Newform orbit 468.2.bd.a

• Label: 468.2.bd.a
• Level: $468$
• Weight: $2$
• Character orbit: 468.bd
• Analytic conductor: $3.737$
• Analytic rank: $0$
• Dimension: $160$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 468 = 2^{2} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	468.bd (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.73699881460\)
Analytic rank:	\(0\)
Dimension:	\(160\)
Relative dimension:	\(80\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 160 q - 3 q^{2} + q^{4} - 12 q^{5} - 3 q^{6} - 2 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
191.1	−1.41421	−	0.00168932i	−1.54587	+	0.781213i	1.99999	+	0.00477812i	3.44158	−	1.98700i	2.18751	−	1.10219i		−	3.54931i	−2.82841	−	0.0101359i	1.77941	−	2.41530i	−4.87048	+	2.80422i
191.2	−1.41045	−	0.103105i	0.0858260	+	1.72992i	1.97874	+	0.290848i	1.88249	−	1.08686i	0.0573096	−	2.44882i			4.10685i	−2.76092	−	0.614243i	−2.98527	+	0.296945i	−2.76722	+	1.33886i
191.3	−1.40141	−	0.189847i	1.40172	+	1.01744i	1.92792	+	0.532107i	−0.224656	+	0.129705i	−1.77123	−	1.69197i		−	1.71023i	−2.60079	−	1.11171i	0.929632	+	2.85233i	0.339459	−	0.139120i
191.4	−1.39961	+	0.202677i	−1.73195	+	0.0186363i	1.91784	−	0.567341i	−3.17826	+	1.83497i	2.42029	−	0.377111i			3.23138i	−2.56926	+	1.18276i	2.99931	−	0.0645544i	4.07644	−	3.21242i
191.5	−1.39604	−	0.226016i	0.596680	−	1.62603i	1.89783	+	0.631054i	−3.27377	+	1.89011i	−1.20050	+	2.13514i		−	4.20595i	−2.50681	−	1.30992i	−2.28795	−	1.94044i	4.99750	−	1.89874i
191.6	−1.39533	−	0.230319i	0.686598	−	1.59015i	1.89391	+	0.642743i	1.97146	−	1.13822i	−1.32428	+	2.06065i			1.68572i	−2.49459	−	1.33304i	−2.05717	−	2.18359i	−3.01300	+	1.13414i
191.7	−1.38456	+	0.288102i	−0.319267	+	1.70237i	1.83399	−	0.797788i	−2.09563	+	1.20991i	−0.0484137	−	2.44901i		−	0.835187i	−2.30942	+	1.63296i	−2.79614	−	1.08702i	2.55294	−	2.27895i
191.8	−1.37354	−	0.336718i	−1.58980	−	0.687410i	1.77324	+	0.924994i	−0.0309949	+	0.0178949i	1.95220	+	1.47950i		−	0.474131i	−2.12416	−	1.86760i	2.05493	+	2.18569i	0.0485983	−	0.0141429i
191.9	−1.35217	+	0.414304i	1.70455	−	0.307446i	1.65670	−	1.12042i	−1.48737	+	0.858734i	−2.17745	+	1.12192i			3.58943i	−1.77594	+	2.20137i	2.81095	−	1.04811i	1.65539	−	1.77737i
191.10	−1.34282	+	0.443647i	1.65417	−	0.513552i	1.60635	−	1.19148i	1.06439	−	0.614528i	−1.99342	+	1.42348i		−	2.76170i	−1.62846	+	2.31260i	2.47253	−	1.69900i	−1.15666	+	1.29742i
191.11	−1.32606	+	0.491480i	−0.246161	−	1.71447i	1.51689	−	1.30347i	−0.388414	+	0.224251i	1.16905	+	2.15251i			1.71872i	−1.37087	+	2.47401i	−2.87881	+	0.844072i	0.404847	−	0.488269i
191.12	−1.26690	+	0.628466i	−1.19506	−	1.25372i	1.21006	−	1.59240i	3.21960	−	1.85884i	2.30195	+	0.837283i			1.43097i	−0.532254	+	2.77790i	−0.143645	+	2.99656i	−2.91069	+	4.37837i
191.13	−1.23703	−	0.685387i	1.36562	+	1.06541i	1.06049	+	1.69569i	−2.65079	+	1.53044i	−0.959095	−	2.25392i			2.68259i	−0.149655	−	2.82447i	0.729821	+	2.90987i	4.32805	−	0.0763773i
191.14	−1.17772	+	0.782933i	1.19506	+	1.25372i	0.774031	−	1.84415i	3.21960	−	1.85884i	−2.38903	−	0.540876i		−	1.43097i	0.532254	+	2.77790i	−0.143645	+	2.99656i	−2.33643	+	4.70992i
191.15	−1.17728	−	0.783587i	1.69723	−	0.345573i	0.771983	+	1.84500i	1.97731	−	1.14160i	−2.26890	−	0.923088i		−	2.23942i	0.536881	−	2.77701i	2.76116	−	1.17303i	−3.22239	−	0.205409i
191.16	−1.16408	−	0.803072i	−0.759437	+	1.55668i	0.710150	+	1.86968i	−2.20444	+	1.27273i	2.13417	−	1.20221i		−	3.92343i	0.674815	−	2.74675i	−1.84651	−	2.36440i	3.58823	+	0.288763i
191.17	−1.13768	−	0.840044i	−1.37000	+	1.05977i	0.588654	+	1.91141i	0.363808	−	0.210044i	2.44888	−	0.0548207i			2.56289i	0.935965	−	2.66908i	0.753793	−	2.90376i	−0.590345	−	0.0666499i
191.18	−1.08867	+	0.902666i	0.246161	+	1.71447i	0.370390	−	1.96540i	−0.388414	+	0.224251i	−1.81558	−	1.64428i		−	1.71872i	1.37087	+	2.47401i	−2.87881	+	0.844072i	0.220430	−	0.594742i
191.19	−1.05562	+	0.941097i	−1.65417	+	0.513552i	0.228674	−	1.98688i	1.06439	−	0.614528i	1.26287	−	2.09885i			2.76170i	1.62846	+	2.31260i	2.47253	−	1.69900i	−0.545267	+	1.65041i
191.20	−1.03488	+	0.963858i	−1.70455	+	0.307446i	0.141957	−	1.99496i	−1.48737	+	0.858734i	1.46767	−	1.96111i		−	3.58943i	1.77594	+	2.20137i	2.81095	−	1.04811i	0.711555	−	2.32230i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
117.u	odd	6	1	inner
468.bd	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	468.2.bd.a	✓	160
4.b	odd	2	1	inner	468.2.bd.a	✓	160
9.d	odd	6	1		468.2.bm.a	yes	160
13.c	even	3	1		468.2.bm.a	yes	160
36.h	even	6	1		468.2.bm.a	yes	160
52.j	odd	6	1		468.2.bm.a	yes	160
117.u	odd	6	1	inner	468.2.bd.a	✓	160
468.bd	even	6	1	inner	468.2.bd.a	✓	160

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
468.2.bd.a	✓	160	1.a	even	1	1	trivial
468.2.bd.a	✓	160	4.b	odd	2	1	inner
468.2.bd.a	✓	160	117.u	odd	6	1	inner
468.2.bd.a	✓	160	468.bd	even	6	1	inner
468.2.bm.a	yes	160	9.d	odd	6	1
468.2.bm.a	yes	160	13.c	even	3	1
468.2.bm.a	yes	160	36.h	even	6	1
468.2.bm.a	yes	160	52.j	odd	6	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(468, [\chi])\).