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Properties

Label 99.3.k.c
Level $99$
Weight $3$
Character orbit 99.k
Analytic conductor $2.698$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [99,3,Mod(19,99)]
 
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, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("99.19");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 99.k (of order \(10\), 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: \(2.69755461717\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2 x^{15} + 3 x^{14} - 4 x^{13} + 77 x^{12} + 88 x^{11} - 577 x^{10} + 578 x^{9} + 1520 x^{8} + \cdots + 83521 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{8} - \beta_{7} + \beta_{5} + \cdots - 1) q^{2}+ \cdots + ( - 3 \beta_{15} - \beta_{14} + \cdots + \beta_{2}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{8} - \beta_{7} + \beta_{5} + \cdots - 1) q^{2}+ \cdots + (47 \beta_{15} + 25 \beta_{14} + \cdots - 27) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 20 q^{4} + 4 q^{5} - 30 q^{7} + 40 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 20 q^{4} + 4 q^{5} - 30 q^{7} + 40 q^{8} + 10 q^{11} + 30 q^{13} + 2 q^{14} + 16 q^{16} + 10 q^{17} - 42 q^{20} + 42 q^{22} - 132 q^{23} - 2 q^{25} - 46 q^{26} - 50 q^{28} - 160 q^{29} + 10 q^{31} - 368 q^{34} + 320 q^{35} - 126 q^{37} + 130 q^{38} + 30 q^{40} + 120 q^{41} + 206 q^{44} + 50 q^{46} + 150 q^{47} + 210 q^{49} - 330 q^{50} + 110 q^{52} - 342 q^{53} + 244 q^{55} - 524 q^{56} + 150 q^{58} - 110 q^{59} - 90 q^{61} - 40 q^{62} - 168 q^{64} + 36 q^{67} - 80 q^{68} + 340 q^{70} + 236 q^{71} - 350 q^{73} + 730 q^{74} + 390 q^{77} + 210 q^{79} + 806 q^{80} + 114 q^{82} + 190 q^{83} + 110 q^{85} - 736 q^{86} + 144 q^{88} - 76 q^{89} + 306 q^{91} + 150 q^{92} - 350 q^{94} - 430 q^{95} - 354 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2 x^{15} + 3 x^{14} - 4 x^{13} + 77 x^{12} + 88 x^{11} - 577 x^{10} + 578 x^{9} + 1520 x^{8} + \cdots + 83521 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 138203247537734 \nu^{15} - 291050501651580 \nu^{14} - 300665341738367 \nu^{13} + \cdots - 10\!\cdots\!14 ) / 10\!\cdots\!99 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1196620054805 \nu^{15} + 7612440593330 \nu^{14} - 13694873409440 \nu^{13} + \cdots + 87\!\cdots\!85 ) / 78\!\cdots\!34 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 79\!\cdots\!69 \nu^{15} + \cdots + 55\!\cdots\!82 ) / 41\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 12\!\cdots\!15 \nu^{15} + \cdots - 57\!\cdots\!30 ) / 41\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 13\!\cdots\!90 \nu^{15} + \cdots - 14\!\cdots\!47 ) / 41\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 17\!\cdots\!19 \nu^{15} + \cdots - 10\!\cdots\!48 ) / 41\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 20\!\cdots\!52 \nu^{15} + \cdots - 12\!\cdots\!26 ) / 41\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 24\!\cdots\!14 \nu^{15} + \cdots + 29\!\cdots\!54 ) / 41\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 29\!\cdots\!12 \nu^{15} + \cdots + 19\!\cdots\!73 ) / 41\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 31\!\cdots\!70 \nu^{15} + \cdots - 62\!\cdots\!81 ) / 41\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 20\!\cdots\!33 \nu^{15} + \cdots - 91\!\cdots\!68 ) / 24\!\cdots\!82 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 35\!\cdots\!76 \nu^{15} + \cdots - 19\!\cdots\!69 ) / 41\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 47\!\cdots\!48 \nu^{15} + \cdots + 91\!\cdots\!78 ) / 41\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 57\!\cdots\!67 \nu^{15} + \cdots + 27\!\cdots\!46 ) / 41\!\cdots\!94 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{15} + \beta_{10} + 3\beta_{8} - 3\beta_{7} - 2\beta_{5} + \beta_{4} - \beta_{3} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 2 \beta_{15} - \beta_{12} - \beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} - 3 \beta_{7} + 6 \beta_{6} + \cdots + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4 \beta_{15} - 9 \beta_{14} - 2 \beta_{13} - 2 \beta_{9} - 27 \beta_{8} - 27 \beta_{7} + 13 \beta_{6} + \cdots - 42 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 4 \beta_{15} - 36 \beta_{14} + 13 \beta_{13} - 13 \beta_{12} - 9 \beta_{11} + 23 \beta_{10} + \cdots - 73 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 7 \beta_{15} + 7 \beta_{14} + 44 \beta_{13} - 76 \beta_{12} - 44 \beta_{11} - 142 \beta_{10} + \cdots + 209 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 449 \beta_{15} + 59 \beta_{14} - 59 \beta_{13} + 157 \beta_{12} + 98 \beta_{11} - 504 \beta_{10} + \cdots - 59 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 11 \beta_{15} + 378 \beta_{14} + 602 \beta_{12} + 390 \beta_{11} + 591 \beta_{10} + 602 \beta_{9} + \cdots + 1028 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 2504 \beta_{15} + 4164 \beta_{14} - 201 \beta_{13} - 367 \beta_{9} + 3660 \beta_{8} + 3660 \beta_{7} + \cdots + 11389 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2504 \beta_{15} + 4696 \beta_{14} - 6668 \beta_{13} + 6668 \beta_{12} + 4164 \beta_{11} + 1972 \beta_{10} + \cdots - 11612 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 22394 \beta_{15} - 22394 \beta_{14} - 3312 \beta_{13} + 5504 \beta_{12} + 3312 \beta_{11} + \cdots - 76812 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 102120 \beta_{15} - 39824 \beta_{14} + 39824 \beta_{13} - 104234 \beta_{12} - 64410 \beta_{11} + \cdots + 39824 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 300533 \beta_{15} - 247371 \beta_{14} - 99867 \beta_{12} - 62296 \beta_{11} - 400400 \beta_{10} + \cdots - 695314 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 594635 \beta_{15} - 966540 \beta_{14} + 338104 \beta_{13} + 547904 \beta_{9} + 786321 \beta_{8} + \cdots - 1445411 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 594635 \beta_{15} + 1832695 \beta_{14} + 1561175 \beta_{13} - 1561175 \beta_{12} - 966540 \beta_{11} + \cdots + 11063785 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
−1.43448 − 2.82504i
−0.797732 − 1.94863i
1.60675 + 1.36085i
2.24350 + 2.23726i
1.64608 − 1.06057i
0.988132 − 0.846795i
−1.29715 − 0.104262i
−1.95510 + 0.109518i
1.64608 + 1.06057i
0.988132 + 0.846795i
−1.29715 + 0.104262i
−1.95510 − 0.109518i
−1.43448 + 2.82504i
−0.797732 + 1.94863i
1.60675 − 1.36085i
2.24350 − 2.23726i
−1.69557 − 2.33376i 0 −1.33538 + 4.10989i −0.356879 − 0.259287i 0 −10.0641 − 3.27002i 0.881730 − 0.286491i 0 1.27251i
19.2 −1.30204 − 1.79211i 0 −0.280267 + 0.862573i 7.03442 + 5.11081i 0 6.34535 + 2.06173i −6.51625 + 2.11726i 0 − 19.2609i
19.3 0.184008 + 0.253266i 0 1.20578 − 3.71102i −5.99919 − 4.35866i 0 −9.53633 − 3.09854i 2.35267 − 0.764430i 0 − 2.32142i
19.4 0.577539 + 0.794915i 0 0.937730 − 2.88604i 0.321645 + 0.233689i 0 6.87311 + 2.23321i 6.57364 − 2.13591i 0 0.390645i
28.1 −2.35440 + 0.764990i 0 1.72190 − 1.25104i 0.789076 − 2.42853i 0 −0.100159 − 0.137856i 2.72337 − 3.74840i 0 6.32135i
28.2 −1.28981 + 0.419086i 0 −1.74808 + 1.27006i −0.708979 + 2.18201i 0 −5.74346 − 7.90520i 4.91103 − 6.75946i 0 − 3.11151i
28.3 2.40785 − 0.782357i 0 1.94958 − 1.41645i 2.61024 − 8.03348i 0 1.43445 + 1.97435i −2.36641 + 3.25708i 0 − 21.3855i
28.4 3.47243 − 1.12826i 0 7.54873 − 5.48447i −1.69033 + 5.20232i 0 −4.20886 − 5.79300i 11.4402 − 15.7461i 0 19.9718i
46.1 −2.35440 − 0.764990i 0 1.72190 + 1.25104i 0.789076 + 2.42853i 0 −0.100159 + 0.137856i 2.72337 + 3.74840i 0 − 6.32135i
46.2 −1.28981 − 0.419086i 0 −1.74808 − 1.27006i −0.708979 − 2.18201i 0 −5.74346 + 7.90520i 4.91103 + 6.75946i 0 3.11151i
46.3 2.40785 + 0.782357i 0 1.94958 + 1.41645i 2.61024 + 8.03348i 0 1.43445 − 1.97435i −2.36641 − 3.25708i 0 21.3855i
46.4 3.47243 + 1.12826i 0 7.54873 + 5.48447i −1.69033 − 5.20232i 0 −4.20886 + 5.79300i 11.4402 + 15.7461i 0 − 19.9718i
73.1 −1.69557 + 2.33376i 0 −1.33538 − 4.10989i −0.356879 + 0.259287i 0 −10.0641 + 3.27002i 0.881730 + 0.286491i 0 − 1.27251i
73.2 −1.30204 + 1.79211i 0 −0.280267 − 0.862573i 7.03442 − 5.11081i 0 6.34535 − 2.06173i −6.51625 − 2.11726i 0 19.2609i
73.3 0.184008 − 0.253266i 0 1.20578 + 3.71102i −5.99919 + 4.35866i 0 −9.53633 + 3.09854i 2.35267 + 0.764430i 0 2.32142i
73.4 0.577539 − 0.794915i 0 0.937730 + 2.88604i 0.321645 − 0.233689i 0 6.87311 − 2.23321i 6.57364 + 2.13591i 0 − 0.390645i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 99.3.k.c 16
3.b odd 2 1 33.3.g.a ✓ 16
11.c even 5 1 1089.3.c.m 16
11.d odd 10 1 inner 99.3.k.c 16
11.d odd 10 1 1089.3.c.m 16
12.b even 2 1 528.3.bf.b 16
33.d even 2 1 363.3.g.f 16
33.f even 10 1 33.3.g.a ✓ 16
33.f even 10 1 363.3.c.e 16
33.f even 10 1 363.3.g.a 16
33.f even 10 1 363.3.g.g 16
33.h odd 10 1 363.3.c.e 16
33.h odd 10 1 363.3.g.a 16
33.h odd 10 1 363.3.g.f 16
33.h odd 10 1 363.3.g.g 16
132.n odd 10 1 528.3.bf.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.3.g.a ✓ 16 3.b odd 2 1
33.3.g.a ✓ 16 33.f even 10 1
99.3.k.c 16 1.a even 1 1 trivial
99.3.k.c 16 11.d odd 10 1 inner
363.3.c.e 16 33.f even 10 1
363.3.c.e 16 33.h odd 10 1
363.3.g.a 16 33.f even 10 1
363.3.g.a 16 33.h odd 10 1
363.3.g.f 16 33.d even 2 1
363.3.g.f 16 33.h odd 10 1
363.3.g.g 16 33.f even 10 1
363.3.g.g 16 33.h odd 10 1
528.3.bf.b 16 12.b even 2 1
528.3.bf.b 16 132.n odd 10 1
1089.3.c.m 16 11.c even 5 1
1089.3.c.m 16 11.d odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} - 18 T_{2}^{14} - 40 T_{2}^{13} + 160 T_{2}^{12} + 720 T_{2}^{11} - 28 T_{2}^{10} + \cdots + 3721 \) acting on \(S_{3}^{\mathrm{new}}(99, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 18 T^{14} + \cdots + 3721 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} - 4 T^{15} + \cdots + 9369721 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 22159001881 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 45\!\cdots\!61 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 47\!\cdots\!16 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 18\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 99\!\cdots\!96 \) Copy content Toggle raw display
$23$ \( (T^{8} + 66 T^{7} + \cdots + 255717136)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 17\!\cdots\!36 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 59\!\cdots\!41 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 30\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 13\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 30\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 41\!\cdots\!41 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 34\!\cdots\!41 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 13\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( (T^{8} - 18 T^{7} + \cdots + 47006885776)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 40\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 63\!\cdots\!36 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 14\!\cdots\!81 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 16\!\cdots\!41 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots - 15121642690304)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 29\!\cdots\!01 \) Copy content Toggle raw display
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