Properties

Label 68.9.d.a
Level $68$
Weight $9$
Character orbit 68.d
Self dual yes
Analytic conductor $27.702$
Analytic rank $0$
Dimension $1$
CM discriminant -68
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [68,9,Mod(67,68)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(68, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("68.67");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 68 = 2^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 68.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: \(27.7017454842\)
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)
 
\(f(q)\) \(=\) \( q + 16 q^{2} - 94 q^{3} + 256 q^{4} - 1504 q^{6} + 706 q^{7} + 4096 q^{8} + 2275 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 16 q^{2} - 94 q^{3} + 256 q^{4} - 1504 q^{6} + 706 q^{7} + 4096 q^{8} + 2275 q^{9} - 13982 q^{11} - 24064 q^{12} + 17954 q^{13} + 11296 q^{14} + 65536 q^{16} + 83521 q^{17} + 36400 q^{18} - 66364 q^{21} - 223712 q^{22} + 190018 q^{23} - 385024 q^{24} + 390625 q^{25} + 287264 q^{26} + 402884 q^{27} + 180736 q^{28} + 1781506 q^{31} + 1048576 q^{32} + 1314308 q^{33} + 1336336 q^{34} + 582400 q^{36} - 1687676 q^{39} - 1061824 q^{42} - 3579392 q^{44} + 3040288 q^{46} - 6160384 q^{48} - 5266365 q^{49} + 6250000 q^{50} - 7850974 q^{51} + 4596224 q^{52} + 1641314 q^{53} + 6446144 q^{54} + 2891776 q^{56} + 28504096 q^{62} + 1606150 q^{63} + 16777216 q^{64} + 21028928 q^{66} + 21381376 q^{68} - 17861692 q^{69} - 11649854 q^{71} + 9318400 q^{72} - 36718750 q^{75} - 9871292 q^{77} - 27002816 q^{78} + 70845826 q^{79} - 52797371 q^{81} - 16989184 q^{84} - 57270272 q^{88} - 125190718 q^{89} + 12675524 q^{91} + 48644608 q^{92} - 167461564 q^{93} - 98566144 q^{96} - 84261840 q^{98} - 31809050 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/68\mathbb{Z}\right)^\times\).

\(n\) \(35\) \(37\)
\(\chi(n)\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
0
16.0000 −94.0000 256.000 0 −1504.00 706.000 4096.00 2275.00 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
68.d odd 2 1 CM by \(\Q(\sqrt{-17}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 68.9.d.a 1
4.b odd 2 1 68.9.d.b yes 1
17.b even 2 1 68.9.d.b yes 1
68.d odd 2 1 CM 68.9.d.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
68.9.d.a 1 1.a even 1 1 trivial
68.9.d.a 1 68.d odd 2 1 CM
68.9.d.b yes 1 4.b odd 2 1
68.9.d.b yes 1 17.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 94 \) acting on \(S_{9}^{\mathrm{new}}(68, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 16 \) Copy content Toggle raw display
$3$ \( T + 94 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T - 706 \) Copy content Toggle raw display
$11$ \( T + 13982 \) Copy content Toggle raw display
$13$ \( T - 17954 \) Copy content Toggle raw display
$17$ \( T - 83521 \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T - 190018 \) Copy content Toggle raw display
$29$ \( T \) Copy content Toggle raw display
$31$ \( T - 1781506 \) Copy content Toggle raw display
$37$ \( T \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T - 1641314 \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T \) Copy content Toggle raw display
$67$ \( T \) Copy content Toggle raw display
$71$ \( T + 11649854 \) Copy content Toggle raw display
$73$ \( T \) Copy content Toggle raw display
$79$ \( T - 70845826 \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T + 125190718 \) Copy content Toggle raw display
$97$ \( T \) Copy content Toggle raw display
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