LMFDB - Embedded newform 2252.4.a.b.1.15

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2252,4,Mod(1,2252)] 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("2252.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2252, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 2252 = 2^{2} \cdot 563 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2252.a (trivial)

Newform invariants

comment:select newform

sage:traces = [75] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	yes
Analytic conductor:	$132.872301333$
Analytic rank:	$0$
Dimension:	$75$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Character	$\chi$	$=$	2252.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-6.71365 q^{3} +6.08579 q^{5} -11.0018 q^{7} +18.0731 q^{9} +32.7744 q^{11} -41.1064 q^{13} -40.8579 q^{15} -94.1075 q^{17} -153.373 q^{19} +73.8622 q^{21} -36.7354 q^{23} -87.9632 q^{25} +59.9319 q^{27} +36.6279 q^{29} -210.992 q^{31} -220.036 q^{33} -66.9546 q^{35} +391.543 q^{37} +275.974 q^{39} +264.513 q^{41} +475.025 q^{43} +109.989 q^{45} -265.144 q^{47} -221.960 q^{49} +631.805 q^{51} -670.151 q^{53} +199.458 q^{55} +1029.70 q^{57} +440.815 q^{59} -667.306 q^{61} -198.837 q^{63} -250.165 q^{65} +33.9180 q^{67} +246.628 q^{69} -1004.80 q^{71} -851.272 q^{73} +590.554 q^{75} -360.577 q^{77} -719.497 q^{79} -890.336 q^{81} +1145.47 q^{83} -572.718 q^{85} -245.907 q^{87} +818.113 q^{89} +452.245 q^{91} +1416.52 q^{93} -933.399 q^{95} -568.529 q^{97} +592.336 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 75 q + 28 q^{3} - 5 q^{5} + 99 q^{7} + 781 q^{9} + 116 q^{11} + 4 q^{13} + 278 q^{15} + 181 q^{17} + 316 q^{19} + 56 q^{21} + 559 q^{23} + 2508 q^{25} + 940 q^{27} - 121 q^{29} + 822 q^{31} + 498 q^{33}+ \cdots + 4658 q^{99}+O(q^{100}) $ 75 * q + 28 * q^3 - 5 * q^5 + 99 * q^7 + 781 * q^9 + 116 * q^11 + 4 * q^13 + 278 * q^15 + 181 * q^17 + 316 * q^19 + 56 * q^21 + 559 * q^23 + 2508 * q^25 + 940 * q^27 - 121 * q^29 + 822 * q^31 + 498 * q^33 + 694 * q^35 + 71 * q^37 + 784 * q^39 + 146 * q^41 + 2503 * q^43 - 163 * q^45 + 997 * q^47 + 5030 * q^49 + 1086 * q^51 - 213 * q^53 + 2916 * q^55 + 1520 * q^57 + 1258 * q^59 + 174 * q^61 + 3011 * q^63 + 856 * q^65 + 4246 * q^67 + 346 * q^69 + 2301 * q^71 + 2828 * q^73 + 3734 * q^75 + 48 * q^77 + 4730 * q^79 + 8991 * q^81 + 2915 * q^83 + 154 * q^85 + 3892 * q^87 + 440 * q^89 + 3846 * q^91 - 240 * q^93 + 4634 * q^95 + 2642 * q^97 + 4658 * q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−6.71365	−1.29204	−0.646022	−	0.763319i	$-0.723568\pi$
$3$	−6.71365	−1.29204	−0.646022	+	0.763319i	$0.723568\pi$
$4$	0	0
$5$	6.08579	0.544330	0.272165	−	0.962251i	$-0.412260\pi$
$5$	6.08579	0.544330	0.272165	+	0.962251i	$0.412260\pi$
$6$	0	0
$7$	−11.0018	−0.594041	−0.297021	−	0.954871i	$-0.595993\pi$
$7$	−11.0018	−0.594041	−0.297021	+	0.954871i	$0.595993\pi$
$8$	0	0
$9$	18.0731	0.669375
$10$	0	0
$11$	32.7744	0.898350	0.449175	−	0.893444i	$-0.351718\pi$
$11$	32.7744	0.898350	0.449175	+	0.893444i	$0.351718\pi$
$12$	0	0
$13$	−41.1064	−0.876991	−0.438495	−	0.898733i	$-0.644488\pi$
$13$	−41.1064	−0.876991	−0.438495	+	0.898733i	$0.644488\pi$
$14$	0	0
$15$	−40.8579	−0.703297
$16$	0	0
$17$	−94.1075	−1.34261	−0.671307	−	0.741180i	$-0.734267\pi$
$17$	−94.1075	−1.34261	−0.671307	+	0.741180i	$0.734267\pi$
$18$	0	0
$19$	−153.373	−1.85191	−0.925955	−	0.377634i	$-0.876738\pi$
$19$	−153.373	−1.85191	−0.925955	+	0.377634i	$0.876738\pi$
$20$	0	0
$21$	73.8622	0.767527
$22$	0	0
$23$	−36.7354	−0.333037	−0.166519	−	0.986038i	$-0.553253\pi$
$23$	−36.7354	−0.333037	−0.166519	+	0.986038i	$0.553253\pi$
$24$	0	0
$25$	−87.9632	−0.703705
$26$	0	0
$27$	59.9319	0.427181
$28$	0	0
$29$	36.6279	0.234539	0.117270	−	0.993100i	$-0.462586\pi$
$29$	36.6279	0.234539	0.117270	+	0.993100i	$0.462586\pi$
$30$	0	0
$31$	−210.992	−1.22243	−0.611213	−	0.791466i	$-0.709318\pi$
$31$	−210.992	−1.22243	−0.611213	+	0.791466i	$0.709318\pi$
$32$	0	0
$33$	−220.036	−1.16071
$34$	0	0
$35$	−66.9546	−0.323354
$36$	0	0
$37$	391.543	1.73971	0.869856	−	0.493306i	$-0.164212\pi$
$37$	391.543	1.73971	0.869856	+	0.493306i	$0.164212\pi$
$38$	0	0
$39$	275.974	1.13311
$40$	0	0
$41$	264.513	1.00756	0.503781	−	0.863831i	$-0.331942\pi$
$41$	264.513	1.00756	0.503781	+	0.863831i	$0.331942\pi$
$42$	0	0
$43$	475.025	1.68467	0.842334	−	0.538956i	$-0.181181\pi$
$43$	475.025	1.68467	0.842334	+	0.538956i	$0.181181\pi$
$44$	0	0
$45$	109.989	0.364361
$46$	0	0
$47$	−265.144	−0.822878	−0.411439	−	0.911437i	$-0.634974\pi$
$47$	−265.144	−0.822878	−0.411439	+	0.911437i	$0.634974\pi$
$48$	0	0
$49$	−221.960	−0.647115
$50$	0	0
$51$	631.805	1.73471
$52$	0	0
$53$	−670.151	−1.73684	−0.868418	−	0.495832i	$-0.834863\pi$
$53$	−670.151	−1.73684	−0.868418	+	0.495832i	$0.834863\pi$
$54$	0	0
$55$	199.458	0.488998
$56$	0	0
$57$	1029.70	2.39275
$58$	0	0
$59$	440.815	0.972700	0.486350	−	0.873764i	$-0.338328\pi$
$59$	440.815	0.972700	0.486350	+	0.873764i	$0.338328\pi$
$60$	0	0
$61$	−667.306	−1.40065	−0.700326	−	0.713823i	$-0.746962\pi$
$61$	−667.306	−1.40065	−0.700326	+	0.713823i	$0.746962\pi$
$62$	0	0
$63$	−198.837	−0.397637
$64$	0	0
$65$	−250.165	−0.477372
$66$	0	0
$67$	33.9180	0.0618468	0.0309234	−	0.999522i	$-0.490155\pi$
$67$	33.9180	0.0618468	0.0309234	+	0.999522i	$0.490155\pi$
$68$	0	0
$69$	246.628	0.430298
$70$	0	0
$71$	−1004.80	−1.67955	−0.839776	−	0.542933i	$-0.817314\pi$
$71$	−1004.80	−1.67955	−0.839776	+	0.542933i	$0.817314\pi$
$72$	0	0
$73$	−851.272	−1.36485	−0.682424	−	0.730957i	$-0.739074\pi$
$73$	−851.272	−1.36485	−0.682424	+	0.730957i	$0.739074\pi$
$74$	0	0
$75$	590.554	0.909218
$76$	0	0
$77$	−360.577	−0.533657
$78$	0	0
$79$	−719.497	−1.02468	−0.512340	−	0.858783i	$-0.671221\pi$
$79$	−719.497	−1.02468	−0.512340	+	0.858783i	$0.671221\pi$
$80$	0	0
$81$	−890.336	−1.22131
$82$	0	0
$83$	1145.47	1.51485	0.757423	−	0.652925i	$-0.226458\pi$
$83$	1145.47	1.51485	0.757423	+	0.652925i	$0.226458\pi$
$84$	0	0
$85$	−572.718	−0.730824
$86$	0	0
$87$	−245.907	−0.303035
$88$	0	0
$89$	818.113	0.974380	0.487190	−	0.873296i	$-0.338022\pi$
$89$	818.113	0.974380	0.487190	+	0.873296i	$0.338022\pi$
$90$	0	0
$91$	452.245	0.520969
$92$	0	0
$93$	1416.52	1.57943
$94$	0	0
$95$	−933.399	−1.00805
$96$	0	0
$97$	−568.529	−0.595107	−0.297553	−	0.954705i	$-0.596171\pi$
$97$	−568.529	−0.595107	−0.297553	+	0.954705i	$0.596171\pi$
$98$	0	0
$99$	592.336	0.601333
$100$	0	0
$101$	−1279.22	−1.26027	−0.630135	−	0.776485i	$-0.717001\pi$
$101$	−1279.22	−1.26027	−0.630135	+	0.776485i	$0.717001\pi$
$102$	0	0
$103$	281.207	0.269011	0.134505	−	0.990913i	$-0.457055\pi$
$103$	281.207	0.269011	0.134505	+	0.990913i	$0.457055\pi$
$104$	0	0
$105$	449.510	0.417787
$106$	0	0
$107$	−160.662	−0.145157	−0.0725786	−	0.997363i	$-0.523123\pi$
$107$	−160.662	−0.145157	−0.0725786	+	0.997363i	$0.523123\pi$
$108$	0	0
$109$	336.598	0.295782	0.147891	−	0.989004i	$-0.452751\pi$
$109$	336.598	0.295782	0.147891	+	0.989004i	$0.452751\pi$
$110$	0	0
$111$	−2628.69	−2.24778
$112$	0	0
$113$	157.176	0.130848	0.0654242	−	0.997858i	$-0.479160\pi$
$113$	157.176	0.130848	0.0654242	+	0.997858i	$0.479160\pi$
$114$	0	0
$115$	−223.564	−0.181282
$116$	0	0
$117$	−742.923	−0.587036
$118$	0	0
$119$	1035.35	0.797568
$120$	0	0
$121$	−256.839	−0.192967
$122$	0	0
$123$	−1775.85	−1.30181
$124$	0	0
$125$	−1296.05	−0.927377
$126$	0	0
$127$	−1050.79	−0.734197	−0.367098	−	0.930182i	$-0.619649\pi$
$127$	−1050.79	−0.734197	−0.367098	+	0.930182i	$0.619649\pi$
$128$	0	0
$129$	−3189.16	−2.17666
$130$	0	0
$131$	2242.33	1.49552	0.747761	−	0.663968i	$-0.231129\pi$
$131$	2242.33	1.49552	0.747761	+	0.663968i	$0.231129\pi$
$132$	0	0
$133$	1687.38	1.10011
$134$	0	0
$135$	364.733	0.232527
$136$	0	0
$137$	1253.26	0.781557	0.390778	−	0.920485i	$-0.372206\pi$
$137$	1253.26	0.781557	0.390778	+	0.920485i	$0.372206\pi$
$138$	0	0
$139$	−836.262	−0.510294	−0.255147	−	0.966902i	$-0.582124\pi$
$139$	−836.262	−0.510294	−0.255147	+	0.966902i	$0.582124\pi$
$140$	0	0
$141$	1780.09	1.06319
$142$	0	0
$143$	−1347.24	−0.787845
$144$	0	0
$145$	222.910	0.127667
$146$	0	0
$147$	1490.17	0.836101
$148$	0	0
$149$	2822.25	1.55173	0.775865	−	0.630898i	$-0.217314\pi$
$149$	2822.25	1.55173	0.775865	+	0.630898i	$0.217314\pi$
$150$	0	0
$151$	342.010	0.184320	0.0921602	−	0.995744i	$-0.470623\pi$
$151$	342.010	0.184320	0.0921602	+	0.995744i	$0.470623\pi$
$152$	0	0
$153$	−1700.82	−0.898712
$154$	0	0
$155$	−1284.05	−0.665403
$156$	0	0
$157$	−2779.16	−1.41275	−0.706374	−	0.707839i	$-0.749670\pi$
$157$	−2779.16	−1.41275	−0.706374	+	0.707839i	$0.749670\pi$
$158$	0	0
$159$	4499.16	2.24407
$160$	0	0
$161$	404.155	0.197838
$162$	0	0
$163$	−3189.45	−1.53262	−0.766309	−	0.642472i	$-0.777909\pi$
$163$	−3189.45	−1.53262	−0.766309	+	0.642472i	$0.777909\pi$
$164$	0	0
$165$	−1339.09	−0.631807
$166$	0	0
$167$	−1490.32	−0.690564	−0.345282	−	0.938499i	$-0.612217\pi$
$167$	−1490.32	−0.690564	−0.345282	+	0.938499i	$0.612217\pi$
$168$	0	0
$169$	−507.260	−0.230887
$170$	0	0
$171$	−2771.94	−1.23962
$172$	0	0
$173$	1247.73	0.548343	0.274171	−	0.961681i	$-0.411596\pi$
$173$	1247.73	0.548343	0.274171	+	0.961681i	$0.411596\pi$
$174$	0	0
$175$	967.753	0.418030
$176$	0	0
$177$	−2959.48	−1.25677
$178$	0	0
$179$	4128.50	1.72390	0.861951	−	0.506991i	$-0.169242\pi$
$179$	4128.50	1.72390	0.861951	+	0.506991i	$0.169242\pi$
$180$	0	0
$181$	1424.75	0.585089	0.292544	−	0.956252i	$-0.405498\pi$
$181$	1424.75	0.585089	0.292544	+	0.956252i	$0.405498\pi$
$182$	0	0
$183$	4480.06	1.80970
$184$	0	0
$185$	2382.85	0.946976
$186$	0	0
$187$	−3084.32	−1.20614
$188$	0	0
$189$	−659.358	−0.253763
$190$	0	0
$191$	4534.79	1.71794	0.858969	−	0.512027i	$-0.171105\pi$
$191$	4534.79	1.71794	0.858969	+	0.512027i	$0.171105\pi$
$192$	0	0
$193$	3846.08	1.43444	0.717219	−	0.696848i	$-0.245415\pi$
$193$	3846.08	1.43444	0.717219	+	0.696848i	$0.245415\pi$
$194$	0	0
$195$	1679.52	0.616785
$196$	0	0
$197$	−1089.04	−0.393861	−0.196931	−	0.980417i	$-0.563097\pi$
$197$	−1089.04	−0.393861	−0.196931	+	0.980417i	$0.563097\pi$
$198$	0	0
$199$	2857.30	1.01783	0.508917	−	0.860816i	$-0.330046\pi$
$199$	2857.30	1.01783	0.508917	+	0.860816i	$0.330046\pi$
$200$	0	0
$201$	−227.713	−0.0799088
$202$	0	0
$203$	−402.973	−0.139326
$204$	0	0
$205$	1609.77	0.548446
$206$	0	0
$207$	−663.923	−0.222927
$208$	0	0
$209$	−5026.72	−1.66366
$210$	0	0
$211$	−452.719	−0.147708	−0.0738542	−	0.997269i	$-0.523530\pi$
$211$	−452.719	−0.147708	−0.0738542	+	0.997269i	$0.523530\pi$
$212$	0	0
$213$	6745.90	2.17005
$214$	0	0
$215$	2890.90	0.917014
$216$	0	0
$217$	2321.29	0.726171
$218$	0	0
$219$	5715.15	1.76344
$220$	0	0
$221$	3868.43	1.17746
$222$	0	0
$223$	−2458.67	−0.738317	−0.369159	−	0.929366i	$-0.620354\pi$
$223$	−2458.67	−0.738317	−0.369159	+	0.929366i	$0.620354\pi$
$224$	0	0
$225$	−1589.77	−0.471043
$226$	0	0
$227$	3791.90	1.10871	0.554355	−	0.832280i	$-0.312965\pi$
$227$	3791.90	1.10871	0.554355	+	0.832280i	$0.312965\pi$
$228$	0	0
$229$	−5728.16	−1.65296	−0.826479	−	0.562968i	$-0.809659\pi$
$229$	−5728.16	−1.65296	−0.826479	+	0.562968i	$0.809659\pi$
$230$	0	0
$231$	2420.79	0.689508
$232$	0	0
$233$	5894.45	1.65733	0.828666	−	0.559743i	$-0.189100\pi$
$233$	5894.45	1.65733	0.828666	+	0.559743i	$0.189100\pi$
$234$	0	0
$235$	−1613.61	−0.447917
$236$	0	0
$237$	4830.45	1.32393
$238$	0	0
$239$	−2526.78	−0.683865	−0.341932	−	0.939725i	$-0.611081\pi$
$239$	−2526.78	−0.683865	−0.341932	+	0.939725i	$0.611081\pi$
$240$	0	0
$241$	−4358.96	−1.16508	−0.582542	−	0.812800i	$-0.697942\pi$
$241$	−4358.96	−1.16508	−0.582542	+	0.812800i	$0.697942\pi$
$242$	0	0
$243$	4359.25	1.15081
$244$	0	0
$245$	−1350.80	−0.352244
$246$	0	0
$247$	6304.64	1.62411
$248$	0	0
$249$	−7690.32	−1.95725
$250$	0	0
$251$	4490.06	1.12912	0.564561	−	0.825391i	$-0.309045\pi$
$251$	4490.06	1.12912	0.564561	+	0.825391i	$0.309045\pi$
$252$	0	0
$253$	−1203.98	−0.299184
$254$	0	0
$255$	3845.03	0.944256
$256$	0	0
$257$	−1188.99	−0.288588	−0.144294	−	0.989535i	$-0.546091\pi$
$257$	−1188.99	−0.288588	−0.144294	+	0.989535i	$0.546091\pi$
$258$	0	0
$259$	−4307.68	−1.03346
$260$	0	0
$261$	661.981	0.156995
$262$	0	0
$263$	2402.25	0.563227	0.281614	−	0.959528i	$-0.409130\pi$
$263$	2402.25	0.563227	0.281614	+	0.959528i	$0.409130\pi$
$264$	0	0
$265$	−4078.40	−0.945411
$266$	0	0
$267$	−5492.53	−1.25894
$268$	0	0
$269$	−610.710	−0.138422	−0.0692112	−	0.997602i	$-0.522048\pi$
$269$	−610.710	−0.138422	−0.0692112	+	0.997602i	$0.522048\pi$
$270$	0	0
$271$	−2649.86	−0.593977	−0.296989	−	0.954881i	$-0.595982\pi$
$271$	−2649.86	−0.593977	−0.296989	+	0.954881i	$0.595982\pi$
$272$	0	0
$273$	−3036.21	−0.673114
$274$	0	0
$275$	−2882.94	−0.632174
$276$	0	0
$277$	−5949.92	−1.29060	−0.645300	−	0.763929i	$-0.723268\pi$
$277$	−5949.92	−1.29060	−0.645300	+	0.763929i	$0.723268\pi$
$278$	0	0
$279$	−3813.28	−0.818262
$280$	0	0
$281$	−7743.59	−1.64393	−0.821964	−	0.569539i	$-0.807122\pi$
$281$	−7743.59	−1.64393	−0.821964	+	0.569539i	$0.807122\pi$
$282$	0	0
$283$	−1623.97	−0.341113	−0.170556	−	0.985348i	$-0.554557\pi$
$283$	−1623.97	−0.341113	−0.170556	+	0.985348i	$0.554557\pi$
$284$	0	0
$285$	6266.51	1.30244
$286$	0	0
$287$	−2910.12	−0.598533
$288$	0	0
$289$	3943.22	0.802610
$290$	0	0
$291$	3816.91	0.768904
$292$	0	0
$293$	3091.97	0.616501	0.308250	−	0.951305i	$-0.400257\pi$
$293$	3091.97	0.616501	0.308250	+	0.951305i	$0.400257\pi$
$294$	0	0
$295$	2682.71	0.529469
$296$	0	0
$297$	1964.23	0.383758
$298$	0	0
$299$	1510.06	0.292070
$300$	0	0
$301$	−5226.13	−1.00076
$302$	0	0
$303$	8588.25	1.62832
$304$	0	0
$305$	−4061.08	−0.762416
$306$	0	0
$307$	7958.68	1.47956	0.739781	−	0.672847i	$-0.234929\pi$
$307$	7958.68	1.47956	0.739781	+	0.672847i	$0.234929\pi$
$308$	0	0
$309$	−1887.92	−0.347574
$310$	0	0
$311$	−1517.54	−0.276693	−0.138346	−	0.990384i	$-0.544179\pi$
$311$	−1517.54	−0.276693	−0.138346	+	0.990384i	$0.544179\pi$
$312$	0	0
$313$	−4888.81	−0.882849	−0.441424	−	0.897298i	$-0.645527\pi$
$313$	−4888.81	−0.882849	−0.441424	+	0.897298i	$0.645527\pi$
$314$	0	0
$315$	−1210.08	−0.216445
$316$	0	0
$317$	9030.85	1.60007	0.800036	−	0.599952i	$-0.204814\pi$
$317$	9030.85	1.60007	0.800036	+	0.599952i	$0.204814\pi$
$318$	0	0
$319$	1200.46	0.210698
$320$	0	0
$321$	1078.63	0.187549
$322$	0	0
$323$	14433.6	2.48640
$324$	0	0
$325$	3615.85	0.617143
$326$	0	0
$327$	−2259.80	−0.382163
$328$	0	0
$329$	2917.06	0.488823
$330$	0	0
$331$	9572.82	1.58964	0.794819	−	0.606847i	$-0.207566\pi$
$331$	9572.82	1.58964	0.794819	+	0.606847i	$0.207566\pi$
$332$	0	0
$333$	7076.42	1.16452
$334$	0	0
$335$	206.418	0.0336651
$336$	0	0
$337$	−5460.13	−0.882588	−0.441294	−	0.897363i	$-0.645480\pi$
$337$	−5460.13	−0.882588	−0.441294	+	0.897363i	$0.645480\pi$
$338$	0	0
$339$	−1055.22	−0.169062
$340$	0	0
$341$	−6915.12	−1.09817
$342$	0	0
$343$	6215.58	0.978454
$344$	0	0
$345$	1500.93	0.234224
$346$	0	0
$347$	5278.46	0.816608	0.408304	−	0.912846i	$-0.366120\pi$
$347$	5278.46	0.816608	0.408304	+	0.912846i	$0.366120\pi$
$348$	0	0
$349$	4710.01	0.722410	0.361205	−	0.932487i	$-0.382366\pi$
$349$	4710.01	0.722410	0.361205	+	0.932487i	$0.382366\pi$
$350$	0	0
$351$	−2463.59	−0.374634
$352$	0	0
$353$	8579.87	1.29366	0.646828	−	0.762636i	$-0.276095\pi$
$353$	8579.87	1.29366	0.646828	+	0.762636i	$0.276095\pi$
$354$	0	0
$355$	−6115.02	−0.914230
$356$	0	0
$357$	−6950.99	−1.03049
$358$	0	0
$359$	383.334	0.0563555	0.0281777	−	0.999603i	$-0.491030\pi$
$359$	383.334	0.0563555	0.0281777	+	0.999603i	$0.491030\pi$
$360$	0	0
$361$	16664.4	2.42957
$362$	0	0
$363$	1724.33	0.249322
$364$	0	0
$365$	−5180.66	−0.742927
$366$	0	0
$367$	13170.9	1.87335	0.936673	−	0.350207i	$-0.113889\pi$
$367$	13170.9	1.87335	0.936673	+	0.350207i	$0.113889\pi$
$368$	0	0
$369$	4780.59	0.674437
$370$	0	0
$371$	7372.87	1.03175
$372$	0	0
$373$	1786.38	0.247977	0.123988	−	0.992284i	$-0.460431\pi$
$373$	1786.38	0.247977	0.123988	+	0.992284i	$0.460431\pi$
$374$	0	0
$375$	8701.22	1.19821
$376$	0	0
$377$	−1505.64	−0.205689
$378$	0	0
$379$	1718.66	0.232933	0.116467	−	0.993195i	$-0.462843\pi$
$379$	1718.66	0.232933	0.116467	+	0.993195i	$0.462843\pi$
$380$	0	0
$381$	7054.67	0.948614
$382$	0	0
$383$	103.144	0.0137609	0.00688043	−	0.999976i	$-0.497810\pi$
$383$	103.144	0.0137609	0.00688043	+	0.999976i	$0.497810\pi$
$384$	0	0
$385$	−2194.40	−0.290485
$386$	0	0
$387$	8585.20	1.12768
$388$	0	0
$389$	−9995.10	−1.30276	−0.651378	−	0.758754i	$-0.725809\pi$
$389$	−9995.10	−1.30276	−0.651378	+	0.758754i	$0.725809\pi$
$390$	0	0
$391$	3457.07	0.447140
$392$	0	0
$393$	−15054.2	−1.93228
$394$	0	0
$395$	−4378.71	−0.557763
$396$	0	0
$397$	5718.09	0.722878	0.361439	−	0.932396i	$-0.382286\pi$
$397$	5718.09	0.722878	0.361439	+	0.932396i	$0.382286\pi$
$398$	0	0
$399$	−11328.5	−1.42139
$400$	0	0
$401$	6494.59	0.808789	0.404394	−	0.914585i	$-0.367482\pi$
$401$	6494.59	0.808789	0.404394	+	0.914585i	$0.367482\pi$
$402$	0	0
$403$	8673.12	1.07206
$404$	0	0
$405$	−5418.40	−0.664796
$406$	0	0
$407$	12832.6	1.56287
$408$	0	0
$409$	−6968.80	−0.842506	−0.421253	−	0.906943i	$-0.638410\pi$
$409$	−6968.80	−0.842506	−0.421253	+	0.906943i	$0.638410\pi$
$410$	0	0
$411$	−8413.95	−1.00980
$412$	0	0
$413$	−4849.76	−0.577824
$414$	0	0
$415$	6971.12	0.824575
$416$	0	0
$417$	5614.37	0.659322
$418$	0	0
$419$	−6984.62	−0.814370	−0.407185	−	0.913346i	$-0.633490\pi$
$419$	−6984.62	−0.814370	−0.407185	+	0.913346i	$0.633490\pi$
$420$	0	0
$421$	−7017.77	−0.812411	−0.406206	−	0.913782i	$-0.633148\pi$
$421$	−7017.77	−0.812411	−0.406206	+	0.913782i	$0.633148\pi$
$422$	0	0
$423$	−4791.99	−0.550814
$424$	0	0
$425$	8278.00	0.944804
$426$	0	0
$427$	7341.56	0.832045
$428$	0	0
$429$	9044.89	1.01793
$430$	0	0
$431$	−4517.07	−0.504825	−0.252413	−	0.967620i	$-0.581224\pi$
$431$	−4517.07	−0.504825	−0.252413	+	0.967620i	$0.581224\pi$
$432$	0	0
$433$	14988.2	1.66348	0.831739	−	0.555166i	$-0.187345\pi$
$433$	14988.2	1.66348	0.831739	+	0.555166i	$0.187345\pi$
$434$	0	0
$435$	−1496.54	−0.164951
$436$	0	0
$437$	5634.23	0.616755
$438$	0	0
$439$	4811.62	0.523112	0.261556	−	0.965188i	$-0.415764\pi$
$439$	4811.62	0.523112	0.261556	+	0.965188i	$0.415764\pi$
$440$	0	0
$441$	−4011.52	−0.433163
$442$	0	0
$443$	7729.32	0.828964	0.414482	−	0.910057i	$-0.363963\pi$
$443$	7729.32	0.828964	0.414482	+	0.910057i	$0.363963\pi$
$444$	0	0
$445$	4978.86	0.530384
$446$	0	0
$447$	−18947.6	−2.00490
$448$	0	0
$449$	1355.07	0.142427	0.0712136	−	0.997461i	$-0.477313\pi$
$449$	1355.07	0.142427	0.0712136	+	0.997461i	$0.477313\pi$
$450$	0	0
$451$	8669.26	0.905144
$452$	0	0
$453$	−2296.14	−0.238150
$454$	0	0
$455$	2752.27	0.283579
$456$	0	0
$457$	−14073.0	−1.44049	−0.720246	−	0.693718i	$-0.755971\pi$
$457$	−14073.0	−1.44049	−0.720246	+	0.693718i	$0.755971\pi$
$458$	0	0
$459$	−5640.04	−0.573539
$460$	0	0
$461$	296.421	0.0299473	0.0149736	−	0.999888i	$-0.495234\pi$
$461$	296.421	0.0299473	0.0149736	+	0.999888i	$0.495234\pi$
$462$	0	0
$463$	−2481.89	−0.249122	−0.124561	−	0.992212i	$-0.539752\pi$
$463$	−2481.89	−0.249122	−0.124561	+	0.992212i	$0.539752\pi$
$464$	0	0
$465$	8620.67	0.859729
$466$	0	0
$467$	−8056.44	−0.798303	−0.399152	−	0.916885i	$-0.630695\pi$
$467$	−8056.44	−0.798303	−0.399152	+	0.916885i	$0.630695\pi$
$468$	0	0
$469$	−373.158	−0.0367396
$470$	0	0
$471$	18658.3	1.82533
$472$	0	0
$473$	15568.7	1.51342
$474$	0	0
$475$	13491.2	1.30320
$476$	0	0
$477$	−12111.7	−1.16260
$478$	0	0
$479$	8958.61	0.854550	0.427275	−	0.904122i	$-0.359474\pi$
$479$	8958.61	0.854550	0.427275	+	0.904122i	$0.359474\pi$
$480$	0	0
$481$	−16095.0	−1.52571
$482$	0	0
$483$	−2713.36	−0.255615
$484$	0	0
$485$	−3459.95	−0.323934
$486$	0	0
$487$	15030.3	1.39854	0.699270	−	0.714858i	$-0.253509\pi$
$487$	15030.3	1.39854	0.699270	+	0.714858i	$0.253509\pi$
$488$	0	0
$489$	21412.8	1.98021
$490$	0	0
$491$	18955.7	1.74228	0.871138	−	0.491038i	$-0.163382\pi$
$491$	18955.7	1.74228	0.871138	+	0.491038i	$0.163382\pi$
$492$	0	0
$493$	−3446.96	−0.314895
$494$	0	0
$495$	3604.83	0.327324
$496$	0	0
$497$	11054.6	0.997723
$498$	0	0
$499$	18450.1	1.65519	0.827596	−	0.561324i	$-0.189708\pi$
$499$	18450.1	1.65519	0.827596	+	0.561324i	$0.189708\pi$
$500$	0	0
$501$	10005.5	0.892238
$502$	0	0
$503$	−7085.19	−0.628058	−0.314029	−	0.949413i	$-0.601679\pi$
$503$	−7085.19	−0.628058	−0.314029	+	0.949413i	$0.601679\pi$
$504$	0	0
$505$	−7785.08	−0.686003
$506$	0	0
$507$	3405.57	0.298317
$508$	0	0
$509$	−2794.98	−0.243390	−0.121695	−	0.992568i	$-0.538833\pi$
$509$	−2794.98	−0.243390	−0.121695	+	0.992568i	$0.538833\pi$
$510$	0	0
$511$	9365.52	0.810775
$512$	0	0
$513$	−9191.96	−0.791101
$514$	0	0
$515$	1711.36	0.146431
$516$	0	0
$517$	−8689.94	−0.739232
$518$	0	0
$519$	−8376.84	−0.708483
$520$	0	0
$521$	7246.51	0.609358	0.304679	−	0.952455i	$-0.401451\pi$
$521$	7246.51	0.609358	0.304679	+	0.952455i	$0.401451\pi$
$522$	0	0
$523$	−17397.3	−1.45455	−0.727275	−	0.686346i	$-0.759214\pi$
$523$	−17397.3	−1.45455	−0.727275	+	0.686346i	$0.759214\pi$
$524$	0	0
$525$	−6497.16	−0.540113
$526$	0	0
$527$	19855.9	1.64125
$528$	0	0
$529$	−10817.5	−0.889086
$530$	0	0
$531$	7966.92	0.651101
$532$	0	0
$533$	−10873.2	−0.883623
$534$	0	0
$535$	−977.758	−0.0790134
$536$	0	0
$537$	−27717.3	−2.22736
$538$	0	0
$539$	−7274.62	−0.581336
$540$	0	0
$541$	−2042.49	−0.162317	−0.0811584	−	0.996701i	$-0.525862\pi$
$541$	−2042.49	−0.162317	−0.0811584	+	0.996701i	$0.525862\pi$
$542$	0	0
$543$	−9565.30	−0.755960
$544$	0	0
$545$	2048.47	0.161003
$546$	0	0
$547$	17232.7	1.34702	0.673509	−	0.739179i	$-0.264786\pi$
$547$	17232.7	1.34702	0.673509	+	0.739179i	$0.264786\pi$
$548$	0	0
$549$	−12060.3	−0.937562
$550$	0	0
$551$	−5617.75	−0.434345
$552$	0	0
$553$	7915.76	0.608702
$554$	0	0
$555$	−15997.6	−1.22353
$556$	0	0
$557$	3269.11	0.248683	0.124342	−	0.992239i	$-0.460318\pi$
$557$	3269.11	0.248683	0.124342	+	0.992239i	$0.460318\pi$
$558$	0	0
$559$	−19526.6	−1.47744
$560$	0	0
$561$	20707.0	1.55838
$562$	0	0
$563$	563.000	0.0421450
$563$	563.000	0.0421450
$564$	0	0
$565$	956.539	0.0712246
$566$	0	0
$567$	9795.30	0.725510
$568$	0	0
$569$	12551.7	0.924774	0.462387	−	0.886678i	$-0.346993\pi$
$569$	12551.7	0.924774	0.462387	+	0.886678i	$0.346993\pi$
$570$	0	0
$571$	−18749.3	−1.37414	−0.687070	−	0.726591i	$-0.741103\pi$
$571$	−18749.3	−1.37414	−0.687070	+	0.726591i	$0.741103\pi$
$572$	0	0
$573$	−30445.0	−2.21965
$574$	0	0
$575$	3231.36	0.234360
$576$	0	0
$577$	−11797.7	−0.851204	−0.425602	−	0.904910i	$-0.639938\pi$
$577$	−11797.7	−0.851204	−0.425602	+	0.904910i	$0.639938\pi$
$578$	0	0
$579$	−25821.2	−1.85336
$580$	0	0
$581$	−12602.3	−0.899880
$582$	0	0
$583$	−21963.8	−1.56029
$584$	0	0
$585$	−4521.27	−0.319541
$586$	0	0
$587$	5138.97	0.361343	0.180671	−	0.983544i	$-0.442173\pi$
$587$	5138.97	0.361343	0.180671	+	0.983544i	$0.442173\pi$
$588$	0	0
$589$	32360.5	2.26382
$590$	0	0
$591$	7311.42	0.508886
$592$	0	0
$593$	27439.8	1.90020	0.950098	−	0.311953i	$-0.100983\pi$
$593$	27439.8	1.90020	0.950098	+	0.311953i	$0.100983\pi$
$594$	0	0
$595$	6300.93	0.434140
$596$	0	0
$597$	−19182.9	−1.31508
$598$	0	0
$599$	−19840.1	−1.35333	−0.676664	−	0.736292i	$-0.736575\pi$
$599$	−19840.1	−1.35333	−0.676664	+	0.736292i	$0.736575\pi$
$600$	0	0
$601$	−20817.3	−1.41290	−0.706451	−	0.707762i	$-0.749705\pi$
$601$	−20817.3	−1.41290	−0.706451	+	0.707762i	$0.749705\pi$
$602$	0	0
$603$	613.004	0.0413988
$604$	0	0
$605$	−1563.07	−0.105038
$606$	0	0
$607$	3720.10	0.248755	0.124377	−	0.992235i	$-0.460307\pi$
$607$	3720.10	0.248755	0.124377	+	0.992235i	$0.460307\pi$
$608$	0	0
$609$	2705.42	0.180015
$610$	0	0
$611$	10899.1	0.721656
$612$	0	0
$613$	7553.08	0.497660	0.248830	−	0.968547i	$-0.419954\pi$
$613$	7553.08	0.497660	0.248830	+	0.968547i	$0.419954\pi$
$614$	0	0
$615$	−10807.5	−0.708616
$616$	0	0
$617$	4603.60	0.300379	0.150190	−	0.988657i	$-0.452012\pi$
$617$	4603.60	0.300379	0.150190	+	0.988657i	$0.452012\pi$
$618$	0	0
$619$	−19776.0	−1.28411	−0.642055	−	0.766659i	$-0.721918\pi$
$619$	−19776.0	−1.28411	−0.642055	+	0.766659i	$0.721918\pi$
$620$	0	0
$621$	−2201.62	−0.142267
$622$	0	0
$623$	−9000.71	−0.578822
$624$	0	0
$625$	3107.92	0.198907
$626$	0	0
$627$	33747.7	2.14952
$628$	0	0
$629$	−36847.2	−2.33576
$630$	0	0
$631$	8212.01	0.518091	0.259045	−	0.965865i	$-0.416592\pi$
$631$	8212.01	0.518091	0.259045	+	0.965865i	$0.416592\pi$
$632$	0	0
$633$	3039.40	0.190846
$634$	0	0
$635$	−6394.92	−0.399645
$636$	0	0
$637$	9124.01	0.567514
$638$	0	0
$639$	−18159.9	−1.12425
$640$	0	0
$641$	11968.2	0.737467	0.368734	−	0.929535i	$-0.379791\pi$
$641$	11968.2	0.737467	0.368734	+	0.929535i	$0.379791\pi$
$642$	0	0
$643$	19860.1	1.21805	0.609025	−	0.793151i	$-0.291561\pi$
$643$	19860.1	1.21805	0.609025	+	0.793151i	$0.291561\pi$
$644$	0	0
$645$	−19408.5	−1.18482
$646$	0	0
$647$	−13797.0	−0.838356	−0.419178	−	0.907904i	$-0.637682\pi$
$647$	−13797.0	−0.838356	−0.419178	+	0.907904i	$0.637682\pi$
$648$	0	0
$649$	14447.5	0.873825
$650$	0	0
$651$	−15584.3	−0.938245
$652$	0	0
$653$	812.272	0.0486779	0.0243389	−	0.999704i	$-0.492252\pi$
$653$	812.272	0.0486779	0.0243389	+	0.999704i	$0.492252\pi$
$654$	0	0
$655$	13646.4	0.814057
$656$	0	0
$657$	−15385.2	−0.913595
$658$	0	0
$659$	−16249.3	−0.960520	−0.480260	−	0.877126i	$-0.659458\pi$
$659$	−16249.3	−0.960520	−0.480260	+	0.877126i	$0.659458\pi$
$660$	0	0
$661$	−27858.2	−1.63927	−0.819636	−	0.572884i	$-0.805825\pi$
$661$	−27858.2	−1.63927	−0.819636	+	0.572884i	$0.805825\pi$
$662$	0	0
$663$	−25971.3	−1.52133
$664$	0	0
$665$	10269.1	0.598823
$666$	0	0
$667$	−1345.54	−0.0781102
$668$	0	0
$669$	16506.7	0.953938
$670$	0	0
$671$	−21870.5	−1.25828
$672$	0	0
$673$	23201.1	1.32888	0.664441	−	0.747341i	$-0.268670\pi$
$673$	23201.1	1.32888	0.664441	+	0.747341i	$0.268670\pi$
$674$	0	0
$675$	−5271.80	−0.300610
$676$	0	0
$677$	17141.2	0.973102	0.486551	−	0.873652i	$-0.338255\pi$
$677$	17141.2	0.973102	0.486551	+	0.873652i	$0.338255\pi$
$678$	0	0
$679$	6254.84	0.353518
$680$	0	0
$681$	−25457.5	−1.43250
$682$	0	0
$683$	17737.8	0.993729	0.496864	−	0.867828i	$-0.334485\pi$
$683$	17737.8	0.993729	0.496864	+	0.867828i	$0.334485\pi$
$684$	0	0
$685$	7627.08	0.425424
$686$	0	0
$687$	38456.9	2.13569
$688$	0	0
$689$	27547.5	1.52319
$690$	0	0
$691$	−14205.3	−0.782050	−0.391025	−	0.920380i	$-0.627879\pi$
$691$	−14205.3	−0.782050	−0.391025	+	0.920380i	$0.627879\pi$
$692$	0	0
$693$	−6516.76	−0.357217
$694$	0	0
$695$	−5089.31	−0.277768
$696$	0	0
$697$	−24892.7	−1.35277
$698$	0	0
$699$	−39573.3	−2.14135
$700$	0	0
$701$	−12811.9	−0.690300	−0.345150	−	0.938547i	$-0.612172\pi$
$701$	−12811.9	−0.690300	−0.345150	+	0.938547i	$0.612172\pi$
$702$	0	0
$703$	−60052.4	−3.22179
$704$	0	0
$705$	10833.2	0.578728
$706$	0	0
$707$	14073.7	0.748653
$708$	0	0
$709$	5329.22	0.282289	0.141145	−	0.989989i	$-0.454922\pi$
$709$	5329.22	0.282289	0.141145	+	0.989989i	$0.454922\pi$
$710$	0	0
$711$	−13003.6	−0.685896
$712$	0	0
$713$	7750.85	0.407113
$714$	0	0
$715$	−8199.01	−0.428847
$716$	0	0
$717$	16963.9	0.883583
$718$	0	0
$719$	1941.47	0.100702	0.0503508	−	0.998732i	$-0.483966\pi$
$719$	1941.47	0.100702	0.0503508	+	0.998732i	$0.483966\pi$
$720$	0	0
$721$	−3093.78	−0.159803
$722$	0	0
$723$	29264.6	1.50534
$724$	0	0
$725$	−3221.91	−0.165046
$726$	0	0
$727$	−24328.5	−1.24112	−0.620559	−	0.784160i	$-0.713094\pi$
$727$	−24328.5	−1.24112	−0.620559	+	0.784160i	$0.713094\pi$
$728$	0	0
$729$	−5227.41	−0.265580
$730$	0	0
$731$	−44703.5	−2.26186
$732$	0	0
$733$	−17957.9	−0.904900	−0.452450	−	0.891790i	$-0.649450\pi$
$733$	−17957.9	−0.904900	−0.452450	+	0.891790i	$0.649450\pi$
$734$	0	0
$735$	9068.83	0.455114
$736$	0	0
$737$	1111.64	0.0555601
$738$	0	0
$739$	−8417.14	−0.418984	−0.209492	−	0.977810i	$-0.567181\pi$
$739$	−8417.14	−0.418984	−0.209492	+	0.977810i	$0.567181\pi$
$740$	0	0
$741$	−42327.2	−2.09842
$742$	0	0
$743$	18351.9	0.906145	0.453073	−	0.891474i	$-0.350328\pi$
$743$	18351.9	0.906145	0.453073	+	0.891474i	$0.350328\pi$
$744$	0	0
$745$	17175.6	0.844653
$746$	0	0
$747$	20702.3	1.01400
$748$	0	0
$749$	1767.58	0.0862294
$750$	0	0
$751$	27617.6	1.34192	0.670958	−	0.741495i	$-0.265883\pi$
$751$	27617.6	1.34192	0.670958	+	0.741495i	$0.265883\pi$
$752$	0	0
$753$	−30144.7	−1.45888
$754$	0	0
$755$	2081.40	0.100331
$756$	0	0
$757$	8875.91	0.426156	0.213078	−	0.977035i	$-0.431651\pi$
$757$	8875.91	0.426156	0.213078	+	0.977035i	$0.431651\pi$
$758$	0	0
$759$	8083.10	0.386558
$760$	0	0
$761$	−9267.19	−0.441439	−0.220720	−	0.975337i	$-0.570841\pi$
$761$	−9267.19	−0.441439	−0.220720	+	0.975337i	$0.570841\pi$
$762$	0	0
$763$	−3703.18	−0.175707
$764$	0	0
$765$	−10350.8	−0.489196
$766$	0	0
$767$	−18120.4	−0.853049
$768$	0	0
$769$	−23159.5	−1.08602	−0.543012	−	0.839725i	$-0.682716\pi$
$769$	−23159.5	−1.08602	−0.543012	+	0.839725i	$0.682716\pi$
$770$	0	0
$771$	7982.47	0.372868
$772$	0	0
$773$	1028.26	0.0478447	0.0239223	−	0.999714i	$-0.492385\pi$
$773$	1028.26	0.0478447	0.0239223	+	0.999714i	$0.492385\pi$
$774$	0	0
$775$	18559.5	0.860228
$776$	0	0
$777$	28920.3	1.33527
$778$	0	0
$779$	−40569.3	−1.86591
$780$	0	0
$781$	−32931.8	−1.50883
$782$	0	0
$783$	2195.18	0.100191
$784$	0	0
$785$	−16913.4	−0.769000
$786$	0	0
$787$	−8703.04	−0.394193	−0.197096	−	0.980384i	$-0.563151\pi$
$787$	−8703.04	−0.394193	−0.197096	+	0.980384i	$0.563151\pi$
$788$	0	0
$789$	−16127.8	−0.727714
$790$	0	0
$791$	−1729.22	−0.0777293
$792$	0	0
$793$	27430.6	1.22836
$794$	0	0
$795$	27381.0	1.22151
$796$	0	0
$797$	14926.6	0.663396	0.331698	−	0.943386i	$-0.392379\pi$
$797$	14926.6	0.663396	0.331698	+	0.943386i	$0.392379\pi$
$798$	0	0
$799$	24952.1	1.10481
$800$	0	0
$801$	14785.9	0.652226
$802$	0	0
$803$	−27899.9	−1.22611
$804$	0	0
$805$	2459.60	0.107689
$806$	0	0
$807$	4100.09	0.178848
$808$	0	0
$809$	−32572.3	−1.41555	−0.707776	−	0.706437i	$-0.750301\pi$
$809$	−32572.3	−1.41555	−0.707776	+	0.706437i	$0.750301\pi$
$810$	0	0
$811$	24742.6	1.07131	0.535653	−	0.844438i	$-0.320065\pi$
$811$	24742.6	1.07131	0.535653	+	0.844438i	$0.320065\pi$
$812$	0	0
$813$	17790.3	0.767444
$814$	0	0
$815$	−19410.3	−0.834250
$816$	0	0
$817$	−72856.3	−3.11985
$818$	0	0
$819$	8173.48	0.348724
$820$	0	0
$821$	11473.0	0.487710	0.243855	−	0.969812i	$-0.421588\pi$
$821$	11473.0	0.487710	0.243855	+	0.969812i	$0.421588\pi$
$822$	0	0
$823$	−6505.04	−0.275518	−0.137759	−	0.990466i	$-0.543990\pi$
$823$	−6505.04	−0.275518	−0.137759	+	0.990466i	$0.543990\pi$
$824$	0	0
$825$	19355.1	0.816796
$826$	0	0
$827$	−29084.3	−1.22292	−0.611462	−	0.791274i	$-0.709418\pi$
$827$	−29084.3	−1.22292	−0.611462	+	0.791274i	$0.709418\pi$
$828$	0	0
$829$	−34516.8	−1.44610	−0.723050	−	0.690796i	$-0.757260\pi$
$829$	−34516.8	−1.44610	−0.723050	+	0.690796i	$0.757260\pi$
$830$	0	0
$831$	39945.7	1.66751
$832$	0	0
$833$	20888.1	0.868825
$834$	0	0
$835$	−9069.75	−0.375894
$836$	0	0
$837$	−12645.1	−0.522197
$838$	0	0
$839$	12050.1	0.495845	0.247923	−	0.968780i	$-0.420252\pi$
$839$	12050.1	0.495845	0.247923	+	0.968780i	$0.420252\pi$
$840$	0	0
$841$	−23047.4	−0.944991
$842$	0	0
$843$	51987.8	2.12403
$844$	0	0
$845$	−3087.08	−0.125679
$846$	0	0
$847$	2825.69	0.114630
$848$	0	0
$849$	10902.8	0.440733
$850$	0	0
$851$	−14383.5	−0.579388
$852$	0	0
$853$	39402.9	1.58163	0.790815	−	0.612055i	$-0.209657\pi$
$853$	39402.9	1.58163	0.790815	+	0.612055i	$0.209657\pi$
$854$	0	0
$855$	−16869.4	−0.674763
$856$	0	0
$857$	−39574.9	−1.57742	−0.788711	−	0.614764i	$-0.789251\pi$
$857$	−39574.9	−1.57742	−0.788711	+	0.614764i	$0.789251\pi$
$858$	0	0
$859$	−12108.4	−0.480947	−0.240474	−	0.970656i	$-0.577303\pi$
$859$	−12108.4	−0.480947	−0.240474	+	0.970656i	$0.577303\pi$
$860$	0	0
$861$	19537.6	0.773331
$862$	0	0
$863$	29594.4	1.16733	0.583665	−	0.811994i	$-0.301618\pi$
$863$	29594.4	1.16733	0.583665	+	0.811994i	$0.301618\pi$
$864$	0	0
$865$	7593.43	0.298479
$866$	0	0
$867$	−26473.4	−1.03701
$868$	0	0
$869$	−23581.1	−0.920521
$870$	0	0
$871$	−1394.25	−0.0542391
$872$	0	0
$873$	−10275.1	−0.398350
$874$	0	0
$875$	14258.9	0.550900
$876$	0	0
$877$	−16838.1	−0.648326	−0.324163	−	0.946001i	$-0.605083\pi$
$877$	−16838.1	−0.648326	−0.324163	+	0.946001i	$0.605083\pi$
$878$	0	0
$879$	−20758.4	−0.796545
$880$	0	0
$881$	−16537.3	−0.632413	−0.316207	−	0.948690i	$-0.602409\pi$
$881$	−16537.3	−0.632413	−0.316207	+	0.948690i	$0.602409\pi$
$882$	0	0
$883$	34065.2	1.29829	0.649143	−	0.760667i	$-0.275128\pi$
$883$	34065.2	1.29829	0.649143	+	0.760667i	$0.275128\pi$
$884$	0	0
$885$	−18010.8	−0.684097
$886$	0	0
$887$	−35515.0	−1.34439	−0.672196	−	0.740373i	$-0.734649\pi$
$887$	−35515.0	−1.34439	−0.672196	+	0.740373i	$0.734649\pi$
$888$	0	0
$889$	11560.6	0.436143
$890$	0	0
$891$	−29180.2	−1.09717
$892$	0	0
$893$	40666.1	1.52390
$894$	0	0
$895$	25125.2	0.938371
$896$	0	0
$897$	−10138.0	−0.377367
$898$	0	0
$899$	−7728.18	−0.286707
$900$	0	0
$901$	63066.3	2.33190
$902$	0	0
$903$	35086.4	1.29303
$904$	0	0
$905$	8670.75	0.318481
$906$	0	0
$907$	−2050.69	−0.0750738	−0.0375369	−	0.999295i	$-0.511951\pi$
$907$	−2050.69	−0.0750738	−0.0375369	+	0.999295i	$0.511951\pi$
$908$	0	0
$909$	−23119.6	−0.843594
$910$	0	0
$911$	17347.9	0.630913	0.315456	−	0.948940i	$-0.397842\pi$
$911$	17347.9	0.630913	0.315456	+	0.948940i	$0.397842\pi$
$912$	0	0
$913$	37542.2	1.36086
$914$	0	0
$915$	27264.7	0.985075
$916$	0	0
$917$	−24669.7	−0.888402
$918$	0	0
$919$	38349.9	1.37655	0.688273	−	0.725452i	$-0.258369\pi$
$919$	38349.9	1.37655	0.688273	+	0.725452i	$0.258369\pi$
$920$	0	0
$921$	−53431.8	−1.91166
$922$	0	0
$923$	41303.9	1.47295
$924$	0	0
$925$	−34441.4	−1.22424
$926$	0	0
$927$	5082.29	0.180069
$928$	0	0
$929$	34315.3	1.21189	0.605946	−	0.795506i	$-0.292795\pi$
$929$	34315.3	1.21189	0.605946	+	0.795506i	$0.292795\pi$
$930$	0	0
$931$	34042.9	1.19840
$932$	0	0
$933$	10188.2	0.357499
$934$	0	0
$935$	−18770.5	−0.656536
$936$	0	0
$937$	−13162.9	−0.458924	−0.229462	−	0.973318i	$-0.573697\pi$
$937$	−13162.9	−0.458924	−0.229462	+	0.973318i	$0.573697\pi$
$938$	0	0
$939$	32821.7	1.14068
$940$	0	0
$941$	34912.8	1.20948	0.604742	−	0.796422i	$-0.293276\pi$
$941$	34912.8	1.20948	0.604742	+	0.796422i	$0.293276\pi$
$942$	0	0
$943$	−9716.99	−0.335555
$944$	0	0
$945$	−4012.71	−0.138131
$946$	0	0
$947$	35185.9	1.20738	0.603690	−	0.797219i	$-0.293696\pi$
$947$	35185.9	1.20738	0.603690	+	0.797219i	$0.293696\pi$
$948$	0	0
$949$	34992.8	1.19696
$950$	0	0
$951$	−60630.0	−2.06736
$952$	0	0
$953$	26228.8	0.891536	0.445768	−	0.895148i	$-0.352931\pi$
$953$	26228.8	0.891536	0.445768	+	0.895148i	$0.352931\pi$
$954$	0	0
$955$	27597.8	0.935124
$956$	0	0
$957$	−8059.45	−0.272231
$958$	0	0
$959$	−13788.1	−0.464277
$960$	0	0
$961$	14726.5	0.494325
$962$	0	0
$963$	−2903.67	−0.0971647
$964$	0	0
$965$	23406.4	0.780807
$966$	0	0
$967$	41496.1	1.37996	0.689982	−	0.723827i	$-0.257619\pi$
$967$	41496.1	1.37996	0.689982	+	0.723827i	$0.257619\pi$
$968$	0	0
$969$	−96902.2	−3.21253
$970$	0	0
$971$	−2862.88	−0.0946181	−0.0473091	−	0.998880i	$-0.515065\pi$
$971$	−2862.88	−0.0946181	−0.0473091	+	0.998880i	$0.515065\pi$
$972$	0	0
$973$	9200.39	0.303135
$974$	0	0
$975$	−24275.6	−0.797375
$976$	0	0
$977$	−27022.6	−0.884882	−0.442441	−	0.896798i	$-0.645887\pi$
$977$	−27022.6	−0.884882	−0.442441	+	0.896798i	$0.645887\pi$
$978$	0	0
$979$	26813.2	0.875334
$980$	0	0
$981$	6083.39	0.197989
$982$	0	0
$983$	−12935.7	−0.419719	−0.209859	−	0.977732i	$-0.567301\pi$
$983$	−12935.7	−0.419719	−0.209859	+	0.977732i	$0.567301\pi$
$984$	0	0
$985$	−6627.65	−0.214390
$986$	0	0
$987$	−19584.1	−0.631581
$988$	0	0
$989$	−17450.2	−0.561057
$990$	0	0
$991$	−39164.8	−1.25541	−0.627705	−	0.778452i	$-0.716005\pi$
$991$	−39164.8	−1.25541	−0.627705	+	0.778452i	$0.716005\pi$
$992$	0	0
$993$	−64268.6	−2.05388
$994$	0	0
$995$	17388.9	0.554037
$996$	0	0
$997$	20397.2	0.647930	0.323965	−	0.946069i	$-0.394984\pi$
$997$	20397.2	0.647930	0.323965	+	0.946069i	$0.394984\pi$
$998$	0	0
$999$	23465.9	0.743172

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2252.4.a.b.1.15	✓	75

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2252.4.a.b.1.15	✓	75	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:	\( N \)	\(=\)	\( 2252 = 2^{2} \cdot 563 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2252.a (trivial)

\(f(q)\)	\(=\)	\(q-6.71365 q^{3} +6.08579 q^{5} -11.0018 q^{7} +18.0731 q^{9} +32.7744 q^{11} -41.1064 q^{13} -40.8579 q^{15} -94.1075 q^{17} -153.373 q^{19} +73.8622 q^{21} -36.7354 q^{23} -87.9632 q^{25} +59.9319 q^{27} +36.6279 q^{29} -210.992 q^{31} -220.036 q^{33} -66.9546 q^{35} +391.543 q^{37} +275.974 q^{39} +264.513 q^{41} +475.025 q^{43} +109.989 q^{45} -265.144 q^{47} -221.960 q^{49} +631.805 q^{51} -670.151 q^{53} +199.458 q^{55} +1029.70 q^{57} +440.815 q^{59} -667.306 q^{61} -198.837 q^{63} -250.165 q^{65} +33.9180 q^{67} +246.628 q^{69} -1004.80 q^{71} -851.272 q^{73} +590.554 q^{75} -360.577 q^{77} -719.497 q^{79} -890.336 q^{81} +1145.47 q^{83} -572.718 q^{85} -245.907 q^{87} +818.113 q^{89} +452.245 q^{91} +1416.52 q^{93} -933.399 q^{95} -568.529 q^{97} +592.336 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 75 q + 28 q^{3} - 5 q^{5} + 99 q^{7} + 781 q^{9} + 116 q^{11} + 4 q^{13} + 278 q^{15} + 181 q^{17} + 316 q^{19} + 56 q^{21} + 559 q^{23} + 2508 q^{25} + 940 q^{27} - 121 q^{29} + 822 q^{31} + 498 q^{33}+ \cdots + 4658 q^{99}+O(q^{100}) \) 75 * q + 28 * q^3 - 5 * q^5 + 99 * q^7 + 781 * q^9 + 116 * q^11 + 4 * q^13 + 278 * q^15 + 181 * q^17 + 316 * q^19 + 56 * q^21 + 559 * q^23 + 2508 * q^25 + 940 * q^27 - 121 * q^29 + 822 * q^31 + 498 * q^33 + 694 * q^35 + 71 * q^37 + 784 * q^39 + 146 * q^41 + 2503 * q^43 - 163 * q^45 + 997 * q^47 + 5030 * q^49 + 1086 * q^51 - 213 * q^53 + 2916 * q^55 + 1520 * q^57 + 1258 * q^59 + 174 * q^61 + 3011 * q^63 + 856 * q^65 + 4246 * q^67 + 346 * q^69 + 2301 * q^71 + 2828 * q^73 + 3734 * q^75 + 48 * q^77 + 4730 * q^79 + 8991 * q^81 + 2915 * q^83 + 154 * q^85 + 3892 * q^87 + 440 * q^89 + 3846 * q^91 - 240 * q^93 + 4634 * q^95 + 2642 * q^97 + 4658 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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