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Properties

Label 245.2.l.d
Level $245$
Weight $2$
Character orbit 245.l
Analytic conductor $1.956$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [245,2,Mod(68,245)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(245, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([9, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("245.68");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 245.l (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: \(1.95633484952\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(12\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 16 q^{11} + 24 q^{16} - 32 q^{18} - 64 q^{22} - 16 q^{23} + 32 q^{25} - 48 q^{30} + 80 q^{32} + 80 q^{36} - 32 q^{37} + 32 q^{43} - 96 q^{46} - 160 q^{50} + 48 q^{51} + 8 q^{53} - 128 q^{57} + 88 q^{58} + 16 q^{60} + 64 q^{65} + 32 q^{67} - 64 q^{71} - 16 q^{72} + 128 q^{78} - 8 q^{81} - 64 q^{86} + 64 q^{92} - 32 q^{93} + 32 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
68.1 −0.566266 − 2.11333i −2.60056 − 0.696819i −2.41347 + 1.39342i −0.521466 + 2.17441i 5.89044i 0 1.21729 + 1.21729i 3.67929 + 2.12424i 4.89055 − 0.129265i
68.2 −0.566266 − 2.11333i 2.60056 + 0.696819i −2.41347 + 1.39342i 0.521466 − 2.17441i − 5.89044i 0 1.21729 + 1.21729i 3.67929 + 2.12424i −4.89055 + 0.129265i
68.3 −0.543253 − 2.02745i −0.387073 − 0.103716i −2.08337 + 1.20284i 2.21036 − 0.338120i 0.841115i 0 0.602098 + 0.602098i −2.45901 − 1.41971i −1.88630 − 4.29770i
68.4 −0.543253 − 2.02745i 0.387073 + 0.103716i −2.08337 + 1.20284i −2.21036 + 0.338120i − 0.841115i 0 0.602098 + 0.602098i −2.45901 − 1.41971i 1.88630 + 4.29770i
68.5 −0.0550662 − 0.205510i −2.73547 − 0.732967i 1.69285 − 0.977367i 2.08567 + 0.806215i 0.602528i 0 −0.594965 − 0.594965i 4.34747 + 2.51002i 0.0508353 − 0.473021i
68.6 −0.0550662 − 0.205510i 2.73547 + 0.732967i 1.69285 − 0.977367i −2.08567 − 0.806215i − 0.602528i 0 −0.594965 − 0.594965i 4.34747 + 2.51002i −0.0508353 + 0.473021i
68.7 0.156306 + 0.583343i −1.17055 − 0.313649i 1.41619 − 0.817640i −1.16008 − 1.91160i − 0.731859i 0 1.55240 + 1.55240i −1.32626 − 0.765714i 0.933789 − 0.975520i
68.8 0.156306 + 0.583343i 1.17055 + 0.313649i 1.41619 − 0.817640i 1.16008 + 1.91160i 0.731859i 0 1.55240 + 1.55240i −1.32626 − 0.765714i −0.933789 + 0.975520i
68.9 0.339500 + 1.26703i −1.30289 − 0.349107i 0.241939 − 0.139683i 0.325562 + 2.21224i − 1.76932i 0 2.11419 + 2.11419i −1.02244 − 0.590307i −2.69245 + 1.16355i
68.10 0.339500 + 1.26703i 1.30289 + 0.349107i 0.241939 − 0.139683i −0.325562 − 2.21224i 1.76932i 0 2.11419 + 2.11419i −1.02244 − 0.590307i 2.69245 − 1.16355i
68.11 0.668779 + 2.49592i −1.09408 − 0.293157i −4.05029 + 2.33843i −2.07570 − 0.831534i − 2.92678i 0 −4.89101 − 4.89101i −1.48701 − 0.858527i 0.687252 − 5.73690i
68.12 0.668779 + 2.49592i 1.09408 + 0.293157i −4.05029 + 2.33843i 2.07570 + 0.831534i 2.92678i 0 −4.89101 − 4.89101i −1.48701 − 0.858527i −0.687252 + 5.73690i
117.1 −2.49592 + 0.668779i −0.293157 + 1.09408i 4.05029 − 2.33843i −1.75798 − 1.38185i − 2.92678i 0 −4.89101 + 4.89101i 1.48701 + 0.858527i 5.31193 + 2.27327i
117.2 −2.49592 + 0.668779i 0.293157 − 1.09408i 4.05029 − 2.33843i 1.75798 + 1.38185i 2.92678i 0 −4.89101 + 4.89101i 1.48701 + 0.858527i −5.31193 − 2.27327i
117.3 −1.26703 + 0.339500i −0.349107 + 1.30289i −0.241939 + 0.139683i 2.07864 − 0.824175i − 1.76932i 0 2.11419 − 2.11419i 1.02244 + 0.590307i −2.35389 + 1.74996i
117.4 −1.26703 + 0.339500i 0.349107 − 1.30289i −0.241939 + 0.139683i −2.07864 + 0.824175i 1.76932i 0 2.11419 − 2.11419i 1.02244 + 0.590307i 2.35389 − 1.74996i
117.5 −0.583343 + 0.156306i −0.313649 + 1.17055i −1.41619 + 0.817640i −2.23553 − 0.0488606i − 0.731859i 0 1.55240 − 1.55240i 1.32626 + 0.765714i 1.31172 − 0.320925i
117.6 −0.583343 + 0.156306i 0.313649 − 1.17055i −1.41619 + 0.817640i 2.23553 + 0.0488606i 0.731859i 0 1.55240 − 1.55240i 1.32626 + 0.765714i −1.31172 + 0.320925i
117.7 0.205510 − 0.0550662i −0.732967 + 2.73547i −1.69285 + 0.977367i 1.74104 + 1.40314i 0.602528i 0 −0.594965 + 0.594965i −4.34747 − 2.51002i 0.435066 + 0.192486i
117.8 0.205510 − 0.0550662i 0.732967 − 2.73547i −1.69285 + 0.977367i −1.74104 − 1.40314i − 0.602528i 0 −0.594965 + 0.594965i −4.34747 − 2.51002i −0.435066 − 0.192486i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 68.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
35.f even 4 1 inner
35.k even 12 1 inner
35.l odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 245.2.l.d 48
5.c odd 4 1 inner 245.2.l.d 48
7.b odd 2 1 inner 245.2.l.d 48
7.c even 3 1 245.2.f.c ✓ 24
7.c even 3 1 inner 245.2.l.d 48
7.d odd 6 1 245.2.f.c ✓ 24
7.d odd 6 1 inner 245.2.l.d 48
35.f even 4 1 inner 245.2.l.d 48
35.k even 12 1 245.2.f.c ✓ 24
35.k even 12 1 inner 245.2.l.d 48
35.l odd 12 1 245.2.f.c ✓ 24
35.l odd 12 1 inner 245.2.l.d 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
245.2.f.c ✓ 24 7.c even 3 1
245.2.f.c ✓ 24 7.d odd 6 1
245.2.f.c ✓ 24 35.k even 12 1
245.2.f.c ✓ 24 35.l odd 12 1
245.2.l.d 48 1.a even 1 1 trivial
245.2.l.d 48 5.c odd 4 1 inner
245.2.l.d 48 7.b odd 2 1 inner
245.2.l.d 48 7.c even 3 1 inner
245.2.l.d 48 7.d odd 6 1 inner
245.2.l.d 48 35.f even 4 1 inner
245.2.l.d 48 35.k even 12 1 inner
245.2.l.d 48 35.l odd 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} - 45 T_{2}^{20} - 8 T_{2}^{19} + 72 T_{2}^{17} + 1737 T_{2}^{16} - 648 T_{2}^{15} + 32 T_{2}^{14} + \cdots + 16 \) acting on \(S_{2}^{\mathrm{new}}(245, [\chi])\). Copy content Toggle raw display