LMFDB - Embedded newform 4598.2.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4598,2,Mod(1,4598)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4598, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4598.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4598 = 2 \cdot 11^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4598.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:	$36.7152148494$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	4598.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} -3.00000 q^{6} -1.00000 q^{8} +6.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} -3.00000 q^{6} -1.00000 q^{8} +6.00000 q^{9} +3.00000 q^{12} -7.00000 q^{13} +1.00000 q^{16} +7.00000 q^{17} -6.00000 q^{18} -1.00000 q^{19} +8.00000 q^{23} -3.00000 q^{24} -5.00000 q^{25} +7.00000 q^{26} +9.00000 q^{27} +9.00000 q^{29} +2.00000 q^{31} -1.00000 q^{32} -7.00000 q^{34} +6.00000 q^{36} +3.00000 q^{37} +1.00000 q^{38} -21.0000 q^{39} -10.0000 q^{41} +10.0000 q^{43} -8.00000 q^{46} +3.00000 q^{47} +3.00000 q^{48} -7.00000 q^{49} +5.00000 q^{50} +21.0000 q^{51} -7.00000 q^{52} +5.00000 q^{53} -9.00000 q^{54} -3.00000 q^{57} -9.00000 q^{58} +4.00000 q^{59} -6.00000 q^{61} -2.00000 q^{62} +1.00000 q^{64} -3.00000 q^{67} +7.00000 q^{68} +24.0000 q^{69} +2.00000 q^{71} -6.00000 q^{72} -3.00000 q^{73} -3.00000 q^{74} -15.0000 q^{75} -1.00000 q^{76} +21.0000 q^{78} +14.0000 q^{79} +9.00000 q^{81} +10.0000 q^{82} -6.00000 q^{83} -10.0000 q^{86} +27.0000 q^{87} +6.00000 q^{89} +8.00000 q^{92} +6.00000 q^{93} -3.00000 q^{94} -3.00000 q^{96} +14.0000 q^{97} +7.00000 q^{98} +O(q^{100})\)

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	3.00000	1.73205	0.866025	−	0.500000i	$-0.166667\pi$
$3$	3.00000	1.73205	0.866025	+	0.500000i	$0.166667\pi$
$4$	1.00000	0.500000
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	−3.00000	−1.22474
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	−1.00000	−0.353553
$9$	6.00000	2.00000
$10$	0	0
$11$	0	0
$11$	0	0
$12$	3.00000	0.866025
$13$	−7.00000	−1.94145	−0.970725	−	0.240192i	$-0.922790\pi$
$13$	−7.00000	−1.94145	−0.970725	+	0.240192i	$0.922790\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	7.00000	1.69775	0.848875	−	0.528594i	$-0.177281\pi$
$17$	7.00000	1.69775	0.848875	+	0.528594i	$0.177281\pi$
$18$	−6.00000	−1.41421
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	8.00000	1.66812	0.834058	−	0.551677i	$-0.186012\pi$
$23$	8.00000	1.66812	0.834058	+	0.551677i	$0.186012\pi$
$24$	−3.00000	−0.612372
$25$	−5.00000	−1.00000
$26$	7.00000	1.37281
$27$	9.00000	1.73205
$28$	0	0
$29$	9.00000	1.67126	0.835629	−	0.549294i	$-0.185103\pi$
$29$	9.00000	1.67126	0.835629	+	0.549294i	$0.185103\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−7.00000	−1.20049
$35$	0	0
$36$	6.00000	1.00000
$37$	3.00000	0.493197	0.246598	−	0.969118i	$-0.420687\pi$
$37$	3.00000	0.493197	0.246598	+	0.969118i	$0.420687\pi$
$38$	1.00000	0.162221
$39$	−21.0000	−3.36269
$40$	0	0
$41$	−10.0000	−1.56174	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000	−1.56174	−0.780869	+	0.624695i	$0.785223\pi$
$42$	0	0
$43$	10.0000	1.52499	0.762493	−	0.646997i	$-0.223975\pi$
$43$	10.0000	1.52499	0.762493	+	0.646997i	$0.223975\pi$
$44$	0	0
$45$	0	0
$46$	−8.00000	−1.17954
$47$	3.00000	0.437595	0.218797	−	0.975770i	$-0.429787\pi$
$47$	3.00000	0.437595	0.218797	+	0.975770i	$0.429787\pi$
$48$	3.00000	0.433013
$49$	−7.00000	−1.00000
$50$	5.00000	0.707107
$51$	21.0000	2.94059
$52$	−7.00000	−0.970725
$53$	5.00000	0.686803	0.343401	−	0.939189i	$-0.388421\pi$
$53$	5.00000	0.686803	0.343401	+	0.939189i	$0.388421\pi$
$54$	−9.00000	−1.22474
$55$	0	0
$56$	0	0
$57$	−3.00000	−0.397360
$58$	−9.00000	−1.18176
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	−2.00000	−0.254000
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−3.00000	−0.366508	−0.183254	−	0.983066i	$-0.558663\pi$
$67$	−3.00000	−0.366508	−0.183254	+	0.983066i	$0.558663\pi$
$68$	7.00000	0.848875
$69$	24.0000	2.88926
$70$	0	0
$71$	2.00000	0.237356	0.118678	−	0.992933i	$-0.462134\pi$
$71$	2.00000	0.237356	0.118678	+	0.992933i	$0.462134\pi$
$72$	−6.00000	−0.707107
$73$	−3.00000	−0.351123	−0.175562	−	0.984468i	$-0.556174\pi$
$73$	−3.00000	−0.351123	−0.175562	+	0.984468i	$0.556174\pi$
$74$	−3.00000	−0.348743
$75$	−15.0000	−1.73205
$76$	−1.00000	−0.114708
$77$	0	0
$78$	21.0000	2.37778
$79$	14.0000	1.57512	0.787562	−	0.616236i	$-0.211343\pi$
$79$	14.0000	1.57512	0.787562	+	0.616236i	$0.211343\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	10.0000	1.10432
$83$	−6.00000	−0.658586	−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000	−0.658586	−0.329293	+	0.944228i	$0.606810\pi$
$84$	0	0
$85$	0	0
$86$	−10.0000	−1.07833
$87$	27.0000	2.89470
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	0	0
$92$	8.00000	0.834058
$93$	6.00000	0.622171
$94$	−3.00000	−0.309426
$95$	0	0
$96$	−3.00000	−0.306186
$97$	14.0000	1.42148	0.710742	−	0.703452i	$-0.248359\pi$
$97$	14.0000	1.42148	0.710742	+	0.703452i	$0.248359\pi$
$98$	7.00000	0.707107
$99$	0	0
$100$	−5.00000	−0.500000
$101$	8.00000	0.796030	0.398015	−	0.917379i	$-0.369699\pi$
$101$	8.00000	0.796030	0.398015	+	0.917379i	$0.369699\pi$
$102$	−21.0000	−2.07931
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	7.00000	0.686406
$105$	0	0
$106$	−5.00000	−0.485643
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	9.00000	0.866025
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	9.00000	0.854242
$112$	0	0
$113$	−20.0000	−1.88144	−0.940721	−	0.339182i	$-0.889850\pi$
$113$	−20.0000	−1.88144	−0.940721	+	0.339182i	$0.889850\pi$
$114$	3.00000	0.280976
$115$	0	0
$116$	9.00000	0.835629
$117$	−42.0000	−3.88290
$118$	−4.00000	−0.368230
$119$	0	0
$120$	0	0
$121$	0	0
$122$	6.00000	0.543214
$123$	−30.0000	−2.70501
$124$	2.00000	0.179605
$125$	0	0
$126$	0	0
$127$	10.0000	0.887357	0.443678	−	0.896186i	$-0.353673\pi$
$127$	10.0000	0.887357	0.443678	+	0.896186i	$0.353673\pi$
$128$	−1.00000	−0.0883883
$129$	30.0000	2.64135
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	0	0
$133$	0	0
$134$	3.00000	0.259161
$135$	0	0
$136$	−7.00000	−0.600245
$137$	−9.00000	−0.768922	−0.384461	−	0.923141i	$-0.625613\pi$
$137$	−9.00000	−0.768922	−0.384461	+	0.923141i	$0.625613\pi$
$138$	−24.0000	−2.04302
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	9.00000	0.757937
$142$	−2.00000	−0.167836
$143$	0	0
$144$	6.00000	0.500000
$145$	0	0
$146$	3.00000	0.248282
$147$	−21.0000	−1.73205
$148$	3.00000	0.246598
$149$	16.0000	1.31077	0.655386	−	0.755295i	$-0.272506\pi$
$149$	16.0000	1.31077	0.655386	+	0.755295i	$0.272506\pi$
$150$	15.0000	1.22474
$151$	6.00000	0.488273	0.244137	−	0.969741i	$-0.421495\pi$
$151$	6.00000	0.488273	0.244137	+	0.969741i	$0.421495\pi$
$152$	1.00000	0.0811107
$153$	42.0000	3.39550
$154$	0	0
$155$	0	0
$156$	−21.0000	−1.68135
$157$	−6.00000	−0.478852	−0.239426	−	0.970915i	$-0.576959\pi$
$157$	−6.00000	−0.478852	−0.239426	+	0.970915i	$0.576959\pi$
$158$	−14.0000	−1.11378
$159$	15.0000	1.18958
$160$	0	0
$161$	0	0
$162$	−9.00000	−0.707107
$163$	−2.00000	−0.156652	−0.0783260	−	0.996928i	$-0.524958\pi$
$163$	−2.00000	−0.156652	−0.0783260	+	0.996928i	$0.524958\pi$
$164$	−10.0000	−0.780869
$165$	0	0
$166$	6.00000	0.465690
$167$	−14.0000	−1.08335	−0.541676	−	0.840587i	$-0.682210\pi$
$167$	−14.0000	−1.08335	−0.541676	+	0.840587i	$0.682210\pi$
$168$	0	0
$169$	36.0000	2.76923
$170$	0	0
$171$	−6.00000	−0.458831
$172$	10.0000	0.762493
$173$	−15.0000	−1.14043	−0.570214	−	0.821496i	$-0.693140\pi$
$173$	−15.0000	−1.14043	−0.570214	+	0.821496i	$0.693140\pi$
$174$	−27.0000	−2.04686
$175$	0	0
$176$	0	0
$177$	12.0000	0.901975
$178$	−6.00000	−0.449719
$179$	−9.00000	−0.672692	−0.336346	−	0.941739i	$-0.609191\pi$
$179$	−9.00000	−0.672692	−0.336346	+	0.941739i	$0.609191\pi$
$180$	0	0
$181$	10.0000	0.743294	0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000	0.743294	0.371647	+	0.928374i	$0.378793\pi$
$182$	0	0
$183$	−18.0000	−1.33060
$184$	−8.00000	−0.589768
$185$	0	0
$186$	−6.00000	−0.439941
$187$	0	0
$188$	3.00000	0.218797
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	3.00000	0.216506
$193$	−16.0000	−1.15171	−0.575853	−	0.817554i	$-0.695330\pi$
$193$	−16.0000	−1.15171	−0.575853	+	0.817554i	$0.695330\pi$
$194$	−14.0000	−1.00514
$195$	0	0
$196$	−7.00000	−0.500000
$197$	8.00000	0.569976	0.284988	−	0.958531i	$-0.408010\pi$
$197$	8.00000	0.569976	0.284988	+	0.958531i	$0.408010\pi$
$198$	0	0
$199$	7.00000	0.496217	0.248108	−	0.968732i	$-0.420191\pi$
$199$	7.00000	0.496217	0.248108	+	0.968732i	$0.420191\pi$
$200$	5.00000	0.353553
$201$	−9.00000	−0.634811
$202$	−8.00000	−0.562878
$203$	0	0
$204$	21.0000	1.47029
$205$	0	0
$206$	4.00000	0.278693
$207$	48.0000	3.33623
$208$	−7.00000	−0.485363
$209$	0	0
$210$	0	0
$211$	−1.00000	−0.0688428	−0.0344214	−	0.999407i	$-0.510959\pi$
$211$	−1.00000	−0.0688428	−0.0344214	+	0.999407i	$0.510959\pi$
$212$	5.00000	0.343401
$213$	6.00000	0.411113
$214$	−12.0000	−0.820303
$215$	0	0
$216$	−9.00000	−0.612372
$217$	0	0
$218$	−2.00000	−0.135457
$219$	−9.00000	−0.608164
$220$	0	0
$221$	−49.0000	−3.29610
$222$	−9.00000	−0.604040
$223$	22.0000	1.47323	0.736614	−	0.676313i	$-0.236423\pi$
$223$	22.0000	1.47323	0.736614	+	0.676313i	$0.236423\pi$
$224$	0	0
$225$	−30.0000	−2.00000
$226$	20.0000	1.33038
$227$	17.0000	1.12833	0.564165	−	0.825662i	$-0.309198\pi$
$227$	17.0000	1.12833	0.564165	+	0.825662i	$0.309198\pi$
$228$	−3.00000	−0.198680
$229$	4.00000	0.264327	0.132164	−	0.991228i	$-0.457808\pi$
$229$	4.00000	0.264327	0.132164	+	0.991228i	$0.457808\pi$
$230$	0	0
$231$	0	0
$232$	−9.00000	−0.590879
$233$	−5.00000	−0.327561	−0.163780	−	0.986497i	$-0.552369\pi$
$233$	−5.00000	−0.327561	−0.163780	+	0.986497i	$0.552369\pi$
$234$	42.0000	2.74563
$235$	0	0
$236$	4.00000	0.260378
$237$	42.0000	2.72819
$238$	0	0
$239$	21.0000	1.35838	0.679189	−	0.733964i	$-0.262332\pi$
$239$	21.0000	1.35838	0.679189	+	0.733964i	$0.262332\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	0	0
$243$	0	0
$244$	−6.00000	−0.384111
$245$	0	0
$246$	30.0000	1.91273
$247$	7.00000	0.445399
$248$	−2.00000	−0.127000
$249$	−18.0000	−1.14070
$250$	0	0
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	0	0
$253$	0	0
$254$	−10.0000	−0.627456
$255$	0	0
$256$	1.00000	0.0625000
$257$	2.00000	0.124757	0.0623783	−	0.998053i	$-0.480131\pi$
$257$	2.00000	0.124757	0.0623783	+	0.998053i	$0.480131\pi$
$258$	−30.0000	−1.86772
$259$	0	0
$260$	0	0
$261$	54.0000	3.34252
$262$	−12.0000	−0.741362
$263$	16.0000	0.986602	0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000	0.986602	0.493301	+	0.869859i	$0.335790\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	18.0000	1.10158
$268$	−3.00000	−0.183254
$269$	−21.0000	−1.28039	−0.640196	−	0.768211i	$-0.721147\pi$
$269$	−21.0000	−1.28039	−0.640196	+	0.768211i	$0.721147\pi$
$270$	0	0
$271$	−29.0000	−1.76162	−0.880812	−	0.473466i	$-0.843003\pi$
$271$	−29.0000	−1.76162	−0.880812	+	0.473466i	$0.843003\pi$
$272$	7.00000	0.424437
$273$	0	0
$274$	9.00000	0.543710
$275$	0	0
$276$	24.0000	1.44463
$277$	8.00000	0.480673	0.240337	−	0.970690i	$-0.422742\pi$
$277$	8.00000	0.480673	0.240337	+	0.970690i	$0.422742\pi$
$278$	4.00000	0.239904
$279$	12.0000	0.718421
$280$	0	0
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	−9.00000	−0.535942
$283$	−16.0000	−0.951101	−0.475551	−	0.879688i	$-0.657751\pi$
$283$	−16.0000	−0.951101	−0.475551	+	0.879688i	$0.657751\pi$
$284$	2.00000	0.118678
$285$	0	0
$286$	0	0
$287$	0	0
$288$	−6.00000	−0.353553
$289$	32.0000	1.88235
$290$	0	0
$291$	42.0000	2.46208
$292$	−3.00000	−0.175562
$293$	−9.00000	−0.525786	−0.262893	−	0.964825i	$-0.584677\pi$
$293$	−9.00000	−0.525786	−0.262893	+	0.964825i	$0.584677\pi$
$294$	21.0000	1.22474
$295$	0	0
$296$	−3.00000	−0.174371
$297$	0	0
$298$	−16.0000	−0.926855
$299$	−56.0000	−3.23856
$300$	−15.0000	−0.866025
$301$	0	0
$302$	−6.00000	−0.345261
$303$	24.0000	1.37876
$304$	−1.00000	−0.0573539
$305$	0	0
$306$	−42.0000	−2.40098
$307$	−19.0000	−1.08439	−0.542194	−	0.840254i	$-0.682406\pi$
$307$	−19.0000	−1.08439	−0.542194	+	0.840254i	$0.682406\pi$
$308$	0	0
$309$	−12.0000	−0.682656
$310$	0	0
$311$	−5.00000	−0.283524	−0.141762	−	0.989901i	$-0.545277\pi$
$311$	−5.00000	−0.283524	−0.141762	+	0.989901i	$0.545277\pi$
$312$	21.0000	1.18889
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	6.00000	0.338600
$315$	0	0
$316$	14.0000	0.787562
$317$	−29.0000	−1.62880	−0.814401	−	0.580302i	$-0.802934\pi$
$317$	−29.0000	−1.62880	−0.814401	+	0.580302i	$0.802934\pi$
$318$	−15.0000	−0.841158
$319$	0	0
$320$	0	0
$321$	36.0000	2.00932
$322$	0	0
$323$	−7.00000	−0.389490
$324$	9.00000	0.500000
$325$	35.0000	1.94145
$326$	2.00000	0.110770
$327$	6.00000	0.331801
$328$	10.0000	0.552158
$329$	0	0
$330$	0	0
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	−6.00000	−0.329293
$333$	18.0000	0.986394
$334$	14.0000	0.766046
$335$	0	0
$336$	0	0
$337$	16.0000	0.871576	0.435788	−	0.900049i	$-0.356470\pi$
$337$	16.0000	0.871576	0.435788	+	0.900049i	$0.356470\pi$
$338$	−36.0000	−1.95814
$339$	−60.0000	−3.25875
$340$	0	0
$341$	0	0
$342$	6.00000	0.324443
$343$	0	0
$344$	−10.0000	−0.539164
$345$	0	0
$346$	15.0000	0.806405
$347$	−8.00000	−0.429463	−0.214731	−	0.976673i	$-0.568888\pi$
$347$	−8.00000	−0.429463	−0.214731	+	0.976673i	$0.568888\pi$
$348$	27.0000	1.44735
$349$	−26.0000	−1.39175	−0.695874	−	0.718164i	$-0.744983\pi$
$349$	−26.0000	−1.39175	−0.695874	+	0.718164i	$0.744983\pi$
$350$	0	0
$351$	−63.0000	−3.36269
$352$	0	0
$353$	−15.0000	−0.798369	−0.399185	−	0.916871i	$-0.630707\pi$
$353$	−15.0000	−0.798369	−0.399185	+	0.916871i	$0.630707\pi$
$354$	−12.0000	−0.637793
$355$	0	0
$356$	6.00000	0.317999
$357$	0	0
$358$	9.00000	0.475665
$359$	−35.0000	−1.84723	−0.923615	−	0.383322i	$-0.874780\pi$
$359$	−35.0000	−1.84723	−0.923615	+	0.383322i	$0.874780\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−10.0000	−0.525588
$363$	0	0
$364$	0	0
$365$	0	0
$366$	18.0000	0.940875
$367$	−23.0000	−1.20059	−0.600295	−	0.799779i	$-0.704950\pi$
$367$	−23.0000	−1.20059	−0.600295	+	0.799779i	$0.704950\pi$
$368$	8.00000	0.417029
$369$	−60.0000	−3.12348
$370$	0	0
$371$	0	0
$372$	6.00000	0.311086
$373$	−5.00000	−0.258890	−0.129445	−	0.991587i	$-0.541320\pi$
$373$	−5.00000	−0.258890	−0.129445	+	0.991587i	$0.541320\pi$
$374$	0	0
$375$	0	0
$376$	−3.00000	−0.154713
$377$	−63.0000	−3.24467
$378$	0	0
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	0	0
$381$	30.0000	1.53695
$382$	0	0
$383$	−30.0000	−1.53293	−0.766464	−	0.642287i	$-0.777986\pi$
$383$	−30.0000	−1.53293	−0.766464	+	0.642287i	$0.777986\pi$
$384$	−3.00000	−0.153093
$385$	0	0
$386$	16.0000	0.814379
$387$	60.0000	3.04997
$388$	14.0000	0.710742
$389$	16.0000	0.811232	0.405616	−	0.914044i	$-0.367057\pi$
$389$	16.0000	0.811232	0.405616	+	0.914044i	$0.367057\pi$
$390$	0	0
$391$	56.0000	2.83204
$392$	7.00000	0.353553
$393$	36.0000	1.81596
$394$	−8.00000	−0.403034
$395$	0	0
$396$	0	0
$397$	−6.00000	−0.301131	−0.150566	−	0.988600i	$-0.548110\pi$
$397$	−6.00000	−0.301131	−0.150566	+	0.988600i	$0.548110\pi$
$398$	−7.00000	−0.350878
$399$	0	0
$400$	−5.00000	−0.250000
$401$	−24.0000	−1.19850	−0.599251	−	0.800561i	$-0.704535\pi$
$401$	−24.0000	−1.19850	−0.599251	+	0.800561i	$0.704535\pi$
$402$	9.00000	0.448879
$403$	−14.0000	−0.697390
$404$	8.00000	0.398015
$405$	0	0
$406$	0	0
$407$	0	0
$408$	−21.0000	−1.03965
$409$	−20.0000	−0.988936	−0.494468	−	0.869196i	$-0.664637\pi$
$409$	−20.0000	−0.988936	−0.494468	+	0.869196i	$0.664637\pi$
$410$	0	0
$411$	−27.0000	−1.33181
$412$	−4.00000	−0.197066
$413$	0	0
$414$	−48.0000	−2.35907
$415$	0	0
$416$	7.00000	0.343203
$417$	−12.0000	−0.587643
$418$	0	0
$419$	14.0000	0.683945	0.341972	−	0.939710i	$-0.388905\pi$
$419$	14.0000	0.683945	0.341972	+	0.939710i	$0.388905\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	1.00000	0.0486792
$423$	18.0000	0.875190
$424$	−5.00000	−0.242821
$425$	−35.0000	−1.69775
$426$	−6.00000	−0.290701
$427$	0	0
$428$	12.0000	0.580042
$429$	0	0
$430$	0	0
$431$	18.0000	0.867029	0.433515	−	0.901146i	$-0.357273\pi$
$431$	18.0000	0.867029	0.433515	+	0.901146i	$0.357273\pi$
$432$	9.00000	0.433013
$433$	−16.0000	−0.768911	−0.384455	−	0.923144i	$-0.625611\pi$
$433$	−16.0000	−0.768911	−0.384455	+	0.923144i	$0.625611\pi$
$434$	0	0
$435$	0	0
$436$	2.00000	0.0957826
$437$	−8.00000	−0.382692
$438$	9.00000	0.430037
$439$	22.0000	1.05000	0.525001	−	0.851101i	$-0.324065\pi$
$439$	22.0000	1.05000	0.525001	+	0.851101i	$0.324065\pi$
$440$	0	0
$441$	−42.0000	−2.00000
$442$	49.0000	2.33069
$443$	−2.00000	−0.0950229	−0.0475114	−	0.998871i	$-0.515129\pi$
$443$	−2.00000	−0.0950229	−0.0475114	+	0.998871i	$0.515129\pi$
$444$	9.00000	0.427121
$445$	0	0
$446$	−22.0000	−1.04173
$447$	48.0000	2.27032
$448$	0	0
$449$	8.00000	0.377543	0.188772	−	0.982021i	$-0.439549\pi$
$449$	8.00000	0.377543	0.188772	+	0.982021i	$0.439549\pi$
$450$	30.0000	1.41421
$451$	0	0
$452$	−20.0000	−0.940721
$453$	18.0000	0.845714
$454$	−17.0000	−0.797850
$455$	0	0
$456$	3.00000	0.140488
$457$	10.0000	0.467780	0.233890	−	0.972263i	$-0.424854\pi$
$457$	10.0000	0.467780	0.233890	+	0.972263i	$0.424854\pi$
$458$	−4.00000	−0.186908
$459$	63.0000	2.94059
$460$	0	0
$461$	18.0000	0.838344	0.419172	−	0.907907i	$-0.362320\pi$
$461$	18.0000	0.838344	0.419172	+	0.907907i	$0.362320\pi$
$462$	0	0
$463$	32.0000	1.48717	0.743583	−	0.668644i	$-0.233125\pi$
$463$	32.0000	1.48717	0.743583	+	0.668644i	$0.233125\pi$
$464$	9.00000	0.417815
$465$	0	0
$466$	5.00000	0.231621
$467$	−32.0000	−1.48078	−0.740392	−	0.672176i	$-0.765360\pi$
$467$	−32.0000	−1.48078	−0.740392	+	0.672176i	$0.765360\pi$
$468$	−42.0000	−1.94145
$469$	0	0
$470$	0	0
$471$	−18.0000	−0.829396
$472$	−4.00000	−0.184115
$473$	0	0
$474$	−42.0000	−1.92912
$475$	5.00000	0.229416
$476$	0	0
$477$	30.0000	1.37361
$478$	−21.0000	−0.960518
$479$	25.0000	1.14228	0.571140	−	0.820853i	$-0.306501\pi$
$479$	25.0000	1.14228	0.571140	+	0.820853i	$0.306501\pi$
$480$	0	0
$481$	−21.0000	−0.957518
$482$	−10.0000	−0.455488
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	6.00000	0.271886	0.135943	−	0.990717i	$-0.456594\pi$
$487$	6.00000	0.271886	0.135943	+	0.990717i	$0.456594\pi$
$488$	6.00000	0.271607
$489$	−6.00000	−0.271329
$490$	0	0
$491$	14.0000	0.631811	0.315906	−	0.948791i	$-0.397692\pi$
$491$	14.0000	0.631811	0.315906	+	0.948791i	$0.397692\pi$
$492$	−30.0000	−1.35250
$493$	63.0000	2.83738
$494$	−7.00000	−0.314945
$495$	0	0
$496$	2.00000	0.0898027
$497$	0	0
$498$	18.0000	0.806599
$499$	−40.0000	−1.79065	−0.895323	−	0.445418i	$-0.853055\pi$
$499$	−40.0000	−1.79065	−0.895323	+	0.445418i	$0.853055\pi$
$500$	0	0
$501$	−42.0000	−1.87642
$502$	−12.0000	−0.535586
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	108.000	4.79645
$508$	10.0000	0.443678
$509$	−5.00000	−0.221621	−0.110811	−	0.993842i	$-0.535345\pi$
$509$	−5.00000	−0.221621	−0.110811	+	0.993842i	$0.535345\pi$
$510$	0	0
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	−9.00000	−0.397360
$514$	−2.00000	−0.0882162
$515$	0	0
$516$	30.0000	1.32068
$517$	0	0
$518$	0	0
$519$	−45.0000	−1.97528
$520$	0	0
$521$	−30.0000	−1.31432	−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000	−1.31432	−0.657162	+	0.753749i	$0.728243\pi$
$522$	−54.0000	−2.36352
$523$	36.0000	1.57417	0.787085	−	0.616844i	$-0.211589\pi$
$523$	36.0000	1.57417	0.787085	+	0.616844i	$0.211589\pi$
$524$	12.0000	0.524222
$525$	0	0
$526$	−16.0000	−0.697633
$527$	14.0000	0.609850
$528$	0	0
$529$	41.0000	1.78261
$530$	0	0
$531$	24.0000	1.04151
$532$	0	0
$533$	70.0000	3.03204
$534$	−18.0000	−0.778936
$535$	0	0
$536$	3.00000	0.129580
$537$	−27.0000	−1.16514
$538$	21.0000	0.905374
$539$	0	0
$540$	0	0
$541$	22.0000	0.945854	0.472927	−	0.881102i	$-0.343197\pi$
$541$	22.0000	0.945854	0.472927	+	0.881102i	$0.343197\pi$
$542$	29.0000	1.24566
$543$	30.0000	1.28742
$544$	−7.00000	−0.300123
$545$	0	0
$546$	0	0
$547$	11.0000	0.470326	0.235163	−	0.971956i	$-0.424438\pi$
$547$	11.0000	0.470326	0.235163	+	0.971956i	$0.424438\pi$
$548$	−9.00000	−0.384461
$549$	−36.0000	−1.53644
$550$	0	0
$551$	−9.00000	−0.383413
$552$	−24.0000	−1.02151
$553$	0	0
$554$	−8.00000	−0.339887
$555$	0	0
$556$	−4.00000	−0.169638
$557$	12.0000	0.508456	0.254228	−	0.967144i	$-0.418179\pi$
$557$	12.0000	0.508456	0.254228	+	0.967144i	$0.418179\pi$
$558$	−12.0000	−0.508001
$559$	−70.0000	−2.96068
$560$	0	0
$561$	0	0
$562$	18.0000	0.759284
$563$	−19.0000	−0.800755	−0.400377	−	0.916350i	$-0.631121\pi$
$563$	−19.0000	−0.800755	−0.400377	+	0.916350i	$0.631121\pi$
$564$	9.00000	0.378968
$565$	0	0
$566$	16.0000	0.672530
$567$	0	0
$568$	−2.00000	−0.0839181
$569$	−38.0000	−1.59304	−0.796521	−	0.604610i	$-0.793329\pi$
$569$	−38.0000	−1.59304	−0.796521	+	0.604610i	$0.793329\pi$
$570$	0	0
$571$	−14.0000	−0.585882	−0.292941	−	0.956131i	$-0.594634\pi$
$571$	−14.0000	−0.585882	−0.292941	+	0.956131i	$0.594634\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−40.0000	−1.66812
$576$	6.00000	0.250000
$577$	−37.0000	−1.54033	−0.770165	−	0.637845i	$-0.779826\pi$
$577$	−37.0000	−1.54033	−0.770165	+	0.637845i	$0.779826\pi$
$578$	−32.0000	−1.33102
$579$	−48.0000	−1.99481
$580$	0	0
$581$	0	0
$582$	−42.0000	−1.74096
$583$	0	0
$584$	3.00000	0.124141
$585$	0	0
$586$	9.00000	0.371787
$587$	46.0000	1.89862	0.949312	−	0.314337i	$-0.101782\pi$
$587$	46.0000	1.89862	0.949312	+	0.314337i	$0.101782\pi$
$588$	−21.0000	−0.866025
$589$	−2.00000	−0.0824086
$590$	0	0
$591$	24.0000	0.987228
$592$	3.00000	0.123299
$593$	−1.00000	−0.0410651	−0.0205325	−	0.999789i	$-0.506536\pi$
$593$	−1.00000	−0.0410651	−0.0205325	+	0.999789i	$0.506536\pi$
$594$	0	0
$595$	0	0
$596$	16.0000	0.655386
$597$	21.0000	0.859473
$598$	56.0000	2.29001
$599$	−30.0000	−1.22577	−0.612883	−	0.790173i	$-0.709990\pi$
$599$	−30.0000	−1.22577	−0.612883	+	0.790173i	$0.709990\pi$
$600$	15.0000	0.612372
$601$	−30.0000	−1.22373	−0.611863	−	0.790964i	$-0.709580\pi$
$601$	−30.0000	−1.22373	−0.611863	+	0.790964i	$0.709580\pi$
$602$	0	0
$603$	−18.0000	−0.733017
$604$	6.00000	0.244137
$605$	0	0
$606$	−24.0000	−0.974933
$607$	−8.00000	−0.324710	−0.162355	−	0.986732i	$-0.551909\pi$
$607$	−8.00000	−0.324710	−0.162355	+	0.986732i	$0.551909\pi$
$608$	1.00000	0.0405554
$609$	0	0
$610$	0	0
$611$	−21.0000	−0.849569
$612$	42.0000	1.69775
$613$	16.0000	0.646234	0.323117	−	0.946359i	$-0.395269\pi$
$613$	16.0000	0.646234	0.323117	+	0.946359i	$0.395269\pi$
$614$	19.0000	0.766778
$615$	0	0
$616$	0	0
$617$	−27.0000	−1.08698	−0.543490	−	0.839416i	$-0.682897\pi$
$617$	−27.0000	−1.08698	−0.543490	+	0.839416i	$0.682897\pi$
$618$	12.0000	0.482711
$619$	26.0000	1.04503	0.522514	−	0.852631i	$-0.324994\pi$
$619$	26.0000	1.04503	0.522514	+	0.852631i	$0.324994\pi$
$620$	0	0
$621$	72.0000	2.88926
$622$	5.00000	0.200482
$623$	0	0
$624$	−21.0000	−0.840673
$625$	25.0000	1.00000
$626$	−10.0000	−0.399680
$627$	0	0
$628$	−6.00000	−0.239426
$629$	21.0000	0.837325
$630$	0	0
$631$	29.0000	1.15447	0.577236	−	0.816577i	$-0.304131\pi$
$631$	29.0000	1.15447	0.577236	+	0.816577i	$0.304131\pi$
$632$	−14.0000	−0.556890
$633$	−3.00000	−0.119239
$634$	29.0000	1.15174
$635$	0	0
$636$	15.0000	0.594789
$637$	49.0000	1.94145
$638$	0	0
$639$	12.0000	0.474713
$640$	0	0
$641$	8.00000	0.315981	0.157991	−	0.987441i	$-0.449498\pi$
$641$	8.00000	0.315981	0.157991	+	0.987441i	$0.449498\pi$
$642$	−36.0000	−1.42081
$643$	46.0000	1.81406	0.907031	−	0.421063i	$-0.138343\pi$
$643$	46.0000	1.81406	0.907031	+	0.421063i	$0.138343\pi$
$644$	0	0
$645$	0	0
$646$	7.00000	0.275411
$647$	21.0000	0.825595	0.412798	−	0.910823i	$-0.364552\pi$
$647$	21.0000	0.825595	0.412798	+	0.910823i	$0.364552\pi$
$648$	−9.00000	−0.353553
$649$	0	0
$650$	−35.0000	−1.37281
$651$	0	0
$652$	−2.00000	−0.0783260
$653$	8.00000	0.313064	0.156532	−	0.987673i	$-0.449969\pi$
$653$	8.00000	0.313064	0.156532	+	0.987673i	$0.449969\pi$
$654$	−6.00000	−0.234619
$655$	0	0
$656$	−10.0000	−0.390434
$657$	−18.0000	−0.702247
$658$	0	0
$659$	−25.0000	−0.973862	−0.486931	−	0.873441i	$-0.661884\pi$
$659$	−25.0000	−0.973862	−0.486931	+	0.873441i	$0.661884\pi$
$660$	0	0
$661$	7.00000	0.272268	0.136134	−	0.990690i	$-0.456532\pi$
$661$	7.00000	0.272268	0.136134	+	0.990690i	$0.456532\pi$
$662$	20.0000	0.777322
$663$	−147.000	−5.70901
$664$	6.00000	0.232845
$665$	0	0
$666$	−18.0000	−0.697486
$667$	72.0000	2.78785
$668$	−14.0000	−0.541676
$669$	66.0000	2.55171
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−30.0000	−1.15642	−0.578208	−	0.815890i	$-0.696248\pi$
$673$	−30.0000	−1.15642	−0.578208	+	0.815890i	$0.696248\pi$
$674$	−16.0000	−0.616297
$675$	−45.0000	−1.73205
$676$	36.0000	1.38462
$677$	2.00000	0.0768662	0.0384331	−	0.999261i	$-0.487763\pi$
$677$	2.00000	0.0768662	0.0384331	+	0.999261i	$0.487763\pi$
$678$	60.0000	2.30429
$679$	0	0
$680$	0	0
$681$	51.0000	1.95432
$682$	0	0
$683$	−24.0000	−0.918334	−0.459167	−	0.888350i	$-0.651852\pi$
$683$	−24.0000	−0.918334	−0.459167	+	0.888350i	$0.651852\pi$
$684$	−6.00000	−0.229416
$685$	0	0
$686$	0	0
$687$	12.0000	0.457829
$688$	10.0000	0.381246
$689$	−35.0000	−1.33339
$690$	0	0
$691$	14.0000	0.532585	0.266293	−	0.963892i	$-0.414201\pi$
$691$	14.0000	0.532585	0.266293	+	0.963892i	$0.414201\pi$
$692$	−15.0000	−0.570214
$693$	0	0
$694$	8.00000	0.303676
$695$	0	0
$696$	−27.0000	−1.02343
$697$	−70.0000	−2.65144
$698$	26.0000	0.984115
$699$	−15.0000	−0.567352
$700$	0	0
$701$	12.0000	0.453234	0.226617	−	0.973984i	$-0.427233\pi$
$701$	12.0000	0.453234	0.226617	+	0.973984i	$0.427233\pi$
$702$	63.0000	2.37778
$703$	−3.00000	−0.113147
$704$	0	0
$705$	0	0
$706$	15.0000	0.564532
$707$	0	0
$708$	12.0000	0.450988
$709$	22.0000	0.826227	0.413114	−	0.910679i	$-0.364441\pi$
$709$	22.0000	0.826227	0.413114	+	0.910679i	$0.364441\pi$
$710$	0	0
$711$	84.0000	3.15025
$712$	−6.00000	−0.224860
$713$	16.0000	0.599205
$714$	0	0
$715$	0	0
$716$	−9.00000	−0.336346
$717$	63.0000	2.35278
$718$	35.0000	1.30619
$719$	−17.0000	−0.633993	−0.316997	−	0.948427i	$-0.602674\pi$
$719$	−17.0000	−0.633993	−0.316997	+	0.948427i	$0.602674\pi$
$720$	0	0
$721$	0	0
$722$	−1.00000	−0.0372161
$723$	30.0000	1.11571
$724$	10.0000	0.371647
$725$	−45.0000	−1.67126
$726$	0	0
$727$	−35.0000	−1.29808	−0.649039	−	0.760755i	$-0.724829\pi$
$727$	−35.0000	−1.29808	−0.649039	+	0.760755i	$0.724829\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	70.0000	2.58904
$732$	−18.0000	−0.665299
$733$	30.0000	1.10808	0.554038	−	0.832492i	$-0.313086\pi$
$733$	30.0000	1.10808	0.554038	+	0.832492i	$0.313086\pi$
$734$	23.0000	0.848945
$735$	0	0
$736$	−8.00000	−0.294884
$737$	0	0
$738$	60.0000	2.20863
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	0	0
$741$	21.0000	0.771454
$742$	0	0
$743$	34.0000	1.24734	0.623670	−	0.781688i	$-0.285641\pi$
$743$	34.0000	1.24734	0.623670	+	0.781688i	$0.285641\pi$
$744$	−6.00000	−0.219971
$745$	0	0
$746$	5.00000	0.183063
$747$	−36.0000	−1.31717
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−10.0000	−0.364905	−0.182453	−	0.983215i	$-0.558404\pi$
$751$	−10.0000	−0.364905	−0.182453	+	0.983215i	$0.558404\pi$
$752$	3.00000	0.109399
$753$	36.0000	1.31191
$754$	63.0000	2.29432
$755$	0	0
$756$	0	0
$757$	−10.0000	−0.363456	−0.181728	−	0.983349i	$-0.558169\pi$
$757$	−10.0000	−0.363456	−0.181728	+	0.983349i	$0.558169\pi$
$758$	20.0000	0.726433
$759$	0	0
$760$	0	0
$761$	22.0000	0.797499	0.398750	−	0.917060i	$-0.369444\pi$
$761$	22.0000	0.797499	0.398750	+	0.917060i	$0.369444\pi$
$762$	−30.0000	−1.08679
$763$	0	0
$764$	0	0
$765$	0	0
$766$	30.0000	1.08394
$767$	−28.0000	−1.01102
$768$	3.00000	0.108253
$769$	5.00000	0.180305	0.0901523	−	0.995928i	$-0.471265\pi$
$769$	5.00000	0.180305	0.0901523	+	0.995928i	$0.471265\pi$
$770$	0	0
$771$	6.00000	0.216085
$772$	−16.0000	−0.575853
$773$	18.0000	0.647415	0.323708	−	0.946157i	$-0.395071\pi$
$773$	18.0000	0.647415	0.323708	+	0.946157i	$0.395071\pi$
$774$	−60.0000	−2.15666
$775$	−10.0000	−0.359211
$776$	−14.0000	−0.502571
$777$	0	0
$778$	−16.0000	−0.573628
$779$	10.0000	0.358287
$780$	0	0
$781$	0	0
$782$	−56.0000	−2.00256
$783$	81.0000	2.89470
$784$	−7.00000	−0.250000
$785$	0	0
$786$	−36.0000	−1.28408
$787$	31.0000	1.10503	0.552515	−	0.833503i	$-0.313668\pi$
$787$	31.0000	1.10503	0.552515	+	0.833503i	$0.313668\pi$
$788$	8.00000	0.284988
$789$	48.0000	1.70885
$790$	0	0
$791$	0	0
$792$	0	0
$793$	42.0000	1.49146
$794$	6.00000	0.212932
$795$	0	0
$796$	7.00000	0.248108
$797$	1.00000	0.0354218	0.0177109	−	0.999843i	$-0.494362\pi$
$797$	1.00000	0.0354218	0.0177109	+	0.999843i	$0.494362\pi$
$798$	0	0
$799$	21.0000	0.742927
$800$	5.00000	0.176777
$801$	36.0000	1.27200
$802$	24.0000	0.847469
$803$	0	0
$804$	−9.00000	−0.317406
$805$	0	0
$806$	14.0000	0.493129
$807$	−63.0000	−2.21771
$808$	−8.00000	−0.281439
$809$	−3.00000	−0.105474	−0.0527372	−	0.998608i	$-0.516795\pi$
$809$	−3.00000	−0.105474	−0.0527372	+	0.998608i	$0.516795\pi$
$810$	0	0
$811$	−47.0000	−1.65039	−0.825197	−	0.564846i	$-0.808936\pi$
$811$	−47.0000	−1.65039	−0.825197	+	0.564846i	$0.808936\pi$
$812$	0	0
$813$	−87.0000	−3.05122
$814$	0	0
$815$	0	0
$816$	21.0000	0.735147
$817$	−10.0000	−0.349856
$818$	20.0000	0.699284
$819$	0	0
$820$	0	0
$821$	−6.00000	−0.209401	−0.104701	−	0.994504i	$-0.533388\pi$
$821$	−6.00000	−0.209401	−0.104701	+	0.994504i	$0.533388\pi$
$822$	27.0000	0.941733
$823$	−25.0000	−0.871445	−0.435723	−	0.900081i	$-0.643507\pi$
$823$	−25.0000	−0.871445	−0.435723	+	0.900081i	$0.643507\pi$
$824$	4.00000	0.139347
$825$	0	0
$826$	0	0
$827$	−3.00000	−0.104320	−0.0521601	−	0.998639i	$-0.516611\pi$
$827$	−3.00000	−0.104320	−0.0521601	+	0.998639i	$0.516611\pi$
$828$	48.0000	1.66812
$829$	−53.0000	−1.84077	−0.920383	−	0.391018i	$-0.872123\pi$
$829$	−53.0000	−1.84077	−0.920383	+	0.391018i	$0.872123\pi$
$830$	0	0
$831$	24.0000	0.832551
$832$	−7.00000	−0.242681
$833$	−49.0000	−1.69775
$834$	12.0000	0.415526
$835$	0	0
$836$	0	0
$837$	18.0000	0.622171
$838$	−14.0000	−0.483622
$839$	−6.00000	−0.207143	−0.103572	−	0.994622i	$-0.533027\pi$
$839$	−6.00000	−0.207143	−0.103572	+	0.994622i	$0.533027\pi$
$840$	0	0
$841$	52.0000	1.79310
$842$	10.0000	0.344623
$843$	−54.0000	−1.85986
$844$	−1.00000	−0.0344214
$845$	0	0
$846$	−18.0000	−0.618853
$847$	0	0
$848$	5.00000	0.171701
$849$	−48.0000	−1.64736
$850$	35.0000	1.20049
$851$	24.0000	0.822709
$852$	6.00000	0.205557
$853$	−36.0000	−1.23262	−0.616308	−	0.787505i	$-0.711372\pi$
$853$	−36.0000	−1.23262	−0.616308	+	0.787505i	$0.711372\pi$
$854$	0	0
$855$	0	0
$856$	−12.0000	−0.410152
$857$	−18.0000	−0.614868	−0.307434	−	0.951569i	$-0.599470\pi$
$857$	−18.0000	−0.614868	−0.307434	+	0.951569i	$0.599470\pi$
$858$	0	0
$859$	−6.00000	−0.204717	−0.102359	−	0.994748i	$-0.532639\pi$
$859$	−6.00000	−0.204717	−0.102359	+	0.994748i	$0.532639\pi$
$860$	0	0
$861$	0	0
$862$	−18.0000	−0.613082
$863$	8.00000	0.272323	0.136162	−	0.990687i	$-0.456523\pi$
$863$	8.00000	0.272323	0.136162	+	0.990687i	$0.456523\pi$
$864$	−9.00000	−0.306186
$865$	0	0
$866$	16.0000	0.543702
$867$	96.0000	3.26033
$868$	0	0
$869$	0	0
$870$	0	0
$871$	21.0000	0.711558
$872$	−2.00000	−0.0677285
$873$	84.0000	2.84297
$874$	8.00000	0.270604
$875$	0	0
$876$	−9.00000	−0.304082
$877$	−33.0000	−1.11433	−0.557165	−	0.830402i	$-0.688111\pi$
$877$	−33.0000	−1.11433	−0.557165	+	0.830402i	$0.688111\pi$
$878$	−22.0000	−0.742464
$879$	−27.0000	−0.910687
$880$	0	0
$881$	−43.0000	−1.44871	−0.724353	−	0.689429i	$-0.757862\pi$
$881$	−43.0000	−1.44871	−0.724353	+	0.689429i	$0.757862\pi$
$882$	42.0000	1.41421
$883$	26.0000	0.874970	0.437485	−	0.899226i	$-0.355869\pi$
$883$	26.0000	0.874970	0.437485	+	0.899226i	$0.355869\pi$
$884$	−49.0000	−1.64805
$885$	0	0
$886$	2.00000	0.0671913
$887$	36.0000	1.20876	0.604381	−	0.796696i	$-0.293421\pi$
$887$	36.0000	1.20876	0.604381	+	0.796696i	$0.293421\pi$
$888$	−9.00000	−0.302020
$889$	0	0
$890$	0	0
$891$	0	0
$892$	22.0000	0.736614
$893$	−3.00000	−0.100391
$894$	−48.0000	−1.60536
$895$	0	0
$896$	0	0
$897$	−168.000	−5.60936
$898$	−8.00000	−0.266963
$899$	18.0000	0.600334
$900$	−30.0000	−1.00000
$901$	35.0000	1.16602
$902$	0	0
$903$	0	0
$904$	20.0000	0.665190
$905$	0	0
$906$	−18.0000	−0.598010
$907$	28.0000	0.929725	0.464862	−	0.885383i	$-0.346104\pi$
$907$	28.0000	0.929725	0.464862	+	0.885383i	$0.346104\pi$
$908$	17.0000	0.564165
$909$	48.0000	1.59206
$910$	0	0
$911$	54.0000	1.78910	0.894550	−	0.446968i	$-0.147496\pi$
$911$	54.0000	1.78910	0.894550	+	0.446968i	$0.147496\pi$
$912$	−3.00000	−0.0993399
$913$	0	0
$914$	−10.0000	−0.330771
$915$	0	0
$916$	4.00000	0.132164
$917$	0	0
$918$	−63.0000	−2.07931
$919$	9.00000	0.296883	0.148441	−	0.988921i	$-0.452574\pi$
$919$	9.00000	0.296883	0.148441	+	0.988921i	$0.452574\pi$
$920$	0	0
$921$	−57.0000	−1.87821
$922$	−18.0000	−0.592798
$923$	−14.0000	−0.460816
$924$	0	0
$925$	−15.0000	−0.493197
$926$	−32.0000	−1.05159
$927$	−24.0000	−0.788263
$928$	−9.00000	−0.295439
$929$	−19.0000	−0.623370	−0.311685	−	0.950186i	$-0.600893\pi$
$929$	−19.0000	−0.623370	−0.311685	+	0.950186i	$0.600893\pi$
$930$	0	0
$931$	7.00000	0.229416
$932$	−5.00000	−0.163780
$933$	−15.0000	−0.491078
$934$	32.0000	1.04707
$935$	0	0
$936$	42.0000	1.37281
$937$	−43.0000	−1.40475	−0.702374	−	0.711808i	$-0.747877\pi$
$937$	−43.0000	−1.40475	−0.702374	+	0.711808i	$0.747877\pi$
$938$	0	0
$939$	30.0000	0.979013
$940$	0	0
$941$	−27.0000	−0.880175	−0.440087	−	0.897955i	$-0.645053\pi$
$941$	−27.0000	−0.880175	−0.440087	+	0.897955i	$0.645053\pi$
$942$	18.0000	0.586472
$943$	−80.0000	−2.60516
$944$	4.00000	0.130189
$945$	0	0
$946$	0	0
$947$	28.0000	0.909878	0.454939	−	0.890523i	$-0.349661\pi$
$947$	28.0000	0.909878	0.454939	+	0.890523i	$0.349661\pi$
$948$	42.0000	1.36410
$949$	21.0000	0.681689
$950$	−5.00000	−0.162221
$951$	−87.0000	−2.82117
$952$	0	0
$953$	28.0000	0.907009	0.453504	−	0.891254i	$-0.350174\pi$
$953$	28.0000	0.907009	0.453504	+	0.891254i	$0.350174\pi$
$954$	−30.0000	−0.971286
$955$	0	0
$956$	21.0000	0.679189
$957$	0	0
$958$	−25.0000	−0.807713
$959$	0	0
$960$	0	0
$961$	−27.0000	−0.870968
$962$	21.0000	0.677067
$963$	72.0000	2.32017
$964$	10.0000	0.322078
$965$	0	0
$966$	0	0
$967$	−43.0000	−1.38279	−0.691393	−	0.722478i	$-0.743003\pi$
$967$	−43.0000	−1.38279	−0.691393	+	0.722478i	$0.743003\pi$
$968$	0	0
$969$	−21.0000	−0.674617
$970$	0	0
$971$	3.00000	0.0962746	0.0481373	−	0.998841i	$-0.484672\pi$
$971$	3.00000	0.0962746	0.0481373	+	0.998841i	$0.484672\pi$
$972$	0	0
$973$	0	0
$974$	−6.00000	−0.192252
$975$	105.000	3.36269
$976$	−6.00000	−0.192055
$977$	42.0000	1.34370	0.671850	−	0.740688i	$-0.265500\pi$
$977$	42.0000	1.34370	0.671850	+	0.740688i	$0.265500\pi$
$978$	6.00000	0.191859
$979$	0	0
$980$	0	0
$981$	12.0000	0.383131
$982$	−14.0000	−0.446758
$983$	−6.00000	−0.191370	−0.0956851	−	0.995412i	$-0.530504\pi$
$983$	−6.00000	−0.191370	−0.0956851	+	0.995412i	$0.530504\pi$
$984$	30.0000	0.956365
$985$	0	0
$986$	−63.0000	−2.00633
$987$	0	0
$988$	7.00000	0.222700
$989$	80.0000	2.54385
$990$	0	0
$991$	32.0000	1.01651	0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000	1.01651	0.508257	+	0.861206i	$0.330290\pi$
$992$	−2.00000	−0.0635001
$993$	−60.0000	−1.90404
$994$	0	0
$995$	0	0
$996$	−18.0000	−0.570352
$997$	−32.0000	−1.01345	−0.506725	−	0.862108i	$-0.669144\pi$
$997$	−32.0000	−1.01345	−0.506725	+	0.862108i	$0.669144\pi$
$998$	40.0000	1.26618
$999$	27.0000	0.854242

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4598.2.a.k.1.1	✓	1
11.10	odd	2		4598.2.a.s.1.1	yes	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4598.2.a.k.1.1	✓	1	1.1	even	1	trivial
4598.2.a.s.1.1	yes	1	11.10	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:	\( N \)	\(=\)	\( 4598 = 2 \cdot 11^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4598.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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