LMFDB - Embedded newform 1575.4.a.bs.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1575,4,Mod(1,1575)]

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

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$92.9280082590$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 50x^{6} + 698x^{4} - 2653x^{2} + 2268 $ x^8 - 50x^6 + 698x^4 - 2653*x^2 + 2268
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
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			$-5.38335$ 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)

$f(q)$	$=$	$q-5.38335 q^{2} +20.9805 q^{4} +7.00000 q^{7} -69.8783 q^{8} +15.8856 q^{11} -40.8573 q^{13} -37.6834 q^{14} +208.336 q^{16} +47.2555 q^{17} +141.113 q^{19} -85.5176 q^{22} -97.3015 q^{23} +219.949 q^{26} +146.863 q^{28} -236.682 q^{29} +95.7947 q^{31} -562.518 q^{32} -254.393 q^{34} +33.8738 q^{37} -759.663 q^{38} -197.295 q^{41} +437.132 q^{43} +333.287 q^{44} +523.808 q^{46} -256.598 q^{47} +49.0000 q^{49} -857.205 q^{52} +64.7048 q^{53} -489.148 q^{56} +1274.14 q^{58} +809.137 q^{59} +318.721 q^{61} -515.696 q^{62} +1361.55 q^{64} -726.284 q^{67} +991.443 q^{68} +68.6547 q^{71} -680.433 q^{73} -182.355 q^{74} +2960.62 q^{76} +111.199 q^{77} +1169.94 q^{79} +1062.11 q^{82} +642.901 q^{83} -2353.24 q^{86} -1110.06 q^{88} +1382.18 q^{89} -286.001 q^{91} -2041.43 q^{92} +1381.36 q^{94} -317.080 q^{97} -263.784 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 36 q^{4} + 56 q^{7} - 38 q^{13} + 320 q^{16} + 68 q^{19} + 110 q^{22} + 252 q^{28} + 534 q^{31} + 118 q^{34} + 14 q^{37} + 816 q^{43} + 1588 q^{46} + 392 q^{49} - 266 q^{52} + 716 q^{58} + 874 q^{61}+ \cdots + 428 q^{97}+O(q^{100}) $ 8 * q + 36 * q^4 + 56 * q^7 - 38 * q^13 + 320 * q^16 + 68 * q^19 + 110 * q^22 + 252 * q^28 + 534 * q^31 + 118 * q^34 + 14 * q^37 + 816 * q^43 + 1588 * q^46 + 392 * q^49 - 266 * q^52 + 716 * q^58 + 874 * q^61 + 2502 * q^64 + 254 * q^67 - 1532 * q^73 + 4304 * q^76 + 5190 * q^79 + 1158 * q^82 - 4300 * q^88 - 266 * q^91 + 6236 * q^94 + 428 * 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 $

$2$	−5.38335	−1.90330	−0.951651	−	0.307182i	$-0.900614\pi$
$2$	−5.38335	−1.90330	−0.951651	+	0.307182i	$0.900614\pi$
$3$	0	0
$3$	0	0
$4$	20.9805	2.62256
$5$	0	0
$5$	0	0
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	−69.8783	−3.08822
$9$	0	0
$10$	0	0
$11$	15.8856	0.435426	0.217713	−	0.976013i	$-0.430140\pi$
$11$	15.8856	0.435426	0.217713	+	0.976013i	$0.430140\pi$
$12$	0	0
$13$	−40.8573	−0.871676	−0.435838	−	0.900025i	$-0.643548\pi$
$13$	−40.8573	−0.871676	−0.435838	+	0.900025i	$0.643548\pi$
$14$	−37.6834	−0.719380
$15$	0	0
$16$	208.336	3.25525
$17$	47.2555	0.674185	0.337093	−	0.941471i	$-0.390556\pi$
$17$	47.2555	0.674185	0.337093	+	0.941471i	$0.390556\pi$
$18$	0	0
$19$	141.113	1.70388	0.851938	−	0.523643i	$-0.175427\pi$
$19$	141.113	1.70388	0.851938	+	0.523643i	$0.175427\pi$
$20$	0	0
$21$	0	0
$22$	−85.5176	−0.828746
$23$	−97.3015	−0.882120	−0.441060	−	0.897478i	$-0.645397\pi$
$23$	−97.3015	−0.882120	−0.441060	+	0.897478i	$0.645397\pi$
$24$	0	0
$25$	0	0
$26$	219.949	1.65906
$27$	0	0
$28$	146.863	0.991233
$29$	−236.682	−1.51555	−0.757773	−	0.652519i	$-0.773712\pi$
$29$	−236.682	−1.51555	−0.757773	+	0.652519i	$0.773712\pi$
$30$	0	0
$31$	95.7947	0.555008	0.277504	−	0.960725i	$-0.410493\pi$
$31$	95.7947	0.555008	0.277504	+	0.960725i	$0.410493\pi$
$32$	−562.518	−3.10750
$33$	0	0
$34$	−254.393	−1.28318
$35$	0	0
$36$	0	0
$37$	33.8738	0.150509	0.0752544	−	0.997164i	$-0.476023\pi$
$37$	33.8738	0.150509	0.0752544	+	0.997164i	$0.476023\pi$
$38$	−759.663	−3.24299
$39$	0	0
$40$	0	0
$41$	−197.295	−0.751520	−0.375760	−	0.926717i	$-0.622618\pi$
$41$	−197.295	−0.751520	−0.375760	+	0.926717i	$0.622618\pi$
$42$	0	0
$43$	437.132	1.55028	0.775140	−	0.631790i	$-0.217679\pi$
$43$	437.132	1.55028	0.775140	+	0.631790i	$0.217679\pi$
$44$	333.287	1.14193
$45$	0	0
$46$	523.808	1.67894
$47$	−256.598	−0.796354	−0.398177	−	0.917309i	$-0.630357\pi$
$47$	−256.598	−0.796354	−0.398177	+	0.917309i	$0.630357\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	0	0
$52$	−857.205	−2.28602
$53$	64.7048	0.167696	0.0838480	−	0.996479i	$-0.473279\pi$
$53$	64.7048	0.167696	0.0838480	+	0.996479i	$0.473279\pi$
$54$	0	0
$55$	0	0
$56$	−489.148	−1.16724
$57$	0	0
$58$	1274.14	2.88454
$59$	809.137	1.78543	0.892717	−	0.450618i	$-0.148796\pi$
$59$	809.137	1.78543	0.892717	+	0.450618i	$0.148796\pi$
$60$	0	0
$61$	318.721	0.668985	0.334492	−	0.942398i	$-0.391435\pi$
$61$	318.721	0.668985	0.334492	+	0.942398i	$0.391435\pi$
$62$	−515.696	−1.05635
$63$	0	0
$64$	1361.55	2.65927
$65$	0	0
$66$	0	0
$67$	−726.284	−1.32432	−0.662162	−	0.749361i	$-0.730361\pi$
$67$	−726.284	−1.32432	−0.662162	+	0.749361i	$0.730361\pi$
$68$	991.443	1.76809
$69$	0	0
$70$	0	0
$71$	68.6547	0.114758	0.0573790	−	0.998352i	$-0.481726\pi$
$71$	68.6547	0.114758	0.0573790	+	0.998352i	$0.481726\pi$
$72$	0	0
$73$	−680.433	−1.09094	−0.545471	−	0.838130i	$-0.683649\pi$
$73$	−680.433	−1.09094	−0.545471	+	0.838130i	$0.683649\pi$
$74$	−182.355	−0.286464
$75$	0	0
$76$	2960.62	4.46851
$77$	111.199	0.164575
$78$	0	0
$79$	1169.94	1.66619	0.833093	−	0.553132i	$-0.186568\pi$
$79$	1169.94	1.66619	0.833093	+	0.553132i	$0.186568\pi$
$80$	0	0
$81$	0	0
$82$	1062.11	1.43037
$83$	642.901	0.850211	0.425105	−	0.905144i	$-0.360237\pi$
$83$	642.901	0.850211	0.425105	+	0.905144i	$0.360237\pi$
$84$	0	0
$85$	0	0
$86$	−2353.24	−2.95065
$87$	0	0
$88$	−1110.06	−1.34469
$89$	1382.18	1.64619	0.823096	−	0.567903i	$-0.192245\pi$
$89$	1382.18	1.64619	0.823096	+	0.567903i	$0.192245\pi$
$90$	0	0
$91$	−286.001	−0.329462
$92$	−2041.43	−2.31341
$93$	0	0
$94$	1381.36	1.51570
$95$	0	0
$96$	0	0
$97$	−317.080	−0.331903	−0.165951	−	0.986134i	$-0.553070\pi$
$97$	−317.080	−0.331903	−0.165951	+	0.986134i	$0.553070\pi$
$98$	−263.784	−0.271900
$99$	0	0
$100$	0	0
$101$	−1140.37	−1.12347	−0.561736	−	0.827316i	$-0.689866\pi$
$101$	−1140.37	−1.12347	−0.561736	+	0.827316i	$0.689866\pi$
$102$	0	0
$103$	970.893	0.928786	0.464393	−	0.885629i	$-0.346273\pi$
$103$	970.893	0.928786	0.464393	+	0.885629i	$0.346273\pi$
$104$	2855.04	2.69192
$105$	0	0
$106$	−348.329	−0.319176
$107$	−707.342	−0.639078	−0.319539	−	0.947573i	$-0.603528\pi$
$107$	−707.342	−0.639078	−0.319539	+	0.947573i	$0.603528\pi$
$108$	0	0
$109$	−1538.49	−1.35193	−0.675967	−	0.736932i	$-0.736274\pi$
$109$	−1538.49	−1.35193	−0.675967	+	0.736932i	$0.736274\pi$
$110$	0	0
$111$	0	0
$112$	1458.35	1.23037
$113$	465.734	0.387722	0.193861	−	0.981029i	$-0.437899\pi$
$113$	465.734	0.387722	0.193861	+	0.981029i	$0.437899\pi$
$114$	0	0
$115$	0	0
$116$	−4965.70	−3.97460
$117$	0	0
$118$	−4355.87	−3.39822
$119$	330.789	0.254818
$120$	0	0
$121$	−1078.65	−0.810405
$122$	−1715.79	−1.27328
$123$	0	0
$124$	2009.82	1.45554
$125$	0	0
$126$	0	0
$127$	964.749	0.674076	0.337038	−	0.941491i	$-0.390575\pi$
$127$	964.749	0.674076	0.337038	+	0.941491i	$0.390575\pi$
$128$	−2829.53	−1.95389
$129$	0	0
$130$	0	0
$131$	2733.39	1.82303	0.911517	−	0.411262i	$-0.134912\pi$
$131$	2733.39	1.82303	0.911517	+	0.411262i	$0.134912\pi$
$132$	0	0
$133$	987.794	0.644005
$134$	3909.84	2.52059
$135$	0	0
$136$	−3302.14	−2.08203
$137$	2707.00	1.68813	0.844067	−	0.536238i	$-0.180155\pi$
$137$	2707.00	1.68813	0.844067	+	0.536238i	$0.180155\pi$
$138$	0	0
$139$	−2459.38	−1.50073	−0.750367	−	0.661021i	$-0.770123\pi$
$139$	−2459.38	−1.50073	−0.750367	+	0.661021i	$0.770123\pi$
$140$	0	0
$141$	0	0
$142$	−369.592	−0.218419
$143$	−649.042	−0.379550
$144$	0	0
$145$	0	0
$146$	3663.01	2.07639
$147$	0	0
$148$	710.689	0.394718
$149$	−3484.84	−1.91604	−0.958018	−	0.286707i	$-0.907439\pi$
$149$	−3484.84	−1.91604	−0.958018	+	0.286707i	$0.907439\pi$
$150$	0	0
$151$	444.052	0.239314	0.119657	−	0.992815i	$-0.461820\pi$
$151$	444.052	0.239314	0.119657	+	0.992815i	$0.461820\pi$
$152$	−9860.77	−5.26194
$153$	0	0
$154$	−598.623	−0.313237
$155$	0	0
$156$	0	0
$157$	−831.556	−0.422709	−0.211355	−	0.977409i	$-0.567788\pi$
$157$	−831.556	−0.422709	−0.211355	+	0.977409i	$0.567788\pi$
$158$	−6298.21	−3.17126
$159$	0	0
$160$	0	0
$161$	−681.110	−0.333410
$162$	0	0
$163$	−419.137	−0.201407	−0.100703	−	0.994916i	$-0.532109\pi$
$163$	−419.137	−0.201407	−0.100703	+	0.994916i	$0.532109\pi$
$164$	−4139.34	−1.97090
$165$	0	0
$166$	−3460.96	−1.61821
$167$	−2441.03	−1.13109	−0.565547	−	0.824716i	$-0.691335\pi$
$167$	−2441.03	−1.13109	−0.565547	+	0.824716i	$0.691335\pi$
$168$	0	0
$169$	−527.678	−0.240181
$170$	0	0
$171$	0	0
$172$	9171.23	4.06570
$173$	2630.72	1.15613	0.578064	−	0.815991i	$-0.303808\pi$
$173$	2630.72	1.15613	0.578064	+	0.815991i	$0.303808\pi$
$174$	0	0
$175$	0	0
$176$	3309.54	1.41742
$177$	0	0
$178$	−7440.77	−3.13320
$179$	739.987	0.308990	0.154495	−	0.987994i	$-0.450625\pi$
$179$	739.987	0.308990	0.154495	+	0.987994i	$0.450625\pi$
$180$	0	0
$181$	−2061.53	−0.846588	−0.423294	−	0.905992i	$-0.639126\pi$
$181$	−2061.53	−0.846588	−0.423294	+	0.905992i	$0.639126\pi$
$182$	1539.65	0.627066
$183$	0	0
$184$	6799.27	2.72418
$185$	0	0
$186$	0	0
$187$	750.681	0.293558
$188$	−5383.54	−2.08848
$189$	0	0
$190$	0	0
$191$	−4047.47	−1.53332	−0.766662	−	0.642051i	$-0.778084\pi$
$191$	−4047.47	−1.53332	−0.766662	+	0.642051i	$0.778084\pi$
$192$	0	0
$193$	−2100.96	−0.783578	−0.391789	−	0.920055i	$-0.628144\pi$
$193$	−2100.96	−0.783578	−0.391789	+	0.920055i	$0.628144\pi$
$194$	1706.95	0.631711
$195$	0	0
$196$	1028.04	0.374651
$197$	−3057.94	−1.10594	−0.552968	−	0.833202i	$-0.686505\pi$
$197$	−3057.94	−1.10594	−0.552968	+	0.833202i	$0.686505\pi$
$198$	0	0
$199$	2915.91	1.03871	0.519356	−	0.854558i	$-0.326172\pi$
$199$	2915.91	1.03871	0.519356	+	0.854558i	$0.326172\pi$
$200$	0	0
$201$	0	0
$202$	6138.99	2.13831
$203$	−1656.78	−0.572822
$204$	0	0
$205$	0	0
$206$	−5226.66	−1.76776
$207$	0	0
$208$	−8512.05	−2.83752
$209$	2241.67	0.741911
$210$	0	0
$211$	3626.89	1.18334	0.591671	−	0.806179i	$-0.298468\pi$
$211$	3626.89	1.18334	0.591671	+	0.806179i	$0.298468\pi$
$212$	1357.54	0.439792
$213$	0	0
$214$	3807.87	1.21636
$215$	0	0
$216$	0	0
$217$	670.563	0.209773
$218$	8282.24	2.57314
$219$	0	0
$220$	0	0
$221$	−1930.74	−0.587671
$222$	0	0
$223$	5661.60	1.70013	0.850064	−	0.526679i	$-0.176563\pi$
$223$	5661.60	1.70013	0.850064	+	0.526679i	$0.176563\pi$
$224$	−3937.63	−1.17453
$225$	0	0
$226$	−2507.21	−0.737951
$227$	5746.85	1.68032	0.840158	−	0.542342i	$-0.182462\pi$
$227$	5746.85	1.68032	0.840158	+	0.542342i	$0.182462\pi$
$228$	0	0
$229$	−1702.78	−0.491367	−0.245684	−	0.969350i	$-0.579012\pi$
$229$	−1702.78	−0.491367	−0.245684	+	0.969350i	$0.579012\pi$
$230$	0	0
$231$	0	0
$232$	16539.0	4.68033
$233$	988.387	0.277903	0.138951	−	0.990299i	$-0.455627\pi$
$233$	988.387	0.277903	0.138951	+	0.990299i	$0.455627\pi$
$234$	0	0
$235$	0	0
$236$	16976.1	4.68240
$237$	0	0
$238$	−1780.75	−0.484996
$239$	−301.963	−0.0817255	−0.0408627	−	0.999165i	$-0.513011\pi$
$239$	−301.963	−0.0817255	−0.0408627	+	0.999165i	$0.513011\pi$
$240$	0	0
$241$	−4054.29	−1.08365	−0.541826	−	0.840491i	$-0.682267\pi$
$241$	−4054.29	−1.08365	−0.541826	+	0.840491i	$0.682267\pi$
$242$	5806.74	1.54244
$243$	0	0
$244$	6686.92	1.75445
$245$	0	0
$246$	0	0
$247$	−5765.52	−1.48523
$248$	−6693.97	−1.71398
$249$	0	0
$250$	0	0
$251$	−6426.79	−1.61616	−0.808079	−	0.589074i	$-0.799493\pi$
$251$	−6426.79	−1.61616	−0.808079	+	0.589074i	$0.799493\pi$
$252$	0	0
$253$	−1545.69	−0.384098
$254$	−5193.58	−1.28297
$255$	0	0
$256$	4339.98	1.05957
$257$	−991.869	−0.240744	−0.120372	−	0.992729i	$-0.538409\pi$
$257$	−991.869	−0.240744	−0.120372	+	0.992729i	$0.538409\pi$
$258$	0	0
$259$	237.117	0.0568870
$260$	0	0
$261$	0	0
$262$	−14714.8	−3.46978
$263$	5394.91	1.26488	0.632442	−	0.774608i	$-0.282053\pi$
$263$	5394.91	1.26488	0.632442	+	0.774608i	$0.282053\pi$
$264$	0	0
$265$	0	0
$266$	−5317.64	−1.22573
$267$	0	0
$268$	−15237.8	−3.47311
$269$	7111.53	1.61189	0.805944	−	0.591992i	$-0.201658\pi$
$269$	7111.53	1.61189	0.805944	+	0.591992i	$0.201658\pi$
$270$	0	0
$271$	609.090	0.136530	0.0682649	−	0.997667i	$-0.478254\pi$
$271$	609.090	0.136530	0.0682649	+	0.997667i	$0.478254\pi$
$272$	9845.02	2.19464
$273$	0	0
$274$	−14572.7	−3.21303
$275$	0	0
$276$	0	0
$277$	3015.06	0.653998	0.326999	−	0.945025i	$-0.393963\pi$
$277$	3015.06	0.653998	0.326999	+	0.945025i	$0.393963\pi$
$278$	13239.7	2.85635
$279$	0	0
$280$	0	0
$281$	−416.706	−0.0884648	−0.0442324	−	0.999021i	$-0.514084\pi$
$281$	−416.706	−0.0884648	−0.0442324	+	0.999021i	$0.514084\pi$
$282$	0	0
$283$	4483.54	0.941763	0.470882	−	0.882196i	$-0.343936\pi$
$283$	4483.54	0.941763	0.470882	+	0.882196i	$0.343936\pi$
$284$	1440.41	0.300959
$285$	0	0
$286$	3494.02	0.722398
$287$	−1381.07	−0.284048
$288$	0	0
$289$	−2679.91	−0.545474
$290$	0	0
$291$	0	0
$292$	−14275.8	−2.86106
$293$	8198.32	1.63464	0.817322	−	0.576181i	$-0.195458\pi$
$293$	8198.32	1.63464	0.817322	+	0.576181i	$0.195458\pi$
$294$	0	0
$295$	0	0
$296$	−2367.05	−0.464804
$297$	0	0
$298$	18760.1	3.64680
$299$	3975.48	0.768923
$300$	0	0
$301$	3059.93	0.585951
$302$	−2390.49	−0.455487
$303$	0	0
$304$	29399.0	5.54654
$305$	0	0
$306$	0	0
$307$	5568.07	1.03514	0.517568	−	0.855642i	$-0.326837\pi$
$307$	5568.07	1.03514	0.517568	+	0.855642i	$0.326837\pi$
$308$	2333.01	0.431608
$309$	0	0
$310$	0	0
$311$	979.150	0.178529	0.0892645	−	0.996008i	$-0.471548\pi$
$311$	979.150	0.178529	0.0892645	+	0.996008i	$0.471548\pi$
$312$	0	0
$313$	−1466.51	−0.264831	−0.132415	−	0.991194i	$-0.542273\pi$
$313$	−1466.51	−0.264831	−0.132415	+	0.991194i	$0.542273\pi$
$314$	4476.56	0.804544
$315$	0	0
$316$	24545.9	4.36967
$317$	9413.22	1.66782	0.833910	−	0.551900i	$-0.186097\pi$
$317$	9413.22	1.66782	0.833910	+	0.551900i	$0.186097\pi$
$318$	0	0
$319$	−3759.84	−0.659907
$320$	0	0
$321$	0	0
$322$	3666.66	0.634580
$323$	6668.39	1.14873
$324$	0	0
$325$	0	0
$326$	2256.36	0.383338
$327$	0	0
$328$	13786.6	2.32085
$329$	−1796.18	−0.300993
$330$	0	0
$331$	6614.38	1.09837	0.549183	−	0.835702i	$-0.314939\pi$
$331$	6614.38	1.09837	0.549183	+	0.835702i	$0.314939\pi$
$332$	13488.3	2.22973
$333$	0	0
$334$	13140.9	2.15281
$335$	0	0
$336$	0	0
$337$	717.673	0.116006	0.0580032	−	0.998316i	$-0.481527\pi$
$337$	717.673	0.116006	0.0580032	+	0.998316i	$0.481527\pi$
$338$	2840.68	0.457137
$339$	0	0
$340$	0	0
$341$	1521.75	0.241665
$342$	0	0
$343$	343.000	0.0539949
$344$	−30546.1	−4.78760
$345$	0	0
$346$	−14162.1	−2.20046
$347$	6994.59	1.08210	0.541051	−	0.840990i	$-0.318027\pi$
$347$	6994.59	1.08210	0.541051	+	0.840990i	$0.318027\pi$
$348$	0	0
$349$	9941.33	1.52478	0.762388	−	0.647120i	$-0.224027\pi$
$349$	9941.33	1.52478	0.762388	+	0.647120i	$0.224027\pi$
$350$	0	0
$351$	0	0
$352$	−8935.93	−1.35309
$353$	1997.26	0.301143	0.150572	−	0.988599i	$-0.451889\pi$
$353$	1997.26	0.301143	0.150572	+	0.988599i	$0.451889\pi$
$354$	0	0
$355$	0	0
$356$	28998.8	4.31723
$357$	0	0
$358$	−3983.61	−0.588101
$359$	3272.07	0.481040	0.240520	−	0.970644i	$-0.422682\pi$
$359$	3272.07	0.481040	0.240520	+	0.970644i	$0.422682\pi$
$360$	0	0
$361$	13054.0	1.90319
$362$	11098.0	1.61131
$363$	0	0
$364$	−6000.44	−0.864034
$365$	0	0
$366$	0	0
$367$	−12744.7	−1.81272	−0.906359	−	0.422509i	$-0.861150\pi$
$367$	−12744.7	−1.81272	−0.906359	+	0.422509i	$0.861150\pi$
$368$	−20271.4	−2.87152
$369$	0	0
$370$	0	0
$371$	452.934	0.0633831
$372$	0	0
$373$	1849.96	0.256803	0.128401	−	0.991722i	$-0.459015\pi$
$373$	1849.96	0.256803	0.128401	+	0.991722i	$0.459015\pi$
$374$	−4041.18	−0.558729
$375$	0	0
$376$	17930.6	2.45931
$377$	9670.21	1.32106
$378$	0	0
$379$	4832.69	0.654983	0.327491	−	0.944854i	$-0.393797\pi$
$379$	4832.69	0.654983	0.327491	+	0.944854i	$0.393797\pi$
$380$	0	0
$381$	0	0
$382$	21789.0	2.91838
$383$	3561.98	0.475218	0.237609	−	0.971361i	$-0.423636\pi$
$383$	3561.98	0.475218	0.237609	+	0.971361i	$0.423636\pi$
$384$	0	0
$385$	0	0
$386$	11310.2	1.49139
$387$	0	0
$388$	−6652.48	−0.870434
$389$	−5095.14	−0.664097	−0.332048	−	0.943262i	$-0.607740\pi$
$389$	−5095.14	−0.664097	−0.332048	+	0.943262i	$0.607740\pi$
$390$	0	0
$391$	−4598.03	−0.594712
$392$	−3424.04	−0.441174
$393$	0	0
$394$	16462.0	2.10493
$395$	0	0
$396$	0	0
$397$	−14511.4	−1.83453	−0.917263	−	0.398283i	$-0.869606\pi$
$397$	−14511.4	−1.83453	−0.917263	+	0.398283i	$0.869606\pi$
$398$	−15697.4	−1.97698
$399$	0	0
$400$	0	0
$401$	−4973.55	−0.619370	−0.309685	−	0.950839i	$-0.600224\pi$
$401$	−4973.55	−0.619370	−0.309685	+	0.950839i	$0.600224\pi$
$402$	0	0
$403$	−3913.92	−0.483787
$404$	−23925.4	−2.94637
$405$	0	0
$406$	8919.01	1.09025
$407$	538.106	0.0655354
$408$	0	0
$409$	2841.63	0.343545	0.171772	−	0.985137i	$-0.445051\pi$
$409$	2841.63	0.343545	0.171772	+	0.985137i	$0.445051\pi$
$410$	0	0
$411$	0	0
$412$	20369.8	2.43579
$413$	5663.96	0.674831
$414$	0	0
$415$	0	0
$416$	22983.0	2.70874
$417$	0	0
$418$	−12067.7	−1.41208
$419$	12983.2	1.51377	0.756886	−	0.653547i	$-0.226720\pi$
$419$	12983.2	1.51377	0.756886	+	0.653547i	$0.226720\pi$
$420$	0	0
$421$	6323.41	0.732029	0.366015	−	0.930609i	$-0.380722\pi$
$421$	6323.41	0.732029	0.366015	+	0.930609i	$0.380722\pi$
$422$	−19524.8	−2.25226
$423$	0	0
$424$	−4521.46	−0.517881
$425$	0	0
$426$	0	0
$427$	2231.05	0.252852
$428$	−14840.4	−1.67602
$429$	0	0
$430$	0	0
$431$	2421.08	0.270579	0.135289	−	0.990806i	$-0.456804\pi$
$431$	2421.08	0.270579	0.135289	+	0.990806i	$0.456804\pi$
$432$	0	0
$433$	3187.12	0.353726	0.176863	−	0.984235i	$-0.443405\pi$
$433$	3187.12	0.353726	0.176863	+	0.984235i	$0.443405\pi$
$434$	−3609.87	−0.399262
$435$	0	0
$436$	−32278.3	−3.54553
$437$	−13730.5	−1.50302
$438$	0	0
$439$	−2492.32	−0.270961	−0.135480	−	0.990780i	$-0.543258\pi$
$439$	−2492.32	−0.270961	−0.135480	+	0.990780i	$0.543258\pi$
$440$	0	0
$441$	0	0
$442$	10393.8	1.11852
$443$	5862.02	0.628698	0.314349	−	0.949308i	$-0.398214\pi$
$443$	5862.02	0.628698	0.314349	+	0.949308i	$0.398214\pi$
$444$	0	0
$445$	0	0
$446$	−30478.4	−3.23586
$447$	0	0
$448$	9530.82	1.00511
$449$	9655.80	1.01489	0.507445	−	0.861684i	$-0.330590\pi$
$449$	9655.80	1.01489	0.507445	+	0.861684i	$0.330590\pi$
$450$	0	0
$451$	−3134.15	−0.327231
$452$	9771.31	1.01682
$453$	0	0
$454$	−30937.3	−3.19815
$455$	0	0
$456$	0	0
$457$	−12608.4	−1.29059	−0.645293	−	0.763935i	$-0.723265\pi$
$457$	−12608.4	−1.29059	−0.645293	+	0.763935i	$0.723265\pi$
$458$	9166.68	0.935220
$459$	0	0
$460$	0	0
$461$	4611.99	0.465947	0.232974	−	0.972483i	$-0.425154\pi$
$461$	4611.99	0.465947	0.232974	+	0.972483i	$0.425154\pi$
$462$	0	0
$463$	11150.2	1.11921	0.559603	−	0.828761i	$-0.310954\pi$
$463$	11150.2	1.11921	0.559603	+	0.828761i	$0.310954\pi$
$464$	−49309.4	−4.93348
$465$	0	0
$466$	−5320.83	−0.528933
$467$	4106.72	0.406930	0.203465	−	0.979082i	$-0.434780\pi$
$467$	4106.72	0.406930	0.203465	+	0.979082i	$0.434780\pi$
$468$	0	0
$469$	−5083.99	−0.500547
$470$	0	0
$471$	0	0
$472$	−56541.1	−5.51380
$473$	6944.10	0.675032
$474$	0	0
$475$	0	0
$476$	6940.10	0.668275
$477$	0	0
$478$	1625.57	0.155548
$479$	−4559.73	−0.434947	−0.217473	−	0.976066i	$-0.569782\pi$
$479$	−4559.73	−0.434947	−0.217473	+	0.976066i	$0.569782\pi$
$480$	0	0
$481$	−1383.99	−0.131195
$482$	21825.7	2.06252
$483$	0	0
$484$	−22630.5	−2.12533
$485$	0	0
$486$	0	0
$487$	3854.56	0.358659	0.179329	−	0.983789i	$-0.442607\pi$
$487$	3854.56	0.358659	0.179329	+	0.983789i	$0.442607\pi$
$488$	−22271.7	−2.06597
$489$	0	0
$490$	0	0
$491$	−10619.9	−0.976110	−0.488055	−	0.872813i	$-0.662293\pi$
$491$	−10619.9	−0.976110	−0.488055	+	0.872813i	$0.662293\pi$
$492$	0	0
$493$	−11184.6	−1.02176
$494$	31037.8	2.82684
$495$	0	0
$496$	19957.5	1.80669
$497$	480.583	0.0433744
$498$	0	0
$499$	−4544.53	−0.407698	−0.203849	−	0.979002i	$-0.565345\pi$
$499$	−4544.53	−0.407698	−0.203849	+	0.979002i	$0.565345\pi$
$500$	0	0
$501$	0	0
$502$	34597.7	3.07604
$503$	−8793.98	−0.779531	−0.389766	−	0.920914i	$-0.627444\pi$
$503$	−8793.98	−0.779531	−0.389766	+	0.920914i	$0.627444\pi$
$504$	0	0
$505$	0	0
$506$	8320.99	0.731054
$507$	0	0
$508$	20240.9	1.76780
$509$	−10573.9	−0.920782	−0.460391	−	0.887716i	$-0.652291\pi$
$509$	−10573.9	−0.920782	−0.460391	+	0.887716i	$0.652291\pi$
$510$	0	0
$511$	−4763.03	−0.412337
$512$	−727.414	−0.0627880
$513$	0	0
$514$	5339.58	0.458208
$515$	0	0
$516$	0	0
$517$	−4076.20	−0.346753
$518$	−1276.48	−0.108273
$519$	0	0
$520$	0	0
$521$	−15708.4	−1.32092	−0.660458	−	0.750863i	$-0.729638\pi$
$521$	−15708.4	−1.32092	−0.660458	+	0.750863i	$0.729638\pi$
$522$	0	0
$523$	2656.84	0.222133	0.111067	−	0.993813i	$-0.464573\pi$
$523$	2656.84	0.222133	0.111067	+	0.993813i	$0.464573\pi$
$524$	57347.8	4.78101
$525$	0	0
$526$	−29042.7	−2.40745
$527$	4526.83	0.374178
$528$	0	0
$529$	−2699.42	−0.221864
$530$	0	0
$531$	0	0
$532$	20724.4	1.68894
$533$	8060.95	0.655081
$534$	0	0
$535$	0	0
$536$	50751.5	4.08980
$537$	0	0
$538$	−38283.9	−3.06791
$539$	778.393	0.0622037
$540$	0	0
$541$	−6340.08	−0.503847	−0.251924	−	0.967747i	$-0.581063\pi$
$541$	−6340.08	−0.503847	−0.251924	+	0.967747i	$0.581063\pi$
$542$	−3278.94	−0.259857
$543$	0	0
$544$	−26582.1	−2.09503
$545$	0	0
$546$	0	0
$547$	22317.5	1.74447	0.872235	−	0.489087i	$-0.162670\pi$
$547$	22317.5	1.74447	0.872235	+	0.489087i	$0.162670\pi$
$548$	56794.0	4.42723
$549$	0	0
$550$	0	0
$551$	−33399.1	−2.58230
$552$	0	0
$553$	8189.59	0.629759
$554$	−16231.1	−1.24476
$555$	0	0
$556$	−51599.0	−3.93576
$557$	−9476.86	−0.720911	−0.360455	−	0.932776i	$-0.617379\pi$
$557$	−9476.86	−0.720911	−0.360455	+	0.932776i	$0.617379\pi$
$558$	0	0
$559$	−17860.1	−1.35134
$560$	0	0
$561$	0	0
$562$	2243.28	0.168375
$563$	21003.7	1.57229	0.786144	−	0.618043i	$-0.212074\pi$
$563$	21003.7	1.57229	0.786144	+	0.618043i	$0.212074\pi$
$564$	0	0
$565$	0	0
$566$	−24136.5	−1.79246
$567$	0	0
$568$	−4797.48	−0.354397
$569$	22905.7	1.68762	0.843811	−	0.536640i	$-0.180307\pi$
$569$	22905.7	1.68762	0.843811	+	0.536640i	$0.180307\pi$
$570$	0	0
$571$	5035.60	0.369060	0.184530	−	0.982827i	$-0.440924\pi$
$571$	5035.60	0.369060	0.184530	+	0.982827i	$0.440924\pi$
$572$	−13617.2	−0.995392
$573$	0	0
$574$	7434.76	0.540628
$575$	0	0
$576$	0	0
$577$	7692.81	0.555036	0.277518	−	0.960720i	$-0.410488\pi$
$577$	7692.81	0.555036	0.277518	+	0.960720i	$0.410488\pi$
$578$	14426.9	1.03820
$579$	0	0
$580$	0	0
$581$	4500.30	0.321350
$582$	0	0
$583$	1027.87	0.0730191
$584$	47547.6	3.36906
$585$	0	0
$586$	−44134.4	−3.11122
$587$	4219.62	0.296699	0.148350	−	0.988935i	$-0.452604\pi$
$587$	4219.62	0.296699	0.148350	+	0.988935i	$0.452604\pi$
$588$	0	0
$589$	13517.9	0.945664
$590$	0	0
$591$	0	0
$592$	7057.14	0.489943
$593$	10710.9	0.741730	0.370865	−	0.928687i	$-0.379061\pi$
$593$	10710.9	0.741730	0.370865	+	0.928687i	$0.379061\pi$
$594$	0	0
$595$	0	0
$596$	−73113.6	−5.02492
$597$	0	0
$598$	−21401.4	−1.46349
$599$	25086.1	1.71117	0.855584	−	0.517663i	$-0.173198\pi$
$599$	25086.1	1.71117	0.855584	+	0.517663i	$0.173198\pi$
$600$	0	0
$601$	25399.6	1.72391	0.861957	−	0.506982i	$-0.169239\pi$
$601$	25399.6	1.72391	0.861957	+	0.506982i	$0.169239\pi$
$602$	−16472.6	−1.11524
$603$	0	0
$604$	9316.42	0.627616
$605$	0	0
$606$	0	0
$607$	−14501.2	−0.969665	−0.484833	−	0.874607i	$-0.661119\pi$
$607$	−14501.2	−0.969665	−0.484833	+	0.874607i	$0.661119\pi$
$608$	−79378.9	−5.29480
$609$	0	0
$610$	0	0
$611$	10483.9	0.694162
$612$	0	0
$613$	−3673.43	−0.242036	−0.121018	−	0.992650i	$-0.538616\pi$
$613$	−3673.43	−0.242036	−0.121018	+	0.992650i	$0.538616\pi$
$614$	−29974.9	−1.97018
$615$	0	0
$616$	−7770.40	−0.508244
$617$	9857.06	0.643161	0.321581	−	0.946882i	$-0.395786\pi$
$617$	9857.06	0.643161	0.321581	+	0.946882i	$0.395786\pi$
$618$	0	0
$619$	29100.2	1.88956	0.944778	−	0.327712i	$-0.106277\pi$
$619$	29100.2	1.88956	0.944778	+	0.327712i	$0.106277\pi$
$620$	0	0
$621$	0	0
$622$	−5271.11	−0.339794
$623$	9675.28	0.622202
$624$	0	0
$625$	0	0
$626$	7894.74	0.504053
$627$	0	0
$628$	−17446.4	−1.10858
$629$	1600.73	0.101471
$630$	0	0
$631$	−5707.06	−0.360055	−0.180027	−	0.983662i	$-0.557619\pi$
$631$	−5707.06	−0.360055	−0.180027	+	0.983662i	$0.557619\pi$
$632$	−81753.6	−5.14554
$633$	0	0
$634$	−50674.7	−3.17437
$635$	0	0
$636$	0	0
$637$	−2002.01	−0.124525
$638$	20240.5	1.25600
$639$	0	0
$640$	0	0
$641$	7664.18	0.472257	0.236129	−	0.971722i	$-0.424121\pi$
$641$	7664.18	0.472257	0.236129	+	0.971722i	$0.424121\pi$
$642$	0	0
$643$	−15108.9	−0.926653	−0.463326	−	0.886188i	$-0.653344\pi$
$643$	−15108.9	−0.926653	−0.463326	+	0.886188i	$0.653344\pi$
$644$	−14290.0	−0.874387
$645$	0	0
$646$	−35898.3	−2.18638
$647$	26190.4	1.59142	0.795712	−	0.605675i	$-0.207097\pi$
$647$	26190.4	1.59142	0.795712	+	0.605675i	$0.207097\pi$
$648$	0	0
$649$	12853.6	0.777424
$650$	0	0
$651$	0	0
$652$	−8793.68	−0.528201
$653$	−609.867	−0.0365482	−0.0182741	−	0.999833i	$-0.505817\pi$
$653$	−609.867	−0.0365482	−0.0182741	+	0.999833i	$0.505817\pi$
$654$	0	0
$655$	0	0
$656$	−41103.6	−2.44638
$657$	0	0
$658$	9669.49	0.572881
$659$	−8456.98	−0.499905	−0.249952	−	0.968258i	$-0.580415\pi$
$659$	−8456.98	−0.499905	−0.249952	+	0.968258i	$0.580415\pi$
$660$	0	0
$661$	11260.1	0.662582	0.331291	−	0.943529i	$-0.392516\pi$
$661$	11260.1	0.662582	0.331291	+	0.943529i	$0.392516\pi$
$662$	−35607.5	−2.09052
$663$	0	0
$664$	−44924.8	−2.62563
$665$	0	0
$666$	0	0
$667$	23029.5	1.33689
$668$	−51213.9	−2.96636
$669$	0	0
$670$	0	0
$671$	5063.07	0.291293
$672$	0	0
$673$	21034.3	1.20477	0.602386	−	0.798205i	$-0.294217\pi$
$673$	21034.3	1.20477	0.602386	+	0.798205i	$0.294217\pi$
$674$	−3863.48	−0.220795
$675$	0	0
$676$	−11070.9	−0.629889
$677$	−6576.39	−0.373340	−0.186670	−	0.982423i	$-0.559770\pi$
$677$	−6576.39	−0.373340	−0.186670	+	0.982423i	$0.559770\pi$
$678$	0	0
$679$	−2219.56	−0.125448
$680$	0	0
$681$	0	0
$682$	−8192.14	−0.459961
$683$	19573.4	1.09657	0.548283	−	0.836293i	$-0.315282\pi$
$683$	19573.4	1.09657	0.548283	+	0.836293i	$0.315282\pi$
$684$	0	0
$685$	0	0
$686$	−1846.49	−0.102769
$687$	0	0
$688$	91070.3	5.04655
$689$	−2643.67	−0.146177
$690$	0	0
$691$	10921.3	0.601251	0.300626	−	0.953742i	$-0.402805\pi$
$691$	10921.3	0.601251	0.300626	+	0.953742i	$0.402805\pi$
$692$	55193.8	3.03201
$693$	0	0
$694$	−37654.3	−2.05957
$695$	0	0
$696$	0	0
$697$	−9323.28	−0.506663
$698$	−53517.6	−2.90211
$699$	0	0
$700$	0	0
$701$	−15067.7	−0.811839	−0.405920	−	0.913909i	$-0.633049\pi$
$701$	−15067.7	−0.811839	−0.405920	+	0.913909i	$0.633049\pi$
$702$	0	0
$703$	4780.05	0.256448
$704$	21628.9	1.15791
$705$	0	0
$706$	−10752.0	−0.573166
$707$	−7982.57	−0.424633
$708$	0	0
$709$	27173.9	1.43940	0.719702	−	0.694283i	$-0.244279\pi$
$709$	27173.9	1.43940	0.719702	+	0.694283i	$0.244279\pi$
$710$	0	0
$711$	0	0
$712$	−96584.6	−5.08379
$713$	−9320.97	−0.489583
$714$	0	0
$715$	0	0
$716$	15525.3	0.810344
$717$	0	0
$718$	−17614.7	−0.915564
$719$	11572.1	0.600233	0.300117	−	0.953902i	$-0.402974\pi$
$719$	11572.1	0.600233	0.300117	+	0.953902i	$0.402974\pi$
$720$	0	0
$721$	6796.25	0.351048
$722$	−70274.2	−3.62235
$723$	0	0
$724$	−43251.9	−2.22023
$725$	0	0
$726$	0	0
$727$	−10390.6	−0.530075	−0.265038	−	0.964238i	$-0.585384\pi$
$727$	−10390.6	−0.530075	−0.265038	+	0.964238i	$0.585384\pi$
$728$	19985.3	1.01745
$729$	0	0
$730$	0	0
$731$	20656.9	1.04518
$732$	0	0
$733$	20807.9	1.04851	0.524255	−	0.851562i	$-0.324344\pi$
$733$	20807.9	1.04851	0.524255	+	0.851562i	$0.324344\pi$
$734$	68609.1	3.45015
$735$	0	0
$736$	54733.9	2.74119
$737$	−11537.4	−0.576644
$738$	0	0
$739$	8201.89	0.408270	0.204135	−	0.978943i	$-0.434562\pi$
$739$	8201.89	0.408270	0.204135	+	0.978943i	$0.434562\pi$
$740$	0	0
$741$	0	0
$742$	−2438.30	−0.120637
$743$	−6711.14	−0.331370	−0.165685	−	0.986179i	$-0.552983\pi$
$743$	−6711.14	−0.331370	−0.165685	+	0.986179i	$0.552983\pi$
$744$	0	0
$745$	0	0
$746$	−9958.99	−0.488773
$747$	0	0
$748$	15749.6	0.769871
$749$	−4951.39	−0.241549
$750$	0	0
$751$	28845.5	1.40158	0.700789	−	0.713368i	$-0.252831\pi$
$751$	28845.5	1.40158	0.700789	+	0.713368i	$0.252831\pi$
$752$	−53458.5	−2.59233
$753$	0	0
$754$	−52058.1	−2.51438
$755$	0	0
$756$	0	0
$757$	−18678.9	−0.896822	−0.448411	−	0.893827i	$-0.648010\pi$
$757$	−18678.9	−0.896822	−0.448411	+	0.893827i	$0.648010\pi$
$758$	−26016.0	−1.24663
$759$	0	0
$760$	0	0
$761$	−22486.7	−1.07115	−0.535573	−	0.844489i	$-0.679904\pi$
$761$	−22486.7	−1.07115	−0.535573	+	0.844489i	$0.679904\pi$
$762$	0	0
$763$	−10769.4	−0.510983
$764$	−84917.8	−4.02123
$765$	0	0
$766$	−19175.4	−0.904483
$767$	−33059.2	−1.55632
$768$	0	0
$769$	3982.98	0.186775	0.0933874	−	0.995630i	$-0.470230\pi$
$769$	3982.98	0.186775	0.0933874	+	0.995630i	$0.470230\pi$
$770$	0	0
$771$	0	0
$772$	−44079.1	−2.05498
$773$	−8594.11	−0.399882	−0.199941	−	0.979808i	$-0.564075\pi$
$773$	−8594.11	−0.399882	−0.199941	+	0.979808i	$0.564075\pi$
$774$	0	0
$775$	0	0
$776$	22157.0	1.02499
$777$	0	0
$778$	27428.9	1.26398
$779$	−27841.0	−1.28050
$780$	0	0
$781$	1090.62	0.0499685
$782$	24752.8	1.13192
$783$	0	0
$784$	10208.5	0.465035
$785$	0	0
$786$	0	0
$787$	−21490.9	−0.973403	−0.486702	−	0.873568i	$-0.661800\pi$
$787$	−21490.9	−0.973403	−0.486702	+	0.873568i	$0.661800\pi$
$788$	−64157.0	−2.90038
$789$	0	0
$790$	0	0
$791$	3260.14	0.146545
$792$	0	0
$793$	−13022.1	−0.583138
$794$	78120.0	3.49165
$795$	0	0
$796$	61177.2	2.72408
$797$	2703.13	0.120138	0.0600688	−	0.998194i	$-0.480868\pi$
$797$	2703.13	0.120138	0.0600688	+	0.998194i	$0.480868\pi$
$798$	0	0
$799$	−12125.7	−0.536890
$800$	0	0
$801$	0	0
$802$	26774.4	1.17885
$803$	−10809.1	−0.475024
$804$	0	0
$805$	0	0
$806$	21070.0	0.920792
$807$	0	0
$808$	79686.9	3.46952
$809$	16331.3	0.709738	0.354869	−	0.934916i	$-0.384526\pi$
$809$	16331.3	0.709738	0.354869	+	0.934916i	$0.384526\pi$
$810$	0	0
$811$	14503.0	0.627953	0.313977	−	0.949431i	$-0.398339\pi$
$811$	14503.0	0.627953	0.313977	+	0.949431i	$0.398339\pi$
$812$	−34759.9	−1.50226
$813$	0	0
$814$	−2896.81	−0.124734
$815$	0	0
$816$	0	0
$817$	61685.2	2.64148
$818$	−15297.5	−0.653869
$819$	0	0
$820$	0	0
$821$	−27641.5	−1.17503	−0.587513	−	0.809215i	$-0.699893\pi$
$821$	−27641.5	−1.17503	−0.587513	+	0.809215i	$0.699893\pi$
$822$	0	0
$823$	20193.2	0.855273	0.427636	−	0.903951i	$-0.359346\pi$
$823$	20193.2	0.855273	0.427636	+	0.903951i	$0.359346\pi$
$824$	−67844.4	−2.86829
$825$	0	0
$826$	−30491.1	−1.28441
$827$	−14333.6	−0.602696	−0.301348	−	0.953514i	$-0.597437\pi$
$827$	−14333.6	−0.602696	−0.301348	+	0.953514i	$0.597437\pi$
$828$	0	0
$829$	3214.10	0.134657	0.0673283	−	0.997731i	$-0.478553\pi$
$829$	3214.10	0.134657	0.0673283	+	0.997731i	$0.478553\pi$
$830$	0	0
$831$	0	0
$832$	−55629.1	−2.31802
$833$	2315.52	0.0963122
$834$	0	0
$835$	0	0
$836$	47031.2	1.94570
$837$	0	0
$838$	−69893.1	−2.88117
$839$	−729.209	−0.0300061	−0.0150030	−	0.999887i	$-0.504776\pi$
$839$	−729.209	−0.0300061	−0.0150030	+	0.999887i	$0.504776\pi$
$840$	0	0
$841$	31629.5	1.29688
$842$	−34041.1	−1.39327
$843$	0	0
$844$	76093.8	3.10338
$845$	0	0
$846$	0	0
$847$	−7550.54	−0.306304
$848$	13480.3	0.545892
$849$	0	0
$850$	0	0
$851$	−3295.98	−0.132767
$852$	0	0
$853$	41938.2	1.68340	0.841698	−	0.539949i	$-0.181557\pi$
$853$	41938.2	1.68340	0.841698	+	0.539949i	$0.181557\pi$
$854$	−12010.5	−0.481254
$855$	0	0
$856$	49427.9	1.97361
$857$	19103.8	0.761464	0.380732	−	0.924685i	$-0.375672\pi$
$857$	19103.8	0.761464	0.380732	+	0.924685i	$0.375672\pi$
$858$	0	0
$859$	13341.8	0.529936	0.264968	−	0.964257i	$-0.414639\pi$
$859$	13341.8	0.529936	0.264968	+	0.964257i	$0.414639\pi$
$860$	0	0
$861$	0	0
$862$	−13033.5	−0.514993
$863$	38561.8	1.52104	0.760522	−	0.649313i	$-0.224943\pi$
$863$	38561.8	1.52104	0.760522	+	0.649313i	$0.224943\pi$
$864$	0	0
$865$	0	0
$866$	−17157.4	−0.673247
$867$	0	0
$868$	14068.7	0.550142
$869$	18585.2	0.725500
$870$	0	0
$871$	29674.0	1.15438
$872$	107507.	4.17507
$873$	0	0
$874$	73916.3	2.86071
$875$	0	0
$876$	0	0
$877$	−28097.0	−1.08183	−0.540917	−	0.841076i	$-0.681923\pi$
$877$	−28097.0	−1.08183	−0.540917	+	0.841076i	$0.681923\pi$
$878$	13417.0	0.515720
$879$	0	0
$880$	0	0
$881$	12677.3	0.484801	0.242400	−	0.970176i	$-0.422065\pi$
$881$	12677.3	0.484801	0.242400	+	0.970176i	$0.422065\pi$
$882$	0	0
$883$	−26590.0	−1.01339	−0.506695	−	0.862125i	$-0.669133\pi$
$883$	−26590.0	−1.01339	−0.506695	+	0.862125i	$0.669133\pi$
$884$	−40507.7	−1.54120
$885$	0	0
$886$	−31557.3	−1.19660
$887$	10754.1	0.407090	0.203545	−	0.979066i	$-0.434754\pi$
$887$	10754.1	0.407090	0.203545	+	0.979066i	$0.434754\pi$
$888$	0	0
$889$	6753.24	0.254777
$890$	0	0
$891$	0	0
$892$	118783.	4.45868
$893$	−36209.4	−1.35689
$894$	0	0
$895$	0	0
$896$	−19806.7	−0.738500
$897$	0	0
$898$	−51980.5	−1.93164
$899$	−22672.9	−0.841139
$900$	0	0
$901$	3057.66	0.113058
$902$	16872.2	0.622819
$903$	0	0
$904$	−32544.7	−1.19737
$905$	0	0
$906$	0	0
$907$	−40275.4	−1.47445	−0.737224	−	0.675649i	$-0.763864\pi$
$907$	−40275.4	−1.47445	−0.737224	+	0.675649i	$0.763864\pi$
$908$	120572.	4.40672
$909$	0	0
$910$	0	0
$911$	19797.9	0.720015	0.360007	−	0.932949i	$-0.382774\pi$
$911$	19797.9	0.720015	0.360007	+	0.932949i	$0.382774\pi$
$912$	0	0
$913$	10212.8	0.370204
$914$	67875.7	2.45638
$915$	0	0
$916$	−35725.2	−1.28864
$917$	19133.7	0.689042
$918$	0	0
$919$	−40237.3	−1.44429	−0.722146	−	0.691740i	$-0.756844\pi$
$919$	−40237.3	−1.44429	−0.722146	+	0.691740i	$0.756844\pi$
$920$	0	0
$921$	0	0
$922$	−24827.9	−0.886838
$923$	−2805.05	−0.100032
$924$	0	0
$925$	0	0
$926$	−60025.3	−2.13019
$927$	0	0
$928$	133138.	4.70956
$929$	13545.9	0.478391	0.239195	−	0.970971i	$-0.423116\pi$
$929$	13545.9	0.478391	0.239195	+	0.970971i	$0.423116\pi$
$930$	0	0
$931$	6914.56	0.243411
$932$	20736.8	0.728816
$933$	0	0
$934$	−22107.9	−0.774511
$935$	0	0
$936$	0	0
$937$	−32849.2	−1.14529	−0.572645	−	0.819803i	$-0.694083\pi$
$937$	−32849.2	−1.14529	−0.572645	+	0.819803i	$0.694083\pi$
$938$	27368.9	0.952692
$939$	0	0
$940$	0	0
$941$	−44538.0	−1.54293	−0.771464	−	0.636272i	$-0.780475\pi$
$941$	−44538.0	−1.54293	−0.771464	+	0.636272i	$0.780475\pi$
$942$	0	0
$943$	19197.1	0.662931
$944$	168572.	5.81203
$945$	0	0
$946$	−37382.5	−1.28479
$947$	−1052.01	−0.0360991	−0.0180496	−	0.999837i	$-0.505746\pi$
$947$	−1052.01	−0.0360991	−0.0180496	+	0.999837i	$0.505746\pi$
$948$	0	0
$949$	27800.7	0.950947
$950$	0	0
$951$	0	0
$952$	−23115.0	−0.786933
$953$	35657.1	1.21201	0.606005	−	0.795461i	$-0.292771\pi$
$953$	35657.1	1.21201	0.606005	+	0.795461i	$0.292771\pi$
$954$	0	0
$955$	0	0
$956$	−6335.33	−0.214330
$957$	0	0
$958$	24546.6	0.827835
$959$	18949.0	0.638055
$960$	0	0
$961$	−20614.4	−0.691967
$962$	7450.53	0.249703
$963$	0	0
$964$	−85060.9	−2.84194
$965$	0	0
$966$	0	0
$967$	−3425.21	−0.113906	−0.0569531	−	0.998377i	$-0.518139\pi$
$967$	−3425.21	−0.113906	−0.0569531	+	0.998377i	$0.518139\pi$
$968$	75374.2	2.50270
$969$	0	0
$970$	0	0
$971$	15544.8	0.513754	0.256877	−	0.966444i	$-0.417306\pi$
$971$	15544.8	0.513754	0.256877	+	0.966444i	$0.417306\pi$
$972$	0	0
$973$	−17215.7	−0.567224
$974$	−20750.5	−0.682636
$975$	0	0
$976$	66401.1	2.17771
$977$	48799.1	1.59798	0.798988	−	0.601347i	$-0.205369\pi$
$977$	48799.1	1.59798	0.798988	+	0.601347i	$0.205369\pi$
$978$	0	0
$979$	21956.8	0.716794
$980$	0	0
$981$	0	0
$982$	57170.7	1.85783
$983$	−39839.1	−1.29264	−0.646322	−	0.763065i	$-0.723694\pi$
$983$	−39839.1	−1.29264	−0.646322	+	0.763065i	$0.723694\pi$
$984$	0	0
$985$	0	0
$986$	60210.4	1.94471
$987$	0	0
$988$	−120963.	−3.89509
$989$	−42533.6	−1.36753
$990$	0	0
$991$	4833.44	0.154934	0.0774668	−	0.996995i	$-0.475317\pi$
$991$	4833.44	0.154934	0.0774668	+	0.996995i	$0.475317\pi$
$992$	−53886.3	−1.72469
$993$	0	0
$994$	−2587.15	−0.0825546
$995$	0	0
$996$	0	0
$997$	2299.66	0.0730502	0.0365251	−	0.999333i	$-0.488371\pi$
$997$	2299.66	0.0730502	0.0365251	+	0.999333i	$0.488371\pi$
$998$	24464.8	0.775971
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.4.a.bs.1.1	yes	8
3.2	odd	2	inner	1575.4.a.bs.1.8	yes	8
5.4	even	2		1575.4.a.br.1.8	yes	8
15.14	odd	2		1575.4.a.br.1.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1575.4.a.br.1.1	✓	8	15.14	odd	2
1575.4.a.br.1.8	yes	8	5.4	even	2
1575.4.a.bs.1.1	yes	8	1.1	even	1	trivial
1575.4.a.bs.1.8	yes	8	3.2	odd	2	inner

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

\(f(q)\)	\(=\)	\(q-5.38335 q^{2} +20.9805 q^{4} +7.00000 q^{7} -69.8783 q^{8} +15.8856 q^{11} -40.8573 q^{13} -37.6834 q^{14} +208.336 q^{16} +47.2555 q^{17} +141.113 q^{19} -85.5176 q^{22} -97.3015 q^{23} +219.949 q^{26} +146.863 q^{28} -236.682 q^{29} +95.7947 q^{31} -562.518 q^{32} -254.393 q^{34} +33.8738 q^{37} -759.663 q^{38} -197.295 q^{41} +437.132 q^{43} +333.287 q^{44} +523.808 q^{46} -256.598 q^{47} +49.0000 q^{49} -857.205 q^{52} +64.7048 q^{53} -489.148 q^{56} +1274.14 q^{58} +809.137 q^{59} +318.721 q^{61} -515.696 q^{62} +1361.55 q^{64} -726.284 q^{67} +991.443 q^{68} +68.6547 q^{71} -680.433 q^{73} -182.355 q^{74} +2960.62 q^{76} +111.199 q^{77} +1169.94 q^{79} +1062.11 q^{82} +642.901 q^{83} -2353.24 q^{86} -1110.06 q^{88} +1382.18 q^{89} -286.001 q^{91} -2041.43 q^{92} +1381.36 q^{94} -317.080 q^{97} -263.784 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 36 q^{4} + 56 q^{7} - 38 q^{13} + 320 q^{16} + 68 q^{19} + 110 q^{22} + 252 q^{28} + 534 q^{31} + 118 q^{34} + 14 q^{37} + 816 q^{43} + 1588 q^{46} + 392 q^{49} - 266 q^{52} + 716 q^{58} + 874 q^{61}+ \cdots + 428 q^{97}+O(q^{100}) \) 8 * q + 36 * q^4 + 56 * q^7 - 38 * q^13 + 320 * q^16 + 68 * q^19 + 110 * q^22 + 252 * q^28 + 534 * q^31 + 118 * q^34 + 14 * q^37 + 816 * q^43 + 1588 * q^46 + 392 * q^49 - 266 * q^52 + 716 * q^58 + 874 * q^61 + 2502 * q^64 + 254 * q^67 - 1532 * q^73 + 4304 * q^76 + 5190 * q^79 + 1158 * q^82 - 4300 * q^88 - 266 * q^91 + 6236 * q^94 + 428 * q^97

Introduction

L-functions

Modular forms

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Database

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