# Properties

 Label 245.2.a.g Level $245$ Weight $2$ Character orbit 245.a Self dual yes Analytic conductor $1.956$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$245 = 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 245.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$1.95633484952$$ 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 35) 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.

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

## 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
−0.414214 −2.41421 −1.82843 −1.00000 1.00000 0 1.58579 2.82843 0.414214
1.2 2.41421 0.414214 3.82843 −1.00000 1.00000 0 4.41421 −2.82843 −2.41421
 $$n$$: 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 245.2.a.g 2
3.b odd 2 1 2205.2.a.q 2
4.b odd 2 1 3920.2.a.bv 2
5.b even 2 1 1225.2.a.m 2
5.c odd 4 2 1225.2.b.h 4
7.b odd 2 1 245.2.a.h 2
7.c even 3 2 245.2.e.e 4
7.d odd 6 2 35.2.e.a 4
21.c even 2 1 2205.2.a.n 2
21.g even 6 2 315.2.j.e 4
28.d even 2 1 3920.2.a.bq 2
28.f even 6 2 560.2.q.k 4
35.c odd 2 1 1225.2.a.k 2
35.f even 4 2 1225.2.b.g 4
35.i odd 6 2 175.2.e.c 4
35.k even 12 4 175.2.k.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.e.a 4 7.d odd 6 2
175.2.e.c 4 35.i odd 6 2
175.2.k.a 8 35.k even 12 4
245.2.a.g 2 1.a even 1 1 trivial
245.2.a.h 2 7.b odd 2 1
245.2.e.e 4 7.c even 3 2
315.2.j.e 4 21.g even 6 2
560.2.q.k 4 28.f even 6 2
1225.2.a.k 2 35.c odd 2 1
1225.2.a.m 2 5.b even 2 1
1225.2.b.g 4 35.f even 4 2
1225.2.b.h 4 5.c odd 4 2
2205.2.a.n 2 21.c even 2 1
2205.2.a.q 2 3.b odd 2 1
3920.2.a.bq 2 28.d even 2 1
3920.2.a.bv 2 4.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(245))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 - 2 T + T^{2}$$
$3$ $$-1 + 2 T + T^{2}$$
$5$ $$( 1 + T )^{2}$$
$7$ $$T^{2}$$
$11$ $$-4 - 4 T + T^{2}$$
$13$ $$-4 - 4 T + T^{2}$$
$17$ $$-4 + 4 T + T^{2}$$
$19$ $$-8 + T^{2}$$
$23$ $$-1 + 2 T + T^{2}$$
$29$ $$( 1 + T )^{2}$$
$31$ $$( -6 + T )^{2}$$
$37$ $$T^{2}$$
$41$ $$17 - 10 T + T^{2}$$
$43$ $$23 - 10 T + T^{2}$$
$47$ $$( 2 + T )^{2}$$
$53$ $$8 + 8 T + T^{2}$$
$59$ $$-56 - 8 T + T^{2}$$
$61$ $$-63 - 6 T + T^{2}$$
$67$ $$119 - 22 T + T^{2}$$
$71$ $$-56 + 8 T + T^{2}$$
$73$ $$-4 + 4 T + T^{2}$$
$79$ $$136 - 24 T + T^{2}$$
$83$ $$-161 + 2 T + T^{2}$$
$89$ $$-23 - 6 T + T^{2}$$
$97$ $$4 + 12 T + T^{2}$$