LMFDB - Embedded newform 1575.4.a.ba.1.2

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 = [3,-3,0,13,0,0,-21,-15,0,0,74] 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:	$3$
Coefficient field:	3.3.14360.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 17x - 14 $ x^3 - 17*x - 14
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			1.2
Root			$-0.861086$ 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)

$f(q)$	$=$	$q-1.86109 q^{2} -4.53636 q^{4} -7.00000 q^{7} +23.3312 q^{8} +36.9807 q^{11} +22.7931 q^{13} +13.0276 q^{14} -7.13061 q^{16} -135.566 q^{17} +6.22620 q^{19} -68.8243 q^{22} -48.7397 q^{23} -42.4199 q^{26} +31.7545 q^{28} +71.1172 q^{29} +124.924 q^{31} -173.379 q^{32} +252.300 q^{34} -84.9919 q^{37} -11.5875 q^{38} -92.5942 q^{41} -299.680 q^{43} -167.758 q^{44} +90.7088 q^{46} -72.9178 q^{47} +49.0000 q^{49} -103.398 q^{52} +362.685 q^{53} -163.319 q^{56} -132.355 q^{58} +375.526 q^{59} +689.610 q^{61} -232.494 q^{62} +379.719 q^{64} +972.591 q^{67} +614.976 q^{68} -281.900 q^{71} +742.980 q^{73} +158.177 q^{74} -28.2443 q^{76} -258.865 q^{77} +592.843 q^{79} +172.326 q^{82} -493.406 q^{83} +557.731 q^{86} +862.806 q^{88} -962.977 q^{89} -159.552 q^{91} +221.101 q^{92} +135.706 q^{94} -740.748 q^{97} -91.1932 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 13 q^{4} - 21 q^{7} - 15 q^{8} + 74 q^{11} - 44 q^{13} + 21 q^{14} - 79 q^{16} - 52 q^{17} + 168 q^{19} - 184 q^{22} - 124 q^{23} + 446 q^{26} - 91 q^{28} - 332 q^{29} + 320 q^{31} - 183 q^{32}+ \cdots - 147 q^{98}+O(q^{100}) $ 3 * q - 3 * q^2 + 13 * q^4 - 21 * q^7 - 15 * q^8 + 74 * q^11 - 44 * q^13 + 21 * q^14 - 79 * q^16 - 52 * q^17 + 168 * q^19 - 184 * q^22 - 124 * q^23 + 446 * q^26 - 91 * q^28 - 332 * q^29 + 320 * q^31 - 183 * q^32 + 582 * q^34 + 54 * q^37 - 460 * q^38 - 362 * q^41 + 16 * q^43 + 264 * q^44 - 336 * q^46 - 730 * q^47 + 147 * q^49 - 1202 * q^52 + 110 * q^53 + 105 * q^56 - 450 * q^58 + 180 * q^59 + 1222 * q^61 + 464 * q^62 - 391 * q^64 - 204 * q^67 + 918 * q^68 + 136 * q^71 - 310 * q^73 - 502 * q^74 + 1796 * q^76 - 518 * q^77 - 1034 * q^79 + 6 * q^82 - 1660 * q^83 - 764 * q^86 + 20 * q^88 - 242 * q^89 + 308 * q^91 + 96 * q^92 - 1108 * q^94 - 100 * q^97 - 147 * 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 $

$2$	−1.86109	−0.657993	−0.328997	−	0.944331i	$-0.606711\pi$
$2$	−1.86109	−0.657993	−0.328997	+	0.944331i	$0.606711\pi$
$3$	0	0
$3$	0	0
$4$	−4.53636	−0.567045
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	23.3312	1.03111
$9$	0	0
$10$	0	0
$11$	36.9807	1.01365	0.506823	−	0.862050i	$-0.330820\pi$
$11$	36.9807	1.01365	0.506823	+	0.862050i	$0.330820\pi$
$12$	0	0
$13$	22.7931	0.486282	0.243141	−	0.969991i	$-0.421822\pi$
$13$	22.7931	0.486282	0.243141	+	0.969991i	$0.421822\pi$
$14$	13.0276	0.248698
$15$	0	0
$16$	−7.13061	−0.111416
$17$	−135.566	−1.93409	−0.967047	−	0.254599i	$-0.918056\pi$
$17$	−135.566	−1.93409	−0.967047	+	0.254599i	$0.918056\pi$
$18$	0	0
$19$	6.22620	0.0751784	0.0375892	−	0.999293i	$-0.488032\pi$
$19$	6.22620	0.0751784	0.0375892	+	0.999293i	$0.488032\pi$
$20$	0	0
$21$	0	0
$22$	−68.8243	−0.666972
$23$	−48.7397	−0.441866	−0.220933	−	0.975289i	$-0.570910\pi$
$23$	−48.7397	−0.441866	−0.220933	+	0.975289i	$0.570910\pi$
$24$	0	0
$25$	0	0
$26$	−42.4199	−0.319970
$27$	0	0
$28$	31.7545	0.214323
$29$	71.1172	0.455384	0.227692	−	0.973733i	$-0.426882\pi$
$29$	71.1172	0.455384	0.227692	+	0.973733i	$0.426882\pi$
$30$	0	0
$31$	124.924	0.723773	0.361886	−	0.932222i	$-0.382133\pi$
$31$	124.924	0.723773	0.361886	+	0.932222i	$0.382133\pi$
$32$	−173.379	−0.957794
$33$	0	0
$34$	252.300	1.27262
$35$	0	0
$36$	0	0
$37$	−84.9919	−0.377638	−0.188819	−	0.982012i	$-0.560466\pi$
$37$	−84.9919	−0.377638	−0.188819	+	0.982012i	$0.560466\pi$
$38$	−11.5875	−0.0494669
$39$	0	0
$40$	0	0
$41$	−92.5942	−0.352702	−0.176351	−	0.984327i	$-0.556429\pi$
$41$	−92.5942	−0.352702	−0.176351	+	0.984327i	$0.556429\pi$
$42$	0	0
$43$	−299.680	−1.06281	−0.531405	−	0.847118i	$-0.678336\pi$
$43$	−299.680	−1.06281	−0.531405	+	0.847118i	$0.678336\pi$
$44$	−167.758	−0.574782
$45$	0	0
$46$	90.7088	0.290745
$47$	−72.9178	−0.226301	−0.113151	−	0.993578i	$-0.536094\pi$
$47$	−72.9178	−0.226301	−0.113151	+	0.993578i	$0.536094\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	0	0
$52$	−103.398	−0.275744
$53$	362.685	0.939974	0.469987	−	0.882673i	$-0.344259\pi$
$53$	362.685	0.939974	0.469987	+	0.882673i	$0.344259\pi$
$54$	0	0
$55$	0	0
$56$	−163.319	−0.389721
$57$	0	0
$58$	−132.355	−0.299640
$59$	375.526	0.828633	0.414316	−	0.910133i	$-0.364021\pi$
$59$	375.526	0.828633	0.414316	+	0.910133i	$0.364021\pi$
$60$	0	0
$61$	689.610	1.44747	0.723734	−	0.690079i	$-0.242424\pi$
$61$	689.610	1.44747	0.723734	+	0.690079i	$0.242424\pi$
$62$	−232.494	−0.476238
$63$	0	0
$64$	379.719	0.741638
$65$	0	0
$66$	0	0
$67$	972.591	1.77345	0.886723	−	0.462301i	$-0.152976\pi$
$67$	972.591	1.77345	0.886723	+	0.462301i	$0.152976\pi$
$68$	614.976	1.09672
$69$	0	0
$70$	0	0
$71$	−281.900	−0.471202	−0.235601	−	0.971850i	$-0.575706\pi$
$71$	−281.900	−0.471202	−0.235601	+	0.971850i	$0.575706\pi$
$72$	0	0
$73$	742.980	1.19122	0.595612	−	0.803273i	$-0.296910\pi$
$73$	742.980	1.19122	0.595612	+	0.803273i	$0.296910\pi$
$74$	158.177	0.248483
$75$	0	0
$76$	−28.2443	−0.0426295
$77$	−258.865	−0.383122
$78$	0	0
$79$	592.843	0.844304	0.422152	−	0.906525i	$-0.361275\pi$
$79$	592.843	0.844304	0.422152	+	0.906525i	$0.361275\pi$
$80$	0	0
$81$	0	0
$82$	172.326	0.232076
$83$	−493.406	−0.652510	−0.326255	−	0.945282i	$-0.605787\pi$
$83$	−493.406	−0.652510	−0.326255	+	0.945282i	$0.605787\pi$
$84$	0	0
$85$	0	0
$86$	557.731	0.699322
$87$	0	0
$88$	862.806	1.04518
$89$	−962.977	−1.14691	−0.573457	−	0.819236i	$-0.694398\pi$
$89$	−962.977	−1.14691	−0.573457	+	0.819236i	$0.694398\pi$
$90$	0	0
$91$	−159.552	−0.183797
$92$	221.101	0.250558
$93$	0	0
$94$	135.706	0.148905
$95$	0	0
$96$	0	0
$97$	−740.748	−0.775377	−0.387689	−	0.921790i	$-0.626726\pi$
$97$	−740.748	−0.775377	−0.387689	+	0.921790i	$0.626726\pi$
$98$	−91.1932	−0.0939991
$99$	0	0
$100$	0	0
$101$	−613.794	−0.604701	−0.302351	−	0.953197i	$-0.597771\pi$
$101$	−613.794	−0.604701	−0.302351	+	0.953197i	$0.597771\pi$
$102$	0	0
$103$	−805.493	−0.770559	−0.385280	−	0.922800i	$-0.625895\pi$
$103$	−805.493	−0.770559	−0.385280	+	0.922800i	$0.625895\pi$
$104$	531.791	0.501408
$105$	0	0
$106$	−674.988	−0.618497
$107$	−1931.30	−1.74491	−0.872457	−	0.488691i	$-0.837474\pi$
$107$	−1931.30	−1.74491	−0.872457	+	0.488691i	$0.837474\pi$
$108$	0	0
$109$	106.462	0.0935522	0.0467761	−	0.998905i	$-0.485105\pi$
$109$	106.462	0.0935522	0.0467761	+	0.998905i	$0.485105\pi$
$110$	0	0
$111$	0	0
$112$	49.9142	0.0421112
$113$	309.076	0.257305	0.128652	−	0.991690i	$-0.458935\pi$
$113$	309.076	0.257305	0.128652	+	0.991690i	$0.458935\pi$
$114$	0	0
$115$	0	0
$116$	−322.613	−0.258223
$117$	0	0
$118$	−698.887	−0.545235
$119$	948.962	0.731019
$120$	0	0
$121$	36.5724	0.0274774
$122$	−1283.42	−0.952425
$123$	0	0
$124$	−566.699	−0.410412
$125$	0	0
$126$	0	0
$127$	199.470	0.139371	0.0696856	−	0.997569i	$-0.477800\pi$
$127$	199.470	0.139371	0.0696856	+	0.997569i	$0.477800\pi$
$128$	680.345	0.469801
$129$	0	0
$130$	0	0
$131$	−601.722	−0.401318	−0.200659	−	0.979661i	$-0.564308\pi$
$131$	−601.722	−0.401318	−0.200659	+	0.979661i	$0.564308\pi$
$132$	0	0
$133$	−43.5834	−0.0284147
$134$	−1810.08	−1.16692
$135$	0	0
$136$	−3162.92	−1.99425
$137$	2092.21	1.30474	0.652370	−	0.757901i	$-0.273775\pi$
$137$	2092.21	1.30474	0.652370	+	0.757901i	$0.273775\pi$
$138$	0	0
$139$	834.466	0.509198	0.254599	−	0.967047i	$-0.418057\pi$
$139$	834.466	0.509198	0.254599	+	0.967047i	$0.418057\pi$
$140$	0	0
$141$	0	0
$142$	524.639	0.310048
$143$	842.905	0.492918
$144$	0	0
$145$	0	0
$146$	−1382.75	−0.783817
$147$	0	0
$148$	385.554	0.214137
$149$	244.258	0.134298	0.0671491	−	0.997743i	$-0.478610\pi$
$149$	244.258	0.134298	0.0671491	+	0.997743i	$0.478610\pi$
$150$	0	0
$151$	−802.158	−0.432309	−0.216155	−	0.976359i	$-0.569352\pi$
$151$	−802.158	−0.432309	−0.216155	+	0.976359i	$0.569352\pi$
$152$	145.265	0.0775168
$153$	0	0
$154$	481.770	0.252092
$155$	0	0
$156$	0	0
$157$	−3541.38	−1.80021	−0.900105	−	0.435673i	$-0.856510\pi$
$157$	−3541.38	−1.80021	−0.900105	+	0.435673i	$0.856510\pi$
$158$	−1103.33	−0.555546
$159$	0	0
$160$	0	0
$161$	341.178	0.167010
$162$	0	0
$163$	2214.57	1.06416	0.532081	−	0.846693i	$-0.321410\pi$
$163$	2214.57	1.06416	0.532081	+	0.846693i	$0.321410\pi$
$164$	420.041	0.199998
$165$	0	0
$166$	918.271	0.429347
$167$	−2617.07	−1.21266	−0.606332	−	0.795212i	$-0.707360\pi$
$167$	−2617.07	−1.21266	−0.606332	+	0.795212i	$0.707360\pi$
$168$	0	0
$169$	−1677.47	−0.763530
$170$	0	0
$171$	0	0
$172$	1359.46	0.602661
$173$	−1634.04	−0.718114	−0.359057	−	0.933316i	$-0.616902\pi$
$173$	−1634.04	−0.718114	−0.359057	+	0.933316i	$0.616902\pi$
$174$	0	0
$175$	0	0
$176$	−263.695	−0.112936
$177$	0	0
$178$	1792.18	0.754662
$179$	−969.160	−0.404684	−0.202342	−	0.979315i	$-0.564855\pi$
$179$	−969.160	−0.404684	−0.202342	+	0.979315i	$0.564855\pi$
$180$	0	0
$181$	4358.20	1.78974	0.894869	−	0.446329i	$-0.147269\pi$
$181$	4358.20	1.78974	0.894869	+	0.446329i	$0.147269\pi$
$182$	296.939	0.120937
$183$	0	0
$184$	−1137.16	−0.455611
$185$	0	0
$186$	0	0
$187$	−5013.33	−1.96049
$188$	330.781	0.128323
$189$	0	0
$190$	0	0
$191$	−2909.97	−1.10240	−0.551199	−	0.834374i	$-0.685830\pi$
$191$	−2909.97	−1.10240	−0.551199	+	0.834374i	$0.685830\pi$
$192$	0	0
$193$	3719.25	1.38714	0.693568	−	0.720391i	$-0.256038\pi$
$193$	3719.25	1.38714	0.693568	+	0.720391i	$0.256038\pi$
$194$	1378.60	0.510193
$195$	0	0
$196$	−222.282	−0.0810064
$197$	−1550.03	−0.560582	−0.280291	−	0.959915i	$-0.590431\pi$
$197$	−1550.03	−0.560582	−0.280291	+	0.959915i	$0.590431\pi$
$198$	0	0
$199$	−3605.14	−1.28423	−0.642115	−	0.766608i	$-0.721943\pi$
$199$	−3605.14	−1.28423	−0.642115	+	0.766608i	$0.721943\pi$
$200$	0	0
$201$	0	0
$202$	1142.32	0.397889
$203$	−497.820	−0.172119
$204$	0	0
$205$	0	0
$206$	1499.09	0.507023
$207$	0	0
$208$	−162.529	−0.0541795
$209$	230.249	0.0762042
$210$	0	0
$211$	−3305.27	−1.07841	−0.539204	−	0.842175i	$-0.681275\pi$
$211$	−3305.27	−1.07841	−0.539204	+	0.842175i	$0.681275\pi$
$212$	−1645.27	−0.533007
$213$	0	0
$214$	3594.31	1.14814
$215$	0	0
$216$	0	0
$217$	−874.466	−0.273560
$218$	−198.135	−0.0615567
$219$	0	0
$220$	0	0
$221$	−3089.97	−0.940515
$222$	0	0
$223$	−3451.37	−1.03642	−0.518209	−	0.855254i	$-0.673401\pi$
$223$	−3451.37	−1.03642	−0.518209	+	0.855254i	$0.673401\pi$
$224$	1213.65	0.362012
$225$	0	0
$226$	−575.218	−0.169305
$227$	2047.24	0.598591	0.299296	−	0.954160i	$-0.403248\pi$
$227$	2047.24	0.598591	0.299296	+	0.954160i	$0.403248\pi$
$228$	0	0
$229$	−1387.42	−0.400362	−0.200181	−	0.979759i	$-0.564153\pi$
$229$	−1387.42	−0.400362	−0.200181	+	0.979759i	$0.564153\pi$
$230$	0	0
$231$	0	0
$232$	1659.25	0.469549
$233$	−374.993	−0.105436	−0.0527181	−	0.998609i	$-0.516788\pi$
$233$	−374.993	−0.105436	−0.0527181	+	0.998609i	$0.516788\pi$
$234$	0	0
$235$	0	0
$236$	−1703.52	−0.469872
$237$	0	0
$238$	−1766.10	−0.481005
$239$	5560.93	1.50505	0.752525	−	0.658564i	$-0.228836\pi$
$239$	5560.93	1.50505	0.752525	+	0.658564i	$0.228836\pi$
$240$	0	0
$241$	4706.19	1.25789	0.628947	−	0.777448i	$-0.283486\pi$
$241$	4706.19	1.25789	0.628947	+	0.777448i	$0.283486\pi$
$242$	−68.0644	−0.0180799
$243$	0	0
$244$	−3128.32	−0.820779
$245$	0	0
$246$	0	0
$247$	141.914	0.0365579
$248$	2914.63	0.746286
$249$	0	0
$250$	0	0
$251$	−589.085	−0.148138	−0.0740692	−	0.997253i	$-0.523599\pi$
$251$	−589.085	−0.148138	−0.0740692	+	0.997253i	$0.523599\pi$
$252$	0	0
$253$	−1802.43	−0.447896
$254$	−371.232	−0.0917053
$255$	0	0
$256$	−4303.93	−1.05076
$257$	−4666.16	−1.13256	−0.566278	−	0.824214i	$-0.691617\pi$
$257$	−4666.16	−1.13256	−0.566278	+	0.824214i	$0.691617\pi$
$258$	0	0
$259$	594.944	0.142734
$260$	0	0
$261$	0	0
$262$	1119.86	0.264065
$263$	−4471.96	−1.04849	−0.524246	−	0.851567i	$-0.675653\pi$
$263$	−4471.96	−1.04849	−0.524246	+	0.851567i	$0.675653\pi$
$264$	0	0
$265$	0	0
$266$	81.1125	0.0186967
$267$	0	0
$268$	−4412.02	−1.00562
$269$	4257.13	0.964913	0.482457	−	0.875920i	$-0.339745\pi$
$269$	4257.13	0.964913	0.482457	+	0.875920i	$0.339745\pi$
$270$	0	0
$271$	−3868.69	−0.867182	−0.433591	−	0.901110i	$-0.642754\pi$
$271$	−3868.69	−0.867182	−0.433591	+	0.901110i	$0.642754\pi$
$272$	966.668	0.215488
$273$	0	0
$274$	−3893.78	−0.858510
$275$	0	0
$276$	0	0
$277$	−8207.30	−1.78025	−0.890124	−	0.455718i	$-0.849382\pi$
$277$	−8207.30	−1.78025	−0.890124	+	0.455718i	$0.849382\pi$
$278$	−1553.01	−0.335049
$279$	0	0
$280$	0	0
$281$	−6471.27	−1.37382	−0.686910	−	0.726743i	$-0.741033\pi$
$281$	−6471.27	−1.37382	−0.686910	+	0.726743i	$0.741033\pi$
$282$	0	0
$283$	−1470.80	−0.308940	−0.154470	−	0.987997i	$-0.549367\pi$
$283$	−1470.80	−0.308940	−0.154470	+	0.987997i	$0.549367\pi$
$284$	1278.80	0.267192
$285$	0	0
$286$	−1568.72	−0.324337
$287$	648.160	0.133309
$288$	0	0
$289$	13465.1	2.74072
$290$	0	0
$291$	0	0
$292$	−3370.42	−0.675477
$293$	−5489.18	−1.09448	−0.547238	−	0.836977i	$-0.684321\pi$
$293$	−5489.18	−1.09448	−0.547238	+	0.836977i	$0.684321\pi$
$294$	0	0
$295$	0	0
$296$	−1982.97	−0.389384
$297$	0	0
$298$	−454.586	−0.0883673
$299$	−1110.93	−0.214872
$300$	0	0
$301$	2097.76	0.401705
$302$	1492.89	0.284457
$303$	0	0
$304$	−44.3966	−0.00837605
$305$	0	0
$306$	0	0
$307$	−1035.35	−0.192477	−0.0962383	−	0.995358i	$-0.530681\pi$
$307$	−1035.35	−0.192477	−0.0962383	+	0.995358i	$0.530681\pi$
$308$	1174.30	0.217247
$309$	0	0
$310$	0	0
$311$	−2544.04	−0.463856	−0.231928	−	0.972733i	$-0.574503\pi$
$311$	−2544.04	−0.463856	−0.231928	+	0.972733i	$0.574503\pi$
$312$	0	0
$313$	−2599.72	−0.469473	−0.234737	−	0.972059i	$-0.575423\pi$
$313$	−2599.72	−0.469473	−0.234737	+	0.972059i	$0.575423\pi$
$314$	6590.82	1.18453
$315$	0	0
$316$	−2689.35	−0.478758
$317$	−2725.13	−0.482835	−0.241417	−	0.970421i	$-0.577612\pi$
$317$	−2725.13	−0.482835	−0.241417	+	0.970421i	$0.577612\pi$
$318$	0	0
$319$	2629.96	0.461598
$320$	0	0
$321$	0	0
$322$	−634.961	−0.109891
$323$	−844.061	−0.145402
$324$	0	0
$325$	0	0
$326$	−4121.50	−0.700212
$327$	0	0
$328$	−2160.34	−0.363673
$329$	510.425	0.0855338
$330$	0	0
$331$	−5178.12	−0.859865	−0.429933	−	0.902861i	$-0.641463\pi$
$331$	−5178.12	−0.859865	−0.429933	+	0.902861i	$0.641463\pi$
$332$	2238.26	0.370002
$333$	0	0
$334$	4870.59	0.797924
$335$	0	0
$336$	0	0
$337$	3656.07	0.590975	0.295488	−	0.955347i	$-0.404518\pi$
$337$	3656.07	0.590975	0.295488	+	0.955347i	$0.404518\pi$
$338$	3121.93	0.502398
$339$	0	0
$340$	0	0
$341$	4619.77	0.733649
$342$	0	0
$343$	−343.000	−0.0539949
$344$	−6991.92	−1.09587
$345$	0	0
$346$	3041.09	0.472514
$347$	−1673.15	−0.258846	−0.129423	−	0.991589i	$-0.541312\pi$
$347$	−1673.15	−0.258846	−0.129423	+	0.991589i	$0.541312\pi$
$348$	0	0
$349$	777.313	0.119222	0.0596112	−	0.998222i	$-0.481014\pi$
$349$	777.313	0.119222	0.0596112	+	0.998222i	$0.481014\pi$
$350$	0	0
$351$	0	0
$352$	−6411.69	−0.970864
$353$	4422.92	0.666879	0.333439	−	0.942772i	$-0.391791\pi$
$353$	4422.92	0.666879	0.333439	+	0.942772i	$0.391791\pi$
$354$	0	0
$355$	0	0
$356$	4368.41	0.650351
$357$	0	0
$358$	1803.69	0.266279
$359$	962.163	0.141451	0.0707256	−	0.997496i	$-0.477469\pi$
$359$	962.163	0.141451	0.0707256	+	0.997496i	$0.477469\pi$
$360$	0	0
$361$	−6820.23	−0.994348
$362$	−8110.99	−1.17764
$363$	0	0
$364$	723.783	0.104221
$365$	0	0
$366$	0	0
$367$	7282.68	1.03584	0.517919	−	0.855430i	$-0.326707\pi$
$367$	7282.68	1.03584	0.517919	+	0.855430i	$0.326707\pi$
$368$	347.543	0.0492308
$369$	0	0
$370$	0	0
$371$	−2538.79	−0.355277
$372$	0	0
$373$	1149.24	0.159532	0.0797658	−	0.996814i	$-0.474583\pi$
$373$	1149.24	0.159532	0.0797658	+	0.996814i	$0.474583\pi$
$374$	9330.23	1.28999
$375$	0	0
$376$	−1701.26	−0.233340
$377$	1620.98	0.221445
$378$	0	0
$379$	−10452.6	−1.41666	−0.708332	−	0.705879i	$-0.750552\pi$
$379$	−10452.6	−1.41666	−0.708332	+	0.705879i	$0.750552\pi$
$380$	0	0
$381$	0	0
$382$	5415.71	0.725371
$383$	−13469.3	−1.79699	−0.898496	−	0.438982i	$-0.855339\pi$
$383$	−13469.3	−1.79699	−0.898496	+	0.438982i	$0.855339\pi$
$384$	0	0
$385$	0	0
$386$	−6921.84	−0.912727
$387$	0	0
$388$	3360.30	0.439674
$389$	−10635.5	−1.38622	−0.693110	−	0.720832i	$-0.743760\pi$
$389$	−10635.5	−1.38622	−0.693110	+	0.720832i	$0.743760\pi$
$390$	0	0
$391$	6607.44	0.854611
$392$	1143.23	0.147301
$393$	0	0
$394$	2884.73	0.368860
$395$	0	0
$396$	0	0
$397$	−1031.93	−0.130456	−0.0652282	−	0.997870i	$-0.520778\pi$
$397$	−1031.93	−0.130456	−0.0652282	+	0.997870i	$0.520778\pi$
$398$	6709.48	0.845015
$399$	0	0
$400$	0	0
$401$	6468.95	0.805596	0.402798	−	0.915289i	$-0.368038\pi$
$401$	6468.95	0.805596	0.402798	+	0.915289i	$0.368038\pi$
$402$	0	0
$403$	2847.40	0.351958
$404$	2784.39	0.342893
$405$	0	0
$406$	926.487	0.113253
$407$	−3143.06	−0.382791
$408$	0	0
$409$	−8652.18	−1.04602	−0.523011	−	0.852326i	$-0.675191\pi$
$409$	−8652.18	−1.04602	−0.523011	+	0.852326i	$0.675191\pi$
$410$	0	0
$411$	0	0
$412$	3654.01	0.436942
$413$	−2628.68	−0.313194
$414$	0	0
$415$	0	0
$416$	−3951.85	−0.465758
$417$	0	0
$418$	−428.514	−0.0501419
$419$	−7303.41	−0.851539	−0.425770	−	0.904832i	$-0.639997\pi$
$419$	−7303.41	−0.851539	−0.425770	+	0.904832i	$0.639997\pi$
$420$	0	0
$421$	−11599.8	−1.34285	−0.671425	−	0.741072i	$-0.734317\pi$
$421$	−11599.8	−1.34285	−0.671425	+	0.741072i	$0.734317\pi$
$422$	6151.39	0.709585
$423$	0	0
$424$	8461.89	0.969212
$425$	0	0
$426$	0	0
$427$	−4827.27	−0.547092
$428$	8761.06	0.989444
$429$	0	0
$430$	0	0
$431$	1506.87	0.168407	0.0842034	−	0.996449i	$-0.473165\pi$
$431$	1506.87	0.168407	0.0842034	+	0.996449i	$0.473165\pi$
$432$	0	0
$433$	−2112.02	−0.234405	−0.117203	−	0.993108i	$-0.537393\pi$
$433$	−2112.02	−0.234405	−0.117203	+	0.993108i	$0.537393\pi$
$434$	1627.46	0.180001
$435$	0	0
$436$	−482.949	−0.0530483
$437$	−303.463	−0.0332188
$438$	0	0
$439$	−3492.88	−0.379740	−0.189870	−	0.981809i	$-0.560807\pi$
$439$	−3492.88	−0.379740	−0.189870	+	0.981809i	$0.560807\pi$
$440$	0	0
$441$	0	0
$442$	5750.70	0.618853
$443$	974.674	0.104533	0.0522666	−	0.998633i	$-0.483355\pi$
$443$	974.674	0.104533	0.0522666	+	0.998633i	$0.483355\pi$
$444$	0	0
$445$	0	0
$446$	6423.30	0.681956
$447$	0	0
$448$	−2658.03	−0.280313
$449$	−6113.63	−0.642584	−0.321292	−	0.946980i	$-0.604117\pi$
$449$	−6113.63	−0.642584	−0.321292	+	0.946980i	$0.604117\pi$
$450$	0	0
$451$	−3424.20	−0.357515
$452$	−1402.08	−0.145903
$453$	0	0
$454$	−3810.09	−0.393869
$455$	0	0
$456$	0	0
$457$	1553.51	0.159015	0.0795077	−	0.996834i	$-0.474665\pi$
$457$	1553.51	0.159015	0.0795077	+	0.996834i	$0.474665\pi$
$458$	2582.10	0.263436
$459$	0	0
$460$	0	0
$461$	−9419.28	−0.951626	−0.475813	−	0.879546i	$-0.657846\pi$
$461$	−9419.28	−0.951626	−0.475813	+	0.879546i	$0.657846\pi$
$462$	0	0
$463$	11458.4	1.15014	0.575070	−	0.818104i	$-0.304975\pi$
$463$	11458.4	1.15014	0.575070	+	0.818104i	$0.304975\pi$
$464$	−507.109	−0.0507369
$465$	0	0
$466$	697.895	0.0693763
$467$	2121.08	0.210175	0.105087	−	0.994463i	$-0.466488\pi$
$467$	2121.08	0.210175	0.105087	+	0.994463i	$0.466488\pi$
$468$	0	0
$469$	−6808.14	−0.670300
$470$	0	0
$471$	0	0
$472$	8761.49	0.854407
$473$	−11082.4	−1.07731
$474$	0	0
$475$	0	0
$476$	−4304.83	−0.414520
$477$	0	0
$478$	−10349.4	−0.990313
$479$	−17261.2	−1.64653	−0.823263	−	0.567660i	$-0.807849\pi$
$479$	−17261.2	−1.64653	−0.823263	+	0.567660i	$0.807849\pi$
$480$	0	0
$481$	−1937.23	−0.183638
$482$	−8758.63	−0.827686
$483$	0	0
$484$	−165.905	−0.0155809
$485$	0	0
$486$	0	0
$487$	18516.8	1.72295	0.861473	−	0.507804i	$-0.169543\pi$
$487$	18516.8	1.72295	0.861473	+	0.507804i	$0.169543\pi$
$488$	16089.5	1.49249
$489$	0	0
$490$	0	0
$491$	−7914.77	−0.727472	−0.363736	−	0.931502i	$-0.618499\pi$
$491$	−7914.77	−0.727472	−0.363736	+	0.931502i	$0.618499\pi$
$492$	0	0
$493$	−9641.07	−0.880755
$494$	−264.115	−0.0240549
$495$	0	0
$496$	−890.782	−0.0806397
$497$	1973.30	0.178097
$498$	0	0
$499$	2388.91	0.214314	0.107157	−	0.994242i	$-0.465825\pi$
$499$	2388.91	0.214314	0.107157	+	0.994242i	$0.465825\pi$
$500$	0	0
$501$	0	0
$502$	1096.34	0.0974740
$503$	18073.0	1.60206	0.801030	−	0.598624i	$-0.204285\pi$
$503$	18073.0	1.60206	0.801030	+	0.598624i	$0.204285\pi$
$504$	0	0
$505$	0	0
$506$	3354.47	0.294713
$507$	0	0
$508$	−904.869	−0.0790296
$509$	−18671.7	−1.62595	−0.812975	−	0.582299i	$-0.802153\pi$
$509$	−18671.7	−1.62595	−0.812975	+	0.582299i	$0.802153\pi$
$510$	0	0
$511$	−5200.86	−0.450240
$512$	2567.23	0.221595
$513$	0	0
$514$	8684.12	0.745214
$515$	0	0
$516$	0	0
$517$	−2696.55	−0.229389
$518$	−1107.24	−0.0939178
$519$	0	0
$520$	0	0
$521$	3526.04	0.296504	0.148252	−	0.988950i	$-0.452635\pi$
$521$	3526.04	0.296504	0.148252	+	0.988950i	$0.452635\pi$
$522$	0	0
$523$	10966.3	0.916865	0.458433	−	0.888729i	$-0.348411\pi$
$523$	10966.3	0.916865	0.458433	+	0.888729i	$0.348411\pi$
$524$	2729.62	0.227565
$525$	0	0
$526$	8322.71	0.689900
$527$	−16935.4	−1.39984
$528$	0	0
$529$	−9791.44	−0.804754
$530$	0	0
$531$	0	0
$532$	197.710	0.0161124
$533$	−2110.51	−0.171513
$534$	0	0
$535$	0	0
$536$	22691.8	1.82861
$537$	0	0
$538$	−7922.88	−0.634907
$539$	1812.05	0.144807
$540$	0	0
$541$	11349.8	0.901971	0.450985	−	0.892531i	$-0.351073\pi$
$541$	11349.8	0.901971	0.450985	+	0.892531i	$0.351073\pi$
$542$	7199.97	0.570600
$543$	0	0
$544$	23504.3	1.85246
$545$	0	0
$546$	0	0
$547$	−11206.1	−0.875940	−0.437970	−	0.898989i	$-0.644302\pi$
$547$	−11206.1	−0.875940	−0.437970	+	0.898989i	$0.644302\pi$
$548$	−9491.00	−0.739845
$549$	0	0
$550$	0	0
$551$	442.790	0.0342350
$552$	0	0
$553$	−4149.90	−0.319117
$554$	15274.5	1.17139
$555$	0	0
$556$	−3785.44	−0.288738
$557$	12631.0	0.960847	0.480424	−	0.877037i	$-0.340483\pi$
$557$	12631.0	0.960847	0.480424	+	0.877037i	$0.340483\pi$
$558$	0	0
$559$	−6830.65	−0.516826
$560$	0	0
$561$	0	0
$562$	12043.6	0.903964
$563$	7000.69	0.524057	0.262028	−	0.965060i	$-0.415609\pi$
$563$	7000.69	0.524057	0.262028	+	0.965060i	$0.415609\pi$
$564$	0	0
$565$	0	0
$566$	2737.29	0.203281
$567$	0	0
$568$	−6577.07	−0.485858
$569$	8659.25	0.637987	0.318994	−	0.947757i	$-0.396655\pi$
$569$	8659.25	0.637987	0.318994	+	0.947757i	$0.396655\pi$
$570$	0	0
$571$	25631.9	1.87857	0.939283	−	0.343145i	$-0.111492\pi$
$571$	25631.9	1.87857	0.939283	+	0.343145i	$0.111492\pi$
$572$	−3823.72	−0.279506
$573$	0	0
$574$	−1206.28	−0.0877164
$575$	0	0
$576$	0	0
$577$	−2546.85	−0.183755	−0.0918775	−	0.995770i	$-0.529287\pi$
$577$	−2546.85	−0.183755	−0.0918775	+	0.995770i	$0.529287\pi$
$578$	−25059.8	−1.80337
$579$	0	0
$580$	0	0
$581$	3453.84	0.246626
$582$	0	0
$583$	13412.3	0.952800
$584$	17334.7	1.22828
$585$	0	0
$586$	10215.8	0.720158
$587$	17798.8	1.25150	0.625752	−	0.780022i	$-0.284792\pi$
$587$	17798.8	1.25150	0.625752	+	0.780022i	$0.284792\pi$
$588$	0	0
$589$	777.800	0.0544121
$590$	0	0
$591$	0	0
$592$	606.044	0.0420748
$593$	3191.29	0.220996	0.110498	−	0.993876i	$-0.464755\pi$
$593$	3191.29	0.220996	0.110498	+	0.993876i	$0.464755\pi$
$594$	0	0
$595$	0	0
$596$	−1108.04	−0.0761531
$597$	0	0
$598$	2067.53	0.141384
$599$	−20511.7	−1.39914	−0.699571	−	0.714563i	$-0.746626\pi$
$599$	−20511.7	−1.39914	−0.699571	+	0.714563i	$0.746626\pi$
$600$	0	0
$601$	22802.2	1.54762	0.773810	−	0.633418i	$-0.218349\pi$
$601$	22802.2	1.54762	0.773810	+	0.633418i	$0.218349\pi$
$602$	−3904.12	−0.264319
$603$	0	0
$604$	3638.87	0.245139
$605$	0	0
$606$	0	0
$607$	−3130.15	−0.209306	−0.104653	−	0.994509i	$-0.533373\pi$
$607$	−3130.15	−0.209306	−0.104653	+	0.994509i	$0.533373\pi$
$608$	−1079.49	−0.0720054
$609$	0	0
$610$	0	0
$611$	−1662.02	−0.110046
$612$	0	0
$613$	12736.1	0.839162	0.419581	−	0.907718i	$-0.362177\pi$
$613$	12736.1	0.839162	0.419581	+	0.907718i	$0.362177\pi$
$614$	1926.87	0.126648
$615$	0	0
$616$	−6039.64	−0.395039
$617$	−16662.4	−1.08720	−0.543600	−	0.839345i	$-0.682939\pi$
$617$	−16662.4	−1.08720	−0.543600	+	0.839345i	$0.682939\pi$
$618$	0	0
$619$	−9967.73	−0.647232	−0.323616	−	0.946188i	$-0.604899\pi$
$619$	−9967.73	−0.647232	−0.323616	+	0.946188i	$0.604899\pi$
$620$	0	0
$621$	0	0
$622$	4734.68	0.305215
$623$	6740.84	0.433493
$624$	0	0
$625$	0	0
$626$	4838.31	0.308910
$627$	0	0
$628$	16065.0	1.02080
$629$	11522.0	0.730386
$630$	0	0
$631$	−6243.76	−0.393914	−0.196957	−	0.980412i	$-0.563106\pi$
$631$	−6243.76	−0.393914	−0.196957	+	0.980412i	$0.563106\pi$
$632$	13831.8	0.870566
$633$	0	0
$634$	5071.70	0.317702
$635$	0	0
$636$	0	0
$637$	1116.86	0.0694689
$638$	−4894.59	−0.303728
$639$	0	0
$640$	0	0
$641$	25915.9	1.59690	0.798452	−	0.602059i	$-0.205653\pi$
$641$	25915.9	1.59690	0.798452	+	0.602059i	$0.205653\pi$
$642$	0	0
$643$	−11833.7	−0.725778	−0.362889	−	0.931832i	$-0.618210\pi$
$643$	−11833.7	−0.725778	−0.362889	+	0.931832i	$0.618210\pi$
$644$	−1547.70	−0.0947020
$645$	0	0
$646$	1570.87	0.0956735
$647$	7268.54	0.441662	0.220831	−	0.975312i	$-0.429123\pi$
$647$	7268.54	0.441662	0.220831	+	0.975312i	$0.429123\pi$
$648$	0	0
$649$	13887.2	0.839940
$650$	0	0
$651$	0	0
$652$	−10046.1	−0.603427
$653$	−30455.8	−1.82515	−0.912577	−	0.408904i	$-0.865911\pi$
$653$	−30455.8	−1.82515	−0.912577	+	0.408904i	$0.865911\pi$
$654$	0	0
$655$	0	0
$656$	660.253	0.0392966
$657$	0	0
$658$	−949.945	−0.0562807
$659$	−16170.2	−0.955843	−0.477922	−	0.878403i	$-0.658610\pi$
$659$	−16170.2	−0.955843	−0.477922	+	0.878403i	$0.658610\pi$
$660$	0	0
$661$	10331.9	0.607962	0.303981	−	0.952678i	$-0.401684\pi$
$661$	10331.9	0.607962	0.303981	+	0.952678i	$0.401684\pi$
$662$	9636.93	0.565786
$663$	0	0
$664$	−11511.8	−0.672806
$665$	0	0
$666$	0	0
$667$	−3466.23	−0.201219
$668$	11872.0	0.687634
$669$	0	0
$670$	0	0
$671$	25502.3	1.46722
$672$	0	0
$673$	−18387.1	−1.05315	−0.526576	−	0.850128i	$-0.676525\pi$
$673$	−18387.1	−1.05315	−0.526576	+	0.850128i	$0.676525\pi$
$674$	−6804.25	−0.388858
$675$	0	0
$676$	7609.63	0.432955
$677$	−1795.92	−0.101954	−0.0509770	−	0.998700i	$-0.516234\pi$
$677$	−1795.92	−0.101954	−0.0509770	+	0.998700i	$0.516234\pi$
$678$	0	0
$679$	5185.24	0.293065
$680$	0	0
$681$	0	0
$682$	−8597.78	−0.482736
$683$	−5203.86	−0.291537	−0.145769	−	0.989319i	$-0.546566\pi$
$683$	−5203.86	−0.291537	−0.145769	+	0.989319i	$0.546566\pi$
$684$	0	0
$685$	0	0
$686$	638.353	0.0355283
$687$	0	0
$688$	2136.90	0.118414
$689$	8266.71	0.457092
$690$	0	0
$691$	−8903.56	−0.490170	−0.245085	−	0.969502i	$-0.578816\pi$
$691$	−8903.56	−0.490170	−0.245085	+	0.969502i	$0.578816\pi$
$692$	7412.58	0.407202
$693$	0	0
$694$	3113.88	0.170319
$695$	0	0
$696$	0	0
$697$	12552.6	0.682159
$698$	−1446.65	−0.0784476
$699$	0	0
$700$	0	0
$701$	8343.11	0.449522	0.224761	−	0.974414i	$-0.427840\pi$
$701$	8343.11	0.449522	0.224761	+	0.974414i	$0.427840\pi$
$702$	0	0
$703$	−529.177	−0.0283902
$704$	14042.3	0.751758
$705$	0	0
$706$	−8231.43	−0.438802
$707$	4296.56	0.228556
$708$	0	0
$709$	−28590.0	−1.51441	−0.757206	−	0.653176i	$-0.773436\pi$
$709$	−28590.0	−1.51441	−0.757206	+	0.653176i	$0.773436\pi$
$710$	0	0
$711$	0	0
$712$	−22467.4	−1.18259
$713$	−6088.74	−0.319811
$714$	0	0
$715$	0	0
$716$	4396.46	0.229474
$717$	0	0
$718$	−1790.67	−0.0930740
$719$	−26730.4	−1.38647	−0.693237	−	0.720710i	$-0.743816\pi$
$719$	−26730.4	−1.38647	−0.693237	+	0.720710i	$0.743816\pi$
$720$	0	0
$721$	5638.45	0.291244
$722$	12693.0	0.654275
$723$	0	0
$724$	−19770.4	−1.01486
$725$	0	0
$726$	0	0
$727$	8903.62	0.454219	0.227109	−	0.973869i	$-0.427073\pi$
$727$	8903.62	0.454219	0.227109	+	0.973869i	$0.427073\pi$
$728$	−3722.54	−0.189514
$729$	0	0
$730$	0	0
$731$	40626.5	2.05557
$732$	0	0
$733$	6022.56	0.303477	0.151738	−	0.988421i	$-0.451513\pi$
$733$	6022.56	0.303477	0.151738	+	0.988421i	$0.451513\pi$
$734$	−13553.7	−0.681575
$735$	0	0
$736$	8450.45	0.423217
$737$	35967.1	1.79765
$738$	0	0
$739$	14078.5	0.700795	0.350398	−	0.936601i	$-0.386046\pi$
$739$	14078.5	0.700795	0.350398	+	0.936601i	$0.386046\pi$
$740$	0	0
$741$	0	0
$742$	4724.92	0.233770
$743$	31431.4	1.55196	0.775980	−	0.630757i	$-0.217256\pi$
$743$	31431.4	1.55196	0.775980	+	0.630757i	$0.217256\pi$
$744$	0	0
$745$	0	0
$746$	−2138.83	−0.104971
$747$	0	0
$748$	22742.2	1.11168
$749$	13519.1	0.659515
$750$	0	0
$751$	−2463.12	−0.119681	−0.0598405	−	0.998208i	$-0.519059\pi$
$751$	−2463.12	−0.119681	−0.0598405	+	0.998208i	$0.519059\pi$
$752$	519.948	0.0252135
$753$	0	0
$754$	−3016.79	−0.145709
$755$	0	0
$756$	0	0
$757$	−37987.2	−1.82387	−0.911935	−	0.410335i	$-0.865412\pi$
$757$	−37987.2	−1.82387	−0.911935	+	0.410335i	$0.865412\pi$
$758$	19453.3	0.932156
$759$	0	0
$760$	0	0
$761$	−18691.9	−0.890384	−0.445192	−	0.895435i	$-0.646865\pi$
$761$	−18691.9	−0.890384	−0.445192	+	0.895435i	$0.646865\pi$
$762$	0	0
$763$	−745.232	−0.0353594
$764$	13200.7	0.625109
$765$	0	0
$766$	25067.5	1.18241
$767$	8559.40	0.402949
$768$	0	0
$769$	−27250.5	−1.27786	−0.638932	−	0.769263i	$-0.720624\pi$
$769$	−27250.5	−1.27786	−0.638932	+	0.769263i	$0.720624\pi$
$770$	0	0
$771$	0	0
$772$	−16871.8	−0.786568
$773$	−1092.67	−0.0508417	−0.0254209	−	0.999677i	$-0.508093\pi$
$773$	−1092.67	−0.0508417	−0.0254209	+	0.999677i	$0.508093\pi$
$774$	0	0
$775$	0	0
$776$	−17282.6	−0.799495
$777$	0	0
$778$	19793.5	0.912124
$779$	−576.511	−0.0265156
$780$	0	0
$781$	−10424.8	−0.477631
$782$	−12297.0	−0.562328
$783$	0	0
$784$	−349.400	−0.0159165
$785$	0	0
$786$	0	0
$787$	12639.1	0.572474	0.286237	−	0.958159i	$-0.407596\pi$
$787$	12639.1	0.572474	0.286237	+	0.958159i	$0.407596\pi$
$788$	7031.47	0.317875
$789$	0	0
$790$	0	0
$791$	−2163.53	−0.0972521
$792$	0	0
$793$	15718.4	0.703878
$794$	1920.51	0.0858394
$795$	0	0
$796$	16354.2	0.728216
$797$	8666.66	0.385180	0.192590	−	0.981279i	$-0.438311\pi$
$797$	8666.66	0.385180	0.192590	+	0.981279i	$0.438311\pi$
$798$	0	0
$799$	9885.18	0.437688
$800$	0	0
$801$	0	0
$802$	−12039.3	−0.530077
$803$	27475.9	1.20748
$804$	0	0
$805$	0	0
$806$	−5299.25	−0.231586
$807$	0	0
$808$	−14320.6	−0.623511
$809$	−3555.20	−0.154504	−0.0772522	−	0.997012i	$-0.524615\pi$
$809$	−3555.20	−0.154504	−0.0772522	+	0.997012i	$0.524615\pi$
$810$	0	0
$811$	21940.0	0.949961	0.474981	−	0.879996i	$-0.342455\pi$
$811$	21940.0	0.949961	0.474981	+	0.879996i	$0.342455\pi$
$812$	2258.29	0.0975991
$813$	0	0
$814$	5849.51	0.251874
$815$	0	0
$816$	0	0
$817$	−1865.87	−0.0799003
$818$	16102.5	0.688275
$819$	0	0
$820$	0	0
$821$	29572.7	1.25712	0.628560	−	0.777761i	$-0.283645\pi$
$821$	29572.7	1.25712	0.628560	+	0.777761i	$0.283645\pi$
$822$	0	0
$823$	19314.7	0.818067	0.409034	−	0.912519i	$-0.365866\pi$
$823$	19314.7	0.818067	0.409034	+	0.912519i	$0.365866\pi$
$824$	−18793.2	−0.794528
$825$	0	0
$826$	4892.21	0.206079
$827$	21107.8	0.887535	0.443768	−	0.896142i	$-0.353642\pi$
$827$	21107.8	0.887535	0.443768	+	0.896142i	$0.353642\pi$
$828$	0	0
$829$	10799.8	0.452463	0.226231	−	0.974074i	$-0.427359\pi$
$829$	10799.8	0.452463	0.226231	+	0.974074i	$0.427359\pi$
$830$	0	0
$831$	0	0
$832$	8654.96	0.360645
$833$	−6642.73	−0.276299
$834$	0	0
$835$	0	0
$836$	−1044.49	−0.0432112
$837$	0	0
$838$	13592.3	0.560307
$839$	11829.3	0.486761	0.243381	−	0.969931i	$-0.421744\pi$
$839$	11829.3	0.486761	0.243381	+	0.969931i	$0.421744\pi$
$840$	0	0
$841$	−19331.3	−0.792626
$842$	21588.2	0.883587
$843$	0	0
$844$	14993.9	0.611506
$845$	0	0
$846$	0	0
$847$	−256.007	−0.0103855
$848$	−2586.16	−0.104728
$849$	0	0
$850$	0	0
$851$	4142.48	0.166865
$852$	0	0
$853$	3426.91	0.137556	0.0687779	−	0.997632i	$-0.478090\pi$
$853$	3426.91	0.137556	0.0687779	+	0.997632i	$0.478090\pi$
$854$	8983.97	0.359983
$855$	0	0
$856$	−45059.6	−1.79919
$857$	7308.16	0.291298	0.145649	−	0.989336i	$-0.453473\pi$
$857$	7308.16	0.291298	0.145649	+	0.989336i	$0.453473\pi$
$858$	0	0
$859$	−22539.8	−0.895282	−0.447641	−	0.894213i	$-0.647736\pi$
$859$	−22539.8	−0.895282	−0.447641	+	0.894213i	$0.647736\pi$
$860$	0	0
$861$	0	0
$862$	−2804.41	−0.110811
$863$	1690.95	0.0666983	0.0333491	−	0.999444i	$-0.489383\pi$
$863$	1690.95	0.0666983	0.0333491	+	0.999444i	$0.489383\pi$
$864$	0	0
$865$	0	0
$866$	3930.66	0.154237
$867$	0	0
$868$	3966.89	0.155121
$869$	21923.7	0.855825
$870$	0	0
$871$	22168.4	0.862395
$872$	2483.89	0.0964621
$873$	0	0
$874$	564.771	0.0218577
$875$	0	0
$876$	0	0
$877$	−33021.5	−1.27145	−0.635723	−	0.771917i	$-0.719298\pi$
$877$	−33021.5	−1.27145	−0.635723	+	0.771917i	$0.719298\pi$
$878$	6500.54	0.249866
$879$	0	0
$880$	0	0
$881$	29413.5	1.12482	0.562410	−	0.826859i	$-0.309874\pi$
$881$	29413.5	1.12482	0.562410	+	0.826859i	$0.309874\pi$
$882$	0	0
$883$	−22413.5	−0.854218	−0.427109	−	0.904200i	$-0.640468\pi$
$883$	−22413.5	−0.854218	−0.427109	+	0.904200i	$0.640468\pi$
$884$	14017.2	0.533314
$885$	0	0
$886$	−1813.95	−0.0687821
$887$	−49069.5	−1.85749	−0.928744	−	0.370723i	$-0.879110\pi$
$887$	−49069.5	−1.85749	−0.928744	+	0.370723i	$0.879110\pi$
$888$	0	0
$889$	−1396.29	−0.0526773
$890$	0	0
$891$	0	0
$892$	15656.7	0.587695
$893$	−454.001	−0.0170130
$894$	0	0
$895$	0	0
$896$	−4762.41	−0.177568
$897$	0	0
$898$	11378.0	0.422816
$899$	8884.22	0.329594
$900$	0	0
$901$	−49167.8	−1.81800
$902$	6372.73	0.235243
$903$	0	0
$904$	7211.13	0.265308
$905$	0	0
$906$	0	0
$907$	42854.4	1.56886	0.784431	−	0.620216i	$-0.212955\pi$
$907$	42854.4	1.56886	0.784431	+	0.620216i	$0.212955\pi$
$908$	−9287.02	−0.339428
$909$	0	0
$910$	0	0
$911$	48846.7	1.77647	0.888234	−	0.459391i	$-0.151932\pi$
$911$	48846.7	1.77647	0.888234	+	0.459391i	$0.151932\pi$
$912$	0	0
$913$	−18246.5	−0.661414
$914$	−2891.21	−0.104631
$915$	0	0
$916$	6293.81	0.227023
$917$	4212.05	0.151684
$918$	0	0
$919$	−22400.9	−0.804067	−0.402033	−	0.915625i	$-0.631696\pi$
$919$	−22400.9	−0.804067	−0.402033	+	0.915625i	$0.631696\pi$
$920$	0	0
$921$	0	0
$922$	17530.1	0.626164
$923$	−6425.36	−0.229137
$924$	0	0
$925$	0	0
$926$	−21325.0	−0.756785
$927$	0	0
$928$	−12330.2	−0.436164
$929$	−2298.05	−0.0811587	−0.0405794	−	0.999176i	$-0.512920\pi$
$929$	−2298.05	−0.0811587	−0.0405794	+	0.999176i	$0.512920\pi$
$930$	0	0
$931$	305.084	0.0107398
$932$	1701.10	0.0597870
$933$	0	0
$934$	−3947.51	−0.138294
$935$	0	0
$936$	0	0
$937$	47163.3	1.64435	0.822176	−	0.569233i	$-0.192760\pi$
$937$	47163.3	1.64435	0.822176	+	0.569233i	$0.192760\pi$
$938$	12670.5	0.441053
$939$	0	0
$940$	0	0
$941$	−40457.0	−1.40155	−0.700776	−	0.713382i	$-0.747163\pi$
$941$	−40457.0	−1.40155	−0.700776	+	0.713382i	$0.747163\pi$
$942$	0	0
$943$	4513.01	0.155847
$944$	−2677.73	−0.0923227
$945$	0	0
$946$	20625.3	0.708865
$947$	−6363.14	−0.218347	−0.109173	−	0.994023i	$-0.534820\pi$
$947$	−6363.14	−0.218347	−0.109173	+	0.994023i	$0.534820\pi$
$948$	0	0
$949$	16934.8	0.579270
$950$	0	0
$951$	0	0
$952$	22140.5	0.753757
$953$	−48268.8	−1.64069	−0.820346	−	0.571867i	$-0.806219\pi$
$953$	−48268.8	−1.64069	−0.820346	+	0.571867i	$0.806219\pi$
$954$	0	0
$955$	0	0
$956$	−25226.4	−0.853430
$957$	0	0
$958$	32124.7	1.08340
$959$	−14645.4	−0.493145
$960$	0	0
$961$	−14185.1	−0.476153
$962$	3605.35	0.120833
$963$	0	0
$964$	−21349.0	−0.713282
$965$	0	0
$966$	0	0
$967$	26795.3	0.891084	0.445542	−	0.895261i	$-0.353011\pi$
$967$	26795.3	0.891084	0.445542	+	0.895261i	$0.353011\pi$
$968$	853.280	0.0283321
$969$	0	0
$970$	0	0
$971$	−21583.8	−0.713345	−0.356672	−	0.934230i	$-0.616089\pi$
$971$	−21583.8	−0.713345	−0.356672	+	0.934230i	$0.616089\pi$
$972$	0	0
$973$	−5841.26	−0.192459
$974$	−34461.3	−1.13369
$975$	0	0
$976$	−4917.34	−0.161271
$977$	51106.3	1.67353	0.836763	−	0.547565i	$-0.184445\pi$
$977$	51106.3	1.67353	0.836763	+	0.547565i	$0.184445\pi$
$978$	0	0
$979$	−35611.6	−1.16256
$980$	0	0
$981$	0	0
$982$	14730.1	0.478672
$983$	−41585.7	−1.34932	−0.674658	−	0.738131i	$-0.735709\pi$
$983$	−41585.7	−1.34932	−0.674658	+	0.738131i	$0.735709\pi$
$984$	0	0
$985$	0	0
$986$	17942.9	0.579531
$987$	0	0
$988$	−643.775	−0.0207300
$989$	14606.3	0.469620
$990$	0	0
$991$	−29082.5	−0.932227	−0.466114	−	0.884725i	$-0.654346\pi$
$991$	−29082.5	−0.932227	−0.466114	+	0.884725i	$0.654346\pi$
$992$	−21659.2	−0.693226
$993$	0	0
$994$	−3672.48	−0.117187
$995$	0	0
$996$	0	0
$997$	−34503.0	−1.09601	−0.548004	−	0.836476i	$-0.684612\pi$
$997$	−34503.0	−1.09601	−0.548004	+	0.836476i	$0.684612\pi$
$998$	−4445.98	−0.141017
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.4.a.ba.1.2		3
3.2	odd	2		175.4.a.f.1.2		3
5.4	even	2		315.4.a.p.1.2		3
15.2	even	4		175.4.b.e.99.4		6
15.8	even	4		175.4.b.e.99.3		6
15.14	odd	2		35.4.a.c.1.2	✓	3
21.20	even	2		1225.4.a.y.1.2		3
35.34	odd	2		2205.4.a.bm.1.2		3
60.59	even	2		560.4.a.u.1.1		3
105.44	odd	6		245.4.e.m.116.2		6
105.59	even	6		245.4.e.n.226.2		6
105.74	odd	6		245.4.e.m.226.2		6
105.89	even	6		245.4.e.n.116.2		6
105.104	even	2		245.4.a.l.1.2		3
120.29	odd	2		2240.4.a.bt.1.1		3
120.59	even	2		2240.4.a.bv.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.4.a.c.1.2	✓	3	15.14	odd	2
175.4.a.f.1.2		3	3.2	odd	2
175.4.b.e.99.3		6	15.8	even	4
175.4.b.e.99.4		6	15.2	even	4
245.4.a.l.1.2		3	105.104	even	2
245.4.e.m.116.2		6	105.44	odd	6
245.4.e.m.226.2		6	105.74	odd	6
245.4.e.n.116.2		6	105.89	even	6
245.4.e.n.226.2		6	105.59	even	6
315.4.a.p.1.2		3	5.4	even	2
560.4.a.u.1.1		3	60.59	even	2
1225.4.a.y.1.2		3	21.20	even	2
1575.4.a.ba.1.2		3	1.1	even	1	trivial
2205.4.a.bm.1.2		3	35.34	odd	2
2240.4.a.bt.1.1		3	120.29	odd	2
2240.4.a.bv.1.3		3	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)

\(f(q)\)	\(=\)	\(q-1.86109 q^{2} -4.53636 q^{4} -7.00000 q^{7} +23.3312 q^{8} +36.9807 q^{11} +22.7931 q^{13} +13.0276 q^{14} -7.13061 q^{16} -135.566 q^{17} +6.22620 q^{19} -68.8243 q^{22} -48.7397 q^{23} -42.4199 q^{26} +31.7545 q^{28} +71.1172 q^{29} +124.924 q^{31} -173.379 q^{32} +252.300 q^{34} -84.9919 q^{37} -11.5875 q^{38} -92.5942 q^{41} -299.680 q^{43} -167.758 q^{44} +90.7088 q^{46} -72.9178 q^{47} +49.0000 q^{49} -103.398 q^{52} +362.685 q^{53} -163.319 q^{56} -132.355 q^{58} +375.526 q^{59} +689.610 q^{61} -232.494 q^{62} +379.719 q^{64} +972.591 q^{67} +614.976 q^{68} -281.900 q^{71} +742.980 q^{73} +158.177 q^{74} -28.2443 q^{76} -258.865 q^{77} +592.843 q^{79} +172.326 q^{82} -493.406 q^{83} +557.731 q^{86} +862.806 q^{88} -962.977 q^{89} -159.552 q^{91} +221.101 q^{92} +135.706 q^{94} -740.748 q^{97} -91.1932 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 13 q^{4} - 21 q^{7} - 15 q^{8} + 74 q^{11} - 44 q^{13} + 21 q^{14} - 79 q^{16} - 52 q^{17} + 168 q^{19} - 184 q^{22} - 124 q^{23} + 446 q^{26} - 91 q^{28} - 332 q^{29} + 320 q^{31} - 183 q^{32}+ \cdots - 147 q^{98}+O(q^{100}) \) 3 * q - 3 * q^2 + 13 * q^4 - 21 * q^7 - 15 * q^8 + 74 * q^11 - 44 * q^13 + 21 * q^14 - 79 * q^16 - 52 * q^17 + 168 * q^19 - 184 * q^22 - 124 * q^23 + 446 * q^26 - 91 * q^28 - 332 * q^29 + 320 * q^31 - 183 * q^32 + 582 * q^34 + 54 * q^37 - 460 * q^38 - 362 * q^41 + 16 * q^43 + 264 * q^44 - 336 * q^46 - 730 * q^47 + 147 * q^49 - 1202 * q^52 + 110 * q^53 + 105 * q^56 - 450 * q^58 + 180 * q^59 + 1222 * q^61 + 464 * q^62 - 391 * q^64 - 204 * q^67 + 918 * q^68 + 136 * q^71 - 310 * q^73 - 502 * q^74 + 1796 * q^76 - 518 * q^77 - 1034 * q^79 + 6 * q^82 - 1660 * q^83 - 764 * q^86 + 20 * q^88 - 242 * q^89 + 308 * q^91 + 96 * q^92 - 1108 * q^94 - 100 * q^97 - 147 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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