Newform orbit 333.3.bg.a

• Label: 333.3.bg.a
• Level: $333$
• Weight: $3$
• Character orbit: 333.bg
• Analytic conductor: $9.074$
• Analytic rank: $0$
• Dimension: $296$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 333 = 3^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	333.bg (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.07359280320\)
Analytic rank:	\(0\)
Dimension:	\(296\)
Relative dimension:	\(74\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 296 q - 2 q^{2} - 6 q^{3} - 6 q^{4} + 4 q^{5} + 12 q^{6} - 4 q^{7} - 12 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
88.1	−1.02248	+	3.81595i	−2.95096	−	0.540215i	−10.0519	−	5.80345i	0.104412	+	0.389670i	5.07873	−	10.7083i	6.05299			21.2496	−	21.2496i	8.41634	+	3.18830i	−1.59372
88.2	−0.997719	+	3.72354i	1.09225	+	2.79410i	−9.40519	−	5.43009i	−1.95532	−	7.29735i	−11.4937	+	1.27930i	7.24106			18.6996	−	18.6996i	−6.61398	+	6.10371i	29.1228
88.3	−0.985256	+	3.67703i	−0.0577468	−	2.99944i	−9.08569	−	5.24562i	−0.864220	−	3.22531i	11.0859	+	2.74288i	−12.7535			17.4729	−	17.4729i	−8.99333	+	0.346416i	12.7110
88.4	−0.975109	+	3.63916i	2.34727	−	1.86824i	−8.82853	−	5.09715i	2.46312	+	9.19249i	4.50999	+	10.3638i	6.97721			16.5019	−	16.5019i	2.01934	−	8.77053i	−35.8547
88.5	−0.924790	+	3.45136i	2.99200	−	0.218932i	−7.59258	−	4.38358i	−0.660436	−	2.46478i	−2.01136	+	10.5290i	−3.30511			12.0446	−	12.0446i	8.90414	−	1.31009i	9.11763
88.6	−0.918427	+	3.42761i	−1.86688	+	2.34836i	−7.44093	−	4.29603i	1.63002	+	6.08331i	−6.33467	−	8.55572i	−2.70320			11.5223	−	11.5223i	−2.02955	−	8.76818i	−22.3483
88.7	−0.843889	+	3.14944i	0.0540759	+	2.99951i	−5.74269	−	3.31555i	0.326614	+	1.21894i	−9.49241	−	2.36095i	−6.54103			6.06610	−	6.06610i	−8.99415	+	0.324403i	−4.11460
88.8	−0.843006	+	3.14614i	−2.29847	+	1.92796i	−5.72345	−	3.30444i	−1.87558	−	6.99975i	−4.12802	−	8.85660i	−5.90420			6.00858	−	6.00858i	1.56593	−	8.86272i	23.6033
88.9	−0.832081	+	3.10537i	−0.901375	−	2.86138i	−5.48686	−	3.16784i	0.442510	+	1.65147i	9.63567	−	0.418199i	4.49844			5.30967	−	5.30967i	−7.37505	+	5.15836i	−5.49663
88.10	−0.820343	+	3.06156i	2.98236	+	0.324820i	−5.23609	−	3.02306i	−0.477706	−	1.78282i	−3.44102	+	8.86422i	−0.433622			4.58578	−	4.58578i	8.78898	+	1.93746i	5.85010
88.11	−0.812805	+	3.03343i	1.62997	+	2.51857i	−5.07695	−	2.93118i	1.23336	+	4.60297i	−8.96476	+	2.89729i	10.8684			4.13559	−	4.13559i	−3.68640	+	8.21039i	−14.9653
88.12	−0.758602	+	2.83114i	−2.59542	−	1.50459i	−3.97578	−	2.29542i	2.29296	+	8.55743i	6.22859	−	6.20662i	−9.56797			1.22454	−	1.22454i	4.47244	+	7.81008i	−25.9667
88.13	−0.745183	+	2.78106i	1.99542	−	2.24015i	−3.71491	−	2.14480i	−0.720689	−	2.68965i	4.74305	+	7.21872i	2.95625			0.589600	−	0.589600i	−1.03658	−	8.94011i	8.01712
88.14	−0.740365	+	2.76308i	−2.21541	−	2.02286i	−3.62238	−	2.09138i	−2.52602	−	9.42724i	7.22954	−	4.62370i	4.33674			0.369674	−	0.369674i	0.816074	+	8.96292i	27.9184
88.15	−0.721912	+	2.69421i	−2.93419	−	0.624946i	−3.27352	−	1.88997i	0.101754	+	0.379751i	3.80196	−	7.45416i	−4.41460			−0.434034	+	0.434034i	8.21888	+	3.66742i	−1.09659
88.16	−0.721020	+	2.69088i	−2.60663	+	1.48509i	−3.25688	−	1.88036i	−0.159399	−	0.594883i	−2.11677	−	8.08491i	11.9856			−0.471357	+	0.471357i	4.58902	−	7.74215i	1.71569
88.17	−0.647014	+	2.41469i	1.93124	−	2.29572i	−1.94800	−	1.12468i	1.63131	+	6.08814i	4.29391	+	6.14871i	−11.2698			−3.09457	+	3.09457i	−1.54064	−	8.86715i	−15.7565
88.18	−0.573540	+	2.14048i	1.52804	+	2.58168i	−0.788605	−	0.455301i	−1.48065	−	5.52585i	−6.40243	+	1.79005i	−9.66044			−4.84090	+	4.84090i	−4.33017	+	7.88985i	12.6772
88.19	−0.571362	+	2.13235i	2.68610	+	1.33599i	−0.756366	−	0.436688i	1.68516	+	6.28911i	−4.38353	+	4.96438i	−3.89649			−4.88063	+	4.88063i	5.43028	+	7.17719i	−14.3734
88.20	−0.542543	+	2.02480i	0.140838	−	2.99669i	−0.341349	−	0.197078i	0.630325	+	2.35240i	5.99128	+	1.91100i	5.45555			−5.34478	+	5.34478i	−8.96033	−	0.844094i	−5.10512

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
333.bg	odd	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	333.3.bg.a	yes	296
9.c	even	3	1		333.3.ba.a	✓	296
37.g	odd	12	1		333.3.ba.a	✓	296
333.bg	odd	12	1	inner	333.3.bg.a	yes	296

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
333.3.ba.a	✓	296	9.c	even	3	1
333.3.ba.a	✓	296	37.g	odd	12	1
333.3.bg.a	yes	296	1.a	even	1	1	trivial
333.3.bg.a	yes	296	333.bg	odd	12	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(333, [\chi])\).