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Properties

Label 370.2.o.c
Level $370$
Weight $2$
Character orbit 370.o
Analytic conductor $2.954$
Analytic rank $0$
Dimension $24$
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] = [370,2,Mod(71,370)] 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("370.71"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(370, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 370 = 2 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 370.o (of order \(9\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24,0,0,0,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.95446487479\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(4\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 6 q^{6} - 3 q^{7} + 12 q^{8} + 12 q^{9} + 12 q^{10} - 15 q^{11} + 3 q^{14} - 6 q^{17} + 6 q^{18} - 6 q^{19} + 36 q^{21} + 9 q^{22} + 21 q^{23} - 6 q^{26} - 12 q^{27} - 3 q^{28} + 12 q^{29} + 30 q^{31}+ \cdots + 63 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
71.1 −0.766044 − 0.642788i −2.54754 + 2.13764i 0.173648 + 0.984808i −0.939693 − 0.342020i 3.32558 −3.31719 − 1.20736i 0.500000 − 0.866025i 1.39951 − 7.93703i 0.500000 + 0.866025i
71.2 −0.766044 − 0.642788i 0.204920 − 0.171948i 0.173648 + 0.984808i −0.939693 − 0.342020i −0.267503 −3.03750 − 1.10556i 0.500000 − 0.866025i −0.508519 + 2.88395i 0.500000 + 0.866025i
71.3 −0.766044 − 0.642788i 0.727578 − 0.610510i 0.173648 + 0.984808i −0.939693 − 0.342020i −0.949785 4.16326 + 1.51530i 0.500000 − 0.866025i −0.364298 + 2.06604i 0.500000 + 0.866025i
71.4 −0.766044 − 0.642788i 2.38109 − 1.99797i 0.173648 + 0.984808i −0.939693 − 0.342020i −3.10829 −0.187965 − 0.0684136i 0.500000 − 0.866025i 1.15675 − 6.56026i 0.500000 + 0.866025i
81.1 0.939693 − 0.342020i −2.67866 − 0.974954i 0.766044 − 0.642788i 0.173648 + 0.984808i −2.85057 −0.751904 − 4.26426i 0.500000 − 0.866025i 3.92657 + 3.29478i 0.500000 + 0.866025i
81.2 0.939693 − 0.342020i −2.36335 − 0.860188i 0.766044 − 0.642788i 0.173648 + 0.984808i −2.51502 0.832620 + 4.72202i 0.500000 − 0.866025i 2.54735 + 2.13748i 0.500000 + 0.866025i
81.3 0.939693 − 0.342020i 1.85873 + 0.676523i 0.766044 − 0.642788i 0.173648 + 0.984808i 1.97802 0.241589 + 1.37012i 0.500000 − 0.866025i 0.699071 + 0.586590i 0.500000 + 0.866025i
81.4 0.939693 − 0.342020i 2.24358 + 0.816598i 0.766044 − 0.642788i 0.173648 + 0.984808i 2.38757 −0.475008 − 2.69391i 0.500000 − 0.866025i 2.06871 + 1.73585i 0.500000 + 0.866025i
181.1 −0.173648 + 0.984808i −0.420953 − 2.38734i −0.939693 − 0.342020i 0.766044 − 0.642788i 2.42417 0.480222 − 0.402954i 0.500000 − 0.866025i −2.70313 + 0.983860i 0.500000 + 0.866025i
181.2 −0.173648 + 0.984808i −0.116582 − 0.661169i −0.939693 − 0.342020i 0.766044 − 0.642788i 0.671368 −1.79162 + 1.50334i 0.500000 − 0.866025i 2.39553 − 0.871900i 0.500000 + 0.866025i
181.3 −0.173648 + 0.984808i 0.160396 + 0.909649i −0.939693 − 0.342020i 0.766044 − 0.642788i −0.923681 1.63147 − 1.36896i 0.500000 − 0.866025i 2.01734 − 0.734253i 0.500000 + 0.866025i
181.4 −0.173648 + 0.984808i 0.550788 + 3.12367i −0.939693 − 0.342020i 0.766044 − 0.642788i −3.17186 0.712016 − 0.597452i 0.500000 − 0.866025i −6.63488 + 2.41490i 0.500000 + 0.866025i
201.1 0.939693 + 0.342020i −2.67866 + 0.974954i 0.766044 + 0.642788i 0.173648 − 0.984808i −2.85057 −0.751904 + 4.26426i 0.500000 + 0.866025i 3.92657 − 3.29478i 0.500000 − 0.866025i
201.2 0.939693 + 0.342020i −2.36335 + 0.860188i 0.766044 + 0.642788i 0.173648 − 0.984808i −2.51502 0.832620 − 4.72202i 0.500000 + 0.866025i 2.54735 − 2.13748i 0.500000 − 0.866025i
201.3 0.939693 + 0.342020i 1.85873 − 0.676523i 0.766044 + 0.642788i 0.173648 − 0.984808i 1.97802 0.241589 − 1.37012i 0.500000 + 0.866025i 0.699071 − 0.586590i 0.500000 − 0.866025i
201.4 0.939693 + 0.342020i 2.24358 − 0.816598i 0.766044 + 0.642788i 0.173648 − 0.984808i 2.38757 −0.475008 + 2.69391i 0.500000 + 0.866025i 2.06871 − 1.73585i 0.500000 − 0.866025i
231.1 −0.173648 − 0.984808i −0.420953 + 2.38734i −0.939693 + 0.342020i 0.766044 + 0.642788i 2.42417 0.480222 + 0.402954i 0.500000 + 0.866025i −2.70313 − 0.983860i 0.500000 − 0.866025i
231.2 −0.173648 − 0.984808i −0.116582 + 0.661169i −0.939693 + 0.342020i 0.766044 + 0.642788i 0.671368 −1.79162 − 1.50334i 0.500000 + 0.866025i 2.39553 + 0.871900i 0.500000 − 0.866025i
231.3 −0.173648 − 0.984808i 0.160396 − 0.909649i −0.939693 + 0.342020i 0.766044 + 0.642788i −0.923681 1.63147 + 1.36896i 0.500000 + 0.866025i 2.01734 + 0.734253i 0.500000 − 0.866025i
231.4 −0.173648 − 0.984808i 0.550788 − 3.12367i −0.939693 + 0.342020i 0.766044 + 0.642788i −3.17186 0.712016 + 0.597452i 0.500000 + 0.866025i −6.63488 − 2.41490i 0.500000 − 0.866025i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 71.4
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.f even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 370.2.o.c ✓ 24
37.f even 9 1 inner 370.2.o.c ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.o.c ✓ 24 1.a even 1 1 trivial
370.2.o.c ✓ 24 37.f even 9 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{24} - 6 T_{3}^{22} + 10 T_{3}^{21} + 54 T_{3}^{20} - 189 T_{3}^{19} + 550 T_{3}^{18} + \cdots + 179776 \) acting on \(S_{2}^{\mathrm{new}}(370, [\chi])\). Copy content Toggle raw display