LMFDB - Embedded newform 675.4.a.r.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [675,4,Mod(1,675)] 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("675.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(675, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 675 = 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	675.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,1,0,23,0,0,-44] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$39.8262892539$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.5637.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 23x + 6 $ x^3 - x^2 - 23*x + 6
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 135)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-4.45938$ of defining polynomial
Character	$\chi$	$=$	675.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-4.45938 q^{2} +11.8861 q^{4} -5.08123 q^{7} -17.3296 q^{8} +58.3007 q^{11} -21.2119 q^{13} +22.6592 q^{14} -17.8095 q^{16} +68.8451 q^{17} -40.8133 q^{19} -259.985 q^{22} +144.318 q^{23} +94.5921 q^{26} -60.3960 q^{28} -220.058 q^{29} +291.545 q^{31} +218.056 q^{32} -307.006 q^{34} -260.637 q^{37} +182.002 q^{38} -169.766 q^{41} +438.596 q^{43} +692.967 q^{44} -643.571 q^{46} +255.481 q^{47} -317.181 q^{49} -252.127 q^{52} -214.714 q^{53} +88.0557 q^{56} +981.322 q^{58} +331.524 q^{59} +54.9647 q^{61} -1300.11 q^{62} -829.920 q^{64} -758.179 q^{67} +818.299 q^{68} -904.348 q^{71} -866.622 q^{73} +1162.28 q^{74} -485.110 q^{76} -296.239 q^{77} +206.961 q^{79} +757.054 q^{82} +463.397 q^{83} -1955.87 q^{86} -1010.33 q^{88} +601.736 q^{89} +107.783 q^{91} +1715.38 q^{92} -1139.29 q^{94} -229.363 q^{97} +1414.43 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} + 23 q^{4} - 44 q^{7} + 36 q^{8} + 38 q^{11} - 28 q^{13} - 108 q^{14} + 191 q^{16} + 19 q^{17} + 187 q^{19} - 122 q^{22} + 81 q^{23} + 416 q^{26} - 410 q^{28} + 160 q^{29} + 227 q^{31} + 569 q^{32}+ \cdots + 2693 q^{98}+O(q^{100}) $ 3 * q + q^2 + 23 * q^4 - 44 * q^7 + 36 * q^8 + 38 * q^11 - 28 * q^13 - 108 * q^14 + 191 * q^16 + 19 * q^17 + 187 * q^19 - 122 * q^22 + 81 * q^23 + 416 * q^26 - 410 * q^28 + 160 * q^29 + 227 * q^31 + 569 * q^32 + 17 * q^34 - 78 * q^37 + 757 * q^38 - 338 * q^41 - 22 * q^43 + 1636 * q^44 - 1425 * q^46 + 472 * q^47 - 197 * q^49 + 1566 * q^52 - 521 * q^53 - 1254 * q^56 + 2096 * q^58 + 140 * q^59 + 595 * q^61 - 1407 * q^62 - 918 * q^64 - 878 * q^67 + 3053 * q^68 - 602 * q^71 - 1294 * q^73 + 2878 * q^74 + 525 * q^76 - 288 * q^77 + 629 * q^79 + 1682 * q^82 + 1287 * q^83 - 3730 * q^86 + 858 * q^88 + 2154 * q^89 - 440 * q^91 - 1959 * q^92 - 1108 * q^94 - 1392 * q^97 + 2693 * q^98

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−4.45938	−1.57663	−0.788315	−	0.615272i	$-0.789046\pi$
$2$	−4.45938	−1.57663	−0.788315	+	0.615272i	$0.789046\pi$
$3$	0	0
$3$	0	0
$4$	11.8861	1.48576
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−5.08123	−0.274361	−0.137180	−	0.990546i	$-0.543804\pi$
$7$	−5.08123	−0.274361	−0.137180	+	0.990546i	$0.543804\pi$
$8$	−17.3296	−0.765867
$9$	0	0
$10$	0	0
$11$	58.3007	1.59803	0.799014	−	0.601312i	$-0.205355\pi$
$11$	58.3007	1.59803	0.799014	+	0.601312i	$0.205355\pi$
$12$	0	0
$13$	−21.2119	−0.452548	−0.226274	−	0.974064i	$-0.572655\pi$
$13$	−21.2119	−0.452548	−0.226274	+	0.974064i	$0.572655\pi$
$14$	22.6592	0.432566
$15$	0	0
$16$	−17.8095	−0.278274
$17$	68.8451	0.982199	0.491099	−	0.871104i	$-0.336595\pi$
$17$	68.8451	0.982199	0.491099	+	0.871104i	$0.336595\pi$
$18$	0	0
$19$	−40.8133	−0.492800	−0.246400	−	0.969168i	$-0.579248\pi$
$19$	−40.8133	−0.492800	−0.246400	+	0.969168i	$0.579248\pi$
$20$	0	0
$21$	0	0
$22$	−259.985	−2.51950
$23$	144.318	1.30837	0.654184	−	0.756336i	$-0.273012\pi$
$23$	144.318	1.30837	0.654184	+	0.756336i	$0.273012\pi$
$24$	0	0
$25$	0	0
$26$	94.5921	0.713501
$27$	0	0
$28$	−60.3960	−0.407635
$29$	−220.058	−1.40909	−0.704547	−	0.709657i	$-0.748850\pi$
$29$	−220.058	−1.40909	−0.704547	+	0.709657i	$0.748850\pi$
$30$	0	0
$31$	291.545	1.68913	0.844566	−	0.535452i	$-0.179859\pi$
$31$	291.545	1.68913	0.844566	+	0.535452i	$0.179859\pi$
$32$	218.056	1.20460
$33$	0	0
$34$	−307.006	−1.54856
$35$	0	0
$36$	0	0
$37$	−260.637	−1.15807	−0.579033	−	0.815304i	$-0.696570\pi$
$37$	−260.637	−1.15807	−0.579033	+	0.815304i	$0.696570\pi$
$38$	182.002	0.776964
$39$	0	0
$40$	0	0
$41$	−169.766	−0.646660	−0.323330	−	0.946286i	$-0.604802\pi$
$41$	−169.766	−0.646660	−0.323330	+	0.946286i	$0.604802\pi$
$42$	0	0
$43$	438.596	1.55547	0.777735	−	0.628592i	$-0.216369\pi$
$43$	438.596	1.55547	0.777735	+	0.628592i	$0.216369\pi$
$44$	692.967	2.37429
$45$	0	0
$46$	−643.571	−2.06281
$47$	255.481	0.792887	0.396444	−	0.918059i	$-0.370244\pi$
$47$	255.481	0.792887	0.396444	+	0.918059i	$0.370244\pi$
$48$	0	0
$49$	−317.181	−0.924726
$50$	0	0
$51$	0	0
$52$	−252.127	−0.672379
$53$	−214.714	−0.556477	−0.278239	−	0.960512i	$-0.589751\pi$
$53$	−214.714	−0.556477	−0.278239	+	0.960512i	$0.589751\pi$
$54$	0	0
$55$	0	0
$56$	88.0557	0.210124
$57$	0	0
$58$	981.322	2.22162
$59$	331.524	0.731537	0.365769	−	0.930706i	$-0.380806\pi$
$59$	331.524	0.731537	0.365769	+	0.930706i	$0.380806\pi$
$60$	0	0
$61$	54.9647	0.115369	0.0576845	−	0.998335i	$-0.481628\pi$
$61$	54.9647	0.115369	0.0576845	+	0.998335i	$0.481628\pi$
$62$	−1300.11	−2.66314
$63$	0	0
$64$	−829.920	−1.62094
$65$	0	0
$66$	0	0
$67$	−758.179	−1.38248	−0.691241	−	0.722624i	$-0.742936\pi$
$67$	−758.179	−1.38248	−0.691241	+	0.722624i	$0.742936\pi$
$68$	818.299	1.45931
$69$	0	0
$70$	0	0
$71$	−904.348	−1.51164	−0.755819	−	0.654780i	$-0.772761\pi$
$71$	−904.348	−1.51164	−0.755819	+	0.654780i	$0.772761\pi$
$72$	0	0
$73$	−866.622	−1.38946	−0.694729	−	0.719271i	$-0.744476\pi$
$73$	−866.622	−1.38946	−0.694729	+	0.719271i	$0.744476\pi$
$74$	1162.28	1.82584
$75$	0	0
$76$	−485.110	−0.732184
$77$	−296.239	−0.438437
$78$	0	0
$79$	206.961	0.294746	0.147373	−	0.989081i	$-0.452918\pi$
$79$	206.961	0.294746	0.147373	+	0.989081i	$0.452918\pi$
$80$	0	0
$81$	0	0
$82$	757.054	1.01954
$83$	463.397	0.612825	0.306412	−	0.951899i	$-0.400871\pi$
$83$	463.397	0.612825	0.306412	+	0.951899i	$0.400871\pi$
$84$	0	0
$85$	0	0
$86$	−1955.87	−2.45240
$87$	0	0
$88$	−1010.33	−1.22388
$89$	601.736	0.716673	0.358337	−	0.933592i	$-0.383344\pi$
$89$	601.736	0.716673	0.358337	+	0.933592i	$0.383344\pi$
$90$	0	0
$91$	107.783	0.124162
$92$	1715.38	1.94392
$93$	0	0
$94$	−1139.29	−1.25009
$95$	0	0
$96$	0	0
$97$	−229.363	−0.240086	−0.120043	−	0.992769i	$-0.538303\pi$
$97$	−229.363	−0.240086	−0.120043	+	0.992769i	$0.538303\pi$
$98$	1414.43	1.45795
$99$	0	0
$100$	0	0
$101$	1345.66	1.32573	0.662863	−	0.748740i	$-0.269341\pi$
$101$	1345.66	1.32573	0.662863	+	0.748740i	$0.269341\pi$
$102$	0	0
$103$	1596.30	1.52707	0.763534	−	0.645768i	$-0.223463\pi$
$103$	1596.30	1.52707	0.763534	+	0.645768i	$0.223463\pi$
$104$	367.594	0.346592
$105$	0	0
$106$	957.494	0.877358
$107$	−958.786	−0.866256	−0.433128	−	0.901333i	$-0.642590\pi$
$107$	−958.786	−0.866256	−0.433128	+	0.901333i	$0.642590\pi$
$108$	0	0
$109$	1690.23	1.48527	0.742635	−	0.669696i	$-0.233576\pi$
$109$	1690.23	1.48527	0.742635	+	0.669696i	$0.233576\pi$
$110$	0	0
$111$	0	0
$112$	90.4943	0.0763474
$113$	11.6211	0.00967456	0.00483728	−	0.999988i	$-0.498460\pi$
$113$	11.6211	0.00967456	0.00483728	+	0.999988i	$0.498460\pi$
$114$	0	0
$115$	0	0
$116$	−2615.63	−2.09358
$117$	0	0
$118$	−1478.39	−1.15336
$119$	−349.818	−0.269477
$120$	0	0
$121$	2067.97	1.55370
$122$	−245.109	−0.181894
$123$	0	0
$124$	3465.34	2.50965
$125$	0	0
$126$	0	0
$127$	−309.141	−0.215999	−0.107999	−	0.994151i	$-0.534444\pi$
$127$	−309.141	−0.215999	−0.107999	+	0.994151i	$0.534444\pi$
$128$	1956.48	1.35102
$129$	0	0
$130$	0	0
$131$	2785.03	1.85747	0.928736	−	0.370742i	$-0.120897\pi$
$131$	2785.03	1.85747	0.928736	+	0.370742i	$0.120897\pi$
$132$	0	0
$133$	207.382	0.135205
$134$	3381.01	2.17966
$135$	0	0
$136$	−1193.06	−0.752233
$137$	2489.41	1.55244	0.776222	−	0.630460i	$-0.217134\pi$
$137$	2489.41	1.55244	0.776222	+	0.630460i	$0.217134\pi$
$138$	0	0
$139$	1786.05	1.08986	0.544931	−	0.838481i	$-0.316556\pi$
$139$	1786.05	1.08986	0.544931	+	0.838481i	$0.316556\pi$
$140$	0	0
$141$	0	0
$142$	4032.83	2.38329
$143$	−1236.67	−0.723185
$144$	0	0
$145$	0	0
$146$	3864.60	2.19066
$147$	0	0
$148$	−3097.95	−1.72061
$149$	1568.83	0.862575	0.431288	−	0.902214i	$-0.358059\pi$
$149$	1568.83	0.862575	0.431288	+	0.902214i	$0.358059\pi$
$150$	0	0
$151$	−438.327	−0.236229	−0.118114	−	0.993000i	$-0.537685\pi$
$151$	−438.327	−0.236229	−0.118114	+	0.993000i	$0.537685\pi$
$152$	707.277	0.377419
$153$	0	0
$154$	1321.04	0.691252
$155$	0	0
$156$	0	0
$157$	44.7479	0.0227469	0.0113735	−	0.999935i	$-0.496380\pi$
$157$	44.7479	0.0227469	0.0113735	+	0.999935i	$0.496380\pi$
$158$	−922.917	−0.464705
$159$	0	0
$160$	0	0
$161$	−733.315	−0.358965
$162$	0	0
$163$	2611.84	1.25506	0.627531	−	0.778591i	$-0.284065\pi$
$163$	2611.84	1.25506	0.627531	+	0.778591i	$0.284065\pi$
$164$	−2017.86	−0.960783
$165$	0	0
$166$	−2066.47	−0.966198
$167$	188.947	0.0875516	0.0437758	−	0.999041i	$-0.486061\pi$
$167$	188.947	0.0875516	0.0437758	+	0.999041i	$0.486061\pi$
$168$	0	0
$169$	−1747.05	−0.795200
$170$	0	0
$171$	0	0
$172$	5213.19	2.31106
$173$	1505.02	0.661413	0.330707	−	0.943734i	$-0.392713\pi$
$173$	1505.02	0.661413	0.330707	+	0.943734i	$0.392713\pi$
$174$	0	0
$175$	0	0
$176$	−1038.31	−0.444689
$177$	0	0
$178$	−2683.37	−1.12993
$179$	−3136.62	−1.30973	−0.654865	−	0.755746i	$-0.727275\pi$
$179$	−3136.62	−1.30973	−0.654865	+	0.755746i	$0.727275\pi$
$180$	0	0
$181$	4512.67	1.85317	0.926586	−	0.376084i	$-0.122730\pi$
$181$	4512.67	1.85317	0.926586	+	0.376084i	$0.122730\pi$
$182$	−480.644	−0.195757
$183$	0	0
$184$	−2500.98	−1.00204
$185$	0	0
$186$	0	0
$187$	4013.71	1.56958
$188$	3036.67	1.17804
$189$	0	0
$190$	0	0
$191$	1207.43	0.457418	0.228709	−	0.973495i	$-0.426550\pi$
$191$	1207.43	0.457418	0.228709	+	0.973495i	$0.426550\pi$
$192$	0	0
$193$	−923.164	−0.344305	−0.172152	−	0.985070i	$-0.555072\pi$
$193$	−923.164	−0.344305	−0.172152	+	0.985070i	$0.555072\pi$
$194$	1022.82	0.378526
$195$	0	0
$196$	−3770.04	−1.37392
$197$	1180.87	0.427075	0.213537	−	0.976935i	$-0.431501\pi$
$197$	1180.87	0.427075	0.213537	+	0.976935i	$0.431501\pi$
$198$	0	0
$199$	−839.805	−0.299157	−0.149578	−	0.988750i	$-0.547792\pi$
$199$	−839.805	−0.299157	−0.149578	+	0.988750i	$0.547792\pi$
$200$	0	0
$201$	0	0
$202$	−6000.82	−2.09018
$203$	1118.17	0.386600
$204$	0	0
$205$	0	0
$206$	−7118.51	−2.40762
$207$	0	0
$208$	377.774	0.125932
$209$	−2379.44	−0.787509
$210$	0	0
$211$	−2589.65	−0.844923	−0.422461	−	0.906381i	$-0.638834\pi$
$211$	−2589.65	−0.844923	−0.422461	+	0.906381i	$0.638834\pi$
$212$	−2552.12	−0.826792
$213$	0	0
$214$	4275.59	1.36576
$215$	0	0
$216$	0	0
$217$	−1481.41	−0.463432
$218$	−7537.38	−2.34172
$219$	0	0
$220$	0	0
$221$	−1460.34	−0.444492
$222$	0	0
$223$	4180.76	1.25544	0.627722	−	0.778437i	$-0.283987\pi$
$223$	4180.76	1.25544	0.627722	+	0.778437i	$0.283987\pi$
$224$	−1107.99	−0.330495
$225$	0	0
$226$	−51.8232	−0.0152532
$227$	2602.67	0.760993	0.380497	−	0.924782i	$-0.375753\pi$
$227$	2602.67	0.760993	0.380497	+	0.924782i	$0.375753\pi$
$228$	0	0
$229$	−1845.35	−0.532508	−0.266254	−	0.963903i	$-0.585786\pi$
$229$	−1845.35	−0.532508	−0.266254	+	0.963903i	$0.585786\pi$
$230$	0	0
$231$	0	0
$232$	3813.51	1.07918
$233$	−240.637	−0.0676594	−0.0338297	−	0.999428i	$-0.510770\pi$
$233$	−240.637	−0.0676594	−0.0338297	+	0.999428i	$0.510770\pi$
$234$	0	0
$235$	0	0
$236$	3940.52	1.08689
$237$	0	0
$238$	1559.97	0.424865
$239$	2567.64	0.694924	0.347462	−	0.937694i	$-0.387044\pi$
$239$	2567.64	0.694924	0.347462	+	0.937694i	$0.387044\pi$
$240$	0	0
$241$	3987.99	1.06593	0.532965	−	0.846137i	$-0.321078\pi$
$241$	3987.99	1.06593	0.532965	+	0.846137i	$0.321078\pi$
$242$	−9221.86	−2.44960
$243$	0	0
$244$	653.315	0.171411
$245$	0	0
$246$	0	0
$247$	865.728	0.223016
$248$	−5052.36	−1.29365
$249$	0	0
$250$	0	0
$251$	967.393	0.243272	0.121636	−	0.992575i	$-0.461186\pi$
$251$	967.393	0.243272	0.121636	+	0.992575i	$0.461186\pi$
$252$	0	0
$253$	8413.86	2.09081
$254$	1378.58	0.340550
$255$	0	0
$256$	−2085.34	−0.509116
$257$	−3743.03	−0.908498	−0.454249	−	0.890875i	$-0.650092\pi$
$257$	−3743.03	−0.908498	−0.454249	+	0.890875i	$0.650092\pi$
$258$	0	0
$259$	1324.36	0.317728
$260$	0	0
$261$	0	0
$262$	−12419.5	−2.92855
$263$	2497.47	0.585554	0.292777	−	0.956181i	$-0.405421\pi$
$263$	2497.47	0.585554	0.292777	+	0.956181i	$0.405421\pi$
$264$	0	0
$265$	0	0
$266$	−924.795	−0.213168
$267$	0	0
$268$	−9011.79	−2.05404
$269$	−2142.91	−0.485709	−0.242855	−	0.970063i	$-0.578084\pi$
$269$	−2142.91	−0.485709	−0.242855	+	0.970063i	$0.578084\pi$
$270$	0	0
$271$	1540.64	0.345341	0.172671	−	0.984980i	$-0.444760\pi$
$271$	1540.64	0.345341	0.172671	+	0.984980i	$0.444760\pi$
$272$	−1226.10	−0.273320
$273$	0	0
$274$	−11101.2	−2.44763
$275$	0	0
$276$	0	0
$277$	−6777.80	−1.47018	−0.735088	−	0.677972i	$-0.762859\pi$
$277$	−6777.80	−1.47018	−0.735088	+	0.677972i	$0.762859\pi$
$278$	−7964.69	−1.71831
$279$	0	0
$280$	0	0
$281$	827.653	0.175707	0.0878535	−	0.996133i	$-0.471999\pi$
$281$	827.653	0.175707	0.0878535	+	0.996133i	$0.471999\pi$
$282$	0	0
$283$	3171.98	0.666270	0.333135	−	0.942879i	$-0.391894\pi$
$283$	3171.98	0.666270	0.333135	+	0.942879i	$0.391894\pi$
$284$	−10749.2	−2.24593
$285$	0	0
$286$	5514.78	1.14020
$287$	862.623	0.177418
$288$	0	0
$289$	−173.358	−0.0352856
$290$	0	0
$291$	0	0
$292$	−10300.8	−2.06440
$293$	−1376.02	−0.274362	−0.137181	−	0.990546i	$-0.543804\pi$
$293$	−1376.02	−0.274362	−0.137181	+	0.990546i	$0.543804\pi$
$294$	0	0
$295$	0	0
$296$	4516.73	0.886923
$297$	0	0
$298$	−6996.02	−1.35996
$299$	−3061.27	−0.592099
$300$	0	0
$301$	−2228.61	−0.426760
$302$	1954.67	0.372445
$303$	0	0
$304$	726.864	0.137133
$305$	0	0
$306$	0	0
$307$	119.504	0.0222165	0.0111083	−	0.999938i	$-0.496464\pi$
$307$	119.504	0.0222165	0.0111083	+	0.999938i	$0.496464\pi$
$308$	−3521.13	−0.651412
$309$	0	0
$310$	0	0
$311$	2139.35	0.390069	0.195035	−	0.980796i	$-0.437518\pi$
$311$	2139.35	0.390069	0.195035	+	0.980796i	$0.437518\pi$
$312$	0	0
$313$	−5163.50	−0.932455	−0.466227	−	0.884665i	$-0.654387\pi$
$313$	−5163.50	−0.932455	−0.466227	+	0.884665i	$0.654387\pi$
$314$	−199.548	−0.0358635
$315$	0	0
$316$	2459.95	0.437922
$317$	8631.69	1.52935	0.764675	−	0.644416i	$-0.222899\pi$
$317$	8631.69	1.52935	0.764675	+	0.644416i	$0.222899\pi$
$318$	0	0
$319$	−12829.5	−2.25177
$320$	0	0
$321$	0	0
$322$	3270.13	0.565955
$323$	−2809.79	−0.484028
$324$	0	0
$325$	0	0
$326$	−11647.2	−1.97877
$327$	0	0
$328$	2941.98	0.495255
$329$	−1298.16	−0.217537
$330$	0	0
$331$	−2942.34	−0.488597	−0.244298	−	0.969700i	$-0.578558\pi$
$331$	−2942.34	−0.488597	−0.244298	+	0.969700i	$0.578558\pi$
$332$	5507.98	0.910512
$333$	0	0
$334$	−842.585	−0.138037
$335$	0	0
$336$	0	0
$337$	−9897.46	−1.59985	−0.799924	−	0.600101i	$-0.795127\pi$
$337$	−9897.46	−1.59985	−0.799924	+	0.600101i	$0.795127\pi$
$338$	7790.79	1.25374
$339$	0	0
$340$	0	0
$341$	16997.3	2.69928
$342$	0	0
$343$	3354.53	0.528070
$344$	−7600.68	−1.19128
$345$	0	0
$346$	−6711.46	−1.04280
$347$	12015.7	1.85890	0.929451	−	0.368947i	$-0.120282\pi$
$347$	12015.7	1.85890	0.929451	+	0.368947i	$0.120282\pi$
$348$	0	0
$349$	7894.62	1.21086	0.605428	−	0.795900i	$-0.293002\pi$
$349$	7894.62	1.21086	0.605428	+	0.795900i	$0.293002\pi$
$350$	0	0
$351$	0	0
$352$	12712.8	1.92499
$353$	−741.154	−0.111750	−0.0558748	−	0.998438i	$-0.517795\pi$
$353$	−741.154	−0.111750	−0.0558748	+	0.998438i	$0.517795\pi$
$354$	0	0
$355$	0	0
$356$	7152.30	1.06481
$357$	0	0
$358$	13987.4	2.06496
$359$	−11564.4	−1.70012	−0.850060	−	0.526685i	$-0.823435\pi$
$359$	−11564.4	−1.70012	−0.850060	+	0.526685i	$0.823435\pi$
$360$	0	0
$361$	−5193.28	−0.757148
$362$	−20123.7	−2.92177
$363$	0	0
$364$	1281.12	0.184474
$365$	0	0
$366$	0	0
$367$	1148.57	0.163365	0.0816823	−	0.996658i	$-0.473971\pi$
$367$	1148.57	0.163365	0.0816823	+	0.996658i	$0.473971\pi$
$368$	−2570.24	−0.364084
$369$	0	0
$370$	0	0
$371$	1091.01	0.152676
$372$	0	0
$373$	4602.98	0.638963	0.319481	−	0.947593i	$-0.396491\pi$
$373$	4602.98	0.638963	0.319481	+	0.947593i	$0.396491\pi$
$374$	−17898.7	−2.47465
$375$	0	0
$376$	−4427.38	−0.607246
$377$	4667.85	0.637683
$378$	0	0
$379$	−3988.46	−0.540563	−0.270282	−	0.962781i	$-0.587117\pi$
$379$	−3988.46	−0.540563	−0.270282	+	0.962781i	$0.587117\pi$
$380$	0	0
$381$	0	0
$382$	−5384.41	−0.721179
$383$	−7788.20	−1.03906	−0.519528	−	0.854454i	$-0.673892\pi$
$383$	−7788.20	−1.03906	−0.519528	+	0.854454i	$0.673892\pi$
$384$	0	0
$385$	0	0
$386$	4116.74	0.542841
$387$	0	0
$388$	−2726.23	−0.356710
$389$	8524.87	1.11113	0.555563	−	0.831474i	$-0.312503\pi$
$389$	8524.87	1.11113	0.555563	+	0.831474i	$0.312503\pi$
$390$	0	0
$391$	9935.60	1.28508
$392$	5496.62	0.708217
$393$	0	0
$394$	−5265.97	−0.673339
$395$	0	0
$396$	0	0
$397$	155.729	0.0196872	0.00984361	−	0.999952i	$-0.496867\pi$
$397$	155.729	0.0196872	0.00984361	+	0.999952i	$0.496867\pi$
$398$	3745.01	0.471659
$399$	0	0
$400$	0	0
$401$	5933.96	0.738972	0.369486	−	0.929236i	$-0.379534\pi$
$401$	5933.96	0.738972	0.369486	+	0.929236i	$0.379534\pi$
$402$	0	0
$403$	−6184.23	−0.764414
$404$	15994.7	1.96971
$405$	0	0
$406$	−4986.33	−0.609526
$407$	−15195.3	−1.85062
$408$	0	0
$409$	14161.4	1.71207	0.856035	−	0.516917i	$-0.172921\pi$
$409$	14161.4	1.71207	0.856035	+	0.516917i	$0.172921\pi$
$410$	0	0
$411$	0	0
$412$	18973.8	2.26886
$413$	−1684.55	−0.200705
$414$	0	0
$415$	0	0
$416$	−4625.39	−0.545140
$417$	0	0
$418$	10610.8	1.24161
$419$	9624.24	1.12214	0.561068	−	0.827770i	$-0.310391\pi$
$419$	9624.24	1.12214	0.561068	+	0.827770i	$0.310391\pi$
$420$	0	0
$421$	−1536.26	−0.177845	−0.0889223	−	0.996039i	$-0.528342\pi$
$421$	−1536.26	−0.177845	−0.0889223	+	0.996039i	$0.528342\pi$
$422$	11548.2	1.33213
$423$	0	0
$424$	3720.91	0.426187
$425$	0	0
$426$	0	0
$427$	−279.288	−0.0316527
$428$	−11396.2	−1.28705
$429$	0	0
$430$	0	0
$431$	11582.2	1.29441	0.647207	−	0.762314i	$-0.275937\pi$
$431$	11582.2	1.29441	0.647207	+	0.762314i	$0.275937\pi$
$432$	0	0
$433$	−14892.6	−1.65287	−0.826437	−	0.563029i	$-0.809636\pi$
$433$	−14892.6	−1.65287	−0.826437	+	0.563029i	$0.809636\pi$
$434$	6606.17	0.730660
$435$	0	0
$436$	20090.2	2.20676
$437$	−5890.10	−0.644764
$438$	0	0
$439$	−1642.51	−0.178571	−0.0892853	−	0.996006i	$-0.528458\pi$
$439$	−1642.51	−0.178571	−0.0892853	+	0.996006i	$0.528458\pi$
$440$	0	0
$441$	0	0
$442$	6512.20	0.700800
$443$	−3916.31	−0.420021	−0.210011	−	0.977699i	$-0.567350\pi$
$443$	−3916.31	−0.420021	−0.210011	+	0.977699i	$0.567350\pi$
$444$	0	0
$445$	0	0
$446$	−18643.6	−1.97937
$447$	0	0
$448$	4217.02	0.444722
$449$	−3985.25	−0.418876	−0.209438	−	0.977822i	$-0.567163\pi$
$449$	−3985.25	−0.418876	−0.209438	+	0.977822i	$0.567163\pi$
$450$	0	0
$451$	−9897.50	−1.03338
$452$	138.130	0.0143741
$453$	0	0
$454$	−11606.3	−1.19980
$455$	0	0
$456$	0	0
$457$	14177.8	1.45122	0.725611	−	0.688105i	$-0.241557\pi$
$457$	14177.8	1.45122	0.725611	+	0.688105i	$0.241557\pi$
$458$	8229.13	0.839567
$459$	0	0
$460$	0	0
$461$	−16394.3	−1.65631	−0.828154	−	0.560500i	$-0.810609\pi$
$461$	−16394.3	−1.65631	−0.828154	+	0.560500i	$0.810609\pi$
$462$	0	0
$463$	−3319.60	−0.333207	−0.166603	−	0.986024i	$-0.553280\pi$
$463$	−3319.60	−0.333207	−0.166603	+	0.986024i	$0.553280\pi$
$464$	3919.12	0.392114
$465$	0	0
$466$	1073.09	0.106674
$467$	2529.79	0.250674	0.125337	−	0.992114i	$-0.459999\pi$
$467$	2529.79	0.250674	0.125337	+	0.992114i	$0.459999\pi$
$468$	0	0
$469$	3852.49	0.379299
$470$	0	0
$471$	0	0
$472$	−5745.17	−0.560260
$473$	25570.4	2.48569
$474$	0	0
$475$	0	0
$476$	−4157.97	−0.400379
$477$	0	0
$478$	−11450.1	−1.09564
$479$	−8646.75	−0.824802	−0.412401	−	0.911002i	$-0.635310\pi$
$479$	−8646.75	−0.824802	−0.412401	+	0.911002i	$0.635310\pi$
$480$	0	0
$481$	5528.60	0.524080
$482$	−17784.0	−1.68058
$483$	0	0
$484$	24580.1	2.30842
$485$	0	0
$486$	0	0
$487$	15251.7	1.41914	0.709569	−	0.704636i	$-0.248890\pi$
$487$	15251.7	1.41914	0.709569	+	0.704636i	$0.248890\pi$
$488$	−952.515	−0.0883572
$489$	0	0
$490$	0	0
$491$	−1766.87	−0.162398	−0.0811992	−	0.996698i	$-0.525875\pi$
$491$	−1766.87	−0.162398	−0.0811992	+	0.996698i	$0.525875\pi$
$492$	0	0
$493$	−15149.9	−1.38401
$494$	−3860.61	−0.351614
$495$	0	0
$496$	−5192.28	−0.470041
$497$	4595.20	0.414734
$498$	0	0
$499$	11733.8	1.05266	0.526332	−	0.850279i	$-0.323567\pi$
$499$	11733.8	1.05266	0.526332	+	0.850279i	$0.323567\pi$
$500$	0	0
$501$	0	0
$502$	−4313.98	−0.383550
$503$	977.608	0.0866589	0.0433294	−	0.999061i	$-0.486203\pi$
$503$	977.608	0.0866589	0.0433294	+	0.999061i	$0.486203\pi$
$504$	0	0
$505$	0	0
$506$	−37520.6	−3.29643
$507$	0	0
$508$	−3674.48	−0.320923
$509$	9674.72	0.842484	0.421242	−	0.906948i	$-0.361594\pi$
$509$	9674.72	0.842484	0.421242	+	0.906948i	$0.361594\pi$
$510$	0	0
$511$	4403.51	0.381213
$512$	−6352.52	−0.548329
$513$	0	0
$514$	16691.6	1.43237
$515$	0	0
$516$	0	0
$517$	14894.7	1.26706
$518$	−5905.81	−0.500939
$519$	0	0
$520$	0	0
$521$	5178.95	0.435497	0.217748	−	0.976005i	$-0.430129\pi$
$521$	5178.95	0.435497	0.217748	+	0.976005i	$0.430129\pi$
$522$	0	0
$523$	−14280.7	−1.19398	−0.596992	−	0.802248i	$-0.703637\pi$
$523$	−14280.7	−1.19398	−0.596992	+	0.802248i	$0.703637\pi$
$524$	33103.1	2.75976
$525$	0	0
$526$	−11137.2	−0.923203
$527$	20071.5	1.65906
$528$	0	0
$529$	8660.78	0.711826
$530$	0	0
$531$	0	0
$532$	2464.96	0.200883
$533$	3601.07	0.292645
$534$	0	0
$535$	0	0
$536$	13138.9	1.05880
$537$	0	0
$538$	9556.08	0.765784
$539$	−18491.9	−1.47774
$540$	0	0
$541$	12923.8	1.02706	0.513529	−	0.858072i	$-0.328338\pi$
$541$	12923.8	1.02706	0.513529	+	0.858072i	$0.328338\pi$
$542$	−6870.32	−0.544475
$543$	0	0
$544$	15012.1	1.18316
$545$	0	0
$546$	0	0
$547$	−13653.3	−1.06723	−0.533614	−	0.845728i	$-0.679166\pi$
$547$	−13653.3	−1.06723	−0.533614	+	0.845728i	$0.679166\pi$
$548$	29589.4	2.30656
$549$	0	0
$550$	0	0
$551$	8981.28	0.694402
$552$	0	0
$553$	−1051.62	−0.0808666
$554$	30224.8	2.31792
$555$	0	0
$556$	21229.2	1.61928
$557$	9313.63	0.708494	0.354247	−	0.935152i	$-0.384737\pi$
$557$	9313.63	0.708494	0.354247	+	0.935152i	$0.384737\pi$
$558$	0	0
$559$	−9303.46	−0.703925
$560$	0	0
$561$	0	0
$562$	−3690.82	−0.277025
$563$	10625.4	0.795394	0.397697	−	0.917517i	$-0.369810\pi$
$563$	10625.4	0.795394	0.397697	+	0.917517i	$0.369810\pi$
$564$	0	0
$565$	0	0
$566$	−14145.1	−1.05046
$567$	0	0
$568$	15672.0	1.15771
$569$	9060.50	0.667550	0.333775	−	0.942653i	$-0.391677\pi$
$569$	9060.50	0.667550	0.333775	+	0.942653i	$0.391677\pi$
$570$	0	0
$571$	−21379.1	−1.56688	−0.783440	−	0.621467i	$-0.786537\pi$
$571$	−21379.1	−1.56688	−0.783440	+	0.621467i	$0.786537\pi$
$572$	−14699.2	−1.07448
$573$	0	0
$574$	−3846.77	−0.279723
$575$	0	0
$576$	0	0
$577$	−6347.76	−0.457991	−0.228996	−	0.973427i	$-0.573544\pi$
$577$	−6347.76	−0.457991	−0.228996	+	0.973427i	$0.573544\pi$
$578$	773.071	0.0556324
$579$	0	0
$580$	0	0
$581$	−2354.63	−0.168135
$582$	0	0
$583$	−12518.0	−0.889266
$584$	15018.2	1.06414
$585$	0	0
$586$	6136.22	0.432568
$587$	−14773.7	−1.03880	−0.519401	−	0.854531i	$-0.673845\pi$
$587$	−14773.7	−1.03880	−0.519401	+	0.854531i	$0.673845\pi$
$588$	0	0
$589$	−11898.9	−0.832405
$590$	0	0
$591$	0	0
$592$	4641.81	0.322259
$593$	−26868.1	−1.86061	−0.930304	−	0.366790i	$-0.880457\pi$
$593$	−26868.1	−1.86061	−0.930304	+	0.366790i	$0.880457\pi$
$594$	0	0
$595$	0	0
$596$	18647.3	1.28158
$597$	0	0
$598$	13651.4	0.933522
$599$	−1749.83	−0.119359	−0.0596794	−	0.998218i	$-0.519008\pi$
$599$	−1749.83	−0.119359	−0.0596794	+	0.998218i	$0.519008\pi$
$600$	0	0
$601$	−17964.0	−1.21924	−0.609622	−	0.792692i	$-0.708679\pi$
$601$	−17964.0	−1.21924	−0.609622	+	0.792692i	$0.708679\pi$
$602$	9938.21	0.672843
$603$	0	0
$604$	−5209.99	−0.350979
$605$	0	0
$606$	0	0
$607$	4418.22	0.295437	0.147718	−	0.989029i	$-0.452807\pi$
$607$	4418.22	0.295437	0.147718	+	0.989029i	$0.452807\pi$
$608$	−8899.58	−0.593628
$609$	0	0
$610$	0	0
$611$	−5419.24	−0.358820
$612$	0	0
$613$	20179.7	1.32961	0.664804	−	0.747018i	$-0.268515\pi$
$613$	20179.7	1.32961	0.664804	+	0.747018i	$0.268515\pi$
$614$	−532.915	−0.0350272
$615$	0	0
$616$	5133.71	0.335784
$617$	4673.01	0.304908	0.152454	−	0.988311i	$-0.451282\pi$
$617$	4673.01	0.304908	0.152454	+	0.988311i	$0.451282\pi$
$618$	0	0
$619$	−19976.8	−1.29715	−0.648574	−	0.761151i	$-0.724634\pi$
$619$	−19976.8	−1.29715	−0.648574	+	0.761151i	$0.724634\pi$
$620$	0	0
$621$	0	0
$622$	−9540.19	−0.614994
$623$	−3057.56	−0.196627
$624$	0	0
$625$	0	0
$626$	23026.0	1.47014
$627$	0	0
$628$	531.877	0.0337965
$629$	−17943.5	−1.13745
$630$	0	0
$631$	10457.5	0.659757	0.329879	−	0.944023i	$-0.392992\pi$
$631$	10457.5	0.659757	0.329879	+	0.944023i	$0.392992\pi$
$632$	−3586.54	−0.225736
$633$	0	0
$634$	−38492.0	−2.41122
$635$	0	0
$636$	0	0
$637$	6728.02	0.418483
$638$	57211.8	3.55021
$639$	0	0
$640$	0	0
$641$	4395.86	0.270867	0.135434	−	0.990786i	$-0.456757\pi$
$641$	4395.86	0.270867	0.135434	+	0.990786i	$0.456757\pi$
$642$	0	0
$643$	−5786.36	−0.354886	−0.177443	−	0.984131i	$-0.556783\pi$
$643$	−5786.36	−0.354886	−0.177443	+	0.984131i	$0.556783\pi$
$644$	−8716.26	−0.533336
$645$	0	0
$646$	12529.9	0.763133
$647$	25367.7	1.54143	0.770717	−	0.637178i	$-0.219898\pi$
$647$	25367.7	1.54143	0.770717	+	0.637178i	$0.219898\pi$
$648$	0	0
$649$	19328.1	1.16902
$650$	0	0
$651$	0	0
$652$	31044.6	1.86472
$653$	21633.2	1.29644	0.648218	−	0.761454i	$-0.275514\pi$
$653$	21633.2	1.29644	0.648218	+	0.761454i	$0.275514\pi$
$654$	0	0
$655$	0	0
$656$	3023.46	0.179948
$657$	0	0
$658$	5788.98	0.342976
$659$	18312.4	1.08248	0.541238	−	0.840870i	$-0.317956\pi$
$659$	18312.4	1.08248	0.541238	+	0.840870i	$0.317956\pi$
$660$	0	0
$661$	−5526.08	−0.325174	−0.162587	−	0.986694i	$-0.551984\pi$
$661$	−5526.08	−0.325174	−0.162587	+	0.986694i	$0.551984\pi$
$662$	13121.0	0.770337
$663$	0	0
$664$	−8030.48	−0.469342
$665$	0	0
$666$	0	0
$667$	−31758.4	−1.84361
$668$	2245.84	0.130081
$669$	0	0
$670$	0	0
$671$	3204.48	0.184363
$672$	0	0
$673$	−1437.24	−0.0823204	−0.0411602	−	0.999153i	$-0.513105\pi$
$673$	−1437.24	−0.0823204	−0.0411602	+	0.999153i	$0.513105\pi$
$674$	44136.6	2.52237
$675$	0	0
$676$	−20765.7	−1.18148
$677$	23405.9	1.32875	0.664373	−	0.747401i	$-0.268699\pi$
$677$	23405.9	1.32875	0.664373	+	0.747401i	$0.268699\pi$
$678$	0	0
$679$	1165.45	0.0658701
$680$	0	0
$681$	0	0
$682$	−75797.4	−4.25577
$683$	−19227.4	−1.07719	−0.538593	−	0.842566i	$-0.681044\pi$
$683$	−19227.4	−1.07719	−0.538593	+	0.842566i	$0.681044\pi$
$684$	0	0
$685$	0	0
$686$	−14959.2	−0.832570
$687$	0	0
$688$	−7811.17	−0.432846
$689$	4554.50	0.251833
$690$	0	0
$691$	−35284.8	−1.94254	−0.971271	−	0.237975i	$-0.923516\pi$
$691$	−35284.8	−1.94254	−0.971271	+	0.237975i	$0.923516\pi$
$692$	17888.8	0.982703
$693$	0	0
$694$	−53582.8	−2.93080
$695$	0	0
$696$	0	0
$697$	−11687.6	−0.635149
$698$	−35205.1	−1.90907
$699$	0	0
$700$	0	0
$701$	9173.00	0.494236	0.247118	−	0.968985i	$-0.420516\pi$
$701$	9173.00	0.494236	0.247118	+	0.968985i	$0.420516\pi$
$702$	0	0
$703$	10637.4	0.570695
$704$	−48384.9	−2.59030
$705$	0	0
$706$	3305.09	0.176188
$707$	−6837.63	−0.363728
$708$	0	0
$709$	−33951.6	−1.79842	−0.899210	−	0.437517i	$-0.855858\pi$
$709$	−33951.6	−1.79842	−0.899210	+	0.437517i	$0.855858\pi$
$710$	0	0
$711$	0	0
$712$	−10427.8	−0.548876
$713$	42075.3	2.21001
$714$	0	0
$715$	0	0
$716$	−37282.1	−1.94595
$717$	0	0
$718$	51569.9	2.68046
$719$	−6727.90	−0.348968	−0.174484	−	0.984660i	$-0.555826\pi$
$719$	−6727.90	−0.348968	−0.174484	+	0.984660i	$0.555826\pi$
$720$	0	0
$721$	−8111.17	−0.418968
$722$	23158.8	1.19374
$723$	0	0
$724$	53638.0	2.75337
$725$	0	0
$726$	0	0
$727$	36726.1	1.87359	0.936793	−	0.349885i	$-0.113779\pi$
$727$	36726.1	1.87359	0.936793	+	0.349885i	$0.113779\pi$
$728$	−1867.83	−0.0950912
$729$	0	0
$730$	0	0
$731$	30195.1	1.52778
$732$	0	0
$733$	−26691.4	−1.34498	−0.672489	−	0.740108i	$-0.734775\pi$
$733$	−26691.4	−1.34498	−0.672489	+	0.740108i	$0.734775\pi$
$734$	−5121.91	−0.257566
$735$	0	0
$736$	31469.5	1.57606
$737$	−44202.4	−2.20925
$738$	0	0
$739$	12207.0	0.607634	0.303817	−	0.952730i	$-0.401739\pi$
$739$	12207.0	0.607634	0.303817	+	0.952730i	$0.401739\pi$
$740$	0	0
$741$	0	0
$742$	−4865.25	−0.240713
$743$	12473.3	0.615882	0.307941	−	0.951405i	$-0.400360\pi$
$743$	12473.3	0.615882	0.307941	+	0.951405i	$0.400360\pi$
$744$	0	0
$745$	0	0
$746$	−20526.4	−1.00741
$747$	0	0
$748$	47707.4	2.33202
$749$	4871.82	0.237667
$750$	0	0
$751$	−15102.6	−0.733825	−0.366913	−	0.930255i	$-0.619585\pi$
$751$	−15102.6	−0.733825	−0.366913	+	0.930255i	$0.619585\pi$
$752$	−4549.99	−0.220640
$753$	0	0
$754$	−20815.7	−1.00539
$755$	0	0
$756$	0	0
$757$	3418.34	0.164124	0.0820618	−	0.996627i	$-0.473850\pi$
$757$	3418.34	0.164124	0.0820618	+	0.996627i	$0.473850\pi$
$758$	17786.1	0.852268
$759$	0	0
$760$	0	0
$761$	−12684.5	−0.604224	−0.302112	−	0.953272i	$-0.597692\pi$
$761$	−12684.5	−0.604224	−0.302112	+	0.953272i	$0.597692\pi$
$762$	0	0
$763$	−8588.45	−0.407500
$764$	14351.7	0.679614
$765$	0	0
$766$	34730.6	1.63821
$767$	−7032.25	−0.331056
$768$	0	0
$769$	−27580.2	−1.29333	−0.646663	−	0.762776i	$-0.723836\pi$
$769$	−27580.2	−1.29333	−0.646663	+	0.762776i	$0.723836\pi$
$770$	0	0
$771$	0	0
$772$	−10972.8	−0.511555
$773$	−17386.0	−0.808966	−0.404483	−	0.914546i	$-0.632548\pi$
$773$	−17386.0	−0.808966	−0.404483	+	0.914546i	$0.632548\pi$
$774$	0	0
$775$	0	0
$776$	3974.77	0.183874
$777$	0	0
$778$	−38015.7	−1.75183
$779$	6928.72	0.318674
$780$	0	0
$781$	−52724.1	−2.41564
$782$	−44306.7	−2.02609
$783$	0	0
$784$	5648.84	0.257327
$785$	0	0
$786$	0	0
$787$	−4680.29	−0.211988	−0.105994	−	0.994367i	$-0.533802\pi$
$787$	−4680.29	−0.211988	−0.105994	+	0.994367i	$0.533802\pi$
$788$	14036.0	0.634531
$789$	0	0
$790$	0	0
$791$	−59.0498	−0.00265432
$792$	0	0
$793$	−1165.91	−0.0522100
$794$	−694.456	−0.0310395
$795$	0	0
$796$	−9982.00	−0.444476
$797$	−7278.62	−0.323490	−0.161745	−	0.986833i	$-0.551712\pi$
$797$	−7278.62	−0.323490	−0.161745	+	0.986833i	$0.551712\pi$
$798$	0	0
$799$	17588.6	0.778773
$800$	0	0
$801$	0	0
$802$	−26461.8	−1.16508
$803$	−50524.7	−2.22039
$804$	0	0
$805$	0	0
$806$	27577.9	1.20520
$807$	0	0
$808$	−23319.8	−1.01533
$809$	29212.5	1.26954	0.634770	−	0.772701i	$-0.281095\pi$
$809$	29212.5	1.26954	0.634770	+	0.772701i	$0.281095\pi$
$810$	0	0
$811$	41992.4	1.81819	0.909094	−	0.416590i	$-0.136775\pi$
$811$	41992.4	1.81819	0.909094	+	0.416590i	$0.136775\pi$
$812$	13290.6	0.574396
$813$	0	0
$814$	67761.6	2.91774
$815$	0	0
$816$	0	0
$817$	−17900.5	−0.766536
$818$	−63151.2	−2.69930
$819$	0	0
$820$	0	0
$821$	−8722.99	−0.370809	−0.185405	−	0.982662i	$-0.559360\pi$
$821$	−8722.99	−0.370809	−0.185405	+	0.982662i	$0.559360\pi$
$822$	0	0
$823$	−13584.8	−0.575379	−0.287690	−	0.957724i	$-0.592887\pi$
$823$	−13584.8	−0.575379	−0.287690	+	0.957724i	$0.592887\pi$
$824$	−27663.2	−1.16953
$825$	0	0
$826$	7512.05	0.316438
$827$	−26573.6	−1.11736	−0.558679	−	0.829384i	$-0.688692\pi$
$827$	−26573.6	−1.11736	−0.558679	+	0.829384i	$0.688692\pi$
$828$	0	0
$829$	−43238.4	−1.81150	−0.905748	−	0.423816i	$-0.860690\pi$
$829$	−43238.4	−1.81150	−0.905748	+	0.423816i	$0.860690\pi$
$830$	0	0
$831$	0	0
$832$	17604.2	0.733552
$833$	−21836.3	−0.908265
$834$	0	0
$835$	0	0
$836$	−28282.3	−1.17005
$837$	0	0
$838$	−42918.2	−1.76919
$839$	−4093.21	−0.168431	−0.0842154	−	0.996448i	$-0.526838\pi$
$839$	−4093.21	−0.168431	−0.0842154	+	0.996448i	$0.526838\pi$
$840$	0	0
$841$	24036.5	0.985546
$842$	6850.75	0.280395
$843$	0	0
$844$	−30780.8	−1.25535
$845$	0	0
$846$	0	0
$847$	−10507.8	−0.426273
$848$	3823.96	0.154853
$849$	0	0
$850$	0	0
$851$	−37614.7	−1.51517
$852$	0	0
$853$	20201.5	0.810886	0.405443	−	0.914120i	$-0.367117\pi$
$853$	20201.5	0.810886	0.405443	+	0.914120i	$0.367117\pi$
$854$	1245.45	0.0499046
$855$	0	0
$856$	16615.4	0.663436
$857$	−4551.65	−0.181425	−0.0907126	−	0.995877i	$-0.528914\pi$
$857$	−4551.65	−0.181425	−0.0907126	+	0.995877i	$0.528914\pi$
$858$	0	0
$859$	11962.6	0.475154	0.237577	−	0.971369i	$-0.423647\pi$
$859$	11962.6	0.475154	0.237577	+	0.971369i	$0.423647\pi$
$860$	0	0
$861$	0	0
$862$	−51649.3	−2.04081
$863$	7164.61	0.282603	0.141301	−	0.989967i	$-0.454871\pi$
$863$	7164.61	0.282603	0.141301	+	0.989967i	$0.454871\pi$
$864$	0	0
$865$	0	0
$866$	66412.0	2.60597
$867$	0	0
$868$	−17608.2	−0.688549
$869$	12065.9	0.471012
$870$	0	0
$871$	16082.4	0.625640
$872$	−29291.0	−1.13752
$873$	0	0
$874$	26266.2	1.01655
$875$	0	0
$876$	0	0
$877$	19218.7	0.739987	0.369994	−	0.929034i	$-0.379360\pi$
$877$	19218.7	0.739987	0.369994	+	0.929034i	$0.379360\pi$
$878$	7324.56	0.281540
$879$	0	0
$880$	0	0
$881$	−45914.5	−1.75585	−0.877923	−	0.478802i	$-0.841071\pi$
$881$	−45914.5	−1.75585	−0.877923	+	0.478802i	$0.841071\pi$
$882$	0	0
$883$	44656.7	1.70194	0.850972	−	0.525211i	$-0.176014\pi$
$883$	44656.7	1.70194	0.850972	+	0.525211i	$0.176014\pi$
$884$	−17357.7	−0.660410
$885$	0	0
$886$	17464.3	0.662218
$887$	−14975.2	−0.566873	−0.283437	−	0.958991i	$-0.591475\pi$
$887$	−14975.2	−0.566873	−0.283437	+	0.958991i	$0.591475\pi$
$888$	0	0
$889$	1570.82	0.0592616
$890$	0	0
$891$	0	0
$892$	49692.9	1.86529
$893$	−10427.0	−0.390735
$894$	0	0
$895$	0	0
$896$	−9941.33	−0.370666
$897$	0	0
$898$	17771.7	0.660413
$899$	−64156.8	−2.38015
$900$	0	0
$901$	−14782.0	−0.546571
$902$	44136.7	1.62926
$903$	0	0
$904$	−201.390	−0.00740943
$905$	0	0
$906$	0	0
$907$	14818.1	0.542479	0.271240	−	0.962512i	$-0.412566\pi$
$907$	14818.1	0.542479	0.271240	+	0.962512i	$0.412566\pi$
$908$	30935.6	1.13065
$909$	0	0
$910$	0	0
$911$	21846.5	0.794518	0.397259	−	0.917707i	$-0.369961\pi$
$911$	21846.5	0.794518	0.397259	+	0.917707i	$0.369961\pi$
$912$	0	0
$913$	27016.4	0.979312
$914$	−63224.2	−2.28804
$915$	0	0
$916$	−21934.0	−0.791179
$917$	−14151.4	−0.509618
$918$	0	0
$919$	28878.9	1.03659	0.518296	−	0.855201i	$-0.326566\pi$
$919$	28878.9	1.03659	0.518296	+	0.855201i	$0.326566\pi$
$920$	0	0
$921$	0	0
$922$	73108.4	2.61139
$923$	19182.9	0.684089
$924$	0	0
$925$	0	0
$926$	14803.4	0.525344
$927$	0	0
$928$	−47985.0	−1.69740
$929$	4608.79	0.162766	0.0813829	−	0.996683i	$-0.474066\pi$
$929$	4608.79	0.162766	0.0813829	+	0.996683i	$0.474066\pi$
$930$	0	0
$931$	12945.2	0.455705
$932$	−2860.23	−0.100526
$933$	0	0
$934$	−11281.3	−0.395220
$935$	0	0
$936$	0	0
$937$	21063.8	0.734391	0.367195	−	0.930144i	$-0.380318\pi$
$937$	21063.8	0.734391	0.367195	+	0.930144i	$0.380318\pi$
$938$	−17179.7	−0.598014
$939$	0	0
$940$	0	0
$941$	−20244.8	−0.701339	−0.350670	−	0.936499i	$-0.614046\pi$
$941$	−20244.8	−0.701339	−0.350670	+	0.936499i	$0.614046\pi$
$942$	0	0
$943$	−24500.4	−0.846069
$944$	−5904.27	−0.203567
$945$	0	0
$946$	−114028.	−3.91901
$947$	29978.3	1.02868	0.514342	−	0.857585i	$-0.328036\pi$
$947$	29978.3	1.02868	0.514342	+	0.857585i	$0.328036\pi$
$948$	0	0
$949$	18382.7	0.628797
$950$	0	0
$951$	0	0
$952$	6062.20	0.206383
$953$	−5782.65	−0.196556	−0.0982782	−	0.995159i	$-0.531334\pi$
$953$	−5782.65	−0.196556	−0.0982782	+	0.995159i	$0.531334\pi$
$954$	0	0
$955$	0	0
$956$	30519.2	1.03249
$957$	0	0
$958$	38559.2	1.30041
$959$	−12649.3	−0.425930
$960$	0	0
$961$	55207.7	1.85317
$962$	−24654.2	−0.826281
$963$	0	0
$964$	47401.6	1.58372
$965$	0	0
$966$	0	0
$967$	26119.2	0.868600	0.434300	−	0.900768i	$-0.356996\pi$
$967$	26119.2	0.868600	0.434300	+	0.900768i	$0.356996\pi$
$968$	−35837.0	−1.18992
$969$	0	0
$970$	0	0
$971$	5101.93	0.168619	0.0843093	−	0.996440i	$-0.473132\pi$
$971$	5101.93	0.168619	0.0843093	+	0.996440i	$0.473132\pi$
$972$	0	0
$973$	−9075.34	−0.299016
$974$	−68013.1	−2.23746
$975$	0	0
$976$	−978.894	−0.0321041
$977$	45902.4	1.50312	0.751559	−	0.659666i	$-0.229302\pi$
$977$	45902.4	1.50312	0.751559	+	0.659666i	$0.229302\pi$
$978$	0	0
$979$	35081.6	1.14526
$980$	0	0
$981$	0	0
$982$	7879.14	0.256042
$983$	−13416.5	−0.435321	−0.217661	−	0.976025i	$-0.569843\pi$
$983$	−13416.5	−0.435321	−0.217661	+	0.976025i	$0.569843\pi$
$984$	0	0
$985$	0	0
$986$	67559.2	2.18207
$987$	0	0
$988$	10290.1	0.331349
$989$	63297.4	2.03513
$990$	0	0
$991$	3806.66	0.122021	0.0610104	−	0.998137i	$-0.480568\pi$
$991$	3806.66	0.122021	0.0610104	+	0.998137i	$0.480568\pi$
$992$	63573.2	2.03473
$993$	0	0
$994$	−20491.8	−0.653883
$995$	0	0
$996$	0	0
$997$	22523.8	0.715483	0.357742	−	0.933821i	$-0.383547\pi$
$997$	22523.8	0.715483	0.357742	+	0.933821i	$0.383547\pi$
$998$	−52325.7	−1.65966
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	675.4.a.r.1.1		3
3.2	odd	2		675.4.a.q.1.3		3
5.2	odd	4		675.4.b.l.649.2		6
5.3	odd	4		675.4.b.l.649.5		6
5.4	even	2		135.4.a.f.1.3	✓	3
15.2	even	4		675.4.b.k.649.5		6
15.8	even	4		675.4.b.k.649.2		6
15.14	odd	2		135.4.a.g.1.1	yes	3
20.19	odd	2		2160.4.a.bm.1.3		3
45.4	even	6		405.4.e.t.136.1		6
45.14	odd	6		405.4.e.r.136.3		6
45.29	odd	6		405.4.e.r.271.3		6
45.34	even	6		405.4.e.t.271.1		6
60.59	even	2		2160.4.a.be.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.4.a.f.1.3	✓	3	5.4	even	2
135.4.a.g.1.1	yes	3	15.14	odd	2
405.4.e.r.136.3		6	45.14	odd	6
405.4.e.r.271.3		6	45.29	odd	6
405.4.e.t.136.1		6	45.4	even	6
405.4.e.t.271.1		6	45.34	even	6
675.4.a.q.1.3		3	3.2	odd	2
675.4.a.r.1.1		3	1.1	even	1	trivial
675.4.b.k.649.2		6	15.8	even	4
675.4.b.k.649.5		6	15.2	even	4
675.4.b.l.649.2		6	5.2	odd	4
675.4.b.l.649.5		6	5.3	odd	4
2160.4.a.be.1.3		3	60.59	even	2
2160.4.a.bm.1.3		3	20.19	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:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	675.a (trivial)

\(f(q)\)	\(=\)	\(q-4.45938 q^{2} +11.8861 q^{4} -5.08123 q^{7} -17.3296 q^{8} +58.3007 q^{11} -21.2119 q^{13} +22.6592 q^{14} -17.8095 q^{16} +68.8451 q^{17} -40.8133 q^{19} -259.985 q^{22} +144.318 q^{23} +94.5921 q^{26} -60.3960 q^{28} -220.058 q^{29} +291.545 q^{31} +218.056 q^{32} -307.006 q^{34} -260.637 q^{37} +182.002 q^{38} -169.766 q^{41} +438.596 q^{43} +692.967 q^{44} -643.571 q^{46} +255.481 q^{47} -317.181 q^{49} -252.127 q^{52} -214.714 q^{53} +88.0557 q^{56} +981.322 q^{58} +331.524 q^{59} +54.9647 q^{61} -1300.11 q^{62} -829.920 q^{64} -758.179 q^{67} +818.299 q^{68} -904.348 q^{71} -866.622 q^{73} +1162.28 q^{74} -485.110 q^{76} -296.239 q^{77} +206.961 q^{79} +757.054 q^{82} +463.397 q^{83} -1955.87 q^{86} -1010.33 q^{88} +601.736 q^{89} +107.783 q^{91} +1715.38 q^{92} -1139.29 q^{94} -229.363 q^{97} +1414.43 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} + 23 q^{4} - 44 q^{7} + 36 q^{8} + 38 q^{11} - 28 q^{13} - 108 q^{14} + 191 q^{16} + 19 q^{17} + 187 q^{19} - 122 q^{22} + 81 q^{23} + 416 q^{26} - 410 q^{28} + 160 q^{29} + 227 q^{31} + 569 q^{32}+ \cdots + 2693 q^{98}+O(q^{100}) \) 3 * q + q^2 + 23 * q^4 - 44 * q^7 + 36 * q^8 + 38 * q^11 - 28 * q^13 - 108 * q^14 + 191 * q^16 + 19 * q^17 + 187 * q^19 - 122 * q^22 + 81 * q^23 + 416 * q^26 - 410 * q^28 + 160 * q^29 + 227 * q^31 + 569 * q^32 + 17 * q^34 - 78 * q^37 + 757 * q^38 - 338 * q^41 - 22 * q^43 + 1636 * q^44 - 1425 * q^46 + 472 * q^47 - 197 * q^49 + 1566 * q^52 - 521 * q^53 - 1254 * q^56 + 2096 * q^58 + 140 * q^59 + 595 * q^61 - 1407 * q^62 - 918 * q^64 - 878 * q^67 + 3053 * q^68 - 602 * q^71 - 1294 * q^73 + 2878 * q^74 + 525 * q^76 - 288 * q^77 + 629 * q^79 + 1682 * q^82 + 1287 * q^83 - 3730 * q^86 + 858 * q^88 + 2154 * q^89 - 440 * q^91 - 1959 * q^92 - 1108 * q^94 - 1392 * q^97 + 2693 * q^98

Introduction

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