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Properties

Label 136.4.n.a
Level $136$
Weight $4$
Character orbit 136.n
Analytic conductor $8.024$
Analytic rank $0$
Dimension $24$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [136,4,Mod(9,136)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(136, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("136.9");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 136 = 2^{3} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 136.n (of order \(8\), degree \(4\), 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: \(8.02425976078\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(6\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 28 q^{5} - 16 q^{9} + 140 q^{11} - 60 q^{15} - 164 q^{17} - 124 q^{19} - 72 q^{23} + 52 q^{25} - 360 q^{27} + 44 q^{29} - 120 q^{31} + 520 q^{33} + 512 q^{35} + 196 q^{37} - 232 q^{39} + 20 q^{41}+ \cdots + 5244 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1 0 −6.48685 − 2.68694i 0 1.11556 − 2.69319i 0 −5.55075 − 13.4007i 0 15.7677 + 15.7677i 0
9.2 0 −5.30585 − 2.19776i 0 0.150111 − 0.362401i 0 10.7876 + 26.0436i 0 4.23006 + 4.23006i 0
9.3 0 −0.101174 − 0.0419078i 0 −7.97306 + 19.2487i 0 −12.0163 − 29.0099i 0 −19.0834 − 19.0834i 0
9.4 0 1.87851 + 0.778104i 0 5.60042 − 13.5206i 0 −3.79791 − 9.16896i 0 −16.1685 − 16.1685i 0
9.5 0 3.15186 + 1.30554i 0 −3.77050 + 9.10280i 0 7.60539 + 18.3610i 0 −10.8621 − 10.8621i 0
9.6 0 8.27773 + 3.42875i 0 1.41301 − 3.41131i 0 −0.563571 − 1.36058i 0 37.6726 + 37.6726i 0
25.1 0 −2.54518 + 6.14461i 0 10.5472 + 4.36879i 0 17.7372 − 7.34700i 0 −12.1864 − 12.1864i 0
25.2 0 −2.28232 + 5.51000i 0 −16.4696 − 6.82195i 0 15.5753 − 6.45152i 0 −6.05928 − 6.05928i 0
25.3 0 −1.80319 + 4.35328i 0 4.35941 + 1.80573i 0 −26.8148 + 11.1071i 0 3.39233 + 3.39233i 0
25.4 0 0.603222 − 1.45631i 0 −10.3784 − 4.29888i 0 −13.2154 + 5.47398i 0 17.3349 + 17.3349i 0
25.5 0 1.34974 − 3.25856i 0 10.0941 + 4.18113i 0 6.43022 − 2.66348i 0 10.2955 + 10.2955i 0
25.6 0 3.26351 − 7.87882i 0 −8.68823 − 3.59878i 0 3.82290 − 1.58350i 0 −32.3334 − 32.3334i 0
49.1 0 −2.54518 − 6.14461i 0 10.5472 − 4.36879i 0 17.7372 + 7.34700i 0 −12.1864 + 12.1864i 0
49.2 0 −2.28232 − 5.51000i 0 −16.4696 + 6.82195i 0 15.5753 + 6.45152i 0 −6.05928 + 6.05928i 0
49.3 0 −1.80319 − 4.35328i 0 4.35941 − 1.80573i 0 −26.8148 − 11.1071i 0 3.39233 − 3.39233i 0
49.4 0 0.603222 + 1.45631i 0 −10.3784 + 4.29888i 0 −13.2154 − 5.47398i 0 17.3349 − 17.3349i 0
49.5 0 1.34974 + 3.25856i 0 10.0941 − 4.18113i 0 6.43022 + 2.66348i 0 10.2955 − 10.2955i 0
49.6 0 3.26351 + 7.87882i 0 −8.68823 + 3.59878i 0 3.82290 + 1.58350i 0 −32.3334 + 32.3334i 0
121.1 0 −6.48685 + 2.68694i 0 1.11556 + 2.69319i 0 −5.55075 + 13.4007i 0 15.7677 − 15.7677i 0
121.2 0 −5.30585 + 2.19776i 0 0.150111 + 0.362401i 0 10.7876 − 26.0436i 0 4.23006 − 4.23006i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.d even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 136.4.n.a ✓ 24
17.d even 8 1 inner 136.4.n.a ✓ 24
17.e odd 16 2 2312.4.a.r 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.4.n.a ✓ 24 1.a even 1 1 trivial
136.4.n.a ✓ 24 17.d even 8 1 inner
2312.4.a.r 24 17.e odd 16 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{24} + 8 T_{3}^{22} - 24 T_{3}^{21} + 32 T_{3}^{20} - 3264 T_{3}^{19} + 34688 T_{3}^{18} + \cdots + 5914779649024 \) acting on \(S_{4}^{\mathrm{new}}(136, [\chi])\). Copy content Toggle raw display