Properties

Label 145.2.b.a
Level $145$
Weight $2$
Character orbit 145.b
Analytic conductor $1.158$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [145,2,Mod(59,145)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(145, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("145.59");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 145 = 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 145.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: \(1.15783082931\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + ( - \beta_{3} - \beta_1) q^{3} - q^{4} + ( - \beta_{3} - \beta_{2} - 1) q^{5} + ( - \beta_{3} + \beta_1 - 2) q^{6} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{7} + \beta_{2} q^{8}+ \cdots + ( - 4 \beta_{3} + 4 \beta_1 - 12) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 3 q^{5} - 6 q^{6} - 2 q^{9} + 9 q^{10} + 14 q^{11} - 12 q^{14} - q^{15} - 20 q^{16} - 16 q^{19} + 3 q^{20} + 16 q^{21} - 6 q^{24} + q^{25} + 18 q^{26} + 4 q^{29} + 21 q^{30} - 2 q^{31}+ \cdots - 40 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 2\nu^{2} - 2\nu - 6 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} - \nu^{2} - 2\nu - 3 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{3} + \beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} + \beta_{2} + 2\beta _1 + 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/145\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(117\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
59.1
−1.18614 + 1.26217i
1.68614 0.396143i
1.68614 + 0.396143i
−1.18614 1.26217i
1.73205i 2.52434i −1.00000 −2.18614 + 0.469882i −4.37228 1.58457i 1.73205i −3.37228 0.813859 + 3.78651i
59.2 1.73205i 0.792287i −1.00000 0.686141 + 2.12819i 1.37228 5.04868i 1.73205i 2.37228 3.68614 1.18843i
59.3 1.73205i 0.792287i −1.00000 0.686141 2.12819i 1.37228 5.04868i 1.73205i 2.37228 3.68614 + 1.18843i
59.4 1.73205i 2.52434i −1.00000 −2.18614 0.469882i −4.37228 1.58457i 1.73205i −3.37228 0.813859 3.78651i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 145.2.b.a 4
3.b odd 2 1 1305.2.c.e 4
4.b odd 2 1 2320.2.d.c 4
5.b even 2 1 inner 145.2.b.a 4
5.c odd 4 2 725.2.a.g 4
15.d odd 2 1 1305.2.c.e 4
15.e even 4 2 6525.2.a.bk 4
20.d odd 2 1 2320.2.d.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.2.b.a 4 1.a even 1 1 trivial
145.2.b.a 4 5.b even 2 1 inner
725.2.a.g 4 5.c odd 4 2
1305.2.c.e 4 3.b odd 2 1
1305.2.c.e 4 15.d odd 2 1
2320.2.d.c 4 4.b odd 2 1
2320.2.d.c 4 20.d odd 2 1
6525.2.a.bk 4 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 3 \) acting on \(S_{2}^{\mathrm{new}}(145, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} + 7T^{2} + 4 \) Copy content Toggle raw display
$5$ \( T^{4} + 3 T^{3} + \cdots + 25 \) Copy content Toggle raw display
$7$ \( T^{4} + 28T^{2} + 64 \) Copy content Toggle raw display
$11$ \( (T^{2} - 7 T + 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 19T^{2} + 16 \) Copy content Toggle raw display
$17$ \( T^{4} + 28T^{2} + 64 \) Copy content Toggle raw display
$19$ \( (T + 4)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$29$ \( (T - 1)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + T - 8)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 112T^{2} + 1024 \) Copy content Toggle raw display
$41$ \( (T^{2} - 2 T - 32)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 151T^{2} + 3844 \) Copy content Toggle raw display
$47$ \( T^{4} + 151T^{2} + 3844 \) Copy content Toggle raw display
$53$ \( T^{4} + 19T^{2} + 16 \) Copy content Toggle raw display
$59$ \( (T^{2} - 10 T - 8)^{2} \) Copy content Toggle raw display
$61$ \( (T - 6)^{4} \) Copy content Toggle raw display
$67$ \( T^{4} + 76T^{2} + 256 \) Copy content Toggle raw display
$71$ \( (T^{2} - 2 T - 32)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 17 T + 64)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 376 T^{2} + 26896 \) Copy content Toggle raw display
$89$ \( (T^{2} + 10 T - 8)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
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