Properties

Label 1305.2.f.f
Level $1305$
Weight $2$
Character orbit 1305.f
Analytic conductor $10.420$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

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: \(\Q(\sqrt{5}, \sqrt{-6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} + 11x^{2} - 10x + 55 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q - q^{2} - q^{4} + \beta_{2} q^{5} + \beta_1 q^{7} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - q^{4} + \beta_{2} q^{5} + \beta_1 q^{7} + 3 q^{8} - \beta_{2} q^{10} - \beta_1 q^{11} - 2 \beta_1 q^{13} - \beta_1 q^{14} - q^{16} - 4 q^{17} + \beta_{3} q^{19} - \beta_{2} q^{20} + \beta_1 q^{22} - \beta_{3} q^{23} + 5 q^{25} + 2 \beta_1 q^{26} - \beta_1 q^{28} + ( - \beta_{2} - 2 \beta_1) q^{29} + \beta_{3} q^{31} - 5 q^{32} + 4 q^{34} - \beta_{3} q^{35} - 2 \beta_{2} q^{37} - \beta_{3} q^{38} + 3 \beta_{2} q^{40} + 2 \beta_1 q^{41} + 4 \beta_{2} q^{43} + \beta_1 q^{44} + \beta_{3} q^{46} + 4 q^{47} + q^{49} - 5 q^{50} + 2 \beta_1 q^{52} + \beta_{3} q^{55} + 3 \beta_1 q^{56} + (\beta_{2} + 2 \beta_1) q^{58} + 4 \beta_{2} q^{59} + 2 \beta_{3} q^{61} - \beta_{3} q^{62} + 7 q^{64} + 2 \beta_{3} q^{65} + \beta_1 q^{67} + 4 q^{68} + \beta_{3} q^{70} - 4 \beta_{2} q^{71} - 4 \beta_{2} q^{73} + 2 \beta_{2} q^{74} - \beta_{3} q^{76} + 6 q^{77} + 3 \beta_{3} q^{79} - \beta_{2} q^{80} - 2 \beta_1 q^{82} + 3 \beta_{3} q^{83} - 4 \beta_{2} q^{85} - 4 \beta_{2} q^{86} - 3 \beta_1 q^{88} + 2 \beta_1 q^{89} + 12 q^{91} + \beta_{3} q^{92} - 4 q^{94} - 5 \beta_1 q^{95} - 4 \beta_{2} q^{97} - q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} - 4 q^{4} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} - 4 q^{4} + 12 q^{8} - 4 q^{16} - 16 q^{17} + 20 q^{25} - 20 q^{32} + 16 q^{34} + 16 q^{47} + 4 q^{49} - 20 q^{50} + 28 q^{64} + 16 q^{68} + 24 q^{77} + 48 q^{91} - 16 q^{94} - 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} + 11x^{2} - 10x + 55 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -2\nu^{3} + 3\nu^{2} - 35\nu + 17 ) / 29 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -4\nu^{3} + 6\nu^{2} - 12\nu + 5 ) / 29 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - \nu + 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - 2\beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{3} + \beta_{2} - 2\beta _1 - 9 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{3} - 16\beta_{2} + 3\beta _1 - 14 ) / 2 \) 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
−0.618034 + 2.44949i
−0.618034 2.44949i
1.61803 + 2.44949i
1.61803 2.44949i
−1.00000 0 −1.00000 −2.23607 0 2.44949i 3.00000 0 2.23607
289.2 −1.00000 0 −1.00000 −2.23607 0 2.44949i 3.00000 0 2.23607
289.3 −1.00000 0 −1.00000 2.23607 0 2.44949i 3.00000 0 −2.23607
289.4 −1.00000 0 −1.00000 2.23607 0 2.44949i 3.00000 0 −2.23607
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Hecke kernels

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

\( T_{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{2} + 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} - 5)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 6)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 6)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 24)^{2} \) Copy content Toggle raw display
$17$ \( (T + 4)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 30)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 30)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + 38T^{2} + 841 \) Copy content Toggle raw display
$31$ \( (T^{2} + 30)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 20)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 24)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 80)^{2} \) Copy content Toggle raw display
$47$ \( (T - 4)^{4} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} - 80)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 120)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 6)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 80)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 80)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 270)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 270)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 24)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 80)^{2} \) Copy content Toggle raw display
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