LMFDB - Embedded newform 380.3.e.a.341.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [380,3,Mod(341,380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(380, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("380.341");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 380 = 2^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	380.e (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:	$10.3542500457$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 62x^{10} + 1445x^{8} + 15924x^{6} + 83244x^{4} + 170640x^{2} + 55600 $ x^12 + 62x^10 + 1445x^8 + 15924x^6 + 83244x^4 + 170640*x^2 + 55600
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{9}\cdot 5 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			341.11
Root			$3.53755i$ of defining polynomial
Character	$\chi$	$=$	380.341
Dual form			380.3.e.a.341.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.53755i q^{3} -2.23607 q^{5} +0.468611 q^{7} -3.51429 q^{9} +O(q^{10})$	$q+3.53755i q^{3} -2.23607 q^{5} +0.468611 q^{7} -3.51429 q^{9} -14.1082 q^{11} +11.0914i q^{13} -7.91021i q^{15} -22.0318 q^{17} +(16.8411 - 8.79644i) q^{19} +1.65774i q^{21} -41.1864 q^{23} +5.00000 q^{25} +19.4060i q^{27} -46.7187i q^{29} +33.9265i q^{31} -49.9085i q^{33} -1.04785 q^{35} -62.6340i q^{37} -39.2365 q^{39} -23.5359i q^{41} -32.8769 q^{43} +7.85820 q^{45} +43.2839 q^{47} -48.7804 q^{49} -77.9388i q^{51} +99.9844i q^{53} +31.5469 q^{55} +(31.1179 + 59.5763i) q^{57} -66.7119i q^{59} -36.0483 q^{61} -1.64684 q^{63} -24.8011i q^{65} +93.5215i q^{67} -145.699i q^{69} +124.406i q^{71} +41.3341 q^{73} +17.6878i q^{75} -6.61125 q^{77} +27.2511i q^{79} -100.278 q^{81} -32.9101 q^{83} +49.2647 q^{85} +165.270 q^{87} +86.3670i q^{89} +5.19755i q^{91} -120.017 q^{93} +(-37.6578 + 19.6694i) q^{95} +20.8071i q^{97} +49.5803 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{7} - 16 q^{9}+O(q^{10}) $ 12 * q - 12 * q^7 - 16 * q^9	$ 12 q - 12 q^{7} - 16 q^{9} - 32 q^{11} - 12 q^{17} + 24 q^{19} + 4 q^{23} + 60 q^{25} + 40 q^{35} + 124 q^{39} - 176 q^{43} - 40 q^{45} - 72 q^{47} - 24 q^{49} + 140 q^{57} + 152 q^{61} + 48 q^{63} - 148 q^{73} + 376 q^{77} - 468 q^{81} - 208 q^{83} - 84 q^{87} - 184 q^{93} + 392 q^{99}+O(q^{100}) $ 12 * q - 12 * q^7 - 16 * q^9 - 32 * q^11 - 12 * q^17 + 24 * q^19 + 4 * q^23 + 60 * q^25 + 40 * q^35 + 124 * q^39 - 176 * q^43 - 40 * q^45 - 72 * q^47 - 24 * q^49 + 140 * q^57 + 152 * q^61 + 48 * q^63 - 148 * q^73 + 376 * q^77 - 468 * q^81 - 208 * q^83 - 84 * q^87 - 184 * q^93 + 392 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/380\mathbb{Z}\right)^\times$.

$n$	$21$	$77$	$191$
$\chi(n)$	$-1$	$1$	$1$

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
$3$			3.53755i			1.17918i	0.807701	+	0.589592i	$0.200712\pi$
$3$			3.53755i			1.17918i	−0.807701	+	0.589592i	$0.799288\pi$
$4$		0			0
$5$	−2.23607			−0.447214
$5$	−2.23607			−0.447214
$6$		0			0
$7$	0.468611			0.0669444			0.0334722	−	0.999440i	$-0.489343\pi$
$7$	0.468611			0.0669444			0.0334722	+	0.999440i	$0.489343\pi$
$8$		0			0
$9$	−3.51429			−0.390477
$10$		0			0
$11$	−14.1082			−1.28256			−0.641281	−	0.767306i	$-0.721597\pi$
$11$	−14.1082			−1.28256			−0.641281	+	0.767306i	$0.721597\pi$
$12$		0			0
$13$			11.0914i			0.853185i	0.904444	+	0.426593i	$0.140286\pi$
$13$			11.0914i			0.853185i	−0.904444	+	0.426593i	$0.859714\pi$
$14$		0			0
$15$		−	7.91021i		−	0.527347i
$16$		0			0
$17$	−22.0318			−1.29599			−0.647995	−	0.761644i	$-0.724392\pi$
$17$	−22.0318			−1.29599			−0.647995	+	0.761644i	$0.724392\pi$
$18$		0			0
$19$	16.8411	−	8.79644i	0.886374	−	0.462971i
$19$	16.8411	−	8.79644i	0.886374	−	0.462971i
$20$		0			0
$21$			1.65774i			0.0789398i
$22$		0			0
$23$	−41.1864			−1.79071			−0.895357	−	0.445348i	$-0.853080\pi$
$23$	−41.1864			−1.79071			−0.895357	+	0.445348i	$0.853080\pi$
$24$		0			0
$25$	5.00000			0.200000
$26$		0			0
$27$			19.4060i			0.718740i
$28$		0			0
$29$		−	46.7187i		−	1.61099i	−0.592604	−	0.805494i	$-0.701900\pi$
$29$		−	46.7187i		−	1.61099i	0.592604	−	0.805494i	$-0.298100\pi$
$30$		0			0
$31$			33.9265i			1.09440i	0.837001	+	0.547201i	$0.184307\pi$
$31$			33.9265i			1.09440i	−0.837001	+	0.547201i	$0.815693\pi$
$32$		0			0
$33$		−	49.9085i		−	1.51238i
$34$		0			0
$35$	−1.04785			−0.0299384
$36$		0			0
$37$		−	62.6340i		−	1.69281i	−0.532540	−	0.846405i	$-0.678762\pi$
$37$		−	62.6340i		−	1.69281i	0.532540	−	0.846405i	$-0.321238\pi$
$38$		0			0
$39$	−39.2365			−1.00606
$40$		0			0
$41$		−	23.5359i		−	0.574046i	−0.957924	−	0.287023i	$-0.907334\pi$
$41$		−	23.5359i		−	0.574046i	0.957924	−	0.287023i	$-0.0926656\pi$
$42$		0			0
$43$	−32.8769			−0.764579			−0.382289	−	0.924043i	$-0.624864\pi$
$43$	−32.8769			−0.764579			−0.382289	+	0.924043i	$0.624864\pi$
$44$		0			0
$45$	7.85820			0.174627
$46$		0			0
$47$	43.2839			0.920935			0.460468	−	0.887677i	$-0.347682\pi$
$47$	43.2839			0.920935			0.460468	+	0.887677i	$0.347682\pi$
$48$		0			0
$49$	−48.7804			−0.995518
$50$		0			0
$51$		−	77.9388i		−	1.52821i
$52$		0			0
$53$			99.9844i			1.88650i	0.332086	+	0.943249i	$0.392248\pi$
$53$			99.9844i			1.88650i	−0.332086	+	0.943249i	$0.607752\pi$
$54$		0			0
$55$	31.5469			0.573579
$56$		0			0
$57$	31.1179	+	59.5763i	0.545928	+	1.04520i
$58$		0			0
$59$		−	66.7119i		−	1.13071i	−0.824848	−	0.565355i	$-0.808739\pi$
$59$		−	66.7119i		−	1.13071i	0.824848	−	0.565355i	$-0.191261\pi$
$60$		0			0
$61$	−36.0483			−0.590956			−0.295478	−	0.955350i	$-0.595479\pi$
$61$	−36.0483			−0.590956			−0.295478	+	0.955350i	$0.595479\pi$
$62$		0			0
$63$	−1.64684			−0.0261402
$64$		0			0
$65$		−	24.8011i		−	0.381556i
$66$		0			0
$67$			93.5215i			1.39584i	0.716174	+	0.697921i	$0.245892\pi$
$67$			93.5215i			1.39584i	−0.716174	+	0.697921i	$0.754108\pi$
$68$		0			0
$69$		−	145.699i		−	2.11158i
$70$		0			0
$71$			124.406i			1.75220i	0.482128	+	0.876101i	$0.339864\pi$
$71$			124.406i			1.75220i	−0.482128	+	0.876101i	$0.660136\pi$
$72$		0			0
$73$	41.3341			0.566220			0.283110	−	0.959087i	$-0.408634\pi$
$73$	41.3341			0.566220			0.283110	+	0.959087i	$0.408634\pi$
$74$		0			0
$75$			17.6878i			0.235837i
$76$		0			0
$77$	−6.61125			−0.0858604
$78$		0			0
$79$			27.2511i			0.344950i	0.985014	+	0.172475i	$0.0551764\pi$
$79$			27.2511i			0.344950i	−0.985014	+	0.172475i	$0.944824\pi$
$80$		0			0
$81$	−100.278			−1.23800
$82$		0			0
$83$	−32.9101			−0.396507			−0.198253	−	0.980151i	$-0.563527\pi$
$83$	−32.9101			−0.396507			−0.198253	+	0.980151i	$0.563527\pi$
$84$		0			0
$85$	49.2647			0.579585
$86$		0			0
$87$	165.270			1.89965
$88$		0			0
$89$			86.3670i			0.970416i	0.874399	+	0.485208i	$0.161256\pi$
$89$			86.3670i			0.970416i	−0.874399	+	0.485208i	$0.838744\pi$
$90$		0			0
$91$			5.19755i			0.0571160i
$92$		0			0
$93$	−120.017			−1.29050
$94$		0			0
$95$	−37.6578	+	19.6694i	−0.396398	+	0.207047i
$96$		0			0
$97$			20.8071i			0.214506i	0.994232	+	0.107253i	$0.0342054\pi$
$97$			20.8071i			0.214506i	−0.994232	+	0.107253i	$0.965795\pi$
$98$		0			0
$99$	49.5803			0.500811
$100$		0			0
$101$	−118.267			−1.17096			−0.585482	−	0.810685i	$-0.699095\pi$
$101$	−118.267			−1.17096			−0.585482	+	0.810685i	$0.699095\pi$
$102$		0			0
$103$			107.875i			1.04733i	0.851924	+	0.523666i	$0.175436\pi$
$103$			107.875i			1.04733i	−0.851924	+	0.523666i	$0.824564\pi$
$104$		0			0
$105$		−	3.70681i		−	0.0353030i
$106$		0			0
$107$		−	16.6498i		−	0.155606i	−0.996969	−	0.0778029i	$-0.975210\pi$
$107$		−	16.6498i		−	0.155606i	0.996969	−	0.0778029i	$-0.0247905\pi$
$108$		0			0
$109$			34.4048i			0.315640i	0.987468	+	0.157820i	$0.0504466\pi$
$109$			34.4048i			0.315640i	−0.987468	+	0.157820i	$0.949553\pi$
$110$		0			0
$111$	221.571			1.99614
$112$		0			0
$113$		−	34.2888i		−	0.303441i	−0.988423	−	0.151720i	$-0.951519\pi$
$113$		−	34.2888i		−	0.303441i	0.988423	−	0.151720i	$-0.0484813\pi$
$114$		0			0
$115$	92.0957			0.800832
$116$		0			0
$117$		−	38.9784i		−	0.333149i
$118$		0			0
$119$	−10.3244			−0.0867593
$120$		0			0
$121$	78.0409			0.644966
$122$		0			0
$123$	83.2594			0.676906
$124$		0			0
$125$	−11.1803			−0.0894427
$126$		0			0
$127$			245.621i			1.93402i	0.254733	+	0.967012i	$0.418013\pi$
$127$			245.621i			1.93402i	−0.254733	+	0.967012i	$0.581987\pi$
$128$		0			0
$129$		−	116.304i		−	0.901580i
$130$		0			0
$131$	196.826			1.50249			0.751245	−	0.660024i	$-0.229454\pi$
$131$	196.826			1.50249			0.751245	+	0.660024i	$0.229454\pi$
$132$		0			0
$133$	7.89192	−	4.12211i	0.0593377	−	0.0309933i
$134$		0			0
$135$		−	43.3931i		−	0.321430i
$136$		0			0
$137$	249.410			1.82051			0.910255	−	0.414049i	$-0.135886\pi$
$137$	249.410			1.82051			0.910255	+	0.414049i	$0.135886\pi$
$138$		0			0
$139$	116.069			0.835030			0.417515	−	0.908670i	$-0.362901\pi$
$139$	116.069			0.835030			0.417515	+	0.908670i	$0.362901\pi$
$140$		0			0
$141$			153.119i			1.08595i
$142$		0			0
$143$		−	156.480i		−	1.09426i
$144$		0			0
$145$			104.466i			0.720456i
$146$		0			0
$147$		−	172.563i		−	1.17390i
$148$		0			0
$149$	−220.978			−1.48308			−0.741538	−	0.670911i	$-0.765903\pi$
$149$	−220.978			−1.48308			−0.741538	+	0.670911i	$0.765903\pi$
$150$		0			0
$151$		−	237.670i		−	1.57397i	−0.616972	−	0.786985i	$-0.711641\pi$
$151$		−	237.670i		−	1.57397i	0.616972	−	0.786985i	$-0.288359\pi$
$152$		0			0
$153$	77.4263			0.506054
$154$		0			0
$155$		−	75.8619i		−	0.489431i
$156$		0			0
$157$	35.3231			0.224988			0.112494	−	0.993652i	$-0.464116\pi$
$157$	35.3231			0.224988			0.112494	+	0.993652i	$0.464116\pi$
$158$		0			0
$159$	−353.700			−2.22453
$160$		0			0
$161$	−19.3004			−0.119878
$162$		0			0
$163$	71.4216			0.438169			0.219085	−	0.975706i	$-0.429693\pi$
$163$	71.4216			0.438169			0.219085	+	0.975706i	$0.429693\pi$
$164$		0			0
$165$			111.599i			0.676356i
$166$		0			0
$167$			185.054i			1.10811i	0.832481	+	0.554054i	$0.186920\pi$
$167$			185.054i			1.10811i	−0.832481	+	0.554054i	$0.813080\pi$
$168$		0			0
$169$	45.9807			0.272075
$170$		0			0
$171$	−59.1845	+	30.9133i	−0.346108	+	0.180779i
$172$		0			0
$173$		−	231.572i		−	1.33857i	−0.743006	−	0.669284i	$-0.766601\pi$
$173$		−	231.572i		−	1.33857i	0.743006	−	0.669284i	$-0.233399\pi$
$174$		0			0
$175$	2.34305			0.0133889
$176$		0			0
$177$	235.997			1.33332
$178$		0			0
$179$			165.841i			0.926487i	0.886231	+	0.463244i	$0.153315\pi$
$179$			165.841i			0.926487i	−0.886231	+	0.463244i	$0.846685\pi$
$180$		0			0
$181$			25.0838i			0.138585i	0.997596	+	0.0692923i	$0.0220741\pi$
$181$			25.0838i			0.138585i	−0.997596	+	0.0692923i	$0.977926\pi$
$182$		0			0
$183$		−	127.523i		−	0.696846i
$184$		0			0
$185$			140.054i			0.757048i
$186$		0			0
$187$	310.829			1.66219
$188$		0			0
$189$			9.09386i			0.0481156i
$190$		0			0
$191$	−52.0646			−0.272590			−0.136295	−	0.990668i	$-0.543519\pi$
$191$	−52.0646			−0.272590			−0.136295	+	0.990668i	$0.543519\pi$
$192$		0			0
$193$		−	70.3184i		−	0.364344i	−0.983267	−	0.182172i	$-0.941687\pi$
$193$		−	70.3184i		−	0.364344i	0.983267	−	0.182172i	$-0.0583128\pi$
$194$		0			0
$195$	87.7354			0.449925
$196$		0			0
$197$	−131.361			−0.666809			−0.333404	−	0.942784i	$-0.608197\pi$
$197$	−131.361			−0.666809			−0.333404	+	0.942784i	$0.608197\pi$
$198$		0			0
$199$	−120.966			−0.607870			−0.303935	−	0.952693i	$-0.598301\pi$
$199$	−120.966			−0.607870			−0.303935	+	0.952693i	$0.598301\pi$
$200$		0			0
$201$	−330.837			−1.64596
$202$		0			0
$203$		−	21.8929i		−	0.107847i
$204$		0			0
$205$			52.6278i			0.256721i
$206$		0			0
$207$	144.741			0.699233
$208$		0			0
$209$	−237.597	+	124.102i	−1.13683	+	0.593789i
$210$		0			0
$211$		−	381.578i		−	1.80843i	−0.427082	−	0.904213i	$-0.640459\pi$
$211$		−	381.578i		−	1.80843i	0.427082	−	0.904213i	$-0.359541\pi$
$212$		0			0
$213$	−440.094			−2.06617
$214$		0			0
$215$	73.5150			0.341930
$216$		0			0
$217$			15.8983i			0.0732641i
$218$		0			0
$219$			146.222i			0.667678i
$220$		0			0
$221$		−	244.364i		−	1.10572i
$222$		0			0
$223$			155.025i			0.695181i	0.937646	+	0.347591i	$0.113000\pi$
$223$			155.025i			0.695181i	−0.937646	+	0.347591i	$0.887000\pi$
$224$		0			0
$225$	−17.5715			−0.0780954
$226$		0			0
$227$			237.583i			1.04662i	0.852142	+	0.523310i	$0.175303\pi$
$227$			237.583i			1.04662i	−0.852142	+	0.523310i	$0.824697\pi$
$228$		0			0
$229$	−275.640			−1.20367			−0.601834	−	0.798621i	$-0.705563\pi$
$229$	−275.640			−1.20367			−0.601834	+	0.798621i	$0.705563\pi$
$230$		0			0
$231$		−	23.3877i		−	0.101245i
$232$		0			0
$233$	−271.591			−1.16562			−0.582812	−	0.812607i	$-0.698048\pi$
$233$	−271.591			−1.16562			−0.582812	+	0.812607i	$0.698048\pi$
$234$		0			0
$235$	−96.7858			−0.411855
$236$		0			0
$237$	−96.4021			−0.406760
$238$		0			0
$239$	−236.246			−0.988476			−0.494238	−	0.869327i	$-0.664553\pi$
$239$	−236.246			−0.988476			−0.494238	+	0.869327i	$0.664553\pi$
$240$		0			0
$241$			346.159i			1.43634i	0.695866	+	0.718171i	$0.255021\pi$
$241$			346.159i			1.43634i	−0.695866	+	0.718171i	$0.744979\pi$
$242$		0			0
$243$		−	180.086i		−	0.741096i
$244$		0			0
$245$	109.076			0.445209
$246$		0			0
$247$	97.5649	+	186.791i	0.395000	+	0.756241i
$248$		0			0
$249$		−	116.421i		−	0.467555i
$250$		0			0
$251$	222.337			0.885805			0.442903	−	0.896570i	$-0.353949\pi$
$251$	222.337			0.885805			0.442903	+	0.896570i	$0.353949\pi$
$252$		0			0
$253$	581.066			2.29670
$254$		0			0
$255$			174.277i			0.683437i
$256$		0			0
$257$		−	143.232i		−	0.557324i	−0.960389	−	0.278662i	$-0.910109\pi$
$257$		−	143.232i		−	0.557324i	0.960389	−	0.278662i	$-0.0898909\pi$
$258$		0			0
$259$		−	29.3510i		−	0.113324i
$260$		0			0
$261$			164.183i			0.629054i
$262$		0			0
$263$	−240.308			−0.913718			−0.456859	−	0.889539i	$-0.651026\pi$
$263$	−240.308			−0.913718			−0.456859	+	0.889539i	$0.651026\pi$
$264$		0			0
$265$		−	223.572i		−	0.843668i
$266$		0			0
$267$	−305.528			−1.14430
$268$		0			0
$269$		−	261.289i		−	0.971335i	−0.874144	−	0.485668i	$-0.838577\pi$
$269$		−	261.289i		−	0.971335i	0.874144	−	0.485668i	$-0.161423\pi$
$270$		0			0
$271$	−234.829			−0.866527			−0.433263	−	0.901267i	$-0.642638\pi$
$271$	−234.829			−0.866527			−0.433263	+	0.901267i	$0.642638\pi$
$272$		0			0
$273$	−18.3866			−0.0673503
$274$		0			0
$275$	−70.5409			−0.256512
$276$		0			0
$277$	−28.6370			−0.103383			−0.0516913	−	0.998663i	$-0.516461\pi$
$277$	−28.6370			−0.103383			−0.0516913	+	0.998663i	$0.516461\pi$
$278$		0			0
$279$		−	119.227i		−	0.427339i
$280$		0			0
$281$			35.2442i			0.125424i	0.998032	+	0.0627121i	$0.0199750\pi$
$281$			35.2442i			0.125424i	−0.998032	+	0.0627121i	$0.980025\pi$
$282$		0			0
$283$	−159.486			−0.563554			−0.281777	−	0.959480i	$-0.590924\pi$
$283$	−159.486			−0.563554			−0.281777	+	0.959480i	$0.590924\pi$
$284$		0			0
$285$	−69.5817	−	133.217i	−0.244146	−	0.467427i
$286$		0			0
$287$		−	11.0292i		−	0.0384291i
$288$		0			0
$289$	196.402			0.679591
$290$		0			0
$291$	−73.6061			−0.252942
$292$		0			0
$293$		−	79.8129i		−	0.272399i	−0.990681	−	0.136199i	$-0.956511\pi$
$293$		−	79.8129i		−	0.272399i	0.990681	−	0.136199i	$-0.0434888\pi$
$294$		0			0
$295$			149.172i			0.505669i
$296$		0			0
$297$		−	273.783i		−	0.921829i
$298$		0			0
$299$		−	456.816i		−	1.52781i
$300$		0			0
$301$	−15.4065			−0.0511843
$302$		0			0
$303$		−	418.377i		−	1.38078i
$304$		0			0
$305$	80.6065			0.264284
$306$		0			0
$307$		−	135.726i		−	0.442105i	−0.975262	−	0.221052i	$-0.929051\pi$
$307$		−	135.726i		−	0.442105i	0.975262	−	0.221052i	$-0.0709492\pi$
$308$		0			0
$309$	−381.614			−1.23500
$310$		0			0
$311$	131.701			0.423475			0.211737	−	0.977327i	$-0.432088\pi$
$311$	131.701			0.423475			0.211737	+	0.977327i	$0.432088\pi$
$312$		0			0
$313$	570.037			1.82120			0.910602	−	0.413285i	$-0.135619\pi$
$313$	570.037			1.82120			0.910602	+	0.413285i	$0.135619\pi$
$314$		0			0
$315$	3.68244			0.0116903
$316$		0			0
$317$		−	334.101i		−	1.05395i	−0.849882	−	0.526974i	$-0.823327\pi$
$317$		−	334.101i		−	1.05395i	0.849882	−	0.526974i	$-0.176673\pi$
$318$		0			0
$319$			659.116i			2.06619i
$320$		0			0
$321$	58.8997			0.183488
$322$		0			0
$323$	−371.040	+	193.802i	−1.14873	+	0.600006i
$324$		0			0
$325$			55.4570i			0.170637i
$326$		0			0
$327$	−121.709			−0.372198
$328$		0			0
$329$	20.2833			0.0616514
$330$		0			0
$331$		−	63.3605i		−	0.191421i	−0.995409	−	0.0957107i	$-0.969488\pi$
$331$		−	63.3605i		−	0.191421i	0.995409	−	0.0957107i	$-0.0305124\pi$
$332$		0			0
$333$			220.114i			0.661003i
$334$		0			0
$335$		−	209.120i		−	0.624240i
$336$		0			0
$337$			153.496i			0.455479i	0.973722	+	0.227739i	$0.0731335\pi$
$337$			153.496i			0.455479i	−0.973722	+	0.227739i	$0.926867\pi$
$338$		0			0
$339$	121.298			0.357812
$340$		0			0
$341$		−	478.641i		−	1.40364i
$342$		0			0
$343$	−45.8210			−0.133589
$344$		0			0
$345$			325.794i			0.944329i
$346$		0			0
$347$	−272.066			−0.784052			−0.392026	−	0.919954i	$-0.628226\pi$
$347$	−272.066			−0.784052			−0.392026	+	0.919954i	$0.628226\pi$
$348$		0			0
$349$	187.833			0.538205			0.269102	−	0.963112i	$-0.413273\pi$
$349$	187.833			0.538205			0.269102	+	0.963112i	$0.413273\pi$
$350$		0			0
$351$	−215.240			−0.613219
$352$		0			0
$353$	−30.8460			−0.0873825			−0.0436913	−	0.999045i	$-0.513912\pi$
$353$	−30.8460			−0.0873825			−0.0436913	+	0.999045i	$0.513912\pi$
$354$		0			0
$355$		−	278.181i		−	0.783609i
$356$		0			0
$357$		−	36.5230i		−	0.102305i
$358$		0			0
$359$	513.758			1.43108			0.715541	−	0.698571i	$-0.246180\pi$
$359$	513.758			1.43108			0.715541	+	0.698571i	$0.246180\pi$
$360$		0			0
$361$	206.245	−	296.284i	0.571316	−	0.820730i
$362$		0			0
$363$			276.074i			0.760535i
$364$		0			0
$365$	−92.4258			−0.253221
$366$		0			0
$367$	−22.6982			−0.0618481			−0.0309240	−	0.999522i	$-0.509845\pi$
$367$	−22.6982			−0.0618481			−0.0309240	+	0.999522i	$0.509845\pi$
$368$		0			0
$369$			82.7119i			0.224152i
$370$		0			0
$371$			46.8538i			0.126290i
$372$		0			0
$373$		−	108.632i		−	0.291238i	−0.989341	−	0.145619i	$-0.953483\pi$
$373$		−	108.632i		−	0.291238i	0.989341	−	0.145619i	$-0.0465174\pi$
$374$		0			0
$375$		−	39.5511i		−	0.105469i
$376$		0			0
$377$	518.176			1.37447
$378$		0			0
$379$		−	203.467i		−	0.536852i	−0.963300	−	0.268426i	$-0.913497\pi$
$379$		−	203.467i		−	0.536852i	0.963300	−	0.268426i	$-0.0865035\pi$
$380$		0			0
$381$	−868.897			−2.28057
$382$		0			0
$383$		−	38.4623i		−	0.100424i	−0.998739	−	0.0502119i	$-0.984010\pi$
$383$		−	38.4623i		−	0.100424i	0.998739	−	0.0502119i	$-0.0159897\pi$
$384$		0			0
$385$	14.7832			0.0383979
$386$		0			0
$387$	115.539			0.298550
$388$		0			0
$389$	521.954			1.34179			0.670893	−	0.741555i	$-0.265911\pi$
$389$	521.954			1.34179			0.670893	+	0.741555i	$0.265911\pi$
$390$		0			0
$391$	907.413			2.32075
$392$		0			0
$393$			696.283i			1.77171i
$394$		0			0
$395$		−	60.9352i		−	0.154266i
$396$		0			0
$397$	369.352			0.930357			0.465179	−	0.885217i	$-0.345990\pi$
$397$	369.352			0.930357			0.465179	+	0.885217i	$0.345990\pi$
$398$		0			0
$399$	14.5822	+	27.9181i	0.0365468	+	0.0699702i
$400$		0			0
$401$			128.575i			0.320636i	0.987065	+	0.160318i	$0.0512520\pi$
$401$			128.575i			0.320636i	−0.987065	+	0.160318i	$0.948748\pi$
$402$		0			0
$403$	−376.292			−0.933727
$404$		0			0
$405$	224.229			0.553653
$406$		0			0
$407$			883.652i			2.17113i
$408$		0			0
$409$		−	611.222i		−	1.49443i	−0.664583	−	0.747215i	$-0.731391\pi$
$409$		−	611.222i		−	1.49443i	0.664583	−	0.747215i	$-0.268609\pi$
$410$		0			0
$411$			882.301i			2.14672i
$412$		0			0
$413$		−	31.2619i		−	0.0756947i
$414$		0			0
$415$	73.5891			0.177323
$416$		0			0
$417$			410.601i			0.984655i
$418$		0			0
$419$	−218.656			−0.521853			−0.260927	−	0.965359i	$-0.584028\pi$
$419$	−218.656			−0.521853			−0.260927	+	0.965359i	$0.584028\pi$
$420$		0			0
$421$			534.615i			1.26987i	0.772566	+	0.634935i	$0.218973\pi$
$421$			534.615i			1.26987i	−0.772566	+	0.634935i	$0.781027\pi$
$422$		0			0
$423$	−152.112			−0.359604
$424$		0			0
$425$	−110.159			−0.259198
$426$		0			0
$427$	−16.8926			−0.0395612
$428$		0			0
$429$	553.555			1.29034
$430$		0			0
$431$			81.6513i			0.189446i	0.995504	+	0.0947231i	$0.0301966\pi$
$431$			81.6513i			0.189446i	−0.995504	+	0.0947231i	$0.969803\pi$
$432$		0			0
$433$			90.7988i			0.209697i	0.994488	+	0.104848i	$0.0334357\pi$
$433$			90.7988i			0.209697i	−0.994488	+	0.104848i	$0.966564\pi$
$434$		0			0
$435$	−369.555			−0.849551
$436$		0			0
$437$	−693.625	+	362.294i	−1.58724	+	0.829048i
$438$		0			0
$439$			89.4005i			0.203646i	0.994803	+	0.101823i	$0.0324675\pi$
$439$			89.4005i			0.203646i	−0.994803	+	0.101823i	$0.967533\pi$
$440$		0			0
$441$	171.429			0.388727
$442$		0			0
$443$	127.590			0.288013			0.144007	−	0.989577i	$-0.454001\pi$
$443$	127.590			0.288013			0.144007	+	0.989577i	$0.454001\pi$
$444$		0			0
$445$		−	193.123i		−	0.433983i
$446$		0			0
$447$		−	781.723i		−	1.74882i
$448$		0			0
$449$			441.576i			0.983465i	0.870746	+	0.491733i	$0.163636\pi$
$449$			441.576i			0.983465i	−0.870746	+	0.491733i	$0.836364\pi$
$450$		0			0
$451$			332.049i			0.736250i
$452$		0			0
$453$	840.769			1.85600
$454$		0			0
$455$		−	11.6221i		−	0.0255430i
$456$		0			0
$457$	166.817			0.365027			0.182513	−	0.983203i	$-0.441577\pi$
$457$	166.817			0.365027			0.182513	+	0.983203i	$0.441577\pi$
$458$		0			0
$459$		−	427.550i		−	0.931481i
$460$		0			0
$461$	−377.705			−0.819317			−0.409659	−	0.912239i	$-0.634352\pi$
$461$	−377.705			−0.819317			−0.409659	+	0.912239i	$0.634352\pi$
$462$		0			0
$463$	−161.036			−0.347810			−0.173905	−	0.984762i	$-0.555639\pi$
$463$	−161.036			−0.347810			−0.173905	+	0.984762i	$0.555639\pi$
$464$		0			0
$465$	268.365			0.577130
$466$		0			0
$467$	−63.1149			−0.135150			−0.0675749	−	0.997714i	$-0.521526\pi$
$467$	−63.1149			−0.135150			−0.0675749	+	0.997714i	$0.521526\pi$
$468$		0			0
$469$			43.8252i			0.0934438i
$470$		0			0
$471$			124.958i			0.265303i
$472$		0			0
$473$	463.833			0.980620
$474$		0			0
$475$	84.2055	−	43.9822i	0.177275	−	0.0925941i
$476$		0			0
$477$		−	351.374i		−	0.736634i
$478$		0			0
$479$	−536.087			−1.11918			−0.559589	−	0.828770i	$-0.689041\pi$
$479$	−536.087			−1.11918			−0.559589	+	0.828770i	$0.689041\pi$
$480$		0			0
$481$	694.699			1.44428
$482$		0			0
$483$		−	68.2763i		−	0.141359i
$484$		0			0
$485$		−	46.5260i		−	0.0959299i
$486$		0			0
$487$		−	247.634i		−	0.508490i	−0.967140	−	0.254245i	$-0.918173\pi$
$487$		−	247.634i		−	0.508490i	0.967140	−	0.254245i	$-0.0818269\pi$
$488$		0			0
$489$			252.658i			0.516683i
$490$		0			0
$491$	895.324			1.82347			0.911735	−	0.410779i	$-0.134743\pi$
$491$	895.324			1.82347			0.911735	+	0.410779i	$0.134743\pi$
$492$		0			0
$493$			1029.30i			2.08783i
$494$		0			0
$495$	−110.865			−0.223969
$496$		0			0
$497$			58.2982i			0.117300i
$498$		0			0
$499$	78.2101			0.156734			0.0783669	−	0.996925i	$-0.475029\pi$
$499$	78.2101			0.156734			0.0783669	+	0.996925i	$0.475029\pi$
$500$		0			0
$501$	−654.638			−1.30666
$502$		0			0
$503$	−568.406			−1.13003			−0.565015	−	0.825080i	$-0.691130\pi$
$503$	−568.406			−1.13003			−0.565015	+	0.825080i	$0.691130\pi$
$504$		0			0
$505$	264.454			0.523671
$506$		0			0
$507$			162.659i			0.320827i
$508$		0			0
$509$		−	528.251i		−	1.03782i	−0.854829	−	0.518911i	$-0.826338\pi$
$509$		−	528.251i		−	1.03782i	0.854829	−	0.518911i	$-0.173662\pi$
$510$		0			0
$511$	19.3696			0.0379053
$512$		0			0
$513$	170.704	+	326.818i	0.332756	+	0.637073i
$514$		0			0
$515$		−	241.216i		−	0.468381i
$516$		0			0
$517$	−610.658			−1.18116
$518$		0			0
$519$	819.200			1.57842
$520$		0			0
$521$		−	1004.23i		−	1.92750i	−0.266807	−	0.963750i	$-0.585969\pi$
$521$		−	1004.23i		−	1.92750i	0.266807	−	0.963750i	$-0.414031\pi$
$522$		0			0
$523$			588.915i			1.12603i	0.826446	+	0.563016i	$0.190359\pi$
$523$			588.915i			1.12603i	−0.826446	+	0.563016i	$0.809641\pi$
$524$		0			0
$525$			8.28868i			0.0157880i
$526$		0			0
$527$		−	747.462i		−	1.41833i
$528$		0			0
$529$	1167.32			2.20666
$530$		0			0
$531$			234.445i			0.441516i
$532$		0			0
$533$	261.046			0.489767
$534$		0			0
$535$			37.2301i			0.0695891i
$536$		0			0
$537$	−586.672			−1.09250
$538$		0			0
$539$	688.203			1.27681
$540$		0			0
$541$	431.404			0.797420			0.398710	−	0.917077i	$-0.369458\pi$
$541$	431.404			0.797420			0.398710	+	0.917077i	$0.369458\pi$
$542$		0			0
$543$	−88.7353			−0.163417
$544$		0			0
$545$		−	76.9314i		−	0.141159i
$546$		0			0
$547$		−	257.771i		−	0.471244i	−0.971845	−	0.235622i	$-0.924287\pi$
$547$		−	257.771i		−	0.471244i	0.971845	−	0.235622i	$-0.0757127\pi$
$548$		0			0
$549$	126.684			0.230755
$550$		0			0
$551$	−410.958	−	786.794i	−0.745840	−	1.42794i
$552$		0			0
$553$			12.7701i			0.0230925i
$554$		0			0
$555$	−495.448			−0.892699
$556$		0			0
$557$	−616.514			−1.10685			−0.553424	−	0.832900i	$-0.686679\pi$
$557$	−616.514			−1.10685			−0.553424	+	0.832900i	$0.686679\pi$
$558$		0			0
$559$		−	364.651i		−	0.652327i
$560$		0			0
$561$			1099.58i			1.96003i
$562$		0			0
$563$			79.5657i			0.141324i	0.997500	+	0.0706622i	$0.0225113\pi$
$563$			79.5657i			0.141324i	−0.997500	+	0.0706622i	$0.977489\pi$
$564$		0			0
$565$			76.6720i			0.135703i
$566$		0			0
$567$	−46.9915			−0.0828775
$568$		0			0
$569$		−	118.993i		−	0.209127i	−0.994518	−	0.104563i	$-0.966655\pi$
$569$		−	118.993i		−	0.209127i	0.994518	−	0.104563i	$-0.0333445\pi$
$570$		0			0
$571$	−1004.19			−1.75865			−0.879323	−	0.476227i	$-0.842004\pi$
$571$	−1004.19			−1.75865			−0.879323	+	0.476227i	$0.842004\pi$
$572$		0			0
$573$		−	184.181i		−	0.321434i
$574$		0			0
$575$	−205.932			−0.358143
$576$		0			0
$577$	−795.249			−1.37825			−0.689123	−	0.724644i	$-0.742004\pi$
$577$	−795.249			−1.37825			−0.689123	+	0.724644i	$0.742004\pi$
$578$		0			0
$579$	248.755			0.429629
$580$		0			0
$581$	−15.4220			−0.0265439
$582$		0			0
$583$		−	1410.60i		−	2.41955i
$584$		0			0
$585$			87.1584i			0.148989i
$586$		0			0
$587$	192.221			0.327463			0.163732	−	0.986505i	$-0.447647\pi$
$587$	192.221			0.327463			0.163732	+	0.986505i	$0.447647\pi$
$588$		0			0
$589$	298.432	+	571.359i	0.506676	+	0.970049i
$590$		0			0
$591$		−	464.698i		−	0.786290i
$592$		0			0
$593$	−83.5456			−0.140886			−0.0704432	−	0.997516i	$-0.522441\pi$
$593$	−83.5456			−0.140886			−0.0704432	+	0.997516i	$0.522441\pi$
$594$		0			0
$595$	23.0860			0.0387999
$596$		0			0
$597$		−	427.924i		−	0.716791i
$598$		0			0
$599$			665.445i			1.11093i	0.831541	+	0.555463i	$0.187459\pi$
$599$			665.445i			1.11093i	−0.831541	+	0.555463i	$0.812541\pi$
$600$		0			0
$601$		−	648.842i		−	1.07960i	−0.841792	−	0.539802i	$-0.818499\pi$
$601$		−	648.842i		−	1.07960i	0.841792	−	0.539802i	$-0.181501\pi$
$602$		0			0
$603$		−	328.662i		−	0.545044i
$604$		0			0
$605$	−174.505			−0.288438
$606$		0			0
$607$		−	212.490i		−	0.350066i	−0.984563	−	0.175033i	$-0.943997\pi$
$607$		−	212.490i		−	0.350066i	0.984563	−	0.175033i	$-0.0560032\pi$
$608$		0			0
$609$	77.4472			0.127171
$610$		0			0
$611$			480.080i			0.785728i
$612$		0			0
$613$	878.692			1.43343			0.716715	−	0.697366i	$-0.245645\pi$
$613$	878.692			1.43343			0.716715	+	0.697366i	$0.245645\pi$
$614$		0			0
$615$	−186.174			−0.302722
$616$		0			0
$617$	−77.9272			−0.126300			−0.0631501	−	0.998004i	$-0.520115\pi$
$617$	−77.9272			−0.126300			−0.0631501	+	0.998004i	$0.520115\pi$
$618$		0			0
$619$	−744.679			−1.20304			−0.601518	−	0.798860i	$-0.705437\pi$
$619$	−744.679			−1.20304			−0.601518	+	0.798860i	$0.705437\pi$
$620$		0			0
$621$		−	799.264i		−	1.28706i
$622$		0			0
$623$			40.4725i			0.0649639i
$624$		0			0
$625$	25.0000			0.0400000
$626$		0			0
$627$	−439.017	−	840.514i	−0.700187	−	1.34053i
$628$		0			0
$629$			1379.94i			2.19387i
$630$		0			0
$631$	583.634			0.924934			0.462467	−	0.886636i	$-0.346964\pi$
$631$	583.634			0.924934			0.462467	+	0.886636i	$0.346964\pi$
$632$		0			0
$633$	1349.85			2.13247
$634$		0			0
$635$		−	549.225i		−	0.864921i
$636$		0			0
$637$		−	541.043i		−	0.849361i
$638$		0			0
$639$		−	437.200i		−	0.684194i
$640$		0			0
$641$			835.068i			1.30276i	0.758753	+	0.651379i	$0.225809\pi$
$641$			835.068i			1.30276i	−0.758753	+	0.651379i	$0.774191\pi$
$642$		0			0
$643$	−348.076			−0.541331			−0.270666	−	0.962673i	$-0.587244\pi$
$643$	−348.076			−0.541331			−0.270666	+	0.962673i	$0.587244\pi$
$644$		0			0
$645$			260.063i			0.403199i
$646$		0			0
$647$	−667.701			−1.03199			−0.515997	−	0.856590i	$-0.672579\pi$
$647$	−667.701			−1.03199			−0.515997	+	0.856590i	$0.672579\pi$
$648$		0			0
$649$			941.184i			1.45021i
$650$		0			0
$651$	−56.2411			−0.0863919
$652$		0			0
$653$	−913.560			−1.39902			−0.699510	−	0.714623i	$-0.746598\pi$
$653$	−913.560			−1.39902			−0.699510	+	0.714623i	$0.746598\pi$
$654$		0			0
$655$	−440.117			−0.671934
$656$		0			0
$657$	−145.260			−0.221096
$658$		0			0
$659$			654.040i			0.992474i	0.868187	+	0.496237i	$0.165285\pi$
$659$			654.040i			0.992474i	−0.868187	+	0.496237i	$0.834715\pi$
$660$		0			0
$661$		−	284.200i		−	0.429955i	−0.976619	−	0.214977i	$-0.931032\pi$
$661$		−	284.200i		−	0.429955i	0.976619	−	0.214977i	$-0.0689678\pi$
$662$		0			0
$663$	864.451			1.30385
$664$		0			0
$665$	−17.6469	+	9.21731i	−0.0265366	+	0.0138606i
$666$		0			0
$667$			1924.18i			2.88482i
$668$		0			0
$669$	−548.411			−0.819747
$670$		0			0
$671$	508.576			0.757938
$672$		0			0
$673$			652.742i			0.969899i	0.874542	+	0.484950i	$0.161162\pi$
$673$			652.742i			0.969899i	−0.874542	+	0.484950i	$0.838838\pi$
$674$		0			0
$675$			97.0300i			0.143748i
$676$		0			0
$677$			548.221i			0.809781i	0.914365	+	0.404890i	$0.132690\pi$
$677$			548.221i			0.809781i	−0.914365	+	0.404890i	$0.867310\pi$
$678$		0			0
$679$			9.75041i			0.0143600i
$680$		0			0
$681$	−840.462			−1.23416
$682$		0			0
$683$		−	183.490i		−	0.268653i	−0.990937	−	0.134327i	$-0.957113\pi$
$683$		−	183.490i		−	0.268653i	0.990937	−	0.134327i	$-0.0428872\pi$
$684$		0			0
$685$	−557.697			−0.814156
$686$		0			0
$687$		−	975.092i		−	1.41935i
$688$		0			0
$689$	−1108.97			−1.60953
$690$		0			0
$691$	−700.503			−1.01375			−0.506876	−	0.862019i	$-0.669200\pi$
$691$	−700.503			−1.01375			−0.506876	+	0.862019i	$0.669200\pi$
$692$		0			0
$693$	23.2339			0.0335265
$694$		0			0
$695$	−259.539			−0.373437
$696$		0			0
$697$			518.539i			0.743958i
$698$		0			0
$699$		−	960.766i		−	1.37449i
$700$		0			0
$701$	353.859			0.504791			0.252396	−	0.967624i	$-0.418782\pi$
$701$	353.859			0.504791			0.252396	+	0.967624i	$0.418782\pi$
$702$		0			0
$703$	−550.956	−	1054.83i	−0.783721	−	1.50046i
$704$		0			0
$705$		−	342.385i		−	0.485653i
$706$		0			0
$707$	−55.4214			−0.0783895
$708$		0			0
$709$	−592.019			−0.835006			−0.417503	−	0.908676i	$-0.637095\pi$
$709$	−592.019			−0.835006			−0.417503	+	0.908676i	$0.637095\pi$
$710$		0			0
$711$		−	95.7682i		−	0.134695i
$712$		0			0
$713$		−	1397.31i		−	1.95976i
$714$		0			0
$715$			349.899i			0.489369i
$716$		0			0
$717$		−	835.732i		−	1.16560i
$718$		0			0
$719$	385.615			0.536321			0.268160	−	0.963374i	$-0.413584\pi$
$719$	385.615			0.536321			0.268160	+	0.963374i	$0.413584\pi$
$720$		0			0
$721$			50.5515i			0.0701130i
$722$		0			0
$723$	−1224.55			−1.69371
$724$		0			0
$725$		−	233.593i		−	0.322198i
$726$		0			0
$727$	−1136.34			−1.56306			−0.781530	−	0.623867i	$-0.785561\pi$
$727$	−1136.34			−1.56306			−0.781530	+	0.623867i	$0.785561\pi$
$728$		0			0
$729$	−265.440			−0.364116
$730$		0			0
$731$	724.338			0.990887
$732$		0			0
$733$	−871.591			−1.18907			−0.594537	−	0.804068i	$-0.702665\pi$
$733$	−871.591			−1.18907			−0.594537	+	0.804068i	$0.702665\pi$
$734$		0			0
$735$			385.863i			0.524984i
$736$		0			0
$737$		−	1319.42i		−	1.79026i
$738$		0			0
$739$	605.269			0.819038			0.409519	−	0.912301i	$-0.365696\pi$
$739$	605.269			0.819038			0.409519	+	0.912301i	$0.365696\pi$
$740$		0			0
$741$	−660.785	+	345.141i	−0.891748	+	0.465778i
$742$		0			0
$743$			490.677i			0.660400i	0.943911	+	0.330200i	$0.107116\pi$
$743$			490.677i			0.660400i	−0.943911	+	0.330200i	$0.892884\pi$
$744$		0			0
$745$	494.122			0.663252
$746$		0			0
$747$	115.656			0.154827
$748$		0			0
$749$		−	7.80229i		−	0.0104169i
$750$		0			0
$751$			1075.66i			1.43230i	0.697948	+	0.716149i	$0.254097\pi$
$751$			1075.66i			1.43230i	−0.697948	+	0.716149i	$0.745903\pi$
$752$		0			0
$753$			786.530i			1.04453i
$754$		0			0
$755$			531.445i			0.703901i
$756$		0			0
$757$	−775.132			−1.02395			−0.511976	−	0.859000i	$-0.671086\pi$
$757$	−775.132			−1.02395			−0.511976	+	0.859000i	$0.671086\pi$
$758$		0			0
$759$			2055.55i			2.70824i
$760$		0			0
$761$	801.361			1.05304			0.526519	−	0.850164i	$-0.323497\pi$
$761$	801.361			1.05304			0.526519	+	0.850164i	$0.323497\pi$
$762$		0			0
$763$			16.1225i			0.0211303i
$764$		0			0
$765$	−173.130			−0.226314
$766$		0			0
$767$	739.929			0.964705
$768$		0			0
$769$	419.799			0.545902			0.272951	−	0.962028i	$-0.412000\pi$
$769$	419.799			0.545902			0.272951	+	0.962028i	$0.412000\pi$
$770$		0			0
$771$	506.692			0.657188
$772$		0			0
$773$			518.154i			0.670316i	0.942162	+	0.335158i	$0.108790\pi$
$773$			518.154i			0.670316i	−0.942162	+	0.335158i	$0.891210\pi$
$774$		0			0
$775$			169.632i			0.218880i
$776$		0			0
$777$	103.831			0.133630
$778$		0			0
$779$	−207.032	−	396.370i	−0.265766	−	0.508819i
$780$		0			0
$781$		−	1755.15i		−	2.24731i
$782$		0			0
$783$	906.622			1.15788
$784$		0			0
$785$	−78.9849			−0.100618
$786$		0			0
$787$		−	393.934i		−	0.500551i	−0.968175	−	0.250276i	$-0.919479\pi$
$787$		−	393.934i		−	0.500551i	0.968175	−	0.250276i	$-0.0805212\pi$
$788$		0			0
$789$		−	850.102i		−	1.07744i
$790$		0			0
$791$		−	16.0681i		−	0.0203136i
$792$		0			0
$793$		−	399.827i		−	0.504195i
$794$		0			0
$795$	790.898			0.994840
$796$		0			0
$797$			305.628i			0.383474i	0.981446	+	0.191737i	$0.0614120\pi$
$797$			305.628i			0.383474i	−0.981446	+	0.191737i	$0.938588\pi$
$798$		0			0
$799$	−953.625			−1.19352
$800$		0			0
$801$		−	303.519i		−	0.378925i
$802$		0			0
$803$	−583.149			−0.726213
$804$		0			0
$805$	43.1570			0.0536112
$806$		0			0
$807$	924.325			1.14538
$808$		0			0
$809$	1186.46			1.46657			0.733285	−	0.679921i	$-0.237986\pi$
$809$	1186.46			1.46657			0.733285	+	0.679921i	$0.237986\pi$
$810$		0			0
$811$		−	1581.99i		−	1.95067i	−0.220740	−	0.975333i	$-0.570847\pi$
$811$		−	1581.99i		−	1.95067i	0.220740	−	0.975333i	$-0.429153\pi$
$812$		0			0
$813$		−	830.719i		−	1.02180i
$814$		0			0
$815$	−159.704			−0.195955
$816$		0			0
$817$	−553.683	+	289.200i	−0.677703	+	0.353978i
$818$		0			0
$819$		−	18.2657i		−	0.0223025i
$820$		0			0
$821$	−95.9928			−0.116922			−0.0584609	−	0.998290i	$-0.518619\pi$
$821$	−95.9928			−0.116922			−0.0584609	+	0.998290i	$0.518619\pi$
$822$		0			0
$823$	1292.87			1.57093			0.785464	−	0.618908i	$-0.212424\pi$
$823$	1292.87			1.57093			0.785464	+	0.618908i	$0.212424\pi$
$824$		0			0
$825$		−	249.542i		−	0.302476i
$826$		0			0
$827$			44.7405i			0.0540998i	0.999634	+	0.0270499i	$0.00861130\pi$
$827$			44.7405i			0.0540998i	−0.999634	+	0.0270499i	$0.991389\pi$
$828$		0			0
$829$			802.912i			0.968530i	0.874921	+	0.484265i	$0.160913\pi$
$829$			802.912i			0.968530i	−0.874921	+	0.484265i	$0.839087\pi$
$830$		0			0
$831$		−	101.305i		−	0.121907i
$832$		0			0
$833$	1074.72			1.29018
$834$		0			0
$835$		−	413.793i		−	0.495561i
$836$		0			0
$837$	−658.377			−0.786591
$838$		0			0
$839$			346.934i			0.413509i	0.978393	+	0.206754i	$0.0662901\pi$
$839$			346.934i			0.413509i	−0.978393	+	0.206754i	$0.933710\pi$
$840$		0			0
$841$	−1341.63			−1.59528
$842$		0			0
$843$	−124.678			−0.147898
$844$		0			0
$845$	−102.816			−0.121676
$846$		0			0
$847$	36.5708			0.0431769
$848$		0			0
$849$		−	564.190i		−	0.664534i
$850$		0			0
$851$			2579.67i			3.03134i
$852$		0			0
$853$	766.503			0.898597			0.449298	−	0.893382i	$-0.351674\pi$
$853$	766.503			0.898597			0.449298	+	0.893382i	$0.351674\pi$
$854$		0			0
$855$	132.341	−	69.1242i	0.154784	−	0.0808470i
$856$		0			0
$857$			1701.81i			1.98577i	0.119063	+	0.992887i	$0.462011\pi$
$857$			1701.81i			1.98577i	−0.119063	+	0.992887i	$0.537989\pi$
$858$		0			0
$859$	436.740			0.508429			0.254214	−	0.967148i	$-0.418183\pi$
$859$	436.740			0.508429			0.254214	+	0.967148i	$0.418183\pi$
$860$		0			0
$861$	39.0163			0.0453151
$862$		0			0
$863$		−	555.776i		−	0.644004i	−0.946739	−	0.322002i	$-0.895644\pi$
$863$		−	555.776i		−	0.644004i	0.946739	−	0.322002i	$-0.104356\pi$
$864$		0			0
$865$			517.812i			0.598626i
$866$		0			0
$867$			694.782i			0.801364i
$868$		0			0
$869$		−	384.463i		−	0.442420i
$870$		0			0
$871$	−1037.28			−1.19091
$872$		0			0
$873$		−	73.1221i		−	0.0837595i
$874$		0			0
$875$	−5.23923			−0.00598769
$876$		0			0
$877$		−	506.241i		−	0.577242i	−0.957443	−	0.288621i	$-0.906803\pi$
$877$		−	506.241i		−	0.577242i	0.957443	−	0.288621i	$-0.0931967\pi$
$878$		0			0
$879$	282.342			0.321209
$880$		0			0
$881$	1208.80			1.37208			0.686038	−	0.727566i	$-0.259348\pi$
$881$	1208.80			1.37208			0.686038	+	0.727566i	$0.259348\pi$
$882$		0			0
$883$	221.477			0.250823			0.125412	−	0.992105i	$-0.459975\pi$
$883$	221.477			0.250823			0.125412	+	0.992105i	$0.459975\pi$
$884$		0			0
$885$	−527.705			−0.596277
$886$		0			0
$887$		−	1533.50i		−	1.72886i	−0.502753	−	0.864430i	$-0.667680\pi$
$887$		−	1533.50i		−	1.72886i	0.502753	−	0.864430i	$-0.332320\pi$
$888$		0			0
$889$			115.101i			0.129472i
$890$		0			0
$891$	1414.75			1.58782
$892$		0			0
$893$	728.949	−	380.745i	0.816292	−	0.426366i
$894$		0			0
$895$		−	370.832i		−	0.414338i
$896$		0			0
$897$	1616.01			1.80157
$898$		0			0
$899$	1585.00			1.76307
$900$		0			0
$901$		−	2202.84i		−	2.44488i
$902$		0			0
$903$		−	54.5012i		−	0.0603557i
$904$		0			0
$905$		−	56.0891i		−	0.0619769i
$906$		0			0
$907$		−	670.343i		−	0.739077i	−0.929215	−	0.369539i	$-0.879516\pi$
$907$		−	670.343i		−	0.739077i	0.929215	−	0.369539i	$-0.120484\pi$
$908$		0			0
$909$	415.626			0.457234
$910$		0			0
$911$			351.217i			0.385529i	0.981245	+	0.192765i	$0.0617454\pi$
$911$			351.217i			0.385529i	−0.981245	+	0.192765i	$0.938255\pi$
$912$		0			0
$913$	464.301			0.508545
$914$		0			0
$915$			285.150i			0.311639i
$916$		0			0
$917$	92.2348			0.100583
$918$		0			0
$919$	−701.167			−0.762967			−0.381484	−	0.924376i	$-0.624587\pi$
$919$	−701.167			−0.762967			−0.381484	+	0.924376i	$0.624587\pi$
$920$		0			0
$921$	480.139			0.521323
$922$		0			0
$923$	−1379.84			−1.49495
$924$		0			0
$925$		−	313.170i		−	0.338562i
$926$		0			0
$927$		−	379.105i		−	0.408959i
$928$		0			0
$929$	−1521.71			−1.63801			−0.819004	−	0.573787i	$-0.805474\pi$
$929$	−1521.71			−1.63801			−0.819004	+	0.573787i	$0.805474\pi$
$930$		0			0
$931$	−821.516	+	429.094i	−0.882401	+	0.460896i
$932$		0			0
$933$			465.898i			0.499355i
$934$		0			0
$935$	−695.035			−0.743353
$936$		0			0
$937$	1462.89			1.56125			0.780624	−	0.625001i	$-0.214902\pi$
$937$	1462.89			1.56125			0.780624	+	0.625001i	$0.214902\pi$
$938$		0			0
$939$			2016.54i			2.14754i
$940$		0			0
$941$			460.939i			0.489840i	0.969543	+	0.244920i	$0.0787617\pi$
$941$			460.939i			0.489840i	−0.969543	+	0.244920i	$0.921238\pi$
$942$		0			0
$943$			969.359i			1.02795i
$944$		0			0
$945$		−	20.3345i		−	0.0215180i
$946$		0			0
$947$	707.825			0.747439			0.373719	−	0.927542i	$-0.378082\pi$
$947$	707.825			0.747439			0.373719	+	0.927542i	$0.378082\pi$
$948$		0			0
$949$			458.453i			0.483091i
$950$		0			0
$951$	1181.90			1.24280
$952$		0			0
$953$			1553.32i			1.62993i	0.579511	+	0.814964i	$0.303243\pi$
$953$			1553.32i			1.62993i	−0.579511	+	0.814964i	$0.696757\pi$
$954$		0			0
$955$	116.420			0.121906
$956$		0			0
$957$	−2331.66			−2.43642
$958$		0			0
$959$	116.876			0.121873
$960$		0			0
$961$	−190.004			−0.197715
$962$		0			0
$963$			58.5124i			0.0607605i
$964$		0			0
$965$			157.237i			0.162940i
$966$		0			0
$967$	155.746			0.161061			0.0805307	−	0.996752i	$-0.474339\pi$
$967$	155.746			0.161061			0.0805307	+	0.996752i	$0.474339\pi$
$968$		0			0
$969$	−685.584	−	1312.58i	−0.707517	−	1.35457i
$970$		0			0
$971$			297.557i			0.306444i	0.988192	+	0.153222i	$0.0489649\pi$
$971$			297.557i			0.306444i	−0.988192	+	0.153222i	$0.951035\pi$
$972$		0			0
$973$	54.3913			0.0559006
$974$		0			0
$975$	−196.182			−0.201213
$976$		0			0
$977$			618.917i			0.633487i	0.948511	+	0.316744i	$0.102589\pi$
$977$			618.917i			0.633487i	−0.948511	+	0.316744i	$0.897411\pi$
$978$		0			0
$979$		−	1218.48i		−	1.24462i
$980$		0			0
$981$		−	120.908i		−	0.123250i
$982$		0			0
$983$		−	632.754i		−	0.643697i	−0.946791	−	0.321848i	$-0.895696\pi$
$983$		−	632.754i		−	0.643697i	0.946791	−	0.321848i	$-0.104304\pi$
$984$		0			0
$985$	293.733			0.298206
$986$		0			0
$987$			71.7534i			0.0726984i
$988$		0			0
$989$	1354.08			1.36914
$990$		0			0
$991$			1876.53i			1.89358i	0.321858	+	0.946788i	$0.395692\pi$
$991$			1876.53i			1.89358i	−0.321858	+	0.946788i	$0.604308\pi$
$992$		0			0
$993$	224.141			0.225721
$994$		0			0
$995$	270.489			0.271848
$996$		0			0
$997$	−270.045			−0.270857			−0.135429	−	0.990787i	$-0.543241\pi$
$997$	−270.045			−0.270857			−0.135429	+	0.990787i	$0.543241\pi$
$998$		0			0
$999$	1215.47			1.21669

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	380.3.e.a.341.11	yes	12
3.2	odd	2		3420.3.o.a.721.10		12
4.3	odd	2		1520.3.h.b.721.2		12
5.2	odd	4		1900.3.g.c.949.22		24
5.3	odd	4		1900.3.g.c.949.3		24
5.4	even	2		1900.3.e.f.1101.2		12
19.18	odd	2	inner	380.3.e.a.341.2	✓	12
57.56	even	2		3420.3.o.a.721.9		12
76.75	even	2		1520.3.h.b.721.11		12
95.18	even	4		1900.3.g.c.949.21		24
95.37	even	4		1900.3.g.c.949.4		24
95.94	odd	2		1900.3.e.f.1101.11		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
380.3.e.a.341.2	✓	12	19.18	odd	2	inner
380.3.e.a.341.11	yes	12	1.1	even	1	trivial
1520.3.h.b.721.2		12	4.3	odd	2
1520.3.h.b.721.11		12	76.75	even	2
1900.3.e.f.1101.2		12	5.4	even	2
1900.3.e.f.1101.11		12	95.94	odd	2
1900.3.g.c.949.3		24	5.3	odd	4
1900.3.g.c.949.4		24	95.37	even	4
1900.3.g.c.949.21		24	95.18	even	4
1900.3.g.c.949.22		24	5.2	odd	4
3420.3.o.a.721.9		12	57.56	even	2
3420.3.o.a.721.10		12	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	380.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.53755i q^{3} -2.23607 q^{5} +0.468611 q^{7} -3.51429 q^{9} +O(q^{10})\)	\(q+3.53755i q^{3} -2.23607 q^{5} +0.468611 q^{7} -3.51429 q^{9} -14.1082 q^{11} +11.0914i q^{13} -7.91021i q^{15} -22.0318 q^{17} +(16.8411 - 8.79644i) q^{19} +1.65774i q^{21} -41.1864 q^{23} +5.00000 q^{25} +19.4060i q^{27} -46.7187i q^{29} +33.9265i q^{31} -49.9085i q^{33} -1.04785 q^{35} -62.6340i q^{37} -39.2365 q^{39} -23.5359i q^{41} -32.8769 q^{43} +7.85820 q^{45} +43.2839 q^{47} -48.7804 q^{49} -77.9388i q^{51} +99.9844i q^{53} +31.5469 q^{55} +(31.1179 + 59.5763i) q^{57} -66.7119i q^{59} -36.0483 q^{61} -1.64684 q^{63} -24.8011i q^{65} +93.5215i q^{67} -145.699i q^{69} +124.406i q^{71} +41.3341 q^{73} +17.6878i q^{75} -6.61125 q^{77} +27.2511i q^{79} -100.278 q^{81} -32.9101 q^{83} +49.2647 q^{85} +165.270 q^{87} +86.3670i q^{89} +5.19755i q^{91} -120.017 q^{93} +(-37.6578 + 19.6694i) q^{95} +20.8071i q^{97} +49.5803 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{7} - 16 q^{9}+O(q^{10}) \) 12 * q - 12 * q^7 - 16 * q^9	\( 12 q - 12 q^{7} - 16 q^{9} - 32 q^{11} - 12 q^{17} + 24 q^{19} + 4 q^{23} + 60 q^{25} + 40 q^{35} + 124 q^{39} - 176 q^{43} - 40 q^{45} - 72 q^{47} - 24 q^{49} + 140 q^{57} + 152 q^{61} + 48 q^{63} - 148 q^{73} + 376 q^{77} - 468 q^{81} - 208 q^{83} - 84 q^{87} - 184 q^{93} + 392 q^{99}+O(q^{100}) \) 12 * q - 12 * q^7 - 16 * q^9 - 32 * q^11 - 12 * q^17 + 24 * q^19 + 4 * q^23 + 60 * q^25 + 40 * q^35 + 124 * q^39 - 176 * q^43 - 40 * q^45 - 72 * q^47 - 24 * q^49 + 140 * q^57 + 152 * q^61 + 48 * q^63 - 148 * q^73 + 376 * q^77 - 468 * q^81 - 208 * q^83 - 84 * q^87 - 184 * q^93 + 392 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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