# Properties

 Label 32.6.a.d Level 32 Weight 6 Character orbit 32.a Self dual yes Analytic conductor 5.132 Analytic rank 0 Dimension 2 CM no 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 16\sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 16 \beta q^{3} + 46 q^{5} -96 \beta q^{7} + 525 q^{9} +O(q^{10})$$ $$q + 16 \beta q^{3} + 46 q^{5} -96 \beta q^{7} + 525 q^{9} + 48 \beta q^{11} -42 q^{13} + 736 \beta q^{15} + 962 q^{17} -1200 \beta q^{19} -4608 q^{21} -1824 \beta q^{23} -1009 q^{25} + 4512 \beta q^{27} -2554 q^{29} + 1152 \beta q^{31} + 2304 q^{33} -4416 \beta q^{35} + 11950 q^{37} -672 \beta q^{39} -5078 q^{41} -7248 \beta q^{43} + 24150 q^{45} + 7104 \beta q^{47} + 10841 q^{49} + 15392 \beta q^{51} -19714 q^{53} + 2208 \beta q^{55} -57600 q^{57} -5136 \beta q^{59} + 29318 q^{61} -50400 \beta q^{63} -1932 q^{65} + 9744 \beta q^{67} -87552 q^{69} + 46752 \beta q^{71} + 37914 q^{73} -16144 \beta q^{75} -13824 q^{77} + 51264 \beta q^{79} + 89001 q^{81} -22704 \beta q^{83} + 44252 q^{85} -40864 \beta q^{87} + 13930 q^{89} + 4032 \beta q^{91} + 55296 q^{93} -55200 \beta q^{95} + 163602 q^{97} + 25200 \beta q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 92q^{5} + 1050q^{9} + O(q^{10})$$ $$2q + 92q^{5} + 1050q^{9} - 84q^{13} + 1924q^{17} - 9216q^{21} - 2018q^{25} - 5108q^{29} + 4608q^{33} + 23900q^{37} - 10156q^{41} + 48300q^{45} + 21682q^{49} - 39428q^{53} - 115200q^{57} + 58636q^{61} - 3864q^{65} - 175104q^{69} + 75828q^{73} - 27648q^{77} + 178002q^{81} + 88504q^{85} + 27860q^{89} + 110592q^{93} + 327204q^{97} + O(q^{100})$$

## 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
 −1.73205 1.73205
0 −27.7128 0 46.0000 0 166.277 0 525.000 0
1.2 0 27.7128 0 46.0000 0 −166.277 0 525.000 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 inner

## Twists

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

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.6.a.d 2 1.a even 1 1 trivial
32.6.a.d 2 4.b odd 2 1 inner
64.6.a.h 2 8.b even 2 1
64.6.a.h 2 8.d odd 2 1
256.6.b.l 4 16.e even 4 2
256.6.b.l 4 16.f odd 4 2
288.6.a.l 2 3.b odd 2 1
288.6.a.l 2 12.b even 2 1
576.6.a.bp 2 24.f even 2 1
576.6.a.bp 2 24.h odd 2 1
800.6.a.k 2 5.b even 2 1
800.6.a.k 2 20.d odd 2 1
800.6.c.d 4 5.c odd 4 2
800.6.c.d 4 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 $$T_{3}^{2} - 768$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(32))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - 282 T^{2} + 59049 T^{4}$$
$5$ $$( 1 - 46 T + 3125 T^{2} )^{2}$$
$7$ $$1 + 5966 T^{2} + 282475249 T^{4}$$
$11$ $$1 + 315190 T^{2} + 25937424601 T^{4}$$
$13$ $$( 1 + 42 T + 371293 T^{2} )^{2}$$
$17$ $$( 1 - 962 T + 1419857 T^{2} )^{2}$$
$19$ $$1 + 632198 T^{2} + 6131066257801 T^{4}$$
$23$ $$1 + 2891758 T^{2} + 41426511213649 T^{4}$$
$29$ $$( 1 + 2554 T + 20511149 T^{2} )^{2}$$
$31$ $$1 + 53276990 T^{2} + 819628286980801 T^{4}$$
$37$ $$( 1 - 11950 T + 69343957 T^{2} )^{2}$$
$41$ $$( 1 + 5078 T + 115856201 T^{2} )^{2}$$
$43$ $$1 + 136416374 T^{2} + 21611482313284249 T^{4}$$
$47$ $$1 + 307289566 T^{2} + 52599132235830049 T^{4}$$
$53$ $$( 1 + 19714 T + 418195493 T^{2} )^{2}$$
$59$ $$1 + 1350713110 T^{2} + 511116753300641401 T^{4}$$
$61$ $$( 1 - 29318 T + 844596301 T^{2} )^{2}$$
$67$ $$1 + 2415413606 T^{2} + 1822837804551761449 T^{4}$$
$71$ $$1 - 2948789810 T^{2} + 3255243551009881201 T^{4}$$
$73$ $$( 1 - 37914 T + 2073071593 T^{2} )^{2}$$
$79$ $$1 - 1729880290 T^{2} + 9468276082626847201 T^{4}$$
$83$ $$1 + 6331666438 T^{2} + 15516041187205853449 T^{4}$$
$89$ $$( 1 - 13930 T + 5584059449 T^{2} )^{2}$$
$97$ $$( 1 - 163602 T + 8587340257 T^{2} )^{2}$$