LMFDB - Embedded newform 6045.2.a.n.1.2

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			1.2
Root			$2.56155$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -2.00000 q^{4} -1.00000 q^{5} +4.12311 q^{7} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{3} -2.00000 q^{4} -1.00000 q^{5} +4.12311 q^{7} +1.00000 q^{9} -4.56155 q^{11} +2.00000 q^{12} -1.00000 q^{13} +1.00000 q^{15} +4.00000 q^{16} +4.56155 q^{17} -2.00000 q^{19} +2.00000 q^{20} -4.12311 q^{21} -4.56155 q^{23} +1.00000 q^{25} -1.00000 q^{27} -8.24621 q^{28} +5.56155 q^{29} -1.00000 q^{31} +4.56155 q^{33} -4.12311 q^{35} -2.00000 q^{36} -9.68466 q^{37} +1.00000 q^{39} -5.00000 q^{41} +3.56155 q^{43} +9.12311 q^{44} -1.00000 q^{45} +11.1231 q^{47} -4.00000 q^{48} +10.0000 q^{49} -4.56155 q^{51} +2.00000 q^{52} +1.43845 q^{53} +4.56155 q^{55} +2.00000 q^{57} -4.68466 q^{59} -2.00000 q^{60} -6.56155 q^{61} +4.12311 q^{63} -8.00000 q^{64} +1.00000 q^{65} +6.43845 q^{67} -9.12311 q^{68} +4.56155 q^{69} +11.6847 q^{71} -11.3693 q^{73} -1.00000 q^{75} +4.00000 q^{76} -18.8078 q^{77} -4.80776 q^{79} -4.00000 q^{80} +1.00000 q^{81} -9.80776 q^{83} +8.24621 q^{84} -4.56155 q^{85} -5.56155 q^{87} +14.8078 q^{89} -4.12311 q^{91} +9.12311 q^{92} +1.00000 q^{93} +2.00000 q^{95} -4.12311 q^{97} -4.56155 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$-0.215629\pi$
$7$	4.12311	1.55839	0.779194	+	0.626783i	$0.215629\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.56155	−1.37536	−0.687680	−	0.726014i	$-0.741371\pi$
$11$	−4.56155	−1.37536	−0.687680	+	0.726014i	$0.741371\pi$
$12$	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
$17$	4.56155	1.10634	0.553170	−	0.833069i	$-0.313418\pi$
$17$	4.56155	1.10634	0.553170	+	0.833069i	$0.313418\pi$
$18$	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
$23$	−4.56155	−0.951150	−0.475575	−	0.879675i	$-0.657760\pi$
$23$	−4.56155	−0.951150	−0.475575	+	0.879675i	$0.657760\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	−8.24621	−1.55839
$29$	5.56155	1.03275	0.516377	−	0.856361i	$-0.327280\pi$
$29$	5.56155	1.03275	0.516377	+	0.856361i	$0.327280\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	0	0
$33$	4.56155	0.794064
$34$	0	0
$35$	−4.12311	−0.696932
$36$	−2.00000	−0.333333
$37$	−9.68466	−1.59215	−0.796074	−	0.605199i	$-0.793093\pi$
$37$	−9.68466	−1.59215	−0.796074	+	0.605199i	$0.793093\pi$
$38$	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
$43$	3.56155	0.543132	0.271566	−	0.962420i	$-0.412459\pi$
$43$	3.56155	0.543132	0.271566	+	0.962420i	$0.412459\pi$
$44$	9.12311	1.37536
$45$	−1.00000	−0.149071
$46$	0	0
$47$	11.1231	1.62247	0.811236	−	0.584719i	$-0.198795\pi$
$47$	11.1231	1.62247	0.811236	+	0.584719i	$0.198795\pi$
$48$	−4.00000	−0.577350
$49$	10.0000	1.42857
$50$	0	0
$51$	−4.56155	−0.638745
$52$	2.00000	0.277350
$53$	1.43845	0.197586	0.0987930	−	0.995108i	$-0.468502\pi$
$53$	1.43845	0.197586	0.0987930	+	0.995108i	$0.468502\pi$
$54$	0	0
$55$	4.56155	0.615080
$56$	0	0
$57$	2.00000	0.264906
$58$	0	0
$59$	−4.68466	−0.609891	−0.304945	−	0.952370i	$-0.598638\pi$
$59$	−4.68466	−0.609891	−0.304945	+	0.952370i	$0.598638\pi$
$60$	−2.00000	−0.258199
$61$	−6.56155	−0.840121	−0.420060	−	0.907496i	$-0.637991\pi$
$61$	−6.56155	−0.840121	−0.420060	+	0.907496i	$0.637991\pi$
$62$	0	0
$63$	4.12311	0.519462
$64$	−8.00000	−1.00000
$65$	1.00000	0.124035
$66$	0	0
$67$	6.43845	0.786582	0.393291	−	0.919414i	$-0.371337\pi$
$67$	6.43845	0.786582	0.393291	+	0.919414i	$0.371337\pi$
$68$	−9.12311	−1.10634
$69$	4.56155	0.549146
$70$	0	0
$71$	11.6847	1.38671	0.693357	−	0.720594i	$-0.256131\pi$
$71$	11.6847	1.38671	0.693357	+	0.720594i	$0.256131\pi$
$72$	0	0
$73$	−11.3693	−1.33068	−0.665339	−	0.746541i	$-0.731713\pi$
$73$	−11.3693	−1.33068	−0.665339	+	0.746541i	$0.731713\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	4.00000	0.458831
$77$	−18.8078	−2.14334
$78$	0	0
$79$	−4.80776	−0.540916	−0.270458	−	0.962732i	$-0.587175\pi$
$79$	−4.80776	−0.540916	−0.270458	+	0.962732i	$0.587175\pi$
$80$	−4.00000	−0.447214
$81$	1.00000	0.111111
$82$	0	0
$83$	−9.80776	−1.07654	−0.538271	−	0.842772i	$-0.680922\pi$
$83$	−9.80776	−1.07654	−0.538271	+	0.842772i	$0.680922\pi$
$84$	8.24621	0.899735
$85$	−4.56155	−0.494770
$86$	0	0
$87$	−5.56155	−0.596261
$88$	0	0
$89$	14.8078	1.56962	0.784810	−	0.619737i	$-0.212761\pi$
$89$	14.8078	1.56962	0.784810	+	0.619737i	$0.212761\pi$
$90$	0	0
$91$	−4.12311	−0.432219
$92$	9.12311	0.951150
$93$	1.00000	0.103695
$94$	0	0
$95$	2.00000	0.205196
$96$	0	0
$97$	−4.12311	−0.418638	−0.209319	−	0.977847i	$-0.567125\pi$
$97$	−4.12311	−0.418638	−0.209319	+	0.977847i	$0.567125\pi$
$98$	0	0
$99$	−4.56155	−0.458453
$100$	−2.00000	−0.200000
$101$	19.3693	1.92732	0.963660	−	0.267133i	$-0.0860765\pi$
$101$	19.3693	1.92732	0.963660	+	0.267133i	$0.0860765\pi$
$102$	0	0
$103$	−6.24621	−0.615457	−0.307729	−	0.951474i	$-0.599569\pi$
$103$	−6.24621	−0.615457	−0.307729	+	0.951474i	$0.599569\pi$
$104$	0	0
$105$	4.12311	0.402374
$106$	0	0
$107$	15.4924	1.49771	0.748855	−	0.662734i	$-0.230604\pi$
$107$	15.4924	1.49771	0.748855	+	0.662734i	$0.230604\pi$
$108$	2.00000	0.192450
$109$	7.36932	0.705853	0.352926	−	0.935651i	$-0.385187\pi$
$109$	7.36932	0.705853	0.352926	+	0.935651i	$0.385187\pi$
$110$	0	0
$111$	9.68466	0.919227
$112$	16.4924	1.55839
$113$	12.4384	1.17011	0.585055	−	0.810993i	$-0.301073\pi$
$113$	12.4384	1.17011	0.585055	+	0.810993i	$0.301073\pi$
$114$	0	0
$115$	4.56155	0.425367
$116$	−11.1231	−1.03275
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	18.8078	1.72410
$120$	0	0
$121$	9.80776	0.891615
$122$	0	0
$123$	5.00000	0.450835
$124$	2.00000	0.179605
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−16.6847	−1.48052	−0.740262	−	0.672318i	$-0.765299\pi$
$127$	−16.6847	−1.48052	−0.740262	+	0.672318i	$0.765299\pi$
$128$	0	0
$129$	−3.56155	−0.313577
$130$	0	0
$131$	−13.1231	−1.14657	−0.573286	−	0.819356i	$-0.694331\pi$
$131$	−13.1231	−1.14657	−0.573286	+	0.819356i	$0.694331\pi$
$132$	−9.12311	−0.794064
$133$	−8.24621	−0.715037
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	3.12311	0.266825	0.133412	−	0.991061i	$-0.457406\pi$
$137$	3.12311	0.266825	0.133412	+	0.991061i	$0.457406\pi$
$138$	0	0
$139$	−2.31534	−0.196385	−0.0981924	−	0.995167i	$-0.531306\pi$
$139$	−2.31534	−0.196385	−0.0981924	+	0.995167i	$0.531306\pi$
$140$	8.24621	0.696932
$141$	−11.1231	−0.936734
$142$	0	0
$143$	4.56155	0.381456
$144$	4.00000	0.333333
$145$	−5.56155	−0.461862
$146$	0	0
$147$	−10.0000	−0.824786
$148$	19.3693	1.59215
$149$	12.1231	0.993164	0.496582	−	0.867990i	$-0.334588\pi$
$149$	12.1231	0.993164	0.496582	+	0.867990i	$0.334588\pi$
$150$	0	0
$151$	−10.9309	−0.889542	−0.444771	−	0.895644i	$-0.646715\pi$
$151$	−10.9309	−0.889542	−0.444771	+	0.895644i	$0.646715\pi$
$152$	0	0
$153$	4.56155	0.368780
$154$	0	0
$155$	1.00000	0.0803219
$156$	−2.00000	−0.160128
$157$	−7.12311	−0.568486	−0.284243	−	0.958752i	$-0.591742\pi$
$157$	−7.12311	−0.568486	−0.284243	+	0.958752i	$0.591742\pi$
$158$	0	0
$159$	−1.43845	−0.114076
$160$	0	0
$161$	−18.8078	−1.48226
$162$	0	0
$163$	−19.6847	−1.54182	−0.770911	−	0.636943i	$-0.780199\pi$
$163$	−19.6847	−1.54182	−0.770911	+	0.636943i	$0.780199\pi$
$164$	10.0000	0.780869
$165$	−4.56155	−0.355116
$166$	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
$172$	−7.12311	−0.543132
$173$	−4.43845	−0.337449	−0.168724	−	0.985663i	$-0.553965\pi$
$173$	−4.43845	−0.337449	−0.168724	+	0.985663i	$0.553965\pi$
$174$	0	0
$175$	4.12311	0.311677
$176$	−18.2462	−1.37536
$177$	4.68466	0.352120
$178$	0	0
$179$	7.80776	0.583580	0.291790	−	0.956482i	$-0.405749\pi$
$179$	7.80776	0.583580	0.291790	+	0.956482i	$0.405749\pi$
$180$	2.00000	0.149071
$181$	−16.8078	−1.24931	−0.624656	−	0.780900i	$-0.714761\pi$
$181$	−16.8078	−1.24931	−0.624656	+	0.780900i	$0.714761\pi$
$182$	0	0
$183$	6.56155	0.485044
$184$	0	0
$185$	9.68466	0.712030
$186$	0	0
$187$	−20.8078	−1.52161
$188$	−22.2462	−1.62247
$189$	−4.12311	−0.299912
$190$	0	0
$191$	0.246211	0.0178152	0.00890761	−	0.999960i	$-0.497165\pi$
$191$	0.246211	0.0178152	0.00890761	+	0.999960i	$0.497165\pi$
$192$	8.00000	0.577350
$193$	13.8769	0.998881	0.499440	−	0.866348i	$-0.333539\pi$
$193$	13.8769	0.998881	0.499440	+	0.866348i	$0.333539\pi$
$194$	0	0
$195$	−1.00000	−0.0716115
$196$	−20.0000	−1.42857
$197$	−11.8078	−0.841268	−0.420634	−	0.907230i	$-0.638192\pi$
$197$	−11.8078	−0.841268	−0.420634	+	0.907230i	$0.638192\pi$
$198$	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
$201$	−6.43845	−0.454133
$202$	0	0
$203$	22.9309	1.60943
$204$	9.12311	0.638745
$205$	5.00000	0.349215
$206$	0	0
$207$	−4.56155	−0.317050
$208$	−4.00000	−0.277350
$209$	9.12311	0.631058
$210$	0	0
$211$	−9.56155	−0.658244	−0.329122	−	0.944287i	$-0.606753\pi$
$211$	−9.56155	−0.658244	−0.329122	+	0.944287i	$0.606753\pi$
$212$	−2.87689	−0.197586
$213$	−11.6847	−0.800620
$214$	0	0
$215$	−3.56155	−0.242896
$216$	0	0
$217$	−4.12311	−0.279895
$218$	0	0
$219$	11.3693	0.768267
$220$	−9.12311	−0.615080
$221$	−4.56155	−0.306843
$222$	0	0
$223$	20.4924	1.37227	0.686137	−	0.727472i	$-0.259305\pi$
$223$	20.4924	1.37227	0.686137	+	0.727472i	$0.259305\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	14.4924	0.961896	0.480948	−	0.876749i	$-0.340293\pi$
$227$	14.4924	0.961896	0.480948	+	0.876749i	$0.340293\pi$
$228$	−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
$231$	18.8078	1.23746
$232$	0	0
$233$	−28.6155	−1.87467	−0.937333	−	0.348435i	$-0.886713\pi$
$233$	−28.6155	−1.87467	−0.937333	+	0.348435i	$0.886713\pi$
$234$	0	0
$235$	−11.1231	−0.725591
$236$	9.36932	0.609891
$237$	4.80776	0.312298
$238$	0	0
$239$	18.5616	1.20065	0.600324	−	0.799757i	$-0.295038\pi$
$239$	18.5616	1.20065	0.600324	+	0.799757i	$0.295038\pi$
$240$	4.00000	0.258199
$241$	8.93087	0.575288	0.287644	−	0.957737i	$-0.407128\pi$
$241$	8.93087	0.575288	0.287644	+	0.957737i	$0.407128\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	13.1231	0.840121
$245$	−10.0000	−0.638877
$246$	0	0
$247$	2.00000	0.127257
$248$	0	0
$249$	9.80776	0.621542
$250$	0	0
$251$	−21.5616	−1.36095	−0.680477	−	0.732770i	$-0.738227\pi$
$251$	−21.5616	−1.36095	−0.680477	+	0.732770i	$0.738227\pi$
$252$	−8.24621	−0.519462
$253$	20.8078	1.30817
$254$	0	0
$255$	4.56155	0.285656
$256$	16.0000	1.00000
$257$	−18.4924	−1.15353	−0.576763	−	0.816912i	$-0.695684\pi$
$257$	−18.4924	−1.15353	−0.576763	+	0.816912i	$0.695684\pi$
$258$	0	0
$259$	−39.9309	−2.48118
$260$	−2.00000	−0.124035
$261$	5.56155	0.344251
$262$	0	0
$263$	−20.8769	−1.28732	−0.643662	−	0.765310i	$-0.722586\pi$
$263$	−20.8769	−1.28732	−0.643662	+	0.765310i	$0.722586\pi$
$264$	0	0
$265$	−1.43845	−0.0883631
$266$	0	0
$267$	−14.8078	−0.906220
$268$	−12.8769	−0.786582
$269$	−19.6155	−1.19598	−0.597990	−	0.801504i	$-0.704034\pi$
$269$	−19.6155	−1.19598	−0.597990	+	0.801504i	$0.704034\pi$
$270$	0	0
$271$	28.6847	1.74247	0.871234	−	0.490867i	$-0.163320\pi$
$271$	28.6847	1.74247	0.871234	+	0.490867i	$0.163320\pi$
$272$	18.2462	1.10634
$273$	4.12311	0.249542
$274$	0	0
$275$	−4.56155	−0.275072
$276$	−9.12311	−0.549146
$277$	−4.68466	−0.281474	−0.140737	−	0.990047i	$-0.544947\pi$
$277$	−4.68466	−0.281474	−0.140737	+	0.990047i	$0.544947\pi$
$278$	0	0
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	−12.4384	−0.742016	−0.371008	−	0.928630i	$-0.620988\pi$
$281$	−12.4384	−0.742016	−0.371008	+	0.928630i	$0.620988\pi$
$282$	0	0
$283$	−28.4924	−1.69370	−0.846849	−	0.531833i	$-0.821503\pi$
$283$	−28.4924	−1.69370	−0.846849	+	0.531833i	$0.821503\pi$
$284$	−23.3693	−1.38671
$285$	−2.00000	−0.118470
$286$	0	0
$287$	−20.6155	−1.21690
$288$	0	0
$289$	3.80776	0.223986
$290$	0	0
$291$	4.12311	0.241701
$292$	22.7386	1.33068
$293$	−11.6155	−0.678586	−0.339293	−	0.940681i	$-0.610188\pi$
$293$	−11.6155	−0.678586	−0.339293	+	0.940681i	$0.610188\pi$
$294$	0	0
$295$	4.68466	0.272751
$296$	0	0
$297$	4.56155	0.264688
$298$	0	0
$299$	4.56155	0.263801
$300$	2.00000	0.115470
$301$	14.6847	0.846410
$302$	0	0
$303$	−19.3693	−1.11274
$304$	−8.00000	−0.458831
$305$	6.56155	0.375713
$306$	0	0
$307$	2.56155	0.146196	0.0730978	−	0.997325i	$-0.476711\pi$
$307$	2.56155	0.146196	0.0730978	+	0.997325i	$0.476711\pi$
$308$	37.6155	2.14334
$309$	6.24621	0.355335
$310$	0	0
$311$	4.49242	0.254742	0.127371	−	0.991855i	$-0.459346\pi$
$311$	4.49242	0.254742	0.127371	+	0.991855i	$0.459346\pi$
$312$	0	0
$313$	16.0540	0.907424	0.453712	−	0.891148i	$-0.350099\pi$
$313$	16.0540	0.907424	0.453712	+	0.891148i	$0.350099\pi$
$314$	0	0
$315$	−4.12311	−0.232311
$316$	9.61553	0.540916
$317$	−25.3693	−1.42488	−0.712441	−	0.701732i	$-0.752411\pi$
$317$	−25.3693	−1.42488	−0.712441	+	0.701732i	$0.752411\pi$
$318$	0	0
$319$	−25.3693	−1.42041
$320$	8.00000	0.447214
$321$	−15.4924	−0.864703
$322$	0	0
$323$	−9.12311	−0.507623
$324$	−2.00000	−0.111111
$325$	−1.00000	−0.0554700
$326$	0	0
$327$	−7.36932	−0.407524
$328$	0	0
$329$	45.8617	2.52844
$330$	0	0
$331$	−10.4384	−0.573749	−0.286874	−	0.957968i	$-0.592616\pi$
$331$	−10.4384	−0.573749	−0.286874	+	0.957968i	$0.592616\pi$
$332$	19.6155	1.07654
$333$	−9.68466	−0.530716
$334$	0	0
$335$	−6.43845	−0.351770
$336$	−16.4924	−0.899735
$337$	−21.3693	−1.16406	−0.582030	−	0.813167i	$-0.697742\pi$
$337$	−21.3693	−1.16406	−0.582030	+	0.813167i	$0.697742\pi$
$338$	0	0
$339$	−12.4384	−0.675564
$340$	9.12311	0.494770
$341$	4.56155	0.247022
$342$	0	0
$343$	12.3693	0.667880
$344$	0	0
$345$	−4.56155	−0.245586
$346$	0	0
$347$	−10.8078	−0.580191	−0.290096	−	0.956998i	$-0.593687\pi$
$347$	−10.8078	−0.580191	−0.290096	+	0.956998i	$0.593687\pi$
$348$	11.1231	0.596261
$349$	20.2462	1.08375	0.541877	−	0.840458i	$-0.317714\pi$
$349$	20.2462	1.08375	0.541877	+	0.840458i	$0.317714\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	−10.0540	−0.535119	−0.267560	−	0.963541i	$-0.586217\pi$
$353$	−10.0540	−0.535119	−0.267560	+	0.963541i	$0.586217\pi$
$354$	0	0
$355$	−11.6847	−0.620157
$356$	−29.6155	−1.56962
$357$	−18.8078	−0.995412
$358$	0	0
$359$	10.4384	0.550920	0.275460	−	0.961313i	$-0.411170\pi$
$359$	10.4384	0.550920	0.275460	+	0.961313i	$0.411170\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	−9.80776	−0.514774
$364$	8.24621	0.432219
$365$	11.3693	0.595097
$366$	0	0
$367$	−10.8769	−0.567769	−0.283885	−	0.958858i	$-0.591623\pi$
$367$	−10.8769	−0.567769	−0.283885	+	0.958858i	$0.591623\pi$
$368$	−18.2462	−0.951150
$369$	−5.00000	−0.260290
$370$	0	0
$371$	5.93087	0.307915
$372$	−2.00000	−0.103695
$373$	8.24621	0.426973	0.213486	−	0.976946i	$-0.431518\pi$
$373$	8.24621	0.426973	0.213486	+	0.976946i	$0.431518\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	−5.56155	−0.286435
$378$	0	0
$379$	−21.1231	−1.08502	−0.542511	−	0.840049i	$-0.682526\pi$
$379$	−21.1231	−1.08502	−0.542511	+	0.840049i	$0.682526\pi$
$380$	−4.00000	−0.205196
$381$	16.6847	0.854781
$382$	0	0
$383$	−4.19224	−0.214213	−0.107107	−	0.994248i	$-0.534159\pi$
$383$	−4.19224	−0.214213	−0.107107	+	0.994248i	$0.534159\pi$
$384$	0	0
$385$	18.8078	0.958532
$386$	0	0
$387$	3.56155	0.181044
$388$	8.24621	0.418638
$389$	−22.0540	−1.11818	−0.559090	−	0.829107i	$-0.688850\pi$
$389$	−22.0540	−1.11818	−0.559090	+	0.829107i	$0.688850\pi$
$390$	0	0
$391$	−20.8078	−1.05229
$392$	0	0
$393$	13.1231	0.661973
$394$	0	0
$395$	4.80776	0.241905
$396$	9.12311	0.458453
$397$	−35.3002	−1.77167	−0.885833	−	0.464005i	$-0.846412\pi$
$397$	−35.3002	−1.77167	−0.885833	+	0.464005i	$0.846412\pi$
$398$	0	0
$399$	8.24621	0.412827
$400$	4.00000	0.200000
$401$	1.50758	0.0752848	0.0376424	−	0.999291i	$-0.488015\pi$
$401$	1.50758	0.0752848	0.0376424	+	0.999291i	$0.488015\pi$
$402$	0	0
$403$	1.00000	0.0498135
$404$	−38.7386	−1.92732
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	44.1771	2.18978
$408$	0	0
$409$	−31.5616	−1.56062	−0.780309	−	0.625394i	$-0.784938\pi$
$409$	−31.5616	−1.56062	−0.780309	+	0.625394i	$0.784938\pi$
$410$	0	0
$411$	−3.12311	−0.154051
$412$	12.4924	0.615457
$413$	−19.3153	−0.950446
$414$	0	0
$415$	9.80776	0.481444
$416$	0	0
$417$	2.31534	0.113383
$418$	0	0
$419$	−22.4924	−1.09883	−0.549413	−	0.835551i	$-0.685149\pi$
$419$	−22.4924	−1.09883	−0.549413	+	0.835551i	$0.685149\pi$
$420$	−8.24621	−0.402374
$421$	−10.2462	−0.499370	−0.249685	−	0.968327i	$-0.580327\pi$
$421$	−10.2462	−0.499370	−0.249685	+	0.968327i	$0.580327\pi$
$422$	0	0
$423$	11.1231	0.540824
$424$	0	0
$425$	4.56155	0.221268
$426$	0	0
$427$	−27.0540	−1.30923
$428$	−30.9848	−1.49771
$429$	−4.56155	−0.220234
$430$	0	0
$431$	−18.0540	−0.869629	−0.434815	−	0.900520i	$-0.643186\pi$
$431$	−18.0540	−0.869629	−0.434815	+	0.900520i	$0.643186\pi$
$432$	−4.00000	−0.192450
$433$	20.6847	0.994041	0.497021	−	0.867739i	$-0.334427\pi$
$433$	20.6847	0.994041	0.497021	+	0.867739i	$0.334427\pi$
$434$	0	0
$435$	5.56155	0.266656
$436$	−14.7386	−0.705853
$437$	9.12311	0.436417
$438$	0	0
$439$	−18.3693	−0.876720	−0.438360	−	0.898800i	$-0.644440\pi$
$439$	−18.3693	−0.876720	−0.438360	+	0.898800i	$0.644440\pi$
$440$	0	0
$441$	10.0000	0.476190
$442$	0	0
$443$	−3.63068	−0.172499	−0.0862495	−	0.996274i	$-0.527488\pi$
$443$	−3.63068	−0.172499	−0.0862495	+	0.996274i	$0.527488\pi$
$444$	−19.3693	−0.919227
$445$	−14.8078	−0.701955
$446$	0	0
$447$	−12.1231	−0.573403
$448$	−32.9848	−1.55839
$449$	2.56155	0.120887	0.0604436	−	0.998172i	$-0.480748\pi$
$449$	2.56155	0.120887	0.0604436	+	0.998172i	$0.480748\pi$
$450$	0	0
$451$	22.8078	1.07398
$452$	−24.8769	−1.17011
$453$	10.9309	0.513577
$454$	0	0
$455$	4.12311	0.193294
$456$	0	0
$457$	34.1771	1.59874	0.799368	−	0.600841i	$-0.205168\pi$
$457$	34.1771	1.59874	0.799368	+	0.600841i	$0.205168\pi$
$458$	0	0
$459$	−4.56155	−0.212915
$460$	−9.12311	−0.425367
$461$	14.5616	0.678199	0.339100	−	0.940750i	$-0.389878\pi$
$461$	14.5616	0.678199	0.339100	+	0.940750i	$0.389878\pi$
$462$	0	0
$463$	21.4384	0.996329	0.498165	−	0.867083i	$-0.334008\pi$
$463$	21.4384	0.996329	0.498165	+	0.867083i	$0.334008\pi$
$464$	22.2462	1.03275
$465$	−1.00000	−0.0463739
$466$	0	0
$467$	29.7386	1.37614	0.688070	−	0.725644i	$-0.258458\pi$
$467$	29.7386	1.37614	0.688070	+	0.725644i	$0.258458\pi$
$468$	2.00000	0.0924500
$469$	26.5464	1.22580
$470$	0	0
$471$	7.12311	0.328215
$472$	0	0
$473$	−16.2462	−0.747002
$474$	0	0
$475$	−2.00000	−0.0917663
$476$	−37.6155	−1.72410
$477$	1.43845	0.0658620
$478$	0	0
$479$	−24.8617	−1.13596	−0.567981	−	0.823042i	$-0.692275\pi$
$479$	−24.8617	−1.13596	−0.567981	+	0.823042i	$0.692275\pi$
$480$	0	0
$481$	9.68466	0.441582
$482$	0	0
$483$	18.8078	0.855783
$484$	−19.6155	−0.891615
$485$	4.12311	0.187221
$486$	0	0
$487$	−30.8078	−1.39603	−0.698017	−	0.716082i	$-0.745934\pi$
$487$	−30.8078	−1.39603	−0.698017	+	0.716082i	$0.745934\pi$
$488$	0	0
$489$	19.6847	0.890171
$490$	0	0
$491$	−33.1771	−1.49726	−0.748630	−	0.662988i	$-0.769288\pi$
$491$	−33.1771	−1.49726	−0.748630	+	0.662988i	$0.769288\pi$
$492$	−10.0000	−0.450835
$493$	25.3693	1.14258
$494$	0	0
$495$	4.56155	0.205027
$496$	−4.00000	−0.179605
$497$	48.1771	2.16104
$498$	0	0
$499$	−4.68466	−0.209714	−0.104857	−	0.994487i	$-0.533439\pi$
$499$	−4.68466	−0.209714	−0.104857	+	0.994487i	$0.533439\pi$
$500$	2.00000	0.0894427
$501$	16.0000	0.714827
$502$	0	0
$503$	22.4384	1.00048	0.500240	−	0.865887i	$-0.333245\pi$
$503$	22.4384	1.00048	0.500240	+	0.865887i	$0.333245\pi$
$504$	0	0
$505$	−19.3693	−0.861923
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	33.3693	1.48052
$509$	1.43845	0.0637581	0.0318790	−	0.999492i	$-0.489851\pi$
$509$	1.43845	0.0637581	0.0318790	+	0.999492i	$0.489851\pi$
$510$	0	0
$511$	−46.8769	−2.07371
$512$	0	0
$513$	2.00000	0.0883022
$514$	0	0
$515$	6.24621	0.275241
$516$	7.12311	0.313577
$517$	−50.7386	−2.23148
$518$	0	0
$519$	4.43845	0.194826
$520$	0	0
$521$	−16.6307	−0.728603	−0.364302	−	0.931281i	$-0.618692\pi$
$521$	−16.6307	−0.728603	−0.364302	+	0.931281i	$0.618692\pi$
$522$	0	0
$523$	22.1922	0.970399	0.485199	−	0.874404i	$-0.338747\pi$
$523$	22.1922	0.970399	0.485199	+	0.874404i	$0.338747\pi$
$524$	26.2462	1.14657
$525$	−4.12311	−0.179947
$526$	0	0
$527$	−4.56155	−0.198704
$528$	18.2462	0.794064
$529$	−2.19224	−0.0953146
$530$	0	0
$531$	−4.68466	−0.203297
$532$	16.4924	0.715037
$533$	5.00000	0.216574
$534$	0	0
$535$	−15.4924	−0.669796
$536$	0	0
$537$	−7.80776	−0.336930
$538$	0	0
$539$	−45.6155	−1.96480
$540$	−2.00000	−0.0860663
$541$	−27.1231	−1.16611	−0.583057	−	0.812431i	$-0.698143\pi$
$541$	−27.1231	−1.16611	−0.583057	+	0.812431i	$0.698143\pi$
$542$	0	0
$543$	16.8078	0.721290
$544$	0	0
$545$	−7.36932	−0.315667
$546$	0	0
$547$	−4.00000	−0.171028	−0.0855138	−	0.996337i	$-0.527253\pi$
$547$	−4.00000	−0.171028	−0.0855138	+	0.996337i	$0.527253\pi$
$548$	−6.24621	−0.266825
$549$	−6.56155	−0.280040
$550$	0	0
$551$	−11.1231	−0.473860
$552$	0	0
$553$	−19.8229	−0.842956
$554$	0	0
$555$	−9.68466	−0.411091
$556$	4.63068	0.196385
$557$	−29.3693	−1.24442	−0.622209	−	0.782851i	$-0.713765\pi$
$557$	−29.3693	−1.24442	−0.622209	+	0.782851i	$0.713765\pi$
$558$	0	0
$559$	−3.56155	−0.150638
$560$	−16.4924	−0.696932
$561$	20.8078	0.878504
$562$	0	0
$563$	29.9309	1.26144	0.630718	−	0.776012i	$-0.282761\pi$
$563$	29.9309	1.26144	0.630718	+	0.776012i	$0.282761\pi$
$564$	22.2462	0.936734
$565$	−12.4384	−0.523289
$566$	0	0
$567$	4.12311	0.173154
$568$	0	0
$569$	−12.2462	−0.513388	−0.256694	−	0.966493i	$-0.582633\pi$
$569$	−12.2462	−0.513388	−0.256694	+	0.966493i	$0.582633\pi$
$570$	0	0
$571$	−26.5616	−1.11157	−0.555783	−	0.831327i	$-0.687582\pi$
$571$	−26.5616	−1.11157	−0.555783	+	0.831327i	$0.687582\pi$
$572$	−9.12311	−0.381456
$573$	−0.246211	−0.0102856
$574$	0	0
$575$	−4.56155	−0.190230
$576$	−8.00000	−0.333333
$577$	2.80776	0.116889	0.0584444	−	0.998291i	$-0.481386\pi$
$577$	2.80776	0.116889	0.0584444	+	0.998291i	$0.481386\pi$
$578$	0	0
$579$	−13.8769	−0.576704
$580$	11.1231	0.461862
$581$	−40.4384	−1.67767
$582$	0	0
$583$	−6.56155	−0.271752
$584$	0	0
$585$	1.00000	0.0413449
$586$	0	0
$587$	10.4384	0.430841	0.215420	−	0.976521i	$-0.430888\pi$
$587$	10.4384	0.430841	0.215420	+	0.976521i	$0.430888\pi$
$588$	20.0000	0.824786
$589$	2.00000	0.0824086
$590$	0	0
$591$	11.8078	0.485707
$592$	−38.7386	−1.59215
$593$	15.1231	0.621032	0.310516	−	0.950568i	$-0.399498\pi$
$593$	15.1231	0.621032	0.310516	+	0.950568i	$0.399498\pi$
$594$	0	0
$595$	−18.8078	−0.771043
$596$	−24.2462	−0.993164
$597$	24.0000	0.982255
$598$	0	0
$599$	−43.2311	−1.76637	−0.883187	−	0.469022i	$-0.844607\pi$
$599$	−43.2311	−1.76637	−0.883187	+	0.469022i	$0.844607\pi$
$600$	0	0
$601$	−2.31534	−0.0944448	−0.0472224	−	0.998884i	$-0.515037\pi$
$601$	−2.31534	−0.0944448	−0.0472224	+	0.998884i	$0.515037\pi$
$602$	0	0
$603$	6.43845	0.262194
$604$	21.8617	0.889542
$605$	−9.80776	−0.398742
$606$	0	0
$607$	45.2311	1.83587	0.917936	−	0.396729i	$-0.129855\pi$
$607$	45.2311	1.83587	0.917936	+	0.396729i	$0.129855\pi$
$608$	0	0
$609$	−22.9309	−0.929206
$610$	0	0
$611$	−11.1231	−0.449993
$612$	−9.12311	−0.368780
$613$	−21.0540	−0.850362	−0.425181	−	0.905108i	$-0.639790\pi$
$613$	−21.0540	−0.850362	−0.425181	+	0.905108i	$0.639790\pi$
$614$	0	0
$615$	−5.00000	−0.201619
$616$	0	0
$617$	24.4924	0.986028	0.493014	−	0.870021i	$-0.335895\pi$
$617$	24.4924	0.986028	0.493014	+	0.870021i	$0.335895\pi$
$618$	0	0
$619$	−11.1771	−0.449245	−0.224622	−	0.974446i	$-0.572115\pi$
$619$	−11.1771	−0.449245	−0.224622	+	0.974446i	$0.572115\pi$
$620$	−2.00000	−0.0803219
$621$	4.56155	0.183049
$622$	0	0
$623$	61.0540	2.44608
$624$	4.00000	0.160128
$625$	1.00000	0.0400000
$626$	0	0
$627$	−9.12311	−0.364342
$628$	14.2462	0.568486
$629$	−44.1771	−1.76146
$630$	0	0
$631$	26.2462	1.04485	0.522423	−	0.852687i	$-0.325028\pi$
$631$	26.2462	1.04485	0.522423	+	0.852687i	$0.325028\pi$
$632$	0	0
$633$	9.56155	0.380038
$634$	0	0
$635$	16.6847	0.662110
$636$	2.87689	0.114076
$637$	−10.0000	−0.396214
$638$	0	0
$639$	11.6847	0.462238
$640$	0	0
$641$	1.31534	0.0519529	0.0259764	−	0.999663i	$-0.491731\pi$
$641$	1.31534	0.0519529	0.0259764	+	0.999663i	$0.491731\pi$
$642$	0	0
$643$	−14.0691	−0.554832	−0.277416	−	0.960750i	$-0.589478\pi$
$643$	−14.0691	−0.554832	−0.277416	+	0.960750i	$0.589478\pi$
$644$	37.6155	1.48226
$645$	3.56155	0.140236
$646$	0	0
$647$	−17.1922	−0.675897	−0.337948	−	0.941165i	$-0.609733\pi$
$647$	−17.1922	−0.675897	−0.337948	+	0.941165i	$0.609733\pi$
$648$	0	0
$649$	21.3693	0.838819
$650$	0	0
$651$	4.12311	0.161597
$652$	39.3693	1.54182
$653$	25.3153	0.990666	0.495333	−	0.868703i	$-0.335046\pi$
$653$	25.3153	0.990666	0.495333	+	0.868703i	$0.335046\pi$
$654$	0	0
$655$	13.1231	0.512762
$656$	−20.0000	−0.780869
$657$	−11.3693	−0.443559
$658$	0	0
$659$	−28.2462	−1.10032	−0.550158	−	0.835061i	$-0.685433\pi$
$659$	−28.2462	−1.10032	−0.550158	+	0.835061i	$0.685433\pi$
$660$	9.12311	0.355116
$661$	−23.1231	−0.899385	−0.449692	−	0.893184i	$-0.648466\pi$
$661$	−23.1231	−0.899385	−0.449692	+	0.893184i	$0.648466\pi$
$662$	0	0
$663$	4.56155	0.177156
$664$	0	0
$665$	8.24621	0.319774
$666$	0	0
$667$	−25.3693	−0.982304
$668$	32.0000	1.23812
$669$	−20.4924	−0.792283
$670$	0	0
$671$	29.9309	1.15547
$672$	0	0
$673$	2.43845	0.0939952	0.0469976	−	0.998895i	$-0.485035\pi$
$673$	2.43845	0.0939952	0.0469976	+	0.998895i	$0.485035\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	−2.00000	−0.0769231
$677$	8.56155	0.329047	0.164524	−	0.986373i	$-0.447391\pi$
$677$	8.56155	0.329047	0.164524	+	0.986373i	$0.447391\pi$
$678$	0	0
$679$	−17.0000	−0.652400
$680$	0	0
$681$	−14.4924	−0.555351
$682$	0	0
$683$	−4.38447	−0.167767	−0.0838836	−	0.996476i	$-0.526732\pi$
$683$	−4.38447	−0.167767	−0.0838836	+	0.996476i	$0.526732\pi$
$684$	4.00000	0.152944
$685$	−3.12311	−0.119328
$686$	0	0
$687$	−14.0000	−0.534133
$688$	14.2462	0.543132
$689$	−1.43845	−0.0548005
$690$	0	0
$691$	12.2462	0.465868	0.232934	−	0.972493i	$-0.425167\pi$
$691$	12.2462	0.465868	0.232934	+	0.972493i	$0.425167\pi$
$692$	8.87689	0.337449
$693$	−18.8078	−0.714448
$694$	0	0
$695$	2.31534	0.0878259
$696$	0	0
$697$	−22.8078	−0.863906
$698$	0	0
$699$	28.6155	1.08234
$700$	−8.24621	−0.311677
$701$	8.38447	0.316677	0.158339	−	0.987385i	$-0.449386\pi$
$701$	8.38447	0.316677	0.158339	+	0.987385i	$0.449386\pi$
$702$	0	0
$703$	19.3693	0.730528
$704$	36.4924	1.37536
$705$	11.1231	0.418920
$706$	0	0
$707$	79.8617	3.00351
$708$	−9.36932	−0.352120
$709$	17.5076	0.657511	0.328755	−	0.944415i	$-0.393371\pi$
$709$	17.5076	0.657511	0.328755	+	0.944415i	$0.393371\pi$
$710$	0	0
$711$	−4.80776	−0.180305
$712$	0	0
$713$	4.56155	0.170831
$714$	0	0
$715$	−4.56155	−0.170592
$716$	−15.6155	−0.583580
$717$	−18.5616	−0.693194
$718$	0	0
$719$	16.0540	0.598712	0.299356	−	0.954141i	$-0.403228\pi$
$719$	16.0540	0.598712	0.299356	+	0.954141i	$0.403228\pi$
$720$	−4.00000	−0.149071
$721$	−25.7538	−0.959121
$722$	0	0
$723$	−8.93087	−0.332143
$724$	33.6155	1.24931
$725$	5.56155	0.206551
$726$	0	0
$727$	−32.8769	−1.21934	−0.609668	−	0.792657i	$-0.708697\pi$
$727$	−32.8769	−1.21934	−0.609668	+	0.792657i	$0.708697\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	16.2462	0.600888
$732$	−13.1231	−0.485044
$733$	−2.12311	−0.0784187	−0.0392093	−	0.999231i	$-0.512484\pi$
$733$	−2.12311	−0.0784187	−0.0392093	+	0.999231i	$0.512484\pi$
$734$	0	0
$735$	10.0000	0.368856
$736$	0	0
$737$	−29.3693	−1.08183
$738$	0	0
$739$	24.4924	0.900968	0.450484	−	0.892784i	$-0.351251\pi$
$739$	24.4924	0.900968	0.450484	+	0.892784i	$0.351251\pi$
$740$	−19.3693	−0.712030
$741$	−2.00000	−0.0734718
$742$	0	0
$743$	26.4384	0.969933	0.484966	−	0.874533i	$-0.338832\pi$
$743$	26.4384	0.969933	0.484966	+	0.874533i	$0.338832\pi$
$744$	0	0
$745$	−12.1231	−0.444156
$746$	0	0
$747$	−9.80776	−0.358847
$748$	41.6155	1.52161
$749$	63.8769	2.33401
$750$	0	0
$751$	25.7386	0.939216	0.469608	−	0.882875i	$-0.344395\pi$
$751$	25.7386	0.939216	0.469608	+	0.882875i	$0.344395\pi$
$752$	44.4924	1.62247
$753$	21.5616	0.785747
$754$	0	0
$755$	10.9309	0.397815
$756$	8.24621	0.299912
$757$	32.9309	1.19689	0.598446	−	0.801163i	$-0.295785\pi$
$757$	32.9309	1.19689	0.598446	+	0.801163i	$0.295785\pi$
$758$	0	0
$759$	−20.8078	−0.755274
$760$	0	0
$761$	−3.86174	−0.139988	−0.0699940	−	0.997547i	$-0.522298\pi$
$761$	−3.86174	−0.139988	−0.0699940	+	0.997547i	$0.522298\pi$
$762$	0	0
$763$	30.3845	1.09999
$764$	−0.492423	−0.0178152
$765$	−4.56155	−0.164923
$766$	0	0
$767$	4.68466	0.169153
$768$	−16.0000	−0.577350
$769$	16.4924	0.594732	0.297366	−	0.954764i	$-0.403892\pi$
$769$	16.4924	0.594732	0.297366	+	0.954764i	$0.403892\pi$
$770$	0	0
$771$	18.4924	0.665988
$772$	−27.7538	−0.998881
$773$	14.0540	0.505486	0.252743	−	0.967533i	$-0.418667\pi$
$773$	14.0540	0.505486	0.252743	+	0.967533i	$0.418667\pi$
$774$	0	0
$775$	−1.00000	−0.0359211
$776$	0	0
$777$	39.9309	1.43251
$778$	0	0
$779$	10.0000	0.358287
$780$	2.00000	0.0716115
$781$	−53.3002	−1.90723
$782$	0	0
$783$	−5.56155	−0.198754
$784$	40.0000	1.42857
$785$	7.12311	0.254235
$786$	0	0
$787$	−52.9848	−1.88871	−0.944353	−	0.328934i	$-0.893311\pi$
$787$	−52.9848	−1.88871	−0.944353	+	0.328934i	$0.893311\pi$
$788$	23.6155	0.841268
$789$	20.8769	0.743237
$790$	0	0
$791$	51.2850	1.82349
$792$	0	0
$793$	6.56155	0.233008
$794$	0	0
$795$	1.43845	0.0510165
$796$	48.0000	1.70131
$797$	−7.68466	−0.272205	−0.136102	−	0.990695i	$-0.543458\pi$
$797$	−7.68466	−0.272205	−0.136102	+	0.990695i	$0.543458\pi$
$798$	0	0
$799$	50.7386	1.79500
$800$	0	0
$801$	14.8078	0.523207
$802$	0	0
$803$	51.8617	1.83016
$804$	12.8769	0.454133
$805$	18.8078	0.662887
$806$	0	0
$807$	19.6155	0.690499
$808$	0	0
$809$	41.5616	1.46123	0.730613	−	0.682792i	$-0.239234\pi$
$809$	41.5616	1.46123	0.730613	+	0.682792i	$0.239234\pi$
$810$	0	0
$811$	14.4924	0.508898	0.254449	−	0.967086i	$-0.418106\pi$
$811$	14.4924	0.508898	0.254449	+	0.967086i	$0.418106\pi$
$812$	−45.8617	−1.60943
$813$	−28.6847	−1.00601
$814$	0	0
$815$	19.6847	0.689524
$816$	−18.2462	−0.638745
$817$	−7.12311	−0.249206
$818$	0	0
$819$	−4.12311	−0.144073
$820$	−10.0000	−0.349215
$821$	−16.0691	−0.560817	−0.280408	−	0.959881i	$-0.590470\pi$
$821$	−16.0691	−0.560817	−0.280408	+	0.959881i	$0.590470\pi$
$822$	0	0
$823$	38.0540	1.32648	0.663239	−	0.748408i	$-0.269181\pi$
$823$	38.0540	1.32648	0.663239	+	0.748408i	$0.269181\pi$
$824$	0	0
$825$	4.56155	0.158813
$826$	0	0
$827$	56.3002	1.95775	0.978875	−	0.204461i	$-0.0655442\pi$
$827$	56.3002	1.95775	0.978875	+	0.204461i	$0.0655442\pi$
$828$	9.12311	0.317050
$829$	11.8617	0.411975	0.205988	−	0.978555i	$-0.433959\pi$
$829$	11.8617	0.411975	0.205988	+	0.978555i	$0.433959\pi$
$830$	0	0
$831$	4.68466	0.162509
$832$	8.00000	0.277350
$833$	45.6155	1.58048
$834$	0	0
$835$	16.0000	0.553703
$836$	−18.2462	−0.631058
$837$	1.00000	0.0345651
$838$	0	0
$839$	−11.6307	−0.401536	−0.200768	−	0.979639i	$-0.564344\pi$
$839$	−11.6307	−0.401536	−0.200768	+	0.979639i	$0.564344\pi$
$840$	0	0
$841$	1.93087	0.0665817
$842$	0	0
$843$	12.4384	0.428403
$844$	19.1231	0.658244
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	40.4384	1.38948
$848$	5.75379	0.197586
$849$	28.4924	0.977857
$850$	0	0
$851$	44.1771	1.51437
$852$	23.3693	0.800620
$853$	16.8617	0.577335	0.288667	−	0.957429i	$-0.406788\pi$
$853$	16.8617	0.577335	0.288667	+	0.957429i	$0.406788\pi$
$854$	0	0
$855$	2.00000	0.0683986
$856$	0	0
$857$	33.5464	1.14592	0.572962	−	0.819582i	$-0.305794\pi$
$857$	33.5464	1.14592	0.572962	+	0.819582i	$0.305794\pi$
$858$	0	0
$859$	5.30019	0.180840	0.0904200	−	0.995904i	$-0.471179\pi$
$859$	5.30019	0.180840	0.0904200	+	0.995904i	$0.471179\pi$
$860$	7.12311	0.242896
$861$	20.6155	0.702575
$862$	0	0
$863$	−38.9309	−1.32522	−0.662611	−	0.748964i	$-0.730552\pi$
$863$	−38.9309	−1.32522	−0.662611	+	0.748964i	$0.730552\pi$
$864$	0	0
$865$	4.43845	0.150912
$866$	0	0
$867$	−3.80776	−0.129318
$868$	8.24621	0.279895
$869$	21.9309	0.743954
$870$	0	0
$871$	−6.43845	−0.218158
$872$	0	0
$873$	−4.12311	−0.139546
$874$	0	0
$875$	−4.12311	−0.139386
$876$	−22.7386	−0.768267
$877$	−26.4924	−0.894586	−0.447293	−	0.894388i	$-0.647612\pi$
$877$	−26.4924	−0.894586	−0.447293	+	0.894388i	$0.647612\pi$
$878$	0	0
$879$	11.6155	0.391782
$880$	18.2462	0.615080
$881$	12.6847	0.427357	0.213679	−	0.976904i	$-0.431456\pi$
$881$	12.6847	0.427357	0.213679	+	0.976904i	$0.431456\pi$
$882$	0	0
$883$	−51.6695	−1.73882	−0.869409	−	0.494094i	$-0.835500\pi$
$883$	−51.6695	−1.73882	−0.869409	+	0.494094i	$0.835500\pi$
$884$	9.12311	0.306843
$885$	−4.68466	−0.157473
$886$	0	0
$887$	1.87689	0.0630199	0.0315100	−	0.999503i	$-0.489968\pi$
$887$	1.87689	0.0630199	0.0315100	+	0.999503i	$0.489968\pi$
$888$	0	0
$889$	−68.7926	−2.30723
$890$	0	0
$891$	−4.56155	−0.152818
$892$	−40.9848	−1.37227
$893$	−22.2462	−0.744441
$894$	0	0
$895$	−7.80776	−0.260985
$896$	0	0
$897$	−4.56155	−0.152306
$898$	0	0
$899$	−5.56155	−0.185488
$900$	−2.00000	−0.0666667
$901$	6.56155	0.218597
$902$	0	0
$903$	−14.6847	−0.488675
$904$	0	0
$905$	16.8078	0.558709
$906$	0	0
$907$	40.3542	1.33994	0.669969	−	0.742389i	$-0.266307\pi$
$907$	40.3542	1.33994	0.669969	+	0.742389i	$0.266307\pi$
$908$	−28.9848	−0.961896
$909$	19.3693	0.642440
$910$	0	0
$911$	−11.9460	−0.395789	−0.197895	−	0.980223i	$-0.563410\pi$
$911$	−11.9460	−0.395789	−0.197895	+	0.980223i	$0.563410\pi$
$912$	8.00000	0.264906
$913$	44.7386	1.48063
$914$	0	0
$915$	−6.56155	−0.216918
$916$	−28.0000	−0.925146
$917$	−54.1080	−1.78680
$918$	0	0
$919$	−46.6155	−1.53770	−0.768852	−	0.639427i	$-0.779172\pi$
$919$	−46.6155	−1.53770	−0.768852	+	0.639427i	$0.779172\pi$
$920$	0	0
$921$	−2.56155	−0.0844060
$922$	0	0
$923$	−11.6847	−0.384605
$924$	−37.6155	−1.23746
$925$	−9.68466	−0.318430
$926$	0	0
$927$	−6.24621	−0.205152
$928$	0	0
$929$	2.31534	0.0759639	0.0379819	−	0.999278i	$-0.487907\pi$
$929$	2.31534	0.0759639	0.0379819	+	0.999278i	$0.487907\pi$
$930$	0	0
$931$	−20.0000	−0.655474
$932$	57.2311	1.87467
$933$	−4.49242	−0.147075
$934$	0	0
$935$	20.8078	0.680487
$936$	0	0
$937$	24.7386	0.808176	0.404088	−	0.914720i	$-0.367589\pi$
$937$	24.7386	0.808176	0.404088	+	0.914720i	$0.367589\pi$
$938$	0	0
$939$	−16.0540	−0.523902
$940$	22.2462	0.725591
$941$	−40.8078	−1.33030	−0.665148	−	0.746712i	$-0.731631\pi$
$941$	−40.8078	−1.33030	−0.665148	+	0.746712i	$0.731631\pi$
$942$	0	0
$943$	22.8078	0.742723
$944$	−18.7386	−0.609891
$945$	4.12311	0.134125
$946$	0	0
$947$	−31.8617	−1.03537	−0.517684	−	0.855572i	$-0.673206\pi$
$947$	−31.8617	−1.03537	−0.517684	+	0.855572i	$0.673206\pi$
$948$	−9.61553	−0.312298
$949$	11.3693	0.369064
$950$	0	0
$951$	25.3693	0.822656
$952$	0	0
$953$	−12.4233	−0.402430	−0.201215	−	0.979547i	$-0.564489\pi$
$953$	−12.4233	−0.402430	−0.201215	+	0.979547i	$0.564489\pi$
$954$	0	0
$955$	−0.246211	−0.00796721
$956$	−37.1231	−1.20065
$957$	25.3693	0.820074
$958$	0	0
$959$	12.8769	0.415817
$960$	−8.00000	−0.258199
$961$	1.00000	0.0322581
$962$	0	0
$963$	15.4924	0.499236
$964$	−17.8617	−0.575288
$965$	−13.8769	−0.446713
$966$	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
$969$	9.12311	0.293076
$970$	0	0
$971$	−44.1080	−1.41549	−0.707746	−	0.706467i	$-0.750288\pi$
$971$	−44.1080	−1.41549	−0.707746	+	0.706467i	$0.750288\pi$
$972$	2.00000	0.0641500
$973$	−9.54640	−0.306043
$974$	0	0
$975$	1.00000	0.0320256
$976$	−26.2462	−0.840121
$977$	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
$979$	−67.5464	−2.15879
$980$	20.0000	0.638877
$981$	7.36932	0.235284
$982$	0	0
$983$	−34.3002	−1.09401	−0.547003	−	0.837131i	$-0.684231\pi$
$983$	−34.3002	−1.09401	−0.547003	+	0.837131i	$0.684231\pi$
$984$	0	0
$985$	11.8078	0.376227
$986$	0	0
$987$	−45.8617	−1.45980
$988$	−4.00000	−0.127257
$989$	−16.2462	−0.516599
$990$	0	0
$991$	31.5464	1.00210	0.501052	−	0.865417i	$-0.332946\pi$
$991$	31.5464	1.00210	0.501052	+	0.865417i	$0.332946\pi$
$992$	0	0
$993$	10.4384	0.331254
$994$	0	0
$995$	24.0000	0.760851
$996$	−19.6155	−0.621542
$997$	−10.4924	−0.332298	−0.166149	−	0.986101i	$-0.553133\pi$
$997$	−10.4924	−0.332298	−0.166149	+	0.986101i	$0.553133\pi$
$998$	0	0
$999$	9.68466	0.306409

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6045.2.a.n.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6045.2.a.n.1.2	✓	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -2.00000 q^{4} -1.00000 q^{5} +4.12311 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -2.00000 q^{4} -1.00000 q^{5} +4.12311 q^{7} +1.00000 q^{9} -4.56155 q^{11} +2.00000 q^{12} -1.00000 q^{13} +1.00000 q^{15} +4.00000 q^{16} +4.56155 q^{17} -2.00000 q^{19} +2.00000 q^{20} -4.12311 q^{21} -4.56155 q^{23} +1.00000 q^{25} -1.00000 q^{27} -8.24621 q^{28} +5.56155 q^{29} -1.00000 q^{31} +4.56155 q^{33} -4.12311 q^{35} -2.00000 q^{36} -9.68466 q^{37} +1.00000 q^{39} -5.00000 q^{41} +3.56155 q^{43} +9.12311 q^{44} -1.00000 q^{45} +11.1231 q^{47} -4.00000 q^{48} +10.0000 q^{49} -4.56155 q^{51} +2.00000 q^{52} +1.43845 q^{53} +4.56155 q^{55} +2.00000 q^{57} -4.68466 q^{59} -2.00000 q^{60} -6.56155 q^{61} +4.12311 q^{63} -8.00000 q^{64} +1.00000 q^{65} +6.43845 q^{67} -9.12311 q^{68} +4.56155 q^{69} +11.6847 q^{71} -11.3693 q^{73} -1.00000 q^{75} +4.00000 q^{76} -18.8078 q^{77} -4.80776 q^{79} -4.00000 q^{80} +1.00000 q^{81} -9.80776 q^{83} +8.24621 q^{84} -4.56155 q^{85} -5.56155 q^{87} +14.8078 q^{89} -4.12311 q^{91} +9.12311 q^{92} +1.00000 q^{93} +2.00000 q^{95} -4.12311 q^{97} -4.56155 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 4 q^{4} - 2 q^{5} + 2 q^{9}+O(q^{10}) \) 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

Properties

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