# Properties

 Label 32.3.c.a Level $32$ Weight $3$ Character orbit 32.c Analytic conductor $0.872$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$32 = 2^{5}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 32.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.871936845953$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 4 i q^{3} + 2 q^{5} -8 i q^{7} -7 q^{9} +O(q^{10})$$ $$q + 4 i q^{3} + 2 q^{5} -8 i q^{7} -7 q^{9} -4 i q^{11} -14 q^{13} + 8 i q^{15} + 18 q^{17} -12 i q^{19} + 32 q^{21} + 40 i q^{23} -21 q^{25} + 8 i q^{27} -14 q^{29} -32 i q^{31} + 16 q^{33} -16 i q^{35} -30 q^{37} -56 i q^{39} -14 q^{41} + 28 i q^{43} -14 q^{45} + 16 i q^{47} -15 q^{49} + 72 i q^{51} + 66 q^{53} -8 i q^{55} + 48 q^{57} -52 i q^{59} + 82 q^{61} + 56 i q^{63} -28 q^{65} + 4 i q^{67} -160 q^{69} + 56 i q^{71} + 66 q^{73} -84 i q^{75} -32 q^{77} -16 i q^{79} -95 q^{81} -140 i q^{83} + 36 q^{85} -56 i q^{87} -30 q^{89} + 112 i q^{91} + 128 q^{93} -24 i q^{95} -14 q^{97} + 28 i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 4q^{5} - 14q^{9} + O(q^{10})$$ $$2q + 4q^{5} - 14q^{9} - 28q^{13} + 36q^{17} + 64q^{21} - 42q^{25} - 28q^{29} + 32q^{33} - 60q^{37} - 28q^{41} - 28q^{45} - 30q^{49} + 132q^{53} + 96q^{57} + 164q^{61} - 56q^{65} - 320q^{69} + 132q^{73} - 64q^{77} - 190q^{81} + 72q^{85} - 60q^{89} + 256q^{93} - 28q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
31.1
 − 1.00000i 1.00000i
0 4.00000i 0 2.00000 0 8.00000i 0 −7.00000 0
31.2 0 4.00000i 0 2.00000 0 8.00000i 0 −7.00000 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
4.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 32.3.c.a 2
3.b odd 2 1 288.3.g.b 2
4.b odd 2 1 inner 32.3.c.a 2
5.b even 2 1 800.3.b.a 2
5.c odd 4 1 800.3.h.a 2
5.c odd 4 1 800.3.h.b 2
7.b odd 2 1 1568.3.d.b 2
8.b even 2 1 64.3.c.b 2
8.d odd 2 1 64.3.c.b 2
12.b even 2 1 288.3.g.b 2
16.e even 4 1 256.3.d.a 2
16.e even 4 1 256.3.d.c 2
16.f odd 4 1 256.3.d.a 2
16.f odd 4 1 256.3.d.c 2
20.d odd 2 1 800.3.b.a 2
20.e even 4 1 800.3.h.a 2
20.e even 4 1 800.3.h.b 2
24.f even 2 1 576.3.g.g 2
24.h odd 2 1 576.3.g.g 2
28.d even 2 1 1568.3.d.b 2
40.e odd 2 1 1600.3.b.e 2
40.f even 2 1 1600.3.b.e 2
40.i odd 4 1 1600.3.h.a 2
40.i odd 4 1 1600.3.h.c 2
40.k even 4 1 1600.3.h.a 2
40.k even 4 1 1600.3.h.c 2
48.i odd 4 1 2304.3.b.c 2
48.i odd 4 1 2304.3.b.g 2
48.k even 4 1 2304.3.b.c 2
48.k even 4 1 2304.3.b.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.3.c.a 2 1.a even 1 1 trivial
32.3.c.a 2 4.b odd 2 1 inner
64.3.c.b 2 8.b even 2 1
64.3.c.b 2 8.d odd 2 1
256.3.d.a 2 16.e even 4 1
256.3.d.a 2 16.f odd 4 1
256.3.d.c 2 16.e even 4 1
256.3.d.c 2 16.f odd 4 1
288.3.g.b 2 3.b odd 2 1
288.3.g.b 2 12.b even 2 1
576.3.g.g 2 24.f even 2 1
576.3.g.g 2 24.h odd 2 1
800.3.b.a 2 5.b even 2 1
800.3.b.a 2 20.d odd 2 1
800.3.h.a 2 5.c odd 4 1
800.3.h.a 2 20.e even 4 1
800.3.h.b 2 5.c odd 4 1
800.3.h.b 2 20.e even 4 1
1568.3.d.b 2 7.b odd 2 1
1568.3.d.b 2 28.d even 2 1
1600.3.b.e 2 40.e odd 2 1
1600.3.b.e 2 40.f even 2 1
1600.3.h.a 2 40.i odd 4 1
1600.3.h.a 2 40.k even 4 1
1600.3.h.c 2 40.i odd 4 1
1600.3.h.c 2 40.k even 4 1
2304.3.b.c 2 48.i odd 4 1
2304.3.b.c 2 48.k even 4 1
2304.3.b.g 2 48.i odd 4 1
2304.3.b.g 2 48.k even 4 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$16 + T^{2}$$
$5$ $$( -2 + T )^{2}$$
$7$ $$64 + T^{2}$$
$11$ $$16 + T^{2}$$
$13$ $$( 14 + T )^{2}$$
$17$ $$( -18 + T )^{2}$$
$19$ $$144 + T^{2}$$
$23$ $$1600 + T^{2}$$
$29$ $$( 14 + T )^{2}$$
$31$ $$1024 + T^{2}$$
$37$ $$( 30 + T )^{2}$$
$41$ $$( 14 + T )^{2}$$
$43$ $$784 + T^{2}$$
$47$ $$256 + T^{2}$$
$53$ $$( -66 + T )^{2}$$
$59$ $$2704 + T^{2}$$
$61$ $$( -82 + T )^{2}$$
$67$ $$16 + T^{2}$$
$71$ $$3136 + T^{2}$$
$73$ $$( -66 + T )^{2}$$
$79$ $$256 + T^{2}$$
$83$ $$19600 + T^{2}$$
$89$ $$( 30 + T )^{2}$$
$97$ $$( 14 + T )^{2}$$