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Properties

Label 8954.2.a.bo
Level $8954$
Weight $2$
Character orbit 8954.a
Self dual yes
Analytic conductor $71.498$
Analytic rank $1$
Dimension $14$
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 = [14,14,-4,14,-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: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 4 x^{13} - 16 x^{12} + 73 x^{11} + 86 x^{10} - 489 x^{9} - 180 x^{8} + 1528 x^{7} + 90 x^{6} + \cdots - 59 \) 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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} - \beta_1 q^{3} + q^{4} + \beta_{4} q^{5} - \beta_1 q^{6} + ( - \beta_{11} + \beta_{7} + \beta_{2}) q^{7} + q^{8} + (\beta_{13} - \beta_{11} + \beta_{10} + \cdots + 1) q^{9} + \beta_{4} q^{10}+ \cdots + (\beta_{13} + \beta_{12} + \beta_{11} + \cdots - 2) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 14 q^{2} - 4 q^{3} + 14 q^{4} - 6 q^{5} - 4 q^{6} - 5 q^{7} + 14 q^{8} + 6 q^{9} - 6 q^{10} - 4 q^{12} - 9 q^{13} - 5 q^{14} - 6 q^{15} + 14 q^{16} - 16 q^{17} + 6 q^{18} + 5 q^{19} - 6 q^{20} - 9 q^{21}+ \cdots - 15 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 4 x^{13} - 16 x^{12} + 73 x^{11} + 86 x^{10} - 489 x^{9} - 180 x^{8} + 1528 x^{7} + 90 x^{6} + \cdots - 59 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 814 \nu^{13} - 666 \nu^{12} + 25877 \nu^{11} + 9614 \nu^{10} - 296940 \nu^{9} - 50833 \nu^{8} + \cdots - 241193 ) / 3230 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 55 \nu^{13} - 346 \nu^{12} - 385 \nu^{11} + 5968 \nu^{10} - 3946 \nu^{9} - 36726 \nu^{8} + 44247 \nu^{7} + \cdots - 10738 ) / 85 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 4530 \nu^{13} + 23132 \nu^{12} + 51005 \nu^{11} - 401831 \nu^{10} - 14463 \nu^{9} + 2489832 \nu^{8} + \cdots + 486596 ) / 3230 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 5921 \nu^{13} + 29188 \nu^{12} + 70228 \nu^{11} - 506920 \nu^{10} - 82048 \nu^{9} + 3137825 \nu^{8} + \cdots + 569059 ) / 3230 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 182 \nu^{13} - 783 \nu^{12} - 2566 \nu^{11} + 13671 \nu^{10} + 9684 \nu^{9} - 85052 \nu^{8} + \cdots - 9944 ) / 85 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 4263 \nu^{13} - 16450 \nu^{12} - 66629 \nu^{11} + 288268 \nu^{10} + 341158 \nu^{9} - 1798356 \nu^{8} + \cdots - 103655 ) / 1615 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 8343 \nu^{13} + 38482 \nu^{12} + 108364 \nu^{11} - 670054 \nu^{10} - 280536 \nu^{9} + \cdots + 620931 ) / 3230 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 10433 \nu^{13} - 53568 \nu^{12} - 116534 \nu^{11} + 930107 \nu^{10} + 17215 \nu^{9} - 5759754 \nu^{8} + \cdots - 1184984 ) / 3230 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 10821 \nu^{13} + 55952 \nu^{12} + 119488 \nu^{11} - 971242 \nu^{10} + 6626 \nu^{9} + \cdots + 1237221 ) / 3230 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 12437 \nu^{13} - 53837 \nu^{12} - 174031 \nu^{11} + 939873 \nu^{10} + 637750 \nu^{9} - 5847216 \nu^{8} + \cdots - 703241 ) / 3230 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 12251 \nu^{13} - 61888 \nu^{12} - 140548 \nu^{11} + 1075495 \nu^{10} + 84833 \nu^{9} - 6665810 \nu^{8} + \cdots - 1302124 ) / 3230 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 15546 \nu^{13} - 76223 \nu^{12} - 186138 \nu^{11} + 1324965 \nu^{10} + 245748 \nu^{9} + \cdots - 1504879 ) / 3230 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{13} - \beta_{11} + \beta_{10} + \beta_{7} + \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{13} + \beta_{11} - \beta_{9} - \beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + 6\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7 \beta_{13} - 7 \beta_{11} + 9 \beta_{10} - \beta_{8} + 7 \beta_{7} + \beta_{6} - \beta_{5} + \beta_{3} + \cdots + 23 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9 \beta_{13} + 2 \beta_{12} + 9 \beta_{11} + 3 \beta_{10} - 8 \beta_{9} - 11 \beta_{8} - 10 \beta_{7} + \cdots + 11 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 47 \beta_{13} - 49 \beta_{11} + 74 \beta_{10} + 3 \beta_{9} - 14 \beta_{8} + 48 \beta_{7} + 14 \beta_{6} + \cdots + 151 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 72 \beta_{13} + 20 \beta_{12} + 72 \beta_{11} + 40 \beta_{10} - 54 \beta_{9} - 100 \beta_{8} - 83 \beta_{7} + \cdots + 99 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 330 \beta_{13} - 6 \beta_{12} - 354 \beta_{11} + 592 \beta_{10} + 45 \beta_{9} - 149 \beta_{8} + \cdots + 1061 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 568 \beta_{13} + 143 \beta_{12} + 556 \beta_{11} + 405 \beta_{10} - 350 \beta_{9} - 859 \beta_{8} + \cdots + 825 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2417 \beta_{13} - 127 \beta_{12} - 2601 \beta_{11} + 4683 \beta_{10} + 492 \beta_{9} - 1423 \beta_{8} + \cdots + 7739 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 4483 \beta_{13} + 856 \beta_{12} + 4229 \beta_{11} + 3738 \beta_{10} - 2220 \beta_{9} - 7207 \beta_{8} + \cdots + 6675 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 18218 \beta_{13} - 1747 \beta_{12} - 19256 \beta_{11} + 36863 \beta_{10} + 4773 \beta_{9} - 12872 \beta_{8} + \cdots + 57677 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 35495 \beta_{13} + 4174 \beta_{12} + 31961 \beta_{11} + 33103 \beta_{10} - 13713 \beta_{9} + \cdots + 53534 \) 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
2.85212
2.81125
2.60061
1.95151
1.20549
1.19337
0.903852
−0.159233
−0.709012
−0.710871
−1.42963
−1.79239
−1.98361
−2.73346
1.00000 −2.85212 1.00000 −2.28752 −2.85212 −3.62358 1.00000 5.13462 −2.28752
1.2 1.00000 −2.81125 1.00000 3.04678 −2.81125 2.46167 1.00000 4.90310 3.04678
1.3 1.00000 −2.60061 1.00000 2.12964 −2.60061 1.60681 1.00000 3.76319 2.12964
1.4 1.00000 −1.95151 1.00000 −2.68819 −1.95151 2.57430 1.00000 0.808399 −2.68819
1.5 1.00000 −1.20549 1.00000 1.53060 −1.20549 −0.340799 1.00000 −1.54679 1.53060
1.6 1.00000 −1.19337 1.00000 −0.0975997 −1.19337 0.998449 1.00000 −1.57587 −0.0975997
1.7 1.00000 −0.903852 1.00000 −4.02378 −0.903852 −4.17654 1.00000 −2.18305 −4.02378
1.8 1.00000 0.159233 1.00000 −3.26846 0.159233 0.694036 1.00000 −2.97464 −3.26846
1.9 1.00000 0.709012 1.00000 −0.170644 0.709012 −2.31398 1.00000 −2.49730 −0.170644
1.10 1.00000 0.710871 1.00000 2.10646 0.710871 −0.807291 1.00000 −2.49466 2.10646
1.11 1.00000 1.42963 1.00000 1.29501 1.42963 −1.57612 1.00000 −0.956161 1.29501
1.12 1.00000 1.79239 1.00000 −1.54691 1.79239 3.54770 1.00000 0.212655 −1.54691
1.13 1.00000 1.98361 1.00000 −0.0356134 1.98361 −3.79809 1.00000 0.934725 −0.0356134
1.14 1.00000 2.73346 1.00000 −1.98979 2.73346 −0.246574 1.00000 4.47181 −1.98979
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
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.bo 14
11.b odd 2 1 8954.2.a.bm 14
11.d odd 10 2 814.2.h.b ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
814.2.h.b ✓ 28 11.d odd 10 2
8954.2.a.bm 14 11.b odd 2 1
8954.2.a.bo 14 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8954))\):

\( T_{3}^{14} + 4 T_{3}^{13} - 16 T_{3}^{12} - 73 T_{3}^{11} + 86 T_{3}^{10} + 489 T_{3}^{9} - 180 T_{3}^{8} + \cdots - 59 \) Copy content Toggle raw display
\( T_{5}^{14} + 6 T_{5}^{13} - 16 T_{5}^{12} - 138 T_{5}^{11} + 47 T_{5}^{10} + 1173 T_{5}^{9} + 337 T_{5}^{8} + \cdots - 4 \) Copy content Toggle raw display
\( T_{7}^{14} + 5 T_{7}^{13} - 29 T_{7}^{12} - 150 T_{7}^{11} + 310 T_{7}^{10} + 1592 T_{7}^{9} + \cdots + 356 \) Copy content Toggle raw display
\( T_{17}^{14} + 16 T_{17}^{13} + 47 T_{17}^{12} - 392 T_{17}^{11} - 1996 T_{17}^{10} + 2793 T_{17}^{9} + \cdots + 20711 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{14} \) Copy content Toggle raw display
$3$ \( T^{14} + 4 T^{13} + \cdots - 59 \) Copy content Toggle raw display
$5$ \( T^{14} + 6 T^{13} + \cdots - 4 \) Copy content Toggle raw display
$7$ \( T^{14} + 5 T^{13} + \cdots + 356 \) Copy content Toggle raw display
$11$ \( T^{14} \) Copy content Toggle raw display
$13$ \( T^{14} + 9 T^{13} + \cdots + 23824 \) Copy content Toggle raw display
$17$ \( T^{14} + 16 T^{13} + \cdots + 20711 \) Copy content Toggle raw display
$19$ \( T^{14} - 5 T^{13} + \cdots - 38401 \) Copy content Toggle raw display
$23$ \( T^{14} + 2 T^{13} + \cdots - 8870336 \) Copy content Toggle raw display
$29$ \( T^{14} + 6 T^{13} + \cdots + 21503396 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots - 285497536 \) Copy content Toggle raw display
$37$ \( (T + 1)^{14} \) Copy content Toggle raw display
$41$ \( T^{14} + 19 T^{13} + \cdots + 9254155 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots - 56091820775 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 19280594500 \) Copy content Toggle raw display
$53$ \( T^{14} + 16 T^{13} + \cdots - 44186564 \) Copy content Toggle raw display
$59$ \( T^{14} + 28 T^{13} + \cdots + 550495 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots - 69063948124 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots - 27351765619 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 34876218436 \) Copy content Toggle raw display
$73$ \( T^{14} - T^{13} + \cdots - 83319269 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots - 65359666900 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 3687050551895 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots - 823924645 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots - 2225523903280 \) Copy content Toggle raw display
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