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Properties

Label 136.2.n.b
Level $136$
Weight $2$
Character orbit 136.n
Analytic conductor $1.086$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [136,2,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, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("136.9");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 136 = 2^{3} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
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: \(1.08596546749\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-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 a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{8}^{3} - \zeta_{8}^{2} + \cdots + 1) q^{3}+ \cdots + ( - 2 \zeta_{8}^{2} + \zeta_{8} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{8}^{3} - \zeta_{8}^{2} + \cdots + 1) q^{3}+ \cdots + (9 \zeta_{8}^{3} - 9 \zeta_{8}^{2} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} - 8 q^{5} + 4 q^{7} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} - 8 q^{5} + 4 q^{7} - 8 q^{9} + 12 q^{11} - 16 q^{15} + 16 q^{19} + 4 q^{23} - 4 q^{25} - 8 q^{27} - 12 q^{29} + 4 q^{31} + 32 q^{33} - 24 q^{35} - 8 q^{37} - 8 q^{39} - 20 q^{41} - 16 q^{43} + 20 q^{45} - 8 q^{49} + 12 q^{51} + 12 q^{53} - 8 q^{57} + 8 q^{59} + 8 q^{61} - 4 q^{63} + 28 q^{65} + 32 q^{69} - 20 q^{71} + 4 q^{73} + 12 q^{75} + 24 q^{77} + 12 q^{79} - 8 q^{83} - 20 q^{85} - 24 q^{87} + 24 q^{91} + 8 q^{93} - 32 q^{95} - 8 q^{97} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(69\) \(103\) \(105\)
\(\chi(n)\) \(1\) \(1\) \(\zeta_{8}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1
0.707107 + 0.707107i
−0.707107 − 0.707107i
−0.707107 + 0.707107i
0.707107 − 0.707107i
0 1.00000 + 0.414214i 0 −1.29289 + 3.12132i 0 1.00000 + 2.41421i 0 −1.29289 − 1.29289i 0
25.1 0 1.00000 − 2.41421i 0 −2.70711 − 1.12132i 0 1.00000 − 0.414214i 0 −2.70711 − 2.70711i 0
49.1 0 1.00000 + 2.41421i 0 −2.70711 + 1.12132i 0 1.00000 + 0.414214i 0 −2.70711 + 2.70711i 0
121.1 0 1.00000 − 0.414214i 0 −1.29289 − 3.12132i 0 1.00000 − 2.41421i 0 −1.29289 + 1.29289i 0
\(n\): e.g. 2-40 or 990-1000
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 136.2.n.b ✓ 4
3.b odd 2 1 1224.2.bq.b 4
4.b odd 2 1 272.2.v.a 4
17.d even 8 1 inner 136.2.n.b ✓ 4
17.e odd 16 2 2312.2.a.t 4
17.e odd 16 2 2312.2.b.i 4
51.g odd 8 1 1224.2.bq.b 4
68.g odd 8 1 272.2.v.a 4
68.i even 16 2 4624.2.a.bo 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.2.n.b ✓ 4 1.a even 1 1 trivial
136.2.n.b ✓ 4 17.d even 8 1 inner
272.2.v.a 4 4.b odd 2 1
272.2.v.a 4 68.g odd 8 1
1224.2.bq.b 4 3.b odd 2 1
1224.2.bq.b 4 51.g odd 8 1
2312.2.a.t 4 17.e odd 16 2
2312.2.b.i 4 17.e odd 16 2
4624.2.a.bo 4 68.i even 16 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 4T_{3}^{3} + 12T_{3}^{2} - 16T_{3} + 8 \) acting on \(S_{2}^{\mathrm{new}}(136, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 4 T^{3} + \cdots + 8 \) Copy content Toggle raw display
$5$ \( T^{4} + 8 T^{3} + \cdots + 98 \) Copy content Toggle raw display
$7$ \( T^{4} - 4 T^{3} + \cdots + 8 \) Copy content Toggle raw display
$11$ \( T^{4} - 12 T^{3} + \cdots + 392 \) Copy content Toggle raw display
$13$ \( T^{4} + 36T^{2} + 196 \) Copy content Toggle raw display
$17$ \( T^{4} + 2T^{2} + 289 \) Copy content Toggle raw display
$19$ \( T^{4} - 16 T^{3} + \cdots + 784 \) Copy content Toggle raw display
$23$ \( T^{4} - 4 T^{3} + \cdots + 8 \) Copy content Toggle raw display
$29$ \( T^{4} + 12 T^{3} + \cdots + 162 \) Copy content Toggle raw display
$31$ \( T^{4} - 4 T^{3} + \cdots + 8 \) Copy content Toggle raw display
$37$ \( T^{4} + 8 T^{3} + \cdots + 2 \) Copy content Toggle raw display
$41$ \( T^{4} + 20 T^{3} + \cdots + 1250 \) Copy content Toggle raw display
$43$ \( T^{4} + 16 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$47$ \( T^{4} + 48T^{2} + 64 \) Copy content Toggle raw display
$53$ \( T^{4} - 12 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$59$ \( T^{4} - 8 T^{3} + \cdots + 8464 \) Copy content Toggle raw display
$61$ \( T^{4} - 8 T^{3} + \cdots + 2 \) Copy content Toggle raw display
$67$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 20 T^{3} + \cdots + 17672 \) Copy content Toggle raw display
$73$ \( T^{4} - 4 T^{3} + \cdots + 12482 \) Copy content Toggle raw display
$79$ \( T^{4} - 12 T^{3} + \cdots + 392 \) Copy content Toggle raw display
$83$ \( T^{4} + 8 T^{3} + \cdots + 8464 \) Copy content Toggle raw display
$89$ \( (T^{2} + 18)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 8 T^{3} + \cdots + 2 \) Copy content Toggle raw display
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