Properties

Label 129.2.a.c
Level $129$
Weight $2$
Character orbit 129.a
Self dual yes
Analytic conductor $1.030$
Analytic rank $0$
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] = [129,2,Mod(1,129)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(129, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("129.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 129 = 3 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 129.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: \(1.03007018607\)
Analytic rank: \(0\)
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.

\(f(q)\) \(=\) \( q + (\beta + 1) q^{2} - q^{3} + (2 \beta + 1) q^{4} + ( - \beta + 1) q^{5} + ( - \beta - 1) q^{6} + ( - 2 \beta + 1) q^{7} + (\beta + 3) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 1) q^{2} - q^{3} + (2 \beta + 1) q^{4} + ( - \beta + 1) q^{5} + ( - \beta - 1) q^{6} + ( - 2 \beta + 1) q^{7} + (\beta + 3) q^{8} + q^{9} - q^{10} + ( - \beta + 3) q^{11} + ( - 2 \beta - 1) q^{12} - 5 q^{13} + ( - \beta - 3) q^{14} + (\beta - 1) q^{15} + 3 q^{16} + ( - 2 \beta - 2) q^{17} + (\beta + 1) q^{18} + (4 \beta - 1) q^{19} + (\beta - 3) q^{20} + (2 \beta - 1) q^{21} + (2 \beta + 1) q^{22} + 6 q^{23} + ( - \beta - 3) q^{24} + ( - 2 \beta - 2) q^{25} + ( - 5 \beta - 5) q^{26} - q^{27} - 7 q^{28} + (3 \beta + 3) q^{29} + q^{30} + 4 q^{31} + (\beta - 3) q^{32} + (\beta - 3) q^{33} + ( - 4 \beta - 6) q^{34} + ( - 3 \beta + 5) q^{35} + (2 \beta + 1) q^{36} + ( - 2 \beta - 4) q^{37} + (3 \beta + 7) q^{38} + 5 q^{39} + ( - 2 \beta + 1) q^{40} + 4 \beta q^{41} + (\beta + 3) q^{42} + q^{43} + (5 \beta - 1) q^{44} + ( - \beta + 1) q^{45} + (6 \beta + 6) q^{46} + (7 \beta - 1) q^{47} - 3 q^{48} + ( - 4 \beta + 2) q^{49} + ( - 4 \beta - 6) q^{50} + (2 \beta + 2) q^{51} + ( - 10 \beta - 5) q^{52} + 8 \beta q^{53} + ( - \beta - 1) q^{54} + ( - 4 \beta + 5) q^{55} + ( - 5 \beta - 1) q^{56} + ( - 4 \beta + 1) q^{57} + (6 \beta + 9) q^{58} + ( - 8 \beta + 2) q^{59} + ( - \beta + 3) q^{60} + (2 \beta - 4) q^{61} + (4 \beta + 4) q^{62} + ( - 2 \beta + 1) q^{63} + ( - 2 \beta - 7) q^{64} + (5 \beta - 5) q^{65} + ( - 2 \beta - 1) q^{66} + ( - 6 \beta - 6) q^{67} + ( - 6 \beta - 10) q^{68} - 6 q^{69} + (2 \beta - 1) q^{70} + ( - 2 \beta + 6) q^{71} + (\beta + 3) q^{72} + (4 \beta + 2) q^{73} + ( - 6 \beta - 8) q^{74} + (2 \beta + 2) q^{75} + (2 \beta + 15) q^{76} + ( - 7 \beta + 7) q^{77} + (5 \beta + 5) q^{78} + (6 \beta + 4) q^{79} + ( - 3 \beta + 3) q^{80} + q^{81} + (4 \beta + 8) q^{82} + ( - \beta - 7) q^{83} + 7 q^{84} + 2 q^{85} + (\beta + 1) q^{86} + ( - 3 \beta - 3) q^{87} + 7 q^{88} - 6 \beta q^{89} - q^{90} + (10 \beta - 5) q^{91} + (12 \beta + 6) q^{92} - 4 q^{93} + (6 \beta + 13) q^{94} + (5 \beta - 9) q^{95} + ( - \beta + 3) q^{96} + (2 \beta - 1) q^{97} + ( - 2 \beta - 6) q^{98} + ( - \beta + 3) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} + 2 q^{7} + 6 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} + 2 q^{7} + 6 q^{8} + 2 q^{9} - 2 q^{10} + 6 q^{11} - 2 q^{12} - 10 q^{13} - 6 q^{14} - 2 q^{15} + 6 q^{16} - 4 q^{17} + 2 q^{18} - 2 q^{19} - 6 q^{20} - 2 q^{21} + 2 q^{22} + 12 q^{23} - 6 q^{24} - 4 q^{25} - 10 q^{26} - 2 q^{27} - 14 q^{28} + 6 q^{29} + 2 q^{30} + 8 q^{31} - 6 q^{32} - 6 q^{33} - 12 q^{34} + 10 q^{35} + 2 q^{36} - 8 q^{37} + 14 q^{38} + 10 q^{39} + 2 q^{40} + 6 q^{42} + 2 q^{43} - 2 q^{44} + 2 q^{45} + 12 q^{46} - 2 q^{47} - 6 q^{48} + 4 q^{49} - 12 q^{50} + 4 q^{51} - 10 q^{52} - 2 q^{54} + 10 q^{55} - 2 q^{56} + 2 q^{57} + 18 q^{58} + 4 q^{59} + 6 q^{60} - 8 q^{61} + 8 q^{62} + 2 q^{63} - 14 q^{64} - 10 q^{65} - 2 q^{66} - 12 q^{67} - 20 q^{68} - 12 q^{69} - 2 q^{70} + 12 q^{71} + 6 q^{72} + 4 q^{73} - 16 q^{74} + 4 q^{75} + 30 q^{76} + 14 q^{77} + 10 q^{78} + 8 q^{79} + 6 q^{80} + 2 q^{81} + 16 q^{82} - 14 q^{83} + 14 q^{84} + 4 q^{85} + 2 q^{86} - 6 q^{87} + 14 q^{88} - 2 q^{90} - 10 q^{91} + 12 q^{92} - 8 q^{93} + 26 q^{94} - 18 q^{95} + 6 q^{96} - 2 q^{97} - 12 q^{98} + 6 q^{99}+O(q^{100}) \) 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
−1.41421
1.41421
−0.414214 −1.00000 −1.82843 2.41421 0.414214 3.82843 1.58579 1.00000 −1.00000
1.2 2.41421 −1.00000 3.82843 −0.414214 −2.41421 −1.82843 4.41421 1.00000 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(43\) \(-1\)

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 129.2.a.c 2
3.b odd 2 1 387.2.a.f 2
4.b odd 2 1 2064.2.a.v 2
5.b even 2 1 3225.2.a.l 2
7.b odd 2 1 6321.2.a.m 2
8.b even 2 1 8256.2.a.ch 2
8.d odd 2 1 8256.2.a.bx 2
12.b even 2 1 6192.2.a.bg 2
15.d odd 2 1 9675.2.a.bm 2
43.b odd 2 1 5547.2.a.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
129.2.a.c 2 1.a even 1 1 trivial
387.2.a.f 2 3.b odd 2 1
2064.2.a.v 2 4.b odd 2 1
3225.2.a.l 2 5.b even 2 1
5547.2.a.f 2 43.b odd 2 1
6192.2.a.bg 2 12.b even 2 1
6321.2.a.m 2 7.b odd 2 1
8256.2.a.bx 2 8.d odd 2 1
8256.2.a.ch 2 8.b even 2 1
9675.2.a.bm 2 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 2T_{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(129))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 2T - 1 \) Copy content Toggle raw display
$3$ \( (T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 2T - 1 \) Copy content Toggle raw display
$7$ \( T^{2} - 2T - 7 \) Copy content Toggle raw display
$11$ \( T^{2} - 6T + 7 \) Copy content Toggle raw display
$13$ \( (T + 5)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 4T - 4 \) Copy content Toggle raw display
$19$ \( T^{2} + 2T - 31 \) Copy content Toggle raw display
$23$ \( (T - 6)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 6T - 9 \) Copy content Toggle raw display
$31$ \( (T - 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 8T + 8 \) Copy content Toggle raw display
$41$ \( T^{2} - 32 \) Copy content Toggle raw display
$43$ \( (T - 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 2T - 97 \) Copy content Toggle raw display
$53$ \( T^{2} - 128 \) Copy content Toggle raw display
$59$ \( T^{2} - 4T - 124 \) Copy content Toggle raw display
$61$ \( T^{2} + 8T + 8 \) Copy content Toggle raw display
$67$ \( T^{2} + 12T - 36 \) Copy content Toggle raw display
$71$ \( T^{2} - 12T + 28 \) Copy content Toggle raw display
$73$ \( T^{2} - 4T - 28 \) Copy content Toggle raw display
$79$ \( T^{2} - 8T - 56 \) Copy content Toggle raw display
$83$ \( T^{2} + 14T + 47 \) Copy content Toggle raw display
$89$ \( T^{2} - 72 \) Copy content Toggle raw display
$97$ \( T^{2} + 2T - 7 \) Copy content Toggle raw display
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