Properties

Label 429.2.a.g
Level $429$
Weight $2$
Character orbit 429.a
Self dual yes
Analytic conductor $3.426$
Analytic rank $0$
Dimension $3$
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] = [429,2,Mod(1,429)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(429, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("429.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 429 = 3 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 429.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: \(3.42558224671\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} + 1) q^{2} + q^{3} + (\beta_{2} - \beta_1 + 2) q^{4} + (\beta_1 + 1) q^{5} + (\beta_{2} + 1) q^{6} + ( - \beta_{2} + \beta_1 - 1) q^{7} + ( - 3 \beta_1 + 4) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + 1) q^{2} + q^{3} + (\beta_{2} - \beta_1 + 2) q^{4} + (\beta_1 + 1) q^{5} + (\beta_{2} + 1) q^{6} + ( - \beta_{2} + \beta_1 - 1) q^{7} + ( - 3 \beta_1 + 4) q^{8} + q^{9} + (\beta_{2} + 2 \beta_1) q^{10} - q^{11} + (\beta_{2} - \beta_1 + 2) q^{12} - q^{13} + ( - \beta_{2} + 3 \beta_1 - 5) q^{14} + (\beta_1 + 1) q^{15} + (2 \beta_{2} - 4 \beta_1 + 3) q^{16} - \beta_{2} q^{17} + (\beta_{2} + 1) q^{18} + (\beta_{2} - \beta_1 + 1) q^{19} + (\beta_1 - 1) q^{20} + ( - \beta_{2} + \beta_1 - 1) q^{21} + ( - \beta_{2} - 1) q^{22} + ( - 3 \beta_{2} + \beta_1 + 1) q^{23} + ( - 3 \beta_1 + 4) q^{24} + (\beta_{2} + 3 \beta_1 - 2) q^{25} + ( - \beta_{2} - 1) q^{26} + q^{27} + ( - 3 \beta_{2} + 5 \beta_1 - 9) q^{28} + ( - 3 \beta_{2} - 4 \beta_1 + 2) q^{29} + (\beta_{2} + 2 \beta_1) q^{30} + ( - \beta_1 + 5) q^{31} + (3 \beta_{2} - 4 \beta_1 + 5) q^{32} - q^{33} + (\beta_1 - 3) q^{34} + 2 q^{35} + (\beta_{2} - \beta_1 + 2) q^{36} - 4 \beta_{2} q^{37} + (\beta_{2} - 3 \beta_1 + 5) q^{38} - q^{39} + ( - 3 \beta_{2} - 2 \beta_1 - 2) q^{40} + (\beta_{2} - \beta_1 - 1) q^{41} + ( - \beta_{2} + 3 \beta_1 - 5) q^{42} + (3 \beta_{2} + 6 \beta_1 - 4) q^{43} + ( - \beta_{2} + \beta_1 - 2) q^{44} + (\beta_1 + 1) q^{45} + (\beta_{2} + 5 \beta_1 - 9) q^{46} + ( - 2 \beta_{2} - 4 \beta_1 + 6) q^{47} + (2 \beta_{2} - 4 \beta_1 + 3) q^{48} + (2 \beta_{2} - 4 \beta_1 + 1) q^{49} + ( - 2 \beta_{2} + 5 \beta_1 - 2) q^{50} - \beta_{2} q^{51} + ( - \beta_{2} + \beta_1 - 2) q^{52} + ( - 2 \beta_{2} + 4 \beta_1 - 2) q^{53} + (\beta_{2} + 1) q^{54} + ( - \beta_1 - 1) q^{55} + ( - 7 \beta_{2} + 7 \beta_1 - 13) q^{56} + (\beta_{2} - \beta_1 + 1) q^{57} + (2 \beta_{2} - 5 \beta_1 - 3) q^{58} + ( - 2 \beta_1 + 10) q^{59} + (\beta_1 - 1) q^{60} + (4 \beta_{2} - 2 \beta_1 - 2) q^{61} + (5 \beta_{2} - 2 \beta_1 + 6) q^{62} + ( - \beta_{2} + \beta_1 - 1) q^{63} + (\beta_{2} - 3 \beta_1 + 12) q^{64} + ( - \beta_1 - 1) q^{65} + ( - \beta_{2} - 1) q^{66} + ( - 2 \beta_{2} - 7 \beta_1 + 3) q^{67} + ( - \beta_{2} + 2 \beta_1 - 4) q^{68} + ( - 3 \beta_{2} + \beta_1 + 1) q^{69} + (2 \beta_{2} + 2) q^{70} + (8 \beta_{2} + 4 \beta_1 + 2) q^{71} + ( - 3 \beta_1 + 4) q^{72} + ( - 5 \beta_{2} - 3 \beta_1 - 7) q^{73} + (4 \beta_1 - 12) q^{74} + (\beta_{2} + 3 \beta_1 - 2) q^{75} + (3 \beta_{2} - 5 \beta_1 + 9) q^{76} + (\beta_{2} - \beta_1 + 1) q^{77} + ( - \beta_{2} - 1) q^{78} + ( - \beta_{2} - 4 \beta_1 - 6) q^{79} + ( - 2 \beta_{2} - 3 \beta_1 - 7) q^{80} + q^{81} + ( - \beta_{2} - 3 \beta_1 + 3) q^{82} + ( - 2 \beta_1 + 6) q^{83} + ( - 3 \beta_{2} + 5 \beta_1 - 9) q^{84} + ( - \beta_{2} - \beta_1 + 1) q^{85} + ( - 4 \beta_{2} + 9 \beta_1 - 1) q^{86} + ( - 3 \beta_{2} - 4 \beta_1 + 2) q^{87} + (3 \beta_1 - 4) q^{88} + ( - \beta_1 + 1) q^{89} + (\beta_{2} + 2 \beta_1) q^{90} + (\beta_{2} - \beta_1 + 1) q^{91} + ( - 3 \beta_{2} + 7 \beta_1 - 13) q^{92} + ( - \beta_1 + 5) q^{93} + (6 \beta_{2} - 6 \beta_1 + 4) q^{94} - 2 q^{95} + (3 \beta_{2} - 4 \beta_1 + 5) q^{96} + (4 \beta_{2} - 4) q^{97} + (\beta_{2} - 10 \beta_1 + 11) q^{98} - q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 3 q^{3} + 5 q^{4} + 4 q^{5} + 3 q^{6} - 2 q^{7} + 9 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} + 3 q^{3} + 5 q^{4} + 4 q^{5} + 3 q^{6} - 2 q^{7} + 9 q^{8} + 3 q^{9} + 2 q^{10} - 3 q^{11} + 5 q^{12} - 3 q^{13} - 12 q^{14} + 4 q^{15} + 5 q^{16} + 3 q^{18} + 2 q^{19} - 2 q^{20} - 2 q^{21} - 3 q^{22} + 4 q^{23} + 9 q^{24} - 3 q^{25} - 3 q^{26} + 3 q^{27} - 22 q^{28} + 2 q^{29} + 2 q^{30} + 14 q^{31} + 11 q^{32} - 3 q^{33} - 8 q^{34} + 6 q^{35} + 5 q^{36} + 12 q^{38} - 3 q^{39} - 8 q^{40} - 4 q^{41} - 12 q^{42} - 6 q^{43} - 5 q^{44} + 4 q^{45} - 22 q^{46} + 14 q^{47} + 5 q^{48} - q^{49} - q^{50} - 5 q^{52} - 2 q^{53} + 3 q^{54} - 4 q^{55} - 32 q^{56} + 2 q^{57} - 14 q^{58} + 28 q^{59} - 2 q^{60} - 8 q^{61} + 16 q^{62} - 2 q^{63} + 33 q^{64} - 4 q^{65} - 3 q^{66} + 2 q^{67} - 10 q^{68} + 4 q^{69} + 6 q^{70} + 10 q^{71} + 9 q^{72} - 24 q^{73} - 32 q^{74} - 3 q^{75} + 22 q^{76} + 2 q^{77} - 3 q^{78} - 22 q^{79} - 24 q^{80} + 3 q^{81} + 6 q^{82} + 16 q^{83} - 22 q^{84} + 2 q^{85} + 6 q^{86} + 2 q^{87} - 9 q^{88} + 2 q^{89} + 2 q^{90} + 2 q^{91} - 32 q^{92} + 14 q^{93} + 6 q^{94} - 6 q^{95} + 11 q^{96} - 12 q^{97} + 23 q^{98} - 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 3x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 2 \) 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
0.311108
2.17009
−1.48119
−1.21432 1.00000 −0.525428 1.31111 −1.21432 1.52543 3.06668 1.00000 −1.59210
1.2 1.53919 1.00000 0.369102 3.17009 1.53919 0.630898 −2.51026 1.00000 4.87936
1.3 2.67513 1.00000 5.15633 −0.481194 2.67513 −4.15633 8.44358 1.00000 −1.28726
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(11\) \(1\)
\(13\) \(1\)

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 429.2.a.g 3
3.b odd 2 1 1287.2.a.h 3
4.b odd 2 1 6864.2.a.bs 3
11.b odd 2 1 4719.2.a.q 3
13.b even 2 1 5577.2.a.j 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.a.g 3 1.a even 1 1 trivial
1287.2.a.h 3 3.b odd 2 1
4719.2.a.q 3 11.b odd 2 1
5577.2.a.j 3 13.b even 2 1
6864.2.a.bs 3 4.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(429))\):

\( T_{2}^{3} - 3T_{2}^{2} - T_{2} + 5 \) Copy content Toggle raw display
\( T_{5}^{3} - 4T_{5}^{2} + 2T_{5} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - 3T^{2} - T + 5 \) Copy content Toggle raw display
$3$ \( (T - 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} - 4 T^{2} + 2 T + 2 \) Copy content Toggle raw display
$7$ \( T^{3} + 2 T^{2} - 8 T + 4 \) Copy content Toggle raw display
$11$ \( (T + 1)^{3} \) Copy content Toggle raw display
$13$ \( (T + 1)^{3} \) Copy content Toggle raw display
$17$ \( T^{3} - 4T - 2 \) Copy content Toggle raw display
$19$ \( T^{3} - 2 T^{2} - 8 T - 4 \) Copy content Toggle raw display
$23$ \( T^{3} - 4 T^{2} - 40 T + 68 \) Copy content Toggle raw display
$29$ \( T^{3} - 2 T^{2} - 64 T + 178 \) Copy content Toggle raw display
$31$ \( T^{3} - 14 T^{2} + 62 T - 86 \) Copy content Toggle raw display
$37$ \( T^{3} - 64T - 128 \) Copy content Toggle raw display
$41$ \( T^{3} + 4 T^{2} - 4 T - 20 \) Copy content Toggle raw display
$43$ \( T^{3} + 6 T^{2} - 108 T - 734 \) Copy content Toggle raw display
$47$ \( T^{3} - 14 T^{2} + 12 T + 296 \) Copy content Toggle raw display
$53$ \( T^{3} + 2 T^{2} - 84 T + 232 \) Copy content Toggle raw display
$59$ \( T^{3} - 28 T^{2} + 248 T - 688 \) Copy content Toggle raw display
$61$ \( T^{3} + 8 T^{2} - 72 T - 368 \) Copy content Toggle raw display
$67$ \( T^{3} - 2 T^{2} - 150 T + 698 \) Copy content Toggle raw display
$71$ \( T^{3} - 10 T^{2} - 212 T + 2056 \) Copy content Toggle raw display
$73$ \( T^{3} + 24 T^{2} + 92 T - 556 \) Copy content Toggle raw display
$79$ \( T^{3} + 22 T^{2} + 112 T + 134 \) Copy content Toggle raw display
$83$ \( T^{3} - 16 T^{2} + 72 T - 80 \) Copy content Toggle raw display
$89$ \( T^{3} - 2 T^{2} - 2 T + 2 \) Copy content Toggle raw display
$97$ \( T^{3} + 12 T^{2} - 16 T - 64 \) Copy content Toggle raw display
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