# Properties

 Label 825.2.k.d Level $825$ Weight $2$ Character orbit 825.k Analytic conductor $6.588$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$825 = 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 825.k (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.58765816676$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(i, \sqrt{6})$$ Defining polynomial: $$x^{4} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} -\beta_{1} q^{3} + \beta_{2} q^{4} -3 \beta_{2} q^{6} + 2 \beta_{3} q^{7} -\beta_{3} q^{8} + 3 \beta_{2} q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} -\beta_{1} q^{3} + \beta_{2} q^{4} -3 \beta_{2} q^{6} + 2 \beta_{3} q^{7} -\beta_{3} q^{8} + 3 \beta_{2} q^{9} -\beta_{2} q^{11} -\beta_{3} q^{12} + 2 \beta_{1} q^{13} -6 q^{14} + 5 q^{16} + 4 \beta_{1} q^{17} + 3 \beta_{3} q^{18} -2 \beta_{2} q^{19} + 6 q^{21} -\beta_{3} q^{22} + 4 \beta_{3} q^{23} -3 q^{24} + 6 \beta_{2} q^{26} -3 \beta_{3} q^{27} -2 \beta_{1} q^{28} + 6 q^{29} + 4 q^{31} + 3 \beta_{1} q^{32} + \beta_{3} q^{33} + 12 \beta_{2} q^{34} -3 q^{36} -2 \beta_{3} q^{38} -6 \beta_{2} q^{39} + 6 \beta_{2} q^{41} + 6 \beta_{1} q^{42} + 6 \beta_{1} q^{43} + q^{44} -12 q^{46} -4 \beta_{1} q^{47} -5 \beta_{1} q^{48} -5 \beta_{2} q^{49} -12 \beta_{2} q^{51} + 2 \beta_{3} q^{52} + 4 \beta_{3} q^{53} + 9 q^{54} + 6 \beta_{2} q^{56} + 2 \beta_{3} q^{57} + 6 \beta_{1} q^{58} + 2 q^{61} + 4 \beta_{1} q^{62} -6 \beta_{1} q^{63} -\beta_{2} q^{64} -3 q^{66} + 2 \beta_{3} q^{67} + 4 \beta_{3} q^{68} + 12 q^{69} -12 \beta_{2} q^{71} + 3 \beta_{1} q^{72} -2 \beta_{1} q^{73} + 2 q^{76} + 2 \beta_{1} q^{77} -6 \beta_{3} q^{78} -10 \beta_{2} q^{79} -9 q^{81} + 6 \beta_{3} q^{82} + 6 \beta_{3} q^{83} + 6 \beta_{2} q^{84} + 18 \beta_{2} q^{86} -6 \beta_{1} q^{87} -\beta_{1} q^{88} -12 q^{89} -12 q^{91} -4 \beta_{1} q^{92} -4 \beta_{1} q^{93} -12 \beta_{2} q^{94} -9 \beta_{2} q^{96} -8 \beta_{3} q^{97} -5 \beta_{3} q^{98} + 3 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + O(q^{10})$$ $$4 q - 24 q^{14} + 20 q^{16} + 24 q^{21} - 12 q^{24} + 24 q^{29} + 16 q^{31} - 12 q^{36} + 4 q^{44} - 48 q^{46} + 36 q^{54} + 8 q^{61} - 12 q^{66} + 48 q^{69} + 8 q^{76} - 36 q^{81} - 48 q^{89} - 48 q^{91} + 12 q^{99} + O(q^{100})$$

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

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

## Character values

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

 $$n$$ $$376$$ $$551$$ $$727$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
518.1
 −1.22474 − 1.22474i 1.22474 + 1.22474i −1.22474 + 1.22474i 1.22474 − 1.22474i
−1.22474 1.22474i 1.22474 + 1.22474i 1.00000i 0 3.00000i 2.44949 2.44949i −1.22474 + 1.22474i 3.00000i 0
518.2 1.22474 + 1.22474i −1.22474 1.22474i 1.00000i 0 3.00000i −2.44949 + 2.44949i 1.22474 1.22474i 3.00000i 0
782.1 −1.22474 + 1.22474i 1.22474 1.22474i 1.00000i 0 3.00000i 2.44949 + 2.44949i −1.22474 1.22474i 3.00000i 0
782.2 1.22474 1.22474i −1.22474 + 1.22474i 1.00000i 0 3.00000i −2.44949 2.44949i 1.22474 + 1.22474i 3.00000i 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
5.b even 2 1 inner
15.e even 4 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 825.2.k.d 4
3.b odd 2 1 825.2.k.e yes 4
5.b even 2 1 inner 825.2.k.d 4
5.c odd 4 2 825.2.k.e yes 4
15.d odd 2 1 825.2.k.e yes 4
15.e even 4 2 inner 825.2.k.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.2.k.d 4 1.a even 1 1 trivial
825.2.k.d 4 5.b even 2 1 inner
825.2.k.d 4 15.e even 4 2 inner
825.2.k.e yes 4 3.b odd 2 1
825.2.k.e yes 4 5.c odd 4 2
825.2.k.e yes 4 15.d odd 2 1

## Hecke kernels

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

 $$T_{2}^{4} + 9$$ $$T_{7}^{4} + 144$$ $$T_{13}^{4} + 144$$ $$T_{29} - 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$9 + T^{4}$$
$3$ $$9 + T^{4}$$
$5$ $$T^{4}$$
$7$ $$144 + T^{4}$$
$11$ $$( 1 + T^{2} )^{2}$$
$13$ $$144 + T^{4}$$
$17$ $$2304 + T^{4}$$
$19$ $$( 4 + T^{2} )^{2}$$
$23$ $$2304 + T^{4}$$
$29$ $$( -6 + T )^{4}$$
$31$ $$( -4 + T )^{4}$$
$37$ $$T^{4}$$
$41$ $$( 36 + T^{2} )^{2}$$
$43$ $$11664 + T^{4}$$
$47$ $$2304 + T^{4}$$
$53$ $$2304 + T^{4}$$
$59$ $$T^{4}$$
$61$ $$( -2 + T )^{4}$$
$67$ $$144 + T^{4}$$
$71$ $$( 144 + T^{2} )^{2}$$
$73$ $$144 + T^{4}$$
$79$ $$( 100 + T^{2} )^{2}$$
$83$ $$11664 + T^{4}$$
$89$ $$( 12 + T )^{4}$$
$97$ $$36864 + T^{4}$$