LMFDB - Embedded newform 380.3.e.a.341.4

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			341.4
Root			$-3.14288i$ of defining polynomial
Character	$\chi$	$=$	380.341
Dual form			380.3.e.a.341.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.14288i q^{3} -2.23607 q^{5} -12.2192 q^{7} -0.877687 q^{9} +O(q^{10})$	$q-3.14288i q^{3} -2.23607 q^{5} -12.2192 q^{7} -0.877687 q^{9} +0.636018 q^{11} +19.9122i q^{13} +7.02769i q^{15} +20.7245 q^{17} +(-4.37892 + 18.4885i) q^{19} +38.4034i q^{21} -6.16098 q^{23} +5.00000 q^{25} -25.5274i q^{27} -11.9924i q^{29} +47.7875i q^{31} -1.99893i q^{33} +27.3229 q^{35} -19.1997i q^{37} +62.5817 q^{39} +22.7335i q^{41} -34.1742 q^{43} +1.96257 q^{45} -75.4198 q^{47} +100.308 q^{49} -65.1347i q^{51} +87.2484i q^{53} -1.42218 q^{55} +(58.1071 + 13.7624i) q^{57} +40.7398i q^{59} -10.2300 q^{61} +10.7246 q^{63} -44.5251i q^{65} +16.8833i q^{67} +19.3632i q^{69} +23.2877i q^{71} -103.642 q^{73} -15.7144i q^{75} -7.77162 q^{77} +123.657i q^{79} -88.1288 q^{81} -82.9966 q^{83} -46.3415 q^{85} -37.6908 q^{87} -34.0374i q^{89} -243.311i q^{91} +150.190 q^{93} +(9.79157 - 41.3416i) q^{95} -66.0861i q^{97} -0.558225 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$		−	3.14288i		−	1.04763i	−0.851833	−	0.523813i	$-0.824509\pi$
$3$		−	3.14288i		−	1.04763i	0.851833	−	0.523813i	$-0.175491\pi$
$4$		0			0
$5$	−2.23607			−0.447214
$5$	−2.23607			−0.447214
$6$		0			0
$7$	−12.2192			−1.74560			−0.872798	−	0.488081i	$-0.837697\pi$
$7$	−12.2192			−1.74560			−0.872798	+	0.488081i	$0.837697\pi$
$8$		0			0
$9$	−0.877687			−0.0975207
$10$		0			0
$11$	0.636018			0.0578199			0.0289099	−	0.999582i	$-0.490796\pi$
$11$	0.636018			0.0578199			0.0289099	+	0.999582i	$0.490796\pi$
$12$		0			0
$13$			19.9122i			1.53171i	0.643014	+	0.765855i	$0.277684\pi$
$13$			19.9122i			1.53171i	−0.643014	+	0.765855i	$0.722316\pi$
$14$		0			0
$15$			7.02769i			0.468513i
$16$		0			0
$17$	20.7245			1.21909			0.609546	−	0.792751i	$-0.291352\pi$
$17$	20.7245			1.21909			0.609546	+	0.792751i	$0.291352\pi$
$18$		0			0
$19$	−4.37892	+	18.4885i	−0.230470	+	0.973080i
$19$	−4.37892	+	18.4885i	−0.230470	+	0.973080i
$20$		0			0
$21$			38.4034i			1.82873i
$22$		0			0
$23$	−6.16098			−0.267869			−0.133934	−	0.990990i	$-0.542761\pi$
$23$	−6.16098			−0.267869			−0.133934	+	0.990990i	$0.542761\pi$
$24$		0			0
$25$	5.00000			0.200000
$26$		0			0
$27$		−	25.5274i		−	0.945461i
$28$		0			0
$29$		−	11.9924i		−	0.413532i	−0.978390	−	0.206766i	$-0.933706\pi$
$29$		−	11.9924i		−	0.413532i	0.978390	−	0.206766i	$-0.0662939\pi$
$30$		0			0
$31$			47.7875i			1.54153i	0.637118	+	0.770767i	$0.280127\pi$
$31$			47.7875i			1.54153i	−0.637118	+	0.770767i	$0.719873\pi$
$32$		0			0
$33$		−	1.99893i		−	0.0605736i
$34$		0			0
$35$	27.3229			0.780655
$36$		0			0
$37$		−	19.1997i		−	0.518910i	−0.965755	−	0.259455i	$-0.916457\pi$
$37$		−	19.1997i		−	0.518910i	0.965755	−	0.259455i	$-0.0835430\pi$
$38$		0			0
$39$	62.5817			1.60466
$40$		0			0
$41$			22.7335i			0.554475i	0.960801	+	0.277238i	$0.0894190\pi$
$41$			22.7335i			0.554475i	−0.960801	+	0.277238i	$0.910581\pi$
$42$		0			0
$43$	−34.1742			−0.794750			−0.397375	−	0.917656i	$-0.630079\pi$
$43$	−34.1742			−0.794750			−0.397375	+	0.917656i	$0.630079\pi$
$44$		0			0
$45$	1.96257			0.0436126
$46$		0			0
$47$	−75.4198			−1.60468			−0.802338	−	0.596870i	$-0.796411\pi$
$47$	−75.4198			−1.60468			−0.802338	+	0.596870i	$0.796411\pi$
$48$		0			0
$49$	100.308			2.04711
$50$		0			0
$51$		−	65.1347i		−	1.27715i
$52$		0			0
$53$			87.2484i			1.64620i	0.567900	+	0.823098i	$0.307756\pi$
$53$			87.2484i			1.64620i	−0.567900	+	0.823098i	$0.692244\pi$
$54$		0			0
$55$	−1.42218			−0.0258578
$56$		0			0
$57$	58.1071	+	13.7624i	1.01942	+	0.241446i
$58$		0			0
$59$			40.7398i			0.690505i	0.938510	+	0.345252i	$0.112207\pi$
$59$			40.7398i			0.690505i	−0.938510	+	0.345252i	$0.887793\pi$
$60$		0			0
$61$	−10.2300			−0.167705			−0.0838525	−	0.996478i	$-0.526722\pi$
$61$	−10.2300			−0.167705			−0.0838525	+	0.996478i	$0.526722\pi$
$62$		0			0
$63$	10.7246			0.170232
$64$		0			0
$65$		−	44.5251i		−	0.685001i
$66$		0			0
$67$			16.8833i			0.251989i	0.992031	+	0.125995i	$0.0402122\pi$
$67$			16.8833i			0.251989i	−0.992031	+	0.125995i	$0.959788\pi$
$68$		0			0
$69$			19.3632i			0.280626i
$70$		0			0
$71$			23.2877i			0.327996i	0.986461	+	0.163998i	$0.0524390\pi$
$71$			23.2877i			0.327996i	−0.986461	+	0.163998i	$0.947561\pi$
$72$		0			0
$73$	−103.642			−1.41975			−0.709875	−	0.704328i	$-0.751249\pi$
$73$	−103.642			−1.41975			−0.709875	+	0.704328i	$0.751249\pi$
$74$		0			0
$75$		−	15.7144i		−	0.209525i
$76$		0			0
$77$	−7.77162			−0.100930
$78$		0			0
$79$			123.657i			1.56528i	0.622473	+	0.782641i	$0.286128\pi$
$79$			123.657i			1.56528i	−0.622473	+	0.782641i	$0.713872\pi$
$80$		0			0
$81$	−88.1288			−1.08801
$82$		0			0
$83$	−82.9966			−0.999959			−0.499979	−	0.866037i	$-0.666659\pi$
$83$	−82.9966			−0.999959			−0.499979	+	0.866037i	$0.666659\pi$
$84$		0			0
$85$	−46.3415			−0.545194
$86$		0			0
$87$	−37.6908			−0.433227
$88$		0			0
$89$		−	34.0374i		−	0.382443i	−0.981547	−	0.191221i	$-0.938755\pi$
$89$		−	34.0374i		−	0.382443i	0.981547	−	0.191221i	$-0.0612448\pi$
$90$		0			0
$91$		−	243.311i		−	2.67375i
$92$		0			0
$93$	150.190			1.61495
$94$		0			0
$95$	9.79157	−	41.3416i	0.103069	−	0.435174i
$96$		0			0
$97$		−	66.0861i		−	0.681300i	−0.940190	−	0.340650i	$-0.889353\pi$
$97$		−	66.0861i		−	0.681300i	0.940190	−	0.340650i	$-0.110647\pi$
$98$		0			0
$99$	−0.558225			−0.00563864
$100$		0			0
$101$	39.1734			0.387855			0.193928	−	0.981016i	$-0.437877\pi$
$101$	39.1734			0.387855			0.193928	+	0.981016i	$0.437877\pi$
$102$		0			0
$103$		−	113.940i		−	1.10621i	−0.833112	−	0.553105i	$-0.813443\pi$
$103$		−	113.940i		−	1.10621i	0.833112	−	0.553105i	$-0.186557\pi$
$104$		0			0
$105$		−	85.8726i		−	0.817834i
$106$		0			0
$107$		−	86.3080i		−	0.806617i	−0.915064	−	0.403308i	$-0.867860\pi$
$107$		−	86.3080i		−	0.806617i	0.915064	−	0.403308i	$-0.132140\pi$
$108$		0			0
$109$			24.4905i			0.224683i	0.993670	+	0.112342i	$0.0358351\pi$
$109$			24.4905i			0.224683i	−0.993670	+	0.112342i	$0.964165\pi$
$110$		0			0
$111$	−60.3423			−0.543624
$112$		0			0
$113$			211.181i			1.86886i	0.356153	+	0.934428i	$0.384088\pi$
$113$			211.181i			1.86886i	−0.356153	+	0.934428i	$0.615912\pi$
$114$		0			0
$115$	13.7764			0.119794
$116$		0			0
$117$		−	17.4767i		−	0.149373i
$118$		0			0
$119$	−253.237			−2.12804
$120$		0			0
$121$	−120.595			−0.996657
$122$		0			0
$123$	71.4486			0.580883
$124$		0			0
$125$	−11.1803			−0.0894427
$126$		0			0
$127$		−	195.613i		−	1.54026i	−0.637885	−	0.770132i	$-0.720190\pi$
$127$		−	195.613i		−	1.54026i	0.637885	−	0.770132i	$-0.279810\pi$
$128$		0			0
$129$			107.405i			0.832600i
$130$		0			0
$131$	247.494			1.88927			0.944635	−	0.328123i	$-0.106416\pi$
$131$	247.494			1.88927			0.944635	+	0.328123i	$0.106416\pi$
$132$		0			0
$133$	53.5068	−	225.914i	0.402307	−	1.69860i
$134$		0			0
$135$			57.0811i			0.422823i
$136$		0			0
$137$	−51.5720			−0.376438			−0.188219	−	0.982127i	$-0.560272\pi$
$137$	−51.5720			−0.376438			−0.188219	+	0.982127i	$0.560272\pi$
$138$		0			0
$139$	114.447			0.823356			0.411678	−	0.911329i	$-0.364943\pi$
$139$	114.447			0.823356			0.411678	+	0.911329i	$0.364943\pi$
$140$		0			0
$141$			237.035i			1.68110i
$142$		0			0
$143$			12.6645i			0.0885632i
$144$		0			0
$145$			26.8159i			0.184937i
$146$		0			0
$147$		−	315.257i		−	2.14460i
$148$		0			0
$149$	141.259			0.948044			0.474022	−	0.880513i	$-0.342802\pi$
$149$	141.259			0.948044			0.474022	+	0.880513i	$0.342802\pi$
$150$		0			0
$151$		−	53.3624i		−	0.353394i	−0.984265	−	0.176697i	$-0.943459\pi$
$151$		−	53.3624i		−	0.353394i	0.984265	−	0.176697i	$-0.0565412\pi$
$152$		0			0
$153$	−18.1897			−0.118887
$154$		0			0
$155$		−	106.856i		−	0.689395i
$156$		0			0
$157$	−124.881			−0.795421			−0.397710	−	0.917511i	$-0.630195\pi$
$157$	−124.881			−0.795421			−0.397710	+	0.917511i	$0.630195\pi$
$158$		0			0
$159$	274.211			1.72460
$160$		0			0
$161$	75.2821			0.467590
$162$		0			0
$163$	9.62929			0.0590754			0.0295377	−	0.999564i	$-0.490596\pi$
$163$	9.62929			0.0590754			0.0295377	+	0.999564i	$0.490596\pi$
$164$		0			0
$165$			4.46974i			0.0270893i
$166$		0			0
$167$		−	225.797i		−	1.35208i	−0.736867	−	0.676038i	$-0.763696\pi$
$167$		−	225.797i		−	1.35208i	0.736867	−	0.676038i	$-0.236304\pi$
$168$		0			0
$169$	−227.497			−1.34613
$170$		0			0
$171$	3.84332	−	16.2271i	0.0224756	−	0.0948954i
$172$		0			0
$173$			227.218i			1.31340i	0.754153	+	0.656698i	$0.228048\pi$
$173$			227.218i			1.31340i	−0.754153	+	0.656698i	$0.771952\pi$
$174$		0			0
$175$	−61.0959			−0.349119
$176$		0			0
$177$	128.040			0.723391
$178$		0			0
$179$		−	146.908i		−	0.820714i	−0.911925	−	0.410357i	$-0.865404\pi$
$179$		−	146.908i		−	0.820714i	0.911925	−	0.410357i	$-0.134596\pi$
$180$		0			0
$181$			13.1154i			0.0724605i	0.999343	+	0.0362303i	$0.0115350\pi$
$181$			13.1154i			0.0724605i	−0.999343	+	0.0362303i	$0.988465\pi$
$182$		0			0
$183$			32.1517i			0.175692i
$184$		0			0
$185$			42.9318i			0.232064i
$186$		0			0
$187$	13.1812			0.0704877
$188$		0			0
$189$			311.924i			1.65039i
$190$		0			0
$191$	−1.77858			−0.00931196			−0.00465598	−	0.999989i	$-0.501482\pi$
$191$	−1.77858			−0.00931196			−0.00465598	+	0.999989i	$0.501482\pi$
$192$		0			0
$193$		−	214.837i		−	1.11314i	−0.830800	−	0.556571i	$-0.812117\pi$
$193$		−	214.837i		−	1.11314i	0.830800	−	0.556571i	$-0.187883\pi$
$194$		0			0
$195$	−139.937			−0.717625
$196$		0			0
$197$	−229.807			−1.16653			−0.583266	−	0.812281i	$-0.698225\pi$
$197$	−229.807			−1.16653			−0.583266	+	0.812281i	$0.698225\pi$
$198$		0			0
$199$	−192.242			−0.966040			−0.483020	−	0.875609i	$-0.660460\pi$
$199$	−192.242			−0.966040			−0.483020	+	0.875609i	$0.660460\pi$
$200$		0			0
$201$	53.0621			0.263991
$202$		0			0
$203$			146.538i			0.721861i
$204$		0			0
$205$		−	50.8336i		−	0.247969i
$206$		0			0
$207$	5.40741			0.0261227
$208$		0			0
$209$	−2.78508	+	11.7590i	−0.0133257	+	0.0562633i
$210$		0			0
$211$			108.302i			0.513280i	0.966507	+	0.256640i	$0.0826154\pi$
$211$			108.302i			0.513280i	−0.966507	+	0.256640i	$0.917385\pi$
$212$		0			0
$213$	73.1904			0.343617
$214$		0			0
$215$	76.4159			0.355423
$216$		0			0
$217$		−	583.924i		−	2.69089i
$218$		0			0
$219$			325.733i			1.48737i
$220$		0			0
$221$			412.672i			1.86729i
$222$		0			0
$223$			248.566i			1.11465i	0.830296	+	0.557323i	$0.188172\pi$
$223$			248.566i			1.11465i	−0.830296	+	0.557323i	$0.811828\pi$
$224$		0			0
$225$	−4.38843			−0.0195041
$226$		0			0
$227$			362.900i			1.59868i	0.600881	+	0.799338i	$0.294817\pi$
$227$			362.900i			1.59868i	−0.600881	+	0.799338i	$0.705183\pi$
$228$		0			0
$229$	−225.882			−0.986385			−0.493193	−	0.869920i	$-0.664170\pi$
$229$	−225.882			−0.986385			−0.493193	+	0.869920i	$0.664170\pi$
$230$		0			0
$231$			24.4253i			0.105737i
$232$		0			0
$233$	327.528			1.40570			0.702850	−	0.711338i	$-0.251911\pi$
$233$	327.528			1.40570			0.702850	+	0.711338i	$0.251911\pi$
$234$		0			0
$235$	168.644			0.717633
$236$		0			0
$237$	388.640			1.63983
$238$		0			0
$239$	−341.632			−1.42942			−0.714711	−	0.699420i	$-0.753442\pi$
$239$	−341.632			−1.42942			−0.714711	+	0.699420i	$0.753442\pi$
$240$		0			0
$241$			2.08810i			0.00866433i	0.999991	+	0.00433217i	$0.00137898\pi$
$241$			2.08810i			0.00866433i	−0.999991	+	0.00433217i	$0.998621\pi$
$242$		0			0
$243$			47.2313i			0.194367i
$244$		0			0
$245$	−224.296			−0.915494
$246$		0			0
$247$	−368.147	−	87.1941i	−1.49048	−	0.353012i
$248$		0			0
$249$			260.848i			1.04758i
$250$		0			0
$251$	20.1361			0.0802234			0.0401117	−	0.999195i	$-0.487229\pi$
$251$	20.1361			0.0802234			0.0401117	+	0.999195i	$0.487229\pi$
$252$		0			0
$253$	−3.91850			−0.0154881
$254$		0			0
$255$			145.646i			0.571160i
$256$		0			0
$257$			347.814i			1.35336i	0.736276	+	0.676682i	$0.236583\pi$
$257$			347.814i			1.35336i	−0.736276	+	0.676682i	$0.763417\pi$
$258$		0			0
$259$			234.604i			0.905808i
$260$		0			0
$261$			10.5256i			0.0403280i
$262$		0			0
$263$	486.729			1.85068			0.925340	−	0.379139i	$-0.123780\pi$
$263$	486.729			1.85068			0.925340	+	0.379139i	$0.123780\pi$
$264$		0			0
$265$		−	195.093i		−	0.736201i
$266$		0			0
$267$	−106.975			−0.400657
$268$		0			0
$269$			201.083i			0.747521i	0.927525	+	0.373760i	$0.121932\pi$
$269$			201.083i			0.747521i	−0.927525	+	0.373760i	$0.878068\pi$
$270$		0			0
$271$	207.523			0.765767			0.382884	−	0.923797i	$-0.374931\pi$
$271$	207.523			0.765767			0.382884	+	0.923797i	$0.374931\pi$
$272$		0			0
$273$	−764.697			−2.80109
$274$		0			0
$275$	3.18009			0.0115640
$276$		0			0
$277$	284.708			1.02783			0.513914	−	0.857842i	$-0.328195\pi$
$277$	284.708			1.02783			0.513914	+	0.857842i	$0.328195\pi$
$278$		0			0
$279$		−	41.9425i		−	0.150331i
$280$		0			0
$281$		−	301.030i		−	1.07128i	−0.844446	−	0.535640i	$-0.820070\pi$
$281$		−	301.030i		−	1.07128i	0.844446	−	0.535640i	$-0.179930\pi$
$282$		0			0
$283$	520.536			1.83935			0.919674	−	0.392683i	$-0.128453\pi$
$283$	520.536			1.83935			0.919674	+	0.392683i	$0.128453\pi$
$284$		0			0
$285$	−129.932	−	30.7737i	−0.455900	−	0.107978i
$286$		0			0
$287$		−	277.785i		−	0.967890i
$288$		0			0
$289$	140.507			0.486183
$290$		0			0
$291$	−207.701			−0.713748
$292$		0			0
$293$		−	42.8491i		−	0.146243i	−0.997323	−	0.0731214i	$-0.976704\pi$
$293$		−	42.8491i		−	0.146243i	0.997323	−	0.0731214i	$-0.0232961\pi$
$294$		0			0
$295$		−	91.0969i		−	0.308803i
$296$		0			0
$297$		−	16.2359i		−	0.0546664i
$298$		0			0
$299$		−	122.679i		−	0.410297i
$300$		0			0
$301$	417.581			1.38731
$302$		0			0
$303$		−	123.117i		−	0.406327i
$304$		0			0
$305$	22.8750			0.0750000
$306$		0			0
$307$			483.543i			1.57506i	0.616277	+	0.787530i	$0.288640\pi$
$307$			483.543i			1.57506i	−0.616277	+	0.787530i	$0.711360\pi$
$308$		0			0
$309$	−358.098			−1.15889
$310$		0			0
$311$	−163.546			−0.525870			−0.262935	−	0.964814i	$-0.584690\pi$
$311$	−163.546			−0.525870			−0.262935	+	0.964814i	$0.584690\pi$
$312$		0			0
$313$	394.650			1.26086			0.630432	−	0.776244i	$-0.282878\pi$
$313$	394.650			1.26086			0.630432	+	0.776244i	$0.282878\pi$
$314$		0			0
$315$	−23.9810			−0.0761300
$316$		0			0
$317$		−	245.190i		−	0.773471i	−0.922191	−	0.386736i	$-0.873603\pi$
$317$		−	245.190i		−	0.773471i	0.922191	−	0.386736i	$-0.126397\pi$
$318$		0			0
$319$		−	7.62741i		−	0.0239104i
$320$		0			0
$321$	−271.255			−0.845033
$322$		0			0
$323$	−90.7512	+	383.166i	−0.280963	+	1.18627i
$324$		0			0
$325$			99.5611i			0.306342i
$326$		0			0
$327$	76.9706			0.235384
$328$		0			0
$329$	921.568			2.80112
$330$		0			0
$331$		−	105.067i		−	0.317423i	−0.987325	−	0.158712i	$-0.949266\pi$
$331$		−	105.067i		−	0.317423i	0.987325	−	0.158712i	$-0.0507340\pi$
$332$		0			0
$333$			16.8513i			0.0506045i
$334$		0			0
$335$		−	37.7522i		−	0.112693i
$336$		0			0
$337$			69.6657i			0.206723i	0.994644	+	0.103362i	$0.0329599\pi$
$337$			69.6657i			0.206723i	−0.994644	+	0.103362i	$0.967040\pi$
$338$		0			0
$339$	663.715			1.95786
$340$		0			0
$341$			30.3937i			0.0891312i
$342$		0			0
$343$	−626.945			−1.82783
$344$		0			0
$345$		−	43.2974i		−	0.125500i
$346$		0			0
$347$	−103.244			−0.297532			−0.148766	−	0.988872i	$-0.547530\pi$
$347$	−103.244			−0.297532			−0.148766	+	0.988872i	$0.547530\pi$
$348$		0			0
$349$	542.273			1.55379			0.776896	−	0.629629i	$-0.216793\pi$
$349$	542.273			1.55379			0.776896	+	0.629629i	$0.216793\pi$
$350$		0			0
$351$	508.308			1.44817
$352$		0			0
$353$	−250.724			−0.710266			−0.355133	−	0.934816i	$-0.615564\pi$
$353$	−250.724			−0.710266			−0.355133	+	0.934816i	$0.615564\pi$
$354$		0			0
$355$		−	52.0729i		−	0.146684i
$356$		0			0
$357$			795.893i			2.22939i
$358$		0			0
$359$	−356.137			−0.992025			−0.496013	−	0.868315i	$-0.665203\pi$
$359$	−356.137			−0.992025			−0.496013	+	0.868315i	$0.665203\pi$
$360$		0			0
$361$	−322.650	−	161.920i	−0.893768	−	0.448530i
$362$		0			0
$363$			379.017i			1.04412i
$364$		0			0
$365$	231.750			0.634932
$366$		0			0
$367$	−247.991			−0.675725			−0.337863	−	0.941195i	$-0.609704\pi$
$367$	−247.991			−0.675725			−0.337863	+	0.941195i	$0.609704\pi$
$368$		0			0
$369$		−	19.9529i		−	0.0540729i
$370$		0			0
$371$		−	1066.10i		−	2.87359i
$372$		0			0
$373$		−	230.389i		−	0.617664i	−0.951117	−	0.308832i	$-0.900062\pi$
$373$		−	230.389i		−	0.617664i	0.951117	−	0.308832i	$-0.0999380\pi$
$374$		0			0
$375$			35.1385i			0.0937025i
$376$		0			0
$377$	238.796			0.633411
$378$		0			0
$379$		−	15.3995i		−	0.0406320i	−0.999794	−	0.0203160i	$-0.993533\pi$
$379$		−	15.3995i		−	0.0406320i	0.999794	−	0.0203160i	$-0.00646722\pi$
$380$		0			0
$381$	−614.789			−1.61362
$382$		0			0
$383$			697.645i			1.82153i	0.412928	+	0.910764i	$0.364506\pi$
$383$			697.645i			1.82153i	−0.412928	+	0.910764i	$0.635494\pi$
$384$		0			0
$385$	17.3779			0.0451373
$386$		0			0
$387$	29.9943			0.0775046
$388$		0			0
$389$	−190.689			−0.490203			−0.245101	−	0.969497i	$-0.578821\pi$
$389$	−190.689			−0.490203			−0.245101	+	0.969497i	$0.578821\pi$
$390$		0			0
$391$	−127.683			−0.326556
$392$		0			0
$393$		−	777.845i		−	1.97925i
$394$		0			0
$395$		−	276.506i		−	0.700016i
$396$		0			0
$397$	251.632			0.633835			0.316917	−	0.948453i	$-0.397352\pi$
$397$	251.632			0.633835			0.316917	+	0.948453i	$0.397352\pi$
$398$		0			0
$399$	−710.021	−	168.165i	−1.77950	−	0.421467i
$400$		0			0
$401$			644.306i			1.60675i	0.595475	+	0.803374i	$0.296964\pi$
$401$			644.306i			1.60675i	−0.595475	+	0.803374i	$0.703036\pi$
$402$		0			0
$403$	−951.556			−2.36118
$404$		0			0
$405$	197.062			0.486573
$406$		0			0
$407$		−	12.2114i		−	0.0300033i
$408$		0			0
$409$		−	216.145i		−	0.528472i	−0.964458	−	0.264236i	$-0.914880\pi$
$409$		−	216.145i		−	0.528472i	0.964458	−	0.264236i	$-0.0851198\pi$
$410$		0			0
$411$			162.085i			0.394367i
$412$		0			0
$413$		−	497.807i		−	1.20534i
$414$		0			0
$415$	185.586			0.447195
$416$		0			0
$417$		−	359.692i		−	0.862570i
$418$		0			0
$419$	165.246			0.394382			0.197191	−	0.980365i	$-0.436818\pi$
$419$	165.246			0.394382			0.197191	+	0.980365i	$0.436818\pi$
$420$		0			0
$421$		−	271.021i		−	0.643755i	−0.946781	−	0.321877i	$-0.895686\pi$
$421$		−	271.021i		−	0.643755i	0.946781	−	0.321877i	$-0.104314\pi$
$422$		0			0
$423$	66.1949			0.156489
$424$		0			0
$425$	103.623			0.243818
$426$		0			0
$427$	125.002			0.292745
$428$		0			0
$429$	39.8031			0.0927812
$430$		0			0
$431$			290.548i			0.674126i	0.941482	+	0.337063i	$0.109434\pi$
$431$			290.548i			0.674126i	−0.941482	+	0.337063i	$0.890566\pi$
$432$		0			0
$433$		−	546.395i		−	1.26188i	−0.775831	−	0.630941i	$-0.782669\pi$
$433$		−	546.395i		−	1.26188i	0.775831	−	0.630941i	$-0.217331\pi$
$434$		0			0
$435$	84.2791			0.193745
$436$		0			0
$437$	26.9784	−	113.907i	0.0617356	−	0.260657i
$438$		0			0
$439$		−	822.087i		−	1.87264i	−0.351153	−	0.936318i	$-0.614210\pi$
$439$		−	822.087i		−	1.87264i	0.351153	−	0.936318i	$-0.385790\pi$
$440$		0			0
$441$	−88.0392			−0.199635
$442$		0			0
$443$	−85.1709			−0.192259			−0.0961297	−	0.995369i	$-0.530646\pi$
$443$	−85.1709			−0.192259			−0.0961297	+	0.995369i	$0.530646\pi$
$444$		0			0
$445$			76.1100i			0.171034i
$446$		0			0
$447$		−	443.958i		−	0.993196i
$448$		0			0
$449$		−	535.082i		−	1.19172i	−0.803089	−	0.595860i	$-0.796811\pi$
$449$		−	535.082i		−	1.19172i	0.803089	−	0.595860i	$-0.203189\pi$
$450$		0			0
$451$			14.4589i			0.0320597i
$452$		0			0
$453$	−167.712			−0.370224
$454$		0			0
$455$			544.060i			1.19574i
$456$		0			0
$457$	−792.130			−1.73333			−0.866663	−	0.498894i	$-0.833740\pi$
$457$	−792.130			−1.73333			−0.866663	+	0.498894i	$0.833740\pi$
$458$		0			0
$459$		−	529.045i		−	1.15260i
$460$		0			0
$461$	−290.229			−0.629565			−0.314782	−	0.949164i	$-0.601932\pi$
$461$	−290.229			−0.629565			−0.314782	+	0.949164i	$0.601932\pi$
$462$		0			0
$463$	−139.806			−0.301957			−0.150978	−	0.988537i	$-0.548242\pi$
$463$	−139.806			−0.301957			−0.150978	+	0.988537i	$0.548242\pi$
$464$		0			0
$465$	−335.836			−0.722228
$466$		0			0
$467$	−252.356			−0.540377			−0.270189	−	0.962807i	$-0.587086\pi$
$467$	−252.356			−0.540377			−0.270189	+	0.962807i	$0.587086\pi$
$468$		0			0
$469$		−	206.300i		−	0.439872i
$470$		0			0
$471$			392.486i			0.833304i
$472$		0			0
$473$	−21.7354			−0.0459523
$474$		0			0
$475$	−21.8946	+	92.4426i	−0.0460939	+	0.194616i
$476$		0			0
$477$		−	76.5767i		−	0.160538i
$478$		0			0
$479$	425.333			0.887960			0.443980	−	0.896037i	$-0.353566\pi$
$479$	425.333			0.887960			0.443980	+	0.896037i	$0.353566\pi$
$480$		0			0
$481$	382.308			0.794820
$482$		0			0
$483$		−	236.602i		−	0.489860i
$484$		0			0
$485$			147.773i			0.304687i
$486$		0			0
$487$		−	359.736i		−	0.738678i	−0.929295	−	0.369339i	$-0.879584\pi$
$487$		−	359.736i		−	0.738678i	0.929295	−	0.369339i	$-0.120416\pi$
$488$		0			0
$489$		−	30.2637i		−	0.0618889i
$490$		0			0
$491$	−466.219			−0.949529			−0.474765	−	0.880113i	$-0.657467\pi$
$491$	−466.219			−0.949529			−0.474765	+	0.880113i	$0.657467\pi$
$492$		0			0
$493$		−	248.538i		−	0.504134i
$494$		0			0
$495$	1.24823			0.00252167
$496$		0			0
$497$		−	284.557i		−	0.572549i
$498$		0			0
$499$	−123.063			−0.246620			−0.123310	−	0.992368i	$-0.539351\pi$
$499$	−123.063			−0.246620			−0.123310	+	0.992368i	$0.539351\pi$
$500$		0			0
$501$	−709.651			−1.41647
$502$		0			0
$503$	794.478			1.57948			0.789740	−	0.613442i	$-0.210215\pi$
$503$	794.478			1.57948			0.789740	+	0.613442i	$0.210215\pi$
$504$		0			0
$505$	−87.5943			−0.173454
$506$		0			0
$507$			714.994i			1.41025i
$508$		0			0
$509$			39.8648i			0.0783199i	0.999233	+	0.0391599i	$0.0124682\pi$
$509$			39.8648i			0.0783199i	−0.999233	+	0.0391599i	$0.987532\pi$
$510$		0			0
$511$	1266.42			2.47831
$512$		0			0
$513$	471.964	+	111.783i	0.920009	+	0.217900i
$514$		0			0
$515$			254.777i			0.494712i
$516$		0			0
$517$	−47.9684			−0.0927822
$518$		0			0
$519$	714.118			1.37595
$520$		0			0
$521$			603.806i			1.15894i	0.814995	+	0.579468i	$0.196740\pi$
$521$			603.806i			1.15894i	−0.814995	+	0.579468i	$0.803260\pi$
$522$		0			0
$523$			461.654i			0.882703i	0.897334	+	0.441352i	$0.145501\pi$
$523$			461.654i			0.882703i	−0.897334	+	0.441352i	$0.854499\pi$
$524$		0			0
$525$			192.017i			0.365747i
$526$		0			0
$527$			990.375i			1.87927i
$528$		0			0
$529$	−491.042			−0.928246
$530$		0			0
$531$		−	35.7568i		−	0.0673385i
$532$		0			0
$533$	−452.674			−0.849295
$534$		0			0
$535$			192.990i			0.360730i
$536$		0			0
$537$	−461.713			−0.859801
$538$		0			0
$539$	63.7979			0.118363
$540$		0			0
$541$	747.259			1.38125			0.690627	−	0.723211i	$-0.257334\pi$
$541$	747.259			1.38125			0.690627	+	0.723211i	$0.257334\pi$
$542$		0			0
$543$	41.2200			0.0759115
$544$		0			0
$545$		−	54.7624i		−	0.100481i
$546$		0			0
$547$			210.917i			0.385588i	0.981239	+	0.192794i	$0.0617549\pi$
$547$			210.917i			0.385588i	−0.981239	+	0.192794i	$0.938245\pi$
$548$		0			0
$549$	8.97874			0.0163547
$550$		0			0
$551$	221.722	+	52.5139i	0.402400	+	0.0953066i
$552$		0			0
$553$		−	1510.99i		−	2.73235i
$554$		0			0
$555$	134.929			0.243116
$556$		0			0
$557$	770.359			1.38305			0.691525	−	0.722353i	$-0.256939\pi$
$557$	770.359			1.38305			0.691525	+	0.722353i	$0.256939\pi$
$558$		0			0
$559$		−	680.485i		−	1.21733i
$560$		0			0
$561$		−	41.4269i		−	0.0738447i
$562$		0			0
$563$			567.202i			1.00746i	0.863860	+	0.503731i	$0.168040\pi$
$563$			567.202i			1.00746i	−0.863860	+	0.503731i	$0.831960\pi$
$564$		0			0
$565$		−	472.214i		−	0.835778i
$566$		0			0
$567$	1076.86			1.89923
$568$		0			0
$569$		−	796.712i		−	1.40020i	−0.714046	−	0.700099i	$-0.753139\pi$
$569$		−	796.712i		−	1.40020i	0.714046	−	0.700099i	$-0.246861\pi$
$570$		0			0
$571$	−808.589			−1.41609			−0.708046	−	0.706166i	$-0.750423\pi$
$571$	−808.589			−1.41609			−0.708046	+	0.706166i	$0.750423\pi$
$572$		0			0
$573$			5.58987i			0.00975545i
$574$		0			0
$575$	−30.8049			−0.0535737
$576$		0			0
$577$	15.4977			0.0268591			0.0134295	−	0.999910i	$-0.495725\pi$
$577$	15.4977			0.0268591			0.0134295	+	0.999910i	$0.495725\pi$
$578$		0			0
$579$	−675.205			−1.16616
$580$		0			0
$581$	1014.15			1.74552
$582$		0			0
$583$			55.4916i			0.0951828i
$584$		0			0
$585$			39.0791i			0.0668018i
$586$		0			0
$587$	645.782			1.10014			0.550070	−	0.835119i	$-0.314601\pi$
$587$	645.782			1.10014			0.550070	+	0.835119i	$0.314601\pi$
$588$		0			0
$589$	−883.520	−	209.258i	−1.50003	−	0.355276i
$590$		0			0
$591$			722.255i			1.22209i
$592$		0			0
$593$	−974.473			−1.64329			−0.821647	−	0.569997i	$-0.806944\pi$
$593$	−974.473			−1.64329			−0.821647	+	0.569997i	$0.806944\pi$
$594$		0			0
$595$	566.255			0.951689
$596$		0			0
$597$			604.193i			1.01205i
$598$		0			0
$599$			853.712i			1.42523i	0.701556	+	0.712614i	$0.252489\pi$
$599$			853.712i			1.42523i	−0.701556	+	0.712614i	$0.747511\pi$
$600$		0			0
$601$		−	655.593i		−	1.09084i	−0.838164	−	0.545419i	$-0.816371\pi$
$601$		−	655.593i		−	1.09084i	0.838164	−	0.545419i	$-0.183629\pi$
$602$		0			0
$603$		−	14.8182i		−	0.0245742i
$604$		0			0
$605$	269.660			0.445718
$606$		0			0
$607$			17.3522i			0.0285868i	0.999898	+	0.0142934i	$0.00454988\pi$
$607$			17.3522i			0.0285868i	−0.999898	+	0.0142934i	$0.995450\pi$
$608$		0			0
$609$	460.550			0.756240
$610$		0			0
$611$		−	1501.78i		−	2.45790i
$612$		0			0
$613$	−562.093			−0.916955			−0.458477	−	0.888706i	$-0.651605\pi$
$613$	−562.093			−0.916955			−0.458477	+	0.888706i	$0.651605\pi$
$614$		0			0
$615$	−159.764			−0.259779
$616$		0			0
$617$	−376.298			−0.609883			−0.304941	−	0.952371i	$-0.598637\pi$
$617$	−376.298			−0.609883			−0.304941	+	0.952371i	$0.598637\pi$
$618$		0			0
$619$	402.209			0.649772			0.324886	−	0.945753i	$-0.394674\pi$
$619$	402.209			0.649772			0.324886	+	0.945753i	$0.394674\pi$
$620$		0			0
$621$			157.274i			0.253259i
$622$		0			0
$623$			415.909i			0.667591i
$624$		0			0
$625$	25.0000			0.0400000
$626$		0			0
$627$	36.9572	+	8.75315i	0.0589429	+	0.0139604i
$628$		0			0
$629$		−	397.905i		−	0.632599i
$630$		0			0
$631$	−1050.52			−1.66485			−0.832427	−	0.554135i	$-0.813049\pi$
$631$	−1050.52			−1.66485			−0.832427	+	0.554135i	$0.813049\pi$
$632$		0			0
$633$	340.380			0.537725
$634$		0			0
$635$			437.405i			0.688827i
$636$		0			0
$637$			1997.36i			3.13557i
$638$		0			0
$639$		−	20.4393i		−	0.0319864i
$640$		0			0
$641$		−	1010.76i		−	1.57685i	−0.615132	−	0.788424i	$-0.710897\pi$
$641$		−	1010.76i		−	1.57685i	0.615132	−	0.788424i	$-0.289103\pi$
$642$		0			0
$643$	546.019			0.849174			0.424587	−	0.905387i	$-0.360419\pi$
$643$	546.019			0.849174			0.424587	+	0.905387i	$0.360419\pi$
$644$		0			0
$645$		−	240.166i		−	0.372350i
$646$		0			0
$647$	120.783			0.186682			0.0933408	−	0.995634i	$-0.470245\pi$
$647$	120.783			0.186682			0.0933408	+	0.995634i	$0.470245\pi$
$648$		0			0
$649$			25.9112i			0.0399249i
$650$		0			0
$651$	−1835.20			−2.81905
$652$		0			0
$653$	−558.984			−0.856024			−0.428012	−	0.903773i	$-0.640786\pi$
$653$	−558.984			−0.856024			−0.428012	+	0.903773i	$0.640786\pi$
$654$		0			0
$655$	−553.414			−0.844907
$656$		0			0
$657$	90.9650			0.138455
$658$		0			0
$659$			932.855i			1.41556i	0.706432	+	0.707781i	$0.250304\pi$
$659$			932.855i			1.41556i	−0.706432	+	0.707781i	$0.749696\pi$
$660$		0			0
$661$			188.164i			0.284665i	0.989819	+	0.142332i	$0.0454602\pi$
$661$			188.164i			0.284665i	−0.989819	+	0.142332i	$0.954540\pi$
$662$		0			0
$663$	1296.98			1.95623
$664$		0			0
$665$	−119.645	+	505.160i	−0.179917	+	0.759639i
$666$		0			0
$667$			73.8851i			0.110772i
$668$		0			0
$669$	781.214			1.16773
$670$		0			0
$671$	−6.50647			−0.00969668
$672$		0			0
$673$			953.791i			1.41722i	0.705599	+	0.708612i	$0.250678\pi$
$673$			953.791i			1.41722i	−0.705599	+	0.708612i	$0.749322\pi$
$674$		0			0
$675$		−	127.637i		−	0.189092i
$676$		0			0
$677$			1189.20i			1.75657i	0.478140	+	0.878284i	$0.341311\pi$
$677$			1189.20i			1.75657i	−0.478140	+	0.878284i	$0.658689\pi$
$678$		0			0
$679$			807.518i			1.18928i
$680$		0			0
$681$	1140.55			1.67482
$682$		0			0
$683$		−	470.293i		−	0.688570i	−0.938865	−	0.344285i	$-0.888121\pi$
$683$		−	470.293i		−	0.688570i	0.938865	−	0.344285i	$-0.111879\pi$
$684$		0			0
$685$	115.319			0.168348
$686$		0			0
$687$			709.920i			1.03336i
$688$		0			0
$689$	−1737.31			−2.52149
$690$		0			0
$691$	707.606			1.02403			0.512016	−	0.858976i	$-0.328899\pi$
$691$	707.606			1.02403			0.512016	+	0.858976i	$0.328899\pi$
$692$		0			0
$693$	6.82105			0.00984278
$694$		0			0
$695$	−255.910			−0.368216
$696$		0			0
$697$			471.141i			0.675956i
$698$		0			0
$699$		−	1029.38i		−	1.47265i
$700$		0			0
$701$	−10.3658			−0.0147871			−0.00739356	−	0.999973i	$-0.502353\pi$
$701$	−10.3658			−0.0147871			−0.00739356	+	0.999973i	$0.502353\pi$
$702$		0			0
$703$	354.974	+	84.0739i	0.504941	+	0.119593i
$704$		0			0
$705$		−	530.027i		−	0.751811i
$706$		0			0
$707$	−478.666			−0.677039
$708$		0			0
$709$	419.639			0.591875			0.295937	−	0.955207i	$-0.404368\pi$
$709$	419.639			0.591875			0.295937	+	0.955207i	$0.404368\pi$
$710$		0			0
$711$		−	108.532i		−	0.152648i
$712$		0			0
$713$		−	294.418i		−	0.412928i
$714$		0			0
$715$		−	28.3188i		−	0.0396067i
$716$		0			0
$717$			1073.71i			1.49750i
$718$		0			0
$719$	−249.933			−0.347612			−0.173806	−	0.984780i	$-0.555607\pi$
$719$	−249.933			−0.347612			−0.173806	+	0.984780i	$0.555607\pi$
$720$		0			0
$721$			1392.25i			1.93100i
$722$		0			0
$723$	6.56266			0.00907698
$724$		0			0
$725$		−	59.9622i		−	0.0827065i
$726$		0			0
$727$	521.188			0.716902			0.358451	−	0.933549i	$-0.383305\pi$
$727$	521.188			0.716902			0.358451	+	0.933549i	$0.383305\pi$
$728$		0			0
$729$	−644.717			−0.884386
$730$		0			0
$731$	−708.245			−0.968872
$732$		0			0
$733$	806.985			1.10093			0.550467	−	0.834857i	$-0.314449\pi$
$733$	806.985			1.10093			0.550467	+	0.834857i	$0.314449\pi$
$734$		0			0
$735$			704.935i			0.959096i
$736$		0			0
$737$			10.7381i			0.0145700i
$738$		0			0
$739$	852.210			1.15319			0.576597	−	0.817029i	$-0.304380\pi$
$739$	852.210			1.15319			0.576597	+	0.817029i	$0.304380\pi$
$740$		0			0
$741$	−274.040	+	1157.04i	−0.369825	+	1.56146i
$742$		0			0
$743$		−	835.888i		−	1.12502i	−0.826792	−	0.562508i	$-0.809836\pi$
$743$		−	835.888i		−	1.12502i	0.826792	−	0.562508i	$-0.190164\pi$
$744$		0			0
$745$	−315.864			−0.423978
$746$		0			0
$747$	72.8450			0.0975167
$748$		0			0
$749$			1054.61i			1.40803i
$750$		0			0
$751$		−	219.641i		−	0.292465i	−0.989250	−	0.146232i	$-0.953285\pi$
$751$		−	219.641i		−	0.292465i	0.989250	−	0.146232i	$-0.0467147\pi$
$752$		0			0
$753$		−	63.2852i		−	0.0840441i
$754$		0			0
$755$			119.322i			0.158042i
$756$		0			0
$757$	−1097.01			−1.44915			−0.724576	−	0.689195i	$-0.757964\pi$
$757$	−1097.01			−1.44915			−0.724576	+	0.689195i	$0.757964\pi$
$758$		0			0
$759$			12.3154i			0.0162258i
$760$		0			0
$761$	637.660			0.837924			0.418962	−	0.908004i	$-0.362394\pi$
$761$	637.660			0.837924			0.418962	+	0.908004i	$0.362394\pi$
$762$		0			0
$763$		−	299.254i		−	0.392206i
$764$		0			0
$765$	40.6733			0.0531677
$766$		0			0
$767$	−811.220			−1.05765
$768$		0			0
$769$	−799.723			−1.03995			−0.519976	−	0.854181i	$-0.674059\pi$
$769$	−799.723			−1.03995			−0.519976	+	0.854181i	$0.674059\pi$
$770$		0			0
$771$	1093.14			1.41782
$772$		0			0
$773$			1229.63i			1.59072i	0.606138	+	0.795359i	$0.292718\pi$
$773$			1229.63i			1.59072i	−0.606138	+	0.795359i	$0.707282\pi$
$774$		0			0
$775$			238.938i			0.308307i
$776$		0			0
$777$	737.333			0.948948
$778$		0			0
$779$	−420.308	−	99.5482i	−0.539549	−	0.127790i
$780$		0			0
$781$			14.8114i			0.0189647i
$782$		0			0
$783$	−306.136			−0.390979
$784$		0			0
$785$	279.243			0.355723
$786$		0			0
$787$		−	211.715i		−	0.269016i	−0.990913	−	0.134508i	$-0.957055\pi$
$787$		−	211.715i		−	0.269016i	0.990913	−	0.134508i	$-0.0429453\pi$
$788$		0			0
$789$		−	1529.73i		−	1.93882i
$790$		0			0
$791$		−	2580.45i		−	3.26227i
$792$		0			0
$793$		−	203.702i		−	0.256875i
$794$		0			0
$795$	−613.154			−0.771263
$796$		0			0
$797$		−	708.469i		−	0.888920i	−0.895799	−	0.444460i	$-0.853396\pi$
$797$		−	708.469i		−	0.888920i	0.895799	−	0.444460i	$-0.146604\pi$
$798$		0			0
$799$	−1563.04			−1.95625
$800$		0			0
$801$			29.8742i			0.0372961i
$802$		0			0
$803$	−65.9181			−0.0820898
$804$		0			0
$805$	−168.336			−0.209113
$806$		0			0
$807$	631.980			0.783122
$808$		0			0
$809$	−784.699			−0.969961			−0.484981	−	0.874525i	$-0.661173\pi$
$809$	−784.699			−0.969961			−0.484981	+	0.874525i	$0.661173\pi$
$810$		0			0
$811$			586.588i			0.723290i	0.932316	+	0.361645i	$0.117785\pi$
$811$			586.588i			0.723290i	−0.932316	+	0.361645i	$0.882215\pi$
$812$		0			0
$813$		−	652.219i		−	0.802238i
$814$		0			0
$815$	−21.5317			−0.0264193
$816$		0			0
$817$	149.646	−	631.831i	0.183166	−	0.773354i
$818$		0			0
$819$			213.551i			0.260746i
$820$		0			0
$821$	1501.33			1.82866			0.914329	−	0.404972i	$-0.132719\pi$
$821$	1501.33			1.82866			0.914329	+	0.404972i	$0.132719\pi$
$822$		0			0
$823$	649.300			0.788943			0.394472	−	0.918908i	$-0.370928\pi$
$823$	649.300			0.788943			0.394472	+	0.918908i	$0.370928\pi$
$824$		0			0
$825$		−	9.99464i		−	0.0121147i
$826$		0			0
$827$		−	167.715i		−	0.202799i	−0.994846	−	0.101400i	$-0.967668\pi$
$827$		−	167.715i		−	0.202799i	0.994846	−	0.101400i	$-0.0323321\pi$
$828$		0			0
$829$		−	77.3323i		−	0.0932838i	−0.998912	−	0.0466419i	$-0.985148\pi$
$829$		−	77.3323i		−	0.0932838i	0.998912	−	0.0466419i	$-0.0148520\pi$
$830$		0			0
$831$		−	894.804i		−	1.07678i
$832$		0			0
$833$	2078.84			2.49561
$834$		0			0
$835$			504.897i			0.604667i
$836$		0			0
$837$	1219.89			1.45746
$838$		0			0
$839$		−	61.7289i		−	0.0735743i	−0.999323	−	0.0367872i	$-0.988288\pi$
$839$		−	61.7289i		−	0.0735743i	0.999323	−	0.0367872i	$-0.0117124\pi$
$840$		0			0
$841$	697.181			0.828991
$842$		0			0
$843$	−946.100			−1.12230
$844$		0			0
$845$	508.698			0.602009
$846$		0			0
$847$	1473.58			1.73976
$848$		0			0
$849$		−	1635.98i		−	1.92695i
$850$		0			0
$851$			118.289i			0.139000i
$852$		0			0
$853$	103.960			0.121876			0.0609381	−	0.998142i	$-0.480591\pi$
$853$	103.960			0.121876			0.0609381	+	0.998142i	$0.480591\pi$
$854$		0			0
$855$	−8.59393	+	36.2849i	−0.0100514	+	0.0424385i
$856$		0			0
$857$		−	733.977i		−	0.856449i	−0.903672	−	0.428225i	$-0.859139\pi$
$857$		−	733.977i		−	0.856449i	0.903672	−	0.428225i	$-0.140861\pi$
$858$		0			0
$859$	158.143			0.184102			0.0920508	−	0.995754i	$-0.470658\pi$
$859$	158.143			0.184102			0.0920508	+	0.995754i	$0.470658\pi$
$860$		0			0
$861$	−873.043			−1.01399
$862$		0			0
$863$			521.014i			0.603725i	0.953352	+	0.301862i	$0.0976083\pi$
$863$			521.014i			0.603725i	−0.953352	+	0.301862i	$0.902392\pi$
$864$		0			0
$865$		−	508.074i		−	0.587369i
$866$		0			0
$867$		−	441.596i		−	0.509338i
$868$		0			0
$869$			78.6483i			0.0905044i
$870$		0			0
$871$	−336.184			−0.385975
$872$		0			0
$873$			58.0029i			0.0664409i
$874$		0			0
$875$	136.615			0.156131
$876$		0			0
$877$			323.145i			0.368466i	0.982883	+	0.184233i	$0.0589802\pi$
$877$			323.145i			0.368466i	−0.982883	+	0.184233i	$0.941020\pi$
$878$		0			0
$879$	−134.670			−0.153208
$880$		0			0
$881$	−827.030			−0.938740			−0.469370	−	0.883002i	$-0.655519\pi$
$881$	−827.030			−0.938740			−0.469370	+	0.883002i	$0.655519\pi$
$882$		0			0
$883$	−1065.03			−1.20615			−0.603075	−	0.797685i	$-0.706058\pi$
$883$	−1065.03			−1.20615			−0.603075	+	0.797685i	$0.706058\pi$
$884$		0			0
$885$	−286.307			−0.323510
$886$		0			0
$887$		−	1471.28i		−	1.65871i	−0.558721	−	0.829356i	$-0.688708\pi$
$887$		−	1471.28i		−	1.65871i	0.558721	−	0.829356i	$-0.311292\pi$
$888$		0			0
$889$			2390.24i			2.68868i
$890$		0			0
$891$	−56.0516			−0.0629086
$892$		0			0
$893$	330.257	−	1394.40i	0.369829	−	1.56148i
$894$		0			0
$895$			328.496i			0.367034i
$896$		0			0
$897$	−385.564			−0.429838
$898$		0			0
$899$	573.089			0.637474
$900$		0			0
$901$			1808.18i			2.00686i
$902$		0			0
$903$		−	1312.41i		−	1.45338i
$904$		0			0
$905$		−	29.3268i		−	0.0324053i
$906$		0			0
$907$			1307.42i			1.44148i	0.693207	+	0.720738i	$0.256197\pi$
$907$			1307.42i			1.44148i	−0.693207	+	0.720738i	$0.743803\pi$
$908$		0			0
$909$	−34.3820			−0.0378239
$910$		0			0
$911$		−	64.9642i		−	0.0713108i	−0.999364	−	0.0356554i	$-0.988648\pi$
$911$		−	64.9642i		−	0.0713108i	0.999364	−	0.0356554i	$-0.0113519\pi$
$912$		0			0
$913$	−52.7873			−0.0578175
$914$		0			0
$915$		−	71.8933i		−	0.0785720i
$916$		0			0
$917$	−3024.18			−3.29790
$918$		0			0
$919$	315.153			0.342930			0.171465	−	0.985190i	$-0.445150\pi$
$919$	315.153			0.342930			0.171465	+	0.985190i	$0.445150\pi$
$920$		0			0
$921$	1519.72			1.65007
$922$		0			0
$923$	−463.710			−0.502394
$924$		0			0
$925$		−	95.9984i		−	0.103782i
$926$		0			0
$927$			100.003i			0.107878i
$928$		0			0
$929$	787.663			0.847861			0.423930	−	0.905695i	$-0.360650\pi$
$929$	787.663			0.847861			0.423930	+	0.905695i	$0.360650\pi$
$930$		0			0
$931$	−439.242	+	1854.55i	−0.471796	+	1.99200i
$932$		0			0
$933$			514.004i			0.550915i
$934$		0			0
$935$	−29.4740			−0.0315230
$936$		0			0
$937$	761.331			0.812520			0.406260	−	0.913758i	$-0.366833\pi$
$937$	761.331			0.812520			0.406260	+	0.913758i	$0.366833\pi$
$938$		0			0
$939$		−	1240.34i		−	1.32091i
$940$		0			0
$941$			38.1755i			0.0405691i	0.999794	+	0.0202845i	$0.00645721\pi$
$941$			38.1755i			0.0405691i	−0.999794	+	0.0202845i	$0.993543\pi$
$942$		0			0
$943$		−	140.061i		−	0.148527i
$944$		0			0
$945$		−	697.484i		−	0.738078i
$946$		0			0
$947$	802.970			0.847909			0.423954	−	0.905684i	$-0.360642\pi$
$947$	802.970			0.847909			0.423954	+	0.905684i	$0.360642\pi$
$948$		0			0
$949$		−	2063.74i		−	2.17464i
$950$		0			0
$951$	−770.604			−0.810309
$952$		0			0
$953$		−	606.643i		−	0.636562i	−0.947996	−	0.318281i	$-0.896895\pi$
$953$		−	606.643i		−	0.636562i	0.947996	−	0.318281i	$-0.103105\pi$
$954$		0			0
$955$	3.97703			0.00416443
$956$		0			0
$957$	−23.9720			−0.0250491
$958$		0			0
$959$	630.168			0.657109
$960$		0			0
$961$	−1322.65			−1.37632
$962$		0			0
$963$			75.7514i			0.0786618i
$964$		0			0
$965$			480.389i			0.497813i
$966$		0			0
$967$	−1086.46			−1.12354			−0.561770	−	0.827293i	$-0.689879\pi$
$967$	−1086.46			−1.12354			−0.561770	+	0.827293i	$0.689879\pi$
$968$		0			0
$969$	1204.24	+	285.220i	1.24277	+	0.294345i
$970$		0			0
$971$		−	1004.12i		−	1.03411i	−0.855952	−	0.517055i	$-0.827028\pi$
$971$		−	1004.12i		−	1.03411i	0.855952	−	0.517055i	$-0.172972\pi$
$972$		0			0
$973$	−1398.44			−1.43725
$974$		0			0
$975$	312.909			0.320932
$976$		0			0
$977$		−	908.123i		−	0.929501i	−0.885442	−	0.464751i	$-0.846144\pi$
$977$		−	908.123i		−	0.929501i	0.885442	−	0.464751i	$-0.153856\pi$
$978$		0			0
$979$		−	21.6484i		−	0.0221128i
$980$		0			0
$981$		−	21.4950i		−	0.0219113i
$982$		0			0
$983$			1162.31i			1.18242i	0.806519	+	0.591208i	$0.201349\pi$
$983$			1162.31i			1.18242i	−0.806519	+	0.591208i	$0.798651\pi$
$984$		0			0
$985$	513.863			0.521689
$986$		0			0
$987$		−	2896.38i		−	2.93452i
$988$		0			0
$989$	210.547			0.212888
$990$		0			0
$991$			157.641i			0.159072i	0.996832	+	0.0795361i	$0.0253439\pi$
$991$			157.641i			0.159072i	−0.996832	+	0.0795361i	$0.974656\pi$
$992$		0			0
$993$	−330.213			−0.332541
$994$		0			0
$995$	429.866			0.432026
$996$		0			0
$997$	987.181			0.990152			0.495076	−	0.868850i	$-0.335140\pi$
$997$	987.181			0.990152			0.495076	+	0.868850i	$0.335140\pi$
$998$		0			0
$999$	−490.119			−0.490609

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	380.3.e.a.341.4	✓	12
3.2	odd	2		3420.3.o.a.721.8		12
4.3	odd	2		1520.3.h.b.721.9		12
5.2	odd	4		1900.3.g.c.949.8		24
5.3	odd	4		1900.3.g.c.949.17		24
5.4	even	2		1900.3.e.f.1101.9		12
19.18	odd	2	inner	380.3.e.a.341.9	yes	12
57.56	even	2		3420.3.o.a.721.7		12
76.75	even	2		1520.3.h.b.721.4		12
95.18	even	4		1900.3.g.c.949.7		24
95.37	even	4		1900.3.g.c.949.18		24
95.94	odd	2		1900.3.e.f.1101.4		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
380.3.e.a.341.4	✓	12	1.1	even	1	trivial
380.3.e.a.341.9	yes	12	19.18	odd	2	inner
1520.3.h.b.721.4		12	76.75	even	2
1520.3.h.b.721.9		12	4.3	odd	2
1900.3.e.f.1101.4		12	95.94	odd	2
1900.3.e.f.1101.9		12	5.4	even	2
1900.3.g.c.949.7		24	95.18	even	4
1900.3.g.c.949.8		24	5.2	odd	4
1900.3.g.c.949.17		24	5.3	odd	4
1900.3.g.c.949.18		24	95.37	even	4
3420.3.o.a.721.7		12	57.56	even	2
3420.3.o.a.721.8		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)

\(f(q)\)	\(=\)	\(q-3.14288i q^{3} -2.23607 q^{5} -12.2192 q^{7} -0.877687 q^{9} +O(q^{10})\)	\(q-3.14288i q^{3} -2.23607 q^{5} -12.2192 q^{7} -0.877687 q^{9} +0.636018 q^{11} +19.9122i q^{13} +7.02769i q^{15} +20.7245 q^{17} +(-4.37892 + 18.4885i) q^{19} +38.4034i q^{21} -6.16098 q^{23} +5.00000 q^{25} -25.5274i q^{27} -11.9924i q^{29} +47.7875i q^{31} -1.99893i q^{33} +27.3229 q^{35} -19.1997i q^{37} +62.5817 q^{39} +22.7335i q^{41} -34.1742 q^{43} +1.96257 q^{45} -75.4198 q^{47} +100.308 q^{49} -65.1347i q^{51} +87.2484i q^{53} -1.42218 q^{55} +(58.1071 + 13.7624i) q^{57} +40.7398i q^{59} -10.2300 q^{61} +10.7246 q^{63} -44.5251i q^{65} +16.8833i q^{67} +19.3632i q^{69} +23.2877i q^{71} -103.642 q^{73} -15.7144i q^{75} -7.77162 q^{77} +123.657i q^{79} -88.1288 q^{81} -82.9966 q^{83} -46.3415 q^{85} -37.6908 q^{87} -34.0374i q^{89} -243.311i q^{91} +150.190 q^{93} +(9.79157 - 41.3416i) q^{95} -66.0861i q^{97} -0.558225 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{7} - 16 q^{9}+O(q^{10}) \) 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

Properties

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