LMFDB - Embedded newform 1575.4.a.bk.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1575,4,Mod(1,1575)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1575.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1575, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1575 = 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1575.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,16,0,0,28,-9,0,0,-21] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$92.9280082590$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 24x^{2} - 3x + 46 $ x^4 - 24x^2 - 3x + 46
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 525)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.52801$ of defining polynomial
Character	$\chi$	$=$	1575.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.52801 q^{2} -5.66519 q^{4} +7.00000 q^{7} -20.8805 q^{8} -51.3481 q^{11} -87.2268 q^{13} +10.6960 q^{14} +13.4160 q^{16} -80.6781 q^{17} -29.8288 q^{19} -78.4602 q^{22} -1.71385 q^{23} -133.283 q^{26} -39.6564 q^{28} +204.810 q^{29} -150.176 q^{31} +187.544 q^{32} -123.277 q^{34} +366.397 q^{37} -45.5786 q^{38} -176.696 q^{41} +394.793 q^{43} +290.897 q^{44} -2.61877 q^{46} -507.757 q^{47} +49.0000 q^{49} +494.157 q^{52} +149.869 q^{53} -146.164 q^{56} +312.952 q^{58} -463.249 q^{59} +380.956 q^{61} -229.471 q^{62} +179.240 q^{64} -797.666 q^{67} +457.057 q^{68} +220.174 q^{71} +1013.61 q^{73} +559.857 q^{74} +168.986 q^{76} -359.437 q^{77} -111.805 q^{79} -269.993 q^{82} -853.761 q^{83} +603.247 q^{86} +1072.17 q^{88} +935.860 q^{89} -610.588 q^{91} +9.70929 q^{92} -775.857 q^{94} +783.811 q^{97} +74.8723 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 16 q^{4} + 28 q^{7} - 9 q^{8} - 21 q^{11} + 5 q^{13} + 72 q^{16} - 99 q^{17} + 72 q^{19} + 221 q^{22} - 102 q^{23} - 129 q^{26} + 112 q^{28} + 240 q^{29} + 351 q^{31} - 72 q^{32} - 285 q^{34} + 399 q^{37}+ \cdots - 372 q^{97}+O(q^{100}) $ 4 * q + 16 * q^4 + 28 * q^7 - 9 * q^8 - 21 * q^11 + 5 * q^13 + 72 * q^16 - 99 * q^17 + 72 * q^19 + 221 * q^22 - 102 * q^23 - 129 * q^26 + 112 * q^28 + 240 * q^29 + 351 * q^31 - 72 * q^32 - 285 * q^34 + 399 * q^37 - 324 * q^38 - 381 * q^41 + 460 * q^43 + 975 * q^44 + 550 * q^46 - 60 * q^47 + 196 * q^49 + 223 * q^52 - 873 * q^53 - 63 * q^56 + 1408 * q^58 + 855 * q^59 + 687 * q^61 + 477 * q^62 - 1285 * q^64 - 503 * q^67 - 1725 * q^68 + 681 * q^71 + 1228 * q^73 + 3369 * q^74 + 1910 * q^76 - 147 * q^77 + 345 * q^79 - 495 * q^82 - 1509 * q^83 + 540 * q^86 + 2266 * q^88 + 198 * q^89 + 35 * q^91 - 2994 * q^92 - 1732 * q^94 - 372 * q^97

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 $

$2$	1.52801	0.540232	0.270116	−	0.962828i	$-0.412938\pi$
$2$	1.52801	0.540232	0.270116	+	0.962828i	$0.412938\pi$
$3$	0	0
$3$	0	0
$4$	−5.66519	−0.708149
$5$	0	0
$5$	0	0
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	−20.8805	−0.922797
$9$	0	0
$10$	0	0
$11$	−51.3481	−1.40746	−0.703729	−	0.710469i	$-0.748483\pi$
$11$	−51.3481	−1.40746	−0.703729	+	0.710469i	$0.748483\pi$
$12$	0	0
$13$	−87.2268	−1.86095	−0.930476	−	0.366353i	$-0.880606\pi$
$13$	−87.2268	−1.86095	−0.930476	+	0.366353i	$0.880606\pi$
$14$	10.6960	0.204189
$15$	0	0
$16$	13.4160	0.209625
$17$	−80.6781	−1.15102	−0.575509	−	0.817795i	$-0.695196\pi$
$17$	−80.6781	−1.15102	−0.575509	+	0.817795i	$0.695196\pi$
$18$	0	0
$19$	−29.8288	−0.360168	−0.180084	−	0.983651i	$-0.557637\pi$
$19$	−29.8288	−0.360168	−0.180084	+	0.983651i	$0.557637\pi$
$20$	0	0
$21$	0	0
$22$	−78.4602	−0.760354
$23$	−1.71385	−0.0155375	−0.00776874	−	0.999970i	$-0.502473\pi$
$23$	−1.71385	−0.0155375	−0.00776874	+	0.999970i	$0.502473\pi$
$24$	0	0
$25$	0	0
$26$	−133.283	−1.00535
$27$	0	0
$28$	−39.6564	−0.267655
$29$	204.810	1.31146	0.655730	−	0.754996i	$-0.272361\pi$
$29$	204.810	1.31146	0.655730	+	0.754996i	$0.272361\pi$
$30$	0	0
$31$	−150.176	−0.870080	−0.435040	−	0.900411i	$-0.643266\pi$
$31$	−150.176	−0.870080	−0.435040	+	0.900411i	$0.643266\pi$
$32$	187.544	1.03604
$33$	0	0
$34$	−123.277	−0.621817
$35$	0	0
$36$	0	0
$37$	366.397	1.62798	0.813990	−	0.580879i	$-0.197291\pi$
$37$	366.397	1.62798	0.813990	+	0.580879i	$0.197291\pi$
$38$	−45.5786	−0.194575
$39$	0	0
$40$	0	0
$41$	−176.696	−0.673057	−0.336528	−	0.941673i	$-0.609253\pi$
$41$	−176.696	−0.673057	−0.336528	+	0.941673i	$0.609253\pi$
$42$	0	0
$43$	394.793	1.40013	0.700063	−	0.714081i	$-0.253155\pi$
$43$	394.793	1.40013	0.700063	+	0.714081i	$0.253155\pi$
$44$	290.897	0.996690
$45$	0	0
$46$	−2.61877	−0.00839385
$47$	−507.757	−1.57583	−0.787915	−	0.615784i	$-0.788839\pi$
$47$	−507.757	−1.57583	−0.787915	+	0.615784i	$0.788839\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	0	0
$52$	494.157	1.31783
$53$	149.869	0.388416	0.194208	−	0.980960i	$-0.437786\pi$
$53$	149.869	0.388416	0.194208	+	0.980960i	$0.437786\pi$
$54$	0	0
$55$	0	0
$56$	−146.164	−0.348785
$57$	0	0
$58$	312.952	0.708493
$59$	−463.249	−1.02220	−0.511101	−	0.859521i	$-0.670762\pi$
$59$	−463.249	−1.02220	−0.511101	+	0.859521i	$0.670762\pi$
$60$	0	0
$61$	380.956	0.799613	0.399807	−	0.916600i	$-0.369077\pi$
$61$	380.956	0.799613	0.399807	+	0.916600i	$0.369077\pi$
$62$	−229.471	−0.470045
$63$	0	0
$64$	179.240	0.350079
$65$	0	0
$66$	0	0
$67$	−797.666	−1.45448	−0.727242	−	0.686381i	$-0.759198\pi$
$67$	−797.666	−1.45448	−0.727242	+	0.686381i	$0.759198\pi$
$68$	457.057	0.815093
$69$	0	0
$70$	0	0
$71$	220.174	0.368025	0.184013	−	0.982924i	$-0.441091\pi$
$71$	220.174	0.368025	0.184013	+	0.982924i	$0.441091\pi$
$72$	0	0
$73$	1013.61	1.62512	0.812559	−	0.582878i	$-0.198074\pi$
$73$	1013.61	1.62512	0.812559	+	0.582878i	$0.198074\pi$
$74$	559.857	0.879487
$75$	0	0
$76$	168.986	0.255053
$77$	−359.437	−0.531969
$78$	0	0
$79$	−111.805	−0.159228	−0.0796140	−	0.996826i	$-0.525369\pi$
$79$	−111.805	−0.159228	−0.0796140	+	0.996826i	$0.525369\pi$
$80$	0	0
$81$	0	0
$82$	−269.993	−0.363607
$83$	−853.761	−1.12907	−0.564533	−	0.825411i	$-0.690944\pi$
$83$	−853.761	−1.12907	−0.564533	+	0.825411i	$0.690944\pi$
$84$	0	0
$85$	0	0
$86$	603.247	0.756393
$87$	0	0
$88$	1072.17	1.29880
$89$	935.860	1.11462	0.557309	−	0.830305i	$-0.311834\pi$
$89$	935.860	1.11462	0.557309	+	0.830305i	$0.311834\pi$
$90$	0	0
$91$	−610.588	−0.703374
$92$	9.70929	0.0110029
$93$	0	0
$94$	−775.857	−0.851314
$95$	0	0
$96$	0	0
$97$	783.811	0.820453	0.410227	−	0.911984i	$-0.365450\pi$
$97$	783.811	0.820453	0.410227	+	0.911984i	$0.365450\pi$
$98$	74.8723	0.0771760
$99$	0	0
$100$	0	0
$101$	−1805.17	−1.77842	−0.889212	−	0.457495i	$-0.848747\pi$
$101$	−1805.17	−1.77842	−0.889212	+	0.457495i	$0.848747\pi$
$102$	0	0
$103$	578.608	0.553514	0.276757	−	0.960940i	$-0.410740\pi$
$103$	578.608	0.553514	0.276757	+	0.960940i	$0.410740\pi$
$104$	1821.34	1.71728
$105$	0	0
$106$	229.000	0.209835
$107$	214.800	0.194070	0.0970349	−	0.995281i	$-0.469064\pi$
$107$	214.800	0.194070	0.0970349	+	0.995281i	$0.469064\pi$
$108$	0	0
$109$	812.409	0.713896	0.356948	−	0.934124i	$-0.383817\pi$
$109$	812.409	0.713896	0.356948	+	0.934124i	$0.383817\pi$
$110$	0	0
$111$	0	0
$112$	93.9119	0.0792307
$113$	−1596.30	−1.32891	−0.664457	−	0.747326i	$-0.731337\pi$
$113$	−1596.30	−1.32891	−0.664457	+	0.747326i	$0.731337\pi$
$114$	0	0
$115$	0	0
$116$	−1160.29	−0.928709
$117$	0	0
$118$	−707.848	−0.552227
$119$	−564.747	−0.435044
$120$	0	0
$121$	1305.63	0.980936
$122$	582.103	0.431977
$123$	0	0
$124$	850.778	0.616146
$125$	0	0
$126$	0	0
$127$	1469.71	1.02690	0.513449	−	0.858120i	$-0.328368\pi$
$127$	1469.71	1.02690	0.513449	+	0.858120i	$0.328368\pi$
$128$	−1226.47	−0.846919
$129$	0	0
$130$	0	0
$131$	1217.36	0.811919	0.405959	−	0.913891i	$-0.366937\pi$
$131$	1217.36	0.811919	0.405959	+	0.913891i	$0.366937\pi$
$132$	0	0
$133$	−208.802	−0.136131
$134$	−1218.84	−0.785759
$135$	0	0
$136$	1684.60	1.06216
$137$	−1637.23	−1.02101	−0.510503	−	0.859876i	$-0.670541\pi$
$137$	−1637.23	−1.02101	−0.510503	+	0.859876i	$0.670541\pi$
$138$	0	0
$139$	44.7736	0.0273212	0.0136606	−	0.999907i	$-0.495652\pi$
$139$	44.7736	0.0273212	0.0136606	+	0.999907i	$0.495652\pi$
$140$	0	0
$141$	0	0
$142$	336.427	0.198819
$143$	4478.93	2.61921
$144$	0	0
$145$	0	0
$146$	1548.80	0.877941
$147$	0	0
$148$	−2075.71	−1.15285
$149$	2153.29	1.18392	0.591960	−	0.805967i	$-0.298354\pi$
$149$	2153.29	1.18392	0.591960	+	0.805967i	$0.298354\pi$
$150$	0	0
$151$	−3037.34	−1.63692	−0.818461	−	0.574562i	$-0.805172\pi$
$151$	−3037.34	−1.63692	−0.818461	+	0.574562i	$0.805172\pi$
$152$	622.841	0.332362
$153$	0	0
$154$	−549.222	−0.287387
$155$	0	0
$156$	0	0
$157$	−1353.92	−0.688245	−0.344122	−	0.938925i	$-0.611823\pi$
$157$	−1353.92	−0.688245	−0.344122	+	0.938925i	$0.611823\pi$
$158$	−170.838	−0.0860201
$159$	0	0
$160$	0	0
$161$	−11.9969	−0.00587262
$162$	0	0
$163$	3174.14	1.52526	0.762631	−	0.646834i	$-0.223907\pi$
$163$	3174.14	1.52526	0.762631	+	0.646834i	$0.223907\pi$
$164$	1001.02	0.476625
$165$	0	0
$166$	−1304.55	−0.609957
$167$	999.272	0.463030	0.231515	−	0.972831i	$-0.425632\pi$
$167$	999.272	0.463030	0.231515	+	0.972831i	$0.425632\pi$
$168$	0	0
$169$	5411.52	2.46314
$170$	0	0
$171$	0	0
$172$	−2236.58	−0.991499
$173$	1208.55	0.531122	0.265561	−	0.964094i	$-0.414443\pi$
$173$	1208.55	0.531122	0.265561	+	0.964094i	$0.414443\pi$
$174$	0	0
$175$	0	0
$176$	−688.885	−0.295038
$177$	0	0
$178$	1430.00	0.602152
$179$	894.985	0.373711	0.186856	−	0.982387i	$-0.440170\pi$
$179$	894.985	0.373711	0.186856	+	0.982387i	$0.440170\pi$
$180$	0	0
$181$	2030.68	0.833917	0.416959	−	0.908925i	$-0.363096\pi$
$181$	2030.68	0.833917	0.416959	+	0.908925i	$0.363096\pi$
$182$	−932.983	−0.379985
$183$	0	0
$184$	35.7861	0.0143379
$185$	0	0
$186$	0	0
$187$	4142.67	1.62001
$188$	2876.54	1.11592
$189$	0	0
$190$	0	0
$191$	4527.59	1.71521	0.857604	−	0.514311i	$-0.171952\pi$
$191$	4527.59	1.71521	0.857604	+	0.514311i	$0.171952\pi$
$192$	0	0
$193$	−2539.92	−0.947294	−0.473647	−	0.880715i	$-0.657063\pi$
$193$	−2539.92	−0.947294	−0.473647	+	0.880715i	$0.657063\pi$
$194$	1197.67	0.443235
$195$	0	0
$196$	−277.595	−0.101164
$197$	−2344.75	−0.848002	−0.424001	−	0.905662i	$-0.639375\pi$
$197$	−2344.75	−0.848002	−0.424001	+	0.905662i	$0.639375\pi$
$198$	0	0
$199$	3253.14	1.15884	0.579420	−	0.815029i	$-0.303279\pi$
$199$	3253.14	1.15884	0.579420	+	0.815029i	$0.303279\pi$
$200$	0	0
$201$	0	0
$202$	−2758.31	−0.960762
$203$	1433.67	0.495685
$204$	0	0
$205$	0	0
$206$	884.117	0.299026
$207$	0	0
$208$	−1170.23	−0.390101
$209$	1531.65	0.506922
$210$	0	0
$211$	2528.43	0.824949	0.412475	−	0.910969i	$-0.364665\pi$
$211$	2528.43	0.824949	0.412475	+	0.910969i	$0.364665\pi$
$212$	−849.035	−0.275056
$213$	0	0
$214$	328.215	0.104843
$215$	0	0
$216$	0	0
$217$	−1051.23	−0.328859
$218$	1241.37	0.385669
$219$	0	0
$220$	0	0
$221$	7037.30	2.14199
$222$	0	0
$223$	−3930.00	−1.18015	−0.590073	−	0.807350i	$-0.700901\pi$
$223$	−3930.00	−1.18015	−0.590073	+	0.807350i	$0.700901\pi$
$224$	1312.81	0.391587
$225$	0	0
$226$	−2439.16	−0.717922
$227$	−512.698	−0.149907	−0.0749536	−	0.997187i	$-0.523881\pi$
$227$	−512.698	−0.149907	−0.0749536	+	0.997187i	$0.523881\pi$
$228$	0	0
$229$	1865.41	0.538296	0.269148	−	0.963099i	$-0.413258\pi$
$229$	1865.41	0.538296	0.269148	+	0.963099i	$0.413258\pi$
$230$	0	0
$231$	0	0
$232$	−4276.55	−1.21021
$233$	1509.17	0.424329	0.212165	−	0.977234i	$-0.431949\pi$
$233$	1509.17	0.424329	0.212165	+	0.977234i	$0.431949\pi$
$234$	0	0
$235$	0	0
$236$	2624.40	0.723872
$237$	0	0
$238$	−862.937	−0.235025
$239$	5852.90	1.58407	0.792035	−	0.610476i	$-0.209022\pi$
$239$	5852.90	1.58407	0.792035	+	0.610476i	$0.209022\pi$
$240$	0	0
$241$	−3050.88	−0.815454	−0.407727	−	0.913104i	$-0.633678\pi$
$241$	−3050.88	−0.815454	−0.407727	+	0.913104i	$0.633678\pi$
$242$	1995.01	0.529933
$243$	0	0
$244$	−2158.19	−0.566246
$245$	0	0
$246$	0	0
$247$	2601.87	0.670256
$248$	3135.76	0.802907
$249$	0	0
$250$	0	0
$251$	−5905.16	−1.48498	−0.742491	−	0.669856i	$-0.766356\pi$
$251$	−5905.16	−1.48498	−0.742491	+	0.669856i	$0.766356\pi$
$252$	0	0
$253$	88.0029	0.0218684
$254$	2245.73	0.554763
$255$	0	0
$256$	−3307.98	−0.807612
$257$	−544.470	−0.132152	−0.0660761	−	0.997815i	$-0.521048\pi$
$257$	−544.470	−0.132152	−0.0660761	+	0.997815i	$0.521048\pi$
$258$	0	0
$259$	2564.78	0.615319
$260$	0	0
$261$	0	0
$262$	1860.14	0.438625
$263$	1706.22	0.400039	0.200019	−	0.979792i	$-0.435900\pi$
$263$	1706.22	0.400039	0.200019	+	0.979792i	$0.435900\pi$
$264$	0	0
$265$	0	0
$266$	−319.051	−0.0735423
$267$	0	0
$268$	4518.93	1.02999
$269$	−289.872	−0.0657019	−0.0328509	−	0.999460i	$-0.510459\pi$
$269$	−289.872	−0.0657019	−0.0328509	+	0.999460i	$0.510459\pi$
$270$	0	0
$271$	−1408.35	−0.315687	−0.157843	−	0.987464i	$-0.550454\pi$
$271$	−1408.35	−0.315687	−0.157843	+	0.987464i	$0.550454\pi$
$272$	−1082.38	−0.241282
$273$	0	0
$274$	−2501.70	−0.551580
$275$	0	0
$276$	0	0
$277$	−6467.17	−1.40280	−0.701398	−	0.712770i	$-0.747440\pi$
$277$	−6467.17	−1.40280	−0.701398	+	0.712770i	$0.747440\pi$
$278$	68.4144	0.0147598
$279$	0	0
$280$	0	0
$281$	4271.00	0.906714	0.453357	−	0.891329i	$-0.350226\pi$
$281$	4271.00	0.906714	0.453357	+	0.891329i	$0.350226\pi$
$282$	0	0
$283$	−3761.13	−0.790021	−0.395011	−	0.918677i	$-0.629259\pi$
$283$	−3761.13	−0.790021	−0.395011	+	0.918677i	$0.629259\pi$
$284$	−1247.33	−0.260617
$285$	0	0
$286$	6843.84	1.41498
$287$	−1236.87	−0.254392
$288$	0	0
$289$	1595.96	0.324843
$290$	0	0
$291$	0	0
$292$	−5742.28	−1.15083
$293$	−468.982	−0.0935093	−0.0467547	−	0.998906i	$-0.514888\pi$
$293$	−468.982	−0.0935093	−0.0467547	+	0.998906i	$0.514888\pi$
$294$	0	0
$295$	0	0
$296$	−7650.56	−1.50230
$297$	0	0
$298$	3290.24	0.639592
$299$	149.494	0.0289145
$300$	0	0
$301$	2763.55	0.529198
$302$	−4641.08	−0.884318
$303$	0	0
$304$	−400.183	−0.0755002
$305$	0	0
$306$	0	0
$307$	4130.28	0.767841	0.383921	−	0.923366i	$-0.374574\pi$
$307$	4130.28	0.767841	0.383921	+	0.923366i	$0.374574\pi$
$308$	2036.28	0.376713
$309$	0	0
$310$	0	0
$311$	3179.75	0.579765	0.289883	−	0.957062i	$-0.406384\pi$
$311$	3179.75	0.579765	0.289883	+	0.957062i	$0.406384\pi$
$312$	0	0
$313$	−4719.06	−0.852194	−0.426097	−	0.904677i	$-0.640112\pi$
$313$	−4719.06	−0.852194	−0.426097	+	0.904677i	$0.640112\pi$
$314$	−2068.80	−0.371812
$315$	0	0
$316$	633.395	0.112757
$317$	−4807.13	−0.851720	−0.425860	−	0.904789i	$-0.640028\pi$
$317$	−4807.13	−0.851720	−0.425860	+	0.904789i	$0.640028\pi$
$318$	0	0
$319$	−10516.6	−1.84582
$320$	0	0
$321$	0	0
$322$	−18.3314	−0.00317258
$323$	2406.53	0.414561
$324$	0	0
$325$	0	0
$326$	4850.10	0.823995
$327$	0	0
$328$	3689.51	0.621095
$329$	−3554.30	−0.595608
$330$	0	0
$331$	−2243.42	−0.372537	−0.186268	−	0.982499i	$-0.559639\pi$
$331$	−2243.42	−0.372537	−0.186268	+	0.982499i	$0.559639\pi$
$332$	4836.72	0.799547
$333$	0	0
$334$	1526.89	0.250144
$335$	0	0
$336$	0	0
$337$	−760.365	−0.122907	−0.0614536	−	0.998110i	$-0.519574\pi$
$337$	−760.365	−0.122907	−0.0614536	+	0.998110i	$0.519574\pi$
$338$	8268.84	1.33067
$339$	0	0
$340$	0	0
$341$	7711.27	1.22460
$342$	0	0
$343$	343.000	0.0539949
$344$	−8243.49	−1.29203
$345$	0	0
$346$	1846.67	0.286929
$347$	2619.75	0.405289	0.202645	−	0.979252i	$-0.435046\pi$
$347$	2619.75	0.405289	0.202645	+	0.979252i	$0.435046\pi$
$348$	0	0
$349$	5950.25	0.912635	0.456318	−	0.889817i	$-0.349168\pi$
$349$	5950.25	0.912635	0.456318	+	0.889817i	$0.349168\pi$
$350$	0	0
$351$	0	0
$352$	−9630.02	−1.45819
$353$	1571.79	0.236991	0.118496	−	0.992955i	$-0.462193\pi$
$353$	1571.79	0.236991	0.118496	+	0.992955i	$0.462193\pi$
$354$	0	0
$355$	0	0
$356$	−5301.83	−0.789315
$357$	0	0
$358$	1367.54	0.201891
$359$	1292.38	0.189998	0.0949988	−	0.995477i	$-0.469715\pi$
$359$	1292.38	0.189998	0.0949988	+	0.995477i	$0.469715\pi$
$360$	0	0
$361$	−5969.24	−0.870279
$362$	3102.89	0.450509
$363$	0	0
$364$	3459.10	0.498094
$365$	0	0
$366$	0	0
$367$	1193.60	0.169770	0.0848849	−	0.996391i	$-0.472948\pi$
$367$	1193.60	0.169770	0.0848849	+	0.996391i	$0.472948\pi$
$368$	−22.9930	−0.00325704
$369$	0	0
$370$	0	0
$371$	1049.08	0.146807
$372$	0	0
$373$	1817.49	0.252295	0.126148	−	0.992011i	$-0.459739\pi$
$373$	1817.49	0.252295	0.126148	+	0.992011i	$0.459739\pi$
$374$	6330.02	0.875181
$375$	0	0
$376$	10602.2	1.45417
$377$	−17865.0	−2.44056
$378$	0	0
$379$	−2201.44	−0.298366	−0.149183	−	0.988810i	$-0.547664\pi$
$379$	−2201.44	−0.298366	−0.149183	+	0.988810i	$0.547664\pi$
$380$	0	0
$381$	0	0
$382$	6918.18	0.926610
$383$	−11487.2	−1.53255	−0.766275	−	0.642512i	$-0.777892\pi$
$383$	−11487.2	−1.53255	−0.766275	+	0.642512i	$0.777892\pi$
$384$	0	0
$385$	0	0
$386$	−3881.02	−0.511758
$387$	0	0
$388$	−4440.44	−0.581003
$389$	284.996	0.0371462	0.0185731	−	0.999828i	$-0.494088\pi$
$389$	284.996	0.0371462	0.0185731	+	0.999828i	$0.494088\pi$
$390$	0	0
$391$	138.270	0.0178839
$392$	−1023.15	−0.131828
$393$	0	0
$394$	−3582.79	−0.458118
$395$	0	0
$396$	0	0
$397$	−9446.19	−1.19418	−0.597092	−	0.802173i	$-0.703677\pi$
$397$	−9446.19	−1.19418	−0.597092	+	0.802173i	$0.703677\pi$
$398$	4970.82	0.626042
$399$	0	0
$400$	0	0
$401$	−11303.8	−1.40769	−0.703845	−	0.710353i	$-0.748535\pi$
$401$	−11303.8	−1.40769	−0.703845	+	0.710353i	$0.748535\pi$
$402$	0	0
$403$	13099.4	1.61918
$404$	10226.6	1.25939
$405$	0	0
$406$	2190.66	0.267785
$407$	−18813.8	−2.29131
$408$	0	0
$409$	10022.2	1.21165	0.605827	−	0.795596i	$-0.292842\pi$
$409$	10022.2	1.21165	0.605827	+	0.795596i	$0.292842\pi$
$410$	0	0
$411$	0	0
$412$	−3277.92	−0.391970
$413$	−3242.75	−0.386356
$414$	0	0
$415$	0	0
$416$	−16358.9	−1.92803
$417$	0	0
$418$	2340.38	0.273855
$419$	−9033.23	−1.05323	−0.526614	−	0.850105i	$-0.676538\pi$
$419$	−9033.23	−1.05323	−0.526614	+	0.850105i	$0.676538\pi$
$420$	0	0
$421$	13162.6	1.52377	0.761883	−	0.647715i	$-0.224275\pi$
$421$	13162.6	1.52377	0.761883	+	0.647715i	$0.224275\pi$
$422$	3863.46	0.445664
$423$	0	0
$424$	−3129.34	−0.358429
$425$	0	0
$426$	0	0
$427$	2666.69	0.302225
$428$	−1216.88	−0.137430
$429$	0	0
$430$	0	0
$431$	15992.1	1.78727	0.893634	−	0.448796i	$-0.148147\pi$
$431$	15992.1	1.78727	0.893634	+	0.448796i	$0.148147\pi$
$432$	0	0
$433$	−1526.80	−0.169453	−0.0847266	−	0.996404i	$-0.527002\pi$
$433$	−1526.80	−0.169453	−0.0847266	+	0.996404i	$0.527002\pi$
$434$	−1606.29	−0.177660
$435$	0	0
$436$	−4602.45	−0.505545
$437$	51.1221	0.00559611
$438$	0	0
$439$	8620.34	0.937190	0.468595	−	0.883413i	$-0.344760\pi$
$439$	8620.34	0.937190	0.468595	+	0.883413i	$0.344760\pi$
$440$	0	0
$441$	0	0
$442$	10753.0	1.15717
$443$	−17733.9	−1.90194	−0.950972	−	0.309278i	$-0.899913\pi$
$443$	−17733.9	−1.90194	−0.950972	+	0.309278i	$0.899913\pi$
$444$	0	0
$445$	0	0
$446$	−6005.07	−0.637552
$447$	0	0
$448$	1254.68	0.132317
$449$	12508.1	1.31469	0.657344	−	0.753591i	$-0.271680\pi$
$449$	12508.1	1.31469	0.657344	+	0.753591i	$0.271680\pi$
$450$	0	0
$451$	9073.02	0.947299
$452$	9043.35	0.941070
$453$	0	0
$454$	−783.406	−0.0809847
$455$	0	0
$456$	0	0
$457$	−6138.41	−0.628321	−0.314160	−	0.949370i	$-0.601723\pi$
$457$	−6138.41	−0.628321	−0.314160	+	0.949370i	$0.601723\pi$
$458$	2850.36	0.290805
$459$	0	0
$460$	0	0
$461$	−534.711	−0.0540216	−0.0270108	−	0.999635i	$-0.508599\pi$
$461$	−534.711	−0.0540216	−0.0270108	+	0.999635i	$0.508599\pi$
$462$	0	0
$463$	5081.20	0.510029	0.255014	−	0.966937i	$-0.417920\pi$
$463$	5081.20	0.510029	0.255014	+	0.966937i	$0.417920\pi$
$464$	2747.73	0.274914
$465$	0	0
$466$	2306.02	0.229236
$467$	17073.9	1.69183	0.845915	−	0.533318i	$-0.179055\pi$
$467$	17073.9	1.69183	0.845915	+	0.533318i	$0.179055\pi$
$468$	0	0
$469$	−5583.66	−0.549743
$470$	0	0
$471$	0	0
$472$	9672.89	0.943285
$473$	−20271.9	−1.97062
$474$	0	0
$475$	0	0
$476$	3199.40	0.308076
$477$	0	0
$478$	8943.27	0.855765
$479$	7075.25	0.674899	0.337449	−	0.941344i	$-0.390436\pi$
$479$	7075.25	0.674899	0.337449	+	0.941344i	$0.390436\pi$
$480$	0	0
$481$	−31959.6	−3.02959
$482$	−4661.76	−0.440534
$483$	0	0
$484$	−7396.63	−0.694649
$485$	0	0
$486$	0	0
$487$	−15770.2	−1.46738	−0.733690	−	0.679485i	$-0.762203\pi$
$487$	−15770.2	−1.46738	−0.733690	+	0.679485i	$0.762203\pi$
$488$	−7954.56	−0.737881
$489$	0	0
$490$	0	0
$491$	−8301.12	−0.762982	−0.381491	−	0.924373i	$-0.624589\pi$
$491$	−8301.12	−0.762982	−0.381491	+	0.924373i	$0.624589\pi$
$492$	0	0
$493$	−16523.7	−1.50951
$494$	3975.68	0.362094
$495$	0	0
$496$	−2014.76	−0.182390
$497$	1541.22	0.139101
$498$	0	0
$499$	−3575.29	−0.320745	−0.160373	−	0.987057i	$-0.551270\pi$
$499$	−3575.29	−0.320745	−0.160373	+	0.987057i	$0.551270\pi$
$500$	0	0
$501$	0	0
$502$	−9023.13	−0.802235
$503$	−11686.3	−1.03592	−0.517960	−	0.855405i	$-0.673308\pi$
$503$	−11686.3	−1.03592	−0.517960	+	0.855405i	$0.673308\pi$
$504$	0	0
$505$	0	0
$506$	134.469	0.0118140
$507$	0	0
$508$	−8326.21	−0.727197
$509$	7736.09	0.673666	0.336833	−	0.941564i	$-0.390644\pi$
$509$	7736.09	0.673666	0.336833	+	0.941564i	$0.390644\pi$
$510$	0	0
$511$	7095.25	0.614237
$512$	4757.14	0.410621
$513$	0	0
$514$	−831.955	−0.0713929
$515$	0	0
$516$	0	0
$517$	26072.4	2.21791
$518$	3919.00	0.332415
$519$	0	0
$520$	0	0
$521$	−47.6564	−0.00400741	−0.00200371	−	0.999998i	$-0.500638\pi$
$521$	−47.6564	−0.00400741	−0.00200371	+	0.999998i	$0.500638\pi$
$522$	0	0
$523$	13429.0	1.12277	0.561385	−	0.827555i	$-0.310269\pi$
$523$	13429.0	1.12277	0.561385	+	0.827555i	$0.310269\pi$
$524$	−6896.59	−0.574960
$525$	0	0
$526$	2607.12	0.216114
$527$	12115.9	1.00148
$528$	0	0
$529$	−12164.1	−0.999759
$530$	0	0
$531$	0	0
$532$	1182.90	0.0964010
$533$	15412.7	1.25253
$534$	0	0
$535$	0	0
$536$	16655.7	1.34219
$537$	0	0
$538$	−442.926	−0.0354943
$539$	−2516.06	−0.201065
$540$	0	0
$541$	13627.6	1.08299	0.541493	−	0.840705i	$-0.317859\pi$
$541$	13627.6	1.08299	0.541493	+	0.840705i	$0.317859\pi$
$542$	−2151.97	−0.170544
$543$	0	0
$544$	−15130.7	−1.19250
$545$	0	0
$546$	0	0
$547$	−1691.42	−0.132212	−0.0661058	−	0.997813i	$-0.521057\pi$
$547$	−1691.42	−0.132212	−0.0661058	+	0.997813i	$0.521057\pi$
$548$	9275.21	0.723024
$549$	0	0
$550$	0	0
$551$	−6109.25	−0.472346
$552$	0	0
$553$	−782.633	−0.0601825
$554$	−9881.88	−0.757836
$555$	0	0
$556$	−253.651	−0.0193475
$557$	4741.73	0.360706	0.180353	−	0.983602i	$-0.442276\pi$
$557$	4741.73	0.360706	0.180353	+	0.983602i	$0.442276\pi$
$558$	0	0
$559$	−34436.6	−2.60557
$560$	0	0
$561$	0	0
$562$	6526.12	0.489836
$563$	−1073.78	−0.0803808	−0.0401904	−	0.999192i	$-0.512796\pi$
$563$	−1073.78	−0.0803808	−0.0401904	+	0.999192i	$0.512796\pi$
$564$	0	0
$565$	0	0
$566$	−5747.03	−0.426795
$567$	0	0
$568$	−4597.34	−0.339613
$569$	−3530.34	−0.260105	−0.130052	−	0.991507i	$-0.541515\pi$
$569$	−3530.34	−0.260105	−0.130052	+	0.991507i	$0.541515\pi$
$570$	0	0
$571$	−2949.59	−0.216176	−0.108088	−	0.994141i	$-0.534473\pi$
$571$	−2949.59	−0.216176	−0.108088	+	0.994141i	$0.534473\pi$
$572$	−25374.0	−1.85479
$573$	0	0
$574$	−1889.95	−0.137430
$575$	0	0
$576$	0	0
$577$	22580.2	1.62916	0.814580	−	0.580051i	$-0.196968\pi$
$577$	22580.2	1.62916	0.814580	+	0.580051i	$0.196968\pi$
$578$	2438.63	0.175491
$579$	0	0
$580$	0	0
$581$	−5976.32	−0.426747
$582$	0	0
$583$	−7695.47	−0.546679
$584$	−21164.6	−1.49965
$585$	0	0
$586$	−716.608	−0.0505168
$587$	18289.4	1.28600	0.643002	−	0.765864i	$-0.277689\pi$
$587$	18289.4	1.28600	0.643002	+	0.765864i	$0.277689\pi$
$588$	0	0
$589$	4479.58	0.313375
$590$	0	0
$591$	0	0
$592$	4915.57	0.341265
$593$	−9131.08	−0.632325	−0.316162	−	0.948705i	$-0.602394\pi$
$593$	−9131.08	−0.632325	−0.316162	+	0.948705i	$0.602394\pi$
$594$	0	0
$595$	0	0
$596$	−12198.8	−0.838393
$597$	0	0
$598$	228.427	0.0156206
$599$	−5457.56	−0.372270	−0.186135	−	0.982524i	$-0.559596\pi$
$599$	−5457.56	−0.372270	−0.186135	+	0.982524i	$0.559596\pi$
$600$	0	0
$601$	6527.79	0.443052	0.221526	−	0.975154i	$-0.428896\pi$
$601$	6527.79	0.443052	0.221526	+	0.975154i	$0.428896\pi$
$602$	4222.73	0.285890
$603$	0	0
$604$	17207.1	1.15919
$605$	0	0
$606$	0	0
$607$	10959.4	0.732833	0.366416	−	0.930451i	$-0.380585\pi$
$607$	10959.4	0.732833	0.366416	+	0.930451i	$0.380585\pi$
$608$	−5594.21	−0.373150
$609$	0	0
$610$	0	0
$611$	44290.1	2.93254
$612$	0	0
$613$	28164.2	1.85569	0.927846	−	0.372963i	$-0.121658\pi$
$613$	28164.2	1.85569	0.927846	+	0.372963i	$0.121658\pi$
$614$	6311.09	0.414813
$615$	0	0
$616$	7505.22	0.490899
$617$	20764.0	1.35483	0.677413	−	0.735603i	$-0.263101\pi$
$617$	20764.0	1.35483	0.677413	+	0.735603i	$0.263101\pi$
$618$	0	0
$619$	−9.21158	−0.000598134 0	−0.000299067	−	1.00000i	$-0.500095\pi$
$619$	−9.21158	−0.000598134 0	−0.000299067		1.00000i	$0.500095\pi$
$620$	0	0
$621$	0	0
$622$	4858.68	0.313208
$623$	6551.02	0.421286
$624$	0	0
$625$	0	0
$626$	−7210.75	−0.460383
$627$	0	0
$628$	7670.21	0.487380
$629$	−29560.2	−1.87384
$630$	0	0
$631$	−15005.5	−0.946684	−0.473342	−	0.880879i	$-0.656953\pi$
$631$	−15005.5	−0.946684	−0.473342	+	0.880879i	$0.656953\pi$
$632$	2334.54	0.146935
$633$	0	0
$634$	−7345.32	−0.460126
$635$	0	0
$636$	0	0
$637$	−4274.12	−0.265850
$638$	−16069.5	−0.997173
$639$	0	0
$640$	0	0
$641$	19610.6	1.20838	0.604191	−	0.796839i	$-0.293496\pi$
$641$	19610.6	1.20838	0.604191	+	0.796839i	$0.293496\pi$
$642$	0	0
$643$	−26668.9	−1.63564	−0.817820	−	0.575474i	$-0.804818\pi$
$643$	−26668.9	−1.63564	−0.817820	+	0.575474i	$0.804818\pi$
$644$	67.9650	0.00415869
$645$	0	0
$646$	3677.20	0.223959
$647$	6895.91	0.419020	0.209510	−	0.977806i	$-0.432813\pi$
$647$	6895.91	0.419020	0.209510	+	0.977806i	$0.432813\pi$
$648$	0	0
$649$	23787.0	1.43871
$650$	0	0
$651$	0	0
$652$	−17982.1	−1.08011
$653$	18809.2	1.12720	0.563598	−	0.826049i	$-0.309417\pi$
$653$	18809.2	1.12720	0.563598	+	0.826049i	$0.309417\pi$
$654$	0	0
$655$	0	0
$656$	−2370.55	−0.141089
$657$	0	0
$658$	−5431.00	−0.321766
$659$	6285.52	0.371547	0.185773	−	0.982593i	$-0.440521\pi$
$659$	6285.52	0.371547	0.185773	+	0.982593i	$0.440521\pi$
$660$	0	0
$661$	1910.45	0.112418	0.0562088	−	0.998419i	$-0.482099\pi$
$661$	1910.45	0.112418	0.0562088	+	0.998419i	$0.482099\pi$
$662$	−3427.97	−0.201256
$663$	0	0
$664$	17827.0	1.04190
$665$	0	0
$666$	0	0
$667$	−351.014	−0.0203768
$668$	−5661.07	−0.327894
$669$	0	0
$670$	0	0
$671$	−19561.4	−1.12542
$672$	0	0
$673$	−22754.2	−1.30328	−0.651641	−	0.758528i	$-0.725919\pi$
$673$	−22754.2	−1.30328	−0.651641	+	0.758528i	$0.725919\pi$
$674$	−1161.84	−0.0663984
$675$	0	0
$676$	−30657.3	−1.74427
$677$	6327.54	0.359213	0.179606	−	0.983739i	$-0.442518\pi$
$677$	6327.54	0.359213	0.179606	+	0.983739i	$0.442518\pi$
$678$	0	0
$679$	5486.68	0.310102
$680$	0	0
$681$	0	0
$682$	11782.9	0.661568
$683$	−12440.7	−0.696969	−0.348484	−	0.937315i	$-0.613304\pi$
$683$	−12440.7	−0.696969	−0.348484	+	0.937315i	$0.613304\pi$
$684$	0	0
$685$	0	0
$686$	524.106	0.0291698
$687$	0	0
$688$	5296.54	0.293501
$689$	−13072.6	−0.722823
$690$	0	0
$691$	−13421.5	−0.738898	−0.369449	−	0.929251i	$-0.620454\pi$
$691$	−13421.5	−0.738898	−0.369449	+	0.929251i	$0.620454\pi$
$692$	−6846.65	−0.376114
$693$	0	0
$694$	4002.99	0.218950
$695$	0	0
$696$	0	0
$697$	14255.5	0.774701
$698$	9092.03	0.493035
$699$	0	0
$700$	0	0
$701$	−3484.08	−0.187720	−0.0938601	−	0.995585i	$-0.529921\pi$
$701$	−3484.08	−0.187720	−0.0938601	+	0.995585i	$0.529921\pi$
$702$	0	0
$703$	−10929.2	−0.586347
$704$	−9203.66	−0.492721
$705$	0	0
$706$	2401.70	0.128030
$707$	−12636.2	−0.672181
$708$	0	0
$709$	22952.1	1.21578	0.607888	−	0.794023i	$-0.292017\pi$
$709$	22952.1	1.21578	0.607888	+	0.794023i	$0.292017\pi$
$710$	0	0
$711$	0	0
$712$	−19541.2	−1.02857
$713$	257.380	0.0135189
$714$	0	0
$715$	0	0
$716$	−5070.26	−0.264643
$717$	0	0
$718$	1974.76	0.102643
$719$	−11393.3	−0.590957	−0.295479	−	0.955349i	$-0.595479\pi$
$719$	−11393.3	−0.590957	−0.295479	+	0.955349i	$0.595479\pi$
$720$	0	0
$721$	4050.25	0.209209
$722$	−9121.04	−0.470152
$723$	0	0
$724$	−11504.2	−0.590538
$725$	0	0
$726$	0	0
$727$	22822.6	1.16430	0.582149	−	0.813082i	$-0.302212\pi$
$727$	22822.6	1.16430	0.582149	+	0.813082i	$0.302212\pi$
$728$	12749.4	0.649071
$729$	0	0
$730$	0	0
$731$	−31851.2	−1.61157
$732$	0	0
$733$	−7923.03	−0.399241	−0.199621	−	0.979873i	$-0.563971\pi$
$733$	−7923.03	−0.399241	−0.199621	+	0.979873i	$0.563971\pi$
$734$	1823.83	0.0917150
$735$	0	0
$736$	−321.422	−0.0160975
$737$	40958.6	2.04712
$738$	0	0
$739$	23588.4	1.17417	0.587087	−	0.809524i	$-0.300275\pi$
$739$	23588.4	1.17417	0.587087	+	0.809524i	$0.300275\pi$
$740$	0	0
$741$	0	0
$742$	1603.00	0.0793101
$743$	−30296.2	−1.49591	−0.747953	−	0.663752i	$-0.768963\pi$
$743$	−30296.2	−1.49591	−0.747953	+	0.663752i	$0.768963\pi$
$744$	0	0
$745$	0	0
$746$	2777.14	0.136298
$747$	0	0
$748$	−23469.0	−1.14721
$749$	1503.60	0.0733515
$750$	0	0
$751$	−3326.15	−0.161615	−0.0808076	−	0.996730i	$-0.525750\pi$
$751$	−3326.15	−0.161615	−0.0808076	+	0.996730i	$0.525750\pi$
$752$	−6812.06	−0.330333
$753$	0	0
$754$	−27297.8	−1.31847
$755$	0	0
$756$	0	0
$757$	−21829.2	−1.04808	−0.524040	−	0.851694i	$-0.675576\pi$
$757$	−21829.2	−1.04808	−0.524040	+	0.851694i	$0.675576\pi$
$758$	−3363.82	−0.161187
$759$	0	0
$760$	0	0
$761$	−32402.3	−1.54347	−0.771736	−	0.635943i	$-0.780611\pi$
$761$	−32402.3	−1.54347	−0.771736	+	0.635943i	$0.780611\pi$
$762$	0	0
$763$	5686.86	0.269827
$764$	−25649.7	−1.21462
$765$	0	0
$766$	−17552.5	−0.827933
$767$	40407.8	1.90227
$768$	0	0
$769$	12942.0	0.606894	0.303447	−	0.952848i	$-0.401863\pi$
$769$	12942.0	0.606894	0.303447	+	0.952848i	$0.401863\pi$
$770$	0	0
$771$	0	0
$772$	14389.2	0.670825
$773$	33087.0	1.53953	0.769766	−	0.638326i	$-0.220373\pi$
$773$	33087.0	1.53953	0.769766	+	0.638326i	$0.220373\pi$
$774$	0	0
$775$	0	0
$776$	−16366.4	−0.757112
$777$	0	0
$778$	435.476	0.0200676
$779$	5270.64	0.242414
$780$	0	0
$781$	−11305.5	−0.517980
$782$	211.278	0.00966148
$783$	0	0
$784$	657.383	0.0299464
$785$	0	0
$786$	0	0
$787$	−30248.7	−1.37008	−0.685038	−	0.728507i	$-0.740215\pi$
$787$	−30248.7	−1.37008	−0.685038	+	0.728507i	$0.740215\pi$
$788$	13283.5	0.600512
$789$	0	0
$790$	0	0
$791$	−11174.1	−0.502282
$792$	0	0
$793$	−33229.6	−1.48804
$794$	−14433.8	−0.645136
$795$	0	0
$796$	−18429.7	−0.820631
$797$	−3123.69	−0.138829	−0.0694145	−	0.997588i	$-0.522113\pi$
$797$	−3123.69	−0.138829	−0.0694145	+	0.997588i	$0.522113\pi$
$798$	0	0
$799$	40964.9	1.81381
$800$	0	0
$801$	0	0
$802$	−17272.3	−0.760480
$803$	−52046.8	−2.28729
$804$	0	0
$805$	0	0
$806$	20016.0	0.874731
$807$	0	0
$808$	37692.8	1.64112
$809$	−35088.2	−1.52489	−0.762444	−	0.647055i	$-0.776001\pi$
$809$	−35088.2	−1.52489	−0.762444	+	0.647055i	$0.776001\pi$
$810$	0	0
$811$	−29394.0	−1.27271	−0.636353	−	0.771398i	$-0.719558\pi$
$811$	−29394.0	−1.27271	−0.636353	+	0.771398i	$0.719558\pi$
$812$	−8122.03	−0.351019
$813$	0	0
$814$	−28747.6	−1.23784
$815$	0	0
$816$	0	0
$817$	−11776.2	−0.504281
$818$	15314.0	0.654575
$819$	0	0
$820$	0	0
$821$	36702.5	1.56020	0.780100	−	0.625654i	$-0.215168\pi$
$821$	36702.5	1.56020	0.780100	+	0.625654i	$0.215168\pi$
$822$	0	0
$823$	−21436.3	−0.907924	−0.453962	−	0.891021i	$-0.649990\pi$
$823$	−21436.3	−0.907924	−0.453962	+	0.891021i	$0.649990\pi$
$824$	−12081.6	−0.510781
$825$	0	0
$826$	−4954.94	−0.208722
$827$	−4079.17	−0.171520	−0.0857598	−	0.996316i	$-0.527332\pi$
$827$	−4079.17	−0.171520	−0.0857598	+	0.996316i	$0.527332\pi$
$828$	0	0
$829$	−35727.1	−1.49681	−0.748404	−	0.663244i	$-0.769179\pi$
$829$	−35727.1	−1.49681	−0.748404	+	0.663244i	$0.769179\pi$
$830$	0	0
$831$	0	0
$832$	−15634.6	−0.651480
$833$	−3953.23	−0.164431
$834$	0	0
$835$	0	0
$836$	−8677.11	−0.358976
$837$	0	0
$838$	−13802.8	−0.568987
$839$	22421.8	0.922628	0.461314	−	0.887237i	$-0.347378\pi$
$839$	22421.8	0.922628	0.461314	+	0.887237i	$0.347378\pi$
$840$	0	0
$841$	17558.3	0.719927
$842$	20112.5	0.823187
$843$	0	0
$844$	−14324.0	−0.584187
$845$	0	0
$846$	0	0
$847$	9139.38	0.370759
$848$	2010.64	0.0814216
$849$	0	0
$850$	0	0
$851$	−627.949	−0.0252947
$852$	0	0
$853$	−43300.2	−1.73807	−0.869034	−	0.494753i	$-0.835259\pi$
$853$	−43300.2	−1.73807	−0.869034	+	0.494753i	$0.835259\pi$
$854$	4074.72	0.163272
$855$	0	0
$856$	−4485.13	−0.179087
$857$	44562.7	1.77623	0.888116	−	0.459619i	$-0.152014\pi$
$857$	44562.7	1.77623	0.888116	+	0.459619i	$0.152014\pi$
$858$	0	0
$859$	−26003.7	−1.03287	−0.516434	−	0.856327i	$-0.672741\pi$
$859$	−26003.7	−1.03287	−0.516434	+	0.856327i	$0.672741\pi$
$860$	0	0
$861$	0	0
$862$	24436.1	0.965540
$863$	28719.8	1.13283	0.566416	−	0.824119i	$-0.308329\pi$
$863$	28719.8	1.13283	0.566416	+	0.824119i	$0.308329\pi$
$864$	0	0
$865$	0	0
$866$	−2332.96	−0.0915441
$867$	0	0
$868$	5955.45	0.232881
$869$	5740.96	0.224107
$870$	0	0
$871$	69577.9	2.70672
$872$	−16963.5	−0.658781
$873$	0	0
$874$	78.1149	0.00302320
$875$	0	0
$876$	0	0
$877$	−4042.01	−0.155632	−0.0778159	−	0.996968i	$-0.524795\pi$
$877$	−4042.01	−0.155632	−0.0778159	+	0.996968i	$0.524795\pi$
$878$	13171.9	0.506300
$879$	0	0
$880$	0	0
$881$	45388.8	1.73574	0.867871	−	0.496790i	$-0.165488\pi$
$881$	45388.8	1.73574	0.867871	+	0.496790i	$0.165488\pi$
$882$	0	0
$883$	10622.5	0.404842	0.202421	−	0.979299i	$-0.435119\pi$
$883$	10622.5	0.404842	0.202421	+	0.979299i	$0.435119\pi$
$884$	−39867.6	−1.51685
$885$	0	0
$886$	−27097.5	−1.02749
$887$	−943.623	−0.0357201	−0.0178601	−	0.999840i	$-0.505685\pi$
$887$	−943.623	−0.0357201	−0.0178601	+	0.999840i	$0.505685\pi$
$888$	0	0
$889$	10288.0	0.388131
$890$	0	0
$891$	0	0
$892$	22264.2	0.835719
$893$	15145.8	0.567564
$894$	0	0
$895$	0	0
$896$	−8585.29	−0.320105
$897$	0	0
$898$	19112.5	0.710236
$899$	−30757.7	−1.14107
$900$	0	0
$901$	−12091.1	−0.447074
$902$	13863.6	0.511761
$903$	0	0
$904$	33331.6	1.22632
$905$	0	0
$906$	0	0
$907$	24087.8	0.881834	0.440917	−	0.897548i	$-0.354653\pi$
$907$	24087.8	0.881834	0.440917	+	0.897548i	$0.354653\pi$
$908$	2904.53	0.106157
$909$	0	0
$910$	0	0
$911$	−44542.8	−1.61994	−0.809972	−	0.586468i	$-0.800518\pi$
$911$	−44542.8	−1.61994	−0.809972	+	0.586468i	$0.800518\pi$
$912$	0	0
$913$	43839.0	1.58911
$914$	−9379.53	−0.339439
$915$	0	0
$916$	−10567.9	−0.381194
$917$	8521.53	0.306876
$918$	0	0
$919$	−22333.9	−0.801661	−0.400830	−	0.916152i	$-0.631278\pi$
$919$	−22333.9	−0.801661	−0.400830	+	0.916152i	$0.631278\pi$
$920$	0	0
$921$	0	0
$922$	−817.042	−0.0291842
$923$	−19205.1	−0.684878
$924$	0	0
$925$	0	0
$926$	7764.11	0.275534
$927$	0	0
$928$	38410.9	1.35873
$929$	48395.4	1.70915	0.854576	−	0.519327i	$-0.173817\pi$
$929$	48395.4	1.70915	0.854576	+	0.519327i	$0.173817\pi$
$930$	0	0
$931$	−1461.61	−0.0514526
$932$	−8549.71	−0.300488
$933$	0	0
$934$	26089.0	0.913981
$935$	0	0
$936$	0	0
$937$	10547.8	0.367749	0.183874	−	0.982950i	$-0.441136\pi$
$937$	10547.8	0.367749	0.183874	+	0.982950i	$0.441136\pi$
$938$	−8531.88	−0.296989
$939$	0	0
$940$	0	0
$941$	47214.8	1.63566	0.817831	−	0.575458i	$-0.195176\pi$
$941$	47214.8	1.63566	0.817831	+	0.575458i	$0.195176\pi$
$942$	0	0
$943$	302.831	0.0104576
$944$	−6214.94	−0.214279
$945$	0	0
$946$	−30975.6	−1.06459
$947$	−30737.7	−1.05474	−0.527370	−	0.849635i	$-0.676822\pi$
$947$	−30737.7	−1.05474	−0.527370	+	0.849635i	$0.676822\pi$
$948$	0	0
$949$	−88413.7	−3.02427
$950$	0	0
$951$	0	0
$952$	11792.2	0.401457
$953$	17032.1	0.578933	0.289466	−	0.957188i	$-0.406522\pi$
$953$	17032.1	0.578933	0.289466	+	0.957188i	$0.406522\pi$
$954$	0	0
$955$	0	0
$956$	−33157.8	−1.12176
$957$	0	0
$958$	10811.0	0.364602
$959$	−11460.6	−0.385904
$960$	0	0
$961$	−7238.05	−0.242961
$962$	−48834.6	−1.63668
$963$	0	0
$964$	17283.8	0.577463
$965$	0	0
$966$	0	0
$967$	7597.47	0.252656	0.126328	−	0.991989i	$-0.459681\pi$
$967$	7597.47	0.252656	0.126328	+	0.991989i	$0.459681\pi$
$968$	−27262.1	−0.905205
$969$	0	0
$970$	0	0
$971$	−19134.9	−0.632410	−0.316205	−	0.948691i	$-0.602409\pi$
$971$	−19134.9	−0.632410	−0.316205	+	0.948691i	$0.602409\pi$
$972$	0	0
$973$	313.415	0.0103265
$974$	−24096.9	−0.792726
$975$	0	0
$976$	5110.90	0.167619
$977$	44846.1	1.46853	0.734266	−	0.678862i	$-0.237527\pi$
$977$	44846.1	1.46853	0.734266	+	0.678862i	$0.237527\pi$
$978$	0	0
$979$	−48054.6	−1.56878
$980$	0	0
$981$	0	0
$982$	−12684.2	−0.412187
$983$	−7385.31	−0.239628	−0.119814	−	0.992796i	$-0.538230\pi$
$983$	−7385.31	−0.239628	−0.119814	+	0.992796i	$0.538230\pi$
$984$	0	0
$985$	0	0
$986$	−25248.3	−0.815488
$987$	0	0
$988$	−14740.1	−0.474641
$989$	−676.616	−0.0217545
$990$	0	0
$991$	56089.0	1.79791	0.898953	−	0.438045i	$-0.144329\pi$
$991$	56089.0	1.79791	0.898953	+	0.438045i	$0.144329\pi$
$992$	−28164.7	−0.901440
$993$	0	0
$994$	2354.99	0.0751466
$995$	0	0
$996$	0	0
$997$	11762.2	0.373633	0.186816	−	0.982395i	$-0.440183\pi$
$997$	11762.2	0.373633	0.186816	+	0.982395i	$0.440183\pi$
$998$	−5463.07	−0.173277
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.4.a.bk.1.3		4
3.2	odd	2		525.4.a.u.1.2	yes	4
5.4	even	2		1575.4.a.bj.1.2		4
15.2	even	4		525.4.d.n.274.3		8
15.8	even	4		525.4.d.n.274.6		8
15.14	odd	2		525.4.a.t.1.3	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
525.4.a.t.1.3	✓	4	15.14	odd	2
525.4.a.u.1.2	yes	4	3.2	odd	2
525.4.d.n.274.3		8	15.2	even	4
525.4.d.n.274.6		8	15.8	even	4
1575.4.a.bj.1.2		4	5.4	even	2
1575.4.a.bk.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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1575.a (trivial)

\(f(q)\)	\(=\)	\(q+1.52801 q^{2} -5.66519 q^{4} +7.00000 q^{7} -20.8805 q^{8} -51.3481 q^{11} -87.2268 q^{13} +10.6960 q^{14} +13.4160 q^{16} -80.6781 q^{17} -29.8288 q^{19} -78.4602 q^{22} -1.71385 q^{23} -133.283 q^{26} -39.6564 q^{28} +204.810 q^{29} -150.176 q^{31} +187.544 q^{32} -123.277 q^{34} +366.397 q^{37} -45.5786 q^{38} -176.696 q^{41} +394.793 q^{43} +290.897 q^{44} -2.61877 q^{46} -507.757 q^{47} +49.0000 q^{49} +494.157 q^{52} +149.869 q^{53} -146.164 q^{56} +312.952 q^{58} -463.249 q^{59} +380.956 q^{61} -229.471 q^{62} +179.240 q^{64} -797.666 q^{67} +457.057 q^{68} +220.174 q^{71} +1013.61 q^{73} +559.857 q^{74} +168.986 q^{76} -359.437 q^{77} -111.805 q^{79} -269.993 q^{82} -853.761 q^{83} +603.247 q^{86} +1072.17 q^{88} +935.860 q^{89} -610.588 q^{91} +9.70929 q^{92} -775.857 q^{94} +783.811 q^{97} +74.8723 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 16 q^{4} + 28 q^{7} - 9 q^{8} - 21 q^{11} + 5 q^{13} + 72 q^{16} - 99 q^{17} + 72 q^{19} + 221 q^{22} - 102 q^{23} - 129 q^{26} + 112 q^{28} + 240 q^{29} + 351 q^{31} - 72 q^{32} - 285 q^{34} + 399 q^{37}+ \cdots - 372 q^{97}+O(q^{100}) \) 4 * q + 16 * q^4 + 28 * q^7 - 9 * q^8 - 21 * q^11 + 5 * q^13 + 72 * q^16 - 99 * q^17 + 72 * q^19 + 221 * q^22 - 102 * q^23 - 129 * q^26 + 112 * q^28 + 240 * q^29 + 351 * q^31 - 72 * q^32 - 285 * q^34 + 399 * q^37 - 324 * q^38 - 381 * q^41 + 460 * q^43 + 975 * q^44 + 550 * q^46 - 60 * q^47 + 196 * q^49 + 223 * q^52 - 873 * q^53 - 63 * q^56 + 1408 * q^58 + 855 * q^59 + 687 * q^61 + 477 * q^62 - 1285 * q^64 - 503 * q^67 - 1725 * q^68 + 681 * q^71 + 1228 * q^73 + 3369 * q^74 + 1910 * q^76 - 147 * q^77 + 345 * q^79 - 495 * q^82 - 1509 * q^83 + 540 * q^86 + 2266 * q^88 + 198 * q^89 + 35 * q^91 - 2994 * q^92 - 1732 * q^94 - 372 * q^97

Introduction

L-functions

Modular forms

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Database

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