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Properties

Label 245.2.l.c
Level $245$
Weight $2$
Character orbit 245.l
Analytic conductor $1.956$
Analytic rank $0$
Dimension $8$
Inner twists $8$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [245,2,Mod(68,245)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(245, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([9, 10])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("245.68"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 245.l (of order \(12\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.95633484952\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: 8.0.3317760000.2
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 25x^{4} + 625 \) 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: $\mathrm{SU}(2)[C_{12}]$

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

\(f(q)\) \(=\) \( q + ( - \beta_{4} - \beta_{2} + 1) q^{2} + \beta_1 q^{3} + \beta_{5} q^{5} + ( - \beta_{5} - \beta_{3} + \beta_1) q^{6} + ( - 2 \beta_{6} - 2) q^{8} + 2 \beta_{2} q^{9} + ( - \beta_{7} + \beta_1) q^{10}+ \cdots + 2 \beta_{6} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{2} - 16 q^{8} + 4 q^{11} - 16 q^{16} - 8 q^{18} + 8 q^{22} - 8 q^{23} + 20 q^{30} + 24 q^{37} - 24 q^{43} + 16 q^{46} + 40 q^{50} - 20 q^{51} - 4 q^{53} + 40 q^{57} - 12 q^{58} + 4 q^{67} - 48 q^{71}+ \cdots + 20 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 25x^{4} + 625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 5 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} ) / 25 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} ) / 25 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} ) / 125 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} ) / 125 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 5\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 5\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 25\beta_{4} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 25\beta_{5} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 125\beta_{6} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 125\beta_{7} \) Copy content Toggle raw display

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

Character values

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

\(n\) \(101\) \(197\)
\(\chi(n)\) \(1 - \beta_{4}\) \(-\beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
68.1
−2.15988 − 0.578737i
2.15988 + 0.578737i
−0.578737 + 2.15988i
0.578737 − 2.15988i
−0.578737 − 2.15988i
0.578737 + 2.15988i
−2.15988 + 0.578737i
2.15988 − 0.578737i
−0.366025 − 1.36603i −2.15988 − 0.578737i 0 −0.578737 − 2.15988i 3.16228i 0 −2.00000 − 2.00000i 1.73205 + 1.00000i −2.73861 + 1.58114i
68.2 −0.366025 − 1.36603i 2.15988 + 0.578737i 0 0.578737 + 2.15988i − 3.16228i 0 −2.00000 − 2.00000i 1.73205 + 1.00000i 2.73861 − 1.58114i
117.1 1.36603 − 0.366025i −0.578737 + 2.15988i 0 −2.15988 + 0.578737i 3.16228i 0 −2.00000 + 2.00000i −1.73205 − 1.00000i −2.73861 + 1.58114i
117.2 1.36603 − 0.366025i 0.578737 − 2.15988i 0 2.15988 − 0.578737i − 3.16228i 0 −2.00000 + 2.00000i −1.73205 − 1.00000i 2.73861 − 1.58114i
178.1 1.36603 + 0.366025i −0.578737 − 2.15988i 0 −2.15988 − 0.578737i − 3.16228i 0 −2.00000 − 2.00000i −1.73205 + 1.00000i −2.73861 − 1.58114i
178.2 1.36603 + 0.366025i 0.578737 + 2.15988i 0 2.15988 + 0.578737i 3.16228i 0 −2.00000 − 2.00000i −1.73205 + 1.00000i 2.73861 + 1.58114i
227.1 −0.366025 + 1.36603i −2.15988 + 0.578737i 0 −0.578737 + 2.15988i − 3.16228i 0 −2.00000 + 2.00000i 1.73205 − 1.00000i −2.73861 − 1.58114i
227.2 −0.366025 + 1.36603i 2.15988 − 0.578737i 0 0.578737 − 2.15988i 3.16228i 0 −2.00000 + 2.00000i 1.73205 − 1.00000i 2.73861 + 1.58114i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 68.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
35.f even 4 1 inner
35.k even 12 1 inner
35.l odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 245.2.l.c 8
5.c odd 4 1 inner 245.2.l.c 8
7.b odd 2 1 inner 245.2.l.c 8
7.c even 3 1 35.2.f.a ✓ 4
7.c even 3 1 inner 245.2.l.c 8
7.d odd 6 1 35.2.f.a ✓ 4
7.d odd 6 1 inner 245.2.l.c 8
21.g even 6 1 315.2.p.c 4
21.h odd 6 1 315.2.p.c 4
28.f even 6 1 560.2.bj.a 4
28.g odd 6 1 560.2.bj.a 4
35.f even 4 1 inner 245.2.l.c 8
35.i odd 6 1 175.2.f.c 4
35.j even 6 1 175.2.f.c 4
35.k even 12 1 35.2.f.a ✓ 4
35.k even 12 1 175.2.f.c 4
35.k even 12 1 inner 245.2.l.c 8
35.l odd 12 1 35.2.f.a ✓ 4
35.l odd 12 1 175.2.f.c 4
35.l odd 12 1 inner 245.2.l.c 8
105.w odd 12 1 315.2.p.c 4
105.x even 12 1 315.2.p.c 4
140.w even 12 1 560.2.bj.a 4
140.x odd 12 1 560.2.bj.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.f.a ✓ 4 7.c even 3 1
35.2.f.a ✓ 4 7.d odd 6 1
35.2.f.a ✓ 4 35.k even 12 1
35.2.f.a ✓ 4 35.l odd 12 1
175.2.f.c 4 35.i odd 6 1
175.2.f.c 4 35.j even 6 1
175.2.f.c 4 35.k even 12 1
175.2.f.c 4 35.l odd 12 1
245.2.l.c 8 1.a even 1 1 trivial
245.2.l.c 8 5.c odd 4 1 inner
245.2.l.c 8 7.b odd 2 1 inner
245.2.l.c 8 7.c even 3 1 inner
245.2.l.c 8 7.d odd 6 1 inner
245.2.l.c 8 35.f even 4 1 inner
245.2.l.c 8 35.k even 12 1 inner
245.2.l.c 8 35.l odd 12 1 inner
315.2.p.c 4 21.g even 6 1
315.2.p.c 4 21.h odd 6 1
315.2.p.c 4 105.w odd 12 1
315.2.p.c 4 105.x even 12 1
560.2.bj.a 4 28.f even 6 1
560.2.bj.a 4 28.g odd 6 1
560.2.bj.a 4 140.w even 12 1
560.2.bj.a 4 140.x odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 2T_{2}^{3} + 2T_{2}^{2} - 4T_{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(245, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 2 T^{3} + 2 T^{2} + \cdots + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} - 25T^{4} + 625 \) Copy content Toggle raw display
$5$ \( T^{8} - 25T^{4} + 625 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} + 25)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} - 25T^{4} + 625 \) Copy content Toggle raw display
$19$ \( (T^{4} + 10 T^{2} + 100)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 4 T^{3} + 8 T^{2} + \cdots + 64)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 9)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} - 10 T^{2} + 100)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 12 T^{3} + \cdots + 5184)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 90)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 6 T + 18)^{4} \) Copy content Toggle raw display
$47$ \( T^{8} - 2025 T^{4} + 4100625 \) Copy content Toggle raw display
$53$ \( (T^{4} + 2 T^{3} + 2 T^{2} + \cdots + 4)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 90 T^{2} + 8100)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 40 T^{2} + 1600)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 2 T^{3} + 2 T^{2} + \cdots + 4)^{2} \) Copy content Toggle raw display
$71$ \( (T + 6)^{8} \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( (T^{4} - 169 T^{2} + 28561)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 400)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 40 T^{2} + 1600)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 25)^{2} \) Copy content Toggle raw display
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