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Properties

Label 912.2.cc.e
Level $912$
Weight $2$
Character orbit 912.cc
Analytic conductor $7.282$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [912,2,Mod(257,912)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(912, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 0, 9, 17]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("912.257");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 912.cc (of order \(18\), degree \(6\), 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: \(7.28235666434\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(4\) over \(\Q(\zeta_{18})\)
Twist minimal: no (minimal twist has level 57)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 9 q^{3} + 6 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 9 q^{3} + 6 q^{7} + 3 q^{9} + 6 q^{13} + 9 q^{15} + 12 q^{19} - 6 q^{21} + 12 q^{25} + 27 q^{27} + 18 q^{31} + 21 q^{33} - 54 q^{43} + 6 q^{45} + 6 q^{49} - 3 q^{51} + 90 q^{55} - 6 q^{57} - 6 q^{61} + 9 q^{63} - 36 q^{69} + 90 q^{73} - 30 q^{79} + 3 q^{81} - 6 q^{85} - 15 q^{87} - 66 q^{91} + 33 q^{93} - 78 q^{97} - 12 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
257.1 0 −1.43743 + 0.966337i 0 −1.09544 + 3.00968i 0 1.23083 − 2.13186i 0 1.13238 − 2.77808i 0
257.2 0 −0.504201 + 1.65704i 0 −0.533860 + 1.46677i 0 −1.49687 + 2.59266i 0 −2.49156 − 1.67096i 0
257.3 0 1.20126 − 1.24779i 0 1.09544 − 3.00968i 0 1.23083 − 2.13186i 0 −0.113935 − 2.99784i 0
257.4 0 1.71942 − 0.208799i 0 0.533860 − 1.46677i 0 −1.49687 + 2.59266i 0 2.91281 − 0.718027i 0
401.1 0 −1.43743 − 0.966337i 0 −1.09544 − 3.00968i 0 1.23083 + 2.13186i 0 1.13238 + 2.77808i 0
401.2 0 −0.504201 − 1.65704i 0 −0.533860 − 1.46677i 0 −1.49687 − 2.59266i 0 −2.49156 + 1.67096i 0
401.3 0 1.20126 + 1.24779i 0 1.09544 + 3.00968i 0 1.23083 + 2.13186i 0 −0.113935 + 2.99784i 0
401.4 0 1.71942 + 0.208799i 0 0.533860 + 1.46677i 0 −1.49687 − 2.59266i 0 2.91281 + 0.718027i 0
497.1 0 −1.40671 + 1.01052i 0 −0.487091 − 0.0858872i 0 0.969730 + 1.67962i 0 0.957685 − 2.84303i 0
497.2 0 −1.09578 − 1.34136i 0 −3.79113 − 0.668479i 0 0.469963 + 0.814000i 0 −0.598514 + 2.93969i 0
497.3 0 −0.0227926 − 1.73190i 0 3.79113 + 0.668479i 0 0.469963 + 0.814000i 0 −2.99896 + 0.0789491i 0
497.4 0 1.72716 − 0.130112i 0 0.487091 + 0.0858872i 0 0.969730 + 1.67962i 0 2.96614 − 0.449448i 0
545.1 0 −1.40671 − 1.01052i 0 −0.487091 + 0.0858872i 0 0.969730 − 1.67962i 0 0.957685 + 2.84303i 0
545.2 0 −1.09578 + 1.34136i 0 −3.79113 + 0.668479i 0 0.469963 − 0.814000i 0 −0.598514 − 2.93969i 0
545.3 0 −0.0227926 + 1.73190i 0 3.79113 − 0.668479i 0 0.469963 − 0.814000i 0 −2.99896 − 0.0789491i 0
545.4 0 1.72716 + 0.130112i 0 0.487091 − 0.0858872i 0 0.969730 − 1.67962i 0 2.96614 + 0.449448i 0
641.1 0 0.292097 + 1.70724i 0 0.485824 − 0.578982i 0 −1.38278 − 2.39504i 0 −2.82936 + 0.997362i 0
641.2 0 0.858393 − 1.50438i 0 −0.485824 + 0.578982i 0 −1.38278 − 2.39504i 0 −1.52632 − 2.58270i 0
641.3 0 1.47284 + 0.911455i 0 1.78018 − 2.12154i 0 1.70913 + 2.96030i 0 1.33850 + 2.68485i 0
641.4 0 1.69575 − 0.352748i 0 −1.78018 + 2.12154i 0 1.70913 + 2.96030i 0 2.75114 − 1.19634i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 257.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
19.f odd 18 1 inner
57.j even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 912.2.cc.e 24
3.b odd 2 1 inner 912.2.cc.e 24
4.b odd 2 1 57.2.j.b ✓ 24
12.b even 2 1 57.2.j.b ✓ 24
19.f odd 18 1 inner 912.2.cc.e 24
57.j even 18 1 inner 912.2.cc.e 24
76.k even 18 1 57.2.j.b ✓ 24
76.k even 18 1 1083.2.d.d 24
76.l odd 18 1 1083.2.d.d 24
228.u odd 18 1 57.2.j.b ✓ 24
228.u odd 18 1 1083.2.d.d 24
228.v even 18 1 1083.2.d.d 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.2.j.b ✓ 24 4.b odd 2 1
57.2.j.b ✓ 24 12.b even 2 1
57.2.j.b ✓ 24 76.k even 18 1
57.2.j.b ✓ 24 228.u odd 18 1
912.2.cc.e 24 1.a even 1 1 trivial
912.2.cc.e 24 3.b odd 2 1 inner
912.2.cc.e 24 19.f odd 18 1 inner
912.2.cc.e 24 57.j even 18 1 inner
1083.2.d.d 24 76.k even 18 1
1083.2.d.d 24 76.l odd 18 1
1083.2.d.d 24 228.u odd 18 1
1083.2.d.d 24 228.v even 18 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(912, [\chi])\):

\( T_{5}^{24} - 6 T_{5}^{22} - 114 T_{5}^{20} - 844 T_{5}^{18} + 15696 T_{5}^{16} + 151632 T_{5}^{14} + \cdots + 157609 \) Copy content Toggle raw display
\( T_{7}^{12} - 3 T_{7}^{11} + 24 T_{7}^{10} - 53 T_{7}^{9} + 333 T_{7}^{8} - 702 T_{7}^{7} + 2637 T_{7}^{6} + \cdots + 16129 \) Copy content Toggle raw display