# Properties

 Label 96.3.g.a Level $96$ Weight $3$ Character orbit 96.g Analytic conductor $2.616$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$96 = 2^{5} \cdot 3$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 96.g (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.61581053786$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + 2 \zeta_{12}^{2} ) q^{3} + ( -2 - 8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{5} + ( -4 + 8 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{7} -3 q^{9} +O(q^{10})$$ $$q + ( -1 + 2 \zeta_{12}^{2} ) q^{3} + ( -2 - 8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{5} + ( -4 + 8 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{7} -3 q^{9} + ( -4 + 8 \zeta_{12}^{2} - 8 \zeta_{12}^{3} ) q^{11} + ( 10 - 16 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{13} + ( 2 - 4 \zeta_{12}^{2} - 12 \zeta_{12}^{3} ) q^{15} + ( -6 + 16 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{17} + ( 4 - 8 \zeta_{12}^{2} + 24 \zeta_{12}^{3} ) q^{19} + ( -12 - 8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{21} + 8 \zeta_{12}^{3} q^{23} + ( 27 + 32 \zeta_{12} - 16 \zeta_{12}^{3} ) q^{25} + ( 3 - 6 \zeta_{12}^{2} ) q^{27} + ( -10 + 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{29} + ( 20 - 40 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{31} + ( -12 + 16 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{33} + ( 24 - 48 \zeta_{12}^{2} - 56 \zeta_{12}^{3} ) q^{35} + ( 18 + 32 \zeta_{12} - 16 \zeta_{12}^{3} ) q^{37} + ( -10 + 20 \zeta_{12}^{2} - 24 \zeta_{12}^{3} ) q^{39} + ( -22 - 16 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{41} + ( -20 + 40 \zeta_{12}^{2} + 40 \zeta_{12}^{3} ) q^{43} + ( 6 + 24 \zeta_{12} - 12 \zeta_{12}^{3} ) q^{45} + ( 8 - 16 \zeta_{12}^{2} + 56 \zeta_{12}^{3} ) q^{47} + ( -15 - 64 \zeta_{12} + 32 \zeta_{12}^{3} ) q^{49} + ( 6 - 12 \zeta_{12}^{2} + 24 \zeta_{12}^{3} ) q^{51} + ( 6 + 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{53} + ( -24 + 48 \zeta_{12}^{2} - 32 \zeta_{12}^{3} ) q^{55} + ( 12 - 48 \zeta_{12} + 24 \zeta_{12}^{3} ) q^{57} + ( -44 + 88 \zeta_{12}^{2} - 32 \zeta_{12}^{3} ) q^{59} -14 q^{61} + ( 12 - 24 \zeta_{12}^{2} - 12 \zeta_{12}^{3} ) q^{63} + ( 76 - 48 \zeta_{12} + 24 \zeta_{12}^{3} ) q^{65} + ( 28 - 56 \zeta_{12}^{2} - 32 \zeta_{12}^{3} ) q^{67} + ( -16 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{69} + ( -48 + 96 \zeta_{12}^{2} + 40 \zeta_{12}^{3} ) q^{71} + ( -30 - 64 \zeta_{12} + 32 \zeta_{12}^{3} ) q^{73} + ( -27 + 54 \zeta_{12}^{2} + 48 \zeta_{12}^{3} ) q^{75} + ( -16 + 32 \zeta_{12} - 16 \zeta_{12}^{3} ) q^{77} + ( -12 + 24 \zeta_{12}^{2} - 76 \zeta_{12}^{3} ) q^{79} + 9 q^{81} + ( 4 - 8 \zeta_{12}^{2} + 56 \zeta_{12}^{3} ) q^{83} + ( -84 + 16 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{85} + ( 10 - 20 \zeta_{12}^{2} + 12 \zeta_{12}^{3} ) q^{87} + ( -78 - 32 \zeta_{12} + 16 \zeta_{12}^{3} ) q^{89} + ( -8 + 16 \zeta_{12}^{2} - 56 \zeta_{12}^{3} ) q^{91} + ( 60 - 8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{93} + ( 88 - 176 \zeta_{12}^{2} ) q^{95} + ( -62 + 96 \zeta_{12} - 48 \zeta_{12}^{3} ) q^{97} + ( 12 - 24 \zeta_{12}^{2} + 24 \zeta_{12}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 8q^{5} - 12q^{9} + O(q^{10})$$ $$4q - 8q^{5} - 12q^{9} + 40q^{13} - 24q^{17} - 48q^{21} + 108q^{25} - 40q^{29} - 48q^{33} + 72q^{37} - 88q^{41} + 24q^{45} - 60q^{49} + 24q^{53} + 48q^{57} - 56q^{61} + 304q^{65} - 120q^{73} - 64q^{77} + 36q^{81} - 336q^{85} - 312q^{89} + 240q^{93} - 248q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/96\mathbb{Z}\right)^\times$$.

 $$n$$ $$31$$ $$37$$ $$65$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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
 0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i −0.866025 − 0.500000i
0 1.73205i 0 −8.92820 0 10.9282i 0 −3.00000 0
31.2 0 1.73205i 0 4.92820 0 2.92820i 0 −3.00000 0
31.3 0 1.73205i 0 −8.92820 0 10.9282i 0 −3.00000 0
31.4 0 1.73205i 0 4.92820 0 2.92820i 0 −3.00000 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 96.3.g.a 4
3.b odd 2 1 288.3.g.d 4
4.b odd 2 1 inner 96.3.g.a 4
5.b even 2 1 2400.3.e.a 4
5.c odd 4 1 2400.3.j.a 4
5.c odd 4 1 2400.3.j.b 4
8.b even 2 1 192.3.g.c 4
8.d odd 2 1 192.3.g.c 4
12.b even 2 1 288.3.g.d 4
16.e even 4 1 768.3.b.a 4
16.e even 4 1 768.3.b.d 4
16.f odd 4 1 768.3.b.a 4
16.f odd 4 1 768.3.b.d 4
20.d odd 2 1 2400.3.e.a 4
20.e even 4 1 2400.3.j.a 4
20.e even 4 1 2400.3.j.b 4
24.f even 2 1 576.3.g.j 4
24.h odd 2 1 576.3.g.j 4
48.i odd 4 1 2304.3.b.k 4
48.i odd 4 1 2304.3.b.o 4
48.k even 4 1 2304.3.b.k 4
48.k even 4 1 2304.3.b.o 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.3.g.a 4 1.a even 1 1 trivial
96.3.g.a 4 4.b odd 2 1 inner
192.3.g.c 4 8.b even 2 1
192.3.g.c 4 8.d odd 2 1
288.3.g.d 4 3.b odd 2 1
288.3.g.d 4 12.b even 2 1
576.3.g.j 4 24.f even 2 1
576.3.g.j 4 24.h odd 2 1
768.3.b.a 4 16.e even 4 1
768.3.b.a 4 16.f odd 4 1
768.3.b.d 4 16.e even 4 1
768.3.b.d 4 16.f odd 4 1
2304.3.b.k 4 48.i odd 4 1
2304.3.b.k 4 48.k even 4 1
2304.3.b.o 4 48.i odd 4 1
2304.3.b.o 4 48.k even 4 1
2400.3.e.a 4 5.b even 2 1
2400.3.e.a 4 20.d odd 2 1
2400.3.j.a 4 5.c odd 4 1
2400.3.j.a 4 20.e even 4 1
2400.3.j.b 4 5.c odd 4 1
2400.3.j.b 4 20.e even 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(96, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 3 + T^{2} )^{2}$$
$5$ $$( -44 + 4 T + T^{2} )^{2}$$
$7$ $$1024 + 128 T^{2} + T^{4}$$
$11$ $$256 + 224 T^{2} + T^{4}$$
$13$ $$( -92 - 20 T + T^{2} )^{2}$$
$17$ $$( -156 + 12 T + T^{2} )^{2}$$
$19$ $$278784 + 1248 T^{2} + T^{4}$$
$23$ $$( 64 + T^{2} )^{2}$$
$29$ $$( 52 + 20 T + T^{2} )^{2}$$
$31$ $$1401856 + 2432 T^{2} + T^{4}$$
$37$ $$( -444 - 36 T + T^{2} )^{2}$$
$41$ $$( 292 + 44 T + T^{2} )^{2}$$
$43$ $$160000 + 5600 T^{2} + T^{4}$$
$47$ $$8667136 + 6656 T^{2} + T^{4}$$
$53$ $$( -12 - 12 T + T^{2} )^{2}$$
$59$ $$22886656 + 13664 T^{2} + T^{4}$$
$61$ $$( 14 + T )^{4}$$
$67$ $$1763584 + 6752 T^{2} + T^{4}$$
$71$ $$28217344 + 17024 T^{2} + T^{4}$$
$73$ $$( -2172 + 60 T + T^{2} )^{2}$$
$79$ $$28558336 + 12416 T^{2} + T^{4}$$
$83$ $$9535744 + 6368 T^{2} + T^{4}$$
$89$ $$( 5316 + 156 T + T^{2} )^{2}$$
$97$ $$( -3068 + 124 T + T^{2} )^{2}$$