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: \(x^{2} - x - 8\)
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.

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

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
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 \( T_{2}^{2} + 13 T_{2} + 34 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(99))\).

Hecke characteristic polynomials

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