# Properties

 Label 825.2.c.g Level $825$ Weight $2$ Character orbit 825.c Analytic conductor $6.588$ Analytic rank $0$ Dimension $6$ 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: $$6$$ Coefficient field: 6.0.350464.1 Defining polynomial: $$x^{6} - 2 x^{5} + 2 x^{4} + 2 x^{3} + 4 x^{2} - 4 x + 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{3}$$ 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 basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + \beta_{3} q^{3} + ( -1 + \beta_{2} + \beta_{4} ) q^{4} + \beta_{2} q^{6} + ( \beta_{1} + \beta_{5} ) q^{7} + ( 2 \beta_{1} - 2 \beta_{3} - \beta_{5} ) q^{8} - q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{2} + \beta_{3} q^{3} + ( -1 + \beta_{2} + \beta_{4} ) q^{4} + \beta_{2} q^{6} + ( \beta_{1} + \beta_{5} ) q^{7} + ( 2 \beta_{1} - 2 \beta_{3} - \beta_{5} ) q^{8} - q^{9} + q^{11} + ( \beta_{1} - \beta_{3} - \beta_{5} ) q^{12} + ( \beta_{1} - \beta_{5} ) q^{13} + ( 4 + \beta_{2} - \beta_{4} ) q^{14} + ( 3 - 4 \beta_{2} ) q^{16} + ( 3 \beta_{1} + 2 \beta_{3} - \beta_{5} ) q^{17} + \beta_{1} q^{18} + ( -2 + 2 \beta_{4} ) q^{19} + ( -\beta_{2} + \beta_{4} ) q^{21} -\beta_{1} q^{22} + ( 2 \beta_{1} + 2 \beta_{5} ) q^{23} + ( 2 - 2 \beta_{2} - \beta_{4} ) q^{24} + ( 2 - 3 \beta_{2} - \beta_{4} ) q^{26} -\beta_{3} q^{27} + ( -3 \beta_{1} - 4 \beta_{3} + \beta_{5} ) q^{28} + ( 4 + 2 \beta_{2} ) q^{29} + ( 4 + 2 \beta_{2} + 2 \beta_{4} ) q^{31} + ( -3 \beta_{1} + 8 \beta_{3} + 2 \beta_{5} ) q^{32} + \beta_{3} q^{33} + ( 8 - 3 \beta_{2} - 3 \beta_{4} ) q^{34} + ( 1 - \beta_{2} - \beta_{4} ) q^{36} + 2 \beta_{3} q^{37} + ( 6 \beta_{1} + 2 \beta_{3} ) q^{38} + ( -\beta_{2} - \beta_{4} ) q^{39} + ( -4 + 2 \beta_{2} ) q^{41} + ( \beta_{1} + 4 \beta_{3} + \beta_{5} ) q^{42} + ( -\beta_{1} + 3 \beta_{5} ) q^{43} + ( -1 + \beta_{2} + \beta_{4} ) q^{44} + ( 8 + 2 \beta_{2} - 2 \beta_{4} ) q^{46} + ( 2 \beta_{1} + 4 \beta_{3} - 2 \beta_{5} ) q^{47} + ( -4 \beta_{1} + 3 \beta_{3} ) q^{48} + ( -5 - 4 \beta_{2} ) q^{49} + ( -2 - 3 \beta_{2} - \beta_{4} ) q^{51} + ( -5 \beta_{1} + 8 \beta_{3} + \beta_{5} ) q^{52} + ( -2 \beta_{1} - 2 \beta_{3} - 2 \beta_{5} ) q^{53} -\beta_{2} q^{54} + ( 3 \beta_{2} + \beta_{4} ) q^{56} + ( -2 \beta_{3} - 2 \beta_{5} ) q^{57} + ( -2 \beta_{1} - 6 \beta_{3} - 2 \beta_{5} ) q^{58} + ( -4 + 2 \beta_{2} - 2 \beta_{4} ) q^{59} + ( -2 + 2 \beta_{2} - 2 \beta_{4} ) q^{61} + ( 2 \beta_{1} - 4 \beta_{3} - 2 \beta_{5} ) q^{62} + ( -\beta_{1} - \beta_{5} ) q^{63} + ( -1 + 7 \beta_{2} + 3 \beta_{4} ) q^{64} + \beta_{2} q^{66} + ( -2 \beta_{1} + 2 \beta_{5} ) q^{67} + ( -11 \beta_{1} + 10 \beta_{3} + \beta_{5} ) q^{68} + ( -2 \beta_{2} + 2 \beta_{4} ) q^{69} + ( 4 + 2 \beta_{2} + 2 \beta_{4} ) q^{71} + ( -2 \beta_{1} + 2 \beta_{3} + \beta_{5} ) q^{72} + ( 3 \beta_{1} - 4 \beta_{3} + \beta_{5} ) q^{73} + 2 \beta_{2} q^{74} + ( 14 - 4 \beta_{2} - 2 \beta_{4} ) q^{76} + ( \beta_{1} + \beta_{5} ) q^{77} + ( -3 \beta_{1} + 2 \beta_{3} + \beta_{5} ) q^{78} + ( -6 - 4 \beta_{2} - 2 \beta_{4} ) q^{79} + q^{81} + ( 6 \beta_{1} - 6 \beta_{3} - 2 \beta_{5} ) q^{82} + ( 3 \beta_{1} + 2 \beta_{3} - 3 \beta_{5} ) q^{83} + ( 4 + 3 \beta_{2} + \beta_{4} ) q^{84} + ( 7 \beta_{2} + \beta_{4} ) q^{86} + ( 2 \beta_{1} + 4 \beta_{3} ) q^{87} + ( 2 \beta_{1} - 2 \beta_{3} - \beta_{5} ) q^{88} + ( 2 - 4 \beta_{2} ) q^{89} + ( 4 + 2 \beta_{2} + 2 \beta_{4} ) q^{91} + ( -6 \beta_{1} - 8 \beta_{3} + 2 \beta_{5} ) q^{92} + ( 2 \beta_{1} + 4 \beta_{3} - 2 \beta_{5} ) q^{93} + ( 4 - 2 \beta_{2} - 2 \beta_{4} ) q^{94} + ( -8 + 3 \beta_{2} + 2 \beta_{4} ) q^{96} + ( 2 \beta_{1} - 6 \beta_{3} - 2 \beta_{5} ) q^{97} + ( \beta_{1} + 12 \beta_{3} + 4 \beta_{5} ) q^{98} - q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 10 q^{4} - 2 q^{6} - 6 q^{9} + O(q^{10})$$ $$6 q - 10 q^{4} - 2 q^{6} - 6 q^{9} + 6 q^{11} + 24 q^{14} + 26 q^{16} - 16 q^{19} + 18 q^{24} + 20 q^{26} + 20 q^{29} + 16 q^{31} + 60 q^{34} + 10 q^{36} + 4 q^{39} - 28 q^{41} - 10 q^{44} + 48 q^{46} - 22 q^{49} - 4 q^{51} + 2 q^{54} - 8 q^{56} - 24 q^{59} - 12 q^{61} - 26 q^{64} - 2 q^{66} + 16 q^{71} - 4 q^{74} + 96 q^{76} - 24 q^{79} + 6 q^{81} + 16 q^{84} - 16 q^{86} + 20 q^{89} + 16 q^{91} + 32 q^{94} - 58 q^{96} - 6 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2 x^{5} + 2 x^{4} + 2 x^{3} + 4 x^{2} - 4 x + 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-3 \nu^{5} + \nu^{4} + 11 \nu^{3} - 26 \nu^{2} + 6 \nu - 1$$$$)/23$$ $$\beta_{2}$$ $$=$$ $$($$$$-4 \nu^{5} + 9 \nu^{4} - 16 \nu^{3} - 4 \nu^{2} + 8 \nu - 9$$$$)/23$$ $$\beta_{3}$$ $$=$$ $$($$$$7 \nu^{5} - 10 \nu^{4} + 5 \nu^{3} + 30 \nu^{2} + 32 \nu - 13$$$$)/23$$ $$\beta_{4}$$ $$=$$ $$($$$$-6 \nu^{5} + 25 \nu^{4} - 24 \nu^{3} - 6 \nu^{2} + 12 \nu + 67$$$$)/23$$ $$\beta_{5}$$ $$=$$ $$($$$$-25 \nu^{5} + 39 \nu^{4} - 31 \nu^{3} - 48 \nu^{2} - 134 \nu + 53$$$$)/23$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} + \beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{5} + 4 \beta_{3} + \beta_{1}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{5} + \beta_{4} + 4 \beta_{3} - 3 \beta_{2} + 3 \beta_{1} - 4$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{4} - 3 \beta_{2} - 7$$ $$\nu^{5}$$ $$=$$ $$($$$$-5 \beta_{5} + 5 \beta_{4} - 18 \beta_{3} - 11 \beta_{2} - 11 \beta_{1} - 18$$$$)/2$$

## 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
 −0.854638 + 0.854638i 1.45161 + 1.45161i 0.403032 − 0.403032i 0.403032 + 0.403032i 1.45161 − 1.45161i −0.854638 − 0.854638i
2.70928i 1.00000i −5.34017 0 −2.70928 1.07838i 9.04945i −1.00000 0
199.2 1.90321i 1.00000i −1.62222 0 1.90321 4.42864i 0.719004i −1.00000 0
199.3 0.193937i 1.00000i 1.96239 0 −0.193937 3.35026i 0.768452i −1.00000 0
199.4 0.193937i 1.00000i 1.96239 0 −0.193937 3.35026i 0.768452i −1.00000 0
199.5 1.90321i 1.00000i −1.62222 0 1.90321 4.42864i 0.719004i −1.00000 0
199.6 2.70928i 1.00000i −5.34017 0 −2.70928 1.07838i 9.04945i −1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 199.6 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.g 6
3.b odd 2 1 2475.2.c.r 6
5.b even 2 1 inner 825.2.c.g 6
5.c odd 4 1 165.2.a.c 3
5.c odd 4 1 825.2.a.k 3
15.d odd 2 1 2475.2.c.r 6
15.e even 4 1 495.2.a.e 3
15.e even 4 1 2475.2.a.bb 3
20.e even 4 1 2640.2.a.be 3
35.f even 4 1 8085.2.a.bk 3
55.e even 4 1 1815.2.a.m 3
55.e even 4 1 9075.2.a.cf 3
60.l odd 4 1 7920.2.a.cj 3
165.l odd 4 1 5445.2.a.z 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.a.c 3 5.c odd 4 1
495.2.a.e 3 15.e even 4 1
825.2.a.k 3 5.c odd 4 1
825.2.c.g 6 1.a even 1 1 trivial
825.2.c.g 6 5.b even 2 1 inner
1815.2.a.m 3 55.e even 4 1
2475.2.a.bb 3 15.e even 4 1
2475.2.c.r 6 3.b odd 2 1
2475.2.c.r 6 15.d odd 2 1
2640.2.a.be 3 20.e even 4 1
5445.2.a.z 3 165.l odd 4 1
7920.2.a.cj 3 60.l odd 4 1
8085.2.a.bk 3 35.f even 4 1
9075.2.a.cf 3 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 $$S_{2}^{\mathrm{new}}(825, [\chi])$$:

 $$T_{2}^{6} + 11 T_{2}^{4} + 27 T_{2}^{2} + 1$$ $$T_{7}^{6} + 32 T_{7}^{4} + 256 T_{7}^{2} + 256$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 27 T^{2} + 11 T^{4} + T^{6}$$
$3$ $$( 1 + T^{2} )^{3}$$
$5$ $$T^{6}$$
$7$ $$256 + 256 T^{2} + 32 T^{4} + T^{6}$$
$11$ $$( -1 + T )^{6}$$
$13$ $$64 + 176 T^{2} + 28 T^{4} + T^{6}$$
$17$ $$33856 + 3440 T^{2} + 108 T^{4} + T^{6}$$
$19$ $$( -160 - 16 T + 8 T^{2} + T^{3} )^{2}$$
$23$ $$16384 + 4096 T^{2} + 128 T^{4} + T^{6}$$
$29$ $$( 40 + 12 T - 10 T^{2} + T^{3} )^{2}$$
$31$ $$( 128 - 32 T - 8 T^{2} + T^{3} )^{2}$$
$37$ $$( 4 + T^{2} )^{3}$$
$41$ $$( 8 + 44 T + 14 T^{2} + T^{3} )^{2}$$
$43$ $$160000 + 9600 T^{2} + 176 T^{4} + T^{6}$$
$47$ $$16384 + 3072 T^{2} + 128 T^{4} + T^{6}$$
$53$ $$64 + 2608 T^{2} + 140 T^{4} + T^{6}$$
$59$ $$( -320 - 16 T + 12 T^{2} + T^{3} )^{2}$$
$61$ $$( -248 - 52 T + 6 T^{2} + T^{3} )^{2}$$
$67$ $$4096 + 2816 T^{2} + 112 T^{4} + T^{6}$$
$71$ $$( 128 - 32 T - 8 T^{2} + T^{3} )^{2}$$
$73$ $$118336 + 9648 T^{2} + 188 T^{4} + T^{6}$$
$79$ $$( -800 - 64 T + 12 T^{2} + T^{3} )^{2}$$
$83$ $$256 + 14400 T^{2} + 240 T^{4} + T^{6}$$
$89$ $$( 200 - 52 T - 10 T^{2} + T^{3} )^{2}$$
$97$ $$64 + 11312 T^{2} + 268 T^{4} + T^{6}$$