# Properties

 Label 315.2.a.b Level $315$ Weight $2$ Character orbit 315.a Self dual yes Analytic conductor $2.515$ Analytic rank $0$ Dimension $1$ 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: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 35) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 2q^{4} + q^{5} + q^{7} + O(q^{10})$$ $$q - 2q^{4} + q^{5} + q^{7} + 3q^{11} + 5q^{13} + 4q^{16} - 3q^{17} + 2q^{19} - 2q^{20} + 6q^{23} + q^{25} - 2q^{28} - 3q^{29} - 4q^{31} + q^{35} + 2q^{37} + 12q^{41} - 10q^{43} - 6q^{44} - 9q^{47} + q^{49} - 10q^{52} - 12q^{53} + 3q^{55} + 8q^{61} - 8q^{64} + 5q^{65} - 4q^{67} + 6q^{68} + 2q^{73} - 4q^{76} + 3q^{77} - q^{79} + 4q^{80} - 12q^{83} - 3q^{85} + 12q^{89} + 5q^{91} - 12q^{92} + 2q^{95} - q^{97} + O(q^{100})$$

## 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
 0
0 0 −2.00000 1.00000 0 1.00000 0 0 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
$$3$$ $$-1$$
$$5$$ $$-1$$
$$7$$ $$-1$$

## 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 315.2.a.b 1
3.b odd 2 1 35.2.a.a 1
4.b odd 2 1 5040.2.a.v 1
5.b even 2 1 1575.2.a.f 1
5.c odd 4 2 1575.2.d.c 2
7.b odd 2 1 2205.2.a.e 1
12.b even 2 1 560.2.a.b 1
15.d odd 2 1 175.2.a.b 1
15.e even 4 2 175.2.b.a 2
21.c even 2 1 245.2.a.c 1
21.g even 6 2 245.2.e.b 2
21.h odd 6 2 245.2.e.a 2
24.f even 2 1 2240.2.a.u 1
24.h odd 2 1 2240.2.a.k 1
33.d even 2 1 4235.2.a.c 1
39.d odd 2 1 5915.2.a.f 1
60.h even 2 1 2800.2.a.z 1
60.l odd 4 2 2800.2.g.l 2
84.h odd 2 1 3920.2.a.ba 1
105.g even 2 1 1225.2.a.e 1
105.k odd 4 2 1225.2.b.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.a.a 1 3.b odd 2 1
175.2.a.b 1 15.d odd 2 1
175.2.b.a 2 15.e even 4 2
245.2.a.c 1 21.c even 2 1
245.2.e.a 2 21.h odd 6 2
245.2.e.b 2 21.g even 6 2
315.2.a.b 1 1.a even 1 1 trivial
560.2.a.b 1 12.b even 2 1
1225.2.a.e 1 105.g even 2 1
1225.2.b.d 2 105.k odd 4 2
1575.2.a.f 1 5.b even 2 1
1575.2.d.c 2 5.c odd 4 2
2205.2.a.e 1 7.b odd 2 1
2240.2.a.k 1 24.h odd 2 1
2240.2.a.u 1 24.f even 2 1
2800.2.a.z 1 60.h even 2 1
2800.2.g.l 2 60.l odd 4 2
3920.2.a.ba 1 84.h odd 2 1
4235.2.a.c 1 33.d even 2 1
5040.2.a.v 1 4.b odd 2 1
5915.2.a.f 1 39.d odd 2 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$-1 + T$$
$7$ $$-1 + T$$
$11$ $$-3 + T$$
$13$ $$-5 + T$$
$17$ $$3 + T$$
$19$ $$-2 + T$$
$23$ $$-6 + T$$
$29$ $$3 + T$$
$31$ $$4 + T$$
$37$ $$-2 + T$$
$41$ $$-12 + T$$
$43$ $$10 + T$$
$47$ $$9 + T$$
$53$ $$12 + T$$
$59$ $$T$$
$61$ $$-8 + T$$
$67$ $$4 + T$$
$71$ $$T$$
$73$ $$-2 + T$$
$79$ $$1 + T$$
$83$ $$12 + T$$
$89$ $$-12 + T$$
$97$ $$1 + T$$