Newform orbit 414.3.h.a

• Label: 414.3.h.a
• Level: $414$
• Weight: $3$
• Character orbit: 414.h
• Analytic conductor: $11.281$
• Analytic rank: $0$
• Dimension: $96$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 414 = 2 \cdot 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	414.h (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.2806829445\)
Analytic rank:	\(0\)
Dimension:	\(96\)
Relative dimension:	\(48\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 96 q + 4 q^{3} - 96 q^{4} + 16 q^{6} + 36 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
229.1	−0.707107	−	1.22474i	−2.89884	−	0.772474i	−1.00000	+	1.73205i	−5.56174	−	3.21107i	1.10371	+	4.09656i	7.31386	−	4.22266i	2.82843			7.80657	+	4.47856i			9.08228i
229.2	−0.707107	−	1.22474i	−2.89884	−	0.772474i	−1.00000	+	1.73205i	5.56174	+	3.21107i	1.10371	+	4.09656i	−7.31386	+	4.22266i	2.82843			7.80657	+	4.47856i		−	9.08228i
229.3	−0.707107	−	1.22474i	−2.69891	+	1.30993i	−1.00000	+	1.73205i	−3.12737	−	1.80559i	3.51274	+	2.37921i	4.42508	−	2.55482i	2.82843			5.56819	−	7.07073i			5.10698i
229.4	−0.707107	−	1.22474i	−2.69891	+	1.30993i	−1.00000	+	1.73205i	3.12737	+	1.80559i	3.51274	+	2.37921i	−4.42508	+	2.55482i	2.82843			5.56819	−	7.07073i		−	5.10698i
229.5	−0.707107	−	1.22474i	−2.61963	−	1.46204i	−1.00000	+	1.73205i	−5.22993	−	3.01950i	0.0617296	+	4.24219i	−11.0196	+	6.36214i	2.82843			4.72488	+	7.65999i			8.54043i
229.6	−0.707107	−	1.22474i	−2.61963	−	1.46204i	−1.00000	+	1.73205i	5.22993	+	3.01950i	0.0617296	+	4.24219i	11.0196	−	6.36214i	2.82843			4.72488	+	7.65999i		−	8.54043i
229.7	−0.707107	−	1.22474i	−1.45535	+	2.62335i	−1.00000	+	1.73205i	−6.63175	−	3.82884i	4.24202	−	0.0725559i	−9.09135	+	5.24889i	2.82843			−4.76392	−	7.63578i			10.8296i
229.8	−0.707107	−	1.22474i	−1.45535	+	2.62335i	−1.00000	+	1.73205i	6.63175	+	3.82884i	4.24202	−	0.0725559i	9.09135	−	5.24889i	2.82843			−4.76392	−	7.63578i		−	10.8296i
229.9	−0.707107	−	1.22474i	−1.38507	−	2.66112i	−1.00000	+	1.73205i	−2.02474	−	1.16898i	−2.27981	+	3.57806i	−2.31483	+	1.33647i	2.82843			−5.16316	+	7.37169i			3.30639i
229.10	−0.707107	−	1.22474i	−1.38507	−	2.66112i	−1.00000	+	1.73205i	2.02474	+	1.16898i	−2.27981	+	3.57806i	2.31483	−	1.33647i	2.82843			−5.16316	+	7.37169i		−	3.30639i
229.11	−0.707107	−	1.22474i	−0.276911	+	2.98719i	−1.00000	+	1.73205i	−3.00967	−	1.73763i	3.85435	−	1.77312i	8.08407	−	4.66734i	2.82843			−8.84664	−	1.65437i			4.91476i
229.12	−0.707107	−	1.22474i	−0.276911	+	2.98719i	−1.00000	+	1.73205i	3.00967	+	1.73763i	3.85435	−	1.77312i	−8.08407	+	4.66734i	2.82843			−8.84664	−	1.65437i		−	4.91476i
229.13	−0.707107	−	1.22474i	0.170504	−	2.99515i	−1.00000	+	1.73205i	−8.47335	−	4.89209i	−3.78886	+	1.90907i	8.68607	−	5.01491i	2.82843			−8.94186	−	1.02137i			13.8369i
229.14	−0.707107	−	1.22474i	0.170504	−	2.99515i	−1.00000	+	1.73205i	8.47335	+	4.89209i	−3.78886	+	1.90907i	−8.68607	+	5.01491i	2.82843			−8.94186	−	1.02137i		−	13.8369i
229.15	−0.707107	−	1.22474i	1.39491	+	2.65598i	−1.00000	+	1.73205i	−3.95291	−	2.28222i	2.26655	−	3.58647i	2.69328	−	1.55496i	2.82843			−5.10845	+	7.40971i			6.45508i
229.16	−0.707107	−	1.22474i	1.39491	+	2.65598i	−1.00000	+	1.73205i	3.95291	+	2.28222i	2.26655	−	3.58647i	−2.69328	+	1.55496i	2.82843			−5.10845	+	7.40971i		−	6.45508i
229.17	−0.707107	−	1.22474i	1.61253	−	2.52977i	−1.00000	+	1.73205i	−0.849237	−	0.490307i	−4.23856	−	0.186126i	3.58678	−	2.07083i	2.82843			−3.79946	−	8.15868i			1.38680i
229.18	−0.707107	−	1.22474i	1.61253	−	2.52977i	−1.00000	+	1.73205i	0.849237	+	0.490307i	−4.23856	−	0.186126i	−3.58678	+	2.07083i	2.82843			−3.79946	−	8.15868i		−	1.38680i
229.19	−0.707107	−	1.22474i	2.13720	+	2.10532i	−1.00000	+	1.73205i	−4.98011	−	2.87527i	1.06725	−	4.10621i	−1.58371	+	0.914355i	2.82843			0.135264	+	8.99898i			8.13249i
229.20	−0.707107	−	1.22474i	2.13720	+	2.10532i	−1.00000	+	1.73205i	4.98011	+	2.87527i	1.06725	−	4.10621i	1.58371	−	0.914355i	2.82843			0.135264	+	8.99898i		−	8.13249i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.c	even	3	1	inner
23.b	odd	2	1	inner
207.f	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	414.3.h.a	✓	96
3.b	odd	2	1		1242.3.h.a		96
9.c	even	3	1	inner	414.3.h.a	✓	96
9.d	odd	6	1		1242.3.h.a		96
23.b	odd	2	1	inner	414.3.h.a	✓	96
69.c	even	2	1		1242.3.h.a		96
207.f	odd	6	1	inner	414.3.h.a	✓	96
207.g	even	6	1		1242.3.h.a		96

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
414.3.h.a	✓	96	1.a	even	1	1	trivial
414.3.h.a	✓	96	9.c	even	3	1	inner
414.3.h.a	✓	96	23.b	odd	2	1	inner
414.3.h.a	✓	96	207.f	odd	6	1	inner
1242.3.h.a		96	3.b	odd	2	1
1242.3.h.a		96	9.d	odd	6	1
1242.3.h.a		96	69.c	even	2	1
1242.3.h.a		96	207.g	even	6	1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(414, [\chi])\).