Properties

Label 570.6.a.a
Level $570$
Weight $6$
Character orbit 570.a
Self dual yes
Analytic conductor $91.419$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [570,6,Mod(1,570)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(570, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("570.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 570.a (trivial)

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: yes
Analytic conductor: \(91.4187772934\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - 4 q^{2} + 9 q^{3} + 16 q^{4} - 25 q^{5} - 36 q^{6} - 82 q^{7} - 64 q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 4 q^{2} + 9 q^{3} + 16 q^{4} - 25 q^{5} - 36 q^{6} - 82 q^{7} - 64 q^{8} + 81 q^{9} + 100 q^{10} + 228 q^{11} + 144 q^{12} - 232 q^{13} + 328 q^{14} - 225 q^{15} + 256 q^{16} - 546 q^{17} - 324 q^{18} + 361 q^{19} - 400 q^{20} - 738 q^{21} - 912 q^{22} + 2622 q^{23} - 576 q^{24} + 625 q^{25} + 928 q^{26} + 729 q^{27} - 1312 q^{28} - 3222 q^{29} + 900 q^{30} + 5408 q^{31} - 1024 q^{32} + 2052 q^{33} + 2184 q^{34} + 2050 q^{35} + 1296 q^{36} - 112 q^{37} - 1444 q^{38} - 2088 q^{39} + 1600 q^{40} - 8946 q^{41} + 2952 q^{42} + 10730 q^{43} + 3648 q^{44} - 2025 q^{45} - 10488 q^{46} + 14478 q^{47} + 2304 q^{48} - 10083 q^{49} - 2500 q^{50} - 4914 q^{51} - 3712 q^{52} + 1044 q^{53} - 2916 q^{54} - 5700 q^{55} + 5248 q^{56} + 3249 q^{57} + 12888 q^{58} - 46284 q^{59} - 3600 q^{60} - 44506 q^{61} - 21632 q^{62} - 6642 q^{63} + 4096 q^{64} + 5800 q^{65} - 8208 q^{66} + 21260 q^{67} - 8736 q^{68} + 23598 q^{69} - 8200 q^{70} + 61560 q^{71} - 5184 q^{72} - 87154 q^{73} + 448 q^{74} + 5625 q^{75} + 5776 q^{76} - 18696 q^{77} + 8352 q^{78} - 57688 q^{79} - 6400 q^{80} + 6561 q^{81} + 35784 q^{82} - 36690 q^{83} - 11808 q^{84} + 13650 q^{85} - 42920 q^{86} - 28998 q^{87} - 14592 q^{88} + 92190 q^{89} + 8100 q^{90} + 19024 q^{91} + 41952 q^{92} + 48672 q^{93} - 57912 q^{94} - 9025 q^{95} - 9216 q^{96} + 114692 q^{97} + 40332 q^{98} + 18468 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
−4.00000 9.00000 16.0000 −25.0000 −36.0000 −82.0000 −64.0000 81.0000 100.000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(5\) \(1\)
\(19\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 570.6.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.6.a.a 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} + 82 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(570))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 4 \) Copy content Toggle raw display
$3$ \( T - 9 \) Copy content Toggle raw display
$5$ \( T + 25 \) Copy content Toggle raw display
$7$ \( T + 82 \) Copy content Toggle raw display
$11$ \( T - 228 \) Copy content Toggle raw display
$13$ \( T + 232 \) Copy content Toggle raw display
$17$ \( T + 546 \) Copy content Toggle raw display
$19$ \( T - 361 \) Copy content Toggle raw display
$23$ \( T - 2622 \) Copy content Toggle raw display
$29$ \( T + 3222 \) Copy content Toggle raw display
$31$ \( T - 5408 \) Copy content Toggle raw display
$37$ \( T + 112 \) Copy content Toggle raw display
$41$ \( T + 8946 \) Copy content Toggle raw display
$43$ \( T - 10730 \) Copy content Toggle raw display
$47$ \( T - 14478 \) Copy content Toggle raw display
$53$ \( T - 1044 \) Copy content Toggle raw display
$59$ \( T + 46284 \) Copy content Toggle raw display
$61$ \( T + 44506 \) Copy content Toggle raw display
$67$ \( T - 21260 \) Copy content Toggle raw display
$71$ \( T - 61560 \) Copy content Toggle raw display
$73$ \( T + 87154 \) Copy content Toggle raw display
$79$ \( T + 57688 \) Copy content Toggle raw display
$83$ \( T + 36690 \) Copy content Toggle raw display
$89$ \( T - 92190 \) Copy content Toggle raw display
$97$ \( T - 114692 \) Copy content Toggle raw display
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