Newform orbit 348.3.o.a

• Label: 348.3.o.a
• Level: $348$
• Weight: $3$
• Character orbit: 348.o
• Analytic conductor: $9.482$
• Analytic rank: $0$
• Dimension: $360$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 348 = 2^{2} \cdot 3 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	348.o (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.48231319974\)
Analytic rank:	\(0\)
Dimension:	\(360\)
Relative dimension:	\(60\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 360 q + 12 q^{4} + 8 q^{5} - 180 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
67.1	−1.99996	+	0.0119931i	1.07992	+	1.35417i	3.99971	−	0.0479716i	−4.03816	+	1.94467i	−2.17603	−	2.69534i	3.71434	−	2.96209i	−7.99871	+	0.143911i	−0.667563	+	2.92478i	8.05285	−	3.93771i
67.2	−1.99855	−	0.0762749i	1.07992	+	1.35417i	3.98836	+	0.304878i	1.12167	−	0.540166i	−2.05497	−	2.78874i	−8.13016	+	6.48359i	−7.94767	−	0.913524i	−0.667563	+	2.92478i	−2.28290	+	0.993991i
67.3	−1.99730	−	0.103862i	−1.07992	−	1.35417i	3.97843	+	0.414887i	−6.06724	+	2.92183i	2.01627	+	2.81685i	−5.92099	+	4.72183i	−7.90302	−	1.24186i	−0.667563	+	2.92478i	12.4216	−	5.20562i
67.4	−1.96561	+	0.369311i	−1.07992	−	1.35417i	3.72722	−	1.45184i	4.91861	−	2.36868i	2.62280	+	2.26294i	−1.64445	+	1.31140i	−6.79006	+	4.23025i	−0.667563	+	2.92478i	−8.79328	+	6.47239i
67.5	−1.94247	−	0.476254i	1.07992	+	1.35417i	3.54637	+	1.85021i	−1.95217	+	0.940114i	−1.45277	−	3.14475i	1.52283	−	1.21442i	−6.00753	−	5.28295i	−0.667563	+	2.92478i	4.23976	−	0.896416i
67.6	−1.94039	+	0.484639i	−1.07992	−	1.35417i	3.53025	−	1.88078i	−5.31448	+	2.55932i	2.75175	+	2.10426i	9.07729	−	7.23890i	−5.93857	+	5.36035i	−0.667563	+	2.92478i	9.07184	−	7.54169i
67.7	−1.87496	+	0.696081i	1.07992	+	1.35417i	3.03094	−	2.61025i	8.05282	−	3.87803i	−2.96741	−	1.78731i	−4.06985	+	3.24560i	−3.86596	+	7.00388i	−0.667563	+	2.92478i	−12.3993	+	12.8766i
67.8	−1.86572	−	0.720486i	−1.07992	−	1.35417i	2.96180	+	2.68845i	−2.61250	+	1.25811i	1.03916	+	3.30457i	0.0564918	−	0.0450507i	−3.58889	−	7.14981i	−0.667563	+	2.92478i	5.78063	−	0.465014i
67.9	−1.85419	+	0.749643i	1.07992	+	1.35417i	2.87607	−	2.77997i	1.86547	−	0.898361i	−3.01752	−	1.70135i	8.97928	−	7.16074i	−3.24881	+	7.31063i	−0.667563	+	2.92478i	−2.78548	+	3.06417i
67.10	−1.77914	+	0.913588i	−1.07992	−	1.35417i	2.33071	−	3.25081i	1.73676	−	0.836379i	3.15848	+	1.42267i	−4.65550	+	3.71264i	−1.17677	+	7.91298i	−0.667563	+	2.92478i	−2.32584	+	3.07472i
67.11	−1.62355	−	1.16794i	−1.07992	−	1.35417i	1.27182	+	3.79242i	8.75363	−	4.21553i	0.171700	+	3.45984i	−9.19318	+	7.33131i	2.36448	−	7.64259i	−0.667563	+	2.92478i	−19.1354	−	3.37963i
67.12	−1.55369	−	1.25938i	1.07992	+	1.35417i	0.827912	+	3.91338i	−8.14727	+	3.92352i	0.0275645	−	3.46399i	−6.59458	+	5.25900i	3.64213	−	7.12284i	−0.667563	+	2.92478i	17.5996	+	4.16460i
67.13	−1.55354	−	1.25957i	1.07992	+	1.35417i	0.826951	+	3.91359i	3.88640	−	1.87159i	0.0279894	−	3.46399i	6.31966	−	5.03976i	3.64475	−	7.12150i	−0.667563	+	2.92478i	−8.39507	−	1.98762i
67.14	−1.48462	−	1.34012i	−1.07992	−	1.35417i	0.408171	+	3.97912i	−0.322259	+	0.155192i	−0.211488	+	3.45764i	0.833833	−	0.664959i	4.72651	−	6.45446i	−0.667563	+	2.92478i	0.686406	+	0.201465i
67.15	−1.40166	+	1.42665i	1.07992	+	1.35417i	−0.0706826	−	3.99938i	0.192941	−	0.0929153i	−3.44561	−	0.357426i	−4.70139	+	3.74923i	5.80480	+	5.50494i	−0.667563	+	2.92478i	−0.137880	+	0.405495i
67.16	−1.37985	−	1.44776i	1.07992	+	1.35417i	−0.192040	+	3.99539i	3.43636	−	1.65486i	0.470401	−	3.43201i	−5.40205	+	4.30799i	6.04936	−	5.23500i	−0.667563	+	2.92478i	−7.13750	−	2.69157i
67.17	−1.37918	−	1.44840i	−1.07992	−	1.35417i	−0.195700	+	3.99521i	3.58690	−	1.72736i	−0.471973	+	3.43180i	8.70457	−	6.94167i	6.05655	−	5.22668i	−0.667563	+	2.92478i	−7.44890	−	2.81290i
67.18	−1.34124	+	1.48360i	−1.07992	−	1.35417i	−0.402162	−	3.97973i	5.28576	−	2.54549i	3.45748	+	0.214099i	8.88786	−	7.08784i	6.44374	+	4.74112i	−0.667563	+	2.92478i	−3.31297	+	11.2561i
67.19	−1.33079	+	1.49298i	1.07992	+	1.35417i	−0.457992	−	3.97369i	−6.09017	+	2.93287i	−3.45890	−	0.189824i	4.78493	−	3.81586i	6.54215	+	4.60438i	−0.667563	+	2.92478i	3.72601	−	12.9955i
67.20	−1.23331	+	1.57447i	−1.07992	−	1.35417i	−0.957881	−	3.88361i	−7.57431	+	3.64759i	3.46397	−	0.0301734i	−9.91631	+	7.90799i	7.29598	+	3.28156i	−0.667563	+	2.92478i	3.59848	−	16.4241i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
29.e	even	14	1	inner
116.h	odd	14	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	348.3.o.a	✓	360
4.b	odd	2	1	inner	348.3.o.a	✓	360
29.e	even	14	1	inner	348.3.o.a	✓	360
116.h	odd	14	1	inner	348.3.o.a	✓	360

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
348.3.o.a	✓	360	1.a	even	1	1	trivial
348.3.o.a	✓	360	4.b	odd	2	1	inner
348.3.o.a	✓	360	29.e	even	14	1	inner
348.3.o.a	✓	360	116.h	odd	14	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(348, [\chi])\).