Properties

Label 3626.2.a.s
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: \(\Q(\sqrt{5}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) 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 74)
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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + \beta q^{3} + q^{4} + ( - 3 \beta + 1) q^{5} + \beta q^{6} + q^{8} + (\beta - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + \beta q^{3} + q^{4} + ( - 3 \beta + 1) q^{5} + \beta q^{6} + q^{8} + (\beta - 2) q^{9} + ( - 3 \beta + 1) q^{10} + (\beta - 3) q^{11} + \beta q^{12} + (3 \beta - 2) q^{13} + ( - 2 \beta - 3) q^{15} + q^{16} + (4 \beta - 2) q^{17} + (\beta - 2) q^{18} + ( - 4 \beta + 2) q^{19} + ( - 3 \beta + 1) q^{20} + (\beta - 3) q^{22} + (3 \beta - 2) q^{23} + \beta q^{24} + (3 \beta + 5) q^{25} + (3 \beta - 2) q^{26} + ( - 4 \beta + 1) q^{27} + ( - 7 \beta + 2) q^{29} + ( - 2 \beta - 3) q^{30} + (\beta - 9) q^{31} + q^{32} + ( - 2 \beta + 1) q^{33} + (4 \beta - 2) q^{34} + (\beta - 2) q^{36} - q^{37} + ( - 4 \beta + 2) q^{38} + (\beta + 3) q^{39} + ( - 3 \beta + 1) q^{40} + ( - \beta - 8) q^{41} + ( - 2 \beta - 2) q^{43} + (\beta - 3) q^{44} + (4 \beta - 5) q^{45} + (3 \beta - 2) q^{46} + (2 \beta - 2) q^{47} + \beta q^{48} + (3 \beta + 5) q^{50} + (2 \beta + 4) q^{51} + (3 \beta - 2) q^{52} + (4 \beta - 6) q^{53} + ( - 4 \beta + 1) q^{54} + (7 \beta - 6) q^{55} + ( - 2 \beta - 4) q^{57} + ( - 7 \beta + 2) q^{58} + ( - 2 \beta + 8) q^{59} + ( - 2 \beta - 3) q^{60} + ( - \beta - 9) q^{61} + (\beta - 9) q^{62} + q^{64} - 11 q^{65} + ( - 2 \beta + 1) q^{66} + (5 \beta - 7) q^{67} + (4 \beta - 2) q^{68} + (\beta + 3) q^{69} + (8 \beta - 10) q^{71} + (\beta - 2) q^{72} + ( - 5 \beta + 1) q^{73} - q^{74} + (8 \beta + 3) q^{75} + ( - 4 \beta + 2) q^{76} + (\beta + 3) q^{78} + ( - 9 \beta + 6) q^{79} + ( - 3 \beta + 1) q^{80} + ( - 6 \beta + 2) q^{81} + ( - \beta - 8) q^{82} + (4 \beta + 8) q^{83} + ( - 2 \beta - 14) q^{85} + ( - 2 \beta - 2) q^{86} + ( - 5 \beta - 7) q^{87} + (\beta - 3) q^{88} + ( - 4 \beta + 8) q^{89} + (4 \beta - 5) q^{90} + (3 \beta - 2) q^{92} + ( - 8 \beta + 1) q^{93} + (2 \beta - 2) q^{94} + (2 \beta + 14) q^{95} + \beta q^{96} + (4 \beta - 6) q^{97} + ( - 4 \beta + 7) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + q^{3} + 2 q^{4} - q^{5} + q^{6} + 2 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + q^{3} + 2 q^{4} - q^{5} + q^{6} + 2 q^{8} - 3 q^{9} - q^{10} - 5 q^{11} + q^{12} - q^{13} - 8 q^{15} + 2 q^{16} - 3 q^{18} - q^{20} - 5 q^{22} - q^{23} + q^{24} + 13 q^{25} - q^{26} - 2 q^{27} - 3 q^{29} - 8 q^{30} - 17 q^{31} + 2 q^{32} - 3 q^{36} - 2 q^{37} + 7 q^{39} - q^{40} - 17 q^{41} - 6 q^{43} - 5 q^{44} - 6 q^{45} - q^{46} - 2 q^{47} + q^{48} + 13 q^{50} + 10 q^{51} - q^{52} - 8 q^{53} - 2 q^{54} - 5 q^{55} - 10 q^{57} - 3 q^{58} + 14 q^{59} - 8 q^{60} - 19 q^{61} - 17 q^{62} + 2 q^{64} - 22 q^{65} - 9 q^{67} + 7 q^{69} - 12 q^{71} - 3 q^{72} - 3 q^{73} - 2 q^{74} + 14 q^{75} + 7 q^{78} + 3 q^{79} - q^{80} - 2 q^{81} - 17 q^{82} + 20 q^{83} - 30 q^{85} - 6 q^{86} - 19 q^{87} - 5 q^{88} + 12 q^{89} - 6 q^{90} - q^{92} - 6 q^{93} - 2 q^{94} + 30 q^{95} + q^{96} - 8 q^{97} + 10 q^{99}+O(q^{100}) \) 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
−0.618034
1.61803
1.00000 −0.618034 1.00000 2.85410 −0.618034 0 1.00000 −2.61803 2.85410
1.2 1.00000 1.61803 1.00000 −3.85410 1.61803 0 1.00000 −0.381966 −3.85410
\(n\): 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.s 2
7.b odd 2 1 74.2.a.b 2
21.c even 2 1 666.2.a.i 2
28.d even 2 1 592.2.a.g 2
35.c odd 2 1 1850.2.a.t 2
35.f even 4 2 1850.2.b.j 4
56.e even 2 1 2368.2.a.u 2
56.h odd 2 1 2368.2.a.y 2
77.b even 2 1 8954.2.a.j 2
84.h odd 2 1 5328.2.a.bc 2
259.b odd 2 1 2738.2.a.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.2.a.b 2 7.b odd 2 1
592.2.a.g 2 28.d even 2 1
666.2.a.i 2 21.c even 2 1
1850.2.a.t 2 35.c odd 2 1
1850.2.b.j 4 35.f even 4 2
2368.2.a.u 2 56.e even 2 1
2368.2.a.y 2 56.h odd 2 1
2738.2.a.g 2 259.b odd 2 1
3626.2.a.s 2 1.a even 1 1 trivial
5328.2.a.bc 2 84.h odd 2 1
8954.2.a.j 2 77.b even 2 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - T - 1 \) Copy content Toggle raw display
$5$ \( T^{2} + T - 11 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 5T + 5 \) Copy content Toggle raw display
$13$ \( T^{2} + T - 11 \) Copy content Toggle raw display
$17$ \( T^{2} - 20 \) Copy content Toggle raw display
$19$ \( T^{2} - 20 \) Copy content Toggle raw display
$23$ \( T^{2} + T - 11 \) Copy content Toggle raw display
$29$ \( T^{2} + 3T - 59 \) Copy content Toggle raw display
$31$ \( T^{2} + 17T + 71 \) Copy content Toggle raw display
$37$ \( (T + 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 17T + 71 \) Copy content Toggle raw display
$43$ \( T^{2} + 6T + 4 \) Copy content Toggle raw display
$47$ \( T^{2} + 2T - 4 \) Copy content Toggle raw display
$53$ \( T^{2} + 8T - 4 \) Copy content Toggle raw display
$59$ \( T^{2} - 14T + 44 \) Copy content Toggle raw display
$61$ \( T^{2} + 19T + 89 \) Copy content Toggle raw display
$67$ \( T^{2} + 9T - 11 \) Copy content Toggle raw display
$71$ \( T^{2} + 12T - 44 \) Copy content Toggle raw display
$73$ \( T^{2} + 3T - 29 \) Copy content Toggle raw display
$79$ \( T^{2} - 3T - 99 \) Copy content Toggle raw display
$83$ \( T^{2} - 20T + 80 \) Copy content Toggle raw display
$89$ \( T^{2} - 12T + 16 \) Copy content Toggle raw display
$97$ \( T^{2} + 8T - 4 \) Copy content Toggle raw display
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