Properties

Label 3800.2.a.s
Level $3800$
Weight $2$
Character orbit 3800.a
Self dual yes
Analytic conductor $30.343$
Analytic rank $1$
Dimension $3$
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: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
Defining polynomial: \(x^{3} - 3 x - 1\)
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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

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
−1.53209
−0.347296
1.87939
0 −1.53209 0 0 0 −2.22668 0 −0.652704 0
1.2 0 −0.347296 0 0 0 3.41147 0 −2.87939 0
1.3 0 1.87939 0 0 0 −1.18479 0 0.532089 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.s 3
4.b odd 2 1 7600.2.a.br 3
5.b even 2 1 3800.2.a.t yes 3
5.c odd 4 2 3800.2.d.o 6
20.d odd 2 1 7600.2.a.bs 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3800.2.a.s 3 1.a even 1 1 trivial
3800.2.a.t yes 3 5.b even 2 1
3800.2.d.o 6 5.c odd 4 2
7600.2.a.br 3 4.b odd 2 1
7600.2.a.bs 3 20.d odd 2 1

Hecke kernels

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

\( T_{3}^{3} - 3 T_{3} - 1 \)
\( T_{7}^{3} - 9 T_{7} - 9 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( -1 - 3 T + T^{3} \)
$5$ \( T^{3} \)
$7$ \( -9 - 9 T + T^{3} \)
$11$ \( -17 - 6 T + 3 T^{2} + T^{3} \)
$13$ \( 17 - 18 T + 3 T^{2} + T^{3} \)
$17$ \( -1 + 9 T - 6 T^{2} + T^{3} \)
$19$ \( ( -1 + T )^{3} \)
$23$ \( -89 - 27 T + 6 T^{2} + T^{3} \)
$29$ \( 51 - 36 T + 3 T^{2} + T^{3} \)
$31$ \( 89 + 66 T + 15 T^{2} + T^{3} \)
$37$ \( 37 - 30 T - 9 T^{2} + T^{3} \)
$41$ \( -109 - 12 T + 9 T^{2} + T^{3} \)
$43$ \( -19 - 9 T + 3 T^{2} + T^{3} \)
$47$ \( 17 - 18 T + 3 T^{2} + T^{3} \)
$53$ \( -111 - 45 T + 3 T^{2} + T^{3} \)
$59$ \( 296 - 72 T - 6 T^{2} + T^{3} \)
$61$ \( -233 - 66 T + 9 T^{2} + T^{3} \)
$67$ \( 17 - 6 T - 3 T^{2} + T^{3} \)
$71$ \( -181 - 21 T + 9 T^{2} + T^{3} \)
$73$ \( 9 + 18 T + 9 T^{2} + T^{3} \)
$79$ \( -71 + 21 T + 12 T^{2} + T^{3} \)
$83$ \( -289 - 102 T + 9 T^{2} + T^{3} \)
$89$ \( 51 - 9 T - 6 T^{2} + T^{3} \)
$97$ \( -107 + 90 T - 21 T^{2} + T^{3} \)
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