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Properties

Label 702.2.bc.a
Level $702$
Weight $2$
Character orbit 702.bc
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(305,702)]
 
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([2, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("702.305");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 702 = 2 \cdot 3^{3} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 702.bc (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.

\(\operatorname{Tr}(f)(q) = \) \( 56 q - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q - 8 q^{7} + 24 q^{11} + 28 q^{16} - 8 q^{19} - 4 q^{28} - 4 q^{31} + 24 q^{35} - 4 q^{37} - 36 q^{38} + 48 q^{41} + 36 q^{47} + 24 q^{50} + 8 q^{52} + 36 q^{65} + 28 q^{67} + 24 q^{71} + 28 q^{73} + 4 q^{76} - 24 q^{77} - 24 q^{79} + 96 q^{83} - 72 q^{85} + 4 q^{91} + 24 q^{92} - 52 q^{97} - 48 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
305.1 −0.258819 + 0.965926i 0 −0.866025 − 0.500000i −1.02684 + 3.83221i 0 1.13597 + 1.13597i 0.707107 − 0.707107i 0 −3.43586 − 1.98370i
305.2 −0.258819 + 0.965926i 0 −0.866025 − 0.500000i −0.406496 + 1.51706i 0 −3.31317 − 3.31317i 0.707107 − 0.707107i 0 −1.36016 − 0.785290i
305.3 −0.258819 + 0.965926i 0 −0.866025 − 0.500000i −0.266463 + 0.994452i 0 −0.248284 − 0.248284i 0.707107 − 0.707107i 0 −0.891601 − 0.514766i
305.4 −0.258819 + 0.965926i 0 −0.866025 − 0.500000i −0.175000 + 0.653109i 0 2.85512 + 2.85512i 0.707107 − 0.707107i 0 −0.585562 − 0.338074i
305.5 −0.258819 + 0.965926i 0 −0.866025 − 0.500000i −0.0908752 + 0.339151i 0 −2.17921 − 2.17921i 0.707107 − 0.707107i 0 −0.304074 − 0.175557i
305.6 −0.258819 + 0.965926i 0 −0.866025 − 0.500000i 0.970449 − 3.62177i 0 2.29680 + 2.29680i 0.707107 − 0.707107i 0 3.24719 + 1.87476i
305.7 −0.258819 + 0.965926i 0 −0.866025 − 0.500000i 0.995221 − 3.71422i 0 −1.54723 − 1.54723i 0.707107 − 0.707107i 0 3.33007 + 1.92262i
305.8 0.258819 − 0.965926i 0 −0.866025 − 0.500000i −0.817032 + 3.04921i 0 −2.08075 − 2.08075i −0.707107 + 0.707107i 0 2.73384 + 1.57838i
305.9 0.258819 − 0.965926i 0 −0.866025 − 0.500000i −0.435860 + 1.62665i 0 −0.212562 − 0.212562i −0.707107 + 0.707107i 0 1.45842 + 0.842018i
305.10 0.258819 − 0.965926i 0 −0.866025 − 0.500000i −0.361873 + 1.35053i 0 −0.715284 − 0.715284i −0.707107 + 0.707107i 0 1.21085 + 0.699086i
305.11 0.258819 − 0.965926i 0 −0.866025 − 0.500000i −0.174335 + 0.650628i 0 1.26694 + 1.26694i −0.707107 + 0.707107i 0 0.583337 + 0.336790i
305.12 0.258819 − 0.965926i 0 −0.866025 − 0.500000i 0.179629 − 0.670386i 0 3.24572 + 3.24572i −0.707107 + 0.707107i 0 −0.601052 − 0.347017i
305.13 0.258819 − 0.965926i 0 −0.866025 − 0.500000i 0.707650 − 2.64098i 0 −2.46440 − 2.46440i −0.707107 + 0.707107i 0 −2.36784 − 1.36707i
305.14 0.258819 − 0.965926i 0 −0.866025 − 0.500000i 0.901822 − 3.36565i 0 −0.0396662 − 0.0396662i −0.707107 + 0.707107i 0 −3.01756 − 1.74219i
557.1 −0.258819 − 0.965926i 0 −0.866025 + 0.500000i −1.02684 − 3.83221i 0 1.13597 − 1.13597i 0.707107 + 0.707107i 0 −3.43586 + 1.98370i
557.2 −0.258819 − 0.965926i 0 −0.866025 + 0.500000i −0.406496 − 1.51706i 0 −3.31317 + 3.31317i 0.707107 + 0.707107i 0 −1.36016 + 0.785290i
557.3 −0.258819 − 0.965926i 0 −0.866025 + 0.500000i −0.266463 − 0.994452i 0 −0.248284 + 0.248284i 0.707107 + 0.707107i 0 −0.891601 + 0.514766i
557.4 −0.258819 − 0.965926i 0 −0.866025 + 0.500000i −0.175000 − 0.653109i 0 2.85512 − 2.85512i 0.707107 + 0.707107i 0 −0.585562 + 0.338074i
557.5 −0.258819 − 0.965926i 0 −0.866025 + 0.500000i −0.0908752 − 0.339151i 0 −2.17921 + 2.17921i 0.707107 + 0.707107i 0 −0.304074 + 0.175557i
557.6 −0.258819 − 0.965926i 0 −0.866025 + 0.500000i 0.970449 + 3.62177i 0 2.29680 − 2.29680i 0.707107 + 0.707107i 0 3.24719 − 1.87476i
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 305.14
Significant digits:
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Inner twists

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

Hecke kernels

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