Properties

Label 4900.2.a.bg
Level $4900$
Weight $2$
Character orbit 4900.a
Self dual yes
Analytic conductor $39.127$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4900 = 2^{2} \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4900.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(39.1266969904\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{5})\)
Defining polynomial: \(x^{4} - 6 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + \beta_{3} q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + \beta_{3} q^{9} + ( -1 - 2 \beta_{3} ) q^{11} + ( -\beta_{1} + \beta_{2} ) q^{13} + 2 \beta_{1} q^{17} + ( -2 \beta_{1} - \beta_{2} ) q^{19} + ( -1 + 2 \beta_{3} ) q^{23} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{27} + ( -2 + \beta_{3} ) q^{29} + ( -3 \beta_{1} - \beta_{2} ) q^{31} + ( -3 \beta_{1} - 4 \beta_{2} ) q^{33} + ( -2 - \beta_{3} ) q^{37} -2 q^{39} + ( -\beta_{1} - 2 \beta_{2} ) q^{41} + ( -6 + \beta_{3} ) q^{43} + ( 4 \beta_{1} - 3 \beta_{2} ) q^{47} + ( 6 + 2 \beta_{3} ) q^{51} + ( -5 - 3 \beta_{3} ) q^{53} + ( -7 - 3 \beta_{3} ) q^{57} + ( -\beta_{1} + 3 \beta_{2} ) q^{59} + ( -7 \beta_{1} + 6 \beta_{2} ) q^{61} + ( 5 + 4 \beta_{3} ) q^{67} + ( \beta_{1} + 4 \beta_{2} ) q^{69} + ( 10 + 3 \beta_{3} ) q^{71} + ( -7 \beta_{1} + 6 \beta_{2} ) q^{73} + ( -7 - 2 \beta_{3} ) q^{79} + ( -4 - 3 \beta_{3} ) q^{81} + ( \beta_{1} + 7 \beta_{2} ) q^{83} + ( -\beta_{1} + 2 \beta_{2} ) q^{87} + ( 3 \beta_{1} - 9 \beta_{2} ) q^{89} + ( -10 - 4 \beta_{3} ) q^{93} + ( 9 \beta_{1} - 3 \beta_{2} ) q^{97} + ( -10 - \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q - 4q^{11} - 4q^{23} - 8q^{29} - 8q^{37} - 8q^{39} - 24q^{43} + 24q^{51} - 20q^{53} - 28q^{57} + 20q^{67} + 40q^{71} - 28q^{79} - 16q^{81} - 40q^{93} - 40q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 6 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} - 4 \nu \)\()/2\)
\(\beta_{3}\)\(=\)\( \nu^{2} - 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 3\)
\(\nu^{3}\)\(=\)\(2 \beta_{2} + 4 \beta_{1}\)

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
−2.28825
−0.874032
0.874032
2.28825
0 −2.28825 0 0 0 0 0 2.23607 0
1.2 0 −0.874032 0 0 0 0 0 −2.23607 0
1.3 0 0.874032 0 0 0 0 0 −2.23607 0
1.4 0 2.28825 0 0 0 0 0 2.23607 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)
\(7\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4900.2.a.bg 4
5.b even 2 1 4900.2.a.bi yes 4
5.c odd 4 2 4900.2.e.u 8
7.b odd 2 1 inner 4900.2.a.bg 4
35.c odd 2 1 4900.2.a.bi yes 4
35.f even 4 2 4900.2.e.u 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4900.2.a.bg 4 1.a even 1 1 trivial
4900.2.a.bg 4 7.b odd 2 1 inner
4900.2.a.bi yes 4 5.b even 2 1
4900.2.a.bi yes 4 35.c odd 2 1
4900.2.e.u 8 5.c odd 4 2
4900.2.e.u 8 35.f even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4900))\):

\( T_{3}^{4} - 6 T_{3}^{2} + 4 \)
\( T_{11}^{2} + 2 T_{11} - 19 \)
\( T_{13}^{4} - 6 T_{13}^{2} + 4 \)
\( T_{19}^{4} - 36 T_{19}^{2} + 4 \)
\( T_{23}^{2} + 2 T_{23} - 19 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 4 - 6 T^{2} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( -19 + 2 T + T^{2} )^{2} \)
$13$ \( 4 - 6 T^{2} + T^{4} \)
$17$ \( 64 - 24 T^{2} + T^{4} \)
$19$ \( 4 - 36 T^{2} + T^{4} \)
$23$ \( ( -19 + 2 T + T^{2} )^{2} \)
$29$ \( ( -1 + 4 T + T^{2} )^{2} \)
$31$ \( 100 - 70 T^{2} + T^{4} \)
$37$ \( ( -1 + 4 T + T^{2} )^{2} \)
$41$ \( 100 - 30 T^{2} + T^{4} \)
$43$ \( ( 31 + 12 T + T^{2} )^{2} \)
$47$ \( 1444 - 84 T^{2} + T^{4} \)
$53$ \( ( -20 + 10 T + T^{2} )^{2} \)
$59$ \( 100 - 30 T^{2} + T^{4} \)
$61$ \( 12100 - 270 T^{2} + T^{4} \)
$67$ \( ( -55 - 10 T + T^{2} )^{2} \)
$71$ \( ( 55 - 20 T + T^{2} )^{2} \)
$73$ \( 12100 - 270 T^{2} + T^{4} \)
$79$ \( ( 29 + 14 T + T^{2} )^{2} \)
$83$ \( 12100 - 230 T^{2} + T^{4} \)
$89$ \( 8100 - 270 T^{2} + T^{4} \)
$97$ \( 39204 - 414 T^{2} + T^{4} \)
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