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Properties

Label 8954.2.a.bq
Level $8954$
Weight $2$
Character orbit 8954.a
Self dual yes
Analytic conductor $71.498$
Analytic rank $1$
Dimension $16$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [8954,2,Mod(1,8954)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8954.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8954, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 8954 = 2 \cdot 11^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8954.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,-16,0,16,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(71.4980499699\)
Analytic rank: \(1\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 35 x^{14} - 6 x^{13} + 477 x^{12} + 154 x^{11} - 3180 x^{10} - 1388 x^{9} + 10627 x^{8} + \cdots + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} - \beta_1 q^{3} + q^{4} - \beta_{10} q^{5} + \beta_1 q^{6} + \beta_{11} q^{7} - q^{8} + (\beta_{2} + 1) q^{9} + \beta_{10} q^{10} - \beta_1 q^{12} + ( - \beta_{3} - 1) q^{13} - \beta_{11} q^{14}+ \cdots + ( - 2 \beta_{14} - \beta_{12} + \cdots - 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{2} + 16 q^{4} + 6 q^{5} - 2 q^{7} - 16 q^{8} + 22 q^{9} - 6 q^{10} - 16 q^{13} + 2 q^{14} + 16 q^{15} + 16 q^{16} - 6 q^{17} - 22 q^{18} - 26 q^{19} + 6 q^{20} + 4 q^{23} + 26 q^{25} + 16 q^{26}+ \cdots - 6 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 35 x^{14} - 6 x^{13} + 477 x^{12} + 154 x^{11} - 3180 x^{10} - 1388 x^{9} + 10627 x^{8} + \cdots + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 92189 \nu^{15} - 554150 \nu^{14} + 4233440 \nu^{13} + 17705972 \nu^{12} - 71252593 \nu^{11} + \cdots - 15921424 ) / 2516688 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 305497 \nu^{15} - 140331 \nu^{14} + 11034938 \nu^{13} + 6045172 \nu^{12} - 155334713 \nu^{11} + \cdots + 13514476 ) / 3355584 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1021375 \nu^{15} + 206749 \nu^{14} - 35370454 \nu^{13} - 13794556 \nu^{12} + 473584175 \nu^{11} + \cdots - 2435092 ) / 10066752 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1314641 \nu^{15} + 173303 \nu^{14} - 45922346 \nu^{13} - 14011700 \nu^{12} + 623042305 \nu^{11} + \cdots - 1128908 ) / 10066752 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1490825 \nu^{15} - 3022295 \nu^{14} + 56949242 \nu^{13} + 104171444 \nu^{12} - 837787705 \nu^{11} + \cdots + 10199660 ) / 10066752 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 755429 \nu^{15} - 673099 \nu^{14} - 24407402 \nu^{13} + 14729884 \nu^{12} + 302470573 \nu^{11} + \cdots - 57225932 ) / 5033376 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 630969 \nu^{15} - 647245 \nu^{14} - 20590314 \nu^{13} + 15746140 \nu^{12} + 259145577 \nu^{11} + \cdots + 20975188 ) / 3355584 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 2326655 \nu^{15} + 1524151 \nu^{14} + 77274590 \nu^{13} - 30877252 \nu^{12} - 991752127 \nu^{11} + \cdots + 23144516 ) / 10066752 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 2334845 \nu^{15} - 387083 \nu^{14} + 80537162 \nu^{13} + 29406596 \nu^{12} - 1073920525 \nu^{11} + \cdots + 52953932 ) / 10066752 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 627037 \nu^{15} - 119293 \nu^{14} + 21665254 \nu^{13} + 8362012 \nu^{12} - 289460909 \nu^{11} + \cdots + 14553124 ) / 2516688 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 4543861 \nu^{15} + 2368037 \nu^{14} + 152679034 \nu^{13} - 42328796 \nu^{12} - 1987269413 \nu^{11} + \cdots - 18531092 ) / 10066752 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 2524877 \nu^{15} + 19157 \nu^{14} - 86982458 \nu^{13} - 18364772 \nu^{12} + 1161106453 \nu^{11} + \cdots - 18023948 ) / 5033376 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 6279689 \nu^{15} - 30565 \nu^{14} - 217317794 \nu^{13} - 41038724 \nu^{12} + 2918863609 \nu^{11} + \cdots - 29436668 ) / 10066752 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} + \beta_{8} + \beta_{7} + \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} + 7\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 2 \beta_{15} + \beta_{14} - \beta_{13} - \beta_{12} + \beta_{11} - \beta_{10} - \beta_{8} + 2 \beta_{6} + \cdots + 29 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{15} + \beta_{14} + \beta_{13} + 14 \beta_{10} + 14 \beta_{8} + 15 \beta_{7} + 7 \beta_{6} + \cdots + 30 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 29 \beta_{15} + 15 \beta_{14} - 13 \beta_{13} - 20 \beta_{12} + 18 \beta_{11} - 11 \beta_{10} + \cdots + 249 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 21 \beta_{15} + 15 \beta_{14} + 17 \beta_{13} - 6 \beta_{12} + 2 \beta_{11} + 157 \beta_{10} + \cdots + 350 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 336 \beta_{15} + 176 \beta_{14} - 142 \beta_{13} - 277 \beta_{12} + 241 \beta_{11} - 87 \beta_{10} + \cdots + 2344 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 313 \beta_{15} + 181 \beta_{14} + 222 \beta_{13} - 136 \beta_{12} + 61 \beta_{11} + 1643 \beta_{10} + \cdots + 3861 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 3635 \beta_{15} + 1899 \beta_{14} - 1495 \beta_{13} - 3356 \beta_{12} + 2899 \beta_{11} + \cdots + 23157 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 4055 \beta_{15} + 2017 \beta_{14} + 2631 \beta_{13} - 2151 \beta_{12} + 1185 \beta_{11} + \cdots + 42068 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 38259 \beta_{15} + 19698 \beta_{14} - 15555 \beta_{13} - 38290 \beta_{12} + 33236 \beta_{11} + \cdots + 234740 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 48888 \beta_{15} + 21490 \beta_{14} + 29707 \beta_{13} - 29540 \beta_{12} + 18923 \beta_{11} + \cdots + 457647 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 397747 \beta_{15} + 199883 \beta_{14} - 160647 \beta_{13} - 424001 \beta_{12} + 372052 \beta_{11} + \cdots + 2414169 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 565724 \beta_{15} + 222012 \beta_{14} + 326015 \beta_{13} - 378323 \beta_{12} + 271760 \beta_{11} + \cdots + 4980311 \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
3.28116
3.22366
2.52535
2.47910
1.58741
0.397406
0.308435
0.202711
−0.0620558
−0.206869
−1.59503
−1.76614
−2.12478
−2.12787
−2.99175
−3.13074
−1.00000 −3.28116 1.00000 −2.64619 3.28116 1.85316 −1.00000 7.76600 2.64619
1.2 −1.00000 −3.22366 1.00000 −0.552141 3.22366 −3.98554 −1.00000 7.39201 0.552141
1.3 −1.00000 −2.52535 1.00000 3.87435 2.52535 1.57017 −1.00000 3.37737 −3.87435
1.4 −1.00000 −2.47910 1.00000 −3.68181 2.47910 −2.00980 −1.00000 3.14596 3.68181
1.5 −1.00000 −1.58741 1.00000 −1.47832 1.58741 −0.0881997 −1.00000 −0.480127 1.47832
1.6 −1.00000 −0.397406 1.00000 3.21229 0.397406 4.98156 −1.00000 −2.84207 −3.21229
1.7 −1.00000 −0.308435 1.00000 3.77258 0.308435 −3.89821 −1.00000 −2.90487 −3.77258
1.8 −1.00000 −0.202711 1.00000 −1.46458 0.202711 0.651059 −1.00000 −2.95891 1.46458
1.9 −1.00000 0.0620558 1.00000 0.0863559 −0.0620558 2.48538 −1.00000 −2.99615 −0.0863559
1.10 −1.00000 0.206869 1.00000 1.61802 −0.206869 −1.03754 −1.00000 −2.95721 −1.61802
1.11 −1.00000 1.59503 1.00000 4.31537 −1.59503 −1.48574 −1.00000 −0.455877 −4.31537
1.12 −1.00000 1.76614 1.00000 −0.807863 −1.76614 5.09241 −1.00000 0.119247 0.807863
1.13 −1.00000 2.12478 1.00000 2.36029 −2.12478 −3.28912 −1.00000 1.51470 −2.36029
1.14 −1.00000 2.12787 1.00000 −3.50101 −2.12787 −2.17195 −1.00000 1.52783 3.50101
1.15 −1.00000 2.99175 1.00000 1.19401 −2.99175 −0.872626 −1.00000 5.95059 −1.19401
1.16 −1.00000 3.13074 1.00000 −0.301345 −3.13074 0.204990 −1.00000 6.80151 0.301345
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(11\) \( +1 \)
\(37\) \( +1 \)

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 8954.2.a.bq ✓ 16
11.b odd 2 1 8954.2.a.br yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8954.2.a.bq ✓ 16 1.a even 1 1 trivial
8954.2.a.br yes 16 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8954))\):

\( T_{3}^{16} - 35 T_{3}^{14} + 6 T_{3}^{13} + 477 T_{3}^{12} - 154 T_{3}^{11} - 3180 T_{3}^{10} + 1388 T_{3}^{9} + \cdots + 4 \) Copy content Toggle raw display
\( T_{5}^{16} - 6 T_{5}^{15} - 35 T_{5}^{14} + 240 T_{5}^{13} + 428 T_{5}^{12} - 3586 T_{5}^{11} + \cdots + 792 \) Copy content Toggle raw display
\( T_{7}^{16} + 2 T_{7}^{15} - 57 T_{7}^{14} - 150 T_{7}^{13} + 1059 T_{7}^{12} + 3460 T_{7}^{11} + \cdots - 648 \) Copy content Toggle raw display
\( T_{17}^{16} + 6 T_{17}^{15} - 163 T_{17}^{14} - 992 T_{17}^{13} + 9580 T_{17}^{12} + 59912 T_{17}^{11} + \cdots - 2870316 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{16} \) Copy content Toggle raw display
$3$ \( T^{16} - 35 T^{14} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( T^{16} - 6 T^{15} + \cdots + 792 \) Copy content Toggle raw display
$7$ \( T^{16} + 2 T^{15} + \cdots - 648 \) Copy content Toggle raw display
$11$ \( T^{16} \) Copy content Toggle raw display
$13$ \( T^{16} + 16 T^{15} + \cdots - 355212 \) Copy content Toggle raw display
$17$ \( T^{16} + 6 T^{15} + \cdots - 2870316 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 2001139893 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 374780736 \) Copy content Toggle raw display
$29$ \( T^{16} + 20 T^{15} + \cdots - 4074624 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots - 939554708711 \) Copy content Toggle raw display
$37$ \( (T + 1)^{16} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 11064233232 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 42084178209 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots - 14513863932 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 4054535946189 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots - 236221410836268 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 163558336896 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots - 47566806188 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 1721884741632 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 749538009 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 129113138398464 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots - 223231907232 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 23314766075712 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots - 4722708982412 \) Copy content Toggle raw display
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