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Properties

Label 912.2.ci.g
Level $912$
Weight $2$
Character orbit 912.ci
Analytic conductor $7.282$
Analytic rank $0$
Dimension $24$
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(79,912)]
 
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([9, 0, 0, 13]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("912.79");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 912.ci (of order \(18\), degree \(6\), 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: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 9 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 9 q^{7} + 9 q^{11} - 9 q^{13} - 6 q^{17} - 3 q^{19} - 6 q^{21} + 15 q^{23} + 6 q^{25} - 12 q^{27} - 6 q^{29} + 12 q^{31} - 3 q^{33} + 30 q^{41} - 9 q^{43} + 3 q^{45} - 15 q^{47} + 27 q^{49} + 3 q^{51} + 6 q^{53} + 21 q^{55} - 9 q^{57} - 36 q^{59} - 21 q^{61} - 3 q^{63} - 9 q^{65} + 45 q^{67} - 36 q^{71} + 42 q^{75} + 108 q^{77} + 36 q^{79} - 27 q^{83} - 9 q^{85} - 9 q^{87} - 27 q^{89} - 36 q^{91} - 18 q^{93} + 30 q^{95} - 51 q^{97} + 3 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1 0 −0.939693 + 0.342020i 0 −0.467043 + 2.64873i 0 −0.480437 + 0.277380i 0 0.766044 − 0.642788i 0
79.2 0 −0.939693 + 0.342020i 0 −0.134718 + 0.764021i 0 −0.911489 + 0.526248i 0 0.766044 − 0.642788i 0
79.3 0 −0.939693 + 0.342020i 0 0.494739 − 2.80581i 0 4.05139 − 2.33907i 0 0.766044 − 0.642788i 0
79.4 0 −0.939693 + 0.342020i 0 0.525770 − 2.98179i 0 −3.04612 + 1.75868i 0 0.766044 − 0.642788i 0
127.1 0 −0.939693 − 0.342020i 0 −0.467043 − 2.64873i 0 −0.480437 − 0.277380i 0 0.766044 + 0.642788i 0
127.2 0 −0.939693 − 0.342020i 0 −0.134718 − 0.764021i 0 −0.911489 − 0.526248i 0 0.766044 + 0.642788i 0
127.3 0 −0.939693 − 0.342020i 0 0.494739 + 2.80581i 0 4.05139 + 2.33907i 0 0.766044 + 0.642788i 0
127.4 0 −0.939693 − 0.342020i 0 0.525770 + 2.98179i 0 −3.04612 − 1.75868i 0 0.766044 + 0.642788i 0
223.1 0 0.173648 − 0.984808i 0 −3.37195 − 2.82940i 0 1.97482 − 1.14016i 0 −0.939693 − 0.342020i 0
223.2 0 0.173648 − 0.984808i 0 −1.40574 − 1.17956i 0 −4.46660 + 2.57879i 0 −0.939693 − 0.342020i 0
223.3 0 0.173648 − 0.984808i 0 1.00588 + 0.844034i 0 3.15949 − 1.82413i 0 −0.939693 − 0.342020i 0
223.4 0 0.173648 − 0.984808i 0 1.30003 + 1.09086i 0 −1.57531 + 0.909508i 0 −0.939693 − 0.342020i 0
319.1 0 0.173648 + 0.984808i 0 −3.37195 + 2.82940i 0 1.97482 + 1.14016i 0 −0.939693 + 0.342020i 0
319.2 0 0.173648 + 0.984808i 0 −1.40574 + 1.17956i 0 −4.46660 − 2.57879i 0 −0.939693 + 0.342020i 0
319.3 0 0.173648 + 0.984808i 0 1.00588 − 0.844034i 0 3.15949 + 1.82413i 0 −0.939693 + 0.342020i 0
319.4 0 0.173648 + 0.984808i 0 1.30003 − 1.09086i 0 −1.57531 − 0.909508i 0 −0.939693 + 0.342020i 0
751.1 0 0.766044 − 0.642788i 0 −2.67232 − 0.972646i 0 −2.07863 − 1.20010i 0 0.173648 − 0.984808i 0
751.2 0 0.766044 − 0.642788i 0 −0.800469 − 0.291347i 0 1.70402 + 0.983816i 0 0.173648 − 0.984808i 0
751.3 0 0.766044 − 0.642788i 0 1.72729 + 0.628683i 0 −3.53090 − 2.03857i 0 0.173648 − 0.984808i 0
751.4 0 0.766044 − 0.642788i 0 3.79853 + 1.38255i 0 0.699779 + 0.404018i 0 0.173648 − 0.984808i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 79.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
76.k 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.ci.g ✓ 24
4.b odd 2 1 912.2.ci.h yes 24
19.f odd 18 1 912.2.ci.h yes 24
76.k even 18 1 inner 912.2.ci.g ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
912.2.ci.g ✓ 24 1.a even 1 1 trivial
912.2.ci.g ✓ 24 76.k even 18 1 inner
912.2.ci.h yes 24 4.b odd 2 1
912.2.ci.h yes 24 19.f odd 18 1

Hecke kernels

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

\( T_{5}^{24} - 3 T_{5}^{22} - 34 T_{5}^{21} + 84 T_{5}^{20} - 549 T_{5}^{19} + 2661 T_{5}^{18} + \cdots + 34012224 \) Copy content Toggle raw display
\( T_{7}^{24} + 9 T_{7}^{23} - 15 T_{7}^{22} - 378 T_{7}^{21} + 36 T_{7}^{20} + 10998 T_{7}^{19} + \cdots + 136118889 \) Copy content Toggle raw display