Properties

Label 1416.1.u.b
Level $1416$
Weight $1$
Character orbit 1416.u
Analytic conductor $0.707$
Analytic rank $0$
Dimension $28$
Projective image $D_{58}$
CM discriminant -8
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1416,1,Mod(11,1416)] 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("1416.11"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1416, base_ring=CyclotomicField(58)) chi = DirichletCharacter(H, H._module([29, 29, 29, 25])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 1416 = 2^{3} \cdot 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1416.u (of order \(58\), degree \(28\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [28,1] 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: \(0.706676057888\)
Analytic rank: \(0\)
Dimension: \(28\)
Coefficient field: \(\Q(\zeta_{58})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{28} - x^{27} + x^{26} - x^{25} + x^{24} - x^{23} + x^{22} - x^{21} + x^{20} - x^{19} + x^{18} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{58}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{58} - \cdots)\)

$q$-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)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q - \zeta_{58}^{2} q^{2} - \zeta_{58}^{9} q^{3} + \zeta_{58}^{4} q^{4} + \zeta_{58}^{11} q^{6} - \zeta_{58}^{6} q^{8} + \zeta_{58}^{18} q^{9} + (\zeta_{58}^{10} - \zeta_{58}^{3}) q^{11} - \zeta_{58}^{13} q^{12} + \cdots + (\zeta_{58}^{28} - \zeta_{58}^{21}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 28 q + q^{2} - q^{3} - q^{4} + q^{6} + q^{8} - q^{9} - 2 q^{11} - q^{12} - q^{16} + q^{18} + 2 q^{19} + 2 q^{22} + q^{24} + q^{25} - q^{27} + q^{32} - 2 q^{33} - q^{36} - 2 q^{38} - 2 q^{44}+ \cdots - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1416\mathbb{Z}\right)^\times\).

\(n\) \(473\) \(709\) \(769\) \(1063\)
\(\chi(n)\) \(-1\) \(-1\) \(-\zeta_{58}^{4}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
0.994138 0.108119i
−0.796093 0.605174i
−0.907575 0.419889i
−0.647386 + 0.762162i
0.725995 0.687699i
−0.907575 + 0.419889i
−0.976621 0.214970i
−0.468408 + 0.883512i
0.561187 0.827689i
0.725995 + 0.687699i
0.161782 0.986827i
0.994138 + 0.108119i
−0.647386 0.762162i
−0.796093 + 0.605174i
−0.976621 + 0.214970i
0.561187 + 0.827689i
0.947653 + 0.319302i
0.370138 + 0.928977i
0.161782 + 0.986827i
−0.267528 0.963550i
−0.976621 + 0.214970i −0.561187 + 0.827689i 0.907575 0.419889i 0 0.370138 0.928977i 0 −0.796093 + 0.605174i −0.370138 0.928977i 0
83.1 −0.267528 0.963550i 0.907575 0.419889i −0.856857 + 0.515554i 0 −0.647386 0.762162i 0 0.725995 + 0.687699i 0.647386 0.762162i 0
131.1 −0.647386 0.762162i −0.725995 0.687699i −0.161782 + 0.986827i 0 −0.0541389 + 0.998533i 0 0.856857 0.515554i 0.0541389 + 0.998533i 0
155.1 0.161782 + 0.986827i 0.0541389 0.998533i −0.947653 + 0.319302i 0 0.994138 0.108119i 0 −0.468408 0.883512i −0.994138 0.108119i 0
179.1 −0.0541389 + 0.998533i −0.856857 + 0.515554i −0.994138 0.108119i 0 −0.468408 0.883512i 0 0.161782 0.986827i 0.468408 0.883512i 0
227.1 −0.647386 + 0.762162i −0.725995 + 0.687699i −0.161782 0.986827i 0 −0.0541389 0.998533i 0 0.856857 + 0.515554i 0.0541389 0.998533i 0
275.1 −0.907575 0.419889i −0.370138 + 0.928977i 0.647386 + 0.762162i 0 0.725995 0.687699i 0 −0.267528 0.963550i −0.725995 0.687699i 0
347.1 0.561187 + 0.827689i −0.947653 + 0.319302i −0.370138 + 0.928977i 0 −0.796093 0.605174i 0 −0.976621 + 0.214970i 0.796093 0.605174i 0
419.1 0.370138 + 0.928977i 0.796093 + 0.605174i −0.725995 + 0.687699i 0 −0.267528 + 0.963550i 0 −0.907575 0.419889i 0.267528 + 0.963550i 0
443.1 −0.0541389 0.998533i −0.856857 0.515554i −0.994138 + 0.108119i 0 −0.468408 + 0.883512i 0 0.161782 + 0.986827i 0.468408 + 0.883512i 0
467.1 0.947653 + 0.319302i −0.994138 + 0.108119i 0.796093 + 0.605174i 0 −0.976621 0.214970i 0 0.561187 + 0.827689i 0.976621 0.214970i 0
515.1 −0.976621 0.214970i −0.561187 0.827689i 0.907575 + 0.419889i 0 0.370138 + 0.928977i 0 −0.796093 0.605174i −0.370138 + 0.928977i 0
539.1 0.161782 0.986827i 0.0541389 + 0.998533i −0.947653 0.319302i 0 0.994138 + 0.108119i 0 −0.468408 + 0.883512i −0.994138 + 0.108119i 0
563.1 −0.267528 + 0.963550i 0.907575 + 0.419889i −0.856857 0.515554i 0 −0.647386 + 0.762162i 0 0.725995 0.687699i 0.647386 + 0.762162i 0
587.1 −0.907575 + 0.419889i −0.370138 0.928977i 0.647386 0.762162i 0 0.725995 + 0.687699i 0 −0.267528 + 0.963550i −0.725995 + 0.687699i 0
659.1 0.370138 0.928977i 0.796093 0.605174i −0.725995 0.687699i 0 −0.267528 0.963550i 0 −0.907575 + 0.419889i 0.267528 0.963550i 0
683.1 −0.796093 0.605174i 0.976621 0.214970i 0.267528 + 0.963550i 0 −0.907575 0.419889i 0 0.370138 0.928977i 0.907575 0.419889i 0
731.1 0.725995 0.687699i 0.267528 + 0.963550i 0.0541389 0.998533i 0 0.856857 + 0.515554i 0 −0.647386 0.762162i −0.856857 + 0.515554i 0
755.1 0.947653 0.319302i −0.994138 0.108119i 0.796093 0.605174i 0 −0.976621 + 0.214970i 0 0.561187 0.827689i 0.976621 + 0.214970i 0
899.1 0.856857 0.515554i 0.647386 0.762162i 0.468408 0.883512i 0 0.161782 0.986827i 0 −0.0541389 0.998533i −0.161782 0.986827i 0
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
177.f even 58 1 inner
1416.u odd 58 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1416.1.u.b yes 28
3.b odd 2 1 1416.1.u.a 28
8.d odd 2 1 CM 1416.1.u.b yes 28
24.f even 2 1 1416.1.u.a 28
59.d odd 58 1 1416.1.u.a 28
177.f even 58 1 inner 1416.1.u.b yes 28
472.l even 58 1 1416.1.u.a 28
1416.u odd 58 1 inner 1416.1.u.b yes 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1416.1.u.a 28 3.b odd 2 1
1416.1.u.a 28 24.f even 2 1
1416.1.u.a 28 59.d odd 58 1
1416.1.u.a 28 472.l even 58 1
1416.1.u.b yes 28 1.a even 1 1 trivial
1416.1.u.b yes 28 8.d odd 2 1 CM
1416.1.u.b yes 28 177.f even 58 1 inner
1416.1.u.b yes 28 1416.u odd 58 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{28} + 2 T_{11}^{27} + 4 T_{11}^{26} + 8 T_{11}^{25} + 16 T_{11}^{24} - 26 T_{11}^{23} - 52 T_{11}^{22} + \cdots + 1 \) acting on \(S_{1}^{\mathrm{new}}(1416, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{28} - T^{27} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{28} + T^{27} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{28} \) Copy content Toggle raw display
$7$ \( T^{28} \) Copy content Toggle raw display
$11$ \( T^{28} + 2 T^{27} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{28} \) Copy content Toggle raw display
$17$ \( T^{28} + 58 T^{23} + \cdots + 29 \) Copy content Toggle raw display
$19$ \( T^{28} - 2 T^{27} + \cdots + 1 \) Copy content Toggle raw display
$23$ \( T^{28} \) Copy content Toggle raw display
$29$ \( T^{28} \) Copy content Toggle raw display
$31$ \( T^{28} \) Copy content Toggle raw display
$37$ \( T^{28} \) Copy content Toggle raw display
$41$ \( T^{28} + 29 T^{25} + \cdots + 29 \) Copy content Toggle raw display
$43$ \( T^{28} + 29 T^{24} + \cdots + 29 \) Copy content Toggle raw display
$47$ \( T^{28} \) Copy content Toggle raw display
$53$ \( T^{28} \) Copy content Toggle raw display
$59$ \( T^{28} - T^{27} + \cdots + 1 \) Copy content Toggle raw display
$61$ \( T^{28} \) Copy content Toggle raw display
$67$ \( T^{28} + 29 T^{22} + \cdots + 29 \) Copy content Toggle raw display
$71$ \( T^{28} \) Copy content Toggle raw display
$73$ \( T^{28} + 29 T^{25} + \cdots + 29 \) Copy content Toggle raw display
$79$ \( T^{28} \) Copy content Toggle raw display
$83$ \( T^{28} - 2 T^{27} + \cdots + 1 \) Copy content Toggle raw display
$89$ \( T^{28} + 2 T^{27} + \cdots + 1 \) Copy content Toggle raw display
$97$ \( T^{28} + 29 T^{22} + \cdots + 29 \) Copy content Toggle raw display
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