LMFDB - Embedded newform 9680.2.a.bj.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 = [2,0,-2,0,2,0,-2,0,0,0,0,0,4,0,-2,0,0] 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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 4840)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ 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)

$f(q)$	$=$	$q-2.41421 q^{3} +1.00000 q^{5} -2.41421 q^{7} +2.82843 q^{9} +2.00000 q^{13} -2.41421 q^{15} +5.65685 q^{17} -4.82843 q^{19} +5.82843 q^{21} -3.65685 q^{23} +1.00000 q^{25} +0.414214 q^{27} +2.00000 q^{29} +5.65685 q^{31} -2.41421 q^{35} -4.00000 q^{37} -4.82843 q^{39} -9.48528 q^{41} -3.58579 q^{43} +2.82843 q^{45} -7.58579 q^{47} -1.17157 q^{49} -13.6569 q^{51} -7.65685 q^{53} +11.6569 q^{57} +11.3137 q^{59} +1.00000 q^{61} -6.82843 q^{63} +2.00000 q^{65} +6.41421 q^{67} +8.82843 q^{69} +4.00000 q^{71} +4.00000 q^{73} -2.41421 q^{75} +14.4853 q^{79} -9.48528 q^{81} +13.3137 q^{83} +5.65685 q^{85} -4.82843 q^{87} +2.65685 q^{89} -4.82843 q^{91} -13.6569 q^{93} -4.82843 q^{95} -17.3137 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{5} - 2 q^{7} + 4 q^{13} - 2 q^{15} - 4 q^{19} + 6 q^{21} + 4 q^{23} + 2 q^{25} - 2 q^{27} + 4 q^{29} - 2 q^{35} - 8 q^{37} - 4 q^{39} - 2 q^{41} - 10 q^{43} - 18 q^{47} - 8 q^{49} - 16 q^{51}+ \cdots - 12 q^{97}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^5 - 2 * q^7 + 4 * q^13 - 2 * q^15 - 4 * q^19 + 6 * q^21 + 4 * q^23 + 2 * q^25 - 2 * q^27 + 4 * q^29 - 2 * q^35 - 8 * q^37 - 4 * q^39 - 2 * q^41 - 10 * q^43 - 18 * q^47 - 8 * q^49 - 16 * q^51 - 4 * q^53 + 12 * q^57 + 2 * q^61 - 8 * q^63 + 4 * q^65 + 10 * q^67 + 12 * q^69 + 8 * q^71 + 8 * q^73 - 2 * q^75 + 12 * q^79 - 2 * q^81 + 4 * q^83 - 4 * q^87 - 6 * q^89 - 4 * q^91 - 16 * q^93 - 4 * q^95 - 12 * q^97

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
$3$	−2.41421	−1.39385	−0.696923	−	0.717146i	$-0.745448\pi$
$3$	−2.41421	−1.39385	−0.696923	+	0.717146i	$0.745448\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−2.41421	−0.912487	−0.456243	−	0.889855i	$-0.650805\pi$
$7$	−2.41421	−0.912487	−0.456243	+	0.889855i	$0.650805\pi$
$8$	0	0
$9$	2.82843	0.942809
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	−2.41421	−0.623347
$16$	0	0
$17$	5.65685	1.37199	0.685994	−	0.727607i	$-0.259367\pi$
$17$	5.65685	1.37199	0.685994	+	0.727607i	$0.259367\pi$
$18$	0	0
$19$	−4.82843	−1.10772	−0.553859	−	0.832611i	$-0.686845\pi$
$19$	−4.82843	−1.10772	−0.553859	+	0.832611i	$0.686845\pi$
$20$	0	0
$21$	5.82843	1.27187
$22$	0	0
$23$	−3.65685	−0.762507	−0.381253	−	0.924471i	$-0.624507\pi$
$23$	−3.65685	−0.762507	−0.381253	+	0.924471i	$0.624507\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0.414214	0.0797154
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	5.65685	1.01600	0.508001	−	0.861357i	$-0.330385\pi$
$31$	5.65685	1.01600	0.508001	+	0.861357i	$0.330385\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.41421	−0.408077
$36$	0	0
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	0	0
$39$	−4.82843	−0.773167
$40$	0	0
$41$	−9.48528	−1.48135	−0.740676	−	0.671862i	$-0.765495\pi$
$41$	−9.48528	−1.48135	−0.740676	+	0.671862i	$0.765495\pi$
$42$	0	0
$43$	−3.58579	−0.546827	−0.273414	−	0.961897i	$-0.588153\pi$
$43$	−3.58579	−0.546827	−0.273414	+	0.961897i	$0.588153\pi$
$44$	0	0
$45$	2.82843	0.421637
$46$	0	0
$47$	−7.58579	−1.10650	−0.553250	−	0.833015i	$-0.686613\pi$
$47$	−7.58579	−1.10650	−0.553250	+	0.833015i	$0.686613\pi$
$48$	0	0
$49$	−1.17157	−0.167368
$50$	0	0
$51$	−13.6569	−1.91234
$52$	0	0
$53$	−7.65685	−1.05175	−0.525875	−	0.850562i	$-0.676262\pi$
$53$	−7.65685	−1.05175	−0.525875	+	0.850562i	$0.676262\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	11.6569	1.54399
$58$	0	0
$59$	11.3137	1.47292	0.736460	−	0.676481i	$-0.236496\pi$
$59$	11.3137	1.47292	0.736460	+	0.676481i	$0.236496\pi$
$60$	0	0
$61$	1.00000	0.128037	0.0640184	−	0.997949i	$-0.479608\pi$
$61$	1.00000	0.128037	0.0640184	+	0.997949i	$0.479608\pi$
$62$	0	0
$63$	−6.82843	−0.860301
$64$	0	0
$65$	2.00000	0.248069
$66$	0	0
$67$	6.41421	0.783621	0.391810	−	0.920046i	$-0.371849\pi$
$67$	6.41421	0.783621	0.391810	+	0.920046i	$0.371849\pi$
$68$	0	0
$69$	8.82843	1.06282
$70$	0	0
$71$	4.00000	0.474713	0.237356	−	0.971423i	$-0.423719\pi$
$71$	4.00000	0.474713	0.237356	+	0.971423i	$0.423719\pi$
$72$	0	0
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	0	0
$75$	−2.41421	−0.278769
$76$	0	0
$77$	0	0
$78$	0	0
$79$	14.4853	1.62972	0.814861	−	0.579657i	$-0.196813\pi$
$79$	14.4853	1.62972	0.814861	+	0.579657i	$0.196813\pi$
$80$	0	0
$81$	−9.48528	−1.05392
$82$	0	0
$83$	13.3137	1.46137	0.730685	−	0.682715i	$-0.239201\pi$
$83$	13.3137	1.46137	0.730685	+	0.682715i	$0.239201\pi$
$84$	0	0
$85$	5.65685	0.613572
$86$	0	0
$87$	−4.82843	−0.517662
$88$	0	0
$89$	2.65685	0.281626	0.140813	−	0.990036i	$-0.455028\pi$
$89$	2.65685	0.281626	0.140813	+	0.990036i	$0.455028\pi$
$90$	0	0
$91$	−4.82843	−0.506157
$92$	0	0
$93$	−13.6569	−1.41615
$94$	0	0
$95$	−4.82843	−0.495386
$96$	0	0
$97$	−17.3137	−1.75794	−0.878970	−	0.476876i	$-0.841769\pi$
$97$	−17.3137	−1.75794	−0.878970	+	0.476876i	$0.841769\pi$
$98$	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.bj.1.1		2
4.3	odd	2		4840.2.a.p.1.2	yes	2
11.10	odd	2		9680.2.a.bk.1.1		2
44.43	even	2		4840.2.a.o.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4840.2.a.o.1.2	✓	2	44.43	even	2
4840.2.a.p.1.2	yes	2	4.3	odd	2
9680.2.a.bj.1.1		2	1.1	even	1	trivial
9680.2.a.bk.1.1		2	11.10	odd	2

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Elliptic curves over $\Q$
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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)

\(f(q)\)	\(=\)	\(q-2.41421 q^{3} +1.00000 q^{5} -2.41421 q^{7} +2.82843 q^{9} +2.00000 q^{13} -2.41421 q^{15} +5.65685 q^{17} -4.82843 q^{19} +5.82843 q^{21} -3.65685 q^{23} +1.00000 q^{25} +0.414214 q^{27} +2.00000 q^{29} +5.65685 q^{31} -2.41421 q^{35} -4.00000 q^{37} -4.82843 q^{39} -9.48528 q^{41} -3.58579 q^{43} +2.82843 q^{45} -7.58579 q^{47} -1.17157 q^{49} -13.6569 q^{51} -7.65685 q^{53} +11.6569 q^{57} +11.3137 q^{59} +1.00000 q^{61} -6.82843 q^{63} +2.00000 q^{65} +6.41421 q^{67} +8.82843 q^{69} +4.00000 q^{71} +4.00000 q^{73} -2.41421 q^{75} +14.4853 q^{79} -9.48528 q^{81} +13.3137 q^{83} +5.65685 q^{85} -4.82843 q^{87} +2.65685 q^{89} -4.82843 q^{91} -13.6569 q^{93} -4.82843 q^{95} -17.3137 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{5} - 2 q^{7} + 4 q^{13} - 2 q^{15} - 4 q^{19} + 6 q^{21} + 4 q^{23} + 2 q^{25} - 2 q^{27} + 4 q^{29} - 2 q^{35} - 8 q^{37} - 4 q^{39} - 2 q^{41} - 10 q^{43} - 18 q^{47} - 8 q^{49} - 16 q^{51}+ \cdots - 12 q^{97}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^5 - 2 * q^7 + 4 * q^13 - 2 * q^15 - 4 * q^19 + 6 * q^21 + 4 * q^23 + 2 * q^25 - 2 * q^27 + 4 * q^29 - 2 * q^35 - 8 * q^37 - 4 * q^39 - 2 * q^41 - 10 * q^43 - 18 * q^47 - 8 * q^49 - 16 * q^51 - 4 * q^53 + 12 * q^57 + 2 * q^61 - 8 * q^63 + 4 * q^65 + 10 * q^67 + 12 * q^69 + 8 * q^71 + 8 * q^73 - 2 * q^75 + 12 * q^79 - 2 * q^81 + 4 * q^83 - 4 * q^87 - 6 * q^89 - 4 * q^91 - 16 * q^93 - 4 * q^95 - 12 * q^97

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