LMFDB - Embedded newform 384.4.f.b.191.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [384,4,Mod(191,384)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("384.191"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(384, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) N = Newforms(chi, 4, names="a")

Level:	$N$	$=$	$384 = 2^{7} \cdot 3$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	384.f (of order $2$ , degree $1$ , minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,-92,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$22.6567334422$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} + 1$ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$2^{3}\cdot 3^{2}$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			191.2
Root			$0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	384.191
Dual form			384.4.f.b.191.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.41421 + 5.00000i) q^{3} +(-23.0000 - 14.1421i) q^{9} -18.0000i q^{11} -107.480i q^{17} +127.279 q^{19} -125.000 q^{25} +(103.238 - 95.0000i) q^{27} +(90.0000 + 25.4558i) q^{33} +56.5685i q^{41} +483.661 q^{43} +343.000 q^{49} +(537.401 + 152.000i) q^{51} +(-180.000 + 636.396i) q^{57} -846.000i q^{59} +1094.60 q^{67} +430.000 q^{73} +(176.777 - 625.000i) q^{75} +(329.000 + 650.538i) q^{81} +1350.00i q^{83} -1329.36i q^{89} -1910.00 q^{97} +(-254.558 + 414.000i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 92 q^{9} - 500 q^{25} + 360 q^{33} + 1372 q^{49} - 720 q^{57} + 1720 q^{73} + 1316 q^{81} - 7640 q^{97}+O(q^{100})$ 4 * q - 92 * q^9 - 500 * q^25 + 360 * q^33 + 1372 * q^49 - 720 * q^57 + 1720 * q^73 + 1316 * q^81 - 7640 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/384\mathbb{Z}\right)^\times$ .

$n$	$127$	$133$	$257$
$\chi(n)$	$-1$	$-1$	$-1$

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
$3$	−1.41421	+	5.00000i	−0.272166	+	0.962250i
$3$	−1.41421	+	5.00000i	−0.272166	+	0.962250i
$4$		0			0
$5$		0			0			−	1.00000i	$-0.5\pi$
$5$		0			0				1.00000i	$0.5\pi$
$6$		0			0
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$		0			0
$9$	−23.0000	−	14.1421i	−0.851852	−	0.523783i
$10$		0			0
$11$		−	18.0000i		−	0.493382i	−0.969094	−	0.246691i	$-0.920657\pi$
$11$		−	18.0000i		−	0.493382i	0.969094	−	0.246691i	$-0.0793433\pi$
$12$		0			0
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	107.480i		−	1.53340i	−0.642006	−	0.766700i	$-0.721898\pi$
$17$		−	107.480i		−	1.53340i	0.642006	−	0.766700i	$-0.278102\pi$
$18$		0			0
$19$	127.279			1.53683			0.768417	−	0.639949i	$-0.221045\pi$
$19$	127.279			1.53683			0.768417	+	0.639949i	$0.221045\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$		0			0
$25$	−125.000			−1.00000
$26$		0			0
$27$	103.238	−	95.0000i	0.735855	−	0.677139i
$28$		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
$33$	90.0000	+	25.4558i	0.474757	+	0.134282i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			56.5685i			0.215476i	0.994179	+	0.107738i	$0.0343608\pi$
$41$			56.5685i			0.215476i	−0.994179	+	0.107738i	$0.965639\pi$
$42$		0			0
$43$	483.661			1.71529			0.857647	−	0.514239i	$-0.171926\pi$
$43$	483.661			1.71529			0.857647	+	0.514239i	$0.171926\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$		0			0
$49$	343.000			1.00000
$50$		0			0
$51$	537.401	+	152.000i	1.47551	+	0.417338i
$52$		0			0
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−180.000	+	636.396i	−0.418273	+	1.47882i
$58$		0			0
$59$		−	846.000i		−	1.86678i	−0.358868	−	0.933388i	$-0.616837\pi$
$59$		−	846.000i		−	1.86678i	0.358868	−	0.933388i	$-0.383163\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	1094.60			1.99592			0.997961	−	0.0638199i	$-0.0203283\pi$
$67$	1094.60			1.99592			0.997961	+	0.0638199i	$0.0203283\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$	430.000			0.689420			0.344710	−	0.938709i	$-0.387977\pi$
$73$	430.000			0.689420			0.344710	+	0.938709i	$0.387977\pi$
$74$		0			0
$75$	176.777	−	625.000i	0.272166	−	0.962250i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$		0			0
$81$	329.000	+	650.538i	0.451303	+	0.892371i
$82$		0			0
$83$			1350.00i			1.78532i	0.450728	+	0.892661i	$0.351164\pi$
$83$			1350.00i			1.78532i	−0.450728	+	0.892661i	$0.648836\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	1329.36i		−	1.58328i	−0.610988	−	0.791640i	$-0.709227\pi$
$89$		−	1329.36i		−	1.58328i	0.610988	−	0.791640i	$-0.290773\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−1910.00			−1.99929			−0.999645	−	0.0266459i	$-0.991517\pi$
$97$	−1910.00			−1.99929			−0.999645	+	0.0266459i	$0.991517\pi$
$98$		0			0
$99$	−254.558	+	414.000i	−0.258425	+	0.420289i
$100$		0			0
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	1710.00i		−	1.54497i	−0.635032	−	0.772486i	$-0.719013\pi$
$107$		−	1710.00i		−	1.54497i	0.635032	−	0.772486i	$-0.280987\pi$
$108$		0			0
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	2387.19i		−	1.98733i	−0.112387	−	0.993665i	$-0.535850\pi$
$113$		−	2387.19i		−	1.98733i	0.112387	−	0.993665i	$-0.464150\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	1007.00			0.756574
$122$		0			0
$123$	−282.843	−	80.0000i	−0.207342	−	0.0586452i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$		0			0
$129$	−684.000	+	2418.31i	−0.466844	+	1.65054i
$130$		0			0
$131$			1242.00i			0.828351i	0.910197	+	0.414176i	$0.135930\pi$
$131$			1242.00i			0.828351i	−0.910197	+	0.414176i	$0.864070\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	2285.37i		−	1.42520i	−0.701571	−	0.712599i	$-0.747518\pi$
$137$		−	2285.37i		−	1.42520i	0.701571	−	0.712599i	$-0.252482\pi$
$138$		0			0
$139$	−2927.42			−1.78634			−0.893168	−	0.449723i	$-0.851523\pi$
$139$	−2927.42			−1.78634			−0.893168	+	0.449723i	$0.851523\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−485.075	+	1715.00i	−0.272166	+	0.962250i
$148$		0			0
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$	−1520.00	+	2472.05i	−0.803168	+	1.30623i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−4047.48			−1.94493			−0.972463	−	0.233056i	$-0.925127\pi$
$163$	−4047.48			−1.94493			−0.972463	+	0.233056i	$0.925127\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	2197.00			1.00000
$170$		0			0
$171$	−2927.42	−	1800.00i	−1.30916	−	0.804967i
$172$		0			0
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	4230.00	+	1196.42i	1.79631	+	0.508072i
$178$		0			0
$179$			3834.00i			1.60093i	0.599379	+	0.800465i	$0.295414\pi$
$179$			3834.00i			1.60093i	−0.599379	+	0.800465i	$0.704586\pi$
$180$		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
$187$	−1934.64			−0.756552
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$		0			0
$193$	−2090.00			−0.779490			−0.389745	−	0.920923i	$-0.627437\pi$
$193$	−2090.00			−0.779490			−0.389745	+	0.920923i	$0.627437\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$		0			0
$201$	−1548.00	+	5473.01i	−0.543221	+	1.92058i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	2291.03i		−	0.758247i
$210$		0			0
$211$	381.838			0.124582			0.0622910	−	0.998058i	$-0.480159\pi$
$211$	381.838			0.124582			0.0622910	+	0.998058i	$0.480159\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−608.112	+	2150.00i	−0.187636	+	0.663395i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		0			0
$225$	2875.00	+	1767.77i	0.851852	+	0.523783i
$226$		0			0
$227$		−	6570.00i		−	1.92100i	−0.278286	−	0.960498i	$-0.589766\pi$
$227$		−	6570.00i		−	1.92100i	0.278286	−	0.960498i	$-0.410234\pi$
$228$		0			0
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	3773.12i		−	1.06088i	−0.847722	−	0.530441i	$-0.822026\pi$
$233$		−	3773.12i		−	1.06088i	0.847722	−	0.530441i	$-0.177974\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	1222.00			0.326622			0.163311	−	0.986575i	$-0.447783\pi$
$241$	1222.00			0.326622			0.163311	+	0.986575i	$0.447783\pi$
$242$		0			0
$243$	−3717.97	+	725.000i	−0.981513	+	0.191394i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$	−6750.00	−	1909.19i	−1.71793	−	0.485903i
$250$		0			0
$251$			4302.00i			1.08183i	0.841077	+	0.540916i	$0.181922\pi$
$251$			4302.00i			1.08183i	−0.841077	+	0.540916i	$0.818078\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	7274.71i		−	1.76570i	−0.469658	−	0.882849i	$-0.655623\pi$
$257$		−	7274.71i		−	1.76570i	0.469658	−	0.882849i	$-0.344377\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	6646.80	+	1880.00i	1.52351	+	0.430914i
$268$		0			0
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			2250.00i			0.493382i
$276$		0			0
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			1216.22i			0.258199i	0.991632	+	0.129099i	$0.0412086\pi$
$281$			1216.22i			0.258199i	−0.991632	+	0.129099i	$0.958791\pi$
$282$		0			0
$283$	5116.62			1.07474			0.537371	−	0.843346i	$-0.319418\pi$
$283$	5116.62			1.07474			0.537371	+	0.843346i	$0.319418\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−6639.00			−1.35131
$290$		0			0
$291$	2701.15	−	9550.00i	0.544138	−	1.92382i
$292$		0			0
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−1710.00	−	1858.28i	−0.334088	−	0.363058i
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	7204.00			1.33926			0.669632	−	0.742693i	$-0.266452\pi$
$307$	7204.00			1.33926			0.669632	+	0.742693i	$0.266452\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$	8390.00			1.51511			0.757557	−	0.652769i	$-0.226393\pi$
$313$	8390.00			1.51511			0.757557	+	0.652769i	$0.226393\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0			−	1.00000i	$-0.5\pi$
$317$		0			0				1.00000i	$0.5\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	8550.00	+	2418.31i	1.48665	+	0.420488i
$322$		0			0
$323$		−	13680.0i		−	2.35658i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	8782.27			1.45836			0.729180	−	0.684322i	$-0.239902\pi$
$331$	8782.27			1.45836			0.729180	+	0.684322i	$0.239902\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−11410.0			−1.84434			−0.922170	−	0.386786i	$-0.873585\pi$
$337$	−11410.0			−1.84434			−0.922170	+	0.386786i	$0.873585\pi$
$338$		0			0
$339$	11936.0	+	3376.00i	1.91231	+	0.540882i
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$			6030.00i			0.932874i	0.884554	+	0.466437i	$0.154463\pi$
$347$			6030.00i			0.932874i	−0.884554	+	0.466437i	$0.845537\pi$
$348$		0			0
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			12377.2i			1.86621i	0.359605	+	0.933104i	$0.382911\pi$
$353$			12377.2i			1.86621i	−0.359605	+	0.933104i	$0.617089\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	9341.00			1.36186
$362$		0			0
$363$	−1424.11	+	5035.00i	−0.205913	+	0.728014i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$		0			0
$369$	800.000	−	1301.08i	0.112863	−	0.183554i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	9036.82			1.22478			0.612389	−	0.790557i	$-0.290209\pi$
$379$	9036.82			1.22478			0.612389	+	0.790557i	$0.290209\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−11124.2	−	6840.00i	−1.46118	−	0.898441i
$388$		0			0
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	−6210.00	−	1756.45i	−0.797082	−	0.225449i
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			14453.3i			1.79990i	0.435989	+	0.899952i	$0.356399\pi$
$401$			14453.3i			1.79990i	−0.435989	+	0.899952i	$0.643601\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	16346.0			1.97618			0.988090	−	0.153877i	$-0.0491758\pi$
$409$	16346.0			1.97618			0.988090	+	0.153877i	$0.0491758\pi$
$410$		0			0
$411$	11426.8	+	3232.00i	1.37140	+	0.387890i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	4140.00	−	14637.1i	0.486179	−	1.71890i
$418$		0			0
$419$		−	16794.0i		−	1.95809i	−0.203639	−	0.979046i	$-0.565277\pi$
$419$		−	16794.0i		−	1.95809i	0.203639	−	0.979046i	$-0.434723\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			13435.0i			1.53340i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		0			0
$433$	−5510.00			−0.611533			−0.305766	−	0.952107i	$-0.598913\pi$
$433$	−5510.00			−0.611533			−0.305766	+	0.952107i	$0.598913\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$	−7889.00	−	4850.75i	−0.851852	−	0.523783i
$442$		0			0
$443$			18270.0i			1.95944i	0.200361	+	0.979722i	$0.435789\pi$
$443$			18270.0i			1.95944i	−0.200361	+	0.979722i	$0.564211\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	7438.76i		−	0.781864i	−0.920420	−	0.390932i	$-0.872153\pi$
$449$		−	7438.76i		−	0.781864i	0.920420	−	0.390932i	$-0.127847\pi$
$450$		0			0
$451$	1018.23			0.106312
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−18070.0			−1.84963			−0.924813	−	0.380422i	$-0.875779\pi$
$457$	−18070.0			−1.84963			−0.924813	+	0.380422i	$0.875779\pi$
$458$		0			0
$459$	−10210.6	−	11096.0i	−1.03832	−	1.12836i
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			15030.0i			1.48931i	0.667452	+	0.744653i	$0.267385\pi$
$467$			15030.0i			1.48931i	−0.667452	+	0.744653i	$0.732615\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	8705.90i		−	0.846295i
$474$		0			0
$475$	−15909.9			−1.53683
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		0			0
$489$	5724.00	−	20237.4i	0.529342	−	1.87151i
$490$		0			0
$491$		−	12222.0i		−	1.12336i	−0.827354	−	0.561681i	$-0.810155\pi$
$491$		−	12222.0i		−	1.12336i	0.827354	−	0.561681i	$-0.189845\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−12855.2			−1.15326			−0.576631	−	0.817005i	$-0.695633\pi$
$499$	−12855.2			−1.15326			−0.576631	+	0.817005i	$0.695633\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−3107.03	+	10985.0i	−0.272166	+	0.962250i
$508$		0			0
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	13140.0	−	12091.5i	1.13089	−	1.04065i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$			21948.6i			1.84565i	0.385215	+	0.922827i	$0.374127\pi$
$521$			21948.6i			1.84565i	−0.385215	+	0.922827i	$0.625873\pi$
$522$		0			0
$523$	−23444.8			−1.96017			−0.980087	−	0.198569i	$-0.936371\pi$
$523$	−23444.8			−1.96017			−0.980087	+	0.198569i	$0.936371\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−12167.0			−1.00000
$530$		0			0
$531$	−11964.2	+	19458.0i	−0.977785	+	1.59022i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−19170.0	−	5422.09i	−1.54050	−	0.435718i
$538$		0			0
$539$		−	6174.00i		−	0.493382i
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−13313.4			−1.04066			−0.520329	−	0.853966i	$-0.674191\pi$
$547$	−13313.4			−1.04066			−0.520329	+	0.853966i	$0.674191\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	2736.00	−	9673.22i	0.205907	−	0.727992i
$562$		0			0
$563$			23670.0i			1.77189i	0.463795	+	0.885943i	$0.346488\pi$
$563$			23670.0i			1.77189i	−0.463795	+	0.885943i	$0.653512\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$			27039.8i			1.99221i	0.0881913	+	0.996104i	$0.471891\pi$
$569$			27039.8i			1.99221i	−0.0881913	+	0.996104i	$0.528109\pi$
$570$		0			0
$571$	3691.10			0.270521			0.135261	−	0.990810i	$-0.456813\pi$
$571$	3691.10			0.270521			0.135261	+	0.990810i	$0.456813\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	19550.0			1.41053			0.705266	−	0.708943i	$-0.250827\pi$
$577$	19550.0			1.41053			0.705266	+	0.708943i	$0.250827\pi$
$578$		0			0
$579$	2955.71	−	10450.0i	0.212150	−	0.750064i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	10350.0i		−	0.727752i	−0.931447	−	0.363876i	$-0.881453\pi$
$587$		−	10350.0i		−	0.727752i	0.931447	−	0.363876i	$-0.118547\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$			12173.6i			0.843015i	0.906825	+	0.421507i	$0.138499\pi$
$593$			12173.6i			0.843015i	−0.906825	+	0.421507i	$0.861501\pi$
$594$		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
$601$	14398.0			0.977216			0.488608	−	0.872503i	$-0.337505\pi$
$601$	14398.0			0.977216			0.488608	+	0.872503i	$0.337505\pi$
$602$		0			0
$603$	−25175.8	−	15480.0i	−1.70023	−	1.04543i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	11206.2i		−	0.731192i	−0.930774	−	0.365596i	$-0.880865\pi$
$617$		−	11206.2i		−	0.731192i	0.930774	−	0.365596i	$-0.119135\pi$
$618$		0			0
$619$	−2418.31			−0.157027			−0.0785136	−	0.996913i	$-0.525017\pi$
$619$	−2418.31			−0.157027			−0.0785136	+	0.996913i	$0.525017\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	15625.0			1.00000
$626$		0			0
$627$	11455.1	+	3240.00i	0.729623	+	0.206369i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$	−540.000	+	1909.19i	−0.0339069	+	0.119879i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$			31763.2i			1.95721i	0.205745	+	0.978606i	$0.434038\pi$
$641$			31763.2i			1.95721i	−0.205745	+	0.978606i	$0.565962\pi$
$642$		0			0
$643$	−15757.2			−0.966411			−0.483205	−	0.875507i	$-0.660528\pi$
$643$	−15757.2			−0.966411			−0.483205	+	0.875507i	$0.660528\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$		0			0
$649$	−15228.0			−0.921034
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0			−	1.00000i	$-0.5\pi$
$653$		0			0				1.00000i	$0.5\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−9890.00	−	6081.12i	−0.587284	−	0.361107i
$658$		0			0
$659$			29754.0i			1.75880i	0.476081	+	0.879402i	$0.342057\pi$
$659$			29754.0i			1.75880i	−0.476081	+	0.879402i	$0.657943\pi$
$660$		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
$671$		0			0
$672$		0			0
$673$	−19190.0			−1.09914			−0.549569	−	0.835448i	$-0.685208\pi$
$673$	−19190.0			−1.09914			−0.549569	+	0.835448i	$0.685208\pi$
$674$		0			0
$675$	−12904.7	+	11875.0i	−0.735855	+	0.677139i
$676$		0			0
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	32850.0	+	9291.38i	1.84848	+	0.522829i
$682$		0			0
$683$		−	11970.0i		−	0.670599i	−0.942112	−	0.335300i	$-0.891162\pi$
$683$		−	11970.0i		−	0.670599i	0.942112	−	0.335300i	$-0.108838\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	36274.6			1.99703			0.998517	−	0.0544477i	$-0.0173398\pi$
$691$	36274.6			1.99703			0.998517	+	0.0544477i	$0.0173398\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	6080.00			0.330411
$698$		0			0
$699$	18865.6	+	5336.00i	1.02083	+	0.288735i
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		0			0		1.00000			$0$
$709$		0			0		−1.00000			$\pi$
$710$		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
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−1728.17	+	6110.00i	−0.0888953	+	0.314292i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	1633.00	−	19615.1i	0.0829650	−	0.996552i
$730$		0			0
$731$		−	51984.0i		−	2.63023i
$732$		0			0
$733$		0			0		1.00000			$0$
$733$		0			0		−1.00000			$\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	19702.8i		−	0.984753i
$738$		0			0
$739$	17691.8			0.880655			0.440327	−	0.897837i	$-0.354862\pi$
$739$	17691.8			0.880655			0.440327	+	0.897837i	$0.354862\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0			−	1.00000i	$-0.5\pi$
$743$		0			0				1.00000i	$0.5\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	19091.9	−	31050.0i	0.935121	−	1.52083i
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$		0			0
$753$	−21510.0	−	6083.95i	−1.04099	−	0.294437i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	24381.0i		−	1.16138i	−0.814124	−	0.580691i	$-0.802782\pi$
$761$		−	24381.0i		−	1.16138i	0.814124	−	0.580691i	$-0.197218\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−40106.0			−1.88070			−0.940351	−	0.340207i	$-0.889503\pi$
$769$	−40106.0			−1.88070			−0.940351	+	0.340207i	$0.889503\pi$
$770$		0			0
$771$	36373.6	+	10288.0i	1.69904	+	0.480562i
$772$		0			0
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$			7200.00i			0.331151i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	43605.9			1.97507			0.987536	−	0.157396i	$-0.0503098\pi$
$787$	43605.9			1.97507			0.987536	+	0.157396i	$0.0503098\pi$
$788$		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			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−18800.0	+	30575.3i	−0.829295	+	1.34872i
$802$		0			0
$803$		−	7740.00i		−	0.340148i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	43727.5i		−	1.90034i	−0.311730	−	0.950171i	$-0.600908\pi$
$809$		−	43727.5i		−	1.90034i	0.311730	−	0.950171i	$-0.399092\pi$
$810$		0			0
$811$	26855.9			1.16281			0.581405	−	0.813614i	$-0.302503\pi$
$811$	26855.9			1.16281			0.581405	+	0.813614i	$0.302503\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	61560.0			2.63612
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0			−	1.00000i	$-0.5\pi$
$821$		0			0				1.00000i	$0.5\pi$
$822$		0			0
$823$		0			0		1.00000			$0$
$823$		0			0		−1.00000			$\pi$
$824$		0			0
$825$	−11250.0	−	3181.98i	−0.474757	−	0.134282i
$826$		0			0
$827$			23490.0i			0.987699i	0.869547	+	0.493850i	$0.164411\pi$
$827$			23490.0i			0.987699i	−0.869547	+	0.493850i	$0.835589\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	36865.7i		−	1.53340i
$834$		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
$841$	−24389.0			−1.00000
$842$		0			0
$843$	−6081.12	−	1720.00i	−0.248452	−	0.0702728i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−7236.00	+	25583.1i	−0.292508	+	1.03417i
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		1.00000			$0$
$853$		0			0		−1.00000			$\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			46589.9i			1.85704i	0.371289	+	0.928518i	$0.378916\pi$
$857$			46589.9i			1.85704i	−0.371289	+	0.928518i	$0.621084\pi$
$858$		0			0
$859$	21255.6			0.844276			0.422138	−	0.906532i	$-0.361280\pi$
$859$	21255.6			0.844276			0.422138	+	0.906532i	$0.361280\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	9388.96	−	33195.0i	0.367781	−	1.30030i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	43930.0	+	27011.5i	1.70310	+	1.04719i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		1.00000			$0$
$877$		0			0		−1.00000			$\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	31508.7i		−	1.20494i	−0.798141	−	0.602471i	$-0.794183\pi$
$881$		−	31508.7i		−	1.20494i	0.798141	−	0.602471i	$-0.205817\pi$
$882$		0			0
$883$	−52209.9			−1.98981			−0.994906	−	0.100806i	$-0.967858\pi$
$883$	−52209.9			−1.98981			−0.994906	+	0.100806i	$0.967858\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	11709.7	−	5922.00i	0.440280	−	0.222665i
$892$		0			0
$893$		0			0
$894$		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
$907$	47933.4			1.75480			0.877399	−	0.479762i	$-0.159277\pi$
$907$	47933.4			1.75480			0.877399	+	0.479762i	$0.159277\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$	24300.0			0.880846
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$	−10188.0	+	36020.0i	−0.364502	+	1.28871i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$			56200.8i			1.98481i	0.123007	+	0.992406i	$0.460746\pi$
$929$			56200.8i			1.98481i	−0.123007	+	0.992406i	$0.539254\pi$
$930$		0			0
$931$	43656.8			1.53683
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−56270.0			−1.96186			−0.980929	−	0.194367i	$-0.937735\pi$
$937$	−56270.0			−1.96186			−0.980929	+	0.194367i	$0.937735\pi$
$938$		0			0
$939$	−11865.3	+	41950.0i	−0.412362	+	1.45792i
$940$		0			0
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	58230.0i		−	1.99812i	−0.0433353	−	0.999061i	$-0.513798\pi$
$947$		−	58230.0i		−	1.99812i	0.0433353	−	0.999061i	$-0.486202\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	36701.7i		−	1.24752i	−0.781617	−	0.623759i	$-0.785605\pi$
$953$		−	36701.7i		−	1.24752i	0.781617	−	0.623759i	$-0.214395\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	29791.0			1.00000
$962$		0			0
$963$	−24183.1	+	39330.0i	−0.809229	+	1.31609i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$		0			0
$969$	68400.0	+	19346.4i	2.26762	+	0.641380i
$970$		0			0
$971$		−	162.000i		−	0.00535410i	−0.999996	−	0.00267705i	$-0.999148\pi$
$971$		−	162.000i		−	0.00535410i	0.999996	−	0.00267705i	$-0.000852132\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	58554.1i		−	1.91741i	−0.284399	−	0.958706i	$-0.591794\pi$
$977$		−	58554.1i		−	1.91741i	0.284399	−	0.958706i	$-0.408206\pi$
$978$		0			0
$979$	−23928.5			−0.781162
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$	−12420.0	+	43911.3i	−0.396915	+	1.40331i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		1.00000			$0$
$997$		0			0		−1.00000			$\pi$
$998$		0			0
$999$		0			0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.4.f.b.191.2	yes	4
3.2	odd	2	inner	384.4.f.b.191.1	✓	4
4.3	odd	2	inner	384.4.f.b.191.3	yes	4
8.3	odd	2	CM	384.4.f.b.191.2	yes	4
8.5	even	2	inner	384.4.f.b.191.3	yes	4
12.11	even	2	inner	384.4.f.b.191.4	yes	4
16.3	odd	4		768.4.c.j.767.2		2
16.5	even	4		768.4.c.j.767.2		2
16.11	odd	4		768.4.c.a.767.1		2
16.13	even	4		768.4.c.a.767.1		2
24.5	odd	2	inner	384.4.f.b.191.4	yes	4
24.11	even	2	inner	384.4.f.b.191.1	✓	4
48.5	odd	4		768.4.c.a.767.2		2
48.11	even	4		768.4.c.j.767.1		2
48.29	odd	4		768.4.c.j.767.1		2
48.35	even	4		768.4.c.a.767.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.4.f.b.191.1	✓	4	3.2	odd	2	inner
384.4.f.b.191.1	✓	4	24.11	even	2	inner
384.4.f.b.191.2	yes	4	1.1	even	1	trivial
384.4.f.b.191.2	yes	4	8.3	odd	2	CM
384.4.f.b.191.3	yes	4	4.3	odd	2	inner
384.4.f.b.191.3	yes	4	8.5	even	2	inner
384.4.f.b.191.4	yes	4	12.11	even	2	inner
384.4.f.b.191.4	yes	4	24.5	odd	2	inner
768.4.c.a.767.1		2	16.11	odd	4
768.4.c.a.767.1		2	16.13	even	4
768.4.c.a.767.2		2	48.5	odd	4
768.4.c.a.767.2		2	48.35	even	4
768.4.c.j.767.1		2	48.11	even	4
768.4.c.j.767.1		2	48.29	odd	4
768.4.c.j.767.2		2	16.3	odd	4
768.4.c.j.767.2		2	16.5	even	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 $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	384.4.f.b.191.2

Level	$384$
Weight	$4$
Character	384.191
Analytic conductor	$22.657$
Analytic rank	$0$
Dimension	$4$
CM discriminant	-8
Inner twists	$8$