# Properties

 Label 315.2.a.a Level $315$ Weight $2$ Character orbit 315.a Self dual yes Analytic conductor $2.515$ Analytic rank $1$ 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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 105) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

## 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
 0
−1.00000 0 −1.00000 −1.00000 0 1.00000 3.00000 0 1.00000
 $$n$$: 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.a 1
3.b odd 2 1 105.2.a.a 1
4.b odd 2 1 5040.2.a.d 1
5.b even 2 1 1575.2.a.h 1
5.c odd 4 2 1575.2.d.b 2
7.b odd 2 1 2205.2.a.b 1
12.b even 2 1 1680.2.a.f 1
15.d odd 2 1 525.2.a.a 1
15.e even 4 2 525.2.d.b 2
21.c even 2 1 735.2.a.f 1
21.g even 6 2 735.2.i.b 2
21.h odd 6 2 735.2.i.a 2
24.f even 2 1 6720.2.a.bk 1
24.h odd 2 1 6720.2.a.p 1
60.h even 2 1 8400.2.a.co 1
105.g even 2 1 3675.2.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.a.a 1 3.b odd 2 1
315.2.a.a 1 1.a even 1 1 trivial
525.2.a.a 1 15.d odd 2 1
525.2.d.b 2 15.e even 4 2
735.2.a.f 1 21.c even 2 1
735.2.i.a 2 21.h odd 6 2
735.2.i.b 2 21.g even 6 2
1575.2.a.h 1 5.b even 2 1
1575.2.d.b 2 5.c odd 4 2
1680.2.a.f 1 12.b even 2 1
2205.2.a.b 1 7.b odd 2 1
3675.2.a.f 1 105.g even 2 1
5040.2.a.d 1 4.b odd 2 1
6720.2.a.p 1 24.h odd 2 1
6720.2.a.bk 1 24.f even 2 1
8400.2.a.co 1 60.h even 2 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T$$
$3$ $$T$$
$5$ $$1 + T$$
$7$ $$-1 + T$$
$11$ $$T$$
$13$ $$6 + T$$
$17$ $$2 + T$$
$19$ $$8 + T$$
$23$ $$8 + T$$
$29$ $$-2 + T$$
$31$ $$-4 + T$$
$37$ $$2 + T$$
$41$ $$-6 + T$$
$43$ $$-4 + T$$
$47$ $$8 + T$$
$53$ $$10 + T$$
$59$ $$4 + T$$
$61$ $$2 + T$$
$67$ $$-4 + T$$
$71$ $$-12 + T$$
$73$ $$2 + T$$
$79$ $$-8 + T$$
$83$ $$-4 + T$$
$89$ $$-6 + T$$
$97$ $$18 + T$$