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Properties

Label 99.6.f.b
Level $99$
Weight $6$
Character orbit 99.f
Analytic conductor $15.878$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [99,6,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, 6, names="a")
 
//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);
 
Level: \( N \) \(=\) \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
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: \(15.8779981615\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(5\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - x^{19} + 78 x^{18} + 79 x^{17} + 10573 x^{16} - 33409 x^{15} + 1262953 x^{14} + \cdots + 25599187870096 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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

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

\(f(q)\) \(=\) \( q + ( - \beta_{6} + \beta_{5}) q^{2} + (\beta_{16} + \beta_{12} + 18 \beta_{7} + \cdots - 12) q^{4}+ \cdots + ( - \beta_{18} + \beta_{17} + \cdots + 25) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{6} + \beta_{5}) q^{2} + (\beta_{16} + \beta_{12} + 18 \beta_{7} + \cdots - 12) q^{4}+ \cdots + ( - 367 \beta_{19} + 367 \beta_{18} + \cdots + 116760) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 6 q^{2} + 8 q^{4} + 11 q^{5} - 139 q^{7} + 76 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 6 q^{2} + 8 q^{4} + 11 q^{5} - 139 q^{7} + 76 q^{8} - 424 q^{10} - 2289 q^{11} - 847 q^{13} - 2022 q^{14} - 7148 q^{16} - 2482 q^{17} + 2958 q^{19} - 8037 q^{20} + 7441 q^{22} - 8140 q^{23} - 13120 q^{25} + 13508 q^{26} + 27819 q^{28} + 20210 q^{29} + 5540 q^{31} + 4626 q^{32} + 16540 q^{34} + 34101 q^{35} - 25173 q^{37} + 55878 q^{38} + 22689 q^{40} - 54349 q^{41} - 21688 q^{43} - 24587 q^{44} + 44155 q^{46} + 48387 q^{47} - 78880 q^{49} - 19956 q^{50} + 109051 q^{52} + 74382 q^{53} - 47630 q^{55} - 168894 q^{56} + 1501 q^{58} + 108412 q^{59} + 16737 q^{61} - 132685 q^{62} - 157030 q^{64} + 134194 q^{65} + 20178 q^{67} - 387013 q^{68} - 411988 q^{70} - 143157 q^{71} + 164980 q^{73} - 286114 q^{74} + 46868 q^{76} + 367461 q^{77} + 369613 q^{79} + 107448 q^{80} - 184129 q^{82} - 267741 q^{83} + 379937 q^{85} + 290531 q^{86} - 740846 q^{88} - 205492 q^{89} + 394559 q^{91} + 483181 q^{92} + 561327 q^{94} - 267881 q^{95} - 300567 q^{97} + 2296174 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - x^{19} + 78 x^{18} + 79 x^{17} + 10573 x^{16} - 33409 x^{15} + 1262953 x^{14} + \cdots + 25599187870096 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 24\!\cdots\!63 \nu^{19} + \cdots - 65\!\cdots\!64 ) / 34\!\cdots\!02 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 98\!\cdots\!84 \nu^{19} + \cdots + 78\!\cdots\!30 ) / 13\!\cdots\!11 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 32\!\cdots\!82 \nu^{19} + \cdots - 50\!\cdots\!28 ) / 27\!\cdots\!22 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 68\!\cdots\!67 \nu^{19} + \cdots + 56\!\cdots\!00 ) / 54\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 30\!\cdots\!65 \nu^{19} + \cdots + 20\!\cdots\!20 ) / 13\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 17\!\cdots\!06 \nu^{19} + \cdots + 32\!\cdots\!16 ) / 69\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 56\!\cdots\!13 \nu^{19} + \cdots - 29\!\cdots\!88 ) / 54\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 16\!\cdots\!73 \nu^{19} + \cdots - 39\!\cdots\!64 ) / 34\!\cdots\!02 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 81\!\cdots\!62 \nu^{19} + \cdots + 18\!\cdots\!84 ) / 15\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 24\!\cdots\!41 \nu^{19} + \cdots + 16\!\cdots\!84 ) / 31\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 56\!\cdots\!84 \nu^{19} + \cdots + 10\!\cdots\!64 ) / 69\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 40\!\cdots\!71 \nu^{19} + \cdots - 24\!\cdots\!12 ) / 40\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 21\!\cdots\!25 \nu^{19} + \cdots + 40\!\cdots\!80 ) / 20\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 23\!\cdots\!85 \nu^{19} + \cdots + 58\!\cdots\!24 ) / 20\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 16\!\cdots\!35 \nu^{19} + \cdots - 10\!\cdots\!84 ) / 13\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 61\!\cdots\!69 \nu^{19} + \cdots - 13\!\cdots\!12 ) / 40\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 62\!\cdots\!69 \nu^{19} + \cdots - 11\!\cdots\!44 ) / 20\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 80\!\cdots\!34 \nu^{19} + \cdots - 26\!\cdots\!28 ) / 20\!\cdots\!16 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{12} + \beta_{9} - \beta_{8} - 49\beta_{7} - 6\beta_{6} + \beta_{4} + 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{19} + \beta_{16} + 2 \beta_{14} + 3 \beta_{13} + \beta_{12} - 2 \beta_{11} + 2 \beta_{9} + \cdots - 35 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 6 \beta_{19} - 3 \beta_{18} + 5 \beta_{17} - 109 \beta_{16} + 3 \beta_{15} - 5 \beta_{14} + \cdots + 2 \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 17 \beta_{19} + 17 \beta_{18} - 249 \beta_{17} - 219 \beta_{16} + 249 \beta_{13} - 408 \beta_{11} + \cdots + 7332 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 391 \beta_{18} - 943 \beta_{17} - 470 \beta_{15} + 472 \beta_{14} + 2129 \beta_{12} - 943 \beta_{11} + \cdots - 129800 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 3378 \beta_{19} + 10216 \beta_{18} - 18712 \beta_{17} + 18726 \beta_{16} + 3378 \beta_{15} + \cdots + 101768 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 103772 \beta_{19} + 1140633 \beta_{16} - 45416 \beta_{15} - 74628 \beta_{14} - 33804 \beta_{13} + \cdots - 24013831 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1450621 \beta_{19} - 960777 \beta_{18} + 2035773 \beta_{17} - 2777337 \beta_{16} + 960777 \beta_{15} + \cdots + 26404325 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 5314715 \beta_{19} + 5314715 \beta_{18} + 4888469 \beta_{17} + 37443407 \beta_{16} + \cdots + 2321917415 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 63162745 \beta_{18} + 291186469 \beta_{17} - 93206829 \beta_{15} + 215238939 \beta_{14} + \cdots - 5229729401 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 628941271 \beta_{19} + 732584482 \beta_{18} - 444888904 \beta_{17} - 4464778965 \beta_{16} + \cdots + 206326475808 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 17032821338 \beta_{19} + 43657317140 \beta_{16} - 7707637614 \beta_{15} + 30587060194 \beta_{14} + \cdots - 1324846024046 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 153953121956 \beta_{19} - 79410379408 \beta_{18} + 53176977104 \beta_{17} - 1277679603561 \beta_{16} + \cdots + 2160057360596 \beta_1 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 913604260816 \beta_{19} + 913604260816 \beta_{18} - 3235932238730 \beta_{17} + 863704227014 \beta_{16} + \cdots + 144241411965627 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 8796134072883 \beta_{18} + 3823389968623 \beta_{17} - 8576511792099 \beta_{15} + \cdots - 26\!\cdots\!65 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 106507268471761 \beta_{19} + 99981415905261 \beta_{18} - 246103968451527 \beta_{17} + \cdots + 14\!\cdots\!77 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 19\!\cdots\!09 \beta_{19} + \cdots - 25\!\cdots\!20 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 22\!\cdots\!26 \beta_{19} + \cdots + 55\!\cdots\!22 \beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(46\) \(56\)
\(\chi(n)\) \(-1 - \beta_{2} + \beta_{6} + \beta_{7}\) \(1\)

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
7.96653 − 5.78803i
4.70404 − 3.41769i
−0.490010 + 0.356013i
−3.89647 + 2.83095i
−8.59311 + 6.24326i
2.93057 + 9.01936i
1.34436 + 4.13753i
0.527694 + 1.62407i
−1.64776 − 5.07128i
−2.34585 − 7.21978i
2.93057 − 9.01936i
1.34436 − 4.13753i
0.527694 − 1.62407i
−1.64776 + 5.07128i
−2.34585 + 7.21978i
7.96653 + 5.78803i
4.70404 + 3.41769i
−0.490010 − 0.356013i
−3.89647 − 2.83095i
−8.59311 − 6.24326i
−8.77555 − 6.37581i 0 26.4708 + 81.4687i −31.3852 + 22.8027i 0 −67.6896 − 208.327i 179.871 − 553.584i 0 420.808
37.2 −5.51306 − 4.00547i 0 4.46148 + 13.7310i 62.5986 − 45.4806i 0 59.3828 + 182.761i −36.9828 + 113.821i 0 −527.281
37.3 −0.319007 − 0.231772i 0 −9.84050 − 30.2859i 17.3054 − 12.5731i 0 −52.3175 − 161.017i −7.77944 + 23.9427i 0 −8.43465
37.4 3.08746 + 2.24317i 0 −5.38796 − 16.5824i −3.77724 + 2.74432i 0 49.4196 + 152.098i 58.2998 − 179.428i 0 −17.8180
37.5 7.78409 + 5.65548i 0 18.7192 + 57.6117i −48.1408 + 34.9763i 0 9.43666 + 29.0431i −84.9654 + 261.497i 0 −572.540
64.1 −2.62155 + 8.06830i 0 −32.3365 − 23.4938i 8.26670 + 25.4423i 0 −58.4588 − 42.4728i 54.7010 − 39.7426i 0 −226.948
64.2 −1.03535 + 3.18647i 0 16.8069 + 12.2109i −32.9083 − 101.281i 0 −32.7853 − 23.8199i −143.049 + 103.931i 0 356.801
64.3 −0.218677 + 0.673018i 0 25.4834 + 18.5148i 21.0138 + 64.6739i 0 98.3565 + 71.4602i −36.3535 + 26.4124i 0 −48.1219
64.4 1.95678 − 6.02234i 0 −6.55106 − 4.75963i −8.65323 − 26.6319i 0 121.539 + 88.3034i 122.450 − 88.9651i 0 −177.319
64.5 2.65487 − 8.17084i 0 −33.8257 − 24.5758i 21.1802 + 65.1858i 0 −196.384 − 142.681i −68.1911 + 49.5438i 0 588.853
82.1 −2.62155 − 8.06830i 0 −32.3365 + 23.4938i 8.26670 − 25.4423i 0 −58.4588 + 42.4728i 54.7010 + 39.7426i 0 −226.948
82.2 −1.03535 − 3.18647i 0 16.8069 − 12.2109i −32.9083 + 101.281i 0 −32.7853 + 23.8199i −143.049 − 103.931i 0 356.801
82.3 −0.218677 − 0.673018i 0 25.4834 − 18.5148i 21.0138 − 64.6739i 0 98.3565 − 71.4602i −36.3535 − 26.4124i 0 −48.1219
82.4 1.95678 + 6.02234i 0 −6.55106 + 4.75963i −8.65323 + 26.6319i 0 121.539 − 88.3034i 122.450 + 88.9651i 0 −177.319
82.5 2.65487 + 8.17084i 0 −33.8257 + 24.5758i 21.1802 − 65.1858i 0 −196.384 + 142.681i −68.1911 − 49.5438i 0 588.853
91.1 −8.77555 + 6.37581i 0 26.4708 − 81.4687i −31.3852 − 22.8027i 0 −67.6896 + 208.327i 179.871 + 553.584i 0 420.808
91.2 −5.51306 + 4.00547i 0 4.46148 − 13.7310i 62.5986 + 45.4806i 0 59.3828 − 182.761i −36.9828 − 113.821i 0 −527.281
91.3 −0.319007 + 0.231772i 0 −9.84050 + 30.2859i 17.3054 + 12.5731i 0 −52.3175 + 161.017i −7.77944 − 23.9427i 0 −8.43465
91.4 3.08746 − 2.24317i 0 −5.38796 + 16.5824i −3.77724 − 2.74432i 0 49.4196 − 152.098i 58.2998 + 179.428i 0 −17.8180
91.5 7.78409 − 5.65548i 0 18.7192 − 57.6117i −48.1408 − 34.9763i 0 9.43666 − 29.0431i −84.9654 − 261.497i 0 −572.540
\(n\): 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.b 20
3.b odd 2 1 33.6.e.b ✓ 20
11.c even 5 1 inner 99.6.f.b 20
11.c even 5 1 1089.6.a.bk 10
11.d odd 10 1 1089.6.a.bi 10
33.f even 10 1 363.6.a.t 10
33.h odd 10 1 33.6.e.b ✓ 20
33.h odd 10 1 363.6.a.r 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.6.e.b ✓ 20 3.b odd 2 1
33.6.e.b ✓ 20 33.h odd 10 1
99.6.f.b 20 1.a even 1 1 trivial
99.6.f.b 20 11.c even 5 1 inner
363.6.a.r 10 33.h odd 10 1
363.6.a.t 10 33.f even 10 1
1089.6.a.bi 10 11.d odd 10 1
1089.6.a.bk 10 11.c even 5 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{20} + 6 T_{2}^{19} + 94 T_{2}^{18} + 488 T_{2}^{17} + 12401 T_{2}^{16} + 30756 T_{2}^{15} + \cdots + 1371559530496 \) acting on \(S_{6}^{\mathrm{new}}(99, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + \cdots + 1371559530496 \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 43\!\cdots\!61 \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 20\!\cdots\!61 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 11\!\cdots\!01 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 14\!\cdots\!76 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 96\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( (T^{10} + \cdots + 12\!\cdots\!56)^{2} \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 91\!\cdots\!01 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 15\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 54\!\cdots\!36 \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots + 51\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 39\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 16\!\cdots\!61 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 17\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 12\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( (T^{10} + \cdots + 33\!\cdots\!36)^{2} \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 31\!\cdots\!36 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 77\!\cdots\!36 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 53\!\cdots\!25 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 75\!\cdots\!61 \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots - 11\!\cdots\!20)^{2} \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 48\!\cdots\!25 \) Copy content Toggle raw display
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