Newform orbit 666.2.g.b

• Label: 666.2.g.b
• Level: $666$
• Weight: $2$
• Character orbit: 666.g
• Analytic conductor: $5.318$
• Analytic rank: $0$
• Dimension: $36$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 666 = 2 \cdot 3^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	666.g (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.31803677462\)
Analytic rank:	\(0\)
Dimension:	\(36\)
Relative dimension:	\(18\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 18 q^{2} - 2 q^{3} - 18 q^{4} + 4 q^{5} + 2 q^{6} - 8 q^{7} - 36 q^{8} - 2 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
211.1	0.500000	−	0.866025i	−1.72970	−	0.0901251i	−0.500000	−	0.866025i	−0.905939	−	1.56913i	−0.942903	+	1.45291i	4.49856			−1.00000			2.98375	+	0.311779i	−1.81188
211.2	0.500000	−	0.866025i	−1.69285	+	0.366421i	−0.500000	−	0.866025i	−1.88299	−	3.26143i	−0.529094	+	1.64926i	−1.57151			−1.00000			2.73147	−	1.24059i	−3.76597
211.3	0.500000	−	0.866025i	−1.65621	+	0.506905i	−0.500000	−	0.866025i	1.31085	+	2.27046i	−0.389115	+	1.68778i	−3.62295			−1.00000			2.48609	−	1.67909i	2.62170
211.4	0.500000	−	0.866025i	−1.46941	−	0.916971i	−0.500000	−	0.866025i	0.773608	+	1.33993i	−1.52882	+	0.814061i	2.91212			−1.00000			1.31833	+	2.69481i	1.54722
211.5	0.500000	−	0.866025i	−1.07177	+	1.36063i	−0.500000	−	0.866025i	0.104797	+	0.181513i	0.642457	+	1.60849i	−0.557903			−1.00000			−0.702629	−	2.91656i	0.209594
211.6	0.500000	−	0.866025i	−0.988272	+	1.42243i	−0.500000	−	0.866025i	−0.236812	−	0.410170i	0.737727	+	1.56709i	1.65252			−1.00000			−1.04663	−	2.81150i	−0.473623
211.7	0.500000	−	0.866025i	−0.883323	−	1.48988i	−0.500000	−	0.866025i	−1.00635	−	1.74305i	−1.73193	+	0.0200411i	−2.39223			−1.00000			−1.43948	+	2.63209i	−2.01270
211.8	0.500000	−	0.866025i	−0.671031	−	1.59678i	−0.500000	−	0.866025i	2.08465	+	3.61071i	−1.71837	−	0.217262i	−1.59042			−1.00000			−2.09943	+	2.14298i	4.16929
211.9	0.500000	−	0.866025i	−0.00919631	−	1.73203i	−0.500000	−	0.866025i	0.388342	+	0.672629i	−1.50458	−	0.858049i	2.71820			−1.00000			−2.99983	+	0.0318565i	0.776685
211.10	0.500000	−	0.866025i	0.0862242	+	1.72990i	−0.500000	−	0.866025i	1.39850	+	2.42227i	1.54125	+	0.790279i	−4.16112			−1.00000			−2.98513	+	0.298319i	2.79700
211.11	0.500000	−	0.866025i	0.211906	+	1.71904i	−0.500000	−	0.866025i	2.03934	+	3.53223i	1.59468	+	0.676003i	4.94838			−1.00000			−2.91019	+	0.728551i	4.07867
211.12	0.500000	−	0.866025i	0.785550	−	1.54367i	−0.500000	−	0.866025i	−1.55895	−	2.70018i	−0.944081	−	1.45214i	−0.992258			−1.00000			−1.76582	−	2.42526i	−3.11790
211.13	0.500000	−	0.866025i	0.822323	+	1.52440i	−0.500000	−	0.866025i	−0.495795	−	0.858743i	1.73133	+	0.0500456i	0.00883149			−1.00000			−1.64757	+	2.50709i	−0.991591
211.14	0.500000	−	0.866025i	1.06203	+	1.36824i	−0.500000	−	0.866025i	−1.08247	−	1.87490i	1.71595	−	0.235623i	−3.03708			−1.00000			−0.744183	+	2.90623i	−2.16495
211.15	0.500000	−	0.866025i	1.08836	−	1.34740i	−0.500000	−	0.866025i	0.563728	+	0.976406i	−0.622701	−	1.61624i	−4.63577			−1.00000			−0.630955	−	2.93290i	1.12746
211.16	0.500000	−	0.866025i	1.65517	+	0.510295i	−0.500000	−	0.866025i	0.798734	+	1.38345i	1.26952	−	1.17827i	0.288095			−1.00000			2.47920	+	1.68925i	1.59747
211.17	0.500000	−	0.866025i	1.72879	−	0.106271i	−0.500000	−	0.866025i	1.39920	+	2.42349i	0.772360	−	1.55031i	0.418231			−1.00000			2.97741	−	0.367441i	2.79841
211.18	0.500000	−	0.866025i	1.73142	+	0.0469029i	−0.500000	−	0.866025i	−1.69244	−	2.93139i	0.906327	−	1.47600i	1.11631			−1.00000			2.99560	+	0.162417i	−3.38488
565.1	0.500000	+	0.866025i	−1.72970	+	0.0901251i	−0.500000	+	0.866025i	−0.905939	+	1.56913i	−0.942903	−	1.45291i	4.49856			−1.00000			2.98375	−	0.311779i	−1.81188
565.2	0.500000	+	0.866025i	−1.69285	−	0.366421i	−0.500000	+	0.866025i	−1.88299	+	3.26143i	−0.529094	−	1.64926i	−1.57151			−1.00000			2.73147	+	1.24059i	−3.76597

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
333.g	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	666.2.g.b	✓	36
3.b	odd	2	1		1998.2.g.b		36
9.c	even	3	1		666.2.h.b	yes	36
9.d	odd	6	1		1998.2.h.b		36
37.c	even	3	1		666.2.h.b	yes	36
111.i	odd	6	1		1998.2.h.b		36
333.g	even	3	1	inner	666.2.g.b	✓	36
333.u	odd	6	1		1998.2.g.b		36

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
666.2.g.b	✓	36	1.a	even	1	1	trivial
666.2.g.b	✓	36	333.g	even	3	1	inner
666.2.h.b	yes	36	9.c	even	3	1
666.2.h.b	yes	36	37.c	even	3	1
1998.2.g.b		36	3.b	odd	2	1
1998.2.g.b		36	333.u	odd	6	1
1998.2.h.b		36	9.d	odd	6	1
1998.2.h.b		36	111.i	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T5^36 - 4*T5^35 + 64*T5^34 - 192*T5^33 + 2166*T5^32 - 5643*T5^31 + 48626*T5^30 - 108731*T5^29 + 788024*T5^28 - 1567644*T5^27 + 9645993*T5^26 - 16962699*T5^25 + 90862650*T5^24 - 143345187*T5^23 + 668030706*T5^22 - 936230952*T5^21 + 3828041418*T5^20 - 4823410107*T5^19 + 17184286673*T5^18 - 19307625431*T5^17 + 59560791560*T5^16 - 60360619059*T5^15 + 158550732663*T5^14 - 140803568652*T5^13 + 312614325505*T5^12 - 240462523483*T5^11 + 444861560248*T5^10 - 269775602178*T5^9 + 410926255779*T5^8 - 189038349360*T5^7 + 256435197228*T5^6 - 80397164250*T5^5 + 80952499737*T5^4 - 8958614904*T5^3 + 13990058208*T5^2 - 2332986624*T5 + 544195584