Properties

Label 3626.2.a.z
Level $3626$
Weight $2$
Character orbit 3626.a
Self dual yes
Analytic conductor $28.954$
Analytic rank $1$
Dimension $4$
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: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.2624.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 3x^{2} + 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 3x^{2} + 2x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2\nu - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 2\nu^{2} - 2\nu + 1 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 2\beta_{2} + 6\beta _1 + 1 \) 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
2.77462
−1.22833
−0.360409
0.814115
1.00000 −2.92391 1.00000 0.774623 −2.92391 0 1.00000 5.54925 0.774623
1.2 1.00000 −0.737118 1.00000 −3.22833 −0.737118 0 1.00000 −2.45666 −3.22833
1.3 1.00000 1.50970 1.00000 −2.36041 1.50970 0 1.00000 −0.720819 −2.36041
1.4 1.00000 2.15133 1.00000 −1.18589 2.15133 0 1.00000 1.62823 −1.18589
\(n\): 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.z 4
7.b odd 2 1 3626.2.a.ba 4
7.c even 3 2 518.2.e.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
518.2.e.g 8 7.c even 3 2
3626.2.a.z 4 1.a even 1 1 trivial
3626.2.a.ba 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3626))\):

\( T_{3}^{4} - 8T_{3}^{2} + 4T_{3} + 7 \) Copy content Toggle raw display
\( T_{5}^{4} + 6T_{5}^{3} + 9T_{5}^{2} - 2T_{5} - 7 \) Copy content Toggle raw display
\( T_{11}^{4} - 2T_{11}^{3} - 35T_{11}^{2} + 86T_{11} - 49 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 8 T^{2} + \cdots + 7 \) Copy content Toggle raw display
$5$ \( T^{4} + 6 T^{3} + \cdots - 7 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 2 T^{3} + \cdots - 49 \) Copy content Toggle raw display
$13$ \( (T^{2} + 6 T + 7)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 16 T^{2} + \cdots - 1 \) Copy content Toggle raw display
$19$ \( T^{4} + 6 T^{3} + \cdots + 17 \) Copy content Toggle raw display
$23$ \( T^{4} + 4 T^{3} + \cdots + 47 \) Copy content Toggle raw display
$29$ \( T^{4} + 4 T^{3} + \cdots - 272 \) Copy content Toggle raw display
$31$ \( T^{4} + 6 T^{3} + \cdots + 167 \) Copy content Toggle raw display
$37$ \( (T - 1)^{4} \) Copy content Toggle raw display
$41$ \( T^{4} + 4 T^{3} + \cdots + 49 \) Copy content Toggle raw display
$43$ \( T^{4} - 14 T^{3} + \cdots - 1951 \) Copy content Toggle raw display
$47$ \( T^{4} + 18 T^{3} + \cdots + 17 \) Copy content Toggle raw display
$53$ \( T^{4} + 6 T^{3} + \cdots + 175 \) Copy content Toggle raw display
$59$ \( T^{4} + 4 T^{3} + \cdots + 1808 \) Copy content Toggle raw display
$61$ \( T^{4} + 18 T^{3} + \cdots + 17 \) Copy content Toggle raw display
$67$ \( T^{4} + 4 T^{3} + \cdots + 20143 \) Copy content Toggle raw display
$71$ \( T^{4} + 16 T^{3} + \cdots - 8176 \) Copy content Toggle raw display
$73$ \( T^{4} + 10 T^{3} + \cdots - 3791 \) Copy content Toggle raw display
$79$ \( T^{4} - 2 T^{3} + \cdots + 1471 \) Copy content Toggle raw display
$83$ \( T^{4} - 10 T^{3} + \cdots - 1841 \) Copy content Toggle raw display
$89$ \( T^{4} - 6 T^{3} + \cdots + 4463 \) Copy content Toggle raw display
$97$ \( T^{4} + 14 T^{3} + \cdots + 2303 \) Copy content Toggle raw display
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