Properties

Label 380.2.a.d
Level $380$
Weight $2$
Character orbit 380.a
Self dual yes
Analytic conductor $3.034$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 380.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.03431527681\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
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 \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \beta ) q^{3} + q^{5} + 2 q^{7} + ( 1 + 2 \beta ) q^{9} +O(q^{10})\) \( q + ( 1 + \beta ) q^{3} + q^{5} + 2 q^{7} + ( 1 + 2 \beta ) q^{9} -2 \beta q^{11} + ( -1 - \beta ) q^{13} + ( 1 + \beta ) q^{15} -2 \beta q^{17} + q^{19} + ( 2 + 2 \beta ) q^{21} -2 \beta q^{23} + q^{25} + 4 q^{27} + 2 \beta q^{29} + ( 2 - 2 \beta ) q^{31} + ( -6 - 2 \beta ) q^{33} + 2 q^{35} + ( 5 + \beta ) q^{37} + ( -4 - 2 \beta ) q^{39} -6 q^{41} + ( 2 - 4 \beta ) q^{43} + ( 1 + 2 \beta ) q^{45} + ( 6 + 4 \beta ) q^{47} -3 q^{49} + ( -6 - 2 \beta ) q^{51} + ( -9 - \beta ) q^{53} -2 \beta q^{55} + ( 1 + \beta ) q^{57} + 4 \beta q^{59} + ( 2 + 6 \beta ) q^{61} + ( 2 + 4 \beta ) q^{63} + ( -1 - \beta ) q^{65} + ( 5 + \beta ) q^{67} + ( -6 - 2 \beta ) q^{69} + ( -6 + 2 \beta ) q^{71} + ( -4 + 2 \beta ) q^{73} + ( 1 + \beta ) q^{75} -4 \beta q^{77} + ( -4 + 4 \beta ) q^{79} + ( 1 - 2 \beta ) q^{81} + 2 \beta q^{83} -2 \beta q^{85} + ( 6 + 2 \beta ) q^{87} + ( -12 - 2 \beta ) q^{89} + ( -2 - 2 \beta ) q^{91} -4 q^{93} + q^{95} + ( -1 - 9 \beta ) q^{97} + ( -12 - 2 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} + 2q^{5} + 4q^{7} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{3} + 2q^{5} + 4q^{7} + 2q^{9} - 2q^{13} + 2q^{15} + 2q^{19} + 4q^{21} + 2q^{25} + 8q^{27} + 4q^{31} - 12q^{33} + 4q^{35} + 10q^{37} - 8q^{39} - 12q^{41} + 4q^{43} + 2q^{45} + 12q^{47} - 6q^{49} - 12q^{51} - 18q^{53} + 2q^{57} + 4q^{61} + 4q^{63} - 2q^{65} + 10q^{67} - 12q^{69} - 12q^{71} - 8q^{73} + 2q^{75} - 8q^{79} + 2q^{81} + 12q^{87} - 24q^{89} - 4q^{91} - 8q^{93} + 2q^{95} - 2q^{97} - 24q^{99} + O(q^{100}) \)

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
−1.73205
1.73205
0 −0.732051 0 1.00000 0 2.00000 0 −2.46410 0
1.2 0 2.73205 0 1.00000 0 2.00000 0 4.46410 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)
\(19\) \(-1\)

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 380.2.a.d 2
3.b odd 2 1 3420.2.a.h 2
4.b odd 2 1 1520.2.a.l 2
5.b even 2 1 1900.2.a.d 2
5.c odd 4 2 1900.2.c.e 4
8.b even 2 1 6080.2.a.z 2
8.d odd 2 1 6080.2.a.bj 2
19.b odd 2 1 7220.2.a.h 2
20.d odd 2 1 7600.2.a.bf 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.a.d 2 1.a even 1 1 trivial
1520.2.a.l 2 4.b odd 2 1
1900.2.a.d 2 5.b even 2 1
1900.2.c.e 4 5.c odd 4 2
3420.2.a.h 2 3.b odd 2 1
6080.2.a.z 2 8.b even 2 1
6080.2.a.bj 2 8.d odd 2 1
7220.2.a.h 2 19.b odd 2 1
7600.2.a.bf 2 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 2 T_{3} - 2 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(380))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -2 - 2 T + T^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( ( -2 + T )^{2} \)
$11$ \( -12 + T^{2} \)
$13$ \( -2 + 2 T + T^{2} \)
$17$ \( -12 + T^{2} \)
$19$ \( ( -1 + T )^{2} \)
$23$ \( -12 + T^{2} \)
$29$ \( -12 + T^{2} \)
$31$ \( -8 - 4 T + T^{2} \)
$37$ \( 22 - 10 T + T^{2} \)
$41$ \( ( 6 + T )^{2} \)
$43$ \( -44 - 4 T + T^{2} \)
$47$ \( -12 - 12 T + T^{2} \)
$53$ \( 78 + 18 T + T^{2} \)
$59$ \( -48 + T^{2} \)
$61$ \( -104 - 4 T + T^{2} \)
$67$ \( 22 - 10 T + T^{2} \)
$71$ \( 24 + 12 T + T^{2} \)
$73$ \( 4 + 8 T + T^{2} \)
$79$ \( -32 + 8 T + T^{2} \)
$83$ \( -12 + T^{2} \)
$89$ \( 132 + 24 T + T^{2} \)
$97$ \( -242 + 2 T + T^{2} \)
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