LMFDB - Embedded newform 585.2.a.m.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:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
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			$1.41421$ 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)

$f(q)$	$=$	$q+2.41421 q^{2} +3.82843 q^{4} -1.00000 q^{5} +4.82843 q^{7} +4.41421 q^{8} +O(q^{10})$	\(q+2.41421 q^{2} +3.82843 q^{4} -1.00000 q^{5} +4.82843 q^{7} +4.41421 q^{8} -2.41421 q^{10} -3.41421 q^{11} -1.00000 q^{13} +11.6569 q^{14} +3.00000 q^{16} -0.828427 q^{17} +0.585786 q^{19} -3.82843 q^{20} -8.24264 q^{22} -1.41421 q^{23} +1.00000 q^{25} -2.41421 q^{26} +18.4853 q^{28} +5.65685 q^{29} +1.75736 q^{31} -1.58579 q^{32} -2.00000 q^{34} -4.82843 q^{35} -8.48528 q^{37} +1.41421 q^{38} -4.41421 q^{40} +3.17157 q^{41} -11.0711 q^{43} -13.0711 q^{44} -3.41421 q^{46} +4.82843 q^{47} +16.3137 q^{49} +2.41421 q^{50} -3.82843 q^{52} -2.48528 q^{53} +3.41421 q^{55} +21.3137 q^{56} +13.6569 q^{58} -1.75736 q^{59} -8.00000 q^{61} +4.24264 q^{62} -9.82843 q^{64} +1.00000 q^{65} -2.00000 q^{67} -3.17157 q^{68} -11.6569 q^{70} -11.8995 q^{71} +8.48528 q^{73} -20.4853 q^{74} +2.24264 q^{76} -16.4853 q^{77} -8.48528 q^{79} -3.00000 q^{80} +7.65685 q^{82} +3.17157 q^{83} +0.828427 q^{85} -26.7279 q^{86} -15.0711 q^{88} -6.00000 q^{89} -4.82843 q^{91} -5.41421 q^{92} +11.6569 q^{94} -0.585786 q^{95} -7.65685 q^{97} +39.3848 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 4 q^{7} + 6 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 + 4 * q^7 + 6 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 4 q^{7} + 6 q^{8} - 2 q^{10} - 4 q^{11} - 2 q^{13} + 12 q^{14} + 6 q^{16} + 4 q^{17} + 4 q^{19} - 2 q^{20} - 8 q^{22} + 2 q^{25} - 2 q^{26} + 20 q^{28} + 12 q^{31} - 6 q^{32} - 4 q^{34} - 4 q^{35} - 6 q^{40} + 12 q^{41} - 8 q^{43} - 12 q^{44} - 4 q^{46} + 4 q^{47} + 10 q^{49} + 2 q^{50} - 2 q^{52} + 12 q^{53} + 4 q^{55} + 20 q^{56} + 16 q^{58} - 12 q^{59} - 16 q^{61} - 14 q^{64} + 2 q^{65} - 4 q^{67} - 12 q^{68} - 12 q^{70} - 4 q^{71} - 24 q^{74} - 4 q^{76} - 16 q^{77} - 6 q^{80} + 4 q^{82} + 12 q^{83} - 4 q^{85} - 28 q^{86} - 16 q^{88} - 12 q^{89} - 4 q^{91} - 8 q^{92} + 12 q^{94} - 4 q^{95} - 4 q^{97} + 42 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 + 4 * q^7 + 6 * q^8 - 2 * q^10 - 4 * q^11 - 2 * q^13 + 12 * q^14 + 6 * q^16 + 4 * q^17 + 4 * q^19 - 2 * q^20 - 8 * q^22 + 2 * q^25 - 2 * q^26 + 20 * q^28 + 12 * q^31 - 6 * q^32 - 4 * q^34 - 4 * q^35 - 6 * q^40 + 12 * q^41 - 8 * q^43 - 12 * q^44 - 4 * q^46 + 4 * q^47 + 10 * q^49 + 2 * q^50 - 2 * q^52 + 12 * q^53 + 4 * q^55 + 20 * q^56 + 16 * q^58 - 12 * q^59 - 16 * q^61 - 14 * q^64 + 2 * q^65 - 4 * q^67 - 12 * q^68 - 12 * q^70 - 4 * q^71 - 24 * q^74 - 4 * q^76 - 16 * q^77 - 6 * q^80 + 4 * q^82 + 12 * q^83 - 4 * q^85 - 28 * q^86 - 16 * q^88 - 12 * q^89 - 4 * q^91 - 8 * q^92 + 12 * q^94 - 4 * q^95 - 4 * q^97 + 42 * 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 $

$2$	2.41421	1.70711	0.853553	−	0.521005i	$-0.174443\pi$
$2$	2.41421	1.70711	0.853553	+	0.521005i	$0.174443\pi$
$3$	0	0
$3$	0	0
$4$	3.82843	1.91421
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	4.82843	1.82497	0.912487	−	0.409106i	$-0.134159\pi$
$7$	4.82843	1.82497	0.912487	+	0.409106i	$0.134159\pi$
$8$	4.41421	1.56066
$9$	0	0
$10$	−2.41421	−0.763441
$11$	−3.41421	−1.02942	−0.514712	−	0.857363i	$-0.672101\pi$
$11$	−3.41421	−1.02942	−0.514712	+	0.857363i	$0.672101\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	11.6569	3.11543
$15$	0	0
$16$	3.00000	0.750000
$17$	−0.828427	−0.200923	−0.100462	−	0.994941i	$-0.532032\pi$
$17$	−0.828427	−0.200923	−0.100462	+	0.994941i	$0.532032\pi$
$18$	0	0
$19$	0.585786	0.134389	0.0671943	−	0.997740i	$-0.478595\pi$
$19$	0.585786	0.134389	0.0671943	+	0.997740i	$0.478595\pi$
$20$	−3.82843	−0.856062
$21$	0	0
$22$	−8.24264	−1.75734
$23$	−1.41421	−0.294884	−0.147442	−	0.989071i	$-0.547104\pi$
$23$	−1.41421	−0.294884	−0.147442	+	0.989071i	$0.547104\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−2.41421	−0.473466
$27$	0	0
$28$	18.4853	3.49339
$29$	5.65685	1.05045	0.525226	−	0.850963i	$-0.323981\pi$
$29$	5.65685	1.05045	0.525226	+	0.850963i	$0.323981\pi$
$30$	0	0
$31$	1.75736	0.315631	0.157816	−	0.987469i	$-0.449555\pi$
$31$	1.75736	0.315631	0.157816	+	0.987469i	$0.449555\pi$
$32$	−1.58579	−0.280330
$33$	0	0
$34$	−2.00000	−0.342997
$35$	−4.82843	−0.816153
$36$	0	0
$37$	−8.48528	−1.39497	−0.697486	−	0.716599i	$-0.745698\pi$
$37$	−8.48528	−1.39497	−0.697486	+	0.716599i	$0.745698\pi$
$38$	1.41421	0.229416
$39$	0	0
$40$	−4.41421	−0.697948
$41$	3.17157	0.495316	0.247658	−	0.968847i	$-0.420339\pi$
$41$	3.17157	0.495316	0.247658	+	0.968847i	$0.420339\pi$
$42$	0	0
$43$	−11.0711	−1.68832	−0.844161	−	0.536090i	$-0.819901\pi$
$43$	−11.0711	−1.68832	−0.844161	+	0.536090i	$0.819901\pi$
$44$	−13.0711	−1.97054
$45$	0	0
$46$	−3.41421	−0.503398
$47$	4.82843	0.704298	0.352149	−	0.935944i	$-0.385451\pi$
$47$	4.82843	0.704298	0.352149	+	0.935944i	$0.385451\pi$
$48$	0	0
$49$	16.3137	2.33053
$50$	2.41421	0.341421
$51$	0	0
$52$	−3.82843	−0.530907
$53$	−2.48528	−0.341380	−0.170690	−	0.985325i	$-0.554600\pi$
$53$	−2.48528	−0.341380	−0.170690	+	0.985325i	$0.554600\pi$
$54$	0	0
$55$	3.41421	0.460372
$56$	21.3137	2.84816
$57$	0	0
$58$	13.6569	1.79323
$59$	−1.75736	−0.228789	−0.114394	−	0.993435i	$-0.536493\pi$
$59$	−1.75736	−0.228789	−0.114394	+	0.993435i	$0.536493\pi$
$60$	0	0
$61$	−8.00000	−1.02430	−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000	−1.02430	−0.512148	+	0.858898i	$0.671150\pi$
$62$	4.24264	0.538816
$63$	0	0
$64$	−9.82843	−1.22855
$65$	1.00000	0.124035
$66$	0	0
$67$	−2.00000	−0.244339	−0.122169	−	0.992509i	$-0.538985\pi$
$67$	−2.00000	−0.244339	−0.122169	+	0.992509i	$0.538985\pi$
$68$	−3.17157	−0.384610
$69$	0	0
$70$	−11.6569	−1.39326
$71$	−11.8995	−1.41221	−0.706105	−	0.708107i	$-0.749549\pi$
$71$	−11.8995	−1.41221	−0.706105	+	0.708107i	$0.749549\pi$
$72$	0	0
$73$	8.48528	0.993127	0.496564	−	0.868000i	$-0.334595\pi$
$73$	8.48528	0.993127	0.496564	+	0.868000i	$0.334595\pi$
$74$	−20.4853	−2.38137
$75$	0	0
$76$	2.24264	0.257249
$77$	−16.4853	−1.87867
$78$	0	0
$79$	−8.48528	−0.954669	−0.477334	−	0.878722i	$-0.658397\pi$
$79$	−8.48528	−0.954669	−0.477334	+	0.878722i	$0.658397\pi$
$80$	−3.00000	−0.335410
$81$	0	0
$82$	7.65685	0.845558
$83$	3.17157	0.348125	0.174063	−	0.984735i	$-0.444310\pi$
$83$	3.17157	0.348125	0.174063	+	0.984735i	$0.444310\pi$
$84$	0	0
$85$	0.828427	0.0898555
$86$	−26.7279	−2.88215
$87$	0	0
$88$	−15.0711	−1.60658
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	−4.82843	−0.506157
$92$	−5.41421	−0.564471
$93$	0	0
$94$	11.6569	1.20231
$95$	−0.585786	−0.0601004
$96$	0	0
$97$	−7.65685	−0.777436	−0.388718	−	0.921357i	$-0.627082\pi$
$97$	−7.65685	−0.777436	−0.388718	+	0.921357i	$0.627082\pi$
$98$	39.3848	3.97846
$99$	0	0
$100$	3.82843	0.382843
$101$	3.65685	0.363871	0.181935	−	0.983311i	$-0.441764\pi$
$101$	3.65685	0.363871	0.181935	+	0.983311i	$0.441764\pi$
$102$	0	0
$103$	14.5858	1.43718	0.718590	−	0.695434i	$-0.244788\pi$
$103$	14.5858	1.43718	0.718590	+	0.695434i	$0.244788\pi$
$104$	−4.41421	−0.432849
$105$	0	0
$106$	−6.00000	−0.582772
$107$	9.41421	0.910106	0.455053	−	0.890464i	$-0.349620\pi$
$107$	9.41421	0.910106	0.455053	+	0.890464i	$0.349620\pi$
$108$	0	0
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	8.24264	0.785905
$111$	0	0
$112$	14.4853	1.36873
$113$	8.82843	0.830509	0.415254	−	0.909705i	$-0.363693\pi$
$113$	8.82843	0.830509	0.415254	+	0.909705i	$0.363693\pi$
$114$	0	0
$115$	1.41421	0.131876
$116$	21.6569	2.01079
$117$	0	0
$118$	−4.24264	−0.390567
$119$	−4.00000	−0.366679
$120$	0	0
$121$	0.656854	0.0597140
$122$	−19.3137	−1.74858
$123$	0	0
$124$	6.72792	0.604185
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−6.58579	−0.584394	−0.292197	−	0.956358i	$-0.594386\pi$
$127$	−6.58579	−0.584394	−0.292197	+	0.956358i	$0.594386\pi$
$128$	−20.5563	−1.81694
$129$	0	0
$130$	2.41421	0.211741
$131$	16.9706	1.48272	0.741362	−	0.671105i	$-0.234180\pi$
$131$	16.9706	1.48272	0.741362	+	0.671105i	$0.234180\pi$
$132$	0	0
$133$	2.82843	0.245256
$134$	−4.82843	−0.417113
$135$	0	0
$136$	−3.65685	−0.313573
$137$	17.3137	1.47921	0.739605	−	0.673041i	$-0.235012\pi$
$137$	17.3137	1.47921	0.739605	+	0.673041i	$0.235012\pi$
$138$	0	0
$139$	4.48528	0.380437	0.190218	−	0.981742i	$-0.439080\pi$
$139$	4.48528	0.380437	0.190218	+	0.981742i	$0.439080\pi$
$140$	−18.4853	−1.56229
$141$	0	0
$142$	−28.7279	−2.41079
$143$	3.41421	0.285511
$144$	0	0
$145$	−5.65685	−0.469776
$146$	20.4853	1.69537
$147$	0	0
$148$	−32.4853	−2.67027
$149$	11.6569	0.954967	0.477483	−	0.878641i	$-0.341549\pi$
$149$	11.6569	0.954967	0.477483	+	0.878641i	$0.341549\pi$
$150$	0	0
$151$	9.75736	0.794043	0.397021	−	0.917809i	$-0.370044\pi$
$151$	9.75736	0.794043	0.397021	+	0.917809i	$0.370044\pi$
$152$	2.58579	0.209735
$153$	0	0
$154$	−39.7990	−3.20709
$155$	−1.75736	−0.141154
$156$	0	0
$157$	18.0000	1.43656	0.718278	−	0.695756i	$-0.244931\pi$
$157$	18.0000	1.43656	0.718278	+	0.695756i	$0.244931\pi$
$158$	−20.4853	−1.62972
$159$	0	0
$160$	1.58579	0.125367
$161$	−6.82843	−0.538155
$162$	0	0
$163$	18.9706	1.48589	0.742945	−	0.669353i	$-0.233429\pi$
$163$	18.9706	1.48589	0.742945	+	0.669353i	$0.233429\pi$
$164$	12.1421	0.948141
$165$	0	0
$166$	7.65685	0.594287
$167$	3.17157	0.245424	0.122712	−	0.992442i	$-0.460841\pi$
$167$	3.17157	0.245424	0.122712	+	0.992442i	$0.460841\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	2.00000	0.153393
$171$	0	0
$172$	−42.3848	−3.23181
$173$	−16.8284	−1.27944	−0.639721	−	0.768607i	$-0.720950\pi$
$173$	−16.8284	−1.27944	−0.639721	+	0.768607i	$0.720950\pi$
$174$	0	0
$175$	4.82843	0.364995
$176$	−10.2426	−0.772068
$177$	0	0
$178$	−14.4853	−1.08572
$179$	−5.65685	−0.422813	−0.211407	−	0.977398i	$-0.567804\pi$
$179$	−5.65685	−0.422813	−0.211407	+	0.977398i	$0.567804\pi$
$180$	0	0
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	−11.6569	−0.864064
$183$	0	0
$184$	−6.24264	−0.460214
$185$	8.48528	0.623850
$186$	0	0
$187$	2.82843	0.206835
$188$	18.4853	1.34818
$189$	0	0
$190$	−1.41421	−0.102598
$191$	−2.34315	−0.169544	−0.0847720	−	0.996400i	$-0.527016\pi$
$191$	−2.34315	−0.169544	−0.0847720	+	0.996400i	$0.527016\pi$
$192$	0	0
$193$	4.34315	0.312626	0.156313	−	0.987708i	$-0.450039\pi$
$193$	4.34315	0.312626	0.156313	+	0.987708i	$0.450039\pi$
$194$	−18.4853	−1.32717
$195$	0	0
$196$	62.4558	4.46113
$197$	−10.9706	−0.781620	−0.390810	−	0.920471i	$-0.627805\pi$
$197$	−10.9706	−0.781620	−0.390810	+	0.920471i	$0.627805\pi$
$198$	0	0
$199$	4.00000	0.283552	0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$200$	4.41421	0.312132
$201$	0	0
$202$	8.82843	0.621166
$203$	27.3137	1.91705
$204$	0	0
$205$	−3.17157	−0.221512
$206$	35.2132	2.45342
$207$	0	0
$208$	−3.00000	−0.208013
$209$	−2.00000	−0.138343
$210$	0	0
$211$	3.31371	0.228125	0.114063	−	0.993474i	$-0.463614\pi$
$211$	3.31371	0.228125	0.114063	+	0.993474i	$0.463614\pi$
$212$	−9.51472	−0.653474
$213$	0	0
$214$	22.7279	1.55365
$215$	11.0711	0.755041
$216$	0	0
$217$	8.48528	0.576018
$218$	−4.82843	−0.327022
$219$	0	0
$220$	13.0711	0.881251
$221$	0.828427	0.0557260
$222$	0	0
$223$	9.51472	0.637153	0.318576	−	0.947897i	$-0.396795\pi$
$223$	9.51472	0.637153	0.318576	+	0.947897i	$0.396795\pi$
$224$	−7.65685	−0.511595
$225$	0	0
$226$	21.3137	1.41777
$227$	16.3431	1.08473	0.542366	−	0.840142i	$-0.317528\pi$
$227$	16.3431	1.08473	0.542366	+	0.840142i	$0.317528\pi$
$228$	0	0
$229$	−4.82843	−0.319071	−0.159536	−	0.987192i	$-0.551000\pi$
$229$	−4.82843	−0.319071	−0.159536	+	0.987192i	$0.551000\pi$
$230$	3.41421	0.225127
$231$	0	0
$232$	24.9706	1.63940
$233$	20.6274	1.35135	0.675674	−	0.737201i	$-0.263853\pi$
$233$	20.6274	1.35135	0.675674	+	0.737201i	$0.263853\pi$
$234$	0	0
$235$	−4.82843	−0.314972
$236$	−6.72792	−0.437950
$237$	0	0
$238$	−9.65685	−0.625961
$239$	3.41421	0.220847	0.110424	−	0.993885i	$-0.464779\pi$
$239$	3.41421	0.220847	0.110424	+	0.993885i	$0.464779\pi$
$240$	0	0
$241$	−14.4853	−0.933079	−0.466539	−	0.884500i	$-0.654499\pi$
$241$	−14.4853	−0.933079	−0.466539	+	0.884500i	$0.654499\pi$
$242$	1.58579	0.101938
$243$	0	0
$244$	−30.6274	−1.96072
$245$	−16.3137	−1.04224
$246$	0	0
$247$	−0.585786	−0.0372727
$248$	7.75736	0.492593
$249$	0	0
$250$	−2.41421	−0.152688
$251$	−19.7990	−1.24970	−0.624851	−	0.780744i	$-0.714840\pi$
$251$	−19.7990	−1.24970	−0.624851	+	0.780744i	$0.714840\pi$
$252$	0	0
$253$	4.82843	0.303561
$254$	−15.8995	−0.997623
$255$	0	0
$256$	−29.9706	−1.87316
$257$	−27.6569	−1.72519	−0.862594	−	0.505898i	$-0.831161\pi$
$257$	−27.6569	−1.72519	−0.862594	+	0.505898i	$0.831161\pi$
$258$	0	0
$259$	−40.9706	−2.54579
$260$	3.82843	0.237429
$261$	0	0
$262$	40.9706	2.53117
$263$	10.5858	0.652748	0.326374	−	0.945241i	$-0.394173\pi$
$263$	10.5858	0.652748	0.326374	+	0.945241i	$0.394173\pi$
$264$	0	0
$265$	2.48528	0.152670
$266$	6.82843	0.418678
$267$	0	0
$268$	−7.65685	−0.467717
$269$	25.3137	1.54340	0.771702	−	0.635984i	$-0.219406\pi$
$269$	25.3137	1.54340	0.771702	+	0.635984i	$0.219406\pi$
$270$	0	0
$271$	26.7279	1.62361	0.811803	−	0.583932i	$-0.198486\pi$
$271$	26.7279	1.62361	0.811803	+	0.583932i	$0.198486\pi$
$272$	−2.48528	−0.150692
$273$	0	0
$274$	41.7990	2.52517
$275$	−3.41421	−0.205885
$276$	0	0
$277$	−12.8284	−0.770785	−0.385393	−	0.922753i	$-0.625934\pi$
$277$	−12.8284	−0.770785	−0.385393	+	0.922753i	$0.625934\pi$
$278$	10.8284	0.649446
$279$	0	0
$280$	−21.3137	−1.27374
$281$	−21.7990	−1.30042	−0.650209	−	0.759755i	$-0.725319\pi$
$281$	−21.7990	−1.30042	−0.650209	+	0.759755i	$0.725319\pi$
$282$	0	0
$283$	−16.7279	−0.994372	−0.497186	−	0.867644i	$-0.665633\pi$
$283$	−16.7279	−0.994372	−0.497186	+	0.867644i	$0.665633\pi$
$284$	−45.5563	−2.70327
$285$	0	0
$286$	8.24264	0.487398
$287$	15.3137	0.903940
$288$	0	0
$289$	−16.3137	−0.959630
$290$	−13.6569	−0.801958
$291$	0	0
$292$	32.4853	1.90106
$293$	−26.1421	−1.52724	−0.763620	−	0.645666i	$-0.776580\pi$
$293$	−26.1421	−1.52724	−0.763620	+	0.645666i	$0.776580\pi$
$294$	0	0
$295$	1.75736	0.102317
$296$	−37.4558	−2.17708
$297$	0	0
$298$	28.1421	1.63023
$299$	1.41421	0.0817861
$300$	0	0
$301$	−53.4558	−3.08114
$302$	23.5563	1.35552
$303$	0	0
$304$	1.75736	0.100791
$305$	8.00000	0.458079
$306$	0	0
$307$	24.8284	1.41703	0.708517	−	0.705694i	$-0.249365\pi$
$307$	24.8284	1.41703	0.708517	+	0.705694i	$0.249365\pi$
$308$	−63.1127	−3.59618
$309$	0	0
$310$	−4.24264	−0.240966
$311$	−8.48528	−0.481156	−0.240578	−	0.970630i	$-0.577337\pi$
$311$	−8.48528	−0.481156	−0.240578	+	0.970630i	$0.577337\pi$
$312$	0	0
$313$	−4.82843	−0.272919	−0.136459	−	0.990646i	$-0.543572\pi$
$313$	−4.82843	−0.272919	−0.136459	+	0.990646i	$0.543572\pi$
$314$	43.4558	2.45236
$315$	0	0
$316$	−32.4853	−1.82744
$317$	2.14214	0.120314	0.0601572	−	0.998189i	$-0.480840\pi$
$317$	2.14214	0.120314	0.0601572	+	0.998189i	$0.480840\pi$
$318$	0	0
$319$	−19.3137	−1.08136
$320$	9.82843	0.549426
$321$	0	0
$322$	−16.4853	−0.918689
$323$	−0.485281	−0.0270018
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	45.7990	2.53657
$327$	0	0
$328$	14.0000	0.773021
$329$	23.3137	1.28533
$330$	0	0
$331$	−26.0416	−1.43138	−0.715689	−	0.698419i	$-0.753887\pi$
$331$	−26.0416	−1.43138	−0.715689	+	0.698419i	$0.753887\pi$
$332$	12.1421	0.666386
$333$	0	0
$334$	7.65685	0.418964
$335$	2.00000	0.109272
$336$	0	0
$337$	12.8284	0.698809	0.349404	−	0.936972i	$-0.386384\pi$
$337$	12.8284	0.698809	0.349404	+	0.936972i	$0.386384\pi$
$338$	2.41421	0.131316
$339$	0	0
$340$	3.17157	0.172003
$341$	−6.00000	−0.324918
$342$	0	0
$343$	44.9706	2.42818
$344$	−48.8701	−2.63490
$345$	0	0
$346$	−40.6274	−2.18414
$347$	4.24264	0.227757	0.113878	−	0.993495i	$-0.463673\pi$
$347$	4.24264	0.227757	0.113878	+	0.993495i	$0.463673\pi$
$348$	0	0
$349$	18.4853	0.989494	0.494747	−	0.869037i	$-0.335261\pi$
$349$	18.4853	0.989494	0.494747	+	0.869037i	$0.335261\pi$
$350$	11.6569	0.623085
$351$	0	0
$352$	5.41421	0.288579
$353$	−14.8284	−0.789238	−0.394619	−	0.918845i	$-0.629123\pi$
$353$	−14.8284	−0.789238	−0.394619	+	0.918845i	$0.629123\pi$
$354$	0	0
$355$	11.8995	0.631560
$356$	−22.9706	−1.21744
$357$	0	0
$358$	−13.6569	−0.721787
$359$	8.10051	0.427528	0.213764	−	0.976885i	$-0.431428\pi$
$359$	8.10051	0.427528	0.213764	+	0.976885i	$0.431428\pi$
$360$	0	0
$361$	−18.6569	−0.981940
$362$	0	0
$363$	0	0
$364$	−18.4853	−0.968892
$365$	−8.48528	−0.444140
$366$	0	0
$367$	35.5563	1.85603	0.928013	−	0.372547i	$-0.121516\pi$
$367$	35.5563	1.85603	0.928013	+	0.372547i	$0.121516\pi$
$368$	−4.24264	−0.221163
$369$	0	0
$370$	20.4853	1.06498
$371$	−12.0000	−0.623009
$372$	0	0
$373$	−2.68629	−0.139091	−0.0695455	−	0.997579i	$-0.522155\pi$
$373$	−2.68629	−0.139091	−0.0695455	+	0.997579i	$0.522155\pi$
$374$	6.82843	0.353090
$375$	0	0
$376$	21.3137	1.09917
$377$	−5.65685	−0.291343
$378$	0	0
$379$	29.0711	1.49328	0.746640	−	0.665228i	$-0.231666\pi$
$379$	29.0711	1.49328	0.746640	+	0.665228i	$0.231666\pi$
$380$	−2.24264	−0.115045
$381$	0	0
$382$	−5.65685	−0.289430
$383$	29.1127	1.48759	0.743795	−	0.668408i	$-0.233024\pi$
$383$	29.1127	1.48759	0.743795	+	0.668408i	$0.233024\pi$
$384$	0	0
$385$	16.4853	0.840168
$386$	10.4853	0.533687
$387$	0	0
$388$	−29.3137	−1.48818
$389$	−28.6274	−1.45147	−0.725734	−	0.687976i	$-0.758500\pi$
$389$	−28.6274	−1.45147	−0.725734	+	0.687976i	$0.758500\pi$
$390$	0	0
$391$	1.17157	0.0592490
$392$	72.0122	3.63717
$393$	0	0
$394$	−26.4853	−1.33431
$395$	8.48528	0.426941
$396$	0	0
$397$	11.7990	0.592174	0.296087	−	0.955161i	$-0.404318\pi$
$397$	11.7990	0.592174	0.296087	+	0.955161i	$0.404318\pi$
$398$	9.65685	0.484054
$399$	0	0
$400$	3.00000	0.150000
$401$	5.31371	0.265354	0.132677	−	0.991159i	$-0.457643\pi$
$401$	5.31371	0.265354	0.132677	+	0.991159i	$0.457643\pi$
$402$	0	0
$403$	−1.75736	−0.0875403
$404$	14.0000	0.696526
$405$	0	0
$406$	65.9411	3.27260
$407$	28.9706	1.43602
$408$	0	0
$409$	7.17157	0.354611	0.177306	−	0.984156i	$-0.443262\pi$
$409$	7.17157	0.354611	0.177306	+	0.984156i	$0.443262\pi$
$410$	−7.65685	−0.378145
$411$	0	0
$412$	55.8406	2.75107
$413$	−8.48528	−0.417533
$414$	0	0
$415$	−3.17157	−0.155686
$416$	1.58579	0.0777496
$417$	0	0
$418$	−4.82843	−0.236166
$419$	−10.8284	−0.529003	−0.264502	−	0.964385i	$-0.585207\pi$
$419$	−10.8284	−0.529003	−0.264502	+	0.964385i	$0.585207\pi$
$420$	0	0
$421$	−34.9706	−1.70436	−0.852180	−	0.523248i	$-0.824720\pi$
$421$	−34.9706	−1.70436	−0.852180	+	0.523248i	$0.824720\pi$
$422$	8.00000	0.389434
$423$	0	0
$424$	−10.9706	−0.532778
$425$	−0.828427	−0.0401846
$426$	0	0
$427$	−38.6274	−1.86931
$428$	36.0416	1.74214
$429$	0	0
$430$	26.7279	1.28893
$431$	−40.3848	−1.94527	−0.972633	−	0.232346i	$-0.925360\pi$
$431$	−40.3848	−1.94527	−0.972633	+	0.232346i	$0.925360\pi$
$432$	0	0
$433$	7.65685	0.367965	0.183982	−	0.982930i	$-0.441101\pi$
$433$	7.65685	0.367965	0.183982	+	0.982930i	$0.441101\pi$
$434$	20.4853	0.983325
$435$	0	0
$436$	−7.65685	−0.366697
$437$	−0.828427	−0.0396290
$438$	0	0
$439$	0.970563	0.0463224	0.0231612	−	0.999732i	$-0.492627\pi$
$439$	0.970563	0.0463224	0.0231612	+	0.999732i	$0.492627\pi$
$440$	15.0711	0.718485
$441$	0	0
$442$	2.00000	0.0951303
$443$	9.41421	0.447283	0.223641	−	0.974671i	$-0.428206\pi$
$443$	9.41421	0.447283	0.223641	+	0.974671i	$0.428206\pi$
$444$	0	0
$445$	6.00000	0.284427
$446$	22.9706	1.08769
$447$	0	0
$448$	−47.4558	−2.24208
$449$	33.1127	1.56268	0.781342	−	0.624103i	$-0.214535\pi$
$449$	33.1127	1.56268	0.781342	+	0.624103i	$0.214535\pi$
$450$	0	0
$451$	−10.8284	−0.509891
$452$	33.7990	1.58977
$453$	0	0
$454$	39.4558	1.85175
$455$	4.82843	0.226360
$456$	0	0
$457$	−18.0000	−0.842004	−0.421002	−	0.907060i	$-0.638322\pi$
$457$	−18.0000	−0.842004	−0.421002	+	0.907060i	$0.638322\pi$
$458$	−11.6569	−0.544689
$459$	0	0
$460$	5.41421	0.252439
$461$	−9.51472	−0.443145	−0.221572	−	0.975144i	$-0.571119\pi$
$461$	−9.51472	−0.443145	−0.221572	+	0.975144i	$0.571119\pi$
$462$	0	0
$463$	−4.34315	−0.201843	−0.100922	−	0.994894i	$-0.532179\pi$
$463$	−4.34315	−0.201843	−0.100922	+	0.994894i	$0.532179\pi$
$464$	16.9706	0.787839
$465$	0	0
$466$	49.7990	2.30689
$467$	13.4142	0.620736	0.310368	−	0.950617i	$-0.399548\pi$
$467$	13.4142	0.620736	0.310368	+	0.950617i	$0.399548\pi$
$468$	0	0
$469$	−9.65685	−0.445912
$470$	−11.6569	−0.537691
$471$	0	0
$472$	−7.75736	−0.357061
$473$	37.7990	1.73800
$474$	0	0
$475$	0.585786	0.0268777
$476$	−15.3137	−0.701903
$477$	0	0
$478$	8.24264	0.377010
$479$	30.7279	1.40399	0.701997	−	0.712180i	$-0.252292\pi$
$479$	30.7279	1.40399	0.701997	+	0.712180i	$0.252292\pi$
$480$	0	0
$481$	8.48528	0.386896
$482$	−34.9706	−1.59287
$483$	0	0
$484$	2.51472	0.114305
$485$	7.65685	0.347680
$486$	0	0
$487$	−10.9706	−0.497124	−0.248562	−	0.968616i	$-0.579958\pi$
$487$	−10.9706	−0.497124	−0.248562	+	0.968616i	$0.579958\pi$
$488$	−35.3137	−1.59858
$489$	0	0
$490$	−39.3848	−1.77922
$491$	−5.17157	−0.233390	−0.116695	−	0.993168i	$-0.537230\pi$
$491$	−5.17157	−0.233390	−0.116695	+	0.993168i	$0.537230\pi$
$492$	0	0
$493$	−4.68629	−0.211060
$494$	−1.41421	−0.0636285
$495$	0	0
$496$	5.27208	0.236723
$497$	−57.4558	−2.57725
$498$	0	0
$499$	41.5563	1.86032	0.930159	−	0.367157i	$-0.119669\pi$
$499$	41.5563	1.86032	0.930159	+	0.367157i	$0.119669\pi$
$500$	−3.82843	−0.171212
$501$	0	0
$502$	−47.7990	−2.13337
$503$	−37.8995	−1.68985	−0.844927	−	0.534881i	$-0.820356\pi$
$503$	−37.8995	−1.68985	−0.844927	+	0.534881i	$0.820356\pi$
$504$	0	0
$505$	−3.65685	−0.162728
$506$	11.6569	0.518210
$507$	0	0
$508$	−25.2132	−1.11866
$509$	−41.1127	−1.82229	−0.911144	−	0.412088i	$-0.864800\pi$
$509$	−41.1127	−1.82229	−0.911144	+	0.412088i	$0.864800\pi$
$510$	0	0
$511$	40.9706	1.81243
$512$	−31.2426	−1.38074
$513$	0	0
$514$	−66.7696	−2.94508
$515$	−14.5858	−0.642727
$516$	0	0
$517$	−16.4853	−0.725022
$518$	−98.9117	−4.34593
$519$	0	0
$520$	4.41421	0.193576
$521$	17.6569	0.773561	0.386780	−	0.922172i	$-0.373587\pi$
$521$	17.6569	0.773561	0.386780	+	0.922172i	$0.373587\pi$
$522$	0	0
$523$	−19.7574	−0.863929	−0.431965	−	0.901891i	$-0.642179\pi$
$523$	−19.7574	−0.863929	−0.431965	+	0.901891i	$0.642179\pi$
$524$	64.9706	2.83825
$525$	0	0
$526$	25.5563	1.11431
$527$	−1.45584	−0.0634176
$528$	0	0
$529$	−21.0000	−0.913043
$530$	6.00000	0.260623
$531$	0	0
$532$	10.8284	0.469472
$533$	−3.17157	−0.137376
$534$	0	0
$535$	−9.41421	−0.407012
$536$	−8.82843	−0.381330
$537$	0	0
$538$	61.1127	2.63476
$539$	−55.6985	−2.39910
$540$	0	0
$541$	−7.17157	−0.308330	−0.154165	−	0.988045i	$-0.549269\pi$
$541$	−7.17157	−0.308330	−0.154165	+	0.988045i	$0.549269\pi$
$542$	64.5269	2.77167
$543$	0	0
$544$	1.31371	0.0563248
$545$	2.00000	0.0856706
$546$	0	0
$547$	13.2132	0.564956	0.282478	−	0.959274i	$-0.408844\pi$
$547$	13.2132	0.564956	0.282478	+	0.959274i	$0.408844\pi$
$548$	66.2843	2.83152
$549$	0	0
$550$	−8.24264	−0.351467
$551$	3.31371	0.141169
$552$	0	0
$553$	−40.9706	−1.74225
$554$	−30.9706	−1.31581
$555$	0	0
$556$	17.1716	0.728237
$557$	35.7990	1.51685	0.758426	−	0.651759i	$-0.225969\pi$
$557$	35.7990	1.51685	0.758426	+	0.651759i	$0.225969\pi$
$558$	0	0
$559$	11.0711	0.468256
$560$	−14.4853	−0.612115
$561$	0	0
$562$	−52.6274	−2.21995
$563$	7.75736	0.326934	0.163467	−	0.986549i	$-0.447732\pi$
$563$	7.75736	0.326934	0.163467	+	0.986549i	$0.447732\pi$
$564$	0	0
$565$	−8.82843	−0.371415
$566$	−40.3848	−1.69750
$567$	0	0
$568$	−52.5269	−2.20398
$569$	10.3431	0.433607	0.216804	−	0.976215i	$-0.430437\pi$
$569$	10.3431	0.433607	0.216804	+	0.976215i	$0.430437\pi$
$570$	0	0
$571$	−11.5147	−0.481876	−0.240938	−	0.970541i	$-0.577455\pi$
$571$	−11.5147	−0.481876	−0.240938	+	0.970541i	$0.577455\pi$
$572$	13.0711	0.546529
$573$	0	0
$574$	36.9706	1.54312
$575$	−1.41421	−0.0589768
$576$	0	0
$577$	−34.8284	−1.44993	−0.724963	−	0.688788i	$-0.758143\pi$
$577$	−34.8284	−1.44993	−0.724963	+	0.688788i	$0.758143\pi$
$578$	−39.3848	−1.63819
$579$	0	0
$580$	−21.6569	−0.899252
$581$	15.3137	0.635320
$582$	0	0
$583$	8.48528	0.351424
$584$	37.4558	1.54993
$585$	0	0
$586$	−63.1127	−2.60716
$587$	−20.3431	−0.839651	−0.419826	−	0.907605i	$-0.637909\pi$
$587$	−20.3431	−0.839651	−0.419826	+	0.907605i	$0.637909\pi$
$588$	0	0
$589$	1.02944	0.0424172
$590$	4.24264	0.174667
$591$	0	0
$592$	−25.4558	−1.04623
$593$	24.6274	1.01133	0.505663	−	0.862731i	$-0.331248\pi$
$593$	24.6274	1.01133	0.505663	+	0.862731i	$0.331248\pi$
$594$	0	0
$595$	4.00000	0.163984
$596$	44.6274	1.82801
$597$	0	0
$598$	3.41421	0.139618
$599$	−25.4558	−1.04010	−0.520049	−	0.854137i	$-0.674086\pi$
$599$	−25.4558	−1.04010	−0.520049	+	0.854137i	$0.674086\pi$
$600$	0	0
$601$	−44.6274	−1.82039	−0.910195	−	0.414180i	$-0.864069\pi$
$601$	−44.6274	−1.82039	−0.910195	+	0.414180i	$0.864069\pi$
$602$	−129.054	−5.25984
$603$	0	0
$604$	37.3553	1.51997
$605$	−0.656854	−0.0267049
$606$	0	0
$607$	31.7574	1.28899	0.644496	−	0.764608i	$-0.277067\pi$
$607$	31.7574	1.28899	0.644496	+	0.764608i	$0.277067\pi$
$608$	−0.928932	−0.0376732
$609$	0	0
$610$	19.3137	0.781989
$611$	−4.82843	−0.195337
$612$	0	0
$613$	−14.6863	−0.593174	−0.296587	−	0.955006i	$-0.595848\pi$
$613$	−14.6863	−0.593174	−0.296587	+	0.955006i	$0.595848\pi$
$614$	59.9411	2.41903
$615$	0	0
$616$	−72.7696	−2.93197
$617$	−10.9706	−0.441658	−0.220829	−	0.975313i	$-0.570876\pi$
$617$	−10.9706	−0.441658	−0.220829	+	0.975313i	$0.570876\pi$
$618$	0	0
$619$	1.75736	0.0706342	0.0353171	−	0.999376i	$-0.488756\pi$
$619$	1.75736	0.0706342	0.0353171	+	0.999376i	$0.488756\pi$
$620$	−6.72792	−0.270200
$621$	0	0
$622$	−20.4853	−0.821385
$623$	−28.9706	−1.16068
$624$	0	0
$625$	1.00000	0.0400000
$626$	−11.6569	−0.465902
$627$	0	0
$628$	68.9117	2.74988
$629$	7.02944	0.280282
$630$	0	0
$631$	−9.75736	−0.388434	−0.194217	−	0.980959i	$-0.562217\pi$
$631$	−9.75736	−0.388434	−0.194217	+	0.980959i	$0.562217\pi$
$632$	−37.4558	−1.48991
$633$	0	0
$634$	5.17157	0.205389
$635$	6.58579	0.261349
$636$	0	0
$637$	−16.3137	−0.646373
$638$	−46.6274	−1.84600
$639$	0	0
$640$	20.5563	0.812561
$641$	−47.6569	−1.88233	−0.941166	−	0.337944i	$-0.890269\pi$
$641$	−47.6569	−1.88233	−0.941166	+	0.337944i	$0.890269\pi$
$642$	0	0
$643$	9.51472	0.375224	0.187612	−	0.982243i	$-0.439925\pi$
$643$	9.51472	0.375224	0.187612	+	0.982243i	$0.439925\pi$
$644$	−26.1421	−1.03014
$645$	0	0
$646$	−1.17157	−0.0460949
$647$	−9.41421	−0.370111	−0.185055	−	0.982728i	$-0.559246\pi$
$647$	−9.41421	−0.370111	−0.185055	+	0.982728i	$0.559246\pi$
$648$	0	0
$649$	6.00000	0.235521
$650$	−2.41421	−0.0946932
$651$	0	0
$652$	72.6274	2.84431
$653$	−46.9706	−1.83810	−0.919050	−	0.394141i	$-0.871042\pi$
$653$	−46.9706	−1.83810	−0.919050	+	0.394141i	$0.871042\pi$
$654$	0	0
$655$	−16.9706	−0.663095
$656$	9.51472	0.371487
$657$	0	0
$658$	56.2843	2.19419
$659$	−17.8579	−0.695644	−0.347822	−	0.937561i	$-0.613079\pi$
$659$	−17.8579	−0.695644	−0.347822	+	0.937561i	$0.613079\pi$
$660$	0	0
$661$	29.5980	1.15123	0.575614	−	0.817722i	$-0.304763\pi$
$661$	29.5980	1.15123	0.575614	+	0.817722i	$0.304763\pi$
$662$	−62.8701	−2.44351
$663$	0	0
$664$	14.0000	0.543305
$665$	−2.82843	−0.109682
$666$	0	0
$667$	−8.00000	−0.309761
$668$	12.1421	0.469793
$669$	0	0
$670$	4.82843	0.186538
$671$	27.3137	1.05443
$672$	0	0
$673$	6.48528	0.249989	0.124995	−	0.992157i	$-0.460109\pi$
$673$	6.48528	0.249989	0.124995	+	0.992157i	$0.460109\pi$
$674$	30.9706	1.19294
$675$	0	0
$676$	3.82843	0.147247
$677$	20.1421	0.774125	0.387063	−	0.922053i	$-0.373490\pi$
$677$	20.1421	0.774125	0.387063	+	0.922053i	$0.373490\pi$
$678$	0	0
$679$	−36.9706	−1.41880
$680$	3.65685	0.140234
$681$	0	0
$682$	−14.4853	−0.554670
$683$	−10.6863	−0.408900	−0.204450	−	0.978877i	$-0.565541\pi$
$683$	−10.6863	−0.408900	−0.204450	+	0.978877i	$0.565541\pi$
$684$	0	0
$685$	−17.3137	−0.661523
$686$	108.569	4.14517
$687$	0	0
$688$	−33.2132	−1.26624
$689$	2.48528	0.0946817
$690$	0	0
$691$	6.92893	0.263589	0.131795	−	0.991277i	$-0.457926\pi$
$691$	6.92893	0.263589	0.131795	+	0.991277i	$0.457926\pi$
$692$	−64.4264	−2.44912
$693$	0	0
$694$	10.2426	0.388805
$695$	−4.48528	−0.170136
$696$	0	0
$697$	−2.62742	−0.0995205
$698$	44.6274	1.68917
$699$	0	0
$700$	18.4853	0.698678
$701$	−14.6863	−0.554694	−0.277347	−	0.960770i	$-0.589455\pi$
$701$	−14.6863	−0.554694	−0.277347	+	0.960770i	$0.589455\pi$
$702$	0	0
$703$	−4.97056	−0.187468
$704$	33.5563	1.26470
$705$	0	0
$706$	−35.7990	−1.34731
$707$	17.6569	0.664054
$708$	0	0
$709$	45.1127	1.69424	0.847121	−	0.531399i	$-0.178334\pi$
$709$	45.1127	1.69424	0.847121	+	0.531399i	$0.178334\pi$
$710$	28.7279	1.07814
$711$	0	0
$712$	−26.4853	−0.992578
$713$	−2.48528	−0.0930745
$714$	0	0
$715$	−3.41421	−0.127684
$716$	−21.6569	−0.809355
$717$	0	0
$718$	19.5563	0.729836
$719$	−28.9706	−1.08042	−0.540210	−	0.841530i	$-0.681655\pi$
$719$	−28.9706	−1.08042	−0.540210	+	0.841530i	$0.681655\pi$
$720$	0	0
$721$	70.4264	2.62282
$722$	−45.0416	−1.67628
$723$	0	0
$724$	0	0
$725$	5.65685	0.210090
$726$	0	0
$727$	−51.3553	−1.90466	−0.952332	−	0.305063i	$-0.901322\pi$
$727$	−51.3553	−1.90466	−0.952332	+	0.305063i	$0.901322\pi$
$728$	−21.3137	−0.789939
$729$	0	0
$730$	−20.4853	−0.758194
$731$	9.17157	0.339223
$732$	0	0
$733$	21.3137	0.787240	0.393620	−	0.919273i	$-0.371223\pi$
$733$	21.3137	0.787240	0.393620	+	0.919273i	$0.371223\pi$
$734$	85.8406	3.16844
$735$	0	0
$736$	2.24264	0.0826648
$737$	6.82843	0.251528
$738$	0	0
$739$	5.27208	0.193937	0.0969683	−	0.995287i	$-0.469085\pi$
$739$	5.27208	0.193937	0.0969683	+	0.995287i	$0.469085\pi$
$740$	32.4853	1.19418
$741$	0	0
$742$	−28.9706	−1.06354
$743$	21.5147	0.789298	0.394649	−	0.918832i	$-0.370866\pi$
$743$	21.5147	0.789298	0.394649	+	0.918832i	$0.370866\pi$
$744$	0	0
$745$	−11.6569	−0.427074
$746$	−6.48528	−0.237443
$747$	0	0
$748$	10.8284	0.395927
$749$	45.4558	1.66092
$750$	0	0
$751$	−27.5147	−1.00403	−0.502013	−	0.864860i	$-0.667407\pi$
$751$	−27.5147	−1.00403	−0.502013	+	0.864860i	$0.667407\pi$
$752$	14.4853	0.528224
$753$	0	0
$754$	−13.6569	−0.497353
$755$	−9.75736	−0.355107
$756$	0	0
$757$	−24.1421	−0.877461	−0.438730	−	0.898619i	$-0.644572\pi$
$757$	−24.1421	−0.877461	−0.438730	+	0.898619i	$0.644572\pi$
$758$	70.1838	2.54919
$759$	0	0
$760$	−2.58579	−0.0937963
$761$	−8.62742	−0.312744	−0.156372	−	0.987698i	$-0.549980\pi$
$761$	−8.62742	−0.312744	−0.156372	+	0.987698i	$0.549980\pi$
$762$	0	0
$763$	−9.65685	−0.349602
$764$	−8.97056	−0.324544
$765$	0	0
$766$	70.2843	2.53947
$767$	1.75736	0.0634546
$768$	0	0
$769$	−22.9706	−0.828340	−0.414170	−	0.910200i	$-0.635928\pi$
$769$	−22.9706	−0.828340	−0.414170	+	0.910200i	$0.635928\pi$
$770$	39.7990	1.43426
$771$	0	0
$772$	16.6274	0.598434
$773$	22.1421	0.796397	0.398199	−	0.917299i	$-0.369635\pi$
$773$	22.1421	0.796397	0.398199	+	0.917299i	$0.369635\pi$
$774$	0	0
$775$	1.75736	0.0631262
$776$	−33.7990	−1.21331
$777$	0	0
$778$	−69.1127	−2.47781
$779$	1.85786	0.0665649
$780$	0	0
$781$	40.6274	1.45376
$782$	2.82843	0.101144
$783$	0	0
$784$	48.9411	1.74790
$785$	−18.0000	−0.642448
$786$	0	0
$787$	22.4853	0.801514	0.400757	−	0.916184i	$-0.368747\pi$
$787$	22.4853	0.801514	0.400757	+	0.916184i	$0.368747\pi$
$788$	−42.0000	−1.49619
$789$	0	0
$790$	20.4853	0.728834
$791$	42.6274	1.51566
$792$	0	0
$793$	8.00000	0.284088
$794$	28.4853	1.01090
$795$	0	0
$796$	15.3137	0.542780
$797$	22.9706	0.813659	0.406830	−	0.913504i	$-0.366634\pi$
$797$	22.9706	0.813659	0.406830	+	0.913504i	$0.366634\pi$
$798$	0	0
$799$	−4.00000	−0.141510
$800$	−1.58579	−0.0560660
$801$	0	0
$802$	12.8284	0.452988
$803$	−28.9706	−1.02235
$804$	0	0
$805$	6.82843	0.240670
$806$	−4.24264	−0.149441
$807$	0	0
$808$	16.1421	0.567878
$809$	−45.2548	−1.59108	−0.795538	−	0.605904i	$-0.792811\pi$
$809$	−45.2548	−1.59108	−0.795538	+	0.605904i	$0.792811\pi$
$810$	0	0
$811$	−28.3848	−0.996724	−0.498362	−	0.866969i	$-0.666065\pi$
$811$	−28.3848	−0.996724	−0.498362	+	0.866969i	$0.666065\pi$
$812$	104.569	3.66964
$813$	0	0
$814$	69.9411	2.45144
$815$	−18.9706	−0.664510
$816$	0	0
$817$	−6.48528	−0.226891
$818$	17.3137	0.605360
$819$	0	0
$820$	−12.1421	−0.424022
$821$	51.2548	1.78881	0.894403	−	0.447262i	$-0.147601\pi$
$821$	51.2548	1.78881	0.894403	+	0.447262i	$0.147601\pi$
$822$	0	0
$823$	−2.38478	−0.0831281	−0.0415640	−	0.999136i	$-0.513234\pi$
$823$	−2.38478	−0.0831281	−0.0415640	+	0.999136i	$0.513234\pi$
$824$	64.3848	2.24295
$825$	0	0
$826$	−20.4853	−0.712774
$827$	−56.1421	−1.95225	−0.976127	−	0.217202i	$-0.930307\pi$
$827$	−56.1421	−1.95225	−0.976127	+	0.217202i	$0.930307\pi$
$828$	0	0
$829$	40.9706	1.42297	0.711483	−	0.702703i	$-0.248024\pi$
$829$	40.9706	1.42297	0.711483	+	0.702703i	$0.248024\pi$
$830$	−7.65685	−0.265773
$831$	0	0
$832$	9.82843	0.340739
$833$	−13.5147	−0.468257
$834$	0	0
$835$	−3.17157	−0.109757
$836$	−7.65685	−0.264818
$837$	0	0
$838$	−26.1421	−0.903065
$839$	−6.72792	−0.232274	−0.116137	−	0.993233i	$-0.537051\pi$
$839$	−6.72792	−0.232274	−0.116137	+	0.993233i	$0.537051\pi$
$840$	0	0
$841$	3.00000	0.103448
$842$	−84.4264	−2.90953
$843$	0	0
$844$	12.6863	0.436680
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	3.17157	0.108977
$848$	−7.45584	−0.256035
$849$	0	0
$850$	−2.00000	−0.0685994
$851$	12.0000	0.411355
$852$	0	0
$853$	−13.4558	−0.460719	−0.230360	−	0.973106i	$-0.573990\pi$
$853$	−13.4558	−0.460719	−0.230360	+	0.973106i	$0.573990\pi$
$854$	−93.2548	−3.19111
$855$	0	0
$856$	41.5563	1.42037
$857$	−11.6569	−0.398191	−0.199095	−	0.979980i	$-0.563800\pi$
$857$	−11.6569	−0.398191	−0.199095	+	0.979980i	$0.563800\pi$
$858$	0	0
$859$	−27.7990	−0.948489	−0.474245	−	0.880393i	$-0.657279\pi$
$859$	−27.7990	−0.948489	−0.474245	+	0.880393i	$0.657279\pi$
$860$	42.3848	1.44531
$861$	0	0
$862$	−97.4975	−3.32078
$863$	31.4558	1.07077	0.535385	−	0.844608i	$-0.320167\pi$
$863$	31.4558	1.07077	0.535385	+	0.844608i	$0.320167\pi$
$864$	0	0
$865$	16.8284	0.572184
$866$	18.4853	0.628155
$867$	0	0
$868$	32.4853	1.10262
$869$	28.9706	0.982759
$870$	0	0
$871$	2.00000	0.0677674
$872$	−8.82843	−0.298968
$873$	0	0
$874$	−2.00000	−0.0676510
$875$	−4.82843	−0.163231
$876$	0	0
$877$	25.3137	0.854783	0.427392	−	0.904067i	$-0.359433\pi$
$877$	25.3137	0.854783	0.427392	+	0.904067i	$0.359433\pi$
$878$	2.34315	0.0790773
$879$	0	0
$880$	10.2426	0.345279
$881$	−19.0294	−0.641118	−0.320559	−	0.947229i	$-0.603871\pi$
$881$	−19.0294	−0.641118	−0.320559	+	0.947229i	$0.603871\pi$
$882$	0	0
$883$	−23.7574	−0.799499	−0.399749	−	0.916624i	$-0.630903\pi$
$883$	−23.7574	−0.799499	−0.399749	+	0.916624i	$0.630903\pi$
$884$	3.17157	0.106672
$885$	0	0
$886$	22.7279	0.763559
$887$	22.3848	0.751607	0.375804	−	0.926699i	$-0.377367\pi$
$887$	22.3848	0.751607	0.375804	+	0.926699i	$0.377367\pi$
$888$	0	0
$889$	−31.7990	−1.06650
$890$	14.4853	0.485548
$891$	0	0
$892$	36.4264	1.21965
$893$	2.82843	0.0946497
$894$	0	0
$895$	5.65685	0.189088
$896$	−99.2548	−3.31587
$897$	0	0
$898$	79.9411	2.66767
$899$	9.94113	0.331555
$900$	0	0
$901$	2.05887	0.0685911
$902$	−26.1421	−0.870438
$903$	0	0
$904$	38.9706	1.29614
$905$	0	0
$906$	0	0
$907$	9.21320	0.305919	0.152960	−	0.988232i	$-0.451120\pi$
$907$	9.21320	0.305919	0.152960	+	0.988232i	$0.451120\pi$
$908$	62.5685	2.07641
$909$	0	0
$910$	11.6569	0.386421
$911$	12.0000	0.397578	0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000	0.397578	0.198789	+	0.980042i	$0.436299\pi$
$912$	0	0
$913$	−10.8284	−0.358369
$914$	−43.4558	−1.43739
$915$	0	0
$916$	−18.4853	−0.610771
$917$	81.9411	2.70593
$918$	0	0
$919$	0.485281	0.0160080	0.00800398	−	0.999968i	$-0.497452\pi$
$919$	0.485281	0.0160080	0.00800398	+	0.999968i	$0.497452\pi$
$920$	6.24264	0.205814
$921$	0	0
$922$	−22.9706	−0.756495
$923$	11.8995	0.391677
$924$	0	0
$925$	−8.48528	−0.278994
$926$	−10.4853	−0.344568
$927$	0	0
$928$	−8.97056	−0.294473
$929$	−16.8284	−0.552123	−0.276061	−	0.961140i	$-0.589029\pi$
$929$	−16.8284	−0.552123	−0.276061	+	0.961140i	$0.589029\pi$
$930$	0	0
$931$	9.55635	0.313197
$932$	78.9706	2.58677
$933$	0	0
$934$	32.3848	1.05966
$935$	−2.82843	−0.0924995
$936$	0	0
$937$	−22.9706	−0.750416	−0.375208	−	0.926941i	$-0.622429\pi$
$937$	−22.9706	−0.750416	−0.375208	+	0.926941i	$0.622429\pi$
$938$	−23.3137	−0.761220
$939$	0	0
$940$	−18.4853	−0.602923
$941$	−18.7696	−0.611870	−0.305935	−	0.952052i	$-0.598969\pi$
$941$	−18.7696	−0.611870	−0.305935	+	0.952052i	$0.598969\pi$
$942$	0	0
$943$	−4.48528	−0.146061
$944$	−5.27208	−0.171592
$945$	0	0
$946$	91.2548	2.96695
$947$	−17.1127	−0.556088	−0.278044	−	0.960568i	$-0.589686\pi$
$947$	−17.1127	−0.556088	−0.278044	+	0.960568i	$0.589686\pi$
$948$	0	0
$949$	−8.48528	−0.275444
$950$	1.41421	0.0458831
$951$	0	0
$952$	−17.6569	−0.572262
$953$	−35.2548	−1.14202	−0.571008	−	0.820944i	$-0.693447\pi$
$953$	−35.2548	−1.14202	−0.571008	+	0.820944i	$0.693447\pi$
$954$	0	0
$955$	2.34315	0.0758224
$956$	13.0711	0.422749
$957$	0	0
$958$	74.1838	2.39677
$959$	83.5980	2.69952
$960$	0	0
$961$	−27.9117	−0.900377
$962$	20.4853	0.660472
$963$	0	0
$964$	−55.4558	−1.78611
$965$	−4.34315	−0.139811
$966$	0	0
$967$	47.9411	1.54168	0.770841	−	0.637027i	$-0.219836\pi$
$967$	47.9411	1.54168	0.770841	+	0.637027i	$0.219836\pi$
$968$	2.89949	0.0931933
$969$	0	0
$970$	18.4853	0.593527
$971$	−44.2843	−1.42115	−0.710575	−	0.703622i	$-0.751565\pi$
$971$	−44.2843	−1.42115	−0.710575	+	0.703622i	$0.751565\pi$
$972$	0	0
$973$	21.6569	0.694287
$974$	−26.4853	−0.848643
$975$	0	0
$976$	−24.0000	−0.768221
$977$	39.5147	1.26419	0.632094	−	0.774892i	$-0.282196\pi$
$977$	39.5147	1.26419	0.632094	+	0.774892i	$0.282196\pi$
$978$	0	0
$979$	20.4853	0.654712
$980$	−62.4558	−1.99508
$981$	0	0
$982$	−12.4853	−0.398421
$983$	1.02944	0.0328339	0.0164170	−	0.999865i	$-0.494774\pi$
$983$	1.02944	0.0328339	0.0164170	+	0.999865i	$0.494774\pi$
$984$	0	0
$985$	10.9706	0.349551
$986$	−11.3137	−0.360302
$987$	0	0
$988$	−2.24264	−0.0713479
$989$	15.6569	0.497859
$990$	0	0
$991$	48.9706	1.55560	0.777801	−	0.628511i	$-0.216335\pi$
$991$	48.9706	1.55560	0.777801	+	0.628511i	$0.216335\pi$
$992$	−2.78680	−0.0884809
$993$	0	0
$994$	−138.711	−4.39964
$995$	−4.00000	−0.126809
$996$	0	0
$997$	28.8284	0.913005	0.456503	−	0.889722i	$-0.349102\pi$
$997$	28.8284	0.913005	0.456503	+	0.889722i	$0.349102\pi$
$998$	100.326	3.17576
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	585.2.a.m.1.2		2
3.2	odd	2		65.2.a.b.1.1	✓	2
4.3	odd	2		9360.2.a.cd.1.1		2
5.2	odd	4		2925.2.c.r.2224.4		4
5.3	odd	4		2925.2.c.r.2224.1		4
5.4	even	2		2925.2.a.u.1.1		2
12.11	even	2		1040.2.a.j.1.2		2
13.12	even	2		7605.2.a.x.1.1		2
15.2	even	4		325.2.b.f.274.1		4
15.8	even	4		325.2.b.f.274.4		4
15.14	odd	2		325.2.a.i.1.2		2
21.20	even	2		3185.2.a.j.1.1		2
24.5	odd	2		4160.2.a.bf.1.2		2
24.11	even	2		4160.2.a.z.1.1		2
33.32	even	2		7865.2.a.j.1.2		2
39.2	even	12		845.2.m.f.316.1		8
39.5	even	4		845.2.c.b.506.4		4
39.8	even	4		845.2.c.b.506.1		4
39.11	even	12		845.2.m.f.316.4		8
39.17	odd	6		845.2.e.c.146.1		4
39.20	even	12		845.2.m.f.361.1		8
39.23	odd	6		845.2.e.c.191.1		4
39.29	odd	6		845.2.e.h.191.2		4
39.32	even	12		845.2.m.f.361.4		8
39.35	odd	6		845.2.e.h.146.2		4
39.38	odd	2		845.2.a.g.1.2		2
60.59	even	2		5200.2.a.bu.1.1		2
195.194	odd	2		4225.2.a.r.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.2.a.b.1.1	✓	2	3.2	odd	2
325.2.a.i.1.2		2	15.14	odd	2
325.2.b.f.274.1		4	15.2	even	4
325.2.b.f.274.4		4	15.8	even	4
585.2.a.m.1.2		2	1.1	even	1	trivial
845.2.a.g.1.2		2	39.38	odd	2
845.2.c.b.506.1		4	39.8	even	4
845.2.c.b.506.4		4	39.5	even	4
845.2.e.c.146.1		4	39.17	odd	6
845.2.e.c.191.1		4	39.23	odd	6
845.2.e.h.146.2		4	39.35	odd	6
845.2.e.h.191.2		4	39.29	odd	6
845.2.m.f.316.1		8	39.2	even	12
845.2.m.f.316.4		8	39.11	even	12
845.2.m.f.361.1		8	39.20	even	12
845.2.m.f.361.4		8	39.32	even	12
1040.2.a.j.1.2		2	12.11	even	2
2925.2.a.u.1.1		2	5.4	even	2
2925.2.c.r.2224.1		4	5.3	odd	4
2925.2.c.r.2224.4		4	5.2	odd	4
3185.2.a.j.1.1		2	21.20	even	2
4160.2.a.z.1.1		2	24.11	even	2
4160.2.a.bf.1.2		2	24.5	odd	2
4225.2.a.r.1.1		2	195.194	odd	2
5200.2.a.bu.1.1		2	60.59	even	2
7605.2.a.x.1.1		2	13.12	even	2
7865.2.a.j.1.2		2	33.32	even	2
9360.2.a.cd.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)

\(f(q)\)	\(=\)	\(q+2.41421 q^{2} +3.82843 q^{4} -1.00000 q^{5} +4.82843 q^{7} +4.41421 q^{8} +O(q^{10})\)	\(q+2.41421 q^{2} +3.82843 q^{4} -1.00000 q^{5} +4.82843 q^{7} +4.41421 q^{8} -2.41421 q^{10} -3.41421 q^{11} -1.00000 q^{13} +11.6569 q^{14} +3.00000 q^{16} -0.828427 q^{17} +0.585786 q^{19} -3.82843 q^{20} -8.24264 q^{22} -1.41421 q^{23} +1.00000 q^{25} -2.41421 q^{26} +18.4853 q^{28} +5.65685 q^{29} +1.75736 q^{31} -1.58579 q^{32} -2.00000 q^{34} -4.82843 q^{35} -8.48528 q^{37} +1.41421 q^{38} -4.41421 q^{40} +3.17157 q^{41} -11.0711 q^{43} -13.0711 q^{44} -3.41421 q^{46} +4.82843 q^{47} +16.3137 q^{49} +2.41421 q^{50} -3.82843 q^{52} -2.48528 q^{53} +3.41421 q^{55} +21.3137 q^{56} +13.6569 q^{58} -1.75736 q^{59} -8.00000 q^{61} +4.24264 q^{62} -9.82843 q^{64} +1.00000 q^{65} -2.00000 q^{67} -3.17157 q^{68} -11.6569 q^{70} -11.8995 q^{71} +8.48528 q^{73} -20.4853 q^{74} +2.24264 q^{76} -16.4853 q^{77} -8.48528 q^{79} -3.00000 q^{80} +7.65685 q^{82} +3.17157 q^{83} +0.828427 q^{85} -26.7279 q^{86} -15.0711 q^{88} -6.00000 q^{89} -4.82843 q^{91} -5.41421 q^{92} +11.6569 q^{94} -0.585786 q^{95} -7.65685 q^{97} +39.3848 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 4 q^{7} + 6 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 + 4 * q^7 + 6 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 4 q^{7} + 6 q^{8} - 2 q^{10} - 4 q^{11} - 2 q^{13} + 12 q^{14} + 6 q^{16} + 4 q^{17} + 4 q^{19} - 2 q^{20} - 8 q^{22} + 2 q^{25} - 2 q^{26} + 20 q^{28} + 12 q^{31} - 6 q^{32} - 4 q^{34} - 4 q^{35} - 6 q^{40} + 12 q^{41} - 8 q^{43} - 12 q^{44} - 4 q^{46} + 4 q^{47} + 10 q^{49} + 2 q^{50} - 2 q^{52} + 12 q^{53} + 4 q^{55} + 20 q^{56} + 16 q^{58} - 12 q^{59} - 16 q^{61} - 14 q^{64} + 2 q^{65} - 4 q^{67} - 12 q^{68} - 12 q^{70} - 4 q^{71} - 24 q^{74} - 4 q^{76} - 16 q^{77} - 6 q^{80} + 4 q^{82} + 12 q^{83} - 4 q^{85} - 28 q^{86} - 16 q^{88} - 12 q^{89} - 4 q^{91} - 8 q^{92} + 12 q^{94} - 4 q^{95} - 4 q^{97} + 42 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 + 4 * q^7 + 6 * q^8 - 2 * q^10 - 4 * q^11 - 2 * q^13 + 12 * q^14 + 6 * q^16 + 4 * q^17 + 4 * q^19 - 2 * q^20 - 8 * q^22 + 2 * q^25 - 2 * q^26 + 20 * q^28 + 12 * q^31 - 6 * q^32 - 4 * q^34 - 4 * q^35 - 6 * q^40 + 12 * q^41 - 8 * q^43 - 12 * q^44 - 4 * q^46 + 4 * q^47 + 10 * q^49 + 2 * q^50 - 2 * q^52 + 12 * q^53 + 4 * q^55 + 20 * q^56 + 16 * q^58 - 12 * q^59 - 16 * q^61 - 14 * q^64 + 2 * q^65 - 4 * q^67 - 12 * q^68 - 12 * q^70 - 4 * q^71 - 24 * q^74 - 4 * q^76 - 16 * q^77 - 6 * q^80 + 4 * q^82 + 12 * q^83 - 4 * q^85 - 28 * q^86 - 16 * q^88 - 12 * q^89 - 4 * q^91 - 8 * q^92 + 12 * q^94 - 4 * q^95 - 4 * q^97 + 42 * q^98

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