Properties

Label 1089.4.a.ba
Level $1089$
Weight $4$
Character orbit 1089.a
Self dual yes
Analytic conductor $64.253$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1089,4,Mod(1,1089)]
 
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([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1089.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1089 = 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1089.a (trivial)

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: \(64.2530799963\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 28x^{2} - 2x + 72 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 363)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{3} + \beta_1 + 6) q^{4} + ( - \beta_{3} + \beta_{2} - 3) q^{5} + ( - \beta_{3} - \beta_{2} + 2 \beta_1 - 6) q^{7} + ( - \beta_{3} - 2 \beta_{2} + \cdots - 18) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{3} + \beta_1 + 6) q^{4} + ( - \beta_{3} + \beta_{2} - 3) q^{5} + ( - \beta_{3} - \beta_{2} + 2 \beta_1 - 6) q^{7} + ( - \beta_{3} - 2 \beta_{2} + \cdots - 18) q^{8}+ \cdots + (31 \beta_{3} + 14 \beta_{2} + \cdots - 150) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{2} + 25 q^{4} - 14 q^{5} - 20 q^{7} - 75 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{2} + 25 q^{4} - 14 q^{5} - 20 q^{7} - 75 q^{8} - 3 q^{10} - 32 q^{13} - 62 q^{14} + 289 q^{16} + 92 q^{17} + 34 q^{19} - 391 q^{20} + 26 q^{23} + 334 q^{25} + 181 q^{26} - 692 q^{28} + 174 q^{29} + 422 q^{31} - 1271 q^{32} - 477 q^{34} - 82 q^{35} + 518 q^{37} + 798 q^{38} - 423 q^{40} + 428 q^{41} + 550 q^{43} + 2004 q^{46} - 556 q^{47} + 282 q^{49} + 2074 q^{50} - 467 q^{52} - 882 q^{53} + 2112 q^{56} - 225 q^{58} + 158 q^{59} - 290 q^{61} + 1142 q^{62} + 3097 q^{64} - 1636 q^{65} + 992 q^{67} - 2033 q^{68} + 948 q^{70} + 42 q^{71} - 1274 q^{73} + 1509 q^{74} - 632 q^{76} - 362 q^{79} - 1423 q^{80} + 2403 q^{82} + 1500 q^{83} + 2388 q^{85} + 1272 q^{86} - 1428 q^{89} + 1750 q^{91} - 896 q^{92} + 1728 q^{94} - 4452 q^{95} + 1052 q^{97} - 615 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 28x^{2} - 2x + 72 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - \nu^{2} - 22\nu - 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - \nu - 14 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 2\beta_{2} + 23\beta _1 + 18 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
5.63177
1.59490
−1.81838
−4.40829
−5.63177 0 23.7168 −5.58204 0 −16.3245 −88.5134 0 31.4368
1.2 −1.59490 0 −5.45630 −8.73607 0 29.0283 21.4614 0 13.9331
1.3 1.81838 0 −4.69349 19.2178 0 −14.1043 −23.0816 0 34.9453
1.4 4.40829 0 11.4330 −18.8997 0 −18.5994 15.1336 0 −83.3152
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(11\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1089.4.a.ba 4
3.b odd 2 1 363.4.a.s yes 4
11.b odd 2 1 1089.4.a.bf 4
33.d even 2 1 363.4.a.q 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
363.4.a.q 4 33.d even 2 1
363.4.a.s yes 4 3.b odd 2 1
1089.4.a.ba 4 1.a even 1 1 trivial
1089.4.a.bf 4 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1089))\):

\( T_{2}^{4} + T_{2}^{3} - 28T_{2}^{2} + 2T_{2} + 72 \) Copy content Toggle raw display
\( T_{5}^{4} + 14T_{5}^{3} - 319T_{5}^{2} - 5216T_{5} - 17712 \) Copy content Toggle raw display
\( T_{7}^{4} + 20T_{7}^{3} - 627T_{7}^{2} - 18830T_{7} - 124312 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + T^{3} + \cdots + 72 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 14 T^{3} + \cdots - 17712 \) Copy content Toggle raw display
$7$ \( T^{4} + 20 T^{3} + \cdots - 124312 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 32 T^{3} + \cdots + 1724492 \) Copy content Toggle raw display
$17$ \( T^{4} - 92 T^{3} + \cdots - 1870416 \) Copy content Toggle raw display
$19$ \( T^{4} - 34 T^{3} + \cdots + 87435072 \) Copy content Toggle raw display
$23$ \( T^{4} - 26 T^{3} + \cdots + 427537152 \) Copy content Toggle raw display
$29$ \( T^{4} - 174 T^{3} + \cdots - 111819852 \) Copy content Toggle raw display
$31$ \( T^{4} - 422 T^{3} + \cdots - 755977024 \) Copy content Toggle raw display
$37$ \( T^{4} - 518 T^{3} + \cdots - 336093777 \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots - 1476203796 \) Copy content Toggle raw display
$43$ \( T^{4} - 550 T^{3} + \cdots - 474515712 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots - 1899747648 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots - 9562089852 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots - 2988982656 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots - 2291291676 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots - 3253123496 \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots + 71278424064 \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 5540854172 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots - 13109080736 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots - 58659282432 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 172456334064 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 840759943813 \) Copy content Toggle raw display
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