Newform orbit 731.2.y.a

• Label: 731.2.y.a
• Level: $731$
• Weight: $2$
• Character orbit: 731.y
• Analytic conductor: $5.837$
• Analytic rank: $0$
• Dimension: $768$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 731 = 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	731.y (of order \(28\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.83706438776\)
Analytic rank:	\(0\)
Dimension:	\(768\)
Relative dimension:	\(64\) over \(\Q(\zeta_{28})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{28}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 768 q - 14 q^{3} + 104 q^{4} - 6 q^{5} - 36 q^{6} - 28 q^{7}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
4.1	−2.19597	−	1.75123i	−2.11738	−	0.238571i	1.31045	+	5.74146i	−0.0617234	+	0.0215980i	4.23192	+	4.23192i	−3.05915	−	3.05915i	4.73957	−	9.84181i	1.50159	+	0.342729i	0.173366	+	0.0606634i
4.2	−2.09603	−	1.67153i	−0.439959	−	0.0495715i	1.15430	+	5.05731i	−3.58688	+	1.25510i	0.839309	+	0.839309i	2.20062	+	2.20062i	3.70758	−	7.69887i	−2.73368	−	0.623944i	9.61617	+	3.36484i
4.3	−2.01642	−	1.60804i	−0.987529	−	0.111268i	1.03511	+	4.53513i	3.97010	−	1.38920i	1.81235	+	1.81235i	2.87738	+	2.87738i	2.96740	−	6.16187i	−1.96195	−	0.447803i	−10.2393	−	3.58289i
4.4	−2.00227	−	1.59676i	3.12141	+	0.351699i	1.01442	+	4.44445i	−2.71628	+	0.950466i	−5.68835	−	5.68835i	−0.896379	−	0.896379i	2.84322	−	5.90401i	6.69474	+	1.52803i	6.95640	+	2.43415i
4.5	−1.99507	−	1.59101i	1.16029	+	0.130733i	1.00393	+	4.39849i	−0.0820838	+	0.0287223i	−2.10685	−	2.10685i	0.499882	+	0.499882i	2.78079	−	5.77437i	−1.59561	−	0.364187i	0.209460	+	0.0732932i
4.6	−1.98792	−	1.58532i	1.88514	+	0.212405i	0.993571	+	4.35312i	0.785952	−	0.275017i	−3.41079	−	3.41079i	−0.526144	−	0.526144i	2.71950	−	5.64710i	0.583864	+	0.133263i	−1.99840	−	0.699271i
4.7	−1.84201	−	1.46896i	−3.11411	−	0.350875i	0.790139	+	3.46182i	−0.601633	+	0.210521i	5.22081	+	5.22081i	2.55945	+	2.55945i	1.58534	−	3.29200i	6.64976	+	1.51776i	1.41746	+	0.495992i
4.8	−1.76597	−	1.40831i	0.0746772	+	0.00841410i	0.690259	+	3.02422i	1.99849	−	0.699300i	−0.120028	−	0.120028i	−2.90895	−	2.90895i	1.08000	−	2.24264i	−2.91928	−	0.666306i	−4.51410	−	1.57955i
4.9	−1.69014	−	1.34784i	−2.41916	−	0.272574i	0.594855	+	2.60623i	−2.17264	+	0.760238i	3.72134	+	3.72134i	−1.13454	−	1.13454i	0.631484	−	1.31129i	2.85326	+	0.651238i	4.69674	+	1.64346i
4.10	−1.59420	−	1.27133i	3.17551	+	0.357794i	0.480150	+	2.10368i	3.26776	−	1.14344i	−4.60753	−	4.60753i	2.24837	+	2.24837i	0.139587	−	0.289855i	7.03107	+	1.60480i	−6.66317	−	2.33155i
4.11	−1.59352	−	1.27079i	−2.77311	−	0.312454i	0.479356	+	2.10019i	3.68453	−	1.28927i	4.02194	+	4.02194i	−1.17641	−	1.17641i	0.136369	−	0.283173i	4.66772	+	1.06538i	−7.50976	−	2.62778i
4.12	−1.57336	−	1.25472i	−1.09566	−	0.123452i	0.456119	+	1.99839i	1.15086	−	0.402703i	1.56898	+	1.56898i	−0.279078	−	0.279078i	0.0434659	−	0.0902579i	−1.73955	−	0.397040i	−2.31600	−	0.810403i
4.13	−1.52642	−	1.21728i	−1.43515	−	0.161703i	0.403150	+	1.76631i	−2.05291	+	0.718343i	1.99381	+	1.99381i	−0.0817540	−	0.0817540i	−0.159477	+	0.331157i	−0.891265	−	0.203425i	4.00802	+	1.40247i
4.14	−1.48420	−	1.18361i	0.454616	+	0.0512230i	0.356876	+	1.56358i	−2.90786	+	1.01751i	−0.614114	−	0.614114i	−3.31960	−	3.31960i	−0.326349	+	0.677670i	−2.72073	−	0.620989i	5.52018	+	1.93160i
4.15	−1.47162	−	1.17358i	1.46588	+	0.165165i	0.343343	+	1.50428i	−1.99219	+	0.697097i	−1.96339	−	1.96339i	3.08138	+	3.08138i	−0.373253	+	0.775068i	−0.803250	−	0.183337i	3.74985	+	1.31213i
4.16	−1.36039	−	1.08488i	2.64691	+	0.298235i	0.228670	+	1.00187i	2.58211	−	0.903520i	−3.27729	−	3.27729i	−2.06532	−	2.06532i	−0.734101	+	1.52438i	3.99239	+	0.911237i	−4.49290	−	1.57213i
4.17	−1.35561	−	1.08107i	1.45849	+	0.164333i	0.223944	+	0.981161i	0.881224	−	0.308354i	−1.79950	−	1.79950i	1.96518	+	1.96518i	−0.747499	+	1.55220i	−0.824584	−	0.188206i	−1.52795	−	0.534653i
4.18	−1.17476	−	0.936838i	2.38910	+	0.269187i	0.0573487	+	0.251261i	−3.52574	+	1.23371i	−2.55443	−	2.55443i	−0.191598	−	0.191598i	−1.13586	+	2.35864i	2.71054	+	0.618664i	5.29767	+	1.85374i
4.19	−0.974593	−	0.777212i	−1.71701	−	0.193461i	−0.0992688	−	0.434925i	1.61232	−	0.564176i	1.52303	+	1.52303i	0.159716	+	0.159716i	−1.32300	+	2.74724i	−0.0140846	−	0.00321473i	−2.00984	−	0.703275i
4.20	−0.921645	−	0.734988i	−0.106119	−	0.0119568i	−0.135818	−	0.595060i	0.980725	−	0.343171i	0.0890163	+	0.0890163i	1.87363	+	1.87363i	−1.33513	+	2.77243i	−2.91367	−	0.665025i	−1.15611	−	0.404539i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
17.c	even	4	1	inner
43.e	even	7	1	inner
731.y	even	28	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	731.2.y.a	✓	768
17.c	even	4	1	inner	731.2.y.a	✓	768
43.e	even	7	1	inner	731.2.y.a	✓	768
731.y	even	28	1	inner	731.2.y.a	✓	768

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
731.2.y.a	✓	768	1.a	even	1	1	trivial
731.2.y.a	✓	768	17.c	even	4	1	inner
731.2.y.a	✓	768	43.e	even	7	1	inner
731.2.y.a	✓	768	731.y	even	28	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(731, [\chi])\).