Properties

Label 99.6.a.d
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: \(\Q(\sqrt{33}) \)
Defining polynomial: \( x^{2} - x - 8 \) Copy content Toggle raw display
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 \(\beta = \frac{1}{2}(1 + \sqrt{33})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 6) q^{2} + (13 \beta + 12) q^{4} + (10 \beta - 34) q^{5} + ( - 62 \beta + 104) q^{7} + ( - 71 \beta + 16) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 6) q^{2} + (13 \beta + 12) q^{4} + (10 \beta - 34) q^{5} + ( - 62 \beta + 104) q^{7} + ( - 71 \beta + 16) q^{8} + ( - 36 \beta + 124) q^{10} + 121 q^{11} + (74 \beta - 102) q^{13} + (330 \beta - 128) q^{14} + (65 \beta + 88) q^{16} + (372 \beta + 178) q^{17} + (852 \beta - 840) q^{19} + ( - 192 \beta + 632) q^{20} + ( - 121 \beta - 726) q^{22} + ( - 330 \beta + 284) q^{23} + ( - 580 \beta - 1169) q^{25} + ( - 416 \beta + 20) q^{26} + ( - 198 \beta - 5200) q^{28} + ( - 1492 \beta + 398) q^{29} + ( - 1600 \beta - 4440) q^{31} + (1729 \beta - 1560) q^{32} + ( - 2782 \beta - 4044) q^{34} + (2528 \beta - 8496) q^{35} + (2816 \beta - 2362) q^{37} + ( - 5124 \beta - 1776) q^{38} + (1864 \beta - 6224) q^{40} + ( - 8 \beta - 18238) q^{41} + (3112 \beta + 3328) q^{43} + (1573 \beta + 1452) q^{44} + (2026 \beta + 936) q^{46} + ( - 390 \beta - 21676) q^{47} + ( - 9052 \beta + 24761) q^{49} + (5229 \beta + 11654) q^{50} + (524 \beta + 6472) q^{52} + ( - 7102 \beta + 9638) q^{53} + (1210 \beta - 4114) q^{55} + ( - 3974 \beta + 36880) q^{56} + (10046 \beta + 9548) q^{58} + (1980 \beta + 404) q^{59} + ( - 2026 \beta - 11638) q^{61} + (15640 \beta + 39440) q^{62} + ( - 12623 \beta - 7288) q^{64} + ( - 2796 \beta + 9388) q^{65} + (12704 \beta - 26612) q^{67} + (11614 \beta + 40824) q^{68} + ( - 9200 \beta + 30752) q^{70} + ( - 4354 \beta - 13516) q^{71} + ( - 5568 \beta - 20606) q^{73} + ( - 17350 \beta - 8356) q^{74} + (10380 \beta + 78528) q^{76} + ( - 7502 \beta + 12584) q^{77} + ( - 11426 \beta - 2712) q^{79} + ( - 680 \beta + 2208) q^{80} + (18294 \beta + 109492) q^{82} + (21960 \beta - 50700) q^{83} + ( - 7148 \beta + 23708) q^{85} + ( - 25112 \beta - 44864) q^{86} + ( - 8591 \beta + 1936) q^{88} + (26704 \beta + 13750) q^{89} + (9432 \beta - 47312) q^{91} + ( - 4558 \beta - 30912) q^{92} + (24406 \beta + 133176) q^{94} + ( - 28848 \beta + 96720) q^{95} + ( - 9924 \beta - 115822) q^{97} + (38603 \beta - 76150) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 13 q^{2} + 37 q^{4} - 58 q^{5} + 146 q^{7} - 39 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 13 q^{2} + 37 q^{4} - 58 q^{5} + 146 q^{7} - 39 q^{8} + 212 q^{10} + 242 q^{11} - 130 q^{13} + 74 q^{14} + 241 q^{16} + 728 q^{17} - 828 q^{19} + 1072 q^{20} - 1573 q^{22} + 238 q^{23} - 2918 q^{25} - 376 q^{26} - 10598 q^{28} - 696 q^{29} - 10480 q^{31} - 1391 q^{32} - 10870 q^{34} - 14464 q^{35} - 1908 q^{37} - 8676 q^{38} - 10584 q^{40} - 36484 q^{41} + 9768 q^{43} + 4477 q^{44} + 3898 q^{46} - 43742 q^{47} + 40470 q^{49} + 28537 q^{50} + 13468 q^{52} + 12174 q^{53} - 7018 q^{55} + 69786 q^{56} + 29142 q^{58} + 2788 q^{59} - 25302 q^{61} + 94520 q^{62} - 27199 q^{64} + 15980 q^{65} - 40520 q^{67} + 93262 q^{68} + 52304 q^{70} - 31386 q^{71} - 46780 q^{73} - 34062 q^{74} + 167436 q^{76} + 17666 q^{77} - 16850 q^{79} + 3736 q^{80} + 237278 q^{82} - 79440 q^{83} + 40268 q^{85} - 114840 q^{86} - 4719 q^{88} + 54204 q^{89} - 85192 q^{91} - 66382 q^{92} + 290758 q^{94} + 164592 q^{95} - 241568 q^{97} - 113697 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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
3.37228
−2.37228
−9.37228 0 55.8397 −0.277187 0 −105.081 −223.432 0 2.59787
1.2 −3.62772 0 −18.8397 −57.7228 0 251.081 184.432 0 209.402
\(n\): 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.d 2
3.b odd 2 1 33.6.a.e 2
11.b odd 2 1 1089.6.a.p 2
12.b even 2 1 528.6.a.o 2
15.d odd 2 1 825.6.a.c 2
33.d even 2 1 363.6.a.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.6.a.e 2 3.b odd 2 1
99.6.a.d 2 1.a even 1 1 trivial
363.6.a.f 2 33.d even 2 1
528.6.a.o 2 12.b even 2 1
825.6.a.c 2 15.d odd 2 1
1089.6.a.p 2 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 13T_{2} + 34 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(99))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 13T + 34 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 58T + 16 \) Copy content Toggle raw display
$7$ \( T^{2} - 146T - 26384 \) Copy content Toggle raw display
$11$ \( (T - 121)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 130T - 40952 \) Copy content Toggle raw display
$17$ \( T^{2} - 728 T - 1009172 \) Copy content Toggle raw display
$19$ \( T^{2} + 828 T - 5817312 \) Copy content Toggle raw display
$23$ \( T^{2} - 238T - 884264 \) Copy content Toggle raw display
$29$ \( T^{2} + 696 T - 18243924 \) Copy content Toggle raw display
$31$ \( T^{2} + 10480 T + 6337600 \) Copy content Toggle raw display
$37$ \( T^{2} + 1908 T - 64511196 \) Copy content Toggle raw display
$41$ \( T^{2} + 36484 T + 332770036 \) Copy content Toggle raw display
$43$ \( T^{2} - 9768 T - 56044032 \) Copy content Toggle raw display
$47$ \( T^{2} + 43742 T + 477085816 \) Copy content Toggle raw display
$53$ \( T^{2} - 12174 T - 379065264 \) Copy content Toggle raw display
$59$ \( T^{2} - 2788 T - 30400064 \) Copy content Toggle raw display
$61$ \( T^{2} + 25302 T + 126184224 \) Copy content Toggle raw display
$67$ \( T^{2} + 40520 T - 921013232 \) Copy content Toggle raw display
$71$ \( T^{2} + 31386 T + 89872392 \) Copy content Toggle raw display
$73$ \( T^{2} + 46780 T + 291320452 \) Copy content Toggle raw display
$79$ \( T^{2} + 16850 T - 1006085552 \) Copy content Toggle raw display
$83$ \( T^{2} + 79440 T - 2400814800 \) Copy content Toggle raw display
$89$ \( T^{2} - 54204 T - 5148586428 \) Copy content Toggle raw display
$97$ \( T^{2} + 241568 T + 13776267004 \) Copy content Toggle raw display
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