Newform orbit 880.3.i.b

• Label: 880.3.i.b
• Level: $880$
• Weight: $3$
• Character orbit: 880.i
• Analytic conductor: $23.978$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -11
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 880 = 2^{4} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	880.i (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(23.9782632637\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-11}) \)
Defining polynomial:	\( x^{2} - x + 3 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 55)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-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 + (-2*b + 1) * q^3 + (3*b - 1) * q^5 - 2 * q^9 - 11 * q^11 + (-b + 17) * q^15 + (18*b - 9) * q^23 + (3*b - 26) * q^25 + (-14*b + 7) * q^27 - 37 * q^31 + (22*b - 11) * q^33 + (42*b - 21) * q^37 + (-6*b + 2) * q^45 + (48*b - 24) * q^47 - 49 * q^49 + (-48*b + 24) * q^53 + (-33*b + 11) * q^55 - 107 * q^59 + (78*b - 39) * q^67 + 99 * q^69 + 133 * q^71 + (49*b - 8) * q^75 - 95 * q^81 + 97 * q^89 + (74*b - 37) * q^93 + (102*b - 51) * q^97 + 22 * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{5} - 4 q^{9} - 22 q^{11} + 33 q^{15} - 49 q^{25} - 74 q^{31} - 2 q^{45} - 98 q^{49} - 11 q^{55} - 214 q^{59} + 198 q^{69} + 266 q^{71} + 33 q^{75} - 190 q^{81} + 194 q^{89} + 44 q^{99}+O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/880\mathbb{Z}\right)^\times\).

\(n\)	\(111\)	\(177\)	\(321\)	\(661\)
\(\chi(n)\)	\(1\)	\(-1\)	\(-1\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

769.1

0.500000	+	1.65831i
0.500000	−	1.65831i

0

−

3.31662i

0

0.500000

+

4.97494i

0

0

0

−2.00000

0

769.2

0

3.31662i

0

0.500000

−

4.97494i

0

0

0

−2.00000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
11.b	odd	2	1	CM by \(\Q(\sqrt{-11}) \)
5.b	even	2	1	inner
55.d	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	880.3.i.b		2
4.b	odd	2	1		55.3.d.a	✓	2
5.b	even	2	1	inner	880.3.i.b		2
11.b	odd	2	1	CM	880.3.i.b		2
12.b	even	2	1		495.3.h.a		2
20.d	odd	2	1		55.3.d.a	✓	2
20.e	even	4	2		275.3.c.d		2
44.c	even	2	1		55.3.d.a	✓	2
55.d	odd	2	1	inner	880.3.i.b		2
60.h	even	2	1		495.3.h.a		2
132.d	odd	2	1		495.3.h.a		2
220.g	even	2	1		55.3.d.a	✓	2
220.i	odd	4	2		275.3.c.d		2
660.g	odd	2	1		495.3.h.a		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
55.3.d.a	✓	2	4.b	odd	2	1
55.3.d.a	✓	2	20.d	odd	2	1
55.3.d.a	✓	2	44.c	even	2	1
55.3.d.a	✓	2	220.g	even	2	1
275.3.c.d		2	20.e	even	4	2
275.3.c.d		2	220.i	odd	4	2
495.3.h.a		2	12.b	even	2	1
495.3.h.a		2	60.h	even	2	1
495.3.h.a		2	132.d	odd	2	1
495.3.h.a		2	660.g	odd	2	1
880.3.i.b		2	1.a	even	1	1	trivial
880.3.i.b		2	5.b	even	2	1	inner
880.3.i.b		2	11.b	odd	2	1	CM
880.3.i.b		2	55.d	odd	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(880, [\chi])\):

\( T_{3}^{2} + 11 \)

\( T_{7} \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} + 11 \)
$5$	\( T^{2} - T + 25 \)
$7$	\( T^{2} \)
$11$	\( (T + 11)^{2} \)