Properties

Label 570.2.u.h
Level $570$
Weight $2$
Character orbit 570.u
Analytic conductor $4.551$
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] = [570,2,Mod(61,570)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(570, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("570.61");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 570.u (of order \(9\), 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: \(4.55147291521\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-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 \(\zeta_{18}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{18}^{4} + \zeta_{18}) q^{2} - \zeta_{18} q^{3} - \zeta_{18}^{5} q^{4} - \zeta_{18}^{4} q^{5} + (\zeta_{18}^{5} - \zeta_{18}^{2}) q^{6} + (\zeta_{18}^{5} + \zeta_{18}^{4} + \cdots + 3) q^{7} + \cdots + \zeta_{18}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{18}^{4} + \zeta_{18}) q^{2} - \zeta_{18} q^{3} - \zeta_{18}^{5} q^{4} - \zeta_{18}^{4} q^{5} + (\zeta_{18}^{5} - \zeta_{18}^{2}) q^{6} + (\zeta_{18}^{5} + \zeta_{18}^{4} + \cdots + 3) q^{7} + \cdots + ( - 3 \zeta_{18}^{5} - \zeta_{18} + 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 9 q^{7} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 9 q^{7} - 3 q^{8} - 9 q^{11} - 3 q^{12} + 9 q^{13} - 3 q^{14} - 9 q^{17} + 6 q^{18} - 9 q^{19} - 6 q^{20} + 6 q^{21} - 3 q^{22} + 3 q^{23} - 3 q^{27} + 6 q^{28} - 9 q^{29} + 3 q^{30} + 18 q^{31} + 6 q^{33} + 9 q^{34} + 3 q^{35} - 6 q^{37} - 6 q^{41} - 3 q^{42} + 21 q^{43} - 3 q^{44} + 3 q^{45} + 3 q^{46} - 12 q^{49} - 3 q^{50} - 9 q^{52} + 12 q^{53} + 3 q^{55} - 18 q^{56} + 18 q^{57} - 6 q^{58} + 9 q^{59} + 3 q^{61} - 24 q^{62} - 3 q^{63} - 3 q^{64} - 3 q^{66} + 36 q^{67} + 6 q^{68} + 3 q^{69} + 3 q^{70} - 36 q^{71} + 21 q^{73} - 21 q^{74} + 6 q^{75} - 12 q^{76} - 60 q^{77} - 9 q^{79} - 6 q^{82} - 3 q^{83} + 9 q^{84} + 9 q^{85} + 3 q^{86} + 3 q^{87} - 9 q^{88} + 3 q^{89} + 27 q^{91} - 15 q^{92} + 3 q^{93} - 18 q^{94} + 18 q^{95} + 6 q^{96} + 15 q^{98} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/570\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(211\) \(457\)
\(\chi(n)\) \(1\) \(-\zeta_{18}^{5}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
61.1
−0.173648 0.984808i
−0.173648 + 0.984808i
0.939693 0.342020i
0.939693 + 0.342020i
−0.766044 0.642788i
−0.766044 + 0.642788i
−0.939693 0.342020i 0.173648 + 0.984808i 0.766044 + 0.642788i −0.766044 + 0.642788i 0.173648 0.984808i 2.43969 4.22567i −0.500000 0.866025i −0.939693 + 0.342020i 0.939693 0.342020i
271.1 −0.939693 + 0.342020i 0.173648 0.984808i 0.766044 0.642788i −0.766044 0.642788i 0.173648 + 0.984808i 2.43969 + 4.22567i −0.500000 + 0.866025i −0.939693 0.342020i 0.939693 + 0.342020i
301.1 0.766044 + 0.642788i −0.939693 + 0.342020i 0.173648 + 0.984808i −0.173648 + 0.984808i −0.939693 0.342020i 0.733956 + 1.27125i −0.500000 + 0.866025i 0.766044 0.642788i −0.766044 + 0.642788i
481.1 0.766044 0.642788i −0.939693 0.342020i 0.173648 0.984808i −0.173648 0.984808i −0.939693 + 0.342020i 0.733956 1.27125i −0.500000 0.866025i 0.766044 + 0.642788i −0.766044 0.642788i
511.1 0.173648 0.984808i 0.766044 + 0.642788i −0.939693 0.342020i 0.939693 0.342020i 0.766044 0.642788i 1.32635 + 2.29731i −0.500000 + 0.866025i 0.173648 + 0.984808i −0.173648 0.984808i
541.1 0.173648 + 0.984808i 0.766044 0.642788i −0.939693 + 0.342020i 0.939693 + 0.342020i 0.766044 + 0.642788i 1.32635 2.29731i −0.500000 0.866025i 0.173648 0.984808i −0.173648 + 0.984808i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 61.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 570.2.u.h 6
19.e even 9 1 inner 570.2.u.h 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.u.h 6 1.a even 1 1 trivial
570.2.u.h 6 19.e even 9 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(570, [\chi])\):

\( T_{7}^{6} - 9T_{7}^{5} + 57T_{7}^{4} - 178T_{7}^{3} + 405T_{7}^{2} - 456T_{7} + 361 \) Copy content Toggle raw display
\( T_{11}^{6} + 9T_{11}^{5} + 57T_{11}^{4} + 182T_{11}^{3} + 423T_{11}^{2} + 408T_{11} + 289 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + T^{3} + 1 \) Copy content Toggle raw display
$3$ \( T^{6} + T^{3} + 1 \) Copy content Toggle raw display
$5$ \( T^{6} - T^{3} + 1 \) Copy content Toggle raw display
$7$ \( T^{6} - 9 T^{5} + \cdots + 361 \) Copy content Toggle raw display
$11$ \( T^{6} + 9 T^{5} + \cdots + 289 \) Copy content Toggle raw display
$13$ \( T^{6} - 9 T^{5} + \cdots + 81 \) Copy content Toggle raw display
$17$ \( T^{6} + 9 T^{5} + \cdots + 361 \) Copy content Toggle raw display
$19$ \( T^{6} + 9 T^{5} + \cdots + 6859 \) Copy content Toggle raw display
$23$ \( T^{6} - 3 T^{5} + \cdots + 3249 \) Copy content Toggle raw display
$29$ \( T^{6} + 9 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$31$ \( T^{6} - 18 T^{5} + \cdots + 83521 \) Copy content Toggle raw display
$37$ \( (T^{3} + 3 T^{2} + \cdots - 163)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} + 6 T^{5} + \cdots + 11881 \) Copy content Toggle raw display
$43$ \( T^{6} - 21 T^{5} + \cdots + 1369 \) Copy content Toggle raw display
$47$ \( T^{6} + 36 T^{4} + \cdots + 81 \) Copy content Toggle raw display
$53$ \( T^{6} - 12 T^{5} + \cdots + 25281 \) Copy content Toggle raw display
$59$ \( T^{6} - 9 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$61$ \( T^{6} - 3 T^{5} + \cdots + 72361 \) Copy content Toggle raw display
$67$ \( T^{6} - 36 T^{5} + \cdots + 908209 \) Copy content Toggle raw display
$71$ \( T^{6} + 36 T^{5} + \cdots + 641601 \) Copy content Toggle raw display
$73$ \( T^{6} - 21 T^{5} + \cdots + 1819801 \) Copy content Toggle raw display
$79$ \( T^{6} + 9 T^{5} + \cdots + 5041 \) Copy content Toggle raw display
$83$ \( T^{6} + 3 T^{5} + \cdots + 5329 \) Copy content Toggle raw display
$89$ \( T^{6} - 3 T^{5} + \cdots + 47961 \) Copy content Toggle raw display
$97$ \( T^{6} + 343 T^{3} + 117649 \) Copy content Toggle raw display
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