Properties

Label 637.2.a.e
Level $637$
Weight $2$
Character orbit 637.a
Self dual yes
Analytic conductor $5.086$
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] = [637,2,Mod(1,637)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(637, 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("637.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.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: \(5.08647060876\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
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.

\(f(q)\) \(=\) \( q + ( - \beta - 1) q^{2} + (2 \beta - 1) q^{3} + 3 \beta q^{4} + ( - 2 \beta + 1) q^{5} + ( - 3 \beta - 1) q^{6} + ( - 4 \beta - 1) q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 1) q^{2} + (2 \beta - 1) q^{3} + 3 \beta q^{4} + ( - 2 \beta + 1) q^{5} + ( - 3 \beta - 1) q^{6} + ( - 4 \beta - 1) q^{8} + 2 q^{9} + (3 \beta + 1) q^{10} - 3 q^{11} + (3 \beta + 6) q^{12} + q^{13} - 5 q^{15} + (3 \beta + 5) q^{16} + ( - 4 \beta + 5) q^{17} + ( - 2 \beta - 2) q^{18} - 3 q^{19} + ( - 3 \beta - 6) q^{20} + (3 \beta + 3) q^{22} + ( - 2 \beta - 5) q^{23} + ( - 6 \beta - 7) q^{24} + ( - \beta - 1) q^{26} + ( - 2 \beta + 1) q^{27} + (4 \beta - 2) q^{29} + (5 \beta + 5) q^{30} - 5 q^{31} + ( - 3 \beta - 6) q^{32} + ( - 6 \beta + 3) q^{33} + (3 \beta - 1) q^{34} + 6 \beta q^{36} + (6 \beta - 5) q^{37} + (3 \beta + 3) q^{38} + (2 \beta - 1) q^{39} + (6 \beta + 7) q^{40} + (4 \beta - 2) q^{41} - 8 q^{43} - 9 \beta q^{44} + ( - 4 \beta + 2) q^{45} + (9 \beta + 7) q^{46} + (4 \beta + 1) q^{47} + (13 \beta + 1) q^{48} + (6 \beta - 13) q^{51} + 3 \beta q^{52} + ( - 4 \beta - 1) q^{53} + (3 \beta + 1) q^{54} + (6 \beta - 3) q^{55} + ( - 6 \beta + 3) q^{57} + ( - 6 \beta - 2) q^{58} + (4 \beta - 5) q^{59} - 15 \beta q^{60} - 3 q^{61} + (5 \beta + 5) q^{62} + (6 \beta - 1) q^{64} + ( - 2 \beta + 1) q^{65} + (9 \beta + 3) q^{66} - 3 q^{67} + (3 \beta - 12) q^{68} + ( - 12 \beta + 1) q^{69} + ( - 8 \beta + 4) q^{71} + ( - 8 \beta - 2) q^{72} + (6 \beta - 7) q^{73} + ( - 7 \beta - 1) q^{74} - 9 \beta q^{76} + ( - 3 \beta - 1) q^{78} + ( - 6 \beta + 7) q^{79} + ( - 13 \beta - 1) q^{80} - 11 q^{81} + ( - 6 \beta - 2) q^{82} + ( - 6 \beta + 13) q^{85} + (8 \beta + 8) q^{86} + 10 q^{87} + (12 \beta + 3) q^{88} + ( - 2 \beta + 1) q^{89} + (6 \beta + 2) q^{90} + ( - 21 \beta - 6) q^{92} + ( - 10 \beta + 5) q^{93} + ( - 9 \beta - 5) q^{94} + (6 \beta - 3) q^{95} - 15 \beta q^{96} + ( - 12 \beta + 10) q^{97} - 6 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{2} + 3 q^{4} - 5 q^{6} - 6 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{2} + 3 q^{4} - 5 q^{6} - 6 q^{8} + 4 q^{9} + 5 q^{10} - 6 q^{11} + 15 q^{12} + 2 q^{13} - 10 q^{15} + 13 q^{16} + 6 q^{17} - 6 q^{18} - 6 q^{19} - 15 q^{20} + 9 q^{22} - 12 q^{23} - 20 q^{24} - 3 q^{26} + 15 q^{30} - 10 q^{31} - 15 q^{32} + q^{34} + 6 q^{36} - 4 q^{37} + 9 q^{38} + 20 q^{40} - 16 q^{43} - 9 q^{44} + 23 q^{46} + 6 q^{47} + 15 q^{48} - 20 q^{51} + 3 q^{52} - 6 q^{53} + 5 q^{54} - 10 q^{58} - 6 q^{59} - 15 q^{60} - 6 q^{61} + 15 q^{62} + 4 q^{64} + 15 q^{66} - 6 q^{67} - 21 q^{68} - 10 q^{69} - 12 q^{72} - 8 q^{73} - 9 q^{74} - 9 q^{76} - 5 q^{78} + 8 q^{79} - 15 q^{80} - 22 q^{81} - 10 q^{82} + 20 q^{85} + 24 q^{86} + 20 q^{87} + 18 q^{88} + 10 q^{90} - 33 q^{92} - 19 q^{94} - 15 q^{96} + 8 q^{97} - 12 q^{99}+O(q^{100}) \) 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
1.61803
−0.618034
−2.61803 2.23607 4.85410 −2.23607 −5.85410 0 −7.47214 2.00000 5.85410
1.2 −0.381966 −2.23607 −1.85410 2.23607 0.854102 0 1.47214 2.00000 −0.854102
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(-1\)
\(13\) \(-1\)

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 637.2.a.e 2
3.b odd 2 1 5733.2.a.w 2
7.b odd 2 1 637.2.a.f 2
7.c even 3 2 637.2.e.h 4
7.d odd 6 2 91.2.e.b 4
13.b even 2 1 8281.2.a.ba 2
21.c even 2 1 5733.2.a.v 2
21.g even 6 2 819.2.j.c 4
28.f even 6 2 1456.2.r.j 4
91.b odd 2 1 8281.2.a.z 2
91.s odd 6 2 1183.2.e.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.e.b 4 7.d odd 6 2
637.2.a.e 2 1.a even 1 1 trivial
637.2.a.f 2 7.b odd 2 1
637.2.e.h 4 7.c even 3 2
819.2.j.c 4 21.g even 6 2
1183.2.e.d 4 91.s odd 6 2
1456.2.r.j 4 28.f even 6 2
5733.2.a.v 2 21.c even 2 1
5733.2.a.w 2 3.b odd 2 1
8281.2.a.z 2 91.b odd 2 1
8281.2.a.ba 2 13.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(637))\):

\( T_{2}^{2} + 3T_{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{2} - 5 \) Copy content Toggle raw display
\( T_{17}^{2} - 6T_{17} - 11 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 3T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} - 5 \) Copy content Toggle raw display
$5$ \( T^{2} - 5 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( (T + 3)^{2} \) Copy content Toggle raw display
$13$ \( (T - 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 6T - 11 \) Copy content Toggle raw display
$19$ \( (T + 3)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 12T + 31 \) Copy content Toggle raw display
$29$ \( T^{2} - 20 \) Copy content Toggle raw display
$31$ \( (T + 5)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 4T - 41 \) Copy content Toggle raw display
$41$ \( T^{2} - 20 \) Copy content Toggle raw display
$43$ \( (T + 8)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 6T - 11 \) Copy content Toggle raw display
$53$ \( T^{2} + 6T - 11 \) Copy content Toggle raw display
$59$ \( T^{2} + 6T - 11 \) Copy content Toggle raw display
$61$ \( (T + 3)^{2} \) Copy content Toggle raw display
$67$ \( (T + 3)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 80 \) Copy content Toggle raw display
$73$ \( T^{2} + 8T - 29 \) Copy content Toggle raw display
$79$ \( T^{2} - 8T - 29 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 5 \) Copy content Toggle raw display
$97$ \( T^{2} - 8T - 164 \) Copy content Toggle raw display
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