Properties

Label 1380.2.a.f
Level $1380$
Weight $2$
Character orbit 1380.a
Self dual yes
Analytic conductor $11.019$
Analytic rank $1$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
Defining polynomial: \(x^{2} - 6\)
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 \(\beta = \sqrt{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} - q^{5} + q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} - q^{5} + q^{7} + q^{9} + ( -2 + \beta ) q^{11} -\beta q^{13} + q^{15} + ( -1 + \beta ) q^{17} -3 \beta q^{19} - q^{21} + q^{23} + q^{25} - q^{27} + ( -7 + \beta ) q^{29} + ( 3 + 2 \beta ) q^{31} + ( 2 - \beta ) q^{33} - q^{35} + ( -5 + 2 \beta ) q^{37} + \beta q^{39} + ( -3 - \beta ) q^{41} + ( 2 + 4 \beta ) q^{43} - q^{45} -3 \beta q^{47} -6 q^{49} + ( 1 - \beta ) q^{51} + ( -3 - 3 \beta ) q^{53} + ( 2 - \beta ) q^{55} + 3 \beta q^{57} + ( -9 - \beta ) q^{59} + ( -2 - \beta ) q^{61} + q^{63} + \beta q^{65} + ( -1 - 4 \beta ) q^{67} - q^{69} + ( -5 - 3 \beta ) q^{71} + 3 \beta q^{73} - q^{75} + ( -2 + \beta ) q^{77} + ( -4 + 4 \beta ) q^{79} + q^{81} + ( -13 + \beta ) q^{83} + ( 1 - \beta ) q^{85} + ( 7 - \beta ) q^{87} + ( -8 - 2 \beta ) q^{89} -\beta q^{91} + ( -3 - 2 \beta ) q^{93} + 3 \beta q^{95} -2 \beta q^{97} + ( -2 + \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} - 2q^{5} + 2q^{7} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} - 2q^{5} + 2q^{7} + 2q^{9} - 4q^{11} + 2q^{15} - 2q^{17} - 2q^{21} + 2q^{23} + 2q^{25} - 2q^{27} - 14q^{29} + 6q^{31} + 4q^{33} - 2q^{35} - 10q^{37} - 6q^{41} + 4q^{43} - 2q^{45} - 12q^{49} + 2q^{51} - 6q^{53} + 4q^{55} - 18q^{59} - 4q^{61} + 2q^{63} - 2q^{67} - 2q^{69} - 10q^{71} - 2q^{75} - 4q^{77} - 8q^{79} + 2q^{81} - 26q^{83} + 2q^{85} + 14q^{87} - 16q^{89} - 6q^{93} - 4q^{99} + O(q^{100}) \)

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
−2.44949
2.44949
0 −1.00000 0 −1.00000 0 1.00000 0 1.00000 0
1.2 0 −1.00000 0 −1.00000 0 1.00000 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.f 2
3.b odd 2 1 4140.2.a.r 2
4.b odd 2 1 5520.2.a.bl 2
5.b even 2 1 6900.2.a.r 2
5.c odd 4 2 6900.2.f.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1380.2.a.f 2 1.a even 1 1 trivial
4140.2.a.r 2 3.b odd 2 1
5520.2.a.bl 2 4.b odd 2 1
6900.2.a.r 2 5.b even 2 1
6900.2.f.h 4 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1380))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( ( -1 + T )^{2} \)
$11$ \( -2 + 4 T + T^{2} \)
$13$ \( -6 + T^{2} \)
$17$ \( -5 + 2 T + T^{2} \)
$19$ \( -54 + T^{2} \)
$23$ \( ( -1 + T )^{2} \)
$29$ \( 43 + 14 T + T^{2} \)
$31$ \( -15 - 6 T + T^{2} \)
$37$ \( 1 + 10 T + T^{2} \)
$41$ \( 3 + 6 T + T^{2} \)
$43$ \( -92 - 4 T + T^{2} \)
$47$ \( -54 + T^{2} \)
$53$ \( -45 + 6 T + T^{2} \)
$59$ \( 75 + 18 T + T^{2} \)
$61$ \( -2 + 4 T + T^{2} \)
$67$ \( -95 + 2 T + T^{2} \)
$71$ \( -29 + 10 T + T^{2} \)
$73$ \( -54 + T^{2} \)
$79$ \( -80 + 8 T + T^{2} \)
$83$ \( 163 + 26 T + T^{2} \)
$89$ \( 40 + 16 T + T^{2} \)
$97$ \( -24 + T^{2} \)
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