Newform orbit 429.2.p.b

• Label: 429.2.p.b
• Level: $429$
• Weight: $2$
• Character orbit: 429.p
• Analytic conductor: $3.426$
• Analytic rank: $0$
• Dimension: $96$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 429 = 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	429.p (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.42558224671\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 96 q + 2 q^{3} - 60 q^{4} - 2 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
230.1	−1.38419	+	2.39749i	−1.43448	−	0.970706i	−2.83198	−	4.90513i			1.85441i	4.31285	−	2.09551i	1.32496	−	0.764967i	10.1432			1.11546	+	2.78491i	−4.44593	−	2.56686i
230.2	−1.38419	+	2.39749i	1.55790	+	0.756943i	−2.83198	−	4.90513i		−	1.85441i	−3.97119	+	2.68729i	−1.32496	+	0.764967i	10.1432			1.85408	+	2.35847i	4.44593	+	2.56686i
230.3	−1.23621	+	2.14118i	−1.16044	+	1.28584i	−2.05645	−	3.56187i			2.31580i	−1.31867	−	4.07429i	−3.20654	+	1.85129i	5.22398			−0.306758	−	2.98428i	−4.95856	−	2.86283i
230.4	−1.23621	+	2.14118i	−0.533348	+	1.64789i	−2.05645	−	3.56187i		−	2.31580i	−2.86910	−	3.17914i	3.20654	−	1.85129i	5.22398			−2.43108	−	1.75780i	4.95856	+	2.86283i
230.5	−1.18009	+	2.04397i	0.425511	−	1.67897i	−1.78521	−	3.09207i			0.303518i	2.92962	+	2.85106i	−1.36283	+	0.786828i	3.70646			−2.63788	−	1.42884i	−0.620383	−	0.358178i
230.6	−1.18009	+	2.04397i	1.24128	−	1.20799i	−1.78521	−	3.09207i		−	0.303518i	1.00428	+	3.96266i	1.36283	−	0.786828i	3.70646			0.0815279	−	2.99889i	0.620383	+	0.358178i
230.7	−1.15197	+	1.99526i	−0.989927	−	1.42128i	−1.65405	−	2.86490i		−	3.64772i	3.97620	−	0.337897i	1.03838	−	0.599507i	3.01377			−1.04009	+	2.81393i	7.27817	+	4.20205i
230.8	−1.15197	+	1.99526i	1.72583	+	0.146661i	−1.65405	−	2.86490i			3.64772i	−2.28072	+	3.27454i	−1.03838	+	0.599507i	3.01377			2.95698	+	0.506224i	−7.27817	−	4.20205i
230.9	−1.00971	+	1.74887i	−1.56002	+	0.752553i	−1.03904	−	1.79966i			1.23164i	0.259053	−	3.48814i	3.16404	−	1.82676i	0.157663			1.86733	−	2.34800i	−2.15397	−	1.24360i
230.10	−1.00971	+	1.74887i	0.128280	+	1.72729i	−1.03904	−	1.79966i		−	1.23164i	−3.15034	−	1.51972i	−3.16404	+	1.82676i	0.157663			−2.96709	+	0.443156i	2.15397	+	1.24360i
230.11	−0.964601	+	1.67074i	−1.68588	−	0.397235i	−0.860909	−	1.49114i		−	1.71348i	2.28988	−	2.43350i	−3.58916	+	2.07221i	−0.536667			2.68441	+	1.33939i	2.86278	+	1.65283i
230.12	−0.964601	+	1.67074i	1.18696	+	1.26140i	−0.860909	−	1.49114i			1.71348i	−3.25241	+	0.766347i	3.58916	−	2.07221i	−0.536667			−0.182262	+	2.99446i	−2.86278	−	1.65283i
230.13	−0.790522	+	1.36922i	−1.05747	−	1.37177i	−0.249851	−	0.432755i			3.89310i	2.71422	−	0.363499i	−2.15619	+	1.24488i	−2.37204			−0.763513	+	2.90121i	−5.33053	−	3.07758i
230.14	−0.790522	+	1.36922i	1.71672	+	0.229910i	−0.249851	−	0.432755i		−	3.89310i	−1.67191	+	2.16883i	2.15619	−	1.24488i	−2.37204			2.89428	+	0.789386i	5.33053	+	3.07758i
230.15	−0.642823	+	1.11340i	−1.71725	+	0.225956i	0.173557	+	0.300610i		−	1.88239i	0.852307	−	2.05724i	0.680391	−	0.392824i	−3.01756			2.89789	−	0.776047i	2.09586	+	1.21004i
230.16	−0.642823	+	1.11340i	0.662940	+	1.60016i	0.173557	+	0.300610i			1.88239i	−2.20777	−	0.290500i	−0.680391	+	0.392824i	−3.01756			−2.12102	+	2.12162i	−2.09586	−	1.21004i
230.17	−0.570464	+	0.988072i	0.0427205	−	1.73152i	0.349142	+	0.604732i			0.927976i	1.68650	+	1.02998i	−1.26517	+	0.730447i	−3.07855			−2.99635	−	0.147943i	−0.916907	−	0.529376i
230.18	−0.570464	+	0.988072i	1.47818	−	0.902759i	0.349142	+	0.604732i		−	0.927976i	0.0487410	+	1.97554i	1.26517	−	0.730447i	−3.07855			1.37005	−	2.66889i	0.916907	+	0.529376i
230.19	−0.400235	+	0.693228i	−1.18152	+	1.26649i	0.679624	+	1.17714i		−	2.49177i	−0.405079	−	1.32596i	−0.533256	+	0.307876i	−2.68898			−0.207999	−	2.99278i	1.72737	+	0.997295i
230.20	−0.400235	+	0.693228i	−0.506051	+	1.65648i	0.679624	+	1.17714i			2.49177i	−0.945776	−	1.01379i	0.533256	−	0.307876i	−2.68898			−2.48782	−	1.67652i	−1.72737	−	0.997295i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
11.b	odd	2	1	inner
13.c	even	3	1	inner
33.d	even	2	1	inner
39.i	odd	6	1	inner
143.k	odd	6	1	inner
429.p	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	429.2.p.b	✓	96
3.b	odd	2	1	inner	429.2.p.b	✓	96
11.b	odd	2	1	inner	429.2.p.b	✓	96
13.c	even	3	1	inner	429.2.p.b	✓	96
33.d	even	2	1	inner	429.2.p.b	✓	96
39.i	odd	6	1	inner	429.2.p.b	✓	96
143.k	odd	6	1	inner	429.2.p.b	✓	96
429.p	even	6	1	inner	429.2.p.b	✓	96

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
429.2.p.b	✓	96	1.a	even	1	1	trivial
429.2.p.b	✓	96	3.b	odd	2	1	inner
429.2.p.b	✓	96	11.b	odd	2	1	inner
429.2.p.b	✓	96	13.c	even	3	1	inner
429.2.p.b	✓	96	33.d	even	2	1	inner
429.2.p.b	✓	96	39.i	odd	6	1	inner
429.2.p.b	✓	96	143.k	odd	6	1	inner
429.2.p.b	✓	96	429.p	even	6	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T2^48 + 39*T2^46 + 859*T2^44 + 12966*T2^42 + 148216*T2^40 + 1339056*T2^38 + 9847180*T2^36 + 59849024*T2^34 + 303967872*T2^32 + 1296076798*T2^30 + 4652133375*T2^28 + 14034397034*T2^26 + 35493321402*T2^24 + 74751778955*T2^22 + 130218713178*T2^20 + 185353047160*T2^18 + 213180171588*T2^16 + 193933575836*T2^14 + 137203371973*T2^12 + 72548816908*T2^10 + 28355839168*T2^8 + 7512602410*T2^6 + 1376634370*T2^4 + 131418672*T2^2 + 8311689