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Properties

Label 1288.1.bi.b
Level $1288$
Weight $1$
Character orbit 1288.bi
Analytic conductor $0.643$
Analytic rank $0$
Dimension $10$
Projective image $D_{11}$
CM discriminant -56
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1288,1,Mod(13,1288)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1288, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([0, 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("1288.13");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1288 = 2^{3} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1288.bi (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: \(0.642795736271\)
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.

\(f(q)\) \(=\) \( q - \zeta_{22}^{3} q^{2} + (\zeta_{22}^{4} + 1) q^{3} + \zeta_{22}^{6} q^{4} + (\zeta_{22}^{8} + \zeta_{22}^{6}) q^{5} + ( - \zeta_{22}^{7} - \zeta_{22}^{3}) q^{6} - \zeta_{22} q^{7} - \zeta_{22}^{9} q^{8} + (\zeta_{22}^{8} + \zeta_{22}^{4} + 1) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{22}^{3} q^{2} + (\zeta_{22}^{4} + 1) q^{3} + \zeta_{22}^{6} q^{4} + (\zeta_{22}^{8} + \zeta_{22}^{6}) q^{5} + ( - \zeta_{22}^{7} - \zeta_{22}^{3}) q^{6} - \zeta_{22} q^{7} - \zeta_{22}^{9} q^{8} + (\zeta_{22}^{8} + \zeta_{22}^{4} + 1) q^{9} + ( - \zeta_{22}^{9} + 1) q^{10} + (\zeta_{22}^{10} + \zeta_{22}^{6}) q^{12} + ( - \zeta_{22}^{7} + \zeta_{22}^{2}) q^{13} + \zeta_{22}^{4} q^{14} + (\zeta_{22}^{10} + \zeta_{22}^{8} + \cdots - \zeta_{22}) q^{15} + \cdots - \zeta_{22}^{5} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - q^{2} + 9 q^{3} - q^{4} - 2 q^{5} - 2 q^{6} - q^{7} - q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - q^{2} + 9 q^{3} - q^{4} - 2 q^{5} - 2 q^{6} - q^{7} - q^{8} + 8 q^{9} + 9 q^{10} - 2 q^{12} - 2 q^{13} - q^{14} - 4 q^{15} - q^{16} + 8 q^{18} - 2 q^{19} - 2 q^{20} - 2 q^{21} - q^{23} - 2 q^{24} - 3 q^{25} - 2 q^{26} + 7 q^{27} - q^{28} + 7 q^{30} - q^{32} - 2 q^{35} - 3 q^{36} - 2 q^{38} + 7 q^{39} - 2 q^{40} - 2 q^{42} - 6 q^{45} - q^{46} - 2 q^{48} - q^{49} - 3 q^{50} - 2 q^{52} + 7 q^{54} - q^{56} - 4 q^{57} - 2 q^{59} - 4 q^{60} - 2 q^{61} - 3 q^{63} - q^{64} - 4 q^{65} - 2 q^{69} - 2 q^{70} + 9 q^{71} - 3 q^{72} - 6 q^{75} - 2 q^{76} - 4 q^{78} - 2 q^{79} - 2 q^{80} + 6 q^{81} - 2 q^{83} + 9 q^{84} + 5 q^{90} - 2 q^{91} - q^{92} - 4 q^{95} - 2 q^{96} - q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1288\mathbb{Z}\right)^\times\).

\(n\) \(185\) \(281\) \(645\) \(967\)
\(\chi(n)\) \(-1\) \(-\zeta_{22}^{3}\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1
0.959493 + 0.281733i
0.142315 + 0.989821i
−0.841254 − 0.540641i
−0.415415 − 0.909632i
0.654861 − 0.755750i
0.654861 + 0.755750i
−0.415415 + 0.909632i
0.142315 − 0.989821i
0.959493 − 0.281733i
−0.841254 + 0.540641i
−0.654861 − 0.755750i 1.41542 + 0.909632i −0.142315 + 0.989821i −0.797176 + 1.74557i −0.239446 − 1.66538i −0.959493 − 0.281733i 0.841254 − 0.540641i 0.760554 + 1.66538i 1.84125 − 0.540641i
349.1 0.415415 + 0.909632i 1.84125 − 0.540641i −0.654861 + 0.755750i −0.239446 − 0.153882i 1.25667 + 1.45027i −0.142315 − 0.989821i −0.959493 − 0.281733i 2.25667 − 1.45027i 0.0405070 − 0.281733i
629.1 −0.142315 + 0.989821i 0.345139 + 0.755750i −0.959493 − 0.281733i −1.10181 − 1.27155i −0.797176 + 0.234072i 0.841254 + 0.540641i 0.415415 − 0.909632i 0.202824 − 0.234072i 1.41542 − 0.909632i
685.1 −0.959493 − 0.281733i 0.857685 − 0.989821i 0.841254 + 0.540641i −0.118239 + 0.822373i −1.10181 + 0.708089i 0.415415 + 0.909632i −0.654861 − 0.755750i −0.101808 − 0.708089i 0.345139 − 0.755750i
853.1 0.841254 + 0.540641i 0.0405070 + 0.281733i 0.415415 + 0.909632i 1.25667 + 0.368991i −0.118239 + 0.258908i −0.654861 + 0.755750i −0.142315 + 0.989821i 0.881761 − 0.258908i 0.857685 + 0.989821i
909.1 0.841254 − 0.540641i 0.0405070 − 0.281733i 0.415415 − 0.909632i 1.25667 − 0.368991i −0.118239 − 0.258908i −0.654861 − 0.755750i −0.142315 − 0.989821i 0.881761 + 0.258908i 0.857685 − 0.989821i
1021.1 −0.959493 + 0.281733i 0.857685 + 0.989821i 0.841254 − 0.540641i −0.118239 − 0.822373i −1.10181 − 0.708089i 0.415415 − 0.909632i −0.654861 + 0.755750i −0.101808 + 0.708089i 0.345139 + 0.755750i
1133.1 0.415415 − 0.909632i 1.84125 + 0.540641i −0.654861 − 0.755750i −0.239446 + 0.153882i 1.25667 − 1.45027i −0.142315 + 0.989821i −0.959493 + 0.281733i 2.25667 + 1.45027i 0.0405070 + 0.281733i
1189.1 −0.654861 + 0.755750i 1.41542 − 0.909632i −0.142315 − 0.989821i −0.797176 − 1.74557i −0.239446 + 1.66538i −0.959493 + 0.281733i 0.841254 + 0.540641i 0.760554 − 1.66538i 1.84125 + 0.540641i
1245.1 −0.142315 − 0.989821i 0.345139 − 0.755750i −0.959493 + 0.281733i −1.10181 + 1.27155i −0.797176 − 0.234072i 0.841254 − 0.540641i 0.415415 + 0.909632i 0.202824 + 0.234072i 1.41542 + 0.909632i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
56.h odd 2 1 CM by \(\Q(\sqrt{-14}) \)
23.c even 11 1 inner
1288.bi odd 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1288.1.bi.b yes 10
7.b odd 2 1 1288.1.bi.a ✓ 10
8.b even 2 1 1288.1.bi.a ✓ 10
23.c even 11 1 inner 1288.1.bi.b yes 10
56.h odd 2 1 CM 1288.1.bi.b yes 10
161.l odd 22 1 1288.1.bi.a ✓ 10
184.p even 22 1 1288.1.bi.a ✓ 10
1288.bi odd 22 1 inner 1288.1.bi.b yes 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1288.1.bi.a ✓ 10 7.b odd 2 1
1288.1.bi.a ✓ 10 8.b even 2 1
1288.1.bi.a ✓ 10 161.l odd 22 1
1288.1.bi.a ✓ 10 184.p even 22 1
1288.1.bi.b yes 10 1.a even 1 1 trivial
1288.1.bi.b yes 10 23.c even 11 1 inner
1288.1.bi.b yes 10 56.h odd 2 1 CM
1288.1.bi.b yes 10 1288.bi odd 22 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(1288, [\chi])\):

\( T_{3}^{10} - 9T_{3}^{9} + 37T_{3}^{8} - 91T_{3}^{7} + 148T_{3}^{6} - 166T_{3}^{5} + 130T_{3}^{4} - 70T_{3}^{3} + 25T_{3}^{2} - 5T_{3} + 1 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + T^{9} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{10} - 9 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{10} + 2 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{10} + T^{9} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{10} \) 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
$37$ \( T^{10} \) Copy content Toggle raw display
$41$ \( T^{10} \) Copy content Toggle raw display
$43$ \( T^{10} \) Copy content Toggle raw display
$47$ \( T^{10} \) Copy content Toggle raw display
$53$ \( T^{10} \) Copy content Toggle raw display
$59$ \( T^{10} + 2 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$61$ \( T^{10} + 2 T^{9} + \cdots + 1024 \) Copy content Toggle raw display
$67$ \( T^{10} \) Copy content Toggle raw display
$71$ \( T^{10} - 9 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$73$ \( T^{10} \) Copy content Toggle raw display
$79$ \( T^{10} + 2 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$83$ \( T^{10} + 2 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$89$ \( T^{10} \) Copy content Toggle raw display
$97$ \( T^{10} \) Copy content Toggle raw display
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