# Properties

 Label 9900.2.c.q 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: $$\Q(i, \sqrt{13})$$ Defining polynomial: $$x^{4} + 7 x^{2} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 660) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{7} +O(q^{10})$$ $$q + \beta_{1} q^{7} - q^{11} + \beta_{1} q^{13} + ( -\beta_{1} + \beta_{2} ) q^{17} -\beta_{3} q^{19} + 8 q^{29} + ( 2 - \beta_{3} ) q^{31} + ( -2 \beta_{1} - 3 \beta_{2} ) q^{37} -8 q^{41} + ( -\beta_{1} - 4 \beta_{2} ) q^{43} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{47} + ( -7 + \beta_{3} ) q^{49} -\beta_{2} q^{53} + 8 q^{59} -\beta_{3} q^{61} + 2 \beta_{2} q^{67} -2 \beta_{3} q^{71} + ( -\beta_{1} - 2 \beta_{2} ) q^{73} -\beta_{1} q^{77} + ( 4 - \beta_{3} ) q^{79} + ( -\beta_{1} + 3 \beta_{2} ) q^{83} + 6 q^{89} + ( -14 + \beta_{3} ) q^{91} + ( -4 \beta_{1} - 3 \beta_{2} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q - 4q^{11} + 32q^{29} + 8q^{31} - 32q^{41} - 28q^{49} + 32q^{59} + 16q^{79} + 24q^{89} - 56q^{91} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 7 x^{2} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu$$ $$\beta_{2}$$ $$=$$ $$($$$$2 \nu^{3} + 8 \nu$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$4 \nu^{2} + 14$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$$$/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} - 14$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$3 \beta_{2} - 4 \beta_{1}$$$$)/2$$

## 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
 − 2.30278i − 1.30278i 1.30278i 2.30278i
0 0 0 0 0 4.60555i 0 0 0
5149.2 0 0 0 0 0 2.60555i 0 0 0
5149.3 0 0 0 0 0 2.60555i 0 0 0
5149.4 0 0 0 0 0 4.60555i 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.q 4
3.b odd 2 1 3300.2.c.l 4
5.b even 2 1 inner 9900.2.c.q 4
5.c odd 4 1 1980.2.a.h 2
5.c odd 4 1 9900.2.a.bl 2
15.d odd 2 1 3300.2.c.l 4
15.e even 4 1 660.2.a.e 2
15.e even 4 1 3300.2.a.w 2
20.e even 4 1 7920.2.a.bo 2
60.l odd 4 1 2640.2.a.bc 2
165.l odd 4 1 7260.2.a.w 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
660.2.a.e 2 15.e even 4 1
1980.2.a.h 2 5.c odd 4 1
2640.2.a.bc 2 60.l odd 4 1
3300.2.a.w 2 15.e even 4 1
3300.2.c.l 4 3.b odd 2 1
3300.2.c.l 4 15.d odd 2 1
7260.2.a.w 2 165.l odd 4 1
7920.2.a.bo 2 20.e even 4 1
9900.2.a.bl 2 5.c odd 4 1
9900.2.c.q 4 1.a even 1 1 trivial
9900.2.c.q 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 $$S_{2}^{\mathrm{new}}(9900, [\chi])$$:

 $$T_{7}^{4} + 28 T_{7}^{2} + 144$$ $$T_{13}^{4} + 28 T_{13}^{2} + 144$$ $$T_{17}^{4} + 44 T_{17}^{2} + 16$$ $$T_{29} - 8$$ $$T_{41} + 8$$ $$T_{59} - 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$144 + 28 T^{2} + T^{4}$$
$11$ $$( 1 + T )^{4}$$
$13$ $$144 + 28 T^{2} + T^{4}$$
$17$ $$16 + 44 T^{2} + T^{4}$$
$19$ $$( -52 + T^{2} )^{2}$$
$23$ $$T^{4}$$
$29$ $$( -8 + T )^{4}$$
$31$ $$( -48 - 4 T + T^{2} )^{2}$$
$37$ $$1296 + 136 T^{2} + T^{4}$$
$41$ $$( 8 + T )^{4}$$
$43$ $$1296 + 124 T^{2} + T^{4}$$
$47$ $$2304 + 112 T^{2} + T^{4}$$
$53$ $$( 4 + T^{2} )^{2}$$
$59$ $$( -8 + T )^{4}$$
$61$ $$( -52 + T^{2} )^{2}$$
$67$ $$( 16 + T^{2} )^{2}$$
$71$ $$( -208 + T^{2} )^{2}$$
$73$ $$16 + 44 T^{2} + T^{4}$$
$79$ $$( -36 - 8 T + T^{2} )^{2}$$
$83$ $$1296 + 124 T^{2} + T^{4}$$
$89$ $$( -6 + T )^{4}$$
$97$ $$41616 + 424 T^{2} + T^{4}$$