Newform orbit 336.2.u.a

• Label: 336.2.u.a
• Level: $336$
• Weight: $2$
• Character orbit: 336.u
• Analytic conductor: $2.683$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	336.u (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.68297350792\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(32\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 64 q - 4 q^{4} + 24 q^{8}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
139.1	−1.38346	+	0.293324i	−0.707107	+	0.707107i	1.82792	−	0.811604i	2.49549	−	2.49549i	0.770842	−	1.18567i	0.134305	+	2.64234i	−2.29079	+	1.65899i		−	1.00000i	−2.72043	+	4.18440i
139.2	−1.38346	+	0.293324i	0.707107	−	0.707107i	1.82792	−	0.811604i	−2.49549	+	2.49549i	−0.770842	+	1.18567i	0.134305	−	2.64234i	−2.29079	+	1.65899i		−	1.00000i	2.72043	−	4.18440i
139.3	−1.27023	−	0.621708i	−0.707107	+	0.707107i	1.22696	+	1.57942i	1.90169	−	1.90169i	1.33780	−	0.458573i	2.22978	−	1.42411i	−0.576579	−	2.76904i		−	1.00000i	−3.59788	+	1.23329i
139.4	−1.27023	−	0.621708i	0.707107	−	0.707107i	1.22696	+	1.57942i	−1.90169	+	1.90169i	−1.33780	+	0.458573i	2.22978	+	1.42411i	−0.576579	−	2.76904i		−	1.00000i	3.59788	−	1.23329i
139.5	−1.20526	+	0.739825i	−0.707107	+	0.707107i	0.905318	−	1.78337i	−1.32544	+	1.32544i	0.329115	−	1.37538i	2.64192	+	0.142388i	0.228233	+	2.81920i		−	1.00000i	0.616912	−	2.57810i
139.6	−1.20526	+	0.739825i	0.707107	−	0.707107i	0.905318	−	1.78337i	1.32544	−	1.32544i	−0.329115	+	1.37538i	2.64192	−	0.142388i	0.228233	+	2.81920i		−	1.00000i	−0.616912	+	2.57810i
139.7	−1.08962	−	0.901509i	−0.707107	+	0.707107i	0.374563	+	1.96461i	0.167468	−	0.167468i	1.40794	−	0.133018i	−2.61783	−	0.383395i	1.36298	−	2.47836i		−	1.00000i	−0.333451	+	0.0315033i
139.8	−1.08962	−	0.901509i	0.707107	−	0.707107i	0.374563	+	1.96461i	−0.167468	+	0.167468i	−1.40794	+	0.133018i	−2.61783	+	0.383395i	1.36298	−	2.47836i		−	1.00000i	0.333451	−	0.0315033i
139.9	−0.777418	−	1.18136i	−0.707107	+	0.707107i	−0.791242	+	1.83683i	−3.13913	+	3.13913i	1.38507	+	0.285633i	1.14645	−	2.38446i	2.78509	−	0.493238i		−	1.00000i	6.14886	+	1.26804i
139.10	−0.777418	−	1.18136i	0.707107	−	0.707107i	−0.791242	+	1.83683i	3.13913	−	3.13913i	−1.38507	−	0.285633i	1.14645	+	2.38446i	2.78509	−	0.493238i		−	1.00000i	−6.14886	−	1.26804i
139.11	−0.618822	+	1.27164i	−0.707107	+	0.707107i	−1.23412	−	1.57383i	1.80101	−	1.80101i	−0.461610	−	1.33676i	−1.67026	+	2.05189i	2.76504	−	0.595430i		−	1.00000i	1.17573	+	3.40473i
139.12	−0.618822	+	1.27164i	0.707107	−	0.707107i	−1.23412	−	1.57383i	−1.80101	+	1.80101i	0.461610	+	1.33676i	−1.67026	−	2.05189i	2.76504	−	0.595430i		−	1.00000i	−1.17573	−	3.40473i
139.13	−0.453489	+	1.33953i	−0.707107	+	0.707107i	−1.58870	−	1.21493i	0.308965	−	0.308965i	−0.626528	−	1.26786i	1.19726	−	2.35936i	2.34789	−	1.57715i		−	1.00000i	0.273757	+	0.553981i
139.14	−0.453489	+	1.33953i	0.707107	−	0.707107i	−1.58870	−	1.21493i	−0.308965	+	0.308965i	0.626528	+	1.26786i	1.19726	+	2.35936i	2.34789	−	1.57715i		−	1.00000i	−0.273757	−	0.553981i
139.15	−0.135637	−	1.40769i	−0.707107	+	0.707107i	−1.96321	+	0.381870i	2.46638	−	2.46638i	1.09130	+	0.899481i	−2.50032	−	0.865115i	0.803838	+	2.71180i		−	1.00000i	−3.80643	−	3.13737i
139.16	−0.135637	−	1.40769i	0.707107	−	0.707107i	−1.96321	+	0.381870i	−2.46638	+	2.46638i	−1.09130	−	0.899481i	−2.50032	+	0.865115i	0.803838	+	2.71180i		−	1.00000i	3.80643	+	3.13737i
139.17	0.0533274	+	1.41321i	−0.707107	+	0.707107i	−1.99431	+	0.150725i	−1.82248	+	1.82248i	−1.03700	−	0.961581i	−1.96782	+	1.76852i	−0.319358	−	2.81034i		−	1.00000i	−2.67273	−	2.47836i
139.18	0.0533274	+	1.41321i	0.707107	−	0.707107i	−1.99431	+	0.150725i	1.82248	−	1.82248i	1.03700	+	0.961581i	−1.96782	−	1.76852i	−0.319358	−	2.81034i		−	1.00000i	2.67273	+	2.47836i
139.19	0.0748365	−	1.41223i	−0.707107	+	0.707107i	−1.98880	−	0.211373i	−0.353074	+	0.353074i	0.945681	+	1.05152i	2.14763	+	1.54521i	−0.447343	+	2.79283i		−	1.00000i	0.472200	+	0.525045i
139.20	0.0748365	−	1.41223i	0.707107	−	0.707107i	−1.98880	−	0.211373i	0.353074	−	0.353074i	−0.945681	−	1.05152i	2.14763	−	1.54521i	−0.447343	+	2.79283i		−	1.00000i	−0.472200	−	0.525045i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	inner
16.f	odd	4	1	inner
112.j	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	336.2.u.a	✓	64
4.b	odd	2	1		1344.2.u.a		64
7.b	odd	2	1	inner	336.2.u.a	✓	64
16.e	even	4	1		1344.2.u.a		64
16.f	odd	4	1	inner	336.2.u.a	✓	64
28.d	even	2	1		1344.2.u.a		64
112.j	even	4	1	inner	336.2.u.a	✓	64
112.l	odd	4	1		1344.2.u.a		64

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
336.2.u.a	✓	64	1.a	even	1	1	trivial
336.2.u.a	✓	64	7.b	odd	2	1	inner
336.2.u.a	✓	64	16.f	odd	4	1	inner
336.2.u.a	✓	64	112.j	even	4	1	inner
1344.2.u.a		64	4.b	odd	2	1
1344.2.u.a		64	16.e	even	4	1
1344.2.u.a		64	28.d	even	2	1
1344.2.u.a		64	112.l	odd	4	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(336, [\chi])\).