# Properties

 Label 1225.2.a.r Level $1225$ Weight $2$ Character orbit 1225.a Self dual yes Analytic conductor $9.782$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1225 = 5^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1225.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$9.78167424761$$ Analytic rank: $$1$$ 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 245) 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.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + ( 1 - \beta ) q^{3} + ( -2 + \beta ) q^{6} -2 \beta q^{8} -2 \beta q^{9} +O(q^{10})$$ $$q + \beta q^{2} + ( 1 - \beta ) q^{3} + ( -2 + \beta ) q^{6} -2 \beta q^{8} -2 \beta q^{9} + ( -3 + 2 \beta ) q^{11} + ( 3 + \beta ) q^{13} -4 q^{16} + ( -1 + 3 \beta ) q^{17} -4 q^{18} -6 q^{19} + ( 4 - 3 \beta ) q^{22} + ( -6 - \beta ) q^{23} + ( 4 - 2 \beta ) q^{24} + ( 2 + 3 \beta ) q^{26} + ( 1 + \beta ) q^{27} + ( -3 - 4 \beta ) q^{29} + ( -6 - 3 \beta ) q^{31} + ( -7 + 5 \beta ) q^{33} + ( 6 - \beta ) q^{34} + ( 2 - 3 \beta ) q^{37} -6 \beta q^{38} + ( 1 - 2 \beta ) q^{39} + ( 2 + 3 \beta ) q^{41} -2 q^{43} + ( -2 - 6 \beta ) q^{46} + ( -3 - 3 \beta ) q^{47} + ( -4 + 4 \beta ) q^{48} + ( -7 + 4 \beta ) q^{51} -3 \beta q^{53} + ( 2 + \beta ) q^{54} + ( -6 + 6 \beta ) q^{57} + ( -8 - 3 \beta ) q^{58} + ( -2 + 3 \beta ) q^{59} + 2 \beta q^{61} + ( -6 - 6 \beta ) q^{62} + 8 q^{64} + ( 10 - 7 \beta ) q^{66} + ( 4 + 3 \beta ) q^{67} + ( -4 + 5 \beta ) q^{69} + ( -6 + 2 \beta ) q^{71} + 8 q^{72} -6 \beta q^{73} + ( -6 + 2 \beta ) q^{74} + ( -4 + \beta ) q^{78} + ( -7 + 6 \beta ) q^{79} + ( -1 + 6 \beta ) q^{81} + ( 6 + 2 \beta ) q^{82} -2 \beta q^{86} + ( 5 - \beta ) q^{87} + ( -8 + 6 \beta ) q^{88} + 8 q^{89} + 3 \beta q^{93} + ( -6 - 3 \beta ) q^{94} + ( 9 + 3 \beta ) q^{97} + ( -8 + 6 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{3} - 4q^{6} + O(q^{10})$$ $$2q + 2q^{3} - 4q^{6} - 6q^{11} + 6q^{13} - 8q^{16} - 2q^{17} - 8q^{18} - 12q^{19} + 8q^{22} - 12q^{23} + 8q^{24} + 4q^{26} + 2q^{27} - 6q^{29} - 12q^{31} - 14q^{33} + 12q^{34} + 4q^{37} + 2q^{39} + 4q^{41} - 4q^{43} - 4q^{46} - 6q^{47} - 8q^{48} - 14q^{51} + 4q^{54} - 12q^{57} - 16q^{58} - 4q^{59} - 12q^{62} + 16q^{64} + 20q^{66} + 8q^{67} - 8q^{69} - 12q^{71} + 16q^{72} - 12q^{74} - 8q^{78} - 14q^{79} - 2q^{81} + 12q^{82} + 10q^{87} - 16q^{88} + 16q^{89} - 12q^{94} + 18q^{97} - 16q^{99} + O(q^{100})$$

## 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
−1.41421 2.41421 0 0 −3.41421 0 2.82843 2.82843 0
1.2 1.41421 −0.414214 0 0 −0.585786 0 −2.82843 −2.82843 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
$$5$$ $$1$$
$$7$$ $$1$$

## 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 1225.2.a.r 2
5.b even 2 1 245.2.a.e 2
5.c odd 4 2 1225.2.b.i 4
7.b odd 2 1 1225.2.a.p 2
15.d odd 2 1 2205.2.a.v 2
20.d odd 2 1 3920.2.a.bw 2
35.c odd 2 1 245.2.a.f yes 2
35.f even 4 2 1225.2.b.j 4
35.i odd 6 2 245.2.e.f 4
35.j even 6 2 245.2.e.g 4
105.g even 2 1 2205.2.a.t 2
140.c even 2 1 3920.2.a.br 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
245.2.a.e 2 5.b even 2 1
245.2.a.f yes 2 35.c odd 2 1
245.2.e.f 4 35.i odd 6 2
245.2.e.g 4 35.j even 6 2
1225.2.a.p 2 7.b odd 2 1
1225.2.a.r 2 1.a even 1 1 trivial
1225.2.b.i 4 5.c odd 4 2
1225.2.b.j 4 35.f even 4 2
2205.2.a.t 2 105.g even 2 1
2205.2.a.v 2 15.d odd 2 1
3920.2.a.br 2 140.c even 2 1
3920.2.a.bw 2 20.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1225))$$:

 $$T_{2}^{2} - 2$$ $$T_{3}^{2} - 2 T_{3} - 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-2 + T^{2}$$
$3$ $$-1 - 2 T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$1 + 6 T + T^{2}$$
$13$ $$7 - 6 T + T^{2}$$
$17$ $$-17 + 2 T + T^{2}$$
$19$ $$( 6 + T )^{2}$$
$23$ $$34 + 12 T + T^{2}$$
$29$ $$-23 + 6 T + T^{2}$$
$31$ $$18 + 12 T + T^{2}$$
$37$ $$-14 - 4 T + T^{2}$$
$41$ $$-14 - 4 T + T^{2}$$
$43$ $$( 2 + T )^{2}$$
$47$ $$-9 + 6 T + T^{2}$$
$53$ $$-18 + T^{2}$$
$59$ $$-14 + 4 T + T^{2}$$
$61$ $$-8 + T^{2}$$
$67$ $$-2 - 8 T + T^{2}$$
$71$ $$28 + 12 T + T^{2}$$
$73$ $$-72 + T^{2}$$
$79$ $$-23 + 14 T + T^{2}$$
$83$ $$T^{2}$$
$89$ $$( -8 + T )^{2}$$
$97$ $$63 - 18 T + T^{2}$$