Newform orbit 320.2.bf.a

• Label: 320.2.bf.a
• Level: $320$
• Weight: $2$
• Character orbit: 320.bf
• Analytic conductor: $2.555$
• Analytic rank: $0$
• Dimension: $368$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	320.bf (of order \(16\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.55521286468\)
Analytic rank:	\(0\)
Dimension:	\(368\)
Relative dimension:	\(46\) over \(\Q(\zeta_{16})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 368 q - 16 q^{4} - 8 q^{5} - 16 q^{6} - 16 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
29.1	−1.41315	+	0.0548766i	0.462030	−	2.32278i	1.99398	−	0.155097i	−1.42528	−	1.72296i	−0.525451	+	3.30779i	0.330301	+	0.797417i	−2.80927	+	0.328598i	−2.41021	−	0.998340i	2.10868	+	2.35658i
29.2	−1.40092	−	0.193431i	−0.612243	+	3.07795i	1.92517	+	0.541963i	−2.22253	+	0.245694i	1.45308	−	4.19355i	1.41327	+	3.41194i	−2.59218	−	1.13164i	−6.32731	−	2.62086i	3.16112	+	0.0857073i
29.3	−1.38805	+	0.270776i	−0.246966	+	1.24158i	1.85336	−	0.751702i	1.74404	+	1.39940i	0.00660982	−	1.79025i	0.414009	+	0.999505i	−2.36901	+	1.54525i	1.29111	+	0.534796i	−2.79974	−	1.47019i
29.4	−1.34695	+	0.430954i	0.329632	−	1.65717i	1.62856	−	1.16095i	−1.38798	+	1.75314i	0.270166	+	2.37419i	0.267328	+	0.645386i	−1.69327	+	2.26557i	0.134078	+	0.0555368i	1.11402	−	2.95955i
29.5	−1.33571	−	0.464641i	0.488587	−	2.45629i	1.56822	+	1.24125i	0.609350	+	2.15144i	−1.79390	+	3.05386i	−0.627512	−	1.51495i	−1.51794	−	2.38660i	−3.02301	−	1.25217i	0.185734	−	3.15682i
29.6	−1.33383	−	0.469997i	0.0916179	−	0.460594i	1.55821	+	1.25379i	1.74536	−	1.39776i	−0.338681	+	0.571295i	1.00872	+	2.43526i	−1.48910	−	2.40470i	2.56789	+	1.06365i	−2.98495	+	1.04406i
29.7	−1.26165	−	0.638935i	−0.547765	+	2.75380i	1.18353	+	1.61222i	1.52882	−	1.63178i	2.45059	−	3.12435i	−1.54675	−	3.73418i	−0.463089	−	2.79026i	−4.51173	−	1.86882i	−2.97144	+	1.08192i
29.8	−1.20469	+	0.740758i	0.318063	−	1.59901i	0.902555	−	1.78477i	2.13894	−	0.651887i	0.801313	+	2.16192i	−1.98639	−	4.79556i	0.234782	+	2.81867i	0.315964	+	0.130877i	−2.09386	+	2.36975i
29.9	−1.15919	−	0.810109i	−0.212827	+	1.06995i	0.687447	+	1.87814i	−0.607268	+	2.15203i	1.11349	−	1.06787i	−0.679902	−	1.64143i	0.724616	−	2.73403i	1.67213	+	0.692620i	2.44732	−	2.00266i
29.10	−1.14869	+	0.824933i	−0.336544	+	1.69192i	0.638972	−	1.89518i	−1.00832	−	1.99582i	−1.00914	−	2.22112i	−0.601901	−	1.45312i	0.829417	+	2.70408i	0.0223079	+	0.00924022i	2.80466	+	1.46078i
29.11	−0.970636	+	1.02853i	−0.0730884	+	0.367440i	−0.115730	−	1.99665i	−2.22727	+	0.198194i	−0.306979	−	0.431824i	1.26909	+	3.06387i	2.16594	+	1.81899i	2.64197	+	1.09434i	1.95802	−	2.48318i
29.12	−0.963775	−	1.03496i	−0.0117336	+	0.0589886i	−0.142276	+	1.99493i	−2.09609	−	0.778715i	0.0723593	−	0.0447080i	0.480951	+	1.16112i	2.20179	−	1.77542i	2.76830	+	1.14667i	1.21422	+	2.91987i
29.13	−0.942250	+	1.05459i	−0.630723	+	3.17086i	−0.224328	−	1.98738i	0.0178005	+	2.23600i	−2.74966	−	3.65290i	−1.29895	−	3.13595i	2.30725	+	1.63603i	−6.88490	−	2.85182i	−2.37484	−	2.08810i
29.14	−0.878389	−	1.10835i	0.572105	−	2.87617i	−0.456865	+	1.94712i	1.03679	−	1.98118i	−3.69032	+	1.89230i	−0.591145	−	1.42715i	2.55939	−	1.20396i	−5.17340	−	2.14289i	−3.10654	+	0.591119i
29.15	−0.713777	+	1.22087i	0.446234	−	2.24337i	−0.981046	−	1.74286i	1.63244	+	1.52812i	2.42035	+	2.14606i	1.38403	+	3.34135i	2.82805	+	0.0462814i	−2.06195	−	0.854087i	−3.03083	+	0.902259i
29.16	−0.635601	+	1.26333i	0.627776	−	3.15604i	−1.19202	−	1.60595i	−1.59499	−	1.56716i	3.58812	+	2.79908i	−0.393699	−	0.950472i	2.78650	−	0.485175i	−6.79487	−	2.81453i	2.99363	−	1.01891i
29.17	−0.628783	−	1.26674i	0.368711	−	1.85364i	−1.20926	+	1.59301i	0.834497	+	2.07452i	−2.57992	+	0.698473i	1.47604	+	3.56349i	2.77829	+	0.530167i	−0.528383	−	0.218863i	2.10316	−	2.36151i
29.18	−0.561336	+	1.29804i	−0.279704	+	1.40617i	−1.36980	−	1.45727i	2.09531	−	0.780827i	−1.66825	−	1.15240i	0.431062	+	1.04068i	2.66051	−	0.960039i	0.872568	+	0.361429i	−0.162628	+	3.15809i
29.19	−0.458870	+	1.33770i	0.110253	−	0.554278i	−1.57888	−	1.22766i	−1.78619	+	1.34519i	0.690865	+	0.401826i	−1.43120	−	3.45521i	2.36674	−	1.54873i	2.47657	+	1.02583i	−0.979827	−	3.00665i
29.20	−0.424621	−	1.34896i	−0.454817	+	2.28652i	−1.63939	+	1.14559i	−0.548285	−	2.16781i	3.27755	−	0.357373i	0.363715	+	0.878084i	2.24148	+	1.72504i	−2.24966	−	0.931842i	−2.69147	+	1.66011i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
64.i	even	16	1	inner
320.bf	even	16	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	320.2.bf.a	✓	368
5.b	even	2	1	inner	320.2.bf.a	✓	368
64.i	even	16	1	inner	320.2.bf.a	✓	368
320.bf	even	16	1	inner	320.2.bf.a	✓	368

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
320.2.bf.a	✓	368	1.a	even	1	1	trivial
320.2.bf.a	✓	368	5.b	even	2	1	inner
320.2.bf.a	✓	368	64.i	even	16	1	inner
320.2.bf.a	✓	368	320.bf	even	16	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(320, [\chi])\).