LMFDB - Embedded newform 315.4.a.a.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [315,4,Mod(1,315)] 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("315.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(315, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 315 = 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	315.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-5,0,17] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	yes
Analytic conductor:	$18.5856016518$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	315.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q-5.00000 q^{2} +17.0000 q^{4} -5.00000 q^{5} +7.00000 q^{7} -45.0000 q^{8} +25.0000 q^{10} -12.0000 q^{11} +30.0000 q^{13} -35.0000 q^{14} +89.0000 q^{16} +134.000 q^{17} -92.0000 q^{19} -85.0000 q^{20} +60.0000 q^{22} -112.000 q^{23} +25.0000 q^{25} -150.000 q^{26} +119.000 q^{28} +58.0000 q^{29} -224.000 q^{31} -85.0000 q^{32} -670.000 q^{34} -35.0000 q^{35} -146.000 q^{37} +460.000 q^{38} +225.000 q^{40} -18.0000 q^{41} +340.000 q^{43} -204.000 q^{44} +560.000 q^{46} -208.000 q^{47} +49.0000 q^{49} -125.000 q^{50} +510.000 q^{52} +754.000 q^{53} +60.0000 q^{55} -315.000 q^{56} -290.000 q^{58} -380.000 q^{59} +718.000 q^{61} +1120.00 q^{62} -287.000 q^{64} -150.000 q^{65} +412.000 q^{67} +2278.00 q^{68} +175.000 q^{70} +960.000 q^{71} +1066.00 q^{73} +730.000 q^{74} -1564.00 q^{76} -84.0000 q^{77} +896.000 q^{79} -445.000 q^{80} +90.0000 q^{82} -436.000 q^{83} -670.000 q^{85} -1700.00 q^{86} +540.000 q^{88} +1038.00 q^{89} +210.000 q^{91} -1904.00 q^{92} +1040.00 q^{94} +460.000 q^{95} -702.000 q^{97} -245.000 q^{98} +O(q^{100})\)

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 $

$2$	−5.00000	−1.76777	−0.883883	−	0.467707i	$-0.845080\pi$
$2$	−5.00000	−1.76777	−0.883883	+	0.467707i	$0.845080\pi$
$3$	0	0
$3$	0	0
$4$	17.0000	2.12500
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	−45.0000	−1.98874
$9$	0	0
$10$	25.0000	0.790569
$11$	−12.0000	−0.328921	−0.164461	−	0.986384i	$-0.552588\pi$
$11$	−12.0000	−0.328921	−0.164461	+	0.986384i	$0.552588\pi$
$12$	0	0
$13$	30.0000	0.640039	0.320019	−	0.947411i	$-0.396311\pi$
$13$	30.0000	0.640039	0.320019	+	0.947411i	$0.396311\pi$
$14$	−35.0000	−0.668153
$15$	0	0
$16$	89.0000	1.39062
$17$	134.000	1.91175	0.955876	−	0.293771i	$-0.0949105\pi$
$17$	134.000	1.91175	0.955876	+	0.293771i	$0.0949105\pi$
$18$	0	0
$19$	−92.0000	−1.11086	−0.555428	−	0.831565i	$-0.687445\pi$
$19$	−92.0000	−1.11086	−0.555428	+	0.831565i	$0.687445\pi$
$20$	−85.0000	−0.950329
$21$	0	0
$22$	60.0000	0.581456
$23$	−112.000	−1.01537	−0.507687	−	0.861541i	$-0.669499\pi$
$23$	−112.000	−1.01537	−0.507687	+	0.861541i	$0.669499\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	−150.000	−1.13144
$27$	0	0
$28$	119.000	0.803175
$29$	58.0000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	58.0000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	−224.000	−1.29779	−0.648897	−	0.760877i	$-0.724769\pi$
$31$	−224.000	−1.29779	−0.648897	+	0.760877i	$0.724769\pi$
$32$	−85.0000	−0.469563
$33$	0	0
$34$	−670.000	−3.37953
$35$	−35.0000	−0.169031
$36$	0	0
$37$	−146.000	−0.648710	−0.324355	−	0.945936i	$-0.605147\pi$
$37$	−146.000	−0.648710	−0.324355	+	0.945936i	$0.605147\pi$
$38$	460.000	1.96373
$39$	0	0
$40$	225.000	0.889391
$41$	−18.0000	−0.0685641	−0.0342820	−	0.999412i	$-0.510914\pi$
$41$	−18.0000	−0.0685641	−0.0342820	+	0.999412i	$0.510914\pi$
$42$	0	0
$43$	340.000	1.20580	0.602901	−	0.797816i	$-0.294011\pi$
$43$	340.000	1.20580	0.602901	+	0.797816i	$0.294011\pi$
$44$	−204.000	−0.698958
$45$	0	0
$46$	560.000	1.79495
$47$	−208.000	−0.645530	−0.322765	−	0.946479i	$-0.604612\pi$
$47$	−208.000	−0.645530	−0.322765	+	0.946479i	$0.604612\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	−125.000	−0.353553
$51$	0	0
$52$	510.000	1.36008
$53$	754.000	1.95415	0.977074	−	0.212899i	$-0.0682905\pi$
$53$	754.000	1.95415	0.977074	+	0.212899i	$0.0682905\pi$
$54$	0	0
$55$	60.0000	0.147098
$56$	−315.000	−0.751672
$57$	0	0
$58$	−290.000	−0.656532
$59$	−380.000	−0.838505	−0.419252	−	0.907870i	$-0.637708\pi$
$59$	−380.000	−0.838505	−0.419252	+	0.907870i	$0.637708\pi$
$60$	0	0
$61$	718.000	1.50706	0.753529	−	0.657415i	$-0.228350\pi$
$61$	718.000	1.50706	0.753529	+	0.657415i	$0.228350\pi$
$62$	1120.00	2.29420
$63$	0	0
$64$	−287.000	−0.560547
$65$	−150.000	−0.286234
$66$	0	0
$67$	412.000	0.751251	0.375625	−	0.926772i	$-0.377428\pi$
$67$	412.000	0.751251	0.375625	+	0.926772i	$0.377428\pi$
$68$	2278.00	4.06247
$69$	0	0
$70$	175.000	0.298807
$71$	960.000	1.60466	0.802331	−	0.596879i	$-0.203593\pi$
$71$	960.000	1.60466	0.802331	+	0.596879i	$0.203593\pi$
$72$	0	0
$73$	1066.00	1.70912	0.854561	−	0.519352i	$-0.173826\pi$
$73$	1066.00	1.70912	0.854561	+	0.519352i	$0.173826\pi$
$74$	730.000	1.14677
$75$	0	0
$76$	−1564.00	−2.36057
$77$	−84.0000	−0.124321
$78$	0	0
$79$	896.000	1.27605	0.638025	−	0.770016i	$-0.279752\pi$
$79$	896.000	1.27605	0.638025	+	0.770016i	$0.279752\pi$
$80$	−445.000	−0.621906
$81$	0	0
$82$	90.0000	0.121205
$83$	−436.000	−0.576593	−0.288296	−	0.957541i	$-0.593089\pi$
$83$	−436.000	−0.576593	−0.288296	+	0.957541i	$0.593089\pi$
$84$	0	0
$85$	−670.000	−0.854961
$86$	−1700.00	−2.13158
$87$	0	0
$88$	540.000	0.654139
$89$	1038.00	1.23627	0.618134	−	0.786073i	$-0.287889\pi$
$89$	1038.00	1.23627	0.618134	+	0.786073i	$0.287889\pi$
$90$	0	0
$91$	210.000	0.241912
$92$	−1904.00	−2.15767
$93$	0	0
$94$	1040.00	1.14115
$95$	460.000	0.496790
$96$	0	0
$97$	−702.000	−0.734818	−0.367409	−	0.930060i	$-0.619755\pi$
$97$	−702.000	−0.734818	−0.367409	+	0.930060i	$0.619755\pi$
$98$	−245.000	−0.252538
$99$	0	0
$100$	425.000	0.425000
$101$	−46.0000	−0.0453185	−0.0226593	−	0.999743i	$-0.507213\pi$
$101$	−46.0000	−0.0453185	−0.0226593	+	0.999743i	$0.507213\pi$
$102$	0	0
$103$	1880.00	1.79847	0.899233	−	0.437471i	$-0.144126\pi$
$103$	1880.00	1.79847	0.899233	+	0.437471i	$0.144126\pi$
$104$	−1350.00	−1.27287
$105$	0	0
$106$	−3770.00	−3.45448
$107$	−732.000	−0.661356	−0.330678	−	0.943744i	$-0.607277\pi$
$107$	−732.000	−0.661356	−0.330678	+	0.943744i	$0.607277\pi$
$108$	0	0
$109$	−378.000	−0.332164	−0.166082	−	0.986112i	$-0.553112\pi$
$109$	−378.000	−0.332164	−0.166082	+	0.986112i	$0.553112\pi$
$110$	−300.000	−0.260035
$111$	0	0
$112$	623.000	0.525607
$113$	−1458.00	−1.21378	−0.606890	−	0.794786i	$-0.707583\pi$
$113$	−1458.00	−1.21378	−0.606890	+	0.794786i	$0.707583\pi$
$114$	0	0
$115$	560.000	0.454089
$116$	986.000	0.789205
$117$	0	0
$118$	1900.00	1.48228
$119$	938.000	0.722574
$120$	0	0
$121$	−1187.00	−0.891811
$122$	−3590.00	−2.66413
$123$	0	0
$124$	−3808.00	−2.75781
$125$	−125.000	−0.0894427
$126$	0	0
$127$	608.000	0.424813	0.212407	−	0.977181i	$-0.431870\pi$
$127$	608.000	0.424813	0.212407	+	0.977181i	$0.431870\pi$
$128$	2115.00	1.46048
$129$	0	0
$130$	750.000	0.505995
$131$	956.000	0.637604	0.318802	−	0.947821i	$-0.396720\pi$
$131$	956.000	0.637604	0.318802	+	0.947821i	$0.396720\pi$
$132$	0	0
$133$	−644.000	−0.419864
$134$	−2060.00	−1.32804
$135$	0	0
$136$	−6030.00	−3.80197
$137$	374.000	0.233233	0.116617	−	0.993177i	$-0.462795\pi$
$137$	374.000	0.233233	0.116617	+	0.993177i	$0.462795\pi$
$138$	0	0
$139$	396.000	0.241642	0.120821	−	0.992674i	$-0.461447\pi$
$139$	396.000	0.241642	0.120821	+	0.992674i	$0.461447\pi$
$140$	−595.000	−0.359191
$141$	0	0
$142$	−4800.00	−2.83667
$143$	−360.000	−0.210522
$144$	0	0
$145$	−290.000	−0.166091
$146$	−5330.00	−3.02133
$147$	0	0
$148$	−2482.00	−1.37851
$149$	1874.00	1.03036	0.515181	−	0.857081i	$-0.327725\pi$
$149$	1874.00	1.03036	0.515181	+	0.857081i	$0.327725\pi$
$150$	0	0
$151$	−1096.00	−0.590670	−0.295335	−	0.955394i	$-0.595431\pi$
$151$	−1096.00	−0.590670	−0.295335	+	0.955394i	$0.595431\pi$
$152$	4140.00	2.20920
$153$	0	0
$154$	420.000	0.219770
$155$	1120.00	0.580391
$156$	0	0
$157$	1918.00	0.974988	0.487494	−	0.873126i	$-0.337911\pi$
$157$	1918.00	0.974988	0.487494	+	0.873126i	$0.337911\pi$
$158$	−4480.00	−2.25576
$159$	0	0
$160$	425.000	0.209995
$161$	−784.000	−0.383776
$162$	0	0
$163$	2316.00	1.11290	0.556451	−	0.830880i	$-0.312163\pi$
$163$	2316.00	1.11290	0.556451	+	0.830880i	$0.312163\pi$
$164$	−306.000	−0.145699
$165$	0	0
$166$	2180.00	1.01928
$167$	1736.00	0.804405	0.402203	−	0.915551i	$-0.368245\pi$
$167$	1736.00	0.804405	0.402203	+	0.915551i	$0.368245\pi$
$168$	0	0
$169$	−1297.00	−0.590350
$170$	3350.00	1.51137
$171$	0	0
$172$	5780.00	2.56233
$173$	2442.00	1.07319	0.536595	−	0.843840i	$-0.319710\pi$
$173$	2442.00	1.07319	0.536595	+	0.843840i	$0.319710\pi$
$174$	0	0
$175$	175.000	0.0755929
$176$	−1068.00	−0.457406
$177$	0	0
$178$	−5190.00	−2.18543
$179$	4092.00	1.70866	0.854331	−	0.519730i	$-0.173967\pi$
$179$	4092.00	1.70866	0.854331	+	0.519730i	$0.173967\pi$
$180$	0	0
$181$	1270.00	0.521538	0.260769	−	0.965401i	$-0.416024\pi$
$181$	1270.00	0.521538	0.260769	+	0.965401i	$0.416024\pi$
$182$	−1050.00	−0.427644
$183$	0	0
$184$	5040.00	2.01931
$185$	730.000	0.290112
$186$	0	0
$187$	−1608.00	−0.628816
$188$	−3536.00	−1.37175
$189$	0	0
$190$	−2300.00	−0.878208
$191$	−4904.00	−1.85781	−0.928903	−	0.370323i	$-0.879247\pi$
$191$	−4904.00	−1.85781	−0.928903	+	0.370323i	$0.879247\pi$
$192$	0	0
$193$	2178.00	0.812310	0.406155	−	0.913804i	$-0.366869\pi$
$193$	2178.00	0.812310	0.406155	+	0.913804i	$0.366869\pi$
$194$	3510.00	1.29899
$195$	0	0
$196$	833.000	0.303571
$197$	2850.00	1.03073	0.515366	−	0.856970i	$-0.327656\pi$
$197$	2850.00	1.03073	0.515366	+	0.856970i	$0.327656\pi$
$198$	0	0
$199$	−1144.00	−0.407518	−0.203759	−	0.979021i	$-0.565316\pi$
$199$	−1144.00	−0.407518	−0.203759	+	0.979021i	$0.565316\pi$
$200$	−1125.00	−0.397748
$201$	0	0
$202$	230.000	0.0801126
$203$	406.000	0.140372
$204$	0	0
$205$	90.0000	0.0306628
$206$	−9400.00	−3.17927
$207$	0	0
$208$	2670.00	0.890054
$209$	1104.00	0.365384
$210$	0	0
$211$	412.000	0.134423	0.0672115	−	0.997739i	$-0.478590\pi$
$211$	412.000	0.134423	0.0672115	+	0.997739i	$0.478590\pi$
$212$	12818.0	4.15257
$213$	0	0
$214$	3660.00	1.16912
$215$	−1700.00	−0.539251
$216$	0	0
$217$	−1568.00	−0.490520
$218$	1890.00	0.587188
$219$	0	0
$220$	1020.00	0.312584
$221$	4020.00	1.22359
$222$	0	0
$223$	−1632.00	−0.490075	−0.245038	−	0.969514i	$-0.578800\pi$
$223$	−1632.00	−0.490075	−0.245038	+	0.969514i	$0.578800\pi$
$224$	−595.000	−0.177478
$225$	0	0
$226$	7290.00	2.14568
$227$	−4084.00	−1.19412	−0.597059	−	0.802198i	$-0.703664\pi$
$227$	−4084.00	−1.19412	−0.597059	+	0.802198i	$0.703664\pi$
$228$	0	0
$229$	−3386.00	−0.977088	−0.488544	−	0.872539i	$-0.662472\pi$
$229$	−3386.00	−0.977088	−0.488544	+	0.872539i	$0.662472\pi$
$230$	−2800.00	−0.802724
$231$	0	0
$232$	−2610.00	−0.738599
$233$	−5322.00	−1.49638	−0.748188	−	0.663486i	$-0.769076\pi$
$233$	−5322.00	−1.49638	−0.748188	+	0.663486i	$0.769076\pi$
$234$	0	0
$235$	1040.00	0.288690
$236$	−6460.00	−1.78182
$237$	0	0
$238$	−4690.00	−1.27734
$239$	−3736.00	−1.01114	−0.505569	−	0.862786i	$-0.668717\pi$
$239$	−3736.00	−1.01114	−0.505569	+	0.862786i	$0.668717\pi$
$240$	0	0
$241$	210.000	0.0561298	0.0280649	−	0.999606i	$-0.491065\pi$
$241$	210.000	0.0561298	0.0280649	+	0.999606i	$0.491065\pi$
$242$	5935.00	1.57651
$243$	0	0
$244$	12206.0	3.20250
$245$	−245.000	−0.0638877
$246$	0	0
$247$	−2760.00	−0.710990
$248$	10080.0	2.58097
$249$	0	0
$250$	625.000	0.158114
$251$	4212.00	1.05920	0.529600	−	0.848248i	$-0.322342\pi$
$251$	4212.00	1.05920	0.529600	+	0.848248i	$0.322342\pi$
$252$	0	0
$253$	1344.00	0.333978
$254$	−3040.00	−0.750971
$255$	0	0
$256$	−8279.00	−2.02124
$257$	−5130.00	−1.24514	−0.622569	−	0.782565i	$-0.713911\pi$
$257$	−5130.00	−1.24514	−0.622569	+	0.782565i	$0.713911\pi$
$258$	0	0
$259$	−1022.00	−0.245189
$260$	−2550.00	−0.608247
$261$	0	0
$262$	−4780.00	−1.12714
$263$	−848.000	−0.198821	−0.0994105	−	0.995047i	$-0.531696\pi$
$263$	−848.000	−0.198821	−0.0994105	+	0.995047i	$0.531696\pi$
$264$	0	0
$265$	−3770.00	−0.873922
$266$	3220.00	0.742221
$267$	0	0
$268$	7004.00	1.59641
$269$	1274.00	0.288763	0.144381	−	0.989522i	$-0.453881\pi$
$269$	1274.00	0.288763	0.144381	+	0.989522i	$0.453881\pi$
$270$	0	0
$271$	864.000	0.193669	0.0968344	−	0.995301i	$-0.469128\pi$
$271$	864.000	0.193669	0.0968344	+	0.995301i	$0.469128\pi$
$272$	11926.0	2.65853
$273$	0	0
$274$	−1870.00	−0.412302
$275$	−300.000	−0.0657843
$276$	0	0
$277$	−8530.00	−1.85025	−0.925123	−	0.379668i	$-0.876038\pi$
$277$	−8530.00	−1.85025	−0.925123	+	0.379668i	$0.876038\pi$
$278$	−1980.00	−0.427167
$279$	0	0
$280$	1575.00	0.336158
$281$	5382.00	1.14257	0.571287	−	0.820750i	$-0.306444\pi$
$281$	5382.00	1.14257	0.571287	+	0.820750i	$0.306444\pi$
$282$	0	0
$283$	6236.00	1.30986	0.654932	−	0.755687i	$-0.272697\pi$
$283$	6236.00	1.30986	0.654932	+	0.755687i	$0.272697\pi$
$284$	16320.0	3.40991
$285$	0	0
$286$	1800.00	0.372155
$287$	−126.000	−0.0259148
$288$	0	0
$289$	13043.0	2.65479
$290$	1450.00	0.293610
$291$	0	0
$292$	18122.0	3.63188
$293$	818.000	0.163099	0.0815496	−	0.996669i	$-0.474013\pi$
$293$	818.000	0.163099	0.0815496	+	0.996669i	$0.474013\pi$
$294$	0	0
$295$	1900.00	0.374991
$296$	6570.00	1.29011
$297$	0	0
$298$	−9370.00	−1.82144
$299$	−3360.00	−0.649879
$300$	0	0
$301$	2380.00	0.455751
$302$	5480.00	1.04417
$303$	0	0
$304$	−8188.00	−1.54478
$305$	−3590.00	−0.673976
$306$	0	0
$307$	−2268.00	−0.421634	−0.210817	−	0.977526i	$-0.567612\pi$
$307$	−2268.00	−0.421634	−0.210817	+	0.977526i	$0.567612\pi$
$308$	−1428.00	−0.264181
$309$	0	0
$310$	−5600.00	−1.02600
$311$	−6648.00	−1.21213	−0.606067	−	0.795414i	$-0.707254\pi$
$311$	−6648.00	−1.21213	−0.606067	+	0.795414i	$0.707254\pi$
$312$	0	0
$313$	9818.00	1.77299	0.886495	−	0.462737i	$-0.153133\pi$
$313$	9818.00	1.77299	0.886495	+	0.462737i	$0.153133\pi$
$314$	−9590.00	−1.72355
$315$	0	0
$316$	15232.0	2.71160
$317$	−934.000	−0.165485	−0.0827424	−	0.996571i	$-0.526368\pi$
$317$	−934.000	−0.165485	−0.0827424	+	0.996571i	$0.526368\pi$
$318$	0	0
$319$	−696.000	−0.122158
$320$	1435.00	0.250684
$321$	0	0
$322$	3920.00	0.678426
$323$	−12328.0	−2.12368
$324$	0	0
$325$	750.000	0.128008
$326$	−11580.0	−1.96735
$327$	0	0
$328$	810.000	0.136356
$329$	−1456.00	−0.243987
$330$	0	0
$331$	2292.00	0.380603	0.190302	−	0.981726i	$-0.439053\pi$
$331$	2292.00	0.380603	0.190302	+	0.981726i	$0.439053\pi$
$332$	−7412.00	−1.22526
$333$	0	0
$334$	−8680.00	−1.42200
$335$	−2060.00	−0.335970
$336$	0	0
$337$	−6062.00	−0.979876	−0.489938	−	0.871757i	$-0.662981\pi$
$337$	−6062.00	−0.979876	−0.489938	+	0.871757i	$0.662981\pi$
$338$	6485.00	1.04360
$339$	0	0
$340$	−11390.0	−1.81679
$341$	2688.00	0.426872
$342$	0	0
$343$	343.000	0.0539949
$344$	−15300.0	−2.39803
$345$	0	0
$346$	−12210.0	−1.89715
$347$	−1484.00	−0.229583	−0.114791	−	0.993390i	$-0.536620\pi$
$347$	−1484.00	−0.229583	−0.114791	+	0.993390i	$0.536620\pi$
$348$	0	0
$349$	254.000	0.0389579	0.0194790	−	0.999810i	$-0.493799\pi$
$349$	254.000	0.0389579	0.0194790	+	0.999810i	$0.493799\pi$
$350$	−875.000	−0.133631
$351$	0	0
$352$	1020.00	0.154449
$353$	10950.0	1.65102	0.825509	−	0.564388i	$-0.190888\pi$
$353$	10950.0	1.65102	0.825509	+	0.564388i	$0.190888\pi$
$354$	0	0
$355$	−4800.00	−0.717627
$356$	17646.0	2.62707
$357$	0	0
$358$	−20460.0	−3.02052
$359$	−11376.0	−1.67243	−0.836215	−	0.548402i	$-0.815236\pi$
$359$	−11376.0	−1.67243	−0.836215	+	0.548402i	$0.815236\pi$
$360$	0	0
$361$	1605.00	0.233999
$362$	−6350.00	−0.921957
$363$	0	0
$364$	3570.00	0.514063
$365$	−5330.00	−0.764342
$366$	0	0
$367$	−1136.00	−0.161577	−0.0807884	−	0.996731i	$-0.525744\pi$
$367$	−1136.00	−0.161577	−0.0807884	+	0.996731i	$0.525744\pi$
$368$	−9968.00	−1.41201
$369$	0	0
$370$	−3650.00	−0.512850
$371$	5278.00	0.738599
$372$	0	0
$373$	−8242.00	−1.14411	−0.572057	−	0.820214i	$-0.693854\pi$
$373$	−8242.00	−1.14411	−0.572057	+	0.820214i	$0.693854\pi$
$374$	8040.00	1.11160
$375$	0	0
$376$	9360.00	1.28379
$377$	1740.00	0.237704
$378$	0	0
$379$	3620.00	0.490625	0.245313	−	0.969444i	$-0.421109\pi$
$379$	3620.00	0.490625	0.245313	+	0.969444i	$0.421109\pi$
$380$	7820.00	1.05568
$381$	0	0
$382$	24520.0	3.28417
$383$	8464.00	1.12922	0.564609	−	0.825359i	$-0.309027\pi$
$383$	8464.00	1.12922	0.564609	+	0.825359i	$0.309027\pi$
$384$	0	0
$385$	420.000	0.0555979
$386$	−10890.0	−1.43598
$387$	0	0
$388$	−11934.0	−1.56149
$389$	−3678.00	−0.479388	−0.239694	−	0.970848i	$-0.577047\pi$
$389$	−3678.00	−0.479388	−0.239694	+	0.970848i	$0.577047\pi$
$390$	0	0
$391$	−15008.0	−1.94114
$392$	−2205.00	−0.284105
$393$	0	0
$394$	−14250.0	−1.82209
$395$	−4480.00	−0.570666
$396$	0	0
$397$	12590.0	1.59162	0.795811	−	0.605545i	$-0.207045\pi$
$397$	12590.0	1.59162	0.795811	+	0.605545i	$0.207045\pi$
$398$	5720.00	0.720396
$399$	0	0
$400$	2225.00	0.278125
$401$	−2850.00	−0.354918	−0.177459	−	0.984128i	$-0.556788\pi$
$401$	−2850.00	−0.354918	−0.177459	+	0.984128i	$0.556788\pi$
$402$	0	0
$403$	−6720.00	−0.830638
$404$	−782.000	−0.0963019
$405$	0	0
$406$	−2030.00	−0.248146
$407$	1752.00	0.213374
$408$	0	0
$409$	1226.00	0.148220	0.0741098	−	0.997250i	$-0.476388\pi$
$409$	1226.00	0.148220	0.0741098	+	0.997250i	$0.476388\pi$
$410$	−450.000	−0.0542047
$411$	0	0
$412$	31960.0	3.82174
$413$	−2660.00	−0.316925
$414$	0	0
$415$	2180.00	0.257860
$416$	−2550.00	−0.300539
$417$	0	0
$418$	−5520.00	−0.645914
$419$	−612.000	−0.0713560	−0.0356780	−	0.999363i	$-0.511359\pi$
$419$	−612.000	−0.0713560	−0.0356780	+	0.999363i	$0.511359\pi$
$420$	0	0
$421$	5182.00	0.599894	0.299947	−	0.953956i	$-0.403031\pi$
$421$	5182.00	0.599894	0.299947	+	0.953956i	$0.403031\pi$
$422$	−2060.00	−0.237629
$423$	0	0
$424$	−33930.0	−3.88629
$425$	3350.00	0.382350
$426$	0	0
$427$	5026.00	0.569614
$428$	−12444.0	−1.40538
$429$	0	0
$430$	8500.00	0.953271
$431$	4984.00	0.557009	0.278504	−	0.960435i	$-0.410161\pi$
$431$	4984.00	0.557009	0.278504	+	0.960435i	$0.410161\pi$
$432$	0	0
$433$	−1694.00	−0.188010	−0.0940051	−	0.995572i	$-0.529967\pi$
$433$	−1694.00	−0.188010	−0.0940051	+	0.995572i	$0.529967\pi$
$434$	7840.00	0.867125
$435$	0	0
$436$	−6426.00	−0.705848
$437$	10304.0	1.12793
$438$	0	0
$439$	13864.0	1.50727	0.753636	−	0.657292i	$-0.228298\pi$
$439$	13864.0	1.50727	0.753636	+	0.657292i	$0.228298\pi$
$440$	−2700.00	−0.292540
$441$	0	0
$442$	−20100.0	−2.16303
$443$	4644.00	0.498066	0.249033	−	0.968495i	$-0.419887\pi$
$443$	4644.00	0.498066	0.249033	+	0.968495i	$0.419887\pi$
$444$	0	0
$445$	−5190.00	−0.552875
$446$	8160.00	0.866339
$447$	0	0
$448$	−2009.00	−0.211867
$449$	4926.00	0.517756	0.258878	−	0.965910i	$-0.416647\pi$
$449$	4926.00	0.517756	0.258878	+	0.965910i	$0.416647\pi$
$450$	0	0
$451$	216.000	0.0225522
$452$	−24786.0	−2.57928
$453$	0	0
$454$	20420.0	2.11092
$455$	−1050.00	−0.108186
$456$	0	0
$457$	−14694.0	−1.50406	−0.752031	−	0.659128i	$-0.770926\pi$
$457$	−14694.0	−1.50406	−0.752031	+	0.659128i	$0.770926\pi$
$458$	16930.0	1.72726
$459$	0	0
$460$	9520.00	0.964940
$461$	−2006.00	−0.202665	−0.101333	−	0.994853i	$-0.532311\pi$
$461$	−2006.00	−0.202665	−0.101333	+	0.994853i	$0.532311\pi$
$462$	0	0
$463$	4896.00	0.491439	0.245720	−	0.969341i	$-0.420976\pi$
$463$	4896.00	0.491439	0.245720	+	0.969341i	$0.420976\pi$
$464$	5162.00	0.516465
$465$	0	0
$466$	26610.0	2.64525
$467$	−2660.00	−0.263576	−0.131788	−	0.991278i	$-0.542072\pi$
$467$	−2660.00	−0.263576	−0.131788	+	0.991278i	$0.542072\pi$
$468$	0	0
$469$	2884.00	0.283946
$470$	−5200.00	−0.510336
$471$	0	0
$472$	17100.0	1.66757
$473$	−4080.00	−0.396614
$474$	0	0
$475$	−2300.00	−0.222171
$476$	15946.0	1.53547
$477$	0	0
$478$	18680.0	1.78745
$479$	5600.00	0.534176	0.267088	−	0.963672i	$-0.413938\pi$
$479$	5600.00	0.534176	0.267088	+	0.963672i	$0.413938\pi$
$480$	0	0
$481$	−4380.00	−0.415199
$482$	−1050.00	−0.0992245
$483$	0	0
$484$	−20179.0	−1.89510
$485$	3510.00	0.328620
$486$	0	0
$487$	−6424.00	−0.597740	−0.298870	−	0.954294i	$-0.596610\pi$
$487$	−6424.00	−0.597740	−0.298870	+	0.954294i	$0.596610\pi$
$488$	−32310.0	−2.99714
$489$	0	0
$490$	1225.00	0.112938
$491$	18900.0	1.73716	0.868579	−	0.495550i	$-0.165033\pi$
$491$	18900.0	1.73716	0.868579	+	0.495550i	$0.165033\pi$
$492$	0	0
$493$	7772.00	0.710007
$494$	13800.0	1.25687
$495$	0	0
$496$	−19936.0	−1.80474
$497$	6720.00	0.606505
$498$	0	0
$499$	−15364.0	−1.37833	−0.689165	−	0.724604i	$-0.742023\pi$
$499$	−15364.0	−1.37833	−0.689165	+	0.724604i	$0.742023\pi$
$500$	−2125.00	−0.190066
$501$	0	0
$502$	−21060.0	−1.87242
$503$	−2216.00	−0.196435	−0.0982173	−	0.995165i	$-0.531314\pi$
$503$	−2216.00	−0.196435	−0.0982173	+	0.995165i	$0.531314\pi$
$504$	0	0
$505$	230.000	0.0202671
$506$	−6720.00	−0.590396
$507$	0	0
$508$	10336.0	0.902728
$509$	3754.00	0.326902	0.163451	−	0.986551i	$-0.447737\pi$
$509$	3754.00	0.326902	0.163451	+	0.986551i	$0.447737\pi$
$510$	0	0
$511$	7462.00	0.645987
$512$	24475.0	2.11260
$513$	0	0
$514$	25650.0	2.20111
$515$	−9400.00	−0.804298
$516$	0	0
$517$	2496.00	0.212329
$518$	5110.00	0.433437
$519$	0	0
$520$	6750.00	0.569244
$521$	4702.00	0.395390	0.197695	−	0.980264i	$-0.436654\pi$
$521$	4702.00	0.395390	0.197695	+	0.980264i	$0.436654\pi$
$522$	0	0
$523$	−22660.0	−1.89456	−0.947278	−	0.320413i	$-0.896178\pi$
$523$	−22660.0	−1.89456	−0.947278	+	0.320413i	$0.896178\pi$
$524$	16252.0	1.35491
$525$	0	0
$526$	4240.00	0.351469
$527$	−30016.0	−2.48106
$528$	0	0
$529$	377.000	0.0309855
$530$	18850.0	1.54489
$531$	0	0
$532$	−10948.0	−0.892211
$533$	−540.000	−0.0438837
$534$	0	0
$535$	3660.00	0.295767
$536$	−18540.0	−1.49404
$537$	0	0
$538$	−6370.00	−0.510465
$539$	−588.000	−0.0469888
$540$	0	0
$541$	−8634.00	−0.686145	−0.343073	−	0.939309i	$-0.611468\pi$
$541$	−8634.00	−0.686145	−0.343073	+	0.939309i	$0.611468\pi$
$542$	−4320.00	−0.342361
$543$	0	0
$544$	−11390.0	−0.897688
$545$	1890.00	0.148548
$546$	0	0
$547$	−19284.0	−1.50736	−0.753679	−	0.657243i	$-0.771722\pi$
$547$	−19284.0	−1.50736	−0.753679	+	0.657243i	$0.771722\pi$
$548$	6358.00	0.495621
$549$	0	0
$550$	1500.00	0.116291
$551$	−5336.00	−0.412561
$552$	0	0
$553$	6272.00	0.482301
$554$	42650.0	3.27080
$555$	0	0
$556$	6732.00	0.513490
$557$	19658.0	1.49540	0.747699	−	0.664038i	$-0.231159\pi$
$557$	19658.0	1.49540	0.747699	+	0.664038i	$0.231159\pi$
$558$	0	0
$559$	10200.0	0.771760
$560$	−3115.00	−0.235059
$561$	0	0
$562$	−26910.0	−2.01980
$563$	25612.0	1.91726	0.958630	−	0.284656i	$-0.0918793\pi$
$563$	25612.0	1.91726	0.958630	+	0.284656i	$0.0918793\pi$
$564$	0	0
$565$	7290.00	0.542819
$566$	−31180.0	−2.31554
$567$	0	0
$568$	−43200.0	−3.19125
$569$	−7002.00	−0.515886	−0.257943	−	0.966160i	$-0.583045\pi$
$569$	−7002.00	−0.515886	−0.257943	+	0.966160i	$0.583045\pi$
$570$	0	0
$571$	−4524.00	−0.331565	−0.165782	−	0.986162i	$-0.553015\pi$
$571$	−4524.00	−0.331565	−0.165782	+	0.986162i	$0.553015\pi$
$572$	−6120.00	−0.447360
$573$	0	0
$574$	630.000	0.0458113
$575$	−2800.00	−0.203075
$576$	0	0
$577$	−6014.00	−0.433910	−0.216955	−	0.976182i	$-0.569612\pi$
$577$	−6014.00	−0.433910	−0.216955	+	0.976182i	$0.569612\pi$
$578$	−65215.0	−4.69306
$579$	0	0
$580$	−4930.00	−0.352943
$581$	−3052.00	−0.217932
$582$	0	0
$583$	−9048.00	−0.642761
$584$	−47970.0	−3.39899
$585$	0	0
$586$	−4090.00	−0.288321
$587$	11748.0	0.826051	0.413025	−	0.910719i	$-0.364472\pi$
$587$	11748.0	0.826051	0.413025	+	0.910719i	$0.364472\pi$
$588$	0	0
$589$	20608.0	1.44166
$590$	−9500.00	−0.662896
$591$	0	0
$592$	−12994.0	−0.902112
$593$	9462.00	0.655241	0.327620	−	0.944809i	$-0.393753\pi$
$593$	9462.00	0.655241	0.327620	+	0.944809i	$0.393753\pi$
$594$	0	0
$595$	−4690.00	−0.323145
$596$	31858.0	2.18952
$597$	0	0
$598$	16800.0	1.14883
$599$	−2320.00	−0.158251	−0.0791257	−	0.996865i	$-0.525213\pi$
$599$	−2320.00	−0.158251	−0.0791257	+	0.996865i	$0.525213\pi$
$600$	0	0
$601$	4650.00	0.315603	0.157802	−	0.987471i	$-0.449559\pi$
$601$	4650.00	0.315603	0.157802	+	0.987471i	$0.449559\pi$
$602$	−11900.0	−0.805661
$603$	0	0
$604$	−18632.0	−1.25517
$605$	5935.00	0.398830
$606$	0	0
$607$	−14656.0	−0.980014	−0.490007	−	0.871718i	$-0.663006\pi$
$607$	−14656.0	−0.980014	−0.490007	+	0.871718i	$0.663006\pi$
$608$	7820.00	0.521617
$609$	0	0
$610$	17950.0	1.19143
$611$	−6240.00	−0.413164
$612$	0	0
$613$	29166.0	1.92170	0.960851	−	0.277065i	$-0.0893616\pi$
$613$	29166.0	1.92170	0.960851	+	0.277065i	$0.0893616\pi$
$614$	11340.0	0.745350
$615$	0	0
$616$	3780.00	0.247241
$617$	−28554.0	−1.86311	−0.931557	−	0.363597i	$-0.881549\pi$
$617$	−28554.0	−1.86311	−0.931557	+	0.363597i	$0.881549\pi$
$618$	0	0
$619$	−3876.00	−0.251679	−0.125840	−	0.992051i	$-0.540163\pi$
$619$	−3876.00	−0.251679	−0.125840	+	0.992051i	$0.540163\pi$
$620$	19040.0	1.23333
$621$	0	0
$622$	33240.0	2.14277
$623$	7266.00	0.467265
$624$	0	0
$625$	625.000	0.0400000
$626$	−49090.0	−3.13423
$627$	0	0
$628$	32606.0	2.07185
$629$	−19564.0	−1.24017
$630$	0	0
$631$	2904.00	0.183211	0.0916057	−	0.995795i	$-0.470800\pi$
$631$	2904.00	0.183211	0.0916057	+	0.995795i	$0.470800\pi$
$632$	−40320.0	−2.53773
$633$	0	0
$634$	4670.00	0.292538
$635$	−3040.00	−0.189982
$636$	0	0
$637$	1470.00	0.0914341
$638$	3480.00	0.215948
$639$	0	0
$640$	−10575.0	−0.653146
$641$	−9330.00	−0.574903	−0.287452	−	0.957795i	$-0.592808\pi$
$641$	−9330.00	−0.574903	−0.287452	+	0.957795i	$0.592808\pi$
$642$	0	0
$643$	−18332.0	−1.12433	−0.562164	−	0.827025i	$-0.690031\pi$
$643$	−18332.0	−1.12433	−0.562164	+	0.827025i	$0.690031\pi$
$644$	−13328.0	−0.815523
$645$	0	0
$646$	61640.0	3.75417
$647$	2088.00	0.126874	0.0634372	−	0.997986i	$-0.479794\pi$
$647$	2088.00	0.126874	0.0634372	+	0.997986i	$0.479794\pi$
$648$	0	0
$649$	4560.00	0.275802
$650$	−3750.00	−0.226288
$651$	0	0
$652$	39372.0	2.36492
$653$	−22.0000	−0.00131842	−0.000659209	−	1.00000i	$-0.500210\pi$
$653$	−22.0000	−0.00131842	−0.000659209		1.00000i	$0.500210\pi$
$654$	0	0
$655$	−4780.00	−0.285145
$656$	−1602.00	−0.0953469
$657$	0	0
$658$	7280.00	0.431313
$659$	−16260.0	−0.961153	−0.480576	−	0.876953i	$-0.659573\pi$
$659$	−16260.0	−0.961153	−0.480576	+	0.876953i	$0.659573\pi$
$660$	0	0
$661$	−23818.0	−1.40153	−0.700766	−	0.713391i	$-0.747158\pi$
$661$	−23818.0	−1.40153	−0.700766	+	0.713391i	$0.747158\pi$
$662$	−11460.0	−0.672818
$663$	0	0
$664$	19620.0	1.14669
$665$	3220.00	0.187769
$666$	0	0
$667$	−6496.00	−0.377101
$668$	29512.0	1.70936
$669$	0	0
$670$	10300.0	0.593916
$671$	−8616.00	−0.495703
$672$	0	0
$673$	31106.0	1.78165	0.890823	−	0.454350i	$-0.150128\pi$
$673$	31106.0	1.78165	0.890823	+	0.454350i	$0.150128\pi$
$674$	30310.0	1.73219
$675$	0	0
$676$	−22049.0	−1.25449
$677$	1090.00	0.0618790	0.0309395	−	0.999521i	$-0.490150\pi$
$677$	1090.00	0.0618790	0.0309395	+	0.999521i	$0.490150\pi$
$678$	0	0
$679$	−4914.00	−0.277735
$680$	30150.0	1.70029
$681$	0	0
$682$	−13440.0	−0.754610
$683$	12372.0	0.693121	0.346560	−	0.938028i	$-0.387350\pi$
$683$	12372.0	0.693121	0.346560	+	0.938028i	$0.387350\pi$
$684$	0	0
$685$	−1870.00	−0.104305
$686$	−1715.00	−0.0954504
$687$	0	0
$688$	30260.0	1.67682
$689$	22620.0	1.25073
$690$	0	0
$691$	3252.00	0.179033	0.0895166	−	0.995985i	$-0.471468\pi$
$691$	3252.00	0.179033	0.0895166	+	0.995985i	$0.471468\pi$
$692$	41514.0	2.28053
$693$	0	0
$694$	7420.00	0.405849
$695$	−1980.00	−0.108066
$696$	0	0
$697$	−2412.00	−0.131077
$698$	−1270.00	−0.0688685
$699$	0	0
$700$	2975.00	0.160635
$701$	5434.00	0.292781	0.146390	−	0.989227i	$-0.453234\pi$
$701$	5434.00	0.292781	0.146390	+	0.989227i	$0.453234\pi$
$702$	0	0
$703$	13432.0	0.720622
$704$	3444.00	0.184376
$705$	0	0
$706$	−54750.0	−2.91862
$707$	−322.000	−0.0171288
$708$	0	0
$709$	−5330.00	−0.282331	−0.141165	−	0.989986i	$-0.545085\pi$
$709$	−5330.00	−0.282331	−0.141165	+	0.989986i	$0.545085\pi$
$710$	24000.0	1.26860
$711$	0	0
$712$	−46710.0	−2.45861
$713$	25088.0	1.31775
$714$	0	0
$715$	1800.00	0.0941485
$716$	69564.0	3.63091
$717$	0	0
$718$	56880.0	2.95647
$719$	7520.00	0.390054	0.195027	−	0.980798i	$-0.437521\pi$
$719$	7520.00	0.390054	0.195027	+	0.980798i	$0.437521\pi$
$720$	0	0
$721$	13160.0	0.679756
$722$	−8025.00	−0.413656
$723$	0	0
$724$	21590.0	1.10827
$725$	1450.00	0.0742781
$726$	0	0
$727$	19336.0	0.986427	0.493214	−	0.869908i	$-0.335822\pi$
$727$	19336.0	0.986427	0.493214	+	0.869908i	$0.335822\pi$
$728$	−9450.00	−0.481099
$729$	0	0
$730$	26650.0	1.35118
$731$	45560.0	2.30519
$732$	0	0
$733$	−22498.0	−1.13367	−0.566837	−	0.823830i	$-0.691833\pi$
$733$	−22498.0	−1.13367	−0.566837	+	0.823830i	$0.691833\pi$
$734$	5680.00	0.285630
$735$	0	0
$736$	9520.00	0.476782
$737$	−4944.00	−0.247103
$738$	0	0
$739$	−18292.0	−0.910531	−0.455265	−	0.890356i	$-0.650456\pi$
$739$	−18292.0	−0.910531	−0.455265	+	0.890356i	$0.650456\pi$
$740$	12410.0	0.616487
$741$	0	0
$742$	−26390.0	−1.30567
$743$	−17904.0	−0.884030	−0.442015	−	0.897008i	$-0.645736\pi$
$743$	−17904.0	−0.884030	−0.442015	+	0.897008i	$0.645736\pi$
$744$	0	0
$745$	−9370.00	−0.460792
$746$	41210.0	2.02253
$747$	0	0
$748$	−27336.0	−1.33623
$749$	−5124.00	−0.249969
$750$	0	0
$751$	5408.00	0.262771	0.131385	−	0.991331i	$-0.458057\pi$
$751$	5408.00	0.262771	0.131385	+	0.991331i	$0.458057\pi$
$752$	−18512.0	−0.897690
$753$	0	0
$754$	−8700.00	−0.420206
$755$	5480.00	0.264156
$756$	0	0
$757$	8318.00	0.399370	0.199685	−	0.979860i	$-0.436008\pi$
$757$	8318.00	0.399370	0.199685	+	0.979860i	$0.436008\pi$
$758$	−18100.0	−0.867311
$759$	0	0
$760$	−20700.0	−0.987984
$761$	−6690.00	−0.318676	−0.159338	−	0.987224i	$-0.550936\pi$
$761$	−6690.00	−0.318676	−0.159338	+	0.987224i	$0.550936\pi$
$762$	0	0
$763$	−2646.00	−0.125546
$764$	−83368.0	−3.94784
$765$	0	0
$766$	−42320.0	−1.99619
$767$	−11400.0	−0.536676
$768$	0	0
$769$	9266.00	0.434513	0.217257	−	0.976115i	$-0.430289\pi$
$769$	9266.00	0.434513	0.217257	+	0.976115i	$0.430289\pi$
$770$	−2100.00	−0.0982841
$771$	0	0
$772$	37026.0	1.72616
$773$	−9678.00	−0.450315	−0.225157	−	0.974322i	$-0.572290\pi$
$773$	−9678.00	−0.450315	−0.225157	+	0.974322i	$0.572290\pi$
$774$	0	0
$775$	−5600.00	−0.259559
$776$	31590.0	1.46136
$777$	0	0
$778$	18390.0	0.847447
$779$	1656.00	0.0761648
$780$	0	0
$781$	−11520.0	−0.527808
$782$	75040.0	3.43149
$783$	0	0
$784$	4361.00	0.198661
$785$	−9590.00	−0.436028
$786$	0	0
$787$	−6860.00	−0.310715	−0.155357	−	0.987858i	$-0.549653\pi$
$787$	−6860.00	−0.310715	−0.155357	+	0.987858i	$0.549653\pi$
$788$	48450.0	2.19030
$789$	0	0
$790$	22400.0	1.00881
$791$	−10206.0	−0.458766
$792$	0	0
$793$	21540.0	0.964575
$794$	−62950.0	−2.81362
$795$	0	0
$796$	−19448.0	−0.865975
$797$	−10950.0	−0.486661	−0.243331	−	0.969943i	$-0.578240\pi$
$797$	−10950.0	−0.486661	−0.243331	+	0.969943i	$0.578240\pi$
$798$	0	0
$799$	−27872.0	−1.23409
$800$	−2125.00	−0.0939126
$801$	0	0
$802$	14250.0	0.627413
$803$	−12792.0	−0.562167
$804$	0	0
$805$	3920.00	0.171630
$806$	33600.0	1.46837
$807$	0	0
$808$	2070.00	0.0901267
$809$	−26010.0	−1.13036	−0.565181	−	0.824967i	$-0.691194\pi$
$809$	−26010.0	−1.13036	−0.565181	+	0.824967i	$0.691194\pi$
$810$	0	0
$811$	−14628.0	−0.633364	−0.316682	−	0.948532i	$-0.602569\pi$
$811$	−14628.0	−0.633364	−0.316682	+	0.948532i	$0.602569\pi$
$812$	6902.00	0.298292
$813$	0	0
$814$	−8760.00	−0.377196
$815$	−11580.0	−0.497705
$816$	0	0
$817$	−31280.0	−1.33947
$818$	−6130.00	−0.262018
$819$	0	0
$820$	1530.00	0.0651584
$821$	−8718.00	−0.370597	−0.185299	−	0.982682i	$-0.559325\pi$
$821$	−8718.00	−0.370597	−0.185299	+	0.982682i	$0.559325\pi$
$822$	0	0
$823$	−7432.00	−0.314779	−0.157390	−	0.987537i	$-0.550308\pi$
$823$	−7432.00	−0.314779	−0.157390	+	0.987537i	$0.550308\pi$
$824$	−84600.0	−3.57668
$825$	0	0
$826$	13300.0	0.560250
$827$	−17388.0	−0.731125	−0.365562	−	0.930787i	$-0.619123\pi$
$827$	−17388.0	−0.731125	−0.365562	+	0.930787i	$0.619123\pi$
$828$	0	0
$829$	7902.00	0.331059	0.165529	−	0.986205i	$-0.447067\pi$
$829$	7902.00	0.331059	0.165529	+	0.986205i	$0.447067\pi$
$830$	−10900.0	−0.455837
$831$	0	0
$832$	−8610.00	−0.358772
$833$	6566.00	0.273107
$834$	0	0
$835$	−8680.00	−0.359741
$836$	18768.0	0.776441
$837$	0	0
$838$	3060.00	0.126141
$839$	31848.0	1.31051	0.655253	−	0.755409i	$-0.272562\pi$
$839$	31848.0	1.31051	0.655253	+	0.755409i	$0.272562\pi$
$840$	0	0
$841$	−21025.0	−0.862069
$842$	−25910.0	−1.06047
$843$	0	0
$844$	7004.00	0.285649
$845$	6485.00	0.264013
$846$	0	0
$847$	−8309.00	−0.337073
$848$	67106.0	2.71749
$849$	0	0
$850$	−16750.0	−0.675906
$851$	16352.0	0.658683
$852$	0	0
$853$	30150.0	1.21022	0.605109	−	0.796142i	$-0.293129\pi$
$853$	30150.0	1.21022	0.605109	+	0.796142i	$0.293129\pi$
$854$	−25130.0	−1.00694
$855$	0	0
$856$	32940.0	1.31526
$857$	4350.00	0.173388	0.0866938	−	0.996235i	$-0.472370\pi$
$857$	4350.00	0.173388	0.0866938	+	0.996235i	$0.472370\pi$
$858$	0	0
$859$	−30676.0	−1.21845	−0.609227	−	0.792996i	$-0.708520\pi$
$859$	−30676.0	−1.21845	−0.609227	+	0.792996i	$0.708520\pi$
$860$	−28900.0	−1.14591
$861$	0	0
$862$	−24920.0	−0.984662
$863$	23688.0	0.934356	0.467178	−	0.884163i	$-0.345271\pi$
$863$	23688.0	0.934356	0.467178	+	0.884163i	$0.345271\pi$
$864$	0	0
$865$	−12210.0	−0.479945
$866$	8470.00	0.332358
$867$	0	0
$868$	−26656.0	−1.04235
$869$	−10752.0	−0.419720
$870$	0	0
$871$	12360.0	0.480830
$872$	17010.0	0.660586
$873$	0	0
$874$	−51520.0	−1.99392
$875$	−875.000	−0.0338062
$876$	0	0
$877$	31910.0	1.22865	0.614324	−	0.789054i	$-0.289429\pi$
$877$	31910.0	1.22865	0.614324	+	0.789054i	$0.289429\pi$
$878$	−69320.0	−2.66451
$879$	0	0
$880$	5340.00	0.204558
$881$	−50250.0	−1.92164	−0.960820	−	0.277172i	$-0.910603\pi$
$881$	−50250.0	−1.92164	−0.960820	+	0.277172i	$0.910603\pi$
$882$	0	0
$883$	5980.00	0.227908	0.113954	−	0.993486i	$-0.463648\pi$
$883$	5980.00	0.227908	0.113954	+	0.993486i	$0.463648\pi$
$884$	68340.0	2.60014
$885$	0	0
$886$	−23220.0	−0.880464
$887$	24568.0	0.930003	0.465002	−	0.885310i	$-0.346054\pi$
$887$	24568.0	0.930003	0.465002	+	0.885310i	$0.346054\pi$
$888$	0	0
$889$	4256.00	0.160564
$890$	25950.0	0.977355
$891$	0	0
$892$	−27744.0	−1.04141
$893$	19136.0	0.717091
$894$	0	0
$895$	−20460.0	−0.764137
$896$	14805.0	0.552009
$897$	0	0
$898$	−24630.0	−0.915271
$899$	−12992.0	−0.481988
$900$	0	0
$901$	101036.	3.73585
$902$	−1080.00	−0.0398670
$903$	0	0
$904$	65610.0	2.41389
$905$	−6350.00	−0.233239
$906$	0	0
$907$	13252.0	0.485144	0.242572	−	0.970133i	$-0.422009\pi$
$907$	13252.0	0.485144	0.242572	+	0.970133i	$0.422009\pi$
$908$	−69428.0	−2.53750
$909$	0	0
$910$	5250.00	0.191248
$911$	6744.00	0.245267	0.122634	−	0.992452i	$-0.460866\pi$
$911$	6744.00	0.245267	0.122634	+	0.992452i	$0.460866\pi$
$912$	0	0
$913$	5232.00	0.189654
$914$	73470.0	2.65883
$915$	0	0
$916$	−57562.0	−2.07631
$917$	6692.00	0.240992
$918$	0	0
$919$	−45336.0	−1.62731	−0.813654	−	0.581349i	$-0.802525\pi$
$919$	−45336.0	−1.62731	−0.813654	+	0.581349i	$0.802525\pi$
$920$	−25200.0	−0.903065
$921$	0	0
$922$	10030.0	0.358265
$923$	28800.0	1.02705
$924$	0	0
$925$	−3650.00	−0.129742
$926$	−24480.0	−0.868750
$927$	0	0
$928$	−4930.00	−0.174391
$929$	−30074.0	−1.06211	−0.531053	−	0.847339i	$-0.678203\pi$
$929$	−30074.0	−1.06211	−0.531053	+	0.847339i	$0.678203\pi$
$930$	0	0
$931$	−4508.00	−0.158694
$932$	−90474.0	−3.17980
$933$	0	0
$934$	13300.0	0.465941
$935$	8040.00	0.281215
$936$	0	0
$937$	21754.0	0.758455	0.379227	−	0.925303i	$-0.376190\pi$
$937$	21754.0	0.758455	0.379227	+	0.925303i	$0.376190\pi$
$938$	−14420.0	−0.501951
$939$	0	0
$940$	17680.0	0.613466
$941$	−14550.0	−0.504056	−0.252028	−	0.967720i	$-0.581097\pi$
$941$	−14550.0	−0.504056	−0.252028	+	0.967720i	$0.581097\pi$
$942$	0	0
$943$	2016.00	0.0696182
$944$	−33820.0	−1.16605
$945$	0	0
$946$	20400.0	0.701122
$947$	−46660.0	−1.60110	−0.800552	−	0.599263i	$-0.795460\pi$
$947$	−46660.0	−1.60110	−0.800552	+	0.599263i	$0.795460\pi$
$948$	0	0
$949$	31980.0	1.09390
$950$	11500.0	0.392747
$951$	0	0
$952$	−42210.0	−1.43701
$953$	−20810.0	−0.707347	−0.353674	−	0.935369i	$-0.615068\pi$
$953$	−20810.0	−0.707347	−0.353674	+	0.935369i	$0.615068\pi$
$954$	0	0
$955$	24520.0	0.830836
$956$	−63512.0	−2.14867
$957$	0	0
$958$	−28000.0	−0.944300
$959$	2618.00	0.0881539
$960$	0	0
$961$	20385.0	0.684267
$962$	21900.0	0.733975
$963$	0	0
$964$	3570.00	0.119276
$965$	−10890.0	−0.363276
$966$	0	0
$967$	2776.00	0.0923166	0.0461583	−	0.998934i	$-0.485302\pi$
$967$	2776.00	0.0923166	0.0461583	+	0.998934i	$0.485302\pi$
$968$	53415.0	1.77358
$969$	0	0
$970$	−17550.0	−0.580924
$971$	−27292.0	−0.902000	−0.451000	−	0.892524i	$-0.648933\pi$
$971$	−27292.0	−0.902000	−0.451000	+	0.892524i	$0.648933\pi$
$972$	0	0
$973$	2772.00	0.0913322
$974$	32120.0	1.05666
$975$	0	0
$976$	63902.0	2.09575
$977$	62.0000	0.00203025	0.00101513	−	0.999999i	$-0.499677\pi$
$977$	62.0000	0.00203025	0.00101513	+	0.999999i	$0.499677\pi$
$978$	0	0
$979$	−12456.0	−0.406635
$980$	−4165.00	−0.135761
$981$	0	0
$982$	−94500.0	−3.07089
$983$	−37912.0	−1.23012	−0.615058	−	0.788481i	$-0.710868\pi$
$983$	−37912.0	−1.23012	−0.615058	+	0.788481i	$0.710868\pi$
$984$	0	0
$985$	−14250.0	−0.460957
$986$	−38860.0	−1.25513
$987$	0	0
$988$	−46920.0	−1.51085
$989$	−38080.0	−1.22434
$990$	0	0
$991$	10656.0	0.341573	0.170787	−	0.985308i	$-0.445369\pi$
$991$	10656.0	0.341573	0.170787	+	0.985308i	$0.445369\pi$
$992$	19040.0	0.609396
$993$	0	0
$994$	−33600.0	−1.07216
$995$	5720.00	0.182247
$996$	0	0
$997$	−29434.0	−0.934989	−0.467495	−	0.883996i	$-0.654843\pi$
$997$	−29434.0	−0.934989	−0.467495	+	0.883996i	$0.654843\pi$
$998$	76820.0	2.43657
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.4.a.a.1.1		1
3.2	odd	2		105.4.a.b.1.1	✓	1
5.4	even	2		1575.4.a.l.1.1		1
7.6	odd	2		2205.4.a.b.1.1		1
12.11	even	2		1680.4.a.u.1.1		1
15.2	even	4		525.4.d.a.274.2		2
15.8	even	4		525.4.d.a.274.1		2
15.14	odd	2		525.4.a.a.1.1		1
21.20	even	2		735.4.a.j.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.a.b.1.1	✓	1	3.2	odd	2
315.4.a.a.1.1		1	1.1	even	1	trivial
525.4.a.a.1.1		1	15.14	odd	2
525.4.d.a.274.1		2	15.8	even	4
525.4.d.a.274.2		2	15.2	even	4
735.4.a.j.1.1		1	21.20	even	2
1575.4.a.l.1.1		1	5.4	even	2
1680.4.a.u.1.1		1	12.11	even	2
2205.4.a.b.1.1		1	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	315.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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