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Properties

Label 245.4.j.b
Level $245$
Weight $4$
Character orbit 245.j
Analytic conductor $14.455$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [245,4,Mod(79,245)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(245, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("245.79");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 245.j (of order \(6\), degree \(2\), not minimal)

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: no
Analytic conductor: \(14.4554679514\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - 4 \beta_{6} + 2 \beta_{4}) q^{3} + \beta_{2} q^{4} + ( - 5 \beta_{7} - 5 \beta_{6} + \cdots + 5 \beta_{4}) q^{5}+ \cdots + ( - 45 \beta_{2} + 45) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - 4 \beta_{6} + 2 \beta_{4}) q^{3} + \beta_{2} q^{4} + ( - 5 \beta_{7} - 5 \beta_{6} + \cdots + 5 \beta_{4}) q^{5}+ \cdots + 1800 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{4} + 180 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{4} + 180 q^{9} + 160 q^{11} + 720 q^{15} + 284 q^{16} + 400 q^{25} + 320 q^{29} - 360 q^{30} + 360 q^{36} - 2448 q^{39} - 160 q^{44} + 144 q^{46} - 1800 q^{50} - 1008 q^{51} + 360 q^{60} - 3464 q^{64} + 3060 q^{65} - 1184 q^{71} + 3816 q^{74} - 2448 q^{79} - 324 q^{81} - 2520 q^{85} + 4320 q^{86} - 1960 q^{95} + 14400 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 3\zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{24}^{4} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 3\zeta_{24}^{6} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \zeta_{24}^{7} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \zeta_{24}^{5} + 2\zeta_{24}^{3} - \zeta_{24} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\zeta_{24}^{7} + 2\zeta_{24} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \zeta_{24}^{7} + \zeta_{24}^{5} - \zeta_{24}^{3} \) Copy content Toggle raw display

Display \(\zeta_{24}^j\) in terms of \(\beta_i\)

\(\zeta_{24}\)\(=\) \( ( \beta_{6} + \beta_{4} ) / 3 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( \beta_1 ) / 3 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( -\beta_{7} + \beta_{5} + \beta_{4} ) / 3 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( 2\beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} ) / 3 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( ( \beta_{3} ) / 3 \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( -\beta_{6} + 2\beta_{4} ) / 3 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{24}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/245\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(197\)
\(\chi(n)\) \(-1 + \beta_{2}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
0.258819 − 0.965926i
−0.258819 + 0.965926i
0.965926 + 0.258819i
−0.965926 − 0.258819i
0.258819 + 0.965926i
−0.258819 − 0.965926i
0.965926 − 0.258819i
−0.965926 + 0.258819i
−2.59808 − 1.50000i −7.34847 + 4.24264i 0.500000 + 0.866025i −10.9534 − 2.24144i 25.4558 0 21.0000i 22.5000 − 38.9711i 25.0955 + 22.2535i
79.2 −2.59808 − 1.50000i 7.34847 − 4.24264i 0.500000 + 0.866025i 10.9534 + 2.24144i −25.4558 0 21.0000i 22.5000 − 38.9711i −25.0955 − 22.2535i
79.3 2.59808 + 1.50000i −7.34847 + 4.24264i 0.500000 + 0.866025i −7.41782 − 8.36516i −25.4558 0 − 21.0000i 22.5000 − 38.9711i −6.72432 − 32.8601i
79.4 2.59808 + 1.50000i 7.34847 − 4.24264i 0.500000 + 0.866025i 7.41782 + 8.36516i 25.4558 0 − 21.0000i 22.5000 − 38.9711i 6.72432 + 32.8601i
214.1 −2.59808 + 1.50000i −7.34847 − 4.24264i 0.500000 − 0.866025i −10.9534 + 2.24144i 25.4558 0 − 21.0000i 22.5000 + 38.9711i 25.0955 − 22.2535i
214.2 −2.59808 + 1.50000i 7.34847 + 4.24264i 0.500000 − 0.866025i 10.9534 − 2.24144i −25.4558 0 − 21.0000i 22.5000 + 38.9711i −25.0955 + 22.2535i
214.3 2.59808 − 1.50000i −7.34847 − 4.24264i 0.500000 − 0.866025i −7.41782 + 8.36516i −25.4558 0 21.0000i 22.5000 + 38.9711i −6.72432 + 32.8601i
214.4 2.59808 − 1.50000i 7.34847 + 4.24264i 0.500000 − 0.866025i 7.41782 − 8.36516i 25.4558 0 21.0000i 22.5000 + 38.9711i 6.72432 − 32.8601i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 79.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
35.c odd 2 1 inner
35.i odd 6 1 inner
35.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 245.4.j.b 8
5.b even 2 1 inner 245.4.j.b 8
7.b odd 2 1 inner 245.4.j.b 8
7.c even 3 1 245.4.b.b ✓ 4
7.c even 3 1 inner 245.4.j.b 8
7.d odd 6 1 245.4.b.b ✓ 4
7.d odd 6 1 inner 245.4.j.b 8
35.c odd 2 1 inner 245.4.j.b 8
35.i odd 6 1 245.4.b.b ✓ 4
35.i odd 6 1 inner 245.4.j.b 8
35.j even 6 1 245.4.b.b ✓ 4
35.j even 6 1 inner 245.4.j.b 8
35.k even 12 1 1225.4.a.n 2
35.k even 12 1 1225.4.a.w 2
35.l odd 12 1 1225.4.a.n 2
35.l odd 12 1 1225.4.a.w 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
245.4.b.b ✓ 4 7.c even 3 1
245.4.b.b ✓ 4 7.d odd 6 1
245.4.b.b ✓ 4 35.i odd 6 1
245.4.b.b ✓ 4 35.j even 6 1
245.4.j.b 8 1.a even 1 1 trivial
245.4.j.b 8 5.b even 2 1 inner
245.4.j.b 8 7.b odd 2 1 inner
245.4.j.b 8 7.c even 3 1 inner
245.4.j.b 8 7.d odd 6 1 inner
245.4.j.b 8 35.c odd 2 1 inner
245.4.j.b 8 35.i odd 6 1 inner
245.4.j.b 8 35.j even 6 1 inner
1225.4.a.n 2 35.k even 12 1
1225.4.a.n 2 35.l odd 12 1
1225.4.a.w 2 35.k even 12 1
1225.4.a.w 2 35.l odd 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(245, [\chi])\):

\( T_{2}^{4} - 9T_{2}^{2} + 81 \) Copy content Toggle raw display
\( T_{19}^{4} + 19208T_{19}^{2} + 368947264 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 9 T^{2} + 81)^{2} \) Copy content Toggle raw display
$3$ \( (T^{4} - 72 T^{2} + 5184)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} - 200 T^{6} + \cdots + 244140625 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{2} - 40 T + 1600)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + 5202)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} - 882 T^{2} + 777924)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 19208 T^{2} + 368947264)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 144 T^{2} + 20736)^{2} \) Copy content Toggle raw display
$29$ \( (T - 40)^{8} \) Copy content Toggle raw display
$31$ \( (T^{4} + 3872 T^{2} + 14992384)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 101124 T^{2} + 10226063376)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 49298)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 129600)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} - 115200 T^{2} + 13271040000)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 20736 T^{2} + 429981696)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 17672 T^{2} + 312299584)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 837218 T^{2} + 700933979524)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 1296 T^{2} + 1679616)^{2} \) Copy content Toggle raw display
$71$ \( (T + 148)^{8} \) Copy content Toggle raw display
$73$ \( (T^{4} - 399618 T^{2} + 159694545924)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 612 T + 374544)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 677448)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} + 960498 T^{2} + 922556408004)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 882)^{4} \) Copy content Toggle raw display
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