Properties

Label 136.3.e.d
Level $136$
Weight $3$
Character orbit 136.e
Analytic conductor $3.706$
Analytic rank $0$
Dimension $28$
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] = [136,3,Mod(67,136)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(136, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("136.67");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 136 = 2^{3} \cdot 17 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 136.e (of order \(2\), degree \(1\), 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: \(3.70573159530\)
Analytic rank: \(0\)
Dimension: \(28\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 28 q - 6 q^{2} - 30 q^{4} - 18 q^{8} - 84 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q - 6 q^{2} - 30 q^{4} - 18 q^{8} - 84 q^{9} - 46 q^{16} + 4 q^{17} + 50 q^{18} - 136 q^{19} + 100 q^{25} + 72 q^{26} + 32 q^{30} + 94 q^{32} - 144 q^{33} + 46 q^{34} + 160 q^{35} + 122 q^{36} + 100 q^{38} - 336 q^{42} + 168 q^{43} + 52 q^{49} + 270 q^{50} + 32 q^{51} + 152 q^{52} + 104 q^{59} - 384 q^{60} - 414 q^{64} + 120 q^{67} - 58 q^{68} - 136 q^{70} + 334 q^{72} + 220 q^{76} - 500 q^{81} + 40 q^{83} + 200 q^{84} - 60 q^{86} + 88 q^{89} - 656 q^{94} - 378 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1 −1.82569 0.816612i 4.38815i 2.66629 + 2.98176i −2.66031 −3.58341 + 8.01140i −7.32916 −2.43288 7.62110i −10.2558 4.85691 + 2.17244i
67.2 −1.82569 0.816612i 4.38815i 2.66629 + 2.98176i 2.66031 3.58341 8.01140i 7.32916 −2.43288 7.62110i −10.2558 −4.85691 2.17244i
67.3 −1.82569 + 0.816612i 4.38815i 2.66629 2.98176i 2.66031 3.58341 + 8.01140i 7.32916 −2.43288 + 7.62110i −10.2558 −4.85691 + 2.17244i
67.4 −1.82569 + 0.816612i 4.38815i 2.66629 2.98176i −2.66031 −3.58341 8.01140i −7.32916 −2.43288 + 7.62110i −10.2558 4.85691 2.17244i
67.5 −1.58917 1.21430i 2.87954i 1.05094 + 3.85947i 4.15587 −3.49664 + 4.57610i 8.29662 3.01643 7.40953i 0.708224 −6.60439 5.04647i
67.6 −1.58917 1.21430i 2.87954i 1.05094 + 3.85947i −4.15587 3.49664 4.57610i −8.29662 3.01643 7.40953i 0.708224 6.60439 + 5.04647i
67.7 −1.58917 + 1.21430i 2.87954i 1.05094 3.85947i −4.15587 3.49664 + 4.57610i −8.29662 3.01643 + 7.40953i 0.708224 6.60439 5.04647i
67.8 −1.58917 + 1.21430i 2.87954i 1.05094 3.85947i 4.15587 −3.49664 4.57610i 8.29662 3.01643 + 7.40953i 0.708224 −6.60439 + 5.04647i
67.9 −0.842089 1.81408i 0.434450i −2.58177 + 3.05523i 5.41022 −0.788127 + 0.365846i −0.0314051 7.71652 + 2.11076i 8.81125 −4.55589 9.81456i
67.10 −0.842089 1.81408i 0.434450i −2.58177 + 3.05523i −5.41022 0.788127 0.365846i 0.0314051 7.71652 + 2.11076i 8.81125 4.55589 + 9.81456i
67.11 −0.842089 + 1.81408i 0.434450i −2.58177 3.05523i −5.41022 0.788127 + 0.365846i 0.0314051 7.71652 2.11076i 8.81125 4.55589 9.81456i
67.12 −0.842089 + 1.81408i 0.434450i −2.58177 3.05523i 5.41022 −0.788127 0.365846i −0.0314051 7.71652 2.11076i 8.81125 −4.55589 + 9.81456i
67.13 −0.203558 1.98961i 5.30587i −3.91713 + 0.810002i −6.46021 −10.5566 + 1.08005i 10.2774 2.40895 + 7.62869i −19.1523 1.31502 + 12.8533i
67.14 −0.203558 1.98961i 5.30587i −3.91713 + 0.810002i 6.46021 10.5566 1.08005i −10.2774 2.40895 + 7.62869i −19.1523 −1.31502 12.8533i
67.15 −0.203558 + 1.98961i 5.30587i −3.91713 0.810002i 6.46021 10.5566 + 1.08005i −10.2774 2.40895 7.62869i −19.1523 −1.31502 + 12.8533i
67.16 −0.203558 + 1.98961i 5.30587i −3.91713 0.810002i −6.46021 −10.5566 1.08005i 10.2774 2.40895 7.62869i −19.1523 1.31502 12.8533i
67.17 0.420841 1.95522i 2.18661i −3.64579 1.64567i −1.66636 −4.27530 0.920213i −10.7711 −4.75195 + 6.43575i 4.21876 −0.701272 + 3.25810i
67.18 0.420841 1.95522i 2.18661i −3.64579 1.64567i 1.66636 4.27530 + 0.920213i 10.7711 −4.75195 + 6.43575i 4.21876 0.701272 3.25810i
67.19 0.420841 + 1.95522i 2.18661i −3.64579 + 1.64567i 1.66636 4.27530 0.920213i 10.7711 −4.75195 6.43575i 4.21876 0.701272 + 3.25810i
67.20 0.420841 + 1.95522i 2.18661i −3.64579 + 1.64567i −1.66636 −4.27530 + 0.920213i −10.7711 −4.75195 6.43575i 4.21876 −0.701272 3.25810i
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 67.28
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
17.b even 2 1 inner
136.e odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 136.3.e.d 28
4.b odd 2 1 544.3.e.d 28
8.b even 2 1 544.3.e.d 28
8.d odd 2 1 inner 136.3.e.d 28
17.b even 2 1 inner 136.3.e.d 28
68.d odd 2 1 544.3.e.d 28
136.e odd 2 1 inner 136.3.e.d 28
136.h even 2 1 544.3.e.d 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.3.e.d 28 1.a even 1 1 trivial
136.3.e.d 28 8.d odd 2 1 inner
136.3.e.d 28 17.b even 2 1 inner
136.3.e.d 28 136.e odd 2 1 inner
544.3.e.d 28 4.b odd 2 1
544.3.e.d 28 8.b even 2 1
544.3.e.d 28 68.d odd 2 1
544.3.e.d 28 136.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(136, [\chi])\):

\( T_{3}^{14} + 84T_{3}^{12} + 2740T_{3}^{10} + 44264T_{3}^{8} + 373072T_{3}^{6} + 1571264T_{3}^{4} + 2682896T_{3}^{2} + 452864 \) Copy content Toggle raw display
\( T_{5}^{14} - 200 T_{5}^{12} + 14400 T_{5}^{10} - 491552 T_{5}^{8} + 8597184 T_{5}^{6} - 76550400 T_{5}^{4} + 315154688 T_{5}^{2} - 442003456 \) Copy content Toggle raw display