Properties

Label 1089.3.b.h
Level $1089$
Weight $3$
Character orbit 1089.b
Analytic conductor $29.673$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1089,3,Mod(485,1089)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1089, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1089.485");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1089 = 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1089.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: \(29.6731007888\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.3317760000.7
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 12x^{6} + 23x^{4} + 12x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
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 a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{7} q^{2} + \beta_{3} q^{4} - \beta_1 q^{5} + (\beta_{4} + \beta_{2}) q^{7} + \beta_{6} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{7} q^{2} + \beta_{3} q^{4} - \beta_1 q^{5} + (\beta_{4} + \beta_{2}) q^{7} + \beta_{6} q^{8} + ( - \beta_{4} + 2 \beta_{2}) q^{10} + ( - 6 \beta_{4} + 5 \beta_{2}) q^{13} + \beta_{5} q^{14} + (4 \beta_{3} - 1) q^{16} + (3 \beta_{7} - 10 \beta_{6}) q^{17} + ( - 2 \beta_{4} - 6 \beta_{2}) q^{19} + ( - 7 \beta_{5} - \beta_1) q^{20} + (13 \beta_{5} - 12 \beta_1) q^{23} + ( - \beta_{3} + 17) q^{25} + ( - 28 \beta_{5} + 11 \beta_1) q^{26} + (3 \beta_{4} + 5 \beta_{2}) q^{28} + 13 \beta_{7} q^{29} + (4 \beta_{3} + 12) q^{31} + (17 \beta_{7} + 8 \beta_{6}) q^{32} + ( - 3 \beta_{3} + 22) q^{34} + ( - \beta_{7} - 5 \beta_{6}) q^{35} + (9 \beta_{3} + 2) q^{37} + (6 \beta_{5} - 4 \beta_1) q^{38} + (2 \beta_{4} + 3 \beta_{2}) q^{40} + (13 \beta_{7} + 26 \beta_{6}) q^{41} + ( - 19 \beta_{4} + 9 \beta_{2}) q^{43} + ( - 25 \beta_{4} + 37 \beta_{2}) q^{46} + ( - 6 \beta_{5} - 10 \beta_1) q^{47} + (2 \beta_{3} - 41) q^{49} + ( - 21 \beta_{7} - \beta_{6}) q^{50} + (15 \beta_{4} - 30 \beta_{2}) q^{52} + ( - 48 \beta_{5} - 3 \beta_1) q^{53} + (3 \beta_{5} + 2 \beta_1) q^{56} + ( - 13 \beta_{3} + 52) q^{58} + ( - 51 \beta_{5} - 6 \beta_1) q^{59} + ( - 2 \beta_{4} + 26 \beta_{2}) q^{61} + (4 \beta_{7} + 4 \beta_{6}) q^{62} + ( - \beta_{3} + 56) q^{64} + (17 \beta_{7} + 8 \beta_{6}) q^{65} + ( - 21 \beta_{3} + 27) q^{67} + ( - 22 \beta_{7} - 43 \beta_{6}) q^{68} + (\beta_{3} + 1) q^{70} + (\beta_{5} + 26 \beta_1) q^{71} + ( - 52 \beta_{4} + 4 \beta_{2}) q^{73} + (34 \beta_{7} + 9 \beta_{6}) q^{74} + ( - 18 \beta_{4} - 10 \beta_{2}) q^{76} + (30 \beta_{4} + 44 \beta_{2}) q^{79} + ( - 28 \beta_{5} - 3 \beta_1) q^{80} + ( - 13 \beta_{3} + 26) q^{82} + ( - 34 \beta_{7} - 20 \beta_{6}) q^{83} + ( - 17 \beta_{4} - 36 \beta_{2}) q^{85} + ( - 75 \beta_{5} + 28 \beta_1) q^{86} + (19 \beta_{5} + 11 \beta_1) q^{89} + ( - \beta_{3} - 15) q^{91} + ( - 97 \beta_{5} + 14 \beta_1) q^{92} + ( - 4 \beta_{4} + 14 \beta_{2}) q^{94} + ( - 2 \beta_{7} + 18 \beta_{6}) q^{95} + ( - 7 \beta_{3} + 44) q^{97} + (49 \beta_{7} + 2 \beta_{6}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{16} + 136 q^{25} + 96 q^{31} + 176 q^{34} + 16 q^{37} - 328 q^{49} + 416 q^{58} + 448 q^{64} + 216 q^{67} + 8 q^{70} + 208 q^{82} - 120 q^{91} + 352 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 12x^{6} + 23x^{4} + 12x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} + 13\nu^{5} + 36\nu^{3} + 48\nu ) / 7 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3\nu^{6} + 32\nu^{4} + 24\nu^{2} - 10 ) / 7 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{6} - 12\nu^{4} - 22\nu^{2} - 6 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 8\nu^{6} + 90\nu^{4} + 120\nu^{2} + 27 ) / 7 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -6\nu^{7} - 71\nu^{5} - 125\nu^{3} - 43\nu ) / 7 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{7} + 11\nu^{5} + 12\nu^{3} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \nu^{7} + 12\nu^{5} + 23\nu^{3} + 11\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{7} - \beta_{5} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{4} + \beta_{3} - 3\beta_{2} - 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 11\beta_{7} + \beta_{6} + 13\beta_{5} - 6\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -21\beta_{4} - 12\beta_{3} + 28\beta_{2} + 49 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -108\beta_{7} - 13\beta_{6} - 132\beta_{5} + 55\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 104\beta_{4} + 60\beta_{3} - 135\beta_{2} - 234 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1056\beta_{7} + 133\beta_{6} + 1296\beta_{5} - 533\beta_1 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1089\mathbb{Z}\right)^\times\).

\(n\) \(244\) \(848\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
485.1
0.319918i
3.12580i
0.837556i
1.19395i
1.19395i
0.837556i
3.12580i
0.319918i
2.80588i 0 −3.87298 2.03151i 0 0.504017 0.356394i 0 −5.70017
485.2 2.80588i 0 −3.87298 2.03151i 0 −0.504017 0.356394i 0 5.70017
485.3 0.356394i 0 3.87298 3.44572i 0 −3.96812 2.80588i 0 −1.22803
485.4 0.356394i 0 3.87298 3.44572i 0 3.96812 2.80588i 0 1.22803
485.5 0.356394i 0 3.87298 3.44572i 0 3.96812 2.80588i 0 1.22803
485.6 0.356394i 0 3.87298 3.44572i 0 −3.96812 2.80588i 0 −1.22803
485.7 2.80588i 0 −3.87298 2.03151i 0 −0.504017 0.356394i 0 5.70017
485.8 2.80588i 0 −3.87298 2.03151i 0 0.504017 0.356394i 0 −5.70017
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 485.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.b odd 2 1 inner
33.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1089.3.b.h 8
3.b odd 2 1 inner 1089.3.b.h 8
11.b odd 2 1 inner 1089.3.b.h 8
33.d even 2 1 inner 1089.3.b.h 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1089.3.b.h 8 1.a even 1 1 trivial
1089.3.b.h 8 3.b odd 2 1 inner
1089.3.b.h 8 11.b odd 2 1 inner
1089.3.b.h 8 33.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1089, [\chi])\):

\( T_{2}^{4} + 8T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{4} - 16T_{7}^{2} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 8 T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} + 16 T^{2} + 49)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} - 16 T^{2} + 4)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( (T^{4} - 510 T^{2} + 11025)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 992 T^{2} + 121801)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 256 T^{2} + 7744)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 2356 T^{2} + 964324)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 1352 T^{2} + 28561)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 24 T - 96)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} - 4 T - 1211)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + 5408 T^{2} + 3455881)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 4096 T^{2} + 2439844)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 1984 T^{2} + 258064)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 9936 T^{2} + 23357889)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 12204 T^{2} + 30935844)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 4096 T^{2} + 4032064)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 54 T - 5886)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + 10924 T^{2} + 21883684)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 27136 T^{2} + \cdots + 181494784)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 20616 T^{2} + 1710864)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 9728 T^{2} + 15085456)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 4216 T^{2} + 85849)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 88 T + 1201)^{4} \) Copy content Toggle raw display
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