# 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 $2$

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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

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