Properties

Label 725.6.a.a
Level $725$
Weight $6$
Character orbit 725.a
Self dual yes
Analytic conductor $116.278$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [725,6,Mod(1,725)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(725, 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("725.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 725 = 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 725.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: \(116.278269364\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.3257317.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 34x^{2} - 27x + 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 29)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + ( - \beta_{2} - \beta_1 + 7) q^{3} + ( - \beta_{3} + 5 \beta_{2} - 3 \beta_1 + 2) q^{4} + (\beta_{3} + 5 \beta_{2} + 3 \beta_1 - 48) q^{6} + (12 \beta_{3} + 4 \beta_{2} + 58) q^{7}+ \cdots + ( - 1972 \beta_{3} + 2714 \beta_{2} + \cdots + 40694) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 28 q^{3} + 10 q^{4} - 194 q^{6} + 208 q^{7} + 504 q^{8} - 280 q^{9} - 124 q^{11} - 20 q^{12} + 460 q^{13} + 768 q^{14} - 414 q^{16} - 184 q^{17} - 3208 q^{18} - 2392 q^{19} + 992 q^{21} - 5538 q^{22}+ \cdots + 166720 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 34x^{2} - 27x + 10 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{2} + 3\nu - 17 ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} - 3\nu - 17 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{3} - \nu^{2} - 67\nu - 25 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{2} + 3\beta _1 + 34 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{3} - 32\beta_{2} + 35\beta _1 + 42 ) / 2 \) 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
0.275208
−1.10057
6.17343
−5.34807
−5.91663 18.2828 3.00648 0 −108.173 −139.558 171.544 91.2623 0
1.2 −4.16235 17.5258 −14.6748 0 −72.9487 220.793 194.277 64.1549 0
1.3 0.863638 −7.07413 −31.2541 0 −6.10949 36.7447 −54.6287 −192.957 0
1.4 9.21534 −0.734546 52.9225 0 −6.76909 90.0205 192.808 −242.460 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \( +1 \)
\(29\) \( +1 \)

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 725.6.a.a 4
5.b even 2 1 29.6.a.a 4
15.d odd 2 1 261.6.a.a 4
20.d odd 2 1 464.6.a.i 4
145.d even 2 1 841.6.a.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.6.a.a 4 5.b even 2 1
261.6.a.a 4 15.d odd 2 1
464.6.a.i 4 20.d odd 2 1
725.6.a.a 4 1.a even 1 1 trivial
841.6.a.a 4 145.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 69T_{2}^{2} - 168T_{2} + 196 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(725))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 69 T^{2} + \cdots + 196 \) Copy content Toggle raw display
$3$ \( T^{4} - 28 T^{3} + \cdots + 1665 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 208 T^{3} + \cdots - 101924272 \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots - 3717303119 \) Copy content Toggle raw display
$13$ \( T^{4} + \cdots - 116053863479 \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 11464717824 \) Copy content Toggle raw display
$19$ \( T^{4} + \cdots + 2620094791680 \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 2033080361984 \) Copy content Toggle raw display
$29$ \( (T + 841)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots - 421127233952247 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 16079593861120 \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 10\!\cdots\!56 \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 47\!\cdots\!37 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots - 35\!\cdots\!87 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 16\!\cdots\!53 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots - 43\!\cdots\!80 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 90\!\cdots\!24 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 20\!\cdots\!92 \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots - 18\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots - 40\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots - 53\!\cdots\!75 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots - 18\!\cdots\!88 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots - 18\!\cdots\!60 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots - 12\!\cdots\!28 \) Copy content Toggle raw display
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