Newform orbit 89.2.e.a

• Label: 89.2.e.a
• Level: $89$
• Weight: $2$
• Character orbit: 89.e
• Analytic conductor: $0.711$
• Analytic rank: $0$
• Dimension: $60$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 89 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	89.e (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.710668577989\)
Analytic rank:	\(0\)
Dimension:	\(60\)
Relative dimension:	\(6\) over \(\Q(\zeta_{11})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 60 q - 7 q^{2} - 16 q^{3} - 13 q^{4} - q^{5} + q^{6} - 9 q^{7} + q^{8} - 10 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
2.1	−1.94635	+	0.571500i	2.12706	+	1.36698i	1.77916	−	1.14339i	2.17320	−	2.50801i	−4.92123	−	1.44500i	−2.08344	+	2.40442i	−0.152616	+	0.176128i	1.40951	+	3.08640i	−2.79649	+	6.12345i
2.2	−1.54771	+	0.454449i	−0.986921	−	0.634256i	0.506380	−	0.325431i	0.907544	−	1.04736i	1.81571	+	0.533139i	1.93063	−	2.22806i	1.47681	−	1.70433i	−0.674512	−	1.47698i	−0.928644	+	2.03345i
2.3	−0.537451	+	0.157810i	−1.53493	−	0.986440i	−1.41856	+	0.911651i	−1.68216	+	1.94131i	0.980620	+	0.287936i	−1.69317	+	1.95403i	1.35217	−	1.56048i	0.136704	+	0.299339i	0.597719	−	1.30882i
2.4	0.678084	−	0.199103i	1.26161	+	0.810786i	−1.26235	+	0.811264i	0.427256	−	0.493080i	1.01691	+	0.298591i	0.716442	−	0.826818i	−1.62005	+	1.86964i	−0.311965	−	0.683108i	0.191542	−	0.419418i
2.5	1.28996	−	0.378768i	−2.73648	−	1.75863i	−0.161964	+	0.104088i	2.40912	−	2.78027i	−4.19608	−	1.23208i	0.821469	−	0.948026i	−1.93032	+	2.22771i	3.14932	+	6.89604i	2.05460	−	4.49894i
2.6	2.14120	−	0.628712i	−0.721717	−	0.463820i	2.50694	−	1.61111i	−1.59296	+	1.83838i	−1.83695	−	0.539376i	−0.549613	+	0.634287i	1.43215	−	1.65279i	−0.940498	−	2.05940i	−2.25503	+	4.93784i
4.1	−2.23131	+	1.43397i	−0.769368	−	1.68468i	2.09162	−	4.58000i	0.512059	+	3.56145i	4.13248	+	2.65579i	0.249247	+	1.73355i	1.14562	+	7.96798i	−0.281643	+	0.325034i	−6.24959	−	7.21241i
4.2	−1.53515	+	0.986583i	1.01864	+	2.23052i	0.552520	−	1.20985i	0.115391	+	0.802560i	−3.76436	−	2.41921i	−0.143079	−	0.995139i	−0.173989	−	1.21012i	−1.97299	+	2.27695i	−0.968934	−	1.11821i
4.3	−1.21656	+	0.781837i	−0.678574	−	1.48587i	0.0379242	−	0.0830424i	−0.393520	−	2.73699i	1.98724	+	1.27712i	−0.148190	−	1.03069i	−0.392823	−	2.73214i	0.217234	−	0.250701i	2.61862	+	3.02205i
4.4	0.175830	−	0.112999i	0.225839	+	0.494518i	−0.812683	+	1.77953i	0.410796	+	2.85715i	0.0955895	+	0.0614317i	−0.387907	−	2.69795i	0.117681	+	0.818492i	1.77104	−	2.04389i	0.395087	+	0.455954i
4.5	0.391938	−	0.251884i	1.02279	+	2.23960i	−0.740660	+	1.62182i	−0.566208	−	3.93806i	0.964991	+	0.620162i	0.304028	+	2.11456i	0.250825	+	1.74452i	−2.00514	+	2.31406i	−1.21385	−	1.40086i
4.6	1.31733	−	0.846597i	−0.712435	−	1.56002i	0.187803	−	0.411231i	−0.0889419	−	0.618604i	−2.25922	−	1.45191i	0.0853945	+	0.593932i	0.344957	+	2.39923i	0.0384984	−	0.0444296i	−0.640874	−	0.739608i
8.1	−1.38057	+	1.59326i	0.236683	−	1.64617i	−0.347882	−	2.41958i	1.51842	+	0.975827i	2.29602	+	2.64975i	2.95389	+	1.89835i	0.788252	+	0.506579i	0.224629	+	0.0659571i	−3.65103	+	1.07204i
8.2	−1.02469	+	1.18256i	−0.295538	+	2.05551i	−0.0638211	−	0.443885i	−0.777686	−	0.499789i	−2.12793	−	2.45577i	0.428157	+	0.275160i	−2.04239	−	1.31256i	−1.25931	−	0.369767i	1.38792	−	0.407531i
8.3	−0.0505740	+	0.0583655i	−0.00961548	+	0.0668771i	0.283781	+	1.97374i	1.24049	+	0.797217i	−0.00341702	−	0.00394346i	−2.15137	−	1.38260i	−0.259488	−	0.166763i	2.87410	+	0.843912i	−0.109267	+	0.0320836i
8.4	0.112683	−	0.130043i	0.471899	−	3.28213i	0.280416	+	1.95034i	−0.873143	−	0.561135i	−0.373643	−	0.431206i	0.667548	+	0.429007i	0.574737	+	0.369361i	−7.67119	−	2.25246i	−0.171360	+	0.0503158i
8.5	1.16763	−	1.34752i	0.0354541	−	0.246589i	−0.167813	−	1.16717i	−2.28673	−	1.46959i	−0.290886	−	0.335700i	0.749104	+	0.481420i	1.23122	+	0.791259i	2.81893	+	0.827712i	−4.65036	+	1.36547i
8.6	1.62756	−	1.87830i	−0.401657	+	2.79358i	−0.594446	−	4.13446i	1.54888	+	0.995405i	4.59348	+	5.30116i	−4.06275	−	2.61097i	−4.55166	−	2.92517i	−4.76431	−	1.39893i	4.39057	−	1.28919i
16.1	−1.12002	+	2.45251i	−0.0795566	+	0.0918132i	−3.45063	−	3.98224i	−2.35741	+	0.692199i	−0.136068	−	0.297946i	−3.58771	+	1.05345i	8.45740	−	2.48332i	0.424844	+	2.95486i	0.942734	−	6.55686i
16.2	−0.633795	+	1.38782i	1.55577	−	1.79546i	−0.214622	−	0.247687i	−1.42773	+	0.419220i	1.50573	+	3.29708i	4.11747	−	1.20900i	−2.44801	+	0.718801i	−0.376295	−	2.61719i	0.323089	−	2.24714i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
89.e	even	11	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	89.2.e.a	✓	60
3.b	odd	2	1		801.2.m.a		60
89.e	even	11	1	inner	89.2.e.a	✓	60
89.e	even	11	1		7921.2.a.q		30
89.f	even	22	1		7921.2.a.p		30
267.k	odd	22	1		801.2.m.a		60

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
89.2.e.a	✓	60	1.a	even	1	1	trivial
89.2.e.a	✓	60	89.e	even	11	1	inner
801.2.m.a		60	3.b	odd	2	1
801.2.m.a		60	267.k	odd	22	1
7921.2.a.p		30	89.f	even	22	1
7921.2.a.q		30	89.e	even	11	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(89, [\chi])\).