# Properties

 Label 96.2.c.a Level 96 Weight 2 Character orbit 96.c Analytic conductor 0.767 Analytic rank 0 Dimension 4 CM no Inner twists 4

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$0.766563859404$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{3}$$ 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 $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{3} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{5} + 2 \zeta_{8}^{2} q^{7} + ( 1 - 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{9} +O(q^{10})$$ $$q + ( -\zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{3} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{5} + 2 \zeta_{8}^{2} q^{7} + ( 1 - 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{9} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{11} -2 q^{13} + ( -2 \zeta_{8} - 4 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{15} -6 \zeta_{8}^{2} q^{19} + ( -2 - 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{21} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{23} -3 q^{25} + ( \zeta_{8} + 5 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{27} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{29} + 2 \zeta_{8}^{2} q^{31} + ( -4 + 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{33} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{35} + 6 q^{37} + ( 2 \zeta_{8} - 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{39} + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{41} + 2 \zeta_{8}^{2} q^{43} + ( 8 + 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{45} + ( -8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{47} + 3 q^{49} + ( -6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{53} + 8 \zeta_{8}^{2} q^{55} + ( 6 + 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{57} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{59} -2 q^{61} + ( 4 \zeta_{8} + 2 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{63} + ( -4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{65} + 2 \zeta_{8}^{2} q^{67} + ( -8 + 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{69} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{71} -6 q^{73} + ( 3 \zeta_{8} - 3 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{75} + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{77} -14 \zeta_{8}^{2} q^{79} + ( -7 - 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{81} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{83} + ( 2 \zeta_{8} + 4 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{87} + ( -12 \zeta_{8} - 12 \zeta_{8}^{3} ) q^{89} -4 \zeta_{8}^{2} q^{91} + ( -2 - 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{93} + ( 12 \zeta_{8} - 12 \zeta_{8}^{3} ) q^{95} + 10 q^{97} + ( 2 \zeta_{8} - 8 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{9} + O(q^{10})$$ $$4q + 4q^{9} - 8q^{13} - 8q^{21} - 12q^{25} - 16q^{33} + 24q^{37} + 32q^{45} + 12q^{49} + 24q^{57} - 8q^{61} - 32q^{69} - 24q^{73} - 28q^{81} - 8q^{93} + 40q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
95.1
 0.707107 − 0.707107i 0.707107 + 0.707107i −0.707107 + 0.707107i −0.707107 − 0.707107i
0 −1.41421 1.00000i 0 2.82843i 0 2.00000i 0 1.00000 + 2.82843i 0
95.2 0 −1.41421 + 1.00000i 0 2.82843i 0 2.00000i 0 1.00000 2.82843i 0
95.3 0 1.41421 1.00000i 0 2.82843i 0 2.00000i 0 1.00000 2.82843i 0
95.4 0 1.41421 + 1.00000i 0 2.82843i 0 2.00000i 0 1.00000 + 2.82843i 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
4.b odd 2 1 inner
12.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 96.2.c.a 4
3.b odd 2 1 inner 96.2.c.a 4
4.b odd 2 1 inner 96.2.c.a 4
5.b even 2 1 2400.2.h.c 4
5.c odd 4 1 2400.2.o.a 4
5.c odd 4 1 2400.2.o.h 4
8.b even 2 1 192.2.c.b 4
8.d odd 2 1 192.2.c.b 4
9.c even 3 2 2592.2.s.e 8
9.d odd 6 2 2592.2.s.e 8
12.b even 2 1 inner 96.2.c.a 4
15.d odd 2 1 2400.2.h.c 4
15.e even 4 1 2400.2.o.a 4
15.e even 4 1 2400.2.o.h 4
16.e even 4 1 768.2.f.a 4
16.e even 4 1 768.2.f.g 4
16.f odd 4 1 768.2.f.a 4
16.f odd 4 1 768.2.f.g 4
20.d odd 2 1 2400.2.h.c 4
20.e even 4 1 2400.2.o.a 4
20.e even 4 1 2400.2.o.h 4
24.f even 2 1 192.2.c.b 4
24.h odd 2 1 192.2.c.b 4
36.f odd 6 2 2592.2.s.e 8
36.h even 6 2 2592.2.s.e 8
48.i odd 4 1 768.2.f.a 4
48.i odd 4 1 768.2.f.g 4
48.k even 4 1 768.2.f.a 4
48.k even 4 1 768.2.f.g 4
60.h even 2 1 2400.2.h.c 4
60.l odd 4 1 2400.2.o.a 4
60.l odd 4 1 2400.2.o.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.2.c.a 4 1.a even 1 1 trivial
96.2.c.a 4 3.b odd 2 1 inner
96.2.c.a 4 4.b odd 2 1 inner
96.2.c.a 4 12.b even 2 1 inner
192.2.c.b 4 8.b even 2 1
192.2.c.b 4 8.d odd 2 1
192.2.c.b 4 24.f even 2 1
192.2.c.b 4 24.h odd 2 1
768.2.f.a 4 16.e even 4 1
768.2.f.a 4 16.f odd 4 1
768.2.f.a 4 48.i odd 4 1
768.2.f.a 4 48.k even 4 1
768.2.f.g 4 16.e even 4 1
768.2.f.g 4 16.f odd 4 1
768.2.f.g 4 48.i odd 4 1
768.2.f.g 4 48.k even 4 1
2400.2.h.c 4 5.b even 2 1
2400.2.h.c 4 15.d odd 2 1
2400.2.h.c 4 20.d odd 2 1
2400.2.h.c 4 60.h even 2 1
2400.2.o.a 4 5.c odd 4 1
2400.2.o.a 4 15.e even 4 1
2400.2.o.a 4 20.e even 4 1
2400.2.o.a 4 60.l odd 4 1
2400.2.o.h 4 5.c odd 4 1
2400.2.o.h 4 15.e even 4 1
2400.2.o.h 4 20.e even 4 1
2400.2.o.h 4 60.l odd 4 1
2592.2.s.e 8 9.c even 3 2
2592.2.s.e 8 9.d odd 6 2
2592.2.s.e 8 36.f odd 6 2
2592.2.s.e 8 36.h even 6 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(96, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - 2 T^{2} + 9 T^{4}$$
$5$ $$( 1 - 2 T^{2} + 25 T^{4} )^{2}$$
$7$ $$( 1 - 10 T^{2} + 49 T^{4} )^{2}$$
$11$ $$( 1 + 14 T^{2} + 121 T^{4} )^{2}$$
$13$ $$( 1 + 2 T + 13 T^{2} )^{4}$$
$17$ $$( 1 - 17 T^{2} )^{4}$$
$19$ $$( 1 - 2 T^{2} + 361 T^{4} )^{2}$$
$23$ $$( 1 + 14 T^{2} + 529 T^{4} )^{2}$$
$29$ $$( 1 - 50 T^{2} + 841 T^{4} )^{2}$$
$31$ $$( 1 - 58 T^{2} + 961 T^{4} )^{2}$$
$37$ $$( 1 - 6 T + 37 T^{2} )^{4}$$
$41$ $$( 1 - 50 T^{2} + 1681 T^{4} )^{2}$$
$43$ $$( 1 - 82 T^{2} + 1849 T^{4} )^{2}$$
$47$ $$( 1 - 34 T^{2} + 2209 T^{4} )^{2}$$
$53$ $$( 1 - 34 T^{2} + 2809 T^{4} )^{2}$$
$59$ $$( 1 + 110 T^{2} + 3481 T^{4} )^{2}$$
$61$ $$( 1 + 2 T + 61 T^{2} )^{4}$$
$67$ $$( 1 - 130 T^{2} + 4489 T^{4} )^{2}$$
$71$ $$( 1 + 110 T^{2} + 5041 T^{4} )^{2}$$
$73$ $$( 1 + 6 T + 73 T^{2} )^{4}$$
$79$ $$( 1 + 38 T^{2} + 6241 T^{4} )^{2}$$
$83$ $$( 1 + 158 T^{2} + 6889 T^{4} )^{2}$$
$89$ $$( 1 + 110 T^{2} + 7921 T^{4} )^{2}$$
$97$ $$( 1 - 10 T + 97 T^{2} )^{4}$$