LMFDB - Embedded newform 9680.2.a.da.1.3

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			1.3
Root			$1.90131$ 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)

$f(q)$	$=$	$q-0.614975 q^{3} +1.00000 q^{5} -2.24598 q^{7} -2.62181 q^{9} -5.19447 q^{13} -0.614975 q^{15} +3.22995 q^{17} -7.19447 q^{19} +1.38122 q^{21} -5.11377 q^{23} +1.00000 q^{25} +3.45727 q^{27} -9.61890 q^{29} -0.491468 q^{31} -2.24598 q^{35} +0.350828 q^{37} +3.19447 q^{39} +4.42090 q^{41} -10.9537 q^{43} -2.62181 q^{45} +1.03420 q^{47} -1.95557 q^{49} -1.98634 q^{51} +3.75348 q^{53} +4.42442 q^{57} -12.7082 q^{59} +6.38983 q^{61} +5.88852 q^{63} -5.19447 q^{65} +8.44020 q^{67} +3.14484 q^{69} +12.4475 q^{71} +12.0273 q^{73} -0.614975 q^{75} +2.13034 q^{79} +5.73928 q^{81} -10.7270 q^{83} +3.22995 q^{85} +5.91538 q^{87} +9.90118 q^{89} +11.6667 q^{91} +0.302241 q^{93} -7.19447 q^{95} -15.9253 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−0.614975	−0.355056	−0.177528	−	0.984116i	$-0.556810\pi$
$3$	−0.614975	−0.355056	−0.177528	+	0.984116i	$0.556810\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−2.24598	−0.848901	−0.424450	−	0.905451i	$-0.639533\pi$
$7$	−2.24598	−0.848901	−0.424450	+	0.905451i	$0.639533\pi$
$8$	0	0
$9$	−2.62181	−0.873935
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−5.19447	−1.44069	−0.720344	−	0.693617i	$-0.756016\pi$
$13$	−5.19447	−1.44069	−0.720344	+	0.693617i	$0.756016\pi$
$14$	0	0
$15$	−0.614975	−0.158786
$16$	0	0
$17$	3.22995	0.783378	0.391689	−	0.920098i	$-0.371891\pi$
$17$	3.22995	0.783378	0.391689	+	0.920098i	$0.371891\pi$
$18$	0	0
$19$	−7.19447	−1.65053	−0.825263	−	0.564749i	$-0.808973\pi$
$19$	−7.19447	−1.65053	−0.825263	+	0.564749i	$0.808973\pi$
$20$	0	0
$21$	1.38122	0.301408
$22$	0	0
$23$	−5.11377	−1.06629	−0.533147	−	0.846023i	$-0.678991\pi$
$23$	−5.11377	−1.06629	−0.533147	+	0.846023i	$0.678991\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	3.45727	0.665352
$28$	0	0
$29$	−9.61890	−1.78618	−0.893092	−	0.449874i	$-0.851469\pi$
$29$	−9.61890	−1.78618	−0.893092	+	0.449874i	$0.851469\pi$
$30$	0	0
$31$	−0.491468	−0.0882703	−0.0441352	−	0.999026i	$-0.514053\pi$
$31$	−0.491468	−0.0882703	−0.0441352	+	0.999026i	$0.514053\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.24598	−0.379640
$36$	0	0
$37$	0.350828	0.0576758	0.0288379	−	0.999584i	$-0.490819\pi$
$37$	0.350828	0.0576758	0.0288379	+	0.999584i	$0.490819\pi$
$38$	0	0
$39$	3.19447	0.511525
$40$	0	0
$41$	4.42090	0.690429	0.345214	−	0.938524i	$-0.387806\pi$
$41$	4.42090	0.690429	0.345214	+	0.938524i	$0.387806\pi$
$42$	0	0
$43$	−10.9537	−1.67042	−0.835211	−	0.549929i	$-0.814655\pi$
$43$	−10.9537	−1.67042	−0.835211	+	0.549929i	$0.814655\pi$
$44$	0	0
$45$	−2.62181	−0.390836
$46$	0	0
$47$	1.03420	0.150853	0.0754265	−	0.997151i	$-0.475968\pi$
$47$	1.03420	0.150853	0.0754265	+	0.997151i	$0.475968\pi$
$48$	0	0
$49$	−1.95557	−0.279367
$50$	0	0
$51$	−1.98634	−0.278143
$52$	0	0
$53$	3.75348	0.515580	0.257790	−	0.966201i	$-0.417006\pi$
$53$	3.75348	0.515580	0.257790	+	0.966201i	$0.417006\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	4.42442	0.586029
$58$	0	0
$59$	−12.7082	−1.65447	−0.827234	−	0.561858i	$-0.810087\pi$
$59$	−12.7082	−1.65447	−0.827234	+	0.561858i	$0.810087\pi$
$60$	0	0
$61$	6.38983	0.818134	0.409067	−	0.912504i	$-0.365854\pi$
$61$	6.38983	0.818134	0.409067	+	0.912504i	$0.365854\pi$
$62$	0	0
$63$	5.88852	0.741884
$64$	0	0
$65$	−5.19447	−0.644295
$66$	0	0
$67$	8.44020	1.03114	0.515568	−	0.856849i	$-0.327581\pi$
$67$	8.44020	1.03114	0.515568	+	0.856849i	$0.327581\pi$
$68$	0	0
$69$	3.14484	0.378594
$70$	0	0
$71$	12.4475	1.47725	0.738625	−	0.674116i	$-0.235475\pi$
$71$	12.4475	1.47725	0.738625	+	0.674116i	$0.235475\pi$
$72$	0	0
$73$	12.0273	1.40769	0.703846	−	0.710353i	$-0.251465\pi$
$73$	12.0273	1.40769	0.703846	+	0.710353i	$0.251465\pi$
$74$	0	0
$75$	−0.614975	−0.0710112
$76$	0	0
$77$	0	0
$78$	0	0
$79$	2.13034	0.239682	0.119841	−	0.992793i	$-0.461762\pi$
$79$	2.13034	0.239682	0.119841	+	0.992793i	$0.461762\pi$
$80$	0	0
$81$	5.73928	0.637698
$82$	0	0
$83$	−10.7270	−1.17744	−0.588719	−	0.808338i	$-0.700367\pi$
$83$	−10.7270	−1.17744	−0.588719	+	0.808338i	$0.700367\pi$
$84$	0	0
$85$	3.22995	0.350337
$86$	0	0
$87$	5.91538	0.634196
$88$	0	0
$89$	9.90118	1.04952	0.524762	−	0.851249i	$-0.324154\pi$
$89$	9.90118	1.04952	0.524762	+	0.851249i	$0.324154\pi$
$90$	0	0
$91$	11.6667	1.22300
$92$	0	0
$93$	0.302241	0.0313409
$94$	0	0
$95$	−7.19447	−0.738137
$96$	0	0
$97$	−15.9253	−1.61697	−0.808484	−	0.588518i	$-0.799712\pi$
$97$	−15.9253	−1.61697	−0.808484	+	0.588518i	$0.799712\pi$
$98$	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.1.3		6
4.3	odd	2		4840.2.a.bd.1.4	yes	6
11.10	odd	2		9680.2.a.db.1.3		6
44.43	even	2		4840.2.a.bc.1.4	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4840.2.a.bc.1.4	✓	6	44.43	even	2
4840.2.a.bd.1.4	yes	6	4.3	odd	2
9680.2.a.da.1.3		6	1.1	even	1	trivial
9680.2.a.db.1.3		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)

\(f(q)\)	\(=\)	\(q-0.614975 q^{3} +1.00000 q^{5} -2.24598 q^{7} -2.62181 q^{9} -5.19447 q^{13} -0.614975 q^{15} +3.22995 q^{17} -7.19447 q^{19} +1.38122 q^{21} -5.11377 q^{23} +1.00000 q^{25} +3.45727 q^{27} -9.61890 q^{29} -0.491468 q^{31} -2.24598 q^{35} +0.350828 q^{37} +3.19447 q^{39} +4.42090 q^{41} -10.9537 q^{43} -2.62181 q^{45} +1.03420 q^{47} -1.95557 q^{49} -1.98634 q^{51} +3.75348 q^{53} +4.42442 q^{57} -12.7082 q^{59} +6.38983 q^{61} +5.88852 q^{63} -5.19447 q^{65} +8.44020 q^{67} +3.14484 q^{69} +12.4475 q^{71} +12.0273 q^{73} -0.614975 q^{75} +2.13034 q^{79} +5.73928 q^{81} -10.7270 q^{83} +3.22995 q^{85} +5.91538 q^{87} +9.90118 q^{89} +11.6667 q^{91} +0.302241 q^{93} -7.19447 q^{95} -15.9253 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{3} + 6 q^{5} - 4 q^{7} + 4 q^{9} + 2 q^{15} + 8 q^{17} - 12 q^{19} - 8 q^{21} + 8 q^{23} + 6 q^{25} + 14 q^{27} + 16 q^{29} + 4 q^{31} - 4 q^{35} + 8 q^{37} - 12 q^{39} + 32 q^{41} + 4 q^{43}+ \cdots - 12 q^{95}+O(q^{100}) \) 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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