Newform orbit 169.8.b.f

• Label: 169.8.b.f
• Level: $169$
• Weight: $8$
• Character orbit: 169.b
• Analytic conductor: $52.793$
• Analytic rank: $0$
• Dimension: $42$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 169 = 13^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	169.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(52.7930693068\)
Analytic rank:	\(0\)
Dimension:	\(42\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 42 q - 52 q^{3} - 2818 q^{4} + 30930 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
168.1		−	21.6353i	79.0768			−340.087					120.676i		−	1710.85i			240.114i			4588.56i	4066.15			2610.86
168.2		−	21.5931i	28.3750			−338.261				−	473.706i		−	612.705i		−	99.1966i			4540.19i	−1381.86			−10228.8
168.3		−	20.8456i	28.0386			−306.538					273.942i		−	584.480i		−	1317.98i			3721.73i	−1400.84			5710.48
168.4		−	20.4423i	−14.6497			−289.887					478.639i			299.473i			1197.76i			3309.33i	−1972.39			9784.47
168.5		−	19.6237i	−72.4635			−257.091					413.760i			1422.00i		−	1005.88i			2533.24i	3063.96			8119.51
168.6		−	17.7683i	58.1116			−187.714				−	15.5333i		−	1032.55i			1688.45i			1061.02i	1189.96			−276.002
168.7		−	16.7260i	−13.7721			−151.758					0.583029i			230.352i			274.333i			397.369i	−1997.33			9.75172
168.8		−	15.3948i	−46.3939			−108.998				−	391.457i			714.223i			1742.39i		−	292.525i	−34.6060			−6026.39
168.9		−	15.0231i	−85.3884			−97.6921				−	439.379i			1282.79i		−	536.518i		−	455.317i	5104.17			−6600.82
168.10		−	13.2906i	38.7978			−48.6390				−	274.097i		−	515.644i		−	774.181i		−	1054.75i	−681.732			−3642.90
168.11		−	13.1620i	−21.4559			−45.2387					344.184i			282.403i		−	168.957i		−	1089.31i	−1726.64			4530.16
168.12		−	11.8381i	−26.1495			−12.1398					422.131i			309.560i		−	1215.99i		−	1371.56i	−1503.20			4997.21
168.13		−	10.5911i	−6.51000			15.8290				−	397.099i			68.9480i		−	1187.35i		−	1523.30i	−2144.62			−4205.71
168.14		−	8.99747i	87.5208			47.0455					190.033i		−	787.466i		−	44.6495i		−	1574.97i	5472.89			1709.81
168.15		−	8.58294i	18.1259			54.3331				−	37.6136i		−	155.574i			711.860i		−	1564.95i	−1858.45			−322.835
168.16		−	6.93821i	−64.5145			79.8613				−	33.5113i			447.615i		−	160.658i		−	1442.18i	1975.12			−232.508
168.17		−	5.77104i	82.1608			94.6950					17.6329i		−	474.154i			888.558i		−	1285.18i	4563.40			101.761
168.18		−	3.19657i	−57.0706			117.782				−	431.499i			182.430i			225.139i		−	785.660i	1070.05			−1379.32
168.19		−	2.91821i	44.2582			119.484				−	197.474i		−	129.155i		−	303.075i		−	722.210i	−228.212			−576.269
168.20		−	2.90053i	−90.5117			119.587					240.795i			262.532i		−	1391.07i		−	718.132i	6005.38			698.433

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	169.8.b.f		42
13.b	even	2	1	inner	169.8.b.f		42
13.d	odd	4	1		169.8.a.h	✓	21
13.d	odd	4	1		169.8.a.i	yes	21

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
169.8.a.h	✓	21	13.d	odd	4	1
169.8.a.i	yes	21	13.d	odd	4	1
169.8.b.f		42	1.a	even	1	1	trivial
169.8.b.f		42	13.b	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T2^42 + 4097*T2^40 + 7721510*T2^38 + 8881814775*T2^36 + 6976509369092*T2^34 + 3967218833649380*T2^32 + 1689625628300670281*T2^30 + 550050352926933121992*T2^28 + 138485512806700789887728*T2^26 + 27111641418548076530727104*T2^24 + 4128654084130308825108474624*T2^22 + 486978369968133670285292699648*T2^20 + 44094385469564453530498432835584*T2^18 + 3022114546175387042223041680539648*T2^16 + 153635941635835735718098277901205504*T2^14 + 5633029727984540306807107661218709504*T2^12 + 143375062019242848599689452167046889472*T2^10 + 2407076775630003845466367237618915082240*T2^8 + 24948473912477922736292742472051627917312*T2^6 + 146787241851159322838507815770944201293824*T2^4 + 420509973853780354680958257104887690035200*T2^2 + 381973312096218670576242075252113092378624