Properties

Label 3479.1.cd.a
Level $3479$
Weight $1$
Character orbit 3479.cd
Analytic conductor $1.736$
Analytic rank $0$
Dimension $8$
Projective image $D_{10}$
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(1292,3479)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3479, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([20, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3479.1292");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3479 = 7^{2} \cdot 71 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3479.cd (of order \(30\), degree \(8\), 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: \(8\)
Coefficient field: \(\Q(\zeta_{15})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{10}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{10} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + ( - \zeta_{30}^{11} - \zeta_{30}^{5}) q^{2} + (\zeta_{30}^{10} - \zeta_{30}^{7} - \zeta_{30}) q^{4} + (\zeta_{30}^{12} + \zeta_{30}^{6} - \zeta_{30}^{3} + 1) q^{8} + \zeta_{30}^{11} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{30}^{11} - \zeta_{30}^{5}) q^{2} + (\zeta_{30}^{10} - \zeta_{30}^{7} - \zeta_{30}) q^{4} + (\zeta_{30}^{12} + \zeta_{30}^{6} - \zeta_{30}^{3} + 1) q^{8} + \zeta_{30}^{11} q^{9} + (\zeta_{30}^{7} + \zeta_{30}^{4}) q^{11} + (\zeta_{30}^{14} - \zeta_{30}^{11} + \zeta_{30}^{8} - \zeta_{30}^{5} + \zeta_{30}^{2}) q^{16} + (\zeta_{30}^{7} + \zeta_{30}) q^{18} + ( - \zeta_{30}^{12} - \zeta_{30}^{9} + \zeta_{30}^{3} + 1) q^{22} + (\zeta_{30}^{8} - \zeta_{30}^{2}) q^{23} + \zeta_{30}^{13} q^{25} + \zeta_{30}^{9} q^{29} + ( - \zeta_{30}^{13} + 2 \zeta_{30}^{10} - \zeta_{30}^{7} + \zeta_{30}^{4} - \zeta_{30}) q^{32} + ( - \zeta_{30}^{12} - \zeta_{30}^{6} + \zeta_{30}^{3}) q^{36} + ( - \zeta_{30}^{14} + \zeta_{30}^{11}) q^{37} + (\zeta_{30}^{12} + \zeta_{30}^{6}) q^{43} + ( - \zeta_{30}^{11} - \zeta_{30}^{8} - \zeta_{30}^{5} - \zeta_{30}^{2}) q^{44} + (\zeta_{30}^{7} + \zeta_{30}^{4}) q^{46} + (\zeta_{30}^{9} + \zeta_{30}^{3}) q^{50} + ( - \zeta_{30}^{7} + \zeta_{30}) q^{53} + ( - 2 \zeta_{30}^{14} + 2 \zeta_{30}^{5}) q^{58} + (\zeta_{30}^{12} - \zeta_{30}^{9} + 2 \zeta_{30}^{6} - \zeta_{30}^{3} - 1) q^{64} + ( - \zeta_{30}^{10} - \zeta_{30}^{7}) q^{67} - \zeta_{30}^{9} q^{71} + ( - \zeta_{30}^{14} + \zeta_{30}^{11} - \zeta_{30}^{8} - \zeta_{30}^{2}) q^{72} + ( - \zeta_{30}^{10} + \zeta_{30}^{7} - \zeta_{30}^{4} + \zeta_{30}) q^{74} + ( - \zeta_{30}^{8} + \zeta_{30}^{5}) q^{79} - \zeta_{30}^{7} q^{81} + ( - \zeta_{30}^{11} + \zeta_{30}^{8} + 2 \zeta_{30}^{2}) q^{86} + (\zeta_{30}^{13} - \zeta_{30}^{4} - \zeta_{30}) q^{88} + ( - \zeta_{30}^{12} - \zeta_{30}^{3} + 1) q^{92} + ( - \zeta_{30}^{3} - 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 3 q^{2} - 2 q^{4} + 2 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 3 q^{2} - 2 q^{4} + 2 q^{8} - q^{9} - 2 q^{18} + 10 q^{22} - q^{25} + 4 q^{29} - 4 q^{32} + 6 q^{36} - 2 q^{37} - 4 q^{43} - 5 q^{44} + 4 q^{50} + 6 q^{58} + 6 q^{64} + 5 q^{67} - 2 q^{71} - 4 q^{72} + q^{74} + 3 q^{79} + q^{81} + 4 q^{86} + 10 q^{92} - 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(640\) \(1569\)
\(\chi(n)\) \(-\zeta_{30}^{5}\) \(-\zeta_{30}^{6}\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1292.1
0.913545 0.406737i
0.669131 + 0.743145i
−0.104528 + 0.994522i
−0.978148 + 0.207912i
−0.978148 0.207912i
−0.104528 0.994522i
0.669131 0.743145i
0.913545 + 0.406737i
−0.604528 + 0.128496i 0 −0.564602 + 0.251377i 0 0 0 0.809017 0.587785i 0.104528 0.994522i 0
1537.1 −1.47815 0.658114i 0 1.08268 + 1.20243i 0 0 0 −0.309017 0.951057i 0.978148 0.207912i 0
1647.1 0.413545 + 0.459289i 0 0.0646021 0.614648i 0 0 0 0.809017 0.587785i −0.913545 + 0.406737i 0
1892.1 0.169131 + 1.60917i 0 −1.58268 + 0.336408i 0 0 0 −0.309017 0.951057i −0.669131 0.743145i 0
2076.1 0.169131 1.60917i 0 −1.58268 0.336408i 0 0 0 −0.309017 + 0.951057i −0.669131 + 0.743145i 0
2125.1 0.413545 0.459289i 0 0.0646021 + 0.614648i 0 0 0 0.809017 + 0.587785i −0.913545 0.406737i 0
2431.1 −1.47815 + 0.658114i 0 1.08268 1.20243i 0 0 0 −0.309017 + 0.951057i 0.978148 + 0.207912i 0
2480.1 −0.604528 0.128496i 0 −0.564602 0.251377i 0 0 0 0.809017 + 0.587785i 0.104528 + 0.994522i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1292.1
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.e odd 10 1 inner
497.m even 10 1 inner
497.s even 30 1 inner
497.u odd 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3479.1.cd.a 8
7.b odd 2 1 CM 3479.1.cd.a 8
7.c even 3 1 3479.1.r.a 4
7.c even 3 1 inner 3479.1.cd.a 8
7.d odd 6 1 3479.1.r.a 4
7.d odd 6 1 inner 3479.1.cd.a 8
71.e odd 10 1 inner 3479.1.cd.a 8
497.m even 10 1 inner 3479.1.cd.a 8
497.s even 30 1 3479.1.r.a 4
497.s even 30 1 inner 3479.1.cd.a 8
497.u odd 30 1 3479.1.r.a 4
497.u odd 30 1 inner 3479.1.cd.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3479.1.r.a 4 7.c even 3 1
3479.1.r.a 4 7.d odd 6 1
3479.1.r.a 4 497.s even 30 1
3479.1.r.a 4 497.u odd 30 1
3479.1.cd.a 8 1.a even 1 1 trivial
3479.1.cd.a 8 7.b odd 2 1 CM
3479.1.cd.a 8 7.c even 3 1 inner
3479.1.cd.a 8 7.d odd 6 1 inner
3479.1.cd.a 8 71.e odd 10 1 inner
3479.1.cd.a 8 497.m even 10 1 inner
3479.1.cd.a 8 497.s even 30 1 inner
3479.1.cd.a 8 497.u odd 30 1 inner

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 3 T^{7} + 5 T^{6} + 8 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} + 10 T^{5} - 5 T^{4} + 25 T^{2} + \cdots + 25 \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( T^{8} - 5 T^{6} + 20 T^{4} - 25 T^{2} + \cdots + 25 \) Copy content Toggle raw display
$29$ \( (T^{4} - 2 T^{3} + 4 T^{2} - 8 T + 16)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( (T^{4} + T^{3} + 2 T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} \) Copy content Toggle raw display
$43$ \( (T^{4} + 2 T^{3} + 4 T^{2} + 3 T + 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} + 10 T^{5} - 5 T^{4} + 25 T^{2} + \cdots + 25 \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( T^{8} - 5 T^{7} + 15 T^{6} - 30 T^{5} + \cdots + 25 \) Copy content Toggle raw display
$71$ \( (T^{4} + T^{3} + T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( T^{8} - 3 T^{7} + 5 T^{6} - 8 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( T^{8} \) Copy content Toggle raw display
$97$ \( T^{8} \) Copy content Toggle raw display
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