Newform orbit 912.2.ch.e

• Label: 912.2.ch.e
• Level: $912$
• Weight: $2$
• Character orbit: 912.ch
• Analytic conductor: $7.282$
• Analytic rank: $0$
• Dimension: $72$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	912.ch (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.28235666434\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(12\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72 q + 6 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
47.1		0		−1.59313	−	0.679660i		0		0.670748	−	1.84286i		0		3.15740	+	1.82293i		0		2.07613	+	2.16557i		0
47.2		0		−1.56389	−	0.744479i		0		−0.923912	+	2.53843i		0		−1.47222	−	0.849986i		0		1.89150	+	2.32857i		0
47.3		0		−1.39442	+	1.02743i		0		−0.806698	+	2.21638i		0		2.23982	+	1.29316i		0		0.888791	−	2.86532i		0
47.4		0		−1.15909	+	1.28705i		0		0.303314	−	0.833349i		0		−3.60249	−	2.07990i		0		−0.312998	−	2.98363i		0
47.5		0		−0.644609	+	1.60763i		0		1.42178	−	3.90632i		0		3.33819	+	1.92731i		0		−2.16896	−	2.07259i		0
47.6		0		−0.461602	−	1.66941i		0		0.923912	−	2.53843i		0		−1.47222	−	0.849986i		0		−2.57385	+	1.54120i		0
47.7		0		−0.392690	−	1.68695i		0		−0.670748	+	1.84286i		0		3.15740	+	1.82293i		0		−2.69159	+	1.32490i		0
47.8		0		−0.0329261	+	1.73174i		0		−0.746079	+	2.04984i		0		−0.320680	−	0.185145i		0		−2.99783	−	0.114039i		0
47.9		0		1.25395	−	1.19482i		0		0.806698	−	2.21638i		0		2.23982	+	1.29316i		0		0.144806	−	2.99650i		0
47.10		0		1.46877	−	0.917992i		0		−0.303314	+	0.833349i		0		−3.60249	−	2.07990i		0		1.31458	−	2.69664i		0
47.11		0		1.69514	−	0.355654i		0		−1.42178	+	3.90632i		0		3.33819	+	1.92731i		0		2.74702	−	1.20577i		0
47.12		0		1.71115	+	0.268287i		0		0.746079	−	2.04984i		0		−0.320680	−	0.185145i		0		2.85604	+	0.918157i		0
479.1		0		−1.73049	−	0.0734549i		0		−2.45060	−	2.92052i		0		−3.57810	−	2.06582i		0		2.98921	+	0.254226i		0
479.2		0		−1.60101	+	0.660888i		0		2.45060	+	2.92052i		0		−3.57810	−	2.06582i		0		2.12645	−	2.11618i		0
479.3		0		−1.05219	−	1.37583i		0		0.636607	+	0.758679i		0		−0.639016	−	0.368936i		0		−0.785799	+	2.89526i		0
479.4		0		−0.810831	−	1.53054i		0		−1.55329	−	1.85114i		0		0.539557	+	0.311514i		0		−1.68510	+	2.48202i		0
479.5		0		−0.518173	+	1.65272i		0		−0.636607	−	0.758679i		0		−0.639016	−	0.368936i		0		−2.46299	−	1.71279i		0
479.6		0		−0.238457	+	1.71556i		0		1.55329	+	1.85114i		0		0.539557	+	0.311514i		0		−2.88628	−	0.818173i		0
479.7		0		0.793686	−	1.53950i		0		−2.08155	−	2.48069i		0		1.70517	+	0.984482i		0		−1.74012	−	2.44376i		0
479.8		0		0.951569	−	1.44724i		0		1.43381	+	1.70875i		0		−3.96599	−	2.28977i		0		−1.18903	−	2.75431i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
76.l	odd	18	1	inner
228.v	even	18	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	912.2.ch.e	✓	72
3.b	odd	2	1	inner	912.2.ch.e	✓	72
4.b	odd	2	1		912.2.ch.f	yes	72
12.b	even	2	1		912.2.ch.f	yes	72
19.e	even	9	1		912.2.ch.f	yes	72
57.l	odd	18	1		912.2.ch.f	yes	72
76.l	odd	18	1	inner	912.2.ch.e	✓	72
228.v	even	18	1	inner	912.2.ch.e	✓	72

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
912.2.ch.e	✓	72	1.a	even	1	1	trivial
912.2.ch.e	✓	72	3.b	odd	2	1	inner
912.2.ch.e	✓	72	76.l	odd	18	1	inner
912.2.ch.e	✓	72	228.v	even	18	1	inner
912.2.ch.f	yes	72	4.b	odd	2	1
912.2.ch.f	yes	72	12.b	even	2	1
912.2.ch.f	yes	72	19.e	even	9	1
912.2.ch.f	yes	72	57.l	odd	18	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(912, [\chi])\):

T5^72 + 6*T5^70 + 48*T5^68 - 4399*T5^66 - 8139*T5^64 - 429600*T5^62 + 15584147*T5^60 - 3710043*T5^58 + 381962649*T5^56 - 15937488344*T5^54 - 43848348081*T5^52 + 133967087520*T5^50 + 12565314071962*T5^48 + 30423759630105*T5^46 - 189957910094808*T5^44 - 4881288789636326*T5^42 - 15205705304689872*T5^40 + 52168420066040754*T5^38 + 1335311114770721529*T5^36 + 6231629113314557115*T5^34 + 13589254446572930397*T5^32 - 65503321304974002830*T5^30 - 391200532617838648965*T5^28 - 909998935760182400952*T5^26 + 3869232982216813234096*T5^24 + 18853144271809713371097*T5^22 + 41017499552951115107667*T5^20 - 27215384300549762277066*T5^18 - 7092737409702937455249*T5^16 - 193231974694207273169637*T5^14 + 78161107269927765782269*T5^12 + 150677419609574892810156*T5^10 + 368807633243562495740160*T5^8 + 259444385378224319650432*T5^6 + 143267267319479399328768*T5^4 - 15356468521924910469120*T5^2 + 431350893116094877696

T7^36 - 72*T7^34 + 3135*T7^32 - 558*T7^31 - 88774*T7^30 + 32139*T7^29 + 1857858*T7^28 - 1170666*T7^27 - 28506309*T7^26 + 26110377*T7^25 + 331777232*T7^24 - 423391950*T7^23 - 2775730266*T7^22 + 4618351188*T7^21 + 16719517671*T7^20 - 36222310107*T7^19 - 61336260261*T7^18 + 170793261936*T7^17 + 162825536454*T7^16 - 572015133981*T7^15 - 275255114784*T7^14 + 1309416823647*T7^13 + 371753577831*T7^12 - 2181885652710*T7^11 - 356145133038*T7^10 + 2540264443452*T7^9 + 376355075859*T7^8 - 2027402980398*T7^7 - 227230923449*T7^6 + 1049977941930*T7^5 + 175706178852*T7^4 - 315149084640*T7^3 - 43529741136*T7^2 + 50129688960*T7 + 15513200704