Properties

Label 920.1.p.b
Level $920$
Weight $1$
Character orbit 920.p
Self dual yes
Analytic conductor $0.459$
Analytic rank $0$
Dimension $2$
Projective image $D_{5}$
CM discriminant -920
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [920,1,Mod(229,920)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(920, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("920.229");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 920 = 2^{3} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 920.p (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: yes
Analytic conductor: \(0.459139811622\)
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: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{5}\)
Projective field: Galois closure of 5.1.846400.1

$q$-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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + ( - \beta + 1) q^{3} + q^{4} + q^{5} + (\beta - 1) q^{6} + \beta q^{7} - q^{8} + ( - \beta + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + ( - \beta + 1) q^{3} + q^{4} + q^{5} + (\beta - 1) q^{6} + \beta q^{7} - q^{8} + ( - \beta + 1) q^{9} - q^{10} + (\beta - 1) q^{11} + ( - \beta + 1) q^{12} + \beta q^{13} - \beta q^{14} + ( - \beta + 1) q^{15} + q^{16} + ( - \beta + 1) q^{17} + (\beta - 1) q^{18} - \beta q^{19} + q^{20} - q^{21} + ( - \beta + 1) q^{22} - q^{23} + (\beta - 1) q^{24} + q^{25} - \beta q^{26} + q^{27} + \beta q^{28} + (\beta - 1) q^{30} - \beta q^{31} - q^{32} + (\beta - 2) q^{33} + (\beta - 1) q^{34} + \beta q^{35} + ( - \beta + 1) q^{36} + \beta q^{38} - q^{39} - q^{40} + (\beta - 1) q^{41} + q^{42} + (\beta - 1) q^{44} + ( - \beta + 1) q^{45} + q^{46} + ( - \beta + 1) q^{48} + \beta q^{49} - q^{50} + ( - \beta + 2) q^{51} + \beta q^{52} - q^{54} + (\beta - 1) q^{55} - \beta q^{56} + q^{57} + ( - \beta + 1) q^{60} + (\beta - 1) q^{61} + \beta q^{62} - q^{63} + q^{64} + \beta q^{65} + ( - \beta + 2) q^{66} + ( - \beta + 1) q^{68} + (\beta - 1) q^{69} - \beta q^{70} + (\beta - 1) q^{71} + (\beta - 1) q^{72} + ( - \beta + 1) q^{75} - \beta q^{76} + q^{77} + q^{78} + q^{80} + ( - \beta + 1) q^{82} - q^{84} + ( - \beta + 1) q^{85} + ( - \beta + 1) q^{88} + (\beta - 1) q^{90} + (\beta + 1) q^{91} - q^{92} + q^{93} - \beta q^{95} + (\beta - 1) q^{96} + ( - \beta + 1) q^{97} - \beta q^{98} + (\beta - 2) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + q^{3} + 2 q^{4} + 2 q^{5} - q^{6} + q^{7} - 2 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + q^{3} + 2 q^{4} + 2 q^{5} - q^{6} + q^{7} - 2 q^{8} + q^{9} - 2 q^{10} - q^{11} + q^{12} + q^{13} - q^{14} + q^{15} + 2 q^{16} + q^{17} - q^{18} - q^{19} + 2 q^{20} - 2 q^{21} + q^{22} - 2 q^{23} - q^{24} + 2 q^{25} - q^{26} + 2 q^{27} + q^{28} - q^{30} - q^{31} - 2 q^{32} - 3 q^{33} - q^{34} + q^{35} + q^{36} + q^{38} - 2 q^{39} - 2 q^{40} - q^{41} + 2 q^{42} - q^{44} + q^{45} + 2 q^{46} + q^{48} + q^{49} - 2 q^{50} + 3 q^{51} + q^{52} - 2 q^{54} - q^{55} - q^{56} + 2 q^{57} + q^{60} - q^{61} + q^{62} - 2 q^{63} + 2 q^{64} + q^{65} + 3 q^{66} + q^{68} - q^{69} - q^{70} - q^{71} - q^{72} + q^{75} - q^{76} + 2 q^{77} + 2 q^{78} + 2 q^{80} + q^{82} - 2 q^{84} + q^{85} + q^{88} - q^{90} + 3 q^{91} - 2 q^{92} + 2 q^{93} - q^{95} - q^{96} + q^{97} - q^{98} - 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/920\mathbb{Z}\right)^\times\).

\(n\) \(231\) \(281\) \(461\) \(737\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
229.1
1.61803
−0.618034
−1.00000 −0.618034 1.00000 1.00000 0.618034 1.61803 −1.00000 −0.618034 −1.00000
229.2 −1.00000 1.61803 1.00000 1.00000 −1.61803 −0.618034 −1.00000 1.61803 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
920.p odd 2 1 CM by \(\Q(\sqrt{-230}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 920.1.p.b yes 2
4.b odd 2 1 3680.1.p.b 2
5.b even 2 1 920.1.p.d yes 2
8.b even 2 1 920.1.p.c yes 2
8.d odd 2 1 3680.1.p.c 2
20.d odd 2 1 3680.1.p.d 2
23.b odd 2 1 920.1.p.a 2
40.e odd 2 1 3680.1.p.a 2
40.f even 2 1 920.1.p.a 2
92.b even 2 1 3680.1.p.a 2
115.c odd 2 1 920.1.p.c yes 2
184.e odd 2 1 920.1.p.d yes 2
184.h even 2 1 3680.1.p.d 2
460.g even 2 1 3680.1.p.c 2
920.b even 2 1 3680.1.p.b 2
920.p odd 2 1 CM 920.1.p.b yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.1.p.a 2 23.b odd 2 1
920.1.p.a 2 40.f even 2 1
920.1.p.b yes 2 1.a even 1 1 trivial
920.1.p.b yes 2 920.p odd 2 1 CM
920.1.p.c yes 2 8.b even 2 1
920.1.p.c yes 2 115.c odd 2 1
920.1.p.d yes 2 5.b even 2 1
920.1.p.d yes 2 184.e odd 2 1
3680.1.p.a 2 40.e odd 2 1
3680.1.p.a 2 92.b even 2 1
3680.1.p.b 2 4.b odd 2 1
3680.1.p.b 2 920.b even 2 1
3680.1.p.c 2 8.d odd 2 1
3680.1.p.c 2 460.g even 2 1
3680.1.p.d 2 20.d odd 2 1
3680.1.p.d 2 184.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(920, [\chi])\):

\( T_{3}^{2} - T_{3} - 1 \) Copy content Toggle raw display
\( T_{7}^{2} - T_{7} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

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