Properties

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

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$99 = 3^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 99.d (of order $$2$$, degree $$1$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$5.84118909057$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ Defining polynomial: $$x^{2} + 2$$ x^2 + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 3 q^{2} + q^{4} + 14 \beta q^{5} + 12 \beta q^{7} - 21 q^{8}+O(q^{10})$$ q + 3 * q^2 + q^4 + 14*b * q^5 + 12*b * q^7 - 21 * q^8 $$q + 3 q^{2} + q^{4} + 14 \beta q^{5} + 12 \beta q^{7} - 21 q^{8} + 42 \beta q^{10} + ( - 11 \beta + 33) q^{11} - 21 \beta q^{13} + 36 \beta q^{14} - 71 q^{16} + 126 q^{17} + 63 \beta q^{19} + 14 \beta q^{20} + ( - 33 \beta + 99) q^{22} - 85 \beta q^{23} - 267 q^{25} - 63 \beta q^{26} + 12 \beta q^{28} - 24 q^{29} - 70 q^{31} - 45 q^{32} + 378 q^{34} - 336 q^{35} + 182 q^{37} + 189 \beta q^{38} - 294 \beta q^{40} + 294 q^{41} + 3 \beta q^{43} + ( - 11 \beta + 33) q^{44} - 255 \beta q^{46} + 77 \beta q^{47} + 55 q^{49} - 801 q^{50} - 21 \beta q^{52} + 104 \beta q^{53} + (462 \beta + 308) q^{55} - 252 \beta q^{56} - 72 q^{58} - 364 \beta q^{59} - 231 \beta q^{61} - 210 q^{62} + 433 q^{64} + 588 q^{65} - 880 q^{67} + 126 q^{68} - 1008 q^{70} + 239 \beta q^{71} - 126 \beta q^{73} + 546 q^{74} + 63 \beta q^{76} + (396 \beta + 264) q^{77} - 546 \beta q^{79} - 994 \beta q^{80} + 882 q^{82} + 1218 q^{83} + 1764 \beta q^{85} + 9 \beta q^{86} + (231 \beta - 693) q^{88} + 1085 \beta q^{89} + 504 q^{91} - 85 \beta q^{92} + 231 \beta q^{94} - 1764 q^{95} - 196 q^{97} + 165 q^{98} +O(q^{100})$$ q + 3 * q^2 + q^4 + 14*b * q^5 + 12*b * q^7 - 21 * q^8 + 42*b * q^10 + (-11*b + 33) * q^11 - 21*b * q^13 + 36*b * q^14 - 71 * q^16 + 126 * q^17 + 63*b * q^19 + 14*b * q^20 + (-33*b + 99) * q^22 - 85*b * q^23 - 267 * q^25 - 63*b * q^26 + 12*b * q^28 - 24 * q^29 - 70 * q^31 - 45 * q^32 + 378 * q^34 - 336 * q^35 + 182 * q^37 + 189*b * q^38 - 294*b * q^40 + 294 * q^41 + 3*b * q^43 + (-11*b + 33) * q^44 - 255*b * q^46 + 77*b * q^47 + 55 * q^49 - 801 * q^50 - 21*b * q^52 + 104*b * q^53 + (462*b + 308) * q^55 - 252*b * q^56 - 72 * q^58 - 364*b * q^59 - 231*b * q^61 - 210 * q^62 + 433 * q^64 + 588 * q^65 - 880 * q^67 + 126 * q^68 - 1008 * q^70 + 239*b * q^71 - 126*b * q^73 + 546 * q^74 + 63*b * q^76 + (396*b + 264) * q^77 - 546*b * q^79 - 994*b * q^80 + 882 * q^82 + 1218 * q^83 + 1764*b * q^85 + 9*b * q^86 + (231*b - 693) * q^88 + 1085*b * q^89 + 504 * q^91 - 85*b * q^92 + 231*b * q^94 - 1764 * q^95 - 196 * q^97 + 165 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{2} + 2 q^{4} - 42 q^{8}+O(q^{10})$$ 2 * q + 6 * q^2 + 2 * q^4 - 42 * q^8 $$2 q + 6 q^{2} + 2 q^{4} - 42 q^{8} + 66 q^{11} - 142 q^{16} + 252 q^{17} + 198 q^{22} - 534 q^{25} - 48 q^{29} - 140 q^{31} - 90 q^{32} + 756 q^{34} - 672 q^{35} + 364 q^{37} + 588 q^{41} + 66 q^{44} + 110 q^{49} - 1602 q^{50} + 616 q^{55} - 144 q^{58} - 420 q^{62} + 866 q^{64} + 1176 q^{65} - 1760 q^{67} + 252 q^{68} - 2016 q^{70} + 1092 q^{74} + 528 q^{77} + 1764 q^{82} + 2436 q^{83} - 1386 q^{88} + 1008 q^{91} - 3528 q^{95} - 392 q^{97} + 330 q^{98}+O(q^{100})$$ 2 * q + 6 * q^2 + 2 * q^4 - 42 * q^8 + 66 * q^11 - 142 * q^16 + 252 * q^17 + 198 * q^22 - 534 * q^25 - 48 * q^29 - 140 * q^31 - 90 * q^32 + 756 * q^34 - 672 * q^35 + 364 * q^37 + 588 * q^41 + 66 * q^44 + 110 * q^49 - 1602 * q^50 + 616 * q^55 - 144 * q^58 - 420 * q^62 + 866 * q^64 + 1176 * q^65 - 1760 * q^67 + 252 * q^68 - 2016 * q^70 + 1092 * q^74 + 528 * q^77 + 1764 * q^82 + 2436 * q^83 - 1386 * q^88 + 1008 * q^91 - 3528 * q^95 - 392 * q^97 + 330 * q^98

Character values

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

 $$n$$ $$46$$ $$56$$ $$\chi(n)$$ $$-1$$ $$-1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
98.1
 − 1.41421i 1.41421i
3.00000 0 1.00000 19.7990i 0 16.9706i −21.0000 0 59.3970i
98.2 3.00000 0 1.00000 19.7990i 0 16.9706i −21.0000 0 59.3970i
 $$n$$: 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
33.d even 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 99.4.d.b yes 2
3.b odd 2 1 99.4.d.a 2
4.b odd 2 1 1584.4.b.a 2
11.b odd 2 1 99.4.d.a 2
12.b even 2 1 1584.4.b.b 2
33.d even 2 1 inner 99.4.d.b yes 2
44.c even 2 1 1584.4.b.b 2
132.d odd 2 1 1584.4.b.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.4.d.a 2 3.b odd 2 1
99.4.d.a 2 11.b odd 2 1
99.4.d.b yes 2 1.a even 1 1 trivial
99.4.d.b yes 2 33.d even 2 1 inner
1584.4.b.a 2 4.b odd 2 1
1584.4.b.a 2 132.d odd 2 1
1584.4.b.b 2 12.b even 2 1
1584.4.b.b 2 44.c even 2 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 3)^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 392$$
$7$ $$T^{2} + 288$$
$11$ $$T^{2} - 66T + 1331$$
$13$ $$T^{2} + 882$$
$17$ $$(T - 126)^{2}$$
$19$ $$T^{2} + 7938$$
$23$ $$T^{2} + 14450$$
$29$ $$(T + 24)^{2}$$
$31$ $$(T + 70)^{2}$$
$37$ $$(T - 182)^{2}$$
$41$ $$(T - 294)^{2}$$
$43$ $$T^{2} + 18$$
$47$ $$T^{2} + 11858$$
$53$ $$T^{2} + 21632$$
$59$ $$T^{2} + 264992$$
$61$ $$T^{2} + 106722$$
$67$ $$(T + 880)^{2}$$
$71$ $$T^{2} + 114242$$
$73$ $$T^{2} + 31752$$
$79$ $$T^{2} + 596232$$
$83$ $$(T - 1218)^{2}$$
$89$ $$T^{2} + 2354450$$
$97$ $$(T + 196)^{2}$$