# Properties

 Label 825.2.f.b Level $825$ Weight $2$ Character orbit 825.f Analytic conductor $6.588$ Analytic rank $0$ Dimension $4$ CM discriminant -11 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$825 = 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 825.f (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.58765816676$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{11})$$ Defining polynomial: $$x^{4} - 5 x^{2} + 9$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 165) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{3} -2 q^{4} + ( 3 + \beta_{2} ) q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{3} -2 q^{4} + ( 3 + \beta_{2} ) q^{9} + ( 1 + 2 \beta_{2} ) q^{11} + 2 \beta_{1} q^{12} + 4 q^{16} + ( -3 \beta_{1} - 3 \beta_{3} ) q^{23} + ( -3 \beta_{1} - \beta_{3} ) q^{27} -5 q^{31} + ( -\beta_{1} - 2 \beta_{3} ) q^{33} + ( -6 - 2 \beta_{2} ) q^{36} + ( -5 \beta_{1} + \beta_{3} ) q^{37} + ( -2 - 4 \beta_{2} ) q^{44} + ( -4 \beta_{1} - 4 \beta_{3} ) q^{47} -4 \beta_{1} q^{48} + 7 q^{49} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{53} + ( -1 - 2 \beta_{2} ) q^{59} -8 q^{64} + ( 5 \beta_{1} - \beta_{3} ) q^{67} + 9 \beta_{2} q^{69} + ( -5 - 10 \beta_{2} ) q^{71} + ( 6 + 5 \beta_{2} ) q^{81} + ( -5 - 10 \beta_{2} ) q^{89} + ( 6 \beta_{1} + 6 \beta_{3} ) q^{92} + 5 \beta_{1} q^{93} + ( 5 \beta_{1} - \beta_{3} ) q^{97} + ( -3 + 5 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 8 q^{4} + 10 q^{9} + O(q^{10})$$ $$4 q - 8 q^{4} + 10 q^{9} + 16 q^{16} - 20 q^{31} - 20 q^{36} + 28 q^{49} - 32 q^{64} - 18 q^{69} + 14 q^{81} - 22 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 5 x^{2} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - 3 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + 3 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/825\mathbb{Z}\right)^\times$$.

 $$n$$ $$376$$ $$551$$ $$727$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
626.1
 1.65831 + 0.500000i 1.65831 − 0.500000i −1.65831 + 0.500000i −1.65831 − 0.500000i
0 −1.65831 0.500000i −2.00000 0 0 0 0 2.50000 + 1.65831i 0
626.2 0 −1.65831 + 0.500000i −2.00000 0 0 0 0 2.50000 1.65831i 0
626.3 0 1.65831 0.500000i −2.00000 0 0 0 0 2.50000 1.65831i 0
626.4 0 1.65831 + 0.500000i −2.00000 0 0 0 0 2.50000 + 1.65831i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 CM by $$\Q(\sqrt{-11})$$
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner
33.d even 2 1 inner
55.d odd 2 1 inner
165.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 825.2.f.b 4
3.b odd 2 1 inner 825.2.f.b 4
5.b even 2 1 inner 825.2.f.b 4
5.c odd 4 1 165.2.d.a 2
5.c odd 4 1 165.2.d.b yes 2
11.b odd 2 1 CM 825.2.f.b 4
15.d odd 2 1 inner 825.2.f.b 4
15.e even 4 1 165.2.d.a 2
15.e even 4 1 165.2.d.b yes 2
33.d even 2 1 inner 825.2.f.b 4
55.d odd 2 1 inner 825.2.f.b 4
55.e even 4 1 165.2.d.a 2
55.e even 4 1 165.2.d.b yes 2
165.d even 2 1 inner 825.2.f.b 4
165.l odd 4 1 165.2.d.a 2
165.l odd 4 1 165.2.d.b yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.d.a 2 5.c odd 4 1
165.2.d.a 2 15.e even 4 1
165.2.d.a 2 55.e even 4 1
165.2.d.a 2 165.l odd 4 1
165.2.d.b yes 2 5.c odd 4 1
165.2.d.b yes 2 15.e even 4 1
165.2.d.b yes 2 55.e even 4 1
165.2.d.b yes 2 165.l odd 4 1
825.2.f.b 4 1.a even 1 1 trivial
825.2.f.b 4 3.b odd 2 1 inner
825.2.f.b 4 5.b even 2 1 inner
825.2.f.b 4 11.b odd 2 1 CM
825.2.f.b 4 15.d odd 2 1 inner
825.2.f.b 4 33.d even 2 1 inner
825.2.f.b 4 55.d odd 2 1 inner
825.2.f.b 4 165.d even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(825, [\chi])$$:

 $$T_{2}$$ $$T_{23}^{2} + 81$$ $$T_{37}^{2} - 99$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$9 - 5 T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$( 11 + T^{2} )^{2}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$( 81 + T^{2} )^{2}$$
$29$ $$T^{4}$$
$31$ $$( 5 + T )^{4}$$
$37$ $$( -99 + T^{2} )^{2}$$
$41$ $$T^{4}$$
$43$ $$T^{4}$$
$47$ $$( 144 + T^{2} )^{2}$$
$53$ $$( 36 + T^{2} )^{2}$$
$59$ $$( 11 + T^{2} )^{2}$$
$61$ $$T^{4}$$
$67$ $$( -99 + T^{2} )^{2}$$
$71$ $$( 275 + T^{2} )^{2}$$
$73$ $$T^{4}$$
$79$ $$T^{4}$$
$83$ $$T^{4}$$
$89$ $$( 275 + T^{2} )^{2}$$
$97$ $$( -99 + T^{2} )^{2}$$