Newform orbit 109.6.h.a

• Label: 109.6.h.a
• Level: $109$
• Weight: $6$
• Character orbit: 109.h
• Analytic conductor: $17.482$
• Analytic rank: $0$
• Dimension: $276$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 109 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	109.h (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(17.4818363596\)
Analytic rank:	\(0\)
Dimension:	\(276\)
Relative dimension:	\(46\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 276 q - 9 q^{2} - 6 q^{3} + 2301 q^{4} - 6 q^{5} - 492 q^{6} + 135 q^{7} - 816 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
4.1	−9.67322	−	5.58484i	−14.1005	+	5.13216i	46.3808	+	80.3339i	−28.1162	−	23.5923i	165.059	+	29.1044i	76.0067	+	63.7772i		−	678.688i	−13.6640	+	11.4655i	140.215	+	385.238i
4.2	−9.66800	−	5.58182i	24.0359	−	8.74834i	46.3135	+	80.2173i	76.3006	+	64.0238i	−281.211	−	49.5850i	73.8353	+	61.9552i		−	676.818i	315.041	−	264.351i	−380.305	−	1044.88i
4.3	−8.68507	−	5.01433i	19.5590	−	7.11891i	34.2870	+	59.3868i	−65.9040	−	55.3000i	−205.568	−	36.2472i	−50.5658	−	42.4297i		−	366.788i	145.728	−	122.280i	295.089	+	810.749i
4.4	−8.33503	−	4.81223i	−25.4622	+	9.26747i	30.3152	+	52.5074i	46.2011	+	38.7674i	256.825	+	45.2852i	−173.932	−	145.946i		−	275.551i	376.287	−	315.742i	−198.530	−	545.458i
4.5	−8.22197	−	4.74695i	2.60615	−	0.948561i	29.0672	+	50.3458i	33.9328	+	28.4730i	−25.9304	−	4.57224i	−136.533	−	114.564i		−	248.117i	−180.257	+	151.253i	−143.834	−	395.182i
4.6	−8.21785	−	4.74458i	2.06000	−	0.749780i	29.0220	+	50.2676i	−15.7276	−	13.1970i	−20.4862	−	3.61227i	10.1004	+	8.47528i		−	247.136i	−182.467	+	153.108i	66.6327	+	183.072i
4.7	−7.55320	−	4.36084i	−18.5150	+	6.73892i	22.0339	+	38.1639i	57.9544	+	48.6295i	169.235	+	29.8407i	176.819	+	148.369i		−	105.252i	111.245	−	93.3452i	−225.675	−	620.038i
4.8	−7.37034	−	4.25527i	−17.2788	+	6.28898i	20.2146	+	35.0127i	−63.7845	−	53.5215i	154.112	+	27.1741i	−26.3747	−	22.1310i		−	71.7372i	72.8581	−	61.1352i	242.365	+	665.892i
4.9	−6.91174	−	3.99049i	7.31651	−	2.66299i	15.8481	+	27.4496i	27.1232	+	22.7590i	−61.1964	−	10.7906i	87.0783	+	73.0674i			2.42534i	−139.709	+	117.230i	−96.6484	−	265.539i
4.10	−6.08228	−	3.51161i	22.7954	−	8.29686i	8.66276	+	15.0043i	−13.2029	−	11.0786i	−167.784	−	29.5848i	116.137	+	97.4509i			103.062i	264.645	−	222.064i	41.4003	+	113.746i
4.11	−6.04333	−	3.48912i	−22.1283	+	8.05404i	8.34787	+	14.4589i	−30.1866	−	25.3295i	161.830	+	28.5350i	−24.6077	−	20.6483i			106.797i	238.645	−	200.247i	94.0496	+	258.399i
4.12	−5.59873	−	3.23243i	23.6350	−	8.60242i	4.89722	+	8.48223i	−6.07573	−	5.09814i	−160.133	−	28.2357i	−130.600	−	109.587i			143.556i	298.460	−	250.438i	17.5370	+	48.1825i
4.13	−4.62801	−	2.67198i	−5.71249	+	2.07918i	−1.72101	−	2.98088i	71.8974	+	60.3291i	31.9930	+	5.64123i	−23.4691	−	19.6929i			189.401i	−157.839	+	132.443i	−171.544	−	471.312i
4.14	−4.53843	−	2.62026i	0.160695	−	0.0584881i	−2.26846	−	3.92908i	−66.5474	−	55.8399i	−0.882556	−	0.155618i	127.339	+	106.850i			191.473i	−186.126	+	156.179i	155.705	+	427.797i
4.15	−4.26230	−	2.46084i	22.2522	−	8.09913i	−3.88853	−	6.73514i	67.3651	+	56.5260i	−114.776	−	20.2381i	−96.2567	−	80.7690i			195.770i	243.415	−	204.249i	−148.029	−	406.706i
4.16	−3.91168	−	2.25841i	−13.3726	+	4.86724i	−5.79920	−	10.0445i	11.4629	+	9.61849i	63.3016	+	11.1618i	−60.9141	−	51.1130i			196.926i	−31.0115	+	26.0217i	−23.1166	−	63.5123i
4.17	−3.87464	−	2.23702i	3.73035	−	1.35774i	−5.99147	−	10.3775i	−55.4057	−	46.4909i	−17.4910	−	3.08414i	−179.798	−	150.869i			196.782i	−174.077	+	146.068i	110.676	+	304.079i
4.18	−2.58640	−	1.49326i	13.5604	−	4.93558i	−11.5404	−	19.9885i	16.5973	+	13.9268i	−42.4427	−	7.48380i	86.1224	+	72.2652i			164.500i	−26.6242	+	22.3403i	−22.1310	−	60.8043i
4.19	−2.28477	−	1.31911i	−28.1871	+	10.2593i	−12.5199	−	21.6851i	−23.0474	−	19.3391i	77.9342	+	13.7419i	146.454	+	122.889i			150.484i	503.111	−	422.160i	27.1477	+	74.5876i
4.20	−2.06981	−	1.19501i	−13.1714	+	4.79398i	−13.1439	−	22.7659i	23.8855	+	20.0423i	32.9910	+	5.81721i	−2.75672	−	2.31316i			139.309i	−35.6464	+	29.9109i	−25.4878	−	70.0271i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
109.h	even	18	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	109.6.h.a	✓	276
109.h	even	18	1	inner	109.6.h.a	✓	276

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
109.6.h.a	✓	276	1.a	even	1	1	trivial
109.6.h.a	✓	276	109.h	even	18	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(109, [\chi])\).