Properties

Label 637.2.a.m
Level $637$
Weight $2$
Character orbit 637.a
Self dual yes
Analytic conductor $5.086$
Analytic rank $1$
Dimension $6$
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: \(6\)
Coefficient field: 6.6.4507648.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 5x^{4} + 8x^{3} + 7x^{2} - 6x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} - 5x^{4} + 8x^{3} + 7x^{2} - 6x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 3\nu + 1 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - \nu^{3} - 4\nu^{2} + 2\nu + 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} - 2\nu^{4} - 4\nu^{3} + 6\nu^{2} + 4\nu - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 4\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + \beta_{3} + 5\beta_{2} + 6\beta _1 + 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} + 2\beta_{4} + 6\beta_{3} + 8\beta_{2} + 18\beta _1 + 8 \) 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
−1.66745
2.35100
−1.20475
1.90903
−0.146243
0.758419
−2.44785 0.667452 3.99195 −0.910286 −1.63382 0 −4.87599 −2.55451 2.22824
1.2 −1.17619 −3.35100 −0.616586 −3.14862 3.94140 0 3.07759 8.22917 3.70337
1.3 −0.656184 0.204753 −1.56942 1.35996 −0.134356 0 2.34220 −2.95808 −0.892385
1.4 0.264627 −2.90903 −1.92997 1.43515 −0.769807 0 −1.03998 5.46247 0.379780
1.5 1.83237 −0.853757 1.35758 −2.62555 −1.56440 0 −1.17715 −2.27110 −4.81098
1.6 2.18322 −1.75842 2.76645 −2.11065 −3.83901 0 1.67333 0.0920365 −4.60802
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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.m 6
3.b odd 2 1 5733.2.a.bu 6
7.b odd 2 1 637.2.a.n yes 6
7.c even 3 2 637.2.e.o 12
7.d odd 6 2 637.2.e.n 12
13.b even 2 1 8281.2.a.cc 6
21.c even 2 1 5733.2.a.br 6
91.b odd 2 1 8281.2.a.cd 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.a.m 6 1.a even 1 1 trivial
637.2.a.n yes 6 7.b odd 2 1
637.2.e.n 12 7.d odd 6 2
637.2.e.o 12 7.c even 3 2
5733.2.a.br 6 21.c even 2 1
5733.2.a.bu 6 3.b odd 2 1
8281.2.a.cc 6 13.b even 2 1
8281.2.a.cd 6 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(637))\):

\( T_{2}^{6} - 8T_{2}^{4} + 14T_{2}^{2} + 4T_{2} - 2 \) Copy content Toggle raw display
\( T_{3}^{6} + 8T_{3}^{5} + 20T_{3}^{4} + 12T_{3}^{3} - 12T_{3}^{2} - 8T_{3} + 2 \) Copy content Toggle raw display
\( T_{17}^{6} + 16T_{17}^{5} + 56T_{17}^{4} - 324T_{17}^{3} - 2792T_{17}^{2} - 6792T_{17} - 5294 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - 8 T^{4} + \cdots - 2 \) Copy content Toggle raw display
$3$ \( T^{6} + 8 T^{5} + \cdots + 2 \) Copy content Toggle raw display
$5$ \( T^{6} + 6 T^{5} + \cdots + 31 \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( T^{6} - 4 T^{5} + \cdots - 562 \) Copy content Toggle raw display
$13$ \( (T + 1)^{6} \) Copy content Toggle raw display
$17$ \( T^{6} + 16 T^{5} + \cdots - 5294 \) Copy content Toggle raw display
$19$ \( T^{6} + 2 T^{5} + \cdots - 73 \) Copy content Toggle raw display
$23$ \( T^{6} + 6 T^{5} + \cdots + 529 \) Copy content Toggle raw display
$29$ \( T^{6} + 6 T^{5} + \cdots + 529 \) Copy content Toggle raw display
$31$ \( T^{6} + 6 T^{5} + \cdots - 44249 \) Copy content Toggle raw display
$37$ \( T^{6} - 82 T^{4} + \cdots + 254 \) Copy content Toggle raw display
$41$ \( T^{6} - 8 T^{5} + \cdots + 28784 \) Copy content Toggle raw display
$43$ \( T^{6} - 2 T^{5} + \cdots + 35153 \) Copy content Toggle raw display
$47$ \( T^{6} + 30 T^{5} + \cdots - 135617 \) Copy content Toggle raw display
$53$ \( T^{6} + 14 T^{5} + \cdots - 1319 \) Copy content Toggle raw display
$59$ \( T^{6} + 24 T^{5} + \cdots + 1532 \) Copy content Toggle raw display
$61$ \( T^{6} - 246 T^{4} + \cdots - 216584 \) Copy content Toggle raw display
$67$ \( T^{6} - 16 T^{5} + \cdots + 6112 \) Copy content Toggle raw display
$71$ \( T^{6} - 8 T^{5} + \cdots - 1206162 \) Copy content Toggle raw display
$73$ \( T^{6} - 6 T^{5} + \cdots - 142657 \) Copy content Toggle raw display
$79$ \( T^{6} + 22 T^{5} + \cdots + 7913 \) Copy content Toggle raw display
$83$ \( T^{6} + 50 T^{5} + \cdots - 167041 \) Copy content Toggle raw display
$89$ \( T^{6} + 26 T^{5} + \cdots + 9959 \) Copy content Toggle raw display
$97$ \( T^{6} - 14 T^{5} + \cdots + 217287 \) Copy content Toggle raw display
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