LMFDB - Embedded newform 405.4.a.k.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [405,4,Mod(1,405)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(405, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("405.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 405 = 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	405.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$23.8957735523$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 38x^{4} + 42x^{3} + 393x^{2} - 72x - 432 $ x^6 - 2x^5 - 38x^4 + 42x^3 + 393x^2 - 72*x - 432
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2\cdot 3^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.07326$ of defining polynomial
Character	$\chi$	$=$	405.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.07326 q^{2} -3.70159 q^{4} -5.00000 q^{5} -4.66112 q^{7} +24.2604 q^{8} +O(q^{10})$	\(q-2.07326 q^{2} -3.70159 q^{4} -5.00000 q^{5} -4.66112 q^{7} +24.2604 q^{8} +10.3663 q^{10} -8.89410 q^{11} +34.6172 q^{13} +9.66372 q^{14} -20.6855 q^{16} -2.66659 q^{17} +125.599 q^{19} +18.5079 q^{20} +18.4398 q^{22} -131.975 q^{23} +25.0000 q^{25} -71.7704 q^{26} +17.2535 q^{28} +71.2106 q^{29} +13.4250 q^{31} -151.197 q^{32} +5.52853 q^{34} +23.3056 q^{35} +283.849 q^{37} -260.400 q^{38} -121.302 q^{40} -383.452 q^{41} -339.175 q^{43} +32.9223 q^{44} +273.618 q^{46} -78.2867 q^{47} -321.274 q^{49} -51.8315 q^{50} -128.138 q^{52} -254.626 q^{53} +44.4705 q^{55} -113.081 q^{56} -147.638 q^{58} +32.8196 q^{59} +186.983 q^{61} -27.8336 q^{62} +478.955 q^{64} -173.086 q^{65} +406.614 q^{67} +9.87060 q^{68} -48.3186 q^{70} -966.124 q^{71} +276.177 q^{73} -588.492 q^{74} -464.916 q^{76} +41.4565 q^{77} -1146.85 q^{79} +103.428 q^{80} +794.996 q^{82} -178.045 q^{83} +13.3329 q^{85} +703.198 q^{86} -215.775 q^{88} -806.486 q^{89} -161.355 q^{91} +488.516 q^{92} +162.309 q^{94} -627.995 q^{95} -1238.05 q^{97} +666.085 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{2} + 34 q^{4} - 30 q^{5} + 40 q^{7} - 66 q^{8}+O(q^{10}) $ 6 * q - 4 * q^2 + 34 * q^4 - 30 * q^5 + 40 * q^7 - 66 * q^8	$ 6 q - 4 q^{2} + 34 q^{4} - 30 q^{5} + 40 q^{7} - 66 q^{8} + 20 q^{10} - 88 q^{11} + 20 q^{13} - 180 q^{14} + 58 q^{16} - 124 q^{17} - 46 q^{19} - 170 q^{20} - 74 q^{22} - 210 q^{23} + 150 q^{25} - 4 q^{26} + 352 q^{28} - 296 q^{29} - 104 q^{31} - 722 q^{32} - 428 q^{34} - 200 q^{35} - 204 q^{37} + 20 q^{38} + 330 q^{40} - 344 q^{41} + 512 q^{43} - 716 q^{44} - 186 q^{46} - 238 q^{47} + 68 q^{49} - 100 q^{50} - 468 q^{52} - 850 q^{53} + 440 q^{55} - 2316 q^{56} + 890 q^{58} - 1840 q^{59} - 364 q^{61} - 1038 q^{62} - 990 q^{64} - 100 q^{65} + 88 q^{67} - 236 q^{68} + 900 q^{70} - 1364 q^{71} + 836 q^{73} - 1316 q^{74} - 2106 q^{76} - 840 q^{77} - 680 q^{79} - 290 q^{80} + 1742 q^{82} - 2148 q^{83} + 620 q^{85} - 2872 q^{86} + 1296 q^{88} - 3000 q^{89} - 3058 q^{91} - 1002 q^{92} - 3662 q^{94} + 230 q^{95} - 612 q^{97} - 1982 q^{98}+O(q^{100}) $ 6 * q - 4 * q^2 + 34 * q^4 - 30 * q^5 + 40 * q^7 - 66 * q^8 + 20 * q^10 - 88 * q^11 + 20 * q^13 - 180 * q^14 + 58 * q^16 - 124 * q^17 - 46 * q^19 - 170 * q^20 - 74 * q^22 - 210 * q^23 + 150 * q^25 - 4 * q^26 + 352 * q^28 - 296 * q^29 - 104 * q^31 - 722 * q^32 - 428 * q^34 - 200 * q^35 - 204 * q^37 + 20 * q^38 + 330 * q^40 - 344 * q^41 + 512 * q^43 - 716 * q^44 - 186 * q^46 - 238 * q^47 + 68 * q^49 - 100 * q^50 - 468 * q^52 - 850 * q^53 + 440 * q^55 - 2316 * q^56 + 890 * q^58 - 1840 * q^59 - 364 * q^61 - 1038 * q^62 - 990 * q^64 - 100 * q^65 + 88 * q^67 - 236 * q^68 + 900 * q^70 - 1364 * q^71 + 836 * q^73 - 1316 * q^74 - 2106 * q^76 - 840 * q^77 - 680 * q^79 - 290 * q^80 + 1742 * q^82 - 2148 * q^83 + 620 * q^85 - 2872 * q^86 + 1296 * q^88 - 3000 * q^89 - 3058 * q^91 - 1002 * q^92 - 3662 * q^94 + 230 * q^95 - 612 * q^97 - 1982 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.07326	−0.733009	−0.366504	−	0.930416i	$-0.619445\pi$
$2$	−2.07326	−0.733009	−0.366504	+	0.930416i	$0.619445\pi$
$3$	0	0
$3$	0	0
$4$	−3.70159	−0.462698
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−4.66112	−0.251677	−0.125838	−	0.992051i	$-0.540162\pi$
$7$	−4.66112	−0.251677	−0.125838	+	0.992051i	$0.540162\pi$
$8$	24.2604	1.07217
$9$	0	0
$10$	10.3663	0.327811
$11$	−8.89410	−0.243788	−0.121894	−	0.992543i	$-0.538897\pi$
$11$	−8.89410	−0.243788	−0.121894	+	0.992543i	$0.538897\pi$
$12$	0	0
$13$	34.6172	0.738544	0.369272	−	0.929321i	$-0.379607\pi$
$13$	34.6172	0.738544	0.369272	+	0.929321i	$0.379607\pi$
$14$	9.66372	0.184481
$15$	0	0
$16$	−20.6855	−0.323212
$17$	−2.66659	−0.0380437	−0.0190218	−	0.999819i	$-0.506055\pi$
$17$	−2.66659	−0.0380437	−0.0190218	+	0.999819i	$0.506055\pi$
$18$	0	0
$19$	125.599	1.51655	0.758273	−	0.651937i	$-0.226043\pi$
$19$	125.599	1.51655	0.758273	+	0.651937i	$0.226043\pi$
$20$	18.5079	0.206925
$21$	0	0
$22$	18.4398	0.178699
$23$	−131.975	−1.19646	−0.598231	−	0.801323i	$-0.704130\pi$
$23$	−131.975	−1.19646	−0.598231	+	0.801323i	$0.704130\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	−71.7704	−0.541359
$27$	0	0
$28$	17.2535	0.116451
$29$	71.2106	0.455982	0.227991	−	0.973663i	$-0.426784\pi$
$29$	71.2106	0.455982	0.227991	+	0.973663i	$0.426784\pi$
$30$	0	0
$31$	13.4250	0.0777810	0.0388905	−	0.999243i	$-0.487618\pi$
$31$	13.4250	0.0777810	0.0388905	+	0.999243i	$0.487618\pi$
$32$	−151.197	−0.835254
$33$	0	0
$34$	5.52853	0.0278863
$35$	23.3056	0.112553
$36$	0	0
$37$	283.849	1.26120	0.630600	−	0.776108i	$-0.282809\pi$
$37$	283.849	1.26120	0.630600	+	0.776108i	$0.282809\pi$
$38$	−260.400	−1.11164
$39$	0	0
$40$	−121.302	−0.479489
$41$	−383.452	−1.46061	−0.730307	−	0.683119i	$-0.760623\pi$
$41$	−383.452	−1.46061	−0.730307	+	0.683119i	$0.760623\pi$
$42$	0	0
$43$	−339.175	−1.20288	−0.601438	−	0.798919i	$-0.705405\pi$
$43$	−339.175	−1.20288	−0.601438	+	0.798919i	$0.705405\pi$
$44$	32.9223	0.112801
$45$	0	0
$46$	273.618	0.877018
$47$	−78.2867	−0.242964	−0.121482	−	0.992594i	$-0.538765\pi$
$47$	−78.2867	−0.242964	−0.121482	+	0.992594i	$0.538765\pi$
$48$	0	0
$49$	−321.274	−0.936659
$50$	−51.8315	−0.146602
$51$	0	0
$52$	−128.138	−0.341723
$53$	−254.626	−0.659915	−0.329958	−	0.943996i	$-0.607034\pi$
$53$	−254.626	−0.659915	−0.329958	+	0.943996i	$0.607034\pi$
$54$	0	0
$55$	44.4705	0.109026
$56$	−113.081	−0.269841
$57$	0	0
$58$	−147.638	−0.334239
$59$	32.8196	0.0724194	0.0362097	−	0.999344i	$-0.488472\pi$
$59$	32.8196	0.0724194	0.0362097	+	0.999344i	$0.488472\pi$
$60$	0	0
$61$	186.983	0.392470	0.196235	−	0.980557i	$-0.437128\pi$
$61$	186.983	0.392470	0.196235	+	0.980557i	$0.437128\pi$
$62$	−27.8336	−0.0570141
$63$	0	0
$64$	478.955	0.935460
$65$	−173.086	−0.330287
$66$	0	0
$67$	406.614	0.741429	0.370715	−	0.928747i	$-0.379113\pi$
$67$	406.614	0.741429	0.370715	+	0.928747i	$0.379113\pi$
$68$	9.87060	0.0176027
$69$	0	0
$70$	−48.3186	−0.0825026
$71$	−966.124	−1.61490	−0.807450	−	0.589936i	$-0.799153\pi$
$71$	−966.124	−1.61490	−0.807450	+	0.589936i	$0.799153\pi$
$72$	0	0
$73$	276.177	0.442796	0.221398	−	0.975184i	$-0.428938\pi$
$73$	276.177	0.442796	0.221398	+	0.975184i	$0.428938\pi$
$74$	−588.492	−0.924471
$75$	0	0
$76$	−464.916	−0.701704
$77$	41.4565	0.0613559
$78$	0	0
$79$	−1146.85	−1.63330	−0.816652	−	0.577130i	$-0.804172\pi$
$79$	−1146.85	−1.63330	−0.816652	+	0.577130i	$0.804172\pi$
$80$	103.428	0.144545
$81$	0	0
$82$	794.996	1.07064
$83$	−178.045	−0.235457	−0.117729	−	0.993046i	$-0.537561\pi$
$83$	−178.045	−0.235457	−0.117729	+	0.993046i	$0.537561\pi$
$84$	0	0
$85$	13.3329	0.0170136
$86$	703.198	0.881719
$87$	0	0
$88$	−215.775	−0.261383
$89$	−806.486	−0.960532	−0.480266	−	0.877123i	$-0.659460\pi$
$89$	−806.486	−0.960532	−0.480266	+	0.877123i	$0.659460\pi$
$90$	0	0
$91$	−161.355	−0.185874
$92$	488.516	0.553602
$93$	0	0
$94$	162.309	0.178094
$95$	−627.995	−0.678220
$96$	0	0
$97$	−1238.05	−1.29592	−0.647961	−	0.761673i	$-0.724378\pi$
$97$	−1238.05	−1.29592	−0.647961	+	0.761673i	$0.724378\pi$
$98$	666.085	0.686579
$99$	0	0
$100$	−92.5397	−0.0925397
$101$	−566.856	−0.558458	−0.279229	−	0.960225i	$-0.590079\pi$
$101$	−566.856	−0.558458	−0.279229	+	0.960225i	$0.590079\pi$
$102$	0	0
$103$	−819.017	−0.783497	−0.391748	−	0.920072i	$-0.628130\pi$
$103$	−819.017	−0.783497	−0.391748	+	0.920072i	$0.628130\pi$
$104$	839.828	0.791845
$105$	0	0
$106$	527.905	0.483724
$107$	543.772	0.491293	0.245647	−	0.969359i	$-0.421000\pi$
$107$	543.772	0.491293	0.245647	+	0.969359i	$0.421000\pi$
$108$	0	0
$109$	−1636.54	−1.43809	−0.719046	−	0.694963i	$-0.755421\pi$
$109$	−1636.54	−1.43809	−0.719046	+	0.694963i	$0.755421\pi$
$110$	−92.1990	−0.0799166
$111$	0	0
$112$	96.4178	0.0813449
$113$	545.798	0.454375	0.227188	−	0.973851i	$-0.427047\pi$
$113$	545.798	0.454375	0.227188	+	0.973851i	$0.427047\pi$
$114$	0	0
$115$	659.874	0.535074
$116$	−263.592	−0.210982
$117$	0	0
$118$	−68.0436	−0.0530841
$119$	12.4293	0.00957471
$120$	0	0
$121$	−1251.89	−0.940567
$122$	−387.664	−0.287684
$123$	0	0
$124$	−49.6940	−0.0359891
$125$	−125.000	−0.0894427
$126$	0	0
$127$	36.0480	0.0251869	0.0125935	−	0.999921i	$-0.495991\pi$
$127$	36.0480	0.0251869	0.0125935	+	0.999921i	$0.495991\pi$
$128$	216.577	0.149554
$129$	0	0
$130$	358.852	0.242103
$131$	−1310.15	−0.873805	−0.436902	−	0.899509i	$-0.643924\pi$
$131$	−1310.15	−0.873805	−0.436902	+	0.899509i	$0.643924\pi$
$132$	0	0
$133$	−585.432	−0.381680
$134$	−843.016	−0.543474
$135$	0	0
$136$	−64.6926	−0.0407893
$137$	2197.20	1.37022	0.685109	−	0.728441i	$-0.259755\pi$
$137$	2197.20	1.37022	0.685109	+	0.728441i	$0.259755\pi$
$138$	0	0
$139$	1726.01	1.05322	0.526612	−	0.850106i	$-0.323462\pi$
$139$	1726.01	1.05322	0.526612	+	0.850106i	$0.323462\pi$
$140$	−86.2677	−0.0520782
$141$	0	0
$142$	2003.03	1.18374
$143$	−307.889	−0.180049
$144$	0	0
$145$	−356.053	−0.203921
$146$	−572.587	−0.324573
$147$	0	0
$148$	−1050.69	−0.583556
$149$	1801.40	0.990445	0.495222	−	0.868766i	$-0.335087\pi$
$149$	1801.40	0.990445	0.495222	+	0.868766i	$0.335087\pi$
$150$	0	0
$151$	3328.19	1.79367	0.896835	−	0.442365i	$-0.145860\pi$
$151$	3328.19	1.79367	0.896835	+	0.442365i	$0.145860\pi$
$152$	3047.09	1.62600
$153$	0	0
$154$	−85.9501	−0.0449744
$155$	−67.1252	−0.0347847
$156$	0	0
$157$	3830.19	1.94702	0.973510	−	0.228643i	$-0.0734288\pi$
$157$	3830.19	1.94702	0.973510	+	0.228643i	$0.0734288\pi$
$158$	2377.73	1.19723
$159$	0	0
$160$	755.985	0.373537
$161$	615.151	0.301122
$162$	0	0
$163$	−2404.51	−1.15543	−0.577717	−	0.816237i	$-0.696056\pi$
$163$	−2404.51	−1.15543	−0.577717	+	0.816237i	$0.696056\pi$
$164$	1419.38	0.675823
$165$	0	0
$166$	369.134	0.172592
$167$	−3570.95	−1.65466	−0.827331	−	0.561715i	$-0.810142\pi$
$167$	−3570.95	−1.65466	−0.827331	+	0.561715i	$0.810142\pi$
$168$	0	0
$169$	−998.653	−0.454553
$170$	−27.6427	−0.0124711
$171$	0	0
$172$	1255.49	0.556569
$173$	159.936	0.0702874	0.0351437	−	0.999382i	$-0.488811\pi$
$173$	159.936	0.0702874	0.0351437	+	0.999382i	$0.488811\pi$
$174$	0	0
$175$	−116.528	−0.0503354
$176$	183.979	0.0787953
$177$	0	0
$178$	1672.06	0.704078
$179$	1120.87	0.468030	0.234015	−	0.972233i	$-0.424813\pi$
$179$	1120.87	0.468030	0.234015	+	0.972233i	$0.424813\pi$
$180$	0	0
$181$	−3856.65	−1.58377	−0.791886	−	0.610669i	$-0.790901\pi$
$181$	−3856.65	−1.58377	−0.791886	+	0.610669i	$0.790901\pi$
$182$	334.531	0.136248
$183$	0	0
$184$	−3201.77	−1.28281
$185$	−1419.24	−0.564026
$186$	0	0
$187$	23.7169	0.00927461
$188$	289.785	0.112419
$189$	0	0
$190$	1302.00	0.497141
$191$	79.0099	0.0299317	0.0149659	−	0.999888i	$-0.495236\pi$
$191$	79.0099	0.0299317	0.0149659	+	0.999888i	$0.495236\pi$
$192$	0	0
$193$	4571.49	1.70499	0.852496	−	0.522734i	$-0.175088\pi$
$193$	4571.49	1.70499	0.852496	+	0.522734i	$0.175088\pi$
$194$	2566.79	0.949922
$195$	0	0
$196$	1189.22	0.433391
$197$	4081.61	1.47615	0.738077	−	0.674717i	$-0.235734\pi$
$197$	4081.61	1.47615	0.738077	+	0.674717i	$0.235734\pi$
$198$	0	0
$199$	−2518.98	−0.897316	−0.448658	−	0.893703i	$-0.648098\pi$
$199$	−2518.98	−0.897316	−0.448658	+	0.893703i	$0.648098\pi$
$200$	606.511	0.214434
$201$	0	0
$202$	1175.24	0.409355
$203$	−331.921	−0.114760
$204$	0	0
$205$	1917.26	0.653206
$206$	1698.04	0.574310
$207$	0	0
$208$	−716.075	−0.238706
$209$	−1117.09	−0.369717
$210$	0	0
$211$	227.244	0.0741428	0.0370714	−	0.999313i	$-0.488197\pi$
$211$	227.244	0.0741428	0.0370714	+	0.999313i	$0.488197\pi$
$212$	942.519	0.305342
$213$	0	0
$214$	−1127.38	−0.360122
$215$	1695.88	0.537943
$216$	0	0
$217$	−62.5758	−0.0195757
$218$	3392.97	1.05413
$219$	0	0
$220$	−164.612	−0.0504459
$221$	−92.3096	−0.0280969
$222$	0	0
$223$	−515.223	−0.154717	−0.0773585	−	0.997003i	$-0.524649\pi$
$223$	−515.223	−0.154717	−0.0773585	+	0.997003i	$0.524649\pi$
$224$	704.748	0.210214
$225$	0	0
$226$	−1131.58	−0.333061
$227$	3956.04	1.15670	0.578352	−	0.815787i	$-0.303696\pi$
$227$	3956.04	1.15670	0.578352	+	0.815787i	$0.303696\pi$
$228$	0	0
$229$	−1676.57	−0.483803	−0.241901	−	0.970301i	$-0.577771\pi$
$229$	−1676.57	−0.483803	−0.241901	+	0.970301i	$0.577771\pi$
$230$	−1368.09	−0.392214
$231$	0	0
$232$	1727.60	0.488890
$233$	−4610.01	−1.29619	−0.648095	−	0.761560i	$-0.724434\pi$
$233$	−4610.01	−1.29619	−0.648095	+	0.761560i	$0.724434\pi$
$234$	0	0
$235$	391.434	0.108657
$236$	−121.485	−0.0335084
$237$	0	0
$238$	−25.7692	−0.00701834
$239$	6908.24	1.86969	0.934846	−	0.355052i	$-0.115537\pi$
$239$	6908.24	1.86969	0.934846	+	0.355052i	$0.115537\pi$
$240$	0	0
$241$	−2980.96	−0.796765	−0.398383	−	0.917219i	$-0.630428\pi$
$241$	−2980.96	−0.796765	−0.398383	+	0.917219i	$0.630428\pi$
$242$	2595.51	0.689444
$243$	0	0
$244$	−692.133	−0.181595
$245$	1606.37	0.418887
$246$	0	0
$247$	4347.88	1.12004
$248$	325.698	0.0833945
$249$	0	0
$250$	259.158	0.0655623
$251$	−7132.90	−1.79372	−0.896862	−	0.442310i	$-0.854159\pi$
$251$	−7132.90	−1.79372	−0.896862	+	0.442310i	$0.854159\pi$
$252$	0	0
$253$	1173.80	0.291684
$254$	−74.7369	−0.0184622
$255$	0	0
$256$	−4280.66	−1.04508
$257$	−6562.56	−1.59285	−0.796423	−	0.604740i	$-0.793277\pi$
$257$	−6562.56	−1.59285	−0.796423	+	0.604740i	$0.793277\pi$
$258$	0	0
$259$	−1323.05	−0.317415
$260$	640.692	0.152823
$261$	0	0
$262$	2716.28	0.640506
$263$	−5663.50	−1.32786	−0.663928	−	0.747797i	$-0.731112\pi$
$263$	−5663.50	−1.32786	−0.663928	+	0.747797i	$0.731112\pi$
$264$	0	0
$265$	1273.13	0.295123
$266$	1213.75	0.279775
$267$	0	0
$268$	−1505.12	−0.343058
$269$	−4018.43	−0.910811	−0.455406	−	0.890284i	$-0.650506\pi$
$269$	−4018.43	−0.910811	−0.455406	+	0.890284i	$0.650506\pi$
$270$	0	0
$271$	−1518.33	−0.340340	−0.170170	−	0.985415i	$-0.554432\pi$
$271$	−1518.33	−0.340340	−0.170170	+	0.985415i	$0.554432\pi$
$272$	55.1598	0.0122962
$273$	0	0
$274$	−4555.38	−1.00438
$275$	−222.353	−0.0487577
$276$	0	0
$277$	2182.44	0.473393	0.236697	−	0.971584i	$-0.423935\pi$
$277$	2182.44	0.473393	0.236697	+	0.971584i	$0.423935\pi$
$278$	−3578.47	−0.772022
$279$	0	0
$280$	565.404	0.120676
$281$	−5102.57	−1.08325	−0.541626	−	0.840620i	$-0.682191\pi$
$281$	−5102.57	−1.08325	−0.541626	+	0.840620i	$0.682191\pi$
$282$	0	0
$283$	−2577.94	−0.541493	−0.270747	−	0.962651i	$-0.587271\pi$
$283$	−2577.94	−0.541493	−0.270747	+	0.962651i	$0.587271\pi$
$284$	3576.19	0.747212
$285$	0	0
$286$	638.333	0.131977
$287$	1787.32	0.367603
$288$	0	0
$289$	−4905.89	−0.998553
$290$	738.191	0.149476
$291$	0	0
$292$	−1022.29	−0.204881
$293$	8339.85	1.66286	0.831432	−	0.555626i	$-0.187521\pi$
$293$	8339.85	1.66286	0.831432	+	0.555626i	$0.187521\pi$
$294$	0	0
$295$	−164.098	−0.0323870
$296$	6886.29	1.35222
$297$	0	0
$298$	−3734.77	−0.726004
$299$	−4568.59	−0.883640
$300$	0	0
$301$	1580.94	0.302736
$302$	−6900.21	−1.31478
$303$	0	0
$304$	−2598.08	−0.490166
$305$	−934.914	−0.175518
$306$	0	0
$307$	7042.31	1.30921	0.654603	−	0.755973i	$-0.272836\pi$
$307$	7042.31	1.30921	0.654603	+	0.755973i	$0.272836\pi$
$308$	−153.455	−0.0283893
$309$	0	0
$310$	139.168	0.0254975
$311$	−2342.92	−0.427186	−0.213593	−	0.976923i	$-0.568517\pi$
$311$	−2342.92	−0.427186	−0.213593	+	0.976923i	$0.568517\pi$
$312$	0	0
$313$	−3833.47	−0.692269	−0.346135	−	0.938185i	$-0.612506\pi$
$313$	−3833.47	−0.692269	−0.346135	+	0.938185i	$0.612506\pi$
$314$	−7940.98	−1.42718
$315$	0	0
$316$	4245.18	0.755727
$317$	−5889.98	−1.04358	−0.521789	−	0.853075i	$-0.674735\pi$
$317$	−5889.98	−1.04358	−0.521789	+	0.853075i	$0.674735\pi$
$318$	0	0
$319$	−633.355	−0.111163
$320$	−2394.78	−0.418350
$321$	0	0
$322$	−1275.37	−0.220725
$323$	−334.921	−0.0576950
$324$	0	0
$325$	865.429	0.147709
$326$	4985.18	0.846943
$327$	0	0
$328$	−9302.72	−1.56603
$329$	364.904	0.0611484
$330$	0	0
$331$	−6338.95	−1.05263	−0.526315	−	0.850290i	$-0.676427\pi$
$331$	−6338.95	−1.05263	−0.526315	+	0.850290i	$0.676427\pi$
$332$	659.049	0.108946
$333$	0	0
$334$	7403.51	1.21288
$335$	−2033.07	−0.331577
$336$	0	0
$337$	968.342	0.156525	0.0782625	−	0.996933i	$-0.475063\pi$
$337$	968.342	0.156525	0.0782625	+	0.996933i	$0.475063\pi$
$338$	2070.47	0.333191
$339$	0	0
$340$	−49.3530	−0.00787219
$341$	−119.404	−0.0189621
$342$	0	0
$343$	3096.26	0.487412
$344$	−8228.54	−1.28969
$345$	0	0
$346$	−331.590	−0.0515213
$347$	−6457.71	−0.999043	−0.499521	−	0.866302i	$-0.666491\pi$
$347$	−6457.71	−0.999043	−0.499521	+	0.866302i	$0.666491\pi$
$348$	0	0
$349$	−6155.18	−0.944066	−0.472033	−	0.881581i	$-0.656480\pi$
$349$	−6155.18	−0.944066	−0.472033	+	0.881581i	$0.656480\pi$
$350$	241.593	0.0368963
$351$	0	0
$352$	1344.76	0.203625
$353$	−3663.53	−0.552380	−0.276190	−	0.961103i	$-0.589072\pi$
$353$	−3663.53	−0.552380	−0.276190	+	0.961103i	$0.589072\pi$
$354$	0	0
$355$	4830.62	0.722205
$356$	2985.28	0.444437
$357$	0	0
$358$	−2323.85	−0.343070
$359$	−12112.6	−1.78073	−0.890364	−	0.455250i	$-0.849550\pi$
$359$	−12112.6	−1.78073	−0.890364	+	0.455250i	$0.849550\pi$
$360$	0	0
$361$	8916.11	1.29991
$362$	7995.85	1.16092
$363$	0	0
$364$	597.269	0.0860038
$365$	−1380.89	−0.198024
$366$	0	0
$367$	11616.2	1.65220	0.826101	−	0.563522i	$-0.190554\pi$
$367$	11616.2	1.65220	0.826101	+	0.563522i	$0.190554\pi$
$368$	2729.97	0.386711
$369$	0	0
$370$	2942.46	0.413436
$371$	1186.84	0.166085
$372$	0	0
$373$	−2215.72	−0.307576	−0.153788	−	0.988104i	$-0.549147\pi$
$373$	−2215.72	−0.307576	−0.153788	+	0.988104i	$0.549147\pi$
$374$	−49.1713	−0.00679837
$375$	0	0
$376$	−1899.27	−0.260499
$377$	2465.11	0.336763
$378$	0	0
$379$	−6539.83	−0.886354	−0.443177	−	0.896434i	$-0.646149\pi$
$379$	−6539.83	−0.886354	−0.443177	+	0.896434i	$0.646149\pi$
$380$	2324.58	0.313811
$381$	0	0
$382$	−163.808	−0.0219402
$383$	−8495.26	−1.13339	−0.566694	−	0.823928i	$-0.691778\pi$
$383$	−8495.26	−1.13339	−0.566694	+	0.823928i	$0.691778\pi$
$384$	0	0
$385$	−207.282	−0.0274392
$386$	−9477.90	−1.24977
$387$	0	0
$388$	4582.74	0.599621
$389$	3536.84	0.460989	0.230494	−	0.973074i	$-0.425966\pi$
$389$	3536.84	0.460989	0.230494	+	0.973074i	$0.425966\pi$
$390$	0	0
$391$	351.922	0.0455178
$392$	−7794.25	−1.00426
$393$	0	0
$394$	−8462.23	−1.08203
$395$	5734.26	0.730436
$396$	0	0
$397$	8586.22	1.08547	0.542733	−	0.839905i	$-0.317390\pi$
$397$	8586.22	1.08547	0.542733	+	0.839905i	$0.317390\pi$
$398$	5222.51	0.657740
$399$	0	0
$400$	−517.139	−0.0646423
$401$	7232.79	0.900719	0.450359	−	0.892847i	$-0.351296\pi$
$401$	7232.79	0.900719	0.450359	+	0.892847i	$0.351296\pi$
$402$	0	0
$403$	464.737	0.0574447
$404$	2098.27	0.258398
$405$	0	0
$406$	688.160	0.0841202
$407$	−2524.58	−0.307466
$408$	0	0
$409$	1230.96	0.148819	0.0744096	−	0.997228i	$-0.476293\pi$
$409$	1230.96	0.148819	0.0744096	+	0.997228i	$0.476293\pi$
$410$	−3974.98	−0.478806
$411$	0	0
$412$	3031.66	0.362523
$413$	−152.976	−0.0182263
$414$	0	0
$415$	890.225	0.105300
$416$	−5234.01	−0.616871
$417$	0	0
$418$	2316.02	0.271005
$419$	582.439	0.0679093	0.0339547	−	0.999423i	$-0.489190\pi$
$419$	582.439	0.0679093	0.0339547	+	0.999423i	$0.489190\pi$
$420$	0	0
$421$	−12562.3	−1.45427	−0.727135	−	0.686494i	$-0.759149\pi$
$421$	−12562.3	−1.45427	−0.727135	+	0.686494i	$0.759149\pi$
$422$	−471.136	−0.0543473
$423$	0	0
$424$	−6177.33	−0.707542
$425$	−66.6647	−0.00760873
$426$	0	0
$427$	−871.549	−0.0987757
$428$	−2012.82	−0.227321
$429$	0	0
$430$	−3515.99	−0.394317
$431$	9612.85	1.07433	0.537163	−	0.843478i	$-0.319496\pi$
$431$	9612.85	1.07433	0.537163	+	0.843478i	$0.319496\pi$
$432$	0	0
$433$	8285.20	0.919541	0.459771	−	0.888038i	$-0.347932\pi$
$433$	8285.20	0.919541	0.459771	+	0.888038i	$0.347932\pi$
$434$	129.736	0.0143491
$435$	0	0
$436$	6057.79	0.665403
$437$	−16575.9	−1.81449
$438$	0	0
$439$	10769.0	1.17079	0.585393	−	0.810750i	$-0.300940\pi$
$439$	10769.0	1.17079	0.585393	+	0.810750i	$0.300940\pi$
$440$	1078.87	0.116894
$441$	0	0
$442$	191.382	0.0205953
$443$	−8809.93	−0.944858	−0.472429	−	0.881369i	$-0.656623\pi$
$443$	−8809.93	−0.944858	−0.472429	+	0.881369i	$0.656623\pi$
$444$	0	0
$445$	4032.43	0.429563
$446$	1068.19	0.113409
$447$	0	0
$448$	−2232.47	−0.235434
$449$	9060.10	0.952277	0.476139	−	0.879370i	$-0.342036\pi$
$449$	9060.10	0.952277	0.476139	+	0.879370i	$0.342036\pi$
$450$	0	0
$451$	3410.46	0.356081
$452$	−2020.32	−0.210239
$453$	0	0
$454$	−8201.91	−0.847874
$455$	806.774	0.0831256
$456$	0	0
$457$	2477.55	0.253600	0.126800	−	0.991928i	$-0.459529\pi$
$457$	2477.55	0.253600	0.126800	+	0.991928i	$0.459529\pi$
$458$	3475.97	0.354631
$459$	0	0
$460$	−2442.58	−0.247578
$461$	15301.5	1.54591	0.772954	−	0.634462i	$-0.218778\pi$
$461$	15301.5	1.54591	0.772954	+	0.634462i	$0.218778\pi$
$462$	0	0
$463$	−683.012	−0.0685578	−0.0342789	−	0.999412i	$-0.510913\pi$
$463$	−683.012	−0.0685578	−0.0342789	+	0.999412i	$0.510913\pi$
$464$	−1473.03	−0.147379
$465$	0	0
$466$	9557.76	0.950118
$467$	−6569.32	−0.650946	−0.325473	−	0.945551i	$-0.605523\pi$
$467$	−6569.32	−0.650946	−0.325473	+	0.945551i	$0.605523\pi$
$468$	0	0
$469$	−1895.28	−0.186601
$470$	−811.544	−0.0796463
$471$	0	0
$472$	796.218	0.0776460
$473$	3016.66	0.293248
$474$	0	0
$475$	3139.98	0.303309
$476$	−46.0081	−0.00443020
$477$	0	0
$478$	−14322.6	−1.37050
$479$	−11288.7	−1.07681	−0.538405	−	0.842686i	$-0.680973\pi$
$479$	−11288.7	−1.07681	−0.538405	+	0.842686i	$0.680973\pi$
$480$	0	0
$481$	9826.03	0.931452
$482$	6180.31	0.584036
$483$	0	0
$484$	4634.00	0.435199
$485$	6190.23	0.579554
$486$	0	0
$487$	−8937.53	−0.831618	−0.415809	−	0.909452i	$-0.636502\pi$
$487$	−8937.53	−0.831618	−0.415809	+	0.909452i	$0.636502\pi$
$488$	4536.28	0.420795
$489$	0	0
$490$	−3330.42	−0.307047
$491$	−3693.14	−0.339448	−0.169724	−	0.985492i	$-0.554288\pi$
$491$	−3693.14	−0.339448	−0.169724	+	0.985492i	$0.554288\pi$
$492$	0	0
$493$	−189.889	−0.0173472
$494$	−9014.29	−0.820996
$495$	0	0
$496$	−277.705	−0.0251397
$497$	4503.22	0.406433
$498$	0	0
$499$	−11186.4	−1.00355	−0.501777	−	0.864997i	$-0.667320\pi$
$499$	−11186.4	−1.00355	−0.501777	+	0.864997i	$0.667320\pi$
$500$	462.698	0.0413850
$501$	0	0
$502$	14788.4	1.31482
$503$	−6553.81	−0.580954	−0.290477	−	0.956882i	$-0.593814\pi$
$503$	−6553.81	−0.580954	−0.290477	+	0.956882i	$0.593814\pi$
$504$	0	0
$505$	2834.28	0.249750
$506$	−2433.59	−0.213807
$507$	0	0
$508$	−133.435	−0.0116540
$509$	15027.7	1.30863	0.654314	−	0.756223i	$-0.272957\pi$
$509$	15027.7	1.30863	0.654314	+	0.756223i	$0.272957\pi$
$510$	0	0
$511$	−1287.29	−0.111441
$512$	7142.32	0.616502
$513$	0	0
$514$	13605.9	1.16757
$515$	4095.09	0.350390
$516$	0	0
$517$	696.290	0.0592318
$518$	2743.03	0.232668
$519$	0	0
$520$	−4199.14	−0.354124
$521$	835.969	0.0702965	0.0351483	−	0.999382i	$-0.488810\pi$
$521$	835.969	0.0702965	0.0351483	+	0.999382i	$0.488810\pi$
$522$	0	0
$523$	−13032.4	−1.08961	−0.544805	−	0.838563i	$-0.683396\pi$
$523$	−13032.4	−1.08961	−0.544805	+	0.838563i	$0.683396\pi$
$524$	4849.64	0.404308
$525$	0	0
$526$	11741.9	0.973330
$527$	−35.7991	−0.00295907
$528$	0	0
$529$	5250.35	0.431524
$530$	−2639.53	−0.216328
$531$	0	0
$532$	2167.03	0.176603
$533$	−13274.0	−1.07873
$534$	0	0
$535$	−2718.86	−0.219713
$536$	9864.63	0.794939
$537$	0	0
$538$	8331.26	0.667633
$539$	2857.44	0.228347
$540$	0	0
$541$	−4182.33	−0.332370	−0.166185	−	0.986095i	$-0.553145\pi$
$541$	−4182.33	−0.332370	−0.166185	+	0.986095i	$0.553145\pi$
$542$	3147.90	0.249472
$543$	0	0
$544$	403.180	0.0317761
$545$	8182.69	0.643134
$546$	0	0
$547$	5013.90	0.391918	0.195959	−	0.980612i	$-0.437218\pi$
$547$	5013.90	0.391918	0.195959	+	0.980612i	$0.437218\pi$
$548$	−8133.14	−0.633997
$549$	0	0
$550$	460.995	0.0357398
$551$	8943.98	0.691518
$552$	0	0
$553$	5345.62	0.411065
$554$	−4524.76	−0.347001
$555$	0	0
$556$	−6388.97	−0.487325
$557$	2611.86	0.198686	0.0993430	−	0.995053i	$-0.468326\pi$
$557$	2611.86	0.198686	0.0993430	+	0.995053i	$0.468326\pi$
$558$	0	0
$559$	−11741.3	−0.888377
$560$	−482.089	−0.0363786
$561$	0	0
$562$	10579.0	0.794033
$563$	18674.0	1.39790	0.698948	−	0.715173i	$-0.253652\pi$
$563$	18674.0	1.39790	0.698948	+	0.715173i	$0.253652\pi$
$564$	0	0
$565$	−2728.99	−0.203203
$566$	5344.74	0.396919
$567$	0	0
$568$	−23438.6	−1.73145
$569$	23671.0	1.74401	0.872004	−	0.489498i	$-0.162820\pi$
$569$	23671.0	1.74401	0.872004	+	0.489498i	$0.162820\pi$
$570$	0	0
$571$	−9355.54	−0.685669	−0.342835	−	0.939396i	$-0.611387\pi$
$571$	−9355.54	−0.685669	−0.342835	+	0.939396i	$0.611387\pi$
$572$	1139.68	0.0833082
$573$	0	0
$574$	−3705.57	−0.269456
$575$	−3299.37	−0.239293
$576$	0	0
$577$	−21695.2	−1.56531	−0.782653	−	0.622459i	$-0.786134\pi$
$577$	−21695.2	−1.56531	−0.782653	+	0.622459i	$0.786134\pi$
$578$	10171.2	0.731948
$579$	0	0
$580$	1317.96	0.0943541
$581$	829.889	0.0592592
$582$	0	0
$583$	2264.67	0.160880
$584$	6700.18	0.474752
$585$	0	0
$586$	−17290.7	−1.21889
$587$	−737.149	−0.0518320	−0.0259160	−	0.999664i	$-0.508250\pi$
$587$	−737.149	−0.0518320	−0.0259160	+	0.999664i	$0.508250\pi$
$588$	0	0
$589$	1686.17	0.117958
$590$	340.218	0.0237399
$591$	0	0
$592$	−5871.56	−0.407635
$593$	21908.1	1.51713	0.758565	−	0.651597i	$-0.225901\pi$
$593$	21908.1	1.51713	0.758565	+	0.651597i	$0.225901\pi$
$594$	0	0
$595$	−62.1464	−0.00428194
$596$	−6668.03	−0.458277
$597$	0	0
$598$	9471.89	0.647716
$599$	7595.84	0.518126	0.259063	−	0.965860i	$-0.416586\pi$
$599$	7595.84	0.518126	0.259063	+	0.965860i	$0.416586\pi$
$600$	0	0
$601$	−6502.52	−0.441337	−0.220668	−	0.975349i	$-0.570824\pi$
$601$	−6502.52	−0.441337	−0.220668	+	0.975349i	$0.570824\pi$
$602$	−3277.69	−0.221908
$603$	0	0
$604$	−12319.6	−0.829929
$605$	6259.47	0.420634
$606$	0	0
$607$	−26156.8	−1.74905	−0.874524	−	0.484981i	$-0.838826\pi$
$607$	−26156.8	−1.74905	−0.874524	+	0.484981i	$0.838826\pi$
$608$	−18990.2	−1.26670
$609$	0	0
$610$	1938.32	0.128656
$611$	−2710.06	−0.179439
$612$	0	0
$613$	18172.8	1.19738	0.598690	−	0.800981i	$-0.295688\pi$
$613$	18172.8	1.19738	0.598690	+	0.800981i	$0.295688\pi$
$614$	−14600.6	−0.959659
$615$	0	0
$616$	1005.75	0.0657840
$617$	26584.8	1.73462	0.867312	−	0.497766i	$-0.165846\pi$
$617$	26584.8	1.73462	0.867312	+	0.497766i	$0.165846\pi$
$618$	0	0
$619$	21687.8	1.40825	0.704124	−	0.710077i	$-0.251340\pi$
$619$	21687.8	1.40825	0.704124	+	0.710077i	$0.251340\pi$
$620$	248.470	0.0160948
$621$	0	0
$622$	4857.48	0.313131
$623$	3759.13	0.241744
$624$	0	0
$625$	625.000	0.0400000
$626$	7947.78	0.507439
$627$	0	0
$628$	−14177.8	−0.900883
$629$	−756.907	−0.0479807
$630$	0	0
$631$	−7685.15	−0.484851	−0.242426	−	0.970170i	$-0.577943\pi$
$631$	−7685.15	−0.484851	−0.242426	+	0.970170i	$0.577943\pi$
$632$	−27823.2	−1.75118
$633$	0	0
$634$	12211.5	0.764951
$635$	−180.240	−0.0112639
$636$	0	0
$637$	−11121.6	−0.691764
$638$	1313.11	0.0814835
$639$	0	0
$640$	−1082.88	−0.0668824
$641$	8055.69	0.496381	0.248191	−	0.968711i	$-0.420164\pi$
$641$	8055.69	0.496381	0.248191	+	0.968711i	$0.420164\pi$
$642$	0	0
$643$	−7122.02	−0.436804	−0.218402	−	0.975859i	$-0.570084\pi$
$643$	−7122.02	−0.436804	−0.218402	+	0.975859i	$0.570084\pi$
$644$	−2277.03	−0.139329
$645$	0	0
$646$	694.378	0.0422909
$647$	−10280.7	−0.624692	−0.312346	−	0.949968i	$-0.601115\pi$
$647$	−10280.7	−0.624692	−0.312346	+	0.949968i	$0.601115\pi$
$648$	0	0
$649$	−291.901	−0.0176550
$650$	−1794.26	−0.108272
$651$	0	0
$652$	8900.51	0.534618
$653$	−8985.03	−0.538456	−0.269228	−	0.963077i	$-0.586768\pi$
$653$	−8985.03	−0.538456	−0.269228	+	0.963077i	$0.586768\pi$
$654$	0	0
$655$	6550.75	0.390777
$656$	7931.92	0.472087
$657$	0	0
$658$	−756.541	−0.0448223
$659$	12538.7	0.741183	0.370591	−	0.928796i	$-0.379155\pi$
$659$	12538.7	0.741183	0.370591	+	0.928796i	$0.379155\pi$
$660$	0	0
$661$	−12479.9	−0.734360	−0.367180	−	0.930150i	$-0.619677\pi$
$661$	−12479.9	−0.734360	−0.367180	+	0.930150i	$0.619677\pi$
$662$	13142.3	0.771586
$663$	0	0
$664$	−4319.45	−0.252451
$665$	2927.16	0.170692
$666$	0	0
$667$	−9398.01	−0.545566
$668$	13218.2	0.765609
$669$	0	0
$670$	4215.08	0.243049
$671$	−1663.04	−0.0956797
$672$	0	0
$673$	13185.6	0.755228	0.377614	−	0.925963i	$-0.376745\pi$
$673$	13185.6	0.755228	0.377614	+	0.925963i	$0.376745\pi$
$674$	−2007.63	−0.114734
$675$	0	0
$676$	3696.60	0.210321
$677$	26568.3	1.50827	0.754136	−	0.656718i	$-0.228056\pi$
$677$	26568.3	1.50827	0.754136	+	0.656718i	$0.228056\pi$
$678$	0	0
$679$	5770.68	0.326154
$680$	323.463	0.0182415
$681$	0	0
$682$	247.555	0.0138994
$683$	−7602.40	−0.425912	−0.212956	−	0.977062i	$-0.568309\pi$
$683$	−7602.40	−0.425912	−0.212956	+	0.977062i	$0.568309\pi$
$684$	0	0
$685$	−10986.0	−0.612780
$686$	−6419.36	−0.357277
$687$	0	0
$688$	7016.02	0.388784
$689$	−8814.41	−0.487376
$690$	0	0
$691$	6257.31	0.344485	0.172243	−	0.985055i	$-0.444899\pi$
$691$	6257.31	0.344485	0.172243	+	0.985055i	$0.444899\pi$
$692$	−592.018	−0.0325219
$693$	0	0
$694$	13388.5	0.732307
$695$	−8630.05	−0.471016
$696$	0	0
$697$	1022.51	0.0555671
$698$	12761.3	0.692009
$699$	0	0
$700$	431.339	0.0232901
$701$	769.271	0.0414479	0.0207239	−	0.999785i	$-0.493403\pi$
$701$	769.271	0.0414479	0.0207239	+	0.999785i	$0.493403\pi$
$702$	0	0
$703$	35651.1	1.91267
$704$	−4259.88	−0.228054
$705$	0	0
$706$	7595.46	0.404900
$707$	2642.18	0.140551
$708$	0	0
$709$	−8450.32	−0.447614	−0.223807	−	0.974633i	$-0.571849\pi$
$709$	−8450.32	−0.447614	−0.223807	+	0.974633i	$0.571849\pi$
$710$	−10015.1	−0.529383
$711$	0	0
$712$	−19565.7	−1.02985
$713$	−1771.77	−0.0930620
$714$	0	0
$715$	1539.44	0.0805201
$716$	−4148.98	−0.216557
$717$	0	0
$718$	25112.7	1.30529
$719$	25350.0	1.31488	0.657438	−	0.753508i	$-0.271640\pi$
$719$	25350.0	1.31488	0.657438	+	0.753508i	$0.271640\pi$
$720$	0	0
$721$	3817.54	0.197188
$722$	−18485.4	−0.952848
$723$	0	0
$724$	14275.7	0.732809
$725$	1780.27	0.0911964
$726$	0	0
$727$	−3825.15	−0.195140	−0.0975701	−	0.995229i	$-0.531107\pi$
$727$	−3825.15	−0.195140	−0.0975701	+	0.995229i	$0.531107\pi$
$728$	−3914.54	−0.199289
$729$	0	0
$730$	2862.94	0.145153
$731$	904.440	0.0457618
$732$	0	0
$733$	−3692.83	−0.186081	−0.0930407	−	0.995662i	$-0.529659\pi$
$733$	−3692.83	−0.186081	−0.0930407	+	0.995662i	$0.529659\pi$
$734$	−24083.3	−1.21108
$735$	0	0
$736$	19954.2	0.999350
$737$	−3616.46	−0.180752
$738$	0	0
$739$	4181.32	0.208136	0.104068	−	0.994570i	$-0.466814\pi$
$739$	4181.32	0.208136	0.104068	+	0.994570i	$0.466814\pi$
$740$	5253.45	0.260974
$741$	0	0
$742$	−2460.63	−0.121742
$743$	927.982	0.0458201	0.0229101	−	0.999738i	$-0.492707\pi$
$743$	927.982	0.0458201	0.0229101	+	0.999738i	$0.492707\pi$
$744$	0	0
$745$	−9006.99	−0.442940
$746$	4593.78	0.225456
$747$	0	0
$748$	−87.7902	−0.00429135
$749$	−2534.59	−0.123647
$750$	0	0
$751$	−22369.5	−1.08692	−0.543458	−	0.839436i	$-0.682885\pi$
$751$	−22369.5	−1.08692	−0.543458	+	0.839436i	$0.682885\pi$
$752$	1619.40	0.0785287
$753$	0	0
$754$	−5110.81	−0.246850
$755$	−16640.9	−0.802154
$756$	0	0
$757$	35390.0	1.69917	0.849586	−	0.527451i	$-0.176852\pi$
$757$	35390.0	1.69917	0.849586	+	0.527451i	$0.176852\pi$
$758$	13558.8	0.649705
$759$	0	0
$760$	−15235.4	−0.727168
$761$	−33434.6	−1.59265	−0.796323	−	0.604872i	$-0.793224\pi$
$761$	−33434.6	−1.59265	−0.796323	+	0.604872i	$0.793224\pi$
$762$	0	0
$763$	7628.10	0.361934
$764$	−292.462	−0.0138494
$765$	0	0
$766$	17612.9	0.830783
$767$	1136.12	0.0534849
$768$	0	0
$769$	25021.4	1.17334	0.586668	−	0.809828i	$-0.300440\pi$
$769$	25021.4	1.17334	0.586668	+	0.809828i	$0.300440\pi$
$770$	429.751	0.0201132
$771$	0	0
$772$	−16921.8	−0.788897
$773$	3065.27	0.142626	0.0713132	−	0.997454i	$-0.477281\pi$
$773$	3065.27	0.142626	0.0713132	+	0.997454i	$0.477281\pi$
$774$	0	0
$775$	335.626	0.0155562
$776$	−30035.5	−1.38945
$777$	0	0
$778$	−7332.79	−0.337909
$779$	−48161.2	−2.21509
$780$	0	0
$781$	8592.81	0.393694
$782$	−729.627	−0.0333650
$783$	0	0
$784$	6645.73	0.302739
$785$	−19150.9	−0.870734
$786$	0	0
$787$	−35683.1	−1.61622	−0.808111	−	0.589031i	$-0.799510\pi$
$787$	−35683.1	−1.61622	−0.808111	+	0.589031i	$0.799510\pi$
$788$	−15108.4	−0.683014
$789$	0	0
$790$	−11888.6	−0.535416
$791$	−2544.03	−0.114356
$792$	0	0
$793$	6472.81	0.289856
$794$	−17801.5	−0.795656
$795$	0	0
$796$	9324.23	0.415187
$797$	4525.07	0.201112	0.100556	−	0.994931i	$-0.467938\pi$
$797$	4525.07	0.201112	0.100556	+	0.994931i	$0.467938\pi$
$798$	0	0
$799$	208.758	0.00924323
$800$	−3779.93	−0.167051
$801$	0	0
$802$	−14995.5	−0.660235
$803$	−2456.35	−0.107948
$804$	0	0
$805$	−3075.75	−0.134666
$806$	−963.521	−0.0421074
$807$	0	0
$808$	−13752.2	−0.598762
$809$	16234.6	0.705535	0.352767	−	0.935711i	$-0.385241\pi$
$809$	16234.6	0.705535	0.352767	+	0.935711i	$0.385241\pi$
$810$	0	0
$811$	2197.06	0.0951286	0.0475643	−	0.998868i	$-0.484854\pi$
$811$	2197.06	0.0951286	0.0475643	+	0.998868i	$0.484854\pi$
$812$	1228.64	0.0530993
$813$	0	0
$814$	5234.11	0.225375
$815$	12022.6	0.516726
$816$	0	0
$817$	−42600.0	−1.82422
$818$	−2552.10	−0.109086
$819$	0	0
$820$	−7096.91	−0.302237
$821$	20141.6	0.856208	0.428104	−	0.903729i	$-0.359182\pi$
$821$	20141.6	0.856208	0.428104	+	0.903729i	$0.359182\pi$
$822$	0	0
$823$	15517.0	0.657217	0.328608	−	0.944466i	$-0.393420\pi$
$823$	15517.0	0.657217	0.328608	+	0.944466i	$0.393420\pi$
$824$	−19869.7	−0.840042
$825$	0	0
$826$	317.159	0.0133600
$827$	−5582.51	−0.234732	−0.117366	−	0.993089i	$-0.537445\pi$
$827$	−5582.51	−0.234732	−0.117366	+	0.993089i	$0.537445\pi$
$828$	0	0
$829$	26037.6	1.09086	0.545430	−	0.838157i	$-0.316367\pi$
$829$	26037.6	1.09086	0.545430	+	0.838157i	$0.316367\pi$
$830$	−1845.67	−0.0771856
$831$	0	0
$832$	16580.1	0.690878
$833$	856.705	0.0356339
$834$	0	0
$835$	17854.8	0.739987
$836$	4135.01	0.171067
$837$	0	0
$838$	−1207.55	−0.0497781
$839$	18088.0	0.744298	0.372149	−	0.928173i	$-0.378621\pi$
$839$	18088.0	0.744298	0.372149	+	0.928173i	$0.378621\pi$
$840$	0	0
$841$	−19318.0	−0.792080
$842$	26044.9	1.06599
$843$	0	0
$844$	−841.164	−0.0343057
$845$	4993.26	0.203282
$846$	0	0
$847$	5835.23	0.236719
$848$	5267.07	0.213292
$849$	0	0
$850$	138.213	0.00557727
$851$	−37460.9	−1.50898
$852$	0	0
$853$	−25349.0	−1.01751	−0.508753	−	0.860912i	$-0.669893\pi$
$853$	−25349.0	−1.01751	−0.508753	+	0.860912i	$0.669893\pi$
$854$	1806.95	0.0724034
$855$	0	0
$856$	13192.1	0.526750
$857$	563.797	0.0224725	0.0112363	−	0.999937i	$-0.496423\pi$
$857$	563.797	0.0224725	0.0112363	+	0.999937i	$0.496423\pi$
$858$	0	0
$859$	−20558.9	−0.816600	−0.408300	−	0.912848i	$-0.633878\pi$
$859$	−20558.9	−0.816600	−0.408300	+	0.912848i	$0.633878\pi$
$860$	−6277.43	−0.248905
$861$	0	0
$862$	−19929.9	−0.787490
$863$	7056.31	0.278331	0.139166	−	0.990269i	$-0.455558\pi$
$863$	7056.31	0.278331	0.139166	+	0.990269i	$0.455558\pi$
$864$	0	0
$865$	−799.681	−0.0314335
$866$	−17177.4	−0.674032
$867$	0	0
$868$	231.630	0.00905763
$869$	10200.2	0.398181
$870$	0	0
$871$	14075.8	0.547578
$872$	−39703.2	−1.54188
$873$	0	0
$874$	34366.2	1.33004
$875$	582.640	0.0225107
$876$	0	0
$877$	2516.30	0.0968864	0.0484432	−	0.998826i	$-0.484574\pi$
$877$	2516.30	0.0968864	0.0484432	+	0.998826i	$0.484574\pi$
$878$	−22326.9	−0.858196
$879$	0	0
$880$	−919.897	−0.0352383
$881$	−31571.2	−1.20733	−0.603666	−	0.797237i	$-0.706294\pi$
$881$	−31571.2	−1.20733	−0.603666	+	0.797237i	$0.706294\pi$
$882$	0	0
$883$	47743.5	1.81959	0.909794	−	0.415059i	$-0.136239\pi$
$883$	47743.5	1.81959	0.909794	+	0.415059i	$0.136239\pi$
$884$	341.692	0.0130004
$885$	0	0
$886$	18265.3	0.692589
$887$	33028.9	1.25028	0.625141	−	0.780512i	$-0.285041\pi$
$887$	33028.9	1.25028	0.625141	+	0.780512i	$0.285041\pi$
$888$	0	0
$889$	−168.024	−0.00633897
$890$	−8360.28	−0.314873
$891$	0	0
$892$	1907.14	0.0715873
$893$	−9832.74	−0.368466
$894$	0	0
$895$	−5604.33	−0.209309
$896$	−1009.49	−0.0376392
$897$	0	0
$898$	−18784.0	−0.698028
$899$	956.006	0.0354667
$900$	0	0
$901$	678.981	0.0251056
$902$	−7070.78	−0.261010
$903$	0	0
$904$	13241.3	0.487168
$905$	19283.3	0.708284
$906$	0	0
$907$	−43330.5	−1.58629	−0.793146	−	0.609032i	$-0.791558\pi$
$907$	−43330.5	−1.58629	−0.793146	+	0.609032i	$0.791558\pi$
$908$	−14643.6	−0.535205
$909$	0	0
$910$	−1672.65	−0.0609318
$911$	−22354.4	−0.812992	−0.406496	−	0.913653i	$-0.633249\pi$
$911$	−22354.4	−0.812992	−0.406496	+	0.913653i	$0.633249\pi$
$912$	0	0
$913$	1583.55	0.0574018
$914$	−5136.62	−0.185891
$915$	0	0
$916$	6205.97	0.223855
$917$	6106.77	0.219916
$918$	0	0
$919$	1306.08	0.0468811	0.0234406	−	0.999725i	$-0.492538\pi$
$919$	1306.08	0.0468811	0.0234406	+	0.999725i	$0.492538\pi$
$920$	16008.8	0.573691
$921$	0	0
$922$	−31724.1	−1.13316
$923$	−33444.5	−1.19267
$924$	0	0
$925$	7096.21	0.252240
$926$	1416.06	0.0502535
$927$	0	0
$928$	−10766.8	−0.380861
$929$	50663.9	1.78927	0.894633	−	0.446802i	$-0.147437\pi$
$929$	50663.9	1.78927	0.894633	+	0.446802i	$0.147437\pi$
$930$	0	0
$931$	−40351.7	−1.42049
$932$	17064.4	0.599745
$933$	0	0
$934$	13619.9	0.477149
$935$	−118.584	−0.00414773
$936$	0	0
$937$	−12652.4	−0.441126	−0.220563	−	0.975373i	$-0.570789\pi$
$937$	−12652.4	−0.441126	−0.220563	+	0.975373i	$0.570789\pi$
$938$	3929.40	0.136780
$939$	0	0
$940$	−1448.93	−0.0502753
$941$	40677.4	1.40919	0.704594	−	0.709610i	$-0.251129\pi$
$941$	40677.4	1.40919	0.704594	+	0.709610i	$0.251129\pi$
$942$	0	0
$943$	50606.0	1.74757
$944$	−678.891	−0.0234068
$945$	0	0
$946$	−6254.32	−0.214953
$947$	−26916.8	−0.923631	−0.461816	−	0.886976i	$-0.652802\pi$
$947$	−26916.8	−0.923631	−0.461816	+	0.886976i	$0.652802\pi$
$948$	0	0
$949$	9560.46	0.327024
$950$	−6509.99	−0.222328
$951$	0	0
$952$	301.540	0.0102657
$953$	29644.9	1.00765	0.503827	−	0.863805i	$-0.331925\pi$
$953$	29644.9	1.00765	0.503827	+	0.863805i	$0.331925\pi$
$954$	0	0
$955$	−395.050	−0.0133859
$956$	−25571.4	−0.865104
$957$	0	0
$958$	23404.3	0.789310
$959$	−10241.4	−0.344852
$960$	0	0
$961$	−29610.8	−0.993950
$962$	−20371.9	−0.682762
$963$	0	0
$964$	11034.3	0.368662
$965$	−22857.5	−0.762495
$966$	0	0
$967$	21452.1	0.713395	0.356697	−	0.934220i	$-0.383903\pi$
$967$	21452.1	0.713395	0.356697	+	0.934220i	$0.383903\pi$
$968$	−30371.5	−1.00845
$969$	0	0
$970$	−12834.0	−0.424818
$971$	−42684.6	−1.41073	−0.705363	−	0.708846i	$-0.749216\pi$
$971$	−42684.6	−1.41073	−0.705363	+	0.708846i	$0.749216\pi$
$972$	0	0
$973$	−8045.14	−0.265072
$974$	18529.8	0.609583
$975$	0	0
$976$	−3867.84	−0.126851
$977$	60386.2	1.97741	0.988704	−	0.149884i	$-0.0478901\pi$
$977$	60386.2	1.97741	0.988704	+	0.149884i	$0.0478901\pi$
$978$	0	0
$979$	7172.97	0.234167
$980$	−5946.12	−0.193818
$981$	0	0
$982$	7656.85	0.248818
$983$	−59009.2	−1.91465	−0.957325	−	0.289014i	$-0.906672\pi$
$983$	−59009.2	−1.91465	−0.957325	+	0.289014i	$0.906672\pi$
$984$	0	0
$985$	−20408.0	−0.660156
$986$	393.690	0.0127157
$987$	0	0
$988$	−16094.1	−0.518239
$989$	44762.6	1.43920
$990$	0	0
$991$	−30899.0	−0.990454	−0.495227	−	0.868764i	$-0.664915\pi$
$991$	−30899.0	−0.990454	−0.495227	+	0.868764i	$0.664915\pi$
$992$	−2029.83	−0.0649668
$993$	0	0
$994$	−9336.36	−0.297919
$995$	12594.9	0.401292
$996$	0	0
$997$	−33866.8	−1.07580	−0.537901	−	0.843008i	$-0.680782\pi$
$997$	−33866.8	−1.07580	−0.537901	+	0.843008i	$0.680782\pi$
$998$	23192.4	0.735614
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.4.a.k.1.3	✓	6
3.2	odd	2		405.4.a.l.1.4	yes	6
5.4	even	2		2025.4.a.z.1.4		6
9.2	odd	6		405.4.e.w.271.3		12
9.4	even	3		405.4.e.x.136.4		12
9.5	odd	6		405.4.e.w.136.3		12
9.7	even	3		405.4.e.x.271.4		12
15.14	odd	2		2025.4.a.y.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
405.4.a.k.1.3	✓	6	1.1	even	1	trivial
405.4.a.l.1.4	yes	6	3.2	odd	2
405.4.e.w.136.3		12	9.5	odd	6
405.4.e.w.271.3		12	9.2	odd	6
405.4.e.x.136.4		12	9.4	even	3
405.4.e.x.271.4		12	9.7	even	3
2025.4.a.y.1.3		6	15.14	odd	2
2025.4.a.z.1.4		6	5.4	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	405.a (trivial)

\(f(q)\)	\(=\)	\(q-2.07326 q^{2} -3.70159 q^{4} -5.00000 q^{5} -4.66112 q^{7} +24.2604 q^{8} +O(q^{10})\)	\(q-2.07326 q^{2} -3.70159 q^{4} -5.00000 q^{5} -4.66112 q^{7} +24.2604 q^{8} +10.3663 q^{10} -8.89410 q^{11} +34.6172 q^{13} +9.66372 q^{14} -20.6855 q^{16} -2.66659 q^{17} +125.599 q^{19} +18.5079 q^{20} +18.4398 q^{22} -131.975 q^{23} +25.0000 q^{25} -71.7704 q^{26} +17.2535 q^{28} +71.2106 q^{29} +13.4250 q^{31} -151.197 q^{32} +5.52853 q^{34} +23.3056 q^{35} +283.849 q^{37} -260.400 q^{38} -121.302 q^{40} -383.452 q^{41} -339.175 q^{43} +32.9223 q^{44} +273.618 q^{46} -78.2867 q^{47} -321.274 q^{49} -51.8315 q^{50} -128.138 q^{52} -254.626 q^{53} +44.4705 q^{55} -113.081 q^{56} -147.638 q^{58} +32.8196 q^{59} +186.983 q^{61} -27.8336 q^{62} +478.955 q^{64} -173.086 q^{65} +406.614 q^{67} +9.87060 q^{68} -48.3186 q^{70} -966.124 q^{71} +276.177 q^{73} -588.492 q^{74} -464.916 q^{76} +41.4565 q^{77} -1146.85 q^{79} +103.428 q^{80} +794.996 q^{82} -178.045 q^{83} +13.3329 q^{85} +703.198 q^{86} -215.775 q^{88} -806.486 q^{89} -161.355 q^{91} +488.516 q^{92} +162.309 q^{94} -627.995 q^{95} -1238.05 q^{97} +666.085 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{2} + 34 q^{4} - 30 q^{5} + 40 q^{7} - 66 q^{8}+O(q^{10}) \) 6 * q - 4 * q^2 + 34 * q^4 - 30 * q^5 + 40 * q^7 - 66 * q^8	\( 6 q - 4 q^{2} + 34 q^{4} - 30 q^{5} + 40 q^{7} - 66 q^{8} + 20 q^{10} - 88 q^{11} + 20 q^{13} - 180 q^{14} + 58 q^{16} - 124 q^{17} - 46 q^{19} - 170 q^{20} - 74 q^{22} - 210 q^{23} + 150 q^{25} - 4 q^{26} + 352 q^{28} - 296 q^{29} - 104 q^{31} - 722 q^{32} - 428 q^{34} - 200 q^{35} - 204 q^{37} + 20 q^{38} + 330 q^{40} - 344 q^{41} + 512 q^{43} - 716 q^{44} - 186 q^{46} - 238 q^{47} + 68 q^{49} - 100 q^{50} - 468 q^{52} - 850 q^{53} + 440 q^{55} - 2316 q^{56} + 890 q^{58} - 1840 q^{59} - 364 q^{61} - 1038 q^{62} - 990 q^{64} - 100 q^{65} + 88 q^{67} - 236 q^{68} + 900 q^{70} - 1364 q^{71} + 836 q^{73} - 1316 q^{74} - 2106 q^{76} - 840 q^{77} - 680 q^{79} - 290 q^{80} + 1742 q^{82} - 2148 q^{83} + 620 q^{85} - 2872 q^{86} + 1296 q^{88} - 3000 q^{89} - 3058 q^{91} - 1002 q^{92} - 3662 q^{94} + 230 q^{95} - 612 q^{97} - 1982 q^{98}+O(q^{100}) \) 6 * q - 4 * q^2 + 34 * q^4 - 30 * q^5 + 40 * q^7 - 66 * q^8 + 20 * q^10 - 88 * q^11 + 20 * q^13 - 180 * q^14 + 58 * q^16 - 124 * q^17 - 46 * q^19 - 170 * q^20 - 74 * q^22 - 210 * q^23 + 150 * q^25 - 4 * q^26 + 352 * q^28 - 296 * q^29 - 104 * q^31 - 722 * q^32 - 428 * q^34 - 200 * q^35 - 204 * q^37 + 20 * q^38 + 330 * q^40 - 344 * q^41 + 512 * q^43 - 716 * q^44 - 186 * q^46 - 238 * q^47 + 68 * q^49 - 100 * q^50 - 468 * q^52 - 850 * q^53 + 440 * q^55 - 2316 * q^56 + 890 * q^58 - 1840 * q^59 - 364 * q^61 - 1038 * q^62 - 990 * q^64 - 100 * q^65 + 88 * q^67 - 236 * q^68 + 900 * q^70 - 1364 * q^71 + 836 * q^73 - 1316 * q^74 - 2106 * q^76 - 840 * q^77 - 680 * q^79 - 290 * q^80 + 1742 * q^82 - 2148 * q^83 + 620 * q^85 - 2872 * q^86 + 1296 * q^88 - 3000 * q^89 - 3058 * q^91 - 1002 * q^92 - 3662 * q^94 + 230 * q^95 - 612 * q^97 - 1982 * q^98

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