Properties

 Label 9900.2.a.bz Level $9900$ Weight $2$ Character orbit 9900.a Self dual yes Analytic conductor $79.052$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$9900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9900.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$79.0518980011$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{21})$$ Defining polynomial: $$x^{2} - x - 5$$ x^2 - x - 5 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1100) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{21})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta + 3) q^{7}+O(q^{10})$$ q + (-b + 3) * q^7 $$q + ( - \beta + 3) q^{7} + q^{11} + q^{13} + (\beta - 2) q^{17} + (2 \beta + 1) q^{19} + ( - \beta - 1) q^{23} + ( - \beta - 4) q^{29} + ( - 2 \beta - 3) q^{31} + (2 \beta - 3) q^{37} + ( - 2 \beta + 7) q^{41} + 10 q^{43} + (2 \beta - 7) q^{47} + ( - 5 \beta + 7) q^{49} + (3 \beta + 3) q^{53} + (2 \beta + 5) q^{59} + ( - \beta + 7) q^{61} + 4 q^{67} + (6 \beta - 6) q^{71} + (\beta + 5) q^{73} + ( - \beta + 3) q^{77} + ( - 7 \beta + 4) q^{79} + ( - 5 \beta + 4) q^{83} + (\beta - 2) q^{89} + ( - \beta + 3) q^{91} + ( - \beta + 9) q^{97}+O(q^{100})$$ q + (-b + 3) * q^7 + q^11 + q^13 + (b - 2) * q^17 + (2*b + 1) * q^19 + (-b - 1) * q^23 + (-b - 4) * q^29 + (-2*b - 3) * q^31 + (2*b - 3) * q^37 + (-2*b + 7) * q^41 + 10 * q^43 + (2*b - 7) * q^47 + (-5*b + 7) * q^49 + (3*b + 3) * q^53 + (2*b + 5) * q^59 + (-b + 7) * q^61 + 4 * q^67 + (6*b - 6) * q^71 + (b + 5) * q^73 + (-b + 3) * q^77 + (-7*b + 4) * q^79 + (-5*b + 4) * q^83 + (b - 2) * q^89 + (-b + 3) * q^91 + (-b + 9) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 5 q^{7}+O(q^{10})$$ 2 * q + 5 * q^7 $$2 q + 5 q^{7} + 2 q^{11} + 2 q^{13} - 3 q^{17} + 4 q^{19} - 3 q^{23} - 9 q^{29} - 8 q^{31} - 4 q^{37} + 12 q^{41} + 20 q^{43} - 12 q^{47} + 9 q^{49} + 9 q^{53} + 12 q^{59} + 13 q^{61} + 8 q^{67} - 6 q^{71} + 11 q^{73} + 5 q^{77} + q^{79} + 3 q^{83} - 3 q^{89} + 5 q^{91} + 17 q^{97}+O(q^{100})$$ 2 * q + 5 * q^7 + 2 * q^11 + 2 * q^13 - 3 * q^17 + 4 * q^19 - 3 * q^23 - 9 * q^29 - 8 * q^31 - 4 * q^37 + 12 * q^41 + 20 * q^43 - 12 * q^47 + 9 * q^49 + 9 * q^53 + 12 * q^59 + 13 * q^61 + 8 * q^67 - 6 * q^71 + 11 * q^73 + 5 * q^77 + q^79 + 3 * q^83 - 3 * q^89 + 5 * q^91 + 17 * q^97

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
 2.79129 −1.79129
0 0 0 0 0 0.208712 0 0 0
1.2 0 0 0 0 0 4.79129 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$1$$
$$11$$ $$-1$$

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 9900.2.a.bz 2
3.b odd 2 1 1100.2.a.h yes 2
5.b even 2 1 9900.2.a.bh 2
5.c odd 4 2 9900.2.c.x 4
12.b even 2 1 4400.2.a.bi 2
15.d odd 2 1 1100.2.a.g 2
15.e even 4 2 1100.2.b.d 4
60.h even 2 1 4400.2.a.bu 2
60.l odd 4 2 4400.2.b.s 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1100.2.a.g 2 15.d odd 2 1
1100.2.a.h yes 2 3.b odd 2 1
1100.2.b.d 4 15.e even 4 2
4400.2.a.bi 2 12.b even 2 1
4400.2.a.bu 2 60.h even 2 1
4400.2.b.s 4 60.l odd 4 2
9900.2.a.bh 2 5.b even 2 1
9900.2.a.bz 2 1.a even 1 1 trivial
9900.2.c.x 4 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(9900))$$:

 $$T_{7}^{2} - 5T_{7} + 1$$ T7^2 - 5*T7 + 1 $$T_{13} - 1$$ T13 - 1 $$T_{17}^{2} + 3T_{17} - 3$$ T17^2 + 3*T17 - 3

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 5T + 1$$
$11$ $$(T - 1)^{2}$$
$13$ $$(T - 1)^{2}$$
$17$ $$T^{2} + 3T - 3$$
$19$ $$T^{2} - 4T - 17$$
$23$ $$T^{2} + 3T - 3$$
$29$ $$T^{2} + 9T + 15$$
$31$ $$T^{2} + 8T - 5$$
$37$ $$T^{2} + 4T - 17$$
$41$ $$T^{2} - 12T + 15$$
$43$ $$(T - 10)^{2}$$
$47$ $$T^{2} + 12T + 15$$
$53$ $$T^{2} - 9T - 27$$
$59$ $$T^{2} - 12T + 15$$
$61$ $$T^{2} - 13T + 37$$
$67$ $$(T - 4)^{2}$$
$71$ $$T^{2} + 6T - 180$$
$73$ $$T^{2} - 11T + 25$$
$79$ $$T^{2} - T - 257$$
$83$ $$T^{2} - 3T - 129$$
$89$ $$T^{2} + 3T - 3$$
$97$ $$T^{2} - 17T + 67$$