# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$96 = 2^{5} \cdot 3$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 96.h (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.61581053786$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-7})$$ Defining polynomial: $$x^{4} + 6 x^{2} + 16$$ 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.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{1} - \beta_{2} ) q^{3} -2 \beta_{1} q^{5} -4 q^{7} + ( -5 + \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( -\beta_{1} - \beta_{2} ) q^{3} -2 \beta_{1} q^{5} -4 q^{7} + ( -5 + \beta_{3} ) q^{9} -3 \beta_{1} q^{11} + ( -2 \beta_{1} - 4 \beta_{2} ) q^{13} + ( 8 + 2 \beta_{3} ) q^{15} -2 \beta_{3} q^{17} + ( \beta_{1} + 2 \beta_{2} ) q^{19} + ( 4 \beta_{1} + 4 \beta_{2} ) q^{21} -4 \beta_{3} q^{23} + 7 q^{25} + ( 10 \beta_{1} + \beta_{2} ) q^{27} + 6 \beta_{1} q^{29} + 4 q^{31} + ( 12 + 3 \beta_{3} ) q^{33} + 8 \beta_{1} q^{35} + ( 10 \beta_{1} + 20 \beta_{2} ) q^{37} + ( -28 + 2 \beta_{3} ) q^{39} -4 \beta_{3} q^{41} + ( -\beta_{1} - 2 \beta_{2} ) q^{43} + ( 2 \beta_{1} - 16 \beta_{2} ) q^{45} -33 q^{49} + ( -10 \beta_{1} + 8 \beta_{2} ) q^{51} -18 \beta_{1} q^{53} + 48 q^{55} + ( 14 - \beta_{3} ) q^{57} -17 \beta_{1} q^{59} + ( -18 \beta_{1} - 36 \beta_{2} ) q^{61} + ( 20 - 4 \beta_{3} ) q^{63} + 8 \beta_{3} q^{65} + ( -9 \beta_{1} - 18 \beta_{2} ) q^{67} + ( -20 \beta_{1} + 16 \beta_{2} ) q^{69} + 12 \beta_{3} q^{71} -6 q^{73} + ( -7 \beta_{1} - 7 \beta_{2} ) q^{75} + 12 \beta_{1} q^{77} -124 q^{79} + ( -31 - 10 \beta_{3} ) q^{81} + \beta_{1} q^{83} + ( 16 \beta_{1} + 32 \beta_{2} ) q^{85} + ( -24 - 6 \beta_{3} ) q^{87} + 14 \beta_{3} q^{89} + ( 8 \beta_{1} + 16 \beta_{2} ) q^{91} + ( -4 \beta_{1} - 4 \beta_{2} ) q^{93} -4 \beta_{3} q^{95} + 118 q^{97} + ( 3 \beta_{1} - 24 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 16q^{7} - 20q^{9} + O(q^{10})$$ $$4q - 16q^{7} - 20q^{9} + 32q^{15} + 28q^{25} + 16q^{31} + 48q^{33} - 112q^{39} - 132q^{49} + 192q^{55} + 56q^{57} + 80q^{63} - 24q^{73} - 496q^{79} - 124q^{81} - 96q^{87} + 472q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 6 x^{2} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{3} - 2 \nu$$$$)/2$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 4 \nu^{2} + 2 \nu + 12$$$$)/4$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} + 10 \nu$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{1}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{2} + \beta_{1} - 6$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{3} - 5 \beta_{1}$$$$)/2$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1
 0.707107 + 1.87083i 0.707107 − 1.87083i −0.707107 − 1.87083i −0.707107 + 1.87083i
0 −1.41421 2.64575i 0 −5.65685 0 −4.00000 0 −5.00000 + 7.48331i 0
17.2 0 −1.41421 + 2.64575i 0 −5.65685 0 −4.00000 0 −5.00000 7.48331i 0
17.3 0 1.41421 2.64575i 0 5.65685 0 −4.00000 0 −5.00000 7.48331i 0
17.4 0 1.41421 + 2.64575i 0 5.65685 0 −4.00000 0 −5.00000 + 7.48331i 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
3.b odd 2 1 inner
8.b even 2 1 inner
24.h 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.h.c 4
3.b odd 2 1 inner 96.3.h.c 4
4.b odd 2 1 24.3.h.c 4
8.b even 2 1 inner 96.3.h.c 4
8.d odd 2 1 24.3.h.c 4
12.b even 2 1 24.3.h.c 4
16.e even 4 2 768.3.e.l 4
16.f odd 4 2 768.3.e.i 4
24.f even 2 1 24.3.h.c 4
24.h odd 2 1 inner 96.3.h.c 4
48.i odd 4 2 768.3.e.l 4
48.k even 4 2 768.3.e.i 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.3.h.c 4 4.b odd 2 1
24.3.h.c 4 8.d odd 2 1
24.3.h.c 4 12.b even 2 1
24.3.h.c 4 24.f even 2 1
96.3.h.c 4 1.a even 1 1 trivial
96.3.h.c 4 3.b odd 2 1 inner
96.3.h.c 4 8.b even 2 1 inner
96.3.h.c 4 24.h odd 2 1 inner
768.3.e.i 4 16.f odd 4 2
768.3.e.i 4 48.k even 4 2
768.3.e.l 4 16.e even 4 2
768.3.e.l 4 48.i odd 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 32$$ acting on $$S_{3}^{\mathrm{new}}(96, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$81 + 10 T^{2} + T^{4}$$
$5$ $$( -32 + T^{2} )^{2}$$
$7$ $$( 4 + T )^{4}$$
$11$ $$( -72 + T^{2} )^{2}$$
$13$ $$( 112 + T^{2} )^{2}$$
$17$ $$( 224 + T^{2} )^{2}$$
$19$ $$( 28 + T^{2} )^{2}$$
$23$ $$( 896 + T^{2} )^{2}$$
$29$ $$( -288 + T^{2} )^{2}$$
$31$ $$( -4 + T )^{4}$$
$37$ $$( 2800 + T^{2} )^{2}$$
$41$ $$( 896 + T^{2} )^{2}$$
$43$ $$( 28 + T^{2} )^{2}$$
$47$ $$T^{4}$$
$53$ $$( -2592 + T^{2} )^{2}$$
$59$ $$( -2312 + T^{2} )^{2}$$
$61$ $$( 9072 + T^{2} )^{2}$$
$67$ $$( 2268 + T^{2} )^{2}$$
$71$ $$( 8064 + T^{2} )^{2}$$
$73$ $$( 6 + T )^{4}$$
$79$ $$( 124 + T )^{4}$$
$83$ $$( -8 + T^{2} )^{2}$$
$89$ $$( 10976 + T^{2} )^{2}$$
$97$ $$( -118 + T )^{4}$$