Properties

Label 558.2.ba.f
Level $558$
Weight $2$
Character orbit 558.ba
Analytic conductor $4.456$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [558,2,Mod(19,558)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(558, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("558.19");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 558 = 2 \cdot 3^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 558.ba (of order \(15\), degree \(8\), 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: \(4.45565243279\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 62)
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-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 \(\zeta_{15}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \zeta_{15}^{6} q^{2} + ( - \zeta_{15}^{7} - \zeta_{15}^{2}) q^{4} + ( - \zeta_{15}^{7} - \zeta_{15}^{6} + \zeta_{15}^{5} - 2 \zeta_{15}^{4} + \zeta_{15}^{3} - \zeta_{15}^{2} - \zeta_{15} + 1) q^{5} + ( - \zeta_{15}^{6} + \zeta_{15}^{5} - \zeta_{15}^{4} + \zeta_{15}^{3} - \zeta_{15} + 1) q^{7} - \zeta_{15}^{3} q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{15}^{6} q^{2} + ( - \zeta_{15}^{7} - \zeta_{15}^{2}) q^{4} + ( - \zeta_{15}^{7} - \zeta_{15}^{6} + \zeta_{15}^{5} - 2 \zeta_{15}^{4} + \zeta_{15}^{3} - \zeta_{15}^{2} - \zeta_{15} + 1) q^{5} + ( - \zeta_{15}^{6} + \zeta_{15}^{5} - \zeta_{15}^{4} + \zeta_{15}^{3} - \zeta_{15} + 1) q^{7} - \zeta_{15}^{3} q^{8} + ( - \zeta_{15}^{7} + \zeta_{15}^{6} - 2 \zeta_{15}^{5} - 2 \zeta_{15}^{2} + \zeta_{15} - 1) q^{10} + ( - 2 \zeta_{15}^{7} + 2 \zeta_{15}^{6} + 2 \zeta_{15}^{5} + \zeta_{15}^{3} - 2 \zeta_{15}^{2} - \zeta_{15} + 4) q^{11} + ( - 4 \zeta_{15}^{7} + 2 \zeta_{15}^{6} - 2 \zeta_{15}^{4} + 2 \zeta_{15}^{3} - 4 \zeta_{15}^{2} + 2 \zeta_{15} + 2) q^{13} + ( - \zeta_{15}^{7} + \zeta_{15}^{6} - \zeta_{15}^{5} + \zeta_{15}^{3} - 2 \zeta_{15}^{2} + \zeta_{15}) q^{14} + (\zeta_{15}^{7} - \zeta_{15}^{6} - \zeta_{15}^{3} + \zeta_{15}^{2} - 1) q^{16} + (4 \zeta_{15}^{7} + \zeta_{15}^{6} - 3 \zeta_{15}^{5} + 5 \zeta_{15}^{4} + \zeta_{15}^{3} - 2 \zeta_{15}^{2} + 3 \zeta_{15} - 2) q^{17} + (3 \zeta_{15}^{7} - 2 \zeta_{15}^{6} - \zeta_{15}^{5} + 2 \zeta_{15}^{4} - 4 \zeta_{15}^{3} + 2 \zeta_{15}^{2} - 2 \zeta_{15} - 3) q^{19} + (\zeta_{15}^{7} - \zeta_{15}^{6} - \zeta_{15}^{5} + \zeta_{15}^{4} - 2 \zeta_{15}^{3} + \zeta_{15}^{2} - \zeta_{15} - 1) q^{20} + (2 \zeta_{15}^{7} - \zeta_{15}^{6} - \zeta_{15}^{3} + \zeta_{15}^{2} + 2 \zeta_{15} + 1) q^{22} + ( - \zeta_{15}^{6} + \zeta_{15}^{5} - \zeta_{15}^{4} - 4 \zeta_{15}^{3} - \zeta_{15}^{2} + \zeta_{15} - 1) q^{23} + ( - 2 \zeta_{15}^{7} + 4 \zeta_{15}^{6} - 2 \zeta_{15}^{5} + 2 \zeta_{15}^{4} - \zeta_{15}^{3} - 5 \zeta_{15}^{2} + 4 \zeta_{15}) q^{25} + ( - 2 \zeta_{15}^{7} - 2 \zeta_{15}^{5} - 2 \zeta_{15}^{3}) q^{26} + ( - \zeta_{15}^{5} + \zeta_{15}^{4} - \zeta_{15}^{3}) q^{28} + ( - 3 \zeta_{15}^{7} - 2 \zeta_{15}^{6} - 6 \zeta_{15}^{4} + 3 \zeta_{15}^{3} + 3 \zeta_{15}^{2} - 3 \zeta_{15}) q^{29} + ( - 3 \zeta_{15}^{7} + 4 \zeta_{15}^{6} - 3 \zeta_{15}^{5} - 3 \zeta_{15}^{4} + 3 \zeta_{15}^{3} - 6 \zeta_{15}^{2} + \cdots + 3) q^{31} + \cdots + ( - \zeta_{15}^{7} + 2 \zeta_{15}^{6} + 4 \zeta_{15}^{5} + 2 \zeta_{15}^{4} - \zeta_{15}^{3}) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{2} - 2 q^{4} - q^{5} + 2 q^{7} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{2} - 2 q^{4} - q^{5} + 2 q^{7} + 2 q^{8} - 4 q^{10} + 13 q^{11} - 2 q^{14} - 2 q^{16} + 2 q^{17} - 3 q^{19} + 4 q^{20} + 17 q^{22} - 3 q^{23} + q^{25} + 10 q^{26} + 7 q^{28} - 11 q^{29} + 11 q^{31} - 8 q^{32} + 18 q^{34} - 16 q^{35} + 6 q^{37} - 12 q^{38} - 4 q^{40} + 19 q^{41} - 26 q^{43} - 7 q^{44} - 17 q^{46} - q^{47} - 23 q^{49} + 19 q^{50} - 17 q^{53} + 7 q^{55} + 3 q^{56} - 19 q^{58} - 8 q^{59} + 48 q^{61} + 19 q^{62} - 2 q^{64} + 30 q^{65} + 12 q^{67} + 17 q^{68} + 16 q^{70} - 9 q^{71} - 33 q^{73} + 19 q^{74} - 18 q^{76} + 13 q^{77} - 19 q^{79} - 11 q^{80} + 21 q^{82} - 40 q^{83} + 29 q^{85} - 19 q^{86} - 3 q^{88} - 8 q^{89} + 20 q^{91} - 28 q^{92} + 26 q^{94} - 36 q^{95} - 23 q^{97} - 17 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/558\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(497\)
\(\chi(n)\) \(-1 + \zeta_{15} - \zeta_{15}^{3} + \zeta_{15}^{4} - \zeta_{15}^{5} + \zeta_{15}^{7}\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
−0.104528 + 0.994522i
−0.104528 0.994522i
0.913545 0.406737i
0.913545 + 0.406737i
−0.978148 0.207912i
0.669131 0.743145i
0.669131 + 0.743145i
−0.978148 + 0.207912i
0.809017 + 0.587785i 0 0.309017 + 0.951057i 0.204489 0.354185i 0 0.809017 0.898504i −0.309017 + 0.951057i 0 0.373619 0.166346i
235.1 0.809017 0.587785i 0 0.309017 0.951057i 0.204489 + 0.354185i 0 0.809017 + 0.898504i −0.309017 0.951057i 0 0.373619 + 0.166346i
289.1 0.809017 + 0.587785i 0 0.309017 + 0.951057i 1.22256 + 2.11754i 0 0.809017 + 0.171962i −0.309017 + 0.951057i 0 −0.255585 + 2.43173i
307.1 0.809017 0.587785i 0 0.309017 0.951057i 1.22256 2.11754i 0 0.809017 0.171962i −0.309017 0.951057i 0 −0.255585 2.43173i
361.1 −0.309017 0.951057i 0 −0.809017 + 0.587785i −1.78716 3.09546i 0 −0.309017 2.94010i 0.809017 + 0.587785i 0 −2.39169 + 2.65624i
379.1 −0.309017 0.951057i 0 −0.809017 + 0.587785i −0.139886 + 0.242290i 0 −0.309017 0.137583i 0.809017 + 0.587785i 0 0.273659 + 0.0581680i
505.1 −0.309017 + 0.951057i 0 −0.809017 0.587785i −0.139886 0.242290i 0 −0.309017 + 0.137583i 0.809017 0.587785i 0 0.273659 0.0581680i
541.1 −0.309017 + 0.951057i 0 −0.809017 0.587785i −1.78716 + 3.09546i 0 −0.309017 + 2.94010i 0.809017 0.587785i 0 −2.39169 2.65624i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.g even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 558.2.ba.f 8
3.b odd 2 1 62.2.g.a 8
12.b even 2 1 496.2.bg.a 8
31.g even 15 1 inner 558.2.ba.f 8
93.o odd 30 1 62.2.g.a 8
93.o odd 30 1 1922.2.a.q 4
93.p even 30 1 1922.2.a.o 4
372.bd even 30 1 496.2.bg.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
62.2.g.a 8 3.b odd 2 1
62.2.g.a 8 93.o odd 30 1
496.2.bg.a 8 12.b even 2 1
496.2.bg.a 8 372.bd even 30 1
558.2.ba.f 8 1.a even 1 1 trivial
558.2.ba.f 8 31.g even 15 1 inner
1922.2.a.o 4 93.p even 30 1
1922.2.a.q 4 93.o odd 30 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + T_{5}^{7} + 10T_{5}^{6} - 11T_{5}^{5} + 79T_{5}^{4} - 11T_{5}^{3} + 10T_{5}^{2} + T_{5} + 1 \) acting on \(S_{2}^{\mathrm{new}}(558, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{3} + T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + T^{7} + 10 T^{6} - 11 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{8} - 2 T^{7} + 10 T^{6} - 22 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{8} - 13 T^{7} + 100 T^{6} + \cdots + 3721 \) Copy content Toggle raw display
$13$ \( T^{8} - 40 T^{6} + 80 T^{5} + \cdots + 6400 \) Copy content Toggle raw display
$17$ \( T^{8} - 2 T^{7} - 60 T^{6} - 272 T^{5} + \cdots + 841 \) Copy content Toggle raw display
$19$ \( T^{8} + 3 T^{7} - 45 T^{6} - 162 T^{5} + \cdots + 81 \) Copy content Toggle raw display
$23$ \( T^{8} + 3 T^{7} + 53 T^{6} + \cdots + 32041 \) Copy content Toggle raw display
$29$ \( T^{8} + 11 T^{7} + 72 T^{6} + \cdots + 2430481 \) Copy content Toggle raw display
$31$ \( T^{8} - 11 T^{7} + 60 T^{6} + \cdots + 923521 \) Copy content Toggle raw display
$37$ \( T^{8} - 6 T^{7} + 70 T^{6} - 114 T^{5} + \cdots + 961 \) Copy content Toggle raw display
$41$ \( T^{8} - 19 T^{7} + 225 T^{6} + \cdots + 201601 \) Copy content Toggle raw display
$43$ \( T^{8} + 26 T^{7} + 240 T^{6} + \cdots + 703921 \) Copy content Toggle raw display
$47$ \( T^{8} + T^{7} + 72 T^{6} - 427 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$53$ \( T^{8} + 17 T^{7} + 100 T^{6} + \cdots + 32761 \) Copy content Toggle raw display
$59$ \( T^{8} + 8 T^{7} - 15 T^{6} + \cdots + 75672601 \) Copy content Toggle raw display
$61$ \( (T^{4} - 24 T^{3} + 141 T^{2} + 36 T - 279)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} - 12 T^{7} + 130 T^{6} + \cdots + 841 \) Copy content Toggle raw display
$71$ \( T^{8} + 9 T^{7} + 75 T^{6} + \cdots + 4626801 \) Copy content Toggle raw display
$73$ \( T^{8} + 33 T^{7} + 695 T^{6} + \cdots + 1515361 \) Copy content Toggle raw display
$79$ \( T^{8} + 19 T^{7} + 315 T^{6} + \cdots + 48427681 \) Copy content Toggle raw display
$83$ \( T^{8} + 40 T^{7} + 615 T^{6} + \cdots + 819025 \) Copy content Toggle raw display
$89$ \( T^{8} + 8 T^{7} + 33 T^{6} + 86 T^{5} + \cdots + 961 \) Copy content Toggle raw display
$97$ \( T^{8} + 23 T^{7} + \cdots + 154977601 \) Copy content Toggle raw display
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