Newform orbit 798.2.r.a

• Label: 798.2.r.a
• Level: $798$
• Weight: $2$
• Character orbit: 798.r
• Analytic conductor: $6.372$
• Analytic rank: $0$
• Dimension: $112$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 798 = 2 \cdot 3 \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	798.r (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.37206208130\)
Analytic rank:	\(0\)
Dimension:	\(112\)
Relative dimension:	\(56\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 112 q + 56 q^{4} + 12 q^{7}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
83.1	−0.866025	+	0.500000i	−1.72743	−	0.126450i	0.500000	−	0.866025i	0.548605	+	0.950212i	1.55922	−	0.754205i	2.60813	−	0.444601i			1.00000i	2.96802	+	0.436868i	−0.950212	−	0.548605i
83.2	−0.866025	+	0.500000i	−1.71344	+	0.253255i	0.500000	−	0.866025i	−1.36799	−	2.36942i	1.35725	−	1.07604i	1.19929	+	2.35833i			1.00000i	2.87172	−	0.867871i	2.36942	+	1.36799i
83.3	−0.866025	+	0.500000i	−1.65763	+	0.502247i	0.500000	−	0.866025i	−0.235680	−	0.408210i	1.18443	−	1.26378i	−2.64463	−	0.0770811i			1.00000i	2.49550	−	1.66508i	0.408210	+	0.235680i
83.4	−0.866025	+	0.500000i	−1.53208	−	0.807918i	0.500000	−	0.866025i	1.05087	+	1.82016i	1.73078	−	0.0663619i	0.156512	+	2.64112i			1.00000i	1.69454	+	2.47559i	−1.82016	−	1.05087i
83.5	−0.866025	+	0.500000i	−1.40912	+	1.00716i	0.500000	−	0.866025i	−1.96496	−	3.40342i	0.716758	−	1.57679i	1.11725	−	2.39828i			1.00000i	0.971261	−	2.83842i	3.40342	+	1.96496i
83.6	−0.866025	+	0.500000i	−1.35420	−	1.07988i	0.500000	−	0.866025i	1.64158	+	2.84330i	1.71271	+	0.258097i	−0.753451	−	2.53620i			1.00000i	0.667738	+	2.92474i	−2.84330	−	1.64158i
83.7	−0.866025	+	0.500000i	−1.18722	+	1.26115i	0.500000	−	0.866025i	1.70754	+	2.95754i	0.397589	−	1.68580i	−1.86706	+	1.87459i			1.00000i	−0.181004	−	2.99453i	−2.95754	−	1.70754i
83.8	−0.866025	+	0.500000i	−1.16399	−	1.28263i	0.500000	−	0.866025i	−1.57739	−	2.73213i	1.64936	+	0.528795i	−2.47741	−	0.928683i			1.00000i	−0.290270	+	2.98592i	2.73213	+	1.57739i
83.9	−0.866025	+	0.500000i	−0.964582	+	1.43860i	0.500000	−	0.866025i	1.36560	+	2.36528i	0.116051	−	1.72816i	2.06333	+	1.65610i			1.00000i	−1.13916	−	2.77530i	−2.36528	−	1.36560i
83.10	−0.866025	+	0.500000i	−0.749166	−	1.56165i	0.500000	−	0.866025i	−0.0444434	−	0.0769782i	1.42962	+	0.977846i	0.496975	−	2.59866i			1.00000i	−1.87750	+	2.33987i	0.0769782	+	0.0444434i
83.11	−0.866025	+	0.500000i	−0.687595	+	1.58972i	0.500000	−	0.866025i	0.154352	+	0.267346i	−0.199386	−	1.72054i	−0.406490	−	2.61434i			1.00000i	−2.05443	−	2.18617i	−0.267346	−	0.154352i
83.12	−0.866025	+	0.500000i	−0.612738	−	1.62005i	0.500000	−	0.866025i	−1.98458	−	3.43740i	1.34067	+	1.09663i	1.50853	+	2.17355i			1.00000i	−2.24910	+	1.98533i	3.43740	+	1.98458i
83.13	−0.866025	+	0.500000i	−0.410850	+	1.68262i	0.500000	−	0.866025i	−0.790910	−	1.36990i	−0.485502	−	1.66261i	−2.14650	+	1.54679i			1.00000i	−2.66240	−	1.38261i	1.36990	+	0.790910i
83.14	−0.866025	+	0.500000i	−0.162861	−	1.72438i	0.500000	−	0.866025i	0.709433	+	1.22877i	1.00323	+	1.41192i	2.64552	+	0.0350878i			1.00000i	−2.94695	+	0.561669i	−1.22877	−	0.709433i
83.15	−0.866025	+	0.500000i	0.162861	+	1.72438i	0.500000	−	0.866025i	−0.709433	−	1.22877i	−1.00323	−	1.41192i	2.64552	−	0.0350878i			1.00000i	−2.94695	+	0.561669i	1.22877	+	0.709433i
83.16	−0.866025	+	0.500000i	0.410850	−	1.68262i	0.500000	−	0.866025i	0.790910	+	1.36990i	0.485502	+	1.66261i	−2.14650	−	1.54679i			1.00000i	−2.66240	−	1.38261i	−1.36990	−	0.790910i
83.17	−0.866025	+	0.500000i	0.612738	+	1.62005i	0.500000	−	0.866025i	1.98458	+	3.43740i	−1.34067	−	1.09663i	1.50853	−	2.17355i			1.00000i	−2.24910	+	1.98533i	−3.43740	−	1.98458i
83.18	−0.866025	+	0.500000i	0.687595	−	1.58972i	0.500000	−	0.866025i	−0.154352	−	0.267346i	0.199386	+	1.72054i	−0.406490	+	2.61434i			1.00000i	−2.05443	−	2.18617i	0.267346	+	0.154352i
83.19	−0.866025	+	0.500000i	0.749166	+	1.56165i	0.500000	−	0.866025i	0.0444434	+	0.0769782i	−1.42962	−	0.977846i	0.496975	+	2.59866i			1.00000i	−1.87750	+	2.33987i	−0.0769782	−	0.0444434i
83.20	−0.866025	+	0.500000i	0.964582	−	1.43860i	0.500000	−	0.866025i	−1.36560	−	2.36528i	−0.116051	+	1.72816i	2.06333	−	1.65610i			1.00000i	−1.13916	−	2.77530i	2.36528	+	1.36560i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
7.b	odd	2	1	inner
19.c	even	3	1	inner
21.c	even	2	1	inner
57.h	odd	6	1	inner
133.m	odd	6	1	inner
399.z	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	798.2.r.a	✓	112
3.b	odd	2	1	inner	798.2.r.a	✓	112
7.b	odd	2	1	inner	798.2.r.a	✓	112
19.c	even	3	1	inner	798.2.r.a	✓	112
21.c	even	2	1	inner	798.2.r.a	✓	112
57.h	odd	6	1	inner	798.2.r.a	✓	112
133.m	odd	6	1	inner	798.2.r.a	✓	112
399.z	even	6	1	inner	798.2.r.a	✓	112

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
798.2.r.a	✓	112	1.a	even	1	1	trivial
798.2.r.a	✓	112	3.b	odd	2	1	inner
798.2.r.a	✓	112	7.b	odd	2	1	inner
798.2.r.a	✓	112	19.c	even	3	1	inner
798.2.r.a	✓	112	21.c	even	2	1	inner
798.2.r.a	✓	112	57.h	odd	6	1	inner
798.2.r.a	✓	112	133.m	odd	6	1	inner
798.2.r.a	✓	112	399.z	even	6	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(798, [\chi])\).