Newform orbit 85.2.l.a

• Label: 85.2.l.a
• Level: $85$
• Weight: $2$
• Character orbit: 85.l
• Analytic conductor: $0.679$
• Analytic rank: $0$
• Dimension: $24$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 85 = 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	85.l (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.678728417181\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(6\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 8 q^{6} - 24 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
26.1	−1.93083	+	1.93083i	−1.42916	−	0.591976i		−	5.45623i	0.382683	−	0.923880i	3.90247	−	1.61646i	−0.483886	−	1.16820i	6.67340	+	6.67340i	−0.429267	−	0.429267i	1.04496	+	2.52275i
26.2	−1.27691	+	1.27691i	0.635552	+	0.263254i		−	1.26102i	−0.382683	+	0.923880i	−1.14770	+	0.475393i	1.66158	+	4.01142i	−0.943613	−	0.943613i	−1.78670	−	1.78670i	−0.691061	−	1.66837i
26.3	−1.09994	+	1.09994i	2.77900	+	1.15110i		−	0.419729i	0.382683	−	0.923880i	−4.32287	+	1.79059i	−1.32205	−	3.19170i	−1.73820	−	1.73820i	4.27649	+	4.27649i	0.595282	+	1.43714i
26.4	0.213325	−	0.213325i	0.980249	+	0.406032i			1.90899i	−0.382683	+	0.923880i	0.295728	−	0.122495i	−0.960473	−	2.31879i	0.833883	+	0.833883i	−1.32529	−	1.32529i	0.115451	+	0.278722i
26.5	1.01710	−	1.01710i	−0.101541	−	0.0420595i		−	0.0689897i	0.382683	−	0.923880i	−0.146056	+	0.0604983i	−0.265997	−	0.642174i	1.96403	+	1.96403i	−2.11278	−	2.11278i	−0.550451	−	1.32891i
26.6	1.66305	−	1.66305i	−1.44989	−	0.600564i		−	3.53144i	−0.382683	+	0.923880i	−3.41000	+	1.41247i	1.37082	+	3.30945i	−2.54686	−	2.54686i	−0.379815	−	0.379815i	0.900034	+	2.17287i
36.1	−1.93083	−	1.93083i	−1.42916	+	0.591976i			5.45623i	0.382683	+	0.923880i	3.90247	+	1.61646i	−0.483886	+	1.16820i	6.67340	−	6.67340i	−0.429267	+	0.429267i	1.04496	−	2.52275i
36.2	−1.27691	−	1.27691i	0.635552	−	0.263254i			1.26102i	−0.382683	−	0.923880i	−1.14770	−	0.475393i	1.66158	−	4.01142i	−0.943613	+	0.943613i	−1.78670	+	1.78670i	−0.691061	+	1.66837i
36.3	−1.09994	−	1.09994i	2.77900	−	1.15110i			0.419729i	0.382683	+	0.923880i	−4.32287	−	1.79059i	−1.32205	+	3.19170i	−1.73820	+	1.73820i	4.27649	−	4.27649i	0.595282	−	1.43714i
36.4	0.213325	+	0.213325i	0.980249	−	0.406032i		−	1.90899i	−0.382683	−	0.923880i	0.295728	+	0.122495i	−0.960473	+	2.31879i	0.833883	−	0.833883i	−1.32529	+	1.32529i	0.115451	−	0.278722i
36.5	1.01710	+	1.01710i	−0.101541	+	0.0420595i			0.0689897i	0.382683	+	0.923880i	−0.146056	−	0.0604983i	−0.265997	+	0.642174i	1.96403	−	1.96403i	−2.11278	+	2.11278i	−0.550451	+	1.32891i
36.6	1.66305	+	1.66305i	−1.44989	+	0.600564i			3.53144i	−0.382683	−	0.923880i	−3.41000	−	1.41247i	1.37082	−	3.30945i	−2.54686	+	2.54686i	−0.379815	+	0.379815i	0.900034	−	2.17287i
66.1	−1.44607	−	1.44607i	−1.22292	−	2.95240i			2.18224i	0.923880	−	0.382683i	−2.50094	+	6.03781i	1.08781	+	0.450584i	0.263530	−	0.263530i	−5.09981	+	5.09981i	−1.88938	−	0.782608i
66.2	−0.528855	−	0.528855i	1.17676	+	2.84096i		−	1.44062i	−0.923880	+	0.382683i	0.880118	−	2.12479i	2.98655	+	1.23707i	−1.81959	+	1.81959i	−4.56494	+	4.56494i	0.690983	+	0.286214i
66.3	−0.254738	−	0.254738i	0.0207557	+	0.0501087i		−	1.87022i	0.923880	−	0.382683i	0.00747733	−	0.0180519i	0.275980	+	0.114315i	−0.985893	+	0.985893i	2.11924	−	2.11924i	−0.332832	−	0.137863i
66.4	0.680853	+	0.680853i	−1.01372	−	2.44733i		−	1.07288i	−0.923880	+	0.382683i	0.976080	−	2.35647i	2.85906	+	1.18426i	2.09218	−	2.09218i	−2.84049	+	2.84049i	−0.889577	−	0.368475i
66.5	1.09631	+	1.09631i	0.436412	+	1.05359i			0.403772i	−0.923880	+	0.382683i	−0.676617	+	1.63350i	−3.45666	−	1.43180i	1.74995	−	1.74995i	1.20172	−	1.20172i	−1.43239	−	0.593316i
66.6	1.86672	+	1.86672i	−0.811501	−	1.95914i			4.96928i	0.923880	−	0.382683i	2.14231	−	5.17200i	−3.75274	−	1.55444i	−5.54282	+	5.54282i	−1.05836	+	1.05836i	2.43899	+	1.01026i
76.1	−1.44607	+	1.44607i	−1.22292	+	2.95240i		−	2.18224i	0.923880	+	0.382683i	−2.50094	−	6.03781i	1.08781	−	0.450584i	0.263530	+	0.263530i	−5.09981	−	5.09981i	−1.88938	+	0.782608i
76.2	−0.528855	+	0.528855i	1.17676	−	2.84096i			1.44062i	−0.923880	−	0.382683i	0.880118	+	2.12479i	2.98655	−	1.23707i	−1.81959	−	1.81959i	−4.56494	−	4.56494i	0.690983	−	0.286214i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
17.d	even	8	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	85.2.l.a	✓	24
3.b	odd	2	1		765.2.be.b		24
5.b	even	2	1		425.2.m.b		24
5.c	odd	4	1		425.2.n.c		24
5.c	odd	4	1		425.2.n.f		24
17.d	even	8	1	inner	85.2.l.a	✓	24
17.e	odd	16	1		1445.2.a.p		12
17.e	odd	16	1		1445.2.a.q		12
17.e	odd	16	2		1445.2.d.j		24
51.g	odd	8	1		765.2.be.b		24
85.k	odd	8	1		425.2.n.c		24
85.m	even	8	1		425.2.m.b		24
85.n	odd	8	1		425.2.n.f		24
85.p	odd	16	1		7225.2.a.bq		12
85.p	odd	16	1		7225.2.a.bs		12

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
85.2.l.a	✓	24	1.a	even	1	1	trivial
85.2.l.a	✓	24	17.d	even	8	1	inner
425.2.m.b		24	5.b	even	2	1
425.2.m.b		24	85.m	even	8	1
425.2.n.c		24	5.c	odd	4	1
425.2.n.c		24	85.k	odd	8	1
425.2.n.f		24	5.c	odd	4	1
425.2.n.f		24	85.n	odd	8	1
765.2.be.b		24	3.b	odd	2	1
765.2.be.b		24	51.g	odd	8	1
1445.2.a.p		12	17.e	odd	16	1
1445.2.a.q		12	17.e	odd	16	1
1445.2.d.j		24	17.e	odd	16	2
7225.2.a.bq		12	85.p	odd	16	1
7225.2.a.bs		12	85.p	odd	16	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(85, [\chi])\).