Newform orbit 588.2.bb.a

• Label: 588.2.bb.a
• Level: $588$
• Weight: $2$
• Character orbit: 588.bb
• Analytic conductor: $4.695$
• Analytic rank: $0$
• Dimension: $1296$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	588.bb (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.69520363885\)
Analytic rank:	\(0\)
Dimension:	\(1296\)
Relative dimension:	\(108\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1296 q - 26 q^{4} - 14 q^{6} - 26 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
11.1	−1.41063	+	0.100644i	0.599574	−	1.62496i	1.97974	−	0.283942i	0.680876	+	0.267224i	−0.682233	+	2.35256i	2.07881	+	1.63662i	−2.76410	+	0.599786i	−2.28102	−	1.94857i	−0.987357	−	0.308428i
11.2	−1.41044	−	0.103214i	−1.17981	−	1.26809i	1.97869	+	0.291155i	0.759132	+	0.297937i	1.53317	+	1.91034i	2.63553	−	0.232337i	−2.76078	−	0.614886i	−0.216109	+	2.99221i	−1.03996	−	0.498577i
11.3	−1.40717	−	0.141002i	−1.31255	+	1.13014i	1.96024	+	0.396827i	2.31688	+	0.909308i	2.00633	−	1.40522i	0.602590	−	2.57622i	−2.70243	−	0.834800i	0.445581	−	2.96673i	−3.13202	−	1.60623i
11.4	−1.40622	−	0.150187i	−0.0976479	+	1.72930i	1.95489	+	0.422390i	−3.24745	−	1.27453i	0.397031	−	2.41710i	−1.51481	−	2.16918i	−2.68556	−	0.887570i	−2.98093	−	0.337724i	4.37520	+	2.27999i
11.5	−1.40394	−	0.170188i	1.73036	−	0.0765510i	1.94207	+	0.477865i	−1.37795	−	0.540805i	−2.44234	−	0.187013i	−2.59523	−	0.514577i	−2.64522	−	1.00141i	2.98828	−	0.264921i	1.84251	+	0.993765i
11.6	−1.40190	−	0.186187i	0.176473	+	1.72304i	1.93067	+	0.522032i	1.98703	+	0.779851i	0.0734094	−	2.44839i	2.20502	+	1.46215i	−2.60942	−	1.09130i	−2.93771	+	0.608138i	−2.64042	−	1.46323i
11.7	−1.39258	+	0.246428i	1.25846	−	1.19008i	1.87855	−	0.686339i	3.11740	+	1.22349i	−1.45923	+	1.96739i	−2.19246	+	1.48092i	−2.44689	+	1.41871i	0.167430	−	2.99532i	−4.64272	−	0.935590i
11.8	−1.38762	+	0.272975i	1.50515	+	0.857035i	1.85097	−	0.757571i	2.13324	+	0.837234i	−2.32253	−	0.778368i	0.587384	−	2.57972i	−2.36164	+	1.55649i	1.53098	+	2.57994i	−3.18866	−	0.579440i
11.9	−1.38269	−	0.296917i	−1.72931	−	0.0974203i	1.82368	+	0.821090i	−1.35901	−	0.533372i	2.36218	+	0.648164i	−1.07157	+	2.41904i	−2.27780	−	1.67680i	2.98102	+	0.336940i	1.72073	+	1.14100i
11.10	−1.37371	+	0.336025i	1.34992	+	1.08522i	1.77417	−	0.923205i	−2.38573	−	0.936329i	−2.21907	−	1.03718i	1.38872	+	2.25199i	−2.12699	+	1.86439i	0.644584	+	2.92993i	3.59193	+	0.484582i
11.11	−1.36429	+	0.372444i	0.0681246	−	1.73071i	1.72257	−	1.01624i	−0.588200	−	0.230852i	0.551651	+	2.38656i	−0.576515	−	2.58218i	−1.97159	+	2.02801i	−2.99072	−	0.235808i	0.888455	+	0.0958769i
11.12	−1.32760	−	0.487327i	1.49812	−	0.869267i	1.52502	+	1.29395i	−2.96320	−	1.16297i	−2.41252	+	0.423960i	2.58249	−	0.575119i	−1.39404	−	2.46103i	1.48875	−	2.60454i	3.36719	+	2.98801i
11.13	−1.31100	+	0.530349i	−0.944256	+	1.45203i	1.43746	−	1.39058i	−0.502702	−	0.197296i	0.467842	−	2.40440i	−1.42404	+	2.22983i	−1.14702	+	2.58541i	−1.21676	−	2.74217i	0.763680	−	0.00795180i
11.14	−1.30545	−	0.543878i	−1.04370	−	1.38227i	1.40839	+	1.42001i	3.82091	+	1.49960i	0.610715	+	2.37214i	−2.49026	−	0.893651i	−1.06627	−	2.61974i	−0.821360	+	2.88537i	−4.17241	−	4.03576i
11.15	−1.29199	+	0.575118i	−1.69922	−	0.335629i	1.33848	−	1.48609i	0.668460	+	0.262351i	2.38840	−	0.543623i	−2.35732	−	1.20127i	−0.874624	+	2.68980i	2.77471	+	1.14062i	−1.01453	+	0.0454879i
11.16	−1.26771	−	0.626835i	0.185502	−	1.72209i	1.21416	+	1.58928i	−1.94354	−	0.762783i	−1.31463	+	2.06682i	−1.63934	+	2.07667i	−0.542975	−	2.77582i	−2.93118	−	0.638903i	1.98570	+	2.18526i
11.17	−1.25082	−	0.659891i	0.332731	+	1.69979i	1.12909	+	1.65081i	1.51264	+	0.593666i	0.705492	−	2.34569i	−2.43954	+	1.02403i	−0.322931	−	2.80993i	−2.77858	+	1.13115i	−1.50028	−	1.74074i
11.18	−1.21463	+	0.724345i	−1.73191	+	0.0218268i	0.950649	−	1.75962i	3.70438	+	1.45386i	2.08782	−	1.28101i	2.05495	+	1.66648i	0.119886	+	2.82589i	2.99905	−	0.0756042i	−5.55255	+	0.917346i
11.19	−1.15709	+	0.813111i	−0.990374	−	1.42097i	0.677700	−	1.88168i	−3.30789	−	1.29825i	2.30136	+	0.838903i	1.42490	+	2.22927i	0.745858	+	2.72831i	−1.03832	+	2.81459i	4.88314	−	1.18749i
11.20	−1.15611	+	0.814502i	1.15094	−	1.29435i	0.673173	−	1.88331i	−3.04354	−	1.19450i	−0.276366	+	2.43385i	0.343034	−	2.62342i	0.755695	+	2.72561i	−0.350665	−	2.97944i	4.49159	−	1.09800i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
4.b	odd	2	1	inner
12.b	even	2	1	inner
49.g	even	21	1	inner
147.n	odd	42	1	inner
196.o	odd	42	1	inner
588.bb	even	42	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	588.2.bb.a	✓	1296
3.b	odd	2	1	inner	588.2.bb.a	✓	1296
4.b	odd	2	1	inner	588.2.bb.a	✓	1296
12.b	even	2	1	inner	588.2.bb.a	✓	1296
49.g	even	21	1	inner	588.2.bb.a	✓	1296
147.n	odd	42	1	inner	588.2.bb.a	✓	1296
196.o	odd	42	1	inner	588.2.bb.a	✓	1296
588.bb	even	42	1	inner	588.2.bb.a	✓	1296

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
588.2.bb.a	✓	1296	1.a	even	1	1	trivial
588.2.bb.a	✓	1296	3.b	odd	2	1	inner
588.2.bb.a	✓	1296	4.b	odd	2	1	inner
588.2.bb.a	✓	1296	12.b	even	2	1	inner
588.2.bb.a	✓	1296	49.g	even	21	1	inner
588.2.bb.a	✓	1296	147.n	odd	42	1	inner
588.2.bb.a	✓	1296	196.o	odd	42	1	inner
588.2.bb.a	✓	1296	588.bb	even	42	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(588, [\chi])\).