Properties

Label 7500.2.d.d
Level $7500$
Weight $2$
Character orbit 7500.d
Analytic conductor $59.888$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 7500 = 2^{2} \cdot 3 \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7500.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(59.8878015160\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.324000000.1
Defining polynomial: \(x^{8} + 9 x^{6} + 26 x^{4} + 24 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 300)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{3} + ( \beta_{1} + 2 \beta_{3} + 2 \beta_{5} + \beta_{7} ) q^{7} - q^{9} +O(q^{10})\) \( q + \beta_{5} q^{3} + ( \beta_{1} + 2 \beta_{3} + 2 \beta_{5} + \beta_{7} ) q^{7} - q^{9} + \beta_{2} q^{11} + ( 2 \beta_{1} + 2 \beta_{5} - \beta_{7} ) q^{13} + ( -\beta_{1} + 3 \beta_{7} ) q^{17} + ( -3 \beta_{2} + \beta_{6} ) q^{19} + ( -1 - \beta_{2} - \beta_{4} ) q^{21} + ( -3 \beta_{1} - 2 \beta_{3} - 4 \beta_{5} ) q^{23} -\beta_{5} q^{27} + ( 2 - 3 \beta_{2} - \beta_{4} - 3 \beta_{6} ) q^{29} + ( -1 + \beta_{2} - 2 \beta_{4} - \beta_{6} ) q^{31} + ( \beta_{3} + \beta_{7} ) q^{33} + ( \beta_{1} - 3 \beta_{3} - 3 \beta_{7} ) q^{37} + ( -1 + \beta_{2} - 2 \beta_{4} + \beta_{6} ) q^{39} + ( 3 + 2 \beta_{2} + 2 \beta_{4} - 6 \beta_{6} ) q^{41} + ( 2 \beta_{1} - \beta_{3} - 8 \beta_{5} - 2 \beta_{7} ) q^{43} + ( -5 \beta_{1} - \beta_{3} - \beta_{7} ) q^{47} + ( 1 - 5 \beta_{2} - 3 \beta_{4} - \beta_{6} ) q^{49} + ( -3 - 3 \beta_{2} + \beta_{4} + 2 \beta_{6} ) q^{51} + ( \beta_{1} + 3 \beta_{7} ) q^{53} + ( -2 \beta_{3} + \beta_{5} - 3 \beta_{7} ) q^{57} + ( -1 - \beta_{2} + 6 \beta_{4} - 2 \beta_{6} ) q^{59} + ( -3 - 2 \beta_{2} + 4 \beta_{4} + 2 \beta_{6} ) q^{61} + ( -\beta_{1} - 2 \beta_{3} - 2 \beta_{5} - \beta_{7} ) q^{63} + ( 4 \beta_{1} - 5 \beta_{3} - 2 \beta_{5} - 2 \beta_{7} ) q^{67} + ( 2 + 3 \beta_{4} - \beta_{6} ) q^{69} + ( -2 + \beta_{2} - 3 \beta_{4} - 4 \beta_{6} ) q^{71} + ( -4 \beta_{1} - 5 \beta_{3} - 5 \beta_{5} - 6 \beta_{7} ) q^{73} + ( \beta_{1} + 4 \beta_{3} + 4 \beta_{5} + 2 \beta_{7} ) q^{77} + ( 6 + 6 \beta_{2} + \beta_{4} - 8 \beta_{6} ) q^{79} + q^{81} + ( 2 \beta_{1} - 3 \beta_{5} ) q^{83} + ( -\beta_{1} - 7 \beta_{3} - 2 \beta_{5} - 3 \beta_{7} ) q^{87} + ( -3 + 3 \beta_{2} - 2 \beta_{4} + \beta_{6} ) q^{89} + ( -3 - \beta_{2} - \beta_{4} + \beta_{6} ) q^{91} + ( -2 \beta_{1} - 2 \beta_{3} - 4 \beta_{5} + \beta_{7} ) q^{93} + ( -2 \beta_{3} + 5 \beta_{5} + 4 \beta_{7} ) q^{97} -\beta_{2} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{9} + O(q^{10}) \) \( 8q - 8q^{9} - 2q^{11} + 10q^{19} - 8q^{21} + 8q^{29} - 18q^{31} - 10q^{39} + 8q^{49} - 8q^{51} - 2q^{59} - 4q^{61} + 18q^{69} - 40q^{71} + 6q^{79} + 8q^{81} - 30q^{89} - 20q^{91} + 2q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 9 x^{6} + 26 x^{4} + 24 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 2 \)
\(\beta_{3}\)\(=\)\( \nu^{3} + 3 \nu \)
\(\beta_{4}\)\(=\)\( \nu^{4} + 4 \nu^{2} + 2 \)
\(\beta_{5}\)\(=\)\( \nu^{5} + 5 \nu^{3} + 5 \nu \)
\(\beta_{6}\)\(=\)\( \nu^{6} + 6 \nu^{4} + 9 \nu^{2} + 2 \)
\(\beta_{7}\)\(=\)\( \nu^{7} + 7 \nu^{5} + 14 \nu^{3} + 7 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 2\)
\(\nu^{3}\)\(=\)\(\beta_{3} - 3 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{4} - 4 \beta_{2} + 6\)
\(\nu^{5}\)\(=\)\(\beta_{5} - 5 \beta_{3} + 10 \beta_{1}\)
\(\nu^{6}\)\(=\)\(\beta_{6} - 6 \beta_{4} + 15 \beta_{2} - 20\)
\(\nu^{7}\)\(=\)\(\beta_{7} - 7 \beta_{5} + 21 \beta_{3} - 35 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7500\mathbb{Z}\right)^\times\).

\(n\) \(2501\) \(3751\) \(6877\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1249.1
0.209057i
1.95630i
1.82709i
1.33826i
1.33826i
1.82709i
1.95630i
0.209057i
0 1.00000i 0 0 0 4.78339i 0 −1.00000 0
1249.2 0 1.00000i 0 0 0 0.511170i 0 −1.00000 0
1249.3 0 1.00000i 0 0 0 0.547318i 0 −1.00000 0
1249.4 0 1.00000i 0 0 0 0.747238i 0 −1.00000 0
1249.5 0 1.00000i 0 0 0 0.747238i 0 −1.00000 0
1249.6 0 1.00000i 0 0 0 0.547318i 0 −1.00000 0
1249.7 0 1.00000i 0 0 0 0.511170i 0 −1.00000 0
1249.8 0 1.00000i 0 0 0 4.78339i 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1249.8
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 7500.2.d.d 8
5.b even 2 1 inner 7500.2.d.d 8
5.c odd 4 1 7500.2.a.d 4
5.c odd 4 1 7500.2.a.g 4
25.d even 5 2 1500.2.o.a 16
25.e even 10 2 1500.2.o.a 16
25.f odd 20 2 300.2.m.a 8
25.f odd 20 2 1500.2.m.b 8
75.l even 20 2 900.2.n.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.2.m.a 8 25.f odd 20 2
900.2.n.a 8 75.l even 20 2
1500.2.m.b 8 25.f odd 20 2
1500.2.o.a 16 25.d even 5 2
1500.2.o.a 16 25.e even 10 2
7500.2.a.d 4 5.c odd 4 1
7500.2.a.g 4 5.c odd 4 1
7500.2.d.d 8 1.a even 1 1 trivial
7500.2.d.d 8 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} + 24 T_{7}^{6} + 26 T_{7}^{4} + 9 T_{7}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(7500, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( 1 + T^{2} )^{4} \)
$5$ \( T^{8} \)
$7$ \( 1 + 9 T^{2} + 26 T^{4} + 24 T^{6} + T^{8} \)
$11$ \( ( 1 - 4 T - 4 T^{2} + T^{3} + T^{4} )^{2} \)
$13$ \( 25 + 300 T^{2} + 290 T^{4} + 45 T^{6} + T^{8} \)
$17$ \( 73441 + 22149 T^{2} + 2186 T^{4} + 84 T^{6} + T^{8} \)
$19$ \( ( 145 + 95 T - 25 T^{2} - 5 T^{3} + T^{4} )^{2} \)
$23$ \( 961 + 16124 T^{2} + 2346 T^{4} + 89 T^{6} + T^{8} \)
$29$ \( ( -599 + 571 T - 94 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$31$ \( ( -279 - 126 T + 6 T^{2} + 9 T^{3} + T^{4} )^{2} \)
$37$ \( 32761 + 16941 T^{2} + 2546 T^{4} + 96 T^{6} + T^{8} \)
$41$ \( ( -155 + 240 T - 70 T^{2} + T^{4} )^{2} \)
$43$ \( 5621641 + 1498794 T^{2} + 40151 T^{4} + 354 T^{6} + T^{8} \)
$47$ \( 2627641 + 419669 T^{2} + 16146 T^{4} + 224 T^{6} + T^{8} \)
$53$ \( 32041 + 22341 T^{2} + 2546 T^{4} + 96 T^{6} + T^{8} \)
$59$ \( ( 2341 + 41 T - 139 T^{2} + T^{3} + T^{4} )^{2} \)
$61$ \( ( -449 + 398 T - 96 T^{2} + 2 T^{3} + T^{4} )^{2} \)
$67$ \( 6046681 + 949894 T^{2} + 30891 T^{4} + 334 T^{6} + T^{8} \)
$71$ \( ( -3875 - 875 T + 50 T^{2} + 20 T^{3} + T^{4} )^{2} \)
$73$ \( 2653641 + 668466 T^{2} + 42111 T^{4} + 426 T^{6} + T^{8} \)
$79$ \( ( -1359 + 1278 T - 186 T^{2} - 3 T^{3} + T^{4} )^{2} \)
$83$ \( 7921 + 11124 T^{2} + 2006 T^{4} + 84 T^{6} + T^{8} \)
$89$ \( ( 45 - 90 T + 30 T^{2} + 15 T^{3} + T^{4} )^{2} \)
$97$ \( 18139081 + 1724316 T^{2} + 44126 T^{4} + 396 T^{6} + T^{8} \)
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