# Properties

 Label 825.2.a.g 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: no (minimal twist has level 165) 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.

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

## 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 4.82843 1.58579 1.00000 0
1.2 2.41421 1.00000 3.82843 0 2.41421 −0.828427 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.g 2
3.b odd 2 1 2475.2.a.m 2
5.b even 2 1 165.2.a.a 2
5.c odd 4 2 825.2.c.e 4
11.b odd 2 1 9075.2.a.v 2
15.d odd 2 1 495.2.a.d 2
15.e even 4 2 2475.2.c.m 4
20.d odd 2 1 2640.2.a.bb 2
35.c odd 2 1 8085.2.a.ba 2
55.d odd 2 1 1815.2.a.k 2
60.h even 2 1 7920.2.a.cg 2
165.d even 2 1 5445.2.a.m 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.a.a 2 5.b even 2 1
495.2.a.d 2 15.d odd 2 1
825.2.a.g 2 1.a even 1 1 trivial
825.2.c.e 4 5.c odd 4 2
1815.2.a.k 2 55.d odd 2 1
2475.2.a.m 2 3.b odd 2 1
2475.2.c.m 4 15.e even 4 2
2640.2.a.bb 2 20.d odd 2 1
5445.2.a.m 2 165.d even 2 1
7920.2.a.cg 2 60.h even 2 1
8085.2.a.ba 2 35.c odd 2 1
9075.2.a.v 2 11.b odd 2 1

## Hecke kernels

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

 $$T_{2}^{2} - 2 T_{2} - 1$$ $$T_{7}^{2} - 4 T_{7} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 - 2 T + T^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$T^{2}$$
$7$ $$-4 - 4 T + T^{2}$$
$11$ $$( 1 + T )^{2}$$
$13$ $$-32 + T^{2}$$
$17$ $$8 - 8 T + T^{2}$$
$19$ $$8 + 8 T + T^{2}$$
$23$ $$( -4 + T )^{2}$$
$29$ $$-4 + 4 T + T^{2}$$
$31$ $$T^{2}$$
$37$ $$4 + 12 T + T^{2}$$
$41$ $$-4 - 4 T + T^{2}$$
$43$ $$28 - 12 T + T^{2}$$
$47$ $$( -4 + T )^{2}$$
$53$ $$-124 - 4 T + T^{2}$$
$59$ $$( 4 + T )^{2}$$
$61$ $$4 + 12 T + T^{2}$$
$67$ $$-32 + T^{2}$$
$71$ $$32 - 16 T + T^{2}$$
$73$ $$-128 + T^{2}$$
$79$ $$-72 + T^{2}$$
$83$ $$( -10 + T )^{2}$$
$89$ $$-28 + 4 T + T^{2}$$
$97$ $$4 + 12 T + T^{2}$$