Properties

Label 1089.6.a.r
Level $1089$
Weight $6$
Character orbit 1089.a
Self dual yes
Analytic conductor $174.658$
Analytic rank $1$
Dimension $3$
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] = [1089,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1089.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1089 = 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 6 \)
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: \(174.657979776\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.54492.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 52x - 38 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 11)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + ( - 4 \beta_{2} - 2 \beta_1 + 28) q^{4} + (\beta_{2} - 3 \beta_1 - 8) q^{5} + (10 \beta_{2} + 10 \beta_1 - 28) q^{7} + (26 \beta_{2} - 188) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + ( - 4 \beta_{2} - 2 \beta_1 + 28) q^{4} + (\beta_{2} - 3 \beta_1 - 8) q^{5} + (10 \beta_{2} + 10 \beta_1 - 28) q^{7} + (26 \beta_{2} - 188) q^{8} + (9 \beta_{2} - 14 \beta_1 + 138) q^{10} + ( - 70 \beta_{2} - 6 \beta_1 - 162) q^{13} + ( - 138 \beta_{2} + 20 \beta_1 + 340) q^{14} + ( - 164 \beta_{2} + 12 \beta_1 + 664) q^{16} + (132 \beta_{2} - 20 \beta_1 + 362) q^{17} + (60 \beta_{2} + 20 \beta_1 - 460) q^{19} + (168 \beta_{2} + 22 \beta_1 + 1160) q^{20} + (21 \beta_{2} - 167 \beta_1 + 1022) q^{23} + (265 \beta_{2} + 85 \beta_1 - 19) q^{25} + (160 \beta_{2} + 116 \beta_1 - 4044) q^{26} + (432 \beta_{2} + 36 \beta_1 - 7904) q^{28} + ( - 182 \beta_{2} - 46 \beta_1 - 1142) q^{29} + ( - 101 \beta_{2} + 23 \beta_1 - 1366) q^{31} + (404 \beta_{2} + 376 \beta_1 - 4136) q^{32} + ( - 26 \beta_{2} - 344 \beta_1 + 8440) q^{34} + ( - 818 \beta_{2} - 306 \beta_1 - 8076) q^{35} + (937 \beta_{2} + 197 \beta_1 + 5908) q^{37} + ( - 840 \beta_{2} - 40 \beta_1 + 3080) q^{38} + (46 \beta_{2} + 200 \beta_1 + 5092) q^{40} + ( - 1378 \beta_{2} + 94 \beta_1 + 1998) q^{41} + ( - 1190 \beta_{2} + 370 \beta_1 + 8736) q^{43} + (2107 \beta_{2} - 710 \beta_1 + 5602) q^{46} + (600 \beta_{2} + 424 \beta_1 + 5744) q^{47} + (340 \beta_{2} + 740 \beta_1 + 16177) q^{49} + ( - 1674 \beta_{2} - 190 \beta_1 + 13690) q^{50} + ( - 3256 \beta_{2} + 336 \beta_1 + 11768) q^{52} + (476 \beta_{2} + 540 \beta_1 - 16862) q^{53} + ( - 5468 \beta_{2} - 1360 \beta_1 + 14104) q^{56} + ( - 92 \beta_{2} + 180 \beta_1 - 9724) q^{58} + ( - 3141 \beta_{2} - 249 \beta_1 + 1246) q^{59} + (1466 \beta_{2} + 1346 \beta_1 - 6162) q^{61} + ( - 1123 \beta_{2} + 294 \beta_1 - 6658) q^{62} + ( - 3136 \beta_{2} + 312 \beta_1 - 6784) q^{64} + ( - 264 \beta_{2} + 1616 \beta_1 - 2556) q^{65} + ( - 6575 \beta_{2} - 907 \beta_1 - 15918) q^{67} + (6728 \beta_{2} - 684 \beta_1 - 4200) q^{68} + ( - 2662 \beta_{2} + 412 \beta_1 - 41124) q^{70} + (3935 \beta_{2} + 891 \beta_1 - 13094) q^{71} + ( - 5370 \beta_{2} + 1174 \beta_1 - 5142) q^{73} + (781 \beta_{2} - 1086 \beta_1 + 51098) q^{74} + (4800 \beta_{2} + 880 \beta_1 - 34640) q^{76} + ( - 3902 \beta_{2} - 454 \beta_1 - 41716) q^{79} + ( - 1868 \beta_{2} + 4 \beta_1 - 39560) q^{80} + (6852 \beta_{2} + 3132 \beta_1 - 85124) q^{82} + (3150 \beta_{2} + 3750 \beta_1 - 47976) q^{83} + (3310 \beta_{2} - 2434 \beta_1 + 34680) q^{85} + (10906 \beta_{2} + 3860 \beta_1 - 81020) q^{86} + (5167 \beta_{2} + 1523 \beta_1 + 35608) q^{89} + (6840 \beta_{2} - 3512 \beta_1 - 36544) q^{91} + (1472 \beta_{2} - 1710 \beta_1 + 112176) q^{92} + (376 \beta_{2} + 496 \beta_1 + 24976) q^{94} + ( - 1680 \beta_{2} + 40 \beta_1 - 7400) q^{95} + (123 \beta_{2} + 2143 \beta_1 + 3228) q^{97} + (9637 \beta_{2} + 2280 \beta_1 + 1160) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 84 q^{4} - 24 q^{5} - 84 q^{7} - 564 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 84 q^{4} - 24 q^{5} - 84 q^{7} - 564 q^{8} + 414 q^{10} - 486 q^{13} + 1020 q^{14} + 1992 q^{16} + 1086 q^{17} - 1380 q^{19} + 3480 q^{20} + 3066 q^{23} - 57 q^{25} - 12132 q^{26} - 23712 q^{28} - 3426 q^{29} - 4098 q^{31} - 12408 q^{32} + 25320 q^{34} - 24228 q^{35} + 17724 q^{37} + 9240 q^{38} + 15276 q^{40} + 5994 q^{41} + 26208 q^{43} + 16806 q^{46} + 17232 q^{47} + 48531 q^{49} + 41070 q^{50} + 35304 q^{52} - 50586 q^{53} + 42312 q^{56} - 29172 q^{58} + 3738 q^{59} - 18486 q^{61} - 19974 q^{62} - 20352 q^{64} - 7668 q^{65} - 47754 q^{67} - 12600 q^{68} - 123372 q^{70} - 39282 q^{71} - 15426 q^{73} + 153294 q^{74} - 103920 q^{76} - 125148 q^{79} - 118680 q^{80} - 255372 q^{82} - 143928 q^{83} + 104040 q^{85} - 243060 q^{86} + 106824 q^{89} - 109632 q^{91} + 336528 q^{92} + 74928 q^{94} - 22200 q^{95} + 9684 q^{97} + 3480 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 52x - 38 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 3\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} - 3\nu - 34 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 3\beta_{2} + \beta _1 + 35 \) 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
−0.749680
8.04796
−6.29828
−10.3963 0 76.0833 −8.64919 0 −164.454 −458.304 0 89.9197
1.2 2.20859 0 −27.1221 −75.2230 0 225.525 −130.577 0 −166.137
1.3 8.18772 0 35.0388 59.8722 0 −145.071 24.8808 0 490.217
\(n\): 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 1089.6.a.r 3
3.b odd 2 1 121.6.a.d 3
11.b odd 2 1 99.6.a.g 3
33.d even 2 1 11.6.a.b 3
132.d odd 2 1 176.6.a.i 3
165.d even 2 1 275.6.a.b 3
165.l odd 4 2 275.6.b.b 6
231.h odd 2 1 539.6.a.e 3
264.m even 2 1 704.6.a.q 3
264.p odd 2 1 704.6.a.t 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.6.a.b 3 33.d even 2 1
99.6.a.g 3 11.b odd 2 1
121.6.a.d 3 3.b odd 2 1
176.6.a.i 3 132.d odd 2 1
275.6.a.b 3 165.d even 2 1
275.6.b.b 6 165.l odd 4 2
539.6.a.e 3 231.h odd 2 1
704.6.a.q 3 264.m even 2 1
704.6.a.t 3 264.p odd 2 1
1089.6.a.r 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(1089))\):

\( T_{2}^{3} - 90T_{2} + 188 \) Copy content Toggle raw display
\( T_{5}^{3} + 24T_{5}^{2} - 4371T_{5} - 38954 \) Copy content Toggle raw display
\( T_{7}^{3} + 84T_{7}^{2} - 45948T_{7} - 5380448 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - 90T + 188 \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( T^{3} + 24 T^{2} + \cdots - 38954 \) Copy content Toggle raw display
$7$ \( T^{3} + 84 T^{2} + \cdots - 5380448 \) Copy content Toggle raw display
$11$ \( T^{3} \) Copy content Toggle raw display
$13$ \( T^{3} + 486 T^{2} + \cdots - 164136608 \) Copy content Toggle raw display
$17$ \( T^{3} - 1086 T^{2} + \cdots + 331752056 \) Copy content Toggle raw display
$19$ \( T^{3} + 1380 T^{2} + \cdots - 57024000 \) Copy content Toggle raw display
$23$ \( T^{3} + \cdots + 17004325928 \) Copy content Toggle raw display
$29$ \( T^{3} + \cdots - 4029189120 \) Copy content Toggle raw display
$31$ \( T^{3} + \cdots + 1094344400 \) Copy content Toggle raw display
$37$ \( T^{3} + \cdots + 541788167034 \) Copy content Toggle raw display
$41$ \( T^{3} + \cdots + 201929821568 \) Copy content Toggle raw display
$43$ \( T^{3} + \cdots + 2443875098544 \) Copy content Toggle raw display
$47$ \( T^{3} + \cdots + 70174939136 \) Copy content Toggle raw display
$53$ \( T^{3} + \cdots + 1850911309656 \) Copy content Toggle raw display
$59$ \( T^{3} + \cdots - 7759637437060 \) Copy content Toggle raw display
$61$ \( T^{3} + \cdots - 15233874751008 \) Copy content Toggle raw display
$67$ \( T^{3} + \cdots - 147288561330212 \) Copy content Toggle raw display
$71$ \( T^{3} + \cdots + 1290398551704 \) Copy content Toggle raw display
$73$ \( T^{3} + \cdots + 34539701265952 \) Copy content Toggle raw display
$79$ \( T^{3} + \cdots - 1279883216320 \) Copy content Toggle raw display
$83$ \( T^{3} + \cdots - 411597824719824 \) Copy content Toggle raw display
$89$ \( T^{3} + \cdots + 90320980174650 \) Copy content Toggle raw display
$97$ \( T^{3} + \cdots - 10221902527106 \) Copy content Toggle raw display
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