LMFDB - Embedded newform 1575.4.a.bg.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1575,4,Mod(1,1575)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1575.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage: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")

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:traces = [4,-4,0,36,0,0,28,-27,0,0,-100] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$92.9280082590$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 32x^{2} - 35x + 120 $ x^4 - 32x^2 - 35x + 120
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 175)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.84167$ 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)

$f(q)$	$=$	$q-4.84167 q^{2} +15.4418 q^{4} +7.00000 q^{7} -36.0308 q^{8} -62.1962 q^{11} +14.0934 q^{13} -33.8917 q^{14} +50.9148 q^{16} -63.5104 q^{17} +48.7094 q^{19} +301.134 q^{22} +99.3784 q^{23} -68.2354 q^{26} +108.093 q^{28} +69.0571 q^{29} -9.68658 q^{31} +41.7334 q^{32} +307.497 q^{34} +240.290 q^{37} -235.835 q^{38} -335.306 q^{41} +51.2582 q^{43} -960.421 q^{44} -481.158 q^{46} +451.564 q^{47} +49.0000 q^{49} +217.627 q^{52} +180.014 q^{53} -252.215 q^{56} -334.352 q^{58} -268.600 q^{59} -323.925 q^{61} +46.8992 q^{62} -609.378 q^{64} -541.910 q^{67} -980.715 q^{68} +161.433 q^{71} +305.751 q^{73} -1163.40 q^{74} +752.161 q^{76} -435.373 q^{77} -504.722 q^{79} +1623.44 q^{82} +513.838 q^{83} -248.176 q^{86} +2240.98 q^{88} -543.158 q^{89} +98.6535 q^{91} +1534.58 q^{92} -2186.32 q^{94} +1863.06 q^{97} -237.242 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 36 q^{4} + 28 q^{7} - 27 q^{8} - 100 q^{11} + 44 q^{13} - 28 q^{14} + 160 q^{16} + 53 q^{17} - 29 q^{19} - 152 q^{22} - 295 q^{23} - 700 q^{26} + 252 q^{28} - 129 q^{29} + 114 q^{31} + 310 q^{32}+ \cdots - 196 q^{98}+O(q^{100}) $ 4 * q - 4 * q^2 + 36 * q^4 + 28 * q^7 - 27 * q^8 - 100 * q^11 + 44 * q^13 - 28 * q^14 + 160 * q^16 + 53 * q^17 - 29 * q^19 - 152 * q^22 - 295 * q^23 - 700 * q^26 + 252 * q^28 - 129 * q^29 + 114 * q^31 + 310 * q^32 + 203 * q^34 + 403 * q^37 - 555 * q^38 - 671 * q^41 - 411 * q^43 - 438 * q^44 - 997 * q^46 + 8 * q^47 + 196 * q^49 + 74 * q^52 - 90 * q^53 - 189 * q^56 + 673 * q^58 - 1018 * q^59 + 50 * q^61 + 1626 * q^62 - 2421 * q^64 + 424 * q^67 + 617 * q^68 - 215 * q^71 - 1207 * q^73 - 623 * q^74 - 3257 * q^76 - 700 * q^77 - 951 * q^79 + 1695 * q^82 + 3035 * q^83 + 99 * q^86 + 163 * q^88 - 2819 * q^89 + 308 * q^91 - 3073 * q^92 - 3056 * q^94 - 1100 * q^97 - 196 * q^98

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 $

$2$	−4.84167	−1.71179	−0.855895	−	0.517150i	$-0.826993\pi$
$2$	−4.84167	−1.71179	−0.855895	+	0.517150i	$0.826993\pi$
$3$	0	0
$3$	0	0
$4$	15.4418	1.93022
$5$	0	0
$5$	0	0
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	−36.0308	−1.59235
$9$	0	0
$10$	0	0
$11$	−62.1962	−1.70481	−0.852403	−	0.522886i	$-0.824855\pi$
$11$	−62.1962	−1.70481	−0.852403	+	0.522886i	$0.824855\pi$
$12$	0	0
$13$	14.0934	0.300676	0.150338	−	0.988635i	$-0.451964\pi$
$13$	14.0934	0.300676	0.150338	+	0.988635i	$0.451964\pi$
$14$	−33.8917	−0.646996
$15$	0	0
$16$	50.9148	0.795543
$17$	−63.5104	−0.906091	−0.453045	−	0.891487i	$-0.649662\pi$
$17$	−63.5104	−0.906091	−0.453045	+	0.891487i	$0.649662\pi$
$18$	0	0
$19$	48.7094	0.588143	0.294071	−	0.955783i	$-0.404990\pi$
$19$	48.7094	0.588143	0.294071	+	0.955783i	$0.404990\pi$
$20$	0	0
$21$	0	0
$22$	301.134	2.91827
$23$	99.3784	0.900949	0.450475	−	0.892789i	$-0.351255\pi$
$23$	99.3784	0.900949	0.450475	+	0.892789i	$0.351255\pi$
$24$	0	0
$25$	0	0
$26$	−68.2354	−0.514695
$27$	0	0
$28$	108.093	0.729556
$29$	69.0571	0.442193	0.221096	−	0.975252i	$-0.429036\pi$
$29$	69.0571	0.442193	0.221096	+	0.975252i	$0.429036\pi$
$30$	0	0
$31$	−9.68658	−0.0561213	−0.0280607	−	0.999606i	$-0.508933\pi$
$31$	−9.68658	−0.0561213	−0.0280607	+	0.999606i	$0.508933\pi$
$32$	41.7334	0.230547
$33$	0	0
$34$	307.497	1.55104
$35$	0	0
$36$	0	0
$37$	240.290	1.06766	0.533829	−	0.845592i	$-0.320752\pi$
$37$	240.290	1.06766	0.533829	+	0.845592i	$0.320752\pi$
$38$	−235.835	−1.00678
$39$	0	0
$40$	0	0
$41$	−335.306	−1.27722	−0.638610	−	0.769531i	$-0.720490\pi$
$41$	−335.306	−1.27722	−0.638610	+	0.769531i	$0.720490\pi$
$42$	0	0
$43$	51.2582	0.181786	0.0908931	−	0.995861i	$-0.471028\pi$
$43$	51.2582	0.181786	0.0908931	+	0.995861i	$0.471028\pi$
$44$	−960.421	−3.29066
$45$	0	0
$46$	−481.158	−1.54224
$47$	451.564	1.40143	0.700716	−	0.713440i	$-0.252864\pi$
$47$	451.564	1.40143	0.700716	+	0.713440i	$0.252864\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	0	0
$52$	217.627	0.580373
$53$	180.014	0.466545	0.233273	−	0.972411i	$-0.425057\pi$
$53$	180.014	0.466545	0.233273	+	0.972411i	$0.425057\pi$
$54$	0	0
$55$	0	0
$56$	−252.215	−0.601852
$57$	0	0
$58$	−334.352	−0.756941
$59$	−268.600	−0.592691	−0.296345	−	0.955081i	$-0.595768\pi$
$59$	−268.600	−0.592691	−0.296345	+	0.955081i	$0.595768\pi$
$60$	0	0
$61$	−323.925	−0.679906	−0.339953	−	0.940442i	$-0.610411\pi$
$61$	−323.925	−0.679906	−0.339953	+	0.940442i	$0.610411\pi$
$62$	46.8992	0.0960679
$63$	0	0
$64$	−609.378	−1.19019
$65$	0	0
$66$	0	0
$67$	−541.910	−0.988132	−0.494066	−	0.869424i	$-0.664490\pi$
$67$	−541.910	−0.988132	−0.494066	+	0.869424i	$0.664490\pi$
$68$	−980.715	−1.74896
$69$	0	0
$70$	0	0
$71$	161.433	0.269839	0.134919	−	0.990857i	$-0.456922\pi$
$71$	161.433	0.269839	0.134919	+	0.990857i	$0.456922\pi$
$72$	0	0
$73$	305.751	0.490212	0.245106	−	0.969496i	$-0.421177\pi$
$73$	305.751	0.490212	0.245106	+	0.969496i	$0.421177\pi$
$74$	−1163.40	−1.82761
$75$	0	0
$76$	752.161	1.13525
$77$	−435.373	−0.644356
$78$	0	0
$79$	−504.722	−0.718805	−0.359403	−	0.933183i	$-0.617020\pi$
$79$	−504.722	−0.718805	−0.359403	+	0.933183i	$0.617020\pi$
$80$	0	0
$81$	0	0
$82$	1623.44	2.18633
$83$	513.838	0.679531	0.339766	−	0.940510i	$-0.389652\pi$
$83$	513.838	0.679531	0.339766	+	0.940510i	$0.389652\pi$
$84$	0	0
$85$	0	0
$86$	−248.176	−0.311180
$87$	0	0
$88$	2240.98	2.71465
$89$	−543.158	−0.646907	−0.323453	−	0.946244i	$-0.604844\pi$
$89$	−543.158	−0.646907	−0.323453	+	0.946244i	$0.604844\pi$
$90$	0	0
$91$	98.6535	0.113645
$92$	1534.58	1.73903
$93$	0	0
$94$	−2186.32	−2.39896
$95$	0	0
$96$	0	0
$97$	1863.06	1.95016	0.975079	−	0.221857i	$-0.0712118\pi$
$97$	1863.06	1.95016	0.975079	+	0.221857i	$0.0712118\pi$
$98$	−237.242	−0.244541
$99$	0	0
$100$	0	0
$101$	1685.70	1.66073	0.830365	−	0.557221i	$-0.188132\pi$
$101$	1685.70	1.66073	0.830365	+	0.557221i	$0.188132\pi$
$102$	0	0
$103$	−1014.19	−0.970203	−0.485102	−	0.874458i	$-0.661217\pi$
$103$	−1014.19	−0.970203	−0.485102	+	0.874458i	$0.661217\pi$
$104$	−507.794	−0.478782
$105$	0	0
$106$	−871.571	−0.798627
$107$	−913.161	−0.825033	−0.412517	−	0.910950i	$-0.635350\pi$
$107$	−913.161	−0.825033	−0.412517	+	0.910950i	$0.635350\pi$
$108$	0	0
$109$	−1397.13	−1.22772	−0.613859	−	0.789416i	$-0.710383\pi$
$109$	−1397.13	−1.22772	−0.613859	+	0.789416i	$0.710383\pi$
$110$	0	0
$111$	0	0
$112$	356.403	0.300687
$113$	−2082.49	−1.73366	−0.866831	−	0.498603i	$-0.833847\pi$
$113$	−2082.49	−1.73366	−0.866831	+	0.498603i	$0.833847\pi$
$114$	0	0
$115$	0	0
$116$	1066.37	0.853531
$117$	0	0
$118$	1300.47	1.01456
$119$	−444.573	−0.342470
$120$	0	0
$121$	2537.37	1.90636
$122$	1568.34	1.16386
$123$	0	0
$124$	−149.578	−0.108327
$125$	0	0
$126$	0	0
$127$	−230.691	−0.161185	−0.0805925	−	0.996747i	$-0.525681\pi$
$127$	−230.691	−0.161185	−0.0805925	+	0.996747i	$0.525681\pi$
$128$	2616.54	1.80681
$129$	0	0
$130$	0	0
$131$	973.877	0.649527	0.324764	−	0.945795i	$-0.394715\pi$
$131$	973.877	0.649527	0.324764	+	0.945795i	$0.394715\pi$
$132$	0	0
$133$	340.966	0.222297
$134$	2623.75	1.69148
$135$	0	0
$136$	2288.33	1.44281
$137$	−695.734	−0.433873	−0.216936	−	0.976186i	$-0.569606\pi$
$137$	−695.734	−0.433873	−0.216936	+	0.976186i	$0.569606\pi$
$138$	0	0
$139$	−298.530	−0.182165	−0.0910826	−	0.995843i	$-0.529033\pi$
$139$	−298.530	−0.182165	−0.0910826	+	0.995843i	$0.529033\pi$
$140$	0	0
$141$	0	0
$142$	−781.605	−0.461908
$143$	−876.553	−0.512595
$144$	0	0
$145$	0	0
$146$	−1480.35	−0.839140
$147$	0	0
$148$	3710.50	2.06082
$149$	1792.02	0.985290	0.492645	−	0.870230i	$-0.336030\pi$
$149$	1792.02	0.985290	0.492645	+	0.870230i	$0.336030\pi$
$150$	0	0
$151$	1201.27	0.647403	0.323701	−	0.946159i	$-0.395073\pi$
$151$	1201.27	0.647403	0.323701	+	0.946159i	$0.395073\pi$
$152$	−1755.04	−0.936529
$153$	0	0
$154$	2107.94	1.10300
$155$	0	0
$156$	0	0
$157$	409.798	0.208315	0.104158	−	0.994561i	$-0.466785\pi$
$157$	409.798	0.208315	0.104158	+	0.994561i	$0.466785\pi$
$158$	2443.70	1.23044
$159$	0	0
$160$	0	0
$161$	695.649	0.340527
$162$	0	0
$163$	−2030.71	−0.975811	−0.487905	−	0.872896i	$-0.662239\pi$
$163$	−2030.71	−0.975811	−0.487905	+	0.872896i	$0.662239\pi$
$164$	−5177.73	−2.46532
$165$	0	0
$166$	−2487.84	−1.16321
$167$	−520.014	−0.240958	−0.120479	−	0.992716i	$-0.538443\pi$
$167$	−520.014	−0.240958	−0.120479	+	0.992716i	$0.538443\pi$
$168$	0	0
$169$	−1998.38	−0.909594
$170$	0	0
$171$	0	0
$172$	791.520	0.350888
$173$	2561.90	1.12588	0.562942	−	0.826497i	$-0.309669\pi$
$173$	2561.90	1.12588	0.562942	+	0.826497i	$0.309669\pi$
$174$	0	0
$175$	0	0
$176$	−3166.71	−1.35625
$177$	0	0
$178$	2629.80	1.10737
$179$	3042.08	1.27025	0.635127	−	0.772408i	$-0.280948\pi$
$179$	3042.08	1.27025	0.635127	+	0.772408i	$0.280948\pi$
$180$	0	0
$181$	−3648.02	−1.49809	−0.749047	−	0.662516i	$-0.769488\pi$
$181$	−3648.02	−1.49809	−0.749047	+	0.662516i	$0.769488\pi$
$182$	−477.648	−0.194536
$183$	0	0
$184$	−3580.68	−1.43463
$185$	0	0
$186$	0	0
$187$	3950.11	1.54471
$188$	6972.95	2.70508
$189$	0	0
$190$	0	0
$191$	4091.12	1.54986	0.774929	−	0.632049i	$-0.217786\pi$
$191$	4091.12	1.54986	0.774929	+	0.632049i	$0.217786\pi$
$192$	0	0
$193$	3051.70	1.13817	0.569084	−	0.822280i	$-0.307298\pi$
$193$	3051.70	1.13817	0.569084	+	0.822280i	$0.307298\pi$
$194$	−9020.34	−3.33826
$195$	0	0
$196$	756.648	0.275746
$197$	−3011.44	−1.08912	−0.544560	−	0.838722i	$-0.683303\pi$
$197$	−3011.44	−1.08912	−0.544560	+	0.838722i	$0.683303\pi$
$198$	0	0
$199$	199.943	0.0712239	0.0356119	−	0.999366i	$-0.488662\pi$
$199$	199.943	0.0712239	0.0356119	+	0.999366i	$0.488662\pi$
$200$	0	0
$201$	0	0
$202$	−8161.62	−2.84282
$203$	483.400	0.167133
$204$	0	0
$205$	0	0
$206$	4910.37	1.66078
$207$	0	0
$208$	717.560	0.239201
$209$	−3029.54	−1.00267
$210$	0	0
$211$	−297.442	−0.0970461	−0.0485231	−	0.998822i	$-0.515451\pi$
$211$	−297.442	−0.0970461	−0.0485231	+	0.998822i	$0.515451\pi$
$212$	2779.75	0.900537
$213$	0	0
$214$	4421.23	1.41228
$215$	0	0
$216$	0	0
$217$	−67.8061	−0.0212119
$218$	6764.47	2.10159
$219$	0	0
$220$	0	0
$221$	−895.075	−0.272440
$222$	0	0
$223$	−6282.68	−1.88663	−0.943317	−	0.331894i	$-0.892312\pi$
$223$	−6282.68	−1.88663	−0.943317	+	0.331894i	$0.892312\pi$
$224$	292.134	0.0871385
$225$	0	0
$226$	10082.7	2.96766
$227$	2357.28	0.689243	0.344621	−	0.938742i	$-0.388007\pi$
$227$	2357.28	0.689243	0.344621	+	0.938742i	$0.388007\pi$
$228$	0	0
$229$	−2476.81	−0.714727	−0.357363	−	0.933965i	$-0.616324\pi$
$229$	−2476.81	−0.714727	−0.357363	+	0.933965i	$0.616324\pi$
$230$	0	0
$231$	0	0
$232$	−2488.18	−0.704125
$233$	5142.47	1.44590	0.722950	−	0.690900i	$-0.242786\pi$
$233$	5142.47	1.44590	0.722950	+	0.690900i	$0.242786\pi$
$234$	0	0
$235$	0	0
$236$	−4147.67	−1.14403
$237$	0	0
$238$	2152.48	0.586237
$239$	831.250	0.224975	0.112488	−	0.993653i	$-0.464118\pi$
$239$	831.250	0.224975	0.112488	+	0.993653i	$0.464118\pi$
$240$	0	0
$241$	4538.85	1.21317	0.606584	−	0.795020i	$-0.292540\pi$
$241$	4538.85	1.21317	0.606584	+	0.795020i	$0.292540\pi$
$242$	−12285.1	−3.26329
$243$	0	0
$244$	−5001.98	−1.31237
$245$	0	0
$246$	0	0
$247$	686.479	0.176841
$248$	349.015	0.0893648
$249$	0	0
$250$	0	0
$251$	−7232.00	−1.81865	−0.909323	−	0.416091i	$-0.863400\pi$
$251$	−7232.00	−1.81865	−0.909323	+	0.416091i	$0.863400\pi$
$252$	0	0
$253$	−6180.96	−1.53594
$254$	1116.93	0.275915
$255$	0	0
$256$	−7793.41	−1.90269
$257$	−4242.66	−1.02977	−0.514883	−	0.857260i	$-0.672165\pi$
$257$	−4242.66	−1.02977	−0.514883	+	0.857260i	$0.672165\pi$
$258$	0	0
$259$	1682.03	0.403537
$260$	0	0
$261$	0	0
$262$	−4715.20	−1.11185
$263$	−6604.75	−1.54854	−0.774271	−	0.632855i	$-0.781883\pi$
$263$	−6604.75	−1.54854	−0.774271	+	0.632855i	$0.781883\pi$
$264$	0	0
$265$	0	0
$266$	−1650.85	−0.380526
$267$	0	0
$268$	−8368.07	−1.90732
$269$	−4612.29	−1.04541	−0.522707	−	0.852512i	$-0.675078\pi$
$269$	−4612.29	−1.04541	−0.522707	+	0.852512i	$0.675078\pi$
$270$	0	0
$271$	−3542.29	−0.794018	−0.397009	−	0.917815i	$-0.629952\pi$
$271$	−3542.29	−0.794018	−0.397009	+	0.917815i	$0.629952\pi$
$272$	−3233.62	−0.720834
$273$	0	0
$274$	3368.52	0.742699
$275$	0	0
$276$	0	0
$277$	19.5351	0.00423737	0.00211868	−	0.999998i	$-0.499326\pi$
$277$	19.5351	0.00423737	0.00211868	+	0.999998i	$0.499326\pi$
$278$	1445.38	0.311829
$279$	0	0
$280$	0	0
$281$	−6769.20	−1.43707	−0.718535	−	0.695491i	$-0.755187\pi$
$281$	−6769.20	−1.43707	−0.718535	+	0.695491i	$0.755187\pi$
$282$	0	0
$283$	1269.89	0.266740	0.133370	−	0.991066i	$-0.457420\pi$
$283$	1269.89	0.266740	0.133370	+	0.991066i	$0.457420\pi$
$284$	2492.81	0.520850
$285$	0	0
$286$	4243.98	0.877455
$287$	−2347.14	−0.482743
$288$	0	0
$289$	−879.424	−0.178999
$290$	0	0
$291$	0	0
$292$	4721.35	0.946220
$293$	−6349.74	−1.26606	−0.633030	−	0.774127i	$-0.718189\pi$
$293$	−6349.74	−1.26606	−0.633030	+	0.774127i	$0.718189\pi$
$294$	0	0
$295$	0	0
$296$	−8657.82	−1.70009
$297$	0	0
$298$	−8676.39	−1.68661
$299$	1400.58	0.270894
$300$	0	0
$301$	358.808	0.0687087
$302$	−5816.15	−1.10822
$303$	0	0
$304$	2480.03	0.467893
$305$	0	0
$306$	0	0
$307$	−4772.37	−0.887211	−0.443605	−	0.896222i	$-0.646301\pi$
$307$	−4772.37	−0.887211	−0.443605	+	0.896222i	$0.646301\pi$
$308$	−6722.95	−1.24375
$309$	0	0
$310$	0	0
$311$	−740.703	−0.135053	−0.0675264	−	0.997717i	$-0.521511\pi$
$311$	−740.703	−0.135053	−0.0675264	+	0.997717i	$0.521511\pi$
$312$	0	0
$313$	−2279.68	−0.411678	−0.205839	−	0.978586i	$-0.565992\pi$
$313$	−2279.68	−0.411678	−0.205839	+	0.978586i	$0.565992\pi$
$314$	−1984.11	−0.356592
$315$	0	0
$316$	−7793.81	−1.38746
$317$	10198.1	1.80689	0.903446	−	0.428702i	$-0.141029\pi$
$317$	10198.1	1.80689	0.903446	+	0.428702i	$0.141029\pi$
$318$	0	0
$319$	−4295.09	−0.753852
$320$	0	0
$321$	0	0
$322$	−3368.10	−0.582910
$323$	−3093.56	−0.532911
$324$	0	0
$325$	0	0
$326$	9832.01	1.67038
$327$	0	0
$328$	12081.3	2.03378
$329$	3160.95	0.529692
$330$	0	0
$331$	−78.0380	−0.0129588	−0.00647939	−	0.999979i	$-0.502062\pi$
$331$	−78.0380	−0.0129588	−0.00647939	+	0.999979i	$0.502062\pi$
$332$	7934.59	1.31165
$333$	0	0
$334$	2517.74	0.412469
$335$	0	0
$336$	0	0
$337$	−4164.73	−0.673196	−0.336598	−	0.941648i	$-0.609276\pi$
$337$	−4164.73	−0.673196	−0.336598	+	0.941648i	$0.609276\pi$
$338$	9675.49	1.55703
$339$	0	0
$340$	0	0
$341$	602.468	0.0956759
$342$	0	0
$343$	343.000	0.0539949
$344$	−1846.87	−0.289467
$345$	0	0
$346$	−12403.9	−1.92728
$347$	−1795.19	−0.277726	−0.138863	−	0.990312i	$-0.544345\pi$
$347$	−1795.19	−0.277726	−0.138863	+	0.990312i	$0.544345\pi$
$348$	0	0
$349$	−11751.4	−1.80241	−0.901203	−	0.433398i	$-0.857314\pi$
$349$	−11751.4	−1.80241	−0.901203	+	0.433398i	$0.857314\pi$
$350$	0	0
$351$	0	0
$352$	−2595.66	−0.393038
$353$	−2882.32	−0.434590	−0.217295	−	0.976106i	$-0.569723\pi$
$353$	−2882.32	−0.434590	−0.217295	+	0.976106i	$0.569723\pi$
$354$	0	0
$355$	0	0
$356$	−8387.34	−1.24868
$357$	0	0
$358$	−14728.7	−2.17441
$359$	1193.78	0.175503	0.0877513	−	0.996142i	$-0.472032\pi$
$359$	1193.78	0.175503	0.0877513	+	0.996142i	$0.472032\pi$
$360$	0	0
$361$	−4486.39	−0.654088
$362$	17662.5	2.56442
$363$	0	0
$364$	1523.39	0.219360
$365$	0	0
$366$	0	0
$367$	8858.80	1.26001	0.630007	−	0.776589i	$-0.283052\pi$
$367$	8858.80	1.26001	0.630007	+	0.776589i	$0.283052\pi$
$368$	5059.83	0.716744
$369$	0	0
$370$	0	0
$371$	1260.10	0.176337
$372$	0	0
$373$	−2287.78	−0.317578	−0.158789	−	0.987313i	$-0.550759\pi$
$373$	−2287.78	−0.317578	−0.158789	+	0.987313i	$0.550759\pi$
$374$	−19125.1	−2.64422
$375$	0	0
$376$	−16270.2	−2.23157
$377$	973.246	0.132957
$378$	0	0
$379$	8870.18	1.20219	0.601095	−	0.799177i	$-0.294731\pi$
$379$	8870.18	1.20219	0.601095	+	0.799177i	$0.294731\pi$
$380$	0	0
$381$	0	0
$382$	−19807.8	−2.65303
$383$	−13359.2	−1.78231	−0.891155	−	0.453700i	$-0.850104\pi$
$383$	−13359.2	−1.78231	−0.891155	+	0.453700i	$0.850104\pi$
$384$	0	0
$385$	0	0
$386$	−14775.3	−1.94830
$387$	0	0
$388$	28769.0	3.76424
$389$	−11324.7	−1.47605	−0.738025	−	0.674773i	$-0.764241\pi$
$389$	−11324.7	−1.47605	−0.738025	+	0.674773i	$0.764241\pi$
$390$	0	0
$391$	−6311.57	−0.816342
$392$	−1765.51	−0.227479
$393$	0	0
$394$	14580.4	1.86434
$395$	0	0
$396$	0	0
$397$	−13242.8	−1.67415	−0.837076	−	0.547087i	$-0.815737\pi$
$397$	−13242.8	−1.67415	−0.837076	+	0.547087i	$0.815737\pi$
$398$	−968.056	−0.121920
$399$	0	0
$400$	0	0
$401$	−1952.96	−0.243207	−0.121604	−	0.992579i	$-0.538804\pi$
$401$	−1952.96	−0.243207	−0.121604	+	0.992579i	$0.538804\pi$
$402$	0	0
$403$	−136.516	−0.0168744
$404$	26030.3	3.20558
$405$	0	0
$406$	−2340.46	−0.286097
$407$	−14945.1	−1.82015
$408$	0	0
$409$	−10597.8	−1.28124	−0.640618	−	0.767860i	$-0.721322\pi$
$409$	−10597.8	−1.28124	−0.640618	+	0.767860i	$0.721322\pi$
$410$	0	0
$411$	0	0
$412$	−15660.9	−1.87271
$413$	−1880.20	−0.224016
$414$	0	0
$415$	0	0
$416$	588.164	0.0693200
$417$	0	0
$418$	14668.1	1.71636
$419$	1631.18	0.190187	0.0950935	−	0.995468i	$-0.469685\pi$
$419$	1631.18	0.190187	0.0950935	+	0.995468i	$0.469685\pi$
$420$	0	0
$421$	13181.2	1.52592	0.762961	−	0.646445i	$-0.223745\pi$
$421$	13181.2	1.52592	0.762961	+	0.646445i	$0.223745\pi$
$422$	1440.12	0.166123
$423$	0	0
$424$	−6486.06	−0.742903
$425$	0	0
$426$	0	0
$427$	−2267.47	−0.256980
$428$	−14100.8	−1.59250
$429$	0	0
$430$	0	0
$431$	−9159.40	−1.02365	−0.511825	−	0.859090i	$-0.671030\pi$
$431$	−9159.40	−1.02365	−0.511825	+	0.859090i	$0.671030\pi$
$432$	0	0
$433$	270.519	0.0300238	0.0150119	−	0.999887i	$-0.495221\pi$
$433$	270.519	0.0300238	0.0150119	+	0.999887i	$0.495221\pi$
$434$	328.295	0.0363103
$435$	0	0
$436$	−21574.3	−2.36977
$437$	4840.67	0.529887
$438$	0	0
$439$	17008.0	1.84908	0.924541	−	0.381082i	$-0.124448\pi$
$439$	17008.0	1.84908	0.924541	+	0.381082i	$0.124448\pi$
$440$	0	0
$441$	0	0
$442$	4333.66	0.466360
$443$	−3968.12	−0.425578	−0.212789	−	0.977098i	$-0.568255\pi$
$443$	−3968.12	−0.425578	−0.212789	+	0.977098i	$0.568255\pi$
$444$	0	0
$445$	0	0
$446$	30418.7	3.22952
$447$	0	0
$448$	−4265.64	−0.449850
$449$	12480.3	1.31177	0.655884	−	0.754862i	$-0.272296\pi$
$449$	12480.3	1.31177	0.655884	+	0.754862i	$0.272296\pi$
$450$	0	0
$451$	20854.8	2.17741
$452$	−32157.3	−3.34636
$453$	0	0
$454$	−11413.2	−1.17984
$455$	0	0
$456$	0	0
$457$	5782.90	0.591932	0.295966	−	0.955199i	$-0.404359\pi$
$457$	5782.90	0.591932	0.295966	+	0.955199i	$0.404359\pi$
$458$	11991.9	1.22346
$459$	0	0
$460$	0	0
$461$	−10049.6	−1.01531	−0.507653	−	0.861562i	$-0.669487\pi$
$461$	−10049.6	−1.01531	−0.507653	+	0.861562i	$0.669487\pi$
$462$	0	0
$463$	−659.842	−0.0662321	−0.0331161	−	0.999452i	$-0.510543\pi$
$463$	−659.842	−0.0662321	−0.0331161	+	0.999452i	$0.510543\pi$
$464$	3516.03	0.351783
$465$	0	0
$466$	−24898.2	−2.47508
$467$	4467.60	0.442690	0.221345	−	0.975196i	$-0.428955\pi$
$467$	4467.60	0.442690	0.221345	+	0.975196i	$0.428955\pi$
$468$	0	0
$469$	−3793.37	−0.373479
$470$	0	0
$471$	0	0
$472$	9677.87	0.943771
$473$	−3188.07	−0.309910
$474$	0	0
$475$	0	0
$476$	−6865.01	−0.661044
$477$	0	0
$478$	−4024.64	−0.385110
$479$	−7633.87	−0.728185	−0.364092	−	0.931363i	$-0.618621\pi$
$479$	−7633.87	−0.728185	−0.364092	+	0.931363i	$0.618621\pi$
$480$	0	0
$481$	3386.49	0.321020
$482$	−21975.6	−2.07669
$483$	0	0
$484$	39181.5	3.67971
$485$	0	0
$486$	0	0
$487$	6289.84	0.585256	0.292628	−	0.956226i	$-0.405470\pi$
$487$	6289.84	0.585256	0.292628	+	0.956226i	$0.405470\pi$
$488$	11671.2	1.08265
$489$	0	0
$490$	0	0
$491$	−3562.54	−0.327445	−0.163722	−	0.986506i	$-0.552350\pi$
$491$	−3562.54	−0.327445	−0.163722	+	0.986506i	$0.552350\pi$
$492$	0	0
$493$	−4385.85	−0.400667
$494$	−3323.71	−0.302714
$495$	0	0
$496$	−493.190	−0.0446469
$497$	1130.03	0.101990
$498$	0	0
$499$	−18916.9	−1.69706	−0.848532	−	0.529144i	$-0.822513\pi$
$499$	−18916.9	−1.69706	−0.848532	+	0.529144i	$0.822513\pi$
$500$	0	0
$501$	0	0
$502$	35015.0	3.11314
$503$	−2565.91	−0.227452	−0.113726	−	0.993512i	$-0.536279\pi$
$503$	−2565.91	−0.227452	−0.113726	+	0.993512i	$0.536279\pi$
$504$	0	0
$505$	0	0
$506$	29926.2	2.62921
$507$	0	0
$508$	−3562.28	−0.311123
$509$	−5447.84	−0.474403	−0.237202	−	0.971460i	$-0.576230\pi$
$509$	−5447.84	−0.474403	−0.237202	+	0.971460i	$0.576230\pi$
$510$	0	0
$511$	2140.26	0.185283
$512$	16800.8	1.45019
$513$	0	0
$514$	20541.6	1.76274
$515$	0	0
$516$	0	0
$517$	−28085.5	−2.38917
$518$	−8143.83	−0.690771
$519$	0	0
$520$	0	0
$521$	4732.95	0.397993	0.198997	−	0.980000i	$-0.436232\pi$
$521$	4732.95	0.397993	0.198997	+	0.980000i	$0.436232\pi$
$522$	0	0
$523$	−9182.96	−0.767769	−0.383884	−	0.923381i	$-0.625414\pi$
$523$	−9182.96	−0.767769	−0.383884	+	0.923381i	$0.625414\pi$
$524$	15038.4	1.25373
$525$	0	0
$526$	31978.0	2.65078
$527$	615.199	0.0508510
$528$	0	0
$529$	−2290.93	−0.188290
$530$	0	0
$531$	0	0
$532$	5265.13	0.429083
$533$	−4725.59	−0.384030
$534$	0	0
$535$	0	0
$536$	19525.4	1.57345
$537$	0	0
$538$	22331.2	1.78953
$539$	−3047.61	−0.243544
$540$	0	0
$541$	14572.9	1.15811	0.579055	−	0.815288i	$-0.303422\pi$
$541$	14572.9	1.15811	0.579055	+	0.815288i	$0.303422\pi$
$542$	17150.6	1.35919
$543$	0	0
$544$	−2650.51	−0.208896
$545$	0	0
$546$	0	0
$547$	11290.3	0.882519	0.441260	−	0.897380i	$-0.354532\pi$
$547$	11290.3	0.882519	0.441260	+	0.897380i	$0.354532\pi$
$548$	−10743.4	−0.837472
$549$	0	0
$550$	0	0
$551$	3363.73	0.260072
$552$	0	0
$553$	−3533.05	−0.271683
$554$	−94.5826	−0.00725349
$555$	0	0
$556$	−4609.84	−0.351620
$557$	−3919.93	−0.298191	−0.149096	−	0.988823i	$-0.547636\pi$
$557$	−3919.93	−0.298191	−0.149096	+	0.988823i	$0.547636\pi$
$558$	0	0
$559$	722.401	0.0546588
$560$	0	0
$561$	0	0
$562$	32774.2	2.45996
$563$	7444.71	0.557295	0.278647	−	0.960393i	$-0.410114\pi$
$563$	7444.71	0.557295	0.278647	+	0.960393i	$0.410114\pi$
$564$	0	0
$565$	0	0
$566$	−6148.41	−0.456602
$567$	0	0
$568$	−5816.55	−0.429678
$569$	−8529.24	−0.628408	−0.314204	−	0.949355i	$-0.601738\pi$
$569$	−8529.24	−0.628408	−0.314204	+	0.949355i	$0.601738\pi$
$570$	0	0
$571$	10324.8	0.756706	0.378353	−	0.925661i	$-0.376491\pi$
$571$	10324.8	0.756706	0.378353	+	0.925661i	$0.376491\pi$
$572$	−13535.6	−0.989423
$573$	0	0
$574$	11364.1	0.826355
$575$	0	0
$576$	0	0
$577$	6467.10	0.466601	0.233301	−	0.972405i	$-0.425047\pi$
$577$	6467.10	0.466601	0.233301	+	0.972405i	$0.425047\pi$
$578$	4257.88	0.306409
$579$	0	0
$580$	0	0
$581$	3596.87	0.256839
$582$	0	0
$583$	−11196.2	−0.795369
$584$	−11016.5	−0.780589
$585$	0	0
$586$	30743.4	2.16723
$587$	−7114.68	−0.500263	−0.250131	−	0.968212i	$-0.580474\pi$
$587$	−7114.68	−0.500263	−0.250131	+	0.968212i	$0.580474\pi$
$588$	0	0
$589$	−471.828	−0.0330073
$590$	0	0
$591$	0	0
$592$	12234.3	0.849369
$593$	21270.8	1.47299	0.736497	−	0.676441i	$-0.236478\pi$
$593$	21270.8	1.47299	0.736497	+	0.676441i	$0.236478\pi$
$594$	0	0
$595$	0	0
$596$	27672.1	1.90183
$597$	0	0
$598$	−6781.13	−0.463714
$599$	4697.43	0.320420	0.160210	−	0.987083i	$-0.448783\pi$
$599$	4697.43	0.320420	0.160210	+	0.987083i	$0.448783\pi$
$600$	0	0
$601$	8719.81	0.591828	0.295914	−	0.955215i	$-0.404376\pi$
$601$	8719.81	0.591828	0.295914	+	0.955215i	$0.404376\pi$
$602$	−1737.23	−0.117615
$603$	0	0
$604$	18549.7	1.24963
$605$	0	0
$606$	0	0
$607$	−23097.1	−1.54445	−0.772227	−	0.635347i	$-0.780857\pi$
$607$	−23097.1	−1.54445	−0.772227	+	0.635347i	$0.780857\pi$
$608$	2032.81	0.135594
$609$	0	0
$610$	0	0
$611$	6364.05	0.421378
$612$	0	0
$613$	−16846.2	−1.10997	−0.554986	−	0.831860i	$-0.687276\pi$
$613$	−16846.2	−1.10997	−0.554986	+	0.831860i	$0.687276\pi$
$614$	23106.3	1.51872
$615$	0	0
$616$	15686.8	1.02604
$617$	−3476.46	−0.226834	−0.113417	−	0.993547i	$-0.536180\pi$
$617$	−3476.46	−0.226834	−0.113417	+	0.993547i	$0.536180\pi$
$618$	0	0
$619$	−5407.16	−0.351102	−0.175551	−	0.984470i	$-0.556171\pi$
$619$	−5407.16	−0.351102	−0.175551	+	0.984470i	$0.556171\pi$
$620$	0	0
$621$	0	0
$622$	3586.24	0.231182
$623$	−3802.11	−0.244508
$624$	0	0
$625$	0	0
$626$	11037.5	0.704706
$627$	0	0
$628$	6328.02	0.402095
$629$	−15260.9	−0.967396
$630$	0	0
$631$	−10498.7	−0.662358	−0.331179	−	0.943568i	$-0.607446\pi$
$631$	−10498.7	−0.662358	−0.331179	+	0.943568i	$0.607446\pi$
$632$	18185.5	1.14459
$633$	0	0
$634$	−49376.1	−3.09302
$635$	0	0
$636$	0	0
$637$	690.574	0.0429538
$638$	20795.4	1.29044
$639$	0	0
$640$	0	0
$641$	−14363.3	−0.885050	−0.442525	−	0.896756i	$-0.645917\pi$
$641$	−14363.3	−0.885050	−0.442525	+	0.896756i	$0.645917\pi$
$642$	0	0
$643$	16883.3	1.03548	0.517739	−	0.855539i	$-0.326774\pi$
$643$	16883.3	1.03548	0.517739	+	0.855539i	$0.326774\pi$
$644$	10742.1	0.657293
$645$	0	0
$646$	14978.0	0.912231
$647$	27365.2	1.66281	0.831404	−	0.555668i	$-0.187538\pi$
$647$	27365.2	1.66281	0.831404	+	0.555668i	$0.187538\pi$
$648$	0	0
$649$	16705.9	1.01042
$650$	0	0
$651$	0	0
$652$	−31357.8	−1.88353
$653$	−28643.3	−1.71654	−0.858269	−	0.513200i	$-0.828460\pi$
$653$	−28643.3	−1.71654	−0.858269	+	0.513200i	$0.828460\pi$
$654$	0	0
$655$	0	0
$656$	−17072.0	−1.01608
$657$	0	0
$658$	−15304.3	−0.906721
$659$	4102.07	0.242480	0.121240	−	0.992623i	$-0.461313\pi$
$659$	4102.07	0.242480	0.121240	+	0.992623i	$0.461313\pi$
$660$	0	0
$661$	−13784.0	−0.811098	−0.405549	−	0.914073i	$-0.632920\pi$
$661$	−13784.0	−0.811098	−0.405549	+	0.914073i	$0.632920\pi$
$662$	377.834	0.0221827
$663$	0	0
$664$	−18514.0	−1.08205
$665$	0	0
$666$	0	0
$667$	6862.79	0.398393
$668$	−8029.95	−0.465102
$669$	0	0
$670$	0	0
$671$	20146.9	1.15911
$672$	0	0
$673$	4676.98	0.267882	0.133941	−	0.990989i	$-0.457237\pi$
$673$	4676.98	0.267882	0.133941	+	0.990989i	$0.457237\pi$
$674$	20164.2	1.15237
$675$	0	0
$676$	−30858.5	−1.75572
$677$	−3849.54	−0.218537	−0.109269	−	0.994012i	$-0.534851\pi$
$677$	−3849.54	−0.218537	−0.109269	+	0.994012i	$0.534851\pi$
$678$	0	0
$679$	13041.4	0.737091
$680$	0	0
$681$	0	0
$682$	−2916.96	−0.163777
$683$	10647.5	0.596508	0.298254	−	0.954487i	$-0.403596\pi$
$683$	10647.5	0.596508	0.298254	+	0.954487i	$0.403596\pi$
$684$	0	0
$685$	0	0
$686$	−1660.69	−0.0924280
$687$	0	0
$688$	2609.80	0.144619
$689$	2537.01	0.140279
$690$	0	0
$691$	22410.8	1.23379	0.616894	−	0.787046i	$-0.288391\pi$
$691$	22410.8	1.23379	0.616894	+	0.787046i	$0.288391\pi$
$692$	39560.4	2.17321
$693$	0	0
$694$	8691.72	0.475408
$695$	0	0
$696$	0	0
$697$	21295.4	1.15728
$698$	56896.6	3.08534
$699$	0	0
$700$	0	0
$701$	−6040.55	−0.325461	−0.162731	−	0.986671i	$-0.552030\pi$
$701$	−6040.55	−0.325461	−0.162731	+	0.986671i	$0.552030\pi$
$702$	0	0
$703$	11704.4	0.627936
$704$	37901.0	2.02904
$705$	0	0
$706$	13955.2	0.743926
$707$	11799.9	0.627697
$708$	0	0
$709$	19582.2	1.03727	0.518634	−	0.854996i	$-0.326441\pi$
$709$	19582.2	1.03727	0.518634	+	0.854996i	$0.326441\pi$
$710$	0	0
$711$	0	0
$712$	19570.4	1.03010
$713$	−962.637	−0.0505625
$714$	0	0
$715$	0	0
$716$	46975.2	2.45188
$717$	0	0
$718$	−5779.91	−0.300424
$719$	−9257.31	−0.480166	−0.240083	−	0.970752i	$-0.577175\pi$
$719$	−9257.31	−0.480166	−0.240083	+	0.970752i	$0.577175\pi$
$720$	0	0
$721$	−7099.32	−0.366702
$722$	21721.6	1.11966
$723$	0	0
$724$	−56332.0	−2.89166
$725$	0	0
$726$	0	0
$727$	11915.5	0.607870	0.303935	−	0.952693i	$-0.401699\pi$
$727$	11915.5	0.607870	0.303935	+	0.952693i	$0.401699\pi$
$728$	−3554.56	−0.180963
$729$	0	0
$730$	0	0
$731$	−3255.43	−0.164715
$732$	0	0
$733$	−15608.0	−0.786486	−0.393243	−	0.919435i	$-0.628647\pi$
$733$	−15608.0	−0.786486	−0.393243	+	0.919435i	$0.628647\pi$
$734$	−42891.4	−2.15688
$735$	0	0
$736$	4147.40	0.207711
$737$	33704.8	1.68457
$738$	0	0
$739$	−26189.9	−1.30367	−0.651835	−	0.758361i	$-0.726000\pi$
$739$	−26189.9	−1.30367	−0.651835	+	0.758361i	$0.726000\pi$
$740$	0	0
$741$	0	0
$742$	−6101.00	−0.301853
$743$	−37914.9	−1.87209	−0.936044	−	0.351882i	$-0.885542\pi$
$743$	−37914.9	−1.87209	−0.936044	+	0.351882i	$0.885542\pi$
$744$	0	0
$745$	0	0
$746$	11076.7	0.543627
$747$	0	0
$748$	60996.8	2.98164
$749$	−6392.12	−0.311833
$750$	0	0
$751$	−21401.6	−1.03989	−0.519944	−	0.854200i	$-0.674047\pi$
$751$	−21401.6	−1.03989	−0.519944	+	0.854200i	$0.674047\pi$
$752$	22991.3	1.11490
$753$	0	0
$754$	−4712.14	−0.227594
$755$	0	0
$756$	0	0
$757$	24094.8	1.15686	0.578428	−	0.815734i	$-0.303667\pi$
$757$	24094.8	1.15686	0.578428	+	0.815734i	$0.303667\pi$
$758$	−42946.5	−2.05790
$759$	0	0
$760$	0	0
$761$	−19101.1	−0.909872	−0.454936	−	0.890524i	$-0.650338\pi$
$761$	−19101.1	−0.909872	−0.454936	+	0.890524i	$0.650338\pi$
$762$	0	0
$763$	−9779.94	−0.464033
$764$	63174.2	2.99157
$765$	0	0
$766$	64681.0	3.05094
$767$	−3785.48	−0.178208
$768$	0	0
$769$	−32718.1	−1.53426	−0.767129	−	0.641493i	$-0.778315\pi$
$769$	−32718.1	−1.53426	−0.767129	+	0.641493i	$0.778315\pi$
$770$	0	0
$771$	0	0
$772$	47123.8	2.19692
$773$	4707.97	0.219061	0.109530	−	0.993983i	$-0.465065\pi$
$773$	4707.97	0.219061	0.109530	+	0.993983i	$0.465065\pi$
$774$	0	0
$775$	0	0
$776$	−67127.6	−3.10533
$777$	0	0
$778$	54830.3	2.52669
$779$	−16332.6	−0.751187
$780$	0	0
$781$	−10040.5	−0.460023
$782$	30558.5	1.39741
$783$	0	0
$784$	2494.82	0.113649
$785$	0	0
$786$	0	0
$787$	3218.47	0.145776	0.0728882	−	0.997340i	$-0.476778\pi$
$787$	3218.47	0.145776	0.0728882	+	0.997340i	$0.476778\pi$
$788$	−46502.1	−2.10224
$789$	0	0
$790$	0	0
$791$	−14577.4	−0.655262
$792$	0	0
$793$	−4565.18	−0.204432
$794$	64117.4	2.86580
$795$	0	0
$796$	3087.47	0.137478
$797$	15548.9	0.691054	0.345527	−	0.938409i	$-0.387700\pi$
$797$	15548.9	0.691054	0.345527	+	0.938409i	$0.387700\pi$
$798$	0	0
$799$	−28679.0	−1.26982
$800$	0	0
$801$	0	0
$802$	9455.58	0.416319
$803$	−19016.6	−0.835717
$804$	0	0
$805$	0	0
$806$	660.968	0.0288854
$807$	0	0
$808$	−60737.1	−2.64446
$809$	3106.83	0.135019	0.0675095	−	0.997719i	$-0.478495\pi$
$809$	3106.83	0.135019	0.0675095	+	0.997719i	$0.478495\pi$
$810$	0	0
$811$	−44061.1	−1.90776	−0.953882	−	0.300183i	$-0.902952\pi$
$811$	−44061.1	−1.90776	−0.953882	+	0.300183i	$0.902952\pi$
$812$	7464.56	0.322604
$813$	0	0
$814$	72359.3	3.11572
$815$	0	0
$816$	0	0
$817$	2496.76	0.106916
$818$	51310.9	2.19321
$819$	0	0
$820$	0	0
$821$	−13977.3	−0.594169	−0.297084	−	0.954851i	$-0.596014\pi$
$821$	−13977.3	−0.594169	−0.297084	+	0.954851i	$0.596014\pi$
$822$	0	0
$823$	3287.14	0.139225	0.0696127	−	0.997574i	$-0.477824\pi$
$823$	3287.14	0.139225	0.0696127	+	0.997574i	$0.477824\pi$
$824$	36542.0	1.54490
$825$	0	0
$826$	9103.32	0.383469
$827$	2454.69	0.103214	0.0516069	−	0.998667i	$-0.483566\pi$
$827$	2454.69	0.103214	0.0516069	+	0.998667i	$0.483566\pi$
$828$	0	0
$829$	−38510.6	−1.61342	−0.806711	−	0.590946i	$-0.798755\pi$
$829$	−38510.6	−1.61342	−0.806711	+	0.590946i	$0.798755\pi$
$830$	0	0
$831$	0	0
$832$	−8588.18	−0.357862
$833$	−3112.01	−0.129442
$834$	0	0
$835$	0	0
$836$	−46781.6	−1.93538
$837$	0	0
$838$	−7897.64	−0.325560
$839$	−32983.5	−1.35723	−0.678616	−	0.734493i	$-0.737420\pi$
$839$	−32983.5	−1.35723	−0.678616	+	0.734493i	$0.737420\pi$
$840$	0	0
$841$	−19620.1	−0.804466
$842$	−63819.1	−2.61206
$843$	0	0
$844$	−4593.04	−0.187321
$845$	0	0
$846$	0	0
$847$	17761.6	0.720537
$848$	9165.39	0.371157
$849$	0	0
$850$	0	0
$851$	23879.6	0.961906
$852$	0	0
$853$	−13620.1	−0.546710	−0.273355	−	0.961913i	$-0.588133\pi$
$853$	−13620.1	−0.546710	−0.273355	+	0.961913i	$0.588133\pi$
$854$	10978.4	0.439897
$855$	0	0
$856$	32901.9	1.31374
$857$	−24493.2	−0.976279	−0.488139	−	0.872766i	$-0.662324\pi$
$857$	−24493.2	−0.976279	−0.488139	+	0.872766i	$0.662324\pi$
$858$	0	0
$859$	3742.34	0.148646	0.0743231	−	0.997234i	$-0.476320\pi$
$859$	3742.34	0.148646	0.0743231	+	0.997234i	$0.476320\pi$
$860$	0	0
$861$	0	0
$862$	44346.8	1.75227
$863$	−165.238	−0.00651770	−0.00325885	−	0.999995i	$-0.501037\pi$
$863$	−165.238	−0.00651770	−0.00325885	+	0.999995i	$0.501037\pi$
$864$	0	0
$865$	0	0
$866$	−1309.77	−0.0513945
$867$	0	0
$868$	−1047.05	−0.0409437
$869$	31391.8	1.22542
$870$	0	0
$871$	−7637.33	−0.297108
$872$	50339.8	1.95495
$873$	0	0
$874$	−23436.9	−0.907055
$875$	0	0
$876$	0	0
$877$	7230.06	0.278383	0.139192	−	0.990265i	$-0.455550\pi$
$877$	7230.06	0.278383	0.139192	+	0.990265i	$0.455550\pi$
$878$	−82347.2	−3.16524
$879$	0	0
$880$	0	0
$881$	18707.9	0.715422	0.357711	−	0.933832i	$-0.383557\pi$
$881$	18707.9	0.715422	0.357711	+	0.933832i	$0.383557\pi$
$882$	0	0
$883$	−28766.9	−1.09636	−0.548179	−	0.836361i	$-0.684679\pi$
$883$	−28766.9	−1.09636	−0.548179	+	0.836361i	$0.684679\pi$
$884$	−13821.6	−0.525871
$885$	0	0
$886$	19212.3	0.728500
$887$	33980.4	1.28630	0.643152	−	0.765738i	$-0.277626\pi$
$887$	33980.4	1.28630	0.643152	+	0.765738i	$0.277626\pi$
$888$	0	0
$889$	−1614.84	−0.0609222
$890$	0	0
$891$	0	0
$892$	−97015.8	−3.64163
$893$	21995.4	0.824242
$894$	0	0
$895$	0	0
$896$	18315.8	0.682910
$897$	0	0
$898$	−60425.7	−2.24547
$899$	−668.927	−0.0248164
$900$	0	0
$901$	−11432.8	−0.422732
$902$	−100972.	−3.72727
$903$	0	0
$904$	75033.5	2.76059
$905$	0	0
$906$	0	0
$907$	−35753.2	−1.30889	−0.654446	−	0.756109i	$-0.727098\pi$
$907$	−35753.2	−1.30889	−0.654446	+	0.756109i	$0.727098\pi$
$908$	36400.6	1.33039
$909$	0	0
$910$	0	0
$911$	−16039.7	−0.583334	−0.291667	−	0.956520i	$-0.594210\pi$
$911$	−16039.7	−0.583334	−0.291667	+	0.956520i	$0.594210\pi$
$912$	0	0
$913$	−31958.8	−1.15847
$914$	−27998.9	−1.01326
$915$	0	0
$916$	−38246.5	−1.37958
$917$	6817.14	0.245498
$918$	0	0
$919$	22104.9	0.793441	0.396720	−	0.917940i	$-0.370148\pi$
$919$	22104.9	0.793441	0.396720	+	0.917940i	$0.370148\pi$
$920$	0	0
$921$	0	0
$922$	48656.8	1.73799
$923$	2275.13	0.0811342
$924$	0	0
$925$	0	0
$926$	3194.74	0.113376
$927$	0	0
$928$	2881.99	0.101946
$929$	−20234.8	−0.714621	−0.357310	−	0.933986i	$-0.616306\pi$
$929$	−20234.8	−0.714621	−0.357310	+	0.933986i	$0.616306\pi$
$930$	0	0
$931$	2386.76	0.0840204
$932$	79409.1	2.79091
$933$	0	0
$934$	−21630.7	−0.757792
$935$	0	0
$936$	0	0
$937$	−37551.2	−1.30923	−0.654614	−	0.755964i	$-0.727169\pi$
$937$	−37551.2	−1.30923	−0.654614	+	0.755964i	$0.727169\pi$
$938$	18366.3	0.639318
$939$	0	0
$940$	0	0
$941$	−19093.3	−0.661449	−0.330725	−	0.943727i	$-0.607293\pi$
$941$	−19093.3	−0.661449	−0.330725	+	0.943727i	$0.607293\pi$
$942$	0	0
$943$	−33322.2	−1.15071
$944$	−13675.7	−0.471511
$945$	0	0
$946$	15435.6	0.530501
$947$	10117.0	0.347158	0.173579	−	0.984820i	$-0.444467\pi$
$947$	10117.0	0.347158	0.173579	+	0.984820i	$0.444467\pi$
$948$	0	0
$949$	4309.06	0.147395
$950$	0	0
$951$	0	0
$952$	16018.3	0.545332
$953$	−6272.48	−0.213206	−0.106603	−	0.994302i	$-0.533997\pi$
$953$	−6272.48	−0.213206	−0.106603	+	0.994302i	$0.533997\pi$
$954$	0	0
$955$	0	0
$956$	12836.0	0.434253
$957$	0	0
$958$	36960.7	1.24650
$959$	−4870.14	−0.163988
$960$	0	0
$961$	−29697.2	−0.996850
$962$	−16396.3	−0.549519
$963$	0	0
$964$	70088.1	2.34169
$965$	0	0
$966$	0	0
$967$	26660.8	0.886613	0.443307	−	0.896370i	$-0.353805\pi$
$967$	26660.8	0.886613	0.443307	+	0.896370i	$0.353805\pi$
$968$	−91423.3	−3.03560
$969$	0	0
$970$	0	0
$971$	461.190	0.0152423	0.00762116	−	0.999971i	$-0.497574\pi$
$971$	461.190	0.0152423	0.00762116	+	0.999971i	$0.497574\pi$
$972$	0	0
$973$	−2089.71	−0.0688520
$974$	−30453.3	−1.00184
$975$	0	0
$976$	−16492.5	−0.540895
$977$	4058.08	0.132886	0.0664429	−	0.997790i	$-0.478835\pi$
$977$	4058.08	0.132886	0.0664429	+	0.997790i	$0.478835\pi$
$978$	0	0
$979$	33782.4	1.10285
$980$	0	0
$981$	0	0
$982$	17248.7	0.560517
$983$	−3259.92	−0.105773	−0.0528867	−	0.998601i	$-0.516842\pi$
$983$	−3259.92	−0.105773	−0.0528867	+	0.998601i	$0.516842\pi$
$984$	0	0
$985$	0	0
$986$	21234.8	0.685857
$987$	0	0
$988$	10600.5	0.341342
$989$	5093.96	0.163780
$990$	0	0
$991$	−21398.9	−0.685933	−0.342967	−	0.939348i	$-0.611432\pi$
$991$	−21398.9	−0.685933	−0.342967	+	0.939348i	$0.611432\pi$
$992$	−404.254	−0.0129386
$993$	0	0
$994$	−5471.24	−0.174585
$995$	0	0
$996$	0	0
$997$	−28609.5	−0.908798	−0.454399	−	0.890798i	$-0.650146\pi$
$997$	−28609.5	−0.908798	−0.454399	+	0.890798i	$0.650146\pi$
$998$	91589.2	2.90502
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.4.a.bg.1.1		4
3.2	odd	2		175.4.a.h.1.4	yes	4
5.4	even	2		1575.4.a.bl.1.4		4
15.2	even	4		175.4.b.f.99.7		8
15.8	even	4		175.4.b.f.99.2		8
15.14	odd	2		175.4.a.g.1.1	✓	4
21.20	even	2		1225.4.a.bd.1.4		4
105.104	even	2		1225.4.a.z.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
175.4.a.g.1.1	✓	4	15.14	odd	2
175.4.a.h.1.4	yes	4	3.2	odd	2
175.4.b.f.99.2		8	15.8	even	4
175.4.b.f.99.7		8	15.2	even	4
1225.4.a.z.1.1		4	105.104	even	2
1225.4.a.bd.1.4		4	21.20	even	2
1575.4.a.bg.1.1		4	1.1	even	1	trivial
1575.4.a.bl.1.4		4	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-4.84167 q^{2} +15.4418 q^{4} +7.00000 q^{7} -36.0308 q^{8} -62.1962 q^{11} +14.0934 q^{13} -33.8917 q^{14} +50.9148 q^{16} -63.5104 q^{17} +48.7094 q^{19} +301.134 q^{22} +99.3784 q^{23} -68.2354 q^{26} +108.093 q^{28} +69.0571 q^{29} -9.68658 q^{31} +41.7334 q^{32} +307.497 q^{34} +240.290 q^{37} -235.835 q^{38} -335.306 q^{41} +51.2582 q^{43} -960.421 q^{44} -481.158 q^{46} +451.564 q^{47} +49.0000 q^{49} +217.627 q^{52} +180.014 q^{53} -252.215 q^{56} -334.352 q^{58} -268.600 q^{59} -323.925 q^{61} +46.8992 q^{62} -609.378 q^{64} -541.910 q^{67} -980.715 q^{68} +161.433 q^{71} +305.751 q^{73} -1163.40 q^{74} +752.161 q^{76} -435.373 q^{77} -504.722 q^{79} +1623.44 q^{82} +513.838 q^{83} -248.176 q^{86} +2240.98 q^{88} -543.158 q^{89} +98.6535 q^{91} +1534.58 q^{92} -2186.32 q^{94} +1863.06 q^{97} -237.242 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 36 q^{4} + 28 q^{7} - 27 q^{8} - 100 q^{11} + 44 q^{13} - 28 q^{14} + 160 q^{16} + 53 q^{17} - 29 q^{19} - 152 q^{22} - 295 q^{23} - 700 q^{26} + 252 q^{28} - 129 q^{29} + 114 q^{31} + 310 q^{32}+ \cdots - 196 q^{98}+O(q^{100}) \) 4 * q - 4 * q^2 + 36 * q^4 + 28 * q^7 - 27 * q^8 - 100 * q^11 + 44 * q^13 - 28 * q^14 + 160 * q^16 + 53 * q^17 - 29 * q^19 - 152 * q^22 - 295 * q^23 - 700 * q^26 + 252 * q^28 - 129 * q^29 + 114 * q^31 + 310 * q^32 + 203 * q^34 + 403 * q^37 - 555 * q^38 - 671 * q^41 - 411 * q^43 - 438 * q^44 - 997 * q^46 + 8 * q^47 + 196 * q^49 + 74 * q^52 - 90 * q^53 - 189 * q^56 + 673 * q^58 - 1018 * q^59 + 50 * q^61 + 1626 * q^62 - 2421 * q^64 + 424 * q^67 + 617 * q^68 - 215 * q^71 - 1207 * q^73 - 623 * q^74 - 3257 * q^76 - 700 * q^77 - 951 * q^79 + 1695 * q^82 + 3035 * q^83 + 99 * q^86 + 163 * q^88 - 2819 * q^89 + 308 * q^91 - 3073 * q^92 - 3056 * q^94 - 1100 * q^97 - 196 * q^98

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