Properties

Label 8.17.d.a
Level $8$
Weight $17$
Character orbit 8.d
Self dual yes
Analytic conductor $12.986$
Analytic rank $0$
Dimension $1$
CM discriminant -8
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8,17,Mod(3,8)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 17, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8.3");
 
S:= CuspForms(chi, 17);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8 = 2^{3} \)
Weight: \( k \) \(=\) \( 17 \)
Character orbit: \([\chi]\) \(=\) 8.d (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(12.9859635085\)
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

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + 256 q^{2} - 11966 q^{3} + 65536 q^{4} - 3063296 q^{6} + 16777216 q^{8} + 100138435 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 256 q^{2} - 11966 q^{3} + 65536 q^{4} - 3063296 q^{6} + 16777216 q^{8} + 100138435 q^{9} + 309273794 q^{11} - 784203776 q^{12} + 4294967296 q^{16} + 12433289474 q^{17} + 25635439360 q^{18} - 28741860286 q^{19} + 79174091264 q^{22} - 200756166656 q^{24} + 152587890625 q^{25} - 683159449724 q^{27} + 1099511627776 q^{32} - 3700770219004 q^{33} + 3182922105344 q^{34} + 6562672476160 q^{36} - 7357916233216 q^{38} + 831999729794 q^{41} + 6069438110402 q^{43} + 20268567363584 q^{44} - 51393578663936 q^{48} + 33232930569601 q^{49} + 39062500000000 q^{50} - 148776741845884 q^{51} - 174888819129344 q^{54} + 343925100182276 q^{57} + 290918580565442 q^{59} + 281474976710656 q^{64} - 947397176065024 q^{66} - 617692243063486 q^{67} + 814828058968064 q^{68} + 16\!\cdots\!60 q^{72}+ \cdots + 30\!\cdots\!90 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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
256.000 −11966.0 65536.0 0 −3.06330e6 0 1.67772e7 1.00138e8 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 8.17.d.a 1
3.b odd 2 1 72.17.b.a 1
4.b odd 2 1 32.17.d.a 1
8.b even 2 1 32.17.d.a 1
8.d odd 2 1 CM 8.17.d.a 1
24.f even 2 1 72.17.b.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.17.d.a 1 1.a even 1 1 trivial
8.17.d.a 1 8.d odd 2 1 CM
32.17.d.a 1 4.b odd 2 1
32.17.d.a 1 8.b even 2 1
72.17.b.a 1 3.b odd 2 1
72.17.b.a 1 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 11966 \) acting on \(S_{17}^{\mathrm{new}}(8, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 256 \) Copy content Toggle raw display
$3$ \( T + 11966 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T - 309273794 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T - 12433289474 \) Copy content Toggle raw display
$19$ \( T + 28741860286 \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T \) Copy content Toggle raw display
$41$ \( T - 831999729794 \) Copy content Toggle raw display
$43$ \( T - 6069438110402 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T \) Copy content Toggle raw display
$59$ \( T - 290918580565442 \) Copy content Toggle raw display
$61$ \( T \) Copy content Toggle raw display
$67$ \( T + 617692243063486 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T + 486139502245246 \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T + 3591943143595966 \) Copy content Toggle raw display
$89$ \( T - 1250855726873474 \) Copy content Toggle raw display
$97$ \( T + 9681283613729278 \) Copy content Toggle raw display
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