Newform orbit 76.2.k.a

• Label: 76.2.k.a
• Level: $76$
• Weight: $2$
• Character orbit: 76.k
• Analytic conductor: $0.607$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 76 = 2^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	76.k (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.606863055362\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(8\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 48q - 6q^{2} - 12q^{5} - 12q^{6} - 9q^{8} - 18q^{9} + O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
3.1	−1.35051	+	0.419681i	−1.23656	+	0.450071i	1.64774	−	1.13356i	−0.503046	+	2.85292i	1.48110	−	1.12678i	−2.71118	+	1.56530i	−1.74954	+	2.22241i	−0.971618	+	0.815285i	−0.517948	−	4.06400i
3.2	−1.32415	−	0.496616i	−1.09089	+	0.397053i	1.50675	+	1.31519i	0.615920	−	3.49306i	1.64169	+	0.0159973i	3.53555	−	2.04125i	−1.34202	−	2.48978i	−1.26573	+	1.06208i	−2.55028	+	4.31946i
3.3	−1.19877	−	0.750298i	2.80443	−	1.02073i	0.874105	+	1.79887i	−0.165124	+	0.936467i	−4.12772	−	0.880539i	−2.67390	+	1.54378i	0.301838	−	2.81228i	4.52481	−	3.79677i	0.900576	−	0.998718i
3.4	−0.178792	+	1.40287i	1.23656	−	0.450071i	−1.93607	−	0.501643i	−0.503046	+	2.85292i	0.410302	+	1.81520i	2.71118	−	1.56530i	1.04989	−	2.62635i	−0.971618	+	0.815285i	−3.91232	−	1.21579i
3.5	0.223323	−	1.39647i	0.855656	−	0.311433i	−1.90025	−	0.623726i	−0.00805719	+	0.0456946i	−0.243820	−	1.26445i	1.20959	−	0.698356i	−1.29538	+	2.51435i	−1.66298	+	1.39540i	0.0620117	+	0.0214562i
3.6	0.719007	+	1.21780i	1.09089	−	0.397053i	−0.966058	+	1.75121i	0.615920	−	3.49306i	1.26789	+	1.04300i	−3.53555	+	2.04125i	−2.82722	+	0.0826699i	−1.26573	+	1.06208i	4.69669	−	1.76147i
3.7	0.947064	+	1.05027i	−2.80443	+	1.02073i	−0.206140	+	1.98935i	−0.165124	+	0.936467i	−3.72802	−	1.97872i	2.67390	−	1.54378i	−2.28458	+	1.66754i	4.52481	−	3.79677i	−1.13993	+	0.713469i
3.8	1.33647	−	0.462424i	−0.855656	+	0.311433i	1.57233	−	1.23604i	−0.00805719	+	0.0456946i	−0.999548	+	0.811899i	−1.20959	+	0.698356i	1.52980	−	2.37901i	−1.66298	+	1.39540i	0.0103621	+	0.0647955i
15.1	−1.37896	+	0.313814i	−0.00846870	−	0.0480284i	1.80304	−	0.865472i	0.579816	−	0.486524i	0.0267499	+	0.0635714i	2.62687	+	1.51662i	−2.21472	+	1.75927i	2.81684	−	1.02525i	−0.646863	+	0.852849i
15.2	−1.06726	−	0.927872i	0.361434	+	2.04979i	0.278109	+	1.98057i	−2.99965	+	2.51700i	1.51620	−	2.52304i	0.0108543	+	0.00626673i	1.54090	−	2.37184i	−1.25194	+	0.455671i	5.53688	+	0.0969785i
15.3	−0.647187	+	1.25744i	0.547993	+	3.10782i	−1.16230	−	1.62759i	1.46633	−	1.23040i	−4.26255	−	1.32228i	−1.58907	−	0.917452i	2.79882	−	0.408157i	−6.53918	+	2.38007i	0.598159	+	2.64012i
15.4	−0.312491	+	1.37926i	−0.547993	−	3.10782i	−1.80470	−	0.862010i	1.46633	−	1.23040i	4.45773	+	0.215343i	1.58907	+	0.917452i	1.75289	−	2.21977i	−6.53918	+	2.38007i	1.23882	+	2.40694i
15.5	−0.0238852	−	1.41401i	−0.306623	−	1.73895i	−1.99886	+	0.0675479i	−0.220151	+	0.184728i	−2.45157	+	0.475104i	0.588321	+	0.339668i	0.143257	+	2.82480i	−0.110838	+	0.0403418i	0.266467	+	0.306884i
15.6	0.854626	+	1.12677i	0.00846870	+	0.0480284i	−0.539229	+	1.92594i	0.579816	−	0.486524i	−0.0468794	+	0.0505886i	−2.62687	−	1.51662i	−2.63093	+	1.03837i	2.81684	−	1.02525i	1.04373	+	0.237525i
15.7	0.927206	−	1.06784i	0.306623	+	1.73895i	−0.280577	−	1.98022i	−0.220151	+	0.184728i	2.14122	+	1.28494i	−0.588321	−	0.339668i	−2.37472	−	1.53646i	−0.110838	+	0.0403418i	−0.00686428	+	0.406368i
15.8	1.41400	−	0.0247662i	−0.361434	−	2.04979i	1.99877	−	0.0700386i	−2.99965	+	2.51700i	−0.561832	−	2.88945i	−0.0108543	−	0.00626673i	2.82452	−	0.148536i	−1.25194	+	0.455671i	−4.17916	+	3.63332i
51.1	−1.35051	−	0.419681i	−1.23656	−	0.450071i	1.64774	+	1.13356i	−0.503046	−	2.85292i	1.48110	+	1.12678i	−2.71118	−	1.56530i	−1.74954	−	2.22241i	−0.971618	−	0.815285i	−0.517948	+	4.06400i
51.2	−1.32415	+	0.496616i	−1.09089	−	0.397053i	1.50675	−	1.31519i	0.615920	+	3.49306i	1.64169	−	0.0159973i	3.53555	+	2.04125i	−1.34202	+	2.48978i	−1.26573	−	1.06208i	−2.55028	−	4.31946i
51.3	−1.19877	+	0.750298i	2.80443	+	1.02073i	0.874105	−	1.79887i	−0.165124	−	0.936467i	−4.12772	+	0.880539i	−2.67390	−	1.54378i	0.301838	+	2.81228i	4.52481	+	3.79677i	0.900576	+	0.998718i
51.4	−0.178792	−	1.40287i	1.23656	+	0.450071i	−1.93607	+	0.501643i	−0.503046	−	2.85292i	0.410302	−	1.81520i	2.71118	+	1.56530i	1.04989	+	2.62635i	−0.971618	−	0.815285i	−3.91232	+	1.21579i

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	76.2.k.a	✓	48
3.b	odd	2	1		684.2.cf.a		48
4.b	odd	2	1	inner	76.2.k.a	✓	48
12.b	even	2	1		684.2.cf.a		48
19.f	odd	18	1	inner	76.2.k.a	✓	48
57.j	even	18	1		684.2.cf.a		48
76.k	even	18	1	inner	76.2.k.a	✓	48
228.u	odd	18	1		684.2.cf.a		48

    

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

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(76, [\chi])\).