Properties

Label 4900.2.a.bd
Level $4900$
Weight $2$
Character orbit 4900.a
Self dual yes
Analytic conductor $39.127$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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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: \(3\)
Coefficient field: 3.3.257.1
Defining polynomial: \(x^{3} - x^{2} - 4 x + 3\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 700)
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.

\(f(q)\) \(=\) \( q -\beta_{2} q^{3} + ( 2 - \beta_{1} - \beta_{2} ) q^{9} +O(q^{10})\) \( q -\beta_{2} q^{3} + ( 2 - \beta_{1} - \beta_{2} ) q^{9} + ( 1 - \beta_{2} ) q^{11} + ( 3 + \beta_{2} ) q^{13} + ( 3 - \beta_{1} ) q^{17} + ( 1 + \beta_{1} ) q^{19} + ( 1 + \beta_{1} - \beta_{2} ) q^{23} + ( 2 - \beta_{1} - 3 \beta_{2} ) q^{27} + ( 1 + \beta_{1} + 2 \beta_{2} ) q^{29} + ( -\beta_{1} - 2 \beta_{2} ) q^{31} + ( 5 - \beta_{1} - 2 \beta_{2} ) q^{33} + ( -1 + 3 \beta_{2} ) q^{37} + ( -5 + \beta_{1} - 2 \beta_{2} ) q^{39} + ( -3 - \beta_{1} + 3 \beta_{2} ) q^{41} + ( 3 - \beta_{1} - \beta_{2} ) q^{43} + ( 2 + \beta_{1} + \beta_{2} ) q^{47} + ( -3 - 6 \beta_{2} ) q^{51} + ( -6 - \beta_{1} - 3 \beta_{2} ) q^{53} + ( 3 + 2 \beta_{2} ) q^{57} + ( 2 + \beta_{2} ) q^{59} + ( -6 + \beta_{1} - 2 \beta_{2} ) q^{61} + ( -6 + 2 \beta_{2} ) q^{67} + ( 8 - \beta_{1} + \beta_{2} ) q^{69} + ( -7 - 2 \beta_{2} ) q^{71} + 4 q^{73} + ( -1 + \beta_{1} - 3 \beta_{2} ) q^{79} + ( 6 - 5 \beta_{2} ) q^{81} + ( 11 + \beta_{1} + 4 \beta_{2} ) q^{83} + ( -7 + 2 \beta_{1} + 4 \beta_{2} ) q^{87} + ( 4 - \beta_{1} - \beta_{2} ) q^{89} + ( 7 - 2 \beta_{1} - 5 \beta_{2} ) q^{93} + ( 6 + 2 \beta_{1} + \beta_{2} ) q^{97} + ( 4 - 2 \beta_{1} - 7 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{3} + 8 q^{9} + O(q^{10}) \) \( 3 q + q^{3} + 8 q^{9} + 4 q^{11} + 8 q^{13} + 10 q^{17} + 2 q^{19} + 3 q^{23} + 10 q^{27} + 3 q^{31} + 18 q^{33} - 6 q^{37} - 14 q^{39} - 11 q^{41} + 11 q^{43} + 4 q^{47} - 3 q^{51} - 14 q^{53} + 7 q^{57} + 5 q^{59} - 17 q^{61} - 20 q^{67} + 24 q^{69} - 19 q^{71} + 12 q^{73} - q^{79} + 23 q^{81} + 28 q^{83} - 27 q^{87} + 14 q^{89} + 28 q^{93} + 15 q^{97} + 21 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 4 x + 3\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} + 2 \nu - 4 \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{2} + \beta_{1} + 1\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{2} + \beta_{1} + 10\)\()/3\)

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
−1.91223
2.19869
0.713538
0 −2.56885 0 0 0 0 0 3.59899 0
1.2 0 0.364448 0 0 0 0 0 −2.86718 0
1.3 0 3.20440 0 0 0 0 0 7.26819 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

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 4900.2.a.bd 3
5.b even 2 1 4900.2.a.bb 3
5.c odd 4 2 4900.2.e.s 6
7.b odd 2 1 4900.2.a.ba 3
7.d odd 6 2 700.2.i.e yes 6
35.c odd 2 1 4900.2.a.bc 3
35.f even 4 2 4900.2.e.t 6
35.i odd 6 2 700.2.i.d 6
35.k even 12 4 700.2.r.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
700.2.i.d 6 35.i odd 6 2
700.2.i.e yes 6 7.d odd 6 2
700.2.r.d 12 35.k even 12 4
4900.2.a.ba 3 7.b odd 2 1
4900.2.a.bb 3 5.b even 2 1
4900.2.a.bc 3 35.c odd 2 1
4900.2.a.bd 3 1.a even 1 1 trivial
4900.2.e.s 6 5.c odd 4 2
4900.2.e.t 6 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4900))\):

\( T_{3}^{3} - T_{3}^{2} - 8 T_{3} + 3 \)
\( T_{11}^{3} - 4 T_{11}^{2} - 3 T_{11} + 9 \)
\( T_{13}^{3} - 8 T_{13}^{2} + 13 T_{13} + 3 \)
\( T_{19}^{3} - 2 T_{19}^{2} - 23 T_{19} - 21 \)
\( T_{23}^{3} - 3 T_{23}^{2} - 36 T_{23} + 81 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( 3 - 8 T - T^{2} + T^{3} \)
$5$ \( T^{3} \)
$7$ \( T^{3} \)
$11$ \( 9 - 3 T - 4 T^{2} + T^{3} \)
$13$ \( 3 + 13 T - 8 T^{2} + T^{3} \)
$17$ \( 81 + 9 T - 10 T^{2} + T^{3} \)
$19$ \( -21 - 23 T - 2 T^{2} + T^{3} \)
$23$ \( 81 - 36 T - 3 T^{2} + T^{3} \)
$29$ \( 81 - 45 T + T^{3} \)
$31$ \( -37 - 42 T - 3 T^{2} + T^{3} \)
$37$ \( -149 - 63 T + 6 T^{2} + T^{3} \)
$41$ \( -873 - 78 T + 11 T^{2} + T^{3} \)
$43$ \( 71 + 14 T - 11 T^{2} + T^{3} \)
$47$ \( 9 - 21 T - 4 T^{2} + T^{3} \)
$53$ \( -549 - 15 T + 14 T^{2} + T^{3} \)
$59$ \( 9 - 5 T^{2} + T^{3} \)
$61$ \( 1 + 26 T + 17 T^{2} + T^{3} \)
$67$ \( 72 + 100 T + 20 T^{2} + T^{3} \)
$71$ \( 45 + 87 T + 19 T^{2} + T^{3} \)
$73$ \( ( -4 + T )^{3} \)
$79$ \( 449 - 118 T + T^{2} + T^{3} \)
$83$ \( 981 + 129 T - 28 T^{2} + T^{3} \)
$89$ \( 45 + 39 T - 14 T^{2} + T^{3} \)
$97$ \( 5 - 18 T - 15 T^{2} + T^{3} \)
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