Properties

Label 570.2.f.a
Level $570$
Weight $2$
Character orbit 570.f
Analytic conductor $4.551$
Analytic rank $0$
Dimension $4$
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(341,570)]
 
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([1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("570.341");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 570.f (of order \(2\), degree \(1\), 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: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + ( - \beta_{2} - 1) q^{3} + q^{4} + \beta_1 q^{5} + (\beta_{2} + 1) q^{6} + (\beta_{3} + 2) q^{7} - q^{8} + (2 \beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + ( - \beta_{2} - 1) q^{3} + q^{4} + \beta_1 q^{5} + (\beta_{2} + 1) q^{6} + (\beta_{3} + 2) q^{7} - q^{8} + (2 \beta_{2} - 1) q^{9} - \beta_1 q^{10} + (\beta_{2} - 4 \beta_1) q^{11} + ( - \beta_{2} - 1) q^{12} + (3 \beta_{2} + 2 \beta_1) q^{13} + ( - \beta_{3} - 2) q^{14} + (\beta_{3} - \beta_1) q^{15} + q^{16} + 2 \beta_{2} q^{17} + ( - 2 \beta_{2} + 1) q^{18} + ( - 3 \beta_{3} - \beta_1) q^{19} + \beta_1 q^{20} + ( - \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 2) q^{21} + ( - \beta_{2} + 4 \beta_1) q^{22} + 6 \beta_1 q^{23} + (\beta_{2} + 1) q^{24} - q^{25} + ( - 3 \beta_{2} - 2 \beta_1) q^{26} + ( - \beta_{2} + 5) q^{27} + (\beta_{3} + 2) q^{28} + (5 \beta_{3} + 2) q^{29} + ( - \beta_{3} + \beta_1) q^{30} + (2 \beta_{2} - 4 \beta_1) q^{31} - q^{32} + ( - 4 \beta_{3} - \beta_{2} + 4 \beta_1 + 2) q^{33} - 2 \beta_{2} q^{34} + (\beta_{2} + 2 \beta_1) q^{35} + (2 \beta_{2} - 1) q^{36} + (3 \beta_{2} + 2 \beta_1) q^{37} + (3 \beta_{3} + \beta_1) q^{38} + (2 \beta_{3} - 3 \beta_{2} - 2 \beta_1 + 6) q^{39} - \beta_1 q^{40} + ( - 5 \beta_{3} + 4) q^{41} + (\beta_{3} + 2 \beta_{2} + 2 \beta_1 + 2) q^{42} + (\beta_{3} + 8) q^{43} + (\beta_{2} - 4 \beta_1) q^{44} + ( - 2 \beta_{3} - \beta_1) q^{45} - 6 \beta_1 q^{46} + ( - 2 \beta_{2} - 6 \beta_1) q^{47} + ( - \beta_{2} - 1) q^{48} + (4 \beta_{3} - 1) q^{49} + q^{50} + ( - 2 \beta_{2} + 4) q^{51} + (3 \beta_{2} + 2 \beta_1) q^{52} + (2 \beta_{3} + 6) q^{53} + (\beta_{2} - 5) q^{54} + ( - \beta_{3} + 4) q^{55} + ( - \beta_{3} - 2) q^{56} + (2 \beta_{3} + 7 \beta_1) q^{57} + ( - 5 \beta_{3} - 2) q^{58} + ( - 2 \beta_{3} + 10) q^{59} + (\beta_{3} - \beta_1) q^{60} + (6 \beta_{3} + 4) q^{61} + ( - 2 \beta_{2} + 4 \beta_1) q^{62} + ( - \beta_{3} + 4 \beta_{2} + 4 \beta_1 - 2) q^{63} + q^{64} + ( - 3 \beta_{3} - 2) q^{65} + (4 \beta_{3} + \beta_{2} - 4 \beta_1 - 2) q^{66} + 2 \beta_{2} q^{68} + (6 \beta_{3} - 6 \beta_1) q^{69} + ( - \beta_{2} - 2 \beta_1) q^{70} - 2 \beta_{3} q^{71} + ( - 2 \beta_{2} + 1) q^{72} + ( - 6 \beta_{3} - 6) q^{73} + ( - 3 \beta_{2} - 2 \beta_1) q^{74} + (\beta_{2} + 1) q^{75} + ( - 3 \beta_{3} - \beta_1) q^{76} + ( - 2 \beta_{2} - 6 \beta_1) q^{77} + ( - 2 \beta_{3} + 3 \beta_{2} + \cdots - 6) q^{78}+ \cdots + (8 \beta_{3} - \beta_{2} + 4 \beta_1 - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 4 q^{6} + 8 q^{7} - 4 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 4 q^{6} + 8 q^{7} - 4 q^{8} - 4 q^{9} - 4 q^{12} - 8 q^{14} + 4 q^{16} + 4 q^{18} - 8 q^{21} + 4 q^{24} - 4 q^{25} + 20 q^{27} + 8 q^{28} + 8 q^{29} - 4 q^{32} + 8 q^{33} - 4 q^{36} + 24 q^{39} + 16 q^{41} + 8 q^{42} + 32 q^{43} - 4 q^{48} - 4 q^{49} + 4 q^{50} + 16 q^{51} + 24 q^{53} - 20 q^{54} + 16 q^{55} - 8 q^{56} - 8 q^{58} + 40 q^{59} + 16 q^{61} - 8 q^{63} + 4 q^{64} - 8 q^{65} - 8 q^{66} + 4 q^{72} - 24 q^{73} + 4 q^{75} - 24 q^{78} - 28 q^{81} - 16 q^{82} - 8 q^{84} - 32 q^{86} - 8 q^{87} - 48 q^{89} + 16 q^{93} + 4 q^{95} + 4 q^{96} + 4 q^{98} - 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{8}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{8}^{3} + \zeta_{8} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{8}^{3} + \zeta_{8} \) Copy content Toggle raw display
\(\zeta_{8}\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{8}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{8}^{3}\)\(=\) \( ( -\beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(191\) \(211\) \(457\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
341.1
−0.707107 + 0.707107i
0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 0.707107i
−1.00000 −1.00000 1.41421i 1.00000 1.00000i 1.00000 + 1.41421i 0.585786 −1.00000 −1.00000 + 2.82843i 1.00000i
341.2 −1.00000 −1.00000 1.41421i 1.00000 1.00000i 1.00000 + 1.41421i 3.41421 −1.00000 −1.00000 + 2.82843i 1.00000i
341.3 −1.00000 −1.00000 + 1.41421i 1.00000 1.00000i 1.00000 1.41421i 3.41421 −1.00000 −1.00000 2.82843i 1.00000i
341.4 −1.00000 −1.00000 + 1.41421i 1.00000 1.00000i 1.00000 1.41421i 0.585786 −1.00000 −1.00000 2.82843i 1.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
57.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 570.2.f.a 4
3.b odd 2 1 570.2.f.b yes 4
19.b odd 2 1 570.2.f.b yes 4
57.d even 2 1 inner 570.2.f.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.f.a 4 1.a even 1 1 trivial
570.2.f.a 4 57.d even 2 1 inner
570.2.f.b yes 4 3.b odd 2 1
570.2.f.b yes 4 19.b odd 2 1

Hecke kernels

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

\( T_{7}^{2} - 4T_{7} + 2 \) Copy content Toggle raw display
\( T_{29}^{2} - 4T_{29} - 46 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 2 T + 3)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 4 T + 2)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 36T^{2} + 196 \) Copy content Toggle raw display
$13$ \( T^{4} + 44T^{2} + 196 \) Copy content Toggle raw display
$17$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} - 34T^{2} + 361 \) Copy content Toggle raw display
$23$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 4 T - 46)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 48T^{2} + 64 \) Copy content Toggle raw display
$37$ \( T^{4} + 44T^{2} + 196 \) Copy content Toggle raw display
$41$ \( (T^{2} - 8 T - 34)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 16 T + 62)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 88T^{2} + 784 \) Copy content Toggle raw display
$53$ \( (T^{2} - 12 T + 28)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 20 T + 92)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 8 T - 56)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 12 T - 36)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 264 T^{2} + 15376 \) Copy content Toggle raw display
$83$ \( T^{4} + 216T^{2} + 8464 \) Copy content Toggle raw display
$89$ \( (T^{2} + 24 T + 142)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 132T^{2} + 3844 \) Copy content Toggle raw display
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