Newform orbit 109.8.b.a

• Label: 109.8.b.a
• Level: $109$
• Weight: $8$
• Character orbit: 109.b
• Analytic conductor: $34.050$
• Analytic rank: $0$
• Dimension: $62$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 109 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	109.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(34.0499677778\)
Analytic rank:	\(0\)
Dimension:	\(62\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 62 q - 56 q^{3} - 3584 q^{4} + 194 q^{5} + 370 q^{7} + 40462 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
108.1		−	22.1186i	68.3096			−361.232			198.755				−	1510.91i	477.809					5158.77i	2479.21				−	4396.19i
108.2		−	21.7464i	−13.5748			−344.905			−10.6141					295.203i	−628.336					4716.91i	−2002.72					230.818i
108.3		−	20.6762i	−63.1733			−299.506			−321.718					1306.18i	710.246					3546.09i	1803.86					6651.91i
108.4		−	20.5650i	−83.1155			−294.921			534.705					1709.27i	−279.478					3432.74i	4721.18				−	10996.2i
108.5		−	19.1107i	70.4891			−237.218			−288.848				−	1347.09i	−109.637					2087.22i	2781.72					5520.08i
108.6		−	18.7221i	−3.68685			−222.515			339.279					69.0254i	552.193					1769.52i	−2173.41				−	6351.99i
108.7		−	18.2217i	−4.72274			−204.029			113.191					86.0562i	1550.44					1385.38i	−2164.70				−	2062.52i
108.8		−	18.1795i	9.33049			−202.493			−443.908				−	169.623i	−988.125					1354.25i	−2099.94					8070.01i
108.9		−	17.9931i	45.7905			−195.751			340.028				−	823.912i	−1712.93					1219.06i	−90.2316				−	6118.15i
108.10		−	17.5675i	39.7962			−180.616			−277.314				−	699.119i	1237.74					924.338i	−603.263					4871.71i
108.11		−	15.8282i	−77.0878			−122.532			−187.933					1220.16i	−1259.33				−	86.5528i	3755.53					2974.64i
108.12		−	15.6070i	−36.0261			−115.577			219.020					562.257i	−1106.83				−	193.886i	−889.123				−	3418.23i
108.13		−	15.1160i	−59.8207			−100.494			62.1458					904.251i	706.973				−	415.786i	1391.52				−	939.396i
108.14		−	14.2491i	65.6723			−75.0363			367.355				−	935.771i	501.747				−	754.684i	2125.86				−	5234.48i
108.15		−	12.0377i	26.3124			−16.9068			54.1389				−	316.741i	−225.072				−	1337.31i	−1494.66				−	651.710i
108.16		−	12.0291i	−31.0591			−16.6993			−170.363					373.613i	−271.091				−	1338.85i	−1222.34					2049.32i
108.17		−	11.2548i	88.3366			1.33012			−147.950				−	994.209i	−1272.73				−	1455.58i	5616.36					1665.14i
108.18		−	10.9727i	−20.6182			7.60069			−529.822					226.236i	1447.54				−	1487.90i	−1761.89					5813.56i
108.19		−	9.83474i	28.1069			31.2778			−189.209				−	276.424i	−141.499				−	1566.46i	−1397.00					1860.82i
108.20		−	8.61539i	−73.6630			53.7750			305.812					634.636i	1462.18				−	1566.06i	3239.24				−	2634.69i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
109.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	109.8.b.a	✓	62
109.b	even	2	1	inner	109.8.b.a	✓	62

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
109.8.b.a	✓	62	1.a	even	1	1	trivial
109.8.b.a	✓	62	109.b	even	2	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(109, [\chi])\).