Properties

Label 4900.2.a.bf
Level $4900$
Weight $2$
Character orbit 4900.a
Self dual yes
Analytic conductor $39.127$
Analytic rank $0$
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: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{3}, \sqrt{19})\)
Defining polynomial: \(x^{4} - 11 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 140)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} +O(q^{10})\) \( q + \beta_{2} q^{3} + ( -2 - \beta_{3} ) q^{11} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{13} + ( -\beta_{1} + 2 \beta_{2} ) q^{17} -\beta_{3} q^{19} + ( 2 \beta_{1} + \beta_{2} ) q^{23} -3 \beta_{2} q^{27} + ( 1 + \beta_{3} ) q^{29} -\beta_{3} q^{31} -3 \beta_{1} q^{33} + ( \beta_{1} + 4 \beta_{2} ) q^{37} + ( -2 + 2 \beta_{3} ) q^{39} + ( 7 - \beta_{3} ) q^{41} + ( -\beta_{1} + 3 \beta_{2} ) q^{43} + ( \beta_{1} + 2 \beta_{2} ) q^{47} + ( 4 - \beta_{3} ) q^{51} + ( 3 \beta_{1} - 2 \beta_{2} ) q^{53} + ( -3 \beta_{1} + 2 \beta_{2} ) q^{57} + \beta_{3} q^{59} + ( 7 + 2 \beta_{3} ) q^{61} + ( -4 \beta_{1} + 5 \beta_{2} ) q^{67} + ( 7 + 2 \beta_{3} ) q^{69} + ( 2 - 2 \beta_{3} ) q^{71} + ( \beta_{1} + 2 \beta_{2} ) q^{73} + ( 4 + \beta_{3} ) q^{79} -9 q^{81} + ( -3 \beta_{1} + \beta_{2} ) q^{83} + ( 3 \beta_{1} - \beta_{2} ) q^{87} + 7 q^{89} + ( -3 \beta_{1} + 2 \beta_{2} ) q^{93} -4 \beta_{2} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + O(q^{10}) \) \( 4 q - 6 q^{11} + 2 q^{19} + 2 q^{29} + 2 q^{31} - 12 q^{39} + 30 q^{41} + 18 q^{51} - 2 q^{59} + 24 q^{61} + 24 q^{69} + 12 q^{71} + 14 q^{79} - 36 q^{81} + 28 q^{89} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 11 x^{2} + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} - 7 \nu \)\()/4\)
\(\beta_{3}\)\(=\)\( \nu^{2} - 6 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 6\)
\(\nu^{3}\)\(=\)\(4 \beta_{2} + 7 \beta_{1}\)

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
−3.04547
1.31342
3.04547
−1.31342
0 −1.73205 0 0 0 0 0 0 0
1.2 0 −1.73205 0 0 0 0 0 0 0
1.3 0 1.73205 0 0 0 0 0 0 0
1.4 0 1.73205 0 0 0 0 0 0 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
5.b even 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.bf 4
5.b even 2 1 inner 4900.2.a.bf 4
5.c odd 4 2 980.2.e.c 4
7.b odd 2 1 4900.2.a.be 4
7.d odd 6 2 700.2.i.f 8
35.c odd 2 1 4900.2.a.be 4
35.f even 4 2 980.2.e.f 4
35.i odd 6 2 700.2.i.f 8
35.k even 12 2 140.2.q.a 4
35.k even 12 2 140.2.q.b yes 4
35.l odd 12 2 980.2.q.b 4
35.l odd 12 2 980.2.q.g 4
105.w odd 12 2 1260.2.bm.a 4
105.w odd 12 2 1260.2.bm.b 4
140.x odd 12 2 560.2.bw.a 4
140.x odd 12 2 560.2.bw.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.q.a 4 35.k even 12 2
140.2.q.b yes 4 35.k even 12 2
560.2.bw.a 4 140.x odd 12 2
560.2.bw.e 4 140.x odd 12 2
700.2.i.f 8 7.d odd 6 2
700.2.i.f 8 35.i odd 6 2
980.2.e.c 4 5.c odd 4 2
980.2.e.f 4 35.f even 4 2
980.2.q.b 4 35.l odd 12 2
980.2.q.g 4 35.l odd 12 2
1260.2.bm.a 4 105.w odd 12 2
1260.2.bm.b 4 105.w odd 12 2
4900.2.a.be 4 7.b odd 2 1
4900.2.a.be 4 35.c odd 2 1
4900.2.a.bf 4 1.a even 1 1 trivial
4900.2.a.bf 4 5.b even 2 1 inner

Hecke kernels

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

\( T_{3}^{2} - 3 \)
\( T_{11}^{2} + 3 T_{11} - 12 \)
\( T_{13}^{4} - 44 T_{13}^{2} + 256 \)
\( T_{19}^{2} - T_{19} - 14 \)
\( T_{23}^{4} - 62 T_{23}^{2} + 49 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( -3 + T^{2} )^{2} \)
$5$ \( T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( -12 + 3 T + T^{2} )^{2} \)
$13$ \( 256 - 44 T^{2} + T^{4} \)
$17$ \( 4 - 23 T^{2} + T^{4} \)
$19$ \( ( -14 - T + T^{2} )^{2} \)
$23$ \( 49 - 62 T^{2} + T^{4} \)
$29$ \( ( -14 - T + T^{2} )^{2} \)
$31$ \( ( -14 - T + T^{2} )^{2} \)
$37$ \( 3136 - 131 T^{2} + T^{4} \)
$41$ \( ( 42 - 15 T + T^{2} )^{2} \)
$43$ \( 196 - 47 T^{2} + T^{4} \)
$47$ \( 196 - 47 T^{2} + T^{4} \)
$53$ \( 1764 - 87 T^{2} + T^{4} \)
$59$ \( ( -14 + T + T^{2} )^{2} \)
$61$ \( ( -21 - 12 T + T^{2} )^{2} \)
$67$ \( 2401 - 206 T^{2} + T^{4} \)
$71$ \( ( -48 - 6 T + T^{2} )^{2} \)
$73$ \( 196 - 47 T^{2} + T^{4} \)
$79$ \( ( -2 - 7 T + T^{2} )^{2} \)
$83$ \( 1764 - 87 T^{2} + T^{4} \)
$89$ \( ( -7 + T )^{4} \)
$97$ \( ( -48 + T^{2} )^{2} \)
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