Properties

Label 245.2.j.d
Level $245$
Weight $2$
Character orbit 245.j
Analytic conductor $1.956$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [245,2,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, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("245.79");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
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: \(1.95633484952\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) 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)
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 primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \zeta_{12} q^{2} + (\zeta_{12}^{3} - \zeta_{12}) q^{3} + 2 \zeta_{12}^{2} q^{4} + (2 \zeta_{12}^{2} + \zeta_{12} - 2) q^{5} - 2 q^{6} + (2 \zeta_{12}^{2} - 2) q^{9} + (4 \zeta_{12}^{3} + \cdots - 4 \zeta_{12}) q^{10}+ \cdots - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{4} - 4 q^{5} - 8 q^{6} - 4 q^{9} + 4 q^{10} + 6 q^{11} - 4 q^{15} + 8 q^{16} - 16 q^{20} - 6 q^{25} + 4 q^{26} + 20 q^{29} + 8 q^{30} + 4 q^{31} + 56 q^{34} - 16 q^{36} + 2 q^{39} - 8 q^{41}+ \cdots - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(197\)
\(\chi(n)\) \(-1 + \zeta_{12}^{2}\) \(-1\)

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
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
−1.73205 1.00000i 0.866025 0.500000i 1.00000 + 1.73205i −1.86603 + 1.23205i −2.00000 0 0 −1.00000 + 1.73205i 4.46410 0.267949i
79.2 1.73205 + 1.00000i −0.866025 + 0.500000i 1.00000 + 1.73205i −0.133975 + 2.23205i −2.00000 0 0 −1.00000 + 1.73205i −2.46410 + 3.73205i
214.1 −1.73205 + 1.00000i 0.866025 + 0.500000i 1.00000 1.73205i −1.86603 1.23205i −2.00000 0 0 −1.00000 1.73205i 4.46410 + 0.267949i
214.2 1.73205 1.00000i −0.866025 0.500000i 1.00000 1.73205i −0.133975 2.23205i −2.00000 0 0 −1.00000 1.73205i −2.46410 3.73205i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.c even 3 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 245.2.j.d 4
5.b even 2 1 inner 245.2.j.d 4
7.b odd 2 1 245.2.j.e 4
7.c even 3 1 245.2.b.a 2
7.c even 3 1 inner 245.2.j.d 4
7.d odd 6 1 35.2.b.a 2
7.d odd 6 1 245.2.j.e 4
21.g even 6 1 315.2.d.a 2
21.h odd 6 1 2205.2.d.b 2
28.f even 6 1 560.2.g.b 2
35.c odd 2 1 245.2.j.e 4
35.i odd 6 1 35.2.b.a 2
35.i odd 6 1 245.2.j.e 4
35.j even 6 1 245.2.b.a 2
35.j even 6 1 inner 245.2.j.d 4
35.k even 12 1 175.2.a.a 1
35.k even 12 1 175.2.a.c 1
35.l odd 12 1 1225.2.a.a 1
35.l odd 12 1 1225.2.a.i 1
56.j odd 6 1 2240.2.g.h 2
56.m even 6 1 2240.2.g.g 2
84.j odd 6 1 5040.2.t.p 2
105.o odd 6 1 2205.2.d.b 2
105.p even 6 1 315.2.d.a 2
105.w odd 12 1 1575.2.a.a 1
105.w odd 12 1 1575.2.a.k 1
140.s even 6 1 560.2.g.b 2
140.x odd 12 1 2800.2.a.l 1
140.x odd 12 1 2800.2.a.w 1
280.ba even 6 1 2240.2.g.g 2
280.bk odd 6 1 2240.2.g.h 2
420.be odd 6 1 5040.2.t.p 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.b.a 2 7.d odd 6 1
35.2.b.a 2 35.i odd 6 1
175.2.a.a 1 35.k even 12 1
175.2.a.c 1 35.k even 12 1
245.2.b.a 2 7.c even 3 1
245.2.b.a 2 35.j even 6 1
245.2.j.d 4 1.a even 1 1 trivial
245.2.j.d 4 5.b even 2 1 inner
245.2.j.d 4 7.c even 3 1 inner
245.2.j.d 4 35.j even 6 1 inner
245.2.j.e 4 7.b odd 2 1
245.2.j.e 4 7.d odd 6 1
245.2.j.e 4 35.c odd 2 1
245.2.j.e 4 35.i odd 6 1
315.2.d.a 2 21.g even 6 1
315.2.d.a 2 105.p even 6 1
560.2.g.b 2 28.f even 6 1
560.2.g.b 2 140.s even 6 1
1225.2.a.a 1 35.l odd 12 1
1225.2.a.i 1 35.l odd 12 1
1575.2.a.a 1 105.w odd 12 1
1575.2.a.k 1 105.w odd 12 1
2205.2.d.b 2 21.h odd 6 1
2205.2.d.b 2 105.o odd 6 1
2240.2.g.g 2 56.m even 6 1
2240.2.g.g 2 280.ba even 6 1
2240.2.g.h 2 56.j odd 6 1
2240.2.g.h 2 280.bk odd 6 1
2800.2.a.l 1 140.x odd 12 1
2800.2.a.w 1 140.x odd 12 1
5040.2.t.p 2 84.j odd 6 1
5040.2.t.p 2 420.be odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(245, [\chi])\):

\( T_{2}^{4} - 4T_{2}^{2} + 16 \) Copy content Toggle raw display
\( T_{3}^{4} - T_{3}^{2} + 1 \) Copy content Toggle raw display
\( T_{31}^{2} - 2T_{31} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$3$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{4} + 4 T^{3} + \cdots + 25 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} - 3 T + 9)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 49T^{2} + 2401 \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$29$ \( (T - 5)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$41$ \( (T + 2)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$53$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$59$ \( (T^{2} + 10 T + 100)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 8 T + 64)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$71$ \( (T + 8)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$79$ \( (T^{2} + 5 T + 25)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 49)^{2} \) Copy content Toggle raw display
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