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Properties

Label 920.1.bh.b
Level $920$
Weight $1$
Character orbit 920.bh
Analytic conductor $0.459$
Analytic rank $0$
Dimension $10$
Projective image $D_{11}$
CM discriminant -40
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [920,1,Mod(59,920)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(920, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([11, 11, 11, 14]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("920.59");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 920 = 2^{3} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 920.bh (of order \(22\), degree \(10\), 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: \(0.459139811622\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\Q(\zeta_{22})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{11}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{11} - \cdots)\)

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + \zeta_{22} q^{2} + \zeta_{22}^{2} q^{4} - \zeta_{22}^{6} q^{5} + (\zeta_{22}^{5} + \zeta_{22}^{3}) q^{7} + \zeta_{22}^{3} q^{8} - \zeta_{22}^{5} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{22} q^{2} + \zeta_{22}^{2} q^{4} - \zeta_{22}^{6} q^{5} + (\zeta_{22}^{5} + \zeta_{22}^{3}) q^{7} + \zeta_{22}^{3} q^{8} - \zeta_{22}^{5} q^{9} - \zeta_{22}^{7} q^{10} + (\zeta_{22}^{8} - \zeta_{22}) q^{11} + ( - \zeta_{22}^{2} + \zeta_{22}) q^{13} + (\zeta_{22}^{6} + \zeta_{22}^{4}) q^{14} + \zeta_{22}^{4} q^{16} - \zeta_{22}^{6} q^{18} + (\zeta_{22}^{10} - \zeta_{22}^{5}) q^{19} - \zeta_{22}^{8} q^{20} + (\zeta_{22}^{9} - \zeta_{22}^{2}) q^{22} - \zeta_{22}^{8} q^{23} - \zeta_{22} q^{25} + ( - \zeta_{22}^{3} + \zeta_{22}^{2}) q^{26} + (\zeta_{22}^{7} + \zeta_{22}^{5}) q^{28} + \zeta_{22}^{5} q^{32} + ( - \zeta_{22}^{9} + 1) q^{35} - \zeta_{22}^{7} q^{36} + (\zeta_{22}^{9} + \zeta_{22}) q^{37} + ( - \zeta_{22}^{6} - 1) q^{38} - \zeta_{22}^{9} q^{40} + ( - \zeta_{22}^{9} - \zeta_{22}^{3}) q^{41} + (\zeta_{22}^{10} - \zeta_{22}^{3}) q^{44} - q^{45} - \zeta_{22}^{9} q^{46} + (\zeta_{22}^{7} - \zeta_{22}^{4}) q^{47} + (\zeta_{22}^{10} + \cdots + \zeta_{22}^{6}) q^{49} + \cdots + (\zeta_{22}^{6} + \zeta_{22}^{2}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + q^{2} - q^{4} + q^{5} + 2 q^{7} + q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + q^{2} - q^{4} + q^{5} + 2 q^{7} + q^{8} - q^{9} - q^{10} - 2 q^{11} + 2 q^{13} - 2 q^{14} - q^{16} + q^{18} - 2 q^{19} + q^{20} + 2 q^{22} + q^{23} - q^{25} - 2 q^{26} + 2 q^{28} + q^{32} + 9 q^{35} - q^{36} + 2 q^{37} - 9 q^{38} - q^{40} - 2 q^{41} - 2 q^{44} - 10 q^{45} - q^{46} + 2 q^{47} - 3 q^{49} + q^{50} + 2 q^{52} + 2 q^{53} + 2 q^{55} - 2 q^{56} + 9 q^{59} + 2 q^{63} - q^{64} - 2 q^{65} + 2 q^{70} + q^{72} - 2 q^{74} - 2 q^{76} - 7 q^{77} + q^{80} - q^{81} + 2 q^{82} - 9 q^{88} - 2 q^{89} - q^{90} - 4 q^{91} + q^{92} - 2 q^{94} - 9 q^{95} - 8 q^{98} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(231\) \(281\) \(461\) \(737\)
\(\chi(n)\) \(-1\) \(-\zeta_{22}\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
59.1
0.654861 + 0.755750i
0.959493 + 0.281733i
−0.841254 + 0.540641i
−0.415415 + 0.909632i
0.654861 − 0.755750i
−0.415415 − 0.909632i
0.959493 − 0.281733i
0.142315 + 0.989821i
0.142315 − 0.989821i
−0.841254 − 0.540641i
0.654861 + 0.755750i 0 −0.142315 + 0.989821i −0.415415 + 0.909632i 0 −1.25667 − 0.368991i −0.841254 + 0.540641i 0.415415 + 0.909632i −0.959493 + 0.281733i
179.1 0.959493 + 0.281733i 0 0.841254 + 0.540641i 0.142315 − 0.989821i 0 0.797176 + 1.74557i 0.654861 + 0.755750i −0.142315 − 0.989821i 0.415415 − 0.909632i
219.1 −0.841254 + 0.540641i 0 0.415415 − 0.909632i 0.959493 − 0.281733i 0 1.10181 + 1.27155i 0.142315 + 0.989821i −0.959493 − 0.281733i −0.654861 + 0.755750i
259.1 −0.415415 + 0.909632i 0 −0.654861 − 0.755750i −0.841254 + 0.540641i 0 0.118239 − 0.822373i 0.959493 − 0.281733i 0.841254 + 0.540641i −0.142315 − 0.989821i
499.1 0.654861 − 0.755750i 0 −0.142315 − 0.989821i −0.415415 − 0.909632i 0 −1.25667 + 0.368991i −0.841254 − 0.540641i 0.415415 − 0.909632i −0.959493 − 0.281733i
579.1 −0.415415 − 0.909632i 0 −0.654861 + 0.755750i −0.841254 − 0.540641i 0 0.118239 + 0.822373i 0.959493 + 0.281733i 0.841254 − 0.540641i −0.142315 + 0.989821i
699.1 0.959493 − 0.281733i 0 0.841254 − 0.540641i 0.142315 + 0.989821i 0 0.797176 − 1.74557i 0.654861 − 0.755750i −0.142315 + 0.989821i 0.415415 + 0.909632i
739.1 0.142315 + 0.989821i 0 −0.959493 + 0.281733i 0.654861 − 0.755750i 0 0.239446 − 0.153882i −0.415415 − 0.909632i −0.654861 − 0.755750i 0.841254 + 0.540641i
859.1 0.142315 − 0.989821i 0 −0.959493 − 0.281733i 0.654861 + 0.755750i 0 0.239446 + 0.153882i −0.415415 + 0.909632i −0.654861 + 0.755750i 0.841254 − 0.540641i
899.1 −0.841254 − 0.540641i 0 0.415415 + 0.909632i 0.959493 + 0.281733i 0 1.10181 − 1.27155i 0.142315 − 0.989821i −0.959493 + 0.281733i −0.654861 − 0.755750i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 59.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
40.e odd 2 1 CM by \(\Q(\sqrt{-10}) \)
23.c even 11 1 inner
920.bh odd 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 920.1.bh.b yes 10
4.b odd 2 1 3680.1.cn.b 10
5.b even 2 1 920.1.bh.a ✓ 10
8.b even 2 1 3680.1.cn.a 10
8.d odd 2 1 920.1.bh.a ✓ 10
20.d odd 2 1 3680.1.cn.a 10
23.c even 11 1 inner 920.1.bh.b yes 10
40.e odd 2 1 CM 920.1.bh.b yes 10
40.f even 2 1 3680.1.cn.b 10
92.g odd 22 1 3680.1.cn.b 10
115.j even 22 1 920.1.bh.a ✓ 10
184.k odd 22 1 920.1.bh.a ✓ 10
184.p even 22 1 3680.1.cn.a 10
460.n odd 22 1 3680.1.cn.a 10
920.bf even 22 1 3680.1.cn.b 10
920.bh odd 22 1 inner 920.1.bh.b yes 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.1.bh.a ✓ 10 5.b even 2 1
920.1.bh.a ✓ 10 8.d odd 2 1
920.1.bh.a ✓ 10 115.j even 22 1
920.1.bh.a ✓ 10 184.k odd 22 1
920.1.bh.b yes 10 1.a even 1 1 trivial
920.1.bh.b yes 10 23.c even 11 1 inner
920.1.bh.b yes 10 40.e odd 2 1 CM
920.1.bh.b yes 10 920.bh odd 22 1 inner
3680.1.cn.a 10 8.b even 2 1
3680.1.cn.a 10 20.d odd 2 1
3680.1.cn.a 10 184.p even 22 1
3680.1.cn.a 10 460.n odd 22 1
3680.1.cn.b 10 4.b odd 2 1
3680.1.cn.b 10 40.f even 2 1
3680.1.cn.b 10 92.g odd 22 1
3680.1.cn.b 10 920.bf even 22 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{10} - 2T_{7}^{9} + 4T_{7}^{8} + 3T_{7}^{7} - 6T_{7}^{6} + 12T_{7}^{5} + 9T_{7}^{4} - 7T_{7}^{3} + 14T_{7}^{2} - 6T_{7} + 1 \) acting on \(S_{1}^{\mathrm{new}}(920, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} - T^{9} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( T^{10} - T^{9} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{10} - 2 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{10} + 2 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{10} - 2 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{10} \) Copy content Toggle raw display
$19$ \( T^{10} + 2 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$23$ \( T^{10} - T^{9} + \cdots + 1 \) Copy content Toggle raw display
$29$ \( T^{10} \) Copy content Toggle raw display
$31$ \( T^{10} \) Copy content Toggle raw display
$37$ \( T^{10} - 2 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( T^{10} + 2 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$43$ \( T^{10} \) Copy content Toggle raw display
$47$ \( (T^{5} - T^{4} - 4 T^{3} + \cdots - 1)^{2} \) Copy content Toggle raw display
$53$ \( T^{10} - 2 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$59$ \( T^{10} - 9 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$61$ \( T^{10} \) Copy content Toggle raw display
$67$ \( T^{10} \) Copy content Toggle raw display
$71$ \( T^{10} \) Copy content Toggle raw display
$73$ \( T^{10} \) Copy content Toggle raw display
$79$ \( T^{10} \) Copy content Toggle raw display
$83$ \( T^{10} \) Copy content Toggle raw display
$89$ \( T^{10} + 2 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$97$ \( T^{10} \) Copy content Toggle raw display
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