Newform orbit 322.3.h.a

• Label: 322.3.h.a
• Level: $322$
• Weight: $3$
• Character orbit: 322.h
• Analytic conductor: $8.774$
• Analytic rank: $0$
• Dimension: $56$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 322 = 2 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	322.h (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.77386451240\)
Analytic rank:	\(0\)
Dimension:	\(56\)
Relative dimension:	\(28\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 56 q + 12 q^{3} - 56 q^{4} - 4 q^{7} + 80 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
47.1	−0.707107	+	1.22474i	−4.33276	+	2.50152i	−1.00000	−	1.73205i	1.21736	+	0.702845i		−	7.07537i	6.79744	+	1.67177i	2.82843			8.01522	−	13.8828i	−1.72161	+	0.993973i
47.2	−0.707107	+	1.22474i	−3.83561	+	2.21449i	−1.00000	−	1.73205i	−5.49758	−	3.17403i		−	6.26352i	−0.781511	−	6.95624i	2.82843			5.30792	−	9.19359i	7.77475	−	4.48875i
47.3	−0.707107	+	1.22474i	−3.06700	+	1.77073i	−1.00000	−	1.73205i	4.80747	+	2.77559i		−	5.00838i	6.67083	−	2.12131i	2.82843			1.77098	−	3.06742i	−6.79879	+	3.92528i
47.4	−0.707107	+	1.22474i	−2.55289	+	1.47391i	−1.00000	−	1.73205i	1.43645	+	0.829334i		−	4.16885i	−2.32604	+	6.60224i	2.82843			−0.155180	+	0.268780i	−2.03145	+	1.17286i
47.5	−0.707107	+	1.22474i	−1.98245	+	1.14457i	−1.00000	−	1.73205i	−5.36395	−	3.09688i		−	3.23733i	−6.39638	+	2.84365i	2.82843			−1.87993	+	3.25613i	7.58578	−	4.37965i
47.6	−0.707107	+	1.22474i	−0.543801	+	0.313964i	−1.00000	−	1.73205i	7.19802	+	4.15578i		−	0.888023i	−5.94482	−	3.69583i	2.82843			−4.30285	+	7.45276i	−10.1795	+	5.87716i
47.7	−0.707107	+	1.22474i	−0.174775	+	0.100906i	−1.00000	−	1.73205i	0.489698	+	0.282727i		−	0.285406i	−4.73142	−	5.15885i	2.82843			−4.47964	+	7.75896i	−0.692537	+	0.399837i
47.8	−0.707107	+	1.22474i	−0.142548	+	0.0823001i	−1.00000	−	1.73205i	−1.73588	−	1.00221i		−	0.232780i	5.04685	−	4.85071i	2.82843			−4.48645	+	7.77077i	2.45491	−	1.41734i
47.9	−0.707107	+	1.22474i	1.39285	−	0.804161i	−1.00000	−	1.73205i	7.16669	+	4.13769i			2.27451i	1.79332	+	6.76639i	2.82843			−3.20665	+	5.55408i	−10.1352	+	5.85157i
47.10	−0.707107	+	1.22474i	2.89183	−	1.66960i	−1.00000	−	1.73205i	1.32786	+	0.766641i			4.72234i	−1.25907	+	6.88584i	2.82843			1.07512	−	1.86217i	−1.87788	+	1.08419i
47.11	−0.707107	+	1.22474i	3.13212	−	1.80833i	−1.00000	−	1.73205i	−6.13932	−	3.54454i			5.11474i	−6.88042	−	1.28836i	2.82843			2.04013	−	3.53361i	8.68231	−	5.01273i
47.12	−0.707107	+	1.22474i	3.51603	−	2.02998i	−1.00000	−	1.73205i	6.22999	+	3.59689i			5.74166i	6.30679	−	3.03716i	2.82843			3.74166	−	6.48074i	−8.81054	+	5.08677i
47.13	−0.707107	+	1.22474i	3.84786	−	2.22156i	−1.00000	−	1.73205i	1.14087	+	0.658680i			6.28352i	−3.44155	−	6.09555i	2.82843			5.37066	−	9.30226i	−1.61343	+	0.931514i
47.14	−0.707107	+	1.22474i	4.85114	−	2.80080i	−1.00000	−	1.73205i	−3.79239	−	2.18954i			7.92187i	5.56018	+	4.25258i	2.82843			11.1890	−	19.3799i	5.36325	−	3.09647i
47.15	0.707107	−	1.22474i	−4.90423	+	2.83146i	−1.00000	−	1.73205i	−5.92198	−	3.41905i			8.00857i	−6.82230	+	1.56724i	−2.82843			11.5343	−	19.9780i	−8.37494	+	4.83527i
47.16	0.707107	−	1.22474i	−3.78100	+	2.18296i	−1.00000	−	1.73205i	−0.607733	−	0.350875i			6.17434i	6.99651	+	0.221121i	−2.82843			5.03062	−	8.71328i	−0.859464	+	0.496212i
47.17	0.707107	−	1.22474i	−2.69391	+	1.55533i	−1.00000	−	1.73205i	−3.08901	−	1.78344i			4.39913i	5.18183	+	4.70623i	−2.82843			0.338087	−	0.585585i	−4.36852	+	2.52217i
47.18	0.707107	−	1.22474i	−2.50244	+	1.44479i	−1.00000	−	1.73205i	4.59492	+	2.65288i			4.08647i	−6.49312	+	2.61521i	−2.82843			−0.325186	+	0.563239i	6.49820	−	3.75174i
47.19	0.707107	−	1.22474i	−1.19086	+	0.687541i	−1.00000	−	1.73205i	−8.55945	−	4.94180i			1.94466i	1.07028	−	6.91770i	−2.82843			−3.55458	+	6.15671i	−12.1049	+	6.98877i
47.20	0.707107	−	1.22474i	−0.935211	+	0.539944i	−1.00000	−	1.73205i	6.24644	+	3.60638i			1.52719i	6.87255	−	1.32966i	−2.82843			−3.91692	+	6.78430i	8.83380	−	5.10020i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.d	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	322.3.h.a	✓	56
7.d	odd	6	1	inner	322.3.h.a	✓	56

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
322.3.h.a	✓	56	1.a	even	1	1	trivial
322.3.h.a	✓	56	7.d	odd	6	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(322, [\chi])\).