Properties

Label 58.2.e.a
Level $58$
Weight $2$
Character orbit 58.e
Analytic conductor $0.463$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [58,2,Mod(5,58)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(58, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([11]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("58.5");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 58 = 2 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 58.e (of order \(14\), degree \(6\), 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: \(0.463132331723\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{14})\)
Coefficient field: \(\Q(\zeta_{28})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

$q$-expansion

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

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{28}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \zeta_{28}^{9} q^{2} + ( - \zeta_{28}^{5} - \zeta_{28}^{3} - \zeta_{28}) q^{3} - \zeta_{28}^{4} q^{4} + ( - \zeta_{28}^{11} + \cdots + \zeta_{28}) q^{5} + \cdots + (\zeta_{28}^{10} + \cdots + \zeta_{28}^{2}) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{28}^{9} q^{2} + ( - \zeta_{28}^{5} - \zeta_{28}^{3} - \zeta_{28}) q^{3} - \zeta_{28}^{4} q^{4} + ( - \zeta_{28}^{11} + \cdots + \zeta_{28}) q^{5} + \cdots + (3 \zeta_{28}^{11} + 2 \zeta_{28}^{10} + \cdots + 2) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{4} - 2 q^{5} - 12 q^{6} + 4 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 2 q^{4} - 2 q^{5} - 12 q^{6} + 4 q^{7} - 4 q^{9} - 26 q^{13} - 14 q^{15} - 2 q^{16} + 2 q^{20} + 14 q^{21} + 4 q^{22} - 16 q^{23} + 12 q^{24} + 22 q^{25} + 14 q^{26} - 4 q^{28} + 18 q^{29} + 16 q^{30} + 28 q^{31} - 10 q^{33} - 4 q^{34} + 4 q^{35} + 18 q^{36} - 28 q^{37} + 22 q^{38} + 28 q^{39} - 14 q^{40} - 4 q^{42} - 28 q^{43} - 28 q^{44} - 4 q^{45} - 14 q^{47} - 8 q^{49} - 28 q^{50} - 4 q^{51} - 2 q^{52} + 30 q^{53} - 32 q^{54} + 28 q^{55} - 20 q^{57} - 10 q^{58} - 48 q^{59} - 14 q^{60} - 28 q^{61} + 10 q^{62} + 22 q^{63} + 2 q^{64} + 2 q^{65} - 16 q^{67} - 14 q^{68} - 28 q^{69} + 30 q^{71} + 42 q^{73} + 2 q^{74} + 28 q^{76} + 14 q^{77} - 16 q^{78} + 28 q^{79} - 2 q^{80} + 26 q^{81} - 20 q^{82} + 28 q^{85} + 52 q^{86} + 4 q^{87} - 4 q^{88} + 14 q^{89} + 70 q^{90} - 4 q^{91} + 16 q^{92} + 66 q^{93} - 8 q^{94} + 2 q^{96} - 28 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/58\mathbb{Z}\right)^\times\).

\(n\) \(31\)
\(\chi(n)\) \(-\zeta_{28}^{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
−0.433884 + 0.900969i
0.433884 0.900969i
−0.974928 + 0.222521i
0.974928 0.222521i
−0.974928 0.222521i
0.974928 + 0.222521i
0.781831 0.623490i
−0.781831 + 0.623490i
−0.433884 0.900969i
0.433884 + 0.900969i
0.781831 + 0.623490i
−0.781831 0.623490i
−0.781831 + 0.623490i 0.240787 0.0549581i 0.222521 0.974928i 2.13946 + 2.68280i −0.153989 + 0.193096i −0.349501 1.53126i 0.433884 + 0.900969i −2.64795 + 1.27518i −3.34540 0.763565i
5.2 0.781831 0.623490i −0.240787 + 0.0549581i 0.222521 0.974928i −0.892482 1.11914i −0.153989 + 0.193096i 0.904459 + 3.96269i −0.433884 0.900969i −2.64795 + 1.27518i −1.39554 0.318523i
9.1 −0.433884 0.900969i 2.19064 1.74698i −0.623490 + 0.781831i −3.43956 + 1.65640i −2.52446 1.21572i 1.36428 + 1.71075i 0.974928 + 0.222521i 1.07942 4.72923i 2.98474 + 2.38025i
9.2 0.433884 + 0.900969i −2.19064 + 1.74698i −0.623490 + 0.781831i 1.63762 0.788637i −2.52446 1.21572i 0.882702 + 1.10687i −0.974928 0.222521i 1.07942 4.72923i 1.42108 + 1.13327i
13.1 −0.433884 + 0.900969i 2.19064 + 1.74698i −0.623490 0.781831i −3.43956 1.65640i −2.52446 + 1.21572i 1.36428 1.71075i 0.974928 0.222521i 1.07942 + 4.72923i 2.98474 2.38025i
13.2 0.433884 0.900969i −2.19064 1.74698i −0.623490 0.781831i 1.63762 + 0.788637i −2.52446 + 1.21572i 0.882702 1.10687i −0.974928 + 0.222521i 1.07942 + 4.72923i 1.42108 1.13327i
33.1 −0.974928 0.222521i 0.626980 + 1.30194i 0.900969 + 0.433884i −0.308457 + 1.35144i −0.321552 1.40881i 1.78967 0.861862i −0.781831 0.623490i 0.568532 0.712916i 0.601447 1.24892i
33.2 0.974928 + 0.222521i −0.626980 1.30194i 0.900969 + 0.433884i −0.136585 + 0.598418i −0.321552 1.40881i −2.59161 + 1.24805i 0.781831 + 0.623490i 0.568532 0.712916i −0.266321 + 0.553021i
35.1 −0.781831 0.623490i 0.240787 + 0.0549581i 0.222521 + 0.974928i 2.13946 2.68280i −0.153989 0.193096i −0.349501 + 1.53126i 0.433884 0.900969i −2.64795 1.27518i −3.34540 + 0.763565i
35.2 0.781831 + 0.623490i −0.240787 0.0549581i 0.222521 + 0.974928i −0.892482 + 1.11914i −0.153989 0.193096i 0.904459 3.96269i −0.433884 + 0.900969i −2.64795 1.27518i −1.39554 + 0.318523i
51.1 −0.974928 + 0.222521i 0.626980 1.30194i 0.900969 0.433884i −0.308457 1.35144i −0.321552 + 1.40881i 1.78967 + 0.861862i −0.781831 + 0.623490i 0.568532 + 0.712916i 0.601447 + 1.24892i
51.2 0.974928 0.222521i −0.626980 + 1.30194i 0.900969 0.433884i −0.136585 0.598418i −0.321552 + 1.40881i −2.59161 1.24805i 0.781831 0.623490i 0.568532 + 0.712916i −0.266321 0.553021i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.e even 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 58.2.e.a 12
3.b odd 2 1 522.2.n.a 12
4.b odd 2 1 464.2.y.c 12
29.d even 7 1 1682.2.b.j 12
29.e even 14 1 inner 58.2.e.a 12
29.e even 14 1 1682.2.b.j 12
29.f odd 28 1 1682.2.a.r 6
29.f odd 28 1 1682.2.a.s 6
87.h odd 14 1 522.2.n.a 12
116.h odd 14 1 464.2.y.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
58.2.e.a 12 1.a even 1 1 trivial
58.2.e.a 12 29.e even 14 1 inner
464.2.y.c 12 4.b odd 2 1
464.2.y.c 12 116.h odd 14 1
522.2.n.a 12 3.b odd 2 1
522.2.n.a 12 87.h odd 14 1
1682.2.a.r 6 29.f odd 28 1
1682.2.a.s 6 29.f odd 28 1
1682.2.b.j 12 29.d even 7 1
1682.2.b.j 12 29.e even 14 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(58, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - T^{10} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{12} - T^{10} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{12} + 2 T^{11} + \cdots + 841 \) Copy content Toggle raw display
$7$ \( T^{12} - 4 T^{11} + \cdots + 12769 \) Copy content Toggle raw display
$11$ \( T^{12} - 4 T^{10} + \cdots + 841 \) Copy content Toggle raw display
$13$ \( T^{12} + 26 T^{11} + \cdots + 38809 \) Copy content Toggle raw display
$17$ \( T^{12} + 94 T^{10} + \cdots + 12769 \) Copy content Toggle raw display
$19$ \( T^{12} + 26 T^{10} + \cdots + 175561 \) Copy content Toggle raw display
$23$ \( T^{12} + 16 T^{11} + \cdots + 729 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 594823321 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 5377435561 \) Copy content Toggle raw display
$37$ \( T^{12} + 28 T^{11} + \cdots + 3736489 \) Copy content Toggle raw display
$41$ \( T^{12} + 236 T^{10} + \cdots + 60171049 \) Copy content Toggle raw display
$43$ \( T^{12} + 28 T^{11} + \cdots + 8637721 \) Copy content Toggle raw display
$47$ \( T^{12} + 14 T^{11} + \cdots + 24571849 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 2202143329 \) Copy content Toggle raw display
$59$ \( (T^{6} + 24 T^{5} + \cdots + 35869)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 105805126729 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 2228500849 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 15150901921 \) Copy content Toggle raw display
$73$ \( T^{12} - 42 T^{11} + \cdots + 5938969 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 4893282304 \) Copy content Toggle raw display
$83$ \( T^{12} - 35 T^{10} + \cdots + 2019241 \) Copy content Toggle raw display
$89$ \( T^{12} - 14 T^{11} + \cdots + 34656769 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 10446475264 \) Copy content Toggle raw display
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