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Properties

Label 637.2.c.d
Level $637$
Weight $2$
Character orbit 637.c
Analytic conductor $5.086$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [637,2,Mod(246,637)]
 
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, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("637.246");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.c (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(5.08647060876\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.350464.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 91)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{4} - \beta_{3}) q^{2} - \beta_1 q^{3} + ( - \beta_{2} - \beta_1 - 1) q^{4} + (\beta_{5} + \beta_{3}) q^{5} + ( - \beta_{5} - \beta_{4} + \beta_{3}) q^{6} + ( - 2 \beta_{5} + 2 \beta_{3}) q^{8} + (\beta_{2} - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{4} - \beta_{3}) q^{2} - \beta_1 q^{3} + ( - \beta_{2} - \beta_1 - 1) q^{4} + (\beta_{5} + \beta_{3}) q^{5} + ( - \beta_{5} - \beta_{4} + \beta_{3}) q^{6} + ( - 2 \beta_{5} + 2 \beta_{3}) q^{8} + (\beta_{2} - \beta_1) q^{9} - \beta_1 q^{10} + ( - \beta_{4} + 3 \beta_{3}) q^{11} + (2 \beta_{2} + 4) q^{12} + (\beta_{4} - 2 \beta_{2} + \beta_1 - 2) q^{13} + (2 \beta_{5} - \beta_{4} + 3 \beta_{3}) q^{15} + (2 \beta_{2} + 2) q^{16} + (2 \beta_{2} - \beta_1 + 2) q^{17} + ( - \beta_{4} - \beta_{3}) q^{18} + ( - 4 \beta_{5} + \beta_{4}) q^{19} + (\beta_{5} - \beta_{4} + 3 \beta_{3}) q^{20} + (3 \beta_{2} + \beta_1 + 5) q^{22} + (3 \beta_{2} - \beta_1 + 2) q^{23} + (2 \beta_{4} - 6 \beta_{3}) q^{24} + ( - \beta_{2} + 3 \beta_1 + 1) q^{25} + ( - \beta_{5} - \beta_{4} + 5 \beta_{3} + \cdots - 2) q^{26}+ \cdots + (2 \beta_{5} - \beta_{4} + \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 4 q^{4} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 4 q^{4} - 2 q^{9} + 20 q^{12} - 8 q^{13} + 8 q^{16} + 8 q^{17} + 24 q^{22} + 6 q^{23} + 8 q^{25} - 12 q^{26} + 12 q^{27} - 14 q^{29} + 16 q^{30} + 4 q^{36} + 4 q^{38} - 8 q^{39} + 20 q^{40} - 26 q^{43} - 8 q^{48} + 8 q^{51} + 20 q^{52} + 2 q^{53} - 12 q^{55} - 28 q^{61} - 16 q^{62} - 8 q^{64} - 6 q^{65} - 28 q^{66} - 20 q^{68} + 4 q^{69} - 24 q^{74} - 44 q^{75} + 16 q^{78} + 26 q^{79} - 26 q^{81} + 56 q^{82} + 40 q^{87} - 40 q^{88} + 12 q^{90} - 36 q^{92} - 20 q^{94} + 58 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

\(\beta_{1}\)\(=\) \( ( -\nu^{5} + 8\nu^{4} - 4\nu^{3} - \nu^{2} + 2\nu + 38 ) / 23 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -5\nu^{5} + 17\nu^{4} - 20\nu^{3} - 5\nu^{2} + 10\nu + 29 ) / 23 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 7\nu^{5} - 10\nu^{4} + 5\nu^{3} + 30\nu^{2} + 32\nu - 13 ) / 23 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -11\nu^{5} + 19\nu^{4} - 21\nu^{3} - 11\nu^{2} - 70\nu + 27 ) / 23 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -14\nu^{5} + 20\nu^{4} - 10\nu^{3} - 37\nu^{2} - 64\nu + 26 ) / 23 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + 2\beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{5} - \beta_{4} + 2\beta_{3} - \beta_{2} + 2\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{2} + 5\beta _1 - 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -8\beta_{5} + 3\beta_{4} - 9\beta_{3} - 3\beta_{2} + 8\beta _1 - 9 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/637\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(248\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
246.1
0.403032 + 0.403032i
−0.854638 + 0.854638i
1.45161 + 1.45161i
1.45161 − 1.45161i
−0.854638 − 0.854638i
0.403032 − 0.403032i
− 2.48119i −1.67513 −4.15633 − 0.675131i 4.15633i 0 5.35026i −0.193937 −1.67513
246.2 − 1.17009i −0.539189 0.630898 − 0.460811i 0.630898i 0 − 3.07838i −2.70928 −0.539189
246.3 − 0.688892i 2.21432 1.52543 3.21432i − 1.52543i 0 − 2.42864i 1.90321 2.21432
246.4 0.688892i 2.21432 1.52543 − 3.21432i 1.52543i 0 2.42864i 1.90321 2.21432
246.5 1.17009i −0.539189 0.630898 0.460811i − 0.630898i 0 3.07838i −2.70928 −0.539189
246.6 2.48119i −1.67513 −4.15633 0.675131i − 4.15633i 0 − 5.35026i −0.193937 −1.67513
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 246.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 637.2.c.d 6
7.b odd 2 1 91.2.c.a ✓ 6
7.c even 3 2 637.2.r.d 12
7.d odd 6 2 637.2.r.e 12
13.b even 2 1 inner 637.2.c.d 6
13.d odd 4 1 8281.2.a.be 3
13.d odd 4 1 8281.2.a.bi 3
21.c even 2 1 819.2.c.b 6
28.d even 2 1 1456.2.k.c 6
91.b odd 2 1 91.2.c.a ✓ 6
91.i even 4 1 1183.2.a.h 3
91.i even 4 1 1183.2.a.j 3
91.r even 6 2 637.2.r.d 12
91.s odd 6 2 637.2.r.e 12
273.g even 2 1 819.2.c.b 6
364.h even 2 1 1456.2.k.c 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.c.a ✓ 6 7.b odd 2 1
91.2.c.a ✓ 6 91.b odd 2 1
637.2.c.d 6 1.a even 1 1 trivial
637.2.c.d 6 13.b even 2 1 inner
637.2.r.d 12 7.c even 3 2
637.2.r.d 12 91.r even 6 2
637.2.r.e 12 7.d odd 6 2
637.2.r.e 12 91.s odd 6 2
819.2.c.b 6 21.c even 2 1
819.2.c.b 6 273.g even 2 1
1183.2.a.h 3 91.i even 4 1
1183.2.a.j 3 91.i even 4 1
1456.2.k.c 6 28.d even 2 1
1456.2.k.c 6 364.h even 2 1
8281.2.a.be 3 13.d odd 4 1
8281.2.a.bi 3 13.d odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(637, [\chi])\):

\( T_{2}^{6} + 8T_{2}^{4} + 12T_{2}^{2} + 4 \) Copy content Toggle raw display
\( T_{3}^{3} - 4T_{3} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 8 T^{4} + \cdots + 4 \) Copy content Toggle raw display
$3$ \( (T^{3} - 4 T - 2)^{2} \) Copy content Toggle raw display
$5$ \( T^{6} + 11 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( T^{6} + 28 T^{4} + \cdots + 100 \) Copy content Toggle raw display
$13$ \( T^{6} + 8 T^{5} + \cdots + 2197 \) Copy content Toggle raw display
$17$ \( (T^{3} - 4 T^{2} - 8 T + 34)^{2} \) Copy content Toggle raw display
$19$ \( T^{6} + 119 T^{4} + \cdots + 37249 \) Copy content Toggle raw display
$23$ \( (T^{3} - 3 T^{2} - 25 T + 79)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} + 7 T^{2} - 21 T + 5)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + 83 T^{4} + \cdots + 4225 \) Copy content Toggle raw display
$37$ \( T^{6} + 108 T^{4} + \cdots + 2916 \) Copy content Toggle raw display
$41$ \( T^{6} + 108 T^{4} + \cdots + 1600 \) Copy content Toggle raw display
$43$ \( (T^{3} + 13 T^{2} + \cdots - 17)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + 151 T^{4} + \cdots + 18769 \) Copy content Toggle raw display
$53$ \( (T^{3} - T^{2} - 9 T + 13)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + 68 T^{4} + \cdots + 2704 \) Copy content Toggle raw display
$61$ \( (T^{3} + 14 T^{2} + \cdots - 152)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 80 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$71$ \( T^{6} + 304 T^{4} + \cdots + 792100 \) Copy content Toggle raw display
$73$ \( T^{6} + 263 T^{4} + \cdots + 961 \) Copy content Toggle raw display
$79$ \( (T^{3} - 13 T^{2} + \cdots + 185)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 227 T^{4} + \cdots + 26569 \) Copy content Toggle raw display
$89$ \( T^{6} + 119 T^{4} + \cdots + 51529 \) Copy content Toggle raw display
$97$ \( T^{6} + 575 T^{4} + \cdots + 4765489 \) Copy content Toggle raw display
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