# Properties

 Label 825.2.bx.c Level $825$ Weight $2$ Character orbit 825.bx Analytic conductor $6.588$ Analytic rank $0$ Dimension $8$ 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.bx (of order $$10$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.58765816676$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{10})$$ Coefficient field: $$\Q(\zeta_{20})$$ Defining polynomial: $$x^{8} - x^{6} + x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{20}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{20} + \zeta_{20}^{3} - \zeta_{20}^{5} ) q^{2} + ( -\zeta_{20} + \zeta_{20}^{3} - \zeta_{20}^{5} + \zeta_{20}^{7} ) q^{3} + ( 1 - \zeta_{20}^{2} - \zeta_{20}^{6} ) q^{4} + ( 1 - \zeta_{20}^{2} + \zeta_{20}^{4} ) q^{6} + ( -2 \zeta_{20} + 2 \zeta_{20}^{5} - 2 \zeta_{20}^{7} ) q^{7} + ( -\zeta_{20} - \zeta_{20}^{3} + \zeta_{20}^{5} + \zeta_{20}^{7} ) q^{8} + ( 1 - \zeta_{20}^{2} + \zeta_{20}^{4} - \zeta_{20}^{6} ) q^{9} +O(q^{10})$$ $$q + ( -\zeta_{20} + \zeta_{20}^{3} - \zeta_{20}^{5} ) q^{2} + ( -\zeta_{20} + \zeta_{20}^{3} - \zeta_{20}^{5} + \zeta_{20}^{7} ) q^{3} + ( 1 - \zeta_{20}^{2} - \zeta_{20}^{6} ) q^{4} + ( 1 - \zeta_{20}^{2} + \zeta_{20}^{4} ) q^{6} + ( -2 \zeta_{20} + 2 \zeta_{20}^{5} - 2 \zeta_{20}^{7} ) q^{7} + ( -\zeta_{20} - \zeta_{20}^{3} + \zeta_{20}^{5} + \zeta_{20}^{7} ) q^{8} + ( 1 - \zeta_{20}^{2} + \zeta_{20}^{4} - \zeta_{20}^{6} ) q^{9} + ( 3 - 2 \zeta_{20}^{2} + 4 \zeta_{20}^{4} - 2 \zeta_{20}^{6} ) q^{11} + ( \zeta_{20}^{3} + \zeta_{20}^{7} ) q^{12} + ( -2 \zeta_{20} + 2 \zeta_{20}^{3} - 2 \zeta_{20}^{5} ) q^{13} + ( 4 \zeta_{20}^{2} - 6 \zeta_{20}^{4} + 4 \zeta_{20}^{6} ) q^{14} + ( -3 + 3 \zeta_{20}^{2} - 3 \zeta_{20}^{4} ) q^{16} -2 \zeta_{20}^{7} q^{17} + ( -\zeta_{20}^{5} + \zeta_{20}^{7} ) q^{18} -5 \zeta_{20}^{4} q^{19} + ( 2 - 2 \zeta_{20}^{4} + 2 \zeta_{20}^{6} ) q^{21} + ( \zeta_{20} - \zeta_{20}^{3} - 3 \zeta_{20}^{5} + 2 \zeta_{20}^{7} ) q^{22} + ( \zeta_{20}^{3} - 4 \zeta_{20}^{5} + \zeta_{20}^{7} ) q^{23} + ( 1 + \zeta_{20}^{2} - \zeta_{20}^{4} - \zeta_{20}^{6} ) q^{24} + ( 2 - 2 \zeta_{20}^{2} + 2 \zeta_{20}^{6} ) q^{26} + \zeta_{20}^{7} q^{27} + ( -2 \zeta_{20} + 2 \zeta_{20}^{3} ) q^{28} + ( 3 - 3 \zeta_{20}^{2} - 4 \zeta_{20}^{6} ) q^{29} + ( -7 + 7 \zeta_{20}^{2} - 7 \zeta_{20}^{4} + 7 \zeta_{20}^{6} ) q^{31} + ( \zeta_{20}^{3} + 4 \zeta_{20}^{5} + \zeta_{20}^{7} ) q^{32} + ( -\zeta_{20} - \zeta_{20}^{3} - \zeta_{20}^{5} + 3 \zeta_{20}^{7} ) q^{33} + ( -2 \zeta_{20}^{4} + 2 \zeta_{20}^{6} ) q^{34} + ( -\zeta_{20}^{2} - \zeta_{20}^{6} ) q^{36} + ( -5 \zeta_{20} - 4 \zeta_{20}^{5} + 4 \zeta_{20}^{7} ) q^{37} + ( -5 \zeta_{20} + 5 \zeta_{20}^{3} ) q^{38} + ( 2 - 2 \zeta_{20}^{2} + 2 \zeta_{20}^{4} ) q^{39} + ( -4 \zeta_{20}^{2} + 9 \zeta_{20}^{4} - 4 \zeta_{20}^{6} ) q^{41} + ( -4 \zeta_{20} + 6 \zeta_{20}^{3} - 4 \zeta_{20}^{5} ) q^{42} + ( 5 \zeta_{20}^{3} - 6 \zeta_{20}^{5} + 5 \zeta_{20}^{7} ) q^{43} + ( 3 - 3 \zeta_{20}^{2} + 2 \zeta_{20}^{4} - 5 \zeta_{20}^{6} ) q^{44} + ( 1 - 5 \zeta_{20}^{2} + 5 \zeta_{20}^{4} - \zeta_{20}^{6} ) q^{46} + ( -2 \zeta_{20} - 3 \zeta_{20}^{3} + 3 \zeta_{20}^{5} + 2 \zeta_{20}^{7} ) q^{47} + ( 3 \zeta_{20}^{5} - 3 \zeta_{20}^{7} ) q^{48} + ( -12 + 13 \zeta_{20}^{2} - 12 \zeta_{20}^{4} ) q^{49} + 2 \zeta_{20}^{6} q^{51} + ( -2 \zeta_{20} + 2 \zeta_{20}^{3} - 2 \zeta_{20}^{5} + 2 \zeta_{20}^{7} ) q^{52} + ( -\zeta_{20} - 6 \zeta_{20}^{3} - \zeta_{20}^{5} ) q^{53} + ( \zeta_{20}^{4} - \zeta_{20}^{6} ) q^{54} + ( -2 + 6 \zeta_{20}^{4} - 6 \zeta_{20}^{6} ) q^{56} + 5 \zeta_{20}^{3} q^{57} + ( -3 \zeta_{20} + 2 \zeta_{20}^{3} - 2 \zeta_{20}^{5} + 3 \zeta_{20}^{7} ) q^{58} + ( 6 - 6 \zeta_{20}^{2} - 3 \zeta_{20}^{6} ) q^{59} -7 \zeta_{20}^{2} q^{61} + ( 7 \zeta_{20}^{5} - 7 \zeta_{20}^{7} ) q^{62} + ( -2 \zeta_{20} + 4 \zeta_{20}^{3} - 4 \zeta_{20}^{5} + 2 \zeta_{20}^{7} ) q^{63} + ( 1 - 3 \zeta_{20}^{2} + 3 \zeta_{20}^{4} - \zeta_{20}^{6} ) q^{64} + ( -1 + \zeta_{20}^{2} + 3 \zeta_{20}^{4} - 2 \zeta_{20}^{6} ) q^{66} + ( \zeta_{20}^{3} - 11 \zeta_{20}^{5} + \zeta_{20}^{7} ) q^{67} + ( -2 \zeta_{20} - 2 \zeta_{20}^{5} ) q^{68} + ( -\zeta_{20}^{2} + 4 \zeta_{20}^{4} - \zeta_{20}^{6} ) q^{69} + 8 \zeta_{20}^{2} q^{71} + ( -2 \zeta_{20} + 2 \zeta_{20}^{3} + \zeta_{20}^{7} ) q^{72} + ( 11 \zeta_{20} + 4 \zeta_{20}^{5} - 4 \zeta_{20}^{7} ) q^{73} + ( \zeta_{20}^{2} + 3 \zeta_{20}^{4} + \zeta_{20}^{6} ) q^{74} + ( -5 - 5 \zeta_{20}^{4} + 5 \zeta_{20}^{6} ) q^{76} + ( -6 \zeta_{20} + 12 \zeta_{20}^{3} - 14 \zeta_{20}^{5} + 6 \zeta_{20}^{7} ) q^{77} + ( 2 \zeta_{20}^{3} - 4 \zeta_{20}^{5} + 2 \zeta_{20}^{7} ) q^{78} + ( -5 + 5 \zeta_{20}^{6} ) q^{79} -\zeta_{20}^{6} q^{81} + ( 9 \zeta_{20} - 9 \zeta_{20}^{3} + 4 \zeta_{20}^{7} ) q^{82} + ( 6 \zeta_{20} - 6 \zeta_{20}^{3} - 2 \zeta_{20}^{7} ) q^{83} + ( 2 - 2 \zeta_{20}^{2} ) q^{84} + ( 5 - 11 \zeta_{20}^{2} + 11 \zeta_{20}^{4} - 5 \zeta_{20}^{6} ) q^{86} + ( 3 \zeta_{20}^{3} + \zeta_{20}^{5} + 3 \zeta_{20}^{7} ) q^{87} + ( -9 \zeta_{20} + 5 \zeta_{20}^{3} - 3 \zeta_{20}^{5} + 3 \zeta_{20}^{7} ) q^{88} + ( -9 - 3 \zeta_{20}^{4} + 3 \zeta_{20}^{6} ) q^{89} + ( 8 \zeta_{20}^{2} - 12 \zeta_{20}^{4} + 8 \zeta_{20}^{6} ) q^{91} + ( -2 \zeta_{20} - 3 \zeta_{20}^{5} + 3 \zeta_{20}^{7} ) q^{92} -7 \zeta_{20}^{7} q^{93} + ( -3 + 8 \zeta_{20}^{2} - 3 \zeta_{20}^{4} ) q^{94} + ( -\zeta_{20}^{2} - 4 \zeta_{20}^{4} - \zeta_{20}^{6} ) q^{96} + ( -4 \zeta_{20} - 5 \zeta_{20}^{3} - 4 \zeta_{20}^{5} ) q^{97} + ( -13 \zeta_{20}^{3} + 25 \zeta_{20}^{5} - 13 \zeta_{20}^{7} ) q^{98} + ( 1 + \zeta_{20}^{2} + \zeta_{20}^{4} - 3 \zeta_{20}^{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 4 q^{4} + 4 q^{6} + 2 q^{9} + O(q^{10})$$ $$8 q + 4 q^{4} + 4 q^{6} + 2 q^{9} + 8 q^{11} + 28 q^{14} - 12 q^{16} + 10 q^{19} + 24 q^{21} + 10 q^{24} + 16 q^{26} + 10 q^{29} - 14 q^{31} + 8 q^{34} - 4 q^{36} + 8 q^{39} - 34 q^{41} + 4 q^{44} - 14 q^{46} - 46 q^{49} + 4 q^{51} - 4 q^{54} - 40 q^{56} + 30 q^{59} - 14 q^{61} - 6 q^{64} - 16 q^{66} - 12 q^{69} + 16 q^{71} - 2 q^{74} - 20 q^{76} - 30 q^{79} - 2 q^{81} + 12 q^{84} - 14 q^{86} - 60 q^{89} + 56 q^{91} - 2 q^{94} + 4 q^{96} + 2 q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$376$$ $$551$$ $$727$$ $$\chi(n)$$ $$-\zeta_{20}^{6}$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 0.587785 + 0.809017i −0.587785 − 0.809017i 0.951057 + 0.309017i −0.951057 − 0.309017i 0.951057 − 0.309017i −0.951057 + 0.309017i 0.587785 − 0.809017i −0.587785 + 0.809017i
−1.53884 + 0.500000i −0.587785 + 0.809017i 0.500000 0.363271i 0 0.500000 1.53884i −3.07768 4.23607i 1.31433 1.80902i −0.309017 0.951057i 0
49.2 1.53884 0.500000i 0.587785 0.809017i 0.500000 0.363271i 0 0.500000 1.53884i 3.07768 + 4.23607i −1.31433 + 1.80902i −0.309017 0.951057i 0
124.1 −0.363271 0.500000i −0.951057 + 0.309017i 0.500000 1.53884i 0 0.500000 + 0.363271i −0.726543 0.236068i −2.12663 + 0.690983i 0.809017 0.587785i 0
124.2 0.363271 + 0.500000i 0.951057 0.309017i 0.500000 1.53884i 0 0.500000 + 0.363271i 0.726543 + 0.236068i 2.12663 0.690983i 0.809017 0.587785i 0
499.1 −0.363271 + 0.500000i −0.951057 0.309017i 0.500000 + 1.53884i 0 0.500000 0.363271i −0.726543 + 0.236068i −2.12663 0.690983i 0.809017 + 0.587785i 0
499.2 0.363271 0.500000i 0.951057 + 0.309017i 0.500000 + 1.53884i 0 0.500000 0.363271i 0.726543 0.236068i 2.12663 + 0.690983i 0.809017 + 0.587785i 0
724.1 −1.53884 0.500000i −0.587785 0.809017i 0.500000 + 0.363271i 0 0.500000 + 1.53884i −3.07768 + 4.23607i 1.31433 + 1.80902i −0.309017 + 0.951057i 0
724.2 1.53884 + 0.500000i 0.587785 + 0.809017i 0.500000 + 0.363271i 0 0.500000 + 1.53884i 3.07768 4.23607i −1.31433 1.80902i −0.309017 + 0.951057i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 724.2 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
11.c even 5 1 inner
55.j even 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 825.2.bx.c 8
5.b even 2 1 inner 825.2.bx.c 8
5.c odd 4 1 825.2.n.b 4
5.c odd 4 1 825.2.n.d yes 4
11.c even 5 1 inner 825.2.bx.c 8
55.j even 10 1 inner 825.2.bx.c 8
55.k odd 20 1 825.2.n.b 4
55.k odd 20 1 825.2.n.d yes 4
55.k odd 20 1 9075.2.a.z 2
55.k odd 20 1 9075.2.a.by 2
55.l even 20 1 9075.2.a.bc 2
55.l even 20 1 9075.2.a.bt 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.2.n.b 4 5.c odd 4 1
825.2.n.b 4 55.k odd 20 1
825.2.n.d yes 4 5.c odd 4 1
825.2.n.d yes 4 55.k odd 20 1
825.2.bx.c 8 1.a even 1 1 trivial
825.2.bx.c 8 5.b even 2 1 inner
825.2.bx.c 8 11.c even 5 1 inner
825.2.bx.c 8 55.j even 10 1 inner
9075.2.a.z 2 55.k odd 20 1
9075.2.a.bc 2 55.l even 20 1
9075.2.a.bt 2 55.l even 20 1
9075.2.a.by 2 55.k odd 20 1

## Hecke kernels

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

 $$T_{2}^{8} - 4 T_{2}^{6} + 6 T_{2}^{4} + T_{2}^{2} + 1$$ $$T_{13}^{8} - 16 T_{13}^{6} + 96 T_{13}^{4} + 64 T_{13}^{2} + 256$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2} + 6 T^{4} - 4 T^{6} + T^{8}$$
$3$ $$1 - T^{2} + T^{4} - T^{6} + T^{8}$$
$5$ $$T^{8}$$
$7$ $$256 - 704 T^{2} + 736 T^{4} + 16 T^{6} + T^{8}$$
$11$ $$( 121 - 44 T + 6 T^{2} - 4 T^{3} + T^{4} )^{2}$$
$13$ $$256 + 64 T^{2} + 96 T^{4} - 16 T^{6} + T^{8}$$
$17$ $$256 - 64 T^{2} + 16 T^{4} - 4 T^{6} + T^{8}$$
$19$ $$( 625 - 125 T + 25 T^{2} - 5 T^{3} + T^{4} )^{2}$$
$23$ $$( 121 + 27 T^{2} + T^{4} )^{2}$$
$29$ $$( 25 - 50 T + 40 T^{2} - 5 T^{3} + T^{4} )^{2}$$
$31$ $$( 2401 + 343 T + 49 T^{2} + 7 T^{3} + T^{4} )^{2}$$
$37$ $$14641 + 3751 T^{2} + 3001 T^{4} - 89 T^{6} + T^{8}$$
$41$ $$( 841 - 87 T + 109 T^{2} + 17 T^{3} + T^{4} )^{2}$$
$43$ $$( 361 + 87 T^{2} + T^{4} )^{2}$$
$47$ $$923521 - 42284 T^{2} + 1086 T^{4} - 19 T^{6} + T^{8}$$
$53$ $$2825761 - 127756 T^{2} + 2526 T^{4} - 11 T^{6} + T^{8}$$
$59$ $$( 2025 - 675 T + 135 T^{2} - 15 T^{3} + T^{4} )^{2}$$
$61$ $$( 2401 + 343 T + 49 T^{2} + 7 T^{3} + T^{4} )^{2}$$
$67$ $$( 11881 + 223 T^{2} + T^{4} )^{2}$$
$71$ $$( 4096 - 512 T + 64 T^{2} - 8 T^{3} + T^{4} )^{2}$$
$73$ $$13845841 + 293959 T^{2} + 29641 T^{4} - 281 T^{6} + T^{8}$$
$79$ $$( 625 + 250 T + 100 T^{2} + 15 T^{3} + T^{4} )^{2}$$
$83$ $$3748096 - 30976 T^{2} + 2656 T^{4} - 76 T^{6} + T^{8}$$
$89$ $$( 45 + 15 T + T^{2} )^{4}$$
$97$ $$707281 - 175769 T^{2} + 16521 T^{4} + 71 T^{6} + T^{8}$$