Properties

Label 99.6.a.f
Level $99$
Weight $6$
Character orbit 99.a
Self dual yes
Analytic conductor $15.878$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 99.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(15.8779981615\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{177}) \)
Defining polynomial: \(x^{2} - x - 44\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{177})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 3 - \beta ) q^{2} + ( 21 - 5 \beta ) q^{4} + ( -34 + 10 \beta ) q^{5} + ( -148 + 10 \beta ) q^{7} + ( 187 + \beta ) q^{8} +O(q^{10})\) \( q + ( 3 - \beta ) q^{2} + ( 21 - 5 \beta ) q^{4} + ( -34 + 10 \beta ) q^{5} + ( -148 + 10 \beta ) q^{7} + ( 187 + \beta ) q^{8} + ( -542 + 54 \beta ) q^{10} + 121 q^{11} + ( -102 + 38 \beta ) q^{13} + ( -884 + 168 \beta ) q^{14} + ( -155 - 25 \beta ) q^{16} + ( 430 - 60 \beta ) q^{17} + ( -732 - 12 \beta ) q^{19} + ( -2914 + 330 \beta ) q^{20} + ( 363 - 121 \beta ) q^{22} + ( 1868 - 366 \beta ) q^{23} + ( 2431 - 580 \beta ) q^{25} + ( -1978 + 178 \beta ) q^{26} + ( -5308 + 900 \beta ) q^{28} + ( -3094 - 412 \beta ) q^{29} + ( -3936 + 344 \beta ) q^{31} + ( -5349 + 73 \beta ) q^{32} + ( 3930 - 550 \beta ) q^{34} + ( 9432 - 1720 \beta ) q^{35} + ( -14890 - 136 \beta ) q^{37} + ( -1668 + 708 \beta ) q^{38} + ( -5918 + 1846 \beta ) q^{40} + ( 2534 + 712 \beta ) q^{41} + ( -7580 - 1496 \beta ) q^{43} + ( 2541 - 605 \beta ) q^{44} + ( 21708 - 2600 \beta ) q^{46} + ( -5188 + 2526 \beta ) q^{47} + ( 9497 - 2860 \beta ) q^{49} + ( 32813 - 3591 \beta ) q^{50} + ( -10502 + 1118 \beta ) q^{52} + ( -5986 - 2206 \beta ) q^{53} + ( -4114 + 1210 \beta ) q^{55} + ( -27236 + 1732 \beta ) q^{56} + ( 8846 + 2270 \beta ) q^{58} + ( -9388 + 1476 \beta ) q^{59} + ( -2638 + 2330 \beta ) q^{61} + ( -26944 + 4624 \beta ) q^{62} + ( -14299 + 6295 \beta ) q^{64} + ( 20188 - 1932 \beta ) q^{65} + ( 14068 + 3200 \beta ) q^{67} + ( 22230 - 3110 \beta ) q^{68} + ( 103976 - 12872 \beta ) q^{70} + ( 15356 + 3098 \beta ) q^{71} + ( 26554 + 7536 \beta ) q^{73} + ( -38686 + 14618 \beta ) q^{74} + ( -12732 + 3468 \beta ) q^{76} + ( -17908 + 1210 \beta ) q^{77} + ( 5676 - 9482 \beta ) q^{79} + ( -5730 - 950 \beta ) q^{80} + ( -23726 - 1110 \beta ) q^{82} + ( 30444 - 2592 \beta ) q^{83} + ( -41020 + 5740 \beta ) q^{85} + ( 43084 + 4588 \beta ) q^{86} + ( 22627 + 121 \beta ) q^{88} + ( -38666 - 15056 \beta ) q^{89} + ( 31816 - 6264 \beta ) q^{91} + ( 119748 - 15196 \beta ) q^{92} + ( -126708 + 10240 \beta ) q^{94} + ( 19608 - 7032 \beta ) q^{95} + ( 14210 - 21300 \beta ) q^{97} + ( 154331 - 15217 \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 5q^{2} + 37q^{4} - 58q^{5} - 286q^{7} + 375q^{8} + O(q^{10}) \) \( 2q + 5q^{2} + 37q^{4} - 58q^{5} - 286q^{7} + 375q^{8} - 1030q^{10} + 242q^{11} - 166q^{13} - 1600q^{14} - 335q^{16} + 800q^{17} - 1476q^{19} - 5498q^{20} + 605q^{22} + 3370q^{23} + 4282q^{25} - 3778q^{26} - 9716q^{28} - 6600q^{29} - 7528q^{31} - 10625q^{32} + 7310q^{34} + 17144q^{35} - 29916q^{37} - 2628q^{38} - 9990q^{40} + 5780q^{41} - 16656q^{43} + 4477q^{44} + 40816q^{46} - 7850q^{47} + 16134q^{49} + 62035q^{50} - 19886q^{52} - 14178q^{53} - 7018q^{55} - 52740q^{56} + 19962q^{58} - 17300q^{59} - 2946q^{61} - 49264q^{62} - 22303q^{64} + 38444q^{65} + 31336q^{67} + 41350q^{68} + 195080q^{70} + 33810q^{71} + 60644q^{73} - 62754q^{74} - 21996q^{76} - 34606q^{77} + 1870q^{79} - 12410q^{80} - 48562q^{82} + 58296q^{83} - 76300q^{85} + 90756q^{86} + 45375q^{88} - 92388q^{89} + 57368q^{91} + 224300q^{92} - 243176q^{94} + 32184q^{95} + 7120q^{97} + 293445q^{98} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
7.15207
−6.15207
−4.15207 0 −14.7603 37.5207 0 −76.4793 194.152 0 −155.788
1.2 9.15207 0 51.7603 −95.5207 0 −209.521 180.848 0 −874.212
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(11\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 99.6.a.f 2
3.b odd 2 1 33.6.a.c 2
11.b odd 2 1 1089.6.a.j 2
12.b even 2 1 528.6.a.s 2
15.d odd 2 1 825.6.a.e 2
33.d even 2 1 363.6.a.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.6.a.c 2 3.b odd 2 1
99.6.a.f 2 1.a even 1 1 trivial
363.6.a.j 2 33.d even 2 1
528.6.a.s 2 12.b even 2 1
825.6.a.e 2 15.d odd 2 1
1089.6.a.j 2 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 5 T_{2} - 38 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(99))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -38 - 5 T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( -3584 + 58 T + T^{2} \)
$7$ \( 16024 + 286 T + T^{2} \)
$11$ \( ( -121 + T )^{2} \)
$13$ \( -57008 + 166 T + T^{2} \)
$17$ \( 700 - 800 T + T^{2} \)
$19$ \( 538272 + 1476 T + T^{2} \)
$23$ \( -3088328 - 3370 T + T^{2} \)
$29$ \( 3378828 + 6600 T + T^{2} \)
$31$ \( 8931328 + 7528 T + T^{2} \)
$37$ \( 222923316 + 29916 T + T^{2} \)
$41$ \( -14080172 - 5780 T + T^{2} \)
$43$ \( -29676624 + 16656 T + T^{2} \)
$47$ \( -266939288 + 7850 T + T^{2} \)
$53$ \( -165085872 + 14178 T + T^{2} \)
$59$ \( -21579488 + 17300 T + T^{2} \)
$61$ \( -238059096 + 2946 T + T^{2} \)
$67$ \( -207633776 - 31336 T + T^{2} \)
$71$ \( -138914952 - 33810 T + T^{2} \)
$73$ \( -1593591164 - 60644 T + T^{2} \)
$79$ \( -3977569112 - 1870 T + T^{2} \)
$83$ \( 552313872 - 58296 T + T^{2} \)
$89$ \( -7896843132 + 92388 T + T^{2} \)
$97$ \( -20063108900 - 7120 T + T^{2} \)
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