Newform orbit 109.6.i.a

• Label: 109.6.i.a
• Level: $109$
• Weight: $6$
• Character orbit: 109.i
• Analytic conductor: $17.482$
• Analytic rank: $0$
• Dimension: $792$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 109 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	109.i (of order \(27\), degree \(18\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(17.4818363596\)
Analytic rank:	\(0\)
Dimension:	\(792\)
Relative dimension:	\(44\) over \(\Q(\zeta_{27})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{27}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 792 q - 18 q^{2} - 18 q^{3} - 18 q^{4} - 18 q^{5} - 18 q^{6} - 18 q^{7} - 1737 q^{8} - 1728 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
3.1	−1.87446	−	10.6306i	22.3842	+	2.61633i	−79.4258	+	28.9086i	−9.30636	+	31.0854i	−14.1451	−	242.861i	55.4909	+	58.8169i	283.483	+	491.007i	257.755	+	61.0892i	347.901	+	40.6638i
3.2	−1.82497	−	10.3499i	−1.91960	−	0.224369i	−73.7198	+	26.8318i	13.8877	−	46.3880i	1.18101	+	20.2772i	−158.261	−	167.747i	244.090	+	422.776i	−232.815	−	55.1783i	−505.456	−	59.0793i
3.3	−1.72006	−	9.75495i	−14.8828	−	1.73955i	−62.1302	+	22.6136i	−25.4339	+	84.9551i	8.63011	+	148.173i	−7.90448	−	8.37826i	168.975	+	292.673i	−17.9777	−	4.26080i	872.481	+	101.978i
3.4	−1.71999	−	9.75457i	−28.1159	−	3.28627i	−62.1231	+	22.6110i	15.6213	−	52.1788i	16.3029	+	279.911i	18.7184	+	19.8403i	168.931	+	292.597i	543.253	+	128.753i	−535.850	−	62.6319i
3.5	−1.68921	−	9.57996i	−12.2191	−	1.42821i	−58.8520	+	21.4204i	−5.53672	+	18.4939i	6.95840	+	119.471i	90.8096	+	96.2526i	148.976	+	258.034i	−89.1830	−	21.1367i	186.524	+	21.8015i
3.6	−1.66943	−	9.46783i	6.20745	+	0.725547i	−56.7827	+	20.6672i	17.3815	−	58.0581i	−3.49358	−	59.9824i	103.641	+	109.853i	136.647	+	236.679i	−198.444	−	47.0321i	−578.702	−	67.6406i
3.7	−1.40469	−	7.96641i	15.5937	+	1.82264i	−31.4204	+	11.4361i	−18.3246	+	61.2085i	−7.38445	−	126.786i	−57.5141	−	60.9614i	5.81182	+	10.0664i	3.39160	+	0.803824i	513.353	+	60.0024i
3.8	−1.40212	−	7.95183i	25.0596	+	2.92905i	−31.1955	+	11.3542i	19.8808	−	66.4065i	−11.8453	−	203.376i	−103.436	−	109.635i	4.83486	+	8.37423i	382.954	+	90.7617i	−555.929	−	64.9787i
3.9	−1.22102	−	6.92475i	2.60928	+	0.304981i	−16.3912	+	5.96590i	19.8934	−	66.4485i	−1.07407	−	18.4410i	62.8824	+	66.6514i	−51.1789	−	88.6445i	−229.735	−	54.4481i	−484.430	−	56.6217i
3.10	−1.11198	−	6.30634i	−12.9401	−	1.51248i	−8.46332	+	3.08040i	−6.83567	+	22.8327i	4.85087	+	83.2863i	−149.023	−	157.955i	−73.6210	−	127.515i	−71.2925	−	16.8966i	151.592	+	17.7186i
3.11	−1.10094	−	6.24374i	3.90639	+	0.456591i	−7.70200	+	2.80330i	−12.5391	+	41.8834i	−1.44986	−	24.8931i	67.2178	+	71.2467i	−75.4584	−	130.698i	−221.399	−	52.4724i	275.313	+	32.1795i
3.12	−1.07281	−	6.08420i	−18.2635	−	2.13470i	−5.79637	+	2.10971i	13.4136	−	44.8047i	6.60532	+	113.409i	−5.25404	−	5.56896i	−79.7946	−	138.208i	92.5496	+	21.9347i	−286.991	−	33.5444i
3.13	−0.941184	−	5.33772i	28.2561	+	3.30267i	2.46474	−	0.897092i	5.60491	−	18.7217i	−8.96550	−	153.932i	132.746	+	140.702i	−93.8292	−	162.517i	551.052	+	130.602i	−105.206	−	12.2969i
3.14	−0.819322	−	4.64661i	−27.3491	−	3.19665i	9.15049	−	3.33051i	−18.8279	+	62.8894i	7.55414	+	129.700i	38.0638	+	40.3452i	−98.4654	−	170.547i	501.304	+	118.811i	307.649	+	35.9590i
3.15	−0.617035	−	3.49938i	16.2399	+	1.89817i	18.2052	−	6.62616i	3.83500	−	12.8098i	−3.37816	−	58.0007i	−75.6228	−	80.1555i	−91.2746	−	158.092i	23.6804	+	5.61235i	−47.1927	−	5.51604i
3.16	−0.609714	−	3.45786i	−1.91042	−	0.223296i	18.4851	−	6.72803i	−20.6855	+	69.0945i	0.392683	+	6.74211i	173.394	+	183.787i	−90.7145	−	157.122i	−232.850	−	55.1865i	251.531	+	29.3998i
3.17	−0.530173	−	3.00676i	4.65466	+	0.544051i	21.3106	−	7.75643i	29.1758	−	97.4538i	−0.831942	−	14.2839i	−35.3410	−	37.4593i	−83.4704	−	144.575i	−215.080	−	50.9749i	−308.489	−	36.0572i
3.18	−0.266168	−	1.50952i	−19.3373	−	2.26020i	27.8624	−	10.1411i	18.0137	−	60.1699i	1.73516	+	29.7915i	150.169	+	159.170i	−47.2491	−	81.8378i	132.372	+	31.3728i	−95.6222	−	11.1766i
3.19	−0.260977	−	1.48007i	28.4239	+	3.32227i	27.9477	−	10.1721i	−30.2452	+	101.026i	−2.50076	−	42.9364i	−50.8447	−	53.8923i	−46.3956	−	80.3596i	560.429	+	132.824i	157.419	+	18.3997i
3.20	−0.256311	−	1.45361i	0.00220241		0.000257425i	28.0229	−	10.1995i	−26.6246	+	88.9323i	−0.000190306	−	0.00326743i	−97.8403	−	103.705i	−45.6252	−	79.0252i	−236.450	−	56.0397i	136.097	+	15.9075i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
109.i	even	27	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	109.6.i.a	✓	792
109.i	even	27	1	inner	109.6.i.a	✓	792

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
109.6.i.a	✓	792	1.a	even	1	1	trivial
109.6.i.a	✓	792	109.i	even	27	1	inner

Hecke kernels

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