Properties

Label 11.3.b.a
Level 11
Weight 3
Character orbit 11.b
Self dual yes
Analytic conductor 0.300
Analytic rank 0
Dimension 1
CM discriminant -11
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 11.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.299728290796\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\) \(=\) \( q - 5q^{3} + 4q^{4} - q^{5} + 16q^{9} + O(q^{10}) \) \( q - 5q^{3} + 4q^{4} - q^{5} + 16q^{9} - 11q^{11} - 20q^{12} + 5q^{15} + 16q^{16} - 4q^{20} + 35q^{23} - 24q^{25} - 35q^{27} - 37q^{31} + 55q^{33} + 64q^{36} - 25q^{37} - 44q^{44} - 16q^{45} + 50q^{47} - 80q^{48} + 49q^{49} - 70q^{53} + 11q^{55} + 107q^{59} + 20q^{60} + 64q^{64} + 35q^{67} - 175q^{69} - 133q^{71} + 120q^{75} - 16q^{80} + 31q^{81} - 97q^{89} + 140q^{92} + 185q^{93} + 95q^{97} - 176q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/11\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1
0
0 −5.00000 4.00000 −1.00000 0 0 0 16.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 11.3.b.a 1
3.b odd 2 1 99.3.c.a 1
4.b odd 2 1 176.3.h.a 1
5.b even 2 1 275.3.c.a 1
5.c odd 4 2 275.3.d.a 2
7.b odd 2 1 539.3.c.a 1
8.b even 2 1 704.3.h.b 1
8.d odd 2 1 704.3.h.a 1
11.b odd 2 1 CM 11.3.b.a 1
11.c even 5 4 121.3.d.b 4
11.d odd 10 4 121.3.d.b 4
12.b even 2 1 1584.3.j.a 1
33.d even 2 1 99.3.c.a 1
44.c even 2 1 176.3.h.a 1
55.d odd 2 1 275.3.c.a 1
55.e even 4 2 275.3.d.a 2
77.b even 2 1 539.3.c.a 1
88.b odd 2 1 704.3.h.b 1
88.g even 2 1 704.3.h.a 1
132.d odd 2 1 1584.3.j.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.3.b.a 1 1.a even 1 1 trivial
11.3.b.a 1 11.b odd 2 1 CM
99.3.c.a 1 3.b odd 2 1
99.3.c.a 1 33.d even 2 1
121.3.d.b 4 11.c even 5 4
121.3.d.b 4 11.d odd 10 4
176.3.h.a 1 4.b odd 2 1
176.3.h.a 1 44.c even 2 1
275.3.c.a 1 5.b even 2 1
275.3.c.a 1 55.d odd 2 1
275.3.d.a 2 5.c odd 4 2
275.3.d.a 2 55.e even 4 2
539.3.c.a 1 7.b odd 2 1
539.3.c.a 1 77.b even 2 1
704.3.h.a 1 8.d odd 2 1
704.3.h.a 1 88.g even 2 1
704.3.h.b 1 8.b even 2 1
704.3.h.b 1 88.b odd 2 1
1584.3.j.a 1 12.b even 2 1
1584.3.j.a 1 132.d odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(11, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 2 T )( 1 + 2 T ) \)
$3$ \( 1 + 5 T + 9 T^{2} \)
$5$ \( 1 + T + 25 T^{2} \)
$7$ \( ( 1 - 7 T )( 1 + 7 T ) \)
$11$ \( 1 + 11 T \)
$13$ \( ( 1 - 13 T )( 1 + 13 T ) \)
$17$ \( ( 1 - 17 T )( 1 + 17 T ) \)
$19$ \( ( 1 - 19 T )( 1 + 19 T ) \)
$23$ \( 1 - 35 T + 529 T^{2} \)
$29$ \( ( 1 - 29 T )( 1 + 29 T ) \)
$31$ \( 1 + 37 T + 961 T^{2} \)
$37$ \( 1 + 25 T + 1369 T^{2} \)
$41$ \( ( 1 - 41 T )( 1 + 41 T ) \)
$43$ \( ( 1 - 43 T )( 1 + 43 T ) \)
$47$ \( 1 - 50 T + 2209 T^{2} \)
$53$ \( 1 + 70 T + 2809 T^{2} \)
$59$ \( 1 - 107 T + 3481 T^{2} \)
$61$ \( ( 1 - 61 T )( 1 + 61 T ) \)
$67$ \( 1 - 35 T + 4489 T^{2} \)
$71$ \( 1 + 133 T + 5041 T^{2} \)
$73$ \( ( 1 - 73 T )( 1 + 73 T ) \)
$79$ \( ( 1 - 79 T )( 1 + 79 T ) \)
$83$ \( ( 1 - 83 T )( 1 + 83 T ) \)
$89$ \( 1 + 97 T + 7921 T^{2} \)
$97$ \( 1 - 95 T + 9409 T^{2} \)
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