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Properties

Label 368.2.m.e
Level $368$
Weight $2$
Character orbit 368.m
Analytic conductor $2.938$
Analytic rank $0$
Dimension $30$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [368,2,Mod(49,368)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("368.49"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(368, base_ring=CyclotomicField(22)) chi = DirichletCharacter(H, H._module([0, 0, 16])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 368 = 2^{4} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 368.m (of order \(11\), degree \(10\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [30,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.93849479438\)
Analytic rank: \(0\)
Dimension: \(30\)
Relative dimension: \(3\) over \(\Q(\zeta_{11})\)
Twist minimal: no (minimal twist has level 184)
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 30 q - 2 q^{3} + 13 q^{7} + 21 q^{9} - 2 q^{11} + 2 q^{15} - 22 q^{17} - 3 q^{19} + 2 q^{21} - q^{23} + 13 q^{25} + 31 q^{27} + 7 q^{29} - 18 q^{31} - 8 q^{33} - 41 q^{35} - 62 q^{37} - 6 q^{39} - 15 q^{41}+ \cdots + 17 q^{99}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1 0 −1.05870 + 2.31822i 0 −0.536040 + 0.618623i 0 −1.79431 + 1.15313i 0 −2.28874 − 2.64134i 0
49.2 0 0.246469 − 0.539692i 0 −1.62555 + 1.87599i 0 3.01490 − 1.93756i 0 1.73406 + 2.00121i 0
49.3 0 0.693988 − 1.51962i 0 2.16159 − 2.49461i 0 −1.24823 + 0.802191i 0 0.136950 + 0.158049i 0
81.1 0 −0.369168 + 2.56762i 0 −3.43820 + 1.00955i 0 2.84293 + 3.28091i 0 −3.57792 − 1.05057i 0
81.2 0 −0.0226391 + 0.157458i 0 0.0359441 − 0.0105542i 0 −2.08148 − 2.40216i 0 2.85420 + 0.838068i 0
81.3 0 0.152362 − 1.05970i 0 3.40226 − 0.998994i 0 2.50777 + 2.89412i 0 1.77873 + 0.522282i 0
177.1 0 −1.67298 + 1.07516i 0 1.60166 + 3.50714i 0 0.512781 − 0.150566i 0 0.396642 − 0.868525i 0
177.2 0 −1.21882 + 0.783290i 0 −1.24491 − 2.72597i 0 −1.62369 + 0.476758i 0 −0.374258 + 0.819511i 0
177.3 0 1.78999 − 1.15036i 0 −0.356751 − 0.781176i 0 3.61448 − 1.06131i 0 0.634502 − 1.38936i 0
193.1 0 −1.32137 − 1.52494i 0 0.0973464 + 0.677059i 0 0.899416 − 1.96945i 0 −0.152486 + 1.06057i 0
193.2 0 0.946512 + 1.09233i 0 0.287791 + 2.00163i 0 −2.03440 + 4.45471i 0 0.129638 − 0.901650i 0
193.3 0 1.63153 + 1.88288i 0 −0.385137 − 2.67869i 0 1.44647 − 3.16732i 0 −0.456422 + 3.17448i 0
209.1 0 −0.369168 − 2.56762i 0 −3.43820 − 1.00955i 0 2.84293 − 3.28091i 0 −3.57792 + 1.05057i 0
209.2 0 −0.0226391 − 0.157458i 0 0.0359441 + 0.0105542i 0 −2.08148 + 2.40216i 0 2.85420 − 0.838068i 0
209.3 0 0.152362 + 1.05970i 0 3.40226 + 0.998994i 0 2.50777 − 2.89412i 0 1.77873 − 0.522282i 0
225.1 0 −1.32137 + 1.52494i 0 0.0973464 − 0.677059i 0 0.899416 + 1.96945i 0 −0.152486 − 1.06057i 0
225.2 0 0.946512 − 1.09233i 0 0.287791 − 2.00163i 0 −2.03440 − 4.45471i 0 0.129638 + 0.901650i 0
225.3 0 1.63153 − 1.88288i 0 −0.385137 + 2.67869i 0 1.44647 + 3.16732i 0 −0.456422 − 3.17448i 0
257.1 0 −3.00781 + 0.883172i 0 −2.52491 − 1.62266i 0 −0.101654 − 0.707018i 0 5.74315 − 3.69090i 0
257.2 0 −0.817468 + 0.240030i 0 2.57662 + 1.65589i 0 0.181823 + 1.26461i 0 −1.91312 + 1.22949i 0
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.3
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.c even 11 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 368.2.m.e 30
4.b odd 2 1 184.2.i.b ✓ 30
23.c even 11 1 inner 368.2.m.e 30
23.c even 11 1 8464.2.a.cg 15
23.d odd 22 1 8464.2.a.ch 15
92.g odd 22 1 184.2.i.b ✓ 30
92.g odd 22 1 4232.2.a.bb 15
92.h even 22 1 4232.2.a.ba 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
184.2.i.b ✓ 30 4.b odd 2 1
184.2.i.b ✓ 30 92.g odd 22 1
368.2.m.e 30 1.a even 1 1 trivial
368.2.m.e 30 23.c even 11 1 inner
4232.2.a.ba 15 92.h even 22 1
4232.2.a.bb 15 92.g odd 22 1
8464.2.a.cg 15 23.c even 11 1
8464.2.a.ch 15 23.d odd 22 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{30} + 2 T_{3}^{29} - 4 T_{3}^{28} - 17 T_{3}^{27} - 21 T_{3}^{26} + 99 T_{3}^{25} + 379 T_{3}^{24} + \cdots + 175561 \) acting on \(S_{2}^{\mathrm{new}}(368, [\chi])\). Copy content Toggle raw display