Properties

Label 3626.2.a.t
Level $3626$
Weight $2$
Character orbit 3626.a
Self dual yes
Analytic conductor $28.954$
Analytic rank $0$
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: \(0\)
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 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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + (\beta + 1) q^{3} + q^{4} + (2 \beta - 1) q^{5} + (\beta + 1) q^{6} + q^{8} + (3 \beta - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + (\beta + 1) q^{3} + q^{4} + (2 \beta - 1) q^{5} + (\beta + 1) q^{6} + q^{8} + (3 \beta - 1) q^{9} + (2 \beta - 1) q^{10} + (\beta - 2) q^{11} + (\beta + 1) q^{12} - q^{13} + (3 \beta + 1) q^{15} + q^{16} + ( - \beta + 5) q^{17} + (3 \beta - 1) q^{18} + ( - 3 \beta + 4) q^{19} + (2 \beta - 1) q^{20} + (\beta - 2) q^{22} + ( - 2 \beta + 4) q^{23} + (\beta + 1) q^{24} - q^{26} + (2 \beta - 1) q^{27} + (8 \beta - 4) q^{29} + (3 \beta + 1) q^{30} + ( - 6 \beta + 5) q^{31} + q^{32} - q^{33} + ( - \beta + 5) q^{34} + (3 \beta - 1) q^{36} + q^{37} + ( - 3 \beta + 4) q^{38} + ( - \beta - 1) q^{39} + (2 \beta - 1) q^{40} + (5 \beta - 1) q^{41} + ( - 6 \beta - 3) q^{43} + (\beta - 2) q^{44} + (\beta + 7) q^{45} + ( - 2 \beta + 4) q^{46} + ( - 7 \beta + 2) q^{47} + (\beta + 1) q^{48} + (3 \beta + 4) q^{51} - q^{52} + (5 \beta - 1) q^{53} + (2 \beta - 1) q^{54} + ( - 3 \beta + 4) q^{55} + ( - 2 \beta + 1) q^{57} + (8 \beta - 4) q^{58} + (\beta + 1) q^{59} + (3 \beta + 1) q^{60} + 2 q^{61} + ( - 6 \beta + 5) q^{62} + q^{64} + ( - 2 \beta + 1) q^{65} - q^{66} + 3 q^{67} + ( - \beta + 5) q^{68} + 2 q^{69} + ( - 5 \beta + 4) q^{71} + (3 \beta - 1) q^{72} + ( - 6 \beta + 4) q^{73} + q^{74} + ( - 3 \beta + 4) q^{76} + ( - \beta - 1) q^{78} + ( - 12 \beta + 7) q^{79} + (2 \beta - 1) q^{80} + ( - 6 \beta + 4) q^{81} + (5 \beta - 1) q^{82} + ( - 4 \beta + 5) q^{83} + (9 \beta - 7) q^{85} + ( - 6 \beta - 3) q^{86} + (12 \beta + 4) q^{87} + (\beta - 2) q^{88} + 12 q^{89} + (\beta + 7) q^{90} + ( - 2 \beta + 4) q^{92} + ( - 7 \beta - 1) q^{93} + ( - 7 \beta + 2) q^{94} + (5 \beta - 10) q^{95} + (\beta + 1) q^{96} + 3 q^{97} + ( - 4 \beta + 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 3 q^{3} + 2 q^{4} + 3 q^{6} + 2 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 3 q^{3} + 2 q^{4} + 3 q^{6} + 2 q^{8} + q^{9} - 3 q^{11} + 3 q^{12} - 2 q^{13} + 5 q^{15} + 2 q^{16} + 9 q^{17} + q^{18} + 5 q^{19} - 3 q^{22} + 6 q^{23} + 3 q^{24} - 2 q^{26} + 5 q^{30} + 4 q^{31} + 2 q^{32} - 2 q^{33} + 9 q^{34} + q^{36} + 2 q^{37} + 5 q^{38} - 3 q^{39} + 3 q^{41} - 12 q^{43} - 3 q^{44} + 15 q^{45} + 6 q^{46} - 3 q^{47} + 3 q^{48} + 11 q^{51} - 2 q^{52} + 3 q^{53} + 5 q^{55} + 3 q^{59} + 5 q^{60} + 4 q^{61} + 4 q^{62} + 2 q^{64} - 2 q^{66} + 6 q^{67} + 9 q^{68} + 4 q^{69} + 3 q^{71} + q^{72} + 2 q^{73} + 2 q^{74} + 5 q^{76} - 3 q^{78} + 2 q^{79} + 2 q^{81} + 3 q^{82} + 6 q^{83} - 5 q^{85} - 12 q^{86} + 20 q^{87} - 3 q^{88} + 24 q^{89} + 15 q^{90} + 6 q^{92} - 9 q^{93} - 3 q^{94} - 15 q^{95} + 3 q^{96} + 6 q^{97} + 6 q^{99}+O(q^{100}) \) 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
−0.618034
1.61803
1.00000 0.381966 1.00000 −2.23607 0.381966 0 1.00000 −2.85410 −2.23607
1.2 1.00000 2.61803 1.00000 2.23607 2.61803 0 1.00000 3.85410 2.23607
\(n\): 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.t 2
7.b odd 2 1 3626.2.a.k 2
7.d odd 6 2 518.2.e.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
518.2.e.b 4 7.d odd 6 2
3626.2.a.k 2 7.b odd 2 1
3626.2.a.t 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3626))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 3T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} - 5 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 3T + 1 \) Copy content Toggle raw display
$13$ \( (T + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 9T + 19 \) Copy content Toggle raw display
$19$ \( T^{2} - 5T - 5 \) Copy content Toggle raw display
$23$ \( T^{2} - 6T + 4 \) Copy content Toggle raw display
$29$ \( T^{2} - 80 \) Copy content Toggle raw display
$31$ \( T^{2} - 4T - 41 \) Copy content Toggle raw display
$37$ \( (T - 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 3T - 29 \) Copy content Toggle raw display
$43$ \( T^{2} + 12T - 9 \) Copy content Toggle raw display
$47$ \( T^{2} + 3T - 59 \) Copy content Toggle raw display
$53$ \( T^{2} - 3T - 29 \) Copy content Toggle raw display
$59$ \( T^{2} - 3T + 1 \) Copy content Toggle raw display
$61$ \( (T - 2)^{2} \) Copy content Toggle raw display
$67$ \( (T - 3)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 3T - 29 \) Copy content Toggle raw display
$73$ \( T^{2} - 2T - 44 \) Copy content Toggle raw display
$79$ \( T^{2} - 2T - 179 \) Copy content Toggle raw display
$83$ \( T^{2} - 6T - 11 \) Copy content Toggle raw display
$89$ \( (T - 12)^{2} \) Copy content Toggle raw display
$97$ \( (T - 3)^{2} \) Copy content Toggle raw display
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