Properties

Label 7500.2.d.c
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: 8.0.6724000000.12
Defining polynomial: \(x^{8} + 16 x^{6} + 86 x^{4} + 181 x^{2} + 121\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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.

\(f(q)\) \(=\) \( q -\beta_{3} q^{3} + ( \beta_{1} - \beta_{3} - \beta_{4} ) q^{7} - q^{9} +O(q^{10})\) \( q -\beta_{3} q^{3} + ( \beta_{1} - \beta_{3} - \beta_{4} ) q^{7} - q^{9} + ( -1 - \beta_{2} + \beta_{5} - \beta_{6} ) q^{11} + ( \beta_{1} + 2 \beta_{3} + \beta_{7} ) q^{13} + ( -\beta_{1} - \beta_{3} + \beta_{4} ) q^{17} + ( 2 - \beta_{2} - \beta_{5} ) q^{19} + ( -1 + \beta_{5} + \beta_{6} ) q^{21} + ( 2 \beta_{1} + 3 \beta_{3} - \beta_{4} + \beta_{7} ) q^{23} + \beta_{3} q^{27} + ( -1 - \beta_{5} ) q^{29} + ( 5 - 3 \beta_{2} - \beta_{5} + 2 \beta_{6} ) q^{31} + ( -\beta_{1} - \beta_{3} - \beta_{4} - \beta_{7} ) q^{33} + ( -3 \beta_{1} - 2 \beta_{3} ) q^{37} + ( 1 - \beta_{2} + \beta_{5} ) q^{39} + ( 1 - 4 \beta_{5} - 2 \beta_{6} ) q^{41} + ( 2 \beta_{1} - \beta_{3} - \beta_{4} ) q^{43} + ( \beta_{1} - 6 \beta_{3} - 6 \beta_{4} + 2 \beta_{7} ) q^{47} + ( 2 \beta_{2} + 3 \beta_{5} + 2 \beta_{6} ) q^{49} + ( -1 - \beta_{5} - \beta_{6} ) q^{51} + ( -\beta_{1} + \beta_{3} + 3 \beta_{4} + 2 \beta_{7} ) q^{53} + ( \beta_{1} - \beta_{3} - \beta_{7} ) q^{57} + ( -4 - \beta_{2} + \beta_{5} - 7 \beta_{6} ) q^{59} + ( -11 + 2 \beta_{2} + 2 \beta_{5} - 8 \beta_{6} ) q^{61} + ( -\beta_{1} + \beta_{3} + \beta_{4} ) q^{63} + ( -2 \beta_{1} - 9 \beta_{3} + \beta_{4} + 2 \beta_{7} ) q^{67} + ( 2 - \beta_{2} + 2 \beta_{5} + \beta_{6} ) q^{69} + ( 4 - \beta_{5} - 3 \beta_{6} ) q^{71} + ( -2 \beta_{1} - 3 \beta_{4} - 2 \beta_{7} ) q^{73} + ( \beta_{3} - 3 \beta_{4} - \beta_{7} ) q^{77} + ( 5 - \beta_{2} - 4 \beta_{5} + 4 \beta_{6} ) q^{79} + q^{81} + ( -6 \beta_{1} - \beta_{3} - 4 \beta_{4} + 2 \beta_{7} ) q^{83} + ( \beta_{1} + 2 \beta_{3} ) q^{87} + ( 1 - 3 \beta_{2} + \beta_{5} - 6 \beta_{6} ) q^{89} + ( -4 - \beta_{5} - 4 \beta_{6} ) q^{91} + ( \beta_{1} - 2 \beta_{3} + 2 \beta_{4} - 3 \beta_{7} ) q^{93} + ( 4 \beta_{1} + 3 \beta_{3} - 2 \beta_{4} - 4 \beta_{7} ) q^{97} + ( 1 + \beta_{2} - \beta_{5} + \beta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{9} + O(q^{10}) \) \( 8q - 8q^{9} - 2q^{11} + 10q^{19} - 8q^{21} - 12q^{29} + 22q^{31} + 10q^{39} + 8q^{49} - 8q^{51} - 2q^{59} - 44q^{61} + 18q^{69} + 40q^{71} + 6q^{79} + 8q^{81} + 30q^{89} - 20q^{91} + 2q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 16 x^{6} + 86 x^{4} + 181 x^{2} + 121\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{6} + 8 \nu^{4} + 4 \nu^{2} - 22 \)\()/9\)
\(\beta_{3}\)\(=\)\((\)\( -5 \nu^{7} - 58 \nu^{5} - 155 \nu^{3} - 25 \nu \)\()/99\)
\(\beta_{4}\)\(=\)\((\)\( -2 \nu^{7} - 43 \nu^{5} - 260 \nu^{3} - 406 \nu \)\()/99\)
\(\beta_{5}\)\(=\)\((\)\( -2 \nu^{6} - 25 \nu^{4} - 80 \nu^{2} - 46 \)\()/9\)
\(\beta_{6}\)\(=\)\((\)\( 2 \nu^{6} + 25 \nu^{4} + 89 \nu^{2} + 82 \)\()/9\)
\(\beta_{7}\)\(=\)\((\)\( 2 \nu^{7} + 25 \nu^{5} + 89 \nu^{3} + 91 \nu \)\()/9\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{6} + \beta_{5} - 4\)
\(\nu^{3}\)\(=\)\(\beta_{7} + \beta_{4} + 4 \beta_{3} - 5 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-8 \beta_{6} - 9 \beta_{5} - 2 \beta_{2} + 22\)
\(\nu^{5}\)\(=\)\(-10 \beta_{7} - 15 \beta_{4} - 38 \beta_{3} + 30 \beta_{1}\)
\(\nu^{6}\)\(=\)\(60 \beta_{6} + 68 \beta_{5} + 25 \beta_{2} - 138\)
\(\nu^{7}\)\(=\)\(85 \beta_{7} + 143 \beta_{4} + 297 \beta_{3} - 198 \beta_{1}\)

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
2.70636i
2.12233i
1.70636i
1.12233i
1.12233i
1.70636i
2.12233i
2.70636i
0 1.00000i 0 0 0 4.32440i 0 −1.00000 0
1249.2 0 1.00000i 0 0 0 1.50430i 0 −1.00000 0
1249.3 0 1.00000i 0 0 0 0.0883282i 0 −1.00000 0
1249.4 0 1.00000i 0 0 0 1.74037i 0 −1.00000 0
1249.5 0 1.00000i 0 0 0 1.74037i 0 −1.00000 0
1249.6 0 1.00000i 0 0 0 0.0883282i 0 −1.00000 0
1249.7 0 1.00000i 0 0 0 1.50430i 0 −1.00000 0
1249.8 0 1.00000i 0 0 0 4.32440i 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.c 8
5.b even 2 1 inner 7500.2.d.c 8
5.c odd 4 1 7500.2.a.e 4
5.c odd 4 1 7500.2.a.f 4
25.d even 5 2 1500.2.o.b 16
25.e even 10 2 1500.2.o.b 16
25.f odd 20 2 300.2.m.b 8
25.f odd 20 2 1500.2.m.a 8
75.l even 20 2 900.2.n.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.2.m.b 8 25.f odd 20 2
900.2.n.b 8 75.l even 20 2
1500.2.m.a 8 25.f odd 20 2
1500.2.o.b 16 25.d even 5 2
1500.2.o.b 16 25.e even 10 2
7500.2.a.e 4 5.c odd 4 1
7500.2.a.f 4 5.c odd 4 1
7500.2.d.c 8 1.a even 1 1 trivial
7500.2.d.c 8 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} + 24 T_{7}^{6} + 106 T_{7}^{4} + 129 T_{7}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(7500, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( 1 + T^{2} )^{4} \)
$5$ \( T^{8} \)
$7$ \( 1 + 129 T^{2} + 106 T^{4} + 24 T^{6} + T^{8} \)
$11$ \( ( -19 + 46 T - 24 T^{2} + T^{3} + T^{4} )^{2} \)
$13$ \( 2025 + 2700 T^{2} + 610 T^{4} + 45 T^{6} + T^{8} \)
$17$ \( 81 + 369 T^{2} + 166 T^{4} + 24 T^{6} + T^{8} \)
$19$ \( ( 5 + 15 T - 5 T^{2} - 5 T^{3} + T^{4} )^{2} \)
$23$ \( 29241 + 20664 T^{2} + 2806 T^{4} + 109 T^{6} + T^{8} \)
$29$ \( ( 1 - 9 T + 6 T^{2} + 6 T^{3} + T^{4} )^{2} \)
$31$ \( ( 981 + 354 T - 34 T^{2} - 11 T^{3} + T^{4} )^{2} \)
$37$ \( 1042441 + 139261 T^{2} + 6666 T^{4} + 136 T^{6} + T^{8} \)
$41$ \( ( 2705 - 160 T - 130 T^{2} + T^{4} )^{2} \)
$43$ \( 17161 + 9074 T^{2} + 1391 T^{4} + 74 T^{6} + T^{8} \)
$47$ \( 16801801 + 1205429 T^{2} + 28766 T^{4} + 284 T^{6} + T^{8} \)
$53$ \( 9801 + 16281 T^{2} + 6286 T^{4} + 156 T^{6} + T^{8} \)
$59$ \( ( 3701 + 211 T - 159 T^{2} + T^{3} + T^{4} )^{2} \)
$61$ \( ( -7909 - 2022 T + 4 T^{2} + 22 T^{3} + T^{4} )^{2} \)
$67$ \( 408321 + 2897334 T^{2} + 65371 T^{4} + 454 T^{6} + T^{8} \)
$71$ \( ( 5 - 215 T + 120 T^{2} - 20 T^{3} + T^{4} )^{2} \)
$73$ \( 121 + 2786 T^{2} + 2831 T^{4} + 266 T^{6} + T^{8} \)
$79$ \( ( -639 + 918 T - 146 T^{2} - 3 T^{3} + T^{4} )^{2} \)
$83$ \( 46908801 + 9083844 T^{2} + 134086 T^{4} + 644 T^{6} + T^{8} \)
$89$ \( ( -9875 + 3000 T - 150 T^{2} - 15 T^{3} + T^{4} )^{2} \)
$97$ \( 64304361 + 6505596 T^{2} + 101566 T^{4} + 556 T^{6} + T^{8} \)
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