Properties

Label 3626.2.a.bc
Level $3626$
Weight $2$
Character orbit 3626.a
Self dual yes
Analytic conductor $28.954$
Analytic rank $0$
Dimension $5$
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: \(5\)
Coefficient field: 5.5.2174276.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 9x^{3} + 10x^{2} + 16x - 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + \beta_1 q^{3} + q^{4} + (\beta_{2} + 1) q^{5} + \beta_1 q^{6} + q^{8} + (\beta_{2} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + \beta_1 q^{3} + q^{4} + (\beta_{2} + 1) q^{5} + \beta_1 q^{6} + q^{8} + (\beta_{2} + 2) q^{9} + (\beta_{2} + 1) q^{10} + (\beta_{3} + 1) q^{11} + \beta_1 q^{12} + ( - \beta_{4} + \beta_{3} - \beta_{2} - 1) q^{13} + ( - \beta_{4} - \beta_{3} + 2 \beta_1 + 2) q^{15} + q^{16} + (\beta_{4} - \beta_{3} + \beta_{2} + \cdots + 1) q^{17}+ \cdots + (\beta_{4} + \beta_{3} + 2 \beta_{2} + \cdots + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 5 q^{2} - q^{3} + 5 q^{4} + 3 q^{5} - q^{6} + 5 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 5 q^{2} - q^{3} + 5 q^{4} + 3 q^{5} - q^{6} + 5 q^{8} + 8 q^{9} + 3 q^{10} + 5 q^{11} - q^{12} - 5 q^{13} + 6 q^{15} + 5 q^{16} + 6 q^{17} + 8 q^{18} - 8 q^{19} + 3 q^{20} + 5 q^{22} + 5 q^{23} - q^{24} + 14 q^{25} - 5 q^{26} + 8 q^{27} - q^{29} + 6 q^{30} + 7 q^{31} + 5 q^{32} - 6 q^{33} + 6 q^{34} + 8 q^{36} + 5 q^{37} - 8 q^{38} - 9 q^{39} + 3 q^{40} + 13 q^{41} + 10 q^{43} + 5 q^{44} + 42 q^{45} + 5 q^{46} - 6 q^{47} - q^{48} + 14 q^{50} - 14 q^{51} - 5 q^{52} - 18 q^{53} + 8 q^{54} + 17 q^{55} - 10 q^{57} - q^{58} + 10 q^{59} + 6 q^{60} + q^{61} + 7 q^{62} + 5 q^{64} - 22 q^{65} - 6 q^{66} + 13 q^{67} + 6 q^{68} + 43 q^{69} + 8 q^{71} + 8 q^{72} + 17 q^{73} + 5 q^{74} + 12 q^{75} - 8 q^{76} - 9 q^{78} + q^{79} + 3 q^{80} - 19 q^{81} + 13 q^{82} + 8 q^{83} + 16 q^{85} + 10 q^{86} - 13 q^{87} + 5 q^{88} + 6 q^{89} + 42 q^{90} + 5 q^{92} - 16 q^{93} - 6 q^{94} - 28 q^{95} - q^{96} - 12 q^{97} + 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - x^{4} - 9x^{3} + 10x^{2} + 16x - 18 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{4} - 8\nu^{2} + 11 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{4} - 8\nu^{2} + 2\nu + 11 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} + \nu^{3} - 8\nu^{2} - 4\nu + 13 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} - \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} + 2\beta_{3} - 3\beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{2} + 8\beta _1 + 21 \) 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
1.11875
−1.54941
1.65252
2.39828
−2.62014
1.00000 −2.74839 1.00000 3.55365 −2.74839 0 1.00000 4.55365 3.55365
1.2 1.00000 −1.59932 1.00000 −1.44216 −1.59932 0 1.00000 −0.442161 −1.44216
1.3 1.00000 −1.26918 1.00000 −2.38918 −1.26918 0 1.00000 −1.38918 −2.38918
1.4 1.00000 1.75175 1.00000 −0.931365 1.75175 0 1.00000 0.0686347 −0.931365
1.5 1.00000 2.86514 1.00000 4.20905 2.86514 0 1.00000 5.20905 4.20905
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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.bc 5
7.b odd 2 1 518.2.a.f 5
21.c even 2 1 4662.2.a.bk 5
28.d even 2 1 4144.2.a.y 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
518.2.a.f 5 7.b odd 2 1
3626.2.a.bc 5 1.a even 1 1 trivial
4144.2.a.y 5 28.d even 2 1
4662.2.a.bk 5 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3626))\):

\( T_{3}^{5} + T_{3}^{4} - 11T_{3}^{3} - 12T_{3}^{2} + 24T_{3} + 28 \) Copy content Toggle raw display
\( T_{5}^{5} - 3T_{5}^{4} - 15T_{5}^{3} + 20T_{5}^{2} + 80T_{5} + 48 \) Copy content Toggle raw display
\( T_{11}^{5} - 5T_{11}^{4} - 23T_{11}^{3} + 104T_{11}^{2} + 32T_{11} - 96 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{5} \) Copy content Toggle raw display
$3$ \( T^{5} + T^{4} + \cdots + 28 \) Copy content Toggle raw display
$5$ \( T^{5} - 3 T^{4} + \cdots + 48 \) Copy content Toggle raw display
$7$ \( T^{5} \) Copy content Toggle raw display
$11$ \( T^{5} - 5 T^{4} + \cdots - 96 \) Copy content Toggle raw display
$13$ \( T^{5} + 5 T^{4} + \cdots - 16 \) Copy content Toggle raw display
$17$ \( T^{5} - 6 T^{4} + \cdots - 144 \) Copy content Toggle raw display
$19$ \( T^{5} + 8 T^{4} + \cdots - 32 \) Copy content Toggle raw display
$23$ \( T^{5} - 5 T^{4} + \cdots + 96 \) Copy content Toggle raw display
$29$ \( T^{5} + T^{4} + \cdots + 456 \) Copy content Toggle raw display
$31$ \( T^{5} - 7 T^{4} + \cdots - 476 \) Copy content Toggle raw display
$37$ \( (T - 1)^{5} \) Copy content Toggle raw display
$41$ \( T^{5} - 13 T^{4} + \cdots - 93432 \) Copy content Toggle raw display
$43$ \( T^{5} - 10 T^{4} + \cdots + 6464 \) Copy content Toggle raw display
$47$ \( T^{5} + 6 T^{4} + \cdots - 768 \) Copy content Toggle raw display
$53$ \( T^{5} + 18 T^{4} + \cdots - 8064 \) Copy content Toggle raw display
$59$ \( T^{5} - 10 T^{4} + \cdots - 384 \) Copy content Toggle raw display
$61$ \( T^{5} - T^{4} + \cdots + 5776 \) Copy content Toggle raw display
$67$ \( T^{5} - 13 T^{4} + \cdots + 448 \) Copy content Toggle raw display
$71$ \( T^{5} - 8 T^{4} + \cdots + 2304 \) Copy content Toggle raw display
$73$ \( T^{5} - 17 T^{4} + \cdots + 30248 \) Copy content Toggle raw display
$79$ \( T^{5} - T^{4} + \cdots - 4064 \) Copy content Toggle raw display
$83$ \( T^{5} - 8 T^{4} + \cdots + 1344 \) Copy content Toggle raw display
$89$ \( T^{5} - 6 T^{4} + \cdots - 34704 \) Copy content Toggle raw display
$97$ \( T^{5} + 12 T^{4} + \cdots + 30352 \) Copy content Toggle raw display
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