LMFDB - Embedded newform 1521.4.a.bm.1.11

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:	$1$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 100 x^{16} + 4066 x^{14} - 86197 x^{12} + 1016064 x^{10} - 6594119 x^{8} + 22251817 x^{6} + \cdots - 6889792 $ x^18 - 100x^16 + 4066x^14 - 86197x^12 + 1016064x^10 - 6594119x^8 + 22251817x^6 - 37096332x^4 + 27824160x^2 - 6889792
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3}\cdot 7\cdot 13^{4} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$1.15064$ 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)

$f(q)$	$=$	$q+1.15064 q^{2} -6.67603 q^{4} +21.2034 q^{5} -31.2668 q^{7} -16.8868 q^{8} +24.3975 q^{10} +17.3811 q^{11} -35.9768 q^{14} +33.9775 q^{16} +89.1071 q^{17} -80.6466 q^{19} -141.555 q^{20} +19.9994 q^{22} -149.683 q^{23} +324.586 q^{25} +208.738 q^{28} +6.30162 q^{29} +78.3099 q^{31} +174.191 q^{32} +102.530 q^{34} -662.964 q^{35} +39.2975 q^{37} -92.7953 q^{38} -358.059 q^{40} +330.647 q^{41} -198.803 q^{43} -116.037 q^{44} -172.231 q^{46} -246.612 q^{47} +634.613 q^{49} +373.482 q^{50} -600.345 q^{53} +368.540 q^{55} +527.997 q^{56} +7.25090 q^{58} -709.534 q^{59} -472.354 q^{61} +90.1066 q^{62} -71.3895 q^{64} -331.545 q^{67} -594.881 q^{68} -762.833 q^{70} -472.765 q^{71} +651.016 q^{73} +45.2173 q^{74} +538.399 q^{76} -543.452 q^{77} +240.920 q^{79} +720.441 q^{80} +380.456 q^{82} +538.031 q^{83} +1889.38 q^{85} -228.751 q^{86} -293.512 q^{88} -673.484 q^{89} +999.288 q^{92} -283.762 q^{94} -1709.99 q^{95} -468.827 q^{97} +730.211 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 56 q^{4} - 94 q^{7} + 156 q^{10} + 88 q^{16} - 448 q^{19} - 256 q^{22} - 16 q^{25} - 1300 q^{28} - 818 q^{31} - 900 q^{34} - 524 q^{37} + 800 q^{40} + 752 q^{43} - 1650 q^{46} + 2948 q^{49} - 464 q^{55}+ \cdots - 4630 q^{97}+O(q^{100}) $ 18 * q + 56 * q^4 - 94 * q^7 + 156 * q^10 + 88 * q^16 - 448 * q^19 - 256 * q^22 - 16 * q^25 - 1300 * q^28 - 818 * q^31 - 900 * q^34 - 524 * q^37 + 800 * q^40 + 752 * q^43 - 1650 * q^46 + 2948 * q^49 - 464 * q^55 - 1626 * q^58 - 1784 * q^61 - 4274 * q^64 - 2872 * q^67 - 5772 * q^70 - 3078 * q^73 - 5726 * q^76 + 3326 * q^79 - 1038 * q^82 - 2340 * q^85 + 780 * q^88 + 1220 * q^94 - 4630 * q^97

Coefficient data

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

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

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

$2$	1.15064	0.406813	0.203406	−	0.979094i	$-0.434799\pi$
$2$	1.15064	0.406813	0.203406	+	0.979094i	$0.434799\pi$
$3$	0	0
$3$	0	0
$4$	−6.67603	−0.834503
$5$	21.2034	1.89649	0.948247	−	0.317534i	$-0.102855\pi$
$5$	21.2034	1.89649	0.948247	+	0.317534i	$0.102855\pi$
$6$	0	0
$7$	−31.2668	−1.68825	−0.844124	−	0.536148i	$-0.819879\pi$
$7$	−31.2668	−1.68825	−0.844124	+	0.536148i	$0.819879\pi$
$8$	−16.8868	−0.746300
$9$	0	0
$10$	24.3975	0.771518
$11$	17.3811	0.476419	0.238209	−	0.971214i	$-0.423440\pi$
$11$	17.3811	0.476419	0.238209	+	0.971214i	$0.423440\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−35.9768	−0.686801
$15$	0	0
$16$	33.9775	0.530899
$17$	89.1071	1.27127	0.635637	−	0.771988i	$-0.280738\pi$
$17$	89.1071	1.27127	0.635637	+	0.771988i	$0.280738\pi$
$18$	0	0
$19$	−80.6466	−0.973769	−0.486884	−	0.873466i	$-0.661867\pi$
$19$	−80.6466	−0.973769	−0.486884	+	0.873466i	$0.661867\pi$
$20$	−141.555	−1.58263
$21$	0	0
$22$	19.9994	0.193813
$23$	−149.683	−1.35700	−0.678502	−	0.734599i	$-0.737370\pi$
$23$	−149.683	−1.35700	−0.678502	+	0.734599i	$0.737370\pi$
$24$	0	0
$25$	324.586	2.59669
$26$	0	0
$27$	0	0
$28$	208.738	1.40885
$29$	6.30162	0.0403511	0.0201755	−	0.999796i	$-0.493577\pi$
$29$	6.30162	0.0403511	0.0201755	+	0.999796i	$0.493577\pi$
$30$	0	0
$31$	78.3099	0.453706	0.226853	−	0.973929i	$-0.427156\pi$
$31$	78.3099	0.453706	0.226853	+	0.973929i	$0.427156\pi$
$32$	174.191	0.962276
$33$	0	0
$34$	102.530	0.517170
$35$	−662.964	−3.20175
$36$	0	0
$37$	39.2975	0.174607	0.0873037	−	0.996182i	$-0.472175\pi$
$37$	39.2975	0.174607	0.0873037	+	0.996182i	$0.472175\pi$
$38$	−92.7953	−0.396142
$39$	0	0
$40$	−358.059	−1.41535
$41$	330.647	1.25947	0.629737	−	0.776809i	$-0.283163\pi$
$41$	330.647	1.25947	0.629737	+	0.776809i	$0.283163\pi$
$42$	0	0
$43$	−198.803	−0.705052	−0.352526	−	0.935802i	$-0.614677\pi$
$43$	−198.803	−0.705052	−0.352526	+	0.935802i	$0.614677\pi$
$44$	−116.037	−0.397573
$45$	0	0
$46$	−172.231	−0.552046
$47$	−246.612	−0.765364	−0.382682	−	0.923880i	$-0.625000\pi$
$47$	−246.612	−0.765364	−0.382682	+	0.923880i	$0.625000\pi$
$48$	0	0
$49$	634.613	1.85018
$50$	373.482	1.05637
$51$	0	0
$52$	0	0
$53$	−600.345	−1.55592	−0.777960	−	0.628314i	$-0.783745\pi$
$53$	−600.345	−1.55592	−0.777960	+	0.628314i	$0.783745\pi$
$54$	0	0
$55$	368.540	0.903525
$56$	527.997	1.25994
$57$	0	0
$58$	7.25090	0.0164153
$59$	−709.534	−1.56565	−0.782826	−	0.622240i	$-0.786223\pi$
$59$	−709.534	−1.56565	−0.782826	+	0.622240i	$0.786223\pi$
$60$	0	0
$61$	−472.354	−0.991455	−0.495727	−	0.868478i	$-0.665098\pi$
$61$	−472.354	−0.991455	−0.495727	+	0.868478i	$0.665098\pi$
$62$	90.1066	0.184573
$63$	0	0
$64$	−71.3895	−0.139433
$65$	0	0
$66$	0	0
$67$	−331.545	−0.604547	−0.302273	−	0.953221i	$-0.597746\pi$
$67$	−331.545	−0.604547	−0.302273	+	0.953221i	$0.597746\pi$
$68$	−594.881	−1.06088
$69$	0	0
$70$	−762.833	−1.30251
$71$	−472.765	−0.790238	−0.395119	−	0.918630i	$-0.629297\pi$
$71$	−472.765	−0.790238	−0.395119	+	0.918630i	$0.629297\pi$
$72$	0	0
$73$	651.016	1.04378	0.521888	−	0.853014i	$-0.325228\pi$
$73$	651.016	1.04378	0.521888	+	0.853014i	$0.325228\pi$
$74$	45.2173	0.0710325
$75$	0	0
$76$	538.399	0.812613
$77$	−543.452	−0.804313
$78$	0	0
$79$	240.920	0.343109	0.171555	−	0.985175i	$-0.445121\pi$
$79$	240.920	0.343109	0.171555	+	0.985175i	$0.445121\pi$
$80$	720.441	1.00685
$81$	0	0
$82$	380.456	0.512370
$83$	538.031	0.711524	0.355762	−	0.934577i	$-0.384221\pi$
$83$	538.031	0.711524	0.355762	+	0.934577i	$0.384221\pi$
$84$	0	0
$85$	1889.38	2.41096
$86$	−228.751	−0.286824
$87$	0	0
$88$	−293.512	−0.355551
$89$	−673.484	−0.802126	−0.401063	−	0.916050i	$-0.631359\pi$
$89$	−673.484	−0.802126	−0.401063	+	0.916050i	$0.631359\pi$
$90$	0	0
$91$	0	0
$92$	999.288	1.13242
$93$	0	0
$94$	−283.762	−0.311360
$95$	−1709.99	−1.84675
$96$	0	0
$97$	−468.827	−0.490744	−0.245372	−	0.969429i	$-0.578910\pi$
$97$	−468.827	−0.490744	−0.245372	+	0.969429i	$0.578910\pi$
$98$	730.211	0.752678
$99$	0	0
$100$	−2166.94	−2.16694
$101$	−305.731	−0.301201	−0.150601	−	0.988595i	$-0.548121\pi$
$101$	−305.731	−0.301201	−0.150601	+	0.988595i	$0.548121\pi$
$102$	0	0
$103$	−1560.85	−1.49315	−0.746576	−	0.665300i	$-0.768304\pi$
$103$	−1560.85	−1.49315	−0.746576	+	0.665300i	$0.768304\pi$
$104$	0	0
$105$	0	0
$106$	−690.781	−0.632968
$107$	−707.803	−0.639494	−0.319747	−	0.947503i	$-0.603598\pi$
$107$	−707.803	−0.639494	−0.319747	+	0.947503i	$0.603598\pi$
$108$	0	0
$109$	−1714.25	−1.50638	−0.753192	−	0.657801i	$-0.771487\pi$
$109$	−1714.25	−1.50638	−0.753192	+	0.657801i	$0.771487\pi$
$110$	424.057	0.367566
$111$	0	0
$112$	−1062.37	−0.896289
$113$	1263.51	1.05187	0.525935	−	0.850525i	$-0.323715\pi$
$113$	1263.51	1.05187	0.525935	+	0.850525i	$0.323715\pi$
$114$	0	0
$115$	−3173.80	−2.57355
$116$	−42.0698	−0.0336731
$117$	0	0
$118$	−816.419	−0.636928
$119$	−2786.09	−2.14623
$120$	0	0
$121$	−1028.90	−0.773025
$122$	−543.510	−0.403337
$123$	0	0
$124$	−522.799	−0.378619
$125$	4231.91	3.02811
$126$	0	0
$127$	893.935	0.624598	0.312299	−	0.949984i	$-0.398901\pi$
$127$	893.935	0.624598	0.312299	+	0.949984i	$0.398901\pi$
$128$	−1475.67	−1.01900
$129$	0	0
$130$	0	0
$131$	−9.87226	−0.00658430	−0.00329215	−	0.999995i	$-0.501048\pi$
$131$	−9.87226	−0.00658430	−0.00329215	+	0.999995i	$0.501048\pi$
$132$	0	0
$133$	2521.56	1.64396
$134$	−381.489	−0.245937
$135$	0	0
$136$	−1504.74	−0.948751
$137$	1486.59	0.927063	0.463532	−	0.886080i	$-0.346582\pi$
$137$	1486.59	0.927063	0.463532	+	0.886080i	$0.346582\pi$
$138$	0	0
$139$	530.179	0.323520	0.161760	−	0.986830i	$-0.448283\pi$
$139$	530.179	0.323520	0.161760	+	0.986830i	$0.448283\pi$
$140$	4425.96	2.67187
$141$	0	0
$142$	−543.983	−0.321479
$143$	0	0
$144$	0	0
$145$	133.616	0.0765255
$146$	749.086	0.424622
$147$	0	0
$148$	−262.351	−0.145710
$149$	−1796.65	−0.987832	−0.493916	−	0.869510i	$-0.664435\pi$
$149$	−1796.65	−0.987832	−0.493916	+	0.869510i	$0.664435\pi$
$150$	0	0
$151$	−1818.69	−0.980150	−0.490075	−	0.871680i	$-0.663031\pi$
$151$	−1818.69	−0.980150	−0.490075	+	0.871680i	$0.663031\pi$
$152$	1361.87	0.726723
$153$	0	0
$154$	−625.318	−0.327205
$155$	1660.44	0.860450
$156$	0	0
$157$	−875.076	−0.444832	−0.222416	−	0.974952i	$-0.571394\pi$
$157$	−875.076	−0.444832	−0.222416	+	0.974952i	$0.571394\pi$
$158$	277.213	0.139581
$159$	0	0
$160$	3693.44	1.82495
$161$	4680.11	2.29096
$162$	0	0
$163$	1777.39	0.854083	0.427042	−	0.904232i	$-0.359556\pi$
$163$	1777.39	0.854083	0.427042	+	0.904232i	$0.359556\pi$
$164$	−2207.41	−1.05103
$165$	0	0
$166$	619.080	0.289457
$167$	−2526.37	−1.17064	−0.585319	−	0.810803i	$-0.699031\pi$
$167$	−2526.37	−1.17064	−0.585319	+	0.810803i	$0.699031\pi$
$168$	0	0
$169$	0	0
$170$	2173.99	0.980810
$171$	0	0
$172$	1327.22	0.588368
$173$	−204.999	−0.0900913	−0.0450457	−	0.998985i	$-0.514343\pi$
$173$	−204.999	−0.0900913	−0.0450457	+	0.998985i	$0.514343\pi$
$174$	0	0
$175$	−10148.8	−4.38385
$176$	590.568	0.252930
$177$	0	0
$178$	−774.939	−0.326315
$179$	−1768.15	−0.738311	−0.369156	−	0.929368i	$-0.620353\pi$
$179$	−1768.15	−0.738311	−0.369156	+	0.929368i	$0.620353\pi$
$180$	0	0
$181$	2542.64	1.04416	0.522079	−	0.852897i	$-0.325157\pi$
$181$	2542.64	1.04416	0.522079	+	0.852897i	$0.325157\pi$
$182$	0	0
$183$	0	0
$184$	2527.67	1.01273
$185$	833.243	0.331142
$186$	0	0
$187$	1548.78	0.605659
$188$	1646.39	0.638699
$189$	0	0
$190$	−1967.58	−0.751280
$191$	40.7190	0.0154258	0.00771289	−	0.999970i	$-0.497545\pi$
$191$	40.7190	0.0154258	0.00771289	+	0.999970i	$0.497545\pi$
$192$	0	0
$193$	−2774.70	−1.03486	−0.517428	−	0.855727i	$-0.673110\pi$
$193$	−2774.70	−1.03486	−0.517428	+	0.855727i	$0.673110\pi$
$194$	−539.451	−0.199641
$195$	0	0
$196$	−4236.69	−1.54398
$197$	−1171.74	−0.423770	−0.211885	−	0.977295i	$-0.567960\pi$
$197$	−1171.74	−0.423770	−0.211885	+	0.977295i	$0.567960\pi$
$198$	0	0
$199$	1940.97	0.691417	0.345708	−	0.938342i	$-0.387639\pi$
$199$	1940.97	0.691417	0.345708	+	0.938342i	$0.387639\pi$
$200$	−5481.23	−1.93791
$201$	0	0
$202$	−351.786	−0.122533
$203$	−197.031	−0.0681226
$204$	0	0
$205$	7010.86	2.38858
$206$	−1795.97	−0.607434
$207$	0	0
$208$	0	0
$209$	−1401.73	−0.463922
$210$	0	0
$211$	1659.92	0.541580	0.270790	−	0.962638i	$-0.412715\pi$
$211$	1659.92	0.541580	0.270790	+	0.962638i	$0.412715\pi$
$212$	4007.92	1.29842
$213$	0	0
$214$	−814.427	−0.260155
$215$	−4215.32	−1.33713
$216$	0	0
$217$	−2448.50	−0.765968
$218$	−1972.49	−0.612816
$219$	0	0
$220$	−2460.38	−0.753995
$221$	0	0
$222$	0	0
$223$	−4619.40	−1.38716	−0.693582	−	0.720378i	$-0.743969\pi$
$223$	−4619.40	−1.38716	−0.693582	+	0.720378i	$0.743969\pi$
$224$	−5446.38	−1.62456
$225$	0	0
$226$	1453.85	0.427914
$227$	1220.49	0.356859	0.178429	−	0.983953i	$-0.442898\pi$
$227$	1220.49	0.356859	0.178429	+	0.983953i	$0.442898\pi$
$228$	0	0
$229$	−3911.47	−1.12872	−0.564361	−	0.825528i	$-0.690878\pi$
$229$	−3911.47	−1.12872	−0.564361	+	0.825528i	$0.690878\pi$
$230$	−3651.90	−1.04695
$231$	0	0
$232$	−106.414	−0.0301140
$233$	−4235.15	−1.19079	−0.595395	−	0.803433i	$-0.703004\pi$
$233$	−4235.15	−1.19079	−0.595395	+	0.803433i	$0.703004\pi$
$234$	0	0
$235$	−5229.03	−1.45151
$236$	4736.87	1.30654
$237$	0	0
$238$	−3205.79	−0.873112
$239$	772.395	0.209046	0.104523	−	0.994522i	$-0.466668\pi$
$239$	772.395	0.209046	0.104523	+	0.994522i	$0.466668\pi$
$240$	0	0
$241$	1319.51	0.352684	0.176342	−	0.984329i	$-0.443574\pi$
$241$	1319.51	0.352684	0.176342	+	0.984329i	$0.443574\pi$
$242$	−1183.89	−0.314477
$243$	0	0
$244$	3153.45	0.827372
$245$	13456.0	3.50886
$246$	0	0
$247$	0	0
$248$	−1322.41	−0.338600
$249$	0	0
$250$	4869.41	1.23187
$251$	−2422.90	−0.609292	−0.304646	−	0.952466i	$-0.598538\pi$
$251$	−2422.90	−0.609292	−0.304646	+	0.952466i	$0.598538\pi$
$252$	0	0
$253$	−2601.66	−0.646502
$254$	1028.60	0.254094
$255$	0	0
$256$	−1126.85	−0.275109
$257$	−3778.25	−0.917047	−0.458523	−	0.888682i	$-0.651621\pi$
$257$	−3778.25	−0.917047	−0.458523	+	0.888682i	$0.651621\pi$
$258$	0	0
$259$	−1228.71	−0.294781
$260$	0	0
$261$	0	0
$262$	−11.3594	−0.00267858
$263$	7343.43	1.72173	0.860866	−	0.508832i	$-0.169923\pi$
$263$	7343.43	1.72173	0.860866	+	0.508832i	$0.169923\pi$
$264$	0	0
$265$	−12729.4	−2.95079
$266$	2901.41	0.668786
$267$	0	0
$268$	2213.40	0.504496
$269$	−7812.34	−1.77073	−0.885365	−	0.464896i	$-0.846092\pi$
$269$	−7812.34	−1.77073	−0.885365	+	0.464896i	$0.846092\pi$
$270$	0	0
$271$	−5592.09	−1.25349	−0.626745	−	0.779225i	$-0.715613\pi$
$271$	−5592.09	−1.25349	−0.626745	+	0.779225i	$0.715613\pi$
$272$	3027.64	0.674918
$273$	0	0
$274$	1710.53	0.377141
$275$	5641.67	1.23711
$276$	0	0
$277$	3897.19	0.845342	0.422671	−	0.906283i	$-0.361093\pi$
$277$	3897.19	0.845342	0.422671	+	0.906283i	$0.361093\pi$
$278$	610.046	0.131612
$279$	0	0
$280$	11195.4	2.38947
$281$	−5921.18	−1.25704	−0.628519	−	0.777794i	$-0.716339\pi$
$281$	−5921.18	−1.25704	−0.628519	+	0.777794i	$0.716339\pi$
$282$	0	0
$283$	−4936.47	−1.03690	−0.518450	−	0.855108i	$-0.673491\pi$
$283$	−4936.47	−1.03690	−0.518450	+	0.855108i	$0.673491\pi$
$284$	3156.19	0.659456
$285$	0	0
$286$	0	0
$287$	−10338.3	−2.12630
$288$	0	0
$289$	3027.07	0.616136
$290$	153.744	0.0311316
$291$	0	0
$292$	−4346.20	−0.871035
$293$	−5575.98	−1.11178	−0.555891	−	0.831255i	$-0.687623\pi$
$293$	−5575.98	−1.11178	−0.555891	+	0.831255i	$0.687623\pi$
$294$	0	0
$295$	−15044.6	−2.96925
$296$	−663.611	−0.130309
$297$	0	0
$298$	−2067.29	−0.401863
$299$	0	0
$300$	0	0
$301$	6215.95	1.19030
$302$	−2092.66	−0.398738
$303$	0	0
$304$	−2740.17	−0.516973
$305$	−10015.5	−1.88029
$306$	0	0
$307$	10168.3	1.89035	0.945175	−	0.326564i	$-0.105891\pi$
$307$	10168.3	1.89035	0.945175	+	0.326564i	$0.105891\pi$
$308$	3628.10	0.671202
$309$	0	0
$310$	1910.57	0.350042
$311$	4680.41	0.853382	0.426691	−	0.904397i	$-0.359679\pi$
$311$	4680.41	0.853382	0.426691	+	0.904397i	$0.359679\pi$
$312$	0	0
$313$	2990.32	0.540009	0.270004	−	0.962859i	$-0.412975\pi$
$313$	2990.32	0.540009	0.270004	+	0.962859i	$0.412975\pi$
$314$	−1006.90	−0.180963
$315$	0	0
$316$	−1608.39	−0.286326
$317$	−9081.93	−1.60912	−0.804561	−	0.593870i	$-0.797599\pi$
$317$	−9081.93	−1.60912	−0.804561	+	0.593870i	$0.797599\pi$
$318$	0	0
$319$	109.529	0.0192240
$320$	−1513.70	−0.264433
$321$	0	0
$322$	5385.12	0.931991
$323$	−7186.19	−1.23793
$324$	0	0
$325$	0	0
$326$	2045.13	0.347452
$327$	0	0
$328$	−5583.58	−0.939945
$329$	7710.78	1.29213
$330$	0	0
$331$	5290.27	0.878488	0.439244	−	0.898368i	$-0.355246\pi$
$331$	5290.27	0.878488	0.439244	+	0.898368i	$0.355246\pi$
$332$	−3591.91	−0.593769
$333$	0	0
$334$	−2906.95	−0.476231
$335$	−7029.89	−1.14652
$336$	0	0
$337$	−3313.51	−0.535603	−0.267802	−	0.963474i	$-0.586297\pi$
$337$	−3313.51	−0.535603	−0.267802	+	0.963474i	$0.586297\pi$
$338$	0	0
$339$	0	0
$340$	−12613.5	−2.01196
$341$	1361.11	0.216154
$342$	0	0
$343$	−9117.79	−1.43532
$344$	3357.16	0.526180
$345$	0	0
$346$	−235.880	−0.0366503
$347$	4218.65	0.652649	0.326324	−	0.945258i	$-0.394190\pi$
$347$	4218.65	0.652649	0.326324	+	0.945258i	$0.394190\pi$
$348$	0	0
$349$	−6552.91	−1.00507	−0.502535	−	0.864557i	$-0.667599\pi$
$349$	−6552.91	−1.00507	−0.502535	+	0.864557i	$0.667599\pi$
$350$	−11677.6	−1.78341
$351$	0	0
$352$	3027.63	0.458447
$353$	−3864.50	−0.582681	−0.291341	−	0.956619i	$-0.594101\pi$
$353$	−3864.50	−0.582681	−0.291341	+	0.956619i	$0.594101\pi$
$354$	0	0
$355$	−10024.2	−1.49868
$356$	4496.20	0.669377
$357$	0	0
$358$	−2034.51	−0.300355
$359$	−588.511	−0.0865192	−0.0432596	−	0.999064i	$-0.513774\pi$
$359$	−588.511	−0.0865192	−0.0432596	+	0.999064i	$0.513774\pi$
$360$	0	0
$361$	−355.122	−0.0517746
$362$	2925.66	0.424777
$363$	0	0
$364$	0	0
$365$	13803.8	1.97952
$366$	0	0
$367$	11082.7	1.57633	0.788167	−	0.615462i	$-0.211030\pi$
$367$	11082.7	1.57633	0.788167	+	0.615462i	$0.211030\pi$
$368$	−5085.86	−0.720432
$369$	0	0
$370$	958.763	0.134713
$371$	18770.9	2.62678
$372$	0	0
$373$	−4106.52	−0.570048	−0.285024	−	0.958520i	$-0.592001\pi$
$373$	−4106.52	−0.570048	−0.285024	+	0.958520i	$0.592001\pi$
$374$	1782.09	0.246390
$375$	0	0
$376$	4164.50	0.571191
$377$	0	0
$378$	0	0
$379$	6169.89	0.836216	0.418108	−	0.908397i	$-0.362693\pi$
$379$	6169.89	0.836216	0.418108	+	0.908397i	$0.362693\pi$
$380$	11415.9	1.54112
$381$	0	0
$382$	46.8530	0.00627541
$383$	9133.11	1.21849	0.609243	−	0.792984i	$-0.291473\pi$
$383$	9133.11	1.21849	0.609243	+	0.792984i	$0.291473\pi$
$384$	0	0
$385$	−11523.1	−1.52538
$386$	−3192.68	−0.420992
$387$	0	0
$388$	3129.90	0.409527
$389$	12369.3	1.61221	0.806104	−	0.591774i	$-0.201572\pi$
$389$	12369.3	1.61221	0.806104	+	0.591774i	$0.201572\pi$
$390$	0	0
$391$	−13337.8	−1.72512
$392$	−10716.6	−1.38079
$393$	0	0
$394$	−1348.25	−0.172395
$395$	5108.34	0.650705
$396$	0	0
$397$	−2520.20	−0.318603	−0.159301	−	0.987230i	$-0.550924\pi$
$397$	−2520.20	−0.318603	−0.159301	+	0.987230i	$0.550924\pi$
$398$	2233.36	0.281277
$399$	0	0
$400$	11028.6	1.37858
$401$	−4400.92	−0.548059	−0.274029	−	0.961721i	$-0.588357\pi$
$401$	−4400.92	−0.548059	−0.274029	+	0.961721i	$0.588357\pi$
$402$	0	0
$403$	0	0
$404$	2041.07	0.251353
$405$	0	0
$406$	−226.712	−0.0277132
$407$	683.035	0.0831862
$408$	0	0
$409$	−6482.35	−0.783696	−0.391848	−	0.920030i	$-0.628164\pi$
$409$	−6482.35	−0.783696	−0.391848	+	0.920030i	$0.628164\pi$
$410$	8066.98	0.971707
$411$	0	0
$412$	10420.2	1.24604
$413$	22184.9	2.64321
$414$	0	0
$415$	11408.1	1.34940
$416$	0	0
$417$	0	0
$418$	−1612.89	−0.188729
$419$	−6293.72	−0.733815	−0.366908	−	0.930257i	$-0.619583\pi$
$419$	−6293.72	−0.733815	−0.366908	+	0.930257i	$0.619583\pi$
$420$	0	0
$421$	8376.89	0.969749	0.484875	−	0.874584i	$-0.338865\pi$
$421$	8376.89	0.969749	0.484875	+	0.874584i	$0.338865\pi$
$422$	1909.97	0.220322
$423$	0	0
$424$	10137.9	1.16118
$425$	28922.9	3.30110
$426$	0	0
$427$	14769.0	1.67382
$428$	4725.31	0.533660
$429$	0	0
$430$	−4850.32	−0.543961
$431$	−3392.05	−0.379094	−0.189547	−	0.981872i	$-0.560702\pi$
$431$	−3392.05	−0.379094	−0.189547	+	0.981872i	$0.560702\pi$
$432$	0	0
$433$	13767.3	1.52798	0.763991	−	0.645227i	$-0.223237\pi$
$433$	13767.3	1.52798	0.763991	+	0.645227i	$0.223237\pi$
$434$	−2817.34	−0.311606
$435$	0	0
$436$	11444.4	1.25708
$437$	12071.4	1.32141
$438$	0	0
$439$	−6355.98	−0.691013	−0.345506	−	0.938416i	$-0.612293\pi$
$439$	−6355.98	−0.691013	−0.345506	+	0.938416i	$0.612293\pi$
$440$	−6223.47	−0.674301
$441$	0	0
$442$	0	0
$443$	20.2187	0.00216844	0.00108422	−	0.999999i	$-0.499655\pi$
$443$	20.2187	0.00216844	0.00108422	+	0.999999i	$0.499655\pi$
$444$	0	0
$445$	−14280.2	−1.52123
$446$	−5315.26	−0.564316
$447$	0	0
$448$	2232.12	0.235397
$449$	−10039.8	−1.05525	−0.527626	−	0.849477i	$-0.676918\pi$
$449$	−10039.8	−1.05525	−0.527626	+	0.849477i	$0.676918\pi$
$450$	0	0
$451$	5747.02	0.600037
$452$	−8435.25	−0.877789
$453$	0	0
$454$	1404.35	0.145175
$455$	0	0
$456$	0	0
$457$	14931.2	1.52834	0.764168	−	0.645017i	$-0.223150\pi$
$457$	14931.2	1.52834	0.764168	+	0.645017i	$0.223150\pi$
$458$	−4500.70	−0.459179
$459$	0	0
$460$	21188.3	2.14763
$461$	952.278	0.0962083	0.0481041	−	0.998842i	$-0.484682\pi$
$461$	952.278	0.0962083	0.0481041	+	0.998842i	$0.484682\pi$
$462$	0	0
$463$	666.672	0.0669176	0.0334588	−	0.999440i	$-0.489348\pi$
$463$	666.672	0.0669176	0.0334588	+	0.999440i	$0.489348\pi$
$464$	214.113	0.0214223
$465$	0	0
$466$	−4873.14	−0.484428
$467$	13525.2	1.34020	0.670098	−	0.742273i	$-0.266252\pi$
$467$	13525.2	1.34020	0.670098	+	0.742273i	$0.266252\pi$
$468$	0	0
$469$	10366.3	1.02062
$470$	−6016.74	−0.590492
$471$	0	0
$472$	11981.8	1.16845
$473$	−3455.43	−0.335900
$474$	0	0
$475$	−26176.8	−2.52857
$476$	18600.0	1.79103
$477$	0	0
$478$	888.749	0.0850427
$479$	−5120.03	−0.488393	−0.244196	−	0.969726i	$-0.578524\pi$
$479$	−5120.03	−0.488393	−0.244196	+	0.969726i	$0.578524\pi$
$480$	0	0
$481$	0	0
$482$	1518.28	0.143476
$483$	0	0
$484$	6868.94	0.645092
$485$	−9940.74	−0.930692
$486$	0	0
$487$	−9776.37	−0.909671	−0.454835	−	0.890575i	$-0.650302\pi$
$487$	−9776.37	−0.909671	−0.454835	+	0.890575i	$0.650302\pi$
$488$	7976.56	0.739922
$489$	0	0
$490$	15483.0	1.42745
$491$	10423.9	0.958097	0.479048	−	0.877788i	$-0.340982\pi$
$491$	10423.9	0.958097	0.479048	+	0.877788i	$0.340982\pi$
$492$	0	0
$493$	561.519	0.0512972
$494$	0	0
$495$	0	0
$496$	2660.78	0.240872
$497$	14781.8	1.33412
$498$	0	0
$499$	10785.0	0.967544	0.483772	−	0.875194i	$-0.339266\pi$
$499$	10785.0	0.967544	0.483772	+	0.875194i	$0.339266\pi$
$500$	−28252.3	−2.52697
$501$	0	0
$502$	−2787.89	−0.247868
$503$	−2135.13	−0.189266	−0.0946330	−	0.995512i	$-0.530168\pi$
$503$	−2135.13	−0.189266	−0.0946330	+	0.995512i	$0.530168\pi$
$504$	0	0
$505$	−6482.54	−0.571226
$506$	−2993.58	−0.263005
$507$	0	0
$508$	−5967.93	−0.521229
$509$	13658.8	1.18942	0.594711	−	0.803939i	$-0.297266\pi$
$509$	13658.8	1.18942	0.594711	+	0.803939i	$0.297266\pi$
$510$	0	0
$511$	−20355.2	−1.76215
$512$	10508.7	0.907081
$513$	0	0
$514$	−4347.41	−0.373066
$515$	−33095.3	−2.83175
$516$	0	0
$517$	−4286.40	−0.364634
$518$	−1413.80	−0.119921
$519$	0	0
$520$	0	0
$521$	14644.5	1.23146	0.615728	−	0.787958i	$-0.288862\pi$
$521$	14644.5	1.23146	0.615728	+	0.787958i	$0.288862\pi$
$522$	0	0
$523$	−810.895	−0.0677973	−0.0338986	−	0.999425i	$-0.510792\pi$
$523$	−810.895	−0.0677973	−0.0338986	+	0.999425i	$0.510792\pi$
$524$	65.9075	0.00549462
$525$	0	0
$526$	8449.65	0.700422
$527$	6977.97	0.576784
$528$	0	0
$529$	10238.0	0.841458
$530$	−14646.9	−1.20042
$531$	0	0
$532$	−16834.0	−1.37189
$533$	0	0
$534$	0	0
$535$	−15007.9	−1.21280
$536$	5598.74	0.451173
$537$	0	0
$538$	−8989.19	−0.720356
$539$	11030.3	0.881462
$540$	0	0
$541$	−23330.4	−1.85407	−0.927036	−	0.374971i	$-0.877653\pi$
$541$	−23330.4	−1.85407	−0.927036	+	0.374971i	$0.877653\pi$
$542$	−6434.49	−0.509935
$543$	0	0
$544$	15521.6	1.22332
$545$	−36348.1	−2.85685
$546$	0	0
$547$	24537.6	1.91801	0.959005	−	0.283390i	$-0.0914589\pi$
$547$	24537.6	1.91801	0.959005	+	0.283390i	$0.0914589\pi$
$548$	−9924.49	−0.773637
$549$	0	0
$550$	6491.54	0.503273
$551$	−508.204	−0.0392926
$552$	0	0
$553$	−7532.81	−0.579254
$554$	4484.27	0.343896
$555$	0	0
$556$	−3539.49	−0.269978
$557$	4219.10	0.320950	0.160475	−	0.987040i	$-0.448697\pi$
$557$	4219.10	0.320950	0.160475	+	0.987040i	$0.448697\pi$
$558$	0	0
$559$	0	0
$560$	−22525.9	−1.69981
$561$	0	0
$562$	−6813.15	−0.511379
$563$	981.849	0.0734991	0.0367496	−	0.999325i	$-0.488300\pi$
$563$	981.849	0.0734991	0.0367496	+	0.999325i	$0.488300\pi$
$564$	0	0
$565$	26790.8	1.99486
$566$	−5680.11	−0.421825
$567$	0	0
$568$	7983.50	0.589754
$569$	−2021.40	−0.148930	−0.0744652	−	0.997224i	$-0.523725\pi$
$569$	−2021.40	−0.148930	−0.0744652	+	0.997224i	$0.523725\pi$
$570$	0	0
$571$	11994.8	0.879099	0.439549	−	0.898218i	$-0.355138\pi$
$571$	11994.8	0.879099	0.439549	+	0.898218i	$0.355138\pi$
$572$	0	0
$573$	0	0
$574$	−11895.6	−0.865008
$575$	−48585.0	−3.52371
$576$	0	0
$577$	−176.997	−0.0127703	−0.00638516	−	0.999980i	$-0.502032\pi$
$577$	−176.997	−0.0127703	−0.00638516	+	0.999980i	$0.502032\pi$
$578$	3483.08	0.250652
$579$	0	0
$580$	−892.024	−0.0638608
$581$	−16822.5	−1.20123
$582$	0	0
$583$	−10434.7	−0.741269
$584$	−10993.6	−0.778970
$585$	0	0
$586$	−6415.95	−0.452288
$587$	10885.6	0.765413	0.382707	−	0.923870i	$-0.374992\pi$
$587$	10885.6	0.765413	0.382707	+	0.923870i	$0.374992\pi$
$588$	0	0
$589$	−6315.43	−0.441804
$590$	−17310.9	−1.20793
$591$	0	0
$592$	1335.23	0.0926989
$593$	6043.37	0.418502	0.209251	−	0.977862i	$-0.432897\pi$
$593$	6043.37	0.418502	0.209251	+	0.977862i	$0.432897\pi$
$594$	0	0
$595$	−59074.8	−4.07030
$596$	11994.5	0.824349
$597$	0	0
$598$	0	0
$599$	−23893.3	−1.62981	−0.814904	−	0.579596i	$-0.803211\pi$
$599$	−23893.3	−1.62981	−0.814904	+	0.579596i	$0.803211\pi$
$600$	0	0
$601$	−25840.1	−1.75381	−0.876905	−	0.480665i	$-0.840395\pi$
$601$	−25840.1	−1.75381	−0.876905	+	0.480665i	$0.840395\pi$
$602$	7152.32	0.484231
$603$	0	0
$604$	12141.6	0.817939
$605$	−21816.1	−1.46604
$606$	0	0
$607$	−395.714	−0.0264605	−0.0132303	−	0.999912i	$-0.504211\pi$
$607$	−395.714	−0.0264605	−0.0132303	+	0.999912i	$0.504211\pi$
$608$	−14047.9	−0.937034
$609$	0	0
$610$	−11524.3	−0.764925
$611$	0	0
$612$	0	0
$613$	20523.7	1.35228	0.676139	−	0.736774i	$-0.263652\pi$
$613$	20523.7	1.35228	0.676139	+	0.736774i	$0.263652\pi$
$614$	11700.1	0.769019
$615$	0	0
$616$	9177.19	0.600259
$617$	−2398.68	−0.156511	−0.0782556	−	0.996933i	$-0.524935\pi$
$617$	−2398.68	−0.156511	−0.0782556	+	0.996933i	$0.524935\pi$
$618$	0	0
$619$	17708.2	1.14984	0.574922	−	0.818208i	$-0.305032\pi$
$619$	17708.2	1.14984	0.574922	+	0.818208i	$0.305032\pi$
$620$	−11085.1	−0.718048
$621$	0	0
$622$	5385.47	0.347167
$623$	21057.7	1.35419
$624$	0	0
$625$	49157.8	3.14610
$626$	3440.78	0.219682
$627$	0	0
$628$	5842.03	0.371214
$629$	3501.69	0.221974
$630$	0	0
$631$	18170.1	1.14634	0.573170	−	0.819437i	$-0.305713\pi$
$631$	18170.1	1.14634	0.573170	+	0.819437i	$0.305713\pi$
$632$	−4068.38	−0.256062
$633$	0	0
$634$	−10450.0	−0.654612
$635$	18954.5	1.18455
$636$	0	0
$637$	0	0
$638$	126.029	0.00782058
$639$	0	0
$640$	−31289.2	−1.93253
$641$	30492.4	1.87891	0.939453	−	0.342678i	$-0.111334\pi$
$641$	30492.4	1.87891	0.939453	+	0.342678i	$0.111334\pi$
$642$	0	0
$643$	22280.9	1.36652	0.683260	−	0.730175i	$-0.260562\pi$
$643$	22280.9	1.36652	0.683260	+	0.730175i	$0.260562\pi$
$644$	−31244.5	−1.91181
$645$	0	0
$646$	−8268.72	−0.503604
$647$	16844.4	1.02353	0.511763	−	0.859127i	$-0.328992\pi$
$647$	16844.4	1.02353	0.511763	+	0.859127i	$0.328992\pi$
$648$	0	0
$649$	−12332.5	−0.745906
$650$	0	0
$651$	0	0
$652$	−11865.9	−0.712735
$653$	5880.96	0.352435	0.176217	−	0.984351i	$-0.443614\pi$
$653$	5880.96	0.352435	0.176217	+	0.984351i	$0.443614\pi$
$654$	0	0
$655$	−209.326	−0.0124871
$656$	11234.6	0.668653
$657$	0	0
$658$	8872.34	0.525653
$659$	−29655.1	−1.75296	−0.876479	−	0.481441i	$-0.840114\pi$
$659$	−29655.1	−1.75296	−0.876479	+	0.481441i	$0.840114\pi$
$660$	0	0
$661$	−18713.4	−1.10116	−0.550581	−	0.834782i	$-0.685594\pi$
$661$	−18713.4	−1.10116	−0.550581	+	0.834782i	$0.685594\pi$
$662$	6087.20	0.357380
$663$	0	0
$664$	−9085.63	−0.531010
$665$	53465.8	3.11777
$666$	0	0
$667$	−943.245	−0.0547565
$668$	16866.1	0.976901
$669$	0	0
$670$	−8088.87	−0.466419
$671$	−8210.05	−0.472348
$672$	0	0
$673$	7932.59	0.454352	0.227176	−	0.973854i	$-0.427051\pi$
$673$	7932.59	0.454352	0.227176	+	0.973854i	$0.427051\pi$
$674$	−3812.66	−0.217890
$675$	0	0
$676$	0	0
$677$	14742.2	0.836910	0.418455	−	0.908238i	$-0.362572\pi$
$677$	14742.2	0.836910	0.418455	+	0.908238i	$0.362572\pi$
$678$	0	0
$679$	14658.7	0.828497
$680$	−31905.6	−1.79930
$681$	0	0
$682$	1566.15	0.0879342
$683$	−3081.46	−0.172634	−0.0863169	−	0.996268i	$-0.527510\pi$
$683$	−3081.46	−0.172634	−0.0863169	+	0.996268i	$0.527510\pi$
$684$	0	0
$685$	31520.8	1.75817
$686$	−10491.3	−0.583906
$687$	0	0
$688$	−6754.85	−0.374312
$689$	0	0
$690$	0	0
$691$	−11563.5	−0.636607	−0.318303	−	0.947989i	$-0.603113\pi$
$691$	−11563.5	−0.636607	−0.318303	+	0.947989i	$0.603113\pi$
$692$	1368.58	0.0751815
$693$	0	0
$694$	4854.15	0.265506
$695$	11241.6	0.613553
$696$	0	0
$697$	29463.0	1.60114
$698$	−7540.04	−0.408875
$699$	0	0
$700$	67753.4	3.65834
$701$	3283.53	0.176914	0.0884572	−	0.996080i	$-0.471806\pi$
$701$	3283.53	0.176914	0.0884572	+	0.996080i	$0.471806\pi$
$702$	0	0
$703$	−3169.21	−0.170027
$704$	−1240.83	−0.0664283
$705$	0	0
$706$	−4446.65	−0.237042
$707$	9559.22	0.508503
$708$	0	0
$709$	−22397.5	−1.18640	−0.593198	−	0.805056i	$-0.702135\pi$
$709$	−22397.5	−1.18640	−0.593198	+	0.805056i	$0.702135\pi$
$710$	−11534.3	−0.609683
$711$	0	0
$712$	11373.0	0.598626
$713$	−11721.7	−0.615680
$714$	0	0
$715$	0	0
$716$	11804.2	0.616123
$717$	0	0
$718$	−677.164	−0.0351971
$719$	21475.6	1.11392	0.556958	−	0.830540i	$-0.311968\pi$
$719$	21475.6	1.11392	0.556958	+	0.830540i	$0.311968\pi$
$720$	0	0
$721$	48802.6	2.52081
$722$	−408.617	−0.0210626
$723$	0	0
$724$	−16974.7	−0.871353
$725$	2045.42	0.104779
$726$	0	0
$727$	−2541.54	−0.129657	−0.0648285	−	0.997896i	$-0.520650\pi$
$727$	−2541.54	−0.129657	−0.0648285	+	0.997896i	$0.520650\pi$
$728$	0	0
$729$	0	0
$730$	15883.2	0.805293
$731$	−17714.8	−0.896314
$732$	0	0
$733$	1921.91	0.0968447	0.0484224	−	0.998827i	$-0.484581\pi$
$733$	1921.91	0.0968447	0.0484224	+	0.998827i	$0.484581\pi$
$734$	12752.3	0.641273
$735$	0	0
$736$	−26073.4	−1.30581
$737$	−5762.62	−0.288017
$738$	0	0
$739$	3501.37	0.174290	0.0871448	−	0.996196i	$-0.472226\pi$
$739$	3501.37	0.174290	0.0871448	+	0.996196i	$0.472226\pi$
$740$	−5562.75	−0.276339
$741$	0	0
$742$	21598.5	1.06861
$743$	−38056.5	−1.87908	−0.939541	−	0.342437i	$-0.888748\pi$
$743$	−38056.5	−1.87908	−0.939541	+	0.342437i	$0.888748\pi$
$744$	0	0
$745$	−38095.1	−1.87342
$746$	−4725.13	−0.231903
$747$	0	0
$748$	−10339.7	−0.505424
$749$	22130.7	1.07963
$750$	0	0
$751$	−9334.53	−0.453557	−0.226779	−	0.973946i	$-0.572819\pi$
$751$	−9334.53	−0.453557	−0.226779	+	0.973946i	$0.572819\pi$
$752$	−8379.28	−0.406331
$753$	0	0
$754$	0	0
$755$	−38562.4	−1.85885
$756$	0	0
$757$	12334.1	0.592193	0.296097	−	0.955158i	$-0.404315\pi$
$757$	12334.1	0.592193	0.296097	+	0.955158i	$0.404315\pi$
$758$	7099.32	0.340183
$759$	0	0
$760$	28876.2	1.37823
$761$	−16234.5	−0.773326	−0.386663	−	0.922221i	$-0.626372\pi$
$761$	−16234.5	−0.773326	−0.386663	+	0.922221i	$0.626372\pi$
$762$	0	0
$763$	53599.2	2.54315
$764$	−271.841	−0.0128729
$765$	0	0
$766$	10508.9	0.495696
$767$	0	0
$768$	0	0
$769$	2020.55	0.0947501	0.0473750	−	0.998877i	$-0.484914\pi$
$769$	2020.55	0.0947501	0.0473750	+	0.998877i	$0.484914\pi$
$770$	−13258.9	−0.620542
$771$	0	0
$772$	18523.9	0.863590
$773$	−31025.1	−1.44359	−0.721795	−	0.692106i	$-0.756683\pi$
$773$	−31025.1	−1.44359	−0.721795	+	0.692106i	$0.756683\pi$
$774$	0	0
$775$	25418.3	1.17813
$776$	7917.00	0.366242
$777$	0	0
$778$	14232.6	0.655867
$779$	−26665.6	−1.22644
$780$	0	0
$781$	−8217.19	−0.376484
$782$	−15347.0	−0.701802
$783$	0	0
$784$	21562.6	0.982260
$785$	−18554.6	−0.843621
$786$	0	0
$787$	−31464.9	−1.42516	−0.712581	−	0.701590i	$-0.752474\pi$
$787$	−31464.9	−1.42516	−0.712581	+	0.701590i	$0.752474\pi$
$788$	7822.54	0.353638
$789$	0	0
$790$	5877.86	0.264715
$791$	−39506.0	−1.77582
$792$	0	0
$793$	0	0
$794$	−2899.85	−0.129612
$795$	0	0
$796$	−12958.0	−0.576989
$797$	15051.4	0.668945	0.334473	−	0.942405i	$-0.391442\pi$
$797$	15051.4	0.668945	0.334473	+	0.942405i	$0.391442\pi$
$798$	0	0
$799$	−21974.9	−0.972987
$800$	56539.8	2.49873
$801$	0	0
$802$	−5063.88	−0.222957
$803$	11315.4	0.497275
$804$	0	0
$805$	99234.4	4.34479
$806$	0	0
$807$	0	0
$808$	5162.82	0.224786
$809$	36468.2	1.58486	0.792432	−	0.609960i	$-0.208815\pi$
$809$	36468.2	1.58486	0.792432	+	0.609960i	$0.208815\pi$
$810$	0	0
$811$	21761.0	0.942210	0.471105	−	0.882077i	$-0.343855\pi$
$811$	21761.0	0.942210	0.471105	+	0.882077i	$0.343855\pi$
$812$	1315.39	0.0568486
$813$	0	0
$814$	785.928	0.0338412
$815$	37686.7	1.61976
$816$	0	0
$817$	16032.8	0.686558
$818$	−7458.86	−0.318818
$819$	0	0
$820$	−46804.7	−1.99328
$821$	−43320.0	−1.84151	−0.920754	−	0.390144i	$-0.872425\pi$
$821$	−43320.0	−1.84151	−0.920754	+	0.390144i	$0.872425\pi$
$822$	0	0
$823$	5819.88	0.246498	0.123249	−	0.992376i	$-0.460669\pi$
$823$	5819.88	0.246498	0.123249	+	0.992376i	$0.460669\pi$
$824$	26357.7	1.11434
$825$	0	0
$826$	25526.8	1.07529
$827$	−9099.01	−0.382592	−0.191296	−	0.981532i	$-0.561269\pi$
$827$	−9099.01	−0.382592	−0.191296	+	0.981532i	$0.561269\pi$
$828$	0	0
$829$	6582.90	0.275794	0.137897	−	0.990447i	$-0.455966\pi$
$829$	6582.90	0.275794	0.137897	+	0.990447i	$0.455966\pi$
$830$	13126.6	0.548954
$831$	0	0
$832$	0	0
$833$	56548.5	2.35209
$834$	0	0
$835$	−53567.8	−2.22011
$836$	9357.98	0.387144
$837$	0	0
$838$	−7241.82	−0.298526
$839$	28867.9	1.18788	0.593939	−	0.804510i	$-0.297572\pi$
$839$	28867.9	1.18788	0.593939	+	0.804510i	$0.297572\pi$
$840$	0	0
$841$	−24349.3	−0.998372
$842$	9638.78	0.394506
$843$	0	0
$844$	−11081.6	−0.451950
$845$	0	0
$846$	0	0
$847$	32170.3	1.30506
$848$	−20398.2	−0.826036
$849$	0	0
$850$	33279.9	1.34293
$851$	−5882.17	−0.236943
$852$	0	0
$853$	17014.4	0.682958	0.341479	−	0.939889i	$-0.389072\pi$
$853$	17014.4	0.682958	0.341479	+	0.939889i	$0.389072\pi$
$854$	16993.8	0.680932
$855$	0	0
$856$	11952.5	0.477254
$857$	6261.29	0.249570	0.124785	−	0.992184i	$-0.460176\pi$
$857$	6261.29	0.249570	0.124785	+	0.992184i	$0.460176\pi$
$858$	0	0
$859$	29513.0	1.17226	0.586129	−	0.810218i	$-0.300651\pi$
$859$	29513.0	1.17226	0.586129	+	0.810218i	$0.300651\pi$
$860$	28141.6	1.11584
$861$	0	0
$862$	−3903.03	−0.154220
$863$	7073.47	0.279008	0.139504	−	0.990222i	$-0.455449\pi$
$863$	7073.47	0.279008	0.139504	+	0.990222i	$0.455449\pi$
$864$	0	0
$865$	−4346.69	−0.170858
$866$	15841.3	0.621603
$867$	0	0
$868$	16346.2	0.639203
$869$	4187.47	0.163464
$870$	0	0
$871$	0	0
$872$	28948.3	1.12421
$873$	0	0
$874$	13889.9	0.537565
$875$	−132318.	−5.11220
$876$	0	0
$877$	−12567.8	−0.483906	−0.241953	−	0.970288i	$-0.577788\pi$
$877$	−12567.8	−0.483906	−0.241953	+	0.970288i	$0.577788\pi$
$878$	−7313.45	−0.281113
$879$	0	0
$880$	12522.1	0.479681
$881$	4722.06	0.180579	0.0902895	−	0.995916i	$-0.471221\pi$
$881$	4722.06	0.180579	0.0902895	+	0.995916i	$0.471221\pi$
$882$	0	0
$883$	20943.3	0.798185	0.399093	−	0.916911i	$-0.369325\pi$
$883$	20943.3	0.798185	0.399093	+	0.916911i	$0.369325\pi$
$884$	0	0
$885$	0	0
$886$	23.2645	0.000882151 0
$887$	36524.2	1.38260	0.691298	−	0.722570i	$-0.257039\pi$
$887$	36524.2	1.38260	0.691298	+	0.722570i	$0.257039\pi$
$888$	0	0
$889$	−27950.5	−1.05448
$890$	−16431.4	−0.618855
$891$	0	0
$892$	30839.2	1.15759
$893$	19888.5	0.745288
$894$	0	0
$895$	−37490.9	−1.40020
$896$	46139.4	1.72032
$897$	0	0
$898$	−11552.2	−0.429290
$899$	493.479	0.0183075
$900$	0	0
$901$	−53495.0	−1.97800
$902$	6612.76	0.244103
$903$	0	0
$904$	−21336.7	−0.785010
$905$	53912.6	1.98024
$906$	0	0
$907$	−19686.4	−0.720701	−0.360351	−	0.932817i	$-0.617343\pi$
$907$	−19686.4	−0.720701	−0.360351	+	0.932817i	$0.617343\pi$
$908$	−8148.04	−0.297800
$909$	0	0
$910$	0	0
$911$	−22339.4	−0.812444	−0.406222	−	0.913774i	$-0.633154\pi$
$911$	−22339.4	−0.812444	−0.406222	+	0.913774i	$0.633154\pi$
$912$	0	0
$913$	9351.58	0.338984
$914$	17180.4	0.621747
$915$	0	0
$916$	26113.1	0.941923
$917$	308.674	0.0111159
$918$	0	0
$919$	34118.9	1.22468	0.612338	−	0.790596i	$-0.290229\pi$
$919$	34118.9	1.22468	0.612338	+	0.790596i	$0.290229\pi$
$920$	53595.4	1.92064
$921$	0	0
$922$	1095.73	0.0391388
$923$	0	0
$924$	0	0
$925$	12755.4	0.453401
$926$	767.100	0.0272230
$927$	0	0
$928$	1097.68	0.0388289
$929$	−19596.3	−0.692071	−0.346035	−	0.938221i	$-0.612472\pi$
$929$	−19596.3	−0.692071	−0.346035	+	0.938221i	$0.612472\pi$
$930$	0	0
$931$	−51179.4	−1.80165
$932$	28274.0	0.993718
$933$	0	0
$934$	15562.6	0.545209
$935$	32839.5	1.14863
$936$	0	0
$937$	52085.3	1.81596	0.907978	−	0.419017i	$-0.137625\pi$
$937$	52085.3	1.81596	0.907978	+	0.419017i	$0.137625\pi$
$938$	11927.9	0.415203
$939$	0	0
$940$	34909.2	1.21129
$941$	1090.90	0.0377921	0.0188960	−	0.999821i	$-0.493985\pi$
$941$	1090.90	0.0377921	0.0188960	+	0.999821i	$0.493985\pi$
$942$	0	0
$943$	−49492.3	−1.70911
$944$	−24108.2	−0.831203
$945$	0	0
$946$	−3975.96	−0.136649
$947$	−25306.1	−0.868361	−0.434181	−	0.900826i	$-0.642962\pi$
$947$	−25306.1	−0.868361	−0.434181	+	0.900826i	$0.642962\pi$
$948$	0	0
$949$	0	0
$950$	−30120.0	−1.02866
$951$	0	0
$952$	47048.3	1.60173
$953$	5321.09	0.180868	0.0904339	−	0.995902i	$-0.471175\pi$
$953$	5321.09	0.180868	0.0904339	+	0.995902i	$0.471175\pi$
$954$	0	0
$955$	863.383	0.0292549
$956$	−5156.53	−0.174450
$957$	0	0
$958$	−5891.31	−0.198684
$959$	−46480.8	−1.56511
$960$	0	0
$961$	−23658.6	−0.794151
$962$	0	0
$963$	0	0
$964$	−8809.06	−0.294316
$965$	−58833.1	−1.96260
$966$	0	0
$967$	−33045.4	−1.09893	−0.549466	−	0.835516i	$-0.685169\pi$
$967$	−33045.4	−1.09893	−0.549466	+	0.835516i	$0.685169\pi$
$968$	17374.8	0.576908
$969$	0	0
$970$	−11438.2	−0.378618
$971$	53559.8	1.77015	0.885076	−	0.465447i	$-0.154106\pi$
$971$	53559.8	1.77015	0.885076	+	0.465447i	$0.154106\pi$
$972$	0	0
$973$	−16577.0	−0.546181
$974$	−11249.1	−0.370066
$975$	0	0
$976$	−16049.4	−0.526362
$977$	−31189.3	−1.02132	−0.510662	−	0.859782i	$-0.670600\pi$
$977$	−31189.3	−1.02132	−0.510662	+	0.859782i	$0.670600\pi$
$978$	0	0
$979$	−11705.9	−0.382148
$980$	−89832.4	−2.92815
$981$	0	0
$982$	11994.2	0.389766
$983$	6235.44	0.202319	0.101160	−	0.994870i	$-0.467745\pi$
$983$	6235.44	0.202319	0.101160	+	0.994870i	$0.467745\pi$
$984$	0	0
$985$	−24844.8	−0.803677
$986$	646.106	0.0208684
$987$	0	0
$988$	0	0
$989$	29757.5	0.956758
$990$	0	0
$991$	24157.5	0.774357	0.387178	−	0.922005i	$-0.373450\pi$
$991$	24157.5	0.774357	0.387178	+	0.922005i	$0.373450\pi$
$992$	13640.8	0.436590
$993$	0	0
$994$	17008.6	0.542736
$995$	41155.3	1.31127
$996$	0	0
$997$	−35679.6	−1.13339	−0.566693	−	0.823929i	$-0.691777\pi$
$997$	−35679.6	−1.13339	−0.566693	+	0.823929i	$0.691777\pi$
$998$	12409.7	0.393609
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.4.a.bm.1.11	yes	18
3.2	odd	2	inner	1521.4.a.bm.1.8	✓	18
13.12	even	2		1521.4.a.bn.1.8	yes	18
39.38	odd	2		1521.4.a.bn.1.11	yes	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1521.4.a.bm.1.8	✓	18	3.2	odd	2	inner
1521.4.a.bm.1.11	yes	18	1.1	even	1	trivial
1521.4.a.bn.1.8	yes	18	13.12	even	2
1521.4.a.bn.1.11	yes	18	39.38	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1521 = 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1521.a (trivial)

\(f(q)\)	\(=\)	\(q+1.15064 q^{2} -6.67603 q^{4} +21.2034 q^{5} -31.2668 q^{7} -16.8868 q^{8} +24.3975 q^{10} +17.3811 q^{11} -35.9768 q^{14} +33.9775 q^{16} +89.1071 q^{17} -80.6466 q^{19} -141.555 q^{20} +19.9994 q^{22} -149.683 q^{23} +324.586 q^{25} +208.738 q^{28} +6.30162 q^{29} +78.3099 q^{31} +174.191 q^{32} +102.530 q^{34} -662.964 q^{35} +39.2975 q^{37} -92.7953 q^{38} -358.059 q^{40} +330.647 q^{41} -198.803 q^{43} -116.037 q^{44} -172.231 q^{46} -246.612 q^{47} +634.613 q^{49} +373.482 q^{50} -600.345 q^{53} +368.540 q^{55} +527.997 q^{56} +7.25090 q^{58} -709.534 q^{59} -472.354 q^{61} +90.1066 q^{62} -71.3895 q^{64} -331.545 q^{67} -594.881 q^{68} -762.833 q^{70} -472.765 q^{71} +651.016 q^{73} +45.2173 q^{74} +538.399 q^{76} -543.452 q^{77} +240.920 q^{79} +720.441 q^{80} +380.456 q^{82} +538.031 q^{83} +1889.38 q^{85} -228.751 q^{86} -293.512 q^{88} -673.484 q^{89} +999.288 q^{92} -283.762 q^{94} -1709.99 q^{95} -468.827 q^{97} +730.211 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 56 q^{4} - 94 q^{7} + 156 q^{10} + 88 q^{16} - 448 q^{19} - 256 q^{22} - 16 q^{25} - 1300 q^{28} - 818 q^{31} - 900 q^{34} - 524 q^{37} + 800 q^{40} + 752 q^{43} - 1650 q^{46} + 2948 q^{49} - 464 q^{55}+ \cdots - 4630 q^{97}+O(q^{100}) \) 18 * q + 56 * q^4 - 94 * q^7 + 156 * q^10 + 88 * q^16 - 448 * q^19 - 256 * q^22 - 16 * q^25 - 1300 * q^28 - 818 * q^31 - 900 * q^34 - 524 * q^37 + 800 * q^40 + 752 * q^43 - 1650 * q^46 + 2948 * q^49 - 464 * q^55 - 1626 * q^58 - 1784 * q^61 - 4274 * q^64 - 2872 * q^67 - 5772 * q^70 - 3078 * q^73 - 5726 * q^76 + 3326 * q^79 - 1038 * q^82 - 2340 * q^85 + 780 * q^88 + 1220 * q^94 - 4630 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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