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Properties

Label 1089.4.a.bi
Level $1089$
Weight $4$
Character orbit 1089.a
Self dual yes
Analytic conductor $64.253$
Analytic rank $1$
Dimension $6$
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: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 30x^{4} + 3x^{3} + 211x^{2} + 208x + 44 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 11 \)
Twist minimal: no (minimal twist has level 33)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} + ( - \beta_{3} + \beta_{2} - \beta_1 + 3) q^{4} + ( - \beta_{5} + 2 \beta_{3} + \beta_1 - 2) q^{5} + (2 \beta_{5} - \beta_{4} + \cdots + \beta_1) q^{7}+ \cdots + (\beta_{5} - 2 \beta_{4} + 3 \beta_{3} + \cdots - 5) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{2} + ( - \beta_{3} + \beta_{2} - \beta_1 + 3) q^{4} + ( - \beta_{5} + 2 \beta_{3} + \beta_1 - 2) q^{5} + (2 \beta_{5} - \beta_{4} + \cdots + \beta_1) q^{7}+ \cdots + ( - 31 \beta_{5} + 81 \beta_{4} + \cdots - 198) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 5 q^{2} + 17 q^{4} - 9 q^{5} + q^{7} - 24 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 5 q^{2} + 17 q^{4} - 9 q^{5} + q^{7} - 24 q^{8} + 50 q^{10} + 66 q^{13} + 42 q^{14} - 71 q^{16} - 80 q^{17} - 90 q^{19} - 455 q^{20} + 196 q^{23} + 351 q^{25} - 93 q^{26} - 52 q^{28} - 579 q^{29} + 129 q^{31} - 657 q^{32} + 530 q^{34} - 618 q^{35} + 192 q^{37} + 769 q^{38} + 1476 q^{40} - 990 q^{41} + 208 q^{43} + 493 q^{46} - 680 q^{47} - 963 q^{49} - 1656 q^{50} + 887 q^{52} - 197 q^{53} + 429 q^{56} + 395 q^{58} + 1229 q^{59} + 928 q^{61} - 2117 q^{62} + 844 q^{64} - 2278 q^{65} - 264 q^{67} - 3279 q^{68} + 665 q^{70} + 786 q^{71} + 593 q^{73} - 1543 q^{74} - 434 q^{76} - 421 q^{79} - 18 q^{80} - 739 q^{82} - 2043 q^{83} + 1900 q^{85} - 2683 q^{86} - 158 q^{89} + 698 q^{91} + 3915 q^{92} - 4934 q^{94} - 1038 q^{95} - 1711 q^{97} - 1405 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} - 30x^{4} + 3x^{3} + 211x^{2} + 208x + 44 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} - 16\nu^{2} - 13\nu - 10 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{4} - 24\nu^{2} - 5\nu + 70 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} - \nu^{4} - 32\nu^{3} + 11\nu^{2} + 243\nu + 114 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2\nu^{5} - 3\nu^{4} - 56\nu^{3} + 38\nu^{2} + 355\nu + 158 ) / 8 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{3} + \beta_{2} + \beta _1 + 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} - 2\beta_{4} + \beta_{2} + 18\beta _1 + 10 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -16\beta_{3} + 24\beta_{2} + 29\beta _1 + 170 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 32\beta_{5} - 56\beta_{4} - 5\beta_{3} + 45\beta_{2} + 351\beta _1 + 266 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
−4.12458
−2.37704
−0.808830
−0.303175
3.80972
4.80390
−5.12458 0 18.2613 −14.9367 0 −12.3403 −52.5849 0 76.5442
1.2 −3.37704 0 3.40437 −14.0063 0 23.3415 15.5196 0 47.2999
1.3 −1.80883 0 −4.72813 21.4614 0 −18.7206 23.0230 0 −38.8200
1.4 −1.30317 0 −6.30174 8.10194 0 0.543555 18.6377 0 −10.5582
1.5 2.80972 0 −0.105474 −12.1997 0 6.70654 −22.7741 0 −34.2777
1.6 3.80390 0 6.46966 2.57942 0 1.46925 −5.82127 0 9.81184
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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.bi 6
3.b odd 2 1 363.4.a.v 6
11.b odd 2 1 1089.4.a.bk 6
11.c even 5 2 99.4.f.d 12
33.d even 2 1 363.4.a.u 6
33.h odd 10 2 33.4.e.c ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.4.e.c ✓ 12 33.h odd 10 2
99.4.f.d 12 11.c even 5 2
363.4.a.u 6 33.d even 2 1
363.4.a.v 6 3.b odd 2 1
1089.4.a.bi 6 1.a even 1 1 trivial
1089.4.a.bk 6 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1089))\):

\( T_{2}^{6} + 5T_{2}^{5} - 20T_{2}^{4} - 107T_{2}^{3} + 45T_{2}^{2} + 520T_{2} + 436 \) Copy content Toggle raw display
\( T_{5}^{6} + 9T_{5}^{5} - 510T_{5}^{4} - 5679T_{5}^{3} + 40162T_{5}^{2} + 386217T_{5} - 1144711 \) Copy content Toggle raw display
\( T_{7}^{6} - T_{7}^{5} - 547T_{7}^{4} - 980T_{7}^{3} + 39913T_{7}^{2} - 74451T_{7} + 28881 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 5 T^{5} + \cdots + 436 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + 9 T^{5} + \cdots - 1144711 \) Copy content Toggle raw display
$7$ \( T^{6} - T^{5} + \cdots + 28881 \) Copy content Toggle raw display
$11$ \( T^{6} \) Copy content Toggle raw display
$13$ \( T^{6} + \cdots + 2222720156 \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots - 5041345424 \) Copy content Toggle raw display
$19$ \( T^{6} + \cdots - 21717954500 \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots - 3716146836 \) Copy content Toggle raw display
$29$ \( T^{6} + \cdots - 979286899120 \) Copy content Toggle raw display
$31$ \( T^{6} + \cdots - 11660646647519 \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots + 1357665091376 \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots + 63361664123636 \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 92003355120644 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 247691916919216 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 189013172221 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots - 150831951297845 \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots - 15\!\cdots\!44 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots - 66\!\cdots\!84 \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots + 11\!\cdots\!36 \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 12\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots - 17\!\cdots\!25 \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 13\!\cdots\!09 \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots - 28\!\cdots\!20 \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots + 10\!\cdots\!11 \) Copy content Toggle raw display
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