Newform orbit 441.2.bd.a

• Label: 441.2.bd.a
• Level: $441$
• Weight: $2$
• Character orbit: 441.bd
• Analytic conductor: $3.521$
• Analytic rank: $0$
• Dimension: $648$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(3.52140272914\)
Analytic rank:	\(0\)
Dimension:	\(648\)
Relative dimension:	\(54\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 648 q - 18 q^{2} - 11 q^{3} - 60 q^{4} - 15 q^{5} - 22 q^{6} - 5 q^{7} + 13 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
47.1	−1.83532	−	1.97800i	0.292304	−	1.70721i	−0.394641	+	5.26612i	−0.817866	−	1.02557i	−3.91333	+	2.55509i	0.0523706	+	2.64523i	6.92143	−	5.51966i	−2.82912	−	0.998047i	−0.527539	+	3.49999i
47.2	−1.83191	−	1.97433i	−1.17858	−	1.26924i	−0.392623	+	5.23919i	1.47204	+	1.84588i	−0.346843	+	4.65202i	−1.31601	−	2.29524i	6.85171	−	5.46406i	−0.221916	+	2.99178i	0.947728	−	6.28776i
47.3	−1.79697	−	1.93667i	−1.59909	+	0.665506i	−0.372140	+	4.96586i	−2.12889	−	2.66954i	4.16239	+	1.90103i	2.41582	−	1.07881i	6.15488	−	4.90835i	2.11420	−	2.12841i	−1.34448	+	8.92003i
47.4	−1.75148	−	1.88764i	0.0116260	+	1.73201i	−0.346064	+	4.61791i	−0.695350	−	0.871941i	3.24905	−	3.05552i	−2.60184	+	0.480046i	5.29657	−	4.22388i	−2.99973	+	0.0402729i	−0.428024	+	2.83975i
47.5	−1.67227	−	1.80228i	−0.998913	+	1.41498i	−0.302261	+	4.03339i	2.61954	+	3.28480i	4.22064	−	0.565913i	2.36845	+	1.17916i	3.93034	−	3.13434i	−1.00435	−	2.82689i	1.53954	−	10.2142i
47.6	−1.59981	−	1.72418i	1.65781	+	0.501676i	−0.263960	+	3.52230i	2.05109	+	2.57199i	−1.78719	−	3.66094i	−2.34210	+	1.23069i	2.81755	−	2.24692i	2.49664	+	1.66336i	1.15322	−	7.65114i
47.7	−1.59975	−	1.72412i	1.51926	−	0.831777i	−0.263930	+	3.52190i	0.503410	+	0.631256i	−3.86451	−	1.28875i	0.617740	−	2.57262i	2.81669	−	2.24623i	1.61629	−	2.52737i	0.283031	−	1.87779i
47.8	−1.58104	−	1.70396i	1.73188	−	0.0242085i	−0.254319	+	3.39365i	−1.54809	−	1.94124i	−2.77943	−	2.91278i	1.62028	+	2.09157i	2.55004	−	2.03359i	2.99883	−	0.0838526i	−0.860202	+	5.70707i
47.9	−1.34426	−	1.44876i	−1.69107	+	0.374554i	−0.142431	+	1.90061i	0.213240	+	0.267394i	2.81587	+	1.94646i	−2.30163	−	1.30480i	−0.145343	+	0.115907i	2.71942	−	1.26679i	0.100742	−	0.668380i
47.10	−1.29221	−	1.39267i	−1.37046	−	1.05916i	−0.120265	+	1.60482i	0.149465	+	0.187424i	0.295858	+	3.27726i	2.64547	−	0.0386573i	−0.580288	+	0.462764i	0.756341	+	2.90309i	0.0678787	−	0.450346i
47.11	−1.29218	−	1.39264i	1.30169	+	1.14263i	−0.120254	+	1.60468i	−2.30579	−	2.89137i	−0.0907452	−	3.28927i	−0.473280	−	2.60308i	−0.580487	+	0.462923i	0.388790	+	2.97470i	−1.04714	+	6.94730i
47.12	−1.22870	−	1.32422i	0.168894	+	1.72380i	−0.0944028	+	1.25972i	0.684410	+	0.858223i	2.07517	−	2.34167i	1.06533	−	2.42179i	−1.04054	+	0.829802i	−2.94295	+	0.582277i	0.295544	−	1.96080i
47.13	−1.22764	−	1.32308i	−1.33805	−	1.09983i	−0.0939836	+	1.25412i	−2.40505	−	3.01584i	0.187488	+	3.12054i	−1.92992	+	1.80981i	−1.04756	+	0.835402i	0.580768	+	2.94325i	−1.03766	+	6.88444i
47.14	−1.18529	−	1.27744i	0.137429	−	1.72659i	−0.0774791	+	1.03389i	2.21900	+	2.78254i	−2.36851	+	1.87096i	−1.21027	+	2.35271i	−1.31233	+	1.04655i	−2.96223	−	0.474566i	0.924366	−	6.13277i
47.15	−1.08388	−	1.16814i	0.336533	−	1.69904i	−0.0403058	+	0.537843i	0.551923	+	0.692089i	−2.34949	+	1.44844i	2.64276	+	0.125839i	−1.81979	+	1.45123i	−2.77349	−	1.14357i	0.210242	−	1.39487i
47.16	−0.950574	−	1.02447i	−0.660575	+	1.60114i	0.00350198	−	0.0467306i	−1.62704	−	2.04025i	2.26825	−	0.845256i	1.86948	+	1.87217i	−2.23650	+	1.78355i	−2.12728	−	2.11534i	−0.543558	+	3.60627i
47.17	−0.932976	−	1.00551i	−1.41782	+	0.994877i	0.00885622	−	0.118178i	0.700700	+	0.878650i	2.32315	+	0.497436i	−1.28096	+	2.31498i	−2.27193	+	1.81180i	1.02044	−	2.82112i	0.229754	−	1.52432i
47.18	−0.742532	−	0.800259i	1.56641	+	0.739164i	0.0603993	−	0.805973i	1.83554	+	2.30169i	−0.571585	−	1.80238i	2.47412	−	0.937401i	−2.39686	+	1.91143i	1.90727	+	2.31567i	0.479004	−	3.17799i
47.19	−0.740834	−	0.798429i	1.60616	−	0.648268i	0.0608061	−	0.811401i	−1.45215	−	1.82094i	−1.70749	−	0.802146i	−1.33635	+	2.28346i	−2.39601	+	1.91076i	2.15950	−	2.08244i	−0.378088	+	2.50845i
47.20	−0.578520	−	0.623496i	−0.623215	−	1.61605i	0.0953980	−	1.27300i	0.0546144	+	0.0684843i	−0.647056	+	1.32349i	−1.63124	−	2.08304i	−2.17887	+	1.73759i	−2.22320	+	2.01429i	0.0111042	−	0.0736714i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	441.2.bd.a	✓	648
9.d	odd	6	1		441.2.bn.a	yes	648
49.h	odd	42	1		441.2.bn.a	yes	648
441.bd	even	42	1	inner	441.2.bd.a	✓	648

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
441.2.bd.a	✓	648	1.a	even	1	1	trivial
441.2.bd.a	✓	648	441.bd	even	42	1	inner
441.2.bn.a	yes	648	9.d	odd	6	1
441.2.bn.a	yes	648	49.h	odd	42	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(441, [\chi])\).