Newform orbit 441.3.bc.a

• Label: 441.3.bc.a
• Level: $441$
• Weight: $3$
• Character orbit: 441.bc
• Analytic conductor: $12.016$
• Analytic rank: $0$
• Dimension: $1320$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	441.bc (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(12.0163796583\)
Analytic rank:	\(0\)
Dimension:	\(1320\)
Relative dimension:	\(110\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 1320 q - 5 q^{2} - 11 q^{3} - 437 q^{4} - 4 q^{5} + 26 q^{6} - 7 q^{7} - 12 q^{8} - 53 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
40.1	−0.876947	+	3.84216i	2.40772	+	1.78967i	−10.3892	−	5.00320i	0.457009	−	3.03205i	−8.98762	+	7.68137i	5.88232	−	3.79451i	18.5053	−	23.2049i	2.59418	+	8.61802i	11.2489	+	4.41485i
40.2	−0.845861	+	3.70596i	1.17067	−	2.76216i	−9.41479	−	4.53392i	−0.243983	+	1.61872i	9.24623	+	6.67487i	2.47583	+	6.54754i	15.2860	−	19.1680i	−6.25906	−	6.46716i	−5.79254	−	2.27340i
40.3	−0.838621	+	3.67424i	−2.67732	+	1.35350i	−9.19288	−	4.42706i	−0.992347	+	6.58379i	−2.72781	−	10.9722i	2.03972	+	6.69623i	14.5763	−	18.2782i	5.33609	−	7.24749i	−23.3582	−	9.16743i
40.4	−0.837274	+	3.66834i	−2.71199	−	1.28263i	−9.15179	−	4.40727i	0.489573	−	3.24811i	6.97578	−	8.87457i	3.90610	−	5.80882i	14.4460	−	18.1147i	5.70974	+	6.95693i	11.5052	+	4.51547i
40.5	−0.819535	+	3.59062i	−2.46817	+	1.70533i	−8.61703	−	4.14974i	0.137804	−	0.914271i	−4.10043	−	10.2598i	−5.24851	−	4.63176i	12.7770	−	16.0218i	3.18372	−	8.41807i	3.16986	+	1.24408i
40.6	−0.803588	+	3.52075i	0.191875	+	2.99386i	−8.14604	−	3.92292i	0.449105	−	2.97962i	−10.6948	−	1.73028i	−6.17849	+	3.29034i	11.3513	−	14.2340i	−8.92637	+	1.14889i	10.1296	+	3.97557i
40.7	−0.801783	+	3.51284i	0.851166	−	2.87672i	−8.09332	−	3.89754i	1.23625	−	8.20199i	9.42301	+	5.29652i	−5.74574	−	3.99831i	11.1943	−	14.0372i	−7.55103	−	4.89713i	27.8211	+	10.9190i
40.8	−0.796897	+	3.49143i	−1.52815	−	2.58162i	−7.95118	−	3.82909i	−1.04179	+	6.91183i	10.2313	−	3.27815i	−6.91808	−	1.06780i	10.7739	−	13.5100i	−4.32951	+	7.89020i	−23.3020	−	9.14535i
40.9	−0.789207	+	3.45774i	2.99921	−	0.0689052i	−7.72926	−	3.72221i	−0.503962	+	3.34357i	−2.12874	+	10.4249i	−4.76466	+	5.12816i	10.1252	−	12.6966i	8.99050	−	0.413322i	−11.1635	−	4.38134i
40.10	−0.763448	+	3.34488i	1.92689	−	2.29936i	−7.00151	−	3.37175i	−0.677466	+	4.49469i	6.22002	+	8.20068i	3.19395	−	6.22886i	8.06686	−	10.1155i	−1.57416	−	8.86127i	−14.5170	−	5.69751i
40.11	−0.755786	+	3.31131i	2.26624	+	1.96575i	−6.78972	−	3.26975i	−1.37731	+	9.13783i	−8.22201	+	6.01854i	−3.61819	−	5.99239i	7.48810	−	9.38978i	1.27166	+	8.90971i	−29.2173	−	11.4669i
40.12	−0.731890	+	3.20662i	2.96993	−	0.423698i	−6.14287	−	2.95825i	1.32570	−	8.79544i	−0.815022	+	9.83353i	1.34106	+	6.87034i	5.77904	−	7.24668i	8.64096	−	2.51671i	27.2333	+	10.6883i
40.13	−0.729085	+	3.19433i	−2.15275	+	2.08942i	−6.06831	−	2.92234i	1.20862	−	8.01870i	−5.10475	−	8.39996i	6.38140	+	2.87711i	5.58785	−	7.00694i	0.268681	−	8.99599i	24.7332	+	9.70707i
40.14	−0.716229	+	3.13800i	0.428968	+	2.96917i	−5.73020	−	2.75952i	−0.769954	+	5.10831i	−9.62451	−	0.780502i	5.96021	+	3.67096i	4.73621	−	5.93901i	−8.63197	+	2.54736i	−15.4784	−	6.07483i
40.15	−0.715720	+	3.13578i	−2.80241	−	1.07075i	−5.71696	−	2.75314i	−1.01120	+	6.70885i	5.36337	−	8.02137i	5.73831	−	4.00896i	4.70336	−	5.89783i	6.70699	+	6.00135i	−20.3137	−	7.97254i
40.16	−0.684125	+	2.99735i	−0.907832	−	2.85934i	−4.91218	−	2.36558i	0.581815	−	3.86009i	9.19151	−	0.764941i	6.83372	+	1.51668i	2.78350	−	3.49040i	−7.35168	+	5.19161i	11.1720	+	4.38468i
40.17	−0.666559	+	2.92039i	−1.70923	−	2.46547i	−4.48049	−	2.15769i	0.358691	−	2.37976i	8.33943	−	3.34823i	−6.26671	+	3.11903i	1.81717	−	2.27865i	−3.15707	+	8.42810i	6.71073	+	2.63377i
40.18	−0.637265	+	2.79204i	2.98080	−	0.338841i	−3.78551	−	1.82300i	0.409729	−	2.71837i	−0.953505	+	8.53845i	−4.33685	−	5.49470i	0.359963	−	0.451380i	8.77037	−	2.02003i	7.32870	+	2.87630i
40.19	−0.629929	+	2.75990i	−2.94850	−	0.553513i	−3.61635	−	1.74154i	0.121060	−	0.803179i	3.38498	−	7.78887i	−1.00933	+	6.92685i	0.0244394	−	0.0306460i	8.38725	+	3.26406i	2.14043	+	0.840058i
40.20	−0.619458	+	2.71402i	2.93146	−	0.637612i	−3.37831	−	1.62691i	−0.511578	+	3.39410i	−0.0854226	+	8.35102i	6.99830	+	0.154228i	−0.434544	+	0.544900i	8.18690	−	3.73827i	−8.89475	−	3.49093i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
441.bc	odd	42	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	441.3.bc.a	✓	1320
9.c	even	3	1		441.3.bl.a	yes	1320
49.h	odd	42	1		441.3.bl.a	yes	1320
441.bc	odd	42	1	inner	441.3.bc.a	✓	1320

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
441.3.bc.a	✓	1320	1.a	even	1	1	trivial
441.3.bc.a	✓	1320	441.bc	odd	42	1	inner
441.3.bl.a	yes	1320	9.c	even	3	1
441.3.bl.a	yes	1320	49.h	odd	42	1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(441, [\chi])\).