# Properties

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

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$0.766563859404$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{3} + 2 i q^{5} + 2 q^{7} - q^{9} +O(q^{10})$$ $$q + i q^{3} + 2 i q^{5} + 2 q^{7} - q^{9} -4 i q^{13} -2 q^{15} -2 q^{17} -4 i q^{19} + 2 i q^{21} -4 q^{23} + q^{25} -i q^{27} -6 i q^{29} -2 q^{31} + 4 i q^{35} + 8 i q^{37} + 4 q^{39} + 2 q^{41} + 4 i q^{43} -2 i q^{45} + 12 q^{47} -3 q^{49} -2 i q^{51} + 6 i q^{53} + 4 q^{57} -4 i q^{59} -2 q^{63} + 8 q^{65} + 12 i q^{67} -4 i q^{69} -12 q^{71} -6 q^{73} + i q^{75} -10 q^{79} + q^{81} -16 i q^{83} -4 i q^{85} + 6 q^{87} -10 q^{89} -8 i q^{91} -2 i q^{93} + 8 q^{95} -2 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 4q^{7} - 2q^{9} + O(q^{10})$$ $$2q + 4q^{7} - 2q^{9} - 4q^{15} - 4q^{17} - 8q^{23} + 2q^{25} - 4q^{31} + 8q^{39} + 4q^{41} + 24q^{47} - 6q^{49} + 8q^{57} - 4q^{63} + 16q^{65} - 24q^{71} - 12q^{73} - 20q^{79} + 2q^{81} + 12q^{87} - 20q^{89} + 16q^{95} - 4q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 − 1.00000i 1.00000i
0 1.00000i 0 2.00000i 0 2.00000 0 −1.00000 0
49.2 0 1.00000i 0 2.00000i 0 2.00000 0 −1.00000 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.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.d.a 2
3.b odd 2 1 288.2.d.b 2
4.b odd 2 1 24.2.d.a 2
5.b even 2 1 2400.2.k.a 2
5.c odd 4 1 2400.2.d.b 2
5.c odd 4 1 2400.2.d.c 2
7.b odd 2 1 4704.2.c.a 2
8.b even 2 1 inner 96.2.d.a 2
8.d odd 2 1 24.2.d.a 2
9.c even 3 2 2592.2.r.f 4
9.d odd 6 2 2592.2.r.g 4
12.b even 2 1 72.2.d.b 2
15.d odd 2 1 7200.2.k.d 2
15.e even 4 1 7200.2.d.d 2
15.e even 4 1 7200.2.d.g 2
16.e even 4 1 768.2.a.d 1
16.e even 4 1 768.2.a.e 1
16.f odd 4 1 768.2.a.a 1
16.f odd 4 1 768.2.a.h 1
20.d odd 2 1 600.2.k.b 2
20.e even 4 1 600.2.d.b 2
20.e even 4 1 600.2.d.c 2
24.f even 2 1 72.2.d.b 2
24.h odd 2 1 288.2.d.b 2
28.d even 2 1 1176.2.c.a 2
36.f odd 6 2 648.2.n.k 4
36.h even 6 2 648.2.n.c 4
40.e odd 2 1 600.2.k.b 2
40.f even 2 1 2400.2.k.a 2
40.i odd 4 1 2400.2.d.b 2
40.i odd 4 1 2400.2.d.c 2
40.k even 4 1 600.2.d.b 2
40.k even 4 1 600.2.d.c 2
48.i odd 4 1 2304.2.a.b 1
48.i odd 4 1 2304.2.a.l 1
48.k even 4 1 2304.2.a.e 1
48.k even 4 1 2304.2.a.o 1
56.e even 2 1 1176.2.c.a 2
56.h odd 2 1 4704.2.c.a 2
60.h even 2 1 1800.2.k.a 2
60.l odd 4 1 1800.2.d.b 2
60.l odd 4 1 1800.2.d.i 2
72.j odd 6 2 2592.2.r.g 4
72.l even 6 2 648.2.n.c 4
72.n even 6 2 2592.2.r.f 4
72.p odd 6 2 648.2.n.k 4
120.i odd 2 1 7200.2.k.d 2
120.m even 2 1 1800.2.k.a 2
120.q odd 4 1 1800.2.d.b 2
120.q odd 4 1 1800.2.d.i 2
120.w even 4 1 7200.2.d.d 2
120.w even 4 1 7200.2.d.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.2.d.a 2 4.b odd 2 1
24.2.d.a 2 8.d odd 2 1
72.2.d.b 2 12.b even 2 1
72.2.d.b 2 24.f even 2 1
96.2.d.a 2 1.a even 1 1 trivial
96.2.d.a 2 8.b even 2 1 inner
288.2.d.b 2 3.b odd 2 1
288.2.d.b 2 24.h odd 2 1
600.2.d.b 2 20.e even 4 1
600.2.d.b 2 40.k even 4 1
600.2.d.c 2 20.e even 4 1
600.2.d.c 2 40.k even 4 1
600.2.k.b 2 20.d odd 2 1
600.2.k.b 2 40.e odd 2 1
648.2.n.c 4 36.h even 6 2
648.2.n.c 4 72.l even 6 2
648.2.n.k 4 36.f odd 6 2
648.2.n.k 4 72.p odd 6 2
768.2.a.a 1 16.f odd 4 1
768.2.a.d 1 16.e even 4 1
768.2.a.e 1 16.e even 4 1
768.2.a.h 1 16.f odd 4 1
1176.2.c.a 2 28.d even 2 1
1176.2.c.a 2 56.e even 2 1
1800.2.d.b 2 60.l odd 4 1
1800.2.d.b 2 120.q odd 4 1
1800.2.d.i 2 60.l odd 4 1
1800.2.d.i 2 120.q odd 4 1
1800.2.k.a 2 60.h even 2 1
1800.2.k.a 2 120.m even 2 1
2304.2.a.b 1 48.i odd 4 1
2304.2.a.e 1 48.k even 4 1
2304.2.a.l 1 48.i odd 4 1
2304.2.a.o 1 48.k even 4 1
2400.2.d.b 2 5.c odd 4 1
2400.2.d.b 2 40.i odd 4 1
2400.2.d.c 2 5.c odd 4 1
2400.2.d.c 2 40.i odd 4 1
2400.2.k.a 2 5.b even 2 1
2400.2.k.a 2 40.f even 2 1
2592.2.r.f 4 9.c even 3 2
2592.2.r.f 4 72.n even 6 2
2592.2.r.g 4 9.d odd 6 2
2592.2.r.g 4 72.j odd 6 2
4704.2.c.a 2 7.b odd 2 1
4704.2.c.a 2 56.h odd 2 1
7200.2.d.d 2 15.e even 4 1
7200.2.d.d 2 120.w even 4 1
7200.2.d.g 2 15.e even 4 1
7200.2.d.g 2 120.w even 4 1
7200.2.k.d 2 15.d odd 2 1
7200.2.k.d 2 120.i odd 2 1

## Hecke kernels

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

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + T^{2}$$
$5$ $$( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} )$$
$7$ $$( 1 - 2 T + 7 T^{2} )^{2}$$
$11$ $$( 1 - 11 T^{2} )^{2}$$
$13$ $$( 1 - 6 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} )$$
$17$ $$( 1 + 2 T + 17 T^{2} )^{2}$$
$19$ $$1 - 22 T^{2} + 361 T^{4}$$
$23$ $$( 1 + 4 T + 23 T^{2} )^{2}$$
$29$ $$1 - 22 T^{2} + 841 T^{4}$$
$31$ $$( 1 + 2 T + 31 T^{2} )^{2}$$
$37$ $$1 - 10 T^{2} + 1369 T^{4}$$
$41$ $$( 1 - 2 T + 41 T^{2} )^{2}$$
$43$ $$1 - 70 T^{2} + 1849 T^{4}$$
$47$ $$( 1 - 12 T + 47 T^{2} )^{2}$$
$53$ $$1 - 70 T^{2} + 2809 T^{4}$$
$59$ $$1 - 102 T^{2} + 3481 T^{4}$$
$61$ $$( 1 - 61 T^{2} )^{2}$$
$67$ $$1 + 10 T^{2} + 4489 T^{4}$$
$71$ $$( 1 + 12 T + 71 T^{2} )^{2}$$
$73$ $$( 1 + 6 T + 73 T^{2} )^{2}$$
$79$ $$( 1 + 10 T + 79 T^{2} )^{2}$$
$83$ $$1 + 90 T^{2} + 6889 T^{4}$$
$89$ $$( 1 + 10 T + 89 T^{2} )^{2}$$
$97$ $$( 1 + 2 T + 97 T^{2} )^{2}$$