# Properties

 Label 825.2.c.e 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: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ 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 $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{2} -\zeta_{8}^{2} q^{3} + ( -1 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{4} + ( 1 + \zeta_{8} - \zeta_{8}^{3} ) q^{6} + ( -2 \zeta_{8} + 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{7} + ( -\zeta_{8} - 3 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{8} - q^{9} +O(q^{10})$$ $$q + ( \zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{2} -\zeta_{8}^{2} q^{3} + ( -1 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{4} + ( 1 + \zeta_{8} - \zeta_{8}^{3} ) q^{6} + ( -2 \zeta_{8} + 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{7} + ( -\zeta_{8} - 3 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{8} - q^{9} - q^{11} + ( 2 \zeta_{8} + \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{12} + ( -4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{13} + 2 q^{14} + 3 q^{16} + ( -2 \zeta_{8} + 4 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{17} + ( -\zeta_{8} - \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{18} + ( 4 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{19} + ( 2 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{21} + ( -\zeta_{8} - \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{22} -4 \zeta_{8}^{2} q^{23} + ( -3 - \zeta_{8} + \zeta_{8}^{3} ) q^{24} + ( 8 + 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{26} + \zeta_{8}^{2} q^{27} + ( -2 \zeta_{8} + 6 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{28} + ( 2 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{29} + ( \zeta_{8} - 3 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{32} + \zeta_{8}^{2} q^{33} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{34} + ( 1 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{36} + ( -4 \zeta_{8} - 6 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{37} + ( 6 \zeta_{8} + 8 \zeta_{8}^{2} + 6 \zeta_{8}^{3} ) q^{38} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{39} + ( 2 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{41} -2 \zeta_{8}^{2} q^{42} + ( -2 \zeta_{8} - 6 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{43} + ( 1 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{44} + ( 4 + 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{46} + 4 \zeta_{8}^{2} q^{47} -3 \zeta_{8}^{2} q^{48} + ( -5 + 8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{49} + ( 4 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{51} + ( 4 \zeta_{8} + 16 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{52} + ( 8 \zeta_{8} - 2 \zeta_{8}^{2} + 8 \zeta_{8}^{3} ) q^{53} + ( -1 - \zeta_{8} + \zeta_{8}^{3} ) q^{54} + ( 2 - 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{56} + ( -2 \zeta_{8} - 4 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{57} + ( 4 \zeta_{8} + 6 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{58} + 4 q^{59} + ( -6 - 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{61} + ( 2 \zeta_{8} - 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{63} + ( 7 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{64} + ( -1 - \zeta_{8} + \zeta_{8}^{3} ) q^{66} + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{67} + ( -6 \zeta_{8} + 4 \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{68} -4 q^{69} + ( 8 - 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{71} + ( \zeta_{8} + 3 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{72} + ( 8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{73} + ( 14 + 10 \zeta_{8} - 10 \zeta_{8}^{3} ) q^{74} + ( -12 - 10 \zeta_{8} + 10 \zeta_{8}^{3} ) q^{76} + ( 2 \zeta_{8} - 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{77} + ( -4 \zeta_{8} - 8 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{78} + ( -6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{79} + q^{81} + ( 4 \zeta_{8} + 6 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{82} -10 \zeta_{8}^{2} q^{83} + ( 6 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{84} + ( 10 + 8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{86} + ( -2 \zeta_{8} - 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{87} + ( \zeta_{8} + 3 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{88} + ( 2 - 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{89} + ( -16 + 8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{91} + ( 8 \zeta_{8} + 4 \zeta_{8}^{2} + 8 \zeta_{8}^{3} ) q^{92} + ( -4 - 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{94} + ( -3 + \zeta_{8} - \zeta_{8}^{3} ) q^{96} + ( -4 \zeta_{8} - 6 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{97} + ( 3 \zeta_{8} + 11 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{98} + q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{4} + 4q^{6} - 4q^{9} + O(q^{10})$$ $$4q - 4q^{4} + 4q^{6} - 4q^{9} - 4q^{11} + 8q^{14} + 12q^{16} + 16q^{19} + 8q^{21} - 12q^{24} + 32q^{26} + 8q^{29} + 4q^{36} + 8q^{41} + 4q^{44} + 16q^{46} - 20q^{49} + 16q^{51} - 4q^{54} + 8q^{56} + 16q^{59} - 24q^{61} + 28q^{64} - 4q^{66} - 16q^{69} + 32q^{71} + 56q^{74} - 48q^{76} + 4q^{81} + 24q^{84} + 40q^{86} + 8q^{89} - 64q^{91} - 16q^{94} - 12q^{96} + 4q^{99} + O(q^{100})$$

## 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
 0.707107 − 0.707107i −0.707107 − 0.707107i −0.707107 + 0.707107i 0.707107 + 0.707107i
2.41421i 1.00000i −3.82843 0 2.41421 0.828427i 4.41421i −1.00000 0
199.2 0.414214i 1.00000i 1.82843 0 −0.414214 4.82843i 1.58579i −1.00000 0
199.3 0.414214i 1.00000i 1.82843 0 −0.414214 4.82843i 1.58579i −1.00000 0
199.4 2.41421i 1.00000i −3.82843 0 2.41421 0.828427i 4.41421i −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.e 4
3.b odd 2 1 2475.2.c.m 4
5.b even 2 1 inner 825.2.c.e 4
5.c odd 4 1 165.2.a.a 2
5.c odd 4 1 825.2.a.g 2
15.d odd 2 1 2475.2.c.m 4
15.e even 4 1 495.2.a.d 2
15.e even 4 1 2475.2.a.m 2
20.e even 4 1 2640.2.a.bb 2
35.f even 4 1 8085.2.a.ba 2
55.e even 4 1 1815.2.a.k 2
55.e even 4 1 9075.2.a.v 2
60.l odd 4 1 7920.2.a.cg 2
165.l odd 4 1 5445.2.a.m 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.a.a 2 5.c odd 4 1
495.2.a.d 2 15.e even 4 1
825.2.a.g 2 5.c odd 4 1
825.2.c.e 4 1.a even 1 1 trivial
825.2.c.e 4 5.b even 2 1 inner
1815.2.a.k 2 55.e even 4 1
2475.2.a.m 2 15.e even 4 1
2475.2.c.m 4 3.b odd 2 1
2475.2.c.m 4 15.d odd 2 1
2640.2.a.bb 2 20.e even 4 1
5445.2.a.m 2 165.l odd 4 1
7920.2.a.cg 2 60.l odd 4 1
8085.2.a.ba 2 35.f even 4 1
9075.2.a.v 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 $$S_{2}^{\mathrm{new}}(825, [\chi])$$:

 $$T_{2}^{4} + 6 T_{2}^{2} + 1$$ $$T_{7}^{4} + 24 T_{7}^{2} + 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 6 T^{2} + T^{4}$$
$3$ $$( 1 + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$16 + 24 T^{2} + T^{4}$$
$11$ $$( 1 + T )^{4}$$
$13$ $$( 32 + T^{2} )^{2}$$
$17$ $$64 + 48 T^{2} + T^{4}$$
$19$ $$( 8 - 8 T + T^{2} )^{2}$$
$23$ $$( 16 + T^{2} )^{2}$$
$29$ $$( -4 - 4 T + T^{2} )^{2}$$
$31$ $$T^{4}$$
$37$ $$16 + 136 T^{2} + T^{4}$$
$41$ $$( -4 - 4 T + T^{2} )^{2}$$
$43$ $$784 + 88 T^{2} + T^{4}$$
$47$ $$( 16 + T^{2} )^{2}$$
$53$ $$15376 + 264 T^{2} + T^{4}$$
$59$ $$( -4 + T )^{4}$$
$61$ $$( 4 + 12 T + T^{2} )^{2}$$
$67$ $$( 32 + T^{2} )^{2}$$
$71$ $$( 32 - 16 T + T^{2} )^{2}$$
$73$ $$( 128 + T^{2} )^{2}$$
$79$ $$( -72 + T^{2} )^{2}$$
$83$ $$( 100 + T^{2} )^{2}$$
$89$ $$( -28 - 4 T + T^{2} )^{2}$$
$97$ $$16 + 136 T^{2} + T^{4}$$