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Properties

Label 135.2.b.b
Level $135$
Weight $2$
Character orbit 135.b
Analytic conductor $1.078$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [135,2,Mod(109,135)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(135, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("135.109");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 135 = 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 135.b (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: \(1.07798042729\)
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: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + \beta_{3} q^{5} - \beta_1 q^{7} + 2 \beta_{2} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + \beta_{3} q^{5} - \beta_1 q^{7} + 2 \beta_{2} q^{8} + (\beta_1 - 1) q^{10} + ( - 2 \beta_{3} + \beta_{2}) q^{11} + \beta_1 q^{13} + (2 \beta_{3} - \beta_{2}) q^{14} - 4 q^{16} - 2 \beta_{2} q^{17} - q^{19} - 2 \beta_1 q^{22} - 5 \beta_{2} q^{23} + (\beta_1 + 4) q^{25} + ( - 2 \beta_{3} + \beta_{2}) q^{26} + ( - 2 \beta_{3} + \beta_{2}) q^{29} + 2 q^{31} + 4 q^{34} + (\beta_{3} - 5 \beta_{2}) q^{35} - 3 \beta_1 q^{37} - \beta_{2} q^{38} + (2 \beta_1 - 2) q^{40} + (2 \beta_{3} - \beta_{2}) q^{41} + 2 \beta_1 q^{43} + 10 q^{46} - 2 \beta_{2} q^{47} - 2 q^{49} + ( - 2 \beta_{3} + 5 \beta_{2}) q^{50} + 7 \beta_{2} q^{53} + ( - \beta_1 - 9) q^{55} + (4 \beta_{3} - 2 \beta_{2}) q^{56} - 2 \beta_1 q^{58} + ( - 4 \beta_{3} + 2 \beta_{2}) q^{59} - 13 q^{61} + 2 \beta_{2} q^{62} - 8 q^{64} + ( - \beta_{3} + 5 \beta_{2}) q^{65} + \beta_1 q^{67} + (\beta_1 + 9) q^{70} + (6 \beta_{3} - 3 \beta_{2}) q^{71} + 3 \beta_1 q^{73} + (6 \beta_{3} - 3 \beta_{2}) q^{74} + 9 \beta_{2} q^{77} + 5 q^{79} - 4 \beta_{3} q^{80} + 2 \beta_1 q^{82} + \beta_{2} q^{83} + ( - 2 \beta_1 + 2) q^{85} + ( - 4 \beta_{3} + 2 \beta_{2}) q^{86} - 4 \beta_1 q^{88} + 9 q^{91} + 4 q^{94} - \beta_{3} q^{95} - \beta_1 q^{97} - 2 \beta_{2} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{10} - 16 q^{16} - 4 q^{19} + 16 q^{25} + 8 q^{31} + 16 q^{34} - 8 q^{40} + 40 q^{46} - 8 q^{49} - 36 q^{55} - 52 q^{61} - 32 q^{64} + 36 q^{70} + 20 q^{79} + 8 q^{85} + 36 q^{91} + 16 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 3\zeta_{8}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{8}^{3} + \zeta_{8} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{8}^{3} + 2\zeta_{8} \) Copy content Toggle raw display

Display \(\zeta_{8}^j\) in terms of \(\beta_i\)

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

Display \(\beta_i\) in terms of \(\zeta_{8}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/135\mathbb{Z}\right)^\times\).

\(n\) \(56\) \(82\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
−0.707107 − 0.707107i
0.707107 − 0.707107i
−0.707107 + 0.707107i
0.707107 + 0.707107i
− 1.41421i 0 0 −2.12132 − 0.707107i 0 − 3.00000i − 2.82843i 0 −1.00000 + 3.00000i
109.2 − 1.41421i 0 0 2.12132 − 0.707107i 0 3.00000i − 2.82843i 0 −1.00000 − 3.00000i
109.3 1.41421i 0 0 −2.12132 + 0.707107i 0 3.00000i 2.82843i 0 −1.00000 − 3.00000i
109.4 1.41421i 0 0 2.12132 + 0.707107i 0 − 3.00000i 2.82843i 0 −1.00000 + 3.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 135.2.b.b ✓ 4
3.b odd 2 1 inner 135.2.b.b ✓ 4
4.b odd 2 1 2160.2.f.k 4
5.b even 2 1 inner 135.2.b.b ✓ 4
5.c odd 4 1 675.2.a.l 2
5.c odd 4 1 675.2.a.m 2
9.c even 3 2 405.2.j.f 8
9.d odd 6 2 405.2.j.f 8
12.b even 2 1 2160.2.f.k 4
15.d odd 2 1 inner 135.2.b.b ✓ 4
15.e even 4 1 675.2.a.l 2
15.e even 4 1 675.2.a.m 2
20.d odd 2 1 2160.2.f.k 4
45.h odd 6 2 405.2.j.f 8
45.j even 6 2 405.2.j.f 8
60.h even 2 1 2160.2.f.k 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.2.b.b ✓ 4 1.a even 1 1 trivial
135.2.b.b ✓ 4 3.b odd 2 1 inner
135.2.b.b ✓ 4 5.b even 2 1 inner
135.2.b.b ✓ 4 15.d odd 2 1 inner
405.2.j.f 8 9.c even 3 2
405.2.j.f 8 9.d odd 6 2
405.2.j.f 8 45.h odd 6 2
405.2.j.f 8 45.j even 6 2
675.2.a.l 2 5.c odd 4 1
675.2.a.l 2 15.e even 4 1
675.2.a.m 2 5.c odd 4 1
675.2.a.m 2 15.e even 4 1
2160.2.f.k 4 4.b odd 2 1
2160.2.f.k 4 12.b even 2 1
2160.2.f.k 4 20.d odd 2 1
2160.2.f.k 4 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 2 \) acting on \(S_{2}^{\mathrm{new}}(135, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 8T^{2} + 25 \) Copy content Toggle raw display
$7$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 18)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$19$ \( (T + 1)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} + 50)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 18)^{2} \) Copy content Toggle raw display
$31$ \( (T - 2)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 81)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 18)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 98)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 72)^{2} \) Copy content Toggle raw display
$61$ \( (T + 13)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 162)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 81)^{2} \) Copy content Toggle raw display
$79$ \( (T - 5)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
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