Properties

Label 7500.2.a.e
Level $7500$
Weight $2$
Character orbit 7500.a
Self dual yes
Analytic conductor $59.888$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7500 = 2^{2} \cdot 3 \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7500.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(59.8878015160\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.5125.1
Defining polynomial: \(x^{4} - 2 x^{3} - 6 x^{2} + 7 x + 11\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 300)
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.

\(f(q)\) \(=\) \( q - q^{3} + ( 1 + \beta_{1} + \beta_{2} ) q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + ( 1 + \beta_{1} + \beta_{2} ) q^{7} + q^{9} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{11} + ( 2 - \beta_{1} + \beta_{3} ) q^{13} + ( 1 - \beta_{1} - \beta_{2} ) q^{17} + ( -1 - \beta_{1} - \beta_{3} ) q^{19} + ( -1 - \beta_{1} - \beta_{2} ) q^{21} + ( 3 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{23} - q^{27} + ( 2 - \beta_{1} ) q^{29} + ( 2 + \beta_{1} - 2 \beta_{2} + 3 \beta_{3} ) q^{31} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{33} + ( 2 - 3 \beta_{1} ) q^{37} + ( -2 + \beta_{1} - \beta_{3} ) q^{39} + ( -1 + 4 \beta_{1} + 2 \beta_{2} ) q^{41} + ( -1 - 2 \beta_{1} - \beta_{2} ) q^{43} + ( 6 + \beta_{1} + 6 \beta_{2} - 2 \beta_{3} ) q^{47} + ( -1 + 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{49} + ( -1 + \beta_{1} + \beta_{2} ) q^{51} + ( 1 + \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{53} + ( 1 + \beta_{1} + \beta_{3} ) q^{57} + ( -4 + \beta_{1} - 7 \beta_{2} - \beta_{3} ) q^{59} + ( -1 - 2 \beta_{1} + 8 \beta_{2} - 2 \beta_{3} ) q^{61} + ( 1 + \beta_{1} + \beta_{2} ) q^{63} + ( 9 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{67} + ( -3 + 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{69} + ( 6 + \beta_{1} + 3 \beta_{2} ) q^{71} + ( 2 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} ) q^{73} + ( -1 + 3 \beta_{2} + \beta_{3} ) q^{77} + ( 3 - 4 \beta_{1} + 4 \beta_{2} - \beta_{3} ) q^{79} + q^{81} + ( -1 + 6 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{83} + ( -2 + \beta_{1} ) q^{87} + ( -8 + \beta_{1} - 6 \beta_{2} - 3 \beta_{3} ) q^{89} + ( -1 + \beta_{1} + 4 \beta_{2} ) q^{91} + ( -2 - \beta_{1} + 2 \beta_{2} - 3 \beta_{3} ) q^{93} + ( -3 + 4 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} ) q^{97} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{3} + 4q^{7} + 4q^{9} + O(q^{10}) \) \( 4q - 4q^{3} + 4q^{7} + 4q^{9} - q^{11} + 5q^{13} + 4q^{17} - 5q^{19} - 4q^{21} + 9q^{23} - 4q^{27} + 6q^{29} + 11q^{31} + q^{33} + 2q^{37} - 5q^{39} - 6q^{43} + 16q^{47} - 4q^{49} - 4q^{51} - 2q^{53} + 5q^{57} + q^{59} - 22q^{61} + 4q^{63} + 36q^{67} - 9q^{69} + 20q^{71} + 12q^{73} - 11q^{77} - 3q^{79} + 4q^{81} + 14q^{83} - 6q^{87} - 15q^{89} - 10q^{91} - 11q^{93} - 12q^{97} - q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{3} - 6 x^{2} + 7 x + 11\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 4 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 4 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{3} + \beta_{2} + 5 \beta_{1} + 4\)

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
−1.12233
−1.70636
2.12233
2.70636
0 −1.00000 0 0 0 −1.74037 0 1.00000 0
1.2 0 −1.00000 0 0 0 −0.0883282 0 1.00000 0
1.3 0 −1.00000 0 0 0 1.50430 0 1.00000 0
1.4 0 −1.00000 0 0 0 4.32440 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(5\) \(1\)

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 7500.2.a.e 4
5.b even 2 1 7500.2.a.f 4
5.c odd 4 2 7500.2.d.c 8
25.d even 5 2 300.2.m.b 8
25.e even 10 2 1500.2.m.a 8
25.f odd 20 4 1500.2.o.b 16
75.j odd 10 2 900.2.n.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.2.m.b 8 25.d even 5 2
900.2.n.b 8 75.j odd 10 2
1500.2.m.a 8 25.e even 10 2
1500.2.o.b 16 25.f odd 20 4
7500.2.a.e 4 1.a even 1 1 trivial
7500.2.a.f 4 5.b even 2 1
7500.2.d.c 8 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} - 4 T_{7}^{3} - 4 T_{7}^{2} + 11 T_{7} + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7500))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 1 + T )^{4} \)
$5$ \( T^{4} \)
$7$ \( 1 + 11 T - 4 T^{2} - 4 T^{3} + T^{4} \)
$11$ \( -19 + 46 T - 24 T^{2} + T^{3} + T^{4} \)
$13$ \( -45 + 60 T - 10 T^{2} - 5 T^{3} + T^{4} \)
$17$ \( -9 + 21 T - 4 T^{2} - 4 T^{3} + T^{4} \)
$19$ \( 5 - 15 T - 5 T^{2} + 5 T^{3} + T^{4} \)
$23$ \( 171 + 126 T - 14 T^{2} - 9 T^{3} + T^{4} \)
$29$ \( 1 + 9 T + 6 T^{2} - 6 T^{3} + T^{4} \)
$31$ \( 981 + 354 T - 34 T^{2} - 11 T^{3} + T^{4} \)
$37$ \( 1021 + 67 T - 66 T^{2} - 2 T^{3} + T^{4} \)
$41$ \( 2705 - 160 T - 130 T^{2} + T^{4} \)
$43$ \( 131 - 64 T - 19 T^{2} + 6 T^{3} + T^{4} \)
$47$ \( -4099 + 1149 T - 14 T^{2} - 16 T^{3} + T^{4} \)
$53$ \( -99 - 177 T - 76 T^{2} + 2 T^{3} + T^{4} \)
$59$ \( 3701 - 211 T - 159 T^{2} - T^{3} + T^{4} \)
$61$ \( -7909 - 2022 T + 4 T^{2} + 22 T^{3} + T^{4} \)
$67$ \( -639 - 1536 T + 421 T^{2} - 36 T^{3} + T^{4} \)
$71$ \( 5 - 215 T + 120 T^{2} - 20 T^{3} + T^{4} \)
$73$ \( 11 - 38 T - 61 T^{2} - 12 T^{3} + T^{4} \)
$79$ \( -639 - 918 T - 146 T^{2} + 3 T^{3} + T^{4} \)
$83$ \( -6849 + 3486 T - 224 T^{2} - 14 T^{3} + T^{4} \)
$89$ \( -9875 - 3000 T - 150 T^{2} + 15 T^{3} + T^{4} \)
$97$ \( -8019 - 3132 T - 206 T^{2} + 12 T^{3} + T^{4} \)
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