LMFDB - Embedded newform 405.4.a.k.1.2

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			1.2
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	$-0.796009\pi$
$2$	−4.53444	−1.60317	−0.801583	+	0.597883i	$0.796009\pi$
$3$	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	$-0.592612\pi$
$11$	−20.9310	−0.573722	−0.286861	+	0.957972i	$0.592612\pi$
$12$	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	$-0.712586\pi$
$17$	−86.8178	−1.23861	−0.619306	+	0.785150i	$0.712586\pi$
$18$	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	$-0.354575\pi$
$23$	97.3186	0.882275	0.441138	+	0.897439i	$0.354575\pi$
$24$	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	$-0.667727\pi$
$29$	−157.070	−1.00576	−0.502882	+	0.864355i	$0.667727\pi$
$30$	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	$-0.353515\pi$
$41$	233.189	0.888245	0.444123	+	0.895966i	$0.353515\pi$
$42$	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	$-0.487991\pi$
$47$	24.3079	0.0754399	0.0377200	+	0.999288i	$0.487991\pi$
$48$	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	$-0.128323\pi$
$53$	709.828	1.83967	0.919834	+	0.392308i	$0.128323\pi$
$54$	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	$-0.567943\pi$
$59$	−191.998	−0.423661	−0.211831	+	0.977306i	$0.567943\pi$
$60$	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	$-0.851619\pi$
$71$	−1068.85	−1.78661	−0.893304	+	0.449452i	$0.851619\pi$
$72$	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	$-0.888052\pi$
$83$	−1419.76	−1.87758	−0.938790	+	0.344489i	$0.888052\pi$
$84$	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	$-0.423081\pi$
$89$	401.847	0.478604	0.239302	+	0.970945i	$0.423081\pi$
$90$	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	$-0.663674\pi$
$101$	−998.463	−0.983671	−0.491835	+	0.870688i	$0.663674\pi$
$102$	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	$-0.332906\pi$
$107$	1109.39	1.00233	0.501163	+	0.865353i	$0.332906\pi$
$108$	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	$-0.649825\pi$
$113$	−1089.50	−0.907003	−0.453501	+	0.891256i	$0.649825\pi$
$114$	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	$-0.471806\pi$
$131$	265.266	0.176919	0.0884597	+	0.996080i	$0.471806\pi$
$132$	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	$-0.526902\pi$
$137$	−270.721	−0.168827	−0.0844134	+	0.996431i	$0.526902\pi$
$138$	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	$-0.269092\pi$
$149$	2413.33	1.32690	0.663450	+	0.748221i	$0.269092\pi$
$150$	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	$-0.241435\pi$
$167$	3133.05	1.45175	0.725876	+	0.687826i	$0.241435\pi$
$168$	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	$-0.483257\pi$
$173$	239.265	0.105150	0.0525751	+	0.998617i	$0.483257\pi$
$174$	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	$-0.477917\pi$
$179$	332.030	0.138643	0.0693215	+	0.997594i	$0.477917\pi$
$180$	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	$-0.973982\pi$
$191$	−5261.72	−1.99332	−0.996661	+	0.0816469i	$0.973982\pi$
$192$	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	$-0.554775\pi$
$197$	−946.927	−0.342466	−0.171233	+	0.985231i	$0.554775\pi$
$198$	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	$-0.595621\pi$
$227$	−2024.04	−0.591807	−0.295903	+	0.955218i	$0.595621\pi$
$228$	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	$-0.697180\pi$
$233$	−4129.88	−1.16119	−0.580595	+	0.814193i	$0.697180\pi$
$234$	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	$-0.632065\pi$
$239$	−2978.73	−0.806185	−0.403093	+	0.915159i	$0.632065\pi$
$240$	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	$-0.404071\pi$
$251$	2360.72	0.593656	0.296828	+	0.954931i	$0.404071\pi$
$252$	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	$-0.597012\pi$
$257$	−2472.64	−0.600151	−0.300076	+	0.953915i	$0.597012\pi$
$258$	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	$-0.418235\pi$
$263$	2167.17	0.508112	0.254056	+	0.967189i	$0.418235\pi$
$264$	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	$-0.336040\pi$
$269$	4346.79	0.985236	0.492618	+	0.870246i	$0.336040\pi$
$270$	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	$-0.421602\pi$
$281$	2296.92	0.487626	0.243813	+	0.969822i	$0.421602\pi$
$282$	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	$-0.633859\pi$
$293$	−4094.98	−0.816490	−0.408245	+	0.912872i	$0.633859\pi$
$294$	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	$-0.695838\pi$
$311$	−6330.89	−1.15432	−0.577158	+	0.816633i	$0.695838\pi$
$312$	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	$-0.592564\pi$
$317$	−3236.48	−0.573435	−0.286718	+	0.958015i	$0.592564\pi$
$318$	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	$-0.580213\pi$
$347$	−3223.38	−0.498675	−0.249337	+	0.968417i	$0.580213\pi$
$348$	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	$-0.443251\pi$
$353$	2352.31	0.354677	0.177339	+	0.984150i	$0.443251\pi$
$354$	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	$-0.657606\pi$
$359$	−6464.01	−0.950299	−0.475150	+	0.879905i	$0.657606\pi$
$360$	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	$-0.745104\pi$
$383$	−10435.9	−1.39229	−0.696146	+	0.717900i	$0.745104\pi$
$384$	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	$-0.429713\pi$
$389$	3360.81	0.438046	0.219023	+	0.975720i	$0.429713\pi$
$390$	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	$-0.162561\pi$
$401$	14010.8	1.74480	0.872402	+	0.488789i	$0.162561\pi$
$402$	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	$-0.484851\pi$
$419$	816.065	0.0951489	0.0475745	+	0.998868i	$0.484851\pi$
$420$	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	$-0.683453\pi$
$431$	−9752.30	−1.08991	−0.544955	+	0.838465i	$0.683453\pi$
$432$	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	$-0.728583\pi$
$443$	−12269.8	−1.31593	−0.657966	+	0.753048i	$0.728583\pi$
$444$	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	$-0.368890\pi$
$449$	7617.88	0.800690	0.400345	+	0.916364i	$0.368890\pi$
$450$	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	$-0.254034\pi$
$461$	13819.5	1.39618	0.698090	+	0.716010i	$0.254034\pi$
$462$	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	$-0.639647\pi$
$467$	−8573.64	−0.849552	−0.424776	+	0.905299i	$0.639647\pi$
$468$	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	$-0.504514\pi$
$479$	−297.354	−0.0283642	−0.0141821	+	0.999899i	$0.504514\pi$
$480$	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	$-0.423629\pi$
$491$	5170.75	0.475260	0.237630	+	0.971356i	$0.423629\pi$
$492$	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	$-0.469749\pi$
$503$	2141.00	0.189786	0.0948930	+	0.995487i	$0.469749\pi$
$504$	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	$-0.414999\pi$
$509$	6060.50	0.527754	0.263877	+	0.964556i	$0.414999\pi$
$510$	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	$-0.381006\pi$
$521$	8685.61	0.730371	0.365186	+	0.930935i	$0.381006\pi$
$522$	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	$-0.375147\pi$
$557$	10050.0	0.764512	0.382256	+	0.924056i	$0.375147\pi$
$558$	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	$-0.529462\pi$
$563$	−2469.40	−0.184854	−0.0924269	+	0.995719i	$0.529462\pi$
$564$	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	$-0.479645\pi$
$569$	1734.73	0.127810	0.0639048	+	0.997956i	$0.479645\pi$
$570$	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	$-0.658729\pi$
$587$	−13603.2	−0.956501	−0.478250	+	0.878224i	$0.658729\pi$
$588$	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	$-0.499087\pi$
$593$	82.8252	0.00573562	0.00286781	+	0.999996i	$0.499087\pi$
$594$	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	$-0.512282\pi$
$599$	−1131.04	−0.0771504	−0.0385752	+	0.999256i	$0.512282\pi$
$600$	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	$-0.490712\pi$
$617$	894.243	0.0583482	0.0291741	+	0.999574i	$0.490712\pi$
$618$	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	$-0.778565\pi$
$641$	−24915.6	−1.53527	−0.767633	+	0.640890i	$0.778565\pi$
$642$	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	$-0.293666\pi$
$647$	19872.6	1.20753	0.603766	+	0.797162i	$0.293666\pi$
$648$	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	$-0.539514\pi$
$653$	−4132.19	−0.247634	−0.123817	+	0.992305i	$0.539514\pi$
$654$	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	$-0.258721\pi$
$659$	23260.1	1.37494	0.687470	+	0.726213i	$0.258721\pi$
$660$	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	$-0.642518\pi$
$677$	−15251.9	−0.865848	−0.432924	+	0.901430i	$0.642518\pi$
$678$	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	$-0.880088\pi$
$683$	−33196.1	−1.85976	−0.929878	+	0.367867i	$0.880088\pi$
$684$	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	$-0.304238\pi$
$701$	21416.8	1.15392	0.576962	+	0.816771i	$0.304238\pi$
$702$	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	$-0.272779\pi$
$719$	25245.9	1.30947	0.654737	+	0.755857i	$0.272779\pi$
$720$	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	$-0.540563\pi$
$743$	−5147.73	−0.254175	−0.127087	+	0.991892i	$0.540563\pi$
$744$	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	$-0.561150\pi$
$761$	−8016.38	−0.381858	−0.190929	+	0.981604i	$0.561150\pi$
$762$	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	$-0.766026\pi$
$773$	−31884.9	−1.48359	−0.741797	+	0.670624i	$0.766026\pi$
$774$	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	$-0.121725\pi$
$797$	41750.0	1.85553	0.927767	+	0.373159i	$0.121725\pi$
$798$	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	$-0.498872\pi$
$809$	163.059	0.00708636	0.00354318	+	0.999994i	$0.498872\pi$
$810$	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	$-0.740616\pi$
$821$	−32273.2	−1.37191	−0.685957	+	0.727642i	$0.740616\pi$
$822$	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	$-0.565807\pi$
$827$	−9763.68	−0.410540	−0.205270	+	0.978705i	$0.565807\pi$
$828$	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	$-0.676897\pi$
$839$	−25642.1	−1.05514	−0.527570	+	0.849512i	$0.676897\pi$
$840$	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	$-0.315826\pi$
$857$	27439.1	1.09370	0.546851	+	0.837230i	$0.315826\pi$
$858$	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	$-0.526389\pi$
$863$	−4198.80	−0.165618	−0.0828092	+	0.996565i	$0.526389\pi$
$864$	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	$-0.552348\pi$
$881$	−8562.17	−0.327431	−0.163716	+	0.986508i	$0.552348\pi$
$882$	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	$-0.450520\pi$
$887$	8179.87	0.309643	0.154821	+	0.987942i	$0.450520\pi$
$888$	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	$-0.198271\pi$
$911$	44665.2	1.62440	0.812198	+	0.583382i	$0.198271\pi$
$912$	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	$-0.268369\pi$
$929$	37667.8	1.33029	0.665146	+	0.746714i	$0.268369\pi$
$930$	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	$-0.443248\pi$
$941$	10238.6	0.354697	0.177348	+	0.984148i	$0.443248\pi$
$942$	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	$-0.556977\pi$
$947$	−10377.2	−0.356088	−0.178044	+	0.984023i	$0.556977\pi$
$948$	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	$-0.125720\pi$
$953$	54309.6	1.84602	0.923012	+	0.384772i	$0.125720\pi$
$954$	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	$-0.519829\pi$
$971$	−3767.34	−0.124511	−0.0622553	+	0.998060i	$0.519829\pi$
$972$	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	$-0.570195\pi$
$977$	−13359.9	−0.437482	−0.218741	+	0.975783i	$0.570195\pi$
$978$	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	$-0.328017\pi$
$983$	31707.1	1.02879	0.514394	+	0.857554i	$0.328017\pi$
$984$	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.k.1.2	✓	6
3.2	odd	2		405.4.a.l.1.5	yes	6
5.4	even	2		2025.4.a.z.1.5		6
9.2	odd	6		405.4.e.w.271.2		12
9.4	even	3		405.4.e.x.136.5		12
9.5	odd	6		405.4.e.w.136.2		12
9.7	even	3		405.4.e.x.271.5		12
15.14	odd	2		2025.4.a.y.1.2		6

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

Overview	Random
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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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