# Properties

 Label 825.2.a.f Level $825$ Weight $2$ Character orbit 825.a Self dual yes Analytic conductor $6.588$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$825 = 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 825.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$6.58765816676$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.41421 1.41421
−0.414214 1.00000 −1.82843 0 −0.414214 −2.41421 1.58579 1.00000 0
1.2 2.41421 1.00000 3.82843 0 2.41421 0.414214 4.41421 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$5$$ $$1$$
$$11$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 825.2.a.f yes 2
3.b odd 2 1 2475.2.a.l 2
5.b even 2 1 825.2.a.d 2
5.c odd 4 2 825.2.c.d 4
11.b odd 2 1 9075.2.a.w 2
15.d odd 2 1 2475.2.a.w 2
15.e even 4 2 2475.2.c.o 4
55.d odd 2 1 9075.2.a.ca 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.2.a.d 2 5.b even 2 1
825.2.a.f yes 2 1.a even 1 1 trivial
825.2.c.d 4 5.c odd 4 2
2475.2.a.l 2 3.b odd 2 1
2475.2.a.w 2 15.d odd 2 1
2475.2.c.o 4 15.e even 4 2
9075.2.a.w 2 11.b odd 2 1
9075.2.a.ca 2 55.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(825))$$:

 $$T_{2}^{2} - 2 T_{2} - 1$$ $$T_{7}^{2} + 2 T_{7} - 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 - 2 T + T^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$T^{2}$$
$7$ $$-1 + 2 T + T^{2}$$
$11$ $$( 1 + T )^{2}$$
$13$ $$-8 + T^{2}$$
$17$ $$-1 - 2 T + T^{2}$$
$19$ $$23 - 10 T + T^{2}$$
$23$ $$( -1 + T )^{2}$$
$29$ $$8 - 8 T + T^{2}$$
$31$ $$-72 + T^{2}$$
$37$ $$1 + 6 T + T^{2}$$
$41$ $$-97 + 2 T + T^{2}$$
$43$ $$4 + 12 T + T^{2}$$
$47$ $$-71 - 2 T + T^{2}$$
$53$ $$-28 - 4 T + T^{2}$$
$59$ $$( -11 + T )^{2}$$
$61$ $$28 - 12 T + T^{2}$$
$67$ $$4 + 12 T + T^{2}$$
$71$ $$17 - 10 T + T^{2}$$
$73$ $$28 + 12 T + T^{2}$$
$79$ $$63 - 18 T + T^{2}$$
$83$ $$-56 - 8 T + T^{2}$$
$89$ $$-28 + 4 T + T^{2}$$
$97$ $$1 - 6 T + T^{2}$$