Properties

Label 1380.2.a.i
Level $1380$
Weight $2$
Character orbit 1380.a
Self dual yes
Analytic conductor $11.019$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{15}) \)
Defining polynomial: \(x^{2} - 15\)
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 \(\beta = \sqrt{15}\). We also show the integral \(q\)-expansion of the trace form.

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

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
−3.87298
3.87298
0 1.00000 0 1.00000 0 3.00000 0 1.00000 0
1.2 0 1.00000 0 1.00000 0 3.00000 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.i 2
3.b odd 2 1 4140.2.a.p 2
4.b odd 2 1 5520.2.a.bj 2
5.b even 2 1 6900.2.a.j 2
5.c odd 4 2 6900.2.f.o 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1380.2.a.i 2 1.a even 1 1 trivial
4140.2.a.p 2 3.b odd 2 1
5520.2.a.bj 2 4.b odd 2 1
6900.2.a.j 2 5.b even 2 1
6900.2.f.o 4 5.c odd 4 2

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( ( -3 + T )^{2} \)
$11$ \( -14 - 2 T + T^{2} \)
$13$ \( -14 - 2 T + T^{2} \)
$17$ \( -15 + T^{2} \)
$19$ \( -14 - 2 T + T^{2} \)
$23$ \( ( 1 + T )^{2} \)
$29$ \( -11 + 4 T + T^{2} \)
$31$ \( ( -3 + T )^{2} \)
$37$ \( ( -1 + T )^{2} \)
$41$ \( -11 + 4 T + T^{2} \)
$43$ \( -44 + 8 T + T^{2} \)
$47$ \( -6 + 6 T + T^{2} \)
$53$ \( -15 + T^{2} \)
$59$ \( -11 - 4 T + T^{2} \)
$61$ \( 10 - 10 T + T^{2} \)
$67$ \( -51 - 6 T + T^{2} \)
$71$ \( -131 + 4 T + T^{2} \)
$73$ \( -14 - 2 T + T^{2} \)
$79$ \( ( -4 + T )^{2} \)
$83$ \( 1 + 8 T + T^{2} \)
$89$ \( -24 + 12 T + T^{2} \)
$97$ \( ( -8 + T )^{2} \)
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