Newform orbit 304.3.j.a

• Label: 304.3.j.a
• Level: $304$
• Weight: $3$
• Character orbit: 304.j
• Analytic conductor: $8.283$
• Analytic rank: $0$
• Dimension: $156$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 304 = 2^{4} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	304.j (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.28340003655\)
Analytic rank:	\(0\)
Dimension:	\(156\)
Relative dimension:	\(78\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 156 q - 4 q^{4} - 4 q^{5} - 4 q^{6}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
37.1	−1.99944	−	0.0473601i	−1.35265	−	1.35265i	3.99551	+	0.189387i	1.06840	+	1.06840i	2.64048	+	2.76860i			13.1300i	−7.97982	−	0.567896i		−	5.34069i	−2.08561	−	2.18681i
37.2	−1.99687	+	0.111904i	0.454956	+	0.454956i	3.97495	−	0.446915i	2.88745	+	2.88745i	−0.959398	−	0.857575i		−	5.60174i	−7.88744	+	1.33724i		−	8.58603i	−6.08897	−	5.44273i
37.3	−1.97492	+	0.315717i	2.29462	+	2.29462i	3.80065	−	1.24704i	−0.749382	−	0.749382i	−5.25614	−	3.80724i		−	2.07330i	−7.11227	+	3.66273i			1.53054i	1.71656	+	1.24338i
37.4	−1.96365	−	0.379582i	−4.06225	−	4.06225i	3.71184	+	1.49073i	−1.33991	−	1.33991i	6.43488	+	9.51879i		−	8.23431i	−6.72289	−	4.33622i			24.0038i	2.12250	+	3.13971i
37.5	−1.94936	−	0.447221i	3.86932	+	3.86932i	3.59999	+	1.74359i	3.65003	+	3.65003i	−5.81225	−	9.27313i			3.29205i	−6.23789	−	5.00886i			20.9433i	−5.48284	−	8.74758i
37.6	−1.93336	+	0.511993i	−3.00693	−	3.00693i	3.47573	−	1.97973i	−1.76587	−	1.76587i	7.35299	+	4.27394i			1.63973i	−5.70621	+	5.60707i			9.08326i	4.31817	+	2.50995i
37.7	−1.91602	−	0.573460i	−0.559375	−	0.559375i	3.34229	+	2.19752i	−5.34137	−	5.34137i	0.750997	+	1.39256i		−	11.5476i	−5.14371	−	6.12717i		−	8.37420i	7.17112	+	13.2972i
37.8	−1.88364	+	0.672248i	−1.43338	−	1.43338i	3.09616	−	2.53254i	−6.70885	−	6.70885i	3.66354	+	1.73637i			1.92877i	−4.12955	+	6.85178i		−	4.89086i	17.1470	+	8.12701i
37.9	−1.87523	−	0.695341i	−0.924333	−	0.924333i	3.03300	+	2.60785i	6.41235	+	6.41235i	1.09061	+	2.37607i		−	5.54807i	−3.87424	−	6.99930i		−	7.29122i	−7.56588	−	16.4834i
37.10	−1.87497	−	0.696061i	2.55145	+	2.55145i	3.03100	+	2.61018i	−3.99349	−	3.99349i	−3.00792	−	6.55985i			5.05204i	−3.86617	−	7.00376i			4.01980i	4.70795	+	10.2674i
37.11	−1.83295	−	0.800177i	0.918920	+	0.918920i	2.71943	+	2.93337i	−3.44458	−	3.44458i	−0.949038	−	2.41964i			6.72205i	−2.63737	−	7.55276i		−	7.31117i	3.55748	+	9.07003i
37.12	−1.78495	+	0.902196i	3.28441	+	3.28441i	2.37209	−	3.22075i	−4.16025	−	4.16025i	−8.82568	−	2.89932i		−	8.16145i	−1.32831	+	7.88895i			12.5747i	11.1792	+	3.67247i
37.13	−1.77751	−	0.916767i	−2.67555	−	2.67555i	2.31908	+	3.25912i	0.271639	+	0.271639i	2.30296	+	7.20866i			4.37468i	−1.13433	−	7.91917i			5.31711i	−0.233811	−	0.731870i
37.14	−1.73439	+	0.995938i	1.52040	+	1.52040i	2.01622	−	3.45469i	5.33979	+	5.33979i	−4.15119	−	1.12274i			9.97335i	−0.0562492	+	7.99980i		−	4.37676i	−14.5794	−	3.94318i
37.15	−1.73185	+	1.00035i	−3.21919	−	3.21919i	1.99862	−	3.46490i	5.92632	+	5.92632i	8.79547	+	2.35486i			0.719658i	0.00478182	+	8.00000i			11.7264i	−16.1919	−	4.33514i
37.16	−1.57793	+	1.22888i	−1.82983	−	1.82983i	0.979712	−	3.87817i	1.12685	+	1.12685i	5.13599	+	0.638701i		−	10.4387i	3.21988	+	7.32341i		−	2.30342i	−3.16285	−	0.393326i
37.17	−1.52570	+	1.29315i	0.383931	+	0.383931i	0.655523	−	3.94592i	0.0541846	+	0.0541846i	−1.08224	−	0.0892829i			2.16156i	4.10254	+	6.86798i		−	8.70519i	−0.152738	−	0.0126006i
37.18	−1.46444	−	1.36214i	−1.96472	−	1.96472i	0.289148	+	3.98954i	1.25228	+	1.25228i	0.200984	+	5.55343i		−	1.30542i	5.01087	−	6.23628i		−	1.27974i	−0.128104	−	3.53968i
37.19	−1.44830	−	1.37929i	2.30806	+	2.30806i	0.195118	+	3.99524i	2.28342	+	2.28342i	−0.159268	−	6.52624i		−	12.0353i	5.22800	−	6.05541i			1.65430i	−0.157568	−	6.45656i
37.20	−1.34139	+	1.48347i	3.86240	+	3.86240i	−0.401363	−	3.97981i	−0.383673	−	0.383673i	−10.9107	+	0.548779i			9.44777i	6.44231	+	4.74306i			20.8362i	1.08382	−	0.0545133i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
16.e	even	4	1	inner
19.b	odd	2	1	inner
304.j	odd	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	304.3.j.a	✓	156
16.e	even	4	1	inner	304.3.j.a	✓	156
19.b	odd	2	1	inner	304.3.j.a	✓	156
304.j	odd	4	1	inner	304.3.j.a	✓	156

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
304.3.j.a	✓	156	1.a	even	1	1	trivial
304.3.j.a	✓	156	16.e	even	4	1	inner
304.3.j.a	✓	156	19.b	odd	2	1	inner
304.3.j.a	✓	156	304.j	odd	4	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(304, [\chi])\).