LMFDB - Embedded newform 1575.4.a.bt.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1575, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1575.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$92.9280082590$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 67x^{8} + 1523x^{6} - 13569x^{4} + 36944x^{2} - 256 $ x^10 - 67x^8 + 1523x^6 - 13569x^4 + 36944x^2 - 256
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{12}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 315)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.0833494$ of defining polynomial
Character	$\chi$	$=$	1575.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.0833494 q^{2} -7.99305 q^{4} -7.00000 q^{7} +1.33301 q^{8} -48.7488 q^{11} -37.1435 q^{13} +0.583446 q^{14} +63.8333 q^{16} +94.0651 q^{17} +83.6312 q^{19} +4.06318 q^{22} +35.0715 q^{23} +3.09589 q^{26} +55.9514 q^{28} +86.8210 q^{29} -105.937 q^{31} -15.9846 q^{32} -7.84026 q^{34} +340.316 q^{37} -6.97061 q^{38} -194.735 q^{41} +201.903 q^{43} +389.652 q^{44} -2.92319 q^{46} +220.564 q^{47} +49.0000 q^{49} +296.890 q^{52} -593.630 q^{53} -9.33108 q^{56} -7.23648 q^{58} +172.639 q^{59} +271.931 q^{61} +8.82978 q^{62} -509.334 q^{64} -260.343 q^{67} -751.867 q^{68} -845.726 q^{71} -882.621 q^{73} -28.3651 q^{74} -668.469 q^{76} +341.242 q^{77} +641.065 q^{79} +16.2310 q^{82} +566.613 q^{83} -16.8285 q^{86} -64.9827 q^{88} -544.829 q^{89} +260.004 q^{91} -280.328 q^{92} -18.3839 q^{94} +148.663 q^{97} -4.08412 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 54 q^{4} - 70 q^{7} - 104 q^{13} + 310 q^{16} + 36 q^{19} - 644 q^{22} - 378 q^{28} - 24 q^{31} + 116 q^{34} - 732 q^{37} - 212 q^{43} - 184 q^{46} + 490 q^{49} - 2692 q^{52} - 3196 q^{58} + 1024 q^{61}+ \cdots - 4024 q^{97}+O(q^{100}) $ 10 * q + 54 * q^4 - 70 * q^7 - 104 * q^13 + 310 * q^16 + 36 * q^19 - 644 * q^22 - 378 * q^28 - 24 * q^31 + 116 * q^34 - 732 * q^37 - 212 * q^43 - 184 * q^46 + 490 * q^49 - 2692 * q^52 - 3196 * q^58 + 1024 * q^61 + 1590 * q^64 - 2388 * q^67 - 3432 * q^73 - 4184 * q^76 - 776 * q^79 - 1860 * q^82 - 10164 * q^88 + 728 * q^91 - 4028 * q^94 - 4024 * 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 $

$2$	−0.0833494	−0.0294685	−0.0147342	−	0.999891i	$-0.504690\pi$
$2$	−0.0833494	−0.0294685	−0.0147342	+	0.999891i	$0.504690\pi$
$3$	0	0
$3$	0	0
$4$	−7.99305	−0.999132
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	1.33301	0.0589113
$9$	0	0
$10$	0	0
$11$	−48.7488	−1.33621	−0.668106	−	0.744066i	$-0.732895\pi$
$11$	−48.7488	−1.33621	−0.668106	+	0.744066i	$0.732895\pi$
$12$	0	0
$13$	−37.1435	−0.792442	−0.396221	−	0.918155i	$-0.629679\pi$
$13$	−37.1435	−0.792442	−0.396221	+	0.918155i	$0.629679\pi$
$14$	0.583446	0.0111380
$15$	0	0
$16$	63.8333	0.997396
$17$	94.0651	1.34201	0.671004	−	0.741454i	$-0.265863\pi$
$17$	94.0651	1.34201	0.671004	+	0.741454i	$0.265863\pi$
$18$	0	0
$19$	83.6312	1.00981	0.504903	−	0.863176i	$-0.331528\pi$
$19$	83.6312	1.00981	0.504903	+	0.863176i	$0.331528\pi$
$20$	0	0
$21$	0	0
$22$	4.06318	0.0393761
$23$	35.0715	0.317953	0.158976	−	0.987282i	$-0.449181\pi$
$23$	35.0715	0.317953	0.158976	+	0.987282i	$0.449181\pi$
$24$	0	0
$25$	0	0
$26$	3.09589	0.0233521
$27$	0	0
$28$	55.9514	0.377636
$29$	86.8210	0.555940	0.277970	−	0.960590i	$-0.410338\pi$
$29$	86.8210	0.555940	0.277970	+	0.960590i	$0.410338\pi$
$30$	0	0
$31$	−105.937	−0.613769	−0.306884	−	0.951747i	$-0.599287\pi$
$31$	−105.937	−0.613769	−0.306884	+	0.951747i	$0.599287\pi$
$32$	−15.9846	−0.0883030
$33$	0	0
$34$	−7.84026	−0.0395469
$35$	0	0
$36$	0	0
$37$	340.316	1.51210	0.756049	−	0.654515i	$-0.227127\pi$
$37$	340.316	1.51210	0.756049	+	0.654515i	$0.227127\pi$
$38$	−6.97061	−0.0297574
$39$	0	0
$40$	0	0
$41$	−194.735	−0.741767	−0.370883	−	0.928679i	$-0.620945\pi$
$41$	−194.735	−0.741767	−0.370883	+	0.928679i	$0.620945\pi$
$42$	0	0
$43$	201.903	0.716044	0.358022	−	0.933713i	$-0.383451\pi$
$43$	201.903	0.716044	0.358022	+	0.933713i	$0.383451\pi$
$44$	389.652	1.33505
$45$	0	0
$46$	−2.92319	−0.00936958
$47$	220.564	0.684522	0.342261	−	0.939605i	$-0.388807\pi$
$47$	220.564	0.684522	0.342261	+	0.939605i	$0.388807\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	0	0
$52$	296.890	0.791754
$53$	−593.630	−1.53852	−0.769258	−	0.638939i	$-0.779374\pi$
$53$	−593.630	−1.53852	−0.769258	+	0.638939i	$0.779374\pi$
$54$	0	0
$55$	0	0
$56$	−9.33108	−0.0222664
$57$	0	0
$58$	−7.23648	−0.0163827
$59$	172.639	0.380944	0.190472	−	0.981693i	$-0.438998\pi$
$59$	172.639	0.380944	0.190472	+	0.981693i	$0.438998\pi$
$60$	0	0
$61$	271.931	0.570773	0.285387	−	0.958412i	$-0.407878\pi$
$61$	271.931	0.570773	0.285387	+	0.958412i	$0.407878\pi$
$62$	8.82978	0.0180868
$63$	0	0
$64$	−509.334	−0.994793
$65$	0	0
$66$	0	0
$67$	−260.343	−0.474716	−0.237358	−	0.971422i	$-0.576281\pi$
$67$	−260.343	−0.474716	−0.237358	+	0.971422i	$0.576281\pi$
$68$	−751.867	−1.34084
$69$	0	0
$70$	0	0
$71$	−845.726	−1.41365	−0.706825	−	0.707388i	$-0.749873\pi$
$71$	−845.726	−1.41365	−0.706825	+	0.707388i	$0.749873\pi$
$72$	0	0
$73$	−882.621	−1.41511	−0.707555	−	0.706659i	$-0.750202\pi$
$73$	−882.621	−1.41511	−0.707555	+	0.706659i	$0.750202\pi$
$74$	−28.3651	−0.0445592
$75$	0	0
$76$	−668.469	−1.00893
$77$	341.242	0.505040
$78$	0	0
$79$	641.065	0.912980	0.456490	−	0.889729i	$-0.349106\pi$
$79$	641.065	0.912980	0.456490	+	0.889729i	$0.349106\pi$
$80$	0	0
$81$	0	0
$82$	16.2310	0.0218587
$83$	566.613	0.749324	0.374662	−	0.927162i	$-0.377759\pi$
$83$	566.613	0.749324	0.374662	+	0.927162i	$0.377759\pi$
$84$	0	0
$85$	0	0
$86$	−16.8285	−0.0211007
$87$	0	0
$88$	−64.9827	−0.0787180
$89$	−544.829	−0.648896	−0.324448	−	0.945903i	$-0.605179\pi$
$89$	−544.829	−0.648896	−0.324448	+	0.945903i	$0.605179\pi$
$90$	0	0
$91$	260.004	0.299515
$92$	−280.328	−0.317677
$93$	0	0
$94$	−18.3839	−0.0201718
$95$	0	0
$96$	0	0
$97$	148.663	0.155613	0.0778066	−	0.996968i	$-0.475208\pi$
$97$	148.663	0.155613	0.0778066	+	0.996968i	$0.475208\pi$
$98$	−4.08412	−0.00420978
$99$	0	0
$100$	0	0
$101$	1846.72	1.81936	0.909682	−	0.415306i	$-0.136325\pi$
$101$	1846.72	1.81936	0.909682	+	0.415306i	$0.136325\pi$
$102$	0	0
$103$	−971.408	−0.929278	−0.464639	−	0.885500i	$-0.653816\pi$
$103$	−971.408	−0.929278	−0.464639	+	0.885500i	$0.653816\pi$
$104$	−49.5127	−0.0466838
$105$	0	0
$106$	49.4787	0.0453377
$107$	−764.765	−0.690959	−0.345479	−	0.938426i	$-0.612284\pi$
$107$	−764.765	−0.690959	−0.345479	+	0.938426i	$0.612284\pi$
$108$	0	0
$109$	−969.755	−0.852163	−0.426081	−	0.904685i	$-0.640106\pi$
$109$	−969.755	−0.852163	−0.426081	+	0.904685i	$0.640106\pi$
$110$	0	0
$111$	0	0
$112$	−446.833	−0.376980
$113$	2076.09	1.72834	0.864170	−	0.503201i	$-0.167844\pi$
$113$	2076.09	1.72834	0.864170	+	0.503201i	$0.167844\pi$
$114$	0	0
$115$	0	0
$116$	−693.965	−0.555457
$117$	0	0
$118$	−14.3894	−0.0112258
$119$	−658.455	−0.507231
$120$	0	0
$121$	1045.45	0.785460
$122$	−22.6653	−0.0168198
$123$	0	0
$124$	846.760	0.613236
$125$	0	0
$126$	0	0
$127$	−360.217	−0.251686	−0.125843	−	0.992050i	$-0.540164\pi$
$127$	−360.217	−0.251686	−0.125843	+	0.992050i	$0.540164\pi$
$128$	170.329	0.117618
$129$	0	0
$130$	0	0
$131$	2341.95	1.56196	0.780980	−	0.624556i	$-0.214720\pi$
$131$	2341.95	1.56196	0.780980	+	0.624556i	$0.214720\pi$
$132$	0	0
$133$	−585.418	−0.381671
$134$	21.6994	0.0139891
$135$	0	0
$136$	125.390	0.0790594
$137$	−939.721	−0.586028	−0.293014	−	0.956108i	$-0.594658\pi$
$137$	−939.721	−0.586028	−0.293014	+	0.956108i	$0.594658\pi$
$138$	0	0
$139$	−518.600	−0.316454	−0.158227	−	0.987403i	$-0.550578\pi$
$139$	−518.600	−0.316454	−0.158227	+	0.987403i	$0.550578\pi$
$140$	0	0
$141$	0	0
$142$	70.4907	0.0416581
$143$	1810.70	1.05887
$144$	0	0
$145$	0	0
$146$	73.5659	0.0417011
$147$	0	0
$148$	−2720.17	−1.51079
$149$	−1174.47	−0.645749	−0.322875	−	0.946442i	$-0.604649\pi$
$149$	−1174.47	−0.645749	−0.322875	+	0.946442i	$0.604649\pi$
$150$	0	0
$151$	−1609.96	−0.867662	−0.433831	−	0.900994i	$-0.642838\pi$
$151$	−1609.96	−0.867662	−0.433831	+	0.900994i	$0.642838\pi$
$152$	111.481	0.0594890
$153$	0	0
$154$	−28.4423	−0.0148828
$155$	0	0
$156$	0	0
$157$	−3492.45	−1.77534	−0.887668	−	0.460484i	$-0.847676\pi$
$157$	−3492.45	−1.77534	−0.887668	+	0.460484i	$0.847676\pi$
$158$	−53.4324	−0.0269041
$159$	0	0
$160$	0	0
$161$	−245.500	−0.120175
$162$	0	0
$163$	−2329.08	−1.11919	−0.559594	−	0.828767i	$-0.689043\pi$
$163$	−2329.08	−1.11919	−0.559594	+	0.828767i	$0.689043\pi$
$164$	1556.52	0.741123
$165$	0	0
$166$	−47.2268	−0.0220814
$167$	1116.49	0.517343	0.258672	−	0.965965i	$-0.416715\pi$
$167$	1116.49	0.517343	0.258672	+	0.965965i	$0.416715\pi$
$168$	0	0
$169$	−817.361	−0.372035
$170$	0	0
$171$	0	0
$172$	−1613.82	−0.715423
$173$	−974.826	−0.428409	−0.214204	−	0.976789i	$-0.568716\pi$
$173$	−974.826	−0.428409	−0.214204	+	0.976789i	$0.568716\pi$
$174$	0	0
$175$	0	0
$176$	−3111.80	−1.33273
$177$	0	0
$178$	45.4112	0.0191220
$179$	−1418.59	−0.592350	−0.296175	−	0.955134i	$-0.595711\pi$
$179$	−1418.59	−0.592350	−0.296175	+	0.955134i	$0.595711\pi$
$180$	0	0
$181$	−153.859	−0.0631838	−0.0315919	−	0.999501i	$-0.510058\pi$
$181$	−153.859	−0.0631838	−0.0315919	+	0.999501i	$0.510058\pi$
$182$	−21.6712	−0.00882625
$183$	0	0
$184$	46.7507	0.0187310
$185$	0	0
$186$	0	0
$187$	−4585.56	−1.79321
$188$	−1762.98	−0.683928
$189$	0	0
$190$	0	0
$191$	4372.37	1.65641	0.828203	−	0.560428i	$-0.189363\pi$
$191$	4372.37	1.65641	0.828203	+	0.560428i	$0.189363\pi$
$192$	0	0
$193$	−4706.42	−1.75531	−0.877657	−	0.479288i	$-0.840895\pi$
$193$	−4706.42	−1.75531	−0.877657	+	0.479288i	$0.840895\pi$
$194$	−12.3910	−0.00458568
$195$	0	0
$196$	−391.660	−0.142733
$197$	−5150.73	−1.86281	−0.931407	−	0.363979i	$-0.881418\pi$
$197$	−5150.73	−1.86281	−0.931407	+	0.363979i	$0.881418\pi$
$198$	0	0
$199$	4677.70	1.66630	0.833148	−	0.553049i	$-0.186536\pi$
$199$	4677.70	1.66630	0.833148	+	0.553049i	$0.186536\pi$
$200$	0	0
$201$	0	0
$202$	−153.923	−0.0536138
$203$	−607.747	−0.210126
$204$	0	0
$205$	0	0
$206$	80.9662	0.0273844
$207$	0	0
$208$	−2370.99	−0.790379
$209$	−4076.92	−1.34931
$210$	0	0
$211$	−834.373	−0.272230	−0.136115	−	0.990693i	$-0.543462\pi$
$211$	−834.373	−0.272230	−0.136115	+	0.990693i	$0.543462\pi$
$212$	4744.91	1.53718
$213$	0	0
$214$	63.7426	0.0203615
$215$	0	0
$216$	0	0
$217$	741.559	0.231983
$218$	80.8285	0.0251119
$219$	0	0
$220$	0	0
$221$	−3493.90	−1.06346
$222$	0	0
$223$	−5912.79	−1.77556	−0.887780	−	0.460267i	$-0.847754\pi$
$223$	−5912.79	−1.77556	−0.887780	+	0.460267i	$0.847754\pi$
$224$	111.892	0.0333754
$225$	0	0
$226$	−173.041	−0.0509315
$227$	−3552.37	−1.03867	−0.519337	−	0.854569i	$-0.673821\pi$
$227$	−3552.37	−1.03867	−0.519337	+	0.854569i	$0.673821\pi$
$228$	0	0
$229$	−1971.38	−0.568875	−0.284438	−	0.958695i	$-0.591807\pi$
$229$	−1971.38	−0.568875	−0.284438	+	0.958695i	$0.591807\pi$
$230$	0	0
$231$	0	0
$232$	115.733	0.0327512
$233$	−4930.95	−1.38643	−0.693213	−	0.720732i	$-0.743806\pi$
$233$	−4930.95	−1.38643	−0.693213	+	0.720732i	$0.743806\pi$
$234$	0	0
$235$	0	0
$236$	−1379.91	−0.380613
$237$	0	0
$238$	54.8819	0.0149473
$239$	5978.13	1.61796	0.808981	−	0.587835i	$-0.200020\pi$
$239$	5978.13	1.61796	0.808981	+	0.587835i	$0.200020\pi$
$240$	0	0
$241$	4920.50	1.31517	0.657587	−	0.753378i	$-0.271577\pi$
$241$	4920.50	1.31517	0.657587	+	0.753378i	$0.271577\pi$
$242$	−87.1374	−0.0231463
$243$	0	0
$244$	−2173.56	−0.570277
$245$	0	0
$246$	0	0
$247$	−3106.35	−0.800213
$248$	−141.215	−0.0361579
$249$	0	0
$250$	0	0
$251$	2295.35	0.577215	0.288608	−	0.957447i	$-0.406808\pi$
$251$	2295.35	0.577215	0.288608	+	0.957447i	$0.406808\pi$
$252$	0	0
$253$	−1709.69	−0.424852
$254$	30.0239	0.00741679
$255$	0	0
$256$	4060.48	0.991327
$257$	4290.88	1.04147	0.520734	−	0.853719i	$-0.325658\pi$
$257$	4290.88	1.04147	0.520734	+	0.853719i	$0.325658\pi$
$258$	0	0
$259$	−2382.21	−0.571520
$260$	0	0
$261$	0	0
$262$	−195.200	−0.0460285
$263$	7577.97	1.77672	0.888361	−	0.459147i	$-0.151845\pi$
$263$	7577.97	1.77672	0.888361	+	0.459147i	$0.151845\pi$
$264$	0	0
$265$	0	0
$266$	48.7943	0.0112472
$267$	0	0
$268$	2080.94	0.474304
$269$	4253.32	0.964050	0.482025	−	0.876157i	$-0.339901\pi$
$269$	4253.32	0.964050	0.482025	+	0.876157i	$0.339901\pi$
$270$	0	0
$271$	−3848.42	−0.862638	−0.431319	−	0.902200i	$-0.641952\pi$
$271$	−3848.42	−0.862638	−0.431319	+	0.902200i	$0.641952\pi$
$272$	6004.48	1.33851
$273$	0	0
$274$	78.3252	0.0172693
$275$	0	0
$276$	0	0
$277$	1742.62	0.377992	0.188996	−	0.981978i	$-0.439477\pi$
$277$	1742.62	0.377992	0.188996	+	0.981978i	$0.439477\pi$
$278$	43.2250	0.00932541
$279$	0	0
$280$	0	0
$281$	−5091.71	−1.08095	−0.540474	−	0.841361i	$-0.681755\pi$
$281$	−5091.71	−1.08095	−0.540474	+	0.841361i	$0.681755\pi$
$282$	0	0
$283$	−5256.26	−1.10407	−0.552036	−	0.833821i	$-0.686149\pi$
$283$	−5256.26	−1.10407	−0.552036	+	0.833821i	$0.686149\pi$
$284$	6759.93	1.41242
$285$	0	0
$286$	−150.921	−0.0312033
$287$	1363.14	0.280361
$288$	0	0
$289$	3935.24	0.800984
$290$	0	0
$291$	0	0
$292$	7054.84	1.41388
$293$	−9361.20	−1.86651	−0.933254	−	0.359217i	$-0.883044\pi$
$293$	−9361.20	−1.86651	−0.933254	+	0.359217i	$0.883044\pi$
$294$	0	0
$295$	0	0
$296$	453.645	0.0890797
$297$	0	0
$298$	97.8917	0.0190292
$299$	−1302.68	−0.251959
$300$	0	0
$301$	−1413.32	−0.270639
$302$	134.189	0.0255687
$303$	0	0
$304$	5338.46	1.00718
$305$	0	0
$306$	0	0
$307$	−5465.91	−1.01614	−0.508072	−	0.861315i	$-0.669642\pi$
$307$	−5465.91	−1.01614	−0.508072	+	0.861315i	$0.669642\pi$
$308$	−2727.56	−0.504602
$309$	0	0
$310$	0	0
$311$	−9924.88	−1.80961	−0.904804	−	0.425827i	$-0.859983\pi$
$311$	−9924.88	−1.80961	−0.904804	+	0.425827i	$0.859983\pi$
$312$	0	0
$313$	−770.801	−0.139196	−0.0695978	−	0.997575i	$-0.522172\pi$
$313$	−770.801	−0.139196	−0.0695978	+	0.997575i	$0.522172\pi$
$314$	291.093	0.0523164
$315$	0	0
$316$	−5124.06	−0.912187
$317$	1948.88	0.345301	0.172650	−	0.984983i	$-0.444767\pi$
$317$	1948.88	0.345301	0.172650	+	0.984983i	$0.444767\pi$
$318$	0	0
$319$	−4232.42	−0.742853
$320$	0	0
$321$	0	0
$322$	20.4623	0.00354137
$323$	7866.77	1.35517
$324$	0	0
$325$	0	0
$326$	194.127	0.0329807
$327$	0	0
$328$	−259.583	−0.0436985
$329$	−1543.95	−0.258725
$330$	0	0
$331$	−715.822	−0.118867	−0.0594337	−	0.998232i	$-0.518929\pi$
$331$	−715.822	−0.118867	−0.0594337	+	0.998232i	$0.518929\pi$
$332$	−4528.97	−0.748673
$333$	0	0
$334$	−93.0585	−0.0152453
$335$	0	0
$336$	0	0
$337$	−3030.12	−0.489796	−0.244898	−	0.969549i	$-0.578754\pi$
$337$	−3030.12	−0.489796	−0.244898	+	0.969549i	$0.578754\pi$
$338$	68.1265	0.0109633
$339$	0	0
$340$	0	0
$341$	5164.30	0.820125
$342$	0	0
$343$	−343.000	−0.0539949
$344$	269.139	0.0421831
$345$	0	0
$346$	81.2512	0.0126245
$347$	4035.15	0.624260	0.312130	−	0.950039i	$-0.398958\pi$
$347$	4035.15	0.624260	0.312130	+	0.950039i	$0.398958\pi$
$348$	0	0
$349$	−3311.37	−0.507890	−0.253945	−	0.967219i	$-0.581728\pi$
$349$	−3311.37	−0.507890	−0.253945	+	0.967219i	$0.581728\pi$
$350$	0	0
$351$	0	0
$352$	779.228	0.117991
$353$	3344.01	0.504204	0.252102	−	0.967701i	$-0.418878\pi$
$353$	3344.01	0.504204	0.252102	+	0.967701i	$0.418878\pi$
$354$	0	0
$355$	0	0
$356$	4354.85	0.648333
$357$	0	0
$358$	118.239	0.0174557
$359$	−8210.80	−1.20710	−0.603551	−	0.797324i	$-0.706248\pi$
$359$	−8210.80	−1.20710	−0.603551	+	0.797324i	$0.706248\pi$
$360$	0	0
$361$	135.176	0.0197078
$362$	12.8241	0.00186193
$363$	0	0
$364$	−2078.23	−0.299255
$365$	0	0
$366$	0	0
$367$	10698.8	1.52173	0.760866	−	0.648909i	$-0.224775\pi$
$367$	10698.8	1.52173	0.760866	+	0.648909i	$0.224775\pi$
$368$	2238.73	0.317125
$369$	0	0
$370$	0	0
$371$	4155.41	0.581504
$372$	0	0
$373$	−12433.1	−1.72590	−0.862950	−	0.505290i	$-0.831386\pi$
$373$	−12433.1	−1.72590	−0.862950	+	0.505290i	$0.831386\pi$
$374$	382.204	0.0528430
$375$	0	0
$376$	294.014	0.0403261
$377$	−3224.84	−0.440550
$378$	0	0
$379$	−12485.5	−1.69218	−0.846088	−	0.533044i	$-0.821048\pi$
$379$	−12485.5	−1.69218	−0.846088	+	0.533044i	$0.821048\pi$
$380$	0	0
$381$	0	0
$382$	−364.434	−0.0488117
$383$	−7645.59	−1.02003	−0.510015	−	0.860166i	$-0.670360\pi$
$383$	−7645.59	−1.02003	−0.510015	+	0.860166i	$0.670360\pi$
$384$	0	0
$385$	0	0
$386$	392.277	0.0517264
$387$	0	0
$388$	−1188.27	−0.155478
$389$	−2070.45	−0.269861	−0.134931	−	0.990855i	$-0.543081\pi$
$389$	−2070.45	−0.269861	−0.134931	+	0.990855i	$0.543081\pi$
$390$	0	0
$391$	3299.00	0.426695
$392$	65.3175	0.00841590
$393$	0	0
$394$	429.310	0.0548943
$395$	0	0
$396$	0	0
$397$	−776.739	−0.0981950	−0.0490975	−	0.998794i	$-0.515635\pi$
$397$	−776.739	−0.0981950	−0.0490975	+	0.998794i	$0.515635\pi$
$398$	−389.883	−0.0491032
$399$	0	0
$400$	0	0
$401$	−6475.92	−0.806464	−0.403232	−	0.915098i	$-0.632113\pi$
$401$	−6475.92	−0.806464	−0.403232	+	0.915098i	$0.632113\pi$
$402$	0	0
$403$	3934.87	0.486376
$404$	−14760.9	−1.81778
$405$	0	0
$406$	50.6553	0.00619208
$407$	−16590.0	−2.02048
$408$	0	0
$409$	15812.6	1.91169	0.955844	−	0.293874i	$-0.0949445\pi$
$409$	15812.6	1.91169	0.955844	+	0.293874i	$0.0949445\pi$
$410$	0	0
$411$	0	0
$412$	7764.51	0.928471
$413$	−1208.47	−0.143983
$414$	0	0
$415$	0	0
$416$	593.722	0.0699751
$417$	0	0
$418$	339.809	0.0397622
$419$	2235.64	0.260664	0.130332	−	0.991470i	$-0.458396\pi$
$419$	2235.64	0.260664	0.130332	+	0.991470i	$0.458396\pi$
$420$	0	0
$421$	−5447.83	−0.630668	−0.315334	−	0.948981i	$-0.602116\pi$
$421$	−5447.83	−0.630668	−0.315334	+	0.948981i	$0.602116\pi$
$422$	69.5444	0.00802220
$423$	0	0
$424$	−791.315	−0.0906360
$425$	0	0
$426$	0	0
$427$	−1903.51	−0.215732
$428$	6112.80	0.690359
$429$	0	0
$430$	0	0
$431$	11751.7	1.31336	0.656681	−	0.754169i	$-0.271960\pi$
$431$	11751.7	1.31336	0.656681	+	0.754169i	$0.271960\pi$
$432$	0	0
$433$	8608.11	0.955380	0.477690	−	0.878528i	$-0.341474\pi$
$433$	8608.11	0.955380	0.477690	+	0.878528i	$0.341474\pi$
$434$	−61.8084	−0.00683618
$435$	0	0
$436$	7751.31	0.851423
$437$	2933.07	0.321071
$438$	0	0
$439$	6662.51	0.724338	0.362169	−	0.932113i	$-0.382036\pi$
$439$	6662.51	0.724338	0.362169	+	0.932113i	$0.382036\pi$
$440$	0	0
$441$	0	0
$442$	291.215	0.0313386
$443$	2433.37	0.260977	0.130489	−	0.991450i	$-0.458345\pi$
$443$	2433.37	0.260977	0.130489	+	0.991450i	$0.458345\pi$
$444$	0	0
$445$	0	0
$446$	492.828	0.0523230
$447$	0	0
$448$	3565.34	0.375997
$449$	13782.3	1.44862	0.724308	−	0.689477i	$-0.242160\pi$
$449$	13782.3	1.44862	0.724308	+	0.689477i	$0.242160\pi$
$450$	0	0
$451$	9493.08	0.991157
$452$	−16594.3	−1.72684
$453$	0	0
$454$	296.088	0.0306081
$455$	0	0
$456$	0	0
$457$	−4736.27	−0.484800	−0.242400	−	0.970176i	$-0.577935\pi$
$457$	−4736.27	−0.484800	−0.242400	+	0.970176i	$0.577935\pi$
$458$	164.313	0.0167639
$459$	0	0
$460$	0	0
$461$	6679.16	0.674793	0.337396	−	0.941363i	$-0.390454\pi$
$461$	6679.16	0.674793	0.337396	+	0.941363i	$0.390454\pi$
$462$	0	0
$463$	1674.24	0.168053	0.0840266	−	0.996464i	$-0.473222\pi$
$463$	1674.24	0.168053	0.0840266	+	0.996464i	$0.473222\pi$
$464$	5542.07	0.554492
$465$	0	0
$466$	410.992	0.0408559
$467$	15602.4	1.54602	0.773010	−	0.634393i	$-0.218750\pi$
$467$	15602.4	1.54602	0.773010	+	0.634393i	$0.218750\pi$
$468$	0	0
$469$	1822.40	0.179426
$470$	0	0
$471$	0	0
$472$	230.130	0.0224419
$473$	−9842.53	−0.956786
$474$	0	0
$475$	0	0
$476$	5263.07	0.506791
$477$	0	0
$478$	−498.273	−0.0476788
$479$	−13133.9	−1.25282	−0.626412	−	0.779492i	$-0.715477\pi$
$479$	−13133.9	−1.25282	−0.626412	+	0.779492i	$0.715477\pi$
$480$	0	0
$481$	−12640.5	−1.19825
$482$	−410.120	−0.0387562
$483$	0	0
$484$	−8356.32	−0.784778
$485$	0	0
$486$	0	0
$487$	−5732.67	−0.533413	−0.266706	−	0.963778i	$-0.585935\pi$
$487$	−5732.67	−0.533413	−0.266706	+	0.963778i	$0.585935\pi$
$488$	362.487	0.0336250
$489$	0	0
$490$	0	0
$491$	749.477	0.0688868	0.0344434	−	0.999407i	$-0.489034\pi$
$491$	749.477	0.0688868	0.0344434	+	0.999407i	$0.489034\pi$
$492$	0	0
$493$	8166.82	0.746076
$494$	258.913	0.0235810
$495$	0	0
$496$	−6762.31	−0.612170
$497$	5920.08	0.534310
$498$	0	0
$499$	15180.5	1.36187	0.680933	−	0.732346i	$-0.261574\pi$
$499$	15180.5	1.36187	0.680933	+	0.732346i	$0.261574\pi$
$500$	0	0
$501$	0	0
$502$	−191.316	−0.0170096
$503$	14069.8	1.24720	0.623598	−	0.781745i	$-0.285670\pi$
$503$	14069.8	1.24720	0.623598	+	0.781745i	$0.285670\pi$
$504$	0	0
$505$	0	0
$506$	142.502	0.0125197
$507$	0	0
$508$	2879.23	0.251467
$509$	8259.05	0.719207	0.359603	−	0.933105i	$-0.382912\pi$
$509$	8259.05	0.719207	0.359603	+	0.933105i	$0.382912\pi$
$510$	0	0
$511$	6178.35	0.534861
$512$	−1701.07	−0.146831
$513$	0	0
$514$	−357.642	−0.0306905
$515$	0	0
$516$	0	0
$517$	−10752.2	−0.914667
$518$	198.556	0.0168418
$519$	0	0
$520$	0	0
$521$	−16129.2	−1.35630	−0.678149	−	0.734924i	$-0.737218\pi$
$521$	−16129.2	−1.35630	−0.678149	+	0.734924i	$0.737218\pi$
$522$	0	0
$523$	−4730.56	−0.395512	−0.197756	−	0.980251i	$-0.563365\pi$
$523$	−4730.56	−0.395512	−0.197756	+	0.980251i	$0.563365\pi$
$524$	−18719.3	−1.56060
$525$	0	0
$526$	−631.619	−0.0523572
$527$	−9964.96	−0.823683
$528$	0	0
$529$	−10937.0	−0.898906
$530$	0	0
$531$	0	0
$532$	4679.28	0.381339
$533$	7233.12	0.587807
$534$	0	0
$535$	0	0
$536$	−347.040	−0.0279661
$537$	0	0
$538$	−354.512	−0.0284091
$539$	−2388.69	−0.190887
$540$	0	0
$541$	14089.0	1.11966	0.559828	−	0.828609i	$-0.310867\pi$
$541$	14089.0	1.11966	0.559828	+	0.828609i	$0.310867\pi$
$542$	320.763	0.0254206
$543$	0	0
$544$	−1503.59	−0.118503
$545$	0	0
$546$	0	0
$547$	−5943.06	−0.464546	−0.232273	−	0.972651i	$-0.574616\pi$
$547$	−5943.06	−0.464546	−0.232273	+	0.972651i	$0.574616\pi$
$548$	7511.24	0.585519
$549$	0	0
$550$	0	0
$551$	7260.94	0.561391
$552$	0	0
$553$	−4487.45	−0.345074
$554$	−145.246	−0.0111388
$555$	0	0
$556$	4145.20	0.316179
$557$	12702.2	0.966268	0.483134	−	0.875546i	$-0.339498\pi$
$557$	12702.2	0.966268	0.483134	+	0.875546i	$0.339498\pi$
$558$	0	0
$559$	−7499.38	−0.567424
$560$	0	0
$561$	0	0
$562$	424.391	0.0318538
$563$	8829.27	0.660940	0.330470	−	0.943816i	$-0.392793\pi$
$563$	8829.27	0.660940	0.330470	+	0.943816i	$0.392793\pi$
$564$	0	0
$565$	0	0
$566$	438.106	0.0325353
$567$	0	0
$568$	−1127.36	−0.0832800
$569$	2414.73	0.177910	0.0889548	−	0.996036i	$-0.471647\pi$
$569$	2414.73	0.177910	0.0889548	+	0.996036i	$0.471647\pi$
$570$	0	0
$571$	−4476.68	−0.328096	−0.164048	−	0.986452i	$-0.552455\pi$
$571$	−4476.68	−0.328096	−0.164048	+	0.986452i	$0.552455\pi$
$572$	−14473.0	−1.05795
$573$	0	0
$574$	−113.617	−0.00826182
$575$	0	0
$576$	0	0
$577$	−15918.8	−1.14854	−0.574272	−	0.818664i	$-0.694715\pi$
$577$	−15918.8	−1.14854	−0.574272	+	0.818664i	$0.694715\pi$
$578$	−327.999	−0.0236038
$579$	0	0
$580$	0	0
$581$	−3966.29	−0.283218
$582$	0	0
$583$	28938.7	2.05578
$584$	−1176.54	−0.0833659
$585$	0	0
$586$	780.250	0.0550031
$587$	9357.52	0.657967	0.328983	−	0.944336i	$-0.393294\pi$
$587$	9357.52	0.657967	0.328983	+	0.944336i	$0.393294\pi$
$588$	0	0
$589$	−8859.63	−0.619787
$590$	0	0
$591$	0	0
$592$	21723.5	1.50816
$593$	−14353.5	−0.993978	−0.496989	−	0.867757i	$-0.665561\pi$
$593$	−14353.5	−0.993978	−0.496989	+	0.867757i	$0.665561\pi$
$594$	0	0
$595$	0	0
$596$	9387.63	0.645188
$597$	0	0
$598$	108.577	0.00742485
$599$	−7717.69	−0.526438	−0.263219	−	0.964736i	$-0.584784\pi$
$599$	−7717.69	−0.526438	−0.263219	+	0.964736i	$0.584784\pi$
$600$	0	0
$601$	−20094.2	−1.36383	−0.681914	−	0.731432i	$-0.738852\pi$
$601$	−20094.2	−1.36383	−0.681914	+	0.731432i	$0.738852\pi$
$602$	117.799	0.00797532
$603$	0	0
$604$	12868.5	0.866908
$605$	0	0
$606$	0	0
$607$	−4577.40	−0.306080	−0.153040	−	0.988220i	$-0.548906\pi$
$607$	−4577.40	−0.306080	−0.153040	+	0.988220i	$0.548906\pi$
$608$	−1336.81	−0.0891689
$609$	0	0
$610$	0	0
$611$	−8192.51	−0.542445
$612$	0	0
$613$	−3199.71	−0.210824	−0.105412	−	0.994429i	$-0.533616\pi$
$613$	−3199.71	−0.210824	−0.105412	+	0.994429i	$0.533616\pi$
$614$	455.581	0.0299442
$615$	0	0
$616$	454.879	0.0297526
$617$	20731.4	1.35270	0.676350	−	0.736581i	$-0.263561\pi$
$617$	20731.4	1.35270	0.676350	+	0.736581i	$0.263561\pi$
$618$	0	0
$619$	6995.16	0.454215	0.227108	−	0.973870i	$-0.427073\pi$
$619$	6995.16	0.454215	0.227108	+	0.973870i	$0.427073\pi$
$620$	0	0
$621$	0	0
$622$	827.233	0.0533264
$623$	3813.80	0.245260
$624$	0	0
$625$	0	0
$626$	64.2458	0.00410188
$627$	0	0
$628$	27915.3	1.77379
$629$	32011.9	2.02925
$630$	0	0
$631$	−11306.5	−0.713317	−0.356658	−	0.934235i	$-0.616084\pi$
$631$	−11306.5	−0.713317	−0.356658	+	0.934235i	$0.616084\pi$
$632$	854.546	0.0537849
$633$	0	0
$634$	−162.438	−0.0101755
$635$	0	0
$636$	0	0
$637$	−1820.03	−0.113206
$638$	352.770	0.0218907
$639$	0	0
$640$	0	0
$641$	−12712.2	−0.783312	−0.391656	−	0.920112i	$-0.628098\pi$
$641$	−12712.2	−0.783312	−0.391656	+	0.920112i	$0.628098\pi$
$642$	0	0
$643$	−15687.5	−0.962138	−0.481069	−	0.876683i	$-0.659751\pi$
$643$	−15687.5	−0.962138	−0.481069	+	0.876683i	$0.659751\pi$
$644$	1962.30	0.120070
$645$	0	0
$646$	−655.691	−0.0399347
$647$	−564.225	−0.0342844	−0.0171422	−	0.999853i	$-0.505457\pi$
$647$	−564.225	−0.0342844	−0.0171422	+	0.999853i	$0.505457\pi$
$648$	0	0
$649$	−8415.96	−0.509022
$650$	0	0
$651$	0	0
$652$	18616.5	1.11822
$653$	−25404.2	−1.52243	−0.761213	−	0.648502i	$-0.775396\pi$
$653$	−25404.2	−1.52243	−0.761213	+	0.648502i	$0.775396\pi$
$654$	0	0
$655$	0	0
$656$	−12430.6	−0.739835
$657$	0	0
$658$	128.687	0.00762423
$659$	3776.09	0.223210	0.111605	−	0.993753i	$-0.464401\pi$
$659$	3776.09	0.223210	0.111605	+	0.993753i	$0.464401\pi$
$660$	0	0
$661$	7423.33	0.436814	0.218407	−	0.975858i	$-0.429914\pi$
$661$	7423.33	0.436814	0.218407	+	0.975858i	$0.429914\pi$
$662$	59.6633	0.00350284
$663$	0	0
$664$	755.301	0.0441436
$665$	0	0
$666$	0	0
$667$	3044.94	0.176763
$668$	−8924.14	−0.516894
$669$	0	0
$670$	0	0
$671$	−13256.3	−0.762673
$672$	0	0
$673$	20555.0	1.17732	0.588661	−	0.808380i	$-0.299655\pi$
$673$	20555.0	1.17732	0.588661	+	0.808380i	$0.299655\pi$
$674$	252.559	0.0144335
$675$	0	0
$676$	6533.21	0.371712
$677$	−994.516	−0.0564584	−0.0282292	−	0.999601i	$-0.508987\pi$
$677$	−994.516	−0.0564584	−0.0282292	+	0.999601i	$0.508987\pi$
$678$	0	0
$679$	−1040.64	−0.0588162
$680$	0	0
$681$	0	0
$682$	−430.441	−0.0241678
$683$	−30061.8	−1.68416	−0.842081	−	0.539352i	$-0.818669\pi$
$683$	−30061.8	−1.68416	−0.842081	+	0.539352i	$0.818669\pi$
$684$	0	0
$685$	0	0
$686$	28.5888	0.00159115
$687$	0	0
$688$	12888.1	0.714179
$689$	22049.5	1.21918
$690$	0	0
$691$	−12973.7	−0.714242	−0.357121	−	0.934058i	$-0.616242\pi$
$691$	−12973.7	−0.714242	−0.357121	+	0.934058i	$0.616242\pi$
$692$	7791.84	0.428037
$693$	0	0
$694$	−336.327	−0.0183960
$695$	0	0
$696$	0	0
$697$	−18317.7	−0.995457
$698$	276.001	0.0149667
$699$	0	0
$700$	0	0
$701$	−13289.0	−0.716003	−0.358002	−	0.933721i	$-0.616542\pi$
$701$	−13289.0	−0.716003	−0.358002	+	0.933721i	$0.616542\pi$
$702$	0	0
$703$	28461.1	1.52693
$704$	24829.4	1.32925
$705$	0	0
$706$	−278.722	−0.0148581
$707$	−12927.1	−0.687655
$708$	0	0
$709$	−16040.2	−0.849650	−0.424825	−	0.905275i	$-0.639664\pi$
$709$	−16040.2	−0.849650	−0.424825	+	0.905275i	$0.639664\pi$
$710$	0	0
$711$	0	0
$712$	−726.263	−0.0382273
$713$	−3715.37	−0.195149
$714$	0	0
$715$	0	0
$716$	11338.9	0.591836
$717$	0	0
$718$	684.365	0.0355714
$719$	9534.95	0.494567	0.247283	−	0.968943i	$-0.420462\pi$
$719$	9534.95	0.494567	0.247283	+	0.968943i	$0.420462\pi$
$720$	0	0
$721$	6799.86	0.351234
$722$	−11.2668	−0.000580757 0
$723$	0	0
$724$	1229.81	0.0631290
$725$	0	0
$726$	0	0
$727$	29066.4	1.48282	0.741411	−	0.671051i	$-0.234157\pi$
$727$	29066.4	1.48282	0.741411	+	0.671051i	$0.234157\pi$
$728$	346.589	0.0176448
$729$	0	0
$730$	0	0
$731$	18992.0	0.960937
$732$	0	0
$733$	1035.33	0.0521701	0.0260851	−	0.999660i	$-0.491696\pi$
$733$	1035.33	0.0521701	0.0260851	+	0.999660i	$0.491696\pi$
$734$	−891.742	−0.0448431
$735$	0	0
$736$	−560.602	−0.0280762
$737$	12691.4	0.634321
$738$	0	0
$739$	18595.2	0.925621	0.462811	−	0.886457i	$-0.346841\pi$
$739$	18595.2	0.925621	0.462811	+	0.886457i	$0.346841\pi$
$740$	0	0
$741$	0	0
$742$	−346.351	−0.0171360
$743$	−24164.2	−1.19314	−0.596568	−	0.802563i	$-0.703469\pi$
$743$	−24164.2	−1.19314	−0.596568	+	0.802563i	$0.703469\pi$
$744$	0	0
$745$	0	0
$746$	1036.29	0.0508596
$747$	0	0
$748$	36652.6	1.79165
$749$	5353.35	0.261158
$750$	0	0
$751$	33783.1	1.64150	0.820748	−	0.571291i	$-0.193557\pi$
$751$	33783.1	1.64150	0.820748	+	0.571291i	$0.193557\pi$
$752$	14079.3	0.682740
$753$	0	0
$754$	268.788	0.0129823
$755$	0	0
$756$	0	0
$757$	15216.7	0.730593	0.365296	−	0.930891i	$-0.380968\pi$
$757$	15216.7	0.730593	0.365296	+	0.930891i	$0.380968\pi$
$758$	1040.65	0.0498658
$759$	0	0
$760$	0	0
$761$	−14324.4	−0.682336	−0.341168	−	0.940002i	$-0.610822\pi$
$761$	−14324.4	−0.682336	−0.341168	+	0.940002i	$0.610822\pi$
$762$	0	0
$763$	6788.29	0.322087
$764$	−34948.6	−1.65497
$765$	0	0
$766$	637.255	0.0300587
$767$	−6412.42	−0.301876
$768$	0	0
$769$	−17998.3	−0.844001	−0.422000	−	0.906596i	$-0.638672\pi$
$769$	−17998.3	−0.844001	−0.422000	+	0.906596i	$0.638672\pi$
$770$	0	0
$771$	0	0
$772$	37618.7	1.75379
$773$	2816.84	0.131067	0.0655335	−	0.997850i	$-0.479125\pi$
$773$	2816.84	0.131067	0.0655335	+	0.997850i	$0.479125\pi$
$774$	0	0
$775$	0	0
$776$	198.170	0.00916738
$777$	0	0
$778$	172.571	0.00795240
$779$	−16285.9	−0.749040
$780$	0	0
$781$	41228.1	1.88894
$782$	−274.970	−0.0125740
$783$	0	0
$784$	3127.83	0.142485
$785$	0	0
$786$	0	0
$787$	−6712.82	−0.304049	−0.152024	−	0.988377i	$-0.548579\pi$
$787$	−6712.82	−0.304049	−0.152024	+	0.988377i	$0.548579\pi$
$788$	41170.1	1.86120
$789$	0	0
$790$	0	0
$791$	−14532.6	−0.653251
$792$	0	0
$793$	−10100.5	−0.452305
$794$	64.7407	0.00289366
$795$	0	0
$796$	−37389.1	−1.66485
$797$	−23506.3	−1.04471	−0.522356	−	0.852727i	$-0.674947\pi$
$797$	−23506.3	−1.04471	−0.522356	+	0.852727i	$0.674947\pi$
$798$	0	0
$799$	20747.4	0.918634
$800$	0	0
$801$	0	0
$802$	539.764	0.0237653
$803$	43026.7	1.89088
$804$	0	0
$805$	0	0
$806$	−327.969	−0.0143328
$807$	0	0
$808$	2461.70	0.107181
$809$	−22544.6	−0.979761	−0.489881	−	0.871790i	$-0.662960\pi$
$809$	−22544.6	−0.979761	−0.489881	+	0.871790i	$0.662960\pi$
$810$	0	0
$811$	−12379.8	−0.536022	−0.268011	−	0.963416i	$-0.586366\pi$
$811$	−12379.8	−0.536022	−0.268011	+	0.963416i	$0.586366\pi$
$812$	4857.75	0.209943
$813$	0	0
$814$	1382.77	0.0595405
$815$	0	0
$816$	0	0
$817$	16885.4	0.723066
$818$	−1317.97	−0.0563345
$819$	0	0
$820$	0	0
$821$	12651.3	0.537798	0.268899	−	0.963168i	$-0.413340\pi$
$821$	12651.3	0.537798	0.268899	+	0.963168i	$0.413340\pi$
$822$	0	0
$823$	38475.4	1.62961	0.814804	−	0.579737i	$-0.196845\pi$
$823$	38475.4	1.62961	0.814804	+	0.579737i	$0.196845\pi$
$824$	−1294.90	−0.0547450
$825$	0	0
$826$	100.726	0.00424297
$827$	3062.97	0.128791	0.0643953	−	0.997924i	$-0.479488\pi$
$827$	3062.97	0.128791	0.0643953	+	0.997924i	$0.479488\pi$
$828$	0	0
$829$	11813.1	0.494917	0.247459	−	0.968898i	$-0.420405\pi$
$829$	11813.1	0.494917	0.247459	+	0.968898i	$0.420405\pi$
$830$	0	0
$831$	0	0
$832$	18918.5	0.788317
$833$	4609.19	0.191715
$834$	0	0
$835$	0	0
$836$	32587.1	1.34814
$837$	0	0
$838$	−186.339	−0.00768136
$839$	−29359.8	−1.20812	−0.604059	−	0.796939i	$-0.706451\pi$
$839$	−29359.8	−1.20812	−0.604059	+	0.796939i	$0.706451\pi$
$840$	0	0
$841$	−16851.1	−0.690931
$842$	454.073	0.0185848
$843$	0	0
$844$	6669.18	0.271994
$845$	0	0
$846$	0	0
$847$	−7318.13	−0.296876
$848$	−37893.4	−1.53451
$849$	0	0
$850$	0	0
$851$	11935.4	0.480776
$852$	0	0
$853$	27515.7	1.10448	0.552239	−	0.833686i	$-0.313774\pi$
$853$	27515.7	1.10448	0.552239	+	0.833686i	$0.313774\pi$
$854$	158.657	0.00635729
$855$	0	0
$856$	−1019.44	−0.0407053
$857$	−9970.63	−0.397421	−0.198711	−	0.980058i	$-0.563675\pi$
$857$	−9970.63	−0.397421	−0.198711	+	0.980058i	$0.563675\pi$
$858$	0	0
$859$	−35051.6	−1.39225	−0.696126	−	0.717919i	$-0.745095\pi$
$859$	−35051.6	−1.39225	−0.696126	+	0.717919i	$0.745095\pi$
$860$	0	0
$861$	0	0
$862$	−979.496	−0.0387027
$863$	27756.4	1.09483	0.547415	−	0.836861i	$-0.315612\pi$
$863$	27756.4	1.09483	0.547415	+	0.836861i	$0.315612\pi$
$864$	0	0
$865$	0	0
$866$	−717.481	−0.0281536
$867$	0	0
$868$	−5927.32	−0.231781
$869$	−31251.2	−1.21993
$870$	0	0
$871$	9670.05	0.376185
$872$	−1292.69	−0.0502020
$873$	0	0
$874$	−244.470	−0.00946145
$875$	0	0
$876$	0	0
$877$	520.779	0.0200518	0.0100259	−	0.999950i	$-0.496809\pi$
$877$	520.779	0.0200518	0.0100259	+	0.999950i	$0.496809\pi$
$878$	−555.316	−0.0213451
$879$	0	0
$880$	0	0
$881$	−16514.4	−0.631536	−0.315768	−	0.948836i	$-0.602262\pi$
$881$	−16514.4	−0.631536	−0.315768	+	0.948836i	$0.602262\pi$
$882$	0	0
$883$	9502.60	0.362161	0.181080	−	0.983468i	$-0.442041\pi$
$883$	9502.60	0.362161	0.181080	+	0.983468i	$0.442041\pi$
$884$	27927.0	1.06254
$885$	0	0
$886$	−202.820	−0.00769060
$887$	−37594.1	−1.42310	−0.711548	−	0.702638i	$-0.752006\pi$
$887$	−37594.1	−1.42310	−0.711548	+	0.702638i	$0.752006\pi$
$888$	0	0
$889$	2521.52	0.0951283
$890$	0	0
$891$	0	0
$892$	47261.3	1.77402
$893$	18446.0	0.691235
$894$	0	0
$895$	0	0
$896$	−1192.30	−0.0444554
$897$	0	0
$898$	−1148.75	−0.0426885
$899$	−9197.55	−0.341219
$900$	0	0
$901$	−55839.8	−2.06470
$902$	−791.243	−0.0292079
$903$	0	0
$904$	2767.45	0.101819
$905$	0	0
$906$	0	0
$907$	17328.4	0.634377	0.317189	−	0.948362i	$-0.397261\pi$
$907$	17328.4	0.634377	0.317189	+	0.948362i	$0.397261\pi$
$908$	28394.3	1.03777
$909$	0	0
$910$	0	0
$911$	37777.9	1.37392	0.686958	−	0.726697i	$-0.258946\pi$
$911$	37777.9	1.37392	0.686958	+	0.726697i	$0.258946\pi$
$912$	0	0
$913$	−27621.7	−1.00125
$914$	394.765	0.0142863
$915$	0	0
$916$	15757.3	0.568381
$917$	−16393.6	−0.590365
$918$	0	0
$919$	17791.9	0.638630	0.319315	−	0.947649i	$-0.396547\pi$
$919$	17791.9	0.638630	0.319315	+	0.947649i	$0.396547\pi$
$920$	0	0
$921$	0	0
$922$	−556.704	−0.0198851
$923$	31413.2	1.12024
$924$	0	0
$925$	0	0
$926$	−139.547	−0.00495227
$927$	0	0
$928$	−1387.80	−0.0490912
$929$	33373.0	1.17861	0.589307	−	0.807909i	$-0.299401\pi$
$929$	33373.0	1.17861	0.589307	+	0.807909i	$0.299401\pi$
$930$	0	0
$931$	4097.93	0.144258
$932$	39413.4	1.38522
$933$	0	0
$934$	−1300.45	−0.0455588
$935$	0	0
$936$	0	0
$937$	−36078.5	−1.25788	−0.628939	−	0.777454i	$-0.716511\pi$
$937$	−36078.5	−1.25788	−0.628939	+	0.777454i	$0.716511\pi$
$938$	−151.896	−0.00528740
$939$	0	0
$940$	0	0
$941$	32456.8	1.12440	0.562201	−	0.827001i	$-0.309955\pi$
$941$	32456.8	1.12440	0.562201	+	0.827001i	$0.309955\pi$
$942$	0	0
$943$	−6829.63	−0.235847
$944$	11020.1	0.379952
$945$	0	0
$946$	820.369	0.0281950
$947$	−3547.41	−0.121727	−0.0608634	−	0.998146i	$-0.519385\pi$
$947$	−3547.41	−0.121727	−0.0608634	+	0.998146i	$0.519385\pi$
$948$	0	0
$949$	32783.6	1.12139
$950$	0	0
$951$	0	0
$952$	−877.728	−0.0298817
$953$	−12278.3	−0.417347	−0.208674	−	0.977985i	$-0.566915\pi$
$953$	−12278.3	−0.417347	−0.208674	+	0.977985i	$0.566915\pi$
$954$	0	0
$955$	0	0
$956$	−47783.5	−1.61656
$957$	0	0
$958$	1094.70	0.0369188
$959$	6578.05	0.221498
$960$	0	0
$961$	−18568.4	−0.623288
$962$	1053.58	0.0353106
$963$	0	0
$964$	−39329.8	−1.31403
$965$	0	0
$966$	0	0
$967$	−32861.0	−1.09280	−0.546401	−	0.837524i	$-0.684002\pi$
$967$	−32861.0	−1.09280	−0.546401	+	0.837524i	$0.684002\pi$
$968$	1393.59	0.0462725
$969$	0	0
$970$	0	0
$971$	27057.8	0.894261	0.447130	−	0.894469i	$-0.352446\pi$
$971$	27057.8	0.894261	0.447130	+	0.894469i	$0.352446\pi$
$972$	0	0
$973$	3630.20	0.119608
$974$	477.814	0.0157188
$975$	0	0
$976$	17358.2	0.569287
$977$	36357.6	1.19057	0.595283	−	0.803516i	$-0.297040\pi$
$977$	36357.6	1.19057	0.595283	+	0.803516i	$0.297040\pi$
$978$	0	0
$979$	26559.8	0.867063
$980$	0	0
$981$	0	0
$982$	−62.4684	−0.00202999
$983$	−41706.4	−1.35323	−0.676616	−	0.736336i	$-0.736554\pi$
$983$	−41706.4	−1.35323	−0.676616	+	0.736336i	$0.736554\pi$
$984$	0	0
$985$	0	0
$986$	−680.700	−0.0219857
$987$	0	0
$988$	24829.3	0.799518
$989$	7081.04	0.227668
$990$	0	0
$991$	61442.3	1.96950	0.984752	−	0.173965i	$-0.0556578\pi$
$991$	61442.3	1.96950	0.984752	+	0.173965i	$0.0556578\pi$
$992$	1693.35	0.0541977
$993$	0	0
$994$	−493.435	−0.0157453
$995$	0	0
$996$	0	0
$997$	1577.35	0.0501056	0.0250528	−	0.999686i	$-0.492025\pi$
$997$	1577.35	0.0501056	0.0250528	+	0.999686i	$0.492025\pi$
$998$	−1265.28	−0.0401321
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.4.a.bt.1.5		10
3.2	odd	2	inner	1575.4.a.bt.1.6		10
5.2	odd	4		315.4.d.d.64.9	✓	20
5.3	odd	4		315.4.d.d.64.11	yes	20
5.4	even	2		1575.4.a.bu.1.6		10
15.2	even	4		315.4.d.d.64.12	yes	20
15.8	even	4		315.4.d.d.64.10	yes	20
15.14	odd	2		1575.4.a.bu.1.5		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.4.d.d.64.9	✓	20	5.2	odd	4
315.4.d.d.64.10	yes	20	15.8	even	4
315.4.d.d.64.11	yes	20	5.3	odd	4
315.4.d.d.64.12	yes	20	15.2	even	4
1575.4.a.bt.1.5		10	1.1	even	1	trivial
1575.4.a.bt.1.6		10	3.2	odd	2	inner
1575.4.a.bu.1.5		10	15.14	odd	2
1575.4.a.bu.1.6		10	5.4	even	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:	\( N \)	\(=\)	\( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1575.a (trivial)

\(f(q)\)	\(=\)	\(q-0.0833494 q^{2} -7.99305 q^{4} -7.00000 q^{7} +1.33301 q^{8} -48.7488 q^{11} -37.1435 q^{13} +0.583446 q^{14} +63.8333 q^{16} +94.0651 q^{17} +83.6312 q^{19} +4.06318 q^{22} +35.0715 q^{23} +3.09589 q^{26} +55.9514 q^{28} +86.8210 q^{29} -105.937 q^{31} -15.9846 q^{32} -7.84026 q^{34} +340.316 q^{37} -6.97061 q^{38} -194.735 q^{41} +201.903 q^{43} +389.652 q^{44} -2.92319 q^{46} +220.564 q^{47} +49.0000 q^{49} +296.890 q^{52} -593.630 q^{53} -9.33108 q^{56} -7.23648 q^{58} +172.639 q^{59} +271.931 q^{61} +8.82978 q^{62} -509.334 q^{64} -260.343 q^{67} -751.867 q^{68} -845.726 q^{71} -882.621 q^{73} -28.3651 q^{74} -668.469 q^{76} +341.242 q^{77} +641.065 q^{79} +16.2310 q^{82} +566.613 q^{83} -16.8285 q^{86} -64.9827 q^{88} -544.829 q^{89} +260.004 q^{91} -280.328 q^{92} -18.3839 q^{94} +148.663 q^{97} -4.08412 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 54 q^{4} - 70 q^{7} - 104 q^{13} + 310 q^{16} + 36 q^{19} - 644 q^{22} - 378 q^{28} - 24 q^{31} + 116 q^{34} - 732 q^{37} - 212 q^{43} - 184 q^{46} + 490 q^{49} - 2692 q^{52} - 3196 q^{58} + 1024 q^{61}+ \cdots - 4024 q^{97}+O(q^{100}) \) 10 * q + 54 * q^4 - 70 * q^7 - 104 * q^13 + 310 * q^16 + 36 * q^19 - 644 * q^22 - 378 * q^28 - 24 * q^31 + 116 * q^34 - 732 * q^37 - 212 * q^43 - 184 * q^46 + 490 * q^49 - 2692 * q^52 - 3196 * q^58 + 1024 * q^61 + 1590 * q^64 - 2388 * q^67 - 3432 * q^73 - 4184 * q^76 - 776 * q^79 - 1860 * q^82 - 10164 * q^88 + 728 * q^91 - 4028 * q^94 - 4024 * q^97

Introduction

L-functions

Modular forms

Varieties

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