Properties

Label 1380.2.a.j
Level $1380$
Weight $2$
Character orbit 1380.a
Self dual yes
Analytic conductor $11.019$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1380.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.0193554789\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.3144.1
Defining polynomial: \(x^{3} - x^{2} - 16 x - 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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.

\(f(q)\) \(=\) \( q + q^{3} - q^{5} + ( 1 - \beta_{1} ) q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} - q^{5} + ( 1 - \beta_{1} ) q^{7} + q^{9} + ( 1 + \beta_{2} ) q^{11} + ( 1 - \beta_{2} ) q^{13} - q^{15} + ( \beta_{1} - \beta_{2} ) q^{17} + ( 3 + \beta_{2} ) q^{19} + ( 1 - \beta_{1} ) q^{21} + q^{23} + q^{25} + q^{27} + ( -\beta_{1} + \beta_{2} ) q^{29} + ( 3 + \beta_{1} ) q^{31} + ( 1 + \beta_{2} ) q^{33} + ( -1 + \beta_{1} ) q^{35} + ( 1 + \beta_{1} - 2 \beta_{2} ) q^{37} + ( 1 - \beta_{2} ) q^{39} + ( 2 + 3 \beta_{1} - \beta_{2} ) q^{41} + ( 6 - 2 \beta_{1} ) q^{43} - q^{45} + ( -1 - 2 \beta_{1} + \beta_{2} ) q^{47} + ( 4 - \beta_{1} + 2 \beta_{2} ) q^{49} + ( \beta_{1} - \beta_{2} ) q^{51} + ( 2 + 3 \beta_{1} - \beta_{2} ) q^{53} + ( -1 - \beta_{2} ) q^{55} + ( 3 + \beta_{2} ) q^{57} + ( \beta_{1} - \beta_{2} ) q^{59} + ( 3 + 2 \beta_{1} - \beta_{2} ) q^{61} + ( 1 - \beta_{1} ) q^{63} + ( -1 + \beta_{2} ) q^{65} + ( 3 + \beta_{1} ) q^{67} + q^{69} + ( -2 - 3 \beta_{1} + \beta_{2} ) q^{71} + ( 5 + 3 \beta_{2} ) q^{73} + q^{75} + ( -3 - 4 \beta_{1} + \beta_{2} ) q^{77} + ( 4 + 2 \beta_{1} ) q^{79} + q^{81} + ( 4 - \beta_{1} - \beta_{2} ) q^{83} + ( -\beta_{1} + \beta_{2} ) q^{85} + ( -\beta_{1} + \beta_{2} ) q^{87} + ( 2 - 2 \beta_{1} - 2 \beta_{2} ) q^{89} + ( 5 + 2 \beta_{1} - \beta_{2} ) q^{91} + ( 3 + \beta_{1} ) q^{93} + ( -3 - \beta_{2} ) q^{95} + ( -6 - 4 \beta_{1} + 2 \beta_{2} ) q^{97} + ( 1 + \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{3} - 3q^{5} + 2q^{7} + 3q^{9} + O(q^{10}) \) \( 3q + 3q^{3} - 3q^{5} + 2q^{7} + 3q^{9} + 4q^{11} + 2q^{13} - 3q^{15} + 10q^{19} + 2q^{21} + 3q^{23} + 3q^{25} + 3q^{27} + 10q^{31} + 4q^{33} - 2q^{35} + 2q^{37} + 2q^{39} + 8q^{41} + 16q^{43} - 3q^{45} - 4q^{47} + 13q^{49} + 8q^{53} - 4q^{55} + 10q^{57} + 10q^{61} + 2q^{63} - 2q^{65} + 10q^{67} + 3q^{69} - 8q^{71} + 18q^{73} + 3q^{75} - 12q^{77} + 14q^{79} + 3q^{81} + 10q^{83} + 2q^{89} + 16q^{91} + 10q^{93} - 10q^{95} - 20q^{97} + 4q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 16 x - 8\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{2} - \nu - 10 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{2} + \beta_{1} + 10\)

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
4.73549
−0.526440
−3.20905
0 1.00000 0 −1.00000 0 −3.73549 0 1.00000 0
1.2 0 1.00000 0 −1.00000 0 1.52644 0 1.00000 0
1.3 0 1.00000 0 −1.00000 0 4.20905 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(1\)
\(23\) \(-1\)

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 1380.2.a.j 3
3.b odd 2 1 4140.2.a.s 3
4.b odd 2 1 5520.2.a.bv 3
5.b even 2 1 6900.2.a.x 3
5.c odd 4 2 6900.2.f.r 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1380.2.a.j 3 1.a even 1 1 trivial
4140.2.a.s 3 3.b odd 2 1
5520.2.a.bv 3 4.b odd 2 1
6900.2.a.x 3 5.b even 2 1
6900.2.f.r 6 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{3} - 2 T_{7}^{2} - 15 T_{7} + 24 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1380))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( ( -1 + T )^{3} \)
$5$ \( ( 1 + T )^{3} \)
$7$ \( 24 - 15 T - 2 T^{2} + T^{3} \)
$11$ \( 48 - 14 T - 4 T^{2} + T^{3} \)
$13$ \( -12 - 18 T - 2 T^{2} + T^{3} \)
$17$ \( 18 - 21 T + T^{3} \)
$19$ \( 52 + 14 T - 10 T^{2} + T^{3} \)
$23$ \( ( -1 + T )^{3} \)
$29$ \( -18 - 21 T + T^{3} \)
$31$ \( 4 + 17 T - 10 T^{2} + T^{3} \)
$37$ \( -108 - 63 T - 2 T^{2} + T^{3} \)
$41$ \( 582 - 101 T - 8 T^{2} + T^{3} \)
$43$ \( 304 + 20 T - 16 T^{2} + T^{3} \)
$47$ \( -216 - 50 T + 4 T^{2} + T^{3} \)
$53$ \( 582 - 101 T - 8 T^{2} + T^{3} \)
$59$ \( 18 - 21 T + T^{3} \)
$61$ \( 292 - 22 T - 10 T^{2} + T^{3} \)
$67$ \( 4 + 17 T - 10 T^{2} + T^{3} \)
$71$ \( -582 - 101 T + 8 T^{2} + T^{3} \)
$73$ \( 1492 - 66 T - 18 T^{2} + T^{3} \)
$79$ \( 96 - 14 T^{2} + T^{3} \)
$83$ \( 228 - 17 T - 10 T^{2} + T^{3} \)
$89$ \( 912 - 200 T - 2 T^{2} + T^{3} \)
$97$ \( -2336 - 88 T + 20 T^{2} + T^{3} \)
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