# Properties

 Label 825.2.n.e Level $825$ Weight $2$ Character orbit 825.n 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.n (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: $$\Z[a_1, a_2]$$ 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.

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

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
301.1
 0.809017 − 0.587785i −0.309017 − 0.951057i −0.309017 + 0.951057i 0.809017 + 0.587785i
1.30902 + 0.951057i −0.309017 + 0.951057i 0.190983 + 0.587785i 0 −1.30902 + 0.951057i −0.927051 2.85317i 0.690983 2.12663i −0.809017 0.587785i 0
526.1 0.190983 0.587785i 0.809017 0.587785i 1.30902 + 0.951057i 0 −0.190983 0.587785i 2.42705 + 1.76336i 1.80902 1.31433i 0.309017 0.951057i 0
676.1 0.190983 + 0.587785i 0.809017 + 0.587785i 1.30902 0.951057i 0 −0.190983 + 0.587785i 2.42705 1.76336i 1.80902 + 1.31433i 0.309017 + 0.951057i 0
751.1 1.30902 0.951057i −0.309017 0.951057i 0.190983 0.587785i 0 −1.30902 0.951057i −0.927051 + 2.85317i 0.690983 + 2.12663i −0.809017 + 0.587785i 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
11.c 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.n.e yes 4
5.b even 2 1 825.2.n.a 4
5.c odd 4 2 825.2.bx.a 8
11.c even 5 1 inner 825.2.n.e yes 4
11.c even 5 1 9075.2.a.y 2
11.d odd 10 1 9075.2.a.bu 2
55.h odd 10 1 9075.2.a.bb 2
55.j even 10 1 825.2.n.a 4
55.j even 10 1 9075.2.a.bz 2
55.k odd 20 2 825.2.bx.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.2.n.a 4 5.b even 2 1
825.2.n.a 4 55.j even 10 1
825.2.n.e yes 4 1.a even 1 1 trivial
825.2.n.e yes 4 11.c even 5 1 inner
825.2.bx.a 8 5.c odd 4 2
825.2.bx.a 8 55.k odd 20 2
9075.2.a.y 2 11.c even 5 1
9075.2.a.bb 2 55.h odd 10 1
9075.2.a.bu 2 11.d odd 10 1
9075.2.a.bz 2 55.j even 10 1

## Hecke kernels

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

 $$T_{2}^{4} - 3 T_{2}^{3} + 4 T_{2}^{2} - 2 T_{2} + 1$$ $$T_{13}^{4} + 6 T_{13}^{3} + 36 T_{13}^{2} + 81 T_{13} + 81$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 2 T + 4 T^{2} - 3 T^{3} + T^{4}$$
$3$ $$1 - T + T^{2} - T^{3} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$81 - 27 T + 9 T^{2} - 3 T^{3} + T^{4}$$
$11$ $$121 + 11 T + 21 T^{2} + T^{3} + T^{4}$$
$13$ $$81 + 81 T + 36 T^{2} + 6 T^{3} + T^{4}$$
$17$ $$121 - 77 T + 69 T^{2} - 13 T^{3} + T^{4}$$
$19$ $$400 + 40 T^{2} - 10 T^{3} + T^{4}$$
$23$ $$( 41 + 13 T + T^{2} )^{2}$$
$29$ $$25 + 25 T + 10 T^{2} + T^{4}$$
$31$ $$361 + 38 T + 64 T^{2} - 13 T^{3} + T^{4}$$
$37$ $$361 - 247 T + 64 T^{2} + 2 T^{3} + T^{4}$$
$41$ $$1 + 3 T + 19 T^{2} + 7 T^{3} + T^{4}$$
$43$ $$( 11 + 8 T + T^{2} )^{2}$$
$47$ $$11881 - 872 T + 114 T^{2} - 13 T^{3} + T^{4}$$
$53$ $$16 + 56 T + 76 T^{2} + 6 T^{3} + T^{4}$$
$59$ $$2025 + 90 T^{2} - 15 T^{3} + T^{4}$$
$61$ $$3481 - 767 T + 79 T^{2} - 3 T^{3} + T^{4}$$
$67$ $$( -41 - 4 T + T^{2} )^{2}$$
$71$ $$121 + 143 T + 69 T^{2} + 7 T^{3} + T^{4}$$
$73$ $$1 + 11 T + 46 T^{2} - 4 T^{3} + T^{4}$$
$79$ $$9025 - 2375 T + 310 T^{2} - 20 T^{3} + T^{4}$$
$83$ $$17161 - 1834 T + 136 T^{2} - 9 T^{3} + T^{4}$$
$89$ $$( -45 + 15 T + T^{2} )^{2}$$
$97$ $$29241 + 5643 T + 549 T^{2} + 27 T^{3} + T^{4}$$