Properties

Label 1305.2.f.g
Level $1305$
Weight $2$
Character orbit 1305.f
Analytic conductor $10.420$
Analytic rank $0$
Dimension $4$
CM discriminant -435
Inner twists $8$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1305,2,Mod(289,1305)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1305, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1305.289");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1305.f (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: \(10.4204774638\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-5}, \sqrt{-29})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 17x^{2} + 36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 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.

\(f(q)\) \(=\) \( q - 2 q^{4} - \beta_1 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{4} - \beta_1 q^{5} - \beta_{2} q^{11} + 4 q^{16} + 2 \beta_1 q^{20} + \beta_1 q^{23} - 5 q^{25} + \beta_{2} q^{29} - \beta_{3} q^{37} - \beta_{2} q^{41} - \beta_{3} q^{43} + 2 \beta_{2} q^{44} + 7 q^{49} - 5 \beta_1 q^{53} - \beta_{3} q^{55} - 8 q^{64} - \beta_{3} q^{73} - 4 \beta_1 q^{80} + 7 \beta_1 q^{83} + 2 \beta_{2} q^{89} - 2 \beta_1 q^{92} - \beta_{3} q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{4} + 16 q^{16} - 20 q^{25} + 28 q^{49} - 32 q^{64}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 17x^{2} + 36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 11\nu ) / 6 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 23\nu ) / 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{2} + 17 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 17 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -11\beta_{2} + 23\beta_1 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1305\mathbb{Z}\right)^\times\).

\(n\) \(146\) \(262\) \(901\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
1.57455i
3.81062i
3.81062i
1.57455i
0 0 −2.00000 2.23607i 0 0 0 0 0
289.2 0 0 −2.00000 2.23607i 0 0 0 0 0
289.3 0 0 −2.00000 2.23607i 0 0 0 0 0
289.4 0 0 −2.00000 2.23607i 0 0 0 0 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
435.b odd 2 1 CM by \(\Q(\sqrt{-435}) \)
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner
29.b even 2 1 inner
87.d odd 2 1 inner
145.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1305.2.f.g 4
3.b odd 2 1 inner 1305.2.f.g 4
5.b even 2 1 inner 1305.2.f.g 4
15.d odd 2 1 inner 1305.2.f.g 4
29.b even 2 1 inner 1305.2.f.g 4
87.d odd 2 1 inner 1305.2.f.g 4
145.d even 2 1 inner 1305.2.f.g 4
435.b odd 2 1 CM 1305.2.f.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1305.2.f.g 4 1.a even 1 1 trivial
1305.2.f.g 4 3.b odd 2 1 inner
1305.2.f.g 4 5.b even 2 1 inner
1305.2.f.g 4 15.d odd 2 1 inner
1305.2.f.g 4 29.b even 2 1 inner
1305.2.f.g 4 87.d odd 2 1 inner
1305.2.f.g 4 145.d even 2 1 inner
1305.2.f.g 4 435.b odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1305, [\chi])\):

\( T_{2} \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + 5)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + 29)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} + 5)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 29)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} - 145)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 29)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 145)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} + 125)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} - 145)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 245)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 116)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 145)^{2} \) Copy content Toggle raw display
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