Newform orbit 728.2.h.a

• Label: 728.2.h.a
• Level: $728$
• Weight: $2$
• Character orbit: 728.h
• Analytic conductor: $5.813$
• Analytic rank: $0$
• Dimension: $48$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 728 = 2^{3} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	728.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.81310926715\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) 48 * q + q^2 + q^4 - 10 * q^6 - 5 * q^8 - 48 * q^9 - 4 * q^11 + 10 * q^12 - 48 * q^13 + 10 * q^14 + 5 * q^16 - 15 * q^18 - 22 * q^20 - 6 * q^22 + 48 * q^25 - q^26 + 4 * q^28 - 26 * q^30 - 19 * q^32 + 36 * q^34 + 12 * q^35 - 5 * q^36 - 10 * q^38 - 48 * q^40 + 3 * q^42 - 4 * q^43 - 30 * q^44 + 24 * q^45 + 2 * q^46 + 48 * q^48 + 11 * q^50 - 40 * q^51 - q^52 + 62 * q^54 + 10 * q^56 + 2 * q^58 + 18 * q^60 - 36 * q^62 + 20 * q^63 - 11 * q^64 + 32 * q^66 - 4 * q^67 + 62 * q^68 - 11 * q^70 + 13 * q^72 - 44 * q^74 - 48 * q^76 - 20 * q^77 + 10 * q^78 + 32 * q^80 + 64 * q^81 - 50 * q^82 + 21 * q^84 + 2 * q^86 - 40 * q^87 - 66 * q^88 + 74 * q^90 - 24 * q^92 - 26 * q^94 - 84 * q^96 - 6 * q^98 + 20 * 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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
27.1	−1.40591	−	0.153058i		−	1.61368i	1.95315	+	0.430370i	−3.74465			−0.246987	+	2.26868i	0.220320	+	2.63656i	−2.68007	−	0.904005i	0.396033			5.26463	+	0.573149i
27.2	−1.40591	+	0.153058i			1.61368i	1.95315	−	0.430370i	−3.74465			−0.246987	−	2.26868i	0.220320	−	2.63656i	−2.68007	+	0.904005i	0.396033			5.26463	−	0.573149i
27.3	−1.39580	−	0.227483i			1.35635i	1.89650	+	0.635042i	2.01767			0.308546	−	1.89318i	−2.18534	−	1.49140i	−2.50267	−	1.31781i	1.16033			−2.81626	−	0.458986i
27.4	−1.39580	+	0.227483i		−	1.35635i	1.89650	−	0.635042i	2.01767			0.308546	+	1.89318i	−2.18534	+	1.49140i	−2.50267	+	1.31781i	1.16033			−2.81626	+	0.458986i
27.5	−1.32721	−	0.488382i			0.610846i	1.52297	+	1.29637i	0.780706			0.298326	−	0.810720i	2.18555	−	1.49110i	−1.38817	−	2.46434i	2.62687			−1.03616	−	0.381283i
27.6	−1.32721	+	0.488382i		−	0.610846i	1.52297	−	1.29637i	0.780706			0.298326	+	0.810720i	2.18555	+	1.49110i	−1.38817	+	2.46434i	2.62687			−1.03616	+	0.381283i
27.7	−1.29766	−	0.562216i		−	3.37415i	1.36783	+	1.45913i	−0.523773			−1.89700	+	4.37848i	−2.30913	−	1.29148i	−0.954623	−	2.66246i	−8.38487			0.679678	+	0.294474i
27.8	−1.29766	+	0.562216i			3.37415i	1.36783	−	1.45913i	−0.523773			−1.89700	−	4.37848i	−2.30913	+	1.29148i	−0.954623	+	2.66246i	−8.38487			0.679678	−	0.294474i
27.9	−1.25982	−	0.642544i		−	2.44281i	1.17427	+	1.61897i	3.85137			−1.56961	+	3.07749i	2.49284	+	0.886436i	−0.439107	−	2.79413i	−2.96733			−4.85202	−	2.47467i
27.10	−1.25982	+	0.642544i			2.44281i	1.17427	−	1.61897i	3.85137			−1.56961	−	3.07749i	2.49284	−	0.886436i	−0.439107	+	2.79413i	−2.96733			−4.85202	+	2.47467i
27.11	−1.10110	−	0.887461i			0.140163i	0.424825	+	1.95436i	−0.986963			0.124389	−	0.154333i	−1.39484	+	2.24820i	1.26665	−	2.52895i	2.98035			1.08674	+	0.875891i
27.12	−1.10110	+	0.887461i		−	0.140163i	0.424825	−	1.95436i	−0.986963			0.124389	+	0.154333i	−1.39484	−	2.24820i	1.26665	+	2.52895i	2.98035			1.08674	−	0.875891i
27.13	−0.879218	−	1.10769i			1.92788i	−0.453951	+	1.94780i	−2.16680			2.13549	−	1.69503i	1.74001	−	1.99308i	2.55668	−	1.20971i	−0.716713			1.90509	+	2.40014i
27.14	−0.879218	+	1.10769i		−	1.92788i	−0.453951	−	1.94780i	−2.16680			2.13549	+	1.69503i	1.74001	+	1.99308i	2.55668	+	1.20971i	−0.716713			1.90509	−	2.40014i
27.15	−0.635480	−	1.26339i		−	1.40360i	−1.19233	+	1.60572i	−3.11493			−1.77331	+	0.891962i	−2.64532	−	0.0479799i	2.78636	+	0.485980i	1.02989			1.97948	+	3.93539i
27.16	−0.635480	+	1.26339i			1.40360i	−1.19233	−	1.60572i	−3.11493			−1.77331	−	0.891962i	−2.64532	+	0.0479799i	2.78636	−	0.485980i	1.02989			1.97948	−	3.93539i
27.17	−0.629285	−	1.26649i			2.38356i	−1.20800	+	1.59397i	−1.68057			3.01876	−	1.49994i	1.10198	+	2.40533i	2.77892	+	0.526865i	−2.68138			1.05755	+	2.12842i
27.18	−0.629285	+	1.26649i		−	2.38356i	−1.20800	−	1.59397i	−1.68057			3.01876	+	1.49994i	1.10198	−	2.40533i	2.77892	−	0.526865i	−2.68138			1.05755	−	2.12842i
27.19	−0.270952	−	1.38801i			0.332595i	−1.85317	+	0.752171i	4.11591			0.461647	−	0.0901174i	−1.93546	+	1.80388i	1.54614	+	2.36843i	2.88938			−1.11521	−	5.71294i
27.20	−0.270952	+	1.38801i		−	0.332595i	−1.85317	−	0.752171i	4.11591			0.461647	+	0.0901174i	−1.93546	−	1.80388i	1.54614	−	2.36843i	2.88938			−1.11521	+	5.71294i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
56.e	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	728.2.h.a	✓	48
4.b	odd	2	1		2912.2.h.a		48
7.b	odd	2	1		728.2.h.b	yes	48
8.b	even	2	1		2912.2.h.b		48
8.d	odd	2	1		728.2.h.b	yes	48
28.d	even	2	1		2912.2.h.b		48
56.e	even	2	1	inner	728.2.h.a	✓	48
56.h	odd	2	1		2912.2.h.a		48

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
728.2.h.a	✓	48	1.a	even	1	1	trivial
728.2.h.a	✓	48	56.e	even	2	1	inner
728.2.h.b	yes	48	7.b	odd	2	1
728.2.h.b	yes	48	8.d	odd	2	1
2912.2.h.a		48	4.b	odd	2	1
2912.2.h.a		48	56.h	odd	2	1
2912.2.h.b		48	8.b	even	2	1
2912.2.h.b		48	28.d	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T5^24 - 72*T5^22 - 4*T5^21 + 2196*T5^20 + 220*T5^19 - 37124*T5^18 - 4932*T5^17 + 382458*T5^16 + 58556*T5^15 - 2488268*T5^14 - 400980*T5^13 + 10254980*T5^12 + 1621636*T5^11 - 26178852*T5^10 - 3885636*T5^9 + 39474333*T5^8 + 5740788*T5^7 - 32683908*T5^6 - 5839920*T5^5 + 13336464*T5^4 + 3514432*T5^3 - 1979328*T5^2 - 818944*T5 - 75264