LMFDB - Embedded newform 9680.2.a.da.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9680,2,Mod(1,9680)] 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("9680.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9680, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 9680 = 2^{4} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9680.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,2,0,6,0,-4,0,4,0,0,0,0,0,2,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$77.2951891566$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.22733568.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 8x^{4} - 2x^{3} + 16x^{2} + 8x - 2 $ x^6 - 8x^4 - 2x^3 + 16x^2 + 8x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 4840)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$0.184585$ of defining polynomial
Character	$\chi$	$=$	9680.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.96593 q^{3} +1.00000 q^{5} -1.02876 q^{7} +5.79673 q^{9} -0.504296 q^{13} +2.96593 q^{15} -3.93186 q^{17} -2.50430 q^{19} -3.05124 q^{21} +5.73921 q^{23} +1.00000 q^{25} +8.29490 q^{27} +6.92327 q^{29} -4.47755 q^{31} -1.02876 q^{35} +4.78329 q^{37} -1.49570 q^{39} +11.8404 q^{41} +7.10060 q^{43} +5.79673 q^{45} +0.182642 q^{47} -5.94165 q^{49} -11.6616 q^{51} +12.4669 q^{53} -7.42756 q^{57} +10.5494 q^{59} -10.9209 q^{61} -5.96346 q^{63} -0.504296 q^{65} -9.76386 q^{67} +17.0221 q^{69} +13.9992 q^{71} -7.32321 q^{73} +2.96593 q^{75} -10.2743 q^{79} +7.21190 q^{81} -4.33368 q^{83} -3.93186 q^{85} +20.5339 q^{87} +17.2789 q^{89} +0.518800 q^{91} -13.2801 q^{93} -2.50430 q^{95} -7.80167 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{3} + 6 q^{5} - 4 q^{7} + 4 q^{9} + 2 q^{15} + 8 q^{17} - 12 q^{19} - 8 q^{21} + 8 q^{23} + 6 q^{25} + 14 q^{27} + 16 q^{29} + 4 q^{31} - 4 q^{35} + 8 q^{37} - 12 q^{39} + 32 q^{41} + 4 q^{43}+ \cdots - 12 q^{95}+O(q^{100}) $ 6 * q + 2 * q^3 + 6 * q^5 - 4 * q^7 + 4 * q^9 + 2 * q^15 + 8 * q^17 - 12 * q^19 - 8 * q^21 + 8 * q^23 + 6 * q^25 + 14 * q^27 + 16 * q^29 + 4 * q^31 - 4 * q^35 + 8 * q^37 - 12 * q^39 + 32 * q^41 + 4 * q^43 + 4 * q^45 + 6 * q^47 + 16 * q^49 - 40 * q^51 + 8 * q^53 - 16 * q^57 - 4 * q^59 + 16 * q^61 - 28 * q^63 + 2 * q^67 + 8 * q^69 + 28 * q^71 + 16 * q^73 + 2 * q^75 - 10 * q^81 - 12 * q^83 + 8 * q^85 + 24 * q^87 + 18 * q^89 + 24 * q^91 + 20 * q^93 - 12 * q^95

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$. You can download additional coefficients here.

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display 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
$3$	2.96593	1.71238	0.856190	−	0.516661i	$-0.172825\pi$
$3$	2.96593	1.71238	0.856190	+	0.516661i	$0.172825\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.02876	−0.388836	−0.194418	−	0.980919i	$-0.562282\pi$
$7$	−1.02876	−0.388836	−0.194418	+	0.980919i	$0.562282\pi$
$8$	0	0
$9$	5.79673	1.93224
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−0.504296	−0.139866	−0.0699332	−	0.997552i	$-0.522279\pi$
$13$	−0.504296	−0.139866	−0.0699332	+	0.997552i	$0.522279\pi$
$14$	0	0
$15$	2.96593	0.765799
$16$	0	0
$17$	−3.93186	−0.953615	−0.476808	−	0.879008i	$-0.658206\pi$
$17$	−3.93186	−0.953615	−0.476808	+	0.879008i	$0.658206\pi$
$18$	0	0
$19$	−2.50430	−0.574525	−0.287262	−	0.957852i	$-0.592745\pi$
$19$	−2.50430	−0.574525	−0.287262	+	0.957852i	$0.592745\pi$
$20$	0	0
$21$	−3.05124	−0.665834
$22$	0	0
$23$	5.73921	1.19671	0.598354	−	0.801232i	$-0.295822\pi$
$23$	5.73921	1.19671	0.598354	+	0.801232i	$0.295822\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	8.29490	1.59636
$28$	0	0
$29$	6.92327	1.28562	0.642809	−	0.766026i	$-0.277769\pi$
$29$	6.92327	1.28562	0.642809	+	0.766026i	$0.277769\pi$
$30$	0	0
$31$	−4.47755	−0.804191	−0.402096	−	0.915598i	$-0.631718\pi$
$31$	−4.47755	−0.804191	−0.402096	+	0.915598i	$0.631718\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.02876	−0.173893
$36$	0	0
$37$	4.78329	0.786367	0.393184	−	0.919460i	$-0.371374\pi$
$37$	4.78329	0.786367	0.393184	+	0.919460i	$0.371374\pi$
$38$	0	0
$39$	−1.49570	−0.239504
$40$	0	0
$41$	11.8404	1.84916	0.924582	−	0.380983i	$-0.124414\pi$
$41$	11.8404	1.84916	0.924582	+	0.380983i	$0.124414\pi$
$42$	0	0
$43$	7.10060	1.08283	0.541416	−	0.840755i	$-0.317889\pi$
$43$	7.10060	1.08283	0.541416	+	0.840755i	$0.317889\pi$
$44$	0	0
$45$	5.79673	0.864126
$46$	0	0
$47$	0.182642	0.0266410	0.0133205	−	0.999911i	$-0.495760\pi$
$47$	0.182642	0.0266410	0.0133205	+	0.999911i	$0.495760\pi$
$48$	0	0
$49$	−5.94165	−0.848807
$50$	0	0
$51$	−11.6616	−1.63295
$52$	0	0
$53$	12.4669	1.71246	0.856232	−	0.516591i	$-0.172799\pi$
$53$	12.4669	1.71246	0.856232	+	0.516591i	$0.172799\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−7.42756	−0.983805
$58$	0	0
$59$	10.5494	1.37341	0.686706	−	0.726935i	$-0.259056\pi$
$59$	10.5494	1.37341	0.686706	+	0.726935i	$0.259056\pi$
$60$	0	0
$61$	−10.9209	−1.39827	−0.699136	−	0.714989i	$-0.746432\pi$
$61$	−10.9209	−1.39827	−0.699136	+	0.714989i	$0.746432\pi$
$62$	0	0
$63$	−5.96346	−0.751325
$64$	0	0
$65$	−0.504296	−0.0625502
$66$	0	0
$67$	−9.76386	−1.19285	−0.596423	−	0.802671i	$-0.703412\pi$
$67$	−9.76386	−1.19285	−0.596423	+	0.802671i	$0.703412\pi$
$68$	0	0
$69$	17.0221	2.04922
$70$	0	0
$71$	13.9992	1.66140	0.830698	−	0.556723i	$-0.187942\pi$
$71$	13.9992	1.66140	0.830698	+	0.556723i	$0.187942\pi$
$72$	0	0
$73$	−7.32321	−0.857117	−0.428559	−	0.903514i	$-0.640978\pi$
$73$	−7.32321	−0.857117	−0.428559	+	0.903514i	$0.640978\pi$
$74$	0	0
$75$	2.96593	0.342476
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−10.2743	−1.15595	−0.577973	−	0.816056i	$-0.696156\pi$
$79$	−10.2743	−1.15595	−0.577973	+	0.816056i	$0.696156\pi$
$80$	0	0
$81$	7.21190	0.801322
$82$	0	0
$83$	−4.33368	−0.475683	−0.237842	−	0.971304i	$-0.576440\pi$
$83$	−4.33368	−0.475683	−0.237842	+	0.971304i	$0.576440\pi$
$84$	0	0
$85$	−3.93186	−0.426470
$86$	0	0
$87$	20.5339	2.20147
$88$	0	0
$89$	17.2789	1.83156	0.915779	−	0.401683i	$-0.131575\pi$
$89$	17.2789	1.83156	0.915779	+	0.401683i	$0.131575\pi$
$90$	0	0
$91$	0.518800	0.0543851
$92$	0	0
$93$	−13.2801	−1.37708
$94$	0	0
$95$	−2.50430	−0.256935
$96$	0	0
$97$	−7.80167	−0.792139	−0.396070	−	0.918220i	$-0.629626\pi$
$97$	−7.80167	−0.792139	−0.396070	+	0.918220i	$0.629626\pi$
$98$	0	0
$99$	0	0

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display only $a_p$

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9680.2.a.da.1.6		6
4.3	odd	2		4840.2.a.bd.1.1	yes	6
11.10	odd	2		9680.2.a.db.1.6		6
44.43	even	2		4840.2.a.bc.1.1	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4840.2.a.bc.1.1	✓	6	44.43	even	2
4840.2.a.bd.1.1	yes	6	4.3	odd	2
9680.2.a.da.1.6		6	1.1	even	1	trivial
9680.2.a.db.1.6		6	11.10	odd	2

Overview	Random
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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

Level:	\( N \)	\(=\)	\( 9680 = 2^{4} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9680.a (trivial)

\(f(q)\)	\(=\)	\(q+2.96593 q^{3} +1.00000 q^{5} -1.02876 q^{7} +5.79673 q^{9} -0.504296 q^{13} +2.96593 q^{15} -3.93186 q^{17} -2.50430 q^{19} -3.05124 q^{21} +5.73921 q^{23} +1.00000 q^{25} +8.29490 q^{27} +6.92327 q^{29} -4.47755 q^{31} -1.02876 q^{35} +4.78329 q^{37} -1.49570 q^{39} +11.8404 q^{41} +7.10060 q^{43} +5.79673 q^{45} +0.182642 q^{47} -5.94165 q^{49} -11.6616 q^{51} +12.4669 q^{53} -7.42756 q^{57} +10.5494 q^{59} -10.9209 q^{61} -5.96346 q^{63} -0.504296 q^{65} -9.76386 q^{67} +17.0221 q^{69} +13.9992 q^{71} -7.32321 q^{73} +2.96593 q^{75} -10.2743 q^{79} +7.21190 q^{81} -4.33368 q^{83} -3.93186 q^{85} +20.5339 q^{87} +17.2789 q^{89} +0.518800 q^{91} -13.2801 q^{93} -2.50430 q^{95} -7.80167 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{3} + 6 q^{5} - 4 q^{7} + 4 q^{9} + 2 q^{15} + 8 q^{17} - 12 q^{19} - 8 q^{21} + 8 q^{23} + 6 q^{25} + 14 q^{27} + 16 q^{29} + 4 q^{31} - 4 q^{35} + 8 q^{37} - 12 q^{39} + 32 q^{41} + 4 q^{43}+ \cdots - 12 q^{95}+O(q^{100}) \) 6 * q + 2 * q^3 + 6 * q^5 - 4 * q^7 + 4 * q^9 + 2 * q^15 + 8 * q^17 - 12 * q^19 - 8 * q^21 + 8 * q^23 + 6 * q^25 + 14 * q^27 + 16 * q^29 + 4 * q^31 - 4 * q^35 + 8 * q^37 - 12 * q^39 + 32 * q^41 + 4 * q^43 + 4 * q^45 + 6 * q^47 + 16 * q^49 - 40 * q^51 + 8 * q^53 - 16 * q^57 - 4 * q^59 + 16 * q^61 - 28 * q^63 + 2 * q^67 + 8 * q^69 + 28 * q^71 + 16 * q^73 + 2 * q^75 - 10 * q^81 - 12 * q^83 + 8 * q^85 + 24 * q^87 + 18 * q^89 + 24 * q^91 + 20 * q^93 - 12 * q^95

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