LMFDB - Embedded newform 585.2.a.k.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:	$\Q(\zeta_{12})^+$
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			1.1
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)

$f(q)$	$=$	$q-1.73205 q^{2} +1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{7} +1.73205 q^{8} +O(q^{10})$	\(q-1.73205 q^{2} +1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{7} +1.73205 q^{8} -1.73205 q^{10} +1.26795 q^{11} +1.00000 q^{13} -3.46410 q^{14} -5.00000 q^{16} -3.46410 q^{17} +4.19615 q^{19} +1.00000 q^{20} -2.19615 q^{22} -4.73205 q^{23} +1.00000 q^{25} -1.73205 q^{26} +2.00000 q^{28} +9.46410 q^{29} -0.196152 q^{31} +5.19615 q^{32} +6.00000 q^{34} +2.00000 q^{35} -4.00000 q^{37} -7.26795 q^{38} +1.73205 q^{40} +3.46410 q^{41} +10.1962 q^{43} +1.26795 q^{44} +8.19615 q^{46} -6.00000 q^{47} -3.00000 q^{49} -1.73205 q^{50} +1.00000 q^{52} +10.3923 q^{53} +1.26795 q^{55} +3.46410 q^{56} -16.3923 q^{58} +15.1244 q^{59} +12.3923 q^{61} +0.339746 q^{62} +1.00000 q^{64} +1.00000 q^{65} -14.3923 q^{67} -3.46410 q^{68} -3.46410 q^{70} -1.26795 q^{71} -4.00000 q^{73} +6.92820 q^{74} +4.19615 q^{76} +2.53590 q^{77} +12.3923 q^{79} -5.00000 q^{80} -6.00000 q^{82} +6.00000 q^{83} -3.46410 q^{85} -17.6603 q^{86} +2.19615 q^{88} -0.928203 q^{89} +2.00000 q^{91} -4.73205 q^{92} +10.3923 q^{94} +4.19615 q^{95} +2.00000 q^{97} +5.19615 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$-0.709785\pi$
$2$	−1.73205	−1.22474	−0.612372	+	0.790569i	$0.709785\pi$
$3$	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
$11$	1.26795	0.382301	0.191151	−	0.981561i	$-0.438778\pi$
$11$	1.26795	0.382301	0.191151	+	0.981561i	$0.438778\pi$
$12$	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	$-0.637999\pi$
$17$	−3.46410	−0.840168	−0.420084	+	0.907485i	$0.637999\pi$
$18$	0	0
$19$	4.19615	0.962663	0.481332	−	0.876539i	$-0.340153\pi$
$19$	4.19615	0.962663	0.481332	+	0.876539i	$0.340153\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	−2.19615	−0.468221
$23$	−4.73205	−0.986701	−0.493350	−	0.869831i	$-0.664228\pi$
$23$	−4.73205	−0.986701	−0.493350	+	0.869831i	$0.664228\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−1.73205	−0.339683
$27$	0	0
$28$	2.00000	0.377964
$29$	9.46410	1.75744	0.878720	−	0.477338i	$-0.158398\pi$
$29$	9.46410	1.75744	0.878720	+	0.477338i	$0.158398\pi$
$30$	0	0
$31$	−0.196152	−0.0352300	−0.0176150	−	0.999845i	$-0.505607\pi$
$31$	−0.196152	−0.0352300	−0.0176150	+	0.999845i	$0.505607\pi$
$32$	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	$0.606644\pi$
$38$	−7.26795	−1.17902
$39$	0	0
$40$	1.73205	0.273861
$41$	3.46410	0.541002	0.270501	−	0.962720i	$-0.412811\pi$
$41$	3.46410	0.541002	0.270501	+	0.962720i	$0.412811\pi$
$42$	0	0
$43$	10.1962	1.55490	0.777449	−	0.628946i	$-0.216513\pi$
$43$	10.1962	1.55490	0.777449	+	0.628946i	$0.216513\pi$
$44$	1.26795	0.191151
$45$	0	0
$46$	8.19615	1.20846
$47$	−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	$-0.246997\pi$
$53$	10.3923	1.42749	0.713746	+	0.700404i	$0.246997\pi$
$54$	0	0
$55$	1.26795	0.170970
$56$	3.46410	0.462910
$57$	0	0
$58$	−16.3923	−2.15242
$59$	15.1244	1.96902	0.984512	−	0.175319i	$-0.0560957\pi$
$59$	15.1244	1.96902	0.984512	+	0.175319i	$0.0560957\pi$
$60$	0	0
$61$	12.3923	1.58667	0.793336	−	0.608784i	$-0.208342\pi$
$61$	12.3923	1.58667	0.793336	+	0.608784i	$0.208342\pi$
$62$	0.339746	0.0431478
$63$	0	0
$64$	1.00000	0.125000
$65$	1.00000	0.124035
$66$	0	0
$67$	−14.3923	−1.75830	−0.879150	−	0.476545i	$-0.841889\pi$
$67$	−14.3923	−1.75830	−0.879150	+	0.476545i	$0.841889\pi$
$68$	−3.46410	−0.420084
$69$	0	0
$70$	−3.46410	−0.414039
$71$	−1.26795	−0.150478	−0.0752389	−	0.997166i	$-0.523972\pi$
$71$	−1.26795	−0.150478	−0.0752389	+	0.997166i	$0.523972\pi$
$72$	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
$76$	4.19615	0.481332
$77$	2.53590	0.288992
$78$	0	0
$79$	12.3923	1.39424	0.697122	−	0.716953i	$-0.254464\pi$
$79$	12.3923	1.39424	0.697122	+	0.716953i	$0.254464\pi$
$80$	−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
$86$	−17.6603	−1.90435
$87$	0	0
$88$	2.19615	0.234111
$89$	−0.928203	−0.0983893	−0.0491947	−	0.998789i	$-0.515665\pi$
$89$	−0.928203	−0.0983893	−0.0491947	+	0.998789i	$0.515665\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	−4.73205	−0.493350
$93$	0	0
$94$	10.3923	1.07188
$95$	4.19615	0.430516
$96$	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
$101$	−12.9282	−1.28640	−0.643202	−	0.765696i	$-0.722395\pi$
$101$	−12.9282	−1.28640	−0.643202	+	0.765696i	$0.722395\pi$
$102$	0	0
$103$	10.1962	1.00466	0.502328	−	0.864677i	$-0.332477\pi$
$103$	10.1962	1.00466	0.502328	+	0.864677i	$0.332477\pi$
$104$	1.73205	0.169842
$105$	0	0
$106$	−18.0000	−1.74831
$107$	−0.339746	−0.0328445	−0.0164222	−	0.999865i	$-0.505228\pi$
$107$	−0.339746	−0.0328445	−0.0164222	+	0.999865i	$0.505228\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	−2.19615	−0.209395
$111$	0	0
$112$	−10.0000	−0.944911
$113$	−15.4641	−1.45474	−0.727370	−	0.686245i	$-0.759258\pi$
$113$	−15.4641	−1.45474	−0.727370	+	0.686245i	$0.759258\pi$
$114$	0	0
$115$	−4.73205	−0.441266
$116$	9.46410	0.878720
$117$	0	0
$118$	−26.1962	−2.41155
$119$	−6.92820	−0.635107
$120$	0	0
$121$	−9.39230	−0.853846
$122$	−21.4641	−1.94327
$123$	0	0
$124$	−0.196152	−0.0176150
$125$	1.00000	0.0894427
$126$	0	0
$127$	5.80385	0.515008	0.257504	−	0.966277i	$-0.417100\pi$
$127$	5.80385	0.515008	0.257504	+	0.966277i	$0.417100\pi$
$128$	−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
$133$	8.39230	0.727705
$134$	24.9282	2.15347
$135$	0	0
$136$	−6.00000	−0.514496
$137$	12.9282	1.10453	0.552265	−	0.833668i	$-0.313763\pi$
$137$	12.9282	1.10453	0.552265	+	0.833668i	$0.313763\pi$
$138$	0	0
$139$	−8.39230	−0.711826	−0.355913	−	0.934519i	$-0.615830\pi$
$139$	−8.39230	−0.711826	−0.355913	+	0.934519i	$0.615830\pi$
$140$	2.00000	0.169031
$141$	0	0
$142$	2.19615	0.184297
$143$	1.26795	0.106031
$144$	0	0
$145$	9.46410	0.785951
$146$	6.92820	0.573382
$147$	0	0
$148$	−4.00000	−0.328798
$149$	−19.8564	−1.62670	−0.813350	−	0.581775i	$-0.802359\pi$
$149$	−19.8564	−1.62670	−0.813350	+	0.581775i	$0.802359\pi$
$150$	0	0
$151$	−12.1962	−0.992509	−0.496254	−	0.868177i	$-0.665292\pi$
$151$	−12.1962	−0.992509	−0.496254	+	0.868177i	$0.665292\pi$
$152$	7.26795	0.589509
$153$	0	0
$154$	−4.39230	−0.353942
$155$	−0.196152	−0.0157553
$156$	0	0
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	−21.4641	−1.70759
$159$	0	0
$160$	5.19615	0.410792
$161$	−9.46410	−0.745876
$162$	0	0
$163$	6.39230	0.500684	0.250342	−	0.968157i	$-0.419457\pi$
$163$	6.39230	0.500684	0.250342	+	0.968157i	$0.419457\pi$
$164$	3.46410	0.270501
$165$	0	0
$166$	−10.3923	−0.806599
$167$	−12.9282	−1.00041	−0.500207	−	0.865906i	$-0.666743\pi$
$167$	−12.9282	−1.00041	−0.500207	+	0.865906i	$0.666743\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	6.00000	0.460179
$171$	0	0
$172$	10.1962	0.777449
$173$	−15.4641	−1.17571	−0.587857	−	0.808965i	$-0.700028\pi$
$173$	−15.4641	−1.17571	−0.587857	+	0.808965i	$0.700028\pi$
$174$	0	0
$175$	2.00000	0.151186
$176$	−6.33975	−0.477876
$177$	0	0
$178$	1.60770	0.120502
$179$	5.07180	0.379084	0.189542	−	0.981873i	$-0.439300\pi$
$179$	5.07180	0.379084	0.189542	+	0.981873i	$0.439300\pi$
$180$	0	0
$181$	−20.3923	−1.51575	−0.757874	−	0.652401i	$-0.773762\pi$
$181$	−20.3923	−1.51575	−0.757874	+	0.652401i	$0.773762\pi$
$182$	−3.46410	−0.256776
$183$	0	0
$184$	−8.19615	−0.604228
$185$	−4.00000	−0.294086
$186$	0	0
$187$	−4.39230	−0.321197
$188$	−6.00000	−0.437595
$189$	0	0
$190$	−7.26795	−0.527272
$191$	18.9282	1.36960	0.684798	−	0.728733i	$-0.259890\pi$
$191$	18.9282	1.36960	0.684798	+	0.728733i	$0.259890\pi$
$192$	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
$197$	−0.928203	−0.0661317	−0.0330659	−	0.999453i	$-0.510527\pi$
$197$	−0.928203	−0.0661317	−0.0330659	+	0.999453i	$0.510527\pi$
$198$	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
$202$	22.3923	1.57552
$203$	18.9282	1.32850
$204$	0	0
$205$	3.46410	0.241943
$206$	−17.6603	−1.23045
$207$	0	0
$208$	−5.00000	−0.346688
$209$	5.32051	0.368027
$210$	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
$214$	0.588457	0.0402261
$215$	10.1962	0.695372
$216$	0	0
$217$	−0.392305	−0.0266314
$218$	−3.46410	−0.234619
$219$	0	0
$220$	1.26795	0.0854851
$221$	−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
$226$	26.7846	1.78169
$227$	−3.46410	−0.229920	−0.114960	−	0.993370i	$-0.536674\pi$
$227$	−3.46410	−0.229920	−0.114960	+	0.993370i	$0.536674\pi$
$228$	0	0
$229$	−14.3923	−0.951070	−0.475535	−	0.879697i	$-0.657746\pi$
$229$	−14.3923	−0.951070	−0.475535	+	0.879697i	$0.657746\pi$
$230$	8.19615	0.540438
$231$	0	0
$232$	16.3923	1.07621
$233$	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
$236$	15.1244	0.984512
$237$	0	0
$238$	12.0000	0.777844
$239$	3.80385	0.246050	0.123025	−	0.992404i	$-0.460740\pi$
$239$	3.80385	0.246050	0.123025	+	0.992404i	$0.460740\pi$
$240$	0	0
$241$	18.3923	1.18475	0.592376	−	0.805661i	$-0.298190\pi$
$241$	18.3923	1.18475	0.592376	+	0.805661i	$0.298190\pi$
$242$	16.2679	1.04574
$243$	0	0
$244$	12.3923	0.793336
$245$	−3.00000	−0.191663
$246$	0	0
$247$	4.19615	0.266995
$248$	−0.339746	−0.0215739
$249$	0	0
$250$	−1.73205	−0.109545
$251$	14.5359	0.917498	0.458749	−	0.888566i	$-0.348298\pi$
$251$	14.5359	0.917498	0.458749	+	0.888566i	$0.348298\pi$
$252$	0	0
$253$	−6.00000	−0.377217
$254$	−10.0526	−0.630754
$255$	0	0
$256$	19.0000	1.18750
$257$	7.85641	0.490069	0.245035	−	0.969514i	$-0.421201\pi$
$257$	7.85641	0.490069	0.245035	+	0.969514i	$0.421201\pi$
$258$	0	0
$259$	−8.00000	−0.497096
$260$	1.00000	0.0620174
$261$	0	0
$262$	0	0
$263$	−4.73205	−0.291791	−0.145895	−	0.989300i	$-0.546606\pi$
$263$	−4.73205	−0.291791	−0.145895	+	0.989300i	$0.546606\pi$
$264$	0	0
$265$	10.3923	0.638394
$266$	−14.5359	−0.891253
$267$	0	0
$268$	−14.3923	−0.879150
$269$	−7.85641	−0.479014	−0.239507	−	0.970895i	$-0.576986\pi$
$269$	−7.85641	−0.479014	−0.239507	+	0.970895i	$0.576986\pi$
$270$	0	0
$271$	−20.9808	−1.27449	−0.637245	−	0.770661i	$-0.719926\pi$
$271$	−20.9808	−1.27449	−0.637245	+	0.770661i	$0.719926\pi$
$272$	17.3205	1.05021
$273$	0	0
$274$	−22.3923	−1.35277
$275$	1.26795	0.0764602
$276$	0	0
$277$	−5.60770	−0.336934	−0.168467	−	0.985707i	$-0.553882\pi$
$277$	−5.60770	−0.336934	−0.168467	+	0.985707i	$0.553882\pi$
$278$	14.5359	0.871805
$279$	0	0
$280$	3.46410	0.207020
$281$	−1.60770	−0.0959071	−0.0479535	−	0.998850i	$-0.515270\pi$
$281$	−1.60770	−0.0959071	−0.0479535	+	0.998850i	$0.515270\pi$
$282$	0	0
$283$	1.41154	0.0839075	0.0419538	−	0.999120i	$-0.486642\pi$
$283$	1.41154	0.0839075	0.0419538	+	0.999120i	$0.486642\pi$
$284$	−1.26795	−0.0752389
$285$	0	0
$286$	−2.19615	−0.129861
$287$	6.92820	0.408959
$288$	0	0
$289$	−5.00000	−0.294118
$290$	−16.3923	−0.962589
$291$	0	0
$292$	−4.00000	−0.234082
$293$	18.9282	1.10580	0.552899	−	0.833248i	$-0.313522\pi$
$293$	18.9282	1.10580	0.552899	+	0.833248i	$0.313522\pi$
$294$	0	0
$295$	15.1244	0.880574
$296$	−6.92820	−0.402694
$297$	0	0
$298$	34.3923	1.99229
$299$	−4.73205	−0.273662
$300$	0	0
$301$	20.3923	1.17539
$302$	21.1244	1.21557
$303$	0	0
$304$	−20.9808	−1.20333
$305$	12.3923	0.709581
$306$	0	0
$307$	22.7846	1.30039	0.650193	−	0.759769i	$-0.274688\pi$
$307$	22.7846	1.30039	0.650193	+	0.759769i	$0.274688\pi$
$308$	2.53590	0.144496
$309$	0	0
$310$	0.339746	0.0192963
$311$	−4.39230	−0.249065	−0.124532	−	0.992216i	$-0.539743\pi$
$311$	−4.39230	−0.249065	−0.124532	+	0.992216i	$0.539743\pi$
$312$	0	0
$313$	6.39230	0.361314	0.180657	−	0.983546i	$-0.442178\pi$
$313$	6.39230	0.361314	0.180657	+	0.983546i	$0.442178\pi$
$314$	17.3205	0.977453
$315$	0	0
$316$	12.3923	0.697122
$317$	−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
$322$	16.3923	0.913507
$323$	−14.5359	−0.808799
$324$	0	0
$325$	1.00000	0.0554700
$326$	−11.0718	−0.613210
$327$	0	0
$328$	6.00000	0.331295
$329$	−12.0000	−0.661581
$330$	0	0
$331$	−28.5885	−1.57136	−0.785682	−	0.618631i	$-0.787688\pi$
$331$	−28.5885	−1.57136	−0.785682	+	0.618631i	$0.787688\pi$
$332$	6.00000	0.329293
$333$	0	0
$334$	22.3923	1.22525
$335$	−14.3923	−0.786336
$336$	0	0
$337$	−5.60770	−0.305471	−0.152735	−	0.988267i	$-0.548808\pi$
$337$	−5.60770	−0.305471	−0.152735	+	0.988267i	$0.548808\pi$
$338$	−1.73205	−0.0942111
$339$	0	0
$340$	−3.46410	−0.187867
$341$	−0.248711	−0.0134685
$342$	0	0
$343$	−20.0000	−1.07990
$344$	17.6603	0.952177
$345$	0	0
$346$	26.7846	1.43995
$347$	−11.6603	−0.625955	−0.312978	−	0.949761i	$-0.601326\pi$
$347$	−11.6603	−0.625955	−0.312978	+	0.949761i	$0.601326\pi$
$348$	0	0
$349$	6.39230	0.342172	0.171086	−	0.985256i	$-0.445272\pi$
$349$	6.39230	0.342172	0.171086	+	0.985256i	$0.445272\pi$
$350$	−3.46410	−0.185164
$351$	0	0
$352$	6.58846	0.351166
$353$	−27.7128	−1.47500	−0.737502	−	0.675345i	$-0.763995\pi$
$353$	−27.7128	−1.47500	−0.737502	+	0.675345i	$0.763995\pi$
$354$	0	0
$355$	−1.26795	−0.0672958
$356$	−0.928203	−0.0491947
$357$	0	0
$358$	−8.78461	−0.464281
$359$	−8.19615	−0.432576	−0.216288	−	0.976330i	$-0.569395\pi$
$359$	−8.19615	−0.432576	−0.216288	+	0.976330i	$0.569395\pi$
$360$	0	0
$361$	−1.39230	−0.0732792
$362$	35.3205	1.85640
$363$	0	0
$364$	2.00000	0.104828
$365$	−4.00000	−0.209370
$366$	0	0
$367$	22.1962	1.15863	0.579315	−	0.815104i	$-0.303320\pi$
$367$	22.1962	1.15863	0.579315	+	0.815104i	$0.303320\pi$
$368$	23.6603	1.23338
$369$	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	$0.583357\pi$
$374$	7.60770	0.393385
$375$	0	0
$376$	−10.3923	−0.535942
$377$	9.46410	0.487426
$378$	0	0
$379$	−32.9808	−1.69411	−0.847054	−	0.531507i	$-0.821626\pi$
$379$	−32.9808	−1.69411	−0.847054	+	0.531507i	$0.821626\pi$
$380$	4.19615	0.215258
$381$	0	0
$382$	−32.7846	−1.67741
$383$	0.928203	0.0474290	0.0237145	−	0.999719i	$-0.492451\pi$
$383$	0.928203	0.0474290	0.0237145	+	0.999719i	$0.492451\pi$
$384$	0	0
$385$	2.53590	0.129241
$386$	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
$391$	16.3923	0.828994
$392$	−5.19615	−0.262445
$393$	0	0
$394$	1.60770	0.0809945
$395$	12.3923	0.623525
$396$	0	0
$397$	−12.7846	−0.641641	−0.320821	−	0.947140i	$-0.603959\pi$
$397$	−12.7846	−0.641641	−0.320821	+	0.947140i	$0.603959\pi$
$398$	−34.6410	−1.73640
$399$	0	0
$400$	−5.00000	−0.250000
$401$	23.0718	1.15215	0.576075	−	0.817397i	$-0.304584\pi$
$401$	23.0718	1.15215	0.576075	+	0.817397i	$0.304584\pi$
$402$	0	0
$403$	−0.196152	−0.00977105
$404$	−12.9282	−0.643202
$405$	0	0
$406$	−32.7846	−1.62707
$407$	−5.07180	−0.251400
$408$	0	0
$409$	−38.3923	−1.89838	−0.949189	−	0.314708i	$-0.898094\pi$
$409$	−38.3923	−1.89838	−0.949189	+	0.314708i	$0.898094\pi$
$410$	−6.00000	−0.296319
$411$	0	0
$412$	10.1962	0.502328
$413$	30.2487	1.48844
$414$	0	0
$415$	6.00000	0.294528
$416$	5.19615	0.254762
$417$	0	0
$418$	−9.21539	−0.450739
$419$	9.46410	0.462352	0.231176	−	0.972912i	$-0.425743\pi$
$419$	9.46410	0.462352	0.231176	+	0.972912i	$0.425743\pi$
$420$	0	0
$421$	10.7846	0.525610	0.262805	−	0.964849i	$-0.415352\pi$
$421$	10.7846	0.525610	0.262805	+	0.964849i	$0.415352\pi$
$422$	−13.8564	−0.674519
$423$	0	0
$424$	18.0000	0.874157
$425$	−3.46410	−0.168034
$426$	0	0
$427$	24.7846	1.19941
$428$	−0.339746	−0.0164222
$429$	0	0
$430$	−17.6603	−0.851653
$431$	−19.5167	−0.940084	−0.470042	−	0.882644i	$-0.655761\pi$
$431$	−19.5167	−0.940084	−0.470042	+	0.882644i	$0.655761\pi$
$432$	0	0
$433$	−6.78461	−0.326048	−0.163024	−	0.986622i	$-0.552125\pi$
$433$	−6.78461	−0.326048	−0.163024	+	0.986622i	$0.552125\pi$
$434$	0.679492	0.0326167
$435$	0	0
$436$	2.00000	0.0957826
$437$	−19.8564	−0.949861
$438$	0	0
$439$	32.0000	1.52728	0.763638	−	0.645644i	$-0.223411\pi$
$439$	32.0000	1.52728	0.763638	+	0.645644i	$0.223411\pi$
$440$	2.19615	0.104697
$441$	0	0
$442$	6.00000	0.285391
$443$	−34.9808	−1.66199	−0.830993	−	0.556283i	$-0.812227\pi$
$443$	−34.9808	−1.66199	−0.830993	+	0.556283i	$0.812227\pi$
$444$	0	0
$445$	−0.928203	−0.0440011
$446$	−3.46410	−0.164030
$447$	0	0
$448$	2.00000	0.0944911
$449$	−27.4641	−1.29611	−0.648056	−	0.761593i	$-0.724418\pi$
$449$	−27.4641	−1.29611	−0.648056	+	0.761593i	$0.724418\pi$
$450$	0	0
$451$	4.39230	0.206826
$452$	−15.4641	−0.727370
$453$	0	0
$454$	6.00000	0.281594
$455$	2.00000	0.0937614
$456$	0	0
$457$	−30.7846	−1.44004	−0.720022	−	0.693952i	$-0.755868\pi$
$457$	−30.7846	−1.44004	−0.720022	+	0.693952i	$0.755868\pi$
$458$	24.9282	1.16482
$459$	0	0
$460$	−4.73205	−0.220633
$461$	−3.46410	−0.161339	−0.0806696	−	0.996741i	$-0.525706\pi$
$461$	−3.46410	−0.161339	−0.0806696	+	0.996741i	$0.525706\pi$
$462$	0	0
$463$	18.3923	0.854763	0.427381	−	0.904071i	$-0.359436\pi$
$463$	18.3923	0.854763	0.427381	+	0.904071i	$0.359436\pi$
$464$	−47.3205	−2.19680
$465$	0	0
$466$	−10.3923	−0.481414
$467$	−38.1962	−1.76751	−0.883754	−	0.467953i	$-0.844992\pi$
$467$	−38.1962	−1.76751	−0.883754	+	0.467953i	$0.844992\pi$
$468$	0	0
$469$	−28.7846	−1.32915
$470$	10.3923	0.479361
$471$	0	0
$472$	26.1962	1.20578
$473$	12.9282	0.594439
$474$	0	0
$475$	4.19615	0.192533
$476$	−6.92820	−0.317554
$477$	0	0
$478$	−6.58846	−0.301349
$479$	−18.3397	−0.837964	−0.418982	−	0.907994i	$-0.637613\pi$
$479$	−18.3397	−0.837964	−0.418982	+	0.907994i	$0.637613\pi$
$480$	0	0
$481$	−4.00000	−0.182384
$482$	−31.8564	−1.45102
$483$	0	0
$484$	−9.39230	−0.426923
$485$	2.00000	0.0908153
$486$	0	0
$487$	−5.60770	−0.254109	−0.127054	−	0.991896i	$-0.540552\pi$
$487$	−5.60770	−0.254109	−0.127054	+	0.991896i	$0.540552\pi$
$488$	21.4641	0.971634
$489$	0	0
$490$	5.19615	0.234738
$491$	9.46410	0.427109	0.213554	−	0.976931i	$-0.431496\pi$
$491$	9.46410	0.427109	0.213554	+	0.976931i	$0.431496\pi$
$492$	0	0
$493$	−32.7846	−1.47654
$494$	−7.26795	−0.327000
$495$	0	0
$496$	0.980762	0.0440375
$497$	−2.53590	−0.113751
$498$	0	0
$499$	12.9808	0.581099	0.290549	−	0.956860i	$-0.406162\pi$
$499$	12.9808	0.581099	0.290549	+	0.956860i	$0.406162\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	−25.1769	−1.12370
$503$	25.5167	1.13773	0.568866	−	0.822430i	$-0.307382\pi$
$503$	25.5167	1.13773	0.568866	+	0.822430i	$0.307382\pi$
$504$	0	0
$505$	−12.9282	−0.575297
$506$	10.3923	0.461994
$507$	0	0
$508$	5.80385	0.257504
$509$	−32.5359	−1.44213	−0.721064	−	0.692868i	$-0.756347\pi$
$509$	−32.5359	−1.44213	−0.721064	+	0.692868i	$0.756347\pi$
$510$	0	0
$511$	−8.00000	−0.353899
$512$	−8.66025	−0.382733
$513$	0	0
$514$	−13.6077	−0.600210
$515$	10.1962	0.449296
$516$	0	0
$517$	−7.60770	−0.334586
$518$	13.8564	0.608816
$519$	0	0
$520$	1.73205	0.0759555
$521$	7.60770	0.333299	0.166650	−	0.986016i	$-0.446705\pi$
$521$	7.60770	0.333299	0.166650	+	0.986016i	$0.446705\pi$
$522$	0	0
$523$	−13.8038	−0.603600	−0.301800	−	0.953371i	$-0.597588\pi$
$523$	−13.8038	−0.603600	−0.301800	+	0.953371i	$0.597588\pi$
$524$	0	0
$525$	0	0
$526$	8.19615	0.357369
$527$	0.679492	0.0295991
$528$	0	0
$529$	−0.607695	−0.0264215
$530$	−18.0000	−0.781870
$531$	0	0
$532$	8.39230	0.363853
$533$	3.46410	0.150047
$534$	0	0
$535$	−0.339746	−0.0146885
$536$	−24.9282	−1.07673
$537$	0	0
$538$	13.6077	0.586669
$539$	−3.80385	−0.163843
$540$	0	0
$541$	−5.60770	−0.241094	−0.120547	−	0.992708i	$-0.538465\pi$
$541$	−5.60770	−0.241094	−0.120547	+	0.992708i	$0.538465\pi$
$542$	36.3397	1.56093
$543$	0	0
$544$	−18.0000	−0.771744
$545$	2.00000	0.0856706
$546$	0	0
$547$	−1.80385	−0.0771270	−0.0385635	−	0.999256i	$-0.512278\pi$
$547$	−1.80385	−0.0771270	−0.0385635	+	0.999256i	$0.512278\pi$
$548$	12.9282	0.552265
$549$	0	0
$550$	−2.19615	−0.0936443
$551$	39.7128	1.69182
$552$	0	0
$553$	24.7846	1.05395
$554$	9.71281	0.412658
$555$	0	0
$556$	−8.39230	−0.355913
$557$	25.8564	1.09557	0.547786	−	0.836619i	$-0.315471\pi$
$557$	25.8564	1.09557	0.547786	+	0.836619i	$0.315471\pi$
$558$	0	0
$559$	10.1962	0.431251
$560$	−10.0000	−0.422577
$561$	0	0
$562$	2.78461	0.117462
$563$	−16.0526	−0.676535	−0.338267	−	0.941050i	$-0.609841\pi$
$563$	−16.0526	−0.676535	−0.338267	+	0.941050i	$0.609841\pi$
$564$	0	0
$565$	−15.4641	−0.650580
$566$	−2.44486	−0.102765
$567$	0	0
$568$	−2.19615	−0.0921485
$569$	9.46410	0.396756	0.198378	−	0.980126i	$-0.436433\pi$
$569$	9.46410	0.396756	0.198378	+	0.980126i	$0.436433\pi$
$570$	0	0
$571$	15.6077	0.653162	0.326581	−	0.945169i	$-0.394103\pi$
$571$	15.6077	0.653162	0.326581	+	0.945169i	$0.394103\pi$
$572$	1.26795	0.0530156
$573$	0	0
$574$	−12.0000	−0.500870
$575$	−4.73205	−0.197340
$576$	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
$580$	9.46410	0.392975
$581$	12.0000	0.497844
$582$	0	0
$583$	13.1769	0.545732
$584$	−6.92820	−0.286691
$585$	0	0
$586$	−32.7846	−1.35432
$587$	15.4641	0.638272	0.319136	−	0.947709i	$-0.396607\pi$
$587$	15.4641	0.638272	0.319136	+	0.947709i	$0.396607\pi$
$588$	0	0
$589$	−0.823085	−0.0339146
$590$	−26.1962	−1.07848
$591$	0	0
$592$	20.0000	0.821995
$593$	14.7846	0.607131	0.303566	−	0.952811i	$-0.401823\pi$
$593$	14.7846	0.607131	0.303566	+	0.952811i	$0.401823\pi$
$594$	0	0
$595$	−6.92820	−0.284029
$596$	−19.8564	−0.813350
$597$	0	0
$598$	8.19615	0.335166
$599$	28.3923	1.16008	0.580039	−	0.814589i	$-0.303037\pi$
$599$	28.3923	1.16008	0.580039	+	0.814589i	$0.303037\pi$
$600$	0	0
$601$	−39.5692	−1.61406	−0.807031	−	0.590509i	$-0.798927\pi$
$601$	−39.5692	−1.61406	−0.807031	+	0.590509i	$0.798927\pi$
$602$	−35.3205	−1.43956
$603$	0	0
$604$	−12.1962	−0.496254
$605$	−9.39230	−0.381851
$606$	0	0
$607$	−26.9808	−1.09512	−0.547558	−	0.836768i	$-0.684442\pi$
$607$	−26.9808	−1.09512	−0.547558	+	0.836768i	$0.684442\pi$
$608$	21.8038	0.884263
$609$	0	0
$610$	−21.4641	−0.869056
$611$	−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	$0.324041\pi$
$614$	−39.4641	−1.59264
$615$	0	0
$616$	4.39230	0.176971
$617$	21.7128	0.874125	0.437062	−	0.899431i	$-0.356019\pi$
$617$	21.7128	0.874125	0.437062	+	0.899431i	$0.356019\pi$
$618$	0	0
$619$	−44.9808	−1.80793	−0.903965	−	0.427607i	$-0.859357\pi$
$619$	−44.9808	−1.80793	−0.903965	+	0.427607i	$0.859357\pi$
$620$	−0.196152	−0.00787767
$621$	0	0
$622$	7.60770	0.305041
$623$	−1.85641	−0.0743754
$624$	0	0
$625$	1.00000	0.0400000
$626$	−11.0718	−0.442518
$627$	0	0
$628$	−10.0000	−0.399043
$629$	13.8564	0.552491
$630$	0	0
$631$	16.1962	0.644759	0.322379	−	0.946611i	$-0.395517\pi$
$631$	16.1962	0.644759	0.322379	+	0.946611i	$0.395517\pi$
$632$	21.4641	0.853796
$633$	0	0
$634$	41.5692	1.65092
$635$	5.80385	0.230319
$636$	0	0
$637$	−3.00000	−0.118864
$638$	−20.7846	−0.822871
$639$	0	0
$640$	−12.1244	−0.479257
$641$	0.928203	0.0366618	0.0183309	−	0.999832i	$-0.494165\pi$
$641$	0.928203	0.0366618	0.0183309	+	0.999832i	$0.494165\pi$
$642$	0	0
$643$	34.7846	1.37177	0.685886	−	0.727709i	$-0.259415\pi$
$643$	34.7846	1.37177	0.685886	+	0.727709i	$0.259415\pi$
$644$	−9.46410	−0.372938
$645$	0	0
$646$	25.1769	0.990572
$647$	16.0526	0.631091	0.315546	−	0.948910i	$-0.397812\pi$
$647$	16.0526	0.631091	0.315546	+	0.948910i	$0.397812\pi$
$648$	0	0
$649$	19.1769	0.752760
$650$	−1.73205	−0.0679366
$651$	0	0
$652$	6.39230	0.250342
$653$	19.8564	0.777041	0.388521	−	0.921440i	$-0.372986\pi$
$653$	19.8564	0.777041	0.388521	+	0.921440i	$0.372986\pi$
$654$	0	0
$655$	0	0
$656$	−17.3205	−0.676252
$657$	0	0
$658$	20.7846	0.810268
$659$	14.5359	0.566238	0.283119	−	0.959085i	$-0.408631\pi$
$659$	14.5359	0.566238	0.283119	+	0.959085i	$0.408631\pi$
$660$	0	0
$661$	−30.7846	−1.19738	−0.598691	−	0.800980i	$-0.704312\pi$
$661$	−30.7846	−1.19738	−0.598691	+	0.800980i	$0.704312\pi$
$662$	49.5167	1.92452
$663$	0	0
$664$	10.3923	0.403300
$665$	8.39230	0.325440
$666$	0	0
$667$	−44.7846	−1.73407
$668$	−12.9282	−0.500207
$669$	0	0
$670$	24.9282	0.963061
$671$	15.7128	0.606586
$672$	0	0
$673$	6.39230	0.246405	0.123203	−	0.992382i	$-0.460683\pi$
$673$	6.39230	0.246405	0.123203	+	0.992382i	$0.460683\pi$
$674$	9.71281	0.374124
$675$	0	0
$676$	1.00000	0.0384615
$677$	10.3923	0.399409	0.199704	−	0.979856i	$-0.436002\pi$
$677$	10.3923	0.399409	0.199704	+	0.979856i	$0.436002\pi$
$678$	0	0
$679$	4.00000	0.153506
$680$	−6.00000	−0.230089
$681$	0	0
$682$	0.430781	0.0164954
$683$	−39.4641	−1.51005	−0.755026	−	0.655695i	$-0.772376\pi$
$683$	−39.4641	−1.51005	−0.755026	+	0.655695i	$0.772376\pi$
$684$	0	0
$685$	12.9282	0.493961
$686$	34.6410	1.32260
$687$	0	0
$688$	−50.9808	−1.94362
$689$	10.3923	0.395915
$690$	0	0
$691$	45.7654	1.74100	0.870498	−	0.492171i	$-0.163797\pi$
$691$	45.7654	1.74100	0.870498	+	0.492171i	$0.163797\pi$
$692$	−15.4641	−0.587857
$693$	0	0
$694$	20.1962	0.766635
$695$	−8.39230	−0.318338
$696$	0	0
$697$	−12.0000	−0.454532
$698$	−11.0718	−0.419074
$699$	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
$703$	−16.7846	−0.633044
$704$	1.26795	0.0477876
$705$	0	0
$706$	48.0000	1.80650
$707$	−25.8564	−0.972430
$708$	0	0
$709$	9.60770	0.360825	0.180412	−	0.983591i	$-0.442257\pi$
$709$	9.60770	0.360825	0.180412	+	0.983591i	$0.442257\pi$
$710$	2.19615	0.0824201
$711$	0	0
$712$	−1.60770	−0.0602509
$713$	0.928203	0.0347615
$714$	0	0
$715$	1.26795	0.0474186
$716$	5.07180	0.189542
$717$	0	0
$718$	14.1962	0.529796
$719$	−1.85641	−0.0692323	−0.0346161	−	0.999401i	$-0.511021\pi$
$719$	−1.85641	−0.0692323	−0.0346161	+	0.999401i	$0.511021\pi$
$720$	0	0
$721$	20.3923	0.759449
$722$	2.41154	0.0897483
$723$	0	0
$724$	−20.3923	−0.757874
$725$	9.46410	0.351488
$726$	0	0
$727$	13.4115	0.497407	0.248703	−	0.968580i	$-0.419996\pi$
$727$	13.4115	0.497407	0.248703	+	0.968580i	$0.419996\pi$
$728$	3.46410	0.128388
$729$	0	0
$730$	6.92820	0.256424
$731$	−35.3205	−1.30638
$732$	0	0
$733$	38.0000	1.40356	0.701781	−	0.712393i	$-0.252388\pi$
$733$	38.0000	1.40356	0.701781	+	0.712393i	$0.252388\pi$
$734$	−38.4449	−1.41903
$735$	0	0
$736$	−24.5885	−0.906343
$737$	−18.2487	−0.672200
$738$	0	0
$739$	−7.80385	−0.287069	−0.143535	−	0.989645i	$-0.545847\pi$
$739$	−7.80385	−0.287069	−0.143535	+	0.989645i	$0.545847\pi$
$740$	−4.00000	−0.147043
$741$	0	0
$742$	−36.0000	−1.32160
$743$	43.8564	1.60894	0.804468	−	0.593996i	$-0.202451\pi$
$743$	43.8564	1.60894	0.804468	+	0.593996i	$0.202451\pi$
$744$	0	0
$745$	−19.8564	−0.727482
$746$	17.3205	0.634149
$747$	0	0
$748$	−4.39230	−0.160599
$749$	−0.679492	−0.0248281
$750$	0	0
$751$	15.6077	0.569533	0.284766	−	0.958597i	$-0.408084\pi$
$751$	15.6077	0.569533	0.284766	+	0.958597i	$0.408084\pi$
$752$	30.0000	1.09399
$753$	0	0
$754$	−16.3923	−0.596973
$755$	−12.1962	−0.443863
$756$	0	0
$757$	18.3923	0.668480	0.334240	−	0.942488i	$-0.391520\pi$
$757$	18.3923	0.668480	0.334240	+	0.942488i	$0.391520\pi$
$758$	57.1244	2.07485
$759$	0	0
$760$	7.26795	0.263636
$761$	−7.85641	−0.284795	−0.142397	−	0.989810i	$-0.545481\pi$
$761$	−7.85641	−0.284795	−0.142397	+	0.989810i	$0.545481\pi$
$762$	0	0
$763$	4.00000	0.144810
$764$	18.9282	0.684798
$765$	0	0
$766$	−1.60770	−0.0580884
$767$	15.1244	0.546109
$768$	0	0
$769$	−6.78461	−0.244659	−0.122330	−	0.992490i	$-0.539037\pi$
$769$	−6.78461	−0.244659	−0.122330	+	0.992490i	$0.539037\pi$
$770$	−4.39230	−0.158288
$771$	0	0
$772$	−10.0000	−0.359908
$773$	6.92820	0.249190	0.124595	−	0.992208i	$-0.460237\pi$
$773$	6.92820	0.249190	0.124595	+	0.992208i	$0.460237\pi$
$774$	0	0
$775$	−0.196152	−0.00704600
$776$	3.46410	0.124354
$777$	0	0
$778$	10.3923	0.372582
$779$	14.5359	0.520803
$780$	0	0
$781$	−1.60770	−0.0575279
$782$	−28.3923	−1.01531
$783$	0	0
$784$	15.0000	0.535714
$785$	−10.0000	−0.356915
$786$	0	0
$787$	−51.5692	−1.83824	−0.919122	−	0.393973i	$-0.871100\pi$
$787$	−51.5692	−1.83824	−0.919122	+	0.393973i	$0.871100\pi$
$788$	−0.928203	−0.0330659
$789$	0	0
$790$	−21.4641	−0.763658
$791$	−30.9282	−1.09968
$792$	0	0
$793$	12.3923	0.440064
$794$	22.1436	0.785847
$795$	0	0
$796$	20.0000	0.708881
$797$	−28.6410	−1.01452	−0.507258	−	0.861794i	$-0.669341\pi$
$797$	−28.6410	−1.01452	−0.507258	+	0.861794i	$0.669341\pi$
$798$	0	0
$799$	20.7846	0.735307
$800$	5.19615	0.183712
$801$	0	0
$802$	−39.9615	−1.41109
$803$	−5.07180	−0.178980
$804$	0	0
$805$	−9.46410	−0.333566
$806$	0.339746	0.0119670
$807$	0	0
$808$	−22.3923	−0.787759
$809$	9.46410	0.332740	0.166370	−	0.986063i	$-0.446795\pi$
$809$	9.46410	0.332740	0.166370	+	0.986063i	$0.446795\pi$
$810$	0	0
$811$	28.1962	0.990101	0.495050	−	0.868864i	$-0.335150\pi$
$811$	28.1962	0.990101	0.495050	+	0.868864i	$0.335150\pi$
$812$	18.9282	0.664250
$813$	0	0
$814$	8.78461	0.307900
$815$	6.39230	0.223913
$816$	0	0
$817$	42.7846	1.49684
$818$	66.4974	2.32503
$819$	0	0
$820$	3.46410	0.120972
$821$	40.6410	1.41838	0.709191	−	0.705017i	$-0.249061\pi$
$821$	40.6410	1.41838	0.709191	+	0.705017i	$0.249061\pi$
$822$	0	0
$823$	−46.5885	−1.62397	−0.811986	−	0.583677i	$-0.801613\pi$
$823$	−46.5885	−1.62397	−0.811986	+	0.583677i	$0.801613\pi$
$824$	17.6603	0.615224
$825$	0	0
$826$	−52.3923	−1.82296
$827$	−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
$829$	−20.3923	−0.708254	−0.354127	−	0.935197i	$-0.615222\pi$
$829$	−20.3923	−0.708254	−0.354127	+	0.935197i	$0.615222\pi$
$830$	−10.3923	−0.360722
$831$	0	0
$832$	1.00000	0.0346688
$833$	10.3923	0.360072
$834$	0	0
$835$	−12.9282	−0.447399
$836$	5.32051	0.184014
$837$	0	0
$838$	−16.3923	−0.566263
$839$	−17.6603	−0.609700	−0.304850	−	0.952400i	$-0.598606\pi$
$839$	−17.6603	−0.609700	−0.304850	+	0.952400i	$0.598606\pi$
$840$	0	0
$841$	60.5692	2.08859
$842$	−18.6795	−0.643738
$843$	0	0
$844$	8.00000	0.275371
$845$	1.00000	0.0344010
$846$	0	0
$847$	−18.7846	−0.645447
$848$	−51.9615	−1.78437
$849$	0	0
$850$	6.00000	0.205798
$851$	18.9282	0.648850
$852$	0	0
$853$	8.00000	0.273915	0.136957	−	0.990577i	$-0.456268\pi$
$853$	8.00000	0.273915	0.136957	+	0.990577i	$0.456268\pi$
$854$	−42.9282	−1.46897
$855$	0	0
$856$	−0.588457	−0.0201131
$857$	−47.5692	−1.62493	−0.812467	−	0.583007i	$-0.801876\pi$
$857$	−47.5692	−1.62493	−0.812467	+	0.583007i	$0.801876\pi$
$858$	0	0
$859$	45.1769	1.54142	0.770708	−	0.637188i	$-0.219903\pi$
$859$	45.1769	1.54142	0.770708	+	0.637188i	$0.219903\pi$
$860$	10.1962	0.347686
$861$	0	0
$862$	33.8038	1.15136
$863$	−2.78461	−0.0947892	−0.0473946	−	0.998876i	$-0.515092\pi$
$863$	−2.78461	−0.0947892	−0.0473946	+	0.998876i	$0.515092\pi$
$864$	0	0
$865$	−15.4641	−0.525795
$866$	11.7513	0.399325
$867$	0	0
$868$	−0.392305	−0.0133157
$869$	15.7128	0.533021
$870$	0	0
$871$	−14.3923	−0.487665
$872$	3.46410	0.117309
$873$	0	0
$874$	34.3923	1.16334
$875$	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
$880$	−6.33975	−0.213713
$881$	12.6795	0.427183	0.213591	−	0.976923i	$-0.431484\pi$
$881$	12.6795	0.427183	0.213591	+	0.976923i	$0.431484\pi$
$882$	0	0
$883$	34.1962	1.15079	0.575396	−	0.817875i	$-0.304848\pi$
$883$	34.1962	1.15079	0.575396	+	0.817875i	$0.304848\pi$
$884$	−3.46410	−0.116510
$885$	0	0
$886$	60.5885	2.03551
$887$	−17.9090	−0.601324	−0.300662	−	0.953731i	$-0.597208\pi$
$887$	−17.9090	−0.601324	−0.300662	+	0.953731i	$0.597208\pi$
$888$	0	0
$889$	11.6077	0.389310
$890$	1.60770	0.0538901
$891$	0	0
$892$	2.00000	0.0669650
$893$	−25.1769	−0.842513
$894$	0	0
$895$	5.07180	0.169531
$896$	−24.2487	−0.810093
$897$	0	0
$898$	47.5692	1.58741
$899$	−1.85641	−0.0619146
$900$	0	0
$901$	−36.0000	−1.19933
$902$	−7.60770	−0.253309
$903$	0	0
$904$	−26.7846	−0.890843
$905$	−20.3923	−0.677863
$906$	0	0
$907$	39.7654	1.32039	0.660194	−	0.751095i	$-0.270474\pi$
$907$	39.7654	1.32039	0.660194	+	0.751095i	$0.270474\pi$
$908$	−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
$913$	7.60770	0.251778
$914$	53.3205	1.76369
$915$	0	0
$916$	−14.3923	−0.475535
$917$	0	0
$918$	0	0
$919$	−53.1769	−1.75414	−0.877072	−	0.480358i	$-0.840507\pi$
$919$	−53.1769	−1.75414	−0.877072	+	0.480358i	$0.840507\pi$
$920$	−8.19615	−0.270219
$921$	0	0
$922$	6.00000	0.197599
$923$	−1.26795	−0.0417351
$924$	0	0
$925$	−4.00000	−0.131519
$926$	−31.8564	−1.04687
$927$	0	0
$928$	49.1769	1.61431
$929$	−51.4641	−1.68848	−0.844241	−	0.535963i	$-0.819948\pi$
$929$	−51.4641	−1.68848	−0.844241	+	0.535963i	$0.819948\pi$
$930$	0	0
$931$	−12.5885	−0.412570
$932$	6.00000	0.196537
$933$	0	0
$934$	66.1577	2.16475
$935$	−4.39230	−0.143644
$936$	0	0
$937$	−6.78461	−0.221644	−0.110822	−	0.993840i	$-0.535348\pi$
$937$	−6.78461	−0.221644	−0.110822	+	0.993840i	$0.535348\pi$
$938$	49.8564	1.62787
$939$	0	0
$940$	−6.00000	−0.195698
$941$	31.1769	1.01634	0.508169	−	0.861257i	$-0.330322\pi$
$941$	31.1769	1.01634	0.508169	+	0.861257i	$0.330322\pi$
$942$	0	0
$943$	−16.3923	−0.533807
$944$	−75.6218	−2.46128
$945$	0	0
$946$	−22.3923	−0.728037
$947$	28.6410	0.930708	0.465354	−	0.885125i	$-0.345927\pi$
$947$	28.6410	0.930708	0.465354	+	0.885125i	$0.345927\pi$
$948$	0	0
$949$	−4.00000	−0.129845
$950$	−7.26795	−0.235803
$951$	0	0
$952$	−12.0000	−0.388922
$953$	12.9282	0.418786	0.209393	−	0.977832i	$-0.432851\pi$
$953$	12.9282	0.418786	0.209393	+	0.977832i	$0.432851\pi$
$954$	0	0
$955$	18.9282	0.612502
$956$	3.80385	0.123025
$957$	0	0
$958$	31.7654	1.02629
$959$	25.8564	0.834947
$960$	0	0
$961$	−30.9615	−0.998759
$962$	6.92820	0.223374
$963$	0	0
$964$	18.3923	0.592376
$965$	−10.0000	−0.321911
$966$	0	0
$967$	−29.6077	−0.952119	−0.476060	−	0.879413i	$-0.657935\pi$
$967$	−29.6077	−0.952119	−0.476060	+	0.879413i	$0.657935\pi$
$968$	−16.2679	−0.522872
$969$	0	0
$970$	−3.46410	−0.111226
$971$	−5.07180	−0.162762	−0.0813809	−	0.996683i	$-0.525933\pi$
$971$	−5.07180	−0.162762	−0.0813809	+	0.996683i	$0.525933\pi$
$972$	0	0
$973$	−16.7846	−0.538090
$974$	9.71281	0.311219
$975$	0	0
$976$	−61.9615	−1.98334
$977$	−39.7128	−1.27053	−0.635263	−	0.772296i	$-0.719108\pi$
$977$	−39.7128	−1.27053	−0.635263	+	0.772296i	$0.719108\pi$
$978$	0	0
$979$	−1.17691	−0.0376144
$980$	−3.00000	−0.0958315
$981$	0	0
$982$	−16.3923	−0.523099
$983$	13.6077	0.434018	0.217009	−	0.976170i	$-0.430370\pi$
$983$	13.6077	0.434018	0.217009	+	0.976170i	$0.430370\pi$
$984$	0	0
$985$	−0.928203	−0.0295750
$986$	56.7846	1.80839
$987$	0	0
$988$	4.19615	0.133497
$989$	−48.2487	−1.53422
$990$	0	0
$991$	8.00000	0.254128	0.127064	−	0.991894i	$-0.459445\pi$
$991$	8.00000	0.254128	0.127064	+	0.991894i	$0.459445\pi$
$992$	−1.01924	−0.0323608
$993$	0	0
$994$	4.39230	0.139315
$995$	20.0000	0.634043
$996$	0	0
$997$	54.3923	1.72262	0.861311	−	0.508078i	$-0.169644\pi$
$997$	54.3923	1.72262	0.861311	+	0.508078i	$0.169644\pi$
$998$	−22.4833	−0.711698
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	585.2.a.k.1.1		2
3.2	odd	2		65.2.a.c.1.2	✓	2
4.3	odd	2		9360.2.a.cm.1.2		2
5.2	odd	4		2925.2.c.v.2224.2		4
5.3	odd	4		2925.2.c.v.2224.3		4
5.4	even	2		2925.2.a.z.1.2		2
12.11	even	2		1040.2.a.h.1.2		2
13.12	even	2		7605.2.a.be.1.2		2
15.2	even	4		325.2.b.e.274.3		4
15.8	even	4		325.2.b.e.274.2		4
15.14	odd	2		325.2.a.g.1.1		2
21.20	even	2		3185.2.a.k.1.2		2
24.5	odd	2		4160.2.a.y.1.2		2
24.11	even	2		4160.2.a.bj.1.1		2
33.32	even	2		7865.2.a.h.1.1		2
39.2	even	12		845.2.m.c.316.2		4
39.5	even	4		845.2.c.e.506.1		4
39.8	even	4		845.2.c.e.506.3		4
39.11	even	12		845.2.m.a.316.2		4
39.17	odd	6		845.2.e.f.146.2		4
39.20	even	12		845.2.m.c.361.2		4
39.23	odd	6		845.2.e.f.191.2		4
39.29	odd	6		845.2.e.e.191.1		4
39.32	even	12		845.2.m.a.361.2		4
39.35	odd	6		845.2.e.e.146.1		4
39.38	odd	2		845.2.a.d.1.1		2
60.59	even	2		5200.2.a.ca.1.1		2
195.194	odd	2		4225.2.a.w.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.2.a.c.1.2	✓	2	3.2	odd	2
325.2.a.g.1.1		2	15.14	odd	2
325.2.b.e.274.2		4	15.8	even	4
325.2.b.e.274.3		4	15.2	even	4
585.2.a.k.1.1		2	1.1	even	1	trivial
845.2.a.d.1.1		2	39.38	odd	2
845.2.c.e.506.1		4	39.5	even	4
845.2.c.e.506.3		4	39.8	even	4
845.2.e.e.146.1		4	39.35	odd	6
845.2.e.e.191.1		4	39.29	odd	6
845.2.e.f.146.2		4	39.17	odd	6
845.2.e.f.191.2		4	39.23	odd	6
845.2.m.a.316.2		4	39.11	even	12
845.2.m.a.361.2		4	39.32	even	12
845.2.m.c.316.2		4	39.2	even	12
845.2.m.c.361.2		4	39.20	even	12
1040.2.a.h.1.2		2	12.11	even	2
2925.2.a.z.1.2		2	5.4	even	2
2925.2.c.v.2224.2		4	5.2	odd	4
2925.2.c.v.2224.3		4	5.3	odd	4
3185.2.a.k.1.2		2	21.20	even	2
4160.2.a.y.1.2		2	24.5	odd	2
4160.2.a.bj.1.1		2	24.11	even	2
4225.2.a.w.1.2		2	195.194	odd	2
5200.2.a.ca.1.1		2	60.59	even	2
7605.2.a.be.1.2		2	13.12	even	2
7865.2.a.h.1.1		2	33.32	even	2
9360.2.a.cm.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)

\(f(q)\)	\(=\)	\(q-1.73205 q^{2} +1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{7} +1.73205 q^{8} +O(q^{10})\)	\(q-1.73205 q^{2} +1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{7} +1.73205 q^{8} -1.73205 q^{10} +1.26795 q^{11} +1.00000 q^{13} -3.46410 q^{14} -5.00000 q^{16} -3.46410 q^{17} +4.19615 q^{19} +1.00000 q^{20} -2.19615 q^{22} -4.73205 q^{23} +1.00000 q^{25} -1.73205 q^{26} +2.00000 q^{28} +9.46410 q^{29} -0.196152 q^{31} +5.19615 q^{32} +6.00000 q^{34} +2.00000 q^{35} -4.00000 q^{37} -7.26795 q^{38} +1.73205 q^{40} +3.46410 q^{41} +10.1962 q^{43} +1.26795 q^{44} +8.19615 q^{46} -6.00000 q^{47} -3.00000 q^{49} -1.73205 q^{50} +1.00000 q^{52} +10.3923 q^{53} +1.26795 q^{55} +3.46410 q^{56} -16.3923 q^{58} +15.1244 q^{59} +12.3923 q^{61} +0.339746 q^{62} +1.00000 q^{64} +1.00000 q^{65} -14.3923 q^{67} -3.46410 q^{68} -3.46410 q^{70} -1.26795 q^{71} -4.00000 q^{73} +6.92820 q^{74} +4.19615 q^{76} +2.53590 q^{77} +12.3923 q^{79} -5.00000 q^{80} -6.00000 q^{82} +6.00000 q^{83} -3.46410 q^{85} -17.6603 q^{86} +2.19615 q^{88} -0.928203 q^{89} +2.00000 q^{91} -4.73205 q^{92} +10.3923 q^{94} +4.19615 q^{95} +2.00000 q^{97} +5.19615 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} + 2 q^{5} + 4 q^{7}+O(q^{10}) \) 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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