Newform orbit 441.2.ba.a

• Label: 441.2.ba.a
• Level: $441$
• Weight: $2$
• Character orbit: 441.ba
• Analytic conductor: $3.521$
• Analytic rank: $0$
• Dimension: $648$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	441.ba (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.52140272914\)
Analytic rank:	\(0\)
Dimension:	\(648\)
Relative dimension:	\(54\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 648 q - 5 q^{2} - 10 q^{3} + 47 q^{4} - 9 q^{5} - 34 q^{6} - 7 q^{7} - 28 q^{8}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
22.1	−2.28558	−	1.55828i	−0.781588	−	1.54568i	2.06496	+	5.26143i	−0.806726	−	0.748533i	−0.622222	+	4.75071i	1.94410	+	1.79457i	2.24807	−	9.84945i	−1.77824	+	2.41617i	0.677414	+	2.96794i
22.2	−2.20796	−	1.50536i	−1.57686	+	0.716598i	1.87830	+	4.78582i	2.79913	+	2.59721i	4.56039	+	0.791525i	−2.53111	−	0.770385i	1.86790	−	8.18382i	1.97297	−	2.25995i	−2.27063	−	9.94826i
22.3	−2.13496	−	1.45559i	1.68474	+	0.402062i	1.70863	+	4.35352i	2.61229	+	2.42385i	−3.01161	−	3.31068i	2.64469	+	0.0747901i	1.53912	−	6.74332i	2.67669	+	1.35474i	−2.04900	−	8.97724i
22.4	−2.12732	−	1.45038i	1.48498	+	0.891527i	1.69120	+	4.30910i	−2.50301	−	2.32245i	−1.86598	−	4.05036i	−2.17833	+	1.50162i	1.50627	−	6.59941i	1.41036	+	2.64781i	1.95625	+	8.57090i
22.5	−2.11562	−	1.44241i	−0.314751	+	1.70321i	1.66464	+	4.24143i	−1.01272	−	0.939663i	3.12262	−	3.14936i	1.62237	−	2.08995i	1.45657	−	6.38167i	−2.80186	−	1.07218i	0.787149	+	3.44872i
22.6	−2.08565	−	1.42197i	1.06450	−	1.36632i	1.59724	+	4.06970i	0.525543	+	0.487633i	−4.16304	+	1.33598i	−2.64570	−	0.0162007i	1.33231	−	5.83723i	−0.733680	−	2.90890i	−0.402698	−	1.76434i
22.7	−1.80236	−	1.22883i	−1.63261	+	0.578434i	1.00781	+	2.56785i	−1.67523	−	1.55438i	3.65335	+	0.963652i	−0.0954394	+	2.64403i	0.368203	−	1.61320i	2.33083	−	1.88871i	1.10930	+	4.86014i
22.8	−1.76857	−	1.20579i	−1.66628	−	0.472785i	0.943225	+	2.40330i	−0.0377034	−	0.0349836i	2.37684	+	2.84533i	1.13677	−	2.38909i	0.277100	−	1.21405i	2.55295	+	1.57558i	0.0244981	+	0.107333i
22.9	−1.71062	−	1.16628i	−0.227483	−	1.71705i	0.835328	+	2.12838i	1.88227	+	1.74649i	−1.61342	+	3.20252i	1.73079	−	2.00109i	0.131960	−	0.578153i	−2.89650	+	0.781197i	−1.18295	−	5.18284i
22.10	−1.58020	−	1.07736i	−0.0140747	−	1.73199i	0.605641	+	1.54315i	−3.06138	−	2.84054i	−1.84374	+	2.75206i	−1.13241	−	2.39116i	−0.145657	+	0.638167i	−2.99960	+	0.0487547i	1.77730	+	7.78684i
22.11	−1.53495	−	1.04651i	1.72901	−	0.102560i	0.530198	+	1.35092i	−0.256118	−	0.237642i	−2.76127	−	1.65200i	−0.648127	−	2.56514i	−0.226850	+	0.993895i	2.97896	−	0.354656i	0.144432	+	0.632798i
22.12	−1.50607	−	1.02682i	−0.181074	+	1.72256i	0.483202	+	1.23118i	0.989195	+	0.917838i	2.04147	−	2.40836i	−2.54327	+	0.729243i	−0.274759	+	1.20380i	−2.93442	−	0.623821i	−0.547340	−	2.39805i
22.13	−1.48693	−	1.01377i	1.13536	−	1.30804i	0.452539	+	1.15305i	1.39969	+	1.29873i	−3.01424	+	0.793970i	−0.229184	+	2.63581i	−0.304877	+	1.33575i	−0.421935	−	2.97018i	−0.764633	−	3.35008i
22.14	−1.31403	−	0.895892i	1.71871	+	0.214588i	0.193375	+	0.492712i	−1.33434	−	1.23808i	−2.06619	−	1.82175i	2.50568	+	0.849447i	−0.520469	+	2.28032i	2.90790	+	0.737628i	0.644172	+	2.82230i
22.15	−1.21486	−	0.828276i	−1.14380	+	1.30067i	0.0591562	+	0.150728i	2.23394	+	2.07279i	2.46686	−	0.632745i	2.42421	+	1.05980i	−0.601388	+	2.63485i	−0.383462	−	2.97539i	−0.997073	−	4.36846i
22.16	−1.20704	−	0.822949i	−1.50963	−	0.849130i	0.0490286	+	0.124923i	−0.415450	−	0.385481i	1.12340	+	2.26728i	−2.62355	+	0.341999i	−0.606532	+	2.65739i	1.55796	+	2.56374i	0.184235	+	0.807187i
22.17	−1.11972	−	0.763412i	0.735246	+	1.56825i	−0.0597084	−	0.152135i	−1.91052	−	1.77271i	0.373953	−	2.31730i	2.60358	+	0.470493i	−0.652406	+	2.85838i	−1.91883	+	2.30610i	0.785945	+	3.44345i
22.18	−0.946246	−	0.645140i	1.51934	+	0.831632i	−0.251505	−	0.640825i	1.33307	+	1.23691i	−0.901149	−	1.76711i	−1.78159	+	1.95600i	−0.685118	+	3.00170i	1.61678	+	2.52706i	−0.463433	−	2.03043i
22.19	−0.912794	−	0.622332i	−0.609005	+	1.62145i	−0.284787	−	0.725626i	−1.69424	−	1.57203i	1.56498	−	1.10105i	−1.84008	−	1.90108i	−0.683292	+	2.99370i	−2.25823	−	1.97495i	0.568171	+	2.48932i
22.20	−0.819051	−	0.558419i	−0.438134	−	1.67572i	−0.371670	−	0.947000i	−1.06857	−	0.991491i	−0.576900	+	1.61716i	0.639861	+	2.56721i	−0.665577	+	2.91608i	−2.61608	+	1.46838i	0.321548	+	1.40879i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.c	even	3	1	inner
49.e	even	7	1	inner
441.ba	even	21	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	441.2.ba.a	✓	648
9.c	even	3	1	inner	441.2.ba.a	✓	648
49.e	even	7	1	inner	441.2.ba.a	✓	648
441.ba	even	21	1	inner	441.2.ba.a	✓	648

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
441.2.ba.a	✓	648	1.a	even	1	1	trivial
441.2.ba.a	✓	648	9.c	even	3	1	inner
441.2.ba.a	✓	648	49.e	even	7	1	inner
441.2.ba.a	✓	648	441.ba	even	21	1	inner

Hecke kernels

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