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