Properties

Label 4029.2.a.c
Level $4029$
Weight $2$
Character orbit 4029.a
Self dual yes
Analytic conductor $32.172$
Analytic rank $1$
Dimension $2$
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] = [4029,2,Mod(1,4029)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4029, 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("4029.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4029 = 3 \cdot 17 \cdot 79 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4029.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: \(32.1717269744\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) 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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + q^{3} + (\beta - 1) q^{5} + \beta q^{6} + q^{7} - 2 \beta q^{8} + q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} + q^{3} + (\beta - 1) q^{5} + \beta q^{6} + q^{7} - 2 \beta q^{8} + q^{9} + ( - \beta + 2) q^{10} - 4 \beta q^{11} + \beta q^{14} + (\beta - 1) q^{15} - 4 q^{16} + q^{17} + \beta q^{18} + (2 \beta - 5) q^{19} + q^{21} - 8 q^{22} + (3 \beta - 3) q^{23} - 2 \beta q^{24} + ( - 2 \beta - 2) q^{25} + q^{27} + (2 \beta + 2) q^{29} + ( - \beta + 2) q^{30} + ( - 4 \beta + 4) q^{31} - 4 \beta q^{33} + \beta q^{34} + (\beta - 1) q^{35} + (4 \beta - 5) q^{37} + ( - 5 \beta + 4) q^{38} + (2 \beta - 4) q^{40} + (4 \beta + 2) q^{41} + \beta q^{42} + ( - 6 \beta + 2) q^{43} + (\beta - 1) q^{45} + ( - 3 \beta + 6) q^{46} + ( - 5 \beta - 3) q^{47} - 4 q^{48} - 6 q^{49} + ( - 2 \beta - 4) q^{50} + q^{51} + ( - \beta - 11) q^{53} + \beta q^{54} + (4 \beta - 8) q^{55} - 2 \beta q^{56} + (2 \beta - 5) q^{57} + (2 \beta + 4) q^{58} + (\beta - 5) q^{59} + ( - 4 \beta + 5) q^{61} + (4 \beta - 8) q^{62} + q^{63} + 8 q^{64} - 8 q^{66} - 10 q^{67} + (3 \beta - 3) q^{69} + ( - \beta + 2) q^{70} + ( - 5 \beta - 6) q^{71} - 2 \beta q^{72} + ( - 9 \beta + 2) q^{73} + ( - 5 \beta + 8) q^{74} + ( - 2 \beta - 2) q^{75} - 4 \beta q^{77} - q^{79} + ( - 4 \beta + 4) q^{80} + q^{81} + (2 \beta + 8) q^{82} + (\beta + 4) q^{83} + (\beta - 1) q^{85} + (2 \beta - 12) q^{86} + (2 \beta + 2) q^{87} + 16 q^{88} + (4 \beta - 12) q^{89} + ( - \beta + 2) q^{90} + ( - 4 \beta + 4) q^{93} + ( - 3 \beta - 10) q^{94} + ( - 7 \beta + 9) q^{95} + (\beta + 16) q^{97} - 6 \beta q^{98} - 4 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9} + 4 q^{10} - 2 q^{15} - 8 q^{16} + 2 q^{17} - 10 q^{19} + 2 q^{21} - 16 q^{22} - 6 q^{23} - 4 q^{25} + 2 q^{27} + 4 q^{29} + 4 q^{30} + 8 q^{31} - 2 q^{35} - 10 q^{37} + 8 q^{38} - 8 q^{40} + 4 q^{41} + 4 q^{43} - 2 q^{45} + 12 q^{46} - 6 q^{47} - 8 q^{48} - 12 q^{49} - 8 q^{50} + 2 q^{51} - 22 q^{53} - 16 q^{55} - 10 q^{57} + 8 q^{58} - 10 q^{59} + 10 q^{61} - 16 q^{62} + 2 q^{63} + 16 q^{64} - 16 q^{66} - 20 q^{67} - 6 q^{69} + 4 q^{70} - 12 q^{71} + 4 q^{73} + 16 q^{74} - 4 q^{75} - 2 q^{79} + 8 q^{80} + 2 q^{81} + 16 q^{82} + 8 q^{83} - 2 q^{85} - 24 q^{86} + 4 q^{87} + 32 q^{88} - 24 q^{89} + 4 q^{90} + 8 q^{93} - 20 q^{94} + 18 q^{95} + 32 q^{97}+O(q^{100}) \) 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
−1.41421
1.41421
−1.41421 1.00000 0 −2.41421 −1.41421 1.00000 2.82843 1.00000 3.41421
1.2 1.41421 1.00000 0 0.414214 1.41421 1.00000 −2.82843 1.00000 0.585786
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(17\) \(-1\)
\(79\) \(1\)

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 4029.2.a.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4029.2.a.c 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4029))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 2 \) Copy content Toggle raw display
$3$ \( (T - 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 2T - 1 \) Copy content Toggle raw display
$7$ \( (T - 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 32 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( (T - 1)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 10T + 17 \) Copy content Toggle raw display
$23$ \( T^{2} + 6T - 9 \) Copy content Toggle raw display
$29$ \( T^{2} - 4T - 4 \) Copy content Toggle raw display
$31$ \( T^{2} - 8T - 16 \) Copy content Toggle raw display
$37$ \( T^{2} + 10T - 7 \) Copy content Toggle raw display
$41$ \( T^{2} - 4T - 28 \) Copy content Toggle raw display
$43$ \( T^{2} - 4T - 68 \) Copy content Toggle raw display
$47$ \( T^{2} + 6T - 41 \) Copy content Toggle raw display
$53$ \( T^{2} + 22T + 119 \) Copy content Toggle raw display
$59$ \( T^{2} + 10T + 23 \) Copy content Toggle raw display
$61$ \( T^{2} - 10T - 7 \) Copy content Toggle raw display
$67$ \( (T + 10)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 12T - 14 \) Copy content Toggle raw display
$73$ \( T^{2} - 4T - 158 \) Copy content Toggle raw display
$79$ \( (T + 1)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 8T + 14 \) Copy content Toggle raw display
$89$ \( T^{2} + 24T + 112 \) Copy content Toggle raw display
$97$ \( T^{2} - 32T + 254 \) Copy content Toggle raw display
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