# Properties

 Label 32.2.g.a Level 32 Weight 2 Character orbit 32.g Analytic conductor 0.256 Analytic rank 0 Dimension 4 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$32 = 2^{5}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 32.g (of order $$8$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.255521286468$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{2} + ( \zeta_{8} + \zeta_{8}^{2} ) q^{3} -2 q^{4} + ( -1 - \zeta_{8} - 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{5} + ( 1 + \zeta_{8} - \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{6} + ( 1 + \zeta_{8}^{2} ) q^{7} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{8} + ( -1 + \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{9} +O(q^{10})$$ $$q + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{2} + ( \zeta_{8} + \zeta_{8}^{2} ) q^{3} -2 q^{4} + ( -1 - \zeta_{8} - 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{5} + ( 1 + \zeta_{8} - \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{6} + ( 1 + \zeta_{8}^{2} ) q^{7} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{8} + ( -1 + \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{9} + ( 1 - \zeta_{8} + 3 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{10} + ( -2 + 2 \zeta_{8} - \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{11} + ( -2 \zeta_{8} - 2 \zeta_{8}^{2} ) q^{12} + ( 1 - \zeta_{8}^{3} ) q^{13} -2 \zeta_{8}^{3} q^{14} + ( -3 \zeta_{8} - 2 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{15} + 4 q^{16} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{17} + ( -1 + 2 \zeta_{8} - \zeta_{8}^{2} ) q^{18} + ( -2 - 3 \zeta_{8} + 3 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{19} + ( 2 + 2 \zeta_{8} + 4 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{20} + ( -1 + \zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{21} + ( 1 + \zeta_{8} - 3 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{22} + ( 3 - 3 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{23} + ( -2 - 2 \zeta_{8} + 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{24} + ( 1 + 5 \zeta_{8} + \zeta_{8}^{2} ) q^{25} + ( -1 - \zeta_{8} - \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{26} + ( 3 + 3 \zeta_{8} - \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{27} + ( -2 - 2 \zeta_{8}^{2} ) q^{28} + ( -1 + 2 \zeta_{8} + 2 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{29} + ( -6 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{30} -4 q^{31} + ( -4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{32} + ( 2 - \zeta_{8} + \zeta_{8}^{3} ) q^{33} -4 q^{34} + ( 1 - 3 \zeta_{8} - 3 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{35} + ( 2 - 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{36} + ( 1 + \zeta_{8} ) q^{37} + ( -1 + 5 \zeta_{8} + 5 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{38} + ( 1 + 2 \zeta_{8} + \zeta_{8}^{2} ) q^{39} + ( -2 + 2 \zeta_{8} - 6 \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{40} + ( -3 + 3 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{41} + ( 2 + 2 \zeta_{8} ) q^{42} + ( 4 - 4 \zeta_{8} - 3 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{43} + ( 4 - 4 \zeta_{8} + 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{44} + ( 2 - 3 \zeta_{8} + 3 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{45} + ( 4 - 6 \zeta_{8} + 4 \zeta_{8}^{2} ) q^{46} + ( 4 \zeta_{8} + 6 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{47} + ( 4 \zeta_{8} + 4 \zeta_{8}^{2} ) q^{48} -5 \zeta_{8}^{2} q^{49} + ( 5 - 5 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{50} + ( 2 + 2 \zeta_{8} - 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{51} + ( -2 + 2 \zeta_{8}^{3} ) q^{52} + ( 1 - \zeta_{8} - 4 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{53} + ( 4 - 4 \zeta_{8} - 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{54} + ( -5 + 5 \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{55} + 4 \zeta_{8}^{3} q^{56} + ( -5 - 4 \zeta_{8} - 5 \zeta_{8}^{2} ) q^{57} + ( 1 + 3 \zeta_{8} - 3 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{58} + ( -4 - 4 \zeta_{8} + \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{59} + ( 6 \zeta_{8} + 4 \zeta_{8}^{2} + 6 \zeta_{8}^{3} ) q^{60} + ( 1 + \zeta_{8}^{3} ) q^{61} + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{62} + ( -2 + \zeta_{8} - \zeta_{8}^{3} ) q^{63} -8 q^{64} + ( -2 - 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{65} + ( -2 \zeta_{8} + 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{66} + ( -2 + 3 \zeta_{8} + 3 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{67} + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{68} + ( -1 - \zeta_{8} + 3 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{69} + ( -2 - 4 \zeta_{8} + 4 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{70} + ( -3 + 4 \zeta_{8} - 3 \zeta_{8}^{2} ) q^{71} + ( 2 - 4 \zeta_{8} + 2 \zeta_{8}^{2} ) q^{72} + ( 7 - 7 \zeta_{8}^{2} ) q^{73} + ( 1 - \zeta_{8} - \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{74} + ( -1 + \zeta_{8} + 6 \zeta_{8}^{2} + 6 \zeta_{8}^{3} ) q^{75} + ( 4 + 6 \zeta_{8} - 6 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{76} + ( -1 + 3 \zeta_{8} - 3 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{77} + ( 2 - 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{78} -6 \zeta_{8}^{2} q^{79} + ( -4 - 4 \zeta_{8} - 8 \zeta_{8}^{2} + 8 \zeta_{8}^{3} ) q^{80} + ( 5 \zeta_{8} + 3 \zeta_{8}^{2} + 5 \zeta_{8}^{3} ) q^{81} + ( 4 + 6 \zeta_{8} + 4 \zeta_{8}^{2} ) q^{82} + ( 4 - \zeta_{8} + \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{83} + ( 2 - 2 \zeta_{8} - 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{84} + ( 2 - 2 \zeta_{8} + 6 \zeta_{8}^{2} + 6 \zeta_{8}^{3} ) q^{85} + ( -7 - 7 \zeta_{8} + \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{86} + ( -1 + \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{87} + ( -2 - 2 \zeta_{8} + 6 \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{88} + ( 3 - 8 \zeta_{8} + 3 \zeta_{8}^{2} ) q^{89} + ( -5 + \zeta_{8} + \zeta_{8}^{2} - 5 \zeta_{8}^{3} ) q^{90} + ( 1 + \zeta_{8} + \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{91} + ( -6 + 6 \zeta_{8}^{2} - 8 \zeta_{8}^{3} ) q^{92} + ( -4 \zeta_{8} - 4 \zeta_{8}^{2} ) q^{93} + ( 8 + 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{94} + ( 16 + 3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{95} + ( 4 + 4 \zeta_{8} - 4 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{96} + ( -10 + 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{97} + ( -5 \zeta_{8} + 5 \zeta_{8}^{3} ) q^{98} + ( 5 - 2 \zeta_{8} - 2 \zeta_{8}^{2} + 5 \zeta_{8}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 8q^{4} - 4q^{5} + 4q^{6} + 4q^{7} - 4q^{9} + O(q^{10})$$ $$4q - 8q^{4} - 4q^{5} + 4q^{6} + 4q^{7} - 4q^{9} + 4q^{10} - 8q^{11} + 4q^{13} + 16q^{16} - 4q^{18} - 8q^{19} + 8q^{20} - 4q^{21} + 4q^{22} + 12q^{23} - 8q^{24} + 4q^{25} - 4q^{26} + 12q^{27} - 8q^{28} - 4q^{29} - 24q^{30} - 16q^{31} + 8q^{33} - 16q^{34} + 4q^{35} + 8q^{36} + 4q^{37} - 4q^{38} + 4q^{39} - 8q^{40} - 12q^{41} + 8q^{42} + 16q^{43} + 16q^{44} + 8q^{45} + 16q^{46} + 20q^{50} + 8q^{51} - 8q^{52} + 4q^{53} + 16q^{54} - 20q^{55} - 20q^{57} + 4q^{58} - 16q^{59} + 4q^{61} - 8q^{63} - 32q^{64} - 8q^{65} - 8q^{67} - 4q^{69} - 8q^{70} - 12q^{71} + 8q^{72} + 28q^{73} + 4q^{74} - 4q^{75} + 16q^{76} - 4q^{77} + 8q^{78} - 16q^{80} + 16q^{82} + 16q^{83} + 8q^{84} + 8q^{85} - 28q^{86} - 4q^{87} - 8q^{88} + 12q^{89} - 20q^{90} + 4q^{91} - 24q^{92} + 32q^{94} + 64q^{95} + 16q^{96} - 40q^{97} + 20q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$5$$ $$31$$ $$\chi(n)$$ $$\zeta_{8}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1
 0.707107 + 0.707107i 0.707107 − 0.707107i −0.707107 − 0.707107i −0.707107 + 0.707107i
1.41421i 0.707107 + 1.70711i −2.00000 −3.12132 1.29289i 2.41421 1.00000i 1.00000 + 1.00000i 2.82843i −0.292893 + 0.292893i −1.82843 + 4.41421i
13.1 1.41421i 0.707107 1.70711i −2.00000 −3.12132 + 1.29289i 2.41421 + 1.00000i 1.00000 1.00000i 2.82843i −0.292893 0.292893i −1.82843 4.41421i
21.1 1.41421i −0.707107 + 0.292893i −2.00000 1.12132 2.70711i −0.414214 1.00000i 1.00000 + 1.00000i 2.82843i −1.70711 + 1.70711i 3.82843 + 1.58579i
29.1 1.41421i −0.707107 0.292893i −2.00000 1.12132 + 2.70711i −0.414214 + 1.00000i 1.00000 1.00000i 2.82843i −1.70711 1.70711i 3.82843 1.58579i
 $$n$$: 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
32.g even 8 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 32.2.g.a 4
3.b odd 2 1 288.2.v.a 4
4.b odd 2 1 128.2.g.a 4
5.b even 2 1 800.2.y.a 4
5.c odd 4 1 800.2.ba.a 4
5.c odd 4 1 800.2.ba.b 4
8.b even 2 1 256.2.g.b 4
8.d odd 2 1 256.2.g.a 4
12.b even 2 1 1152.2.v.a 4
16.e even 4 1 512.2.g.a 4
16.e even 4 1 512.2.g.d 4
16.f odd 4 1 512.2.g.b 4
16.f odd 4 1 512.2.g.c 4
32.g even 8 1 inner 32.2.g.a 4
32.g even 8 1 256.2.g.b 4
32.g even 8 1 512.2.g.a 4
32.g even 8 1 512.2.g.d 4
32.h odd 8 1 128.2.g.a 4
32.h odd 8 1 256.2.g.a 4
32.h odd 8 1 512.2.g.b 4
32.h odd 8 1 512.2.g.c 4
64.i even 16 2 4096.2.a.e 4
64.j odd 16 2 4096.2.a.f 4
96.o even 8 1 1152.2.v.a 4
96.p odd 8 1 288.2.v.a 4
160.v odd 8 1 800.2.ba.b 4
160.z even 8 1 800.2.y.a 4
160.bb odd 8 1 800.2.ba.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.2.g.a 4 1.a even 1 1 trivial
32.2.g.a 4 32.g even 8 1 inner
128.2.g.a 4 4.b odd 2 1
128.2.g.a 4 32.h odd 8 1
256.2.g.a 4 8.d odd 2 1
256.2.g.a 4 32.h odd 8 1
256.2.g.b 4 8.b even 2 1
256.2.g.b 4 32.g even 8 1
288.2.v.a 4 3.b odd 2 1
288.2.v.a 4 96.p odd 8 1
512.2.g.a 4 16.e even 4 1
512.2.g.a 4 32.g even 8 1
512.2.g.b 4 16.f odd 4 1
512.2.g.b 4 32.h odd 8 1
512.2.g.c 4 16.f odd 4 1
512.2.g.c 4 32.h odd 8 1
512.2.g.d 4 16.e even 4 1
512.2.g.d 4 32.g even 8 1
800.2.y.a 4 5.b even 2 1
800.2.y.a 4 160.z even 8 1
800.2.ba.a 4 5.c odd 4 1
800.2.ba.a 4 160.bb odd 8 1
800.2.ba.b 4 5.c odd 4 1
800.2.ba.b 4 160.v odd 8 1
1152.2.v.a 4 12.b even 2 1
1152.2.v.a 4 96.o even 8 1
4096.2.a.e 4 64.i even 16 2
4096.2.a.f 4 64.j odd 16 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 2 T_{3}^{2} + 4 T_{3} + 2$$ acting on $$S_{2}^{\mathrm{new}}(32, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + 2 T^{2} )^{2}$$
$3$ $$( 1 + 2 T + 3 T^{2} )^{2}( 1 - 4 T + 8 T^{2} - 12 T^{3} + 9 T^{4} )$$
$5$ $$( 1 + 2 T + 5 T^{2} )^{2}( 1 - 8 T^{2} + 25 T^{4} )$$
$7$ $$( 1 - 2 T + 2 T^{2} - 14 T^{3} + 49 T^{4} )^{2}$$
$11$ $$( 1 + 6 T + 11 T^{2} )^{2}( 1 - 4 T + 8 T^{2} - 44 T^{3} + 121 T^{4} )$$
$13$ $$1 - 4 T + 6 T^{2} - 4 T^{3} + 2 T^{4} - 52 T^{5} + 1014 T^{6} - 8788 T^{7} + 28561 T^{8}$$
$17$ $$( 1 - 26 T^{2} + 289 T^{4} )^{2}$$
$19$ $$1 + 8 T + 18 T^{2} - 160 T^{3} - 1246 T^{4} - 3040 T^{5} + 6498 T^{6} + 54872 T^{7} + 130321 T^{8}$$
$23$ $$1 - 12 T + 72 T^{2} - 300 T^{3} + 1246 T^{4} - 6900 T^{5} + 38088 T^{6} - 146004 T^{7} + 279841 T^{8}$$
$29$ $$1 + 4 T + 6 T^{2} - 204 T^{3} - 830 T^{4} - 5916 T^{5} + 5046 T^{6} + 97556 T^{7} + 707281 T^{8}$$
$31$ $$( 1 + 4 T + 31 T^{2} )^{4}$$
$37$ $$1 - 4 T + 6 T^{2} - 4 T^{3} + 2 T^{4} - 148 T^{5} + 8214 T^{6} - 202612 T^{7} + 1874161 T^{8}$$
$41$ $$1 + 12 T + 72 T^{2} + 516 T^{3} + 3694 T^{4} + 21156 T^{5} + 121032 T^{6} + 827052 T^{7} + 2825761 T^{8}$$
$43$ $$1 - 16 T + 162 T^{2} - 1384 T^{3} + 10178 T^{4} - 59512 T^{5} + 299538 T^{6} - 1272112 T^{7} + 3418801 T^{8}$$
$47$ $$1 - 52 T^{2} + 486 T^{4} - 114868 T^{6} + 4879681 T^{8}$$
$53$ $$1 - 4 T + 54 T^{2} - 708 T^{3} + 3490 T^{4} - 37524 T^{5} + 151686 T^{6} - 595508 T^{7} + 7890481 T^{8}$$
$59$ $$1 + 16 T + 114 T^{2} + 696 T^{3} + 4834 T^{4} + 41064 T^{5} + 396834 T^{6} + 3286064 T^{7} + 12117361 T^{8}$$
$61$ $$1 - 4 T + 6 T^{2} - 4 T^{3} + 2 T^{4} - 244 T^{5} + 22326 T^{6} - 907924 T^{7} + 13845841 T^{8}$$
$67$ $$1 + 8 T + 18 T^{2} - 736 T^{3} - 5854 T^{4} - 49312 T^{5} + 80802 T^{6} + 2406104 T^{7} + 20151121 T^{8}$$
$71$ $$1 + 12 T + 72 T^{2} + 876 T^{3} + 10654 T^{4} + 62196 T^{5} + 362952 T^{6} + 4294932 T^{7} + 25411681 T^{8}$$
$73$ $$( 1 - 14 T + 98 T^{2} - 1022 T^{3} + 5329 T^{4} )^{2}$$
$79$ $$( 1 - 122 T^{2} + 6241 T^{4} )^{2}$$
$83$ $$1 - 16 T + 114 T^{2} - 792 T^{3} + 6370 T^{4} - 65736 T^{5} + 785346 T^{6} - 9148592 T^{7} + 47458321 T^{8}$$
$89$ $$1 - 12 T + 72 T^{2} - 516 T^{3} + 1582 T^{4} - 45924 T^{5} + 570312 T^{6} - 8459628 T^{7} + 62742241 T^{8}$$
$97$ $$( 1 + 20 T + 222 T^{2} + 1940 T^{3} + 9409 T^{4} )^{2}$$