Properties

Label 637.2.a.h
Level $637$
Weight $2$
Character orbit 637.a
Self dual yes
Analytic conductor $5.086$
Analytic rank $1$
Dimension $3$
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: \(3\)
Coefficient field: 3.3.404.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 5x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 5x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 3 \) 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
−1.65544
−0.210756
2.86620
−2.65544 −2.39593 5.05137 −3.65544 6.36226 0 −8.10275 2.74049 9.70682
1.2 −1.21076 1.74483 −0.534070 −2.21076 −2.11256 0 3.06814 0.0444180 2.67669
1.3 1.86620 −3.34889 1.48270 0.866198 −6.24970 0 −0.965392 8.21509 1.61650
\(n\): 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.h 3
3.b odd 2 1 5733.2.a.be 3
7.b odd 2 1 637.2.a.i yes 3
7.c even 3 2 637.2.e.l 6
7.d odd 6 2 637.2.e.k 6
13.b even 2 1 8281.2.a.bh 3
21.c even 2 1 5733.2.a.bd 3
91.b odd 2 1 8281.2.a.bk 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.a.h 3 1.a even 1 1 trivial
637.2.a.i yes 3 7.b odd 2 1
637.2.e.k 6 7.d odd 6 2
637.2.e.l 6 7.c even 3 2
5733.2.a.bd 3 21.c even 2 1
5733.2.a.be 3 3.b odd 2 1
8281.2.a.bh 3 13.b even 2 1
8281.2.a.bk 3 91.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(637))\):

\( T_{2}^{3} + 2T_{2}^{2} - 4T_{2} - 6 \) Copy content Toggle raw display
\( T_{3}^{3} + 4T_{3}^{2} - 2T_{3} - 14 \) Copy content Toggle raw display
\( T_{17}^{3} + 4T_{17}^{2} - 2T_{17} - 14 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} + 2 T^{2} + \cdots - 6 \) Copy content Toggle raw display
$3$ \( T^{3} + 4 T^{2} + \cdots - 14 \) Copy content Toggle raw display
$5$ \( T^{3} + 5 T^{2} + \cdots - 7 \) Copy content Toggle raw display
$7$ \( T^{3} \) Copy content Toggle raw display
$11$ \( T^{3} + 4T^{2} - 2 \) Copy content Toggle raw display
$13$ \( (T - 1)^{3} \) Copy content Toggle raw display
$17$ \( T^{3} + 4 T^{2} + \cdots - 14 \) Copy content Toggle raw display
$19$ \( T^{3} + 7 T^{2} + \cdots - 63 \) Copy content Toggle raw display
$23$ \( T^{3} - T^{2} + \cdots - 43 \) Copy content Toggle raw display
$29$ \( T^{3} + 7 T^{2} + \cdots - 3 \) Copy content Toggle raw display
$31$ \( T^{3} - 3 T^{2} + \cdots + 49 \) Copy content Toggle raw display
$37$ \( T^{3} + 10 T^{2} + \cdots - 82 \) Copy content Toggle raw display
$41$ \( T^{3} + 6 T^{2} + \cdots - 504 \) Copy content Toggle raw display
$43$ \( T^{3} - 9 T^{2} + \cdots + 101 \) Copy content Toggle raw display
$47$ \( T^{3} + 17 T^{2} + \cdots + 147 \) Copy content Toggle raw display
$53$ \( T^{3} - 13 T^{2} + \cdots + 9 \) Copy content Toggle raw display
$59$ \( T^{3} + 22 T^{2} + \cdots + 252 \) Copy content Toggle raw display
$61$ \( T^{3} - 24 T^{2} + \cdots - 224 \) Copy content Toggle raw display
$67$ \( T^{3} + 14 T^{2} + \cdots - 648 \) Copy content Toggle raw display
$71$ \( T^{3} - 4 T^{2} + \cdots + 194 \) Copy content Toggle raw display
$73$ \( T^{3} + 5 T^{2} + \cdots - 1561 \) Copy content Toggle raw display
$79$ \( T^{3} - T^{2} + \cdots - 99 \) Copy content Toggle raw display
$83$ \( T^{3} + 23 T^{2} + \cdots + 203 \) Copy content Toggle raw display
$89$ \( T^{3} - 11 T^{2} + \cdots - 21 \) Copy content Toggle raw display
$97$ \( T^{3} - 3 T^{2} + \cdots + 7 \) Copy content Toggle raw display
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