Newform orbit 85.2.m.a

• Label: 85.2.m.a
• Level: $85$
• Weight: $2$
• Character orbit: 85.m
• Analytic conductor: $0.679$
• Analytic rank: $0$
• Dimension: $24$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 85 = 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	85.m (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.

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

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
9.1	−1.48843	−	1.48843i	−0.690961	+	1.66813i			2.43082i	1.70647	+	1.44498i	3.51133	−	1.45444i	2.47912	−	1.02689i	0.641241	−	0.641241i	−0.183901	−	0.183901i	−0.389210	−	4.69069i
9.2	−0.672981	−	0.672981i	0.947078	−	2.28645i		−	1.09419i	−0.973741	+	2.01292i	−2.17610	+	0.901371i	−0.492777	+	0.204115i	−2.08233	+	2.08233i	−2.20957	−	2.20957i	2.00996	−	0.699345i
9.3	−0.352960	−	0.352960i	−0.0655465	+	0.158243i		−	1.75084i	−1.40760	−	1.73743i	0.0789888	−	0.0327182i	3.57820	−	1.48214i	−1.32390	+	1.32390i	2.10058	+	2.10058i	−0.116419	+	1.11007i
9.4	0.352960	+	0.352960i	0.0655465	−	0.158243i		−	1.75084i	2.22387	−	0.233230i	0.0789888	−	0.0327182i	−3.57820	+	1.48214i	1.32390	−	1.32390i	2.10058	+	2.10058i	0.867258	+	0.702616i
9.5	0.672981	+	0.672981i	−0.947078	+	2.28645i		−	1.09419i	−0.734807	+	2.11188i	−2.17610	+	0.901371i	0.492777	−	0.204115i	2.08233	−	2.08233i	−2.20957	−	2.20957i	−1.91577	+	0.926747i
9.6	1.48843	+	1.48843i	0.690961	−	1.66813i			2.43082i	−2.22841	−	0.184902i	3.51133	−	1.45444i	−2.47912	+	1.02689i	−0.641241	+	0.641241i	−0.183901	−	0.183901i	−3.04161	−	3.59203i
19.1	−1.48843	+	1.48843i	−0.690961	−	1.66813i		−	2.43082i	1.70647	−	1.44498i	3.51133	+	1.45444i	2.47912	+	1.02689i	0.641241	+	0.641241i	−0.183901	+	0.183901i	−0.389210	+	4.69069i
19.2	−0.672981	+	0.672981i	0.947078	+	2.28645i			1.09419i	−0.973741	−	2.01292i	−2.17610	−	0.901371i	−0.492777	−	0.204115i	−2.08233	−	2.08233i	−2.20957	+	2.20957i	2.00996	+	0.699345i
19.3	−0.352960	+	0.352960i	−0.0655465	−	0.158243i			1.75084i	−1.40760	+	1.73743i	0.0789888	+	0.0327182i	3.57820	+	1.48214i	−1.32390	−	1.32390i	2.10058	−	2.10058i	−0.116419	−	1.11007i
19.4	0.352960	−	0.352960i	0.0655465	+	0.158243i			1.75084i	2.22387	+	0.233230i	0.0789888	+	0.0327182i	−3.57820	−	1.48214i	1.32390	+	1.32390i	2.10058	−	2.10058i	0.867258	−	0.702616i
19.5	0.672981	−	0.672981i	−0.947078	−	2.28645i			1.09419i	−0.734807	−	2.11188i	−2.17610	−	0.901371i	0.492777	+	0.204115i	2.08233	+	2.08233i	−2.20957	+	2.20957i	−1.91577	−	0.926747i
19.6	1.48843	−	1.48843i	0.690961	+	1.66813i		−	2.43082i	−2.22841	+	0.184902i	3.51133	+	1.45444i	−2.47912	−	1.02689i	−0.641241	−	0.641241i	−0.183901	+	0.183901i	−3.04161	+	3.59203i
49.1	−1.63043	+	1.63043i	1.73474	−	0.718554i		−	3.31660i	2.22935	+	0.173214i	−1.65683	+	3.99993i	0.591528	−	1.42808i	2.14663	+	2.14663i	0.371694	−	0.371694i	−3.91721	+	3.35238i
49.2	−1.23200	+	1.23200i	−0.553124	+	0.229112i		−	1.03565i	−1.73956	+	1.40496i	0.399184	−	0.963715i	−0.397218	+	0.958968i	−1.18808	−	1.18808i	−1.86787	+	1.86787i	0.412231	−	3.87406i
49.3	−0.176012	+	0.176012i	1.51856	−	0.629010i			1.93804i	0.0665446	−	2.23508i	−0.156572	+	0.377999i	−0.547653	+	1.32215i	−0.693142	−	0.693142i	−0.210935	+	0.210935i	0.381688	+	0.405113i
49.4	0.176012	−	0.176012i	−1.51856	+	0.629010i			1.93804i	1.62749	+	1.53338i	−0.156572	+	0.377999i	0.547653	−	1.32215i	0.693142	+	0.693142i	−0.210935	+	0.210935i	0.556352	−	0.0165642i
49.5	1.23200	−	1.23200i	0.553124	−	0.229112i		−	1.03565i	−2.22352	+	0.236600i	0.399184	−	0.963715i	0.397218	−	0.958968i	1.18808	+	1.18808i	−1.86787	+	1.86787i	−2.44788	+	3.03086i
49.6	1.63043	−	1.63043i	−1.73474	+	0.718554i		−	3.31660i	1.45391	−	1.69887i	−1.65683	+	3.99993i	−0.591528	+	1.42808i	−2.14663	−	2.14663i	0.371694	−	0.371694i	−0.399393	−	5.14038i
59.1	−1.63043	−	1.63043i	1.73474	+	0.718554i			3.31660i	2.22935	−	0.173214i	−1.65683	−	3.99993i	0.591528	+	1.42808i	2.14663	−	2.14663i	0.371694	+	0.371694i	−3.91721	−	3.35238i
59.2	−1.23200	−	1.23200i	−0.553124	−	0.229112i			1.03565i	−1.73956	−	1.40496i	0.399184	+	0.963715i	−0.397218	−	0.958968i	−1.18808	+	1.18808i	−1.86787	−	1.86787i	0.412231	+	3.87406i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
17.d	even	8	1	inner
85.m	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.m.a	✓	24
3.b	odd	2	1		765.2.bh.b		24
5.b	even	2	1	inner	85.2.m.a	✓	24
5.c	odd	4	2		425.2.m.e		24
15.d	odd	2	1		765.2.bh.b		24
17.d	even	8	1	inner	85.2.m.a	✓	24
17.e	odd	16	2		1445.2.b.i		24
51.g	odd	8	1		765.2.bh.b		24
85.k	odd	8	1		425.2.m.e		24
85.m	even	8	1	inner	85.2.m.a	✓	24
85.n	odd	8	1		425.2.m.e		24
85.o	even	16	2		7225.2.a.by		24
85.p	odd	16	2		1445.2.b.i		24
85.r	even	16	2		7225.2.a.by		24
255.y	odd	8	1		765.2.bh.b		24

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
85.2.m.a	✓	24	1.a	even	1	1	trivial
85.2.m.a	✓	24	5.b	even	2	1	inner
85.2.m.a	✓	24	17.d	even	8	1	inner
85.2.m.a	✓	24	85.m	even	8	1	inner
425.2.m.e		24	5.c	odd	4	2
425.2.m.e		24	85.k	odd	8	1
425.2.m.e		24	85.n	odd	8	1
765.2.bh.b		24	3.b	odd	2	1
765.2.bh.b		24	15.d	odd	2	1
765.2.bh.b		24	51.g	odd	8	1
765.2.bh.b		24	255.y	odd	8	1
1445.2.b.i		24	17.e	odd	16	2
1445.2.b.i		24	85.p	odd	16	2
7225.2.a.by		24	85.o	even	16	2
7225.2.a.by		24	85.r	even	16	2

Hecke kernels

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