Properties

Label 3479.1.dc.a
Level $3479$
Weight $1$
Character orbit 3479.dc
Analytic conductor $1.736$
Analytic rank $0$
Dimension $12$
Projective image $D_{14}$
CM discriminant -7
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3479,1,Mod(165,3479)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3479, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([28, 9]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3479.165");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3479 = 7^{2} \cdot 71 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3479.dc (of order \(42\), degree \(12\), 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: \(1.73624717895\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\Q(\zeta_{21})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} + x^{9} - x^{8} + x^{6} - x^{4} + x^{3} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{14}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{14} - \cdots)\)

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

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

\(f(q)\) \(=\) \( q + (\zeta_{42}^{4} - \zeta_{42}) q^{2} + (\zeta_{42}^{8} - \zeta_{42}^{5} + \zeta_{42}^{2}) q^{4} + (\zeta_{42}^{12} - \zeta_{42}^{9} + \zeta_{42}^{6} - \zeta_{42}^{3}) q^{8} - \zeta_{42}^{10} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{42}^{4} - \zeta_{42}) q^{2} + (\zeta_{42}^{8} - \zeta_{42}^{5} + \zeta_{42}^{2}) q^{4} + (\zeta_{42}^{12} - \zeta_{42}^{9} + \zeta_{42}^{6} - \zeta_{42}^{3}) q^{8} - \zeta_{42}^{10} q^{9} + ( - \zeta_{42}^{8} - \zeta_{42}^{5}) q^{11} + (\zeta_{42}^{16} - \zeta_{42}^{13} + \zeta_{42}^{10} - \zeta_{42}^{7} + \zeta_{42}^{4}) q^{16} + ( - \zeta_{42}^{14} + \zeta_{42}^{11}) q^{18} + ( - \zeta_{42}^{12} + \zeta_{42}^{6}) q^{22} + ( - \zeta_{42}^{13} - \zeta_{42}^{10}) q^{23} - \zeta_{42}^{14} q^{25} + (\zeta_{42}^{3} - 1) q^{29} + (\zeta_{42}^{20} - \zeta_{42}^{17} + \zeta_{42}^{14} - \zeta_{42}^{11} + \zeta_{42}^{8} - \zeta_{42}^{5}) q^{32} + ( - \zeta_{42}^{18} + \zeta_{42}^{15} - \zeta_{42}^{12}) q^{36} + ( - \zeta_{42}^{4} + \zeta_{42}) q^{37} + (\zeta_{42}^{12} + 1) q^{43} + ( - \zeta_{42}^{16} - \zeta_{42}^{7}) q^{44} + ( - \zeta_{42}^{17} + \zeta_{42}^{11}) q^{46} + ( - \zeta_{42}^{18} + \zeta_{42}^{15}) q^{50} + ( - \zeta_{42}^{20} - \zeta_{42}^{5}) q^{53} + (\zeta_{42}^{7} - 2 \zeta_{42}^{4} + \zeta_{42}) q^{58} + (\zeta_{42}^{18} - \zeta_{42}^{15} + \zeta_{42}^{12} - \zeta_{42}^{9} + \zeta_{42}^{6} - \zeta_{42}^{3} - 1) q^{64} + ( - \zeta_{42}^{20} - \zeta_{42}^{11}) q^{67} + \zeta_{42}^{6} q^{71} + (\zeta_{42}^{19} - \zeta_{42}^{16} + \zeta_{42}^{13} + \zeta_{42}) q^{72} + ( - \zeta_{42}^{8} + 2 \zeta_{42}^{5} - \zeta_{42}^{2}) q^{74} + (\zeta_{42}^{19} - \zeta_{42}^{10}) q^{79} + \zeta_{42}^{20} q^{81} + (\zeta_{42}^{16} - \zeta_{42}^{13} + \zeta_{42}^{4} - \zeta_{42}) q^{86} + ( - \zeta_{42}^{20} + \zeta_{42}^{8}) q^{88} + ( - \zeta_{42}^{12} + 1) q^{92} + (\zeta_{42}^{18} + \zeta_{42}^{15}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{2} + 3 q^{4} - 8 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 2 q^{2} + 3 q^{4} - 8 q^{8} - q^{9} - 2 q^{16} + 5 q^{18} + 6 q^{25} - 10 q^{29} - q^{32} + 6 q^{36} - 2 q^{37} + 10 q^{43} - 7 q^{44} + 4 q^{50} + 3 q^{58} - 2 q^{71} - 4 q^{72} - 4 q^{74} - 2 q^{79} + q^{81} + 4 q^{86} + 14 q^{92}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3479\mathbb{Z}\right)^\times\).

\(n\) \(640\) \(1569\)
\(\chi(n)\) \(-\zeta_{42}^{7}\) \(\zeta_{42}^{9}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
165.1
−0.988831 0.149042i
0.365341 + 0.930874i
0.0747301 + 0.997204i
0.826239 0.563320i
0.826239 + 0.563320i
0.955573 0.294755i
0.0747301 0.997204i
0.365341 0.930874i
−0.733052 + 0.680173i
−0.733052 0.680173i
−0.988831 + 0.149042i
0.955573 + 0.294755i
−0.162592 + 0.414278i 0 0.587862 + 0.545456i 0 0 0 −0.722521 + 0.347948i −0.0747301 0.997204i 0
520.1 0.440071 0.0663300i 0 −0.766310 + 0.236375i 0 0 0 −0.722521 + 0.347948i −0.826239 + 0.563320i 0
949.1 1.03030 + 0.702449i 0 0.202749 + 0.516596i 0 0 0 0.123490 0.541044i 0.733052 0.680173i 0
1304.1 0.0931869 1.24349i 0 −0.548760 0.0827122i 0 0 0 0.123490 0.541044i −0.955573 0.294755i 0
1390.1 0.0931869 + 1.24349i 0 −0.548760 + 0.0827122i 0 0 0 0.123490 + 0.541044i −0.955573 + 0.294755i 0
1684.1 1.32091 1.22563i 0 0.167917 2.24070i 0 0 0 −1.40097 1.75676i 0.988831 + 0.149042i 0
1745.1 1.03030 0.702449i 0 0.202749 0.516596i 0 0 0 0.123490 + 0.541044i 0.733052 + 0.680173i 0
1880.1 0.440071 + 0.0663300i 0 −0.766310 0.236375i 0 0 0 −0.722521 0.347948i −0.826239 0.563320i 0
2027.1 −1.72188 + 0.531130i 0 1.85654 1.26577i 0 0 0 −1.40097 + 1.75676i −0.365341 + 0.930874i 0
2039.1 −1.72188 0.531130i 0 1.85654 + 1.26577i 0 0 0 −1.40097 1.75676i −0.365341 0.930874i 0
2235.1 −0.162592 0.414278i 0 0.587862 0.545456i 0 0 0 −0.722521 0.347948i −0.0747301 + 0.997204i 0
2382.1 1.32091 + 1.22563i 0 0.167917 + 2.24070i 0 0 0 −1.40097 + 1.75676i 0.988831 0.149042i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 165.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
7.c even 3 1 inner
7.d odd 6 1 inner
71.f odd 14 1 inner
497.p even 14 1 inner
497.w even 42 1 inner
497.y odd 42 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3479.1.dc.a 12
7.b odd 2 1 CM 3479.1.dc.a 12
7.c even 3 1 3479.1.bb.a 6
7.c even 3 1 inner 3479.1.dc.a 12
7.d odd 6 1 3479.1.bb.a 6
7.d odd 6 1 inner 3479.1.dc.a 12
71.f odd 14 1 inner 3479.1.dc.a 12
497.p even 14 1 inner 3479.1.dc.a 12
497.w even 42 1 3479.1.bb.a 6
497.w even 42 1 inner 3479.1.dc.a 12
497.y odd 42 1 3479.1.bb.a 6
497.y odd 42 1 inner 3479.1.dc.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3479.1.bb.a 6 7.c even 3 1
3479.1.bb.a 6 7.d odd 6 1
3479.1.bb.a 6 497.w even 42 1
3479.1.bb.a 6 497.y odd 42 1
3479.1.dc.a 12 1.a even 1 1 trivial
3479.1.dc.a 12 7.b odd 2 1 CM
3479.1.dc.a 12 7.c even 3 1 inner
3479.1.dc.a 12 7.d odd 6 1 inner
3479.1.dc.a 12 71.f odd 14 1 inner
3479.1.dc.a 12 497.p even 14 1 inner
3479.1.dc.a 12 497.w even 42 1 inner
3479.1.dc.a 12 497.y odd 42 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(3479, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 2 T^{11} + 8 T^{9} - 9 T^{8} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( T^{12} - 14 T^{9} - 7 T^{7} + 63 T^{6} + \cdots + 49 \) Copy content Toggle raw display
$13$ \( T^{12} \) Copy content Toggle raw display
$17$ \( T^{12} \) Copy content Toggle raw display
$19$ \( T^{12} \) Copy content Toggle raw display
$23$ \( T^{12} - 7 T^{8} - 14 T^{7} + 14 T^{6} + \cdots + 49 \) Copy content Toggle raw display
$29$ \( (T^{6} + 5 T^{5} + 11 T^{4} + 13 T^{3} + \cdots + 1)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} \) Copy content Toggle raw display
$37$ \( T^{12} + 2 T^{11} - 8 T^{9} - 9 T^{8} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( T^{12} \) Copy content Toggle raw display
$43$ \( (T^{6} - 5 T^{5} + 11 T^{4} - 13 T^{3} + \cdots + 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} \) Copy content Toggle raw display
$53$ \( T^{12} - 7 T^{8} - 14 T^{7} + 14 T^{6} + \cdots + 49 \) Copy content Toggle raw display
$59$ \( T^{12} \) Copy content Toggle raw display
$61$ \( T^{12} \) Copy content Toggle raw display
$67$ \( T^{12} + 14 T^{9} + 7 T^{7} + 63 T^{6} + \cdots + 49 \) Copy content Toggle raw display
$71$ \( (T^{6} + T^{5} + T^{4} + T^{3} + T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} \) Copy content Toggle raw display
$79$ \( T^{12} + 2 T^{11} - 8 T^{9} - 9 T^{8} + \cdots + 1 \) Copy content Toggle raw display
$83$ \( T^{12} \) Copy content Toggle raw display
$89$ \( T^{12} \) Copy content Toggle raw display
$97$ \( T^{12} \) Copy content Toggle raw display
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