Newform orbit 132.3.e.a

• Label: 132.3.e.a
• Level: $132$
• Weight: $3$
• Character orbit: 132.e
• Analytic conductor: $3.597$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 132 = 2^{2} \cdot 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	132.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.59673948956\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-11}) \)
Defining polynomial:	\( x^{2} - x + 3 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{-11})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + (-b + 3) * q^3 + (-2*b + 1) * q^5 + 2 * q^7 + (-5*b + 6) * q^9 + (-2*b + 1) * q^11 + (-5*b - 3) * q^15 + (4*b - 2) * q^17 + 4 * q^19 + (-2*b + 6) * q^21 + (2*b - 1) * q^23 + 14 * q^25 + (-16*b + 3) * q^27 + (24*b - 12) * q^29 + 9 * q^31 + (-5*b - 3) * q^33 + (-4*b + 2) * q^35 - 51 * q^37 + (48*b - 24) * q^41 + 22 * q^43 + (-7*b - 24) * q^45 + (24*b - 12) * q^47 - 45 * q^49 + (10*b + 6) * q^51 + (40*b - 20) * q^53 - 11 * q^55 + (-4*b + 12) * q^57 + (-42*b + 21) * q^59 - 80 * q^61 + (-10*b + 12) * q^63 + q^67 + (5*b + 3) * q^69 + (-30*b + 15) * q^71 + 108 * q^73 + (-14*b + 42) * q^75 + (-4*b + 2) * q^77 + 122 * q^79 + (-35*b - 39) * q^81 + (-52*b + 26) * q^83 + 22 * q^85 + (60*b + 36) * q^87 + (-58*b + 29) * q^89 + (-9*b + 27) * q^93 + (-8*b + 4) * q^95 + 79 * q^97 + (-7*b - 24) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + 5 * q^3 + 4 * q^7 + 7 * q^9 - 11 * q^15 + 8 * q^19 + 10 * q^21 + 28 * q^25 - 10 * q^27 + 18 * q^31 - 11 * q^33 - 102 * q^37 + 44 * q^43 - 55 * q^45 - 90 * q^49 + 22 * q^51 - 22 * q^55 + 20 * q^57 - 160 * q^61 + 14 * q^63 + 2 * q^67 + 11 * q^69 + 216 * q^73 + 70 * q^75 + 244 * q^79 - 113 * q^81 + 44 * q^85 + 132 * q^87 + 45 * q^93 + 158 * q^97 - 55 * q^99

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/132\mathbb{Z}\right)^\times\).

\(n\)	\(13\)	\(67\)	\(89\)
\(\chi(n)\)	\(1\)	\(1\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

89.1

0.500000	+	1.65831i
0.500000	−	1.65831i

0

2.50000

−

1.65831i

0

−

3.31662i

0

2.00000

0

3.50000

−

8.29156i

0

89.2

0

2.50000

+

1.65831i

0

3.31662i

0

2.00000

0

3.50000

+

8.29156i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	132.3.e.a	✓	2
3.b	odd	2	1	inner	132.3.e.a	✓	2
4.b	odd	2	1		528.3.i.b		2
11.b	odd	2	1		1452.3.e.d		2
12.b	even	2	1		528.3.i.b		2
33.d	even	2	1		1452.3.e.d		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
132.3.e.a	✓	2	1.a	even	1	1	trivial
132.3.e.a	✓	2	3.b	odd	2	1	inner
528.3.i.b		2	4.b	odd	2	1
528.3.i.b		2	12.b	even	2	1
1452.3.e.d		2	11.b	odd	2	1
1452.3.e.d		2	33.d	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 11 \) acting on \(S_{3}^{\mathrm{new}}(132, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} - 5T + 9 \)
$5$	\( T^{2} + 11 \)
$7$	\( (T - 2)^{2} \)
$11$	\( T^{2} + 11 \)