# Properties

 Label 3600.3.l.r Level $3600$ Weight $3$ Character orbit 3600.l Analytic conductor $98.093$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3600 = 2^{4} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 3600.l (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$98.0928951697$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-23})$$ Defining polynomial: $$x^{4} - 2x^{3} + 17x^{2} - 16x + 18$$ x^4 - 2*x^3 + 17*x^2 - 16*x + 18 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 180) 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 $$q - \beta_{2} q^{7} - 7 \beta_1 q^{11} - 3 \beta_{2} q^{13} - 2 \beta_{3} q^{17} + 12 q^{19} - \beta_{3} q^{23} + 6 \beta_1 q^{29} + 38 q^{31} + \beta_{2} q^{37} + 49 \beta_1 q^{41} - 10 \beta_{2} q^{43} - 8 \beta_{3} q^{47} - 3 q^{49} - 59 \beta_1 q^{59} - 70 q^{61} + 16 \beta_{2} q^{67} - 84 \beta_1 q^{71} - 2 \beta_{2} q^{73} - 7 \beta_{3} q^{77} + 30 q^{79} + 14 \beta_{3} q^{83} - 23 \beta_1 q^{89} + 138 q^{91} + 14 \beta_{2} q^{97}+O(q^{100})$$ q - b2 * q^7 - 7*b1 * q^11 - 3*b2 * q^13 - 2*b3 * q^17 + 12 * q^19 - b3 * q^23 + 6*b1 * q^29 + 38 * q^31 + b2 * q^37 + 49*b1 * q^41 - 10*b2 * q^43 - 8*b3 * q^47 - 3 * q^49 - 59*b1 * q^59 - 70 * q^61 + 16*b2 * q^67 - 84*b1 * q^71 - 2*b2 * q^73 - 7*b3 * q^77 + 30 * q^79 + 14*b3 * q^83 - 23*b1 * q^89 + 138 * q^91 + 14*b2 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q + 48 q^{19} + 152 q^{31} - 12 q^{49} - 280 q^{61} + 120 q^{79} + 552 q^{91}+O(q^{100})$$ 4 * q + 48 * q^19 + 152 * q^31 - 12 * q^49 - 280 * q^61 + 120 * q^79 + 552 * q^91

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2x^{3} + 17x^{2} - 16x + 18$$ :

 $$\beta_{1}$$ $$=$$ $$( -2\nu^{3} + 3\nu^{2} - 25\nu + 12 ) / 15$$ (-2*v^3 + 3*v^2 - 25*v + 12) / 15 $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu + 8$$ v^2 - v + 8 $$\beta_{3}$$ $$=$$ $$( 8\nu^{3} - 12\nu^{2} + 160\nu - 78 ) / 15$$ (8*v^3 - 12*v^2 + 160*v - 78) / 15
 $$\nu$$ $$=$$ $$( \beta_{3} + 4\beta _1 + 2 ) / 4$$ (b3 + 4*b1 + 2) / 4 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + 4\beta_{2} + 4\beta _1 - 30 ) / 4$$ (b3 + 4*b2 + 4*b1 - 30) / 4 $$\nu^{3}$$ $$=$$ $$( -11\beta_{3} + 6\beta_{2} - 74\beta _1 - 46 ) / 4$$ (-11*b3 + 6*b2 - 74*b1 - 46) / 4

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3600\mathbb{Z}\right)^\times$$.

 $$n$$ $$577$$ $$901$$ $$2801$$ $$3151$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1601.1
 0.5 − 0.983702i 0.5 + 0.983702i 0.5 + 3.81213i 0.5 − 3.81213i
0 0 0 0 0 −6.78233 0 0 0
1601.2 0 0 0 0 0 −6.78233 0 0 0
1601.3 0 0 0 0 0 6.78233 0 0 0
1601.4 0 0 0 0 0 6.78233 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
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3600.3.l.r 4
3.b odd 2 1 inner 3600.3.l.r 4
4.b odd 2 1 900.3.g.c 4
5.b even 2 1 inner 3600.3.l.r 4
5.c odd 4 2 720.3.c.b 4
12.b even 2 1 900.3.g.c 4
15.d odd 2 1 inner 3600.3.l.r 4
15.e even 4 2 720.3.c.b 4
20.d odd 2 1 900.3.g.c 4
20.e even 4 2 180.3.b.a 4
40.i odd 4 2 2880.3.c.f 4
40.k even 4 2 2880.3.c.c 4
60.h even 2 1 900.3.g.c 4
60.l odd 4 2 180.3.b.a 4
120.q odd 4 2 2880.3.c.c 4
120.w even 4 2 2880.3.c.f 4
180.v odd 12 4 1620.3.t.c 8
180.x even 12 4 1620.3.t.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.3.b.a 4 20.e even 4 2
180.3.b.a 4 60.l odd 4 2
720.3.c.b 4 5.c odd 4 2
720.3.c.b 4 15.e even 4 2
900.3.g.c 4 4.b odd 2 1
900.3.g.c 4 12.b even 2 1
900.3.g.c 4 20.d odd 2 1
900.3.g.c 4 60.h even 2 1
1620.3.t.c 8 180.v odd 12 4
1620.3.t.c 8 180.x even 12 4
2880.3.c.c 4 40.k even 4 2
2880.3.c.c 4 120.q odd 4 2
2880.3.c.f 4 40.i odd 4 2
2880.3.c.f 4 120.w even 4 2
3600.3.l.r 4 1.a even 1 1 trivial
3600.3.l.r 4 3.b odd 2 1 inner
3600.3.l.r 4 5.b even 2 1 inner
3600.3.l.r 4 15.d odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(3600, [\chi])$$:

 $$T_{7}^{2} - 46$$ T7^2 - 46 $$T_{11}^{2} + 98$$ T11^2 + 98 $$T_{13}^{2} - 414$$ T13^2 - 414

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$(T^{2} - 46)^{2}$$
$11$ $$(T^{2} + 98)^{2}$$
$13$ $$(T^{2} - 414)^{2}$$
$17$ $$(T^{2} + 368)^{2}$$
$19$ $$(T - 12)^{4}$$
$23$ $$(T^{2} + 92)^{2}$$
$29$ $$(T^{2} + 72)^{2}$$
$31$ $$(T - 38)^{4}$$
$37$ $$(T^{2} - 46)^{2}$$
$41$ $$(T^{2} + 4802)^{2}$$
$43$ $$(T^{2} - 4600)^{2}$$
$47$ $$(T^{2} + 5888)^{2}$$
$53$ $$T^{4}$$
$59$ $$(T^{2} + 6962)^{2}$$
$61$ $$(T + 70)^{4}$$
$67$ $$(T^{2} - 11776)^{2}$$
$71$ $$(T^{2} + 14112)^{2}$$
$73$ $$(T^{2} - 184)^{2}$$
$79$ $$(T - 30)^{4}$$
$83$ $$(T^{2} + 18032)^{2}$$
$89$ $$(T^{2} + 1058)^{2}$$
$97$ $$(T^{2} - 9016)^{2}$$