# Properties

 Label 32.6.a.b Level 32 Weight 6 Character orbit 32.a Self dual yes Analytic conductor 5.132 Analytic rank 1 Dimension 1 CM discriminant -4 Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$32 = 2^{5}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 32.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$5.13228223402$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $N(\mathrm{U}(1))$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 82q^{5} - 243q^{9} + O(q^{10})$$ $$q - 82q^{5} - 243q^{9} - 1194q^{13} + 2242q^{17} + 3599q^{25} + 2950q^{29} - 12242q^{37} - 20950q^{41} + 19926q^{45} - 16807q^{49} + 7294q^{53} + 18950q^{61} + 97908q^{65} - 88806q^{73} + 59049q^{81} - 183844q^{85} + 51050q^{89} - 92142q^{97} + O(q^{100})$$

## 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 0 −82.0000 0 0 0 −243.000 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
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 32.6.a.b 1
3.b odd 2 1 288.6.a.i 1
4.b odd 2 1 CM 32.6.a.b 1
5.b even 2 1 800.6.a.d 1
5.c odd 4 2 800.6.c.c 2
8.b even 2 1 64.6.a.d 1
8.d odd 2 1 64.6.a.d 1
12.b even 2 1 288.6.a.i 1
16.e even 4 2 256.6.b.e 2
16.f odd 4 2 256.6.b.e 2
20.d odd 2 1 800.6.a.d 1
20.e even 4 2 800.6.c.c 2
24.f even 2 1 576.6.a.e 1
24.h odd 2 1 576.6.a.e 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.6.a.b 1 1.a even 1 1 trivial
32.6.a.b 1 4.b odd 2 1 CM
64.6.a.d 1 8.b even 2 1
64.6.a.d 1 8.d odd 2 1
256.6.b.e 2 16.e even 4 2
256.6.b.e 2 16.f odd 4 2
288.6.a.i 1 3.b odd 2 1
288.6.a.i 1 12.b even 2 1
576.6.a.e 1 24.f even 2 1
576.6.a.e 1 24.h odd 2 1
800.6.a.d 1 5.b even 2 1
800.6.a.d 1 20.d odd 2 1
800.6.c.c 2 5.c odd 4 2
800.6.c.c 2 20.e even 4 2

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(32))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + 243 T^{2}$$
$5$ $$1 + 82 T + 3125 T^{2}$$
$7$ $$1 + 16807 T^{2}$$
$11$ $$1 + 161051 T^{2}$$
$13$ $$1 + 1194 T + 371293 T^{2}$$
$17$ $$1 - 2242 T + 1419857 T^{2}$$
$19$ $$1 + 2476099 T^{2}$$
$23$ $$1 + 6436343 T^{2}$$
$29$ $$1 - 2950 T + 20511149 T^{2}$$
$31$ $$1 + 28629151 T^{2}$$
$37$ $$1 + 12242 T + 69343957 T^{2}$$
$41$ $$1 + 20950 T + 115856201 T^{2}$$
$43$ $$1 + 147008443 T^{2}$$
$47$ $$1 + 229345007 T^{2}$$
$53$ $$1 - 7294 T + 418195493 T^{2}$$
$59$ $$1 + 714924299 T^{2}$$
$61$ $$1 - 18950 T + 844596301 T^{2}$$
$67$ $$1 + 1350125107 T^{2}$$
$71$ $$1 + 1804229351 T^{2}$$
$73$ $$1 + 88806 T + 2073071593 T^{2}$$
$79$ $$1 + 3077056399 T^{2}$$
$83$ $$1 + 3939040643 T^{2}$$
$89$ $$1 - 51050 T + 5584059449 T^{2}$$
$97$ $$1 + 92142 T + 8587340257 T^{2}$$