# Properties

 Label 9801.2.a.bl Level 9801 Weight 2 Character orbit 9801.a Self dual yes Analytic conductor 78.261 Analytic rank 1 Dimension 4 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$9801 = 3^{4} \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9801.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$78.2613790211$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.22545.1 Defining polynomial: $$x^{4} - x^{3} - 6 x^{2} + 5 x + 4$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 99) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + ( 3 + \beta_{3} ) q^{4} + ( -1 + \beta_{1} + \beta_{3} ) q^{5} -\beta_{1} q^{7} + ( -1 + \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{8} +O(q^{10})$$ $$q + \beta_{2} q^{2} + ( 3 + \beta_{3} ) q^{4} + ( -1 + \beta_{1} + \beta_{3} ) q^{5} -\beta_{1} q^{7} + ( -1 + \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{8} + ( -1 + 3 \beta_{1} ) q^{10} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{13} + ( -2 \beta_{1} - \beta_{3} ) q^{14} + ( 5 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{16} + ( -1 - 2 \beta_{1} + \beta_{2} ) q^{17} + ( -2 - 2 \beta_{1} - \beta_{3} ) q^{19} + ( 2 + 4 \beta_{1} - \beta_{2} + \beta_{3} ) q^{20} + ( -4 + \beta_{1} - \beta_{3} ) q^{23} + ( 3 - \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{25} + ( -6 - \beta_{1} - 3 \beta_{3} ) q^{26} + ( 1 - 3 \beta_{1} - \beta_{2} - \beta_{3} ) q^{28} + ( -1 - \beta_{2} + \beta_{3} ) q^{29} + ( -\beta_{2} + \beta_{3} ) q^{31} + ( -10 + 2 \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{32} + ( 5 - 4 \beta_{1} - \beta_{2} - \beta_{3} ) q^{34} + ( -2 + \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{35} + ( 1 - \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{37} + ( 1 - 5 \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{38} + ( -4 + 3 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{40} + ( -1 + 3 \beta_{1} - \beta_{2} ) q^{41} + ( 5 + \beta_{2} ) q^{43} + ( 1 + \beta_{1} - 5 \beta_{2} + 2 \beta_{3} ) q^{46} + ( 2 - 2 \beta_{1} + \beta_{2} ) q^{47} + ( -4 + \beta_{2} ) q^{49} + ( 7 - 4 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{50} + ( 5 - 3 \beta_{1} - 7 \beta_{2} ) q^{52} + ( 1 + 3 \beta_{1} + \beta_{3} ) q^{53} + ( -4 - 3 \beta_{1} - \beta_{3} ) q^{56} + ( -6 + \beta_{1} - 2 \beta_{3} ) q^{58} + ( -1 + \beta_{1} - \beta_{3} ) q^{59} + ( -4 + \beta_{1} ) q^{61} + ( -6 + \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{62} + ( 6 + \beta_{1} - 7 \beta_{2} + 2 \beta_{3} ) q^{64} + ( 5 - 2 \beta_{1} - 3 \beta_{2} - 4 \beta_{3} ) q^{65} + ( 4 - 3 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{67} + ( -2 - 5 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} ) q^{68} + ( -9 + \beta_{1} - 3 \beta_{2} ) q^{70} + ( 1 - 3 \beta_{1} + 2 \beta_{2} ) q^{71} + ( -6 + \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{73} + ( -8 - 4 \beta_{1} - \beta_{2} - \beta_{3} ) q^{74} + ( -10 - 7 \beta_{1} - 5 \beta_{3} ) q^{76} + ( -2 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{79} + ( 9 + 2 \beta_{3} ) q^{80} + ( -5 + 6 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{82} + ( 5 + 2 \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{83} + ( -4 + 4 \beta_{1} - 4 \beta_{2} - 3 \beta_{3} ) q^{85} + ( 5 + 5 \beta_{2} + \beta_{3} ) q^{86} + ( 4 \beta_{1} + 2 \beta_{2} ) q^{89} + ( 4 + 3 \beta_{1} ) q^{91} + ( -19 + 2 \beta_{1} + 3 \beta_{2} - 4 \beta_{3} ) q^{92} + ( 5 - 4 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{94} + ( -7 - \beta_{1} - 3 \beta_{2} - 2 \beta_{3} ) q^{95} + ( 7 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{97} + ( 5 - 4 \beta_{2} + \beta_{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + q^{2} + 11q^{4} - 4q^{5} - q^{7} + O(q^{10})$$ $$4q + q^{2} + 11q^{4} - 4q^{5} - q^{7} - q^{10} - 7q^{13} - q^{14} + 17q^{16} - 5q^{17} - 9q^{19} + 10q^{20} - 14q^{23} + 14q^{25} - 22q^{26} + q^{28} - 6q^{29} - 2q^{31} - 34q^{32} + 16q^{34} - 8q^{35} + 3q^{37} - 3q^{38} - 12q^{40} - 2q^{41} + 21q^{43} - 2q^{46} + 7q^{47} - 15q^{49} + 23q^{50} + 10q^{52} + 6q^{53} - 18q^{56} - 21q^{58} - 2q^{59} - 15q^{61} - 20q^{62} + 16q^{64} + 19q^{65} + 14q^{67} - 7q^{68} - 38q^{70} + 3q^{71} - 22q^{73} - 36q^{74} - 42q^{76} - 11q^{79} + 34q^{80} - 17q^{82} + 18q^{83} - 13q^{85} + 24q^{86} + 6q^{89} + 19q^{91} - 67q^{92} + 19q^{94} - 30q^{95} + 26q^{97} + 15q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 6 x^{2} + 5 x + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - 5 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + 5 \beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −0.519120 1.45106 −2.27060 2.33866
−2.73051 0 5.45571 0.936586 0 0.519120 −9.43585 0 −2.55736
1.2 −0.894434 0 −1.19999 −3.74893 0 −1.45106 2.86218 0 3.35317
1.3 2.15561 0 2.64667 −3.62393 0 2.27060 1.39396 0 −7.81179
1.4 2.46934 0 4.09762 2.43628 0 −2.33866 5.17972 0 6.01598
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 9801.2.a.bl 4
3.b odd 2 1 9801.2.a.bi 4
9.d odd 6 2 1089.2.e.i 8
11.b odd 2 1 891.2.a.p 4
33.d even 2 1 891.2.a.q 4
99.g even 6 2 99.2.e.e 8
99.h odd 6 2 297.2.e.e 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.2.e.e 8 99.g even 6 2
297.2.e.e 8 99.h odd 6 2
891.2.a.p 4 11.b odd 2 1
891.2.a.q 4 33.d even 2 1
1089.2.e.i 8 9.d odd 6 2
9801.2.a.bi 4 3.b odd 2 1
9801.2.a.bl 4 1.a even 1 1 trivial

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$11$$ $$-1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(9801))$$:

 $$T_{2}^{4} - T_{2}^{3} - 9 T_{2}^{2} + 8 T_{2} + 13$$ $$T_{5}^{4} + 4 T_{5}^{3} - 9 T_{5}^{2} - 29 T_{5} + 31$$ $$T_{7}^{4} + T_{7}^{3} - 6 T_{7}^{2} - 5 T_{7} + 4$$ $$T_{17}^{4} + 5 T_{17}^{3} - 24 T_{17}^{2} - 169 T_{17} - 236$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T - T^{2} + 2 T^{3} + T^{4} + 4 T^{5} - 4 T^{6} - 8 T^{7} + 16 T^{8}$$
$3$ 1
$5$ $$1 + 4 T + 11 T^{2} + 31 T^{3} + 91 T^{4} + 155 T^{5} + 275 T^{6} + 500 T^{7} + 625 T^{8}$$
$7$ $$1 + T + 22 T^{2} + 16 T^{3} + 214 T^{4} + 112 T^{5} + 1078 T^{6} + 343 T^{7} + 2401 T^{8}$$
$11$ 1
$13$ $$1 + 7 T + 37 T^{2} + 118 T^{3} + 466 T^{4} + 1534 T^{5} + 6253 T^{6} + 15379 T^{7} + 28561 T^{8}$$
$17$ $$1 + 5 T + 44 T^{2} + 86 T^{3} + 682 T^{4} + 1462 T^{5} + 12716 T^{6} + 24565 T^{7} + 83521 T^{8}$$
$19$ $$1 + 9 T + 76 T^{2} + 432 T^{3} + 2112 T^{4} + 8208 T^{5} + 27436 T^{6} + 61731 T^{7} + 130321 T^{8}$$
$23$ $$1 + 14 T + 143 T^{2} + 953 T^{3} + 5332 T^{4} + 21919 T^{5} + 75647 T^{6} + 170338 T^{7} + 279841 T^{8}$$
$29$ $$1 + 6 T + 107 T^{2} + 417 T^{3} + 4374 T^{4} + 12093 T^{5} + 89987 T^{6} + 146334 T^{7} + 707281 T^{8}$$
$31$ $$1 + 2 T + 103 T^{2} + 113 T^{3} + 4405 T^{4} + 3503 T^{5} + 98983 T^{6} + 59582 T^{7} + 923521 T^{8}$$
$37$ $$1 - 3 T + 67 T^{2} - 189 T^{3} + 2277 T^{4} - 6993 T^{5} + 91723 T^{6} - 151959 T^{7} + 1874161 T^{8}$$
$41$ $$1 + 2 T + 101 T^{2} + 407 T^{3} + 4894 T^{4} + 16687 T^{5} + 169781 T^{6} + 137842 T^{7} + 2825761 T^{8}$$
$43$ $$1 - 21 T + 328 T^{2} - 3186 T^{3} + 25008 T^{4} - 136998 T^{5} + 606472 T^{6} - 1669647 T^{7} + 3418801 T^{8}$$
$47$ $$1 - 7 T + 173 T^{2} - 985 T^{3} + 11845 T^{4} - 46295 T^{5} + 382157 T^{6} - 726761 T^{7} + 4879681 T^{8}$$
$53$ $$1 - 6 T + 167 T^{2} - 789 T^{3} + 11961 T^{4} - 41817 T^{5} + 469103 T^{6} - 893262 T^{7} + 7890481 T^{8}$$
$59$ $$1 + 2 T + 215 T^{2} + 305 T^{3} + 18421 T^{4} + 17995 T^{5} + 748415 T^{6} + 410758 T^{7} + 12117361 T^{8}$$
$61$ $$1 + 15 T + 322 T^{2} + 2910 T^{3} + 31962 T^{4} + 177510 T^{5} + 1198162 T^{6} + 3404715 T^{7} + 13845841 T^{8}$$
$67$ $$1 - 14 T + 247 T^{2} - 1979 T^{3} + 21949 T^{4} - 132593 T^{5} + 1108783 T^{6} - 4210682 T^{7} + 20151121 T^{8}$$
$71$ $$1 - 3 T + 197 T^{2} - 909 T^{3} + 17697 T^{4} - 64539 T^{5} + 993077 T^{6} - 1073733 T^{7} + 25411681 T^{8}$$
$73$ $$1 + 22 T + 421 T^{2} + 4807 T^{3} + 49780 T^{4} + 350911 T^{5} + 2243509 T^{6} + 8558374 T^{7} + 28398241 T^{8}$$
$79$ $$1 + 11 T + 292 T^{2} + 2582 T^{3} + 33694 T^{4} + 203978 T^{5} + 1822372 T^{6} + 5423429 T^{7} + 38950081 T^{8}$$
$83$ $$1 - 18 T + 329 T^{2} - 3771 T^{3} + 42384 T^{4} - 312993 T^{5} + 2266481 T^{6} - 10292166 T^{7} + 47458321 T^{8}$$
$89$ $$1 - 6 T + 224 T^{2} - 1554 T^{3} + 26094 T^{4} - 138306 T^{5} + 1774304 T^{6} - 4229814 T^{7} + 62742241 T^{8}$$
$97$ $$1 - 26 T + 544 T^{2} - 7460 T^{3} + 85489 T^{4} - 723620 T^{5} + 5118496 T^{6} - 23729498 T^{7} + 88529281 T^{8}$$