# Properties

 Label 315.4.a.d Level 315 Weight 4 Character orbit 315.a Self dual yes Analytic conductor 18.586 Analytic rank 0 Dimension 1 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$315 = 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 315.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$18.5856016518$$ Analytic rank: $$0$$ 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

 $$f(q)$$ $$=$$ $$q - 8q^{4} - 5q^{5} + 7q^{7} + O(q^{10})$$ $$q - 8q^{4} - 5q^{5} + 7q^{7} - 42q^{11} + 20q^{13} + 64q^{16} - 66q^{17} + 38q^{19} + 40q^{20} - 12q^{23} + 25q^{25} - 56q^{28} + 258q^{29} + 146q^{31} - 35q^{35} + 434q^{37} + 282q^{41} + 20q^{43} + 336q^{44} + 72q^{47} + 49q^{49} - 160q^{52} - 336q^{53} + 210q^{55} + 360q^{59} - 682q^{61} - 512q^{64} - 100q^{65} + 812q^{67} + 528q^{68} - 810q^{71} - 124q^{73} - 304q^{76} - 294q^{77} + 1136q^{79} - 320q^{80} - 156q^{83} + 330q^{85} + 1038q^{89} + 140q^{91} + 96q^{92} - 190q^{95} + 1208q^{97} + O(q^{100})$$

## 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 −8.00000 −5.00000 0 7.00000 0 0 0
 $$n$$: 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 315.4.a.d 1
3.b odd 2 1 105.4.a.a 1
5.b even 2 1 1575.4.a.f 1
7.b odd 2 1 2205.4.a.o 1
12.b even 2 1 1680.4.a.s 1
15.d odd 2 1 525.4.a.e 1
15.e even 4 2 525.4.d.f 2
21.c even 2 1 735.4.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.a 1 3.b odd 2 1
315.4.a.d 1 1.a even 1 1 trivial
525.4.a.e 1 15.d odd 2 1
525.4.d.f 2 15.e even 4 2
735.4.a.c 1 21.c even 2 1
1575.4.a.f 1 5.b even 2 1
1680.4.a.s 1 12.b even 2 1
2205.4.a.o 1 7.b odd 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 $$T_{2}$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(315))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 8 T^{2}$$
$3$ 1
$5$ $$1 + 5 T$$
$7$ $$1 - 7 T$$
$11$ $$1 + 42 T + 1331 T^{2}$$
$13$ $$1 - 20 T + 2197 T^{2}$$
$17$ $$1 + 66 T + 4913 T^{2}$$
$19$ $$1 - 38 T + 6859 T^{2}$$
$23$ $$1 + 12 T + 12167 T^{2}$$
$29$ $$1 - 258 T + 24389 T^{2}$$
$31$ $$1 - 146 T + 29791 T^{2}$$
$37$ $$1 - 434 T + 50653 T^{2}$$
$41$ $$1 - 282 T + 68921 T^{2}$$
$43$ $$1 - 20 T + 79507 T^{2}$$
$47$ $$1 - 72 T + 103823 T^{2}$$
$53$ $$1 + 336 T + 148877 T^{2}$$
$59$ $$1 - 360 T + 205379 T^{2}$$
$61$ $$1 + 682 T + 226981 T^{2}$$
$67$ $$1 - 812 T + 300763 T^{2}$$
$71$ $$1 + 810 T + 357911 T^{2}$$
$73$ $$1 + 124 T + 389017 T^{2}$$
$79$ $$1 - 1136 T + 493039 T^{2}$$
$83$ $$1 + 156 T + 571787 T^{2}$$
$89$ $$1 - 1038 T + 704969 T^{2}$$
$97$ $$1 - 1208 T + 912673 T^{2}$$