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Properties

Label 676.6.a.f
Level $676$
Weight $6$
Character orbit 676.a
Self dual yes
Analytic conductor $108.419$
Analytic rank $0$
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] = [676,6,Mod(1,676)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(676, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("676.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 676 = 2^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 676.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: \(108.419462194\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} - 1057x^{4} + 1387x^{3} + 247152x^{2} - 816400x - 5374200 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 52)
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.

\(f(q)\) \(=\) \( q + (\beta_1 + 1) q^{3} + ( - \beta_{3} - 2) q^{5} + (\beta_{2} - 2 \beta_1) q^{7} + (\beta_{4} + \beta_{3} - \beta_{2} + \cdots + 111) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 1) q^{3} + ( - \beta_{3} - 2) q^{5} + (\beta_{2} - 2 \beta_1) q^{7} + (\beta_{4} + \beta_{3} - \beta_{2} + \cdots + 111) q^{9}+ \cdots + ( - 41 \beta_{5} + 35 \beta_{4} + \cdots + 40519) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 9 q^{3} - 11 q^{5} - 7 q^{7} + 677 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 9 q^{3} - 11 q^{5} - 7 q^{7} + 677 q^{9} + 459 q^{11} - 260 q^{15} - 358 q^{17} + 4295 q^{19} - 3995 q^{21} + 725 q^{23} + 2977 q^{25} + 7389 q^{27} + 5996 q^{29} + 954 q^{31} - 6893 q^{33} + 4102 q^{35} - 1384 q^{37} + 7802 q^{41} + 2813 q^{43} - 30373 q^{45} + 28158 q^{47} - 1959 q^{49} - 26381 q^{51} - 3751 q^{53} - 8042 q^{55} - 2495 q^{57} + 4965 q^{59} + 55892 q^{61} - 98766 q^{63} + 102273 q^{67} + 90543 q^{69} + 32579 q^{71} + 16671 q^{73} - 59563 q^{75} + 76893 q^{77} + 130296 q^{79} + 99530 q^{81} - 71664 q^{83} - 172409 q^{85} + 304959 q^{87} + 21239 q^{89} + 247188 q^{93} - 78940 q^{95} + 62543 q^{97} + 235226 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3x^{5} - 1057x^{4} + 1387x^{3} + 247152x^{2} - 816400x - 5374200 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} + 20\nu^{4} - 1641\nu^{3} - 13388\nu^{2} + 511340\nu + 78468 ) / 13572 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -7\nu^{5} + 556\nu^{4} + 699\nu^{3} - 346504\nu^{2} + 755656\nu + 15642120 ) / 271440 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 9\nu^{5} - 52\nu^{4} - 11173\nu^{3} + 116728\nu^{2} + 2976088\nu - 36630360 ) / 90480 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 109\nu^{5} - 372\nu^{4} - 94073\nu^{3} - 71352\nu^{2} + 13285048\nu + 1979640 ) / 90480 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{3} - \beta_{2} + 2\beta _1 + 353 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{5} + 5\beta_{4} + 27\beta_{3} - 30\beta_{2} + 597\beta _1 + 598 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 31\beta_{5} + 710\beta_{4} + 1441\beta_{3} - 961\beta_{2} + 4290\beta _1 + 209278 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2662\beta_{5} + 7393\beta_{4} + 28875\beta_{3} - 29826\beta_{2} + 409313\beta _1 + 1443254 \) Copy content Toggle raw display

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

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
−23.5470
−21.1630
−3.37118
7.67325
14.3451
29.0628
0 −22.5470 0 −69.0549 0 117.022 0 265.366 0
1.2 0 −20.1630 0 75.0770 0 −62.1926 0 163.545 0
1.3 0 −2.37118 0 −35.9109 0 −120.909 0 −237.377 0
1.4 0 8.67325 0 −13.4052 0 173.895 0 −167.775 0
1.5 0 15.3451 0 84.4519 0 64.8034 0 −7.52770 0
1.6 0 30.0628 0 −52.1580 0 −179.619 0 660.769 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(13\) \(1\)

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 676.6.a.f 6
13.b even 2 1 676.6.a.g 6
13.c even 3 2 52.6.e.a ✓ 12
13.d odd 4 2 676.6.d.f 12
39.i odd 6 2 468.6.l.f 12
52.j odd 6 2 208.6.i.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
52.6.e.a ✓ 12 13.c even 3 2
208.6.i.d 12 52.j odd 6 2
468.6.l.f 12 39.i odd 6 2
676.6.a.f 6 1.a even 1 1 trivial
676.6.a.g 6 13.b even 2 1
676.6.d.f 12 13.d odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(676))\):

\( T_{3}^{6} - 9T_{3}^{5} - 1027T_{3}^{4} + 5565T_{3}^{3} + 236694T_{3}^{2} - 1302336T_{3} - 4313088 \) Copy content Toggle raw display
\( T_{5}^{6} + 11T_{5}^{5} - 10803T_{5}^{4} - 287807T_{5}^{3} + 27878978T_{5}^{2} + 1219577196T_{5} + 10993328424 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - 9 T^{5} + \cdots - 4313088 \) Copy content Toggle raw display
$5$ \( T^{6} + \cdots + 10993328424 \) Copy content Toggle raw display
$7$ \( T^{6} + \cdots - 1781161641920 \) Copy content Toggle raw display
$11$ \( T^{6} + \cdots + 324135720230400 \) Copy content Toggle raw display
$13$ \( T^{6} \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots + 13\!\cdots\!17 \) Copy content Toggle raw display
$19$ \( T^{6} + \cdots + 13\!\cdots\!20 \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots - 20\!\cdots\!48 \) Copy content Toggle raw display
$29$ \( T^{6} + \cdots + 10\!\cdots\!75 \) Copy content Toggle raw display
$31$ \( T^{6} + \cdots - 20\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots + 48\!\cdots\!19 \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots - 10\!\cdots\!95 \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 25\!\cdots\!40 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots - 93\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 37\!\cdots\!12 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots - 14\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots - 16\!\cdots\!25 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 14\!\cdots\!60 \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots + 49\!\cdots\!28 \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 83\!\cdots\!32 \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots + 48\!\cdots\!48 \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots - 44\!\cdots\!60 \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots - 24\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots - 18\!\cdots\!56 \) Copy content Toggle raw display
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