Properties

Label 3626.2.a.w
Level $3626$
Weight $2$
Character orbit 3626.a
Self dual yes
Analytic conductor $28.954$
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] = [3626,2,Mod(1,3626)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3626, 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("3626.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3626 = 2 \cdot 7^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3626.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: \(28.9537557729\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 4x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 518)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - 4x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) 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
2.11491
−1.86081
−0.254102
−1.00000 −1.47283 1.00000 3.11491 1.47283 0 −1.00000 −0.830760 −3.11491
1.2 −1.00000 −0.462598 1.00000 −0.860806 0.462598 0 −1.00000 −2.78600 0.860806
1.3 −1.00000 2.93543 1.00000 0.745898 −2.93543 0 −1.00000 5.61676 −0.745898
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(7\) \(-1\)
\(37\) \(1\)

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 3626.2.a.w 3
7.b odd 2 1 518.2.a.d 3
21.c even 2 1 4662.2.a.be 3
28.d even 2 1 4144.2.a.n 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
518.2.a.d 3 7.b odd 2 1
3626.2.a.w 3 1.a even 1 1 trivial
4144.2.a.n 3 28.d even 2 1
4662.2.a.be 3 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3626))\):

\( T_{3}^{3} - T_{3}^{2} - 5T_{3} - 2 \) Copy content Toggle raw display
\( T_{5}^{3} - 3T_{5}^{2} - T_{5} + 2 \) Copy content Toggle raw display
\( T_{11}^{3} - 5T_{11}^{2} - 31T_{11} + 148 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{3} - T^{2} - 5T - 2 \) Copy content Toggle raw display
$5$ \( T^{3} - 3T^{2} - T + 2 \) Copy content Toggle raw display
$7$ \( T^{3} \) Copy content Toggle raw display
$11$ \( T^{3} - 5 T^{2} + \cdots + 148 \) Copy content Toggle raw display
$13$ \( T^{3} - 11 T^{2} + \cdots - 26 \) Copy content Toggle raw display
$17$ \( T^{3} - 2 T^{2} + \cdots - 16 \) Copy content Toggle raw display
$19$ \( T^{3} - 10 T^{2} + \cdots + 64 \) Copy content Toggle raw display
$23$ \( T^{3} - 7 T^{2} + \cdots + 424 \) Copy content Toggle raw display
$29$ \( T^{3} + 7 T^{2} + \cdots - 106 \) Copy content Toggle raw display
$31$ \( T^{3} - 5 T^{2} + \cdots + 446 \) Copy content Toggle raw display
$37$ \( (T + 1)^{3} \) Copy content Toggle raw display
$41$ \( T^{3} - T^{2} - 5T - 2 \) Copy content Toggle raw display
$43$ \( T^{3} + 14 T^{2} + \cdots - 848 \) Copy content Toggle raw display
$47$ \( T^{3} + 14 T^{2} + \cdots + 32 \) Copy content Toggle raw display
$53$ \( T^{3} + 14 T^{2} + \cdots - 32 \) Copy content Toggle raw display
$59$ \( T^{3} + 8 T^{2} + \cdots - 448 \) Copy content Toggle raw display
$61$ \( T^{3} - 21 T^{2} + \cdots - 314 \) Copy content Toggle raw display
$67$ \( T^{3} + 23 T^{2} + \cdots - 548 \) Copy content Toggle raw display
$71$ \( T^{3} - 12 T^{2} + \cdots + 128 \) Copy content Toggle raw display
$73$ \( T^{3} - 5 T^{2} + \cdots + 386 \) Copy content Toggle raw display
$79$ \( T^{3} - 3 T^{2} + \cdots + 148 \) Copy content Toggle raw display
$83$ \( T^{3} - 6 T^{2} + \cdots + 1184 \) Copy content Toggle raw display
$89$ \( T^{3} + 18 T^{2} + \cdots - 864 \) Copy content Toggle raw display
$97$ \( T^{3} - 8 T^{2} + \cdots + 1568 \) Copy content Toggle raw display
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