LMFDB - Embedded newform 2352.4.a.cg.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2352,4,Mod(1,2352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2352, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2352.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2352 = 2^{4} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2352.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:	$138.772492334$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.57516.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 24x + 6 $ x^3 - x^2 - 24*x + 6
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 21)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$5.30829$ of defining polynomial
Character	$\chi$	$=$	2352.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} -5.56140 q^{5} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} -5.56140 q^{5} +9.00000 q^{9} +13.9174 q^{11} -38.6718 q^{13} +16.6842 q^{15} -43.4788 q^{17} -109.028 q^{19} +74.8778 q^{23} -94.0708 q^{25} -27.0000 q^{27} -72.3589 q^{29} +64.0431 q^{31} -41.7521 q^{33} +188.727 q^{37} +116.015 q^{39} +24.7923 q^{41} +243.881 q^{43} -50.0526 q^{45} -620.549 q^{47} +130.436 q^{51} -287.839 q^{53} -77.4001 q^{55} +327.083 q^{57} +525.051 q^{59} +383.436 q^{61} +215.069 q^{65} -198.117 q^{67} -224.634 q^{69} -785.432 q^{71} +331.141 q^{73} +282.213 q^{75} -437.647 q^{79} +81.0000 q^{81} +241.241 q^{83} +241.803 q^{85} +217.077 q^{87} -1585.54 q^{89} -192.129 q^{93} +606.347 q^{95} -79.2754 q^{97} +125.256 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 9 q^{3} - 11 q^{5} + 27 q^{9}+O(q^{10}) $ 3 * q - 9 * q^3 - 11 * q^5 + 27 * q^9	$ 3 q - 9 q^{3} - 11 q^{5} + 27 q^{9} - 35 q^{11} - 62 q^{13} + 33 q^{15} - 48 q^{17} - 202 q^{19} - 216 q^{23} + 130 q^{25} - 81 q^{27} + 53 q^{29} - 95 q^{31} + 105 q^{33} + 262 q^{37} + 186 q^{39} - 244 q^{41} - 360 q^{43} - 99 q^{45} - 210 q^{47} + 144 q^{51} + 393 q^{53} - 1031 q^{55} + 606 q^{57} + 1143 q^{59} + 70 q^{61} - 472 q^{65} + 628 q^{67} + 648 q^{69} - 318 q^{71} - 988 q^{73} - 390 q^{75} - 861 q^{79} + 243 q^{81} + 519 q^{83} + 1800 q^{85} - 159 q^{87} - 1766 q^{89} + 285 q^{93} + 736 q^{95} - 19 q^{97} - 315 q^{99}+O(q^{100}) $ 3 * q - 9 * q^3 - 11 * q^5 + 27 * q^9 - 35 * q^11 - 62 * q^13 + 33 * q^15 - 48 * q^17 - 202 * q^19 - 216 * q^23 + 130 * q^25 - 81 * q^27 + 53 * q^29 - 95 * q^31 + 105 * q^33 + 262 * q^37 + 186 * q^39 - 244 * q^41 - 360 * q^43 - 99 * q^45 - 210 * q^47 + 144 * q^51 + 393 * q^53 - 1031 * q^55 + 606 * q^57 + 1143 * q^59 + 70 * q^61 - 472 * q^65 + 628 * q^67 + 648 * q^69 - 318 * q^71 - 988 * q^73 - 390 * q^75 - 861 * q^79 + 243 * q^81 + 519 * q^83 + 1800 * q^85 - 159 * q^87 - 1766 * q^89 + 285 * q^93 + 736 * q^95 - 19 * q^97 - 315 * q^99

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
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	−5.56140	−0.497427	−0.248713	−	0.968577i	$-0.580008\pi$
$5$	−5.56140	−0.497427	−0.248713	+	0.968577i	$0.580008\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	13.9174	0.381477	0.190738	−	0.981641i	$-0.438912\pi$
$11$	13.9174	0.381477	0.190738	+	0.981641i	$0.438912\pi$
$12$	0	0
$13$	−38.6718	−0.825048	−0.412524	−	0.910947i	$-0.635353\pi$
$13$	−38.6718	−0.825048	−0.412524	+	0.910947i	$0.635353\pi$
$14$	0	0
$15$	16.6842	0.287189
$16$	0	0
$17$	−43.4788	−0.620303	−0.310152	−	0.950687i	$-0.600380\pi$
$17$	−43.4788	−0.620303	−0.310152	+	0.950687i	$0.600380\pi$
$18$	0	0
$19$	−109.028	−1.31646	−0.658228	−	0.752818i	$-0.728694\pi$
$19$	−109.028	−1.31646	−0.658228	+	0.752818i	$0.728694\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	74.8778	0.678831	0.339415	−	0.940637i	$-0.389771\pi$
$23$	74.8778	0.678831	0.339415	+	0.940637i	$0.389771\pi$
$24$	0	0
$25$	−94.0708	−0.752567
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	−72.3589	−0.463335	−0.231667	−	0.972795i	$-0.574418\pi$
$29$	−72.3589	−0.463335	−0.231667	+	0.972795i	$0.574418\pi$
$30$	0	0
$31$	64.0431	0.371048	0.185524	−	0.982640i	$-0.440602\pi$
$31$	64.0431	0.371048	0.185524	+	0.982640i	$0.440602\pi$
$32$	0	0
$33$	−41.7521	−0.220246
$34$	0	0
$35$	0	0
$36$	0	0
$37$	188.727	0.838556	0.419278	−	0.907858i	$-0.362283\pi$
$37$	188.727	0.838556	0.419278	+	0.907858i	$0.362283\pi$
$38$	0	0
$39$	116.015	0.476342
$40$	0	0
$41$	24.7923	0.0944367	0.0472184	−	0.998885i	$-0.484964\pi$
$41$	24.7923	0.0944367	0.0472184	+	0.998885i	$0.484964\pi$
$42$	0	0
$43$	243.881	0.864920	0.432460	−	0.901653i	$-0.357646\pi$
$43$	243.881	0.864920	0.432460	+	0.901653i	$0.357646\pi$
$44$	0	0
$45$	−50.0526	−0.165809
$46$	0	0
$47$	−620.549	−1.92588	−0.962940	−	0.269717i	$-0.913070\pi$
$47$	−620.549	−1.92588	−0.962940	+	0.269717i	$0.913070\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	130.436	0.358132
$52$	0	0
$53$	−287.839	−0.745995	−0.372997	−	0.927832i	$-0.621670\pi$
$53$	−287.839	−0.745995	−0.372997	+	0.927832i	$0.621670\pi$
$54$	0	0
$55$	−77.4001	−0.189757
$56$	0	0
$57$	327.083	0.760057
$58$	0	0
$59$	525.051	1.15857	0.579287	−	0.815124i	$-0.303331\pi$
$59$	525.051	1.15857	0.579287	+	0.815124i	$0.303331\pi$
$60$	0	0
$61$	383.436	0.804818	0.402409	−	0.915460i	$-0.368173\pi$
$61$	383.436	0.804818	0.402409	+	0.915460i	$0.368173\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	215.069	0.410401
$66$	0	0
$67$	−198.117	−0.361251	−0.180625	−	0.983552i	$-0.557812\pi$
$67$	−198.117	−0.361251	−0.180625	+	0.983552i	$0.557812\pi$
$68$	0	0
$69$	−224.634	−0.391923
$70$	0	0
$71$	−785.432	−1.31287	−0.656434	−	0.754384i	$-0.727936\pi$
$71$	−785.432	−1.31287	−0.656434	+	0.754384i	$0.727936\pi$
$72$	0	0
$73$	331.141	0.530919	0.265459	−	0.964122i	$-0.414476\pi$
$73$	331.141	0.530919	0.265459	+	0.964122i	$0.414476\pi$
$74$	0	0
$75$	282.213	0.434495
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−437.647	−0.623280	−0.311640	−	0.950200i	$-0.600878\pi$
$79$	−437.647	−0.623280	−0.311640	+	0.950200i	$0.600878\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	241.241	0.319032	0.159516	−	0.987195i	$-0.449007\pi$
$83$	241.241	0.319032	0.159516	+	0.987195i	$0.449007\pi$
$84$	0	0
$85$	241.803	0.308555
$86$	0	0
$87$	217.077	0.267506
$88$	0	0
$89$	−1585.54	−1.88840	−0.944198	−	0.329378i	$-0.893161\pi$
$89$	−1585.54	−1.88840	−0.944198	+	0.329378i	$0.893161\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−192.129	−0.214224
$94$	0	0
$95$	606.347	0.654841
$96$	0	0
$97$	−79.2754	−0.0829814	−0.0414907	−	0.999139i	$-0.513211\pi$
$97$	−79.2754	−0.0829814	−0.0414907	+	0.999139i	$0.513211\pi$
$98$	0	0
$99$	125.256	0.127159
$100$	0	0
$101$	−1154.97	−1.13786	−0.568931	−	0.822385i	$-0.692643\pi$
$101$	−1154.97	−1.13786	−0.568931	+	0.822385i	$0.692643\pi$
$102$	0	0
$103$	−1444.86	−1.38220	−0.691098	−	0.722761i	$-0.742873\pi$
$103$	−1444.86	−1.38220	−0.691098	+	0.722761i	$0.742873\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−990.960	−0.895325	−0.447662	−	0.894203i	$-0.647743\pi$
$107$	−990.960	−0.895325	−0.447662	+	0.894203i	$0.647743\pi$
$108$	0	0
$109$	1953.17	1.71633	0.858164	−	0.513376i	$-0.171605\pi$
$109$	1953.17	1.71633	0.858164	+	0.513376i	$0.171605\pi$
$110$	0	0
$111$	−566.182	−0.484141
$112$	0	0
$113$	672.882	0.560172	0.280086	−	0.959975i	$-0.409637\pi$
$113$	672.882	0.560172	0.280086	+	0.959975i	$0.409637\pi$
$114$	0	0
$115$	−416.426	−0.337669
$116$	0	0
$117$	−348.046	−0.275016
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−1137.31	−0.854475
$122$	0	0
$123$	−74.3769	−0.0545231
$124$	0	0
$125$	1218.34	0.871773
$126$	0	0
$127$	−175.815	−0.122843	−0.0614216	−	0.998112i	$-0.519563\pi$
$127$	−175.815	−0.122843	−0.0614216	+	0.998112i	$0.519563\pi$
$128$	0	0
$129$	−731.644	−0.499362
$130$	0	0
$131$	1125.93	0.750939	0.375470	−	0.926835i	$-0.377481\pi$
$131$	1125.93	0.750939	0.375470	+	0.926835i	$0.377481\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	150.158	0.0957298
$136$	0	0
$137$	1868.70	1.16536	0.582678	−	0.812703i	$-0.302005\pi$
$137$	1868.70	1.16536	0.582678	+	0.812703i	$0.302005\pi$
$138$	0	0
$139$	−2817.19	−1.71907	−0.859537	−	0.511074i	$-0.829248\pi$
$139$	−2817.19	−1.71907	−0.859537	+	0.511074i	$0.829248\pi$
$140$	0	0
$141$	1861.65	1.11191
$142$	0	0
$143$	−538.210	−0.314737
$144$	0	0
$145$	402.417	0.230475
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1800.33	0.989855	0.494928	−	0.868934i	$-0.335195\pi$
$149$	1800.33	0.989855	0.494928	+	0.868934i	$0.335195\pi$
$150$	0	0
$151$	452.984	0.244128	0.122064	−	0.992522i	$-0.461049\pi$
$151$	452.984	0.244128	0.122064	+	0.992522i	$0.461049\pi$
$152$	0	0
$153$	−391.309	−0.206768
$154$	0	0
$155$	−356.169	−0.184569
$156$	0	0
$157$	1863.66	0.947364	0.473682	−	0.880696i	$-0.342925\pi$
$157$	1863.66	0.947364	0.473682	+	0.880696i	$0.342925\pi$
$158$	0	0
$159$	863.517	0.430700
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−2321.14	−1.11537	−0.557686	−	0.830052i	$-0.688311\pi$
$163$	−2321.14	−1.11537	−0.557686	+	0.830052i	$0.688311\pi$
$164$	0	0
$165$	232.200	0.109556
$166$	0	0
$167$	3211.62	1.48816	0.744079	−	0.668092i	$-0.232889\pi$
$167$	3211.62	1.48816	0.744079	+	0.668092i	$0.232889\pi$
$168$	0	0
$169$	−701.494	−0.319296
$170$	0	0
$171$	−981.250	−0.438819
$172$	0	0
$173$	−214.277	−0.0941687	−0.0470844	−	0.998891i	$-0.514993\pi$
$173$	−214.277	−0.0941687	−0.0470844	+	0.998891i	$0.514993\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−1575.15	−0.668903
$178$	0	0
$179$	2437.22	1.01769	0.508845	−	0.860858i	$-0.330073\pi$
$179$	2437.22	1.01769	0.508845	+	0.860858i	$0.330073\pi$
$180$	0	0
$181$	248.631	0.102103	0.0510514	−	0.998696i	$-0.483743\pi$
$181$	248.631	0.102103	0.0510514	+	0.998696i	$0.483743\pi$
$182$	0	0
$183$	−1150.31	−0.464662
$184$	0	0
$185$	−1049.59	−0.417120
$186$	0	0
$187$	−605.110	−0.236631
$188$	0	0
$189$	0	0
$190$	0	0
$191$	4313.08	1.63394	0.816972	−	0.576677i	$-0.195651\pi$
$191$	4313.08	1.63394	0.816972	+	0.576677i	$0.195651\pi$
$192$	0	0
$193$	2060.85	0.768618	0.384309	−	0.923205i	$-0.374440\pi$
$193$	2060.85	0.768618	0.384309	+	0.923205i	$0.374440\pi$
$194$	0	0
$195$	−645.208	−0.236945
$196$	0	0
$197$	−1666.09	−0.602557	−0.301279	−	0.953536i	$-0.597413\pi$
$197$	−1666.09	−0.602557	−0.301279	+	0.953536i	$0.597413\pi$
$198$	0	0
$199$	−1087.53	−0.387403	−0.193702	−	0.981061i	$-0.562049\pi$
$199$	−1087.53	−0.387403	−0.193702	+	0.981061i	$0.562049\pi$
$200$	0	0
$201$	594.350	0.208568
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−137.880	−0.0469754
$206$	0	0
$207$	673.901	0.226277
$208$	0	0
$209$	−1517.38	−0.502198
$210$	0	0
$211$	4676.47	1.52579	0.762895	−	0.646522i	$-0.223777\pi$
$211$	4676.47	1.52579	0.762895	+	0.646522i	$0.223777\pi$
$212$	0	0
$213$	2356.29	0.757984
$214$	0	0
$215$	−1356.32	−0.430234
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−993.422	−0.306526
$220$	0	0
$221$	1681.40	0.511780
$222$	0	0
$223$	3246.03	0.974754	0.487377	−	0.873192i	$-0.337954\pi$
$223$	3246.03	0.974754	0.487377	+	0.873192i	$0.337954\pi$
$224$	0	0
$225$	−846.638	−0.250856
$226$	0	0
$227$	5138.16	1.50234	0.751171	−	0.660108i	$-0.229490\pi$
$227$	5138.16	1.50234	0.751171	+	0.660108i	$0.229490\pi$
$228$	0	0
$229$	−614.806	−0.177413	−0.0887064	−	0.996058i	$-0.528273\pi$
$229$	−614.806	−0.177413	−0.0887064	+	0.996058i	$0.528273\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	2827.42	0.794979	0.397490	−	0.917607i	$-0.369881\pi$
$233$	2827.42	0.794979	0.397490	+	0.917607i	$0.369881\pi$
$234$	0	0
$235$	3451.12	0.957984
$236$	0	0
$237$	1312.94	0.359851
$238$	0	0
$239$	3432.45	0.928983	0.464491	−	0.885578i	$-0.346237\pi$
$239$	3432.45	0.928983	0.464491	+	0.885578i	$0.346237\pi$
$240$	0	0
$241$	2636.11	0.704593	0.352296	−	0.935888i	$-0.385401\pi$
$241$	2636.11	0.704593	0.352296	+	0.935888i	$0.385401\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	4216.30	1.08614
$248$	0	0
$249$	−723.724	−0.184193
$250$	0	0
$251$	2057.57	0.517422	0.258711	−	0.965955i	$-0.416702\pi$
$251$	2057.57	0.517422	0.258711	+	0.965955i	$0.416702\pi$
$252$	0	0
$253$	1042.10	0.258958
$254$	0	0
$255$	−725.408	−0.178144
$256$	0	0
$257$	−2150.21	−0.521892	−0.260946	−	0.965353i	$-0.584034\pi$
$257$	−2150.21	−0.521892	−0.260946	+	0.965353i	$0.584034\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−651.230	−0.154445
$262$	0	0
$263$	4590.15	1.07620	0.538100	−	0.842881i	$-0.319142\pi$
$263$	4590.15	1.07620	0.538100	+	0.842881i	$0.319142\pi$
$264$	0	0
$265$	1600.79	0.371078
$266$	0	0
$267$	4756.63	1.09027
$268$	0	0
$269$	379.378	0.0859891	0.0429945	−	0.999075i	$-0.486310\pi$
$269$	379.378	0.0859891	0.0429945	+	0.999075i	$0.486310\pi$
$270$	0	0
$271$	−5368.84	−1.20345	−0.601723	−	0.798705i	$-0.705519\pi$
$271$	−5368.84	−1.20345	−0.601723	+	0.798705i	$0.705519\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1309.22	−0.287087
$276$	0	0
$277$	−4781.60	−1.03718	−0.518590	−	0.855023i	$-0.673543\pi$
$277$	−4781.60	−1.03718	−0.518590	+	0.855023i	$0.673543\pi$
$278$	0	0
$279$	576.388	0.123683
$280$	0	0
$281$	−2076.57	−0.440845	−0.220423	−	0.975404i	$-0.570744\pi$
$281$	−2076.57	−0.440845	−0.220423	+	0.975404i	$0.570744\pi$
$282$	0	0
$283$	2557.62	0.537224	0.268612	−	0.963248i	$-0.413435\pi$
$283$	2557.62	0.537224	0.268612	+	0.963248i	$0.413435\pi$
$284$	0	0
$285$	−1819.04	−0.378073
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−3022.60	−0.615224
$290$	0	0
$291$	237.826	0.0479093
$292$	0	0
$293$	−560.049	−0.111667	−0.0558335	−	0.998440i	$-0.517782\pi$
$293$	−560.049	−0.111667	−0.0558335	+	0.998440i	$0.517782\pi$
$294$	0	0
$295$	−2920.02	−0.576305
$296$	0	0
$297$	−375.769	−0.0734153
$298$	0	0
$299$	−2895.66	−0.560068
$300$	0	0
$301$	0	0
$302$	0	0
$303$	3464.92	0.656945
$304$	0	0
$305$	−2132.44	−0.400338
$306$	0	0
$307$	3653.02	0.679117	0.339558	−	0.940585i	$-0.389722\pi$
$307$	3653.02	0.679117	0.339558	+	0.940585i	$0.389722\pi$
$308$	0	0
$309$	4334.58	0.798011
$310$	0	0
$311$	3492.27	0.636747	0.318374	−	0.947965i	$-0.396863\pi$
$311$	3492.27	0.636747	0.318374	+	0.947965i	$0.396863\pi$
$312$	0	0
$313$	−8712.09	−1.57328	−0.786640	−	0.617412i	$-0.788181\pi$
$313$	−8712.09	−1.57328	−0.786640	+	0.617412i	$0.788181\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1940.33	0.343785	0.171892	−	0.985116i	$-0.445012\pi$
$317$	1940.33	0.343785	0.171892	+	0.985116i	$0.445012\pi$
$318$	0	0
$319$	−1007.05	−0.176752
$320$	0	0
$321$	2972.88	0.516916
$322$	0	0
$323$	4740.39	0.816602
$324$	0	0
$325$	3637.89	0.620904
$326$	0	0
$327$	−5859.51	−0.990922
$328$	0	0
$329$	0	0
$330$	0	0
$331$	5731.51	0.951759	0.475879	−	0.879510i	$-0.342130\pi$
$331$	5731.51	0.951759	0.475879	+	0.879510i	$0.342130\pi$
$332$	0	0
$333$	1698.55	0.279519
$334$	0	0
$335$	1101.81	0.179696
$336$	0	0
$337$	2403.74	0.388547	0.194273	−	0.980947i	$-0.437765\pi$
$337$	2403.74	0.388547	0.194273	+	0.980947i	$0.437765\pi$
$338$	0	0
$339$	−2018.65	−0.323416
$340$	0	0
$341$	891.312	0.141546
$342$	0	0
$343$	0	0
$344$	0	0
$345$	1249.28	0.194953
$346$	0	0
$347$	3336.44	0.516166	0.258083	−	0.966123i	$-0.416909\pi$
$347$	3336.44	0.516166	0.258083	+	0.966123i	$0.416909\pi$
$348$	0	0
$349$	2424.54	0.371870	0.185935	−	0.982562i	$-0.440469\pi$
$349$	2424.54	0.371870	0.185935	+	0.982562i	$0.440469\pi$
$350$	0	0
$351$	1044.14	0.158781
$352$	0	0
$353$	12403.1	1.87012	0.935059	−	0.354491i	$-0.115346\pi$
$353$	12403.1	1.87012	0.935059	+	0.354491i	$0.115346\pi$
$354$	0	0
$355$	4368.10	0.653055
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−1353.84	−0.199034	−0.0995168	−	0.995036i	$-0.531730\pi$
$359$	−1353.84	−0.199034	−0.0995168	+	0.995036i	$0.531730\pi$
$360$	0	0
$361$	5028.05	0.733059
$362$	0	0
$363$	3411.92	0.493332
$364$	0	0
$365$	−1841.61	−0.264093
$366$	0	0
$367$	−1378.06	−0.196006	−0.0980031	−	0.995186i	$-0.531246\pi$
$367$	−1378.06	−0.196006	−0.0980031	+	0.995186i	$0.531246\pi$
$368$	0	0
$369$	223.131	0.0314789
$370$	0	0
$371$	0	0
$372$	0	0
$373$	5456.92	0.757503	0.378752	−	0.925498i	$-0.376353\pi$
$373$	5456.92	0.757503	0.378752	+	0.925498i	$0.376353\pi$
$374$	0	0
$375$	−3655.02	−0.503319
$376$	0	0
$377$	2798.25	0.382273
$378$	0	0
$379$	−554.675	−0.0751761	−0.0375881	−	0.999293i	$-0.511967\pi$
$379$	−554.675	−0.0751761	−0.0375881	+	0.999293i	$0.511967\pi$
$380$	0	0
$381$	527.446	0.0709235
$382$	0	0
$383$	5860.66	0.781895	0.390948	−	0.920413i	$-0.372147\pi$
$383$	5860.66	0.781895	0.390948	+	0.920413i	$0.372147\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	2194.93	0.288307
$388$	0	0
$389$	7778.86	1.01389	0.506946	−	0.861978i	$-0.330774\pi$
$389$	7778.86	1.01389	0.506946	+	0.861978i	$0.330774\pi$
$390$	0	0
$391$	−3255.60	−0.421081
$392$	0	0
$393$	−3377.79	−0.433555
$394$	0	0
$395$	2433.93	0.310036
$396$	0	0
$397$	−8027.88	−1.01488	−0.507440	−	0.861687i	$-0.669408\pi$
$397$	−8027.88	−1.01488	−0.507440	+	0.861687i	$0.669408\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	779.980	0.0971330	0.0485665	−	0.998820i	$-0.484535\pi$
$401$	779.980	0.0971330	0.0485665	+	0.998820i	$0.484535\pi$
$402$	0	0
$403$	−2476.66	−0.306132
$404$	0	0
$405$	−450.473	−0.0552696
$406$	0	0
$407$	2626.59	0.319890
$408$	0	0
$409$	−14692.5	−1.77628	−0.888139	−	0.459575i	$-0.848002\pi$
$409$	−14692.5	−1.77628	−0.888139	+	0.459575i	$0.848002\pi$
$410$	0	0
$411$	−5606.10	−0.672819
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−1341.64	−0.158695
$416$	0	0
$417$	8451.58	0.992508
$418$	0	0
$419$	3370.31	0.392960	0.196480	−	0.980508i	$-0.437049\pi$
$419$	3370.31	0.392960	0.196480	+	0.980508i	$0.437049\pi$
$420$	0	0
$421$	15651.0	1.81184	0.905919	−	0.423450i	$-0.139181\pi$
$421$	15651.0	1.81184	0.905919	+	0.423450i	$0.139181\pi$
$422$	0	0
$423$	−5584.94	−0.641960
$424$	0	0
$425$	4090.08	0.466819
$426$	0	0
$427$	0	0
$428$	0	0
$429$	1614.63	0.181713
$430$	0	0
$431$	−4888.12	−0.546294	−0.273147	−	0.961972i	$-0.588064\pi$
$431$	−4888.12	−0.546294	−0.273147	+	0.961972i	$0.588064\pi$
$432$	0	0
$433$	5255.73	0.583313	0.291656	−	0.956523i	$-0.405794\pi$
$433$	5255.73	0.583313	0.291656	+	0.956523i	$0.405794\pi$
$434$	0	0
$435$	−1207.25	−0.133065
$436$	0	0
$437$	−8163.76	−0.893651
$438$	0	0
$439$	−824.977	−0.0896902	−0.0448451	−	0.998994i	$-0.514279\pi$
$439$	−824.977	−0.0896902	−0.0448451	+	0.998994i	$0.514279\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−13027.3	−1.39717	−0.698583	−	0.715529i	$-0.746186\pi$
$443$	−13027.3	−1.39717	−0.698583	+	0.715529i	$0.746186\pi$
$444$	0	0
$445$	8817.84	0.939339
$446$	0	0
$447$	−5400.98	−0.571493
$448$	0	0
$449$	16526.1	1.73700	0.868500	−	0.495689i	$-0.165084\pi$
$449$	16526.1	1.73700	0.868500	+	0.495689i	$0.165084\pi$
$450$	0	0
$451$	345.044	0.0360254
$452$	0	0
$453$	−1358.95	−0.140947
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−3710.82	−0.379836	−0.189918	−	0.981800i	$-0.560822\pi$
$457$	−3710.82	−0.379836	−0.189918	+	0.981800i	$0.560822\pi$
$458$	0	0
$459$	1173.93	0.119377
$460$	0	0
$461$	9714.00	0.981401	0.490701	−	0.871328i	$-0.336741\pi$
$461$	9714.00	0.981401	0.490701	+	0.871328i	$0.336741\pi$
$462$	0	0
$463$	43.2780	0.00434406	0.00217203	−	0.999998i	$-0.499309\pi$
$463$	43.2780	0.00434406	0.00217203	+	0.999998i	$0.499309\pi$
$464$	0	0
$465$	1068.51	0.106561
$466$	0	0
$467$	−1533.24	−0.151927	−0.0759633	−	0.997111i	$-0.524203\pi$
$467$	−1533.24	−0.151927	−0.0759633	+	0.997111i	$0.524203\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−5590.98	−0.546961
$472$	0	0
$473$	3394.19	0.329947
$474$	0	0
$475$	10256.3	0.990722
$476$	0	0
$477$	−2590.55	−0.248665
$478$	0	0
$479$	−7035.37	−0.671095	−0.335547	−	0.942023i	$-0.608921\pi$
$479$	−7035.37	−0.671095	−0.335547	+	0.942023i	$0.608921\pi$
$480$	0	0
$481$	−7298.42	−0.691849
$482$	0	0
$483$	0	0
$484$	0	0
$485$	440.882	0.0412772
$486$	0	0
$487$	15371.3	1.43026	0.715132	−	0.698989i	$-0.246366\pi$
$487$	15371.3	1.43026	0.715132	+	0.698989i	$0.246366\pi$
$488$	0	0
$489$	6963.41	0.643960
$490$	0	0
$491$	2393.35	0.219980	0.109990	−	0.993933i	$-0.464918\pi$
$491$	2393.35	0.219980	0.109990	+	0.993933i	$0.464918\pi$
$492$	0	0
$493$	3146.08	0.287408
$494$	0	0
$495$	−696.601	−0.0632523
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−693.520	−0.0622169	−0.0311084	−	0.999516i	$-0.509904\pi$
$499$	−693.520	−0.0622169	−0.0311084	+	0.999516i	$0.509904\pi$
$500$	0	0
$501$	−9634.85	−0.859188
$502$	0	0
$503$	−8646.95	−0.766498	−0.383249	−	0.923645i	$-0.625195\pi$
$503$	−8646.95	−0.766498	−0.383249	+	0.923645i	$0.625195\pi$
$504$	0	0
$505$	6423.27	0.566003
$506$	0	0
$507$	2104.48	0.184346
$508$	0	0
$509$	15500.9	1.34983	0.674916	−	0.737895i	$-0.264180\pi$
$509$	15500.9	1.34983	0.674916	+	0.737895i	$0.264180\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	2943.75	0.253352
$514$	0	0
$515$	8035.44	0.687541
$516$	0	0
$517$	−8636.41	−0.734678
$518$	0	0
$519$	642.831	0.0543683
$520$	0	0
$521$	−864.707	−0.0727131	−0.0363565	−	0.999339i	$-0.511575\pi$
$521$	−864.707	−0.0727131	−0.0363565	+	0.999339i	$0.511575\pi$
$522$	0	0
$523$	−6255.61	−0.523019	−0.261509	−	0.965201i	$-0.584220\pi$
$523$	−6255.61	−0.523019	−0.261509	+	0.965201i	$0.584220\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−2784.51	−0.230162
$528$	0	0
$529$	−6560.31	−0.539189
$530$	0	0
$531$	4725.46	0.386191
$532$	0	0
$533$	−958.762	−0.0779148
$534$	0	0
$535$	5511.13	0.445358
$536$	0	0
$537$	−7311.67	−0.587564
$538$	0	0
$539$	0	0
$540$	0	0
$541$	143.871	0.0114334	0.00571671	−	0.999984i	$-0.498180\pi$
$541$	143.871	0.0114334	0.00571671	+	0.999984i	$0.498180\pi$
$542$	0	0
$543$	−745.893	−0.0589490
$544$	0	0
$545$	−10862.4	−0.853747
$546$	0	0
$547$	−5455.65	−0.426448	−0.213224	−	0.977003i	$-0.568396\pi$
$547$	−5455.65	−0.426448	−0.213224	+	0.977003i	$0.568396\pi$
$548$	0	0
$549$	3450.92	0.268273
$550$	0	0
$551$	7889.13	0.609960
$552$	0	0
$553$	0	0
$554$	0	0
$555$	3148.76	0.240824
$556$	0	0
$557$	24809.9	1.88730	0.943652	−	0.330940i	$-0.107366\pi$
$557$	24809.9	1.88730	0.943652	+	0.330940i	$0.107366\pi$
$558$	0	0
$559$	−9431.33	−0.713600
$560$	0	0
$561$	1815.33	0.136619
$562$	0	0
$563$	16369.8	1.22541	0.612705	−	0.790312i	$-0.290081\pi$
$563$	16369.8	1.22541	0.612705	+	0.790312i	$0.290081\pi$
$564$	0	0
$565$	−3742.17	−0.278645
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−18450.6	−1.35938	−0.679691	−	0.733498i	$-0.737886\pi$
$569$	−18450.6	−1.35938	−0.679691	+	0.733498i	$0.737886\pi$
$570$	0	0
$571$	7108.69	0.520997	0.260499	−	0.965474i	$-0.416113\pi$
$571$	7108.69	0.520997	0.260499	+	0.965474i	$0.416113\pi$
$572$	0	0
$573$	−12939.2	−0.943358
$574$	0	0
$575$	−7043.82	−0.510865
$576$	0	0
$577$	−7594.17	−0.547919	−0.273960	−	0.961741i	$-0.588333\pi$
$577$	−7594.17	−0.547919	−0.273960	+	0.961741i	$0.588333\pi$
$578$	0	0
$579$	−6182.55	−0.443762
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−4005.96	−0.284580
$584$	0	0
$585$	1935.62	0.136800
$586$	0	0
$587$	−1763.34	−0.123988	−0.0619939	−	0.998077i	$-0.519746\pi$
$587$	−1763.34	−0.123988	−0.0619939	+	0.998077i	$0.519746\pi$
$588$	0	0
$589$	−6982.47	−0.488468
$590$	0	0
$591$	4998.26	0.347887
$592$	0	0
$593$	12316.1	0.852889	0.426445	−	0.904514i	$-0.359766\pi$
$593$	12316.1	0.852889	0.426445	+	0.904514i	$0.359766\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	3262.60	0.223667
$598$	0	0
$599$	8903.40	0.607317	0.303659	−	0.952781i	$-0.401792\pi$
$599$	8903.40	0.607317	0.303659	+	0.952781i	$0.401792\pi$
$600$	0	0
$601$	19157.1	1.30022	0.650112	−	0.759838i	$-0.274722\pi$
$601$	19157.1	1.30022	0.650112	+	0.759838i	$0.274722\pi$
$602$	0	0
$603$	−1783.05	−0.120417
$604$	0	0
$605$	6325.02	0.425039
$606$	0	0
$607$	−7569.93	−0.506184	−0.253092	−	0.967442i	$-0.581448\pi$
$607$	−7569.93	−0.506184	−0.253092	+	0.967442i	$0.581448\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	23997.7	1.58894
$612$	0	0
$613$	2907.13	0.191546	0.0957730	−	0.995403i	$-0.469468\pi$
$613$	2907.13	0.191546	0.0957730	+	0.995403i	$0.469468\pi$
$614$	0	0
$615$	413.640	0.0271212
$616$	0	0
$617$	−12510.9	−0.816320	−0.408160	−	0.912910i	$-0.633829\pi$
$617$	−12510.9	−0.816320	−0.408160	+	0.912910i	$0.633829\pi$
$618$	0	0
$619$	10065.6	0.653585	0.326792	−	0.945096i	$-0.394032\pi$
$619$	10065.6	0.653585	0.326792	+	0.945096i	$0.394032\pi$
$620$	0	0
$621$	−2021.70	−0.130641
$622$	0	0
$623$	0	0
$624$	0	0
$625$	4983.18	0.318923
$626$	0	0
$627$	4552.14	0.289944
$628$	0	0
$629$	−8205.63	−0.520159
$630$	0	0
$631$	25146.6	1.58648	0.793242	−	0.608907i	$-0.208392\pi$
$631$	25146.6	1.58648	0.793242	+	0.608907i	$0.208392\pi$
$632$	0	0
$633$	−14029.4	−0.880915
$634$	0	0
$635$	977.779	0.0611055
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−7068.88	−0.437622
$640$	0	0
$641$	−28958.9	−1.78441	−0.892206	−	0.451629i	$-0.850843\pi$
$641$	−28958.9	−1.78441	−0.892206	+	0.451629i	$0.850843\pi$
$642$	0	0
$643$	7341.90	0.450290	0.225145	−	0.974325i	$-0.427714\pi$
$643$	7341.90	0.450290	0.225145	+	0.974325i	$0.427714\pi$
$644$	0	0
$645$	4068.97	0.248396
$646$	0	0
$647$	6071.98	0.368955	0.184478	−	0.982837i	$-0.440941\pi$
$647$	6071.98	0.368955	0.184478	+	0.982837i	$0.440941\pi$
$648$	0	0
$649$	7307.33	0.441969
$650$	0	0
$651$	0	0
$652$	0	0
$653$	26262.1	1.57384	0.786920	−	0.617056i	$-0.211675\pi$
$653$	26262.1	1.57384	0.786920	+	0.617056i	$0.211675\pi$
$654$	0	0
$655$	−6261.75	−0.373537
$656$	0	0
$657$	2980.27	0.176973
$658$	0	0
$659$	−26130.1	−1.54459	−0.772296	−	0.635263i	$-0.780892\pi$
$659$	−26130.1	−1.54459	−0.772296	+	0.635263i	$0.780892\pi$
$660$	0	0
$661$	11925.5	0.701737	0.350868	−	0.936425i	$-0.385886\pi$
$661$	11925.5	0.701737	0.350868	+	0.936425i	$0.385886\pi$
$662$	0	0
$663$	−5044.20	−0.295476
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−5418.08	−0.314526
$668$	0	0
$669$	−9738.09	−0.562775
$670$	0	0
$671$	5336.42	0.307019
$672$	0	0
$673$	−6359.85	−0.364271	−0.182135	−	0.983273i	$-0.558301\pi$
$673$	−6359.85	−0.364271	−0.182135	+	0.983273i	$0.558301\pi$
$674$	0	0
$675$	2539.91	0.144832
$676$	0	0
$677$	−8561.61	−0.486041	−0.243020	−	0.970021i	$-0.578138\pi$
$677$	−8561.61	−0.486041	−0.243020	+	0.970021i	$0.578138\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−15414.5	−0.867377
$682$	0	0
$683$	6705.88	0.375686	0.187843	−	0.982199i	$-0.439850\pi$
$683$	6705.88	0.375686	0.187843	+	0.982199i	$0.439850\pi$
$684$	0	0
$685$	−10392.6	−0.579679
$686$	0	0
$687$	1844.42	0.102429
$688$	0	0
$689$	11131.2	0.615481
$690$	0	0
$691$	−25330.6	−1.39453	−0.697267	−	0.716811i	$-0.745601\pi$
$691$	−25330.6	−1.39453	−0.697267	+	0.716811i	$0.745601\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	15667.5	0.855113
$696$	0	0
$697$	−1077.94	−0.0585794
$698$	0	0
$699$	−8482.25	−0.458981
$700$	0	0
$701$	−27184.1	−1.46467	−0.732333	−	0.680947i	$-0.761568\pi$
$701$	−27184.1	−1.46467	−0.732333	+	0.680947i	$0.761568\pi$
$702$	0	0
$703$	−20576.5	−1.10392
$704$	0	0
$705$	−10353.4	−0.553092
$706$	0	0
$707$	0	0
$708$	0	0
$709$	16145.5	0.855228	0.427614	−	0.903961i	$-0.359354\pi$
$709$	16145.5	0.855228	0.427614	+	0.903961i	$0.359354\pi$
$710$	0	0
$711$	−3938.82	−0.207760
$712$	0	0
$713$	4795.41	0.251879
$714$	0	0
$715$	2993.20	0.156558
$716$	0	0
$717$	−10297.4	−0.536348
$718$	0	0
$719$	17297.5	0.897200	0.448600	−	0.893733i	$-0.351923\pi$
$719$	17297.5	0.897200	0.448600	+	0.893733i	$0.351923\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−7908.34	−0.406797
$724$	0	0
$725$	6806.86	0.348690
$726$	0	0
$727$	3514.71	0.179303	0.0896516	−	0.995973i	$-0.471425\pi$
$727$	3514.71	0.179303	0.0896516	+	0.995973i	$0.471425\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−10603.7	−0.536513
$732$	0	0
$733$	27511.2	1.38629	0.693144	−	0.720799i	$-0.256225\pi$
$733$	27511.2	1.38629	0.693144	+	0.720799i	$0.256225\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−2757.26	−0.137809
$738$	0	0
$739$	−16101.6	−0.801497	−0.400749	−	0.916188i	$-0.631250\pi$
$739$	−16101.6	−0.801497	−0.400749	+	0.916188i	$0.631250\pi$
$740$	0	0
$741$	−12648.9	−0.627083
$742$	0	0
$743$	−14682.4	−0.724961	−0.362480	−	0.931991i	$-0.618070\pi$
$743$	−14682.4	−0.724961	−0.362480	+	0.931991i	$0.618070\pi$
$744$	0	0
$745$	−10012.3	−0.492380
$746$	0	0
$747$	2171.17	0.106344
$748$	0	0
$749$	0	0
$750$	0	0
$751$	7273.06	0.353393	0.176696	−	0.984265i	$-0.443459\pi$
$751$	7273.06	0.353393	0.176696	+	0.984265i	$0.443459\pi$
$752$	0	0
$753$	−6172.72	−0.298734
$754$	0	0
$755$	−2519.23	−0.121436
$756$	0	0
$757$	8505.93	0.408393	0.204196	−	0.978930i	$-0.434542\pi$
$757$	8505.93	0.408393	0.204196	+	0.978930i	$0.434542\pi$
$758$	0	0
$759$	−3126.31	−0.149510
$760$	0	0
$761$	−14217.7	−0.677256	−0.338628	−	0.940920i	$-0.609963\pi$
$761$	−14217.7	−0.677256	−0.338628	+	0.940920i	$0.609963\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	2176.23	0.102852
$766$	0	0
$767$	−20304.7	−0.955879
$768$	0	0
$769$	−16379.1	−0.768068	−0.384034	−	0.923319i	$-0.625466\pi$
$769$	−16379.1	−0.768068	−0.384034	+	0.923319i	$0.625466\pi$
$770$	0	0
$771$	6450.62	0.301314
$772$	0	0
$773$	39896.7	1.85638	0.928192	−	0.372102i	$-0.121363\pi$
$773$	39896.7	1.85638	0.928192	+	0.372102i	$0.121363\pi$
$774$	0	0
$775$	−6024.59	−0.279238
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2703.05	−0.124322
$780$	0	0
$781$	−10931.1	−0.500829
$782$	0	0
$783$	1953.69	0.0891688
$784$	0	0
$785$	−10364.5	−0.471244
$786$	0	0
$787$	−33128.5	−1.50051	−0.750257	−	0.661146i	$-0.770070\pi$
$787$	−33128.5	−1.50051	−0.750257	+	0.661146i	$0.770070\pi$
$788$	0	0
$789$	−13770.5	−0.621345
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−14828.1	−0.664013
$794$	0	0
$795$	−4802.36	−0.214242
$796$	0	0
$797$	17851.5	0.793390	0.396695	−	0.917951i	$-0.370157\pi$
$797$	17851.5	0.793390	0.396695	+	0.917951i	$0.370157\pi$
$798$	0	0
$799$	26980.7	1.19463
$800$	0	0
$801$	−14269.9	−0.629465
$802$	0	0
$803$	4608.61	0.202533
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−1138.13	−0.0496458
$808$	0	0
$809$	−5057.03	−0.219772	−0.109886	−	0.993944i	$-0.535049\pi$
$809$	−5057.03	−0.219772	−0.109886	+	0.993944i	$0.535049\pi$
$810$	0	0
$811$	−17535.4	−0.759251	−0.379626	−	0.925140i	$-0.623947\pi$
$811$	−17535.4	−0.759251	−0.379626	+	0.925140i	$0.623947\pi$
$812$	0	0
$813$	16106.5	0.694810
$814$	0	0
$815$	12908.8	0.554815
$816$	0	0
$817$	−26589.8	−1.13863
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−18700.8	−0.794959	−0.397480	−	0.917611i	$-0.630115\pi$
$821$	−18700.8	−0.794959	−0.397480	+	0.917611i	$0.630115\pi$
$822$	0	0
$823$	−22222.6	−0.941230	−0.470615	−	0.882339i	$-0.655968\pi$
$823$	−22222.6	−0.941230	−0.470615	+	0.882339i	$0.655968\pi$
$824$	0	0
$825$	3927.66	0.165750
$826$	0	0
$827$	−25178.9	−1.05872	−0.529358	−	0.848399i	$-0.677567\pi$
$827$	−25178.9	−1.05872	−0.529358	+	0.848399i	$0.677567\pi$
$828$	0	0
$829$	−12278.9	−0.514432	−0.257216	−	0.966354i	$-0.582805\pi$
$829$	−12278.9	−0.514432	−0.257216	+	0.966354i	$0.582805\pi$
$830$	0	0
$831$	14344.8	0.598816
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−17861.1	−0.740249
$836$	0	0
$837$	−1729.16	−0.0714082
$838$	0	0
$839$	−25765.0	−1.06020	−0.530098	−	0.847936i	$-0.677845\pi$
$839$	−25765.0	−1.06020	−0.530098	+	0.847936i	$0.677845\pi$
$840$	0	0
$841$	−19153.2	−0.785321
$842$	0	0
$843$	6229.70	0.254522
$844$	0	0
$845$	3901.29	0.158826
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−7672.85	−0.310167
$850$	0	0
$851$	14131.5	0.569238
$852$	0	0
$853$	−37864.5	−1.51988	−0.759939	−	0.649995i	$-0.774771\pi$
$853$	−37864.5	−1.51988	−0.759939	+	0.649995i	$0.774771\pi$
$854$	0	0
$855$	5457.12	0.218280
$856$	0	0
$857$	29208.7	1.16424	0.582118	−	0.813104i	$-0.302224\pi$
$857$	29208.7	1.16424	0.582118	+	0.813104i	$0.302224\pi$
$858$	0	0
$859$	34902.9	1.38635	0.693173	−	0.720771i	$-0.256212\pi$
$859$	34902.9	1.38635	0.693173	+	0.720771i	$0.256212\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	13589.3	0.536021	0.268011	−	0.963416i	$-0.413634\pi$
$863$	13589.3	0.536021	0.268011	+	0.963416i	$0.413634\pi$
$864$	0	0
$865$	1191.68	0.0468420
$866$	0	0
$867$	9067.79	0.355200
$868$	0	0
$869$	−6090.89	−0.237767
$870$	0	0
$871$	7661.52	0.298049
$872$	0	0
$873$	−713.478	−0.0276605
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−2379.75	−0.0916288	−0.0458144	−	0.998950i	$-0.514588\pi$
$877$	−2379.75	−0.0916288	−0.0458144	+	0.998950i	$0.514588\pi$
$878$	0	0
$879$	1680.15	0.0644709
$880$	0	0
$881$	24235.5	0.926803	0.463401	−	0.886148i	$-0.346629\pi$
$881$	24235.5	0.926803	0.463401	+	0.886148i	$0.346629\pi$
$882$	0	0
$883$	9844.13	0.375177	0.187589	−	0.982248i	$-0.439933\pi$
$883$	9844.13	0.375177	0.187589	+	0.982248i	$0.439933\pi$
$884$	0	0
$885$	8760.06	0.332730
$886$	0	0
$887$	28609.9	1.08300	0.541502	−	0.840699i	$-0.317856\pi$
$887$	28609.9	1.08300	0.541502	+	0.840699i	$0.317856\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	1127.31	0.0423863
$892$	0	0
$893$	67657.0	2.53534
$894$	0	0
$895$	−13554.4	−0.506226
$896$	0	0
$897$	8686.98	0.323355
$898$	0	0
$899$	−4634.09	−0.171919
$900$	0	0
$901$	12514.9	0.462743
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1382.74	−0.0507886
$906$	0	0
$907$	−44578.3	−1.63197	−0.815986	−	0.578071i	$-0.803806\pi$
$907$	−44578.3	−1.63197	−0.815986	+	0.578071i	$0.803806\pi$
$908$	0	0
$909$	−10394.8	−0.379288
$910$	0	0
$911$	−45870.6	−1.66823	−0.834116	−	0.551589i	$-0.814022\pi$
$911$	−45870.6	−1.66823	−0.834116	+	0.551589i	$0.814022\pi$
$912$	0	0
$913$	3357.45	0.121703
$914$	0	0
$915$	6397.31	0.231135
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−31088.3	−1.11590	−0.557948	−	0.829876i	$-0.688411\pi$
$919$	−31088.3	−1.11590	−0.557948	+	0.829876i	$0.688411\pi$
$920$	0	0
$921$	−10959.1	−0.392088
$922$	0	0
$923$	30374.0	1.08318
$924$	0	0
$925$	−17753.7	−0.631069
$926$	0	0
$927$	−13003.7	−0.460732
$928$	0	0
$929$	42094.5	1.48663	0.743313	−	0.668943i	$-0.233253\pi$
$929$	42094.5	1.48663	0.743313	+	0.668943i	$0.233253\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−10476.8	−0.367626
$934$	0	0
$935$	3365.26	0.117707
$936$	0	0
$937$	44385.1	1.54749	0.773745	−	0.633497i	$-0.218381\pi$
$937$	44385.1	1.54749	0.773745	+	0.633497i	$0.218381\pi$
$938$	0	0
$939$	26136.3	0.908334
$940$	0	0
$941$	−40991.6	−1.42007	−0.710036	−	0.704165i	$-0.751321\pi$
$941$	−40991.6	−1.42007	−0.710036	+	0.704165i	$0.751321\pi$
$942$	0	0
$943$	1856.39	0.0641066
$944$	0	0
$945$	0	0
$946$	0	0
$947$	52622.4	1.80570	0.902851	−	0.429955i	$-0.141470\pi$
$947$	52622.4	1.80570	0.902851	+	0.429955i	$0.141470\pi$
$948$	0	0
$949$	−12805.8	−0.438034
$950$	0	0
$951$	−5820.99	−0.198484
$952$	0	0
$953$	−10798.1	−0.367035	−0.183517	−	0.983016i	$-0.558748\pi$
$953$	−10798.1	−0.367035	−0.183517	+	0.983016i	$0.558748\pi$
$954$	0	0
$955$	−23986.8	−0.812768
$956$	0	0
$957$	3021.14	0.102048
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−25689.5	−0.862324
$962$	0	0
$963$	−8918.64	−0.298442
$964$	0	0
$965$	−11461.2	−0.382331
$966$	0	0
$967$	−15648.9	−0.520408	−0.260204	−	0.965554i	$-0.583790\pi$
$967$	−15648.9	−0.520408	−0.260204	+	0.965554i	$0.583790\pi$
$968$	0	0
$969$	−14221.2	−0.471466
$970$	0	0
$971$	47259.8	1.56194	0.780968	−	0.624571i	$-0.214726\pi$
$971$	47259.8	1.56194	0.780968	+	0.624571i	$0.214726\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−10913.7	−0.358479
$976$	0	0
$977$	−48966.5	−1.60346	−0.801728	−	0.597689i	$-0.796086\pi$
$977$	−48966.5	−1.60346	−0.801728	+	0.597689i	$0.796086\pi$
$978$	0	0
$979$	−22066.6	−0.720380
$980$	0	0
$981$	17578.5	0.572109
$982$	0	0
$983$	−19111.3	−0.620097	−0.310049	−	0.950721i	$-0.600345\pi$
$983$	−19111.3	−0.620097	−0.310049	+	0.950721i	$0.600345\pi$
$984$	0	0
$985$	9265.77	0.299728
$986$	0	0
$987$	0	0
$988$	0	0
$989$	18261.3	0.587134
$990$	0	0
$991$	54102.5	1.73423	0.867115	−	0.498107i	$-0.165971\pi$
$991$	54102.5	1.73423	0.867115	+	0.498107i	$0.165971\pi$
$992$	0	0
$993$	−17194.5	−0.549498
$994$	0	0
$995$	6048.21	0.192705
$996$	0	0
$997$	−9192.80	−0.292015	−0.146008	−	0.989283i	$-0.546642\pi$
$997$	−9192.80	−0.292015	−0.146008	+	0.989283i	$0.546642\pi$
$998$	0	0
$999$	−5095.64	−0.161380

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.4.a.cg.1.2		3
4.3	odd	2		147.4.a.m.1.3		3
7.3	odd	6		336.4.q.k.289.2		6
7.5	odd	6		336.4.q.k.193.2		6
7.6	odd	2		2352.4.a.ci.1.2		3
12.11	even	2		441.4.a.t.1.1		3
28.3	even	6		21.4.e.b.16.1	yes	6
28.11	odd	6		147.4.e.n.79.1		6
28.19	even	6		21.4.e.b.4.1	✓	6
28.23	odd	6		147.4.e.n.67.1		6
28.27	even	2		147.4.a.l.1.3		3
84.11	even	6		441.4.e.w.226.3		6
84.23	even	6		441.4.e.w.361.3		6
84.47	odd	6		63.4.e.c.46.3		6
84.59	odd	6		63.4.e.c.37.3		6
84.83	odd	2		441.4.a.s.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.4.e.b.4.1	✓	6	28.19	even	6
21.4.e.b.16.1	yes	6	28.3	even	6
63.4.e.c.37.3		6	84.59	odd	6
63.4.e.c.46.3		6	84.47	odd	6
147.4.a.l.1.3		3	28.27	even	2
147.4.a.m.1.3		3	4.3	odd	2
147.4.e.n.67.1		6	28.23	odd	6
147.4.e.n.79.1		6	28.11	odd	6
336.4.q.k.193.2		6	7.5	odd	6
336.4.q.k.289.2		6	7.3	odd	6
441.4.a.s.1.1		3	84.83	odd	2
441.4.a.t.1.1		3	12.11	even	2
441.4.e.w.226.3		6	84.11	even	6
441.4.e.w.361.3		6	84.23	even	6
2352.4.a.cg.1.2		3	1.1	even	1	trivial
2352.4.a.ci.1.2		3	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2352.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} -5.56140 q^{5} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} -5.56140 q^{5} +9.00000 q^{9} +13.9174 q^{11} -38.6718 q^{13} +16.6842 q^{15} -43.4788 q^{17} -109.028 q^{19} +74.8778 q^{23} -94.0708 q^{25} -27.0000 q^{27} -72.3589 q^{29} +64.0431 q^{31} -41.7521 q^{33} +188.727 q^{37} +116.015 q^{39} +24.7923 q^{41} +243.881 q^{43} -50.0526 q^{45} -620.549 q^{47} +130.436 q^{51} -287.839 q^{53} -77.4001 q^{55} +327.083 q^{57} +525.051 q^{59} +383.436 q^{61} +215.069 q^{65} -198.117 q^{67} -224.634 q^{69} -785.432 q^{71} +331.141 q^{73} +282.213 q^{75} -437.647 q^{79} +81.0000 q^{81} +241.241 q^{83} +241.803 q^{85} +217.077 q^{87} -1585.54 q^{89} -192.129 q^{93} +606.347 q^{95} -79.2754 q^{97} +125.256 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 9 q^{3} - 11 q^{5} + 27 q^{9}+O(q^{10}) \) 3 * q - 9 * q^3 - 11 * q^5 + 27 * q^9	\( 3 q - 9 q^{3} - 11 q^{5} + 27 q^{9} - 35 q^{11} - 62 q^{13} + 33 q^{15} - 48 q^{17} - 202 q^{19} - 216 q^{23} + 130 q^{25} - 81 q^{27} + 53 q^{29} - 95 q^{31} + 105 q^{33} + 262 q^{37} + 186 q^{39} - 244 q^{41} - 360 q^{43} - 99 q^{45} - 210 q^{47} + 144 q^{51} + 393 q^{53} - 1031 q^{55} + 606 q^{57} + 1143 q^{59} + 70 q^{61} - 472 q^{65} + 628 q^{67} + 648 q^{69} - 318 q^{71} - 988 q^{73} - 390 q^{75} - 861 q^{79} + 243 q^{81} + 519 q^{83} + 1800 q^{85} - 159 q^{87} - 1766 q^{89} + 285 q^{93} + 736 q^{95} - 19 q^{97} - 315 q^{99}+O(q^{100}) \) 3 * q - 9 * q^3 - 11 * q^5 + 27 * q^9 - 35 * q^11 - 62 * q^13 + 33 * q^15 - 48 * q^17 - 202 * q^19 - 216 * q^23 + 130 * q^25 - 81 * q^27 + 53 * q^29 - 95 * q^31 + 105 * q^33 + 262 * q^37 + 186 * q^39 - 244 * q^41 - 360 * q^43 - 99 * q^45 - 210 * q^47 + 144 * q^51 + 393 * q^53 - 1031 * q^55 + 606 * q^57 + 1143 * q^59 + 70 * q^61 - 472 * q^65 + 628 * q^67 + 648 * q^69 - 318 * q^71 - 988 * q^73 - 390 * q^75 - 861 * q^79 + 243 * q^81 + 519 * q^83 + 1800 * q^85 - 159 * q^87 - 1766 * q^89 + 285 * q^93 + 736 * q^95 - 19 * q^97 - 315 * q^99

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