Newform orbit 561.2.z.a

• Label: 561.2.z.a
• Level: $561$
• Weight: $2$
• Character orbit: 561.z
• Analytic conductor: $4.480$
• Analytic rank: $0$
• Dimension: $480$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 561 = 3 \cdot 11 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	561.z (of order \(16\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.47960755339\)
Analytic rank:	\(0\)
Dimension:	\(480\)
Relative dimension:	\(60\) over \(\Q(\zeta_{16})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 480 q+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
23.1	−1.05612	+	2.54971i	−1.59799	−	0.668157i	−3.97139	−	3.97139i	−1.83853	−	1.22847i	3.39128	−	3.36874i	2.61220	+	3.90943i	9.22074	−	3.81935i	2.10713	+	2.13541i	5.07395	−	3.39031i
23.2	−1.03899	+	2.50834i	−0.105610	−	1.72883i	−3.79804	−	3.79804i	1.70884	+	1.14181i	4.44621	+	1.53132i	−1.16702	−	1.74658i	8.45622	−	3.50268i	−2.97769	+	0.365165i	−4.63951	+	3.10002i
23.3	−1.01582	+	2.45241i	1.27318	+	1.17432i	−3.56822	−	3.56822i	−2.50981	−	1.67700i	−4.17323	+	1.92946i	−0.769883	−	1.15221i	7.47059	−	3.09442i	0.241959	+	2.99023i	6.66222	−	4.45155i
23.4	−1.00655	+	2.43003i	1.72728	−	0.128432i	−3.47769	−	3.47769i	1.21924	+	0.814671i	−1.42651	+	4.32662i	−0.454154	−	0.679689i	7.09129	−	2.93731i	2.96701	−	0.443677i	−3.20690	+	2.14278i
23.5	−0.979830	+	2.36552i	0.0539671	+	1.73121i	−3.22140	−	3.22140i	1.92270	+	1.28471i	−4.14809	−	1.56863i	2.11493	+	3.16521i	6.04566	−	2.50419i	−2.99418	+	0.186857i	−4.92291	+	3.28938i
23.6	−0.974477	+	2.35259i	−1.57334	+	0.724299i	−3.17088	−	3.17088i	2.86037	+	1.91124i	−0.170802	−	4.40724i	−1.46637	−	2.19458i	5.84456	−	2.42090i	1.95078	−	2.27913i	−7.28373	+	4.86683i
23.7	−0.907823	+	2.19168i	−1.73168	+	0.0357757i	−2.56510	−	2.56510i	−1.88639	−	1.26044i	1.49365	−	3.82777i	−2.42276	−	3.62592i	3.56718	−	1.47757i	2.99744	−	0.123904i	4.47499	−	2.99009i
23.8	−0.891396	+	2.15202i	1.18256	−	1.26553i	−2.42239	−	2.42239i	−0.847631	−	0.566369i	1.66931	+	3.67297i	0.416056	+	0.622671i	3.06829	−	1.27093i	−0.203114	−	2.99312i	1.97441	−	1.31926i
23.9	−0.841186	+	2.03080i	0.00163296	−	1.73205i	−2.00235	−	2.00235i	−2.88189	−	1.92562i	3.51608	+	1.46029i	0.541085	+	0.809791i	1.68912	−	0.699655i	−2.99999	−	0.00565673i	6.33475	−	4.23274i
23.10	−0.828119	+	1.99926i	0.283448	+	1.70870i	−1.89703	−	1.89703i	0.588202	+	0.393024i	−3.65086	−	0.848323i	−1.52907	−	2.28842i	1.36511	−	0.565446i	−2.83931	+	0.968654i	−1.27286	+	0.850497i
23.11	−0.759895	+	1.83455i	−0.503075	+	1.65738i	−1.37391	−	1.37391i	−3.49931	−	2.33816i	−2.65826	−	2.18235i	0.428628	+	0.641488i	−0.104555	+	0.0433080i	−2.49383	−	1.66757i	6.94857	−	4.64289i
23.12	−0.710800	+	1.71602i	−1.24310	−	1.20611i	−1.02529	−	1.02529i	2.17975	+	1.45646i	2.95331	−	1.27589i	0.757363	+	1.13347i	−0.943856	+	0.390958i	0.0906017	+	2.99863i	−4.04869	+	2.70525i
23.13	−0.697106	+	1.68296i	1.68258	+	0.410997i	−0.932196	−	0.932196i	2.87762	+	1.92276i	−1.86463	+	2.54522i	−0.0162799	−	0.0243646i	−1.14724	+	0.475201i	2.66216	+	1.38307i	−5.24195	+	3.50256i
23.14	−0.676535	+	1.63330i	0.322378	−	1.70179i	−0.795758	−	0.795758i	2.39246	+	1.59859i	2.56143	+	1.67786i	2.75394	+	4.12156i	−1.42853	+	0.591717i	−2.79214	−	1.09724i	−4.22957	+	2.82611i
23.15	−0.674599	+	1.62863i	1.06786	−	1.36370i	−0.783126	−	0.783126i	0.442604	+	0.295739i	1.50058	+	2.65909i	−1.99834	−	2.99072i	−1.45354	+	0.602075i	−0.719355	−	2.91248i	−0.780228	+	0.521332i
23.16	−0.635722	+	1.53477i	1.39652	+	1.02457i	−0.537157	−	0.537157i	−0.216567	−	0.144705i	−2.46027	+	1.49199i	2.35280	+	3.52121i	−1.90364	+	0.788514i	0.900532	+	2.86165i	0.359766	−	0.240388i
23.17	−0.548937	+	1.32525i	−1.70618	−	0.298223i	−0.0407443	−	0.0407443i	0.940391	+	0.628349i	1.33181	−	2.09742i	−0.657182	−	0.983542i	−2.57414	+	1.06624i	2.82213	+	1.01765i	−1.34894	+	0.901330i
23.18	−0.530222	+	1.28007i	1.72519	−	0.153984i	0.0567722	+	0.0567722i	−2.51508	−	1.68052i	−0.717625	+	2.29001i	−2.78305	−	4.16512i	−2.66291	+	1.10301i	2.95258	−	0.531304i	3.48474	−	2.32843i
23.19	−0.446393	+	1.07769i	−1.42324	−	0.987112i	0.452067	+	0.452067i	−2.04606	−	1.36713i	1.69912	−	1.09317i	0.513563	+	0.768601i	−2.84437	+	1.17817i	1.05122	+	2.80979i	2.38669	−	1.59474i
23.20	−0.423290	+	1.02191i	1.19639	+	1.25246i	0.549085	+	0.549085i	−1.86139	−	1.24374i	−1.78632	+	0.692451i	0.220877	+	0.330566i	−2.83736	+	1.17527i	−0.137308	+	2.99686i	2.05890	−	1.37571i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
17.e	odd	16	1	inner
51.i	even	16	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	561.2.z.a	✓	480
3.b	odd	2	1	inner	561.2.z.a	✓	480
17.e	odd	16	1	inner	561.2.z.a	✓	480
51.i	even	16	1	inner	561.2.z.a	✓	480

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
561.2.z.a	✓	480	1.a	even	1	1	trivial
561.2.z.a	✓	480	3.b	odd	2	1	inner
561.2.z.a	✓	480	17.e	odd	16	1	inner
561.2.z.a	✓	480	51.i	even	16	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(561, [\chi])\).