Newform orbit 89.3.d.a

• Label: 89.3.d.a
• Level: $89$
• Weight: $3$
• Character orbit: 89.d
• Analytic conductor: $2.425$
• Analytic rank: $0$
• Dimension: $56$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 89 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	89.d (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.42507435281\)
Analytic rank:	\(0\)
Dimension:	\(56\)
Relative dimension:	\(14\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 56 q - 8 q^{2} + 4 q^{3} + 96 q^{4} + 16 q^{5} - 36 q^{6} - 4 q^{7} - 40 q^{8} + 32 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
12.1	−3.78121			−0.938447	−	2.26561i	10.2975			0.392266	+	0.392266i	3.54846	+	8.56674i	2.19157	+	5.29092i	−23.8122			2.11165	−	2.11165i	−1.48324	−	1.48324i
12.2	−3.22198			1.38997	+	3.35568i	6.38113			3.41899	+	3.41899i	−4.47845	−	10.8119i	−1.30398	−	3.14808i	−7.67195			−2.96463	+	2.96463i	−11.0159	−	11.0159i
12.3	−2.50532			0.983271	+	2.37383i	2.27664			−5.24182	−	5.24182i	−2.46341	−	5.94720i	0.0958833	+	0.231483i	4.31757			1.69573	−	1.69573i	13.1325	+	13.1325i
12.4	−2.21601			−2.02048	−	4.87787i	0.910682			−1.64094	−	1.64094i	4.47740	+	10.8094i	−1.43103	−	3.45482i	6.84595			−13.3473	+	13.3473i	3.63633	+	3.63633i
12.5	−1.69812			−0.670500	−	1.61873i	−1.11640			5.78266	+	5.78266i	1.13859	+	2.74879i	−3.10615	−	7.49891i	8.68824			4.19324	−	4.19324i	−9.81963	−	9.81963i
12.6	−1.40043			−0.273663	−	0.660681i	−2.03881			0.0380197	+	0.0380197i	0.383245	+	0.925236i	5.17903	+	12.5033i	8.45690			6.00235	−	6.00235i	−0.0532438	−	0.0532438i
12.7	−0.287508			2.13011	+	5.14254i	−3.91734			0.266731	+	0.266731i	−0.612423	−	1.47852i	0.343871	+	0.830178i	2.27630			−15.5443	+	15.5443i	−0.0766874	−	0.0766874i
12.8	0.256471			−0.112248	−	0.270992i	−3.93422			−3.71524	−	3.71524i	−0.0287885	−	0.0695015i	−3.28160	−	7.92248i	−2.03490			6.30312	−	6.30312i	−0.952851	−	0.952851i
12.9	0.995285			0.640743	+	1.54689i	−3.00941			5.08304	+	5.08304i	0.637722	+	1.53960i	1.08410	+	2.61725i	−6.97636			4.38164	−	4.38164i	5.05907	+	5.05907i
12.10	1.24726			−1.61704	−	3.90387i	−2.44434			−2.08808	−	2.08808i	−2.01687	−	4.86915i	0.925889	+	2.23529i	−8.03778			−6.26145	+	6.26145i	−2.60439	−	2.60439i
12.11	2.52359			0.799955	+	1.93126i	2.36851			0.889737	+	0.889737i	2.01876	+	4.87372i	1.62619	+	3.92597i	−4.11721			3.27411	−	3.27411i	2.24533	+	2.24533i
12.12	2.96024			−1.37777	−	3.32622i	4.76302			4.64200	+	4.64200i	−4.07852	−	9.84642i	−2.68343	−	6.47838i	2.25873			−2.80155	+	2.80155i	13.7414	+	13.7414i
12.13	3.09123			1.62117	+	3.91384i	5.55568			−3.16843	−	3.16843i	5.01139	+	12.0986i	−3.83240	−	9.25223i	4.80895			−6.32600	+	6.32600i	−9.79432	−	9.79432i
12.14	3.45070			−0.969285	−	2.34006i	7.90733			−4.90157	−	4.90157i	−3.34471	−	8.07484i	3.89917	+	9.41343i	13.4830			1.82759	−	1.82759i	−16.9138	−	16.9138i
37.1	−3.80007			4.75462	+	1.96943i	10.4405			3.85964	−	3.85964i	−18.0679	−	7.48397i	−1.33750	−	0.554011i	−24.4744			12.3638	+	12.3638i	−14.6669	+	14.6669i
37.2	−3.34177			−1.30469	−	0.540422i	7.16741			−3.99321	+	3.99321i	4.35998	+	1.80596i	3.25837	+	1.34966i	−10.5847			−4.95379	−	4.95379i	13.3444	−	13.3444i
37.3	−3.23244			−3.00477	−	1.24462i	6.44864			4.32771	−	4.32771i	9.71274	+	4.02315i	−6.53999	−	2.70895i	−7.91507			1.11563	+	1.11563i	−13.9891	+	13.9891i
37.4	−2.19259			3.26194	+	1.35114i	0.807462			−5.59757	+	5.59757i	−7.15210	−	2.96250i	−7.79196	−	3.22754i	6.99994			2.45069	+	2.45069i	12.2732	−	12.2732i
37.5	−1.67635			2.62114	+	1.08571i	−1.18986			2.50096	−	2.50096i	−4.39394	−	1.82003i	6.01507	+	2.49152i	8.70001			−0.672345	−	0.672345i	−4.19248	+	4.19248i
37.6	−1.38722			−4.41884	−	1.83034i	−2.07561			−0.797063	+	0.797063i	6.12992	+	2.53910i	8.30334	+	3.43936i	8.42823			9.81205	+	9.81205i	1.10570	−	1.10570i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	89.3.d.a	✓	56
89.d	odd	8	1	inner	89.3.d.a	✓	56

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
89.3.d.a	✓	56	1.a	even	1	1	trivial
89.3.d.a	✓	56	89.d	odd	8	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(89, [\chi])\).