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Properties

Label 442.2.m.b
Level $442$
Weight $2$
Character orbit 442.m
Analytic conductor $3.529$
Analytic rank $0$
Dimension $24$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [442,2,Mod(237,442)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(442, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("442.237");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 442 = 2 \cdot 13 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 442.m (of order \(6\), degree \(2\), 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: \(3.52938776934\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 12 q^{2} - 12 q^{4} - 24 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 12 q^{2} - 12 q^{4} - 24 q^{8} + 12 q^{9} + 12 q^{13} - 12 q^{16} + 5 q^{17} + 24 q^{18} + 8 q^{19} - 4 q^{21} - 48 q^{25} - 6 q^{26} + 12 q^{32} - 6 q^{33} + 10 q^{34} - 10 q^{35} + 12 q^{36} + 16 q^{38} - 2 q^{42} - 4 q^{43} + 12 q^{47} + 20 q^{49} - 24 q^{50} + 16 q^{51} - 18 q^{52} - 44 q^{53} - 12 q^{55} + 14 q^{59} + 24 q^{64} - 12 q^{66} + 6 q^{67} + 5 q^{68} + 10 q^{69} - 20 q^{70} - 12 q^{72} + 8 q^{76} - 28 q^{77} - 4 q^{81} - 4 q^{83} + 2 q^{84} + 13 q^{85} - 8 q^{86} - 12 q^{87} + 20 q^{89} + 6 q^{94} - 20 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
237.1 0.500000 + 0.866025i −2.65587 + 1.53337i −0.500000 + 0.866025i − 3.52205i −2.65587 − 1.53337i 0.842681 + 0.486522i −1.00000 3.20242 − 5.54676i 3.05019 − 1.76103i
237.2 0.500000 + 0.866025i −2.19681 + 1.26833i −0.500000 + 0.866025i 3.07158i −2.19681 − 1.26833i −2.78270 − 1.60659i −1.00000 1.71730 − 2.97445i −2.66007 + 1.53579i
237.3 0.500000 + 0.866025i −2.03326 + 1.17390i −0.500000 + 0.866025i 2.91757i −2.03326 − 1.17390i 4.35220 + 2.51274i −1.00000 1.25611 − 2.17564i −2.52669 + 1.45878i
237.4 0.500000 + 0.866025i −1.16408 + 0.672080i −0.500000 + 0.866025i − 2.19243i −1.16408 − 0.672080i −2.48223 − 1.43312i −1.00000 −0.596616 + 1.03337i 1.89870 − 1.09622i
237.5 0.500000 + 0.866025i −0.717811 + 0.414429i −0.500000 + 0.866025i − 1.92016i −0.717811 − 0.414429i −0.823994 − 0.475733i −1.00000 −1.15650 + 2.00311i 1.66291 − 0.960079i
237.6 0.500000 + 0.866025i −0.340479 + 0.196576i −0.500000 + 0.866025i 1.77611i −0.340479 − 0.196576i −2.18274 − 1.26021i −1.00000 −1.42272 + 2.46422i −1.53816 + 0.888055i
237.7 0.500000 + 0.866025i 0.340479 − 0.196576i −0.500000 + 0.866025i − 1.77611i 0.340479 + 0.196576i 2.18274 + 1.26021i −1.00000 −1.42272 + 2.46422i 1.53816 − 0.888055i
237.8 0.500000 + 0.866025i 0.717811 − 0.414429i −0.500000 + 0.866025i 1.92016i 0.717811 + 0.414429i 0.823994 + 0.475733i −1.00000 −1.15650 + 2.00311i −1.66291 + 0.960079i
237.9 0.500000 + 0.866025i 1.16408 − 0.672080i −0.500000 + 0.866025i 2.19243i 1.16408 + 0.672080i 2.48223 + 1.43312i −1.00000 −0.596616 + 1.03337i −1.89870 + 1.09622i
237.10 0.500000 + 0.866025i 2.03326 − 1.17390i −0.500000 + 0.866025i − 2.91757i 2.03326 + 1.17390i −4.35220 − 2.51274i −1.00000 1.25611 − 2.17564i 2.52669 − 1.45878i
237.11 0.500000 + 0.866025i 2.19681 − 1.26833i −0.500000 + 0.866025i − 3.07158i 2.19681 + 1.26833i 2.78270 + 1.60659i −1.00000 1.71730 − 2.97445i 2.66007 − 1.53579i
237.12 0.500000 + 0.866025i 2.65587 − 1.53337i −0.500000 + 0.866025i 3.52205i 2.65587 + 1.53337i −0.842681 − 0.486522i −1.00000 3.20242 − 5.54676i −3.05019 + 1.76103i
373.1 0.500000 − 0.866025i −2.65587 − 1.53337i −0.500000 − 0.866025i 3.52205i −2.65587 + 1.53337i 0.842681 − 0.486522i −1.00000 3.20242 + 5.54676i 3.05019 + 1.76103i
373.2 0.500000 − 0.866025i −2.19681 − 1.26833i −0.500000 − 0.866025i − 3.07158i −2.19681 + 1.26833i −2.78270 + 1.60659i −1.00000 1.71730 + 2.97445i −2.66007 − 1.53579i
373.3 0.500000 − 0.866025i −2.03326 − 1.17390i −0.500000 − 0.866025i − 2.91757i −2.03326 + 1.17390i 4.35220 − 2.51274i −1.00000 1.25611 + 2.17564i −2.52669 − 1.45878i
373.4 0.500000 − 0.866025i −1.16408 − 0.672080i −0.500000 − 0.866025i 2.19243i −1.16408 + 0.672080i −2.48223 + 1.43312i −1.00000 −0.596616 − 1.03337i 1.89870 + 1.09622i
373.5 0.500000 − 0.866025i −0.717811 − 0.414429i −0.500000 − 0.866025i 1.92016i −0.717811 + 0.414429i −0.823994 + 0.475733i −1.00000 −1.15650 − 2.00311i 1.66291 + 0.960079i
373.6 0.500000 − 0.866025i −0.340479 − 0.196576i −0.500000 − 0.866025i − 1.77611i −0.340479 + 0.196576i −2.18274 + 1.26021i −1.00000 −1.42272 − 2.46422i −1.53816 − 0.888055i
373.7 0.500000 − 0.866025i 0.340479 + 0.196576i −0.500000 − 0.866025i 1.77611i 0.340479 − 0.196576i 2.18274 − 1.26021i −1.00000 −1.42272 − 2.46422i 1.53816 + 0.888055i
373.8 0.500000 − 0.866025i 0.717811 + 0.414429i −0.500000 − 0.866025i − 1.92016i 0.717811 − 0.414429i 0.823994 − 0.475733i −1.00000 −1.15650 − 2.00311i −1.66291 − 0.960079i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 237.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner
17.b even 2 1 inner
221.l even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 442.2.m.b ✓ 24
13.c even 3 1 inner 442.2.m.b ✓ 24
17.b even 2 1 inner 442.2.m.b ✓ 24
221.l even 6 1 inner 442.2.m.b ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
442.2.m.b ✓ 24 1.a even 1 1 trivial
442.2.m.b ✓ 24 13.c even 3 1 inner
442.2.m.b ✓ 24 17.b even 2 1 inner
442.2.m.b ✓ 24 221.l even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{24} - 24 T_{3}^{22} + 370 T_{3}^{20} - 3424 T_{3}^{18} + 23068 T_{3}^{16} - 102987 T_{3}^{14} + \cdots + 4096 \) acting on \(S_{2}^{\mathrm{new}}(442, [\chi])\). Copy content Toggle raw display