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Properties

Label 2340.2.r.f
Level $2340$
Weight $2$
Character orbit 2340.r
Analytic conductor $18.685$
Analytic rank $0$
Dimension $26$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2340,2,Mod(781,2340)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2340, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2340.781");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2340 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2340.r (of order \(3\), 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: \(18.6849940730\)
Analytic rank: \(0\)
Dimension: \(26\)
Relative dimension: \(13\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 26 q - q^{3} + 13 q^{5} - q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 26 q - q^{3} + 13 q^{5} - q^{9} + q^{11} + 13 q^{13} - 2 q^{15} - 2 q^{17} + 6 q^{19} + 9 q^{21} - 13 q^{25} + 14 q^{27} + 6 q^{29} + 4 q^{31} - 4 q^{33} + 20 q^{37} - 2 q^{39} + 11 q^{41} + 2 q^{43} + q^{45} + 5 q^{47} - 21 q^{49} - 9 q^{51} + 2 q^{53} + 2 q^{55} + 5 q^{57} - 7 q^{59} - 2 q^{61} + 22 q^{63} - 13 q^{65} - 7 q^{67} - 18 q^{69} - 36 q^{71} + 28 q^{73} - q^{75} + 38 q^{77} + 8 q^{79} - 25 q^{81} - 9 q^{83} - q^{85} - 32 q^{87} - 32 q^{89} + 5 q^{93} + 3 q^{95} - 29 q^{97} + 52 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
781.1 0 −1.72809 − 0.117002i 0 0.500000 + 0.866025i 0 1.87998 − 3.25623i 0 2.97262 + 0.404382i 0
781.2 0 −1.54366 − 0.785562i 0 0.500000 + 0.866025i 0 −1.37020 + 2.37326i 0 1.76579 + 2.42528i 0
781.3 0 −1.20392 − 1.24522i 0 0.500000 + 0.866025i 0 1.86401 − 3.22856i 0 −0.101170 + 2.99829i 0
781.4 0 −1.14379 + 1.30067i 0 0.500000 + 0.866025i 0 1.63870 − 2.83832i 0 −0.383504 − 2.97539i 0
781.5 0 −0.961635 + 1.44058i 0 0.500000 + 0.866025i 0 −1.25670 + 2.17667i 0 −1.15052 − 2.77062i 0
781.6 0 −0.774583 − 1.54920i 0 0.500000 + 0.866025i 0 −2.09678 + 3.63173i 0 −1.80004 + 2.39997i 0
781.7 0 −0.230114 + 1.71670i 0 0.500000 + 0.866025i 0 1.17123 − 2.02863i 0 −2.89409 − 0.790073i 0
781.8 0 0.564945 + 1.63733i 0 0.500000 + 0.866025i 0 −1.40578 + 2.43489i 0 −2.36167 + 1.85000i 0
781.9 0 0.861706 − 1.50249i 0 0.500000 + 0.866025i 0 −0.540503 + 0.936179i 0 −1.51493 − 2.58940i 0
781.10 0 0.875611 + 1.49443i 0 0.500000 + 0.866025i 0 −0.00230079 + 0.00398509i 0 −1.46661 + 2.61707i 0
781.11 0 1.35022 − 1.08486i 0 0.500000 + 0.866025i 0 −2.05659 + 3.56213i 0 0.646168 − 2.92958i 0
781.12 0 1.70783 − 0.288621i 0 0.500000 + 0.866025i 0 1.41733 − 2.45490i 0 2.83340 − 0.985833i 0
781.13 0 1.72548 − 0.150719i 0 0.500000 + 0.866025i 0 0.757602 − 1.31221i 0 2.95457 − 0.520126i 0
1561.1 0 −1.72809 + 0.117002i 0 0.500000 − 0.866025i 0 1.87998 + 3.25623i 0 2.97262 − 0.404382i 0
1561.2 0 −1.54366 + 0.785562i 0 0.500000 − 0.866025i 0 −1.37020 − 2.37326i 0 1.76579 − 2.42528i 0
1561.3 0 −1.20392 + 1.24522i 0 0.500000 − 0.866025i 0 1.86401 + 3.22856i 0 −0.101170 − 2.99829i 0
1561.4 0 −1.14379 − 1.30067i 0 0.500000 − 0.866025i 0 1.63870 + 2.83832i 0 −0.383504 + 2.97539i 0
1561.5 0 −0.961635 − 1.44058i 0 0.500000 − 0.866025i 0 −1.25670 − 2.17667i 0 −1.15052 + 2.77062i 0
1561.6 0 −0.774583 + 1.54920i 0 0.500000 − 0.866025i 0 −2.09678 − 3.63173i 0 −1.80004 − 2.39997i 0
1561.7 0 −0.230114 − 1.71670i 0 0.500000 − 0.866025i 0 1.17123 + 2.02863i 0 −2.89409 + 0.790073i 0
See all 26 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 781.13
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2340.2.r.f ✓ 26
3.b odd 2 1 7020.2.r.f 26
9.c even 3 1 inner 2340.2.r.f ✓ 26
9.d odd 6 1 7020.2.r.f 26
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2340.2.r.f ✓ 26 1.a even 1 1 trivial
2340.2.r.f ✓ 26 9.c even 3 1 inner
7020.2.r.f 26 3.b odd 2 1
7020.2.r.f 26 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{26} + 56 T_{7}^{24} - 16 T_{7}^{23} + 1909 T_{7}^{22} - 737 T_{7}^{21} + 42194 T_{7}^{20} + \cdots + 589824 \) acting on \(S_{2}^{\mathrm{new}}(2340, [\chi])\). Copy content Toggle raw display