Newform orbit 441.2.bn.a

• Label: 441.2.bn.a
• Level: $441$
• Weight: $2$
• Character orbit: 441.bn
• 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.bn (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.

\(\operatorname{Tr}(f)(q) = \) \( 648 q - 21 q^{2} - 11 q^{3} + 99 q^{4} - 18 q^{5} - 34 q^{6} - 5 q^{7} - 23 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
5.1	−2.16353	+	1.72536i	0.581018	−	1.63169i	1.25896	−	5.51585i	2.91211	+	1.98544i	1.55820	+	4.53267i	−2.07232	+	1.64484i	4.39168	+	9.11943i	−2.32484	−	1.89608i	−9.72603	+	0.728865i
5.2	−2.15589	+	1.71927i	−1.44111	−	0.960829i	1.24695	−	5.46324i	−1.16954	−	0.797380i	4.75880	−	0.406215i	1.71430	−	2.01524i	4.31162	+	8.95316i	1.15362	+	2.76933i	3.89231	−	0.291689i
5.3	−2.01930	+	1.61034i	0.161619	+	1.72449i	1.03935	−	4.55368i	0.846045	+	0.576824i	−3.10338	−	3.22201i	−1.90950	−	1.83134i	2.99296	+	6.21494i	−2.94776	+	0.557422i	−2.63730	+	0.197639i
5.4	−1.98600	+	1.58378i	1.73017	+	0.0806248i	0.990793	−	4.34095i	−3.40470	−	2.32129i	−3.56382	+	2.58010i	−2.11613	−	1.58809i	2.70311	+	5.61307i	2.98700	+	0.278990i	10.4382	−	0.782232i
5.5	−1.96009	+	1.56312i	1.60753	+	0.644869i	0.953564	−	4.17784i	0.688287	+	0.469266i	−4.15890	+	1.24876i	1.84507	+	1.89623i	2.48585	+	5.16192i	2.16829	+	2.07329i	−2.08262	+	0.156071i
5.6	−1.86842	+	1.49001i	−1.41768	+	0.995078i	0.825803	−	3.61808i	−1.88505	−	1.28520i	1.16614	−	3.97158i	−0.922050	+	2.47988i	1.77425	+	3.68427i	1.01964	−	2.82141i	5.43702	−	0.407448i
5.7	−1.70698	+	1.36127i	0.921293	−	1.46670i	0.615683	−	2.69748i	−1.79313	−	1.22254i	0.423952	+	3.75777i	2.16539	+	1.52022i	0.726444	+	1.50848i	−1.30244	−	2.70253i	4.72505	−	0.354094i
5.8	−1.65706	+	1.32146i	−1.64577	−	0.539842i	0.554552	−	2.42965i	0.208441	+	0.142112i	3.44053	−	1.28028i	−2.12758	+	1.57270i	0.452565	+	0.939762i	2.41714	+	1.77691i	−0.533196	+	0.0399575i
5.9	−1.53265	+	1.22225i	−0.923591	+	1.46526i	0.410090	−	1.79672i	−2.34879	−	1.60137i	−0.375366	−	3.37459i	1.90530	−	1.83571i	−0.133600	−	0.277423i	−1.29396	−	2.70660i	5.55716	−	0.416451i
5.10	−1.52262	+	1.21425i	0.820482	+	1.52539i	0.398931	−	1.74783i	0.733166	+	0.499864i	−3.10149	−	1.32632i	2.50828	−	0.841742i	−0.175099	−	0.363598i	−1.65362	+	2.50311i	−1.72330	+	0.129143i
5.11	−1.50680	+	1.20163i	−0.277954	−	1.70960i	0.381481	−	1.67138i	0.918605	+	0.626294i	2.47313	+	2.24203i	0.418961	−	2.61237i	−0.238856	−	0.495990i	−2.84548	+	0.950381i	−2.13673	+	0.160126i
5.12	−1.50264	+	1.19832i	−1.36206	−	1.06995i	0.376930	−	1.65144i	2.30957	+	1.57464i	3.32883	−	0.0244174i	1.77434	+	1.96258i	−0.255253	−	0.530039i	0.710396	+	2.91468i	−5.35737	+	0.401480i
5.13	−1.47430	+	1.17571i	1.71213	−	0.261949i	0.346209	−	1.51684i	3.14585	+	2.14480i	−2.21621	+	2.39916i	0.361407	−	2.62095i	−0.363391	−	0.754589i	2.86277	−	0.896980i	−7.15958	+	0.536537i
5.14	−1.33541	+	1.06496i	1.50168	−	0.863111i	0.204153	−	0.894452i	0.337281	+	0.229955i	−1.08619	+	2.75183i	−2.64519	+	0.0547315i	−0.802272	−	1.66593i	1.51008	−	2.59223i	−0.695301	+	0.0521056i
5.15	−1.24435	+	0.992336i	−0.323588	+	1.70156i	0.118634	−	0.519770i	3.37511	+	2.30111i	−1.28586	−	2.43844i	−0.700800	+	2.55125i	−1.01296	−	2.10343i	−2.79058	−	1.10121i	−6.48329	+	0.485855i
5.16	−1.14829	+	0.915734i	−0.441504	−	1.67484i	0.0349690	−	0.153209i	−1.40572	−	0.958401i	2.04068	+	1.51890i	−2.52036	−	0.804862i	−1.17437	−	2.43860i	−2.61015	+	1.47889i	2.49182	−	0.186736i
5.17	−1.08970	+	0.869006i	0.851533	+	1.50827i	−0.0127693	+	0.0559462i	−2.47373	−	1.68656i	−2.23861	−	0.903578i	−0.888042	+	2.49226i	−1.24418	−	2.58356i	−1.54978	+	2.56869i	4.16125	−	0.311843i
5.18	−0.920093	+	0.733749i	−1.68844	+	0.386212i	−0.136860	+	0.599621i	1.62477	+	1.10775i	1.27014	−	1.59424i	1.62874	−	2.08500i	−1.33527	−	2.77272i	2.70168	−	1.30419i	−2.30775	+	0.172942i
5.19	−0.781758	+	0.623431i	−1.03674	−	1.38750i	−0.222563	+	0.975111i	−3.13042	−	2.13429i	1.67549	+	0.438356i	2.20194	+	1.46679i	−1.30161	−	2.70282i	−0.850336	+	2.87697i	3.77782	−	0.283108i
5.20	−0.744514	+	0.593730i	−1.04233	+	1.38331i	−0.243256	+	1.06577i	0.397639	+	0.271106i	−0.0452820	−	1.64876i	−2.38053	−	1.15459i	−1.27802	−	2.65384i	−0.827090	−	2.88373i	−0.457012	+	0.0342483i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
441.bn	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.bn.a	yes	648
9.d	odd	6	1		441.2.bd.a	✓	648
49.h	odd	42	1		441.2.bd.a	✓	648
441.bn	even	42	1	inner	441.2.bn.a	yes	648

    

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

Hecke kernels

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