Properties

Label 51.4.a.c
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)
 
\(f(q)\) \(=\) \( q + q^{2} - 3 q^{3} - 7 q^{4} - 10 q^{5} - 3 q^{6} - 8 q^{7} - 15 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - 3 q^{3} - 7 q^{4} - 10 q^{5} - 3 q^{6} - 8 q^{7} - 15 q^{8} + 9 q^{9} - 10 q^{10} + 12 q^{11} + 21 q^{12} - 26 q^{13} - 8 q^{14} + 30 q^{15} + 41 q^{16} + 17 q^{17} + 9 q^{18} - 148 q^{19} + 70 q^{20} + 24 q^{21} + 12 q^{22} + 152 q^{23} + 45 q^{24} - 25 q^{25} - 26 q^{26} - 27 q^{27} + 56 q^{28} - 66 q^{29} + 30 q^{30} - 32 q^{31} + 161 q^{32} - 36 q^{33} + 17 q^{34} + 80 q^{35} - 63 q^{36} - 266 q^{37} - 148 q^{38} + 78 q^{39} + 150 q^{40} - 6 q^{41} + 24 q^{42} - 92 q^{43} - 84 q^{44} - 90 q^{45} + 152 q^{46} - 288 q^{47} - 123 q^{48} - 279 q^{49} - 25 q^{50} - 51 q^{51} + 182 q^{52} - 546 q^{53} - 27 q^{54} - 120 q^{55} + 120 q^{56} + 444 q^{57} - 66 q^{58} + 420 q^{59} - 210 q^{60} + 350 q^{61} - 32 q^{62} - 72 q^{63} - 167 q^{64} + 260 q^{65} - 36 q^{66} + 940 q^{67} - 119 q^{68} - 456 q^{69} + 80 q^{70} + 424 q^{71} - 135 q^{72} + 378 q^{73} - 266 q^{74} + 75 q^{75} + 1036 q^{76} - 96 q^{77} + 78 q^{78} + 288 q^{79} - 410 q^{80} + 81 q^{81} - 6 q^{82} + 748 q^{83} - 168 q^{84} - 170 q^{85} - 92 q^{86} + 198 q^{87} - 180 q^{88} - 1558 q^{89} - 90 q^{90} + 208 q^{91} - 1064 q^{92} + 96 q^{93} - 288 q^{94} + 1480 q^{95} - 483 q^{96} + 530 q^{97} - 279 q^{98} + 108 q^{99}+O(q^{100}) \) 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 −10.0000 −3.00000 −8.00000 −15.0000 9.00000 −10.0000
\(n\): 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.c 1
3.b odd 2 1 153.4.a.a 1
4.b odd 2 1 816.4.a.f 1
5.b even 2 1 1275.4.a.d 1
7.b odd 2 1 2499.4.a.i 1
12.b even 2 1 2448.4.a.n 1
17.b even 2 1 867.4.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
51.4.a.c 1 1.a even 1 1 trivial
153.4.a.a 1 3.b odd 2 1
816.4.a.f 1 4.b odd 2 1
867.4.a.e 1 17.b even 2 1
1275.4.a.d 1 5.b even 2 1
2448.4.a.n 1 12.b even 2 1
2499.4.a.i 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
\( T_{5} + 10 \) 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
$5$ \( T + 10 \) Copy content Toggle raw display
$7$ \( T + 8 \) Copy content Toggle raw display
$11$ \( T - 12 \) Copy content Toggle raw display
$13$ \( T + 26 \) Copy content Toggle raw display
$17$ \( T - 17 \) Copy content Toggle raw display
$19$ \( T + 148 \) Copy content Toggle raw display
$23$ \( T - 152 \) Copy content Toggle raw display
$29$ \( T + 66 \) Copy content Toggle raw display
$31$ \( T + 32 \) Copy content Toggle raw display
$37$ \( T + 266 \) Copy content Toggle raw display
$41$ \( T + 6 \) Copy content Toggle raw display
$43$ \( T + 92 \) Copy content Toggle raw display
$47$ \( T + 288 \) Copy content Toggle raw display
$53$ \( T + 546 \) Copy content Toggle raw display
$59$ \( T - 420 \) Copy content Toggle raw display
$61$ \( T - 350 \) Copy content Toggle raw display
$67$ \( T - 940 \) Copy content Toggle raw display
$71$ \( T - 424 \) Copy content Toggle raw display
$73$ \( T - 378 \) Copy content Toggle raw display
$79$ \( T - 288 \) Copy content Toggle raw display
$83$ \( T - 748 \) Copy content Toggle raw display
$89$ \( T + 1558 \) Copy content Toggle raw display
$97$ \( T - 530 \) Copy content Toggle raw display
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