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Properties

Label 1216.2.k.b
Level $1216$
Weight $2$
Character orbit 1216.k
Analytic conductor $9.710$
Analytic rank $0$
Dimension $68$
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] = [1216,2,Mod(305,1216)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1216, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1216.305");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1216.k (of order \(4\), degree \(2\), 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: \(9.70980888579\)
Analytic rank: \(0\)
Dimension: \(68\)
Relative dimension: \(34\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 304)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 68 q - 4 q^{3} + 8 q^{5}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 68 q - 4 q^{3} + 8 q^{5} - 4 q^{11} + 16 q^{15} + 4 q^{17} + 16 q^{21} - 28 q^{27} - 32 q^{31} - 16 q^{33} + 24 q^{35} - 24 q^{37} + 16 q^{43} + 24 q^{47} - 56 q^{49} + 52 q^{51} + 32 q^{53} - 28 q^{59} - 24 q^{63} - 16 q^{65} - 28 q^{67} - 8 q^{69} - 44 q^{75} - 12 q^{77} - 8 q^{79} - 76 q^{81} - 40 q^{83} - 8 q^{85} + 4 q^{91} - 84 q^{93} - 32 q^{95} - 16 q^{97} + 24 q^{99}+O(q^{100}) \) 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 −2.26925 + 2.26925i 0 −1.08984 − 1.08984i 0 − 4.17557i 0 − 7.29898i 0
305.2 0 −2.23665 + 2.23665i 0 3.10689 + 3.10689i 0 − 0.488348i 0 − 7.00520i 0
305.3 0 −2.08795 + 2.08795i 0 −2.31417 − 2.31417i 0 1.60156i 0 − 5.71908i 0
305.4 0 −2.07405 + 2.07405i 0 −0.193650 − 0.193650i 0 0.804888i 0 − 5.60338i 0
305.5 0 −1.66448 + 1.66448i 0 −2.15868 − 2.15868i 0 1.65346i 0 − 2.54100i 0
305.6 0 −1.62771 + 1.62771i 0 1.14169 + 1.14169i 0 2.54896i 0 − 2.29890i 0
305.7 0 −1.51572 + 1.51572i 0 1.66420 + 1.66420i 0 − 3.87356i 0 − 1.59482i 0
305.8 0 −1.38476 + 1.38476i 0 −0.0854723 − 0.0854723i 0 − 0.914325i 0 − 0.835133i 0
305.9 0 −1.26193 + 1.26193i 0 0.588594 + 0.588594i 0 − 3.53958i 0 − 0.184946i 0
305.10 0 −1.14883 + 1.14883i 0 0.889280 + 0.889280i 0 1.83745i 0 0.360387i 0
305.11 0 −1.00943 + 1.00943i 0 1.26480 + 1.26480i 0 4.00933i 0 0.962122i 0
305.12 0 −0.982629 + 0.982629i 0 −2.22243 − 2.22243i 0 0.518576i 0 1.06888i 0
305.13 0 −0.794278 + 0.794278i 0 −2.69752 − 2.69752i 0 1.16480i 0 1.73824i 0
305.14 0 −0.641204 + 0.641204i 0 2.46245 + 2.46245i 0 0.804840i 0 2.17771i 0
305.15 0 −0.504056 + 0.504056i 0 −0.312442 − 0.312442i 0 4.86969i 0 2.49185i 0
305.16 0 −0.464161 + 0.464161i 0 0.625824 + 0.625824i 0 − 3.89874i 0 2.56911i 0
305.17 0 −0.227569 + 0.227569i 0 −2.67518 − 2.67518i 0 − 4.39636i 0 2.89642i 0
305.18 0 −0.0736275 + 0.0736275i 0 1.57485 + 1.57485i 0 − 2.89934i 0 2.98916i 0
305.19 0 0.0648127 − 0.0648127i 0 2.89124 + 2.89124i 0 2.82413i 0 2.99160i 0
305.20 0 0.0665185 − 0.0665185i 0 −0.733023 − 0.733023i 0 − 2.20223i 0 2.99115i 0
See all 68 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 305.34
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1216.2.k.b 68
4.b odd 2 1 304.2.k.b ✓ 68
16.e even 4 1 inner 1216.2.k.b 68
16.f odd 4 1 304.2.k.b ✓ 68
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
304.2.k.b ✓ 68 4.b odd 2 1
304.2.k.b ✓ 68 16.f odd 4 1
1216.2.k.b 68 1.a even 1 1 trivial
1216.2.k.b 68 16.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{68} + 4 T_{3}^{67} + 8 T_{3}^{66} + 12 T_{3}^{65} + 503 T_{3}^{64} + 1984 T_{3}^{63} + \cdots + 4194304 \) acting on \(S_{2}^{\mathrm{new}}(1216, [\chi])\). Copy content Toggle raw display