# Properties

 Label 825.2.r.a Level $825$ Weight $2$ Character orbit 825.r 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.r (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.

 $$f(q)$$ $$=$$ $$q + ( -1 - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{2} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{3} + 3 q^{4} + ( 2 - 2 \zeta_{10} - \zeta_{10}^{3} ) q^{5} + ( 1 + \zeta_{10} - \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{6} + ( 1 - \zeta_{10} - 2 \zeta_{10}^{3} ) q^{7} + ( -1 - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{8} -\zeta_{10}^{3} q^{9} +O(q^{10})$$ $$q + ( -1 - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{2} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{3} + 3 q^{4} + ( 2 - 2 \zeta_{10} - \zeta_{10}^{3} ) q^{5} + ( 1 + \zeta_{10} - \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{6} + ( 1 - \zeta_{10} - 2 \zeta_{10}^{3} ) q^{7} + ( -1 - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{8} -\zeta_{10}^{3} q^{9} + 5 \zeta_{10}^{3} q^{10} + ( -4 + 3 \zeta_{10} - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{11} + ( -3 + 3 \zeta_{10} - 3 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{12} + 6 \zeta_{10}^{3} q^{13} + ( -3 + 3 \zeta_{10} + 4 \zeta_{10}^{3} ) q^{14} + ( 2 \zeta_{10} - \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{15} - q^{16} + ( -5 + 3 \zeta_{10} - 3 \zeta_{10}^{2} + 5 \zeta_{10}^{3} ) q^{17} + ( -2 + 2 \zeta_{10} + \zeta_{10}^{3} ) q^{18} + ( 6 + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{19} + ( 6 - 6 \zeta_{10} - 3 \zeta_{10}^{3} ) q^{20} + ( \zeta_{10} + \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{21} + ( 2 + 3 \zeta_{10} - 6 \zeta_{10}^{3} ) q^{22} + ( -4 + \zeta_{10} - 4 \zeta_{10}^{2} ) q^{23} + ( 1 + \zeta_{10} - \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{24} -5 \zeta_{10} q^{25} + ( 12 - 12 \zeta_{10} - 6 \zeta_{10}^{3} ) q^{26} + \zeta_{10}^{2} q^{27} + ( 3 - 3 \zeta_{10} - 6 \zeta_{10}^{3} ) q^{28} + ( 4 + 5 \zeta_{10}^{2} - 5 \zeta_{10}^{3} ) q^{29} -5 \zeta_{10}^{2} q^{30} + ( 4 \zeta_{10} + \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{31} + ( 3 + 6 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{32} + ( 1 - 2 \zeta_{10} + 2 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{33} + ( 9 - \zeta_{10} + \zeta_{10}^{2} - 9 \zeta_{10}^{3} ) q^{34} + ( -3 - \zeta_{10} - 3 \zeta_{10}^{2} ) q^{35} -3 \zeta_{10}^{3} q^{36} + ( -3 \zeta_{10} + 6 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{37} + ( -8 - 11 \zeta_{10}^{2} + 11 \zeta_{10}^{3} ) q^{38} -6 \zeta_{10}^{2} q^{39} + 5 \zeta_{10}^{3} q^{40} + ( 2 - 5 \zeta_{10} + 2 \zeta_{10}^{2} ) q^{41} + ( -3 \zeta_{10} - \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{42} + ( 3 + 8 \zeta_{10}^{2} - 8 \zeta_{10}^{3} ) q^{43} + ( -12 + 9 \zeta_{10} - 6 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{44} + ( -2 + \zeta_{10} - 2 \zeta_{10}^{2} ) q^{45} + ( 2 + 9 \zeta_{10} + 2 \zeta_{10}^{2} ) q^{46} + ( -5 + 5 \zeta_{10}^{3} ) q^{47} + ( 1 - \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{48} + ( -3 + 5 \zeta_{10} - 3 \zeta_{10}^{2} ) q^{49} + ( 10 - 5 \zeta_{10} + 10 \zeta_{10}^{2} ) q^{50} + ( 2 - 2 \zeta_{10} - 5 \zeta_{10}^{3} ) q^{51} + 18 \zeta_{10}^{3} q^{52} + \zeta_{10} q^{53} + ( -2 \zeta_{10} + \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{54} + ( -3 + 9 \zeta_{10} - 3 \zeta_{10}^{2} + 5 \zeta_{10}^{3} ) q^{55} + ( -3 + 3 \zeta_{10} + 4 \zeta_{10}^{3} ) q^{56} + ( -6 + 5 \zeta_{10} - 5 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{57} + ( -14 - 3 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{58} + ( 6 \zeta_{10} + 2 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{59} + ( 6 \zeta_{10} - 3 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{60} + ( -7 \zeta_{10} + 6 \zeta_{10}^{2} - 7 \zeta_{10}^{3} ) q^{61} + ( -6 \zeta_{10} - 7 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{62} + ( -1 - \zeta_{10} - \zeta_{10}^{2} ) q^{63} -13 q^{64} + ( 12 - 6 \zeta_{10} + 12 \zeta_{10}^{2} ) q^{65} + ( -5 + 2 \zeta_{10} + 4 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{66} + ( -3 + 3 \zeta_{10} + 4 \zeta_{10}^{3} ) q^{67} + ( -15 + 9 \zeta_{10} - 9 \zeta_{10}^{2} + 15 \zeta_{10}^{3} ) q^{68} + ( 3 + 4 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{69} + ( 5 + 5 \zeta_{10} + 5 \zeta_{10}^{2} ) q^{70} + ( 4 - 4 \zeta_{10} - 5 \zeta_{10}^{3} ) q^{71} + ( -2 + 2 \zeta_{10} + \zeta_{10}^{3} ) q^{72} + ( 2 - 2 \zeta_{10}^{3} ) q^{73} + ( -9 \zeta_{10} + 12 \zeta_{10}^{2} - 9 \zeta_{10}^{3} ) q^{74} + 5 q^{75} + ( 18 + 3 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{76} + ( 3 \zeta_{10} + 3 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{77} + ( 12 \zeta_{10} - 6 \zeta_{10}^{2} + 12 \zeta_{10}^{3} ) q^{78} + ( 8 + 8 \zeta_{10}^{2} ) q^{79} + ( -2 + 2 \zeta_{10} + \zeta_{10}^{3} ) q^{80} -\zeta_{10} q^{81} + ( 8 - 9 \zeta_{10} + 8 \zeta_{10}^{2} ) q^{82} + ( 3 + 6 \zeta_{10} - 6 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{83} + ( 3 \zeta_{10} + 3 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{84} + ( 8 \zeta_{10} + \zeta_{10}^{2} + 8 \zeta_{10}^{3} ) q^{85} + ( -19 + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{86} + ( -4 - \zeta_{10} + \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{87} + ( 2 + 3 \zeta_{10} - 6 \zeta_{10}^{3} ) q^{88} + ( -9 + 11 \zeta_{10} - 9 \zeta_{10}^{2} ) q^{89} + 5 \zeta_{10} q^{90} + ( 6 + 6 \zeta_{10} + 6 \zeta_{10}^{2} ) q^{91} + ( -12 + 3 \zeta_{10} - 12 \zeta_{10}^{2} ) q^{92} + ( -4 - \zeta_{10} - 4 \zeta_{10}^{2} ) q^{93} + ( 15 - 10 \zeta_{10} + 10 \zeta_{10}^{2} - 15 \zeta_{10}^{3} ) q^{94} + ( 11 - 11 \zeta_{10} - 8 \zeta_{10}^{3} ) q^{95} + ( -3 - 3 \zeta_{10} + 3 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{96} + ( 4 - 9 \zeta_{10} + 4 \zeta_{10}^{2} ) q^{97} + ( -7 + 11 \zeta_{10} - 7 \zeta_{10}^{2} ) q^{98} + ( 1 - \zeta_{10} + 3 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - q^{3} + 12 q^{4} + 5 q^{5} + 5 q^{6} + q^{7} - q^{9} + O(q^{10})$$ $$4 q - q^{3} + 12 q^{4} + 5 q^{5} + 5 q^{6} + q^{7} - q^{9} + 5 q^{10} - 9 q^{11} - 3 q^{12} + 6 q^{13} - 5 q^{14} + 5 q^{15} - 4 q^{16} - 9 q^{17} - 5 q^{18} + 22 q^{19} + 15 q^{20} + q^{21} + 5 q^{22} - 11 q^{23} + 5 q^{24} - 5 q^{25} + 30 q^{26} - q^{27} + 3 q^{28} + 6 q^{29} + 5 q^{30} + 7 q^{31} - 4 q^{33} + 25 q^{34} - 10 q^{35} - 3 q^{36} - 12 q^{37} - 10 q^{38} + 6 q^{39} + 5 q^{40} + q^{41} - 5 q^{42} - 4 q^{43} - 27 q^{44} - 5 q^{45} + 15 q^{46} - 15 q^{47} + q^{48} - 4 q^{49} + 25 q^{50} + q^{51} + 18 q^{52} + q^{53} - 5 q^{54} + 5 q^{55} - 5 q^{56} - 8 q^{57} - 50 q^{58} + 10 q^{59} + 15 q^{60} - 20 q^{61} - 5 q^{62} - 4 q^{63} - 52 q^{64} + 30 q^{65} - 20 q^{66} - 5 q^{67} - 27 q^{68} + 4 q^{69} + 20 q^{70} + 7 q^{71} - 5 q^{72} + 6 q^{73} - 30 q^{74} + 20 q^{75} + 66 q^{76} + 4 q^{77} + 30 q^{78} + 24 q^{79} - 5 q^{80} - q^{81} + 15 q^{82} + 21 q^{83} + 3 q^{84} + 15 q^{85} - 80 q^{86} - 14 q^{87} + 5 q^{88} - 16 q^{89} + 5 q^{90} + 24 q^{91} - 33 q^{92} - 13 q^{93} + 25 q^{94} + 25 q^{95} - 15 q^{96} + 3 q^{97} - 10 q^{98} + q^{99} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
31.1
 −0.309017 − 0.951057i 0.809017 + 0.587785i 0.809017 − 0.587785i −0.309017 + 0.951057i
2.23607 0.309017 0.951057i 3.00000 1.80902 + 1.31433i 0.690983 2.12663i −0.309017 0.224514i 2.23607 −0.809017 0.587785i 4.04508 + 2.93893i
91.1 −2.23607 −0.809017 + 0.587785i 3.00000 0.690983 2.12663i 1.80902 1.31433i 0.809017 2.48990i −2.23607 0.309017 0.951057i −1.54508 + 4.75528i
136.1 −2.23607 −0.809017 0.587785i 3.00000 0.690983 + 2.12663i 1.80902 + 1.31433i 0.809017 + 2.48990i −2.23607 0.309017 + 0.951057i −1.54508 4.75528i
346.1 2.23607 0.309017 + 0.951057i 3.00000 1.80902 1.31433i 0.690983 + 2.12663i −0.309017 + 0.224514i 2.23607 −0.809017 + 0.587785i 4.04508 2.93893i
 $$n$$: 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.l 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.r.a yes 4
11.c even 5 1 825.2.p.a 4
25.d even 5 1 825.2.p.a 4
275.l even 5 1 inner 825.2.r.a yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.2.p.a 4 11.c even 5 1
825.2.p.a 4 25.d even 5 1
825.2.r.a yes 4 1.a even 1 1 trivial
825.2.r.a yes 4 275.l even 5 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 5$$ acting on $$S_{2}^{\mathrm{new}}(825, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -5 + T^{2} )^{2}$$
$3$ $$1 + T + T^{2} + T^{3} + T^{4}$$
$5$ $$25 - 25 T + 15 T^{2} - 5 T^{3} + T^{4}$$
$7$ $$1 + 4 T + 6 T^{2} - T^{3} + T^{4}$$
$11$ $$121 + 99 T + 41 T^{2} + 9 T^{3} + T^{4}$$
$13$ $$1296 - 216 T + 36 T^{2} - 6 T^{3} + T^{4}$$
$17$ $$121 - 11 T + 31 T^{2} + 9 T^{3} + T^{4}$$
$19$ $$( 29 - 11 T + T^{2} )^{2}$$
$23$ $$361 + 171 T + 61 T^{2} + 11 T^{3} + T^{4}$$
$29$ $$( -29 - 3 T + T^{2} )^{2}$$
$31$ $$121 - 143 T + 69 T^{2} - 7 T^{3} + T^{4}$$
$37$ $$81 - 27 T + 54 T^{2} + 12 T^{3} + T^{4}$$
$41$ $$121 + 99 T + 31 T^{2} - T^{3} + T^{4}$$
$43$ $$( -79 + 2 T + T^{2} )^{2}$$
$47$ $$625 + 250 T + 100 T^{2} + 15 T^{3} + T^{4}$$
$53$ $$1 - T + T^{2} - T^{3} + T^{4}$$
$59$ $$400 - 400 T + 160 T^{2} - 10 T^{3} + T^{4}$$
$61$ $$3025 + 825 T + 190 T^{2} + 20 T^{3} + T^{4}$$
$67$ $$25 + 50 T + 40 T^{2} + 5 T^{3} + T^{4}$$
$71$ $$121 - 143 T + 69 T^{2} - 7 T^{3} + T^{4}$$
$73$ $$16 - 16 T + 16 T^{2} - 6 T^{3} + T^{4}$$
$79$ $$4096 - 1024 T + 256 T^{2} - 24 T^{3} + T^{4}$$
$83$ $$9801 - 2376 T + 306 T^{2} - 21 T^{3} + T^{4}$$
$89$ $$3481 + 1711 T + 346 T^{2} + 16 T^{3} + T^{4}$$
$97$ $$841 + 493 T + 109 T^{2} - 3 T^{3} + T^{4}$$