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Properties

Label 702.2.bg.a
Level $702$
Weight $2$
Character orbit 702.bg
Analytic conductor $5.605$
Analytic rank $0$
Dimension $56$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [702,2,Mod(125,702)]
 
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, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("702.125");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 702 = 2 \cdot 3^{3} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 702.bg (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.

\(\operatorname{Tr}(f)(q) = \) \( 56 q + 4 q^{7} - 24 q^{11} + 28 q^{16} - 8 q^{19} + 8 q^{28} - 4 q^{31} + 8 q^{37} + 48 q^{41} + 24 q^{47} + 24 q^{50} - 4 q^{52} - 48 q^{65} + 28 q^{67} - 56 q^{73} + 48 q^{74} + 4 q^{76} + 48 q^{79} + 24 q^{83}+ \cdots + 20 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
125.1 −0.965926 − 0.258819i 0 0.866025 + 0.500000i −1.08814 − 4.06098i 0 −1.10431 − 0.295899i −0.707107 − 0.707107i 0 4.20424i
125.2 −0.965926 − 0.258819i 0 0.866025 + 0.500000i −0.603112 − 2.25085i 0 −1.18901 − 0.318594i −0.707107 − 0.707107i 0 2.33025i
125.3 −0.965926 − 0.258819i 0 0.866025 + 0.500000i −0.320826 − 1.19734i 0 0.0195586 + 0.00524071i −0.707107 − 0.707107i 0 1.23958i
125.4 −0.965926 − 0.258819i 0 0.866025 + 0.500000i −0.0365170 − 0.136283i 0 −2.57622 − 0.690296i −0.707107 − 0.707107i 0 0.141091i
125.5 −0.965926 − 0.258819i 0 0.866025 + 0.500000i 0.437284 + 1.63197i 0 3.59443 + 0.963125i −0.707107 − 0.707107i 0 − 1.68954i
125.6 −0.965926 − 0.258819i 0 0.866025 + 0.500000i 0.795294 + 2.96808i 0 4.26716 + 1.14338i −0.707107 − 0.707107i 0 − 3.07278i
125.7 −0.965926 − 0.258819i 0 0.866025 + 0.500000i 0.816014 + 3.04540i 0 −1.64558 − 0.440933i −0.707107 − 0.707107i 0 − 3.15283i
125.8 0.965926 + 0.258819i 0 0.866025 + 0.500000i −0.945094 − 3.52714i 0 −3.38290 − 0.906445i 0.707107 + 0.707107i 0 − 3.65156i
125.9 0.965926 + 0.258819i 0 0.866025 + 0.500000i −0.586717 − 2.18966i 0 0.211451 + 0.0566582i 0.707107 + 0.707107i 0 − 2.26690i
125.10 0.965926 + 0.258819i 0 0.866025 + 0.500000i −0.137818 − 0.514345i 0 2.18767 + 0.586183i 0.707107 + 0.707107i 0 − 0.532489i
125.11 0.965926 + 0.258819i 0 0.866025 + 0.500000i −0.100432 − 0.374818i 0 1.26805 + 0.339772i 0.707107 + 0.707107i 0 − 0.388040i
125.12 0.965926 + 0.258819i 0 0.866025 + 0.500000i 0.372850 + 1.39150i 0 −4.70013 − 1.25940i 0.707107 + 0.707107i 0 1.44058i
125.13 0.965926 + 0.258819i 0 0.866025 + 0.500000i 0.470946 + 1.75759i 0 1.35773 + 0.363802i 0.707107 + 0.707107i 0 1.81960i
125.14 0.965926 + 0.258819i 0 0.866025 + 0.500000i 0.926266 + 3.45687i 0 4.42416 + 1.18545i 0.707107 + 0.707107i 0 3.57882i
359.1 −0.258819 − 0.965926i 0 −0.866025 + 0.500000i −4.06098 − 1.08814i 0 0.295899 + 1.10431i 0.707107 + 0.707107i 0 4.20424i
359.2 −0.258819 − 0.965926i 0 −0.866025 + 0.500000i −2.25085 − 0.603112i 0 0.318594 + 1.18901i 0.707107 + 0.707107i 0 2.33025i
359.3 −0.258819 − 0.965926i 0 −0.866025 + 0.500000i −1.19734 − 0.320826i 0 −0.00524071 − 0.0195586i 0.707107 + 0.707107i 0 1.23958i
359.4 −0.258819 − 0.965926i 0 −0.866025 + 0.500000i −0.136283 − 0.0365170i 0 0.690296 + 2.57622i 0.707107 + 0.707107i 0 0.141091i
359.5 −0.258819 − 0.965926i 0 −0.866025 + 0.500000i 1.63197 + 0.437284i 0 −0.963125 − 3.59443i 0.707107 + 0.707107i 0 − 1.68954i
359.6 −0.258819 − 0.965926i 0 −0.866025 + 0.500000i 2.96808 + 0.795294i 0 −1.14338 − 4.26716i 0.707107 + 0.707107i 0 − 3.07278i
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 125.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner
13.d odd 4 1 inner
117.z 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.bg.a 56
3.b odd 2 1 234.2.bd.a ✓ 56
9.c even 3 1 234.2.bd.a ✓ 56
9.d odd 6 1 inner 702.2.bg.a 56
13.d odd 4 1 inner 702.2.bg.a 56
39.f even 4 1 234.2.bd.a ✓ 56
117.y odd 12 1 234.2.bd.a ✓ 56
117.z even 12 1 inner 702.2.bg.a 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
234.2.bd.a ✓ 56 3.b odd 2 1
234.2.bd.a ✓ 56 9.c even 3 1
234.2.bd.a ✓ 56 39.f even 4 1
234.2.bd.a ✓ 56 117.y odd 12 1
702.2.bg.a 56 1.a even 1 1 trivial
702.2.bg.a 56 9.d odd 6 1 inner
702.2.bg.a 56 13.d odd 4 1 inner
702.2.bg.a 56 117.z even 12 1 inner

Hecke kernels

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