Properties

Label 136.3.e.a
Level $136$
Weight $3$
Character orbit 136.e
Self dual yes
Analytic conductor $3.706$
Analytic rank $0$
Dimension $2$
CM discriminant -136
Inner twists $4$

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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: yes
Analytic conductor: \(3.70573159530\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{17}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 q^{2} + 4 q^{4} + \beta q^{5} - \beta q^{7} - 8 q^{8} + 9 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} + 4 q^{4} + \beta q^{5} - \beta q^{7} - 8 q^{8} + 9 q^{9} - 2 \beta q^{10} + 2 \beta q^{14} + 16 q^{16} + 17 q^{17} - 18 q^{18} + 30 q^{19} + 4 \beta q^{20} - \beta q^{23} + 43 q^{25} - 4 \beta q^{28} - 7 \beta q^{29} + 7 \beta q^{31} - 32 q^{32} - 34 q^{34} - 68 q^{35} + 36 q^{36} + \beta q^{37} - 60 q^{38} - 8 \beta q^{40} - 50 q^{43} + 9 \beta q^{45} + 2 \beta q^{46} + 19 q^{49} - 86 q^{50} + 8 \beta q^{56} + 14 \beta q^{58} - 18 q^{59} - 7 \beta q^{61} - 14 \beta q^{62} - 9 \beta q^{63} + 64 q^{64} - 66 q^{67} + 68 q^{68} + 136 q^{70} - 17 \beta q^{71} - 72 q^{72} - 2 \beta q^{74} + 120 q^{76} + 7 \beta q^{79} + 16 \beta q^{80} + 81 q^{81} + 30 q^{83} + 17 \beta q^{85} + 100 q^{86} - 110 q^{89} - 18 \beta q^{90} - 4 \beta q^{92} + 30 \beta q^{95} - 38 q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{2} + 8 q^{4} - 16 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{2} + 8 q^{4} - 16 q^{8} + 18 q^{9} + 32 q^{16} + 34 q^{17} - 36 q^{18} + 60 q^{19} + 86 q^{25} - 64 q^{32} - 68 q^{34} - 136 q^{35} + 72 q^{36} - 120 q^{38} - 100 q^{43} + 38 q^{49} - 172 q^{50} - 36 q^{59} + 128 q^{64} - 132 q^{67} + 136 q^{68} + 272 q^{70} - 144 q^{72} + 240 q^{76} + 162 q^{81} + 60 q^{83} + 200 q^{86} - 220 q^{89} - 76 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/136\mathbb{Z}\right)^\times\).

\(n\) \(69\) \(103\) \(105\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
−1.56155
2.56155
−2.00000 0 4.00000 −8.24621 0 8.24621 −8.00000 9.00000 16.4924
67.2 −2.00000 0 4.00000 8.24621 0 −8.24621 −8.00000 9.00000 −16.4924
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
136.e odd 2 1 CM by \(\Q(\sqrt{-34}) \)
8.d odd 2 1 inner
17.b even 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.a 2
4.b odd 2 1 544.3.e.c 2
8.b even 2 1 544.3.e.c 2
8.d odd 2 1 inner 136.3.e.a 2
17.b even 2 1 inner 136.3.e.a 2
68.d odd 2 1 544.3.e.c 2
136.e odd 2 1 CM 136.3.e.a 2
136.h even 2 1 544.3.e.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.3.e.a 2 1.a even 1 1 trivial
136.3.e.a 2 8.d odd 2 1 inner
136.3.e.a 2 17.b even 2 1 inner
136.3.e.a 2 136.e odd 2 1 CM
544.3.e.c 2 4.b odd 2 1
544.3.e.c 2 8.b even 2 1
544.3.e.c 2 68.d odd 2 1
544.3.e.c 2 136.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 \(S_{3}^{\mathrm{new}}(136, [\chi])\):

\( T_{3} \) Copy content Toggle raw display
\( T_{5}^{2} - 68 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 68 \) Copy content Toggle raw display
$7$ \( T^{2} - 68 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( (T - 17)^{2} \) Copy content Toggle raw display
$19$ \( (T - 30)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 68 \) Copy content Toggle raw display
$29$ \( T^{2} - 3332 \) Copy content Toggle raw display
$31$ \( T^{2} - 3332 \) Copy content Toggle raw display
$37$ \( T^{2} - 68 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( (T + 50)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( (T + 18)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 3332 \) Copy content Toggle raw display
$67$ \( (T + 66)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 19652 \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 3332 \) Copy content Toggle raw display
$83$ \( (T - 30)^{2} \) Copy content Toggle raw display
$89$ \( (T + 110)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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