LMFDB - Embedded newform 1521.4.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1521,4,Mod(1,1521)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1521, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1521.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1521 = 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1521.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:	$89.7419051187$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 39)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	1521.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.00000 q^{4} -5.19615 q^{5} +10.3923 q^{7} +51.9615 q^{11} +64.0000 q^{16} -117.000 q^{17} +24.2487 q^{19} +41.5692 q^{20} +18.0000 q^{23} -98.0000 q^{25} -83.1384 q^{28} +99.0000 q^{29} +193.990 q^{31} -54.0000 q^{35} -112.583 q^{37} +36.3731 q^{41} +82.0000 q^{43} -415.692 q^{44} -72.7461 q^{47} -235.000 q^{49} +261.000 q^{53} -270.000 q^{55} +789.815 q^{59} -719.000 q^{61} -512.000 q^{64} -703.213 q^{67} +936.000 q^{68} -467.654 q^{71} -684.160 q^{73} -193.990 q^{76} +540.000 q^{77} -440.000 q^{79} -332.554 q^{80} -1195.12 q^{83} +607.950 q^{85} +1517.28 q^{89} -144.000 q^{92} -126.000 q^{95} +1157.01 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 16 q^{4} + 128 q^{16} - 234 q^{17} + 36 q^{23} - 196 q^{25} + 198 q^{29} - 108 q^{35} + 164 q^{43} - 470 q^{49} + 522 q^{53} - 540 q^{55} - 1438 q^{61} - 1024 q^{64} + 1872 q^{68} + 1080 q^{77} - 880 q^{79}+ \cdots - 252 q^{95}+O(q^{100}) $ 2 * q - 16 * q^4 + 128 * q^16 - 234 * q^17 + 36 * q^23 - 196 * q^25 + 198 * q^29 - 108 * q^35 + 164 * q^43 - 470 * q^49 + 522 * q^53 - 540 * q^55 - 1438 * q^61 - 1024 * q^64 + 1872 * q^68 + 1080 * q^77 - 880 * q^79 - 288 * q^92 - 252 * q^95

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		−	1.00000i	$-0.5\pi$
$2$	0	0			1.00000i	$0.5\pi$
$3$	0	0
$3$	0	0
$4$	−8.00000	−1.00000
$5$	−5.19615	−0.464758	−0.232379	−	0.972625i	$-0.574651\pi$
$5$	−5.19615	−0.464758	−0.232379	+	0.972625i	$0.574651\pi$
$6$	0	0
$7$	10.3923	0.561132	0.280566	−	0.959835i	$-0.409478\pi$
$7$	10.3923	0.561132	0.280566	+	0.959835i	$0.409478\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	51.9615	1.42427	0.712136	−	0.702042i	$-0.247728\pi$
$11$	51.9615	1.42427	0.712136	+	0.702042i	$0.247728\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	0	0
$16$	64.0000	1.00000
$17$	−117.000	−1.66922	−0.834608	−	0.550845i	$-0.814306\pi$
$17$	−117.000	−1.66922	−0.834608	+	0.550845i	$0.814306\pi$
$18$	0	0
$19$	24.2487	0.292791	0.146396	−	0.989226i	$-0.453233\pi$
$19$	24.2487	0.292791	0.146396	+	0.989226i	$0.453233\pi$
$20$	41.5692	0.464758
$21$	0	0
$22$	0	0
$23$	18.0000	0.163185	0.0815926	−	0.996666i	$-0.473999\pi$
$23$	18.0000	0.163185	0.0815926	+	0.996666i	$0.473999\pi$
$24$	0	0
$25$	−98.0000	−0.784000
$26$	0	0
$27$	0	0
$28$	−83.1384	−0.561132
$29$	99.0000	0.633925	0.316963	−	0.948438i	$-0.397337\pi$
$29$	99.0000	0.633925	0.316963	+	0.948438i	$0.397337\pi$
$30$	0	0
$31$	193.990	1.12392	0.561961	−	0.827164i	$-0.310047\pi$
$31$	193.990	1.12392	0.561961	+	0.827164i	$0.310047\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−54.0000	−0.260790
$36$	0	0
$37$	−112.583	−0.500232	−0.250116	−	0.968216i	$-0.580469\pi$
$37$	−112.583	−0.500232	−0.250116	+	0.968216i	$0.580469\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	36.3731	0.138549	0.0692746	−	0.997598i	$-0.477932\pi$
$41$	36.3731	0.138549	0.0692746	+	0.997598i	$0.477932\pi$
$42$	0	0
$43$	82.0000	0.290811	0.145406	−	0.989372i	$-0.453551\pi$
$43$	82.0000	0.290811	0.145406	+	0.989372i	$0.453551\pi$
$44$	−415.692	−1.42427
$45$	0	0
$46$	0	0
$47$	−72.7461	−0.225768	−0.112884	−	0.993608i	$-0.536009\pi$
$47$	−72.7461	−0.225768	−0.112884	+	0.993608i	$0.536009\pi$
$48$	0	0
$49$	−235.000	−0.685131
$50$	0	0
$51$	0	0
$52$	0	0
$53$	261.000	0.676436	0.338218	−	0.941068i	$-0.390176\pi$
$53$	261.000	0.676436	0.338218	+	0.941068i	$0.390176\pi$
$54$	0	0
$55$	−270.000	−0.661942
$56$	0	0
$57$	0	0
$58$	0	0
$59$	789.815	1.74280	0.871400	−	0.490574i	$-0.163213\pi$
$59$	789.815	1.74280	0.871400	+	0.490574i	$0.163213\pi$
$60$	0	0
$61$	−719.000	−1.50916	−0.754578	−	0.656210i	$-0.772158\pi$
$61$	−719.000	−1.50916	−0.754578	+	0.656210i	$0.772158\pi$
$62$	0	0
$63$	0	0
$64$	−512.000	−1.00000
$65$	0	0
$66$	0	0
$67$	−703.213	−1.28226	−0.641128	−	0.767434i	$-0.721533\pi$
$67$	−703.213	−1.28226	−0.641128	+	0.767434i	$0.721533\pi$
$68$	936.000	1.66922
$69$	0	0
$70$	0	0
$71$	−467.654	−0.781694	−0.390847	−	0.920456i	$-0.627818\pi$
$71$	−467.654	−0.781694	−0.390847	+	0.920456i	$0.627818\pi$
$72$	0	0
$73$	−684.160	−1.09692	−0.548458	−	0.836178i	$-0.684785\pi$
$73$	−684.160	−1.09692	−0.548458	+	0.836178i	$0.684785\pi$
$74$	0	0
$75$	0	0
$76$	−193.990	−0.292791
$77$	540.000	0.799204
$78$	0	0
$79$	−440.000	−0.626631	−0.313316	−	0.949649i	$-0.601440\pi$
$79$	−440.000	−0.626631	−0.313316	+	0.949649i	$0.601440\pi$
$80$	−332.554	−0.464758
$81$	0	0
$82$	0	0
$83$	−1195.12	−1.58049	−0.790247	−	0.612789i	$-0.790048\pi$
$83$	−1195.12	−1.58049	−0.790247	+	0.612789i	$0.790048\pi$
$84$	0	0
$85$	607.950	0.775781
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1517.28	1.80709	0.903545	−	0.428493i	$-0.140955\pi$
$89$	1517.28	1.80709	0.903545	+	0.428493i	$0.140955\pi$
$90$	0	0
$91$	0	0
$92$	−144.000	−0.163185
$93$	0	0
$94$	0	0
$95$	−126.000	−0.136077
$96$	0	0
$97$	1157.01	1.21110	0.605549	−	0.795808i	$-0.292953\pi$
$97$	1157.01	1.21110	0.605549	+	0.795808i	$0.292953\pi$
$98$	0	0
$99$	0	0
$100$	784.000	0.784000
$101$	1575.00	1.55167	0.775833	−	0.630938i	$-0.217330\pi$
$101$	1575.00	1.55167	0.775833	+	0.630938i	$0.217330\pi$
$102$	0	0
$103$	−794.000	−0.759565	−0.379782	−	0.925076i	$-0.624001\pi$
$103$	−794.000	−0.759565	−0.379782	+	0.925076i	$0.624001\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−450.000	−0.406571	−0.203286	−	0.979119i	$-0.565162\pi$
$107$	−450.000	−0.406571	−0.203286	+	0.979119i	$0.565162\pi$
$108$	0	0
$109$	595.825	0.523576	0.261788	−	0.965125i	$-0.415688\pi$
$109$	595.825	0.523576	0.261788	+	0.965125i	$0.415688\pi$
$110$	0	0
$111$	0	0
$112$	665.108	0.561132
$113$	1701.00	1.41608	0.708038	−	0.706174i	$-0.249580\pi$
$113$	1701.00	1.41608	0.708038	+	0.706174i	$0.249580\pi$
$114$	0	0
$115$	−93.5307	−0.0758416
$116$	−792.000	−0.633925
$117$	0	0
$118$	0	0
$119$	−1215.90	−0.936650
$120$	0	0
$121$	1369.00	1.02855
$122$	0	0
$123$	0	0
$124$	−1551.92	−1.12392
$125$	1158.74	0.829128
$126$	0	0
$127$	1664.00	1.16265	0.581323	−	0.813673i	$-0.302535\pi$
$127$	1664.00	1.16265	0.581323	+	0.813673i	$0.302535\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1476.00	0.984418	0.492209	−	0.870477i	$-0.336190\pi$
$131$	1476.00	0.984418	0.492209	+	0.870477i	$0.336190\pi$
$132$	0	0
$133$	252.000	0.164295
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1013.25	−0.631882	−0.315941	−	0.948779i	$-0.602320\pi$
$137$	−1013.25	−0.631882	−0.315941	+	0.948779i	$0.602320\pi$
$138$	0	0
$139$	1124.00	0.685874	0.342937	−	0.939358i	$-0.388578\pi$
$139$	1124.00	0.685874	0.342937	+	0.939358i	$0.388578\pi$
$140$	432.000	0.260790
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−514.419	−0.294622
$146$	0	0
$147$	0	0
$148$	900.666	0.500232
$149$	3268.38	1.79702	0.898510	−	0.438952i	$-0.144650\pi$
$149$	3268.38	1.79702	0.898510	+	0.438952i	$0.144650\pi$
$150$	0	0
$151$	1638.52	0.883052	0.441526	−	0.897248i	$-0.354437\pi$
$151$	1638.52	0.883052	0.441526	+	0.897248i	$0.354437\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−1008.00	−0.522352
$156$	0	0
$157$	1259.00	0.639995	0.319997	−	0.947418i	$-0.396318\pi$
$157$	1259.00	0.639995	0.319997	+	0.947418i	$0.396318\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	187.061	0.0915684
$162$	0	0
$163$	2951.41	1.41824	0.709118	−	0.705089i	$-0.249093\pi$
$163$	2951.41	1.41824	0.709118	+	0.705089i	$0.249093\pi$
$164$	−290.985	−0.138549
$165$	0	0
$166$	0	0
$167$	−3138.48	−1.45427	−0.727133	−	0.686496i	$-0.759148\pi$
$167$	−3138.48	−1.45427	−0.727133	+	0.686496i	$0.759148\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	0	0
$172$	−656.000	−0.290811
$173$	4266.00	1.87479	0.937393	−	0.348273i	$-0.113232\pi$
$173$	4266.00	1.87479	0.937393	+	0.348273i	$0.113232\pi$
$174$	0	0
$175$	−1018.45	−0.439927
$176$	3325.54	1.42427
$177$	0	0
$178$	0	0
$179$	3006.00	1.25519	0.627595	−	0.778540i	$-0.284039\pi$
$179$	3006.00	1.25519	0.627595	+	0.778540i	$0.284039\pi$
$180$	0	0
$181$	1873.00	0.769166	0.384583	−	0.923090i	$-0.374345\pi$
$181$	1873.00	0.769166	0.384583	+	0.923090i	$0.374345\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	585.000	0.232487
$186$	0	0
$187$	−6079.50	−2.37742
$188$	581.969	0.225768
$189$	0	0
$190$	0	0
$191$	2736.00	1.03649	0.518246	−	0.855232i	$-0.326585\pi$
$191$	2736.00	1.03649	0.518246	+	0.855232i	$0.326585\pi$
$192$	0	0
$193$	2603.27	0.970920	0.485460	−	0.874259i	$-0.338652\pi$
$193$	2603.27	0.970920	0.485460	+	0.874259i	$0.338652\pi$
$194$	0	0
$195$	0	0
$196$	1880.00	0.685131
$197$	−3720.45	−1.34554	−0.672768	−	0.739853i	$-0.734895\pi$
$197$	−3720.45	−1.34554	−0.672768	+	0.739853i	$0.734895\pi$
$198$	0	0
$199$	−1198.00	−0.426754	−0.213377	−	0.976970i	$-0.568446\pi$
$199$	−1198.00	−0.426754	−0.213377	+	0.976970i	$0.568446\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1028.84	0.355716
$204$	0	0
$205$	−189.000	−0.0643919
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1260.00	0.417014
$210$	0	0
$211$	2392.00	0.780436	0.390218	−	0.920722i	$-0.372400\pi$
$211$	2392.00	0.780436	0.390218	+	0.920722i	$0.372400\pi$
$212$	−2088.00	−0.676436
$213$	0	0
$214$	0	0
$215$	−426.084	−0.135157
$216$	0	0
$217$	2016.00	0.630668
$218$	0	0
$219$	0	0
$220$	2160.00	0.661942
$221$	0	0
$222$	0	0
$223$	−2036.89	−0.611661	−0.305830	−	0.952086i	$-0.598934\pi$
$223$	−2036.89	−0.611661	−0.305830	+	0.952086i	$0.598934\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2151.21	0.628990	0.314495	−	0.949259i	$-0.398165\pi$
$227$	2151.21	0.628990	0.314495	+	0.949259i	$0.398165\pi$
$228$	0	0
$229$	−3471.03	−1.00162	−0.500812	−	0.865556i	$-0.666965\pi$
$229$	−3471.03	−1.00162	−0.500812	+	0.865556i	$0.666965\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1854.00	−0.521286	−0.260643	−	0.965435i	$-0.583935\pi$
$233$	−1854.00	−0.521286	−0.260643	+	0.965435i	$0.583935\pi$
$234$	0	0
$235$	378.000	0.104928
$236$	−6318.52	−1.74280
$237$	0	0
$238$	0	0
$239$	4458.30	1.20662	0.603312	−	0.797505i	$-0.293847\pi$
$239$	4458.30	1.20662	0.603312	+	0.797505i	$0.293847\pi$
$240$	0	0
$241$	−417.424	−0.111571	−0.0557856	−	0.998443i	$-0.517766\pi$
$241$	−417.424	−0.111571	−0.0557856	+	0.998443i	$0.517766\pi$
$242$	0	0
$243$	0	0
$244$	5752.00	1.50916
$245$	1221.10	0.318420
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	4104.00	1.03204	0.516020	−	0.856576i	$-0.327413\pi$
$251$	4104.00	1.03204	0.516020	+	0.856576i	$0.327413\pi$
$252$	0	0
$253$	935.307	0.232420
$254$	0	0
$255$	0	0
$256$	4096.00	1.00000
$257$	1989.00	0.482764	0.241382	−	0.970430i	$-0.422399\pi$
$257$	1989.00	0.482764	0.241382	+	0.970430i	$0.422399\pi$
$258$	0	0
$259$	−1170.00	−0.280696
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−738.000	−0.173031	−0.0865153	−	0.996251i	$-0.527573\pi$
$263$	−738.000	−0.173031	−0.0865153	+	0.996251i	$0.527573\pi$
$264$	0	0
$265$	−1356.20	−0.314379
$266$	0	0
$267$	0	0
$268$	5625.70	1.28226
$269$	2106.00	0.477342	0.238671	−	0.971100i	$-0.423288\pi$
$269$	2106.00	0.477342	0.238671	+	0.971100i	$0.423288\pi$
$270$	0	0
$271$	−685.892	−0.153745	−0.0768727	−	0.997041i	$-0.524493\pi$
$271$	−685.892	−0.153745	−0.0768727	+	0.997041i	$0.524493\pi$
$272$	−7488.00	−1.66922
$273$	0	0
$274$	0	0
$275$	−5092.23	−1.11663
$276$	0	0
$277$	−3665.00	−0.794977	−0.397488	−	0.917607i	$-0.630118\pi$
$277$	−3665.00	−0.794977	−0.397488	+	0.917607i	$0.630118\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−1719.93	−0.365132	−0.182566	−	0.983194i	$-0.558440\pi$
$281$	−1719.93	−0.365132	−0.182566	+	0.983194i	$0.558440\pi$
$282$	0	0
$283$	1826.00	0.383549	0.191775	−	0.981439i	$-0.438576\pi$
$283$	1826.00	0.383549	0.191775	+	0.981439i	$0.438576\pi$
$284$	3741.23	0.781694
$285$	0	0
$286$	0	0
$287$	378.000	0.0777444
$288$	0	0
$289$	8776.00	1.78628
$290$	0	0
$291$	0	0
$292$	5473.28	1.09692
$293$	504.027	0.100497	0.0502484	−	0.998737i	$-0.483999\pi$
$293$	504.027	0.100497	0.0502484	+	0.998737i	$0.483999\pi$
$294$	0	0
$295$	−4104.00	−0.809980
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	852.169	0.163183
$302$	0	0
$303$	0	0
$304$	1551.92	0.292791
$305$	3736.03	0.701392
$306$	0	0
$307$	−1950.29	−0.362570	−0.181285	−	0.983431i	$-0.558026\pi$
$307$	−1950.29	−0.362570	−0.181285	+	0.983431i	$0.558026\pi$
$308$	−4320.00	−0.799204
$309$	0	0
$310$	0	0
$311$	−3798.00	−0.692491	−0.346246	−	0.938144i	$-0.612544\pi$
$311$	−3798.00	−0.692491	−0.346246	+	0.938144i	$0.612544\pi$
$312$	0	0
$313$	1378.00	0.248847	0.124424	−	0.992229i	$-0.460292\pi$
$313$	1378.00	0.248847	0.124424	+	0.992229i	$0.460292\pi$
$314$	0	0
$315$	0	0
$316$	3520.00	0.626631
$317$	−7103.14	−1.25852	−0.629262	−	0.777193i	$-0.716643\pi$
$317$	−7103.14	−1.25852	−0.629262	+	0.777193i	$0.716643\pi$
$318$	0	0
$319$	5144.19	0.902882
$320$	2660.43	0.464758
$321$	0	0
$322$	0	0
$323$	−2837.10	−0.488732
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−756.000	−0.126686
$330$	0	0
$331$	−10073.6	−1.67280	−0.836398	−	0.548122i	$-0.815343\pi$
$331$	−10073.6	−1.67280	−0.836398	+	0.548122i	$0.815343\pi$
$332$	9560.92	1.58049
$333$	0	0
$334$	0	0
$335$	3654.00	0.595938
$336$	0	0
$337$	−9001.00	−1.45494	−0.727471	−	0.686138i	$-0.759305\pi$
$337$	−9001.00	−1.45494	−0.727471	+	0.686138i	$0.759305\pi$
$338$	0	0
$339$	0	0
$340$	−4863.60	−0.775781
$341$	10080.0	1.60077
$342$	0	0
$343$	−6006.75	−0.945581
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−3294.00	−0.509600	−0.254800	−	0.966994i	$-0.582010\pi$
$347$	−3294.00	−0.509600	−0.254800	+	0.966994i	$0.582010\pi$
$348$	0	0
$349$	10544.7	1.61732	0.808662	−	0.588273i	$-0.200192\pi$
$349$	10544.7	1.61732	0.808662	+	0.588273i	$0.200192\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	2478.56	0.373713	0.186856	−	0.982387i	$-0.440170\pi$
$353$	2478.56	0.373713	0.186856	+	0.982387i	$0.440170\pi$
$354$	0	0
$355$	2430.00	0.363299
$356$	−12138.2	−1.80709
$357$	0	0
$358$	0	0
$359$	−5414.39	−0.795991	−0.397995	−	0.917387i	$-0.630294\pi$
$359$	−5414.39	−0.795991	−0.397995	+	0.917387i	$0.630294\pi$
$360$	0	0
$361$	−6271.00	−0.914273
$362$	0	0
$363$	0	0
$364$	0	0
$365$	3555.00	0.509801
$366$	0	0
$367$	−9946.00	−1.41465	−0.707326	−	0.706888i	$-0.750099\pi$
$367$	−9946.00	−1.41465	−0.707326	+	0.706888i	$0.750099\pi$
$368$	1152.00	0.163185
$369$	0	0
$370$	0	0
$371$	2712.39	0.379570
$372$	0	0
$373$	−7301.00	−1.01349	−0.506745	−	0.862096i	$-0.669151\pi$
$373$	−7301.00	−1.01349	−0.506745	+	0.862096i	$0.669151\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−3422.53	−0.463862	−0.231931	−	0.972732i	$-0.574504\pi$
$379$	−3422.53	−0.463862	−0.231931	+	0.972732i	$0.574504\pi$
$380$	1008.00	0.136077
$381$	0	0
$382$	0	0
$383$	−5778.12	−0.770883	−0.385442	−	0.922732i	$-0.625951\pi$
$383$	−5778.12	−0.770883	−0.385442	+	0.922732i	$0.625951\pi$
$384$	0	0
$385$	−2805.92	−0.371436
$386$	0	0
$387$	0	0
$388$	−9256.08	−1.21110
$389$	9153.00	1.19300	0.596498	−	0.802614i	$-0.296558\pi$
$389$	9153.00	1.19300	0.596498	+	0.802614i	$0.296558\pi$
$390$	0	0
$391$	−2106.00	−0.272391
$392$	0	0
$393$	0	0
$394$	0	0
$395$	2286.31	0.291232
$396$	0	0
$397$	2023.04	0.255751	0.127876	−	0.991790i	$-0.459184\pi$
$397$	2023.04	0.255751	0.127876	+	0.991790i	$0.459184\pi$
$398$	0	0
$399$	0	0
$400$	−6272.00	−0.784000
$401$	8308.65	1.03470	0.517349	−	0.855774i	$-0.326919\pi$
$401$	8308.65	1.03470	0.517349	+	0.855774i	$0.326919\pi$
$402$	0	0
$403$	0	0
$404$	−12600.0	−1.55167
$405$	0	0
$406$	0	0
$407$	−5850.00	−0.712466
$408$	0	0
$409$	10418.3	1.25954	0.629769	−	0.776782i	$-0.283150\pi$
$409$	10418.3	1.25954	0.629769	+	0.776782i	$0.283150\pi$
$410$	0	0
$411$	0	0
$412$	6352.00	0.759565
$413$	8208.00	0.977940
$414$	0	0
$415$	6210.00	0.734547
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−4176.00	−0.486900	−0.243450	−	0.969913i	$-0.578279\pi$
$419$	−4176.00	−0.486900	−0.243450	+	0.969913i	$0.578279\pi$
$420$	0	0
$421$	14471.3	1.67527	0.837633	−	0.546233i	$-0.183939\pi$
$421$	14471.3	1.67527	0.837633	+	0.546233i	$0.183939\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	11466.0	1.30867
$426$	0	0
$427$	−7472.07	−0.846835
$428$	3600.00	0.406571
$429$	0	0
$430$	0	0
$431$	6578.33	0.735190	0.367595	−	0.929986i	$-0.380181\pi$
$431$	6578.33	0.735190	0.367595	+	0.929986i	$0.380181\pi$
$432$	0	0
$433$	6605.00	0.733062	0.366531	−	0.930406i	$-0.380545\pi$
$433$	6605.00	0.733062	0.366531	+	0.930406i	$0.380545\pi$
$434$	0	0
$435$	0	0
$436$	−4766.60	−0.523576
$437$	436.477	0.0477792
$438$	0	0
$439$	8542.00	0.928673	0.464336	−	0.885659i	$-0.346293\pi$
$439$	8542.00	0.928673	0.464336	+	0.885659i	$0.346293\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	14328.0	1.53667	0.768334	−	0.640049i	$-0.221086\pi$
$443$	14328.0	1.53667	0.768334	+	0.640049i	$0.221086\pi$
$444$	0	0
$445$	−7884.00	−0.839859
$446$	0	0
$447$	0	0
$448$	−5320.86	−0.561132
$449$	−3013.77	−0.316767	−0.158384	−	0.987378i	$-0.550628\pi$
$449$	−3013.77	−0.316767	−0.158384	+	0.987378i	$0.550628\pi$
$450$	0	0
$451$	1890.00	0.197332
$452$	−13608.0	−1.41608
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−2887.33	−0.295544	−0.147772	−	0.989021i	$-0.547210\pi$
$457$	−2887.33	−0.295544	−0.147772	+	0.989021i	$0.547210\pi$
$458$	0	0
$459$	0	0
$460$	748.246	0.0758416
$461$	3600.93	0.363801	0.181900	−	0.983317i	$-0.441775\pi$
$461$	3600.93	0.363801	0.181900	+	0.983317i	$0.441775\pi$
$462$	0	0
$463$	−2677.75	−0.268781	−0.134391	−	0.990928i	$-0.542908\pi$
$463$	−2677.75	−0.268781	−0.134391	+	0.990928i	$0.542908\pi$
$464$	6336.00	0.633925
$465$	0	0
$466$	0	0
$467$	13878.0	1.37515	0.687577	−	0.726111i	$-0.258674\pi$
$467$	13878.0	1.37515	0.687577	+	0.726111i	$0.258674\pi$
$468$	0	0
$469$	−7308.00	−0.719514
$470$	0	0
$471$	0	0
$472$	0	0
$473$	4260.84	0.414194
$474$	0	0
$475$	−2376.37	−0.229548
$476$	9727.20	0.936650
$477$	0	0
$478$	0	0
$479$	1101.58	0.105079	0.0525393	−	0.998619i	$-0.483269\pi$
$479$	1101.58	0.105079	0.0525393	+	0.998619i	$0.483269\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	−10952.0	−1.02855
$485$	−6012.00	−0.562868
$486$	0	0
$487$	17123.1	1.59326	0.796632	−	0.604464i	$-0.206613\pi$
$487$	17123.1	1.59326	0.796632	+	0.604464i	$0.206613\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	450.000	0.0413609	0.0206805	−	0.999786i	$-0.493417\pi$
$491$	450.000	0.0413609	0.0206805	+	0.999786i	$0.493417\pi$
$492$	0	0
$493$	−11583.0	−1.05816
$494$	0	0
$495$	0	0
$496$	12415.3	1.12392
$497$	−4860.00	−0.438633
$498$	0	0
$499$	13219.0	1.18590	0.592950	−	0.805239i	$-0.297963\pi$
$499$	13219.0	1.18590	0.592950	+	0.805239i	$0.297963\pi$
$500$	−9269.94	−0.829128
$501$	0	0
$502$	0	0
$503$	−5346.00	−0.473889	−0.236945	−	0.971523i	$-0.576146\pi$
$503$	−5346.00	−0.473889	−0.236945	+	0.971523i	$0.576146\pi$
$504$	0	0
$505$	−8183.94	−0.721150
$506$	0	0
$507$	0	0
$508$	−13312.0	−1.16265
$509$	5866.46	0.510857	0.255428	−	0.966828i	$-0.417783\pi$
$509$	5866.46	0.510857	0.255428	+	0.966828i	$0.417783\pi$
$510$	0	0
$511$	−7110.00	−0.615514
$512$	0	0
$513$	0	0
$514$	0	0
$515$	4125.75	0.353014
$516$	0	0
$517$	−3780.00	−0.321556
$518$	0	0
$519$	0	0
$520$	0	0
$521$	9657.00	0.812055	0.406028	−	0.913861i	$-0.366914\pi$
$521$	9657.00	0.812055	0.406028	+	0.913861i	$0.366914\pi$
$522$	0	0
$523$	21626.0	1.80811	0.904053	−	0.427421i	$-0.140578\pi$
$523$	21626.0	1.80811	0.904053	+	0.427421i	$0.140578\pi$
$524$	−11808.0	−0.984418
$525$	0	0
$526$	0	0
$527$	−22696.8	−1.87607
$528$	0	0
$529$	−11843.0	−0.973371
$530$	0	0
$531$	0	0
$532$	−2016.00	−0.164295
$533$	0	0
$534$	0	0
$535$	2338.27	0.188957
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−12211.0	−0.975813
$540$	0	0
$541$	5371.09	0.426841	0.213421	−	0.976960i	$-0.431540\pi$
$541$	5371.09	0.426841	0.213421	+	0.976960i	$0.431540\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−3096.00	−0.243336
$546$	0	0
$547$	16946.0	1.32460	0.662302	−	0.749237i	$-0.269579\pi$
$547$	16946.0	1.32460	0.662302	+	0.749237i	$0.269579\pi$
$548$	8106.00	0.631882
$549$	0	0
$550$	0	0
$551$	2400.62	0.185608
$552$	0	0
$553$	−4572.61	−0.351623
$554$	0	0
$555$	0	0
$556$	−8992.00	−0.685874
$557$	3860.74	0.293689	0.146845	−	0.989160i	$-0.453088\pi$
$557$	3860.74	0.293689	0.146845	+	0.989160i	$0.453088\pi$
$558$	0	0
$559$	0	0
$560$	−3456.00	−0.260790
$561$	0	0
$562$	0	0
$563$	21672.0	1.62232	0.811160	−	0.584825i	$-0.198837\pi$
$563$	21672.0	1.62232	0.811160	+	0.584825i	$0.198837\pi$
$564$	0	0
$565$	−8838.66	−0.658133
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−1386.00	−0.102116	−0.0510581	−	0.998696i	$-0.516259\pi$
$569$	−1386.00	−0.102116	−0.0510581	+	0.998696i	$0.516259\pi$
$570$	0	0
$571$	−1162.00	−0.0851632	−0.0425816	−	0.999093i	$-0.513558\pi$
$571$	−1162.00	−0.0851632	−0.0425816	+	0.999093i	$0.513558\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1764.00	−0.127937
$576$	0	0
$577$	−8045.38	−0.580474	−0.290237	−	0.956955i	$-0.593734\pi$
$577$	−8045.38	−0.580474	−0.290237	+	0.956955i	$0.593734\pi$
$578$	0	0
$579$	0	0
$580$	4115.35	0.294622
$581$	−12420.0	−0.886865
$582$	0	0
$583$	13562.0	0.963429
$584$	0	0
$585$	0	0
$586$	0	0
$587$	27622.7	1.94227	0.971135	−	0.238530i	$-0.0766654\pi$
$587$	27622.7	1.94227	0.971135	+	0.238530i	$0.0766654\pi$
$588$	0	0
$589$	4704.00	0.329075
$590$	0	0
$591$	0	0
$592$	−7205.33	−0.500232
$593$	−275.396	−0.0190711	−0.00953555	−	0.999955i	$-0.503035\pi$
$593$	−275.396	−0.0190711	−0.00953555	+	0.999955i	$0.503035\pi$
$594$	0	0
$595$	6318.00	0.435316
$596$	−26147.0	−1.79702
$597$	0	0
$598$	0	0
$599$	−22356.0	−1.52494	−0.762472	−	0.647021i	$-0.776014\pi$
$599$	−22356.0	−1.52494	−0.762472	+	0.647021i	$0.776014\pi$
$600$	0	0
$601$	−18083.0	−1.22732	−0.613661	−	0.789569i	$-0.710304\pi$
$601$	−18083.0	−1.22732	−0.613661	+	0.789569i	$0.710304\pi$
$602$	0	0
$603$	0	0
$604$	−13108.2	−0.883052
$605$	−7113.53	−0.478027
$606$	0	0
$607$	5480.00	0.366435	0.183218	−	0.983072i	$-0.441349\pi$
$607$	5480.00	0.366435	0.183218	+	0.983072i	$0.441349\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	17737.9	1.16872	0.584362	−	0.811493i	$-0.301345\pi$
$613$	17737.9	1.16872	0.584362	+	0.811493i	$0.301345\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	9867.49	0.643842	0.321921	−	0.946767i	$-0.395672\pi$
$617$	9867.49	0.643842	0.321921	+	0.946767i	$0.395672\pi$
$618$	0	0
$619$	−4115.35	−0.267221	−0.133611	−	0.991034i	$-0.542657\pi$
$619$	−4115.35	−0.267221	−0.133611	+	0.991034i	$0.542657\pi$
$620$	8064.00	0.522352
$621$	0	0
$622$	0	0
$623$	15768.0	1.01402
$624$	0	0
$625$	6229.00	0.398656
$626$	0	0
$627$	0	0
$628$	−10072.0	−0.639995
$629$	13172.2	0.834995
$630$	0	0
$631$	−12664.8	−0.799011	−0.399506	−	0.916731i	$-0.630818\pi$
$631$	−12664.8	−0.799011	−0.399506	+	0.916731i	$0.630818\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−8646.40	−0.540349
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−3789.00	−0.233473	−0.116737	−	0.993163i	$-0.537243\pi$
$641$	−3789.00	−0.233473	−0.116737	+	0.993163i	$0.537243\pi$
$642$	0	0
$643$	−16911.7	−1.03722	−0.518611	−	0.855010i	$-0.673551\pi$
$643$	−16911.7	−1.03722	−0.518611	+	0.855010i	$0.673551\pi$
$644$	−1496.49	−0.0915684
$645$	0	0
$646$	0	0
$647$	−27792.0	−1.68874	−0.844371	−	0.535759i	$-0.820026\pi$
$647$	−27792.0	−1.68874	−0.844371	+	0.535759i	$0.820026\pi$
$648$	0	0
$649$	41040.0	2.48222
$650$	0	0
$651$	0	0
$652$	−23611.3	−1.41824
$653$	594.000	0.0355973	0.0177986	−	0.999842i	$-0.494334\pi$
$653$	594.000	0.0355973	0.0177986	+	0.999842i	$0.494334\pi$
$654$	0	0
$655$	−7669.52	−0.457516
$656$	2327.88	0.138549
$657$	0	0
$658$	0	0
$659$	−17748.0	−1.04911	−0.524555	−	0.851376i	$-0.675768\pi$
$659$	−17748.0	−1.04911	−0.524555	+	0.851376i	$0.675768\pi$
$660$	0	0
$661$	−15791.1	−0.929203	−0.464601	−	0.885520i	$-0.653802\pi$
$661$	−15791.1	−0.929203	−0.464601	+	0.885520i	$0.653802\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1309.43	−0.0763572
$666$	0	0
$667$	1782.00	0.103447
$668$	25107.8	1.45427
$669$	0	0
$670$	0	0
$671$	−37360.3	−2.14945
$672$	0	0
$673$	−20933.0	−1.19897	−0.599486	−	0.800385i	$-0.704628\pi$
$673$	−20933.0	−1.19897	−0.599486	+	0.800385i	$0.704628\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−3402.00	−0.193131	−0.0965653	−	0.995327i	$-0.530786\pi$
$677$	−3402.00	−0.193131	−0.0965653	+	0.995327i	$0.530786\pi$
$678$	0	0
$679$	12024.0	0.679586
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−24983.1	−1.39964	−0.699818	−	0.714321i	$-0.746736\pi$
$683$	−24983.1	−1.39964	−0.699818	+	0.714321i	$0.746736\pi$
$684$	0	0
$685$	5265.00	0.293672
$686$	0	0
$687$	0	0
$688$	5248.00	0.290811
$689$	0	0
$690$	0	0
$691$	13866.8	0.763412	0.381706	−	0.924284i	$-0.375337\pi$
$691$	13866.8	0.763412	0.381706	+	0.924284i	$0.375337\pi$
$692$	−34128.0	−1.87479
$693$	0	0
$694$	0	0
$695$	−5840.48	−0.318765
$696$	0	0
$697$	−4255.65	−0.231269
$698$	0	0
$699$	0	0
$700$	8147.57	0.439927
$701$	−21906.0	−1.18028	−0.590141	−	0.807300i	$-0.700928\pi$
$701$	−21906.0	−1.18028	−0.590141	+	0.807300i	$0.700928\pi$
$702$	0	0
$703$	−2730.00	−0.146464
$704$	−26604.3	−1.42427
$705$	0	0
$706$	0	0
$707$	16367.9	0.870690
$708$	0	0
$709$	−13057.9	−0.691680	−0.345840	−	0.938294i	$-0.612406\pi$
$709$	−13057.9	−0.691680	−0.345840	+	0.938294i	$0.612406\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	3491.81	0.183407
$714$	0	0
$715$	0	0
$716$	−24048.0	−1.25519
$717$	0	0
$718$	0	0
$719$	−14220.0	−0.737575	−0.368788	−	0.929514i	$-0.620227\pi$
$719$	−14220.0	−0.737575	−0.368788	+	0.929514i	$0.620227\pi$
$720$	0	0
$721$	−8251.49	−0.426216
$722$	0	0
$723$	0	0
$724$	−14984.0	−0.769166
$725$	−9702.00	−0.496998
$726$	0	0
$727$	5282.00	0.269462	0.134731	−	0.990882i	$-0.456983\pi$
$727$	5282.00	0.269462	0.134731	+	0.990882i	$0.456983\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−9594.00	−0.485427
$732$	0	0
$733$	11419.4	0.575424	0.287712	−	0.957717i	$-0.407105\pi$
$733$	11419.4	0.575424	0.287712	+	0.957717i	$0.407105\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−36540.0	−1.82628
$738$	0	0
$739$	20535.2	1.02219	0.511096	−	0.859524i	$-0.329240\pi$
$739$	20535.2	1.02219	0.511096	+	0.859524i	$0.329240\pi$
$740$	−4680.00	−0.232487
$741$	0	0
$742$	0	0
$743$	20826.2	1.02832	0.514158	−	0.857696i	$-0.328105\pi$
$743$	20826.2	1.02832	0.514158	+	0.857696i	$0.328105\pi$
$744$	0	0
$745$	−16983.0	−0.835180
$746$	0	0
$747$	0	0
$748$	48636.0	2.37742
$749$	−4676.54	−0.228140
$750$	0	0
$751$	4834.00	0.234880	0.117440	−	0.993080i	$-0.462531\pi$
$751$	4834.00	0.234880	0.117440	+	0.993080i	$0.462531\pi$
$752$	−4655.75	−0.225768
$753$	0	0
$754$	0	0
$755$	−8514.00	−0.410406
$756$	0	0
$757$	9046.00	0.434323	0.217161	−	0.976136i	$-0.430320\pi$
$757$	9046.00	0.434323	0.217161	+	0.976136i	$0.430320\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	12034.3	0.573249	0.286625	−	0.958043i	$-0.407467\pi$
$761$	12034.3	0.573249	0.286625	+	0.958043i	$0.407467\pi$
$762$	0	0
$763$	6192.00	0.293795
$764$	−21888.0	−1.03649
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	37543.9	1.76056	0.880279	−	0.474457i	$-0.157355\pi$
$769$	37543.9	1.76056	0.880279	+	0.474457i	$0.157355\pi$
$770$	0	0
$771$	0	0
$772$	−20826.2	−0.970920
$773$	−15713.2	−0.731130	−0.365565	−	0.930786i	$-0.619124\pi$
$773$	−15713.2	−0.731130	−0.365565	+	0.930786i	$0.619124\pi$
$774$	0	0
$775$	−19011.0	−0.881155
$776$	0	0
$777$	0	0
$778$	0	0
$779$	882.000	0.0405660
$780$	0	0
$781$	−24300.0	−1.11334
$782$	0	0
$783$	0	0
$784$	−15040.0	−0.685131
$785$	−6541.96	−0.297443
$786$	0	0
$787$	−3755.09	−0.170082	−0.0850409	−	0.996377i	$-0.527102\pi$
$787$	−3755.09	−0.170082	−0.0850409	+	0.996377i	$0.527102\pi$
$788$	29763.6	1.34554
$789$	0	0
$790$	0	0
$791$	17677.3	0.794605
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	9584.00	0.426754
$797$	−7830.00	−0.347996	−0.173998	−	0.984746i	$-0.555669\pi$
$797$	−7830.00	−0.347996	−0.173998	+	0.984746i	$0.555669\pi$
$798$	0	0
$799$	8511.30	0.376856
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−35550.0	−1.56231
$804$	0	0
$805$	−972.000	−0.0425571
$806$	0	0
$807$	0	0
$808$	0	0
$809$	6165.00	0.267923	0.133962	−	0.990987i	$-0.457230\pi$
$809$	6165.00	0.267923	0.133962	+	0.990987i	$0.457230\pi$
$810$	0	0
$811$	−29839.8	−1.29201	−0.646003	−	0.763335i	$-0.723560\pi$
$811$	−29839.8	−1.29201	−0.646003	+	0.763335i	$0.723560\pi$
$812$	−8230.71	−0.355716
$813$	0	0
$814$	0	0
$815$	−15336.0	−0.659137
$816$	0	0
$817$	1988.39	0.0851470
$818$	0	0
$819$	0	0
$820$	1512.00	0.0643919
$821$	29763.6	1.26523	0.632616	−	0.774466i	$-0.281981\pi$
$821$	29763.6	1.26523	0.632616	+	0.774466i	$0.281981\pi$
$822$	0	0
$823$	8920.00	0.377803	0.188901	−	0.981996i	$-0.439507\pi$
$823$	8920.00	0.377803	0.188901	+	0.981996i	$0.439507\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−18041.0	−0.758583	−0.379292	−	0.925277i	$-0.623832\pi$
$827$	−18041.0	−0.758583	−0.379292	+	0.925277i	$0.623832\pi$
$828$	0	0
$829$	21023.0	0.880771	0.440385	−	0.897809i	$-0.354842\pi$
$829$	21023.0	0.880771	0.440385	+	0.897809i	$0.354842\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	27495.0	1.14363
$834$	0	0
$835$	16308.0	0.675882
$836$	−10080.0	−0.417014
$837$	0	0
$838$	0	0
$839$	22073.3	0.908288	0.454144	−	0.890928i	$-0.349945\pi$
$839$	22073.3	0.908288	0.454144	+	0.890928i	$0.349945\pi$
$840$	0	0
$841$	−14588.0	−0.598139
$842$	0	0
$843$	0	0
$844$	−19136.0	−0.780436
$845$	0	0
$846$	0	0
$847$	14227.1	0.577152
$848$	16704.0	0.676436
$849$	0	0
$850$	0	0
$851$	−2026.50	−0.0816304
$852$	0	0
$853$	26609.5	1.06810	0.534051	−	0.845452i	$-0.320669\pi$
$853$	26609.5	1.06810	0.534051	+	0.845452i	$0.320669\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−12771.0	−0.509042	−0.254521	−	0.967067i	$-0.581918\pi$
$857$	−12771.0	−0.509042	−0.254521	+	0.967067i	$0.581918\pi$
$858$	0	0
$859$	17134.0	0.680564	0.340282	−	0.940323i	$-0.389477\pi$
$859$	17134.0	0.680564	0.340282	+	0.940323i	$0.389477\pi$
$860$	3408.68	0.135157
$861$	0	0
$862$	0	0
$863$	7929.33	0.312766	0.156383	−	0.987696i	$-0.450017\pi$
$863$	7929.33	0.312766	0.156383	+	0.987696i	$0.450017\pi$
$864$	0	0
$865$	−22166.8	−0.871322
$866$	0	0
$867$	0	0
$868$	−16128.0	−0.630668
$869$	−22863.1	−0.892493
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	12042.0	0.465250
$876$	0	0
$877$	−9864.03	−0.379800	−0.189900	−	0.981803i	$-0.560816\pi$
$877$	−9864.03	−0.379800	−0.189900	+	0.981803i	$0.560816\pi$
$878$	0	0
$879$	0	0
$880$	−17280.0	−0.661942
$881$	29169.0	1.11547	0.557735	−	0.830019i	$-0.311671\pi$
$881$	29169.0	1.11547	0.557735	+	0.830019i	$0.311671\pi$
$882$	0	0
$883$	−928.000	−0.0353677	−0.0176839	−	0.999844i	$-0.505629\pi$
$883$	−928.000	−0.0353677	−0.0176839	+	0.999844i	$0.505629\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	14400.0	0.545101	0.272551	−	0.962141i	$-0.412133\pi$
$887$	14400.0	0.545101	0.272551	+	0.962141i	$0.412133\pi$
$888$	0	0
$889$	17292.8	0.652398
$890$	0	0
$891$	0	0
$892$	16295.1	0.611661
$893$	−1764.00	−0.0661030
$894$	0	0
$895$	−15619.6	−0.583360
$896$	0	0
$897$	0	0
$898$	0	0
$899$	19205.0	0.712483
$900$	0	0
$901$	−30537.0	−1.12912
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−9732.39	−0.357476
$906$	0	0
$907$	−19684.0	−0.720614	−0.360307	−	0.932834i	$-0.617328\pi$
$907$	−19684.0	−0.720614	−0.360307	+	0.932834i	$0.617328\pi$
$908$	−17209.7	−0.628990
$909$	0	0
$910$	0	0
$911$	−24480.0	−0.890295	−0.445147	−	0.895457i	$-0.646849\pi$
$911$	−24480.0	−0.890295	−0.445147	+	0.895457i	$0.646849\pi$
$912$	0	0
$913$	−62100.0	−2.25105
$914$	0	0
$915$	0	0
$916$	27768.2	1.00162
$917$	15339.0	0.552388
$918$	0	0
$919$	38608.0	1.38581	0.692906	−	0.721028i	$-0.256330\pi$
$919$	38608.0	1.38581	0.692906	+	0.721028i	$0.256330\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	11033.2	0.392182
$926$	0	0
$927$	0	0
$928$	0	0
$929$	10210.4	0.360596	0.180298	−	0.983612i	$-0.442294\pi$
$929$	10210.4	0.360596	0.180298	+	0.983612i	$0.442294\pi$
$930$	0	0
$931$	−5698.45	−0.200600
$932$	14832.0	0.521286
$933$	0	0
$934$	0	0
$935$	31590.0	1.10492
$936$	0	0
$937$	28495.0	0.993480	0.496740	−	0.867899i	$-0.334530\pi$
$937$	28495.0	0.993480	0.496740	+	0.867899i	$0.334530\pi$
$938$	0	0
$939$	0	0
$940$	−3024.00	−0.104928
$941$	−10724.9	−0.371541	−0.185771	−	0.982593i	$-0.559478\pi$
$941$	−10724.9	−0.371541	−0.185771	+	0.982593i	$0.559478\pi$
$942$	0	0
$943$	654.715	0.0226092
$944$	50548.2	1.74280
$945$	0	0
$946$	0	0
$947$	33962.1	1.16538	0.582692	−	0.812693i	$-0.301999\pi$
$947$	33962.1	1.16538	0.582692	+	0.812693i	$0.301999\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	23814.0	0.809456	0.404728	−	0.914437i	$-0.367366\pi$
$953$	23814.0	0.809456	0.404728	+	0.914437i	$0.367366\pi$
$954$	0	0
$955$	−14216.7	−0.481718
$956$	−35666.4	−1.20662
$957$	0	0
$958$	0	0
$959$	−10530.0	−0.354569
$960$	0	0
$961$	7841.00	0.263200
$962$	0	0
$963$	0	0
$964$	3339.39	0.111571
$965$	−13527.0	−0.451243
$966$	0	0
$967$	51549.3	1.71429	0.857143	−	0.515079i	$-0.172238\pi$
$967$	51549.3	1.71429	0.857143	+	0.515079i	$0.172238\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	12312.0	0.406911	0.203456	−	0.979084i	$-0.434783\pi$
$971$	12312.0	0.406911	0.203456	+	0.979084i	$0.434783\pi$
$972$	0	0
$973$	11681.0	0.384865
$974$	0	0
$975$	0	0
$976$	−46016.0	−1.50916
$977$	−21538.1	−0.705285	−0.352642	−	0.935758i	$-0.614717\pi$
$977$	−21538.1	−0.705285	−0.352642	+	0.935758i	$0.614717\pi$
$978$	0	0
$979$	78840.0	2.57379
$980$	−9768.77	−0.318420
$981$	0	0
$982$	0	0
$983$	−32611.1	−1.05812	−0.529060	−	0.848585i	$-0.677455\pi$
$983$	−32611.1	−1.05812	−0.529060	+	0.848585i	$0.677455\pi$
$984$	0	0
$985$	19332.0	0.625349
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1476.00	0.0474561
$990$	0	0
$991$	−22330.0	−0.715778	−0.357889	−	0.933764i	$-0.616503\pi$
$991$	−22330.0	−0.715778	−0.357889	+	0.933764i	$0.616503\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	6224.99	0.198337
$996$	0	0
$997$	−24931.0	−0.791949	−0.395974	−	0.918262i	$-0.629593\pi$
$997$	−24931.0	−0.791949	−0.395974	+	0.918262i	$0.629593\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.4.a.m.1.1		2
3.2	odd	2		507.4.a.g.1.2		2
13.2	odd	12		117.4.q.b.82.1		2
13.7	odd	12		117.4.q.b.10.1		2
13.12	even	2	inner	1521.4.a.m.1.2		2
39.2	even	12		39.4.j.a.4.1	✓	2
39.5	even	4		507.4.b.a.337.1		2
39.8	even	4		507.4.b.a.337.2		2
39.20	even	12		39.4.j.a.10.1	yes	2
39.38	odd	2		507.4.a.g.1.1		2
156.59	odd	12		624.4.bv.a.49.1		2
156.119	odd	12		624.4.bv.a.433.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.j.a.4.1	✓	2	39.2	even	12
39.4.j.a.10.1	yes	2	39.20	even	12
117.4.q.b.10.1		2	13.7	odd	12
117.4.q.b.82.1		2	13.2	odd	12
507.4.a.g.1.1		2	39.38	odd	2
507.4.a.g.1.2		2	3.2	odd	2
507.4.b.a.337.1		2	39.5	even	4
507.4.b.a.337.2		2	39.8	even	4
624.4.bv.a.49.1		2	156.59	odd	12
624.4.bv.a.433.1		2	156.119	odd	12
1521.4.a.m.1.1		2	1.1	even	1	trivial
1521.4.a.m.1.2		2	13.12	even	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:	\( N \)	\(=\)	\( 1521 = 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1521.a (trivial)

\(f(q)\)	\(=\)	\(q-8.00000 q^{4} -5.19615 q^{5} +10.3923 q^{7} +51.9615 q^{11} +64.0000 q^{16} -117.000 q^{17} +24.2487 q^{19} +41.5692 q^{20} +18.0000 q^{23} -98.0000 q^{25} -83.1384 q^{28} +99.0000 q^{29} +193.990 q^{31} -54.0000 q^{35} -112.583 q^{37} +36.3731 q^{41} +82.0000 q^{43} -415.692 q^{44} -72.7461 q^{47} -235.000 q^{49} +261.000 q^{53} -270.000 q^{55} +789.815 q^{59} -719.000 q^{61} -512.000 q^{64} -703.213 q^{67} +936.000 q^{68} -467.654 q^{71} -684.160 q^{73} -193.990 q^{76} +540.000 q^{77} -440.000 q^{79} -332.554 q^{80} -1195.12 q^{83} +607.950 q^{85} +1517.28 q^{89} -144.000 q^{92} -126.000 q^{95} +1157.01 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 16 q^{4} + 128 q^{16} - 234 q^{17} + 36 q^{23} - 196 q^{25} + 198 q^{29} - 108 q^{35} + 164 q^{43} - 470 q^{49} + 522 q^{53} - 540 q^{55} - 1438 q^{61} - 1024 q^{64} + 1872 q^{68} + 1080 q^{77} - 880 q^{79}+ \cdots - 252 q^{95}+O(q^{100}) \) 2 * q - 16 * q^4 + 128 * q^16 - 234 * q^17 + 36 * q^23 - 196 * q^25 + 198 * q^29 - 108 * q^35 + 164 * q^43 - 470 * q^49 + 522 * q^53 - 540 * q^55 - 1438 * q^61 - 1024 * q^64 + 1872 * q^68 + 1080 * q^77 - 880 * q^79 - 288 * q^92 - 252 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more