Properties

Label 4900.2.a.bj
Level $4900$
Weight $2$
Character orbit 4900.a
Self dual yes
Analytic conductor $39.127$
Analytic rank $0$
Dimension $4$
CM discriminant -35
Inner twists $4$

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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: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{5}, \sqrt{21})\)
Defining polynomial: \(x^{4} - 13 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 980)
Fricke sign: \(-1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + ( 4 + \beta_{3} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + ( 4 + \beta_{3} ) q^{9} + ( 1 - \beta_{3} ) q^{11} -\beta_{2} q^{13} + ( 2 \beta_{1} - \beta_{2} ) q^{17} + ( 5 \beta_{1} + 2 \beta_{2} ) q^{27} + ( 5 + \beta_{3} ) q^{29} + ( -3 \beta_{1} - 2 \beta_{2} ) q^{33} + ( 1 - \beta_{3} ) q^{39} + ( -\beta_{1} + 2 \beta_{2} ) q^{47} + ( 15 + \beta_{3} ) q^{51} -12 q^{71} + ( 3 \beta_{1} + 3 \beta_{2} ) q^{73} + ( -1 - 3 \beta_{3} ) q^{79} + ( 21 + 4 \beta_{3} ) q^{81} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{83} + ( 9 \beta_{1} + 2 \beta_{2} ) q^{87} + ( 6 \beta_{1} - \beta_{2} ) q^{97} + ( -22 - 2 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 14q^{9} + O(q^{10}) \) \( 4q + 14q^{9} + 6q^{11} + 18q^{29} + 6q^{39} + 58q^{51} - 48q^{71} + 2q^{79} + 76q^{81} - 84q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 13 x^{2} + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} - 11 \nu \)\()/2\)
\(\beta_{3}\)\(=\)\( \nu^{2} - 7 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 7\)
\(\nu^{3}\)\(=\)\(2 \beta_{2} + 11 \beta_{1}\)

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
−3.40932
−1.17325
1.17325
3.40932
0 −3.40932 0 0 0 0 0 8.62348 0
1.2 0 −1.17325 0 0 0 0 0 −1.62348 0
1.3 0 1.17325 0 0 0 0 0 −1.62348 0
1.4 0 3.40932 0 0 0 0 0 8.62348 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
35.c odd 2 1 CM by \(\Q(\sqrt{-35}) \)
5.b even 2 1 inner
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.bj 4
5.b even 2 1 inner 4900.2.a.bj 4
5.c odd 4 2 980.2.e.d 4
7.b odd 2 1 inner 4900.2.a.bj 4
35.c odd 2 1 CM 4900.2.a.bj 4
35.f even 4 2 980.2.e.d 4
35.k even 12 4 980.2.q.i 8
35.l odd 12 4 980.2.q.i 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
980.2.e.d 4 5.c odd 4 2
980.2.e.d 4 35.f even 4 2
980.2.q.i 8 35.k even 12 4
980.2.q.i 8 35.l odd 12 4
4900.2.a.bj 4 1.a even 1 1 trivial
4900.2.a.bj 4 5.b even 2 1 inner
4900.2.a.bj 4 7.b odd 2 1 inner
4900.2.a.bj 4 35.c odd 2 1 CM

Hecke kernels

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

\( T_{3}^{4} - 13 T_{3}^{2} + 16 \)
\( T_{11}^{2} - 3 T_{11} - 24 \)
\( T_{13}^{4} - 33 T_{13}^{2} + 36 \)
\( T_{19} \)
\( T_{23} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 16 - 13 T^{2} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( -24 - 3 T + T^{2} )^{2} \)
$13$ \( 36 - 33 T^{2} + T^{4} \)
$17$ \( 2116 - 97 T^{2} + T^{4} \)
$19$ \( T^{4} \)
$23$ \( T^{4} \)
$29$ \( ( -6 - 9 T + T^{2} )^{2} \)
$31$ \( T^{4} \)
$37$ \( T^{4} \)
$41$ \( T^{4} \)
$43$ \( T^{4} \)
$47$ \( 256 - 157 T^{2} + T^{4} \)
$53$ \( T^{4} \)
$59$ \( T^{4} \)
$61$ \( T^{4} \)
$67$ \( T^{4} \)
$71$ \( ( 12 + T )^{4} \)
$73$ \( ( -180 + T^{2} )^{2} \)
$79$ \( ( -236 - T + T^{2} )^{2} \)
$83$ \( ( -80 + T^{2} )^{2} \)
$89$ \( T^{4} \)
$97$ \( 60516 - 537 T^{2} + T^{4} \)
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