# Properties

 Label 825.2.m.b Level $825$ Weight $2$ Character orbit 825.m 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.m (of order $$5$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.58765816676$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{10}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\zeta_{10} q^{3} -2 \zeta_{10}^{2} q^{4} + ( 2 - \zeta_{10} + 2 \zeta_{10}^{2} ) q^{5} + ( 1 - \zeta_{10} + 3 \zeta_{10}^{3} ) q^{7} + \zeta_{10}^{2} q^{9} +O(q^{10})$$ $$q -\zeta_{10} q^{3} -2 \zeta_{10}^{2} q^{4} + ( 2 - \zeta_{10} + 2 \zeta_{10}^{2} ) q^{5} + ( 1 - \zeta_{10} + 3 \zeta_{10}^{3} ) q^{7} + \zeta_{10}^{2} q^{9} + ( 2 + 2 \zeta_{10} - \zeta_{10}^{2} ) q^{11} + 2 \zeta_{10}^{3} q^{12} + ( \zeta_{10} - 4 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{13} + ( -2 \zeta_{10} + \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{15} + ( -4 + 4 \zeta_{10} - 4 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{16} + ( 2 \zeta_{10} + 3 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{17} + ( -4 + 4 \zeta_{10} - \zeta_{10}^{3} ) q^{19} + ( 4 - 4 \zeta_{10} - 2 \zeta_{10}^{3} ) q^{20} + ( 3 - 4 \zeta_{10} + 4 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{21} + ( \zeta_{10} + 3 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{23} + 5 \zeta_{10}^{2} q^{25} -\zeta_{10}^{3} q^{27} + ( 6 - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{28} + ( -6 \zeta_{10} + 2 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{29} -2 q^{31} + ( -2 \zeta_{10} - 2 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{33} + ( -1 - 6 \zeta_{10} + 6 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{35} + ( 2 - 2 \zeta_{10} + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{36} + ( 2 + 7 \zeta_{10} + 2 \zeta_{10}^{2} ) q^{37} + ( 1 - \zeta_{10} + 3 \zeta_{10}^{3} ) q^{39} + ( 5 + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{41} + ( 5 + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{43} + ( -2 + 2 \zeta_{10} - 6 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{44} + ( -2 + 2 \zeta_{10} + \zeta_{10}^{3} ) q^{45} + ( 1 - 3 \zeta_{10} + \zeta_{10}^{2} ) q^{47} + 4 q^{48} + ( 7 - 10 \zeta_{10} + 7 \zeta_{10}^{2} ) q^{49} + ( 2 - 2 \zeta_{10} - 5 \zeta_{10}^{3} ) q^{51} + ( -6 + 8 \zeta_{10} - 8 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{52} -9 \zeta_{10}^{3} q^{53} + ( 6 + 2 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{55} + ( -1 + 5 \zeta_{10} - 5 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{57} + ( 2 + 9 \zeta_{10} + 2 \zeta_{10}^{2} ) q^{59} + ( -2 - 2 \zeta_{10} + 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{60} + ( 8 - 8 \zeta_{10} - 9 \zeta_{10}^{3} ) q^{61} + ( -3 + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{63} + 8 \zeta_{10} q^{64} + ( 7 - 7 \zeta_{10} - \zeta_{10}^{3} ) q^{65} + ( -2 + \zeta_{10} - 2 \zeta_{10}^{2} ) q^{67} + ( 10 - 6 \zeta_{10} + 6 \zeta_{10}^{2} - 10 \zeta_{10}^{3} ) q^{68} + ( 1 - \zeta_{10} - 4 \zeta_{10}^{3} ) q^{69} + ( -9 - 4 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{71} + ( -4 - 6 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{73} -5 \zeta_{10}^{3} q^{75} + ( -2 + 8 \zeta_{10}^{2} - 8 \zeta_{10}^{3} ) q^{76} + ( -1 + 6 \zeta_{10} - 9 \zeta_{10}^{2} + 13 \zeta_{10}^{3} ) q^{77} + ( 5 - 5 \zeta_{10} + 4 \zeta_{10}^{3} ) q^{79} + ( -4 - 8 \zeta_{10}^{2} + 8 \zeta_{10}^{3} ) q^{80} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{81} + ( -5 \zeta_{10} + 9 \zeta_{10}^{2} - 5 \zeta_{10}^{3} ) q^{83} + ( 2 - 8 \zeta_{10} + 2 \zeta_{10}^{2} ) q^{84} + ( -8 + 8 \zeta_{10} + 9 \zeta_{10}^{3} ) q^{85} + ( -6 + 6 \zeta_{10} + 4 \zeta_{10}^{3} ) q^{87} + ( -6 \zeta_{10} + 9 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{89} + ( 10 - 7 \zeta_{10}^{2} + 7 \zeta_{10}^{3} ) q^{91} + ( 8 - 6 \zeta_{10} + 6 \zeta_{10}^{2} - 8 \zeta_{10}^{3} ) q^{92} + 2 \zeta_{10} q^{93} + ( -7 + 13 \zeta_{10} - 13 \zeta_{10}^{2} + 7 \zeta_{10}^{3} ) q^{95} + ( 1 - \zeta_{10} + 5 \zeta_{10}^{3} ) q^{97} + ( 1 - \zeta_{10} + 3 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - q^{3} + 2 q^{4} + 5 q^{5} + 6 q^{7} - q^{9} + O(q^{10})$$ $$4 q - q^{3} + 2 q^{4} + 5 q^{5} + 6 q^{7} - q^{9} + 11 q^{11} + 2 q^{12} + 6 q^{13} - 5 q^{15} - 4 q^{16} + q^{17} - 13 q^{19} + 10 q^{20} + q^{21} - q^{23} - 5 q^{25} - q^{27} + 28 q^{28} - 14 q^{29} - 8 q^{31} + q^{33} - 15 q^{35} + 2 q^{36} + 13 q^{37} + 6 q^{39} + 16 q^{41} + 16 q^{43} - 2 q^{44} - 5 q^{45} + 16 q^{48} + 11 q^{49} + q^{51} - 2 q^{52} - 9 q^{53} + 25 q^{55} + 7 q^{57} + 15 q^{59} - 10 q^{60} + 15 q^{61} - 14 q^{63} + 8 q^{64} + 20 q^{65} - 5 q^{67} + 18 q^{68} - q^{69} - 28 q^{71} - 4 q^{73} - 5 q^{75} - 24 q^{76} + 24 q^{77} + 19 q^{79} - q^{81} - 19 q^{83} - 2 q^{84} - 15 q^{85} - 14 q^{87} - 21 q^{89} + 54 q^{91} + 12 q^{92} + 2 q^{93} + 5 q^{95} + 8 q^{97} + q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$376$$ $$551$$ $$727$$ $$\chi(n)$$ $$-\zeta_{10}^{3}$$ $$1$$ $$-1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
16.1
 −0.309017 + 0.951057i 0.809017 + 0.587785i −0.309017 − 0.951057i 0.809017 − 0.587785i
0 0.309017 0.951057i 1.61803 + 1.17557i 0.690983 2.12663i 0 3.73607 2.71441i 0 −0.809017 0.587785i 0
256.1 0 −0.809017 0.587785i −0.618034 1.90211i 1.80902 + 1.31433i 0 −0.736068 + 2.26538i 0 0.309017 + 0.951057i 0
361.1 0 0.309017 + 0.951057i 1.61803 1.17557i 0.690983 + 2.12663i 0 3.73607 + 2.71441i 0 −0.809017 + 0.587785i 0
796.1 0 −0.809017 + 0.587785i −0.618034 + 1.90211i 1.80902 1.31433i 0 −0.736068 2.26538i 0 0.309017 0.951057i 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
275.g even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 825.2.m.b 4
11.c even 5 1 825.2.o.a yes 4
25.d even 5 1 825.2.o.a yes 4
275.g even 5 1 inner 825.2.m.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.2.m.b 4 1.a even 1 1 trivial
825.2.m.b 4 275.g even 5 1 inner
825.2.o.a yes 4 11.c even 5 1
825.2.o.a yes 4 25.d even 5 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{2}^{\mathrm{new}}(825, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$1 + T + T^{2} + T^{3} + T^{4}$$
$5$ $$25 - 25 T + 15 T^{2} - 5 T^{3} + T^{4}$$
$7$ $$121 - 11 T + 16 T^{2} - 6 T^{3} + T^{4}$$
$11$ $$121 - 121 T + 51 T^{2} - 11 T^{3} + T^{4}$$
$13$ $$121 - 11 T + 16 T^{2} - 6 T^{3} + T^{4}$$
$17$ $$121 + 99 T + 31 T^{2} - T^{3} + T^{4}$$
$19$ $$121 + 77 T + 69 T^{2} + 13 T^{3} + T^{4}$$
$23$ $$121 + 66 T + 16 T^{2} + T^{3} + T^{4}$$
$29$ $$1936 + 704 T + 136 T^{2} + 14 T^{3} + T^{4}$$
$31$ $$( 2 + T )^{4}$$
$37$ $$3481 - 177 T + 79 T^{2} - 13 T^{3} + T^{4}$$
$41$ $$( 11 - 8 T + T^{2} )^{2}$$
$43$ $$( 11 - 8 T + T^{2} )^{2}$$
$47$ $$25 + 25 T + 10 T^{2} + T^{4}$$
$53$ $$6561 + 729 T + 81 T^{2} + 9 T^{3} + T^{4}$$
$59$ $$9025 - 475 T + 115 T^{2} - 15 T^{3} + T^{4}$$
$61$ $$3025 - 1375 T + 265 T^{2} - 15 T^{3} + T^{4}$$
$67$ $$25 + 25 T + 15 T^{2} + 5 T^{3} + T^{4}$$
$71$ $$( 29 + 14 T + T^{2} )^{2}$$
$73$ $$( -44 + 2 T + T^{2} )^{2}$$
$79$ $$121 + 66 T + 136 T^{2} - 19 T^{3} + T^{4}$$
$83$ $$121 - 66 T + 136 T^{2} + 19 T^{3} + T^{4}$$
$89$ $$81 + 81 T + 171 T^{2} + 21 T^{3} + T^{4}$$
$97$ $$841 - 87 T + 34 T^{2} - 8 T^{3} + T^{4}$$