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Properties

Label 338.4.e.i
Level $338$
Weight $4$
Character orbit 338.e
Analytic conductor $19.943$
Analytic rank $0$
Dimension $24$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [338,4,Mod(23,338)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(338, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("338.23");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 338 = 2 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 338.e (of order \(6\), degree \(2\), 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: \(19.9426455819\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 18 q^{3} + 48 q^{4} - 226 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 18 q^{3} + 48 q^{4} - 226 q^{9} + 72 q^{10} - 144 q^{12} + 200 q^{14} - 192 q^{16} + 198 q^{17} - 148 q^{22} + 534 q^{23} - 1472 q^{25} + 2676 q^{27} + 238 q^{29} + 472 q^{30} - 1228 q^{35} + 904 q^{36} + 648 q^{38} + 576 q^{40} + 104 q^{42} + 856 q^{43} - 288 q^{48} + 1798 q^{49} - 1156 q^{51} + 356 q^{53} - 2252 q^{55} + 400 q^{56} - 3408 q^{61} - 2500 q^{62} - 1536 q^{64} + 6096 q^{66} - 792 q^{68} + 2336 q^{69} - 1096 q^{74} + 3596 q^{75} - 11160 q^{77} - 3500 q^{79} - 6676 q^{81} + 4560 q^{82} - 3204 q^{87} + 592 q^{88} + 9720 q^{90} + 4272 q^{92} + 3944 q^{94} - 8186 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1 −1.73205 + 1.00000i −5.00428 − 8.66767i 2.00000 − 3.46410i 13.8136i 17.3353 + 10.0086i 0.227146 + 0.131143i 8.00000i −36.5857 + 63.3682i −13.8136 − 23.9258i
23.2 −1.73205 + 1.00000i 0.622927 + 1.07894i 2.00000 − 3.46410i − 20.5002i −2.15788 − 1.24585i −18.4070 − 10.6273i 8.00000i 12.7239 − 22.0385i 20.5002 + 35.5074i
23.3 −1.73205 + 1.00000i 1.96372 + 3.40126i 2.00000 − 3.46410i 0.152827i −6.80251 − 3.92743i −29.4373 − 16.9956i 8.00000i 5.78763 − 10.0245i −0.152827 − 0.264703i
23.4 −1.73205 + 1.00000i −4.47729 − 7.75489i 2.00000 − 3.46410i − 17.3267i 15.5098 + 8.95458i 14.5068 + 8.37550i 8.00000i −26.5923 + 46.0591i 17.3267 + 30.0108i
23.5 −1.73205 + 1.00000i 4.07401 + 7.05638i 2.00000 − 3.46410i 12.6646i −14.1128 − 8.14801i 24.9160 + 14.3853i 8.00000i −19.6950 + 34.1128i −12.6646 − 21.9357i
23.6 −1.73205 + 1.00000i −1.67908 − 2.90825i 2.00000 − 3.46410i − 6.80407i 5.81649 + 3.35815i −13.4563 − 7.76902i 8.00000i 7.86140 − 13.6163i 6.80407 + 11.7850i
23.7 1.73205 − 1.00000i −4.47729 − 7.75489i 2.00000 − 3.46410i 17.3267i −15.5098 − 8.95458i −14.5068 − 8.37550i − 8.00000i −26.5923 + 46.0591i 17.3267 + 30.0108i
23.8 1.73205 − 1.00000i 0.622927 + 1.07894i 2.00000 − 3.46410i 20.5002i 2.15788 + 1.24585i 18.4070 + 10.6273i − 8.00000i 12.7239 − 22.0385i 20.5002 + 35.5074i
23.9 1.73205 − 1.00000i −1.67908 − 2.90825i 2.00000 − 3.46410i 6.80407i −5.81649 − 3.35815i 13.4563 + 7.76902i − 8.00000i 7.86140 − 13.6163i 6.80407 + 11.7850i
23.10 1.73205 − 1.00000i −5.00428 − 8.66767i 2.00000 − 3.46410i − 13.8136i −17.3353 − 10.0086i −0.227146 − 0.131143i − 8.00000i −36.5857 + 63.3682i −13.8136 − 23.9258i
23.11 1.73205 − 1.00000i 1.96372 + 3.40126i 2.00000 − 3.46410i − 0.152827i 6.80251 + 3.92743i 29.4373 + 16.9956i − 8.00000i 5.78763 − 10.0245i −0.152827 − 0.264703i
23.12 1.73205 − 1.00000i 4.07401 + 7.05638i 2.00000 − 3.46410i − 12.6646i 14.1128 + 8.14801i −24.9160 − 14.3853i − 8.00000i −19.6950 + 34.1128i −12.6646 − 21.9357i
147.1 −1.73205 − 1.00000i −5.00428 + 8.66767i 2.00000 + 3.46410i − 13.8136i 17.3353 − 10.0086i 0.227146 − 0.131143i − 8.00000i −36.5857 − 63.3682i −13.8136 + 23.9258i
147.2 −1.73205 − 1.00000i 0.622927 − 1.07894i 2.00000 + 3.46410i 20.5002i −2.15788 + 1.24585i −18.4070 + 10.6273i − 8.00000i 12.7239 + 22.0385i 20.5002 − 35.5074i
147.3 −1.73205 − 1.00000i 1.96372 − 3.40126i 2.00000 + 3.46410i − 0.152827i −6.80251 + 3.92743i −29.4373 + 16.9956i − 8.00000i 5.78763 + 10.0245i −0.152827 + 0.264703i
147.4 −1.73205 − 1.00000i −4.47729 + 7.75489i 2.00000 + 3.46410i 17.3267i 15.5098 − 8.95458i 14.5068 − 8.37550i − 8.00000i −26.5923 − 46.0591i 17.3267 − 30.0108i
147.5 −1.73205 − 1.00000i 4.07401 − 7.05638i 2.00000 + 3.46410i − 12.6646i −14.1128 + 8.14801i 24.9160 − 14.3853i − 8.00000i −19.6950 − 34.1128i −12.6646 + 21.9357i
147.6 −1.73205 − 1.00000i −1.67908 + 2.90825i 2.00000 + 3.46410i 6.80407i 5.81649 − 3.35815i −13.4563 + 7.76902i − 8.00000i 7.86140 + 13.6163i 6.80407 − 11.7850i
147.7 1.73205 + 1.00000i −4.47729 + 7.75489i 2.00000 + 3.46410i − 17.3267i −15.5098 + 8.95458i −14.5068 + 8.37550i 8.00000i −26.5923 − 46.0591i 17.3267 − 30.0108i
147.8 1.73205 + 1.00000i 0.622927 − 1.07894i 2.00000 + 3.46410i − 20.5002i 2.15788 − 1.24585i 18.4070 − 10.6273i 8.00000i 12.7239 + 22.0385i 20.5002 − 35.5074i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 23.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner
13.c even 3 1 inner
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 338.4.e.i 24
13.b even 2 1 inner 338.4.e.i 24
13.c even 3 1 338.4.b.h 12
13.c even 3 1 inner 338.4.e.i 24
13.d odd 4 1 338.4.c.o 12
13.d odd 4 1 338.4.c.p 12
13.e even 6 1 338.4.b.h 12
13.e even 6 1 inner 338.4.e.i 24
13.f odd 12 1 338.4.a.n ✓ 6
13.f odd 12 1 338.4.a.o yes 6
13.f odd 12 1 338.4.c.o 12
13.f odd 12 1 338.4.c.p 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
338.4.a.n ✓ 6 13.f odd 12 1
338.4.a.o yes 6 13.f odd 12 1
338.4.b.h 12 13.c even 3 1
338.4.b.h 12 13.e even 6 1
338.4.c.o 12 13.d odd 4 1
338.4.c.o 12 13.f odd 12 1
338.4.c.p 12 13.d odd 4 1
338.4.c.p 12 13.f odd 12 1
338.4.e.i 24 1.a even 1 1 trivial
338.4.e.i 24 13.b even 2 1 inner
338.4.e.i 24 13.c even 3 1 inner
338.4.e.i 24 13.e even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(338, [\chi])\):

\( T_{3}^{12} + 9 T_{3}^{11} + 178 T_{3}^{10} + 589 T_{3}^{9} + 13675 T_{3}^{8} + 37320 T_{3}^{7} + \cdots + 143976001 \) Copy content Toggle raw display
\( T_{5}^{12} + 1118T_{5}^{10} + 459449T_{5}^{8} + 85344405T_{5}^{6} + 6935622164T_{5}^{4} + 178926938144T_{5}^{2} + 4175227456 \) Copy content Toggle raw display