Properties

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

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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: \(0\)
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: \( x^{10} - 3 x^{9} - 228 x^{8} + 523 x^{7} + 17396 x^{6} - 31445 x^{5} - 508100 x^{4} + 960757 x^{3} + 4870759 x^{2} - 11540360 x + 5059564 \) 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.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} + ( - \beta_{3} + \beta_{2} - \beta_1 + 15) q^{4} + ( - \beta_{8} + \beta_{2} + \beta_1 + 3) q^{5} + (\beta_{7} - \beta_{6} - \beta_{5} + \beta_{3} + 4 \beta_1 + 7) q^{7} + (\beta_{6} + \beta_{5} - \beta_{4} - 2 \beta_{3} + \beta_{2} - 15 \beta_1 + 48) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{2} + ( - \beta_{3} + \beta_{2} - \beta_1 + 15) q^{4} + ( - \beta_{8} + \beta_{2} + \beta_1 + 3) q^{5} + (\beta_{7} - \beta_{6} - \beta_{5} + \beta_{3} + 4 \beta_1 + 7) q^{7} + (\beta_{6} + \beta_{5} - \beta_{4} - 2 \beta_{3} + \beta_{2} - 15 \beta_1 + 48) q^{8} + (\beta_{9} - \beta_{8} + \beta_{5} - 2 \beta_{4} + 11 \beta_{3} + \beta_{2} - 25 \beta_1 - 8) q^{10} + (\beta_{9} - \beta_{8} + 4 \beta_{6} + 3 \beta_{4} - 12 \beta_{3} + 2 \beta_{2} + 17 \beta_1 - 112) q^{13} + (4 \beta_{9} - 4 \beta_{8} + 7 \beta_{7} - 6 \beta_{6} - 8 \beta_{5} + 2 \beta_{4} + \cdots - 144) q^{14}+ \cdots + (367 \beta_{9} - 274 \beta_{8} + 309 \beta_{7} - 113 \beta_{6} + \cdots + 117918) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 7 q^{2} + 149 q^{4} + 33 q^{5} + 78 q^{7} + 438 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 7 q^{2} + 149 q^{4} + 33 q^{5} + 78 q^{7} + 438 q^{8} - 212 q^{10} - 1016 q^{13} - 1566 q^{14} + 2361 q^{16} + 1669 q^{17} + 2929 q^{19} + 10189 q^{20} - 4070 q^{23} + 2425 q^{25} - 8481 q^{26} + 3272 q^{28} + 11940 q^{29} - 16085 q^{31} + 2313 q^{32} + 8270 q^{34} - 6987 q^{35} + 16136 q^{37} - 10721 q^{38} + 9332 q^{40} + 16278 q^{41} - 10844 q^{43} - 25995 q^{46} + 22411 q^{47} + 75150 q^{49} - 738 q^{50} - 8677 q^{52} + 27511 q^{53} - 84447 q^{56} + 16853 q^{58} + 39641 q^{59} - 3509 q^{61} + 227845 q^{62} - 22980 q^{64} + 67097 q^{65} + 10089 q^{67} + 273621 q^{68} - 38919 q^{70} - 60681 q^{71} - 133740 q^{73} + 317933 q^{74} + 23434 q^{76} - 12386 q^{79} + 289014 q^{80} + 385033 q^{82} + 187242 q^{83} - 191504 q^{85} + 43793 q^{86} - 102746 q^{89} - 435248 q^{91} - 30867 q^{92} - 404734 q^{94} + 648147 q^{95} - 120631 q^{97} + 1148087 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 3 x^{9} - 228 x^{8} + 523 x^{7} + 17396 x^{6} - 31445 x^{5} - 508100 x^{4} + 960757 x^{3} + 4870759 x^{2} - 11540360 x + 5059564 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 10753 \nu^{9} - 296252 \nu^{8} - 1630248 \nu^{7} + 57573155 \nu^{6} + 73965785 \nu^{5} - 3358085302 \nu^{4} - 1435937310 \nu^{3} + \cdots - 247021531028 ) / 3973778688 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 10753 \nu^{9} - 296252 \nu^{8} - 1630248 \nu^{7} + 57573155 \nu^{6} + 73965785 \nu^{5} - 3358085302 \nu^{4} - 1435937310 \nu^{3} + \cdots - 64227711380 ) / 3973778688 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 519533 \nu^{9} - 14486692 \nu^{8} - 212554608 \nu^{7} + 3386645791 \nu^{6} + 29011252981 \nu^{5} - 231974286038 \nu^{4} + \cdots - 9403256470084 ) / 115239581952 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 48247 \nu^{9} - 4026100 \nu^{8} + 18089376 \nu^{7} + 853512643 \nu^{6} - 1526550335 \nu^{5} - 54131712926 \nu^{4} + \cdots - 749819187028 ) / 10476325632 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 454029 \nu^{9} + 7069700 \nu^{8} - 152938312 \nu^{7} - 1444123929 \nu^{6} + 15982771477 \nu^{5} + 88696694130 \nu^{4} + \cdots - 3502807289892 ) / 38413193984 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1679081 \nu^{9} + 25969108 \nu^{8} + 188783208 \nu^{7} - 4628495563 \nu^{6} + 6141870071 \nu^{5} + 236898347246 \nu^{4} + \cdots - 24594764837372 ) / 57619790976 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 2254471 \nu^{9} + 653136 \nu^{8} + 491336420 \nu^{7} + 263165775 \nu^{6} - 33881768431 \nu^{5} - 37321971170 \nu^{4} + \cdots + 2117426070700 ) / 28809895488 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 18645735 \nu^{9} - 4735412 \nu^{8} + 4158442448 \nu^{7} + 3646561091 \nu^{6} - 294057013535 \nu^{5} - 336750092382 \nu^{4} + \cdots + 58402093950476 ) / 115239581952 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{3} + \beta_{2} + \beta _1 + 46 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + 2\beta_{2} + 79\beta _1 + 27 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3 \beta_{9} - 6 \beta_{8} - 2 \beta_{7} + 4 \beta_{6} + 4 \beta_{5} + \beta_{4} - 141 \beta_{3} + 113 \beta_{2} + 189 \beta _1 + 3545 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 17 \beta_{9} - 50 \beta_{8} + 22 \beta_{7} - 168 \beta_{6} - 124 \beta_{5} + 131 \beta_{4} - 186 \beta_{3} + 298 \beta_{2} + 7311 \beta _1 + 6463 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 391 \beta_{9} - 886 \beta_{8} - 234 \beta_{7} + 293 \beta_{6} + 709 \beta_{5} + 236 \beta_{4} - 17026 \beta_{3} + 11797 \beta_{2} + 25281 \beta _1 + 321948 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 3378 \beta_{9} - 10236 \beta_{8} + 4192 \beta_{7} - 22566 \beta_{6} - 12442 \beta_{5} + 14408 \beta_{4} - 34284 \beta_{3} + 38786 \beta_{2} + 723351 \beta _1 + 914046 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 45416 \beta_{9} - 115856 \beta_{8} - 19212 \beta_{7} - 2084 \beta_{6} + 92880 \beta_{5} + 39144 \beta_{4} - 1964949 \beta_{3} + 1225193 \beta_{2} + 3066769 \beta _1 + 31352122 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 489844 \beta_{9} - 1529824 \beta_{8} + 583196 \beta_{7} - 2776141 \beta_{6} - 1154921 \beta_{5} + 1543973 \beta_{4} - 5524037 \beta_{3} + 4811154 \beta_{2} + 73875371 \beta _1 + 113088743 \) 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
10.6217
9.48352
4.81631
4.35046
1.70765
0.605686
−5.33226
−5.81452
−7.59133
−9.84718
−9.62167 0 60.5765 59.5053 0 30.5377 −274.954 0 −572.540
1.2 −8.48352 0 39.9701 26.7516 0 72.2591 −67.6142 0 −226.948
1.3 −3.81631 0 −17.4358 4.66892 0 159.925 188.662 0 −17.8180
1.4 −3.35046 0 −20.7744 −106.493 0 40.5248 176.818 0 356.801
1.5 −0.707653 0 −31.4992 68.0022 0 −121.575 44.9354 0 −48.1219
1.6 0.394314 0 −31.8445 −21.3907 0 −169.303 −25.1748 0 −8.43465
1.7 6.33226 0 8.09756 −28.0024 0 −150.231 −151.357 0 −177.319
1.8 6.81452 0 14.4377 −77.3762 0 192.167 −119.679 0 −527.281
1.9 8.59133 0 41.8109 68.5404 0 242.743 84.2889 0 588.853
1.10 10.8472 0 85.6613 38.7942 0 −219.048 582.073 0 420.808
\(n\): 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.bk 10
3.b odd 2 1 363.6.a.r 10
11.b odd 2 1 1089.6.a.bi 10
11.c even 5 2 99.6.f.b 20
33.d even 2 1 363.6.a.t 10
33.h odd 10 2 33.6.e.b 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.6.e.b 20 33.h odd 10 2
99.6.f.b 20 11.c even 5 2
363.6.a.r 10 3.b odd 2 1
363.6.a.t 10 33.d even 2 1
1089.6.a.bi 10 11.b odd 2 1
1089.6.a.bk 10 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(1089))\):

\( T_{2}^{10} - 7 T_{2}^{9} - 210 T_{2}^{8} + 1289 T_{2}^{7} + 14631 T_{2}^{6} - 71020 T_{2}^{5} - 402208 T_{2}^{4} + 1032768 T_{2}^{3} + 4655456 T_{2}^{2} + 1000000 T_{2} - 1171136 \) Copy content Toggle raw display
\( T_{5}^{10} - 33 T_{5}^{9} - 16293 T_{5}^{8} + 808962 T_{5}^{7} + 68061617 T_{5}^{6} - 4418342163 T_{5}^{5} - 34870674703 T_{5}^{4} + 4847138355822 T_{5}^{3} + \cdots + 66\!\cdots\!81 \) Copy content Toggle raw display
\( T_{7}^{10} - 78 T_{7}^{9} - 118568 T_{7}^{8} + 7602586 T_{7}^{7} + 4823696363 T_{7}^{6} - 263550403828 T_{7}^{5} - 77804205272685 T_{7}^{4} + \cdots + 45\!\cdots\!81 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} - 7 T^{9} - 210 T^{8} + \cdots - 1171136 \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( T^{10} - 33 T^{9} + \cdots + 66\!\cdots\!81 \) Copy content Toggle raw display
$7$ \( T^{10} - 78 T^{9} + \cdots + 45\!\cdots\!81 \) Copy content Toggle raw display
$11$ \( T^{10} \) Copy content Toggle raw display
$13$ \( T^{10} + 1016 T^{9} + \cdots - 38\!\cdots\!24 \) Copy content Toggle raw display
$17$ \( T^{10} - 1669 T^{9} + \cdots - 31\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{10} - 2929 T^{9} + \cdots - 14\!\cdots\!20 \) Copy content Toggle raw display
$23$ \( T^{10} + 4070 T^{9} + \cdots + 12\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( T^{10} - 11940 T^{9} + \cdots + 36\!\cdots\!80 \) Copy content Toggle raw display
$31$ \( T^{10} + 16085 T^{9} + \cdots + 95\!\cdots\!01 \) Copy content Toggle raw display
$37$ \( T^{10} - 16136 T^{9} + \cdots + 38\!\cdots\!04 \) Copy content Toggle raw display
$41$ \( T^{10} - 16278 T^{9} + \cdots - 73\!\cdots\!44 \) Copy content Toggle raw display
$43$ \( T^{10} + 10844 T^{9} + \cdots + 51\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{10} - 22411 T^{9} + \cdots + 19\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{10} - 27511 T^{9} + \cdots - 40\!\cdots\!69 \) Copy content Toggle raw display
$59$ \( T^{10} - 39641 T^{9} + \cdots + 13\!\cdots\!05 \) Copy content Toggle raw display
$61$ \( T^{10} + 3509 T^{9} + \cdots - 35\!\cdots\!96 \) Copy content Toggle raw display
$67$ \( T^{10} - 10089 T^{9} + \cdots + 33\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( T^{10} + 60681 T^{9} + \cdots - 17\!\cdots\!44 \) Copy content Toggle raw display
$73$ \( T^{10} + 133740 T^{9} + \cdots + 27\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( T^{10} + 12386 T^{9} + \cdots + 23\!\cdots\!45 \) Copy content Toggle raw display
$83$ \( T^{10} - 187242 T^{9} + \cdots + 27\!\cdots\!19 \) Copy content Toggle raw display
$89$ \( T^{10} + 102746 T^{9} + \cdots - 11\!\cdots\!20 \) Copy content Toggle raw display
$97$ \( T^{10} + 120631 T^{9} + \cdots - 69\!\cdots\!75 \) Copy content Toggle raw display
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