500229	−	0.865893i	$-0.333249\pi$
$59$	7.68466	1.00046	0.500229	+	0.865893i	$0.333249\pi$
$60$	−2.00000	−0.258199
$61$	−2.43845	−0.312211	−0.156106	−	0.987740i	$-0.549894\pi$
$61$	−2.43845	−0.312211	−0.156106	+	0.987740i	$0.549894\pi$
$62$	0	0
$63$	−4.12311	−0.519462
$64$	−8.00000	−1.00000
$65$	1.00000	0.124035
$66$	0	0
$67$	10.5616	1.29030	0.645150	−	0.764056i	$-0.276795\pi$
$67$	10.5616	1.29030	0.645150	+	0.764056i	$0.276795\pi$
$68$	−0.876894	−0.106339
$69$	0.438447	0.0527828
$70$	0	0
$71$	−0.684658	−0.0812540	−0.0406270	−	0.999174i	$-0.512936\pi$
$71$	−0.684658	−0.0812540	−0.0406270	+	0.999174i	$0.512936\pi$
$72$	0	0
$73$	13.3693	1.56476	0.782380	−	0.622801i	$-0.214005\pi$
$73$	13.3693	1.56476	0.782380	+	0.622801i	$0.214005\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	4.00000	0.458831
$77$	1.80776	0.206014
$78$	0	0
$79$	15.8078	1.77851	0.889256	−	0.457409i	$-0.151223\pi$
$79$	15.8078	1.77851	0.889256	+	0.457409i	$0.151223\pi$
$80$	−4.00000	−0.447214
$81$	1.00000	0.111111
$82$	0	0
$83$	10.8078	1.18631	0.593153	−	0.805090i	$-0.297883\pi$
$83$	10.8078	1.18631	0.593153	+	0.805090i	$0.297883\pi$
$84$	−8.24621	−0.899735
$85$	−0.438447	−0.0475563
$86$	0	0
$87$	−1.43845	−0.154218
$88$	0	0
$89$	−5.80776	−0.615622	−0.307811	−	0.951448i	$-0.599596\pi$
$89$	−5.80776	−0.615622	−0.307811	+	0.951448i	$0.599596\pi$
$90$	0	0
$91$	4.12311	0.432219
$92$	0.876894	0.0914226
$93$	1.00000	0.103695
$94$	0	0
$95$	2.00000	0.205196
$96$	0	0
$97$	4.12311	0.418638	0.209319	−	0.977847i	$-0.432875\pi$
$97$	4.12311	0.418638	0.209319	+	0.977847i	$0.432875\pi$
$98$	0	0
$99$	−0.438447	−0.0440656
$100$	−2.00000	−0.200000
$101$	−5.36932	−0.534267	−0.267133	−	0.963660i	$-0.586076\pi$
$101$	−5.36932	−0.534267	−0.267133	+	0.963660i	$0.586076\pi$
$102$	0	0
$103$	10.2462	1.00959	0.504795	−	0.863239i	$-0.331568\pi$
$103$	10.2462	1.00959	0.504795	+	0.863239i	$0.331568\pi$
$104$	0	0
$105$	−4.12311	−0.402374
$106$	0	0
$107$	−17.4924	−1.69106	−0.845528	−	0.533931i	$-0.820714\pi$
$107$	−17.4924	−1.69106	−0.845528	+	0.533931i	$0.820714\pi$
$108$	2.00000	0.192450
$109$	−17.3693	−1.66368	−0.831839	−	0.555016i	$-0.812712\pi$
$109$	−17.3693	−1.66368	−0.831839	+	0.555016i	$0.812712\pi$
$110$	0	0
$111$	−2.68466	−0.254817
$112$	−16.4924	−1.55839
$113$	16.5616	1.55798	0.778990	−	0.627036i	$-0.215732\pi$
$113$	16.5616	1.55798	0.778990	+	0.627036i	$0.215732\pi$
$114$	0	0
$115$	0.438447	0.0408854
$116$	−2.87689	−0.267113
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	−1.80776	−0.165717
$120$	0	0
$121$	−10.8078	−0.982524
$122$	0	0
$123$	5.00000	0.450835
$124$	2.00000	0.179605
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−4.31534	−0.382925	−0.191462	−	0.981500i	$-0.561323\pi$
$127$	−4.31534	−0.382925	−0.191462	+	0.981500i	$0.561323\pi$
$128$	0	0
$129$	0.561553	0.0494420
$130$	0	0
$131$	−4.87689	−0.426096	−0.213048	−	0.977042i	$-0.568339\pi$
$131$	−4.87689	−0.426096	−0.213048	+	0.977042i	$0.568339\pi$
$132$	−0.876894	−0.0763239
$133$	8.24621	0.715037
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	−5.12311	−0.437696	−0.218848	−	0.975759i	$-0.570230\pi$
$137$	−5.12311	−0.437696	−0.218848	+	0.975759i	$0.570230\pi$
$138$	0	0
$139$	−14.6847	−1.24554	−0.622768	−	0.782406i	$-0.713992\pi$
$139$	−14.6847	−1.24554	−0.622768	+	0.782406i	$0.713992\pi$
$140$	−8.24621	−0.696932
$141$	−2.87689	−0.242278
$142$	0	0
$143$	0.438447	0.0366648
$144$	4.00000	0.333333
$145$	−1.43845	−0.119457
$146$	0	0
$147$	−10.0000	−0.824786
$148$	−5.36932	−0.441355
$149$	3.87689	0.317608	0.158804	−	0.987310i	$-0.449236\pi$
$149$	3.87689	0.317608	0.158804	+	0.987310i	$0.449236\pi$
$150$	0	0
$151$	17.9309	1.45919	0.729597	−	0.683878i	$-0.239708\pi$
$151$	17.9309	1.45919	0.729597	+	0.683878i	$0.239708\pi$
$152$	0	0
$153$	0.438447	0.0354464
$154$	0	0
$155$	1.00000	0.0803219
$156$	−2.00000	−0.160128
$157$	1.12311	0.0896336	0.0448168	−	0.998995i	$-0.485730\pi$
$157$	1.12311	0.0896336	0.0448168	+	0.998995i	$0.485730\pi$
$158$	0	0
$159$	−5.56155	−0.441060
$160$	0	0
$161$	1.80776	0.142472
$162$	0	0
$163$	−7.31534	−0.572982	−0.286491	−	0.958083i	$-0.592489\pi$
$163$	−7.31534	−0.572982	−0.286491	+	0.958083i	$0.592489\pi$
$164$	10.0000	0.780869
$165$	−0.438447	−0.0341331
$166$	0	0
$167$	−16.0000	−1.23812	−0.619059	−	0.785345i	$-0.712486\pi$
$167$	−16.0000	−1.23812	−0.619059	+	0.785345i	$0.712486\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−2.00000	−0.152944
$172$	1.12311	0.0856360
$173$	−8.56155	−0.650923	−0.325461	−	0.945555i	$-0.605520\pi$
$173$	−8.56155	−0.650923	−0.325461	+	0.945555i	$0.605520\pi$
$174$	0	0
$175$	−4.12311	−0.311677
$176$	−1.75379	−0.132197
$177$	−7.68466	−0.577614
$178$	0	0
$179$	−12.8078	−0.957297	−0.478649	−	0.878007i	$-0.658873\pi$
$179$	−12.8078	−0.957297	−0.478649	+	0.878007i	$0.658873\pi$
$180$	2.00000	0.149071
$181$	3.80776	0.283029	0.141514	−	0.989936i	$-0.454803\pi$
$181$	3.80776	0.283029	0.141514	+	0.989936i	$0.454803\pi$
$182$	0	0
$183$	2.43845	0.180255
$184$	0	0
$185$	−2.68466	−0.197380
$186$	0	0
$187$	−0.192236	−0.0140577
$188$	−5.75379	−0.419638
$189$	4.12311	0.299912
$190$	0	0
$191$	−16.2462	−1.17553	−0.587767	−	0.809030i	$-0.699993\pi$
$191$	−16.2462	−1.17553	−0.587767	+	0.809030i	$0.699993\pi$
$192$	8.00000	0.577350
$193$	22.1231	1.59246	0.796228	−	0.604997i	$-0.206826\pi$
$193$	22.1231	1.59246	0.796228	+	0.604997i	$0.206826\pi$
$194$	0	0
$195$	−1.00000	−0.0716115
$196$	−20.0000	−1.42857
$197$	8.80776	0.627527	0.313764	−	0.949501i	$-0.398410\pi$
$197$	8.80776	0.627527	0.313764	+	0.949501i	$0.398410\pi$
$198$	0	0
$199$	−24.0000	−1.70131	−0.850657	−	0.525720i	$-0.823796\pi$
$199$	−24.0000	−1.70131	−0.850657	+	0.525720i	$0.823796\pi$
$200$	0	0
$201$	−10.5616	−0.744954
$202$	0	0
$203$	−5.93087	−0.416265
$204$	0.876894	0.0613949
$205$	5.00000	0.349215
$206$	0	0
$207$	−0.438447	−0.0304742
$208$	−4.00000	−0.277350
$209$	0.876894	0.0606561
$210$	0	0
$211$	−5.43845	−0.374398	−0.187199	−	0.982322i	$-0.559941\pi$
$211$	−5.43845	−0.374398	−0.187199	+	0.982322i	$0.559941\pi$
$212$	−11.1231	−0.763938
$213$	0.684658	0.0469120
$214$	0	0
$215$	0.561553	0.0382976
$216$	0	0
$217$	4.12311	0.279895
$218$	0	0
$219$	−13.3693	−0.903415
$220$	−0.876894	−0.0591202
$221$	−0.438447	−0.0294931
$222$	0	0
$223$	−12.4924	−0.836554	−0.418277	−	0.908319i	$-0.637366\pi$
$223$	−12.4924	−0.836554	−0.418277	+	0.908319i	$0.637366\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−18.4924	−1.22739	−0.613693	−	0.789545i	$-0.710317\pi$
$227$	−18.4924	−1.22739	−0.613693	+	0.789545i	$0.710317\pi$
$228$	−4.00000	−0.264906
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	0	0
$231$	−1.80776	−0.118942
$232$	0	0
$233$	12.6155	0.826471	0.413235	−	0.910624i	$-0.364399\pi$
$233$	12.6155	0.826471	0.413235	+	0.910624i	$0.364399\pi$
$234$	0	0
$235$	−2.87689	−0.187668
$236$	−15.3693	−1.00046
$237$	−15.8078	−1.02682
$238$	0	0
$239$	14.4384	0.933946	0.466973	−	0.884272i	$-0.345345\pi$
$239$	14.4384	0.933946	0.466973	+	0.884272i	$0.345345\pi$
$240$	4.00000	0.258199
$241$	−19.9309	−1.28386	−0.641930	−	0.766763i	$-0.721866\pi$
$241$	−19.9309	−1.28386	−0.641930	+	0.766763i	$0.721866\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	4.87689	0.312211
$245$	−10.0000	−0.638877
$246$	0	0
$247$	2.00000	0.127257
$248$	0	0
$249$	−10.8078	−0.684914
$250$	0	0
$251$	−17.4384	−1.10071	−0.550353	−	0.834932i	$-0.685507\pi$
$251$	−17.4384	−1.10071	−0.550353	+	0.834932i	$0.685507\pi$
$252$	8.24621	0.519462
$253$	0.192236	0.0120858
$254$	0	0
$255$	0.438447	0.0274566
$256$	16.0000	1.00000
$257$	14.4924	0.904012	0.452006	−	0.892015i	$-0.350708\pi$
$257$	14.4924	0.904012	0.452006	+	0.892015i	$0.350708\pi$
$258$	0	0
$259$	−11.0691	−0.687802
$260$	−2.00000	−0.124035
$261$	1.43845	0.0890376
$262$	0	0
$263$	−29.1231	−1.79581	−0.897904	−	0.440192i	$-0.854910\pi$
$263$	−29.1231	−1.79581	−0.897904	+	0.440192i	$0.854910\pi$
$264$	0	0
$265$	−5.56155	−0.341643
$266$	0	0
$267$	5.80776	0.355429
$268$	−21.1231	−1.29030
$269$	21.6155	1.31792	0.658961	−	0.752177i	$-0.270996\pi$
$269$	21.6155	1.31792	0.658961	+	0.752177i	$0.270996\pi$
$270$	0	0
$271$	16.3153	0.991086	0.495543	−	0.868583i	$-0.334969\pi$
$271$	16.3153	0.991086	0.495543	+	0.868583i	$0.334969\pi$
$272$	1.75379	0.106339
$273$	−4.12311	−0.249542
$274$	0	0
$275$	−0.438447	−0.0264394
$276$	−0.876894	−0.0527828
$277$	7.68466	0.461726	0.230863	−	0.972986i	$-0.425845\pi$
$277$	7.68466	0.461726	0.230863	+	0.972986i	$0.425845\pi$
$278$	0	0
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	−16.5616	−0.987979	−0.493990	−	0.869468i	$-0.664462\pi$
$281$	−16.5616	−0.987979	−0.493990	+	0.869468i	$0.664462\pi$
$282$	0	0
$283$	4.49242	0.267047	0.133523	−	0.991046i	$-0.457371\pi$
$283$	4.49242	0.267047	0.133523	+	0.991046i	$0.457371\pi$
$284$	1.36932	0.0812540
$285$	−2.00000	−0.118470
$286$	0	0
$287$	20.6155	1.21690
$288$	0	0
$289$	−16.8078	−0.988692
$290$	0	0
$291$	−4.12311	−0.241701
$292$	−26.7386	−1.56476
$293$	29.6155	1.73016	0.865079	−	0.501636i	$-0.167268\pi$
$293$	29.6155	1.73016	0.865079	+	0.501636i	$0.167268\pi$
$294$	0	0
$295$	−7.68466	−0.447418
$296$	0	0
$297$	0.438447	0.0254413
$298$	0	0
$299$	0.438447	0.0253561
$300$	2.00000	0.115470
$301$	2.31534	0.133454
$302$	0	0
$303$	5.36932	0.308459
$304$	−8.00000	−0.458831
$305$	2.43845	0.139625
$306$	0	0
$307$	−1.56155	−0.0891225	−0.0445613	−	0.999007i	$-0.514189\pi$
$307$	−1.56155	−0.0891225	−0.0445613	+	0.999007i	$0.514189\pi$
$308$	−3.61553	−0.206014
$309$	−10.2462	−0.582887
$310$	0	0
$311$	−28.4924	−1.61566	−0.807829	−	0.589418i	$-0.799357\pi$
$311$	−28.4924	−1.61566	−0.807829	+	0.589418i	$0.799357\pi$
$312$	0	0
$313$	−21.0540	−1.19004	−0.595021	−	0.803711i	$-0.702856\pi$
$313$	−21.0540	−1.19004	−0.595021	+	0.803711i	$0.702856\pi$
$314$	0	0
$315$	4.12311	0.232311
$316$	−31.6155	−1.77851
$317$	−0.630683	−0.0354227	−0.0177113	−	0.999843i	$-0.505638\pi$
$317$	−0.630683	−0.0354227	−0.0177113	+	0.999843i	$0.505638\pi$
$318$	0	0
$319$	−0.630683	−0.0353115
$320$	8.00000	0.447214
$321$	17.4924	0.976332
$322$	0	0
$323$	−0.876894	−0.0487917
$324$	−2.00000	−0.111111
$325$	−1.00000	−0.0554700
$326$	0	0
$327$	17.3693	0.960525
$328$	0	0
$329$	−11.8617	−0.653959
$330$	0	0
$331$	−14.5616	−0.800375	−0.400188	−	0.916433i	$-0.631055\pi$
$331$	−14.5616	−0.800375	−0.400188	+	0.916433i	$0.631055\pi$
$332$	−21.6155	−1.18631
$333$	2.68466	0.147118
$334$	0	0
$335$	−10.5616	−0.577039
$336$	16.4924	0.899735
$337$	3.36932	0.183538	0.0917692	−	0.995780i	$-0.470748\pi$
$337$	3.36932	0.183538	0.0917692	+	0.995780i	$0.470748\pi$
$338$	0	0
$339$	−16.5616	−0.899500
$340$	0.876894	0.0475563
$341$	0.438447	0.0237432
$342$	0	0
$343$	−12.3693	−0.667880
$344$	0	0
$345$	−0.438447	−0.0236052
$346$	0	0
$347$	9.80776	0.526508	0.263254	−	0.964727i	$-0.415204\pi$
$347$	9.80776	0.526508	0.263254	+	0.964727i	$0.415204\pi$
$348$	2.87689	0.154218
$349$	3.75379	0.200936	0.100468	−	0.994940i	$-0.467966\pi$
$349$	3.75379	0.200936	0.100468	+	0.994940i	$0.467966\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	27.0540	1.43994	0.719969	−	0.694006i	$-0.244156\pi$
$353$	27.0540	1.43994	0.719969	+	0.694006i	$0.244156\pi$
$354$	0	0
$355$	0.684658	0.0363379
$356$	11.6155	0.615622
$357$	1.80776	0.0956770
$358$	0	0
$359$	14.5616	0.768529	0.384265	−	0.923223i	$-0.374455\pi$
$359$	14.5616	0.768529	0.384265	+	0.923223i	$0.374455\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	10.8078	0.567260
$364$	−8.24621	−0.432219
$365$	−13.3693	−0.699782
$366$	0	0
$367$	−19.1231	−0.998218	−0.499109	−	0.866539i	$-0.666339\pi$
$367$	−19.1231	−0.998218	−0.499109	+	0.866539i	$0.666339\pi$
$368$	−1.75379	−0.0914226
$369$	−5.00000	−0.260290
$370$	0	0
$371$	−22.9309	−1.19051
$372$	−2.00000	−0.103695
$373$	−8.24621	−0.426973	−0.213486	−	0.976946i	$-0.568482\pi$
$373$	−8.24621	−0.426973	−0.213486	+	0.976946i	$0.568482\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	−1.43845	−0.0740838
$378$	0	0
$379$	−12.8769	−0.661442	−0.330721	−	0.943729i	$-0.607292\pi$
$379$	−12.8769	−0.661442	−0.330721	+	0.943729i	$0.607292\pi$
$380$	−4.00000	−0.205196
$381$	4.31534	0.221082
$382$	0	0
$383$	−24.8078	−1.26762	−0.633809	−	0.773490i	$-0.718509\pi$
$383$	−24.8078	−1.26762	−0.633809	+	0.773490i	$0.718509\pi$
$384$	0	0
$385$	−1.80776	−0.0921322
$386$	0	0
$387$	−0.561553	−0.0285453
$388$	−8.24621	−0.418638
$389$	15.0540	0.763267	0.381633	−	0.924314i	$-0.375362\pi$
$389$	15.0540	0.763267	0.381633	+	0.924314i	$0.375362\pi$
$390$	0	0
$391$	−0.192236	−0.00972179
$392$	0	0
$393$	4.87689	0.246007
$394$	0	0
$395$	−15.8078	−0.795375
$396$	0.876894	0.0440656
$397$	18.3002	0.918460	0.459230	−	0.888317i	$-0.348125\pi$
$397$	18.3002	0.918460	0.459230	+	0.888317i	$0.348125\pi$
$398$	0	0
$399$	−8.24621	−0.412827
$400$	4.00000	0.200000
$401$	34.4924	1.72247	0.861235	−	0.508207i	$-0.169692\pi$
$401$	34.4924	1.72247	0.861235	+	0.508207i	$0.169692\pi$
$402$	0	0
$403$	1.00000	0.0498135
$404$	10.7386	0.534267
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−1.17708	−0.0583457
$408$	0	0
$409$	−27.4384	−1.35674	−0.678372	−	0.734719i	$-0.737314\pi$
$409$	−27.4384	−1.35674	−0.678372	+	0.734719i	$0.737314\pi$
$410$	0	0
$411$	5.12311	0.252704
$412$	−20.4924	−1.00959
$413$	−31.6847	−1.55910
$414$	0	0
$415$	−10.8078	−0.530532
$416$	0	0
$417$	14.6847	0.719111
$418$	0	0
$419$	10.4924	0.512588	0.256294	−	0.966599i	$-0.417498\pi$
$419$	10.4924	0.512588	0.256294	+	0.966599i	$0.417498\pi$
$420$	8.24621	0.402374
$421$	6.24621	0.304422	0.152211	−	0.988348i	$-0.451361\pi$
$421$	6.24621	0.304422	0.152211	+	0.988348i	$0.451361\pi$
$422$	0	0
$423$	2.87689	0.139879
$424$	0	0
$425$	0.438447	0.0212678
$426$	0	0
$427$	10.0540	0.486546
$428$	34.9848	1.69106
$429$	−0.438447	−0.0211684
$430$	0	0
$431$	19.0540	0.917798	0.458899	−	0.888489i	$-0.348244\pi$
$431$	19.0540	0.917798	0.458899	+	0.888489i	$0.348244\pi$
$432$	−4.00000	−0.192450
$433$	8.31534	0.399610	0.199805	−	0.979836i	$-0.435969\pi$
$433$	8.31534	0.399610	0.199805	+	0.979836i	$0.435969\pi$
$434$	0	0
$435$	1.43845	0.0689683
$436$	34.7386	1.66368
$437$	0.876894	0.0419475
$438$	0	0
$439$	6.36932	0.303991	0.151995	−	0.988381i	$-0.451430\pi$
$439$	6.36932	0.303991	0.151995	+	0.988381i	$0.451430\pi$
$440$	0	0
$441$	10.0000	0.476190
$442$	0	0
$443$	−28.3693	−1.34787	−0.673933	−	0.738792i	$-0.735397\pi$
$443$	−28.3693	−1.34787	−0.673933	+	0.738792i	$0.735397\pi$
$444$	5.36932	0.254817
$445$	5.80776	0.275314
$446$	0	0
$447$	−3.87689	−0.183371
$448$	32.9848	1.55839
$449$	−1.56155	−0.0736942	−0.0368471	−	0.999321i	$-0.511731\pi$
$449$	−1.56155	−0.0736942	−0.0368471	+	0.999321i	$0.511731\pi$
$450$	0	0
$451$	2.19224	0.103228
$452$	−33.1231	−1.55798
$453$	−17.9309	−0.842466
$454$	0	0
$455$	−4.12311	−0.193294
$456$	0	0
$457$	−11.1771	−0.522842	−0.261421	−	0.965225i	$-0.584191\pi$
$457$	−11.1771	−0.522842	−0.261421	+	0.965225i	$0.584191\pi$
$458$	0	0
$459$	−0.438447	−0.0204650
$460$	−0.876894	−0.0408854
$461$	10.4384	0.486167	0.243083	−	0.970005i	$-0.421841\pi$
$461$	10.4384	0.486167	0.243083	+	0.970005i	$0.421841\pi$
$462$	0	0
$463$	25.5616	1.18795	0.593973	−	0.804485i	$-0.297559\pi$
$463$	25.5616	1.18795	0.593973	+	0.804485i	$0.297559\pi$
$464$	5.75379	0.267113
$465$	−1.00000	−0.0463739
$466$	0	0
$467$	−19.7386	−0.913395	−0.456698	−	0.889622i	$-0.650968\pi$
$467$	−19.7386	−0.913395	−0.456698	+	0.889622i	$0.650968\pi$
$468$	2.00000	0.0924500
$469$	−43.5464	−2.01079
$470$	0	0
$471$	−1.12311	−0.0517500
$472$	0	0
$473$	0.246211	0.0113208
$474$	0	0
$475$	−2.00000	−0.0917663
$476$	3.61553	0.165717
$477$	5.56155	0.254646
$478$	0	0
$479$	32.8617	1.50149	0.750746	−	0.660591i	$-0.229694\pi$
$479$	32.8617	1.50149	0.750746	+	0.660591i	$0.229694\pi$
$480$	0	0
$481$	−2.68466	−0.122410
$482$	0	0
$483$	−1.80776	−0.0822561
$484$	21.6155	0.982524
$485$	−4.12311	−0.187221
$486$	0	0
$487$	−10.1922	−0.461854	−0.230927	−	0.972971i	$-0.574176\pi$
$487$	−10.1922	−0.461854	−0.230927	+	0.972971i	$0.574176\pi$
$488$	0	0
$489$	7.31534	0.330811
$490$	0	0
$491$	12.1771	0.549544	0.274772	−	0.961509i	$-0.411398\pi$
$491$	12.1771	0.549544	0.274772	+	0.961509i	$0.411398\pi$
$492$	−10.0000	−0.450835
$493$	0.630683	0.0284045
$494$	0	0
$495$	0.438447	0.0197067
$496$	−4.00000	−0.179605
$497$	2.82292	0.126625
$498$	0	0
$499$	7.68466	0.344013	0.172006	−	0.985096i	$-0.444975\pi$
$499$	7.68466	0.344013	0.172006	+	0.985096i	$0.444975\pi$
$500$	2.00000	0.0894427
$501$	16.0000	0.714827
$502$	0	0
$503$	26.5616	1.18432	0.592161	−	0.805820i	$-0.298275\pi$
$503$	26.5616	1.18432	0.592161	+	0.805820i	$0.298275\pi$
$504$	0	0
$505$	5.36932	0.238931
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	8.63068	0.382925
$509$	5.56155	0.246511	0.123256	−	0.992375i	$-0.460666\pi$
$509$	5.56155	0.246511	0.123256	+	0.992375i	$0.460666\pi$
$510$	0	0
$511$	−55.1231	−2.43850
$512$	0	0
$513$	2.00000	0.0883022
$514$	0	0
$515$	−10.2462	−0.451502
$516$	−1.12311	−0.0494420
$517$	−1.26137	−0.0554748
$518$	0	0
$519$	8.56155	0.375810
$520$	0	0
$521$	−41.3693	−1.81242	−0.906211	−	0.422825i	$-0.861038\pi$
$521$	−41.3693	−1.81242	−0.906211	+	0.422825i	$0.861038\pi$
$522$	0	0
$523$	42.8078	1.87185	0.935926	−	0.352196i	$-0.114565\pi$
$523$	42.8078	1.87185	0.935926	+	0.352196i	$0.114565\pi$
$524$	9.75379	0.426096
$525$	4.12311	0.179947
$526$	0	0
$527$	−0.438447	−0.0190991
$528$	1.75379	0.0763239
$529$	−22.8078	−0.991642
$530$	0	0
$531$	7.68466	0.333486
$532$	−16.4924	−0.715037
$533$	5.00000	0.216574
$534$	0	0
$535$	17.4924	0.756263
$536$	0	0
$537$	12.8078	0.552696
$538$	0	0
$539$	−4.38447	−0.188853
$540$	−2.00000	−0.0860663
$541$	−18.8769	−0.811581	−0.405791	−	0.913966i	$-0.633004\pi$
$541$	−18.8769	−0.811581	−0.405791	+	0.913966i	$0.633004\pi$
$542$	0	0
$543$	−3.80776	−0.163407
$544$	0	0
$545$	17.3693	0.744020
$546$	0	0
$547$	−4.00000	−0.171028	−0.0855138	−	0.996337i	$-0.527253\pi$
$547$	−4.00000	−0.171028	−0.0855138	+	0.996337i	$0.527253\pi$
$548$	10.2462	0.437696
$549$	−2.43845	−0.104070
$550$	0	0
$551$	−2.87689	−0.122560
$552$	0	0
$553$	−65.1771	−2.77161
$554$	0	0
$555$	2.68466	0.113957
$556$	29.3693	1.24554
$557$	−4.63068	−0.196208	−0.0981042	−	0.995176i	$-0.531278\pi$
$557$	−4.63068	−0.196208	−0.0981042	+	0.995176i	$0.531278\pi$
$558$	0	0
$559$	0.561553	0.0237512
$560$	16.4924	0.696932
$561$	0.192236	0.00811621
$562$	0	0
$563$	1.06913	0.0450585	0.0225292	−	0.999746i	$-0.492828\pi$
$563$	1.06913	0.0450585	0.0225292	+	0.999746i	$0.492828\pi$
$564$	5.75379	0.242278
$565$	−16.5616	−0.696750
$566$	0	0
$567$	−4.12311	−0.173154
$568$	0	0
$569$	4.24621	0.178010	0.0890052	−	0.996031i	$-0.471631\pi$
$569$	4.24621	0.178010	0.0890052	+	0.996031i	$0.471631\pi$
$570$	0	0
$571$	−22.4384	−0.939020	−0.469510	−	0.882927i	$-0.655569\pi$
$571$	−22.4384	−0.939020	−0.469510	+	0.882927i	$0.655569\pi$
$572$	−0.876894	−0.0366648
$573$	16.2462	0.678695
$574$	0	0
$575$	−0.438447	−0.0182845
$576$	−8.00000	−0.333333
$577$	−17.8078	−0.741347	−0.370673	−	0.928763i	$-0.620873\pi$
$577$	−17.8078	−0.741347	−0.370673	+	0.928763i	$0.620873\pi$
$578$	0	0
$579$	−22.1231	−0.919405
$580$	2.87689	0.119457
$581$	−44.5616	−1.84872
$582$	0	0
$583$	−2.43845	−0.100990
$584$	0	0
$585$	1.00000	0.0413449
$586$	0	0
$587$	14.5616	0.601020	0.300510	−	0.953779i	$-0.402843\pi$
$587$	14.5616	0.601020	0.300510	+	0.953779i	$0.402843\pi$
$588$	20.0000	0.824786
$589$	2.00000	0.0824086
$590$	0	0
$591$	−8.80776	−0.362303
$592$	10.7386	0.441355
$593$	6.87689	0.282400	0.141200	−	0.989981i	$-0.454904\pi$
$593$	6.87689	0.282400	0.141200	+	0.989981i	$0.454904\pi$
$594$	0	0
$595$	1.80776	0.0741111
$596$	−7.75379	−0.317608
$597$	24.0000	0.982255
$598$	0	0
$599$	39.2311	1.60294	0.801469	−	0.598037i	$-0.204052\pi$
$599$	39.2311	1.60294	0.801469	+	0.598037i	$0.204052\pi$
$600$	0	0
$601$	−14.6847	−0.599000	−0.299500	−	0.954096i	$-0.596820\pi$
$601$	−14.6847	−0.599000	−0.299500	+	0.954096i	$0.596820\pi$
$602$	0	0
$603$	10.5616	0.430100
$604$	−35.8617	−1.45919
$605$	10.8078	0.439398
$606$	0	0
$607$	−37.2311	−1.51116	−0.755581	−	0.655055i	$-0.772645\pi$
$607$	−37.2311	−1.51116	−0.755581	+	0.655055i	$0.772645\pi$
$608$	0	0
$609$	5.93087	0.240331
$610$	0	0
$611$	−2.87689	−0.116387
$612$	−0.876894	−0.0354464
$613$	16.0540	0.648414	0.324207	−	0.945986i	$-0.394903\pi$
$613$	16.0540	0.648414	0.324207	+	0.945986i	$0.394903\pi$
$614$	0	0
$615$	−5.00000	−0.201619
$616$	0	0
$617$	−8.49242	−0.341892	−0.170946	−	0.985280i	$-0.554682\pi$
$617$	−8.49242	−0.341892	−0.170946	+	0.985280i	$0.554682\pi$
$618$	0	0
$619$	34.1771	1.37369	0.686846	−	0.726803i	$-0.258994\pi$
$619$	34.1771	1.37369	0.686846	+	0.726803i	$0.258994\pi$
$620$	−2.00000	−0.0803219
$621$	0.438447	0.0175943
$622$	0	0
$623$	23.9460	0.959377
$624$	4.00000	0.160128
$625$	1.00000	0.0400000
$626$	0	0
$627$	−0.876894	−0.0350198
$628$	−2.24621	−0.0896336
$629$	1.17708	0.0469333
$630$	0	0
$631$	9.75379	0.388292	0.194146	−	0.980973i	$-0.437806\pi$
$631$	9.75379	0.388292	0.194146	+	0.980973i	$0.437806\pi$
$632$	0	0
$633$	5.43845	0.216159
$634$	0	0
$635$	4.31534	0.171249
$636$	11.1231	0.441060
$637$	−10.0000	−0.396214
$638$	0	0
$639$	−0.684658	−0.0270847
$640$	0	0
$641$	13.6847	0.540512	0.270256	−	0.962789i	$-0.412892\pi$
$641$	13.6847	0.540512	0.270256	+	0.962789i	$0.412892\pi$
$642$	0	0
$643$	−42.9309	−1.69303	−0.846514	−	0.532366i	$-0.821303\pi$
$643$	−42.9309	−1.69303	−0.846514	+	0.532366i	$0.821303\pi$
$644$	−3.61553	−0.142472
$645$	−0.561553	−0.0221111
$646$	0	0
$647$	−37.8078	−1.48638	−0.743188	−	0.669082i	$-0.766687\pi$
$647$	−37.8078	−1.48638	−0.743188	+	0.669082i	$0.766687\pi$
$648$	0	0
$649$	−3.36932	−0.132257
$650$	0	0
$651$	−4.12311	−0.161597
$652$	14.6307	0.572982
$653$	37.6847	1.47471	0.737357	−	0.675503i	$-0.236073\pi$
$653$	37.6847	1.47471	0.737357	+	0.675503i	$0.236073\pi$
$654$	0	0
$655$	4.87689	0.190556
$656$	−20.0000	−0.780869
$657$	13.3693	0.521587
$658$	0	0
$659$	−11.7538	−0.457863	−0.228931	−	0.973443i	$-0.573523\pi$
$659$	−11.7538	−0.457863	−0.228931	+	0.973443i	$0.573523\pi$
$660$	0.876894	0.0341331
$661$	−14.8769	−0.578644	−0.289322	−	0.957232i	$-0.593430\pi$
$661$	−14.8769	−0.578644	−0.289322	+	0.957232i	$0.593430\pi$
$662$	0	0
$663$	0.438447	0.0170279
$664$	0	0
$665$	−8.24621	−0.319774
$666$	0	0
$667$	−0.630683	−0.0244201
$668$	32.0000	1.23812
$669$	12.4924	0.482985
$670$	0	0
$671$	1.06913	0.0412733
$672$	0	0
$673$	6.56155	0.252929	0.126465	−	0.991971i	$-0.459637\pi$
$673$	6.56155	0.252929	0.126465	+	0.991971i	$0.459637\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	−2.00000	−0.0769231
$677$	4.43845	0.170583	0.0852917	−	0.996356i	$-0.472818\pi$
$677$	4.43845	0.170583	0.0852917	+	0.996356i	$0.472818\pi$
$678$	0	0
$679$	−17.0000	−0.652400
$680$	0	0
$681$	18.4924	0.708631
$682$	0	0
$683$	−45.6155	−1.74543	−0.872715	−	0.488230i	$-0.837643\pi$
$683$	−45.6155	−1.74543	−0.872715	+	0.488230i	$0.837643\pi$
$684$	4.00000	0.152944
$685$	5.12311	0.195744
$686$	0	0
$687$	−14.0000	−0.534133
$688$	−2.24621	−0.0856360
$689$	−5.56155	−0.211878
$690$	0	0
$691$	−4.24621	−0.161533	−0.0807667	−	0.996733i	$-0.525737\pi$
$691$	−4.24621	−0.161533	−0.0807667	+	0.996733i	$0.525737\pi$
$692$	17.1231	0.650923
$693$	1.80776	0.0686713
$694$	0	0
$695$	14.6847	0.557021
$696$	0	0
$697$	−2.19224	−0.0830369
$698$	0	0
$699$	−12.6155	−0.477163
$700$	8.24621	0.311677
$701$	49.6155	1.87395	0.936976	−	0.349393i	$-0.113612\pi$
$701$	49.6155	1.87395	0.936976	+	0.349393i	$0.113612\pi$
$702$	0	0
$703$	−5.36932	−0.202508
$704$	3.50758	0.132197
$705$	2.87689	0.108350
$706$	0	0
$707$	22.1383	0.832595
$708$	15.3693	0.577614
$709$	50.4924	1.89628	0.948141	−	0.317849i	$-0.102960\pi$
$709$	50.4924	1.89628	0.948141	+	0.317849i	$0.102960\pi$
$710$	0	0
$711$	15.8078	0.592837
$712$	0	0
$713$	0.438447	0.0164200
$714$	0	0
$715$	−0.438447	−0.0163970
$716$	25.6155	0.957297
$717$	−14.4384	−0.539214
$718$	0	0
$719$	−21.0540	−0.785181	−0.392590	−	0.919713i	$-0.628421\pi$
$719$	−21.0540	−0.785181	−0.392590	+	0.919713i	$0.628421\pi$
$720$	−4.00000	−0.149071
$721$	−42.2462	−1.57333
$722$	0	0
$723$	19.9309	0.741237
$724$	−7.61553	−0.283029
$725$	1.43845	0.0534226
$726$	0	0
$727$	−41.1231	−1.52517	−0.762586	−	0.646887i	$-0.776070\pi$
$727$	−41.1231	−1.52517	−0.762586	+	0.646887i	$0.776070\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−0.246211	−0.00910645
$732$	−4.87689	−0.180255
$733$	6.12311	0.226162	0.113081	−	0.993586i	$-0.463928\pi$
$733$	6.12311	0.226162	0.113081	+	0.993586i	$0.463928\pi$
$734$	0	0
$735$	10.0000	0.368856
$736$	0	0
$737$	−4.63068	−0.170573
$738$	0	0
$739$	−8.49242	−0.312399	−0.156199	−	0.987726i	$-0.549924\pi$
$739$	−8.49242	−0.312399	−0.156199	+	0.987726i	$0.549924\pi$
$740$	5.36932	0.197380
$741$	−2.00000	−0.0734718
$742$	0	0
$743$	30.5616	1.12119	0.560597	−	0.828089i	$-0.310572\pi$
$743$	30.5616	1.12119	0.560597	+	0.828089i	$0.310572\pi$
$744$	0	0
$745$	−3.87689	−0.142038
$746$	0	0
$747$	10.8078	0.395435
$748$	0.384472	0.0140577
$749$	72.1231	2.63532
$750$	0	0
$751$	−23.7386	−0.866235	−0.433118	−	0.901337i	$-0.642586\pi$
$751$	−23.7386	−0.866235	−0.433118	+	0.901337i	$0.642586\pi$
$752$	11.5076	0.419638
$753$	17.4384	0.635492
$754$	0	0
$755$	−17.9309	−0.652571
$756$	−8.24621	−0.299912
$757$	4.06913	0.147895	0.0739475	−	0.997262i	$-0.476440\pi$
$757$	4.06913	0.147895	0.0739475	+	0.997262i	$0.476440\pi$
$758$	0	0
$759$	−0.192236	−0.00697772
$760$	0	0
$761$	53.8617	1.95249	0.976243	−	0.216677i	$-0.0695220\pi$
$761$	53.8617	1.95249	0.976243	+	0.216677i	$0.0695220\pi$
$762$	0	0
$763$	71.6155	2.59266
$764$	32.4924	1.17553
$765$	−0.438447	−0.0158521
$766$	0	0
$767$	−7.68466	−0.277477
$768$	−16.0000	−0.577350
$769$	−16.4924	−0.594732	−0.297366	−	0.954764i	$-0.596108\pi$
$769$	−16.4924	−0.594732	−0.297366	+	0.954764i	$0.596108\pi$
$770$	0	0
$771$	−14.4924	−0.521932
$772$	−44.2462	−1.59246
$773$	−23.0540	−0.829194	−0.414597	−	0.910005i	$-0.636077\pi$
$773$	−23.0540	−0.829194	−0.414597	+	0.910005i	$0.636077\pi$
$774$	0	0
$775$	−1.00000	−0.0359211
$776$	0	0
$777$	11.0691	0.397103
$778$	0	0
$779$	10.0000	0.358287
$780$	2.00000	0.0716115
$781$	0.300187	0.0107415
$782$	0	0
$783$	−1.43845	−0.0514059
$784$	40.0000	1.42857
$785$	−1.12311	−0.0400854
$786$	0	0
$787$	12.9848	0.462860	0.231430	−	0.972852i	$-0.425660\pi$
$787$	12.9848	0.462860	0.231430	+	0.972852i	$0.425660\pi$
$788$	−17.6155	−0.627527
$789$	29.1231	1.03681
$790$	0	0
$791$	−68.2850	−2.42794
$792$	0	0
$793$	2.43845	0.0865918
$794$	0	0
$795$	5.56155	0.197248
$796$	48.0000	1.70131
$797$	4.68466	0.165939	0.0829696	−	0.996552i	$-0.473560\pi$
$797$	4.68466	0.165939	0.0829696	+	0.996552i	$0.473560\pi$
$798$	0	0
$799$	1.26137	0.0446239
$800$	0	0
$801$	−5.80776	−0.205207
$802$	0	0
$803$	−5.86174	−0.206856
$804$	21.1231	0.744954
$805$	−1.80776	−0.0637153
$806$	0	0
$807$	−21.6155	−0.760903
$808$	0	0
$809$	37.4384	1.31627	0.658133	−	0.752902i	$-0.271347\pi$
$809$	37.4384	1.31627	0.658133	+	0.752902i	$0.271347\pi$
$810$	0	0
$811$	−18.4924	−0.649357	−0.324678	−	0.945824i	$-0.605256\pi$
$811$	−18.4924	−0.649357	−0.324678	+	0.945824i	$0.605256\pi$
$812$	11.8617	0.416265
$813$	−16.3153	−0.572204
$814$	0	0
$815$	7.31534	0.256245
$816$	−1.75379	−0.0613949
$817$	1.12311	0.0392925
$818$	0	0
$819$	4.12311	0.144073
$820$	−10.0000	−0.349215
$821$	−44.9309	−1.56810	−0.784049	−	0.620699i	$-0.786849\pi$
$821$	−44.9309	−1.56810	−0.784049	+	0.620699i	$0.786849\pi$
$822$	0	0
$823$	0.946025	0.0329763	0.0164882	−	0.999864i	$-0.494751\pi$
$823$	0.946025	0.0329763	0.0164882	+	0.999864i	$0.494751\pi$
$824$	0	0
$825$	0.438447	0.0152648
$826$	0	0
$827$	2.69981	0.0938817	0.0469409	−	0.998898i	$-0.485053\pi$
$827$	2.69981	0.0938817	0.0469409	+	0.998898i	$0.485053\pi$
$828$	0.876894	0.0304742
$829$	−45.8617	−1.59284	−0.796422	−	0.604741i	$-0.793277\pi$
$829$	−45.8617	−1.59284	−0.796422	+	0.604741i	$0.793277\pi$
$830$	0	0
$831$	−7.68466	−0.266578
$832$	8.00000	0.277350
$833$	4.38447	0.151913
$834$	0	0
$835$	16.0000	0.553703
$836$	−1.75379	−0.0606561
$837$	1.00000	0.0345651
$838$	0	0
$839$	−36.3693	−1.25561	−0.627804	−	0.778371i	$-0.716046\pi$
$839$	−36.3693	−1.25561	−0.627804	+	0.778371i	$0.716046\pi$
$840$	0	0
$841$	−26.9309	−0.928651
$842$	0	0
$843$	16.5616	0.570410
$844$	10.8769	0.374398
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	44.5616	1.53115
$848$	22.2462	0.763938
$849$	−4.49242	−0.154180
$850$	0	0
$851$	−1.17708	−0.0403498
$852$	−1.36932	−0.0469120
$853$	−40.8617	−1.39908	−0.699540	−	0.714594i	$-0.746612\pi$
$853$	−40.8617	−1.39908	−0.699540	+	0.714594i	$0.746612\pi$
$854$	0	0
$855$	2.00000	0.0683986
$856$	0	0
$857$	−36.5464	−1.24840	−0.624201	−	0.781264i	$-0.714575\pi$
$857$	−36.5464	−1.24840	−0.624201	+	0.781264i	$0.714575\pi$
$858$	0	0
$859$	−48.3002	−1.64798	−0.823991	−	0.566604i	$-0.808257\pi$
$859$	−48.3002	−1.64798	−0.823991	+	0.566604i	$0.808257\pi$
$860$	−1.12311	−0.0382976
$861$	−20.6155	−0.702575
$862$	0	0
$863$	−10.0691	−0.342757	−0.171379	−	0.985205i	$-0.554822\pi$
$863$	−10.0691	−0.342757	−0.171379	+	0.985205i	$0.554822\pi$
$864$	0	0
$865$	8.56155	0.291102
$866$	0	0
$867$	16.8078	0.570822
$868$	−8.24621	−0.279895
$869$	−6.93087	−0.235114
$870$	0	0
$871$	−10.5616	−0.357865
$872$	0	0
$873$	4.12311	0.139546
$874$	0	0
$875$	4.12311	0.139386
$876$	26.7386	0.903415
$877$	6.49242	0.219234	0.109617	−	0.993974i	$-0.465038\pi$
$877$	6.49242	0.219234	0.109617	+	0.993974i	$0.465038\pi$
$878$	0	0
$879$	−29.6155	−0.998907
$880$	1.75379	0.0591202
$881$	0.315342	0.0106241	0.00531206	−	0.999986i	$-0.498309\pi$
$881$	0.315342	0.0106241	0.00531206	+	0.999986i	$0.498309\pi$
$882$	0	0
$883$	26.6695	0.897500	0.448750	−	0.893657i	$-0.351869\pi$
$883$	26.6695	0.897500	0.448750	+	0.893657i	$0.351869\pi$
$884$	0.876894	0.0294931
$885$	7.68466	0.258317
$886$	0	0
$887$	10.1231	0.339901	0.169950	−	0.985453i	$-0.445639\pi$
$887$	10.1231	0.339901	0.169950	+	0.985453i	$0.445639\pi$
$888$	0	0
$889$	17.7926	0.596745
$890$	0	0
$891$	−0.438447	−0.0146885
$892$	24.9848	0.836554
$893$	−5.75379	−0.192543
$894$	0	0
$895$	12.8078	0.428116
$896$	0	0
$897$	−0.438447	−0.0146393
$898$	0	0
$899$	−1.43845	−0.0479749
$900$	−2.00000	−0.0666667
$901$	2.43845	0.0812365
$902$	0	0
$903$	−2.31534	−0.0770497
$904$	0	0
$905$	−3.80776	−0.126574
$906$	0	0
$907$	−50.3542	−1.67198	−0.835991	−	0.548743i	$-0.815107\pi$
$907$	−50.3542	−1.67198	−0.835991	+	0.548743i	$0.815107\pi$
$908$	36.9848	1.22739
$909$	−5.36932	−0.178089
$910$	0	0
$911$	−49.0540	−1.62523	−0.812615	−	0.582800i	$-0.801957\pi$
$911$	−49.0540	−1.62523	−0.812615	+	0.582800i	$0.801957\pi$
$912$	8.00000	0.264906
$913$	−4.73863	−0.156826
$914$	0	0
$915$	−2.43845	−0.0806126
$916$	−28.0000	−0.925146
$917$	20.1080	0.664023
$918$	0	0
$919$	−5.38447	−0.177617	−0.0888087	−	0.996049i	$-0.528306\pi$
$919$	−5.38447	−0.177617	−0.0888087	+	0.996049i	$0.528306\pi$
$920$	0	0
$921$	1.56155	0.0514549
$922$	0	0
$923$	0.684658	0.0225358
$924$	3.61553	0.118942
$925$	2.68466	0.0882710
$926$	0	0
$927$	10.2462	0.336530
$928$	0	0
$929$	14.6847	0.481788	0.240894	−	0.970551i	$-0.422559\pi$
$929$	14.6847	0.481788	0.240894	+	0.970551i	$0.422559\pi$
$930$	0	0
$931$	−20.0000	−0.655474
$932$	−25.2311	−0.826471
$933$	28.4924	0.932800
$934$	0	0
$935$	0.192236	0.00628679
$936$	0	0
$937$	−24.7386	−0.808176	−0.404088	−	0.914720i	$-0.632411\pi$
$937$	−24.7386	−0.808176	−0.404088	+	0.914720i	$0.632411\pi$
$938$	0	0
$939$	21.0540	0.687071
$940$	5.75379	0.187668
$941$	−20.1922	−0.658248	−0.329124	−	0.944287i	$-0.606753\pi$
$941$	−20.1922	−0.658248	−0.329124	+	0.944287i	$0.606753\pi$
$942$	0	0
$943$	2.19224	0.0713890
$944$	30.7386	1.00046
$945$	−4.12311	−0.134125
$946$	0	0
$947$	25.8617	0.840394	0.420197	−	0.907433i	$-0.361961\pi$
$947$	25.8617	0.840394	0.420197	+	0.907433i	$0.361961\pi$
$948$	31.6155	1.02682
$949$	−13.3693	−0.433986
$950$	0	0
$951$	0.630683	0.0204513
$952$	0	0
$953$	49.4233	1.60098	0.800489	−	0.599348i	$-0.204573\pi$
$953$	49.4233	1.60098	0.800489	+	0.599348i	$0.204573\pi$
$954$	0	0
$955$	16.2462	0.525715
$956$	−28.8769	−0.933946
$957$	0.630683	0.0203871
$958$	0	0
$959$	21.1231	0.682101
$960$	−8.00000	−0.258199
$961$	1.00000	0.0322581
$962$	0	0
$963$	−17.4924	−0.563685
$964$	39.8617	1.28386
$965$	−22.1231	−0.712168
$966$	0	0
$967$	48.0000	1.54358	0.771788	−	0.635880i	$-0.219363\pi$
$967$	48.0000	1.54358	0.771788	+	0.635880i	$0.219363\pi$
$968$	0	0
$969$	0.876894	0.0281699
$970$	0	0
$971$	30.1080	0.966210	0.483105	−	0.875562i	$-0.339509\pi$
$971$	30.1080	0.966210	0.483105	+	0.875562i	$0.339509\pi$
$972$	2.00000	0.0641500
$973$	60.5464	1.94103
$974$	0	0
$975$	1.00000	0.0320256
$976$	−9.75379	−0.312211
$977$	28.0000	0.895799	0.447900	−	0.894084i	$-0.352172\pi$
$977$	28.0000	0.895799	0.447900	+	0.894084i	$0.352172\pi$
$978$	0	0
$979$	2.54640	0.0813832
$980$	20.0000	0.638877
$981$	−17.3693	−0.554560
$982$	0	0
$983$	19.3002	0.615580	0.307790	−	0.951454i	$-0.400411\pi$
$983$	19.3002	0.615580	0.307790	+	0.951454i	$0.400411\pi$
$984$	0	0
$985$	−8.80776	−0.280639
$986$	0	0
$987$	11.8617	0.377563
$988$	−4.00000	−0.127257
$989$	0.246211	0.00782906
$990$	0	0
$991$	−38.5464	−1.22447	−0.612233	−	0.790677i	$-0.709729\pi$
$991$	−38.5464	−1.22447	−0.612233	+	0.790677i	$0.709729\pi$
$992$	0	0
$993$	14.5616	0.462097
$994$	0	0
$995$	24.0000	0.760851
$996$	21.6155	0.684914
$997$	22.4924	0.712342	0.356171	−	0.934421i	$-0.384082\pi$
$997$	22.4924	0.712342	0.356171	+	0.934421i	$0.384082\pi$
$998$	0	0
$999$	−2.68466	−0.0849388

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6045.2.a.n.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6045.2.a.n.1.1	✓	2	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:	\( N \)	\(=\)	\( 6045 = 3 \cdot 5 \cdot 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6045.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -2.00000 q^{4} -1.00000 q^{5} -4.12311 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -2.00000 q^{4} -1.00000 q^{5} -4.12311 q^{7} +1.00000 q^{9} -0.438447 q^{11} +2.00000 q^{12} -1.00000 q^{13} +1.00000 q^{15} +4.00000 q^{16} +0.438447 q^{17} -2.00000 q^{19} +2.00000 q^{20} +4.12311 q^{21} -0.438447 q^{23} +1.00000 q^{25} -1.00000 q^{27} +8.24621 q^{28} +1.43845 q^{29} -1.00000 q^{31} +0.438447 q^{33} +4.12311 q^{35} -2.00000 q^{36} +2.68466 q^{37} +1.00000 q^{39} -5.00000 q^{41} -0.561553 q^{43} +0.876894 q^{44} -1.00000 q^{45} +2.87689 q^{47} -4.00000 q^{48} +10.0000 q^{49} -0.438447 q^{51} +2.00000 q^{52} +5.56155 q^{53} +0.438447 q^{55} +2.00000 q^{57} +7.68466 q^{59} -2.00000 q^{60} -2.43845 q^{61} -4.12311 q^{63} -8.00000 q^{64} +1.00000 q^{65} +10.5616 q^{67} -0.876894 q^{68} +0.438447 q^{69} -0.684658 q^{71} +13.3693 q^{73} -1.00000 q^{75} +4.00000 q^{76} +1.80776 q^{77} +15.8078 q^{79} -4.00000 q^{80} +1.00000 q^{81} +10.8078 q^{83} -8.24621 q^{84} -0.438447 q^{85} -1.43845 q^{87} -5.80776 q^{89} +4.12311 q^{91} +0.876894 q^{92} +1.00000 q^{93} +2.00000 q^{95} +4.12311 q^{97} -0.438447 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 4 q^{4} - 2 q^{5} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 4 * q^4 - 2 * q^5 + 2 * q^9	\( 2 q - 2 q^{3} - 4 q^{4} - 2 q^{5} + 2 q^{9} - 5 q^{11} + 4 q^{12} - 2 q^{13} + 2 q^{15} + 8 q^{16} + 5 q^{17} - 4 q^{19} + 4 q^{20} - 5 q^{23} + 2 q^{25} - 2 q^{27} + 7 q^{29} - 2 q^{31} + 5 q^{33} - 4 q^{36} - 7 q^{37} + 2 q^{39} - 10 q^{41} + 3 q^{43} + 10 q^{44} - 2 q^{45} + 14 q^{47} - 8 q^{48} + 20 q^{49} - 5 q^{51} + 4 q^{52} + 7 q^{53} + 5 q^{55} + 4 q^{57} + 3 q^{59} - 4 q^{60} - 9 q^{61} - 16 q^{64} + 2 q^{65} + 17 q^{67} - 10 q^{68} + 5 q^{69} + 11 q^{71} + 2 q^{73} - 2 q^{75} + 8 q^{76} - 17 q^{77} + 11 q^{79} - 8 q^{80} + 2 q^{81} + q^{83} - 5 q^{85} - 7 q^{87} + 9 q^{89} + 10 q^{92} + 2 q^{93} + 4 q^{95} - 5 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 4 * q^4 - 2 * q^5 + 2 * q^9 - 5 * q^11 + 4 * q^12 - 2 * q^13 + 2 * q^15 + 8 * q^16 + 5 * q^17 - 4 * q^19 + 4 * q^20 - 5 * q^23 + 2 * q^25 - 2 * q^27 + 7 * q^29 - 2 * q^31 + 5 * q^33 - 4 * q^36 - 7 * q^37 + 2 * q^39 - 10 * q^41 + 3 * q^43 + 10 * q^44 - 2 * q^45 + 14 * q^47 - 8 * q^48 + 20 * q^49 - 5 * q^51 + 4 * q^52 + 7 * q^53 + 5 * q^55 + 4 * q^57 + 3 * q^59 - 4 * q^60 - 9 * q^61 - 16 * q^64 + 2 * q^65 + 17 * q^67 - 10 * q^68 + 5 * q^69 + 11 * q^71 + 2 * q^73 - 2 * q^75 + 8 * q^76 - 17 * q^77 + 11 * q^79 - 8 * q^80 + 2 * q^81 + q^83 - 5 * q^85 - 7 * q^87 + 9 * q^89 + 10 * q^92 + 2 * q^93 + 4 * q^95 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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