Newform orbit 660.2.i.a

• Label: 660.2.i.a
• Level: $660$
• Weight: $2$
• Character orbit: 660.i
• Analytic conductor: $5.270$
• Analytic rank: $0$
• Dimension: $36$
• 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.i (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.27012653340\)
Analytic rank:	\(0\)
Dimension:	\(36\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 36 q^{3} + 36 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
439.1	−1.40632	−	0.149231i	−1.00000			1.95546	+	0.419733i	1.38929	−	1.75210i	1.40632	+	0.149231i			1.65784i	−2.68736	−	0.882094i	1.00000			−2.21526	+	2.25669i
439.2	−1.40632	+	0.149231i	−1.00000			1.95546	−	0.419733i	1.38929	+	1.75210i	1.40632	−	0.149231i		−	1.65784i	−2.68736	+	0.882094i	1.00000			−2.21526	−	2.25669i
439.3	−1.39913	−	0.205967i	−1.00000			1.91516	+	0.576351i	−1.69611	+	1.45712i	1.39913	+	0.205967i			4.68645i	−2.56085	−	1.20085i	1.00000			2.67321	−	1.68937i
439.4	−1.39913	+	0.205967i	−1.00000			1.91516	−	0.576351i	−1.69611	−	1.45712i	1.39913	−	0.205967i		−	4.68645i	−2.56085	+	1.20085i	1.00000			2.67321	+	1.68937i
439.5	−1.27125	−	0.619623i	−1.00000			1.23213	+	1.57539i	−2.09155	−	0.790837i	1.27125	+	0.619623i		−	1.28548i	−0.590198	−	2.76616i	1.00000			2.16885	+	2.30132i
439.6	−1.27125	+	0.619623i	−1.00000			1.23213	−	1.57539i	−2.09155	+	0.790837i	1.27125	−	0.619623i			1.28548i	−0.590198	+	2.76616i	1.00000			2.16885	−	2.30132i
439.7	−1.13115	−	0.848819i	−1.00000			0.559014	+	1.92029i	−0.432292	+	2.19388i	1.13115	+	0.848819i		−	1.63179i	0.997645	−	2.64664i	1.00000			2.35120	−	2.11468i
439.8	−1.13115	+	0.848819i	−1.00000			0.559014	−	1.92029i	−0.432292	−	2.19388i	1.13115	−	0.848819i			1.63179i	0.997645	+	2.64664i	1.00000			2.35120	+	2.11468i
439.9	−1.06618	−	0.929121i	−1.00000			0.273470	+	1.98122i	2.23191	+	0.136381i	1.06618	+	0.929121i		−	2.40844i	1.54922	−	2.36641i	1.00000			−2.25289	−	2.21912i
439.10	−1.06618	+	0.929121i	−1.00000			0.273470	−	1.98122i	2.23191	−	0.136381i	1.06618	−	0.929121i			2.40844i	1.54922	+	2.36641i	1.00000			−2.25289	+	2.21912i
439.11	−0.862133	−	1.12104i	−1.00000			−0.513455	+	1.93297i	0.699431	−	2.12386i	0.862133	+	1.12104i			2.14050i	2.60960	−	1.09087i	1.00000			−2.98394	+	1.04696i
439.12	−0.862133	+	1.12104i	−1.00000			−0.513455	−	1.93297i	0.699431	+	2.12386i	0.862133	−	1.12104i		−	2.14050i	2.60960	+	1.09087i	1.00000			−2.98394	−	1.04696i
439.13	−0.388391	−	1.35984i	−1.00000			−1.69830	+	1.05630i	0.0421170	−	2.23567i	0.388391	+	1.35984i		−	4.45086i	2.09599	+	1.89916i	1.00000			−3.05650	+	0.811042i
439.14	−0.388391	+	1.35984i	−1.00000			−1.69830	−	1.05630i	0.0421170	+	2.23567i	0.388391	−	1.35984i			4.45086i	2.09599	−	1.89916i	1.00000			−3.05650	−	0.811042i
439.15	−0.350224	−	1.37016i	−1.00000			−1.75469	+	0.959727i	1.81924	+	1.30014i	0.350224	+	1.37016i			3.49286i	1.92952	+	2.06808i	1.00000			1.14425	−	2.94800i
439.16	−0.350224	+	1.37016i	−1.00000			−1.75469	−	0.959727i	1.81924	−	1.30014i	0.350224	−	1.37016i		−	3.49286i	1.92952	−	2.06808i	1.00000			1.14425	+	2.94800i
439.17	−0.124926	−	1.40869i	−1.00000			−1.96879	+	0.351962i	−1.96204	+	1.07257i	0.124926	+	1.40869i			0.762247i	0.741756	+	2.72943i	1.00000			1.75603	+	2.62990i
439.18	−0.124926	+	1.40869i	−1.00000			−1.96879	−	0.351962i	−1.96204	−	1.07257i	0.124926	−	1.40869i		−	0.762247i	0.741756	−	2.72943i	1.00000			1.75603	−	2.62990i
439.19	0.124926	−	1.40869i	−1.00000			−1.96879	−	0.351962i	−1.96204	−	1.07257i	−0.124926	+	1.40869i			0.762247i	−0.741756	+	2.72943i	1.00000			−1.75603	+	2.62990i
439.20	0.124926	+	1.40869i	−1.00000			−1.96879	+	0.351962i	−1.96204	+	1.07257i	−0.124926	−	1.40869i		−	0.762247i	−0.741756	−	2.72943i	1.00000			−1.75603	−	2.62990i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
11.b	odd	2	1	inner
20.d	odd	2	1	inner
220.g	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.i.a	✓	36
4.b	odd	2	1		660.2.i.b	yes	36
5.b	even	2	1		660.2.i.b	yes	36
11.b	odd	2	1	inner	660.2.i.a	✓	36
20.d	odd	2	1	inner	660.2.i.a	✓	36
44.c	even	2	1		660.2.i.b	yes	36
55.d	odd	2	1		660.2.i.b	yes	36
220.g	even	2	1	inner	660.2.i.a	✓	36

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
660.2.i.a	✓	36	1.a	even	1	1	trivial
660.2.i.a	✓	36	11.b	odd	2	1	inner
660.2.i.a	✓	36	20.d	odd	2	1	inner
660.2.i.a	✓	36	220.g	even	2	1	inner
660.2.i.b	yes	36	4.b	odd	2	1
660.2.i.b	yes	36	5.b	even	2	1
660.2.i.b	yes	36	44.c	even	2	1
660.2.i.b	yes	36	55.d	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T23^9 - 132*T23^7 + 228*T23^6 + 5536*T23^5 - 18672*T23^4 - 56672*T23^3 + 351488*T23^2 - 564224*T23 + 299008