# Properties

 Label 1800.4.f.s Level $1800$ Weight $4$ Character orbit 1800.f Analytic conductor $106.203$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1800 = 2^{3} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1800.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$106.203438010$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{37}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 360) 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.

 $$f(q)$$ $$=$$ $$q + 9 \beta q^{7}+O(q^{10})$$ q + 9*b * q^7 $$q + 9 \beta q^{7} + 34 q^{11} + 6 \beta q^{13} + 51 \beta q^{17} - 164 q^{19} + 24 \beta q^{23} - 146 q^{29} + 100 q^{31} - 164 \beta q^{37} - 288 q^{41} + 60 \beta q^{43} - 8 \beta q^{47} + 19 q^{49} - 63 \beta q^{53} - 642 q^{59} + 602 q^{61} - 218 \beta q^{67} + 652 q^{71} + 531 \beta q^{73} + 306 \beta q^{77} - 388 q^{79} - 222 \beta q^{83} + 820 q^{89} - 216 q^{91} + 383 \beta q^{97} +O(q^{100})$$ q + 9*b * q^7 + 34 * q^11 + 6*b * q^13 + 51*b * q^17 - 164 * q^19 + 24*b * q^23 - 146 * q^29 + 100 * q^31 - 164*b * q^37 - 288 * q^41 + 60*b * q^43 - 8*b * q^47 + 19 * q^49 - 63*b * q^53 - 642 * q^59 + 602 * q^61 - 218*b * q^67 + 652 * q^71 + 531*b * q^73 + 306*b * q^77 - 388 * q^79 - 222*b * q^83 + 820 * q^89 - 216 * q^91 + 383*b * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q+O(q^{10})$$ 2 * q $$2 q + 68 q^{11} - 328 q^{19} - 292 q^{29} + 200 q^{31} - 576 q^{41} + 38 q^{49} - 1284 q^{59} + 1204 q^{61} + 1304 q^{71} - 776 q^{79} + 1640 q^{89} - 432 q^{91}+O(q^{100})$$ 2 * q + 68 * q^11 - 328 * q^19 - 292 * q^29 + 200 * q^31 - 576 * q^41 + 38 * q^49 - 1284 * q^59 + 1204 * q^61 + 1304 * q^71 - 776 * q^79 + 1640 * q^89 - 432 * q^91

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times$$.

 $$n$$ $$577$$ $$901$$ $$1001$$ $$1351$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
649.1
 − 1.00000i 1.00000i
0 0 0 0 0 18.0000i 0 0 0
649.2 0 0 0 0 0 18.0000i 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 1800.4.f.s 2
3.b odd 2 1 1800.4.f.e 2
5.b even 2 1 inner 1800.4.f.s 2
5.c odd 4 1 360.4.a.j yes 1
5.c odd 4 1 1800.4.a.be 1
15.d odd 2 1 1800.4.f.e 2
15.e even 4 1 360.4.a.a 1
15.e even 4 1 1800.4.a.bc 1
20.e even 4 1 720.4.a.z 1
60.l odd 4 1 720.4.a.m 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.4.a.a 1 15.e even 4 1
360.4.a.j yes 1 5.c odd 4 1
720.4.a.m 1 60.l odd 4 1
720.4.a.z 1 20.e even 4 1
1800.4.a.bc 1 15.e even 4 1
1800.4.a.be 1 5.c odd 4 1
1800.4.f.e 2 3.b odd 2 1
1800.4.f.e 2 15.d odd 2 1
1800.4.f.s 2 1.a even 1 1 trivial
1800.4.f.s 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 $$S_{4}^{\mathrm{new}}(1800, [\chi])$$:

 $$T_{7}^{2} + 324$$ T7^2 + 324 $$T_{11} - 34$$ T11 - 34

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 324$$
$11$ $$(T - 34)^{2}$$
$13$ $$T^{2} + 144$$
$17$ $$T^{2} + 10404$$
$19$ $$(T + 164)^{2}$$
$23$ $$T^{2} + 2304$$
$29$ $$(T + 146)^{2}$$
$31$ $$(T - 100)^{2}$$
$37$ $$T^{2} + 107584$$
$41$ $$(T + 288)^{2}$$
$43$ $$T^{2} + 14400$$
$47$ $$T^{2} + 256$$
$53$ $$T^{2} + 15876$$
$59$ $$(T + 642)^{2}$$
$61$ $$(T - 602)^{2}$$
$67$ $$T^{2} + 190096$$
$71$ $$(T - 652)^{2}$$
$73$ $$T^{2} + 1127844$$
$79$ $$(T + 388)^{2}$$
$83$ $$T^{2} + 197136$$
$89$ $$(T - 820)^{2}$$
$97$ $$T^{2} + 586756$$