Newform orbit 89.6.f.a

• Label: 89.6.f.a
• Level: $89$
• Weight: $6$
• Character orbit: 89.f
• Analytic conductor: $14.274$
• Analytic rank: $0$
• Dimension: $360$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 89 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	89.f (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(14.2741599634\)
Analytic rank:	\(0\)
Dimension:	\(360\)
Relative dimension:	\(36\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 360 q - q^{2} - 553 q^{4} - 17 q^{5} - 11 q^{6} - 11 q^{7} + 439 q^{8} + 1858 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
11.1	−7.26528	−	8.38458i	−7.63612	+	1.09791i	−12.9628	+	90.1584i	71.8160	−	46.1533i	64.6840	+	56.0490i	−64.9556	−	101.073i	551.456	−	354.399i	−176.052	+	51.6935i	−908.740	−	266.830i
11.2	−6.68108	−	7.71037i	−1.00653	+	0.144717i	−10.2590	+	71.3529i	−49.1507	+	31.5872i	7.84052	+	6.79385i	115.071	+	179.054i	344.052	−	221.109i	−232.165	+	68.1697i	571.929	+	167.933i
11.3	−6.56293	−	7.57402i	20.2823	−	2.91615i	−9.73972	+	67.7412i	−30.4301	+	19.5563i	−155.198	−	134.480i	−58.9765	−	91.7692i	307.205	−	197.428i	169.709	−	49.8312i	347.830	+	102.132i
11.4	−6.00572	−	6.93097i	−25.4359	+	3.65713i	−7.41560	+	51.5766i	−15.4832	+	9.95045i	178.109	+	154.332i	12.4787	+	19.4173i	155.128	−	99.6948i	400.455	−	117.584i	161.954	+	47.5540i
11.5	−5.60658	−	6.47034i	23.5056	−	3.37959i	−5.87746	+	40.8786i	63.7382	−	40.9620i	−153.653	−	133.141i	29.1741	+	45.3958i	66.9747	−	43.0420i	307.934	−	90.4176i	−622.391	−	182.751i
11.6	−5.56870	−	6.42663i	0.369057	−	0.0530624i	−5.73699	+	39.9017i	−0.636155	+	0.408832i	−2.39618	−	2.07630i	−12.9300	−	20.1195i	59.4616	−	38.2137i	−233.023	+	68.4218i	6.16997	+	1.81167i
11.7	−4.74767	−	5.47911i	−0.836846	+	0.120320i	−2.92614	+	20.3517i	72.6194	−	46.6697i	4.63232	+	4.01393i	116.033	+	180.550i	−69.7664	+	44.8362i	−232.471	+	68.2596i	−600.481	−	176.317i
11.8	−4.66027	−	5.37824i	−6.44351	+	0.926436i	−2.65326	+	18.4539i	−92.0083	+	59.1302i	35.0111	+	30.3373i	−119.985	−	186.700i	−79.9610	+	51.3878i	−192.496	+	56.5220i	746.800	+	219.280i
11.9	−4.26793	−	4.92545i	−16.2346	+	2.33419i	−1.49079	+	10.3686i	35.1948	−	22.6183i	80.7851	+	70.0007i	−69.7330	−	108.507i	−118.014	+	75.8429i	24.9576	−	7.32821i	−261.614	−	76.8169i
11.10	−3.58104	−	4.13274i	25.1654	−	3.61823i	0.298381	−	2.07529i	−72.1964	+	46.3978i	−105.071	−	91.0448i	83.4876	+	129.909i	−156.855	+	100.805i	387.046	−	113.647i	450.288	+	132.217i
11.11	−3.45431	−	3.98649i	15.5491	−	2.23563i	0.594250	−	4.13310i	−11.6828	+	7.50809i	−62.6239	−	54.2639i	−1.55790	−	2.42414i	−160.530	+	103.166i	3.62084	−	1.06318i	70.2871	+	20.6382i
11.12	−2.45880	−	2.83760i	−2.53998	+	0.365194i	2.54777	−	17.7201i	7.45165	−	4.78889i	7.28157	+	6.30952i	70.4673	+	109.649i	−157.624	+	101.299i	−226.839	+	66.6058i	−31.9110	−	9.36992i
11.13	−2.44398	−	2.82050i	12.2716	−	1.76438i	2.57188	−	17.8878i	57.3607	−	36.8635i	−34.9679	−	30.2998i	−126.621	−	197.025i	−157.206	+	101.030i	−85.6787	+	25.1575i	−244.162	−	71.6924i
11.14	−2.21280	−	2.55371i	−18.5026	+	2.66028i	2.92914	−	20.3726i	−62.0665	+	39.8877i	47.7362	+	41.3637i	73.4585	+	114.304i	−149.471	+	96.0594i	102.114	−	29.9833i	239.202	+	70.2361i
11.15	−2.12866	−	2.45661i	−23.7404	+	3.41335i	3.05036	−	21.2157i	61.5317	−	39.5440i	58.9206	+	51.0550i	41.3998	+	64.4194i	−146.117	+	93.9039i	318.798	−	93.6075i	−228.125	−	66.9835i
11.16	−0.867452	−	1.00109i	−0.174883	+	0.0251444i	4.30436	−	29.9375i	−32.4896	+	20.8798i	0.176874	+	0.153262i	−17.7099	−	27.5571i	−69.3634	+	44.5771i	−233.127	+	68.4522i	49.0857	+	14.4129i
11.17	−0.284969	−	0.328872i	28.7663	−	4.13598i	4.52713	−	31.4868i	45.5661	−	29.2835i	−9.55773	−	8.28182i	15.9840	+	24.8717i	−23.3598	+	15.0124i	577.240	−	169.493i	−22.6155	−	6.64050i
11.18	0.194195	+	0.224113i	19.9890	−	2.87398i	4.54156	−	31.5872i	−59.2305	+	38.0651i	4.52585	+	3.92167i	−59.5362	−	92.6401i	15.9440	−	10.2466i	158.143	−	46.4349i	−20.0331	−	5.88226i
11.19	0.443220	+	0.511504i	−24.2552	+	3.48738i	4.48888	−	31.2209i	−13.0438	+	8.38273i	−12.5342	−	10.8610i	−111.321	−	173.219i	36.1791	−	23.2509i	342.998	−	100.713i	−10.0691	−	2.95655i
11.20	0.757731	+	0.874469i	13.9206	−	2.00148i	4.36354	−	30.3491i	28.6041	−	18.3827i	12.2983	+	10.6566i	133.934	+	208.405i	60.9946	−	39.1989i	−43.3791	+	12.7372i	37.7493	+	11.0842i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
89.f	even	22	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	89.6.f.a	✓	360
89.f	even	22	1	inner	89.6.f.a	✓	360

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
89.6.f.a	✓	360	1.a	even	1	1	trivial
89.6.f.a	✓	360	89.f	even	22	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(89, [\chi])\).