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Properties

Label 912.2.cc.h
Level $912$
Weight $2$
Character orbit 912.cc
Analytic conductor $7.282$
Analytic rank $0$
Dimension $60$
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] = [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: \(60\)
Relative dimension: \(10\) over \(\Q(\zeta_{18})\)
Twist minimal: no (minimal twist has level 456)
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.

\(\operatorname{Tr}(f)(q) = \) \( 60 q + 3 q^{3} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 60 q + 3 q^{3} - 3 q^{9} + 3 q^{13} + 9 q^{15} + 6 q^{17} - 3 q^{19} + 6 q^{25} - 6 q^{27} - 6 q^{29} + 45 q^{33} + 24 q^{35} - 18 q^{39} - 3 q^{41} + 21 q^{43} - 45 q^{45} - 18 q^{47} - 30 q^{49} + 6 q^{51} + 36 q^{53} - 18 q^{55} + 48 q^{57} - 21 q^{59} - 6 q^{61} - 78 q^{63} - 24 q^{65} + 48 q^{67} + 21 q^{69} + 36 q^{71} - 57 q^{73} + 36 q^{79} - 39 q^{81} - 36 q^{83} + 6 q^{85} + 90 q^{87} + 24 q^{89} + 18 q^{91} - 54 q^{93} + 30 q^{95} + 15 q^{97} + 3 q^{99}+O(q^{100}) \) 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 −1.55640 + 0.760012i 0 −0.293373 + 0.806036i 0 −0.627281 + 1.08648i 0 1.84476 − 2.36577i 0
257.2 0 −1.52796 − 0.815675i 0 0.621356 − 1.70716i 0 −0.144681 + 0.250595i 0 1.66935 + 2.49264i 0
257.3 0 −0.828110 − 1.52126i 0 −1.42044 + 3.90263i 0 0.0388591 − 0.0673059i 0 −1.62847 + 2.51954i 0
257.4 0 −0.602283 + 1.62396i 0 1.11883 − 3.07396i 0 −0.429476 + 0.743873i 0 −2.27451 − 1.95617i 0
257.5 0 −0.427096 + 1.67857i 0 −0.774768 + 2.12866i 0 2.02663 − 3.51023i 0 −2.63518 − 1.43382i 0
257.6 0 0.230961 − 1.71658i 0 0.555150 − 1.52526i 0 1.81489 − 3.14348i 0 −2.89331 − 0.792928i 0
257.7 0 0.848284 − 1.51010i 0 −0.0679758 + 0.186762i 0 −2.39129 + 4.14183i 0 −1.56083 − 2.56199i 0
257.8 0 1.30244 + 1.14178i 0 −0.978409 + 2.68816i 0 −1.73645 + 3.00762i 0 0.392677 + 2.97419i 0
257.9 0 1.33217 + 1.10694i 0 0.0199730 − 0.0548754i 0 1.38520 − 2.39923i 0 0.549363 + 2.94927i 0
257.10 0 1.72800 + 0.118384i 0 1.21966 − 3.35098i 0 −0.702445 + 1.21667i 0 2.97197 + 0.409135i 0
401.1 0 −1.55640 − 0.760012i 0 −0.293373 − 0.806036i 0 −0.627281 − 1.08648i 0 1.84476 + 2.36577i 0
401.2 0 −1.52796 + 0.815675i 0 0.621356 + 1.70716i 0 −0.144681 − 0.250595i 0 1.66935 − 2.49264i 0
401.3 0 −0.828110 + 1.52126i 0 −1.42044 − 3.90263i 0 0.0388591 + 0.0673059i 0 −1.62847 − 2.51954i 0
401.4 0 −0.602283 − 1.62396i 0 1.11883 + 3.07396i 0 −0.429476 − 0.743873i 0 −2.27451 + 1.95617i 0
401.5 0 −0.427096 − 1.67857i 0 −0.774768 − 2.12866i 0 2.02663 + 3.51023i 0 −2.63518 + 1.43382i 0
401.6 0 0.230961 + 1.71658i 0 0.555150 + 1.52526i 0 1.81489 + 3.14348i 0 −2.89331 + 0.792928i 0
401.7 0 0.848284 + 1.51010i 0 −0.0679758 − 0.186762i 0 −2.39129 − 4.14183i 0 −1.56083 + 2.56199i 0
401.8 0 1.30244 − 1.14178i 0 −0.978409 − 2.68816i 0 −1.73645 − 3.00762i 0 0.392677 − 2.97419i 0
401.9 0 1.33217 − 1.10694i 0 0.0199730 + 0.0548754i 0 1.38520 + 2.39923i 0 0.549363 − 2.94927i 0
401.10 0 1.72800 − 0.118384i 0 1.21966 + 3.35098i 0 −0.702445 − 1.21667i 0 2.97197 − 0.409135i 0
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 257.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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.h 60
3.b odd 2 1 912.2.cc.g 60
4.b odd 2 1 456.2.bm.a ✓ 60
12.b even 2 1 456.2.bm.b yes 60
19.f odd 18 1 912.2.cc.g 60
57.j even 18 1 inner 912.2.cc.h 60
76.k even 18 1 456.2.bm.b yes 60
228.u odd 18 1 456.2.bm.a ✓ 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
456.2.bm.a ✓ 60 4.b odd 2 1
456.2.bm.a ✓ 60 228.u odd 18 1
456.2.bm.b yes 60 12.b even 2 1
456.2.bm.b yes 60 76.k even 18 1
912.2.cc.g 60 3.b odd 2 1
912.2.cc.g 60 19.f odd 18 1
912.2.cc.h 60 1.a even 1 1 trivial
912.2.cc.h 60 57.j even 18 1 inner

Hecke kernels

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

\( T_{5}^{60} - 3 T_{5}^{58} - 36 T_{5}^{57} + 3 T_{5}^{56} + 78 T_{5}^{55} - 3824 T_{5}^{54} + \cdots + 66324791296 \) Copy content Toggle raw display
\( T_{7}^{60} + 120 T_{7}^{58} + 14 T_{7}^{57} + 8106 T_{7}^{56} + 1581 T_{7}^{55} + 375711 T_{7}^{54} + \cdots + 1671828544 \) Copy content Toggle raw display