LMFDB - Embedded newform 608.3.e.b.417.19

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [608,3,Mod(417,608)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("608.417"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(608, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 608 = 2^{5} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	608.e (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [20,0,0,0,0,0,0,0,-52,0,0,0,0,0,0,0,56] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	no
Analytic conductor:	$16.5668000731$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 8 x^{19} - 196 x^{18} + 1676 x^{17} + 16346 x^{16} - 161824 x^{15} - 667200 x^{14} + \cdots + 1135285065792 $ x^20 - 8x^19 - 196x^18 + 1676x^17 + 16346x^16 - 161824x^15 - 667200x^14 + 8850712x^13 + 9918758x^12 - 288078272x^11 + 224256138x^10 + 5383493580x^9 - 11116604703x^8 - 53755955160x^7 + 166723956672x^6 + 262105513072x^5 - 1053812084224x^4 - 855143026944x^3 + 3318150008320x^2 + 3262428344448*x + 1135285065792
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{27} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			417.19
Root			$-7.02110 - 2.30731i$ of defining polynomial
Character	$\chi$	$=$	608.417
Dual form			608.3.e.b.417.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+4.61461i q^{3} +3.04266 q^{5} -10.5588 q^{7} -12.2947 q^{9} +0.744974 q^{11} -9.55468i q^{13} +14.0407i q^{15} +9.60456 q^{17} +(-18.6445 - 3.65821i) q^{19} -48.7248i q^{21} -33.8269 q^{23} -15.7422 q^{25} -15.2036i q^{27} -14.7653i q^{29} -32.7915i q^{31} +3.43777i q^{33} -32.1269 q^{35} -26.3076i q^{37} +44.0912 q^{39} +35.4097i q^{41} -6.65208 q^{43} -37.4085 q^{45} +83.1646 q^{47} +62.4884 q^{49} +44.3214i q^{51} -1.90270i q^{53} +2.26670 q^{55} +(16.8812 - 86.0372i) q^{57} +86.1200i q^{59} -81.2923 q^{61} +129.817 q^{63} -29.0717i q^{65} -76.4505i q^{67} -156.098i q^{69} +72.4205i q^{71} -49.5794 q^{73} -72.6442i q^{75} -7.86604 q^{77} -83.0953i q^{79} -40.4932 q^{81} -22.0785 q^{83} +29.2234 q^{85} +68.1360 q^{87} -143.280i q^{89} +100.886i q^{91} +151.320 q^{93} +(-56.7289 - 11.1307i) q^{95} +27.2590i q^{97} -9.15921 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 52 q^{9} + 56 q^{17} + 36 q^{25} + 64 q^{45} + 332 q^{49} + 88 q^{57} - 32 q^{61} - 152 q^{73} + 360 q^{77} - 476 q^{81} - 552 q^{85} + 336 q^{93}+O(q^{100}) $ 20 * q - 52 * q^9 + 56 * q^17 + 36 * q^25 + 64 * q^45 + 332 * q^49 + 88 * q^57 - 32 * q^61 - 152 * q^73 + 360 * q^77 - 476 * q^81 - 552 * q^85 + 336 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/608\mathbb{Z}\right)^\times$.

$n$	$97$	$191$	$229$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$			4.61461i			1.53820i	0.639126	+	0.769102i	$0.279296\pi$
$3$			4.61461i			1.53820i	−0.639126	+	0.769102i	$0.720704\pi$
$4$		0			0
$5$	3.04266			0.608532			0.304266	−	0.952587i	$-0.401589\pi$
$5$	3.04266			0.608532			0.304266	+	0.952587i	$0.401589\pi$
$6$		0			0
$7$	−10.5588			−1.50840			−0.754200	−	0.656644i	$-0.771975\pi$
$7$	−10.5588			−1.50840			−0.754200	+	0.656644i	$0.771975\pi$
$8$		0			0
$9$	−12.2947			−1.36607
$10$		0			0
$11$	0.744974			0.0677249			0.0338625	−	0.999427i	$-0.489219\pi$
$11$	0.744974			0.0677249			0.0338625	+	0.999427i	$0.489219\pi$
$12$		0			0
$13$		−	9.55468i		−	0.734975i	−0.930028	−	0.367488i	$-0.880218\pi$
$13$		−	9.55468i		−	0.734975i	0.930028	−	0.367488i	$-0.119782\pi$
$14$		0			0
$15$			14.0407i			0.936048i
$16$		0			0
$17$	9.60456			0.564974			0.282487	−	0.959271i	$-0.408841\pi$
$17$	9.60456			0.564974			0.282487	+	0.959271i	$0.408841\pi$
$18$		0			0
$19$	−18.6445	−	3.65821i	−0.981290	−	0.192537i
$19$	−18.6445	−	3.65821i	−0.981290	−	0.192537i
$20$		0			0
$21$		−	48.7248i		−	2.32023i
$22$		0			0
$23$	−33.8269			−1.47073			−0.735367	−	0.677670i	$-0.762990\pi$
$23$	−33.8269			−1.47073			−0.735367	+	0.677670i	$0.762990\pi$
$24$		0			0
$25$	−15.7422			−0.629688
$26$		0			0
$27$		−	15.2036i		−	0.563097i
$28$		0			0
$29$		−	14.7653i		−	0.509147i	−0.967053	−	0.254573i	$-0.918065\pi$
$29$		−	14.7653i		−	0.509147i	0.967053	−	0.254573i	$-0.0819350\pi$
$30$		0			0
$31$		−	32.7915i		−	1.05779i	−0.848687	−	0.528895i	$-0.822606\pi$
$31$		−	32.7915i		−	1.05779i	0.848687	−	0.528895i	$-0.177394\pi$
$32$		0			0
$33$			3.43777i			0.104175i
$34$		0			0
$35$	−32.1269			−0.917911
$36$		0			0
$37$		−	26.3076i		−	0.711016i	−0.934673	−	0.355508i	$-0.884308\pi$
$37$		−	26.3076i		−	0.711016i	0.934673	−	0.355508i	$-0.115692\pi$
$38$		0			0
$39$	44.0912			1.13054
$40$		0			0
$41$			35.4097i			0.863651i	0.901957	+	0.431825i	$0.142130\pi$
$41$			35.4097i			0.863651i	−0.901957	+	0.431825i	$0.857870\pi$
$42$		0			0
$43$	−6.65208			−0.154699			−0.0773497	−	0.997004i	$-0.524646\pi$
$43$	−6.65208			−0.154699			−0.0773497	+	0.997004i	$0.524646\pi$
$44$		0			0
$45$	−37.4085			−0.831301
$46$		0			0
$47$	83.1646			1.76946			0.884730	−	0.466104i	$-0.154343\pi$
$47$	83.1646			1.76946			0.884730	+	0.466104i	$0.154343\pi$
$48$		0			0
$49$	62.4884			1.27527
$50$		0			0
$51$			44.3214i			0.869046i
$52$		0			0
$53$		−	1.90270i		−	0.0359000i	−0.999839	−	0.0179500i	$-0.994286\pi$
$53$		−	1.90270i		−	0.0359000i	0.999839	−	0.0179500i	$-0.00571397\pi$
$54$		0			0
$55$	2.26670			0.0412128
$56$		0			0
$57$	16.8812	−	86.0372i	0.296162	−	1.50942i
$58$		0			0
$59$			86.1200i			1.45966i	0.683628	+	0.729830i	$0.260401\pi$
$59$			86.1200i			1.45966i	−0.683628	+	0.729830i	$0.739599\pi$
$60$		0			0
$61$	−81.2923			−1.33266			−0.666330	−	0.745657i	$-0.732136\pi$
$61$	−81.2923			−1.33266			−0.666330	+	0.745657i	$0.732136\pi$
$62$		0			0
$63$	129.817			2.06059
$64$		0			0
$65$		−	29.0717i		−	0.447256i
$66$		0			0
$67$		−	76.4505i		−	1.14105i	−0.821280	−	0.570526i	$-0.806739\pi$
$67$		−	76.4505i		−	1.14105i	0.821280	−	0.570526i	$-0.193261\pi$
$68$		0			0
$69$		−	156.098i		−	2.26229i
$70$		0			0
$71$			72.4205i			1.02001i	0.860172	+	0.510004i	$0.170356\pi$
$71$			72.4205i			1.02001i	−0.860172	+	0.510004i	$0.829644\pi$
$72$		0			0
$73$	−49.5794			−0.679169			−0.339585	−	0.940575i	$-0.610287\pi$
$73$	−49.5794			−0.679169			−0.339585	+	0.940575i	$0.610287\pi$
$74$		0			0
$75$		−	72.6442i		−	0.968589i
$76$		0			0
$77$	−7.86604			−0.102156
$78$		0			0
$79$		−	83.0953i		−	1.05184i	−0.850534	−	0.525920i	$-0.823721\pi$
$79$		−	83.0953i		−	1.05184i	0.850534	−	0.525920i	$-0.176279\pi$
$80$		0			0
$81$	−40.4932			−0.499915
$82$		0			0
$83$	−22.0785			−0.266006			−0.133003	−	0.991116i	$-0.542462\pi$
$83$	−22.0785			−0.266006			−0.133003	+	0.991116i	$0.542462\pi$
$84$		0			0
$85$	29.2234			0.343805
$86$		0			0
$87$	68.1360			0.783172
$88$		0			0
$89$		−	143.280i		−	1.60989i	−0.593347	−	0.804946i	$-0.702194\pi$
$89$		−	143.280i		−	1.60989i	0.593347	−	0.804946i	$-0.297806\pi$
$90$		0			0
$91$			100.886i			1.10864i
$92$		0			0
$93$	151.320			1.62710
$94$		0			0
$95$	−56.7289	−	11.1307i	−0.597147	−	0.117165i
$96$		0			0
$97$			27.2590i			0.281021i	0.990079	+	0.140510i	$0.0448743\pi$
$97$			27.2590i			0.281021i	−0.990079	+	0.140510i	$0.955126\pi$
$98$		0			0
$99$	−9.15921			−0.0925173
$100$		0			0
$101$	−37.0675			−0.367005			−0.183502	−	0.983019i	$-0.558744\pi$
$101$	−37.0675			−0.367005			−0.183502	+	0.983019i	$0.558744\pi$
$102$		0			0
$103$		−	133.682i		−	1.29789i	−0.760837	−	0.648943i	$-0.775211\pi$
$103$		−	133.682i		−	1.29789i	0.760837	−	0.648943i	$-0.224789\pi$
$104$		0			0
$105$		−	148.253i		−	1.41193i
$106$		0			0
$107$		−	89.8803i		−	0.840003i	−0.907523	−	0.420001i	$-0.862030\pi$
$107$		−	89.8803i		−	0.840003i	0.907523	−	0.420001i	$-0.137970\pi$
$108$		0			0
$109$			89.3900i			0.820092i	0.912065	+	0.410046i	$0.134487\pi$
$109$			89.3900i			0.820092i	−0.912065	+	0.410046i	$0.865513\pi$
$110$		0			0
$111$	121.399			1.09369
$112$		0			0
$113$			215.912i			1.91073i	0.295428	+	0.955365i	$0.404538\pi$
$113$			215.912i			1.91073i	−0.295428	+	0.955365i	$0.595462\pi$
$114$		0			0
$115$	−102.924			−0.894989
$116$		0			0
$117$			117.472i			1.00403i
$118$		0			0
$119$	−101.413			−0.852208
$120$		0			0
$121$	−120.445			−0.995413
$122$		0			0
$123$	−163.402			−1.32847
$124$		0			0
$125$	−123.965			−0.991718
$126$		0			0
$127$			149.423i			1.17656i	0.808659	+	0.588278i	$0.200194\pi$
$127$			149.423i			1.17656i	−0.808659	+	0.588278i	$0.799806\pi$
$128$		0			0
$129$		−	30.6968i		−	0.237959i
$130$		0			0
$131$	−57.6652			−0.440193			−0.220096	−	0.975478i	$-0.570637\pi$
$131$	−57.6652			−0.440193			−0.220096	+	0.975478i	$0.570637\pi$
$132$		0			0
$133$	196.864	+	38.6263i	1.48018	+	0.290424i
$134$		0			0
$135$		−	46.2595i		−	0.342663i
$136$		0			0
$137$	5.25298			0.0383429			0.0191715	−	0.999816i	$-0.493897\pi$
$137$	5.25298			0.0383429			0.0191715	+	0.999816i	$0.493897\pi$
$138$		0			0
$139$	−233.771			−1.68180			−0.840902	−	0.541188i	$-0.817975\pi$
$139$	−233.771			−1.68180			−0.840902	+	0.541188i	$0.817975\pi$
$140$		0			0
$141$			383.773i			2.72179i
$142$		0			0
$143$		−	7.11799i		−	0.0497761i
$144$		0			0
$145$		−	44.9257i		−	0.309832i
$146$		0			0
$147$			288.360i			1.96163i
$148$		0			0
$149$	−37.4417			−0.251286			−0.125643	−	0.992075i	$-0.540099\pi$
$149$	−37.4417			−0.251286			−0.125643	+	0.992075i	$0.540099\pi$
$150$		0			0
$151$			191.702i			1.26955i	0.772696	+	0.634776i	$0.218908\pi$
$151$			191.702i			1.26955i	−0.772696	+	0.634776i	$0.781092\pi$
$152$		0			0
$153$	−118.085			−0.771797
$154$		0			0
$155$		−	99.7735i		−	0.643700i
$156$		0			0
$157$	212.399			1.35286			0.676431	−	0.736506i	$-0.263526\pi$
$157$	212.399			1.35286			0.676431	+	0.736506i	$0.263526\pi$
$158$		0			0
$159$	8.78023			0.0552215
$160$		0			0
$161$	357.171			2.21845
$162$		0			0
$163$	−176.541			−1.08307			−0.541536	−	0.840677i	$-0.682157\pi$
$163$	−176.541			−1.08307			−0.541536	+	0.840677i	$0.682157\pi$
$164$		0			0
$165$			10.4600i			0.0633938i
$166$		0			0
$167$			88.7285i			0.531309i	0.964068	+	0.265654i	$0.0855880\pi$
$167$			88.7285i			0.531309i	−0.964068	+	0.265654i	$0.914412\pi$
$168$		0			0
$169$	77.7081			0.459811
$170$		0			0
$171$	229.228	+	44.9765i	1.34051	+	0.263020i
$172$		0			0
$173$			65.5625i			0.378974i	0.981883	+	0.189487i	$0.0606825\pi$
$173$			65.5625i			0.378974i	−0.981883	+	0.189487i	$0.939318\pi$
$174$		0			0
$175$	166.219			0.949822
$176$		0			0
$177$	−397.411			−2.24526
$178$		0			0
$179$			8.36841i			0.0467509i	0.999727	+	0.0233755i	$0.00744132\pi$
$179$			8.36841i			0.0467509i	−0.999727	+	0.0233755i	$0.992559\pi$
$180$		0			0
$181$		−	135.567i		−	0.748988i	−0.927229	−	0.374494i	$-0.877816\pi$
$181$		−	135.567i		−	0.748988i	0.927229	−	0.374494i	$-0.122184\pi$
$182$		0			0
$183$		−	375.133i		−	2.04991i
$184$		0			0
$185$		−	80.0451i		−	0.432676i
$186$		0			0
$187$	7.15515			0.0382628
$188$		0			0
$189$			160.532i			0.849376i
$190$		0			0
$191$	−307.043			−1.60755			−0.803777	−	0.594931i	$-0.797179\pi$
$191$	−307.043			−1.60755			−0.803777	+	0.594931i	$0.797179\pi$
$192$		0			0
$193$			329.860i			1.70912i	0.519353	+	0.854560i	$0.326173\pi$
$193$			329.860i			1.70912i	−0.519353	+	0.854560i	$0.673827\pi$
$194$		0			0
$195$	134.155			0.687972
$196$		0			0
$197$	−295.125			−1.49809			−0.749047	−	0.662516i	$-0.769488\pi$
$197$	−295.125			−1.49809			−0.749047	+	0.662516i	$0.769488\pi$
$198$		0			0
$199$	−12.7280			−0.0639600			−0.0319800	−	0.999489i	$-0.510181\pi$
$199$	−12.7280			−0.0639600			−0.0319800	+	0.999489i	$0.510181\pi$
$200$		0			0
$201$	352.789			1.75517
$202$		0			0
$203$			155.903i			0.767997i
$204$		0			0
$205$			107.740i			0.525560i
$206$		0			0
$207$	415.890			2.00913
$208$		0			0
$209$	−13.8897	−	2.72527i	−0.0664578	−	0.0130396i
$210$		0			0
$211$		−	61.9360i		−	0.293536i	−0.989171	−	0.146768i	$-0.953113\pi$
$211$		−	61.9360i		−	0.293536i	0.989171	−	0.146768i	$-0.0468870\pi$
$212$		0			0
$213$	−334.193			−1.56898
$214$		0			0
$215$	−20.2400			−0.0941396
$216$		0			0
$217$			346.239i			1.59557i
$218$		0			0
$219$		−	228.790i		−	1.04470i
$220$		0			0
$221$		−	91.7685i		−	0.415242i
$222$		0			0
$223$		−	138.551i		−	0.621304i	−0.950524	−	0.310652i	$-0.899453\pi$
$223$		−	138.551i		−	0.621304i	0.950524	−	0.310652i	$-0.100547\pi$
$224$		0			0
$225$	193.545			0.860201
$226$		0			0
$227$		−	224.696i		−	0.989851i	−0.868935	−	0.494926i	$-0.835195\pi$
$227$		−	224.696i		−	0.989851i	0.868935	−	0.494926i	$-0.164805\pi$
$228$		0			0
$229$	−80.7129			−0.352458			−0.176229	−	0.984349i	$-0.556390\pi$
$229$	−80.7129			−0.352458			−0.176229	+	0.984349i	$0.556390\pi$
$230$		0			0
$231$		−	36.2987i		−	0.157137i
$232$		0			0
$233$	228.785			0.981908			0.490954	−	0.871186i	$-0.336648\pi$
$233$	228.785			0.981908			0.490954	+	0.871186i	$0.336648\pi$
$234$		0			0
$235$	253.042			1.07677
$236$		0			0
$237$	383.453			1.61794
$238$		0			0
$239$	−56.9787			−0.238404			−0.119202	−	0.992870i	$-0.538034\pi$
$239$	−56.9787			−0.238404			−0.119202	+	0.992870i	$0.538034\pi$
$240$		0			0
$241$			373.328i			1.54908i	0.632525	+	0.774540i	$0.282019\pi$
$241$			373.328i			1.54908i	−0.632525	+	0.774540i	$0.717981\pi$
$242$		0			0
$243$		−	323.693i		−	1.33207i
$244$		0			0
$245$	190.131			0.776045
$246$		0			0
$247$	−34.9530	+	178.142i	−0.141510	+	0.721224i
$248$		0			0
$249$		−	101.884i		−	0.409172i
$250$		0			0
$251$	186.042			0.741201			0.370601	−	0.928792i	$-0.379152\pi$
$251$	186.042			0.741201			0.370601	+	0.928792i	$0.379152\pi$
$252$		0			0
$253$	−25.2001			−0.0996053
$254$		0			0
$255$			134.855i			0.528843i
$256$		0			0
$257$		−	43.9460i		−	0.170996i	−0.996338	−	0.0854980i	$-0.972752\pi$
$257$		−	43.9460i		−	0.170996i	0.996338	−	0.0854980i	$-0.0272481\pi$
$258$		0			0
$259$			277.777i			1.07250i
$260$		0			0
$261$			181.534i			0.695532i
$262$		0			0
$263$	202.871			0.771371			0.385686	−	0.922630i	$-0.373965\pi$
$263$	202.871			0.771371			0.385686	+	0.922630i	$0.373965\pi$
$264$		0			0
$265$		−	5.78927i		−	0.0218463i
$266$		0			0
$267$	661.184			2.47635
$268$		0			0
$269$		−	15.4568i		−	0.0574601i	−0.999587	−	0.0287300i	$-0.990854\pi$
$269$		−	15.4568i		−	0.0574601i	0.999587	−	0.0287300i	$-0.00914631\pi$
$270$		0			0
$271$	41.7084			0.153906			0.0769528	−	0.997035i	$-0.475481\pi$
$271$	41.7084			0.153906			0.0769528	+	0.997035i	$0.475481\pi$
$272$		0			0
$273$	−465.550			−1.70531
$274$		0			0
$275$	−11.7275			−0.0426456
$276$		0			0
$277$	−315.358			−1.13848			−0.569238	−	0.822173i	$-0.692762\pi$
$277$	−315.358			−1.13848			−0.569238	+	0.822173i	$0.692762\pi$
$278$		0			0
$279$			403.161i			1.44502i
$280$		0			0
$281$			325.209i			1.15733i	0.815566	+	0.578664i	$0.196426\pi$
$281$			325.209i			1.15733i	−0.815566	+	0.578664i	$0.803574\pi$
$282$		0			0
$283$	−393.336			−1.38988			−0.694941	−	0.719067i	$-0.744569\pi$
$283$	−393.336			−1.38988			−0.694941	+	0.719067i	$0.744569\pi$
$284$		0			0
$285$	51.3639	−	261.782i	0.180224	−	0.918534i
$286$		0			0
$287$		−	373.884i		−	1.30273i
$288$		0			0
$289$	−196.752			−0.680804
$290$		0			0
$291$	−125.790			−0.432267
$292$		0			0
$293$		−	552.119i		−	1.88437i	−0.335099	−	0.942183i	$-0.608770\pi$
$293$		−	552.119i		−	1.88437i	0.335099	−	0.942183i	$-0.391230\pi$
$294$		0			0
$295$			262.034i			0.888251i
$296$		0			0
$297$		−	11.3263i		−	0.0381357i
$298$		0			0
$299$			323.205i			1.08095i
$300$		0			0
$301$	70.2380			0.233349
$302$		0			0
$303$		−	171.052i		−	0.564529i
$304$		0			0
$305$	−247.345			−0.810967
$306$		0			0
$307$			79.4760i			0.258880i	0.991587	+	0.129440i	$0.0413179\pi$
$307$			79.4760i			0.258880i	−0.991587	+	0.129440i	$0.958682\pi$
$308$		0			0
$309$	616.892			1.99641
$310$		0			0
$311$	−339.941			−1.09306			−0.546529	−	0.837440i	$-0.684051\pi$
$311$	−339.941			−1.09306			−0.546529	+	0.837440i	$0.684051\pi$
$312$		0			0
$313$	374.341			1.19598			0.597989	−	0.801505i	$-0.295967\pi$
$313$	374.341			1.19598			0.597989	+	0.801505i	$0.295967\pi$
$314$		0			0
$315$	394.989			1.25393
$316$		0			0
$317$		−	574.694i		−	1.81292i	−0.422296	−	0.906458i	$-0.638776\pi$
$317$		−	574.694i		−	1.81292i	0.422296	−	0.906458i	$-0.361224\pi$
$318$		0			0
$319$		−	10.9997i		−	0.0344819i
$320$		0			0
$321$	414.763			1.29210
$322$		0			0
$323$	−179.072	−	35.1355i	−0.554403	−	0.108779i
$324$		0			0
$325$			150.412i			0.462805i
$326$		0			0
$327$	−412.500			−1.26147
$328$		0			0
$329$	−878.119			−2.66905
$330$		0			0
$331$			393.365i			1.18841i	0.804312	+	0.594207i	$0.202534\pi$
$331$			393.365i			1.18841i	−0.804312	+	0.594207i	$0.797466\pi$
$332$		0			0
$333$			323.443i			0.971300i
$334$		0			0
$335$		−	232.613i		−	0.694367i
$336$		0			0
$337$			518.395i			1.53826i	0.639091	+	0.769131i	$0.279311\pi$
$337$			518.395i			1.53826i	−0.639091	+	0.769131i	$0.720689\pi$
$338$		0			0
$339$	−996.353			−2.93909
$340$		0			0
$341$		−	24.4288i		−	0.0716388i
$342$		0			0
$343$	−142.421			−0.415221
$344$		0			0
$345$		−	474.953i		−	1.37668i
$346$		0			0
$347$	486.736			1.40270			0.701349	−	0.712818i	$-0.252581\pi$
$347$	486.736			1.40270			0.701349	+	0.712818i	$0.252581\pi$
$348$		0			0
$349$	−437.910			−1.25476			−0.627378	−	0.778715i	$-0.715872\pi$
$349$	−437.910			−1.25476			−0.627378	+	0.778715i	$0.715872\pi$
$350$		0			0
$351$	−145.266			−0.413862
$352$		0			0
$353$	395.142			1.11938			0.559691	−	0.828701i	$-0.310920\pi$
$353$	395.142			1.11938			0.559691	+	0.828701i	$0.310920\pi$
$354$		0			0
$355$			220.351i			0.620708i
$356$		0			0
$357$		−	467.981i		−	1.31087i
$358$		0			0
$359$	−610.714			−1.70115			−0.850577	−	0.525851i	$-0.823747\pi$
$359$	−610.714			−1.70115			−0.850577	+	0.525851i	$0.823747\pi$
$360$		0			0
$361$	334.235	+	136.411i	0.925859	+	0.377870i
$362$		0			0
$363$		−	555.807i		−	1.53115i
$364$		0			0
$365$	−150.853			−0.413297
$366$		0			0
$367$	417.827			1.13849			0.569246	−	0.822167i	$-0.307235\pi$
$367$	417.827			1.13849			0.569246	+	0.822167i	$0.307235\pi$
$368$		0			0
$369$		−	435.350i		−	1.17981i
$370$		0			0
$371$			20.0902i			0.0541516i
$372$		0			0
$373$			380.137i			1.01913i	0.860431	+	0.509567i	$0.170194\pi$
$373$			380.137i			1.01913i	−0.860431	+	0.509567i	$0.829806\pi$
$374$		0			0
$375$		−	572.050i		−	1.52547i
$376$		0			0
$377$	−141.077			−0.374210
$378$		0			0
$379$		−	686.740i		−	1.81198i	−0.423300	−	0.905989i	$-0.639128\pi$
$379$		−	686.740i		−	1.81198i	0.423300	−	0.905989i	$-0.360872\pi$
$380$		0			0
$381$	−689.528			−1.80978
$382$		0			0
$383$			678.393i			1.77126i	0.464391	+	0.885630i	$0.346273\pi$
$383$			678.393i			1.77126i	−0.464391	+	0.885630i	$0.653727\pi$
$384$		0			0
$385$	−23.9337			−0.0621654
$386$		0			0
$387$	81.7851			0.211331
$388$		0			0
$389$	325.724			0.837338			0.418669	−	0.908139i	$-0.362497\pi$
$389$	325.724			0.837338			0.418669	+	0.908139i	$0.362497\pi$
$390$		0			0
$391$	−324.892			−0.830926
$392$		0			0
$393$		−	266.103i		−	0.677107i
$394$		0			0
$395$		−	252.831i		−	0.640078i
$396$		0			0
$397$	415.385			1.04631			0.523155	−	0.852238i	$-0.324755\pi$
$397$	415.385			1.04631			0.523155	+	0.852238i	$0.324755\pi$
$398$		0			0
$399$	−178.246	+	908.450i	−0.446731	+	2.27682i
$400$		0			0
$401$		−	546.926i		−	1.36391i	−0.731396	−	0.681953i	$-0.761131\pi$
$401$		−	546.926i		−	1.36391i	0.731396	−	0.681953i	$-0.238869\pi$
$402$		0			0
$403$	−313.312			−0.777450
$404$		0			0
$405$	−123.207			−0.304215
$406$		0			0
$407$		−	19.5985i		−	0.0481535i
$408$		0			0
$409$			336.646i			0.823095i	0.911388	+	0.411547i	$0.135012\pi$
$409$			336.646i			0.823095i	−0.911388	+	0.411547i	$0.864988\pi$
$410$		0			0
$411$			24.2405i			0.0589793i
$412$		0			0
$413$		−	909.324i		−	2.20175i
$414$		0			0
$415$	−67.1774			−0.161873
$416$		0			0
$417$		−	1078.76i		−	2.58696i
$418$		0			0
$419$	379.191			0.904990			0.452495	−	0.891767i	$-0.350534\pi$
$419$	379.191			0.904990			0.452495	+	0.891767i	$0.350534\pi$
$420$		0			0
$421$			210.111i			0.499076i	0.968365	+	0.249538i	$0.0802787\pi$
$421$			210.111i			0.499076i	−0.968365	+	0.249538i	$0.919721\pi$
$422$		0			0
$423$	−1022.48			−2.41721
$424$		0			0
$425$	−151.197			−0.355758
$426$		0			0
$427$	858.350			2.01019
$428$		0			0
$429$	32.8468			0.0765659
$430$		0			0
$431$		−	419.971i		−	0.974410i	−0.873288	−	0.487205i	$-0.838017\pi$
$431$		−	419.971i		−	0.974410i	0.873288	−	0.487205i	$-0.161983\pi$
$432$		0			0
$433$		−	560.709i		−	1.29494i	−0.762091	−	0.647470i	$-0.775827\pi$
$433$		−	560.709i		−	1.29494i	0.762091	−	0.647470i	$-0.224173\pi$
$434$		0			0
$435$	207.315			0.476586
$436$		0			0
$437$	630.685	+	123.746i	1.44322	+	0.283171i
$438$		0			0
$439$		−	595.774i		−	1.35712i	−0.734547	−	0.678558i	$-0.762605\pi$
$439$		−	595.774i		−	1.35712i	0.734547	−	0.678558i	$-0.237395\pi$
$440$		0			0
$441$	−768.274			−1.74212
$442$		0			0
$443$	203.741			0.459913			0.229956	−	0.973201i	$-0.426142\pi$
$443$	203.741			0.459913			0.229956	+	0.973201i	$0.426142\pi$
$444$		0			0
$445$		−	435.954i		−	0.979672i
$446$		0			0
$447$		−	172.779i		−	0.386530i
$448$		0			0
$449$		−	267.956i		−	0.596784i	−0.954443	−	0.298392i	$-0.903550\pi$
$449$		−	267.956i		−	0.596784i	0.954443	−	0.298392i	$-0.0964503\pi$
$450$		0			0
$451$			26.3793i			0.0584907i
$452$		0			0
$453$	−884.633			−1.95283
$454$		0			0
$455$			306.962i			0.674642i
$456$		0			0
$457$	542.354			1.18677			0.593385	−	0.804918i	$-0.297791\pi$
$457$	542.354			1.18677			0.593385	+	0.804918i	$0.297791\pi$
$458$		0			0
$459$		−	146.024i		−	0.318135i
$460$		0			0
$461$	−374.095			−0.811486			−0.405743	−	0.913987i	$-0.632987\pi$
$461$	−374.095			−0.811486			−0.405743	+	0.913987i	$0.632987\pi$
$462$		0			0
$463$	95.0496			0.205291			0.102645	−	0.994718i	$-0.467269\pi$
$463$	95.0496			0.205291			0.102645	+	0.994718i	$0.467269\pi$
$464$		0			0
$465$	460.416			0.990143
$466$		0			0
$467$	233.057			0.499051			0.249525	−	0.968368i	$-0.419725\pi$
$467$	233.057			0.499051			0.249525	+	0.968368i	$0.419725\pi$
$468$		0			0
$469$			807.226i			1.72116i
$470$		0			0
$471$			980.141i			2.08098i
$472$		0			0
$473$	−4.95562			−0.0104770
$474$		0			0
$475$	293.506	+	57.5883i	0.617907	+	0.121239i
$476$		0			0
$477$			23.3931i			0.0490421i
$478$		0			0
$479$	−108.804			−0.227148			−0.113574	−	0.993530i	$-0.536230\pi$
$479$	−108.804			−0.227148			−0.113574	+	0.993530i	$0.536230\pi$
$480$		0			0
$481$	−251.360			−0.522579
$482$		0			0
$483$			1648.21i			3.41244i
$484$		0			0
$485$			82.9399i			0.171010i
$486$		0			0
$487$		−	280.774i		−	0.576538i	−0.957550	−	0.288269i	$-0.906920\pi$
$487$		−	280.774i		−	0.576538i	0.957550	−	0.288269i	$-0.0930797\pi$
$488$		0			0
$489$		−	814.668i		−	1.66599i
$490$		0			0
$491$	795.085			1.61932			0.809659	−	0.586901i	$-0.199652\pi$
$491$	795.085			1.61932			0.809659	+	0.586901i	$0.199652\pi$
$492$		0			0
$493$		−	141.814i		−	0.287655i
$494$		0			0
$495$	−27.8684			−0.0562998
$496$		0			0
$497$		−	764.674i		−	1.53858i
$498$		0			0
$499$	598.787			1.19997			0.599987	−	0.800010i	$-0.295173\pi$
$499$	598.787			1.19997			0.599987	+	0.800010i	$0.295173\pi$
$500$		0			0
$501$	−409.448			−0.817262
$502$		0			0
$503$	377.735			0.750965			0.375482	−	0.926830i	$-0.377477\pi$
$503$	377.735			0.750965			0.375482	+	0.926830i	$0.377477\pi$
$504$		0			0
$505$	−112.784			−0.223334
$506$		0			0
$507$			358.593i			0.707284i
$508$		0			0
$509$			302.308i			0.593926i	0.954889	+	0.296963i	$0.0959738\pi$
$509$			302.308i			0.593926i	−0.954889	+	0.296963i	$0.904026\pi$
$510$		0			0
$511$	523.499			1.02446
$512$		0			0
$513$	−55.6181	+	283.464i	−0.108417	+	0.552561i
$514$		0			0
$515$		−	406.750i		−	0.789805i
$516$		0			0
$517$	61.9555			0.119837
$518$		0			0
$519$	−302.546			−0.582940
$520$		0			0
$521$			235.449i			0.451918i	0.974137	+	0.225959i	$0.0725515\pi$
$521$			235.449i			0.451918i	−0.974137	+	0.225959i	$0.927449\pi$
$522$		0			0
$523$		−	394.033i		−	0.753409i	−0.926333	−	0.376705i	$-0.877057\pi$
$523$		−	394.033i		−	0.753409i	0.926333	−	0.376705i	$-0.122943\pi$
$524$		0			0
$525$			767.036i			1.46102i
$526$		0			0
$527$		−	314.948i		−	0.597625i
$528$		0			0
$529$	615.257			1.16306
$530$		0			0
$531$		−	1058.82i		−	1.99401i
$532$		0			0
$533$	338.328			0.634762
$534$		0			0
$535$		−	273.475i		−	0.511169i
$536$		0			0
$537$	−38.6170			−0.0719125
$538$		0			0
$539$	46.5522			0.0863677
$540$		0			0
$541$	275.902			0.509985			0.254993	−	0.966943i	$-0.417927\pi$
$541$	275.902			0.509985			0.254993	+	0.966943i	$0.417927\pi$
$542$		0			0
$543$	625.589			1.15210
$544$		0			0
$545$			271.984i			0.499053i
$546$		0			0
$547$		−	953.772i		−	1.74364i	−0.489825	−	0.871821i	$-0.662939\pi$
$547$		−	953.772i		−	1.74364i	0.489825	−	0.871821i	$-0.337061\pi$
$548$		0			0
$549$	999.462			1.82051
$550$		0			0
$551$	−54.0144	+	275.291i	−0.0980298	+	0.499620i
$552$		0			0
$553$			877.387i			1.58659i
$554$		0			0
$555$	369.377			0.665544
$556$		0			0
$557$	21.1674			0.0380026			0.0190013	−	0.999819i	$-0.493951\pi$
$557$	21.1674			0.0380026			0.0190013	+	0.999819i	$0.493951\pi$
$558$		0			0
$559$			63.5584i			0.113700i
$560$		0			0
$561$			33.0183i			0.0588561i
$562$		0			0
$563$			492.820i			0.875346i	0.899134	+	0.437673i	$0.144197\pi$
$563$			492.820i			0.875346i	−0.899134	+	0.437673i	$0.855803\pi$
$564$		0			0
$565$			656.949i			1.16274i
$566$		0			0
$567$	427.559			0.754073
$568$		0			0
$569$			227.031i			0.399000i	0.979898	+	0.199500i	$0.0639317\pi$
$569$			227.031i			0.399000i	−0.979898	+	0.199500i	$0.936068\pi$
$570$		0			0
$571$	−605.159			−1.05982			−0.529911	−	0.848053i	$-0.677775\pi$
$571$	−605.159			−1.05982			−0.529911	+	0.848053i	$0.677775\pi$
$572$		0			0
$573$		−	1416.88i		−	2.47275i
$574$		0			0
$575$	532.509			0.926103
$576$		0			0
$577$	−580.531			−1.00612			−0.503060	−	0.864252i	$-0.667792\pi$
$577$	−580.531			−1.00612			−0.503060	+	0.864252i	$0.667792\pi$
$578$		0			0
$579$	−1522.18			−2.62898
$580$		0			0
$581$	233.123			0.401244
$582$		0			0
$583$		−	1.41746i		−	0.00243132i
$584$		0			0
$585$			357.426i			0.610985i
$586$		0			0
$587$	326.970			0.557019			0.278510	−	0.960433i	$-0.410160\pi$
$587$	326.970			0.557019			0.278510	+	0.960433i	$0.410160\pi$
$588$		0			0
$589$	−119.958	+	611.381i	−0.203664	+	1.03800i
$590$		0			0
$591$		−	1361.89i		−	2.30438i
$592$		0			0
$593$	−395.107			−0.666285			−0.333143	−	0.942876i	$-0.608109\pi$
$593$	−395.107			−0.666285			−0.333143	+	0.942876i	$0.608109\pi$
$594$		0			0
$595$	−308.565			−0.518596
$596$		0			0
$597$		−	58.7350i		−	0.0983836i
$598$		0			0
$599$			516.663i			0.862543i	0.902222	+	0.431272i	$0.141935\pi$
$599$			516.663i			0.862543i	−0.902222	+	0.431272i	$0.858065\pi$
$600$		0			0
$601$			711.606i			1.18404i	0.805925	+	0.592018i	$0.201669\pi$
$601$			711.606i			1.18404i	−0.805925	+	0.592018i	$0.798331\pi$
$602$		0			0
$603$			939.933i			1.55876i
$604$		0			0
$605$	−366.474			−0.605741
$606$		0			0
$607$		−	416.619i		−	0.686358i	−0.939270	−	0.343179i	$-0.888496\pi$
$607$		−	416.619i		−	0.686358i	0.939270	−	0.343179i	$-0.111504\pi$
$608$		0			0
$609$	−719.434			−1.18134
$610$		0			0
$611$		−	794.611i		−	1.30051i
$612$		0			0
$613$	487.985			0.796060			0.398030	−	0.917372i	$-0.369694\pi$
$613$	487.985			0.796060			0.398030	+	0.917372i	$0.369694\pi$
$614$		0			0
$615$	−497.177			−0.808418
$616$		0			0
$617$	−850.573			−1.37856			−0.689281	−	0.724494i	$-0.742073\pi$
$617$	−850.573			−1.37856			−0.689281	+	0.724494i	$0.742073\pi$
$618$		0			0
$619$	−1113.88			−1.79948			−0.899740	−	0.436425i	$-0.856244\pi$
$619$	−1113.88			−1.79948			−0.899740	+	0.436425i	$0.856244\pi$
$620$		0			0
$621$			514.291i			0.828166i
$622$		0			0
$623$			1512.87i			2.42836i
$624$		0			0
$625$	16.3722			0.0261954
$626$		0			0
$627$	12.5761	−	64.0955i	0.0200575	−	0.102226i
$628$		0			0
$629$		−	252.673i		−	0.401706i
$630$		0			0
$631$	633.935			1.00465			0.502326	−	0.864678i	$-0.332478\pi$
$631$	633.935			1.00465			0.502326	+	0.864678i	$0.332478\pi$
$632$		0			0
$633$	285.811			0.451518
$634$		0			0
$635$			454.642i			0.715972i
$636$		0			0
$637$		−	597.056i		−	0.937294i
$638$		0			0
$639$		−	890.386i		−	1.39341i
$640$		0			0
$641$		−	230.942i		−	0.360283i	−0.983641	−	0.180142i	$-0.942344\pi$
$641$		−	230.942i		−	0.360283i	0.983641	−	0.180142i	$-0.0576556\pi$
$642$		0			0
$643$	665.863			1.03556			0.517778	−	0.855515i	$-0.326759\pi$
$643$	665.863			1.03556			0.517778	+	0.855515i	$0.326759\pi$
$644$		0			0
$645$		−	93.3999i		−	0.144806i
$646$		0			0
$647$	−418.479			−0.646800			−0.323400	−	0.946262i	$-0.604826\pi$
$647$	−418.479			−0.646800			−0.323400	+	0.946262i	$0.604826\pi$
$648$		0			0
$649$			64.1572i			0.0988554i
$650$		0			0
$651$	−1597.76			−2.45432
$652$		0			0
$653$	−456.869			−0.699647			−0.349823	−	0.936816i	$-0.613758\pi$
$653$	−456.869			−0.699647			−0.349823	+	0.936816i	$0.613758\pi$
$654$		0			0
$655$	−175.456			−0.267872
$656$		0			0
$657$	609.562			0.927796
$658$		0			0
$659$			638.685i			0.969173i	0.874743	+	0.484587i	$0.161030\pi$
$659$			638.685i			0.969173i	−0.874743	+	0.484587i	$0.838970\pi$
$660$		0			0
$661$			264.809i			0.400619i	0.979733	+	0.200310i	$0.0641948\pi$
$661$			264.809i			0.400619i	−0.979733	+	0.200310i	$0.935805\pi$
$662$		0			0
$663$	423.476			0.638727
$664$		0			0
$665$	598.990	+	117.527i	0.900736	+	0.176732i
$666$		0			0
$667$			499.462i			0.748819i
$668$		0			0
$669$	639.359			0.955693
$670$		0			0
$671$	−60.5607			−0.0902543
$672$		0			0
$673$		−	912.276i		−	1.35554i	−0.735276	−	0.677768i	$-0.762947\pi$
$673$		−	912.276i		−	1.35554i	0.735276	−	0.677768i	$-0.237053\pi$
$674$		0			0
$675$			239.339i			0.354576i
$676$		0			0
$677$		−	81.3790i		−	0.120205i	−0.998192	−	0.0601027i	$-0.980857\pi$
$677$		−	81.3790i		−	0.120205i	0.998192	−	0.0601027i	$-0.0191428\pi$
$678$		0			0
$679$		−	287.822i		−	0.423892i
$680$		0			0
$681$	1036.89			1.52259
$682$		0			0
$683$		−	1192.27i		−	1.74563i	−0.488051	−	0.872815i	$-0.662292\pi$
$683$		−	1192.27i		−	1.74563i	0.488051	−	0.872815i	$-0.337708\pi$
$684$		0			0
$685$	15.9830			0.0233329
$686$		0			0
$687$		−	372.459i		−	0.542153i
$688$		0			0
$689$	−18.1797			−0.0263856
$690$		0			0
$691$	274.429			0.397148			0.198574	−	0.980086i	$-0.436369\pi$
$691$	274.429			0.397148			0.198574	+	0.980086i	$0.436369\pi$
$692$		0			0
$693$	96.7103			0.139553
$694$		0			0
$695$	−711.285			−1.02343
$696$		0			0
$697$			340.095i			0.487941i
$698$		0			0
$699$			1055.75i			1.51038i
$700$		0			0
$701$	39.0692			0.0557335			0.0278667	−	0.999612i	$-0.491129\pi$
$701$	39.0692			0.0557335			0.0278667	+	0.999612i	$0.491129\pi$
$702$		0			0
$703$	−96.2387	+	490.492i	−0.136897	+	0.697712i
$704$		0			0
$705$			1167.69i			1.65630i
$706$		0			0
$707$	391.388			0.553590
$708$		0			0
$709$	−1179.61			−1.66377			−0.831885	−	0.554948i	$-0.812738\pi$
$709$	−1179.61			−1.66377			−0.831885	+	0.554948i	$0.812738\pi$
$710$		0			0
$711$			1021.63i			1.43689i
$712$		0			0
$713$			1109.23i			1.55573i
$714$		0			0
$715$		−	21.6576i		−	0.0302904i
$716$		0			0
$717$		−	262.935i		−	0.366715i
$718$		0			0
$719$	−541.922			−0.753716			−0.376858	−	0.926271i	$-0.622996\pi$
$719$	−541.922			−0.753716			−0.376858	+	0.926271i	$0.622996\pi$
$720$		0			0
$721$			1411.52i			1.95773i
$722$		0			0
$723$	−1722.77			−2.38280
$724$		0			0
$725$			232.438i			0.320604i
$726$		0			0
$727$	−74.6095			−0.102627			−0.0513133	−	0.998683i	$-0.516341\pi$
$727$	−74.6095			−0.102627			−0.0513133	+	0.998683i	$0.516341\pi$
$728$		0			0
$729$	1129.28			1.54908
$730$		0			0
$731$	−63.8903			−0.0874012
$732$		0			0
$733$	450.558			0.614677			0.307338	−	0.951600i	$-0.400562\pi$
$733$	450.558			0.614677			0.307338	+	0.951600i	$0.400562\pi$
$734$		0			0
$735$			877.381i			1.19372i
$736$		0			0
$737$		−	56.9536i		−	0.0772776i
$738$		0			0
$739$	−147.137			−0.199102			−0.0995511	−	0.995032i	$-0.531741\pi$
$739$	−147.137			−0.199102			−0.0995511	+	0.995032i	$0.531741\pi$
$740$		0			0
$741$	−822.058	−	161.295i	−1.10939	−	0.217672i
$742$		0			0
$743$		−	246.029i		−	0.331130i	−0.986199	−	0.165565i	$-0.947055\pi$
$743$		−	246.029i		−	0.331130i	0.986199	−	0.165565i	$-0.0529447\pi$
$744$		0			0
$745$	−113.922			−0.152916
$746$		0			0
$747$	271.448			0.363384
$748$		0			0
$749$			949.029i			1.26706i
$750$		0			0
$751$			759.697i			1.01158i	0.862656	+	0.505790i	$0.168799\pi$
$751$			759.697i			1.01158i	−0.862656	+	0.505790i	$0.831201\pi$
$752$		0			0
$753$			858.510i			1.14012i
$754$		0			0
$755$			583.286i			0.772564i
$756$		0			0
$757$	−883.598			−1.16724			−0.583618	−	0.812028i	$-0.698364\pi$
$757$	−883.598			−1.16724			−0.583618	+	0.812028i	$0.698364\pi$
$758$		0			0
$759$		−	116.289i		−	0.153213i
$760$		0			0
$761$	967.768			1.27171			0.635853	−	0.771810i	$-0.280648\pi$
$761$	967.768			1.27171			0.635853	+	0.771810i	$0.280648\pi$
$762$		0			0
$763$		−	943.852i		−	1.23703i
$764$		0			0
$765$	−359.293			−0.469663
$766$		0			0
$767$	822.849			1.07281
$768$		0			0
$769$	−172.159			−0.223874			−0.111937	−	0.993715i	$-0.535705\pi$
$769$	−172.159			−0.223874			−0.111937	+	0.993715i	$0.535705\pi$
$770$		0			0
$771$	202.794			0.263027
$772$		0			0
$773$			231.818i			0.299894i	0.988694	+	0.149947i	$0.0479103\pi$
$773$			231.818i			0.299894i	−0.988694	+	0.149947i	$0.952090\pi$
$774$		0			0
$775$			516.211i			0.666078i
$776$		0			0
$777$	−1281.83			−1.64972
$778$		0			0
$779$	129.536	−	660.196i	0.166285	−	0.847492i
$780$		0			0
$781$			53.9514i			0.0690799i
$782$		0			0
$783$	−224.485			−0.286699
$784$		0			0
$785$	646.259			0.823260
$786$		0			0
$787$			1399.98i			1.77888i	0.457047	+	0.889442i	$0.348907\pi$
$787$			1399.98i			1.77888i	−0.457047	+	0.889442i	$0.651093\pi$
$788$		0			0
$789$			936.170i			1.18653i
$790$		0			0
$791$		−	2279.78i		−	2.88215i
$792$		0			0
$793$			776.722i			0.979473i
$794$		0			0
$795$	26.7153			0.0336041
$796$		0			0
$797$		−	1091.52i		−	1.36954i	−0.728759	−	0.684770i	$-0.759902\pi$
$797$		−	1091.52i		−	1.36954i	0.728759	−	0.684770i	$-0.240098\pi$
$798$		0			0
$799$	798.760			0.999699
$800$		0			0
$801$			1761.59i			2.19923i
$802$		0			0
$803$	−36.9353			−0.0459967
$804$		0			0
$805$	1086.75			1.35000
$806$		0			0
$807$	71.3270			0.0883853
$808$		0			0
$809$	−961.211			−1.18815			−0.594074	−	0.804411i	$-0.702481\pi$
$809$	−961.211			−1.18815			−0.594074	+	0.804411i	$0.702481\pi$
$810$		0			0
$811$			269.569i			0.332391i	0.986093	+	0.166195i	$0.0531483\pi$
$811$			269.569i			0.332391i	−0.986093	+	0.166195i	$0.946852\pi$
$812$		0			0
$813$			192.468i			0.236738i
$814$		0			0
$815$	−537.154			−0.659085
$816$		0			0
$817$	124.025	+	24.3347i	0.151805	+	0.0297854i
$818$		0			0
$819$		−	1240.36i		−	1.51448i
$820$		0			0
$821$	978.779			1.19218			0.596089	−	0.802918i	$-0.296720\pi$
$821$	978.779			1.19218			0.596089	+	0.802918i	$0.296720\pi$
$822$		0			0
$823$	821.505			0.998184			0.499092	−	0.866549i	$-0.333667\pi$
$823$	821.505			0.998184			0.499092	+	0.866549i	$0.333667\pi$
$824$		0			0
$825$		−	54.1181i		−	0.0655976i
$826$		0			0
$827$		−	1042.24i		−	1.26026i	−0.776489	−	0.630130i	$-0.783002\pi$
$827$		−	1042.24i		−	1.26026i	0.776489	−	0.630130i	$-0.216998\pi$
$828$		0			0
$829$		−	686.208i		−	0.827754i	−0.910333	−	0.413877i	$-0.864174\pi$
$829$		−	686.208i		−	0.827754i	0.910333	−	0.413877i	$-0.135826\pi$
$830$		0			0
$831$		−	1455.26i		−	1.75121i
$832$		0			0
$833$	600.173			0.720496
$834$		0			0
$835$			269.971i			0.323319i
$836$		0			0
$837$	−498.550			−0.595639
$838$		0			0
$839$		−	1278.82i		−	1.52422i	−0.647448	−	0.762110i	$-0.724164\pi$
$839$		−	1278.82i		−	1.52422i	0.647448	−	0.762110i	$-0.275836\pi$
$840$		0			0
$841$	622.987			0.740770
$842$		0			0
$843$	−1500.72			−1.78021
$844$		0			0
$845$	236.440			0.279810
$846$		0			0
$847$	1271.76			1.50148
$848$		0			0
$849$		−	1815.10i		−	2.13792i
$850$		0			0
$851$			889.903i			1.04571i
$852$		0			0
$853$	−624.553			−0.732185			−0.366092	−	0.930579i	$-0.619305\pi$
$853$	−624.553			−0.732185			−0.366092	+	0.930579i	$0.619305\pi$
$854$		0			0
$855$	697.463	+	136.848i	0.815747	+	0.160056i
$856$		0			0
$857$		−	449.009i		−	0.523931i	−0.965077	−	0.261965i	$-0.915629\pi$
$857$		−	449.009i		−	0.523931i	0.965077	−	0.261965i	$-0.0843706\pi$
$858$		0			0
$859$	−29.9823			−0.0349037			−0.0174519	−	0.999848i	$-0.505555\pi$
$859$	−29.9823			−0.0349037			−0.0174519	+	0.999848i	$0.505555\pi$
$860$		0			0
$861$	1725.33			2.00387
$862$		0			0
$863$		−	398.762i		−	0.462065i	−0.972946	−	0.231033i	$-0.925790\pi$
$863$		−	398.762i		−	0.462065i	0.972946	−	0.231033i	$-0.0742104\pi$
$864$		0			0
$865$			199.485i			0.230618i
$866$		0			0
$867$		−	907.936i		−	1.04722i
$868$		0			0
$869$		−	61.9038i		−	0.0712357i
$870$		0			0
$871$	−730.460			−0.838645
$872$		0			0
$873$		−	335.140i		−	0.383895i
$874$		0			0
$875$	1308.92			1.49591
$876$		0			0
$877$			1265.59i			1.44309i	0.692369	+	0.721543i	$0.256567\pi$
$877$			1265.59i			1.44309i	−0.692369	+	0.721543i	$0.743433\pi$
$878$		0			0
$879$	2547.82			2.89854
$880$		0			0
$881$	396.725			0.450312			0.225156	−	0.974323i	$-0.427711\pi$
$881$	396.725			0.450312			0.225156	+	0.974323i	$0.427711\pi$
$882$		0			0
$883$	1183.69			1.34053			0.670267	−	0.742120i	$-0.266180\pi$
$883$	1183.69			1.34053			0.670267	+	0.742120i	$0.266180\pi$
$884$		0			0
$885$	−1209.19			−1.36631
$886$		0			0
$887$			1461.28i			1.64744i	0.566996	+	0.823721i	$0.308106\pi$
$887$			1461.28i			1.64744i	−0.566996	+	0.823721i	$0.691894\pi$
$888$		0			0
$889$		−	1577.72i		−	1.77472i
$890$		0			0
$891$	−30.1664			−0.0338567
$892$		0			0
$893$	−1550.56	−	304.234i	−1.73635	−	0.340687i
$894$		0			0
$895$			25.4623i			0.0284494i
$896$		0			0
$897$	−1491.47			−1.66273
$898$		0			0
$899$	−484.175			−0.538571
$900$		0			0
$901$		−	18.2746i		−	0.0202826i
$902$		0			0
$903$			324.121i			0.358938i
$904$		0			0
$905$		−	412.484i		−	0.455784i
$906$		0			0
$907$			1236.34i			1.36310i	0.731769	+	0.681552i	$0.238695\pi$
$907$			1236.34i			1.36310i	−0.731769	+	0.681552i	$0.761305\pi$
$908$		0			0
$909$	455.732			0.501356
$910$		0			0
$911$		−	78.5080i		−	0.0861778i	−0.999071	−	0.0430889i	$-0.986280\pi$
$911$		−	78.5080i		−	0.0861778i	0.999071	−	0.0430889i	$-0.0137199\pi$
$912$		0			0
$913$	−16.4479			−0.0180152
$914$		0			0
$915$		−	1141.40i		−	1.24743i
$916$		0			0
$917$	608.876			0.663987
$918$		0			0
$919$	525.085			0.571366			0.285683	−	0.958324i	$-0.407780\pi$
$919$	525.085			0.571366			0.285683	+	0.958324i	$0.407780\pi$
$920$		0			0
$921$	−366.751			−0.398210
$922$		0			0
$923$	691.955			0.749680
$924$		0			0
$925$			414.139i			0.447718i
$926$		0			0
$927$			1643.58i			1.77301i
$928$		0			0
$929$	−987.073			−1.06251			−0.531256	−	0.847212i	$-0.678280\pi$
$929$	−987.073			−1.06251			−0.531256	+	0.847212i	$0.678280\pi$
$930$		0			0
$931$	−1165.06	−	228.596i	−1.25141	−	0.245538i
$932$		0			0
$933$		−	1568.70i		−	1.68135i
$934$		0			0
$935$	21.7707			0.0232842
$936$		0			0
$937$	645.896			0.689323			0.344662	−	0.938727i	$-0.387994\pi$
$937$	645.896			0.689323			0.344662	+	0.938727i	$0.387994\pi$
$938$		0			0
$939$			1727.44i			1.83966i
$940$		0			0
$941$		−	75.1337i		−	0.0798446i	−0.999203	−	0.0399223i	$-0.987289\pi$
$941$		−	75.1337i		−	0.0798446i	0.999203	−	0.0399223i	$-0.0127110\pi$
$942$		0			0
$943$		−	1197.80i		−	1.27020i
$944$		0			0
$945$			488.445i			0.516873i
$946$		0			0
$947$	10.5722			0.0111639			0.00558193	−	0.999984i	$-0.498223\pi$
$947$	10.5722			0.0111639			0.00558193	+	0.999984i	$0.498223\pi$
$948$		0			0
$949$			473.715i			0.499173i
$950$		0			0
$951$	2651.99			2.78864
$952$		0			0
$953$		−	810.970i		−	0.850965i	−0.904967	−	0.425483i	$-0.860104\pi$
$953$		−	810.970i		−	0.850965i	0.904967	−	0.425483i	$-0.139896\pi$
$954$		0			0
$955$	−934.227			−0.978248
$956$		0			0
$957$	50.7595			0.0530403
$958$		0			0
$959$	−55.4652			−0.0578365
$960$		0			0
$961$	−114.283			−0.118921
$962$		0			0
$963$			1105.05i			1.14751i
$964$		0			0
$965$			1003.65i			1.04006i
$966$		0			0
$967$	−658.111			−0.680569			−0.340285	−	0.940322i	$-0.610523\pi$
$967$	−658.111			−0.680569			−0.340285	+	0.940322i	$0.610523\pi$
$968$		0			0
$969$	162.137	−	826.350i	0.167324	−	0.852786i
$970$		0			0
$971$			766.876i			0.789779i	0.918729	+	0.394890i	$0.129217\pi$
$971$			766.876i			0.789779i	−0.918729	+	0.394890i	$0.870783\pi$
$972$		0			0
$973$	2468.34			2.53683
$974$		0			0
$975$	−694.092			−0.711889
$976$		0			0
$977$			619.720i			0.634310i	0.948374	+	0.317155i	$0.102727\pi$
$977$			619.720i			0.634310i	−0.948374	+	0.317155i	$0.897273\pi$
$978$		0			0
$979$		−	106.740i		−	0.109030i
$980$		0			0
$981$		−	1099.02i		−	1.12031i
$982$		0			0
$983$			1020.53i			1.03818i	0.854721	+	0.519088i	$0.173728\pi$
$983$			1020.53i			1.03818i	−0.854721	+	0.519088i	$0.826272\pi$
$984$		0			0
$985$	−897.965			−0.911639
$986$		0			0
$987$		−	4052.18i		−	4.10555i
$988$		0			0
$989$	225.019			0.227522
$990$		0			0
$991$			139.383i			0.140649i	0.997524	+	0.0703243i	$0.0224034\pi$
$991$			139.383i			0.140649i	−0.997524	+	0.0703243i	$0.977597\pi$
$992$		0			0
$993$	−1815.23			−1.82802
$994$		0			0
$995$	−38.7271			−0.0389218
$996$		0			0
$997$	−478.232			−0.479671			−0.239836	−	0.970813i	$-0.577094\pi$
$997$	−478.232			−0.479671			−0.239836	+	0.970813i	$0.577094\pi$
$998$		0			0
$999$	−399.970			−0.400371

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	608.3.e.b.417.19	yes	20
4.3	odd	2	inner	608.3.e.b.417.2	yes	20
8.3	odd	2		1216.3.e.p.1025.20		20
8.5	even	2		1216.3.e.p.1025.1		20
19.18	odd	2	inner	608.3.e.b.417.1	✓	20
76.75	even	2	inner	608.3.e.b.417.20	yes	20
152.37	odd	2		1216.3.e.p.1025.19		20
152.75	even	2		1216.3.e.p.1025.2		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
608.3.e.b.417.1	✓	20	19.18	odd	2	inner
608.3.e.b.417.2	yes	20	4.3	odd	2	inner
608.3.e.b.417.19	yes	20	1.1	even	1	trivial
608.3.e.b.417.20	yes	20	76.75	even	2	inner
1216.3.e.p.1025.1		20	8.5	even	2
1216.3.e.p.1025.2		20	152.75	even	2
1216.3.e.p.1025.19		20	152.37	odd	2
1216.3.e.p.1025.20		20	8.3	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:	\( N \)	\(=\)	\( 608 = 2^{5} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	608.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+4.61461i q^{3} +3.04266 q^{5} -10.5588 q^{7} -12.2947 q^{9} +0.744974 q^{11} -9.55468i q^{13} +14.0407i q^{15} +9.60456 q^{17} +(-18.6445 - 3.65821i) q^{19} -48.7248i q^{21} -33.8269 q^{23} -15.7422 q^{25} -15.2036i q^{27} -14.7653i q^{29} -32.7915i q^{31} +3.43777i q^{33} -32.1269 q^{35} -26.3076i q^{37} +44.0912 q^{39} +35.4097i q^{41} -6.65208 q^{43} -37.4085 q^{45} +83.1646 q^{47} +62.4884 q^{49} +44.3214i q^{51} -1.90270i q^{53} +2.26670 q^{55} +(16.8812 - 86.0372i) q^{57} +86.1200i q^{59} -81.2923 q^{61} +129.817 q^{63} -29.0717i q^{65} -76.4505i q^{67} -156.098i q^{69} +72.4205i q^{71} -49.5794 q^{73} -72.6442i q^{75} -7.86604 q^{77} -83.0953i q^{79} -40.4932 q^{81} -22.0785 q^{83} +29.2234 q^{85} +68.1360 q^{87} -143.280i q^{89} +100.886i q^{91} +151.320 q^{93} +(-56.7289 - 11.1307i) q^{95} +27.2590i q^{97} -9.15921 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 52 q^{9} + 56 q^{17} + 36 q^{25} + 64 q^{45} + 332 q^{49} + 88 q^{57} - 32 q^{61} - 152 q^{73} + 360 q^{77} - 476 q^{81} - 552 q^{85} + 336 q^{93}+O(q^{100}) \) 20 * q - 52 * q^9 + 56 * q^17 + 36 * q^25 + 64 * q^45 + 332 * q^49 + 88 * q^57 - 32 * q^61 - 152 * q^73 + 360 * q^77 - 476 * q^81 - 552 * q^85 + 336 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more