Newform orbit 561.2.i.a

• Label: 561.2.i.a
• Level: $561$
• Weight: $2$
• Character orbit: 561.i
• Analytic conductor: $4.480$
• Analytic rank: $0$
• Dimension: $136$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 561 = 3 \cdot 11 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	561.i (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.47960755339\)
Analytic rank:	\(0\)
Dimension:	\(136\)
Relative dimension:	\(68\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 136 q - 8 q^{3} - 144 q^{4}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
98.1		−	2.74981i	−0.946940	+	1.45028i	−5.56145			−2.34423	+	2.34423i	3.98799	+	2.60391i	−0.203045	−	0.203045i			9.79331i	−1.20661	−	2.74665i	6.44620	+	6.44620i
98.2		−	2.74981i	1.45028	−	0.946940i	−5.56145			2.34423	−	2.34423i	−2.60391	−	3.98799i	0.203045	+	0.203045i			9.79331i	1.20661	−	2.74665i	−6.44620	−	6.44620i
98.3		−	2.58390i	−1.56343	−	0.745434i	−4.67653			0.222487	−	0.222487i	−1.92613	+	4.03976i	2.35102	+	2.35102i			6.91588i	1.88865	+	2.33088i	−0.574885	−	0.574885i
98.4		−	2.58390i	−0.745434	−	1.56343i	−4.67653			−0.222487	+	0.222487i	−4.03976	+	1.92613i	−2.35102	−	2.35102i			6.91588i	−1.88865	+	2.33088i	0.574885	+	0.574885i
98.5		−	2.44823i	−1.68153	+	0.415274i	−3.99384			2.13017	−	2.13017i	1.01669	+	4.11678i	−1.37603	−	1.37603i			4.88138i	2.65510	−	1.39659i	−5.21516	−	5.21516i
98.6		−	2.44823i	0.415274	−	1.68153i	−3.99384			−2.13017	+	2.13017i	−4.11678	−	1.01669i	1.37603	+	1.37603i			4.88138i	−2.65510	−	1.39659i	5.21516	+	5.21516i
98.7		−	2.39662i	0.236615	+	1.71581i	−3.74378			1.87345	−	1.87345i	4.11215	−	0.567075i	−2.66025	−	2.66025i			4.17919i	−2.88803	+	0.811973i	−4.48994	−	4.48994i
98.8		−	2.39662i	1.71581	+	0.236615i	−3.74378			−1.87345	+	1.87345i	0.567075	−	4.11215i	2.66025	+	2.66025i			4.17919i	2.88803	+	0.811973i	4.48994	+	4.48994i
98.9		−	2.30711i	−0.892959	+	1.48412i	−3.32275			0.540350	−	0.540350i	3.42404	+	2.06015i	1.50218	+	1.50218i			3.05173i	−1.40525	−	2.65052i	−1.24665	−	1.24665i
98.10		−	2.30711i	1.48412	−	0.892959i	−3.32275			−0.540350	+	0.540350i	−2.06015	−	3.42404i	−1.50218	−	1.50218i			3.05173i	1.40525	−	2.65052i	1.24665	+	1.24665i
98.11		−	2.15161i	0.929363	+	1.46160i	−2.62943			−1.90997	+	1.90997i	3.14480	−	1.99963i	−1.44274	−	1.44274i			1.35428i	−1.27257	+	2.71672i	4.10950	+	4.10950i
98.12		−	2.15161i	1.46160	+	0.929363i	−2.62943			1.90997	−	1.90997i	1.99963	−	3.14480i	1.44274	+	1.44274i			1.35428i	1.27257	+	2.71672i	−4.10950	−	4.10950i
98.13		−	1.88266i	−1.63746	−	0.564570i	−1.54439			−2.61250	+	2.61250i	−1.06289	+	3.08276i	−2.91241	−	2.91241i		−	0.857755i	2.36252	+	1.84892i	4.91843	+	4.91843i
98.14		−	1.88266i	−0.564570	−	1.63746i	−1.54439			2.61250	−	2.61250i	−3.08276	+	1.06289i	2.91241	+	2.91241i		−	0.857755i	−2.36252	+	1.84892i	−4.91843	−	4.91843i
98.15		−	1.76007i	−1.72446	−	0.161940i	−1.09784			0.411171	−	0.411171i	−0.285025	+	3.03517i	0.842559	+	0.842559i		−	1.58787i	2.94755	+	0.558519i	−0.723688	−	0.723688i
98.16		−	1.76007i	−0.161940	−	1.72446i	−1.09784			−0.411171	+	0.411171i	−3.03517	+	0.285025i	−0.842559	−	0.842559i		−	1.58787i	−2.94755	+	0.558519i	0.723688	+	0.723688i
98.17		−	1.70260i	−1.58078	+	0.707900i	−0.898853			−2.18340	+	2.18340i	1.20527	+	2.69145i	1.95813	+	1.95813i		−	1.87481i	1.99775	−	2.23807i	3.71747	+	3.71747i
98.18		−	1.70260i	0.707900	−	1.58078i	−0.898853			2.18340	−	2.18340i	−2.69145	−	1.20527i	−1.95813	−	1.95813i		−	1.87481i	−1.99775	−	2.23807i	−3.71747	−	3.71747i
98.19		−	1.46242i	0.493710	+	1.66020i	−0.138662			−1.30974	+	1.30974i	2.42790	−	0.722010i	1.13205	+	1.13205i		−	2.72205i	−2.51250	+	1.63931i	1.91539	+	1.91539i
98.20		−	1.46242i	1.66020	+	0.493710i	−0.138662			1.30974	−	1.30974i	0.722010	−	2.42790i	−1.13205	−	1.13205i		−	2.72205i	2.51250	+	1.63931i	−1.91539	−	1.91539i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
11.b	odd	2	1	inner
17.c	even	4	1	inner
33.d	even	2	1	inner
51.f	odd	4	1	inner
187.f	odd	4	1	inner
561.i	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	561.2.i.a	✓	136
3.b	odd	2	1	inner	561.2.i.a	✓	136
11.b	odd	2	1	inner	561.2.i.a	✓	136
17.c	even	4	1	inner	561.2.i.a	✓	136
33.d	even	2	1	inner	561.2.i.a	✓	136
51.f	odd	4	1	inner	561.2.i.a	✓	136
187.f	odd	4	1	inner	561.2.i.a	✓	136
561.i	even	4	1	inner	561.2.i.a	✓	136

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
561.2.i.a	✓	136	1.a	even	1	1	trivial
561.2.i.a	✓	136	3.b	odd	2	1	inner
561.2.i.a	✓	136	11.b	odd	2	1	inner
561.2.i.a	✓	136	17.c	even	4	1	inner
561.2.i.a	✓	136	33.d	even	2	1	inner
561.2.i.a	✓	136	51.f	odd	4	1	inner
561.2.i.a	✓	136	187.f	odd	4	1	inner
561.2.i.a	✓	136	561.i	even	4	1	inner

Hecke kernels

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