Newform orbit 99.6.j.a

• Label: 99.6.j.a
• Level: $99$
• Weight: $6$
• Character orbit: 99.j
• Analytic conductor: $15.878$
• Analytic rank: $0$
• Dimension: $80$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 99 = 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	99.j (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(15.8779981615\)
Analytic rank:	\(0\)
Dimension:	\(80\)
Relative dimension:	\(20\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 80 q - 320 q^{4}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
8.1	−8.28842	+	6.02189i		0		22.5462	−	69.3900i	−11.2524	+	15.4876i		0		68.7398	+	22.3349i	129.678	+	399.108i		0			−	196.129i
8.2	−7.78338	+	5.65495i		0		18.7139	−	57.5955i	−54.9091	+	75.5758i		0		141.091	+	45.8432i	84.9069	+	261.316i		0			−	898.743i
8.3	−7.03416	+	5.11062i		0		13.4725	−	41.4640i	10.3670	−	14.2689i		0		−228.418	−	74.2174i	31.1612	+	95.9043i		0				153.352i
8.4	−6.86099	+	4.98480i		0		12.3364	−	37.9675i	47.9367	−	65.9792i		0		186.838	+	60.7074i	20.7594	+	63.8909i		0				691.637i
8.5	−6.39034	+	4.64285i		0		9.39178	−	28.9049i	38.1976	−	52.5745i		0		−173.240	−	56.2891i	−3.92387	−	12.0764i		0				513.315i
8.6	−4.66817	+	3.39162i		0		0.400136	−	1.23149i	−6.21029	+	8.54773i		0		130.340	+	42.3502i	−54.7497	−	168.502i		0			−	60.9652i
8.7	−3.44317	+	2.50161i		0		−4.29118	+	13.2069i	−15.5337	+	21.3802i		0		−53.6955	−	17.4467i	−60.3488	−	185.734i		0			−	112.475i
8.8	−3.25694	+	2.36630i		0		−4.88030	+	15.0200i	−61.8682	+	85.1542i		0		−181.666	−	59.0269i	−59.4563	−	182.988i		0			−	423.741i
8.9	−1.43236	+	1.04067i		0		−8.91988	+	27.4526i	51.8959	−	71.4286i		0		1.81017	+	0.588159i	−33.3003	−	102.488i		0				156.318i
8.10	−1.09077	+	0.792494i		0		−9.32680	+	28.7049i	17.2505	−	23.7433i		0		65.7147	+	21.3520i	−25.9075	−	79.7350i		0				39.5695i
8.11	1.09077	−	0.792494i		0		−9.32680	+	28.7049i	−17.2505	+	23.7433i		0		65.7147	+	21.3520i	25.9075	+	79.7350i		0				39.5695i
8.12	1.43236	−	1.04067i		0		−8.91988	+	27.4526i	−51.8959	+	71.4286i		0		1.81017	+	0.588159i	33.3003	+	102.488i		0				156.318i
8.13	3.25694	−	2.36630i		0		−4.88030	+	15.0200i	61.8682	−	85.1542i		0		−181.666	−	59.0269i	59.4563	+	182.988i		0			−	423.741i
8.14	3.44317	−	2.50161i		0		−4.29118	+	13.2069i	15.5337	−	21.3802i		0		−53.6955	−	17.4467i	60.3488	+	185.734i		0			−	112.475i
8.15	4.66817	−	3.39162i		0		0.400136	−	1.23149i	6.21029	−	8.54773i		0		130.340	+	42.3502i	54.7497	+	168.502i		0			−	60.9652i
8.16	6.39034	−	4.64285i		0		9.39178	−	28.9049i	−38.1976	+	52.5745i		0		−173.240	−	56.2891i	3.92387	+	12.0764i		0				513.315i
8.17	6.86099	−	4.98480i		0		12.3364	−	37.9675i	−47.9367	+	65.9792i		0		186.838	+	60.7074i	−20.7594	−	63.8909i		0				691.637i
8.18	7.03416	−	5.11062i		0		13.4725	−	41.4640i	−10.3670	+	14.2689i		0		−228.418	−	74.2174i	−31.1612	−	95.9043i		0				153.352i
8.19	7.78338	−	5.65495i		0		18.7139	−	57.5955i	54.9091	−	75.5758i		0		141.091	+	45.8432i	−84.9069	−	261.316i		0			−	898.743i
8.20	8.28842	−	6.02189i		0		22.5462	−	69.3900i	11.2524	−	15.4876i		0		68.7398	+	22.3349i	−129.678	−	399.108i		0			−	196.129i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
11.d	odd	10	1	inner
33.f	even	10	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	99.6.j.a	✓	80
3.b	odd	2	1	inner	99.6.j.a	✓	80
11.d	odd	10	1	inner	99.6.j.a	✓	80
33.f	even	10	1	inner	99.6.j.a	✓	80

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
99.6.j.a	✓	80	1.a	even	1	1	trivial
99.6.j.a	✓	80	3.b	odd	2	1	inner
99.6.j.a	✓	80	11.d	odd	10	1	inner
99.6.j.a	✓	80	33.f	even	10	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(99, [\chi])\).