LMFDB - Embedded newform 405.4.a.l.1.5

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:	$0$
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			1.5
Root			$-3.53444$ 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)

$f(q)$	$=$	$q+4.53444 q^{2} +12.5612 q^{4} +5.00000 q^{5} -2.63618 q^{7} +20.6823 q^{8} +O(q^{10})$	\(q+4.53444 q^{2} +12.5612 q^{4} +5.00000 q^{5} -2.63618 q^{7} +20.6823 q^{8} +22.6722 q^{10} +20.9310 q^{11} +60.9451 q^{13} -11.9536 q^{14} -6.70667 q^{16} +86.8178 q^{17} +41.8063 q^{19} +62.8058 q^{20} +94.9105 q^{22} -97.3186 q^{23} +25.0000 q^{25} +276.352 q^{26} -33.1135 q^{28} +157.070 q^{29} +95.3961 q^{31} -195.869 q^{32} +393.670 q^{34} -13.1809 q^{35} -160.833 q^{37} +189.568 q^{38} +103.411 q^{40} -233.189 q^{41} +487.694 q^{43} +262.918 q^{44} -441.285 q^{46} -24.3079 q^{47} -336.051 q^{49} +113.361 q^{50} +765.540 q^{52} -709.828 q^{53} +104.655 q^{55} -54.5222 q^{56} +712.225 q^{58} +191.998 q^{59} -744.633 q^{61} +432.568 q^{62} -834.504 q^{64} +304.725 q^{65} -823.567 q^{67} +1090.53 q^{68} -59.7680 q^{70} +1068.85 q^{71} -132.416 q^{73} -729.289 q^{74} +525.135 q^{76} -55.1780 q^{77} -704.577 q^{79} -33.5333 q^{80} -1057.38 q^{82} +1419.76 q^{83} +434.089 q^{85} +2211.42 q^{86} +432.901 q^{88} -401.847 q^{89} -160.662 q^{91} -1222.43 q^{92} -110.223 q^{94} +209.031 q^{95} +530.880 q^{97} -1523.80 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	4.53444	1.60317	0.801583	−	0.597883i	$-0.203991\pi$
$2$	4.53444	1.60317	0.801583	+	0.597883i	$0.203991\pi$
$3$	0	0
$3$	0	0
$4$	12.5612	1.57014
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	−2.63618	−0.142340	−0.0711702	−	0.997464i	$-0.522673\pi$
$7$	−2.63618	−0.142340	−0.0711702	+	0.997464i	$0.522673\pi$
$8$	20.6823	0.914036
$9$	0	0
$10$	22.6722	0.716958
$11$	20.9310	0.573722	0.286861	−	0.957972i	$-0.407388\pi$
$11$	20.9310	0.573722	0.286861	+	0.957972i	$0.407388\pi$
$12$	0	0
$13$	60.9451	1.30024	0.650120	−	0.759832i	$-0.274719\pi$
$13$	60.9451	1.30024	0.650120	+	0.759832i	$0.274719\pi$
$14$	−11.9536	−0.228195
$15$	0	0
$16$	−6.70667	−0.104792
$17$	86.8178	1.23861	0.619306	−	0.785150i	$-0.287414\pi$
$17$	86.8178	1.23861	0.619306	+	0.785150i	$0.287414\pi$
$18$	0	0
$19$	41.8063	0.504791	0.252395	−	0.967624i	$-0.418782\pi$
$19$	41.8063	0.504791	0.252395	+	0.967624i	$0.418782\pi$
$20$	62.8058	0.702190
$21$	0	0
$22$	94.9105	0.919772
$23$	−97.3186	−0.882275	−0.441138	−	0.897439i	$-0.645425\pi$
$23$	−97.3186	−0.882275	−0.441138	+	0.897439i	$0.645425\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	276.352	2.08450
$27$	0	0
$28$	−33.1135	−0.223495
$29$	157.070	1.00576	0.502882	−	0.864355i	$-0.332273\pi$
$29$	157.070	1.00576	0.502882	+	0.864355i	$0.332273\pi$
$30$	0	0
$31$	95.3961	0.552698	0.276349	−	0.961057i	$-0.410875\pi$
$31$	95.3961	0.552698	0.276349	+	0.961057i	$0.410875\pi$
$32$	−195.869	−1.08203
$33$	0	0
$34$	393.670	1.98570
$35$	−13.1809	−0.0636566
$36$	0	0
$37$	−160.833	−0.714617	−0.357309	−	0.933986i	$-0.616306\pi$
$37$	−160.833	−0.714617	−0.357309	+	0.933986i	$0.616306\pi$
$38$	189.568	0.809263
$39$	0	0
$40$	103.411	0.408769
$41$	−233.189	−0.888245	−0.444123	−	0.895966i	$-0.646485\pi$
$41$	−233.189	−0.888245	−0.444123	+	0.895966i	$0.646485\pi$
$42$	0	0
$43$	487.694	1.72960	0.864798	−	0.502121i	$-0.167447\pi$
$43$	487.694	1.72960	0.864798	+	0.502121i	$0.167447\pi$
$44$	262.918	0.900826
$45$	0	0
$46$	−441.285	−1.41443
$47$	−24.3079	−0.0754399	−0.0377200	−	0.999288i	$-0.512009\pi$
$47$	−24.3079	−0.0754399	−0.0377200	+	0.999288i	$0.512009\pi$
$48$	0	0
$49$	−336.051	−0.979739
$50$	113.361	0.320633
$51$	0	0
$52$	765.540	2.04156
$53$	−709.828	−1.83967	−0.919834	−	0.392308i	$-0.871677\pi$
$53$	−709.828	−1.83967	−0.919834	+	0.392308i	$0.871677\pi$
$54$	0	0
$55$	104.655	0.256576
$56$	−54.5222	−0.130104
$57$	0	0
$58$	712.225	1.61241
$59$	191.998	0.423661	0.211831	−	0.977306i	$-0.432057\pi$
$59$	191.998	0.423661	0.211831	+	0.977306i	$0.432057\pi$
$60$	0	0
$61$	−744.633	−1.56296	−0.781480	−	0.623931i	$-0.785535\pi$
$61$	−744.633	−1.56296	−0.781480	+	0.623931i	$0.785535\pi$
$62$	432.568	0.886067
$63$	0	0
$64$	−834.504	−1.62989
$65$	304.725	0.581485
$66$	0	0
$67$	−823.567	−1.50171	−0.750856	−	0.660466i	$-0.770359\pi$
$67$	−823.567	−1.50171	−0.750856	+	0.660466i	$0.770359\pi$
$68$	1090.53	1.94480
$69$	0	0
$70$	−59.7680	−0.102052
$71$	1068.85	1.78661	0.893304	−	0.449452i	$-0.148381\pi$
$71$	1068.85	1.78661	0.893304	+	0.449452i	$0.148381\pi$
$72$	0	0
$73$	−132.416	−0.212302	−0.106151	−	0.994350i	$-0.533853\pi$
$73$	−132.416	−0.212302	−0.106151	+	0.994350i	$0.533853\pi$
$74$	−729.289	−1.14565
$75$	0	0
$76$	525.135	0.792594
$77$	−55.1780	−0.0816638
$78$	0	0
$79$	−704.577	−1.00343	−0.501716	−	0.865033i	$-0.667298\pi$
$79$	−704.577	−1.00343	−0.501716	+	0.865033i	$0.667298\pi$
$80$	−33.5333	−0.0468642
$81$	0	0
$82$	−1057.38	−1.42401
$83$	1419.76	1.87758	0.938790	−	0.344489i	$-0.111948\pi$
$83$	1419.76	1.87758	0.938790	+	0.344489i	$0.111948\pi$
$84$	0	0
$85$	434.089	0.553924
$86$	2211.42	2.77283
$87$	0	0
$88$	432.901	0.524402
$89$	−401.847	−0.478604	−0.239302	−	0.970945i	$-0.576919\pi$
$89$	−401.847	−0.478604	−0.239302	+	0.970945i	$0.576919\pi$
$90$	0	0
$91$	−160.662	−0.185077
$92$	−1222.43	−1.38530
$93$	0	0
$94$	−110.223	−0.120943
$95$	209.031	0.225749
$96$	0	0
$97$	530.880	0.555698	0.277849	−	0.960625i	$-0.410378\pi$
$97$	530.880	0.555698	0.277849	+	0.960625i	$0.410378\pi$
$98$	−1523.80	−1.57069
$99$	0	0
$100$	314.029	0.314029
$101$	998.463	0.983671	0.491835	−	0.870688i	$-0.336326\pi$
$101$	998.463	0.983671	0.491835	+	0.870688i	$0.336326\pi$
$102$	0	0
$103$	−837.473	−0.801152	−0.400576	−	0.916264i	$-0.631190\pi$
$103$	−837.473	−0.801152	−0.400576	+	0.916264i	$0.631190\pi$
$104$	1260.48	1.18847
$105$	0	0
$106$	−3218.67	−2.94929
$107$	−1109.39	−1.00233	−0.501163	−	0.865353i	$-0.667094\pi$
$107$	−1109.39	−1.00233	−0.501163	+	0.865353i	$0.667094\pi$
$108$	0	0
$109$	−1071.73	−0.941769	−0.470885	−	0.882195i	$-0.656065\pi$
$109$	−1071.73	−0.941769	−0.470885	+	0.882195i	$0.656065\pi$
$110$	474.552	0.411334
$111$	0	0
$112$	17.6800	0.0149161
$113$	1089.50	0.907003	0.453501	−	0.891256i	$-0.350175\pi$
$113$	1089.50	0.907003	0.453501	+	0.891256i	$0.350175\pi$
$114$	0	0
$115$	−486.593	−0.394565
$116$	1972.98	1.57920
$117$	0	0
$118$	870.604	0.679200
$119$	−228.867	−0.176305
$120$	0	0
$121$	−892.892	−0.670843
$122$	−3376.50	−2.50568
$123$	0	0
$124$	1198.28	0.867816
$125$	125.000	0.0894427
$126$	0	0
$127$	−371.442	−0.259529	−0.129764	−	0.991545i	$-0.541422\pi$
$127$	−371.442	−0.259529	−0.129764	+	0.991545i	$0.541422\pi$
$128$	−2217.05	−1.53095
$129$	0	0
$130$	1381.76	0.932217
$131$	−265.266	−0.176919	−0.0884597	−	0.996080i	$-0.528194\pi$
$131$	−265.266	−0.176919	−0.0884597	+	0.996080i	$0.528194\pi$
$132$	0	0
$133$	−110.209	−0.0718521
$134$	−3734.42	−2.40749
$135$	0	0
$136$	1795.59	1.13214
$137$	270.721	0.168827	0.0844134	−	0.996431i	$-0.473098\pi$
$137$	270.721	0.168827	0.0844134	+	0.996431i	$0.473098\pi$
$138$	0	0
$139$	−1212.09	−0.739627	−0.369814	−	0.929106i	$-0.620578\pi$
$139$	−1212.09	−0.739627	−0.369814	+	0.929106i	$0.620578\pi$
$140$	−165.567	−0.0999500
$141$	0	0
$142$	4846.64	2.86423
$143$	1275.64	0.745976
$144$	0	0
$145$	785.350	0.449792
$146$	−600.431	−0.340356
$147$	0	0
$148$	−2020.25	−1.12205
$149$	−2413.33	−1.32690	−0.663450	−	0.748221i	$-0.730908\pi$
$149$	−2413.33	−1.32690	−0.663450	+	0.748221i	$0.730908\pi$
$150$	0	0
$151$	−1169.19	−0.630114	−0.315057	−	0.949073i	$-0.602024\pi$
$151$	−1169.19	−0.630114	−0.315057	+	0.949073i	$0.602024\pi$
$152$	864.649	0.461397
$153$	0	0
$154$	−250.201	−0.130921
$155$	476.980	0.247174
$156$	0	0
$157$	−2928.23	−1.48852	−0.744262	−	0.667888i	$-0.767199\pi$
$157$	−2928.23	−1.48852	−0.744262	+	0.667888i	$0.767199\pi$
$158$	−3194.86	−1.60867
$159$	0	0
$160$	−979.346	−0.483901
$161$	256.549	0.125583
$162$	0	0
$163$	−936.208	−0.449874	−0.224937	−	0.974373i	$-0.572218\pi$
$163$	−936.208	−0.449874	−0.224937	+	0.974373i	$0.572218\pi$
$164$	−2929.13	−1.39467
$165$	0	0
$166$	6437.83	3.01008
$167$	−3133.05	−1.45175	−0.725876	−	0.687826i	$-0.758565\pi$
$167$	−3133.05	−1.45175	−0.725876	+	0.687826i	$0.758565\pi$
$168$	0	0
$169$	1517.30	0.690624
$170$	1968.35	0.888033
$171$	0	0
$172$	6125.99	2.71571
$173$	−239.265	−0.105150	−0.0525751	−	0.998617i	$-0.516743\pi$
$173$	−239.265	−0.105150	−0.0525751	+	0.998617i	$0.516743\pi$
$174$	0	0
$175$	−65.9045	−0.0284681
$176$	−140.377	−0.0601212
$177$	0	0
$178$	−1822.15	−0.767282
$179$	−332.030	−0.138643	−0.0693215	−	0.997594i	$-0.522083\pi$
$179$	−332.030	−0.138643	−0.0693215	+	0.997594i	$0.522083\pi$
$180$	0	0
$181$	3405.54	1.39852	0.699259	−	0.714869i	$-0.253514\pi$
$181$	3405.54	1.39852	0.699259	+	0.714869i	$0.253514\pi$
$182$	−728.513	−0.296709
$183$	0	0
$184$	−2012.77	−0.806431
$185$	−804.167	−0.319587
$186$	0	0
$187$	1817.18	0.710619
$188$	−305.336	−0.118452
$189$	0	0
$190$	947.841	0.361914
$191$	5261.72	1.99332	0.996661	−	0.0816469i	$-0.0260180\pi$
$191$	5261.72	1.99332	0.996661	+	0.0816469i	$0.0260180\pi$
$192$	0	0
$193$	408.973	0.152531	0.0762656	−	0.997088i	$-0.475700\pi$
$193$	408.973	0.152531	0.0762656	+	0.997088i	$0.475700\pi$
$194$	2407.24	0.890877
$195$	0	0
$196$	−4221.18	−1.53833
$197$	946.927	0.342466	0.171233	−	0.985231i	$-0.445225\pi$
$197$	946.927	0.342466	0.171233	+	0.985231i	$0.445225\pi$
$198$	0	0
$199$	4917.06	1.75156	0.875782	−	0.482707i	$-0.160346\pi$
$199$	4917.06	1.75156	0.875782	+	0.482707i	$0.160346\pi$
$200$	517.057	0.182807
$201$	0	0
$202$	4527.47	1.57699
$203$	−414.065	−0.143161
$204$	0	0
$205$	−1165.95	−0.397235
$206$	−3797.47	−1.28438
$207$	0	0
$208$	−408.738	−0.136254
$209$	875.048	0.289609
$210$	0	0
$211$	2449.40	0.799163	0.399582	−	0.916698i	$-0.369155\pi$
$211$	2449.40	0.799163	0.399582	+	0.916698i	$0.369155\pi$
$212$	−8916.26	−2.88854
$213$	0	0
$214$	−5030.47	−1.60690
$215$	2438.47	0.773498
$216$	0	0
$217$	−251.481	−0.0786713
$218$	−4859.68	−1.50981
$219$	0	0
$220$	1314.59	0.402862
$221$	5291.11	1.61049
$222$	0	0
$223$	−1006.44	−0.302227	−0.151113	−	0.988516i	$-0.548286\pi$
$223$	−1006.44	−0.302227	−0.151113	+	0.988516i	$0.548286\pi$
$224$	516.347	0.154017
$225$	0	0
$226$	4940.26	1.45408
$227$	2024.04	0.591807	0.295903	−	0.955218i	$-0.404379\pi$
$227$	2024.04	0.591807	0.295903	+	0.955218i	$0.404379\pi$
$228$	0	0
$229$	−6787.92	−1.95877	−0.979385	−	0.202004i	$-0.935255\pi$
$229$	−6787.92	−1.95877	−0.979385	+	0.202004i	$0.935255\pi$
$230$	−2206.43	−0.632554
$231$	0	0
$232$	3248.57	0.919305
$233$	4129.88	1.16119	0.580595	−	0.814193i	$-0.302820\pi$
$233$	4129.88	1.16119	0.580595	+	0.814193i	$0.302820\pi$
$234$	0	0
$235$	−121.540	−0.0337378
$236$	2411.72	0.665209
$237$	0	0
$238$	−1037.79	−0.282646
$239$	2978.73	0.806185	0.403093	−	0.915159i	$-0.367935\pi$
$239$	2978.73	0.806185	0.403093	+	0.915159i	$0.367935\pi$
$240$	0	0
$241$	−1843.94	−0.492858	−0.246429	−	0.969161i	$-0.579257\pi$
$241$	−1843.94	−0.492858	−0.246429	+	0.969161i	$0.579257\pi$
$242$	−4048.77	−1.07547
$243$	0	0
$244$	−9353.45	−2.45407
$245$	−1680.25	−0.438153
$246$	0	0
$247$	2547.89	0.656349
$248$	1973.01	0.505186
$249$	0	0
$250$	566.805	0.143392
$251$	−2360.72	−0.593656	−0.296828	−	0.954931i	$-0.595929\pi$
$251$	−2360.72	−0.593656	−0.296828	+	0.954931i	$0.595929\pi$
$252$	0	0
$253$	−2036.98	−0.506181
$254$	−1684.28	−0.416068
$255$	0	0
$256$	−3377.07	−0.824481
$257$	2472.64	0.600151	0.300076	−	0.953915i	$-0.402988\pi$
$257$	2472.64	0.600151	0.300076	+	0.953915i	$0.402988\pi$
$258$	0	0
$259$	423.986	0.101719
$260$	3827.70	0.913015
$261$	0	0
$262$	−1202.83	−0.283631
$263$	−2167.17	−0.508112	−0.254056	−	0.967189i	$-0.581765\pi$
$263$	−2167.17	−0.508112	−0.254056	+	0.967189i	$0.581765\pi$
$264$	0	0
$265$	−3549.14	−0.822725
$266$	−499.736	−0.115191
$267$	0	0
$268$	−10344.9	−2.35790
$269$	−4346.79	−0.985236	−0.492618	−	0.870246i	$-0.663960\pi$
$269$	−4346.79	−0.985236	−0.492618	+	0.870246i	$0.663960\pi$
$270$	0	0
$271$	−3923.45	−0.879456	−0.439728	−	0.898131i	$-0.644925\pi$
$271$	−3923.45	−0.879456	−0.439728	+	0.898131i	$0.644925\pi$
$272$	−582.258	−0.129796
$273$	0	0
$274$	1227.57	0.270657
$275$	523.275	0.114744
$276$	0	0
$277$	6035.89	1.30925	0.654624	−	0.755955i	$-0.272827\pi$
$277$	6035.89	1.30925	0.654624	+	0.755955i	$0.272827\pi$
$278$	−5496.15	−1.18575
$279$	0	0
$280$	−272.611	−0.0581844
$281$	−2296.92	−0.487626	−0.243813	−	0.969822i	$-0.578398\pi$
$281$	−2296.92	−0.487626	−0.243813	+	0.969822i	$0.578398\pi$
$282$	0	0
$283$	−5496.61	−1.15456	−0.577279	−	0.816547i	$-0.695885\pi$
$283$	−5496.61	−1.15456	−0.577279	+	0.816547i	$0.695885\pi$
$284$	13426.0	2.80523
$285$	0	0
$286$	5784.32	1.19592
$287$	614.729	0.126433
$288$	0	0
$289$	2624.32	0.534159
$290$	3561.12	0.721091
$291$	0	0
$292$	−1663.29	−0.333345
$293$	4094.98	0.816490	0.408245	−	0.912872i	$-0.366141\pi$
$293$	4094.98	0.816490	0.408245	+	0.912872i	$0.366141\pi$
$294$	0	0
$295$	959.990	0.189467
$296$	−3326.40	−0.653186
$297$	0	0
$298$	−10943.1	−2.12724
$299$	−5931.09	−1.14717
$300$	0	0
$301$	−1285.65	−0.246191
$302$	−5301.62	−1.01018
$303$	0	0
$304$	−280.381	−0.0528978
$305$	−3723.17	−0.698977
$306$	0	0
$307$	7535.90	1.40097	0.700483	−	0.713669i	$-0.252968\pi$
$307$	7535.90	1.40097	0.700483	+	0.713669i	$0.252968\pi$
$308$	−693.099	−0.128224
$309$	0	0
$310$	2162.84	0.396261
$311$	6330.89	1.15432	0.577158	−	0.816633i	$-0.304162\pi$
$311$	6330.89	1.15432	0.577158	+	0.816633i	$0.304162\pi$
$312$	0	0
$313$	8981.97	1.62202	0.811008	−	0.585036i	$-0.198920\pi$
$313$	8981.97	1.62202	0.811008	+	0.585036i	$0.198920\pi$
$314$	−13277.9	−2.38635
$315$	0	0
$316$	−8850.29	−1.57553
$317$	3236.48	0.573435	0.286718	−	0.958015i	$-0.407436\pi$
$317$	3236.48	0.573435	0.286718	+	0.958015i	$0.407436\pi$
$318$	0	0
$319$	3287.64	0.577029
$320$	−4172.52	−0.728909
$321$	0	0
$322$	1163.31	0.201331
$323$	3629.53	0.625240
$324$	0	0
$325$	1523.63	0.260048
$326$	−4245.18	−0.721223
$327$	0	0
$328$	−4822.89	−0.811888
$329$	64.0801	0.0107381
$330$	0	0
$331$	6615.59	1.09857	0.549283	−	0.835636i	$-0.314901\pi$
$331$	6615.59	1.09857	0.549283	+	0.835636i	$0.314901\pi$
$332$	17833.9	2.94807
$333$	0	0
$334$	−14206.6	−2.32740
$335$	−4117.83	−0.671586
$336$	0	0
$337$	2028.29	0.327858	0.163929	−	0.986472i	$-0.447583\pi$
$337$	2028.29	0.327858	0.163929	+	0.986472i	$0.447583\pi$
$338$	6880.11	1.10718
$339$	0	0
$340$	5452.66	0.869741
$341$	1996.74	0.317095
$342$	0	0
$343$	1790.10	0.281797
$344$	10086.6	1.58091
$345$	0	0
$346$	−1084.93	−0.168573
$347$	3223.38	0.498675	0.249337	−	0.968417i	$-0.419787\pi$
$347$	3223.38	0.498675	0.249337	+	0.968417i	$0.419787\pi$
$348$	0	0
$349$	−9628.52	−1.47680	−0.738399	−	0.674364i	$-0.764418\pi$
$349$	−9628.52	−1.47680	−0.738399	+	0.674364i	$0.764418\pi$
$350$	−298.840	−0.0456391
$351$	0	0
$352$	−4099.74	−0.620787
$353$	−2352.31	−0.354677	−0.177339	−	0.984150i	$-0.556749\pi$
$353$	−2352.31	−0.354677	−0.177339	+	0.984150i	$0.556749\pi$
$354$	0	0
$355$	5344.25	0.798996
$356$	−5047.67	−0.751477
$357$	0	0
$358$	−1505.57	−0.222268
$359$	6464.01	0.950299	0.475150	−	0.879905i	$-0.342394\pi$
$359$	6464.01	0.950299	0.475150	+	0.879905i	$0.342394\pi$
$360$	0	0
$361$	−5111.23	−0.745187
$362$	15442.2	2.24206
$363$	0	0
$364$	−2018.10	−0.290597
$365$	−662.078	−0.0949445
$366$	0	0
$367$	−9156.74	−1.30239	−0.651196	−	0.758910i	$-0.725732\pi$
$367$	−9156.74	−1.30239	−0.651196	+	0.758910i	$0.725732\pi$
$368$	652.683	0.0924551
$369$	0	0
$370$	−3646.45	−0.512351
$371$	1871.24	0.261859
$372$	0	0
$373$	7929.65	1.10075	0.550377	−	0.834916i	$-0.314484\pi$
$373$	7929.65	1.10075	0.550377	+	0.834916i	$0.314484\pi$
$374$	8239.91	1.13924
$375$	0	0
$376$	−502.743	−0.0689548
$377$	9572.64	1.30774
$378$	0	0
$379$	9302.25	1.26075	0.630375	−	0.776290i	$-0.282901\pi$
$379$	9302.25	1.26075	0.630375	+	0.776290i	$0.282901\pi$
$380$	2625.68	0.354459
$381$	0	0
$382$	23859.0	3.19563
$383$	10435.9	1.39229	0.696146	−	0.717900i	$-0.254896\pi$
$383$	10435.9	1.39229	0.696146	+	0.717900i	$0.254896\pi$
$384$	0	0
$385$	−275.890	−0.0365212
$386$	1854.47	0.244533
$387$	0	0
$388$	6668.47	0.872526
$389$	−3360.81	−0.438046	−0.219023	−	0.975720i	$-0.570287\pi$
$389$	−3360.81	−0.438046	−0.219023	+	0.975720i	$0.570287\pi$
$390$	0	0
$391$	−8448.98	−1.09280
$392$	−6950.29	−0.895517
$393$	0	0
$394$	4293.78	0.549030
$395$	−3522.88	−0.448748
$396$	0	0
$397$	−1324.66	−0.167463	−0.0837314	−	0.996488i	$-0.526684\pi$
$397$	−1324.66	−0.167463	−0.0837314	+	0.996488i	$0.526684\pi$
$398$	22296.1	2.80805
$399$	0	0
$400$	−167.667	−0.0209583
$401$	−14010.8	−1.74480	−0.872402	−	0.488789i	$-0.837439\pi$
$401$	−14010.8	−1.74480	−0.872402	+	0.488789i	$0.837439\pi$
$402$	0	0
$403$	5813.92	0.718640
$404$	12541.8	1.54450
$405$	0	0
$406$	−1877.55	−0.229511
$407$	−3366.41	−0.409991
$408$	0	0
$409$	3616.01	0.437165	0.218582	−	0.975818i	$-0.429857\pi$
$409$	3616.01	0.437165	0.218582	+	0.975818i	$0.429857\pi$
$410$	−5286.91	−0.636834
$411$	0	0
$412$	−10519.6	−1.25792
$413$	−506.142	−0.0603041
$414$	0	0
$415$	7098.81	0.839680
$416$	−11937.3	−1.40690
$417$	0	0
$418$	3967.85	0.464292
$419$	−816.065	−0.0951489	−0.0475745	−	0.998868i	$-0.515149\pi$
$419$	−816.065	−0.0951489	−0.0475745	+	0.998868i	$0.515149\pi$
$420$	0	0
$421$	14965.7	1.73250	0.866252	−	0.499607i	$-0.166522\pi$
$421$	14965.7	1.73250	0.866252	+	0.499607i	$0.166522\pi$
$422$	11106.6	1.28119
$423$	0	0
$424$	−14680.9	−1.68152
$425$	2170.44	0.247722
$426$	0	0
$427$	1962.99	0.222472
$428$	−13935.2	−1.57380
$429$	0	0
$430$	11057.1	1.24005
$431$	9752.30	1.08991	0.544955	−	0.838465i	$-0.316547\pi$
$431$	9752.30	1.08991	0.544955	+	0.838465i	$0.316547\pi$
$432$	0	0
$433$	4546.03	0.504545	0.252273	−	0.967656i	$-0.418822\pi$
$433$	4546.03	0.504545	0.252273	+	0.967656i	$0.418822\pi$
$434$	−1140.33	−0.126123
$435$	0	0
$436$	−13462.1	−1.47871
$437$	−4068.53	−0.445364
$438$	0	0
$439$	−8618.63	−0.937004	−0.468502	−	0.883462i	$-0.655206\pi$
$439$	−8618.63	−0.937004	−0.468502	+	0.883462i	$0.655206\pi$
$440$	2164.51	0.234520
$441$	0	0
$442$	23992.2	2.58189
$443$	12269.8	1.31593	0.657966	−	0.753048i	$-0.271417\pi$
$443$	12269.8	1.31593	0.657966	+	0.753048i	$0.271417\pi$
$444$	0	0
$445$	−2009.24	−0.214038
$446$	−4563.66	−0.484520
$447$	0	0
$448$	2199.90	0.231999
$449$	−7617.88	−0.800690	−0.400345	−	0.916364i	$-0.631110\pi$
$449$	−7617.88	−0.800690	−0.400345	+	0.916364i	$0.631110\pi$
$450$	0	0
$451$	−4880.89	−0.509606
$452$	13685.3	1.42412
$453$	0	0
$454$	9177.88	0.948765
$455$	−803.311	−0.0827688
$456$	0	0
$457$	6062.05	0.620505	0.310253	−	0.950654i	$-0.399586\pi$
$457$	6062.05	0.620505	0.310253	+	0.950654i	$0.399586\pi$
$458$	−30779.4	−3.14023
$459$	0	0
$460$	−6112.17	−0.619525
$461$	−13819.5	−1.39618	−0.698090	−	0.716010i	$-0.745966\pi$
$461$	−13819.5	−1.39618	−0.698090	+	0.716010i	$0.745966\pi$
$462$	0	0
$463$	16845.0	1.69083	0.845415	−	0.534110i	$-0.179353\pi$
$463$	16845.0	1.69083	0.845415	+	0.534110i	$0.179353\pi$
$464$	−1053.42	−0.105396
$465$	0	0
$466$	18726.7	1.86158
$467$	8573.64	0.849552	0.424776	−	0.905299i	$-0.360353\pi$
$467$	8573.64	0.849552	0.424776	+	0.905299i	$0.360353\pi$
$468$	0	0
$469$	2171.07	0.213754
$470$	−551.114	−0.0540873
$471$	0	0
$472$	3970.96	0.387242
$473$	10207.9	0.992306
$474$	0	0
$475$	1045.16	0.100958
$476$	−2874.84	−0.276823
$477$	0	0
$478$	13506.9	1.29245
$479$	297.354	0.0283642	0.0141821	−	0.999899i	$-0.495486\pi$
$479$	297.354	0.0283642	0.0141821	+	0.999899i	$0.495486\pi$
$480$	0	0
$481$	−9802.00	−0.929174
$482$	−8361.24	−0.790133
$483$	0	0
$484$	−11215.8	−1.05332
$485$	2654.40	0.248516
$486$	0	0
$487$	−6827.58	−0.635292	−0.317646	−	0.948209i	$-0.602892\pi$
$487$	−6827.58	−0.635292	−0.317646	+	0.948209i	$0.602892\pi$
$488$	−15400.7	−1.42860
$489$	0	0
$490$	−7619.01	−0.702432
$491$	−5170.75	−0.475260	−0.237630	−	0.971356i	$-0.576371\pi$
$491$	−5170.75	−0.475260	−0.237630	+	0.971356i	$0.576371\pi$
$492$	0	0
$493$	13636.5	1.24575
$494$	11553.2	1.05224
$495$	0	0
$496$	−639.790	−0.0579181
$497$	−2817.68	−0.254307
$498$	0	0
$499$	−10754.0	−0.964757	−0.482379	−	0.875963i	$-0.660227\pi$
$499$	−10754.0	−0.964757	−0.482379	+	0.875963i	$0.660227\pi$
$500$	1570.14	0.140438
$501$	0	0
$502$	−10704.6	−0.951730
$503$	−2141.00	−0.189786	−0.0948930	−	0.995487i	$-0.530251\pi$
$503$	−2141.00	−0.189786	−0.0948930	+	0.995487i	$0.530251\pi$
$504$	0	0
$505$	4992.31	0.439911
$506$	−9236.55	−0.811492
$507$	0	0
$508$	−4665.74	−0.407498
$509$	−6060.50	−0.527754	−0.263877	−	0.964556i	$-0.585001\pi$
$509$	−6060.50	−0.527754	−0.263877	+	0.964556i	$0.585001\pi$
$510$	0	0
$511$	349.071	0.0302192
$512$	2423.30	0.209172
$513$	0	0
$514$	11212.0	0.962143
$515$	−4187.36	−0.358286
$516$	0	0
$517$	−508.790	−0.0432815
$518$	1922.54	0.163072
$519$	0	0
$520$	6302.41	0.531498
$521$	−8685.61	−0.730371	−0.365186	−	0.930935i	$-0.618994\pi$
$521$	−8685.61	−0.730371	−0.365186	+	0.930935i	$0.618994\pi$
$522$	0	0
$523$	−20944.4	−1.75112	−0.875561	−	0.483108i	$-0.839508\pi$
$523$	−20944.4	−1.75112	−0.875561	+	0.483108i	$0.839508\pi$
$524$	−3332.05	−0.277789
$525$	0	0
$526$	−9826.91	−0.814589
$527$	8282.07	0.684578
$528$	0	0
$529$	−2696.09	−0.221590
$530$	−16093.4	−1.31896
$531$	0	0
$532$	−1384.35	−0.112818
$533$	−14211.7	−1.15493
$534$	0	0
$535$	−5546.96	−0.448254
$536$	−17033.2	−1.37262
$537$	0	0
$538$	−19710.3	−1.57950
$539$	−7033.88	−0.562098
$540$	0	0
$541$	−17178.0	−1.36513	−0.682567	−	0.730823i	$-0.739137\pi$
$541$	−17178.0	−1.36513	−0.682567	+	0.730823i	$0.739137\pi$
$542$	−17790.7	−1.40992
$543$	0	0
$544$	−17004.9	−1.34022
$545$	−5358.64	−0.421172
$546$	0	0
$547$	−22293.5	−1.74260	−0.871300	−	0.490751i	$-0.836722\pi$
$547$	−22293.5	−1.74260	−0.871300	+	0.490751i	$0.836722\pi$
$548$	3400.57	0.265082
$549$	0	0
$550$	2372.76	0.183954
$551$	6566.52	0.507701
$552$	0	0
$553$	1857.39	0.142829
$554$	27369.4	2.09894
$555$	0	0
$556$	−15225.3	−1.16132
$557$	−10050.0	−0.764512	−0.382256	−	0.924056i	$-0.624853\pi$
$557$	−10050.0	−0.764512	−0.382256	+	0.924056i	$0.624853\pi$
$558$	0	0
$559$	29722.5	2.24889
$560$	88.3999	0.00667068
$561$	0	0
$562$	−10415.2	−0.781745
$563$	2469.40	0.184854	0.0924269	−	0.995719i	$-0.470538\pi$
$563$	2469.40	0.184854	0.0924269	+	0.995719i	$0.470538\pi$
$564$	0	0
$565$	5447.49	0.405624
$566$	−24924.1	−1.85095
$567$	0	0
$568$	22106.3	1.63303
$569$	−1734.73	−0.127810	−0.0639048	−	0.997956i	$-0.520355\pi$
$569$	−1734.73	−0.127810	−0.0639048	+	0.997956i	$0.520355\pi$
$570$	0	0
$571$	5164.53	0.378509	0.189255	−	0.981928i	$-0.439393\pi$
$571$	5164.53	0.378509	0.189255	+	0.981928i	$0.439393\pi$
$572$	16023.5	1.17129
$573$	0	0
$574$	2787.45	0.202693
$575$	−2432.96	−0.176455
$576$	0	0
$577$	−3796.17	−0.273894	−0.136947	−	0.990578i	$-0.543729\pi$
$577$	−3796.17	−0.273894	−0.136947	+	0.990578i	$0.543729\pi$
$578$	11899.8	0.856346
$579$	0	0
$580$	9864.90	0.706238
$581$	−3742.75	−0.267256
$582$	0	0
$583$	−14857.4	−1.05546
$584$	−2738.66	−0.194052
$585$	0	0
$586$	18568.5	1.30897
$587$	13603.2	0.956501	0.478250	−	0.878224i	$-0.341271\pi$
$587$	13603.2	0.956501	0.478250	+	0.878224i	$0.341271\pi$
$588$	0	0
$589$	3988.16	0.278997
$590$	4353.02	0.303747
$591$	0	0
$592$	1078.66	0.0748859
$593$	−82.8252	−0.00573562	−0.00286781	−	0.999996i	$-0.500913\pi$
$593$	−82.8252	−0.00573562	−0.00286781	+	0.999996i	$0.500913\pi$
$594$	0	0
$595$	−1144.34	−0.0788458
$596$	−30314.3	−2.08342
$597$	0	0
$598$	−26894.2	−1.83910
$599$	1131.04	0.0771504	0.0385752	−	0.999256i	$-0.487718\pi$
$599$	1131.04	0.0771504	0.0385752	+	0.999256i	$0.487718\pi$
$600$	0	0
$601$	−2722.67	−0.184792	−0.0923960	−	0.995722i	$-0.529453\pi$
$601$	−2722.67	−0.184792	−0.0923960	+	0.995722i	$0.529453\pi$
$602$	−5829.70	−0.394686
$603$	0	0
$604$	−14686.4	−0.989370
$605$	−4464.46	−0.300010
$606$	0	0
$607$	11017.7	0.736732	0.368366	−	0.929681i	$-0.379917\pi$
$607$	11017.7	0.736732	0.368366	+	0.929681i	$0.379917\pi$
$608$	−8188.56	−0.546201
$609$	0	0
$610$	−16882.5	−1.12058
$611$	−1481.45	−0.0980900
$612$	0	0
$613$	−12198.6	−0.803750	−0.401875	−	0.915695i	$-0.631641\pi$
$613$	−12198.6	−0.803750	−0.401875	+	0.915695i	$0.631641\pi$
$614$	34171.1	2.24598
$615$	0	0
$616$	−1141.21	−0.0746437
$617$	−894.243	−0.0583482	−0.0291741	−	0.999574i	$-0.509288\pi$
$617$	−894.243	−0.0583482	−0.0291741	+	0.999574i	$0.509288\pi$
$618$	0	0
$619$	−27671.1	−1.79676	−0.898381	−	0.439218i	$-0.855256\pi$
$619$	−27671.1	−1.79676	−0.898381	+	0.439218i	$0.855256\pi$
$620$	5991.42	0.388099
$621$	0	0
$622$	28707.1	1.85056
$623$	1059.34	0.0681247
$624$	0	0
$625$	625.000	0.0400000
$626$	40728.2	2.60036
$627$	0	0
$628$	−36782.0	−2.33720
$629$	−13963.2	−0.885133
$630$	0	0
$631$	−4377.81	−0.276193	−0.138097	−	0.990419i	$-0.544098\pi$
$631$	−4377.81	−0.276193	−0.138097	+	0.990419i	$0.544098\pi$
$632$	−14572.2	−0.917172
$633$	0	0
$634$	14675.6	0.919312
$635$	−1857.21	−0.116065
$636$	0	0
$637$	−20480.6	−1.27390
$638$	14907.6	0.925074
$639$	0	0
$640$	−11085.3	−0.684662
$641$	24915.6	1.53527	0.767633	−	0.640890i	$-0.221435\pi$
$641$	24915.6	1.53527	0.767633	+	0.640890i	$0.221435\pi$
$642$	0	0
$643$	10773.0	0.660727	0.330363	−	0.943854i	$-0.392829\pi$
$643$	10773.0	0.660727	0.330363	+	0.943854i	$0.392829\pi$
$644$	3222.56	0.197184
$645$	0	0
$646$	16457.9	1.00236
$647$	−19872.6	−1.20753	−0.603766	−	0.797162i	$-0.706334\pi$
$647$	−19872.6	−1.20753	−0.603766	+	0.797162i	$0.706334\pi$
$648$	0	0
$649$	4018.71	0.243064
$650$	6908.79	0.416900
$651$	0	0
$652$	−11759.9	−0.706367
$653$	4132.19	0.247634	0.123817	−	0.992305i	$-0.460486\pi$
$653$	4132.19	0.247634	0.123817	+	0.992305i	$0.460486\pi$
$654$	0	0
$655$	−1326.33	−0.0791207
$656$	1563.92	0.0930807
$657$	0	0
$658$	290.567	0.0172150
$659$	−23260.1	−1.37494	−0.687470	−	0.726213i	$-0.741279\pi$
$659$	−23260.1	−1.37494	−0.687470	+	0.726213i	$0.741279\pi$
$660$	0	0
$661$	11316.5	0.665900	0.332950	−	0.942945i	$-0.391956\pi$
$661$	11316.5	0.665900	0.332950	+	0.942945i	$0.391956\pi$
$662$	29998.0	1.76119
$663$	0	0
$664$	29363.9	1.71618
$665$	−551.045	−0.0321332
$666$	0	0
$667$	−15285.8	−0.887361
$668$	−39354.7	−2.27946
$669$	0	0
$670$	−18672.1	−1.07666
$671$	−15585.9	−0.896704
$672$	0	0
$673$	26072.2	1.49332	0.746662	−	0.665203i	$-0.231655\pi$
$673$	26072.2	1.49332	0.746662	+	0.665203i	$0.231655\pi$
$674$	9197.16	0.525611
$675$	0	0
$676$	19059.0	1.08438
$677$	15251.9	0.865848	0.432924	−	0.901430i	$-0.357482\pi$
$677$	15251.9	0.865848	0.432924	+	0.901430i	$0.357482\pi$
$678$	0	0
$679$	−1399.50	−0.0790983
$680$	8977.94	0.506307
$681$	0	0
$682$	9054.09	0.508356
$683$	33196.1	1.85976	0.929878	−	0.367867i	$-0.119912\pi$
$683$	33196.1	1.85976	0.929878	+	0.367867i	$0.119912\pi$
$684$	0	0
$685$	1353.61	0.0755016
$686$	8117.10	0.451767
$687$	0	0
$688$	−3270.80	−0.181247
$689$	−43260.5	−2.39201
$690$	0	0
$691$	−31327.4	−1.72467	−0.862337	−	0.506334i	$-0.831000\pi$
$691$	−31327.4	−1.72467	−0.862337	+	0.506334i	$0.831000\pi$
$692$	−3005.45	−0.165101
$693$	0	0
$694$	14616.2	0.799459
$695$	−6060.45	−0.330771
$696$	0	0
$697$	−20245.0	−1.10019
$698$	−43659.9	−2.36755
$699$	0	0
$700$	−827.837	−0.0446990
$701$	−21416.8	−1.15392	−0.576962	−	0.816771i	$-0.695762\pi$
$701$	−21416.8	−1.15392	−0.576962	+	0.816771i	$0.695762\pi$
$702$	0	0
$703$	−6723.85	−0.360732
$704$	−17467.0	−0.935104
$705$	0	0
$706$	−10666.4	−0.568607
$707$	−2632.13	−0.140016
$708$	0	0
$709$	6678.58	0.353765	0.176883	−	0.984232i	$-0.443399\pi$
$709$	6678.58	0.353765	0.176883	+	0.984232i	$0.443399\pi$
$710$	24233.2	1.28092
$711$	0	0
$712$	−8311.12	−0.437461
$713$	−9283.81	−0.487632
$714$	0	0
$715$	6378.21	0.333611
$716$	−4170.68	−0.217689
$717$	0	0
$718$	29310.7	1.52349
$719$	−25245.9	−1.30947	−0.654737	−	0.755857i	$-0.727221\pi$
$719$	−25245.9	−1.30947	−0.654737	+	0.755857i	$0.727221\pi$
$720$	0	0
$721$	2207.73	0.114036
$722$	−23176.6	−1.19466
$723$	0	0
$724$	42777.5	2.19587
$725$	3926.75	0.201153
$726$	0	0
$727$	35186.7	1.79505	0.897527	−	0.440960i	$-0.145362\pi$
$727$	35186.7	1.79505	0.897527	+	0.440960i	$0.145362\pi$
$728$	−3322.86	−0.169167
$729$	0	0
$730$	−3002.15	−0.152212
$731$	42340.5	2.14230
$732$	0	0
$733$	−17068.6	−0.860087	−0.430043	−	0.902808i	$-0.641502\pi$
$733$	−17068.6	−0.860087	−0.430043	+	0.902808i	$0.641502\pi$
$734$	−41520.7	−2.08795
$735$	0	0
$736$	19061.7	0.954652
$737$	−17238.1	−0.861565
$738$	0	0
$739$	−38144.8	−1.89875	−0.949376	−	0.314141i	$-0.898283\pi$
$739$	−38144.8	−1.89875	−0.949376	+	0.314141i	$0.898283\pi$
$740$	−10101.3	−0.501797
$741$	0	0
$742$	8485.01	0.419804
$743$	5147.73	0.254175	0.127087	−	0.991892i	$-0.459437\pi$
$743$	5147.73	0.254175	0.127087	+	0.991892i	$0.459437\pi$
$744$	0	0
$745$	−12066.7	−0.593408
$746$	35956.5	1.76469
$747$	0	0
$748$	22825.9	1.11577
$749$	2924.56	0.142672
$750$	0	0
$751$	23597.8	1.14660	0.573300	−	0.819346i	$-0.305663\pi$
$751$	23597.8	1.14660	0.573300	+	0.819346i	$0.305663\pi$
$752$	163.025	0.00790547
$753$	0	0
$754$	43406.6	2.09652
$755$	−5845.94	−0.281796
$756$	0	0
$757$	1343.38	0.0644995	0.0322497	−	0.999480i	$-0.489733\pi$
$757$	1343.38	0.0644995	0.0322497	+	0.999480i	$0.489733\pi$
$758$	42180.5	2.02119
$759$	0	0
$760$	4323.25	0.206343
$761$	8016.38	0.381858	0.190929	−	0.981604i	$-0.438850\pi$
$761$	8016.38	0.381858	0.190929	+	0.981604i	$0.438850\pi$
$762$	0	0
$763$	2825.27	0.134052
$764$	66093.3	3.12980
$765$	0	0
$766$	47320.8	2.23208
$767$	11701.3	0.550861
$768$	0	0
$769$	−23044.0	−1.08061	−0.540305	−	0.841469i	$-0.681691\pi$
$769$	−23044.0	−1.08061	−0.540305	+	0.841469i	$0.681691\pi$
$770$	−1251.01	−0.0585495
$771$	0	0
$772$	5137.18	0.239496
$773$	31884.9	1.48359	0.741797	−	0.670624i	$-0.233974\pi$
$773$	31884.9	1.48359	0.741797	+	0.670624i	$0.233974\pi$
$774$	0	0
$775$	2384.90	0.110540
$776$	10979.8	0.507928
$777$	0	0
$778$	−15239.4	−0.702261
$779$	−9748.78	−0.448378
$780$	0	0
$781$	22372.1	1.02502
$782$	−38311.4	−1.75194
$783$	0	0
$784$	2253.78	0.102668
$785$	−14641.2	−0.665688
$786$	0	0
$787$	17033.0	0.771488	0.385744	−	0.922606i	$-0.373945\pi$
$787$	17033.0	0.771488	0.385744	+	0.922606i	$0.373945\pi$
$788$	11894.5	0.537720
$789$	0	0
$790$	−15974.3	−0.719418
$791$	−2872.11	−0.129103
$792$	0	0
$793$	−45381.7	−2.03222
$794$	−6006.59	−0.268471
$795$	0	0
$796$	61764.0	2.75021
$797$	−41750.0	−1.85553	−0.927767	−	0.373159i	$-0.878275\pi$
$797$	−41750.0	−1.85553	−0.927767	+	0.373159i	$0.878275\pi$
$798$	0	0
$799$	−2110.36	−0.0934408
$800$	−4896.73	−0.216407
$801$	0	0
$802$	−63531.2	−2.79721
$803$	−2771.59	−0.121802
$804$	0	0
$805$	1282.75	0.0561626
$806$	26362.9	1.15210
$807$	0	0
$808$	20650.5	0.899111
$809$	−163.059	−0.00708636	−0.00354318	−	0.999994i	$-0.501128\pi$
$809$	−163.059	−0.00708636	−0.00354318	+	0.999994i	$0.501128\pi$
$810$	0	0
$811$	9958.10	0.431167	0.215583	−	0.976485i	$-0.430835\pi$
$811$	9958.10	0.431167	0.215583	+	0.976485i	$0.430835\pi$
$812$	−5201.14	−0.224783
$813$	0	0
$814$	−15264.8	−0.657285
$815$	−4681.04	−0.201190
$816$	0	0
$817$	20388.7	0.873083
$818$	16396.6	0.700848
$819$	0	0
$820$	−14645.6	−0.623717
$821$	32273.2	1.37191	0.685957	−	0.727642i	$-0.259384\pi$
$821$	32273.2	1.37191	0.685957	+	0.727642i	$0.259384\pi$
$822$	0	0
$823$	32710.6	1.38544	0.692722	−	0.721205i	$-0.256411\pi$
$823$	32710.6	1.38544	0.692722	+	0.721205i	$0.256411\pi$
$824$	−17320.8	−0.732282
$825$	0	0
$826$	−2295.07	−0.0966776
$827$	9763.68	0.410540	0.205270	−	0.978705i	$-0.434193\pi$
$827$	9763.68	0.410540	0.205270	+	0.978705i	$0.434193\pi$
$828$	0	0
$829$	34886.9	1.46160	0.730802	−	0.682589i	$-0.239146\pi$
$829$	34886.9	1.46160	0.730802	+	0.682589i	$0.239146\pi$
$830$	32189.2	1.34615
$831$	0	0
$832$	−50858.9	−2.11925
$833$	−29175.2	−1.21352
$834$	0	0
$835$	−15665.2	−0.649243
$836$	10991.6	0.454728
$837$	0	0
$838$	−3700.40	−0.152540
$839$	25642.1	1.05514	0.527570	−	0.849512i	$-0.323103\pi$
$839$	25642.1	1.05514	0.527570	+	0.849512i	$0.323103\pi$
$840$	0	0
$841$	282.003	0.0115627
$842$	67861.1	2.77749
$843$	0	0
$844$	30767.2	1.25480
$845$	7586.50	0.308856
$846$	0	0
$847$	2353.83	0.0954881
$848$	4760.58	0.192782
$849$	0	0
$850$	9841.75	0.397140
$851$	15652.1	0.630489
$852$	0	0
$853$	−4502.72	−0.180739	−0.0903694	−	0.995908i	$-0.528805\pi$
$853$	−4502.72	−0.180739	−0.0903694	+	0.995908i	$0.528805\pi$
$854$	8901.05	0.356660
$855$	0	0
$856$	−22944.8	−0.916163
$857$	−27439.1	−1.09370	−0.546851	−	0.837230i	$-0.684174\pi$
$857$	−27439.1	−1.09370	−0.546851	+	0.837230i	$0.684174\pi$
$858$	0	0
$859$	26746.7	1.06238	0.531190	−	0.847253i	$-0.321745\pi$
$859$	26746.7	1.06238	0.531190	+	0.847253i	$0.321745\pi$
$860$	30630.0	1.21450
$861$	0	0
$862$	44221.2	1.74731
$863$	4198.80	0.165618	0.0828092	−	0.996565i	$-0.473611\pi$
$863$	4198.80	0.165618	0.0828092	+	0.996565i	$0.473611\pi$
$864$	0	0
$865$	−1196.33	−0.0470246
$866$	20613.7	0.808870
$867$	0	0
$868$	−3158.90	−0.123525
$869$	−14747.5	−0.575690
$870$	0	0
$871$	−50192.3	−1.95259
$872$	−22165.8	−0.860811
$873$	0	0
$874$	−18448.5	−0.713993
$875$	−329.523	−0.0127313
$876$	0	0
$877$	−19404.6	−0.747147	−0.373574	−	0.927601i	$-0.621868\pi$
$877$	−19404.6	−0.747147	−0.373574	+	0.927601i	$0.621868\pi$
$878$	−39080.7	−1.50217
$879$	0	0
$880$	−701.887	−0.0268870
$881$	8562.17	0.327431	0.163716	−	0.986508i	$-0.447652\pi$
$881$	8562.17	0.327431	0.163716	+	0.986508i	$0.447652\pi$
$882$	0	0
$883$	44660.0	1.70207	0.851036	−	0.525107i	$-0.175975\pi$
$883$	44660.0	1.70207	0.851036	+	0.525107i	$0.175975\pi$
$884$	66462.5	2.52871
$885$	0	0
$886$	55636.9	2.10966
$887$	−8179.87	−0.309643	−0.154821	−	0.987942i	$-0.549480\pi$
$887$	−8179.87	−0.309643	−0.154821	+	0.987942i	$0.549480\pi$
$888$	0	0
$889$	979.189	0.0369415
$890$	−9110.77	−0.343139
$891$	0	0
$892$	−12642.1	−0.474539
$893$	−1016.22	−0.0380814
$894$	0	0
$895$	−1660.15	−0.0620030
$896$	5844.56	0.217916
$897$	0	0
$898$	−34542.8	−1.28364
$899$	14983.9	0.555884
$900$	0	0
$901$	−61625.7	−2.27863
$902$	−22132.1	−0.816983
$903$	0	0
$904$	22533.3	0.829033
$905$	17027.7	0.625436
$906$	0	0
$907$	−33509.0	−1.22673	−0.613366	−	0.789798i	$-0.710185\pi$
$907$	−33509.0	−1.22673	−0.613366	+	0.789798i	$0.710185\pi$
$908$	25424.3	0.929222
$909$	0	0
$910$	−3642.57	−0.132692
$911$	−44665.2	−1.62440	−0.812198	−	0.583382i	$-0.801729\pi$
$911$	−44665.2	−1.62440	−0.812198	+	0.583382i	$0.801729\pi$
$912$	0	0
$913$	29717.1	1.07721
$914$	27488.0	0.994773
$915$	0	0
$916$	−85264.1	−3.07555
$917$	699.290	0.0251828
$918$	0	0
$919$	13156.3	0.472237	0.236118	−	0.971724i	$-0.424125\pi$
$919$	13156.3	0.472237	0.236118	+	0.971724i	$0.424125\pi$
$920$	−10063.9	−0.360647
$921$	0	0
$922$	−62663.7	−2.23831
$923$	65141.2	2.32302
$924$	0	0
$925$	−4020.83	−0.142923
$926$	76382.7	2.71068
$927$	0	0
$928$	−30765.2	−1.08827
$929$	−37667.8	−1.33029	−0.665146	−	0.746714i	$-0.731631\pi$
$929$	−37667.8	−1.33029	−0.665146	+	0.746714i	$0.731631\pi$
$930$	0	0
$931$	−14049.0	−0.494563
$932$	51876.0	1.82323
$933$	0	0
$934$	38876.7	1.36197
$935$	9085.92	0.317798
$936$	0	0
$937$	−37999.6	−1.32486	−0.662429	−	0.749125i	$-0.730474\pi$
$937$	−37999.6	−1.32486	−0.662429	+	0.749125i	$0.730474\pi$
$938$	9844.60	0.342684
$939$	0	0
$940$	−1526.68	−0.0529731
$941$	−10238.6	−0.354697	−0.177348	−	0.984148i	$-0.556752\pi$
$941$	−10238.6	−0.354697	−0.177348	+	0.984148i	$0.556752\pi$
$942$	0	0
$943$	22693.7	0.783677
$944$	−1287.67	−0.0443962
$945$	0	0
$946$	46287.2	1.59083
$947$	10377.2	0.356088	0.178044	−	0.984023i	$-0.443023\pi$
$947$	10377.2	0.356088	0.178044	+	0.984023i	$0.443023\pi$
$948$	0	0
$949$	−8070.07	−0.276044
$950$	4739.20	0.161853
$951$	0	0
$952$	−4733.50	−0.161149
$953$	−54309.6	−1.84602	−0.923012	−	0.384772i	$-0.874280\pi$
$953$	−54309.6	−1.84602	−0.923012	+	0.384772i	$0.874280\pi$
$954$	0	0
$955$	26308.6	0.891441
$956$	37416.3	1.26583
$957$	0	0
$958$	1348.33	0.0454725
$959$	−713.670	−0.0240309
$960$	0	0
$961$	−20690.6	−0.694525
$962$	−44446.6	−1.48962
$963$	0	0
$964$	−23162.0	−0.773858
$965$	2044.87	0.0682141
$966$	0	0
$967$	−12701.3	−0.422386	−0.211193	−	0.977444i	$-0.567735\pi$
$967$	−12701.3	−0.422386	−0.211193	+	0.977444i	$0.567735\pi$
$968$	−18467.0	−0.613175
$969$	0	0
$970$	12036.2	0.398412
$971$	3767.34	0.124511	0.0622553	−	0.998060i	$-0.480171\pi$
$971$	3767.34	0.124511	0.0622553	+	0.998060i	$0.480171\pi$
$972$	0	0
$973$	3195.29	0.105279
$974$	−30959.2	−1.01848
$975$	0	0
$976$	4994.01	0.163785
$977$	13359.9	0.437482	0.218741	−	0.975783i	$-0.429805\pi$
$977$	13359.9	0.437482	0.218741	+	0.975783i	$0.429805\pi$
$978$	0	0
$979$	−8411.07	−0.274585
$980$	−21105.9	−0.687963
$981$	0	0
$982$	−23446.5	−0.761921
$983$	−31707.1	−1.02879	−0.514394	−	0.857554i	$-0.671983\pi$
$983$	−31707.1	−1.02879	−0.514394	+	0.857554i	$0.671983\pi$
$984$	0	0
$985$	4734.63	0.153155
$986$	61833.8	1.99715
$987$	0	0
$988$	32004.4	1.03056
$989$	−47461.7	−1.52598
$990$	0	0
$991$	−19393.7	−0.621657	−0.310829	−	0.950466i	$-0.600606\pi$
$991$	−19393.7	−0.621657	−0.310829	+	0.950466i	$0.600606\pi$
$992$	−18685.2	−0.598039
$993$	0	0
$994$	−12776.6	−0.407696
$995$	24585.3	0.783323
$996$	0	0
$997$	41543.1	1.31964	0.659821	−	0.751423i	$-0.270632\pi$
$997$	41543.1	1.31964	0.659821	+	0.751423i	$0.270632\pi$
$998$	−48763.3	−1.54667
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.4.a.l.1.5	yes	6
3.2	odd	2		405.4.a.k.1.2	✓	6
5.4	even	2		2025.4.a.y.1.2		6
9.2	odd	6		405.4.e.x.271.5		12
9.4	even	3		405.4.e.w.136.2		12
9.5	odd	6		405.4.e.x.136.5		12
9.7	even	3		405.4.e.w.271.2		12
15.14	odd	2		2025.4.a.z.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
405.4.a.k.1.2	✓	6	3.2	odd	2
405.4.a.l.1.5	yes	6	1.1	even	1	trivial
405.4.e.w.136.2		12	9.4	even	3
405.4.e.w.271.2		12	9.7	even	3
405.4.e.x.136.5		12	9.5	odd	6
405.4.e.x.271.5		12	9.2	odd	6
2025.4.a.y.1.2		6	5.4	even	2
2025.4.a.z.1.5		6	15.14	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+4.53444 q^{2} +12.5612 q^{4} +5.00000 q^{5} -2.63618 q^{7} +20.6823 q^{8} +O(q^{10})\)	\(q+4.53444 q^{2} +12.5612 q^{4} +5.00000 q^{5} -2.63618 q^{7} +20.6823 q^{8} +22.6722 q^{10} +20.9310 q^{11} +60.9451 q^{13} -11.9536 q^{14} -6.70667 q^{16} +86.8178 q^{17} +41.8063 q^{19} +62.8058 q^{20} +94.9105 q^{22} -97.3186 q^{23} +25.0000 q^{25} +276.352 q^{26} -33.1135 q^{28} +157.070 q^{29} +95.3961 q^{31} -195.869 q^{32} +393.670 q^{34} -13.1809 q^{35} -160.833 q^{37} +189.568 q^{38} +103.411 q^{40} -233.189 q^{41} +487.694 q^{43} +262.918 q^{44} -441.285 q^{46} -24.3079 q^{47} -336.051 q^{49} +113.361 q^{50} +765.540 q^{52} -709.828 q^{53} +104.655 q^{55} -54.5222 q^{56} +712.225 q^{58} +191.998 q^{59} -744.633 q^{61} +432.568 q^{62} -834.504 q^{64} +304.725 q^{65} -823.567 q^{67} +1090.53 q^{68} -59.7680 q^{70} +1068.85 q^{71} -132.416 q^{73} -729.289 q^{74} +525.135 q^{76} -55.1780 q^{77} -704.577 q^{79} -33.5333 q^{80} -1057.38 q^{82} +1419.76 q^{83} +434.089 q^{85} +2211.42 q^{86} +432.901 q^{88} -401.847 q^{89} -160.662 q^{91} -1222.43 q^{92} -110.223 q^{94} +209.031 q^{95} +530.880 q^{97} -1523.80 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{2} + 34 q^{4} + 30 q^{5} + 40 q^{7} + 66 q^{8}+O(q^{10}) \) 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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