# Properties

 Label 825.2.c.c Level $825$ Weight $2$ Character orbit 825.c 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.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.58765816676$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 165) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + 2 \zeta_{12}^{2} ) q^{2} + \zeta_{12}^{3} q^{3} - q^{4} + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{6} -2 \zeta_{12}^{3} q^{7} + ( -1 + 2 \zeta_{12}^{2} ) q^{8} - q^{9} +O(q^{10})$$ $$q + ( -1 + 2 \zeta_{12}^{2} ) q^{2} + \zeta_{12}^{3} q^{3} - q^{4} + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{6} -2 \zeta_{12}^{3} q^{7} + ( -1 + 2 \zeta_{12}^{2} ) q^{8} - q^{9} - q^{11} -\zeta_{12}^{3} q^{12} + ( -2 + 4 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{13} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{14} -5 q^{16} + ( 1 - 2 \zeta_{12}^{2} ) q^{18} + ( -2 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{19} + 2 q^{21} + ( 1 - 2 \zeta_{12}^{2} ) q^{22} + ( -4 + 8 \zeta_{12}^{2} ) q^{23} + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{24} + ( -6 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{26} -\zeta_{12}^{3} q^{27} + 2 \zeta_{12}^{3} q^{28} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{29} + ( -4 - 8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{31} + ( 3 - 6 \zeta_{12}^{2} ) q^{32} -\zeta_{12}^{3} q^{33} + q^{36} + ( -4 + 8 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{37} + ( 2 - 4 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{38} + ( -2 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{39} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{41} + ( -2 + 4 \zeta_{12}^{2} ) q^{42} + ( 4 - 8 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{43} + q^{44} -12 q^{46} + ( -4 + 8 \zeta_{12}^{2} ) q^{47} -5 \zeta_{12}^{3} q^{48} + 3 q^{49} + ( 2 - 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{52} + ( -4 + 8 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{53} + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{54} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{56} + ( 2 - 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{57} + 6 \zeta_{12}^{3} q^{58} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{59} + 2 q^{61} + ( 4 - 8 \zeta_{12}^{2} - 12 \zeta_{12}^{3} ) q^{62} + 2 \zeta_{12}^{3} q^{63} - q^{64} + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{66} -8 \zeta_{12}^{3} q^{67} + ( -8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{69} + ( 16 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{71} + ( 1 - 2 \zeta_{12}^{2} ) q^{72} + ( 6 - 12 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{73} + ( -12 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{74} + ( 2 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{76} + 2 \zeta_{12}^{3} q^{77} + ( 2 - 4 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{78} + ( 10 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{79} + q^{81} + 6 \zeta_{12}^{3} q^{82} + ( 2 - 4 \zeta_{12}^{2} + 12 \zeta_{12}^{3} ) q^{83} -2 q^{84} + ( 12 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{86} + ( -2 + 4 \zeta_{12}^{2} ) q^{87} + ( 1 - 2 \zeta_{12}^{2} ) q^{88} + ( 6 - 8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{89} + ( 4 + 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{91} + ( 4 - 8 \zeta_{12}^{2} ) q^{92} + ( 4 - 8 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{93} -12 q^{94} + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{96} + 10 \zeta_{12}^{3} q^{97} + ( -3 + 6 \zeta_{12}^{2} ) q^{98} + q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{4} - 4q^{9} + O(q^{10})$$ $$4q - 4q^{4} - 4q^{9} - 4q^{11} - 20q^{16} - 8q^{19} + 8q^{21} - 24q^{26} - 16q^{31} + 4q^{36} - 8q^{39} + 4q^{44} - 48q^{46} + 12q^{49} + 8q^{61} - 4q^{64} - 48q^{74} + 8q^{76} + 40q^{79} + 4q^{81} - 8q^{84} + 48q^{86} + 24q^{89} + 16q^{91} - 48q^{94} + 4q^{99} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
199.1
 0.866025 − 0.500000i −0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i
1.73205i 1.00000i −1.00000 0 −1.73205 2.00000i 1.73205i −1.00000 0
199.2 1.73205i 1.00000i −1.00000 0 1.73205 2.00000i 1.73205i −1.00000 0
199.3 1.73205i 1.00000i −1.00000 0 1.73205 2.00000i 1.73205i −1.00000 0
199.4 1.73205i 1.00000i −1.00000 0 −1.73205 2.00000i 1.73205i −1.00000 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 825.2.c.c 4
3.b odd 2 1 2475.2.c.n 4
5.b even 2 1 inner 825.2.c.c 4
5.c odd 4 1 165.2.a.b 2
5.c odd 4 1 825.2.a.e 2
15.d odd 2 1 2475.2.c.n 4
15.e even 4 1 495.2.a.c 2
15.e even 4 1 2475.2.a.r 2
20.e even 4 1 2640.2.a.x 2
35.f even 4 1 8085.2.a.bd 2
55.e even 4 1 1815.2.a.i 2
55.e even 4 1 9075.2.a.bh 2
60.l odd 4 1 7920.2.a.bz 2
165.l odd 4 1 5445.2.a.s 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.a.b 2 5.c odd 4 1
495.2.a.c 2 15.e even 4 1
825.2.a.e 2 5.c odd 4 1
825.2.c.c 4 1.a even 1 1 trivial
825.2.c.c 4 5.b even 2 1 inner
1815.2.a.i 2 55.e even 4 1
2475.2.a.r 2 15.e even 4 1
2475.2.c.n 4 3.b odd 2 1
2475.2.c.n 4 15.d odd 2 1
2640.2.a.x 2 20.e even 4 1
5445.2.a.s 2 165.l odd 4 1
7920.2.a.bz 2 60.l odd 4 1
8085.2.a.bd 2 35.f even 4 1
9075.2.a.bh 2 55.e even 4 1

## Hecke kernels

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

 $$T_{2}^{2} + 3$$ $$T_{7}^{2} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 3 + T^{2} )^{2}$$
$3$ $$( 1 + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$( 4 + T^{2} )^{2}$$
$11$ $$( 1 + T )^{4}$$
$13$ $$64 + 32 T^{2} + T^{4}$$
$17$ $$T^{4}$$
$19$ $$( -8 + 4 T + T^{2} )^{2}$$
$23$ $$( 48 + T^{2} )^{2}$$
$29$ $$( -12 + T^{2} )^{2}$$
$31$ $$( -32 + 8 T + T^{2} )^{2}$$
$37$ $$1936 + 104 T^{2} + T^{4}$$
$41$ $$( -12 + T^{2} )^{2}$$
$43$ $$1936 + 104 T^{2} + T^{4}$$
$47$ $$( 48 + T^{2} )^{2}$$
$53$ $$144 + 168 T^{2} + T^{4}$$
$59$ $$( -48 + T^{2} )^{2}$$
$61$ $$( -2 + T )^{4}$$
$67$ $$( 64 + T^{2} )^{2}$$
$71$ $$( -192 + T^{2} )^{2}$$
$73$ $$10816 + 224 T^{2} + T^{4}$$
$79$ $$( 88 - 20 T + T^{2} )^{2}$$
$83$ $$17424 + 312 T^{2} + T^{4}$$
$89$ $$( -12 - 12 T + T^{2} )^{2}$$
$97$ $$( 100 + T^{2} )^{2}$$