Properties

Label 51.4.a.b
Level $51$
Weight $4$
Character orbit 51.a
Self dual yes
Analytic conductor $3.009$
Analytic rank $1$
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] = [51,4,Mod(1,51)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(51, 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("51.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 51 = 3 \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 51.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: \(3.00909741029\)
Analytic rank: \(1\)
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)
 
\(f(q)\) \(=\) \( q - q^{2} + 3 q^{3} - 7 q^{4} - 20 q^{5} - 3 q^{6} - 2 q^{7} + 15 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + 3 q^{3} - 7 q^{4} - 20 q^{5} - 3 q^{6} - 2 q^{7} + 15 q^{8} + 9 q^{9} + 20 q^{10} - 48 q^{11} - 21 q^{12} - 14 q^{13} + 2 q^{14} - 60 q^{15} + 41 q^{16} - 17 q^{17} - 9 q^{18} + 92 q^{19} + 140 q^{20} - 6 q^{21} + 48 q^{22} - 122 q^{23} + 45 q^{24} + 275 q^{25} + 14 q^{26} + 27 q^{27} + 14 q^{28} - 36 q^{29} + 60 q^{30} - 182 q^{31} - 161 q^{32} - 144 q^{33} + 17 q^{34} + 40 q^{35} - 63 q^{36} + 76 q^{37} - 92 q^{38} - 42 q^{39} - 300 q^{40} + 294 q^{41} + 6 q^{42} - 428 q^{43} + 336 q^{44} - 180 q^{45} + 122 q^{46} - 12 q^{47} + 123 q^{48} - 339 q^{49} - 275 q^{50} - 51 q^{51} + 98 q^{52} - 234 q^{53} - 27 q^{54} + 960 q^{55} - 30 q^{56} + 276 q^{57} + 36 q^{58} - 540 q^{59} + 420 q^{60} - 820 q^{61} + 182 q^{62} - 18 q^{63} - 167 q^{64} + 280 q^{65} + 144 q^{66} + 700 q^{67} + 119 q^{68} - 366 q^{69} - 40 q^{70} + 794 q^{71} + 135 q^{72} - 1038 q^{73} - 76 q^{74} + 825 q^{75} - 644 q^{76} + 96 q^{77} + 42 q^{78} + 858 q^{79} - 820 q^{80} + 81 q^{81} - 294 q^{82} + 1052 q^{83} + 42 q^{84} + 340 q^{85} + 428 q^{86} - 108 q^{87} - 720 q^{88} + 1102 q^{89} + 180 q^{90} + 28 q^{91} + 854 q^{92} - 546 q^{93} + 12 q^{94} - 1840 q^{95} - 483 q^{96} + 710 q^{97} + 339 q^{98} - 432 q^{99}+O(q^{100}) \) 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
−1.00000 3.00000 −7.00000 −20.0000 −3.00000 −2.00000 15.0000 9.00000 20.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)
\(17\) \( +1 \)

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 51.4.a.b 1
3.b odd 2 1 153.4.a.c 1
4.b odd 2 1 816.4.a.a 1
5.b even 2 1 1275.4.a.e 1
7.b odd 2 1 2499.4.a.d 1
12.b even 2 1 2448.4.a.r 1
17.b even 2 1 867.4.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
51.4.a.b 1 1.a even 1 1 trivial
153.4.a.c 1 3.b odd 2 1
816.4.a.a 1 4.b odd 2 1
867.4.a.c 1 17.b even 2 1
1275.4.a.e 1 5.b even 2 1
2448.4.a.r 1 12.b even 2 1
2499.4.a.d 1 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(51))\):

\( T_{2} + 1 \) Copy content Toggle raw display
\( T_{5} + 20 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 1 \) Copy content Toggle raw display
$3$ \( T - 3 \) Copy content Toggle raw display
$5$ \( T + 20 \) Copy content Toggle raw display
$7$ \( T + 2 \) Copy content Toggle raw display
$11$ \( T + 48 \) Copy content Toggle raw display
$13$ \( T + 14 \) Copy content Toggle raw display
$17$ \( T + 17 \) Copy content Toggle raw display
$19$ \( T - 92 \) Copy content Toggle raw display
$23$ \( T + 122 \) Copy content Toggle raw display
$29$ \( T + 36 \) Copy content Toggle raw display
$31$ \( T + 182 \) Copy content Toggle raw display
$37$ \( T - 76 \) Copy content Toggle raw display
$41$ \( T - 294 \) Copy content Toggle raw display
$43$ \( T + 428 \) Copy content Toggle raw display
$47$ \( T + 12 \) Copy content Toggle raw display
$53$ \( T + 234 \) Copy content Toggle raw display
$59$ \( T + 540 \) Copy content Toggle raw display
$61$ \( T + 820 \) Copy content Toggle raw display
$67$ \( T - 700 \) Copy content Toggle raw display
$71$ \( T - 794 \) Copy content Toggle raw display
$73$ \( T + 1038 \) Copy content Toggle raw display
$79$ \( T - 858 \) Copy content Toggle raw display
$83$ \( T - 1052 \) Copy content Toggle raw display
$89$ \( T - 1102 \) Copy content Toggle raw display
$97$ \( T - 710 \) Copy content Toggle raw display
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