LMFDB - Embedded newform 252.4.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [252,4,Mod(1,252)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(252, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("252.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 252 = 2^{2} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	252.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:	$14.8684813214$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{7}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 7 $ x^2 - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.64575$ of defining polynomial
Character	$\chi$	$=$	252.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-15.8745 q^{5} +7.00000 q^{7} +O(q^{10})$	$q-15.8745 q^{5} +7.00000 q^{7} -15.8745 q^{11} +26.0000 q^{13} +79.3725 q^{17} +68.0000 q^{19} -47.6235 q^{23} +127.000 q^{25} +253.992 q^{29} +212.000 q^{31} -111.122 q^{35} +218.000 q^{37} -396.863 q^{41} +260.000 q^{43} -412.737 q^{47} +49.0000 q^{49} +476.235 q^{53} +252.000 q^{55} +285.741 q^{59} -322.000 q^{61} -412.737 q^{65} +356.000 q^{67} +1127.09 q^{71} -226.000 q^{73} -111.122 q^{77} +440.000 q^{79} +253.992 q^{83} -1260.00 q^{85} -206.369 q^{89} +182.000 q^{91} -1079.47 q^{95} -1330.00 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 14 q^{7}+O(q^{10}) $ 2 * q + 14 * q^7	$ 2 q + 14 q^{7} + 52 q^{13} + 136 q^{19} + 254 q^{25} + 424 q^{31} + 436 q^{37} + 520 q^{43} + 98 q^{49} + 504 q^{55} - 644 q^{61} + 712 q^{67} - 452 q^{73} + 880 q^{79} - 2520 q^{85} + 364 q^{91} - 2660 q^{97}+O(q^{100}) $ 2 * q + 14 * q^7 + 52 * q^13 + 136 * q^19 + 254 * q^25 + 424 * q^31 + 436 * q^37 + 520 * q^43 + 98 * q^49 + 504 * q^55 - 644 * q^61 + 712 * q^67 - 452 * q^73 + 880 * q^79 - 2520 * q^85 + 364 * q^91 - 2660 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−15.8745	−1.41986	−0.709930	−	0.704273i	$-0.751273\pi$
$5$	−15.8745	−1.41986	−0.709930	+	0.704273i	$0.751273\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−15.8745	−0.435122	−0.217561	−	0.976047i	$-0.569810\pi$
$11$	−15.8745	−0.435122	−0.217561	+	0.976047i	$0.569810\pi$
$12$	0	0
$13$	26.0000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	26.0000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	79.3725	1.13239	0.566196	−	0.824271i	$-0.308414\pi$
$17$	79.3725	1.13239	0.566196	+	0.824271i	$0.308414\pi$
$18$	0	0
$19$	68.0000	0.821067	0.410533	−	0.911846i	$-0.365343\pi$
$19$	68.0000	0.821067	0.410533	+	0.911846i	$0.365343\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−47.6235	−0.431747	−0.215874	−	0.976421i	$-0.569260\pi$
$23$	−47.6235	−0.431747	−0.215874	+	0.976421i	$0.569260\pi$
$24$	0	0
$25$	127.000	1.01600
$26$	0	0
$27$	0	0
$28$	0	0
$29$	253.992	1.62638	0.813192	−	0.581995i	$-0.197728\pi$
$29$	253.992	1.62638	0.813192	+	0.581995i	$0.197728\pi$
$30$	0	0
$31$	212.000	1.22827	0.614134	−	0.789202i	$-0.289505\pi$
$31$	212.000	1.22827	0.614134	+	0.789202i	$0.289505\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−111.122	−0.536656
$36$	0	0
$37$	218.000	0.968621	0.484311	−	0.874896i	$-0.339070\pi$
$37$	218.000	0.968621	0.484311	+	0.874896i	$0.339070\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−396.863	−1.51170	−0.755848	−	0.654747i	$-0.772775\pi$
$41$	−396.863	−1.51170	−0.755848	+	0.654747i	$0.772775\pi$
$42$	0	0
$43$	260.000	0.922084	0.461042	−	0.887378i	$-0.347476\pi$
$43$	260.000	0.922084	0.461042	+	0.887378i	$0.347476\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−412.737	−1.28093	−0.640467	−	0.767986i	$-0.721259\pi$
$47$	−412.737	−1.28093	−0.640467	+	0.767986i	$0.721259\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	476.235	1.23426	0.617132	−	0.786860i	$-0.288295\pi$
$53$	476.235	1.23426	0.617132	+	0.786860i	$0.288295\pi$
$54$	0	0
$55$	252.000	0.617812
$56$	0	0
$57$	0	0
$58$	0	0
$59$	285.741	0.630514	0.315257	−	0.949006i	$-0.397909\pi$
$59$	285.741	0.630514	0.315257	+	0.949006i	$0.397909\pi$
$60$	0	0
$61$	−322.000	−0.675867	−0.337933	−	0.941170i	$-0.609728\pi$
$61$	−322.000	−0.675867	−0.337933	+	0.941170i	$0.609728\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−412.737	−0.787596
$66$	0	0
$67$	356.000	0.649139	0.324570	−	0.945862i	$-0.394781\pi$
$67$	356.000	0.649139	0.324570	+	0.945862i	$0.394781\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	1127.09	1.88396	0.941979	−	0.335673i	$-0.108964\pi$
$71$	1127.09	1.88396	0.941979	+	0.335673i	$0.108964\pi$
$72$	0	0
$73$	−226.000	−0.362347	−0.181173	−	0.983451i	$-0.557989\pi$
$73$	−226.000	−0.362347	−0.181173	+	0.983451i	$0.557989\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−111.122	−0.164461
$78$	0	0
$79$	440.000	0.626631	0.313316	−	0.949649i	$-0.398560\pi$
$79$	440.000	0.626631	0.313316	+	0.949649i	$0.398560\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	253.992	0.335895	0.167947	−	0.985796i	$-0.446286\pi$
$83$	253.992	0.335895	0.167947	+	0.985796i	$0.446286\pi$
$84$	0	0
$85$	−1260.00	−1.60784
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−206.369	−0.245787	−0.122893	−	0.992420i	$-0.539217\pi$
$89$	−206.369	−0.245787	−0.122893	+	0.992420i	$0.539217\pi$
$90$	0	0
$91$	182.000	0.209657
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1079.47	−1.16580
$96$	0	0
$97$	−1330.00	−1.39218	−0.696088	−	0.717957i	$-0.745078\pi$
$97$	−1330.00	−1.39218	−0.696088	+	0.717957i	$0.745078\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−238.118	−0.234590	−0.117295	−	0.993097i	$-0.537422\pi$
$101$	−238.118	−0.234590	−0.117295	+	0.993097i	$0.537422\pi$
$102$	0	0
$103$	−1804.00	−1.72576	−0.862881	−	0.505408i	$-0.831342\pi$
$103$	−1804.00	−1.72576	−0.862881	+	0.505408i	$0.831342\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	174.620	0.157767	0.0788837	−	0.996884i	$-0.474864\pi$
$107$	174.620	0.157767	0.0788837	+	0.996884i	$0.474864\pi$
$108$	0	0
$109$	−454.000	−0.398948	−0.199474	−	0.979903i	$-0.563923\pi$
$109$	−454.000	−0.398948	−0.199474	+	0.979903i	$0.563923\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1777.94	−1.48013	−0.740066	−	0.672534i	$-0.765206\pi$
$113$	−1777.94	−1.48013	−0.740066	+	0.672534i	$0.765206\pi$
$114$	0	0
$115$	756.000	0.613021
$116$	0	0
$117$	0	0
$118$	0	0
$119$	555.608	0.428004
$120$	0	0
$121$	−1079.00	−0.810669
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−31.7490	−0.0227177
$126$	0	0
$127$	−904.000	−0.631630	−0.315815	−	0.948821i	$-0.602278\pi$
$127$	−904.000	−0.631630	−0.315815	+	0.948821i	$0.602278\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1333.46	0.889350	0.444675	−	0.895692i	$-0.353319\pi$
$131$	1333.46	0.889350	0.444675	+	0.895692i	$0.353319\pi$
$132$	0	0
$133$	476.000	0.310334
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1619.20	1.00976	0.504882	−	0.863189i	$-0.331536\pi$
$137$	1619.20	1.00976	0.504882	+	0.863189i	$0.331536\pi$
$138$	0	0
$139$	−880.000	−0.536983	−0.268491	−	0.963282i	$-0.586525\pi$
$139$	−880.000	−0.536983	−0.268491	+	0.963282i	$0.586525\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−412.737	−0.241362
$144$	0	0
$145$	−4032.00	−2.30924
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2571.67	1.41396	0.706978	−	0.707236i	$-0.250058\pi$
$149$	2571.67	1.41396	0.706978	+	0.707236i	$0.250058\pi$
$150$	0	0
$151$	3512.00	1.89273	0.946366	−	0.323097i	$-0.104724\pi$
$151$	3512.00	1.89273	0.946366	+	0.323097i	$0.104724\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−3365.40	−1.74397
$156$	0	0
$157$	−322.000	−0.163684	−0.0818420	−	0.996645i	$-0.526080\pi$
$157$	−322.000	−0.163684	−0.0818420	+	0.996645i	$0.526080\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−333.365	−0.163185
$162$	0	0
$163$	3404.00	1.63572	0.817858	−	0.575419i	$-0.195161\pi$
$163$	3404.00	1.63572	0.817858	+	0.575419i	$0.195161\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2762.16	1.27990	0.639948	−	0.768418i	$-0.278956\pi$
$167$	2762.16	1.27990	0.639948	+	0.768418i	$0.278956\pi$
$168$	0	0
$169$	−1521.00	−0.692308
$170$	0	0
$171$	0	0
$172$	0	0
$173$	3508.27	1.54178	0.770892	−	0.636966i	$-0.219811\pi$
$173$	3508.27	1.54178	0.770892	+	0.636966i	$0.219811\pi$
$174$	0	0
$175$	889.000	0.384012
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−3603.51	−1.50469	−0.752344	−	0.658770i	$-0.771077\pi$
$179$	−3603.51	−1.50469	−0.752344	+	0.658770i	$0.771077\pi$
$180$	0	0
$181$	−502.000	−0.206151	−0.103076	−	0.994674i	$-0.532868\pi$
$181$	−502.000	−0.206151	−0.103076	+	0.994674i	$0.532868\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−3460.64	−1.37531
$186$	0	0
$187$	−1260.00	−0.492729
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−2492.30	−0.944169	−0.472085	−	0.881553i	$-0.656498\pi$
$191$	−2492.30	−0.944169	−0.472085	+	0.881553i	$0.656498\pi$
$192$	0	0
$193$	1310.00	0.488580	0.244290	−	0.969702i	$-0.421445\pi$
$193$	1310.00	0.488580	0.244290	+	0.969702i	$0.421445\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1174.71	−0.424847	−0.212424	−	0.977178i	$-0.568136\pi$
$197$	−1174.71	−0.424847	−0.212424	+	0.977178i	$0.568136\pi$
$198$	0	0
$199$	4592.00	1.63577	0.817885	−	0.575382i	$-0.195146\pi$
$199$	4592.00	1.63577	0.817885	+	0.575382i	$0.195146\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1777.94	0.614716
$204$	0	0
$205$	6300.00	2.14640
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−1079.47	−0.357264
$210$	0	0
$211$	−1780.00	−0.580759	−0.290380	−	0.956911i	$-0.593782\pi$
$211$	−1780.00	−0.580759	−0.290380	+	0.956911i	$0.593782\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−4127.37	−1.30923
$216$	0	0
$217$	1484.00	0.464242
$218$	0	0
$219$	0	0
$220$	0	0
$221$	2063.69	0.628138
$222$	0	0
$223$	4088.00	1.22759	0.613795	−	0.789465i	$-0.289642\pi$
$223$	4088.00	1.22759	0.613795	+	0.789465i	$0.289642\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4857.60	1.42031	0.710155	−	0.704046i	$-0.248625\pi$
$227$	4857.60	1.42031	0.710155	+	0.704046i	$0.248625\pi$
$228$	0	0
$229$	−6382.00	−1.84164	−0.920818	−	0.389994i	$-0.872477\pi$
$229$	−6382.00	−1.84164	−0.920818	+	0.389994i	$0.872477\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	2190.68	0.615950	0.307975	−	0.951394i	$-0.400349\pi$
$233$	2190.68	0.615950	0.307975	+	0.951394i	$0.400349\pi$
$234$	0	0
$235$	6552.00	1.81875
$236$	0	0
$237$	0	0
$238$	0	0
$239$	5349.71	1.44788	0.723941	−	0.689862i	$-0.242329\pi$
$239$	5349.71	1.44788	0.723941	+	0.689862i	$0.242329\pi$
$240$	0	0
$241$	4214.00	1.12634	0.563169	−	0.826342i	$-0.309582\pi$
$241$	4214.00	1.12634	0.563169	+	0.826342i	$0.309582\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−777.851	−0.202837
$246$	0	0
$247$	1768.00	0.455446
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1428.71	0.359279	0.179640	−	0.983732i	$-0.442507\pi$
$251$	1428.71	0.359279	0.179640	+	0.983732i	$0.442507\pi$
$252$	0	0
$253$	756.000	0.187863
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−841.349	−0.204210	−0.102105	−	0.994774i	$-0.532558\pi$
$257$	−841.349	−0.204210	−0.102105	+	0.994774i	$0.532558\pi$
$258$	0	0
$259$	1526.00	0.366104
$260$	0	0
$261$	0	0
$262$	0	0
$263$	269.867	0.0632726	0.0316363	−	0.999499i	$-0.489928\pi$
$263$	269.867	0.0632726	0.0316363	+	0.999499i	$0.489928\pi$
$264$	0	0
$265$	−7560.00	−1.75248
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−2746.29	−0.622469	−0.311235	−	0.950333i	$-0.600743\pi$
$269$	−2746.29	−0.622469	−0.311235	+	0.950333i	$0.600743\pi$
$270$	0	0
$271$	−6244.00	−1.39962	−0.699808	−	0.714331i	$-0.746731\pi$
$271$	−6244.00	−1.39962	−0.699808	+	0.714331i	$0.746731\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−2016.06	−0.442084
$276$	0	0
$277$	8126.00	1.76261	0.881307	−	0.472544i	$-0.156664\pi$
$277$	8126.00	1.76261	0.881307	+	0.472544i	$0.156664\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	3079.65	0.653796	0.326898	−	0.945060i	$-0.393997\pi$
$281$	3079.65	0.653796	0.326898	+	0.945060i	$0.393997\pi$
$282$	0	0
$283$	−604.000	−0.126870	−0.0634348	−	0.997986i	$-0.520205\pi$
$283$	−604.000	−0.126870	−0.0634348	+	0.997986i	$0.520205\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2778.04	−0.571367
$288$	0	0
$289$	1387.00	0.282312
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−7667.39	−1.52878	−0.764392	−	0.644752i	$-0.776961\pi$
$293$	−7667.39	−1.52878	−0.764392	+	0.644752i	$0.776961\pi$
$294$	0	0
$295$	−4536.00	−0.895241
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1238.21	−0.239490
$300$	0	0
$301$	1820.00	0.348515
$302$	0	0
$303$	0	0
$304$	0	0
$305$	5111.59	0.959636
$306$	0	0
$307$	−1636.00	−0.304142	−0.152071	−	0.988370i	$-0.548594\pi$
$307$	−1636.00	−0.304142	−0.152071	+	0.988370i	$0.548594\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	3270.15	0.596248	0.298124	−	0.954527i	$-0.403639\pi$
$311$	3270.15	0.596248	0.298124	+	0.954527i	$0.403639\pi$
$312$	0	0
$313$	−5038.00	−0.909791	−0.454896	−	0.890545i	$-0.650323\pi$
$313$	−5038.00	−0.909791	−0.454896	+	0.890545i	$0.650323\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−9873.94	−1.74945	−0.874725	−	0.484619i	$-0.838958\pi$
$317$	−9873.94	−1.74945	−0.874725	+	0.484619i	$0.838958\pi$
$318$	0	0
$319$	−4032.00	−0.707676
$320$	0	0
$321$	0	0
$322$	0	0
$323$	5397.33	0.929770
$324$	0	0
$325$	3302.00	0.563575
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−2889.16	−0.484148
$330$	0	0
$331$	428.000	0.0710725	0.0355363	−	0.999368i	$-0.488686\pi$
$331$	428.000	0.0710725	0.0355363	+	0.999368i	$0.488686\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−5651.32	−0.921686
$336$	0	0
$337$	−9106.00	−1.47192	−0.735958	−	0.677028i	$-0.763268\pi$
$337$	−9106.00	−1.47192	−0.735958	+	0.677028i	$0.763268\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−3365.40	−0.534447
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−1952.56	−0.302072	−0.151036	−	0.988528i	$-0.548261\pi$
$347$	−1952.56	−0.302072	−0.151036	+	0.988528i	$0.548261\pi$
$348$	0	0
$349$	−1834.00	−0.281294	−0.140647	−	0.990060i	$-0.544918\pi$
$349$	−1834.00	−0.281294	−0.140647	+	0.990060i	$0.544918\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	10493.0	1.58212	0.791060	−	0.611738i	$-0.209529\pi$
$353$	10493.0	1.58212	0.791060	+	0.611738i	$0.209529\pi$
$354$	0	0
$355$	−17892.0	−2.67495
$356$	0	0
$357$	0	0
$358$	0	0
$359$	13382.2	1.96737	0.983685	−	0.179898i	$-0.0575767\pi$
$359$	13382.2	1.96737	0.983685	+	0.179898i	$0.0575767\pi$
$360$	0	0
$361$	−2235.00	−0.325849
$362$	0	0
$363$	0	0
$364$	0	0
$365$	3587.64	0.514481
$366$	0	0
$367$	−208.000	−0.0295845	−0.0147923	−	0.999891i	$-0.504709\pi$
$367$	−208.000	−0.0295845	−0.0147923	+	0.999891i	$0.504709\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	3333.65	0.466508
$372$	0	0
$373$	5774.00	0.801518	0.400759	−	0.916183i	$-0.368746\pi$
$373$	5774.00	0.801518	0.400759	+	0.916183i	$0.368746\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	6603.80	0.902156
$378$	0	0
$379$	−12004.0	−1.62692	−0.813462	−	0.581618i	$-0.802420\pi$
$379$	−12004.0	−1.62692	−0.813462	+	0.581618i	$0.802420\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−7048.28	−0.940340	−0.470170	−	0.882576i	$-0.655807\pi$
$383$	−7048.28	−0.940340	−0.470170	+	0.882576i	$0.655807\pi$
$384$	0	0
$385$	1764.00	0.233511
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−1206.46	−0.157250	−0.0786248	−	0.996904i	$-0.525053\pi$
$389$	−1206.46	−0.157250	−0.0786248	+	0.996904i	$0.525053\pi$
$390$	0	0
$391$	−3780.00	−0.488907
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−6984.78	−0.889728
$396$	0	0
$397$	3470.00	0.438676	0.219338	−	0.975649i	$-0.429610\pi$
$397$	3470.00	0.438676	0.219338	+	0.975649i	$0.429610\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−1365.21	−0.170013	−0.0850065	−	0.996380i	$-0.527091\pi$
$401$	−1365.21	−0.170013	−0.0850065	+	0.996380i	$0.527091\pi$
$402$	0	0
$403$	5512.00	0.681321
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−3460.64	−0.421469
$408$	0	0
$409$	−11842.0	−1.43166	−0.715830	−	0.698274i	$-0.753952\pi$
$409$	−11842.0	−1.43166	−0.715830	+	0.698274i	$0.753952\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	2000.19	0.238312
$414$	0	0
$415$	−4032.00	−0.476923
$416$	0	0
$417$	0	0
$418$	0	0
$419$	3460.64	0.403493	0.201746	−	0.979438i	$-0.435338\pi$
$419$	3460.64	0.403493	0.201746	+	0.979438i	$0.435338\pi$
$420$	0	0
$421$	15518.0	1.79644	0.898220	−	0.439547i	$-0.144861\pi$
$421$	15518.0	1.79644	0.898220	+	0.439547i	$0.144861\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	10080.3	1.15051
$426$	0	0
$427$	−2254.00	−0.255454
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−6937.16	−0.775293	−0.387646	−	0.921808i	$-0.626712\pi$
$431$	−6937.16	−0.775293	−0.387646	+	0.921808i	$0.626712\pi$
$432$	0	0
$433$	−4006.00	−0.444610	−0.222305	−	0.974977i	$-0.571358\pi$
$433$	−4006.00	−0.444610	−0.222305	+	0.974977i	$0.571358\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−3238.40	−0.354494
$438$	0	0
$439$	−7504.00	−0.815823	−0.407912	−	0.913021i	$-0.633743\pi$
$439$	−7504.00	−0.815823	−0.407912	+	0.913021i	$0.633743\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−18271.6	−1.95961	−0.979806	−	0.199951i	$-0.935922\pi$
$443$	−18271.6	−1.95961	−0.979806	+	0.199951i	$0.935922\pi$
$444$	0	0
$445$	3276.00	0.348983
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−8445.24	−0.887651	−0.443826	−	0.896113i	$-0.646379\pi$
$449$	−8445.24	−0.887651	−0.443826	+	0.896113i	$0.646379\pi$
$450$	0	0
$451$	6300.00	0.657773
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−2889.16	−0.297683
$456$	0	0
$457$	10214.0	1.04549	0.522747	−	0.852488i	$-0.324907\pi$
$457$	10214.0	1.04549	0.522747	+	0.852488i	$0.324907\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	1285.84	0.129907	0.0649537	−	0.997888i	$-0.479310\pi$
$461$	1285.84	0.129907	0.0649537	+	0.997888i	$0.479310\pi$
$462$	0	0
$463$	−1408.00	−0.141329	−0.0706645	−	0.997500i	$-0.522512\pi$
$463$	−1408.00	−0.141329	−0.0706645	+	0.997500i	$0.522512\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	11397.9	1.12940	0.564702	−	0.825295i	$-0.308991\pi$
$467$	11397.9	1.12940	0.564702	+	0.825295i	$0.308991\pi$
$468$	0	0
$469$	2492.00	0.245352
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−4127.37	−0.401219
$474$	0	0
$475$	8636.00	0.834204
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−10762.9	−1.02666	−0.513330	−	0.858191i	$-0.671588\pi$
$479$	−10762.9	−1.02666	−0.513330	+	0.858191i	$0.671588\pi$
$480$	0	0
$481$	5668.00	0.537294
$482$	0	0
$483$	0	0
$484$	0	0
$485$	21113.1	1.97669
$486$	0	0
$487$	4712.00	0.438442	0.219221	−	0.975675i	$-0.429648\pi$
$487$	4712.00	0.438442	0.219221	+	0.975675i	$0.429648\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−15795.1	−1.45178	−0.725891	−	0.687810i	$-0.758572\pi$
$491$	−15795.1	−1.45178	−0.725891	+	0.687810i	$0.758572\pi$
$492$	0	0
$493$	20160.0	1.84171
$494$	0	0
$495$	0	0
$496$	0	0
$497$	7889.63	0.712069
$498$	0	0
$499$	−16540.0	−1.48383	−0.741916	−	0.670493i	$-0.766083\pi$
$499$	−16540.0	−1.48383	−0.741916	+	0.670493i	$0.766083\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	10921.7	0.968137	0.484068	−	0.875030i	$-0.339159\pi$
$503$	10921.7	0.968137	0.484068	+	0.875030i	$0.339159\pi$
$504$	0	0
$505$	3780.00	0.333085
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−2174.81	−0.189384	−0.0946922	−	0.995507i	$-0.530187\pi$
$509$	−2174.81	−0.189384	−0.0946922	+	0.995507i	$0.530187\pi$
$510$	0	0
$511$	−1582.00	−0.136954
$512$	0	0
$513$	0	0
$514$	0	0
$515$	28637.6	2.45034
$516$	0	0
$517$	6552.00	0.557363
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−11731.3	−0.986480	−0.493240	−	0.869893i	$-0.664187\pi$
$521$	−11731.3	−0.986480	−0.493240	+	0.869893i	$0.664187\pi$
$522$	0	0
$523$	15440.0	1.29091	0.645453	−	0.763800i	$-0.276669\pi$
$523$	15440.0	1.29091	0.645453	+	0.763800i	$0.276669\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	16827.0	1.39088
$528$	0	0
$529$	−9899.00	−0.813594
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−10318.4	−0.838538
$534$	0	0
$535$	−2772.00	−0.224007
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−777.851	−0.0621603
$540$	0	0
$541$	19046.0	1.51359	0.756794	−	0.653653i	$-0.226764\pi$
$541$	19046.0	1.51359	0.756794	+	0.653653i	$0.226764\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	7207.03	0.566450
$546$	0	0
$547$	1892.00	0.147890	0.0739452	−	0.997262i	$-0.476441\pi$
$547$	1892.00	0.147890	0.0739452	+	0.997262i	$0.476441\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	17271.5	1.33537
$552$	0	0
$553$	3080.00	0.236844
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−23653.0	−1.79930	−0.899650	−	0.436611i	$-0.856179\pi$
$557$	−23653.0	−1.79930	−0.899650	+	0.436611i	$0.856179\pi$
$558$	0	0
$559$	6760.00	0.511480
$560$	0	0
$561$	0	0
$562$	0	0
$563$	6953.03	0.520489	0.260245	−	0.965543i	$-0.416197\pi$
$563$	6953.03	0.520489	0.260245	+	0.965543i	$0.416197\pi$
$564$	0	0
$565$	28224.0	2.10158
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−11016.9	−0.811692	−0.405846	−	0.913941i	$-0.633023\pi$
$569$	−11016.9	−0.811692	−0.405846	+	0.913941i	$0.633023\pi$
$570$	0	0
$571$	7580.00	0.555540	0.277770	−	0.960648i	$-0.410405\pi$
$571$	7580.00	0.555540	0.277770	+	0.960648i	$0.410405\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−6048.19	−0.438655
$576$	0	0
$577$	−6718.00	−0.484704	−0.242352	−	0.970188i	$-0.577919\pi$
$577$	−6718.00	−0.484704	−0.242352	+	0.970188i	$0.577919\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	1777.94	0.126956
$582$	0	0
$583$	−7560.00	−0.537055
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−24288.0	−1.70779	−0.853895	−	0.520445i	$-0.825766\pi$
$587$	−24288.0	−1.70779	−0.853895	+	0.520445i	$0.825766\pi$
$588$	0	0
$589$	14416.0	1.00849
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−20113.0	−1.39282	−0.696410	−	0.717644i	$-0.745220\pi$
$593$	−20113.0	−1.39282	−0.696410	+	0.717644i	$0.745220\pi$
$594$	0	0
$595$	−8820.00	−0.607705
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−5984.69	−0.408227	−0.204113	−	0.978947i	$-0.565431\pi$
$599$	−5984.69	−0.408227	−0.204113	+	0.978947i	$0.565431\pi$
$600$	0	0
$601$	−8230.00	−0.558584	−0.279292	−	0.960206i	$-0.590100\pi$
$601$	−8230.00	−0.558584	−0.279292	+	0.960206i	$0.590100\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	17128.6	1.15104
$606$	0	0
$607$	8696.00	0.581482	0.290741	−	0.956802i	$-0.406098\pi$
$607$	8696.00	0.581482	0.290741	+	0.956802i	$0.406098\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−10731.2	−0.710534
$612$	0	0
$613$	−5794.00	−0.381758	−0.190879	−	0.981614i	$-0.561134\pi$
$613$	−5794.00	−0.381758	−0.190879	+	0.981614i	$0.561134\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	13493.3	0.880423	0.440212	−	0.897894i	$-0.354903\pi$
$617$	13493.3	0.880423	0.440212	+	0.897894i	$0.354903\pi$
$618$	0	0
$619$	−19336.0	−1.25554	−0.627770	−	0.778399i	$-0.716032\pi$
$619$	−19336.0	−1.25554	−0.627770	+	0.778399i	$0.716032\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−1444.58	−0.0928987
$624$	0	0
$625$	−15371.0	−0.983744
$626$	0	0
$627$	0	0
$628$	0	0
$629$	17303.2	1.09686
$630$	0	0
$631$	632.000	0.0398725	0.0199362	−	0.999801i	$-0.493654\pi$
$631$	632.000	0.0398725	0.0199362	+	0.999801i	$0.493654\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	14350.6	0.896826
$636$	0	0
$637$	1274.00	0.0792429
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−18890.7	−1.16402	−0.582010	−	0.813182i	$-0.697733\pi$
$641$	−18890.7	−1.16402	−0.582010	+	0.813182i	$0.697733\pi$
$642$	0	0
$643$	28460.0	1.74549	0.872747	−	0.488173i	$-0.162336\pi$
$643$	28460.0	1.74549	0.872747	+	0.488173i	$0.162336\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−10254.9	−0.623127	−0.311563	−	0.950225i	$-0.600853\pi$
$647$	−10254.9	−0.623127	−0.311563	+	0.950225i	$0.600853\pi$
$648$	0	0
$649$	−4536.00	−0.274351
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−5968.81	−0.357700	−0.178850	−	0.983876i	$-0.557238\pi$
$653$	−5968.81	−0.357700	−0.178850	+	0.983876i	$0.557238\pi$
$654$	0	0
$655$	−21168.0	−1.26275
$656$	0	0
$657$	0	0
$658$	0	0
$659$	8048.38	0.475751	0.237876	−	0.971296i	$-0.423549\pi$
$659$	8048.38	0.475751	0.237876	+	0.971296i	$0.423549\pi$
$660$	0	0
$661$	−12178.0	−0.716595	−0.358298	−	0.933607i	$-0.616643\pi$
$661$	−12178.0	−0.716595	−0.358298	+	0.933607i	$0.616643\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−7556.27	−0.440631
$666$	0	0
$667$	−12096.0	−0.702187
$668$	0	0
$669$	0	0
$670$	0	0
$671$	5111.59	0.294085
$672$	0	0
$673$	12902.0	0.738983	0.369491	−	0.929234i	$-0.379532\pi$
$673$	12902.0	0.738983	0.369491	+	0.929234i	$0.379532\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−14080.7	−0.799357	−0.399679	−	0.916655i	$-0.630878\pi$
$677$	−14080.7	−0.799357	−0.399679	+	0.916655i	$0.630878\pi$
$678$	0	0
$679$	−9310.00	−0.526193
$680$	0	0
$681$	0	0
$682$	0	0
$683$	19954.3	1.11790	0.558952	−	0.829200i	$-0.311204\pi$
$683$	19954.3	1.11790	0.558952	+	0.829200i	$0.311204\pi$
$684$	0	0
$685$	−25704.0	−1.43372
$686$	0	0
$687$	0	0
$688$	0	0
$689$	12382.1	0.684646
$690$	0	0
$691$	2168.00	0.119355	0.0596777	−	0.998218i	$-0.480993\pi$
$691$	2168.00	0.119355	0.0596777	+	0.998218i	$0.480993\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	13969.6	0.762440
$696$	0	0
$697$	−31500.0	−1.71183
$698$	0	0
$699$	0	0
$700$	0	0
$701$	30987.0	1.66956	0.834782	−	0.550581i	$-0.185594\pi$
$701$	30987.0	1.66956	0.834782	+	0.550581i	$0.185594\pi$
$702$	0	0
$703$	14824.0	0.795303
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−1666.82	−0.0886667
$708$	0	0
$709$	−19294.0	−1.02200	−0.511002	−	0.859579i	$-0.670726\pi$
$709$	−19294.0	−1.02200	−0.511002	+	0.859579i	$0.670726\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−10096.2	−0.530302
$714$	0	0
$715$	6552.00	0.342701
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−28320.1	−1.46893	−0.734466	−	0.678645i	$-0.762567\pi$
$719$	−28320.1	−1.46893	−0.734466	+	0.678645i	$0.762567\pi$
$720$	0	0
$721$	−12628.0	−0.652276
$722$	0	0
$723$	0	0
$724$	0	0
$725$	32257.0	1.65241
$726$	0	0
$727$	15860.0	0.809099	0.404549	−	0.914516i	$-0.367428\pi$
$727$	15860.0	0.809099	0.404549	+	0.914516i	$0.367428\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	20636.9	1.04416
$732$	0	0
$733$	38498.0	1.93991	0.969956	−	0.243279i	$-0.0782230\pi$
$733$	38498.0	1.93991	0.969956	+	0.243279i	$0.0782230\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−5651.32	−0.282455
$738$	0	0
$739$	−26356.0	−1.31194	−0.655968	−	0.754788i	$-0.727740\pi$
$739$	−26356.0	−1.31194	−0.655968	+	0.754788i	$0.727740\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	9000.85	0.444427	0.222213	−	0.974998i	$-0.428672\pi$
$743$	9000.85	0.444427	0.222213	+	0.974998i	$0.428672\pi$
$744$	0	0
$745$	−40824.0	−2.00762
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1222.34	0.0596305
$750$	0	0
$751$	−17800.0	−0.864888	−0.432444	−	0.901661i	$-0.642349\pi$
$751$	−17800.0	−0.864888	−0.432444	+	0.901661i	$0.642349\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−55751.3	−2.68741
$756$	0	0
$757$	2882.00	0.138373	0.0691863	−	0.997604i	$-0.477960\pi$
$757$	2882.00	0.138373	0.0691863	+	0.997604i	$0.477960\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−8397.61	−0.400017	−0.200009	−	0.979794i	$-0.564097\pi$
$761$	−8397.61	−0.400017	−0.200009	+	0.979794i	$0.564097\pi$
$762$	0	0
$763$	−3178.00	−0.150788
$764$	0	0
$765$	0	0
$766$	0	0
$767$	7429.27	0.349746
$768$	0	0
$769$	−40630.0	−1.90527	−0.952637	−	0.304111i	$-0.901641\pi$
$769$	−40630.0	−1.90527	−0.952637	+	0.304111i	$0.901641\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	4524.23	0.210512	0.105256	−	0.994445i	$-0.466434\pi$
$773$	4524.23	0.210512	0.105256	+	0.994445i	$0.466434\pi$
$774$	0	0
$775$	26924.0	1.24792
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−26986.7	−1.24120
$780$	0	0
$781$	−17892.0	−0.819752
$782$	0	0
$783$	0	0
$784$	0	0
$785$	5111.59	0.232408
$786$	0	0
$787$	−6496.00	−0.294228	−0.147114	−	0.989120i	$-0.546998\pi$
$787$	−6496.00	−0.294228	−0.147114	+	0.989120i	$0.546998\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−12445.6	−0.559438
$792$	0	0
$793$	−8372.00	−0.374903
$794$	0	0
$795$	0	0
$796$	0	0
$797$	2301.80	0.102301	0.0511506	−	0.998691i	$-0.483711\pi$
$797$	2301.80	0.102301	0.0511506	+	0.998691i	$0.483711\pi$
$798$	0	0
$799$	−32760.0	−1.45052
$800$	0	0
$801$	0	0
$802$	0	0
$803$	3587.64	0.157665
$804$	0	0
$805$	5292.00	0.231700
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−7492.77	−0.325626	−0.162813	−	0.986657i	$-0.552057\pi$
$809$	−7492.77	−0.325626	−0.162813	+	0.986657i	$0.552057\pi$
$810$	0	0
$811$	38864.0	1.68274	0.841368	−	0.540462i	$-0.181751\pi$
$811$	38864.0	1.68274	0.841368	+	0.540462i	$0.181751\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−54036.8	−2.32249
$816$	0	0
$817$	17680.0	0.757093
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−15715.8	−0.668068	−0.334034	−	0.942561i	$-0.608410\pi$
$821$	−15715.8	−0.668068	−0.334034	+	0.942561i	$0.608410\pi$
$822$	0	0
$823$	36416.0	1.54238	0.771192	−	0.636603i	$-0.219661\pi$
$823$	36416.0	1.54238	0.771192	+	0.636603i	$0.219661\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	24684.9	1.03794	0.518970	−	0.854792i	$-0.326316\pi$
$827$	24684.9	1.03794	0.518970	+	0.854792i	$0.326316\pi$
$828$	0	0
$829$	−17302.0	−0.724877	−0.362439	−	0.932008i	$-0.618056\pi$
$829$	−17302.0	−0.724877	−0.362439	+	0.932008i	$0.618056\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	3889.25	0.161770
$834$	0	0
$835$	−43848.0	−1.81727
$836$	0	0
$837$	0	0
$838$	0	0
$839$	29240.8	1.20323	0.601613	−	0.798788i	$-0.294525\pi$
$839$	29240.8	1.20323	0.601613	+	0.798788i	$0.294525\pi$
$840$	0	0
$841$	40123.0	1.64513
$842$	0	0
$843$	0	0
$844$	0	0
$845$	24145.1	0.982979
$846$	0	0
$847$	−7553.00	−0.306404
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−10381.9	−0.418200
$852$	0	0
$853$	−82.0000	−0.00329147	−0.00164574	−	0.999999i	$-0.500524\pi$
$853$	−82.0000	−0.00329147	−0.00164574	+	0.999999i	$0.500524\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	10651.8	0.424572	0.212286	−	0.977208i	$-0.431909\pi$
$857$	10651.8	0.424572	0.212286	+	0.977208i	$0.431909\pi$
$858$	0	0
$859$	25508.0	1.01318	0.506590	−	0.862187i	$-0.330906\pi$
$859$	25508.0	1.01318	0.506590	+	0.862187i	$0.330906\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	34177.8	1.34812	0.674059	−	0.738677i	$-0.264549\pi$
$863$	34177.8	1.34812	0.674059	+	0.738677i	$0.264549\pi$
$864$	0	0
$865$	−55692.0	−2.18912
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−6984.78	−0.272661
$870$	0	0
$871$	9256.00	0.360078
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−222.243	−0.00858650
$876$	0	0
$877$	10838.0	0.417301	0.208651	−	0.977990i	$-0.433093\pi$
$877$	10838.0	0.417301	0.208651	+	0.977990i	$0.433093\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	6175.18	0.236149	0.118074	−	0.993005i	$-0.462328\pi$
$881$	6175.18	0.236149	0.118074	+	0.993005i	$0.462328\pi$
$882$	0	0
$883$	22100.0	0.842270	0.421135	−	0.906998i	$-0.361632\pi$
$883$	22100.0	0.842270	0.421135	+	0.906998i	$0.361632\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−20732.1	−0.784798	−0.392399	−	0.919795i	$-0.628355\pi$
$887$	−20732.1	−0.784798	−0.392399	+	0.919795i	$0.628355\pi$
$888$	0	0
$889$	−6328.00	−0.238734
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−28066.1	−1.05173
$894$	0	0
$895$	57204.0	2.13645
$896$	0	0
$897$	0	0
$898$	0	0
$899$	53846.3	1.99764
$900$	0	0
$901$	37800.0	1.39767
$902$	0	0
$903$	0	0
$904$	0	0
$905$	7969.00	0.292706
$906$	0	0
$907$	6812.00	0.249381	0.124691	−	0.992196i	$-0.460206\pi$
$907$	6812.00	0.249381	0.124691	+	0.992196i	$0.460206\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−11636.0	−0.423182	−0.211591	−	0.977358i	$-0.567864\pi$
$911$	−11636.0	−0.423182	−0.211591	+	0.977358i	$0.567864\pi$
$912$	0	0
$913$	−4032.00	−0.146155
$914$	0	0
$915$	0	0
$916$	0	0
$917$	9334.21	0.336143
$918$	0	0
$919$	−1168.00	−0.0419247	−0.0209623	−	0.999780i	$-0.506673\pi$
$919$	−1168.00	−0.0419247	−0.0209623	+	0.999780i	$0.506673\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	29304.3	1.04503
$924$	0	0
$925$	27686.0	0.984119
$926$	0	0
$927$	0	0
$928$	0	0
$929$	9635.83	0.340303	0.170151	−	0.985418i	$-0.445574\pi$
$929$	9635.83	0.340303	0.170151	+	0.985418i	$0.445574\pi$
$930$	0	0
$931$	3332.00	0.117295
$932$	0	0
$933$	0	0
$934$	0	0
$935$	20001.9	0.699606
$936$	0	0
$937$	−32254.0	−1.12454	−0.562269	−	0.826954i	$-0.690071\pi$
$937$	−32254.0	−1.12454	−0.562269	+	0.826954i	$0.690071\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	4111.50	0.142435	0.0712173	−	0.997461i	$-0.477312\pi$
$941$	4111.50	0.142435	0.0712173	+	0.997461i	$0.477312\pi$
$942$	0	0
$943$	18900.0	0.652671
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−6873.66	−0.235865	−0.117932	−	0.993022i	$-0.537627\pi$
$947$	−6873.66	−0.235865	−0.117932	+	0.993022i	$0.537627\pi$
$948$	0	0
$949$	−5876.00	−0.200994
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−33971.4	−1.15471	−0.577357	−	0.816492i	$-0.695916\pi$
$953$	−33971.4	−1.15471	−0.577357	+	0.816492i	$0.695916\pi$
$954$	0	0
$955$	39564.0	1.34059
$956$	0	0
$957$	0	0
$958$	0	0
$959$	11334.4	0.381655
$960$	0	0
$961$	15153.0	0.508644
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−20795.6	−0.693714
$966$	0	0
$967$	43952.0	1.46163	0.730817	−	0.682573i	$-0.239139\pi$
$967$	43952.0	1.46163	0.730817	+	0.682573i	$0.239139\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	3936.88	0.130114	0.0650569	−	0.997882i	$-0.479277\pi$
$971$	3936.88	0.130114	0.0650569	+	0.997882i	$0.479277\pi$
$972$	0	0
$973$	−6160.00	−0.202960
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−12159.9	−0.398187	−0.199094	−	0.979980i	$-0.563800\pi$
$977$	−12159.9	−0.398187	−0.199094	+	0.979980i	$0.563800\pi$
$978$	0	0
$979$	3276.00	0.106947
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−39432.3	−1.27944	−0.639722	−	0.768606i	$-0.720951\pi$
$983$	−39432.3	−1.27944	−0.639722	+	0.768606i	$0.720951\pi$
$984$	0	0
$985$	18648.0	0.603223
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−12382.1	−0.398108
$990$	0	0
$991$	10760.0	0.344907	0.172453	−	0.985018i	$-0.444831\pi$
$991$	10760.0	0.344907	0.172453	+	0.985018i	$0.444831\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−72895.7	−2.32256
$996$	0	0
$997$	−9970.00	−0.316703	−0.158352	−	0.987383i	$-0.550618\pi$
$997$	−9970.00	−0.316703	−0.158352	+	0.987383i	$0.550618\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.4.a.f.1.1	✓	2
3.2	odd	2	inner	252.4.a.f.1.2	yes	2
4.3	odd	2		1008.4.a.bc.1.1		2
7.2	even	3		1764.4.k.v.361.2		4
7.3	odd	6		1764.4.k.u.1549.1		4
7.4	even	3		1764.4.k.v.1549.2		4
7.5	odd	6		1764.4.k.u.361.1		4
7.6	odd	2		1764.4.a.s.1.2		2
12.11	even	2		1008.4.a.bc.1.2		2
21.2	odd	6		1764.4.k.v.361.1		4
21.5	even	6		1764.4.k.u.361.2		4
21.11	odd	6		1764.4.k.v.1549.1		4
21.17	even	6		1764.4.k.u.1549.2		4
21.20	even	2		1764.4.a.s.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.4.a.f.1.1	✓	2	1.1	even	1	trivial
252.4.a.f.1.2	yes	2	3.2	odd	2	inner
1008.4.a.bc.1.1		2	4.3	odd	2
1008.4.a.bc.1.2		2	12.11	even	2
1764.4.a.s.1.1		2	21.20	even	2
1764.4.a.s.1.2		2	7.6	odd	2
1764.4.k.u.361.1		4	7.5	odd	6
1764.4.k.u.361.2		4	21.5	even	6
1764.4.k.u.1549.1		4	7.3	odd	6
1764.4.k.u.1549.2		4	21.17	even	6
1764.4.k.v.361.1		4	21.2	odd	6
1764.4.k.v.361.2		4	7.2	even	3
1764.4.k.v.1549.1		4	21.11	odd	6
1764.4.k.v.1549.2		4	7.4	even	3

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:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	252.a (trivial)

\(f(q)\)	\(=\)	\(q-15.8745 q^{5} +7.00000 q^{7} +O(q^{10})\)	\(q-15.8745 q^{5} +7.00000 q^{7} -15.8745 q^{11} +26.0000 q^{13} +79.3725 q^{17} +68.0000 q^{19} -47.6235 q^{23} +127.000 q^{25} +253.992 q^{29} +212.000 q^{31} -111.122 q^{35} +218.000 q^{37} -396.863 q^{41} +260.000 q^{43} -412.737 q^{47} +49.0000 q^{49} +476.235 q^{53} +252.000 q^{55} +285.741 q^{59} -322.000 q^{61} -412.737 q^{65} +356.000 q^{67} +1127.09 q^{71} -226.000 q^{73} -111.122 q^{77} +440.000 q^{79} +253.992 q^{83} -1260.00 q^{85} -206.369 q^{89} +182.000 q^{91} -1079.47 q^{95} -1330.00 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 14 q^{7}+O(q^{10}) \) 2 * q + 14 * q^7	\( 2 q + 14 q^{7} + 52 q^{13} + 136 q^{19} + 254 q^{25} + 424 q^{31} + 436 q^{37} + 520 q^{43} + 98 q^{49} + 504 q^{55} - 644 q^{61} + 712 q^{67} - 452 q^{73} + 880 q^{79} - 2520 q^{85} + 364 q^{91} - 2660 q^{97}+O(q^{100}) \) 2 * q + 14 * q^7 + 52 * q^13 + 136 * q^19 + 254 * q^25 + 424 * q^31 + 436 * q^37 + 520 * q^43 + 98 * q^49 + 504 * q^55 - 644 * q^61 + 712 * q^67 - 452 * q^73 + 880 * q^79 - 2520 * q^85 + 364 * q^91 - 2660 * q^97

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