Newform orbit 348.2.c.b

• Label: 348.2.c.b
• Level: $348$
• Weight: $2$
• Character orbit: 348.c
• Analytic conductor: $2.779$
• Analytic rank: $0$
• Dimension: $40$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 348 = 2^{2} \cdot 3 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	348.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.77879399034\)
Analytic rank:	\(0\)
Dimension:	\(40\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 6 q^{6} - 4 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
59.1	−1.41245	−	0.0705195i	−0.494212	−	1.66005i	1.99005	+	0.199211i		−	3.75368i	0.580986	+	2.37959i		−	4.16013i	−2.79681	−	0.421714i	−2.51151	+	1.64083i	−0.264707	+	5.30190i
59.2	−1.41245	+	0.0705195i	−0.494212	+	1.66005i	1.99005	−	0.199211i			3.75368i	0.580986	−	2.37959i			4.16013i	−2.79681	+	0.421714i	−2.51151	−	1.64083i	−0.264707	−	5.30190i
59.3	−1.38479	−	0.286955i	1.43171	+	0.974781i	1.83531	+	0.794747i		−	1.32510i	−1.70291	−	1.76071i			2.06888i	−2.31348	−	1.62721i	1.09961	+	2.79121i	−0.380244	+	1.83499i
59.4	−1.38479	+	0.286955i	1.43171	−	0.974781i	1.83531	−	0.794747i			1.32510i	−1.70291	+	1.76071i		−	2.06888i	−2.31348	+	1.62721i	1.09961	−	2.79121i	−0.380244	−	1.83499i
59.5	−1.34502	−	0.436932i	−0.312286	−	1.70367i	1.61818	+	1.17537i			2.54452i	−0.324355	+	2.42792i		−	0.190467i	−1.66294	−	2.28794i	−2.80496	+	1.06406i	1.11178	−	3.42245i
59.6	−1.34502	+	0.436932i	−0.312286	+	1.70367i	1.61818	−	1.17537i		−	2.54452i	−0.324355	−	2.42792i			0.190467i	−1.66294	+	2.28794i	−2.80496	−	1.06406i	1.11178	+	3.42245i
59.7	−1.30790	−	0.537964i	−1.67797	+	0.429454i	1.42119	+	1.40720i			1.34318i	2.42564	+	0.341005i		−	1.97166i	−1.10174	−	2.60503i	2.63114	−	1.44122i	0.722585	−	1.75675i
59.8	−1.30790	+	0.537964i	−1.67797	−	0.429454i	1.42119	−	1.40720i		−	1.34318i	2.42564	−	0.341005i			1.97166i	−1.10174	+	2.60503i	2.63114	+	1.44122i	0.722585	+	1.75675i
59.9	−1.10940	−	0.877053i	−1.15966	−	1.28654i	0.461555	+	1.94601i		−	1.35780i	0.158175	+	2.44438i			3.91076i	1.19471	−	2.56372i	−0.310357	+	2.98390i	−1.19086	+	1.50634i
59.10	−1.10940	+	0.877053i	−1.15966	+	1.28654i	0.461555	−	1.94601i			1.35780i	0.158175	−	2.44438i		−	3.91076i	1.19471	+	2.56372i	−0.310357	−	2.98390i	−1.19086	−	1.50634i
59.11	−0.946094	−	1.05115i	1.21284	−	1.23653i	−0.209812	+	1.98896i		−	2.17467i	−2.44724	−	0.104998i			0.0947864i	2.28919	−	1.66120i	−0.0580256	−	2.99944i	−2.28589	+	2.05744i
59.12	−0.946094	+	1.05115i	1.21284	+	1.23653i	−0.209812	−	1.98896i			2.17467i	−2.44724	+	0.104998i		−	0.0947864i	2.28919	+	1.66120i	−0.0580256	+	2.99944i	−2.28589	−	2.05744i
59.13	−0.479912	−	1.33029i	−1.72606	+	0.143940i	−1.53937	+	1.27685i		−	4.13273i	1.01984	+	2.22709i		−	2.24214i	2.43735	+	1.43504i	2.95856	−	0.496898i	−5.49775	+	1.98335i
59.14	−0.479912	+	1.33029i	−1.72606	−	0.143940i	−1.53937	−	1.27685i			4.13273i	1.01984	−	2.22709i			2.24214i	2.43735	−	1.43504i	2.95856	+	0.496898i	−5.49775	−	1.98335i
59.15	−0.377291	−	1.36296i	1.65882	+	0.498301i	−1.71530	+	1.02846i			0.328095i	0.0533041	−	2.44891i		−	4.37145i	2.04892	+	1.94986i	2.50339	+	1.65319i	0.447179	−	0.123787i
59.16	−0.377291	+	1.36296i	1.65882	−	0.498301i	−1.71530	−	1.02846i		−	0.328095i	0.0533041	+	2.44891i			4.37145i	2.04892	−	1.94986i	2.50339	−	1.65319i	0.447179	+	0.123787i
59.17	−0.254328	−	1.39116i	0.774651	−	1.54917i	−1.87063	+	0.707619i			3.96336i	−2.35215	−	0.683665i			3.59138i	1.46016	+	2.42238i	−1.79983	−	2.40013i	5.51365	−	1.00799i
59.18	−0.254328	+	1.39116i	0.774651	+	1.54917i	−1.87063	−	0.707619i		−	3.96336i	−2.35215	+	0.683665i		−	3.59138i	1.46016	−	2.42238i	−1.79983	+	2.40013i	5.51365	+	1.00799i
59.19	−0.0664333	−	1.41265i	−0.382087	+	1.68938i	−1.99117	+	0.187694i		−	1.18625i	2.41189	+	0.427525i			1.07303i	0.397427	+	2.80037i	−2.70802	−	1.29098i	−1.67575	+	0.0788063i
59.20	−0.0664333	+	1.41265i	−0.382087	−	1.68938i	−1.99117	−	0.187694i			1.18625i	2.41189	−	0.427525i		−	1.07303i	0.397427	−	2.80037i	−2.70802	+	1.29098i	−1.67575	−	0.0788063i

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	348.2.c.b	✓	40
3.b	odd	2	1	inner	348.2.c.b	✓	40
4.b	odd	2	1	inner	348.2.c.b	✓	40
12.b	even	2	1	inner	348.2.c.b	✓	40

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
348.2.c.b	✓	40	1.a	even	1	1	trivial
348.2.c.b	✓	40	3.b	odd	2	1	inner
348.2.c.b	✓	40	4.b	odd	2	1	inner
348.2.c.b	✓	40	12.b	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T5^20 + 65*T5^18 + 1706*T5^16 + 23366*T5^14 + 181893*T5^12 + 833653*T5^10 + 2270080*T5^8 + 3615636*T5^6 + 3156768*T5^4 + 1251904*T5^2 + 102400