# Properties

 Label 8100.2.d.n Level $8100$ Weight $2$ Character orbit 8100.d Analytic conductor $64.679$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$8100 = 2^{2} \cdot 3^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8100.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$64.6788256372$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ Defining polynomial: $$x^{4} + 3x^{2} + 1$$ x^4 + 3*x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{7}+O(q^{10})$$ q - b2 * q^7 $$q - \beta_{2} q^{7} + ( - \beta_{3} + 2) q^{11} + ( - \beta_{2} + 2 \beta_1) q^{13} + (\beta_{2} - 4 \beta_1) q^{17} - 2 q^{19} + ( - \beta_{2} - 2 \beta_1) q^{23} + (2 \beta_{3} - 4) q^{29} + (2 \beta_{3} - 5) q^{31} + ( - \beta_{2} - 3 \beta_1) q^{37} + 3 \beta_{3} q^{41} + ( - \beta_{2} + 2 \beta_1) q^{43} + (2 \beta_{2} + \beta_1) q^{47} + (\beta_{3} - 5) q^{49} + (\beta_{2} + 2 \beta_1) q^{53} + ( - \beta_{3} + 5) q^{59} + (\beta_{3} - 3) q^{61} - 7 \beta_1 q^{67} - 9 q^{71} + ( - 2 \beta_{2} + 3 \beta_1) q^{73} + ( - 2 \beta_{2} + 11 \beta_1) q^{77} + ( - 2 \beta_{3} - 4) q^{79} + (2 \beta_{2} + \beta_1) q^{83} - 12 q^{89} + (3 \beta_{3} - 14) q^{91} - 10 \beta_1 q^{97}+O(q^{100})$$ q - b2 * q^7 + (-b3 + 2) * q^11 + (-b2 + 2*b1) * q^13 + (b2 - 4*b1) * q^17 - 2 * q^19 + (-b2 - 2*b1) * q^23 + (2*b3 - 4) * q^29 + (2*b3 - 5) * q^31 + (-b2 - 3*b1) * q^37 + 3*b3 * q^41 + (-b2 + 2*b1) * q^43 + (2*b2 + b1) * q^47 + (b3 - 5) * q^49 + (b2 + 2*b1) * q^53 + (-b3 + 5) * q^59 + (b3 - 3) * q^61 - 7*b1 * q^67 - 9 * q^71 + (-2*b2 + 3*b1) * q^73 + (-2*b2 + 11*b1) * q^77 + (-2*b3 - 4) * q^79 + (2*b2 + b1) * q^83 - 12 * q^89 + (3*b3 - 14) * q^91 - 10*b1 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q + 6 q^{11} - 8 q^{19} - 12 q^{29} - 16 q^{31} + 6 q^{41} - 18 q^{49} + 18 q^{59} - 10 q^{61} - 36 q^{71} - 20 q^{79} - 48 q^{89} - 50 q^{91}+O(q^{100})$$ 4 * q + 6 * q^11 - 8 * q^19 - 12 * q^29 - 16 * q^31 + 6 * q^41 - 18 * q^49 + 18 * q^59 - 10 * q^61 - 36 * q^71 - 20 * q^79 - 48 * q^89 - 50 * q^91

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 3x^{2} + 1$$ :

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/8100\mathbb{Z}\right)^\times$$.

 $$n$$ $$4051$$ $$6401$$ $$7777$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
649.1
 1.61803i 0.618034i − 0.618034i − 1.61803i
0 0 0 0 0 3.85410i 0 0 0
649.2 0 0 0 0 0 2.85410i 0 0 0
649.3 0 0 0 0 0 2.85410i 0 0 0
649.4 0 0 0 0 0 3.85410i 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 8100.2.d.n 4
3.b odd 2 1 8100.2.d.k 4
5.b even 2 1 inner 8100.2.d.n 4
5.c odd 4 1 8100.2.a.p yes 2
5.c odd 4 1 8100.2.a.r yes 2
15.d odd 2 1 8100.2.d.k 4
15.e even 4 1 8100.2.a.o 2
15.e even 4 1 8100.2.a.q yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8100.2.a.o 2 15.e even 4 1
8100.2.a.p yes 2 5.c odd 4 1
8100.2.a.q yes 2 15.e even 4 1
8100.2.a.r yes 2 5.c odd 4 1
8100.2.d.k 4 3.b odd 2 1
8100.2.d.k 4 15.d odd 2 1
8100.2.d.n 4 1.a even 1 1 trivial
8100.2.d.n 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 $$S_{2}^{\mathrm{new}}(8100, [\chi])$$:

 $$T_{7}^{4} + 23T_{7}^{2} + 121$$ T7^4 + 23*T7^2 + 121 $$T_{11}^{2} - 3T_{11} - 9$$ T11^2 - 3*T11 - 9 $$T_{29}^{2} + 6T_{29} - 36$$ T29^2 + 6*T29 - 36

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 23T^{2} + 121$$
$11$ $$(T^{2} - 3 T - 9)^{2}$$
$13$ $$T^{4} + 35T^{2} + 25$$
$17$ $$T^{4} + 63T^{2} + 81$$
$19$ $$(T + 2)^{4}$$
$23$ $$T^{4} + 27T^{2} + 81$$
$29$ $$(T^{2} + 6 T - 36)^{2}$$
$31$ $$(T^{2} + 8 T - 29)^{2}$$
$37$ $$T^{4} + 35T^{2} + 25$$
$41$ $$(T^{2} - 3 T - 99)^{2}$$
$43$ $$T^{4} + 35T^{2} + 25$$
$47$ $$(T^{2} + 45)^{2}$$
$53$ $$T^{4} + 27T^{2} + 81$$
$59$ $$(T^{2} - 9 T + 9)^{2}$$
$61$ $$(T^{2} + 5 T - 5)^{2}$$
$67$ $$(T^{2} + 49)^{2}$$
$71$ $$(T + 9)^{4}$$
$73$ $$T^{4} + 122T^{2} + 841$$
$79$ $$(T^{2} + 10 T - 20)^{2}$$
$83$ $$(T^{2} + 45)^{2}$$
$89$ $$(T + 12)^{4}$$
$97$ $$(T^{2} + 100)^{2}$$