Properties

Label 49.6.a.g
Level $49$
Weight $6$
Character orbit 49.a
Self dual yes
Analytic conductor $7.859$
Analytic rank $1$
Dimension $4$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [49,6,Mod(1,49)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(49, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("49.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 49.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: \(7.85880717084\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{113})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 59x^{2} + 60x + 674 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 7 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 2) q^{2} + ( - \beta_{3} + 2 \beta_{2}) q^{3} - 5 \beta_1 q^{4} + (2 \beta_{3} - 8 \beta_{2}) q^{5} + (11 \beta_{3} - 11 \beta_{2}) q^{6} + ( - 17 \beta_1 - 76) q^{8} + ( - 48 \beta_1 + 31) q^{9}+ \cdots + (22764 \beta_1 + 814) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 10 q^{2} + 10 q^{4} - 270 q^{8} + 220 q^{9} - 1952 q^{11} - 4096 q^{15} - 1566 q^{16} - 5974 q^{18} + 3524 q^{22} - 7136 q^{23} + 2764 q^{25} - 3352 q^{29} + 25608 q^{30} + 27810 q^{32} + 27670 q^{36}+ \cdots - 42272 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 59x^{2} + 60x + 674 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -2\nu^{3} + 3\nu^{2} + 172\nu - 139 ) / 105 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 51\nu^{2} - 86\nu - 1558 ) / 105 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{3} - 3\nu^{2} - 67\nu + 34 ) / 15 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + 7\beta _1 + 7 ) / 7 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} + \beta _1 + 31 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 86\beta_{3} + 21\beta_{2} + 245\beta _1 + 441 ) / 7 \) 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
−3.40086
−6.22929
4.40086
7.22929
−7.81507 −23.5186 29.0754 74.2753 183.799 0 22.8562 310.123 −580.467
1.2 −7.81507 23.5186 29.0754 −74.2753 −183.799 0 22.8562 310.123 580.467
1.3 2.81507 −6.54802 −24.0754 45.9910 −18.4331 0 −157.856 −200.123 129.468
1.4 2.81507 6.54802 −24.0754 −45.9910 18.4331 0 −157.856 −200.123 −129.468
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \( +1 \)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 49.6.a.g 4
3.b odd 2 1 441.6.a.z 4
4.b odd 2 1 784.6.a.bf 4
7.b odd 2 1 inner 49.6.a.g 4
7.c even 3 2 49.6.c.h 8
7.d odd 6 2 49.6.c.h 8
21.c even 2 1 441.6.a.z 4
28.d even 2 1 784.6.a.bf 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.6.a.g 4 1.a even 1 1 trivial
49.6.a.g 4 7.b odd 2 1 inner
49.6.c.h 8 7.c even 3 2
49.6.c.h 8 7.d odd 6 2
441.6.a.z 4 3.b odd 2 1
441.6.a.z 4 21.c even 2 1
784.6.a.bf 4 4.b odd 2 1
784.6.a.bf 4 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(49))\):

\( T_{2}^{2} + 5T_{2} - 22 \) Copy content Toggle raw display
\( T_{3}^{4} - 596T_{3}^{2} + 23716 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 5 T - 22)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} - 596 T^{2} + 23716 \) Copy content Toggle raw display
$5$ \( T^{4} - 7632 T^{2} + 11669056 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + 976 T + 234076)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + \cdots + 76158337024 \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 1700202150724 \) Copy content Toggle raw display
$19$ \( T^{4} - 1359892 T^{2} + 56942116 \) Copy content Toggle raw display
$23$ \( (T^{2} + 3568 T - 160336)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 1676 T - 16661788)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 614334295349824 \) Copy content Toggle raw display
$37$ \( (T^{2} + 4604 T + 5234116)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 14\!\cdots\!56 \) Copy content Toggle raw display
$43$ \( (T^{2} - 10224 T - 998756)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 17\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( (T^{2} + 51460 T + 472236292)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 11\!\cdots\!76 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 11\!\cdots\!24 \) Copy content Toggle raw display
$67$ \( (T^{2} + 11448 T - 3789557552)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 76912 T + 953601968)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 20\!\cdots\!44 \) Copy content Toggle raw display
$79$ \( (T^{2} + 45344 T + 386672)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 13\!\cdots\!96 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 11\!\cdots\!44 \) Copy content Toggle raw display
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