Newform orbit 468.2.cf.a

• Label: 468.2.cf.a
• Level: $468$
• Weight: $2$
• Character orbit: 468.cf
• Analytic conductor: $3.737$
• Analytic rank: $0$
• Dimension: $320$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 468 = 2^{2} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	468.cf (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.73699881460\)
Analytic rank:	\(0\)
Dimension:	\(320\)
Relative dimension:	\(80\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 320 q - 2 q^{2} - 6 q^{4} - 4 q^{5} - 14 q^{6} - 14 q^{8} - 4 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
7.1	−1.41240	−	0.0715581i	−0.357529	−	1.69475i	1.98976	+	0.202138i	0.469660	−	1.75280i	0.383702	+	2.41925i	−2.91036	−	2.91036i	−2.79587	−	0.427883i	−2.74435	+	1.21184i	−0.788775	+	2.44204i
7.2	−1.40609	−	0.151328i	1.36607	+	1.06483i	1.95420	+	0.425562i	0.173938	−	0.649145i	−1.75969	−	1.70397i	−1.53809	−	1.53809i	−2.68339	−	0.894105i	0.732295	+	2.90925i	−0.342807	+	0.886437i
7.3	−1.40419	+	0.168079i	0.379069	−	1.69006i	1.94350	−	0.472030i	0.316460	−	1.18104i	−0.248221	+	2.43688i	0.508751	+	0.508751i	−2.64970	+	0.989482i	−2.71261	−	1.28130i	−0.245861	+	1.71160i
7.4	−1.39980	+	0.201395i	−0.244546	+	1.71470i	1.91888	−	0.563825i	0.694645	−	2.59245i	−0.00301709	−	2.44949i	2.17859	+	2.17859i	−2.57250	+	1.17570i	−2.88039	−	0.838645i	−0.450258	+	3.76881i
7.5	−1.37245	+	0.341155i	−1.57588	+	0.718755i	1.76723	−	0.936436i	−0.767258	+	2.86345i	1.91760	−	1.52407i	−0.0682650	−	0.0682650i	−2.10596	+	1.88811i	1.96678	−	2.26534i	0.0761416	−	4.19168i
7.6	−1.36138	−	0.382947i	−1.71253	−	0.259309i	1.70670	+	1.04267i	0.481639	−	1.79750i	2.23210	+	1.00882i	2.16072	+	2.16072i	−1.92418	−	2.07305i	2.86552	+	0.888148i	−1.34404	+	2.26264i
7.7	−1.35433	−	0.407166i	1.39549	+	1.02597i	1.66843	+	1.10288i	−0.868183	+	3.24010i	−1.47222	−	1.95770i	2.59721	+	2.59721i	−1.81056	−	2.17299i	0.894774	+	2.86346i	2.49507	−	4.03468i
7.8	−1.34623	+	0.433190i	0.389529	+	1.68768i	1.62469	−	1.16635i	−0.451553	+	1.68522i	−1.25548	−	2.10327i	−2.05260	−	2.05260i	−1.68196	+	2.27398i	−2.69653	+	1.31480i	−0.122124	−	2.46431i
7.9	−1.33778	+	0.458630i	1.72397	−	0.167114i	1.57932	−	1.22709i	−0.0363043	+	0.135489i	−2.22965	+	1.01423i	3.46374	+	3.46374i	−1.55000	+	2.36590i	2.94415	−	0.576198i	−0.0135723	−	0.197905i
7.10	−1.33489	−	0.466976i	−1.19126	+	1.25734i	1.56387	+	1.24672i	−0.298162	+	1.11275i	2.17735	−	1.12212i	−0.822480	−	0.822480i	−1.50540	−	2.39453i	−0.161797	−	2.99563i	0.917642	−	1.34617i
7.11	−1.30925	−	0.534663i	0.964386	−	1.43874i	1.42827	+	1.40002i	−1.09591	+	4.08998i	−2.03186	+	1.36804i	−0.288756	−	0.288756i	−1.12142	−	2.59661i	−1.13992	−	2.77499i	3.62158	−	4.76886i
7.12	−1.30627	+	0.541893i	1.60446	−	0.652469i	1.41270	−	1.41572i	−0.627423	+	2.34158i	−1.74229	+	1.72175i	−2.31433	−	2.31433i	−1.07821	+	2.61485i	2.14857	−	2.09372i	−0.449297	−	3.39874i
7.13	−1.25808	+	0.645934i	−1.68278	−	0.410186i	1.16554	−	1.62528i	0.665935	−	2.48530i	2.38203	−	0.570916i	−0.645435	−	0.645435i	−0.416523	+	2.79759i	2.66349	+	1.38051i	0.767541	+	3.55686i
7.14	−1.24379	−	0.673048i	1.25925	−	1.18924i	1.09401	+	1.67426i	0.845905	−	3.15696i	−2.36666	+	0.631619i	2.25688	+	2.25688i	−0.233864	−	2.81874i	0.171434	−	2.99510i	−3.17691	+	3.35725i
7.15	−1.17622	−	0.785179i	1.73188	+	0.0242841i	0.766989	+	1.84709i	0.173656	−	0.648092i	−2.01801	−	1.38840i	−1.49547	−	1.49547i	0.548145	−	2.77480i	2.99882	+	0.0841145i	−0.713126	+	0.625949i
7.16	−1.14835	−	0.825408i	−0.0588040	+	1.73105i	0.637404	+	1.89571i	0.884828	−	3.30222i	1.49635	−	1.93931i	−2.03279	−	2.03279i	0.832773	−	2.70305i	−2.99308	−	0.203586i	−3.74177	+	3.06175i
7.17	−1.12187	+	0.861056i	0.0263223	−	1.73185i	0.517166	−	1.93198i	−0.502832	+	1.87660i	1.46169	+	1.96557i	1.56840	+	1.56840i	1.08335	+	2.61273i	−2.99861	−	0.0911725i	−1.05174	−	2.53826i
7.18	−1.10108	−	0.887486i	−0.677419	−	1.59408i	0.424738	+	1.95438i	−0.238385	+	0.889665i	−0.668837	+	2.35641i	0.401256	+	0.401256i	1.26682	−	2.52887i	−2.08221	+	2.15973i	1.05204	−	0.768025i
7.19	−1.06856	+	0.926385i	1.47582	−	0.906622i	0.283621	−	1.97979i	1.06048	−	3.95777i	−0.737111	+	2.33595i	−2.01510	−	2.01510i	1.53098	+	2.37825i	1.35607	−	2.67602i	2.53324	+	5.21151i
7.20	−1.03383	+	0.964987i	−1.36594	−	1.06499i	0.137601	−	1.99526i	−0.407546	+	1.52098i	2.43985	−	0.217096i	−1.65468	−	1.65468i	1.78314	+	2.19554i	0.731585	+	2.90943i	−1.04640	−	1.96571i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
117.bb	odd	12	1	inner
468.cf	even	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	468.2.cf.a	yes	320
4.b	odd	2	1	inner	468.2.cf.a	yes	320
9.c	even	3	1		468.2.cc.a	✓	320
13.f	odd	12	1		468.2.cc.a	✓	320
36.f	odd	6	1		468.2.cc.a	✓	320
52.l	even	12	1		468.2.cc.a	✓	320
117.bb	odd	12	1	inner	468.2.cf.a	yes	320
468.cf	even	12	1	inner	468.2.cf.a	yes	320

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
468.2.cc.a	✓	320	9.c	even	3	1
468.2.cc.a	✓	320	13.f	odd	12	1
468.2.cc.a	✓	320	36.f	odd	6	1
468.2.cc.a	✓	320	52.l	even	12	1
468.2.cf.a	yes	320	1.a	even	1	1	trivial
468.2.cf.a	yes	320	4.b	odd	2	1	inner
468.2.cf.a	yes	320	117.bb	odd	12	1	inner
468.2.cf.a	yes	320	468.cf	even	12	1	inner

Hecke kernels

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