Properties

Label 99.2.f.a
Level 99
Weight 2
Character orbit 99.f
Analytic conductor 0.791
Analytic rank 0
Dimension 4
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 99.f (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.790518980011\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + 2 \zeta_{10} - 2 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{2} + ( -3 + 3 \zeta_{10} ) q^{4} + ( 1 + \zeta_{10}^{2} ) q^{5} + \zeta_{10}^{3} q^{7} + ( -4 \zeta_{10} + 5 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{8} +O(q^{10})\) \( q + ( -1 + 2 \zeta_{10} - 2 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{2} + ( -3 + 3 \zeta_{10} ) q^{4} + ( 1 + \zeta_{10}^{2} ) q^{5} + \zeta_{10}^{3} q^{7} + ( -4 \zeta_{10} + 5 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{8} + ( -\zeta_{10}^{2} + \zeta_{10}^{3} ) q^{10} + ( 2 + \zeta_{10} + 2 \zeta_{10}^{3} ) q^{11} + ( 3 - \zeta_{10} + \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{13} + ( \zeta_{10} - 2 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{14} + ( 3 - 8 \zeta_{10} + 3 \zeta_{10}^{2} ) q^{16} + ( -3 - 3 \zeta_{10} - 3 \zeta_{10}^{2} ) q^{17} + ( -3 \zeta_{10} + 4 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{19} + ( -3 + 3 \zeta_{10} - 3 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{20} + ( -3 + 6 \zeta_{10} - 7 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{22} + ( 3 + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{23} + ( \zeta_{10} - 4 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{25} + ( -1 + \zeta_{10} + \zeta_{10}^{3} ) q^{26} + ( -3 + 3 \zeta_{10} - 3 \zeta_{10}^{2} ) q^{28} -6 \zeta_{10}^{3} q^{29} + ( -2 - 3 \zeta_{10} + 3 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{31} + ( 6 - 3 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{32} + 3 q^{34} + ( -1 + \zeta_{10}^{3} ) q^{35} + ( 2 - 2 \zeta_{10} + 3 \zeta_{10}^{3} ) q^{37} + ( 7 - 11 \zeta_{10} + 7 \zeta_{10}^{2} ) q^{38} + ( -1 + \zeta_{10} - 3 \zeta_{10}^{3} ) q^{40} + ( 2 \zeta_{10} + \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{41} + ( 3 + 6 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{43} + ( -12 + 9 \zeta_{10} - 3 \zeta_{10}^{2} ) q^{44} + ( -1 + \zeta_{10}^{3} ) q^{46} + ( -5 \zeta_{10} + 7 \zeta_{10}^{2} - 5 \zeta_{10}^{3} ) q^{47} + 6 \zeta_{10} q^{49} + ( -5 + 9 \zeta_{10} - 5 \zeta_{10}^{2} ) q^{50} + ( 3 \zeta_{10} + 3 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{52} + ( -2 + \zeta_{10} - \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{53} + ( \zeta_{10} + 2 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{55} + ( -1 + 4 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{56} + ( -6 \zeta_{10} + 12 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{58} + ( -1 + \zeta_{10} + 9 \zeta_{10}^{3} ) q^{59} + ( -9 + 6 \zeta_{10} - 9 \zeta_{10}^{2} ) q^{61} + ( 8 - 8 \zeta_{10} + 3 \zeta_{10}^{3} ) q^{62} + ( 1 + 5 \zeta_{10} - 5 \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{64} + ( 5 + 3 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{65} + ( -3 - 3 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{67} + ( 9 - 9 \zeta_{10}^{3} ) q^{68} + ( 1 - \zeta_{10} ) q^{70} + ( -7 + 6 \zeta_{10} - 7 \zeta_{10}^{2} ) q^{71} + ( 6 - 6 \zeta_{10} - 4 \zeta_{10}^{3} ) q^{73} + ( 7 \zeta_{10} - 12 \zeta_{10}^{2} + 7 \zeta_{10}^{3} ) q^{74} + ( 9 - 12 \zeta_{10}^{2} + 12 \zeta_{10}^{3} ) q^{76} + ( -1 - \zeta_{10} - \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{77} + ( -11 + 11 \zeta_{10} - 11 \zeta_{10}^{2} + 11 \zeta_{10}^{3} ) q^{79} + ( -5 \zeta_{10} + 3 \zeta_{10}^{2} - 5 \zeta_{10}^{3} ) q^{80} + ( -1 - \zeta_{10}^{2} ) q^{82} + ( -4 - \zeta_{10} - 4 \zeta_{10}^{2} ) q^{83} + ( -6 \zeta_{10} - 3 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{85} + ( 3 - 12 \zeta_{10} + 12 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{86} + ( 2 - 12 \zeta_{10} + 18 \zeta_{10}^{2} - 15 \zeta_{10}^{3} ) q^{88} + ( 5 - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{89} + ( 2 \zeta_{10} + \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{91} + ( -3 + 3 \zeta_{10} + 6 \zeta_{10}^{3} ) q^{92} + ( 12 - 19 \zeta_{10} + 12 \zeta_{10}^{2} ) q^{94} + ( -1 + \zeta_{10} - 2 \zeta_{10}^{3} ) q^{95} + ( -3 + 6 \zeta_{10} - 6 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{97} + ( -6 + 6 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + q^{2} - 9q^{4} + 3q^{5} + q^{7} - 13q^{8} + O(q^{10}) \) \( 4q + q^{2} - 9q^{4} + 3q^{5} + q^{7} - 13q^{8} + 2q^{10} + 11q^{11} + 7q^{13} + 4q^{14} + q^{16} - 12q^{17} - 10q^{19} - 3q^{20} + 4q^{22} + 8q^{23} + 6q^{25} - 2q^{26} - 6q^{28} - 6q^{29} - 12q^{31} + 30q^{32} + 12q^{34} - 3q^{35} + 9q^{37} + 10q^{38} - 6q^{40} + 3q^{41} - 36q^{44} - 3q^{46} - 17q^{47} + 6q^{49} - 6q^{50} + 3q^{52} - 4q^{53} + 2q^{55} - 12q^{56} - 24q^{58} + 6q^{59} - 21q^{61} + 27q^{62} + 13q^{64} + 14q^{65} - 6q^{67} + 27q^{68} + 3q^{70} - 15q^{71} + 14q^{73} + 26q^{74} + 60q^{76} - q^{77} - 11q^{79} - 13q^{80} - 3q^{82} - 13q^{83} - 9q^{85} - 15q^{86} - 37q^{88} + 24q^{89} + 3q^{91} - 3q^{92} + 17q^{94} - 5q^{95} + 3q^{97} - 36q^{98} + O(q^{100}) \)

Character values

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

\(n\) \(46\) \(56\)
\(\chi(n)\) \(-\zeta_{10}^{3}\) \(1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
0.809017 0.587785i
−0.309017 0.951057i
−0.309017 + 0.951057i
0.809017 + 0.587785i
−0.309017 0.224514i 0 −0.572949 1.76336i 1.30902 0.951057i 0 −0.309017 0.951057i −0.454915 + 1.40008i 0 −0.618034
64.1 0.809017 2.48990i 0 −3.92705 2.85317i 0.190983 + 0.587785i 0 0.809017 + 0.587785i −6.04508 + 4.39201i 0 1.61803
82.1 0.809017 + 2.48990i 0 −3.92705 + 2.85317i 0.190983 0.587785i 0 0.809017 0.587785i −6.04508 4.39201i 0 1.61803
91.1 −0.309017 + 0.224514i 0 −0.572949 + 1.76336i 1.30902 + 0.951057i 0 −0.309017 + 0.951057i −0.454915 1.40008i 0 −0.618034
\(n\): e.g. 2-40 or 990-1000
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 99.2.f.a 4
3.b odd 2 1 33.2.e.b 4
9.c even 3 2 891.2.n.b 8
9.d odd 6 2 891.2.n.c 8
11.c even 5 1 inner 99.2.f.a 4
11.c even 5 1 1089.2.a.t 2
11.d odd 10 1 1089.2.a.l 2
12.b even 2 1 528.2.y.b 4
15.d odd 2 1 825.2.n.c 4
15.e even 4 2 825.2.bx.d 8
33.d even 2 1 363.2.e.f 4
33.f even 10 1 363.2.a.i 2
33.f even 10 2 363.2.e.b 4
33.f even 10 1 363.2.e.f 4
33.h odd 10 1 33.2.e.b 4
33.h odd 10 1 363.2.a.d 2
33.h odd 10 2 363.2.e.k 4
99.m even 15 2 891.2.n.b 8
99.n odd 30 2 891.2.n.c 8
132.n odd 10 1 5808.2.a.ci 2
132.o even 10 1 528.2.y.b 4
132.o even 10 1 5808.2.a.cj 2
165.o odd 10 1 825.2.n.c 4
165.o odd 10 1 9075.2.a.cb 2
165.r even 10 1 9075.2.a.u 2
165.v even 20 2 825.2.bx.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.2.e.b 4 3.b odd 2 1
33.2.e.b 4 33.h odd 10 1
99.2.f.a 4 1.a even 1 1 trivial
99.2.f.a 4 11.c even 5 1 inner
363.2.a.d 2 33.h odd 10 1
363.2.a.i 2 33.f even 10 1
363.2.e.b 4 33.f even 10 2
363.2.e.f 4 33.d even 2 1
363.2.e.f 4 33.f even 10 1
363.2.e.k 4 33.h odd 10 2
528.2.y.b 4 12.b even 2 1
528.2.y.b 4 132.o even 10 1
825.2.n.c 4 15.d odd 2 1
825.2.n.c 4 165.o odd 10 1
825.2.bx.d 8 15.e even 4 2
825.2.bx.d 8 165.v even 20 2
891.2.n.b 8 9.c even 3 2
891.2.n.b 8 99.m even 15 2
891.2.n.c 8 9.d odd 6 2
891.2.n.c 8 99.n odd 30 2
1089.2.a.l 2 11.d odd 10 1
1089.2.a.t 2 11.c even 5 1
5808.2.a.ci 2 132.n odd 10 1
5808.2.a.cj 2 132.o even 10 1
9075.2.a.u 2 165.r even 10 1
9075.2.a.cb 2 165.o odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - T_{2}^{3} + 6 T_{2}^{2} + 4 T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(99, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + 4 T^{2} - 2 T^{3} + 9 T^{4} - 4 T^{5} + 16 T^{6} - 8 T^{7} + 16 T^{8} \)
$3$ \( \)
$5$ \( 1 - 3 T - T^{2} + 3 T^{3} + 16 T^{4} + 15 T^{5} - 25 T^{6} - 375 T^{7} + 625 T^{8} \)
$7$ \( 1 - T - 6 T^{2} + 13 T^{3} + 29 T^{4} + 91 T^{5} - 294 T^{6} - 343 T^{7} + 2401 T^{8} \)
$11$ \( 1 - 11 T + 51 T^{2} - 121 T^{3} + 121 T^{4} \)
$13$ \( 1 - 7 T + 6 T^{2} + 49 T^{3} - 181 T^{4} + 637 T^{5} + 1014 T^{6} - 15379 T^{7} + 28561 T^{8} \)
$17$ \( 1 + 12 T + 37 T^{2} - 180 T^{3} - 1619 T^{4} - 3060 T^{5} + 10693 T^{6} + 58956 T^{7} + 83521 T^{8} \)
$19$ \( 1 + 10 T + 21 T^{2} - 70 T^{3} - 469 T^{4} - 1330 T^{5} + 7581 T^{6} + 68590 T^{7} + 130321 T^{8} \)
$23$ \( ( 1 - 4 T + 45 T^{2} - 92 T^{3} + 529 T^{4} )^{2} \)
$29$ \( 1 + 6 T + 7 T^{2} - 132 T^{3} - 995 T^{4} - 3828 T^{5} + 5887 T^{6} + 146334 T^{7} + 707281 T^{8} \)
$31$ \( 1 + 12 T + 63 T^{2} + 434 T^{3} + 3255 T^{4} + 13454 T^{5} + 60543 T^{6} + 357492 T^{7} + 923521 T^{8} \)
$37$ \( 1 - 9 T - 6 T^{2} + 307 T^{3} - 1581 T^{4} + 11359 T^{5} - 8214 T^{6} - 455877 T^{7} + 1874161 T^{8} \)
$41$ \( 1 - 3 T - 22 T^{2} - 171 T^{3} + 2215 T^{4} - 7011 T^{5} - 36982 T^{6} - 206763 T^{7} + 2825761 T^{8} \)
$43$ \( ( 1 + 41 T^{2} + 1849 T^{4} )^{2} \)
$47$ \( 1 + 17 T + 67 T^{2} - 335 T^{3} - 4344 T^{4} - 15745 T^{5} + 148003 T^{6} + 1764991 T^{7} + 4879681 T^{8} \)
$53$ \( 1 + 4 T - 47 T^{2} - 160 T^{3} + 2121 T^{4} - 8480 T^{5} - 132023 T^{6} + 595508 T^{7} + 7890481 T^{8} \)
$59$ \( 1 - 6 T + 17 T^{2} + 222 T^{3} - 1685 T^{4} + 13098 T^{5} + 59177 T^{6} - 1232274 T^{7} + 12117361 T^{8} \)
$61$ \( 1 + 21 T + 245 T^{2} + 2559 T^{3} + 23404 T^{4} + 156099 T^{5} + 911645 T^{6} + 4766601 T^{7} + 13845841 T^{8} \)
$67$ \( ( 1 + 3 T + 125 T^{2} + 201 T^{3} + 4489 T^{4} )^{2} \)
$71$ \( 1 + 15 T + 119 T^{2} + 1455 T^{3} + 17296 T^{4} + 103305 T^{5} + 599879 T^{6} + 5368665 T^{7} + 25411681 T^{8} \)
$73$ \( 1 - 14 T + 63 T^{2} - 850 T^{3} + 12521 T^{4} - 62050 T^{5} + 335727 T^{6} - 5446238 T^{7} + 28398241 T^{8} \)
$79$ \( 1 + 11 T + 42 T^{2} - 407 T^{3} - 7795 T^{4} - 32153 T^{5} + 262122 T^{6} + 5423429 T^{7} + 38950081 T^{8} \)
$83$ \( 1 + 13 T - 14 T^{2} - 421 T^{3} + 1449 T^{4} - 34943 T^{5} - 96446 T^{6} + 7433231 T^{7} + 47458321 T^{8} \)
$89$ \( ( 1 - 12 T + 209 T^{2} - 1068 T^{3} + 7921 T^{4} )^{2} \)
$97$ \( 1 - 3 T - 43 T^{2} - 765 T^{3} + 11236 T^{4} - 74205 T^{5} - 404587 T^{6} - 2738019 T^{7} + 88529281 T^{8} \)
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