Properties

Label 9800.2.a.cu
Level $9800$
Weight $2$
Character orbit 9800.a
Self dual yes
Analytic conductor $78.253$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 9800 = 2^{3} \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9800.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(78.2533939809\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.43449.1
Defining polynomial: \(x^{4} - x^{3} - 7 x^{2} + 2 x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1400)
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.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{1} ) q^{3} + ( 2 - \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( 1 - \beta_{1} ) q^{3} + ( 2 - \beta_{1} + \beta_{2} ) q^{9} + ( -\beta_{2} + \beta_{3} ) q^{11} + ( -2 - \beta_{2} - \beta_{3} ) q^{13} + ( -2 - \beta_{1} - \beta_{3} ) q^{17} + ( 4 + \beta_{2} + 2 \beta_{3} ) q^{19} + ( -1 - 2 \beta_{1} - 3 \beta_{3} ) q^{23} + ( 4 - 2 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{27} + ( 2 + \beta_{2} + 2 \beta_{3} ) q^{29} + ( 3 - \beta_{2} ) q^{31} + ( 4 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{33} + ( -2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{37} + ( -4 + 4 \beta_{1} - \beta_{2} ) q^{39} + ( 3 - 2 \beta_{1} + 3 \beta_{3} ) q^{41} + ( -3 - \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{43} + ( -4 + 3 \beta_{1} + \beta_{2} ) q^{47} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{51} + ( 2 - \beta_{3} ) q^{53} + ( 7 - 5 \beta_{1} + \beta_{2} + \beta_{3} ) q^{57} + ( 1 - 3 \beta_{1} ) q^{59} + ( 5 - \beta_{1} - 3 \beta_{3} ) q^{61} + ( -4 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{67} + ( 4 - 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} ) q^{69} + ( -5 - 4 \beta_{1} - 2 \beta_{3} ) q^{71} + ( -2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{73} + ( -1 - 3 \beta_{3} ) q^{79} + ( 7 - 8 \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{81} + ( 6 + \beta_{2} - 2 \beta_{3} ) q^{83} + ( 5 - 3 \beta_{1} + \beta_{2} + \beta_{3} ) q^{87} + ( 4 - 3 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{89} + ( 2 - \beta_{2} + \beta_{3} ) q^{93} + ( -1 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{97} + ( -15 + 5 \beta_{1} - 2 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 3q^{3} + 5q^{9} + O(q^{10}) \) \( 4q + 3q^{3} + 5q^{9} - 4q^{13} - 7q^{17} + 10q^{19} + 12q^{27} + 2q^{29} + 14q^{31} + 2q^{33} + 2q^{37} - 10q^{39} + 4q^{41} - 15q^{43} - 15q^{47} + 5q^{51} + 10q^{53} + 19q^{57} + q^{59} + 25q^{61} + 4q^{67} + 16q^{69} - 20q^{71} - 2q^{73} + 2q^{79} + 24q^{81} + 26q^{83} + 13q^{87} + 19q^{89} + 8q^{93} - 6q^{97} - 51q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 7 x^{2} + 2 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 4 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 7 \nu + 1 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{3} + \beta_{2} + 8 \beta_{1} + 3\)

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
3.04673
0.534166
−0.265362
−2.31553
0 −2.04673 0 0 0 0 0 1.18910 0
1.2 0 0.465834 0 0 0 0 0 −2.78300 0
1.3 0 1.26536 0 0 0 0 0 −1.39886 0
1.4 0 3.31553 0 0 0 0 0 7.99276 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 9800.2.a.cu 4
5.b even 2 1 9800.2.a.ck 4
7.b odd 2 1 9800.2.a.cj 4
7.d odd 6 2 1400.2.q.m yes 8
35.c odd 2 1 9800.2.a.ct 4
35.i odd 6 2 1400.2.q.l 8
35.k even 12 4 1400.2.bh.j 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1400.2.q.l 8 35.i odd 6 2
1400.2.q.m yes 8 7.d odd 6 2
1400.2.bh.j 16 35.k even 12 4
9800.2.a.cj 4 7.b odd 2 1
9800.2.a.ck 4 5.b even 2 1
9800.2.a.ct 4 35.c odd 2 1
9800.2.a.cu 4 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9800))\):

\( T_{3}^{4} - 3 T_{3}^{3} - 4 T_{3}^{2} + 11 T_{3} - 4 \)
\( T_{11}^{4} - 37 T_{11}^{2} - 49 T_{11} + 134 \)
\( T_{13}^{4} + 4 T_{13}^{3} - 23 T_{13}^{2} - 105 T_{13} - 84 \)
\( T_{19}^{4} - 10 T_{19}^{3} - 13 T_{19}^{2} + 359 T_{19} - 824 \)
\( T_{23}^{4} - 79 T_{23}^{2} + 91 T_{23} + 953 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( -4 + 11 T - 4 T^{2} - 3 T^{3} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( T^{4} \)
$11$ \( 134 - 49 T - 37 T^{2} + T^{4} \)
$13$ \( -84 - 105 T - 23 T^{2} + 4 T^{3} + T^{4} \)
$17$ \( 5 - 18 T + 7 T^{2} + 7 T^{3} + T^{4} \)
$19$ \( -824 + 359 T - 13 T^{2} - 10 T^{3} + T^{4} \)
$23$ \( 953 + 91 T - 79 T^{2} + T^{4} \)
$29$ \( -222 + 219 T - 49 T^{2} - 2 T^{3} + T^{4} \)
$31$ \( -25 + 3 T + 49 T^{2} - 14 T^{3} + T^{4} \)
$37$ \( -92 + 127 T - 43 T^{2} - 2 T^{3} + T^{4} \)
$41$ \( 3319 + 125 T - 127 T^{2} - 4 T^{3} + T^{4} \)
$43$ \( -1956 - 621 T + 4 T^{2} + 15 T^{3} + T^{4} \)
$47$ \( -3015 - 816 T + T^{2} + 15 T^{3} + T^{4} \)
$53$ \( -32 - 13 T + 29 T^{2} - 10 T^{3} + T^{4} \)
$59$ \( 70 + 77 T - 66 T^{2} - T^{3} + T^{4} \)
$61$ \( -2130 + 57 T + 164 T^{2} - 25 T^{3} + T^{4} \)
$67$ \( -1472 + 1016 T - 172 T^{2} - 4 T^{3} + T^{4} \)
$71$ \( -1035 - 732 T + 34 T^{2} + 20 T^{3} + T^{4} \)
$73$ \( 1152 - 384 T - 184 T^{2} + 2 T^{3} + T^{4} \)
$79$ \( -149 + 265 T - 75 T^{2} - 2 T^{3} + T^{4} \)
$83$ \( -2826 + 15 T + 187 T^{2} - 26 T^{3} + T^{4} \)
$89$ \( -1437 + 474 T + 53 T^{2} - 19 T^{3} + T^{4} \)
$97$ \( -27 - 81 T - 57 T^{2} + 6 T^{3} + T^{4} \)
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