Newform orbit 869.2.b.b

• Label: 869.2.b.b
• Level: $869$
• Weight: $2$
• Character orbit: 869.b
• Analytic conductor: $6.939$
• Analytic rank: $0$
• Dimension: $68$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 869 = 11 \cdot 79 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	869.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.93899993565\)
Analytic rank:	\(0\)
Dimension:	\(68\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 68 q - 64 q^{4} - 4 q^{5} - 116 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
868.1		−	2.70559i		−	3.15250i	−5.32020			−3.69878			−8.52936			3.63397					8.98309i	−6.93826					10.0074i
868.2		−	2.70559i			3.15250i	−5.32020			−3.69878			8.52936			−3.63397					8.98309i	−6.93826					10.0074i
868.3		−	2.55585i		−	1.46281i	−4.53237			2.37731			−3.73873			4.15317					6.47235i	0.860178				−	6.07605i
868.4		−	2.55585i			1.46281i	−4.53237			2.37731			3.73873			−4.15317					6.47235i	0.860178				−	6.07605i
868.5		−	2.45467i		−	2.78231i	−4.02540			2.98327			−6.82966			−1.85480					4.97168i	−4.74127				−	7.32294i
868.6		−	2.45467i			2.78231i	−4.02540			2.98327			6.82966			1.85480					4.97168i	−4.74127				−	7.32294i
868.7		−	2.39707i		−	2.47469i	−3.74596			0.686239			−5.93201			−3.19480					4.18519i	−3.12408				−	1.64497i
868.8		−	2.39707i			2.47469i	−3.74596			0.686239			5.93201			3.19480					4.18519i	−3.12408				−	1.64497i
868.9		−	2.21862i		−	0.392107i	−2.92225			−0.991219			−0.869935			1.95494					2.04613i	2.84625					2.19913i
868.10		−	2.21862i			0.392107i	−2.92225			−0.991219			0.869935			−1.95494					2.04613i	2.84625					2.19913i
868.11		−	2.01103i		−	2.12850i	−2.04424			−1.34068			−4.28049			0.465746					0.0889744i	−1.53053					2.69614i
868.12		−	2.01103i			2.12850i	−2.04424			−1.34068			4.28049			−0.465746					0.0889744i	−1.53053					2.69614i
868.13		−	1.91985i		−	1.62647i	−1.68584			0.00491305			−3.12259			3.56665				−	0.603141i	0.354582				−	0.00943234i
868.14		−	1.91985i			1.62647i	−1.68584			0.00491305			3.12259			−3.56665				−	0.603141i	0.354582				−	0.00943234i
868.15		−	1.58723i		−	3.29792i	−0.519300			−1.49780			−5.23456			−0.187276				−	2.35021i	−7.87628					2.37735i
868.16		−	1.58723i			3.29792i	−0.519300			−1.49780			5.23456			0.187276				−	2.35021i	−7.87628					2.37735i
868.17		−	1.57033i		−	1.43165i	−0.465933			3.04233			−2.24816			−2.09874				−	2.40899i	0.950375				−	4.77746i
868.18		−	1.57033i			1.43165i	−0.465933			3.04233			2.24816			2.09874				−	2.40899i	0.950375				−	4.77746i
868.19		−	1.53693i		−	1.88825i	−0.362146			−4.29536			−2.90210			−4.33784				−	2.51726i	−0.565484					6.60166i
868.20		−	1.53693i			1.88825i	−0.362146			−4.29536			2.90210			4.33784				−	2.51726i	−0.565484					6.60166i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
11.b	odd	2	1	inner
79.b	odd	2	1	inner
869.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	869.2.b.b	✓	68
11.b	odd	2	1	inner	869.2.b.b	✓	68
79.b	odd	2	1	inner	869.2.b.b	✓	68
869.b	even	2	1	inner	869.2.b.b	✓	68

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
869.2.b.b	✓	68	1.a	even	1	1	trivial
869.2.b.b	✓	68	11.b	odd	2	1	inner
869.2.b.b	✓	68	79.b	odd	2	1	inner
869.2.b.b	✓	68	869.b	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T2^34 + 50*T2^32 + 1129*T2^30 + 15238*T2^28 + 137157*T2^26 + 869911*T2^24 + 4005749*T2^22 + 13600596*T2^20 + 34241239*T2^18 + 63787740*T2^16 + 87120165*T2^14 + 85800139*T2^12 + 59415900*T2^10 + 27897225*T2^8 + 8425647*T2^6 + 1511621*T2^4 + 140725*T2^2 + 5000