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Properties

Label 1089.6.a.bo
Level $1089$
Weight $6$
Character orbit 1089.a
Self dual yes
Analytic conductor $174.658$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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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: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 469 x^{18} + 90890 x^{16} - 9428033 x^{14} + 567050771 x^{12} - 20037943016 x^{10} + \cdots + 14645220356096 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{8}\cdot 5^{2}\cdot 11^{9}\cdot 31^{2} \)
Twist minimal: no (minimal twist has level 99)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 15) q^{4} + (\beta_{8} + \beta_1) q^{5} + ( - \beta_{7} + \beta_{4} + 23) q^{7} + (\beta_{3} + 17 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + 15) q^{4} + (\beta_{8} + \beta_1) q^{5} + ( - \beta_{7} + \beta_{4} + 23) q^{7} + (\beta_{3} + 17 \beta_1) q^{8} + ( - \beta_{10} - 5 \beta_{4} + \cdots + 44) q^{10}+ \cdots + (36 \beta_{19} - \beta_{12} + \cdots - 3334 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 298 q^{4} + 472 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 298 q^{4} + 472 q^{7} + 836 q^{10} + 1816 q^{13} + 6794 q^{16} + 6478 q^{19} + 15250 q^{25} + 12264 q^{28} + 1226 q^{31} + 27548 q^{34} - 1032 q^{37} + 15224 q^{40} + 6128 q^{43} + 6710 q^{46} + 10268 q^{49} + 180082 q^{52} + 27898 q^{58} + 103962 q^{61} + 16948 q^{64} + 182882 q^{67} - 160154 q^{70} + 132276 q^{73} + 448700 q^{76} + 365012 q^{79} - 492270 q^{82} + 451616 q^{85} + 312092 q^{91} + 830808 q^{94} - 233530 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 469 x^{18} + 90890 x^{16} - 9428033 x^{14} + 567050771 x^{12} - 20037943016 x^{10} + \cdots + 14645220356096 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 47 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 81\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 79776697385 \nu^{18} + 36939765919801 \nu^{16} + \cdots + 15\!\cdots\!96 ) / 18\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 18\!\cdots\!63 \nu^{19} + \cdots + 25\!\cdots\!84 \nu ) / 73\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 79776697385 \nu^{19} - 36939765919801 \nu^{17} + \cdots - 15\!\cdots\!96 \nu ) / 18\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 41\!\cdots\!91 \nu^{18} + \cdots + 31\!\cdots\!20 ) / 18\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 63\!\cdots\!81 \nu^{19} + \cdots + 16\!\cdots\!92 \nu ) / 73\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 87\!\cdots\!29 \nu^{19} + \cdots + 58\!\cdots\!08 \nu ) / 73\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 58\!\cdots\!69 \nu^{18} + \cdots + 10\!\cdots\!56 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 80\!\cdots\!89 \nu^{19} + \cdots - 24\!\cdots\!16 \nu ) / 18\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 13\!\cdots\!12 \nu^{19} + \cdots - 84\!\cdots\!48 \nu ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 1091333207777 \nu^{18} - 504599617216753 \nu^{16} + \cdots - 20\!\cdots\!84 ) / 62\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 23\!\cdots\!83 \nu^{18} + \cdots - 54\!\cdots\!16 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 27\!\cdots\!89 \nu^{19} + \cdots + 12\!\cdots\!44 \nu ) / 24\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 12\!\cdots\!79 \nu^{18} + \cdots + 28\!\cdots\!20 ) / 36\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 13\!\cdots\!97 \nu^{18} + \cdots - 30\!\cdots\!28 ) / 36\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 26\!\cdots\!95 \nu^{18} + \cdots + 49\!\cdots\!56 ) / 36\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 22\!\cdots\!53 \nu^{19} + \cdots - 48\!\cdots\!96 \nu ) / 73\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 47 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 81\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{17} + \beta_{14} - \beta_{10} - 3\beta_{7} - 13\beta_{4} + 110\beta_{2} + 3835 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 4 \beta_{19} - 5 \beta_{15} + 7 \beta_{11} + 5 \beta_{9} + 27 \beta_{8} + 13 \beta_{6} + \cdots + 7541 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 5 \beta_{18} - 143 \beta_{17} + 9 \beta_{16} + 179 \beta_{14} - 31 \beta_{13} - 146 \beta_{10} + \cdots + 358743 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 750 \beta_{19} - 969 \beta_{15} - 65 \beta_{12} + 1323 \beta_{11} + 926 \beta_{9} + \cdots + 749411 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 530 \beta_{18} - 16622 \beta_{17} + 2740 \beta_{16} + 24242 \beta_{14} - 8260 \beta_{13} + \cdots + 35789161 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 106000 \beta_{19} - 138902 \beta_{15} - 15124 \beta_{12} + 183882 \beta_{11} + 131114 \beta_{9} + \cdots + 77363097 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 15650 \beta_{18} - 1841033 \beta_{17} + 496762 \beta_{16} + 2984417 \beta_{14} - 1484742 \beta_{13} + \cdots + 3706528755 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 13617496 \beta_{19} - 17869401 \beta_{15} - 2321818 \beta_{12} + 22805619 \beta_{11} + \cdots + 8177801409 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 5151517 \beta_{18} - 202168445 \beta_{17} + 73770915 \beta_{16} + 351931025 \beta_{14} + \cdots + 392876227851 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 1677798458 \beta_{19} - 2187435755 \beta_{15} - 299506027 \beta_{12} + 2677517889 \beta_{11} + \cdots + 877984912215 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 1421414658 \beta_{18} - 22255120288 \beta_{17} + 9941830820 \beta_{16} + 40587605740 \beta_{14} + \cdots + 42278189766453 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 202405148504 \beta_{19} - 261020770816 \beta_{15} - 35229616644 \beta_{12} + \cdots + 95278787627905 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 253509857076 \beta_{18} - 2461575336085 \beta_{17} + 1270430587100 \beta_{16} + 4623366550645 \beta_{14} + \cdots + 45\!\cdots\!23 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 24128749990476 \beta_{19} - 30712646679953 \beta_{15} - 3918851954860 \beta_{12} + \cdots + 10\!\cdots\!73 \beta_1 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 38537247549859 \beta_{18} - 273494967101659 \beta_{17} + 157308009046969 \beta_{16} + \cdots + 50\!\cdots\!31 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 28\!\cdots\!38 \beta_{19} + \cdots + 11\!\cdots\!99 \beta_1 \) 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
−10.6821
−10.2155
−9.05157
−7.90110
−6.67673
−6.22592
−2.91854
−2.72296
−2.56323
−0.579109
0.579109
2.56323
2.72296
2.91854
6.22592
6.67673
7.90110
9.05157
10.2155
10.6821
−10.6821 0 82.1066 −69.8551 0 −0.319155 −535.243 0 746.198
1.2 −10.2155 0 72.3564 71.8022 0 199.773 −412.261 0 −733.495
1.3 −9.05157 0 49.9308 −79.2093 0 −58.3107 −162.302 0 716.968
1.4 −7.90110 0 30.4274 52.8238 0 −65.5054 12.4256 0 −417.366
1.5 −6.67673 0 12.5788 36.0382 0 −85.7719 129.670 0 −240.618
1.6 −6.22592 0 6.76211 −7.30063 0 136.582 157.129 0 45.4532
1.7 −2.91854 0 −23.4821 −84.8960 0 219.333 161.927 0 247.772
1.8 −2.72296 0 −24.5855 29.0218 0 −69.8168 154.080 0 −79.0253
1.9 −2.56323 0 −25.4299 −29.5179 0 −171.330 147.206 0 75.6611
1.10 −0.579109 0 −31.6646 −97.4796 0 131.366 36.8688 0 56.4513
1.11 0.579109 0 −31.6646 97.4796 0 131.366 −36.8688 0 56.4513
1.12 2.56323 0 −25.4299 29.5179 0 −171.330 −147.206 0 75.6611
1.13 2.72296 0 −24.5855 −29.0218 0 −69.8168 −154.080 0 −79.0253
1.14 2.91854 0 −23.4821 84.8960 0 219.333 −161.927 0 247.772
1.15 6.22592 0 6.76211 7.30063 0 136.582 −157.129 0 45.4532
1.16 6.67673 0 12.5788 −36.0382 0 −85.7719 −129.670 0 −240.618
1.17 7.90110 0 30.4274 −52.8238 0 −65.5054 −12.4256 0 −417.366
1.18 9.05157 0 49.9308 79.2093 0 −58.3107 162.302 0 716.968
1.19 10.2155 0 72.3564 −71.8022 0 199.773 412.261 0 −733.495
1.20 10.6821 0 82.1066 69.8551 0 −0.319155 535.243 0 746.198
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.20
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1089.6.a.bo 20
3.b odd 2 1 inner 1089.6.a.bo 20
11.b odd 2 1 1089.6.a.bn 20
11.d odd 10 2 99.6.f.d ✓ 40
33.d even 2 1 1089.6.a.bn 20
33.f even 10 2 99.6.f.d ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.6.f.d ✓ 40 11.d odd 10 2
99.6.f.d ✓ 40 33.f even 10 2
1089.6.a.bn 20 11.b odd 2 1
1089.6.a.bn 20 33.d even 2 1
1089.6.a.bo 20 1.a even 1 1 trivial
1089.6.a.bo 20 3.b odd 2 1 inner

Hecke kernels

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

\( T_{2}^{20} - 469 T_{2}^{18} + 90890 T_{2}^{16} - 9428033 T_{2}^{14} + 567050771 T_{2}^{12} + \cdots + 14645220356096 \) Copy content Toggle raw display
\( T_{5}^{20} - 38875 T_{5}^{18} + 634163165 T_{5}^{16} - 5655530404490 T_{5}^{14} + \cdots + 15\!\cdots\!51 \) Copy content Toggle raw display
\( T_{7}^{10} - 236 T_{7}^{9} - 58754 T_{7}^{8} + 12627620 T_{7}^{7} + 1413190217 T_{7}^{6} + \cdots + 98\!\cdots\!19 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + \cdots + 14645220356096 \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 15\!\cdots\!51 \) Copy content Toggle raw display
$7$ \( (T^{10} + \cdots + 98\!\cdots\!19)^{2} \) Copy content Toggle raw display
$11$ \( T^{20} \) Copy content Toggle raw display
$13$ \( (T^{10} + \cdots + 70\!\cdots\!96)^{2} \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 37\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( (T^{10} + \cdots - 71\!\cdots\!36)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 11\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 29\!\cdots\!56 \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots + 12\!\cdots\!31)^{2} \) Copy content Toggle raw display
$37$ \( (T^{10} + \cdots + 14\!\cdots\!36)^{2} \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 32\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots + 13\!\cdots\!76)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 16\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 31\!\cdots\!75 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 89\!\cdots\!71 \) Copy content Toggle raw display
$61$ \( (T^{10} + \cdots + 13\!\cdots\!96)^{2} \) Copy content Toggle raw display
$67$ \( (T^{10} + \cdots - 49\!\cdots\!84)^{2} \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots + 22\!\cdots\!96)^{2} \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots - 27\!\cdots\!75)^{2} \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 54\!\cdots\!71 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots - 12\!\cdots\!71)^{2} \) Copy content Toggle raw display
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