LMFDB - Embedded newform 1710.4.f.a.341.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1710,4,Mod(341,1710)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1710, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1710.341"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 1710 = 2 \cdot 3^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1710.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [40,-80] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	no
Analytic conductor:	$100.893266110$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			341.3
Character	$\chi$	$=$	1710.341
Dual form			1710.4.f.a.341.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.00000 q^{2} +4.00000 q^{4} -5.00000i q^{5} +32.6046 q^{7} -8.00000 q^{8} +10.0000i q^{10} -52.9133i q^{11} +20.1863i q^{13} -65.2091 q^{14} +16.0000 q^{16} -21.3488i q^{17} +(50.1729 + 65.8914i) q^{19} -20.0000i q^{20} +105.827i q^{22} +24.2406i q^{23} -25.0000 q^{25} -40.3725i q^{26} +130.418 q^{28} +186.987 q^{29} +118.246i q^{31} -32.0000 q^{32} +42.6976i q^{34} -163.023i q^{35} +229.120i q^{37} +(-100.346 - 131.783i) q^{38} +40.0000i q^{40} +79.7794 q^{41} +85.5228 q^{43} -211.653i q^{44} -48.4811i q^{46} +216.581i q^{47} +720.057 q^{49} +50.0000 q^{50} +80.7450i q^{52} +475.719 q^{53} -264.567 q^{55} -260.836 q^{56} -373.974 q^{58} +418.563 q^{59} -16.3957 q^{61} -236.492i q^{62} +64.0000 q^{64} +100.931 q^{65} -769.606i q^{67} -85.3952i q^{68} +326.046i q^{70} +358.177 q^{71} -173.467 q^{73} -458.239i q^{74} +(200.692 + 263.566i) q^{76} -1725.22i q^{77} +475.363i q^{79} -80.0000i q^{80} -159.559 q^{82} +1275.17i q^{83} -106.744 q^{85} -171.046 q^{86} +423.307i q^{88} -933.329 q^{89} +658.164i q^{91} +96.9623i q^{92} -433.162i q^{94} +(329.457 - 250.864i) q^{95} +324.190i q^{97} -1440.11 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 80 q^{2} + 160 q^{4} - 56 q^{7} - 320 q^{8} + 112 q^{14} + 640 q^{16} - 76 q^{19} - 1000 q^{25} - 224 q^{28} - 120 q^{29} - 1280 q^{32} + 152 q^{38} - 312 q^{41} + 56 q^{43} + 2112 q^{49} + 2000 q^{50}+ \cdots - 4224 q^{98}+O(q^{100}) $ 40 * q - 80 * q^2 + 160 * q^4 - 56 * q^7 - 320 * q^8 + 112 * q^14 + 640 * q^16 - 76 * q^19 - 1000 * q^25 - 224 * q^28 - 120 * q^29 - 1280 * q^32 + 152 * q^38 - 312 * q^41 + 56 * q^43 + 2112 * q^49 + 2000 * q^50 + 288 * q^53 + 448 * q^56 + 240 * q^58 + 912 * q^59 - 848 * q^61 + 2560 * q^64 + 240 * q^65 - 1248 * q^71 + 512 * q^73 - 304 * q^76 + 624 * q^82 - 112 * q^86 - 552 * q^89 - 60 * q^95 - 4224 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1710\mathbb{Z}\right)^\times$.

$n$	$191$	$1027$	$1351$
$\chi(n)$	$-1$	$1$	$-1$

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
$5$		−	5.00000i		−	0.447214i
$5$		−	5.00000i		−	0.447214i
$6$		0			0
$7$	32.6046			1.76048			0.880240	−	0.474528i	$-0.157381\pi$
$7$	32.6046			1.76048			0.880240	+	0.474528i	$0.157381\pi$
$8$	−8.00000			−0.353553
$9$		0			0
$10$			10.0000i			0.316228i
$11$		−	52.9133i		−	1.45036i	−0.688559	−	0.725180i	$-0.741756\pi$
$11$		−	52.9133i		−	1.45036i	0.688559	−	0.725180i	$-0.258244\pi$
$12$		0			0
$13$			20.1863i			0.430666i	0.976541	+	0.215333i	$0.0690837\pi$
$13$			20.1863i			0.430666i	−0.976541	+	0.215333i	$0.930916\pi$
$14$	−65.2091			−1.24485
$15$		0			0
$16$	16.0000			0.250000
$17$		−	21.3488i		−	0.304579i	−0.988336	−	0.152290i	$-0.951335\pi$
$17$		−	21.3488i		−	0.304579i	0.988336	−	0.152290i	$-0.0486646\pi$
$18$		0			0
$19$	50.1729	+	65.8914i	0.605813	+	0.795607i
$19$	50.1729	+	65.8914i	0.605813	+	0.795607i
$20$		−	20.0000i		−	0.223607i
$21$		0			0
$22$			105.827i			1.02556i
$23$			24.2406i			0.219761i	0.993945	+	0.109881i	$0.0350469\pi$
$23$			24.2406i			0.219761i	−0.993945	+	0.109881i	$0.964953\pi$
$24$		0			0
$25$	−25.0000			−0.200000
$26$		−	40.3725i		−	0.304527i
$27$		0			0
$28$	130.418			0.880240
$29$	186.987			1.19733			0.598666	−	0.800999i	$-0.295698\pi$
$29$	186.987			1.19733			0.598666	+	0.800999i	$0.295698\pi$
$30$		0			0
$31$			118.246i			0.685085i	0.939502	+	0.342543i	$0.111288\pi$
$31$			118.246i			0.685085i	−0.939502	+	0.342543i	$0.888712\pi$
$32$	−32.0000			−0.176777
$33$		0			0
$34$			42.6976i			0.215370i
$35$		−	163.023i		−	0.787311i
$36$		0			0
$37$			229.120i			1.01803i	0.860758	+	0.509014i	$0.169990\pi$
$37$			229.120i			1.01803i	−0.860758	+	0.509014i	$0.830010\pi$
$38$	−100.346	−	131.783i	−0.428374	−	0.562579i
$39$		0			0
$40$			40.0000i			0.158114i
$41$	79.7794			0.303889			0.151945	−	0.988389i	$-0.451446\pi$
$41$	79.7794			0.303889			0.151945	+	0.988389i	$0.451446\pi$
$42$		0			0
$43$	85.5228			0.303305			0.151652	−	0.988434i	$-0.451541\pi$
$43$	85.5228			0.303305			0.151652	+	0.988434i	$0.451541\pi$
$44$		−	211.653i		−	0.725180i
$45$		0			0
$46$		−	48.4811i		−	0.155395i
$47$			216.581i			0.672161i	0.941833	+	0.336081i	$0.109102\pi$
$47$			216.581i			0.672161i	−0.941833	+	0.336081i	$0.890898\pi$
$48$		0			0
$49$	720.057			2.09929
$50$	50.0000			0.141421
$51$		0			0
$52$			80.7450i			0.215333i
$53$	475.719			1.23293			0.616463	−	0.787384i	$-0.288565\pi$
$53$	475.719			1.23293			0.616463	+	0.787384i	$0.288565\pi$
$54$		0			0
$55$	−264.567			−0.648621
$56$	−260.836			−0.622424
$57$		0			0
$58$	−373.974			−0.846641
$59$	418.563			0.923597			0.461799	−	0.886985i	$-0.347204\pi$
$59$	418.563			0.923597			0.461799	+	0.886985i	$0.347204\pi$
$60$		0			0
$61$	−16.3957			−0.0344140			−0.0172070	−	0.999852i	$-0.505477\pi$
$61$	−16.3957			−0.0344140			−0.0172070	+	0.999852i	$0.505477\pi$
$62$		−	236.492i		−	0.484429i
$63$		0			0
$64$	64.0000			0.125000
$65$	100.931			0.192600
$66$		0			0
$67$		−	769.606i		−	1.40332i	−0.712513	−	0.701659i	$-0.752443\pi$
$67$		−	769.606i		−	1.40332i	0.712513	−	0.701659i	$-0.247557\pi$
$68$		−	85.3952i		−	0.152290i
$69$		0			0
$70$			326.046i			0.556713i
$71$	358.177			0.598701			0.299351	−	0.954143i	$-0.403230\pi$
$71$	358.177			0.598701			0.299351	+	0.954143i	$0.403230\pi$
$72$		0			0
$73$	−173.467			−0.278120			−0.139060	−	0.990284i	$-0.544408\pi$
$73$	−173.467			−0.278120			−0.139060	+	0.990284i	$0.544408\pi$
$74$		−	458.239i		−	0.719855i
$75$		0			0
$76$	200.692	+	263.566i	0.302907	+	0.397804i
$77$		−	1725.22i		−	2.55333i
$78$		0			0
$79$			475.363i			0.676994i	0.940967	+	0.338497i	$0.109919\pi$
$79$			475.363i			0.676994i	−0.940967	+	0.338497i	$0.890081\pi$
$80$		−	80.0000i		−	0.111803i
$81$		0			0
$82$	−159.559			−0.214882
$83$			1275.17i			1.68636i	0.537633	+	0.843179i	$0.319319\pi$
$83$			1275.17i			1.68636i	−0.537633	+	0.843179i	$0.680681\pi$
$84$		0			0
$85$	−106.744			−0.136212
$86$	−171.046			−0.214469
$87$		0			0
$88$			423.307i			0.512780i
$89$	−933.329			−1.11160			−0.555802	−	0.831315i	$-0.687589\pi$
$89$	−933.329			−1.11160			−0.555802	+	0.831315i	$0.687589\pi$
$90$		0			0
$91$			658.164i			0.758179i
$92$			96.9623i			0.109881i
$93$		0			0
$94$		−	433.162i		−	0.475290i
$95$	329.457	−	250.864i	0.355806	−	0.270928i
$96$		0			0
$97$			324.190i			0.339345i	0.985501	+	0.169672i	$0.0542710\pi$
$97$			324.190i			0.339345i	−0.985501	+	0.169672i	$0.945729\pi$
$98$	−1440.11			−1.48442
$99$		0			0
$100$	−100.000			−0.100000
$101$			685.379i			0.675225i	0.941285	+	0.337613i	$0.109619\pi$
$101$			685.379i			0.675225i	−0.941285	+	0.337613i	$0.890381\pi$
$102$		0			0
$103$		−	133.983i		−	0.128173i	−0.997944	−	0.0640863i	$-0.979587\pi$
$103$		−	133.983i		−	0.128173i	0.997944	−	0.0640863i	$-0.0204133\pi$
$104$		−	161.490i		−	0.152263i
$105$		0			0
$106$	−951.438			−0.871810
$107$	736.340			0.665277			0.332638	−	0.943054i	$-0.392061\pi$
$107$	736.340			0.665277			0.332638	+	0.943054i	$0.392061\pi$
$108$		0			0
$109$			884.479i			0.777227i	0.921401	+	0.388613i	$0.127046\pi$
$109$			884.479i			0.777227i	−0.921401	+	0.388613i	$0.872954\pi$
$110$	529.133			0.458644
$111$		0			0
$112$	521.673			0.440120
$113$	−2128.70			−1.77214			−0.886068	−	0.463555i	$-0.846574\pi$
$113$	−2128.70			−1.77214			−0.886068	+	0.463555i	$0.846574\pi$
$114$		0			0
$115$	121.203			0.0982802
$116$	747.948			0.598666
$117$		0			0
$118$	−837.126			−0.653082
$119$		−	696.068i		−	0.536205i
$120$		0			0
$121$	−1468.82			−1.10355
$122$	32.7914			0.0243344
$123$		0			0
$124$			472.985i			0.342543i
$125$			125.000i			0.0894427i
$126$		0			0
$127$		−	1919.74i		−	1.34133i	−0.741760	−	0.670666i	$-0.766008\pi$
$127$		−	1919.74i		−	1.34133i	0.741760	−	0.670666i	$-0.233992\pi$
$128$	−128.000			−0.0883883
$129$		0			0
$130$	−201.863			−0.136189
$131$		−	602.975i		−	0.402154i	−0.979575	−	0.201077i	$-0.935556\pi$
$131$		−	602.975i		−	0.402154i	0.979575	−	0.201077i	$-0.0644442\pi$
$132$		0			0
$133$	1635.86	+	2148.36i	1.06652	+	1.40065i
$134$			1539.21i			0.992296i
$135$		0			0
$136$			170.790i			0.107685i
$137$			390.538i			0.243547i	0.992558	+	0.121773i	$0.0388581\pi$
$137$			390.538i			0.243547i	−0.992558	+	0.121773i	$0.961142\pi$
$138$		0			0
$139$	2712.81			1.65538			0.827689	−	0.561186i	$-0.189655\pi$
$139$	2712.81			1.65538			0.827689	+	0.561186i	$0.189655\pi$
$140$		−	652.091i		−	0.393655i
$141$		0			0
$142$	−716.354			−0.423346
$143$	1068.12			0.624621
$144$		0			0
$145$		−	934.935i		−	0.535463i
$146$	346.934			0.196661
$147$		0			0
$148$			916.479i			0.509014i
$149$		−	2172.84i		−	1.19467i	−0.801991	−	0.597336i	$-0.796226\pi$
$149$		−	2172.84i		−	1.19467i	0.801991	−	0.597336i	$-0.203774\pi$
$150$		0			0
$151$		−	316.539i		−	0.170593i	−0.996356	−	0.0852965i	$-0.972816\pi$
$151$		−	316.539i		−	0.170593i	0.996356	−	0.0852965i	$-0.0271838\pi$
$152$	−401.383	−	527.132i	−0.214187	−	0.281290i
$153$		0			0
$154$			3450.43i			1.80548i
$155$	591.231			0.306379
$156$		0			0
$157$	−2883.16			−1.46561			−0.732806	−	0.680438i	$-0.761790\pi$
$157$	−2883.16			−1.46561			−0.732806	+	0.680438i	$0.761790\pi$
$158$		−	950.727i		−	0.478707i
$159$		0			0
$160$			160.000i			0.0790569i
$161$			790.353i			0.386885i
$162$		0			0
$163$	−387.370			−0.186142			−0.0930709	−	0.995659i	$-0.529668\pi$
$163$	−387.370			−0.186142			−0.0930709	+	0.995659i	$0.529668\pi$
$164$	319.118			0.151945
$165$		0			0
$166$		−	2550.33i		−	1.19244i
$167$	−1191.80			−0.552242			−0.276121	−	0.961123i	$-0.589049\pi$
$167$	−1191.80			−0.552242			−0.276121	+	0.961123i	$0.589049\pi$
$168$		0			0
$169$	1789.52			0.814527
$170$	213.488			0.0963164
$171$		0			0
$172$	342.091			0.151652
$173$	−1079.15			−0.474254			−0.237127	−	0.971479i	$-0.576206\pi$
$173$	−1079.15			−0.474254			−0.237127	+	0.971479i	$0.576206\pi$
$174$		0			0
$175$	−815.114			−0.352096
$176$		−	846.613i		−	0.362590i
$177$		0			0
$178$	1866.66			0.786022
$179$	−2093.09			−0.873995			−0.436998	−	0.899463i	$-0.643958\pi$
$179$	−2093.09			−0.873995			−0.436998	+	0.899463i	$0.643958\pi$
$180$		0			0
$181$			1400.49i			0.575125i	0.957762	+	0.287563i	$0.0928450\pi$
$181$			1400.49i			0.575125i	−0.957762	+	0.287563i	$0.907155\pi$
$182$		−	1316.33i		−	0.536114i
$183$		0			0
$184$		−	193.925i		−	0.0776973i
$185$	1145.60			0.455276
$186$		0			0
$187$	−1129.64			−0.441749
$188$			866.324i			0.336081i
$189$		0			0
$190$	−658.914	+	501.729i	−0.251593	+	0.191575i
$191$		−	3957.48i		−	1.49923i	−0.661874	−	0.749615i	$-0.730239\pi$
$191$		−	3957.48i		−	1.49923i	0.661874	−	0.749615i	$-0.269761\pi$
$192$		0			0
$193$			193.248i			0.0720739i	0.999350	+	0.0360370i	$0.0114734\pi$
$193$			193.248i			0.0720739i	−0.999350	+	0.0360370i	$0.988527\pi$
$194$		−	648.379i		−	0.239953i
$195$		0			0
$196$	2880.23			1.04965
$197$		−	3971.35i		−	1.43628i	−0.695898	−	0.718140i	$-0.744994\pi$
$197$		−	3971.35i		−	1.43628i	0.695898	−	0.718140i	$-0.255006\pi$
$198$		0			0
$199$	4340.04			1.54602			0.773008	−	0.634396i	$-0.218751\pi$
$199$	4340.04			1.54602			0.773008	+	0.634396i	$0.218751\pi$
$200$	200.000			0.0707107
$201$		0			0
$202$		−	1370.76i		−	0.477456i
$203$	6096.63			2.10788
$204$		0			0
$205$		−	398.897i		−	0.135903i
$206$			267.967i			0.0906316i
$207$		0			0
$208$			322.980i			0.107667i
$209$	3486.54	−	2654.81i	1.15392	−	0.878647i
$210$		0			0
$211$			3071.11i			1.00201i	0.865445	+	0.501005i	$0.167036\pi$
$211$			3071.11i			1.00201i	−0.865445	+	0.501005i	$0.832964\pi$
$212$	1902.88			0.616463
$213$		0			0
$214$	−1472.68			−0.470422
$215$		−	427.614i		−	0.135642i
$216$		0			0
$217$			3855.36i			1.20608i
$218$		−	1768.96i		−	0.549582i
$219$		0			0
$220$	−1058.27			−0.324310
$221$	430.952			0.131172
$222$		0			0
$223$			1531.70i			0.459956i	0.973196	+	0.229978i	$0.0738655\pi$
$223$			1531.70i			0.459956i	−0.973196	+	0.229978i	$0.926135\pi$
$224$	−1043.35			−0.311212
$225$		0			0
$226$	4257.40			1.25309
$227$	5558.67			1.62529			0.812647	−	0.582756i	$-0.198026\pi$
$227$	5558.67			1.62529			0.812647	+	0.582756i	$0.198026\pi$
$228$		0			0
$229$	−3306.38			−0.954114			−0.477057	−	0.878872i	$-0.658296\pi$
$229$	−3306.38			−0.954114			−0.477057	+	0.878872i	$0.658296\pi$
$230$	−242.406			−0.0694946
$231$		0			0
$232$	−1495.90			−0.423321
$233$		−	3124.50i		−	0.878510i	−0.898362	−	0.439255i	$-0.855242\pi$
$233$		−	3124.50i		−	0.878510i	0.898362	−	0.439255i	$-0.144758\pi$
$234$		0			0
$235$	1082.91			0.300600
$236$	1674.25			0.461799
$237$		0			0
$238$			1392.14i			0.379154i
$239$		−	2597.12i		−	0.702903i	−0.936206	−	0.351451i	$-0.885688\pi$
$239$		−	2597.12i		−	0.702903i	0.936206	−	0.351451i	$-0.114312\pi$
$240$		0			0
$241$		−	2709.02i		−	0.724081i	−0.932162	−	0.362041i	$-0.882080\pi$
$241$		−	2709.02i		−	0.724081i	0.932162	−	0.362041i	$-0.117920\pi$
$242$	2937.64			0.780325
$243$		0			0
$244$	−65.5828			−0.0172070
$245$		−	3600.28i		−	0.938832i
$246$		0			0
$247$	−1330.10	+	1012.80i	−0.342641	+	0.260903i
$248$		−	945.970i		−	0.242214i
$249$		0			0
$250$		−	250.000i		−	0.0632456i
$251$			170.585i			0.0428973i	0.999770	+	0.0214487i	$0.00682785\pi$
$251$			170.585i			0.0428973i	−0.999770	+	0.0214487i	$0.993172\pi$
$252$		0			0
$253$	1282.65			0.318733
$254$			3839.47i			0.948465i
$255$		0			0
$256$	256.000			0.0625000
$257$	4928.58			1.19625			0.598125	−	0.801403i	$-0.295913\pi$
$257$	4928.58			1.19625			0.598125	+	0.801403i	$0.295913\pi$
$258$		0			0
$259$			7470.34i			1.79222i
$260$	403.725			0.0962999
$261$		0			0
$262$			1205.95i			0.284366i
$263$			6370.31i			1.49358i	0.665062	+	0.746788i	$0.268405\pi$
$263$			6370.31i			1.49358i	−0.665062	+	0.746788i	$0.731595\pi$
$264$		0			0
$265$		−	2378.60i		−	0.551381i
$266$	−3271.73	−	4296.72i	−0.754145	−	0.990410i
$267$		0			0
$268$		−	3078.42i		−	0.701659i
$269$	−6479.43			−1.46862			−0.734308	−	0.678816i	$-0.762493\pi$
$269$	−6479.43			−1.46862			−0.734308	+	0.678816i	$0.762493\pi$
$270$		0			0
$271$	3026.25			0.678345			0.339173	−	0.940724i	$-0.389853\pi$
$271$	3026.25			0.678345			0.339173	+	0.940724i	$0.389853\pi$
$272$		−	341.581i		−	0.0761448i
$273$		0			0
$274$		−	781.076i		−	0.172214i
$275$			1322.83i			0.290072i
$276$		0			0
$277$	989.863			0.214712			0.107356	−	0.994221i	$-0.465762\pi$
$277$	989.863			0.214712			0.107356	+	0.994221i	$0.465762\pi$
$278$	−5425.62			−1.17053
$279$		0			0
$280$			1304.18i			0.278356i
$281$	−7382.21			−1.56721			−0.783605	−	0.621260i	$-0.786621\pi$
$281$	−7382.21			−1.56721			−0.783605	+	0.621260i	$0.786621\pi$
$282$		0			0
$283$	3104.66			0.652129			0.326065	−	0.945347i	$-0.394277\pi$
$283$	3104.66			0.652129			0.326065	+	0.945347i	$0.394277\pi$
$284$	1432.71			0.299351
$285$		0			0
$286$	−2136.24			−0.441674
$287$	2601.17			0.534991
$288$		0			0
$289$	4457.23			0.907232
$290$			1869.87i			0.378630i
$291$		0			0
$292$	−693.868			−0.139060
$293$	9085.97			1.81163			0.905816	−	0.423672i	$-0.139259\pi$
$293$	9085.97			1.81163			0.905816	+	0.423672i	$0.139259\pi$
$294$		0			0
$295$		−	2092.81i		−	0.413045i
$296$		−	1832.96i		−	0.359927i
$297$		0			0
$298$			4345.68i			0.844760i
$299$	−489.326			−0.0946437
$300$		0			0
$301$	2788.43			0.533962
$302$			633.077i			0.120627i
$303$		0			0
$304$	802.766	+	1054.26i	0.151453	+	0.198902i
$305$			81.9785i			0.0153904i
$306$		0			0
$307$			2544.01i			0.472946i	0.971638	+	0.236473i	$0.0759915\pi$
$307$			2544.01i			0.472946i	−0.971638	+	0.236473i	$0.924009\pi$
$308$		−	6900.86i		−	1.27667i
$309$		0			0
$310$	−1182.46			−0.216643
$311$		−	5169.11i		−	0.942488i	−0.882003	−	0.471244i	$-0.843805\pi$
$311$		−	5169.11i		−	0.942488i	0.882003	−	0.471244i	$-0.156195\pi$
$312$		0			0
$313$	6584.44			1.18906			0.594528	−	0.804075i	$-0.297339\pi$
$313$	6584.44			1.18906			0.594528	+	0.804075i	$0.297339\pi$
$314$	5766.31			1.03634
$315$		0			0
$316$			1901.45i			0.338497i
$317$	−3044.97			−0.539503			−0.269752	−	0.962930i	$-0.586942\pi$
$317$	−3044.97			−0.539503			−0.269752	+	0.962930i	$0.586942\pi$
$318$		0			0
$319$		−	9894.10i		−	1.73656i
$320$		−	320.000i		−	0.0559017i
$321$		0			0
$322$		−	1580.71i		−	0.273569i
$323$	1406.70	−	1071.13i	0.242325	−	0.184518i
$324$		0			0
$325$		−	504.656i		−	0.0861332i
$326$	774.739			0.131622
$327$		0			0
$328$	−638.236			−0.107441
$329$			7061.53i			1.18333i
$330$		0			0
$331$		−	9909.79i		−	1.64559i	−0.568337	−	0.822796i	$-0.692413\pi$
$331$		−	9909.79i		−	1.64559i	0.568337	−	0.822796i	$-0.307587\pi$
$332$			5100.67i			0.843179i
$333$		0			0
$334$	2383.60			0.390494
$335$	−3848.03			−0.627583
$336$		0			0
$337$			6770.17i			1.09435i	0.837019	+	0.547173i	$0.184296\pi$
$337$			6770.17i			1.09435i	−0.837019	+	0.547173i	$0.815704\pi$
$338$	−3579.03			−0.575957
$339$		0			0
$340$	−426.976			−0.0681059
$341$	6256.80			0.993621
$342$		0			0
$343$	12293.8			1.93528
$344$	−684.182			−0.107234
$345$		0			0
$346$	2158.29			0.335348
$347$			10131.6i			1.56742i	0.621129	+	0.783709i	$0.286674\pi$
$347$			10131.6i			1.56742i	−0.621129	+	0.783709i	$0.713326\pi$
$348$		0			0
$349$	313.776			0.0481262			0.0240631	−	0.999710i	$-0.492340\pi$
$349$	313.776			0.0481262			0.0240631	+	0.999710i	$0.492340\pi$
$350$	1630.23			0.248970
$351$		0			0
$352$			1693.23i			0.256390i
$353$			808.934i			0.121969i	0.998139	+	0.0609847i	$0.0194241\pi$
$353$			808.934i			0.121969i	−0.998139	+	0.0609847i	$0.980576\pi$
$354$		0			0
$355$		−	1790.89i		−	0.267747i
$356$	−3733.32			−0.555802
$357$		0			0
$358$	4186.19			0.618008
$359$			3535.56i			0.519776i	0.965639	+	0.259888i	$0.0836857\pi$
$359$			3535.56i			0.519776i	−0.965639	+	0.259888i	$0.916314\pi$
$360$		0			0
$361$	−1824.36	+	6611.93i	−0.265981	+	0.963978i
$362$		−	2800.98i		−	0.406675i
$363$		0			0
$364$			2632.65i			0.379090i
$365$			867.335i			0.124379i
$366$		0			0
$367$	12642.8			1.79822			0.899110	−	0.437724i	$-0.144215\pi$
$367$	12642.8			1.79822			0.899110	+	0.437724i	$0.144215\pi$
$368$			387.849i			0.0549403i
$369$		0			0
$370$	−2291.20			−0.321929
$371$	15510.6			2.17054
$372$		0			0
$373$		−	12884.5i		−	1.78856i	−0.447509	−	0.894279i	$-0.647689\pi$
$373$		−	12884.5i		−	1.78856i	0.447509	−	0.894279i	$-0.352311\pi$
$374$	2259.27			0.312364
$375$		0			0
$376$		−	1732.65i		−	0.237645i
$377$			3774.57i			0.515650i
$378$		0			0
$379$		−	14537.8i		−	1.97033i	−0.171602	−	0.985166i	$-0.554894\pi$
$379$		−	14537.8i		−	1.97033i	0.171602	−	0.985166i	$-0.445106\pi$
$380$	1317.83	−	1003.46i	0.177903	−	0.135464i
$381$		0			0
$382$			7914.95i			1.06012i
$383$	−8218.59			−1.09648			−0.548238	−	0.836322i	$-0.684701\pi$
$383$	−8218.59			−1.09648			−0.548238	+	0.836322i	$0.684701\pi$
$384$		0			0
$385$	−8626.08			−1.14188
$386$		−	386.495i		−	0.0509640i
$387$		0			0
$388$			1296.76i			0.169672i
$389$		−	8759.62i		−	1.14172i	−0.821046	−	0.570861i	$-0.806609\pi$
$389$		−	8759.62i		−	1.14172i	0.821046	−	0.570861i	$-0.193391\pi$
$390$		0			0
$391$	517.507			0.0669347
$392$	−5760.46			−0.742212
$393$		0			0
$394$			7942.71i			1.01560i
$395$	2376.82			0.302761
$396$		0			0
$397$	9334.89			1.18011			0.590056	−	0.807362i	$-0.299106\pi$
$397$	9334.89			1.18011			0.590056	+	0.807362i	$0.299106\pi$
$398$	−8680.08			−1.09320
$399$		0			0
$400$	−400.000			−0.0500000
$401$	12194.4			1.51860			0.759298	−	0.650743i	$-0.225542\pi$
$401$	12194.4			1.51860			0.759298	+	0.650743i	$0.225542\pi$
$402$		0			0
$403$	−2386.95			−0.295043
$404$			2741.52i			0.337613i
$405$		0			0
$406$	−12193.3			−1.49050
$407$	12123.5			1.47651
$408$		0			0
$409$		−	12954.3i		−	1.56613i	−0.621941	−	0.783064i	$-0.713656\pi$
$409$		−	12954.3i		−	1.56613i	0.621941	−	0.783064i	$-0.286344\pi$
$410$			797.794i			0.0960982i
$411$		0			0
$412$		−	535.933i		−	0.0640863i
$413$	13647.1			1.62597
$414$		0			0
$415$	6375.83			0.754162
$416$		−	645.960i		−	0.0761317i
$417$		0			0
$418$	−6973.07	+	5309.63i	−0.815943	+	0.621297i
$419$		−	1466.91i		−	0.171034i	−0.996337	−	0.0855171i	$-0.972746\pi$
$419$		−	1466.91i		−	0.171034i	0.996337	−	0.0855171i	$-0.0272542\pi$
$420$		0			0
$421$			7089.04i			0.820662i	0.911937	+	0.410331i	$0.134587\pi$
$421$			7089.04i			0.820662i	−0.911937	+	0.410331i	$0.865413\pi$
$422$		−	6142.22i		−	0.708527i
$423$		0			0
$424$	−3805.75			−0.435905
$425$			533.720i			0.0609158i
$426$		0			0
$427$	−534.575			−0.0605852
$428$	2945.36			0.332638
$429$		0			0
$430$			855.228i			0.0959134i
$431$	−6662.99			−0.744651			−0.372326	−	0.928102i	$-0.621440\pi$
$431$	−6662.99			−0.744651			−0.372326	+	0.928102i	$0.621440\pi$
$432$		0			0
$433$			10340.0i			1.14760i	0.818996	+	0.573799i	$0.194531\pi$
$433$			10340.0i			1.14760i	−0.818996	+	0.573799i	$0.805469\pi$
$434$		−	7710.73i		−	0.852827i
$435$		0			0
$436$			3537.92i			0.388613i
$437$	−1597.25	+	1216.22i	−0.174844	+	0.133134i
$438$		0			0
$439$		−	6619.90i		−	0.719705i	−0.933009	−	0.359853i	$-0.882827\pi$
$439$		−	6619.90i		−	0.719705i	0.933009	−	0.359853i	$-0.117173\pi$
$440$	2116.53			0.229322
$441$		0			0
$442$	−861.904			−0.0927525
$443$		−	3409.67i		−	0.365685i	−0.983142	−	0.182842i	$-0.941470\pi$
$443$		−	3409.67i		−	0.365685i	0.983142	−	0.182842i	$-0.0585298\pi$
$444$		0			0
$445$			4666.65i			0.497124i
$446$		−	3063.40i		−	0.325238i
$447$		0			0
$448$	2086.69			0.220060
$449$	−15642.3			−1.64411			−0.822057	−	0.569405i	$-0.807173\pi$
$449$	−15642.3			−1.64411			−0.822057	+	0.569405i	$0.807173\pi$
$450$		0			0
$451$		−	4221.40i		−	0.440749i
$452$	−8514.81			−0.886068
$453$		0			0
$454$	−11117.3			−1.14926
$455$	3290.82			0.339068
$456$		0			0
$457$	1073.59			0.109892			0.0549460	−	0.998489i	$-0.482501\pi$
$457$	1073.59			0.109892			0.0549460	+	0.998489i	$0.482501\pi$
$458$	6612.77			0.674660
$459$		0			0
$460$	484.811			0.0491401
$461$		−	3384.99i		−	0.341984i	−0.985272	−	0.170992i	$-0.945303\pi$
$461$		−	3384.99i		−	0.341984i	0.985272	−	0.170992i	$-0.0546972\pi$
$462$		0			0
$463$	6722.43			0.674768			0.337384	−	0.941367i	$-0.390458\pi$
$463$	6722.43			0.674768			0.337384	+	0.941367i	$0.390458\pi$
$464$	2991.79			0.299333
$465$		0			0
$466$			6249.00i			0.621201i
$467$			785.347i			0.0778191i	0.999243	+	0.0389096i	$0.0123884\pi$
$467$			785.347i			0.0778191i	−0.999243	+	0.0389096i	$0.987612\pi$
$468$		0			0
$469$		−	25092.7i		−	2.47051i
$470$	−2165.81			−0.212556
$471$		0			0
$472$	−3348.50			−0.326541
$473$		−	4525.29i		−	0.439901i
$474$		0			0
$475$	−1254.32	−	1647.29i	−0.121163	−	0.159121i
$476$		−	2784.27i		−	0.268103i
$477$		0			0
$478$			5194.24i			0.497027i
$479$		−	8127.14i		−	0.775237i	−0.921820	−	0.387619i	$-0.873298\pi$
$479$		−	8127.14i		−	0.775237i	0.921820	−	0.387619i	$-0.126702\pi$
$480$		0			0
$481$	−4625.07			−0.438430
$482$			5418.05i			0.512003i
$483$		0			0
$484$	−5875.28			−0.551773
$485$	1620.95			0.151760
$486$		0			0
$487$		−	6703.97i		−	0.623790i	−0.950117	−	0.311895i	$-0.899036\pi$
$487$		−	6703.97i		−	0.623790i	0.950117	−	0.311895i	$-0.100964\pi$
$488$	131.166			0.0121672
$489$		0			0
$490$			7200.57i			0.663854i
$491$			1359.32i			0.124940i	0.998047	+	0.0624698i	$0.0198977\pi$
$491$			1359.32i			0.124940i	−0.998047	+	0.0624698i	$0.980102\pi$
$492$		0			0
$493$		−	3991.95i		−	0.364682i
$494$	2660.20	−	2025.60i	0.242284	−	0.184486i
$495$		0			0
$496$			1891.94i			0.171271i
$497$	11678.2			1.05400
$498$		0			0
$499$	−9449.65			−0.847744			−0.423872	−	0.905722i	$-0.639329\pi$
$499$	−9449.65			−0.847744			−0.423872	+	0.905722i	$0.639329\pi$
$500$			500.000i			0.0447214i
$501$		0			0
$502$		−	341.170i		−	0.0303330i
$503$			3954.44i			0.350537i	0.984521	+	0.175268i	$0.0560793\pi$
$503$			3954.44i			0.350537i	−0.984521	+	0.175268i	$0.943921\pi$
$504$		0			0
$505$	3426.89			0.301970
$506$	−2565.30			−0.225378
$507$		0			0
$508$		−	7678.95i		−	0.670666i
$509$	9984.26			0.869439			0.434719	−	0.900566i	$-0.356848\pi$
$509$	9984.26			0.869439			0.434719	+	0.900566i	$0.356848\pi$
$510$		0			0
$511$	−5655.81			−0.489625
$512$	−512.000			−0.0441942
$513$		0			0
$514$	−9857.16			−0.845877
$515$	−669.917			−0.0573205
$516$		0			0
$517$	11460.0			0.974877
$518$		−	14940.7i		−	1.26729i
$519$		0			0
$520$	−807.450			−0.0680943
$521$	1419.65			0.119378			0.0596892	−	0.998217i	$-0.480989\pi$
$521$	1419.65			0.119378			0.0596892	+	0.998217i	$0.480989\pi$
$522$		0			0
$523$			1225.63i			0.102472i	0.998687	+	0.0512362i	$0.0163161\pi$
$523$			1225.63i			0.102472i	−0.998687	+	0.0512362i	$0.983684\pi$
$524$		−	2411.90i		−	0.201077i
$525$		0			0
$526$		−	12740.6i		−	1.05612i
$527$	2524.41			0.208663
$528$		0			0
$529$	11579.4			0.951705
$530$			4757.19i			0.389885i
$531$		0			0
$532$	6543.46	+	8593.44i	0.533261	+	0.700325i
$533$			1610.45i			0.130875i
$534$		0			0
$535$		−	3681.70i		−	0.297521i
$536$			6156.85i			0.496148i
$537$		0			0
$538$	12958.9			1.03847
$539$		−	38100.6i		−	3.04473i
$540$		0			0
$541$	13317.1			1.05831			0.529155	−	0.848525i	$-0.322509\pi$
$541$	13317.1			1.05831			0.529155	+	0.848525i	$0.322509\pi$
$542$	−6052.50			−0.479663
$543$		0			0
$544$			683.161i			0.0538425i
$545$	4422.40			0.347586
$546$		0			0
$547$			17683.3i			1.38224i	0.722740	+	0.691120i	$0.242882\pi$
$547$			17683.3i			1.38224i	−0.722740	+	0.691120i	$0.757118\pi$
$548$			1562.15i			0.121773i
$549$		0			0
$550$		−	2645.67i		−	0.205112i
$551$	9381.68	+	12320.8i	0.725359	+	0.952606i
$552$		0			0
$553$			15499.0i			1.19184i
$554$	−1979.73			−0.151824
$555$		0			0
$556$	10851.2			0.827689
$557$			12618.3i			0.959882i	0.877301	+	0.479941i	$0.159342\pi$
$557$			12618.3i			0.959882i	−0.877301	+	0.479941i	$0.840658\pi$
$558$		0			0
$559$			1726.38i			0.130623i
$560$		−	2608.36i		−	0.196828i
$561$		0			0
$562$	14764.4			1.10818
$563$	−20701.3			−1.54966			−0.774829	−	0.632171i	$-0.782164\pi$
$563$	−20701.3			−1.54966			−0.774829	+	0.632171i	$0.782164\pi$
$564$		0			0
$565$			10643.5i			0.792523i
$566$	−6209.31			−0.461125
$567$		0			0
$568$	−2865.42			−0.211673
$569$	1423.11			0.104850			0.0524251	−	0.998625i	$-0.483305\pi$
$569$	1423.11			0.104850			0.0524251	+	0.998625i	$0.483305\pi$
$570$		0			0
$571$	14162.1			1.03795			0.518973	−	0.854791i	$-0.326315\pi$
$571$	14162.1			1.03795			0.518973	+	0.854791i	$0.326315\pi$
$572$	4272.49			0.312311
$573$		0			0
$574$	−5202.35			−0.378296
$575$		−	606.014i		−	0.0439522i
$576$		0			0
$577$	−13148.9			−0.948695			−0.474348	−	0.880338i	$-0.657316\pi$
$577$	−13148.9			−0.948695			−0.474348	+	0.880338i	$0.657316\pi$
$578$	−8914.46			−0.641510
$579$		0			0
$580$		−	3739.74i		−	0.267732i
$581$			41576.2i			2.96880i
$582$		0			0
$583$		−	25171.9i		−	1.78819i
$584$	1387.74			0.0983303
$585$		0			0
$586$	−18171.9			−1.28102
$587$			13282.4i			0.933944i	0.884272	+	0.466972i	$0.154655\pi$
$587$			13282.4i			0.933944i	−0.884272	+	0.466972i	$0.845345\pi$
$588$		0			0
$589$	−7791.41	+	5932.75i	−0.545059	+	0.415034i
$590$			4185.63i			0.292067i
$591$		0			0
$592$			3665.91i			0.254507i
$593$			8070.49i			0.558879i	0.960163	+	0.279440i	$0.0901487\pi$
$593$			8070.49i			0.558879i	−0.960163	+	0.279440i	$0.909851\pi$
$594$		0			0
$595$	−3480.34			−0.239798
$596$		−	8691.36i		−	0.597336i
$597$		0			0
$598$	978.652			0.0669232
$599$	6265.24			0.427363			0.213682	−	0.976903i	$-0.431454\pi$
$599$	6265.24			0.427363			0.213682	+	0.976903i	$0.431454\pi$
$600$		0			0
$601$		−	15518.5i		−	1.05327i	−0.850093	−	0.526633i	$-0.823454\pi$
$601$		−	15518.5i		−	1.05327i	0.850093	−	0.526633i	$-0.176546\pi$
$602$	−5576.86			−0.377568
$603$		0			0
$604$		−	1266.15i		−	0.0852965i
$605$			7344.10i			0.493521i
$606$		0			0
$607$		−	14506.8i		−	0.970039i	−0.874503	−	0.485020i	$-0.838812\pi$
$607$		−	14506.8i		−	0.970039i	0.874503	−	0.485020i	$-0.161188\pi$
$608$	−1605.53	−	2108.53i	−0.107094	−	0.140645i
$609$		0			0
$610$		−	163.957i		−	0.0108827i
$611$	−4371.96			−0.289477
$612$		0			0
$613$	−7997.78			−0.526962			−0.263481	−	0.964665i	$-0.584871\pi$
$613$	−7997.78			−0.526962			−0.263481	+	0.964665i	$0.584871\pi$
$614$		−	5088.02i		−	0.334423i
$615$		0			0
$616$			13801.7i			0.902739i
$617$			15350.5i			1.00160i	0.865562	+	0.500802i	$0.166961\pi$
$617$			15350.5i			1.00160i	−0.865562	+	0.500802i	$0.833039\pi$
$618$		0			0
$619$	−23894.0			−1.55151			−0.775753	−	0.631037i	$-0.782630\pi$
$619$	−23894.0			−1.55151			−0.775753	+	0.631037i	$0.782630\pi$
$620$	2364.92			0.153190
$621$		0			0
$622$			10338.2i			0.666439i
$623$	−30430.8			−1.95696
$624$		0			0
$625$	625.000			0.0400000
$626$	−13168.9			−0.840790
$627$		0			0
$628$	−11532.6			−0.732806
$629$	4891.43			0.310070
$630$		0			0
$631$	2.64044			0.000166584			8.32918e−5	−	1.00000i	$-0.499973\pi$
$631$	2.64044			0.000166584			8.32918e−5		1.00000i	$0.499973\pi$
$632$		−	3802.91i		−	0.239354i
$633$		0			0
$634$	6089.94			0.381487
$635$	−9598.69			−0.599862
$636$		0			0
$637$			14535.2i			0.904094i
$638$			19788.2i			1.22794i
$639$		0			0
$640$			640.000i			0.0395285i
$641$	8327.48			0.513129			0.256565	−	0.966527i	$-0.417409\pi$
$641$	8327.48			0.513129			0.256565	+	0.966527i	$0.417409\pi$
$642$		0			0
$643$	1852.11			0.113593			0.0567963	−	0.998386i	$-0.481911\pi$
$643$	1852.11			0.113593			0.0567963	+	0.998386i	$0.481911\pi$
$644$			3161.41i			0.193443i
$645$		0			0
$646$	−2813.41	+	2142.26i	−0.171350	+	0.130474i
$647$			18096.3i			1.09960i	0.835298	+	0.549798i	$0.185295\pi$
$647$			18096.3i			1.09960i	−0.835298	+	0.549798i	$0.814705\pi$
$648$		0			0
$649$		−	22147.5i		−	1.33955i
$650$			1009.31i			0.0609054i
$651$		0			0
$652$	−1549.48			−0.0930709
$653$			25855.0i			1.54944i	0.632304	+	0.774720i	$0.282109\pi$
$653$			25855.0i			1.54944i	−0.632304	+	0.774720i	$0.717891\pi$
$654$		0			0
$655$	−3014.88			−0.179849
$656$	1276.47			0.0759723
$657$		0			0
$658$		−	14123.1i		−	0.836739i
$659$	−16207.3			−0.958039			−0.479020	−	0.877804i	$-0.659008\pi$
$659$	−16207.3			−0.958039			−0.479020	+	0.877804i	$0.659008\pi$
$660$		0			0
$661$			3696.48i			0.217513i	0.994068	+	0.108757i	$0.0346869\pi$
$661$			3696.48i			0.217513i	−0.994068	+	0.108757i	$0.965313\pi$
$662$			19819.6i			1.16361i
$663$		0			0
$664$		−	10201.3i		−	0.596218i
$665$	10741.8	−	8179.32i	0.626390	−	0.476963i
$666$		0			0
$667$			4532.67i			0.263127i
$668$	−4767.21			−0.276121
$669$		0			0
$670$	7696.06			0.443768
$671$			867.551i			0.0499127i
$672$		0			0
$673$		−	10038.0i		−	0.574942i	−0.957789	−	0.287471i	$-0.907186\pi$
$673$		−	10038.0i		−	0.574942i	0.957789	−	0.287471i	$-0.0928144\pi$
$674$		−	13540.3i		−	0.773820i
$675$		0			0
$676$	7158.06			0.407263
$677$	−998.108			−0.0566624			−0.0283312	−	0.999599i	$-0.509019\pi$
$677$	−998.108			−0.0566624			−0.0283312	+	0.999599i	$0.509019\pi$
$678$		0			0
$679$			10570.1i			0.597410i
$680$	853.952			0.0481582
$681$		0			0
$682$	−12513.6			−0.702596
$683$	−26244.3			−1.47029			−0.735145	−	0.677909i	$-0.762886\pi$
$683$	−26244.3			−1.47029			−0.735145	+	0.677909i	$0.762886\pi$
$684$		0			0
$685$	1952.69			0.108917
$686$	−24587.5			−1.36845
$687$		0			0
$688$	1368.36			0.0758262
$689$			9602.99i			0.530979i
$690$		0			0
$691$	−19900.1			−1.09557			−0.547783	−	0.836621i	$-0.684528\pi$
$691$	−19900.1			−1.09557			−0.547783	+	0.836621i	$0.684528\pi$
$692$	−4316.58			−0.237127
$693$		0			0
$694$		−	20263.2i		−	1.10833i
$695$		−	13564.1i		−	0.740308i
$696$		0			0
$697$		−	1703.20i		−	0.0925583i
$698$	−627.553			−0.0340304
$699$		0			0
$700$	−3260.46			−0.176048
$701$		−	2772.66i		−	0.149389i	−0.997206	−	0.0746946i	$-0.976202\pi$
$701$		−	2772.66i		−	0.149389i	0.997206	−	0.0746946i	$-0.0237982\pi$
$702$		0			0
$703$	−15097.0	+	11495.6i	−0.809950	+	0.616735i
$704$		−	3386.45i		−	0.181295i
$705$		0			0
$706$		−	1617.87i		−	0.0862454i
$707$			22346.5i			1.18872i
$708$		0			0
$709$	−24784.7			−1.31285			−0.656423	−	0.754393i	$-0.727931\pi$
$709$	−24784.7			−1.31285			−0.656423	+	0.754393i	$0.727931\pi$
$710$			3581.77i			0.189326i
$711$		0			0
$712$	7466.63			0.393011
$713$	−2866.36			−0.150555
$714$		0			0
$715$		−	5340.61i		−	0.279339i
$716$	−8372.38			−0.436998
$717$		0			0
$718$		−	7071.12i		−	0.367537i
$719$		−	27114.7i		−	1.40641i	−0.710989	−	0.703203i	$-0.751752\pi$
$719$		−	27114.7i		−	1.40641i	0.710989	−	0.703203i	$-0.248248\pi$
$720$		0			0
$721$		−	4368.47i		−	0.225645i
$722$	3648.73	−	13223.9i	0.188077	−	0.681636i
$723$		0			0
$724$			5601.97i			0.287563i
$725$	−4674.68			−0.239466
$726$		0			0
$727$	11250.7			0.573954			0.286977	−	0.957937i	$-0.407350\pi$
$727$	11250.7			0.573954			0.286977	+	0.957937i	$0.407350\pi$
$728$		−	5265.31i		−	0.268057i
$729$		0			0
$730$		−	1734.67i		−	0.0879493i
$731$		−	1825.81i		−	0.0923802i
$732$		0			0
$733$	−28013.1			−1.41158			−0.705790	−	0.708421i	$-0.749408\pi$
$733$	−28013.1			−1.41158			−0.705790	+	0.708421i	$0.749408\pi$
$734$	−25285.5			−1.27153
$735$		0			0
$736$		−	775.698i		−	0.0388487i
$737$	−40722.4			−2.03532
$738$		0			0
$739$	−21386.6			−1.06457			−0.532286	−	0.846565i	$-0.678667\pi$
$739$	−21386.6			−1.06457			−0.532286	+	0.846565i	$0.678667\pi$
$740$	4582.39			0.227638
$741$		0			0
$742$	−31021.2			−1.53480
$743$	16319.5			0.805793			0.402897	−	0.915245i	$-0.368003\pi$
$743$	16319.5			0.805793			0.402897	+	0.915245i	$0.368003\pi$
$744$		0			0
$745$	−10864.2			−0.534273
$746$			25768.9i			1.26470i
$747$		0			0
$748$	−4518.54			−0.220875
$749$	24008.0			1.17121
$750$		0			0
$751$			3692.61i			0.179421i	0.995968	+	0.0897105i	$0.0285942\pi$
$751$			3692.61i			0.179421i	−0.995968	+	0.0897105i	$0.971406\pi$
$752$			3465.30i			0.168040i
$753$		0			0
$754$		−	7549.13i		−	0.364620i
$755$	−1582.69			−0.0762915
$756$		0			0
$757$	−31829.4			−1.52822			−0.764109	−	0.645088i	$-0.776821\pi$
$757$	−31829.4			−1.52822			−0.764109	+	0.645088i	$0.776821\pi$
$758$			29075.6i			1.39324i
$759$		0			0
$760$	−2635.66	+	2006.92i	−0.125797	+	0.0957874i
$761$			20092.5i			0.957102i	0.878060	+	0.478551i	$0.158838\pi$
$761$			20092.5i			0.957102i	−0.878060	+	0.478551i	$0.841162\pi$
$762$		0			0
$763$			28838.0i			1.36829i
$764$		−	15829.9i		−	0.749615i
$765$		0			0
$766$	16437.2			0.775326
$767$			8449.21i			0.397762i
$768$		0			0
$769$	15880.6			0.744694			0.372347	−	0.928094i	$-0.378553\pi$
$769$	15880.6			0.744694			0.372347	+	0.928094i	$0.378553\pi$
$770$	17252.2			0.807434
$771$		0			0
$772$			772.990i			0.0360370i
$773$	−15034.9			−0.699568			−0.349784	−	0.936830i	$-0.613745\pi$
$773$	−15034.9			−0.699568			−0.349784	+	0.936830i	$0.613745\pi$
$774$		0			0
$775$		−	2956.16i		−	0.137017i
$776$		−	2593.52i		−	0.119977i
$777$		0			0
$778$			17519.2i			0.807320i
$779$	4002.76	+	5256.78i	0.184100	+	0.241776i
$780$		0			0
$781$		−	18952.3i		−	0.868333i
$782$	−1035.01			−0.0473300
$783$		0			0
$784$	11520.9			0.524823
$785$			14415.8i			0.655441i
$786$		0			0
$787$			4385.91i			0.198654i	0.995055	+	0.0993271i	$0.0316690\pi$
$787$			4385.91i			0.198654i	−0.995055	+	0.0993271i	$0.968331\pi$
$788$		−	15885.4i		−	0.718140i
$789$		0			0
$790$	−4753.63			−0.214084
$791$	−69405.4			−3.11981
$792$		0			0
$793$		−	330.968i		−	0.0148209i
$794$	−18669.8			−0.834466
$795$		0			0
$796$	17360.2			0.773008
$797$	−37878.4			−1.68347			−0.841733	−	0.539894i	$-0.818464\pi$
$797$	−37878.4			−1.68347			−0.841733	+	0.539894i	$0.818464\pi$
$798$		0			0
$799$	4623.74			0.204726
$800$	800.000			0.0353553
$801$		0			0
$802$	−24388.7			−1.07381
$803$			9178.71i			0.403374i
$804$		0			0
$805$	3951.76			0.173020
$806$	4773.90			0.208627
$807$		0			0
$808$		−	5483.03i		−	0.238728i
$809$		−	24331.6i		−	1.05742i	−0.848802	−	0.528710i	$-0.822676\pi$
$809$		−	24331.6i		−	1.05742i	0.848802	−	0.528710i	$-0.177324\pi$
$810$		0			0
$811$			28135.4i			1.21821i	0.793090	+	0.609105i	$0.208471\pi$
$811$			28135.4i			1.21821i	−0.793090	+	0.609105i	$0.791529\pi$
$812$	24386.5			1.05394
$813$		0			0
$814$	−24247.0			−1.04405
$815$			1936.85i			0.0832452i
$816$		0			0
$817$	4290.92	+	5635.22i	0.183746	+	0.241311i
$818$			25908.5i			1.10742i
$819$		0			0
$820$		−	1595.59i		−	0.0679517i
$821$			7287.14i			0.309772i	0.987932	+	0.154886i	$0.0495011\pi$
$821$			7287.14i			0.309772i	−0.987932	+	0.154886i	$0.950499\pi$
$822$		0			0
$823$	23616.7			1.00028			0.500138	−	0.865946i	$-0.333283\pi$
$823$	23616.7			1.00028			0.500138	+	0.865946i	$0.333283\pi$
$824$			1071.87i			0.0453158i
$825$		0			0
$826$	−27294.1			−1.14974
$827$	45536.2			1.91469			0.957345	−	0.288949i	$-0.0933058\pi$
$827$	45536.2			1.91469			0.957345	+	0.288949i	$0.0933058\pi$
$828$		0			0
$829$		−	23319.7i		−	0.976991i	−0.872567	−	0.488495i	$-0.837546\pi$
$829$		−	23319.7i		−	0.976991i	0.872567	−	0.488495i	$-0.162454\pi$
$830$	−12751.7			−0.533273
$831$		0			0
$832$			1291.92i			0.0538333i
$833$		−	15372.3i		−	0.639400i
$834$		0			0
$835$			5959.01i			0.246970i
$836$	13946.1	−	10619.3i	0.576959	−	0.439324i
$837$		0			0
$838$			2933.82i			0.120939i
$839$	12589.5			0.518042			0.259021	−	0.965872i	$-0.416600\pi$
$839$	12589.5			0.518042			0.259021	+	0.965872i	$0.416600\pi$
$840$		0			0
$841$	10575.1			0.433603
$842$		−	14178.1i		−	0.580295i
$843$		0			0
$844$			12284.4i			0.501005i
$845$		−	8947.58i		−	0.364267i
$846$		0			0
$847$	−47890.2			−1.94277
$848$	7611.51			0.308231
$849$		0			0
$850$		−	1067.44i		−	0.0430740i
$851$	−5553.99			−0.223723
$852$		0			0
$853$	35803.8			1.43716			0.718581	−	0.695443i	$-0.244792\pi$
$853$	35803.8			1.43716			0.718581	+	0.695443i	$0.244792\pi$
$854$	1069.15			0.0428402
$855$		0			0
$856$	−5890.72			−0.235211
$857$	517.971			0.0206459			0.0103230	−	0.999947i	$-0.496714\pi$
$857$	517.971			0.0206459			0.0103230	+	0.999947i	$0.496714\pi$
$858$		0			0
$859$	−17684.4			−0.702425			−0.351212	−	0.936296i	$-0.614230\pi$
$859$	−17684.4			−0.702425			−0.351212	+	0.936296i	$0.614230\pi$
$860$		−	1710.46i		−	0.0678210i
$861$		0			0
$862$	13326.0			0.526548
$863$	−31064.6			−1.22532			−0.612659	−	0.790347i	$-0.709900\pi$
$863$	−31064.6			−1.22532			−0.612659	+	0.790347i	$0.709900\pi$
$864$		0			0
$865$			5395.73i			0.212093i
$866$		−	20680.0i		−	0.811474i
$867$		0			0
$868$			15421.5i			0.603040i
$869$	25153.0			0.981886
$870$		0			0
$871$	15535.5			0.604361
$872$		−	7075.83i		−	0.274791i
$873$		0			0
$874$	3194.49	−	2432.44i	0.123633	−	0.0941401i
$875$			4075.57i			0.157462i
$876$		0			0
$877$		−	39558.3i		−	1.52313i	−0.648086	−	0.761567i	$-0.724430\pi$
$877$		−	39558.3i		−	1.52313i	0.648086	−	0.761567i	$-0.275570\pi$
$878$			13239.8i			0.508909i
$879$		0			0
$880$	−4233.07			−0.162155
$881$		−	6220.93i		−	0.237898i	−0.992900	−	0.118949i	$-0.962047\pi$
$881$		−	6220.93i		−	0.237898i	0.992900	−	0.118949i	$-0.0379526\pi$
$882$		0			0
$883$	−46769.7			−1.78247			−0.891237	−	0.453538i	$-0.850162\pi$
$883$	−46769.7			−1.78247			−0.891237	+	0.453538i	$0.850162\pi$
$884$	1723.81			0.0655859
$885$		0			0
$886$			6819.34i			0.258578i
$887$	20419.6			0.772970			0.386485	−	0.922296i	$-0.373689\pi$
$887$	20419.6			0.772970			0.386485	+	0.922296i	$0.373689\pi$
$888$		0			0
$889$		−	62592.2i		−	2.36139i
$890$		−	9333.29i		−	0.351520i
$891$		0			0
$892$			6126.80i			0.229978i
$893$	−14270.8	+	10866.5i	−0.534776	+	0.407204i
$894$		0			0
$895$			10465.5i			0.390863i
$896$	−4173.38			−0.155606
$897$		0			0
$898$	31284.7			1.16256
$899$			22110.5i			0.820274i
$900$		0			0
$901$		−	10156.0i		−	0.375523i
$902$			8442.79i			0.311657i
$903$		0			0
$904$	17029.6			0.626545
$905$	7002.46			0.257204
$906$		0			0
$907$			27572.3i			1.00940i	0.863296	+	0.504699i	$0.168396\pi$
$907$			27572.3i			1.00940i	−0.863296	+	0.504699i	$0.831604\pi$
$908$	22234.7			0.812647
$909$		0			0
$910$	−6581.64			−0.239757
$911$	−10690.2			−0.388785			−0.194392	−	0.980924i	$-0.562273\pi$
$911$	−10690.2			−0.388785			−0.194392	+	0.980924i	$0.562273\pi$
$912$		0			0
$913$	67473.3			2.44583
$914$	−2147.19			−0.0777053
$915$		0			0
$916$	−13225.5			−0.477057
$917$		−	19659.7i		−	0.707984i
$918$		0			0
$919$	−1917.86			−0.0688404			−0.0344202	−	0.999407i	$-0.510958\pi$
$919$	−1917.86			−0.0688404			−0.0344202	+	0.999407i	$0.510958\pi$
$920$	−969.623			−0.0347473
$921$		0			0
$922$			6769.98i			0.241819i
$923$			7230.25i			0.257840i
$924$		0			0
$925$		−	5727.99i		−	0.203606i
$926$	−13444.9			−0.477133
$927$		0			0
$928$	−5983.58			−0.211660
$929$			4510.71i			0.159302i	0.996823	+	0.0796510i	$0.0253806\pi$
$929$			4510.71i			0.159302i	−0.996823	+	0.0796510i	$0.974619\pi$
$930$		0			0
$931$	36127.3	+	47445.6i	1.27178	+	1.67021i
$932$		−	12498.0i		−	0.439255i
$933$		0			0
$934$		−	1570.69i		−	0.0550264i
$935$			5648.18i			0.197556i
$936$		0			0
$937$	−11740.7			−0.409341			−0.204670	−	0.978831i	$-0.565612\pi$
$937$	−11740.7			−0.409341			−0.204670	+	0.978831i	$0.565612\pi$
$938$			50185.3i			1.74692i
$939$		0			0
$940$	4331.62			0.150300
$941$	5038.12			0.174536			0.0872678	−	0.996185i	$-0.472186\pi$
$941$	5038.12			0.174536			0.0872678	+	0.996185i	$0.472186\pi$
$942$		0			0
$943$			1933.90i			0.0667831i
$944$	6697.01			0.230899
$945$		0			0
$946$			9050.59i			0.311057i
$947$		−	44851.5i		−	1.53905i	−0.638619	−	0.769523i	$-0.720494\pi$
$947$		−	44851.5i		−	1.53905i	0.638619	−	0.769523i	$-0.279506\pi$
$948$		0			0
$949$		−	3501.65i		−	0.119777i
$950$	2508.64	+	3294.57i	0.0856749	+	0.112516i
$951$		0			0
$952$			5568.54i			0.189577i
$953$	−3551.34			−0.120713			−0.0603564	−	0.998177i	$-0.519224\pi$
$953$	−3551.34			−0.120713			−0.0603564	+	0.998177i	$0.519224\pi$
$954$		0			0
$955$	−19787.4			−0.670476
$956$		−	10388.5i		−	0.351451i
$957$		0			0
$958$			16254.3i			0.548175i
$959$			12733.3i			0.428759i
$960$		0			0
$961$	15808.8			0.530658
$962$	9250.13			0.310017
$963$		0			0
$964$		−	10836.1i		−	0.362041i
$965$	966.238			0.0322324
$966$		0			0
$967$	1399.51			0.0465411			0.0232706	−	0.999729i	$-0.492592\pi$
$967$	1399.51			0.0465411			0.0232706	+	0.999729i	$0.492592\pi$
$968$	11750.6			0.390162
$969$		0			0
$970$	−3241.90			−0.107310
$971$	−30479.8			−1.00736			−0.503678	−	0.863892i	$-0.668020\pi$
$971$	−30479.8			−1.00736			−0.503678	+	0.863892i	$0.668020\pi$
$972$		0			0
$973$	88450.0			2.91426
$974$			13407.9i			0.441086i
$975$		0			0
$976$	−262.331			−0.00860350
$977$	47276.1			1.54810			0.774052	−	0.633122i	$-0.218227\pi$
$977$	47276.1			1.54810			0.774052	+	0.633122i	$0.218227\pi$
$978$		0			0
$979$			49385.6i			1.61223i
$980$		−	14401.1i		−	0.469416i
$981$		0			0
$982$		−	2718.64i		−	0.0883457i
$983$	−29707.9			−0.963921			−0.481960	−	0.876193i	$-0.660075\pi$
$983$	−29707.9			−0.963921			−0.481960	+	0.876193i	$0.660075\pi$
$984$		0			0
$985$	−19856.8			−0.642324
$986$			7983.90i			0.257869i
$987$		0			0
$988$	−5320.41	+	4051.21i	−0.171320	+	0.130452i
$989$			2073.12i			0.0666546i
$990$		0			0
$991$		−	51601.0i		−	1.65405i	−0.562168	−	0.827023i	$-0.690033\pi$
$991$		−	51601.0i		−	1.65405i	0.562168	−	0.827023i	$-0.309967\pi$
$992$		−	3783.88i		−	0.121107i
$993$		0			0
$994$	−23356.4			−0.745292
$995$		−	21700.2i		−	0.691399i
$996$		0			0
$997$	21421.7			0.680472			0.340236	−	0.940340i	$-0.389493\pi$
$997$	21421.7			0.680472			0.340236	+	0.940340i	$0.389493\pi$
$998$	18899.3			0.599446
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1710.4.f.a.341.3	✓	40
3.2	odd	2		1710.4.f.b.341.4	yes	40
19.18	odd	2		1710.4.f.b.341.3	yes	40
57.56	even	2	inner	1710.4.f.a.341.4	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1710.4.f.a.341.3	✓	40	1.1	even	1	trivial
1710.4.f.a.341.4	yes	40	57.56	even	2	inner
1710.4.f.b.341.3	yes	40	19.18	odd	2
1710.4.f.b.341.4	yes	40	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1710 = 2 \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1710.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +4.00000 q^{4} -5.00000i q^{5} +32.6046 q^{7} -8.00000 q^{8} +10.0000i q^{10} -52.9133i q^{11} +20.1863i q^{13} -65.2091 q^{14} +16.0000 q^{16} -21.3488i q^{17} +(50.1729 + 65.8914i) q^{19} -20.0000i q^{20} +105.827i q^{22} +24.2406i q^{23} -25.0000 q^{25} -40.3725i q^{26} +130.418 q^{28} +186.987 q^{29} +118.246i q^{31} -32.0000 q^{32} +42.6976i q^{34} -163.023i q^{35} +229.120i q^{37} +(-100.346 - 131.783i) q^{38} +40.0000i q^{40} +79.7794 q^{41} +85.5228 q^{43} -211.653i q^{44} -48.4811i q^{46} +216.581i q^{47} +720.057 q^{49} +50.0000 q^{50} +80.7450i q^{52} +475.719 q^{53} -264.567 q^{55} -260.836 q^{56} -373.974 q^{58} +418.563 q^{59} -16.3957 q^{61} -236.492i q^{62} +64.0000 q^{64} +100.931 q^{65} -769.606i q^{67} -85.3952i q^{68} +326.046i q^{70} +358.177 q^{71} -173.467 q^{73} -458.239i q^{74} +(200.692 + 263.566i) q^{76} -1725.22i q^{77} +475.363i q^{79} -80.0000i q^{80} -159.559 q^{82} +1275.17i q^{83} -106.744 q^{85} -171.046 q^{86} +423.307i q^{88} -933.329 q^{89} +658.164i q^{91} +96.9623i q^{92} -433.162i q^{94} +(329.457 - 250.864i) q^{95} +324.190i q^{97} -1440.11 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 80 q^{2} + 160 q^{4} - 56 q^{7} - 320 q^{8} + 112 q^{14} + 640 q^{16} - 76 q^{19} - 1000 q^{25} - 224 q^{28} - 120 q^{29} - 1280 q^{32} + 152 q^{38} - 312 q^{41} + 56 q^{43} + 2112 q^{49} + 2000 q^{50}+ \cdots - 4224 q^{98}+O(q^{100}) \) 40 * q - 80 * q^2 + 160 * q^4 - 56 * q^7 - 320 * q^8 + 112 * q^14 + 640 * q^16 - 76 * q^19 - 1000 * q^25 - 224 * q^28 - 120 * q^29 - 1280 * q^32 + 152 * q^38 - 312 * q^41 + 56 * q^43 + 2112 * q^49 + 2000 * q^50 + 288 * q^53 + 448 * q^56 + 240 * q^58 + 912 * q^59 - 848 * q^61 + 2560 * q^64 + 240 * q^65 - 1248 * q^71 + 512 * q^73 - 304 * q^76 + 624 * q^82 - 112 * q^86 - 552 * q^89 - 60 * q^95 - 4224 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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