Newform orbit 109.5.l.a

• Label: 109.5.l.a
• Level: $109$
• Weight: $5$
• Character orbit: 109.l
• Analytic conductor: $11.267$
• Analytic rank: $0$
• Dimension: $1296$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 109 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	109.l (of order \(108\), degree \(36\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.2673259761\)
Analytic rank:	\(0\)
Dimension:	\(1296\)
Relative dimension:	\(36\) over \(\Q(\zeta_{108})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{108}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1296 q - 36 q^{2} - 36 q^{3} - 36 q^{4} - 36 q^{5} - 36 q^{6} - 36 q^{7} - 36 q^{8} - 36 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
6.1	−0.637911	+	7.29136i	−0.402324	−	6.90764i	−37.0000	−	6.52410i	−17.7911	+	23.8976i	50.6227	+	1.47297i	−16.6530	−	38.6060i	40.8627	−	152.502i	32.8987	−	3.84531i	−162.897	−	144.966i
6.2	−0.630326	+	7.20466i	−0.482689	−	8.28746i	−35.7529	−	6.30420i	6.96253	−	9.35229i	60.0126	+	1.74619i	27.6433	+	64.0845i	38.0063	−	141.842i	12.0033	−	1.40299i	62.9914	+	56.0576i
6.3	−0.622749	+	7.11805i	0.261686	+	4.49297i	−34.5219	−	6.08714i	26.0366	−	34.9732i	−32.1441	−	0.935299i	−11.5555	−	26.7886i	35.2379	−	131.509i	60.3340	−	7.05204i	232.727	+	207.109i
6.4	−0.598337	+	6.83902i	0.789023	+	13.5470i	−30.6572	−	5.40570i	−3.64489	+	4.89593i	−93.1203	−	2.70952i	14.5904	+	33.8243i	26.8837	−	100.331i	−102.447	+	11.9743i	−31.3025	−	27.8569i
6.5	−0.542911	+	6.20550i	0.562043	+	9.64990i	−22.4566	−	3.95970i	−13.6215	+	18.2968i	−60.1876	−	1.75128i	−32.0505	−	74.3014i	10.9680	−	40.9333i	−12.3523	+	1.44378i	−106.146	−	94.4617i
6.6	−0.520352	+	5.94765i	−0.895870	−	15.3815i	−19.3468	−	3.41137i	12.8834	−	17.3055i	91.9499	+	2.67547i	−9.73319	−	22.5641i	5.63289	−	21.0222i	−155.335	+	18.1561i	96.2229	+	85.6311i
6.7	−0.449390	+	5.13656i	0.0902502	+	1.54954i	−10.4253	−	1.83827i	−20.8251	+	27.9729i	−7.99984	−	0.232772i	27.9300	+	64.7490i	−7.22485	+	26.9635i	78.0594	−	9.12383i	−134.326	−	119.540i
6.8	−0.427952	+	4.89151i	0.0332947	+	0.571647i	−7.98680	−	1.40829i	6.99992	−	9.40253i	−2.81047	−	0.0817762i	3.95081	+	9.15902i	−10.0270	+	37.4213i	80.1266	−	9.36546i	42.9969	+	38.2640i
6.9	−0.358573	+	4.09850i	−0.740576	−	12.7152i	−0.912226	−	0.160850i	−3.73687	+	5.01949i	52.3788	+	1.52407i	5.35317	+	12.4100i	−16.0508	+	59.9025i	−80.6756	+	9.42963i	−19.2325	−	17.1154i
6.10	−0.325183	+	3.71686i	0.0491095	+	0.843177i	2.04763	+	0.361053i	10.4717	−	14.0659i	−3.14994	−	0.0916539i	−17.7268	−	41.0953i	−17.4585	+	65.1561i	79.7438	−	9.32071i	48.8759	+	43.4959i
6.11	−0.319544	+	3.65241i	0.809775	+	13.9033i	2.51894	+	0.444157i	24.8938	−	33.4381i	−51.0393	−	1.48509i	29.5831	+	68.5813i	−17.6099	+	65.7212i	−112.194	+	13.1136i	114.175	+	101.607i
6.12	−0.278219	+	3.18006i	−0.640868	−	11.0033i	5.72158	+	1.00887i	−20.2639	+	27.2191i	35.1693	+	1.02332i	−17.1880	−	39.8462i	−18.0194	+	67.2491i	−40.2092	+	4.69979i	−80.9205	−	72.0131i
6.13	−0.266413	+	3.04511i	0.922705	+	15.8422i	6.55520	+	1.15586i	6.35995	−	8.54289i	−48.4872	−	1.41083i	−22.5101	−	52.1842i	−17.9244	+	66.8948i	−169.673	+	19.8319i	24.3197	+	21.6427i
6.14	−0.164552	+	1.88083i	0.703703	+	12.0821i	12.2465	+	2.15938i	−23.6317	+	31.7429i	−22.8402	−	0.664583i	0.456425	+	1.05811i	−13.8951	+	51.8572i	−65.0300	+	7.60091i	−55.8145	−	49.6707i
6.15	−0.154384	+	1.76462i	−0.587526	−	10.0874i	12.6669	+	2.23351i	22.6772	−	30.4608i	17.8912	+	0.520581i	27.8451	+	64.5522i	−13.2323	+	49.3835i	−20.9589	+	2.44974i	50.2507	+	44.7194i
6.16	−0.0585313	+	0.669015i	0.381109	+	6.54339i	15.3128	+	2.70005i	3.26346	−	4.38358i	−4.39994	−	0.128025i	16.2141	+	37.5884i	−5.48370	+	20.4654i	37.7815	−	4.41603i	2.74167	+	2.43988i
6.17	−0.0566615	+	0.647644i	−0.551870	−	9.47525i	15.3407	+	2.70498i	26.6959	−	35.8588i	6.16785	+	0.179466i	−25.9279	−	60.1075i	−5.31329	+	19.8295i	−9.02340	+	1.05468i	21.7111	+	19.3212i
6.18	−0.0446465	+	0.510312i	−0.697373	−	11.9734i	15.4985	+	2.73280i	−13.4136	+	18.0175i	6.14132	+	0.178694i	30.4713	+	70.6404i	−4.20786	+	15.7039i	−62.4245	+	7.29637i	−8.59569	−	7.64951i
6.19	0.0230272	−	0.263202i	0.0874998	+	1.50231i	15.6882	+	2.76625i	−6.58289	+	8.84236i	0.397426	+	0.0115639i	2.26605	+	5.25328i	2.18344	−	8.14872i	78.2030	−	9.14062i	2.17574	+	1.93624i
6.20	0.0373606	−	0.427033i	−0.204494	−	3.51102i	15.5760	+	2.74646i	−10.8300	+	14.5472i	−1.50696	−	0.0438481i	−33.3841	−	77.3932i	3.52990	−	13.1738i	68.1668	−	7.96756i	5.80753	+	5.16826i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
109.l	odd	108	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	109.5.l.a	✓	1296
109.l	odd	108	1	inner	109.5.l.a	✓	1296

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
109.5.l.a	✓	1296	1.a	even	1	1	trivial
109.5.l.a	✓	1296	109.l	odd	108	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(109, [\chi])\).