Properties

Label 245.6.a.d
Level $245$
Weight $6$
Character orbit 245.a
Self dual yes
Analytic conductor $39.294$
Analytic rank $1$
Dimension $3$
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] = [245,6,Mod(1,245)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(245, 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("245.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 245.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: \(39.2940358542\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.577880.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 98x - 232 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 2) q^{2} + ( - \beta_{2} - 9) q^{3} + (2 \beta_{2} + 38) q^{4} + 25 q^{5} + (6 \beta_{2} - 17 \beta_1 + 34) q^{6} + ( - 12 \beta_{2} + 22 \beta_1 - 44) q^{8} + (9 \beta_{2} - 24 \beta_1 + 166) q^{9}+ \cdots + ( - 1198 \beta_{2} + 3328 \beta_1 - 20650) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 6 q^{2} - 26 q^{3} + 112 q^{4} + 75 q^{5} + 96 q^{6} - 120 q^{8} + 489 q^{9} - 150 q^{10} - 194 q^{11} - 2956 q^{12} - 1892 q^{13} - 650 q^{15} + 1496 q^{16} + 184 q^{17} - 6078 q^{18} - 1212 q^{19}+ \cdots - 60752 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - 98x - 232 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} - 4\nu - 66 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} + 4\beta _1 + 66 \) 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
−8.38673
−2.53323
10.9200
−10.3867 −27.9421 75.8842 25.0000 290.227 0 −455.813 537.760 −259.668
1.2 −4.53323 15.7249 −11.4498 25.0000 −71.2847 0 196.968 4.27317 −113.331
1.3 8.91996 −13.7828 47.5657 25.0000 −122.942 0 138.845 −53.0335 222.999
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \( -1 \)
\(7\) \( -1 \)

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 245.6.a.d 3
7.b odd 2 1 35.6.a.c 3
21.c even 2 1 315.6.a.i 3
28.d even 2 1 560.6.a.q 3
35.c odd 2 1 175.6.a.e 3
35.f even 4 2 175.6.b.e 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.6.a.c 3 7.b odd 2 1
175.6.a.e 3 35.c odd 2 1
175.6.b.e 6 35.f even 4 2
245.6.a.d 3 1.a even 1 1 trivial
315.6.a.i 3 21.c even 2 1
560.6.a.q 3 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(245))\):

\( T_{2}^{3} + 6T_{2}^{2} - 86T_{2} - 420 \) Copy content Toggle raw display
\( T_{3}^{3} + 26T_{3}^{2} - 271T_{3} - 6056 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} + 6 T^{2} + \cdots - 420 \) Copy content Toggle raw display
$3$ \( T^{3} + 26 T^{2} + \cdots - 6056 \) Copy content Toggle raw display
$5$ \( (T - 25)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} \) Copy content Toggle raw display
$11$ \( T^{3} + 194 T^{2} + \cdots - 3144468 \) Copy content Toggle raw display
$13$ \( T^{3} + 1892 T^{2} + \cdots + 171071930 \) Copy content Toggle raw display
$17$ \( T^{3} - 184 T^{2} + \cdots - 132716862 \) Copy content Toggle raw display
$19$ \( T^{3} + 1212 T^{2} + \cdots + 140259040 \) Copy content Toggle raw display
$23$ \( T^{3} - 3188 T^{2} + \cdots + 125424384 \) Copy content Toggle raw display
$29$ \( T^{3} + \cdots + 51669601050 \) Copy content Toggle raw display
$31$ \( T^{3} + \cdots - 8818293376 \) Copy content Toggle raw display
$37$ \( T^{3} + \cdots + 1043894647208 \) Copy content Toggle raw display
$41$ \( T^{3} + \cdots - 4748673370848 \) Copy content Toggle raw display
$43$ \( T^{3} + \cdots + 4763987701360 \) Copy content Toggle raw display
$47$ \( T^{3} + \cdots + 1072310433384 \) Copy content Toggle raw display
$53$ \( T^{3} + \cdots + 515502653472 \) Copy content Toggle raw display
$59$ \( T^{3} + \cdots - 7000620748800 \) Copy content Toggle raw display
$61$ \( T^{3} + \cdots + 1590789613952 \) Copy content Toggle raw display
$67$ \( T^{3} + \cdots + 26595134017984 \) Copy content Toggle raw display
$71$ \( T^{3} + \cdots - 77522711777280 \) Copy content Toggle raw display
$73$ \( T^{3} + \cdots - 11550435576632 \) Copy content Toggle raw display
$79$ \( T^{3} + \cdots - 40598762145400 \) Copy content Toggle raw display
$83$ \( T^{3} + \cdots + 529374252288 \) Copy content Toggle raw display
$89$ \( T^{3} + \cdots + 31660963259040 \) Copy content Toggle raw display
$97$ \( T^{3} + \cdots - 117394067785166 \) Copy content Toggle raw display
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