Properties

Label 3800.2.a.j
Level $3800$
Weight $2$
Character orbit 3800.a
Self dual yes
Analytic conductor $30.343$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3800.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(30.3431527681\)
Analytic rank: \(1\)
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: no (minimal twist has level 760)
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.

\(f(q)\) \(=\) \( q + ( -1 + \beta ) q^{3} + ( 1 - 2 \beta ) q^{9} +O(q^{10})\) \( q + ( -1 + \beta ) q^{3} + ( 1 - 2 \beta ) q^{9} + 2 q^{11} + ( -1 - \beta ) q^{13} + ( -4 + 2 \beta ) q^{17} + q^{19} + ( -2 - 2 \beta ) q^{23} -4 q^{27} + 2 \beta q^{29} + 4 q^{31} + ( -2 + 2 \beta ) q^{33} + ( 1 + 5 \beta ) q^{37} -2 q^{39} + ( 4 + 2 \beta ) q^{41} + ( -4 - 4 \beta ) q^{43} + ( -4 - 4 \beta ) q^{47} -7 q^{49} + ( 10 - 6 \beta ) q^{51} + ( 3 - 5 \beta ) q^{53} + ( -1 + \beta ) q^{57} + ( -2 - 2 \beta ) q^{59} + ( -10 - 2 \beta ) q^{61} + ( -1 - 3 \beta ) q^{67} -4 q^{69} + ( 4 - 4 \beta ) q^{71} -6 \beta q^{73} + ( 2 + 6 \beta ) q^{79} + ( 1 + 2 \beta ) q^{81} + ( 2 - 2 \beta ) q^{83} + ( 6 - 2 \beta ) q^{87} + 2 \beta q^{89} + ( -4 + 4 \beta ) q^{93} + ( 7 - 5 \beta ) q^{97} + ( 2 - 4 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{3} + 2 q^{9} + 4 q^{11} - 2 q^{13} - 8 q^{17} + 2 q^{19} - 4 q^{23} - 8 q^{27} + 8 q^{31} - 4 q^{33} + 2 q^{37} - 4 q^{39} + 8 q^{41} - 8 q^{43} - 8 q^{47} - 14 q^{49} + 20 q^{51} + 6 q^{53} - 2 q^{57} - 4 q^{59} - 20 q^{61} - 2 q^{67} - 8 q^{69} + 8 q^{71} + 4 q^{79} + 2 q^{81} + 4 q^{83} + 12 q^{87} - 8 q^{93} + 14 q^{97} + 4 q^{99} + O(q^{100}) \)

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 −2.73205 0 0 0 0 0 4.46410 0
1.2 0 0.732051 0 0 0 0 0 −2.46410 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 3800.2.a.j 2
4.b odd 2 1 7600.2.a.bd 2
5.b even 2 1 760.2.a.g 2
5.c odd 4 2 3800.2.d.h 4
15.d odd 2 1 6840.2.a.y 2
20.d odd 2 1 1520.2.a.k 2
40.e odd 2 1 6080.2.a.bk 2
40.f even 2 1 6080.2.a.ba 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.2.a.g 2 5.b even 2 1
1520.2.a.k 2 20.d odd 2 1
3800.2.a.j 2 1.a even 1 1 trivial
3800.2.d.h 4 5.c odd 4 2
6080.2.a.ba 2 40.f even 2 1
6080.2.a.bk 2 40.e odd 2 1
6840.2.a.y 2 15.d odd 2 1
7600.2.a.bd 2 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3800))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -2 + 2 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( -2 + T )^{2} \)
$13$ \( -2 + 2 T + T^{2} \)
$17$ \( 4 + 8 T + T^{2} \)
$19$ \( ( -1 + T )^{2} \)
$23$ \( -8 + 4 T + T^{2} \)
$29$ \( -12 + T^{2} \)
$31$ \( ( -4 + T )^{2} \)
$37$ \( -74 - 2 T + T^{2} \)
$41$ \( 4 - 8 T + T^{2} \)
$43$ \( -32 + 8 T + T^{2} \)
$47$ \( -32 + 8 T + T^{2} \)
$53$ \( -66 - 6 T + T^{2} \)
$59$ \( -8 + 4 T + T^{2} \)
$61$ \( 88 + 20 T + T^{2} \)
$67$ \( -26 + 2 T + T^{2} \)
$71$ \( -32 - 8 T + T^{2} \)
$73$ \( -108 + T^{2} \)
$79$ \( -104 - 4 T + T^{2} \)
$83$ \( -8 - 4 T + T^{2} \)
$89$ \( -12 + T^{2} \)
$97$ \( -26 - 14 T + T^{2} \)
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