Properties

Label 1300.2.a.i
Level $1300$
Weight $2$
Character orbit 1300.a
Self dual yes
Analytic conductor $10.381$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1300,2,Mod(1,1300)]
 
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, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1300.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1300 = 2^{2} \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1300.a (trivial)

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: yes
Analytic conductor: \(10.3805522628\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.564.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 5x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 260)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - 1) q^{3} + (\beta_1 + 1) q^{7} + (\beta_1 + 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - 1) q^{3} + (\beta_1 + 1) q^{7} + (\beta_1 + 4) q^{9} + ( - \beta_{2} + \beta_1) q^{11} - q^{13} + 2 \beta_{2} q^{17} + ( - \beta_{2} - \beta_1 + 2) q^{19} + ( - 2 \beta_{2} - 2 \beta_1 - 4) q^{21} + ( - \beta_{2} + 3) q^{23} + ( - 2 \beta_{2} - 2 \beta_1 - 4) q^{27} + ( - \beta_1 + 3) q^{29} + ( - \beta_{2} + \beta_1 - 4) q^{31} + ( - 2 \beta_{2} - \beta_1 + 3) q^{33} + (2 \beta_{2} - \beta_1 + 1) q^{37} + (\beta_{2} + 1) q^{39} + 2 \beta_{2} q^{41} + (\beta_{2} + 1) q^{43} + ( - \beta_1 + 3) q^{47} + (4 \beta_{2} + 9) q^{49} + (2 \beta_{2} - 2 \beta_1 - 12) q^{51} + ( - 2 \beta_{2} + 2 \beta_1 + 6) q^{53} + ( - 2 \beta_{2} + 3 \beta_1 + 7) q^{57} + ( - 3 \beta_{2} + \beta_1 - 6) q^{59} + (\beta_1 + 5) q^{61} + (4 \beta_{2} + 3 \beta_1 + 19) q^{63} + ( - 2 \beta_{2} + \beta_1 - 5) q^{67} + ( - 4 \beta_{2} + \beta_1 + 3) q^{69} + ( - \beta_{2} + \beta_1) q^{71} + (2 \beta_{2} - 3 \beta_1 - 5) q^{73} + (2 \beta_{2} - 2 \beta_1 + 12) q^{77} + ( - 2 \beta_{2} + 2) q^{79} + (4 \beta_{2} + 3 \beta_1 + 10) q^{81} + (4 \beta_{2} - \beta_1 + 3) q^{83} + ( - 2 \beta_{2} + 2 \beta_1) q^{87} + (4 \beta_{2} - 2 \beta_1) q^{89} + ( - \beta_1 - 1) q^{91} + (2 \beta_{2} - \beta_1 + 7) q^{93} + (2 \beta_1 - 8) q^{97} + ( - \beta_{2} + \beta_1 + 12) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{3} + 2 q^{7} + 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 2 q^{3} + 2 q^{7} + 11 q^{9} - 3 q^{13} - 2 q^{17} + 8 q^{19} - 8 q^{21} + 10 q^{23} - 8 q^{27} + 10 q^{29} - 12 q^{31} + 12 q^{33} + 2 q^{37} + 2 q^{39} - 2 q^{41} + 2 q^{43} + 10 q^{47} + 23 q^{49} - 36 q^{51} + 18 q^{53} + 20 q^{57} - 16 q^{59} + 14 q^{61} + 50 q^{63} - 14 q^{67} + 12 q^{69} - 14 q^{73} + 36 q^{77} + 8 q^{79} + 23 q^{81} + 6 q^{83} - 2 q^{89} - 2 q^{91} + 20 q^{93} - 26 q^{97} + 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 5x + 3 \) : Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.51414
−2.08613
0.571993
0 −3.32088 0 0 0 5.02827 0 8.02827 0
1.2 0 −1.35194 0 0 0 −4.17226 0 −1.17226 0
1.3 0 2.67282 0 0 0 1.14399 0 4.14399 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(5\) \( +1 \)
\(13\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1300.2.a.i 3
4.b odd 2 1 5200.2.a.ci 3
5.b even 2 1 260.2.a.b 3
5.c odd 4 2 1300.2.c.f 6
15.d odd 2 1 2340.2.a.n 3
20.d odd 2 1 1040.2.a.o 3
40.e odd 2 1 4160.2.a.br 3
40.f even 2 1 4160.2.a.bo 3
60.h even 2 1 9360.2.a.da 3
65.d even 2 1 3380.2.a.o 3
65.g odd 4 2 3380.2.f.h 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.a.b 3 5.b even 2 1
1040.2.a.o 3 20.d odd 2 1
1300.2.a.i 3 1.a even 1 1 trivial
1300.2.c.f 6 5.c odd 4 2
2340.2.a.n 3 15.d odd 2 1
3380.2.a.o 3 65.d even 2 1
3380.2.f.h 6 65.g odd 4 2
4160.2.a.bo 3 40.f even 2 1
4160.2.a.br 3 40.e odd 2 1
5200.2.a.ci 3 4.b odd 2 1
9360.2.a.da 3 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1300))\):

\( T_{3}^{3} + 2T_{3}^{2} - 8T_{3} - 12 \) Copy content Toggle raw display
\( T_{7}^{3} - 2T_{7}^{2} - 20T_{7} + 24 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} + 2 T^{2} + \cdots - 12 \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 2 T^{2} + \cdots + 24 \) Copy content Toggle raw display
$11$ \( T^{3} - 24T + 36 \) Copy content Toggle raw display
$13$ \( (T + 1)^{3} \) Copy content Toggle raw display
$17$ \( T^{3} + 2 T^{2} + \cdots + 24 \) Copy content Toggle raw display
$19$ \( T^{3} - 8 T^{2} + \cdots + 164 \) Copy content Toggle raw display
$23$ \( T^{3} - 10 T^{2} + \cdots - 12 \) Copy content Toggle raw display
$29$ \( T^{3} - 10 T^{2} + \cdots + 24 \) Copy content Toggle raw display
$31$ \( T^{3} + 12 T^{2} + \cdots + 4 \) Copy content Toggle raw display
$37$ \( T^{3} - 2 T^{2} + \cdots + 72 \) Copy content Toggle raw display
$41$ \( T^{3} + 2 T^{2} + \cdots + 24 \) Copy content Toggle raw display
$43$ \( T^{3} - 2 T^{2} + \cdots + 12 \) Copy content Toggle raw display
$47$ \( T^{3} - 10 T^{2} + \cdots + 24 \) Copy content Toggle raw display
$53$ \( T^{3} - 18 T^{2} + \cdots + 648 \) Copy content Toggle raw display
$59$ \( T^{3} + 16T^{2} - 564 \) Copy content Toggle raw display
$61$ \( T^{3} - 14 T^{2} + \cdots + 8 \) Copy content Toggle raw display
$67$ \( T^{3} + 14 T^{2} + \cdots - 152 \) Copy content Toggle raw display
$71$ \( T^{3} - 24T + 36 \) Copy content Toggle raw display
$73$ \( T^{3} + 14 T^{2} + \cdots - 1784 \) Copy content Toggle raw display
$79$ \( T^{3} - 8 T^{2} + \cdots + 32 \) Copy content Toggle raw display
$83$ \( T^{3} - 6 T^{2} + \cdots + 936 \) Copy content Toggle raw display
$89$ \( T^{3} + 2 T^{2} + \cdots + 216 \) Copy content Toggle raw display
$97$ \( T^{3} + 26 T^{2} + \cdots - 8 \) Copy content Toggle raw display
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