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Properties

Label 1470.3.f.a
Level $1470$
Weight $3$
Character orbit 1470.f
Analytic conductor $40.055$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1470,3,Mod(391,1470)] 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("1470.391"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1470, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1470.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,16,0,0,0,0,-24,0,8,0,0,0,0,32,0,0,0,0,0,-48] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(22)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(40.0545988610\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.3317760000.3
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{6} + 7x^{4} - 36x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 210)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{5} q^{3} + 2 q^{4} + \beta_{6} q^{5} + \beta_{3} q^{6} + 2 \beta_1 q^{8} - 3 q^{9} - \beta_{7} q^{10} + (\beta_{4} + 2 \beta_{2} - 3 \beta_1 + 1) q^{11} - 2 \beta_{5} q^{12}+ \cdots + ( - 3 \beta_{4} - 6 \beta_{2} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 16 q^{4} - 24 q^{9} + 8 q^{11} + 32 q^{16} - 48 q^{22} - 24 q^{23} - 40 q^{25} + 72 q^{29} - 48 q^{36} + 192 q^{37} - 48 q^{39} - 112 q^{43} + 16 q^{44} - 16 q^{46} - 168 q^{51} - 64 q^{53} + 216 q^{57}+ \cdots - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{6} + 7x^{4} - 36x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -4\nu^{7} + 7\nu^{5} + 35\nu^{3} + 81\nu ) / 189 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -10\nu^{7} + 49\nu^{5} - 133\nu^{3} + 801\nu ) / 189 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{7} + \nu^{5} + 5\nu^{3} - 63\nu ) / 27 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{6} + 4\nu^{4} + 2\nu^{2} + 18 ) / 9 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -8\nu^{6} + 14\nu^{4} - 56\nu^{2} + 225 ) / 63 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{6} + 22 ) / 7 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 4\nu^{7} - 7\nu^{5} + 19\nu^{3} - 81\nu ) / 27 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} - \beta_{3} + \beta_{2} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} - 2\beta_{5} + \beta_{4} + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} + 7\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -4\beta_{6} + \beta_{5} + 4\beta_{4} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -5\beta_{7} + 19\beta_{3} + 5\beta_{2} + 19\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -7\beta_{6} + 22 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 29\beta_{7} + 13\beta_{3} + 29\beta_{2} - 13\beta_1 ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1470\mathbb{Z}\right)^\times\).

\(n\) \(491\) \(1081\) \(1177\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
391.1
1.01575 − 1.40294i
−1.72286 + 0.178197i
−1.72286 − 0.178197i
1.01575 + 1.40294i
−1.01575 + 1.40294i
1.72286 − 0.178197i
1.72286 + 0.178197i
−1.01575 − 1.40294i
−1.41421 − 1.73205i 2.00000 − 2.23607i 2.44949i 0 −2.82843 −3.00000 3.16228i
391.2 −1.41421 − 1.73205i 2.00000 2.23607i 2.44949i 0 −2.82843 −3.00000 − 3.16228i
391.3 −1.41421 1.73205i 2.00000 − 2.23607i − 2.44949i 0 −2.82843 −3.00000 3.16228i
391.4 −1.41421 1.73205i 2.00000 2.23607i − 2.44949i 0 −2.82843 −3.00000 − 3.16228i
391.5 1.41421 − 1.73205i 2.00000 − 2.23607i − 2.44949i 0 2.82843 −3.00000 − 3.16228i
391.6 1.41421 − 1.73205i 2.00000 2.23607i − 2.44949i 0 2.82843 −3.00000 3.16228i
391.7 1.41421 1.73205i 2.00000 − 2.23607i 2.44949i 0 2.82843 −3.00000 − 3.16228i
391.8 1.41421 1.73205i 2.00000 2.23607i 2.44949i 0 2.82843 −3.00000 3.16228i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 391.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1470.3.f.a 8
7.b odd 2 1 inner 1470.3.f.a 8
7.c even 3 1 210.3.o.a ✓ 8
7.d odd 6 1 210.3.o.a ✓ 8
21.g even 6 1 630.3.v.b 8
21.h odd 6 1 630.3.v.b 8
35.i odd 6 1 1050.3.p.b 8
35.j even 6 1 1050.3.p.b 8
35.k even 12 2 1050.3.q.c 16
35.l odd 12 2 1050.3.q.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.3.o.a ✓ 8 7.c even 3 1
210.3.o.a ✓ 8 7.d odd 6 1
630.3.v.b 8 21.g even 6 1
630.3.v.b 8 21.h odd 6 1
1050.3.p.b 8 35.i odd 6 1
1050.3.p.b 8 35.j even 6 1
1050.3.q.c 16 35.k even 12 2
1050.3.q.c 16 35.l odd 12 2
1470.3.f.a 8 1.a even 1 1 trivial
1470.3.f.a 8 7.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{4} - 4T_{11}^{3} - 300T_{11}^{2} + 2048T_{11} + 4744 \) acting on \(S_{3}^{\mathrm{new}}(1470, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2)^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3)^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + 5)^{4} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} - 4 T^{3} + \cdots + 4744)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 492 T^{6} + \cdots + 33189121 \) Copy content Toggle raw display
$17$ \( T^{8} + 952 T^{6} + \cdots + 15808576 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 3571138081 \) Copy content Toggle raw display
$23$ \( (T^{4} + 12 T^{3} + \cdots - 1016)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 36 T^{3} + \cdots + 20104)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 2556 T^{6} + \cdots + 56085121 \) Copy content Toggle raw display
$37$ \( (T^{4} - 96 T^{3} + \cdots + 26209)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 6360766467136 \) Copy content Toggle raw display
$43$ \( (T^{4} + 56 T^{3} + \cdots - 877199)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 17622397993216 \) Copy content Toggle raw display
$53$ \( (T^{4} + 32 T^{3} + \cdots - 38336)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 19263180552256 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 981652934656 \) Copy content Toggle raw display
$67$ \( (T^{4} - 120 T^{3} + \cdots + 921249)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 4 T^{3} + \cdots - 2872184)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 52818460352161 \) Copy content Toggle raw display
$79$ \( (T^{4} + 12 T^{3} + \cdots + 108899161)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 89\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 91315148362816 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 34\!\cdots\!56 \) Copy content Toggle raw display
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