Newform orbit 109.2.f.a

• Label: 109.2.f.a
• Level: $109$
• Weight: $2$
• Character orbit: 109.f
• Analytic conductor: $0.870$
• Analytic rank: $0$
• Dimension: $42$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 109 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	109.f (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.870369382032\)
Analytic rank:	\(0\)
Dimension:	\(42\)
Relative dimension:	\(7\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 42 q - 6 q^{3} - 12 q^{4} - 6 q^{5} + 12 q^{6} + 3 q^{7} - 12 q^{8} - 12 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
16.1	−1.22244	−	2.11733i	−1.03698	+	0.870129i	−1.98872	+	3.44456i	0.575007	+	3.26103i	3.10999	+	1.13194i	0.0336150	+	0.190640i	4.83459			−0.202743	+	1.14981i	6.20175	−	5.20389i
16.2	−0.603099	−	1.04460i	−1.82067	+	1.52773i	0.272543	−	0.472059i	−0.272178	−	1.54360i	2.69391	+	0.980502i	−0.703227	−	3.98820i	−3.06988			0.459961	−	2.60857i	−1.44829	+	1.21526i
16.3	−0.529087	−	0.916406i	1.30845	−	1.09792i	0.440133	−	0.762333i	0.109661	+	0.621921i	−1.69843	−	0.618177i	0.171837	+	0.974534i	−3.04783			−0.0143299	+	0.0812691i	0.511912	−	0.429545i
16.4	0.219432	+	0.380068i	−0.0615666	+	0.0516605i	0.903699	−	1.56525i	−0.640473	−	3.63230i	−0.0331442	−	0.0120635i	0.228918	+	1.29826i	1.67093			−0.519823	+	2.94806i	1.23998	−	1.04047i
16.5	0.503663	+	0.872370i	−0.352094	+	0.295442i	0.492647	−	0.853289i	0.517671	+	2.93586i	−0.435072	−	0.158353i	−0.307807	−	1.74566i	3.00717			−0.484260	+	2.74638i	−2.30042	+	1.93028i
16.6	1.05944	+	1.83500i	−1.78471	+	1.49755i	−1.24481	+	2.15608i	−0.222139	−	1.25981i	−4.63879	−	1.68838i	0.247600	+	1.40421i	−1.03746			0.421589	−	2.39095i	2.07641	−	1.74232i
16.7	1.33814	+	2.31772i	1.04183	−	0.874202i	−2.58123	+	4.47082i	−0.127857	−	0.725111i	3.42028	+	1.24488i	−0.876671	−	4.97185i	−8.46360			−0.199757	+	1.13288i	1.50952	−	1.26663i
27.1	−1.13355	−	1.96336i	1.99711	+	0.726890i	−1.56986	+	2.71908i	0.184738	−	0.155013i	−0.836675	−	4.74502i	2.60110	−	2.18258i	2.58386			1.16196	+	0.974999i	−0.513756	−	0.186992i
27.2	−0.754385	−	1.30663i	−0.821044	−	0.298836i	−0.138193	+	0.239358i	1.83302	−	1.53808i	0.228915	+	1.29824i	−0.324966	+	0.272679i	−2.60054			−1.71332	−	1.43765i	−3.39251	−	1.23477i
27.3	−0.641080	−	1.11038i	−2.04885	−	0.745722i	0.178032	−	0.308361i	−2.94251	+	2.46906i	0.485442	+	2.75308i	0.512859	−	0.430340i	−3.02085			1.34357	+	1.12739i	4.62799	+	1.68445i
27.4	−0.304565	−	0.527522i	1.79458	+	0.653175i	0.814480	−	1.41072i	0.460326	−	0.386260i	−0.202003	−	1.14562i	−2.95565	+	2.48009i	−2.21051			0.495757	+	0.415990i	−0.343960	−	0.125191i
27.5	0.368017	+	0.637423i	1.41819	+	0.516179i	0.729128	−	1.26289i	−2.11121	+	1.77151i	0.192893	+	1.09395i	1.03302	−	0.866806i	2.54539			−0.553313	−	0.464285i	−1.90616	−	0.693787i
27.6	0.384791	+	0.666477i	−2.14187	−	0.779577i	0.703872	−	1.21914i	1.33144	−	1.11721i	−0.304602	−	1.72748i	0.847872	−	0.711449i	2.62254			1.68174	+	1.41114i	1.25692	+	0.457482i
27.7	1.14108	+	1.97641i	−0.0847756	−	0.0308558i	−1.60412	+	2.77841i	0.0705498	−	0.0591983i	−0.0357520	−	0.202760i	−0.100894	+	0.0846602i	−2.75738			−2.29190	−	1.92313i	0.197503	+	0.0718851i
38.1	−1.09927	+	1.90400i	−0.271232	+	1.53823i	−1.41681	−	2.45399i	−1.85814	+	0.676309i	−2.63064	−	2.20737i	−0.0942741	+	0.0343130i	1.83275			0.526478	+	0.191622i	0.754918	−	4.28135i
38.2	−0.892046	+	1.54507i	0.172558	−	0.978624i	−0.591492	−	1.02449i	3.13823	−	1.14222i	1.35811	+	1.13959i	−0.0641258	+	0.0233399i	−1.45763			1.89115	+	0.688322i	−1.03463	+	5.86769i
38.3	−0.253598	+	0.439245i	0.420069	−	2.38233i	0.871376	+	1.50927i	−1.20157	+	0.437336i	0.939898	+	0.788668i	4.19577	−	1.52714i	−1.89831			−2.67995	−	0.975422i	0.112619	−	0.638692i
38.4	−0.0775572	+	0.134333i	−0.124689	+	0.707146i	0.987970	+	1.71121i	−0.326444	+	0.118816i	−0.0853226	−	0.0715941i	−1.80571	+	0.657225i	−0.616725			2.33457	+	0.849714i	0.00935720	−	0.0530673i
38.5	0.682493	−	1.18211i	−0.498285	+	2.82591i	0.0684068	+	0.118484i	−3.58556	+	1.30504i	3.00047	+	2.51769i	4.26974	−	1.55406i	2.91672			−4.91842	−	1.79016i	−0.904418	+	5.12921i
38.6	0.809610	−	1.40229i	0.309749	−	1.75667i	−0.310937	−	0.538559i	−1.08704	+	0.395650i	−2.21258	−	1.85658i	−1.51205	+	0.550341i	2.23149			−0.170882	−	0.0621961i	−0.325263	+	1.84466i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
109.f	even	9	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	109.2.f.a	✓	42
3.b	odd	2	1		981.2.w.a		42
109.f	even	9	1	inner	109.2.f.a	✓	42
327.o	odd	18	1		981.2.w.a		42

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
109.2.f.a	✓	42	1.a	even	1	1	trivial
109.2.f.a	✓	42	109.f	even	9	1	inner
981.2.w.a		42	3.b	odd	2	1
981.2.w.a		42	327.o	odd	18	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(109, [\chi])\).