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Properties

Label 99.5.k.a
Level $99$
Weight $5$
Character orbit 99.k
Analytic conductor $10.234$
Analytic rank $0$
Dimension $12$
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] = [99,5,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, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("99.19");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 5 \)
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: \(10.2336263453\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(3\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 115x^{10} + 5030x^{8} + 102975x^{6} + 953170x^{4} + 2910655x^{2} + 73205 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 11)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{7} + \beta_{4} + 2 \beta_{3} + \cdots + 1) q^{2}+ \cdots + ( - \beta_{11} - 6 \beta_{10} + \cdots + 19) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{7} + \beta_{4} + 2 \beta_{3} + \cdots + 1) q^{2}+ \cdots + (92 \beta_{11} + 313 \beta_{10} + \cdots - 630) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 5 q^{2} + 7 q^{4} + 18 q^{5} - 80 q^{7} + 245 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 5 q^{2} + 7 q^{4} + 18 q^{5} - 80 q^{7} + 245 q^{8} + 43 q^{11} + 250 q^{13} - 610 q^{14} - 633 q^{16} + 1250 q^{17} - 1025 q^{19} - 752 q^{20} - 35 q^{22} - 1684 q^{23} + 197 q^{25} - 3490 q^{26} - 3580 q^{28} + 2690 q^{29} - 1136 q^{31} + 2370 q^{34} - 3610 q^{35} - 336 q^{37} - 1900 q^{38} - 2340 q^{40} + 4550 q^{41} + 6268 q^{44} + 4150 q^{46} - 24 q^{47} + 827 q^{49} - 8895 q^{50} + 14070 q^{52} - 414 q^{53} - 2738 q^{55} + 21340 q^{56} + 2980 q^{58} + 10011 q^{59} + 9460 q^{61} + 6200 q^{62} - 2633 q^{64} + 12154 q^{67} + 9400 q^{68} - 9380 q^{70} - 17574 q^{71} + 27950 q^{73} - 43270 q^{74} - 4090 q^{77} - 41540 q^{79} + 2308 q^{80} - 28175 q^{82} + 18665 q^{83} - 4230 q^{85} + 10125 q^{86} - 15125 q^{88} - 5554 q^{89} + 7390 q^{91} - 3904 q^{92} + 18920 q^{94} - 14110 q^{95} + 20769 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 115x^{10} + 5030x^{8} + 102975x^{6} + 953170x^{4} + 2910655x^{2} + 73205 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 13052 \nu^{11} - 117304 \nu^{10} - 1219776 \nu^{9} - 10477907 \nu^{8} - 40135806 \nu^{7} + \cdots - 1427754383 ) / 1339827192 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 13052 \nu^{11} - 117304 \nu^{10} + 1219776 \nu^{9} - 10477907 \nu^{8} + 40135806 \nu^{7} + \cdots - 1427754383 ) / 1339827192 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 31452 \nu^{11} + 117304 \nu^{10} - 2737673 \nu^{9} + 10477907 \nu^{8} - 79586472 \nu^{7} + \cdots + 757840787 ) / 1339827192 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 45643 \nu^{10} + 4016197 \nu^{8} + 120327945 \nu^{6} + 1386775659 \nu^{4} + 4387794733 \nu^{2} - 676101899 ) / 60901236 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 10664 \nu^{11} + 25564 \nu^{10} + 952537 \nu^{9} + 2319614 \nu^{8} + 29205906 \nu^{7} + \cdots + 86861060 ) / 121802472 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 10664 \nu^{11} - 25564 \nu^{10} + 952537 \nu^{9} - 2319614 \nu^{8} + 29205906 \nu^{7} + \cdots - 86861060 ) / 121802472 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 10664 \nu^{11} - 79937 \nu^{10} + 952537 \nu^{9} - 7147008 \nu^{8} + 29205906 \nu^{7} + \cdots - 209313060 ) / 121802472 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 169940 \nu^{11} - 259930 \nu^{10} - 15878131 \nu^{9} - 23961014 \nu^{8} - 530879748 \nu^{7} + \cdots + 4416548158 ) / 1339827192 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 169940 \nu^{11} - 259930 \nu^{10} + 15878131 \nu^{9} - 23961014 \nu^{8} + 530879748 \nu^{7} + \cdots + 4416548158 ) / 1339827192 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 380623 \nu^{11} + 502073 \nu^{10} - 35561311 \nu^{9} + 44178167 \nu^{8} - 1181014629 \nu^{7} + \cdots - 7437120889 ) / 1339827192 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} + \beta_{9} - 2\beta_{8} + \beta_{7} + \beta_{6} - \beta_{5} + \beta_{3} + \beta_{2} - \beta _1 - 19 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} - \beta_{9} - 5\beta_{7} - 5\beta_{6} - 24\beta_{4} - 9\beta_{3} - 15\beta_{2} - 25\beta _1 - 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 31 \beta_{10} - 31 \beta_{9} + 90 \beta_{8} - 51 \beta_{7} - 39 \beta_{6} + 33 \beta_{5} - 113 \beta_{3} + \cdots + 476 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 32 \beta_{11} - 33 \beta_{10} + 33 \beta_{9} + 271 \beta_{7} + 271 \beta_{6} - 16 \beta_{5} + 1300 \beta_{4} + \cdots + 650 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 954 \beta_{10} + 954 \beta_{9} - 3628 \beta_{8} + 2292 \beta_{7} + 1336 \beta_{6} - 1142 \beta_{5} + \cdots - 12905 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 2096 \beta_{11} + 1056 \beta_{10} - 1056 \beta_{9} - 11834 \beta_{7} - 11834 \beta_{6} + 1048 \beta_{5} + \cdots - 28374 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 30477 \beta_{10} - 30477 \beta_{9} + 140790 \beta_{8} - 95925 \beta_{7} - 44865 \beta_{6} + \cdots + 369011 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 99264 \beta_{11} - 36579 \beta_{10} + 36579 \beta_{9} + 478665 \beta_{7} + 478665 \beta_{6} + \cdots + 1150758 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1012445 \beta_{10} + 1012445 \beta_{9} - 5366098 \beta_{8} + 3851627 \beta_{7} + 1514471 \beta_{6} + \cdots - 11069600 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 4178328 \beta_{11} + 1343903 \beta_{10} - 1343903 \beta_{9} - 18668485 \beta_{7} - 18668485 \beta_{6} + \cdots - 45055758 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(46\) \(56\)
\(\chi(n)\) \(-\beta_{4}\) \(1\)

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
5.08417i
− 0.159251i
− 6.10049i
5.04186i
− 2.38108i
− 4.56289i
− 5.04186i
2.38108i
4.56289i
− 5.08417i
0.159251i
6.10049i
−4.10644 − 5.65202i 0 −10.1383 + 31.2024i 10.0543 + 7.30486i 0 23.8119 + 7.73696i 111.679 − 36.2869i 0 − 86.8239i
19.2 −1.02443 − 1.41001i 0 4.00561 − 12.3280i 8.06057 + 5.85635i 0 −56.0338 − 18.2065i −48.0070 + 15.5984i 0 − 17.3649i
19.3 2.46775 + 3.39656i 0 −0.502581 + 1.54679i −14.7329 − 10.7041i 0 47.9990 + 15.5958i 57.3923 − 18.6479i 0 − 76.4560i
28.1 −3.67706 + 1.19475i 0 −0.850950 + 0.618251i 1.76709 − 5.43854i 0 −2.52826 − 3.47985i 38.7511 − 53.3363i 0 22.1090i
28.2 3.38257 − 1.09906i 0 −2.71041 + 1.96923i −5.48545 + 16.8825i 0 −19.6356 − 27.0262i −40.4526 + 55.6782i 0 63.1351i
28.3 5.45760 − 1.77328i 0 13.6966 − 9.95117i 9.33639 − 28.7345i 0 −33.6132 − 46.2646i 3.13669 − 4.31729i 0 − 173.377i
46.1 −3.67706 − 1.19475i 0 −0.850950 − 0.618251i 1.76709 + 5.43854i 0 −2.52826 + 3.47985i 38.7511 + 53.3363i 0 − 22.1090i
46.2 3.38257 + 1.09906i 0 −2.71041 − 1.96923i −5.48545 − 16.8825i 0 −19.6356 + 27.0262i −40.4526 − 55.6782i 0 − 63.1351i
46.3 5.45760 + 1.77328i 0 13.6966 + 9.95117i 9.33639 + 28.7345i 0 −33.6132 + 46.2646i 3.13669 + 4.31729i 0 173.377i
73.1 −4.10644 + 5.65202i 0 −10.1383 − 31.2024i 10.0543 − 7.30486i 0 23.8119 − 7.73696i 111.679 + 36.2869i 0 86.8239i
73.2 −1.02443 + 1.41001i 0 4.00561 + 12.3280i 8.06057 − 5.85635i 0 −56.0338 + 18.2065i −48.0070 − 15.5984i 0 17.3649i
73.3 2.46775 − 3.39656i 0 −0.502581 − 1.54679i −14.7329 + 10.7041i 0 47.9990 − 15.5958i 57.3923 + 18.6479i 0 76.4560i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.3
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 99.5.k.a 12
3.b odd 2 1 11.5.d.a ✓ 12
11.d odd 10 1 inner 99.5.k.a 12
12.b even 2 1 176.5.n.a 12
33.d even 2 1 121.5.d.d 12
33.f even 10 1 11.5.d.a ✓ 12
33.f even 10 1 121.5.b.b 12
33.f even 10 1 121.5.d.c 12
33.f even 10 1 121.5.d.e 12
33.h odd 10 1 121.5.b.b 12
33.h odd 10 1 121.5.d.c 12
33.h odd 10 1 121.5.d.d 12
33.h odd 10 1 121.5.d.e 12
132.n odd 10 1 176.5.n.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.5.d.a ✓ 12 3.b odd 2 1
11.5.d.a ✓ 12 33.f even 10 1
99.5.k.a 12 1.a even 1 1 trivial
99.5.k.a 12 11.d odd 10 1 inner
121.5.b.b 12 33.f even 10 1
121.5.b.b 12 33.h odd 10 1
121.5.d.c 12 33.f even 10 1
121.5.d.c 12 33.h odd 10 1
121.5.d.d 12 33.d even 2 1
121.5.d.d 12 33.h odd 10 1
121.5.d.e 12 33.f even 10 1
121.5.d.e 12 33.h odd 10 1
176.5.n.a 12 12.b even 2 1
176.5.n.a 12 132.n odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} - 5 T_{2}^{11} - 15 T_{2}^{10} - 125 T_{2}^{9} + 2135 T_{2}^{8} - 2050 T_{2}^{7} + \cdots + 16272080 \) acting on \(S_{5}^{\mathrm{new}}(99, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 5 T^{11} + \cdots + 16272080 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 47826408560896 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 98\!\cdots\!41 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 10\!\cdots\!80 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 16\!\cdots\!05 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 31\!\cdots\!05 \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots - 10\!\cdots\!96)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 61\!\cdots\!80 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 41\!\cdots\!36 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 49\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 72\!\cdots\!05 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 84\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 11\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 38\!\cdots\!41 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 15\!\cdots\!80 \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 15\!\cdots\!84)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 56\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 14\!\cdots\!05 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 36\!\cdots\!80 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 18\!\cdots\!05 \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 66\!\cdots\!44)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 18\!\cdots\!01 \) Copy content Toggle raw display
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