Newform orbit 208.4.bj.a

• Label: 208.4.bj.a
• Level: $208$
• Weight: $4$
• Character orbit: 208.bj
• Analytic conductor: $12.272$
• Analytic rank: $0$
• Dimension: $328$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 208 = 2^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	208.bj (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(12.2723972812\)
Analytic rank:	\(0\)
Dimension:	\(328\)
Relative dimension:	\(82\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 328 q - 2 q^{2} - 2 q^{3} - 2 q^{4} - 8 q^{5} - 32 q^{6} - 8 q^{8}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
29.1	−2.82743	−	0.0751137i	1.86791	−	6.97115i	7.98872	+	0.424757i	−14.5060	+	14.5060i	−5.80503	+	19.5701i	4.34431	+	2.50819i	−22.5556	−	1.80103i	−21.7252	−	12.5430i	42.1043	−	39.9251i
29.2	−2.82680	−	0.0958661i	0.827065	−	3.08665i	7.98162	+	0.541989i	6.57725	−	6.57725i	−2.63385	+	8.64606i	2.35488	+	1.35959i	−22.5105	−	2.29726i	14.5393	+	8.39428i	−19.2231	+	17.9621i
29.3	−2.82653	−	0.103711i	2.49970	−	9.32900i	7.97849	+	0.586285i	10.2468	−	10.2468i	−8.03298	+	26.1094i	−4.76543	−	2.75132i	−22.4906	−	2.48461i	−57.3990	−	33.1393i	−30.0256	+	27.9001i
29.4	−2.82568	−	0.124735i	−0.973419	+	3.63285i	7.96888	+	0.704923i	2.38856	−	2.38856i	3.20371	−	10.1438i	24.6824	+	14.2504i	−22.4295	−	2.98588i	11.1326	+	6.42742i	−7.04723	+	6.45136i
29.5	−2.82162	−	0.196106i	−1.54466	+	5.76475i	7.92308	+	1.10668i	9.14283	−	9.14283i	5.48895	−	15.9630i	−13.1500	−	7.59215i	−22.1389	−	4.67639i	−7.46365	−	4.30914i	−27.5906	+	24.0046i
29.6	−2.71413	+	0.795923i	−2.61812	+	9.77095i	6.73301	−	4.32048i	−9.96776	+	9.96776i	−0.671006	−	28.6035i	20.5901	+	11.8877i	−14.8355	+	17.0853i	−65.2343	−	37.6630i	19.1202	−	34.9874i
29.7	−2.70833	+	0.815459i	−0.386786	+	1.44350i	6.67005	−	4.41706i	−8.75753	+	8.75753i	−0.129576	−	4.22489i	1.05470	+	0.608931i	−14.4628	+	17.4020i	21.4486	+	12.3833i	16.5768	−	30.8597i
29.8	−2.70493	−	0.826660i	−1.31458	+	4.90609i	6.63327	+	4.47211i	−8.54681	+	8.54681i	7.61152	−	12.1839i	−13.3654	−	7.71654i	−14.2456	−	17.5802i	1.04108	+	0.601069i	30.1838	−	16.0532i
29.9	−2.70276	+	0.833716i	1.43105	−	5.34075i	6.60984	−	4.50667i	−3.03215	+	3.03215i	0.584883	+	15.6279i	−3.59013	−	2.07276i	−14.1075	+	17.6912i	−3.09300	−	1.78574i	5.66722	−	10.7231i
29.10	−2.59425	−	1.12689i	0.851666	−	3.17846i	5.46026	+	5.84685i	−3.85745	+	3.85745i	−5.79120	+	7.28599i	−27.2059	−	15.7073i	−7.57653	−	21.3213i	14.0054	+	8.08602i	14.3541	−	5.66027i
29.11	−2.54136	+	1.24157i	−0.234516	+	0.875227i	4.91703	−	6.31053i	−4.35636	+	4.35636i	−0.490661	−	2.51543i	−28.2753	−	16.3247i	−4.66101	+	22.1422i	22.6717	+	13.0895i	5.66237	−	16.4798i
29.12	−2.51943	−	1.28549i	−1.12356	+	4.19319i	4.69503	+	6.47740i	−9.97130	+	9.97130i	8.22104	−	9.12012i	7.61443	+	4.39619i	−3.50217	−	22.3547i	7.06220	+	4.07736i	37.9400	−	12.3040i
29.13	−2.51941	+	1.28552i	−2.04076	+	7.61622i	4.69486	−	6.47752i	6.09557	−	6.09557i	−4.64932	−	21.8118i	−9.51260	−	5.49210i	−3.50126	+	22.3549i	−30.4594	−	17.5857i	−7.52124	+	23.1932i
29.14	−2.50186	−	1.31936i	1.14915	−	4.28869i	4.51857	+	6.60170i	−2.87179	+	2.87179i	−8.53335	+	9.21355i	27.7347	+	16.0126i	−2.59479	−	22.4781i	6.31033	+	3.64327i	10.9737	−	3.39588i
29.15	−2.48604	+	1.34893i	0.631322	−	2.35613i	4.36079	−	6.70698i	9.42829	−	9.42829i	1.60875	+	6.70903i	19.6222	+	11.3289i	−1.79387	+	22.5562i	18.2299	+	10.5250i	−10.7210	+	36.1572i
29.16	−2.40912	−	1.48194i	0.801963	−	2.99297i	3.60770	+	7.14034i	8.47558	−	8.47558i	−6.36742	+	6.02195i	−10.3520	−	5.97673i	1.89017	−	22.5483i	15.0680	+	8.69951i	−32.9790	+	7.85837i
29.17	−2.23941	−	1.72772i	−2.59673	+	9.69113i	2.02995	+	7.73817i	0.794114	−	0.794114i	22.5587	−	17.2160i	−9.79255	−	5.65373i	8.82350	−	20.8362i	−63.7923	−	36.8305i	−3.15036	+	0.406342i
29.18	−2.15473	−	1.83225i	2.14172	−	7.99299i	1.28575	+	7.89600i	0.339556	−	0.339556i	−19.2600	+	13.2986i	23.3362	+	13.4732i	11.6970	−	19.3696i	−35.9183	−	20.7374i	−1.35380	+	0.109503i
29.19	−2.08405	−	1.91226i	−0.677118	+	2.52704i	0.686539	+	7.97049i	15.5066	−	15.5066i	6.24350	−	3.97165i	11.8012	+	6.81342i	13.8108	−	17.9237i	17.4552	+	10.0778i	−61.9693	+	2.66394i
29.20	−2.01672	+	1.98314i	−1.50360	+	5.61152i	0.134302	−	7.99887i	10.0630	−	10.0630i	−8.09610	−	14.2987i	6.05213	+	3.49420i	15.5920	+	16.3978i	−5.84570	−	3.37501i	−0.337881	+	40.2504i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
13.c	even	3	1	inner
16.e	even	4	1	inner
208.bj	even	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	208.4.bj.a	✓	328
13.c	even	3	1	inner	208.4.bj.a	✓	328
16.e	even	4	1	inner	208.4.bj.a	✓	328
208.bj	even	12	1	inner	208.4.bj.a	✓	328

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
208.4.bj.a	✓	328	1.a	even	1	1	trivial
208.4.bj.a	✓	328	13.c	even	3	1	inner
208.4.bj.a	✓	328	16.e	even	4	1	inner
208.4.bj.a	✓	328	208.bj	even	12	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(208, [\chi])\).