LMFDB - Embedded newform 5202.2.a.bt.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5202,2,Mod(1,5202)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5202, 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("5202.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5202 = 2 \cdot 3^{2} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5202.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:	$41.5381791315$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{16})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 2 $ x^4 - 4*x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 102)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.765367$ of defining polynomial
Character	$\chi$	$=$	5202.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -0.433546 q^{5} +3.41421 q^{7} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} -0.433546 q^{5} +3.41421 q^{7} -1.00000 q^{8} +0.433546 q^{10} +1.69552 q^{11} -0.331821 q^{13} -3.41421 q^{14} +1.00000 q^{16} +2.14386 q^{19} -0.433546 q^{20} -1.69552 q^{22} -1.88348 q^{23} -4.81204 q^{25} +0.331821 q^{26} +3.41421 q^{28} -8.53874 q^{29} +2.58579 q^{31} -1.00000 q^{32} -1.48022 q^{35} -10.6143 q^{37} -2.14386 q^{38} +0.433546 q^{40} -12.1177 q^{41} -10.0933 q^{43} +1.69552 q^{44} +1.88348 q^{46} -2.38009 q^{47} +4.65685 q^{49} +4.81204 q^{50} -0.331821 q^{52} +3.13066 q^{53} -0.735084 q^{55} -3.41421 q^{56} +8.53874 q^{58} +7.67459 q^{59} +8.03988 q^{61} -2.58579 q^{62} +1.00000 q^{64} +0.143860 q^{65} -15.7711 q^{67} +1.48022 q^{70} -0.226626 q^{71} -12.1387 q^{73} +10.6143 q^{74} +2.14386 q^{76} +5.78886 q^{77} +12.5394 q^{79} -0.433546 q^{80} +12.1177 q^{82} +1.04373 q^{83} +10.0933 q^{86} -1.69552 q^{88} -13.8281 q^{89} -1.13291 q^{91} -1.88348 q^{92} +2.38009 q^{94} -0.929461 q^{95} -3.75539 q^{97} -4.65685 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 4 q^{4} + 8 q^{7} - 4 q^{8}+O(q^{10}) $ 4 * q - 4 * q^2 + 4 * q^4 + 8 * q^7 - 4 * q^8	$ 4 q - 4 q^{2} + 4 q^{4} + 8 q^{7} - 4 q^{8} - 8 q^{11} - 8 q^{14} + 4 q^{16} - 8 q^{19} + 8 q^{22} - 8 q^{23} - 4 q^{25} + 8 q^{28} - 8 q^{29} + 16 q^{31} - 4 q^{32} + 8 q^{35} + 8 q^{37} + 8 q^{38} - 8 q^{41} - 16 q^{43} - 8 q^{44} + 8 q^{46} - 4 q^{49} + 4 q^{50} - 16 q^{55} - 8 q^{56} + 8 q^{58} + 8 q^{59} - 16 q^{62} + 4 q^{64} - 16 q^{65} - 24 q^{67} - 8 q^{70} - 24 q^{71} - 16 q^{73} - 8 q^{74} - 8 q^{76} - 16 q^{77} + 8 q^{79} + 8 q^{82} - 8 q^{83} + 16 q^{86} + 8 q^{88} - 8 q^{91} - 8 q^{92} - 8 q^{95} + 16 q^{97} + 4 q^{98}+O(q^{100}) $ 4 * q - 4 * q^2 + 4 * q^4 + 8 * q^7 - 4 * q^8 - 8 * q^11 - 8 * q^14 + 4 * q^16 - 8 * q^19 + 8 * q^22 - 8 * q^23 - 4 * q^25 + 8 * q^28 - 8 * q^29 + 16 * q^31 - 4 * q^32 + 8 * q^35 + 8 * q^37 + 8 * q^38 - 8 * q^41 - 16 * q^43 - 8 * q^44 + 8 * q^46 - 4 * q^49 + 4 * q^50 - 16 * q^55 - 8 * q^56 + 8 * q^58 + 8 * q^59 - 16 * q^62 + 4 * q^64 - 16 * q^65 - 24 * q^67 - 8 * q^70 - 24 * q^71 - 16 * q^73 - 8 * q^74 - 8 * q^76 - 16 * q^77 + 8 * q^79 + 8 * q^82 - 8 * q^83 + 16 * q^86 + 8 * q^88 - 8 * q^91 - 8 * q^92 - 8 * q^95 + 16 * q^97 + 4 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−0.433546	−0.193887	−0.0969437	−	0.995290i	$-0.530907\pi$
$5$	−0.433546	−0.193887	−0.0969437	+	0.995290i	$0.530907\pi$
$6$	0	0
$7$	3.41421	1.29045	0.645226	−	0.763992i	$-0.276763\pi$
$7$	3.41421	1.29045	0.645226	+	0.763992i	$0.276763\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	0.433546	0.137099
$11$	1.69552	0.511218	0.255609	−	0.966780i	$-0.417724\pi$
$11$	1.69552	0.511218	0.255609	+	0.966780i	$0.417724\pi$
$12$	0	0
$13$	−0.331821	−0.0920307	−0.0460153	−	0.998941i	$-0.514652\pi$
$13$	−0.331821	−0.0920307	−0.0460153	+	0.998941i	$0.514652\pi$
$14$	−3.41421	−0.912487
$15$	0	0
$16$	1.00000	0.250000
$17$	0	0
$17$	0	0
$18$	0	0
$19$	2.14386	0.491835	0.245918	−	0.969291i	$-0.420911\pi$
$19$	2.14386	0.491835	0.245918	+	0.969291i	$0.420911\pi$
$20$	−0.433546	−0.0969437
$21$	0	0
$22$	−1.69552	−0.361486
$23$	−1.88348	−0.392733	−0.196366	−	0.980531i	$-0.562914\pi$
$23$	−1.88348	−0.392733	−0.196366	+	0.980531i	$0.562914\pi$
$24$	0	0
$25$	−4.81204	−0.962408
$26$	0.331821	0.0650755
$27$	0	0
$28$	3.41421	0.645226
$29$	−8.53874	−1.58560	−0.792802	−	0.609479i	$-0.791379\pi$
$29$	−8.53874	−1.58560	−0.792802	+	0.609479i	$0.791379\pi$
$30$	0	0
$31$	2.58579	0.464421	0.232210	−	0.972666i	$-0.425404\pi$
$31$	2.58579	0.464421	0.232210	+	0.972666i	$0.425404\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	0	0
$35$	−1.48022	−0.250202
$36$	0	0
$37$	−10.6143	−1.74499	−0.872494	−	0.488625i	$-0.837499\pi$
$37$	−10.6143	−1.74499	−0.872494	+	0.488625i	$0.837499\pi$
$38$	−2.14386	−0.347780
$39$	0	0
$40$	0.433546	0.0685496
$41$	−12.1177	−1.89247	−0.946236	−	0.323476i	$-0.895149\pi$
$41$	−12.1177	−1.89247	−0.946236	+	0.323476i	$0.895149\pi$
$42$	0	0
$43$	−10.0933	−1.53922	−0.769610	−	0.638514i	$-0.779549\pi$
$43$	−10.0933	−1.53922	−0.769610	+	0.638514i	$0.779549\pi$
$44$	1.69552	0.255609
$45$	0	0
$46$	1.88348	0.277704
$47$	−2.38009	−0.347171	−0.173586	−	0.984819i	$-0.555535\pi$
$47$	−2.38009	−0.347171	−0.173586	+	0.984819i	$0.555535\pi$
$48$	0	0
$49$	4.65685	0.665265
$50$	4.81204	0.680525
$51$	0	0
$52$	−0.331821	−0.0460153
$53$	3.13066	0.430029	0.215014	−	0.976611i	$-0.431020\pi$
$53$	3.13066	0.430029	0.215014	+	0.976611i	$0.431020\pi$
$54$	0	0
$55$	−0.735084	−0.0991187
$56$	−3.41421	−0.456243
$57$	0	0
$58$	8.53874	1.12119
$59$	7.67459	0.999147	0.499573	−	0.866272i	$-0.333490\pi$
$59$	7.67459	0.999147	0.499573	+	0.866272i	$0.333490\pi$
$60$	0	0
$61$	8.03988	1.02940	0.514701	−	0.857370i	$-0.327903\pi$
$61$	8.03988	1.02940	0.514701	+	0.857370i	$0.327903\pi$
$62$	−2.58579	−0.328395
$63$	0	0
$64$	1.00000	0.125000
$65$	0.143860	0.0178436
$66$	0	0
$67$	−15.7711	−1.92675	−0.963375	−	0.268159i	$-0.913585\pi$
$67$	−15.7711	−1.92675	−0.963375	+	0.268159i	$0.913585\pi$
$68$	0	0
$69$	0	0
$70$	1.48022	0.176920
$71$	−0.226626	−0.0268955	−0.0134478	−	0.999910i	$-0.504281\pi$
$71$	−0.226626	−0.0268955	−0.0134478	+	0.999910i	$0.504281\pi$
$72$	0	0
$73$	−12.1387	−1.42072	−0.710362	−	0.703837i	$-0.751469\pi$
$73$	−12.1387	−1.42072	−0.710362	+	0.703837i	$0.751469\pi$
$74$	10.6143	1.23389
$75$	0	0
$76$	2.14386	0.245918
$77$	5.78886	0.659702
$78$	0	0
$79$	12.5394	1.41080	0.705398	−	0.708811i	$-0.250768\pi$
$79$	12.5394	1.41080	0.705398	+	0.708811i	$0.250768\pi$
$80$	−0.433546	−0.0484719
$81$	0	0
$82$	12.1177	1.33818
$83$	1.04373	0.114564	0.0572820	−	0.998358i	$-0.481757\pi$
$83$	1.04373	0.114564	0.0572820	+	0.998358i	$0.481757\pi$
$84$	0	0
$85$	0	0
$86$	10.0933	1.08839
$87$	0	0
$88$	−1.69552	−0.180743
$89$	−13.8281	−1.46577	−0.732885	−	0.680352i	$-0.761827\pi$
$89$	−13.8281	−1.46577	−0.732885	+	0.680352i	$0.761827\pi$
$90$	0	0
$91$	−1.13291	−0.118761
$92$	−1.88348	−0.196366
$93$	0	0
$94$	2.38009	0.245487
$95$	−0.929461	−0.0953607
$96$	0	0
$97$	−3.75539	−0.381302	−0.190651	−	0.981658i	$-0.561060\pi$
$97$	−3.75539	−0.381302	−0.190651	+	0.981658i	$0.561060\pi$
$98$	−4.65685	−0.470413
$99$	0	0
$100$	−4.81204	−0.481204
$101$	−0.636303	−0.0633145	−0.0316573	−	0.999499i	$-0.510079\pi$
$101$	−0.636303	−0.0633145	−0.0316573	+	0.999499i	$0.510079\pi$
$102$	0	0
$103$	8.59955	0.847339	0.423669	−	0.905817i	$-0.360742\pi$
$103$	8.59955	0.847339	0.423669	+	0.905817i	$0.360742\pi$
$104$	0.331821	0.0325378
$105$	0	0
$106$	−3.13066	−0.304076
$107$	−2.16478	−0.209278	−0.104639	−	0.994510i	$-0.533369\pi$
$107$	−2.16478	−0.209278	−0.104639	+	0.994510i	$0.533369\pi$
$108$	0	0
$109$	−7.91376	−0.758001	−0.379000	−	0.925396i	$-0.623732\pi$
$109$	−7.91376	−0.758001	−0.379000	+	0.925396i	$0.623732\pi$
$110$	0.735084	0.0700875
$111$	0	0
$112$	3.41421	0.322613
$113$	9.18828	0.864361	0.432180	−	0.901787i	$-0.357744\pi$
$113$	9.18828	0.864361	0.432180	+	0.901787i	$0.357744\pi$
$114$	0	0
$115$	0.816574	0.0761459
$116$	−8.53874	−0.792802
$117$	0	0
$118$	−7.67459	−0.706504
$119$	0	0
$120$	0	0
$121$	−8.12522	−0.738656
$122$	−8.03988	−0.727897
$123$	0	0
$124$	2.58579	0.232210
$125$	4.25397	0.380486
$126$	0	0
$127$	−12.0843	−1.07231	−0.536153	−	0.844121i	$-0.680123\pi$
$127$	−12.0843	−1.07231	−0.536153	+	0.844121i	$0.680123\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−0.143860	−0.0126173
$131$	−0.896683	−0.0783436	−0.0391718	−	0.999232i	$-0.512472\pi$
$131$	−0.896683	−0.0783436	−0.0391718	+	0.999232i	$0.512472\pi$
$132$	0	0
$133$	7.31959	0.634689
$134$	15.7711	1.36242
$135$	0	0
$136$	0	0
$137$	−12.6109	−1.07742	−0.538710	−	0.842491i	$-0.681088\pi$
$137$	−12.6109	−1.07742	−0.538710	+	0.842491i	$0.681088\pi$
$138$	0	0
$139$	17.0447	1.44571	0.722857	−	0.690998i	$-0.242829\pi$
$139$	17.0447	1.44571	0.722857	+	0.690998i	$0.242829\pi$
$140$	−1.48022	−0.125101
$141$	0	0
$142$	0.226626	0.0190180
$143$	−0.562609	−0.0470477
$144$	0	0
$145$	3.70193	0.307429
$146$	12.1387	1.00460
$147$	0	0
$148$	−10.6143	−0.872494
$149$	−17.5004	−1.43369	−0.716844	−	0.697234i	$-0.754414\pi$
$149$	−17.5004	−1.43369	−0.716844	+	0.697234i	$0.754414\pi$
$150$	0	0
$151$	9.64009	0.784500	0.392250	−	0.919859i	$-0.371697\pi$
$151$	9.64009	0.784500	0.392250	+	0.919859i	$0.371697\pi$
$152$	−2.14386	−0.173890
$153$	0	0
$154$	−5.78886	−0.466480
$155$	−1.12106	−0.0900454
$156$	0	0
$157$	−4.78976	−0.382265	−0.191132	−	0.981564i	$-0.561216\pi$
$157$	−4.78976	−0.382265	−0.191132	+	0.981564i	$0.561216\pi$
$158$	−12.5394	−0.997584
$159$	0	0
$160$	0.433546	0.0342748
$161$	−6.43060	−0.506802
$162$	0	0
$163$	17.8049	1.39459	0.697293	−	0.716786i	$-0.254388\pi$
$163$	17.8049	1.39459	0.697293	+	0.716786i	$0.254388\pi$
$164$	−12.1177	−0.946236
$165$	0	0
$166$	−1.04373	−0.0810090
$167$	17.5950	1.36154	0.680771	−	0.732496i	$-0.261645\pi$
$167$	17.5950	1.36154	0.680771	+	0.732496i	$0.261645\pi$
$168$	0	0
$169$	−12.8899	−0.991530
$170$	0	0
$171$	0	0
$172$	−10.0933	−0.769610
$173$	19.3462	1.47087	0.735434	−	0.677597i	$-0.236978\pi$
$173$	19.3462	1.47087	0.735434	+	0.677597i	$0.236978\pi$
$174$	0	0
$175$	−16.4293	−1.24194
$176$	1.69552	0.127804
$177$	0	0
$178$	13.8281	1.03646
$179$	9.11933	0.681611	0.340805	−	0.940134i	$-0.389300\pi$
$179$	9.11933	0.681611	0.340805	+	0.940134i	$0.389300\pi$
$180$	0	0
$181$	−0.776691	−0.0577310	−0.0288655	−	0.999583i	$-0.509189\pi$
$181$	−0.776691	−0.0577310	−0.0288655	+	0.999583i	$0.509189\pi$
$182$	1.13291	0.0839768
$183$	0	0
$184$	1.88348	0.138852
$185$	4.60180	0.338331
$186$	0	0
$187$	0	0
$188$	−2.38009	−0.173586
$189$	0	0
$190$	0.929461	0.0674302
$191$	13.4248	0.971384	0.485692	−	0.874130i	$-0.338568\pi$
$191$	13.4248	0.971384	0.485692	+	0.874130i	$0.338568\pi$
$192$	0	0
$193$	5.08905	0.366318	0.183159	−	0.983083i	$-0.441368\pi$
$193$	5.08905	0.366318	0.183159	+	0.983083i	$0.441368\pi$
$194$	3.75539	0.269621
$195$	0	0
$196$	4.65685	0.332632
$197$	4.15449	0.295995	0.147998	−	0.988988i	$-0.452717\pi$
$197$	4.15449	0.295995	0.147998	+	0.988988i	$0.452717\pi$
$198$	0	0
$199$	−10.7087	−0.759121	−0.379561	−	0.925167i	$-0.623925\pi$
$199$	−10.7087	−0.759121	−0.379561	+	0.925167i	$0.623925\pi$
$200$	4.81204	0.340262
$201$	0	0
$202$	0.636303	0.0447701
$203$	−29.1531	−2.04615
$204$	0	0
$205$	5.25359	0.366927
$206$	−8.59955	−0.599159
$207$	0	0
$208$	−0.331821	−0.0230077
$209$	3.63495	0.251435
$210$	0	0
$211$	−4.72151	−0.325042	−0.162521	−	0.986705i	$-0.551963\pi$
$211$	−4.72151	−0.325042	−0.162521	+	0.986705i	$0.551963\pi$
$212$	3.13066	0.215014
$213$	0	0
$214$	2.16478	0.147982
$215$	4.37592	0.298435
$216$	0	0
$217$	8.82843	0.599313
$218$	7.91376	0.535988
$219$	0	0
$220$	−0.735084	−0.0495594
$221$	0	0
$222$	0	0
$223$	−15.6359	−1.04706	−0.523530	−	0.852007i	$-0.675385\pi$
$223$	−15.6359	−1.04706	−0.523530	+	0.852007i	$0.675385\pi$
$224$	−3.41421	−0.228122
$225$	0	0
$226$	−9.18828	−0.611195
$227$	11.0615	0.734175	0.367088	−	0.930186i	$-0.380355\pi$
$227$	11.0615	0.734175	0.367088	+	0.930186i	$0.380355\pi$
$228$	0	0
$229$	−15.3852	−1.01668	−0.508340	−	0.861157i	$-0.669741\pi$
$229$	−15.3852	−1.01668	−0.508340	+	0.861157i	$0.669741\pi$
$230$	−0.816574	−0.0538433
$231$	0	0
$232$	8.53874	0.560596
$233$	−10.3776	−0.679859	−0.339929	−	0.940451i	$-0.610403\pi$
$233$	−10.3776	−0.679859	−0.339929	+	0.940451i	$0.610403\pi$
$234$	0	0
$235$	1.03188	0.0673121
$236$	7.67459	0.499573
$237$	0	0
$238$	0	0
$239$	24.7803	1.60291	0.801454	−	0.598057i	$-0.204060\pi$
$239$	24.7803	1.60291	0.801454	+	0.598057i	$0.204060\pi$
$240$	0	0
$241$	−4.30517	−0.277321	−0.138660	−	0.990340i	$-0.544280\pi$
$241$	−4.30517	−0.277321	−0.138660	+	0.990340i	$0.544280\pi$
$242$	8.12522	0.522309
$243$	0	0
$244$	8.03988	0.514701
$245$	−2.01896	−0.128987
$246$	0	0
$247$	−0.711378	−0.0452639
$248$	−2.58579	−0.164198
$249$	0	0
$250$	−4.25397	−0.269044
$251$	19.0639	1.20330	0.601652	−	0.798759i	$-0.294510\pi$
$251$	19.0639	1.20330	0.601652	+	0.798759i	$0.294510\pi$
$252$	0	0
$253$	−3.19347	−0.200772
$254$	12.0843	0.758235
$255$	0	0
$256$	1.00000	0.0625000
$257$	26.7594	1.66921	0.834603	−	0.550851i	$-0.185697\pi$
$257$	26.7594	1.66921	0.834603	+	0.550851i	$0.185697\pi$
$258$	0	0
$259$	−36.2396	−2.25182
$260$	0.143860	0.00892180
$261$	0	0
$262$	0.896683	0.0553973
$263$	−27.5122	−1.69648	−0.848239	−	0.529614i	$-0.822337\pi$
$263$	−27.5122	−1.69648	−0.848239	+	0.529614i	$0.822337\pi$
$264$	0	0
$265$	−1.35728	−0.0833772
$266$	−7.31959	−0.448793
$267$	0	0
$268$	−15.7711	−0.963375
$269$	10.0148	0.610613	0.305306	−	0.952254i	$-0.401241\pi$
$269$	10.0148	0.610613	0.305306	+	0.952254i	$0.401241\pi$
$270$	0	0
$271$	9.20949	0.559437	0.279718	−	0.960082i	$-0.409759\pi$
$271$	9.20949	0.559437	0.279718	+	0.960082i	$0.409759\pi$
$272$	0	0
$273$	0	0
$274$	12.6109	0.761851
$275$	−8.15890	−0.492000
$276$	0	0
$277$	−8.36529	−0.502622	−0.251311	−	0.967906i	$-0.580862\pi$
$277$	−8.36529	−0.502622	−0.251311	+	0.967906i	$0.580862\pi$
$278$	−17.0447	−1.02227
$279$	0	0
$280$	1.48022	0.0884599
$281$	−10.9996	−0.656183	−0.328091	−	0.944646i	$-0.606405\pi$
$281$	−10.9996	−0.656183	−0.328091	+	0.944646i	$0.606405\pi$
$282$	0	0
$283$	−21.1380	−1.25653	−0.628263	−	0.778001i	$-0.716234\pi$
$283$	−21.1380	−1.25653	−0.628263	+	0.778001i	$0.716234\pi$
$284$	−0.226626	−0.0134478
$285$	0	0
$286$	0.562609	0.0332678
$287$	−41.3725	−2.44214
$288$	0	0
$289$	0	0
$290$	−3.70193	−0.217385
$291$	0	0
$292$	−12.1387	−0.710362
$293$	−25.1277	−1.46797	−0.733987	−	0.679164i	$-0.762343\pi$
$293$	−25.1277	−1.46797	−0.733987	+	0.679164i	$0.762343\pi$
$294$	0	0
$295$	−3.32729	−0.193722
$296$	10.6143	0.616946
$297$	0	0
$298$	17.5004	1.01377
$299$	0.624979	0.0361435
$300$	0	0
$301$	−34.4608	−1.98629
$302$	−9.64009	−0.554725
$303$	0	0
$304$	2.14386	0.122959
$305$	−3.48566	−0.199588
$306$	0	0
$307$	15.3433	0.875688	0.437844	−	0.899051i	$-0.355742\pi$
$307$	15.3433	0.875688	0.437844	+	0.899051i	$0.355742\pi$
$308$	5.78886	0.329851
$309$	0	0
$310$	1.12106	0.0636717
$311$	−21.6825	−1.22950	−0.614750	−	0.788722i	$-0.710743\pi$
$311$	−21.6825	−1.22950	−0.614750	+	0.788722i	$0.710743\pi$
$312$	0	0
$313$	−8.38847	−0.474144	−0.237072	−	0.971492i	$-0.576188\pi$
$313$	−8.38847	−0.474144	−0.237072	+	0.971492i	$0.576188\pi$
$314$	4.78976	0.270302
$315$	0	0
$316$	12.5394	0.705398
$317$	23.9803	1.34687	0.673434	−	0.739248i	$-0.264819\pi$
$317$	23.9803	1.34687	0.673434	+	0.739248i	$0.264819\pi$
$318$	0	0
$319$	−14.4776	−0.810589
$320$	−0.433546	−0.0242359
$321$	0	0
$322$	6.43060	0.358363
$323$	0	0
$324$	0	0
$325$	1.59674	0.0885710
$326$	−17.8049	−0.986121
$327$	0	0
$328$	12.1177	0.669090
$329$	−8.12612	−0.448008
$330$	0	0
$331$	−4.07733	−0.224110	−0.112055	−	0.993702i	$-0.535743\pi$
$331$	−4.07733	−0.224110	−0.112055	+	0.993702i	$0.535743\pi$
$332$	1.04373	0.0572820
$333$	0	0
$334$	−17.5950	−0.962756
$335$	6.83750	0.373572
$336$	0	0
$337$	10.0167	0.545645	0.272822	−	0.962064i	$-0.412043\pi$
$337$	10.0167	0.545645	0.272822	+	0.962064i	$0.412043\pi$
$338$	12.8899	0.701118
$339$	0	0
$340$	0	0
$341$	4.38425	0.237420
$342$	0	0
$343$	−8.00000	−0.431959
$344$	10.0933	0.544197
$345$	0	0
$346$	−19.3462	−1.04006
$347$	8.64418	0.464044	0.232022	−	0.972711i	$-0.425466\pi$
$347$	8.64418	0.464044	0.232022	+	0.972711i	$0.425466\pi$
$348$	0	0
$349$	20.8126	1.11407	0.557036	−	0.830488i	$-0.311938\pi$
$349$	20.8126	1.11407	0.557036	+	0.830488i	$0.311938\pi$
$350$	16.4293	0.878184
$351$	0	0
$352$	−1.69552	−0.0903714
$353$	20.1049	1.07007	0.535037	−	0.844829i	$-0.320298\pi$
$353$	20.1049	1.07007	0.535037	+	0.844829i	$0.320298\pi$
$354$	0	0
$355$	0.0982525	0.00521470
$356$	−13.8281	−0.732885
$357$	0	0
$358$	−9.11933	−0.481972
$359$	−25.3001	−1.33529	−0.667645	−	0.744480i	$-0.732698\pi$
$359$	−25.3001	−1.33529	−0.667645	+	0.744480i	$0.732698\pi$
$360$	0	0
$361$	−14.4039	−0.758098
$362$	0.776691	0.0408220
$363$	0	0
$364$	−1.13291	−0.0593806
$365$	5.26266	0.275460
$366$	0	0
$367$	10.3200	0.538698	0.269349	−	0.963043i	$-0.413191\pi$
$367$	10.3200	0.538698	0.269349	+	0.963043i	$0.413191\pi$
$368$	−1.88348	−0.0981832
$369$	0	0
$370$	−4.60180	−0.239236
$371$	10.6887	0.554931
$372$	0	0
$373$	18.2719	0.946083	0.473041	−	0.881040i	$-0.343156\pi$
$373$	18.2719	0.946083	0.473041	+	0.881040i	$0.343156\pi$
$374$	0	0
$375$	0	0
$376$	2.38009	0.122744
$377$	2.83334	0.145924
$378$	0	0
$379$	20.9319	1.07520	0.537600	−	0.843200i	$-0.319331\pi$
$379$	20.9319	1.07520	0.537600	+	0.843200i	$0.319331\pi$
$380$	−0.929461	−0.0476803
$381$	0	0
$382$	−13.4248	−0.686872
$383$	−22.9974	−1.17511	−0.587555	−	0.809184i	$-0.699910\pi$
$383$	−22.9974	−1.17511	−0.587555	+	0.809184i	$0.699910\pi$
$384$	0	0
$385$	−2.50973	−0.127908
$386$	−5.08905	−0.259026
$387$	0	0
$388$	−3.75539	−0.190651
$389$	9.07748	0.460247	0.230123	−	0.973161i	$-0.426087\pi$
$389$	9.07748	0.460247	0.230123	+	0.973161i	$0.426087\pi$
$390$	0	0
$391$	0	0
$392$	−4.65685	−0.235207
$393$	0	0
$394$	−4.15449	−0.209300
$395$	−5.43641	−0.273536
$396$	0	0
$397$	−7.80857	−0.391901	−0.195950	−	0.980614i	$-0.562779\pi$
$397$	−7.80857	−0.391901	−0.195950	+	0.980614i	$0.562779\pi$
$398$	10.7087	0.536780
$399$	0	0
$400$	−4.81204	−0.240602
$401$	−17.3608	−0.866955	−0.433477	−	0.901164i	$-0.642714\pi$
$401$	−17.3608	−0.866955	−0.433477	+	0.901164i	$0.642714\pi$
$402$	0	0
$403$	−0.858019	−0.0427410
$404$	−0.636303	−0.0316573
$405$	0	0
$406$	29.1531	1.44684
$407$	−17.9968	−0.892069
$408$	0	0
$409$	4.42153	0.218631	0.109315	−	0.994007i	$-0.465134\pi$
$409$	4.42153	0.218631	0.109315	+	0.994007i	$0.465134\pi$
$410$	−5.25359	−0.259456
$411$	0	0
$412$	8.59955	0.423669
$413$	26.2027	1.28935
$414$	0	0
$415$	−0.452504	−0.0222125
$416$	0.331821	0.0162689
$417$	0	0
$418$	−3.63495	−0.177791
$419$	23.1663	1.13175	0.565874	−	0.824491i	$-0.308539\pi$
$419$	23.1663	1.13175	0.565874	+	0.824491i	$0.308539\pi$
$420$	0	0
$421$	−20.0524	−0.977295	−0.488648	−	0.872481i	$-0.662510\pi$
$421$	−20.0524	−0.977295	−0.488648	+	0.872481i	$0.662510\pi$
$422$	4.72151	0.229839
$423$	0	0
$424$	−3.13066	−0.152038
$425$	0	0
$426$	0	0
$427$	27.4499	1.32839
$428$	−2.16478	−0.104639
$429$	0	0
$430$	−4.37592	−0.211026
$431$	−40.1467	−1.93380	−0.966900	−	0.255154i	$-0.917874\pi$
$431$	−40.1467	−1.93380	−0.966900	+	0.255154i	$0.917874\pi$
$432$	0	0
$433$	−21.8499	−1.05004	−0.525020	−	0.851090i	$-0.675942\pi$
$433$	−21.8499	−1.05004	−0.525020	+	0.851090i	$0.675942\pi$
$434$	−8.82843	−0.423778
$435$	0	0
$436$	−7.91376	−0.379000
$437$	−4.03792	−0.193160
$438$	0	0
$439$	−0.838775	−0.0400325	−0.0200163	−	0.999800i	$-0.506372\pi$
$439$	−0.838775	−0.0400325	−0.0200163	+	0.999800i	$0.506372\pi$
$440$	0.735084	0.0350438
$441$	0	0
$442$	0	0
$443$	3.21267	0.152639	0.0763194	−	0.997083i	$-0.475683\pi$
$443$	3.21267	0.152639	0.0763194	+	0.997083i	$0.475683\pi$
$444$	0	0
$445$	5.99509	0.284195
$446$	15.6359	0.740383
$447$	0	0
$448$	3.41421	0.161306
$449$	28.7483	1.35671	0.678357	−	0.734732i	$-0.262692\pi$
$449$	28.7483	1.35671	0.678357	+	0.734732i	$0.262692\pi$
$450$	0	0
$451$	−20.5458	−0.967466
$452$	9.18828	0.432180
$453$	0	0
$454$	−11.0615	−0.519140
$455$	0.491168	0.0230263
$456$	0	0
$457$	−7.45097	−0.348542	−0.174271	−	0.984698i	$-0.555757\pi$
$457$	−7.45097	−0.348542	−0.174271	+	0.984698i	$0.555757\pi$
$458$	15.3852	0.718901
$459$	0	0
$460$	0.816574	0.0380730
$461$	−25.6378	−1.19407	−0.597037	−	0.802214i	$-0.703655\pi$
$461$	−25.6378	−1.19407	−0.597037	+	0.802214i	$0.703655\pi$
$462$	0	0
$463$	−10.7747	−0.500744	−0.250372	−	0.968150i	$-0.580553\pi$
$463$	−10.7747	−0.500744	−0.250372	+	0.968150i	$0.580553\pi$
$464$	−8.53874	−0.396401
$465$	0	0
$466$	10.3776	0.480733
$467$	−35.6492	−1.64965	−0.824823	−	0.565391i	$-0.808725\pi$
$467$	−35.6492	−1.64965	−0.824823	+	0.565391i	$0.808725\pi$
$468$	0	0
$469$	−53.8460	−2.48638
$470$	−1.03188	−0.0475969
$471$	0	0
$472$	−7.67459	−0.353252
$473$	−17.1134	−0.786877
$474$	0	0
$475$	−10.3163	−0.473346
$476$	0	0
$477$	0	0
$478$	−24.7803	−1.13343
$479$	14.0702	0.642882	0.321441	−	0.946930i	$-0.395833\pi$
$479$	14.0702	0.642882	0.321441	+	0.946930i	$0.395833\pi$
$480$	0	0
$481$	3.52207	0.160592
$482$	4.30517	0.196095
$483$	0	0
$484$	−8.12522	−0.369328
$485$	1.62813	0.0739297
$486$	0	0
$487$	24.1325	1.09355	0.546775	−	0.837280i	$-0.315855\pi$
$487$	24.1325	1.09355	0.546775	+	0.837280i	$0.315855\pi$
$488$	−8.03988	−0.363948
$489$	0	0
$490$	2.01896	0.0912072
$491$	−26.5200	−1.19683	−0.598416	−	0.801186i	$-0.704203\pi$
$491$	−26.5200	−1.19683	−0.598416	+	0.801186i	$0.704203\pi$
$492$	0	0
$493$	0	0
$494$	0.711378	0.0320064
$495$	0	0
$496$	2.58579	0.116105
$497$	−0.773748	−0.0347073
$498$	0	0
$499$	−4.53996	−0.203237	−0.101618	−	0.994823i	$-0.532402\pi$
$499$	−4.53996	−0.203237	−0.101618	+	0.994823i	$0.532402\pi$
$500$	4.25397	0.190243
$501$	0	0
$502$	−19.0639	−0.850864
$503$	3.99734	0.178233	0.0891164	−	0.996021i	$-0.471596\pi$
$503$	3.99734	0.178233	0.0891164	+	0.996021i	$0.471596\pi$
$504$	0	0
$505$	0.275866	0.0122759
$506$	3.19347	0.141967
$507$	0	0
$508$	−12.0843	−0.536153
$509$	22.9751	1.01835	0.509176	−	0.860662i	$-0.329950\pi$
$509$	22.9751	1.01835	0.509176	+	0.860662i	$0.329950\pi$
$510$	0	0
$511$	−41.4440	−1.83337
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−26.7594	−1.18031
$515$	−3.72830	−0.164288
$516$	0	0
$517$	−4.03548	−0.177480
$518$	36.2396	1.59228
$519$	0	0
$520$	−0.143860	−0.00630866
$521$	−16.8828	−0.739650	−0.369825	−	0.929101i	$-0.620582\pi$
$521$	−16.8828	−0.739650	−0.369825	+	0.929101i	$0.620582\pi$
$522$	0	0
$523$	−15.5903	−0.681717	−0.340859	−	0.940115i	$-0.610718\pi$
$523$	−15.5903	−0.681717	−0.340859	+	0.940115i	$0.610718\pi$
$524$	−0.896683	−0.0391718
$525$	0	0
$526$	27.5122	1.19959
$527$	0	0
$528$	0	0
$529$	−19.4525	−0.845761
$530$	1.35728	0.0589566
$531$	0	0
$532$	7.31959	0.317345
$533$	4.02092	0.174166
$534$	0	0
$535$	0.938533	0.0405763
$536$	15.7711	0.681209
$537$	0	0
$538$	−10.0148	−0.431769
$539$	7.89578	0.340095
$540$	0	0
$541$	−15.1306	−0.650515	−0.325258	−	0.945625i	$-0.605451\pi$
$541$	−15.1306	−0.650515	−0.325258	+	0.945625i	$0.605451\pi$
$542$	−9.20949	−0.395581
$543$	0	0
$544$	0	0
$545$	3.43098	0.146967
$546$	0	0
$547$	−2.64688	−0.113172	−0.0565862	−	0.998398i	$-0.518022\pi$
$547$	−2.64688	−0.113172	−0.0565862	+	0.998398i	$0.518022\pi$
$548$	−12.6109	−0.538710
$549$	0	0
$550$	8.15890	0.347897
$551$	−18.3059	−0.779856
$552$	0	0
$553$	42.8123	1.82056
$554$	8.36529	0.355407
$555$	0	0
$556$	17.0447	0.722857
$557$	12.4900	0.529217	0.264609	−	0.964356i	$-0.414757\pi$
$557$	12.4900	0.529217	0.264609	+	0.964356i	$0.414757\pi$
$558$	0	0
$559$	3.34919	0.141656
$560$	−1.48022	−0.0625506
$561$	0	0
$562$	10.9996	0.463991
$563$	−1.98226	−0.0835423	−0.0417712	−	0.999127i	$-0.513300\pi$
$563$	−1.98226	−0.0835423	−0.0417712	+	0.999127i	$0.513300\pi$
$564$	0	0
$565$	−3.98354	−0.167589
$566$	21.1380	0.888498
$567$	0	0
$568$	0.226626	0.00950900
$569$	−16.7666	−0.702892	−0.351446	−	0.936208i	$-0.614310\pi$
$569$	−16.7666	−0.702892	−0.351446	+	0.936208i	$0.614310\pi$
$570$	0	0
$571$	30.1229	1.26061	0.630303	−	0.776349i	$-0.282931\pi$
$571$	30.1229	1.26061	0.630303	+	0.776349i	$0.282931\pi$
$572$	−0.562609	−0.0235239
$573$	0	0
$574$	41.3725	1.72686
$575$	9.06338	0.377969
$576$	0	0
$577$	−19.8204	−0.825132	−0.412566	−	0.910928i	$-0.635367\pi$
$577$	−19.8204	−0.825132	−0.412566	+	0.910928i	$0.635367\pi$
$578$	0	0
$579$	0	0
$580$	3.70193	0.153714
$581$	3.56351	0.147839
$582$	0	0
$583$	5.30808	0.219838
$584$	12.1387	0.502301
$585$	0	0
$586$	25.1277	1.03801
$587$	−12.8612	−0.530839	−0.265419	−	0.964133i	$-0.585510\pi$
$587$	−12.8612	−0.530839	−0.265419	+	0.964133i	$0.585510\pi$
$588$	0	0
$589$	5.54356	0.228419
$590$	3.32729	0.136982
$591$	0	0
$592$	−10.6143	−0.436247
$593$	35.8300	1.47136	0.735681	−	0.677328i	$-0.236862\pi$
$593$	35.8300	1.47136	0.735681	+	0.677328i	$0.236862\pi$
$594$	0	0
$595$	0	0
$596$	−17.5004	−0.716844
$597$	0	0
$598$	−0.624979	−0.0255573
$599$	21.1148	0.862728	0.431364	−	0.902178i	$-0.358032\pi$
$599$	21.1148	0.862728	0.431364	+	0.902178i	$0.358032\pi$
$600$	0	0
$601$	15.2346	0.621434	0.310717	−	0.950503i	$-0.399431\pi$
$601$	15.2346	0.621434	0.310717	+	0.950503i	$0.399431\pi$
$602$	34.4608	1.40452
$603$	0	0
$604$	9.64009	0.392250
$605$	3.52265	0.143216
$606$	0	0
$607$	38.3848	1.55799	0.778995	−	0.627030i	$-0.215730\pi$
$607$	38.3848	1.55799	0.778995	+	0.627030i	$0.215730\pi$
$608$	−2.14386	−0.0869450
$609$	0	0
$610$	3.48566	0.141130
$611$	0.789763	0.0319504
$612$	0	0
$613$	−1.39554	−0.0563654	−0.0281827	−	0.999603i	$-0.508972\pi$
$613$	−1.39554	−0.0563654	−0.0281827	+	0.999603i	$0.508972\pi$
$614$	−15.3433	−0.619205
$615$	0	0
$616$	−5.78886	−0.233240
$617$	−8.41487	−0.338770	−0.169385	−	0.985550i	$-0.554178\pi$
$617$	−8.41487	−0.338770	−0.169385	+	0.985550i	$0.554178\pi$
$618$	0	0
$619$	−5.34330	−0.214765	−0.107383	−	0.994218i	$-0.534247\pi$
$619$	−5.34330	−0.214765	−0.107383	+	0.994218i	$0.534247\pi$
$620$	−1.12106	−0.0450227
$621$	0	0
$622$	21.6825	0.869388
$623$	−47.2119	−1.89151
$624$	0	0
$625$	22.2159	0.888636
$626$	8.38847	0.335271
$627$	0	0
$628$	−4.78976	−0.191132
$629$	0	0
$630$	0	0
$631$	−1.46282	−0.0582340	−0.0291170	−	0.999576i	$-0.509270\pi$
$631$	−1.46282	−0.0582340	−0.0291170	+	0.999576i	$0.509270\pi$
$632$	−12.5394	−0.498792
$633$	0	0
$634$	−23.9803	−0.952379
$635$	5.23908	0.207907
$636$	0	0
$637$	−1.54524	−0.0612248
$638$	14.4776	0.573173
$639$	0	0
$640$	0.433546	0.0171374
$641$	−0.479372	−0.0189340	−0.00946702	−	0.999955i	$-0.503013\pi$
$641$	−0.479372	−0.0189340	−0.00946702	+	0.999955i	$0.503013\pi$
$642$	0	0
$643$	32.1049	1.26609	0.633046	−	0.774114i	$-0.281804\pi$
$643$	32.1049	1.26609	0.633046	+	0.774114i	$0.281804\pi$
$644$	−6.43060	−0.253401
$645$	0	0
$646$	0	0
$647$	−23.4802	−0.923103	−0.461551	−	0.887114i	$-0.652707\pi$
$647$	−23.4802	−0.923103	−0.461551	+	0.887114i	$0.652707\pi$
$648$	0	0
$649$	13.0124	0.510782
$650$	−1.59674	−0.0626292
$651$	0	0
$652$	17.8049	0.697293
$653$	−45.3689	−1.77542	−0.887710	−	0.460402i	$-0.847705\pi$
$653$	−45.3689	−1.77542	−0.887710	+	0.460402i	$0.847705\pi$
$654$	0	0
$655$	0.388753	0.0151898
$656$	−12.1177	−0.473118
$657$	0	0
$658$	8.12612	0.316789
$659$	−12.8417	−0.500240	−0.250120	−	0.968215i	$-0.580470\pi$
$659$	−12.8417	−0.500240	−0.250120	+	0.968215i	$0.580470\pi$
$660$	0	0
$661$	−27.1878	−1.05748	−0.528741	−	0.848783i	$-0.677336\pi$
$661$	−27.1878	−1.05748	−0.528741	+	0.848783i	$0.677336\pi$
$662$	4.07733	0.158470
$663$	0	0
$664$	−1.04373	−0.0405045
$665$	−3.17338	−0.123058
$666$	0	0
$667$	16.0825	0.622719
$668$	17.5950	0.680771
$669$	0	0
$670$	−6.83750	−0.264156
$671$	13.6318	0.526249
$672$	0	0
$673$	−16.6772	−0.642857	−0.321429	−	0.946934i	$-0.604163\pi$
$673$	−16.6772	−0.642857	−0.321429	+	0.946934i	$0.604163\pi$
$674$	−10.0167	−0.385829
$675$	0	0
$676$	−12.8899	−0.495765
$677$	2.60284	0.100035	0.0500175	−	0.998748i	$-0.484072\pi$
$677$	2.60284	0.100035	0.0500175	+	0.998748i	$0.484072\pi$
$678$	0	0
$679$	−12.8217	−0.492052
$680$	0	0
$681$	0	0
$682$	−4.38425	−0.167882
$683$	−12.2427	−0.468453	−0.234227	−	0.972182i	$-0.575256\pi$
$683$	−12.2427	−0.468453	−0.234227	+	0.972182i	$0.575256\pi$
$684$	0	0
$685$	5.46739	0.208898
$686$	8.00000	0.305441
$687$	0	0
$688$	−10.0933	−0.384805
$689$	−1.03882	−0.0395758
$690$	0	0
$691$	15.6736	0.596252	0.298126	−	0.954526i	$-0.403638\pi$
$691$	15.6736	0.596252	0.298126	+	0.954526i	$0.403638\pi$
$692$	19.3462	0.735434
$693$	0	0
$694$	−8.64418	−0.328129
$695$	−7.38966	−0.280306
$696$	0	0
$697$	0	0
$698$	−20.8126	−0.787768
$699$	0	0
$700$	−16.4293	−0.620970
$701$	−30.8222	−1.16414	−0.582070	−	0.813139i	$-0.697757\pi$
$701$	−30.8222	−1.16414	−0.582070	+	0.813139i	$0.697757\pi$
$702$	0	0
$703$	−22.7557	−0.858246
$704$	1.69552	0.0639022
$705$	0	0
$706$	−20.1049	−0.756656
$707$	−2.17248	−0.0817043
$708$	0	0
$709$	−0.394882	−0.0148301	−0.00741505	−	0.999973i	$-0.502360\pi$
$709$	−0.394882	−0.0148301	−0.00741505	+	0.999973i	$0.502360\pi$
$710$	−0.0982525	−0.00368735
$711$	0	0
$712$	13.8281	0.518228
$713$	−4.87028	−0.182393
$714$	0	0
$715$	0.243917	0.00912197
$716$	9.11933	0.340805
$717$	0	0
$718$	25.3001	0.944193
$719$	11.0962	0.413817	0.206908	−	0.978360i	$-0.433660\pi$
$719$	11.0962	0.413817	0.206908	+	0.978360i	$0.433660\pi$
$720$	0	0
$721$	29.3607	1.09345
$722$	14.4039	0.536056
$723$	0	0
$724$	−0.776691	−0.0288655
$725$	41.0888	1.52600
$726$	0	0
$727$	27.2866	1.01200	0.506001	−	0.862533i	$-0.331123\pi$
$727$	27.2866	1.01200	0.506001	+	0.862533i	$0.331123\pi$
$728$	1.13291	0.0419884
$729$	0	0
$730$	−5.26266	−0.194780
$731$	0	0
$732$	0	0
$733$	−29.2870	−1.08174	−0.540870	−	0.841106i	$-0.681905\pi$
$733$	−29.2870	−1.08174	−0.540870	+	0.841106i	$0.681905\pi$
$734$	−10.3200	−0.380917
$735$	0	0
$736$	1.88348	0.0694260
$737$	−26.7402	−0.984989
$738$	0	0
$739$	37.0534	1.36303	0.681516	−	0.731803i	$-0.261321\pi$
$739$	37.0534	1.36303	0.681516	+	0.731803i	$0.261321\pi$
$740$	4.60180	0.169166
$741$	0	0
$742$	−10.6887	−0.392396
$743$	14.4872	0.531483	0.265742	−	0.964044i	$-0.414383\pi$
$743$	14.4872	0.531483	0.265742	+	0.964044i	$0.414383\pi$
$744$	0	0
$745$	7.58722	0.277974
$746$	−18.2719	−0.668981
$747$	0	0
$748$	0	0
$749$	−7.39104	−0.270063
$750$	0	0
$751$	43.9632	1.60424	0.802121	−	0.597162i	$-0.203705\pi$
$751$	43.9632	1.60424	0.802121	+	0.597162i	$0.203705\pi$
$752$	−2.38009	−0.0867928
$753$	0	0
$754$	−2.83334	−0.103184
$755$	−4.17942	−0.152105
$756$	0	0
$757$	−37.8588	−1.37600	−0.688001	−	0.725710i	$-0.741511\pi$
$757$	−37.8588	−1.37600	−0.688001	+	0.725710i	$0.741511\pi$
$758$	−20.9319	−0.760281
$759$	0	0
$760$	0.929461	0.0337151
$761$	9.82880	0.356294	0.178147	−	0.984004i	$-0.442990\pi$
$761$	9.82880	0.356294	0.178147	+	0.984004i	$0.442990\pi$
$762$	0	0
$763$	−27.0193	−0.978163
$764$	13.4248	0.485692
$765$	0	0
$766$	22.9974	0.830929
$767$	−2.54659	−0.0919522
$768$	0	0
$769$	41.1522	1.48398	0.741992	−	0.670408i	$-0.233881\pi$
$769$	41.1522	1.48398	0.741992	+	0.670408i	$0.233881\pi$
$770$	2.50973	0.0904446
$771$	0	0
$772$	5.08905	0.183159
$773$	−21.7456	−0.782136	−0.391068	−	0.920362i	$-0.627894\pi$
$773$	−21.7456	−0.782136	−0.391068	+	0.920362i	$0.627894\pi$
$774$	0	0
$775$	−12.4429	−0.446962
$776$	3.75539	0.134811
$777$	0	0
$778$	−9.07748	−0.325444
$779$	−25.9787	−0.930785
$780$	0	0
$781$	−0.384248	−0.0137495
$782$	0	0
$783$	0	0
$784$	4.65685	0.166316
$785$	2.07658	0.0741163
$786$	0	0
$787$	47.6021	1.69683	0.848415	−	0.529331i	$-0.177557\pi$
$787$	47.6021	1.69683	0.848415	+	0.529331i	$0.177557\pi$
$788$	4.15449	0.147998
$789$	0	0
$790$	5.43641	0.193419
$791$	31.3707	1.11542
$792$	0	0
$793$	−2.66780	−0.0947365
$794$	7.80857	0.277116
$795$	0	0
$796$	−10.7087	−0.379561
$797$	37.8984	1.34243	0.671215	−	0.741263i	$-0.265773\pi$
$797$	37.8984	1.34243	0.671215	+	0.741263i	$0.265773\pi$
$798$	0	0
$799$	0	0
$800$	4.81204	0.170131
$801$	0	0
$802$	17.3608	0.613030
$803$	−20.5813	−0.726299
$804$	0	0
$805$	2.78796	0.0982626
$806$	0.858019	0.0302224
$807$	0	0
$808$	0.636303	0.0223851
$809$	−38.5731	−1.35616	−0.678079	−	0.734989i	$-0.737187\pi$
$809$	−38.5731	−1.35616	−0.678079	+	0.734989i	$0.737187\pi$
$810$	0	0
$811$	19.4393	0.682608	0.341304	−	0.939953i	$-0.389131\pi$
$811$	19.4393	0.682608	0.341304	+	0.939953i	$0.389131\pi$
$812$	−29.1531	−1.02307
$813$	0	0
$814$	17.9968	0.630788
$815$	−7.71922	−0.270393
$816$	0	0
$817$	−21.6387	−0.757043
$818$	−4.42153	−0.154595
$819$	0	0
$820$	5.25359	0.183463
$821$	−0.231821	−0.00809062	−0.00404531	−	0.999992i	$-0.501288\pi$
$821$	−0.231821	−0.00809062	−0.00404531	+	0.999992i	$0.501288\pi$
$822$	0	0
$823$	−22.4931	−0.784059	−0.392030	−	0.919953i	$-0.628227\pi$
$823$	−22.4931	−0.784059	−0.392030	+	0.919953i	$0.628227\pi$
$824$	−8.59955	−0.299579
$825$	0	0
$826$	−26.2027	−0.911709
$827$	−52.3121	−1.81907	−0.909534	−	0.415629i	$-0.863561\pi$
$827$	−52.3121	−1.81907	−0.909534	+	0.415629i	$0.863561\pi$
$828$	0	0
$829$	23.3260	0.810144	0.405072	−	0.914285i	$-0.367246\pi$
$829$	23.3260	0.810144	0.405072	+	0.914285i	$0.367246\pi$
$830$	0.452504	0.0157066
$831$	0	0
$832$	−0.331821	−0.0115038
$833$	0	0
$834$	0	0
$835$	−7.62824	−0.263986
$836$	3.63495	0.125717
$837$	0	0
$838$	−23.1663	−0.800267
$839$	13.4212	0.463350	0.231675	−	0.972793i	$-0.425579\pi$
$839$	13.4212	0.463350	0.231675	+	0.972793i	$0.425579\pi$
$840$	0	0
$841$	43.9101	1.51414
$842$	20.0524	0.691052
$843$	0	0
$844$	−4.72151	−0.162521
$845$	5.58836	0.192245
$846$	0	0
$847$	−27.7412	−0.953200
$848$	3.13066	0.107507
$849$	0	0
$850$	0	0
$851$	19.9919	0.685314
$852$	0	0
$853$	−49.9350	−1.70974	−0.854871	−	0.518841i	$-0.826364\pi$
$853$	−49.9350	−1.70974	−0.854871	+	0.518841i	$0.826364\pi$
$854$	−27.4499	−0.939315
$855$	0	0
$856$	2.16478	0.0739908
$857$	−17.1760	−0.586722	−0.293361	−	0.956002i	$-0.594774\pi$
$857$	−17.1760	−0.586722	−0.293361	+	0.956002i	$0.594774\pi$
$858$	0	0
$859$	8.76362	0.299011	0.149505	−	0.988761i	$-0.452232\pi$
$859$	8.76362	0.299011	0.149505	+	0.988761i	$0.452232\pi$
$860$	4.37592	0.149218
$861$	0	0
$862$	40.1467	1.36740
$863$	−35.9177	−1.22265	−0.611326	−	0.791379i	$-0.709364\pi$
$863$	−35.9177	−1.22265	−0.611326	+	0.791379i	$0.709364\pi$
$864$	0	0
$865$	−8.38748	−0.285183
$866$	21.8499	0.742490
$867$	0	0
$868$	8.82843	0.299656
$869$	21.2608	0.721224
$870$	0	0
$871$	5.23320	0.177320
$872$	7.91376	0.267994
$873$	0	0
$874$	4.03792	0.136585
$875$	14.5239	0.490999
$876$	0	0
$877$	4.22519	0.142674	0.0713372	−	0.997452i	$-0.477273\pi$
$877$	4.22519	0.142674	0.0713372	+	0.997452i	$0.477273\pi$
$878$	0.838775	0.0283073
$879$	0	0
$880$	−0.735084	−0.0247797
$881$	−29.9958	−1.01058	−0.505292	−	0.862949i	$-0.668615\pi$
$881$	−29.9958	−1.01058	−0.505292	+	0.862949i	$0.668615\pi$
$882$	0	0
$883$	13.9714	0.470175	0.235087	−	0.971974i	$-0.424462\pi$
$883$	13.9714	0.470175	0.235087	+	0.971974i	$0.424462\pi$
$884$	0	0
$885$	0	0
$886$	−3.21267	−0.107932
$887$	22.0276	0.739613	0.369807	−	0.929109i	$-0.379424\pi$
$887$	22.0276	0.739613	0.369807	+	0.929109i	$0.379424\pi$
$888$	0	0
$889$	−41.2583	−1.38376
$890$	−5.99509	−0.200956
$891$	0	0
$892$	−15.6359	−0.523530
$893$	−5.10257	−0.170751
$894$	0	0
$895$	−3.95365	−0.132156
$896$	−3.41421	−0.114061
$897$	0	0
$898$	−28.7483	−0.959342
$899$	−22.0794	−0.736388
$900$	0	0
$901$	0	0
$902$	20.5458	0.684102
$903$	0	0
$904$	−9.18828	−0.305598
$905$	0.336731	0.0111933
$906$	0	0
$907$	−26.9106	−0.893553	−0.446776	−	0.894646i	$-0.647428\pi$
$907$	−26.9106	−0.893553	−0.446776	+	0.894646i	$0.647428\pi$
$908$	11.0615	0.367088
$909$	0	0
$910$	−0.491168	−0.0162820
$911$	−35.4849	−1.17567	−0.587834	−	0.808982i	$-0.700019\pi$
$911$	−35.4849	−1.17567	−0.587834	+	0.808982i	$0.700019\pi$
$912$	0	0
$913$	1.76966	0.0585672
$914$	7.45097	0.246456
$915$	0	0
$916$	−15.3852	−0.508340
$917$	−3.06147	−0.101099
$918$	0	0
$919$	−38.6107	−1.27365	−0.636824	−	0.771009i	$-0.719752\pi$
$919$	−38.6107	−1.27365	−0.636824	+	0.771009i	$0.719752\pi$
$920$	−0.816574	−0.0269217
$921$	0	0
$922$	25.6378	0.844337
$923$	0.0751992	0.00247521
$924$	0	0
$925$	51.0766	1.67939
$926$	10.7747	0.354079
$927$	0	0
$928$	8.53874	0.280298
$929$	22.8949	0.751157	0.375579	−	0.926791i	$-0.377444\pi$
$929$	22.8949	0.751157	0.375579	+	0.926791i	$0.377444\pi$
$930$	0	0
$931$	9.98364	0.327201
$932$	−10.3776	−0.339929
$933$	0	0
$934$	35.6492	1.16648
$935$	0	0
$936$	0	0
$937$	39.6793	1.29627	0.648133	−	0.761527i	$-0.275550\pi$
$937$	39.6793	1.29627	0.648133	+	0.761527i	$0.275550\pi$
$938$	53.8460	1.75813
$939$	0	0
$940$	1.03188	0.0336561
$941$	52.6651	1.71683	0.858416	−	0.512953i	$-0.171449\pi$
$941$	52.6651	1.71683	0.858416	+	0.512953i	$0.171449\pi$
$942$	0	0
$943$	22.8235	0.743236
$944$	7.67459	0.249787
$945$	0	0
$946$	17.1134	0.556406
$947$	−1.52698	−0.0496201	−0.0248100	−	0.999692i	$-0.507898\pi$
$947$	−1.52698	−0.0496201	−0.0248100	+	0.999692i	$0.507898\pi$
$948$	0	0
$949$	4.02787	0.130750
$950$	10.3163	0.334706
$951$	0	0
$952$	0	0
$953$	−9.95360	−0.322429	−0.161214	−	0.986919i	$-0.551541\pi$
$953$	−9.95360	−0.322429	−0.161214	+	0.986919i	$0.551541\pi$
$954$	0	0
$955$	−5.82026	−0.188339
$956$	24.7803	0.801454
$957$	0	0
$958$	−14.0702	−0.454586
$959$	−43.0562	−1.39036
$960$	0	0
$961$	−24.3137	−0.784313
$962$	−3.52207	−0.113556
$963$	0	0
$964$	−4.30517	−0.138660
$965$	−2.20633	−0.0710244
$966$	0	0
$967$	5.88843	0.189359	0.0946796	−	0.995508i	$-0.469817\pi$
$967$	5.88843	0.189359	0.0946796	+	0.995508i	$0.469817\pi$
$968$	8.12522	0.261154
$969$	0	0
$970$	−1.62813	−0.0522762
$971$	−18.9670	−0.608681	−0.304341	−	0.952563i	$-0.598436\pi$
$971$	−18.9670	−0.608681	−0.304341	+	0.952563i	$0.598436\pi$
$972$	0	0
$973$	58.1943	1.86562
$974$	−24.1325	−0.773256
$975$	0	0
$976$	8.03988	0.257350
$977$	−32.3172	−1.03392	−0.516959	−	0.856010i	$-0.672936\pi$
$977$	−32.3172	−1.03392	−0.516959	+	0.856010i	$0.672936\pi$
$978$	0	0
$979$	−23.4457	−0.749328
$980$	−2.01896	−0.0644933
$981$	0	0
$982$	26.5200	0.846288
$983$	−32.4363	−1.03456	−0.517278	−	0.855817i	$-0.673055\pi$
$983$	−32.4363	−1.03456	−0.517278	+	0.855817i	$0.673055\pi$
$984$	0	0
$985$	−1.80116	−0.0573898
$986$	0	0
$987$	0	0
$988$	−0.711378	−0.0226320
$989$	19.0106	0.604502
$990$	0	0
$991$	46.9010	1.48986	0.744930	−	0.667142i	$-0.232483\pi$
$991$	46.9010	1.48986	0.744930	+	0.667142i	$0.232483\pi$
$992$	−2.58579	−0.0820988
$993$	0	0
$994$	0.773748	0.0245418
$995$	4.64272	0.147184
$996$	0	0
$997$	47.4571	1.50298	0.751490	−	0.659744i	$-0.229335\pi$
$997$	47.4571	1.50298	0.751490	+	0.659744i	$0.229335\pi$
$998$	4.53996	0.143710
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5202.2.a.bt.1.3		4
3.2	odd	2		1734.2.a.w.1.2		4
17.10	odd	16		306.2.l.d.253.2		8
17.12	odd	16		306.2.l.d.127.2		8
17.16	even	2		5202.2.a.br.1.2		4
51.2	odd	8		1734.2.f.m.1483.3		8
51.8	odd	8		1734.2.f.j.829.2		8
51.26	odd	8		1734.2.f.m.829.3		8
51.29	even	16		102.2.h.a.25.2	✓	8
51.32	odd	8		1734.2.f.j.1483.2		8
51.38	odd	4		1734.2.b.k.577.3		8
51.44	even	16		102.2.h.a.49.2	yes	8
51.47	odd	4		1734.2.b.k.577.6		8
51.50	odd	2		1734.2.a.v.1.3		4
204.95	odd	16		816.2.bq.b.49.1		8
204.131	odd	16		816.2.bq.b.433.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
102.2.h.a.25.2	✓	8	51.29	even	16
102.2.h.a.49.2	yes	8	51.44	even	16
306.2.l.d.127.2		8	17.12	odd	16
306.2.l.d.253.2		8	17.10	odd	16
816.2.bq.b.49.1		8	204.95	odd	16
816.2.bq.b.433.1		8	204.131	odd	16
1734.2.a.v.1.3		4	51.50	odd	2
1734.2.a.w.1.2		4	3.2	odd	2
1734.2.b.k.577.3		8	51.38	odd	4
1734.2.b.k.577.6		8	51.47	odd	4
1734.2.f.j.829.2		8	51.8	odd	8
1734.2.f.j.1483.2		8	51.32	odd	8
1734.2.f.m.829.3		8	51.26	odd	8
1734.2.f.m.1483.3		8	51.2	odd	8
5202.2.a.br.1.2		4	17.16	even	2
5202.2.a.bt.1.3		4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 5202 = 2 \cdot 3^{2} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5202.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -0.433546 q^{5} +3.41421 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -0.433546 q^{5} +3.41421 q^{7} -1.00000 q^{8} +0.433546 q^{10} +1.69552 q^{11} -0.331821 q^{13} -3.41421 q^{14} +1.00000 q^{16} +2.14386 q^{19} -0.433546 q^{20} -1.69552 q^{22} -1.88348 q^{23} -4.81204 q^{25} +0.331821 q^{26} +3.41421 q^{28} -8.53874 q^{29} +2.58579 q^{31} -1.00000 q^{32} -1.48022 q^{35} -10.6143 q^{37} -2.14386 q^{38} +0.433546 q^{40} -12.1177 q^{41} -10.0933 q^{43} +1.69552 q^{44} +1.88348 q^{46} -2.38009 q^{47} +4.65685 q^{49} +4.81204 q^{50} -0.331821 q^{52} +3.13066 q^{53} -0.735084 q^{55} -3.41421 q^{56} +8.53874 q^{58} +7.67459 q^{59} +8.03988 q^{61} -2.58579 q^{62} +1.00000 q^{64} +0.143860 q^{65} -15.7711 q^{67} +1.48022 q^{70} -0.226626 q^{71} -12.1387 q^{73} +10.6143 q^{74} +2.14386 q^{76} +5.78886 q^{77} +12.5394 q^{79} -0.433546 q^{80} +12.1177 q^{82} +1.04373 q^{83} +10.0933 q^{86} -1.69552 q^{88} -13.8281 q^{89} -1.13291 q^{91} -1.88348 q^{92} +2.38009 q^{94} -0.929461 q^{95} -3.75539 q^{97} -4.65685 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 4 q^{4} + 8 q^{7} - 4 q^{8}+O(q^{10}) \) 4 * q - 4 * q^2 + 4 * q^4 + 8 * q^7 - 4 * q^8	\( 4 q - 4 q^{2} + 4 q^{4} + 8 q^{7} - 4 q^{8} - 8 q^{11} - 8 q^{14} + 4 q^{16} - 8 q^{19} + 8 q^{22} - 8 q^{23} - 4 q^{25} + 8 q^{28} - 8 q^{29} + 16 q^{31} - 4 q^{32} + 8 q^{35} + 8 q^{37} + 8 q^{38} - 8 q^{41} - 16 q^{43} - 8 q^{44} + 8 q^{46} - 4 q^{49} + 4 q^{50} - 16 q^{55} - 8 q^{56} + 8 q^{58} + 8 q^{59} - 16 q^{62} + 4 q^{64} - 16 q^{65} - 24 q^{67} - 8 q^{70} - 24 q^{71} - 16 q^{73} - 8 q^{74} - 8 q^{76} - 16 q^{77} + 8 q^{79} + 8 q^{82} - 8 q^{83} + 16 q^{86} + 8 q^{88} - 8 q^{91} - 8 q^{92} - 8 q^{95} + 16 q^{97} + 4 q^{98}+O(q^{100}) \) 4 * q - 4 * q^2 + 4 * q^4 + 8 * q^7 - 4 * q^8 - 8 * q^11 - 8 * q^14 + 4 * q^16 - 8 * q^19 + 8 * q^22 - 8 * q^23 - 4 * q^25 + 8 * q^28 - 8 * q^29 + 16 * q^31 - 4 * q^32 + 8 * q^35 + 8 * q^37 + 8 * q^38 - 8 * q^41 - 16 * q^43 - 8 * q^44 + 8 * q^46 - 4 * q^49 + 4 * q^50 - 16 * q^55 - 8 * q^56 + 8 * q^58 + 8 * q^59 - 16 * q^62 + 4 * q^64 - 16 * q^65 - 24 * q^67 - 8 * q^70 - 24 * q^71 - 16 * q^73 - 8 * q^74 - 8 * q^76 - 16 * q^77 + 8 * q^79 + 8 * q^82 - 8 * q^83 + 16 * q^86 + 8 * q^88 - 8 * q^91 - 8 * q^92 - 8 * q^95 + 16 * q^97 + 4 * q^98

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