Newform orbit 429.2.y.a

• Label: 429.2.y.a
• Level: $429$
• Weight: $2$
• Character orbit: 429.y
• Analytic conductor: $3.426$
• Analytic rank: $0$
• Dimension: $32$
• CM discriminant: -39
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 429 = 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	429.y (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.42558224671\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(8\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 16 q^{4} - 24 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
116.1	−2.60814	+	0.847436i	−1.40126	−	1.01807i	4.46622	−	3.24490i	−0.309156	+	0.951485i	4.51743	+	1.46780i		0		−5.67484	+	7.81075i	0.927051	+	2.85317i		−	2.74360i
116.2	−2.06579	+	0.671217i	1.40126	+	1.01807i	2.19893	−	1.59762i	−0.710253	+	2.18593i	−3.57806	−	1.16258i		0		−0.916732	+	1.26177i	0.927051	+	2.85317i		−	4.99242i
116.3	−1.72295	+	0.559822i	−1.40126	−	1.01807i	1.03713	−	0.753522i	1.18548	−	3.64854i	2.98424	+	0.969639i		0		0.764591	−	1.05237i	0.927051	+	2.85317i			6.94993i
116.4	−0.658535	+	0.213971i	1.40126	+	1.01807i	−1.23015	+	0.893756i	−1.34694	+	4.14546i	−1.14062	−	0.370608i		0		1.43285	−	1.97215i	0.927051	+	2.85317i		−	3.01814i
116.5	0.658535	−	0.213971i	1.40126	+	1.01807i	−1.23015	+	0.893756i	1.34694	−	4.14546i	1.14062	+	0.370608i		0		−1.43285	+	1.97215i	0.927051	+	2.85317i		−	3.01814i
116.6	1.72295	−	0.559822i	−1.40126	−	1.01807i	1.03713	−	0.753522i	−1.18548	+	3.64854i	−2.98424	−	0.969639i		0		−0.764591	+	1.05237i	0.927051	+	2.85317i			6.94993i
116.7	2.06579	−	0.671217i	1.40126	+	1.01807i	2.19893	−	1.59762i	0.710253	−	2.18593i	3.57806	+	1.16258i		0		0.916732	−	1.26177i	0.927051	+	2.85317i		−	4.99242i
116.8	2.60814	−	0.847436i	−1.40126	−	1.01807i	4.46622	−	3.24490i	0.309156	−	0.951485i	−4.51743	−	1.46780i		0		5.67484	−	7.81075i	0.927051	+	2.85317i		−	2.74360i
194.1	−1.65880	+	2.28314i	−0.535233	+	1.64728i	−1.84308	−	5.67240i	−2.37713	+	1.72708i	−2.87312	−	3.95451i		0		10.6402	+	3.45720i	−2.42705	−	1.76336i		−	8.29218i
194.2	−1.54330	+	2.12417i	0.535233	−	1.64728i	−1.51228	−	4.65433i	3.60386	−	2.61836i	2.67307	+	3.67916i		0		7.22625	+	2.34795i	−2.42705	−	1.76336i			11.6961i
194.3	−0.618195	+	0.850873i	−0.535233	+	1.64728i	0.276215	+	0.850102i	−0.319931	+	0.232444i	−1.07075	−	1.47376i		0		−2.89461	−	0.940514i	−2.42705	−	1.76336i		−	0.415916i
194.4	−0.111034	+	0.152825i	0.535233	−	1.64728i	0.607007	+	1.86818i	−2.72753	+	1.98167i	0.192316	+	0.264700i		0		−0.712214	−	0.231412i	−2.42705	−	1.76336i		−	0.636866i
194.5	0.111034	−	0.152825i	0.535233	−	1.64728i	0.607007	+	1.86818i	2.72753	−	1.98167i	−0.192316	−	0.264700i		0		0.712214	+	0.231412i	−2.42705	−	1.76336i		−	0.636866i
194.6	0.618195	−	0.850873i	−0.535233	+	1.64728i	0.276215	+	0.850102i	0.319931	−	0.232444i	1.07075	+	1.47376i		0		2.89461	+	0.940514i	−2.42705	−	1.76336i		−	0.415916i
194.7	1.54330	−	2.12417i	0.535233	−	1.64728i	−1.51228	−	4.65433i	−3.60386	+	2.61836i	−2.67307	−	3.67916i		0		−7.22625	−	2.34795i	−2.42705	−	1.76336i			11.6961i
194.8	1.65880	−	2.28314i	−0.535233	+	1.64728i	−1.84308	−	5.67240i	2.37713	−	1.72708i	2.87312	+	3.95451i		0		−10.6402	−	3.45720i	−2.42705	−	1.76336i		−	8.29218i
233.1	−2.60814	−	0.847436i	−1.40126	+	1.01807i	4.46622	+	3.24490i	−0.309156	−	0.951485i	4.51743	−	1.46780i		0		−5.67484	−	7.81075i	0.927051	−	2.85317i			2.74360i
233.2	−2.06579	−	0.671217i	1.40126	−	1.01807i	2.19893	+	1.59762i	−0.710253	−	2.18593i	−3.57806	+	1.16258i		0		−0.916732	−	1.26177i	0.927051	−	2.85317i			4.99242i
233.3	−1.72295	−	0.559822i	−1.40126	+	1.01807i	1.03713	+	0.753522i	1.18548	+	3.64854i	2.98424	−	0.969639i		0		0.764591	+	1.05237i	0.927051	−	2.85317i		−	6.94993i
233.4	−0.658535	−	0.213971i	1.40126	−	1.01807i	−1.23015	−	0.893756i	−1.34694	−	4.14546i	−1.14062	+	0.370608i		0		1.43285	+	1.97215i	0.927051	−	2.85317i			3.01814i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
39.d	odd	2	1	CM by \(\Q(\sqrt{-39}) \)
3.b	odd	2	1	inner
11.d	odd	10	1	inner
13.b	even	2	1	inner
33.f	even	10	1	inner
143.l	odd	10	1	inner
429.y	even	10	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	429.2.y.a	✓	32
3.b	odd	2	1	inner	429.2.y.a	✓	32
11.d	odd	10	1	inner	429.2.y.a	✓	32
13.b	even	2	1	inner	429.2.y.a	✓	32
33.f	even	10	1	inner	429.2.y.a	✓	32
39.d	odd	2	1	CM	429.2.y.a	✓	32
143.l	odd	10	1	inner	429.2.y.a	✓	32
429.y	even	10	1	inner	429.2.y.a	✓	32

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
429.2.y.a	✓	32	1.a	even	1	1	trivial
429.2.y.a	✓	32	3.b	odd	2	1	inner
429.2.y.a	✓	32	11.d	odd	10	1	inner
429.2.y.a	✓	32	13.b	even	2	1	inner
429.2.y.a	✓	32	33.f	even	10	1	inner
429.2.y.a	✓	32	39.d	odd	2	1	CM
429.2.y.a	✓	32	143.l	odd	10	1	inner
429.2.y.a	✓	32	429.y	even	10	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T2^32 - 16*T2^30 + 191*T2^28 - 2024*T2^26 + 18683*T2^24 - 126536*T2^22 + 757749*T2^20 - 3996952*T2^18 + 16262568*T2^16 - 41315816*T2^14 + 60590301*T2^12 - 48946192*T2^10 + 69605843*T2^8 - 41048032*T2^6 + 10649879*T2^4 + 199408*T2^2 + 14641