Properties

Label 555.2.c.a
Level $555$
Weight $2$
Character orbit 555.c
Analytic conductor $4.432$
Analytic rank $0$
Dimension $2$
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] = [555,2,Mod(334,555)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(555, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("555.334");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 555 = 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 555.c (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: \(4.43169731218\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 i q^{2} + i q^{3} - 2 q^{4} + (2 i - 1) q^{5} - 2 q^{6} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 i q^{2} + i q^{3} - 2 q^{4} + (2 i - 1) q^{5} - 2 q^{6} - q^{9} + ( - 2 i - 4) q^{10} - 2 q^{11} - 2 i q^{12} - i q^{13} + ( - i - 2) q^{15} - 4 q^{16} - 2 i q^{18} + 2 q^{19} + ( - 4 i + 2) q^{20} - 4 i q^{22} + 6 i q^{23} + ( - 4 i - 3) q^{25} + 2 q^{26} - i q^{27} + 3 q^{29} + ( - 4 i + 2) q^{30} - 2 q^{31} - 8 i q^{32} - 2 i q^{33} + 2 q^{36} + i q^{37} + 4 i q^{38} + q^{39} - 2 q^{41} - 5 i q^{43} + 4 q^{44} + ( - 2 i + 1) q^{45} - 12 q^{46} + 3 i q^{47} - 4 i q^{48} + 7 q^{49} + ( - 6 i + 8) q^{50} + 2 i q^{52} + 7 i q^{53} + 2 q^{54} + ( - 4 i + 2) q^{55} + 2 i q^{57} + 6 i q^{58} + 3 q^{59} + (2 i + 4) q^{60} - 8 q^{61} - 4 i q^{62} + 8 q^{64} + (i + 2) q^{65} + 4 q^{66} + 10 i q^{67} - 6 q^{69} + 4 q^{71} + 10 i q^{73} - 2 q^{74} + ( - 3 i + 4) q^{75} - 4 q^{76} + 2 i q^{78} + 2 q^{79} + ( - 8 i + 4) q^{80} + q^{81} - 4 i q^{82} + 7 i q^{83} + 10 q^{86} + 3 i q^{87} + 5 q^{89} + (2 i + 4) q^{90} - 12 i q^{92} - 2 i q^{93} - 6 q^{94} + (4 i - 2) q^{95} + 8 q^{96} + 2 i q^{97} + 14 i q^{98} + 2 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{4} - 2 q^{5} - 4 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{4} - 2 q^{5} - 4 q^{6} - 2 q^{9} - 8 q^{10} - 4 q^{11} - 4 q^{15} - 8 q^{16} + 4 q^{19} + 4 q^{20} - 6 q^{25} + 4 q^{26} + 6 q^{29} + 4 q^{30} - 4 q^{31} + 4 q^{36} + 2 q^{39} - 4 q^{41} + 8 q^{44} + 2 q^{45} - 24 q^{46} + 14 q^{49} + 16 q^{50} + 4 q^{54} + 4 q^{55} + 6 q^{59} + 8 q^{60} - 16 q^{61} + 16 q^{64} + 4 q^{65} + 8 q^{66} - 12 q^{69} + 8 q^{71} - 4 q^{74} + 8 q^{75} - 8 q^{76} + 4 q^{79} + 8 q^{80} + 2 q^{81} + 20 q^{86} + 10 q^{89} + 8 q^{90} - 12 q^{94} - 4 q^{95} + 16 q^{96} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/555\mathbb{Z}\right)^\times\).

\(n\) \(76\) \(112\) \(371\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
334.1
1.00000i
1.00000i
2.00000i 1.00000i −2.00000 −1.00000 2.00000i −2.00000 0 0 −1.00000 −4.00000 + 2.00000i
334.2 2.00000i 1.00000i −2.00000 −1.00000 + 2.00000i −2.00000 0 0 −1.00000 −4.00000 2.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 555.2.c.a 2
3.b odd 2 1 1665.2.c.a 2
5.b even 2 1 inner 555.2.c.a 2
5.c odd 4 1 2775.2.a.a 1
5.c odd 4 1 2775.2.a.j 1
15.d odd 2 1 1665.2.c.a 2
15.e even 4 1 8325.2.a.d 1
15.e even 4 1 8325.2.a.bc 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
555.2.c.a 2 1.a even 1 1 trivial
555.2.c.a 2 5.b even 2 1 inner
1665.2.c.a 2 3.b odd 2 1
1665.2.c.a 2 15.d odd 2 1
2775.2.a.a 1 5.c odd 4 1
2775.2.a.j 1 5.c odd 4 1
8325.2.a.d 1 15.e even 4 1
8325.2.a.bc 1 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(555, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 4 \) Copy content Toggle raw display
$3$ \( T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{2} + 2T + 5 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( (T + 2)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 1 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( (T - 2)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 36 \) Copy content Toggle raw display
$29$ \( (T - 3)^{2} \) Copy content Toggle raw display
$31$ \( (T + 2)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 1 \) Copy content Toggle raw display
$41$ \( (T + 2)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 25 \) Copy content Toggle raw display
$47$ \( T^{2} + 9 \) Copy content Toggle raw display
$53$ \( T^{2} + 49 \) Copy content Toggle raw display
$59$ \( (T - 3)^{2} \) Copy content Toggle raw display
$61$ \( (T + 8)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 100 \) Copy content Toggle raw display
$71$ \( (T - 4)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 100 \) Copy content Toggle raw display
$79$ \( (T - 2)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 49 \) Copy content Toggle raw display
$89$ \( (T - 5)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 4 \) Copy content Toggle raw display
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