Properties

Label 136.3.j.a
Level $136$
Weight $3$
Character orbit 136.j
Analytic conductor $3.706$
Analytic rank $0$
Dimension $4$
CM discriminant -8
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [136,3,Mod(115,136)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(136, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 2, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("136.115");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 136 = 2^{3} \cdot 17 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 136.j (of order \(4\), degree \(2\), 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: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
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: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

$q$-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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \beta_{2} q^{2} + (\beta_{2} + \beta_1 + 1) q^{3} - 4 q^{4} + (2 \beta_{3} + 2 \beta_{2} - 2) q^{6} - 8 \beta_{2} q^{8} + (2 \beta_{3} + 9 \beta_{2} + 2 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 \beta_{2} q^{2} + (\beta_{2} + \beta_1 + 1) q^{3} - 4 q^{4} + (2 \beta_{3} + 2 \beta_{2} - 2) q^{6} - 8 \beta_{2} q^{8} + (2 \beta_{3} + 9 \beta_{2} + 2 \beta_1) q^{9} + (3 \beta_{3} - 7 \beta_{2} + 7) q^{11} + ( - 4 \beta_{2} - 4 \beta_1 - 4) q^{12} + 16 q^{16} + ( - 3 \beta_{3} - \beta_{2} + 3 \beta_1) q^{17} + (4 \beta_{3} - 4 \beta_1 - 18) q^{18} - 34 \beta_{2} q^{19} + (14 \beta_{2} - 6 \beta_1 + 14) q^{22} + ( - 8 \beta_{3} - 8 \beta_{2} + 8) q^{24} + 25 \beta_{2} q^{25} + (4 \beta_{3} + 32 \beta_{2} - 32) q^{27} + 32 \beta_{2} q^{32} + ( - 4 \beta_{3} + 4 \beta_1 - 34) q^{33} + (6 \beta_{3} + 6 \beta_1 + 2) q^{34} + ( - 8 \beta_{3} - 36 \beta_{2} - 8 \beta_1) q^{36} + 68 q^{38} + ( - 12 \beta_{3} - 23 \beta_{2} + 23) q^{41} + (15 \beta_{3} + 15 \beta_1) q^{43} + ( - 12 \beta_{3} + 28 \beta_{2} - 28) q^{44} + (16 \beta_{2} + 16 \beta_1 + 16) q^{48} - 49 \beta_{2} q^{49} - 50 q^{50} + ( - \beta_{3} + 47 \beta_{2} + 6 \beta_1 + 49) q^{51} + ( - 64 \beta_{2} - 8 \beta_1 - 64) q^{54} + ( - 34 \beta_{3} - 34 \beta_{2} + 34) q^{57} + ( - 15 \beta_{3} - 15 \beta_1) q^{59} - 64 q^{64} + (8 \beta_{3} - 68 \beta_{2} + 8 \beta_1) q^{66} + (21 \beta_{3} - 21 \beta_1) q^{67} + (12 \beta_{3} + 4 \beta_{2} - 12 \beta_1) q^{68} + ( - 16 \beta_{3} + 16 \beta_1 + 72) q^{72} + (71 \beta_{2} + 6 \beta_1 + 71) q^{73} + (25 \beta_{3} + 25 \beta_{2} - 25) q^{75} + 136 \beta_{2} q^{76} + (18 \beta_{3} - 18 \beta_1 - 47) q^{81} + (46 \beta_{2} + 24 \beta_1 + 46) q^{82} + ( - 9 \beta_{3} - 9 \beta_1) q^{83} + (30 \beta_{3} - 30 \beta_1) q^{86} + ( - 56 \beta_{2} + 24 \beta_1 - 56) q^{88} + (18 \beta_{3} - 18 \beta_1) q^{89} + (32 \beta_{3} + 32 \beta_{2} - 32) q^{96} + ( - 47 \beta_{2} - 30 \beta_1 - 47) q^{97} + 98 q^{98} + ( - 33 \beta_{2} + \beta_1 - 33) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} - 16 q^{4} - 8 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} - 16 q^{4} - 8 q^{6} + 28 q^{11} - 16 q^{12} + 64 q^{16} - 72 q^{18} + 56 q^{22} + 32 q^{24} - 128 q^{27} - 136 q^{33} + 8 q^{34} + 272 q^{38} + 92 q^{41} - 112 q^{44} + 64 q^{48} - 200 q^{50} + 196 q^{51} - 256 q^{54} + 136 q^{57} - 256 q^{64} + 288 q^{72} + 284 q^{73} - 100 q^{75} - 188 q^{81} + 184 q^{82} - 224 q^{88} - 128 q^{96} - 188 q^{97} + 392 q^{98} - 132 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 4\zeta_{8} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{8}^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 4\zeta_{8}^{3} \) Copy content Toggle raw display
\(\zeta_{8}\)\(=\) \( ( \beta_1 ) / 4 \) Copy content Toggle raw display
\(\zeta_{8}^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{8}^{3}\)\(=\) \( ( \beta_{3} ) / 4 \) Copy content Toggle raw display

Character values

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

\(n\) \(69\) \(103\) \(105\)
\(\chi(n)\) \(-1\) \(-1\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
115.1
−0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 + 0.707107i
0.707107 0.707107i
2.00000i −1.82843 1.82843i −4.00000 0 3.65685 3.65685i 0 8.00000i 2.31371i 0
115.2 2.00000i 3.82843 + 3.82843i −4.00000 0 −7.65685 + 7.65685i 0 8.00000i 20.3137i 0
123.1 2.00000i −1.82843 + 1.82843i −4.00000 0 3.65685 + 3.65685i 0 8.00000i 2.31371i 0
123.2 2.00000i 3.82843 3.82843i −4.00000 0 −7.65685 7.65685i 0 8.00000i 20.3137i 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
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
17.c even 4 1 inner
136.j odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 136.3.j.a 4
4.b odd 2 1 544.3.n.a 4
8.b even 2 1 544.3.n.a 4
8.d odd 2 1 CM 136.3.j.a 4
17.c even 4 1 inner 136.3.j.a 4
68.f odd 4 1 544.3.n.a 4
136.i even 4 1 544.3.n.a 4
136.j odd 4 1 inner 136.3.j.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.3.j.a 4 1.a even 1 1 trivial
136.3.j.a 4 8.d odd 2 1 CM
136.3.j.a 4 17.c even 4 1 inner
136.3.j.a 4 136.j odd 4 1 inner
544.3.n.a 4 4.b odd 2 1
544.3.n.a 4 8.b even 2 1
544.3.n.a 4 68.f odd 4 1
544.3.n.a 4 136.i even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 4T_{3}^{3} + 8T_{3}^{2} + 56T_{3} + 196 \) acting on \(S_{3}^{\mathrm{new}}(136, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} - 4 T^{3} + 8 T^{2} + 56 T + 196 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 28 T^{3} + 392 T^{2} + \cdots + 2116 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} - 574 T^{2} + 83521 \) Copy content Toggle raw display
$19$ \( (T^{2} + 1156)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( T^{4} - 92 T^{3} + 4232 T^{2} + \cdots + 1552516 \) Copy content Toggle raw display
$43$ \( (T^{2} + 7200)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} + 7200)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} - 14112)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} - 284 T^{3} + \cdots + 90364036 \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 2592)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 10368)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 188 T^{3} + \cdots + 99640324 \) Copy content Toggle raw display
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