LMFDB - Embedded newform 5075.2.a.n.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5075,2,Mod(1,5075)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5075.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5075, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5075 = 5^{2} \cdot 7 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5075.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,2,0,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$40.5240790258$
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 1015)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	5075.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205 q^{2} +1.00000 q^{4} -1.00000 q^{7} -1.73205 q^{8} -3.00000 q^{9} +1.46410 q^{11} +2.00000 q^{13} -1.73205 q^{14} -5.00000 q^{16} -3.46410 q^{17} -5.19615 q^{18} +2.00000 q^{19} +2.53590 q^{22} +6.92820 q^{23} +3.46410 q^{26} -1.00000 q^{28} +1.00000 q^{29} +4.92820 q^{31} -5.19615 q^{32} -6.00000 q^{34} -3.00000 q^{36} +3.46410 q^{38} -1.46410 q^{41} +3.46410 q^{43} +1.46410 q^{44} +12.0000 q^{46} +4.00000 q^{47} +1.00000 q^{49} +2.00000 q^{52} +8.92820 q^{53} +1.73205 q^{56} +1.73205 q^{58} -8.00000 q^{59} +5.46410 q^{61} +8.53590 q^{62} +3.00000 q^{63} +1.00000 q^{64} +4.00000 q^{67} -3.46410 q^{68} -8.00000 q^{71} +5.19615 q^{72} +6.39230 q^{73} +2.00000 q^{76} -1.46410 q^{77} +5.46410 q^{79} +9.00000 q^{81} -2.53590 q^{82} +14.9282 q^{83} +6.00000 q^{86} -2.53590 q^{88} -10.5359 q^{89} -2.00000 q^{91} +6.92820 q^{92} +6.92820 q^{94} +4.53590 q^{97} +1.73205 q^{98} -4.39230 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4} - 2 q^{7} - 6 q^{9} - 4 q^{11} + 4 q^{13} - 10 q^{16} + 4 q^{19} + 12 q^{22} - 2 q^{28} + 2 q^{29} - 4 q^{31} - 12 q^{34} - 6 q^{36} + 4 q^{41} - 4 q^{44} + 24 q^{46} + 8 q^{47} + 2 q^{49}+ \cdots + 12 q^{99}+O(q^{100}) $ 2 * q + 2 * q^4 - 2 * q^7 - 6 * q^9 - 4 * q^11 + 4 * q^13 - 10 * q^16 + 4 * q^19 + 12 * q^22 - 2 * q^28 + 2 * q^29 - 4 * q^31 - 12 * q^34 - 6 * q^36 + 4 * q^41 - 4 * q^44 + 24 * q^46 + 8 * q^47 + 2 * q^49 + 4 * q^52 + 4 * q^53 - 16 * q^59 + 4 * q^61 + 24 * q^62 + 6 * q^63 + 2 * q^64 + 8 * q^67 - 16 * q^71 - 8 * q^73 + 4 * q^76 + 4 * q^77 + 4 * q^79 + 18 * q^81 - 12 * q^82 + 16 * q^83 + 12 * q^86 - 12 * q^88 - 28 * q^89 - 4 * q^91 + 16 * q^97 + 12 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.73205	1.22474	0.612372	−	0.790569i	$-0.290215\pi$
$2$	1.73205	1.22474	0.612372	+	0.790569i	$0.290215\pi$
$3$	0	0		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.73205	−0.612372
$9$	−3.00000	−1.00000
$10$	0	0
$11$	1.46410	0.441443	0.220722	−	0.975337i	$-0.429159\pi$
$11$	1.46410	0.441443	0.220722	+	0.975337i	$0.429159\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	−1.73205	−0.462910
$15$	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	$0.637999\pi$
$18$	−5.19615	−1.22474
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	0	0
$21$	0	0
$22$	2.53590	0.540655
$23$	6.92820	1.44463	0.722315	−	0.691564i	$-0.243078\pi$
$23$	6.92820	1.44463	0.722315	+	0.691564i	$0.243078\pi$
$24$	0	0
$25$	0	0
$26$	3.46410	0.679366
$27$	0	0
$28$	−1.00000	−0.188982
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	4.92820	0.885131	0.442566	−	0.896736i	$-0.354068\pi$
$31$	4.92820	0.885131	0.442566	+	0.896736i	$0.354068\pi$
$32$	−5.19615	−0.918559
$33$	0	0
$34$	−6.00000	−1.02899
$35$	0	0
$36$	−3.00000	−0.500000
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	3.46410	0.561951
$39$	0	0
$40$	0	0
$41$	−1.46410	−0.228654	−0.114327	−	0.993443i	$-0.536471\pi$
$41$	−1.46410	−0.228654	−0.114327	+	0.993443i	$0.536471\pi$
$42$	0	0
$43$	3.46410	0.528271	0.264135	−	0.964486i	$-0.414913\pi$
$43$	3.46410	0.528271	0.264135	+	0.964486i	$0.414913\pi$
$44$	1.46410	0.220722
$45$	0	0
$46$	12.0000	1.76930
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	2.00000	0.277350
$53$	8.92820	1.22638	0.613192	−	0.789934i	$-0.289885\pi$
$53$	8.92820	1.22638	0.613192	+	0.789934i	$0.289885\pi$
$54$	0	0
$55$	0	0
$56$	1.73205	0.231455
$57$	0	0
$58$	1.73205	0.227429
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	0	0
$61$	5.46410	0.699607	0.349803	−	0.936823i	$-0.386248\pi$
$61$	5.46410	0.699607	0.349803	+	0.936823i	$0.386248\pi$
$62$	8.53590	1.08406
$63$	3.00000	0.377964
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	−3.46410	−0.420084
$69$	0	0
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	5.19615	0.612372
$73$	6.39230	0.748163	0.374081	−	0.927396i	$-0.377958\pi$
$73$	6.39230	0.748163	0.374081	+	0.927396i	$0.377958\pi$
$74$	0	0
$75$	0	0
$76$	2.00000	0.229416
$77$	−1.46410	−0.166850
$78$	0	0
$79$	5.46410	0.614759	0.307380	−	0.951587i	$-0.400548\pi$
$79$	5.46410	0.614759	0.307380	+	0.951587i	$0.400548\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	−2.53590	−0.280043
$83$	14.9282	1.63858	0.819292	−	0.573377i	$-0.194367\pi$
$83$	14.9282	1.63858	0.819292	+	0.573377i	$0.194367\pi$
$84$	0	0
$85$	0	0
$86$	6.00000	0.646997
$87$	0	0
$88$	−2.53590	−0.270328
$89$	−10.5359	−1.11680	−0.558401	−	0.829571i	$-0.688585\pi$
$89$	−10.5359	−1.11680	−0.558401	+	0.829571i	$0.688585\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	6.92820	0.722315
$93$	0	0
$94$	6.92820	0.714590
$95$	0	0
$96$	0	0
$97$	4.53590	0.460551	0.230275	−	0.973126i	$-0.426037\pi$
$97$	4.53590	0.460551	0.230275	+	0.973126i	$0.426037\pi$
$98$	1.73205	0.174964
$99$	−4.39230	−0.441443
$100$	0	0
$101$	−6.53590	−0.650346	−0.325173	−	0.945655i	$-0.605423\pi$
$101$	−6.53590	−0.650346	−0.325173	+	0.945655i	$0.605423\pi$
$102$	0	0
$103$	16.0000	1.57653	0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000	1.57653	0.788263	+	0.615338i	$0.210980\pi$
$104$	−3.46410	−0.339683
$105$	0	0
$106$	15.4641	1.50201
$107$	2.92820	0.283080	0.141540	−	0.989933i	$-0.454795\pi$
$107$	2.92820	0.283080	0.141540	+	0.989933i	$0.454795\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	0	0
$112$	5.00000	0.472456
$113$	16.0000	1.50515	0.752577	−	0.658505i	$-0.228811\pi$
$113$	16.0000	1.50515	0.752577	+	0.658505i	$0.228811\pi$
$114$	0	0
$115$	0	0
$116$	1.00000	0.0928477
$117$	−6.00000	−0.554700
$118$	−13.8564	−1.27559
$119$	3.46410	0.317554
$120$	0	0
$121$	−8.85641	−0.805128
$122$	9.46410	0.856840
$123$	0	0
$124$	4.92820	0.442566
$125$	0	0
$126$	5.19615	0.462910
$127$	−15.4641	−1.37222	−0.686109	−	0.727499i	$-0.740683\pi$
$127$	−15.4641	−1.37222	−0.686109	+	0.727499i	$0.740683\pi$
$128$	12.1244	1.07165
$129$	0	0
$130$	0	0
$131$	8.92820	0.780061	0.390030	−	0.920802i	$-0.372465\pi$
$131$	8.92820	0.780061	0.390030	+	0.920802i	$0.372465\pi$
$132$	0	0
$133$	−2.00000	−0.173422
$134$	6.92820	0.598506
$135$	0	0
$136$	6.00000	0.514496
$137$	5.07180	0.433313	0.216656	−	0.976248i	$-0.430485\pi$
$137$	5.07180	0.433313	0.216656	+	0.976248i	$0.430485\pi$
$138$	0	0
$139$	−9.85641	−0.836009	−0.418005	−	0.908445i	$-0.637270\pi$
$139$	−9.85641	−0.836009	−0.418005	+	0.908445i	$0.637270\pi$
$140$	0	0
$141$	0	0
$142$	−13.8564	−1.16280
$143$	2.92820	0.244869
$144$	15.0000	1.25000
$145$	0	0
$146$	11.0718	0.916308
$147$	0	0
$148$	0	0
$149$	−3.85641	−0.315929	−0.157965	−	0.987445i	$-0.550493\pi$
$149$	−3.85641	−0.315929	−0.157965	+	0.987445i	$0.550493\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	−3.46410	−0.280976
$153$	10.3923	0.840168
$154$	−2.53590	−0.204349
$155$	0	0
$156$	0	0
$157$	4.53590	0.362004	0.181002	−	0.983483i	$-0.442066\pi$
$157$	4.53590	0.362004	0.181002	+	0.983483i	$0.442066\pi$
$158$	9.46410	0.752923
$159$	0	0
$160$	0	0
$161$	−6.92820	−0.546019
$162$	15.5885	1.22474
$163$	17.3205	1.35665	0.678323	−	0.734763i	$-0.262707\pi$
$163$	17.3205	1.35665	0.678323	+	0.734763i	$0.262707\pi$
$164$	−1.46410	−0.114327
$165$	0	0
$166$	25.8564	2.00685
$167$	21.8564	1.69130	0.845650	−	0.533738i	$-0.179213\pi$
$167$	21.8564	1.69130	0.845650	+	0.533738i	$0.179213\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	−6.00000	−0.458831
$172$	3.46410	0.264135
$173$	−12.9282	−0.982913	−0.491457	−	0.870902i	$-0.663535\pi$
$173$	−12.9282	−0.982913	−0.491457	+	0.870902i	$0.663535\pi$
$174$	0	0
$175$	0	0
$176$	−7.32051	−0.551804
$177$	0	0
$178$	−18.2487	−1.36780
$179$	6.92820	0.517838	0.258919	−	0.965899i	$-0.416634\pi$
$179$	6.92820	0.517838	0.258919	+	0.965899i	$0.416634\pi$
$180$	0	0
$181$	0.928203	0.0689928	0.0344964	−	0.999405i	$-0.489017\pi$
$181$	0.928203	0.0689928	0.0344964	+	0.999405i	$0.489017\pi$
$182$	−3.46410	−0.256776
$183$	0	0
$184$	−12.0000	−0.884652
$185$	0	0
$186$	0	0
$187$	−5.07180	−0.370887
$188$	4.00000	0.291730
$189$	0	0
$190$	0	0
$191$	−12.3923	−0.896676	−0.448338	−	0.893864i	$-0.647984\pi$
$191$	−12.3923	−0.896676	−0.448338	+	0.893864i	$0.647984\pi$
$192$	0	0
$193$	−16.0000	−1.15171	−0.575853	−	0.817554i	$-0.695330\pi$
$193$	−16.0000	−1.15171	−0.575853	+	0.817554i	$0.695330\pi$
$194$	7.85641	0.564057
$195$	0	0
$196$	1.00000	0.0714286
$197$	−22.7846	−1.62334	−0.811668	−	0.584119i	$-0.801440\pi$
$197$	−22.7846	−1.62334	−0.811668	+	0.584119i	$0.801440\pi$
$198$	−7.60770	−0.540655
$199$	4.00000	0.283552	0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$200$	0	0
$201$	0	0
$202$	−11.3205	−0.796508
$203$	−1.00000	−0.0701862
$204$	0	0
$205$	0	0
$206$	27.7128	1.93084
$207$	−20.7846	−1.44463
$208$	−10.0000	−0.693375
$209$	2.92820	0.202548
$210$	0	0
$211$	20.3923	1.40386	0.701932	−	0.712244i	$-0.252321\pi$
$211$	20.3923	1.40386	0.701932	+	0.712244i	$0.252321\pi$
$212$	8.92820	0.613192
$213$	0	0
$214$	5.07180	0.346701
$215$	0	0
$216$	0	0
$217$	−4.92820	−0.334548
$218$	17.3205	1.17309
$219$	0	0
$220$	0	0
$221$	−6.92820	−0.466041
$222$	0	0
$223$	−18.9282	−1.26753	−0.633763	−	0.773527i	$-0.718491\pi$
$223$	−18.9282	−1.26753	−0.633763	+	0.773527i	$0.718491\pi$
$224$	5.19615	0.347183
$225$	0	0
$226$	27.7128	1.84343
$227$	17.8564	1.18517	0.592586	−	0.805507i	$-0.298107\pi$
$227$	17.8564	1.18517	0.592586	+	0.805507i	$0.298107\pi$
$228$	0	0
$229$	−22.2487	−1.47024	−0.735118	−	0.677939i	$-0.762873\pi$
$229$	−22.2487	−1.47024	−0.735118	+	0.677939i	$0.762873\pi$
$230$	0	0
$231$	0	0
$232$	−1.73205	−0.113715
$233$	−23.8564	−1.56289	−0.781443	−	0.623977i	$-0.785516\pi$
$233$	−23.8564	−1.56289	−0.781443	+	0.623977i	$0.785516\pi$
$234$	−10.3923	−0.679366
$235$	0	0
$236$	−8.00000	−0.520756
$237$	0	0
$238$	6.00000	0.388922
$239$	16.7846	1.08571	0.542853	−	0.839828i	$-0.317344\pi$
$239$	16.7846	1.08571	0.542853	+	0.839828i	$0.317344\pi$
$240$	0	0
$241$	27.8564	1.79439	0.897194	−	0.441636i	$-0.145602\pi$
$241$	27.8564	1.79439	0.897194	+	0.441636i	$0.145602\pi$
$242$	−15.3397	−0.986076
$243$	0	0
$244$	5.46410	0.349803
$245$	0	0
$246$	0	0
$247$	4.00000	0.254514
$248$	−8.53590	−0.542030
$249$	0	0
$250$	0	0
$251$	0.928203	0.0585877	0.0292938	−	0.999571i	$-0.490674\pi$
$251$	0.928203	0.0585877	0.0292938	+	0.999571i	$0.490674\pi$
$252$	3.00000	0.188982
$253$	10.1436	0.637722
$254$	−26.7846	−1.68062
$255$	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
$259$	0	0
$260$	0	0
$261$	−3.00000	−0.185695
$262$	15.4641	0.955375
$263$	−21.3205	−1.31468	−0.657339	−	0.753595i	$-0.728318\pi$
$263$	−21.3205	−1.31468	−0.657339	+	0.753595i	$0.728318\pi$
$264$	0	0
$265$	0	0
$266$	−3.46410	−0.212398
$267$	0	0
$268$	4.00000	0.244339
$269$	−7.60770	−0.463849	−0.231925	−	0.972734i	$-0.574502\pi$
$269$	−7.60770	−0.463849	−0.231925	+	0.972734i	$0.574502\pi$
$270$	0	0
$271$	27.8564	1.69216	0.846078	−	0.533059i	$-0.178958\pi$
$271$	27.8564	1.69216	0.846078	+	0.533059i	$0.178958\pi$
$272$	17.3205	1.05021
$273$	0	0
$274$	8.78461	0.530698
$275$	0	0
$276$	0	0
$277$	0.143594	0.00862770	0.00431385	−	0.999991i	$-0.498627\pi$
$277$	0.143594	0.00862770	0.00431385	+	0.999991i	$0.498627\pi$
$278$	−17.0718	−1.02390
$279$	−14.7846	−0.885131
$280$	0	0
$281$	0.143594	0.00856607	0.00428304	−	0.999991i	$-0.498637\pi$
$281$	0.143594	0.00856607	0.00428304	+	0.999991i	$0.498637\pi$
$282$	0	0
$283$	−6.92820	−0.411839	−0.205919	−	0.978569i	$-0.566018\pi$
$283$	−6.92820	−0.411839	−0.205919	+	0.978569i	$0.566018\pi$
$284$	−8.00000	−0.474713
$285$	0	0
$286$	5.07180	0.299902
$287$	1.46410	0.0864232
$288$	15.5885	0.918559
$289$	−5.00000	−0.294118
$290$	0	0
$291$	0	0
$292$	6.39230	0.374081
$293$	−4.53590	−0.264990	−0.132495	−	0.991184i	$-0.542299\pi$
$293$	−4.53590	−0.264990	−0.132495	+	0.991184i	$0.542299\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	−6.67949	−0.386933
$299$	13.8564	0.801337
$300$	0	0
$301$	−3.46410	−0.199667
$302$	0	0
$303$	0	0
$304$	−10.0000	−0.573539
$305$	0	0
$306$	18.0000	1.02899
$307$	−6.92820	−0.395413	−0.197707	−	0.980261i	$-0.563349\pi$
$307$	−6.92820	−0.395413	−0.197707	+	0.980261i	$0.563349\pi$
$308$	−1.46410	−0.0834249
$309$	0	0
$310$	0	0
$311$	10.7846	0.611539	0.305770	−	0.952106i	$-0.401086\pi$
$311$	10.7846	0.611539	0.305770	+	0.952106i	$0.401086\pi$
$312$	0	0
$313$	16.9282	0.956839	0.478419	−	0.878132i	$-0.341210\pi$
$313$	16.9282	0.956839	0.478419	+	0.878132i	$0.341210\pi$
$314$	7.85641	0.443363
$315$	0	0
$316$	5.46410	0.307380
$317$	9.07180	0.509523	0.254761	−	0.967004i	$-0.418003\pi$
$317$	9.07180	0.509523	0.254761	+	0.967004i	$0.418003\pi$
$318$	0	0
$319$	1.46410	0.0819740
$320$	0	0
$321$	0	0
$322$	−12.0000	−0.668734
$323$	−6.92820	−0.385496
$324$	9.00000	0.500000
$325$	0	0
$326$	30.0000	1.66155
$327$	0	0
$328$	2.53590	0.140022
$329$	−4.00000	−0.220527
$330$	0	0
$331$	11.6077	0.638017	0.319008	−	0.947752i	$-0.396650\pi$
$331$	11.6077	0.638017	0.319008	+	0.947752i	$0.396650\pi$
$332$	14.9282	0.819292
$333$	0	0
$334$	37.8564	2.07141
$335$	0	0
$336$	0	0
$337$	21.0718	1.14785	0.573927	−	0.818906i	$-0.305419\pi$
$337$	21.0718	1.14785	0.573927	+	0.818906i	$0.305419\pi$
$338$	−15.5885	−0.847900
$339$	0	0
$340$	0	0
$341$	7.21539	0.390735
$342$	−10.3923	−0.561951
$343$	−1.00000	−0.0539949
$344$	−6.00000	−0.323498
$345$	0	0
$346$	−22.3923	−1.20382
$347$	−4.00000	−0.214731	−0.107366	−	0.994220i	$-0.534242\pi$
$347$	−4.00000	−0.214731	−0.107366	+	0.994220i	$0.534242\pi$
$348$	0	0
$349$	−3.07180	−0.164430	−0.0822148	−	0.996615i	$-0.526199\pi$
$349$	−3.07180	−0.164430	−0.0822148	+	0.996615i	$0.526199\pi$
$350$	0	0
$351$	0	0
$352$	−7.60770	−0.405492
$353$	8.14359	0.433440	0.216720	−	0.976234i	$-0.430464\pi$
$353$	8.14359	0.433440	0.216720	+	0.976234i	$0.430464\pi$
$354$	0	0
$355$	0	0
$356$	−10.5359	−0.558401
$357$	0	0
$358$	12.0000	0.634220
$359$	−4.39230	−0.231817	−0.115908	−	0.993260i	$-0.536978\pi$
$359$	−4.39230	−0.231817	−0.115908	+	0.993260i	$0.536978\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	1.60770	0.0844986
$363$	0	0
$364$	−2.00000	−0.104828
$365$	0	0
$366$	0	0
$367$	5.07180	0.264746	0.132373	−	0.991200i	$-0.457740\pi$
$367$	5.07180	0.264746	0.132373	+	0.991200i	$0.457740\pi$
$368$	−34.6410	−1.80579
$369$	4.39230	0.228654
$370$	0	0
$371$	−8.92820	−0.463529
$372$	0	0
$373$	−3.07180	−0.159052	−0.0795258	−	0.996833i	$-0.525341\pi$
$373$	−3.07180	−0.159052	−0.0795258	+	0.996833i	$0.525341\pi$
$374$	−8.78461	−0.454241
$375$	0	0
$376$	−6.92820	−0.357295
$377$	2.00000	0.103005
$378$	0	0
$379$	−1.46410	−0.0752058	−0.0376029	−	0.999293i	$-0.511972\pi$
$379$	−1.46410	−0.0752058	−0.0376029	+	0.999293i	$0.511972\pi$
$380$	0	0
$381$	0	0
$382$	−21.4641	−1.09820
$383$	16.0000	0.817562	0.408781	−	0.912633i	$-0.365954\pi$
$383$	16.0000	0.817562	0.408781	+	0.912633i	$0.365954\pi$
$384$	0	0
$385$	0	0
$386$	−27.7128	−1.41055
$387$	−10.3923	−0.528271
$388$	4.53590	0.230275
$389$	2.00000	0.101404	0.0507020	−	0.998714i	$-0.483854\pi$
$389$	2.00000	0.101404	0.0507020	+	0.998714i	$0.483854\pi$
$390$	0	0
$391$	−24.0000	−1.21373
$392$	−1.73205	−0.0874818
$393$	0	0
$394$	−39.4641	−1.98817
$395$	0	0
$396$	−4.39230	−0.220722
$397$	−12.9282	−0.648848	−0.324424	−	0.945912i	$-0.605170\pi$
$397$	−12.9282	−0.648848	−0.324424	+	0.945912i	$0.605170\pi$
$398$	6.92820	0.347279
$399$	0	0
$400$	0	0
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	0	0
$403$	9.85641	0.490983
$404$	−6.53590	−0.325173
$405$	0	0
$406$	−1.73205	−0.0859602
$407$	0	0
$408$	0	0
$409$	−34.2487	−1.69349	−0.846745	−	0.531999i	$-0.821441\pi$
$409$	−34.2487	−1.69349	−0.846745	+	0.531999i	$0.821441\pi$
$410$	0	0
$411$	0	0
$412$	16.0000	0.788263
$413$	8.00000	0.393654
$414$	−36.0000	−1.76930
$415$	0	0
$416$	−10.3923	−0.509525
$417$	0	0
$418$	5.07180	0.248070
$419$	24.0000	1.17248	0.586238	−	0.810139i	$-0.300608\pi$
$419$	24.0000	1.17248	0.586238	+	0.810139i	$0.300608\pi$
$420$	0	0
$421$	−22.7846	−1.11045	−0.555227	−	0.831699i	$-0.687369\pi$
$421$	−22.7846	−1.11045	−0.555227	+	0.831699i	$0.687369\pi$
$422$	35.3205	1.71938
$423$	−12.0000	−0.583460
$424$	−15.4641	−0.751003
$425$	0	0
$426$	0	0
$427$	−5.46410	−0.264426
$428$	2.92820	0.141540
$429$	0	0
$430$	0	0
$431$	−21.8564	−1.05279	−0.526393	−	0.850241i	$-0.676456\pi$
$431$	−21.8564	−1.05279	−0.526393	+	0.850241i	$0.676456\pi$
$432$	0	0
$433$	33.3205	1.60128	0.800641	−	0.599145i	$-0.204493\pi$
$433$	33.3205	1.60128	0.800641	+	0.599145i	$0.204493\pi$
$434$	−8.53590	−0.409736
$435$	0	0
$436$	10.0000	0.478913
$437$	13.8564	0.662842
$438$	0	0
$439$	6.14359	0.293218	0.146609	−	0.989195i	$-0.453164\pi$
$439$	6.14359	0.293218	0.146609	+	0.989195i	$0.453164\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	−12.0000	−0.570782
$443$	−29.3205	−1.39306	−0.696530	−	0.717528i	$-0.745274\pi$
$443$	−29.3205	−1.39306	−0.696530	+	0.717528i	$0.745274\pi$
$444$	0	0
$445$	0	0
$446$	−32.7846	−1.55240
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	−19.8564	−0.937082	−0.468541	−	0.883442i	$-0.655220\pi$
$449$	−19.8564	−0.937082	−0.468541	+	0.883442i	$0.655220\pi$
$450$	0	0
$451$	−2.14359	−0.100938
$452$	16.0000	0.752577
$453$	0	0
$454$	30.9282	1.45153
$455$	0	0
$456$	0	0
$457$	22.7846	1.06582	0.532910	−	0.846172i	$-0.321099\pi$
$457$	22.7846	1.06582	0.532910	+	0.846172i	$0.321099\pi$
$458$	−38.5359	−1.80066
$459$	0	0
$460$	0	0
$461$	−19.3205	−0.899846	−0.449923	−	0.893067i	$-0.648549\pi$
$461$	−19.3205	−0.899846	−0.449923	+	0.893067i	$0.648549\pi$
$462$	0	0
$463$	−20.7846	−0.965943	−0.482971	−	0.875636i	$-0.660442\pi$
$463$	−20.7846	−0.965943	−0.482971	+	0.875636i	$0.660442\pi$
$464$	−5.00000	−0.232119
$465$	0	0
$466$	−41.3205	−1.91414
$467$	−13.8564	−0.641198	−0.320599	−	0.947215i	$-0.603884\pi$
$467$	−13.8564	−0.641198	−0.320599	+	0.947215i	$0.603884\pi$
$468$	−6.00000	−0.277350
$469$	−4.00000	−0.184703
$470$	0	0
$471$	0	0
$472$	13.8564	0.637793
$473$	5.07180	0.233201
$474$	0	0
$475$	0	0
$476$	3.46410	0.158777
$477$	−26.7846	−1.22638
$478$	29.0718	1.32971
$479$	−3.85641	−0.176204	−0.0881019	−	0.996111i	$-0.528080\pi$
$479$	−3.85641	−0.176204	−0.0881019	+	0.996111i	$0.528080\pi$
$480$	0	0
$481$	0	0
$482$	48.2487	2.19767
$483$	0	0
$484$	−8.85641	−0.402564
$485$	0	0
$486$	0	0
$487$	−20.7846	−0.941841	−0.470920	−	0.882176i	$-0.656078\pi$
$487$	−20.7846	−0.941841	−0.470920	+	0.882176i	$0.656078\pi$
$488$	−9.46410	−0.428420
$489$	0	0
$490$	0	0
$491$	−21.4641	−0.968661	−0.484331	−	0.874885i	$-0.660937\pi$
$491$	−21.4641	−0.968661	−0.484331	+	0.874885i	$0.660937\pi$
$492$	0	0
$493$	−3.46410	−0.156015
$494$	6.92820	0.311715
$495$	0	0
$496$	−24.6410	−1.10641
$497$	8.00000	0.358849
$498$	0	0
$499$	9.07180	0.406109	0.203055	−	0.979167i	$-0.434913\pi$
$499$	9.07180	0.406109	0.203055	+	0.979167i	$0.434913\pi$
$500$	0	0
$501$	0	0
$502$	1.60770	0.0717549
$503$	2.14359	0.0955781	0.0477891	−	0.998857i	$-0.484782\pi$
$503$	2.14359	0.0955781	0.0477891	+	0.998857i	$0.484782\pi$
$504$	−5.19615	−0.231455
$505$	0	0
$506$	17.5692	0.781047
$507$	0	0
$508$	−15.4641	−0.686109
$509$	−19.8564	−0.880120	−0.440060	−	0.897968i	$-0.645043\pi$
$509$	−19.8564	−0.880120	−0.440060	+	0.897968i	$0.645043\pi$
$510$	0	0
$511$	−6.39230	−0.282779
$512$	8.66025	0.382733
$513$	0	0
$514$	13.6077	0.600210
$515$	0	0
$516$	0	0
$517$	5.85641	0.257564
$518$	0	0
$519$	0	0
$520$	0	0
$521$	26.7846	1.17346	0.586728	−	0.809784i	$-0.300416\pi$
$521$	26.7846	1.17346	0.586728	+	0.809784i	$0.300416\pi$
$522$	−5.19615	−0.227429
$523$	44.7846	1.95829	0.979147	−	0.203152i	$-0.0651187\pi$
$523$	44.7846	1.95829	0.979147	+	0.203152i	$0.0651187\pi$
$524$	8.92820	0.390030
$525$	0	0
$526$	−36.9282	−1.61015
$527$	−17.0718	−0.743659
$528$	0	0
$529$	25.0000	1.08696
$530$	0	0
$531$	24.0000	1.04151
$532$	−2.00000	−0.0867110
$533$	−2.92820	−0.126835
$534$	0	0
$535$	0	0
$536$	−6.92820	−0.299253
$537$	0	0
$538$	−13.1769	−0.568097
$539$	1.46410	0.0630633
$540$	0	0
$541$	36.6410	1.57532	0.787660	−	0.616110i	$-0.211292\pi$
$541$	36.6410	1.57532	0.787660	+	0.616110i	$0.211292\pi$
$542$	48.2487	2.07246
$543$	0	0
$544$	18.0000	0.771744
$545$	0	0
$546$	0	0
$547$	−40.7846	−1.74382	−0.871912	−	0.489663i	$-0.837120\pi$
$547$	−40.7846	−1.74382	−0.871912	+	0.489663i	$0.837120\pi$
$548$	5.07180	0.216656
$549$	−16.3923	−0.699607
$550$	0	0
$551$	2.00000	0.0852029
$552$	0	0
$553$	−5.46410	−0.232357
$554$	0.248711	0.0105667
$555$	0	0
$556$	−9.85641	−0.418005
$557$	41.7128	1.76743	0.883714	−	0.468027i	$-0.155035\pi$
$557$	41.7128	1.76743	0.883714	+	0.468027i	$0.155035\pi$
$558$	−25.6077	−1.08406
$559$	6.92820	0.293032
$560$	0	0
$561$	0	0
$562$	0.248711	0.0104913
$563$	25.8564	1.08972	0.544859	−	0.838528i	$-0.316583\pi$
$563$	25.8564	1.08972	0.544859	+	0.838528i	$0.316583\pi$
$564$	0	0
$565$	0	0
$566$	−12.0000	−0.504398
$567$	−9.00000	−0.377964
$568$	13.8564	0.581402
$569$	6.78461	0.284426	0.142213	−	0.989836i	$-0.454578\pi$
$569$	6.78461	0.284426	0.142213	+	0.989836i	$0.454578\pi$
$570$	0	0
$571$	−25.8564	−1.08206	−0.541028	−	0.841004i	$-0.681965\pi$
$571$	−25.8564	−1.08206	−0.541028	+	0.841004i	$0.681965\pi$
$572$	2.92820	0.122434
$573$	0	0
$574$	2.53590	0.105846
$575$	0	0
$576$	−3.00000	−0.125000
$577$	24.5359	1.02144	0.510721	−	0.859746i	$-0.329378\pi$
$577$	24.5359	1.02144	0.510721	+	0.859746i	$0.329378\pi$
$578$	−8.66025	−0.360219
$579$	0	0
$580$	0	0
$581$	−14.9282	−0.619326
$582$	0	0
$583$	13.0718	0.541379
$584$	−11.0718	−0.458154
$585$	0	0
$586$	−7.85641	−0.324545
$587$	−6.92820	−0.285958	−0.142979	−	0.989726i	$-0.545668\pi$
$587$	−6.92820	−0.285958	−0.142979	+	0.989726i	$0.545668\pi$
$588$	0	0
$589$	9.85641	0.406126
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−0.928203	−0.0381167	−0.0190584	−	0.999818i	$-0.506067\pi$
$593$	−0.928203	−0.0381167	−0.0190584	+	0.999818i	$0.506067\pi$
$594$	0	0
$595$	0	0
$596$	−3.85641	−0.157965
$597$	0	0
$598$	24.0000	0.981433
$599$	7.60770	0.310842	0.155421	−	0.987848i	$-0.450327\pi$
$599$	7.60770	0.310842	0.155421	+	0.987848i	$0.450327\pi$
$600$	0	0
$601$	13.4641	0.549212	0.274606	−	0.961557i	$-0.411453\pi$
$601$	13.4641	0.549212	0.274606	+	0.961557i	$0.411453\pi$
$602$	−6.00000	−0.244542
$603$	−12.0000	−0.488678
$604$	0	0
$605$	0	0
$606$	0	0
$607$	4.00000	0.162355	0.0811775	−	0.996700i	$-0.474132\pi$
$607$	4.00000	0.162355	0.0811775	+	0.996700i	$0.474132\pi$
$608$	−10.3923	−0.421464
$609$	0	0
$610$	0	0
$611$	8.00000	0.323645
$612$	10.3923	0.420084
$613$	−18.0000	−0.727013	−0.363507	−	0.931592i	$-0.618421\pi$
$613$	−18.0000	−0.727013	−0.363507	+	0.931592i	$0.618421\pi$
$614$	−12.0000	−0.484281
$615$	0	0
$616$	2.53590	0.102174
$617$	−5.07180	−0.204183	−0.102091	−	0.994775i	$-0.532553\pi$
$617$	−5.07180	−0.204183	−0.102091	+	0.994775i	$0.532553\pi$
$618$	0	0
$619$	40.9282	1.64504	0.822522	−	0.568734i	$-0.192567\pi$
$619$	40.9282	1.64504	0.822522	+	0.568734i	$0.192567\pi$
$620$	0	0
$621$	0	0
$622$	18.6795	0.748979
$623$	10.5359	0.422112
$624$	0	0
$625$	0	0
$626$	29.3205	1.17188
$627$	0	0
$628$	4.53590	0.181002
$629$	0	0
$630$	0	0
$631$	5.07180	0.201905	0.100953	−	0.994891i	$-0.467811\pi$
$631$	5.07180	0.201905	0.100953	+	0.994891i	$0.467811\pi$
$632$	−9.46410	−0.376462
$633$	0	0
$634$	15.7128	0.624036
$635$	0	0
$636$	0	0
$637$	2.00000	0.0792429
$638$	2.53590	0.100397
$639$	24.0000	0.949425
$640$	0	0
$641$	26.7846	1.05793	0.528964	−	0.848644i	$-0.322581\pi$
$641$	26.7846	1.05793	0.528964	+	0.848644i	$0.322581\pi$
$642$	0	0
$643$	−25.0718	−0.988735	−0.494368	−	0.869253i	$-0.664600\pi$
$643$	−25.0718	−0.988735	−0.494368	+	0.869253i	$0.664600\pi$
$644$	−6.92820	−0.273009
$645$	0	0
$646$	−12.0000	−0.472134
$647$	32.7846	1.28890	0.644448	−	0.764648i	$-0.277087\pi$
$647$	32.7846	1.28890	0.644448	+	0.764648i	$0.277087\pi$
$648$	−15.5885	−0.612372
$649$	−11.7128	−0.459768
$650$	0	0
$651$	0	0
$652$	17.3205	0.678323
$653$	−27.7128	−1.08449	−0.542243	−	0.840222i	$-0.682425\pi$
$653$	−27.7128	−1.08449	−0.542243	+	0.840222i	$0.682425\pi$
$654$	0	0
$655$	0	0
$656$	7.32051	0.285818
$657$	−19.1769	−0.748163
$658$	−6.92820	−0.270089
$659$	41.4641	1.61521	0.807606	−	0.589722i	$-0.200763\pi$
$659$	41.4641	1.61521	0.807606	+	0.589722i	$0.200763\pi$
$660$	0	0
$661$	−0.928203	−0.0361029	−0.0180515	−	0.999837i	$-0.505746\pi$
$661$	−0.928203	−0.0361029	−0.0180515	+	0.999837i	$0.505746\pi$
$662$	20.1051	0.781408
$663$	0	0
$664$	−25.8564	−1.00342
$665$	0	0
$666$	0	0
$667$	6.92820	0.268261
$668$	21.8564	0.845650
$669$	0	0
$670$	0	0
$671$	8.00000	0.308837
$672$	0	0
$673$	1.21539	0.0468499	0.0234249	−	0.999726i	$-0.492543\pi$
$673$	1.21539	0.0468499	0.0234249	+	0.999726i	$0.492543\pi$
$674$	36.4974	1.40583
$675$	0	0
$676$	−9.00000	−0.346154
$677$	16.2487	0.624489	0.312244	−	0.950002i	$-0.398919\pi$
$677$	16.2487	0.624489	0.312244	+	0.950002i	$0.398919\pi$
$678$	0	0
$679$	−4.53590	−0.174072
$680$	0	0
$681$	0	0
$682$	12.4974	0.478551
$683$	−5.07180	−0.194067	−0.0970335	−	0.995281i	$-0.530935\pi$
$683$	−5.07180	−0.194067	−0.0970335	+	0.995281i	$0.530935\pi$
$684$	−6.00000	−0.229416
$685$	0	0
$686$	−1.73205	−0.0661300
$687$	0	0
$688$	−17.3205	−0.660338
$689$	17.8564	0.680275
$690$	0	0
$691$	−31.7128	−1.20641	−0.603206	−	0.797585i	$-0.706110\pi$
$691$	−31.7128	−1.20641	−0.603206	+	0.797585i	$0.706110\pi$
$692$	−12.9282	−0.491457
$693$	4.39230	0.166850
$694$	−6.92820	−0.262991
$695$	0	0
$696$	0	0
$697$	5.07180	0.192108
$698$	−5.32051	−0.201384
$699$	0	0
$700$	0	0
$701$	−8.14359	−0.307579	−0.153790	−	0.988104i	$-0.549148\pi$
$701$	−8.14359	−0.307579	−0.153790	+	0.988104i	$0.549148\pi$
$702$	0	0
$703$	0	0
$704$	1.46410	0.0551804
$705$	0	0
$706$	14.1051	0.530853
$707$	6.53590	0.245808
$708$	0	0
$709$	29.7128	1.11589	0.557944	−	0.829879i	$-0.311590\pi$
$709$	29.7128	1.11589	0.557944	+	0.829879i	$0.311590\pi$
$710$	0	0
$711$	−16.3923	−0.614759
$712$	18.2487	0.683899
$713$	34.1436	1.27869
$714$	0	0
$715$	0	0
$716$	6.92820	0.258919
$717$	0	0
$718$	−7.60770	−0.283917
$719$	−39.7128	−1.48104	−0.740519	−	0.672035i	$-0.765420\pi$
$719$	−39.7128	−1.48104	−0.740519	+	0.672035i	$0.765420\pi$
$720$	0	0
$721$	−16.0000	−0.595871
$722$	−25.9808	−0.966904
$723$	0	0
$724$	0.928203	0.0344964
$725$	0	0
$726$	0	0
$727$	−9.85641	−0.365554	−0.182777	−	0.983154i	$-0.558509\pi$
$727$	−9.85641	−0.365554	−0.182777	+	0.983154i	$0.558509\pi$
$728$	3.46410	0.128388
$729$	−27.0000	−1.00000
$730$	0	0
$731$	−12.0000	−0.443836
$732$	0	0
$733$	28.5359	1.05400	0.526999	−	0.849866i	$-0.323317\pi$
$733$	28.5359	1.05400	0.526999	+	0.849866i	$0.323317\pi$
$734$	8.78461	0.324246
$735$	0	0
$736$	−36.0000	−1.32698
$737$	5.85641	0.215724
$738$	7.60770	0.280043
$739$	−3.32051	−0.122147	−0.0610734	−	0.998133i	$-0.519452\pi$
$739$	−3.32051	−0.122147	−0.0610734	+	0.998133i	$0.519452\pi$
$740$	0	0
$741$	0	0
$742$	−15.4641	−0.567705
$743$	−47.4641	−1.74129	−0.870645	−	0.491913i	$-0.836298\pi$
$743$	−47.4641	−1.74129	−0.870645	+	0.491913i	$0.836298\pi$
$744$	0	0
$745$	0	0
$746$	−5.32051	−0.194798
$747$	−44.7846	−1.63858
$748$	−5.07180	−0.185443
$749$	−2.92820	−0.106994
$750$	0	0
$751$	−4.39230	−0.160277	−0.0801387	−	0.996784i	$-0.525536\pi$
$751$	−4.39230	−0.160277	−0.0801387	+	0.996784i	$0.525536\pi$
$752$	−20.0000	−0.729325
$753$	0	0
$754$	3.46410	0.126155
$755$	0	0
$756$	0	0
$757$	−27.7128	−1.00724	−0.503620	−	0.863925i	$-0.667999\pi$
$757$	−27.7128	−1.00724	−0.503620	+	0.863925i	$0.667999\pi$
$758$	−2.53590	−0.0921080
$759$	0	0
$760$	0	0
$761$	−9.21539	−0.334058	−0.167029	−	0.985952i	$-0.553417\pi$
$761$	−9.21539	−0.334058	−0.167029	+	0.985952i	$0.553417\pi$
$762$	0	0
$763$	−10.0000	−0.362024
$764$	−12.3923	−0.448338
$765$	0	0
$766$	27.7128	1.00130
$767$	−16.0000	−0.577727
$768$	0	0
$769$	51.3205	1.85066	0.925332	−	0.379157i	$-0.123786\pi$
$769$	51.3205	1.85066	0.925332	+	0.379157i	$0.123786\pi$
$770$	0	0
$771$	0	0
$772$	−16.0000	−0.575853
$773$	15.4641	0.556205	0.278103	−	0.960551i	$-0.410294\pi$
$773$	15.4641	0.556205	0.278103	+	0.960551i	$0.410294\pi$
$774$	−18.0000	−0.646997
$775$	0	0
$776$	−7.85641	−0.282029
$777$	0	0
$778$	3.46410	0.124194
$779$	−2.92820	−0.104914
$780$	0	0
$781$	−11.7128	−0.419117
$782$	−41.5692	−1.48651
$783$	0	0
$784$	−5.00000	−0.178571
$785$	0	0
$786$	0	0
$787$	−12.7846	−0.455722	−0.227861	−	0.973694i	$-0.573173\pi$
$787$	−12.7846	−0.455722	−0.227861	+	0.973694i	$0.573173\pi$
$788$	−22.7846	−0.811668
$789$	0	0
$790$	0	0
$791$	−16.0000	−0.568895
$792$	7.60770	0.270328
$793$	10.9282	0.388072
$794$	−22.3923	−0.794673
$795$	0	0
$796$	4.00000	0.141776
$797$	10.6795	0.378287	0.189144	−	0.981949i	$-0.439429\pi$
$797$	10.6795	0.378287	0.189144	+	0.981949i	$0.439429\pi$
$798$	0	0
$799$	−13.8564	−0.490204
$800$	0	0
$801$	31.6077	1.11680
$802$	−10.3923	−0.366965
$803$	9.35898	0.330271
$804$	0	0
$805$	0	0
$806$	17.0718	0.601328
$807$	0	0
$808$	11.3205	0.398254
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	0	0
$811$	−4.00000	−0.140459	−0.0702295	−	0.997531i	$-0.522373\pi$
$811$	−4.00000	−0.140459	−0.0702295	+	0.997531i	$0.522373\pi$
$812$	−1.00000	−0.0350931
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	6.92820	0.242387
$818$	−59.3205	−2.07409
$819$	6.00000	0.209657
$820$	0	0
$821$	47.5692	1.66018	0.830089	−	0.557632i	$-0.188290\pi$
$821$	47.5692	1.66018	0.830089	+	0.557632i	$0.188290\pi$
$822$	0	0
$823$	35.1769	1.22619	0.613095	−	0.790009i	$-0.289924\pi$
$823$	35.1769	1.22619	0.613095	+	0.790009i	$0.289924\pi$
$824$	−27.7128	−0.965422
$825$	0	0
$826$	13.8564	0.482126
$827$	−43.4641	−1.51139	−0.755697	−	0.654921i	$-0.772702\pi$
$827$	−43.4641	−1.51139	−0.755697	+	0.654921i	$0.772702\pi$
$828$	−20.7846	−0.722315
$829$	11.6077	0.403152	0.201576	−	0.979473i	$-0.435394\pi$
$829$	11.6077	0.403152	0.201576	+	0.979473i	$0.435394\pi$
$830$	0	0
$831$	0	0
$832$	2.00000	0.0693375
$833$	−3.46410	−0.120024
$834$	0	0
$835$	0	0
$836$	2.92820	0.101274
$837$	0	0
$838$	41.5692	1.43598
$839$	14.7846	0.510421	0.255211	−	0.966885i	$-0.417855\pi$
$839$	14.7846	0.510421	0.255211	+	0.966885i	$0.417855\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−39.4641	−1.36002
$843$	0	0
$844$	20.3923	0.701932
$845$	0	0
$846$	−20.7846	−0.714590
$847$	8.85641	0.304310
$848$	−44.6410	−1.53298
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−22.6795	−0.776531	−0.388266	−	0.921548i	$-0.626926\pi$
$853$	−22.6795	−0.776531	−0.388266	+	0.921548i	$0.626926\pi$
$854$	−9.46410	−0.323855
$855$	0	0
$856$	−5.07180	−0.173350
$857$	37.7128	1.28825	0.644123	−	0.764922i	$-0.277223\pi$
$857$	37.7128	1.28825	0.644123	+	0.764922i	$0.277223\pi$
$858$	0	0
$859$	−50.7846	−1.73275	−0.866374	−	0.499395i	$-0.833556\pi$
$859$	−50.7846	−1.73275	−0.866374	+	0.499395i	$0.833556\pi$
$860$	0	0
$861$	0	0
$862$	−37.8564	−1.28939
$863$	11.7128	0.398709	0.199354	−	0.979927i	$-0.436116\pi$
$863$	11.7128	0.398709	0.199354	+	0.979927i	$0.436116\pi$
$864$	0	0
$865$	0	0
$866$	57.7128	1.96116
$867$	0	0
$868$	−4.92820	−0.167274
$869$	8.00000	0.271381
$870$	0	0
$871$	8.00000	0.271070
$872$	−17.3205	−0.586546
$873$	−13.6077	−0.460551
$874$	24.0000	0.811812
$875$	0	0
$876$	0	0
$877$	−12.9282	−0.436554	−0.218277	−	0.975887i	$-0.570044\pi$
$877$	−12.9282	−0.436554	−0.218277	+	0.975887i	$0.570044\pi$
$878$	10.6410	0.359117
$879$	0	0
$880$	0	0
$881$	−37.1769	−1.25252	−0.626261	−	0.779613i	$-0.715416\pi$
$881$	−37.1769	−1.25252	−0.626261	+	0.779613i	$0.715416\pi$
$882$	−5.19615	−0.174964
$883$	−16.7846	−0.564847	−0.282424	−	0.959290i	$-0.591138\pi$
$883$	−16.7846	−0.564847	−0.282424	+	0.959290i	$0.591138\pi$
$884$	−6.92820	−0.233021
$885$	0	0
$886$	−50.7846	−1.70614
$887$	20.7846	0.697879	0.348939	−	0.937145i	$-0.386542\pi$
$887$	20.7846	0.697879	0.348939	+	0.937145i	$0.386542\pi$
$888$	0	0
$889$	15.4641	0.518649
$890$	0	0
$891$	13.1769	0.441443
$892$	−18.9282	−0.633763
$893$	8.00000	0.267710
$894$	0	0
$895$	0	0
$896$	−12.1244	−0.405046
$897$	0	0
$898$	−34.3923	−1.14769
$899$	4.92820	0.164365
$900$	0	0
$901$	−30.9282	−1.03037
$902$	−3.71281	−0.123623
$903$	0	0
$904$	−27.7128	−0.921714
$905$	0	0
$906$	0	0
$907$	42.3923	1.40761	0.703807	−	0.710392i	$-0.251482\pi$
$907$	42.3923	1.40761	0.703807	+	0.710392i	$0.251482\pi$
$908$	17.8564	0.592586
$909$	19.6077	0.650346
$910$	0	0
$911$	12.6795	0.420090	0.210045	−	0.977692i	$-0.432639\pi$
$911$	12.6795	0.420090	0.210045	+	0.977692i	$0.432639\pi$
$912$	0	0
$913$	21.8564	0.723341
$914$	39.4641	1.30536
$915$	0	0
$916$	−22.2487	−0.735118
$917$	−8.92820	−0.294835
$918$	0	0
$919$	−2.92820	−0.0965925	−0.0482963	−	0.998833i	$-0.515379\pi$
$919$	−2.92820	−0.0965925	−0.0482963	+	0.998833i	$0.515379\pi$
$920$	0	0
$921$	0	0
$922$	−33.4641	−1.10208
$923$	−16.0000	−0.526646
$924$	0	0
$925$	0	0
$926$	−36.0000	−1.18303
$927$	−48.0000	−1.57653
$928$	−5.19615	−0.170572
$929$	9.71281	0.318667	0.159334	−	0.987225i	$-0.449065\pi$
$929$	9.71281	0.318667	0.159334	+	0.987225i	$0.449065\pi$
$930$	0	0
$931$	2.00000	0.0655474
$932$	−23.8564	−0.781443
$933$	0	0
$934$	−24.0000	−0.785304
$935$	0	0
$936$	10.3923	0.339683
$937$	−28.6410	−0.935661	−0.467831	−	0.883818i	$-0.654964\pi$
$937$	−28.6410	−0.935661	−0.467831	+	0.883818i	$0.654964\pi$
$938$	−6.92820	−0.226214
$939$	0	0
$940$	0	0
$941$	−19.0718	−0.621723	−0.310861	−	0.950455i	$-0.600618\pi$
$941$	−19.0718	−0.621723	−0.310861	+	0.950455i	$0.600618\pi$
$942$	0	0
$943$	−10.1436	−0.330321
$944$	40.0000	1.30189
$945$	0	0
$946$	8.78461	0.285612
$947$	−54.3923	−1.76751	−0.883756	−	0.467948i	$-0.844994\pi$
$947$	−54.3923	−1.76751	−0.883756	+	0.467948i	$0.844994\pi$
$948$	0	0
$949$	12.7846	0.415006
$950$	0	0
$951$	0	0
$952$	−6.00000	−0.194461
$953$	−42.7846	−1.38593	−0.692965	−	0.720971i	$-0.743696\pi$
$953$	−42.7846	−1.38593	−0.692965	+	0.720971i	$0.743696\pi$
$954$	−46.3923	−1.50201
$955$	0	0
$956$	16.7846	0.542853
$957$	0	0
$958$	−6.67949	−0.215805
$959$	−5.07180	−0.163777
$960$	0	0
$961$	−6.71281	−0.216542
$962$	0	0
$963$	−8.78461	−0.283080
$964$	27.8564	0.897194
$965$	0	0
$966$	0	0
$967$	−17.6077	−0.566225	−0.283113	−	0.959087i	$-0.591367\pi$
$967$	−17.6077	−0.566225	−0.283113	+	0.959087i	$0.591367\pi$
$968$	15.3397	0.493038
$969$	0	0
$970$	0	0
$971$	−42.0000	−1.34784	−0.673922	−	0.738802i	$-0.735392\pi$
$971$	−42.0000	−1.34784	−0.673922	+	0.738802i	$0.735392\pi$
$972$	0	0
$973$	9.85641	0.315982
$974$	−36.0000	−1.15351
$975$	0	0
$976$	−27.3205	−0.874508
$977$	18.0000	0.575871	0.287936	−	0.957650i	$-0.407031\pi$
$977$	18.0000	0.575871	0.287936	+	0.957650i	$0.407031\pi$
$978$	0	0
$979$	−15.4256	−0.493005
$980$	0	0
$981$	−30.0000	−0.957826
$982$	−37.1769	−1.18636
$983$	28.7846	0.918086	0.459043	−	0.888414i	$-0.348192\pi$
$983$	28.7846	0.918086	0.459043	+	0.888414i	$0.348192\pi$
$984$	0	0
$985$	0	0
$986$	−6.00000	−0.191079
$987$	0	0
$988$	4.00000	0.127257
$989$	24.0000	0.763156
$990$	0	0
$991$	−19.7128	−0.626198	−0.313099	−	0.949720i	$-0.601367\pi$
$991$	−19.7128	−0.626198	−0.313099	+	0.949720i	$0.601367\pi$
$992$	−25.6077	−0.813045
$993$	0	0
$994$	13.8564	0.439499
$995$	0	0
$996$	0	0
$997$	26.1051	0.826757	0.413379	−	0.910559i	$-0.364349\pi$
$997$	26.1051	0.826757	0.413379	+	0.910559i	$0.364349\pi$
$998$	15.7128	0.497380
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5075.2.a.n.1.2		2
5.4	even	2		1015.2.a.d.1.1	✓	2
15.14	odd	2		9135.2.a.r.1.2		2
35.34	odd	2		7105.2.a.g.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1015.2.a.d.1.1	✓	2	5.4	even	2
5075.2.a.n.1.2		2	1.1	even	1	trivial
7105.2.a.g.1.1		2	35.34	odd	2
9135.2.a.r.1.2		2	15.14	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 5075 = 5^{2} \cdot 7 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5075.a (trivial)

\(f(q)\)	\(=\)	\(q+1.73205 q^{2} +1.00000 q^{4} -1.00000 q^{7} -1.73205 q^{8} -3.00000 q^{9} +1.46410 q^{11} +2.00000 q^{13} -1.73205 q^{14} -5.00000 q^{16} -3.46410 q^{17} -5.19615 q^{18} +2.00000 q^{19} +2.53590 q^{22} +6.92820 q^{23} +3.46410 q^{26} -1.00000 q^{28} +1.00000 q^{29} +4.92820 q^{31} -5.19615 q^{32} -6.00000 q^{34} -3.00000 q^{36} +3.46410 q^{38} -1.46410 q^{41} +3.46410 q^{43} +1.46410 q^{44} +12.0000 q^{46} +4.00000 q^{47} +1.00000 q^{49} +2.00000 q^{52} +8.92820 q^{53} +1.73205 q^{56} +1.73205 q^{58} -8.00000 q^{59} +5.46410 q^{61} +8.53590 q^{62} +3.00000 q^{63} +1.00000 q^{64} +4.00000 q^{67} -3.46410 q^{68} -8.00000 q^{71} +5.19615 q^{72} +6.39230 q^{73} +2.00000 q^{76} -1.46410 q^{77} +5.46410 q^{79} +9.00000 q^{81} -2.53590 q^{82} +14.9282 q^{83} +6.00000 q^{86} -2.53590 q^{88} -10.5359 q^{89} -2.00000 q^{91} +6.92820 q^{92} +6.92820 q^{94} +4.53590 q^{97} +1.73205 q^{98} -4.39230 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} - 2 q^{7} - 6 q^{9} - 4 q^{11} + 4 q^{13} - 10 q^{16} + 4 q^{19} + 12 q^{22} - 2 q^{28} + 2 q^{29} - 4 q^{31} - 12 q^{34} - 6 q^{36} + 4 q^{41} - 4 q^{44} + 24 q^{46} + 8 q^{47} + 2 q^{49}+ \cdots + 12 q^{99}+O(q^{100}) \) 2 * q + 2 * q^4 - 2 * q^7 - 6 * q^9 - 4 * q^11 + 4 * q^13 - 10 * q^16 + 4 * q^19 + 12 * q^22 - 2 * q^28 + 2 * q^29 - 4 * q^31 - 12 * q^34 - 6 * q^36 + 4 * q^41 - 4 * q^44 + 24 * q^46 + 8 * q^47 + 2 * q^49 + 4 * q^52 + 4 * q^53 - 16 * q^59 + 4 * q^61 + 24 * q^62 + 6 * q^63 + 2 * q^64 + 8 * q^67 - 16 * q^71 - 8 * q^73 + 4 * q^76 + 4 * q^77 + 4 * q^79 + 18 * q^81 - 12 * q^82 + 16 * q^83 + 12 * q^86 - 12 * q^88 - 28 * q^89 - 4 * q^91 + 16 * q^97 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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