Properties

Label 3800.2.a.y
Level $3800$
Weight $2$
Character orbit 3800.a
Self dual yes
Analytic conductor $30.343$
Analytic rank $0$
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: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.568.1
Defining polynomial: \(x^{3} - x^{2} - 6 x - 2\)
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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + ( 2 + \beta_{2} ) q^{7} + ( 1 + 2 \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + ( 2 + \beta_{2} ) q^{7} + ( 1 + 2 \beta_{1} + \beta_{2} ) q^{9} + ( -2 + 2 \beta_{1} ) q^{11} + ( -2 - \beta_{1} - 2 \beta_{2} ) q^{13} + ( 4 + \beta_{2} ) q^{17} - q^{19} + ( -2 + 2 \beta_{1} - \beta_{2} ) q^{21} + ( 2 + 2 \beta_{1} - \beta_{2} ) q^{23} + ( 6 + 2 \beta_{1} + \beta_{2} ) q^{27} -3 \beta_{2} q^{29} + ( -4 + 2 \beta_{2} ) q^{31} + ( 8 + 2 \beta_{1} + 2 \beta_{2} ) q^{33} + ( -4 - \beta_{1} + \beta_{2} ) q^{37} + ( -4 \beta_{1} + \beta_{2} ) q^{39} + ( -2 - 2 \beta_{1} + 2 \beta_{2} ) q^{41} + ( 4 + 2 \beta_{1} + 4 \beta_{2} ) q^{43} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{47} + ( 3 - 2 \beta_{1} + 3 \beta_{2} ) q^{49} + ( -2 + 4 \beta_{1} - \beta_{2} ) q^{51} + ( 2 - \beta_{1} - 2 \beta_{2} ) q^{53} -\beta_{1} q^{57} + ( -2 + \beta_{2} ) q^{59} + ( 8 - 4 \beta_{1} ) q^{61} + ( 4 + 2 \beta_{1} ) q^{63} + ( -3 \beta_{1} - 4 \beta_{2} ) q^{67} + ( 10 + 6 \beta_{1} + 3 \beta_{2} ) q^{69} + ( -4 + 4 \beta_{1} ) q^{71} + ( 4 + 4 \beta_{1} + 3 \beta_{2} ) q^{73} + ( -8 + 4 \beta_{1} - 4 \beta_{2} ) q^{77} + ( 4 - 2 \beta_{1} - 4 \beta_{2} ) q^{79} + ( 3 + 4 \beta_{1} - 2 \beta_{2} ) q^{81} + ( 8 - 2 \beta_{2} ) q^{83} + ( 6 + 3 \beta_{2} ) q^{87} + ( 6 - 2 \beta_{1} - 2 \beta_{2} ) q^{89} + ( -14 + 2 \beta_{1} - 3 \beta_{2} ) q^{91} + ( -4 - 4 \beta_{1} - 2 \beta_{2} ) q^{93} + ( 4 - 7 \beta_{1} - \beta_{2} ) q^{97} + ( 10 + 6 \beta_{1} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{3} + 5 q^{7} + 4 q^{9} + O(q^{10}) \) \( 3 q + q^{3} + 5 q^{7} + 4 q^{9} - 4 q^{11} - 5 q^{13} + 11 q^{17} - 3 q^{19} - 3 q^{21} + 9 q^{23} + 19 q^{27} + 3 q^{29} - 14 q^{31} + 24 q^{33} - 14 q^{37} - 5 q^{39} - 10 q^{41} + 10 q^{43} + 4 q^{49} - q^{51} + 7 q^{53} - q^{57} - 7 q^{59} + 20 q^{61} + 14 q^{63} + q^{67} + 33 q^{69} - 8 q^{71} + 13 q^{73} - 16 q^{77} + 14 q^{79} + 15 q^{81} + 26 q^{83} + 15 q^{87} + 18 q^{89} - 37 q^{91} - 14 q^{93} + 6 q^{97} + 36 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 6 x - 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 2 \nu - 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 2 \beta_{1} + 4\)

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
−1.76156
−0.363328
3.12489
0 −1.76156 0 0 0 4.62620 0 0.103084 0
1.2 0 −0.363328 0 0 0 −1.14134 0 −2.86799 0
1.3 0 3.12489 0 0 0 1.51514 0 6.76491 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.y 3
4.b odd 2 1 7600.2.a.bo 3
5.b even 2 1 760.2.a.h 3
5.c odd 4 2 3800.2.d.k 6
15.d odd 2 1 6840.2.a.bj 3
20.d odd 2 1 1520.2.a.r 3
40.e odd 2 1 6080.2.a.bs 3
40.f even 2 1 6080.2.a.bw 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.2.a.h 3 5.b even 2 1
1520.2.a.r 3 20.d odd 2 1
3800.2.a.y 3 1.a even 1 1 trivial
3800.2.d.k 6 5.c odd 4 2
6080.2.a.bs 3 40.e odd 2 1
6080.2.a.bw 3 40.f even 2 1
6840.2.a.bj 3 15.d odd 2 1
7600.2.a.bo 3 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3800))\):

\( T_{3}^{3} - T_{3}^{2} - 6 T_{3} - 2 \)
\( T_{7}^{3} - 5 T_{7}^{2} + 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( -2 - 6 T - T^{2} + T^{3} \)
$5$ \( T^{3} \)
$7$ \( 8 - 5 T^{2} + T^{3} \)
$11$ \( -64 - 20 T + 4 T^{2} + T^{3} \)
$13$ \( -106 - 22 T + 5 T^{2} + T^{3} \)
$17$ \( -20 + 32 T - 11 T^{2} + T^{3} \)
$19$ \( ( 1 + T )^{3} \)
$23$ \( 160 - 16 T - 9 T^{2} + T^{3} \)
$29$ \( 108 - 72 T - 3 T^{2} + T^{3} \)
$31$ \( -64 + 32 T + 14 T^{2} + T^{3} \)
$37$ \( -20 + 46 T + 14 T^{2} + T^{3} \)
$41$ \( -472 - 44 T + 10 T^{2} + T^{3} \)
$43$ \( 848 - 88 T - 10 T^{2} + T^{3} \)
$47$ \( -64 - 40 T + T^{3} \)
$53$ \( -2 - 14 T - 7 T^{2} + T^{3} \)
$59$ \( -8 + 8 T + 7 T^{2} + T^{3} \)
$61$ \( 640 + 32 T - 20 T^{2} + T^{3} \)
$67$ \( -530 - 134 T - T^{2} + T^{3} \)
$71$ \( -512 - 80 T + 8 T^{2} + T^{3} \)
$73$ \( 500 - 64 T - 13 T^{2} + T^{3} \)
$79$ \( -16 - 56 T - 14 T^{2} + T^{3} \)
$83$ \( -352 + 192 T - 26 T^{2} + T^{3} \)
$89$ \( -40 + 68 T - 18 T^{2} + T^{3} \)
$97$ \( 2308 - 274 T - 6 T^{2} + T^{3} \)
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