# Properties

 Label 315.2.a.c Level 315 Weight 2 Character orbit 315.a Self dual yes Analytic conductor 2.515 Analytic rank 1 Dimension 2 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$315 = 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 315.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$2.51528766367$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta ) q^{2} + ( 1 - 2 \beta ) q^{4} - q^{5} - q^{7} + ( -3 + \beta ) q^{8} +O(q^{10})$$ $$q + ( -1 + \beta ) q^{2} + ( 1 - 2 \beta ) q^{4} - q^{5} - q^{7} + ( -3 + \beta ) q^{8} + ( 1 - \beta ) q^{10} + ( -2 - 2 \beta ) q^{11} + ( -2 + 2 \beta ) q^{13} + ( 1 - \beta ) q^{14} + 3 q^{16} + ( -2 - 4 \beta ) q^{17} -2 \beta q^{19} + ( -1 + 2 \beta ) q^{20} -2 q^{22} + ( -2 + 4 \beta ) q^{23} + q^{25} + ( 6 - 4 \beta ) q^{26} + ( -1 + 2 \beta ) q^{28} -8 q^{29} + 6 \beta q^{31} + ( 3 + \beta ) q^{32} + ( -6 + 2 \beta ) q^{34} + q^{35} -6 q^{37} + ( -4 + 2 \beta ) q^{38} + ( 3 - \beta ) q^{40} + ( 2 + 4 \beta ) q^{41} + ( -4 + 4 \beta ) q^{43} + ( 6 + 2 \beta ) q^{44} + ( 10 - 6 \beta ) q^{46} + 4 q^{47} + q^{49} + ( -1 + \beta ) q^{50} + ( -10 + 6 \beta ) q^{52} + ( -8 + 2 \beta ) q^{53} + ( 2 + 2 \beta ) q^{55} + ( 3 - \beta ) q^{56} + ( 8 - 8 \beta ) q^{58} + 4 q^{59} + 6 q^{61} + ( 12 - 6 \beta ) q^{62} + ( -7 + 2 \beta ) q^{64} + ( 2 - 2 \beta ) q^{65} + ( -4 - 8 \beta ) q^{67} + 14 q^{68} + ( -1 + \beta ) q^{70} + ( -2 - 6 \beta ) q^{71} + ( 2 - 10 \beta ) q^{73} + ( 6 - 6 \beta ) q^{74} + ( 8 - 2 \beta ) q^{76} + ( 2 + 2 \beta ) q^{77} + 4 \beta q^{79} -3 q^{80} + ( 6 - 2 \beta ) q^{82} + 8 q^{83} + ( 2 + 4 \beta ) q^{85} + ( 12 - 8 \beta ) q^{86} + ( 2 + 4 \beta ) q^{88} + ( -6 + 8 \beta ) q^{89} + ( 2 - 2 \beta ) q^{91} + ( -18 + 8 \beta ) q^{92} + ( -4 + 4 \beta ) q^{94} + 2 \beta q^{95} + ( -6 + 2 \beta ) q^{97} + ( -1 + \beta ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} + 2q^{4} - 2q^{5} - 2q^{7} - 6q^{8} + O(q^{10})$$ $$2q - 2q^{2} + 2q^{4} - 2q^{5} - 2q^{7} - 6q^{8} + 2q^{10} - 4q^{11} - 4q^{13} + 2q^{14} + 6q^{16} - 4q^{17} - 2q^{20} - 4q^{22} - 4q^{23} + 2q^{25} + 12q^{26} - 2q^{28} - 16q^{29} + 6q^{32} - 12q^{34} + 2q^{35} - 12q^{37} - 8q^{38} + 6q^{40} + 4q^{41} - 8q^{43} + 12q^{44} + 20q^{46} + 8q^{47} + 2q^{49} - 2q^{50} - 20q^{52} - 16q^{53} + 4q^{55} + 6q^{56} + 16q^{58} + 8q^{59} + 12q^{61} + 24q^{62} - 14q^{64} + 4q^{65} - 8q^{67} + 28q^{68} - 2q^{70} - 4q^{71} + 4q^{73} + 12q^{74} + 16q^{76} + 4q^{77} - 6q^{80} + 12q^{82} + 16q^{83} + 4q^{85} + 24q^{86} + 4q^{88} - 12q^{89} + 4q^{91} - 36q^{92} - 8q^{94} - 12q^{97} - 2q^{98} + O(q^{100})$$

## 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
−2.41421 0 3.82843 −1.00000 0 −1.00000 −4.41421 0 2.41421
1.2 0.414214 0 −1.82843 −1.00000 0 −1.00000 −1.58579 0 −0.414214
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 315.2.a.c 2
3.b odd 2 1 315.2.a.f yes 2
4.b odd 2 1 5040.2.a.bu 2
5.b even 2 1 1575.2.a.u 2
5.c odd 4 2 1575.2.d.h 4
7.b odd 2 1 2205.2.a.p 2
12.b even 2 1 5040.2.a.bx 2
15.d odd 2 1 1575.2.a.m 2
15.e even 4 2 1575.2.d.j 4
21.c even 2 1 2205.2.a.y 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.a.c 2 1.a even 1 1 trivial
315.2.a.f yes 2 3.b odd 2 1
1575.2.a.m 2 15.d odd 2 1
1575.2.a.u 2 5.b even 2 1
1575.2.d.h 4 5.c odd 4 2
1575.2.d.j 4 15.e even 4 2
2205.2.a.p 2 7.b odd 2 1
2205.2.a.y 2 21.c even 2 1
5040.2.a.bu 2 4.b odd 2 1
5040.2.a.bx 2 12.b even 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$5$$ $$1$$
$$7$$ $$1$$

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 2 T_{2} - 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(315))$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + 2 T + 3 T^{2} + 4 T^{3} + 4 T^{4}$$
$3$ 
$5$ $$( 1 + T )^{2}$$
$7$ $$( 1 + T )^{2}$$
$11$ $$1 + 4 T + 18 T^{2} + 44 T^{3} + 121 T^{4}$$
$13$ $$1 + 4 T + 22 T^{2} + 52 T^{3} + 169 T^{4}$$
$17$ $$1 + 4 T + 6 T^{2} + 68 T^{3} + 289 T^{4}$$
$19$ $$1 + 30 T^{2} + 361 T^{4}$$
$23$ $$1 + 4 T + 18 T^{2} + 92 T^{3} + 529 T^{4}$$
$29$ $$( 1 + 8 T + 29 T^{2} )^{2}$$
$31$ $$1 - 10 T^{2} + 961 T^{4}$$
$37$ $$( 1 + 6 T + 37 T^{2} )^{2}$$
$41$ $$1 - 4 T + 54 T^{2} - 164 T^{3} + 1681 T^{4}$$
$43$ $$1 + 8 T + 70 T^{2} + 344 T^{3} + 1849 T^{4}$$
$47$ $$( 1 - 4 T + 47 T^{2} )^{2}$$
$53$ $$1 + 16 T + 162 T^{2} + 848 T^{3} + 2809 T^{4}$$
$59$ $$( 1 - 4 T + 59 T^{2} )^{2}$$
$61$ $$( 1 - 6 T + 61 T^{2} )^{2}$$
$67$ $$1 + 8 T + 22 T^{2} + 536 T^{3} + 4489 T^{4}$$
$71$ $$1 + 4 T + 74 T^{2} + 284 T^{3} + 5041 T^{4}$$
$73$ $$1 - 4 T - 50 T^{2} - 292 T^{3} + 5329 T^{4}$$
$79$ $$1 + 126 T^{2} + 6241 T^{4}$$
$83$ $$( 1 - 8 T + 83 T^{2} )^{2}$$
$89$ $$1 + 12 T + 86 T^{2} + 1068 T^{3} + 7921 T^{4}$$
$97$ $$1 + 12 T + 222 T^{2} + 1164 T^{3} + 9409 T^{4}$$