Newform orbit 465.2.bf.a

• Label: 465.2.bf.a
• Level: $465$
• Weight: $2$
• Character orbit: 465.bf
• Analytic conductor: $3.713$
• Analytic rank: $0$
• Dimension: $128$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 465 = 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	465.bf (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.71304369399\)
Analytic rank:	\(0\)
Dimension:	\(128\)
Relative dimension:	\(32\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 128 q + 12 q^{6} + 4 q^{7} - 24 q^{8}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
37.1	−1.97490	−	1.97490i	0.258819	−	0.965926i			5.80046i	0.791657	+	2.09124i	−2.41875	+	1.39647i	−0.344903	+	1.28720i	7.50554	−	7.50554i	−0.866025	−	0.500000i	2.56654	−	5.69343i
37.2	−1.78101	−	1.78101i	−0.258819	+	0.965926i			4.34402i	2.13220	+	0.673586i	2.18129	−	1.25937i	0.501482	−	1.87156i	4.17474	−	4.17474i	−0.866025	−	0.500000i	−2.59781	−	4.99715i
37.3	−1.74184	−	1.74184i	−0.258819	+	0.965926i			4.06803i	1.11323	−	1.93926i	2.13331	−	1.23167i	−0.821469	+	3.06577i	3.60218	−	3.60218i	−0.866025	−	0.500000i	−5.31695	+	1.43882i
37.4	−1.63560	−	1.63560i	−0.258819	+	0.965926i			3.35039i	−1.33975	+	1.79027i	2.00320	−	1.15655i	−0.511609	+	1.90935i	2.20870	−	2.20870i	−0.866025	−	0.500000i	5.11947	−	0.736861i
37.5	−1.52609	−	1.52609i	0.258819	−	0.965926i			2.65792i	−0.00670335	−	2.23606i	−1.86908	+	1.07911i	−0.401868	+	1.49979i	1.00405	−	1.00405i	−0.866025	−	0.500000i	−3.40220	+	3.42266i
37.6	−1.49087	−	1.49087i	−0.258819	+	0.965926i			2.44541i	−1.20184	−	1.88563i	1.82594	−	1.05421i	1.01426	−	3.78526i	0.664048	−	0.664048i	−0.866025	−	0.500000i	−1.01944	+	4.60302i
37.7	−1.32442	−	1.32442i	0.258819	−	0.965926i			1.50820i	1.95093	+	1.09264i	−1.62208	+	0.936509i	−0.431023	+	1.60860i	−0.651354	+	0.651354i	−0.866025	−	0.500000i	−1.13675	−	4.03098i
37.8	−1.27552	−	1.27552i	0.258819	−	0.965926i			1.25390i	−2.23416	+	0.0923541i	−1.56219	+	0.901928i	−1.00371	+	3.74588i	−0.951668	+	0.951668i	−0.866025	−	0.500000i	2.96751	+	2.73191i
37.9	−0.947704	−	0.947704i	−0.258819	+	0.965926i		−	0.203713i	−0.0222221	+	2.23596i	1.16070	−	0.670128i	0.586650	−	2.18941i	−2.08847	+	2.08847i	−0.866025	−	0.500000i	2.14009	−	2.09797i
37.10	−0.855487	−	0.855487i	−0.258819	+	0.965926i		−	0.536283i	1.73559	+	1.40987i	1.04775	−	0.604921i	−0.855524	+	3.19286i	−2.16976	+	2.16976i	−0.866025	−	0.500000i	−0.278655	−	2.69090i
37.11	−0.818748	−	0.818748i	0.258819	−	0.965926i		−	0.659305i	1.87231	−	1.22247i	−1.00276	+	0.578942i	0.759389	−	2.83408i	−2.17730	+	2.17730i	−0.866025	−	0.500000i	−2.53385	−	0.532058i
37.12	−0.484320	−	0.484320i	−0.258819	+	0.965926i		−	1.53087i	−2.22070	+	0.261702i	0.593168	−	0.342466i	−0.147138	+	0.549125i	−1.71007	+	1.71007i	−0.866025	−	0.500000i	1.20228	+	0.948782i
37.13	−0.481256	−	0.481256i	0.258819	−	0.965926i		−	1.53679i	−1.11263	−	1.93960i	−0.589415	+	0.340299i	1.04816	−	3.91180i	−1.70210	+	1.70210i	−0.866025	−	0.500000i	−0.397981	+	1.46890i
37.14	−0.431245	−	0.431245i	0.258819	−	0.965926i		−	1.62806i	−0.683600	+	2.12901i	−0.528165	+	0.304936i	0.510043	−	1.90351i	−1.56458	+	1.56458i	−0.866025	−	0.500000i	1.21293	−	0.623327i
37.15	−0.366734	−	0.366734i	0.258819	−	0.965926i		−	1.73101i	−1.11960	+	1.93559i	−0.449156	+	0.259320i	−0.953184	+	3.55733i	−1.36829	+	1.36829i	−0.866025	−	0.500000i	1.12044	−	0.299251i
37.16	−0.163038	−	0.163038i	−0.258819	+	0.965926i		−	1.94684i	1.01346	−	1.99321i	0.199681	−	0.115286i	0.856607	−	3.19690i	−0.643486	+	0.643486i	−0.866025	−	0.500000i	−0.490203	+	0.159738i
37.17	−0.133936	−	0.133936i	−0.258819	+	0.965926i		−	1.96412i	−1.28179	−	1.83221i	0.164038	−	0.0947072i	−1.05066	+	3.92114i	−0.530940	+	0.530940i	−0.866025	−	0.500000i	−0.0737211	+	0.417078i
37.18	0.115348	+	0.115348i	−0.258819	+	0.965926i		−	1.97339i	2.23568	+	0.0415101i	−0.141271	+	0.0815631i	0.0931331	−	0.347578i	0.458321	−	0.458321i	−0.866025	−	0.500000i	0.253093	+	0.262669i
37.19	0.264712	+	0.264712i	0.258819	−	0.965926i		−	1.85986i	1.93907	−	1.11355i	0.324205	−	0.187180i	−0.184213	+	0.687493i	1.02175	−	1.02175i	−0.866025	−	0.500000i	0.808066	+	0.218526i
37.20	0.400867	+	0.400867i	0.258819	−	0.965926i		−	1.67861i	−2.13134	−	0.676305i	0.490960	−	0.283456i	−0.161769	+	0.603730i	1.47463	−	1.47463i	−0.866025	−	0.500000i	−0.583276	−	1.12549i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.c	odd	4	1	inner
31.e	odd	6	1	inner
155.p	even	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	465.2.bf.a	✓	128
5.c	odd	4	1	inner	465.2.bf.a	✓	128
31.e	odd	6	1	inner	465.2.bf.a	✓	128
155.p	even	12	1	inner	465.2.bf.a	✓	128

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
465.2.bf.a	✓	128	1.a	even	1	1	trivial
465.2.bf.a	✓	128	5.c	odd	4	1	inner
465.2.bf.a	✓	128	31.e	odd	6	1	inner
465.2.bf.a	✓	128	155.p	even	12	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(465, [\chi])\).