Newform orbit 861.2.bh.a

• Label: 861.2.bh.a
• Level: $861$
• Weight: $2$
• Character orbit: 861.bh
• Analytic conductor: $6.875$
• Analytic rank: $0$
• Dimension: $432$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 861 = 3 \cdot 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	861.bh (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.87511961403\)
Analytic rank:	\(0\)
Dimension:	\(432\)
Relative dimension:	\(108\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 432 q - 6 q^{3} - 216 q^{4} - 8 q^{7}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
173.1	−1.40608	+	2.43541i	1.19260	−	1.25607i	−2.95415	−	5.11673i	1.01122	+	0.583826i	1.38214	+	4.67061i	−2.17839	+	1.50154i	10.9908			−0.155409	−	2.99597i	−2.84371	+	1.64182i
173.2	−1.36378	+	2.36214i	−0.471240	+	1.66671i	−2.71981	−	4.71085i	−3.31557	−	1.91424i	−3.29435	−	3.38617i	2.60728	+	0.449525i	9.38180			−2.55587	−	1.57084i	9.04344	−	5.22123i
173.3	−1.35549	+	2.34777i	1.31761	+	1.12424i	−2.67469	−	4.63270i	0.0648062	+	0.0374159i	−4.42545	+	1.56956i	−0.766268	−	2.53236i	9.08007			0.472188	+	2.96261i	−0.175688	+	0.101433i
173.4	−1.32025	+	2.28673i	−1.07522	−	1.35791i	−2.48610	−	4.30605i	−2.22764	−	1.28613i	4.52472	−	0.665967i	−0.517101	+	2.59473i	7.84806			−0.687814	+	2.92009i	5.88206	−	3.39601i
173.5	−1.31583	+	2.27909i	−1.61798	+	0.618166i	−2.46283	−	4.26574i	1.35478	+	0.782184i	0.720139	−	4.50093i	−0.238938	+	2.63494i	7.69934			2.23574	−	2.00037i	−3.56533	+	2.05845i
173.6	−1.27574	+	2.20964i	−1.45623	−	0.937763i	−2.25500	−	3.90578i	−1.67046	−	0.964441i	3.92988	−	2.02140i	−1.07350	−	2.41818i	6.40420			1.24120	+	2.73119i	4.26213	−	2.46074i
173.7	−1.24912	+	2.16354i	1.73066	+	0.0693311i	−2.12062	−	3.67301i	−0.128936	−	0.0744415i	−2.31181	+	3.65776i	2.43910	+	1.02508i	5.59915			2.99039	+	0.239977i	0.322115	−	0.185973i
173.8	−1.24690	+	2.15969i	−0.845767	+	1.51152i	−2.10952	−	3.65379i	3.02330	+	1.74550i	−2.20982	−	3.71131i	0.564554	−	2.58482i	5.53384			−1.56936	−	2.55678i	−7.53950	+	4.35293i
173.9	−1.23592	+	2.14068i	0.681878	+	1.59218i	−2.05501	−	3.55938i	2.85889	+	1.65058i	−4.25110	−	0.508131i	0.860008	+	2.50208i	5.21564			−2.07009	+	2.17135i	−7.06674	+	4.07999i
173.10	−1.22794	+	2.12686i	0.0282489	−	1.73182i	−2.01569	−	3.49128i	0.537051	+	0.310067i	3.64865	+	2.18666i	0.704325	−	2.55028i	4.98886			−2.99840	−	0.0978442i	−1.31894	+	0.761490i
173.11	−1.22412	+	2.12024i	0.534775	−	1.64743i	−1.99695	−	3.45882i	−2.48558	−	1.43505i	2.83831	+	3.15050i	2.60131	−	0.482872i	4.88154			−2.42803	−	1.76200i	6.08529	−	3.51335i
173.12	−1.19533	+	2.07038i	−0.634645	+	1.61159i	−1.85765	−	3.21754i	−1.40870	−	0.813315i	−2.57799	−	3.24035i	−2.55351	−	0.692504i	4.10070			−2.19445	−	2.04558i	3.36774	−	1.94437i
173.13	−1.18208	+	2.04742i	−1.46004	−	0.931815i	−1.79462	−	3.10837i	3.35949	+	1.93960i	3.63370	−	1.88784i	−2.62391	−	0.339238i	3.75720			1.26344	+	2.72098i	−7.94237	+	4.58553i
173.14	−1.13443	+	1.96489i	1.36819	−	1.06210i	−1.57386	−	2.72600i	3.39629	+	1.96085i	0.534807	+	3.89322i	1.14760	−	2.38391i	2.60400			0.743867	−	2.90631i	−7.70570	+	4.44889i
173.15	−1.10433	+	1.91275i	1.05281	+	1.37535i	−1.43907	−	2.49254i	−1.95853	−	1.13076i	−3.79334	+	0.494937i	−2.28045	+	1.34147i	1.93950			−0.783163	+	2.89597i	4.32572	−	2.49745i
173.16	−1.04616	+	1.81201i	−1.58350	−	0.701801i	−1.18891	−	2.05926i	0.769448	+	0.444241i	2.92827	−	2.13512i	2.63787	+	0.204066i	0.790543			2.01495	+	2.22261i	−1.60994	+	0.929497i
173.17	−1.03557	+	1.79366i	−1.27647	+	1.17074i	−1.14481	−	1.98287i	0.850668	+	0.491133i	−0.778041	−	3.50193i	−2.64500	+	0.0631591i	0.599853			0.258732	−	2.98882i	−1.76185	+	1.01721i
173.18	−1.02381	+	1.77328i	1.63061	−	0.584034i	−1.09636	−	1.89895i	−2.10816	−	1.21714i	−0.633774	+	3.48948i	−0.735444	+	2.54148i	0.394606			2.31781	−	1.90467i	4.31669	−	2.49224i
173.19	−1.01997	+	1.76664i	1.72708	+	0.131165i	−1.08068	−	1.87179i	1.11214	+	0.642094i	−1.99329	+	2.91734i	−1.96143	−	1.77561i	0.329161			2.96559	+	0.453063i	−2.26870	+	1.30983i
173.20	−1.00057	+	1.73305i	−1.70339	+	0.313804i	−1.00230	−	1.73603i	−2.08758	−	1.20526i	1.16053	−	3.26603i	−0.140043	+	2.64204i	0.00919927			2.80305	−	1.06906i	4.17755	−	2.41191i

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
41.c	even	4	1	inner
123.f	odd	4	1	inner
287.q	odd	12	1	inner
861.bh	even	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	861.2.bh.a	✓	432
3.b	odd	2	1	inner	861.2.bh.a	✓	432
7.d	odd	6	1	inner	861.2.bh.a	✓	432
21.g	even	6	1	inner	861.2.bh.a	✓	432
41.c	even	4	1	inner	861.2.bh.a	✓	432
123.f	odd	4	1	inner	861.2.bh.a	✓	432
287.q	odd	12	1	inner	861.2.bh.a	✓	432
861.bh	even	12	1	inner	861.2.bh.a	✓	432

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
861.2.bh.a	✓	432	1.a	even	1	1	trivial
861.2.bh.a	✓	432	3.b	odd	2	1	inner
861.2.bh.a	✓	432	7.d	odd	6	1	inner
861.2.bh.a	✓	432	21.g	even	6	1	inner
861.2.bh.a	✓	432	41.c	even	4	1	inner
861.2.bh.a	✓	432	123.f	odd	4	1	inner
861.2.bh.a	✓	432	287.q	odd	12	1	inner
861.2.bh.a	✓	432	861.bh	even	12	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(861, [\chi])\).