LMFDB - Embedded newform 1520.2.g.f.1519.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1520,2,Mod(1519,1520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1520, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1520.1519");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1520 = 2^{4} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1520.g (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$12.1372611072$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 32x^{14} + 380x^{12} - 1752x^{10} + 1904x^{8} + 7824x^{6} + 7352x^{4} + 2992x^{2} + 484 $ x^16 - 32x^14 + 380x^12 - 1752x^10 + 1904x^8 + 7824x^6 + 7352x^4 + 2992*x^2 + 484
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{15} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1519.10
Root			$0.213388 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1520.1519
Dual form			1520.2.g.f.1519.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.80775i q^{3} +(1.73205 - 1.41421i) q^{5} +2.12976 q^{7} -0.267949 q^{9} +O(q^{10})$	$q+1.80775i q^{3} +(1.73205 - 1.41421i) q^{5} +2.12976 q^{7} -0.267949 q^{9} -4.11439i q^{11} -2.72241 q^{13} +(2.55654 + 3.13111i) q^{15} -1.79315i q^{17} +(3.49230 + 2.60842i) q^{19} +3.85008i q^{21} +3.68886 q^{23} +(1.00000 - 4.89898i) q^{25} +4.93886i q^{27} -10.5186i q^{29} +4.42806 q^{31} +7.43778 q^{33} +(3.68886 - 3.01194i) q^{35} -10.1602 q^{37} -4.92144i q^{39} -3.85008i q^{41} +7.94839 q^{43} +(-0.464102 + 0.378937i) q^{45} -6.38929 q^{47} -2.46410 q^{49} +3.24156 q^{51} +4.71536 q^{53} +(-5.81863 - 7.12633i) q^{55} +(-4.71536 + 6.31319i) q^{57} -4.42806 q^{59} +8.19615 q^{61} -0.570669 q^{63} +(-4.71536 + 3.85008i) q^{65} +3.13111i q^{67} +6.66853i q^{69} +16.5257 q^{71} +13.3843i q^{73} +(8.85612 + 1.80775i) q^{75} -8.76268i q^{77} -6.29958 q^{79} -9.73205 q^{81} -13.7670 q^{83} +(-2.53590 - 3.10583i) q^{85} +19.0150 q^{87} +10.5186i q^{89} -5.79810 q^{91} +8.00481i q^{93} +(9.73770 - 0.420943i) q^{95} +10.1602 q^{97} +1.10245i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 32 q^{9}+O(q^{10}) $ 16 * q - 32 * q^9	$ 16 q - 32 q^{9} + 16 q^{25} + 48 q^{45} + 16 q^{49} + 48 q^{61} - 128 q^{81} - 96 q^{85}+O(q^{100}) $ 16 * q - 32 * q^9 + 16 * q^25 + 48 * q^45 + 16 * q^49 + 48 * q^61 - 128 * q^81 - 96 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1520\mathbb{Z}\right)^\times$.

$n$	$191$	$401$	$1141$	$1217$
$\chi(n)$	$-1$	$-1$	$1$	$-1$

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
$3$			1.80775i			1.04370i	0.853036	+	0.521852i	$0.174759\pi$
$3$			1.80775i			1.04370i	−0.853036	+	0.521852i	$0.825241\pi$
$4$		0			0
$5$	1.73205	−	1.41421i	0.774597	−	0.632456i
$5$	1.73205	−	1.41421i	0.774597	−	0.632456i
$6$		0			0
$7$	2.12976			0.804975			0.402488	−	0.915425i	$-0.368146\pi$
$7$	2.12976			0.804975			0.402488	+	0.915425i	$0.368146\pi$
$8$		0			0
$9$	−0.267949			−0.0893164
$10$		0			0
$11$		−	4.11439i		−	1.24054i	−0.784390	−	0.620268i	$-0.787024\pi$
$11$		−	4.11439i		−	1.24054i	0.784390	−	0.620268i	$-0.212976\pi$
$12$		0			0
$13$	−2.72241			−0.755062			−0.377531	−	0.925997i	$-0.623227\pi$
$13$	−2.72241			−0.755062			−0.377531	+	0.925997i	$0.623227\pi$
$14$		0			0
$15$	2.55654	+	3.13111i	0.660096	+	0.808449i
$16$		0			0
$17$		−	1.79315i		−	0.434903i	−0.976071	−	0.217451i	$-0.930226\pi$
$17$		−	1.79315i		−	0.434903i	0.976071	−	0.217451i	$-0.0697744\pi$
$18$		0			0
$19$	3.49230	+	2.60842i	0.801188	+	0.598412i
$19$	3.49230	+	2.60842i	0.801188	+	0.598412i
$20$		0			0
$21$			3.85008i			0.840155i
$22$		0			0
$23$	3.68886			0.769181			0.384590	−	0.923087i	$-0.374343\pi$
$23$	3.68886			0.769181			0.384590	+	0.923087i	$0.374343\pi$
$24$		0			0
$25$	1.00000	−	4.89898i	0.200000	−	0.979796i
$26$		0			0
$27$			4.93886i			0.950483i
$28$		0			0
$29$		−	10.5186i		−	1.95326i	−0.214938	−	0.976628i	$-0.568955\pi$
$29$		−	10.5186i		−	1.95326i	0.214938	−	0.976628i	$-0.431045\pi$
$30$		0			0
$31$	4.42806			0.795303			0.397651	−	0.917537i	$-0.369825\pi$
$31$	4.42806			0.795303			0.397651	+	0.917537i	$0.369825\pi$
$32$		0			0
$33$	7.43778			1.29475
$34$		0			0
$35$	3.68886	−	3.01194i	0.623531	−	0.509111i
$36$		0			0
$37$	−10.1602			−1.67033			−0.835163	−	0.550003i	$-0.814626\pi$
$37$	−10.1602			−1.67033			−0.835163	+	0.550003i	$0.814626\pi$
$38$		0			0
$39$		−	4.92144i		−	0.788061i
$40$		0			0
$41$		−	3.85008i		−	0.601281i	−0.953738	−	0.300640i	$-0.902800\pi$
$41$		−	3.85008i		−	0.601281i	0.953738	−	0.300640i	$-0.0972004\pi$
$42$		0			0
$43$	7.94839			1.21212			0.606059	−	0.795420i	$-0.292749\pi$
$43$	7.94839			1.21212			0.606059	+	0.795420i	$0.292749\pi$
$44$		0			0
$45$	−0.464102	+	0.378937i	−0.0691842	+	0.0564886i
$46$		0			0
$47$	−6.38929			−0.931974			−0.465987	−	0.884791i	$-0.654301\pi$
$47$	−6.38929			−0.931974			−0.465987	+	0.884791i	$0.654301\pi$
$48$		0			0
$49$	−2.46410			−0.352015
$50$		0			0
$51$	3.24156			0.453910
$52$		0			0
$53$	4.71536			0.647705			0.323852	−	0.946108i	$-0.395022\pi$
$53$	4.71536			0.647705			0.323852	+	0.946108i	$0.395022\pi$
$54$		0			0
$55$	−5.81863	−	7.12633i	−0.784583	−	0.960914i
$56$		0			0
$57$	−4.71536	+	6.31319i	−0.624565	+	0.836203i
$58$		0			0
$59$	−4.42806			−0.576484			−0.288242	−	0.957558i	$-0.593071\pi$
$59$	−4.42806			−0.576484			−0.288242	+	0.957558i	$0.593071\pi$
$60$		0			0
$61$	8.19615			1.04941			0.524705	−	0.851284i	$-0.324176\pi$
$61$	8.19615			1.04941			0.524705	+	0.851284i	$0.324176\pi$
$62$		0			0
$63$	−0.570669			−0.0718975
$64$		0			0
$65$	−4.71536	+	3.85008i	−0.584869	+	0.477543i
$66$		0			0
$67$			3.13111i			0.382526i	0.981539	+	0.191263i	$0.0612583\pi$
$67$			3.13111i			0.382526i	−0.981539	+	0.191263i	$0.938742\pi$
$68$		0			0
$69$			6.66853i			0.802796i
$70$		0			0
$71$	16.5257			1.96124			0.980622	−	0.195909i	$-0.0627658\pi$
$71$	16.5257			1.96124			0.980622	+	0.195909i	$0.0627658\pi$
$72$		0			0
$73$			13.3843i			1.56651i	0.621701	+	0.783255i	$0.286442\pi$
$73$			13.3843i			1.56651i	−0.621701	+	0.783255i	$0.713558\pi$
$74$		0			0
$75$	8.85612	+	1.80775i	1.02262	+	0.208741i
$76$		0			0
$77$		−	8.76268i		−	0.998600i
$78$		0			0
$79$	−6.29958			−0.708758			−0.354379	−	0.935102i	$-0.615308\pi$
$79$	−6.29958			−0.708758			−0.354379	+	0.935102i	$0.615308\pi$
$80$		0			0
$81$	−9.73205			−1.08134
$82$		0			0
$83$	−13.7670			−1.51113			−0.755563	−	0.655076i	$-0.772637\pi$
$83$	−13.7670			−1.51113			−0.755563	+	0.655076i	$0.772637\pi$
$84$		0			0
$85$	−2.53590	−	3.10583i	−0.275057	−	0.336874i
$86$		0			0
$87$	19.0150			2.03862
$88$		0			0
$89$			10.5186i			1.11497i	0.830187	+	0.557485i	$0.188234\pi$
$89$			10.5186i			1.11497i	−0.830187	+	0.557485i	$0.811766\pi$
$90$		0			0
$91$	−5.79810			−0.607806
$92$		0			0
$93$			8.00481i			0.830060i
$94$		0			0
$95$	9.73770	−	0.420943i	0.999067	−	0.0431879i
$96$		0			0
$97$	10.1602			1.03161			0.515806	−	0.856706i	$-0.327493\pi$
$97$	10.1602			1.03161			0.515806	+	0.856706i	$0.327493\pi$
$98$		0			0
$99$			1.10245i			0.110800i
$100$		0			0
$101$	−2.19615			−0.218525			−0.109263	−	0.994013i	$-0.534849\pi$
$101$	−2.19615			−0.218525			−0.109263	+	0.994013i	$0.534849\pi$
$102$		0			0
$103$			9.39333i			0.925552i	0.886475	+	0.462776i	$0.153147\pi$
$103$			9.39333i			0.925552i	−0.886475	+	0.462776i	$0.846853\pi$
$104$		0			0
$105$	5.44483	+	6.66853i	0.531361	+	0.650782i
$106$		0			0
$107$		−	0.484384i		−	0.0468272i	−0.999726	−	0.0234136i	$-0.992547\pi$
$107$		−	0.484384i		−	0.0468272i	0.999726	−	0.0234136i	$-0.00745346\pi$
$108$		0			0
$109$		−	6.66853i		−	0.638729i	−0.947632	−	0.319365i	$-0.896531\pi$
$109$		−	6.66853i		−	0.638729i	0.947632	−	0.319365i	$-0.103469\pi$
$110$		0			0
$111$		−	18.3671i		−	1.74332i
$112$		0			0
$113$	17.5980			1.65548			0.827739	−	0.561114i	$-0.189627\pi$
$113$	17.5980			1.65548			0.827739	+	0.561114i	$0.189627\pi$
$114$		0			0
$115$	6.38929	−	5.21684i	0.595805	−	0.486473i
$116$		0			0
$117$	0.729469			0.0674394
$118$		0			0
$119$		−	3.81899i		−	0.350086i
$120$		0			0
$121$	−5.92820			−0.538928
$122$		0			0
$123$	6.95996			0.627559
$124$		0			0
$125$	−5.19615	−	9.89949i	−0.464758	−	0.885438i
$126$		0			0
$127$		−	0.838978i		−	0.0744473i	−0.999307	−	0.0372236i	$-0.988149\pi$
$127$		−	0.838978i		−	0.0744473i	0.999307	−	0.0372236i	$-0.0118514\pi$
$128$		0			0
$129$			14.3687i			1.26509i
$130$		0			0
$131$			2.20489i			0.192643i	0.995350	+	0.0963213i	$0.0307076\pi$
$131$			2.20489i			0.192643i	−0.995350	+	0.0963213i	$0.969292\pi$
$132$		0			0
$133$	7.43778	+	5.55532i	0.644937	+	0.481707i
$134$		0			0
$135$	6.98460	+	8.55435i	0.601138	+	0.736241i
$136$		0			0
$137$			14.4195i			1.23194i	0.787768	+	0.615972i	$0.211237\pi$
$137$			14.4195i			1.23194i	−0.787768	+	0.615972i	$0.788763\pi$
$138$		0			0
$139$			16.1622i			1.37086i	0.728140	+	0.685428i	$0.240385\pi$
$139$			16.1622i			1.37086i	−0.728140	+	0.685428i	$0.759615\pi$
$140$		0			0
$141$		−	11.5502i		−	0.972705i
$142$		0			0
$143$			11.2011i			0.936681i
$144$		0			0
$145$	−14.8756	−	18.2188i	−1.23535	−	1.51299i
$146$		0			0
$147$		−	4.45447i		−	0.367399i
$148$		0			0
$149$	5.66025			0.463706			0.231853	−	0.972751i	$-0.425521\pi$
$149$	5.66025			0.463706			0.231853	+	0.972751i	$0.425521\pi$
$150$		0			0
$151$	3.24156			0.263795			0.131897	−	0.991263i	$-0.457893\pi$
$151$	3.24156			0.263795			0.131897	+	0.991263i	$0.457893\pi$
$152$		0			0
$153$			0.480473i			0.0388440i
$154$		0			0
$155$	7.66962	−	6.26222i	0.616039	−	0.502994i
$156$		0			0
$157$		−	6.69213i		−	0.534090i	−0.963684	−	0.267045i	$-0.913953\pi$
$157$		−	6.69213i		−	0.534090i	0.963684	−	0.267045i	$-0.0860473\pi$
$158$		0			0
$159$			8.52418i			0.676011i
$160$		0			0
$161$	7.85641			0.619172
$162$		0			0
$163$	−0.570669			−0.0446982			−0.0223491	−	0.999750i	$-0.507115\pi$
$163$	−0.570669			−0.0446982			−0.0223491	+	0.999750i	$0.507115\pi$
$164$		0			0
$165$	12.8826	−	10.5186i	1.00291	−	0.818872i
$166$		0			0
$167$			17.9477i			1.38883i	0.719573	+	0.694417i	$0.244337\pi$
$167$			17.9477i			1.38883i	−0.719573	+	0.694417i	$0.755663\pi$
$168$		0			0
$169$	−5.58846			−0.429881
$170$		0			0
$171$	−0.935759	−	0.698924i	−0.0715593	−	0.0534480i
$172$		0			0
$173$	−8.16724			−0.620944			−0.310472	−	0.950583i	$-0.600487\pi$
$173$	−8.16724			−0.620944			−0.310472	+	0.950583i	$0.600487\pi$
$174$		0			0
$175$	2.12976	−	10.4337i	0.160995	−	0.788712i
$176$		0			0
$177$		−	8.00481i		−	0.601678i
$178$		0			0
$179$	−19.7673			−1.47748			−0.738739	−	0.673992i	$-0.764578\pi$
$179$	−19.7673			−1.47748			−0.738739	+	0.673992i	$0.764578\pi$
$180$		0			0
$181$			18.2188i			1.35419i	0.735896	+	0.677095i	$0.236761\pi$
$181$			18.2188i			1.35419i	−0.735896	+	0.677095i	$0.763239\pi$
$182$		0			0
$183$			14.8166i			1.09527i
$184$		0			0
$185$	−17.5980	+	14.3687i	−1.29383	+	1.05641i
$186$		0			0
$187$	−7.37772			−0.539512
$188$		0			0
$189$			10.5186i			0.765116i
$190$		0			0
$191$		−	11.2407i		−	0.813350i	−0.913573	−	0.406675i	$-0.866688\pi$
$191$		−	11.2407i		−	0.813350i	0.913573	−	0.406675i	$-0.133312\pi$
$192$		0			0
$193$	−15.6050			−1.12327			−0.561637	−	0.827384i	$-0.689828\pi$
$193$	−15.6050			−1.12327			−0.561637	+	0.827384i	$0.689828\pi$
$194$		0			0
$195$	−6.95996	−	8.52418i	−0.498413	−	0.610429i
$196$		0			0
$197$		−	6.69213i		−	0.476795i	−0.971168	−	0.238397i	$-0.923378\pi$
$197$		−	6.69213i		−	0.476795i	0.971168	−	0.238397i	$-0.0766220\pi$
$198$		0			0
$199$			19.4695i			1.38016i	0.723735	+	0.690078i	$0.242424\pi$
$199$			19.4695i			1.38016i	−0.723735	+	0.690078i	$0.757576\pi$
$200$		0			0
$201$	−5.66025			−0.399244
$202$		0			0
$203$		−	22.4022i		−	1.57232i
$204$		0			0
$205$	−5.44483	−	6.66853i	−0.380283	−	0.465750i
$206$		0			0
$207$	−0.988427			−0.0687004
$208$		0			0
$209$	10.7321	−	14.3687i	0.742351	−	0.993902i
$210$		0			0
$211$	−6.98460			−0.480840			−0.240420	−	0.970669i	$-0.577285\pi$
$211$	−6.98460			−0.480840			−0.240420	+	0.970669i	$0.577285\pi$
$212$		0			0
$213$			29.8744i			2.04696i
$214$		0			0
$215$	13.7670	−	11.2407i	0.938903	−	0.766611i
$216$		0			0
$217$	9.43072			0.640199
$218$		0			0
$219$	−24.1954			−1.63497
$220$		0			0
$221$			4.88170i			0.328379i
$222$		0			0
$223$		−	24.2099i		−	1.62121i	−0.585590	−	0.810607i	$-0.699137\pi$
$223$		−	24.2099i		−	1.62121i	0.585590	−	0.810607i	$-0.300863\pi$
$224$		0			0
$225$	−0.267949	+	1.31268i	−0.0178633	+	0.0875118i
$226$		0			0
$227$		−	13.9776i		−	0.927725i	−0.885907	−	0.463863i	$-0.846463\pi$
$227$		−	13.9776i		−	0.927725i	0.885907	−	0.463863i	$-0.153537\pi$
$228$		0			0
$229$	9.12436			0.602954			0.301477	−	0.953473i	$-0.402520\pi$
$229$	9.12436			0.602954			0.301477	+	0.953473i	$0.402520\pi$
$230$		0			0
$231$	15.8407			1.04224
$232$		0			0
$233$		−	5.93426i		−	0.388766i	−0.980926	−	0.194383i	$-0.937730\pi$
$233$		−	5.93426i		−	0.388766i	0.980926	−	0.194383i	$-0.0622705\pi$
$234$		0			0
$235$	−11.0666	+	9.03583i	−0.721904	+	0.589432i
$236$		0			0
$237$		−	11.3880i		−	0.739733i
$238$		0			0
$239$			16.4576i			1.06455i	0.846571	+	0.532276i	$0.178663\pi$
$239$			16.4576i			1.06455i	−0.846571	+	0.532276i	$0.821337\pi$
$240$		0			0
$241$		−	6.66853i		−	0.429558i	−0.976663	−	0.214779i	$-0.931097\pi$
$241$		−	6.66853i		−	0.429558i	0.976663	−	0.214779i	$-0.0689031\pi$
$242$		0			0
$243$		−	2.77652i		−	0.178114i
$244$		0			0
$245$	−4.26795	+	3.48477i	−0.272669	+	0.222634i
$246$		0			0
$247$	−9.50749	−	7.10120i	−0.604947	−	0.451838i
$248$		0			0
$249$		−	24.8873i		−	1.57717i
$250$		0			0
$251$			25.4934i			1.60913i	0.593866	+	0.804564i	$0.297601\pi$
$251$			25.4934i			1.60913i	−0.593866	+	0.804564i	$0.702399\pi$
$252$		0			0
$253$		−	15.1774i		−	0.954196i
$254$		0			0
$255$	5.61455	−	4.58426i	0.351597	−	0.287078i
$256$		0			0
$257$	−21.0499			−1.31305			−0.656527	−	0.754303i	$-0.727975\pi$
$257$	−21.0499			−1.31305			−0.656527	+	0.754303i	$0.727975\pi$
$258$		0			0
$259$	−21.6388			−1.34457
$260$		0			0
$261$			2.81845i			0.174458i
$262$		0			0
$263$	16.4675			1.01543			0.507713	−	0.861526i	$-0.330491\pi$
$263$	16.4675			1.01543			0.507713	+	0.861526i	$0.330491\pi$
$264$		0			0
$265$	8.16724	−	6.66853i	0.501710	−	0.409644i
$266$		0			0
$267$	−19.0150			−1.16370
$268$		0			0
$269$			3.85008i			0.234743i	0.993088	+	0.117372i	$0.0374469\pi$
$269$			3.85008i			0.234743i	−0.993088	+	0.117372i	$0.962553\pi$
$270$		0			0
$271$			16.1622i			0.981781i	0.871221	+	0.490891i	$0.163329\pi$
$271$			16.1622i			0.981781i	−0.871221	+	0.490891i	$0.836671\pi$
$272$		0			0
$273$		−	10.4815i		−	0.634369i
$274$		0			0
$275$	−20.1563	−	4.11439i	−1.21547	−	0.248107i
$276$		0			0
$277$		−	23.6627i		−	1.42175i	−0.703317	−	0.710877i	$-0.748298\pi$
$277$		−	23.6627i		−	1.42175i	0.703317	−	0.710877i	$-0.251702\pi$
$278$		0			0
$279$	−1.18649			−0.0710336
$280$		0			0
$281$		−	14.3687i		−	0.857164i	−0.903503	−	0.428582i	$-0.859013\pi$
$281$		−	14.3687i		−	0.857164i	0.903503	−	0.428582i	$-0.140987\pi$
$282$		0			0
$283$	−25.4043			−1.51013			−0.755063	−	0.655652i	$-0.772394\pi$
$283$	−25.4043			−1.51013			−0.755063	+	0.655652i	$0.772394\pi$
$284$		0			0
$285$	0.760959	+	17.6033i	0.0450753	+	1.04273i
$286$		0			0
$287$		−	8.19976i		−	0.484016i
$288$		0			0
$289$	13.7846			0.810859
$290$		0			0
$291$			18.3671i			1.07670i
$292$		0			0
$293$	−4.71536			−0.275474			−0.137737	−	0.990469i	$-0.543983\pi$
$293$	−4.71536			−0.275474			−0.137737	+	0.990469i	$0.543983\pi$
$294$		0			0
$295$	−7.66962	+	6.26222i	−0.446543	+	0.364601i
$296$		0			0
$297$	20.3204			1.17911
$298$		0			0
$299$	−10.0426			−0.580779
$300$		0			0
$301$	16.9282			0.975725
$302$		0			0
$303$		−	3.97009i		−	0.228076i
$304$		0			0
$305$	14.1962	−	11.5911i	0.812869	−	0.663705i
$306$		0			0
$307$		−	32.7642i		−	1.86995i	−0.354708	−	0.934977i	$-0.615420\pi$
$307$		−	32.7642i		−	1.86995i	0.354708	−	0.934977i	$-0.384580\pi$
$308$		0			0
$309$	−16.9808			−0.966002
$310$		0			0
$311$		−	31.0056i		−	1.75817i	−0.476667	−	0.879084i	$-0.658155\pi$
$311$		−	31.0056i		−	1.75817i	0.476667	−	0.879084i	$-0.341845\pi$
$312$		0			0
$313$		−	16.4901i		−	0.932075i	−0.884765	−	0.466037i	$-0.845681\pi$
$313$		−	16.4901i		−	0.932075i	0.884765	−	0.466037i	$-0.154319\pi$
$314$		0			0
$315$	−0.988427	+	0.807048i	−0.0556916	+	0.0454720i
$316$		0			0
$317$	−8.16724			−0.458718			−0.229359	−	0.973342i	$-0.573663\pi$
$317$	−8.16724			−0.458718			−0.229359	+	0.973342i	$0.573663\pi$
$318$		0			0
$319$	−43.2776			−2.42308
$320$		0			0
$321$	0.875644			0.0488737
$322$		0			0
$323$	4.67729	−	6.26222i	0.260251	−	0.348439i
$324$		0			0
$325$	−2.72241	+	13.3371i	−0.151012	+	0.739807i
$326$		0			0
$327$	12.0550			0.666644
$328$		0			0
$329$	−13.6077			−0.750217
$330$		0			0
$331$	−24.1954			−1.32990			−0.664949	−	0.746889i	$-0.731547\pi$
$331$	−24.1954			−1.32990			−0.664949	+	0.746889i	$0.731547\pi$
$332$		0			0
$333$	2.72241			0.149187
$334$		0			0
$335$	4.42806	+	5.42324i	0.241931	+	0.296303i
$336$		0			0
$337$	12.1531			0.662024			0.331012	−	0.943627i	$-0.392610\pi$
$337$	12.1531			0.662024			0.331012	+	0.943627i	$0.392610\pi$
$338$		0			0
$339$			31.8127i			1.72783i
$340$		0			0
$341$		−	18.2188i		−	0.986601i
$342$		0			0
$343$	−20.1563			−1.08834
$344$		0			0
$345$	9.43072	+	11.5502i	0.507733	+	0.621843i
$346$		0			0
$347$	−3.68886			−0.198028			−0.0990142	−	0.995086i	$-0.531569\pi$
$347$	−3.68886			−0.198028			−0.0990142	+	0.995086i	$0.531569\pi$
$348$		0			0
$349$	−20.0000			−1.07058			−0.535288	−	0.844670i	$-0.679797\pi$
$349$	−20.0000			−1.07058			−0.535288	+	0.844670i	$0.679797\pi$
$350$		0			0
$351$		−	13.4456i		−	0.717674i
$352$		0			0
$353$		−	12.1459i		−	0.646462i	−0.946320	−	0.323231i	$-0.895231\pi$
$353$		−	12.1459i		−	0.646462i	0.946320	−	0.323231i	$-0.104769\pi$
$354$		0			0
$355$	28.6234	−	23.3709i	1.51917	−	1.24040i
$356$		0			0
$357$	6.90377			0.365386
$358$		0			0
$359$			6.31928i			0.333519i	0.985998	+	0.166760i	$0.0533304\pi$
$359$			6.31928i			0.333519i	−0.985998	+	0.166760i	$0.946670\pi$
$360$		0			0
$361$	5.39230	+	18.2188i	0.283806	+	0.958882i
$362$		0			0
$363$		−	10.7167i		−	0.562480i
$364$		0			0
$365$	18.9282	+	23.1822i	0.990747	+	1.21341i
$366$		0			0
$367$	−6.80705			−0.355325			−0.177663	−	0.984091i	$-0.556854\pi$
$367$	−6.80705			−0.355325			−0.177663	+	0.984091i	$0.556854\pi$
$368$		0			0
$369$			1.03162i			0.0537042i
$370$		0			0
$371$	10.0426			0.521386
$372$		0			0
$373$	10.1602			0.526075			0.263037	−	0.964786i	$-0.415276\pi$
$373$	10.1602			0.526075			0.263037	+	0.964786i	$0.415276\pi$
$374$		0			0
$375$	17.8958	−	9.39333i	0.924134	−	0.485069i
$376$		0			0
$377$			28.6360i			1.47483i
$378$		0			0
$379$	−20.9538			−1.07632			−0.538162	−	0.842841i	$-0.680881\pi$
$379$	−20.9538			−1.07632			−0.538162	+	0.842841i	$0.680881\pi$
$380$		0			0
$381$	1.51666			0.0777009
$382$		0			0
$383$			0.484384i			0.0247509i	0.999923	+	0.0123754i	$0.00393933\pi$
$383$			0.484384i			0.0247509i	−0.999923	+	0.0123754i	$0.996061\pi$
$384$		0			0
$385$	−12.3923	−	15.1774i	−0.631570	−	0.773513i
$386$		0			0
$387$	−2.12976			−0.108262
$388$		0			0
$389$	14.5359			0.736999			0.368500	−	0.929628i	$-0.379872\pi$
$389$	14.5359			0.736999			0.368500	+	0.929628i	$0.379872\pi$
$390$		0			0
$391$		−	6.61468i		−	0.334519i
$392$		0			0
$393$	−3.98589			−0.201062
$394$		0			0
$395$	−10.9112	+	8.90894i	−0.549001	+	0.448258i
$396$		0			0
$397$			18.2832i			0.917610i	0.888537	+	0.458805i	$0.151722\pi$
$397$			18.2832i			0.917610i	−0.888537	+	0.458805i	$0.848278\pi$
$398$		0			0
$399$	−10.0426	+	13.4456i	−0.502759	+	0.673123i
$400$		0			0
$401$			21.0372i			1.05055i	0.850933	+	0.525274i	$0.176037\pi$
$401$			21.0372i			1.05055i	−0.850933	+	0.525274i	$0.823963\pi$
$402$		0			0
$403$	−12.0550			−0.600503
$404$		0			0
$405$	−16.8564	+	13.7632i	−0.837602	+	0.683899i
$406$		0			0
$407$			41.8030i			2.07210i
$408$		0			0
$409$		−	11.5502i		−	0.571122i	−0.958361	−	0.285561i	$-0.907820\pi$
$409$		−	11.5502i		−	0.571122i	0.958361	−	0.285561i	$-0.0921799\pi$
$410$		0			0
$411$	−26.0669			−1.28578
$412$		0			0
$413$	−9.43072			−0.464055
$414$		0			0
$415$	−23.8452	+	19.4695i	−1.17051	+	0.955720i
$416$		0			0
$417$	−29.2171			−1.43077
$418$		0			0
$419$			18.6625i			0.911721i	0.890051	+	0.455860i	$0.150668\pi$
$419$			18.6625i			0.911721i	−0.890051	+	0.455860i	$0.849332\pi$
$420$		0			0
$421$		−	11.5502i		−	0.562924i	−0.959572	−	0.281462i	$-0.909181\pi$
$421$		−	11.5502i		−	0.562924i	0.959572	−	0.281462i	$-0.0908193\pi$
$422$		0			0
$423$	1.71201			0.0832406
$424$		0			0
$425$	−8.78461	−	1.79315i	−0.426116	−	0.0869806i
$426$		0			0
$427$	17.4559			0.844749
$428$		0			0
$429$	−20.2487			−0.977617
$430$		0			0
$431$	29.8099			1.43589			0.717946	−	0.696098i	$-0.245082\pi$
$431$	29.8099			1.43589			0.717946	+	0.696098i	$0.245082\pi$
$432$		0			0
$433$	−12.1531			−0.584042			−0.292021	−	0.956412i	$-0.594328\pi$
$433$	−12.1531			−0.584042			−0.292021	+	0.956412i	$0.594328\pi$
$434$		0			0
$435$	32.9349	−	26.8912i	1.57911	−	1.28934i
$436$		0			0
$437$	12.8826	+	9.62209i	0.616259	+	0.460287i
$438$		0			0
$439$	−9.03967			−0.431440			−0.215720	−	0.976455i	$-0.569210\pi$
$439$	−9.03967			−0.431440			−0.215720	+	0.976455i	$0.569210\pi$
$440$		0			0
$441$	0.660254			0.0314407
$442$		0			0
$443$	3.68886			0.175263			0.0876315	−	0.996153i	$-0.472070\pi$
$443$	3.68886			0.175263			0.0876315	+	0.996153i	$0.472070\pi$
$444$		0			0
$445$	14.8756	+	18.2188i	0.705169	+	0.863652i
$446$		0			0
$447$			10.2323i			0.483972i
$448$		0			0
$449$			14.3687i			0.678100i	0.940768	+	0.339050i	$0.110106\pi$
$449$			14.3687i			0.678100i	−0.940768	+	0.339050i	$0.889894\pi$
$450$		0			0
$451$	−15.8407			−0.745910
$452$		0			0
$453$			5.85993i			0.275323i
$454$		0			0
$455$	−10.0426	+	8.19976i	−0.470805	+	0.384411i
$456$		0			0
$457$			25.4558i			1.19077i	0.803439	+	0.595387i	$0.203001\pi$
$457$			25.4558i			1.19077i	−0.803439	+	0.595387i	$0.796999\pi$
$458$		0			0
$459$	8.85612			0.413368
$460$		0			0
$461$	−24.2487			−1.12938			−0.564688	−	0.825305i	$-0.691003\pi$
$461$	−24.2487			−1.12938			−0.564688	+	0.825305i	$0.691003\pi$
$462$		0			0
$463$	28.1047			1.30614			0.653068	−	0.757299i	$-0.273482\pi$
$463$	28.1047			1.30614			0.653068	+	0.757299i	$0.273482\pi$
$464$		0			0
$465$	11.3205	+	13.8647i	0.524976	+	0.642962i
$466$		0			0
$467$	−11.0666			−0.512100			−0.256050	−	0.966663i	$-0.582421\pi$
$467$	−11.0666			−0.512100			−0.256050	+	0.966663i	$0.582421\pi$
$468$		0			0
$469$			6.66853i			0.307924i
$470$		0			0
$471$	12.0977			0.557432
$472$		0			0
$473$		−	32.7028i		−	1.50368i
$474$		0			0
$475$	16.2709	−	14.5003i	0.746560	−	0.665319i
$476$		0			0
$477$	−1.26348			−0.0578506
$478$		0			0
$479$			4.11439i			0.187991i	0.995573	+	0.0939956i	$0.0299640\pi$
$479$			4.11439i			0.187991i	−0.995573	+	0.0939956i	$0.970036\pi$
$480$		0			0
$481$	27.6603			1.26120
$482$		0			0
$483$			14.2024i			0.646231i
$484$		0			0
$485$	17.5980	−	14.3687i	0.799082	−	0.652448i
$486$		0			0
$487$		−	24.8241i		−	1.12489i	−0.826836	−	0.562443i	$-0.809862\pi$
$487$		−	24.8241i		−	1.12489i	0.826836	−	0.562443i	$-0.190138\pi$
$488$		0			0
$489$		−	1.03162i		−	0.0466517i
$490$		0			0
$491$		−	17.8554i		−	0.805803i	−0.915243	−	0.402902i	$-0.868002\pi$
$491$		−	17.8554i		−	0.805803i	0.915243	−	0.402902i	$-0.131998\pi$
$492$		0			0
$493$	−18.8614			−0.849477
$494$		0			0
$495$	1.55910	+	1.90949i	0.0700762	+	0.0858254i
$496$		0			0
$497$	35.1959			1.57875
$498$		0			0
$499$			16.1622i			0.723518i	0.932272	+	0.361759i	$0.117824\pi$
$499$			16.1622i			0.723518i	−0.932272	+	0.361759i	$0.882176\pi$
$500$		0			0
$501$	−32.4449			−1.44953
$502$		0			0
$503$	16.4675			0.734247			0.367124	−	0.930172i	$-0.380343\pi$
$503$	16.4675			0.734247			0.367124	+	0.930172i	$0.380343\pi$
$504$		0			0
$505$	−3.80385	+	3.10583i	−0.169269	+	0.138208i
$506$		0			0
$507$		−	10.1025i		−	0.448669i
$508$		0			0
$509$		−	15.4003i		−	0.682606i	−0.939953	−	0.341303i	$-0.889132\pi$
$509$		−	15.4003i		−	0.682606i	0.939953	−	0.341303i	$-0.110868\pi$
$510$		0			0
$511$			28.5053i			1.26100i
$512$		0			0
$513$	−12.8826	+	17.2480i	−0.568781	+	0.761516i
$514$		0			0
$515$	13.2842	+	16.2697i	0.585371	+	0.716930i
$516$		0			0
$517$			26.2880i			1.15615i
$518$		0			0
$519$		−	14.7643i		−	0.648081i
$520$		0			0
$521$			10.5186i			0.460828i	0.973093	+	0.230414i	$0.0740081\pi$
$521$			10.5186i			0.460828i	−0.973093	+	0.230414i	$0.925992\pi$
$522$		0			0
$523$			0.838978i			0.0366860i	0.999832	+	0.0183430i	$0.00583908\pi$
$523$			0.838978i			0.0366860i	−0.999832	+	0.0183430i	$0.994161\pi$
$524$		0			0
$525$	18.8614	+	3.85008i	0.823181	+	0.168031i
$526$		0			0
$527$		−	7.94018i		−	0.345879i
$528$		0			0
$529$	−9.39230			−0.408361
$530$		0			0
$531$	1.18649			0.0514895
$532$		0			0
$533$			10.4815i			0.454004i
$534$		0			0
$535$	−0.685023	−	0.838978i	−0.0296161	−	0.0362722i
$536$		0			0
$537$		−	35.7343i		−	1.54205i
$538$		0			0
$539$			10.1383i			0.436686i
$540$		0			0
$541$	1.80385			0.0775535			0.0387767	−	0.999248i	$-0.487654\pi$
$541$	1.80385			0.0775535			0.0387767	+	0.999248i	$0.487654\pi$
$542$		0			0
$543$	−32.9349			−1.41337
$544$		0			0
$545$	−9.43072	−	11.5502i	−0.403968	−	0.494757i
$546$		0			0
$547$			35.0564i			1.49890i	0.662059	+	0.749451i	$0.269683\pi$
$547$			35.0564i			1.49890i	−0.662059	+	0.749451i	$0.730317\pi$
$548$		0			0
$549$	−2.19615			−0.0937295
$550$		0			0
$551$	27.4369	−	36.7341i	1.16885	−	1.56493i
$552$		0			0
$553$	−13.4166			−0.570532
$554$		0			0
$555$	−25.9749	−	31.8127i	−1.10257	−	1.35037i
$556$		0			0
$557$		−	11.0363i		−	0.467623i	−0.972282	−	0.233812i	$-0.924880\pi$
$557$		−	11.0363i		−	0.467623i	0.972282	−	0.233812i	$-0.0751199\pi$
$558$		0			0
$559$	−21.6388			−0.915224
$560$		0			0
$561$		−	13.3371i		−	0.563091i
$562$		0			0
$563$			17.9477i			0.756405i	0.925723	+	0.378202i	$0.123458\pi$
$563$			17.9477i			0.756405i	−0.925723	+	0.378202i	$0.876542\pi$
$564$		0			0
$565$	30.4806	−	24.8873i	1.28233	−	1.04702i
$566$		0			0
$567$	−20.7270			−0.870451
$568$		0			0
$569$			1.03162i			0.0432480i	0.999766	+	0.0216240i	$0.00688366\pi$
$569$			1.03162i			0.0432480i	−0.999766	+	0.0216240i	$0.993116\pi$
$570$		0			0
$571$			30.4148i			1.27282i	0.771351	+	0.636410i	$0.219581\pi$
$571$			30.4148i			1.27282i	−0.771351	+	0.636410i	$0.780419\pi$
$572$		0			0
$573$	20.3204			0.848896
$574$		0			0
$575$	3.68886	−	18.0717i	0.153836	−	0.753640i
$576$		0			0
$577$			8.00481i			0.333245i	0.986021	+	0.166622i	$0.0532860\pi$
$577$			8.00481i			0.333245i	−0.986021	+	0.166622i	$0.946714\pi$
$578$		0			0
$579$		−	28.2099i		−	1.17236i
$580$		0			0
$581$	−29.3205			−1.21642
$582$		0			0
$583$		−	19.4008i		−	0.803500i
$584$		0			0
$585$	1.26348	−	1.03162i	0.0522384	−	0.0426524i
$586$		0			0
$587$	8.36615			0.345308			0.172654	−	0.984983i	$-0.444766\pi$
$587$	8.36615			0.345308			0.172654	+	0.984983i	$0.444766\pi$
$588$		0			0
$589$	15.4641	+	11.5502i	0.637187	+	0.475919i
$590$		0			0
$591$	12.0977			0.497632
$592$		0			0
$593$		−	11.8685i		−	0.487381i	−0.969853	−	0.243691i	$-0.921642\pi$
$593$		−	11.8685i		−	0.487381i	0.969853	−	0.243691i	$-0.0783582\pi$
$594$		0			0
$595$	−5.40087	−	6.61468i	−0.221414	−	0.271176i
$596$		0			0
$597$	−35.1959			−1.44047
$598$		0			0
$599$	−16.5257			−0.675223			−0.337612	−	0.941285i	$-0.609619\pi$
$599$	−16.5257			−0.675223			−0.337612	+	0.941285i	$0.609619\pi$
$600$		0			0
$601$			6.66853i			0.272015i	0.990708	+	0.136007i	$0.0434271\pi$
$601$			6.66853i			0.272015i	−0.990708	+	0.136007i	$0.956573\pi$
$602$		0			0
$603$		−	0.838978i		−	0.0341658i
$604$		0			0
$605$	−10.2679	+	8.38375i	−0.417451	+	0.340848i
$606$		0			0
$607$		−	31.0863i		−	1.26175i	−0.775883	−	0.630877i	$-0.782695\pi$
$607$		−	31.0863i		−	1.26175i	0.775883	−	0.630877i	$-0.217305\pi$
$608$		0			0
$609$	40.4974			1.64104
$610$		0			0
$611$	17.3943			0.703698
$612$		0			0
$613$		−	29.0421i		−	1.17300i	−0.809949	−	0.586501i	$-0.800505\pi$
$613$		−	29.0421i		−	1.17300i	0.809949	−	0.586501i	$-0.199495\pi$
$614$		0			0
$615$	12.0550	−	9.84287i	0.486105	−	0.396903i
$616$		0			0
$617$			37.8792i			1.52496i	0.647013	+	0.762479i	$0.276018\pi$
$617$			37.8792i			1.52496i	−0.647013	+	0.762479i	$0.723982\pi$
$618$		0			0
$619$			22.7768i			0.915479i	0.889086	+	0.457739i	$0.151341\pi$
$619$			22.7768i			0.915479i	−0.889086	+	0.457739i	$0.848659\pi$
$620$		0			0
$621$			18.2188i			0.731093i
$622$		0			0
$623$			22.4022i			0.897523i
$624$		0			0
$625$	−23.0000	−	9.79796i	−0.920000	−	0.391918i
$626$		0			0
$627$	25.9749	+	19.4008i	1.03734	+	0.774795i
$628$		0			0
$629$			18.2188i			0.726429i
$630$		0			0
$631$			8.52418i			0.339342i	0.985501	+	0.169671i	$0.0542705\pi$
$631$			8.52418i			0.339342i	−0.985501	+	0.169671i	$0.945729\pi$
$632$		0			0
$633$		−	12.6264i		−	0.501854i
$634$		0			0
$635$	−1.18649	−	1.45315i	−0.0470846	−	0.0576666i
$636$		0			0
$637$	6.70831			0.265793
$638$		0			0
$639$	−4.42806			−0.175171
$640$		0			0
$641$		−	27.7057i		−	1.09431i	−0.837031	−	0.547155i	$-0.815711\pi$
$641$		−	27.7057i		−	1.09431i	0.837031	−	0.547155i	$-0.184289\pi$
$642$		0			0
$643$	−6.80705			−0.268444			−0.134222	−	0.990951i	$-0.542854\pi$
$643$	−6.80705			−0.268444			−0.134222	+	0.990951i	$0.542854\pi$
$644$		0			0
$645$	20.3204	+	24.8873i	0.800114	+	0.979936i
$646$		0			0
$647$	46.7019			1.83604			0.918021	−	0.396532i	$-0.129786\pi$
$647$	46.7019			1.83604			0.918021	+	0.396532i	$0.129786\pi$
$648$		0			0
$649$			18.2188i			0.715149i
$650$		0			0
$651$			17.0484i			0.668178i
$652$		0			0
$653$		−	45.0518i		−	1.76301i	−0.472173	−	0.881506i	$-0.656530\pi$
$653$		−	45.0518i		−	1.76301i	0.472173	−	0.881506i	$-0.343470\pi$
$654$		0			0
$655$	3.11819	+	3.81899i	0.121838	+	0.149220i
$656$		0			0
$657$		−	3.58630i		−	0.139915i
$658$		0			0
$659$	23.0089			0.896298			0.448149	−	0.893959i	$-0.352083\pi$
$659$	23.0089			0.896298			0.448149	+	0.893959i	$0.352083\pi$
$660$		0			0
$661$			24.8873i			0.968003i	0.875067	+	0.484002i	$0.160817\pi$
$661$			24.8873i			0.968003i	−0.875067	+	0.484002i	$0.839183\pi$
$662$		0			0
$663$	−8.82488			−0.342730
$664$		0			0
$665$	20.7390	−	0.896510i	0.804224	−	0.0347652i
$666$		0			0
$667$		−	38.8017i		−	1.50241i
$668$		0			0
$669$	43.7654			1.69207
$670$		0			0
$671$		−	33.7222i		−	1.30183i
$672$		0			0
$673$	12.1531			0.468469			0.234234	−	0.972180i	$-0.424742\pi$
$673$	12.1531			0.468469			0.234234	+	0.972180i	$0.424742\pi$
$674$		0			0
$675$	24.1954	+	4.93886i	0.931280	+	0.190097i
$676$		0			0
$677$	−30.4806			−1.17146			−0.585732	−	0.810505i	$-0.699193\pi$
$677$	−30.4806			−1.17146			−0.585732	+	0.810505i	$0.699193\pi$
$678$		0			0
$679$	21.6388			0.830422
$680$		0			0
$681$	25.2679			0.968270
$682$		0			0
$683$			25.1787i			0.963435i	0.876327	+	0.481717i	$0.159987\pi$
$683$			25.1787i			0.963435i	−0.876327	+	0.481717i	$0.840013\pi$
$684$		0			0
$685$	20.3923	+	24.9754i	0.779150	+	0.954260i
$686$		0			0
$687$			16.4945i			0.629305i
$688$		0			0
$689$	−12.8372			−0.489057
$690$		0			0
$691$		−	19.9811i		−	0.760119i	−0.924962	−	0.380059i	$-0.875904\pi$
$691$		−	19.9811i		−	0.760119i	0.924962	−	0.380059i	$-0.124096\pi$
$692$		0			0
$693$			2.34795i			0.0891914i
$694$		0			0
$695$	22.8567	+	27.9937i	0.867006	+	1.06186i
$696$		0			0
$697$	−6.90377			−0.261499
$698$		0			0
$699$	10.7276			0.405756
$700$		0			0
$701$	7.51666			0.283900			0.141950	−	0.989874i	$-0.454663\pi$
$701$	7.51666			0.283900			0.141950	+	0.989874i	$0.454663\pi$
$702$		0			0
$703$	−35.4824	−	26.5020i	−1.33824	−	0.999543i
$704$		0			0
$705$	−16.3345	−	20.0056i	−0.615192	−	0.753454i
$706$		0			0
$707$	−4.67729			−0.175908
$708$		0			0
$709$	−7.60770			−0.285713			−0.142856	−	0.989743i	$-0.545629\pi$
$709$	−7.60770			−0.285713			−0.142856	+	0.989743i	$0.545629\pi$
$710$		0			0
$711$	1.68797			0.0633037
$712$		0			0
$713$	16.3345			0.611731
$714$		0			0
$715$	15.8407	+	19.4008i	0.592409	+	0.725550i
$716$		0			0
$717$	−29.7511			−1.11108
$718$		0			0
$719$		−	3.52359i		−	0.131408i	−0.997839	−	0.0657039i	$-0.979071\pi$
$719$		−	3.52359i		−	0.131408i	0.997839	−	0.0657039i	$-0.0209293\pi$
$720$		0			0
$721$			20.0056i			0.745047i
$722$		0			0
$723$	12.0550			0.448331
$724$		0			0
$725$	−51.5304	−	10.5186i	−1.91379	−	0.390651i
$726$		0			0
$727$	−48.2610			−1.78990			−0.894951	−	0.446165i	$-0.852790\pi$
$727$	−48.2610			−1.78990			−0.894951	+	0.446165i	$0.852790\pi$
$728$		0			0
$729$	−24.1769			−0.895441
$730$		0			0
$731$		−	14.2527i		−	0.527154i
$732$		0			0
$733$		−	20.0764i		−	0.741538i	−0.928725	−	0.370769i	$-0.879094\pi$
$733$		−	20.0764i		−	0.741538i	0.928725	−	0.370769i	$-0.120906\pi$
$734$		0			0
$735$	−6.29958	−	7.71537i	−0.232363	−	0.284586i
$736$		0			0
$737$	12.8826			0.474537
$738$		0			0
$739$		−	33.7222i		−	1.24049i	−0.784408	−	0.620245i	$-0.787033\pi$
$739$		−	33.7222i		−	1.24049i	0.784408	−	0.620245i	$-0.212967\pi$
$740$		0			0
$741$	12.8372	−	17.1871i	0.471585	−	0.631385i
$742$		0			0
$743$			44.9341i			1.64847i	0.566246	+	0.824236i	$0.308395\pi$
$743$			44.9341i			1.64847i	−0.566246	+	0.824236i	$0.691605\pi$
$744$		0			0
$745$	9.80385	−	8.00481i	0.359185	−	0.293273i
$746$		0			0
$747$	3.68886			0.134968
$748$		0			0
$749$		−	1.03162i		−	0.0376947i
$750$		0			0
$751$	−27.4369			−1.00119			−0.500594	−	0.865682i	$-0.666885\pi$
$751$	−27.4369			−1.00119			−0.500594	+	0.865682i	$0.666885\pi$
$752$		0			0
$753$	−46.0856			−1.67945
$754$		0			0
$755$	5.61455	−	4.58426i	0.204334	−	0.166838i
$756$		0			0
$757$		−	24.0144i		−	0.872819i	−0.899748	−	0.436410i	$-0.856250\pi$
$757$		−	24.0144i		−	0.872819i	0.899748	−	0.436410i	$-0.143750\pi$
$758$		0			0
$759$	27.4369			0.995897
$760$		0			0
$761$	32.4449			1.17613			0.588063	−	0.808815i	$-0.299891\pi$
$761$	32.4449			1.17613			0.588063	+	0.808815i	$0.299891\pi$
$762$		0			0
$763$		−	14.2024i		−	0.514161i
$764$		0			0
$765$	0.679492	+	0.832204i	0.0245671	+	0.0300884i
$766$		0			0
$767$	12.0550			0.435281
$768$		0			0
$769$	13.8038			0.497779			0.248890	−	0.968532i	$-0.419934\pi$
$769$	13.8038			0.497779			0.248890	+	0.968532i	$0.419934\pi$
$770$		0			0
$771$		−	38.0528i		−	1.37044i
$772$		0			0
$773$	17.5980			0.632955			0.316477	−	0.948600i	$-0.397500\pi$
$773$	17.5980			0.632955			0.316477	+	0.948600i	$0.397500\pi$
$774$		0			0
$775$	4.42806	−	21.6930i	0.159061	−	0.779234i
$776$		0			0
$777$		−	39.1175i		−	1.40333i
$778$		0			0
$779$	10.0426	−	13.4456i	0.359814	−	0.481739i
$780$		0			0
$781$		−	67.9933i		−	2.43299i
$782$		0			0
$783$	51.9499			1.85654
$784$		0			0
$785$	−9.46410	−	11.5911i	−0.337788	−	0.413704i
$786$		0			0
$787$			44.2249i			1.57645i	0.615389	+	0.788224i	$0.288999\pi$
$787$			44.2249i			1.57645i	−0.615389	+	0.788224i	$0.711001\pi$
$788$		0			0
$789$			29.7690i			1.05980i
$790$		0			0
$791$	37.4795			1.33262
$792$		0			0
$793$	−22.3133			−0.792369
$794$		0			0
$795$	12.0550	+	14.7643i	0.427547	+	0.523636i
$796$		0			0
$797$	49.3420			1.74778			0.873892	−	0.486120i	$-0.161588\pi$
$797$	49.3420			1.74778			0.873892	+	0.486120i	$0.161588\pi$
$798$		0			0
$799$			11.4570i			0.405318i
$800$		0			0
$801$		−	2.81845i		−	0.0995851i
$802$		0			0
$803$	55.0681			1.94331
$804$		0			0
$805$	13.6077	−	11.1106i	0.479608	−	0.391598i
$806$		0			0
$807$	−6.95996			−0.245002
$808$		0			0
$809$	−9.46410			−0.332740			−0.166370	−	0.986063i	$-0.553205\pi$
$809$	−9.46410			−0.332740			−0.166370	+	0.986063i	$0.553205\pi$
$810$		0			0
$811$	−4.92953			−0.173099			−0.0865496	−	0.996248i	$-0.527584\pi$
$811$	−4.92953			−0.173099			−0.0865496	+	0.996248i	$0.527584\pi$
$812$		0			0
$813$	−29.2171			−1.02469
$814$		0			0
$815$	−0.988427	+	0.807048i	−0.0346231	+	0.0282696i
$816$		0			0
$817$	27.7582	+	20.7327i	0.971135	+	0.725346i
$818$		0			0
$819$	1.55360			0.0542871
$820$		0			0
$821$	35.3205			1.23269			0.616347	−	0.787475i	$-0.288612\pi$
$821$	35.3205			1.23269			0.616347	+	0.787475i	$0.288612\pi$
$822$		0			0
$823$	0.570669			0.0198923			0.00994613	−	0.999951i	$-0.496834\pi$
$823$	0.570669			0.0198923			0.00994613	+	0.999951i	$0.496834\pi$
$824$		0			0
$825$	7.43778	−	36.4375i	0.258950	−	1.26859i
$826$		0			0
$827$			11.6855i			0.406343i	0.979143	+	0.203172i	$0.0651249\pi$
$827$			11.6855i			0.406343i	−0.979143	+	0.203172i	$0.934875\pi$
$828$		0			0
$829$		−	43.1060i		−	1.49713i	−0.663059	−	0.748567i	$-0.730742\pi$
$829$		−	43.1060i		−	1.49713i	0.663059	−	0.748567i	$-0.269258\pi$
$830$		0			0
$831$	42.7762			1.48389
$832$		0			0
$833$			4.41851i			0.153092i
$834$		0			0
$835$	25.3819	+	31.0863i	0.878375	+	1.07579i
$836$		0			0
$837$			21.8695i			0.755922i
$838$		0			0
$839$	−4.42806			−0.152874			−0.0764368	−	0.997074i	$-0.524354\pi$
$839$	−4.42806			−0.152874			−0.0764368	+	0.997074i	$0.524354\pi$
$840$		0			0
$841$	−81.6410			−2.81521
$842$		0			0
$843$	25.9749			0.894624
$844$		0			0
$845$	−9.67949	+	7.90327i	−0.332985	+	0.271881i
$846$		0			0
$847$	−12.6257			−0.433823
$848$		0			0
$849$		−	45.9245i		−	1.57612i
$850$		0			0
$851$	−37.4795			−1.28478
$852$		0			0
$853$			9.79796i			0.335476i	0.985832	+	0.167738i	$0.0536462\pi$
$853$			9.79796i			0.335476i	−0.985832	+	0.167738i	$0.946354\pi$
$854$		0			0
$855$	−2.60921	+	0.112791i	−0.0892331	+	0.00385738i
$856$		0			0
$857$	4.71536			0.161074			0.0805368	−	0.996752i	$-0.474337\pi$
$857$	4.71536			0.161074			0.0805368	+	0.996752i	$0.474337\pi$
$858$		0			0
$859$		−	28.5053i		−	0.972589i	−0.873795	−	0.486295i	$-0.838348\pi$
$859$		−	28.5053i		−	0.972589i	0.873795	−	0.486295i	$-0.161652\pi$
$860$		0			0
$861$	14.8231			0.505169
$862$		0			0
$863$		−	15.3010i		−	0.520851i	−0.965494	−	0.260425i	$-0.916137\pi$
$863$		−	15.3010i		−	0.520851i	0.965494	−	0.260425i	$-0.0838628\pi$
$864$		0			0
$865$	−14.1461	+	11.5502i	−0.480981	+	0.392720i
$866$		0			0
$867$			24.9191i			0.846297i
$868$		0			0
$869$			25.9189i			0.879239i
$870$		0			0
$871$		−	8.52418i		−	0.288831i
$872$		0			0
$873$	−2.72241			−0.0921398
$874$		0			0
$875$	−11.0666	−	21.0836i	−0.374119	−	0.712756i
$876$		0			0
$877$	−19.5909			−0.661538			−0.330769	−	0.943712i	$-0.607308\pi$
$877$	−19.5909			−0.661538			−0.330769	+	0.943712i	$0.607308\pi$
$878$		0			0
$879$		−	8.52418i		−	0.287513i
$880$		0			0
$881$	−52.3013			−1.76207			−0.881037	−	0.473047i	$-0.843154\pi$
$881$	−52.3013			−1.76207			−0.881037	+	0.473047i	$0.843154\pi$
$882$		0			0
$883$	22.2861			0.749986			0.374993	−	0.927028i	$-0.377645\pi$
$883$	22.2861			0.749986			0.374993	+	0.927028i	$0.377645\pi$
$884$		0			0
$885$	−11.3205	−	13.8647i	−0.380535	−	0.466058i
$886$		0			0
$887$			51.8105i			1.73963i	0.493382	+	0.869813i	$0.335761\pi$
$887$			51.8105i			1.73963i	−0.493382	+	0.869813i	$0.664239\pi$
$888$		0			0
$889$		−	1.78683i		−	0.0599282i
$890$		0			0
$891$			40.0415i			1.34144i
$892$		0			0
$893$	−22.3133	−	16.6660i	−0.746687	−	0.557705i
$894$		0			0
$895$	−34.2380	+	27.9552i	−1.14445	+	0.934439i
$896$		0			0
$897$		−	18.1545i		−	0.606161i
$898$		0			0
$899$		−	46.5770i		−	1.55343i
$900$		0			0
$901$		−	8.45535i		−	0.281689i
$902$		0			0
$903$			30.6019i			1.01837i
$904$		0			0
$905$	25.7652	+	31.5558i	0.856465	+	1.04895i
$906$		0			0
$907$			9.39333i			0.311900i	0.987765	+	0.155950i	$0.0498439\pi$
$907$			9.39333i			0.311900i	−0.987765	+	0.155950i	$0.950156\pi$
$908$		0			0
$909$	0.588457			0.0195179
$910$		0			0
$911$	29.8099			0.987646			0.493823	−	0.869562i	$-0.335599\pi$
$911$	29.8099			0.987646			0.493823	+	0.869562i	$0.335599\pi$
$912$		0			0
$913$			56.6429i			1.87461i
$914$		0			0
$915$	20.9538	+	25.6631i	0.692711	+	0.848394i
$916$		0			0
$917$			4.69591i			0.155073i
$918$		0			0
$919$			18.0717i			0.596129i	0.954546	+	0.298064i	$0.0963410\pi$
$919$			18.0717i			0.596129i	−0.954546	+	0.298064i	$0.903659\pi$
$920$		0			0
$921$	59.2295			1.95168
$922$		0			0
$923$	−44.9899			−1.48086
$924$		0			0
$925$	−10.1602	+	49.7746i	−0.334065	+	1.63658i
$926$		0			0
$927$		−	2.51693i		−	0.0826670i
$928$		0			0
$929$	−36.2487			−1.18928			−0.594641	−	0.803991i	$-0.702706\pi$
$929$	−36.2487			−1.18928			−0.594641	+	0.803991i	$0.702706\pi$
$930$		0			0
$931$	−8.60538	−	6.42741i	−0.282030	−	0.210650i
$932$		0			0
$933$	56.0503			1.83501
$934$		0			0
$935$	−12.7786	+	10.4337i	−0.417905	+	0.341218i
$936$		0			0
$937$			46.0127i			1.50317i	0.659636	+	0.751585i	$0.270710\pi$
$937$			46.0127i			1.50317i	−0.659636	+	0.751585i	$0.729290\pi$
$938$		0			0
$939$	29.8099			0.972810
$940$		0			0
$941$			5.63690i			0.183758i	0.995770	+	0.0918789i	$0.0292873\pi$
$941$			5.63690i			0.183758i	−0.995770	+	0.0918789i	$0.970713\pi$
$942$		0			0
$943$		−	14.2024i		−	0.462494i
$944$		0			0
$945$	14.8756	+	18.2188i	0.483902	+	0.592656i
$946$		0			0
$947$	33.1997			1.07885			0.539423	−	0.842035i	$-0.318642\pi$
$947$	33.1997			1.07885			0.539423	+	0.842035i	$0.318642\pi$
$948$		0			0
$949$		−	36.4375i		−	1.18281i
$950$		0			0
$951$		−	14.7643i		−	0.478766i
$952$		0			0
$953$	−4.71536			−0.152746			−0.0763728	−	0.997079i	$-0.524334\pi$
$953$	−4.71536			−0.152746			−0.0763728	+	0.997079i	$0.524334\pi$
$954$		0			0
$955$	−15.8968	−	19.4695i	−0.514408	−	0.630018i
$956$		0			0
$957$		−	78.2350i		−	2.52898i
$958$		0			0
$959$			30.7102i			0.991685i
$960$		0			0
$961$	−11.3923			−0.367494
$962$		0			0
$963$			0.129790i			0.00418244i
$964$		0			0
$965$	−27.0287	+	22.0688i	−0.870084	+	0.710421i
$966$		0			0
$967$	47.8433			1.53854			0.769268	−	0.638927i	$-0.220621\pi$
$967$	47.8433			1.53854			0.769268	+	0.638927i	$0.220621\pi$
$968$		0			0
$969$	11.3205	+	8.45535i	0.363667	+	0.271625i
$970$		0			0
$971$	25.3819			0.814542			0.407271	−	0.913307i	$-0.366480\pi$
$971$	25.3819			0.814542			0.407271	+	0.913307i	$0.366480\pi$
$972$		0			0
$973$			34.4216i			1.10351i
$974$		0			0
$975$	−24.1100	−	4.92144i	−0.772139	−	0.157612i
$976$		0			0
$977$	8.16724			0.261293			0.130647	−	0.991429i	$-0.458295\pi$
$977$	8.16724			0.261293			0.130647	+	0.991429i	$0.458295\pi$
$978$		0			0
$979$	43.2776			1.38316
$980$		0			0
$981$			1.78683i			0.0570490i
$982$		0			0
$983$		−	7.71537i		−	0.246082i	−0.992402	−	0.123041i	$-0.960735\pi$
$983$		−	7.71537i		−	0.246082i	0.992402	−	0.123041i	$-0.0392647\pi$
$984$		0			0
$985$	−9.46410	−	11.5911i	−0.301551	−	0.369323i
$986$		0			0
$987$		−	24.5993i		−	0.783003i
$988$		0			0
$989$	29.3205			0.932338
$990$		0			0
$991$	−34.9230			−1.10937			−0.554683	−	0.832062i	$-0.687160\pi$
$991$	−34.9230			−1.10937			−0.554683	+	0.832062i	$0.687160\pi$
$992$		0			0
$993$		−	43.7391i		−	1.38802i
$994$		0			0
$995$	27.5340	+	33.7222i	0.872888	+	1.06906i
$996$		0			0
$997$			17.8028i			0.563819i	0.959441	+	0.281910i	$0.0909678\pi$
$997$			17.8028i			0.563819i	−0.959441	+	0.281910i	$0.909032\pi$
$998$		0			0
$999$		−	50.1797i		−	1.58762i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1520.2.g.f.1519.10	yes	16
4.3	odd	2	inner	1520.2.g.f.1519.5	✓	16
5.4	even	2	inner	1520.2.g.f.1519.7	yes	16
19.18	odd	2	inner	1520.2.g.f.1519.6	yes	16
20.19	odd	2	inner	1520.2.g.f.1519.12	yes	16
76.75	even	2	inner	1520.2.g.f.1519.9	yes	16
95.94	odd	2	inner	1520.2.g.f.1519.11	yes	16
380.379	even	2	inner	1520.2.g.f.1519.8	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1520.2.g.f.1519.5	✓	16	4.3	odd	2	inner
1520.2.g.f.1519.6	yes	16	19.18	odd	2	inner
1520.2.g.f.1519.7	yes	16	5.4	even	2	inner
1520.2.g.f.1519.8	yes	16	380.379	even	2	inner
1520.2.g.f.1519.9	yes	16	76.75	even	2	inner
1520.2.g.f.1519.10	yes	16	1.1	even	1	trivial
1520.2.g.f.1519.11	yes	16	95.94	odd	2	inner
1520.2.g.f.1519.12	yes	16	20.19	odd	2	inner

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:	\( N \)	\(=\)	\( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1520.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.80775i q^{3} +(1.73205 - 1.41421i) q^{5} +2.12976 q^{7} -0.267949 q^{9} +O(q^{10})\)	\(q+1.80775i q^{3} +(1.73205 - 1.41421i) q^{5} +2.12976 q^{7} -0.267949 q^{9} -4.11439i q^{11} -2.72241 q^{13} +(2.55654 + 3.13111i) q^{15} -1.79315i q^{17} +(3.49230 + 2.60842i) q^{19} +3.85008i q^{21} +3.68886 q^{23} +(1.00000 - 4.89898i) q^{25} +4.93886i q^{27} -10.5186i q^{29} +4.42806 q^{31} +7.43778 q^{33} +(3.68886 - 3.01194i) q^{35} -10.1602 q^{37} -4.92144i q^{39} -3.85008i q^{41} +7.94839 q^{43} +(-0.464102 + 0.378937i) q^{45} -6.38929 q^{47} -2.46410 q^{49} +3.24156 q^{51} +4.71536 q^{53} +(-5.81863 - 7.12633i) q^{55} +(-4.71536 + 6.31319i) q^{57} -4.42806 q^{59} +8.19615 q^{61} -0.570669 q^{63} +(-4.71536 + 3.85008i) q^{65} +3.13111i q^{67} +6.66853i q^{69} +16.5257 q^{71} +13.3843i q^{73} +(8.85612 + 1.80775i) q^{75} -8.76268i q^{77} -6.29958 q^{79} -9.73205 q^{81} -13.7670 q^{83} +(-2.53590 - 3.10583i) q^{85} +19.0150 q^{87} +10.5186i q^{89} -5.79810 q^{91} +8.00481i q^{93} +(9.73770 - 0.420943i) q^{95} +10.1602 q^{97} +1.10245i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 32 q^{9}+O(q^{10}) \) 16 * q - 32 * q^9	\( 16 q - 32 q^{9} + 16 q^{25} + 48 q^{45} + 16 q^{49} + 48 q^{61} - 128 q^{81} - 96 q^{85}+O(q^{100}) \) 16 * q - 32 * q^9 + 16 * q^25 + 48 * q^45 + 16 * q^49 + 48 * q^61 - 128 * q^81 - 96 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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