# Properties

 Label 99.4.f.a Level $99$ Weight $4$ Character orbit 99.f Analytic conductor $5.841$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$99 = 3^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 99.f (of order $$5$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.84118909057$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ x^4 - x^3 + x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 33) Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{10}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} - 2) q^{2} - 12 \zeta_{10}^{3} q^{4} + (11 \zeta_{10}^{2} - 12 \zeta_{10} + 11) q^{5} + (19 \zeta_{10}^{3} - 6 \zeta_{10} + 6) q^{7} + (16 \zeta_{10}^{3} - 8 \zeta_{10}^{2} + 16 \zeta_{10}) q^{8} +O(q^{10})$$ q + (2*z^3 + 2*z^2 - 2*z - 2) * q^2 - 12*z^3 * q^4 + (11*z^2 - 12*z + 11) * q^5 + (19*z^3 - 6*z + 6) * q^7 + (16*z^3 - 8*z^2 + 16*z) * q^8 $$q + (2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} - 2) q^{2} - 12 \zeta_{10}^{3} q^{4} + (11 \zeta_{10}^{2} - 12 \zeta_{10} + 11) q^{5} + (19 \zeta_{10}^{3} - 6 \zeta_{10} + 6) q^{7} + (16 \zeta_{10}^{3} - 8 \zeta_{10}^{2} + 16 \zeta_{10}) q^{8} + ( - 26 \zeta_{10}^{3} + 26 \zeta_{10}^{2} - 42) q^{10} + (10 \zeta_{10}^{3} + 6 \zeta_{10}^{2} + 35 \zeta_{10} - 20) q^{11} + (33 \zeta_{10}^{3} - 11 \zeta_{10}^{2} + 11 \zeta_{10} - 33) q^{13} + ( - 88 \zeta_{10}^{3} + 74 \zeta_{10}^{2} - 88 \zeta_{10}) q^{14} + 16 \zeta_{10} q^{16} + ( - 57 \zeta_{10}^{2} - 21 \zeta_{10} - 57) q^{17} + ( - 45 \zeta_{10}^{3} - 38 \zeta_{10}^{2} - 45 \zeta_{10}) q^{19} + (12 \zeta_{10}^{3} - 144 \zeta_{10}^{2} + 144 \zeta_{10} - 12) q^{20} + (60 \zeta_{10}^{3} - 184 \zeta_{10}^{2} + 12 \zeta_{10} - 54) q^{22} + (44 \zeta_{10}^{3} - 44 \zeta_{10}^{2} + 15) q^{23} + ( - 143 \zeta_{10}^{3} + 140 \zeta_{10}^{2} - 143 \zeta_{10}) q^{25} + ( - 154 \zeta_{10}^{3} - 88 \zeta_{10} + 88) q^{26} + ( - 72 \zeta_{10}^{2} + 300 \zeta_{10} - 72) q^{28} + (162 \zeta_{10}^{3} + 192 \zeta_{10} - 192) q^{29} + (44 \zeta_{10}^{3} + 33 \zeta_{10}^{2} - 33 \zeta_{10} - 44) q^{31} + (192 \zeta_{10}^{3} - 192 \zeta_{10}^{2} - 96) q^{32} + ( - 198 \zeta_{10}^{3} + 198 \zeta_{10}^{2} + 384) q^{34} + ( - 85 \zeta_{10}^{3} + 366 \zeta_{10}^{2} - 366 \zeta_{10} + 85) q^{35} + (267 \zeta_{10}^{3} + 22 \zeta_{10} - 22) q^{37} + (242 \zeta_{10}^{2} + 104 \zeta_{10} + 242) q^{38} + (168 \zeta_{10}^{3} - 104 \zeta_{10} + 104) q^{40} + (34 \zeta_{10}^{3} - 61 \zeta_{10}^{2} + 34 \zeta_{10}) q^{41} + (84 \zeta_{10}^{3} - 84 \zeta_{10}^{2} - 75) q^{43} + ( - 180 \zeta_{10}^{3} + 420 \zeta_{10}^{2} - 300 \zeta_{10} + 492) q^{44} + ( - 146 \zeta_{10}^{3} + 294 \zeta_{10}^{2} - 294 \zeta_{10} + 146) q^{46} + ( - 55 \zeta_{10}^{3} - 145 \zeta_{10}^{2} - 55 \zeta_{10}) q^{47} + (264 \zeta_{10}^{2} - 318 \zeta_{10} + 264) q^{49} + ( - 274 \zeta_{10}^{2} + 852 \zeta_{10} - 274) q^{50} + (264 \zeta_{10}^{3} + 132 \zeta_{10}^{2} + 264 \zeta_{10}) q^{52} + ( - 98 \zeta_{10}^{3} + 241 \zeta_{10}^{2} - 241 \zeta_{10} + 98) q^{53} + (369 \zeta_{10}^{3} - 520 \zeta_{10}^{2} + 571 \zeta_{10} - 276) q^{55} + (352 \zeta_{10}^{3} - 352 \zeta_{10}^{2} - 56) q^{56} + ( - 264 \zeta_{10}^{3} - 828 \zeta_{10}^{2} - 264 \zeta_{10}) q^{58} + ( - 33 \zeta_{10}^{3} - 451 \zeta_{10} + 451) q^{59} + (279 \zeta_{10}^{2} + 438 \zeta_{10} + 279) q^{61} + ( - 396 \zeta_{10}^{3} - 22 \zeta_{10} + 22) q^{62} + ( - 832 \zeta_{10}^{3} + 832 \zeta_{10}^{2} - 832 \zeta_{10} + 832) q^{64} + (99 \zeta_{10}^{3} - 99 \zeta_{10}^{2} - 209) q^{65} + (561 \zeta_{10}^{3} - 561 \zeta_{10}^{2} - 243) q^{67} + (936 \zeta_{10}^{3} - 252 \zeta_{10}^{2} + 252 \zeta_{10} - 936) q^{68} + ( - 954 \zeta_{10}^{3} + 902 \zeta_{10} - 902) q^{70} + ( - 275 \zeta_{10}^{2} + 708 \zeta_{10} - 275) q^{71} + ( - 28 \zeta_{10}^{3} + 318 \zeta_{10} - 318) q^{73} + ( - 1024 \zeta_{10}^{3} + 402 \zeta_{10}^{2} - 1024 \zeta_{10}) q^{74} + (540 \zeta_{10}^{3} - 540 \zeta_{10}^{2} - 996) q^{76} + (249 \zeta_{10}^{3} - 779 \zeta_{10}^{2} + 745 \zeta_{10} - 839) q^{77} + (65 \zeta_{10}^{3} + 637 \zeta_{10}^{2} - 637 \zeta_{10} - 65) q^{79} + (176 \zeta_{10}^{3} - 192 \zeta_{10}^{2} + 176 \zeta_{10}) q^{80} + (176 \zeta_{10}^{2} - 258 \zeta_{10} + 176) q^{82} + (466 \zeta_{10}^{2} - 431 \zeta_{10} + 466) q^{83} + ( - 174 \zeta_{10}^{3} - 375 \zeta_{10}^{2} - 174 \zeta_{10}) q^{85} + ( - 486 \zeta_{10}^{3} + 354 \zeta_{10}^{2} - 354 \zeta_{10} + 486) q^{86} + (168 \zeta_{10}^{3} + 48 \zeta_{10}^{2} + 192 \zeta_{10} - 688) q^{88} + (154 \zeta_{10}^{3} - 154 \zeta_{10}^{2} + 1279) q^{89} + ( - 352 \zeta_{10}^{3} - 143 \zeta_{10}^{2} - 352 \zeta_{10}) q^{91} + ( - 180 \zeta_{10}^{3} + 528 \zeta_{10} - 528) q^{92} + (690 \zeta_{10}^{2} - 70 \zeta_{10} + 690) q^{94} + ( - 412 \zeta_{10}^{3} - 373 \zeta_{10} + 373) q^{95} + ( - 609 \zeta_{10}^{3} - 282 \zeta_{10}^{2} + 282 \zeta_{10} + 609) q^{97} + ( - 744 \zeta_{10}^{3} + 744 \zeta_{10}^{2} - 948) q^{98} +O(q^{100})$$ q + (2*z^3 + 2*z^2 - 2*z - 2) * q^2 - 12*z^3 * q^4 + (11*z^2 - 12*z + 11) * q^5 + (19*z^3 - 6*z + 6) * q^7 + (16*z^3 - 8*z^2 + 16*z) * q^8 + (-26*z^3 + 26*z^2 - 42) * q^10 + (10*z^3 + 6*z^2 + 35*z - 20) * q^11 + (33*z^3 - 11*z^2 + 11*z - 33) * q^13 + (-88*z^3 + 74*z^2 - 88*z) * q^14 + 16*z * q^16 + (-57*z^2 - 21*z - 57) * q^17 + (-45*z^3 - 38*z^2 - 45*z) * q^19 + (12*z^3 - 144*z^2 + 144*z - 12) * q^20 + (60*z^3 - 184*z^2 + 12*z - 54) * q^22 + (44*z^3 - 44*z^2 + 15) * q^23 + (-143*z^3 + 140*z^2 - 143*z) * q^25 + (-154*z^3 - 88*z + 88) * q^26 + (-72*z^2 + 300*z - 72) * q^28 + (162*z^3 + 192*z - 192) * q^29 + (44*z^3 + 33*z^2 - 33*z - 44) * q^31 + (192*z^3 - 192*z^2 - 96) * q^32 + (-198*z^3 + 198*z^2 + 384) * q^34 + (-85*z^3 + 366*z^2 - 366*z + 85) * q^35 + (267*z^3 + 22*z - 22) * q^37 + (242*z^2 + 104*z + 242) * q^38 + (168*z^3 - 104*z + 104) * q^40 + (34*z^3 - 61*z^2 + 34*z) * q^41 + (84*z^3 - 84*z^2 - 75) * q^43 + (-180*z^3 + 420*z^2 - 300*z + 492) * q^44 + (-146*z^3 + 294*z^2 - 294*z + 146) * q^46 + (-55*z^3 - 145*z^2 - 55*z) * q^47 + (264*z^2 - 318*z + 264) * q^49 + (-274*z^2 + 852*z - 274) * q^50 + (264*z^3 + 132*z^2 + 264*z) * q^52 + (-98*z^3 + 241*z^2 - 241*z + 98) * q^53 + (369*z^3 - 520*z^2 + 571*z - 276) * q^55 + (352*z^3 - 352*z^2 - 56) * q^56 + (-264*z^3 - 828*z^2 - 264*z) * q^58 + (-33*z^3 - 451*z + 451) * q^59 + (279*z^2 + 438*z + 279) * q^61 + (-396*z^3 - 22*z + 22) * q^62 + (-832*z^3 + 832*z^2 - 832*z + 832) * q^64 + (99*z^3 - 99*z^2 - 209) * q^65 + (561*z^3 - 561*z^2 - 243) * q^67 + (936*z^3 - 252*z^2 + 252*z - 936) * q^68 + (-954*z^3 + 902*z - 902) * q^70 + (-275*z^2 + 708*z - 275) * q^71 + (-28*z^3 + 318*z - 318) * q^73 + (-1024*z^3 + 402*z^2 - 1024*z) * q^74 + (540*z^3 - 540*z^2 - 996) * q^76 + (249*z^3 - 779*z^2 + 745*z - 839) * q^77 + (65*z^3 + 637*z^2 - 637*z - 65) * q^79 + (176*z^3 - 192*z^2 + 176*z) * q^80 + (176*z^2 - 258*z + 176) * q^82 + (466*z^2 - 431*z + 466) * q^83 + (-174*z^3 - 375*z^2 - 174*z) * q^85 + (-486*z^3 + 354*z^2 - 354*z + 486) * q^86 + (168*z^3 + 48*z^2 + 192*z - 688) * q^88 + (154*z^3 - 154*z^2 + 1279) * q^89 + (-352*z^3 - 143*z^2 - 352*z) * q^91 + (-180*z^3 + 528*z - 528) * q^92 + (690*z^2 - 70*z + 690) * q^94 + (-412*z^3 - 373*z + 373) * q^95 + (-609*z^3 - 282*z^2 + 282*z + 609) * q^97 + (-744*z^3 + 744*z^2 - 948) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 10 q^{2} - 12 q^{4} + 21 q^{5} + 37 q^{7} + 40 q^{8}+O(q^{10})$$ 4 * q - 10 * q^2 - 12 * q^4 + 21 * q^5 + 37 * q^7 + 40 * q^8 $$4 q - 10 q^{2} - 12 q^{4} + 21 q^{5} + 37 q^{7} + 40 q^{8} - 220 q^{10} - 41 q^{11} - 77 q^{13} - 250 q^{14} + 16 q^{16} - 192 q^{17} - 52 q^{19} + 252 q^{20} + 40 q^{22} + 148 q^{23} - 426 q^{25} + 110 q^{26} + 84 q^{28} - 414 q^{29} - 198 q^{31} + 1140 q^{34} - 477 q^{35} + 201 q^{37} + 830 q^{38} + 480 q^{40} + 129 q^{41} - 132 q^{43} + 1068 q^{44} - 150 q^{46} + 35 q^{47} + 474 q^{49} + 30 q^{50} + 396 q^{52} - 188 q^{53} + 356 q^{55} + 480 q^{56} + 300 q^{58} + 1320 q^{59} + 1275 q^{61} - 330 q^{62} + 832 q^{64} - 638 q^{65} + 150 q^{67} - 2304 q^{68} - 3660 q^{70} - 117 q^{71} - 982 q^{73} - 2450 q^{74} - 2904 q^{76} - 1583 q^{77} - 1469 q^{79} + 544 q^{80} + 270 q^{82} + 967 q^{83} + 27 q^{85} + 750 q^{86} - 2440 q^{88} + 5424 q^{89} - 561 q^{91} - 1764 q^{92} + 2000 q^{94} + 707 q^{95} + 2391 q^{97} - 5280 q^{98}+O(q^{100})$$ 4 * q - 10 * q^2 - 12 * q^4 + 21 * q^5 + 37 * q^7 + 40 * q^8 - 220 * q^10 - 41 * q^11 - 77 * q^13 - 250 * q^14 + 16 * q^16 - 192 * q^17 - 52 * q^19 + 252 * q^20 + 40 * q^22 + 148 * q^23 - 426 * q^25 + 110 * q^26 + 84 * q^28 - 414 * q^29 - 198 * q^31 + 1140 * q^34 - 477 * q^35 + 201 * q^37 + 830 * q^38 + 480 * q^40 + 129 * q^41 - 132 * q^43 + 1068 * q^44 - 150 * q^46 + 35 * q^47 + 474 * q^49 + 30 * q^50 + 396 * q^52 - 188 * q^53 + 356 * q^55 + 480 * q^56 + 300 * q^58 + 1320 * q^59 + 1275 * q^61 - 330 * q^62 + 832 * q^64 - 638 * q^65 + 150 * q^67 - 2304 * q^68 - 3660 * q^70 - 117 * q^71 - 982 * q^73 - 2450 * q^74 - 2904 * q^76 - 1583 * q^77 - 1469 * q^79 + 544 * q^80 + 270 * q^82 + 967 * q^83 + 27 * q^85 + 750 * q^86 - 2440 * q^88 + 5424 * q^89 - 561 * q^91 - 1764 * q^92 + 2000 * q^94 + 707 * q^95 + 2391 * q^97 - 5280 * q^98

## Character values

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

 $$n$$ $$46$$ $$56$$ $$\chi(n)$$ $$-\zeta_{10}^{3}$$ $$1$$

## 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
 0.809017 − 0.587785i −0.309017 − 0.951057i −0.309017 + 0.951057i 0.809017 + 0.587785i
−3.61803 2.62866i 0 3.70820 + 11.4127i 4.69098 3.40820i 0 −4.72542 14.5434i 5.52786 17.0130i 0 −25.9311
64.1 −1.38197 + 4.25325i 0 −9.70820 7.05342i 5.80902 + 17.8783i 0 23.2254 + 16.8743i 14.4721 10.5146i 0 −84.0689
82.1 −1.38197 4.25325i 0 −9.70820 + 7.05342i 5.80902 17.8783i 0 23.2254 16.8743i 14.4721 + 10.5146i 0 −84.0689
91.1 −3.61803 + 2.62866i 0 3.70820 11.4127i 4.69098 + 3.40820i 0 −4.72542 + 14.5434i 5.52786 + 17.0130i 0 −25.9311
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 99.4.f.a 4
3.b odd 2 1 33.4.e.a 4
11.c even 5 1 inner 99.4.f.a 4
11.c even 5 1 1089.4.a.p 2
11.d odd 10 1 1089.4.a.q 2
33.f even 10 1 363.4.a.o 2
33.h odd 10 1 33.4.e.a 4
33.h odd 10 1 363.4.a.n 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.4.e.a 4 3.b odd 2 1
33.4.e.a 4 33.h odd 10 1
99.4.f.a 4 1.a even 1 1 trivial
99.4.f.a 4 11.c even 5 1 inner
363.4.a.n 2 33.h odd 10 1
363.4.a.o 2 33.f even 10 1
1089.4.a.p 2 11.c even 5 1
1089.4.a.q 2 11.d odd 10 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 10T_{2}^{3} + 60T_{2}^{2} + 200T_{2} + 400$$ acting on $$S_{4}^{\mathrm{new}}(99, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 10 T^{3} + 60 T^{2} + \cdots + 400$$
$3$ $$T^{4}$$
$5$ $$T^{4} - 21 T^{3} + 496 T^{2} + \cdots + 11881$$
$7$ $$T^{4} - 37 T^{3} + 619 T^{2} + \cdots + 192721$$
$11$ $$T^{4} + 41 T^{3} + 1881 T^{2} + \cdots + 1771561$$
$13$ $$T^{4} + 77 T^{3} + 2299 T^{2} + \cdots + 14641$$
$17$ $$T^{4} + 192 T^{3} + 14634 T^{2} + \cdots + 2595321$$
$19$ $$T^{4} + 52 T^{3} + 11254 T^{2} + \cdots + 1274641$$
$23$ $$(T^{2} - 74 T - 1051)^{2}$$
$29$ $$T^{4} + 414 T^{3} + \cdots + 1740892176$$
$31$ $$T^{4} + 198 T^{3} + \cdots + 54479161$$
$37$ $$T^{4} - 201 T^{3} + \cdots + 4216034761$$
$41$ $$T^{4} - 129 T^{3} + 6271 T^{2} + \cdots + 241081$$
$43$ $$(T^{2} + 66 T - 7731)^{2}$$
$47$ $$T^{4} - 35 T^{3} + \cdots + 674700625$$
$53$ $$T^{4} + 188 T^{3} + \cdots + 10042561$$
$59$ $$T^{4} - 1320 T^{3} + \cdots + 47173668025$$
$61$ $$T^{4} - 1275 T^{3} + \cdots + 55792802025$$
$67$ $$(T^{2} - 75 T - 391995)^{2}$$
$71$ $$T^{4} + 117 T^{3} + \cdots + 53332821721$$
$73$ $$T^{4} + 982 T^{3} + \cdots + 8360542096$$
$79$ $$T^{4} + 1469 T^{3} + \cdots + 285379255681$$
$83$ $$T^{4} - 967 T^{3} + \cdots + 53935882081$$
$89$ $$(T^{2} - 2712 T + 1809091)^{2}$$
$97$ $$T^{4} - 2391 T^{3} + \cdots + 932420053161$$