LMFDB - Embedded newform 380.2.l.a.37.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [380,2,Mod(37,380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(380, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 1, 2]))

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

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

chi := DirichletCharacter("380.37");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 380 = 2^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	380.l (of order $4$, degree $2$, 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:	$3.03431527681$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	8.0.2702336256.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 9x^{6} + 56x^{4} + 225x^{2} + 625 $ x^8 + 9x^6 + 56x^4 + 225*x^2 + 625
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{2}\cdot 5 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			37.1
Root			$-1.52274 - 1.63746i$ of defining polynomial
Character	$\chi$	$=$	380.37
Dual form			380.2.l.a.113.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.13746 - 0.656712i) q^{5} +(3.52622 + 3.52622i) q^{7} -3.00000i q^{9} +O(q^{10})$	$q+(-2.13746 - 0.656712i) q^{5} +(3.52622 + 3.52622i) q^{7} -3.00000i q^{9} +2.15068 q^{11} +(3.98336 + 3.98336i) q^{17} +4.35890i q^{19} +(6.35890 - 6.35890i) q^{23} +(4.13746 + 2.80739i) q^{25} +(-5.22144 - 9.85286i) q^{35} +(-1.71019 + 1.71019i) q^{43} +(-1.97014 + 6.41238i) q^{45} +(-9.67690 - 9.67690i) q^{47} +17.8685i q^{49} +(-4.59698 - 1.41238i) q^{55} -15.1698 q^{61} +(10.5787 - 10.5787i) q^{63} +(6.91841 - 6.91841i) q^{73} +(7.58377 + 7.58377i) q^{77} -9.00000 q^{81} +(-3.64110 + 3.64110i) q^{83} +(-5.89834 - 11.1302i) q^{85} +(2.86254 - 9.31697i) q^{95} -6.45203i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{5} - 6 q^{7}+O(q^{10}) $ 8 * q - 2 * q^5 - 6 * q^7	$ 8 q - 2 q^{5} - 6 q^{7} + 14 q^{17} + 16 q^{23} + 18 q^{25} - 22 q^{35} + 2 q^{43} - 26 q^{47} - 18 q^{63} + 22 q^{73} + 26 q^{77} - 72 q^{81} - 64 q^{83} + 24 q^{85} + 38 q^{95}+O(q^{100}) $ 8 * q - 2 * q^5 - 6 * q^7 + 14 * q^17 + 16 * q^23 + 18 * q^25 - 22 * q^35 + 2 * q^43 - 26 * q^47 - 18 * q^63 + 22 * q^73 + 26 * q^77 - 72 * q^81 - 64 * q^83 + 24 * q^85 + 38 * q^95

Display 100 coefficients 10 coefficients

Character values

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

$n$	$21$	$77$	$191$
$\chi(n)$	$-1$	$e\left(\frac{1}{4}\right)$	$1$

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
$3$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$3$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$4$		0			0
$5$	−2.13746	−	0.656712i	−0.955901	−	0.293691i
$5$	−2.13746	−	0.656712i	−0.955901	−	0.293691i
$6$		0			0
$7$	3.52622	+	3.52622i	1.33279	+	1.33279i	0.902867	+	0.429919i	$0.141458\pi$
$7$	3.52622	+	3.52622i	1.33279	+	1.33279i	0.429919	+	0.902867i	$0.358542\pi$
$8$		0			0
$9$		−	3.00000i		−	1.00000i
$10$		0			0
$11$	2.15068			0.648454			0.324227	−	0.945979i	$-0.394896\pi$
$11$	2.15068			0.648454			0.324227	+	0.945979i	$0.394896\pi$
$12$		0			0
$13$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$13$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.98336	+	3.98336i	0.966106	+	0.966106i	0.999444	−	0.0333386i	$-0.0106140\pi$
$17$	3.98336	+	3.98336i	0.966106	+	0.966106i	−0.0333386	+	0.999444i	$0.510614\pi$
$18$		0			0
$19$			4.35890i			1.00000i
$19$			4.35890i			1.00000i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	6.35890	−	6.35890i	1.32592	−	1.32592i	0.417029	−	0.908893i	$-0.363071\pi$
$23$	6.35890	−	6.35890i	1.32592	−	1.32592i	0.908893	−	0.417029i	$-0.136929\pi$
$24$		0			0
$25$	4.13746	+	2.80739i	0.827492	+	0.561478i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−5.22144	−	9.85286i	−0.882585	−	1.66544i
$36$		0			0
$37$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$37$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$	−1.71019	+	1.71019i	−0.260801	+	0.260801i	−0.825380	−	0.564578i	$-0.809039\pi$
$43$	−1.71019	+	1.71019i	−0.260801	+	0.260801i	0.564578	+	0.825380i	$0.309039\pi$
$44$		0			0
$45$	−1.97014	+	6.41238i	−0.293691	+	0.955901i
$46$		0			0
$47$	−9.67690	−	9.67690i	−1.41152	−	1.41152i	−0.749375	−	0.662145i	$-0.769646\pi$
$47$	−9.67690	−	9.67690i	−1.41152	−	1.41152i	−0.662145	−	0.749375i	$-0.730354\pi$
$48$		0			0
$49$			17.8685i			2.55264i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$53$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$54$		0			0
$55$	−4.59698	−	1.41238i	−0.619857	−	0.190445i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$	−15.1698			−1.94230			−0.971149	−	0.238474i	$-0.923353\pi$
$61$	−15.1698			−1.94230			−0.971149	+	0.238474i	$0.923353\pi$
$62$		0			0
$63$	10.5787	−	10.5787i	1.33279	−	1.33279i
$64$		0			0
$65$		0			0
$66$		0			0
$67$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$67$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	6.91841	−	6.91841i	0.809739	−	0.809739i	−0.174855	−	0.984594i	$-0.555946\pi$
$73$	6.91841	−	6.91841i	0.809739	−	0.809739i	0.984594	+	0.174855i	$0.0559458\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	7.58377	+	7.58377i	0.864250	+	0.864250i
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$	−9.00000			−1.00000
$82$		0			0
$83$	−3.64110	+	3.64110i	−0.399663	+	0.399663i	−0.878114	−	0.478451i	$-0.841198\pi$
$83$	−3.64110	+	3.64110i	−0.399663	+	0.399663i	0.478451	+	0.878114i	$0.341198\pi$
$84$		0			0
$85$	−5.89834	−	11.1302i	−0.639765	−	1.20724i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	2.86254	−	9.31697i	0.293691	−	0.955901i
$96$		0			0
$97$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$97$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$98$		0			0
$99$		−	6.45203i		−	0.648454i
$100$		0			0
$101$	−17.4356			−1.73491			−0.867453	−	0.497519i	$-0.834245\pi$
$101$	−17.4356			−1.73491			−0.867453	+	0.497519i	$0.834245\pi$
$102$		0			0
$103$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$103$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$107$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$108$		0			0
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$113$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$114$		0			0
$115$	−17.7678	+	9.41592i	−1.65686	+	0.878039i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			28.0924i			2.57522i
$120$		0			0
$121$	−6.37459			−0.579508
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−7.00000	−	8.71780i	−0.626099	−	0.779744i
$126$		0			0
$127$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$127$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	15.3746			1.34328			0.671642	−	0.740876i	$-0.265589\pi$
$131$	15.3746			1.34328			0.671642	+	0.740876i	$0.265589\pi$
$132$		0			0
$133$	−15.3704	+	15.3704i	−1.33279	+	1.33279i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	7.18648	+	7.18648i	0.613982	+	0.613982i	0.943981	−	0.329999i	$-0.107048\pi$
$137$	7.18648	+	7.18648i	0.613982	+	0.613982i	−0.329999	+	0.943981i	$0.607048\pi$
$138$		0			0
$139$		−	23.3746i		−	1.98261i	−0.131597	−	0.991303i	$-0.542011\pi$
$139$		−	23.3746i		−	1.98261i	0.131597	−	0.991303i	$-0.457989\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	13.3746i		−	1.09569i	−0.836580	−	0.547844i	$-0.815449\pi$
$149$		−	13.3746i		−	1.09569i	0.836580	−	0.547844i	$-0.184551\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$	11.9501	−	11.9501i	0.966106	−	0.966106i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	17.7178	+	17.7178i	1.41403	+	1.41403i	0.718278	+	0.695756i	$0.244931\pi$
$157$	17.7178	+	17.7178i	1.41403	+	1.41403i	0.695756	+	0.718278i	$0.255069\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	44.8458			3.53434
$162$		0			0
$163$	16.3589	−	16.3589i	1.28133	−	1.28133i	0.341415	−	0.939913i	$-0.389094\pi$
$163$	16.3589	−	16.3589i	1.28133	−	1.28133i	0.939913	−	0.341415i	$-0.110906\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$167$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$168$		0			0
$169$		−	13.0000i		−	1.00000i
$170$		0			0
$171$	13.0767			1.00000
$172$		0			0
$173$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$173$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$174$		0			0
$175$	4.69012	+	24.4891i	0.354540	+	1.85120i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	8.56691	+	8.56691i	0.626475	+	0.626475i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−27.3746			−1.98076			−0.990378	−	0.138390i	$-0.955807\pi$
$191$	−27.3746			−1.98076			−0.990378	+	0.138390i	$0.955807\pi$
$192$		0			0
$193$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$193$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−2.28220	−	2.28220i	−0.162600	−	0.162600i	0.621117	−	0.783718i	$-0.286679\pi$
$197$	−2.28220	−	2.28220i	−0.162600	−	0.162600i	−0.783718	+	0.621117i	$0.786679\pi$
$198$		0			0
$199$			28.1890i			1.99826i	0.0416556	+	0.999132i	$0.486737\pi$
$199$			28.1890i			1.99826i	−0.0416556	+	0.999132i	$0.513263\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−19.0767	−	19.0767i	−1.32592	−	1.32592i
$208$		0			0
$209$			9.37459i			0.648454i
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	4.77856	−	2.53236i	0.325895	−	0.172705i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$223$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$224$		0			0
$225$	8.42217	−	12.4124i	0.561478	−	0.827492i
$226$		0			0
$227$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$227$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$228$		0			0
$229$			29.3746i			1.94113i	0.240845	+	0.970564i	$0.422576\pi$
$229$			29.3746i			1.94113i	−0.240845	+	0.970564i	$0.577424\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−21.0233	+	21.0233i	−1.37728	+	1.37728i	−0.528099	+	0.849183i	$0.677095\pi$
$233$	−21.0233	+	21.0233i	−1.37728	+	1.37728i	−0.849183	+	0.528099i	$0.822905\pi$
$234$		0			0
$235$	14.3290	+	27.0389i	0.934723	+	1.76382i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	19.5863i		−	1.26693i	−0.773771	−	0.633465i	$-0.781632\pi$
$239$		−	19.5863i		−	1.26693i	0.773771	−	0.633465i	$-0.218368\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	11.7344	−	38.1931i	0.749686	−	2.44007i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−7.37459			−0.465480			−0.232740	−	0.972539i	$-0.574769\pi$
$251$	−7.37459			−0.465480			−0.232740	+	0.972539i	$0.574769\pi$
$252$		0			0
$253$	13.6759	−	13.6759i	0.859799	−	0.859799i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$257$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	13.0973	−	13.0973i	0.807613	−	0.807613i	−0.176659	−	0.984272i	$-0.556529\pi$
$263$	13.0973	−	13.0973i	0.807613	−	0.807613i	0.984272	+	0.176659i	$0.0565291\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$	26.1534			1.58871			0.794353	−	0.607457i	$-0.207810\pi$
$271$	26.1534			1.58871			0.794353	+	0.607457i	$0.207810\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	8.89834	+	6.03779i	0.536590	+	0.364092i
$276$		0			0
$277$	−23.3372	−	23.3372i	−1.40219	−	1.40219i	−0.793093	−	0.609101i	$-0.791530\pi$
$277$	−23.3372	−	23.3372i	−1.40219	−	1.40219i	−0.609101	−	0.793093i	$-0.708470\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$	3.25816	−	3.25816i	0.193677	−	0.193677i	−0.603606	−	0.797283i	$-0.706270\pi$
$283$	3.25816	−	3.25816i	0.193677	−	0.193677i	0.797283	+	0.603606i	$0.206270\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$			14.7342i			0.866720i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$293$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−12.0610			−0.695185
$302$		0			0
$303$		0			0
$304$		0			0
$305$	32.4249	+	9.96221i	1.85664	+	0.570435i
$306$		0			0
$307$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$307$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−32.4903			−1.84236			−0.921179	−	0.389139i	$-0.872773\pi$
$311$	−32.4903			−1.84236			−0.921179	+	0.389139i	$0.872773\pi$
$312$		0			0
$313$	−20.4356	+	20.4356i	−1.15509	+	1.15509i	−0.169570	+	0.985518i	$0.554238\pi$
$313$	−20.4356	+	20.4356i	−1.15509	+	1.15509i	−0.985518	+	0.169570i	$0.945762\pi$
$314$		0			0
$315$	−29.5586	+	15.6643i	−1.66544	+	0.882585i
$316$		0			0
$317$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$317$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−17.3630	+	17.3630i	−0.966106	+	0.966106i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	68.2458i		−	3.76251i
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$337$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−38.3247	+	38.3247i	−2.06934	+	2.06934i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−3.79429	−	3.79429i	−0.203688	−	0.203688i	0.597890	−	0.801578i	$-0.296006\pi$
$347$	−3.79429	−	3.79429i	−0.203688	−	0.203688i	−0.801578	+	0.597890i	$0.796006\pi$
$348$		0			0
$349$		−	23.7725i		−	1.27251i	−0.771477	−	0.636257i	$-0.780482\pi$
$349$		−	23.7725i		−	1.27251i	0.771477	−	0.636257i	$-0.219518\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−10.4356	+	10.4356i	−0.555431	+	0.555431i	−0.928003	−	0.372572i	$-0.878476\pi$
$353$	−10.4356	+	10.4356i	−0.555431	+	0.555431i	0.372572	+	0.928003i	$0.378476\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	3.37459i		−	0.178104i	−0.996027	−	0.0890519i	$-0.971616\pi$
$359$		−	3.37459i		−	0.178104i	0.996027	−	0.0890519i	$-0.0283837\pi$
$360$		0			0
$361$	−19.0000			−1.00000
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−19.3312	+	10.2444i	−1.01184	+	0.536217i
$366$		0			0
$367$	0.923303	+	0.923303i	0.0481960	+	0.0481960i	0.730794	−	0.682598i	$-0.239150\pi$
$367$	0.923303	+	0.923303i	0.0481960	+	0.0481960i	−0.682598	+	0.730794i	$0.739150\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$373$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$383$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$384$		0			0
$385$	−11.2296	−	21.1903i	−0.572315	−	1.07996i
$386$		0			0
$387$	5.13057	+	5.13057i	0.260801	+	0.260801i
$388$		0			0
$389$		−	36.9068i		−	1.87125i	−0.352998	−	0.935624i	$-0.614838\pi$
$389$		−	36.9068i		−	1.87125i	0.352998	−	0.935624i	$-0.385162\pi$
$390$		0			0
$391$	50.6595			2.56196
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−28.0757	−	28.0757i	−1.40908	−	1.40908i	−0.764730	−	0.644351i	$-0.777127\pi$
$397$	−28.0757	−	28.0757i	−1.40908	−	1.40908i	−0.644351	−	0.764730i	$-0.722873\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	19.2371	+	5.91041i	0.955901	+	0.293691i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0			−	1.00000i	$-0.5\pi$
$409$		0			0				1.00000i	$0.5\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	10.1739	−	5.39155i	0.499415	−	0.264661i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			8.71780i			0.425892i	0.977064	+	0.212946i	$0.0683059\pi$
$419$			8.71780i			0.425892i	−0.977064	+	0.212946i	$0.931694\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$	−29.0307	+	29.0307i	−1.41152	+	1.41152i
$424$		0			0
$425$	5.29814	+	27.6638i	0.256997	+	1.34189i
$426$		0			0
$427$	−53.4922	−	53.4922i	−2.58867	−	2.58867i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		0			0
$433$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$433$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	27.7178	+	27.7178i	1.32592	+	1.32592i
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	53.6054			2.55264
$442$		0			0
$443$	−14.2232	+	14.2232i	−0.675766	+	0.675766i	−0.959039	−	0.283273i	$-0.908580\pi$
$443$	−14.2232	+	14.2232i	−0.675766	+	0.675766i	0.283273	+	0.959039i	$0.408580\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	18.7908	+	18.7908i	0.878998	+	0.878998i	0.993431	−	0.114433i	$-0.0365053\pi$
$457$	18.7908	+	18.7908i	0.878998	+	0.878998i	−0.114433	+	0.993431i	$0.536505\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−37.3746			−1.74071			−0.870354	−	0.492427i	$-0.836110\pi$
$461$	−37.3746			−1.74071			−0.870354	+	0.492427i	$0.836110\pi$
$462$		0			0
$463$	25.6103	−	25.6103i	1.19021	−	1.19021i	0.213205	−	0.977007i	$-0.431610\pi$
$463$	25.6103	−	25.6103i	1.19021	−	1.19021i	0.977007	−	0.213205i	$-0.0683902\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	10.8467	+	10.8467i	0.501927	+	0.501927i	0.912036	−	0.410110i	$-0.134510\pi$
$467$	10.8467	+	10.8467i	0.501927	+	0.501927i	−0.410110	+	0.912036i	$0.634510\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−3.67806	+	3.67806i	−0.169118	+	0.169118i
$474$		0			0
$475$	−12.2371	+	18.0348i	−0.561478	+	0.827492i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			4.00000i			0.182765i	0.995816	+	0.0913823i	$0.0291285\pi$
$479$			4.00000i			0.182765i	−0.995816	+	0.0913823i	$0.970871\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$487$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−8.00000			−0.361035			−0.180517	−	0.983572i	$-0.557777\pi$
$491$	−8.00000			−0.361035			−0.180517	+	0.983572i	$0.557777\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	−4.23713	+	13.7910i	−0.190445	+	0.619857i
$496$		0			0
$497$		0			0
$498$		0			0
$499$		−	0.625414i		−	0.0279974i	−0.999902	−	0.0139987i	$-0.995544\pi$
$499$		−	0.625414i		−	0.0279974i	0.999902	−	0.0139987i	$-0.00445607\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	26.3589	−	26.3589i	1.17529	−	1.17529i	0.194354	−	0.980932i	$-0.437739\pi$
$503$	26.3589	−	26.3589i	1.17529	−	1.17529i	0.980932	−	0.194354i	$-0.0622609\pi$
$504$		0			0
$505$	37.2679	+	11.4502i	1.65840	+	0.509526i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$	48.7917			2.15842
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−20.8119	−	20.8119i	−0.915306	−	0.915306i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$523$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$		−	57.8712i		−	2.51614i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$			38.4293i			1.65527i
$540$		0			0
$541$	−2.03559			−0.0875168			−0.0437584	−	0.999042i	$-0.513933\pi$
$541$	−2.03559			−0.0875168			−0.0437584	+	0.999042i	$0.513933\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$547$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$548$		0			0
$549$			45.5095i			1.94230i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	31.3039	+	31.3039i	1.32639	+	1.32639i	0.908498	+	0.417889i	$0.137230\pi$
$557$	31.3039	+	31.3039i	1.32639	+	1.32639i	0.417889	+	0.908498i	$0.362770\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$563$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−31.7360	−	31.7360i	−1.33279	−	1.33279i
$568$		0			0
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$	26.1534			1.09449			0.547243	−	0.836974i	$-0.315677\pi$
$571$	26.1534			1.09449			0.547243	+	0.836974i	$0.315677\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	44.1616	−	8.45777i	1.84167	−	0.352714i
$576$		0			0
$577$	−10.8241	−	10.8241i	−0.450614	−	0.450614i	0.444945	−	0.895558i	$-0.353223\pi$
$577$	−10.8241	−	10.8241i	−0.450614	−	0.450614i	−0.895558	+	0.444945i	$0.853223\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−25.6787			−1.06533
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	31.4679	+	31.4679i	1.29882	+	1.29882i	0.929172	+	0.369649i	$0.120522\pi$
$587$	31.4679	+	31.4679i	1.29882	+	1.29882i	0.369649	+	0.929172i	$0.379478\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−0.435596	+	0.435596i	−0.0178878	+	0.0178878i	−0.715994	−	0.698106i	$-0.754026\pi$
$593$	−0.435596	+	0.435596i	−0.0178878	+	0.0178878i	0.698106	+	0.715994i	$0.254026\pi$
$594$		0			0
$595$	18.4486	−	60.0463i	0.756319	−	2.46166i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	13.6254	+	4.18627i	0.553952	+	0.170196i
$606$		0			0
$607$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$607$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−13.7028	+	13.7028i	−0.553450	+	0.553450i	−0.927435	−	0.373985i	$-0.877991\pi$
$613$	−13.7028	+	13.7028i	−0.553450	+	0.553450i	0.373985	+	0.927435i	$0.377991\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	35.1282	+	35.1282i	1.41421	+	1.41421i	0.709199	+	0.705008i	$0.249057\pi$
$617$	35.1282	+	35.1282i	1.41421	+	1.41421i	0.705008	+	0.709199i	$0.250943\pi$
$618$		0			0
$619$			24.0000i			0.964641i	0.875995	+	0.482321i	$0.160206\pi$
$619$			24.0000i			0.964641i	−0.875995	+	0.482321i	$0.839794\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	9.23713	+	23.2309i	0.369485	+	0.929237i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−10.9836			−0.437249			−0.218624	−	0.975809i	$-0.570157\pi$
$631$	−10.9836			−0.437249			−0.218624	+	0.975809i	$0.570157\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$		0			0
$643$	17.8992	−	17.8992i	0.705874	−	0.705874i	−0.259791	−	0.965665i	$-0.583654\pi$
$643$	17.8992	−	17.8992i	0.705874	−	0.705874i	0.965665	+	0.259791i	$0.0836535\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	2.83614	+	2.83614i	0.111500	+	0.111500i	0.760656	−	0.649155i	$-0.224878\pi$
$647$	2.83614	+	2.83614i	0.111500	+	0.111500i	−0.649155	+	0.760656i	$0.724878\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−28.3438	+	28.3438i	−1.10918	+	1.10918i	−0.115920	+	0.993259i	$0.536982\pi$
$653$	−28.3438	+	28.3438i	−1.10918	+	1.10918i	−0.993259	+	0.115920i	$0.963018\pi$
$654$		0			0
$655$	−32.8625	−	10.0967i	−1.28405	−	0.394510i
$656$		0			0
$657$	−20.7552	−	20.7552i	−0.809739	−	0.809739i
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	42.9476	−	22.7597i	1.66544	−	0.882585i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−32.6254			−1.25949
$672$		0			0
$673$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$673$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$677$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$683$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$684$		0			0
$685$	−10.6413	−	20.0802i	−0.406585	−	0.767226i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	49.9259			1.89927			0.949636	−	0.313355i	$-0.101453\pi$
$691$	49.9259			1.89927			0.949636	+	0.313355i	$0.101453\pi$
$692$		0			0
$693$	22.7513	−	22.7513i	0.864250	−	0.864250i
$694$		0			0
$695$	−15.3504	+	49.9622i	−0.582273	+	1.89517i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−17.4356			−0.658533			−0.329267	−	0.944237i	$-0.606802\pi$
$701$	−17.4356			−0.658533			−0.329267	+	0.944237i	$0.606802\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−61.4818	−	61.4818i	−2.31226	−	2.31226i
$708$		0			0
$709$			52.3068i			1.96442i	0.187779	+	0.982211i	$0.439871\pi$
$709$			52.3068i			1.96442i	−0.187779	+	0.982211i	$0.560129\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	43.3746i		−	1.61760i	−0.588084	−	0.808800i	$-0.700118\pi$
$719$		−	43.3746i		−	1.61760i	0.588084	−	0.808800i	$-0.299882\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	32.4511	+	32.4511i	1.20354	+	1.20354i	0.973081	+	0.230463i	$0.0740239\pi$
$727$	32.4511	+	32.4511i	1.20354	+	1.20354i	0.230463	+	0.973081i	$0.425976\pi$
$728$		0			0
$729$			27.0000i			1.00000i
$730$		0			0
$731$	−13.6246			−0.503923
$732$		0			0
$733$	33.1534	−	33.1534i	1.22455	−	1.22455i	0.258551	−	0.965998i	$-0.416755\pi$
$733$	33.1534	−	33.1534i	1.22455	−	1.22455i	0.965998	−	0.258551i	$-0.0832450\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		−	54.2273i		−	1.99478i	−0.0721811	−	0.997392i	$-0.522996\pi$
$739$		−	54.2273i		−	1.99478i	0.0721811	−	0.997392i	$-0.477004\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$743$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$744$		0			0
$745$	−8.78325	+	28.5876i	−0.321793	+	1.04737i
$746$		0			0
$747$	10.9233	+	10.9233i	0.399663	+	0.399663i
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	27.8077	+	27.8077i	1.01069	+	1.01069i	0.999942	+	0.0107448i	$0.00342025\pi$
$757$	27.8077	+	27.8077i	1.01069	+	1.01069i	0.0107448	+	0.999942i	$0.496580\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−49.8108			−1.80564			−0.902821	−	0.430017i	$-0.858508\pi$
$761$	−49.8108			−1.80564			−0.902821	+	0.430017i	$0.858508\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−33.3905	+	17.6950i	−1.20724	+	0.639765i
$766$		0			0
$767$		0			0
$768$		0			0
$769$			6.62541i			0.238919i	0.992839	+	0.119459i	$0.0381161\pi$
$769$			6.62541i			0.238919i	−0.992839	+	0.119459i	$0.961884\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$773$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−26.2356	−	49.5066i	−0.936388	−	1.76696i
$786$		0			0
$787$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$787$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$797$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$798$		0			0
$799$		−	77.0930i		−	2.72736i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	14.8793	−	14.8793i	0.525078	−	0.525078i
$804$		0			0
$805$	−95.8560	−	29.4508i	−3.37848	−	1.03800i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			24.0027i			0.843891i	0.906621	+	0.421945i	$0.138653\pi$
$809$			24.0027i			0.843891i	−0.906621	+	0.421945i	$0.861347\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−45.7096	+	24.2234i	−1.60114	+	0.848508i
$816$		0			0
$817$	−7.45454	−	7.45454i	−0.260801	−	0.260801i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	45.3746			1.58358			0.791792	−	0.610791i	$-0.209148\pi$
$821$	45.3746			1.58358			0.791792	+	0.610791i	$0.209148\pi$
$822$		0			0
$823$	−16.5177	+	16.5177i	−0.575769	+	0.575769i	−0.933735	−	0.357966i	$-0.883471\pi$
$823$	−16.5177	+	16.5177i	−0.575769	+	0.575769i	0.357966	+	0.933735i	$0.383471\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$827$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$828$		0			0
$829$		0			0			−	1.00000i	$-0.5\pi$
$829$		0			0				1.00000i	$0.5\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−71.1765	+	71.1765i	−2.46612	+	2.46612i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−8.53726	+	27.7870i	−0.293691	+	0.955901i
$846$		0			0
$847$	−22.4782	−	22.4782i	−0.772360	−	0.772360i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	13.1534	−	13.1534i	0.450364	−	0.450364i	−0.445112	−	0.895475i	$-0.646836\pi$
$853$	13.1534	−	13.1534i	0.450364	−	0.450364i	0.895475	+	0.445112i	$0.146836\pi$
$854$		0			0
$855$	−27.9509	−	8.58762i	−0.955901	−	0.293691i
$856$		0			0
$857$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$857$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$858$		0			0
$859$			41.3232i			1.40993i	0.709242	+	0.704965i	$0.249037\pi$
$859$			41.3232i			1.40993i	−0.709242	+	0.704965i	$0.750963\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$863$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$	6.05734	−	55.4244i	0.204775	−	1.87369i
$876$		0			0
$877$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$877$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	6.33694			0.213497			0.106749	−	0.994286i	$-0.465956\pi$
$881$	6.33694			0.213497			0.106749	+	0.994286i	$0.465956\pi$
$882$		0			0
$883$	40.4178	−	40.4178i	1.36017	−	1.36017i	0.486469	−	0.873698i	$-0.338285\pi$
$883$	40.4178	−	40.4178i	1.36017	−	1.36017i	0.873698	−	0.486469i	$-0.161715\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$887$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−19.3561			−0.648454
$892$		0			0
$893$	42.1806	−	42.1806i	1.41152	−	1.41152i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$907$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$908$		0			0
$909$			52.3068i			1.73491i
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$	−7.83083	+	7.83083i	−0.259163	+	0.259163i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	54.2142	+	54.2142i	1.79031	+	1.79031i
$918$		0			0
$919$			8.71780i			0.287574i	0.989609	+	0.143787i	$0.0459280\pi$
$919$			8.71780i			0.287574i	−0.989609	+	0.143787i	$0.954072\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	34.8712i		−	1.14409i	−0.820223	−	0.572043i	$-0.806151\pi$
$929$		−	34.8712i		−	1.14409i	0.820223	−	0.572043i	$-0.193849\pi$
$930$		0			0
$931$	−77.8869			−2.55264
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−12.6854	−	23.9374i	−0.414858	−	0.782837i
$936$		0			0
$937$	−35.3962	−	35.3962i	−1.15634	−	1.15634i	−0.985255	−	0.171089i	$-0.945271\pi$
$937$	−35.3962	−	35.3962i	−1.15634	−	1.15634i	−0.171089	−	0.985255i	$-0.554729\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	34.5123	+	34.5123i	1.12150	+	1.12150i	0.991516	+	0.129983i	$0.0414921\pi$
$947$	34.5123	+	34.5123i	1.12150	+	1.12150i	0.129983	+	0.991516i	$0.458508\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$953$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$954$		0			0
$955$	58.5120	+	17.9772i	1.89341	+	0.581729i
$956$		0			0
$957$		0			0
$958$		0			0
$959$			50.6822i			1.63661i
$960$		0			0
$961$	31.0000			1.00000
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	24.5123	+	24.5123i	0.788262	+	0.788262i	0.981209	−	0.192947i	$-0.0618045\pi$
$967$	24.5123	+	24.5123i	0.788262	+	0.788262i	−0.192947	+	0.981209i	$0.561805\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$		0			0
$973$	82.4240	−	82.4240i	2.64239	−	2.64239i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$977$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$983$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$984$		0			0
$985$	3.37936	+	6.37686i	0.107675	+	0.203184i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			21.7498i			0.691605i
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	18.5120	−	60.2528i	0.586871	−	1.91014i
$996$		0			0
$997$	14.5070	+	14.5070i	0.459441	+	0.459441i	0.898472	−	0.439031i	$-0.144678\pi$
$997$	14.5070	+	14.5070i	0.459441	+	0.459441i	−0.439031	+	0.898472i	$0.644678\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	380.2.l.a.37.1	✓	8
3.2	odd	2		3420.2.bb.c.37.4		8
5.2	odd	4		1900.2.l.a.493.1		8
5.3	odd	4	inner	380.2.l.a.113.2	yes	8
5.4	even	2		1900.2.l.a.1557.1		8
15.8	even	4		3420.2.bb.c.2773.3		8
19.18	odd	2	CM	380.2.l.a.37.1	✓	8
57.56	even	2		3420.2.bb.c.37.4		8
95.18	even	4	inner	380.2.l.a.113.2	yes	8
95.37	even	4		1900.2.l.a.493.1		8
95.94	odd	2		1900.2.l.a.1557.1		8
285.113	odd	4		3420.2.bb.c.2773.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
380.2.l.a.37.1	✓	8	1.1	even	1	trivial
380.2.l.a.37.1	✓	8	19.18	odd	2	CM
380.2.l.a.113.2	yes	8	5.3	odd	4	inner
380.2.l.a.113.2	yes	8	95.18	even	4	inner
1900.2.l.a.493.1		8	5.2	odd	4
1900.2.l.a.493.1		8	95.37	even	4
1900.2.l.a.1557.1		8	5.4	even	2
1900.2.l.a.1557.1		8	95.94	odd	2
3420.2.bb.c.37.4		8	3.2	odd	2
3420.2.bb.c.37.4		8	57.56	even	2
3420.2.bb.c.2773.3		8	15.8	even	4
3420.2.bb.c.2773.3		8	285.113	odd	4

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:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	380.l (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.13746 - 0.656712i) q^{5} +(3.52622 + 3.52622i) q^{7} -3.00000i q^{9} +O(q^{10})\)	\(q+(-2.13746 - 0.656712i) q^{5} +(3.52622 + 3.52622i) q^{7} -3.00000i q^{9} +2.15068 q^{11} +(3.98336 + 3.98336i) q^{17} +4.35890i q^{19} +(6.35890 - 6.35890i) q^{23} +(4.13746 + 2.80739i) q^{25} +(-5.22144 - 9.85286i) q^{35} +(-1.71019 + 1.71019i) q^{43} +(-1.97014 + 6.41238i) q^{45} +(-9.67690 - 9.67690i) q^{47} +17.8685i q^{49} +(-4.59698 - 1.41238i) q^{55} -15.1698 q^{61} +(10.5787 - 10.5787i) q^{63} +(6.91841 - 6.91841i) q^{73} +(7.58377 + 7.58377i) q^{77} -9.00000 q^{81} +(-3.64110 + 3.64110i) q^{83} +(-5.89834 - 11.1302i) q^{85} +(2.86254 - 9.31697i) q^{95} -6.45203i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{5} - 6 q^{7}+O(q^{10}) \) 8 * q - 2 * q^5 - 6 * q^7	\( 8 q - 2 q^{5} - 6 q^{7} + 14 q^{17} + 16 q^{23} + 18 q^{25} - 22 q^{35} + 2 q^{43} - 26 q^{47} - 18 q^{63} + 22 q^{73} + 26 q^{77} - 72 q^{81} - 64 q^{83} + 24 q^{85} + 38 q^{95}+O(q^{100}) \) 8 * q - 2 * q^5 - 6 * q^7 + 14 * q^17 + 16 * q^23 + 18 * q^25 - 22 * q^35 + 2 * q^43 - 26 * q^47 - 18 * q^63 + 22 * q^73 + 26 * q^77 - 72 * q^81 - 64 * q^83 + 24 * q^85 + 38 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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