LMFDB - Embedded newform 9680.2.a.da.1.1

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			1.1
Root			$2.27997$ 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)

$f(q)$	$=$	$q-2.19828 q^{3} +1.00000 q^{5} -2.58485 q^{7} +1.83244 q^{9} +1.66906 q^{13} -2.19828 q^{15} +6.39656 q^{17} -0.330943 q^{19} +5.68224 q^{21} -1.33727 q^{23} +1.00000 q^{25} +2.56662 q^{27} +0.941551 q^{29} -5.78263 q^{31} -2.58485 q^{35} -7.41429 q^{37} -3.66906 q^{39} +5.49063 q^{41} +4.54413 q^{43} +1.83244 q^{45} +7.21601 q^{47} -0.318527 q^{49} -14.0614 q^{51} +6.55577 q^{53} +0.727506 q^{57} +7.74191 q^{59} +3.88821 q^{61} -4.73660 q^{63} +1.66906 q^{65} -9.68736 q^{67} +2.93969 q^{69} +11.4882 q^{71} -12.1229 q^{73} -2.19828 q^{75} -7.61507 q^{79} -11.1395 q^{81} -9.29886 q^{83} +6.39656 q^{85} -2.06979 q^{87} -17.7651 q^{89} -4.31427 q^{91} +12.7118 q^{93} -0.330943 q^{95} +4.03731 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.19828	−1.26918	−0.634589	−	0.772850i	$-0.718831\pi$
$3$	−2.19828	−1.26918	−0.634589	+	0.772850i	$0.718831\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−2.58485	−0.976983	−0.488492	−	0.872569i	$-0.662453\pi$
$7$	−2.58485	−0.976983	−0.488492	+	0.872569i	$0.662453\pi$
$8$	0	0
$9$	1.83244	0.610814
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	1.66906	0.462913	0.231457	−	0.972845i	$-0.425651\pi$
$13$	1.66906	0.462913	0.231457	+	0.972845i	$0.425651\pi$
$14$	0	0
$15$	−2.19828	−0.567594
$16$	0	0
$17$	6.39656	1.55139	0.775697	−	0.631105i	$-0.217398\pi$
$17$	6.39656	1.55139	0.775697	+	0.631105i	$0.217398\pi$
$18$	0	0
$19$	−0.330943	−0.0759236	−0.0379618	−	0.999279i	$-0.512087\pi$
$19$	−0.330943	−0.0759236	−0.0379618	+	0.999279i	$0.512087\pi$
$20$	0	0
$21$	5.68224	1.23997
$22$	0	0
$23$	−1.33727	−0.278839	−0.139420	−	0.990233i	$-0.544524\pi$
$23$	−1.33727	−0.278839	−0.139420	+	0.990233i	$0.544524\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	2.56662	0.493947
$28$	0	0
$29$	0.941551	0.174842	0.0874208	−	0.996171i	$-0.472138\pi$
$29$	0.941551	0.174842	0.0874208	+	0.996171i	$0.472138\pi$
$30$	0	0
$31$	−5.78263	−1.03859	−0.519295	−	0.854595i	$-0.673806\pi$
$31$	−5.78263	−1.03859	−0.519295	+	0.854595i	$0.673806\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.58485	−0.436920
$36$	0	0
$37$	−7.41429	−1.21890	−0.609451	−	0.792824i	$-0.708610\pi$
$37$	−7.41429	−1.21890	−0.609451	+	0.792824i	$0.708610\pi$
$38$	0	0
$39$	−3.66906	−0.587519
$40$	0	0
$41$	5.49063	0.857492	0.428746	−	0.903425i	$-0.358955\pi$
$41$	5.49063	0.857492	0.428746	+	0.903425i	$0.358955\pi$
$42$	0	0
$43$	4.54413	0.692973	0.346487	−	0.938055i	$-0.387375\pi$
$43$	4.54413	0.692973	0.346487	+	0.938055i	$0.387375\pi$
$44$	0	0
$45$	1.83244	0.273164
$46$	0	0
$47$	7.21601	1.05256	0.526281	−	0.850310i	$-0.323586\pi$
$47$	7.21601	1.05256	0.526281	+	0.850310i	$0.323586\pi$
$48$	0	0
$49$	−0.318527	−0.0455039
$50$	0	0
$51$	−14.0614	−1.96900
$52$	0	0
$53$	6.55577	0.900505	0.450252	−	0.892901i	$-0.351334\pi$
$53$	6.55577	0.900505	0.450252	+	0.892901i	$0.351334\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0.727506	0.0963606
$58$	0	0
$59$	7.74191	1.00791	0.503955	−	0.863730i	$-0.331878\pi$
$59$	7.74191	1.00791	0.503955	+	0.863730i	$0.331878\pi$
$60$	0	0
$61$	3.88821	0.497834	0.248917	−	0.968525i	$-0.419925\pi$
$61$	3.88821	0.497834	0.248917	+	0.968525i	$0.419925\pi$
$62$	0	0
$63$	−4.73660	−0.596755
$64$	0	0
$65$	1.66906	0.207021
$66$	0	0
$67$	−9.68736	−1.18350	−0.591750	−	0.806122i	$-0.701563\pi$
$67$	−9.68736	−1.18350	−0.591750	+	0.806122i	$0.701563\pi$
$68$	0	0
$69$	2.93969	0.353897
$70$	0	0
$71$	11.4882	1.36340	0.681701	−	0.731631i	$-0.261240\pi$
$71$	11.4882	1.36340	0.681701	+	0.731631i	$0.261240\pi$
$72$	0	0
$73$	−12.1229	−1.41888	−0.709439	−	0.704767i	$-0.751051\pi$
$73$	−12.1229	−1.41888	−0.709439	+	0.704767i	$0.751051\pi$
$74$	0	0
$75$	−2.19828	−0.253836
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−7.61507	−0.856762	−0.428381	−	0.903598i	$-0.640916\pi$
$79$	−7.61507	−0.856762	−0.428381	+	0.903598i	$0.640916\pi$
$80$	0	0
$81$	−11.1395	−1.23772
$82$	0	0
$83$	−9.29886	−1.02068	−0.510341	−	0.859972i	$-0.670481\pi$
$83$	−9.29886	−1.02068	−0.510341	+	0.859972i	$0.670481\pi$
$84$	0	0
$85$	6.39656	0.693805
$86$	0	0
$87$	−2.06979	−0.221905
$88$	0	0
$89$	−17.7651	−1.88309	−0.941546	−	0.336885i	$-0.890627\pi$
$89$	−17.7651	−1.88309	−0.941546	+	0.336885i	$0.890627\pi$
$90$	0	0
$91$	−4.31427	−0.452258
$92$	0	0
$93$	12.7118	1.31816
$94$	0	0
$95$	−0.330943	−0.0339541
$96$	0	0
$97$	4.03731	0.409927	0.204963	−	0.978770i	$-0.434292\pi$
$97$	4.03731	0.409927	0.204963	+	0.978770i	$0.434292\pi$
$98$	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.1.1		6
4.3	odd	2		4840.2.a.bd.1.6	yes	6
11.10	odd	2		9680.2.a.db.1.1		6
44.43	even	2		4840.2.a.bc.1.6	✓	6

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-2.19828 q^{3} +1.00000 q^{5} -2.58485 q^{7} +1.83244 q^{9} +1.66906 q^{13} -2.19828 q^{15} +6.39656 q^{17} -0.330943 q^{19} +5.68224 q^{21} -1.33727 q^{23} +1.00000 q^{25} +2.56662 q^{27} +0.941551 q^{29} -5.78263 q^{31} -2.58485 q^{35} -7.41429 q^{37} -3.66906 q^{39} +5.49063 q^{41} +4.54413 q^{43} +1.83244 q^{45} +7.21601 q^{47} -0.318527 q^{49} -14.0614 q^{51} +6.55577 q^{53} +0.727506 q^{57} +7.74191 q^{59} +3.88821 q^{61} -4.73660 q^{63} +1.66906 q^{65} -9.68736 q^{67} +2.93969 q^{69} +11.4882 q^{71} -12.1229 q^{73} -2.19828 q^{75} -7.61507 q^{79} -11.1395 q^{81} -9.29886 q^{83} +6.39656 q^{85} -2.06979 q^{87} -17.7651 q^{89} -4.31427 q^{91} +12.7118 q^{93} -0.330943 q^{95} +4.03731 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{3} + 6 q^{5} - 4 q^{7} + 4 q^{9} + 2 q^{15} + 8 q^{17} - 12 q^{19} - 8 q^{21} + 8 q^{23} + 6 q^{25} + 14 q^{27} + 16 q^{29} + 4 q^{31} - 4 q^{35} + 8 q^{37} - 12 q^{39} + 32 q^{41} + 4 q^{43}+ \cdots - 12 q^{95}+O(q^{100}) \) 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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