Properties

Label 4900.2.e.s
Level $4900$
Weight $2$
Character orbit 4900.e
Analytic conductor $39.127$
Analytic rank $0$
Dimension $6$
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.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(39.1266969904\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.4227136.2
Defining polynomial: \(x^{6} + 9 x^{4} + 22 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 700)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + ( -2 - \beta_{4} - \beta_{5} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + ( -2 - \beta_{4} - \beta_{5} ) q^{9} + ( 1 + \beta_{5} ) q^{11} + ( -\beta_{1} + 3 \beta_{2} ) q^{13} + ( -3 \beta_{2} - \beta_{3} ) q^{17} + ( -1 + \beta_{4} ) q^{19} + ( \beta_{1} + \beta_{2} - \beta_{3} ) q^{23} + ( -3 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{27} + ( -1 + \beta_{4} + 2 \beta_{5} ) q^{29} + ( \beta_{4} + 2 \beta_{5} ) q^{31} + ( 2 \beta_{1} + 5 \beta_{2} + \beta_{3} ) q^{33} + ( 3 \beta_{1} + \beta_{2} ) q^{37} + ( 5 + \beta_{4} - 2 \beta_{5} ) q^{39} + ( -3 + \beta_{4} - 3 \beta_{5} ) q^{41} + ( \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{43} + ( \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{47} + ( -3 + 6 \beta_{5} ) q^{51} + ( 3 \beta_{1} - 6 \beta_{2} + \beta_{3} ) q^{53} + ( 2 \beta_{1} - 3 \beta_{2} ) q^{57} + ( -2 + \beta_{5} ) q^{59} + ( -6 - \beta_{4} + 2 \beta_{5} ) q^{61} + ( 2 \beta_{1} + 6 \beta_{2} ) q^{67} + ( -8 - \beta_{4} + \beta_{5} ) q^{69} + ( -7 + 2 \beta_{5} ) q^{71} + 4 \beta_{2} q^{73} + ( 1 + \beta_{4} - 3 \beta_{5} ) q^{79} + ( 6 + 5 \beta_{5} ) q^{81} + ( -4 \beta_{1} + 11 \beta_{2} - \beta_{3} ) q^{83} + ( 4 \beta_{1} + 7 \beta_{2} + 2 \beta_{3} ) q^{87} + ( -4 - \beta_{4} - \beta_{5} ) q^{89} + ( 5 \beta_{1} + 7 \beta_{2} + 2 \beta_{3} ) q^{93} + ( \beta_{1} - 6 \beta_{2} + 2 \beta_{3} ) q^{97} + ( -4 - 2 \beta_{4} - 7 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 16 q^{9} + O(q^{10}) \) \( 6 q - 16 q^{9} + 8 q^{11} - 4 q^{19} + 6 q^{31} + 28 q^{39} - 22 q^{41} - 6 q^{51} - 10 q^{59} - 34 q^{61} - 48 q^{69} - 38 q^{71} + 2 q^{79} + 46 q^{81} - 28 q^{89} - 42 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} + 9 x^{4} + 22 x^{2} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} + 5 \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{5} + 6 \nu^{3} + 7 \nu \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{5} + 9 \nu^{3} + 13 \nu \)\()/3\)
\(\beta_{4}\)\(=\)\( -2 \nu^{4} - 9 \nu^{2} - 2 \)
\(\beta_{5}\)\(=\)\( \nu^{4} + 6 \nu^{2} + 6 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{3} + \beta_{2} + \beta_{1}\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{5} + \beta_{4} - 10\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(5 \beta_{3} - 5 \beta_{2} - 2 \beta_{1}\)\()/3\)
\(\nu^{4}\)\(=\)\(-3 \beta_{5} - 2 \beta_{4} + 14\)
\(\nu^{5}\)\(=\)\((\)\(-23 \beta_{3} + 32 \beta_{2} + 5 \beta_{1}\)\()/3\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4900\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(1177\) \(2451\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2549.1
0.713538i
1.91223i
2.19869i
2.19869i
1.91223i
0.713538i
0 3.20440i 0 0 0 0 0 −7.26819 0
2549.2 0 2.56885i 0 0 0 0 0 −3.59899 0
2549.3 0 0.364448i 0 0 0 0 0 2.86718 0
2549.4 0 0.364448i 0 0 0 0 0 2.86718 0
2549.5 0 2.56885i 0 0 0 0 0 −3.59899 0
2549.6 0 3.20440i 0 0 0 0 0 −7.26819 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2549.6
Significant digits:
Format:

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.e.s 6
5.b even 2 1 inner 4900.2.e.s 6
5.c odd 4 1 4900.2.a.bb 3
5.c odd 4 1 4900.2.a.bd 3
7.b odd 2 1 4900.2.e.t 6
7.d odd 6 2 700.2.r.d 12
35.c odd 2 1 4900.2.e.t 6
35.f even 4 1 4900.2.a.ba 3
35.f even 4 1 4900.2.a.bc 3
35.i odd 6 2 700.2.r.d 12
35.k even 12 2 700.2.i.d 6
35.k even 12 2 700.2.i.e yes 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
700.2.i.d 6 35.k even 12 2
700.2.i.e yes 6 35.k even 12 2
700.2.r.d 12 7.d odd 6 2
700.2.r.d 12 35.i odd 6 2
4900.2.a.ba 3 35.f even 4 1
4900.2.a.bb 3 5.c odd 4 1
4900.2.a.bc 3 35.f even 4 1
4900.2.a.bd 3 5.c odd 4 1
4900.2.e.s 6 1.a even 1 1 trivial
4900.2.e.s 6 5.b even 2 1 inner
4900.2.e.t 6 7.b odd 2 1
4900.2.e.t 6 35.c odd 2 1

Hecke kernels

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

\( T_{3}^{6} + 17 T_{3}^{4} + 70 T_{3}^{2} + 9 \)
\( T_{11}^{3} - 4 T_{11}^{2} - 3 T_{11} + 9 \)
\( T_{19}^{3} + 2 T_{19}^{2} - 23 T_{19} + 21 \)
\( T_{31}^{3} - 3 T_{31}^{2} - 42 T_{31} - 37 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( 9 + 70 T^{2} + 17 T^{4} + T^{6} \)
$5$ \( T^{6} \)
$7$ \( T^{6} \)
$11$ \( ( 9 - 3 T - 4 T^{2} + T^{3} )^{2} \)
$13$ \( 9 + 217 T^{2} + 38 T^{4} + T^{6} \)
$17$ \( 6561 + 1701 T^{2} + 82 T^{4} + T^{6} \)
$19$ \( ( 21 - 23 T + 2 T^{2} + T^{3} )^{2} \)
$23$ \( 6561 + 1782 T^{2} + 81 T^{4} + T^{6} \)
$29$ \( ( -81 - 45 T + T^{3} )^{2} \)
$31$ \( ( -37 - 42 T - 3 T^{2} + T^{3} )^{2} \)
$37$ \( 22201 + 5757 T^{2} + 162 T^{4} + T^{6} \)
$41$ \( ( -873 - 78 T + 11 T^{2} + T^{3} )^{2} \)
$43$ \( 5041 + 1758 T^{2} + 93 T^{4} + T^{6} \)
$47$ \( 81 + 513 T^{2} + 58 T^{4} + T^{6} \)
$53$ \( 301401 + 15597 T^{2} + 226 T^{4} + T^{6} \)
$59$ \( ( -9 + 5 T^{2} + T^{3} )^{2} \)
$61$ \( ( 1 + 26 T + 17 T^{2} + T^{3} )^{2} \)
$67$ \( 5184 + 7120 T^{2} + 200 T^{4} + T^{6} \)
$71$ \( ( 45 + 87 T + 19 T^{2} + T^{3} )^{2} \)
$73$ \( ( 16 + T^{2} )^{3} \)
$79$ \( ( -449 - 118 T - T^{2} + T^{3} )^{2} \)
$83$ \( 962361 + 71577 T^{2} + 526 T^{4} + T^{6} \)
$89$ \( ( -45 + 39 T + 14 T^{2} + T^{3} )^{2} \)
$97$ \( 25 + 474 T^{2} + 261 T^{4} + T^{6} \)
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