Properties

Label 2300.2.a.k
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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Newspace parameters

Level: \( N \) \(=\) \( 2300 = 2^{2} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2300.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(18.3655924649\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.321.1
Defining polynomial: \(x^{3} - x^{2} - 4 x + 1\)
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

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.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 3\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.46050
0.239123
−1.69963
0 −1.46050 0 0 0 4.92101 0 −0.866926 0
1.2 0 0.760877 0 0 0 0.478247 0 −2.42107 0
1.3 0 2.69963 0 0 0 −3.39926 0 4.28799 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.k yes 3
4.b odd 2 1 9200.2.a.cc 3
5.b even 2 1 2300.2.a.j 3
5.c odd 4 2 2300.2.c.i 6
20.d odd 2 1 9200.2.a.cg 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2300.2.a.j 3 5.b even 2 1
2300.2.a.k yes 3 1.a even 1 1 trivial
2300.2.c.i 6 5.c odd 4 2
9200.2.a.cc 3 4.b odd 2 1
9200.2.a.cg 3 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2300))\):

\( T_{3}^{3} - 2 T_{3}^{2} - 3 T_{3} + 3 \)
\( T_{7}^{3} - 2 T_{7}^{2} - 16 T_{7} + 8 \)

Hecke characteristic polynomials

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