Properties

Label 3626.2.a.b
Level $3626$
Weight $2$
Character orbit 3626.a
Self dual yes
Analytic conductor $28.954$
Analytic rank $1$
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] = [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: \(2\)
Coefficient field: \(\Q(\sqrt{21}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 5 \) 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 \(\beta = \frac{1}{2}(1 + \sqrt{21})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} - q^{3} + q^{4} + (\beta + 1) q^{5} + q^{6} - q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - q^{3} + q^{4} + (\beta + 1) q^{5} + q^{6} - q^{8} - 2 q^{9} + ( - \beta - 1) q^{10} + ( - \beta - 1) q^{11} - q^{12} + q^{13} + ( - \beta - 1) q^{15} + q^{16} + (2 \beta - 1) q^{17} + 2 q^{18} + ( - 3 \beta + 1) q^{19} + (\beta + 1) q^{20} + (\beta + 1) q^{22} + ( - 2 \beta + 1) q^{23} + q^{24} + (3 \beta + 1) q^{25} - q^{26} + 5 q^{27} + (\beta + 1) q^{30} + (3 \beta - 5) q^{31} - q^{32} + (\beta + 1) q^{33} + ( - 2 \beta + 1) q^{34} - 2 q^{36} + q^{37} + (3 \beta - 1) q^{38} - q^{39} + ( - \beta - 1) q^{40} + (2 \beta + 5) q^{41} + ( - 3 \beta + 5) q^{43} + ( - \beta - 1) q^{44} + ( - 2 \beta - 2) q^{45} + (2 \beta - 1) q^{46} + ( - \beta - 4) q^{47} - q^{48} + ( - 3 \beta - 1) q^{50} + ( - 2 \beta + 1) q^{51} + q^{52} + ( - 5 \beta + 1) q^{53} - 5 q^{54} + ( - 3 \beta - 6) q^{55} + (3 \beta - 1) q^{57} + ( - 4 \beta + 8) q^{59} + ( - \beta - 1) q^{60} + (3 \beta - 8) q^{61} + ( - 3 \beta + 5) q^{62} + q^{64} + (\beta + 1) q^{65} + ( - \beta - 1) q^{66} - 13 q^{67} + (2 \beta - 1) q^{68} + (2 \beta - 1) q^{69} + ( - 2 \beta + 10) q^{71} + 2 q^{72} + ( - 3 \beta - 8) q^{73} - q^{74} + ( - 3 \beta - 1) q^{75} + ( - 3 \beta + 1) q^{76} + q^{78} + (3 \beta - 4) q^{79} + (\beta + 1) q^{80} + q^{81} + ( - 2 \beta - 5) q^{82} + (\beta - 11) q^{83} + (3 \beta + 9) q^{85} + (3 \beta - 5) q^{86} + (\beta + 1) q^{88} + ( - 3 \beta - 3) q^{89} + (2 \beta + 2) q^{90} + ( - 2 \beta + 1) q^{92} + ( - 3 \beta + 5) q^{93} + (\beta + 4) q^{94} + ( - 5 \beta - 14) q^{95} + q^{96} + (3 \beta - 8) q^{97} + (2 \beta + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 3 q^{5} + 2 q^{6} - 2 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 3 q^{5} + 2 q^{6} - 2 q^{8} - 4 q^{9} - 3 q^{10} - 3 q^{11} - 2 q^{12} + 2 q^{13} - 3 q^{15} + 2 q^{16} + 4 q^{18} - q^{19} + 3 q^{20} + 3 q^{22} + 2 q^{24} + 5 q^{25} - 2 q^{26} + 10 q^{27} + 3 q^{30} - 7 q^{31} - 2 q^{32} + 3 q^{33} - 4 q^{36} + 2 q^{37} + q^{38} - 2 q^{39} - 3 q^{40} + 12 q^{41} + 7 q^{43} - 3 q^{44} - 6 q^{45} - 9 q^{47} - 2 q^{48} - 5 q^{50} + 2 q^{52} - 3 q^{53} - 10 q^{54} - 15 q^{55} + q^{57} + 12 q^{59} - 3 q^{60} - 13 q^{61} + 7 q^{62} + 2 q^{64} + 3 q^{65} - 3 q^{66} - 26 q^{67} + 18 q^{71} + 4 q^{72} - 19 q^{73} - 2 q^{74} - 5 q^{75} - q^{76} + 2 q^{78} - 5 q^{79} + 3 q^{80} + 2 q^{81} - 12 q^{82} - 21 q^{83} + 21 q^{85} - 7 q^{86} + 3 q^{88} - 9 q^{89} + 6 q^{90} + 7 q^{93} + 9 q^{94} - 33 q^{95} + 2 q^{96} - 13 q^{97} + 6 q^{99}+O(q^{100}) \) 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
−1.79129
2.79129
−1.00000 −1.00000 1.00000 −0.791288 1.00000 0 −1.00000 −2.00000 0.791288
1.2 −1.00000 −1.00000 1.00000 3.79129 1.00000 0 −1.00000 −2.00000 −3.79129
\(n\): 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.b 2
7.b odd 2 1 3626.2.a.j 2
7.d odd 6 2 518.2.e.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
518.2.e.c 4 7.d odd 6 2
3626.2.a.b 2 1.a even 1 1 trivial
3626.2.a.j 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3626))\):

\( T_{3} + 1 \) Copy content Toggle raw display
\( T_{5}^{2} - 3T_{5} - 3 \) Copy content Toggle raw display
\( T_{11}^{2} + 3T_{11} - 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 3T - 3 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 3T - 3 \) Copy content Toggle raw display
$13$ \( (T - 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 21 \) Copy content Toggle raw display
$19$ \( T^{2} + T - 47 \) Copy content Toggle raw display
$23$ \( T^{2} - 21 \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 7T - 35 \) Copy content Toggle raw display
$37$ \( (T - 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 12T + 15 \) Copy content Toggle raw display
$43$ \( T^{2} - 7T - 35 \) Copy content Toggle raw display
$47$ \( T^{2} + 9T + 15 \) Copy content Toggle raw display
$53$ \( T^{2} + 3T - 129 \) Copy content Toggle raw display
$59$ \( T^{2} - 12T - 48 \) Copy content Toggle raw display
$61$ \( T^{2} + 13T - 5 \) Copy content Toggle raw display
$67$ \( (T + 13)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 18T + 60 \) Copy content Toggle raw display
$73$ \( T^{2} + 19T + 43 \) Copy content Toggle raw display
$79$ \( T^{2} + 5T - 41 \) Copy content Toggle raw display
$83$ \( T^{2} + 21T + 105 \) Copy content Toggle raw display
$89$ \( T^{2} + 9T - 27 \) Copy content Toggle raw display
$97$ \( T^{2} + 13T - 5 \) Copy content Toggle raw display
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