Properties

Label 784.2.bg.a
Level $784$
Weight $2$
Character orbit 784.bg
Analytic conductor $6.260$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [784,2,Mod(65,784)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(784, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([0, 0, 20]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("784.65");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 784.bg (of order \(21\), degree \(12\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.26027151847\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(2\) over \(\Q(\zeta_{21})\)
Twist minimal: no (minimal twist has level 98)
Sato-Tate group: $\mathrm{SU}(2)[C_{21}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 7 q^{3} - 33 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 7 q^{3} - 33 q^{9} + 7 q^{11} + 14 q^{13} + 7 q^{15} - 7 q^{17} + 7 q^{21} + 21 q^{23} + 4 q^{25} + 35 q^{27} - 11 q^{29} - 28 q^{31} + 14 q^{33} - 21 q^{35} - 24 q^{37} + 40 q^{39} + 28 q^{41} - 10 q^{43} + 7 q^{45} + 70 q^{47} + 84 q^{49} - 60 q^{51} + 26 q^{53} - 56 q^{55} - 33 q^{57} + 7 q^{59} + 14 q^{61} + 14 q^{63} + 36 q^{67} - 35 q^{69} - 7 q^{73} + 28 q^{75} - 91 q^{77} + 26 q^{79} + 55 q^{81} + 7 q^{83} + 49 q^{85} - 35 q^{87} - 56 q^{89} - 7 q^{91} + 72 q^{93} + 14 q^{95} - 126 q^{97} + 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1 0 −0.0519131 0.692733i 0 1.52046 1.03663i 0 −2.03934 + 1.68555i 0 2.48931 0.375203i 0
65.2 0 0.250133 + 3.33779i 0 −1.09780 + 0.748464i 0 −2.60370 0.469838i 0 −8.11181 + 1.22266i 0
81.1 0 −1.52870 0.471543i 0 −0.144243 + 0.133838i 0 −2.44775 + 1.00425i 0 −0.364133 0.248262i 0
81.2 0 1.08331 + 0.334156i 0 2.80532 2.60296i 0 2.61464 + 0.404557i 0 −1.41682 0.965972i 0
193.1 0 −0.0519131 + 0.692733i 0 1.52046 + 1.03663i 0 −2.03934 1.68555i 0 2.48931 + 0.375203i 0
193.2 0 0.250133 3.33779i 0 −1.09780 0.748464i 0 −2.60370 + 0.469838i 0 −8.11181 1.22266i 0
289.1 0 −0.520118 0.354611i 0 −0.0807922 + 1.07810i 0 2.64324 0.115218i 0 −0.951249 2.42374i 0
289.2 0 1.46985 + 1.00212i 0 0.169008 2.25526i 0 −2.40601 1.10050i 0 0.0601727 + 0.153317i 0
305.1 0 −1.17911 + 0.177723i 0 −1.10349 + 2.81164i 0 −1.42353 + 2.23015i 0 −1.50800 + 0.465156i 0
305.2 0 −0.532240 + 0.0802223i 0 0.459870 1.17173i 0 1.68832 2.03705i 0 −2.58987 + 0.798871i 0
401.1 0 −1.17911 0.177723i 0 −1.10349 2.81164i 0 −1.42353 2.23015i 0 −1.50800 0.465156i 0
401.2 0 −0.532240 0.0802223i 0 0.459870 + 1.17173i 0 1.68832 + 2.03705i 0 −2.58987 0.798871i 0
417.1 0 −1.04951 2.67410i 0 0.926902 + 0.139708i 0 2.29875 + 1.30987i 0 −3.85019 + 3.57245i 0
417.2 0 0.692327 + 1.76402i 0 −4.01065 0.604508i 0 2.17740 1.50297i 0 −0.433294 + 0.402038i 0
513.1 0 −1.52870 + 0.471543i 0 −0.144243 0.133838i 0 −2.44775 1.00425i 0 −0.364133 + 0.248262i 0
513.2 0 1.08331 0.334156i 0 2.80532 + 2.60296i 0 2.61464 0.404557i 0 −1.41682 + 0.965972i 0
529.1 0 −0.520118 + 0.354611i 0 −0.0807922 1.07810i 0 2.64324 + 0.115218i 0 −0.951249 + 2.42374i 0
529.2 0 1.46985 1.00212i 0 0.169008 + 2.25526i 0 −2.40601 + 1.10050i 0 0.0601727 0.153317i 0
625.1 0 −2.11865 + 1.96582i 0 −0.596780 0.184082i 0 2.13962 + 1.55628i 0 0.400039 5.33815i 0
625.2 0 −0.0153718 + 0.0142629i 0 1.15218 + 0.355400i 0 −2.64164 0.147471i 0 −0.224157 + 2.99117i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 65.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
49.g even 21 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 784.2.bg.a 24
4.b odd 2 1 98.2.g.a 24
12.b even 2 1 882.2.z.d 24
28.d even 2 1 686.2.g.b 24
28.f even 6 1 686.2.e.e 24
28.f even 6 1 686.2.g.c 24
28.g odd 6 1 686.2.e.f 24
28.g odd 6 1 686.2.g.a 24
49.g even 21 1 inner 784.2.bg.a 24
196.j even 14 1 686.2.g.c 24
196.k odd 14 1 686.2.g.a 24
196.o odd 42 1 98.2.g.a 24
196.o odd 42 1 686.2.e.f 24
196.o odd 42 1 4802.2.a.i 12
196.p even 42 1 686.2.e.e 24
196.p even 42 1 686.2.g.b 24
196.p even 42 1 4802.2.a.k 12
588.bb even 42 1 882.2.z.d 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.2.g.a 24 4.b odd 2 1
98.2.g.a 24 196.o odd 42 1
686.2.e.e 24 28.f even 6 1
686.2.e.e 24 196.p even 42 1
686.2.e.f 24 28.g odd 6 1
686.2.e.f 24 196.o odd 42 1
686.2.g.a 24 28.g odd 6 1
686.2.g.a 24 196.k odd 14 1
686.2.g.b 24 28.d even 2 1
686.2.g.b 24 196.p even 42 1
686.2.g.c 24 28.f even 6 1
686.2.g.c 24 196.j even 14 1
784.2.bg.a 24 1.a even 1 1 trivial
784.2.bg.a 24 49.g even 21 1 inner
882.2.z.d 24 12.b even 2 1
882.2.z.d 24 588.bb even 42 1
4802.2.a.i 12 196.o odd 42 1
4802.2.a.k 12 196.p even 42 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{24} + 7 T_{3}^{23} + 38 T_{3}^{22} + 140 T_{3}^{21} + 413 T_{3}^{20} + 861 T_{3}^{19} + 1387 T_{3}^{18} + 1778 T_{3}^{17} + 3090 T_{3}^{16} + 7469 T_{3}^{15} + 9807 T_{3}^{14} + 13650 T_{3}^{13} + 40489 T_{3}^{12} + 47299 T_{3}^{11} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(784, [\chi])\). Copy content Toggle raw display