↑

Properties

Label 1089.6.a.bh
Level $1089$
Weight $6$
Character orbit 1089.a
Self dual yes
Analytic conductor $174.658$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} - 252 x^{8} + 45 x^{7} + 21644 x^{6} + 14121 x^{5} - 727612 x^{4} - 1049829 x^{3} + \cdots - 5072980 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 11^{4} \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} + (\beta_{2} - \beta_1 + 19) q^{4} + (\beta_{7} + 2 \beta_1 - 1) q^{5} + (\beta_{8} - \beta_{7} + \beta_{5} + \cdots + 46) q^{7}+ \cdots + (\beta_{9} + 2 \beta_{7} + \beta_{5} + \cdots - 34) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{2} + (\beta_{2} - \beta_1 + 19) q^{4} + (\beta_{7} + 2 \beta_1 - 1) q^{5} + (\beta_{8} - \beta_{7} + \beta_{5} + \cdots + 46) q^{7}+ \cdots + (247 \beta_{9} + 174 \beta_{8} + \cdots + 5829) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 9 q^{2} + 193 q^{4} - 11 q^{5} + 470 q^{7} - 324 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 9 q^{2} + 193 q^{4} - 11 q^{5} + 470 q^{7} - 324 q^{8} + 976 q^{10} + 2308 q^{13} - 540 q^{14} + 2801 q^{16} - 3093 q^{17} + 5305 q^{19} + 2229 q^{20} - 700 q^{23} + 8381 q^{25} + 15559 q^{26} + 24656 q^{28} - 10392 q^{29} - 101 q^{31} - 34557 q^{32} - 4542 q^{34} - 19867 q^{35} - 1284 q^{37} + 35769 q^{38} + 66596 q^{40} - 17944 q^{41} + 31812 q^{43} + 36417 q^{46} - 8787 q^{47} - 23810 q^{49} + 910 q^{50} + 51663 q^{52} + 3261 q^{53} - 84819 q^{56} + 53125 q^{58} - 49375 q^{59} + 63175 q^{61} - 28399 q^{62} + 124764 q^{64} + 14105 q^{65} + 5365 q^{67} + 53313 q^{68} + 40297 q^{70} + 236675 q^{71} + 200912 q^{73} + 180329 q^{74} + 39606 q^{76} + 210802 q^{79} - 270298 q^{80} - 369223 q^{82} + 178968 q^{83} + 107352 q^{85} - 465999 q^{86} - 90816 q^{89} + 7500 q^{91} + 136407 q^{92} + 71890 q^{94} - 335807 q^{95} + 271521 q^{97} + 75285 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} - 252 x^{8} + 45 x^{7} + 21644 x^{6} + 14121 x^{5} - 727612 x^{4} - 1049829 x^{3} + \cdots - 5072980 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 50 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 841691 \nu^{9} + 5068686 \nu^{8} + 181569926 \nu^{7} - 978567189 \nu^{6} + \cdots + 4876453545436 ) / 395536590592 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 5156577 \nu^{9} + 25451154 \nu^{8} + 1261949082 \nu^{7} - 3662022167 \nu^{6} + \cdots - 65006231435948 ) / 471601319552 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 133133015 \nu^{9} + 1020373306 \nu^{8} + 30067762042 \nu^{7} - 201647414909 \nu^{6} + \cdots + 740426530922188 ) / 6130817154176 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 52924412 \nu^{9} + 393062283 \nu^{8} + 11865701963 \nu^{7} - 80616808825 \nu^{6} + \cdots + 146978875604572 ) / 1532704288544 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1035550053 \nu^{9} + 4298056786 \nu^{8} + 242682292218 \nu^{7} - 810627771947 \nu^{6} + \cdots + 257946287954916 ) / 12261634308352 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 820567249 \nu^{9} + 3369750322 \nu^{8} + 196956527714 \nu^{7} - 653216718863 \nu^{6} + \cdots + 10\!\cdots\!68 ) / 6130817154176 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 175505569 \nu^{9} - 803578306 \nu^{8} - 40736571962 \nu^{7} + 155411118919 \nu^{6} + \cdots - 368483419473620 ) / 943202639104 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 50 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} + 2\beta_{7} + \beta_{5} - \beta_{4} + 3\beta_{3} + 82\beta _1 + 53 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} + \beta_{8} + \beta_{7} - 9 \beta_{6} + 14 \beta_{5} - 7 \beta_{4} + 24 \beta_{3} + \cdots + 4054 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 133 \beta_{9} + 13 \beta_{8} + 241 \beta_{7} - 13 \beta_{6} + 134 \beta_{5} - 131 \beta_{4} + \cdots + 7174 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 234 \beta_{9} + 117 \beta_{8} + 351 \beta_{7} - 1549 \beta_{6} + 2427 \beta_{5} - 944 \beta_{4} + \cdots + 378533 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 15308 \beta_{9} + 3622 \beta_{8} + 23926 \beta_{7} - 2958 \beta_{6} + 16162 \beta_{5} - 13976 \beta_{4} + \cdots + 905864 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 38080 \beta_{9} + 15400 \beta_{8} + 53756 \beta_{7} - 203024 \beta_{6} + 332528 \beta_{5} + \cdots + 37289562 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1712341 \beta_{9} + 628148 \beta_{8} + 2255074 \beta_{7} - 479436 \beta_{6} + 1967041 \beta_{5} + \cdots + 111982649 \) 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
−9.96090
−8.80939
−5.30415
−3.80353
−3.21290
0.269818
4.80749
6.64018
9.71912
10.6543
−10.9609 0 88.1414 −27.1711 0 136.430 −615.361 0 297.820
1.2 −9.80939 0 64.2242 −46.0745 0 120.590 −316.100 0 451.963
1.3 −6.30415 0 7.74233 67.6152 0 −122.486 152.924 0 −426.256
1.4 −4.80353 0 −8.92614 −59.7896 0 −72.0300 196.590 0 287.201
1.5 −4.21290 0 −14.2515 87.2845 0 140.921 194.853 0 −367.721
1.6 −0.730182 0 −31.4668 −95.8360 0 95.7058 46.3424 0 69.9777
1.7 3.80749 0 −17.5030 25.4239 0 −87.0774 −188.482 0 96.8014
1.8 5.64018 0 −0.188416 −68.1325 0 131.688 −181.548 0 −384.279
1.9 8.71912 0 44.0230 74.6070 0 −57.2872 104.830 0 650.507
1.10 9.65427 0 61.2049 31.0730 0 183.547 281.952 0 299.987
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
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.bh 10
3.b odd 2 1 363.6.a.u 10
11.b odd 2 1 1089.6.a.bl 10
11.c even 5 2 99.6.f.c 20
33.d even 2 1 363.6.a.q 10
33.h odd 10 2 33.6.e.a ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.6.e.a ✓ 20 33.h odd 10 2
99.6.f.c 20 11.c even 5 2
363.6.a.q 10 33.d even 2 1
363.6.a.u 10 3.b odd 2 1
1089.6.a.bh 10 1.a even 1 1 trivial
1089.6.a.bl 10 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(1089))\):

\( T_{2}^{10} + 9 T_{2}^{9} - 216 T_{2}^{8} - 1887 T_{2}^{7} + 15029 T_{2}^{6} + 130944 T_{2}^{5} + \cdots + 18105536 \) Copy content Toggle raw display
\( T_{5}^{10} + 11 T_{5}^{9} - 19755 T_{5}^{8} - 186994 T_{5}^{7} + 136714125 T_{5}^{6} + \cdots - 17\!\cdots\!41 \) Copy content Toggle raw display
\( T_{7}^{10} - 470 T_{7}^{9} + 38320 T_{7}^{8} + 12138682 T_{7}^{7} - 1934695405 T_{7}^{6} + \cdots + 23\!\cdots\!45 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 9 T^{9} + \cdots + 18105536 \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots - 17\!\cdots\!41 \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 23\!\cdots\!45 \) Copy content Toggle raw display
$11$ \( T^{10} \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 12\!\cdots\!96 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots - 17\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots - 11\!\cdots\!20 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots - 37\!\cdots\!20 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 11\!\cdots\!16 \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots + 20\!\cdots\!25 \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 10\!\cdots\!20 \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots - 17\!\cdots\!20 \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots - 17\!\cdots\!80 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 28\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 27\!\cdots\!25 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots - 11\!\cdots\!29 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots - 20\!\cdots\!24 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots - 38\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots - 81\!\cdots\!20 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots - 29\!\cdots\!20 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots - 27\!\cdots\!95 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots - 16\!\cdots\!75 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots - 89\!\cdots\!36 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 21\!\cdots\!69 \) Copy content Toggle raw display
show more
show less