Properties

Label 245.6.a.c
Level $245$
Weight $6$
Character orbit 245.a
Self dual yes
Analytic conductor $39.294$
Analytic rank $1$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{65}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{65})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + (3 \beta - 3) q^{3} + (\beta - 16) q^{4} + 25 q^{5} + 48 q^{6} + ( - 47 \beta + 16) q^{8} + ( - 9 \beta - 90) q^{9} + 25 \beta q^{10} + (97 \beta - 349) q^{11} + ( - 48 \beta + 96) q^{12}+ \cdots + ( - 6462 \beta + 17442) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - 3 q^{3} - 31 q^{4} + 50 q^{5} + 96 q^{6} - 15 q^{8} - 189 q^{9} + 25 q^{10} - 601 q^{11} + 144 q^{12} + 577 q^{13} - 75 q^{15} - 543 q^{16} - 41 q^{17} - 387 q^{18} - 630 q^{19} - 775 q^{20}+ \cdots + 28422 q^{99}+O(q^{100}) \) 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
−3.53113
4.53113
−3.53113 −13.5934 −19.5311 25.0000 48.0000 0 181.963 −58.2198 −88.2782
1.2 4.53113 10.5934 −11.4689 25.0000 48.0000 0 −196.963 −130.780 113.278
\(n\): 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.c 2
7.b odd 2 1 35.6.a.b 2
21.c even 2 1 315.6.a.c 2
28.d even 2 1 560.6.a.l 2
35.c odd 2 1 175.6.a.d 2
35.f even 4 2 175.6.b.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.6.a.b 2 7.b odd 2 1
175.6.a.d 2 35.c odd 2 1
175.6.b.d 4 35.f even 4 2
245.6.a.c 2 1.a even 1 1 trivial
315.6.a.c 2 21.c even 2 1
560.6.a.l 2 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(245))\):

\( T_{2}^{2} - T_{2} - 16 \) Copy content Toggle raw display
\( T_{3}^{2} + 3T_{3} - 144 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T - 16 \) Copy content Toggle raw display
$3$ \( T^{2} + 3T - 144 \) Copy content Toggle raw display
$5$ \( (T - 25)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 601T - 62596 \) Copy content Toggle raw display
$13$ \( T^{2} - 577T + 37586 \) Copy content Toggle raw display
$17$ \( T^{2} + 41T - 1023346 \) Copy content Toggle raw display
$19$ \( T^{2} + 630T - 20960 \) Copy content Toggle raw display
$23$ \( T^{2} + 442 T - 13172224 \) Copy content Toggle raw display
$29$ \( T^{2} - 5885 T - 5853350 \) Copy content Toggle raw display
$31$ \( T^{2} - 396 T - 13834656 \) Copy content Toggle raw display
$37$ \( T^{2} + 8904 T - 5978196 \) Copy content Toggle raw display
$41$ \( T^{2} + 1774 T - 236091496 \) Copy content Toggle raw display
$43$ \( T^{2} + 27122 T + 170086856 \) Copy content Toggle raw display
$47$ \( T^{2} - 21289 T + 72995224 \) Copy content Toggle raw display
$53$ \( T^{2} + 55582 T + 768990296 \) Copy content Toggle raw display
$59$ \( T^{2} + 59600 T + 451339840 \) Copy content Toggle raw display
$61$ \( T^{2} - 51846 T + 142927344 \) Copy content Toggle raw display
$67$ \( T^{2} + 45344 T + 174936944 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots + 1172746624 \) Copy content Toggle raw display
$73$ \( T^{2} - 13532 T - 239780284 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots - 4382959400 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots + 2765333536 \) Copy content Toggle raw display
$89$ \( T^{2} - 37650 T - 177665240 \) Copy content Toggle raw display
$97$ \( T^{2} - 96339 T - 838817066 \) Copy content Toggle raw display
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