# Properties

 Label 825.2.m.a Level $825$ Weight $2$ Character orbit 825.m Analytic conductor $6.588$ Analytic rank $1$ Dimension $4$ CM no Inner twists $2$

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## 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: $$1$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ 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.

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

## 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
−0.500000 + 1.53884i −0.309017 + 0.951057i −0.500000 0.363271i −1.80902 + 1.31433i −1.30902 0.951057i 2.42705 1.76336i −1.80902 + 1.31433i −0.809017 0.587785i −1.11803 3.44095i
256.1 −0.500000 0.363271i 0.809017 + 0.587785i −0.500000 1.53884i −0.690983 + 2.12663i −0.190983 0.587785i −0.927051 + 2.85317i −0.690983 + 2.12663i 0.309017 + 0.951057i 1.11803 0.812299i
361.1 −0.500000 1.53884i −0.309017 0.951057i −0.500000 + 0.363271i −1.80902 1.31433i −1.30902 + 0.951057i 2.42705 + 1.76336i −1.80902 1.31433i −0.809017 + 0.587785i −1.11803 + 3.44095i
796.1 −0.500000 + 0.363271i 0.809017 0.587785i −0.500000 + 1.53884i −0.690983 2.12663i −0.190983 + 0.587785i −0.927051 2.85317i −0.690983 2.12663i 0.309017 0.951057i 1.11803 + 0.812299i
 $$n$$: 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.a 4
11.c even 5 1 825.2.o.b yes 4
25.d even 5 1 825.2.o.b yes 4
275.g even 5 1 inner 825.2.m.a 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 2 T_{2}^{3} + 4 T_{2}^{2} + 3 T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(825, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 3 T + 4 T^{2} + 2 T^{3} + T^{4}$$
$3$ $$1 - T + T^{2} - T^{3} + T^{4}$$
$5$ $$25 + 25 T + 15 T^{2} + 5 T^{3} + T^{4}$$
$7$ $$81 - 27 T + 9 T^{2} - 3 T^{3} + T^{4}$$
$11$ $$121 + 121 T + 51 T^{2} + 11 T^{3} + T^{4}$$
$13$ $$16 + 16 T + 16 T^{2} + 6 T^{3} + T^{4}$$
$17$ $$16 - 32 T + 24 T^{2} + 2 T^{3} + T^{4}$$
$19$ $$25 - 25 T + 10 T^{2} + T^{4}$$
$23$ $$81 + 81 T + 36 T^{2} + 6 T^{3} + T^{4}$$
$29$ $$3025 + 1100 T + 190 T^{2} + 15 T^{3} + T^{4}$$
$31$ $$( 59 + 16 T + T^{2} )^{2}$$
$37$ $$361 - 247 T + 64 T^{2} + 2 T^{3} + T^{4}$$
$41$ $$( -1 + 11 T + T^{2} )^{2}$$
$43$ $$( -6 + T )^{4}$$
$47$ $$121 - 187 T + 114 T^{2} - 8 T^{3} + T^{4}$$
$53$ $$5776 - 1064 T + 96 T^{2} - 4 T^{3} + T^{4}$$
$59$ $$6400 + 160 T^{2} - 20 T^{3} + T^{4}$$
$61$ $$1681 + 533 T + 139 T^{2} + 17 T^{3} + T^{4}$$
$67$ $$81 - 27 T + 9 T^{2} - 3 T^{3} + T^{4}$$
$71$ $$( 109 + 21 T + T^{2} )^{2}$$
$73$ $$( -99 + 3 T + T^{2} )^{2}$$
$79$ $$625 - 125 T + 25 T^{2} - 5 T^{3} + T^{4}$$
$83$ $$1681 - 369 T + 46 T^{2} - 4 T^{3} + T^{4}$$
$89$ $$25 - 25 T + 15 T^{2} - 5 T^{3} + T^{4}$$
$97$ $$841 + 493 T + 109 T^{2} - 3 T^{3} + T^{4}$$
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