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Properties

Label 368.3.r.a
Level $368$
Weight $3$
Character orbit 368.r
Analytic conductor $10.027$
Analytic rank $0$
Dimension $80$
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] = [368,3,Mod(31,368)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(368, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([11, 0, 6]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("368.31");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 368 = 2^{4} \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 368.r (of order \(22\), degree \(10\), 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: \(10.0272737285\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(8\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 80 q + 8 q^{5} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 80 q + 8 q^{5} + 4 q^{9} + 4 q^{13} + 114 q^{17} + 70 q^{21} - 140 q^{25} - 182 q^{29} - 246 q^{33} + 48 q^{37} - 4 q^{41} - 84 q^{45} - 48 q^{49} + 36 q^{53} - 726 q^{57} - 420 q^{61} - 506 q^{65} + 110 q^{69} + 480 q^{73} + 830 q^{77} + 1984 q^{81} + 748 q^{85} + 56 q^{89} - 220 q^{93} + 974 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1 0 −5.00974 + 2.28787i 0 −2.34734 − 2.70898i 0 1.26664 − 1.97094i 0 13.9694 − 16.1215i 0
31.2 0 −2.68858 + 1.22783i 0 4.54287 + 5.24275i 0 −3.65615 + 5.68908i 0 −0.172854 + 0.199484i 0
31.3 0 −1.64249 + 0.750101i 0 0.362178 + 0.417975i 0 5.45972 − 8.49548i 0 −3.75862 + 4.33768i 0
31.4 0 −1.12057 + 0.511748i 0 −3.66096 − 4.22498i 0 −2.96881 + 4.61956i 0 −4.89995 + 5.65484i 0
31.5 0 1.12057 − 0.511748i 0 −3.66096 − 4.22498i 0 2.96881 − 4.61956i 0 −4.89995 + 5.65484i 0
31.6 0 1.64249 − 0.750101i 0 0.362178 + 0.417975i 0 −5.45972 + 8.49548i 0 −3.75862 + 4.33768i 0
31.7 0 2.68858 − 1.22783i 0 4.54287 + 5.24275i 0 3.65615 − 5.68908i 0 −0.172854 + 0.199484i 0
31.8 0 5.00974 − 2.28787i 0 −2.34734 − 2.70898i 0 −1.26664 + 1.97094i 0 13.9694 − 16.1215i 0
95.1 0 −5.00974 − 2.28787i 0 −2.34734 + 2.70898i 0 1.26664 + 1.97094i 0 13.9694 + 16.1215i 0
95.2 0 −2.68858 − 1.22783i 0 4.54287 − 5.24275i 0 −3.65615 − 5.68908i 0 −0.172854 − 0.199484i 0
95.3 0 −1.64249 − 0.750101i 0 0.362178 − 0.417975i 0 5.45972 + 8.49548i 0 −3.75862 − 4.33768i 0
95.4 0 −1.12057 − 0.511748i 0 −3.66096 + 4.22498i 0 −2.96881 − 4.61956i 0 −4.89995 − 5.65484i 0
95.5 0 1.12057 + 0.511748i 0 −3.66096 + 4.22498i 0 2.96881 + 4.61956i 0 −4.89995 − 5.65484i 0
95.6 0 1.64249 + 0.750101i 0 0.362178 − 0.417975i 0 −5.45972 − 8.49548i 0 −3.75862 − 4.33768i 0
95.7 0 2.68858 + 1.22783i 0 4.54287 − 5.24275i 0 3.65615 + 5.68908i 0 −0.172854 − 0.199484i 0
95.8 0 5.00974 + 2.28787i 0 −2.34734 + 2.70898i 0 −1.26664 − 1.97094i 0 13.9694 + 16.1215i 0
127.1 0 −5.78100 − 0.831182i 0 2.57174 − 0.755130i 0 −4.04879 + 3.50830i 0 24.0936 + 7.07452i 0
127.2 0 −2.76561 − 0.397635i 0 1.45039 − 0.425874i 0 1.14549 − 0.992571i 0 −1.14493 − 0.336182i 0
127.3 0 −2.69424 − 0.387373i 0 −5.85332 + 1.71869i 0 2.98524 − 2.58673i 0 −1.52656 − 0.448237i 0
127.4 0 −0.307698 − 0.0442403i 0 6.94735 − 2.03993i 0 8.70081 − 7.53930i 0 −8.54272 − 2.50837i 0
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.8
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
23.c even 11 1 inner
92.g odd 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 368.3.r.a ✓ 80
4.b odd 2 1 inner 368.3.r.a ✓ 80
23.c even 11 1 inner 368.3.r.a ✓ 80
92.g odd 22 1 inner 368.3.r.a ✓ 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
368.3.r.a ✓ 80 1.a even 1 1 trivial
368.3.r.a ✓ 80 4.b odd 2 1 inner
368.3.r.a ✓ 80 23.c even 11 1 inner
368.3.r.a ✓ 80 92.g odd 22 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{80} - 38 T_{3}^{78} + 577 T_{3}^{76} - 10765 T_{3}^{74} + 632570 T_{3}^{72} + \cdots + 35\!\cdots\!49 \) acting on \(S_{3}^{\mathrm{new}}(368, [\chi])\). Copy content Toggle raw display