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Properties

Label 8954.2.a.bn
Level $8954$
Weight $2$
Character orbit 8954.a
Self dual yes
Analytic conductor $71.498$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $1$

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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,-14,0,14,-10] 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: \(0\)
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: \( x^{14} - 27 x^{12} - 2 x^{11} + 285 x^{10} + 30 x^{9} - 1512 x^{8} - 160 x^{7} + 4335 x^{6} + \cdots - 1499 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + \beta_1 q^{3} + q^{4} + (\beta_{5} - 1) q^{5} - \beta_1 q^{6} + (\beta_{6} + 1) q^{7} - q^{8} + (\beta_{2} + 1) q^{9} + ( - \beta_{5} + 1) q^{10} + \beta_1 q^{12} + (\beta_{8} + \beta_{7} + 1) q^{13}+ \cdots + (\beta_{13} - \beta_{12} + \beta_{10} + \cdots - 3) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 14 q^{2} + 14 q^{4} - 10 q^{5} + 14 q^{7} - 14 q^{8} + 12 q^{9} + 10 q^{10} + 12 q^{13} - 14 q^{14} - 12 q^{15} + 14 q^{16} + 6 q^{17} - 12 q^{18} + 20 q^{19} - 10 q^{20} + 8 q^{21} + 12 q^{25} - 12 q^{26}+ \cdots - 20 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 27 x^{12} - 2 x^{11} + 285 x^{10} + 30 x^{9} - 1512 x^{8} - 160 x^{7} + 4335 x^{6} + \cdots - 1499 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 46475 \nu^{13} - 282767 \nu^{12} + 1110410 \nu^{11} + 7295480 \nu^{10} - 9440239 \nu^{9} + \cdots - 245505993 ) / 2214544 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 30363 \nu^{13} - 3014 \nu^{12} + 738794 \nu^{11} + 48874 \nu^{10} - 6644889 \nu^{9} + \cdots + 44835478 ) / 1107272 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 70827 \nu^{13} + 28375 \nu^{12} + 1754546 \nu^{11} - 695332 \nu^{10} - 16480099 \nu^{9} + \cdots + 73672937 ) / 2214544 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 99317 \nu^{13} - 132507 \nu^{12} - 2507318 \nu^{11} + 3230336 \nu^{10} + 24044665 \nu^{9} + \cdots - 152586077 ) / 2214544 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 116113 \nu^{13} + 417007 \nu^{12} - 2936494 \nu^{11} - 10326420 \nu^{10} + 27033113 \nu^{9} + \cdots + 282272905 ) / 2214544 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 117417 \nu^{13} - 308043 \nu^{12} + 2780742 \nu^{11} + 7770740 \nu^{10} - 23564641 \nu^{9} + \cdots - 205733221 ) / 2214544 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 2179 \nu^{13} - 2899 \nu^{12} + 51898 \nu^{11} + 76720 \nu^{10} - 450127 \nu^{9} - 726401 \nu^{8} + \cdots - 2107077 ) / 36304 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 76875 \nu^{13} - 215183 \nu^{12} + 1884694 \nu^{11} + 5330716 \nu^{10} - 16730555 \nu^{9} + \cdots - 102445037 ) / 1107272 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 26636 \nu^{13} + 23084 \nu^{12} + 691455 \nu^{11} - 552959 \nu^{10} - 6851767 \nu^{9} + \cdots + 25235137 ) / 276818 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 107921 \nu^{13} + 317790 \nu^{12} - 2590310 \nu^{11} - 8018342 \nu^{10} + 22219379 \nu^{9} + \cdots + 206869170 ) / 1107272 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 297035 \nu^{13} - 146057 \nu^{12} - 7451346 \nu^{11} + 3262880 \nu^{10} + 70908967 \nu^{9} + \cdots - 184326503 ) / 2214544 \) 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
\(\nu^{3}\)\(=\) \( \beta_{12} + \beta_{10} + \beta_{9} - \beta_{5} + \beta_{2} + 6\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{12} + 2\beta_{9} - \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} + 2\beta_{4} + 10\beta_{2} + 2\beta _1 + 25 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{13} + 12 \beta_{12} - \beta_{11} + 8 \beta_{10} + 14 \beta_{9} - 2 \beta_{8} - 4 \beta_{7} + \cdots + 17 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{13} + 18 \beta_{12} - \beta_{10} + 31 \beta_{9} - 11 \beta_{8} - 17 \beta_{7} + 12 \beta_{6} + \cdots + 192 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 21 \beta_{13} + 124 \beta_{12} - 10 \beta_{11} + 56 \beta_{10} + 157 \beta_{9} - 27 \beta_{8} + \cdots + 216 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 25 \beta_{13} + 235 \beta_{12} - 21 \beta_{10} + 369 \beta_{9} - 96 \beta_{8} - 215 \beta_{7} + \cdots + 1618 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 292 \beta_{13} + 1238 \beta_{12} - 70 \beta_{11} + 376 \beta_{10} + 1642 \beta_{9} - 278 \beta_{8} + \cdots + 2462 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 406 \beta_{13} + 2715 \beta_{12} + 5 \beta_{11} - 295 \beta_{10} + 4041 \beta_{9} - 813 \beta_{8} + \cdots + 14301 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 3426 \beta_{13} + 12218 \beta_{12} - 400 \beta_{11} + 2417 \beta_{10} + 16637 \beta_{9} - 2650 \beta_{8} + \cdots + 26541 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 5466 \beta_{13} + 29515 \beta_{12} + 142 \beta_{11} - 3510 \beta_{10} + 42636 \beta_{9} - 7039 \beta_{8} + \cdots + 130057 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 36935 \beta_{13} + 119980 \beta_{12} - 1712 \beta_{11} + 14446 \beta_{10} + 165769 \beta_{9} + \cdots + 276956 \) 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
−2.75244
−2.49365
−2.42577
−1.67174
−1.35373
−1.18951
−0.915817
0.836391
1.07002
1.58125
1.66755
1.86062
2.65973
3.12711
−1.00000 −2.75244 1.00000 −2.80070 2.75244 5.04704 −1.00000 4.57595 2.80070
1.2 −1.00000 −2.49365 1.00000 1.57691 2.49365 0.676532 −1.00000 3.21830 −1.57691
1.3 −1.00000 −2.42577 1.00000 2.90400 2.42577 1.30961 −1.00000 2.88437 −2.90400
1.4 −1.00000 −1.67174 1.00000 0.104493 1.67174 −3.21691 −1.00000 −0.205288 −0.104493
1.5 −1.00000 −1.35373 1.00000 −2.02983 1.35373 −4.00766 −1.00000 −1.16741 2.02983
1.6 −1.00000 −1.18951 1.00000 −4.21161 1.18951 2.74459 −1.00000 −1.58506 4.21161
1.7 −1.00000 −0.915817 1.00000 0.457814 0.915817 2.69568 −1.00000 −2.16128 −0.457814
1.8 −1.00000 0.836391 1.00000 −0.682500 −0.836391 −2.38191 −1.00000 −2.30045 0.682500
1.9 −1.00000 1.07002 1.00000 −0.846301 −1.07002 3.02778 −1.00000 −1.85506 0.846301
1.10 −1.00000 1.58125 1.00000 2.48122 −1.58125 2.67280 −1.00000 −0.499637 −2.48122
1.11 −1.00000 1.66755 1.00000 2.11463 −1.66755 1.43839 −1.00000 −0.219280 −2.11463
1.12 −1.00000 1.86062 1.00000 −4.13032 −1.86062 −1.14448 −1.00000 0.461899 4.13032
1.13 −1.00000 2.65973 1.00000 −2.68941 −2.65973 0.269321 −1.00000 4.07416 2.68941
1.14 −1.00000 3.12711 1.00000 −2.24839 −3.12711 4.86919 −1.00000 6.77879 2.24839
\(n\): 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.bn ✓ 14
11.b odd 2 1 8954.2.a.bp yes 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8954.2.a.bn ✓ 14 1.a even 1 1 trivial
8954.2.a.bp yes 14 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8954))\):

\( T_{3}^{14} - 27 T_{3}^{12} - 2 T_{3}^{11} + 285 T_{3}^{10} + 30 T_{3}^{9} - 1512 T_{3}^{8} - 160 T_{3}^{7} + \cdots - 1499 \) Copy content Toggle raw display
\( T_{5}^{14} + 10 T_{5}^{13} + 9 T_{5}^{12} - 192 T_{5}^{11} - 504 T_{5}^{10} + 1138 T_{5}^{9} + \cdots + 397 \) Copy content Toggle raw display
\( T_{7}^{14} - 14 T_{7}^{13} + 39 T_{7}^{12} + 294 T_{7}^{11} - 1861 T_{7}^{10} + 688 T_{7}^{9} + \cdots + 17749 \) Copy content Toggle raw display
\( T_{17}^{14} - 6 T_{17}^{13} - 131 T_{17}^{12} + 964 T_{17}^{11} + 5192 T_{17}^{10} - 54020 T_{17}^{9} + \cdots + 57360493 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{14} \) Copy content Toggle raw display
$3$ \( T^{14} - 27 T^{12} + \cdots - 1499 \) Copy content Toggle raw display
$5$ \( T^{14} + 10 T^{13} + \cdots + 397 \) Copy content Toggle raw display
$7$ \( T^{14} - 14 T^{13} + \cdots + 17749 \) Copy content Toggle raw display
$11$ \( T^{14} \) Copy content Toggle raw display
$13$ \( T^{14} - 12 T^{13} + \cdots + 4869 \) Copy content Toggle raw display
$17$ \( T^{14} - 6 T^{13} + \cdots + 57360493 \) Copy content Toggle raw display
$19$ \( T^{14} - 20 T^{13} + \cdots - 282347 \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots - 906466752 \) Copy content Toggle raw display
$29$ \( T^{14} - 8 T^{13} + \cdots + 21894417 \) Copy content Toggle raw display
$31$ \( T^{14} + 4 T^{13} + \cdots - 419783 \) Copy content Toggle raw display
$37$ \( (T - 1)^{14} \) Copy content Toggle raw display
$41$ \( T^{14} - 40 T^{13} + \cdots + 1750329 \) Copy content Toggle raw display
$43$ \( T^{14} + 2 T^{13} + \cdots - 217503 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots - 1127392883 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots - 5948514359 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 810017816233 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots - 198300464 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 70559281957 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 3961407744 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 161446624261 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots - 209305580976 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots - 3892637219 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots - 12338058096 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots - 3627795051 \) Copy content Toggle raw display
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