LMFDB - Embedded newform 2070.4.a.bj.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2070,4,Mod(1,2070)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2070, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2070.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2070.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:	$122.133953712$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 68x^{2} - 111x + 342 $ x^4 - 68x^2 - 111x + 342
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 230)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-6.57209$ of defining polynomial
Character	$\chi$	$=$	2070.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{2} +4.00000 q^{4} +5.00000 q^{5} -35.4229 q^{7} +8.00000 q^{8} +O(q^{10})$	\(q+2.00000 q^{2} +4.00000 q^{4} +5.00000 q^{5} -35.4229 q^{7} +8.00000 q^{8} +10.0000 q^{10} +16.6298 q^{11} -79.9132 q^{13} -70.8458 q^{14} +16.0000 q^{16} +46.8219 q^{17} -110.653 q^{19} +20.0000 q^{20} +33.2597 q^{22} +23.0000 q^{23} +25.0000 q^{25} -159.826 q^{26} -141.692 q^{28} +0.836422 q^{29} -119.836 q^{31} +32.0000 q^{32} +93.6438 q^{34} -177.114 q^{35} +368.201 q^{37} -221.307 q^{38} +40.0000 q^{40} +95.7927 q^{41} +331.961 q^{43} +66.5193 q^{44} +46.0000 q^{46} +535.037 q^{47} +911.782 q^{49} +50.0000 q^{50} -319.653 q^{52} -409.345 q^{53} +83.1492 q^{55} -283.383 q^{56} +1.67284 q^{58} +352.950 q^{59} -507.223 q^{61} -239.672 q^{62} +64.0000 q^{64} -399.566 q^{65} -820.056 q^{67} +187.288 q^{68} -354.229 q^{70} +733.770 q^{71} -91.4599 q^{73} +736.402 q^{74} -442.613 q^{76} -589.077 q^{77} +329.381 q^{79} +80.0000 q^{80} +191.585 q^{82} +753.834 q^{83} +234.110 q^{85} +663.922 q^{86} +133.039 q^{88} +1050.14 q^{89} +2830.76 q^{91} +92.0000 q^{92} +1070.07 q^{94} -553.267 q^{95} -271.928 q^{97} +1823.56 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{2} + 16 q^{4} + 20 q^{5} - q^{7} + 32 q^{8}+O(q^{10}) $ 4 * q + 8 * q^2 + 16 * q^4 + 20 * q^5 - q^7 + 32 * q^8	$ 4 q + 8 q^{2} + 16 q^{4} + 20 q^{5} - q^{7} + 32 q^{8} + 40 q^{10} + 39 q^{11} - 20 q^{13} - 2 q^{14} + 64 q^{16} + 23 q^{17} + 53 q^{19} + 80 q^{20} + 78 q^{22} + 92 q^{23} + 100 q^{25} - 40 q^{26} - 4 q^{28} - 161 q^{29} + 388 q^{31} + 128 q^{32} + 46 q^{34} - 5 q^{35} + 466 q^{37} + 106 q^{38} + 160 q^{40} - 484 q^{41} + 894 q^{43} + 156 q^{44} + 184 q^{46} + 265 q^{47} + 1643 q^{49} + 200 q^{50} - 80 q^{52} - 576 q^{53} + 195 q^{55} - 8 q^{56} - 322 q^{58} + 94 q^{59} + 1153 q^{61} + 776 q^{62} + 256 q^{64} - 100 q^{65} - 1472 q^{67} + 92 q^{68} - 10 q^{70} - 200 q^{71} + 1147 q^{73} + 932 q^{74} + 212 q^{76} + 2176 q^{77} - 908 q^{79} + 320 q^{80} - 968 q^{82} + 1048 q^{83} + 115 q^{85} + 1788 q^{86} + 312 q^{88} + 1784 q^{89} + 2329 q^{91} + 368 q^{92} + 530 q^{94} + 265 q^{95} - 2047 q^{97} + 3286 q^{98}+O(q^{100}) $ 4 * q + 8 * q^2 + 16 * q^4 + 20 * q^5 - q^7 + 32 * q^8 + 40 * q^10 + 39 * q^11 - 20 * q^13 - 2 * q^14 + 64 * q^16 + 23 * q^17 + 53 * q^19 + 80 * q^20 + 78 * q^22 + 92 * q^23 + 100 * q^25 - 40 * q^26 - 4 * q^28 - 161 * q^29 + 388 * q^31 + 128 * q^32 + 46 * q^34 - 5 * q^35 + 466 * q^37 + 106 * q^38 + 160 * q^40 - 484 * q^41 + 894 * q^43 + 156 * q^44 + 184 * q^46 + 265 * q^47 + 1643 * q^49 + 200 * q^50 - 80 * q^52 - 576 * q^53 + 195 * q^55 - 8 * q^56 - 322 * q^58 + 94 * q^59 + 1153 * q^61 + 776 * q^62 + 256 * q^64 - 100 * q^65 - 1472 * q^67 + 92 * q^68 - 10 * q^70 - 200 * q^71 + 1147 * q^73 + 932 * q^74 + 212 * q^76 + 2176 * q^77 - 908 * q^79 + 320 * q^80 - 968 * q^82 + 1048 * q^83 + 115 * q^85 + 1788 * q^86 + 312 * q^88 + 1784 * q^89 + 2329 * q^91 + 368 * q^92 + 530 * q^94 + 265 * q^95 - 2047 * q^97 + 3286 * 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
$7$	−35.4229	−1.91266	−0.956328	−	0.292294i	$-0.905581\pi$
$7$	−35.4229	−1.91266	−0.956328	+	0.292294i	$0.905581\pi$
$8$	8.00000	0.353553
$9$	0	0
$10$	10.0000	0.316228
$11$	16.6298	0.455826	0.227913	−	0.973682i	$-0.426810\pi$
$11$	16.6298	0.455826	0.227913	+	0.973682i	$0.426810\pi$
$12$	0	0
$13$	−79.9132	−1.70492	−0.852459	−	0.522795i	$-0.824889\pi$
$13$	−79.9132	−1.70492	−0.852459	+	0.522795i	$0.824889\pi$
$14$	−70.8458	−1.35245
$15$	0	0
$16$	16.0000	0.250000
$17$	46.8219	0.667999	0.333999	−	0.942573i	$-0.391602\pi$
$17$	46.8219	0.667999	0.333999	+	0.942573i	$0.391602\pi$
$18$	0	0
$19$	−110.653	−1.33608	−0.668042	−	0.744123i	$-0.732868\pi$
$19$	−110.653	−1.33608	−0.668042	+	0.744123i	$0.732868\pi$
$20$	20.0000	0.223607
$21$	0	0
$22$	33.2597	0.322318
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	25.0000	0.200000
$26$	−159.826	−1.20556
$27$	0	0
$28$	−141.692	−0.956328
$29$	0.836422	0.00535585	0.00267793	−	0.999996i	$-0.499148\pi$
$29$	0.836422	0.00535585	0.00267793	+	0.999996i	$0.499148\pi$
$30$	0	0
$31$	−119.836	−0.694295	−0.347147	−	0.937811i	$-0.612850\pi$
$31$	−119.836	−0.694295	−0.347147	+	0.937811i	$0.612850\pi$
$32$	32.0000	0.176777
$33$	0	0
$34$	93.6438	0.472346
$35$	−177.114	−0.855366
$36$	0	0
$37$	368.201	1.63600	0.817998	−	0.575221i	$-0.195084\pi$
$37$	368.201	1.63600	0.817998	+	0.575221i	$0.195084\pi$
$38$	−221.307	−0.944755
$39$	0	0
$40$	40.0000	0.158114
$41$	95.7927	0.364886	0.182443	−	0.983216i	$-0.441600\pi$
$41$	95.7927	0.364886	0.182443	+	0.983216i	$0.441600\pi$
$42$	0	0
$43$	331.961	1.17729	0.588647	−	0.808390i	$-0.299661\pi$
$43$	331.961	1.17729	0.588647	+	0.808390i	$0.299661\pi$
$44$	66.5193	0.227913
$45$	0	0
$46$	46.0000	0.147442
$47$	535.037	1.66049	0.830246	−	0.557397i	$-0.188200\pi$
$47$	535.037	1.66049	0.830246	+	0.557397i	$0.188200\pi$
$48$	0	0
$49$	911.782	2.65826
$50$	50.0000	0.141421
$51$	0	0
$52$	−319.653	−0.852459
$53$	−409.345	−1.06090	−0.530452	−	0.847715i	$-0.677978\pi$
$53$	−409.345	−1.06090	−0.530452	+	0.847715i	$0.677978\pi$
$54$	0	0
$55$	83.1492	0.203852
$56$	−283.383	−0.676226
$57$	0	0
$58$	1.67284	0.00378716
$59$	352.950	0.778817	0.389408	−	0.921065i	$-0.372679\pi$
$59$	352.950	0.778817	0.389408	+	0.921065i	$0.372679\pi$
$60$	0	0
$61$	−507.223	−1.06464	−0.532322	−	0.846542i	$-0.678680\pi$
$61$	−507.223	−1.06464	−0.532322	+	0.846542i	$0.678680\pi$
$62$	−239.672	−0.490941
$63$	0	0
$64$	64.0000	0.125000
$65$	−399.566	−0.762462
$66$	0	0
$67$	−820.056	−1.49531	−0.747655	−	0.664087i	$-0.768820\pi$
$67$	−820.056	−1.49531	−0.747655	+	0.664087i	$0.768820\pi$
$68$	187.288	0.333999
$69$	0	0
$70$	−354.229	−0.604835
$71$	733.770	1.22651	0.613257	−	0.789884i	$-0.289859\pi$
$71$	733.770	1.22651	0.613257	+	0.789884i	$0.289859\pi$
$72$	0	0
$73$	−91.4599	−0.146638	−0.0733190	−	0.997309i	$-0.523359\pi$
$73$	−91.4599	−0.146638	−0.0733190	+	0.997309i	$0.523359\pi$
$74$	736.402	1.15682
$75$	0	0
$76$	−442.613	−0.668042
$77$	−589.077	−0.871838
$78$	0	0
$79$	329.381	0.469092	0.234546	−	0.972105i	$-0.424640\pi$
$79$	329.381	0.469092	0.234546	+	0.972105i	$0.424640\pi$
$80$	80.0000	0.111803
$81$	0	0
$82$	191.585	0.258013
$83$	753.834	0.996916	0.498458	−	0.866914i	$-0.333900\pi$
$83$	753.834	0.996916	0.498458	+	0.866914i	$0.333900\pi$
$84$	0	0
$85$	234.110	0.298738
$86$	663.922	0.832472
$87$	0	0
$88$	133.039	0.161159
$89$	1050.14	1.25073	0.625365	−	0.780332i	$-0.284950\pi$
$89$	1050.14	1.25073	0.625365	+	0.780332i	$0.284950\pi$
$90$	0	0
$91$	2830.76	3.26092
$92$	92.0000	0.104257
$93$	0	0
$94$	1070.07	1.17415
$95$	−553.267	−0.597515
$96$	0	0
$97$	−271.928	−0.284641	−0.142320	−	0.989821i	$-0.545456\pi$
$97$	−271.928	−0.284641	−0.142320	+	0.989821i	$0.545456\pi$
$98$	1823.56	1.87967
$99$	0	0
$100$	100.000	0.100000
$101$	−1658.68	−1.63410	−0.817052	−	0.576563i	$-0.804393\pi$
$101$	−1658.68	−1.63410	−0.817052	+	0.576563i	$0.804393\pi$
$102$	0	0
$103$	1735.52	1.66025	0.830123	−	0.557580i	$-0.188270\pi$
$103$	1735.52	1.66025	0.830123	+	0.557580i	$0.188270\pi$
$104$	−639.305	−0.602779
$105$	0	0
$106$	−818.691	−0.750172
$107$	1629.88	1.47258	0.736292	−	0.676664i	$-0.236575\pi$
$107$	1629.88	1.47258	0.736292	+	0.676664i	$0.236575\pi$
$108$	0	0
$109$	−432.151	−0.379748	−0.189874	−	0.981808i	$-0.560808\pi$
$109$	−432.151	−0.379748	−0.189874	+	0.981808i	$0.560808\pi$
$110$	166.298	0.144145
$111$	0	0
$112$	−566.766	−0.478164
$113$	240.115	0.199895	0.0999476	−	0.994993i	$-0.468132\pi$
$113$	240.115	0.199895	0.0999476	+	0.994993i	$0.468132\pi$
$114$	0	0
$115$	115.000	0.0932505
$116$	3.34569	0.00267793
$117$	0	0
$118$	705.900	0.550707
$119$	−1658.57	−1.27765
$120$	0	0
$121$	−1054.45	−0.792223
$122$	−1014.45	−0.752817
$123$	0	0
$124$	−479.343	−0.347147
$125$	125.000	0.0894427
$126$	0	0
$127$	−210.993	−0.147422	−0.0737110	−	0.997280i	$-0.523484\pi$
$127$	−210.993	−0.147422	−0.0737110	+	0.997280i	$0.523484\pi$
$128$	128.000	0.0883883
$129$	0	0
$130$	−799.132	−0.539142
$131$	746.178	0.497663	0.248832	−	0.968547i	$-0.419953\pi$
$131$	746.178	0.497663	0.248832	+	0.968547i	$0.419953\pi$
$132$	0	0
$133$	3919.66	2.55547
$134$	−1640.11	−1.05734
$135$	0	0
$136$	374.575	0.236173
$137$	2420.07	1.50920	0.754601	−	0.656184i	$-0.227830\pi$
$137$	2420.07	1.50920	0.754601	+	0.656184i	$0.227830\pi$
$138$	0	0
$139$	924.030	0.563850	0.281925	−	0.959436i	$-0.409027\pi$
$139$	924.030	0.563850	0.281925	+	0.959436i	$0.409027\pi$
$140$	−708.458	−0.427683
$141$	0	0
$142$	1467.54	0.867276
$143$	−1328.94	−0.777145
$144$	0	0
$145$	4.18211	0.00239521
$146$	−182.920	−0.103689
$147$	0	0
$148$	1472.80	0.817998
$149$	−430.614	−0.236760	−0.118380	−	0.992968i	$-0.537770\pi$
$149$	−430.614	−0.236760	−0.118380	+	0.992968i	$0.537770\pi$
$150$	0	0
$151$	−25.4118	−0.0136953	−0.00684763	−	0.999977i	$-0.502180\pi$
$151$	−25.4118	−0.0136953	−0.00684763	+	0.999977i	$0.502180\pi$
$152$	−885.226	−0.472377
$153$	0	0
$154$	−1178.15	−0.616483
$155$	−599.179	−0.310498
$156$	0	0
$157$	−1580.29	−0.803317	−0.401658	−	0.915790i	$-0.631566\pi$
$157$	−1580.29	−0.803317	−0.401658	+	0.915790i	$0.631566\pi$
$158$	658.762	0.331698
$159$	0	0
$160$	160.000	0.0790569
$161$	−814.727	−0.398817
$162$	0	0
$163$	458.739	0.220437	0.110218	−	0.993907i	$-0.464845\pi$
$163$	458.739	0.220437	0.110218	+	0.993907i	$0.464845\pi$
$164$	383.171	0.182443
$165$	0	0
$166$	1507.67	0.704926
$167$	561.275	0.260076	0.130038	−	0.991509i	$-0.458490\pi$
$167$	561.275	0.260076	0.130038	+	0.991509i	$0.458490\pi$
$168$	0	0
$169$	4189.11	1.90674
$170$	468.219	0.211240
$171$	0	0
$172$	1327.84	0.588647
$173$	−1573.95	−0.691705	−0.345852	−	0.938289i	$-0.612410\pi$
$173$	−1573.95	−0.691705	−0.345852	+	0.938289i	$0.612410\pi$
$174$	0	0
$175$	−885.572	−0.382531
$176$	266.077	0.113956
$177$	0	0
$178$	2100.29	0.884400
$179$	−2215.59	−0.925145	−0.462572	−	0.886581i	$-0.653073\pi$
$179$	−2215.59	−0.925145	−0.462572	+	0.886581i	$0.653073\pi$
$180$	0	0
$181$	873.497	0.358710	0.179355	−	0.983784i	$-0.442599\pi$
$181$	873.497	0.358710	0.179355	+	0.983784i	$0.442599\pi$
$182$	5661.51	2.30582
$183$	0	0
$184$	184.000	0.0737210
$185$	1841.00	0.731640
$186$	0	0
$187$	778.641	0.304491
$188$	2140.15	0.830246
$189$	0	0
$190$	−1106.53	−0.422507
$191$	2497.74	0.946230	0.473115	−	0.881001i	$-0.343130\pi$
$191$	2497.74	0.946230	0.473115	+	0.881001i	$0.343130\pi$
$192$	0	0
$193$	909.155	0.339080	0.169540	−	0.985523i	$-0.445772\pi$
$193$	909.155	0.339080	0.169540	+	0.985523i	$0.445772\pi$
$194$	−543.857	−0.201271
$195$	0	0
$196$	3647.13	1.32913
$197$	−608.627	−0.220116	−0.110058	−	0.993925i	$-0.535104\pi$
$197$	−608.627	−0.220116	−0.110058	+	0.993925i	$0.535104\pi$
$198$	0	0
$199$	−2304.98	−0.821083	−0.410542	−	0.911842i	$-0.634660\pi$
$199$	−2304.98	−0.821083	−0.410542	+	0.911842i	$0.634660\pi$
$200$	200.000	0.0707107
$201$	0	0
$202$	−3317.36	−1.15549
$203$	−29.6285	−0.0102439
$204$	0	0
$205$	478.963	0.163182
$206$	3471.03	1.17397
$207$	0	0
$208$	−1278.61	−0.426229
$209$	−1840.15	−0.609022
$210$	0	0
$211$	−4373.37	−1.42690	−0.713448	−	0.700708i	$-0.752868\pi$
$211$	−4373.37	−1.42690	−0.713448	+	0.700708i	$0.752868\pi$
$212$	−1637.38	−0.530452
$213$	0	0
$214$	3259.76	1.04127
$215$	1659.81	0.526502
$216$	0	0
$217$	4244.93	1.32795
$218$	−864.303	−0.268523
$219$	0	0
$220$	332.597	0.101926
$221$	−3741.69	−1.13888
$222$	0	0
$223$	5439.50	1.63343	0.816717	−	0.577038i	$-0.195792\pi$
$223$	5439.50	1.63343	0.816717	+	0.577038i	$0.195792\pi$
$224$	−1133.53	−0.338113
$225$	0	0
$226$	480.231	0.141347
$227$	−216.498	−0.0633017	−0.0316508	−	0.999499i	$-0.510076\pi$
$227$	−216.498	−0.0633017	−0.0316508	+	0.999499i	$0.510076\pi$
$228$	0	0
$229$	−924.664	−0.266828	−0.133414	−	0.991060i	$-0.542594\pi$
$229$	−924.664	−0.266828	−0.133414	+	0.991060i	$0.542594\pi$
$230$	230.000	0.0659380
$231$	0	0
$232$	6.69138	0.00189358
$233$	2369.69	0.666280	0.333140	−	0.942877i	$-0.391892\pi$
$233$	2369.69	0.666280	0.333140	+	0.942877i	$0.391892\pi$
$234$	0	0
$235$	2675.18	0.742595
$236$	1411.80	0.389408
$237$	0	0
$238$	−3317.14	−0.903437
$239$	2769.60	0.749583	0.374791	−	0.927109i	$-0.377714\pi$
$239$	2769.60	0.749583	0.374791	+	0.927109i	$0.377714\pi$
$240$	0	0
$241$	−5334.10	−1.42572	−0.712862	−	0.701305i	$-0.752601\pi$
$241$	−5334.10	−1.42572	−0.712862	+	0.701305i	$0.752601\pi$
$242$	−2108.90	−0.560186
$243$	0	0
$244$	−2028.89	−0.532322
$245$	4558.91	1.18881
$246$	0	0
$247$	8842.66	2.27791
$248$	−958.686	−0.245470
$249$	0	0
$250$	250.000	0.0632456
$251$	−4677.13	−1.17617	−0.588084	−	0.808800i	$-0.700118\pi$
$251$	−4677.13	−1.17617	−0.588084	+	0.808800i	$0.700118\pi$
$252$	0	0
$253$	382.486	0.0950463
$254$	−421.986	−0.104243
$255$	0	0
$256$	256.000	0.0625000
$257$	2602.99	0.631789	0.315895	−	0.948794i	$-0.397695\pi$
$257$	2602.99	0.631789	0.315895	+	0.948794i	$0.397695\pi$
$258$	0	0
$259$	−13042.7	−3.12910
$260$	−1598.26	−0.381231
$261$	0	0
$262$	1492.36	0.351901
$263$	3411.30	0.799809	0.399904	−	0.916557i	$-0.369043\pi$
$263$	3411.30	0.799809	0.399904	+	0.916557i	$0.369043\pi$
$264$	0	0
$265$	−2046.73	−0.474451
$266$	7839.32	1.80699
$267$	0	0
$268$	−3280.22	−0.747655
$269$	−4366.56	−0.989718	−0.494859	−	0.868973i	$-0.664780\pi$
$269$	−4366.56	−0.989718	−0.494859	+	0.868973i	$0.664780\pi$
$270$	0	0
$271$	5682.45	1.27374	0.636872	−	0.770970i	$-0.280228\pi$
$271$	5682.45	1.27374	0.636872	+	0.770970i	$0.280228\pi$
$272$	749.150	0.167000
$273$	0	0
$274$	4840.14	1.06717
$275$	415.746	0.0911652
$276$	0	0
$277$	−1884.62	−0.408794	−0.204397	−	0.978888i	$-0.565523\pi$
$277$	−1884.62	−0.408794	−0.204397	+	0.978888i	$0.565523\pi$
$278$	1848.06	0.398702
$279$	0	0
$280$	−1416.92	−0.302418
$281$	2706.42	0.574561	0.287281	−	0.957846i	$-0.407249\pi$
$281$	2706.42	0.574561	0.287281	+	0.957846i	$0.407249\pi$
$282$	0	0
$283$	−5963.64	−1.25266	−0.626328	−	0.779560i	$-0.715443\pi$
$283$	−5963.64	−1.25266	−0.626328	+	0.779560i	$0.715443\pi$
$284$	2935.08	0.613257
$285$	0	0
$286$	−2657.89	−0.549525
$287$	−3393.26	−0.697901
$288$	0	0
$289$	−2720.71	−0.553778
$290$	8.36422	0.00169367
$291$	0	0
$292$	−365.840	−0.0733190
$293$	4735.85	0.944272	0.472136	−	0.881526i	$-0.343483\pi$
$293$	4735.85	0.944272	0.472136	+	0.881526i	$0.343483\pi$
$294$	0	0
$295$	1764.75	0.348297
$296$	2945.61	0.578412
$297$	0	0
$298$	−861.227	−0.167415
$299$	−1838.00	−0.355500
$300$	0	0
$301$	−11759.0	−2.25176
$302$	−50.8236	−0.00968401
$303$	0	0
$304$	−1770.45	−0.334021
$305$	−2536.12	−0.476123
$306$	0	0
$307$	−769.241	−0.143006	−0.0715031	−	0.997440i	$-0.522780\pi$
$307$	−769.241	−0.143006	−0.0715031	+	0.997440i	$0.522780\pi$
$308$	−2356.31	−0.435919
$309$	0	0
$310$	−1198.36	−0.219555
$311$	5592.71	1.01972	0.509861	−	0.860257i	$-0.329697\pi$
$311$	5592.71	1.01972	0.509861	+	0.860257i	$0.329697\pi$
$312$	0	0
$313$	9777.19	1.76562	0.882811	−	0.469729i	$-0.155648\pi$
$313$	9777.19	1.76562	0.882811	+	0.469729i	$0.155648\pi$
$314$	−3160.58	−0.568031
$315$	0	0
$316$	1317.52	0.234546
$317$	−1868.51	−0.331060	−0.165530	−	0.986205i	$-0.552934\pi$
$317$	−1868.51	−0.331060	−0.165530	+	0.986205i	$0.552934\pi$
$318$	0	0
$319$	13.9096	0.00244134
$320$	320.000	0.0559017
$321$	0	0
$322$	−1629.45	−0.282006
$323$	−5181.00	−0.892503
$324$	0	0
$325$	−1997.83	−0.340983
$326$	917.477	0.155872
$327$	0	0
$328$	766.342	0.129007
$329$	−18952.6	−3.17595
$330$	0	0
$331$	6966.91	1.15691	0.578454	−	0.815715i	$-0.303656\pi$
$331$	6966.91	1.15691	0.578454	+	0.815715i	$0.303656\pi$
$332$	3015.34	0.498458
$333$	0	0
$334$	1122.55	0.183902
$335$	−4100.28	−0.668723
$336$	0	0
$337$	−17.5487	−0.00283661	−0.00141830	−	0.999999i	$-0.500451\pi$
$337$	−17.5487	−0.00283661	−0.00141830	+	0.999999i	$0.500451\pi$
$338$	8378.23	1.34827
$339$	0	0
$340$	936.438	0.149369
$341$	−1992.85	−0.316477
$342$	0	0
$343$	−20147.9	−3.17167
$344$	2655.69	0.416236
$345$	0	0
$346$	−3147.89	−0.489109
$347$	9859.30	1.52529	0.762644	−	0.646818i	$-0.223901\pi$
$347$	9859.30	1.52529	0.762644	+	0.646818i	$0.223901\pi$
$348$	0	0
$349$	7200.25	1.10436	0.552178	−	0.833726i	$-0.313797\pi$
$349$	7200.25	1.10436	0.552178	+	0.833726i	$0.313797\pi$
$350$	−1771.14	−0.270491
$351$	0	0
$352$	532.155	0.0805794
$353$	6054.58	0.912897	0.456449	−	0.889750i	$-0.349121\pi$
$353$	6054.58	0.912897	0.456449	+	0.889750i	$0.349121\pi$
$354$	0	0
$355$	3668.85	0.548513
$356$	4200.57	0.625365
$357$	0	0
$358$	−4431.18	−0.654176
$359$	−2734.26	−0.401974	−0.200987	−	0.979594i	$-0.564415\pi$
$359$	−2734.26	−0.401974	−0.200987	+	0.979594i	$0.564415\pi$
$360$	0	0
$361$	5385.16	0.785123
$362$	1746.99	0.253646
$363$	0	0
$364$	11323.0	1.63046
$365$	−457.300	−0.0655785
$366$	0	0
$367$	−2465.50	−0.350676	−0.175338	−	0.984508i	$-0.556102\pi$
$367$	−2465.50	−0.350676	−0.175338	+	0.984508i	$0.556102\pi$
$368$	368.000	0.0521286
$369$	0	0
$370$	3682.01	0.517347
$371$	14500.2	2.02915
$372$	0	0
$373$	5704.45	0.791863	0.395932	−	0.918280i	$-0.370422\pi$
$373$	5704.45	0.791863	0.395932	+	0.918280i	$0.370422\pi$
$374$	1557.28	0.215308
$375$	0	0
$376$	4280.29	0.587073
$377$	−66.8411	−0.00913128
$378$	0	0
$379$	10231.9	1.38675	0.693376	−	0.720576i	$-0.256122\pi$
$379$	10231.9	1.38675	0.693376	+	0.720576i	$0.256122\pi$
$380$	−2213.07	−0.298758
$381$	0	0
$382$	4995.47	0.669086
$383$	−6321.34	−0.843356	−0.421678	−	0.906746i	$-0.638559\pi$
$383$	−6321.34	−0.843356	−0.421678	+	0.906746i	$0.638559\pi$
$384$	0	0
$385$	−2945.38	−0.389898
$386$	1818.31	0.239766
$387$	0	0
$388$	−1087.71	−0.142320
$389$	925.594	0.120641	0.0603207	−	0.998179i	$-0.480788\pi$
$389$	925.594	0.120641	0.0603207	+	0.998179i	$0.480788\pi$
$390$	0	0
$391$	1076.90	0.139287
$392$	7294.25	0.939835
$393$	0	0
$394$	−1217.25	−0.155646
$395$	1646.91	0.209784
$396$	0	0
$397$	−2706.75	−0.342187	−0.171093	−	0.985255i	$-0.554730\pi$
$397$	−2706.75	−0.342187	−0.171093	+	0.985255i	$0.554730\pi$
$398$	−4609.96	−0.580594
$399$	0	0
$400$	400.000	0.0500000
$401$	14095.3	1.75532	0.877661	−	0.479281i	$-0.159103\pi$
$401$	14095.3	1.75532	0.877661	+	0.479281i	$0.159103\pi$
$402$	0	0
$403$	9576.45	1.18372
$404$	−6634.71	−0.817052
$405$	0	0
$406$	−59.2570	−0.00724353
$407$	6123.12	0.745729
$408$	0	0
$409$	−10846.0	−1.31125	−0.655625	−	0.755087i	$-0.727595\pi$
$409$	−10846.0	−1.31125	−0.655625	+	0.755087i	$0.727595\pi$
$410$	957.927	0.115387
$411$	0	0
$412$	6942.06	0.830123
$413$	−12502.5	−1.48961
$414$	0	0
$415$	3769.17	0.445834
$416$	−2557.22	−0.301390
$417$	0	0
$418$	−3680.29	−0.430644
$419$	5626.55	0.656026	0.328013	−	0.944673i	$-0.393621\pi$
$419$	5626.55	0.656026	0.328013	+	0.944673i	$0.393621\pi$
$420$	0	0
$421$	−7109.09	−0.822983	−0.411491	−	0.911414i	$-0.634992\pi$
$421$	−7109.09	−0.822983	−0.411491	+	0.911414i	$0.634992\pi$
$422$	−8746.74	−1.00897
$423$	0	0
$424$	−3274.76	−0.375086
$425$	1170.55	0.133600
$426$	0	0
$427$	17967.3	2.03630
$428$	6519.52	0.736292
$429$	0	0
$430$	3319.61	0.372293
$431$	−4464.14	−0.498909	−0.249455	−	0.968387i	$-0.580251\pi$
$431$	−4464.14	−0.498909	−0.249455	+	0.968387i	$0.580251\pi$
$432$	0	0
$433$	11009.4	1.22189	0.610947	−	0.791672i	$-0.290789\pi$
$433$	11009.4	1.22189	0.610947	+	0.791672i	$0.290789\pi$
$434$	8489.86	0.939001
$435$	0	0
$436$	−1728.61	−0.189874
$437$	−2545.03	−0.278593
$438$	0	0
$439$	−3052.70	−0.331885	−0.165943	−	0.986135i	$-0.553067\pi$
$439$	−3052.70	−0.331885	−0.165943	+	0.986135i	$0.553067\pi$
$440$	665.193	0.0720724
$441$	0	0
$442$	−7483.37	−0.805312
$443$	4465.82	0.478956	0.239478	−	0.970902i	$-0.423024\pi$
$443$	4465.82	0.478956	0.239478	+	0.970902i	$0.423024\pi$
$444$	0	0
$445$	5250.72	0.559343
$446$	10879.0	1.15501
$447$	0	0
$448$	−2267.07	−0.239082
$449$	−9040.14	−0.950179	−0.475090	−	0.879937i	$-0.657584\pi$
$449$	−9040.14	−0.950179	−0.475090	+	0.879937i	$0.657584\pi$
$450$	0	0
$451$	1593.02	0.166324
$452$	960.462	0.0999476
$453$	0	0
$454$	−432.996	−0.0447610
$455$	14153.8	1.45833
$456$	0	0
$457$	7756.88	0.793986	0.396993	−	0.917822i	$-0.370054\pi$
$457$	7756.88	0.793986	0.396993	+	0.917822i	$0.370054\pi$
$458$	−1849.33	−0.188676
$459$	0	0
$460$	460.000	0.0466252
$461$	−7064.19	−0.713692	−0.356846	−	0.934163i	$-0.616148\pi$
$461$	−7064.19	−0.713692	−0.356846	+	0.934163i	$0.616148\pi$
$462$	0	0
$463$	12599.7	1.26470	0.632350	−	0.774682i	$-0.282090\pi$
$463$	12599.7	1.26470	0.632350	+	0.774682i	$0.282090\pi$
$464$	13.3828	0.00133896
$465$	0	0
$466$	4739.37	0.471131
$467$	−14746.0	−1.46117	−0.730583	−	0.682824i	$-0.760751\pi$
$467$	−14746.0	−1.46117	−0.730583	+	0.682824i	$0.760751\pi$
$468$	0	0
$469$	29048.8	2.86002
$470$	5350.37	0.525094
$471$	0	0
$472$	2823.60	0.275353
$473$	5520.46	0.536641
$474$	0	0
$475$	−2766.33	−0.267217
$476$	−6634.27	−0.638826
$477$	0	0
$478$	5539.19	0.530035
$479$	17411.2	1.66084	0.830418	−	0.557141i	$-0.188102\pi$
$479$	17411.2	1.66084	0.830418	+	0.557141i	$0.188102\pi$
$480$	0	0
$481$	−29424.1	−2.78924
$482$	−10668.2	−1.00814
$483$	0	0
$484$	−4217.79	−0.396111
$485$	−1359.64	−0.127295
$486$	0	0
$487$	1320.34	0.122855	0.0614276	−	0.998112i	$-0.480435\pi$
$487$	1320.34	0.122855	0.0614276	+	0.998112i	$0.480435\pi$
$488$	−4057.78	−0.376408
$489$	0	0
$490$	9117.82	0.840614
$491$	−17115.8	−1.57317	−0.786583	−	0.617485i	$-0.788152\pi$
$491$	−17115.8	−1.57317	−0.786583	+	0.617485i	$0.788152\pi$
$492$	0	0
$493$	39.1629	0.00357770
$494$	17685.3	1.61073
$495$	0	0
$496$	−1917.37	−0.173574
$497$	−25992.3	−2.34590
$498$	0	0
$499$	17540.6	1.57360	0.786798	−	0.617210i	$-0.211737\pi$
$499$	17540.6	1.57360	0.786798	+	0.617210i	$0.211737\pi$
$500$	500.000	0.0447214
$501$	0	0
$502$	−9354.27	−0.831676
$503$	4934.98	0.437455	0.218727	−	0.975786i	$-0.429809\pi$
$503$	4934.98	0.437455	0.218727	+	0.975786i	$0.429809\pi$
$504$	0	0
$505$	−8293.39	−0.730794
$506$	764.972	0.0672079
$507$	0	0
$508$	−843.972	−0.0737110
$509$	−11927.3	−1.03865	−0.519323	−	0.854578i	$-0.673816\pi$
$509$	−11927.3	−1.03865	−0.519323	+	0.854578i	$0.673816\pi$
$510$	0	0
$511$	3239.78	0.280468
$512$	512.000	0.0441942
$513$	0	0
$514$	5205.97	0.446743
$515$	8677.58	0.742485
$516$	0	0
$517$	8897.57	0.756895
$518$	−26085.5	−2.21261
$519$	0	0
$520$	−3196.53	−0.269571
$521$	9592.76	0.806653	0.403327	−	0.915056i	$-0.367854\pi$
$521$	9592.76	0.806653	0.403327	+	0.915056i	$0.367854\pi$
$522$	0	0
$523$	6525.20	0.545558	0.272779	−	0.962077i	$-0.412057\pi$
$523$	6525.20	0.545558	0.272779	+	0.962077i	$0.412057\pi$
$524$	2984.71	0.248832
$525$	0	0
$526$	6822.60	0.565550
$527$	−5610.94	−0.463788
$528$	0	0
$529$	529.000	0.0434783
$530$	−4093.45	−0.335487
$531$	0	0
$532$	15678.6	1.27774
$533$	−7655.10	−0.622100
$534$	0	0
$535$	8149.40	0.658560
$536$	−6560.45	−0.528672
$537$	0	0
$538$	−8733.13	−0.699836
$539$	15162.8	1.21170
$540$	0	0
$541$	5634.05	0.447739	0.223870	−	0.974619i	$-0.428131\pi$
$541$	5634.05	0.447739	0.223870	+	0.974619i	$0.428131\pi$
$542$	11364.9	0.900673
$543$	0	0
$544$	1498.30	0.118087
$545$	−2160.76	−0.169829
$546$	0	0
$547$	6093.08	0.476273	0.238136	−	0.971232i	$-0.423463\pi$
$547$	6093.08	0.476273	0.238136	+	0.971232i	$0.423463\pi$
$548$	9680.29	0.754601
$549$	0	0
$550$	831.492	0.0644635
$551$	−92.5529	−0.00715587
$552$	0	0
$553$	−11667.6	−0.897212
$554$	−3769.24	−0.289061
$555$	0	0
$556$	3696.12	0.281925
$557$	−21461.9	−1.63262	−0.816311	−	0.577612i	$-0.803985\pi$
$557$	−21461.9	−1.63262	−0.816311	+	0.577612i	$0.803985\pi$
$558$	0	0
$559$	−26528.1	−2.00719
$560$	−2833.83	−0.213842
$561$	0	0
$562$	5412.85	0.406276
$563$	11038.5	0.826316	0.413158	−	0.910659i	$-0.364426\pi$
$563$	11038.5	0.826316	0.413158	+	0.910659i	$0.364426\pi$
$564$	0	0
$565$	1200.58	0.0893959
$566$	−11927.3	−0.885761
$567$	0	0
$568$	5870.16	0.433638
$569$	19210.3	1.41535	0.707677	−	0.706536i	$-0.249743\pi$
$569$	19210.3	1.41535	0.707677	+	0.706536i	$0.249743\pi$
$570$	0	0
$571$	−11967.0	−0.877066	−0.438533	−	0.898715i	$-0.644502\pi$
$571$	−11967.0	−0.877066	−0.438533	+	0.898715i	$0.644502\pi$
$572$	−5315.77	−0.388573
$573$	0	0
$574$	−6786.51	−0.493490
$575$	575.000	0.0417029
$576$	0	0
$577$	−14982.3	−1.08097	−0.540487	−	0.841352i	$-0.681760\pi$
$577$	−14982.3	−1.08097	−0.540487	+	0.841352i	$0.681760\pi$
$578$	−5441.42	−0.391580
$579$	0	0
$580$	16.7284	0.00119760
$581$	−26703.0	−1.90676
$582$	0	0
$583$	−6807.35	−0.483587
$584$	−731.679	−0.0518444
$585$	0	0
$586$	9471.71	0.667701
$587$	−24002.9	−1.68774	−0.843871	−	0.536547i	$-0.819729\pi$
$587$	−24002.9	−1.68774	−0.843871	+	0.536547i	$0.819729\pi$
$588$	0	0
$589$	13260.2	0.927637
$590$	3529.50	0.246283
$591$	0	0
$592$	5891.21	0.408999
$593$	−5124.67	−0.354882	−0.177441	−	0.984131i	$-0.556782\pi$
$593$	−5124.67	−0.354882	−0.177441	+	0.984131i	$0.556782\pi$
$594$	0	0
$595$	−8292.84	−0.571384
$596$	−1722.45	−0.118380
$597$	0	0
$598$	−3676.01	−0.251376
$599$	23776.8	1.62186	0.810928	−	0.585146i	$-0.198963\pi$
$599$	23776.8	1.62186	0.810928	+	0.585146i	$0.198963\pi$
$600$	0	0
$601$	−25435.3	−1.72633	−0.863166	−	0.504920i	$-0.831522\pi$
$601$	−25435.3	−1.72633	−0.863166	+	0.504920i	$0.831522\pi$
$602$	−23518.1	−1.59223
$603$	0	0
$604$	−101.647	−0.00684763
$605$	−5272.24	−0.354293
$606$	0	0
$607$	−7445.94	−0.497894	−0.248947	−	0.968517i	$-0.580084\pi$
$607$	−7445.94	−0.497894	−0.248947	+	0.968517i	$0.580084\pi$
$608$	−3540.91	−0.236189
$609$	0	0
$610$	−5072.23	−0.336670
$611$	−42756.5	−2.83100
$612$	0	0
$613$	−12874.4	−0.848275	−0.424138	−	0.905598i	$-0.639423\pi$
$613$	−12874.4	−0.848275	−0.424138	+	0.905598i	$0.639423\pi$
$614$	−1538.48	−0.101121
$615$	0	0
$616$	−4712.62	−0.308241
$617$	19247.2	1.25585	0.627927	−	0.778272i	$-0.283904\pi$
$617$	19247.2	1.25585	0.627927	+	0.778272i	$0.283904\pi$
$618$	0	0
$619$	−14496.4	−0.941293	−0.470647	−	0.882322i	$-0.655979\pi$
$619$	−14496.4	−0.941293	−0.470647	+	0.882322i	$0.655979\pi$
$620$	−2396.72	−0.155249
$621$	0	0
$622$	11185.4	0.721052
$623$	−37199.1	−2.39222
$624$	0	0
$625$	625.000	0.0400000
$626$	19554.4	1.24848
$627$	0	0
$628$	−6321.15	−0.401658
$629$	17239.9	1.09284
$630$	0	0
$631$	−18030.9	−1.13756	−0.568779	−	0.822490i	$-0.692584\pi$
$631$	−18030.9	−1.13756	−0.568779	+	0.822490i	$0.692584\pi$
$632$	2635.05	0.165849
$633$	0	0
$634$	−3737.03	−0.234095
$635$	−1054.96	−0.0659291
$636$	0	0
$637$	−72863.4	−4.53211
$638$	27.8191	0.00172628
$639$	0	0
$640$	640.000	0.0395285
$641$	−11776.4	−0.725646	−0.362823	−	0.931858i	$-0.618187\pi$
$641$	−11776.4	−0.725646	−0.362823	+	0.931858i	$0.618187\pi$
$642$	0	0
$643$	20207.4	1.23935	0.619676	−	0.784858i	$-0.287264\pi$
$643$	20207.4	1.23935	0.619676	+	0.784858i	$0.287264\pi$
$644$	−3258.91	−0.199408
$645$	0	0
$646$	−10362.0	−0.631095
$647$	−21452.8	−1.30355	−0.651776	−	0.758412i	$-0.725976\pi$
$647$	−21452.8	−1.30355	−0.651776	+	0.758412i	$0.725976\pi$
$648$	0	0
$649$	5869.50	0.355005
$650$	−3995.66	−0.241112
$651$	0	0
$652$	1834.95	0.110218
$653$	15095.2	0.904624	0.452312	−	0.891860i	$-0.350599\pi$
$653$	15095.2	0.904624	0.452312	+	0.891860i	$0.350599\pi$
$654$	0	0
$655$	3730.89	0.222562
$656$	1532.68	0.0912214
$657$	0	0
$658$	−37905.1	−2.24574
$659$	−6964.68	−0.411693	−0.205846	−	0.978584i	$-0.565995\pi$
$659$	−6964.68	−0.411693	−0.205846	+	0.978584i	$0.565995\pi$
$660$	0	0
$661$	−17380.4	−1.02272	−0.511361	−	0.859366i	$-0.670858\pi$
$661$	−17380.4	−1.02272	−0.511361	+	0.859366i	$0.670858\pi$
$662$	13933.8	0.818057
$663$	0	0
$664$	6030.67	0.352463
$665$	19598.3	1.14284
$666$	0	0
$667$	19.2377	0.00111677
$668$	2245.10	0.130038
$669$	0	0
$670$	−8200.56	−0.472859
$671$	−8435.04	−0.485292
$672$	0	0
$673$	12383.2	0.709271	0.354635	−	0.935005i	$-0.384605\pi$
$673$	12383.2	0.709271	0.354635	+	0.935005i	$0.384605\pi$
$674$	−35.0973	−0.00200578
$675$	0	0
$676$	16756.5	0.953371
$677$	8375.74	0.475489	0.237744	−	0.971328i	$-0.423592\pi$
$677$	8375.74	0.475489	0.237744	+	0.971328i	$0.423592\pi$
$678$	0	0
$679$	9632.49	0.544420
$680$	1872.88	0.105620
$681$	0	0
$682$	−3985.70	−0.223783
$683$	4434.66	0.248444	0.124222	−	0.992254i	$-0.460356\pi$
$683$	4434.66	0.248444	0.124222	+	0.992254i	$0.460356\pi$
$684$	0	0
$685$	12100.4	0.674936
$686$	−40295.8	−2.24271
$687$	0	0
$688$	5311.38	0.294323
$689$	32712.1	1.80875
$690$	0	0
$691$	11032.8	0.607389	0.303694	−	0.952770i	$-0.401780\pi$
$691$	11032.8	0.607389	0.303694	+	0.952770i	$0.401780\pi$
$692$	−6295.79	−0.345852
$693$	0	0
$694$	19718.6	1.07854
$695$	4620.15	0.252162
$696$	0	0
$697$	4485.20	0.243743
$698$	14400.5	0.780898
$699$	0	0
$700$	−3542.29	−0.191266
$701$	−3708.40	−0.199806	−0.0999032	−	0.994997i	$-0.531853\pi$
$701$	−3708.40	−0.199806	−0.0999032	+	0.994997i	$0.531853\pi$
$702$	0	0
$703$	−40742.6	−2.18583
$704$	1064.31	0.0569782
$705$	0	0
$706$	12109.2	0.645516
$707$	58755.2	3.12548
$708$	0	0
$709$	−6315.04	−0.334508	−0.167254	−	0.985914i	$-0.553490\pi$
$709$	−6315.04	−0.334508	−0.167254	+	0.985914i	$0.553490\pi$
$710$	7337.70	0.387858
$711$	0	0
$712$	8401.15	0.442200
$713$	−2756.22	−0.144770
$714$	0	0
$715$	−6644.71	−0.347550
$716$	−8862.36	−0.462572
$717$	0	0
$718$	−5468.52	−0.284239
$719$	−24736.9	−1.28307	−0.641536	−	0.767093i	$-0.721703\pi$
$719$	−24736.9	−1.28307	−0.641536	+	0.767093i	$0.721703\pi$
$720$	0	0
$721$	−61477.0	−3.17548
$722$	10770.3	0.555165
$723$	0	0
$724$	3493.99	0.179355
$725$	20.9106	0.00107117
$726$	0	0
$727$	33283.6	1.69797	0.848983	−	0.528420i	$-0.177215\pi$
$727$	33283.6	1.69797	0.848983	+	0.528420i	$0.177215\pi$
$728$	22646.0	1.15291
$729$	0	0
$730$	−914.599	−0.0463710
$731$	15543.1	0.786430
$732$	0	0
$733$	−33636.5	−1.69494	−0.847471	−	0.530841i	$-0.821876\pi$
$733$	−33636.5	−1.69494	−0.847471	+	0.530841i	$0.821876\pi$
$734$	−4931.00	−0.247965
$735$	0	0
$736$	736.000	0.0368605
$737$	−13637.4	−0.681601
$738$	0	0
$739$	36575.5	1.82064	0.910319	−	0.413907i	$-0.135836\pi$
$739$	36575.5	1.82064	0.910319	+	0.413907i	$0.135836\pi$
$740$	7364.02	0.365820
$741$	0	0
$742$	29000.4	1.43482
$743$	17704.7	0.874190	0.437095	−	0.899415i	$-0.356007\pi$
$743$	17704.7	0.874190	0.437095	+	0.899415i	$0.356007\pi$
$744$	0	0
$745$	−2153.07	−0.105882
$746$	11408.9	0.559932
$747$	0	0
$748$	3114.56	0.152246
$749$	−57735.1	−2.81655
$750$	0	0
$751$	8858.37	0.430421	0.215211	−	0.976568i	$-0.430956\pi$
$751$	8858.37	0.430421	0.215211	+	0.976568i	$0.430956\pi$
$752$	8560.59	0.415123
$753$	0	0
$754$	−133.682	−0.00645679
$755$	−127.059	−0.00612470
$756$	0	0
$757$	7899.26	0.379265	0.189633	−	0.981855i	$-0.439270\pi$
$757$	7899.26	0.379265	0.189633	+	0.981855i	$0.439270\pi$
$758$	20463.9	0.980582
$759$	0	0
$760$	−4426.13	−0.211254
$761$	−23437.4	−1.11643	−0.558216	−	0.829696i	$-0.688514\pi$
$761$	−23437.4	−1.11643	−0.558216	+	0.829696i	$0.688514\pi$
$762$	0	0
$763$	15308.1	0.726329
$764$	9990.95	0.473115
$765$	0	0
$766$	−12642.7	−0.596343
$767$	−28205.4	−1.32782
$768$	0	0
$769$	31447.7	1.47468	0.737342	−	0.675519i	$-0.236081\pi$
$769$	31447.7	1.47468	0.737342	+	0.675519i	$0.236081\pi$
$770$	−5890.77	−0.275699
$771$	0	0
$772$	3636.62	0.169540
$773$	−2397.42	−0.111551	−0.0557756	−	0.998443i	$-0.517763\pi$
$773$	−2397.42	−0.111551	−0.0557756	+	0.998443i	$0.517763\pi$
$774$	0	0
$775$	−2995.89	−0.138859
$776$	−2175.43	−0.100636
$777$	0	0
$778$	1851.19	0.0853063
$779$	−10599.8	−0.487518
$780$	0	0
$781$	12202.5	0.559077
$782$	2153.81	0.0984911
$783$	0	0
$784$	14588.5	0.664564
$785$	−7901.44	−0.359254
$786$	0	0
$787$	−19407.0	−0.879014	−0.439507	−	0.898239i	$-0.644847\pi$
$787$	−19407.0	−0.879014	−0.439507	+	0.898239i	$0.644847\pi$
$788$	−2434.51	−0.110058
$789$	0	0
$790$	3293.81	0.148340
$791$	−8505.58	−0.382331
$792$	0	0
$793$	40533.8	1.81513
$794$	−5413.51	−0.241962
$795$	0	0
$796$	−9219.91	−0.410542
$797$	28631.1	1.27248	0.636240	−	0.771491i	$-0.280489\pi$
$797$	28631.1	1.27248	0.636240	+	0.771491i	$0.280489\pi$
$798$	0	0
$799$	25051.4	1.10921
$800$	800.000	0.0353553
$801$	0	0
$802$	28190.6	1.24120
$803$	−1520.96	−0.0668414
$804$	0	0
$805$	−4073.63	−0.178356
$806$	19152.9	0.837013
$807$	0	0
$808$	−13269.4	−0.577743
$809$	9191.25	0.399440	0.199720	−	0.979853i	$-0.435997\pi$
$809$	9191.25	0.399440	0.199720	+	0.979853i	$0.435997\pi$
$810$	0	0
$811$	−567.477	−0.0245706	−0.0122853	−	0.999925i	$-0.503911\pi$
$811$	−567.477	−0.0245706	−0.0122853	+	0.999925i	$0.503911\pi$
$812$	−118.514	−0.00512195
$813$	0	0
$814$	12246.2	0.527310
$815$	2293.69	0.0985823
$816$	0	0
$817$	−36732.6	−1.57296
$818$	−21692.0	−0.927193
$819$	0	0
$820$	1915.85	0.0815909
$821$	6057.35	0.257494	0.128747	−	0.991677i	$-0.458904\pi$
$821$	6057.35	0.257494	0.128747	+	0.991677i	$0.458904\pi$
$822$	0	0
$823$	−21327.0	−0.903294	−0.451647	−	0.892197i	$-0.649163\pi$
$823$	−21327.0	−0.903294	−0.451647	+	0.892197i	$0.649163\pi$
$824$	13884.1	0.586986
$825$	0	0
$826$	−25005.0	−1.05331
$827$	27553.8	1.15857	0.579286	−	0.815124i	$-0.303332\pi$
$827$	27553.8	1.15857	0.579286	+	0.815124i	$0.303332\pi$
$828$	0	0
$829$	11181.6	0.468461	0.234231	−	0.972181i	$-0.424743\pi$
$829$	11181.6	0.468461	0.234231	+	0.972181i	$0.424743\pi$
$830$	7538.34	0.315253
$831$	0	0
$832$	−5114.44	−0.213115
$833$	42691.4	1.77571
$834$	0	0
$835$	2806.37	0.116310
$836$	−7360.59	−0.304511
$837$	0	0
$838$	11253.1	0.463881
$839$	−9811.83	−0.403745	−0.201872	−	0.979412i	$-0.564703\pi$
$839$	−9811.83	−0.403745	−0.201872	+	0.979412i	$0.564703\pi$
$840$	0	0
$841$	−24388.3	−0.999971
$842$	−14218.2	−0.581937
$843$	0	0
$844$	−17493.5	−0.713448
$845$	20945.6	0.852721
$846$	0	0
$847$	37351.6	1.51525
$848$	−6549.53	−0.265226
$849$	0	0
$850$	2341.10	0.0944693
$851$	8468.62	0.341129
$852$	0	0
$853$	−24030.8	−0.964594	−0.482297	−	0.876008i	$-0.660197\pi$
$853$	−24030.8	−0.964594	−0.482297	+	0.876008i	$0.660197\pi$
$854$	35934.6	1.43988
$855$	0	0
$856$	13039.0	0.520637
$857$	16934.8	0.675008	0.337504	−	0.941324i	$-0.390417\pi$
$857$	16934.8	0.675008	0.337504	+	0.941324i	$0.390417\pi$
$858$	0	0
$859$	−31057.0	−1.23359	−0.616793	−	0.787125i	$-0.711568\pi$
$859$	−31057.0	−1.23359	−0.616793	+	0.787125i	$0.711568\pi$
$860$	6639.22	0.263251
$861$	0	0
$862$	−8928.27	−0.352782
$863$	16157.8	0.637332	0.318666	−	0.947867i	$-0.396765\pi$
$863$	16157.8	0.637332	0.318666	+	0.947867i	$0.396765\pi$
$864$	0	0
$865$	−7869.73	−0.309340
$866$	22018.9	0.864009
$867$	0	0
$868$	16979.7	0.663974
$869$	5477.55	0.213824
$870$	0	0
$871$	65533.3	2.54938
$872$	−3457.21	−0.134261
$873$	0	0
$874$	−5090.05	−0.196995
$875$	−4427.86	−0.171073
$876$	0	0
$877$	−32473.9	−1.25036	−0.625180	−	0.780480i	$-0.714975\pi$
$877$	−32473.9	−1.25036	−0.625180	+	0.780480i	$0.714975\pi$
$878$	−6105.41	−0.234678
$879$	0	0
$880$	1330.39	0.0509629
$881$	15703.2	0.600517	0.300258	−	0.953858i	$-0.402927\pi$
$881$	15703.2	0.600517	0.300258	+	0.953858i	$0.402927\pi$
$882$	0	0
$883$	−14330.4	−0.546155	−0.273078	−	0.961992i	$-0.588042\pi$
$883$	−14330.4	−0.546155	−0.273078	+	0.961992i	$0.588042\pi$
$884$	−14966.7	−0.569441
$885$	0	0
$886$	8931.65	0.338673
$887$	−23238.2	−0.879664	−0.439832	−	0.898080i	$-0.644962\pi$
$887$	−23238.2	−0.879664	−0.439832	+	0.898080i	$0.644962\pi$
$888$	0	0
$889$	7473.98	0.281968
$890$	10501.4	0.395515
$891$	0	0
$892$	21758.0	0.816717
$893$	−59203.6	−2.21856
$894$	0	0
$895$	−11077.9	−0.413737
$896$	−4534.13	−0.169057
$897$	0	0
$898$	−18080.3	−0.671878
$899$	−100.233	−0.00371854
$900$	0	0
$901$	−19166.3	−0.708683
$902$	3186.03	0.117609
$903$	0	0
$904$	1920.92	0.0706736
$905$	4367.48	0.160420
$906$	0	0
$907$	−3667.46	−0.134263	−0.0671313	−	0.997744i	$-0.521385\pi$
$907$	−3667.46	−0.134263	−0.0671313	+	0.997744i	$0.521385\pi$
$908$	−865.992	−0.0316508
$909$	0	0
$910$	28307.6	1.03119
$911$	−21457.9	−0.780385	−0.390192	−	0.920733i	$-0.627591\pi$
$911$	−21457.9	−0.780385	−0.390192	+	0.920733i	$0.627591\pi$
$912$	0	0
$913$	12536.1	0.454420
$914$	15513.8	0.561433
$915$	0	0
$916$	−3698.66	−0.133414
$917$	−26431.8	−0.951859
$918$	0	0
$919$	−11109.3	−0.398762	−0.199381	−	0.979922i	$-0.563893\pi$
$919$	−11109.3	−0.398762	−0.199381	+	0.979922i	$0.563893\pi$
$920$	920.000	0.0329690
$921$	0	0
$922$	−14128.4	−0.504657
$923$	−58637.9	−2.09110
$924$	0	0
$925$	9205.02	0.327199
$926$	25199.3	0.894279
$927$	0	0
$928$	26.7655	0.000946790 0
$929$	20804.3	0.734732	0.367366	−	0.930076i	$-0.380260\pi$
$929$	20804.3	0.734732	0.367366	+	0.930076i	$0.380260\pi$
$930$	0	0
$931$	−100892.	−3.55166
$932$	9478.75	0.333140
$933$	0	0
$934$	−29492.0	−1.03320
$935$	3893.20	0.136173
$936$	0	0
$937$	−35550.5	−1.23947	−0.619735	−	0.784811i	$-0.712760\pi$
$937$	−35550.5	−1.23947	−0.619735	+	0.784811i	$0.712760\pi$
$938$	58097.5	2.02234
$939$	0	0
$940$	10700.7	0.371297
$941$	−49674.4	−1.72087	−0.860436	−	0.509558i	$-0.829809\pi$
$941$	−49674.4	−1.72087	−0.860436	+	0.509558i	$0.829809\pi$
$942$	0	0
$943$	2203.23	0.0760839
$944$	5647.20	0.194704
$945$	0	0
$946$	11040.9	0.379462
$947$	−33466.2	−1.14837	−0.574184	−	0.818726i	$-0.694681\pi$
$947$	−33466.2	−1.14837	−0.574184	+	0.818726i	$0.694681\pi$
$948$	0	0
$949$	7308.85	0.250006
$950$	−5532.67	−0.188951
$951$	0	0
$952$	−13268.5	−0.451718
$953$	17298.4	0.587984	0.293992	−	0.955808i	$-0.405016\pi$
$953$	17298.4	0.587984	0.293992	+	0.955808i	$0.405016\pi$
$954$	0	0
$955$	12488.7	0.423167
$956$	11078.4	0.374791
$957$	0	0
$958$	34822.5	1.17439
$959$	−85726.0	−2.88659
$960$	0	0
$961$	−15430.4	−0.517955
$962$	−58848.2	−1.97229
$963$	0	0
$964$	−21336.4	−0.712862
$965$	4545.78	0.151641
$966$	0	0
$967$	−14869.7	−0.494495	−0.247248	−	0.968952i	$-0.579526\pi$
$967$	−14869.7	−0.494495	−0.247248	+	0.968952i	$0.579526\pi$
$968$	−8435.59	−0.280093
$969$	0	0
$970$	−2719.28	−0.0900113
$971$	−26634.9	−0.880283	−0.440142	−	0.897928i	$-0.645072\pi$
$971$	−26634.9	−0.880283	−0.440142	+	0.897928i	$0.645072\pi$
$972$	0	0
$973$	−32731.8	−1.07845
$974$	2640.69	0.0868717
$975$	0	0
$976$	−8115.57	−0.266161
$977$	−40803.6	−1.33615	−0.668077	−	0.744092i	$-0.732882\pi$
$977$	−40803.6	−1.33615	−0.668077	+	0.744092i	$0.732882\pi$
$978$	0	0
$979$	17463.7	0.570115
$980$	18235.6	0.594404
$981$	0	0
$982$	−34231.6	−1.11240
$983$	18615.8	0.604021	0.302011	−	0.953305i	$-0.402342\pi$
$983$	18615.8	0.604021	0.302011	+	0.953305i	$0.402342\pi$
$984$	0	0
$985$	−3043.14	−0.0984389
$986$	78.3257	0.00252982
$987$	0	0
$988$	35370.6	1.13896
$989$	7635.11	0.245483
$990$	0	0
$991$	46185.1	1.48044	0.740220	−	0.672364i	$-0.234721\pi$
$991$	46185.1	1.48044	0.740220	+	0.672364i	$0.234721\pi$
$992$	−3834.74	−0.122735
$993$	0	0
$994$	−51984.5	−1.65880
$995$	−11524.9	−0.367200
$996$	0	0
$997$	16544.3	0.525541	0.262771	−	0.964858i	$-0.415364\pi$
$997$	16544.3	0.525541	0.262771	+	0.964858i	$0.415364\pi$
$998$	35081.2	1.11270
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2070.4.a.bj.1.1		4
3.2	odd	2		230.4.a.h.1.1	✓	4
12.11	even	2		1840.4.a.m.1.4		4
15.2	even	4		1150.4.b.n.599.4		8
15.8	even	4		1150.4.b.n.599.5		8
15.14	odd	2		1150.4.a.p.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.4.a.h.1.1	✓	4	3.2	odd	2
1150.4.a.p.1.4		4	15.14	odd	2
1150.4.b.n.599.4		8	15.2	even	4
1150.4.b.n.599.5		8	15.8	even	4
1840.4.a.m.1.4		4	12.11	even	2
2070.4.a.bj.1.1		4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2070.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +4.00000 q^{4} +5.00000 q^{5} -35.4229 q^{7} +8.00000 q^{8} +O(q^{10})\)	\(q+2.00000 q^{2} +4.00000 q^{4} +5.00000 q^{5} -35.4229 q^{7} +8.00000 q^{8} +10.0000 q^{10} +16.6298 q^{11} -79.9132 q^{13} -70.8458 q^{14} +16.0000 q^{16} +46.8219 q^{17} -110.653 q^{19} +20.0000 q^{20} +33.2597 q^{22} +23.0000 q^{23} +25.0000 q^{25} -159.826 q^{26} -141.692 q^{28} +0.836422 q^{29} -119.836 q^{31} +32.0000 q^{32} +93.6438 q^{34} -177.114 q^{35} +368.201 q^{37} -221.307 q^{38} +40.0000 q^{40} +95.7927 q^{41} +331.961 q^{43} +66.5193 q^{44} +46.0000 q^{46} +535.037 q^{47} +911.782 q^{49} +50.0000 q^{50} -319.653 q^{52} -409.345 q^{53} +83.1492 q^{55} -283.383 q^{56} +1.67284 q^{58} +352.950 q^{59} -507.223 q^{61} -239.672 q^{62} +64.0000 q^{64} -399.566 q^{65} -820.056 q^{67} +187.288 q^{68} -354.229 q^{70} +733.770 q^{71} -91.4599 q^{73} +736.402 q^{74} -442.613 q^{76} -589.077 q^{77} +329.381 q^{79} +80.0000 q^{80} +191.585 q^{82} +753.834 q^{83} +234.110 q^{85} +663.922 q^{86} +133.039 q^{88} +1050.14 q^{89} +2830.76 q^{91} +92.0000 q^{92} +1070.07 q^{94} -553.267 q^{95} -271.928 q^{97} +1823.56 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{2} + 16 q^{4} + 20 q^{5} - q^{7} + 32 q^{8}+O(q^{10}) \) 4 * q + 8 * q^2 + 16 * q^4 + 20 * q^5 - q^7 + 32 * q^8	\( 4 q + 8 q^{2} + 16 q^{4} + 20 q^{5} - q^{7} + 32 q^{8} + 40 q^{10} + 39 q^{11} - 20 q^{13} - 2 q^{14} + 64 q^{16} + 23 q^{17} + 53 q^{19} + 80 q^{20} + 78 q^{22} + 92 q^{23} + 100 q^{25} - 40 q^{26} - 4 q^{28} - 161 q^{29} + 388 q^{31} + 128 q^{32} + 46 q^{34} - 5 q^{35} + 466 q^{37} + 106 q^{38} + 160 q^{40} - 484 q^{41} + 894 q^{43} + 156 q^{44} + 184 q^{46} + 265 q^{47} + 1643 q^{49} + 200 q^{50} - 80 q^{52} - 576 q^{53} + 195 q^{55} - 8 q^{56} - 322 q^{58} + 94 q^{59} + 1153 q^{61} + 776 q^{62} + 256 q^{64} - 100 q^{65} - 1472 q^{67} + 92 q^{68} - 10 q^{70} - 200 q^{71} + 1147 q^{73} + 932 q^{74} + 212 q^{76} + 2176 q^{77} - 908 q^{79} + 320 q^{80} - 968 q^{82} + 1048 q^{83} + 115 q^{85} + 1788 q^{86} + 312 q^{88} + 1784 q^{89} + 2329 q^{91} + 368 q^{92} + 530 q^{94} + 265 q^{95} - 2047 q^{97} + 3286 q^{98}+O(q^{100}) \) 4 * q + 8 * q^2 + 16 * q^4 + 20 * q^5 - q^7 + 32 * q^8 + 40 * q^10 + 39 * q^11 - 20 * q^13 - 2 * q^14 + 64 * q^16 + 23 * q^17 + 53 * q^19 + 80 * q^20 + 78 * q^22 + 92 * q^23 + 100 * q^25 - 40 * q^26 - 4 * q^28 - 161 * q^29 + 388 * q^31 + 128 * q^32 + 46 * q^34 - 5 * q^35 + 466 * q^37 + 106 * q^38 + 160 * q^40 - 484 * q^41 + 894 * q^43 + 156 * q^44 + 184 * q^46 + 265 * q^47 + 1643 * q^49 + 200 * q^50 - 80 * q^52 - 576 * q^53 + 195 * q^55 - 8 * q^56 - 322 * q^58 + 94 * q^59 + 1153 * q^61 + 776 * q^62 + 256 * q^64 - 100 * q^65 - 1472 * q^67 + 92 * q^68 - 10 * q^70 - 200 * q^71 + 1147 * q^73 + 932 * q^74 + 212 * q^76 + 2176 * q^77 - 908 * q^79 + 320 * q^80 - 968 * q^82 + 1048 * q^83 + 115 * q^85 + 1788 * q^86 + 312 * q^88 + 1784 * q^89 + 2329 * q^91 + 368 * q^92 + 530 * q^94 + 265 * q^95 - 2047 * q^97 + 3286 * q^98

Introduction

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