LMFDB - Embedded newform 1575.4.a.z.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 = [2,8,0,20,0,0,14,48,0,0,14] 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:	$\Q(\zeta_{8})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 35)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ 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)

$f(q)$	$=$	$q+2.58579 q^{2} -1.31371 q^{4} +7.00000 q^{7} -24.0833 q^{8} -38.2548 q^{11} -19.3431 q^{13} +18.1005 q^{14} -51.7645 q^{16} -87.2254 q^{17} -44.2254 q^{19} -98.9188 q^{22} +218.167 q^{23} -50.0172 q^{26} -9.19596 q^{28} +46.9411 q^{29} +194.558 q^{31} +58.8141 q^{32} -225.546 q^{34} -366.853 q^{37} -114.357 q^{38} +339.362 q^{41} +226.167 q^{43} +50.2557 q^{44} +564.132 q^{46} +11.6762 q^{47} +49.0000 q^{49} +25.4113 q^{52} -209.019 q^{53} -168.583 q^{56} +121.380 q^{58} +616.000 q^{59} +320.735 q^{61} +503.087 q^{62} +566.197 q^{64} -14.5097 q^{67} +114.589 q^{68} +952.000 q^{71} -824.489 q^{73} -948.603 q^{74} +58.0993 q^{76} -267.784 q^{77} +156.275 q^{79} +877.519 q^{82} -1036.53 q^{83} +584.818 q^{86} +921.301 q^{88} +170.225 q^{89} -135.402 q^{91} -286.607 q^{92} +30.1921 q^{94} -1059.87 q^{97} +126.704 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 8 q^{2} + 20 q^{4} + 14 q^{7} + 48 q^{8} + 14 q^{11} - 50 q^{13} + 56 q^{14} + 168 q^{16} - 50 q^{17} + 36 q^{19} + 184 q^{22} + 244 q^{23} - 216 q^{26} + 140 q^{28} + 26 q^{29} - 120 q^{31} + 672 q^{32}+ \cdots + 392 q^{98}+O(q^{100}) $ 2 * q + 8 * q^2 + 20 * q^4 + 14 * q^7 + 48 * q^8 + 14 * q^11 - 50 * q^13 + 56 * q^14 + 168 * q^16 - 50 * q^17 + 36 * q^19 + 184 * q^22 + 244 * q^23 - 216 * q^26 + 140 * q^28 + 26 * q^29 - 120 * q^31 + 672 * q^32 - 24 * q^34 - 564 * q^37 + 320 * q^38 + 328 * q^41 + 260 * q^43 + 1164 * q^44 + 704 * q^46 - 350 * q^47 + 98 * q^49 - 628 * q^52 - 56 * q^53 + 336 * q^56 + 8 * q^58 + 1232 * q^59 + 336 * q^61 - 1200 * q^62 + 2128 * q^64 + 152 * q^67 + 908 * q^68 + 1904 * q^71 - 676 * q^73 - 2016 * q^74 + 1768 * q^76 + 98 * q^77 + 1014 * q^79 + 816 * q^82 - 376 * q^83 + 768 * q^86 + 4688 * q^88 + 216 * q^89 - 350 * q^91 + 264 * q^92 - 1928 * q^94 - 2742 * q^97 + 392 * 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 $

$2$	2.58579	0.914214	0.457107	−	0.889412i	$-0.348886\pi$
$2$	2.58579	0.914214	0.457107	+	0.889412i	$0.348886\pi$
$3$	0	0
$3$	0	0
$4$	−1.31371	−0.164214
$5$	0	0
$5$	0	0
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	−24.0833	−1.06434
$9$	0	0
$10$	0	0
$11$	−38.2548	−1.04857	−0.524285	−	0.851543i	$-0.675667\pi$
$11$	−38.2548	−1.04857	−0.524285	+	0.851543i	$0.675667\pi$
$12$	0	0
$13$	−19.3431	−0.412679	−0.206339	−	0.978480i	$-0.566155\pi$
$13$	−19.3431	−0.412679	−0.206339	+	0.978480i	$0.566155\pi$
$14$	18.1005	0.345540
$15$	0	0
$16$	−51.7645	−0.808820
$17$	−87.2254	−1.24443	−0.622214	−	0.782847i	$-0.713767\pi$
$17$	−87.2254	−1.24443	−0.622214	+	0.782847i	$0.713767\pi$
$18$	0	0
$19$	−44.2254	−0.534000	−0.267000	−	0.963697i	$-0.586032\pi$
$19$	−44.2254	−0.534000	−0.267000	+	0.963697i	$0.586032\pi$
$20$	0	0
$21$	0	0
$22$	−98.9188	−0.958617
$23$	218.167	1.97786	0.988932	−	0.148371i	$-0.0474028\pi$
$23$	218.167	1.97786	0.988932	+	0.148371i	$0.0474028\pi$
$24$	0	0
$25$	0	0
$26$	−50.0172	−0.377276
$27$	0	0
$28$	−9.19596	−0.0620669
$29$	46.9411	0.300578	0.150289	−	0.988642i	$-0.451980\pi$
$29$	46.9411	0.300578	0.150289	+	0.988642i	$0.451980\pi$
$30$	0	0
$31$	194.558	1.12722	0.563609	−	0.826042i	$-0.309413\pi$
$31$	194.558	1.12722	0.563609	+	0.826042i	$0.309413\pi$
$32$	58.8141	0.324905
$33$	0	0
$34$	−225.546	−1.13767
$35$	0	0
$36$	0	0
$37$	−366.853	−1.63001	−0.815003	−	0.579457i	$-0.803265\pi$
$37$	−366.853	−1.63001	−0.815003	+	0.579457i	$0.803265\pi$
$38$	−114.357	−0.488190
$39$	0	0
$40$	0	0
$41$	339.362	1.29267	0.646336	−	0.763053i	$-0.276301\pi$
$41$	339.362	1.29267	0.646336	+	0.763053i	$0.276301\pi$
$42$	0	0
$43$	226.167	0.802095	0.401047	−	0.916057i	$-0.368646\pi$
$43$	226.167	0.802095	0.401047	+	0.916057i	$0.368646\pi$
$44$	50.2557	0.172189
$45$	0	0
$46$	564.132	1.80819
$47$	11.6762	0.0362372	0.0181186	−	0.999836i	$-0.494232\pi$
$47$	11.6762	0.0362372	0.0181186	+	0.999836i	$0.494232\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	0	0
$52$	25.4113	0.0677674
$53$	−209.019	−0.541717	−0.270859	−	0.962619i	$-0.587308\pi$
$53$	−209.019	−0.541717	−0.270859	+	0.962619i	$0.587308\pi$
$54$	0	0
$55$	0	0
$56$	−168.583	−0.402283
$57$	0	0
$58$	121.380	0.274792
$59$	616.000	1.35926	0.679630	−	0.733555i	$-0.262140\pi$
$59$	616.000	1.35926	0.679630	+	0.733555i	$0.262140\pi$
$60$	0	0
$61$	320.735	0.673212	0.336606	−	0.941646i	$-0.390721\pi$
$61$	320.735	0.673212	0.336606	+	0.941646i	$0.390721\pi$
$62$	503.087	1.03052
$63$	0	0
$64$	566.197	1.10585
$65$	0	0
$66$	0	0
$67$	−14.5097	−0.0264573	−0.0132286	−	0.999912i	$-0.504211\pi$
$67$	−14.5097	−0.0264573	−0.0132286	+	0.999912i	$0.504211\pi$
$68$	114.589	0.204352
$69$	0	0
$70$	0	0
$71$	952.000	1.59129	0.795645	−	0.605763i	$-0.207132\pi$
$71$	952.000	1.59129	0.795645	+	0.605763i	$0.207132\pi$
$72$	0	0
$73$	−824.489	−1.32191	−0.660953	−	0.750427i	$-0.729848\pi$
$73$	−824.489	−1.32191	−0.660953	+	0.750427i	$0.729848\pi$
$74$	−948.603	−1.49017
$75$	0	0
$76$	58.0993	0.0876901
$77$	−267.784	−0.396322
$78$	0	0
$79$	156.275	0.222561	0.111280	−	0.993789i	$-0.464505\pi$
$79$	156.275	0.222561	0.111280	+	0.993789i	$0.464505\pi$
$80$	0	0
$81$	0	0
$82$	877.519	1.18178
$83$	−1036.53	−1.37077	−0.685384	−	0.728182i	$-0.740366\pi$
$83$	−1036.53	−1.37077	−0.685384	+	0.728182i	$0.740366\pi$
$84$	0	0
$85$	0	0
$86$	584.818	0.733286
$87$	0	0
$88$	921.301	1.11603
$89$	170.225	0.202740	0.101370	−	0.994849i	$-0.467677\pi$
$89$	170.225	0.202740	0.101370	+	0.994849i	$0.467677\pi$
$90$	0	0
$91$	−135.402	−0.155978
$92$	−286.607	−0.324792
$93$	0	0
$94$	30.1921	0.0331285
$95$	0	0
$96$	0	0
$97$	−1059.87	−1.10942	−0.554710	−	0.832044i	$-0.687171\pi$
$97$	−1059.87	−1.10942	−0.554710	+	0.832044i	$0.687171\pi$
$98$	126.704	0.130602
$99$	0	0
$100$	0	0
$101$	241.833	0.238251	0.119125	−	0.992879i	$-0.461991\pi$
$101$	241.833	0.238251	0.119125	+	0.992879i	$0.461991\pi$
$102$	0	0
$103$	1679.58	1.60673	0.803367	−	0.595484i	$-0.203040\pi$
$103$	1679.58	1.60673	0.803367	+	0.595484i	$0.203040\pi$
$104$	465.846	0.439230
$105$	0	0
$106$	−540.479	−0.495245
$107$	1506.88	1.36146	0.680728	−	0.732537i	$-0.261664\pi$
$107$	1506.88	1.36146	0.680728	+	0.732537i	$0.261664\pi$
$108$	0	0
$109$	−1252.41	−1.10054	−0.550271	−	0.834986i	$-0.685476\pi$
$109$	−1252.41	−1.10054	−0.550271	+	0.834986i	$0.685476\pi$
$110$	0	0
$111$	0	0
$112$	−362.352	−0.305705
$113$	1370.20	1.14069	0.570345	−	0.821405i	$-0.306810\pi$
$113$	1370.20	1.14069	0.570345	+	0.821405i	$0.306810\pi$
$114$	0	0
$115$	0	0
$116$	−61.6670	−0.0493589
$117$	0	0
$118$	1592.84	1.24265
$119$	−610.578	−0.470349
$120$	0	0
$121$	132.432	0.0994984
$122$	829.352	0.615459
$123$	0	0
$124$	−255.593	−0.185104
$125$	0	0
$126$	0	0
$127$	−1213.49	−0.847873	−0.423936	−	0.905692i	$-0.639352\pi$
$127$	−1213.49	−0.847873	−0.423936	+	0.905692i	$0.639352\pi$
$128$	993.551	0.686081
$129$	0	0
$130$	0	0
$131$	1982.42	1.32217	0.661087	−	0.750309i	$-0.270096\pi$
$131$	1982.42	1.32217	0.661087	+	0.750309i	$0.270096\pi$
$132$	0	0
$133$	−309.578	−0.201833
$134$	−37.5189	−0.0241876
$135$	0	0
$136$	2100.67	1.32449
$137$	2210.95	1.37879	0.689394	−	0.724386i	$-0.257877\pi$
$137$	2210.95	1.37879	0.689394	+	0.724386i	$0.257877\pi$
$138$	0	0
$139$	528.039	0.322213	0.161107	−	0.986937i	$-0.448494\pi$
$139$	528.039	0.322213	0.161107	+	0.986937i	$0.448494\pi$
$140$	0	0
$141$	0	0
$142$	2461.67	1.45478
$143$	739.969	0.432722
$144$	0	0
$145$	0	0
$146$	−2131.95	−1.20851
$147$	0	0
$148$	481.938	0.267669
$149$	328.372	0.180545	0.0902727	−	0.995917i	$-0.471226\pi$
$149$	328.372	0.180545	0.0902727	+	0.995917i	$0.471226\pi$
$150$	0	0
$151$	1029.43	0.554793	0.277396	−	0.960756i	$-0.410528\pi$
$151$	1029.43	0.554793	0.277396	+	0.960756i	$0.410528\pi$
$152$	1065.09	0.568358
$153$	0	0
$154$	−692.432	−0.362323
$155$	0	0
$156$	0	0
$157$	−525.098	−0.266926	−0.133463	−	0.991054i	$-0.542610\pi$
$157$	−525.098	−0.266926	−0.133463	+	0.991054i	$0.542610\pi$
$158$	404.094	0.203468
$159$	0	0
$160$	0	0
$161$	1527.17	0.747562
$162$	0	0
$163$	−1002.63	−0.481790	−0.240895	−	0.970551i	$-0.577441\pi$
$163$	−1002.63	−0.481790	−0.240895	+	0.970551i	$0.577441\pi$
$164$	−445.823	−0.212274
$165$	0	0
$166$	−2680.24	−1.25317
$167$	−359.422	−0.166544	−0.0832722	−	0.996527i	$-0.526537\pi$
$167$	−359.422	−0.166544	−0.0832722	+	0.996527i	$0.526537\pi$
$168$	0	0
$169$	−1822.84	−0.829696
$170$	0	0
$171$	0	0
$172$	−297.117	−0.131715
$173$	3293.65	1.44747	0.723733	−	0.690080i	$-0.242425\pi$
$173$	3293.65	1.44747	0.723733	+	0.690080i	$0.242425\pi$
$174$	0	0
$175$	0	0
$176$	1980.24	0.848104
$177$	0	0
$178$	440.167	0.185348
$179$	−2978.82	−1.24384	−0.621921	−	0.783080i	$-0.713647\pi$
$179$	−2978.82	−1.24384	−0.621921	+	0.783080i	$0.713647\pi$
$180$	0	0
$181$	1462.31	0.600514	0.300257	−	0.953858i	$-0.402928\pi$
$181$	1462.31	0.600514	0.300257	+	0.953858i	$0.402928\pi$
$182$	−350.121	−0.142597
$183$	0	0
$184$	−5254.16	−2.10512
$185$	0	0
$186$	0	0
$187$	3336.79	1.30487
$188$	−15.3391	−0.00595064
$189$	0	0
$190$	0	0
$191$	374.923	0.142034	0.0710169	−	0.997475i	$-0.477376\pi$
$191$	374.923	0.142034	0.0710169	+	0.997475i	$0.477376\pi$
$192$	0	0
$193$	−733.028	−0.273391	−0.136696	−	0.990613i	$-0.543648\pi$
$193$	−733.028	−0.273391	−0.136696	+	0.990613i	$0.543648\pi$
$194$	−2740.61	−1.01425
$195$	0	0
$196$	−64.3717	−0.0234591
$197$	−2093.24	−0.757043	−0.378521	−	0.925593i	$-0.623567\pi$
$197$	−2093.24	−0.757043	−0.378521	+	0.925593i	$0.623567\pi$
$198$	0	0
$199$	2865.04	1.02059	0.510295	−	0.860000i	$-0.329536\pi$
$199$	2865.04	1.02059	0.510295	+	0.860000i	$0.329536\pi$
$200$	0	0
$201$	0	0
$202$	625.330	0.217812
$203$	328.588	0.113608
$204$	0	0
$205$	0	0
$206$	4343.03	1.46890
$207$	0	0
$208$	1001.29	0.333783
$209$	1691.84	0.559936
$210$	0	0
$211$	5643.65	1.84135	0.920674	−	0.390331i	$-0.127640\pi$
$211$	5643.65	1.84135	0.920674	+	0.390331i	$0.127640\pi$
$212$	274.590	0.0889573
$213$	0	0
$214$	3896.47	1.24466
$215$	0	0
$216$	0	0
$217$	1361.91	0.426048
$218$	−3238.46	−1.00613
$219$	0	0
$220$	0	0
$221$	1687.21	0.513549
$222$	0	0
$223$	6369.16	1.91260	0.956302	−	0.292381i	$-0.0944477\pi$
$223$	6369.16	1.91260	0.956302	+	0.292381i	$0.0944477\pi$
$224$	411.699	0.122803
$225$	0	0
$226$	3543.05	1.04283
$227$	−1015.67	−0.296972	−0.148486	−	0.988914i	$-0.547440\pi$
$227$	−1015.67	−0.296972	−0.148486	+	0.988914i	$0.547440\pi$
$228$	0	0
$229$	4108.35	1.18554	0.592768	−	0.805373i	$-0.298035\pi$
$229$	4108.35	1.18554	0.592768	+	0.805373i	$0.298035\pi$
$230$	0	0
$231$	0	0
$232$	−1130.50	−0.319917
$233$	608.431	0.171071	0.0855357	−	0.996335i	$-0.472740\pi$
$233$	608.431	0.171071	0.0855357	+	0.996335i	$0.472740\pi$
$234$	0	0
$235$	0	0
$236$	−809.244	−0.223209
$237$	0	0
$238$	−1578.82	−0.430000
$239$	5054.44	1.36797	0.683985	−	0.729496i	$-0.260245\pi$
$239$	5054.44	1.36797	0.683985	+	0.729496i	$0.260245\pi$
$240$	0	0
$241$	4.86782	0.00130109	0.000650547	−	1.00000i	$-0.499793\pi$
$241$	4.86782	0.00130109	0.000650547		1.00000i	$0.499793\pi$
$242$	342.442	0.0909628
$243$	0	0
$244$	−421.352	−0.110551
$245$	0	0
$246$	0	0
$247$	855.458	0.220370
$248$	−4685.60	−1.19974
$249$	0	0
$250$	0	0
$251$	547.921	0.137787	0.0688934	−	0.997624i	$-0.478053\pi$
$251$	547.921	0.137787	0.0688934	+	0.997624i	$0.478053\pi$
$252$	0	0
$253$	−8345.92	−2.07393
$254$	−3137.83	−0.775137
$255$	0	0
$256$	−1960.46	−0.478629
$257$	−1774.61	−0.430729	−0.215364	−	0.976534i	$-0.569094\pi$
$257$	−1774.61	−0.430729	−0.215364	+	0.976534i	$0.569094\pi$
$258$	0	0
$259$	−2567.97	−0.616084
$260$	0	0
$261$	0	0
$262$	5126.11	1.20875
$263$	−1199.09	−0.281138	−0.140569	−	0.990071i	$-0.544893\pi$
$263$	−1199.09	−0.281138	−0.140569	+	0.990071i	$0.544893\pi$
$264$	0	0
$265$	0	0
$266$	−800.502	−0.184519
$267$	0	0
$268$	19.0615	0.00434464
$269$	−3250.29	−0.736706	−0.368353	−	0.929686i	$-0.620078\pi$
$269$	−3250.29	−0.736706	−0.368353	+	0.929686i	$0.620078\pi$
$270$	0	0
$271$	−896.143	−0.200874	−0.100437	−	0.994943i	$-0.532024\pi$
$271$	−896.143	−0.200874	−0.100437	+	0.994943i	$0.532024\pi$
$272$	4515.18	1.00652
$273$	0	0
$274$	5717.04	1.26051
$275$	0	0
$276$	0	0
$277$	386.562	0.0838492	0.0419246	−	0.999121i	$-0.486651\pi$
$277$	386.562	0.0838492	0.0419246	+	0.999121i	$0.486651\pi$
$278$	1365.40	0.294572
$279$	0	0
$280$	0	0
$281$	3335.10	0.708025	0.354013	−	0.935241i	$-0.384817\pi$
$281$	3335.10	0.708025	0.354013	+	0.935241i	$0.384817\pi$
$282$	0	0
$283$	−5412.26	−1.13684	−0.568419	−	0.822739i	$-0.692445\pi$
$283$	−5412.26	−1.13684	−0.568419	+	0.822739i	$0.692445\pi$
$284$	−1250.65	−0.261311
$285$	0	0
$286$	1913.40	0.395601
$287$	2375.54	0.488584
$288$	0	0
$289$	2695.27	0.548600
$290$	0	0
$291$	0	0
$292$	1083.14	0.217075
$293$	282.211	0.0562695	0.0281347	−	0.999604i	$-0.491043\pi$
$293$	282.211	0.0562695	0.0281347	+	0.999604i	$0.491043\pi$
$294$	0	0
$295$	0	0
$296$	8835.01	1.73488
$297$	0	0
$298$	849.099	0.165057
$299$	−4220.03	−0.816222
$300$	0	0
$301$	1583.17	0.303163
$302$	2661.88	0.507199
$303$	0	0
$304$	2289.31	0.431910
$305$	0	0
$306$	0	0
$307$	−1919.67	−0.356878	−0.178439	−	0.983951i	$-0.557105\pi$
$307$	−1919.67	−0.356878	−0.178439	+	0.983951i	$0.557105\pi$
$308$	351.790	0.0650815
$309$	0	0
$310$	0	0
$311$	−1213.31	−0.221223	−0.110612	−	0.993864i	$-0.535281\pi$
$311$	−1213.31	−0.221223	−0.110612	+	0.993864i	$0.535281\pi$
$312$	0	0
$313$	1434.00	0.258960	0.129480	−	0.991582i	$-0.458669\pi$
$313$	1434.00	0.258960	0.129480	+	0.991582i	$0.458669\pi$
$314$	−1357.79	−0.244028
$315$	0	0
$316$	−205.300	−0.0365475
$317$	6496.95	1.15112	0.575560	−	0.817760i	$-0.304784\pi$
$317$	6496.95	1.15112	0.575560	+	0.817760i	$0.304784\pi$
$318$	0	0
$319$	−1795.72	−0.315176
$320$	0	0
$321$	0	0
$322$	3948.92	0.683432
$323$	3857.58	0.664524
$324$	0	0
$325$	0	0
$326$	−2592.58	−0.440459
$327$	0	0
$328$	−8172.96	−1.37584
$329$	81.7333	0.0136964
$330$	0	0
$331$	−9683.88	−1.60808	−0.804039	−	0.594576i	$-0.797320\pi$
$331$	−9683.88	−1.60808	−0.804039	+	0.594576i	$0.797320\pi$
$332$	1361.70	0.225099
$333$	0	0
$334$	−929.389	−0.152257
$335$	0	0
$336$	0	0
$337$	−29.1319	−0.00470895	−0.00235447	−	0.999997i	$-0.500749\pi$
$337$	−29.1319	−0.00470895	−0.00235447	+	0.999997i	$0.500749\pi$
$338$	−4713.48	−0.758520
$339$	0	0
$340$	0	0
$341$	−7442.80	−1.18197
$342$	0	0
$343$	343.000	0.0539949
$344$	−5446.83	−0.853701
$345$	0	0
$346$	8516.68	1.32329
$347$	−7848.58	−1.21422	−0.607110	−	0.794618i	$-0.707671\pi$
$347$	−7848.58	−1.21422	−0.607110	+	0.794618i	$0.707671\pi$
$348$	0	0
$349$	−10269.6	−1.57513	−0.787567	−	0.616229i	$-0.788659\pi$
$349$	−10269.6	−1.57513	−0.787567	+	0.616229i	$0.788659\pi$
$350$	0	0
$351$	0	0
$352$	−2249.93	−0.340686
$353$	2799.93	0.422168	0.211084	−	0.977468i	$-0.432301\pi$
$353$	2799.93	0.422168	0.211084	+	0.977468i	$0.432301\pi$
$354$	0	0
$355$	0	0
$356$	−223.627	−0.0332927
$357$	0	0
$358$	−7702.60	−1.13714
$359$	3163.29	0.465048	0.232524	−	0.972591i	$-0.425302\pi$
$359$	3163.29	0.465048	0.232524	+	0.972591i	$0.425302\pi$
$360$	0	0
$361$	−4903.11	−0.714844
$362$	3781.23	0.548998
$363$	0	0
$364$	177.879	0.0256137
$365$	0	0
$366$	0	0
$367$	−3182.85	−0.452706	−0.226353	−	0.974045i	$-0.572680\pi$
$367$	−3182.85	−0.452706	−0.226353	+	0.974045i	$0.572680\pi$
$368$	−11293.3	−1.59974
$369$	0	0
$370$	0	0
$371$	−1463.14	−0.204750
$372$	0	0
$373$	2615.14	0.363021	0.181510	−	0.983389i	$-0.441901\pi$
$373$	2615.14	0.363021	0.181510	+	0.983389i	$0.441901\pi$
$374$	8628.23	1.19293
$375$	0	0
$376$	−281.201	−0.0385687
$377$	−907.989	−0.124042
$378$	0	0
$379$	−672.434	−0.0911362	−0.0455681	−	0.998961i	$-0.514510\pi$
$379$	−672.434	−0.0911362	−0.0455681	+	0.998961i	$0.514510\pi$
$380$	0	0
$381$	0	0
$382$	969.470	0.129849
$383$	1169.86	0.156075	0.0780377	−	0.996950i	$-0.475135\pi$
$383$	1169.86	0.156075	0.0780377	+	0.996950i	$0.475135\pi$
$384$	0	0
$385$	0	0
$386$	−1895.45	−0.249938
$387$	0	0
$388$	1392.36	0.182182
$389$	1122.22	0.146269	0.0731347	−	0.997322i	$-0.476700\pi$
$389$	1122.22	0.146269	0.0731347	+	0.997322i	$0.476700\pi$
$390$	0	0
$391$	−19029.7	−2.46131
$392$	−1180.08	−0.152049
$393$	0	0
$394$	−5412.68	−0.692099
$395$	0	0
$396$	0	0
$397$	1985.93	0.251060	0.125530	−	0.992090i	$-0.459937\pi$
$397$	1985.93	0.251060	0.125530	+	0.992090i	$0.459937\pi$
$398$	7408.38	0.933037
$399$	0	0
$400$	0	0
$401$	4172.38	0.519597	0.259799	−	0.965663i	$-0.416344\pi$
$401$	4172.38	0.519597	0.259799	+	0.965663i	$0.416344\pi$
$402$	0	0
$403$	−3763.37	−0.465178
$404$	−317.699	−0.0391240
$405$	0	0
$406$	849.658	0.103862
$407$	14033.9	1.70918
$408$	0	0
$409$	−11700.8	−1.41459	−0.707295	−	0.706919i	$-0.750085\pi$
$409$	−11700.8	−1.41459	−0.707295	+	0.706919i	$0.750085\pi$
$410$	0	0
$411$	0	0
$412$	−2206.47	−0.263848
$413$	4312.00	0.513752
$414$	0	0
$415$	0	0
$416$	−1137.65	−0.134082
$417$	0	0
$418$	4374.72	0.511901
$419$	−2733.20	−0.318677	−0.159339	−	0.987224i	$-0.550936\pi$
$419$	−2733.20	−0.318677	−0.159339	+	0.987224i	$0.550936\pi$
$420$	0	0
$421$	13549.4	1.56854	0.784272	−	0.620417i	$-0.213037\pi$
$421$	13549.4	1.56854	0.784272	+	0.620417i	$0.213037\pi$
$422$	14593.3	1.68339
$423$	0	0
$424$	5033.87	0.576571
$425$	0	0
$426$	0	0
$427$	2245.15	0.254450
$428$	−1979.60	−0.223569
$429$	0	0
$430$	0	0
$431$	6429.25	0.718530	0.359265	−	0.933236i	$-0.383027\pi$
$431$	6429.25	0.718530	0.359265	+	0.933236i	$0.383027\pi$
$432$	0	0
$433$	−8022.03	−0.890333	−0.445166	−	0.895448i	$-0.646855\pi$
$433$	−8022.03	−0.890333	−0.445166	+	0.895448i	$0.646855\pi$
$434$	3521.61	0.389499
$435$	0	0
$436$	1645.30	0.180724
$437$	−9648.50	−1.05618
$438$	0	0
$439$	−5569.88	−0.605549	−0.302774	−	0.953062i	$-0.597913\pi$
$439$	−5569.88	−0.605549	−0.302774	+	0.953062i	$0.597913\pi$
$440$	0	0
$441$	0	0
$442$	4362.77	0.469493
$443$	−5486.21	−0.588392	−0.294196	−	0.955745i	$-0.595052\pi$
$443$	−5486.21	−0.588392	−0.294196	+	0.955745i	$0.595052\pi$
$444$	0	0
$445$	0	0
$446$	16469.3	1.74853
$447$	0	0
$448$	3963.38	0.417973
$449$	7232.67	0.760203	0.380101	−	0.924945i	$-0.375889\pi$
$449$	7232.67	0.760203	0.380101	+	0.924945i	$0.375889\pi$
$450$	0	0
$451$	−12982.3	−1.35546
$452$	−1800.05	−0.187317
$453$	0	0
$454$	−2626.32	−0.271496
$455$	0	0
$456$	0	0
$457$	2900.51	0.296893	0.148446	−	0.988920i	$-0.452573\pi$
$457$	2900.51	0.296893	0.148446	+	0.988920i	$0.452573\pi$
$458$	10623.3	1.08383
$459$	0	0
$460$	0	0
$461$	−6073.57	−0.613611	−0.306805	−	0.951772i	$-0.599260\pi$
$461$	−6073.57	−0.613611	−0.306805	+	0.951772i	$0.599260\pi$
$462$	0	0
$463$	18922.8	1.89939	0.949693	−	0.313183i	$-0.101395\pi$
$463$	18922.8	1.89939	0.949693	+	0.313183i	$0.101395\pi$
$464$	−2429.88	−0.243113
$465$	0	0
$466$	1573.27	0.156396
$467$	−6776.71	−0.671496	−0.335748	−	0.941952i	$-0.608989\pi$
$467$	−6776.71	−0.671496	−0.335748	+	0.941952i	$0.608989\pi$
$468$	0	0
$469$	−101.568	−0.00999991
$470$	0	0
$471$	0	0
$472$	−14835.3	−1.44672
$473$	−8651.96	−0.841052
$474$	0	0
$475$	0	0
$476$	802.121	0.0772377
$477$	0	0
$478$	13069.7	1.25062
$479$	−2397.32	−0.228677	−0.114338	−	0.993442i	$-0.536475\pi$
$479$	−2397.32	−0.228677	−0.114338	+	0.993442i	$0.536475\pi$
$480$	0	0
$481$	7096.09	0.672669
$482$	12.5871	0.00118948
$483$	0	0
$484$	−173.977	−0.0163390
$485$	0	0
$486$	0	0
$487$	−5586.17	−0.519781	−0.259890	−	0.965638i	$-0.583686\pi$
$487$	−5586.17	−0.519781	−0.259890	+	0.965638i	$0.583686\pi$
$488$	−7724.35	−0.716526
$489$	0	0
$490$	0	0
$491$	−537.392	−0.0493934	−0.0246967	−	0.999695i	$-0.507862\pi$
$491$	−537.392	−0.0493934	−0.0246967	+	0.999695i	$0.507862\pi$
$492$	0	0
$493$	−4094.46	−0.374047
$494$	2212.03	0.201466
$495$	0	0
$496$	−10071.2	−0.911716
$497$	6664.00	0.601451
$498$	0	0
$499$	598.965	0.0537342	0.0268671	−	0.999639i	$-0.491447\pi$
$499$	598.965	0.0537342	0.0268671	+	0.999639i	$0.491447\pi$
$500$	0	0
$501$	0	0
$502$	1416.81	0.125966
$503$	4426.76	0.392405	0.196202	−	0.980563i	$-0.437139\pi$
$503$	4426.76	0.392405	0.196202	+	0.980563i	$0.437139\pi$
$504$	0	0
$505$	0	0
$506$	−21580.8	−1.89601
$507$	0	0
$508$	1594.17	0.139232
$509$	−17727.7	−1.54374	−0.771872	−	0.635779i	$-0.780679\pi$
$509$	−17727.7	−1.54374	−0.771872	+	0.635779i	$0.780679\pi$
$510$	0	0
$511$	−5771.43	−0.499634
$512$	−13017.7	−1.12365
$513$	0	0
$514$	−4588.77	−0.393778
$515$	0	0
$516$	0	0
$517$	−446.671	−0.0379972
$518$	−6640.22	−0.563233
$519$	0	0
$520$	0	0
$521$	−8662.79	−0.728453	−0.364226	−	0.931310i	$-0.618667\pi$
$521$	−8662.79	−0.728453	−0.364226	+	0.931310i	$0.618667\pi$
$522$	0	0
$523$	7770.40	0.649667	0.324833	−	0.945771i	$-0.394692\pi$
$523$	7770.40	0.649667	0.324833	+	0.945771i	$0.394692\pi$
$524$	−2604.32	−0.217119
$525$	0	0
$526$	−3100.60	−0.257020
$527$	−16970.4	−1.40274
$528$	0	0
$529$	35429.6	2.91194
$530$	0	0
$531$	0	0
$532$	406.695	0.0331437
$533$	−6564.34	−0.533458
$534$	0	0
$535$	0	0
$536$	349.440	0.0281595
$537$	0	0
$538$	−8404.56	−0.673506
$539$	−1874.49	−0.149796
$540$	0	0
$541$	21641.0	1.71981	0.859906	−	0.510453i	$-0.170522\pi$
$541$	21641.0	1.71981	0.859906	+	0.510453i	$0.170522\pi$
$542$	−2317.23	−0.183642
$543$	0	0
$544$	−5130.09	−0.404321
$545$	0	0
$546$	0	0
$547$	−7489.29	−0.585409	−0.292705	−	0.956203i	$-0.594555\pi$
$547$	−7489.29	−0.585409	−0.292705	+	0.956203i	$0.594555\pi$
$548$	−2904.54	−0.226416
$549$	0	0
$550$	0	0
$551$	−2075.99	−0.160508
$552$	0	0
$553$	1093.93	0.0841201
$554$	999.566	0.0766561
$555$	0	0
$556$	−693.689	−0.0529118
$557$	25297.9	1.92443	0.962214	−	0.272295i	$-0.0877826\pi$
$557$	25297.9	1.92443	0.962214	+	0.272295i	$0.0877826\pi$
$558$	0	0
$559$	−4374.77	−0.331007
$560$	0	0
$561$	0	0
$562$	8623.85	0.647286
$563$	15661.3	1.17237	0.586186	−	0.810177i	$-0.300629\pi$
$563$	15661.3	1.17237	0.586186	+	0.810177i	$0.300629\pi$
$564$	0	0
$565$	0	0
$566$	−13994.9	−1.03931
$567$	0	0
$568$	−22927.3	−1.69367
$569$	9982.75	0.735498	0.367749	−	0.929925i	$-0.380129\pi$
$569$	9982.75	0.735498	0.367749	+	0.929925i	$0.380129\pi$
$570$	0	0
$571$	−11583.6	−0.848966	−0.424483	−	0.905436i	$-0.639544\pi$
$571$	−11583.6	−0.848966	−0.424483	+	0.905436i	$0.639544\pi$
$572$	−972.103	−0.0710589
$573$	0	0
$574$	6142.63	0.446670
$575$	0	0
$576$	0	0
$577$	595.378	0.0429565	0.0214783	−	0.999769i	$-0.493163\pi$
$577$	595.378	0.0429565	0.0214783	+	0.999769i	$0.493163\pi$
$578$	6969.39	0.501537
$579$	0	0
$580$	0	0
$581$	−7255.70	−0.518102
$582$	0	0
$583$	7996.00	0.568028
$584$	19856.4	1.40696
$585$	0	0
$586$	729.738	0.0514423
$587$	−15750.3	−1.10747	−0.553736	−	0.832693i	$-0.686798\pi$
$587$	−15750.3	−1.10747	−0.553736	+	0.832693i	$0.686798\pi$
$588$	0	0
$589$	−8604.42	−0.601934
$590$	0	0
$591$	0	0
$592$	18990.0	1.31838
$593$	−417.878	−0.0289379	−0.0144690	−	0.999895i	$-0.504606\pi$
$593$	−417.878	−0.0289379	−0.0144690	+	0.999895i	$0.504606\pi$
$594$	0	0
$595$	0	0
$596$	−431.385	−0.0296480
$597$	0	0
$598$	−10912.1	−0.746201
$599$	19997.3	1.36406	0.682028	−	0.731326i	$-0.261098\pi$
$599$	19997.3	1.36406	0.682028	+	0.731326i	$0.261098\pi$
$600$	0	0
$601$	−15992.6	−1.08545	−0.542723	−	0.839912i	$-0.682607\pi$
$601$	−15992.6	−1.08545	−0.542723	+	0.839912i	$0.682607\pi$
$602$	4093.73	0.277156
$603$	0	0
$604$	−1352.37	−0.0911045
$605$	0	0
$606$	0	0
$607$	−14159.2	−0.946793	−0.473396	−	0.880850i	$-0.656972\pi$
$607$	−14159.2	−0.946793	−0.473396	+	0.880850i	$0.656972\pi$
$608$	−2601.08	−0.173499
$609$	0	0
$610$	0	0
$611$	−225.854	−0.0149543
$612$	0	0
$613$	4629.41	0.305025	0.152512	−	0.988302i	$-0.451264\pi$
$613$	4629.41	0.305025	0.152512	+	0.988302i	$0.451264\pi$
$614$	−4963.87	−0.326263
$615$	0	0
$616$	6449.11	0.421821
$617$	−23165.3	−1.51151	−0.755753	−	0.654857i	$-0.772729\pi$
$617$	−23165.3	−1.51151	−0.755753	+	0.654857i	$0.772729\pi$
$618$	0	0
$619$	12370.6	0.803258	0.401629	−	0.915803i	$-0.368444\pi$
$619$	12370.6	0.803258	0.401629	+	0.915803i	$0.368444\pi$
$620$	0	0
$621$	0	0
$622$	−3137.36	−0.202245
$623$	1191.58	0.0766285
$624$	0	0
$625$	0	0
$626$	3708.02	0.236745
$627$	0	0
$628$	689.826	0.0438329
$629$	31998.9	2.02842
$630$	0	0
$631$	13980.2	0.882002	0.441001	−	0.897507i	$-0.354623\pi$
$631$	13980.2	0.882002	0.441001	+	0.897507i	$0.354623\pi$
$632$	−3763.61	−0.236880
$633$	0	0
$634$	16799.7	1.05237
$635$	0	0
$636$	0	0
$637$	−947.814	−0.0589541
$638$	−4643.36	−0.288139
$639$	0	0
$640$	0	0
$641$	16060.9	0.989655	0.494828	−	0.868991i	$-0.335231\pi$
$641$	16060.9	0.989655	0.494828	+	0.868991i	$0.335231\pi$
$642$	0	0
$643$	4502.17	0.276125	0.138063	−	0.990424i	$-0.455913\pi$
$643$	4502.17	0.276125	0.138063	+	0.990424i	$0.455913\pi$
$644$	−2006.25	−0.122760
$645$	0	0
$646$	9974.87	0.607517
$647$	29414.8	1.78735	0.893675	−	0.448715i	$-0.148118\pi$
$647$	29414.8	1.78735	0.893675	+	0.448715i	$0.148118\pi$
$648$	0	0
$649$	−23565.0	−1.42528
$650$	0	0
$651$	0	0
$652$	1317.16	0.0791164
$653$	13013.6	0.779882	0.389941	−	0.920840i	$-0.372495\pi$
$653$	13013.6	0.779882	0.389941	+	0.920840i	$0.372495\pi$
$654$	0	0
$655$	0	0
$656$	−17566.9	−1.04554
$657$	0	0
$658$	211.345	0.0125214
$659$	−23474.2	−1.38759	−0.693797	−	0.720171i	$-0.744063\pi$
$659$	−23474.2	−1.38759	−0.693797	+	0.720171i	$0.744063\pi$
$660$	0	0
$661$	−9266.36	−0.545264	−0.272632	−	0.962118i	$-0.587894\pi$
$661$	−9266.36	−0.545264	−0.272632	+	0.962118i	$0.587894\pi$
$662$	−25040.4	−1.47013
$663$	0	0
$664$	24963.0	1.45896
$665$	0	0
$666$	0	0
$667$	10241.0	0.594501
$668$	472.176	0.0273489
$669$	0	0
$670$	0	0
$671$	−12269.7	−0.705909
$672$	0	0
$673$	25067.2	1.43576	0.717882	−	0.696164i	$-0.245112\pi$
$673$	25067.2	1.43576	0.717882	+	0.696164i	$0.245112\pi$
$674$	−75.3288	−0.00430498
$675$	0	0
$676$	2394.68	0.136247
$677$	−22409.6	−1.27219	−0.636093	−	0.771613i	$-0.719450\pi$
$677$	−22409.6	−1.27219	−0.636093	+	0.771613i	$0.719450\pi$
$678$	0	0
$679$	−7419.11	−0.419322
$680$	0	0
$681$	0	0
$682$	−19245.5	−1.08057
$683$	−8757.53	−0.490626	−0.245313	−	0.969444i	$-0.578891\pi$
$683$	−8757.53	−0.490626	−0.245313	+	0.969444i	$0.578891\pi$
$684$	0	0
$685$	0	0
$686$	886.925	0.0493629
$687$	0	0
$688$	−11707.4	−0.648750
$689$	4043.09	0.223555
$690$	0	0
$691$	8468.42	0.466214	0.233107	−	0.972451i	$-0.425111\pi$
$691$	8468.42	0.466214	0.233107	+	0.972451i	$0.425111\pi$
$692$	−4326.90	−0.237694
$693$	0	0
$694$	−20294.8	−1.11006
$695$	0	0
$696$	0	0
$697$	−29601.0	−1.60864
$698$	−26555.1	−1.44001
$699$	0	0
$700$	0	0
$701$	−15996.9	−0.861906	−0.430953	−	0.902374i	$-0.641823\pi$
$701$	−15996.9	−0.861906	−0.430953	+	0.902374i	$0.641823\pi$
$702$	0	0
$703$	16224.2	0.870423
$704$	−21659.8	−1.15956
$705$	0	0
$706$	7240.03	0.385952
$707$	1692.83	0.0900503
$708$	0	0
$709$	19903.0	1.05426	0.527131	−	0.849784i	$-0.323268\pi$
$709$	19903.0	1.05426	0.527131	+	0.849784i	$0.323268\pi$
$710$	0	0
$711$	0	0
$712$	−4099.58	−0.215784
$713$	42446.1	2.22948
$714$	0	0
$715$	0	0
$716$	3913.30	0.204256
$717$	0	0
$718$	8179.59	0.425153
$719$	11073.1	0.574347	0.287174	−	0.957879i	$-0.407284\pi$
$719$	11073.1	0.574347	0.287174	+	0.957879i	$0.407284\pi$
$720$	0	0
$721$	11757.0	0.607288
$722$	−12678.4	−0.653520
$723$	0	0
$724$	−1921.06	−0.0986125
$725$	0	0
$726$	0	0
$727$	−31652.7	−1.61476	−0.807382	−	0.590029i	$-0.799116\pi$
$727$	−31652.7	−1.61476	−0.807382	+	0.590029i	$0.799116\pi$
$728$	3260.92	0.166013
$729$	0	0
$730$	0	0
$731$	−19727.5	−0.998149
$732$	0	0
$733$	−16958.3	−0.854528	−0.427264	−	0.904127i	$-0.640523\pi$
$733$	−16958.3	−0.854528	−0.427264	+	0.904127i	$0.640523\pi$
$734$	−8230.16	−0.413870
$735$	0	0
$736$	12831.3	0.642618
$737$	555.065	0.0277423
$738$	0	0
$739$	−11616.6	−0.578245	−0.289123	−	0.957292i	$-0.593364\pi$
$739$	−11616.6	−0.578245	−0.289123	+	0.957292i	$0.593364\pi$
$740$	0	0
$741$	0	0
$742$	−3783.36	−0.187185
$743$	15928.0	0.786464	0.393232	−	0.919439i	$-0.371357\pi$
$743$	15928.0	0.786464	0.393232	+	0.919439i	$0.371357\pi$
$744$	0	0
$745$	0	0
$746$	6762.19	0.331879
$747$	0	0
$748$	−4383.57	−0.214277
$749$	10548.2	0.514582
$750$	0	0
$751$	25571.9	1.24252	0.621260	−	0.783604i	$-0.286621\pi$
$751$	25571.9	1.24252	0.621260	+	0.783604i	$0.286621\pi$
$752$	−604.412	−0.0293094
$753$	0	0
$754$	−2347.87	−0.113401
$755$	0	0
$756$	0	0
$757$	−6202.41	−0.297794	−0.148897	−	0.988853i	$-0.547572\pi$
$757$	−6202.41	−0.297794	−0.148897	+	0.988853i	$0.547572\pi$
$758$	−1738.77	−0.0833179
$759$	0	0
$760$	0	0
$761$	29199.1	1.39089	0.695444	−	0.718580i	$-0.255208\pi$
$761$	29199.1	1.39089	0.695444	+	0.718580i	$0.255208\pi$
$762$	0	0
$763$	−8766.87	−0.415966
$764$	−492.539	−0.0233239
$765$	0	0
$766$	3025.00	0.142686
$767$	−11915.4	−0.560938
$768$	0	0
$769$	21838.2	1.02407	0.512033	−	0.858966i	$-0.328892\pi$
$769$	21838.2	1.02407	0.512033	+	0.858966i	$0.328892\pi$
$770$	0	0
$771$	0	0
$772$	962.985	0.0448945
$773$	−25544.8	−1.18859	−0.594296	−	0.804246i	$-0.702569\pi$
$773$	−25544.8	−1.18859	−0.594296	+	0.804246i	$0.702569\pi$
$774$	0	0
$775$	0	0
$776$	25525.2	1.18080
$777$	0	0
$778$	2901.82	0.133721
$779$	−15008.4	−0.690286
$780$	0	0
$781$	−36418.6	−1.66858
$782$	−49206.6	−2.25016
$783$	0	0
$784$	−2536.46	−0.115546
$785$	0	0
$786$	0	0
$787$	37223.2	1.68598	0.842989	−	0.537931i	$-0.180794\pi$
$787$	37223.2	1.68598	0.842989	+	0.537931i	$0.180794\pi$
$788$	2749.91	0.124317
$789$	0	0
$790$	0	0
$791$	9591.42	0.431140
$792$	0	0
$793$	−6204.03	−0.277820
$794$	5135.18	0.229522
$795$	0	0
$796$	−3763.83	−0.167595
$797$	−40384.6	−1.79485	−0.897425	−	0.441168i	$-0.854564\pi$
$797$	−40384.6	−1.79485	−0.897425	+	0.441168i	$0.854564\pi$
$798$	0	0
$799$	−1018.46	−0.0450945
$800$	0	0
$801$	0	0
$802$	10788.9	0.475023
$803$	31540.7	1.38611
$804$	0	0
$805$	0	0
$806$	−9731.28	−0.425272
$807$	0	0
$808$	−5824.14	−0.253580
$809$	1955.76	0.0849948	0.0424974	−	0.999097i	$-0.486469\pi$
$809$	1955.76	0.0849948	0.0424974	+	0.999097i	$0.486469\pi$
$810$	0	0
$811$	−34301.8	−1.48520	−0.742600	−	0.669735i	$-0.766408\pi$
$811$	−34301.8	−1.48520	−0.742600	+	0.669735i	$0.766408\pi$
$812$	−431.669	−0.0186559
$813$	0	0
$814$	36288.7	1.56255
$815$	0	0
$816$	0	0
$817$	−10002.3	−0.428319
$818$	−30255.8	−1.29324
$819$	0	0
$820$	0	0
$821$	13665.6	0.580918	0.290459	−	0.956887i	$-0.406192\pi$
$821$	13665.6	0.580918	0.290459	+	0.956887i	$0.406192\pi$
$822$	0	0
$823$	21519.5	0.911449	0.455724	−	0.890121i	$-0.349380\pi$
$823$	21519.5	0.911449	0.455724	+	0.890121i	$0.349380\pi$
$824$	−40449.7	−1.71011
$825$	0	0
$826$	11149.9	0.469679
$827$	35220.6	1.48094	0.740471	−	0.672088i	$-0.234602\pi$
$827$	35220.6	1.48094	0.740471	+	0.672088i	$0.234602\pi$
$828$	0	0
$829$	31365.5	1.31408	0.657039	−	0.753857i	$-0.271809\pi$
$829$	31365.5	1.31408	0.657039	+	0.753857i	$0.271809\pi$
$830$	0	0
$831$	0	0
$832$	−10952.0	−0.456362
$833$	−4274.04	−0.177775
$834$	0	0
$835$	0	0
$836$	−2222.58	−0.0919491
$837$	0	0
$838$	−7067.48	−0.291339
$839$	28287.1	1.16398	0.581990	−	0.813196i	$-0.302274\pi$
$839$	28287.1	1.16398	0.581990	+	0.813196i	$0.302274\pi$
$840$	0	0
$841$	−22185.5	−0.909653
$842$	35035.8	1.43398
$843$	0	0
$844$	−7414.11	−0.302374
$845$	0	0
$846$	0	0
$847$	927.026	0.0376068
$848$	10819.8	0.438152
$849$	0	0
$850$	0	0
$851$	−80035.0	−3.22393
$852$	0	0
$853$	−9405.41	−0.377533	−0.188766	−	0.982022i	$-0.560449\pi$
$853$	−9405.41	−0.377533	−0.188766	+	0.982022i	$0.560449\pi$
$854$	5805.47	0.232622
$855$	0	0
$856$	−36290.6	−1.44905
$857$	−27966.9	−1.11474	−0.557369	−	0.830265i	$-0.688189\pi$
$857$	−27966.9	−1.11474	−0.557369	+	0.830265i	$0.688189\pi$
$858$	0	0
$859$	6281.11	0.249486	0.124743	−	0.992189i	$-0.460189\pi$
$859$	6281.11	0.249486	0.124743	+	0.992189i	$0.460189\pi$
$860$	0	0
$861$	0	0
$862$	16624.7	0.656890
$863$	−4757.13	−0.187642	−0.0938208	−	0.995589i	$-0.529908\pi$
$863$	−4757.13	−0.187642	−0.0938208	+	0.995589i	$0.529908\pi$
$864$	0	0
$865$	0	0
$866$	−20743.3	−0.813954
$867$	0	0
$868$	−1789.15	−0.0699629
$869$	−5978.28	−0.233371
$870$	0	0
$871$	280.663	0.0109184
$872$	30162.1	1.17135
$873$	0	0
$874$	−24949.0	−0.965574
$875$	0	0
$876$	0	0
$877$	30240.5	1.16437	0.582184	−	0.813057i	$-0.302198\pi$
$877$	30240.5	1.16437	0.582184	+	0.813057i	$0.302198\pi$
$878$	−14402.5	−0.553601
$879$	0	0
$880$	0	0
$881$	−44875.5	−1.71611	−0.858056	−	0.513556i	$-0.828328\pi$
$881$	−44875.5	−1.71611	−0.858056	+	0.513556i	$0.828328\pi$
$882$	0	0
$883$	−4892.13	−0.186448	−0.0932238	−	0.995645i	$-0.529717\pi$
$883$	−4892.13	−0.186448	−0.0932238	+	0.995645i	$0.529717\pi$
$884$	−2216.51	−0.0843317
$885$	0	0
$886$	−14186.2	−0.537916
$887$	1761.40	0.0666765	0.0333382	−	0.999444i	$-0.489386\pi$
$887$	1761.40	0.0666765	0.0333382	+	0.999444i	$0.489386\pi$
$888$	0	0
$889$	−8494.43	−0.320466
$890$	0	0
$891$	0	0
$892$	−8367.22	−0.314075
$893$	−516.384	−0.0193507
$894$	0	0
$895$	0	0
$896$	6954.86	0.259314
$897$	0	0
$898$	18702.2	0.694988
$899$	9132.79	0.338816
$900$	0	0
$901$	18231.8	0.674128
$902$	−33569.3	−1.23918
$903$	0	0
$904$	−32999.0	−1.21408
$905$	0	0
$906$	0	0
$907$	−23689.1	−0.867238	−0.433619	−	0.901096i	$-0.642764\pi$
$907$	−23689.1	−0.867238	−0.433619	+	0.901096i	$0.642764\pi$
$908$	1334.30	0.0487669
$909$	0	0
$910$	0	0
$911$	13877.3	0.504692	0.252346	−	0.967637i	$-0.418798\pi$
$911$	13877.3	0.504692	0.252346	+	0.967637i	$0.418798\pi$
$912$	0	0
$913$	39652.2	1.43735
$914$	7500.09	0.271423
$915$	0	0
$916$	−5397.18	−0.194681
$917$	13876.9	0.499735
$918$	0	0
$919$	14331.6	0.514426	0.257213	−	0.966355i	$-0.417196\pi$
$919$	14331.6	0.514426	0.257213	+	0.966355i	$0.417196\pi$
$920$	0	0
$921$	0	0
$922$	−15705.0	−0.560971
$923$	−18414.7	−0.656692
$924$	0	0
$925$	0	0
$926$	48930.2	1.73644
$927$	0	0
$928$	2760.80	0.0976592
$929$	−16668.4	−0.588668	−0.294334	−	0.955703i	$-0.595098\pi$
$929$	−16668.4	−0.588668	−0.294334	+	0.955703i	$0.595098\pi$
$930$	0	0
$931$	−2167.04	−0.0762857
$932$	−799.300	−0.0280922
$933$	0	0
$934$	−17523.1	−0.613891
$935$	0	0
$936$	0	0
$937$	−30384.9	−1.05937	−0.529685	−	0.848194i	$-0.677690\pi$
$937$	−30384.9	−1.05937	−0.529685	+	0.848194i	$0.677690\pi$
$938$	−262.632	−0.00914206
$939$	0	0
$940$	0	0
$941$	1196.35	0.0414452	0.0207226	−	0.999785i	$-0.493403\pi$
$941$	1196.35	0.0414452	0.0207226	+	0.999785i	$0.493403\pi$
$942$	0	0
$943$	74037.5	2.55673
$944$	−31886.9	−1.09940
$945$	0	0
$946$	−22372.1	−0.768901
$947$	1788.41	0.0613681	0.0306840	−	0.999529i	$-0.490231\pi$
$947$	1788.41	0.0613681	0.0306840	+	0.999529i	$0.490231\pi$
$948$	0	0
$949$	15948.2	0.545523
$950$	0	0
$951$	0	0
$952$	14704.7	0.500612
$953$	8578.60	0.291593	0.145796	−	0.989315i	$-0.453426\pi$
$953$	8578.60	0.291593	0.145796	+	0.989315i	$0.453426\pi$
$954$	0	0
$955$	0	0
$956$	−6640.06	−0.224639
$957$	0	0
$958$	−6198.95	−0.209059
$959$	15476.6	0.521133
$960$	0	0
$961$	8061.99	0.270618
$962$	18349.0	0.614963
$963$	0	0
$964$	−6.39489	−0.000213657 0
$965$	0	0
$966$	0	0
$967$	55459.3	1.84431	0.922156	−	0.386818i	$-0.126426\pi$
$967$	55459.3	1.84431	0.922156	+	0.386818i	$0.126426\pi$
$968$	−3189.40	−0.105900
$969$	0	0
$970$	0	0
$971$	−22047.3	−0.728662	−0.364331	−	0.931270i	$-0.618702\pi$
$971$	−22047.3	−0.728662	−0.364331	+	0.931270i	$0.618702\pi$
$972$	0	0
$973$	3696.27	0.121785
$974$	−14444.6	−0.475191
$975$	0	0
$976$	−16602.7	−0.544507
$977$	14402.3	0.471617	0.235809	−	0.971800i	$-0.424226\pi$
$977$	14402.3	0.471617	0.235809	+	0.971800i	$0.424226\pi$
$978$	0	0
$979$	−6511.94	−0.212587
$980$	0	0
$981$	0	0
$982$	−1389.58	−0.0451561
$983$	7817.11	0.253639	0.126819	−	0.991926i	$-0.459523\pi$
$983$	7817.11	0.253639	0.126819	+	0.991926i	$0.459523\pi$
$984$	0	0
$985$	0	0
$986$	−10587.4	−0.341959
$987$	0	0
$988$	−1123.82	−0.0361878
$989$	49342.0	1.58643
$990$	0	0
$991$	24501.6	0.785386	0.392693	−	0.919670i	$-0.371543\pi$
$991$	24501.6	0.785386	0.392693	+	0.919670i	$0.371543\pi$
$992$	11442.8	0.366239
$993$	0	0
$994$	17231.7	0.549855
$995$	0	0
$996$	0	0
$997$	50696.0	1.61039	0.805195	−	0.593010i	$-0.202060\pi$
$997$	50696.0	1.61039	0.805195	+	0.593010i	$0.202060\pi$
$998$	1548.80	0.0491245
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.4.a.z.1.1		2
3.2	odd	2		175.4.a.c.1.2		2
5.4	even	2		315.4.a.f.1.2		2
15.2	even	4		175.4.b.c.99.2		4
15.8	even	4		175.4.b.c.99.3		4
15.14	odd	2		35.4.a.b.1.1	✓	2
21.20	even	2		1225.4.a.m.1.2		2
35.34	odd	2		2205.4.a.u.1.2		2
60.59	even	2		560.4.a.r.1.1		2
105.44	odd	6		245.4.e.h.116.2		4
105.59	even	6		245.4.e.i.226.2		4
105.74	odd	6		245.4.e.h.226.2		4
105.89	even	6		245.4.e.i.116.2		4
105.104	even	2		245.4.a.k.1.1		2
120.29	odd	2		2240.4.a.bn.1.1		2
120.59	even	2		2240.4.a.bo.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.4.a.b.1.1	✓	2	15.14	odd	2
175.4.a.c.1.2		2	3.2	odd	2
175.4.b.c.99.2		4	15.2	even	4
175.4.b.c.99.3		4	15.8	even	4
245.4.a.k.1.1		2	105.104	even	2
245.4.e.h.116.2		4	105.44	odd	6
245.4.e.h.226.2		4	105.74	odd	6
245.4.e.i.116.2		4	105.89	even	6
245.4.e.i.226.2		4	105.59	even	6
315.4.a.f.1.2		2	5.4	even	2
560.4.a.r.1.1		2	60.59	even	2
1225.4.a.m.1.2		2	21.20	even	2
1575.4.a.z.1.1		2	1.1	even	1	trivial
2205.4.a.u.1.2		2	35.34	odd	2
2240.4.a.bn.1.1		2	120.29	odd	2
2240.4.a.bo.1.2		2	120.59	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)

\(f(q)\)	\(=\)	\(q+2.58579 q^{2} -1.31371 q^{4} +7.00000 q^{7} -24.0833 q^{8} -38.2548 q^{11} -19.3431 q^{13} +18.1005 q^{14} -51.7645 q^{16} -87.2254 q^{17} -44.2254 q^{19} -98.9188 q^{22} +218.167 q^{23} -50.0172 q^{26} -9.19596 q^{28} +46.9411 q^{29} +194.558 q^{31} +58.8141 q^{32} -225.546 q^{34} -366.853 q^{37} -114.357 q^{38} +339.362 q^{41} +226.167 q^{43} +50.2557 q^{44} +564.132 q^{46} +11.6762 q^{47} +49.0000 q^{49} +25.4113 q^{52} -209.019 q^{53} -168.583 q^{56} +121.380 q^{58} +616.000 q^{59} +320.735 q^{61} +503.087 q^{62} +566.197 q^{64} -14.5097 q^{67} +114.589 q^{68} +952.000 q^{71} -824.489 q^{73} -948.603 q^{74} +58.0993 q^{76} -267.784 q^{77} +156.275 q^{79} +877.519 q^{82} -1036.53 q^{83} +584.818 q^{86} +921.301 q^{88} +170.225 q^{89} -135.402 q^{91} -286.607 q^{92} +30.1921 q^{94} -1059.87 q^{97} +126.704 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 8 q^{2} + 20 q^{4} + 14 q^{7} + 48 q^{8} + 14 q^{11} - 50 q^{13} + 56 q^{14} + 168 q^{16} - 50 q^{17} + 36 q^{19} + 184 q^{22} + 244 q^{23} - 216 q^{26} + 140 q^{28} + 26 q^{29} - 120 q^{31} + 672 q^{32}+ \cdots + 392 q^{98}+O(q^{100}) \) 2 * q + 8 * q^2 + 20 * q^4 + 14 * q^7 + 48 * q^8 + 14 * q^11 - 50 * q^13 + 56 * q^14 + 168 * q^16 - 50 * q^17 + 36 * q^19 + 184 * q^22 + 244 * q^23 - 216 * q^26 + 140 * q^28 + 26 * q^29 - 120 * q^31 + 672 * q^32 - 24 * q^34 - 564 * q^37 + 320 * q^38 + 328 * q^41 + 260 * q^43 + 1164 * q^44 + 704 * q^46 - 350 * q^47 + 98 * q^49 - 628 * q^52 - 56 * q^53 + 336 * q^56 + 8 * q^58 + 1232 * q^59 + 336 * q^61 - 1200 * q^62 + 2128 * q^64 + 152 * q^67 + 908 * q^68 + 1904 * q^71 - 676 * q^73 - 2016 * q^74 + 1768 * q^76 + 98 * q^77 + 1014 * q^79 + 816 * q^82 - 376 * q^83 + 768 * q^86 + 4688 * q^88 + 216 * q^89 - 350 * q^91 + 264 * q^92 - 1928 * q^94 - 2742 * q^97 + 392 * q^98

Introduction

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