LMFDB - Embedded newform 1575.4.a.w.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 = [2,1,0,17,0,0,14,33,0,0,22] 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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{65}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 16 $ x^2 - x - 16
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.53113$ 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)

$f(q)$	$=$	$q-3.53113 q^{2} +4.46887 q^{4} +7.00000 q^{7} +12.4689 q^{8} +2.93774 q^{11} +19.0623 q^{13} -24.7179 q^{14} -79.7802 q^{16} +122.498 q^{17} +107.436 q^{19} -10.3735 q^{22} +210.623 q^{23} -67.3113 q^{26} +31.2821 q^{28} -95.4942 q^{29} -94.3074 q^{31} +181.963 q^{32} -432.556 q^{34} -97.1206 q^{37} -379.370 q^{38} +491.113 q^{41} +43.0039 q^{43} +13.1284 q^{44} -743.735 q^{46} +473.494 q^{47} +49.0000 q^{49} +85.1868 q^{52} -183.677 q^{53} +87.2821 q^{56} +337.202 q^{58} +760.615 q^{59} -198.747 q^{61} +333.012 q^{62} -4.29373 q^{64} +309.992 q^{67} +547.428 q^{68} -665.693 q^{71} -621.288 q^{73} +342.945 q^{74} +480.117 q^{76} +20.5642 q^{77} -24.7626 q^{79} -1734.18 q^{82} -406.724 q^{83} -151.852 q^{86} +36.6303 q^{88} -261.751 q^{89} +133.436 q^{91} +941.245 q^{92} -1671.97 q^{94} +1004.77 q^{97} -173.025 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} + 17 q^{4} + 14 q^{7} + 33 q^{8} + 22 q^{11} + 22 q^{13} + 7 q^{14} - 87 q^{16} + 116 q^{17} + 102 q^{19} + 76 q^{22} + 260 q^{23} - 54 q^{26} + 119 q^{28} + 196 q^{29} + 150 q^{31} - 15 q^{32}+ \cdots + 49 q^{98}+O(q^{100}) $ 2 * q + q^2 + 17 * q^4 + 14 * q^7 + 33 * q^8 + 22 * q^11 + 22 * q^13 + 7 * q^14 - 87 * q^16 + 116 * q^17 + 102 * q^19 + 76 * q^22 + 260 * q^23 - 54 * q^26 + 119 * q^28 + 196 * q^29 + 150 * q^31 - 15 * q^32 - 462 * q^34 + 96 * q^37 - 404 * q^38 + 176 * q^41 + 344 * q^43 + 252 * q^44 - 520 * q^46 + 560 * q^47 + 98 * q^49 + 122 * q^52 + 326 * q^53 + 231 * q^56 + 1658 * q^58 + 844 * q^59 - 204 * q^61 + 1440 * q^62 - 839 * q^64 + 104 * q^67 + 466 * q^68 - 1670 * q^71 + 386 * q^73 + 1218 * q^74 + 412 * q^76 + 154 * q^77 - 888 * q^79 - 3162 * q^82 + 928 * q^83 + 1212 * q^86 + 428 * q^88 - 588 * q^89 + 154 * q^91 + 1560 * q^92 - 1280 * q^94 - 522 * q^97 + 49 * 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 $

$2$	−3.53113	−1.24844	−0.624221	−	0.781248i	$-0.714584\pi$
$2$	−3.53113	−1.24844	−0.624221	+	0.781248i	$0.714584\pi$
$3$	0	0
$3$	0	0
$4$	4.46887	0.558609
$5$	0	0
$5$	0	0
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	12.4689	0.551051
$9$	0	0
$10$	0	0
$11$	2.93774	0.0805239	0.0402619	−	0.999189i	$-0.487181\pi$
$11$	2.93774	0.0805239	0.0402619	+	0.999189i	$0.487181\pi$
$12$	0	0
$13$	19.0623	0.406686	0.203343	−	0.979108i	$-0.434819\pi$
$13$	19.0623	0.406686	0.203343	+	0.979108i	$0.434819\pi$
$14$	−24.7179	−0.471867
$15$	0	0
$16$	−79.7802	−1.24656
$17$	122.498	1.74766	0.873828	−	0.486236i	$-0.161630\pi$
$17$	122.498	1.74766	0.873828	+	0.486236i	$0.161630\pi$
$18$	0	0
$19$	107.436	1.29723	0.648617	−	0.761115i	$-0.275348\pi$
$19$	107.436	1.29723	0.648617	+	0.761115i	$0.275348\pi$
$20$	0	0
$21$	0	0
$22$	−10.3735	−0.100529
$23$	210.623	1.90947	0.954736	−	0.297455i	$-0.0961379\pi$
$23$	210.623	1.90947	0.954736	+	0.297455i	$0.0961379\pi$
$24$	0	0
$25$	0	0
$26$	−67.3113	−0.507724
$27$	0	0
$28$	31.2821	0.211134
$29$	−95.4942	−0.611477	−0.305738	−	0.952116i	$-0.598903\pi$
$29$	−95.4942	−0.611477	−0.305738	+	0.952116i	$0.598903\pi$
$30$	0	0
$31$	−94.3074	−0.546391	−0.273195	−	0.961959i	$-0.588081\pi$
$31$	−94.3074	−0.546391	−0.273195	+	0.961959i	$0.588081\pi$
$32$	181.963	1.00521
$33$	0	0
$34$	−432.556	−2.18185
$35$	0	0
$36$	0	0
$37$	−97.1206	−0.431528	−0.215764	−	0.976446i	$-0.569224\pi$
$37$	−97.1206	−0.431528	−0.215764	+	0.976446i	$0.569224\pi$
$38$	−379.370	−1.61952
$39$	0	0
$40$	0	0
$41$	491.113	1.87071	0.935353	−	0.353716i	$-0.115082\pi$
$41$	491.113	1.87071	0.935353	+	0.353716i	$0.115082\pi$
$42$	0	0
$43$	43.0039	0.152512	0.0762562	−	0.997088i	$-0.475703\pi$
$43$	43.0039	0.152512	0.0762562	+	0.997088i	$0.475703\pi$
$44$	13.1284	0.0449814
$45$	0	0
$46$	−743.735	−2.38387
$47$	473.494	1.46949	0.734747	−	0.678341i	$-0.237301\pi$
$47$	473.494	1.46949	0.734747	+	0.678341i	$0.237301\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	0	0
$52$	85.1868	0.227178
$53$	−183.677	−0.476038	−0.238019	−	0.971261i	$-0.576498\pi$
$53$	−183.677	−0.476038	−0.238019	+	0.971261i	$0.576498\pi$
$54$	0	0
$55$	0	0
$56$	87.2821	0.208278
$57$	0	0
$58$	337.202	0.763394
$59$	760.615	1.67837	0.839183	−	0.543849i	$-0.183034\pi$
$59$	760.615	1.67837	0.839183	+	0.543849i	$0.183034\pi$
$60$	0	0
$61$	−198.747	−0.417163	−0.208582	−	0.978005i	$-0.566885\pi$
$61$	−198.747	−0.417163	−0.208582	+	0.978005i	$0.566885\pi$
$62$	333.012	0.682137
$63$	0	0
$64$	−4.29373	−0.00838618
$65$	0	0
$66$	0	0
$67$	309.992	0.565247	0.282624	−	0.959231i	$-0.408795\pi$
$67$	309.992	0.565247	0.282624	+	0.959231i	$0.408795\pi$
$68$	547.428	0.976256
$69$	0	0
$70$	0	0
$71$	−665.693	−1.11272	−0.556360	−	0.830941i	$-0.687803\pi$
$71$	−665.693	−1.11272	−0.556360	+	0.830941i	$0.687803\pi$
$72$	0	0
$73$	−621.288	−0.996113	−0.498057	−	0.867145i	$-0.665953\pi$
$73$	−621.288	−0.996113	−0.498057	+	0.867145i	$0.665953\pi$
$74$	342.945	0.538738
$75$	0	0
$76$	480.117	0.724647
$77$	20.5642	0.0304352
$78$	0	0
$79$	−24.7626	−0.0352659	−0.0176330	−	0.999845i	$-0.505613\pi$
$79$	−24.7626	−0.0352659	−0.0176330	+	0.999845i	$0.505613\pi$
$80$	0	0
$81$	0	0
$82$	−1734.18	−2.33547
$83$	−406.724	−0.537876	−0.268938	−	0.963157i	$-0.586673\pi$
$83$	−406.724	−0.537876	−0.268938	+	0.963157i	$0.586673\pi$
$84$	0	0
$85$	0	0
$86$	−151.852	−0.190403
$87$	0	0
$88$	36.6303	0.0443728
$89$	−261.751	−0.311748	−0.155874	−	0.987777i	$-0.549819\pi$
$89$	−261.751	−0.311748	−0.155874	+	0.987777i	$0.549819\pi$
$90$	0	0
$91$	133.436	0.153713
$92$	941.245	1.06665
$93$	0	0
$94$	−1671.97	−1.83458
$95$	0	0
$96$	0	0
$97$	1004.77	1.05175	0.525873	−	0.850563i	$-0.323739\pi$
$97$	1004.77	1.05175	0.525873	+	0.850563i	$0.323739\pi$
$98$	−173.025	−0.178349
$99$	0	0
$100$	0	0
$101$	128.872	0.126962	0.0634812	−	0.997983i	$-0.479780\pi$
$101$	128.872	0.126962	0.0634812	+	0.997983i	$0.479780\pi$
$102$	0	0
$103$	−806.008	−0.771051	−0.385526	−	0.922697i	$-0.625980\pi$
$103$	−806.008	−0.771051	−0.385526	+	0.922697i	$0.625980\pi$
$104$	237.685	0.224105
$105$	0	0
$106$	648.587	0.594305
$107$	−769.712	−0.695429	−0.347714	−	0.937600i	$-0.613042\pi$
$107$	−769.712	−0.695429	−0.347714	+	0.937600i	$0.613042\pi$
$108$	0	0
$109$	−780.856	−0.686169	−0.343085	−	0.939304i	$-0.611472\pi$
$109$	−780.856	−0.686169	−0.343085	+	0.939304i	$0.611472\pi$
$110$	0	0
$111$	0	0
$112$	−558.461	−0.471157
$113$	−1115.65	−0.928771	−0.464386	−	0.885633i	$-0.653725\pi$
$113$	−1115.65	−0.928771	−0.464386	+	0.885633i	$0.653725\pi$
$114$	0	0
$115$	0	0
$116$	−426.751	−0.341576
$117$	0	0
$118$	−2685.83	−2.09534
$119$	857.486	0.660552
$120$	0	0
$121$	−1322.37	−0.993516
$122$	701.802	0.520804
$123$	0	0
$124$	−421.448	−0.305219
$125$	0	0
$126$	0	0
$127$	1875.98	1.31076	0.655381	−	0.755299i	$-0.272508\pi$
$127$	1875.98	1.31076	0.655381	+	0.755299i	$0.272508\pi$
$128$	−1440.54	−0.994744
$129$	0	0
$130$	0	0
$131$	−364.203	−0.242905	−0.121452	−	0.992597i	$-0.538755\pi$
$131$	−364.203	−0.242905	−0.121452	+	0.992597i	$0.538755\pi$
$132$	0	0
$133$	752.051	0.490309
$134$	−1094.62	−0.705679
$135$	0	0
$136$	1527.41	0.963048
$137$	1603.13	0.999743	0.499872	−	0.866099i	$-0.333380\pi$
$137$	1603.13	0.999743	0.499872	+	0.866099i	$0.333380\pi$
$138$	0	0
$139$	2431.12	1.48349	0.741746	−	0.670681i	$-0.233998\pi$
$139$	2431.12	1.48349	0.741746	+	0.670681i	$0.233998\pi$
$140$	0	0
$141$	0	0
$142$	2350.65	1.38917
$143$	56.0000	0.0327479
$144$	0	0
$145$	0	0
$146$	2193.85	1.24359
$147$	0	0
$148$	−434.020	−0.241055
$149$	−2341.57	−1.28744	−0.643722	−	0.765260i	$-0.722611\pi$
$149$	−2341.57	−1.28744	−0.643722	+	0.765260i	$0.722611\pi$
$150$	0	0
$151$	−2104.07	−1.13395	−0.566976	−	0.823734i	$-0.691887\pi$
$151$	−2104.07	−1.13395	−0.566976	+	0.823734i	$0.691887\pi$
$152$	1339.60	0.714843
$153$	0	0
$154$	−72.6148	−0.0379966
$155$	0	0
$156$	0	0
$157$	593.467	0.301680	0.150840	−	0.988558i	$-0.451802\pi$
$157$	593.467	0.301680	0.150840	+	0.988558i	$0.451802\pi$
$158$	87.4399	0.0440275
$159$	0	0
$160$	0	0
$161$	1474.36	0.721712
$162$	0	0
$163$	−2178.71	−1.04693	−0.523465	−	0.852047i	$-0.675361\pi$
$163$	−2178.71	−1.04693	−0.523465	+	0.852047i	$0.675361\pi$
$164$	2194.72	1.04499
$165$	0	0
$166$	1436.19	0.671508
$167$	799.502	0.370463	0.185231	−	0.982695i	$-0.440697\pi$
$167$	799.502	0.370463	0.185231	+	0.982695i	$0.440697\pi$
$168$	0	0
$169$	−1833.63	−0.834606
$170$	0	0
$171$	0	0
$172$	192.179	0.0851947
$173$	−1444.36	−0.634754	−0.317377	−	0.948299i	$-0.602802\pi$
$173$	−1444.36	−0.634754	−0.317377	+	0.948299i	$0.602802\pi$
$174$	0	0
$175$	0	0
$176$	−234.374	−0.100378
$177$	0	0
$178$	924.276	0.389199
$179$	−3343.49	−1.39611	−0.698056	−	0.716043i	$-0.745952\pi$
$179$	−3343.49	−1.39611	−0.698056	+	0.716043i	$0.745952\pi$
$180$	0	0
$181$	2251.81	0.924729	0.462365	−	0.886690i	$-0.347001\pi$
$181$	2251.81	0.924729	0.462365	+	0.886690i	$0.347001\pi$
$182$	−471.179	−0.191902
$183$	0	0
$184$	2626.23	1.05222
$185$	0	0
$186$	0	0
$187$	359.868	0.140728
$188$	2115.98	0.820873
$189$	0	0
$190$	0	0
$191$	1001.93	0.379565	0.189782	−	0.981826i	$-0.439222\pi$
$191$	1001.93	0.379565	0.189782	+	0.981826i	$0.439222\pi$
$192$	0	0
$193$	4054.97	1.51235	0.756173	−	0.654372i	$-0.227067\pi$
$193$	4054.97	1.51235	0.756173	+	0.654372i	$0.227067\pi$
$194$	−3547.99	−1.31304
$195$	0	0
$196$	218.975	0.0798013
$197$	−5140.23	−1.85902	−0.929508	−	0.368802i	$-0.879768\pi$
$197$	−5140.23	−1.85902	−0.929508	+	0.368802i	$0.879768\pi$
$198$	0	0
$199$	585.631	0.208614	0.104307	−	0.994545i	$-0.466737\pi$
$199$	585.631	0.208614	0.104307	+	0.994545i	$0.466737\pi$
$200$	0	0
$201$	0	0
$202$	−455.062	−0.158505
$203$	−668.459	−0.231116
$204$	0	0
$205$	0	0
$206$	2846.12	0.962614
$207$	0	0
$208$	−1520.79	−0.506961
$209$	315.619	0.104458
$210$	0	0
$211$	−1055.16	−0.344266	−0.172133	−	0.985074i	$-0.555066\pi$
$211$	−1055.16	−0.344266	−0.172133	+	0.985074i	$0.555066\pi$
$212$	−820.829	−0.265919
$213$	0	0
$214$	2717.95	0.868203
$215$	0	0
$216$	0	0
$217$	−660.152	−0.206516
$218$	2757.30	0.856643
$219$	0	0
$220$	0	0
$221$	2335.09	0.710747
$222$	0	0
$223$	−4675.85	−1.40412	−0.702059	−	0.712119i	$-0.747736\pi$
$223$	−4675.85	−1.40412	−0.702059	+	0.712119i	$0.747736\pi$
$224$	1273.74	0.379935
$225$	0	0
$226$	3939.49	1.15952
$227$	5443.11	1.59151	0.795754	−	0.605621i	$-0.207075\pi$
$227$	5443.11	1.59151	0.795754	+	0.605621i	$0.207075\pi$
$228$	0	0
$229$	−536.303	−0.154759	−0.0773797	−	0.997002i	$-0.524655\pi$
$229$	−536.303	−0.154759	−0.0773797	+	0.997002i	$0.524655\pi$
$230$	0	0
$231$	0	0
$232$	−1190.70	−0.336955
$233$	−183.490	−0.0515916	−0.0257958	−	0.999667i	$-0.508212\pi$
$233$	−183.490	−0.0515916	−0.0257958	+	0.999667i	$0.508212\pi$
$234$	0	0
$235$	0	0
$236$	3399.09	0.937550
$237$	0	0
$238$	−3027.90	−0.824661
$239$	−643.218	−0.174085	−0.0870425	−	0.996205i	$-0.527742\pi$
$239$	−643.218	−0.174085	−0.0870425	+	0.996205i	$0.527742\pi$
$240$	0	0
$241$	−5755.61	−1.53839	−0.769194	−	0.639015i	$-0.779342\pi$
$241$	−5755.61	−1.53839	−0.769194	+	0.639015i	$0.779342\pi$
$242$	4669.46	1.24035
$243$	0	0
$244$	−888.175	−0.233031
$245$	0	0
$246$	0	0
$247$	2047.97	0.527567
$248$	−1175.91	−0.301089
$249$	0	0
$250$	0	0
$251$	5132.27	1.29062	0.645311	−	0.763920i	$-0.276728\pi$
$251$	5132.27	1.29062	0.645311	+	0.763920i	$0.276728\pi$
$252$	0	0
$253$	618.755	0.153758
$254$	−6624.34	−1.63641
$255$	0	0
$256$	5121.09	1.25027
$257$	5041.74	1.22372	0.611859	−	0.790967i	$-0.290422\pi$
$257$	5041.74	1.22372	0.611859	+	0.790967i	$0.290422\pi$
$258$	0	0
$259$	−679.844	−0.163102
$260$	0	0
$261$	0	0
$262$	1286.05	0.303253
$263$	7577.00	1.77649	0.888246	−	0.459367i	$-0.151924\pi$
$263$	7577.00	1.77649	0.888246	+	0.459367i	$0.151924\pi$
$264$	0	0
$265$	0	0
$266$	−2655.59	−0.612122
$267$	0	0
$268$	1385.32	0.315752
$269$	−1023.10	−0.231893	−0.115947	−	0.993255i	$-0.536990\pi$
$269$	−1023.10	−0.231893	−0.115947	+	0.993255i	$0.536990\pi$
$270$	0	0
$271$	−2251.98	−0.504790	−0.252395	−	0.967624i	$-0.581218\pi$
$271$	−2251.98	−0.504790	−0.252395	+	0.967624i	$0.581218\pi$
$272$	−9772.91	−2.17857
$273$	0	0
$274$	−5660.87	−1.24812
$275$	0	0
$276$	0	0
$277$	8630.72	1.87209	0.936047	−	0.351875i	$-0.114456\pi$
$277$	8630.72	1.87209	0.936047	+	0.351875i	$0.114456\pi$
$278$	−8584.61	−1.85205
$279$	0	0
$280$	0	0
$281$	7521.62	1.59680	0.798402	−	0.602124i	$-0.205679\pi$
$281$	7521.62	1.59680	0.798402	+	0.602124i	$0.205679\pi$
$282$	0	0
$283$	−14.8169	−0.00311226	−0.00155613	−	0.999999i	$-0.500495\pi$
$283$	−14.8169	−0.00311226	−0.00155613	+	0.999999i	$0.500495\pi$
$284$	−2974.89	−0.621576
$285$	0	0
$286$	−197.743	−0.0408839
$287$	3437.79	0.707060
$288$	0	0
$289$	10092.8	2.05430
$290$	0	0
$291$	0	0
$292$	−2776.46	−0.556438
$293$	6913.39	1.37844	0.689222	−	0.724550i	$-0.257952\pi$
$293$	6913.39	1.37844	0.689222	+	0.724550i	$0.257952\pi$
$294$	0	0
$295$	0	0
$296$	−1210.98	−0.237794
$297$	0	0
$298$	8268.39	1.60730
$299$	4014.94	0.776555
$300$	0	0
$301$	301.027	0.0576442
$302$	7429.74	1.41567
$303$	0	0
$304$	−8571.25	−1.61709
$305$	0	0
$306$	0	0
$307$	7644.12	1.42108	0.710542	−	0.703655i	$-0.248450\pi$
$307$	7644.12	1.42108	0.710542	+	0.703655i	$0.248450\pi$
$308$	91.8987	0.0170014
$309$	0	0
$310$	0	0
$311$	−7593.99	−1.38462	−0.692308	−	0.721602i	$-0.743406\pi$
$311$	−7593.99	−1.38462	−0.692308	+	0.721602i	$0.743406\pi$
$312$	0	0
$313$	9127.84	1.64836	0.824179	−	0.566329i	$-0.191637\pi$
$313$	9127.84	1.64836	0.824179	+	0.566329i	$0.191637\pi$
$314$	−2095.61	−0.376631
$315$	0	0
$316$	−110.661	−0.0196999
$317$	−4929.81	−0.873456	−0.436728	−	0.899593i	$-0.643863\pi$
$317$	−4929.81	−0.873456	−0.436728	+	0.899593i	$0.643863\pi$
$318$	0	0
$319$	−280.537	−0.0492385
$320$	0	0
$321$	0	0
$322$	−5206.15	−0.901016
$323$	13160.7	2.26712
$324$	0	0
$325$	0	0
$326$	7693.30	1.30703
$327$	0	0
$328$	6123.62	1.03086
$329$	3314.46	0.555417
$330$	0	0
$331$	1221.67	0.202867	0.101433	−	0.994842i	$-0.467657\pi$
$331$	1221.67	0.202867	0.101433	+	0.994842i	$0.467657\pi$
$332$	−1817.60	−0.300463
$333$	0	0
$334$	−2823.14	−0.462502
$335$	0	0
$336$	0	0
$337$	8744.83	1.41354	0.706768	−	0.707446i	$-0.250153\pi$
$337$	8744.83	1.41354	0.706768	+	0.707446i	$0.250153\pi$
$338$	6474.79	1.04196
$339$	0	0
$340$	0	0
$341$	−277.051	−0.0439975
$342$	0	0
$343$	343.000	0.0539949
$344$	536.210	0.0840421
$345$	0	0
$346$	5100.21	0.792454
$347$	4589.56	0.710031	0.355015	−	0.934860i	$-0.384476\pi$
$347$	4589.56	0.710031	0.355015	+	0.934860i	$0.384476\pi$
$348$	0	0
$349$	−3989.89	−0.611960	−0.305980	−	0.952038i	$-0.598984\pi$
$349$	−3989.89	−0.611960	−0.305980	+	0.952038i	$0.598984\pi$
$350$	0	0
$351$	0	0
$352$	534.561	0.0809437
$353$	2416.35	0.364333	0.182166	−	0.983268i	$-0.441689\pi$
$353$	2416.35	0.364333	0.182166	+	0.983268i	$0.441689\pi$
$354$	0	0
$355$	0	0
$356$	−1169.73	−0.174145
$357$	0	0
$358$	11806.3	1.74297
$359$	2756.24	0.405206	0.202603	−	0.979261i	$-0.435060\pi$
$359$	2756.24	0.405206	0.202603	+	0.979261i	$0.435060\pi$
$360$	0	0
$361$	4683.45	0.682818
$362$	−7951.44	−1.15447
$363$	0	0
$364$	596.307	0.0858654
$365$	0	0
$366$	0	0
$367$	−11112.8	−1.58061	−0.790307	−	0.612711i	$-0.790079\pi$
$367$	−11112.8	−1.58061	−0.790307	+	0.612711i	$0.790079\pi$
$368$	−16803.5	−2.38028
$369$	0	0
$370$	0	0
$371$	−1285.74	−0.179925
$372$	0	0
$373$	−6091.09	−0.845535	−0.422768	−	0.906238i	$-0.638941\pi$
$373$	−6091.09	−0.845535	−0.422768	+	0.906238i	$0.638941\pi$
$374$	−1270.74	−0.175691
$375$	0	0
$376$	5903.94	0.809767
$377$	−1820.33	−0.248679
$378$	0	0
$379$	3984.29	0.539998	0.269999	−	0.962861i	$-0.412977\pi$
$379$	3984.29	0.539998	0.269999	+	0.962861i	$0.412977\pi$
$380$	0	0
$381$	0	0
$382$	−3537.93	−0.473865
$383$	318.475	0.0424890	0.0212445	−	0.999774i	$-0.493237\pi$
$383$	318.475	0.0424890	0.0212445	+	0.999774i	$0.493237\pi$
$384$	0	0
$385$	0	0
$386$	−14318.6	−1.88808
$387$	0	0
$388$	4490.21	0.587515
$389$	−3885.46	−0.506429	−0.253214	−	0.967410i	$-0.581488\pi$
$389$	−3885.46	−0.506429	−0.253214	+	0.967410i	$0.581488\pi$
$390$	0	0
$391$	25800.9	3.33710
$392$	610.975	0.0787216
$393$	0	0
$394$	18150.8	2.32088
$395$	0	0
$396$	0	0
$397$	−4806.04	−0.607578	−0.303789	−	0.952739i	$-0.598252\pi$
$397$	−4806.04	−0.607578	−0.303789	+	0.952739i	$0.598252\pi$
$398$	−2067.94	−0.260443
$399$	0	0
$400$	0	0
$401$	−3618.59	−0.450633	−0.225316	−	0.974286i	$-0.572342\pi$
$401$	−3618.59	−0.450633	−0.225316	+	0.974286i	$0.572342\pi$
$402$	0	0
$403$	−1797.71	−0.222209
$404$	575.911	0.0709223
$405$	0	0
$406$	2360.42	0.288536
$407$	−285.315	−0.0347483
$408$	0	0
$409$	−2109.05	−0.254978	−0.127489	−	0.991840i	$-0.540692\pi$
$409$	−2109.05	−0.254978	−0.127489	+	0.991840i	$0.540692\pi$
$410$	0	0
$411$	0	0
$412$	−3601.94	−0.430716
$413$	5324.30	0.634363
$414$	0	0
$415$	0	0
$416$	3468.63	0.408806
$417$	0	0
$418$	−1114.49	−0.130410
$419$	6905.91	0.805193	0.402597	−	0.915377i	$-0.368108\pi$
$419$	6905.91	0.805193	0.402597	+	0.915377i	$0.368108\pi$
$420$	0	0
$421$	−9647.54	−1.11685	−0.558423	−	0.829556i	$-0.688593\pi$
$421$	−9647.54	−1.11685	−0.558423	+	0.829556i	$0.688593\pi$
$422$	3725.91	0.429797
$423$	0	0
$424$	−2290.25	−0.262321
$425$	0	0
$426$	0	0
$427$	−1391.23	−0.157673
$428$	−3439.74	−0.388473
$429$	0	0
$430$	0	0
$431$	13002.7	1.45318	0.726589	−	0.687073i	$-0.241105\pi$
$431$	13002.7	1.45318	0.726589	+	0.687073i	$0.241105\pi$
$432$	0	0
$433$	−7356.07	−0.816420	−0.408210	−	0.912888i	$-0.633847\pi$
$433$	−7356.07	−0.816420	−0.408210	+	0.912888i	$0.633847\pi$
$434$	2331.08	0.257824
$435$	0	0
$436$	−3489.55	−0.383300
$437$	22628.4	2.47703
$438$	0	0
$439$	6909.21	0.751159	0.375579	−	0.926790i	$-0.377444\pi$
$439$	6909.21	0.751159	0.375579	+	0.926790i	$0.377444\pi$
$440$	0	0
$441$	0	0
$442$	−8245.50	−0.887327
$443$	−14812.6	−1.58864	−0.794318	−	0.607502i	$-0.792172\pi$
$443$	−14812.6	−1.58864	−0.794318	+	0.607502i	$0.792172\pi$
$444$	0	0
$445$	0	0
$446$	16511.0	1.75296
$447$	0	0
$448$	−30.0561	−0.00316968
$449$	10654.5	1.11986	0.559932	−	0.828538i	$-0.310827\pi$
$449$	10654.5	1.11986	0.559932	+	0.828538i	$0.310827\pi$
$450$	0	0
$451$	1442.76	0.150636
$452$	−4985.68	−0.518820
$453$	0	0
$454$	−19220.3	−1.98691
$455$	0	0
$456$	0	0
$457$	5855.16	0.599328	0.299664	−	0.954045i	$-0.403125\pi$
$457$	5855.16	0.599328	0.299664	+	0.954045i	$0.403125\pi$
$458$	1893.76	0.193208
$459$	0	0
$460$	0	0
$461$	−3204.74	−0.323774	−0.161887	−	0.986809i	$-0.551758\pi$
$461$	−3204.74	−0.323774	−0.161887	+	0.986809i	$0.551758\pi$
$462$	0	0
$463$	−371.658	−0.0373054	−0.0186527	−	0.999826i	$-0.505938\pi$
$463$	−371.658	−0.0373054	−0.0186527	+	0.999826i	$0.505938\pi$
$464$	7618.54	0.762245
$465$	0	0
$466$	647.927	0.0644091
$467$	−19752.3	−1.95723	−0.978614	−	0.205703i	$-0.934052\pi$
$467$	−19752.3	−1.95723	−0.978614	+	0.205703i	$0.934052\pi$
$468$	0	0
$469$	2169.95	0.213643
$470$	0	0
$471$	0	0
$472$	9484.01	0.924866
$473$	126.334	0.0122809
$474$	0	0
$475$	0	0
$476$	3832.00	0.368990
$477$	0	0
$478$	2271.28	0.217335
$479$	20762.0	1.98046	0.990232	−	0.139433i	$-0.0445279\pi$
$479$	20762.0	1.98046	0.990232	+	0.139433i	$0.0445279\pi$
$480$	0	0
$481$	−1851.34	−0.175496
$482$	20323.8	1.92059
$483$	0	0
$484$	−5909.50	−0.554987
$485$	0	0
$486$	0	0
$487$	−17647.6	−1.64207	−0.821035	−	0.570878i	$-0.806603\pi$
$487$	−17647.6	−1.64207	−0.821035	+	0.570878i	$0.806603\pi$
$488$	−2478.15	−0.229878
$489$	0	0
$490$	0	0
$491$	5637.46	0.518157	0.259078	−	0.965856i	$-0.416581\pi$
$491$	5637.46	0.518157	0.259078	+	0.965856i	$0.416581\pi$
$492$	0	0
$493$	−11697.9	−1.06865
$494$	−7231.64	−0.658638
$495$	0	0
$496$	7523.86	0.681112
$497$	−4659.85	−0.420569
$498$	0	0
$499$	−17474.1	−1.56764	−0.783818	−	0.620991i	$-0.786730\pi$
$499$	−17474.1	−1.56764	−0.783818	+	0.620991i	$0.786730\pi$
$500$	0	0
$501$	0	0
$502$	−18122.7	−1.61127
$503$	7444.81	0.659936	0.329968	−	0.943992i	$-0.392962\pi$
$503$	7444.81	0.659936	0.329968	+	0.943992i	$0.392962\pi$
$504$	0	0
$505$	0	0
$506$	−2184.90	−0.191958
$507$	0	0
$508$	8383.53	0.732203
$509$	3384.48	0.294724	0.147362	−	0.989083i	$-0.452922\pi$
$509$	3384.48	0.294724	0.147362	+	0.989083i	$0.452922\pi$
$510$	0	0
$511$	−4349.02	−0.376495
$512$	−6558.89	−0.566142
$513$	0	0
$514$	−17803.0	−1.52774
$515$	0	0
$516$	0	0
$517$	1391.00	0.118329
$518$	2400.62	0.203624
$519$	0	0
$520$	0	0
$521$	−2973.12	−0.250009	−0.125005	−	0.992156i	$-0.539895\pi$
$521$	−2973.12	−0.250009	−0.125005	+	0.992156i	$0.539895\pi$
$522$	0	0
$523$	−2689.02	−0.224823	−0.112412	−	0.993662i	$-0.535858\pi$
$523$	−2689.02	−0.224823	−0.112412	+	0.993662i	$0.535858\pi$
$524$	−1627.57	−0.135689
$525$	0	0
$526$	−26755.3	−2.21785
$527$	−11552.5	−0.954903
$528$	0	0
$529$	32194.9	2.64608
$530$	0	0
$531$	0	0
$532$	3360.82	0.273891
$533$	9361.72	0.760790
$534$	0	0
$535$	0	0
$536$	3865.25	0.311480
$537$	0	0
$538$	3612.69	0.289506
$539$	143.949	0.0115034
$540$	0	0
$541$	−14429.5	−1.14671	−0.573356	−	0.819306i	$-0.694359\pi$
$541$	−14429.5	−1.14671	−0.573356	+	0.819306i	$0.694359\pi$
$542$	7952.03	0.630201
$543$	0	0
$544$	22290.1	1.75677
$545$	0	0
$546$	0	0
$547$	−13811.2	−1.07957	−0.539784	−	0.841804i	$-0.681494\pi$
$547$	−13811.2	−1.07957	−0.539784	+	0.841804i	$0.681494\pi$
$548$	7164.19	0.558466
$549$	0	0
$550$	0	0
$551$	−10259.5	−0.793229
$552$	0	0
$553$	−173.338	−0.0133293
$554$	−30476.2	−2.33720
$555$	0	0
$556$	10864.4	0.828692
$557$	−6033.26	−0.458954	−0.229477	−	0.973314i	$-0.573702\pi$
$557$	−6033.26	−0.458954	−0.229477	+	0.973314i	$0.573702\pi$
$558$	0	0
$559$	819.751	0.0620246
$560$	0	0
$561$	0	0
$562$	−26559.8	−1.99352
$563$	−6958.47	−0.520896	−0.260448	−	0.965488i	$-0.583870\pi$
$563$	−6958.47	−0.520896	−0.260448	+	0.965488i	$0.583870\pi$
$564$	0	0
$565$	0	0
$566$	52.3202	0.00388548
$567$	0	0
$568$	−8300.44	−0.613166
$569$	13396.4	0.987009	0.493505	−	0.869743i	$-0.335716\pi$
$569$	13396.4	0.987009	0.493505	+	0.869743i	$0.335716\pi$
$570$	0	0
$571$	−8055.84	−0.590414	−0.295207	−	0.955433i	$-0.595389\pi$
$571$	−8055.84	−0.590414	−0.295207	+	0.955433i	$0.595389\pi$
$572$	250.257	0.0182933
$573$	0	0
$574$	−12139.3	−0.882724
$575$	0	0
$576$	0	0
$577$	21456.9	1.54812	0.774059	−	0.633114i	$-0.218223\pi$
$577$	21456.9	1.54812	0.774059	+	0.633114i	$0.218223\pi$
$578$	−35638.9	−2.56468
$579$	0	0
$580$	0	0
$581$	−2847.07	−0.203298
$582$	0	0
$583$	−539.596	−0.0383324
$584$	−7746.76	−0.548910
$585$	0	0
$586$	−24412.1	−1.72091
$587$	20156.3	1.41728	0.708638	−	0.705572i	$-0.249310\pi$
$587$	20156.3	1.41728	0.708638	+	0.705572i	$0.249310\pi$
$588$	0	0
$589$	−10132.0	−0.708797
$590$	0	0
$591$	0	0
$592$	7748.30	0.537928
$593$	599.307	0.0415018	0.0207509	−	0.999785i	$-0.493394\pi$
$593$	599.307	0.0415018	0.0207509	+	0.999785i	$0.493394\pi$
$594$	0	0
$595$	0	0
$596$	−10464.2	−0.719177
$597$	0	0
$598$	−14177.3	−0.969485
$599$	5493.05	0.374691	0.187346	−	0.982294i	$-0.440012\pi$
$599$	5493.05	0.374691	0.187346	+	0.982294i	$0.440012\pi$
$600$	0	0
$601$	24292.8	1.64879	0.824396	−	0.566014i	$-0.191515\pi$
$601$	24292.8	1.64879	0.824396	+	0.566014i	$0.191515\pi$
$602$	−1062.97	−0.0719655
$603$	0	0
$604$	−9402.82	−0.633436
$605$	0	0
$606$	0	0
$607$	−3029.50	−0.202576	−0.101288	−	0.994857i	$-0.532296\pi$
$607$	−3029.50	−0.202576	−0.101288	+	0.994857i	$0.532296\pi$
$608$	19549.3	1.30400
$609$	0	0
$610$	0	0
$611$	9025.87	0.597623
$612$	0	0
$613$	19339.6	1.27426	0.637129	−	0.770757i	$-0.280122\pi$
$613$	19339.6	1.27426	0.637129	+	0.770757i	$0.280122\pi$
$614$	−26992.4	−1.77414
$615$	0	0
$616$	256.412	0.0167713
$617$	−5743.91	−0.374783	−0.187391	−	0.982285i	$-0.560003\pi$
$617$	−5743.91	−0.374783	−0.187391	+	0.982285i	$0.560003\pi$
$618$	0	0
$619$	−8243.35	−0.535264	−0.267632	−	0.963521i	$-0.586241\pi$
$619$	−8243.35	−0.535264	−0.267632	+	0.963521i	$0.586241\pi$
$620$	0	0
$621$	0	0
$622$	26815.4	1.72861
$623$	−1832.26	−0.117830
$624$	0	0
$625$	0	0
$626$	−32231.6	−2.05788
$627$	0	0
$628$	2652.13	0.168521
$629$	−11897.1	−0.754162
$630$	0	0
$631$	−4376.56	−0.276114	−0.138057	−	0.990424i	$-0.544086\pi$
$631$	−4376.56	−0.276114	−0.138057	+	0.990424i	$0.544086\pi$
$632$	−308.762	−0.0194334
$633$	0	0
$634$	17407.8	1.09046
$635$	0	0
$636$	0	0
$637$	934.051	0.0580980
$638$	990.613	0.0614714
$639$	0	0
$640$	0	0
$641$	−11836.6	−0.729357	−0.364678	−	0.931133i	$-0.618821\pi$
$641$	−11836.6	−0.729357	−0.364678	+	0.931133i	$0.618821\pi$
$642$	0	0
$643$	−1448.21	−0.0888209	−0.0444104	−	0.999013i	$-0.514141\pi$
$643$	−1448.21	−0.0888209	−0.0444104	+	0.999013i	$0.514141\pi$
$644$	6588.72	0.403155
$645$	0	0
$646$	−46472.0	−2.83037
$647$	−8732.95	−0.530646	−0.265323	−	0.964160i	$-0.585479\pi$
$647$	−8732.95	−0.530646	−0.265323	+	0.964160i	$0.585479\pi$
$648$	0	0
$649$	2234.49	0.135149
$650$	0	0
$651$	0	0
$652$	−9736.37	−0.584824
$653$	−21978.4	−1.31712	−0.658562	−	0.752527i	$-0.728835\pi$
$653$	−21978.4	−1.31712	−0.658562	+	0.752527i	$0.728835\pi$
$654$	0	0
$655$	0	0
$656$	−39181.1	−2.33196
$657$	0	0
$658$	−11703.8	−0.693406
$659$	27761.7	1.64103	0.820516	−	0.571623i	$-0.193686\pi$
$659$	27761.7	1.64103	0.820516	+	0.571623i	$0.193686\pi$
$660$	0	0
$661$	−8573.72	−0.504507	−0.252254	−	0.967661i	$-0.581172\pi$
$661$	−8573.72	−0.504507	−0.252254	+	0.967661i	$0.581172\pi$
$662$	−4313.86	−0.253267
$663$	0	0
$664$	−5071.39	−0.296398
$665$	0	0
$666$	0	0
$667$	−20113.2	−1.16760
$668$	3572.87	0.206944
$669$	0	0
$670$	0	0
$671$	−583.868	−0.0335916
$672$	0	0
$673$	−27159.2	−1.55559	−0.777795	−	0.628518i	$-0.783662\pi$
$673$	−27159.2	−1.55559	−0.777795	+	0.628518i	$0.783662\pi$
$674$	−30879.1	−1.76472
$675$	0	0
$676$	−8194.26	−0.466219
$677$	−1392.30	−0.0790404	−0.0395202	−	0.999219i	$-0.512583\pi$
$677$	−1392.30	−0.0790404	−0.0395202	+	0.999219i	$0.512583\pi$
$678$	0	0
$679$	7033.42	0.397523
$680$	0	0
$681$	0	0
$682$	978.302	0.0549283
$683$	−8675.09	−0.486007	−0.243004	−	0.970025i	$-0.578133\pi$
$683$	−8675.09	−0.486007	−0.243004	+	0.970025i	$0.578133\pi$
$684$	0	0
$685$	0	0
$686$	−1211.18	−0.0674096
$687$	0	0
$688$	−3430.86	−0.190117
$689$	−3501.30	−0.193598
$690$	0	0
$691$	−21426.0	−1.17957	−0.589785	−	0.807561i	$-0.700787\pi$
$691$	−21426.0	−1.17957	−0.589785	+	0.807561i	$0.700787\pi$
$692$	−6454.65	−0.354579
$693$	0	0
$694$	−16206.3	−0.886433
$695$	0	0
$696$	0	0
$697$	60160.4	3.26935
$698$	14088.8	0.763997
$699$	0	0
$700$	0	0
$701$	−24840.5	−1.33839	−0.669197	−	0.743085i	$-0.733362\pi$
$701$	−24840.5	−1.33839	−0.669197	+	0.743085i	$0.733362\pi$
$702$	0	0
$703$	−10434.2	−0.559793
$704$	−12.6139	−0.000675288 0
$705$	0	0
$706$	−8532.45	−0.454848
$707$	902.101	0.0479873
$708$	0	0
$709$	12525.0	0.663450	0.331725	−	0.943376i	$-0.392369\pi$
$709$	12525.0	0.663450	0.331725	+	0.943376i	$0.392369\pi$
$710$	0	0
$711$	0	0
$712$	−3263.74	−0.171789
$713$	−19863.3	−1.04332
$714$	0	0
$715$	0	0
$716$	−14941.6	−0.779881
$717$	0	0
$718$	−9732.66	−0.505877
$719$	−28085.0	−1.45674	−0.728369	−	0.685185i	$-0.759721\pi$
$719$	−28085.0	−1.45674	−0.728369	+	0.685185i	$0.759721\pi$
$720$	0	0
$721$	−5642.05	−0.291430
$722$	−16537.9	−0.852460
$723$	0	0
$724$	10063.1	0.516562
$725$	0	0
$726$	0	0
$727$	14326.2	0.730851	0.365426	−	0.930841i	$-0.380923\pi$
$727$	14326.2	0.730851	0.365426	+	0.930841i	$0.380923\pi$
$728$	1663.79	0.0847037
$729$	0	0
$730$	0	0
$731$	5267.89	0.266539
$732$	0	0
$733$	−6727.85	−0.339016	−0.169508	−	0.985529i	$-0.554218\pi$
$733$	−6727.85	−0.339016	−0.169508	+	0.985529i	$0.554218\pi$
$734$	39240.8	1.97331
$735$	0	0
$736$	38325.5	1.91943
$737$	910.677	0.0455159
$738$	0	0
$739$	−3418.51	−0.170165	−0.0850826	−	0.996374i	$-0.527115\pi$
$739$	−3418.51	−0.170165	−0.0850826	+	0.996374i	$0.527115\pi$
$740$	0	0
$741$	0	0
$742$	4540.11	0.224626
$743$	8095.50	0.399724	0.199862	−	0.979824i	$-0.435951\pi$
$743$	8095.50	0.399724	0.199862	+	0.979824i	$0.435951\pi$
$744$	0	0
$745$	0	0
$746$	21508.4	1.05560
$747$	0	0
$748$	1608.20	0.0786119
$749$	−5387.99	−0.262847
$750$	0	0
$751$	13446.8	0.653371	0.326686	−	0.945133i	$-0.394068\pi$
$751$	13446.8	0.653371	0.326686	+	0.945133i	$0.394068\pi$
$752$	−37775.4	−1.83182
$753$	0	0
$754$	6427.84	0.310462
$755$	0	0
$756$	0	0
$757$	2593.24	0.124508	0.0622541	−	0.998060i	$-0.480171\pi$
$757$	2593.24	0.124508	0.0622541	+	0.998060i	$0.480171\pi$
$758$	−14069.0	−0.674156
$759$	0	0
$760$	0	0
$761$	−27079.4	−1.28992	−0.644959	−	0.764217i	$-0.723125\pi$
$761$	−27079.4	−1.28992	−0.644959	+	0.764217i	$0.723125\pi$
$762$	0	0
$763$	−5465.99	−0.259348
$764$	4477.48	0.212028
$765$	0	0
$766$	−1124.58	−0.0530451
$767$	14499.0	0.682568
$768$	0	0
$769$	2138.72	0.100292	0.0501458	−	0.998742i	$-0.484031\pi$
$769$	2138.72	0.100292	0.0501458	+	0.998742i	$0.484031\pi$
$770$	0	0
$771$	0	0
$772$	18121.1	0.844810
$773$	25864.0	1.20345	0.601724	−	0.798704i	$-0.294481\pi$
$773$	25864.0	1.20345	0.601724	+	0.798704i	$0.294481\pi$
$774$	0	0
$775$	0	0
$776$	12528.4	0.579566
$777$	0	0
$778$	13720.1	0.632247
$779$	52763.1	2.42675
$780$	0	0
$781$	−1955.63	−0.0896006
$782$	−91106.2	−4.16618
$783$	0	0
$784$	−3909.23	−0.178081
$785$	0	0
$786$	0	0
$787$	32371.3	1.46621	0.733107	−	0.680113i	$-0.238069\pi$
$787$	32371.3	1.46621	0.733107	+	0.680113i	$0.238069\pi$
$788$	−22971.0	−1.03846
$789$	0	0
$790$	0	0
$791$	−7809.52	−0.351043
$792$	0	0
$793$	−3788.57	−0.169654
$794$	16970.8	0.758526
$795$	0	0
$796$	2617.11	0.116534
$797$	2024.33	0.0899691	0.0449845	−	0.998988i	$-0.485676\pi$
$797$	2024.33	0.0899691	0.0449845	+	0.998988i	$0.485676\pi$
$798$	0	0
$799$	58002.1	2.56817
$800$	0	0
$801$	0	0
$802$	12777.7	0.562589
$803$	−1825.18	−0.0802109
$804$	0	0
$805$	0	0
$806$	6347.95	0.277416
$807$	0	0
$808$	1606.88	0.0699628
$809$	−12391.7	−0.538526	−0.269263	−	0.963067i	$-0.586780\pi$
$809$	−12391.7	−0.538526	−0.269263	+	0.963067i	$0.586780\pi$
$810$	0	0
$811$	14654.5	0.634511	0.317256	−	0.948340i	$-0.397239\pi$
$811$	14654.5	0.634511	0.317256	+	0.948340i	$0.397239\pi$
$812$	−2987.26	−0.129104
$813$	0	0
$814$	1007.49	0.0433813
$815$	0	0
$816$	0	0
$817$	4620.16	0.197844
$818$	7447.33	0.318325
$819$	0	0
$820$	0	0
$821$	−23887.9	−1.01546	−0.507731	−	0.861516i	$-0.669515\pi$
$821$	−23887.9	−1.01546	−0.507731	+	0.861516i	$0.669515\pi$
$822$	0	0
$823$	4008.41	0.169774	0.0848871	−	0.996391i	$-0.472947\pi$
$823$	4008.41	0.169774	0.0848871	+	0.996391i	$0.472947\pi$
$824$	−10050.0	−0.424889
$825$	0	0
$826$	−18800.8	−0.791966
$827$	−45110.4	−1.89679	−0.948394	−	0.317096i	$-0.897292\pi$
$827$	−45110.4	−1.89679	−0.948394	+	0.317096i	$0.897292\pi$
$828$	0	0
$829$	16165.4	0.677260	0.338630	−	0.940920i	$-0.390036\pi$
$829$	16165.4	0.677260	0.338630	+	0.940920i	$0.390036\pi$
$830$	0	0
$831$	0	0
$832$	−81.8481	−0.00341054
$833$	6002.41	0.249665
$834$	0	0
$835$	0	0
$836$	1410.46	0.0583514
$837$	0	0
$838$	−24385.7	−1.00524
$839$	25244.4	1.03878	0.519388	−	0.854538i	$-0.326160\pi$
$839$	25244.4	1.03878	0.519388	+	0.854538i	$0.326160\pi$
$840$	0	0
$841$	−15269.9	−0.626096
$842$	34066.7	1.39432
$843$	0	0
$844$	−4715.37	−0.192310
$845$	0	0
$846$	0	0
$847$	−9256.59	−0.375514
$848$	14653.8	0.593412
$849$	0	0
$850$	0	0
$851$	−20455.8	−0.823990
$852$	0	0
$853$	30168.1	1.21094	0.605472	−	0.795867i	$-0.292984\pi$
$853$	30168.1	1.21094	0.605472	+	0.795867i	$0.292984\pi$
$854$	4912.61	0.196846
$855$	0	0
$856$	−9597.44	−0.383217
$857$	−13393.6	−0.533857	−0.266929	−	0.963716i	$-0.586009\pi$
$857$	−13393.6	−0.533857	−0.266929	+	0.963716i	$0.586009\pi$
$858$	0	0
$859$	19060.4	0.757081	0.378541	−	0.925585i	$-0.376426\pi$
$859$	19060.4	0.757081	0.378541	+	0.925585i	$0.376426\pi$
$860$	0	0
$861$	0	0
$862$	−45914.3	−1.81421
$863$	−9466.86	−0.373413	−0.186707	−	0.982416i	$-0.559781\pi$
$863$	−9466.86	−0.373413	−0.186707	+	0.982416i	$0.559781\pi$
$864$	0	0
$865$	0	0
$866$	25975.2	1.01925
$867$	0	0
$868$	−2950.13	−0.115362
$869$	−72.7461	−0.00283975
$870$	0	0
$871$	5909.15	0.229878
$872$	−9736.39	−0.378115
$873$	0	0
$874$	−79903.8	−3.09243
$875$	0	0
$876$	0	0
$877$	−37740.6	−1.45315	−0.726573	−	0.687090i	$-0.758888\pi$
$877$	−37740.6	−1.45315	−0.726573	+	0.687090i	$0.758888\pi$
$878$	−24397.3	−0.937778
$879$	0	0
$880$	0	0
$881$	−25991.5	−0.993957	−0.496979	−	0.867763i	$-0.665557\pi$
$881$	−25991.5	−0.993957	−0.496979	+	0.867763i	$0.665557\pi$
$882$	0	0
$883$	−39420.3	−1.50238	−0.751189	−	0.660087i	$-0.770519\pi$
$883$	−39420.3	−1.50238	−0.751189	+	0.660087i	$0.770519\pi$
$884$	10435.2	0.397030
$885$	0	0
$886$	52305.0	1.98332
$887$	46005.2	1.74149	0.870745	−	0.491735i	$-0.163637\pi$
$887$	46005.2	1.74149	0.870745	+	0.491735i	$0.163637\pi$
$888$	0	0
$889$	13131.9	0.495421
$890$	0	0
$891$	0	0
$892$	−20895.8	−0.784353
$893$	50870.2	1.90628
$894$	0	0
$895$	0	0
$896$	−10083.8	−0.375978
$897$	0	0
$898$	−37622.6	−1.39809
$899$	9005.81	0.334105
$900$	0	0
$901$	−22500.1	−0.831950
$902$	−5094.58	−0.188061
$903$	0	0
$904$	−13910.8	−0.511801
$905$	0	0
$906$	0	0
$907$	2838.97	0.103932	0.0519661	−	0.998649i	$-0.483451\pi$
$907$	2838.97	0.103932	0.0519661	+	0.998649i	$0.483451\pi$
$908$	24324.6	0.889030
$909$	0	0
$910$	0	0
$911$	−39890.9	−1.45076	−0.725382	−	0.688347i	$-0.758337\pi$
$911$	−39890.9	−1.45076	−0.725382	+	0.688347i	$0.758337\pi$
$912$	0	0
$913$	−1194.85	−0.0433119
$914$	−20675.3	−0.748226
$915$	0	0
$916$	−2396.67	−0.0864500
$917$	−2549.42	−0.0918094
$918$	0	0
$919$	−646.475	−0.0232048	−0.0116024	−	0.999933i	$-0.503693\pi$
$919$	−646.475	−0.0232048	−0.0116024	+	0.999933i	$0.503693\pi$
$920$	0	0
$921$	0	0
$922$	11316.3	0.404213
$923$	−12689.6	−0.452528
$924$	0	0
$925$	0	0
$926$	1312.37	0.0465737
$927$	0	0
$928$	−17376.4	−0.614665
$929$	−51188.2	−1.80778	−0.903892	−	0.427760i	$-0.859303\pi$
$929$	−51188.2	−1.80778	−0.903892	+	0.427760i	$0.859303\pi$
$930$	0	0
$931$	5264.35	0.185319
$932$	−819.993	−0.0288195
$933$	0	0
$934$	69747.8	2.44349
$935$	0	0
$936$	0	0
$937$	−29786.1	−1.03849	−0.519247	−	0.854624i	$-0.673788\pi$
$937$	−29786.1	−1.03849	−0.519247	+	0.854624i	$0.673788\pi$
$938$	−7662.36	−0.266722
$939$	0	0
$940$	0	0
$941$	44817.4	1.55261	0.776304	−	0.630358i	$-0.217092\pi$
$941$	44817.4	1.55261	0.776304	+	0.630358i	$0.217092\pi$
$942$	0	0
$943$	103439.	3.57206
$944$	−60682.0	−2.09219
$945$	0	0
$946$	−446.103	−0.0153320
$947$	54697.1	1.87689	0.938446	−	0.345425i	$-0.112265\pi$
$947$	54697.1	1.87689	0.938446	+	0.345425i	$0.112265\pi$
$948$	0	0
$949$	−11843.2	−0.405105
$950$	0	0
$951$	0	0
$952$	10691.9	0.363998
$953$	−7577.51	−0.257565	−0.128783	−	0.991673i	$-0.541107\pi$
$953$	−7577.51	−0.257565	−0.128783	+	0.991673i	$0.541107\pi$
$954$	0	0
$955$	0	0
$956$	−2874.46	−0.0972454
$957$	0	0
$958$	−73313.4	−2.47249
$959$	11221.9	0.377868
$960$	0	0
$961$	−20897.1	−0.701457
$962$	6537.32	0.219097
$963$	0	0
$964$	−25721.1	−0.859357
$965$	0	0
$966$	0	0
$967$	−50779.0	−1.68867	−0.844334	−	0.535817i	$-0.820004\pi$
$967$	−50779.0	−1.68867	−0.844334	+	0.535817i	$0.820004\pi$
$968$	−16488.5	−0.547478
$969$	0	0
$970$	0	0
$971$	15313.2	0.506102	0.253051	−	0.967453i	$-0.418566\pi$
$971$	15313.2	0.506102	0.253051	+	0.967453i	$0.418566\pi$
$972$	0	0
$973$	17017.9	0.560707
$974$	62315.9	2.05003
$975$	0	0
$976$	15856.1	0.520021
$977$	46620.4	1.52663	0.763316	−	0.646025i	$-0.223570\pi$
$977$	46620.4	1.52663	0.763316	+	0.646025i	$0.223570\pi$
$978$	0	0
$979$	−768.957	−0.0251031
$980$	0	0
$981$	0	0
$982$	−19906.6	−0.646889
$983$	−2824.37	−0.0916414	−0.0458207	−	0.998950i	$-0.514590\pi$
$983$	−2824.37	−0.0916414	−0.0458207	+	0.998950i	$0.514590\pi$
$984$	0	0
$985$	0	0
$986$	41306.6	1.33415
$987$	0	0
$988$	9152.11	0.294704
$989$	9057.59	0.291218
$990$	0	0
$991$	16951.4	0.543370	0.271685	−	0.962386i	$-0.412419\pi$
$991$	16951.4	0.543370	0.271685	+	0.962386i	$0.412419\pi$
$992$	−17160.5	−0.549239
$993$	0	0
$994$	16454.5	0.525056
$995$	0	0
$996$	0	0
$997$	−23847.8	−0.757540	−0.378770	−	0.925491i	$-0.623653\pi$
$997$	−23847.8	−0.757540	−0.378770	+	0.925491i	$0.623653\pi$
$998$	61703.4	1.95710
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.4.a.w.1.1		2
3.2	odd	2		525.4.a.k.1.2		2
5.4	even	2		315.4.a.i.1.2		2
15.2	even	4		525.4.d.h.274.3		4
15.8	even	4		525.4.d.h.274.2		4
15.14	odd	2		105.4.a.f.1.1	✓	2
35.34	odd	2		2205.4.a.z.1.2		2
60.59	even	2		1680.4.a.bg.1.1		2
105.104	even	2		735.4.a.p.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.a.f.1.1	✓	2	15.14	odd	2
315.4.a.i.1.2		2	5.4	even	2
525.4.a.k.1.2		2	3.2	odd	2
525.4.d.h.274.2		4	15.8	even	4
525.4.d.h.274.3		4	15.2	even	4
735.4.a.p.1.1		2	105.104	even	2
1575.4.a.w.1.1		2	1.1	even	1	trivial
1680.4.a.bg.1.1		2	60.59	even	2
2205.4.a.z.1.2		2	35.34	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-3.53113 q^{2} +4.46887 q^{4} +7.00000 q^{7} +12.4689 q^{8} +2.93774 q^{11} +19.0623 q^{13} -24.7179 q^{14} -79.7802 q^{16} +122.498 q^{17} +107.436 q^{19} -10.3735 q^{22} +210.623 q^{23} -67.3113 q^{26} +31.2821 q^{28} -95.4942 q^{29} -94.3074 q^{31} +181.963 q^{32} -432.556 q^{34} -97.1206 q^{37} -379.370 q^{38} +491.113 q^{41} +43.0039 q^{43} +13.1284 q^{44} -743.735 q^{46} +473.494 q^{47} +49.0000 q^{49} +85.1868 q^{52} -183.677 q^{53} +87.2821 q^{56} +337.202 q^{58} +760.615 q^{59} -198.747 q^{61} +333.012 q^{62} -4.29373 q^{64} +309.992 q^{67} +547.428 q^{68} -665.693 q^{71} -621.288 q^{73} +342.945 q^{74} +480.117 q^{76} +20.5642 q^{77} -24.7626 q^{79} -1734.18 q^{82} -406.724 q^{83} -151.852 q^{86} +36.6303 q^{88} -261.751 q^{89} +133.436 q^{91} +941.245 q^{92} -1671.97 q^{94} +1004.77 q^{97} -173.025 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} + 17 q^{4} + 14 q^{7} + 33 q^{8} + 22 q^{11} + 22 q^{13} + 7 q^{14} - 87 q^{16} + 116 q^{17} + 102 q^{19} + 76 q^{22} + 260 q^{23} - 54 q^{26} + 119 q^{28} + 196 q^{29} + 150 q^{31} - 15 q^{32}+ \cdots + 49 q^{98}+O(q^{100}) \) 2 * q + q^2 + 17 * q^4 + 14 * q^7 + 33 * q^8 + 22 * q^11 + 22 * q^13 + 7 * q^14 - 87 * q^16 + 116 * q^17 + 102 * q^19 + 76 * q^22 + 260 * q^23 - 54 * q^26 + 119 * q^28 + 196 * q^29 + 150 * q^31 - 15 * q^32 - 462 * q^34 + 96 * q^37 - 404 * q^38 + 176 * q^41 + 344 * q^43 + 252 * q^44 - 520 * q^46 + 560 * q^47 + 98 * q^49 + 122 * q^52 + 326 * q^53 + 231 * q^56 + 1658 * q^58 + 844 * q^59 - 204 * q^61 + 1440 * q^62 - 839 * q^64 + 104 * q^67 + 466 * q^68 - 1670 * q^71 + 386 * q^73 + 1218 * q^74 + 412 * q^76 + 154 * q^77 - 888 * q^79 - 3162 * q^82 + 928 * q^83 + 1212 * q^86 + 428 * q^88 - 588 * q^89 + 154 * q^91 + 1560 * q^92 - 1280 * q^94 - 522 * q^97 + 49 * q^98

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