Properties

Label 11.9.b.a
Level 11
Weight 9
Character orbit 11.b
Self dual yes
Analytic conductor 4.481
Analytic rank 0
Dimension 1
CM discriminant -11
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 11 \)
Weight: \( k \) = \( 9 \)
Character orbit: \([\chi]\) = 11.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(4.48116471067\)
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

\(f(q)\) \(=\) \( q - 113q^{3} + 256q^{4} + 1151q^{5} + 6208q^{9} + O(q^{10}) \) \( q - 113q^{3} + 256q^{4} + 1151q^{5} + 6208q^{9} + 14641q^{11} - 28928q^{12} - 130063q^{15} + 65536q^{16} + 294656q^{20} - 531793q^{23} + 934176q^{25} + 39889q^{27} - 1541233q^{31} - 1654433q^{33} + 1589248q^{36} + 716447q^{37} + 3748096q^{44} + 7145408q^{45} - 6080638q^{47} - 7405568q^{48} + 5764801q^{49} - 15265438q^{53} + 16851791q^{55} - 4101553q^{59} - 33296128q^{60} + 16777216q^{64} + 19806767q^{67} + 60092609q^{69} + 7043087q^{71} - 105561888q^{75} + 75431936q^{80} - 45238145q^{81} - 84100993q^{89} - 136139008q^{92} + 174159329q^{93} - 81155713q^{97} + 90891328q^{99} + O(q^{100}) \)

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 −113.000 256.000 1151.00 0 0 0 6208.00 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 11.9.b.a 1
3.b odd 2 1 99.9.c.a 1
4.b odd 2 1 176.9.h.a 1
11.b odd 2 1 CM 11.9.b.a 1
33.d even 2 1 99.9.c.a 1
44.c even 2 1 176.9.h.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.9.b.a 1 1.a even 1 1 trivial
11.9.b.a 1 11.b odd 2 1 CM
99.9.c.a 1 3.b odd 2 1
99.9.c.a 1 33.d even 2 1
176.9.h.a 1 4.b odd 2 1
176.9.h.a 1 44.c even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{9}^{\mathrm{new}}(11, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 16 T )( 1 + 16 T ) \)
$3$ \( 1 + 113 T + 6561 T^{2} \)
$5$ \( 1 - 1151 T + 390625 T^{2} \)
$7$ \( ( 1 - 2401 T )( 1 + 2401 T ) \)
$11$ \( 1 - 14641 T \)
$13$ \( ( 1 - 28561 T )( 1 + 28561 T ) \)
$17$ \( ( 1 - 83521 T )( 1 + 83521 T ) \)
$19$ \( ( 1 - 130321 T )( 1 + 130321 T ) \)
$23$ \( 1 + 531793 T + 78310985281 T^{2} \)
$29$ \( ( 1 - 707281 T )( 1 + 707281 T ) \)
$31$ \( 1 + 1541233 T + 852891037441 T^{2} \)
$37$ \( 1 - 716447 T + 3512479453921 T^{2} \)
$41$ \( ( 1 - 2825761 T )( 1 + 2825761 T ) \)
$43$ \( ( 1 - 3418801 T )( 1 + 3418801 T ) \)
$47$ \( 1 + 6080638 T + 23811286661761 T^{2} \)
$53$ \( 1 + 15265438 T + 62259690411361 T^{2} \)
$59$ \( 1 + 4101553 T + 146830437604321 T^{2} \)
$61$ \( ( 1 - 13845841 T )( 1 + 13845841 T ) \)
$67$ \( 1 - 19806767 T + 406067677556641 T^{2} \)
$71$ \( 1 - 7043087 T + 645753531245761 T^{2} \)
$73$ \( ( 1 - 28398241 T )( 1 + 28398241 T ) \)
$79$ \( ( 1 - 38950081 T )( 1 + 38950081 T ) \)
$83$ \( ( 1 - 47458321 T )( 1 + 47458321 T ) \)
$89$ \( 1 + 84100993 T + 3936588805702081 T^{2} \)
$97$ \( 1 + 81155713 T + 7837433594376961 T^{2} \)
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