# Properties

 Label 9900.2.c.j Level $9900$ Weight $2$ Character orbit 9900.c Analytic conductor $79.052$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2$$ 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 \beta q^{7}+O(q^{10})$$ q + 2*b * q^7 $$q + 2 \beta q^{7} + q^{11} - 2 \beta q^{13} - 3 \beta q^{17} - 2 q^{19} - 4 q^{31} + 5 \beta q^{37} - 2 \beta q^{43} + 6 \beta q^{47} - 9 q^{49} - 3 \beta q^{53} + 12 q^{59} - 10 q^{61} + 2 \beta q^{67} + 4 \beta q^{73} + 2 \beta q^{77} + 10 q^{79} + 3 \beta q^{83} - 6 q^{89} + 16 q^{91} + 5 \beta q^{97} +O(q^{100})$$ q + 2*b * q^7 + q^11 - 2*b * q^13 - 3*b * q^17 - 2 * q^19 - 4 * q^31 + 5*b * q^37 - 2*b * q^43 + 6*b * q^47 - 9 * q^49 - 3*b * q^53 + 12 * q^59 - 10 * q^61 + 2*b * q^67 + 4*b * q^73 + 2*b * q^77 + 10 * q^79 + 3*b * q^83 - 6 * q^89 + 16 * q^91 + 5*b * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q+O(q^{10})$$ 2 * q $$2 q + 2 q^{11} - 4 q^{19} - 8 q^{31} - 18 q^{49} + 24 q^{59} - 20 q^{61} + 20 q^{79} - 12 q^{89} + 32 q^{91}+O(q^{100})$$ 2 * q + 2 * q^11 - 4 * q^19 - 8 * q^31 - 18 * q^49 + 24 * q^59 - 20 * q^61 + 20 * q^79 - 12 * q^89 + 32 * q^91

## 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
 − 1.00000i 1.00000i
0 0 0 0 0 4.00000i 0 0 0
5149.2 0 0 0 0 0 4.00000i 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.j 2
3.b odd 2 1 3300.2.c.b 2
5.b even 2 1 inner 9900.2.c.j 2
5.c odd 4 1 1980.2.a.c 1
5.c odd 4 1 9900.2.a.bc 1
15.d odd 2 1 3300.2.c.b 2
15.e even 4 1 660.2.a.c 1
15.e even 4 1 3300.2.a.h 1
20.e even 4 1 7920.2.a.bl 1
60.l odd 4 1 2640.2.a.f 1
165.l odd 4 1 7260.2.a.p 1

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

 $$T_{7}^{2} + 16$$ T7^2 + 16 $$T_{13}^{2} + 16$$ T13^2 + 16 $$T_{17}^{2} + 36$$ T17^2 + 36 $$T_{29}$$ T29 $$T_{41}$$ T41 $$T_{59} - 12$$ T59 - 12

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 16$$
$11$ $$(T - 1)^{2}$$
$13$ $$T^{2} + 16$$
$17$ $$T^{2} + 36$$
$19$ $$(T + 2)^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$(T + 4)^{2}$$
$37$ $$T^{2} + 100$$
$41$ $$T^{2}$$
$43$ $$T^{2} + 16$$
$47$ $$T^{2} + 144$$
$53$ $$T^{2} + 36$$
$59$ $$(T - 12)^{2}$$
$61$ $$(T + 10)^{2}$$
$67$ $$T^{2} + 16$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 64$$
$79$ $$(T - 10)^{2}$$
$83$ $$T^{2} + 36$$
$89$ $$(T + 6)^{2}$$
$97$ $$T^{2} + 100$$