LMFDB - Embedded newform 675.4.a.q.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:	$1$
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			$5.20067$ 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)

$f(q)$	$=$	$q-5.20067 q^{2} +19.0470 q^{4} -24.4013 q^{7} -57.4517 q^{8} -28.9839 q^{11} +65.3919 q^{13} +126.903 q^{14} +146.411 q^{16} -68.1718 q^{17} +104.424 q^{19} +150.736 q^{22} +154.807 q^{23} -340.082 q^{26} -464.772 q^{28} -205.658 q^{29} -18.2497 q^{31} -301.824 q^{32} +354.539 q^{34} +337.613 q^{37} -543.076 q^{38} -195.969 q^{41} -334.882 q^{43} -552.055 q^{44} -805.098 q^{46} +5.00398 q^{47} +252.425 q^{49} +1245.52 q^{52} +319.965 q^{53} +1401.90 q^{56} +1069.56 q^{58} +430.611 q^{59} +594.581 q^{61} +94.9106 q^{62} +398.396 q^{64} -195.876 q^{67} -1298.47 q^{68} +425.955 q^{71} -929.193 q^{73} -1755.82 q^{74} +1988.96 q^{76} +707.245 q^{77} +24.4296 q^{79} +1019.17 q^{82} +545.859 q^{83} +1741.61 q^{86} +1665.17 q^{88} -84.1332 q^{89} -1595.65 q^{91} +2948.60 q^{92} -26.0241 q^{94} -827.613 q^{97} -1312.78 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−5.20067	−1.83871	−0.919357	−	0.393424i	$-0.871291\pi$
$2$	−5.20067	−1.83871	−0.919357	+	0.393424i	$0.871291\pi$
$3$	0	0
$3$	0	0
$4$	19.0470	2.38087
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−24.4013	−1.31755	−0.658774	−	0.752341i	$-0.728925\pi$
$7$	−24.4013	−1.31755	−0.658774	+	0.752341i	$0.728925\pi$
$8$	−57.4517	−2.53903
$9$	0	0
$10$	0	0
$11$	−28.9839	−0.794451	−0.397226	−	0.917721i	$-0.630027\pi$
$11$	−28.9839	−0.794451	−0.397226	+	0.917721i	$0.630027\pi$
$12$	0	0
$13$	65.3919	1.39511	0.697556	−	0.716530i	$-0.254271\pi$
$13$	65.3919	1.39511	0.697556	+	0.716530i	$0.254271\pi$
$14$	126.903	2.42260
$15$	0	0
$16$	146.411	2.28768
$17$	−68.1718	−0.972593	−0.486296	−	0.873794i	$-0.661653\pi$
$17$	−68.1718	−0.972593	−0.486296	+	0.873794i	$0.661653\pi$
$18$	0	0
$19$	104.424	1.26087	0.630435	−	0.776242i	$-0.282876\pi$
$19$	104.424	1.26087	0.630435	+	0.776242i	$0.282876\pi$
$20$	0	0
$21$	0	0
$22$	150.736	1.46077
$23$	154.807	1.40345	0.701727	−	0.712446i	$-0.252413\pi$
$23$	154.807	1.40345	0.701727	+	0.712446i	$0.252413\pi$
$24$	0	0
$25$	0	0
$26$	−340.082	−2.56521
$27$	0	0
$28$	−464.772	−3.13691
$29$	−205.658	−1.31689	−0.658443	−	0.752631i	$-0.728785\pi$
$29$	−205.658	−1.31689	−0.658443	+	0.752631i	$0.728785\pi$
$30$	0	0
$31$	−18.2497	−0.105734	−0.0528668	−	0.998602i	$-0.516836\pi$
$31$	−18.2497	−0.105734	−0.0528668	+	0.998602i	$0.516836\pi$
$32$	−301.824	−1.66736
$33$	0	0
$34$	354.539	1.78832
$35$	0	0
$36$	0	0
$37$	337.613	1.50009	0.750044	−	0.661387i	$-0.230032\pi$
$37$	337.613	1.50009	0.750044	+	0.661387i	$0.230032\pi$
$38$	−543.076	−2.31838
$39$	0	0
$40$	0	0
$41$	−195.969	−0.746469	−0.373234	−	0.927737i	$-0.621751\pi$
$41$	−195.969	−0.746469	−0.373234	+	0.927737i	$0.621751\pi$
$42$	0	0
$43$	−334.882	−1.18765	−0.593826	−	0.804594i	$-0.702383\pi$
$43$	−334.882	−1.18765	−0.593826	+	0.804594i	$0.702383\pi$
$44$	−552.055	−1.89149
$45$	0	0
$46$	−805.098	−2.58055
$47$	5.00398	0.0155299	0.00776496	−	0.999970i	$-0.497528\pi$
$47$	5.00398	0.0155299	0.00776496	+	0.999970i	$0.497528\pi$
$48$	0	0
$49$	252.425	0.735934
$50$	0	0
$51$	0	0
$52$	1245.52	3.32158
$53$	319.965	0.829256	0.414628	−	0.909991i	$-0.363912\pi$
$53$	319.965	0.829256	0.414628	+	0.909991i	$0.363912\pi$
$54$	0	0
$55$	0	0
$56$	1401.90	3.34529
$57$	0	0
$58$	1069.56	2.42138
$59$	430.611	0.950182	0.475091	−	0.879937i	$-0.342415\pi$
$59$	430.611	0.950182	0.475091	+	0.879937i	$0.342415\pi$
$60$	0	0
$61$	594.581	1.24800	0.624002	−	0.781422i	$-0.285505\pi$
$61$	594.581	1.24800	0.624002	+	0.781422i	$0.285505\pi$
$62$	94.9106	0.194414
$63$	0	0
$64$	398.396	0.778118
$65$	0	0
$66$	0	0
$67$	−195.876	−0.357166	−0.178583	−	0.983925i	$-0.557151\pi$
$67$	−195.876	−0.357166	−0.178583	+	0.983925i	$0.557151\pi$
$68$	−1298.47	−2.31562
$69$	0	0
$70$	0	0
$71$	425.955	0.711994	0.355997	−	0.934487i	$-0.384141\pi$
$71$	425.955	0.711994	0.355997	+	0.934487i	$0.384141\pi$
$72$	0	0
$73$	−929.193	−1.48978	−0.744889	−	0.667188i	$-0.767498\pi$
$73$	−929.193	−1.48978	−0.744889	+	0.667188i	$0.767498\pi$
$74$	−1755.82	−2.75824
$75$	0	0
$76$	1988.96	3.00197
$77$	707.245	1.04673
$78$	0	0
$79$	24.4296	0.0347917	0.0173959	−	0.999849i	$-0.494462\pi$
$79$	24.4296	0.0347917	0.0173959	+	0.999849i	$0.494462\pi$
$80$	0	0
$81$	0	0
$82$	1019.17	1.37254
$83$	545.859	0.721877	0.360938	−	0.932590i	$-0.382456\pi$
$83$	545.859	0.721877	0.360938	+	0.932590i	$0.382456\pi$
$84$	0	0
$85$	0	0
$86$	1741.61	2.18375
$87$	0	0
$88$	1665.17	2.01714
$89$	−84.1332	−0.100203	−0.0501017	−	0.998744i	$-0.515955\pi$
$89$	−84.1332	−0.100203	−0.0501017	+	0.998744i	$0.515955\pi$
$90$	0	0
$91$	−1595.65	−1.83813
$92$	2948.60	3.34144
$93$	0	0
$94$	−26.0241	−0.0285551
$95$	0	0
$96$	0	0
$97$	−827.613	−0.866303	−0.433152	−	0.901321i	$-0.642599\pi$
$97$	−827.613	−0.866303	−0.433152	+	0.901321i	$0.642599\pi$
$98$	−1312.78	−1.35317
$99$	0	0
$100$	0	0
$101$	823.576	0.811375	0.405688	−	0.914012i	$-0.367032\pi$
$101$	823.576	0.811375	0.405688	+	0.914012i	$0.367032\pi$
$102$	0	0
$103$	−1171.19	−1.12040	−0.560198	−	0.828359i	$-0.689275\pi$
$103$	−1171.19	−1.12040	−0.560198	+	0.828359i	$0.689275\pi$
$104$	−3756.87	−3.54223
$105$	0	0
$106$	−1664.03	−1.52477
$107$	−1023.21	−0.924460	−0.462230	−	0.886760i	$-0.652951\pi$
$107$	−1023.21	−0.924460	−0.462230	+	0.886760i	$0.652951\pi$
$108$	0	0
$109$	−403.647	−0.354700	−0.177350	−	0.984148i	$-0.556753\pi$
$109$	−403.647	−0.354700	−0.177350	+	0.984148i	$0.556753\pi$
$110$	0	0
$111$	0	0
$112$	−3572.63	−3.01413
$113$	−1082.20	−0.900931	−0.450465	−	0.892794i	$-0.648742\pi$
$113$	−1082.20	−0.900931	−0.450465	+	0.892794i	$0.648742\pi$
$114$	0	0
$115$	0	0
$116$	−3917.16	−3.13534
$117$	0	0
$118$	−2239.46	−1.74711
$119$	1663.48	1.28144
$120$	0	0
$121$	−490.935	−0.368847
$122$	−3092.22	−2.29473
$123$	0	0
$124$	−347.601	−0.251738
$125$	0	0
$126$	0	0
$127$	774.132	0.540890	0.270445	−	0.962735i	$-0.412829\pi$
$127$	774.132	0.540890	0.270445	+	0.962735i	$0.412829\pi$
$128$	342.664	0.236621
$129$	0	0
$130$	0	0
$131$	−1214.04	−0.809702	−0.404851	−	0.914383i	$-0.632677\pi$
$131$	−1214.04	−0.809702	−0.404851	+	0.914383i	$0.632677\pi$
$132$	0	0
$133$	−2548.09	−1.66126
$134$	1018.69	0.656726
$135$	0	0
$136$	3916.58	2.46944
$137$	−2300.15	−1.43441	−0.717207	−	0.696860i	$-0.754580\pi$
$137$	−2300.15	−1.43441	−0.717207	+	0.696860i	$0.754580\pi$
$138$	0	0
$139$	−1355.93	−0.827396	−0.413698	−	0.910414i	$-0.635763\pi$
$139$	−1355.93	−0.827396	−0.413698	+	0.910414i	$0.635763\pi$
$140$	0	0
$141$	0	0
$142$	−2215.25	−1.30915
$143$	−1895.31	−1.10835
$144$	0	0
$145$	0	0
$146$	4832.43	2.73928
$147$	0	0
$148$	6430.51	3.57152
$149$	259.845	0.142868	0.0714340	−	0.997445i	$-0.477242\pi$
$149$	259.845	0.142868	0.0714340	+	0.997445i	$0.477242\pi$
$150$	0	0
$151$	−508.304	−0.273941	−0.136971	−	0.990575i	$-0.543737\pi$
$151$	−508.304	−0.273941	−0.136971	+	0.990575i	$0.543737\pi$
$152$	−5999.34	−3.20139
$153$	0	0
$154$	−3678.15	−1.92463
$155$	0	0
$156$	0	0
$157$	23.3052	0.0118468	0.00592342	−	0.999982i	$-0.498115\pi$
$157$	23.3052	0.0118468	0.00592342	+	0.999982i	$0.498115\pi$
$158$	−127.050	−0.0639721
$159$	0	0
$160$	0	0
$161$	−3777.49	−1.84912
$162$	0	0
$163$	4032.10	1.93754	0.968769	−	0.247964i	$-0.0797616\pi$
$163$	4032.10	1.93754	0.968769	+	0.247964i	$0.0797616\pi$
$164$	−3732.62	−1.77725
$165$	0	0
$166$	−2838.83	−1.32733
$167$	671.911	0.311341	0.155671	−	0.987809i	$-0.450246\pi$
$167$	671.911	0.311341	0.155671	+	0.987809i	$0.450246\pi$
$168$	0	0
$169$	2079.10	0.946337
$170$	0	0
$171$	0	0
$172$	−6378.49	−2.82765
$173$	−1633.53	−0.717889	−0.358944	−	0.933359i	$-0.616863\pi$
$173$	−1633.53	−0.717889	−0.358944	+	0.933359i	$0.616863\pi$
$174$	0	0
$175$	0	0
$176$	−4243.57	−1.81745
$177$	0	0
$178$	437.549	0.184245
$179$	341.260	0.142497	0.0712485	−	0.997459i	$-0.477302\pi$
$179$	341.260	0.142497	0.0712485	+	0.997459i	$0.477302\pi$
$180$	0	0
$181$	−1695.92	−0.696447	−0.348223	−	0.937412i	$-0.613215\pi$
$181$	−1695.92	−0.696447	−0.348223	+	0.937412i	$0.613215\pi$
$182$	8298.45	3.37979
$183$	0	0
$184$	−8893.90	−3.56341
$185$	0	0
$186$	0	0
$187$	1975.88	0.772678
$188$	95.3107	0.0369747
$189$	0	0
$190$	0	0
$191$	−726.451	−0.275205	−0.137603	−	0.990488i	$-0.543940\pi$
$191$	−726.451	−0.275205	−0.137603	+	0.990488i	$0.543940\pi$
$192$	0	0
$193$	−4247.26	−1.58406	−0.792032	−	0.610479i	$-0.790977\pi$
$193$	−4247.26	−1.58406	−0.792032	+	0.610479i	$0.790977\pi$
$194$	4304.14	1.59288
$195$	0	0
$196$	4807.94	1.75216
$197$	−2678.52	−0.968713	−0.484357	−	0.874871i	$-0.660946\pi$
$197$	−2678.52	−0.968713	−0.484357	+	0.874871i	$0.660946\pi$
$198$	0	0
$199$	1486.48	0.529517	0.264759	−	0.964315i	$-0.414708\pi$
$199$	1486.48	0.529517	0.264759	+	0.964315i	$0.414708\pi$
$200$	0	0
$201$	0	0
$202$	−4283.15	−1.49189
$203$	5018.32	1.73506
$204$	0	0
$205$	0	0
$206$	6090.97	2.06009
$207$	0	0
$208$	9574.12	3.19157
$209$	−3026.62	−1.00170
$210$	0	0
$211$	−4827.41	−1.57504	−0.787519	−	0.616291i	$-0.788635\pi$
$211$	−4827.41	−1.57504	−0.787519	+	0.616291i	$0.788635\pi$
$212$	6094.37	1.97435
$213$	0	0
$214$	5321.37	1.69982
$215$	0	0
$216$	0	0
$217$	445.317	0.139309
$218$	2099.23	0.652193
$219$	0	0
$220$	0	0
$221$	−4457.88	−1.35688
$222$	0	0
$223$	−2774.48	−0.833153	−0.416576	−	0.909101i	$-0.636770\pi$
$223$	−2774.48	−0.833153	−0.416576	+	0.909101i	$0.636770\pi$
$224$	7364.91	2.19683
$225$	0	0
$226$	5628.19	1.65655
$227$	5101.34	1.49158	0.745788	−	0.666184i	$-0.232073\pi$
$227$	5101.34	1.49158	0.745788	+	0.666184i	$0.232073\pi$
$228$	0	0
$229$	−4097.83	−1.18250	−0.591249	−	0.806489i	$-0.701365\pi$
$229$	−4097.83	−1.18250	−0.591249	+	0.806489i	$0.701365\pi$
$230$	0	0
$231$	0	0
$232$	11815.4	3.34361
$233$	−357.613	−0.100549	−0.0502747	−	0.998735i	$-0.516010\pi$
$233$	−357.613	−0.100549	−0.0502747	+	0.998735i	$0.516010\pi$
$234$	0	0
$235$	0	0
$236$	8201.83	2.26226
$237$	0	0
$238$	−8651.22	−2.35620
$239$	−351.682	−0.0951818	−0.0475909	−	0.998867i	$-0.515154\pi$
$239$	−351.682	−0.0951818	−0.0475909	+	0.998867i	$0.515154\pi$
$240$	0	0
$241$	−6165.53	−1.64795	−0.823976	−	0.566624i	$-0.808249\pi$
$241$	−6165.53	−1.64795	−0.823976	+	0.566624i	$0.808249\pi$
$242$	2553.19	0.678204
$243$	0	0
$244$	11325.0	2.97134
$245$	0	0
$246$	0	0
$247$	6828.50	1.75906
$248$	1048.48	0.268461
$249$	0	0
$250$	0	0
$251$	3245.53	0.816160	0.408080	−	0.912946i	$-0.366198\pi$
$251$	3245.53	0.816160	0.408080	+	0.912946i	$0.366198\pi$
$252$	0	0
$253$	−4486.90	−1.11498
$254$	−4026.00	−0.994543
$255$	0	0
$256$	−4969.25	−1.21320
$257$	−3552.19	−0.862178	−0.431089	−	0.902309i	$-0.641871\pi$
$257$	−3552.19	−0.862178	−0.431089	+	0.902309i	$0.641871\pi$
$258$	0	0
$259$	−8238.22	−1.97644
$260$	0	0
$261$	0	0
$262$	6313.81	1.48881
$263$	−4416.59	−1.03551	−0.517754	−	0.855530i	$-0.673232\pi$
$263$	−4416.59	−1.03551	−0.517754	+	0.855530i	$0.673232\pi$
$264$	0	0
$265$	0	0
$266$	13251.8	3.05458
$267$	0	0
$268$	−3730.85	−0.850366
$269$	−3419.93	−0.775155	−0.387578	−	0.921837i	$-0.626688\pi$
$269$	−3419.93	−0.775155	−0.387578	+	0.921837i	$0.626688\pi$
$270$	0	0
$271$	716.407	0.160585	0.0802927	−	0.996771i	$-0.474415\pi$
$271$	716.407	0.160585	0.0802927	+	0.996771i	$0.474415\pi$
$272$	−9981.12	−2.22498
$273$	0	0
$274$	11962.3	2.63748
$275$	0	0
$276$	0	0
$277$	−657.529	−0.142625	−0.0713124	−	0.997454i	$-0.522719\pi$
$277$	−657.529	−0.142625	−0.0713124	+	0.997454i	$0.522719\pi$
$278$	7051.72	1.52135
$279$	0	0
$280$	0	0
$281$	1513.91	0.321397	0.160698	−	0.987004i	$-0.448625\pi$
$281$	1513.91	0.321397	0.160698	+	0.987004i	$0.448625\pi$
$282$	0	0
$283$	−3906.38	−0.820532	−0.410266	−	0.911966i	$-0.634564\pi$
$283$	−3906.38	−0.820532	−0.410266	+	0.911966i	$0.634564\pi$
$284$	8113.16	1.69517
$285$	0	0
$286$	9856.89	2.03794
$287$	4781.91	0.983509
$288$	0	0
$289$	−265.611	−0.0540629
$290$	0	0
$291$	0	0
$292$	−17698.3	−3.54697
$293$	8048.76	1.60483	0.802413	−	0.596770i	$-0.203549\pi$
$293$	8048.76	1.60483	0.802413	+	0.596770i	$0.203549\pi$
$294$	0	0
$295$	0	0
$296$	−19396.4	−3.80877
$297$	0	0
$298$	−1351.37	−0.262693
$299$	10123.1	1.95797
$300$	0	0
$301$	8171.57	1.56479
$302$	2643.52	0.503700
$303$	0	0
$304$	15288.9	2.88447
$305$	0	0
$306$	0	0
$307$	−101.564	−0.0188814	−0.00944068	−	0.999955i	$-0.503005\pi$
$307$	−101.564	−0.0188814	−0.00944068	+	0.999955i	$0.503005\pi$
$308$	13470.9	2.49213
$309$	0	0
$310$	0	0
$311$	−7684.59	−1.40113	−0.700567	−	0.713586i	$-0.747070\pi$
$311$	−7684.59	−1.40113	−0.700567	+	0.713586i	$0.747070\pi$
$312$	0	0
$313$	1345.15	0.242915	0.121457	−	0.992597i	$-0.461243\pi$
$313$	1345.15	0.242915	0.121457	+	0.992597i	$0.461243\pi$
$314$	−121.202	−0.0217830
$315$	0	0
$316$	465.310	0.0828346
$317$	−7622.33	−1.35051	−0.675257	−	0.737583i	$-0.735967\pi$
$317$	−7622.33	−1.35051	−0.675257	+	0.737583i	$0.735967\pi$
$318$	0	0
$319$	5960.76	1.04620
$320$	0	0
$321$	0	0
$322$	19645.5	3.40000
$323$	−7118.78	−1.22631
$324$	0	0
$325$	0	0
$326$	−20969.6	−3.56258
$327$	0	0
$328$	11258.7	1.89531
$329$	−122.104	−0.0204614
$330$	0	0
$331$	−6585.09	−1.09350	−0.546751	−	0.837295i	$-0.684136\pi$
$331$	−6585.09	−1.09350	−0.546751	+	0.837295i	$0.684136\pi$
$332$	10397.0	1.71870
$333$	0	0
$334$	−3494.39	−0.572468
$335$	0	0
$336$	0	0
$337$	2946.94	0.476351	0.238175	−	0.971222i	$-0.423451\pi$
$337$	2946.94	0.476351	0.238175	+	0.971222i	$0.423451\pi$
$338$	−10812.7	−1.74004
$339$	0	0
$340$	0	0
$341$	528.947	0.0840002
$342$	0	0
$343$	2210.14	0.347919
$344$	19239.5	3.01548
$345$	0	0
$346$	8495.44	1.31999
$347$	−8493.48	−1.31399	−0.656994	−	0.753896i	$-0.728172\pi$
$347$	−8493.48	−1.31399	−0.656994	+	0.753896i	$0.728172\pi$
$348$	0	0
$349$	5646.54	0.866053	0.433027	−	0.901381i	$-0.357446\pi$
$349$	5646.54	0.866053	0.433027	+	0.901381i	$0.357446\pi$
$350$	0	0
$351$	0	0
$352$	8748.03	1.32464
$353$	1221.93	0.184240	0.0921202	−	0.995748i	$-0.470636\pi$
$353$	1221.93	0.184240	0.0921202	+	0.995748i	$0.470636\pi$
$354$	0	0
$355$	0	0
$356$	−1602.48	−0.238571
$357$	0	0
$358$	−1774.78	−0.262011
$359$	−4151.44	−0.610319	−0.305160	−	0.952301i	$-0.598710\pi$
$359$	−4151.44	−0.610319	−0.305160	+	0.952301i	$0.598710\pi$
$360$	0	0
$361$	4045.41	0.589796
$362$	8819.93	1.28057
$363$	0	0
$364$	−30392.3	−4.37635
$365$	0	0
$366$	0	0
$367$	7038.71	1.00114	0.500569	−	0.865696i	$-0.333124\pi$
$367$	7038.71	1.00114	0.500569	+	0.865696i	$0.333124\pi$
$368$	22665.5	3.21065
$369$	0	0
$370$	0	0
$371$	−7807.58	−1.09259
$372$	0	0
$373$	7119.57	0.988303	0.494152	−	0.869376i	$-0.335479\pi$
$373$	7119.57	0.988303	0.494152	+	0.869376i	$0.335479\pi$
$374$	−10275.9	−1.42073
$375$	0	0
$376$	−287.487	−0.0394309
$377$	−13448.4	−1.83720
$378$	0	0
$379$	3372.29	0.457053	0.228526	−	0.973538i	$-0.426609\pi$
$379$	3372.29	0.457053	0.228526	+	0.973538i	$0.426609\pi$
$380$	0	0
$381$	0	0
$382$	3778.03	0.506024
$383$	−3958.63	−0.528138	−0.264069	−	0.964504i	$-0.585065\pi$
$383$	−3958.63	−0.528138	−0.264069	+	0.964504i	$0.585065\pi$
$384$	0	0
$385$	0	0
$386$	22088.6	2.91264
$387$	0	0
$388$	−15763.5	−2.06256
$389$	9654.01	1.25830	0.629148	−	0.777285i	$-0.283404\pi$
$389$	9654.01	1.25830	0.629148	+	0.777285i	$0.283404\pi$
$390$	0	0
$391$	−10553.4	−1.36499
$392$	−14502.3	−1.86856
$393$	0	0
$394$	13930.1	1.78119
$395$	0	0
$396$	0	0
$397$	−10928.3	−1.38155	−0.690776	−	0.723068i	$-0.742731\pi$
$397$	−10928.3	−1.38155	−0.690776	+	0.723068i	$0.742731\pi$
$398$	−7730.70	−0.973631
$399$	0	0
$400$	0	0
$401$	4085.57	0.508787	0.254393	−	0.967101i	$-0.418124\pi$
$401$	4085.57	0.508787	0.254393	+	0.967101i	$0.418124\pi$
$402$	0	0
$403$	−1193.38	−0.147510
$404$	15686.6	1.93178
$405$	0	0
$406$	−26098.7	−3.19028
$407$	−9785.34	−1.19175
$408$	0	0
$409$	10156.3	1.22786	0.613930	−	0.789361i	$-0.289588\pi$
$409$	10156.3	1.22786	0.613930	+	0.789361i	$0.289588\pi$
$410$	0	0
$411$	0	0
$412$	−22307.6	−2.66752
$413$	−10507.5	−1.25191
$414$	0	0
$415$	0	0
$416$	−19736.9	−2.32615
$417$	0	0
$418$	15740.4	1.84184
$419$	15878.8	1.85139	0.925693	−	0.378275i	$-0.123483\pi$
$419$	15878.8	1.85139	0.925693	+	0.378275i	$0.123483\pi$
$420$	0	0
$421$	−2279.85	−0.263926	−0.131963	−	0.991255i	$-0.542128\pi$
$421$	−2279.85	−0.263926	−0.131963	+	0.991255i	$0.542128\pi$
$422$	25105.8	2.89604
$423$	0	0
$424$	−18382.5	−2.10551
$425$	0	0
$426$	0	0
$427$	−14508.6	−1.64431
$428$	−19489.0	−2.20102
$429$	0	0
$430$	0	0
$431$	−1947.38	−0.217638	−0.108819	−	0.994062i	$-0.534707\pi$
$431$	−1947.38	−0.217638	−0.108819	+	0.994062i	$0.534707\pi$
$432$	0	0
$433$	−12636.2	−1.40244	−0.701219	−	0.712946i	$-0.747361\pi$
$433$	−12636.2	−1.40244	−0.701219	+	0.712946i	$0.747361\pi$
$434$	−2315.95	−0.256150
$435$	0	0
$436$	−7688.25	−0.844496
$437$	16165.6	1.76957
$438$	0	0
$439$	−15849.8	−1.72317	−0.861585	−	0.507614i	$-0.830528\pi$
$439$	−15849.8	−1.72317	−0.861585	+	0.507614i	$0.830528\pi$
$440$	0	0
$441$	0	0
$442$	23184.0	2.49491
$443$	17455.6	1.87210	0.936048	−	0.351872i	$-0.114455\pi$
$443$	17455.6	1.87210	0.936048	+	0.351872i	$0.114455\pi$
$444$	0	0
$445$	0	0
$446$	14429.2	1.53193
$447$	0	0
$448$	−9721.41	−1.02521
$449$	−16068.1	−1.68887	−0.844435	−	0.535658i	$-0.820064\pi$
$449$	−16068.1	−1.68887	−0.844435	+	0.535658i	$0.820064\pi$
$450$	0	0
$451$	5679.94	0.593033
$452$	−20612.7	−2.14500
$453$	0	0
$454$	−26530.4	−2.74258
$455$	0	0
$456$	0	0
$457$	−11891.7	−1.21722	−0.608612	−	0.793468i	$-0.708273\pi$
$457$	−11891.7	−1.21722	−0.608612	+	0.793468i	$0.708273\pi$
$458$	21311.4	2.17428
$459$	0	0
$460$	0	0
$461$	−2802.23	−0.283108	−0.141554	−	0.989931i	$-0.545210\pi$
$461$	−2802.23	−0.283108	−0.141554	+	0.989931i	$0.545210\pi$
$462$	0	0
$463$	12933.3	1.29819	0.649096	−	0.760707i	$-0.275147\pi$
$463$	12933.3	1.29819	0.649096	+	0.760707i	$0.275147\pi$
$464$	−30110.6	−3.01261
$465$	0	0
$466$	1859.83	0.184882
$467$	−5748.11	−0.569573	−0.284787	−	0.958591i	$-0.591923\pi$
$467$	−5748.11	−0.569573	−0.284787	+	0.958591i	$0.591923\pi$
$468$	0	0
$469$	4779.65	0.470583
$470$	0	0
$471$	0	0
$472$	−24739.3	−2.41254
$473$	9706.17	0.943531
$474$	0	0
$475$	0	0
$476$	31684.3	3.05094
$477$	0	0
$478$	1828.98	0.175012
$479$	−11217.3	−1.07000	−0.535002	−	0.844851i	$-0.679689\pi$
$479$	−11217.3	−1.07000	−0.535002	+	0.844851i	$0.679689\pi$
$480$	0	0
$481$	22077.2	2.09279
$482$	32064.9	3.03012
$483$	0	0
$484$	−9350.83	−0.878177
$485$	0	0
$486$	0	0
$487$	8905.12	0.828603	0.414301	−	0.910140i	$-0.364026\pi$
$487$	8905.12	0.828603	0.414301	+	0.910140i	$0.364026\pi$
$488$	−34159.7	−3.16872
$489$	0	0
$490$	0	0
$491$	−6553.10	−0.602316	−0.301158	−	0.953574i	$-0.597373\pi$
$491$	−6553.10	−0.602316	−0.301158	+	0.953574i	$0.597373\pi$
$492$	0	0
$493$	14020.1	1.28079
$494$	−35512.8	−3.23440
$495$	0	0
$496$	−2671.96	−0.241884
$497$	−10393.9	−0.938087
$498$	0	0
$499$	−4610.09	−0.413579	−0.206789	−	0.978385i	$-0.566302\pi$
$499$	−4610.09	−0.413579	−0.206789	+	0.978385i	$0.566302\pi$
$500$	0	0
$501$	0	0
$502$	−16878.9	−1.50069
$503$	−13069.1	−1.15850	−0.579249	−	0.815151i	$-0.696654\pi$
$503$	−13069.1	−1.15850	−0.579249	+	0.815151i	$0.696654\pi$
$504$	0	0
$505$	0	0
$506$	23334.9	2.05012
$507$	0	0
$508$	14744.9	1.28779
$509$	15930.8	1.38727	0.693635	−	0.720327i	$-0.256008\pi$
$509$	15930.8	1.38727	0.693635	+	0.720327i	$0.256008\pi$
$510$	0	0
$511$	22673.6	1.96286
$512$	23102.1	1.99410
$513$	0	0
$514$	18473.8	1.58530
$515$	0	0
$516$	0	0
$517$	−145.035	−0.0123378
$518$	42844.3	3.63411
$519$	0	0
$520$	0	0
$521$	3654.38	0.307296	0.153648	−	0.988126i	$-0.450898\pi$
$521$	3654.38	0.307296	0.153648	+	0.988126i	$0.450898\pi$
$522$	0	0
$523$	5138.66	0.429633	0.214816	−	0.976654i	$-0.431085\pi$
$523$	5138.66	0.429633	0.214816	+	0.976654i	$0.431085\pi$
$524$	−23123.7	−1.92780
$525$	0	0
$526$	22969.2	1.90400
$527$	1244.11	0.102836
$528$	0	0
$529$	11798.1	0.969681
$530$	0	0
$531$	0	0
$532$	−48533.4	−3.95524
$533$	−12814.8	−1.04141
$534$	0	0
$535$	0	0
$536$	11253.4	0.906854
$537$	0	0
$538$	17785.9	1.42529
$539$	−7316.27	−0.584664
$540$	0	0
$541$	6932.06	0.550892	0.275446	−	0.961317i	$-0.411175\pi$
$541$	6932.06	0.550892	0.275446	+	0.961317i	$0.411175\pi$
$542$	−3725.80	−0.295271
$543$	0	0
$544$	20575.9	1.62166
$545$	0	0
$546$	0	0
$547$	3423.11	0.267572	0.133786	−	0.991010i	$-0.457287\pi$
$547$	3423.11	0.267572	0.133786	+	0.991010i	$0.457287\pi$
$548$	−43810.8	−3.41516
$549$	0	0
$550$	0	0
$551$	−21475.6	−1.66042
$552$	0	0
$553$	−596.115	−0.0458398
$554$	3419.59	0.262246
$555$	0	0
$556$	−25826.3	−1.96992
$557$	24489.2	1.86291	0.931455	−	0.363856i	$-0.118540\pi$
$557$	24489.2	1.86291	0.931455	+	0.363856i	$0.118540\pi$
$558$	0	0
$559$	−21898.6	−1.65691
$560$	0	0
$561$	0	0
$562$	−7873.37	−0.590957
$563$	10053.1	0.752552	0.376276	−	0.926508i	$-0.377204\pi$
$563$	10053.1	0.752552	0.376276	+	0.926508i	$0.377204\pi$
$564$	0	0
$565$	0	0
$566$	20315.8	1.50872
$567$	0	0
$568$	−24471.8	−1.80777
$569$	−6670.45	−0.491459	−0.245729	−	0.969338i	$-0.579027\pi$
$569$	−6670.45	−0.491459	−0.245729	+	0.969338i	$0.579027\pi$
$570$	0	0
$571$	4633.55	0.339594	0.169797	−	0.985479i	$-0.445689\pi$
$571$	4633.55	0.339594	0.169797	+	0.985479i	$0.445689\pi$
$572$	−36099.9	−2.63884
$573$	0	0
$574$	−24869.1	−1.80839
$575$	0	0
$576$	0	0
$577$	−7045.15	−0.508307	−0.254154	−	0.967164i	$-0.581797\pi$
$577$	−7045.15	−0.508307	−0.254154	+	0.967164i	$0.581797\pi$
$578$	1381.35	0.0994062
$579$	0	0
$580$	0	0
$581$	−13319.7	−0.951108
$582$	0	0
$583$	−9273.82	−0.658804
$584$	53383.7	3.78259
$585$	0	0
$586$	−41859.0	−2.95082
$587$	8001.06	0.562588	0.281294	−	0.959622i	$-0.409236\pi$
$587$	8001.06	0.562588	0.281294	+	0.959622i	$0.409236\pi$
$588$	0	0
$589$	−1905.71	−0.133316
$590$	0	0
$591$	0	0
$592$	49430.4	3.43172
$593$	−6747.53	−0.467265	−0.233632	−	0.972325i	$-0.575061\pi$
$593$	−6747.53	−0.467265	−0.233632	+	0.972325i	$0.575061\pi$
$594$	0	0
$595$	0	0
$596$	4949.26	0.340150
$597$	0	0
$598$	−52646.9	−3.60016
$599$	21547.3	1.46978	0.734890	−	0.678186i	$-0.237234\pi$
$599$	21547.3	1.46978	0.734890	+	0.678186i	$0.237234\pi$
$600$	0	0
$601$	−12155.1	−0.824983	−0.412492	−	0.910961i	$-0.635341\pi$
$601$	−12155.1	−0.824983	−0.412492	+	0.910961i	$0.635341\pi$
$602$	−42497.6	−2.87720
$603$	0	0
$604$	−9681.64	−0.652219
$605$	0	0
$606$	0	0
$607$	−16348.9	−1.09322	−0.546608	−	0.837388i	$-0.684081\pi$
$607$	−16348.9	−1.09322	−0.546608	+	0.837388i	$0.684081\pi$
$608$	−31517.7	−2.10232
$609$	0	0
$610$	0	0
$611$	327.220	0.0216660
$612$	0	0
$613$	29955.5	1.97372	0.986859	−	0.161581i	$-0.0516593\pi$
$613$	29955.5	1.97372	0.986859	+	0.161581i	$0.0516593\pi$
$614$	528.202	0.0347174
$615$	0	0
$616$	−40632.4	−2.65767
$617$	−2159.74	−0.140921	−0.0704603	−	0.997515i	$-0.522447\pi$
$617$	−2159.74	−0.140921	−0.0704603	+	0.997515i	$0.522447\pi$
$618$	0	0
$619$	22100.8	1.43507	0.717535	−	0.696523i	$-0.245270\pi$
$619$	22100.8	1.43507	0.717535	+	0.696523i	$0.245270\pi$
$620$	0	0
$621$	0	0
$622$	39965.0	2.57629
$623$	2052.96	0.132023
$624$	0	0
$625$	0	0
$626$	−6995.68	−0.446651
$627$	0	0
$628$	443.893	0.0282058
$629$	−23015.7	−1.45898
$630$	0	0
$631$	18360.1	1.15833	0.579164	−	0.815211i	$-0.303379\pi$
$631$	18360.1	1.15833	0.579164	+	0.815211i	$0.303379\pi$
$632$	−1403.52	−0.0883372
$633$	0	0
$634$	39641.2	2.48321
$635$	0	0
$636$	0	0
$637$	16506.6	1.02671
$638$	−30999.9	−1.92367
$639$	0	0
$640$	0	0
$641$	−21064.5	−1.29797	−0.648985	−	0.760802i	$-0.724806\pi$
$641$	−21064.5	−1.29797	−0.648985	+	0.760802i	$0.724806\pi$
$642$	0	0
$643$	10539.1	0.646381	0.323190	−	0.946334i	$-0.395245\pi$
$643$	10539.1	0.646381	0.323190	+	0.946334i	$0.395245\pi$
$644$	−71949.8	−4.40251
$645$	0	0
$646$	37022.4	2.25484
$647$	−22553.4	−1.37043	−0.685213	−	0.728343i	$-0.740291\pi$
$647$	−22553.4	−1.37043	−0.685213	+	0.728343i	$0.740291\pi$
$648$	0	0
$649$	−12480.8	−0.754873
$650$	0	0
$651$	0	0
$652$	76799.4	4.61303
$653$	−22624.0	−1.35582	−0.677908	−	0.735147i	$-0.737113\pi$
$653$	−22624.0	−1.35582	−0.677908	+	0.735147i	$0.737113\pi$
$654$	0	0
$655$	0	0
$656$	−28692.1	−1.70768
$657$	0	0
$658$	635.022	0.0376227
$659$	6376.60	0.376930	0.188465	−	0.982080i	$-0.439649\pi$
$659$	6376.60	0.376930	0.188465	+	0.982080i	$0.439649\pi$
$660$	0	0
$661$	−22097.5	−1.30029	−0.650146	−	0.759809i	$-0.725292\pi$
$661$	−22097.5	−1.30029	−0.650146	+	0.759809i	$0.725292\pi$
$662$	34246.9	2.01064
$663$	0	0
$664$	−31360.5	−1.83287
$665$	0	0
$666$	0	0
$667$	−31837.2	−1.84819
$668$	12797.9	0.741264
$669$	0	0
$670$	0	0
$671$	−17233.3	−0.991479
$672$	0	0
$673$	−24033.0	−1.37653	−0.688264	−	0.725461i	$-0.741627\pi$
$673$	−24033.0	−1.37653	−0.688264	+	0.725461i	$0.741627\pi$
$674$	−15326.1	−0.875873
$675$	0	0
$676$	39600.6	2.25311
$677$	−179.638	−0.0101980	−0.00509901	−	0.999987i	$-0.501623\pi$
$677$	−179.638	−0.0101980	−0.00509901	+	0.999987i	$0.501623\pi$
$678$	0	0
$679$	20194.9	1.14140
$680$	0	0
$681$	0	0
$682$	−2750.88	−0.154452
$683$	30434.2	1.70502	0.852511	−	0.522709i	$-0.175079\pi$
$683$	30434.2	1.70502	0.852511	+	0.522709i	$0.175079\pi$
$684$	0	0
$685$	0	0
$686$	−11494.2	−0.639725
$687$	0	0
$688$	−49030.5	−2.71696
$689$	20923.1	1.15691
$690$	0	0
$691$	9792.73	0.539122	0.269561	−	0.962983i	$-0.413121\pi$
$691$	9792.73	0.539122	0.269561	+	0.962983i	$0.413121\pi$
$692$	−31113.7	−1.70920
$693$	0	0
$694$	44171.8	2.41605
$695$	0	0
$696$	0	0
$697$	13359.6	0.726010
$698$	−29365.8	−1.59242
$699$	0	0
$700$	0	0
$701$	8130.47	0.438065	0.219032	−	0.975718i	$-0.429710\pi$
$701$	8130.47	0.438065	0.219032	+	0.975718i	$0.429710\pi$
$702$	0	0
$703$	35255.0	1.89142
$704$	−11547.1	−0.618177
$705$	0	0
$706$	−6354.86	−0.338765
$707$	−20096.4	−1.06903
$708$	0	0
$709$	−4859.95	−0.257432	−0.128716	−	0.991681i	$-0.541086\pi$
$709$	−4859.95	−0.257432	−0.128716	+	0.991681i	$0.541086\pi$
$710$	0	0
$711$	0	0
$712$	4833.59	0.254419
$713$	−2825.17	−0.148392
$714$	0	0
$715$	0	0
$716$	6499.96	0.339267
$717$	0	0
$718$	21590.3	1.12220
$719$	19463.0	1.00952	0.504762	−	0.863259i	$-0.331580\pi$
$719$	19463.0	1.00952	0.504762	+	0.863259i	$0.331580\pi$
$720$	0	0
$721$	28578.6	1.47618
$722$	−21038.8	−1.08447
$723$	0	0
$724$	−32302.2	−1.65815
$725$	0	0
$726$	0	0
$727$	−2432.66	−0.124102	−0.0620512	−	0.998073i	$-0.519764\pi$
$727$	−2432.66	−0.124102	−0.0620512	+	0.998073i	$0.519764\pi$
$728$	91672.8	4.66706
$729$	0	0
$730$	0	0
$731$	22829.5	1.15510
$732$	0	0
$733$	17967.6	0.905386	0.452693	−	0.891666i	$-0.350463\pi$
$733$	17967.6	0.905386	0.452693	+	0.891666i	$0.350463\pi$
$734$	−36606.0	−1.84081
$735$	0	0
$736$	−46724.4	−2.34006
$737$	5677.26	0.283751
$738$	0	0
$739$	23473.0	1.16843	0.584214	−	0.811599i	$-0.301403\pi$
$739$	23473.0	1.16843	0.584214	+	0.811599i	$0.301403\pi$
$740$	0	0
$741$	0	0
$742$	40604.6	2.00895
$743$	33559.2	1.65702	0.828512	−	0.559971i	$-0.189188\pi$
$743$	33559.2	1.65702	0.828512	+	0.559971i	$0.189188\pi$
$744$	0	0
$745$	0	0
$746$	−37026.5	−1.81721
$747$	0	0
$748$	37634.6	1.83965
$749$	24967.6	1.21802
$750$	0	0
$751$	−7782.75	−0.378158	−0.189079	−	0.981962i	$-0.560550\pi$
$751$	−7782.75	−0.378158	−0.189079	+	0.981962i	$0.560550\pi$
$752$	732.640	0.0355274
$753$	0	0
$754$	69940.5	3.37809
$755$	0	0
$756$	0	0
$757$	38154.3	1.83189	0.915946	−	0.401300i	$-0.131442\pi$
$757$	38154.3	1.83189	0.915946	+	0.401300i	$0.131442\pi$
$758$	−17538.2	−0.840390
$759$	0	0
$760$	0	0
$761$	19867.1	0.946363	0.473182	−	0.880965i	$-0.343105\pi$
$761$	19867.1	0.946363	0.473182	+	0.880965i	$0.343105\pi$
$762$	0	0
$763$	9849.52	0.467335
$764$	−13836.7	−0.655228
$765$	0	0
$766$	20587.5	0.971094
$767$	28158.5	1.32561
$768$	0	0
$769$	−15710.8	−0.736730	−0.368365	−	0.929681i	$-0.620082\pi$
$769$	−15710.8	−0.736730	−0.368365	+	0.929681i	$0.620082\pi$
$770$	0	0
$771$	0	0
$772$	−80897.5	−3.77146
$773$	−25811.9	−1.20102	−0.600510	−	0.799617i	$-0.705036\pi$
$773$	−25811.9	−1.20102	−0.600510	+	0.799617i	$0.705036\pi$
$774$	0	0
$775$	0	0
$776$	47547.8	2.19957
$777$	0	0
$778$	−50207.3	−2.31365
$779$	−20463.9	−0.941201
$780$	0	0
$781$	−12345.8	−0.565645
$782$	54885.0	2.50982
$783$	0	0
$784$	36958.0	1.68358
$785$	0	0
$786$	0	0
$787$	29242.6	1.32450	0.662252	−	0.749281i	$-0.269601\pi$
$787$	29242.6	1.32450	0.662252	+	0.749281i	$0.269601\pi$
$788$	−51017.7	−2.30638
$789$	0	0
$790$	0	0
$791$	26407.2	1.18702
$792$	0	0
$793$	38880.8	1.74111
$794$	56834.6	2.54028
$795$	0	0
$796$	28313.0	1.26071
$797$	−32573.7	−1.44771	−0.723853	−	0.689955i	$-0.757630\pi$
$797$	−32573.7	−1.44771	−0.723853	+	0.689955i	$0.757630\pi$
$798$	0	0
$799$	−341.130	−0.0151043
$800$	0	0
$801$	0	0
$802$	−21247.7	−0.935514
$803$	26931.6	1.18356
$804$	0	0
$805$	0	0
$806$	6206.39	0.271229
$807$	0	0
$808$	−47315.8	−2.06010
$809$	−34644.9	−1.50562	−0.752812	−	0.658236i	$-0.771303\pi$
$809$	−34644.9	−1.50562	−0.752812	+	0.658236i	$0.771303\pi$
$810$	0	0
$811$	−29057.9	−1.25815	−0.629077	−	0.777343i	$-0.716567\pi$
$811$	−29057.9	−1.25815	−0.629077	+	0.777343i	$0.716567\pi$
$812$	95583.9	4.13096
$813$	0	0
$814$	50890.3	2.19128
$815$	0	0
$816$	0	0
$817$	−34969.8	−1.49748
$818$	−52819.3	−2.25768
$819$	0	0
$820$	0	0
$821$	46709.5	1.98560	0.992798	−	0.119802i	$-0.0382259\pi$
$821$	46709.5	1.98560	0.992798	+	0.119802i	$0.0382259\pi$
$822$	0	0
$823$	3468.10	0.146890	0.0734450	−	0.997299i	$-0.476601\pi$
$823$	3468.10	0.146890	0.0734450	+	0.997299i	$0.476601\pi$
$824$	67286.8	2.84472
$825$	0	0
$826$	54645.9	2.30191
$827$	−42454.9	−1.78513	−0.892564	−	0.450920i	$-0.851096\pi$
$827$	−42454.9	−1.78513	−0.892564	+	0.450920i	$0.851096\pi$
$828$	0	0
$829$	−3933.47	−0.164795	−0.0823975	−	0.996600i	$-0.526258\pi$
$829$	−3933.47	−0.164795	−0.0823975	+	0.996600i	$0.526258\pi$
$830$	0	0
$831$	0	0
$832$	26051.9	1.08556
$833$	−17208.3	−0.715764
$834$	0	0
$835$	0	0
$836$	−57647.9	−2.38492
$837$	0	0
$838$	−82580.5	−3.40417
$839$	−32959.6	−1.35625	−0.678123	−	0.734948i	$-0.737206\pi$
$839$	−32959.6	−1.35625	−0.678123	+	0.734948i	$0.737206\pi$
$840$	0	0
$841$	17906.1	0.734188
$842$	11856.7	0.485285
$843$	0	0
$844$	−91947.6	−3.74996
$845$	0	0
$846$	0	0
$847$	11979.5	0.485974
$848$	46846.5	1.89707
$849$	0	0
$850$	0	0
$851$	52264.8	2.10530
$852$	0	0
$853$	−38845.8	−1.55927	−0.779634	−	0.626235i	$-0.784595\pi$
$853$	−38845.8	−1.55927	−0.779634	+	0.626235i	$0.784595\pi$
$854$	75454.3	3.02341
$855$	0	0
$856$	58785.0	2.34723
$857$	29305.3	1.16809	0.584043	−	0.811723i	$-0.301470\pi$
$857$	29305.3	1.16809	0.584043	+	0.811723i	$0.301470\pi$
$858$	0	0
$859$	−909.659	−0.0361318	−0.0180659	−	0.999837i	$-0.505751\pi$
$859$	−909.659	−0.0361318	−0.0180659	+	0.999837i	$0.505751\pi$
$860$	0	0
$861$	0	0
$862$	10127.7	0.400173
$863$	−47998.0	−1.89324	−0.946622	−	0.322346i	$-0.895529\pi$
$863$	−47998.0	−1.89324	−0.946622	+	0.322346i	$0.895529\pi$
$864$	0	0
$865$	0	0
$866$	65716.6	2.57868
$867$	0	0
$868$	8481.94	0.331677
$869$	−708.065	−0.0276403
$870$	0	0
$871$	−12808.7	−0.498286
$872$	23190.2	0.900594
$873$	0	0
$874$	−84071.7	−3.25374
$875$	0	0
$876$	0	0
$877$	−3258.01	−0.125445	−0.0627225	−	0.998031i	$-0.519978\pi$
$877$	−3258.01	−0.125445	−0.0627225	+	0.998031i	$0.519978\pi$
$878$	82429.8	3.16842
$879$	0	0
$880$	0	0
$881$	−33380.2	−1.27651	−0.638256	−	0.769824i	$-0.720344\pi$
$881$	−33380.2	−1.27651	−0.638256	+	0.769824i	$0.720344\pi$
$882$	0	0
$883$	−33714.6	−1.28492	−0.642460	−	0.766319i	$-0.722086\pi$
$883$	−33714.6	−1.28492	−0.642460	+	0.766319i	$0.722086\pi$
$884$	−84909.2	−3.23055
$885$	0	0
$886$	−90780.6	−3.44225
$887$	6218.80	0.235408	0.117704	−	0.993049i	$-0.462447\pi$
$887$	6218.80	0.235408	0.117704	+	0.993049i	$0.462447\pi$
$888$	0	0
$889$	−18889.8	−0.712649
$890$	0	0
$891$	0	0
$892$	−52845.5	−1.98363
$893$	522.537	0.0195812
$894$	0	0
$895$	0	0
$896$	−8361.46	−0.311760
$897$	0	0
$898$	83565.1	3.10535
$899$	3753.19	0.139239
$900$	0	0
$901$	−21812.6	−0.806529
$902$	−29539.5	−1.09042
$903$	0	0
$904$	62174.4	2.28749
$905$	0	0
$906$	0	0
$907$	22878.6	0.837566	0.418783	−	0.908086i	$-0.362457\pi$
$907$	22878.6	0.837566	0.418783	+	0.908086i	$0.362457\pi$
$908$	97165.0	3.55125
$909$	0	0
$910$	0	0
$911$	−30144.8	−1.09631	−0.548157	−	0.836376i	$-0.684670\pi$
$911$	−30144.8	−1.09631	−0.548157	+	0.836376i	$0.684670\pi$
$912$	0	0
$913$	−15821.1	−0.573496
$914$	61844.9	2.23813
$915$	0	0
$916$	−78051.2	−2.81538
$917$	29624.1	1.06682
$918$	0	0
$919$	3803.52	0.136525	0.0682625	−	0.997667i	$-0.478254\pi$
$919$	3803.52	0.136525	0.0682625	+	0.997667i	$0.478254\pi$
$920$	0	0
$921$	0	0
$922$	14573.5	0.520555
$923$	27854.0	0.993312
$924$	0	0
$925$	0	0
$926$	−67262.0	−2.38700
$927$	0	0
$928$	62072.5	2.19572
$929$	125.985	0.00444934	0.00222467	−	0.999998i	$-0.499292\pi$
$929$	125.985	0.00444934	0.00222467	+	0.999998i	$0.499292\pi$
$930$	0	0
$931$	26359.3	0.927918
$932$	−6811.45	−0.239395
$933$	0	0
$934$	29894.0	1.04728
$935$	0	0
$936$	0	0
$937$	−28107.9	−0.979984	−0.489992	−	0.871727i	$-0.663000\pi$
$937$	−28107.9	−0.979984	−0.489992	+	0.871727i	$0.663000\pi$
$938$	−24857.4	−0.865268
$939$	0	0
$940$	0	0
$941$	−49194.9	−1.70426	−0.852130	−	0.523330i	$-0.824689\pi$
$941$	−49194.9	−1.70426	−0.852130	+	0.523330i	$0.824689\pi$
$942$	0	0
$943$	−30337.3	−1.04763
$944$	63046.3	2.17371
$945$	0	0
$946$	−50478.6	−1.73488
$947$	14498.0	0.497490	0.248745	−	0.968569i	$-0.419982\pi$
$947$	14498.0	0.497490	0.248745	+	0.968569i	$0.419982\pi$
$948$	0	0
$949$	−60761.7	−2.07841
$950$	0	0
$951$	0	0
$952$	−95569.8	−3.25361
$953$	−3201.79	−0.108831	−0.0544155	−	0.998518i	$-0.517330\pi$
$953$	−3201.79	−0.108831	−0.0544155	+	0.998518i	$0.517330\pi$
$954$	0	0
$955$	0	0
$956$	−6698.48	−0.226616
$957$	0	0
$958$	58337.6	1.96743
$959$	56126.7	1.88991
$960$	0	0
$961$	−29457.9	−0.988820
$962$	−114816.	−3.84805
$963$	0	0
$964$	−117435.	−3.92356
$965$	0	0
$966$	0	0
$967$	−57781.9	−1.92155	−0.960776	−	0.277326i	$-0.910552\pi$
$967$	−57781.9	−1.92155	−0.960776	+	0.277326i	$0.910552\pi$
$968$	28205.1	0.936513
$969$	0	0
$970$	0	0
$971$	2611.60	0.0863133	0.0431566	−	0.999068i	$-0.486259\pi$
$971$	2611.60	0.0863133	0.0431566	+	0.999068i	$0.486259\pi$
$972$	0	0
$973$	33086.4	1.09013
$974$	−46312.6	−1.52356
$975$	0	0
$976$	87053.4	2.85503
$977$	33598.3	1.10021	0.550104	−	0.835096i	$-0.314588\pi$
$977$	33598.3	1.10021	0.550104	+	0.835096i	$0.314588\pi$
$978$	0	0
$979$	2438.51	0.0796067
$980$	0	0
$981$	0	0
$982$	34080.5	1.10749
$983$	39484.8	1.28115	0.640575	−	0.767895i	$-0.278696\pi$
$983$	39484.8	1.28115	0.640575	+	0.767895i	$0.278696\pi$
$984$	0	0
$985$	0	0
$986$	−72913.7	−2.35501
$987$	0	0
$988$	130062.	4.18809
$989$	−51841.9	−1.66681
$990$	0	0
$991$	39918.6	1.27957	0.639786	−	0.768553i	$-0.279023\pi$
$991$	39918.6	1.27957	0.639786	+	0.768553i	$0.279023\pi$
$992$	5508.20	0.176296
$993$	0	0
$994$	54055.2	1.72487
$995$	0	0
$996$	0	0
$997$	25670.3	0.815432	0.407716	−	0.913109i	$-0.366325\pi$
$997$	25670.3	0.815432	0.407716	+	0.913109i	$0.366325\pi$
$998$	23975.6	0.760454
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	675.4.a.q.1.1		3
3.2	odd	2		675.4.a.r.1.3		3
5.2	odd	4		675.4.b.k.649.1		6
5.3	odd	4		675.4.b.k.649.6		6
5.4	even	2		135.4.a.g.1.3	yes	3
15.2	even	4		675.4.b.l.649.6		6
15.8	even	4		675.4.b.l.649.1		6
15.14	odd	2		135.4.a.f.1.1	✓	3
20.19	odd	2		2160.4.a.be.1.1		3
45.4	even	6		405.4.e.r.136.1		6
45.14	odd	6		405.4.e.t.136.3		6
45.29	odd	6		405.4.e.t.271.3		6
45.34	even	6		405.4.e.r.271.1		6
60.59	even	2		2160.4.a.bm.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.4.a.f.1.1	✓	3	15.14	odd	2
135.4.a.g.1.3	yes	3	5.4	even	2
405.4.e.r.136.1		6	45.4	even	6
405.4.e.r.271.1		6	45.34	even	6
405.4.e.t.136.3		6	45.14	odd	6
405.4.e.t.271.3		6	45.29	odd	6
675.4.a.q.1.1		3	1.1	even	1	trivial
675.4.a.r.1.3		3	3.2	odd	2
675.4.b.k.649.1		6	5.2	odd	4
675.4.b.k.649.6		6	5.3	odd	4
675.4.b.l.649.1		6	15.8	even	4
675.4.b.l.649.6		6	15.2	even	4
2160.4.a.be.1.1		3	20.19	odd	2
2160.4.a.bm.1.1		3	60.59	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:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	675.a (trivial)

\(f(q)\)	\(=\)	\(q-5.20067 q^{2} +19.0470 q^{4} -24.4013 q^{7} -57.4517 q^{8} -28.9839 q^{11} +65.3919 q^{13} +126.903 q^{14} +146.411 q^{16} -68.1718 q^{17} +104.424 q^{19} +150.736 q^{22} +154.807 q^{23} -340.082 q^{26} -464.772 q^{28} -205.658 q^{29} -18.2497 q^{31} -301.824 q^{32} +354.539 q^{34} +337.613 q^{37} -543.076 q^{38} -195.969 q^{41} -334.882 q^{43} -552.055 q^{44} -805.098 q^{46} +5.00398 q^{47} +252.425 q^{49} +1245.52 q^{52} +319.965 q^{53} +1401.90 q^{56} +1069.56 q^{58} +430.611 q^{59} +594.581 q^{61} +94.9106 q^{62} +398.396 q^{64} -195.876 q^{67} -1298.47 q^{68} +425.955 q^{71} -929.193 q^{73} -1755.82 q^{74} +1988.96 q^{76} +707.245 q^{77} +24.4296 q^{79} +1019.17 q^{82} +545.859 q^{83} +1741.61 q^{86} +1665.17 q^{88} -84.1332 q^{89} -1595.65 q^{91} +2948.60 q^{92} -26.0241 q^{94} -827.613 q^{97} -1312.78 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + 23 q^{4} - 44 q^{7} - 36 q^{8} - 38 q^{11} - 28 q^{13} + 108 q^{14} + 191 q^{16} - 19 q^{17} + 187 q^{19} - 122 q^{22} - 81 q^{23} - 416 q^{26} - 410 q^{28} - 160 q^{29} + 227 q^{31} - 569 q^{32}+ \cdots - 2693 q^{98}+O(q^{100}) \) 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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