# Properties

 Label 315.5.e.b Level $315$ Weight $5$ Character orbit 315.e Self dual yes Analytic conductor $32.562$ Analytic rank $0$ Dimension $1$ CM discriminant -35 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$315 = 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 315.e (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$32.5615383714$$ 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) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 16q^{4} + 25q^{5} - 49q^{7} + O(q^{10})$$ $$q + 16q^{4} + 25q^{5} - 49q^{7} + 73q^{11} - 23q^{13} + 256q^{16} + 263q^{17} + 400q^{20} + 625q^{25} - 784q^{28} + 1153q^{29} - 1225q^{35} + 1168q^{44} - 3457q^{47} + 2401q^{49} - 368q^{52} + 1825q^{55} + 4096q^{64} - 575q^{65} + 4208q^{68} + 10078q^{71} + 9502q^{73} - 3577q^{77} + 12167q^{79} + 6400q^{80} - 6382q^{83} + 6575q^{85} + 1127q^{91} - 3383q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/315\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$136$$ $$281$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
244.1
 0
0 0 16.0000 25.0000 0 −49.0000 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
35.c odd 2 1 CM by $$\Q(\sqrt{-35})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.5.e.b 1
3.b odd 2 1 35.5.c.b yes 1
5.b even 2 1 315.5.e.a 1
7.b odd 2 1 315.5.e.a 1
12.b even 2 1 560.5.p.a 1
15.d odd 2 1 35.5.c.a 1
15.e even 4 2 175.5.d.c 2
21.c even 2 1 35.5.c.a 1
35.c odd 2 1 CM 315.5.e.b 1
60.h even 2 1 560.5.p.b 1
84.h odd 2 1 560.5.p.b 1
105.g even 2 1 35.5.c.b yes 1
105.k odd 4 2 175.5.d.c 2
420.o odd 2 1 560.5.p.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.5.c.a 1 15.d odd 2 1
35.5.c.a 1 21.c even 2 1
35.5.c.b yes 1 3.b odd 2 1
35.5.c.b yes 1 105.g even 2 1
175.5.d.c 2 15.e even 4 2
175.5.d.c 2 105.k odd 4 2
315.5.e.a 1 5.b even 2 1
315.5.e.a 1 7.b odd 2 1
315.5.e.b 1 1.a even 1 1 trivial
315.5.e.b 1 35.c odd 2 1 CM
560.5.p.a 1 12.b even 2 1
560.5.p.a 1 420.o odd 2 1
560.5.p.b 1 60.h even 2 1
560.5.p.b 1 84.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{5}^{\mathrm{new}}(315, [\chi])$$:

 $$T_{2}$$ $$T_{13} + 23$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$-25 + T$$
$7$ $$49 + T$$
$11$ $$-73 + T$$
$13$ $$23 + T$$
$17$ $$-263 + T$$
$19$ $$T$$
$23$ $$T$$
$29$ $$-1153 + T$$
$31$ $$T$$
$37$ $$T$$
$41$ $$T$$
$43$ $$T$$
$47$ $$3457 + T$$
$53$ $$T$$
$59$ $$T$$
$61$ $$T$$
$67$ $$T$$
$71$ $$-10078 + T$$
$73$ $$-9502 + T$$
$79$ $$-12167 + T$$
$83$ $$6382 + T$$
$89$ $$T$$
$97$ $$3383 + T$$