Properties

Label 11.5.b.a
Level 11
Weight 5
Character orbit 11.b
Self dual yes
Analytic conductor 1.137
Analytic rank 0
Dimension 1
CM discriminant -11
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 11 \)
Weight: \( k \) = \( 5 \)
Character orbit: \([\chi]\) = 11.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(1.13706959392\)
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

\(f(q)\) \(=\) \( q + 7q^{3} + 16q^{4} - 49q^{5} - 32q^{9} + O(q^{10}) \) \( q + 7q^{3} + 16q^{4} - 49q^{5} - 32q^{9} + 121q^{11} + 112q^{12} - 343q^{15} + 256q^{16} - 784q^{20} + 167q^{23} + 1776q^{25} - 791q^{27} - 553q^{31} + 847q^{33} - 512q^{36} - 2113q^{37} + 1936q^{44} + 1568q^{45} - 1918q^{47} + 1792q^{48} + 2401q^{49} - 718q^{53} - 5929q^{55} + 4487q^{59} - 5488q^{60} + 4096q^{64} - 7753q^{67} + 1169q^{69} + 7607q^{71} + 12432q^{75} - 12544q^{80} - 2945q^{81} - 6433q^{89} + 2672q^{92} - 3871q^{93} - 9793q^{97} - 3872q^{99} + O(q^{100}) \)

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 7.00000 16.0000 −49.0000 0 0 0 −32.0000 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 11.5.b.a 1
3.b odd 2 1 99.5.c.a 1
4.b odd 2 1 176.5.h.a 1
5.b even 2 1 275.5.c.a 1
5.c odd 4 2 275.5.d.a 2
8.b even 2 1 704.5.h.a 1
8.d odd 2 1 704.5.h.b 1
11.b odd 2 1 CM 11.5.b.a 1
11.c even 5 4 121.5.d.a 4
11.d odd 10 4 121.5.d.a 4
33.d even 2 1 99.5.c.a 1
44.c even 2 1 176.5.h.a 1
55.d odd 2 1 275.5.c.a 1
55.e even 4 2 275.5.d.a 2
88.b odd 2 1 704.5.h.a 1
88.g even 2 1 704.5.h.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.5.b.a 1 1.a even 1 1 trivial
11.5.b.a 1 11.b odd 2 1 CM
99.5.c.a 1 3.b odd 2 1
99.5.c.a 1 33.d even 2 1
121.5.d.a 4 11.c even 5 4
121.5.d.a 4 11.d odd 10 4
176.5.h.a 1 4.b odd 2 1
176.5.h.a 1 44.c even 2 1
275.5.c.a 1 5.b even 2 1
275.5.c.a 1 55.d odd 2 1
275.5.d.a 2 5.c odd 4 2
275.5.d.a 2 55.e even 4 2
704.5.h.a 1 8.b even 2 1
704.5.h.a 1 88.b odd 2 1
704.5.h.b 1 8.d odd 2 1
704.5.h.b 1 88.g even 2 1

Hecke kernels

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 4 T )( 1 + 4 T ) \)
$3$ \( 1 - 7 T + 81 T^{2} \)
$5$ \( 1 + 49 T + 625 T^{2} \)
$7$ \( ( 1 - 49 T )( 1 + 49 T ) \)
$11$ \( 1 - 121 T \)
$13$ \( ( 1 - 169 T )( 1 + 169 T ) \)
$17$ \( ( 1 - 289 T )( 1 + 289 T ) \)
$19$ \( ( 1 - 361 T )( 1 + 361 T ) \)
$23$ \( 1 - 167 T + 279841 T^{2} \)
$29$ \( ( 1 - 841 T )( 1 + 841 T ) \)
$31$ \( 1 + 553 T + 923521 T^{2} \)
$37$ \( 1 + 2113 T + 1874161 T^{2} \)
$41$ \( ( 1 - 1681 T )( 1 + 1681 T ) \)
$43$ \( ( 1 - 1849 T )( 1 + 1849 T ) \)
$47$ \( 1 + 1918 T + 4879681 T^{2} \)
$53$ \( 1 + 718 T + 7890481 T^{2} \)
$59$ \( 1 - 4487 T + 12117361 T^{2} \)
$61$ \( ( 1 - 3721 T )( 1 + 3721 T ) \)
$67$ \( 1 + 7753 T + 20151121 T^{2} \)
$71$ \( 1 - 7607 T + 25411681 T^{2} \)
$73$ \( ( 1 - 5329 T )( 1 + 5329 T ) \)
$79$ \( ( 1 - 6241 T )( 1 + 6241 T ) \)
$83$ \( ( 1 - 6889 T )( 1 + 6889 T ) \)
$89$ \( 1 + 6433 T + 62742241 T^{2} \)
$97$ \( 1 + 9793 T + 88529281 T^{2} \)
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