LMFDB - Embedded newform 675.4.a.n.1.2

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 = [2,0,0,20,0,0,-22] 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:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 27)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ 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)

$f(q)$	$=$	$q+4.24264 q^{2} +10.0000 q^{4} -11.0000 q^{7} +8.48528 q^{8} -16.9706 q^{11} -29.0000 q^{13} -46.6690 q^{14} -44.0000 q^{16} -50.9117 q^{17} +29.0000 q^{19} -72.0000 q^{22} +84.8528 q^{23} -123.037 q^{26} -110.000 q^{28} -271.529 q^{29} -268.000 q^{31} -254.558 q^{32} -216.000 q^{34} -83.0000 q^{37} +123.037 q^{38} +271.529 q^{41} +232.000 q^{43} -169.706 q^{44} +360.000 q^{46} -390.323 q^{47} -222.000 q^{49} -290.000 q^{52} +305.470 q^{53} -93.3381 q^{56} -1152.00 q^{58} -288.500 q^{59} +767.000 q^{61} -1137.03 q^{62} -728.000 q^{64} +511.000 q^{67} -509.117 q^{68} -712.764 q^{71} -137.000 q^{73} -352.139 q^{74} +290.000 q^{76} +186.676 q^{77} -475.000 q^{79} +1152.00 q^{82} +576.999 q^{83} +984.293 q^{86} -144.000 q^{88} +254.558 q^{89} +319.000 q^{91} +848.528 q^{92} -1656.00 q^{94} -821.000 q^{97} -941.866 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 20 q^{4} - 22 q^{7} - 58 q^{13} - 88 q^{16} + 58 q^{19} - 144 q^{22} - 220 q^{28} - 536 q^{31} - 432 q^{34} - 166 q^{37} + 464 q^{43} + 720 q^{46} - 444 q^{49} - 580 q^{52} - 2304 q^{58} + 1534 q^{61}+ \cdots - 1642 q^{97}+O(q^{100}) $ 2 * q + 20 * q^4 - 22 * q^7 - 58 * q^13 - 88 * q^16 + 58 * q^19 - 144 * q^22 - 220 * q^28 - 536 * q^31 - 432 * q^34 - 166 * q^37 + 464 * q^43 + 720 * q^46 - 444 * q^49 - 580 * q^52 - 2304 * q^58 + 1534 * q^61 - 1456 * q^64 + 1022 * q^67 - 274 * q^73 + 580 * q^76 - 950 * q^79 + 2304 * q^82 - 288 * q^88 + 638 * q^91 - 3312 * q^94 - 1642 * q^97

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 $

$2$	4.24264	1.50000	0.750000	−	0.661438i	$-0.230053\pi$
$2$	4.24264	1.50000	0.750000	+	0.661438i	$0.230053\pi$
$3$	0	0
$3$	0	0
$4$	10.0000	1.25000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−11.0000	−0.593944	−0.296972	−	0.954886i	$-0.595977\pi$
$7$	−11.0000	−0.593944	−0.296972	+	0.954886i	$0.595977\pi$
$8$	8.48528	0.375000
$9$	0	0
$10$	0	0
$11$	−16.9706	−0.465165	−0.232583	−	0.972577i	$-0.574718\pi$
$11$	−16.9706	−0.465165	−0.232583	+	0.972577i	$0.574718\pi$
$12$	0	0
$13$	−29.0000	−0.618704	−0.309352	−	0.950948i	$-0.600112\pi$
$13$	−29.0000	−0.618704	−0.309352	+	0.950948i	$0.600112\pi$
$14$	−46.6690	−0.890916
$15$	0	0
$16$	−44.0000	−0.687500
$17$	−50.9117	−0.726347	−0.363173	−	0.931722i	$-0.618307\pi$
$17$	−50.9117	−0.726347	−0.363173	+	0.931722i	$0.618307\pi$
$18$	0	0
$19$	29.0000	0.350161	0.175080	−	0.984554i	$-0.443981\pi$
$19$	29.0000	0.350161	0.175080	+	0.984554i	$0.443981\pi$
$20$	0	0
$21$	0	0
$22$	−72.0000	−0.697748
$23$	84.8528	0.769262	0.384631	−	0.923070i	$-0.374329\pi$
$23$	84.8528	0.769262	0.384631	+	0.923070i	$0.374329\pi$
$24$	0	0
$25$	0	0
$26$	−123.037	−0.928056
$27$	0	0
$28$	−110.000	−0.742430
$29$	−271.529	−1.73868	−0.869339	−	0.494216i	$-0.835455\pi$
$29$	−271.529	−1.73868	−0.869339	+	0.494216i	$0.835455\pi$
$30$	0	0
$31$	−268.000	−1.55272	−0.776358	−	0.630292i	$-0.782935\pi$
$31$	−268.000	−1.55272	−0.776358	+	0.630292i	$0.782935\pi$
$32$	−254.558	−1.40625
$33$	0	0
$34$	−216.000	−1.08952
$35$	0	0
$36$	0	0
$37$	−83.0000	−0.368787	−0.184393	−	0.982853i	$-0.559032\pi$
$37$	−83.0000	−0.368787	−0.184393	+	0.982853i	$0.559032\pi$
$38$	123.037	0.525241
$39$	0	0
$40$	0	0
$41$	271.529	1.03429	0.517143	−	0.855899i	$-0.326996\pi$
$41$	271.529	1.03429	0.517143	+	0.855899i	$0.326996\pi$
$42$	0	0
$43$	232.000	0.822783	0.411391	−	0.911459i	$-0.365043\pi$
$43$	232.000	0.822783	0.411391	+	0.911459i	$0.365043\pi$
$44$	−169.706	−0.581456
$45$	0	0
$46$	360.000	1.15389
$47$	−390.323	−1.21137	−0.605686	−	0.795704i	$-0.707101\pi$
$47$	−390.323	−1.21137	−0.605686	+	0.795704i	$0.707101\pi$
$48$	0	0
$49$	−222.000	−0.647230
$50$	0	0
$51$	0	0
$52$	−290.000	−0.773380
$53$	305.470	0.791690	0.395845	−	0.918317i	$-0.370452\pi$
$53$	305.470	0.791690	0.395845	+	0.918317i	$0.370452\pi$
$54$	0	0
$55$	0	0
$56$	−93.3381	−0.222729
$57$	0	0
$58$	−1152.00	−2.60802
$59$	−288.500	−0.636601	−0.318300	−	0.947990i	$-0.603112\pi$
$59$	−288.500	−0.636601	−0.318300	+	0.947990i	$0.603112\pi$
$60$	0	0
$61$	767.000	1.60991	0.804953	−	0.593338i	$-0.202190\pi$
$61$	767.000	1.60991	0.804953	+	0.593338i	$0.202190\pi$
$62$	−1137.03	−2.32908
$63$	0	0
$64$	−728.000	−1.42188
$65$	0	0
$66$	0	0
$67$	511.000	0.931770	0.465885	−	0.884845i	$-0.345736\pi$
$67$	511.000	0.931770	0.465885	+	0.884845i	$0.345736\pi$
$68$	−509.117	−0.907934
$69$	0	0
$70$	0	0
$71$	−712.764	−1.19140	−0.595701	−	0.803207i	$-0.703126\pi$
$71$	−712.764	−1.19140	−0.595701	+	0.803207i	$0.703126\pi$
$72$	0	0
$73$	−137.000	−0.219653	−0.109826	−	0.993951i	$-0.535029\pi$
$73$	−137.000	−0.219653	−0.109826	+	0.993951i	$0.535029\pi$
$74$	−352.139	−0.553180
$75$	0	0
$76$	290.000	0.437701
$77$	186.676	0.276282
$78$	0	0
$79$	−475.000	−0.676477	−0.338238	−	0.941060i	$-0.609831\pi$
$79$	−475.000	−0.676477	−0.338238	+	0.941060i	$0.609831\pi$
$80$	0	0
$81$	0	0
$82$	1152.00	1.55143
$83$	576.999	0.763059	0.381529	−	0.924357i	$-0.375397\pi$
$83$	576.999	0.763059	0.381529	+	0.924357i	$0.375397\pi$
$84$	0	0
$85$	0	0
$86$	984.293	1.23417
$87$	0	0
$88$	−144.000	−0.174437
$89$	254.558	0.303181	0.151591	−	0.988443i	$-0.451560\pi$
$89$	254.558	0.303181	0.151591	+	0.988443i	$0.451560\pi$
$90$	0	0
$91$	319.000	0.367476
$92$	848.528	0.961578
$93$	0	0
$94$	−1656.00	−1.81706
$95$	0	0
$96$	0	0
$97$	−821.000	−0.859381	−0.429690	−	0.902976i	$-0.641377\pi$
$97$	−821.000	−0.859381	−0.429690	+	0.902976i	$0.641377\pi$
$98$	−941.866	−0.970845
$99$	0	0
$100$	0	0
$101$	543.058	0.535013	0.267506	−	0.963556i	$-0.413800\pi$
$101$	543.058	0.535013	0.267506	+	0.963556i	$0.413800\pi$
$102$	0	0
$103$	−839.000	−0.802613	−0.401306	−	0.915944i	$-0.631444\pi$
$103$	−839.000	−0.802613	−0.401306	+	0.915944i	$0.631444\pi$
$104$	−246.073	−0.232014
$105$	0	0
$106$	1296.00	1.18753
$107$	−763.675	−0.689975	−0.344987	−	0.938607i	$-0.612117\pi$
$107$	−763.675	−0.689975	−0.344987	+	0.938607i	$0.612117\pi$
$108$	0	0
$109$	218.000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	218.000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	0	0
$112$	484.000	0.408337
$113$	1510.38	1.25739	0.628693	−	0.777654i	$-0.283590\pi$
$113$	1510.38	1.25739	0.628693	+	0.777654i	$0.283590\pi$
$114$	0	0
$115$	0	0
$116$	−2715.29	−2.17335
$117$	0	0
$118$	−1224.00	−0.954901
$119$	560.029	0.431410
$120$	0	0
$121$	−1043.00	−0.783621
$122$	3254.11	2.41486
$123$	0	0
$124$	−2680.00	−1.94090
$125$	0	0
$126$	0	0
$127$	−1244.00	−0.869190	−0.434595	−	0.900626i	$-0.643109\pi$
$127$	−1244.00	−0.869190	−0.434595	+	0.900626i	$0.643109\pi$
$128$	−1052.17	−0.726562
$129$	0	0
$130$	0	0
$131$	2511.64	1.67514	0.837570	−	0.546330i	$-0.183976\pi$
$131$	2511.64	1.67514	0.837570	+	0.546330i	$0.183976\pi$
$132$	0	0
$133$	−319.000	−0.207976
$134$	2167.99	1.39765
$135$	0	0
$136$	−432.000	−0.272380
$137$	1340.67	0.836070	0.418035	−	0.908431i	$-0.362719\pi$
$137$	1340.67	0.836070	0.418035	+	0.908431i	$0.362719\pi$
$138$	0	0
$139$	947.000	0.577867	0.288933	−	0.957349i	$-0.406699\pi$
$139$	947.000	0.577867	0.288933	+	0.957349i	$0.406699\pi$
$140$	0	0
$141$	0	0
$142$	−3024.00	−1.78710
$143$	492.146	0.287800
$144$	0	0
$145$	0	0
$146$	−581.242	−0.329479
$147$	0	0
$148$	−830.000	−0.460984
$149$	576.999	0.317246	0.158623	−	0.987339i	$-0.449295\pi$
$149$	576.999	0.317246	0.158623	+	0.987339i	$0.449295\pi$
$150$	0	0
$151$	−2311.00	−1.24547	−0.622737	−	0.782431i	$-0.713979\pi$
$151$	−2311.00	−1.24547	−0.622737	+	0.782431i	$0.713979\pi$
$152$	246.073	0.131310
$153$	0	0
$154$	792.000	0.414423
$155$	0	0
$156$	0	0
$157$	−1622.00	−0.824520	−0.412260	−	0.911066i	$-0.635261\pi$
$157$	−1622.00	−0.824520	−0.412260	+	0.911066i	$0.635261\pi$
$158$	−2015.25	−1.01472
$159$	0	0
$160$	0	0
$161$	−933.381	−0.456899
$162$	0	0
$163$	−2243.00	−1.07782	−0.538912	−	0.842362i	$-0.681164\pi$
$163$	−2243.00	−1.07782	−0.538912	+	0.842362i	$0.681164\pi$
$164$	2715.29	1.29286
$165$	0	0
$166$	2448.00	1.14459
$167$	2732.26	1.26604	0.633020	−	0.774135i	$-0.281815\pi$
$167$	2732.26	1.26604	0.633020	+	0.774135i	$0.281815\pi$
$168$	0	0
$169$	−1356.00	−0.617205
$170$	0	0
$171$	0	0
$172$	2320.00	1.02848
$173$	−1357.65	−0.596646	−0.298323	−	0.954465i	$-0.596427\pi$
$173$	−1357.65	−0.596646	−0.298323	+	0.954465i	$0.596427\pi$
$174$	0	0
$175$	0	0
$176$	746.705	0.319801
$177$	0	0
$178$	1080.00	0.454772
$179$	2341.94	0.977903	0.488952	−	0.872311i	$-0.337379\pi$
$179$	2341.94	0.977903	0.488952	+	0.872311i	$0.337379\pi$
$180$	0	0
$181$	−1591.00	−0.653360	−0.326680	−	0.945135i	$-0.605930\pi$
$181$	−1591.00	−0.653360	−0.326680	+	0.945135i	$0.605930\pi$
$182$	1353.40	0.551214
$183$	0	0
$184$	720.000	0.288473
$185$	0	0
$186$	0	0
$187$	864.000	0.337871
$188$	−3903.23	−1.51421
$189$	0	0
$190$	0	0
$191$	4768.73	1.80656	0.903280	−	0.429051i	$-0.141152\pi$
$191$	4768.73	1.80656	0.903280	+	0.429051i	$0.141152\pi$
$192$	0	0
$193$	3481.00	1.29828	0.649140	−	0.760669i	$-0.275129\pi$
$193$	3481.00	1.29828	0.649140	+	0.760669i	$0.275129\pi$
$194$	−3483.21	−1.28907
$195$	0	0
$196$	−2220.00	−0.809038
$197$	−2392.85	−0.865398	−0.432699	−	0.901538i	$-0.642439\pi$
$197$	−2392.85	−0.865398	−0.432699	+	0.901538i	$0.642439\pi$
$198$	0	0
$199$	2351.00	0.837477	0.418739	−	0.908107i	$-0.362472\pi$
$199$	2351.00	0.837477	0.418739	+	0.908107i	$0.362472\pi$
$200$	0	0
$201$	0	0
$202$	2304.00	0.802519
$203$	2986.82	1.03268
$204$	0	0
$205$	0	0
$206$	−3559.58	−1.20392
$207$	0	0
$208$	1276.00	0.425359
$209$	−492.146	−0.162883
$210$	0	0
$211$	1703.00	0.555637	0.277818	−	0.960634i	$-0.410389\pi$
$211$	1703.00	0.555637	0.277818	+	0.960634i	$0.410389\pi$
$212$	3054.70	0.989612
$213$	0	0
$214$	−3240.00	−1.03496
$215$	0	0
$216$	0	0
$217$	2948.00	0.922227
$218$	924.896	0.287348
$219$	0	0
$220$	0	0
$221$	1476.44	0.449394
$222$	0	0
$223$	−1388.00	−0.416804	−0.208402	−	0.978043i	$-0.566826\pi$
$223$	−1388.00	−0.416804	−0.208402	+	0.978043i	$0.566826\pi$
$224$	2800.14	0.835234
$225$	0	0
$226$	6408.00	1.88608
$227$	−4717.82	−1.37944	−0.689719	−	0.724077i	$-0.742266\pi$
$227$	−4717.82	−1.37944	−0.689719	+	0.724077i	$0.742266\pi$
$228$	0	0
$229$	434.000	0.125238	0.0626191	−	0.998038i	$-0.480055\pi$
$229$	434.000	0.125238	0.0626191	+	0.998038i	$0.480055\pi$
$230$	0	0
$231$	0	0
$232$	−2304.00	−0.652004
$233$	3461.99	0.973403	0.486701	−	0.873568i	$-0.338200\pi$
$233$	3461.99	0.973403	0.486701	+	0.873568i	$0.338200\pi$
$234$	0	0
$235$	0	0
$236$	−2885.00	−0.795751
$237$	0	0
$238$	2376.00	0.647114
$239$	2817.11	0.762443	0.381222	−	0.924484i	$-0.375503\pi$
$239$	2817.11	0.762443	0.381222	+	0.924484i	$0.375503\pi$
$240$	0	0
$241$	−2095.00	−0.559962	−0.279981	−	0.960006i	$-0.590328\pi$
$241$	−2095.00	−0.559962	−0.279981	+	0.960006i	$0.590328\pi$
$242$	−4425.07	−1.17543
$243$	0	0
$244$	7670.00	2.01238
$245$	0	0
$246$	0	0
$247$	−841.000	−0.216646
$248$	−2274.06	−0.582269
$249$	0	0
$250$	0	0
$251$	−203.647	−0.0512114	−0.0256057	−	0.999672i	$-0.508151\pi$
$251$	−203.647	−0.0512114	−0.0256057	+	0.999672i	$0.508151\pi$
$252$	0	0
$253$	−1440.00	−0.357834
$254$	−5277.85	−1.30379
$255$	0	0
$256$	1360.00	0.332031
$257$	−1798.88	−0.436619	−0.218309	−	0.975880i	$-0.570054\pi$
$257$	−1798.88	−0.436619	−0.218309	+	0.975880i	$0.570054\pi$
$258$	0	0
$259$	913.000	0.219039
$260$	0	0
$261$	0	0
$262$	10656.0	2.51271
$263$	373.352	0.0875357	0.0437679	−	0.999042i	$-0.486064\pi$
$263$	373.352	0.0875357	0.0437679	+	0.999042i	$0.486064\pi$
$264$	0	0
$265$	0	0
$266$	−1353.40	−0.311964
$267$	0	0
$268$	5110.00	1.16471
$269$	−7484.02	−1.69631	−0.848157	−	0.529744i	$-0.822288\pi$
$269$	−7484.02	−1.69631	−0.848157	+	0.529744i	$0.822288\pi$
$270$	0	0
$271$	−3319.00	−0.743966	−0.371983	−	0.928239i	$-0.621322\pi$
$271$	−3319.00	−0.743966	−0.371983	+	0.928239i	$0.621322\pi$
$272$	2240.11	0.499364
$273$	0	0
$274$	5688.00	1.25410
$275$	0	0
$276$	0	0
$277$	−8354.00	−1.81207	−0.906035	−	0.423203i	$-0.860906\pi$
$277$	−8354.00	−1.81207	−0.906035	+	0.423203i	$0.860906\pi$
$278$	4017.78	0.866800
$279$	0	0
$280$	0	0
$281$	2579.53	0.547621	0.273811	−	0.961784i	$-0.411716\pi$
$281$	2579.53	0.547621	0.273811	+	0.961784i	$0.411716\pi$
$282$	0	0
$283$	6208.00	1.30398	0.651992	−	0.758226i	$-0.273934\pi$
$283$	6208.00	1.30398	0.651992	+	0.758226i	$0.273934\pi$
$284$	−7127.64	−1.48925
$285$	0	0
$286$	2088.00	0.431699
$287$	−2986.82	−0.614308
$288$	0	0
$289$	−2321.00	−0.472420
$290$	0	0
$291$	0	0
$292$	−1370.00	−0.274566
$293$	6194.26	1.23506	0.617529	−	0.786548i	$-0.288134\pi$
$293$	6194.26	1.23506	0.617529	+	0.786548i	$0.288134\pi$
$294$	0	0
$295$	0	0
$296$	−704.278	−0.138295
$297$	0	0
$298$	2448.00	0.475869
$299$	−2460.73	−0.475946
$300$	0	0
$301$	−2552.00	−0.488687
$302$	−9804.74	−1.86821
$303$	0	0
$304$	−1276.00	−0.240736
$305$	0	0
$306$	0	0
$307$	2320.00	0.431301	0.215650	−	0.976471i	$-0.430813\pi$
$307$	2320.00	0.431301	0.215650	+	0.976471i	$0.430813\pi$
$308$	1866.76	0.345353
$309$	0	0
$310$	0	0
$311$	−797.616	−0.145430	−0.0727149	−	0.997353i	$-0.523166\pi$
$311$	−797.616	−0.145430	−0.0727149	+	0.997353i	$0.523166\pi$
$312$	0	0
$313$	−1307.00	−0.236026	−0.118013	−	0.993012i	$-0.537652\pi$
$313$	−1307.00	−0.236026	−0.118013	+	0.993012i	$0.537652\pi$
$314$	−6881.56	−1.23678
$315$	0	0
$316$	−4750.00	−0.845596
$317$	−2070.41	−0.366832	−0.183416	−	0.983035i	$-0.558716\pi$
$317$	−2070.41	−0.366832	−0.183416	+	0.983035i	$0.558716\pi$
$318$	0	0
$319$	4608.00	0.808773
$320$	0	0
$321$	0	0
$322$	−3960.00	−0.685348
$323$	−1476.44	−0.254338
$324$	0	0
$325$	0	0
$326$	−9516.24	−1.61674
$327$	0	0
$328$	2304.00	0.387857
$329$	4293.55	0.719487
$330$	0	0
$331$	−5173.00	−0.859014	−0.429507	−	0.903063i	$-0.641313\pi$
$331$	−5173.00	−0.859014	−0.429507	+	0.903063i	$0.641313\pi$
$332$	5769.99	0.953824
$333$	0	0
$334$	11592.0	1.89906
$335$	0	0
$336$	0	0
$337$	−2621.00	−0.423665	−0.211832	−	0.977306i	$-0.567943\pi$
$337$	−2621.00	−0.423665	−0.211832	+	0.977306i	$0.567943\pi$
$338$	−5753.02	−0.925808
$339$	0	0
$340$	0	0
$341$	4548.11	0.722270
$342$	0	0
$343$	6215.00	0.978363
$344$	1968.59	0.308544
$345$	0	0
$346$	−5760.00	−0.894970
$347$	−7704.64	−1.19195	−0.595975	−	0.803003i	$-0.703234\pi$
$347$	−7704.64	−1.19195	−0.595975	+	0.803003i	$0.703234\pi$
$348$	0	0
$349$	1955.00	0.299853	0.149927	−	0.988697i	$-0.452096\pi$
$349$	1955.00	0.299853	0.149927	+	0.988697i	$0.452096\pi$
$350$	0	0
$351$	0	0
$352$	4320.00	0.654139
$353$	1187.94	0.179115	0.0895576	−	0.995982i	$-0.471455\pi$
$353$	1187.94	0.179115	0.0895576	+	0.995982i	$0.471455\pi$
$354$	0	0
$355$	0	0
$356$	2545.58	0.378977
$357$	0	0
$358$	9936.00	1.46685
$359$	560.029	0.0823320	0.0411660	−	0.999152i	$-0.486893\pi$
$359$	560.029	0.0823320	0.0411660	+	0.999152i	$0.486893\pi$
$360$	0	0
$361$	−6018.00	−0.877387
$362$	−6750.04	−0.980039
$363$	0	0
$364$	3190.00	0.459345
$365$	0	0
$366$	0	0
$367$	−7895.00	−1.12293	−0.561465	−	0.827500i	$-0.689762\pi$
$367$	−7895.00	−1.12293	−0.561465	+	0.827500i	$0.689762\pi$
$368$	−3733.52	−0.528868
$369$	0	0
$370$	0	0
$371$	−3360.17	−0.470219
$372$	0	0
$373$	−9803.00	−1.36080	−0.680402	−	0.732839i	$-0.738195\pi$
$373$	−9803.00	−1.36080	−0.680402	+	0.732839i	$0.738195\pi$
$374$	3665.64	0.506807
$375$	0	0
$376$	−3312.00	−0.454264
$377$	7874.34	1.07573
$378$	0	0
$379$	10505.0	1.42376	0.711881	−	0.702300i	$-0.247844\pi$
$379$	10505.0	1.42376	0.711881	+	0.702300i	$0.247844\pi$
$380$	0	0
$381$	0	0
$382$	20232.0	2.70984
$383$	−1086.12	−0.144903	−0.0724516	−	0.997372i	$-0.523082\pi$
$383$	−1086.12	−0.144903	−0.0724516	+	0.997372i	$0.523082\pi$
$384$	0	0
$385$	0	0
$386$	14768.6	1.94742
$387$	0	0
$388$	−8210.00	−1.07423
$389$	−2155.26	−0.280915	−0.140458	−	0.990087i	$-0.544857\pi$
$389$	−2155.26	−0.280915	−0.140458	+	0.990087i	$0.544857\pi$
$390$	0	0
$391$	−4320.00	−0.558751
$392$	−1883.73	−0.242711
$393$	0	0
$394$	−10152.0	−1.29810
$395$	0	0
$396$	0	0
$397$	−12422.0	−1.57038	−0.785192	−	0.619253i	$-0.787436\pi$
$397$	−12422.0	−1.57038	−0.785192	+	0.619253i	$0.787436\pi$
$398$	9974.45	1.25622
$399$	0	0
$400$	0	0
$401$	−15511.1	−1.93164	−0.965819	−	0.259216i	$-0.916536\pi$
$401$	−15511.1	−1.93164	−0.965819	+	0.259216i	$0.916536\pi$
$402$	0	0
$403$	7772.00	0.960672
$404$	5430.58	0.668766
$405$	0	0
$406$	12672.0	1.54902
$407$	1408.56	0.171547
$408$	0	0
$409$	7265.00	0.878316	0.439158	−	0.898410i	$-0.355277\pi$
$409$	7265.00	0.878316	0.439158	+	0.898410i	$0.355277\pi$
$410$	0	0
$411$	0	0
$412$	−8390.00	−1.00327
$413$	3173.50	0.378105
$414$	0	0
$415$	0	0
$416$	7382.19	0.870053
$417$	0	0
$418$	−2088.00	−0.244324
$419$	3173.50	0.370013	0.185006	−	0.982737i	$-0.440769\pi$
$419$	3173.50	0.370013	0.185006	+	0.982737i	$0.440769\pi$
$420$	0	0
$421$	3413.00	0.395106	0.197553	−	0.980292i	$-0.436701\pi$
$421$	3413.00	0.395106	0.197553	+	0.980292i	$0.436701\pi$
$422$	7225.22	0.833455
$423$	0	0
$424$	2592.00	0.296884
$425$	0	0
$426$	0	0
$427$	−8437.00	−0.956194
$428$	−7636.75	−0.862468
$429$	0	0
$430$	0	0
$431$	12677.0	1.41678	0.708388	−	0.705824i	$-0.249423\pi$
$431$	12677.0	1.41678	0.708388	+	0.705824i	$0.249423\pi$
$432$	0	0
$433$	−8642.00	−0.959141	−0.479570	−	0.877503i	$-0.659208\pi$
$433$	−8642.00	−0.959141	−0.479570	+	0.877503i	$0.659208\pi$
$434$	12507.3	1.38334
$435$	0	0
$436$	2180.00	0.239457
$437$	2460.73	0.269366
$438$	0	0
$439$	524.000	0.0569685	0.0284842	−	0.999594i	$-0.490932\pi$
$439$	524.000	0.0569685	0.0284842	+	0.999594i	$0.490932\pi$
$440$	0	0
$441$	0	0
$442$	6264.00	0.674091
$443$	−18158.5	−1.94749	−0.973743	−	0.227649i	$-0.926896\pi$
$443$	−18158.5	−1.94749	−0.973743	+	0.227649i	$0.926896\pi$
$444$	0	0
$445$	0	0
$446$	−5888.79	−0.625206
$447$	0	0
$448$	8008.00	0.844514
$449$	−3309.26	−0.347825	−0.173913	−	0.984761i	$-0.555641\pi$
$449$	−3309.26	−0.347825	−0.173913	+	0.984761i	$0.555641\pi$
$450$	0	0
$451$	−4608.00	−0.481114
$452$	15103.8	1.57173
$453$	0	0
$454$	−20016.0	−2.06916
$455$	0	0
$456$	0	0
$457$	9466.00	0.968930	0.484465	−	0.874811i	$-0.339014\pi$
$457$	9466.00	0.968930	0.484465	+	0.874811i	$0.339014\pi$
$458$	1841.31	0.187857
$459$	0	0
$460$	0	0
$461$	3241.38	0.327475	0.163738	−	0.986504i	$-0.447645\pi$
$461$	3241.38	0.327475	0.163738	+	0.986504i	$0.447645\pi$
$462$	0	0
$463$	−11315.0	−1.13575	−0.567875	−	0.823115i	$-0.692234\pi$
$463$	−11315.0	−1.13575	−0.567875	+	0.823115i	$0.692234\pi$
$464$	11947.3	1.19534
$465$	0	0
$466$	14688.0	1.46010
$467$	17462.7	1.73036	0.865180	−	0.501462i	$-0.167204\pi$
$467$	17462.7	1.73036	0.865180	+	0.501462i	$0.167204\pi$
$468$	0	0
$469$	−5621.00	−0.553419
$470$	0	0
$471$	0	0
$472$	−2448.00	−0.238725
$473$	−3937.17	−0.382730
$474$	0	0
$475$	0	0
$476$	5600.29	0.539262
$477$	0	0
$478$	11952.0	1.14366
$479$	−8926.52	−0.851488	−0.425744	−	0.904844i	$-0.639988\pi$
$479$	−8926.52	−0.851488	−0.425744	+	0.904844i	$0.639988\pi$
$480$	0	0
$481$	2407.00	0.228170
$482$	−8888.33	−0.839943
$483$	0	0
$484$	−10430.0	−0.979527
$485$	0	0
$486$	0	0
$487$	18493.0	1.72073	0.860367	−	0.509674i	$-0.170234\pi$
$487$	18493.0	1.72073	0.860367	+	0.509674i	$0.170234\pi$
$488$	6508.21	0.603715
$489$	0	0
$490$	0	0
$491$	−12812.8	−1.17766	−0.588831	−	0.808256i	$-0.700412\pi$
$491$	−12812.8	−1.17766	−0.588831	+	0.808256i	$0.700412\pi$
$492$	0	0
$493$	13824.0	1.26288
$494$	−3568.06	−0.324969
$495$	0	0
$496$	11792.0	1.06749
$497$	7840.40	0.707626
$498$	0	0
$499$	−12256.0	−1.09951	−0.549753	−	0.835327i	$-0.685278\pi$
$499$	−12256.0	−1.09951	−0.549753	+	0.835327i	$0.685278\pi$
$500$	0	0
$501$	0	0
$502$	−864.000	−0.0768171
$503$	−7382.19	−0.654385	−0.327193	−	0.944958i	$-0.606103\pi$
$503$	−7382.19	−0.654385	−0.327193	+	0.944958i	$0.606103\pi$
$504$	0	0
$505$	0	0
$506$	−6109.40	−0.536751
$507$	0	0
$508$	−12440.0	−1.08649
$509$	20992.6	1.82806	0.914028	−	0.405652i	$-0.132956\pi$
$509$	20992.6	1.82806	0.914028	+	0.405652i	$0.132956\pi$
$510$	0	0
$511$	1507.00	0.130461
$512$	14187.4	1.22461
$513$	0	0
$514$	−7632.00	−0.654928
$515$	0	0
$516$	0	0
$517$	6624.00	0.563488
$518$	3873.53	0.328558
$519$	0	0
$520$	0	0
$521$	−12269.7	−1.03176	−0.515879	−	0.856661i	$-0.672535\pi$
$521$	−12269.7	−1.03176	−0.515879	+	0.856661i	$0.672535\pi$
$522$	0	0
$523$	−6833.00	−0.571293	−0.285646	−	0.958335i	$-0.592208\pi$
$523$	−6833.00	−0.571293	−0.285646	+	0.958335i	$0.592208\pi$
$524$	25116.4	2.09392
$525$	0	0
$526$	1584.00	0.131304
$527$	13644.3	1.12781
$528$	0	0
$529$	−4967.00	−0.408235
$530$	0	0
$531$	0	0
$532$	−3190.00	−0.259970
$533$	−7874.34	−0.639917
$534$	0	0
$535$	0	0
$536$	4335.98	0.349414
$537$	0	0
$538$	−31752.0	−2.54447
$539$	3767.46	0.301069
$540$	0	0
$541$	−10555.0	−0.838808	−0.419404	−	0.907800i	$-0.637761\pi$
$541$	−10555.0	−0.838808	−0.419404	+	0.907800i	$0.637761\pi$
$542$	−14081.3	−1.11595
$543$	0	0
$544$	12960.0	1.02143
$545$	0	0
$546$	0	0
$547$	−17291.0	−1.35157	−0.675786	−	0.737098i	$-0.736196\pi$
$547$	−17291.0	−1.35157	−0.675786	+	0.737098i	$0.736196\pi$
$548$	13406.7	1.04509
$549$	0	0
$550$	0	0
$551$	−7874.34	−0.608817
$552$	0	0
$553$	5225.00	0.401790
$554$	−35443.0	−2.71810
$555$	0	0
$556$	9470.00	0.722334
$557$	10335.1	0.786196	0.393098	−	0.919497i	$-0.371403\pi$
$557$	10335.1	0.786196	0.393098	+	0.919497i	$0.371403\pi$
$558$	0	0
$559$	−6728.00	−0.509059
$560$	0	0
$561$	0	0
$562$	10944.0	0.821432
$563$	16427.5	1.22973	0.614864	−	0.788633i	$-0.289211\pi$
$563$	16427.5	1.22973	0.614864	+	0.788633i	$0.289211\pi$
$564$	0	0
$565$	0	0
$566$	26338.3	1.95598
$567$	0	0
$568$	−6048.00	−0.446775
$569$	−15697.8	−1.15656	−0.578282	−	0.815837i	$-0.696277\pi$
$569$	−15697.8	−1.15656	−0.578282	+	0.815837i	$0.696277\pi$
$570$	0	0
$571$	−4075.00	−0.298658	−0.149329	−	0.988788i	$-0.547711\pi$
$571$	−4075.00	−0.298658	−0.149329	+	0.988788i	$0.547711\pi$
$572$	4921.46	0.359749
$573$	0	0
$574$	−12672.0	−0.921462
$575$	0	0
$576$	0	0
$577$	−6995.00	−0.504689	−0.252345	−	0.967637i	$-0.581202\pi$
$577$	−6995.00	−0.504689	−0.252345	+	0.967637i	$0.581202\pi$
$578$	−9847.17	−0.708630
$579$	0	0
$580$	0	0
$581$	−6346.99	−0.453214
$582$	0	0
$583$	−5184.00	−0.368266
$584$	−1162.48	−0.0823697
$585$	0	0
$586$	26280.0	1.85259
$587$	−5583.32	−0.392586	−0.196293	−	0.980545i	$-0.562890\pi$
$587$	−5583.32	−0.392586	−0.196293	+	0.980545i	$0.562890\pi$
$588$	0	0
$589$	−7772.00	−0.543701
$590$	0	0
$591$	0	0
$592$	3652.00	0.253541
$593$	−14968.0	−1.03653	−0.518266	−	0.855219i	$-0.673422\pi$
$593$	−14968.0	−1.03653	−0.518266	+	0.855219i	$0.673422\pi$
$594$	0	0
$595$	0	0
$596$	5769.99	0.396557
$597$	0	0
$598$	−10440.0	−0.713919
$599$	18192.4	1.24094	0.620470	−	0.784230i	$-0.286942\pi$
$599$	18192.4	1.24094	0.620470	+	0.784230i	$0.286942\pi$
$600$	0	0
$601$	−6550.00	−0.444559	−0.222280	−	0.974983i	$-0.571350\pi$
$601$	−6550.00	−0.444559	−0.222280	+	0.974983i	$0.571350\pi$
$602$	−10827.2	−0.733031
$603$	0	0
$604$	−23110.0	−1.55684
$605$	0	0
$606$	0	0
$607$	−12827.0	−0.857713	−0.428857	−	0.903373i	$-0.641083\pi$
$607$	−12827.0	−0.857713	−0.428857	+	0.903373i	$0.641083\pi$
$608$	−7382.19	−0.492414
$609$	0	0
$610$	0	0
$611$	11319.4	0.749480
$612$	0	0
$613$	−18767.0	−1.23653	−0.618264	−	0.785970i	$-0.712164\pi$
$613$	−18767.0	−1.23653	−0.618264	+	0.785970i	$0.712164\pi$
$614$	9842.93	0.646951
$615$	0	0
$616$	1584.00	0.103606
$617$	−7551.90	−0.492752	−0.246376	−	0.969174i	$-0.579240\pi$
$617$	−7551.90	−0.492752	−0.246376	+	0.969174i	$0.579240\pi$
$618$	0	0
$619$	24581.0	1.59611	0.798056	−	0.602583i	$-0.205862\pi$
$619$	24581.0	1.59611	0.798056	+	0.602583i	$0.205862\pi$
$620$	0	0
$621$	0	0
$622$	−3384.00	−0.218145
$623$	−2800.14	−0.180073
$624$	0	0
$625$	0	0
$626$	−5545.13	−0.354038
$627$	0	0
$628$	−16220.0	−1.03065
$629$	4225.67	0.267867
$630$	0	0
$631$	−18223.0	−1.14968	−0.574838	−	0.818267i	$-0.694935\pi$
$631$	−18223.0	−1.14968	−0.574838	+	0.818267i	$0.694935\pi$
$632$	−4030.51	−0.253679
$633$	0	0
$634$	−8784.00	−0.550248
$635$	0	0
$636$	0	0
$637$	6438.00	0.400444
$638$	19550.1	1.21316
$639$	0	0
$640$	0	0
$641$	−1595.23	−0.0982963	−0.0491481	−	0.998792i	$-0.515651\pi$
$641$	−1595.23	−0.0982963	−0.0491481	+	0.998792i	$0.515651\pi$
$642$	0	0
$643$	26296.0	1.61277	0.806386	−	0.591389i	$-0.201420\pi$
$643$	26296.0	1.61277	0.806386	+	0.591389i	$0.201420\pi$
$644$	−9333.81	−0.571124
$645$	0	0
$646$	−6264.00	−0.381507
$647$	−25659.5	−1.55916	−0.779582	−	0.626301i	$-0.784568\pi$
$647$	−25659.5	−1.55916	−0.779582	+	0.626301i	$0.784568\pi$
$648$	0	0
$649$	4896.00	0.296125
$650$	0	0
$651$	0	0
$652$	−22430.0	−1.34728
$653$	441.235	0.0264423	0.0132212	−	0.999913i	$-0.495791\pi$
$653$	441.235	0.0264423	0.0132212	+	0.999913i	$0.495791\pi$
$654$	0	0
$655$	0	0
$656$	−11947.3	−0.711071
$657$	0	0
$658$	18216.0	1.07923
$659$	−27051.1	−1.59903	−0.799515	−	0.600647i	$-0.794910\pi$
$659$	−27051.1	−1.59903	−0.799515	+	0.600647i	$0.794910\pi$
$660$	0	0
$661$	623.000	0.0366594	0.0183297	−	0.999832i	$-0.494165\pi$
$661$	623.000	0.0366594	0.0183297	+	0.999832i	$0.494165\pi$
$662$	−21947.2	−1.28852
$663$	0	0
$664$	4896.00	0.286147
$665$	0	0
$666$	0	0
$667$	−23040.0	−1.33750
$668$	27322.6	1.58255
$669$	0	0
$670$	0	0
$671$	−13016.4	−0.748872
$672$	0	0
$673$	−27875.0	−1.59659	−0.798293	−	0.602269i	$-0.794263\pi$
$673$	−27875.0	−1.59659	−0.798293	+	0.602269i	$0.794263\pi$
$674$	−11120.0	−0.635497
$675$	0	0
$676$	−13560.0	−0.771507
$677$	−6601.55	−0.374768	−0.187384	−	0.982287i	$-0.560001\pi$
$677$	−6601.55	−0.374768	−0.187384	+	0.982287i	$0.560001\pi$
$678$	0	0
$679$	9031.00	0.510424
$680$	0	0
$681$	0	0
$682$	19296.0	1.08340
$683$	−15680.8	−0.878491	−0.439245	−	0.898367i	$-0.644754\pi$
$683$	−15680.8	−0.878491	−0.439245	+	0.898367i	$0.644754\pi$
$684$	0	0
$685$	0	0
$686$	26368.0	1.46754
$687$	0	0
$688$	−10208.0	−0.565663
$689$	−8858.63	−0.489822
$690$	0	0
$691$	−10600.0	−0.583564	−0.291782	−	0.956485i	$-0.594248\pi$
$691$	−10600.0	−0.583564	−0.291782	+	0.956485i	$0.594248\pi$
$692$	−13576.5	−0.745808
$693$	0	0
$694$	−32688.0	−1.78792
$695$	0	0
$696$	0	0
$697$	−13824.0	−0.751250
$698$	8294.36	0.449780
$699$	0	0
$700$	0	0
$701$	−13593.4	−0.732406	−0.366203	−	0.930535i	$-0.619342\pi$
$701$	−13593.4	−0.732406	−0.366203	+	0.930535i	$0.619342\pi$
$702$	0	0
$703$	−2407.00	−0.129135
$704$	12354.6	0.661407
$705$	0	0
$706$	5040.00	0.268673
$707$	−5973.64	−0.317768
$708$	0	0
$709$	−33523.0	−1.77572	−0.887858	−	0.460117i	$-0.847807\pi$
$709$	−33523.0	−1.77572	−0.887858	+	0.460117i	$0.847807\pi$
$710$	0	0
$711$	0	0
$712$	2160.00	0.113693
$713$	−22740.6	−1.19445
$714$	0	0
$715$	0	0
$716$	23419.4	1.22238
$717$	0	0
$718$	2376.00	0.123498
$719$	−31870.7	−1.65310	−0.826549	−	0.562865i	$-0.809699\pi$
$719$	−31870.7	−1.65310	−0.826549	+	0.562865i	$0.809699\pi$
$720$	0	0
$721$	9229.00	0.476707
$722$	−25532.2	−1.31608
$723$	0	0
$724$	−15910.0	−0.816700
$725$	0	0
$726$	0	0
$727$	13084.0	0.667481	0.333741	−	0.942665i	$-0.391689\pi$
$727$	13084.0	0.667481	0.333741	+	0.942665i	$0.391689\pi$
$728$	2706.80	0.137803
$729$	0	0
$730$	0	0
$731$	−11811.5	−0.597626
$732$	0	0
$733$	19222.0	0.968596	0.484298	−	0.874903i	$-0.339075\pi$
$733$	19222.0	0.968596	0.484298	+	0.874903i	$0.339075\pi$
$734$	−33495.6	−1.68440
$735$	0	0
$736$	−21600.0	−1.08178
$737$	−8671.96	−0.433427
$738$	0	0
$739$	6320.00	0.314594	0.157297	−	0.987551i	$-0.449722\pi$
$739$	6320.00	0.314594	0.157297	+	0.987551i	$0.449722\pi$
$740$	0	0
$741$	0	0
$742$	−14256.0	−0.705329
$743$	4157.79	0.205295	0.102648	−	0.994718i	$-0.467269\pi$
$743$	4157.79	0.205295	0.102648	+	0.994718i	$0.467269\pi$
$744$	0	0
$745$	0	0
$746$	−41590.6	−2.04121
$747$	0	0
$748$	8640.00	0.422339
$749$	8400.43	0.409806
$750$	0	0
$751$	20333.0	0.987965	0.493982	−	0.869472i	$-0.335541\pi$
$751$	20333.0	0.987965	0.493982	+	0.869472i	$0.335541\pi$
$752$	17174.2	0.832818
$753$	0	0
$754$	33408.0	1.61359
$755$	0	0
$756$	0	0
$757$	14011.0	0.672706	0.336353	−	0.941736i	$-0.390806\pi$
$757$	14011.0	0.672706	0.336353	+	0.941736i	$0.390806\pi$
$758$	44568.9	2.13564
$759$	0	0
$760$	0	0
$761$	25981.9	1.23764	0.618820	−	0.785533i	$-0.287611\pi$
$761$	25981.9	1.23764	0.618820	+	0.785533i	$0.287611\pi$
$762$	0	0
$763$	−2398.00	−0.113779
$764$	47687.3	2.25820
$765$	0	0
$766$	−4608.00	−0.217355
$767$	8366.49	0.393867
$768$	0	0
$769$	−6289.00	−0.294912	−0.147456	−	0.989069i	$-0.547108\pi$
$769$	−6289.00	−0.294912	−0.147456	+	0.989069i	$0.547108\pi$
$770$	0	0
$771$	0	0
$772$	34810.0	1.62285
$773$	−7229.46	−0.336385	−0.168192	−	0.985754i	$-0.553793\pi$
$773$	−7229.46	−0.336385	−0.168192	+	0.985754i	$0.553793\pi$
$774$	0	0
$775$	0	0
$776$	−6966.42	−0.322268
$777$	0	0
$778$	−9144.00	−0.421373
$779$	7874.34	0.362166
$780$	0	0
$781$	12096.0	0.554198
$782$	−18328.2	−0.838127
$783$	0	0
$784$	9768.00	0.444971
$785$	0	0
$786$	0	0
$787$	25675.0	1.16292	0.581458	−	0.813576i	$-0.302482\pi$
$787$	25675.0	1.16292	0.581458	+	0.813576i	$0.302482\pi$
$788$	−23928.5	−1.08175
$789$	0	0
$790$	0	0
$791$	−16614.2	−0.746817
$792$	0	0
$793$	−22243.0	−0.996056
$794$	−52702.1	−2.35558
$795$	0	0
$796$	23510.0	1.04685
$797$	−3326.23	−0.147831	−0.0739154	−	0.997265i	$-0.523549\pi$
$797$	−3326.23	−0.147831	−0.0739154	+	0.997265i	$0.523549\pi$
$798$	0	0
$799$	19872.0	0.879876
$800$	0	0
$801$	0	0
$802$	−65808.0	−2.89746
$803$	2324.97	0.102175
$804$	0	0
$805$	0	0
$806$	32973.8	1.44101
$807$	0	0
$808$	4608.00	0.200630
$809$	−10080.5	−0.438087	−0.219043	−	0.975715i	$-0.570294\pi$
$809$	−10080.5	−0.438087	−0.219043	+	0.975715i	$0.570294\pi$
$810$	0	0
$811$	14312.0	0.619682	0.309841	−	0.950788i	$-0.399724\pi$
$811$	14312.0	0.619682	0.309841	+	0.950788i	$0.399724\pi$
$812$	29868.2	1.29085
$813$	0	0
$814$	5976.00	0.257320
$815$	0	0
$816$	0	0
$817$	6728.00	0.288106
$818$	30822.8	1.31747
$819$	0	0
$820$	0	0
$821$	−2766.20	−0.117590	−0.0587948	−	0.998270i	$-0.518726\pi$
$821$	−2766.20	−0.117590	−0.0587948	+	0.998270i	$0.518726\pi$
$822$	0	0
$823$	33343.0	1.41223	0.706114	−	0.708098i	$-0.250447\pi$
$823$	33343.0	1.41223	0.706114	+	0.708098i	$0.250447\pi$
$824$	−7119.15	−0.300980
$825$	0	0
$826$	13464.0	0.567158
$827$	−18379.1	−0.772799	−0.386399	−	0.922332i	$-0.626281\pi$
$827$	−18379.1	−0.772799	−0.386399	+	0.922332i	$0.626281\pi$
$828$	0	0
$829$	3593.00	0.150531	0.0752654	−	0.997164i	$-0.476020\pi$
$829$	3593.00	0.150531	0.0752654	+	0.997164i	$0.476020\pi$
$830$	0	0
$831$	0	0
$832$	21112.0	0.879720
$833$	11302.4	0.470114
$834$	0	0
$835$	0	0
$836$	−4921.46	−0.203603
$837$	0	0
$838$	13464.0	0.555019
$839$	17140.3	0.705301	0.352651	−	0.935755i	$-0.385280\pi$
$839$	17140.3	0.705301	0.352651	+	0.935755i	$0.385280\pi$
$840$	0	0
$841$	49339.0	2.02300
$842$	14480.1	0.592658
$843$	0	0
$844$	17030.0	0.694546
$845$	0	0
$846$	0	0
$847$	11473.0	0.465427
$848$	−13440.7	−0.544287
$849$	0	0
$850$	0	0
$851$	−7042.78	−0.283694
$852$	0	0
$853$	4741.00	0.190303	0.0951517	−	0.995463i	$-0.469666\pi$
$853$	4741.00	0.190303	0.0951517	+	0.995463i	$0.469666\pi$
$854$	−35795.2	−1.43429
$855$	0	0
$856$	−6480.00	−0.258740
$857$	11981.2	0.477562	0.238781	−	0.971073i	$-0.423252\pi$
$857$	11981.2	0.477562	0.238781	+	0.971073i	$0.423252\pi$
$858$	0	0
$859$	6887.00	0.273552	0.136776	−	0.990602i	$-0.456326\pi$
$859$	6887.00	0.273552	0.136776	+	0.990602i	$0.456326\pi$
$860$	0	0
$861$	0	0
$862$	53784.0	2.12516
$863$	−8400.43	−0.331349	−0.165674	−	0.986181i	$-0.552980\pi$
$863$	−8400.43	−0.331349	−0.165674	+	0.986181i	$0.552980\pi$
$864$	0	0
$865$	0	0
$866$	−36664.9	−1.43871
$867$	0	0
$868$	29480.0	1.15278
$869$	8061.02	0.314674
$870$	0	0
$871$	−14819.0	−0.576490
$872$	1849.79	0.0718370
$873$	0	0
$874$	10440.0	0.404048
$875$	0	0
$876$	0	0
$877$	−13475.0	−0.518835	−0.259418	−	0.965765i	$-0.583531\pi$
$877$	−13475.0	−0.518835	−0.259418	+	0.965765i	$0.583531\pi$
$878$	2223.14	0.0854527
$879$	0	0
$880$	0	0
$881$	5243.90	0.200535	0.100268	−	0.994961i	$-0.468030\pi$
$881$	5243.90	0.200535	0.100268	+	0.994961i	$0.468030\pi$
$882$	0	0
$883$	7909.00	0.301426	0.150713	−	0.988578i	$-0.451843\pi$
$883$	7909.00	0.301426	0.150713	+	0.988578i	$0.451843\pi$
$884$	14764.4	0.561742
$885$	0	0
$886$	−77040.0	−2.92123
$887$	35672.1	1.35034	0.675171	−	0.737662i	$-0.264070\pi$
$887$	35672.1	1.35034	0.675171	+	0.737662i	$0.264070\pi$
$888$	0	0
$889$	13684.0	0.516250
$890$	0	0
$891$	0	0
$892$	−13880.0	−0.521005
$893$	−11319.4	−0.424175
$894$	0	0
$895$	0	0
$896$	11573.9	0.431538
$897$	0	0
$898$	−14040.0	−0.521738
$899$	72769.8	2.69967
$900$	0	0
$901$	−15552.0	−0.575041
$902$	−19550.1	−0.721670
$903$	0	0
$904$	12816.0	0.471520
$905$	0	0
$906$	0	0
$907$	16999.0	0.622318	0.311159	−	0.950358i	$-0.399283\pi$
$907$	16999.0	0.622318	0.311159	+	0.950358i	$0.399283\pi$
$908$	−47178.2	−1.72430
$909$	0	0
$910$	0	0
$911$	39032.3	1.41954	0.709768	−	0.704435i	$-0.248800\pi$
$911$	39032.3	1.41954	0.709768	+	0.704435i	$0.248800\pi$
$912$	0	0
$913$	−9792.00	−0.354948
$914$	40160.8	1.45339
$915$	0	0
$916$	4340.00	0.156548
$917$	−27628.1	−0.994939
$918$	0	0
$919$	−28348.0	−1.01753	−0.508767	−	0.860904i	$-0.669899\pi$
$919$	−28348.0	−1.01753	−0.508767	+	0.860904i	$0.669899\pi$
$920$	0	0
$921$	0	0
$922$	13752.0	0.491213
$923$	20670.1	0.737125
$924$	0	0
$925$	0	0
$926$	−48005.5	−1.70363
$927$	0	0
$928$	69120.0	2.44502
$929$	−33160.5	−1.17111	−0.585554	−	0.810633i	$-0.699123\pi$
$929$	−33160.5	−1.17111	−0.585554	+	0.810633i	$0.699123\pi$
$930$	0	0
$931$	−6438.00	−0.226635
$932$	34619.9	1.21675
$933$	0	0
$934$	74088.0	2.59554
$935$	0	0
$936$	0	0
$937$	133.000	0.00463706	0.00231853	−	0.999997i	$-0.499262\pi$
$937$	133.000	0.00463706	0.00231853	+	0.999997i	$0.499262\pi$
$938$	−23847.9	−0.830129
$939$	0	0
$940$	0	0
$941$	48790.4	1.69024	0.845122	−	0.534573i	$-0.179527\pi$
$941$	48790.4	1.69024	0.845122	+	0.534573i	$0.179527\pi$
$942$	0	0
$943$	23040.0	0.795637
$944$	12694.0	0.437663
$945$	0	0
$946$	−16704.0	−0.574095
$947$	45328.4	1.55541	0.777705	−	0.628629i	$-0.216383\pi$
$947$	45328.4	1.55541	0.777705	+	0.628629i	$0.216383\pi$
$948$	0	0
$949$	3973.00	0.135900
$950$	0	0
$951$	0	0
$952$	4752.00	0.161779
$953$	43122.2	1.46576	0.732878	−	0.680360i	$-0.238177\pi$
$953$	43122.2	1.46576	0.732878	+	0.680360i	$0.238177\pi$
$954$	0	0
$955$	0	0
$956$	28171.1	0.953054
$957$	0	0
$958$	−37872.0	−1.27723
$959$	−14747.4	−0.496579
$960$	0	0
$961$	42033.0	1.41093
$962$	10212.0	0.342255
$963$	0	0
$964$	−20950.0	−0.699952
$965$	0	0
$966$	0	0
$967$	−22061.0	−0.733644	−0.366822	−	0.930291i	$-0.619554\pi$
$967$	−22061.0	−0.733644	−0.366822	+	0.930291i	$0.619554\pi$
$968$	−8850.15	−0.293858
$969$	0	0
$970$	0	0
$971$	−33449.0	−1.10549	−0.552744	−	0.833351i	$-0.686419\pi$
$971$	−33449.0	−1.10549	−0.552744	+	0.833351i	$0.686419\pi$
$972$	0	0
$973$	−10417.0	−0.343221
$974$	78459.2	2.58110
$975$	0	0
$976$	−33748.0	−1.10681
$977$	20602.3	0.674642	0.337321	−	0.941390i	$-0.390479\pi$
$977$	20602.3	0.674642	0.337321	+	0.941390i	$0.390479\pi$
$978$	0	0
$979$	−4320.00	−0.141029
$980$	0	0
$981$	0	0
$982$	−54360.0	−1.76649
$983$	11930.3	0.387098	0.193549	−	0.981091i	$-0.438000\pi$
$983$	11930.3	0.387098	0.193549	+	0.981091i	$0.438000\pi$
$984$	0	0
$985$	0	0
$986$	58650.3	1.89433
$987$	0	0
$988$	−8410.00	−0.270807
$989$	19685.9	0.632936
$990$	0	0
$991$	−35017.0	−1.12245	−0.561227	−	0.827662i	$-0.689670\pi$
$991$	−35017.0	−1.12245	−0.561227	+	0.827662i	$0.689670\pi$
$992$	68221.7	2.18351
$993$	0	0
$994$	33264.0	1.06144
$995$	0	0
$996$	0	0
$997$	−13646.0	−0.433474	−0.216737	−	0.976230i	$-0.569541\pi$
$997$	−13646.0	−0.433474	−0.216737	+	0.976230i	$0.569541\pi$
$998$	−51997.8	−1.64926
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	675.4.a.n.1.2		2
3.2	odd	2	inner	675.4.a.n.1.1		2
5.2	odd	4		675.4.b.i.649.3		4
5.3	odd	4		675.4.b.i.649.2		4
5.4	even	2		27.4.a.c.1.1	✓	2
15.2	even	4		675.4.b.i.649.1		4
15.8	even	4		675.4.b.i.649.4		4
15.14	odd	2		27.4.a.c.1.2	yes	2
20.19	odd	2		432.4.a.q.1.2		2
35.34	odd	2		1323.4.a.t.1.1		2
40.19	odd	2		1728.4.a.bk.1.1		2
40.29	even	2		1728.4.a.bp.1.1		2
45.4	even	6		81.4.c.e.55.2		4
45.14	odd	6		81.4.c.e.55.1		4
45.29	odd	6		81.4.c.e.28.1		4
45.34	even	6		81.4.c.e.28.2		4
60.59	even	2		432.4.a.q.1.1		2
105.104	even	2		1323.4.a.t.1.2		2
120.29	odd	2		1728.4.a.bp.1.2		2
120.59	even	2		1728.4.a.bk.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.4.a.c.1.1	✓	2	5.4	even	2
27.4.a.c.1.2	yes	2	15.14	odd	2
81.4.c.e.28.1		4	45.29	odd	6
81.4.c.e.28.2		4	45.34	even	6
81.4.c.e.55.1		4	45.14	odd	6
81.4.c.e.55.2		4	45.4	even	6
432.4.a.q.1.1		2	60.59	even	2
432.4.a.q.1.2		2	20.19	odd	2
675.4.a.n.1.1		2	3.2	odd	2	inner
675.4.a.n.1.2		2	1.1	even	1	trivial
675.4.b.i.649.1		4	15.2	even	4
675.4.b.i.649.2		4	5.3	odd	4
675.4.b.i.649.3		4	5.2	odd	4
675.4.b.i.649.4		4	15.8	even	4
1323.4.a.t.1.1		2	35.34	odd	2
1323.4.a.t.1.2		2	105.104	even	2
1728.4.a.bk.1.1		2	40.19	odd	2
1728.4.a.bk.1.2		2	120.59	even	2
1728.4.a.bp.1.1		2	40.29	even	2
1728.4.a.bp.1.2		2	120.29	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)

\(f(q)\)	\(=\)	\(q+4.24264 q^{2} +10.0000 q^{4} -11.0000 q^{7} +8.48528 q^{8} -16.9706 q^{11} -29.0000 q^{13} -46.6690 q^{14} -44.0000 q^{16} -50.9117 q^{17} +29.0000 q^{19} -72.0000 q^{22} +84.8528 q^{23} -123.037 q^{26} -110.000 q^{28} -271.529 q^{29} -268.000 q^{31} -254.558 q^{32} -216.000 q^{34} -83.0000 q^{37} +123.037 q^{38} +271.529 q^{41} +232.000 q^{43} -169.706 q^{44} +360.000 q^{46} -390.323 q^{47} -222.000 q^{49} -290.000 q^{52} +305.470 q^{53} -93.3381 q^{56} -1152.00 q^{58} -288.500 q^{59} +767.000 q^{61} -1137.03 q^{62} -728.000 q^{64} +511.000 q^{67} -509.117 q^{68} -712.764 q^{71} -137.000 q^{73} -352.139 q^{74} +290.000 q^{76} +186.676 q^{77} -475.000 q^{79} +1152.00 q^{82} +576.999 q^{83} +984.293 q^{86} -144.000 q^{88} +254.558 q^{89} +319.000 q^{91} +848.528 q^{92} -1656.00 q^{94} -821.000 q^{97} -941.866 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 20 q^{4} - 22 q^{7} - 58 q^{13} - 88 q^{16} + 58 q^{19} - 144 q^{22} - 220 q^{28} - 536 q^{31} - 432 q^{34} - 166 q^{37} + 464 q^{43} + 720 q^{46} - 444 q^{49} - 580 q^{52} - 2304 q^{58} + 1534 q^{61}+ \cdots - 1642 q^{97}+O(q^{100}) \) 2 * q + 20 * q^4 - 22 * q^7 - 58 * q^13 - 88 * q^16 + 58 * q^19 - 144 * q^22 - 220 * q^28 - 536 * q^31 - 432 * q^34 - 166 * q^37 + 464 * q^43 + 720 * q^46 - 444 * q^49 - 580 * q^52 - 2304 * q^58 + 1534 * q^61 - 1456 * q^64 + 1022 * q^67 - 274 * q^73 + 580 * q^76 - 950 * q^79 + 2304 * q^82 - 288 * q^88 + 638 * q^91 - 3312 * q^94 - 1642 * q^97

Introduction

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