Properties

Label 1300.2.d.a
Level $1300$
Weight $2$
Character orbit 1300.d
Analytic conductor $10.381$
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] = [1300,2,Mod(649,1300)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1300, 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("1300.649");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1300 = 2^{2} \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1300.d (of order \(2\), degree \(1\), not 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: \(10.3805522628\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - 2 \beta_1 q^{3} - \beta_{3} q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 \beta_1 q^{3} - \beta_{3} q^{7} - q^{9} + 3 \beta_{2} q^{11} + (2 \beta_{3} + \beta_1) q^{13} + 3 \beta_1 q^{17} + 2 \beta_{2} q^{19} + 2 \beta_{2} q^{21} - 4 \beta_1 q^{27} + 3 q^{29} + 5 \beta_{2} q^{31} + 6 \beta_{3} q^{33} + ( - 4 \beta_{2} + 2) q^{39} + 4 \beta_1 q^{43} + 3 \beta_{3} q^{47} - 4 q^{49} + 6 q^{51} - 9 \beta_1 q^{53} + 4 \beta_{3} q^{57} - 3 \beta_{2} q^{59} + 13 q^{61} + \beta_{3} q^{63} + 5 \beta_{3} q^{67} + 6 \beta_{2} q^{71} - 9 \beta_1 q^{77} - 10 q^{79} - 11 q^{81} - 9 \beta_{3} q^{83} - 6 \beta_1 q^{87} - 6 \beta_{2} q^{89} + ( - \beta_{2} - 6) q^{91} + 10 \beta_{3} q^{93} + 10 \beta_{3} q^{97} - 3 \beta_{2} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{9} + 12 q^{29} + 8 q^{39} - 16 q^{49} + 24 q^{51} + 52 q^{61} - 40 q^{79} - 44 q^{81} - 24 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\zeta_{12}^{2} - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( \beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1300\mathbb{Z}\right)^\times\).

\(n\) \(301\) \(651\) \(677\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
649.1
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
0 2.00000i 0 0 0 −1.73205 0 −1.00000 0
649.2 0 2.00000i 0 0 0 1.73205 0 −1.00000 0
649.3 0 2.00000i 0 0 0 −1.73205 0 −1.00000 0
649.4 0 2.00000i 0 0 0 1.73205 0 −1.00000 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
5.b even 2 1 inner
13.b even 2 1 inner
65.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1300.2.d.a 4
5.b even 2 1 inner 1300.2.d.a 4
5.c odd 4 1 1300.2.f.a 2
5.c odd 4 1 1300.2.f.d yes 2
13.b even 2 1 inner 1300.2.d.a 4
65.d even 2 1 inner 1300.2.d.a 4
65.h odd 4 1 1300.2.f.a 2
65.h odd 4 1 1300.2.f.d yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1300.2.d.a 4 1.a even 1 1 trivial
1300.2.d.a 4 5.b even 2 1 inner
1300.2.d.a 4 13.b even 2 1 inner
1300.2.d.a 4 65.d even 2 1 inner
1300.2.f.a 2 5.c odd 4 1
1300.2.f.a 2 65.h odd 4 1
1300.2.f.d yes 2 5.c odd 4 1
1300.2.f.d yes 2 65.h odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1300, [\chi])\):

\( T_{3}^{2} + 4 \) Copy content Toggle raw display
\( T_{7}^{2} - 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 27)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} - 22T^{2} + 169 \) Copy content Toggle raw display
$17$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( (T - 3)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 75)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 27)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 81)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 27)^{2} \) Copy content Toggle raw display
$61$ \( (T - 13)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} - 75)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 108)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( (T + 10)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 243)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 108)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 300)^{2} \) Copy content Toggle raw display
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