# Properties

 Label 99.2.f.c Level 99 Weight 2 Character orbit 99.f Analytic conductor 0.791 Analytic rank 0 Dimension 8 CM no Inner twists 4

# Related objects

## 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: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{5})$$ Coefficient field: 8.0.484000000.9 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( 1 + 2 \beta_{2} + \beta_{3} ) q^{4} + ( \beta_{4} + \beta_{7} ) q^{5} + ( -2 - 3 \beta_{2} - 2 \beta_{3} ) q^{7} + ( \beta_{1} + \beta_{4} ) q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( 1 + 2 \beta_{2} + \beta_{3} ) q^{4} + ( \beta_{4} + \beta_{7} ) q^{5} + ( -2 - 3 \beta_{2} - 2 \beta_{3} ) q^{7} + ( \beta_{1} + \beta_{4} ) q^{8} + ( -4 - \beta_{2} + \beta_{5} ) q^{10} + ( -\beta_{1} - 2 \beta_{4} + \beta_{6} ) q^{11} + ( 1 + \beta_{2} + 3 \beta_{3} - 3 \beta_{5} ) q^{13} + ( \beta_{1} - 2 \beta_{4} - 3 \beta_{6} - 3 \beta_{7} ) q^{14} + ( -1 - 4 \beta_{3} + \beta_{5} ) q^{16} + ( -3 \beta_{1} + \beta_{4} + 3 \beta_{6} + \beta_{7} ) q^{17} + ( -\beta_{2} - \beta_{3} + 4 \beta_{5} ) q^{19} + ( -\beta_{1} + \beta_{4} - \beta_{6} ) q^{20} + ( 7 + 5 \beta_{2} + 6 \beta_{3} - 3 \beta_{5} ) q^{22} + ( -3 \beta_{1} + 3 \beta_{7} ) q^{23} + ( -\beta_{2} - \beta_{3} + 2 \beta_{5} ) q^{25} + ( \beta_{6} - 2 \beta_{7} ) q^{26} + ( 5 - 3 \beta_{3} - 5 \beta_{5} ) q^{28} + ( -2 \beta_{6} - 2 \beta_{7} ) q^{29} + ( 4 + 4 \beta_{2} + 3 \beta_{3} - 3 \beta_{5} ) q^{31} + ( \beta_{1} - 3 \beta_{4} - \beta_{7} ) q^{32} + ( -7 - 10 \beta_{2} + 10 \beta_{5} ) q^{34} + ( \beta_{1} - 2 \beta_{4} + 2 \beta_{6} ) q^{35} + ( -2 + 3 \beta_{2} - 2 \beta_{3} ) q^{37} + ( \beta_{1} + 3 \beta_{4} - \beta_{6} + 3 \beta_{7} ) q^{38} + ( -3 - \beta_{2} - 3 \beta_{3} ) q^{40} + ( 3 \beta_{1} + \beta_{4} - 2 \beta_{6} - 2 \beta_{7} ) q^{41} + ( -1 + 6 \beta_{2} - 6 \beta_{5} ) q^{43} + ( \beta_{4} + 3 \beta_{6} + 2 \beta_{7} ) q^{44} + ( -3 - 3 \beta_{2} + 6 \beta_{3} - 6 \beta_{5} ) q^{46} + ( \beta_{4} + \beta_{6} + \beta_{7} ) q^{47} + ( -8 - 2 \beta_{3} + 8 \beta_{5} ) q^{49} + ( \beta_{1} + \beta_{4} - \beta_{6} + \beta_{7} ) q^{50} + ( -\beta_{2} - \beta_{3} + \beta_{5} ) q^{52} + ( 4 \beta_{1} + 3 \beta_{4} - 3 \beta_{6} ) q^{53} + ( -2 - 3 \beta_{2} + 3 \beta_{3} + 4 \beta_{5} ) q^{55} + ( \beta_{1} - 2 \beta_{4} - \beta_{7} ) q^{56} + ( -8 \beta_{2} - 8 \beta_{3} - 2 \beta_{5} ) q^{58} + ( \beta_{6} + 4 \beta_{7} ) q^{59} + ( 7 - 7 \beta_{5} ) q^{61} + ( 4 \beta_{6} + \beta_{7} ) q^{62} + ( 7 + 7 \beta_{2} - \beta_{3} + \beta_{5} ) q^{64} + ( 2 \beta_{1} + \beta_{4} - 2 \beta_{7} ) q^{65} + ( -1 - 3 \beta_{2} + 3 \beta_{5} ) q^{67} + ( -\beta_{1} + 4 \beta_{4} - 4 \beta_{6} ) q^{68} + ( 9 + 14 \beta_{2} + 9 \beta_{3} ) q^{70} + ( -5 \beta_{4} - 5 \beta_{7} ) q^{71} + ( 4 + 4 \beta_{3} ) q^{73} + ( -5 \beta_{1} - 2 \beta_{4} + 3 \beta_{6} + 3 \beta_{7} ) q^{74} + ( -5 - 2 \beta_{2} + 2 \beta_{5} ) q^{76} + ( \beta_{1} - \beta_{4} - 5 \beta_{6} - 4 \beta_{7} ) q^{77} + ( 7 + 7 \beta_{2} + 3 \beta_{3} - 3 \beta_{5} ) q^{79} + ( -4 \beta_{1} - \beta_{4} + 3 \beta_{6} + 3 \beta_{7} ) q^{80} + ( -1 - 8 \beta_{3} + \beta_{5} ) q^{82} + ( 3 \beta_{1} - 2 \beta_{4} - 3 \beta_{6} - 2 \beta_{7} ) q^{83} + ( -10 \beta_{2} - 10 \beta_{3} + 3 \beta_{5} ) q^{85} + ( -7 \beta_{1} - 6 \beta_{4} + 6 \beta_{6} ) q^{86} + ( 2 + 3 \beta_{2} + 8 \beta_{3} - 4 \beta_{5} ) q^{88} + ( 3 \beta_{1} + 6 \beta_{4} - 3 \beta_{7} ) q^{89} + \beta_{5} q^{91} + ( -3 \beta_{6} - 3 \beta_{7} ) q^{92} + ( -4 + \beta_{3} + 4 \beta_{5} ) q^{94} + ( -3 \beta_{6} + \beta_{7} ) q^{95} + ( -5 - 5 \beta_{2} + 6 \beta_{3} - 6 \beta_{5} ) q^{97} + ( -8 \beta_{1} + 6 \beta_{4} + 8 \beta_{7} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 2q^{4} - 6q^{7} + O(q^{10})$$ $$8q + 2q^{4} - 6q^{7} - 28q^{10} - 6q^{13} + 2q^{16} + 12q^{19} + 28q^{22} + 8q^{25} + 36q^{28} + 12q^{31} - 16q^{34} - 18q^{37} - 16q^{40} - 32q^{43} - 42q^{46} - 44q^{49} + 6q^{52} - 8q^{55} + 28q^{58} + 42q^{61} + 46q^{64} + 4q^{67} + 26q^{70} + 24q^{73} - 32q^{76} + 30q^{79} + 10q^{82} + 46q^{85} - 14q^{88} + 2q^{91} - 26q^{94} - 54q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + x^{6} + 16 x^{4} + 66 x^{2} + 121$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$7 \nu^{6} - 37 \nu^{4} + 629 \nu^{2} - 363$$$$)/1991$$ $$\beta_{3}$$ $$=$$ $$($$$$-28 \nu^{6} + 148 \nu^{4} - 525 \nu^{2} - 539$$$$)/1991$$ $$\beta_{4}$$ $$=$$ $$($$$$-28 \nu^{7} + 148 \nu^{5} - 525 \nu^{3} - 539 \nu$$$$)/1991$$ $$\beta_{5}$$ $$=$$ $$($$$$40 \nu^{6} + 73 \nu^{4} + 750 \nu^{2} + 2761$$$$)/1991$$ $$\beta_{6}$$ $$=$$ $$($$$$-61 \nu^{7} + 38 \nu^{5} - 646 \nu^{3} - 1672 \nu$$$$)/1991$$ $$\beta_{7}$$ $$=$$ $$($$$$68 \nu^{7} - 75 \nu^{5} + 1275 \nu^{3} + 3300 \nu$$$$)/1991$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + 4 \beta_{2} + 1$$ $$\nu^{3}$$ $$=$$ $$4 \beta_{7} + 4 \beta_{6} + \beta_{4} - 3 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$7 \beta_{5} + 10 \beta_{3} - 7$$ $$\nu^{5}$$ $$=$$ $$7 \beta_{7} + 17 \beta_{4} - 7 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$37 \beta_{5} - 37 \beta_{3} - 75 \beta_{2} - 75$$ $$\nu^{7}$$ $$=$$ $$-38 \beta_{7} - 75 \beta_{6}$$

## Character values

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

 $$n$$ $$46$$ $$56$$ $$\chi(n)$$ $$\beta_{2}$$ $$1$$

## 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
 −1.73855 − 1.26313i 1.73855 + 1.26313i −0.476925 + 1.46782i 0.476925 − 1.46782i −0.476925 − 1.46782i 0.476925 + 1.46782i −1.73855 + 1.26313i 1.73855 − 1.26313i
−1.73855 1.26313i 0 0.809017 + 2.48990i 1.73855 1.26313i 0 −1.30902 4.02874i 0.410415 1.26313i 0 −4.61803
37.2 1.73855 + 1.26313i 0 0.809017 + 2.48990i −1.73855 + 1.26313i 0 −1.30902 4.02874i −0.410415 + 1.26313i 0 −4.61803
64.1 −0.476925 + 1.46782i 0 −0.309017 0.224514i 0.476925 + 1.46782i 0 −0.190983 0.138757i −2.02029 + 1.46782i 0 −2.38197
64.2 0.476925 1.46782i 0 −0.309017 0.224514i −0.476925 1.46782i 0 −0.190983 0.138757i 2.02029 1.46782i 0 −2.38197
82.1 −0.476925 1.46782i 0 −0.309017 + 0.224514i 0.476925 1.46782i 0 −0.190983 + 0.138757i −2.02029 1.46782i 0 −2.38197
82.2 0.476925 + 1.46782i 0 −0.309017 + 0.224514i −0.476925 + 1.46782i 0 −0.190983 + 0.138757i 2.02029 + 1.46782i 0 −2.38197
91.1 −1.73855 + 1.26313i 0 0.809017 2.48990i 1.73855 + 1.26313i 0 −1.30902 + 4.02874i 0.410415 + 1.26313i 0 −4.61803
91.2 1.73855 1.26313i 0 0.809017 2.48990i −1.73855 1.26313i 0 −1.30902 + 4.02874i −0.410415 1.26313i 0 −4.61803
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 91.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.c even 5 1 inner
33.h odd 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 99.2.f.c 8
3.b odd 2 1 inner 99.2.f.c 8
9.c even 3 2 891.2.n.e 16
9.d odd 6 2 891.2.n.e 16
11.c even 5 1 inner 99.2.f.c 8
11.c even 5 1 1089.2.a.v 4
11.d odd 10 1 1089.2.a.w 4
33.f even 10 1 1089.2.a.w 4
33.h odd 10 1 inner 99.2.f.c 8
33.h odd 10 1 1089.2.a.v 4
99.m even 15 2 891.2.n.e 16
99.n odd 30 2 891.2.n.e 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.2.f.c 8 1.a even 1 1 trivial
99.2.f.c 8 3.b odd 2 1 inner
99.2.f.c 8 11.c even 5 1 inner
99.2.f.c 8 33.h odd 10 1 inner
891.2.n.e 16 9.c even 3 2
891.2.n.e 16 9.d odd 6 2
891.2.n.e 16 99.m even 15 2
891.2.n.e 16 99.n odd 30 2
1089.2.a.v 4 11.c even 5 1
1089.2.a.v 4 33.h odd 10 1
1089.2.a.w 4 11.d odd 10 1
1089.2.a.w 4 33.f even 10 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} + T_{2}^{6} + 16 T_{2}^{4} + 66 T_{2}^{2} + 121$$ acting on $$S_{2}^{\mathrm{new}}(99, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 3 T^{2} + 2 T^{6} + 9 T^{8} + 8 T^{10} - 192 T^{14} + 256 T^{16}$$
$3$ 1
$5$ $$1 - 9 T^{2} + 21 T^{4} + 161 T^{6} - 1644 T^{8} + 4025 T^{10} + 13125 T^{12} - 140625 T^{14} + 390625 T^{16}$$
$7$ $$( 1 + 3 T + 12 T^{2} + 35 T^{3} + 141 T^{4} + 245 T^{5} + 588 T^{6} + 1029 T^{7} + 2401 T^{8} )^{2}$$
$11$ $$1 - 19 T^{2} + 231 T^{4} - 2299 T^{6} + 14641 T^{8}$$
$13$ $$( 1 + 3 T + 6 T^{2} + 59 T^{3} + 339 T^{4} + 767 T^{5} + 1014 T^{6} + 6591 T^{7} + 28561 T^{8} )^{2}$$
$17$ $$1 - 279 T^{4} - 1420 T^{6} + 82101 T^{8} - 410380 T^{10} - 23302359 T^{12} + 6975757441 T^{16}$$
$19$ $$( 1 - 6 T - 3 T^{2} + 122 T^{3} - 525 T^{4} + 2318 T^{5} - 1083 T^{6} - 41154 T^{7} + 130321 T^{8} )^{2}$$
$23$ $$( 1 + 20 T^{2} + 753 T^{4} + 10580 T^{6} + 279841 T^{8} )^{2}$$
$29$ $$1 + 18 T^{2} + 1683 T^{4} + 10856 T^{6} + 1510005 T^{8} + 9129896 T^{10} + 1190353923 T^{12} + 10706819778 T^{14} + 500246412961 T^{16}$$
$31$ $$( 1 - 6 T - 15 T^{2} + 206 T^{3} - 561 T^{4} + 6386 T^{5} - 14415 T^{6} - 178746 T^{7} + 923521 T^{8} )^{2}$$
$37$ $$( 1 + 9 T - 6 T^{2} - 307 T^{3} - 1581 T^{4} - 11359 T^{5} - 8214 T^{6} + 455877 T^{7} + 1874161 T^{8} )^{2}$$
$41$ $$1 - 87 T^{2} + 2538 T^{4} + 70841 T^{6} - 7729545 T^{8} + 119083721 T^{10} + 7171781418 T^{12} - 413259068967 T^{14} + 7984925229121 T^{16}$$
$43$ $$( 1 + 8 T + 57 T^{2} + 344 T^{3} + 1849 T^{4} )^{4}$$
$47$ $$1 - 75 T^{2} + 5541 T^{4} - 375925 T^{6} + 17183556 T^{8} - 830418325 T^{10} + 27038312421 T^{12} - 808441149675 T^{14} + 23811286661761 T^{16}$$
$53$ $$1 + 24 T^{2} + 5817 T^{4} - 20908 T^{6} + 15682005 T^{8} - 58730572 T^{10} + 45898927977 T^{12} + 531944667096 T^{14} + 62259690411361 T^{16}$$
$59$ $$1 - 72 T^{2} - 327 T^{4} + 271796 T^{6} - 16865475 T^{8} + 946121876 T^{10} - 3962377047 T^{12} - 3036998422152 T^{14} + 146830437604321 T^{16}$$
$61$ $$( 1 - 21 T + 135 T^{2} - 259 T^{3} + 144 T^{4} - 15799 T^{5} + 502335 T^{6} - 4766601 T^{7} + 13845841 T^{8} )^{2}$$
$67$ $$( 1 - T + 123 T^{2} - 67 T^{3} + 4489 T^{4} )^{4}$$
$71$ $$1 - 117 T^{2} + 273 T^{4} + 566981 T^{6} - 45119220 T^{8} + 2858151221 T^{10} + 6937388913 T^{12} - 14987733218757 T^{14} + 645753531245761 T^{16}$$
$73$ $$( 1 - 12 T - 9 T^{2} + 164 T^{3} + 3249 T^{4} + 11972 T^{5} - 47961 T^{6} - 4668204 T^{7} + 28398241 T^{8} )^{2}$$
$79$ $$( 1 - 15 T + 6 T^{2} + 815 T^{3} - 5979 T^{4} + 64385 T^{5} + 37446 T^{6} - 7395585 T^{7} + 38950081 T^{8} )^{2}$$
$83$ $$1 - 177 T^{2} + 9810 T^{4} + 724223 T^{6} - 144542001 T^{8} + 4989172247 T^{10} + 465566129010 T^{12} - 57868446086313 T^{14} + 2252292232139041 T^{16}$$
$89$ $$( 1 - 4 T^{2} + 5721 T^{4} - 31684 T^{6} + 62742241 T^{8} )^{2}$$
$97$ $$( 1 + 27 T + 357 T^{2} + 4325 T^{3} + 49476 T^{4} + 419525 T^{5} + 3359013 T^{6} + 24642171 T^{7} + 88529281 T^{8} )^{2}$$