LMFDB - Embedded newform 7920.2.a.by.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7920,2,Mod(1,7920)] 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("7920.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7920, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 7920 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7920.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,2,0,-5,0,0,0,-2,0,6,0,0,0,3,0,-7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	yes
Analytic conductor:	$63.2415184009$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 440)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	7920.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} -0.438447 q^{7} -1.00000 q^{11} +7.12311 q^{13} -4.68466 q^{17} -5.56155 q^{19} -7.12311 q^{23} +1.00000 q^{25} -4.43845 q^{29} +5.56155 q^{31} -0.438447 q^{35} +11.5616 q^{37} -4.24621 q^{41} -5.12311 q^{43} +13.3693 q^{47} -6.80776 q^{49} +2.68466 q^{53} -1.00000 q^{55} -7.12311 q^{59} -8.43845 q^{61} +7.12311 q^{65} -8.68466 q^{71} -7.12311 q^{73} +0.438447 q^{77} -13.3693 q^{79} -6.00000 q^{83} -4.68466 q^{85} +2.68466 q^{89} -3.12311 q^{91} -5.56155 q^{95} -13.1231 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} - 5 q^{7} - 2 q^{11} + 6 q^{13} + 3 q^{17} - 7 q^{19} - 6 q^{23} + 2 q^{25} - 13 q^{29} + 7 q^{31} - 5 q^{35} + 19 q^{37} + 8 q^{41} - 2 q^{43} + 2 q^{47} + 7 q^{49} - 7 q^{53} - 2 q^{55}+ \cdots - 18 q^{97}+O(q^{100}) $ 2 * q + 2 * q^5 - 5 * q^7 - 2 * q^11 + 6 * q^13 + 3 * q^17 - 7 * q^19 - 6 * q^23 + 2 * q^25 - 13 * q^29 + 7 * q^31 - 5 * q^35 + 19 * q^37 + 8 * q^41 - 2 * q^43 + 2 * q^47 + 7 * q^49 - 7 * q^53 - 2 * q^55 - 6 * q^59 - 21 * q^61 + 6 * q^65 - 5 * q^71 - 6 * q^73 + 5 * q^77 - 2 * q^79 - 12 * q^83 + 3 * q^85 - 7 * q^89 + 2 * q^91 - 7 * q^95 - 18 * 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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−0.438447	−0.165717	−0.0828587	−	0.996561i	$-0.526405\pi$
$7$	−0.438447	−0.165717	−0.0828587	+	0.996561i	$0.526405\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	7.12311	1.97559	0.987797	−	0.155747i	$-0.0497784\pi$
$13$	7.12311	1.97559	0.987797	+	0.155747i	$0.0497784\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.68466	−1.13620	−0.568098	−	0.822961i	$-0.692321\pi$
$17$	−4.68466	−1.13620	−0.568098	+	0.822961i	$0.692321\pi$
$18$	0	0
$19$	−5.56155	−1.27591	−0.637954	−	0.770075i	$-0.720219\pi$
$19$	−5.56155	−1.27591	−0.637954	+	0.770075i	$0.720219\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−7.12311	−1.48527	−0.742635	−	0.669696i	$-0.766424\pi$
$23$	−7.12311	−1.48527	−0.742635	+	0.669696i	$0.766424\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−4.43845	−0.824199	−0.412099	−	0.911139i	$-0.635204\pi$
$29$	−4.43845	−0.824199	−0.412099	+	0.911139i	$0.635204\pi$
$30$	0	0
$31$	5.56155	0.998884	0.499442	−	0.866347i	$-0.333538\pi$
$31$	5.56155	0.998884	0.499442	+	0.866347i	$0.333538\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−0.438447	−0.0741111
$36$	0	0
$37$	11.5616	1.90071	0.950354	−	0.311171i	$-0.100721\pi$
$37$	11.5616	1.90071	0.950354	+	0.311171i	$0.100721\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−4.24621	−0.663147	−0.331573	−	0.943429i	$-0.607579\pi$
$41$	−4.24621	−0.663147	−0.331573	+	0.943429i	$0.607579\pi$
$42$	0	0
$43$	−5.12311	−0.781266	−0.390633	−	0.920546i	$-0.627744\pi$
$43$	−5.12311	−0.781266	−0.390633	+	0.920546i	$0.627744\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	13.3693	1.95012	0.975058	−	0.221952i	$-0.0712428\pi$
$47$	13.3693	1.95012	0.975058	+	0.221952i	$0.0712428\pi$
$48$	0	0
$49$	−6.80776	−0.972538
$50$	0	0
$51$	0	0
$52$	0	0
$53$	2.68466	0.368766	0.184383	−	0.982854i	$-0.440971\pi$
$53$	2.68466	0.368766	0.184383	+	0.982854i	$0.440971\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−7.12311	−0.927349	−0.463675	−	0.886006i	$-0.653469\pi$
$59$	−7.12311	−0.927349	−0.463675	+	0.886006i	$0.653469\pi$
$60$	0	0
$61$	−8.43845	−1.08043	−0.540216	−	0.841526i	$-0.681658\pi$
$61$	−8.43845	−1.08043	−0.540216	+	0.841526i	$0.681658\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	7.12311	0.883513
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−8.68466	−1.03068	−0.515340	−	0.856986i	$-0.672334\pi$
$71$	−8.68466	−1.03068	−0.515340	+	0.856986i	$0.672334\pi$
$72$	0	0
$73$	−7.12311	−0.833696	−0.416848	−	0.908976i	$-0.636865\pi$
$73$	−7.12311	−0.833696	−0.416848	+	0.908976i	$0.636865\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.438447	0.0499657
$78$	0	0
$79$	−13.3693	−1.50417	−0.752083	−	0.659069i	$-0.770951\pi$
$79$	−13.3693	−1.50417	−0.752083	+	0.659069i	$0.770951\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−6.00000	−0.658586	−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000	−0.658586	−0.329293	+	0.944228i	$0.606810\pi$
$84$	0	0
$85$	−4.68466	−0.508123
$86$	0	0
$87$	0	0
$88$	0	0
$89$	2.68466	0.284573	0.142287	−	0.989825i	$-0.454555\pi$
$89$	2.68466	0.284573	0.142287	+	0.989825i	$0.454555\pi$
$90$	0	0
$91$	−3.12311	−0.327390
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−5.56155	−0.570603
$96$	0	0
$97$	−13.1231	−1.33245	−0.666225	−	0.745751i	$-0.732091\pi$
$97$	−13.1231	−1.33245	−0.666225	+	0.745751i	$0.732091\pi$
$98$	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	7920.2.a.by.1.2		2
3.2	odd	2		880.2.a.k.1.2		2
4.3	odd	2		3960.2.a.bf.1.1		2
12.11	even	2		440.2.a.g.1.1	✓	2
15.2	even	4		4400.2.b.w.4049.2		4
15.8	even	4		4400.2.b.w.4049.3		4
15.14	odd	2		4400.2.a.bt.1.1		2
24.5	odd	2		3520.2.a.br.1.1		2
24.11	even	2		3520.2.a.bm.1.2		2
33.32	even	2		9680.2.a.bm.1.2		2
60.23	odd	4		2200.2.b.f.1849.2		4
60.47	odd	4		2200.2.b.f.1849.3		4
60.59	even	2		2200.2.a.l.1.2		2
132.131	odd	2		4840.2.a.m.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
440.2.a.g.1.1	✓	2	12.11	even	2
880.2.a.k.1.2		2	3.2	odd	2
2200.2.a.l.1.2		2	60.59	even	2
2200.2.b.f.1849.2		4	60.23	odd	4
2200.2.b.f.1849.3		4	60.47	odd	4
3520.2.a.bm.1.2		2	24.11	even	2
3520.2.a.br.1.1		2	24.5	odd	2
3960.2.a.bf.1.1		2	4.3	odd	2
4400.2.a.bt.1.1		2	15.14	odd	2
4400.2.b.w.4049.2		4	15.2	even	4
4400.2.b.w.4049.3		4	15.8	even	4
4840.2.a.m.1.1		2	132.131	odd	2
7920.2.a.by.1.2		2	1.1	even	1	trivial
9680.2.a.bm.1.2		2	33.32	even	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:	\( N \)	\(=\)	\( 7920 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7920.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} -0.438447 q^{7} -1.00000 q^{11} +7.12311 q^{13} -4.68466 q^{17} -5.56155 q^{19} -7.12311 q^{23} +1.00000 q^{25} -4.43845 q^{29} +5.56155 q^{31} -0.438447 q^{35} +11.5616 q^{37} -4.24621 q^{41} -5.12311 q^{43} +13.3693 q^{47} -6.80776 q^{49} +2.68466 q^{53} -1.00000 q^{55} -7.12311 q^{59} -8.43845 q^{61} +7.12311 q^{65} -8.68466 q^{71} -7.12311 q^{73} +0.438447 q^{77} -13.3693 q^{79} -6.00000 q^{83} -4.68466 q^{85} +2.68466 q^{89} -3.12311 q^{91} -5.56155 q^{95} -13.1231 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} - 5 q^{7} - 2 q^{11} + 6 q^{13} + 3 q^{17} - 7 q^{19} - 6 q^{23} + 2 q^{25} - 13 q^{29} + 7 q^{31} - 5 q^{35} + 19 q^{37} + 8 q^{41} - 2 q^{43} + 2 q^{47} + 7 q^{49} - 7 q^{53} - 2 q^{55}+ \cdots - 18 q^{97}+O(q^{100}) \) 2 * q + 2 * q^5 - 5 * q^7 - 2 * q^11 + 6 * q^13 + 3 * q^17 - 7 * q^19 - 6 * q^23 + 2 * q^25 - 13 * q^29 + 7 * q^31 - 5 * q^35 + 19 * q^37 + 8 * q^41 - 2 * q^43 + 2 * q^47 + 7 * q^49 - 7 * q^53 - 2 * q^55 - 6 * q^59 - 21 * q^61 + 6 * q^65 - 5 * q^71 - 6 * q^73 + 5 * q^77 - 2 * q^79 - 12 * q^83 + 3 * q^85 - 7 * q^89 + 2 * q^91 - 7 * q^95 - 18 * q^97

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