# Properties

 Label 32.3.d.a Level $32$ Weight $3$ Character orbit 32.d Self dual yes Analytic conductor $0.872$ Analytic rank $0$ Dimension $1$ CM discriminant -8 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$32 = 2^{5}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 32.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.871936845953$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 8) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 2q^{3} - 5q^{9} + O(q^{10})$$ $$q + 2q^{3} - 5q^{9} - 14q^{11} + 2q^{17} + 34q^{19} + 25q^{25} - 28q^{27} - 28q^{33} - 46q^{41} - 14q^{43} + 49q^{49} + 4q^{51} + 68q^{57} + 82q^{59} - 62q^{67} - 142q^{73} + 50q^{75} - 11q^{81} - 158q^{83} + 146q^{89} - 94q^{97} + 70q^{99} + O(q^{100})$$

## Expression as an eta quotient

$$f(z) = \dfrac{\eta(4z)^{5}\eta(8z)^{5}}{\eta(2z)^{2}\eta(16z)^{2}}=q\prod_{n=1}^\infty(1 - q^{2n})^{-2}(1 - q^{4n})^{5}(1 - q^{8n})^{5}(1 - q^{16n})^{-2}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/32\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$31$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
15.1
 0
0 2.00000 0 0 0 0 0 −5.00000 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
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 32.3.d.a 1
3.b odd 2 1 288.3.b.a 1
4.b odd 2 1 8.3.d.a 1
5.b even 2 1 800.3.g.a 1
5.c odd 4 2 800.3.e.a 2
7.b odd 2 1 1568.3.g.a 1
8.b even 2 1 8.3.d.a 1
8.d odd 2 1 CM 32.3.d.a 1
12.b even 2 1 72.3.b.a 1
16.e even 4 2 256.3.c.e 2
16.f odd 4 2 256.3.c.e 2
20.d odd 2 1 200.3.g.a 1
20.e even 4 2 200.3.e.a 2
24.f even 2 1 288.3.b.a 1
24.h odd 2 1 72.3.b.a 1
28.d even 2 1 392.3.g.a 1
28.f even 6 2 392.3.k.b 2
28.g odd 6 2 392.3.k.d 2
40.e odd 2 1 800.3.g.a 1
40.f even 2 1 200.3.g.a 1
40.i odd 4 2 200.3.e.a 2
40.k even 4 2 800.3.e.a 2
48.i odd 4 2 2304.3.g.j 2
48.k even 4 2 2304.3.g.j 2
56.e even 2 1 1568.3.g.a 1
56.h odd 2 1 392.3.g.a 1
56.j odd 6 2 392.3.k.b 2
56.p even 6 2 392.3.k.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.3.d.a 1 4.b odd 2 1
8.3.d.a 1 8.b even 2 1
32.3.d.a 1 1.a even 1 1 trivial
32.3.d.a 1 8.d odd 2 1 CM
72.3.b.a 1 12.b even 2 1
72.3.b.a 1 24.h odd 2 1
200.3.e.a 2 20.e even 4 2
200.3.e.a 2 40.i odd 4 2
200.3.g.a 1 20.d odd 2 1
200.3.g.a 1 40.f even 2 1
256.3.c.e 2 16.e even 4 2
256.3.c.e 2 16.f odd 4 2
288.3.b.a 1 3.b odd 2 1
288.3.b.a 1 24.f even 2 1
392.3.g.a 1 28.d even 2 1
392.3.g.a 1 56.h odd 2 1
392.3.k.b 2 28.f even 6 2
392.3.k.b 2 56.j odd 6 2
392.3.k.d 2 28.g odd 6 2
392.3.k.d 2 56.p even 6 2
800.3.e.a 2 5.c odd 4 2
800.3.e.a 2 40.k even 4 2
800.3.g.a 1 5.b even 2 1
800.3.g.a 1 40.e odd 2 1
1568.3.g.a 1 7.b odd 2 1
1568.3.g.a 1 56.e even 2 1
2304.3.g.j 2 48.i odd 4 2
2304.3.g.j 2 48.k even 4 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(32, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$-2 + T$$
$5$ $$T$$
$7$ $$T$$
$11$ $$14 + T$$
$13$ $$T$$
$17$ $$-2 + T$$
$19$ $$-34 + T$$
$23$ $$T$$
$29$ $$T$$
$31$ $$T$$
$37$ $$T$$
$41$ $$46 + T$$
$43$ $$14 + T$$
$47$ $$T$$
$53$ $$T$$
$59$ $$-82 + T$$
$61$ $$T$$
$67$ $$62 + T$$
$71$ $$T$$
$73$ $$142 + T$$
$79$ $$T$$
$83$ $$158 + T$$
$89$ $$-146 + T$$
$97$ $$94 + T$$