Newform orbit 85.4.r.a

• Label: 85.4.r.a
• Level: $85$
• Weight: $4$
• Character orbit: 85.r
• Analytic conductor: $5.015$
• Analytic rank: $0$
• Dimension: $200$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 85 = 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	85.r (of order \(16\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.01516235049\)
Analytic rank:	\(0\)
Dimension:	\(200\)
Relative dimension:	\(25\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 200 q - 8 q^{2} - 8 q^{3} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 72 q^{8}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
12.1	−2.04131	+	4.92815i	0.434698	−	2.18537i	−14.4629	−	14.4629i	−6.74285	+	8.91818i	9.88249	+	6.60327i	−16.8370	−	11.2501i	61.3732	−	25.4216i	20.3579	+	8.43250i	−30.1859	−	51.4345i
12.2	−2.03338	+	4.90900i	−1.74644	+	8.77995i	−14.3068	−	14.3068i	10.7085	−	3.21367i	−39.5496	−	26.4262i	−13.0168	−	8.69758i	60.0515	−	24.8741i	−49.0927	−	20.3349i	−5.99852	+	59.1027i
12.3	−1.78644	+	4.31285i	0.459726	−	2.31120i	−9.75244	−	9.75244i	0.486014	−	11.1698i	9.14658	+	6.11155i	14.4990	+	9.68795i	24.9802	−	10.3471i	19.8144	+	8.20741i	47.3053	+	22.0502i
12.4	−1.56343	+	3.77446i	0.501843	−	2.52294i	−6.14535	−	6.14535i	9.04488	+	6.57192i	8.73812	+	5.83862i	4.58649	+	3.06459i	2.60756	−	1.08009i	18.8314	+	7.80021i	−38.9465	+	23.8648i
12.5	−1.41524	+	3.41669i	1.83901	−	9.24530i	−4.01401	−	4.01401i	−11.1788	−	0.185557i	28.9857	+	19.3676i	3.04969	+	2.03774i	−7.93810	+	3.28807i	−57.1490	−	23.6719i	16.4547	−	37.9319i
12.6	−1.39176	+	3.36002i	−1.33676	+	6.72035i	−3.69585	−	3.69585i	−4.90250	+	10.0482i	−20.7200	−	13.8447i	16.5490	+	11.0577i	−9.31826	+	3.85975i	−18.4314	−	7.63453i	−26.9388	−	30.4572i
12.7	−1.21562	+	2.93477i	−0.979596	+	4.92476i	−1.47829	−	1.47829i	−7.29757	−	8.47027i	−13.2622	−	8.86154i	−16.4250	−	10.9748i	−17.3427	+	7.18358i	1.65108	+	0.683900i	33.7294	−	11.1201i
12.8	−1.00623	+	2.42926i	1.44386	−	7.25875i	0.768062	+	0.768062i	8.84332	−	6.84074i	16.1805	+	10.8115i	−23.6408	−	15.7963i	−22.0727	+	9.14283i	−25.6600	−	10.6287i	7.71950	+	28.3661i
12.9	−0.748031	+	1.80591i	−0.976537	+	4.90938i	2.95511	+	2.95511i	11.0296	−	1.82965i	−8.13541	−	5.43590i	17.8434	+	11.9226i	−21.9944	+	9.11038i	1.79632	+	0.744058i	−4.94632	+	21.2871i
12.10	−0.507196	+	1.22448i	0.0167418	−	0.0841667i	4.41475	+	4.41475i	−7.51707	+	8.27609i	0.0945690	+	0.0631890i	−13.0093	−	8.69252i	−17.4408	+	7.22420i	24.9379	+	10.3296i	−6.32127	−	13.4021i
12.11	−0.353628	+	0.853732i	1.35559	−	6.81500i	5.05305	+	5.05305i	3.82263	+	10.5065i	5.33881	+	3.56728i	6.46588	+	4.32036i	−12.9307	+	5.35607i	−19.6619	−	8.14421i	−10.3216	−	0.451899i
12.12	−0.274072	+	0.661668i	0.520324	−	2.61585i	5.29417	+	5.29417i	−10.1647	−	4.65615i	1.58821	+	1.06121i	24.3329	+	16.2587i	−10.2473	+	4.24457i	18.3728	+	7.61028i	5.86667	−	5.44951i
12.13	−0.0229419	+	0.0553867i	−0.312794	+	1.57252i	5.65431	+	5.65431i	8.26108	−	7.53356i	−0.0799207	−	0.0534013i	−6.74608	−	4.50758i	−0.885988	+	0.366988i	22.5698	+	9.34870i	0.227734	+	0.630388i
12.14	0.148897	−	0.359470i	−1.66056	+	8.34818i	5.54981	+	5.54981i	5.58084	+	9.68784i	2.75367	+	1.83994i	−21.2306	−	14.1858i	5.69710	−	2.35982i	−41.9900	−	17.3928i	4.31346	−	0.563650i
12.15	0.351698	−	0.849073i	−1.87156	+	9.40896i	5.05962	+	5.05962i	−8.27758	−	7.51543i	7.33067	+	4.89820i	15.5563	+	10.3944i	12.8680	−	5.33011i	−60.0810	−	24.8864i	−9.29236	+	4.38511i
12.16	0.665575	−	1.60684i	1.08698	−	5.46463i	3.51791	+	3.51791i	−8.49133	−	7.27305i	−8.05732	−	5.38373i	−22.3480	−	14.9324i	20.8489	−	8.63589i	−3.73585	−	1.54744i	−17.3383	+	8.80346i
12.17	0.684013	−	1.65135i	1.72391	−	8.66667i	3.39776	+	3.39776i	8.28751	−	7.50447i	−13.1326	−	8.77490i	20.5331	+	13.7198i	21.1458	−	8.75889i	−47.1946	−	19.5487i	−6.72377	−	18.8188i
12.18	0.884793	−	2.13608i	0.271041	−	1.36261i	1.87688	+	1.87688i	7.62959	+	8.17247i	−2.67084	−	1.78460i	0.276141	+	0.184511i	22.7584	−	9.42686i	23.1615	+	9.59380i	24.2077	−	9.06647i
12.19	1.08910	−	2.62932i	−0.533800	+	2.68359i	−0.0703143	−	0.0703143i	−7.46888	+	8.31960i	6.47465	+	4.32623i	11.5751	+	7.73424i	20.7731	−	8.60449i	18.0280	+	7.46745i	13.7405	+	28.6989i
12.20	1.16376	−	2.80956i	−0.780312	+	3.92289i	−0.882452	−	0.882452i	2.38753	−	10.9224i	10.1135	+	6.75764i	−1.26824	−	0.847409i	18.9702	−	7.85773i	10.1645	+	4.21029i	−27.9088	−	19.4190i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	85.4.r.a	yes	200
5.c	odd	4	1		85.4.o.a	✓	200
17.e	odd	16	1		85.4.o.a	✓	200
85.r	even	16	1	inner	85.4.r.a	yes	200

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
85.4.o.a	✓	200	5.c	odd	4	1
85.4.o.a	✓	200	17.e	odd	16	1
85.4.r.a	yes	200	1.a	even	1	1	trivial
85.4.r.a	yes	200	85.r	even	16	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(85, [\chi])\).