# Properties

 Label 99.6.a.e Level $99$ Weight $6$ Character orbit 99.a Self dual yes Analytic conductor $15.878$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{313})$$ Defining polynomial: $$x^{2} - x - 78$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{313})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta q^{2} + ( 46 + \beta ) q^{4} + ( 24 - 10 \beta ) q^{5} + ( -10 + 2 \beta ) q^{7} + ( -78 - 15 \beta ) q^{8} +O(q^{10})$$ $$q -\beta q^{2} + ( 46 + \beta ) q^{4} + ( 24 - 10 \beta ) q^{5} + ( -10 + 2 \beta ) q^{7} + ( -78 - 15 \beta ) q^{8} + ( 780 - 14 \beta ) q^{10} -121 q^{11} + ( 20 - 106 \beta ) q^{13} + ( -156 + 8 \beta ) q^{14} + ( -302 + 61 \beta ) q^{16} + ( 462 - 4 \beta ) q^{17} + ( -1468 + 4 \beta ) q^{19} + ( 324 - 446 \beta ) q^{20} + 121 \beta q^{22} + ( -2610 - 26 \beta ) q^{23} + ( 5251 - 380 \beta ) q^{25} + ( 8268 + 86 \beta ) q^{26} + ( -304 + 84 \beta ) q^{28} + ( 6234 + 132 \beta ) q^{29} + ( 4664 + 608 \beta ) q^{31} + ( -2262 + 721 \beta ) q^{32} + ( 312 - 458 \beta ) q^{34} + ( -1800 + 128 \beta ) q^{35} + ( 3158 - 320 \beta ) q^{37} + ( -312 + 1464 \beta ) q^{38} + ( 9828 + 570 \beta ) q^{40} + ( -12486 + 728 \beta ) q^{41} + ( 9560 + 1240 \beta ) q^{43} + ( -5566 - 121 \beta ) q^{44} + ( 2028 + 2636 \beta ) q^{46} + ( 2514 + 778 \beta ) q^{47} + ( -16395 - 36 \beta ) q^{49} + ( 29640 - 4871 \beta ) q^{50} + ( -7348 - 4962 \beta ) q^{52} + ( -20088 - 594 \beta ) q^{53} + ( -2904 + 1210 \beta ) q^{55} + ( -1560 - 36 \beta ) q^{56} + ( -10296 - 6366 \beta ) q^{58} + ( -10944 + 3676 \beta ) q^{59} + ( -7072 + 2746 \beta ) q^{61} + ( -47424 - 5272 \beta ) q^{62} + ( -46574 - 411 \beta ) q^{64} + ( 83160 - 1684 \beta ) q^{65} + ( 32300 + 768 \beta ) q^{67} + ( 20940 + 274 \beta ) q^{68} + ( -9984 + 1672 \beta ) q^{70} + ( -32274 + 3102 \beta ) q^{71} + ( 26546 + 320 \beta ) q^{73} + ( 24960 - 2838 \beta ) q^{74} + ( -67216 - 1280 \beta ) q^{76} + ( 1210 - 242 \beta ) q^{77} + ( 9626 - 2130 \beta ) q^{79} + ( -54828 + 3874 \beta ) q^{80} + ( -56784 + 11758 \beta ) q^{82} + ( 5388 + 3528 \beta ) q^{83} + ( 14208 - 4676 \beta ) q^{85} + ( -96720 - 10800 \beta ) q^{86} + ( 9438 + 1815 \beta ) q^{88} + ( 30582 - 3024 \beta ) q^{89} + ( -16736 + 888 \beta ) q^{91} + ( -122088 - 3832 \beta ) q^{92} + ( -60684 - 3292 \beta ) q^{94} + ( -38352 + 14736 \beta ) q^{95} + ( -92074 + 1092 \beta ) q^{97} + ( 2808 + 16431 \beta ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} + 93q^{4} + 38q^{5} - 18q^{7} - 171q^{8} + O(q^{10})$$ $$2q - q^{2} + 93q^{4} + 38q^{5} - 18q^{7} - 171q^{8} + 1546q^{10} - 242q^{11} - 66q^{13} - 304q^{14} - 543q^{16} + 920q^{17} - 2932q^{19} + 202q^{20} + 121q^{22} - 5246q^{23} + 10122q^{25} + 16622q^{26} - 524q^{28} + 12600q^{29} + 9936q^{31} - 3803q^{32} + 166q^{34} - 3472q^{35} + 5996q^{37} + 840q^{38} + 20226q^{40} - 24244q^{41} + 20360q^{43} - 11253q^{44} + 6692q^{46} + 5806q^{47} - 32826q^{49} + 54409q^{50} - 19658q^{52} - 40770q^{53} - 4598q^{55} - 3156q^{56} - 26958q^{58} - 18212q^{59} - 11398q^{61} - 100120q^{62} - 93559q^{64} + 164636q^{65} + 65368q^{67} + 42154q^{68} - 18296q^{70} - 61446q^{71} + 53412q^{73} + 47082q^{74} - 135712q^{76} + 2178q^{77} + 17122q^{79} - 105782q^{80} - 101810q^{82} + 14304q^{83} + 23740q^{85} - 204240q^{86} + 20691q^{88} + 58140q^{89} - 32584q^{91} - 248008q^{92} - 124660q^{94} - 61968q^{95} - 183056q^{97} + 22047q^{98} + O(q^{100})$$

## 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
 9.34590 −8.34590
−9.34590 0 55.3459 −69.4590 0 8.69181 −218.189 0 649.157
1.2 8.34590 0 37.6541 107.459 0 −26.6918 47.1885 0 896.843
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.e 2
3.b odd 2 1 33.6.a.d 2
11.b odd 2 1 1089.6.a.o 2
12.b even 2 1 528.6.a.q 2
15.d odd 2 1 825.6.a.d 2
33.d even 2 1 363.6.a.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.6.a.d 2 3.b odd 2 1
99.6.a.e 2 1.a even 1 1 trivial
363.6.a.g 2 33.d even 2 1
528.6.a.q 2 12.b even 2 1
825.6.a.d 2 15.d odd 2 1
1089.6.a.o 2 11.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + T_{2} - 78$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(99))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-78 + T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$-7464 - 38 T + T^{2}$$
$7$ $$-232 + 18 T + T^{2}$$
$11$ $$( 121 + T )^{2}$$
$13$ $$-878128 + 66 T + T^{2}$$
$17$ $$210348 - 920 T + T^{2}$$
$19$ $$2147904 + 2932 T + T^{2}$$
$23$ $$6827232 + 5246 T + T^{2}$$
$29$ $$38326572 - 12600 T + T^{2}$$
$31$ $$-4245184 - 9936 T + T^{2}$$
$37$ $$975204 - 5996 T + T^{2}$$
$41$ $$105471636 + 24244 T + T^{2}$$
$43$ $$-16684800 - 20360 T + T^{2}$$
$47$ $$-38936064 - 5806 T + T^{2}$$
$53$ $$387938808 + 40770 T + T^{2}$$
$59$ $$-974471136 + 18212 T + T^{2}$$
$61$ $$-557566776 + 11398 T + T^{2}$$
$67$ $$1022090128 - 65368 T + T^{2}$$
$71$ $$190949616 + 61446 T + T^{2}$$
$73$ $$705197636 - 53412 T + T^{2}$$
$79$ $$-281721704 - 17122 T + T^{2}$$
$83$ $$-922809744 - 14304 T + T^{2}$$
$89$ $$129501828 - 58140 T + T^{2}$$
$97$ $$8284064476 + 183056 T + T^{2}$$