Newform orbit 96.6.a.h

• Label: 96.6.a.h
• Level: $96$
• Weight: $6$
• Character orbit: 96.a
• Self dual: yes
• Analytic conductor: $15.397$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	$N$	$=$	$96 = 2^{5} \cdot 3$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	96.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$15.3968467020$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{31})$
Defining polynomial:	$x^{2} - 31$
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$2^{4}$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $q$-expansion are expressed in terms of $\beta = 16\sqrt{31}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	q + 9 * q^3 + (b + 18) * q^5 + (b + 60) * q^7 + 81 * q^9 + (-6*b + 100) * q^11 + (-6*b + 142) * q^13 + (9*b + 162) * q^15 + (-6*b + 1338) * q^17 + (10*b - 36) * q^19 + (9*b + 540) * q^21 + (-30*b - 1920) * q^23 + (36*b + 5135) * q^25 + 729 * q^27 + (19*b + 5106) * q^29 + (5*b - 5244) * q^31 + (-54*b + 900) * q^33 + (78*b + 9016) * q^35 + (-84*b + 6574) * q^37 + (-54*b + 1278) * q^39 + (-50*b + 2082) * q^41 + (-130*b + 2916) * q^43 + (81*b + 1458) * q^45 + (198*b - 760) * q^47 + (120*b - 5271) * q^49 + (-54*b + 12042) * q^51 + (71*b + 4506) * q^53 + (-8*b - 45816) * q^55 + (90*b - 324) * q^57 + (-216*b + 27548) * q^59 + (-48*b - 31722) * q^61 + (81*b + 4860) * q^63 + (34*b - 45060) * q^65 + (488*b - 18396) * q^67 + (-270*b - 17280) * q^69 + (-438*b - 18832) * q^71 + (-552*b - 18918) * q^73 + (324*b + 46215) * q^75 + (-260*b - 41616) * q^77 + (-295*b - 72444) * q^79 + 6561 * q^81 + (-222*b + 54636) * q^83 + (1230*b - 23532) * q^85 + (171*b + 45954) * q^87 + (572*b - 16278) * q^89 + (-218*b - 39096) * q^91 + (45*b - 47196) * q^93 + (144*b + 78712) * q^95 + (60*b + 34546) * q^97 + (-486*b + 8100) * q^99
$\operatorname{Tr}(f)(q)$	$=$	2 * q + 18 * q^3 + 36 * q^5 + 120 * q^7 + 162 * q^9 + 200 * q^11 + 284 * q^13 + 324 * q^15 + 2676 * q^17 - 72 * q^19 + 1080 * q^21 - 3840 * q^23 + 10270 * q^25 + 1458 * q^27 + 10212 * q^29 - 10488 * q^31 + 1800 * q^33 + 18032 * q^35 + 13148 * q^37 + 2556 * q^39 + 4164 * q^41 + 5832 * q^43 + 2916 * q^45 - 1520 * q^47 - 10542 * q^49 + 24084 * q^51 + 9012 * q^53 - 91632 * q^55 - 648 * q^57 + 55096 * q^59 - 63444 * q^61 + 9720 * q^63 - 90120 * q^65 - 36792 * q^67 - 34560 * q^69 - 37664 * q^71 - 37836 * q^73 + 92430 * q^75 - 83232 * q^77 - 144888 * q^79 + 13122 * q^81 + 109272 * q^83 - 47064 * q^85 + 91908 * q^87 - 32556 * q^89 - 78192 * q^91 - 94392 * q^93 + 157424 * q^95 + 69092 * q^97 + 16200 * q^99

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  

$\iota_m(\nu)$

$a_{2}$

$a_{3}$

$a_{4}$

$a_{5}$

$a_{6}$

$a_{7}$

$a_{8}$

$a_{9}$

$a_{10}$

1.1

−5.56776
5.56776

0

9.00000

0

−71.0842

0

−29.0842

0

81.0000

0

1.2

0

9.00000

0

107.084

0

149.084

0

81.0000

0

Atkin-Lehner signs

$p$	Sign
$2$	$+1$
$3$	$-1$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	96.6.a.h	yes	2
3.b	odd	2	1		288.6.a.o		2
4.b	odd	2	1		96.6.a.g	✓	2
8.b	even	2	1		192.6.a.q		2
8.d	odd	2	1		192.6.a.r		2
12.b	even	2	1		288.6.a.n		2
16.e	even	4	2		768.6.d.s		4
16.f	odd	4	2		768.6.d.z		4
24.f	even	2	1		576.6.a.bm		2
24.h	odd	2	1		576.6.a.bn		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
96.6.a.g	✓	2	4.b	odd	2	1
96.6.a.h	yes	2	1.a	even	1	1	trivial
192.6.a.q		2	8.b	even	2	1
192.6.a.r		2	8.d	odd	2	1
288.6.a.n		2	12.b	even	2	1
288.6.a.o		2	3.b	odd	2	1
576.6.a.bm		2	24.f	even	2	1
576.6.a.bn		2	24.h	odd	2	1
768.6.d.s		4	16.e	even	4	2
768.6.d.z		4	16.f	odd	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{6}^{\mathrm{new}}(\Gamma_0(96))$:

$T_{5}^{2} - 36T_{5} - 7612$

$T_{7}^{2} - 120T_{7} - 4336$

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$
$3$	$(T - 9)^{2}$
$5$	$T^{2} - 36T - 7612$
$7$	$T^{2} - 120T - 4336$
$11$	$T^{2} - 200T - 275696$