LMFDB - Embedded newform 1900.2.l.a.1557.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1900,2,Mod(493,1900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1900, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3, 2]))

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

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

chi := DirichletCharacter("1900.493");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1900 = 2^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1900.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:	$15.1715763840$
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^{3}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 380)
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			1557.2
Root			$0.656712 + 2.13746i$ of defining polynomial
Character	$\chi$	$=$	1900.1557
Dual form			1900.2.l.a.493.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.25130 + 1.25130i) q^{7} -3.00000i q^{9} +O(q^{10})$	$q+(1.25130 + 1.25130i) q^{7} -3.00000i q^{9} -2.15068 q^{11} +(4.25827 + 4.25827i) q^{17} -4.35890i q^{19} +(2.35890 - 2.35890i) q^{23} +(9.11456 - 9.11456i) q^{43} +(0.598018 + 0.598018i) q^{47} -3.86848i q^{49} +15.1698 q^{61} +(3.75391 - 3.75391i) q^{63} +(-9.90634 + 9.90634i) q^{73} +(-2.69115 - 2.69115i) q^{77} -9.00000 q^{81} +(12.3589 - 12.3589i) q^{83} +6.45203i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 6 q^{7}+O(q^{10}) $ 8 * q + 6 * q^7	$ 8 q + 6 q^{7} - 14 q^{17} - 16 q^{23} - 2 q^{43} + 26 q^{47} + 18 q^{63} - 22 q^{73} - 26 q^{77} - 72 q^{81} + 64 q^{83}+O(q^{100}) $ 8 * q + 6 * q^7 - 14 * q^17 - 16 * q^23 - 2 * q^43 + 26 * q^47 + 18 * q^63 - 22 * q^73 - 26 * q^77 - 72 * q^81 + 64 * q^83

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times$.

$n$	$77$	$401$	$951$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$-1$	$1$

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
$3$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$3$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	1.25130	+	1.25130i	0.472949	+	0.472949i	0.902867	−	0.429919i	$-0.141458\pi$
$7$	1.25130	+	1.25130i	0.472949	+	0.472949i	−0.429919	+	0.902867i	$0.641458\pi$
$8$		0			0
$9$		−	3.00000i		−	1.00000i
$10$		0			0
$11$	−2.15068			−0.648454			−0.324227	−	0.945979i	$-0.605104\pi$
$11$	−2.15068			−0.648454			−0.324227	+	0.945979i	$0.605104\pi$
$12$		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
$17$	4.25827	+	4.25827i	1.03278	+	1.03278i	0.999444	+	0.0333386i	$0.0106140\pi$
$17$	4.25827	+	4.25827i	1.03278	+	1.03278i	0.0333386	+	0.999444i	$0.489386\pi$
$18$		0			0
$19$		−	4.35890i		−	1.00000i
$19$		−	4.35890i		−	1.00000i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.35890	−	2.35890i	0.491864	−	0.491864i	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	2.35890	−	2.35890i	0.491864	−	0.491864i	0.908893	+	0.417029i	$0.136929\pi$
$24$		0			0
$25$		0			0
$26$		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
$35$		0			0
$36$		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
$43$	9.11456	−	9.11456i	1.38996	−	1.38996i	0.564578	−	0.825380i	$-0.309039\pi$
$43$	9.11456	−	9.11456i	1.38996	−	1.38996i	0.825380	−	0.564578i	$-0.190961\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	0.598018	+	0.598018i	0.0872299	+	0.0872299i	0.749375	−	0.662145i	$-0.230354\pi$
$47$	0.598018	+	0.598018i	0.0872299	+	0.0872299i	−0.662145	+	0.749375i	$0.730354\pi$
$48$		0			0
$49$		−	3.86848i		−	0.552639i
$50$		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
$55$		0			0
$56$		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	$-0.0766472\pi$
$61$	15.1698			1.94230			0.971149	+	0.238474i	$0.0766472\pi$
$62$		0			0
$63$	3.75391	−	3.75391i	0.472949	−	0.472949i
$64$		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
$73$	−9.90634	+	9.90634i	−1.15945	+	1.15945i	−0.174855	+	0.984594i	$0.555946\pi$
$73$	−9.90634	+	9.90634i	−1.15945	+	1.15945i	−0.984594	+	0.174855i	$0.944054\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−2.69115	−	2.69115i	−0.306685	−	0.306685i
$78$		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
$83$	12.3589	−	12.3589i	1.35657	−	1.35657i	0.478451	−	0.878114i	$-0.341198\pi$
$83$	12.3589	−	12.3589i	1.35657	−	1.35657i	0.878114	−	0.478451i	$-0.158802\pi$
$84$		0			0
$85$		0			0
$86$		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
$95$		0			0
$96$		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	$-0.165755\pi$
$101$	17.4356			1.73491			0.867453	+	0.497519i	$0.165755\pi$
$102$		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
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$			10.6568i			0.976906i
$120$		0			0
$121$	−6.37459			−0.579508
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		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
$133$	5.45431	−	5.45431i	0.472949	−	0.472949i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	14.9116	+	14.9116i	1.27398	+	1.27398i	0.943981	+	0.329999i	$0.107048\pi$
$137$	14.9116	+	14.9116i	1.27398	+	1.27398i	0.329999	+	0.943981i	$0.392952\pi$
$138$		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
$153$	12.7748	−	12.7748i	1.03278	−	1.03278i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−0.282202	−	0.282202i	−0.0225222	−	0.0225222i	0.695756	−	0.718278i	$-0.255069\pi$
$157$	−0.282202	−	0.282202i	−0.0225222	−	0.0225222i	−0.718278	+	0.695756i	$0.755069\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	5.90340			0.465253
$162$		0			0
$163$	−7.64110	+	7.64110i	−0.598497	+	0.598497i	−0.939913	−	0.341415i	$-0.889094\pi$
$163$	−7.64110	+	7.64110i	−0.598497	+	0.598497i	0.341415	+	0.939913i	$0.389094\pi$
$164$		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
$175$		0			0
$176$		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
$187$	−9.15817	−	9.15817i	−0.669712	−	0.669712i
$188$		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
$197$	19.7178	+	19.7178i	1.40483	+	1.40483i	0.783718	+	0.621117i	$0.213321\pi$
$197$	19.7178	+	19.7178i	1.40483	+	1.40483i	0.621117	+	0.783718i	$0.286679\pi$
$198$		0			0
$199$		−	28.1890i		−	1.99826i	−0.0416556	−	0.999132i	$-0.513263\pi$
$199$		−	28.1890i		−	1.99826i	0.0416556	−	0.999132i	$-0.486737\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−7.07670	−	7.07670i	−0.491864	−	0.491864i
$208$		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
$215$		0			0
$216$		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
$225$		0			0
$226$		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
$233$	4.90112	−	4.90112i	0.321083	−	0.321083i	−0.528099	−	0.849183i	$-0.677095\pi$
$233$	4.90112	−	4.90112i	0.321083	−	0.321083i	0.849183	+	0.528099i	$0.177095\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$			19.5863i			1.26693i	0.773771	+	0.633465i	$0.218368\pi$
$239$			19.5863i			1.26693i	−0.773771	+	0.633465i	$0.781632\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		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
$253$	−5.07323	+	5.07323i	−0.318951	+	0.318951i
$254$		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
$263$	−18.8271	+	18.8271i	−1.16093	+	1.16093i	−0.176659	+	0.984272i	$0.556529\pi$
$263$	−18.8271	+	18.8271i	−1.16093	+	1.16093i	−0.984272	+	0.176659i	$0.943471\pi$
$264$		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	$-0.792190\pi$
$271$	−26.1534			−1.58871			−0.794353	+	0.607457i	$0.792190\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−3.06224	−	3.06224i	−0.183992	−	0.183992i	0.609101	−	0.793093i	$-0.291530\pi$
$277$	−3.06224	−	3.06224i	−0.183992	−	0.183992i	−0.793093	+	0.609101i	$0.791530\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$	−23.5666	+	23.5666i	−1.40089	+	1.40089i	−0.603606	+	0.797283i	$0.706270\pi$
$283$	−23.5666	+	23.5666i	−1.40089	+	1.40089i	−0.797283	+	0.603606i	$0.793730\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$			19.2658i			1.13328i
$290$		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
$301$	22.8102			1.31476
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		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	$-0.127227\pi$
$311$	32.4903			1.84236			0.921179	+	0.389139i	$0.127227\pi$
$312$		0			0
$313$	−14.4356	+	14.4356i	−0.815948	+	0.815948i	−0.985518	−	0.169570i	$-0.945762\pi$
$313$	−14.4356	+	14.4356i	−0.815948	+	0.815948i	0.169570	+	0.985518i	$0.445762\pi$
$314$		0			0
$315$		0			0
$316$		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
$323$	18.5614	−	18.5614i	1.03278	−	1.03278i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$			1.49661i			0.0825105i
$330$		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
$343$	13.5998	−	13.5998i	0.734319	−	0.734319i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−26.0692	−	26.0692i	−1.39947	−	1.39947i	−0.801578	−	0.597890i	$-0.796006\pi$
$347$	−26.0692	−	26.0692i	−1.39947	−	1.39947i	−0.597890	−	0.801578i	$-0.703994\pi$
$348$		0			0
$349$			23.7725i			1.27251i	0.771477	+	0.636257i	$0.219518\pi$
$349$			23.7725i			1.27251i	−0.771477	+	0.636257i	$0.780482\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−24.4356	+	24.4356i	−1.30058	+	1.30058i	−0.372572	+	0.928003i	$0.621524\pi$
$353$	−24.4356	+	24.4356i	−1.30058	+	1.30058i	−0.928003	+	0.372572i	$0.878476\pi$
$354$		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
$365$		0			0
$366$		0			0
$367$	−27.0767	−	27.0767i	−1.41339	−	1.41339i	−0.730794	−	0.682598i	$-0.760850\pi$
$367$	−27.0767	−	27.0767i	−1.41339	−	1.41339i	−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
$385$		0			0
$386$		0			0
$387$	−27.3437	−	27.3437i	−1.38996	−	1.38996i
$388$		0			0
$389$			36.9068i			1.87125i	0.352998	+	0.935624i	$0.385162\pi$
$389$			36.9068i			1.87125i	−0.352998	+	0.935624i	$0.614838\pi$
$390$		0			0
$391$	20.0897			1.01598
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	2.39852	+	2.39852i	0.120378	+	0.120378i	0.764730	−	0.644351i	$-0.222873\pi$
$397$	2.39852	+	2.39852i	0.120378	+	0.120378i	−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
$405$		0			0
$406$		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
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	8.71780i		−	0.425892i	−0.977064	−	0.212946i	$-0.931694\pi$
$419$		−	8.71780i		−	0.425892i	0.977064	−	0.212946i	$-0.0683059\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$	1.79405	−	1.79405i	0.0872299	−	0.0872299i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	18.9821	+	18.9821i	0.918607	+	0.918607i
$428$		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
$437$	−10.2822	−	10.2822i	−0.491864	−	0.491864i
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	−11.6054			−0.552639
$442$		0			0
$443$	−26.1477	+	26.1477i	−1.24231	+	1.24231i	−0.283273	+	0.959039i	$0.591420\pi$
$443$	−26.1477	+	26.1477i	−1.24231	+	1.24231i	−0.959039	+	0.283273i	$0.908580\pi$
$444$		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
$457$	−23.6834	−	23.6834i	−1.10786	−	1.10786i	−0.993431	−	0.114433i	$-0.963495\pi$
$457$	−23.6834	−	23.6834i	−1.10786	−	1.10786i	−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
$463$	16.4351	−	16.4351i	0.763803	−	0.763803i	−0.213205	−	0.977007i	$-0.568390\pi$
$463$	16.4351	−	16.4351i	0.763803	−	0.763803i	0.977007	+	0.213205i	$0.0683902\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	28.5718	+	28.5718i	1.32215	+	1.32215i	0.912036	+	0.410110i	$0.134510\pi$
$467$	28.5718	+	28.5718i	1.32215	+	1.32215i	0.410110	+	0.912036i	$0.365490\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−19.6025	+	19.6025i	−0.901323	+	0.901323i
$474$		0			0
$475$		0			0
$476$		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
$495$		0			0
$496$		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
$503$	−17.6411	+	17.6411i	−0.786578	+	0.786578i	−0.980932	−	0.194354i	$-0.937739\pi$
$503$	−17.6411	+	17.6411i	−0.786578	+	0.786578i	0.194354	+	0.980932i	$0.437739\pi$
$504$		0			0
$505$		0			0
$506$		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
$511$	−24.7917			−1.09672
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−1.28614	−	1.28614i	−0.0565646	−	0.0565646i
$518$		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
$529$			11.8712i			0.516139i
$530$		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
$539$			8.31984i			0.358361i
$540$		0			0
$541$	2.03559			0.0875168			0.0437584	−	0.999042i	$-0.486067\pi$
$541$	2.03559			0.0875168			0.0437584	+	0.999042i	$0.486067\pi$
$542$		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
$557$	11.5788	+	11.5788i	0.490609	+	0.490609i	0.908498	−	0.417889i	$-0.137230\pi$
$557$	11.5788	+	11.5788i	0.490609	+	0.490609i	−0.417889	+	0.908498i	$0.637230\pi$
$558$		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
$567$	−11.2617	−	11.2617i	−0.472949	−	0.472949i
$568$		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	$-0.684323\pi$
$571$	−26.1534			−1.09449			−0.547243	+	0.836974i	$0.684323\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	32.2000	+	32.2000i	1.34050	+	1.34050i	0.895558	+	0.444945i	$0.146777\pi$
$577$	32.2000	+	32.2000i	1.34050	+	1.34050i	0.444945	+	0.895558i	$0.353223\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	30.9295			1.28317
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−13.5562	−	13.5562i	−0.559523	−	0.559523i	0.369649	−	0.929172i	$-0.379478\pi$
$587$	−13.5562	−	13.5562i	−0.559523	−	0.559523i	−0.929172	+	0.369649i	$0.879478\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−34.4356	+	34.4356i	−1.41410	+	1.41410i	−0.698106	+	0.715994i	$0.745974\pi$
$593$	−34.4356	+	34.4356i	−1.41410	+	1.41410i	−0.715994	+	0.698106i	$0.754026\pi$
$594$		0			0
$595$		0			0
$596$		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
$605$		0			0
$606$		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
$613$	32.2216	−	32.2216i	1.30142	−	1.30142i	0.373985	−	0.927435i	$-0.377991\pi$
$613$	32.2216	−	32.2216i	1.30142	−	1.30142i	0.927435	−	0.373985i	$-0.122009\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	0.104093	+	0.104093i	0.00419061	+	0.00419061i	0.709199	−	0.705008i	$-0.249057\pi$
$617$	0.104093	+	0.104093i	0.00419061	+	0.00419061i	−0.705008	+	0.709199i	$0.749057\pi$
$618$		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
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	10.9836			0.437249			0.218624	−	0.975809i	$-0.429843\pi$
$631$	10.9836			0.437249			0.218624	+	0.975809i	$0.429843\pi$
$632$		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
$643$	31.0744	−	31.0744i	1.22546	−	1.22546i	0.259791	−	0.965665i	$-0.416346\pi$
$643$	31.0744	−	31.0744i	1.22546	−	1.22546i	0.965665	−	0.259791i	$-0.0836535\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	35.8602	+	35.8602i	1.40981	+	1.40981i	0.760656	+	0.649155i	$0.224878\pi$
$647$	35.8602	+	35.8602i	1.40981	+	1.40981i	0.649155	+	0.760656i	$0.275122\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−22.4194	+	22.4194i	−0.877338	+	0.877338i	−0.993259	−	0.115920i	$-0.963018\pi$
$653$	−22.4194	+	22.4194i	−0.877338	+	0.877338i	0.115920	+	0.993259i	$0.463018\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	29.7190	+	29.7190i	1.15945	+	1.15945i
$658$		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
$665$		0			0
$666$		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
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−49.9259			−1.89927			−0.949636	−	0.313355i	$-0.898547\pi$
$691$	−49.9259			−1.89927			−0.949636	+	0.313355i	$0.898547\pi$
$692$		0			0
$693$	−8.07346	+	8.07346i	−0.306685	+	0.306685i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	17.4356			0.658533			0.329267	−	0.944237i	$-0.393198\pi$
$701$	17.4356			0.658533			0.329267	+	0.944237i	$0.393198\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$	21.8172	+	21.8172i	0.820522	+	0.820522i
$708$		0			0
$709$		−	52.3068i		−	1.96442i	−0.187779	−	0.982211i	$-0.560129\pi$
$709$		−	52.3068i		−	1.96442i	0.187779	−	0.982211i	$-0.439871\pi$
$710$		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
$727$	−20.0232	−	20.0232i	−0.742619	−	0.742619i	0.230463	−	0.973081i	$-0.425976\pi$
$727$	−20.0232	−	20.0232i	−0.742619	−	0.742619i	−0.973081	+	0.230463i	$0.925976\pi$
$728$		0			0
$729$			27.0000i			1.00000i
$730$		0			0
$731$	77.6246			2.87105
$732$		0			0
$733$	19.1534	−	19.1534i	0.707447	−	0.707447i	−0.258551	−	0.965998i	$-0.583245\pi$
$733$	19.1534	−	19.1534i	0.707447	−	0.707447i	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	$0.477004\pi$
$739$			54.2273i			1.99478i	−0.0721811	+	0.997392i	$0.522996\pi$
$740$		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
$745$		0			0
$746$		0			0
$747$	−37.0767	−	37.0767i	−1.35657	−	1.35657i
$748$		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
$757$	−27.2164	−	27.2164i	−0.989197	−	0.989197i	0.0107448	−	0.999942i	$-0.496580\pi$
$757$	−27.2164	−	27.2164i	−0.989197	−	0.989197i	−0.999942	+	0.0107448i	$0.996580\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	49.8108			1.80564			0.902821	−	0.430017i	$-0.141492\pi$
$761$	49.8108			1.80564			0.902821	+	0.430017i	$0.141492\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		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
$785$		0			0
$786$		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
$799$			5.09305i			0.180179i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	21.3053	−	21.3053i	0.751849	−	0.751849i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	24.0027i		−	0.843891i	−0.906621	−	0.421945i	$-0.861347\pi$
$809$		−	24.0027i		−	0.843891i	0.906621	−	0.421945i	$-0.138653\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−39.7295	−	39.7295i	−1.38996	−	1.38996i
$818$		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
$823$	37.0563	−	37.0563i	1.29170	−	1.29170i	0.357966	−	0.933735i	$-0.383471\pi$
$823$	37.0563	−	37.0563i	1.29170	−	1.29170i	0.933735	−	0.357966i	$-0.116529\pi$
$824$		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
$833$	16.4730	−	16.4730i	0.570756	−	0.570756i
$834$		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
$845$		0			0
$846$		0			0
$847$	−7.97655	−	7.97655i	−0.274077	−	0.274077i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	39.1534	−	39.1534i	1.34059	−	1.34059i	0.445112	−	0.895475i	$-0.353164\pi$
$853$	39.1534	−	39.1534i	1.34059	−	1.34059i	0.895475	−	0.445112i	$-0.146836\pi$
$854$		0			0
$855$		0			0
$856$		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	$-0.750963\pi$
$859$		−	41.3232i		−	1.40993i	0.709242	−	0.704965i	$-0.249037\pi$
$860$		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
$875$		0			0
$876$		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	$-0.534044\pi$
$881$	−6.33694			−0.213497			−0.106749	+	0.994286i	$0.534044\pi$
$882$		0			0
$883$	−11.5066	+	11.5066i	−0.387229	+	0.387229i	−0.873698	−	0.486469i	$-0.838285\pi$
$883$	−11.5066	+	11.5066i	−0.387229	+	0.387229i	0.486469	+	0.873698i	$0.338285\pi$
$884$		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
$893$	2.60670	−	2.60670i	0.0872299	−	0.0872299i
$894$		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
$913$	−26.5800	+	26.5800i	−0.879670	+	0.879670i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	19.2383	+	19.2383i	0.635304	+	0.635304i
$918$		0			0
$919$		−	8.71780i		−	0.287574i	−0.989609	−	0.143787i	$-0.954072\pi$
$919$		−	8.71780i		−	0.287574i	0.989609	−	0.143787i	$-0.0459280\pi$
$920$		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	$0.193849\pi$
$929$			34.8712i			1.14409i	−0.820223	+	0.572043i	$0.806151\pi$
$930$		0			0
$931$	−16.8623			−0.552639
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−24.9220	−	24.9220i	−0.814166	−	0.814166i	0.171089	−	0.985255i	$-0.445271\pi$
$937$	−24.9220	−	24.9220i	−0.814166	−	0.814166i	−0.985255	+	0.171089i	$0.945271\pi$
$938$		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
$947$	26.5123	+	26.5123i	0.861534	+	0.861534i	0.991516	−	0.129983i	$-0.0414921\pi$
$947$	26.5123	+	26.5123i	0.861534	+	0.861534i	−0.129983	+	0.991516i	$0.541492\pi$
$948$		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
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$			37.3178i			1.20505i
$960$		0			0
$961$	31.0000			1.00000
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	36.5123	+	36.5123i	1.17416	+	1.17416i	0.981209	+	0.192947i	$0.0618045\pi$
$967$	36.5123	+	36.5123i	1.17416	+	1.17416i	0.192947	+	0.981209i	$0.438195\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$		0			0
$973$	29.2487	−	29.2487i	0.937671	−	0.937671i
$974$		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
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	43.0007i		−	1.36734i
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	42.2321	+	42.2321i	1.33750	+	1.33750i	0.898472	+	0.439031i	$0.144678\pi$
$997$	42.2321	+	42.2321i	1.33750	+	1.33750i	0.439031	+	0.898472i	$0.355322\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1900.2.l.a.1557.2		8
5.2	odd	4		380.2.l.a.113.1	yes	8
5.3	odd	4	inner	1900.2.l.a.493.2		8
5.4	even	2		380.2.l.a.37.2	✓	8
15.2	even	4		3420.2.bb.c.2773.4		8
15.14	odd	2		3420.2.bb.c.37.3		8
19.18	odd	2	CM	1900.2.l.a.1557.2		8
95.18	even	4	inner	1900.2.l.a.493.2		8
95.37	even	4		380.2.l.a.113.1	yes	8
95.94	odd	2		380.2.l.a.37.2	✓	8
285.227	odd	4		3420.2.bb.c.2773.4		8
285.284	even	2		3420.2.bb.c.37.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
380.2.l.a.37.2	✓	8	5.4	even	2
380.2.l.a.37.2	✓	8	95.94	odd	2
380.2.l.a.113.1	yes	8	5.2	odd	4
380.2.l.a.113.1	yes	8	95.37	even	4
1900.2.l.a.493.2		8	5.3	odd	4	inner
1900.2.l.a.493.2		8	95.18	even	4	inner
1900.2.l.a.1557.2		8	1.1	even	1	trivial
1900.2.l.a.1557.2		8	19.18	odd	2	CM
3420.2.bb.c.37.3		8	15.14	odd	2
3420.2.bb.c.37.3		8	285.284	even	2
3420.2.bb.c.2773.4		8	15.2	even	4
3420.2.bb.c.2773.4		8	285.227	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:	\( N \)	\(=\)	\( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1900.l (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.25130 + 1.25130i) q^{7} -3.00000i q^{9} +O(q^{10})\)	\(q+(1.25130 + 1.25130i) q^{7} -3.00000i q^{9} -2.15068 q^{11} +(4.25827 + 4.25827i) q^{17} -4.35890i q^{19} +(2.35890 - 2.35890i) q^{23} +(9.11456 - 9.11456i) q^{43} +(0.598018 + 0.598018i) q^{47} -3.86848i q^{49} +15.1698 q^{61} +(3.75391 - 3.75391i) q^{63} +(-9.90634 + 9.90634i) q^{73} +(-2.69115 - 2.69115i) q^{77} -9.00000 q^{81} +(12.3589 - 12.3589i) q^{83} +6.45203i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 6 q^{7}+O(q^{10}) \) 8 * q + 6 * q^7	\( 8 q + 6 q^{7} - 14 q^{17} - 16 q^{23} - 2 q^{43} + 26 q^{47} + 18 q^{63} - 22 q^{73} - 26 q^{77} - 72 q^{81} + 64 q^{83}+O(q^{100}) \) 8 * q + 6 * q^7 - 14 * q^17 - 16 * q^23 - 2 * q^43 + 26 * q^47 + 18 * q^63 - 22 * q^73 - 26 * q^77 - 72 * q^81 + 64 * q^83

Introduction

L-functions

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