Properties

Label 68.5.d.a
Level $68$
Weight $5$
Character orbit 68.d
Self dual yes
Analytic conductor $7.029$
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] = [68,5,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, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("68.67");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 68 = 2^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 5 \)
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: \(7.02915748970\)
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)
 
\(f(q)\) \(=\) \( q + 4 q^{2} - 16 q^{3} + 16 q^{4} - 64 q^{6} + 64 q^{7} + 64 q^{8} + 175 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 4 q^{2} - 16 q^{3} + 16 q^{4} - 64 q^{6} + 64 q^{7} + 64 q^{8} + 175 q^{9} + 208 q^{11} - 256 q^{12} - 274 q^{13} + 256 q^{14} + 256 q^{16} + 289 q^{17} + 700 q^{18} - 1024 q^{21} + 832 q^{22} - 608 q^{23} - 1024 q^{24} + 625 q^{25} - 1096 q^{26} - 1504 q^{27} + 1024 q^{28} + 256 q^{31} + 1024 q^{32} - 3328 q^{33} + 1156 q^{34} + 2800 q^{36} + 4384 q^{39} - 4096 q^{42} + 3328 q^{44} - 2432 q^{46} - 4096 q^{48} + 1695 q^{49} + 2500 q^{50} - 4624 q^{51} - 4384 q^{52} - 4174 q^{53} - 6016 q^{54} + 4096 q^{56} + 1024 q^{62} + 11200 q^{63} + 4096 q^{64} - 13312 q^{66} + 4624 q^{68} + 9728 q^{69} - 7904 q^{71} + 11200 q^{72} - 10000 q^{75} + 13312 q^{77} + 17536 q^{78} + 2656 q^{79} + 9889 q^{81} - 16384 q^{84} + 13312 q^{88} + 542 q^{89} - 17536 q^{91} - 9728 q^{92} - 4096 q^{93} - 16384 q^{96} + 6780 q^{98} + 36400 q^{99}+O(q^{100}) \) 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
4.00000 −16.0000 16.0000 0 −64.0000 64.0000 64.0000 175.000 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 68.5.d.a 1
4.b odd 2 1 68.5.d.b yes 1
17.b even 2 1 68.5.d.b yes 1
68.d odd 2 1 CM 68.5.d.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
68.5.d.a 1 1.a even 1 1 trivial
68.5.d.a 1 68.d odd 2 1 CM
68.5.d.b yes 1 4.b odd 2 1
68.5.d.b yes 1 17.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 16 \) acting on \(S_{5}^{\mathrm{new}}(68, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 4 \) Copy content Toggle raw display
$3$ \( T + 16 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T - 64 \) Copy content Toggle raw display
$11$ \( T - 208 \) Copy content Toggle raw display
$13$ \( T + 274 \) Copy content Toggle raw display
$17$ \( T - 289 \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T + 608 \) Copy content Toggle raw display
$29$ \( T \) Copy content Toggle raw display
$31$ \( T - 256 \) 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
$53$ \( T + 4174 \) 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
$71$ \( T + 7904 \) Copy content Toggle raw display
$73$ \( T \) Copy content Toggle raw display
$79$ \( T - 2656 \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T - 542 \) Copy content Toggle raw display
$97$ \( T \) Copy content Toggle raw display
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