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Properties

Label 99.6.f.c
Level $99$
Weight $6$
Character orbit 99.f
Analytic conductor $15.878$
Analytic rank $0$
Dimension $20$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [99,6,Mod(37,99)] 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("99.37"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(99, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 2])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 99.f (of order \(5\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [20,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(15.8779981615\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(5\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 3 x^{19} + 142 x^{18} - 389 x^{17} + 14927 x^{16} - 6599 x^{15} + 1399353 x^{14} + \cdots + 25735126080400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4}\cdot 11^{2} \)
Twist minimal: no (minimal twist has level 33)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{7} + \beta_1) q^{2} + ( - \beta_{15} - 2 \beta_{9} + \cdots - \beta_{6}) q^{4} + (\beta_{13} - \beta_{12} + \beta_{11} + \cdots + 3) q^{5} + (2 \beta_{19} - 2 \beta_{15} + \cdots - 1) q^{7}+ \cdots + ( - 204 \beta_{18} + 55 \beta_{17} + \cdots + 17699) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 2 q^{2} - 124 q^{4} + 33 q^{5} - 335 q^{7} + 472 q^{8} + 1952 q^{10} + 835 q^{11} - 959 q^{13} + 3020 q^{14} - 1428 q^{16} + 3144 q^{17} - 930 q^{19} - 3177 q^{20} + 2713 q^{22} - 1400 q^{23} - 9388 q^{25}+ \cdots + 150570 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 3 x^{19} + 142 x^{18} - 389 x^{17} + 14927 x^{16} - 6599 x^{15} + 1399353 x^{14} + \cdots + 25735126080400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 43\!\cdots\!67 \nu^{19} + \cdots - 33\!\cdots\!00 ) / 78\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 11\!\cdots\!83 \nu^{19} + \cdots + 41\!\cdots\!00 ) / 86\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 45\!\cdots\!18 \nu^{19} + \cdots - 64\!\cdots\!50 ) / 21\!\cdots\!95 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 52\!\cdots\!85 \nu^{19} + \cdots + 76\!\cdots\!80 ) / 17\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 24\!\cdots\!56 \nu^{19} + \cdots - 14\!\cdots\!00 ) / 43\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 65\!\cdots\!47 \nu^{19} + \cdots + 39\!\cdots\!00 ) / 86\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 46\!\cdots\!17 \nu^{19} + \cdots - 67\!\cdots\!40 ) / 39\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 59\!\cdots\!67 \nu^{19} + \cdots - 62\!\cdots\!80 ) / 36\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 35\!\cdots\!52 \nu^{19} + \cdots + 21\!\cdots\!00 ) / 19\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 18\!\cdots\!63 \nu^{19} + \cdots - 73\!\cdots\!40 ) / 36\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 62\!\cdots\!25 \nu^{19} + \cdots + 12\!\cdots\!00 ) / 19\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 10\!\cdots\!65 \nu^{19} + \cdots + 49\!\cdots\!80 ) / 19\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 21\!\cdots\!75 \nu^{19} + \cdots - 32\!\cdots\!00 ) / 39\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 61\!\cdots\!65 \nu^{19} + \cdots + 29\!\cdots\!00 ) / 99\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 31\!\cdots\!87 \nu^{19} + \cdots + 33\!\cdots\!00 ) / 39\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 34\!\cdots\!31 \nu^{19} + \cdots - 22\!\cdots\!60 ) / 39\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 18\!\cdots\!43 \nu^{19} + \cdots + 22\!\cdots\!40 ) / 19\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 18\!\cdots\!89 \nu^{19} + \cdots + 14\!\cdots\!20 ) / 19\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 40\!\cdots\!63 \nu^{19} + \cdots + 33\!\cdots\!80 ) / 39\!\cdots\!40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_{6} + \beta_{5} - \beta_{3} + \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{15} + \beta_{14} + \beta_{13} - 49\beta_{9} - 2\beta_{7} + \beta_{5} + \beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{18} - \beta_{16} + 2 \beta_{15} - 3 \beta_{12} - 16 \beta_{9} + 44 \beta_{8} + 16 \beta_{7} + \cdots + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 15 \beta_{19} - \beta_{18} + 4 \beta_{17} + \beta_{16} + 7 \beta_{15} - 9 \beta_{14} - 118 \beta_{13} + \cdots + 111 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 110 \beta_{18} - 395 \beta_{17} - 494 \beta_{16} - 505 \beta_{15} - 150 \beta_{14} - 161 \beta_{13} + \cdots - 10803 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2122 \beta_{19} + 1654 \beta_{18} - 2122 \beta_{17} + 827 \beta_{16} - 1053 \beta_{15} - 9744 \beta_{14} + \cdots - 378939 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 3254 \beta_{19} - 9950 \beta_{18} + 40006 \beta_{17} + 40970 \beta_{16} - 10342 \beta_{15} + \cdots - 922644 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 177348 \beta_{19} - 31076 \beta_{18} + 50584 \beta_{16} - 1303965 \beta_{15} + 124144 \beta_{14} + \cdots - 78920 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 4524927 \beta_{19} + 1170881 \beta_{18} - 3886103 \beta_{17} + 2336386 \beta_{16} - 6312007 \beta_{15} + \cdots + 126112226 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 3689918 \beta_{18} + 25938091 \beta_{17} + 15102918 \beta_{16} + 22248173 \beta_{15} + \cdots + 3935325780 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 467464551 \beta_{19} - 77745526 \beta_{18} + 467464551 \beta_{17} - 38872763 \beta_{16} + \cdots + 22616158914 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 2521311739 \beta_{19} - 285290813 \beta_{18} - 230635103 \beta_{17} - 2111953519 \beta_{16} + \cdots + 51454975884 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 14716314606 \beta_{19} + 10210150996 \beta_{18} - 16215654966 \beta_{16} + 94634925546 \beta_{15} + \cdots + 4355405318 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 292352283784 \beta_{19} - 34854111888 \beta_{18} - 5557400036 \beta_{17} - 252545818884 \beta_{16} + \cdots - 7544573958818 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 479735928081 \beta_{18} - 4998525075371 \beta_{17} - 5009638076492 \beta_{16} + \cdots - 316478871362913 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 31257017635655 \beta_{19} + 10444761992334 \beta_{18} - 31257017635655 \beta_{17} + \cdots - 47\!\cdots\!43 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 253472038415379 \beta_{19} - 38850817478382 \beta_{18} + 267287994097648 \beta_{17} + \cdots - 25\!\cdots\!41 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 40\!\cdots\!51 \beta_{19} - 311297496607090 \beta_{18} + \cdots - 479901689565643 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 54\!\cdots\!64 \beta_{19} + \cdots + 29\!\cdots\!94 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(46\) \(56\)
\(\chi(n)\) \(-\beta_{8}\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
8.61948 − 6.26242i
3.88934 − 2.82577i
0.218287 − 0.158595i
−4.29115 + 3.11770i
−7.12695 + 5.17803i
3.07809 + 9.47338i
1.17535 + 3.61737i
0.992840 + 3.05565i
−2.05193 − 6.31518i
−3.00337 − 9.24343i
3.07809 − 9.47338i
1.17535 − 3.61737i
0.992840 − 3.05565i
−2.05193 + 6.31518i
−3.00337 + 9.24343i
8.61948 + 6.26242i
3.88934 + 2.82577i
0.218287 + 0.158595i
−4.29115 − 3.11770i
−7.12695 − 5.17803i
−7.81047 − 5.67464i 0 18.9134 + 58.2093i −25.1385 + 18.2642i 0 56.7190 + 174.563i 87.1280 − 268.152i 0 299.987
37.2 −3.08033 − 2.23799i 0 −5.40872 − 16.6463i −20.5684 + 14.9438i 0 −26.9084 − 82.8155i −58.2442 + 179.257i 0 96.8014
37.3 0.590730 + 0.429190i 0 −9.72379 − 29.9267i 77.5329 − 56.3310i 0 29.5747 + 91.0216i 14.3206 − 44.0742i 0 69.9777
37.4 5.10017 + 3.70549i 0 2.39251 + 7.36340i −54.7018 + 39.7432i 0 −37.8504 − 116.492i 47.2561 − 145.439i 0 −426.256
37.5 7.93597 + 5.76582i 0 19.8464 + 61.0809i 37.2750 − 27.0819i 0 37.2644 + 114.688i −97.6803 + 300.629i 0 451.963
64.1 −3.38711 + 10.4244i 0 −71.3079 − 51.8082i −8.39633 − 25.8412i 0 −110.374 − 80.1917i 497.837 − 361.700i 0 297.820
64.2 −1.48437 + 4.56842i 0 7.22140 + 5.24666i −18.4760 − 56.8633i 0 58.2735 + 42.3382i −159.044 + 115.553i 0 287.201
64.3 −1.30186 + 4.00670i 0 11.5297 + 8.37681i 26.9724 + 83.0125i 0 −114.007 − 82.8310i −157.639 + 114.532i 0 −367.721
64.4 1.74291 − 5.36413i 0 0.152432 + 0.110748i −21.0541 − 64.7978i 0 −106.538 − 77.4041i 146.876 − 106.711i 0 −384.279
64.5 2.69436 − 8.29237i 0 −35.6154 − 25.8761i 23.0548 + 70.9555i 0 46.3463 + 33.6726i −84.8093 + 61.6176i 0 650.507
82.1 −3.38711 − 10.4244i 0 −71.3079 + 51.8082i −8.39633 + 25.8412i 0 −110.374 + 80.1917i 497.837 + 361.700i 0 297.820
82.2 −1.48437 − 4.56842i 0 7.22140 − 5.24666i −18.4760 + 56.8633i 0 58.2735 − 42.3382i −159.044 − 115.553i 0 287.201
82.3 −1.30186 − 4.00670i 0 11.5297 − 8.37681i 26.9724 − 83.0125i 0 −114.007 + 82.8310i −157.639 − 114.532i 0 −367.721
82.4 1.74291 + 5.36413i 0 0.152432 − 0.110748i −21.0541 + 64.7978i 0 −106.538 + 77.4041i 146.876 + 106.711i 0 −384.279
82.5 2.69436 + 8.29237i 0 −35.6154 + 25.8761i 23.0548 − 70.9555i 0 46.3463 − 33.6726i −84.8093 − 61.6176i 0 650.507
91.1 −7.81047 + 5.67464i 0 18.9134 − 58.2093i −25.1385 − 18.2642i 0 56.7190 − 174.563i 87.1280 + 268.152i 0 299.987
91.2 −3.08033 + 2.23799i 0 −5.40872 + 16.6463i −20.5684 − 14.9438i 0 −26.9084 + 82.8155i −58.2442 − 179.257i 0 96.8014
91.3 0.590730 − 0.429190i 0 −9.72379 + 29.9267i 77.5329 + 56.3310i 0 29.5747 − 91.0216i 14.3206 + 44.0742i 0 69.9777
91.4 5.10017 − 3.70549i 0 2.39251 − 7.36340i −54.7018 − 39.7432i 0 −37.8504 + 116.492i 47.2561 + 145.439i 0 −426.256
91.5 7.93597 − 5.76582i 0 19.8464 − 61.0809i 37.2750 + 27.0819i 0 37.2644 − 114.688i −97.6803 − 300.629i 0 451.963
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.5
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 99.6.f.c 20
3.b odd 2 1 33.6.e.a ✓ 20
11.c even 5 1 inner 99.6.f.c 20
11.c even 5 1 1089.6.a.bh 10
11.d odd 10 1 1089.6.a.bl 10
33.f even 10 1 363.6.a.q 10
33.h odd 10 1 33.6.e.a ✓ 20
33.h odd 10 1 363.6.a.u 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.6.e.a ✓ 20 3.b odd 2 1
33.6.e.a ✓ 20 33.h odd 10 1
99.6.f.c 20 1.a even 1 1 trivial
99.6.f.c 20 11.c even 5 1 inner
363.6.a.q 10 33.f even 10 1
363.6.a.u 10 33.h odd 10 1
1089.6.a.bh 10 11.c even 5 1
1089.6.a.bl 10 11.d odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{20} - 2 T_{2}^{19} + 144 T_{2}^{18} - 592 T_{2}^{17} + 15553 T_{2}^{16} - 24816 T_{2}^{15} + \cdots + 327810433847296 \) acting on \(S_{6}^{\mathrm{new}}(99, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + \cdots + 327810433847296 \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 28\!\cdots\!81 \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 55\!\cdots\!25 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 11\!\cdots\!01 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 14\!\cdots\!16 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 29\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( (T^{10} + \cdots - 37\!\cdots\!20)^{2} \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 14\!\cdots\!56 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 41\!\cdots\!25 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots - 17\!\cdots\!80)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 82\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 76\!\cdots\!25 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 12\!\cdots\!41 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 41\!\cdots\!76 \) Copy content Toggle raw display
$67$ \( (T^{10} + \cdots - 38\!\cdots\!56)^{2} \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 65\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 84\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 77\!\cdots\!25 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 28\!\cdots\!25 \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots - 89\!\cdots\!36)^{2} \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 46\!\cdots\!61 \) Copy content Toggle raw display
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