# Properties

 Label 96.3.b.a Level $96$ Weight $3$ Character orbit 96.b 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.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.61581053786$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.0.4752.1 Defining polynomial: $$x^{4} + 3 x^{2} - 6 x + 6$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 24) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{3} -\beta_{2} q^{5} + ( -\beta_{2} - \beta_{3} ) q^{7} + 3 q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{3} -\beta_{2} q^{5} + ( -\beta_{2} - \beta_{3} ) q^{7} + 3 q^{9} + 8 q^{11} -2 \beta_{3} q^{13} + ( -\beta_{2} + \beta_{3} ) q^{15} + ( -2 - 8 \beta_{1} ) q^{17} + ( -8 + 4 \beta_{1} ) q^{19} + ( \beta_{2} + 2 \beta_{3} ) q^{21} + ( 2 \beta_{2} + 2 \beta_{3} ) q^{23} + ( -11 + 16 \beta_{1} ) q^{25} -3 \beta_{1} q^{27} + ( 3 \beta_{2} + 4 \beta_{3} ) q^{29} + ( 5 \beta_{2} - 3 \beta_{3} ) q^{31} -8 \beta_{1} q^{33} + ( -24 - 4 \beta_{1} ) q^{35} + ( 2 \beta_{2} - 2 \beta_{3} ) q^{37} + ( 4 \beta_{2} + 2 \beta_{3} ) q^{39} + ( 10 + 24 \beta_{1} ) q^{41} + ( -8 + 12 \beta_{1} ) q^{43} -3 \beta_{2} q^{45} + ( -6 \beta_{2} + 2 \beta_{3} ) q^{47} + ( -11 - 32 \beta_{1} ) q^{49} + ( 24 + 2 \beta_{1} ) q^{51} + ( \beta_{2} - 4 \beta_{3} ) q^{53} -8 \beta_{2} q^{55} + ( -12 + 8 \beta_{1} ) q^{57} + ( 32 - 12 \beta_{1} ) q^{59} + ( -6 \beta_{2} + 2 \beta_{3} ) q^{61} + ( -3 \beta_{2} - 3 \beta_{3} ) q^{63} + ( 24 - 40 \beta_{1} ) q^{65} + ( 64 - 12 \beta_{1} ) q^{67} + ( -2 \beta_{2} - 4 \beta_{3} ) q^{69} + ( -2 \beta_{2} - 10 \beta_{3} ) q^{71} + ( 50 + 32 \beta_{1} ) q^{73} + ( -48 + 11 \beta_{1} ) q^{75} + ( -8 \beta_{2} - 8 \beta_{3} ) q^{77} + ( \beta_{2} + 9 \beta_{3} ) q^{79} + 9 q^{81} + ( -40 + 16 \beta_{1} ) q^{83} + ( -6 \beta_{2} + 8 \beta_{3} ) q^{85} + ( -5 \beta_{2} - 7 \beta_{3} ) q^{87} + ( -50 + 48 \beta_{1} ) q^{89} + ( -72 - 56 \beta_{1} ) q^{91} + ( 11 \beta_{2} - 2 \beta_{3} ) q^{93} + ( 12 \beta_{2} - 4 \beta_{3} ) q^{95} + ( 14 - 48 \beta_{1} ) q^{97} + 24 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 12q^{9} + O(q^{10})$$ $$4q + 12q^{9} + 32q^{11} - 8q^{17} - 32q^{19} - 44q^{25} - 96q^{35} + 40q^{41} - 32q^{43} - 44q^{49} + 96q^{51} - 48q^{57} + 128q^{59} + 96q^{65} + 256q^{67} + 200q^{73} - 192q^{75} + 36q^{81} - 160q^{83} - 200q^{89} - 288q^{91} + 56q^{97} + 96q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 3 x^{2} - 6 x + 6$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{3} + \nu^{2} + 6$$$$)/4$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} - \nu^{2} + 8 \nu - 6$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$($$$$3 \nu^{3} + 5 \nu^{2} + 8 \nu - 6$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + 2 \beta_{1}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} - \beta_{2} + 4 \beta_{1} - 6$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{3} - \beta_{2} - 12 \beta_{1} + 18$$$$)/4$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
79.1
 0.866025 + 0.719687i 0.866025 − 0.719687i −0.866025 + 1.99551i −0.866025 − 1.99551i
0 −1.73205 0 2.87875i 0 10.7436i 0 3.00000 0
79.2 0 −1.73205 0 2.87875i 0 10.7436i 0 3.00000 0
79.3 0 1.73205 0 7.98203i 0 2.13878i 0 3.00000 0
79.4 0 1.73205 0 7.98203i 0 2.13878i 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
8.d 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.b.a 4
3.b odd 2 1 288.3.b.b 4
4.b odd 2 1 24.3.b.a 4
5.b even 2 1 2400.3.g.a 4
5.c odd 4 2 2400.3.p.a 8
8.b even 2 1 24.3.b.a 4
8.d odd 2 1 inner 96.3.b.a 4
12.b even 2 1 72.3.b.b 4
16.e even 4 2 768.3.g.h 8
16.f odd 4 2 768.3.g.h 8
20.d odd 2 1 600.3.g.a 4
20.e even 4 2 600.3.p.a 8
24.f even 2 1 288.3.b.b 4
24.h odd 2 1 72.3.b.b 4
40.e odd 2 1 2400.3.g.a 4
40.f even 2 1 600.3.g.a 4
40.i odd 4 2 600.3.p.a 8
40.k even 4 2 2400.3.p.a 8
48.i odd 4 2 2304.3.g.z 8
48.k even 4 2 2304.3.g.z 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.3.b.a 4 4.b odd 2 1
24.3.b.a 4 8.b even 2 1
72.3.b.b 4 12.b even 2 1
72.3.b.b 4 24.h odd 2 1
96.3.b.a 4 1.a even 1 1 trivial
96.3.b.a 4 8.d odd 2 1 inner
288.3.b.b 4 3.b odd 2 1
288.3.b.b 4 24.f even 2 1
600.3.g.a 4 20.d odd 2 1
600.3.g.a 4 40.f even 2 1
600.3.p.a 8 20.e even 4 2
600.3.p.a 8 40.i odd 4 2
768.3.g.h 8 16.e even 4 2
768.3.g.h 8 16.f odd 4 2
2304.3.g.z 8 48.i odd 4 2
2304.3.g.z 8 48.k even 4 2
2400.3.g.a 4 5.b even 2 1
2400.3.g.a 4 40.e odd 2 1
2400.3.p.a 8 5.c odd 4 2
2400.3.p.a 8 40.k even 4 2

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( -3 + T^{2} )^{2}$$
$5$ $$528 + 72 T^{2} + T^{4}$$
$7$ $$528 + 120 T^{2} + T^{4}$$
$11$ $$( -8 + T )^{4}$$
$13$ $$33792 + 384 T^{2} + T^{4}$$
$17$ $$( -188 + 4 T + T^{2} )^{2}$$
$19$ $$( 16 + 16 T + T^{2} )^{2}$$
$23$ $$8448 + 480 T^{2} + T^{4}$$
$29$ $$528 + 1608 T^{2} + T^{4}$$
$31$ $$279312 + 3384 T^{2} + T^{4}$$
$37$ $$76032 + 864 T^{2} + T^{4}$$
$41$ $$( -1628 - 20 T + T^{2} )^{2}$$
$43$ $$( -368 + 16 T + T^{2} )^{2}$$
$47$ $$8448 + 3552 T^{2} + T^{4}$$
$53$ $$803088 + 1800 T^{2} + T^{4}$$
$59$ $$( 592 - 64 T + T^{2} )^{2}$$
$61$ $$8448 + 3552 T^{2} + T^{4}$$
$67$ $$( 3664 - 128 T + T^{2} )^{2}$$
$71$ $$12849408 + 8928 T^{2} + T^{4}$$
$73$ $$( -572 - 100 T + T^{2} )^{2}$$
$79$ $$10797072 + 7416 T^{2} + T^{4}$$
$83$ $$( 832 + 80 T + T^{2} )^{2}$$
$89$ $$( -4412 + 100 T + T^{2} )^{2}$$
$97$ $$( -6716 - 28 T + T^{2} )^{2}$$