Newform orbit 684.2.z.b

• Label: 684.2.z.b
• Level: $684$
• Weight: $2$
• Character orbit: 684.z
• Analytic conductor: $5.462$
• Analytic rank: $0$
• Dimension: $72$
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	684.z (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.46176749826\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(36\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) 72 * q - 12 * q^4 + 12 * q^10 + 4 * q^13 + 20 * q^16 + 28 * q^25 - 20 * q^34 - 40 * q^37 - 40 * q^40 - 32 * q^46 - 32 * q^49 - 8 * q^52 + 96 * q^58 + 28 * q^61 + 48 * q^64 - 72 * q^70 - 20 * q^73 - 36 * q^76 - 4 * q^82 + 48 * q^85 - 56 * q^88 - 112 * q^94

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
467.1	−1.40394	−	0.170195i		0		1.94207	+	0.477886i	−1.69671	−	0.979598i		0			−	0.187958i	−2.64520	−	1.00145i		0		2.21535	+	1.66407i
467.2	−1.40322	+	0.175978i		0		1.93806	−	0.493874i	2.75201	+	1.58887i		0			−	2.00261i	−2.63262	+	1.03407i		0		−4.14128	−	1.74525i
467.3	−1.39040	+	0.258456i		0		1.86640	−	0.718711i	−0.458047	−	0.264454i		0				4.98089i	−2.40928	+	1.48167i		0		0.705216	+	0.249310i
467.4	−1.38455	−	0.288112i		0		1.83398	+	0.797813i	−2.41325	−	1.39329i		0			−	2.82389i	−2.30939	−	1.63301i		0		2.93986	+	2.62438i
467.5	−1.30794	+	0.537849i		0		1.42144	−	1.40695i	−0.345026	−	0.199201i		0				2.46529i	−1.10243	+	2.60473i		0		0.558414	+	0.0749718i
467.6	−1.16717	+	0.798572i		0		0.724564	−	1.86414i	−0.436776	−	0.252172i		0			−	1.18882i	0.642960	+	2.75438i		0		0.711169	−	0.0544691i
467.7	−1.13615	+	0.842123i		0		0.581656	−	1.91355i	3.01008	+	1.73787i		0			−	0.522990i	0.950599	+	2.66390i		0		−4.88339	+	0.560383i
467.8	−0.941790	−	1.05500i		0		−0.226065	+	1.98718i	−2.41325	−	1.39329i		0				2.82389i	2.30939	−	1.63301i		0		0.802849	+	3.85818i
467.9	−0.929883	+	1.06551i		0		−0.270635	−	1.98160i	−3.60678	−	2.08238i		0			−	4.46262i	2.36308	+	1.55430i		0		5.57268	−	1.90671i
467.10	−0.918370	+	1.07545i		0		−0.313194	−	1.97533i	0.367115	+	0.211954i		0			−	1.57832i	2.41200	+	1.47725i		0		−0.565094	+	0.200163i
467.11	−0.849361	−	1.13075i		0		−0.557172	+	1.92082i	−1.69671	−	0.979598i		0				0.187958i	2.64520	−	1.00145i		0		0.333446	+	2.75059i
467.12	−0.549209	−	1.30321i		0		−1.39674	+	1.43148i	2.75201	+	1.58887i		0				2.00261i	2.63262	+	1.03407i		0		0.559214	−	4.45908i
467.13	−0.472184	+	1.33306i		0		−1.55408	−	1.25890i	−0.367115	−	0.211954i		0				1.57832i	2.41200	−	1.47725i		0		0.455893	−	0.389305i
467.14	−0.471369	−	1.33335i		0		−1.55562	+	1.25700i	−0.458047	−	0.264454i		0			−	4.98089i	2.40928	+	1.48167i		0		−0.136699	+	0.735390i
467.15	−0.457820	+	1.33806i		0		−1.58080	−	1.22518i	3.60678	+	2.08238i		0				4.46262i	2.36308	−	1.55430i		0		−4.43760	+	3.87273i
467.16	−0.188182	−	1.40164i		0		−1.92918	+	0.527525i	−0.345026	−	0.199201i		0			−	2.46529i	1.10243	+	2.60473i		0		−0.214279	+	0.521087i
467.17	−0.161227	+	1.40499i		0		−1.94801	−	0.453046i	−3.01008	−	1.73787i		0				0.522990i	0.950599	−	2.66390i		0		2.92700	−	3.94895i
467.18	−0.108000	+	1.41008i		0		−1.97667	−	0.304577i	0.436776	+	0.252172i		0				1.18882i	0.642960	−	2.75438i		0		−0.402756	+	0.588655i
467.19	0.108000	−	1.41008i		0		−1.97667	−	0.304577i	−0.436776	−	0.252172i		0				1.18882i	−0.642960	+	2.75438i		0		−0.402756	+	0.588655i
467.20	0.161227	−	1.40499i		0		−1.94801	−	0.453046i	3.01008	+	1.73787i		0				0.522990i	−0.950599	+	2.66390i		0		2.92700	−	3.94895i

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
19.c	even	3	1	inner
57.h	odd	6	1	inner
76.g	odd	6	1	inner
228.m	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	684.2.z.b	✓	72
3.b	odd	2	1	inner	684.2.z.b	✓	72
4.b	odd	2	1	inner	684.2.z.b	✓	72
12.b	even	2	1	inner	684.2.z.b	✓	72
19.c	even	3	1	inner	684.2.z.b	✓	72
57.h	odd	6	1	inner	684.2.z.b	✓	72
76.g	odd	6	1	inner	684.2.z.b	✓	72
228.m	even	6	1	inner	684.2.z.b	✓	72

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
684.2.z.b	✓	72	1.a	even	1	1	trivial
684.2.z.b	✓	72	3.b	odd	2	1	inner
684.2.z.b	✓	72	4.b	odd	2	1	inner
684.2.z.b	✓	72	12.b	even	2	1	inner
684.2.z.b	✓	72	19.c	even	3	1	inner
684.2.z.b	✓	72	57.h	odd	6	1	inner
684.2.z.b	✓	72	76.g	odd	6	1	inner
684.2.z.b	✓	72	228.m	even	6	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T5^36 - 52*T5^34 + 1664*T5^32 - 33968*T5^30 + 510748*T5^28 - 5572760*T5^26 + 46184000*T5^24 - 276980128*T5^22 + 1222656336*T5^20 - 3489901280*T5^18 + 6679092480*T5^16 - 4848667520*T5^14 + 2371900160*T5^12 - 773989888*T5^10 + 187995136*T5^8 - 32288768*T5^6 + 4087808*T5^4 - 327680*T5^2 + 16384