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Properties

Label 1300.2.f.e
Level $1300$
Weight $2$
Character orbit 1300.f
Analytic conductor $10.381$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1300,2,Mod(701,1300)]
 
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, 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("1300.701");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1300 = 2^{2} \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1300.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: \(10.3805522628\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.9144576.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 12x^{4} + 36x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: no (minimal twist has level 260)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} + \beta_{5} q^{7} + ( - \beta_{4} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{3} + \beta_{5} q^{7} + ( - \beta_{4} + 1) q^{9} + ( - \beta_{5} + \beta_{2}) q^{11} + ( - \beta_{4} - \beta_{3} + \beta_1) q^{13} + ( - \beta_{5} + \beta_{2} - 2 \beta_1) q^{19} + 2 \beta_1 q^{21} + ( - 2 \beta_{4} - \beta_{3} + 2) q^{23} - 2 q^{27} + ( - \beta_{4} - 2 \beta_{3} - 2) q^{29} + ( - \beta_{5} - \beta_{2}) q^{31} + ( - \beta_{5} + 2 \beta_{2} - 2 \beta_1) q^{33} + ( - \beta_{5} - 2 \beta_{2}) q^{37} + (\beta_{5} - \beta_{4} - 2 \beta_{3} + \cdots + 2) q^{39}+ \cdots + ( - \beta_{5} + 3 \beta_{2} + 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 6 q^{9} + 12 q^{23} - 12 q^{27} - 12 q^{29} + 12 q^{39} - 12 q^{43} - 6 q^{49} + 12 q^{53} - 12 q^{61} + 36 q^{77} - 24 q^{79} - 18 q^{81} + 36 q^{87} + 12 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 12x^{4} + 36x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 4\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} + 7\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{4} + 6\nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{2} + 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} + 10\nu^{3} + 24\nu ) / 2 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{2} - 2\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -4\beta_{2} + 14\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -6\beta_{4} + 2\beta_{3} + 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 6\beta_{5} + 16\beta_{2} - 92\beta_1 ) / 3 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
701.1
− 2.60168i
2.60168i
− 0.339877i
0.339877i
2.26180i
− 2.26180i
0 −2.60168 0 0 0 − 2.76873i 0 3.76873 0
701.2 0 −2.60168 0 0 0 2.76873i 0 3.76873 0
701.3 0 0.339877 0 0 0 − 3.88448i 0 −2.88448 0
701.4 0 0.339877 0 0 0 3.88448i 0 −2.88448 0
701.5 0 2.26180 0 0 0 − 1.11575i 0 2.11575 0
701.6 0 2.26180 0 0 0 1.11575i 0 2.11575 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 701.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b 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.f.e 6
5.b even 2 1 260.2.f.a ✓ 6
5.c odd 4 1 1300.2.d.c 6
5.c odd 4 1 1300.2.d.d 6
13.b even 2 1 inner 1300.2.f.e 6
15.d odd 2 1 2340.2.c.d 6
20.d odd 2 1 1040.2.k.c 6
65.d even 2 1 260.2.f.a ✓ 6
65.g odd 4 1 3380.2.a.m 3
65.g odd 4 1 3380.2.a.n 3
65.h odd 4 1 1300.2.d.c 6
65.h odd 4 1 1300.2.d.d 6
195.e odd 2 1 2340.2.c.d 6
260.g odd 2 1 1040.2.k.c 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.f.a ✓ 6 5.b even 2 1
260.2.f.a ✓ 6 65.d even 2 1
1040.2.k.c 6 20.d odd 2 1
1040.2.k.c 6 260.g odd 2 1
1300.2.d.c 6 5.c odd 4 1
1300.2.d.c 6 65.h odd 4 1
1300.2.d.d 6 5.c odd 4 1
1300.2.d.d 6 65.h odd 4 1
1300.2.f.e 6 1.a even 1 1 trivial
1300.2.f.e 6 13.b even 2 1 inner
2340.2.c.d 6 15.d odd 2 1
2340.2.c.d 6 195.e odd 2 1
3380.2.a.m 3 65.g odd 4 1
3380.2.a.n 3 65.g odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} - 6T_{3} + 2 \) acting on \(S_{2}^{\mathrm{new}}(1300, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( (T^{3} - 6 T + 2)^{2} \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 24 T^{4} + \cdots + 144 \) Copy content Toggle raw display
$11$ \( T^{6} + 36 T^{4} + \cdots + 324 \) Copy content Toggle raw display
$13$ \( T^{6} - 9 T^{4} + \cdots + 2197 \) Copy content Toggle raw display
$17$ \( T^{6} \) Copy content Toggle raw display
$19$ \( T^{6} + 96 T^{4} + \cdots + 12996 \) Copy content Toggle raw display
$23$ \( (T^{3} - 6 T^{2} + \cdots + 174)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} + 6 T^{2} - 12 T - 84)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + 60 T^{4} + \cdots + 4356 \) Copy content Toggle raw display
$37$ \( T^{6} + 144 T^{4} + \cdots + 63504 \) Copy content Toggle raw display
$41$ \( T^{6} + 108 T^{4} + \cdots + 5184 \) Copy content Toggle raw display
$43$ \( (T^{3} + 6 T^{2} - 42 T - 46)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + 216 T^{4} + \cdots + 219024 \) Copy content Toggle raw display
$53$ \( (T^{3} - 6 T^{2} - 84 T - 24)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + 216 T^{4} + \cdots + 171396 \) Copy content Toggle raw display
$61$ \( (T^{3} + 6 T^{2} + \cdots - 532)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 240 T^{4} + \cdots + 345744 \) Copy content Toggle raw display
$71$ \( T^{6} + 432 T^{4} + \cdots + 492804 \) Copy content Toggle raw display
$73$ \( T^{6} + 288 T^{4} + \cdots + 63504 \) Copy content Toggle raw display
$79$ \( (T^{3} + 12 T^{2} + \cdots - 1696)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 144 T^{4} + \cdots + 63504 \) Copy content Toggle raw display
$89$ \( T^{6} + 576 T^{4} + \cdots + 4064256 \) Copy content Toggle raw display
$97$ \( T^{6} + 204 T^{4} + \cdots + 576 \) Copy content Toggle raw display
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