Properties

Label 58.2.d.a
Level $58$
Weight $2$
Character orbit 58.d
Analytic conductor $0.463$
Analytic rank $0$
Dimension $6$
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(7,58)]
 
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([6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("58.7");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 58 = 2 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 58.d (of order \(7\), 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: \(6\)
Coefficient field: \(\Q(\zeta_{14})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-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 \(\zeta_{14}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \zeta_{14}^{4} q^{2} + ( - \zeta_{14}^{5} + \zeta_{14}^{4} + \cdots + 1) q^{3} + \cdots + (\zeta_{14}^{5} + \zeta_{14}^{3} + \cdots - 1) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{14}^{4} q^{2} + ( - \zeta_{14}^{5} + \zeta_{14}^{4} + \cdots + 1) q^{3} + \cdots + (3 \zeta_{14}^{5} - 3 \zeta_{14}^{4} + \cdots + 6) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + q^{2} + 3 q^{3} - q^{4} - 4 q^{5} + 4 q^{6} - 5 q^{7} + q^{8} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + q^{2} + 3 q^{3} - q^{4} - 4 q^{5} + 4 q^{6} - 5 q^{7} + q^{8} - 6 q^{9} - 3 q^{10} - 6 q^{11} - 4 q^{12} + 11 q^{13} - 9 q^{14} - 9 q^{15} - q^{16} - 10 q^{17} + 6 q^{18} - 6 q^{19} - 4 q^{20} + 15 q^{21} + 6 q^{22} - 7 q^{23} + 4 q^{24} + 17 q^{25} + 3 q^{26} + 9 q^{27} + 2 q^{28} + 15 q^{29} + 2 q^{30} + 4 q^{31} + q^{32} - 3 q^{33} + 10 q^{34} + 22 q^{35} + q^{36} - 13 q^{37} - 15 q^{38} + 2 q^{39} + 4 q^{40} - 12 q^{41} - 15 q^{42} - 7 q^{43} + 15 q^{44} - 3 q^{45} + 3 q^{47} + 3 q^{48} - 18 q^{49} + 4 q^{50} - 5 q^{51} - 10 q^{52} - 22 q^{53} + 12 q^{54} - 24 q^{55} - 9 q^{56} + 18 q^{57} - 15 q^{58} - 38 q^{59} - 9 q^{60} + 31 q^{61} - 18 q^{62} - 2 q^{63} - q^{64} + 9 q^{65} - 18 q^{66} + 11 q^{67} + 11 q^{68} - 28 q^{69} + 6 q^{70} + 23 q^{71} - 8 q^{72} + 5 q^{73} + 13 q^{74} + 26 q^{75} + 15 q^{76} + 12 q^{77} + 19 q^{78} + 3 q^{80} - 23 q^{81} - 16 q^{82} + 35 q^{83} + q^{84} - 26 q^{85} + 42 q^{86} + 11 q^{87} + 6 q^{88} + 13 q^{89} + 10 q^{90} - q^{91} - 7 q^{92} - 12 q^{93} + 4 q^{94} + 25 q^{95} - 3 q^{96} + 20 q^{97} + 18 q^{98} + 48 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(31\)
\(\chi(n)\) \(-\zeta_{14}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
0.900969 0.433884i
0.222521 + 0.974928i
0.900969 + 0.433884i
−0.623490 0.781831i
−0.623490 + 0.781831i
0.222521 0.974928i
0.222521 + 0.974928i 0.500000 + 0.240787i −0.900969 + 0.433884i −0.0440730 0.193096i −0.123490 + 0.541044i −0.0990311 0.0476909i −0.623490 0.781831i −1.67845 2.10471i 0.178448 0.0859360i
23.1 −0.623490 + 0.781831i 0.500000 2.19064i −0.222521 0.974928i 0.969501 1.21572i 1.40097 + 1.75676i −0.777479 + 3.40636i 0.900969 + 0.433884i −1.84601 0.888992i 0.346011 + 1.51597i
25.1 0.222521 0.974928i 0.500000 0.240787i −0.900969 0.433884i −0.0440730 + 0.193096i −0.123490 0.541044i −0.0990311 + 0.0476909i −0.623490 + 0.781831i −1.67845 + 2.10471i 0.178448 + 0.0859360i
45.1 0.900969 + 0.433884i 0.500000 0.626980i 0.623490 + 0.781831i −2.92543 1.40881i 0.722521 0.347948i −1.62349 + 2.03579i 0.222521 + 0.974928i 0.524459 + 2.29780i −2.02446 2.53859i
49.1 0.900969 0.433884i 0.500000 + 0.626980i 0.623490 0.781831i −2.92543 + 1.40881i 0.722521 + 0.347948i −1.62349 2.03579i 0.222521 0.974928i 0.524459 2.29780i −2.02446 + 2.53859i
53.1 −0.623490 0.781831i 0.500000 + 2.19064i −0.222521 + 0.974928i 0.969501 + 1.21572i 1.40097 1.75676i −0.777479 3.40636i 0.900969 0.433884i −1.84601 + 0.888992i 0.346011 1.51597i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.d even 7 1 inner

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} - 3T_{3}^{5} + 9T_{3}^{4} - 13T_{3}^{3} + 11T_{3}^{2} - 5T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(58, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - T^{5} + T^{4} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{6} - 3 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{6} + 4 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{6} + 5 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{6} + 6 T^{5} + \cdots + 729 \) Copy content Toggle raw display
$13$ \( T^{6} - 11 T^{5} + \cdots + 841 \) Copy content Toggle raw display
$17$ \( (T^{3} + 5 T^{2} - 8 T - 41)^{2} \) Copy content Toggle raw display
$19$ \( T^{6} + 6 T^{5} + \cdots + 1681 \) Copy content Toggle raw display
$23$ \( T^{6} + 7 T^{5} + \cdots + 8281 \) Copy content Toggle raw display
$29$ \( T^{6} - 15 T^{5} + \cdots + 24389 \) Copy content Toggle raw display
$31$ \( T^{6} - 4 T^{5} + \cdots + 1849 \) Copy content Toggle raw display
$37$ \( T^{6} + 13 T^{5} + \cdots + 19321 \) Copy content Toggle raw display
$41$ \( (T^{3} + 6 T^{2} - 37 T + 1)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 7 T^{5} + \cdots + 41209 \) Copy content Toggle raw display
$47$ \( T^{6} - 3 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$53$ \( T^{6} + 22 T^{5} + \cdots + 113569 \) Copy content Toggle raw display
$59$ \( (T^{3} + 19 T^{2} + \cdots + 169)^{2} \) Copy content Toggle raw display
$61$ \( T^{6} - 31 T^{5} + \cdots + 169 \) Copy content Toggle raw display
$67$ \( T^{6} - 11 T^{5} + \cdots + 1771561 \) Copy content Toggle raw display
$71$ \( T^{6} - 23 T^{5} + \cdots + 63001 \) Copy content Toggle raw display
$73$ \( T^{6} - 5 T^{5} + \cdots + 241081 \) Copy content Toggle raw display
$79$ \( T^{6} + 56 T^{4} + \cdots + 3136 \) Copy content Toggle raw display
$83$ \( T^{6} - 35 T^{5} + \cdots + 790321 \) Copy content Toggle raw display
$89$ \( T^{6} - 13 T^{5} + \cdots + 253009 \) Copy content Toggle raw display
$97$ \( T^{6} - 20 T^{5} + \cdots + 10816 \) Copy content Toggle raw display
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