Properties

Label 2300.2.a.j
Level $2300$
Weight $2$
Character orbit 2300.a
Self dual yes
Analytic conductor $18.366$
Analytic rank $1$
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] = [2300,2,Mod(1,2300)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2300, 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("2300.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2300 = 2^{2} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2300.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: \(18.3655924649\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.321.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{3} - 2 \beta_1 q^{7} + (\beta_{2} - \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{3} - 2 \beta_1 q^{7} + (\beta_{2} - \beta_1 + 1) q^{9} + (2 \beta_1 - 2) q^{11} + ( - \beta_{2} + \beta_1 + 2) q^{13} + (2 \beta_{2} + 2) q^{17} + ( - 2 \beta_{2} - 2 \beta_1 - 2) q^{19} + ( - 2 \beta_{2} - 6) q^{21} + q^{23} + ( - 2 \beta_{2} - \beta_1 - 2) q^{27} + ( - 3 \beta_{2} - 2 \beta_1 - 3) q^{29} + (\beta_{2} + 2 \beta_1 - 7) q^{31} + (2 \beta_{2} - 2 \beta_1 + 8) q^{33} + 2 \beta_1 q^{37} + (2 \beta_{2} + \beta_1 + 2) q^{39} + (4 \beta_{2} - \beta_1 + 3) q^{41} + (2 \beta_{2} - 2 \beta_1) q^{43} + ( - 3 \beta_{2} - 3 \beta_1 - 2) q^{47} + (4 \beta_{2} + 4 \beta_1 + 5) q^{49} + ( - 2 \beta_{2} + 4 \beta_1 - 4) q^{51} + ( - 4 \beta_{2} - 2 \beta_1 + 2) q^{53} + ( - 4 \beta_1 - 2) q^{57} + (2 \beta_{2} - 4 \beta_1 - 1) q^{59} + (2 \beta_{2} - 2 \beta_1 - 6) q^{61} + (2 \beta_{2} - 2 \beta_1 + 8) q^{63} + (6 \beta_{2} + 4 \beta_1 + 2) q^{67} + (\beta_1 - 1) q^{69} + ( - \beta_{2} - \beta_1 - 10) q^{71} + ( - \beta_{2} - 2 \beta_1 - 5) q^{73} + ( - 4 \beta_{2} - 12) q^{77} + ( - 2 \beta_{2} + 2 \beta_1 - 8) q^{79} + ( - 2 \beta_{2} - \beta_1 - 2) q^{81} + ( - 2 \beta_{2} + 2) q^{83} + (\beta_{2} - 6 \beta_1) q^{87} + (2 \beta_{2} - 4 \beta_1 + 4) q^{89} + ( - 2 \beta_{2} - 4 \beta_1 - 8) q^{91} + (\beta_{2} - 6 \beta_1 + 12) q^{93} + ( - 2 \beta_{2} + 2 \beta_1 - 4) q^{97} + ( - 4 \beta_{2} + 4 \beta_1 - 10) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{3} - 2 q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 2 q^{3} - 2 q^{7} + q^{9} - 4 q^{11} + 8 q^{13} + 4 q^{17} - 6 q^{19} - 16 q^{21} + 3 q^{23} - 5 q^{27} - 8 q^{29} - 20 q^{31} + 20 q^{33} + 2 q^{37} + 5 q^{39} + 4 q^{41} - 4 q^{43} - 6 q^{47} + 15 q^{49} - 6 q^{51} + 8 q^{53} - 10 q^{57} - 9 q^{59} - 22 q^{61} + 20 q^{63} + 4 q^{67} - 2 q^{69} - 30 q^{71} - 16 q^{73} - 32 q^{77} - 20 q^{79} - 5 q^{81} + 8 q^{83} - 7 q^{87} + 6 q^{89} - 26 q^{91} + 29 q^{93} - 8 q^{97} - 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 4x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 3 \) 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
−1.69963
0.239123
2.46050
0 −2.69963 0 0 0 3.39926 0 4.28799 0
1.2 0 −0.760877 0 0 0 −0.478247 0 −2.42107 0
1.3 0 1.46050 0 0 0 −4.92101 0 −0.866926 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)
\(23\) \(-1\)

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 2300.2.a.j 3
4.b odd 2 1 9200.2.a.cg 3
5.b even 2 1 2300.2.a.k yes 3
5.c odd 4 2 2300.2.c.i 6
20.d odd 2 1 9200.2.a.cc 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2300.2.a.j 3 1.a even 1 1 trivial
2300.2.a.k yes 3 5.b even 2 1
2300.2.c.i 6 5.c odd 4 2
9200.2.a.cc 3 20.d odd 2 1
9200.2.a.cg 3 4.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2300))\):

\( T_{3}^{3} + 2T_{3}^{2} - 3T_{3} - 3 \) Copy content Toggle raw display
\( T_{7}^{3} + 2T_{7}^{2} - 16T_{7} - 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} + 2 T^{2} - 3 T - 3 \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( T^{3} + 2 T^{2} - 16 T - 8 \) Copy content Toggle raw display
$11$ \( T^{3} + 4 T^{2} - 12 T - 24 \) Copy content Toggle raw display
$13$ \( T^{3} - 8 T^{2} + 9 T + 27 \) Copy content Toggle raw display
$17$ \( T^{3} - 4 T^{2} - 20 T + 72 \) Copy content Toggle raw display
$19$ \( T^{3} + 6 T^{2} - 24 T - 56 \) Copy content Toggle raw display
$23$ \( (T - 1)^{3} \) Copy content Toggle raw display
$29$ \( T^{3} + 8 T^{2} - 43 T - 257 \) Copy content Toggle raw display
$31$ \( T^{3} + 20 T^{2} + 113 T + 127 \) Copy content Toggle raw display
$37$ \( T^{3} - 2 T^{2} - 16 T + 8 \) Copy content Toggle raw display
$41$ \( T^{3} - 4 T^{2} - 107 T + 321 \) Copy content Toggle raw display
$43$ \( T^{3} + 4 T^{2} - 44 T - 168 \) Copy content Toggle raw display
$47$ \( T^{3} + 6 T^{2} - 69 T - 127 \) Copy content Toggle raw display
$53$ \( T^{3} - 8 T^{2} - 84 T - 72 \) Copy content Toggle raw display
$59$ \( T^{3} + 9 T^{2} - 81 T - 721 \) Copy content Toggle raw display
$61$ \( T^{3} + 22 T^{2} + 112 T - 72 \) Copy content Toggle raw display
$67$ \( T^{3} - 4 T^{2} - 252 T + 1176 \) Copy content Toggle raw display
$71$ \( T^{3} + 30 T^{2} + 291 T + 911 \) Copy content Toggle raw display
$73$ \( T^{3} + 16 T^{2} + 65 T + 77 \) Copy content Toggle raw display
$79$ \( T^{3} + 20 T^{2} + 84 T + 72 \) Copy content Toggle raw display
$83$ \( T^{3} - 8 T^{2} - 4 T + 8 \) Copy content Toggle raw display
$89$ \( T^{3} - 6 T^{2} - 96 T - 216 \) Copy content Toggle raw display
$97$ \( T^{3} + 8 T^{2} - 28 T - 8 \) Copy content Toggle raw display
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