Newform orbit 109.4.k.a

• Label: 109.4.k.a
• Level: $109$
• Weight: $4$
• Character orbit: 109.k
• Analytic conductor: $6.431$
• Analytic rank: $0$
• Dimension: $468$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 109 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	109.k (of order \(54\), degree \(18\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.43120819063\)
Analytic rank:	\(0\)
Dimension:	\(468\)
Relative dimension:	\(26\) over \(\Q(\zeta_{54})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{54}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 468 q - 18 q^{2} - 18 q^{3} - 18 q^{4} - 18 q^{5} - 18 q^{6} - 18 q^{7} - 27 q^{8} - 144 q^{9}+O(q^{10}) \)

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	−1.83850	+	5.05125i	−5.21072	+	1.23496i	−16.0066	−	13.4312i	12.8303	−	8.43859i	3.34182	−	28.5911i	0.634985	+	10.9023i	60.0303	−	34.6585i	1.49837	−	0.752512i	19.0369	+	80.3231i
12.2	−1.72784	+	4.74721i	5.33150	−	1.26359i	−13.4222	−	11.2626i	−13.1655	+	8.65908i	−3.21348	+	27.4930i	0.612182	+	10.5108i	41.6568	−	24.0506i	2.70017	−	1.35607i	−18.3586	−	77.4608i
12.3	−1.61850	+	4.44679i	7.14128	−	1.69251i	−11.0261	−	9.25196i	10.7970	−	7.10131i	−4.03190	+	34.4951i	−1.31379	−	22.5569i	26.2017	−	15.1276i	24.0052	−	12.0559i	14.1031	+	59.5055i
12.4	−1.61205	+	4.42906i	−5.83374	+	1.38262i	−10.8895	−	9.13739i	−11.1887	+	7.35893i	3.28054	−	28.0668i	−0.417935	−	7.17567i	25.3697	−	14.6472i	7.99280	−	4.01413i	−14.5564	−	61.4184i
12.5	−1.48613	+	4.08312i	2.34584	−	0.555974i	−8.33491	−	6.99382i	6.16915	−	4.05751i	−1.21612	+	10.4046i	0.668207	+	11.4727i	10.8392	−	6.25801i	−18.9342	+	9.50913i	7.39913	+	31.2194i
12.6	−1.16044	+	3.18828i	−1.21799	+	0.288668i	−2.69016	−	2.25731i	−1.10360	+	0.725851i	0.493044	−	4.21826i	−1.96720	−	33.7756i	−13.1880	+	7.61408i	−22.7279	+	11.4144i	−1.03355	−	4.36090i
12.7	−0.972375	+	2.67158i	−2.89771	+	0.686769i	−0.0634628	−	0.0532516i	5.58382	−	3.67254i	0.982900	−	8.40925i	0.992739	+	17.0447i	−19.4931	+	11.2544i	−16.2030	+	8.13747i	4.38190	+	18.4887i
12.8	−0.919894	+	2.52739i	7.90497	−	1.87351i	0.586876	+	0.492448i	−1.94752	+	1.28091i	−2.53664	+	21.7024i	0.591496	+	10.1556i	−20.4185	+	11.7886i	34.8504	−	17.5025i	−1.44583	−	6.10044i
12.9	−0.865714	+	2.37853i	−9.86167	+	2.33726i	1.22042	+	1.02405i	13.7058	−	9.01444i	2.97814	−	25.4797i	−1.23873	−	21.2683i	−21.0288	+	12.1410i	67.6616	−	33.9810i	9.57581	+	40.4035i
12.10	−0.800440	+	2.19919i	−8.22189	+	1.94862i	1.93262	+	1.62166i	−10.6628	+	7.01303i	2.29573	−	19.6413i	1.76563	+	30.3146i	−21.3276	+	12.3135i	39.6742	−	19.9252i	−6.88807	−	29.0631i
12.11	−0.515665	+	1.41678i	2.50812	−	0.594435i	4.38700	+	3.68113i	−13.5824	+	8.93327i	−0.451167	+	3.85998i	−0.274535	−	4.71359i	−17.9233	+	10.3480i	−18.1908	+	9.13575i	−5.65250	−	23.8498i
12.12	−0.458114	+	1.25866i	6.76311	−	1.60289i	4.75401	+	3.98908i	11.1691	−	7.34605i	−1.08079	+	9.24674i	1.02238	+	17.5536i	−16.4786	+	9.51395i	19.0423	−	9.56339i	4.12943	+	17.4234i
12.13	0.0368186	−	0.101158i	−1.74498	+	0.413569i	6.11948	+	5.13485i	12.4845	−	8.21116i	−0.0224119	+	0.191746i	0.282524	+	4.85075i	1.49057	−	0.860579i	−21.2542	+	10.6742i	−0.370966	−	1.56523i
12.14	0.0754171	−	0.207207i	5.53781	−	1.31249i	6.09111	+	5.11105i	7.89713	−	5.19402i	0.145690	−	1.24646i	−1.88672	−	32.3938i	3.04612	−	1.75868i	4.81669	−	2.41903i	−0.480658	−	2.02806i
12.15	0.0790895	−	0.217297i	−5.16211	+	1.22344i	6.08739	+	5.10793i	−0.211176	+	0.138893i	−0.142419	+	1.21847i	−0.243151	−	4.17475i	3.19348	−	1.84376i	1.02245	−	0.513495i	0.0134791	+	0.0568729i
12.16	0.449662	−	1.23544i	−7.43960	+	1.76322i	4.80425	+	4.03124i	−10.8096	+	7.10957i	−1.16696	+	9.98400i	−1.18302	−	20.3117i	16.2493	−	9.38154i	28.1106	−	14.1177i	3.92276	+	16.5514i
12.17	0.577463	−	1.58657i	0.890597	−	0.211075i	3.94463	+	3.30994i	−7.73330	+	5.08627i	0.179402	−	1.53488i	1.73011	+	29.7049i	19.2268	−	11.1006i	−23.3795	+	11.7416i	3.60401	+	15.2065i
12.18	0.630920	−	1.73344i	8.84275	−	2.09577i	3.52161	+	2.95498i	−9.21252	+	6.05917i	1.94618	−	16.6506i	−0.0509433	−	0.874663i	20.1245	−	11.6189i	49.6739	−	24.9472i	4.69084	+	19.7922i
12.19	1.10418	−	3.03372i	1.94510	−	0.460997i	−1.85587	−	1.55726i	1.32591	−	0.872067i	0.749211	−	6.40991i	−1.07172	−	18.4008i	15.5936	−	9.00297i	−20.5572	+	10.3242i	−1.18156	−	4.98537i
12.20	1.13003	−	3.10474i	4.99556	−	1.18397i	−2.23408	−	1.87462i	13.5612	−	8.91935i	1.96923	−	16.8478i	1.21203	+	20.8098i	14.5460	−	8.39812i	−0.574265	+	0.288406i	−12.3676	−	52.1832i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
109.k	even	54	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	109.4.k.a	✓	468
109.k	even	54	1	inner	109.4.k.a	✓	468

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
109.4.k.a	✓	468	1.a	even	1	1	trivial
109.4.k.a	✓	468	109.k	even	54	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(109, [\chi])\).