Newform orbit 109.6.e.a

• Label: 109.6.e.a
• Level: $109$
• Weight: $6$
• Character orbit: 109.e
• Analytic conductor: $17.482$
• Analytic rank: $0$
• Dimension: $92$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(17.4818363596\)
Analytic rank:	\(0\)
Dimension:	\(92\)
Relative dimension:	\(46\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 92 q - 19 q^{3} - 1542 q^{4} + 47 q^{5} + 483 q^{6} - 146 q^{7} - 4189 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
46.1		−	11.2456i	−3.20136	−	5.54491i	−94.4635			20.5119	+	35.5277i	−62.3559	+	36.0012i	31.0525	+	53.7846i			702.439i	101.003	−	174.942i	399.531	−	230.669i
46.2		−	10.7928i	5.98861	+	10.3726i	−84.4849			−51.0305	−	88.3873i	111.949	−	64.6340i	−36.3646	−	62.9853i			566.460i	49.7731	−	86.2095i	−953.948	+	550.762i
46.3		−	10.2603i	14.2834	+	24.7396i	−73.2733			46.6938	+	80.8760i	253.835	−	146.552i	−69.0928	−	119.672i			423.476i	−286.532	+	496.288i	829.810	−	479.091i
46.4		−	9.95700i	−12.5404	−	21.7205i	−67.1418			1.44778	+	2.50763i	−216.271	+	124.864i	−111.937	−	193.880i			349.906i	−193.021	+	334.322i	24.9685	−	14.4156i
46.5		−	9.41803i	−10.3812	−	17.9808i	−56.6993			−32.4674	−	56.2353i	−169.343	+	97.7704i	64.8316	+	112.292i			232.618i	−94.0385	+	162.879i	−529.625	+	305.779i
46.6		−	9.02758i	7.53490	+	13.0508i	−49.4972			−0.662501	−	1.14749i	117.817	−	68.0219i	−7.56298	−	13.0995i			157.957i	7.95048	−	13.7706i	−10.3590	+	5.98078i
46.7		−	9.02122i	1.22169	+	2.11604i	−49.3824			26.0046	+	45.0413i	19.0892	−	11.0212i	57.2040	+	99.0803i			156.811i	118.515	−	205.274i	406.327	−	234.593i
46.8		−	8.29750i	13.5064	+	23.3937i	−36.8485			−22.4201	−	38.8328i	194.109	−	112.069i	118.842	+	205.841i			40.2300i	−243.343	+	421.482i	−322.215	+	186.031i
46.9		−	8.06610i	−2.54613	−	4.41003i	−33.0619			26.9552	+	46.6877i	−35.5717	+	20.5373i	−119.424	−	206.848i			8.56541i	108.534	−	187.987i	376.588	−	217.423i
46.10		−	7.96258i	−10.7438	−	18.6089i	−31.4027			38.1279	+	66.0395i	−148.174	+	85.5486i	68.5157	+	118.673i		−	4.75619i	−109.360	+	189.416i	525.845	−	303.597i
46.11		−	7.58084i	−1.27535	−	2.20896i	−25.4692			−17.7106	−	30.6756i	−16.7458	+	9.66819i	−2.49714	−	4.32517i		−	49.5093i	118.247	−	204.810i	−232.547	+	134.261i
46.12		−	5.79452i	−4.24708	−	7.35615i	−1.57648			−46.2644	−	80.1323i	−42.6254	+	24.6098i	−82.4845	−	142.867i		−	176.290i	85.4247	−	147.960i	−464.329	+	268.080i
46.13		−	5.30418i	11.8475	+	20.5204i	3.86565			−19.1116	−	33.1023i	108.844	−	62.8411i	−80.1154	−	138.764i		−	190.238i	−159.225	+	275.785i	−175.581	+	101.372i
46.14		−	4.86704i	−13.7996	−	23.9017i	8.31190			−11.2747	−	19.5284i	−116.330	+	67.1634i	6.11704	+	10.5950i		−	196.200i	−259.359	+	449.223i	−95.0454	+	54.8745i
46.15		−	4.79563i	7.31806	+	12.6753i	9.00192			55.2069	+	95.6211i	60.7858	−	35.0947i	56.8545	+	98.4749i		−	196.630i	14.3920	−	24.9277i	458.563	−	264.752i
46.16		−	4.41142i	−8.72775	−	15.1169i	12.5394			34.0251	+	58.9331i	−66.6869	+	38.5017i	−4.41612	−	7.64894i		−	196.482i	−30.8471	+	53.4288i	259.979	−	150.099i
46.17		−	4.28930i	7.86712	+	13.6263i	13.6019			19.5205	+	33.8105i	58.4471	−	33.7445i	4.55474	+	7.88904i		−	195.600i	−2.28328	+	3.95476i	145.024	−	83.7294i
46.18		−	4.13796i	−1.17636	−	2.03751i	14.8773			−27.7657	−	48.0917i	−8.43114	+	4.86772i	120.211	+	208.211i		−	193.976i	118.732	−	205.650i	−199.001	+	114.894i
46.19		−	1.82493i	5.23473	+	9.06682i	28.6696			−34.1398	−	59.1319i	16.5463	−	9.55302i	−8.85047	−	15.3295i		−	110.718i	66.6952	−	115.519i	−107.912	+	62.3028i
46.20		−	1.15541i	−6.37574	−	11.0431i	30.6650			11.8192	+	20.4715i	−12.7593	+	7.36661i	−22.3237	−	38.6657i		−	72.4040i	40.1999	−	69.6283i	23.6530	−	13.6561i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
109.e	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	109.6.e.a	✓	92
109.e	even	6	1	inner	109.6.e.a	✓	92

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
109.6.e.a	✓	92	1.a	even	1	1	trivial
109.6.e.a	✓	92	109.e	even	6	1	inner

Hecke kernels

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