Newform orbit 85.2.o.a

• Label: 85.2.o.a
• Level: $85$
• Weight: $2$
• Character orbit: 85.o
• Analytic conductor: $0.679$
• Analytic rank: $0$
• Dimension: $56$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 85 = 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	85.o (of order \(16\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.678728417181\)
Analytic rank:	\(0\)
Dimension:	\(56\)
Relative dimension:	\(7\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 56 q - 8 q^{2} - 8 q^{3} - 8 q^{5} - 16 q^{6} - 8 q^{7} + 8 q^{8}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
3.1	−1.03863	+	2.50747i	1.10781	+	1.65796i	−3.79444	−	3.79444i	1.28952	+	1.82678i	−5.30789	+	1.05580i	−0.312113	−	1.56910i	8.44052	−	3.49618i	−0.373528	+	0.901777i	−5.91993	+	1.33609i
3.2	−0.857790	+	2.07089i	−1.60184	−	2.39732i	−2.13856	−	2.13856i	−2.22090	−	0.259984i	6.33863	−	1.26083i	−0.672494	−	3.38086i	2.12139	−	0.878709i	−2.03321	+	4.90861i	2.44347	−	4.37623i
3.3	−0.510242	+	1.23183i	−0.774396	−	1.15897i	0.157149	+	0.157149i	2.19536	+	0.424711i	1.82278	−	0.362574i	0.668895	+	3.36276i	−2.73743	+	1.13388i	0.404539	−	0.976643i	−1.64334	+	2.48761i
3.4	−0.448053	+	1.08170i	1.13243	+	1.69480i	0.444899	+	0.444899i	−0.658958	−	2.13677i	−2.34064	+	0.465583i	−0.161470	−	0.811766i	−2.84398	+	1.17801i	−0.441894	+	1.06683i	2.60658	+	0.244593i
3.5	0.0288415	−	0.0696296i	0.258477	+	0.386838i	1.41020	+	1.41020i	−1.45017	+	1.70206i	0.0343903	−	0.00684065i	−0.328800	−	1.65299i	0.278123	−	0.115202i	1.06522	−	2.57166i	0.0766887	+	0.150065i
3.6	0.667727	−	1.61204i	−1.59173	−	2.38220i	−0.738587	−	0.738587i	0.486638	+	2.18247i	−4.90303	+	0.975273i	−0.155366	−	0.781079i	1.54027	−	0.638000i	−1.99320	+	4.81200i	3.84316	+	0.672817i
3.7	0.775461	−	1.87213i	−0.0719451	−	0.107673i	−1.48931	−	1.48931i	−1.67302	−	1.48357i	−0.257369	+	0.0511939i	0.726716	+	3.65345i	−0.198830	+	0.0823582i	1.14163	−	2.75615i	−4.07481	+	1.98166i
7.1	−2.30461	+	0.954602i	−0.0638551	−	0.321021i	2.98576	−	2.98576i	−1.76196	−	1.37677i	0.453609	+	0.678874i	1.90843	−	1.27517i	−2.12161	+	5.12202i	2.67266	−	1.10705i	5.37491	+	1.49095i
7.2	−1.80686	+	0.748424i	0.551762	+	2.77389i	1.29037	−	1.29037i	2.15481	−	0.597315i	−3.07300	−	4.59907i	−3.34126	+	2.23256i	0.131075	−	0.316444i	−4.61840	+	1.91301i	−3.44639	+	2.69197i
7.3	−1.14715	+	0.475165i	−0.281905	−	1.41723i	−0.324044	+	0.324044i	2.23301	+	0.116981i	0.996805	+	1.49182i	2.92293	−	1.95304i	1.16808	−	2.82000i	0.842564	−	0.349001i	−2.61718	+	0.926851i
7.4	−0.346799	+	0.143649i	0.208197	+	1.04667i	−1.31458	+	1.31458i	−1.10424	+	1.94439i	−0.222556	−	0.333079i	−0.0532956	+	0.0356110i	0.554355	−	1.33833i	1.71946	−	0.712222i	0.103641	−	0.832936i
7.5	0.833790	−	0.345367i	0.458863	+	2.30686i	−0.838286	+	0.838286i	−0.423720	−	2.19555i	1.17931	+	1.76496i	1.90814	−	1.27498i	−1.10017	+	2.65605i	−2.33941	+	0.969017i	−1.11157	−	1.68429i
7.6	1.15118	−	0.476835i	−0.324751	−	1.63263i	−0.316366	+	0.316366i	1.84089	−	1.26930i	−1.15234	−	1.72460i	−3.16858	+	2.11718i	−1.16701	+	2.81741i	0.211613	−	0.0876532i	1.51395	−	2.33900i
7.7	1.69657	−	0.702741i	−0.241748	−	1.21535i	0.970277	−	0.970277i	−1.60069	+	1.56135i	−1.26422	−	1.89203i	0.671398	−	0.448614i	−0.441195	+	1.06514i	1.35301	−	0.560435i	−1.61845	+	3.77379i
27.1	−2.46604	+	1.02147i	−0.00877934	+	0.00174632i	3.62373	−	3.62373i	0.664061	+	2.13519i	0.0198664	−	0.0132743i	0.497790	+	0.744996i	−3.19180	+	7.70568i	−2.77156	+	1.14802i	−3.81862	−	4.58713i
27.2	−1.56354	+	0.647639i	−2.30010	+	0.457519i	0.611002	−	0.611002i	1.25339	−	1.85176i	3.29999	−	2.20498i	−1.48373	−	2.22056i	0.735661	−	1.77604i	2.30951	−	0.956631i	−0.760460	+	3.70704i
27.3	−0.536705	+	0.222310i	2.64177	−	0.525480i	−1.17558	+	1.17558i	0.252040	+	2.22182i	−1.30103	+	0.869319i	−1.45803	−	2.18210i	0.814217	−	1.96569i	3.93116	−	1.62834i	−0.629204	−	1.13643i
27.4	−0.248110	+	0.102770i	−1.76664	+	0.351407i	−1.36322	+	1.36322i	−1.90795	+	1.16608i	0.402207	−	0.268746i	−0.383694	−	0.574238i	0.403670	−	0.974545i	0.225900	−	0.0935709i	0.353541	−	0.485396i
27.5	0.812918	−	0.336722i	1.42996	−	0.284437i	−0.866759	+	0.866759i	−0.721559	−	2.11645i	1.06666	−	0.712723i	0.619405	+	0.927005i	−1.08619	+	2.62230i	−0.807756	+	0.334584i	−1.29922	−	1.47753i
27.6	1.58620	−	0.657027i	−0.777798	+	0.154714i	0.670142	−	0.670142i	2.14144	+	0.643596i	−1.13209	+	0.756441i	−2.17207	−	3.25074i	−0.691374	+	1.66912i	−2.19061	+	0.907379i	3.81962	−	0.386112i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
85.o	even	16	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	85.2.o.a	✓	56
3.b	odd	2	1		765.2.bx.a		56
5.b	even	2	1		425.2.s.b		56
5.c	odd	4	1		85.2.r.a	yes	56
5.c	odd	4	1		425.2.v.b		56
15.e	even	4	1		765.2.cc.a		56
17.e	odd	16	1		85.2.r.a	yes	56
51.i	even	16	1		765.2.cc.a		56
85.o	even	16	1	inner	85.2.o.a	✓	56
85.p	odd	16	1		425.2.v.b		56
85.r	even	16	1		425.2.s.b		56
255.bc	odd	16	1		765.2.bx.a		56

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
85.2.o.a	✓	56	1.a	even	1	1	trivial
85.2.o.a	✓	56	85.o	even	16	1	inner
85.2.r.a	yes	56	5.c	odd	4	1
85.2.r.a	yes	56	17.e	odd	16	1
425.2.s.b		56	5.b	even	2	1
425.2.s.b		56	85.r	even	16	1
425.2.v.b		56	5.c	odd	4	1
425.2.v.b		56	85.p	odd	16	1
765.2.bx.a		56	3.b	odd	2	1
765.2.bx.a		56	255.bc	odd	16	1
765.2.cc.a		56	15.e	even	4	1
765.2.cc.a		56	51.i	even	16	1

Hecke kernels

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