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Properties

Label 3360.2.e.d
Level $3360$
Weight $2$
Character orbit 3360.e
Analytic conductor $26.830$
Analytic rank $0$
Dimension $44$
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] = [3360,2,Mod(911,3360)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3360, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1, 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("3360.911");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3360.e (of order \(2\), degree \(1\), not 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: \(26.8297350792\)
Analytic rank: \(0\)
Dimension: \(44\)
Twist minimal: no (minimal twist has level 840)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 44 q - 4 q^{3} + 44 q^{5} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 44 q - 4 q^{3} + 44 q^{5} + 4 q^{9} - 4 q^{15} - 16 q^{19} + 16 q^{23} + 44 q^{25} - 4 q^{27} - 16 q^{29} + 24 q^{33} - 16 q^{39} - 24 q^{43} + 4 q^{45} - 24 q^{47} - 44 q^{49} + 32 q^{53} - 24 q^{67} + 8 q^{69} + 8 q^{71} + 8 q^{73} - 4 q^{75} - 8 q^{77} + 20 q^{81} - 16 q^{87} + 16 q^{91} - 16 q^{93} - 16 q^{95} + 24 q^{97} + 64 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
911.1 0 −1.72227 − 0.183849i 0 1.00000 0 1.00000i 0 2.93240 + 0.633274i 0
911.2 0 −1.72227 + 0.183849i 0 1.00000 0 − 1.00000i 0 2.93240 − 0.633274i 0
911.3 0 −1.70474 − 0.306378i 0 1.00000 0 − 1.00000i 0 2.81226 + 1.04459i 0
911.4 0 −1.70474 + 0.306378i 0 1.00000 0 1.00000i 0 2.81226 − 1.04459i 0
911.5 0 −1.67255 − 0.450097i 0 1.00000 0 1.00000i 0 2.59483 + 1.50562i 0
911.6 0 −1.67255 + 0.450097i 0 1.00000 0 − 1.00000i 0 2.59483 − 1.50562i 0
911.7 0 −1.52280 − 0.825275i 0 1.00000 0 1.00000i 0 1.63784 + 2.51346i 0
911.8 0 −1.52280 + 0.825275i 0 1.00000 0 − 1.00000i 0 1.63784 − 2.51346i 0
911.9 0 −1.51567 − 0.838298i 0 1.00000 0 1.00000i 0 1.59451 + 2.54117i 0
911.10 0 −1.51567 + 0.838298i 0 1.00000 0 − 1.00000i 0 1.59451 − 2.54117i 0
911.11 0 −1.39892 − 1.02128i 0 1.00000 0 − 1.00000i 0 0.913962 + 2.85739i 0
911.12 0 −1.39892 + 1.02128i 0 1.00000 0 1.00000i 0 0.913962 − 2.85739i 0
911.13 0 −1.20864 − 1.24064i 0 1.00000 0 − 1.00000i 0 −0.0783833 + 2.99898i 0
911.14 0 −1.20864 + 1.24064i 0 1.00000 0 1.00000i 0 −0.0783833 − 2.99898i 0
911.15 0 −1.08178 − 1.35269i 0 1.00000 0 − 1.00000i 0 −0.659524 + 2.92661i 0
911.16 0 −1.08178 + 1.35269i 0 1.00000 0 1.00000i 0 −0.659524 − 2.92661i 0
911.17 0 −0.663478 − 1.59994i 0 1.00000 0 1.00000i 0 −2.11959 + 2.12305i 0
911.18 0 −0.663478 + 1.59994i 0 1.00000 0 − 1.00000i 0 −2.11959 − 2.12305i 0
911.19 0 −0.467013 − 1.66790i 0 1.00000 0 − 1.00000i 0 −2.56380 + 1.55787i 0
911.20 0 −0.467013 + 1.66790i 0 1.00000 0 1.00000i 0 −2.56380 − 1.55787i 0
See all 44 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 911.44
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3360.2.e.d 44
3.b odd 2 1 3360.2.e.c 44
4.b odd 2 1 840.2.e.c ✓ 44
8.b even 2 1 840.2.e.d yes 44
8.d odd 2 1 3360.2.e.c 44
12.b even 2 1 840.2.e.d yes 44
24.f even 2 1 inner 3360.2.e.d 44
24.h odd 2 1 840.2.e.c ✓ 44
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.e.c ✓ 44 4.b odd 2 1
840.2.e.c ✓ 44 24.h odd 2 1
840.2.e.d yes 44 8.b even 2 1
840.2.e.d yes 44 12.b even 2 1
3360.2.e.c 44 3.b odd 2 1
3360.2.e.c 44 8.d odd 2 1
3360.2.e.d 44 1.a even 1 1 trivial
3360.2.e.d 44 24.f even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3360, [\chi])\):

\( T_{11}^{44} + 264 T_{11}^{42} + 31844 T_{11}^{40} + 2327928 T_{11}^{38} + 115382518 T_{11}^{36} + \cdots + 202012780134400 \) Copy content Toggle raw display
\( T_{23}^{22} - 8 T_{23}^{21} - 216 T_{23}^{20} + 1776 T_{23}^{19} + 18240 T_{23}^{18} - 153728 T_{23}^{17} + \cdots - 527433728 \) Copy content Toggle raw display