Newform orbit 348.2.t.a

• Label: 348.2.t.a
• Level: $348$
• Weight: $2$
• Character orbit: 348.t
• Analytic conductor: $2.779$
• Analytic rank: $0$
• Dimension: $336$
• CM: no
• Inner twists: $8$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(2.77879399034\)
Analytic rank:	\(0\)
Dimension:	\(336\)
Relative dimension:	\(56\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 336 q - 10 q^{4} - 3 q^{6} - 10 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
35.1	−1.41321	−	0.0532282i	−1.71646	+	0.231871i	1.99433	+	0.150445i	2.31270	+	1.84432i	2.43806	−	0.236318i	1.11294	+	0.254022i	−2.81041	−	0.318766i	2.89247	−	0.795994i	−3.17016	−	2.72951i
35.2	−1.40439	+	0.166373i	1.72952	+	0.0935630i	1.94464	−	0.467305i	−1.09448	−	0.872821i	−2.44450	+	0.156346i	0.256689	+	0.0585876i	−2.65329	+	0.979815i	2.98249	+	0.323638i	1.68230	+	1.04369i
35.3	−1.39037	+	0.258621i	−0.520219	−	1.65208i	1.86623	−	0.719156i	1.14035	+	0.909396i	1.15056	+	2.16246i	−3.79504	−	0.866193i	−2.40875	+	1.48254i	−2.45874	+	1.71889i	−1.82069	−	0.969474i
35.4	−1.37991	−	0.309609i	0.578538	−	1.63257i	1.80828	+	0.854463i	−2.25006	−	1.79437i	−1.30379	+	2.07368i	0.00145454		0.000331989i	−2.23071	−	1.73894i	−2.33059	−	1.88901i	2.54933	+	3.17270i
35.5	−1.32649	+	0.490331i	−1.67868	+	0.426655i	1.51915	−	1.30084i	−2.71854	−	2.16797i	2.01755	−	1.38906i	−4.49236	−	1.02535i	−1.37730	+	2.47043i	2.63593	−	1.43244i	4.66914	+	1.54280i
35.6	−1.31443	−	0.521790i	−0.468526	+	1.66748i	1.45547	+	1.37172i	−0.227131	−	0.181131i	1.48592	−	1.94732i	−2.72634	−	0.622270i	−1.19737	−	2.56248i	−2.56097	−	1.56251i	0.204036	+	0.356598i
35.7	−1.29359	−	0.571517i	1.49175	+	0.880152i	1.34674	+	1.47861i	1.76879	+	1.41056i	−1.42669	−	1.99112i	4.04660	+	0.923610i	−0.897070	−	2.68240i	1.45066	+	2.62594i	−1.48192	−	2.83558i
35.8	−1.28664	+	0.586989i	0.441588	+	1.67481i	1.31089	−	1.51049i	2.87932	+	2.29618i	−1.55126	−	1.89568i	−0.962659	−	0.219721i	−0.800003	+	2.71293i	−2.61000	+	1.47916i	−5.05248	−	1.26423i
35.9	−1.23942	+	0.681054i	1.07334	−	1.35939i	1.07233	−	1.68823i	1.51787	+	1.21046i	−0.404494	+	2.41586i	4.22479	+	0.964281i	−0.179298	+	2.82274i	−0.695904	−	2.91817i	−2.70566	−	0.466519i
35.10	−1.21474	−	0.724167i	1.36947	−	1.06044i	0.951165	+	1.75934i	3.09364	+	2.46710i	−2.43149	+	0.296432i	−2.60358	−	0.594250i	0.118644	−	2.82594i	0.750915	−	2.90450i	−1.97137	−	5.23718i
35.11	−1.19538	−	0.755688i	−1.14525	−	1.29939i	0.857871	+	1.80667i	0.278916	+	0.222428i	0.387077	+	2.41871i	0.870753	+	0.198744i	0.339796	−	2.80794i	−0.376812	+	2.97624i	−0.165325	−	0.476660i
35.12	−1.18856	+	0.766379i	0.725197	+	1.57292i	0.825327	−	1.82177i	−2.26138	−	1.80339i	−2.06739	−	1.31373i	−0.188704	−	0.0430704i	0.415217	+	2.79778i	−1.94818	+	2.28136i	4.06986	+	0.410356i
35.13	−1.03132	+	0.967665i	−1.50908	−	0.850111i	0.127251	−	1.99595i	−0.0130888	−	0.0104380i	2.37897	−	0.583541i	1.19888	+	0.273637i	1.80017	+	2.18160i	1.55462	+	2.56576i	0.0235993	−	0.00190066i
35.14	−1.00274	−	0.997254i	−1.59562	+	0.673802i	0.0109695	+	1.99997i	−0.278916	−	0.222428i	2.27194	+	0.915587i	−0.870753	−	0.198744i	1.98348	−	2.01639i	2.09198	−	2.15026i	0.0578626	+	0.501188i
35.15	−0.976315	−	1.02314i	0.773743	+	1.54962i	−0.0936199	+	1.99781i	−3.09364	−	2.46710i	0.830057	−	2.30456i	2.60358	+	0.594250i	2.13543	−	1.85470i	−1.80264	+	2.39802i	0.496188	+	5.57388i
35.16	−0.927728	+	1.06739i	−1.21862	+	1.23084i	−0.278641	−	1.98049i	−0.252365	−	0.201255i	−0.183246	−	2.44263i	3.46126	+	0.790009i	2.37246	+	1.53994i	−0.0299545	−	2.99985i	0.448943	−	0.0826624i
35.17	−0.845038	−	1.13398i	1.72591	−	0.145741i	−0.571822	+	1.91651i	−1.76879	−	1.41056i	−1.62373	−	1.83399i	−4.04660	−	0.923610i	2.65650	−	0.971091i	2.95752	−	0.503073i	−0.104856	+	3.19775i
35.18	−0.834189	+	1.14198i	1.21862	−	1.23084i	−0.608258	−	1.90526i	−0.252365	−	0.201255i	0.389050	+	2.41840i	−3.46126	−	0.790009i	2.68318	+	0.894727i	−0.0299545	−	2.99985i	0.440350	−	0.120313i
35.19	−0.801196	−	1.16537i	0.301364	−	1.70563i	−0.716169	+	1.86738i	0.227131	+	0.181131i	−2.22914	+	1.01535i	2.72634	+	0.622270i	2.74998	−	0.661535i	−2.81836	−	1.02803i	0.0291078	−	0.409812i
35.20	−0.713912	+	1.22079i	1.50908	+	0.850111i	−0.980658	−	1.74307i	−0.0130888	−	0.0104380i	−2.11516	+	1.23536i	−1.19888	−	0.273637i	2.82803	+	0.0472247i	1.55462	+	2.56576i	0.0220869	−	0.00852690i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	348.2.t.a	✓	336
3.b	odd	2	1	inner	348.2.t.a	✓	336
4.b	odd	2	1	inner	348.2.t.a	✓	336
12.b	even	2	1	inner	348.2.t.a	✓	336
29.e	even	14	1	inner	348.2.t.a	✓	336
87.h	odd	14	1	inner	348.2.t.a	✓	336
116.h	odd	14	1	inner	348.2.t.a	✓	336
348.t	even	14	1	inner	348.2.t.a	✓	336

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
348.2.t.a	✓	336	1.a	even	1	1	trivial
348.2.t.a	✓	336	3.b	odd	2	1	inner
348.2.t.a	✓	336	4.b	odd	2	1	inner
348.2.t.a	✓	336	12.b	even	2	1	inner
348.2.t.a	✓	336	29.e	even	14	1	inner
348.2.t.a	✓	336	87.h	odd	14	1	inner
348.2.t.a	✓	336	116.h	odd	14	1	inner
348.2.t.a	✓	336	348.t	even	14	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(348, [\chi])\).