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Properties

Label 702.2.bb.a
Level $702$
Weight $2$
Character orbit 702.bb
Analytic conductor $5.605$
Analytic rank $0$
Dimension $56$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [702,2,Mod(71,702)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(702, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([10, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("702.71");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 702 = 2 \cdot 3^{3} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 702.bb (of order \(12\), degree \(4\), 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: \(5.60549822189\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(14\) over \(\Q(\zeta_{12})\)
Twist minimal: no (minimal twist has level 234)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 56 q + 4 q^{7} - 56 q^{16} - 8 q^{19} - 4 q^{28} + 8 q^{31} - 24 q^{35} - 4 q^{37} + 36 q^{38} + 48 q^{41} + 12 q^{43} - 60 q^{47} + 24 q^{50} - 4 q^{52} + 120 q^{65} - 56 q^{67} - 24 q^{71} + 28 q^{73}+ \cdots + 48 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
71.1 −0.707107 − 0.707107i 0 1.00000i −3.04921 + 0.817032i 0 2.84235 − 0.761606i 0.707107 − 0.707107i 0 2.73384 + 1.57838i
71.2 −0.707107 − 0.707107i 0 1.00000i −1.62665 + 0.435860i 0 0.290365 − 0.0778030i 0.707107 − 0.707107i 0 1.45842 + 0.842018i
71.3 −0.707107 − 0.707107i 0 1.00000i −1.35053 + 0.361873i 0 0.977097 − 0.261812i 0.707107 − 0.707107i 0 1.21085 + 0.699086i
71.4 −0.707107 − 0.707107i 0 1.00000i −0.650628 + 0.174335i 0 −1.73068 + 0.463733i 0.707107 − 0.707107i 0 0.583337 + 0.336790i
71.5 −0.707107 − 0.707107i 0 1.00000i 0.670386 − 0.179629i 0 −4.43374 + 1.18802i 0.707107 − 0.707107i 0 −0.601052 − 0.347017i
71.6 −0.707107 − 0.707107i 0 1.00000i 2.64098 − 0.707650i 0 3.36644 − 0.902034i 0.707107 − 0.707107i 0 −2.36784 − 1.36707i
71.7 −0.707107 − 0.707107i 0 1.00000i 3.36565 − 0.901822i 0 0.0541850 − 0.0145188i 0.707107 − 0.707107i 0 −3.01756 − 1.74219i
71.8 0.707107 + 0.707107i 0 1.00000i −3.83221 + 1.02684i 0 −1.55176 + 0.415793i −0.707107 + 0.707107i 0 −3.43586 − 1.98370i
71.9 0.707107 + 0.707107i 0 1.00000i −1.51706 + 0.406496i 0 4.52587 − 1.21270i −0.707107 + 0.707107i 0 −1.36016 − 0.785290i
71.10 0.707107 + 0.707107i 0 1.00000i −0.994452 + 0.266463i 0 0.339163 − 0.0908784i −0.707107 + 0.707107i 0 −0.891601 − 0.514766i
71.11 0.707107 + 0.707107i 0 1.00000i −0.653109 + 0.175000i 0 −3.90017 + 1.04505i −0.707107 + 0.707107i 0 −0.585562 − 0.338074i
71.12 0.707107 + 0.707107i 0 1.00000i −0.339151 + 0.0908752i 0 2.97685 − 0.797644i −0.707107 + 0.707107i 0 −0.304074 − 0.175557i
71.13 0.707107 + 0.707107i 0 1.00000i 3.62177 − 0.970449i 0 −3.13748 + 0.840685i −0.707107 + 0.707107i 0 3.24719 + 1.87476i
71.14 0.707107 + 0.707107i 0 1.00000i 3.71422 − 0.995221i 0 2.11355 − 0.566325i −0.707107 + 0.707107i 0 3.33007 + 1.92262i
89.1 −0.707107 + 0.707107i 0 − 1.00000i −3.04921 − 0.817032i 0 2.84235 + 0.761606i 0.707107 + 0.707107i 0 2.73384 − 1.57838i
89.2 −0.707107 + 0.707107i 0 − 1.00000i −1.62665 − 0.435860i 0 0.290365 + 0.0778030i 0.707107 + 0.707107i 0 1.45842 − 0.842018i
89.3 −0.707107 + 0.707107i 0 − 1.00000i −1.35053 − 0.361873i 0 0.977097 + 0.261812i 0.707107 + 0.707107i 0 1.21085 − 0.699086i
89.4 −0.707107 + 0.707107i 0 − 1.00000i −0.650628 − 0.174335i 0 −1.73068 − 0.463733i 0.707107 + 0.707107i 0 0.583337 − 0.336790i
89.5 −0.707107 + 0.707107i 0 − 1.00000i 0.670386 + 0.179629i 0 −4.43374 − 1.18802i 0.707107 + 0.707107i 0 −0.601052 + 0.347017i
89.6 −0.707107 + 0.707107i 0 − 1.00000i 2.64098 + 0.707650i 0 3.36644 + 0.902034i 0.707107 + 0.707107i 0 −2.36784 + 1.36707i
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 71.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
117.x even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 702.2.bb.a 56
3.b odd 2 1 234.2.y.a ✓ 56
9.c even 3 1 234.2.z.a yes 56
9.d odd 6 1 702.2.bc.a 56
13.f odd 12 1 702.2.bc.a 56
39.k even 12 1 234.2.z.a yes 56
117.w odd 12 1 234.2.y.a ✓ 56
117.x even 12 1 inner 702.2.bb.a 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
234.2.y.a ✓ 56 3.b odd 2 1
234.2.y.a ✓ 56 117.w odd 12 1
234.2.z.a yes 56 9.c even 3 1
234.2.z.a yes 56 39.k even 12 1
702.2.bb.a 56 1.a even 1 1 trivial
702.2.bb.a 56 117.x even 12 1 inner
702.2.bc.a 56 9.d odd 6 1
702.2.bc.a 56 13.f odd 12 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(702, [\chi])\).