Properties

Label 8.11.d.a
Level 8
Weight 11
Character orbit 8.d
Self dual yes
Analytic conductor 5.083
Analytic rank 0
Dimension 1
CM discriminant -8
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 8 = 2^{3} \)
Weight: \( k \) = \( 11 \)
Character orbit: \([\chi]\) = 8.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(5.08285802139\)
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

\(f(q)\) \(=\) \( q - 32q^{2} - 482q^{3} + 1024q^{4} + 15424q^{6} - 32768q^{8} + 173275q^{9} + O(q^{10}) \) \( q - 32q^{2} - 482q^{3} + 1024q^{4} + 15424q^{6} - 32768q^{8} + 173275q^{9} - 97426q^{11} - 493568q^{12} + 1048576q^{16} + 823682q^{17} - 5544800q^{18} + 3353726q^{19} + 3117632q^{22} + 15794176q^{24} + 9765625q^{25} - 55056932q^{27} - 33554432q^{32} + 46959332q^{33} - 26357824q^{34} + 177433600q^{36} - 107319232q^{38} - 37778926q^{41} + 214485614q^{43} - 99764224q^{44} - 505413632q^{48} + 282475249q^{49} - 312500000q^{50} - 397014724q^{51} + 1761821824q^{54} - 1616495932q^{57} + 921043598q^{59} + 1073741824q^{64} - 1502698624q^{66} + 1813708382q^{67} + 843450368q^{68} - 5677875200q^{72} - 1605781582q^{73} - 4707031250q^{75} + 3434215424q^{76} + 16305725749q^{81} + 1208925632q^{82} + 96051518q^{83} - 6863539648q^{86} + 3192455168q^{88} - 11116019374q^{89} + 16173236224q^{96} - 9872978014q^{97} - 9039207968q^{98} - 16881490150q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/8\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(7\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
0
−32.0000 −482.000 1024.00 0 15424.0 0 −32768.0 173275. 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
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8.11.d.a 1
3.b odd 2 1 72.11.b.a 1
4.b odd 2 1 32.11.d.a 1
8.b even 2 1 32.11.d.a 1
8.d odd 2 1 CM 8.11.d.a 1
12.b even 2 1 288.11.b.a 1
16.e even 4 2 256.11.c.c 2
16.f odd 4 2 256.11.c.c 2
24.f even 2 1 72.11.b.a 1
24.h odd 2 1 288.11.b.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.11.d.a 1 1.a even 1 1 trivial
8.11.d.a 1 8.d odd 2 1 CM
32.11.d.a 1 4.b odd 2 1
32.11.d.a 1 8.b even 2 1
72.11.b.a 1 3.b odd 2 1
72.11.b.a 1 24.f even 2 1
256.11.c.c 2 16.e even 4 2
256.11.c.c 2 16.f odd 4 2
288.11.b.a 1 12.b even 2 1
288.11.b.a 1 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 482 \) acting on \(S_{11}^{\mathrm{new}}(8, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 32 T \)
$3$ \( 1 + 482 T + 59049 T^{2} \)
$5$ \( ( 1 - 3125 T )( 1 + 3125 T ) \)
$7$ \( ( 1 - 16807 T )( 1 + 16807 T ) \)
$11$ \( 1 + 97426 T + 25937424601 T^{2} \)
$13$ \( ( 1 - 371293 T )( 1 + 371293 T ) \)
$17$ \( 1 - 823682 T + 2015993900449 T^{2} \)
$19$ \( 1 - 3353726 T + 6131066257801 T^{2} \)
$23$ \( ( 1 - 6436343 T )( 1 + 6436343 T ) \)
$29$ \( ( 1 - 20511149 T )( 1 + 20511149 T ) \)
$31$ \( ( 1 - 28629151 T )( 1 + 28629151 T ) \)
$37$ \( ( 1 - 69343957 T )( 1 + 69343957 T ) \)
$41$ \( 1 + 37778926 T + 13422659310152401 T^{2} \)
$43$ \( 1 - 214485614 T + 21611482313284249 T^{2} \)
$47$ \( ( 1 - 229345007 T )( 1 + 229345007 T ) \)
$53$ \( ( 1 - 418195493 T )( 1 + 418195493 T ) \)
$59$ \( 1 - 921043598 T + 511116753300641401 T^{2} \)
$61$ \( ( 1 - 844596301 T )( 1 + 844596301 T ) \)
$67$ \( 1 - 1813708382 T + 1822837804551761449 T^{2} \)
$71$ \( ( 1 - 1804229351 T )( 1 + 1804229351 T ) \)
$73$ \( 1 + 1605781582 T + 4297625829703557649 T^{2} \)
$79$ \( ( 1 - 3077056399 T )( 1 + 3077056399 T ) \)
$83$ \( 1 - 96051518 T + 15516041187205853449 T^{2} \)
$89$ \( 1 + 11116019374 T + 31181719929966183601 T^{2} \)
$97$ \( 1 + 9872978014 T + 73742412689492826049 T^{2} \)
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