# Properties

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

# 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: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 165) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{2} + ( 1 - \zeta_{8} - \zeta_{8}^{3} ) q^{3} + ( -2 \zeta_{8} - \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{4} + ( -2 \zeta_{8} - 2 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{6} + ( 2 - 2 \zeta_{8} + 2 \zeta_{8}^{2} ) q^{7} + ( -1 - 3 \zeta_{8} - \zeta_{8}^{2} ) q^{8} + ( -1 - 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{9} +O(q^{10})$$ $$q + ( 1 - \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{2} + ( 1 - \zeta_{8} - \zeta_{8}^{3} ) q^{3} + ( -2 \zeta_{8} - \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{4} + ( -2 \zeta_{8} - 2 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{6} + ( 2 - 2 \zeta_{8} + 2 \zeta_{8}^{2} ) q^{7} + ( -1 - 3 \zeta_{8} - \zeta_{8}^{2} ) q^{8} + ( -1 - 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{9} -\zeta_{8}^{2} q^{11} + ( -4 - 3 \zeta_{8} - \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{12} + ( 2 - 2 \zeta_{8}^{2} ) q^{13} + 2 q^{14} -3 q^{16} + 4 \zeta_{8}^{3} q^{17} + ( -3 - 4 \zeta_{8} - \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{18} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{19} + ( -2 \zeta_{8} + 4 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{21} + ( -1 - \zeta_{8} - \zeta_{8}^{2} ) q^{22} + ( 1 + 6 \zeta_{8} + \zeta_{8}^{2} ) q^{23} + ( -4 - 3 \zeta_{8} + 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{24} + ( -2 \zeta_{8} - 4 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{26} + ( -5 - \zeta_{8} - \zeta_{8}^{3} ) q^{27} + ( -2 + 2 \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{28} + ( -2 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{29} + ( -6 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{31} + ( -1 + \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{32} + ( -\zeta_{8} - \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{33} + ( 4 \zeta_{8} + 4 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{34} + ( -8 + \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{36} + ( 3 + 4 \zeta_{8} + 3 \zeta_{8}^{2} ) q^{37} + ( 2 + 4 \zeta_{8} + 2 \zeta_{8}^{2} ) q^{38} + ( 2 - 4 \zeta_{8} - 2 \zeta_{8}^{2} ) q^{39} + ( 4 \zeta_{8} - 2 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{41} + ( 2 - 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{42} + ( 4 - 4 \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{43} + ( -1 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{44} + ( 8 + 7 \zeta_{8} - 7 \zeta_{8}^{3} ) q^{46} + ( 3 - 3 \zeta_{8}^{2} + 6 \zeta_{8}^{3} ) q^{47} + ( -3 + 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{48} + ( -8 \zeta_{8} + 5 \zeta_{8}^{2} - 8 \zeta_{8}^{3} ) q^{49} + ( 4 + 4 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{51} + ( -2 - 8 \zeta_{8} - 2 \zeta_{8}^{2} ) q^{52} + ( -1 - \zeta_{8}^{2} ) q^{53} + ( -6 - 2 \zeta_{8} + 4 \zeta_{8}^{2} + 5 \zeta_{8}^{3} ) q^{54} + ( -4 \zeta_{8} + 2 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{56} + ( 4 + 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{57} + ( -4 + 4 \zeta_{8}^{2} + 6 \zeta_{8}^{3} ) q^{58} -4 q^{59} + ( 2 + 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{61} + ( -8 + 8 \zeta_{8}^{2} + 10 \zeta_{8}^{3} ) q^{62} + ( -6 + 2 \zeta_{8} + 2 \zeta_{8}^{2} - 8 \zeta_{8}^{3} ) q^{63} + ( -2 \zeta_{8} - 7 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{64} + ( -2 - \zeta_{8} + 2 \zeta_{8}^{3} ) q^{66} + ( 5 - 2 \zeta_{8} + 5 \zeta_{8}^{2} ) q^{67} + ( 8 + 4 \zeta_{8} + 8 \zeta_{8}^{2} ) q^{68} + ( 7 + 6 \zeta_{8} - 5 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{69} + ( 6 \zeta_{8} + 6 \zeta_{8}^{2} + 6 \zeta_{8}^{3} ) q^{71} + ( -5 + 3 \zeta_{8} + 7 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{72} + ( 10 + 7 \zeta_{8} - 7 \zeta_{8}^{3} ) q^{74} + ( 8 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{76} + ( 2 - 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{77} + ( -4 - 6 \zeta_{8} - 4 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{78} + ( -2 \zeta_{8} + 8 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{79} + ( -7 + 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{81} + ( 2 + 6 \zeta_{8} + 2 \zeta_{8}^{2} ) q^{82} + ( 2 - 10 \zeta_{8} + 2 \zeta_{8}^{2} ) q^{83} + ( -8 + 4 \zeta_{8} - 4 \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{84} + ( -10 \zeta_{8} - 14 \zeta_{8}^{2} - 10 \zeta_{8}^{3} ) q^{86} + ( -2 + 4 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{87} + ( -1 + \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{88} + ( -4 + 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{89} + ( 8 - 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{91} + ( 13 - 13 \zeta_{8}^{2} - 10 \zeta_{8}^{3} ) q^{92} + ( -6 + 4 \zeta_{8} + 4 \zeta_{8}^{2} + 8 \zeta_{8}^{3} ) q^{93} + ( 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{94} + ( -4 + 2 \zeta_{8} - 2 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{96} + ( -5 + 8 \zeta_{8} - 5 \zeta_{8}^{2} ) q^{97} + ( -3 - 11 \zeta_{8} - 3 \zeta_{8}^{2} ) q^{98} + ( -2 \zeta_{8} + \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{2} + 4 q^{3} + 8 q^{7} - 4 q^{8} - 4 q^{9} + O(q^{10})$$ $$4 q + 4 q^{2} + 4 q^{3} + 8 q^{7} - 4 q^{8} - 4 q^{9} - 16 q^{12} + 8 q^{13} + 8 q^{14} - 12 q^{16} - 12 q^{18} - 4 q^{22} + 4 q^{23} - 16 q^{24} - 20 q^{27} - 8 q^{28} - 8 q^{29} - 24 q^{31} - 4 q^{32} - 32 q^{36} + 12 q^{37} + 8 q^{38} + 8 q^{39} + 8 q^{42} + 16 q^{43} - 4 q^{44} + 32 q^{46} + 12 q^{47} - 12 q^{48} + 16 q^{51} - 8 q^{52} - 4 q^{53} - 24 q^{54} + 16 q^{57} - 16 q^{58} - 16 q^{59} + 8 q^{61} - 32 q^{62} - 24 q^{63} - 8 q^{66} + 20 q^{67} + 32 q^{68} + 28 q^{69} - 20 q^{72} + 40 q^{74} + 32 q^{76} + 8 q^{77} - 16 q^{78} - 28 q^{81} + 8 q^{82} + 8 q^{83} - 32 q^{84} - 8 q^{87} - 4 q^{88} - 16 q^{89} + 32 q^{91} + 52 q^{92} - 24 q^{93} - 16 q^{96} - 20 q^{97} - 12 q^{98} + O(q^{100})$$

## Character values

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

 $$n$$ $$376$$ $$551$$ $$727$$ $$\chi(n)$$ $$1$$ $$-1$$ $$\zeta_{8}^{2}$$

## 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
 −0.707107 + 0.707107i 0.707107 − 0.707107i −0.707107 − 0.707107i 0.707107 + 0.707107i
0.292893 + 0.292893i 1.00000 1.41421i 1.82843i 0 0.707107 0.121320i 3.41421 3.41421i 1.12132 1.12132i −1.00000 2.82843i 0
518.2 1.70711 + 1.70711i 1.00000 + 1.41421i 3.82843i 0 −0.707107 + 4.12132i 0.585786 0.585786i −3.12132 + 3.12132i −1.00000 + 2.82843i 0
782.1 0.292893 0.292893i 1.00000 + 1.41421i 1.82843i 0 0.707107 + 0.121320i 3.41421 + 3.41421i 1.12132 + 1.12132i −1.00000 + 2.82843i 0
782.2 1.70711 1.70711i 1.00000 1.41421i 3.82843i 0 −0.707107 4.12132i 0.585786 + 0.585786i −3.12132 3.12132i −1.00000 2.82843i 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
15.e even 4 1 inner

## Twists

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

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

## Hecke kernels

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

 $$T_{2}^{4} - 4 T_{2}^{3} + 8 T_{2}^{2} - 4 T_{2} + 1$$ $$T_{7}^{4} - 8 T_{7}^{3} + 32 T_{7}^{2} - 32 T_{7} + 16$$ $$T_{13}^{2} - 4 T_{13} + 8$$ $$T_{29}^{2} + 4 T_{29} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 4 T + 8 T^{2} - 4 T^{3} + T^{4}$$
$3$ $$( 3 - 2 T + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$16 - 32 T + 32 T^{2} - 8 T^{3} + T^{4}$$
$11$ $$( 1 + T^{2} )^{2}$$
$13$ $$( 8 - 4 T + T^{2} )^{2}$$
$17$ $$256 + T^{4}$$
$19$ $$( 8 + T^{2} )^{2}$$
$23$ $$1156 + 136 T + 8 T^{2} - 4 T^{3} + T^{4}$$
$29$ $$( -4 + 4 T + T^{2} )^{2}$$
$31$ $$( 28 + 12 T + T^{2} )^{2}$$
$37$ $$4 - 24 T + 72 T^{2} - 12 T^{3} + T^{4}$$
$41$ $$784 + 72 T^{2} + T^{4}$$
$43$ $$16 + 64 T + 128 T^{2} - 16 T^{3} + T^{4}$$
$47$ $$324 + 216 T + 72 T^{2} - 12 T^{3} + T^{4}$$
$53$ $$( 2 + 2 T + T^{2} )^{2}$$
$59$ $$( 4 + T )^{4}$$
$61$ $$( -68 - 4 T + T^{2} )^{2}$$
$67$ $$2116 - 920 T + 200 T^{2} - 20 T^{3} + T^{4}$$
$71$ $$1296 + 216 T^{2} + T^{4}$$
$73$ $$T^{4}$$
$79$ $$3136 + 144 T^{2} + T^{4}$$
$83$ $$8464 + 736 T + 32 T^{2} - 8 T^{3} + T^{4}$$
$89$ $$( -16 + 8 T + T^{2} )^{2}$$
$97$ $$196 - 280 T + 200 T^{2} + 20 T^{3} + T^{4}$$