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Properties

Label 8954.2.a.bk
Level $8954$
Weight $2$
Character orbit 8954.a
Self dual yes
Analytic conductor $71.498$
Analytic rank $1$
Dimension $12$
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 = [12,-12,-7,12,-4] 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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 5 x^{11} - 7 x^{10} + 57 x^{9} + 4 x^{8} - 225 x^{7} + 46 x^{6} + 361 x^{5} - 70 x^{4} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 814)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + (\beta_1 - 1) q^{3} + q^{4} + \beta_{5} q^{5} + ( - \beta_1 + 1) q^{6} + \beta_{11} q^{7} - q^{8} + ( - \beta_{5} + \beta_{4} - \beta_1 + 1) q^{9} - \beta_{5} q^{10} + (\beta_1 - 1) q^{12}+ \cdots + ( - \beta_{10} + \beta_{9} + \cdots + \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{2} - 7 q^{3} + 12 q^{4} - 4 q^{5} + 7 q^{6} + q^{7} - 12 q^{8} + 5 q^{9} + 4 q^{10} - 7 q^{12} + 7 q^{13} - q^{14} + 12 q^{16} + 17 q^{17} - 5 q^{18} - 8 q^{19} - 4 q^{20} + q^{21} - 18 q^{23}+ \cdots + 21 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 5 x^{11} - 7 x^{10} + 57 x^{9} + 4 x^{8} - 225 x^{7} + 46 x^{6} + 361 x^{5} - 70 x^{4} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 43 \nu^{11} + 1835 \nu^{10} - 409 \nu^{9} - 47645 \nu^{8} + 34411 \nu^{7} + 382771 \nu^{6} + \cdots - 173859 ) / 84062 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 146 \nu^{11} + 5253 \nu^{10} - 20938 \nu^{9} - 32746 \nu^{8} + 210674 \nu^{7} - 1365 \nu^{6} + \cdots + 195570 ) / 84062 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 3126 \nu^{11} + 27834 \nu^{10} - 39508 \nu^{9} - 246843 \nu^{8} + 615092 \nu^{7} + 633482 \nu^{6} + \cdots - 183307 ) / 84062 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 3126 \nu^{11} + 27834 \nu^{10} - 39508 \nu^{9} - 246843 \nu^{8} + 615092 \nu^{7} + 633482 \nu^{6} + \cdots + 68879 ) / 84062 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2373 \nu^{11} - 14272 \nu^{10} - 2843 \nu^{9} + 140713 \nu^{8} - 120998 \nu^{7} - 450231 \nu^{6} + \cdots + 7613 ) / 42031 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 8611 \nu^{11} + 42951 \nu^{10} + 68625 \nu^{9} - 539712 \nu^{8} - 35315 \nu^{7} + 2401913 \nu^{6} + \cdots + 11784 ) / 84062 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 7613 \nu^{11} - 40438 \nu^{10} - 39019 \nu^{9} + 436784 \nu^{8} - 110261 \nu^{7} - 1591927 \nu^{6} + \cdots + 55491 ) / 42031 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 19722 \nu^{11} - 95819 \nu^{10} - 153547 \nu^{9} + 1121377 \nu^{8} + 212594 \nu^{7} - 4596783 \nu^{6} + \cdots + 162265 ) / 84062 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 27403 \nu^{11} + 135249 \nu^{10} + 197784 \nu^{9} - 1537715 \nu^{8} - 176941 \nu^{7} + \cdots - 249307 ) / 84062 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 45462 \nu^{11} - 228523 \nu^{10} - 306546 \nu^{9} + 2576864 \nu^{8} + 50850 \nu^{7} - 9950285 \nu^{6} + \cdots + 321732 ) / 84062 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} + \beta_{4} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} + \beta_{10} - \beta_{9} + \beta_{6} - \beta_{5} + 2\beta_{4} - \beta_{2} + 6\beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} - 4 \beta_{9} + \beta_{8} - \beta_{7} + 4 \beta_{6} - 8 \beta_{5} + 11 \beta_{4} + \beta_{3} + \cdots + 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9 \beta_{11} + 6 \beta_{10} - 18 \beta_{9} + 2 \beta_{8} - 4 \beta_{7} + 20 \beta_{6} - 14 \beta_{5} + \cdots + 58 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 18 \beta_{11} - \beta_{10} - 67 \beta_{9} + 7 \beta_{8} - 20 \beta_{7} + 75 \beta_{6} - 68 \beta_{5} + \cdots + 234 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 84 \beta_{11} + 20 \beta_{10} - 248 \beta_{9} + 12 \beta_{8} - 75 \beta_{7} + 301 \beta_{6} - 162 \beta_{5} + \cdots + 689 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 225 \beta_{11} - 54 \beta_{10} - 890 \beta_{9} + 11 \beta_{8} - 301 \beta_{7} + 1096 \beta_{6} + \cdots + 2464 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 840 \beta_{11} - 129 \beta_{10} - 3141 \beta_{9} - 54 \beta_{8} - 1096 \beta_{7} + 4075 \beta_{6} + \cdots + 7841 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2538 \beta_{11} - 1168 \beta_{10} - 11032 \beta_{9} - 510 \beta_{8} - 4075 \beta_{7} + 14544 \beta_{6} + \cdots + 26937 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 8755 \beta_{11} - 4177 \beta_{10} - 38297 \beta_{9} - 2481 \beta_{8} - 14544 \beta_{7} + 52150 \beta_{6} + \cdots + 88577 \) 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
−2.16391
−1.85219
−1.20453
−1.01645
−0.217426
−0.0636383
0.406211
1.04829
1.65093
2.15754
2.85973
3.39543
−1.00000 −3.16391 1.00000 −3.99732 3.16391 −0.312884 −1.00000 7.01032 3.99732
1.2 −1.00000 −2.85219 1.00000 −1.71023 2.85219 −2.84010 −1.00000 5.13498 1.71023
1.3 −1.00000 −2.20453 1.00000 3.01904 2.20453 2.70805 −1.00000 1.85995 −3.01904
1.4 −1.00000 −2.01645 1.00000 −0.904437 2.01645 3.37811 −1.00000 1.06606 0.904437
1.5 −1.00000 −1.21743 1.00000 2.27612 1.21743 −3.34577 −1.00000 −1.51787 −2.27612
1.6 −1.00000 −1.06364 1.00000 1.32222 1.06364 2.40325 −1.00000 −1.86867 −1.32222
1.7 −1.00000 −0.593789 1.00000 −0.795738 0.593789 −3.10316 −1.00000 −2.64741 0.795738
1.8 −1.00000 0.0482939 1.00000 3.10147 −0.0482939 2.48609 −1.00000 −2.99767 −3.10147
1.9 −1.00000 0.650929 1.00000 −3.38118 −0.650929 −1.06173 −1.00000 −2.57629 3.38118
1.10 −1.00000 1.15754 1.00000 0.270677 −1.15754 0.100148 −1.00000 −1.66011 −0.270677
1.11 −1.00000 1.85973 1.00000 −0.777219 −1.85973 −0.706150 −1.00000 0.458614 0.777219
1.12 −1.00000 2.39543 1.00000 −2.42341 −2.39543 1.29415 −1.00000 2.73810 2.42341
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
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.bk 12
11.b odd 2 1 8954.2.a.bl 12
11.d odd 10 2 814.2.h.a ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
814.2.h.a ✓ 24 11.d odd 10 2
8954.2.a.bk 12 1.a even 1 1 trivial
8954.2.a.bl 12 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8954))\):

\( T_{3}^{12} + 7 T_{3}^{11} + 4 T_{3}^{10} - 68 T_{3}^{9} - 128 T_{3}^{8} + 161 T_{3}^{7} + 515 T_{3}^{6} + \cdots - 5 \) Copy content Toggle raw display
\( T_{5}^{12} + 4 T_{5}^{11} - 24 T_{5}^{10} - 100 T_{5}^{9} + 177 T_{5}^{8} + 855 T_{5}^{7} - 267 T_{5}^{6} + \cdots - 239 \) Copy content Toggle raw display
\( T_{7}^{12} - T_{7}^{11} - 31 T_{7}^{10} + 32 T_{7}^{9} + 344 T_{7}^{8} - 350 T_{7}^{7} - 1615 T_{7}^{6} + \cdots + 49 \) Copy content Toggle raw display
\( T_{17}^{12} - 17 T_{17}^{11} + 75 T_{17}^{10} + 204 T_{17}^{9} - 2323 T_{17}^{8} + 3266 T_{17}^{7} + \cdots + 2151 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 7 T^{11} + \cdots - 5 \) Copy content Toggle raw display
$5$ \( T^{12} + 4 T^{11} + \cdots - 239 \) Copy content Toggle raw display
$7$ \( T^{12} - T^{11} + \cdots + 49 \) Copy content Toggle raw display
$11$ \( T^{12} \) Copy content Toggle raw display
$13$ \( T^{12} - 7 T^{11} + \cdots - 1 \) Copy content Toggle raw display
$17$ \( T^{12} - 17 T^{11} + \cdots + 2151 \) Copy content Toggle raw display
$19$ \( T^{12} + 8 T^{11} + \cdots + 975789 \) Copy content Toggle raw display
$23$ \( T^{12} + 18 T^{11} + \cdots - 1069631 \) Copy content Toggle raw display
$29$ \( T^{12} - 4 T^{11} + \cdots - 121 \) Copy content Toggle raw display
$31$ \( T^{12} + 24 T^{11} + \cdots + 13869999 \) Copy content Toggle raw display
$37$ \( (T - 1)^{12} \) Copy content Toggle raw display
$41$ \( T^{12} - 26 T^{11} + \cdots - 51024559 \) Copy content Toggle raw display
$43$ \( T^{12} + 4 T^{11} + \cdots - 7090775 \) Copy content Toggle raw display
$47$ \( T^{12} + 6 T^{11} + \cdots - 665789 \) Copy content Toggle raw display
$53$ \( T^{12} - 6 T^{11} + \cdots + 18021125 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots - 2975104359 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 1696345405 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 25702388921 \) Copy content Toggle raw display
$71$ \( T^{12} - 87 T^{10} + \cdots + 3249 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 1538913629 \) Copy content Toggle raw display
$79$ \( T^{12} - 15 T^{11} + \cdots - 10692391 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 1747382881 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 497599475 \) Copy content Toggle raw display
$97$ \( T^{12} + 11 T^{11} + \cdots + 6631920 \) Copy content Toggle raw display
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