Properties

Label 3626.2.a.bf
Level $3626$
Weight $2$
Character orbit 3626.a
Self dual yes
Analytic conductor $28.954$
Analytic rank $0$
Dimension $6$
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: \(6\)
Coefficient field: 6.6.24635632.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 10x^{4} + 13x^{3} + 10x^{2} - 13x - 1 \) 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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} - 10x^{4} + 13x^{3} + 10x^{2} - 13x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} - 9\nu^{2} + 4\nu + 5 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} - 9\nu^{3} + 4\nu^{2} + 5\nu - 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{5} - \nu^{4} + 29\nu^{3} - \nu^{2} - 35\nu - 1 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -4\nu^{5} - \nu^{4} + 38\nu^{3} - 3\nu^{2} - 38\nu - 7 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - \beta_{4} + \beta_{3} - \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{5} + 2\beta_{4} + 2\beta_{3} + \beta_{2} + 9\beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 9\beta_{5} - 9\beta_{4} + 9\beta_{3} + 2\beta_{2} - 13\beta _1 + 31 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -13\beta_{5} + 22\beta_{4} + 16\beta_{3} + 9\beta_{2} + 80\beta _1 - 41 \) 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
−1.02721
1.35453
−3.10233
1.19066
2.65756
−0.0732141
−1.00000 −2.74591 1.00000 1.58067 2.74591 0 −1.00000 4.54005 −1.58067
1.2 −1.00000 −0.364177 1.00000 −1.58067 0.364177 0 −1.00000 −2.86737 1.58067
1.3 −1.00000 0.300349 1.00000 −3.29499 −0.300349 0 −1.00000 −2.90979 3.29499
1.4 −1.00000 0.506692 1.00000 3.55591 −0.506692 0 −1.00000 −2.74326 −3.55591
1.5 −1.00000 1.97359 1.00000 −3.55591 −1.97359 0 −1.00000 0.895047 3.55591
1.6 −1.00000 3.32946 1.00000 3.29499 −3.32946 0 −1.00000 8.08534 −3.29499
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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.bf 6
7.b odd 2 1 3626.2.a.bd 6
7.d odd 6 2 518.2.e.h 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
518.2.e.h 12 7.d odd 6 2
3626.2.a.bd 6 7.b odd 2 1
3626.2.a.bf 6 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3626))\):

\( T_{3}^{6} - 3T_{3}^{5} - 7T_{3}^{4} + 22T_{3}^{3} - 7T_{3}^{2} - 3T_{3} + 1 \) Copy content Toggle raw display
\( T_{5}^{6} - 26T_{5}^{4} + 196T_{5}^{2} - 343 \) Copy content Toggle raw display
\( T_{11}^{6} + T_{11}^{5} - 32T_{11}^{4} - 11T_{11}^{3} + 110T_{11}^{2} + 31T_{11} - 47 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - 3 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{6} - 26 T^{4} + \cdots - 343 \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( T^{6} + T^{5} + \cdots - 47 \) Copy content Toggle raw display
$13$ \( T^{6} - 6 T^{5} + \cdots + 67 \) Copy content Toggle raw display
$17$ \( T^{6} + 3 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{6} - 13 T^{5} + \cdots - 2267 \) Copy content Toggle raw display
$23$ \( T^{6} + 8 T^{5} + \cdots - 28 \) Copy content Toggle raw display
$29$ \( (T^{3} + 4 T^{2} - 44 T - 64)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} - 26 T^{5} + \cdots - 4679 \) Copy content Toggle raw display
$37$ \( (T + 1)^{6} \) Copy content Toggle raw display
$41$ \( T^{6} + 13 T^{5} + \cdots - 593 \) Copy content Toggle raw display
$43$ \( (T^{3} + 8 T^{2} - 14 T - 59)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + 9 T^{5} + \cdots + 99397 \) Copy content Toggle raw display
$53$ \( T^{6} - 23 T^{5} + \cdots - 104713 \) Copy content Toggle raw display
$59$ \( T^{6} - 23 T^{5} + \cdots - 61904 \) Copy content Toggle raw display
$61$ \( T^{6} - 20 T^{5} + \cdots - 27292 \) Copy content Toggle raw display
$67$ \( T^{6} - 24 T^{5} + \cdots + 23969 \) Copy content Toggle raw display
$71$ \( T^{6} + 15 T^{5} + \cdots + 30272 \) Copy content Toggle raw display
$73$ \( T^{6} - 201 T^{4} + \cdots + 24668 \) Copy content Toggle raw display
$79$ \( T^{6} + 6 T^{5} + \cdots - 81961 \) Copy content Toggle raw display
$83$ \( T^{6} + 10 T^{5} + \cdots - 227 \) Copy content Toggle raw display
$89$ \( T^{6} + 4 T^{5} + \cdots + 38448 \) Copy content Toggle raw display
$97$ \( T^{6} - 22 T^{5} + \cdots - 17533 \) Copy content Toggle raw display
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