Properties

Label 229.2.b.a
Level $229$
Weight $2$
Character orbit 229.b
Analytic conductor $1.829$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [229,2,Mod(228,229)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(229, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("229.228");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 229 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 229.b (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: \(1.82857420629\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-5}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(\beta = \sqrt{-5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + q^{3} - 3 q^{4} + 3 q^{5} + \beta q^{6} - \beta q^{8} - 2 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} + q^{3} - 3 q^{4} + 3 q^{5} + \beta q^{6} - \beta q^{8} - 2 q^{9} + 3 \beta q^{10} + 3 q^{11} - 3 q^{12} + 3 q^{15} - q^{16} - 3 q^{17} - 2 \beta q^{18} - q^{19} - 9 q^{20} + 3 \beta q^{22} - 2 \beta q^{23} - \beta q^{24} + 4 q^{25} - 5 q^{27} - 2 \beta q^{29} + 3 \beta q^{30} - 3 \beta q^{32} + 3 q^{33} - 3 \beta q^{34} + 6 q^{36} + 2 q^{37} - \beta q^{38} - 3 \beta q^{40} - 2 \beta q^{41} - q^{43} - 9 q^{44} - 6 q^{45} + 10 q^{46} + 4 \beta q^{47} - q^{48} + 7 q^{49} + 4 \beta q^{50} - 3 q^{51} + 6 q^{53} - 5 \beta q^{54} + 9 q^{55} - q^{57} + 10 q^{58} + 4 \beta q^{59} - 9 q^{60} + 5 q^{61} + 13 q^{64} + 3 \beta q^{66} + 6 \beta q^{67} + 9 q^{68} - 2 \beta q^{69} - 15 q^{71} + 2 \beta q^{72} - 6 \beta q^{73} + 2 \beta q^{74} + 4 q^{75} + 3 q^{76} - 6 \beta q^{79} - 3 q^{80} + q^{81} + 10 q^{82} - 9 q^{83} - 9 q^{85} - \beta q^{86} - 2 \beta q^{87} - 3 \beta q^{88} - 8 \beta q^{89} - 6 \beta q^{90} + 6 \beta q^{92} - 20 q^{94} - 3 q^{95} - 3 \beta q^{96} - 7 q^{97} + 7 \beta q^{98} - 6 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 6 q^{4} + 6 q^{5} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - 6 q^{4} + 6 q^{5} - 4 q^{9} + 6 q^{11} - 6 q^{12} + 6 q^{15} - 2 q^{16} - 6 q^{17} - 2 q^{19} - 18 q^{20} + 8 q^{25} - 10 q^{27} + 6 q^{33} + 12 q^{36} + 4 q^{37} - 2 q^{43} - 18 q^{44} - 12 q^{45} + 20 q^{46} - 2 q^{48} + 14 q^{49} - 6 q^{51} + 12 q^{53} + 18 q^{55} - 2 q^{57} + 20 q^{58} - 18 q^{60} + 10 q^{61} + 26 q^{64} + 18 q^{68} - 30 q^{71} + 8 q^{75} + 6 q^{76} - 6 q^{80} + 2 q^{81} + 20 q^{82} - 18 q^{83} - 18 q^{85} - 40 q^{94} - 6 q^{95} - 14 q^{97} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/229\mathbb{Z}\right)^\times\).

\(n\) \(6\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
228.1
2.23607i
2.23607i
2.23607i 1.00000 −3.00000 3.00000 2.23607i 0 2.23607i −2.00000 6.70820i
228.2 2.23607i 1.00000 −3.00000 3.00000 2.23607i 0 2.23607i −2.00000 6.70820i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 229.2.b.a 2
3.b odd 2 1 2061.2.c.a 2
229.b even 2 1 inner 229.2.b.a 2
687.d odd 2 1 2061.2.c.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
229.2.b.a 2 1.a even 1 1 trivial
229.2.b.a 2 229.b even 2 1 inner
2061.2.c.a 2 3.b odd 2 1
2061.2.c.a 2 687.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 5 \) acting on \(S_{2}^{\mathrm{new}}(229, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 5 \) Copy content Toggle raw display
$3$ \( (T - 1)^{2} \) Copy content Toggle raw display
$5$ \( (T - 3)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( (T - 3)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( (T + 3)^{2} \) Copy content Toggle raw display
$19$ \( (T + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 20 \) Copy content Toggle raw display
$29$ \( T^{2} + 20 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( (T - 2)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 20 \) Copy content Toggle raw display
$43$ \( (T + 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 80 \) Copy content Toggle raw display
$53$ \( (T - 6)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 80 \) Copy content Toggle raw display
$61$ \( (T - 5)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 180 \) Copy content Toggle raw display
$71$ \( (T + 15)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 180 \) Copy content Toggle raw display
$79$ \( T^{2} + 180 \) Copy content Toggle raw display
$83$ \( (T + 9)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 320 \) Copy content Toggle raw display
$97$ \( (T + 7)^{2} \) Copy content Toggle raw display
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