# Properties

 Label 9900.2.c.x Level $9900$ Weight $2$ Character orbit 9900.c Analytic conductor $79.052$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$9900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9900.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$79.0518980011$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{21})$$ Defining polynomial: $$x^{4} + 11x^{2} + 25$$ x^4 + 11*x^2 + 25 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1100) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (3 \beta_{2} + \beta_1) q^{7}+O(q^{10})$$ q + (3*b2 + b1) * q^7 $$q + (3 \beta_{2} + \beta_1) q^{7} + q^{11} - \beta_{2} q^{13} + ( - 2 \beta_{2} - \beta_1) q^{17} + (2 \beta_{3} - 3) q^{19} + (\beta_{2} - \beta_1) q^{23} + ( - \beta_{3} + 5) q^{29} + (2 \beta_{3} - 5) q^{31} + ( - 3 \beta_{2} - 2 \beta_1) q^{37} + (2 \beta_{3} + 5) q^{41} - 10 \beta_{2} q^{43} + ( - 7 \beta_{2} - 2 \beta_1) q^{47} + ( - 5 \beta_{3} - 2) q^{49} + ( - 3 \beta_{2} + 3 \beta_1) q^{53} + (2 \beta_{3} - 7) q^{59} + (\beta_{3} + 6) q^{61} + 4 \beta_{2} q^{67} - 6 \beta_{3} q^{71} + ( - 5 \beta_{2} + \beta_1) q^{73} + (3 \beta_{2} + \beta_1) q^{77} + ( - 7 \beta_{3} + 3) q^{79} + ( - 4 \beta_{2} - 5 \beta_1) q^{83} + (\beta_{3} + 1) q^{89} + (\beta_{3} + 2) q^{91} + (9 \beta_{2} + \beta_1) q^{97}+O(q^{100})$$ q + (3*b2 + b1) * q^7 + q^11 - b2 * q^13 + (-2*b2 - b1) * q^17 + (2*b3 - 3) * q^19 + (b2 - b1) * q^23 + (-b3 + 5) * q^29 + (2*b3 - 5) * q^31 + (-3*b2 - 2*b1) * q^37 + (2*b3 + 5) * q^41 - 10*b2 * q^43 + (-7*b2 - 2*b1) * q^47 + (-5*b3 - 2) * q^49 + (-3*b2 + 3*b1) * q^53 + (2*b3 - 7) * q^59 + (b3 + 6) * q^61 + 4*b2 * q^67 - 6*b3 * q^71 + (-5*b2 + b1) * q^73 + (3*b2 + b1) * q^77 + (-7*b3 + 3) * q^79 + (-4*b2 - 5*b1) * q^83 + (b3 + 1) * q^89 + (b3 + 2) * q^91 + (9*b2 + b1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q + 4 q^{11} - 8 q^{19} + 18 q^{29} - 16 q^{31} + 24 q^{41} - 18 q^{49} - 24 q^{59} + 26 q^{61} - 12 q^{71} - 2 q^{79} + 6 q^{89} + 10 q^{91}+O(q^{100})$$ 4 * q + 4 * q^11 - 8 * q^19 + 18 * q^29 - 16 * q^31 + 24 * q^41 - 18 * q^49 - 24 * q^59 + 26 * q^61 - 12 * q^71 - 2 * q^79 + 6 * q^89 + 10 * q^91

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 11x^{2} + 25$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 6\nu ) / 5$$ (v^3 + 6*v) / 5 $$\beta_{3}$$ $$=$$ $$\nu^{2} + 6$$ v^2 + 6
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} - 6$$ b3 - 6 $$\nu^{3}$$ $$=$$ $$5\beta_{2} - 6\beta_1$$ 5*b2 - 6*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/9900\mathbb{Z}\right)^\times$$.

 $$n$$ $$2377$$ $$4501$$ $$4951$$ $$5501$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5149.1
 − 1.79129i 2.79129i − 2.79129i 1.79129i
0 0 0 0 0 4.79129i 0 0 0
5149.2 0 0 0 0 0 0.208712i 0 0 0
5149.3 0 0 0 0 0 0.208712i 0 0 0
5149.4 0 0 0 0 0 4.79129i 0 0 0
 $$n$$: 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
5.b even 2 1 inner

## Twists

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

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

## Hecke kernels

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

 $$T_{7}^{4} + 23T_{7}^{2} + 1$$ T7^4 + 23*T7^2 + 1 $$T_{13}^{2} + 1$$ T13^2 + 1 $$T_{17}^{4} + 15T_{17}^{2} + 9$$ T17^4 + 15*T17^2 + 9 $$T_{29}^{2} - 9T_{29} + 15$$ T29^2 - 9*T29 + 15 $$T_{41}^{2} - 12T_{41} + 15$$ T41^2 - 12*T41 + 15 $$T_{59}^{2} + 12T_{59} + 15$$ T59^2 + 12*T59 + 15

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 23T^{2} + 1$$
$11$ $$(T - 1)^{4}$$
$13$ $$(T^{2} + 1)^{2}$$
$17$ $$T^{4} + 15T^{2} + 9$$
$19$ $$(T^{2} + 4 T - 17)^{2}$$
$23$ $$T^{4} + 15T^{2} + 9$$
$29$ $$(T^{2} - 9 T + 15)^{2}$$
$31$ $$(T^{2} + 8 T - 5)^{2}$$
$37$ $$T^{4} + 50T^{2} + 289$$
$41$ $$(T^{2} - 12 T + 15)^{2}$$
$43$ $$(T^{2} + 100)^{2}$$
$47$ $$T^{4} + 114T^{2} + 225$$
$53$ $$T^{4} + 135T^{2} + 729$$
$59$ $$(T^{2} + 12 T + 15)^{2}$$
$61$ $$(T^{2} - 13 T + 37)^{2}$$
$67$ $$(T^{2} + 16)^{2}$$
$71$ $$(T^{2} + 6 T - 180)^{2}$$
$73$ $$T^{4} + 71T^{2} + 625$$
$79$ $$(T^{2} + T - 257)^{2}$$
$83$ $$T^{4} + 267 T^{2} + 16641$$
$89$ $$(T^{2} - 3 T - 3)^{2}$$
$97$ $$T^{4} + 155T^{2} + 4489$$