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Properties

Label 2380.2.m.d
Level $2380$
Weight $2$
Character orbit 2380.m
Analytic conductor $19.004$
Analytic rank $0$
Dimension $26$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2380,2,Mod(169,2380)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2380, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 1])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2380.169"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 2380 = 2^{2} \cdot 5 \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2380.m (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [26,0,6] 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: \(19.0043956811\)
Analytic rank: \(0\)
Dimension: \(26\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 26 q + 6 q^{3} - 4 q^{5} + 26 q^{7} + 32 q^{9} + 10 q^{15} - 11 q^{17} + 6 q^{21} - 16 q^{23} + 4 q^{25} + 6 q^{27} - 4 q^{35} - 20 q^{37} - 18 q^{45} + 26 q^{49} + 9 q^{51} - 14 q^{55} + 32 q^{59} + 32 q^{63}+ \cdots - 46 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
169.1 0 −3.24912 0 −1.49895 − 1.65926i 0 1.00000 0 7.55679 0
169.2 0 −3.24912 0 −1.49895 + 1.65926i 0 1.00000 0 7.55679 0
169.3 0 −2.69877 0 1.83179 + 1.28240i 0 1.00000 0 4.28335 0
169.4 0 −2.69877 0 1.83179 − 1.28240i 0 1.00000 0 4.28335 0
169.5 0 −1.69804 0 −2.03720 + 0.921846i 0 1.00000 0 −0.116652 0
169.6 0 −1.69804 0 −2.03720 − 0.921846i 0 1.00000 0 −0.116652 0
169.7 0 −1.20796 0 −1.45998 + 1.69366i 0 1.00000 0 −1.54083 0
169.8 0 −1.20796 0 −1.45998 − 1.69366i 0 1.00000 0 −1.54083 0
169.9 0 −1.20753 0 −0.390259 + 2.20175i 0 1.00000 0 −1.54186 0
169.10 0 −1.20753 0 −0.390259 − 2.20175i 0 1.00000 0 −1.54186 0
169.11 0 −0.0401658 0 1.78015 + 1.35317i 0 1.00000 0 −2.99839 0
169.12 0 −0.0401658 0 1.78015 − 1.35317i 0 1.00000 0 −2.99839 0
169.13 0 0.355118 0 0.778855 − 2.09604i 0 1.00000 0 −2.87389 0
169.14 0 0.355118 0 0.778855 + 2.09604i 0 1.00000 0 −2.87389 0
169.15 0 0.807362 0 −2.19195 + 0.442004i 0 1.00000 0 −2.34817 0
169.16 0 0.807362 0 −2.19195 − 0.442004i 0 1.00000 0 −2.34817 0
169.17 0 1.60963 0 −0.593519 − 2.15586i 0 1.00000 0 −0.409099 0
169.18 0 1.60963 0 −0.593519 + 2.15586i 0 1.00000 0 −0.409099 0
169.19 0 1.69521 0 1.71782 + 1.43146i 0 1.00000 0 −0.126273 0
169.20 0 1.69521 0 1.71782 − 1.43146i 0 1.00000 0 −0.126273 0
See all 26 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 169.26
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
85.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2380.2.m.d yes 26
5.b even 2 1 2380.2.m.c ✓ 26
17.b even 2 1 2380.2.m.c ✓ 26
85.c even 2 1 inner 2380.2.m.d yes 26
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2380.2.m.c ✓ 26 5.b even 2 1
2380.2.m.c ✓ 26 17.b even 2 1
2380.2.m.d yes 26 1.a even 1 1 trivial
2380.2.m.d yes 26 85.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{13} - 3 T_{3}^{12} - 23 T_{3}^{11} + 71 T_{3}^{10} + 176 T_{3}^{9} - 574 T_{3}^{8} - 522 T_{3}^{7} + \cdots - 16 \) acting on \(S_{2}^{\mathrm{new}}(2380, [\chi])\). Copy content Toggle raw display