LMFDB - Embedded newform 810.4.a.o.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [810,4,Mod(1,810)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(810, 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("810.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 810 = 2 \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	810.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:	$47.7915471046$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 6 $ x^2 - 6
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 90)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.44949$ of defining polynomial
Character	$\chi$	$=$	810.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{2} +4.00000 q^{4} +5.00000 q^{5} +16.1464 q^{7} +8.00000 q^{8} +O(q^{10})$	\(q+2.00000 q^{2} +4.00000 q^{4} +5.00000 q^{5} +16.1464 q^{7} +8.00000 q^{8} +10.0000 q^{10} -63.1918 q^{11} -59.5959 q^{13} +32.2929 q^{14} +16.0000 q^{16} -73.5959 q^{17} -30.3133 q^{19} +20.0000 q^{20} -126.384 q^{22} -169.429 q^{23} +25.0000 q^{25} -119.192 q^{26} +64.5857 q^{28} +68.2827 q^{29} -206.677 q^{31} +32.0000 q^{32} -147.192 q^{34} +80.7321 q^{35} +10.3633 q^{37} -60.6265 q^{38} +40.0000 q^{40} -311.636 q^{41} +229.010 q^{43} -252.767 q^{44} -338.858 q^{46} +548.853 q^{47} -82.2929 q^{49} +50.0000 q^{50} -238.384 q^{52} -351.798 q^{53} -315.959 q^{55} +129.171 q^{56} +136.565 q^{58} +608.797 q^{59} +284.514 q^{61} -413.353 q^{62} +64.0000 q^{64} -297.980 q^{65} -94.0444 q^{67} -294.384 q^{68} +161.464 q^{70} -15.0918 q^{71} +1027.21 q^{73} +20.7265 q^{74} -121.253 q^{76} -1020.32 q^{77} +511.544 q^{79} +80.0000 q^{80} -623.271 q^{82} -976.215 q^{83} -367.980 q^{85} +458.020 q^{86} -505.535 q^{88} +572.394 q^{89} -962.261 q^{91} -677.716 q^{92} +1097.71 q^{94} -151.566 q^{95} -663.371 q^{97} -164.586 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{2} + 8 q^{4} + 10 q^{5} - 2 q^{7} + 16 q^{8}+O(q^{10}) $ 2 * q + 4 * q^2 + 8 * q^4 + 10 * q^5 - 2 * q^7 + 16 * q^8	$ 2 q + 4 q^{2} + 8 q^{4} + 10 q^{5} - 2 q^{7} + 16 q^{8} + 20 q^{10} - 48 q^{11} - 80 q^{13} - 4 q^{14} + 32 q^{16} - 108 q^{17} - 188 q^{19} + 40 q^{20} - 96 q^{22} - 138 q^{23} + 50 q^{25} - 160 q^{26} - 8 q^{28} - 30 q^{29} - 188 q^{31} + 64 q^{32} - 216 q^{34} - 10 q^{35} - 332 q^{37} - 376 q^{38} + 80 q^{40} - 6 q^{41} + 556 q^{43} - 192 q^{44} - 276 q^{46} + 162 q^{47} - 96 q^{49} + 100 q^{50} - 320 q^{52} - 684 q^{53} - 240 q^{55} - 16 q^{56} - 60 q^{58} + 228 q^{59} - 254 q^{61} - 376 q^{62} + 128 q^{64} - 400 q^{65} + 826 q^{67} - 432 q^{68} - 20 q^{70} - 912 q^{71} + 232 q^{73} - 664 q^{74} - 752 q^{76} - 1296 q^{77} - 476 q^{79} + 160 q^{80} - 12 q^{82} + 306 q^{83} - 540 q^{85} + 1112 q^{86} - 384 q^{88} + 1086 q^{89} - 592 q^{91} - 552 q^{92} + 324 q^{94} - 940 q^{95} + 868 q^{97} - 192 q^{98}+O(q^{100}) $ 2 * q + 4 * q^2 + 8 * q^4 + 10 * q^5 - 2 * q^7 + 16 * q^8 + 20 * q^10 - 48 * q^11 - 80 * q^13 - 4 * q^14 + 32 * q^16 - 108 * q^17 - 188 * q^19 + 40 * q^20 - 96 * q^22 - 138 * q^23 + 50 * q^25 - 160 * q^26 - 8 * q^28 - 30 * q^29 - 188 * q^31 + 64 * q^32 - 216 * q^34 - 10 * q^35 - 332 * q^37 - 376 * q^38 + 80 * q^40 - 6 * q^41 + 556 * q^43 - 192 * q^44 - 276 * q^46 + 162 * q^47 - 96 * q^49 + 100 * q^50 - 320 * q^52 - 684 * q^53 - 240 * q^55 - 16 * q^56 - 60 * q^58 + 228 * q^59 - 254 * q^61 - 376 * q^62 + 128 * q^64 - 400 * q^65 + 826 * q^67 - 432 * q^68 - 20 * q^70 - 912 * q^71 + 232 * q^73 - 664 * q^74 - 752 * q^76 - 1296 * q^77 - 476 * q^79 + 160 * q^80 - 12 * q^82 + 306 * q^83 - 540 * q^85 + 1112 * q^86 - 384 * q^88 + 1086 * q^89 - 592 * q^91 - 552 * q^92 + 324 * q^94 - 940 * q^95 + 868 * q^97 - 192 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	16.1464	0.871825	0.435913	−	0.899989i	$-0.356426\pi$
$7$	16.1464	0.871825	0.435913	+	0.899989i	$0.356426\pi$
$8$	8.00000	0.353553
$9$	0	0
$10$	10.0000	0.316228
$11$	−63.1918	−1.73210	−0.866048	−	0.499961i	$-0.833348\pi$
$11$	−63.1918	−1.73210	−0.866048	+	0.499961i	$0.833348\pi$
$12$	0	0
$13$	−59.5959	−1.27146	−0.635728	−	0.771913i	$-0.719300\pi$
$13$	−59.5959	−1.27146	−0.635728	+	0.771913i	$0.719300\pi$
$14$	32.2929	0.616473
$15$	0	0
$16$	16.0000	0.250000
$17$	−73.5959	−1.04998	−0.524989	−	0.851109i	$-0.675931\pi$
$17$	−73.5959	−1.04998	−0.524989	+	0.851109i	$0.675931\pi$
$18$	0	0
$19$	−30.3133	−0.366018	−0.183009	−	0.983111i	$-0.558584\pi$
$19$	−30.3133	−0.366018	−0.183009	+	0.983111i	$0.558584\pi$
$20$	20.0000	0.223607
$21$	0	0
$22$	−126.384	−1.22478
$23$	−169.429	−1.53602	−0.768009	−	0.640439i	$-0.778752\pi$
$23$	−169.429	−1.53602	−0.768009	+	0.640439i	$0.778752\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	−119.192	−0.899055
$27$	0	0
$28$	64.5857	0.435913
$29$	68.2827	0.437233	0.218617	−	0.975811i	$-0.429846\pi$
$29$	68.2827	0.437233	0.218617	+	0.975811i	$0.429846\pi$
$30$	0	0
$31$	−206.677	−1.19743	−0.598713	−	0.800964i	$-0.704321\pi$
$31$	−206.677	−1.19743	−0.598713	+	0.800964i	$0.704321\pi$
$32$	32.0000	0.176777
$33$	0	0
$34$	−147.192	−0.742447
$35$	80.7321	0.389892
$36$	0	0
$37$	10.3633	0.0460462	0.0230231	−	0.999735i	$-0.492671\pi$
$37$	10.3633	0.0460462	0.0230231	+	0.999735i	$0.492671\pi$
$38$	−60.6265	−0.258814
$39$	0	0
$40$	40.0000	0.158114
$41$	−311.636	−1.18706	−0.593528	−	0.804813i	$-0.702265\pi$
$41$	−311.636	−1.18706	−0.593528	+	0.804813i	$0.702265\pi$
$42$	0	0
$43$	229.010	0.812180	0.406090	−	0.913833i	$-0.366892\pi$
$43$	229.010	0.812180	0.406090	+	0.913833i	$0.366892\pi$
$44$	−252.767	−0.866048
$45$	0	0
$46$	−338.858	−1.08613
$47$	548.853	1.70337	0.851685	−	0.524054i	$-0.175581\pi$
$47$	548.853	1.70337	0.851685	+	0.524054i	$0.175581\pi$
$48$	0	0
$49$	−82.2929	−0.239921
$50$	50.0000	0.141421
$51$	0	0
$52$	−238.384	−0.635728
$53$	−351.798	−0.911758	−0.455879	−	0.890042i	$-0.650675\pi$
$53$	−351.798	−0.911758	−0.455879	+	0.890042i	$0.650675\pi$
$54$	0	0
$55$	−315.959	−0.774617
$56$	129.171	0.308237
$57$	0	0
$58$	136.565	0.309171
$59$	608.797	1.34337	0.671683	−	0.740839i	$-0.265572\pi$
$59$	608.797	1.34337	0.671683	+	0.740839i	$0.265572\pi$
$60$	0	0
$61$	284.514	0.597186	0.298593	−	0.954381i	$-0.403483\pi$
$61$	284.514	0.597186	0.298593	+	0.954381i	$0.403483\pi$
$62$	−413.353	−0.846708
$63$	0	0
$64$	64.0000	0.125000
$65$	−297.980	−0.568613
$66$	0	0
$67$	−94.0444	−0.171483	−0.0857414	−	0.996317i	$-0.527326\pi$
$67$	−94.0444	−0.171483	−0.0857414	+	0.996317i	$0.527326\pi$
$68$	−294.384	−0.524989
$69$	0	0
$70$	161.464	0.275695
$71$	−15.0918	−0.0252264	−0.0126132	−	0.999920i	$-0.504015\pi$
$71$	−15.0918	−0.0252264	−0.0126132	+	0.999920i	$0.504015\pi$
$72$	0	0
$73$	1027.21	1.64693	0.823465	−	0.567367i	$-0.192038\pi$
$73$	1027.21	1.64693	0.823465	+	0.567367i	$0.192038\pi$
$74$	20.7265	0.0325596
$75$	0	0
$76$	−121.253	−0.183009
$77$	−1020.32	−1.51008
$78$	0	0
$79$	511.544	0.728521	0.364261	−	0.931297i	$-0.381322\pi$
$79$	511.544	0.728521	0.364261	+	0.931297i	$0.381322\pi$
$80$	80.0000	0.111803
$81$	0	0
$82$	−623.271	−0.839376
$83$	−976.215	−1.29101	−0.645503	−	0.763758i	$-0.723352\pi$
$83$	−976.215	−1.29101	−0.645503	+	0.763758i	$0.723352\pi$
$84$	0	0
$85$	−367.980	−0.469565
$86$	458.020	0.574298
$87$	0	0
$88$	−505.535	−0.612388
$89$	572.394	0.681726	0.340863	−	0.940113i	$-0.389281\pi$
$89$	572.394	0.681726	0.340863	+	0.940113i	$0.389281\pi$
$90$	0	0
$91$	−962.261	−1.10849
$92$	−677.716	−0.768009
$93$	0	0
$94$	1097.71	1.20446
$95$	−151.566	−0.163688
$96$	0	0
$97$	−663.371	−0.694383	−0.347192	−	0.937794i	$-0.612865\pi$
$97$	−663.371	−0.694383	−0.347192	+	0.937794i	$0.612865\pi$
$98$	−164.586	−0.169650
$99$	0	0
$100$	100.000	0.100000
$101$	−664.343	−0.654501	−0.327250	−	0.944938i	$-0.606122\pi$
$101$	−664.343	−0.654501	−0.327250	+	0.944938i	$0.606122\pi$
$102$	0	0
$103$	−1932.18	−1.84838	−0.924191	−	0.381931i	$-0.875259\pi$
$103$	−1932.18	−1.84838	−0.924191	+	0.381931i	$0.875259\pi$
$104$	−476.767	−0.449528
$105$	0	0
$106$	−703.596	−0.644710
$107$	−1009.70	−0.912257	−0.456129	−	0.889914i	$-0.650764\pi$
$107$	−1009.70	−0.912257	−0.456129	+	0.889914i	$0.650764\pi$
$108$	0	0
$109$	−209.504	−0.184100	−0.0920498	−	0.995754i	$-0.529342\pi$
$109$	−209.504	−0.184100	−0.0920498	+	0.995754i	$0.529342\pi$
$110$	−631.918	−0.547737
$111$	0	0
$112$	258.343	0.217956
$113$	−842.325	−0.701232	−0.350616	−	0.936519i	$-0.614028\pi$
$113$	−842.325	−0.701232	−0.350616	+	0.936519i	$0.614028\pi$
$114$	0	0
$115$	−847.145	−0.686928
$116$	273.131	0.218617
$117$	0	0
$118$	1217.59	0.949903
$119$	−1188.31	−0.915398
$120$	0	0
$121$	2662.21	2.00016
$122$	569.029	0.422274
$123$	0	0
$124$	−826.706	−0.598713
$125$	125.000	0.0894427
$126$	0	0
$127$	886.925	0.619700	0.309850	−	0.950785i	$-0.399721\pi$
$127$	886.925	0.619700	0.309850	+	0.950785i	$0.399721\pi$
$128$	128.000	0.0883883
$129$	0	0
$130$	−595.959	−0.402070
$131$	−1730.96	−1.15446	−0.577231	−	0.816581i	$-0.695867\pi$
$131$	−1730.96	−1.15446	−0.577231	+	0.816581i	$0.695867\pi$
$132$	0	0
$133$	−489.451	−0.319104
$134$	−188.089	−0.121257
$135$	0	0
$136$	−588.767	−0.371223
$137$	−1412.49	−0.880853	−0.440426	−	0.897789i	$-0.645173\pi$
$137$	−1412.49	−0.880853	−0.440426	+	0.897789i	$0.645173\pi$
$138$	0	0
$139$	1338.79	0.816939	0.408469	−	0.912772i	$-0.366063\pi$
$139$	1338.79	0.816939	0.408469	+	0.912772i	$0.366063\pi$
$140$	322.929	0.194946
$141$	0	0
$142$	−30.1837	−0.0178377
$143$	3765.98	2.20228
$144$	0	0
$145$	341.413	0.195537
$146$	2054.42	1.16456
$147$	0	0
$148$	41.4530	0.0230231
$149$	221.241	0.121643	0.0608213	−	0.998149i	$-0.480628\pi$
$149$	221.241	0.121643	0.0608213	+	0.998149i	$0.480628\pi$
$150$	0	0
$151$	653.453	0.352167	0.176084	−	0.984375i	$-0.443657\pi$
$151$	653.453	0.352167	0.176084	+	0.984375i	$0.443657\pi$
$152$	−242.506	−0.129407
$153$	0	0
$154$	−2040.64	−1.06779
$155$	−1033.38	−0.535505
$156$	0	0
$157$	3106.22	1.57900	0.789501	−	0.613750i	$-0.210340\pi$
$157$	3106.22	1.57900	0.789501	+	0.613750i	$0.210340\pi$
$158$	1023.09	0.515142
$159$	0	0
$160$	160.000	0.0790569
$161$	−2735.67	−1.33914
$162$	0	0
$163$	1373.29	0.659905	0.329953	−	0.943998i	$-0.392967\pi$
$163$	1373.29	0.659905	0.329953	+	0.943998i	$0.392967\pi$
$164$	−1246.54	−0.593528
$165$	0	0
$166$	−1952.43	−0.912879
$167$	−1678.15	−0.777601	−0.388801	−	0.921322i	$-0.627111\pi$
$167$	−1678.15	−0.777601	−0.388801	+	0.921322i	$0.627111\pi$
$168$	0	0
$169$	1354.67	0.616601
$170$	−735.959	−0.332032
$171$	0	0
$172$	916.041	0.406090
$173$	−2215.80	−0.973779	−0.486890	−	0.873463i	$-0.661869\pi$
$173$	−2215.80	−0.973779	−0.486890	+	0.873463i	$0.661869\pi$
$174$	0	0
$175$	403.661	0.174365
$176$	−1011.07	−0.433024
$177$	0	0
$178$	1144.79	0.482053
$179$	−3195.54	−1.33433	−0.667167	−	0.744909i	$-0.732493\pi$
$179$	−3195.54	−1.33433	−0.667167	+	0.744909i	$0.732493\pi$
$180$	0	0
$181$	2329.66	0.956697	0.478348	−	0.878170i	$-0.341236\pi$
$181$	2329.66	0.956697	0.478348	+	0.878170i	$0.341236\pi$
$182$	−1924.52	−0.783819
$183$	0	0
$184$	−1355.43	−0.543064
$185$	51.8163	0.0205925
$186$	0	0
$187$	4650.66	1.81866
$188$	2195.41	0.851685
$189$	0	0
$190$	−303.133	−0.115745
$191$	3509.22	1.32941	0.664707	−	0.747105i	$-0.268557\pi$
$191$	3509.22	1.32941	0.664707	+	0.747105i	$0.268557\pi$
$192$	0	0
$193$	−536.059	−0.199929	−0.0999647	−	0.994991i	$-0.531873\pi$
$193$	−536.059	−0.199929	−0.0999647	+	0.994991i	$0.531873\pi$
$194$	−1326.74	−0.491003
$195$	0	0
$196$	−329.171	−0.119960
$197$	−3688.16	−1.33386	−0.666930	−	0.745120i	$-0.732392\pi$
$197$	−3688.16	−1.33386	−0.666930	+	0.745120i	$0.732392\pi$
$198$	0	0
$199$	−1142.65	−0.407038	−0.203519	−	0.979071i	$-0.565238\pi$
$199$	−1142.65	−0.407038	−0.203519	+	0.979071i	$0.565238\pi$
$200$	200.000	0.0707107
$201$	0	0
$202$	−1328.69	−0.462802
$203$	1102.52	0.381191
$204$	0	0
$205$	−1558.18	−0.530868
$206$	−3864.36	−1.30700
$207$	0	0
$208$	−953.535	−0.317864
$209$	1915.55	0.633978
$210$	0	0
$211$	3366.59	1.09841	0.549207	−	0.835686i	$-0.314930\pi$
$211$	3366.59	1.09841	0.549207	+	0.835686i	$0.314930\pi$
$212$	−1407.19	−0.455879
$213$	0	0
$214$	−2019.40	−0.645063
$215$	1145.05	0.363218
$216$	0	0
$217$	−3337.09	−1.04395
$218$	−419.008	−0.130178
$219$	0	0
$220$	−1263.84	−0.387308
$221$	4386.02	1.33500
$222$	0	0
$223$	4056.13	1.21802	0.609010	−	0.793163i	$-0.291567\pi$
$223$	4056.13	1.21802	0.609010	+	0.793163i	$0.291567\pi$
$224$	516.686	0.154118
$225$	0	0
$226$	−1684.65	−0.495846
$227$	4876.48	1.42583	0.712914	−	0.701251i	$-0.247375\pi$
$227$	4876.48	1.42583	0.712914	+	0.701251i	$0.247375\pi$
$228$	0	0
$229$	−4126.16	−1.19067	−0.595337	−	0.803476i	$-0.702981\pi$
$229$	−4126.16	−1.19067	−0.595337	+	0.803476i	$0.702981\pi$
$230$	−1694.29	−0.485731
$231$	0	0
$232$	546.261	0.154585
$233$	−1207.14	−0.339411	−0.169705	−	0.985495i	$-0.554282\pi$
$233$	−1207.14	−0.339411	−0.169705	+	0.985495i	$0.554282\pi$
$234$	0	0
$235$	2744.26	0.761770
$236$	2435.19	0.671683
$237$	0	0
$238$	−2376.62	−0.647284
$239$	273.965	0.0741479	0.0370739	−	0.999313i	$-0.488196\pi$
$239$	273.965	0.0741479	0.0370739	+	0.999313i	$0.488196\pi$
$240$	0	0
$241$	−2870.04	−0.767118	−0.383559	−	0.923516i	$-0.625302\pi$
$241$	−2870.04	−0.767118	−0.383559	+	0.923516i	$0.625302\pi$
$242$	5324.42	1.41432
$243$	0	0
$244$	1138.06	0.298593
$245$	−411.464	−0.107296
$246$	0	0
$247$	1806.55	0.465376
$248$	−1653.41	−0.423354
$249$	0	0
$250$	250.000	0.0632456
$251$	−3846.18	−0.967207	−0.483603	−	0.875287i	$-0.660672\pi$
$251$	−3846.18	−0.967207	−0.483603	+	0.875287i	$0.660672\pi$
$252$	0	0
$253$	10706.5	2.66053
$254$	1773.85	0.438194
$255$	0	0
$256$	256.000	0.0625000
$257$	883.802	0.214514	0.107257	−	0.994231i	$-0.465793\pi$
$257$	883.802	0.214514	0.107257	+	0.994231i	$0.465793\pi$
$258$	0	0
$259$	167.330	0.0401442
$260$	−1191.92	−0.284306
$261$	0	0
$262$	−3461.92	−0.816328
$263$	−946.759	−0.221976	−0.110988	−	0.993822i	$-0.535402\pi$
$263$	−946.759	−0.221976	−0.110988	+	0.993822i	$0.535402\pi$
$264$	0	0
$265$	−1758.99	−0.407751
$266$	−978.902	−0.225640
$267$	0	0
$268$	−376.178	−0.0857414
$269$	5283.51	1.19755	0.598775	−	0.800917i	$-0.295654\pi$
$269$	5283.51	1.19755	0.598775	+	0.800917i	$0.295654\pi$
$270$	0	0
$271$	−6518.81	−1.46122	−0.730608	−	0.682797i	$-0.760763\pi$
$271$	−6518.81	−1.46122	−0.730608	+	0.682797i	$0.760763\pi$
$272$	−1177.53	−0.262495
$273$	0	0
$274$	−2824.97	−0.622857
$275$	−1579.80	−0.346419
$276$	0	0
$277$	1571.65	0.340908	0.170454	−	0.985366i	$-0.445477\pi$
$277$	1571.65	0.340908	0.170454	+	0.985366i	$0.445477\pi$
$278$	2677.58	0.577663
$279$	0	0
$280$	645.857	0.137848
$281$	−1489.92	−0.316302	−0.158151	−	0.987415i	$-0.550553\pi$
$281$	−1489.92	−0.316302	−0.158151	+	0.987415i	$0.550553\pi$
$282$	0	0
$283$	4677.49	0.982502	0.491251	−	0.871018i	$-0.336540\pi$
$283$	4677.49	0.982502	0.491251	+	0.871018i	$0.336540\pi$
$284$	−60.3674	−0.0126132
$285$	0	0
$286$	7531.95	1.55725
$287$	−5031.80	−1.03491
$288$	0	0
$289$	503.359	0.102455
$290$	682.827	0.138265
$291$	0	0
$292$	4108.84	0.823465
$293$	7019.68	1.39964	0.699819	−	0.714320i	$-0.253264\pi$
$293$	7019.68	1.39964	0.699819	+	0.714320i	$0.253264\pi$
$294$	0	0
$295$	3043.98	0.600772
$296$	82.9061	0.0162798
$297$	0	0
$298$	442.482	0.0860143
$299$	10097.3	1.95298
$300$	0	0
$301$	3697.70	0.708079
$302$	1306.91	0.249020
$303$	0	0
$304$	−485.012	−0.0915045
$305$	1422.57	0.267070
$306$	0	0
$307$	7401.73	1.37602	0.688012	−	0.725699i	$-0.258484\pi$
$307$	7401.73	1.37602	0.688012	+	0.725699i	$0.258484\pi$
$308$	−4081.29	−0.755042
$309$	0	0
$310$	−2066.77	−0.378659
$311$	−2045.68	−0.372990	−0.186495	−	0.982456i	$-0.559713\pi$
$311$	−2045.68	−0.372990	−0.186495	+	0.982456i	$0.559713\pi$
$312$	0	0
$313$	−143.551	−0.0259233	−0.0129616	−	0.999916i	$-0.504126\pi$
$313$	−143.551	−0.0259233	−0.0129616	+	0.999916i	$0.504126\pi$
$314$	6212.44	1.11652
$315$	0	0
$316$	2046.18	0.364261
$317$	−4211.98	−0.746272	−0.373136	−	0.927777i	$-0.621718\pi$
$317$	−4211.98	−0.746272	−0.373136	+	0.927777i	$0.621718\pi$
$318$	0	0
$319$	−4314.91	−0.757330
$320$	320.000	0.0559017
$321$	0	0
$322$	−5471.35	−0.946914
$323$	2230.93	0.384311
$324$	0	0
$325$	−1489.90	−0.254291
$326$	2746.58	0.466623
$327$	0	0
$328$	−2493.09	−0.419688
$329$	8862.01	1.48504
$330$	0	0
$331$	−8692.93	−1.44353	−0.721763	−	0.692141i	$-0.756668\pi$
$331$	−8692.93	−1.44353	−0.721763	+	0.692141i	$0.756668\pi$
$332$	−3904.86	−0.645503
$333$	0	0
$334$	−3356.31	−0.549847
$335$	−470.222	−0.0766895
$336$	0	0
$337$	−9593.42	−1.55070	−0.775352	−	0.631530i	$-0.782427\pi$
$337$	−9593.42	−1.55070	−0.775352	+	0.631530i	$0.782427\pi$
$338$	2709.35	0.436003
$339$	0	0
$340$	−1471.92	−0.234782
$341$	13060.3	2.07406
$342$	0	0
$343$	−6866.96	−1.08099
$344$	1832.08	0.287149
$345$	0	0
$346$	−4431.59	−0.688566
$347$	−6714.97	−1.03884	−0.519421	−	0.854518i	$-0.673852\pi$
$347$	−6714.97	−1.03884	−0.519421	+	0.854518i	$0.673852\pi$
$348$	0	0
$349$	−12278.0	−1.88318	−0.941589	−	0.336765i	$-0.890667\pi$
$349$	−12278.0	−1.88318	−0.941589	+	0.336765i	$0.890667\pi$
$350$	807.321	0.123295
$351$	0	0
$352$	−2022.14	−0.306194
$353$	−1252.36	−0.188828	−0.0944141	−	0.995533i	$-0.530098\pi$
$353$	−1252.36	−0.188828	−0.0944141	+	0.995533i	$0.530098\pi$
$354$	0	0
$355$	−75.4592	−0.0112816
$356$	2289.58	0.340863
$357$	0	0
$358$	−6391.07	−0.943516
$359$	8606.47	1.26527	0.632635	−	0.774450i	$-0.281973\pi$
$359$	8606.47	1.26527	0.632635	+	0.774450i	$0.281973\pi$
$360$	0	0
$361$	−5940.11	−0.866031
$362$	4659.32	0.676487
$363$	0	0
$364$	−3849.04	−0.554244
$365$	5136.05	0.736529
$366$	0	0
$367$	5283.44	0.751481	0.375741	−	0.926725i	$-0.377388\pi$
$367$	5283.44	0.751481	0.375741	+	0.926725i	$0.377388\pi$
$368$	−2710.87	−0.384004
$369$	0	0
$370$	103.633	0.0145611
$371$	−5680.28	−0.794893
$372$	0	0
$373$	7374.07	1.02363	0.511816	−	0.859095i	$-0.328973\pi$
$373$	7374.07	1.02363	0.511816	+	0.859095i	$0.328973\pi$
$374$	9301.32	1.28599
$375$	0	0
$376$	4390.82	0.602232
$377$	−4069.37	−0.555923
$378$	0	0
$379$	−5156.81	−0.698911	−0.349456	−	0.936953i	$-0.613633\pi$
$379$	−5156.81	−0.698911	−0.349456	+	0.936953i	$0.613633\pi$
$380$	−606.265	−0.0818441
$381$	0	0
$382$	7018.43	0.940037
$383$	5419.62	0.723055	0.361527	−	0.932362i	$-0.382255\pi$
$383$	5419.62	0.723055	0.361527	+	0.932362i	$0.382255\pi$
$384$	0	0
$385$	−5101.61	−0.675330
$386$	−1072.12	−0.141371
$387$	0	0
$388$	−2653.49	−0.347192
$389$	7949.29	1.03610	0.518052	−	0.855349i	$-0.326657\pi$
$389$	7949.29	1.03610	0.518052	+	0.855349i	$0.326657\pi$
$390$	0	0
$391$	12469.3	1.61279
$392$	−658.343	−0.0848248
$393$	0	0
$394$	−7376.32	−0.943182
$395$	2557.72	0.325805
$396$	0	0
$397$	−6312.82	−0.798064	−0.399032	−	0.916937i	$-0.630654\pi$
$397$	−6312.82	−0.798064	−0.399032	+	0.916937i	$0.630654\pi$
$398$	−2285.31	−0.287819
$399$	0	0
$400$	400.000	0.0500000
$401$	−84.6735	−0.0105446	−0.00527231	−	0.999986i	$-0.501678\pi$
$401$	−84.6735	−0.0105446	−0.00527231	+	0.999986i	$0.501678\pi$
$402$	0	0
$403$	12317.1	1.52247
$404$	−2657.37	−0.327250
$405$	0	0
$406$	2205.04	0.269543
$407$	−654.874	−0.0797565
$408$	0	0
$409$	−2916.84	−0.352636	−0.176318	−	0.984333i	$-0.556419\pi$
$409$	−2916.84	−0.352636	−0.176318	+	0.984333i	$0.556419\pi$
$410$	−3116.36	−0.375380
$411$	0	0
$412$	−7728.72	−0.924191
$413$	9829.90	1.17118
$414$	0	0
$415$	−4881.07	−0.577355
$416$	−1907.07	−0.224764
$417$	0	0
$418$	3831.10	0.448290
$419$	−13761.6	−1.60453	−0.802267	−	0.596965i	$-0.796373\pi$
$419$	−13761.6	−1.60453	−0.802267	+	0.596965i	$0.796373\pi$
$420$	0	0
$421$	14136.3	1.63649	0.818243	−	0.574872i	$-0.194948\pi$
$421$	14136.3	1.63649	0.818243	+	0.574872i	$0.194948\pi$
$422$	6733.18	0.776696
$423$	0	0
$424$	−2814.38	−0.322355
$425$	−1839.90	−0.209996
$426$	0	0
$427$	4593.89	0.520641
$428$	−4038.81	−0.456129
$429$	0	0
$430$	2290.10	0.256834
$431$	11023.8	1.23201	0.616005	−	0.787742i	$-0.288750\pi$
$431$	11023.8	1.23201	0.616005	+	0.787742i	$0.288750\pi$
$432$	0	0
$433$	7714.94	0.856250	0.428125	−	0.903719i	$-0.359174\pi$
$433$	7714.94	0.856250	0.428125	+	0.903719i	$0.359174\pi$
$434$	−6674.18	−0.738181
$435$	0	0
$436$	−838.016	−0.0920498
$437$	5135.95	0.562210
$438$	0	0
$439$	−9976.79	−1.08466	−0.542331	−	0.840165i	$-0.682458\pi$
$439$	−9976.79	−1.08466	−0.542331	+	0.840165i	$0.682458\pi$
$440$	−2527.67	−0.273868
$441$	0	0
$442$	8772.03	0.943989
$443$	17184.0	1.84297	0.921484	−	0.388416i	$-0.126978\pi$
$443$	17184.0	1.84297	0.921484	+	0.388416i	$0.126978\pi$
$444$	0	0
$445$	2861.97	0.304877
$446$	8112.25	0.861270
$447$	0	0
$448$	1033.37	0.108978
$449$	−18329.8	−1.92658	−0.963291	−	0.268458i	$-0.913486\pi$
$449$	−18329.8	−1.92658	−0.963291	+	0.268458i	$0.913486\pi$
$450$	0	0
$451$	19692.8	2.05610
$452$	−3369.30	−0.350616
$453$	0	0
$454$	9752.95	1.00821
$455$	−4811.31	−0.495731
$456$	0	0
$457$	−12115.9	−1.24017	−0.620086	−	0.784534i	$-0.712902\pi$
$457$	−12115.9	−1.24017	−0.620086	+	0.784534i	$0.712902\pi$
$458$	−8252.32	−0.841933
$459$	0	0
$460$	−3388.58	−0.343464
$461$	−14370.1	−1.45181	−0.725903	−	0.687797i	$-0.758578\pi$
$461$	−14370.1	−1.45181	−0.725903	+	0.687797i	$0.758578\pi$
$462$	0	0
$463$	3390.55	0.340329	0.170164	−	0.985416i	$-0.445570\pi$
$463$	3390.55	0.340329	0.170164	+	0.985416i	$0.445570\pi$
$464$	1092.52	0.109308
$465$	0	0
$466$	−2414.29	−0.240000
$467$	1474.30	0.146087	0.0730435	−	0.997329i	$-0.476729\pi$
$467$	1474.30	0.146087	0.0730435	+	0.997329i	$0.476729\pi$
$468$	0	0
$469$	−1518.48	−0.149503
$470$	5488.53	0.538653
$471$	0	0
$472$	4870.38	0.474952
$473$	−14471.6	−1.40677
$474$	0	0
$475$	−757.832	−0.0732036
$476$	−4753.24	−0.457699
$477$	0	0
$478$	547.931	0.0524305
$479$	−3705.50	−0.353462	−0.176731	−	0.984259i	$-0.556552\pi$
$479$	−3705.50	−0.353462	−0.176731	+	0.984259i	$0.556552\pi$
$480$	0	0
$481$	−617.608	−0.0585457
$482$	−5740.08	−0.542435
$483$	0	0
$484$	10648.8	1.00008
$485$	−3316.86	−0.310538
$486$	0	0
$487$	4952.80	0.460848	0.230424	−	0.973090i	$-0.425989\pi$
$487$	4952.80	0.460848	0.230424	+	0.973090i	$0.425989\pi$
$488$	2276.11	0.211137
$489$	0	0
$490$	−822.929	−0.0758696
$491$	7804.85	0.717368	0.358684	−	0.933459i	$-0.383226\pi$
$491$	7804.85	0.717368	0.358684	+	0.933459i	$0.383226\pi$
$492$	0	0
$493$	−5025.32	−0.459086
$494$	3613.09	0.329070
$495$	0	0
$496$	−3306.82	−0.299356
$497$	−243.679	−0.0219930
$498$	0	0
$499$	−17793.3	−1.59627	−0.798134	−	0.602479i	$-0.794180\pi$
$499$	−17793.3	−1.59627	−0.798134	+	0.602479i	$0.794180\pi$
$500$	500.000	0.0447214
$501$	0	0
$502$	−7692.37	−0.683918
$503$	1136.09	0.100708	0.0503538	−	0.998731i	$-0.483965\pi$
$503$	1136.09	0.100708	0.0503538	+	0.998731i	$0.483965\pi$
$504$	0	0
$505$	−3321.71	−0.292702
$506$	21413.1	1.88128
$507$	0	0
$508$	3547.70	0.309850
$509$	860.550	0.0749375	0.0374688	−	0.999298i	$-0.488071\pi$
$509$	860.550	0.0749375	0.0374688	+	0.999298i	$0.488071\pi$
$510$	0	0
$511$	16585.8	1.43583
$512$	512.000	0.0441942
$513$	0	0
$514$	1767.60	0.151684
$515$	−9660.90	−0.826621
$516$	0	0
$517$	−34683.0	−2.95040
$518$	334.659	0.0283863
$519$	0	0
$520$	−2383.84	−0.201035
$521$	11689.8	0.982991	0.491495	−	0.870880i	$-0.336451\pi$
$521$	11689.8	0.982991	0.491495	+	0.870880i	$0.336451\pi$
$522$	0	0
$523$	−1314.55	−0.109907	−0.0549535	−	0.998489i	$-0.517501\pi$
$523$	−1314.55	−0.109907	−0.0549535	+	0.998489i	$0.517501\pi$
$524$	−6923.83	−0.577231
$525$	0	0
$526$	−1893.52	−0.156961
$527$	15210.5	1.25727
$528$	0	0
$529$	16539.2	1.35935
$530$	−3517.98	−0.288323
$531$	0	0
$532$	−1957.80	−0.159552
$533$	18572.2	1.50929
$534$	0	0
$535$	−5048.51	−0.407974
$536$	−752.355	−0.0606283
$537$	0	0
$538$	10567.0	0.846796
$539$	5200.24	0.415566
$540$	0	0
$541$	15481.3	1.23030	0.615150	−	0.788410i	$-0.289096\pi$
$541$	15481.3	1.23030	0.615150	+	0.788410i	$0.289096\pi$
$542$	−13037.6	−1.03324
$543$	0	0
$544$	−2355.07	−0.185612
$545$	−1047.52	−0.0823318
$546$	0	0
$547$	20903.9	1.63398	0.816990	−	0.576652i	$-0.195641\pi$
$547$	20903.9	1.63398	0.816990	+	0.576652i	$0.195641\pi$
$548$	−5649.94	−0.440426
$549$	0	0
$550$	−3159.59	−0.244955
$551$	−2069.87	−0.160035
$552$	0	0
$553$	8259.61	0.635143
$554$	3143.31	0.241058
$555$	0	0
$556$	5355.15	0.408469
$557$	2306.65	0.175468	0.0877342	−	0.996144i	$-0.472037\pi$
$557$	2306.65	0.175468	0.0877342	+	0.996144i	$0.472037\pi$
$558$	0	0
$559$	−13648.1	−1.03265
$560$	1291.71	0.0974730
$561$	0	0
$562$	−2979.83	−0.223659
$563$	9515.25	0.712291	0.356146	−	0.934430i	$-0.384091\pi$
$563$	9515.25	0.712291	0.356146	+	0.934430i	$0.384091\pi$
$564$	0	0
$565$	−4211.62	−0.313601
$566$	9354.98	0.694734
$567$	0	0
$568$	−120.735	−0.00891887
$569$	−14983.3	−1.10392	−0.551960	−	0.833870i	$-0.686120\pi$
$569$	−14983.3	−1.10392	−0.551960	+	0.833870i	$0.686120\pi$
$570$	0	0
$571$	−13997.6	−1.02589	−0.512943	−	0.858423i	$-0.671445\pi$
$571$	−13997.6	−1.02589	−0.512943	+	0.858423i	$0.671445\pi$
$572$	15063.9	1.10114
$573$	0	0
$574$	−10063.6	−0.731789
$575$	−4235.73	−0.307204
$576$	0	0
$577$	4949.53	0.357108	0.178554	−	0.983930i	$-0.442858\pi$
$577$	4949.53	0.357108	0.178554	+	0.983930i	$0.442858\pi$
$578$	1006.72	0.0724463
$579$	0	0
$580$	1365.65	0.0977684
$581$	−15762.4	−1.12553
$582$	0	0
$583$	22230.8	1.57925
$584$	8217.68	0.582278
$585$	0	0
$586$	14039.4	0.989694
$587$	16919.9	1.18971	0.594855	−	0.803833i	$-0.297209\pi$
$587$	16919.9	1.18971	0.594855	+	0.803833i	$0.297209\pi$
$588$	0	0
$589$	6265.04	0.438279
$590$	6087.97	0.424810
$591$	0	0
$592$	165.812	0.0115116
$593$	−26868.0	−1.86060	−0.930301	−	0.366797i	$-0.880454\pi$
$593$	−26868.0	−1.86060	−0.930301	+	0.366797i	$0.880454\pi$
$594$	0	0
$595$	−5941.56	−0.409378
$596$	884.963	0.0608213
$597$	0	0
$598$	20194.6	1.38097
$599$	−7358.04	−0.501905	−0.250953	−	0.967999i	$-0.580744\pi$
$599$	−7358.04	−0.501905	−0.250953	+	0.967999i	$0.580744\pi$
$600$	0	0
$601$	−29242.3	−1.98472	−0.992361	−	0.123369i	$-0.960630\pi$
$601$	−29242.3	−1.98472	−0.992361	+	0.123369i	$0.960630\pi$
$602$	7395.39	0.500687
$603$	0	0
$604$	2613.81	0.176084
$605$	13311.0	0.894497
$606$	0	0
$607$	871.723	0.0582902	0.0291451	−	0.999575i	$-0.490722\pi$
$607$	871.723	0.0582902	0.0291451	+	0.999575i	$0.490722\pi$
$608$	−970.025	−0.0647034
$609$	0	0
$610$	2845.14	0.188847
$611$	−32709.4	−2.16576
$612$	0	0
$613$	−1899.22	−0.125136	−0.0625682	−	0.998041i	$-0.519929\pi$
$613$	−1899.22	−0.125136	−0.0625682	+	0.998041i	$0.519929\pi$
$614$	14803.5	0.972996
$615$	0	0
$616$	−8162.58	−0.533896
$617$	−21989.7	−1.43480	−0.717399	−	0.696663i	$-0.754667\pi$
$617$	−21989.7	−1.43480	−0.717399	+	0.696663i	$0.754667\pi$
$618$	0	0
$619$	−6399.42	−0.415532	−0.207766	−	0.978179i	$-0.566619\pi$
$619$	−6399.42	−0.415532	−0.207766	+	0.978179i	$0.566619\pi$
$620$	−4133.53	−0.267753
$621$	0	0
$622$	−4091.37	−0.263744
$623$	9242.12	0.594346
$624$	0	0
$625$	625.000	0.0400000
$626$	−287.102	−0.0183305
$627$	0	0
$628$	12424.9	0.789501
$629$	−762.694	−0.0483475
$630$	0	0
$631$	27523.2	1.73642	0.868210	−	0.496197i	$-0.165271\pi$
$631$	27523.2	1.73642	0.868210	+	0.496197i	$0.165271\pi$
$632$	4092.35	0.257571
$633$	0	0
$634$	−8423.96	−0.527694
$635$	4434.63	0.277138
$636$	0	0
$637$	4904.32	0.305049
$638$	−8629.81	−0.535513
$639$	0	0
$640$	640.000	0.0395285
$641$	−10976.0	−0.676328	−0.338164	−	0.941087i	$-0.609806\pi$
$641$	−10976.0	−0.676328	−0.338164	+	0.941087i	$0.609806\pi$
$642$	0	0
$643$	22407.0	1.37425	0.687126	−	0.726538i	$-0.258872\pi$
$643$	22407.0	1.37425	0.687126	+	0.726538i	$0.258872\pi$
$644$	−10942.7	−0.669569
$645$	0	0
$646$	4461.87	0.271749
$647$	2359.95	0.143399	0.0716996	−	0.997426i	$-0.477158\pi$
$647$	2359.95	0.143399	0.0716996	+	0.997426i	$0.477158\pi$
$648$	0	0
$649$	−38471.0	−2.32684
$650$	−2979.80	−0.179811
$651$	0	0
$652$	5493.17	0.329953
$653$	2622.37	0.157154	0.0785768	−	0.996908i	$-0.474962\pi$
$653$	2622.37	0.157154	0.0785768	+	0.996908i	$0.474962\pi$
$654$	0	0
$655$	−8654.79	−0.516291
$656$	−4986.17	−0.296764
$657$	0	0
$658$	17724.0	1.05008
$659$	−15393.6	−0.909938	−0.454969	−	0.890507i	$-0.650350\pi$
$659$	−15393.6	−0.909938	−0.454969	+	0.890507i	$0.650350\pi$
$660$	0	0
$661$	5134.08	0.302107	0.151053	−	0.988526i	$-0.451733\pi$
$661$	5134.08	0.302107	0.151053	+	0.988526i	$0.451733\pi$
$662$	−17385.9	−1.02073
$663$	0	0
$664$	−7809.72	−0.456440
$665$	−2447.25	−0.142707
$666$	0	0
$667$	−11569.1	−0.671598
$668$	−6712.61	−0.388801
$669$	0	0
$670$	−940.444	−0.0542276
$671$	−17979.0	−1.03438
$672$	0	0
$673$	−12193.8	−0.698420	−0.349210	−	0.937044i	$-0.613550\pi$
$673$	−12193.8	−0.698420	−0.349210	+	0.937044i	$0.613550\pi$
$674$	−19186.8	−1.09651
$675$	0	0
$676$	5418.69	0.308301
$677$	−20290.4	−1.15188	−0.575942	−	0.817491i	$-0.695364\pi$
$677$	−20290.4	−1.15188	−0.575942	+	0.817491i	$0.695364\pi$
$678$	0	0
$679$	−10711.1	−0.605381
$680$	−2943.84	−0.166016
$681$	0	0
$682$	26120.5	1.46658
$683$	7760.27	0.434756	0.217378	−	0.976087i	$-0.430250\pi$
$683$	7760.27	0.434756	0.217378	+	0.976087i	$0.430250\pi$
$684$	0	0
$685$	−7062.43	−0.393929
$686$	−13733.9	−0.764378
$687$	0	0
$688$	3664.16	0.203045
$689$	20965.7	1.15926
$690$	0	0
$691$	−8572.49	−0.471943	−0.235972	−	0.971760i	$-0.575827\pi$
$691$	−8572.49	−0.471943	−0.235972	+	0.971760i	$0.575827\pi$
$692$	−8863.18	−0.486890
$693$	0	0
$694$	−13429.9	−0.734573
$695$	6693.94	0.365346
$696$	0	0
$697$	22935.1	1.24638
$698$	−24556.1	−1.33161
$699$	0	0
$700$	1614.64	0.0871825
$701$	9858.62	0.531177	0.265588	−	0.964086i	$-0.414434\pi$
$701$	9858.62	0.531177	0.265588	+	0.964086i	$0.414434\pi$
$702$	0	0
$703$	−314.144	−0.0168537
$704$	−4044.28	−0.216512
$705$	0	0
$706$	−2504.72	−0.133522
$707$	−10726.8	−0.570610
$708$	0	0
$709$	−17484.4	−0.926151	−0.463076	−	0.886319i	$-0.653254\pi$
$709$	−17484.4	−0.926151	−0.463076	+	0.886319i	$0.653254\pi$
$710$	−150.918	−0.00797728
$711$	0	0
$712$	4579.15	0.241027
$713$	35017.0	1.83927
$714$	0	0
$715$	18829.9	0.984892
$716$	−12782.1	−0.667167
$717$	0	0
$718$	17212.9	0.894681
$719$	−13179.5	−0.683605	−0.341803	−	0.939772i	$-0.611037\pi$
$719$	−13179.5	−0.683605	−0.341803	+	0.939772i	$0.611037\pi$
$720$	0	0
$721$	−31197.8	−1.61147
$722$	−11880.2	−0.612376
$723$	0	0
$724$	9318.63	0.478348
$725$	1707.07	0.0874467
$726$	0	0
$727$	−14854.3	−0.757791	−0.378895	−	0.925439i	$-0.623696\pi$
$727$	−14854.3	−0.757791	−0.378895	+	0.925439i	$0.623696\pi$
$728$	−7698.09	−0.391910
$729$	0	0
$730$	10272.1	0.520805
$731$	−16854.2	−0.852771
$732$	0	0
$733$	−28060.1	−1.41395	−0.706973	−	0.707241i	$-0.749940\pi$
$733$	−28060.1	−1.41395	−0.706973	+	0.707241i	$0.749940\pi$
$734$	10566.9	0.531377
$735$	0	0
$736$	−5421.73	−0.271532
$737$	5942.84	0.297025
$738$	0	0
$739$	26616.3	1.32490	0.662448	−	0.749108i	$-0.269518\pi$
$739$	26616.3	1.32490	0.662448	+	0.749108i	$0.269518\pi$
$740$	207.265	0.0102962
$741$	0	0
$742$	−11360.6	−0.562075
$743$	37.9054	0.00187162	0.000935811	−	1.00000i	$-0.499702\pi$
$743$	37.9054	0.00187162	0.000935811		1.00000i	$0.499702\pi$
$744$	0	0
$745$	1106.20	0.0544002
$746$	14748.1	0.723817
$747$	0	0
$748$	18602.6	0.909332
$749$	−16303.1	−0.795329
$750$	0	0
$751$	4461.90	0.216801	0.108400	−	0.994107i	$-0.465427\pi$
$751$	4461.90	0.216801	0.108400	+	0.994107i	$0.465427\pi$
$752$	8781.64	0.425842
$753$	0	0
$754$	−8138.73	−0.393097
$755$	3267.27	0.157494
$756$	0	0
$757$	−772.025	−0.0370670	−0.0185335	−	0.999828i	$-0.505900\pi$
$757$	−772.025	−0.0370670	−0.0185335	+	0.999828i	$0.505900\pi$
$758$	−10313.6	−0.494205
$759$	0	0
$760$	−1212.53	−0.0578725
$761$	3213.38	0.153068	0.0765340	−	0.997067i	$-0.475615\pi$
$761$	3213.38	0.153068	0.0765340	+	0.997067i	$0.475615\pi$
$762$	0	0
$763$	−3382.74	−0.160503
$764$	14036.9	0.664707
$765$	0	0
$766$	10839.2	0.511277
$767$	−36281.8	−1.70803
$768$	0	0
$769$	−20076.7	−0.941462	−0.470731	−	0.882277i	$-0.656010\pi$
$769$	−20076.7	−0.941462	−0.470731	+	0.882277i	$0.656010\pi$
$770$	−10203.2	−0.477531
$771$	0	0
$772$	−2144.24	−0.0999647
$773$	−1477.71	−0.0687576	−0.0343788	−	0.999409i	$-0.510945\pi$
$773$	−1477.71	−0.0687576	−0.0343788	+	0.999409i	$0.510945\pi$
$774$	0	0
$775$	−5166.91	−0.239485
$776$	−5306.97	−0.245502
$777$	0	0
$778$	15898.6	0.732637
$779$	9446.70	0.434484
$780$	0	0
$781$	953.681	0.0436945
$782$	24938.6	1.14041
$783$	0	0
$784$	−1316.69	−0.0599802
$785$	15531.1	0.706151
$786$	0	0
$787$	16975.0	0.768862	0.384431	−	0.923154i	$-0.374398\pi$
$787$	16975.0	0.768862	0.384431	+	0.923154i	$0.374398\pi$
$788$	−14752.6	−0.666930
$789$	0	0
$790$	5115.44	0.230379
$791$	−13600.5	−0.611352
$792$	0	0
$793$	−16955.9	−0.759295
$794$	−12625.6	−0.564317
$795$	0	0
$796$	−4570.62	−0.203519
$797$	−10184.8	−0.452652	−0.226326	−	0.974052i	$-0.572671\pi$
$797$	−10184.8	−0.452652	−0.226326	+	0.974052i	$0.572671\pi$
$798$	0	0
$799$	−40393.3	−1.78850
$800$	800.000	0.0353553
$801$	0	0
$802$	−169.347	−0.00745618
$803$	−64911.3	−2.85264
$804$	0	0
$805$	−13678.4	−0.598881
$806$	24634.2	1.07655
$807$	0	0
$808$	−5314.74	−0.231401
$809$	8005.96	0.347929	0.173964	−	0.984752i	$-0.444342\pi$
$809$	8005.96	0.347929	0.173964	+	0.984752i	$0.444342\pi$
$810$	0	0
$811$	635.515	0.0275166	0.0137583	−	0.999905i	$-0.495620\pi$
$811$	635.515	0.0275166	0.0137583	+	0.999905i	$0.495620\pi$
$812$	4410.08	0.190596
$813$	0	0
$814$	−1309.75	−0.0563963
$815$	6866.46	0.295118
$816$	0	0
$817$	−6942.05	−0.297272
$818$	−5833.67	−0.249352
$819$	0	0
$820$	−6232.71	−0.265434
$821$	−16843.0	−0.715987	−0.357993	−	0.933724i	$-0.616539\pi$
$821$	−16843.0	−0.715987	−0.357993	+	0.933724i	$0.616539\pi$
$822$	0	0
$823$	15277.7	0.647080	0.323540	−	0.946214i	$-0.395127\pi$
$823$	15277.7	0.647080	0.323540	+	0.946214i	$0.395127\pi$
$824$	−15457.4	−0.653502
$825$	0	0
$826$	19659.8	0.828150
$827$	−948.282	−0.0398730	−0.0199365	−	0.999801i	$-0.506346\pi$
$827$	−948.282	−0.0398730	−0.0199365	+	0.999801i	$0.506346\pi$
$828$	0	0
$829$	−13177.9	−0.552094	−0.276047	−	0.961144i	$-0.589025\pi$
$829$	−13177.9	−0.552094	−0.276047	+	0.961144i	$0.589025\pi$
$830$	−9762.15	−0.408252
$831$	0	0
$832$	−3814.14	−0.158932
$833$	6056.42	0.251912
$834$	0	0
$835$	−8390.77	−0.347754
$836$	7662.20	0.316989
$837$	0	0
$838$	−27523.3	−1.13458
$839$	−7888.87	−0.324617	−0.162309	−	0.986740i	$-0.551894\pi$
$839$	−7888.87	−0.324617	−0.162309	+	0.986740i	$0.551894\pi$
$840$	0	0
$841$	−19726.5	−0.808827
$842$	28272.6	1.15717
$843$	0	0
$844$	13466.4	0.549207
$845$	6773.37	0.275753
$846$	0	0
$847$	42985.2	1.74379
$848$	−5628.77	−0.227939
$849$	0	0
$850$	−3679.80	−0.148489
$851$	−1755.84	−0.0707278
$852$	0	0
$853$	−16114.6	−0.646837	−0.323419	−	0.946256i	$-0.604832\pi$
$853$	−16114.6	−0.646837	−0.323419	+	0.946256i	$0.604832\pi$
$854$	9187.78	0.368149
$855$	0	0
$856$	−8077.61	−0.322532
$857$	36872.6	1.46971	0.734857	−	0.678222i	$-0.237249\pi$
$857$	36872.6	1.46971	0.734857	+	0.678222i	$0.237249\pi$
$858$	0	0
$859$	758.784	0.0301390	0.0150695	−	0.999886i	$-0.495203\pi$
$859$	758.784	0.0301390	0.0150695	+	0.999886i	$0.495203\pi$
$860$	4580.20	0.181609
$861$	0	0
$862$	22047.5	0.871163
$863$	−28729.9	−1.13323	−0.566614	−	0.823983i	$-0.691747\pi$
$863$	−28729.9	−1.13323	−0.566614	+	0.823983i	$0.691747\pi$
$864$	0	0
$865$	−11079.0	−0.435487
$866$	15429.9	0.605460
$867$	0	0
$868$	−13348.4	−0.521973
$869$	−32325.4	−1.26187
$870$	0	0
$871$	5604.66	0.218033
$872$	−1676.03	−0.0650890
$873$	0	0
$874$	10271.9	0.397542
$875$	2018.30	0.0779784
$876$	0	0
$877$	15696.0	0.604352	0.302176	−	0.953252i	$-0.402287\pi$
$877$	15696.0	0.604352	0.302176	+	0.953252i	$0.402287\pi$
$878$	−19953.6	−0.766971
$879$	0	0
$880$	−5055.35	−0.193654
$881$	2354.49	0.0900396	0.0450198	−	0.998986i	$-0.485665\pi$
$881$	2354.49	0.0900396	0.0450198	+	0.998986i	$0.485665\pi$
$882$	0	0
$883$	−20289.8	−0.773280	−0.386640	−	0.922231i	$-0.626364\pi$
$883$	−20289.8	−0.773280	−0.386640	+	0.922231i	$0.626364\pi$
$884$	17544.1	0.667501
$885$	0	0
$886$	34367.9	1.30318
$887$	30092.3	1.13912	0.569561	−	0.821949i	$-0.307113\pi$
$887$	30092.3	1.13912	0.569561	+	0.821949i	$0.307113\pi$
$888$	0	0
$889$	14320.7	0.540270
$890$	5723.94	0.215581
$891$	0	0
$892$	16224.5	0.609010
$893$	−16637.5	−0.623464
$894$	0	0
$895$	−15977.7	−0.596732
$896$	2066.74	0.0770592
$897$	0	0
$898$	−36659.6	−1.36230
$899$	−14112.4	−0.523555
$900$	0	0
$901$	25890.9	0.957326
$902$	39385.7	1.45388
$903$	0	0
$904$	−6738.60	−0.247923
$905$	11648.3	0.427848
$906$	0	0
$907$	−23075.4	−0.844771	−0.422386	−	0.906416i	$-0.638807\pi$
$907$	−23075.4	−0.844771	−0.422386	+	0.906416i	$0.638807\pi$
$908$	19505.9	0.712914
$909$	0	0
$910$	−9622.61	−0.350535
$911$	−23123.8	−0.840971	−0.420485	−	0.907299i	$-0.638140\pi$
$911$	−23123.8	−0.840971	−0.420485	+	0.907299i	$0.638140\pi$
$912$	0	0
$913$	61688.8	2.23615
$914$	−24231.8	−0.876934
$915$	0	0
$916$	−16504.6	−0.595337
$917$	−27948.8	−1.00649
$918$	0	0
$919$	1940.85	0.0696658	0.0348329	−	0.999393i	$-0.488910\pi$
$919$	1940.85	0.0696658	0.0348329	+	0.999393i	$0.488910\pi$
$920$	−6777.16	−0.242866
$921$	0	0
$922$	−28740.2	−1.02658
$923$	899.412	0.0320742
$924$	0	0
$925$	259.082	0.00920924
$926$	6781.10	0.240649
$927$	0	0
$928$	2185.04	0.0772927
$929$	38961.4	1.37597	0.687987	−	0.725723i	$-0.258494\pi$
$929$	38961.4	1.37597	0.687987	+	0.725723i	$0.258494\pi$
$930$	0	0
$931$	2494.57	0.0878153
$932$	−4828.58	−0.169705
$933$	0	0
$934$	2948.61	0.103299
$935$	23253.3	0.813331
$936$	0	0
$937$	−9568.52	−0.333607	−0.166804	−	0.985990i	$-0.553345\pi$
$937$	−9568.52	−0.333607	−0.166804	+	0.985990i	$0.553345\pi$
$938$	−3036.96	−0.105715
$939$	0	0
$940$	10977.1	0.380885
$941$	5570.74	0.192987	0.0964936	−	0.995334i	$-0.469237\pi$
$941$	5570.74	0.192987	0.0964936	+	0.995334i	$0.469237\pi$
$942$	0	0
$943$	52800.2	1.82334
$944$	9740.75	0.335842
$945$	0	0
$946$	−28943.2	−0.994739
$947$	38473.9	1.32020	0.660102	−	0.751176i	$-0.270513\pi$
$947$	38473.9	1.32020	0.660102	+	0.751176i	$0.270513\pi$
$948$	0	0
$949$	−61217.5	−2.09400
$950$	−1515.66	−0.0517627
$951$	0	0
$952$	−9506.49	−0.323642
$953$	35982.1	1.22306	0.611530	−	0.791222i	$-0.290555\pi$
$953$	35982.1	1.22306	0.611530	+	0.791222i	$0.290555\pi$
$954$	0	0
$955$	17546.1	0.594532
$956$	1095.86	0.0370739
$957$	0	0
$958$	−7410.99	−0.249936
$959$	−22806.6	−0.767949
$960$	0	0
$961$	12924.2	0.433829
$962$	−1235.22	−0.0413981
$963$	0	0
$964$	−11480.2	−0.383559
$965$	−2680.30	−0.0894112
$966$	0	0
$967$	−7058.70	−0.234739	−0.117369	−	0.993088i	$-0.537446\pi$
$967$	−7058.70	−0.234739	−0.117369	+	0.993088i	$0.537446\pi$
$968$	21297.7	0.707162
$969$	0	0
$970$	−6633.71	−0.219583
$971$	−6866.60	−0.226941	−0.113471	−	0.993541i	$-0.536197\pi$
$971$	−6866.60	−0.226941	−0.113471	+	0.993541i	$0.536197\pi$
$972$	0	0
$973$	21616.6	0.712228
$974$	9905.61	0.325869
$975$	0	0
$976$	4552.23	0.149296
$977$	−12268.4	−0.401741	−0.200870	−	0.979618i	$-0.564377\pi$
$977$	−12268.4	−0.401741	−0.200870	+	0.979618i	$0.564377\pi$
$978$	0	0
$979$	−36170.6	−1.18082
$980$	−1645.86	−0.0536479
$981$	0	0
$982$	15609.7	0.507256
$983$	33652.9	1.09192	0.545962	−	0.837810i	$-0.316164\pi$
$983$	33652.9	1.09192	0.545962	+	0.837810i	$0.316164\pi$
$984$	0	0
$985$	−18440.8	−0.596520
$986$	−10050.6	−0.324623
$987$	0	0
$988$	7226.19	0.232688
$989$	−38801.0	−1.24752
$990$	0	0
$991$	−45767.8	−1.46707	−0.733533	−	0.679654i	$-0.762130\pi$
$991$	−45767.8	−1.46707	−0.733533	+	0.679654i	$0.762130\pi$
$992$	−6613.65	−0.211677
$993$	0	0
$994$	−487.359	−0.0155514
$995$	−5713.27	−0.182033
$996$	0	0
$997$	1887.66	0.0599627	0.0299813	−	0.999550i	$-0.490455\pi$
$997$	1887.66	0.0599627	0.0299813	+	0.999550i	$0.490455\pi$
$998$	−35586.6	−1.12873
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	810.4.a.o.1.2		2
3.2	odd	2		810.4.a.i.1.2		2
9.2	odd	6		90.4.e.c.31.1	✓	4
9.4	even	3		270.4.e.b.181.1		4
9.5	odd	6		90.4.e.c.61.1	yes	4
9.7	even	3		270.4.e.b.91.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
90.4.e.c.31.1	✓	4	9.2	odd	6
90.4.e.c.61.1	yes	4	9.5	odd	6
270.4.e.b.91.1		4	9.7	even	3
270.4.e.b.181.1		4	9.4	even	3
810.4.a.i.1.2		2	3.2	odd	2
810.4.a.o.1.2		2	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	810.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +4.00000 q^{4} +5.00000 q^{5} +16.1464 q^{7} +8.00000 q^{8} +O(q^{10})\)	\(q+2.00000 q^{2} +4.00000 q^{4} +5.00000 q^{5} +16.1464 q^{7} +8.00000 q^{8} +10.0000 q^{10} -63.1918 q^{11} -59.5959 q^{13} +32.2929 q^{14} +16.0000 q^{16} -73.5959 q^{17} -30.3133 q^{19} +20.0000 q^{20} -126.384 q^{22} -169.429 q^{23} +25.0000 q^{25} -119.192 q^{26} +64.5857 q^{28} +68.2827 q^{29} -206.677 q^{31} +32.0000 q^{32} -147.192 q^{34} +80.7321 q^{35} +10.3633 q^{37} -60.6265 q^{38} +40.0000 q^{40} -311.636 q^{41} +229.010 q^{43} -252.767 q^{44} -338.858 q^{46} +548.853 q^{47} -82.2929 q^{49} +50.0000 q^{50} -238.384 q^{52} -351.798 q^{53} -315.959 q^{55} +129.171 q^{56} +136.565 q^{58} +608.797 q^{59} +284.514 q^{61} -413.353 q^{62} +64.0000 q^{64} -297.980 q^{65} -94.0444 q^{67} -294.384 q^{68} +161.464 q^{70} -15.0918 q^{71} +1027.21 q^{73} +20.7265 q^{74} -121.253 q^{76} -1020.32 q^{77} +511.544 q^{79} +80.0000 q^{80} -623.271 q^{82} -976.215 q^{83} -367.980 q^{85} +458.020 q^{86} -505.535 q^{88} +572.394 q^{89} -962.261 q^{91} -677.716 q^{92} +1097.71 q^{94} -151.566 q^{95} -663.371 q^{97} -164.586 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} + 8 q^{4} + 10 q^{5} - 2 q^{7} + 16 q^{8}+O(q^{10}) \) 2 * q + 4 * q^2 + 8 * q^4 + 10 * q^5 - 2 * q^7 + 16 * q^8	\( 2 q + 4 q^{2} + 8 q^{4} + 10 q^{5} - 2 q^{7} + 16 q^{8} + 20 q^{10} - 48 q^{11} - 80 q^{13} - 4 q^{14} + 32 q^{16} - 108 q^{17} - 188 q^{19} + 40 q^{20} - 96 q^{22} - 138 q^{23} + 50 q^{25} - 160 q^{26} - 8 q^{28} - 30 q^{29} - 188 q^{31} + 64 q^{32} - 216 q^{34} - 10 q^{35} - 332 q^{37} - 376 q^{38} + 80 q^{40} - 6 q^{41} + 556 q^{43} - 192 q^{44} - 276 q^{46} + 162 q^{47} - 96 q^{49} + 100 q^{50} - 320 q^{52} - 684 q^{53} - 240 q^{55} - 16 q^{56} - 60 q^{58} + 228 q^{59} - 254 q^{61} - 376 q^{62} + 128 q^{64} - 400 q^{65} + 826 q^{67} - 432 q^{68} - 20 q^{70} - 912 q^{71} + 232 q^{73} - 664 q^{74} - 752 q^{76} - 1296 q^{77} - 476 q^{79} + 160 q^{80} - 12 q^{82} + 306 q^{83} - 540 q^{85} + 1112 q^{86} - 384 q^{88} + 1086 q^{89} - 592 q^{91} - 552 q^{92} + 324 q^{94} - 940 q^{95} + 868 q^{97} - 192 q^{98}+O(q^{100}) \) 2 * q + 4 * q^2 + 8 * q^4 + 10 * q^5 - 2 * q^7 + 16 * q^8 + 20 * q^10 - 48 * q^11 - 80 * q^13 - 4 * q^14 + 32 * q^16 - 108 * q^17 - 188 * q^19 + 40 * q^20 - 96 * q^22 - 138 * q^23 + 50 * q^25 - 160 * q^26 - 8 * q^28 - 30 * q^29 - 188 * q^31 + 64 * q^32 - 216 * q^34 - 10 * q^35 - 332 * q^37 - 376 * q^38 + 80 * q^40 - 6 * q^41 + 556 * q^43 - 192 * q^44 - 276 * q^46 + 162 * q^47 - 96 * q^49 + 100 * q^50 - 320 * q^52 - 684 * q^53 - 240 * q^55 - 16 * q^56 - 60 * q^58 + 228 * q^59 - 254 * q^61 - 376 * q^62 + 128 * q^64 - 400 * q^65 + 826 * q^67 - 432 * q^68 - 20 * q^70 - 912 * q^71 + 232 * q^73 - 664 * q^74 - 752 * q^76 - 1296 * q^77 - 476 * q^79 + 160 * q^80 - 12 * q^82 + 306 * q^83 - 540 * q^85 + 1112 * q^86 - 384 * q^88 + 1086 * q^89 - 592 * q^91 - 552 * q^92 + 324 * q^94 - 940 * q^95 + 868 * q^97 - 192 * q^98

Introduction

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