↑

Properties

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

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [245,2,Mod(116,245)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("245.116"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(245, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 245.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,1,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content 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 \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{17})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 5x^{2} + 4x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{3} - \beta_{2} + \beta_1 + 1) q^{3} + (\beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{4} - \beta_{2} q^{5} - 4 q^{6} + (\beta_{3} - 4) q^{8} + ( - 2 \beta_{2} + \beta_1) q^{9} + ( - \beta_{3} - \beta_1) q^{10}+ \cdots + (2 \beta_{3} + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} + q^{3} - 5 q^{4} - 2 q^{5} - 16 q^{6} - 18 q^{8} - 3 q^{9} + q^{10} - q^{11} - 6 q^{12} + 10 q^{13} - 2 q^{15} - 3 q^{16} + 5 q^{17} - 7 q^{18} + 6 q^{19} + 10 q^{20} + 16 q^{22} + 2 q^{23}+ \cdots + 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 5x^{2} + 4x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 5\nu^{2} - 5\nu + 16 ) / 20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 4 ) / 5 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 4\beta_{2} + \beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 5\beta_{3} - 4 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

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

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
116.1
−0.780776 − 1.35234i
1.28078 + 2.21837i
−0.780776 + 1.35234i
1.28078 − 2.21837i
−0.780776 − 1.35234i 1.28078 − 2.21837i −0.219224 + 0.379706i −0.500000 − 0.866025i −4.00000 0 −2.43845 −1.78078 − 3.08440i −0.780776 + 1.35234i
116.2 1.28078 + 2.21837i −0.780776 + 1.35234i −2.28078 + 3.95042i −0.500000 − 0.866025i −4.00000 0 −6.56155 0.280776 + 0.486319i 1.28078 − 2.21837i
226.1 −0.780776 + 1.35234i 1.28078 + 2.21837i −0.219224 − 0.379706i −0.500000 + 0.866025i −4.00000 0 −2.43845 −1.78078 + 3.08440i −0.780776 − 1.35234i
226.2 1.28078 − 2.21837i −0.780776 − 1.35234i −2.28078 − 3.95042i −0.500000 + 0.866025i −4.00000 0 −6.56155 0.280776 − 0.486319i 1.28078 + 2.21837i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 245.2.e.i 4
7.b odd 2 1 245.2.e.h 4
7.c even 3 1 35.2.a.b ✓ 2
7.c even 3 1 inner 245.2.e.i 4
7.d odd 6 1 245.2.a.d 2
7.d odd 6 1 245.2.e.h 4
21.g even 6 1 2205.2.a.x 2
21.h odd 6 1 315.2.a.e 2
28.f even 6 1 3920.2.a.bs 2
28.g odd 6 1 560.2.a.i 2
35.i odd 6 1 1225.2.a.s 2
35.j even 6 1 175.2.a.f 2
35.k even 12 2 1225.2.b.f 4
35.l odd 12 2 175.2.b.b 4
56.k odd 6 1 2240.2.a.bd 2
56.p even 6 1 2240.2.a.bh 2
77.h odd 6 1 4235.2.a.m 2
84.n even 6 1 5040.2.a.bt 2
91.r even 6 1 5915.2.a.l 2
105.o odd 6 1 1575.2.a.p 2
105.x even 12 2 1575.2.d.e 4
140.p odd 6 1 2800.2.a.bi 2
140.w even 12 2 2800.2.g.t 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.a.b ✓ 2 7.c even 3 1
175.2.a.f 2 35.j even 6 1
175.2.b.b 4 35.l odd 12 2
245.2.a.d 2 7.d odd 6 1
245.2.e.h 4 7.b odd 2 1
245.2.e.h 4 7.d odd 6 1
245.2.e.i 4 1.a even 1 1 trivial
245.2.e.i 4 7.c even 3 1 inner
315.2.a.e 2 21.h odd 6 1
560.2.a.i 2 28.g odd 6 1
1225.2.a.s 2 35.i odd 6 1
1225.2.b.f 4 35.k even 12 2
1575.2.a.p 2 105.o odd 6 1
1575.2.d.e 4 105.x even 12 2
2205.2.a.x 2 21.g even 6 1
2240.2.a.bd 2 56.k odd 6 1
2240.2.a.bh 2 56.p even 6 1
2800.2.a.bi 2 140.p odd 6 1
2800.2.g.t 4 140.w even 12 2
3920.2.a.bs 2 28.f even 6 1
4235.2.a.m 2 77.h odd 6 1
5040.2.a.bt 2 84.n even 6 1
5915.2.a.l 2 91.r even 6 1

Hecke kernels

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

\( T_{2}^{4} - T_{2}^{3} + 5T_{2}^{2} + 4T_{2} + 16 \) Copy content Toggle raw display
\( T_{3}^{4} - T_{3}^{3} + 5T_{3}^{2} + 4T_{3} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{3} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( T^{4} - T^{3} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + T^{3} + \cdots + 16 \) Copy content Toggle raw display
$13$ \( (T^{2} - 5 T + 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 5 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$19$ \( T^{4} - 6 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$23$ \( T^{4} - 2 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$29$ \( (T^{2} - T - 38)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 2 T - 16)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 10 T + 8)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 5 T^{3} + \cdots + 1024 \) Copy content Toggle raw display
$53$ \( T^{4} - 2 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$59$ \( (T^{2} - 4 T + 16)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} + 6 T^{3} + \cdots + 20736 \) Copy content Toggle raw display
$67$ \( T^{4} + 4 T^{3} + \cdots + 4096 \) Copy content Toggle raw display
$71$ \( (T - 8)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} - 8 T^{3} + \cdots + 2704 \) Copy content Toggle raw display
$79$ \( T^{4} - 9 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$83$ \( (T - 4)^{4} \) Copy content Toggle raw display
$89$ \( T^{4} + 6 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$97$ \( (T^{2} + 9 T - 86)^{2} \) Copy content Toggle raw display
show more
show less