Properties

Label 99.6.f.a
Level $99$
Weight $6$
Character orbit 99.f
Analytic conductor $15.878$
Analytic rank $0$
Dimension $16$
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: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 5 x^{15} + 86 x^{14} - 146 x^{13} + 7205 x^{12} - 23732 x^{11} + 774165 x^{10} - 1228996 x^{9} + 44649817 x^{8} - 92720004 x^{7} + 943915592 x^{6} + \cdots + 393784336 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 11)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{7} - \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{2} + ( - \beta_{14} + \beta_{8} - 3 \beta_{7} - 9 \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + \cdots - 3) q^{4}+ \cdots + ( - 2 \beta_{15} - 2 \beta_{14} + 3 \beta_{13} + 4 \beta_{11} - 4 \beta_{10} + 41 \beta_{8} + \cdots - 41) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{7} - \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{2} + ( - \beta_{14} + \beta_{8} - 3 \beta_{7} - 9 \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + \cdots - 3) q^{4}+ \cdots + ( - 1294 \beta_{14} + 1294 \beta_{13} - 1110 \beta_{12} + \cdots - 49597) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + q^{2} - 73 q^{4} + 10 q^{5} + 196 q^{7} - 527 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + q^{2} - 73 q^{4} + 10 q^{5} + 196 q^{7} - 527 q^{8} - 672 q^{10} + 692 q^{11} + 1162 q^{13} - 560 q^{14} + 6399 q^{16} + 22 q^{17} - 3236 q^{19} + 5514 q^{20} - 13059 q^{22} + 10848 q^{23} + 15686 q^{25} - 16216 q^{26} + 8838 q^{28} - 13070 q^{29} - 14764 q^{31} + 58812 q^{32} - 26966 q^{34} - 43368 q^{35} + 4638 q^{37} - 68144 q^{38} + 52284 q^{40} + 14806 q^{41} - 24376 q^{43} + 104960 q^{44} + 50452 q^{46} - 40364 q^{47} + 32246 q^{49} - 10839 q^{50} - 80654 q^{52} + 11654 q^{53} + 7052 q^{55} - 70632 q^{56} + 10276 q^{58} - 70804 q^{59} - 31446 q^{61} + 153388 q^{62} + 17695 q^{64} + 41284 q^{65} - 64200 q^{67} + 94114 q^{68} - 103768 q^{70} + 184380 q^{71} - 1750 q^{73} - 306048 q^{74} + 174806 q^{76} - 384646 q^{77} + 24324 q^{79} + 386444 q^{80} - 316255 q^{82} + 46028 q^{83} + 63914 q^{85} - 22931 q^{86} + 211871 q^{88} - 148364 q^{89} - 258448 q^{91} + 58658 q^{92} - 55370 q^{94} + 4716 q^{95} + 484296 q^{97} - 743692 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 5 x^{15} + 86 x^{14} - 146 x^{13} + 7205 x^{12} - 23732 x^{11} + 774165 x^{10} - 1228996 x^{9} + 44649817 x^{8} - 92720004 x^{7} + 943915592 x^{6} + \cdots + 393784336 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 25\!\cdots\!53 \nu^{15} + \cdots + 31\!\cdots\!52 ) / 22\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 19\!\cdots\!81 \nu^{15} + \cdots + 38\!\cdots\!44 ) / 12\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 21\!\cdots\!19 \nu^{15} + \cdots - 37\!\cdots\!20 ) / 12\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 35\!\cdots\!61 \nu^{15} + \cdots + 32\!\cdots\!44 ) / 12\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 48\!\cdots\!69 \nu^{15} + \cdots + 18\!\cdots\!36 ) / 99\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 78\!\cdots\!15 \nu^{15} + \cdots + 19\!\cdots\!24 ) / 99\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 39\!\cdots\!39 \nu^{15} + \cdots + 65\!\cdots\!40 ) / 49\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 47\!\cdots\!26 \nu^{15} + \cdots - 20\!\cdots\!72 ) / 12\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 28\!\cdots\!59 \nu^{15} + \cdots + 15\!\cdots\!04 ) / 49\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 64\!\cdots\!06 \nu^{15} + \cdots + 17\!\cdots\!80 ) / 49\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 76\!\cdots\!69 \nu^{15} + \cdots + 11\!\cdots\!76 ) / 49\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 10\!\cdots\!27 \nu^{15} + \cdots - 17\!\cdots\!96 ) / 49\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 26\!\cdots\!26 \nu^{15} + \cdots - 64\!\cdots\!38 ) / 12\!\cdots\!99 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 27\!\cdots\!65 \nu^{15} + \cdots + 68\!\cdots\!96 ) / 81\!\cdots\!36 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{15} - 43\beta_{8} + \beta_{7} - 2\beta_{6} - \beta_{5} + \beta_{4} - \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 2 \beta_{14} - 4 \beta_{12} + 4 \beta_{11} - 4 \beta_{10} - 12 \beta_{8} + 34 \beta_{7} - 28 \beta_{6} - 77 \beta_{3} + 16 \beta_{2} + 16 \beta _1 - 16 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2 \beta_{15} - 87 \beta_{14} + 2 \beta_{13} + 4 \beta_{11} - 12 \beta_{10} - 4 \beta_{9} + 135 \beta_{8} - 833 \beta_{7} - 2576 \beta_{6} + 87 \beta_{5} - 107 \beta_{4} - 135 \beta_{3} + 135 \beta_{2} - 833 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 14 \beta_{14} + 14 \beta_{13} + 336 \beta_{12} - 336 \beta_{11} + 16 \beta_{9} + 9007 \beta_{8} - 9007 \beta_{7} + 262 \beta_{5} + 1934 \beta_{4} + 4483 \beta_{2} - 1934 \beta _1 - 3705 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 724 \beta_{15} - 724 \beta_{14} + 7321 \beta_{13} + 1008 \beta_{12} - 1608 \beta_{11} + 1608 \beta_{10} + 1008 \beta_{9} + 272598 \beta_{8} - 191468 \beta_{7} + 191468 \beta_{6} + 8045 \beta_{5} + \cdots - 272598 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 4252 \beta_{15} + 25030 \beta_{13} + 3904 \beta_{12} + 25788 \beta_{10} + 25788 \beta_{9} - 282911 \beta_{8} + 353785 \beta_{7} + 518476 \beta_{6} + 4252 \beta_{5} + 353785 \beta_{4} + \cdots - 164691 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 622115 \beta_{15} + 105294 \beta_{14} - 57324 \beta_{12} + 161348 \beta_{11} - 57324 \beta_{10} + 6672519 \beta_{8} + 16558939 \beta_{7} + 8139610 \beta_{6} + 824020 \beta_{3} + \cdots + 1467091 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 2363932 \beta_{15} + 710642 \beta_{14} - 2363932 \beta_{13} + 2009960 \beta_{11} - 1484760 \beta_{10} - 2009960 \beta_{9} - 18818226 \beta_{8} + 72285022 \beta_{7} + \cdots + 72285022 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 53543269 \beta_{14} - 53543269 \beta_{13} + 4603200 \beta_{12} - 4603200 \beta_{11} - 9980928 \beta_{9} - 738647557 \beta_{8} + 738647557 \beta_{7} - 65728725 \beta_{5} + \cdots + 1965217305 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 217581914 \beta_{15} + 217581914 \beta_{14} - 94872624 \beta_{13} - 58722752 \beta_{12} - 102105300 \beta_{11} + 102105300 \beta_{10} - 58722752 \beta_{9} + \cdots + 2995735314 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 4657355935 \beta_{15} - 1286346842 \beta_{13} - 929050408 \beta_{12} + 330009108 \beta_{10} + 330009108 \beta_{9} - 176806829869 \beta_{8} + 2019861251 \beta_{7} + \cdots + 75162756573 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 11293502550 \beta_{15} - 19835222816 \beta_{14} - 13154027264 \beta_{12} + 7079589488 \beta_{11} - 13154027264 \beta_{10} - 344191010582 \beta_{8} + \cdots - 165692995286 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 129838068780 \beta_{15} - 408574340305 \beta_{14} + 129838068780 \beta_{13} + 21012853800 \beta_{11} - 106428182840 \beta_{10} - 21012853800 \beta_{9} + \cdots - 8094359895355 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 1256105108580 \beta_{14} + 1256105108580 \beta_{13} + 1093932232300 \beta_{12} - 1093932232300 \beta_{11} + 604767604160 \beta_{9} + \cdots - 60157769831531 \) Copy content Toggle raw display

Character values

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

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

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
−2.13151 + 6.56011i
−1.19154 + 3.66717i
1.27432 3.92196i
2.73971 8.43195i
−7.26917 + 5.28136i
0.136182 0.0989419i
1.13051 0.821361i
7.81150 5.67539i
−7.26917 5.28136i
0.136182 + 0.0989419i
1.13051 + 0.821361i
7.81150 + 5.67539i
−2.13151 6.56011i
−1.19154 3.66717i
1.27432 + 3.92196i
2.73971 + 8.43195i
−6.38938 4.64216i 0 9.38602 + 28.8872i 51.9930 37.7751i 0 −33.5385 103.221i −3.96879 + 12.2147i 0 −507.561
37.2 −3.92850 2.85422i 0 −2.60202 8.00819i −68.6567 + 49.8820i 0 19.5777 + 60.2541i −60.6528 + 186.670i 0 412.092
37.3 2.52720 + 1.83612i 0 −6.87313 21.1533i −23.3906 + 16.9942i 0 27.3683 + 84.2310i 52.3600 161.147i 0 −90.3160
37.4 6.36363 + 4.62345i 0 9.23097 + 28.4100i 59.3249 43.1020i 0 −51.6143 158.852i 5.17243 15.9191i 0 576.801
64.1 −2.46756 + 7.59437i 0 −25.6970 18.6700i 2.24156 + 6.89880i 0 136.674 + 99.2995i −1.52919 + 1.11102i 0 −57.9232
64.2 0.361034 1.11115i 0 24.7842 + 18.0068i −1.31139 4.03604i 0 −146.551 106.476i 59.2025 43.0132i 0 −4.95810
64.3 0.740832 2.28005i 0 21.2388 + 15.4309i 5.23817 + 16.1214i 0 87.7560 + 63.7585i 112.982 82.0864i 0 40.6382
64.4 3.29275 10.1340i 0 −65.9678 47.9284i −20.4388 62.9043i 0 58.3278 + 42.3777i −427.066 + 310.282i 0 −704.774
82.1 −2.46756 7.59437i 0 −25.6970 + 18.6700i 2.24156 6.89880i 0 136.674 99.2995i −1.52919 1.11102i 0 −57.9232
82.2 0.361034 + 1.11115i 0 24.7842 18.0068i −1.31139 + 4.03604i 0 −146.551 + 106.476i 59.2025 + 43.0132i 0 −4.95810
82.3 0.740832 + 2.28005i 0 21.2388 15.4309i 5.23817 16.1214i 0 87.7560 63.7585i 112.982 + 82.0864i 0 40.6382
82.4 3.29275 + 10.1340i 0 −65.9678 + 47.9284i −20.4388 + 62.9043i 0 58.3278 42.3777i −427.066 310.282i 0 −704.774
91.1 −6.38938 + 4.64216i 0 9.38602 28.8872i 51.9930 + 37.7751i 0 −33.5385 + 103.221i −3.96879 12.2147i 0 −507.561
91.2 −3.92850 + 2.85422i 0 −2.60202 + 8.00819i −68.6567 49.8820i 0 19.5777 60.2541i −60.6528 186.670i 0 412.092
91.3 2.52720 1.83612i 0 −6.87313 + 21.1533i −23.3906 16.9942i 0 27.3683 84.2310i 52.3600 + 161.147i 0 −90.3160
91.4 6.36363 4.62345i 0 9.23097 28.4100i 59.3249 + 43.1020i 0 −51.6143 + 158.852i 5.17243 + 15.9191i 0 576.801
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.4
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.a 16
3.b odd 2 1 11.6.c.a 16
11.c even 5 1 inner 99.6.f.a 16
11.c even 5 1 1089.6.a.bg 8
11.d odd 10 1 1089.6.a.bb 8
33.f even 10 1 121.6.a.i 8
33.h odd 10 1 11.6.c.a 16
33.h odd 10 1 121.6.a.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.6.c.a 16 3.b odd 2 1
11.6.c.a 16 33.h odd 10 1
99.6.f.a 16 1.a even 1 1 trivial
99.6.f.a 16 11.c even 5 1 inner
121.6.a.g 8 33.h odd 10 1
121.6.a.i 8 33.f even 10 1
1089.6.a.bb 8 11.d odd 10 1
1089.6.a.bg 8 11.c even 5 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} - T_{2}^{15} + 101 T_{2}^{14} + 267 T_{2}^{13} + 5493 T_{2}^{12} - 11892 T_{2}^{11} + 277876 T_{2}^{10} + 638544 T_{2}^{9} + 28681664 T_{2}^{8} + 3917248 T_{2}^{7} - 33746048 T_{2}^{6} + \cdots + 50434379776 \) acting on \(S_{6}^{\mathrm{new}}(99, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - T^{15} + 101 T^{14} + \cdots + 50434379776 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} - 10 T^{15} + \cdots + 15\!\cdots\!96 \) Copy content Toggle raw display
$7$ \( T^{16} - 196 T^{15} + \cdots + 59\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{16} - 692 T^{15} + \cdots + 45\!\cdots\!01 \) Copy content Toggle raw display
$13$ \( T^{16} - 1162 T^{15} + \cdots + 18\!\cdots\!36 \) Copy content Toggle raw display
$17$ \( T^{16} - 22 T^{15} + \cdots + 64\!\cdots\!61 \) Copy content Toggle raw display
$19$ \( T^{16} + 3236 T^{15} + \cdots + 11\!\cdots\!25 \) Copy content Toggle raw display
$23$ \( (T^{8} - 5424 T^{7} + \cdots - 13\!\cdots\!36)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + 13070 T^{15} + \cdots + 90\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{16} + 14764 T^{15} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{16} - 4638 T^{15} + \cdots + 18\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( T^{16} - 14806 T^{15} + \cdots + 80\!\cdots\!41 \) Copy content Toggle raw display
$43$ \( (T^{8} + 12188 T^{7} + \cdots - 31\!\cdots\!84)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + 40364 T^{15} + \cdots + 32\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{16} - 11654 T^{15} + \cdots + 46\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( T^{16} + 70804 T^{15} + \cdots + 97\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{16} + 31446 T^{15} + \cdots + 64\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{8} + 32100 T^{7} + \cdots - 81\!\cdots\!36)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} - 184380 T^{15} + \cdots + 12\!\cdots\!76 \) Copy content Toggle raw display
$73$ \( T^{16} + 1750 T^{15} + \cdots + 39\!\cdots\!41 \) Copy content Toggle raw display
$79$ \( T^{16} - 24324 T^{15} + \cdots + 89\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{16} - 46028 T^{15} + \cdots + 35\!\cdots\!41 \) Copy content Toggle raw display
$89$ \( (T^{8} + 74182 T^{7} + \cdots - 32\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} - 484296 T^{15} + \cdots + 72\!\cdots\!25 \) Copy content Toggle raw display
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