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Properties

Label 99.4.f.d
Level $99$
Weight $4$
Character orbit 99.f
Analytic conductor $5.841$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [99,4,Mod(37,99)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(99, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("99.37");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 99.f (of order \(5\), degree \(4\), minimal)

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: no
Analytic conductor: \(5.84118909057\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(3\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3 x^{11} + 21 x^{10} - 26 x^{9} + 281 x^{8} + 486 x^{7} + 3506 x^{6} + 15102 x^{5} + \cdots + 1936 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{7} + \beta_{6} + \beta_{5} + \cdots + 1) q^{2} + ( - \beta_{9} - \beta_{8} + 2 \beta_{5} + \cdots - 1) q^{4} + (\beta_{11} - 2 \beta_{10} + \beta_{9} + \cdots - 4) q^{5} + (2 \beta_{10} - \beta_{9} - \beta_{8} + \cdots + 2) q^{7}+ \cdots + ( - 31 \beta_{11} - 41 \beta_{9} + \cdots - 400) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 16 q^{4} - 28 q^{5} + 12 q^{7} + 112 q^{8} + 100 q^{10} + 54 q^{11} - 18 q^{13} - 156 q^{14} + 308 q^{16} + 80 q^{17} - 280 q^{19} + 15 q^{20} - 193 q^{22} + 392 q^{23} + 77 q^{25} - 406 q^{26} - 429 q^{28}+ \cdots - 2810 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 3 x^{11} + 21 x^{10} - 26 x^{9} + 281 x^{8} + 486 x^{7} + 3506 x^{6} + 15102 x^{5} + \cdots + 1936 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 69\!\cdots\!13 \nu^{11} + \cdots - 78\!\cdots\!98 ) / 18\!\cdots\!89 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 16\!\cdots\!34 \nu^{11} + \cdots + 11\!\cdots\!56 ) / 37\!\cdots\!78 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 46\!\cdots\!57 \nu^{11} + \cdots + 12\!\cdots\!44 ) / 74\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 12\!\cdots\!25 \nu^{11} + \cdots - 10\!\cdots\!24 ) / 14\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 66\!\cdots\!54 \nu^{11} + \cdots - 29\!\cdots\!64 ) / 74\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 83\!\cdots\!37 \nu^{11} + \cdots + 12\!\cdots\!00 ) / 74\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 11\!\cdots\!73 \nu^{11} + \cdots + 12\!\cdots\!84 ) / 74\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 53\!\cdots\!95 \nu^{11} + \cdots - 13\!\cdots\!12 ) / 14\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 55\!\cdots\!27 \nu^{11} + \cdots - 42\!\cdots\!12 ) / 74\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 18\!\cdots\!45 \nu^{11} + \cdots + 40\!\cdots\!04 ) / 14\!\cdots\!12 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} + \beta_{8} - \beta_{7} + 10\beta_{6} + \beta_{5} - 2\beta_{4} + \beta_{3} - \beta_{2} + 2\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{10} - 16\beta_{5} - 17\beta_{4} - 11\beta_{3} - \beta_{2} + \beta _1 - 11 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 16 \beta_{11} - 16 \beta_{10} - 16 \beta_{8} + 21 \beta_{7} - 162 \beta_{6} - 154 \beta_{5} + \cdots - 154 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -32\beta_{11} - 5\beta_{9} + 61\beta_{7} - 315\beta_{6} - 315\beta_{5} - 279\beta_{2} - 61\beta _1 - 429 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 29 \beta_{11} + 274 \beta_{10} - 29 \beta_{9} + 245 \beta_{8} + 807 \beta_{7} - 253 \beta_{6} + \cdots - 253 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 168 \beta_{11} + 778 \beta_{10} - 778 \beta_{9} - 168 \beta_{8} + 5121 \beta_{7} - 2604 \beta_{6} + \cdots - 7243 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1344 \beta_{10} - 4953 \beta_{9} - 4953 \beta_{8} + 64453 \beta_{5} + 28886 \beta_{4} + 13703 \beta_{3} + \cdots + 13703 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 4788 \beta_{11} + 4788 \beta_{10} - 13013 \beta_{8} - 98104 \beta_{7} + 62485 \beta_{6} + 238564 \beta_{5} + \cdots + 238564 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 43012 \beta_{11} + 93316 \beta_{9} - 444424 \beta_{7} + 445776 \beta_{6} + 445776 \beta_{5} + \cdots + 1400254 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 401412 \beta_{11} - 129437 \beta_{10} + 401412 \beta_{9} + 271975 \beta_{8} - 1593097 \beta_{7} + \cdots + 4013999 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/99\mathbb{Z}\right)^\times\).

\(n\) \(46\) \(56\)
\(\chi(n)\) \(\beta_{5}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
3.88644 − 2.82366i
−0.654357 + 0.475418i
−1.92306 + 1.39719i
1.27457 + 3.92271i
0.0936861 + 0.288336i
−1.17727 − 3.62326i
1.27457 − 3.92271i
0.0936861 − 0.288336i
−1.17727 + 3.62326i
3.88644 + 2.82366i
−0.654357 − 0.475418i
−1.92306 − 1.39719i
−3.07742 − 2.23588i 0 1.99923 + 6.15301i −2.08679 + 1.51614i 0 0.454025 + 1.39734i −1.79887 + 5.53636i 0 9.81184
37.2 1.46337 + 1.06320i 0 −1.46107 − 4.49672i −17.3626 + 12.6147i 0 −5.78498 − 17.8043i 7.11451 − 21.8962i 0 −38.8200
37.3 2.73208 + 1.98497i 0 1.05201 + 3.23775i 11.3314 − 8.23272i 0 7.21292 + 22.1991i 4.79582 − 14.7600i 0 47.2999
64.1 −1.58358 + 4.87376i 0 −14.7737 − 10.7337i −4.61569 − 14.2056i 0 9.98349 + 7.25343i 42.5421 − 30.9086i 0 76.5442
64.2 −0.402703 + 1.23939i 0 5.09821 + 3.70407i 2.50364 + 7.70540i 0 −0.439746 − 0.319494i −15.0782 + 10.9549i 0 −10.5582
64.3 0.868251 − 2.67220i 0 0.0853305 + 0.0619962i −3.76992 − 11.6026i 0 −5.42571 − 3.94201i 18.4246 − 13.3863i 0 −34.2777
82.1 −1.58358 − 4.87376i 0 −14.7737 + 10.7337i −4.61569 + 14.2056i 0 9.98349 − 7.25343i 42.5421 + 30.9086i 0 76.5442
82.2 −0.402703 − 1.23939i 0 5.09821 − 3.70407i 2.50364 − 7.70540i 0 −0.439746 + 0.319494i −15.0782 − 10.9549i 0 −10.5582
82.3 0.868251 + 2.67220i 0 0.0853305 − 0.0619962i −3.76992 + 11.6026i 0 −5.42571 + 3.94201i 18.4246 + 13.3863i 0 −34.2777
91.1 −3.07742 + 2.23588i 0 1.99923 − 6.15301i −2.08679 − 1.51614i 0 0.454025 − 1.39734i −1.79887 − 5.53636i 0 9.81184
91.2 1.46337 − 1.06320i 0 −1.46107 + 4.49672i −17.3626 − 12.6147i 0 −5.78498 + 17.8043i 7.11451 + 21.8962i 0 −38.8200
91.3 2.73208 − 1.98497i 0 1.05201 − 3.23775i 11.3314 + 8.23272i 0 7.21292 − 22.1991i 4.79582 + 14.7600i 0 47.2999
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 99.4.f.d 12
3.b odd 2 1 33.4.e.c ✓ 12
11.c even 5 1 inner 99.4.f.d 12
11.c even 5 1 1089.4.a.bi 6
11.d odd 10 1 1089.4.a.bk 6
33.f even 10 1 363.4.a.u 6
33.h odd 10 1 33.4.e.c ✓ 12
33.h odd 10 1 363.4.a.v 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.4.e.c ✓ 12 3.b odd 2 1
33.4.e.c ✓ 12 33.h odd 10 1
99.4.f.d 12 1.a even 1 1 trivial
99.4.f.d 12 11.c even 5 1 inner
363.4.a.u 6 33.f even 10 1
363.4.a.v 6 33.h odd 10 1
1089.4.a.bi 6 11.c even 5 1
1089.4.a.bk 6 11.d odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 20 T_{2}^{10} - 64 T_{2}^{9} + 225 T_{2}^{8} - 70 T_{2}^{7} + 3191 T_{2}^{6} + \cdots + 190096 \) acting on \(S_{4}^{\mathrm{new}}(99, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 20 T^{10} + \cdots + 190096 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 1310363273521 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 834112161 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 55\!\cdots\!81 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 49\!\cdots\!36 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 25\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 47\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( (T^{6} - 196 T^{5} + \cdots - 3716146836)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 95\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 13\!\cdots\!61 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 18\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 40\!\cdots\!96 \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 92003355120644)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 61\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 35\!\cdots\!41 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 22\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 24\!\cdots\!36 \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots - 66\!\cdots\!84)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 12\!\cdots\!96 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 16\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 32\!\cdots\!25 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 18\!\cdots\!81 \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots - 28\!\cdots\!20)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 11\!\cdots\!21 \) Copy content Toggle raw display
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