LMFDB - Embedded newform 405.2.a.h.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [405,2,Mod(1,405)] 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("405.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(405, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 405 = 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	405.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,2,0,4,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$3.23394128186$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ 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)

$f(q)$	$=$	\(q+2.73205 q^{2} +5.46410 q^{4} -1.00000 q^{5} -1.26795 q^{7} +9.46410 q^{8} -2.73205 q^{10} +2.26795 q^{11} -5.46410 q^{13} -3.46410 q^{14} +14.9282 q^{16} -0.732051 q^{17} -2.46410 q^{19} -5.46410 q^{20} +6.19615 q^{22} -3.46410 q^{23} +1.00000 q^{25} -14.9282 q^{26} -6.92820 q^{28} +7.19615 q^{29} -3.00000 q^{31} +21.8564 q^{32} -2.00000 q^{34} +1.26795 q^{35} +0.732051 q^{37} -6.73205 q^{38} -9.46410 q^{40} -3.19615 q^{41} -10.1962 q^{43} +12.3923 q^{44} -9.46410 q^{46} +5.26795 q^{47} -5.39230 q^{49} +2.73205 q^{50} -29.8564 q^{52} -3.26795 q^{53} -2.26795 q^{55} -12.0000 q^{56} +19.6603 q^{58} +11.7321 q^{59} +4.00000 q^{61} -8.19615 q^{62} +29.8564 q^{64} +5.46410 q^{65} -3.46410 q^{67} -4.00000 q^{68} +3.46410 q^{70} +0.267949 q^{71} +9.66025 q^{73} +2.00000 q^{74} -13.4641 q^{76} -2.87564 q^{77} +8.53590 q^{79} -14.9282 q^{80} -8.73205 q^{82} +8.19615 q^{83} +0.732051 q^{85} -27.8564 q^{86} +21.4641 q^{88} +5.19615 q^{89} +6.92820 q^{91} -18.9282 q^{92} +14.3923 q^{94} +2.46410 q^{95} +7.66025 q^{97} -14.7321 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 4 q^{4} - 2 q^{5} - 6 q^{7} + 12 q^{8} - 2 q^{10} + 8 q^{11} - 4 q^{13} + 16 q^{16} + 2 q^{17} + 2 q^{19} - 4 q^{20} + 2 q^{22} + 2 q^{25} - 16 q^{26} + 4 q^{29} - 6 q^{31} + 16 q^{32} - 4 q^{34}+ \cdots - 26 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 4 * q^4 - 2 * q^5 - 6 * q^7 + 12 * q^8 - 2 * q^10 + 8 * q^11 - 4 * q^13 + 16 * q^16 + 2 * q^17 + 2 * q^19 - 4 * q^20 + 2 * q^22 + 2 * q^25 - 16 * q^26 + 4 * q^29 - 6 * q^31 + 16 * q^32 - 4 * q^34 + 6 * q^35 - 2 * q^37 - 10 * q^38 - 12 * q^40 + 4 * q^41 - 10 * q^43 + 4 * q^44 - 12 * q^46 + 14 * q^47 + 10 * q^49 + 2 * q^50 - 32 * q^52 - 10 * q^53 - 8 * q^55 - 24 * q^56 + 22 * q^58 + 20 * q^59 + 8 * q^61 - 6 * q^62 + 32 * q^64 + 4 * q^65 - 8 * q^68 + 4 * q^71 + 2 * q^73 + 4 * q^74 - 20 * q^76 - 30 * q^77 + 24 * q^79 - 16 * q^80 - 14 * q^82 + 6 * q^83 - 2 * q^85 - 28 * q^86 + 36 * q^88 - 24 * q^92 + 8 * q^94 - 2 * q^95 - 2 * q^97 - 26 * 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 $

$2$	2.73205	1.93185	0.965926	−	0.258819i	$-0.0833333\pi$
$2$	2.73205	1.93185	0.965926	+	0.258819i	$0.0833333\pi$
$3$	0	0
$3$	0	0
$4$	5.46410	2.73205
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.26795	−0.479240	−0.239620	−	0.970867i	$-0.577023\pi$
$7$	−1.26795	−0.479240	−0.239620	+	0.970867i	$0.577023\pi$
$8$	9.46410	3.34607
$9$	0	0
$10$	−2.73205	−0.863950
$11$	2.26795	0.683812	0.341906	−	0.939734i	$-0.388927\pi$
$11$	2.26795	0.683812	0.341906	+	0.939734i	$0.388927\pi$
$12$	0	0
$13$	−5.46410	−1.51547	−0.757735	−	0.652563i	$-0.773694\pi$
$13$	−5.46410	−1.51547	−0.757735	+	0.652563i	$0.773694\pi$
$14$	−3.46410	−0.925820
$15$	0	0
$16$	14.9282	3.73205
$17$	−0.732051	−0.177548	−0.0887742	−	0.996052i	$-0.528295\pi$
$17$	−0.732051	−0.177548	−0.0887742	+	0.996052i	$0.528295\pi$
$18$	0	0
$19$	−2.46410	−0.565304	−0.282652	−	0.959223i	$-0.591214\pi$
$19$	−2.46410	−0.565304	−0.282652	+	0.959223i	$0.591214\pi$
$20$	−5.46410	−1.22181
$21$	0	0
$22$	6.19615	1.32102
$23$	−3.46410	−0.722315	−0.361158	−	0.932505i	$-0.617618\pi$
$23$	−3.46410	−0.722315	−0.361158	+	0.932505i	$0.617618\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−14.9282	−2.92766
$27$	0	0
$28$	−6.92820	−1.30931
$29$	7.19615	1.33629	0.668146	−	0.744030i	$-0.267088\pi$
$29$	7.19615	1.33629	0.668146	+	0.744030i	$0.267088\pi$
$30$	0	0
$31$	−3.00000	−0.538816	−0.269408	−	0.963026i	$-0.586828\pi$
$31$	−3.00000	−0.538816	−0.269408	+	0.963026i	$0.586828\pi$
$32$	21.8564	3.86370
$33$	0	0
$34$	−2.00000	−0.342997
$35$	1.26795	0.214323
$36$	0	0
$37$	0.732051	0.120348	0.0601742	−	0.998188i	$-0.480834\pi$
$37$	0.732051	0.120348	0.0601742	+	0.998188i	$0.480834\pi$
$38$	−6.73205	−1.09208
$39$	0	0
$40$	−9.46410	−1.49641
$41$	−3.19615	−0.499155	−0.249578	−	0.968355i	$-0.580292\pi$
$41$	−3.19615	−0.499155	−0.249578	+	0.968355i	$0.580292\pi$
$42$	0	0
$43$	−10.1962	−1.55490	−0.777449	−	0.628946i	$-0.783487\pi$
$43$	−10.1962	−1.55490	−0.777449	+	0.628946i	$0.783487\pi$
$44$	12.3923	1.86821
$45$	0	0
$46$	−9.46410	−1.39541
$47$	5.26795	0.768409	0.384205	−	0.923248i	$-0.374476\pi$
$47$	5.26795	0.768409	0.384205	+	0.923248i	$0.374476\pi$
$48$	0	0
$49$	−5.39230	−0.770329
$50$	2.73205	0.386370
$51$	0	0
$52$	−29.8564	−4.14034
$53$	−3.26795	−0.448887	−0.224444	−	0.974487i	$-0.572056\pi$
$53$	−3.26795	−0.448887	−0.224444	+	0.974487i	$0.572056\pi$
$54$	0	0
$55$	−2.26795	−0.305810
$56$	−12.0000	−1.60357
$57$	0	0
$58$	19.6603	2.58152
$59$	11.7321	1.52738	0.763691	−	0.645581i	$-0.223385\pi$
$59$	11.7321	1.52738	0.763691	+	0.645581i	$0.223385\pi$
$60$	0	0
$61$	4.00000	0.512148	0.256074	−	0.966657i	$-0.417571\pi$
$61$	4.00000	0.512148	0.256074	+	0.966657i	$0.417571\pi$
$62$	−8.19615	−1.04091
$63$	0	0
$64$	29.8564	3.73205
$65$	5.46410	0.677738
$66$	0	0
$67$	−3.46410	−0.423207	−0.211604	−	0.977356i	$-0.567869\pi$
$67$	−3.46410	−0.423207	−0.211604	+	0.977356i	$0.567869\pi$
$68$	−4.00000	−0.485071
$69$	0	0
$70$	3.46410	0.414039
$71$	0.267949	0.0317997	0.0158999	−	0.999874i	$-0.494939\pi$
$71$	0.267949	0.0317997	0.0158999	+	0.999874i	$0.494939\pi$
$72$	0	0
$73$	9.66025	1.13065	0.565324	−	0.824869i	$-0.308751\pi$
$73$	9.66025	1.13065	0.565324	+	0.824869i	$0.308751\pi$
$74$	2.00000	0.232495
$75$	0	0
$76$	−13.4641	−1.54444
$77$	−2.87564	−0.327710
$78$	0	0
$79$	8.53590	0.960364	0.480182	−	0.877169i	$-0.340571\pi$
$79$	8.53590	0.960364	0.480182	+	0.877169i	$0.340571\pi$
$80$	−14.9282	−1.66902
$81$	0	0
$82$	−8.73205	−0.964294
$83$	8.19615	0.899645	0.449822	−	0.893118i	$-0.351487\pi$
$83$	8.19615	0.899645	0.449822	+	0.893118i	$0.351487\pi$
$84$	0	0
$85$	0.732051	0.0794021
$86$	−27.8564	−3.00383
$87$	0	0
$88$	21.4641	2.28808
$89$	5.19615	0.550791	0.275396	−	0.961331i	$-0.411191\pi$
$89$	5.19615	0.550791	0.275396	+	0.961331i	$0.411191\pi$
$90$	0	0
$91$	6.92820	0.726273
$92$	−18.9282	−1.97340
$93$	0	0
$94$	14.3923	1.48445
$95$	2.46410	0.252811
$96$	0	0
$97$	7.66025	0.777781	0.388890	−	0.921284i	$-0.372858\pi$
$97$	7.66025	0.777781	0.388890	+	0.921284i	$0.372858\pi$
$98$	−14.7321	−1.48816
$99$	0	0
$100$	5.46410	0.546410
$101$	−14.6603	−1.45875	−0.729375	−	0.684114i	$-0.760189\pi$
$101$	−14.6603	−1.45875	−0.729375	+	0.684114i	$0.760189\pi$
$102$	0	0
$103$	−7.46410	−0.735460	−0.367730	−	0.929933i	$-0.619865\pi$
$103$	−7.46410	−0.735460	−0.367730	+	0.929933i	$0.619865\pi$
$104$	−51.7128	−5.07086
$105$	0	0
$106$	−8.92820	−0.867184
$107$	15.4641	1.49497	0.747486	−	0.664278i	$-0.231261\pi$
$107$	15.4641	1.49497	0.747486	+	0.664278i	$0.231261\pi$
$108$	0	0
$109$	−19.9282	−1.90878	−0.954388	−	0.298570i	$-0.903490\pi$
$109$	−19.9282	−1.90878	−0.954388	+	0.298570i	$0.903490\pi$
$110$	−6.19615	−0.590780
$111$	0	0
$112$	−18.9282	−1.78855
$113$	−5.12436	−0.482059	−0.241029	−	0.970518i	$-0.577485\pi$
$113$	−5.12436	−0.482059	−0.241029	+	0.970518i	$0.577485\pi$
$114$	0	0
$115$	3.46410	0.323029
$116$	39.3205	3.65082
$117$	0	0
$118$	32.0526	2.95068
$119$	0.928203	0.0850883
$120$	0	0
$121$	−5.85641	−0.532401
$122$	10.9282	0.989393
$123$	0	0
$124$	−16.3923	−1.47207
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	16.5885	1.47199	0.735994	−	0.676988i	$-0.236715\pi$
$127$	16.5885	1.47199	0.735994	+	0.676988i	$0.236715\pi$
$128$	37.8564	3.34607
$129$	0	0
$130$	14.9282	1.30929
$131$	15.5885	1.36197	0.680985	−	0.732297i	$-0.261552\pi$
$131$	15.5885	1.36197	0.680985	+	0.732297i	$0.261552\pi$
$132$	0	0
$133$	3.12436	0.270916
$134$	−9.46410	−0.817574
$135$	0	0
$136$	−6.92820	−0.594089
$137$	−9.46410	−0.808573	−0.404286	−	0.914632i	$-0.632480\pi$
$137$	−9.46410	−0.808573	−0.404286	+	0.914632i	$0.632480\pi$
$138$	0	0
$139$	21.3923	1.81447	0.907236	−	0.420622i	$-0.138188\pi$
$139$	21.3923	1.81447	0.907236	+	0.420622i	$0.138188\pi$
$140$	6.92820	0.585540
$141$	0	0
$142$	0.732051	0.0614323
$143$	−12.3923	−1.03630
$144$	0	0
$145$	−7.19615	−0.597608
$146$	26.3923	2.18424
$147$	0	0
$148$	4.00000	0.328798
$149$	8.00000	0.655386	0.327693	−	0.944784i	$-0.393729\pi$
$149$	8.00000	0.655386	0.327693	+	0.944784i	$0.393729\pi$
$150$	0	0
$151$	−15.3923	−1.25261	−0.626304	−	0.779579i	$-0.715433\pi$
$151$	−15.3923	−1.25261	−0.626304	+	0.779579i	$0.715433\pi$
$152$	−23.3205	−1.89154
$153$	0	0
$154$	−7.85641	−0.633087
$155$	3.00000	0.240966
$156$	0	0
$157$	5.12436	0.408968	0.204484	−	0.978870i	$-0.434448\pi$
$157$	5.12436	0.408968	0.204484	+	0.978870i	$0.434448\pi$
$158$	23.3205	1.85528
$159$	0	0
$160$	−21.8564	−1.72790
$161$	4.39230	0.346162
$162$	0	0
$163$	9.26795	0.725922	0.362961	−	0.931804i	$-0.381766\pi$
$163$	9.26795	0.725922	0.362961	+	0.931804i	$0.381766\pi$
$164$	−17.4641	−1.36372
$165$	0	0
$166$	22.3923	1.73798
$167$	0.339746	0.0262903	0.0131452	−	0.999914i	$-0.495816\pi$
$167$	0.339746	0.0262903	0.0131452	+	0.999914i	$0.495816\pi$
$168$	0	0
$169$	16.8564	1.29665
$170$	2.00000	0.153393
$171$	0	0
$172$	−55.7128	−4.24806
$173$	−15.4641	−1.17571	−0.587857	−	0.808965i	$-0.700028\pi$
$173$	−15.4641	−1.17571	−0.587857	+	0.808965i	$0.700028\pi$
$174$	0	0
$175$	−1.26795	−0.0958479
$176$	33.8564	2.55202
$177$	0	0
$178$	14.1962	1.06405
$179$	16.1244	1.20519	0.602595	−	0.798047i	$-0.294133\pi$
$179$	16.1244	1.20519	0.602595	+	0.798047i	$0.294133\pi$
$180$	0	0
$181$	19.5359	1.45209	0.726046	−	0.687646i	$-0.241356\pi$
$181$	19.5359	1.45209	0.726046	+	0.687646i	$0.241356\pi$
$182$	18.9282	1.40305
$183$	0	0
$184$	−32.7846	−2.41691
$185$	−0.732051	−0.0538214
$186$	0	0
$187$	−1.66025	−0.121410
$188$	28.7846	2.09933
$189$	0	0
$190$	6.73205	0.488394
$191$	−16.1244	−1.16672	−0.583359	−	0.812215i	$-0.698262\pi$
$191$	−16.1244	−1.16672	−0.583359	+	0.812215i	$0.698262\pi$
$192$	0	0
$193$	−8.73205	−0.628547	−0.314273	−	0.949333i	$-0.601761\pi$
$193$	−8.73205	−0.628547	−0.314273	+	0.949333i	$0.601761\pi$
$194$	20.9282	1.50256
$195$	0	0
$196$	−29.4641	−2.10458
$197$	−13.8564	−0.987228	−0.493614	−	0.869681i	$-0.664324\pi$
$197$	−13.8564	−0.987228	−0.493614	+	0.869681i	$0.664324\pi$
$198$	0	0
$199$	−2.00000	−0.141776	−0.0708881	−	0.997484i	$-0.522583\pi$
$199$	−2.00000	−0.141776	−0.0708881	+	0.997484i	$0.522583\pi$
$200$	9.46410	0.669213
$201$	0	0
$202$	−40.0526	−2.81809
$203$	−9.12436	−0.640404
$204$	0	0
$205$	3.19615	0.223229
$206$	−20.3923	−1.42080
$207$	0	0
$208$	−81.5692	−5.65581
$209$	−5.58846	−0.386562
$210$	0	0
$211$	18.8564	1.29813	0.649064	−	0.760734i	$-0.275161\pi$
$211$	18.8564	1.29813	0.649064	+	0.760734i	$0.275161\pi$
$212$	−17.8564	−1.22638
$213$	0	0
$214$	42.2487	2.88806
$215$	10.1962	0.695372
$216$	0	0
$217$	3.80385	0.258222
$218$	−54.4449	−3.68747
$219$	0	0
$220$	−12.3923	−0.835489
$221$	4.00000	0.269069
$222$	0	0
$223$	−24.7846	−1.65970	−0.829850	−	0.557986i	$-0.811574\pi$
$223$	−24.7846	−1.65970	−0.829850	+	0.557986i	$0.811574\pi$
$224$	−27.7128	−1.85164
$225$	0	0
$226$	−14.0000	−0.931266
$227$	−20.0526	−1.33094	−0.665468	−	0.746427i	$-0.731768\pi$
$227$	−20.0526	−1.33094	−0.665468	+	0.746427i	$0.731768\pi$
$228$	0	0
$229$	−12.0000	−0.792982	−0.396491	−	0.918039i	$-0.629772\pi$
$229$	−12.0000	−0.792982	−0.396491	+	0.918039i	$0.629772\pi$
$230$	9.46410	0.624044
$231$	0	0
$232$	68.1051	4.47132
$233$	−10.0526	−0.658565	−0.329283	−	0.944231i	$-0.606807\pi$
$233$	−10.0526	−0.658565	−0.329283	+	0.944231i	$0.606807\pi$
$234$	0	0
$235$	−5.26795	−0.343643
$236$	64.1051	4.17289
$237$	0	0
$238$	2.53590	0.164378
$239$	7.46410	0.482813	0.241406	−	0.970424i	$-0.422391\pi$
$239$	7.46410	0.482813	0.241406	+	0.970424i	$0.422391\pi$
$240$	0	0
$241$	18.3205	1.18013	0.590064	−	0.807357i	$-0.299103\pi$
$241$	18.3205	1.18013	0.590064	+	0.807357i	$0.299103\pi$
$242$	−16.0000	−1.02852
$243$	0	0
$244$	21.8564	1.39921
$245$	5.39230	0.344502
$246$	0	0
$247$	13.4641	0.856700
$248$	−28.3923	−1.80291
$249$	0	0
$250$	−2.73205	−0.172790
$251$	−10.3923	−0.655956	−0.327978	−	0.944685i	$-0.606367\pi$
$251$	−10.3923	−0.655956	−0.327978	+	0.944685i	$0.606367\pi$
$252$	0	0
$253$	−7.85641	−0.493928
$254$	45.3205	2.84366
$255$	0	0
$256$	43.7128	2.73205
$257$	10.3923	0.648254	0.324127	−	0.946014i	$-0.394929\pi$
$257$	10.3923	0.648254	0.324127	+	0.946014i	$0.394929\pi$
$258$	0	0
$259$	−0.928203	−0.0576757
$260$	29.8564	1.85162
$261$	0	0
$262$	42.5885	2.63112
$263$	13.3205	0.821378	0.410689	−	0.911776i	$-0.365288\pi$
$263$	13.3205	0.821378	0.410689	+	0.911776i	$0.365288\pi$
$264$	0	0
$265$	3.26795	0.200749
$266$	8.53590	0.523370
$267$	0	0
$268$	−18.9282	−1.15622
$269$	6.66025	0.406083	0.203041	−	0.979170i	$-0.434917\pi$
$269$	6.66025	0.406083	0.203041	+	0.979170i	$0.434917\pi$
$270$	0	0
$271$	10.9282	0.663841	0.331921	−	0.943307i	$-0.392303\pi$
$271$	10.9282	0.663841	0.331921	+	0.943307i	$0.392303\pi$
$272$	−10.9282	−0.662620
$273$	0	0
$274$	−25.8564	−1.56204
$275$	2.26795	0.136762
$276$	0	0
$277$	14.1962	0.852964	0.426482	−	0.904496i	$-0.359753\pi$
$277$	14.1962	0.852964	0.426482	+	0.904496i	$0.359753\pi$
$278$	58.4449	3.50529
$279$	0	0
$280$	12.0000	0.717137
$281$	−8.53590	−0.509209	−0.254605	−	0.967045i	$-0.581945\pi$
$281$	−8.53590	−0.509209	−0.254605	+	0.967045i	$0.581945\pi$
$282$	0	0
$283$	5.32051	0.316271	0.158136	−	0.987417i	$-0.449452\pi$
$283$	5.32051	0.316271	0.158136	+	0.987417i	$0.449452\pi$
$284$	1.46410	0.0868784
$285$	0	0
$286$	−33.8564	−2.00197
$287$	4.05256	0.239215
$288$	0	0
$289$	−16.4641	−0.968477
$290$	−19.6603	−1.15449
$291$	0	0
$292$	52.7846	3.08899
$293$	−25.2679	−1.47617	−0.738085	−	0.674708i	$-0.764270\pi$
$293$	−25.2679	−1.47617	−0.738085	+	0.674708i	$0.764270\pi$
$294$	0	0
$295$	−11.7321	−0.683066
$296$	6.92820	0.402694
$297$	0	0
$298$	21.8564	1.26611
$299$	18.9282	1.09465
$300$	0	0
$301$	12.9282	0.745169
$302$	−42.0526	−2.41985
$303$	0	0
$304$	−36.7846	−2.10974
$305$	−4.00000	−0.229039
$306$	0	0
$307$	−24.0526	−1.37275	−0.686376	−	0.727247i	$-0.740800\pi$
$307$	−24.0526	−1.37275	−0.686376	+	0.727247i	$0.740800\pi$
$308$	−15.7128	−0.895321
$309$	0	0
$310$	8.19615	0.465510
$311$	−16.2679	−0.922471	−0.461235	−	0.887278i	$-0.652594\pi$
$311$	−16.2679	−0.922471	−0.461235	+	0.887278i	$0.652594\pi$
$312$	0	0
$313$	−22.9282	−1.29598	−0.647989	−	0.761649i	$-0.724390\pi$
$313$	−22.9282	−1.29598	−0.647989	+	0.761649i	$0.724390\pi$
$314$	14.0000	0.790066
$315$	0	0
$316$	46.6410	2.62376
$317$	4.19615	0.235679	0.117840	−	0.993033i	$-0.462403\pi$
$317$	4.19615	0.235679	0.117840	+	0.993033i	$0.462403\pi$
$318$	0	0
$319$	16.3205	0.913773
$320$	−29.8564	−1.66902
$321$	0	0
$322$	12.0000	0.668734
$323$	1.80385	0.100369
$324$	0	0
$325$	−5.46410	−0.303094
$326$	25.3205	1.40237
$327$	0	0
$328$	−30.2487	−1.67021
$329$	−6.67949	−0.368252
$330$	0	0
$331$	6.46410	0.355299	0.177650	−	0.984094i	$-0.443151\pi$
$331$	6.46410	0.355299	0.177650	+	0.984094i	$0.443151\pi$
$332$	44.7846	2.45787
$333$	0	0
$334$	0.928203	0.0507890
$335$	3.46410	0.189264
$336$	0	0
$337$	−7.32051	−0.398773	−0.199387	−	0.979921i	$-0.563895\pi$
$337$	−7.32051	−0.398773	−0.199387	+	0.979921i	$0.563895\pi$
$338$	46.0526	2.50493
$339$	0	0
$340$	4.00000	0.216930
$341$	−6.80385	−0.368449
$342$	0	0
$343$	15.7128	0.848412
$344$	−96.4974	−5.20279
$345$	0	0
$346$	−42.2487	−2.27130
$347$	−2.58846	−0.138956	−0.0694778	−	0.997583i	$-0.522133\pi$
$347$	−2.58846	−0.138956	−0.0694778	+	0.997583i	$0.522133\pi$
$348$	0	0
$349$	8.85641	0.474073	0.237036	−	0.971501i	$-0.423824\pi$
$349$	8.85641	0.474073	0.237036	+	0.971501i	$0.423824\pi$
$350$	−3.46410	−0.185164
$351$	0	0
$352$	49.5692	2.64205
$353$	19.5167	1.03877	0.519384	−	0.854541i	$-0.326162\pi$
$353$	19.5167	1.03877	0.519384	+	0.854541i	$0.326162\pi$
$354$	0	0
$355$	−0.267949	−0.0142213
$356$	28.3923	1.50479
$357$	0	0
$358$	44.0526	2.32825
$359$	−18.1244	−0.956567	−0.478283	−	0.878206i	$-0.658741\pi$
$359$	−18.1244	−0.956567	−0.478283	+	0.878206i	$0.658741\pi$
$360$	0	0
$361$	−12.9282	−0.680432
$362$	53.3731	2.80523
$363$	0	0
$364$	37.8564	1.98421
$365$	−9.66025	−0.505641
$366$	0	0
$367$	−31.1769	−1.62742	−0.813711	−	0.581270i	$-0.802556\pi$
$367$	−31.1769	−1.62742	−0.813711	+	0.581270i	$0.802556\pi$
$368$	−51.7128	−2.69572
$369$	0	0
$370$	−2.00000	−0.103975
$371$	4.14359	0.215125
$372$	0	0
$373$	−18.0526	−0.934726	−0.467363	−	0.884065i	$-0.654796\pi$
$373$	−18.0526	−0.934726	−0.467363	+	0.884065i	$0.654796\pi$
$374$	−4.53590	−0.234546
$375$	0	0
$376$	49.8564	2.57115
$377$	−39.3205	−2.02511
$378$	0	0
$379$	−18.3923	−0.944749	−0.472375	−	0.881398i	$-0.656603\pi$
$379$	−18.3923	−0.944749	−0.472375	+	0.881398i	$0.656603\pi$
$380$	13.4641	0.690694
$381$	0	0
$382$	−44.0526	−2.25392
$383$	−9.46410	−0.483593	−0.241797	−	0.970327i	$-0.577737\pi$
$383$	−9.46410	−0.483593	−0.241797	+	0.970327i	$0.577737\pi$
$384$	0	0
$385$	2.87564	0.146556
$386$	−23.8564	−1.21426
$387$	0	0
$388$	41.8564	2.12494
$389$	−20.5359	−1.04121	−0.520606	−	0.853797i	$-0.674294\pi$
$389$	−20.5359	−1.04121	−0.520606	+	0.853797i	$0.674294\pi$
$390$	0	0
$391$	2.53590	0.128246
$392$	−51.0333	−2.57757
$393$	0	0
$394$	−37.8564	−1.90718
$395$	−8.53590	−0.429488
$396$	0	0
$397$	−6.39230	−0.320821	−0.160410	−	0.987050i	$-0.551282\pi$
$397$	−6.39230	−0.320821	−0.160410	+	0.987050i	$0.551282\pi$
$398$	−5.46410	−0.273891
$399$	0	0
$400$	14.9282	0.746410
$401$	−11.0718	−0.552899	−0.276450	−	0.961028i	$-0.589158\pi$
$401$	−11.0718	−0.552899	−0.276450	+	0.961028i	$0.589158\pi$
$402$	0	0
$403$	16.3923	0.816559
$404$	−80.1051	−3.98538
$405$	0	0
$406$	−24.9282	−1.23717
$407$	1.66025	0.0822957
$408$	0	0
$409$	9.85641	0.487368	0.243684	−	0.969855i	$-0.421644\pi$
$409$	9.85641	0.487368	0.243684	+	0.969855i	$0.421644\pi$
$410$	8.73205	0.431245
$411$	0	0
$412$	−40.7846	−2.00931
$413$	−14.8756	−0.731983
$414$	0	0
$415$	−8.19615	−0.402333
$416$	−119.426	−5.85532
$417$	0	0
$418$	−15.2679	−0.746780
$419$	0.392305	0.0191653	0.00958267	−	0.999954i	$-0.496950\pi$
$419$	0.392305	0.0191653	0.00958267	+	0.999954i	$0.496950\pi$
$420$	0	0
$421$	−7.78461	−0.379399	−0.189699	−	0.981842i	$-0.560751\pi$
$421$	−7.78461	−0.379399	−0.189699	+	0.981842i	$0.560751\pi$
$422$	51.5167	2.50779
$423$	0	0
$424$	−30.9282	−1.50201
$425$	−0.732051	−0.0355097
$426$	0	0
$427$	−5.07180	−0.245441
$428$	84.4974	4.08434
$429$	0	0
$430$	27.8564	1.34336
$431$	38.6603	1.86220	0.931099	−	0.364765i	$-0.118851\pi$
$431$	38.6603	1.86220	0.931099	+	0.364765i	$0.118851\pi$
$432$	0	0
$433$	−28.5359	−1.37135	−0.685674	−	0.727909i	$-0.740492\pi$
$433$	−28.5359	−1.37135	−0.685674	+	0.727909i	$0.740492\pi$
$434$	10.3923	0.498847
$435$	0	0
$436$	−108.890	−5.21487
$437$	8.53590	0.408327
$438$	0	0
$439$	−15.3923	−0.734635	−0.367317	−	0.930096i	$-0.619724\pi$
$439$	−15.3923	−0.734635	−0.367317	+	0.930096i	$0.619724\pi$
$440$	−21.4641	−1.02326
$441$	0	0
$442$	10.9282	0.519802
$443$	17.6603	0.839064	0.419532	−	0.907741i	$-0.362194\pi$
$443$	17.6603	0.839064	0.419532	+	0.907741i	$0.362194\pi$
$444$	0	0
$445$	−5.19615	−0.246321
$446$	−67.7128	−3.20629
$447$	0	0
$448$	−37.8564	−1.78855
$449$	16.1244	0.760955	0.380478	−	0.924790i	$-0.375760\pi$
$449$	16.1244	0.760955	0.380478	+	0.924790i	$0.375760\pi$
$450$	0	0
$451$	−7.24871	−0.341328
$452$	−28.0000	−1.31701
$453$	0	0
$454$	−54.7846	−2.57117
$455$	−6.92820	−0.324799
$456$	0	0
$457$	−0.732051	−0.0342439	−0.0171219	−	0.999853i	$-0.505450\pi$
$457$	−0.732051	−0.0342439	−0.0171219	+	0.999853i	$0.505450\pi$
$458$	−32.7846	−1.53192
$459$	0	0
$460$	18.9282	0.882532
$461$	−1.05256	−0.0490226	−0.0245113	−	0.999700i	$-0.507803\pi$
$461$	−1.05256	−0.0490226	−0.0245113	+	0.999700i	$0.507803\pi$
$462$	0	0
$463$	10.3923	0.482971	0.241486	−	0.970404i	$-0.422365\pi$
$463$	10.3923	0.482971	0.241486	+	0.970404i	$0.422365\pi$
$464$	107.426	4.98711
$465$	0	0
$466$	−27.4641	−1.27225
$467$	35.3731	1.63687	0.818435	−	0.574599i	$-0.194842\pi$
$467$	35.3731	1.63687	0.818435	+	0.574599i	$0.194842\pi$
$468$	0	0
$469$	4.39230	0.202818
$470$	−14.3923	−0.663868
$471$	0	0
$472$	111.033	5.11072
$473$	−23.1244	−1.06326
$474$	0	0
$475$	−2.46410	−0.113061
$476$	5.07180	0.232465
$477$	0	0
$478$	20.3923	0.932722
$479$	36.1244	1.65056	0.825282	−	0.564721i	$-0.191016\pi$
$479$	36.1244	1.65056	0.825282	+	0.564721i	$0.191016\pi$
$480$	0	0
$481$	−4.00000	−0.182384
$482$	50.0526	2.27983
$483$	0	0
$484$	−32.0000	−1.45455
$485$	−7.66025	−0.347834
$486$	0	0
$487$	3.60770	0.163480	0.0817401	−	0.996654i	$-0.473952\pi$
$487$	3.60770	0.163480	0.0817401	+	0.996654i	$0.473952\pi$
$488$	37.8564	1.71368
$489$	0	0
$490$	14.7321	0.665526
$491$	38.1244	1.72053	0.860264	−	0.509849i	$-0.170299\pi$
$491$	38.1244	1.72053	0.860264	+	0.509849i	$0.170299\pi$
$492$	0	0
$493$	−5.26795	−0.237256
$494$	36.7846	1.65502
$495$	0	0
$496$	−44.7846	−2.01089
$497$	−0.339746	−0.0152397
$498$	0	0
$499$	−10.3205	−0.462009	−0.231005	−	0.972953i	$-0.574201\pi$
$499$	−10.3205	−0.462009	−0.231005	+	0.972953i	$0.574201\pi$
$500$	−5.46410	−0.244362
$501$	0	0
$502$	−28.3923	−1.26721
$503$	−27.3205	−1.21816	−0.609081	−	0.793108i	$-0.708461\pi$
$503$	−27.3205	−1.21816	−0.609081	+	0.793108i	$0.708461\pi$
$504$	0	0
$505$	14.6603	0.652373
$506$	−21.4641	−0.954196
$507$	0	0
$508$	90.6410	4.02154
$509$	−34.7846	−1.54180	−0.770900	−	0.636956i	$-0.780193\pi$
$509$	−34.7846	−1.54180	−0.770900	+	0.636956i	$0.780193\pi$
$510$	0	0
$511$	−12.2487	−0.541851
$512$	43.7128	1.93185
$513$	0	0
$514$	28.3923	1.25233
$515$	7.46410	0.328908
$516$	0	0
$517$	11.9474	0.525448
$518$	−2.53590	−0.111421
$519$	0	0
$520$	51.7128	2.26776
$521$	−12.5359	−0.549208	−0.274604	−	0.961557i	$-0.588547\pi$
$521$	−12.5359	−0.549208	−0.274604	+	0.961557i	$0.588547\pi$
$522$	0	0
$523$	−26.2487	−1.14778	−0.573888	−	0.818934i	$-0.694566\pi$
$523$	−26.2487	−1.14778	−0.573888	+	0.818934i	$0.694566\pi$
$524$	85.1769	3.72097
$525$	0	0
$526$	36.3923	1.58678
$527$	2.19615	0.0956659
$528$	0	0
$529$	−11.0000	−0.478261
$530$	8.92820	0.387816
$531$	0	0
$532$	17.0718	0.740156
$533$	17.4641	0.756454
$534$	0	0
$535$	−15.4641	−0.668571
$536$	−32.7846	−1.41608
$537$	0	0
$538$	18.1962	0.784492
$539$	−12.2295	−0.526761
$540$	0	0
$541$	−17.5359	−0.753927	−0.376964	−	0.926228i	$-0.623032\pi$
$541$	−17.5359	−0.753927	−0.376964	+	0.926228i	$0.623032\pi$
$542$	29.8564	1.28244
$543$	0	0
$544$	−16.0000	−0.685994
$545$	19.9282	0.853630
$546$	0	0
$547$	−6.14359	−0.262681	−0.131341	−	0.991337i	$-0.541928\pi$
$547$	−6.14359	−0.262681	−0.131341	+	0.991337i	$0.541928\pi$
$548$	−51.7128	−2.20906
$549$	0	0
$550$	6.19615	0.264205
$551$	−17.7321	−0.755411
$552$	0	0
$553$	−10.8231	−0.460244
$554$	38.7846	1.64780
$555$	0	0
$556$	116.890	4.95723
$557$	9.46410	0.401007	0.200503	−	0.979693i	$-0.435742\pi$
$557$	9.46410	0.401007	0.200503	+	0.979693i	$0.435742\pi$
$558$	0	0
$559$	55.7128	2.35640
$560$	18.9282	0.799863
$561$	0	0
$562$	−23.3205	−0.983716
$563$	13.2679	0.559177	0.279589	−	0.960120i	$-0.409802\pi$
$563$	13.2679	0.559177	0.279589	+	0.960120i	$0.409802\pi$
$564$	0	0
$565$	5.12436	0.215583
$566$	14.5359	0.610989
$567$	0	0
$568$	2.53590	0.106404
$569$	−32.9090	−1.37962	−0.689808	−	0.723993i	$-0.742305\pi$
$569$	−32.9090	−1.37962	−0.689808	+	0.723993i	$0.742305\pi$
$570$	0	0
$571$	1.78461	0.0746836	0.0373418	−	0.999303i	$-0.488111\pi$
$571$	1.78461	0.0746836	0.0373418	+	0.999303i	$0.488111\pi$
$572$	−67.7128	−2.83121
$573$	0	0
$574$	11.0718	0.462128
$575$	−3.46410	−0.144463
$576$	0	0
$577$	18.7321	0.779825	0.389913	−	0.920852i	$-0.372505\pi$
$577$	18.7321	0.779825	0.389913	+	0.920852i	$0.372505\pi$
$578$	−44.9808	−1.87095
$579$	0	0
$580$	−39.3205	−1.63270
$581$	−10.3923	−0.431145
$582$	0	0
$583$	−7.41154	−0.306955
$584$	91.4256	3.78322
$585$	0	0
$586$	−69.0333	−2.85174
$587$	−5.66025	−0.233624	−0.116812	−	0.993154i	$-0.537267\pi$
$587$	−5.66025	−0.233624	−0.116812	+	0.993154i	$0.537267\pi$
$588$	0	0
$589$	7.39230	0.304595
$590$	−32.0526	−1.31958
$591$	0	0
$592$	10.9282	0.449146
$593$	27.8564	1.14393	0.571963	−	0.820280i	$-0.306182\pi$
$593$	27.8564	1.14393	0.571963	+	0.820280i	$0.306182\pi$
$594$	0	0
$595$	−0.928203	−0.0380526
$596$	43.7128	1.79055
$597$	0	0
$598$	51.7128	2.11469
$599$	−16.8038	−0.686587	−0.343293	−	0.939228i	$-0.611542\pi$
$599$	−16.8038	−0.686587	−0.343293	+	0.939228i	$0.611542\pi$
$600$	0	0
$601$	17.2487	0.703590	0.351795	−	0.936077i	$-0.385571\pi$
$601$	17.2487	0.703590	0.351795	+	0.936077i	$0.385571\pi$
$602$	35.3205	1.43956
$603$	0	0
$604$	−84.1051	−3.42219
$605$	5.85641	0.238097
$606$	0	0
$607$	−5.80385	−0.235571	−0.117785	−	0.993039i	$-0.537580\pi$
$607$	−5.80385	−0.235571	−0.117785	+	0.993039i	$0.537580\pi$
$608$	−53.8564	−2.18417
$609$	0	0
$610$	−10.9282	−0.442470
$611$	−28.7846	−1.16450
$612$	0	0
$613$	−5.46410	−0.220693	−0.110346	−	0.993893i	$-0.535196\pi$
$613$	−5.46410	−0.220693	−0.110346	+	0.993893i	$0.535196\pi$
$614$	−65.7128	−2.65195
$615$	0	0
$616$	−27.2154	−1.09654
$617$	−6.92820	−0.278919	−0.139459	−	0.990228i	$-0.544536\pi$
$617$	−6.92820	−0.278919	−0.139459	+	0.990228i	$0.544536\pi$
$618$	0	0
$619$	15.8564	0.637323	0.318661	−	0.947869i	$-0.396767\pi$
$619$	15.8564	0.637323	0.318661	+	0.947869i	$0.396767\pi$
$620$	16.3923	0.658331
$621$	0	0
$622$	−44.4449	−1.78208
$623$	−6.58846	−0.263961
$624$	0	0
$625$	1.00000	0.0400000
$626$	−62.6410	−2.50364
$627$	0	0
$628$	28.0000	1.11732
$629$	−0.535898	−0.0213677
$630$	0	0
$631$	22.7128	0.904183	0.452091	−	0.891972i	$-0.350678\pi$
$631$	22.7128	0.904183	0.452091	+	0.891972i	$0.350678\pi$
$632$	80.7846	3.21344
$633$	0	0
$634$	11.4641	0.455298
$635$	−16.5885	−0.658293
$636$	0	0
$637$	29.4641	1.16741
$638$	44.5885	1.76527
$639$	0	0
$640$	−37.8564	−1.49641
$641$	−26.6603	−1.05302	−0.526508	−	0.850170i	$-0.676499\pi$
$641$	−26.6603	−1.05302	−0.526508	+	0.850170i	$0.676499\pi$
$642$	0	0
$643$	24.3397	0.959866	0.479933	−	0.877305i	$-0.340661\pi$
$643$	24.3397	0.959866	0.479933	+	0.877305i	$0.340661\pi$
$644$	24.0000	0.945732
$645$	0	0
$646$	4.92820	0.193898
$647$	−4.53590	−0.178325	−0.0891623	−	0.996017i	$-0.528419\pi$
$647$	−4.53590	−0.178325	−0.0891623	+	0.996017i	$0.528419\pi$
$648$	0	0
$649$	26.6077	1.04444
$650$	−14.9282	−0.585532
$651$	0	0
$652$	50.6410	1.98326
$653$	10.5359	0.412302	0.206151	−	0.978520i	$-0.433906\pi$
$653$	10.5359	0.412302	0.206151	+	0.978520i	$0.433906\pi$
$654$	0	0
$655$	−15.5885	−0.609091
$656$	−47.7128	−1.86287
$657$	0	0
$658$	−18.2487	−0.711409
$659$	−5.46410	−0.212851	−0.106426	−	0.994321i	$-0.533941\pi$
$659$	−5.46410	−0.212851	−0.106426	+	0.994321i	$0.533941\pi$
$660$	0	0
$661$	−16.3205	−0.634794	−0.317397	−	0.948293i	$-0.602809\pi$
$661$	−16.3205	−0.634794	−0.317397	+	0.948293i	$0.602809\pi$
$662$	17.6603	0.686385
$663$	0	0
$664$	77.5692	3.01027
$665$	−3.12436	−0.121157
$666$	0	0
$667$	−24.9282	−0.965224
$668$	1.85641	0.0718265
$669$	0	0
$670$	9.46410	0.365630
$671$	9.07180	0.350213
$672$	0	0
$673$	10.3923	0.400594	0.200297	−	0.979735i	$-0.435809\pi$
$673$	10.3923	0.400594	0.200297	+	0.979735i	$0.435809\pi$
$674$	−20.0000	−0.770371
$675$	0	0
$676$	92.1051	3.54250
$677$	18.0000	0.691796	0.345898	−	0.938272i	$-0.387574\pi$
$677$	18.0000	0.691796	0.345898	+	0.938272i	$0.387574\pi$
$678$	0	0
$679$	−9.71281	−0.372744
$680$	6.92820	0.265684
$681$	0	0
$682$	−18.5885	−0.711789
$683$	40.3923	1.54557	0.772784	−	0.634669i	$-0.218863\pi$
$683$	40.3923	1.54557	0.772784	+	0.634669i	$0.218863\pi$
$684$	0	0
$685$	9.46410	0.361605
$686$	42.9282	1.63901
$687$	0	0
$688$	−152.210	−5.80296
$689$	17.8564	0.680275
$690$	0	0
$691$	37.7128	1.43466	0.717332	−	0.696732i	$-0.245363\pi$
$691$	37.7128	1.43466	0.717332	+	0.696732i	$0.245363\pi$
$692$	−84.4974	−3.21211
$693$	0	0
$694$	−7.07180	−0.268442
$695$	−21.3923	−0.811456
$696$	0	0
$697$	2.33975	0.0886242
$698$	24.1962	0.915838
$699$	0	0
$700$	−6.92820	−0.261861
$701$	31.1962	1.17826	0.589131	−	0.808037i	$-0.299470\pi$
$701$	31.1962	1.17826	0.589131	+	0.808037i	$0.299470\pi$
$702$	0	0
$703$	−1.80385	−0.0680334
$704$	67.7128	2.55202
$705$	0	0
$706$	53.3205	2.00674
$707$	18.5885	0.699091
$708$	0	0
$709$	−29.4641	−1.10655	−0.553274	−	0.832999i	$-0.686622\pi$
$709$	−29.4641	−1.10655	−0.553274	+	0.832999i	$0.686622\pi$
$710$	−0.732051	−0.0274734
$711$	0	0
$712$	49.1769	1.84298
$713$	10.3923	0.389195
$714$	0	0
$715$	12.3923	0.463446
$716$	88.1051	3.29264
$717$	0	0
$718$	−49.5167	−1.84795
$719$	−39.5885	−1.47640	−0.738200	−	0.674582i	$-0.764324\pi$
$719$	−39.5885	−1.47640	−0.738200	+	0.674582i	$0.764324\pi$
$720$	0	0
$721$	9.46410	0.352462
$722$	−35.3205	−1.31449
$723$	0	0
$724$	106.746	3.96719
$725$	7.19615	0.267258
$726$	0	0
$727$	12.3923	0.459605	0.229803	−	0.973237i	$-0.426192\pi$
$727$	12.3923	0.459605	0.229803	+	0.973237i	$0.426192\pi$
$728$	65.5692	2.43016
$729$	0	0
$730$	−26.3923	−0.976823
$731$	7.46410	0.276070
$732$	0	0
$733$	−6.78461	−0.250595	−0.125298	−	0.992119i	$-0.539989\pi$
$733$	−6.78461	−0.250595	−0.125298	+	0.992119i	$0.539989\pi$
$734$	−85.1769	−3.14394
$735$	0	0
$736$	−75.7128	−2.79081
$737$	−7.85641	−0.289394
$738$	0	0
$739$	15.5359	0.571497	0.285749	−	0.958305i	$-0.407758\pi$
$739$	15.5359	0.571497	0.285749	+	0.958305i	$0.407758\pi$
$740$	−4.00000	−0.147043
$741$	0	0
$742$	11.3205	0.415589
$743$	45.9090	1.68424	0.842118	−	0.539293i	$-0.181308\pi$
$743$	45.9090	1.68424	0.842118	+	0.539293i	$0.181308\pi$
$744$	0	0
$745$	−8.00000	−0.293097
$746$	−49.3205	−1.80575
$747$	0	0
$748$	−9.07180	−0.331698
$749$	−19.6077	−0.716450
$750$	0	0
$751$	7.21539	0.263293	0.131647	−	0.991297i	$-0.457974\pi$
$751$	7.21539	0.263293	0.131647	+	0.991297i	$0.457974\pi$
$752$	78.6410	2.86774
$753$	0	0
$754$	−107.426	−3.91221
$755$	15.3923	0.560183
$756$	0	0
$757$	53.1769	1.93275	0.966374	−	0.257141i	$-0.0827805\pi$
$757$	53.1769	1.93275	0.966374	+	0.257141i	$0.0827805\pi$
$758$	−50.2487	−1.82512
$759$	0	0
$760$	23.3205	0.845924
$761$	35.4449	1.28488	0.642438	−	0.766338i	$-0.277923\pi$
$761$	35.4449	1.28488	0.642438	+	0.766338i	$0.277923\pi$
$762$	0	0
$763$	25.2679	0.914761
$764$	−88.1051	−3.18753
$765$	0	0
$766$	−25.8564	−0.934230
$767$	−64.1051	−2.31470
$768$	0	0
$769$	−16.4641	−0.593711	−0.296855	−	0.954922i	$-0.595938\pi$
$769$	−16.4641	−0.593711	−0.296855	+	0.954922i	$0.595938\pi$
$770$	7.85641	0.283125
$771$	0	0
$772$	−47.7128	−1.71722
$773$	43.5167	1.56519	0.782593	−	0.622534i	$-0.213897\pi$
$773$	43.5167	1.56519	0.782593	+	0.622534i	$0.213897\pi$
$774$	0	0
$775$	−3.00000	−0.107763
$776$	72.4974	2.60251
$777$	0	0
$778$	−56.1051	−2.01147
$779$	7.87564	0.282174
$780$	0	0
$781$	0.607695	0.0217450
$782$	6.92820	0.247752
$783$	0	0
$784$	−80.4974	−2.87491
$785$	−5.12436	−0.182896
$786$	0	0
$787$	−9.94744	−0.354588	−0.177294	−	0.984158i	$-0.556734\pi$
$787$	−9.94744	−0.354588	−0.177294	+	0.984158i	$0.556734\pi$
$788$	−75.7128	−2.69716
$789$	0	0
$790$	−23.3205	−0.829706
$791$	6.49742	0.231022
$792$	0	0
$793$	−21.8564	−0.776144
$794$	−17.4641	−0.619778
$795$	0	0
$796$	−10.9282	−0.387340
$797$	36.3923	1.28908	0.644541	−	0.764570i	$-0.277049\pi$
$797$	36.3923	1.28908	0.644541	+	0.764570i	$0.277049\pi$
$798$	0	0
$799$	−3.85641	−0.136430
$800$	21.8564	0.772741
$801$	0	0
$802$	−30.2487	−1.06812
$803$	21.9090	0.773151
$804$	0	0
$805$	−4.39230	−0.154808
$806$	44.7846	1.57747
$807$	0	0
$808$	−138.746	−4.88107
$809$	13.4449	0.472696	0.236348	−	0.971668i	$-0.424049\pi$
$809$	13.4449	0.472696	0.236348	+	0.971668i	$0.424049\pi$
$810$	0	0
$811$	−11.5359	−0.405080	−0.202540	−	0.979274i	$-0.564920\pi$
$811$	−11.5359	−0.405080	−0.202540	+	0.979274i	$0.564920\pi$
$812$	−49.8564	−1.74962
$813$	0	0
$814$	4.53590	0.158983
$815$	−9.26795	−0.324642
$816$	0	0
$817$	25.1244	0.878990
$818$	26.9282	0.941523
$819$	0	0
$820$	17.4641	0.609873
$821$	34.2679	1.19596	0.597980	−	0.801511i	$-0.295970\pi$
$821$	34.2679	1.19596	0.597980	+	0.801511i	$0.295970\pi$
$822$	0	0
$823$	−23.8564	−0.831582	−0.415791	−	0.909460i	$-0.636495\pi$
$823$	−23.8564	−0.831582	−0.415791	+	0.909460i	$0.636495\pi$
$824$	−70.6410	−2.46090
$825$	0	0
$826$	−40.6410	−1.41408
$827$	32.3923	1.12639	0.563195	−	0.826324i	$-0.309572\pi$
$827$	32.3923	1.12639	0.563195	+	0.826324i	$0.309572\pi$
$828$	0	0
$829$	17.7846	0.617685	0.308843	−	0.951113i	$-0.400058\pi$
$829$	17.7846	0.617685	0.308843	+	0.951113i	$0.400058\pi$
$830$	−22.3923	−0.777248
$831$	0	0
$832$	−163.138	−5.65581
$833$	3.94744	0.136771
$834$	0	0
$835$	−0.339746	−0.0117574
$836$	−30.5359	−1.05611
$837$	0	0
$838$	1.07180	0.0370246
$839$	−22.8038	−0.787276	−0.393638	−	0.919265i	$-0.628784\pi$
$839$	−22.8038	−0.787276	−0.393638	+	0.919265i	$0.628784\pi$
$840$	0	0
$841$	22.7846	0.785676
$842$	−21.2679	−0.732942
$843$	0	0
$844$	103.033	3.54655
$845$	−16.8564	−0.579878
$846$	0	0
$847$	7.42563	0.255148
$848$	−48.7846	−1.67527
$849$	0	0
$850$	−2.00000	−0.0685994
$851$	−2.53590	−0.0869295
$852$	0	0
$853$	55.5167	1.90085	0.950427	−	0.310947i	$-0.100646\pi$
$853$	55.5167	1.90085	0.950427	+	0.310947i	$0.100646\pi$
$854$	−13.8564	−0.474156
$855$	0	0
$856$	146.354	5.00227
$857$	52.4974	1.79328	0.896639	−	0.442763i	$-0.146002\pi$
$857$	52.4974	1.79328	0.896639	+	0.442763i	$0.146002\pi$
$858$	0	0
$859$	−11.0000	−0.375315	−0.187658	−	0.982235i	$-0.560090\pi$
$859$	−11.0000	−0.375315	−0.187658	+	0.982235i	$0.560090\pi$
$860$	55.7128	1.89979
$861$	0	0
$862$	105.622	3.59749
$863$	−1.12436	−0.0382735	−0.0191368	−	0.999817i	$-0.506092\pi$
$863$	−1.12436	−0.0382735	−0.0191368	+	0.999817i	$0.506092\pi$
$864$	0	0
$865$	15.4641	0.525795
$866$	−77.9615	−2.64924
$867$	0	0
$868$	20.7846	0.705476
$869$	19.3590	0.656709
$870$	0	0
$871$	18.9282	0.641358
$872$	−188.603	−6.38689
$873$	0	0
$874$	23.3205	0.788828
$875$	1.26795	0.0428645
$876$	0	0
$877$	26.5359	0.896054	0.448027	−	0.894020i	$-0.352127\pi$
$877$	26.5359	0.896054	0.448027	+	0.894020i	$0.352127\pi$
$878$	−42.0526	−1.41921
$879$	0	0
$880$	−33.8564	−1.14130
$881$	8.94744	0.301447	0.150723	−	0.988576i	$-0.451840\pi$
$881$	8.94744	0.301447	0.150723	+	0.988576i	$0.451840\pi$
$882$	0	0
$883$	−19.8038	−0.666453	−0.333226	−	0.942847i	$-0.608137\pi$
$883$	−19.8038	−0.666453	−0.333226	+	0.942847i	$0.608137\pi$
$884$	21.8564	0.735111
$885$	0	0
$886$	48.2487	1.62095
$887$	−44.1962	−1.48396	−0.741981	−	0.670421i	$-0.766113\pi$
$887$	−44.1962	−1.48396	−0.741981	+	0.670421i	$0.766113\pi$
$888$	0	0
$889$	−21.0333	−0.705435
$890$	−14.1962	−0.475856
$891$	0	0
$892$	−135.426	−4.53439
$893$	−12.9808	−0.434385
$894$	0	0
$895$	−16.1244	−0.538978
$896$	−48.0000	−1.60357
$897$	0	0
$898$	44.0526	1.47005
$899$	−21.5885	−0.720015
$900$	0	0
$901$	2.39230	0.0796992
$902$	−19.8038	−0.659396
$903$	0	0
$904$	−48.4974	−1.61300
$905$	−19.5359	−0.649395
$906$	0	0
$907$	−30.0000	−0.996134	−0.498067	−	0.867139i	$-0.665957\pi$
$907$	−30.0000	−0.996134	−0.498067	+	0.867139i	$0.665957\pi$
$908$	−109.569	−3.63618
$909$	0	0
$910$	−18.9282	−0.627464
$911$	7.58846	0.251417	0.125708	−	0.992067i	$-0.459880\pi$
$911$	7.58846	0.251417	0.125708	+	0.992067i	$0.459880\pi$
$912$	0	0
$913$	18.5885	0.615188
$914$	−2.00000	−0.0661541
$915$	0	0
$916$	−65.5692	−2.16647
$917$	−19.7654	−0.652710
$918$	0	0
$919$	46.9615	1.54912	0.774559	−	0.632502i	$-0.217972\pi$
$919$	46.9615	1.54912	0.774559	+	0.632502i	$0.217972\pi$
$920$	32.7846	1.08088
$921$	0	0
$922$	−2.87564	−0.0947043
$923$	−1.46410	−0.0481915
$924$	0	0
$925$	0.732051	0.0240697
$926$	28.3923	0.933029
$927$	0	0
$928$	157.282	5.16304
$929$	−28.3731	−0.930890	−0.465445	−	0.885077i	$-0.654106\pi$
$929$	−28.3731	−0.930890	−0.465445	+	0.885077i	$0.654106\pi$
$930$	0	0
$931$	13.2872	0.435470
$932$	−54.9282	−1.79923
$933$	0	0
$934$	96.6410	3.16219
$935$	1.66025	0.0542961
$936$	0	0
$937$	−31.8564	−1.04070	−0.520352	−	0.853952i	$-0.674199\pi$
$937$	−31.8564	−1.04070	−0.520352	+	0.853952i	$0.674199\pi$
$938$	12.0000	0.391814
$939$	0	0
$940$	−28.7846	−0.938850
$941$	27.1769	0.885942	0.442971	−	0.896536i	$-0.353924\pi$
$941$	27.1769	0.885942	0.442971	+	0.896536i	$0.353924\pi$
$942$	0	0
$943$	11.0718	0.360547
$944$	175.138	5.70027
$945$	0	0
$946$	−63.1769	−2.05406
$947$	2.28719	0.0743236	0.0371618	−	0.999309i	$-0.488168\pi$
$947$	2.28719	0.0743236	0.0371618	+	0.999309i	$0.488168\pi$
$948$	0	0
$949$	−52.7846	−1.71346
$950$	−6.73205	−0.218417
$951$	0	0
$952$	8.78461	0.284711
$953$	−15.6077	−0.505583	−0.252791	−	0.967521i	$-0.581349\pi$
$953$	−15.6077	−0.505583	−0.252791	+	0.967521i	$0.581349\pi$
$954$	0	0
$955$	16.1244	0.521772
$956$	40.7846	1.31907
$957$	0	0
$958$	98.6936	3.18864
$959$	12.0000	0.387500
$960$	0	0
$961$	−22.0000	−0.709677
$962$	−10.9282	−0.352339
$963$	0	0
$964$	100.105	3.22417
$965$	8.73205	0.281095
$966$	0	0
$967$	−36.1962	−1.16399	−0.581995	−	0.813192i	$-0.697728\pi$
$967$	−36.1962	−1.16399	−0.581995	+	0.813192i	$0.697728\pi$
$968$	−55.4256	−1.78145
$969$	0	0
$970$	−20.9282	−0.671964
$971$	41.4449	1.33003	0.665014	−	0.746830i	$-0.268425\pi$
$971$	41.4449	1.33003	0.665014	+	0.746830i	$0.268425\pi$
$972$	0	0
$973$	−27.1244	−0.869567
$974$	9.85641	0.315820
$975$	0	0
$976$	59.7128	1.91136
$977$	1.46410	0.0468408	0.0234204	−	0.999726i	$-0.492544\pi$
$977$	1.46410	0.0468408	0.0234204	+	0.999726i	$0.492544\pi$
$978$	0	0
$979$	11.7846	0.376638
$980$	29.4641	0.941196
$981$	0	0
$982$	104.158	3.32381
$983$	−17.4115	−0.555342	−0.277671	−	0.960676i	$-0.589563\pi$
$983$	−17.4115	−0.555342	−0.277671	+	0.960676i	$0.589563\pi$
$984$	0	0
$985$	13.8564	0.441502
$986$	−14.3923	−0.458344
$987$	0	0
$988$	73.5692	2.34055
$989$	35.3205	1.12313
$990$	0	0
$991$	3.14359	0.0998595	0.0499298	−	0.998753i	$-0.484100\pi$
$991$	3.14359	0.0998595	0.0499298	+	0.998753i	$0.484100\pi$
$992$	−65.5692	−2.08182
$993$	0	0
$994$	−0.928203	−0.0294408
$995$	2.00000	0.0634043
$996$	0	0
$997$	−19.5167	−0.618099	−0.309049	−	0.951046i	$-0.600011\pi$
$997$	−19.5167	−0.618099	−0.309049	+	0.951046i	$0.600011\pi$
$998$	−28.1962	−0.892534
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.2.a.h.1.2	yes	2
3.2	odd	2		405.2.a.g.1.1	✓	2
4.3	odd	2		6480.2.a.bi.1.1		2
5.2	odd	4		2025.2.b.h.649.4		4
5.3	odd	4		2025.2.b.h.649.1		4
5.4	even	2		2025.2.a.g.1.1		2
9.2	odd	6		405.2.e.l.271.2		4
9.4	even	3		405.2.e.i.136.1		4
9.5	odd	6		405.2.e.l.136.2		4
9.7	even	3		405.2.e.i.271.1		4
12.11	even	2		6480.2.a.br.1.1		2
15.2	even	4		2025.2.b.g.649.1		4
15.8	even	4		2025.2.b.g.649.4		4
15.14	odd	2		2025.2.a.m.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
405.2.a.g.1.1	✓	2	3.2	odd	2
405.2.a.h.1.2	yes	2	1.1	even	1	trivial
405.2.e.i.136.1		4	9.4	even	3
405.2.e.i.271.1		4	9.7	even	3
405.2.e.l.136.2		4	9.5	odd	6
405.2.e.l.271.2		4	9.2	odd	6
2025.2.a.g.1.1		2	5.4	even	2
2025.2.a.m.1.2		2	15.14	odd	2
2025.2.b.g.649.1		4	15.2	even	4
2025.2.b.g.649.4		4	15.8	even	4
2025.2.b.h.649.1		4	5.3	odd	4
2025.2.b.h.649.4		4	5.2	odd	4
6480.2.a.bi.1.1		2	4.3	odd	2
6480.2.a.br.1.1		2	12.11	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:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	405.a (trivial)

\(f(q)\)	\(=\)	\(q+2.73205 q^{2} +5.46410 q^{4} -1.00000 q^{5} -1.26795 q^{7} +9.46410 q^{8} -2.73205 q^{10} +2.26795 q^{11} -5.46410 q^{13} -3.46410 q^{14} +14.9282 q^{16} -0.732051 q^{17} -2.46410 q^{19} -5.46410 q^{20} +6.19615 q^{22} -3.46410 q^{23} +1.00000 q^{25} -14.9282 q^{26} -6.92820 q^{28} +7.19615 q^{29} -3.00000 q^{31} +21.8564 q^{32} -2.00000 q^{34} +1.26795 q^{35} +0.732051 q^{37} -6.73205 q^{38} -9.46410 q^{40} -3.19615 q^{41} -10.1962 q^{43} +12.3923 q^{44} -9.46410 q^{46} +5.26795 q^{47} -5.39230 q^{49} +2.73205 q^{50} -29.8564 q^{52} -3.26795 q^{53} -2.26795 q^{55} -12.0000 q^{56} +19.6603 q^{58} +11.7321 q^{59} +4.00000 q^{61} -8.19615 q^{62} +29.8564 q^{64} +5.46410 q^{65} -3.46410 q^{67} -4.00000 q^{68} +3.46410 q^{70} +0.267949 q^{71} +9.66025 q^{73} +2.00000 q^{74} -13.4641 q^{76} -2.87564 q^{77} +8.53590 q^{79} -14.9282 q^{80} -8.73205 q^{82} +8.19615 q^{83} +0.732051 q^{85} -27.8564 q^{86} +21.4641 q^{88} +5.19615 q^{89} +6.92820 q^{91} -18.9282 q^{92} +14.3923 q^{94} +2.46410 q^{95} +7.66025 q^{97} -14.7321 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 4 q^{4} - 2 q^{5} - 6 q^{7} + 12 q^{8} - 2 q^{10} + 8 q^{11} - 4 q^{13} + 16 q^{16} + 2 q^{17} + 2 q^{19} - 4 q^{20} + 2 q^{22} + 2 q^{25} - 16 q^{26} + 4 q^{29} - 6 q^{31} + 16 q^{32} - 4 q^{34}+ \cdots - 26 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 4 * q^4 - 2 * q^5 - 6 * q^7 + 12 * q^8 - 2 * q^10 + 8 * q^11 - 4 * q^13 + 16 * q^16 + 2 * q^17 + 2 * q^19 - 4 * q^20 + 2 * q^22 + 2 * q^25 - 16 * q^26 + 4 * q^29 - 6 * q^31 + 16 * q^32 - 4 * q^34 + 6 * q^35 - 2 * q^37 - 10 * q^38 - 12 * q^40 + 4 * q^41 - 10 * q^43 + 4 * q^44 - 12 * q^46 + 14 * q^47 + 10 * q^49 + 2 * q^50 - 32 * q^52 - 10 * q^53 - 8 * q^55 - 24 * q^56 + 22 * q^58 + 20 * q^59 + 8 * q^61 - 6 * q^62 + 32 * q^64 + 4 * q^65 - 8 * q^68 + 4 * q^71 + 2 * q^73 + 4 * q^74 - 20 * q^76 - 30 * q^77 + 24 * q^79 - 16 * q^80 - 14 * q^82 + 6 * q^83 - 2 * q^85 - 28 * q^86 + 36 * q^88 - 24 * q^92 + 8 * q^94 - 2 * q^95 - 2 * q^97 - 26 * q^98

Introduction

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