Newform orbit 672.2.cl.a

• Label: 672.2.cl.a
• Level: $672$
• Weight: $2$
• Character orbit: 672.cl
• Analytic conductor: $5.366$
• Analytic rank: $0$
• Dimension: $992$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	672.cl (of order \(24\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.36594701583\)
Analytic rank:	\(0\)
Dimension:	\(992\)
Relative dimension:	\(124\) over \(\Q(\zeta_{24})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{24}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 992 q - 12 q^{3} - 8 q^{4} - 16 q^{7} - 4 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
5.1	−1.41420	−	0.00627207i	1.70828	+	0.286000i	1.99992	+	0.0177399i	2.85386	−	2.18984i	−2.41405	−	0.415175i	1.02470	−	2.43926i	−2.82818	−	0.0376314i	2.83641	+	0.977132i	−4.04966	+	3.07897i
5.2	−1.41095	+	0.0959980i	1.14857	−	1.29645i	1.98157	−	0.270897i	−0.943490	+	0.723965i	−1.49612	+	1.93949i	2.59706	−	0.505248i	−2.76989	+	0.572449i	−0.361572	−	2.97813i	1.26172	−	1.11205i
5.3	−1.40961	−	0.113975i	−1.53695	−	0.798620i	1.97402	+	0.321321i	1.34762	−	1.03406i	2.07548	+	1.30092i	−1.72361	+	2.00728i	−2.74598	−	0.677926i	1.72441	+	2.45487i	−2.01748	+	1.30404i
5.4	−1.40796	+	0.132797i	−1.50196	+	0.862626i	1.96473	−	0.373948i	1.45654	−	1.11764i	2.00015	−	1.41400i	0.612663	−	2.57384i	−2.71661	+	0.787417i	1.51175	−	2.59126i	−1.90233	+	1.76702i
5.5	−1.40471	−	0.163696i	−0.492524	+	1.66055i	1.94641	+	0.459890i	2.75037	−	2.11044i	0.963677	−	2.25196i	1.12926	+	2.39265i	−2.65885	−	0.964629i	−2.51484	−	1.63572i	−4.20894	+	2.51432i
5.6	−1.40255	+	0.181281i	1.06633	+	1.36489i	1.93427	−	0.508510i	−0.619389	+	0.475274i	−1.74301	−	1.72102i	−0.389030	+	2.61699i	−2.62073	+	1.06386i	−0.725861	+	2.91086i	0.782564	−	0.778877i
5.7	−1.40141	+	0.189876i	−1.45571	−	0.938572i	1.92789	−	0.532187i	−2.63125	+	2.01903i	2.21825	+	1.03892i	2.38849	−	1.13803i	−2.60072	+	1.11187i	1.23816	+	2.73257i	3.30409	−	3.32909i
5.8	−1.40039	+	0.197272i	−0.343286	+	1.69769i	1.92217	−	0.552515i	−1.11245	+	0.853611i	0.145826	−	2.44515i	−2.23299	−	1.41907i	−2.58278	+	1.15293i	−2.76431	−	1.16559i	1.38946	−	1.41484i
5.9	−1.37985	−	0.309881i	−0.795286	−	1.53868i	1.80795	+	0.855176i	−0.314549	+	0.241362i	0.620565	+	2.36958i	−1.00424	−	2.44775i	−2.22969	−	1.74026i	−1.73504	+	2.44737i	0.508823	−	0.235569i
5.10	−1.37901	−	0.313576i	−1.60439	+	0.652629i	1.80334	+	0.864850i	−2.50546	+	1.92250i	2.41712	−	0.396883i	−2.58895	+	0.545310i	−2.21563	−	1.75812i	2.14815	−	2.09415i	4.05790	−	1.86550i
5.11	−1.37037	+	0.349411i	0.847014	−	1.51082i	1.75582	−	0.957644i	3.07582	−	2.36016i	−0.632825	+	2.36633i	−2.61637	+	0.393209i	−2.07152	+	1.92583i	−1.56514	−	2.55936i	−3.39035	+	4.30902i
5.12	−1.35484	−	0.405475i	1.73060	+	0.0709126i	1.67118	+	1.09871i	−3.18327	+	2.44261i	−2.31593	−	0.797790i	−0.600237	−	2.57676i	−1.81868	−	2.16619i	2.98994	+	0.245443i	5.30324	−	2.01861i
5.13	−1.35453	+	0.406491i	1.70394	−	0.310785i	1.66953	−	1.10121i	0.463851	−	0.355926i	−2.18172	+	1.11361i	2.00155	+	1.73026i	−1.81380	+	2.17028i	2.80683	−	1.05912i	−0.483622	+	0.670666i
5.14	−1.32344	−	0.498498i	1.71920	+	0.210627i	1.50300	+	1.31947i	0.873713	−	0.670423i	−2.17026	−	1.13577i	−2.22939	+	1.42472i	−1.33138	−	2.49548i	2.91127	+	0.724217i	−1.49051	+	0.451722i
5.15	−1.31601	−	0.517795i	0.850481	−	1.50887i	1.46378	+	1.36285i	−1.67532	+	1.28552i	−1.90053	+	1.54531i	−0.689207	+	2.55441i	−1.22067	−	2.55146i	−1.55336	−	2.56653i	2.87038	−	0.824287i
5.16	−1.28684	−	0.586545i	−0.137590	−	1.72658i	1.31193	+	1.50958i	2.29808	−	1.76338i	−0.835659	+	2.30254i	2.60977	+	0.434876i	−0.802810	−	2.71210i	−2.96214	+	0.475118i	−3.99157	+	0.921264i
5.17	−1.28211	+	0.596819i	−1.70808	+	0.287184i	1.28761	−	1.53038i	−0.995230	+	0.763667i	2.01855	−	1.38761i	0.767863	+	2.53187i	−0.737507	+	2.73058i	2.83505	−	0.981063i	0.820224	−	1.57308i
5.18	−1.27614	+	0.609486i	1.45112	+	0.945653i	1.25705	−	1.55558i	−0.911522	+	0.699435i	−2.42819	−	0.322347i	−2.20024	−	1.46933i	−0.656072	+	2.75128i	1.21148	+	2.74451i	0.736931	−	1.44813i
5.19	−1.27455	+	0.612801i	−0.459613	−	1.66996i	1.24895	−	1.56209i	0.761923	−	0.584644i	1.60915	+	1.84679i	1.26453	+	2.32400i	−0.634599	+	2.75632i	−2.57751	+	1.53507i	−0.612838	+	1.21206i
5.20	−1.26440	+	0.633474i	−0.0774681	−	1.73032i	1.19742	−	1.60193i	−3.19508	+	2.45167i	1.19406	+	2.13874i	−2.54901	+	0.708900i	−0.499238	+	2.78402i	−2.98800	+	0.268089i	2.48679	−	5.12390i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
7.d	odd	6	1	inner
21.g	even	6	1	inner
32.g	even	8	1	inner
96.p	odd	8	1	inner
224.bc	odd	24	1	inner
672.cl	even	24	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	672.2.cl.a	✓	992
3.b	odd	2	1	inner	672.2.cl.a	✓	992
7.d	odd	6	1	inner	672.2.cl.a	✓	992
21.g	even	6	1	inner	672.2.cl.a	✓	992
32.g	even	8	1	inner	672.2.cl.a	✓	992
96.p	odd	8	1	inner	672.2.cl.a	✓	992
224.bc	odd	24	1	inner	672.2.cl.a	✓	992
672.cl	even	24	1	inner	672.2.cl.a	✓	992

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
672.2.cl.a	✓	992	1.a	even	1	1	trivial
672.2.cl.a	✓	992	3.b	odd	2	1	inner
672.2.cl.a	✓	992	7.d	odd	6	1	inner
672.2.cl.a	✓	992	21.g	even	6	1	inner
672.2.cl.a	✓	992	32.g	even	8	1	inner
672.2.cl.a	✓	992	96.p	odd	8	1	inner
672.2.cl.a	✓	992	224.bc	odd	24	1	inner
672.2.cl.a	✓	992	672.cl	even	24	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(672, [\chi])\).