LMFDB - Embedded newform 585.2.a.k.1.2

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:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 65)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ 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)

$f(q)$	$=$	$q+1.73205 q^{2} +1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{7} -1.73205 q^{8} +O(q^{10})$	\(q+1.73205 q^{2} +1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{7} -1.73205 q^{8} +1.73205 q^{10} +4.73205 q^{11} +1.00000 q^{13} +3.46410 q^{14} -5.00000 q^{16} +3.46410 q^{17} -6.19615 q^{19} +1.00000 q^{20} +8.19615 q^{22} -1.26795 q^{23} +1.00000 q^{25} +1.73205 q^{26} +2.00000 q^{28} +2.53590 q^{29} +10.1962 q^{31} -5.19615 q^{32} +6.00000 q^{34} +2.00000 q^{35} -4.00000 q^{37} -10.7321 q^{38} -1.73205 q^{40} -3.46410 q^{41} -0.196152 q^{43} +4.73205 q^{44} -2.19615 q^{46} -6.00000 q^{47} -3.00000 q^{49} +1.73205 q^{50} +1.00000 q^{52} -10.3923 q^{53} +4.73205 q^{55} -3.46410 q^{56} +4.39230 q^{58} -9.12436 q^{59} -8.39230 q^{61} +17.6603 q^{62} +1.00000 q^{64} +1.00000 q^{65} +6.39230 q^{67} +3.46410 q^{68} +3.46410 q^{70} -4.73205 q^{71} -4.00000 q^{73} -6.92820 q^{74} -6.19615 q^{76} +9.46410 q^{77} -8.39230 q^{79} -5.00000 q^{80} -6.00000 q^{82} +6.00000 q^{83} +3.46410 q^{85} -0.339746 q^{86} -8.19615 q^{88} +12.9282 q^{89} +2.00000 q^{91} -1.26795 q^{92} -10.3923 q^{94} -6.19615 q^{95} +2.00000 q^{97} -5.19615 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4} + 2 q^{5} + 4 q^{7}+O(q^{10}) $ 2 * q + 2 * q^4 + 2 * q^5 + 4 * q^7	$ 2 q + 2 q^{4} + 2 q^{5} + 4 q^{7} + 6 q^{11} + 2 q^{13} - 10 q^{16} - 2 q^{19} + 2 q^{20} + 6 q^{22} - 6 q^{23} + 2 q^{25} + 4 q^{28} + 12 q^{29} + 10 q^{31} + 12 q^{34} + 4 q^{35} - 8 q^{37} - 18 q^{38} + 10 q^{43} + 6 q^{44} + 6 q^{46} - 12 q^{47} - 6 q^{49} + 2 q^{52} + 6 q^{55} - 12 q^{58} + 6 q^{59} + 4 q^{61} + 18 q^{62} + 2 q^{64} + 2 q^{65} - 8 q^{67} - 6 q^{71} - 8 q^{73} - 2 q^{76} + 12 q^{77} + 4 q^{79} - 10 q^{80} - 12 q^{82} + 12 q^{83} - 18 q^{86} - 6 q^{88} + 12 q^{89} + 4 q^{91} - 6 q^{92} - 2 q^{95} + 4 q^{97}+O(q^{100}) $ 2 * q + 2 * q^4 + 2 * q^5 + 4 * q^7 + 6 * q^11 + 2 * q^13 - 10 * q^16 - 2 * q^19 + 2 * q^20 + 6 * q^22 - 6 * q^23 + 2 * q^25 + 4 * q^28 + 12 * q^29 + 10 * q^31 + 12 * q^34 + 4 * q^35 - 8 * q^37 - 18 * q^38 + 10 * q^43 + 6 * q^44 + 6 * q^46 - 12 * q^47 - 6 * q^49 + 2 * q^52 + 6 * q^55 - 12 * q^58 + 6 * q^59 + 4 * q^61 + 18 * q^62 + 2 * q^64 + 2 * q^65 - 8 * q^67 - 6 * q^71 - 8 * q^73 - 2 * q^76 + 12 * q^77 + 4 * q^79 - 10 * q^80 - 12 * q^82 + 12 * q^83 - 18 * q^86 - 6 * q^88 + 12 * q^89 + 4 * q^91 - 6 * q^92 - 2 * q^95 + 4 * q^97

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 $

$2$	1.73205	1.22474	0.612372	−	0.790569i	$-0.290215\pi$
$2$	1.73205	1.22474	0.612372	+	0.790569i	$0.290215\pi$
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	−1.73205	−0.612372
$9$	0	0
$10$	1.73205	0.547723
$11$	4.73205	1.42677	0.713384	−	0.700774i	$-0.247162\pi$
$11$	4.73205	1.42677	0.713384	+	0.700774i	$0.247162\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	3.46410	0.925820
$15$	0	0
$16$	−5.00000	−1.25000
$17$	3.46410	0.840168	0.420084	−	0.907485i	$-0.362001\pi$
$17$	3.46410	0.840168	0.420084	+	0.907485i	$0.362001\pi$
$18$	0	0
$19$	−6.19615	−1.42149	−0.710747	−	0.703447i	$-0.751643\pi$
$19$	−6.19615	−1.42149	−0.710747	+	0.703447i	$0.751643\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	8.19615	1.74743
$23$	−1.26795	−0.264386	−0.132193	−	0.991224i	$-0.542202\pi$
$23$	−1.26795	−0.264386	−0.132193	+	0.991224i	$0.542202\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	1.73205	0.339683
$27$	0	0
$28$	2.00000	0.377964
$29$	2.53590	0.470905	0.235452	−	0.971886i	$-0.424343\pi$
$29$	2.53590	0.470905	0.235452	+	0.971886i	$0.424343\pi$
$30$	0	0
$31$	10.1962	1.83128	0.915642	−	0.401996i	$-0.131683\pi$
$31$	10.1962	1.83128	0.915642	+	0.401996i	$0.131683\pi$
$32$	−5.19615	−0.918559
$33$	0	0
$34$	6.00000	1.02899
$35$	2.00000	0.338062
$36$	0	0
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	−10.7321	−1.74097
$39$	0	0
$40$	−1.73205	−0.273861
$41$	−3.46410	−0.541002	−0.270501	−	0.962720i	$-0.587189\pi$
$41$	−3.46410	−0.541002	−0.270501	+	0.962720i	$0.587189\pi$
$42$	0	0
$43$	−0.196152	−0.0299130	−0.0149565	−	0.999888i	$-0.504761\pi$
$43$	−0.196152	−0.0299130	−0.0149565	+	0.999888i	$0.504761\pi$
$44$	4.73205	0.713384
$45$	0	0
$46$	−2.19615	−0.323805
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	1.73205	0.244949
$51$	0	0
$52$	1.00000	0.138675
$53$	−10.3923	−1.42749	−0.713746	−	0.700404i	$-0.753003\pi$
$53$	−10.3923	−1.42749	−0.713746	+	0.700404i	$0.753003\pi$
$54$	0	0
$55$	4.73205	0.638070
$56$	−3.46410	−0.462910
$57$	0	0
$58$	4.39230	0.576738
$59$	−9.12436	−1.18789	−0.593945	−	0.804506i	$-0.702430\pi$
$59$	−9.12436	−1.18789	−0.593945	+	0.804506i	$0.702430\pi$
$60$	0	0
$61$	−8.39230	−1.07452	−0.537262	−	0.843415i	$-0.680541\pi$
$61$	−8.39230	−1.07452	−0.537262	+	0.843415i	$0.680541\pi$
$62$	17.6603	2.24285
$63$	0	0
$64$	1.00000	0.125000
$65$	1.00000	0.124035
$66$	0	0
$67$	6.39230	0.780944	0.390472	−	0.920615i	$-0.372312\pi$
$67$	6.39230	0.780944	0.390472	+	0.920615i	$0.372312\pi$
$68$	3.46410	0.420084
$69$	0	0
$70$	3.46410	0.414039
$71$	−4.73205	−0.561591	−0.280796	−	0.959768i	$-0.590598\pi$
$71$	−4.73205	−0.561591	−0.280796	+	0.959768i	$0.590598\pi$
$72$	0	0
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	−6.92820	−0.805387
$75$	0	0
$76$	−6.19615	−0.710747
$77$	9.46410	1.07853
$78$	0	0
$79$	−8.39230	−0.944208	−0.472104	−	0.881543i	$-0.656505\pi$
$79$	−8.39230	−0.944208	−0.472104	+	0.881543i	$0.656505\pi$
$80$	−5.00000	−0.559017
$81$	0	0
$82$	−6.00000	−0.662589
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	0	0
$85$	3.46410	0.375735
$86$	−0.339746	−0.0366357
$87$	0	0
$88$	−8.19615	−0.873713
$89$	12.9282	1.37039	0.685193	−	0.728361i	$-0.259718\pi$
$89$	12.9282	1.37039	0.685193	+	0.728361i	$0.259718\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	−1.26795	−0.132193
$93$	0	0
$94$	−10.3923	−1.07188
$95$	−6.19615	−0.635712
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	−5.19615	−0.524891
$99$	0	0
$100$	1.00000	0.100000
$101$	0.928203	0.0923597	0.0461798	−	0.998933i	$-0.485295\pi$
$101$	0.928203	0.0923597	0.0461798	+	0.998933i	$0.485295\pi$
$102$	0	0
$103$	−0.196152	−0.0193275	−0.00966374	−	0.999953i	$-0.503076\pi$
$103$	−0.196152	−0.0193275	−0.00966374	+	0.999953i	$0.503076\pi$
$104$	−1.73205	−0.169842
$105$	0	0
$106$	−18.0000	−1.74831
$107$	−17.6603	−1.70728	−0.853641	−	0.520862i	$-0.825610\pi$
$107$	−17.6603	−1.70728	−0.853641	+	0.520862i	$0.825610\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	8.19615	0.781472
$111$	0	0
$112$	−10.0000	−0.944911
$113$	−8.53590	−0.802990	−0.401495	−	0.915861i	$-0.631509\pi$
$113$	−8.53590	−0.802990	−0.401495	+	0.915861i	$0.631509\pi$
$114$	0	0
$115$	−1.26795	−0.118237
$116$	2.53590	0.235452
$117$	0	0
$118$	−15.8038	−1.45486
$119$	6.92820	0.635107
$120$	0	0
$121$	11.3923	1.03566
$122$	−14.5359	−1.31602
$123$	0	0
$124$	10.1962	0.915642
$125$	1.00000	0.0894427
$126$	0	0
$127$	16.1962	1.43718	0.718588	−	0.695436i	$-0.244789\pi$
$127$	16.1962	1.43718	0.718588	+	0.695436i	$0.244789\pi$
$128$	12.1244	1.07165
$129$	0	0
$130$	1.73205	0.151911
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	−12.3923	−1.07455
$134$	11.0718	0.956458
$135$	0	0
$136$	−6.00000	−0.514496
$137$	−0.928203	−0.0793018	−0.0396509	−	0.999214i	$-0.512625\pi$
$137$	−0.928203	−0.0793018	−0.0396509	+	0.999214i	$0.512625\pi$
$138$	0	0
$139$	12.3923	1.05110	0.525551	−	0.850762i	$-0.323859\pi$
$139$	12.3923	1.05110	0.525551	+	0.850762i	$0.323859\pi$
$140$	2.00000	0.169031
$141$	0	0
$142$	−8.19615	−0.687806
$143$	4.73205	0.395714
$144$	0	0
$145$	2.53590	0.210595
$146$	−6.92820	−0.573382
$147$	0	0
$148$	−4.00000	−0.328798
$149$	7.85641	0.643622	0.321811	−	0.946804i	$-0.395708\pi$
$149$	7.85641	0.643622	0.321811	+	0.946804i	$0.395708\pi$
$150$	0	0
$151$	−1.80385	−0.146795	−0.0733975	−	0.997303i	$-0.523384\pi$
$151$	−1.80385	−0.146795	−0.0733975	+	0.997303i	$0.523384\pi$
$152$	10.7321	0.870484
$153$	0	0
$154$	16.3923	1.32093
$155$	10.1962	0.818975
$156$	0	0
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	−14.5359	−1.15641
$159$	0	0
$160$	−5.19615	−0.410792
$161$	−2.53590	−0.199857
$162$	0	0
$163$	−14.3923	−1.12729	−0.563646	−	0.826016i	$-0.690602\pi$
$163$	−14.3923	−1.12729	−0.563646	+	0.826016i	$0.690602\pi$
$164$	−3.46410	−0.270501
$165$	0	0
$166$	10.3923	0.806599
$167$	0.928203	0.0718265	0.0359133	−	0.999355i	$-0.488566\pi$
$167$	0.928203	0.0718265	0.0359133	+	0.999355i	$0.488566\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	6.00000	0.460179
$171$	0	0
$172$	−0.196152	−0.0149565
$173$	−8.53590	−0.648972	−0.324486	−	0.945890i	$-0.605191\pi$
$173$	−8.53590	−0.648972	−0.324486	+	0.945890i	$0.605191\pi$
$174$	0	0
$175$	2.00000	0.151186
$176$	−23.6603	−1.78346
$177$	0	0
$178$	22.3923	1.67837
$179$	18.9282	1.41476	0.707380	−	0.706833i	$-0.249877\pi$
$179$	18.9282	1.41476	0.707380	+	0.706833i	$0.249877\pi$
$180$	0	0
$181$	0.392305	0.0291598	0.0145799	−	0.999894i	$-0.495359\pi$
$181$	0.392305	0.0291598	0.0145799	+	0.999894i	$0.495359\pi$
$182$	3.46410	0.256776
$183$	0	0
$184$	2.19615	0.161903
$185$	−4.00000	−0.294086
$186$	0	0
$187$	16.3923	1.19872
$188$	−6.00000	−0.437595
$189$	0	0
$190$	−10.7321	−0.778585
$191$	5.07180	0.366982	0.183491	−	0.983021i	$-0.441260\pi$
$191$	5.07180	0.366982	0.183491	+	0.983021i	$0.441260\pi$
$192$	0	0
$193$	−10.0000	−0.719816	−0.359908	−	0.932988i	$-0.617192\pi$
$193$	−10.0000	−0.719816	−0.359908	+	0.932988i	$0.617192\pi$
$194$	3.46410	0.248708
$195$	0	0
$196$	−3.00000	−0.214286
$197$	12.9282	0.921096	0.460548	−	0.887635i	$-0.347653\pi$
$197$	12.9282	0.921096	0.460548	+	0.887635i	$0.347653\pi$
$198$	0	0
$199$	20.0000	1.41776	0.708881	−	0.705328i	$-0.249200\pi$
$199$	20.0000	1.41776	0.708881	+	0.705328i	$0.249200\pi$
$200$	−1.73205	−0.122474
$201$	0	0
$202$	1.60770	0.113117
$203$	5.07180	0.355970
$204$	0	0
$205$	−3.46410	−0.241943
$206$	−0.339746	−0.0236712
$207$	0	0
$208$	−5.00000	−0.346688
$209$	−29.3205	−2.02814
$210$	0	0
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	−10.3923	−0.713746
$213$	0	0
$214$	−30.5885	−2.09098
$215$	−0.196152	−0.0133775
$216$	0	0
$217$	20.3923	1.38432
$218$	3.46410	0.234619
$219$	0	0
$220$	4.73205	0.319035
$221$	3.46410	0.233021
$222$	0	0
$223$	2.00000	0.133930	0.0669650	−	0.997755i	$-0.478668\pi$
$223$	2.00000	0.133930	0.0669650	+	0.997755i	$0.478668\pi$
$224$	−10.3923	−0.694365
$225$	0	0
$226$	−14.7846	−0.983458
$227$	3.46410	0.229920	0.114960	−	0.993370i	$-0.463326\pi$
$227$	3.46410	0.229920	0.114960	+	0.993370i	$0.463326\pi$
$228$	0	0
$229$	6.39230	0.422415	0.211208	−	0.977441i	$-0.432260\pi$
$229$	6.39230	0.422415	0.211208	+	0.977441i	$0.432260\pi$
$230$	−2.19615	−0.144810
$231$	0	0
$232$	−4.39230	−0.288369
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	−6.00000	−0.391397
$236$	−9.12436	−0.593945
$237$	0	0
$238$	12.0000	0.777844
$239$	14.1962	0.918273	0.459136	−	0.888366i	$-0.348159\pi$
$239$	14.1962	0.918273	0.459136	+	0.888366i	$0.348159\pi$
$240$	0	0
$241$	−2.39230	−0.154102	−0.0770510	−	0.997027i	$-0.524550\pi$
$241$	−2.39230	−0.154102	−0.0770510	+	0.997027i	$0.524550\pi$
$242$	19.7321	1.26842
$243$	0	0
$244$	−8.39230	−0.537262
$245$	−3.00000	−0.191663
$246$	0	0
$247$	−6.19615	−0.394252
$248$	−17.6603	−1.12143
$249$	0	0
$250$	1.73205	0.109545
$251$	21.4641	1.35480	0.677401	−	0.735614i	$-0.263106\pi$
$251$	21.4641	1.35480	0.677401	+	0.735614i	$0.263106\pi$
$252$	0	0
$253$	−6.00000	−0.377217
$254$	28.0526	1.76017
$255$	0	0
$256$	19.0000	1.18750
$257$	−19.8564	−1.23861	−0.619304	−	0.785151i	$-0.712585\pi$
$257$	−19.8564	−1.23861	−0.619304	+	0.785151i	$0.712585\pi$
$258$	0	0
$259$	−8.00000	−0.497096
$260$	1.00000	0.0620174
$261$	0	0
$262$	0	0
$263$	−1.26795	−0.0781851	−0.0390925	−	0.999236i	$-0.512447\pi$
$263$	−1.26795	−0.0781851	−0.0390925	+	0.999236i	$0.512447\pi$
$264$	0	0
$265$	−10.3923	−0.638394
$266$	−21.4641	−1.31605
$267$	0	0
$268$	6.39230	0.390472
$269$	19.8564	1.21067	0.605333	−	0.795972i	$-0.293040\pi$
$269$	19.8564	1.21067	0.605333	+	0.795972i	$0.293040\pi$
$270$	0	0
$271$	30.9808	1.88195	0.940974	−	0.338480i	$-0.109913\pi$
$271$	30.9808	1.88195	0.940974	+	0.338480i	$0.109913\pi$
$272$	−17.3205	−1.05021
$273$	0	0
$274$	−1.60770	−0.0971244
$275$	4.73205	0.285353
$276$	0	0
$277$	−26.3923	−1.58576	−0.792880	−	0.609378i	$-0.791419\pi$
$277$	−26.3923	−1.58576	−0.792880	+	0.609378i	$0.791419\pi$
$278$	21.4641	1.28733
$279$	0	0
$280$	−3.46410	−0.207020
$281$	−22.3923	−1.33581	−0.667906	−	0.744245i	$-0.732809\pi$
$281$	−22.3923	−1.33581	−0.667906	+	0.744245i	$0.732809\pi$
$282$	0	0
$283$	32.5885	1.93718	0.968591	−	0.248658i	$-0.0799895\pi$
$283$	32.5885	1.93718	0.968591	+	0.248658i	$0.0799895\pi$
$284$	−4.73205	−0.280796
$285$	0	0
$286$	8.19615	0.484649
$287$	−6.92820	−0.408959
$288$	0	0
$289$	−5.00000	−0.294118
$290$	4.39230	0.257925
$291$	0	0
$292$	−4.00000	−0.234082
$293$	5.07180	0.296298	0.148149	−	0.988965i	$-0.452669\pi$
$293$	5.07180	0.296298	0.148149	+	0.988965i	$0.452669\pi$
$294$	0	0
$295$	−9.12436	−0.531241
$296$	6.92820	0.402694
$297$	0	0
$298$	13.6077	0.788273
$299$	−1.26795	−0.0733274
$300$	0	0
$301$	−0.392305	−0.0226121
$302$	−3.12436	−0.179786
$303$	0	0
$304$	30.9808	1.77687
$305$	−8.39230	−0.480542
$306$	0	0
$307$	−18.7846	−1.07209	−0.536047	−	0.844188i	$-0.680083\pi$
$307$	−18.7846	−1.07209	−0.536047	+	0.844188i	$0.680083\pi$
$308$	9.46410	0.539267
$309$	0	0
$310$	17.6603	1.00304
$311$	16.3923	0.929522	0.464761	−	0.885436i	$-0.346140\pi$
$311$	16.3923	0.929522	0.464761	+	0.885436i	$0.346140\pi$
$312$	0	0
$313$	−14.3923	−0.813501	−0.406751	−	0.913539i	$-0.633338\pi$
$313$	−14.3923	−0.813501	−0.406751	+	0.913539i	$0.633338\pi$
$314$	−17.3205	−0.977453
$315$	0	0
$316$	−8.39230	−0.472104
$317$	−24.0000	−1.34797	−0.673987	−	0.738743i	$-0.735420\pi$
$317$	−24.0000	−1.34797	−0.673987	+	0.738743i	$0.735420\pi$
$318$	0	0
$319$	12.0000	0.671871
$320$	1.00000	0.0559017
$321$	0	0
$322$	−4.39230	−0.244774
$323$	−21.4641	−1.19429
$324$	0	0
$325$	1.00000	0.0554700
$326$	−24.9282	−1.38065
$327$	0	0
$328$	6.00000	0.331295
$329$	−12.0000	−0.661581
$330$	0	0
$331$	2.58846	0.142274	0.0711372	−	0.997467i	$-0.477337\pi$
$331$	2.58846	0.142274	0.0711372	+	0.997467i	$0.477337\pi$
$332$	6.00000	0.329293
$333$	0	0
$334$	1.60770	0.0879692
$335$	6.39230	0.349249
$336$	0	0
$337$	−26.3923	−1.43768	−0.718840	−	0.695175i	$-0.755327\pi$
$337$	−26.3923	−1.43768	−0.718840	+	0.695175i	$0.755327\pi$
$338$	1.73205	0.0942111
$339$	0	0
$340$	3.46410	0.187867
$341$	48.2487	2.61281
$342$	0	0
$343$	−20.0000	−1.07990
$344$	0.339746	0.0183179
$345$	0	0
$346$	−14.7846	−0.794826
$347$	5.66025	0.303858	0.151929	−	0.988391i	$-0.451451\pi$
$347$	5.66025	0.303858	0.151929	+	0.988391i	$0.451451\pi$
$348$	0	0
$349$	−14.3923	−0.770402	−0.385201	−	0.922833i	$-0.625868\pi$
$349$	−14.3923	−0.770402	−0.385201	+	0.922833i	$0.625868\pi$
$350$	3.46410	0.185164
$351$	0	0
$352$	−24.5885	−1.31057
$353$	27.7128	1.47500	0.737502	−	0.675345i	$-0.236005\pi$
$353$	27.7128	1.47500	0.737502	+	0.675345i	$0.236005\pi$
$354$	0	0
$355$	−4.73205	−0.251151
$356$	12.9282	0.685193
$357$	0	0
$358$	32.7846	1.73272
$359$	2.19615	0.115908	0.0579542	−	0.998319i	$-0.481542\pi$
$359$	2.19615	0.115908	0.0579542	+	0.998319i	$0.481542\pi$
$360$	0	0
$361$	19.3923	1.02065
$362$	0.679492	0.0357133
$363$	0	0
$364$	2.00000	0.104828
$365$	−4.00000	−0.209370
$366$	0	0
$367$	11.8038	0.616156	0.308078	−	0.951361i	$-0.400314\pi$
$367$	11.8038	0.616156	0.308078	+	0.951361i	$0.400314\pi$
$368$	6.33975	0.330482
$369$	0	0
$370$	−6.92820	−0.360180
$371$	−20.7846	−1.07908
$372$	0	0
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	28.3923	1.46813
$375$	0	0
$376$	10.3923	0.535942
$377$	2.53590	0.130605
$378$	0	0
$379$	18.9808	0.974976	0.487488	−	0.873130i	$-0.337913\pi$
$379$	18.9808	0.974976	0.487488	+	0.873130i	$0.337913\pi$
$380$	−6.19615	−0.317856
$381$	0	0
$382$	8.78461	0.449460
$383$	−12.9282	−0.660600	−0.330300	−	0.943876i	$-0.607150\pi$
$383$	−12.9282	−0.660600	−0.330300	+	0.943876i	$0.607150\pi$
$384$	0	0
$385$	9.46410	0.482335
$386$	−17.3205	−0.881591
$387$	0	0
$388$	2.00000	0.101535
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	−4.39230	−0.222128
$392$	5.19615	0.262445
$393$	0	0
$394$	22.3923	1.12811
$395$	−8.39230	−0.422263
$396$	0	0
$397$	28.7846	1.44466	0.722329	−	0.691549i	$-0.243072\pi$
$397$	28.7846	1.44466	0.722329	+	0.691549i	$0.243072\pi$
$398$	34.6410	1.73640
$399$	0	0
$400$	−5.00000	−0.250000
$401$	36.9282	1.84411	0.922053	−	0.387063i	$-0.126510\pi$
$401$	36.9282	1.84411	0.922053	+	0.387063i	$0.126510\pi$
$402$	0	0
$403$	10.1962	0.507907
$404$	0.928203	0.0461798
$405$	0	0
$406$	8.78461	0.435973
$407$	−18.9282	−0.938236
$408$	0	0
$409$	−17.6077	−0.870644	−0.435322	−	0.900275i	$-0.643366\pi$
$409$	−17.6077	−0.870644	−0.435322	+	0.900275i	$0.643366\pi$
$410$	−6.00000	−0.296319
$411$	0	0
$412$	−0.196152	−0.00966374
$413$	−18.2487	−0.897960
$414$	0	0
$415$	6.00000	0.294528
$416$	−5.19615	−0.254762
$417$	0	0
$418$	−50.7846	−2.48396
$419$	2.53590	0.123887	0.0619434	−	0.998080i	$-0.480270\pi$
$419$	2.53590	0.123887	0.0619434	+	0.998080i	$0.480270\pi$
$420$	0	0
$421$	−30.7846	−1.50035	−0.750175	−	0.661239i	$-0.770031\pi$
$421$	−30.7846	−1.50035	−0.750175	+	0.661239i	$0.770031\pi$
$422$	13.8564	0.674519
$423$	0	0
$424$	18.0000	0.874157
$425$	3.46410	0.168034
$426$	0	0
$427$	−16.7846	−0.812264
$428$	−17.6603	−0.853641
$429$	0	0
$430$	−0.339746	−0.0163840
$431$	25.5167	1.22909	0.614547	−	0.788880i	$-0.289339\pi$
$431$	25.5167	1.22909	0.614547	+	0.788880i	$0.289339\pi$
$432$	0	0
$433$	34.7846	1.67164	0.835821	−	0.549002i	$-0.184992\pi$
$433$	34.7846	1.67164	0.835821	+	0.549002i	$0.184992\pi$
$434$	35.3205	1.69544
$435$	0	0
$436$	2.00000	0.0957826
$437$	7.85641	0.375823
$438$	0	0
$439$	32.0000	1.52728	0.763638	−	0.645644i	$-0.223411\pi$
$439$	32.0000	1.52728	0.763638	+	0.645644i	$0.223411\pi$
$440$	−8.19615	−0.390736
$441$	0	0
$442$	6.00000	0.285391
$443$	16.9808	0.806780	0.403390	−	0.915028i	$-0.367832\pi$
$443$	16.9808	0.806780	0.403390	+	0.915028i	$0.367832\pi$
$444$	0	0
$445$	12.9282	0.612856
$446$	3.46410	0.164030
$447$	0	0
$448$	2.00000	0.0944911
$449$	−20.5359	−0.969149	−0.484574	−	0.874750i	$-0.661026\pi$
$449$	−20.5359	−0.969149	−0.484574	+	0.874750i	$0.661026\pi$
$450$	0	0
$451$	−16.3923	−0.771883
$452$	−8.53590	−0.401495
$453$	0	0
$454$	6.00000	0.281594
$455$	2.00000	0.0937614
$456$	0	0
$457$	10.7846	0.504483	0.252241	−	0.967664i	$-0.418832\pi$
$457$	10.7846	0.504483	0.252241	+	0.967664i	$0.418832\pi$
$458$	11.0718	0.517351
$459$	0	0
$460$	−1.26795	−0.0591184
$461$	3.46410	0.161339	0.0806696	−	0.996741i	$-0.474294\pi$
$461$	3.46410	0.161339	0.0806696	+	0.996741i	$0.474294\pi$
$462$	0	0
$463$	−2.39230	−0.111180	−0.0555899	−	0.998454i	$-0.517704\pi$
$463$	−2.39230	−0.111180	−0.0555899	+	0.998454i	$0.517704\pi$
$464$	−12.6795	−0.588631
$465$	0	0
$466$	10.3923	0.481414
$467$	−27.8038	−1.28661	−0.643304	−	0.765611i	$-0.722437\pi$
$467$	−27.8038	−1.28661	−0.643304	+	0.765611i	$0.722437\pi$
$468$	0	0
$469$	12.7846	0.590338
$470$	−10.3923	−0.479361
$471$	0	0
$472$	15.8038	0.727431
$473$	−0.928203	−0.0426788
$474$	0	0
$475$	−6.19615	−0.284299
$476$	6.92820	0.317554
$477$	0	0
$478$	24.5885	1.12465
$479$	−35.6603	−1.62936	−0.814679	−	0.579912i	$-0.803087\pi$
$479$	−35.6603	−1.62936	−0.814679	+	0.579912i	$0.803087\pi$
$480$	0	0
$481$	−4.00000	−0.182384
$482$	−4.14359	−0.188736
$483$	0	0
$484$	11.3923	0.517832
$485$	2.00000	0.0908153
$486$	0	0
$487$	−26.3923	−1.19595	−0.597975	−	0.801515i	$-0.704028\pi$
$487$	−26.3923	−1.19595	−0.597975	+	0.801515i	$0.704028\pi$
$488$	14.5359	0.658009
$489$	0	0
$490$	−5.19615	−0.234738
$491$	2.53590	0.114443	0.0572217	−	0.998361i	$-0.481776\pi$
$491$	2.53590	0.114443	0.0572217	+	0.998361i	$0.481776\pi$
$492$	0	0
$493$	8.78461	0.395639
$494$	−10.7321	−0.482858
$495$	0	0
$496$	−50.9808	−2.28910
$497$	−9.46410	−0.424523
$498$	0	0
$499$	−38.9808	−1.74502	−0.872509	−	0.488598i	$-0.837509\pi$
$499$	−38.9808	−1.74502	−0.872509	+	0.488598i	$0.837509\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	37.1769	1.65929
$503$	−19.5167	−0.870205	−0.435102	−	0.900381i	$-0.643288\pi$
$503$	−19.5167	−0.870205	−0.435102	+	0.900381i	$0.643288\pi$
$504$	0	0
$505$	0.928203	0.0413045
$506$	−10.3923	−0.461994
$507$	0	0
$508$	16.1962	0.718588
$509$	−39.4641	−1.74922	−0.874608	−	0.484831i	$-0.838881\pi$
$509$	−39.4641	−1.74922	−0.874608	+	0.484831i	$0.838881\pi$
$510$	0	0
$511$	−8.00000	−0.353899
$512$	8.66025	0.382733
$513$	0	0
$514$	−34.3923	−1.51698
$515$	−0.196152	−0.00864351
$516$	0	0
$517$	−28.3923	−1.24869
$518$	−13.8564	−0.608816
$519$	0	0
$520$	−1.73205	−0.0759555
$521$	28.3923	1.24389	0.621945	−	0.783061i	$-0.286343\pi$
$521$	28.3923	1.24389	0.621945	+	0.783061i	$0.286343\pi$
$522$	0	0
$523$	−24.1962	−1.05802	−0.529012	−	0.848614i	$-0.677437\pi$
$523$	−24.1962	−1.05802	−0.529012	+	0.848614i	$0.677437\pi$
$524$	0	0
$525$	0	0
$526$	−2.19615	−0.0957568
$527$	35.3205	1.53859
$528$	0	0
$529$	−21.3923	−0.930100
$530$	−18.0000	−0.781870
$531$	0	0
$532$	−12.3923	−0.537275
$533$	−3.46410	−0.150047
$534$	0	0
$535$	−17.6603	−0.763519
$536$	−11.0718	−0.478229
$537$	0	0
$538$	34.3923	1.48276
$539$	−14.1962	−0.611472
$540$	0	0
$541$	−26.3923	−1.13469	−0.567347	−	0.823479i	$-0.692030\pi$
$541$	−26.3923	−1.13469	−0.567347	+	0.823479i	$0.692030\pi$
$542$	53.6603	2.30491
$543$	0	0
$544$	−18.0000	−0.771744
$545$	2.00000	0.0856706
$546$	0	0
$547$	−12.1962	−0.521470	−0.260735	−	0.965410i	$-0.583965\pi$
$547$	−12.1962	−0.521470	−0.260735	+	0.965410i	$0.583965\pi$
$548$	−0.928203	−0.0396509
$549$	0	0
$550$	8.19615	0.349485
$551$	−15.7128	−0.669388
$552$	0	0
$553$	−16.7846	−0.713754
$554$	−45.7128	−1.94215
$555$	0	0
$556$	12.3923	0.525551
$557$	−1.85641	−0.0786585	−0.0393292	−	0.999226i	$-0.512522\pi$
$557$	−1.85641	−0.0786585	−0.0393292	+	0.999226i	$0.512522\pi$
$558$	0	0
$559$	−0.196152	−0.00829636
$560$	−10.0000	−0.422577
$561$	0	0
$562$	−38.7846	−1.63603
$563$	22.0526	0.929405	0.464702	−	0.885467i	$-0.346161\pi$
$563$	22.0526	0.929405	0.464702	+	0.885467i	$0.346161\pi$
$564$	0	0
$565$	−8.53590	−0.359108
$566$	56.4449	2.37255
$567$	0	0
$568$	8.19615	0.343903
$569$	2.53590	0.106310	0.0531552	−	0.998586i	$-0.483072\pi$
$569$	2.53590	0.106310	0.0531552	+	0.998586i	$0.483072\pi$
$570$	0	0
$571$	36.3923	1.52297	0.761485	−	0.648182i	$-0.224470\pi$
$571$	36.3923	1.52297	0.761485	+	0.648182i	$0.224470\pi$
$572$	4.73205	0.197857
$573$	0	0
$574$	−12.0000	−0.500870
$575$	−1.26795	−0.0528771
$576$	0	0
$577$	−4.00000	−0.166522	−0.0832611	−	0.996528i	$-0.526534\pi$
$577$	−4.00000	−0.166522	−0.0832611	+	0.996528i	$0.526534\pi$
$578$	−8.66025	−0.360219
$579$	0	0
$580$	2.53590	0.105297
$581$	12.0000	0.497844
$582$	0	0
$583$	−49.1769	−2.03670
$584$	6.92820	0.286691
$585$	0	0
$586$	8.78461	0.362889
$587$	8.53590	0.352314	0.176157	−	0.984362i	$-0.443633\pi$
$587$	8.53590	0.352314	0.176157	+	0.984362i	$0.443633\pi$
$588$	0	0
$589$	−63.1769	−2.60316
$590$	−15.8038	−0.650634
$591$	0	0
$592$	20.0000	0.821995
$593$	−26.7846	−1.09991	−0.549956	−	0.835194i	$-0.685356\pi$
$593$	−26.7846	−1.09991	−0.549956	+	0.835194i	$0.685356\pi$
$594$	0	0
$595$	6.92820	0.284029
$596$	7.85641	0.321811
$597$	0	0
$598$	−2.19615	−0.0898074
$599$	7.60770	0.310842	0.155421	−	0.987848i	$-0.450327\pi$
$599$	7.60770	0.310842	0.155421	+	0.987848i	$0.450327\pi$
$600$	0	0
$601$	43.5692	1.77723	0.888613	−	0.458658i	$-0.151670\pi$
$601$	43.5692	1.77723	0.888613	+	0.458658i	$0.151670\pi$
$602$	−0.679492	−0.0276940
$603$	0	0
$604$	−1.80385	−0.0733975
$605$	11.3923	0.463163
$606$	0	0
$607$	24.9808	1.01394	0.506969	−	0.861964i	$-0.330766\pi$
$607$	24.9808	1.01394	0.506969	+	0.861964i	$0.330766\pi$
$608$	32.1962	1.30573
$609$	0	0
$610$	−14.5359	−0.588541
$611$	−6.00000	−0.242734
$612$	0	0
$613$	26.0000	1.05013	0.525065	−	0.851062i	$-0.324041\pi$
$613$	26.0000	1.05013	0.525065	+	0.851062i	$0.324041\pi$
$614$	−32.5359	−1.31304
$615$	0	0
$616$	−16.3923	−0.660465
$617$	−33.7128	−1.35723	−0.678613	−	0.734496i	$-0.737419\pi$
$617$	−33.7128	−1.35723	−0.678613	+	0.734496i	$0.737419\pi$
$618$	0	0
$619$	6.98076	0.280581	0.140290	−	0.990110i	$-0.455196\pi$
$619$	6.98076	0.280581	0.140290	+	0.990110i	$0.455196\pi$
$620$	10.1962	0.409487
$621$	0	0
$622$	28.3923	1.13843
$623$	25.8564	1.03592
$624$	0	0
$625$	1.00000	0.0400000
$626$	−24.9282	−0.996331
$627$	0	0
$628$	−10.0000	−0.399043
$629$	−13.8564	−0.552491
$630$	0	0
$631$	5.80385	0.231048	0.115524	−	0.993305i	$-0.463145\pi$
$631$	5.80385	0.231048	0.115524	+	0.993305i	$0.463145\pi$
$632$	14.5359	0.578207
$633$	0	0
$634$	−41.5692	−1.65092
$635$	16.1962	0.642725
$636$	0	0
$637$	−3.00000	−0.118864
$638$	20.7846	0.822871
$639$	0	0
$640$	12.1244	0.479257
$641$	−12.9282	−0.510633	−0.255317	−	0.966857i	$-0.582180\pi$
$641$	−12.9282	−0.510633	−0.255317	+	0.966857i	$0.582180\pi$
$642$	0	0
$643$	−6.78461	−0.267559	−0.133779	−	0.991011i	$-0.542711\pi$
$643$	−6.78461	−0.267559	−0.133779	+	0.991011i	$0.542711\pi$
$644$	−2.53590	−0.0999284
$645$	0	0
$646$	−37.1769	−1.46271
$647$	−22.0526	−0.866976	−0.433488	−	0.901159i	$-0.642717\pi$
$647$	−22.0526	−0.866976	−0.433488	+	0.901159i	$0.642717\pi$
$648$	0	0
$649$	−43.1769	−1.69484
$650$	1.73205	0.0679366
$651$	0	0
$652$	−14.3923	−0.563646
$653$	−7.85641	−0.307445	−0.153722	−	0.988114i	$-0.549126\pi$
$653$	−7.85641	−0.307445	−0.153722	+	0.988114i	$0.549126\pi$
$654$	0	0
$655$	0	0
$656$	17.3205	0.676252
$657$	0	0
$658$	−20.7846	−0.810268
$659$	21.4641	0.836123	0.418061	−	0.908419i	$-0.362710\pi$
$659$	21.4641	0.836123	0.418061	+	0.908419i	$0.362710\pi$
$660$	0	0
$661$	10.7846	0.419473	0.209736	−	0.977758i	$-0.432739\pi$
$661$	10.7846	0.419473	0.209736	+	0.977758i	$0.432739\pi$
$662$	4.48334	0.174250
$663$	0	0
$664$	−10.3923	−0.403300
$665$	−12.3923	−0.480553
$666$	0	0
$667$	−3.21539	−0.124500
$668$	0.928203	0.0359133
$669$	0	0
$670$	11.0718	0.427741
$671$	−39.7128	−1.53310
$672$	0	0
$673$	−14.3923	−0.554783	−0.277391	−	0.960757i	$-0.589470\pi$
$673$	−14.3923	−0.554783	−0.277391	+	0.960757i	$0.589470\pi$
$674$	−45.7128	−1.76079
$675$	0	0
$676$	1.00000	0.0384615
$677$	−10.3923	−0.399409	−0.199704	−	0.979856i	$-0.563998\pi$
$677$	−10.3923	−0.399409	−0.199704	+	0.979856i	$0.563998\pi$
$678$	0	0
$679$	4.00000	0.153506
$680$	−6.00000	−0.230089
$681$	0	0
$682$	83.5692	3.20003
$683$	−32.5359	−1.24495	−0.622476	−	0.782639i	$-0.713873\pi$
$683$	−32.5359	−1.24495	−0.622476	+	0.782639i	$0.713873\pi$
$684$	0	0
$685$	−0.928203	−0.0354648
$686$	−34.6410	−1.32260
$687$	0	0
$688$	0.980762	0.0373912
$689$	−10.3923	−0.395915
$690$	0	0
$691$	−47.7654	−1.81708	−0.908540	−	0.417797i	$-0.862802\pi$
$691$	−47.7654	−1.81708	−0.908540	+	0.417797i	$0.862802\pi$
$692$	−8.53590	−0.324486
$693$	0	0
$694$	9.80385	0.372149
$695$	12.3923	0.470067
$696$	0	0
$697$	−12.0000	−0.454532
$698$	−24.9282	−0.943546
$699$	0	0
$700$	2.00000	0.0755929
$701$	−42.0000	−1.58632	−0.793159	−	0.609015i	$-0.791565\pi$
$701$	−42.0000	−1.58632	−0.793159	+	0.609015i	$0.791565\pi$
$702$	0	0
$703$	24.7846	0.934769
$704$	4.73205	0.178346
$705$	0	0
$706$	48.0000	1.80650
$707$	1.85641	0.0698174
$708$	0	0
$709$	30.3923	1.14141	0.570703	−	0.821156i	$-0.306671\pi$
$709$	30.3923	1.14141	0.570703	+	0.821156i	$0.306671\pi$
$710$	−8.19615	−0.307596
$711$	0	0
$712$	−22.3923	−0.839187
$713$	−12.9282	−0.484165
$714$	0	0
$715$	4.73205	0.176969
$716$	18.9282	0.707380
$717$	0	0
$718$	3.80385	0.141958
$719$	25.8564	0.964281	0.482141	−	0.876094i	$-0.339859\pi$
$719$	25.8564	0.964281	0.482141	+	0.876094i	$0.339859\pi$
$720$	0	0
$721$	−0.392305	−0.0146102
$722$	33.5885	1.25003
$723$	0	0
$724$	0.392305	0.0145799
$725$	2.53590	0.0941809
$726$	0	0
$727$	44.5885	1.65369	0.826847	−	0.562427i	$-0.190132\pi$
$727$	44.5885	1.65369	0.826847	+	0.562427i	$0.190132\pi$
$728$	−3.46410	−0.128388
$729$	0	0
$730$	−6.92820	−0.256424
$731$	−0.679492	−0.0251319
$732$	0	0
$733$	38.0000	1.40356	0.701781	−	0.712393i	$-0.252388\pi$
$733$	38.0000	1.40356	0.701781	+	0.712393i	$0.252388\pi$
$734$	20.4449	0.754634
$735$	0	0
$736$	6.58846	0.242854
$737$	30.2487	1.11423
$738$	0	0
$739$	−18.1962	−0.669356	−0.334678	−	0.942332i	$-0.608628\pi$
$739$	−18.1962	−0.669356	−0.334678	+	0.942332i	$0.608628\pi$
$740$	−4.00000	−0.147043
$741$	0	0
$742$	−36.0000	−1.32160
$743$	16.1436	0.592251	0.296126	−	0.955149i	$-0.404305\pi$
$743$	16.1436	0.592251	0.296126	+	0.955149i	$0.404305\pi$
$744$	0	0
$745$	7.85641	0.287836
$746$	−17.3205	−0.634149
$747$	0	0
$748$	16.3923	0.599362
$749$	−35.3205	−1.29058
$750$	0	0
$751$	36.3923	1.32797	0.663987	−	0.747744i	$-0.268863\pi$
$751$	36.3923	1.32797	0.663987	+	0.747744i	$0.268863\pi$
$752$	30.0000	1.09399
$753$	0	0
$754$	4.39230	0.159958
$755$	−1.80385	−0.0656487
$756$	0	0
$757$	−2.39230	−0.0869498	−0.0434749	−	0.999055i	$-0.513843\pi$
$757$	−2.39230	−0.0869498	−0.0434749	+	0.999055i	$0.513843\pi$
$758$	32.8756	1.19410
$759$	0	0
$760$	10.7321	0.389292
$761$	19.8564	0.719794	0.359897	−	0.932992i	$-0.382812\pi$
$761$	19.8564	0.719794	0.359897	+	0.932992i	$0.382812\pi$
$762$	0	0
$763$	4.00000	0.144810
$764$	5.07180	0.183491
$765$	0	0
$766$	−22.3923	−0.809067
$767$	−9.12436	−0.329461
$768$	0	0
$769$	34.7846	1.25437	0.627183	−	0.778872i	$-0.284208\pi$
$769$	34.7846	1.25437	0.627183	+	0.778872i	$0.284208\pi$
$770$	16.3923	0.590738
$771$	0	0
$772$	−10.0000	−0.359908
$773$	−6.92820	−0.249190	−0.124595	−	0.992208i	$-0.539763\pi$
$773$	−6.92820	−0.249190	−0.124595	+	0.992208i	$0.539763\pi$
$774$	0	0
$775$	10.1962	0.366257
$776$	−3.46410	−0.124354
$777$	0	0
$778$	−10.3923	−0.372582
$779$	21.4641	0.769031
$780$	0	0
$781$	−22.3923	−0.801260
$782$	−7.60770	−0.272051
$783$	0	0
$784$	15.0000	0.535714
$785$	−10.0000	−0.356915
$786$	0	0
$787$	31.5692	1.12532	0.562661	−	0.826688i	$-0.309778\pi$
$787$	31.5692	1.12532	0.562661	+	0.826688i	$0.309778\pi$
$788$	12.9282	0.460548
$789$	0	0
$790$	−14.5359	−0.517164
$791$	−17.0718	−0.607003
$792$	0	0
$793$	−8.39230	−0.298019
$794$	49.8564	1.76934
$795$	0	0
$796$	20.0000	0.708881
$797$	40.6410	1.43958	0.719789	−	0.694193i	$-0.244238\pi$
$797$	40.6410	1.43958	0.719789	+	0.694193i	$0.244238\pi$
$798$	0	0
$799$	−20.7846	−0.735307
$800$	−5.19615	−0.183712
$801$	0	0
$802$	63.9615	2.25856
$803$	−18.9282	−0.667962
$804$	0	0
$805$	−2.53590	−0.0893787
$806$	17.6603	0.622056
$807$	0	0
$808$	−1.60770	−0.0565585
$809$	2.53590	0.0891574	0.0445787	−	0.999006i	$-0.485805\pi$
$809$	2.53590	0.0891574	0.0445787	+	0.999006i	$0.485805\pi$
$810$	0	0
$811$	17.8038	0.625178	0.312589	−	0.949889i	$-0.398804\pi$
$811$	17.8038	0.625178	0.312589	+	0.949889i	$0.398804\pi$
$812$	5.07180	0.177985
$813$	0	0
$814$	−32.7846	−1.14910
$815$	−14.3923	−0.504140
$816$	0	0
$817$	1.21539	0.0425211
$818$	−30.4974	−1.06632
$819$	0	0
$820$	−3.46410	−0.120972
$821$	−28.6410	−0.999578	−0.499789	−	0.866147i	$-0.666589\pi$
$821$	−28.6410	−0.999578	−0.499789	+	0.866147i	$0.666589\pi$
$822$	0	0
$823$	−15.4115	−0.537213	−0.268606	−	0.963250i	$-0.586563\pi$
$823$	−15.4115	−0.537213	−0.268606	+	0.963250i	$0.586563\pi$
$824$	0.339746	0.0118356
$825$	0	0
$826$	−31.6077	−1.09977
$827$	−18.0000	−0.625921	−0.312961	−	0.949766i	$-0.601321\pi$
$827$	−18.0000	−0.625921	−0.312961	+	0.949766i	$0.601321\pi$
$828$	0	0
$829$	0.392305	0.0136253	0.00681266	−	0.999977i	$-0.497831\pi$
$829$	0.392305	0.0136253	0.00681266	+	0.999977i	$0.497831\pi$
$830$	10.3923	0.360722
$831$	0	0
$832$	1.00000	0.0346688
$833$	−10.3923	−0.360072
$834$	0	0
$835$	0.928203	0.0321218
$836$	−29.3205	−1.01407
$837$	0	0
$838$	4.39230	0.151730
$839$	−0.339746	−0.0117293	−0.00586467	−	0.999983i	$-0.501867\pi$
$839$	−0.339746	−0.0117293	−0.00586467	+	0.999983i	$0.501867\pi$
$840$	0	0
$841$	−22.5692	−0.778249
$842$	−53.3205	−1.83755
$843$	0	0
$844$	8.00000	0.275371
$845$	1.00000	0.0344010
$846$	0	0
$847$	22.7846	0.782888
$848$	51.9615	1.78437
$849$	0	0
$850$	6.00000	0.205798
$851$	5.07180	0.173859
$852$	0	0
$853$	8.00000	0.273915	0.136957	−	0.990577i	$-0.456268\pi$
$853$	8.00000	0.273915	0.136957	+	0.990577i	$0.456268\pi$
$854$	−29.0718	−0.994816
$855$	0	0
$856$	30.5885	1.04549
$857$	35.5692	1.21502	0.607511	−	0.794311i	$-0.292168\pi$
$857$	35.5692	1.21502	0.607511	+	0.794311i	$0.292168\pi$
$858$	0	0
$859$	−17.1769	−0.586069	−0.293034	−	0.956102i	$-0.594665\pi$
$859$	−17.1769	−0.586069	−0.293034	+	0.956102i	$0.594665\pi$
$860$	−0.196152	−0.00668874
$861$	0	0
$862$	44.1962	1.50533
$863$	38.7846	1.32024	0.660122	−	0.751159i	$-0.270505\pi$
$863$	38.7846	1.32024	0.660122	+	0.751159i	$0.270505\pi$
$864$	0	0
$865$	−8.53590	−0.290229
$866$	60.2487	2.04733
$867$	0	0
$868$	20.3923	0.692160
$869$	−39.7128	−1.34716
$870$	0	0
$871$	6.39230	0.216595
$872$	−3.46410	−0.117309
$873$	0	0
$874$	13.6077	0.460287
$875$	2.00000	0.0676123
$876$	0	0
$877$	2.00000	0.0675352	0.0337676	−	0.999430i	$-0.489249\pi$
$877$	2.00000	0.0675352	0.0337676	+	0.999430i	$0.489249\pi$
$878$	55.4256	1.87052
$879$	0	0
$880$	−23.6603	−0.797587
$881$	47.3205	1.59427	0.797134	−	0.603802i	$-0.206348\pi$
$881$	47.3205	1.59427	0.797134	+	0.603802i	$0.206348\pi$
$882$	0	0
$883$	23.8038	0.801063	0.400532	−	0.916283i	$-0.368825\pi$
$883$	23.8038	0.801063	0.400532	+	0.916283i	$0.368825\pi$
$884$	3.46410	0.116510
$885$	0	0
$886$	29.4115	0.988100
$887$	47.9090	1.60863	0.804313	−	0.594206i	$-0.202534\pi$
$887$	47.9090	1.60863	0.804313	+	0.594206i	$0.202534\pi$
$888$	0	0
$889$	32.3923	1.08640
$890$	22.3923	0.750592
$891$	0	0
$892$	2.00000	0.0669650
$893$	37.1769	1.24408
$894$	0	0
$895$	18.9282	0.632700
$896$	24.2487	0.810093
$897$	0	0
$898$	−35.5692	−1.18696
$899$	25.8564	0.862359
$900$	0	0
$901$	−36.0000	−1.19933
$902$	−28.3923	−0.945360
$903$	0	0
$904$	14.7846	0.491729
$905$	0.392305	0.0130407
$906$	0	0
$907$	−53.7654	−1.78525	−0.892625	−	0.450800i	$-0.851139\pi$
$907$	−53.7654	−1.78525	−0.892625	+	0.450800i	$0.851139\pi$
$908$	3.46410	0.114960
$909$	0	0
$910$	3.46410	0.114834
$911$	36.0000	1.19273	0.596367	−	0.802712i	$-0.296610\pi$
$911$	36.0000	1.19273	0.596367	+	0.802712i	$0.296610\pi$
$912$	0	0
$913$	28.3923	0.939648
$914$	18.6795	0.617863
$915$	0	0
$916$	6.39230	0.211208
$917$	0	0
$918$	0	0
$919$	9.17691	0.302718	0.151359	−	0.988479i	$-0.451635\pi$
$919$	9.17691	0.302718	0.151359	+	0.988479i	$0.451635\pi$
$920$	2.19615	0.0724050
$921$	0	0
$922$	6.00000	0.197599
$923$	−4.73205	−0.155757
$924$	0	0
$925$	−4.00000	−0.131519
$926$	−4.14359	−0.136167
$927$	0	0
$928$	−13.1769	−0.432553
$929$	−44.5359	−1.46118	−0.730588	−	0.682819i	$-0.760754\pi$
$929$	−44.5359	−1.46118	−0.730588	+	0.682819i	$0.760754\pi$
$930$	0	0
$931$	18.5885	0.609212
$932$	6.00000	0.196537
$933$	0	0
$934$	−48.1577	−1.57577
$935$	16.3923	0.536086
$936$	0	0
$937$	34.7846	1.13636	0.568182	−	0.822903i	$-0.307647\pi$
$937$	34.7846	1.13636	0.568182	+	0.822903i	$0.307647\pi$
$938$	22.1436	0.723014
$939$	0	0
$940$	−6.00000	−0.195698
$941$	−31.1769	−1.01634	−0.508169	−	0.861257i	$-0.669678\pi$
$941$	−31.1769	−1.01634	−0.508169	+	0.861257i	$0.669678\pi$
$942$	0	0
$943$	4.39230	0.143033
$944$	45.6218	1.48486
$945$	0	0
$946$	−1.60770	−0.0522707
$947$	−40.6410	−1.32066	−0.660328	−	0.750978i	$-0.729583\pi$
$947$	−40.6410	−1.32066	−0.660328	+	0.750978i	$0.729583\pi$
$948$	0	0
$949$	−4.00000	−0.129845
$950$	−10.7321	−0.348194
$951$	0	0
$952$	−12.0000	−0.388922
$953$	−0.928203	−0.0300675	−0.0150337	−	0.999887i	$-0.504786\pi$
$953$	−0.928203	−0.0300675	−0.0150337	+	0.999887i	$0.504786\pi$
$954$	0	0
$955$	5.07180	0.164119
$956$	14.1962	0.459136
$957$	0	0
$958$	−61.7654	−1.99555
$959$	−1.85641	−0.0599465
$960$	0	0
$961$	72.9615	2.35360
$962$	−6.92820	−0.223374
$963$	0	0
$964$	−2.39230	−0.0770510
$965$	−10.0000	−0.321911
$966$	0	0
$967$	−50.3923	−1.62051	−0.810254	−	0.586079i	$-0.800671\pi$
$967$	−50.3923	−1.62051	−0.810254	+	0.586079i	$0.800671\pi$
$968$	−19.7321	−0.634212
$969$	0	0
$970$	3.46410	0.111226
$971$	−18.9282	−0.607435	−0.303717	−	0.952762i	$-0.598228\pi$
$971$	−18.9282	−0.607435	−0.303717	+	0.952762i	$0.598228\pi$
$972$	0	0
$973$	24.7846	0.794558
$974$	−45.7128	−1.46473
$975$	0	0
$976$	41.9615	1.34316
$977$	15.7128	0.502697	0.251349	−	0.967897i	$-0.419126\pi$
$977$	15.7128	0.502697	0.251349	+	0.967897i	$0.419126\pi$
$978$	0	0
$979$	61.1769	1.95522
$980$	−3.00000	−0.0958315
$981$	0	0
$982$	4.39230	0.140164
$983$	34.3923	1.09694	0.548472	−	0.836169i	$-0.315210\pi$
$983$	34.3923	1.09694	0.548472	+	0.836169i	$0.315210\pi$
$984$	0	0
$985$	12.9282	0.411927
$986$	15.2154	0.484557
$987$	0	0
$988$	−6.19615	−0.197126
$989$	0.248711	0.00790856
$990$	0	0
$991$	8.00000	0.254128	0.127064	−	0.991894i	$-0.459445\pi$
$991$	8.00000	0.254128	0.127064	+	0.991894i	$0.459445\pi$
$992$	−52.9808	−1.68214
$993$	0	0
$994$	−16.3923	−0.519932
$995$	20.0000	0.634043
$996$	0	0
$997$	33.6077	1.06437	0.532183	−	0.846629i	$-0.321372\pi$
$997$	33.6077	1.06437	0.532183	+	0.846629i	$0.321372\pi$
$998$	−67.5167	−2.13720
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	585.2.a.k.1.2		2
3.2	odd	2		65.2.a.c.1.1	✓	2
4.3	odd	2		9360.2.a.cm.1.1		2
5.2	odd	4		2925.2.c.v.2224.4		4
5.3	odd	4		2925.2.c.v.2224.1		4
5.4	even	2		2925.2.a.z.1.1		2
12.11	even	2		1040.2.a.h.1.1		2
13.12	even	2		7605.2.a.be.1.1		2
15.2	even	4		325.2.b.e.274.1		4
15.8	even	4		325.2.b.e.274.4		4
15.14	odd	2		325.2.a.g.1.2		2
21.20	even	2		3185.2.a.k.1.1		2
24.5	odd	2		4160.2.a.y.1.1		2
24.11	even	2		4160.2.a.bj.1.2		2
33.32	even	2		7865.2.a.h.1.2		2
39.2	even	12		845.2.m.a.316.1		4
39.5	even	4		845.2.c.e.506.4		4
39.8	even	4		845.2.c.e.506.2		4
39.11	even	12		845.2.m.c.316.1		4
39.17	odd	6		845.2.e.f.146.1		4
39.20	even	12		845.2.m.a.361.1		4
39.23	odd	6		845.2.e.f.191.1		4
39.29	odd	6		845.2.e.e.191.2		4
39.32	even	12		845.2.m.c.361.1		4
39.35	odd	6		845.2.e.e.146.2		4
39.38	odd	2		845.2.a.d.1.2		2
60.59	even	2		5200.2.a.ca.1.2		2
195.194	odd	2		4225.2.a.w.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.2.a.c.1.1	✓	2	3.2	odd	2
325.2.a.g.1.2		2	15.14	odd	2
325.2.b.e.274.1		4	15.2	even	4
325.2.b.e.274.4		4	15.8	even	4
585.2.a.k.1.2		2	1.1	even	1	trivial
845.2.a.d.1.2		2	39.38	odd	2
845.2.c.e.506.2		4	39.8	even	4
845.2.c.e.506.4		4	39.5	even	4
845.2.e.e.146.2		4	39.35	odd	6
845.2.e.e.191.2		4	39.29	odd	6
845.2.e.f.146.1		4	39.17	odd	6
845.2.e.f.191.1		4	39.23	odd	6
845.2.m.a.316.1		4	39.2	even	12
845.2.m.a.361.1		4	39.20	even	12
845.2.m.c.316.1		4	39.11	even	12
845.2.m.c.361.1		4	39.32	even	12
1040.2.a.h.1.1		2	12.11	even	2
2925.2.a.z.1.1		2	5.4	even	2
2925.2.c.v.2224.1		4	5.3	odd	4
2925.2.c.v.2224.4		4	5.2	odd	4
3185.2.a.k.1.1		2	21.20	even	2
4160.2.a.y.1.1		2	24.5	odd	2
4160.2.a.bj.1.2		2	24.11	even	2
4225.2.a.w.1.1		2	195.194	odd	2
5200.2.a.ca.1.2		2	60.59	even	2
7605.2.a.be.1.1		2	13.12	even	2
7865.2.a.h.1.2		2	33.32	even	2
9360.2.a.cm.1.1		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)

\(f(q)\)	\(=\)	\(q+1.73205 q^{2} +1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{7} -1.73205 q^{8} +O(q^{10})\)	\(q+1.73205 q^{2} +1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{7} -1.73205 q^{8} +1.73205 q^{10} +4.73205 q^{11} +1.00000 q^{13} +3.46410 q^{14} -5.00000 q^{16} +3.46410 q^{17} -6.19615 q^{19} +1.00000 q^{20} +8.19615 q^{22} -1.26795 q^{23} +1.00000 q^{25} +1.73205 q^{26} +2.00000 q^{28} +2.53590 q^{29} +10.1962 q^{31} -5.19615 q^{32} +6.00000 q^{34} +2.00000 q^{35} -4.00000 q^{37} -10.7321 q^{38} -1.73205 q^{40} -3.46410 q^{41} -0.196152 q^{43} +4.73205 q^{44} -2.19615 q^{46} -6.00000 q^{47} -3.00000 q^{49} +1.73205 q^{50} +1.00000 q^{52} -10.3923 q^{53} +4.73205 q^{55} -3.46410 q^{56} +4.39230 q^{58} -9.12436 q^{59} -8.39230 q^{61} +17.6603 q^{62} +1.00000 q^{64} +1.00000 q^{65} +6.39230 q^{67} +3.46410 q^{68} +3.46410 q^{70} -4.73205 q^{71} -4.00000 q^{73} -6.92820 q^{74} -6.19615 q^{76} +9.46410 q^{77} -8.39230 q^{79} -5.00000 q^{80} -6.00000 q^{82} +6.00000 q^{83} +3.46410 q^{85} -0.339746 q^{86} -8.19615 q^{88} +12.9282 q^{89} +2.00000 q^{91} -1.26795 q^{92} -10.3923 q^{94} -6.19615 q^{95} +2.00000 q^{97} -5.19615 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} + 2 q^{5} + 4 q^{7}+O(q^{10}) \) 2 * q + 2 * q^4 + 2 * q^5 + 4 * q^7	\( 2 q + 2 q^{4} + 2 q^{5} + 4 q^{7} + 6 q^{11} + 2 q^{13} - 10 q^{16} - 2 q^{19} + 2 q^{20} + 6 q^{22} - 6 q^{23} + 2 q^{25} + 4 q^{28} + 12 q^{29} + 10 q^{31} + 12 q^{34} + 4 q^{35} - 8 q^{37} - 18 q^{38} + 10 q^{43} + 6 q^{44} + 6 q^{46} - 12 q^{47} - 6 q^{49} + 2 q^{52} + 6 q^{55} - 12 q^{58} + 6 q^{59} + 4 q^{61} + 18 q^{62} + 2 q^{64} + 2 q^{65} - 8 q^{67} - 6 q^{71} - 8 q^{73} - 2 q^{76} + 12 q^{77} + 4 q^{79} - 10 q^{80} - 12 q^{82} + 12 q^{83} - 18 q^{86} - 6 q^{88} + 12 q^{89} + 4 q^{91} - 6 q^{92} - 2 q^{95} + 4 q^{97}+O(q^{100}) \) 2 * q + 2 * q^4 + 2 * q^5 + 4 * q^7 + 6 * q^11 + 2 * q^13 - 10 * q^16 - 2 * q^19 + 2 * q^20 + 6 * q^22 - 6 * q^23 + 2 * q^25 + 4 * q^28 + 12 * q^29 + 10 * q^31 + 12 * q^34 + 4 * q^35 - 8 * q^37 - 18 * q^38 + 10 * q^43 + 6 * q^44 + 6 * q^46 - 12 * q^47 - 6 * q^49 + 2 * q^52 + 6 * q^55 - 12 * q^58 + 6 * q^59 + 4 * q^61 + 18 * q^62 + 2 * q^64 + 2 * q^65 - 8 * q^67 - 6 * q^71 - 8 * q^73 - 2 * q^76 + 12 * q^77 + 4 * q^79 - 10 * q^80 - 12 * q^82 + 12 * q^83 - 18 * q^86 - 6 * q^88 + 12 * q^89 + 4 * q^91 - 6 * q^92 - 2 * q^95 + 4 * q^97

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