LMFDB - Embedded newform 585.2.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [585,2,Mod(1,585)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(585, 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("585.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 585 = 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	585.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:	$4.67124851824$
Analytic rank:	$0$
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:	$\Z[a_1, a_2]$
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	$\chi$	$=$	585.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.56155 q^{2} +0.438447 q^{4} +1.00000 q^{5} -4.56155 q^{7} +2.43845 q^{8} +O(q^{10})$	\(q-1.56155 q^{2} +0.438447 q^{4} +1.00000 q^{5} -4.56155 q^{7} +2.43845 q^{8} -1.56155 q^{10} +2.56155 q^{11} -1.00000 q^{13} +7.12311 q^{14} -4.68466 q^{16} -2.56155 q^{17} +3.12311 q^{19} +0.438447 q^{20} -4.00000 q^{22} +6.56155 q^{23} +1.00000 q^{25} +1.56155 q^{26} -2.00000 q^{28} +1.12311 q^{29} +6.00000 q^{31} +2.43845 q^{32} +4.00000 q^{34} -4.56155 q^{35} +1.68466 q^{37} -4.87689 q^{38} +2.43845 q^{40} -0.561553 q^{41} -5.12311 q^{43} +1.12311 q^{44} -10.2462 q^{46} +2.87689 q^{47} +13.8078 q^{49} -1.56155 q^{50} -0.438447 q^{52} +7.68466 q^{53} +2.56155 q^{55} -11.1231 q^{56} -1.75379 q^{58} +12.0000 q^{59} +5.68466 q^{61} -9.36932 q^{62} +5.56155 q^{64} -1.00000 q^{65} +13.3693 q^{67} -1.12311 q^{68} +7.12311 q^{70} +14.5616 q^{71} -6.00000 q^{73} -2.63068 q^{74} +1.36932 q^{76} -11.6847 q^{77} -7.68466 q^{79} -4.68466 q^{80} +0.876894 q^{82} +16.4924 q^{83} -2.56155 q^{85} +8.00000 q^{86} +6.24621 q^{88} -1.68466 q^{89} +4.56155 q^{91} +2.87689 q^{92} -4.49242 q^{94} +3.12311 q^{95} -11.9309 q^{97} -21.5616 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} + 5 q^{4} + 2 q^{5} - 5 q^{7} + 9 q^{8}+O(q^{10}) $ 2 * q + q^2 + 5 * q^4 + 2 * q^5 - 5 * q^7 + 9 * q^8	$ 2 q + q^{2} + 5 q^{4} + 2 q^{5} - 5 q^{7} + 9 q^{8} + q^{10} + q^{11} - 2 q^{13} + 6 q^{14} + 3 q^{16} - q^{17} - 2 q^{19} + 5 q^{20} - 8 q^{22} + 9 q^{23} + 2 q^{25} - q^{26} - 4 q^{28} - 6 q^{29} + 12 q^{31} + 9 q^{32} + 8 q^{34} - 5 q^{35} - 9 q^{37} - 18 q^{38} + 9 q^{40} + 3 q^{41} - 2 q^{43} - 6 q^{44} - 4 q^{46} + 14 q^{47} + 7 q^{49} + q^{50} - 5 q^{52} + 3 q^{53} + q^{55} - 14 q^{56} - 20 q^{58} + 24 q^{59} - q^{61} + 6 q^{62} + 7 q^{64} - 2 q^{65} + 2 q^{67} + 6 q^{68} + 6 q^{70} + 25 q^{71} - 12 q^{73} - 30 q^{74} - 22 q^{76} - 11 q^{77} - 3 q^{79} + 3 q^{80} + 10 q^{82} - q^{85} + 16 q^{86} - 4 q^{88} + 9 q^{89} + 5 q^{91} + 14 q^{92} + 24 q^{94} - 2 q^{95} + 5 q^{97} - 39 q^{98}+O(q^{100}) $ 2 * q + q^2 + 5 * q^4 + 2 * q^5 - 5 * q^7 + 9 * q^8 + q^10 + q^11 - 2 * q^13 + 6 * q^14 + 3 * q^16 - q^17 - 2 * q^19 + 5 * q^20 - 8 * q^22 + 9 * q^23 + 2 * q^25 - q^26 - 4 * q^28 - 6 * q^29 + 12 * q^31 + 9 * q^32 + 8 * q^34 - 5 * q^35 - 9 * q^37 - 18 * q^38 + 9 * q^40 + 3 * q^41 - 2 * q^43 - 6 * q^44 - 4 * q^46 + 14 * q^47 + 7 * q^49 + q^50 - 5 * q^52 + 3 * q^53 + q^55 - 14 * q^56 - 20 * q^58 + 24 * q^59 - q^61 + 6 * q^62 + 7 * q^64 - 2 * q^65 + 2 * q^67 + 6 * q^68 + 6 * q^70 + 25 * q^71 - 12 * q^73 - 30 * q^74 - 22 * q^76 - 11 * q^77 - 3 * q^79 + 3 * q^80 + 10 * q^82 - q^85 + 16 * q^86 - 4 * q^88 + 9 * q^89 + 5 * q^91 + 14 * q^92 + 24 * q^94 - 2 * q^95 + 5 * q^97 - 39 * 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 $

$2$	−1.56155	−1.10418	−0.552092	−	0.833783i	$-0.686170\pi$
$2$	−1.56155	−1.10418	−0.552092	+	0.833783i	$0.686170\pi$
$3$	0	0
$3$	0	0
$4$	0.438447	0.219224
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−4.56155	−1.72410	−0.862052	−	0.506819i	$-0.830821\pi$
$7$	−4.56155	−1.72410	−0.862052	+	0.506819i	$0.830821\pi$
$8$	2.43845	0.862121
$9$	0	0
$10$	−1.56155	−0.493806
$11$	2.56155	0.772337	0.386169	−	0.922428i	$-0.373798\pi$
$11$	2.56155	0.772337	0.386169	+	0.922428i	$0.373798\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	7.12311	1.90373
$15$	0	0
$16$	−4.68466	−1.17116
$17$	−2.56155	−0.621268	−0.310634	−	0.950530i	$-0.600541\pi$
$17$	−2.56155	−0.621268	−0.310634	+	0.950530i	$0.600541\pi$
$18$	0	0
$19$	3.12311	0.716490	0.358245	−	0.933628i	$-0.383375\pi$
$19$	3.12311	0.716490	0.358245	+	0.933628i	$0.383375\pi$
$20$	0.438447	0.0980398
$21$	0	0
$22$	−4.00000	−0.852803
$23$	6.56155	1.36818	0.684089	−	0.729398i	$-0.260200\pi$
$23$	6.56155	1.36818	0.684089	+	0.729398i	$0.260200\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	1.56155	0.306246
$27$	0	0
$28$	−2.00000	−0.377964
$29$	1.12311	0.208555	0.104278	−	0.994548i	$-0.466747\pi$
$29$	1.12311	0.208555	0.104278	+	0.994548i	$0.466747\pi$
$30$	0	0
$31$	6.00000	1.07763	0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000	1.07763	0.538816	+	0.842424i	$0.318872\pi$
$32$	2.43845	0.431061
$33$	0	0
$34$	4.00000	0.685994
$35$	−4.56155	−0.771043
$36$	0	0
$37$	1.68466	0.276956	0.138478	−	0.990366i	$-0.455779\pi$
$37$	1.68466	0.276956	0.138478	+	0.990366i	$0.455779\pi$
$38$	−4.87689	−0.791137
$39$	0	0
$40$	2.43845	0.385552
$41$	−0.561553	−0.0876998	−0.0438499	−	0.999038i	$-0.513962\pi$
$41$	−0.561553	−0.0876998	−0.0438499	+	0.999038i	$0.513962\pi$
$42$	0	0
$43$	−5.12311	−0.781266	−0.390633	−	0.920546i	$-0.627744\pi$
$43$	−5.12311	−0.781266	−0.390633	+	0.920546i	$0.627744\pi$
$44$	1.12311	0.169315
$45$	0	0
$46$	−10.2462	−1.51072
$47$	2.87689	0.419638	0.209819	−	0.977740i	$-0.432712\pi$
$47$	2.87689	0.419638	0.209819	+	0.977740i	$0.432712\pi$
$48$	0	0
$49$	13.8078	1.97254
$50$	−1.56155	−0.220837
$51$	0	0
$52$	−0.438447	−0.0608017
$53$	7.68466	1.05557	0.527785	−	0.849378i	$-0.323023\pi$
$53$	7.68466	1.05557	0.527785	+	0.849378i	$0.323023\pi$
$54$	0	0
$55$	2.56155	0.345400
$56$	−11.1231	−1.48639
$57$	0	0
$58$	−1.75379	−0.230284
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	0	0
$61$	5.68466	0.727846	0.363923	−	0.931429i	$-0.381437\pi$
$61$	5.68466	0.727846	0.363923	+	0.931429i	$0.381437\pi$
$62$	−9.36932	−1.18990
$63$	0	0
$64$	5.56155	0.695194
$65$	−1.00000	−0.124035
$66$	0	0
$67$	13.3693	1.63332	0.816661	−	0.577118i	$-0.195823\pi$
$67$	13.3693	1.63332	0.816661	+	0.577118i	$0.195823\pi$
$68$	−1.12311	−0.136197
$69$	0	0
$70$	7.12311	0.851374
$71$	14.5616	1.72814	0.864069	−	0.503373i	$-0.167908\pi$
$71$	14.5616	1.72814	0.864069	+	0.503373i	$0.167908\pi$
$72$	0	0
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	−2.63068	−0.305811
$75$	0	0
$76$	1.36932	0.157071
$77$	−11.6847	−1.33159
$78$	0	0
$79$	−7.68466	−0.864592	−0.432296	−	0.901732i	$-0.642296\pi$
$79$	−7.68466	−0.864592	−0.432296	+	0.901732i	$0.642296\pi$
$80$	−4.68466	−0.523761
$81$	0	0
$82$	0.876894	0.0968368
$83$	16.4924	1.81028	0.905139	−	0.425115i	$-0.139766\pi$
$83$	16.4924	1.81028	0.905139	+	0.425115i	$0.139766\pi$
$84$	0	0
$85$	−2.56155	−0.277839
$86$	8.00000	0.862662
$87$	0	0
$88$	6.24621	0.665848
$89$	−1.68466	−0.178573	−0.0892867	−	0.996006i	$-0.528459\pi$
$89$	−1.68466	−0.178573	−0.0892867	+	0.996006i	$0.528459\pi$
$90$	0	0
$91$	4.56155	0.478181
$92$	2.87689	0.299937
$93$	0	0
$94$	−4.49242	−0.463358
$95$	3.12311	0.320424
$96$	0	0
$97$	−11.9309	−1.21140	−0.605698	−	0.795695i	$-0.707106\pi$
$97$	−11.9309	−1.21140	−0.605698	+	0.795695i	$0.707106\pi$
$98$	−21.5616	−2.17805
$99$	0	0
$100$	0.438447	0.0438447
$101$	−6.24621	−0.621521	−0.310761	−	0.950488i	$-0.600584\pi$
$101$	−6.24621	−0.621521	−0.310761	+	0.950488i	$0.600584\pi$
$102$	0	0
$103$	−6.87689	−0.677601	−0.338800	−	0.940858i	$-0.610021\pi$
$103$	−6.87689	−0.677601	−0.338800	+	0.940858i	$0.610021\pi$
$104$	−2.43845	−0.239109
$105$	0	0
$106$	−12.0000	−1.16554
$107$	−17.9309	−1.73344	−0.866721	−	0.498793i	$-0.833777\pi$
$107$	−17.9309	−1.73344	−0.866721	+	0.498793i	$0.833777\pi$
$108$	0	0
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	−4.00000	−0.381385
$111$	0	0
$112$	21.3693	2.01921
$113$	−13.1231	−1.23452	−0.617259	−	0.786760i	$-0.711757\pi$
$113$	−13.1231	−1.23452	−0.617259	+	0.786760i	$0.711757\pi$
$114$	0	0
$115$	6.56155	0.611868
$116$	0.492423	0.0457203
$117$	0	0
$118$	−18.7386	−1.72503
$119$	11.6847	1.07113
$120$	0	0
$121$	−4.43845	−0.403495
$122$	−8.87689	−0.803676
$123$	0	0
$124$	2.63068	0.236242
$125$	1.00000	0.0894427
$126$	0	0
$127$	18.2462	1.61909	0.809545	−	0.587058i	$-0.199714\pi$
$127$	18.2462	1.61909	0.809545	+	0.587058i	$0.199714\pi$
$128$	−13.5616	−1.19868
$129$	0	0
$130$	1.56155	0.136957
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	−14.2462	−1.23530
$134$	−20.8769	−1.80349
$135$	0	0
$136$	−6.24621	−0.535608
$137$	−7.12311	−0.608568	−0.304284	−	0.952581i	$-0.598417\pi$
$137$	−7.12311	−0.608568	−0.304284	+	0.952581i	$0.598417\pi$
$138$	0	0
$139$	15.6847	1.33036	0.665178	−	0.746685i	$-0.268356\pi$
$139$	15.6847	1.33036	0.665178	+	0.746685i	$0.268356\pi$
$140$	−2.00000	−0.169031
$141$	0	0
$142$	−22.7386	−1.90818
$143$	−2.56155	−0.214208
$144$	0	0
$145$	1.12311	0.0932688
$146$	9.36932	0.775410
$147$	0	0
$148$	0.738634	0.0607153
$149$	2.80776	0.230021	0.115010	−	0.993364i	$-0.463310\pi$
$149$	2.80776	0.230021	0.115010	+	0.993364i	$0.463310\pi$
$150$	0	0
$151$	−13.3693	−1.08798	−0.543990	−	0.839092i	$-0.683087\pi$
$151$	−13.3693	−1.08798	−0.543990	+	0.839092i	$0.683087\pi$
$152$	7.61553	0.617701
$153$	0	0
$154$	18.2462	1.47032
$155$	6.00000	0.481932
$156$	0	0
$157$	−21.3693	−1.70546	−0.852729	−	0.522354i	$-0.825054\pi$
$157$	−21.3693	−1.70546	−0.852729	+	0.522354i	$0.825054\pi$
$158$	12.0000	0.954669
$159$	0	0
$160$	2.43845	0.192776
$161$	−29.9309	−2.35888
$162$	0	0
$163$	−21.0540	−1.64907	−0.824537	−	0.565808i	$-0.808564\pi$
$163$	−21.0540	−1.64907	−0.824537	+	0.565808i	$0.808564\pi$
$164$	−0.246211	−0.0192259
$165$	0	0
$166$	−25.7538	−1.99888
$167$	13.1231	1.01550	0.507748	−	0.861506i	$-0.330478\pi$
$167$	13.1231	1.01550	0.507748	+	0.861506i	$0.330478\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	4.00000	0.306786
$171$	0	0
$172$	−2.24621	−0.171272
$173$	21.1231	1.60596	0.802980	−	0.596006i	$-0.203247\pi$
$173$	21.1231	1.60596	0.802980	+	0.596006i	$0.203247\pi$
$174$	0	0
$175$	−4.56155	−0.344821
$176$	−12.0000	−0.904534
$177$	0	0
$178$	2.63068	0.197178
$179$	−13.1231	−0.980867	−0.490433	−	0.871479i	$-0.663162\pi$
$179$	−13.1231	−0.980867	−0.490433	+	0.871479i	$0.663162\pi$
$180$	0	0
$181$	25.6847	1.90913	0.954563	−	0.298010i	$-0.0963228\pi$
$181$	25.6847	1.90913	0.954563	+	0.298010i	$0.0963228\pi$
$182$	−7.12311	−0.528000
$183$	0	0
$184$	16.0000	1.17954
$185$	1.68466	0.123859
$186$	0	0
$187$	−6.56155	−0.479828
$188$	1.26137	0.0919946
$189$	0	0
$190$	−4.87689	−0.353807
$191$	−21.6155	−1.56404	−0.782022	−	0.623250i	$-0.785812\pi$
$191$	−21.6155	−1.56404	−0.782022	+	0.623250i	$0.785812\pi$
$192$	0	0
$193$	15.4384	1.11128	0.555642	−	0.831422i	$-0.312473\pi$
$193$	15.4384	1.11128	0.555642	+	0.831422i	$0.312473\pi$
$194$	18.6307	1.33761
$195$	0	0
$196$	6.05398	0.432427
$197$	21.3693	1.52250	0.761250	−	0.648458i	$-0.224586\pi$
$197$	21.3693	1.52250	0.761250	+	0.648458i	$0.224586\pi$
$198$	0	0
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	2.43845	0.172424
$201$	0	0
$202$	9.75379	0.686274
$203$	−5.12311	−0.359572
$204$	0	0
$205$	−0.561553	−0.0392205
$206$	10.7386	0.748196
$207$	0	0
$208$	4.68466	0.324823
$209$	8.00000	0.553372
$210$	0	0
$211$	−10.2462	−0.705378	−0.352689	−	0.935741i	$-0.614733\pi$
$211$	−10.2462	−0.705378	−0.352689	+	0.935741i	$0.614733\pi$
$212$	3.36932	0.231406
$213$	0	0
$214$	28.0000	1.91404
$215$	−5.12311	−0.349393
$216$	0	0
$217$	−27.3693	−1.85795
$218$	3.12311	0.211523
$219$	0	0
$220$	1.12311	0.0757198
$221$	2.56155	0.172309
$222$	0	0
$223$	9.36932	0.627416	0.313708	−	0.949520i	$-0.398429\pi$
$223$	9.36932	0.627416	0.313708	+	0.949520i	$0.398429\pi$
$224$	−11.1231	−0.743194
$225$	0	0
$226$	20.4924	1.36314
$227$	22.2462	1.47653	0.738266	−	0.674509i	$-0.235645\pi$
$227$	22.2462	1.47653	0.738266	+	0.674509i	$0.235645\pi$
$228$	0	0
$229$	8.87689	0.586602	0.293301	−	0.956020i	$-0.405246\pi$
$229$	8.87689	0.586602	0.293301	+	0.956020i	$0.405246\pi$
$230$	−10.2462	−0.675615
$231$	0	0
$232$	2.73863	0.179800
$233$	−7.19224	−0.471179	−0.235590	−	0.971853i	$-0.575702\pi$
$233$	−7.19224	−0.471179	−0.235590	+	0.971853i	$0.575702\pi$
$234$	0	0
$235$	2.87689	0.187668
$236$	5.26137	0.342486
$237$	0	0
$238$	−18.2462	−1.18273
$239$	−5.93087	−0.383636	−0.191818	−	0.981431i	$-0.561438\pi$
$239$	−5.93087	−0.383636	−0.191818	+	0.981431i	$0.561438\pi$
$240$	0	0
$241$	−24.7386	−1.59356	−0.796778	−	0.604272i	$-0.793464\pi$
$241$	−24.7386	−1.59356	−0.796778	+	0.604272i	$0.793464\pi$
$242$	6.93087	0.445533
$243$	0	0
$244$	2.49242	0.159561
$245$	13.8078	0.882146
$246$	0	0
$247$	−3.12311	−0.198718
$248$	14.6307	0.929049
$249$	0	0
$250$	−1.56155	−0.0987613
$251$	9.75379	0.615654	0.307827	−	0.951442i	$-0.400398\pi$
$251$	9.75379	0.615654	0.307827	+	0.951442i	$0.400398\pi$
$252$	0	0
$253$	16.8078	1.05670
$254$	−28.4924	−1.78777
$255$	0	0
$256$	10.0540	0.628373
$257$	−21.1231	−1.31762	−0.658812	−	0.752308i	$-0.728941\pi$
$257$	−21.1231	−1.31762	−0.658812	+	0.752308i	$0.728941\pi$
$258$	0	0
$259$	−7.68466	−0.477501
$260$	−0.438447	−0.0271913
$261$	0	0
$262$	0	0
$263$	14.2462	0.878459	0.439230	−	0.898375i	$-0.355251\pi$
$263$	14.2462	0.878459	0.439230	+	0.898375i	$0.355251\pi$
$264$	0	0
$265$	7.68466	0.472065
$266$	22.2462	1.36400
$267$	0	0
$268$	5.86174	0.358063
$269$	−9.12311	−0.556246	−0.278123	−	0.960546i	$-0.589712\pi$
$269$	−9.12311	−0.556246	−0.278123	+	0.960546i	$0.589712\pi$
$270$	0	0
$271$	5.36932	0.326163	0.163081	−	0.986613i	$-0.447857\pi$
$271$	5.36932	0.326163	0.163081	+	0.986613i	$0.447857\pi$
$272$	12.0000	0.727607
$273$	0	0
$274$	11.1231	0.671971
$275$	2.56155	0.154467
$276$	0	0
$277$	4.24621	0.255130	0.127565	−	0.991830i	$-0.459284\pi$
$277$	4.24621	0.255130	0.127565	+	0.991830i	$0.459284\pi$
$278$	−24.4924	−1.46896
$279$	0	0
$280$	−11.1231	−0.664733
$281$	−12.2462	−0.730548	−0.365274	−	0.930900i	$-0.619025\pi$
$281$	−12.2462	−0.730548	−0.365274	+	0.930900i	$0.619025\pi$
$282$	0	0
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	6.38447	0.378849
$285$	0	0
$286$	4.00000	0.236525
$287$	2.56155	0.151204
$288$	0	0
$289$	−10.4384	−0.614026
$290$	−1.75379	−0.102986
$291$	0	0
$292$	−2.63068	−0.153949
$293$	−3.75379	−0.219299	−0.109649	−	0.993970i	$-0.534973\pi$
$293$	−3.75379	−0.219299	−0.109649	+	0.993970i	$0.534973\pi$
$294$	0	0
$295$	12.0000	0.698667
$296$	4.10795	0.238770
$297$	0	0
$298$	−4.38447	−0.253986
$299$	−6.56155	−0.379464
$300$	0	0
$301$	23.3693	1.34699
$302$	20.8769	1.20133
$303$	0	0
$304$	−14.6307	−0.839127
$305$	5.68466	0.325503
$306$	0	0
$307$	3.43845	0.196243	0.0981213	−	0.995174i	$-0.468717\pi$
$307$	3.43845	0.196243	0.0981213	+	0.995174i	$0.468717\pi$
$308$	−5.12311	−0.291916
$309$	0	0
$310$	−9.36932	−0.532141
$311$	27.3693	1.55197	0.775986	−	0.630750i	$-0.217253\pi$
$311$	27.3693	1.55197	0.775986	+	0.630750i	$0.217253\pi$
$312$	0	0
$313$	20.8769	1.18003	0.590016	−	0.807392i	$-0.299121\pi$
$313$	20.8769	1.18003	0.590016	+	0.807392i	$0.299121\pi$
$314$	33.3693	1.88314
$315$	0	0
$316$	−3.36932	−0.189539
$317$	34.4924	1.93729	0.968644	−	0.248454i	$-0.0799224\pi$
$317$	34.4924	1.93729	0.968644	+	0.248454i	$0.0799224\pi$
$318$	0	0
$319$	2.87689	0.161075
$320$	5.56155	0.310900
$321$	0	0
$322$	46.7386	2.60464
$323$	−8.00000	−0.445132
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	32.8769	1.82088
$327$	0	0
$328$	−1.36932	−0.0756079
$329$	−13.1231	−0.723500
$330$	0	0
$331$	20.8769	1.14750	0.573749	−	0.819031i	$-0.305489\pi$
$331$	20.8769	1.14750	0.573749	+	0.819031i	$0.305489\pi$
$332$	7.23106	0.396856
$333$	0	0
$334$	−20.4924	−1.12130
$335$	13.3693	0.730444
$336$	0	0
$337$	2.49242	0.135771	0.0678855	−	0.997693i	$-0.478375\pi$
$337$	2.49242	0.135771	0.0678855	+	0.997693i	$0.478375\pi$
$338$	−1.56155	−0.0849373
$339$	0	0
$340$	−1.12311	−0.0609090
$341$	15.3693	0.832295
$342$	0	0
$343$	−31.0540	−1.67676
$344$	−12.4924	−0.673546
$345$	0	0
$346$	−32.9848	−1.77328
$347$	11.0540	0.593408	0.296704	−	0.954969i	$-0.404112\pi$
$347$	11.0540	0.593408	0.296704	+	0.954969i	$0.404112\pi$
$348$	0	0
$349$	1.36932	0.0732979	0.0366489	−	0.999328i	$-0.488332\pi$
$349$	1.36932	0.0732979	0.0366489	+	0.999328i	$0.488332\pi$
$350$	7.12311	0.380746
$351$	0	0
$352$	6.24621	0.332924
$353$	10.4924	0.558455	0.279228	−	0.960225i	$-0.409922\pi$
$353$	10.4924	0.558455	0.279228	+	0.960225i	$0.409922\pi$
$354$	0	0
$355$	14.5616	0.772847
$356$	−0.738634	−0.0391475
$357$	0	0
$358$	20.4924	1.08306
$359$	2.24621	0.118550	0.0592752	−	0.998242i	$-0.481121\pi$
$359$	2.24621	0.118550	0.0592752	+	0.998242i	$0.481121\pi$
$360$	0	0
$361$	−9.24621	−0.486643
$362$	−40.1080	−2.10803
$363$	0	0
$364$	2.00000	0.104828
$365$	−6.00000	−0.314054
$366$	0	0
$367$	−18.2462	−0.952444	−0.476222	−	0.879325i	$-0.657994\pi$
$367$	−18.2462	−0.952444	−0.476222	+	0.879325i	$0.657994\pi$
$368$	−30.7386	−1.60236
$369$	0	0
$370$	−2.63068	−0.136763
$371$	−35.0540	−1.81991
$372$	0	0
$373$	−4.24621	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$373$	−4.24621	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$374$	10.2462	0.529819
$375$	0	0
$376$	7.01515	0.361779
$377$	−1.12311	−0.0578429
$378$	0	0
$379$	2.49242	0.128027	0.0640136	−	0.997949i	$-0.479610\pi$
$379$	2.49242	0.128027	0.0640136	+	0.997949i	$0.479610\pi$
$380$	1.36932	0.0702445
$381$	0	0
$382$	33.7538	1.72699
$383$	36.4924	1.86468	0.932338	−	0.361588i	$-0.117765\pi$
$383$	36.4924	1.86468	0.932338	+	0.361588i	$0.117765\pi$
$384$	0	0
$385$	−11.6847	−0.595505
$386$	−24.1080	−1.22706
$387$	0	0
$388$	−5.23106	−0.265567
$389$	−19.8617	−1.00703	−0.503515	−	0.863986i	$-0.667960\pi$
$389$	−19.8617	−1.00703	−0.503515	+	0.863986i	$0.667960\pi$
$390$	0	0
$391$	−16.8078	−0.850005
$392$	33.6695	1.70057
$393$	0	0
$394$	−33.3693	−1.68112
$395$	−7.68466	−0.386657
$396$	0	0
$397$	−8.56155	−0.429692	−0.214846	−	0.976648i	$-0.568925\pi$
$397$	−8.56155	−0.429692	−0.214846	+	0.976648i	$0.568925\pi$
$398$	12.4924	0.626189
$399$	0	0
$400$	−4.68466	−0.234233
$401$	20.2462	1.01105	0.505524	−	0.862813i	$-0.331299\pi$
$401$	20.2462	1.01105	0.505524	+	0.862813i	$0.331299\pi$
$402$	0	0
$403$	−6.00000	−0.298881
$404$	−2.73863	−0.136252
$405$	0	0
$406$	8.00000	0.397033
$407$	4.31534	0.213904
$408$	0	0
$409$	−15.1231	−0.747789	−0.373895	−	0.927471i	$-0.621978\pi$
$409$	−15.1231	−0.747789	−0.373895	+	0.927471i	$0.621978\pi$
$410$	0.876894	0.0433067
$411$	0	0
$412$	−3.01515	−0.148546
$413$	−54.7386	−2.69351
$414$	0	0
$415$	16.4924	0.809581
$416$	−2.43845	−0.119555
$417$	0	0
$418$	−12.4924	−0.611024
$419$	−33.6155	−1.64223	−0.821113	−	0.570766i	$-0.806646\pi$
$419$	−33.6155	−1.64223	−0.821113	+	0.570766i	$0.806646\pi$
$420$	0	0
$421$	14.6307	0.713056	0.356528	−	0.934285i	$-0.383960\pi$
$421$	14.6307	0.713056	0.356528	+	0.934285i	$0.383960\pi$
$422$	16.0000	0.778868
$423$	0	0
$424$	18.7386	0.910029
$425$	−2.56155	−0.124254
$426$	0	0
$427$	−25.9309	−1.25488
$428$	−7.86174	−0.380012
$429$	0	0
$430$	8.00000	0.385794
$431$	−36.4924	−1.75778	−0.878889	−	0.477026i	$-0.841715\pi$
$431$	−36.4924	−1.75778	−0.878889	+	0.477026i	$0.841715\pi$
$432$	0	0
$433$	31.6155	1.51935	0.759673	−	0.650306i	$-0.225359\pi$
$433$	31.6155	1.51935	0.759673	+	0.650306i	$0.225359\pi$
$434$	42.7386	2.05152
$435$	0	0
$436$	−0.876894	−0.0419956
$437$	20.4924	0.980286
$438$	0	0
$439$	−15.0540	−0.718487	−0.359244	−	0.933244i	$-0.616965\pi$
$439$	−15.0540	−0.718487	−0.359244	+	0.933244i	$0.616965\pi$
$440$	6.24621	0.297776
$441$	0	0
$442$	−4.00000	−0.190261
$443$	12.8078	0.608515	0.304258	−	0.952590i	$-0.401592\pi$
$443$	12.8078	0.608515	0.304258	+	0.952590i	$0.401592\pi$
$444$	0	0
$445$	−1.68466	−0.0798605
$446$	−14.6307	−0.692783
$447$	0	0
$448$	−25.3693	−1.19859
$449$	10.1771	0.480286	0.240143	−	0.970738i	$-0.422806\pi$
$449$	10.1771	0.480286	0.240143	+	0.970738i	$0.422806\pi$
$450$	0	0
$451$	−1.43845	−0.0677338
$452$	−5.75379	−0.270635
$453$	0	0
$454$	−34.7386	−1.63036
$455$	4.56155	0.213849
$456$	0	0
$457$	−25.6847	−1.20148	−0.600739	−	0.799445i	$-0.705127\pi$
$457$	−25.6847	−1.20148	−0.600739	+	0.799445i	$0.705127\pi$
$458$	−13.8617	−0.647717
$459$	0	0
$460$	2.87689	0.134136
$461$	−29.0540	−1.35318	−0.676589	−	0.736361i	$-0.736543\pi$
$461$	−29.0540	−1.35318	−0.676589	+	0.736361i	$0.736543\pi$
$462$	0	0
$463$	−0.0691303	−0.00321276	−0.00160638	−	0.999999i	$-0.500511\pi$
$463$	−0.0691303	−0.00321276	−0.00160638	+	0.999999i	$0.500511\pi$
$464$	−5.26137	−0.244253
$465$	0	0
$466$	11.2311	0.520269
$467$	−21.9309	−1.01484	−0.507420	−	0.861699i	$-0.669401\pi$
$467$	−21.9309	−1.01484	−0.507420	+	0.861699i	$0.669401\pi$
$468$	0	0
$469$	−60.9848	−2.81602
$470$	−4.49242	−0.207220
$471$	0	0
$472$	29.2614	1.34686
$473$	−13.1231	−0.603401
$474$	0	0
$475$	3.12311	0.143298
$476$	5.12311	0.234817
$477$	0	0
$478$	9.26137	0.423605
$479$	−11.0540	−0.505069	−0.252535	−	0.967588i	$-0.581264\pi$
$479$	−11.0540	−0.505069	−0.252535	+	0.967588i	$0.581264\pi$
$480$	0	0
$481$	−1.68466	−0.0768138
$482$	38.6307	1.75958
$483$	0	0
$484$	−1.94602	−0.0884557
$485$	−11.9309	−0.541753
$486$	0	0
$487$	5.05398	0.229017	0.114509	−	0.993422i	$-0.463471\pi$
$487$	5.05398	0.229017	0.114509	+	0.993422i	$0.463471\pi$
$488$	13.8617	0.627491
$489$	0	0
$490$	−21.5616	−0.974052
$491$	13.6155	0.614460	0.307230	−	0.951635i	$-0.400598\pi$
$491$	13.6155	0.614460	0.307230	+	0.951635i	$0.400598\pi$
$492$	0	0
$493$	−2.87689	−0.129569
$494$	4.87689	0.219422
$495$	0	0
$496$	−28.1080	−1.26208
$497$	−66.4233	−2.97949
$498$	0	0
$499$	−30.9848	−1.38707	−0.693536	−	0.720422i	$-0.743948\pi$
$499$	−30.9848	−1.38707	−0.693536	+	0.720422i	$0.743948\pi$
$500$	0.438447	0.0196080
$501$	0	0
$502$	−15.2311	−0.679795
$503$	30.2462	1.34861	0.674306	−	0.738452i	$-0.264443\pi$
$503$	30.2462	1.34861	0.674306	+	0.738452i	$0.264443\pi$
$504$	0	0
$505$	−6.24621	−0.277953
$506$	−26.2462	−1.16679
$507$	0	0
$508$	8.00000	0.354943
$509$	−19.4384	−0.861594	−0.430797	−	0.902449i	$-0.641768\pi$
$509$	−19.4384	−0.861594	−0.430797	+	0.902449i	$0.641768\pi$
$510$	0	0
$511$	27.3693	1.21075
$512$	11.4233	0.504843
$513$	0	0
$514$	32.9848	1.45490
$515$	−6.87689	−0.303032
$516$	0	0
$517$	7.36932	0.324102
$518$	12.0000	0.527250
$519$	0	0
$520$	−2.43845	−0.106933
$521$	31.8617	1.39589	0.697944	−	0.716152i	$-0.254098\pi$
$521$	31.8617	1.39589	0.697944	+	0.716152i	$0.254098\pi$
$522$	0	0
$523$	18.7386	0.819383	0.409692	−	0.912224i	$-0.365636\pi$
$523$	18.7386	0.819383	0.409692	+	0.912224i	$0.365636\pi$
$524$	0	0
$525$	0	0
$526$	−22.2462	−0.969981
$527$	−15.3693	−0.669498
$528$	0	0
$529$	20.0540	0.871912
$530$	−12.0000	−0.521247
$531$	0	0
$532$	−6.24621	−0.270808
$533$	0.561553	0.0243236
$534$	0	0
$535$	−17.9309	−0.775219
$536$	32.6004	1.40812
$537$	0	0
$538$	14.2462	0.614198
$539$	35.3693	1.52346
$540$	0	0
$541$	27.1231	1.16611	0.583057	−	0.812431i	$-0.301857\pi$
$541$	27.1231	1.16611	0.583057	+	0.812431i	$0.301857\pi$
$542$	−8.38447	−0.360144
$543$	0	0
$544$	−6.24621	−0.267804
$545$	−2.00000	−0.0856706
$546$	0	0
$547$	19.3693	0.828172	0.414086	−	0.910238i	$-0.364101\pi$
$547$	19.3693	0.828172	0.414086	+	0.910238i	$0.364101\pi$
$548$	−3.12311	−0.133412
$549$	0	0
$550$	−4.00000	−0.170561
$551$	3.50758	0.149428
$552$	0	0
$553$	35.0540	1.49065
$554$	−6.63068	−0.281711
$555$	0	0
$556$	6.87689	0.291645
$557$	−26.4924	−1.12252	−0.561260	−	0.827640i	$-0.689683\pi$
$557$	−26.4924	−1.12252	−0.561260	+	0.827640i	$0.689683\pi$
$558$	0	0
$559$	5.12311	0.216684
$560$	21.3693	0.903018
$561$	0	0
$562$	19.1231	0.806660
$563$	−31.6847	−1.33535	−0.667675	−	0.744453i	$-0.732710\pi$
$563$	−31.6847	−1.33535	−0.667675	+	0.744453i	$0.732710\pi$
$564$	0	0
$565$	−13.1231	−0.552093
$566$	6.24621	0.262548
$567$	0	0
$568$	35.5076	1.48986
$569$	−41.1231	−1.72397	−0.861985	−	0.506934i	$-0.830779\pi$
$569$	−41.1231	−1.72397	−0.861985	+	0.506934i	$0.830779\pi$
$570$	0	0
$571$	15.0540	0.629989	0.314995	−	0.949093i	$-0.397997\pi$
$571$	15.0540	0.629989	0.314995	+	0.949093i	$0.397997\pi$
$572$	−1.12311	−0.0469594
$573$	0	0
$574$	−4.00000	−0.166957
$575$	6.56155	0.273636
$576$	0	0
$577$	28.5616	1.18903	0.594517	−	0.804083i	$-0.297343\pi$
$577$	28.5616	1.18903	0.594517	+	0.804083i	$0.297343\pi$
$578$	16.3002	0.677998
$579$	0	0
$580$	0.492423	0.0204467
$581$	−75.2311	−3.12111
$582$	0	0
$583$	19.6847	0.815255
$584$	−14.6307	−0.605422
$585$	0	0
$586$	5.86174	0.242146
$587$	0.492423	0.0203245	0.0101622	−	0.999948i	$-0.496765\pi$
$587$	0.492423	0.0203245	0.0101622	+	0.999948i	$0.496765\pi$
$588$	0	0
$589$	18.7386	0.772112
$590$	−18.7386	−0.771457
$591$	0	0
$592$	−7.89205	−0.324361
$593$	7.75379	0.318410	0.159205	−	0.987246i	$-0.449107\pi$
$593$	7.75379	0.318410	0.159205	+	0.987246i	$0.449107\pi$
$594$	0	0
$595$	11.6847	0.479024
$596$	1.23106	0.0504260
$597$	0	0
$598$	10.2462	0.418999
$599$	15.3693	0.627973	0.313987	−	0.949427i	$-0.398335\pi$
$599$	15.3693	0.627973	0.313987	+	0.949427i	$0.398335\pi$
$600$	0	0
$601$	41.5464	1.69471	0.847356	−	0.531025i	$-0.178193\pi$
$601$	41.5464	1.69471	0.847356	+	0.531025i	$0.178193\pi$
$602$	−36.4924	−1.48732
$603$	0	0
$604$	−5.86174	−0.238511
$605$	−4.43845	−0.180449
$606$	0	0
$607$	−24.0000	−0.974130	−0.487065	−	0.873366i	$-0.661933\pi$
$607$	−24.0000	−0.974130	−0.487065	+	0.873366i	$0.661933\pi$
$608$	7.61553	0.308850
$609$	0	0
$610$	−8.87689	−0.359415
$611$	−2.87689	−0.116387
$612$	0	0
$613$	37.5464	1.51648	0.758242	−	0.651973i	$-0.226058\pi$
$613$	37.5464	1.51648	0.758242	+	0.651973i	$0.226058\pi$
$614$	−5.36932	−0.216688
$615$	0	0
$616$	−28.4924	−1.14799
$617$	−48.7386	−1.96214	−0.981072	−	0.193645i	$-0.937969\pi$
$617$	−48.7386	−1.96214	−0.981072	+	0.193645i	$0.937969\pi$
$618$	0	0
$619$	−33.3693	−1.34123	−0.670613	−	0.741807i	$-0.733969\pi$
$619$	−33.3693	−1.34123	−0.670613	+	0.741807i	$0.733969\pi$
$620$	2.63068	0.105651
$621$	0	0
$622$	−42.7386	−1.71366
$623$	7.68466	0.307879
$624$	0	0
$625$	1.00000	0.0400000
$626$	−32.6004	−1.30297
$627$	0	0
$628$	−9.36932	−0.373876
$629$	−4.31534	−0.172064
$630$	0	0
$631$	16.7386	0.666354	0.333177	−	0.942864i	$-0.391879\pi$
$631$	16.7386	0.666354	0.333177	+	0.942864i	$0.391879\pi$
$632$	−18.7386	−0.745383
$633$	0	0
$634$	−53.8617	−2.13912
$635$	18.2462	0.724079
$636$	0	0
$637$	−13.8078	−0.547084
$638$	−4.49242	−0.177857
$639$	0	0
$640$	−13.5616	−0.536067
$641$	32.9848	1.30282	0.651412	−	0.758725i	$-0.274177\pi$
$641$	32.9848	1.30282	0.651412	+	0.758725i	$0.274177\pi$
$642$	0	0
$643$	1.68466	0.0664364	0.0332182	−	0.999448i	$-0.489424\pi$
$643$	1.68466	0.0664364	0.0332182	+	0.999448i	$0.489424\pi$
$644$	−13.1231	−0.517123
$645$	0	0
$646$	12.4924	0.491508
$647$	11.1922	0.440012	0.220006	−	0.975498i	$-0.429392\pi$
$647$	11.1922	0.440012	0.220006	+	0.975498i	$0.429392\pi$
$648$	0	0
$649$	30.7386	1.20660
$650$	1.56155	0.0612491
$651$	0	0
$652$	−9.23106	−0.361516
$653$	−8.63068	−0.337745	−0.168872	−	0.985638i	$-0.554013\pi$
$653$	−8.63068	−0.337745	−0.168872	+	0.985638i	$0.554013\pi$
$654$	0	0
$655$	0	0
$656$	2.63068	0.102711
$657$	0	0
$658$	20.4924	0.798878
$659$	9.12311	0.355386	0.177693	−	0.984086i	$-0.443137\pi$
$659$	9.12311	0.355386	0.177693	+	0.984086i	$0.443137\pi$
$660$	0	0
$661$	−26.4924	−1.03044	−0.515218	−	0.857059i	$-0.672289\pi$
$661$	−26.4924	−1.03044	−0.515218	+	0.857059i	$0.672289\pi$
$662$	−32.6004	−1.26705
$663$	0	0
$664$	40.2159	1.56068
$665$	−14.2462	−0.552444
$666$	0	0
$667$	7.36932	0.285341
$668$	5.75379	0.222621
$669$	0	0
$670$	−20.8769	−0.806545
$671$	14.5616	0.562143
$672$	0	0
$673$	−32.7386	−1.26198	−0.630991	−	0.775790i	$-0.717351\pi$
$673$	−32.7386	−1.26198	−0.630991	+	0.775790i	$0.717351\pi$
$674$	−3.89205	−0.149916
$675$	0	0
$676$	0.438447	0.0168634
$677$	−18.5616	−0.713378	−0.356689	−	0.934223i	$-0.616094\pi$
$677$	−18.5616	−0.713378	−0.356689	+	0.934223i	$0.616094\pi$
$678$	0	0
$679$	54.4233	2.08857
$680$	−6.24621	−0.239531
$681$	0	0
$682$	−24.0000	−0.919007
$683$	−0.492423	−0.0188420	−0.00942101	−	0.999956i	$-0.502999\pi$
$683$	−0.492423	−0.0188420	−0.00942101	+	0.999956i	$0.502999\pi$
$684$	0	0
$685$	−7.12311	−0.272160
$686$	48.4924	1.85145
$687$	0	0
$688$	24.0000	0.914991
$689$	−7.68466	−0.292762
$690$	0	0
$691$	19.6155	0.746210	0.373105	−	0.927789i	$-0.378293\pi$
$691$	19.6155	0.746210	0.373105	+	0.927789i	$0.378293\pi$
$692$	9.26137	0.352064
$693$	0	0
$694$	−17.2614	−0.655233
$695$	15.6847	0.594953
$696$	0	0
$697$	1.43845	0.0544851
$698$	−2.13826	−0.0809344
$699$	0	0
$700$	−2.00000	−0.0755929
$701$	16.9848	0.641509	0.320754	−	0.947162i	$-0.396064\pi$
$701$	16.9848	0.641509	0.320754	+	0.947162i	$0.396064\pi$
$702$	0	0
$703$	5.26137	0.198436
$704$	14.2462	0.536924
$705$	0	0
$706$	−16.3845	−0.616638
$707$	28.4924	1.07157
$708$	0	0
$709$	0.876894	0.0329325	0.0164662	−	0.999864i	$-0.494758\pi$
$709$	0.876894	0.0329325	0.0164662	+	0.999864i	$0.494758\pi$
$710$	−22.7386	−0.853366
$711$	0	0
$712$	−4.10795	−0.153952
$713$	39.3693	1.47439
$714$	0	0
$715$	−2.56155	−0.0957966
$716$	−5.75379	−0.215029
$717$	0	0
$718$	−3.50758	−0.130902
$719$	36.0000	1.34257	0.671287	−	0.741198i	$-0.265742\pi$
$719$	36.0000	1.34257	0.671287	+	0.741198i	$0.265742\pi$
$720$	0	0
$721$	31.3693	1.16825
$722$	14.4384	0.537343
$723$	0	0
$724$	11.2614	0.418525
$725$	1.12311	0.0417111
$726$	0	0
$727$	21.1231	0.783413	0.391706	−	0.920090i	$-0.371885\pi$
$727$	21.1231	0.783413	0.391706	+	0.920090i	$0.371885\pi$
$728$	11.1231	0.412250
$729$	0	0
$730$	9.36932	0.346774
$731$	13.1231	0.485376
$732$	0	0
$733$	−20.5616	−0.759458	−0.379729	−	0.925098i	$-0.623983\pi$
$733$	−20.5616	−0.759458	−0.379729	+	0.925098i	$0.623983\pi$
$734$	28.4924	1.05167
$735$	0	0
$736$	16.0000	0.589768
$737$	34.2462	1.26148
$738$	0	0
$739$	−0.738634	−0.0271711	−0.0135855	−	0.999908i	$-0.504325\pi$
$739$	−0.738634	−0.0271711	−0.0135855	+	0.999908i	$0.504325\pi$
$740$	0.738634	0.0271527
$741$	0	0
$742$	54.7386	2.00952
$743$	−30.7386	−1.12769	−0.563846	−	0.825880i	$-0.690679\pi$
$743$	−30.7386	−1.12769	−0.563846	+	0.825880i	$0.690679\pi$
$744$	0	0
$745$	2.80776	0.102869
$746$	6.63068	0.242767
$747$	0	0
$748$	−2.87689	−0.105190
$749$	81.7926	2.98864
$750$	0	0
$751$	−23.0540	−0.841252	−0.420626	−	0.907234i	$-0.638189\pi$
$751$	−23.0540	−0.841252	−0.420626	+	0.907234i	$0.638189\pi$
$752$	−13.4773	−0.491465
$753$	0	0
$754$	1.75379	0.0638692
$755$	−13.3693	−0.486559
$756$	0	0
$757$	18.4924	0.672119	0.336059	−	0.941841i	$-0.390906\pi$
$757$	18.4924	0.672119	0.336059	+	0.941841i	$0.390906\pi$
$758$	−3.89205	−0.141366
$759$	0	0
$760$	7.61553	0.276244
$761$	−37.2311	−1.34962	−0.674812	−	0.737989i	$-0.735775\pi$
$761$	−37.2311	−1.34962	−0.674812	+	0.737989i	$0.735775\pi$
$762$	0	0
$763$	9.12311	0.330279
$764$	−9.47727	−0.342876
$765$	0	0
$766$	−56.9848	−2.05895
$767$	−12.0000	−0.433295
$768$	0	0
$769$	−30.0000	−1.08183	−0.540914	−	0.841078i	$-0.681921\pi$
$769$	−30.0000	−1.08183	−0.540914	+	0.841078i	$0.681921\pi$
$770$	18.2462	0.657548
$771$	0	0
$772$	6.76894	0.243620
$773$	−0.876894	−0.0315397	−0.0157698	−	0.999876i	$-0.505020\pi$
$773$	−0.876894	−0.0315397	−0.0157698	+	0.999876i	$0.505020\pi$
$774$	0	0
$775$	6.00000	0.215526
$776$	−29.0928	−1.04437
$777$	0	0
$778$	31.0152	1.11195
$779$	−1.75379	−0.0628360
$780$	0	0
$781$	37.3002	1.33471
$782$	26.2462	0.938563
$783$	0	0
$784$	−64.6847	−2.31017
$785$	−21.3693	−0.762704
$786$	0	0
$787$	−13.3693	−0.476565	−0.238282	−	0.971196i	$-0.576584\pi$
$787$	−13.3693	−0.476565	−0.238282	+	0.971196i	$0.576584\pi$
$788$	9.36932	0.333768
$789$	0	0
$790$	12.0000	0.426941
$791$	59.8617	2.12844
$792$	0	0
$793$	−5.68466	−0.201868
$794$	13.3693	0.474459
$795$	0	0
$796$	−3.50758	−0.124323
$797$	−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
$799$	−7.36932	−0.260708
$800$	2.43845	0.0862121
$801$	0	0
$802$	−31.6155	−1.11638
$803$	−15.3693	−0.542371
$804$	0	0
$805$	−29.9309	−1.05492
$806$	9.36932	0.330020
$807$	0	0
$808$	−15.2311	−0.535827
$809$	40.4924	1.42364	0.711819	−	0.702363i	$-0.247872\pi$
$809$	40.4924	1.42364	0.711819	+	0.702363i	$0.247872\pi$
$810$	0	0
$811$	−12.2462	−0.430023	−0.215011	−	0.976612i	$-0.568979\pi$
$811$	−12.2462	−0.430023	−0.215011	+	0.976612i	$0.568979\pi$
$812$	−2.24621	−0.0788266
$813$	0	0
$814$	−6.73863	−0.236189
$815$	−21.0540	−0.737489
$816$	0	0
$817$	−16.0000	−0.559769
$818$	23.6155	0.825698
$819$	0	0
$820$	−0.246211	−0.00859807
$821$	−21.5464	−0.751974	−0.375987	−	0.926625i	$-0.622696\pi$
$821$	−21.5464	−0.751974	−0.375987	+	0.926625i	$0.622696\pi$
$822$	0	0
$823$	−46.2462	−1.61204	−0.806021	−	0.591887i	$-0.798383\pi$
$823$	−46.2462	−1.61204	−0.806021	+	0.591887i	$0.798383\pi$
$824$	−16.7689	−0.584174
$825$	0	0
$826$	85.4773	2.97413
$827$	−1.12311	−0.0390542	−0.0195271	−	0.999809i	$-0.506216\pi$
$827$	−1.12311	−0.0390542	−0.0195271	+	0.999809i	$0.506216\pi$
$828$	0	0
$829$	−48.7386	−1.69276	−0.846381	−	0.532577i	$-0.821224\pi$
$829$	−48.7386	−1.69276	−0.846381	+	0.532577i	$0.821224\pi$
$830$	−25.7538	−0.893927
$831$	0	0
$832$	−5.56155	−0.192812
$833$	−35.3693	−1.22547
$834$	0	0
$835$	13.1231	0.454144
$836$	3.50758	0.121312
$837$	0	0
$838$	52.4924	1.81332
$839$	41.7926	1.44284	0.721421	−	0.692497i	$-0.243490\pi$
$839$	41.7926	1.44284	0.721421	+	0.692497i	$0.243490\pi$
$840$	0	0
$841$	−27.7386	−0.956505
$842$	−22.8466	−0.787345
$843$	0	0
$844$	−4.49242	−0.154636
$845$	1.00000	0.0344010
$846$	0	0
$847$	20.2462	0.695668
$848$	−36.0000	−1.23625
$849$	0	0
$850$	4.00000	0.137199
$851$	11.0540	0.378925
$852$	0	0
$853$	−20.4233	−0.699280	−0.349640	−	0.936884i	$-0.613696\pi$
$853$	−20.4233	−0.699280	−0.349640	+	0.936884i	$0.613696\pi$
$854$	40.4924	1.38562
$855$	0	0
$856$	−43.7235	−1.49444
$857$	6.56155	0.224138	0.112069	−	0.993700i	$-0.464252\pi$
$857$	6.56155	0.224138	0.112069	+	0.993700i	$0.464252\pi$
$858$	0	0
$859$	−16.8078	−0.573474	−0.286737	−	0.958009i	$-0.592571\pi$
$859$	−16.8078	−0.573474	−0.286737	+	0.958009i	$0.592571\pi$
$860$	−2.24621	−0.0765952
$861$	0	0
$862$	56.9848	1.94091
$863$	15.3693	0.523178	0.261589	−	0.965179i	$-0.415754\pi$
$863$	15.3693	0.523178	0.261589	+	0.965179i	$0.415754\pi$
$864$	0	0
$865$	21.1231	0.718207
$866$	−49.3693	−1.67764
$867$	0	0
$868$	−12.0000	−0.407307
$869$	−19.6847	−0.667756
$870$	0	0
$871$	−13.3693	−0.453002
$872$	−4.87689	−0.165152
$873$	0	0
$874$	−32.0000	−1.08242
$875$	−4.56155	−0.154209
$876$	0	0
$877$	−18.9848	−0.641073	−0.320536	−	0.947236i	$-0.603863\pi$
$877$	−18.9848	−0.641073	−0.320536	+	0.947236i	$0.603863\pi$
$878$	23.5076	0.793342
$879$	0	0
$880$	−12.0000	−0.404520
$881$	3.36932	0.113515	0.0567576	−	0.998388i	$-0.481924\pi$
$881$	3.36932	0.113515	0.0567576	+	0.998388i	$0.481924\pi$
$882$	0	0
$883$	18.1080	0.609381	0.304691	−	0.952451i	$-0.401447\pi$
$883$	18.1080	0.609381	0.304691	+	0.952451i	$0.401447\pi$
$884$	1.12311	0.0377741
$885$	0	0
$886$	−20.0000	−0.671913
$887$	1.43845	0.0482983	0.0241492	−	0.999708i	$-0.492312\pi$
$887$	1.43845	0.0482983	0.0241492	+	0.999708i	$0.492312\pi$
$888$	0	0
$889$	−83.2311	−2.79148
$890$	2.63068	0.0881807
$891$	0	0
$892$	4.10795	0.137544
$893$	8.98485	0.300666
$894$	0	0
$895$	−13.1231	−0.438657
$896$	61.8617	2.06666
$897$	0	0
$898$	−15.8920	−0.530325
$899$	6.73863	0.224746
$900$	0	0
$901$	−19.6847	−0.655791
$902$	2.24621	0.0747907
$903$	0	0
$904$	−32.0000	−1.06430
$905$	25.6847	0.853787
$906$	0	0
$907$	−27.3693	−0.908783	−0.454392	−	0.890802i	$-0.650143\pi$
$907$	−27.3693	−0.908783	−0.454392	+	0.890802i	$0.650143\pi$
$908$	9.75379	0.323691
$909$	0	0
$910$	−7.12311	−0.236129
$911$	−12.0000	−0.397578	−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000	−0.397578	−0.198789	+	0.980042i	$0.563701\pi$
$912$	0	0
$913$	42.2462	1.39815
$914$	40.1080	1.32665
$915$	0	0
$916$	3.89205	0.128597
$917$	0	0
$918$	0	0
$919$	39.0540	1.28827	0.644136	−	0.764911i	$-0.277217\pi$
$919$	39.0540	1.28827	0.644136	+	0.764911i	$0.277217\pi$
$920$	16.0000	0.527504
$921$	0	0
$922$	45.3693	1.49416
$923$	−14.5616	−0.479299
$924$	0	0
$925$	1.68466	0.0553912
$926$	0.107951	0.00354748
$927$	0	0
$928$	2.73863	0.0899001
$929$	−38.6695	−1.26871	−0.634353	−	0.773044i	$-0.718733\pi$
$929$	−38.6695	−1.26871	−0.634353	+	0.773044i	$0.718733\pi$
$930$	0	0
$931$	43.1231	1.41330
$932$	−3.15342	−0.103294
$933$	0	0
$934$	34.2462	1.12057
$935$	−6.56155	−0.214586
$936$	0	0
$937$	−2.63068	−0.0859407	−0.0429703	−	0.999076i	$-0.513682\pi$
$937$	−2.63068	−0.0859407	−0.0429703	+	0.999076i	$0.513682\pi$
$938$	95.2311	3.10940
$939$	0	0
$940$	1.26137	0.0411412
$941$	−21.1922	−0.690847	−0.345424	−	0.938447i	$-0.612265\pi$
$941$	−21.1922	−0.690847	−0.345424	+	0.938447i	$0.612265\pi$
$942$	0	0
$943$	−3.68466	−0.119989
$944$	−56.2159	−1.82967
$945$	0	0
$946$	20.4924	0.666266
$947$	−46.6004	−1.51431	−0.757154	−	0.653236i	$-0.773411\pi$
$947$	−46.6004	−1.51431	−0.757154	+	0.653236i	$0.773411\pi$
$948$	0	0
$949$	6.00000	0.194768
$950$	−4.87689	−0.158227
$951$	0	0
$952$	28.4924	0.923445
$953$	25.4384	0.824032	0.412016	−	0.911177i	$-0.364825\pi$
$953$	25.4384	0.824032	0.412016	+	0.911177i	$0.364825\pi$
$954$	0	0
$955$	−21.6155	−0.699462
$956$	−2.60037	−0.0841021
$957$	0	0
$958$	17.2614	0.557689
$959$	32.4924	1.04924
$960$	0	0
$961$	5.00000	0.161290
$962$	2.63068	0.0848166
$963$	0	0
$964$	−10.8466	−0.349345
$965$	15.4384	0.496981
$966$	0	0
$967$	17.3693	0.558560	0.279280	−	0.960210i	$-0.409904\pi$
$967$	17.3693	0.558560	0.279280	+	0.960210i	$0.409904\pi$
$968$	−10.8229	−0.347862
$969$	0	0
$970$	18.6307	0.598195
$971$	8.49242	0.272535	0.136267	−	0.990672i	$-0.456489\pi$
$971$	8.49242	0.272535	0.136267	+	0.990672i	$0.456489\pi$
$972$	0	0
$973$	−71.5464	−2.29367
$974$	−7.89205	−0.252878
$975$	0	0
$976$	−26.6307	−0.852427
$977$	55.4773	1.77488	0.887438	−	0.460928i	$-0.152483\pi$
$977$	55.4773	1.77488	0.887438	+	0.460928i	$0.152483\pi$
$978$	0	0
$979$	−4.31534	−0.137919
$980$	6.05398	0.193387
$981$	0	0
$982$	−21.2614	−0.678477
$983$	−6.73863	−0.214929	−0.107465	−	0.994209i	$-0.534273\pi$
$983$	−6.73863	−0.214929	−0.107465	+	0.994209i	$0.534273\pi$
$984$	0	0
$985$	21.3693	0.680883
$986$	4.49242	0.143068
$987$	0	0
$988$	−1.36932	−0.0435638
$989$	−33.6155	−1.06891
$990$	0	0
$991$	−27.0540	−0.859398	−0.429699	−	0.902972i	$-0.641380\pi$
$991$	−27.0540	−0.859398	−0.429699	+	0.902972i	$0.641380\pi$
$992$	14.6307	0.464525
$993$	0	0
$994$	103.723	3.28991
$995$	−8.00000	−0.253617
$996$	0	0
$997$	11.7538	0.372246	0.186123	−	0.982526i	$-0.440408\pi$
$997$	11.7538	0.372246	0.186123	+	0.982526i	$0.440408\pi$
$998$	48.3845	1.53158
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	585.2.a.l.1.1	yes	2
3.2	odd	2		585.2.a.j.1.2	✓	2
4.3	odd	2		9360.2.a.cw.1.2		2
5.2	odd	4		2925.2.c.p.2224.2		4
5.3	odd	4		2925.2.c.p.2224.3		4
5.4	even	2		2925.2.a.x.1.2		2
12.11	even	2		9360.2.a.cl.1.2		2
13.12	even	2		7605.2.a.bd.1.2		2
15.2	even	4		2925.2.c.o.2224.3		4
15.8	even	4		2925.2.c.o.2224.2		4
15.14	odd	2		2925.2.a.bc.1.1		2
39.38	odd	2		7605.2.a.bi.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
585.2.a.j.1.2	✓	2	3.2	odd	2
585.2.a.l.1.1	yes	2	1.1	even	1	trivial
2925.2.a.x.1.2		2	5.4	even	2
2925.2.a.bc.1.1		2	15.14	odd	2
2925.2.c.o.2224.2		4	15.8	even	4
2925.2.c.o.2224.3		4	15.2	even	4
2925.2.c.p.2224.2		4	5.2	odd	4
2925.2.c.p.2224.3		4	5.3	odd	4
7605.2.a.bd.1.2		2	13.12	even	2
7605.2.a.bi.1.1		2	39.38	odd	2
9360.2.a.cl.1.2		2	12.11	even	2
9360.2.a.cw.1.2		2	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	585.a (trivial)

\(f(q)\)	\(=\)	\(q-1.56155 q^{2} +0.438447 q^{4} +1.00000 q^{5} -4.56155 q^{7} +2.43845 q^{8} +O(q^{10})\)	\(q-1.56155 q^{2} +0.438447 q^{4} +1.00000 q^{5} -4.56155 q^{7} +2.43845 q^{8} -1.56155 q^{10} +2.56155 q^{11} -1.00000 q^{13} +7.12311 q^{14} -4.68466 q^{16} -2.56155 q^{17} +3.12311 q^{19} +0.438447 q^{20} -4.00000 q^{22} +6.56155 q^{23} +1.00000 q^{25} +1.56155 q^{26} -2.00000 q^{28} +1.12311 q^{29} +6.00000 q^{31} +2.43845 q^{32} +4.00000 q^{34} -4.56155 q^{35} +1.68466 q^{37} -4.87689 q^{38} +2.43845 q^{40} -0.561553 q^{41} -5.12311 q^{43} +1.12311 q^{44} -10.2462 q^{46} +2.87689 q^{47} +13.8078 q^{49} -1.56155 q^{50} -0.438447 q^{52} +7.68466 q^{53} +2.56155 q^{55} -11.1231 q^{56} -1.75379 q^{58} +12.0000 q^{59} +5.68466 q^{61} -9.36932 q^{62} +5.56155 q^{64} -1.00000 q^{65} +13.3693 q^{67} -1.12311 q^{68} +7.12311 q^{70} +14.5616 q^{71} -6.00000 q^{73} -2.63068 q^{74} +1.36932 q^{76} -11.6847 q^{77} -7.68466 q^{79} -4.68466 q^{80} +0.876894 q^{82} +16.4924 q^{83} -2.56155 q^{85} +8.00000 q^{86} +6.24621 q^{88} -1.68466 q^{89} +4.56155 q^{91} +2.87689 q^{92} -4.49242 q^{94} +3.12311 q^{95} -11.9309 q^{97} -21.5616 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} + 5 q^{4} + 2 q^{5} - 5 q^{7} + 9 q^{8}+O(q^{10}) \) 2 * q + q^2 + 5 * q^4 + 2 * q^5 - 5 * q^7 + 9 * q^8	\( 2 q + q^{2} + 5 q^{4} + 2 q^{5} - 5 q^{7} + 9 q^{8} + q^{10} + q^{11} - 2 q^{13} + 6 q^{14} + 3 q^{16} - q^{17} - 2 q^{19} + 5 q^{20} - 8 q^{22} + 9 q^{23} + 2 q^{25} - q^{26} - 4 q^{28} - 6 q^{29} + 12 q^{31} + 9 q^{32} + 8 q^{34} - 5 q^{35} - 9 q^{37} - 18 q^{38} + 9 q^{40} + 3 q^{41} - 2 q^{43} - 6 q^{44} - 4 q^{46} + 14 q^{47} + 7 q^{49} + q^{50} - 5 q^{52} + 3 q^{53} + q^{55} - 14 q^{56} - 20 q^{58} + 24 q^{59} - q^{61} + 6 q^{62} + 7 q^{64} - 2 q^{65} + 2 q^{67} + 6 q^{68} + 6 q^{70} + 25 q^{71} - 12 q^{73} - 30 q^{74} - 22 q^{76} - 11 q^{77} - 3 q^{79} + 3 q^{80} + 10 q^{82} - q^{85} + 16 q^{86} - 4 q^{88} + 9 q^{89} + 5 q^{91} + 14 q^{92} + 24 q^{94} - 2 q^{95} + 5 q^{97} - 39 q^{98}+O(q^{100}) \) 2 * q + q^2 + 5 * q^4 + 2 * q^5 - 5 * q^7 + 9 * q^8 + q^10 + q^11 - 2 * q^13 + 6 * q^14 + 3 * q^16 - q^17 - 2 * q^19 + 5 * q^20 - 8 * q^22 + 9 * q^23 + 2 * q^25 - q^26 - 4 * q^28 - 6 * q^29 + 12 * q^31 + 9 * q^32 + 8 * q^34 - 5 * q^35 - 9 * q^37 - 18 * q^38 + 9 * q^40 + 3 * q^41 - 2 * q^43 - 6 * q^44 - 4 * q^46 + 14 * q^47 + 7 * q^49 + q^50 - 5 * q^52 + 3 * q^53 + q^55 - 14 * q^56 - 20 * q^58 + 24 * q^59 - q^61 + 6 * q^62 + 7 * q^64 - 2 * q^65 + 2 * q^67 + 6 * q^68 + 6 * q^70 + 25 * q^71 - 12 * q^73 - 30 * q^74 - 22 * q^76 - 11 * q^77 - 3 * q^79 + 3 * q^80 + 10 * q^82 - q^85 + 16 * q^86 - 4 * q^88 + 9 * q^89 + 5 * q^91 + 14 * q^92 + 24 * q^94 - 2 * q^95 + 5 * q^97 - 39 * q^98

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