# Properties

 Label 1800.4.f.r 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: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 120) 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.

 $$f(q)$$ $$=$$ $$q + 8 \beta q^{7}+O(q^{10})$$ q + 8*b * q^7 $$q + 8 \beta q^{7} + 28 q^{11} - 13 \beta q^{13} - 31 \beta q^{17} + 68 q^{19} + 104 \beta q^{23} - 58 q^{29} + 160 q^{31} - 135 \beta q^{37} - 282 q^{41} + 38 \beta q^{43} - 140 \beta q^{47} + 87 q^{49} + 105 \beta q^{53} + 196 q^{59} + 742 q^{61} - 418 \beta q^{67} + 504 q^{71} - 531 \beta q^{73} + 224 \beta q^{77} - 768 q^{79} + 526 \beta q^{83} - 726 q^{89} + 416 q^{91} + 703 \beta q^{97} +O(q^{100})$$ q + 8*b * q^7 + 28 * q^11 - 13*b * q^13 - 31*b * q^17 + 68 * q^19 + 104*b * q^23 - 58 * q^29 + 160 * q^31 - 135*b * q^37 - 282 * q^41 + 38*b * q^43 - 140*b * q^47 + 87 * q^49 + 105*b * q^53 + 196 * q^59 + 742 * q^61 - 418*b * q^67 + 504 * q^71 - 531*b * q^73 + 224*b * q^77 - 768 * q^79 + 526*b * q^83 - 726 * q^89 + 416 * q^91 + 703*b * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q+O(q^{10})$$ 2 * q $$2 q + 56 q^{11} + 136 q^{19} - 116 q^{29} + 320 q^{31} - 564 q^{41} + 174 q^{49} + 392 q^{59} + 1484 q^{61} + 1008 q^{71} - 1536 q^{79} - 1452 q^{89} + 832 q^{91}+O(q^{100})$$ 2 * q + 56 * q^11 + 136 * q^19 - 116 * q^29 + 320 * q^31 - 564 * q^41 + 174 * q^49 + 392 * q^59 + 1484 * q^61 + 1008 * q^71 - 1536 * q^79 - 1452 * q^89 + 832 * 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 16.0000i 0 0 0
649.2 0 0 0 0 0 16.0000i 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.r 2
3.b odd 2 1 600.4.f.c 2
5.b even 2 1 inner 1800.4.f.r 2
5.c odd 4 1 360.4.a.b 1
5.c odd 4 1 1800.4.a.bb 1
12.b even 2 1 1200.4.f.o 2
15.d odd 2 1 600.4.f.c 2
15.e even 4 1 120.4.a.c 1
15.e even 4 1 600.4.a.q 1
20.e even 4 1 720.4.a.l 1
60.h even 2 1 1200.4.f.o 2
60.l odd 4 1 240.4.a.l 1
60.l odd 4 1 1200.4.a.c 1
120.q odd 4 1 960.4.a.h 1
120.w even 4 1 960.4.a.u 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.4.a.c 1 15.e even 4 1
240.4.a.l 1 60.l odd 4 1
360.4.a.b 1 5.c odd 4 1
600.4.a.q 1 15.e even 4 1
600.4.f.c 2 3.b odd 2 1
600.4.f.c 2 15.d odd 2 1
720.4.a.l 1 20.e even 4 1
960.4.a.h 1 120.q odd 4 1
960.4.a.u 1 120.w even 4 1
1200.4.a.c 1 60.l odd 4 1
1200.4.f.o 2 12.b even 2 1
1200.4.f.o 2 60.h even 2 1
1800.4.a.bb 1 5.c odd 4 1
1800.4.f.r 2 1.a even 1 1 trivial
1800.4.f.r 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 $$S_{4}^{\mathrm{new}}(1800, [\chi])$$:

 $$T_{7}^{2} + 256$$ T7^2 + 256 $$T_{11} - 28$$ T11 - 28

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 256$$
$11$ $$(T - 28)^{2}$$
$13$ $$T^{2} + 676$$
$17$ $$T^{2} + 3844$$
$19$ $$(T - 68)^{2}$$
$23$ $$T^{2} + 43264$$
$29$ $$(T + 58)^{2}$$
$31$ $$(T - 160)^{2}$$
$37$ $$T^{2} + 72900$$
$41$ $$(T + 282)^{2}$$
$43$ $$T^{2} + 5776$$
$47$ $$T^{2} + 78400$$
$53$ $$T^{2} + 44100$$
$59$ $$(T - 196)^{2}$$
$61$ $$(T - 742)^{2}$$
$67$ $$T^{2} + 698896$$
$71$ $$(T - 504)^{2}$$
$73$ $$T^{2} + 1127844$$
$79$ $$(T + 768)^{2}$$
$83$ $$T^{2} + 1106704$$
$89$ $$(T + 726)^{2}$$
$97$ $$T^{2} + 1976836$$