Properties

Label 66.4.a.b
Level $66$
Weight $4$
Character orbit 66.a
Self dual yes
Analytic conductor $3.894$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [66,4,Mod(1,66)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(66, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("66.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 66 = 2 \cdot 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 66.a (trivial)

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: \(3.89412606038\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + 2 q^{2} - 3 q^{3} + 4 q^{4} + 10 q^{5} - 6 q^{6} + 16 q^{7} + 8 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} - 3 q^{3} + 4 q^{4} + 10 q^{5} - 6 q^{6} + 16 q^{7} + 8 q^{8} + 9 q^{9} + 20 q^{10} + 11 q^{11} - 12 q^{12} + 10 q^{13} + 32 q^{14} - 30 q^{15} + 16 q^{16} - 10 q^{17} + 18 q^{18} - 144 q^{19} + 40 q^{20} - 48 q^{21} + 22 q^{22} - 84 q^{23} - 24 q^{24} - 25 q^{25} + 20 q^{26} - 27 q^{27} + 64 q^{28} + 218 q^{29} - 60 q^{30} - 176 q^{31} + 32 q^{32} - 33 q^{33} - 20 q^{34} + 160 q^{35} + 36 q^{36} + 46 q^{37} - 288 q^{38} - 30 q^{39} + 80 q^{40} - 26 q^{41} - 96 q^{42} - 488 q^{43} + 44 q^{44} + 90 q^{45} - 168 q^{46} + 404 q^{47} - 48 q^{48} - 87 q^{49} - 50 q^{50} + 30 q^{51} + 40 q^{52} + 194 q^{53} - 54 q^{54} + 110 q^{55} + 128 q^{56} + 432 q^{57} + 436 q^{58} + 444 q^{59} - 120 q^{60} + 202 q^{61} - 352 q^{62} + 144 q^{63} + 64 q^{64} + 100 q^{65} - 66 q^{66} - 84 q^{67} - 40 q^{68} + 252 q^{69} + 320 q^{70} - 764 q^{71} + 72 q^{72} + 354 q^{73} + 92 q^{74} + 75 q^{75} - 576 q^{76} + 176 q^{77} - 60 q^{78} + 1312 q^{79} + 160 q^{80} + 81 q^{81} - 52 q^{82} - 1252 q^{83} - 192 q^{84} - 100 q^{85} - 976 q^{86} - 654 q^{87} + 88 q^{88} - 1222 q^{89} + 180 q^{90} + 160 q^{91} - 336 q^{92} + 528 q^{93} + 808 q^{94} - 1440 q^{95} - 96 q^{96} - 1358 q^{97} - 174 q^{98} + 99 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
2.00000 −3.00000 4.00000 10.0000 −6.00000 16.0000 8.00000 9.00000 20.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(11\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 66.4.a.b 1
3.b odd 2 1 198.4.a.a 1
4.b odd 2 1 528.4.a.j 1
5.b even 2 1 1650.4.a.e 1
5.c odd 4 2 1650.4.c.e 2
8.b even 2 1 2112.4.a.r 1
8.d odd 2 1 2112.4.a.d 1
11.b odd 2 1 726.4.a.b 1
12.b even 2 1 1584.4.a.e 1
33.d even 2 1 2178.4.a.m 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
66.4.a.b 1 1.a even 1 1 trivial
198.4.a.a 1 3.b odd 2 1
528.4.a.j 1 4.b odd 2 1
726.4.a.b 1 11.b odd 2 1
1584.4.a.e 1 12.b even 2 1
1650.4.a.e 1 5.b even 2 1
1650.4.c.e 2 5.c odd 4 2
2112.4.a.d 1 8.d odd 2 1
2112.4.a.r 1 8.b even 2 1
2178.4.a.m 1 33.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 10 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(66))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 2 \) Copy content Toggle raw display
$3$ \( T + 3 \) Copy content Toggle raw display
$5$ \( T - 10 \) Copy content Toggle raw display
$7$ \( T - 16 \) Copy content Toggle raw display
$11$ \( T - 11 \) Copy content Toggle raw display
$13$ \( T - 10 \) Copy content Toggle raw display
$17$ \( T + 10 \) Copy content Toggle raw display
$19$ \( T + 144 \) Copy content Toggle raw display
$23$ \( T + 84 \) Copy content Toggle raw display
$29$ \( T - 218 \) Copy content Toggle raw display
$31$ \( T + 176 \) Copy content Toggle raw display
$37$ \( T - 46 \) Copy content Toggle raw display
$41$ \( T + 26 \) Copy content Toggle raw display
$43$ \( T + 488 \) Copy content Toggle raw display
$47$ \( T - 404 \) Copy content Toggle raw display
$53$ \( T - 194 \) Copy content Toggle raw display
$59$ \( T - 444 \) Copy content Toggle raw display
$61$ \( T - 202 \) Copy content Toggle raw display
$67$ \( T + 84 \) Copy content Toggle raw display
$71$ \( T + 764 \) Copy content Toggle raw display
$73$ \( T - 354 \) Copy content Toggle raw display
$79$ \( T - 1312 \) Copy content Toggle raw display
$83$ \( T + 1252 \) Copy content Toggle raw display
$89$ \( T + 1222 \) Copy content Toggle raw display
$97$ \( T + 1358 \) Copy content Toggle raw display
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