Newform orbit 660.2.l.a

• Label: 660.2.l.a
• Level: $660$
• Weight: $2$
• Character orbit: 660.l
• Analytic conductor: $5.270$
• Analytic rank: $0$
• Dimension: $60$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 660 = 2^{2} \cdot 3 \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	660.l (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.27012653340\)
Analytic rank:	\(0\)
Dimension:	\(60\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 60 q + 2 q^{4}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
419.1	−1.40092	−	0.193477i	1.35050	+	1.08450i	1.92513	+	0.542091i	−1.91130	−	1.16057i	−1.68211	−	1.78059i	−2.97756			−2.59207	−	1.13189i	0.647709	+	2.92924i	2.45303	+	1.99566i
419.2	−1.40092	+	0.193477i	1.35050	−	1.08450i	1.92513	−	0.542091i	−1.91130	+	1.16057i	−1.68211	+	1.78059i	−2.97756			−2.59207	+	1.13189i	0.647709	−	2.92924i	2.45303	−	1.99566i
419.3	−1.39451	−	0.235277i	−1.68374	+	0.406232i	1.88929	+	0.656191i	1.79493	+	1.33351i	2.44356	−	0.170346i	−0.561324			−2.48024	−	1.35957i	2.66995	−	1.36798i	−2.18929	−	2.28189i
419.4	−1.39451	+	0.235277i	−1.68374	−	0.406232i	1.88929	−	0.656191i	1.79493	−	1.33351i	2.44356	+	0.170346i	−0.561324			−2.48024	+	1.35957i	2.66995	+	1.36798i	−2.18929	+	2.28189i
419.5	−1.39001	−	0.260540i	−0.722480	+	1.57417i	1.86424	+	0.724305i	−0.565386	−	2.16341i	1.41439	−	1.99988i	0.863256			−2.40259	−	1.49250i	−1.95605	−	2.27462i	0.222236	+	3.15446i
419.6	−1.39001	+	0.260540i	−0.722480	−	1.57417i	1.86424	−	0.724305i	−0.565386	+	2.16341i	1.41439	+	1.99988i	0.863256			−2.40259	+	1.49250i	−1.95605	+	2.27462i	0.222236	−	3.15446i
419.7	−1.36051	−	0.386021i	0.295073	−	1.70673i	1.70198	+	1.05037i	1.82303	−	1.29481i	−1.06028	+	2.20812i	4.81816			−1.91009	−	2.08604i	−2.82586	−	1.00722i	−2.98008	+	1.05788i
419.8	−1.36051	+	0.386021i	0.295073	+	1.70673i	1.70198	−	1.05037i	1.82303	+	1.29481i	−1.06028	−	2.20812i	4.81816			−1.91009	+	2.08604i	−2.82586	+	1.00722i	−2.98008	−	1.05788i
419.9	−1.21181	−	0.729051i	−0.768338	−	1.55231i	0.936970	+	1.76694i	−1.81161	−	1.31075i	−0.200630	+	2.44126i	−1.08462			0.152760	−	2.82430i	−1.81931	+	2.38539i	1.23973	+	2.90914i
419.10	−1.21181	+	0.729051i	−0.768338	+	1.55231i	0.936970	−	1.76694i	−1.81161	+	1.31075i	−0.200630	−	2.44126i	−1.08462			0.152760	+	2.82430i	−1.81931	−	2.38539i	1.23973	−	2.90914i
419.11	−1.18563	−	0.770892i	1.53393	−	0.804394i	0.811450	+	1.82799i	1.33670	−	1.79255i	−2.43878	−	0.228781i	−4.50056			0.447101	−	2.79287i	1.70590	−	2.46777i	−2.96670	+	1.09486i
419.12	−1.18563	+	0.770892i	1.53393	+	0.804394i	0.811450	−	1.82799i	1.33670	+	1.79255i	−2.43878	+	0.228781i	−4.50056			0.447101	+	2.79287i	1.70590	+	2.46777i	−2.96670	−	1.09486i
419.13	−1.14827	−	0.825510i	1.72338	+	0.173113i	0.637067	+	1.89582i	0.210438	+	2.22614i	−1.83600	−	1.62145i	1.54251			0.833494	−	2.70283i	2.94006	+	0.596678i	1.59606	−	2.72994i
419.14	−1.14827	+	0.825510i	1.72338	−	0.173113i	0.637067	−	1.89582i	0.210438	−	2.22614i	−1.83600	+	1.62145i	1.54251			0.833494	+	2.70283i	2.94006	−	0.596678i	1.59606	+	2.72994i
419.15	−1.06294	−	0.932828i	−1.21747	+	1.23198i	0.259664	+	1.98307i	−0.0516122	+	2.23547i	2.44331	−	0.173832i	−4.05669			1.57386	−	2.35010i	−0.0355578	−	2.99979i	2.14017	−	2.32802i
419.16	−1.06294	+	0.932828i	−1.21747	−	1.23198i	0.259664	−	1.98307i	−0.0516122	−	2.23547i	2.44331	+	0.173832i	−4.05669			1.57386	+	2.35010i	−0.0355578	+	2.99979i	2.14017	+	2.32802i
419.17	−0.877374	−	1.10915i	−1.70802	+	0.287526i	−0.460431	+	1.94628i	−2.22842	+	0.184775i	1.81748	+	1.64218i	2.27349			2.56269	−	1.19693i	2.83466	−	0.982199i	2.16010	+	2.30954i
419.18	−0.877374	+	1.10915i	−1.70802	−	0.287526i	−0.460431	−	1.94628i	−2.22842	−	0.184775i	1.81748	−	1.64218i	2.27349			2.56269	+	1.19693i	2.83466	+	0.982199i	2.16010	−	2.30954i
419.19	−0.729321	−	1.21165i	−1.21847	−	1.23099i	−0.936182	+	1.76736i	2.19888	−	0.406107i	−0.602866	+	2.37414i	−0.565325			2.82420	−	0.154649i	−0.0306548	+	2.99984i	−2.09575	−	2.36809i
419.20	−0.729321	+	1.21165i	−1.21847	+	1.23099i	−0.936182	−	1.76736i	2.19888	+	0.406107i	−0.602866	−	2.37414i	−0.565325			2.82420	+	0.154649i	−0.0306548	−	2.99984i	−2.09575	+	2.36809i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
12.b	even	2	1	inner
60.h	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	660.2.l.a	✓	60
3.b	odd	2	1		660.2.l.b	yes	60
4.b	odd	2	1		660.2.l.b	yes	60
5.b	even	2	1	inner	660.2.l.a	✓	60
12.b	even	2	1	inner	660.2.l.a	✓	60
15.d	odd	2	1		660.2.l.b	yes	60
20.d	odd	2	1		660.2.l.b	yes	60
60.h	even	2	1	inner	660.2.l.a	✓	60

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
660.2.l.a	✓	60	1.a	even	1	1	trivial
660.2.l.a	✓	60	5.b	even	2	1	inner
660.2.l.a	✓	60	12.b	even	2	1	inner
660.2.l.a	✓	60	60.h	even	2	1	inner
660.2.l.b	yes	60	3.b	odd	2	1
660.2.l.b	yes	60	4.b	odd	2	1
660.2.l.b	yes	60	15.d	odd	2	1
660.2.l.b	yes	60	20.d	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T59^15 - 435*T59^13 - 162*T59^12 + 66100*T59^11 + 56936*T59^10 - 4371120*T59^9 - 5642144*T59^8 + 133881664*T59^7 + 213628544*T59^6 - 1876452352*T59^5 - 3259062272*T59^4 + 10660929536*T59^3 + 17365991424*T59^2 - 19097321472*T59 - 22649241600