Newform orbit 684.2.cf.a

• Label: 684.2.cf.a
• Level: $684$
• Weight: $2$
• Character orbit: 684.cf
• Analytic conductor: $5.462$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	684.cf (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.46176749826\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(8\) over \(\Q(\zeta_{18})\)
Twist minimal:	no (minimal twist has level 76)
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 6 q^{2} + 12 q^{5} + 9 q^{8}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
91.1	−1.41400	+	0.0247662i		0		1.99877	−	0.0700386i	2.99965	−	2.51700i		0		−0.0108543	−	0.00626673i	−2.82452	+	0.148536i		0		−4.17916	+	3.63332i
91.2	−0.927206	+	1.06784i		0		−0.280577	−	1.98022i	0.220151	−	0.184728i		0		−0.588321	−	0.339668i	2.37472	+	1.53646i		0		−0.00686428	+	0.406368i
91.3	−0.854626	−	1.12677i		0		−0.539229	+	1.92594i	−0.579816	+	0.486524i		0		−2.62687	−	1.51662i	2.63093	−	1.03837i		0		1.04373	+	0.237525i
91.4	0.0238852	+	1.41401i		0		−1.99886	+	0.0675479i	0.220151	−	0.184728i		0		0.588321	+	0.339668i	−0.143257	−	2.82480i		0		0.266467	+	0.306884i
91.5	0.312491	−	1.37926i		0		−1.80470	−	0.862010i	−1.46633	+	1.23040i		0		1.58907	+	0.917452i	−1.75289	+	2.21977i		0		1.23882	+	2.40694i
91.6	0.647187	−	1.25744i		0		−1.16230	−	1.62759i	−1.46633	+	1.23040i		0		−1.58907	−	0.917452i	−2.79882	+	0.408157i		0		0.598159	+	2.64012i
91.7	1.06726	+	0.927872i		0		0.278109	+	1.98057i	2.99965	−	2.51700i		0		0.0108543	+	0.00626673i	−1.54090	+	2.37184i		0		5.53688	+	0.0969785i
91.8	1.37896	−	0.313814i		0		1.80304	−	0.865472i	−0.579816	+	0.486524i		0		2.62687	+	1.51662i	2.21472	−	1.75927i		0		−0.646863	+	0.852849i
127.1	−1.33647	−	0.462424i		0		1.57233	+	1.23604i	0.00805719	+	0.0456946i		0		−1.20959	−	0.698356i	−1.52980	−	2.37901i		0		0.0103621	−	0.0647955i
127.2	−0.947064	+	1.05027i		0		−0.206140	−	1.98935i	0.165124	+	0.936467i		0		2.67390	+	1.54378i	2.28458	+	1.66754i		0		−1.13993	−	0.713469i
127.3	−0.719007	+	1.21780i		0		−0.966058	−	1.75121i	−0.615920	−	3.49306i		0		−3.53555	−	2.04125i	2.82722	+	0.0826699i		0		4.69669	+	1.76147i
127.4	−0.223323	−	1.39647i		0		−1.90025	+	0.623726i	0.00805719	+	0.0456946i		0		1.20959	+	0.698356i	1.29538	+	2.51435i		0		0.0620117	−	0.0214562i
127.5	0.178792	+	1.40287i		0		−1.93607	+	0.501643i	0.503046	+	2.85292i		0		2.71118	+	1.56530i	−1.04989	−	2.62635i		0		−3.91232	+	1.21579i
127.6	1.19877	−	0.750298i		0		0.874105	−	1.79887i	0.165124	+	0.936467i		0		−2.67390	−	1.54378i	−0.301838	−	2.81228i		0		0.900576	+	0.998718i
127.7	1.32415	−	0.496616i		0		1.50675	−	1.31519i	−0.615920	−	3.49306i		0		3.53555	+	2.04125i	1.34202	−	2.48978i		0		−2.55028	−	4.31946i
127.8	1.35051	+	0.419681i		0		1.64774	+	1.13356i	0.503046	+	2.85292i		0		−2.71118	−	1.56530i	1.74954	+	2.22241i		0		−0.517948	+	4.06400i
307.1	−1.33647	+	0.462424i		0		1.57233	−	1.23604i	0.00805719	−	0.0456946i		0		−1.20959	+	0.698356i	−1.52980	+	2.37901i		0		0.0103621	+	0.0647955i
307.2	−0.947064	−	1.05027i		0		−0.206140	+	1.98935i	0.165124	−	0.936467i		0		2.67390	−	1.54378i	2.28458	−	1.66754i		0		−1.13993	+	0.713469i
307.3	−0.719007	−	1.21780i		0		−0.966058	+	1.75121i	−0.615920	+	3.49306i		0		−3.53555	+	2.04125i	2.82722	−	0.0826699i		0		4.69669	−	1.76147i
307.4	−0.223323	+	1.39647i		0		−1.90025	−	0.623726i	0.00805719	−	0.0456946i		0		1.20959	−	0.698356i	1.29538	−	2.51435i		0		0.0620117	+	0.0214562i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
19.f	odd	18	1	inner
76.k	even	18	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	684.2.cf.a		48
3.b	odd	2	1		76.2.k.a	✓	48
4.b	odd	2	1	inner	684.2.cf.a		48
12.b	even	2	1		76.2.k.a	✓	48
19.f	odd	18	1	inner	684.2.cf.a		48
57.j	even	18	1		76.2.k.a	✓	48
76.k	even	18	1	inner	684.2.cf.a		48
228.u	odd	18	1		76.2.k.a	✓	48

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
76.2.k.a	✓	48	3.b	odd	2	1
76.2.k.a	✓	48	12.b	even	2	1
76.2.k.a	✓	48	57.j	even	18	1
76.2.k.a	✓	48	228.u	odd	18	1
684.2.cf.a		48	1.a	even	1	1	trivial
684.2.cf.a		48	4.b	odd	2	1	inner
684.2.cf.a		48	19.f	odd	18	1	inner
684.2.cf.a		48	76.k	even	18	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(684, [\chi])\):

T5^24 - 6*T5^23 + 21*T5^22 - 49*T5^21 + 102*T5^20 + 243*T5^19 - 1111*T5^18 - 198*T5^17 + 4194*T5^16 - 17758*T5^15 + 37455*T5^14 + 126168*T5^13 - 95762*T5^12 - 698997*T5^11 + 150219*T5^10 + 1027540*T5^9 + 1275903*T5^8 + 1549953*T5^7 + 1435283*T5^6 + 524166*T5^5 - 59952*T5^4 - 42992*T5^3 + 30336*T5^2 - 576*T5 + 64

T7^48 - 75*T7^46 + 3246*T7^44 - 94779*T7^42 + 2073579*T7^40 - 35170146*T7^38 + 476790445*T7^36 - 5223502719*T7^34 + 46769734377*T7^32 - 342738020964*T7^30 + 2063279470374*T7^28 - 10176265590120*T7^26 + 41222100851757*T7^24 - 136650441075507*T7^22 + 370360001025873*T7^20 - 812771933531010*T7^18 + 1432836089416575*T7^16 - 1983745654793055*T7^14 + 2117008715830237*T7^12 - 1650110658117276*T7^10 + 910028738155632*T7^8 - 295687702799616*T7^6 + 59968205664000*T7^4 - 9420244992*T7^2 + 1478656