# Properties

 Label 1800.2.k.a Level $1800$ Weight $2$ Character orbit 1800.k Analytic conductor $14.373$ 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: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1800.k (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.3730723638$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 24) 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.

 $$f(q)$$ $$=$$ $$q + ( -1 - i ) q^{2} + 2 i q^{4} + 2 q^{7} + ( 2 - 2 i ) q^{8} +O(q^{10})$$ $$q + ( -1 - i ) q^{2} + 2 i q^{4} + 2 q^{7} + ( 2 - 2 i ) q^{8} + 4 i q^{13} + ( -2 - 2 i ) q^{14} -4 q^{16} -2 q^{17} + 4 i q^{19} + 4 q^{23} + ( 4 - 4 i ) q^{26} + 4 i q^{28} + 6 i q^{29} + 2 q^{31} + ( 4 + 4 i ) q^{32} + ( 2 + 2 i ) q^{34} -8 i q^{37} + ( 4 - 4 i ) q^{38} -2 q^{41} + 4 i q^{43} + ( -4 - 4 i ) q^{46} -12 q^{47} -3 q^{49} -8 q^{52} + 6 i q^{53} + ( 4 - 4 i ) q^{56} + ( 6 - 6 i ) q^{58} -4 i q^{59} + ( -2 - 2 i ) q^{62} -8 i q^{64} + 12 i q^{67} -4 i q^{68} -12 q^{71} + 6 q^{73} + ( -8 + 8 i ) q^{74} -8 q^{76} + 10 q^{79} + ( 2 + 2 i ) q^{82} + 16 i q^{83} + ( 4 - 4 i ) q^{86} + 10 q^{89} + 8 i q^{91} + 8 i q^{92} + ( 12 + 12 i ) q^{94} + 2 q^{97} + ( 3 + 3 i ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 4 q^{7} + 4 q^{8} + O(q^{10})$$ $$2 q - 2 q^{2} + 4 q^{7} + 4 q^{8} - 4 q^{14} - 8 q^{16} - 4 q^{17} + 8 q^{23} + 8 q^{26} + 4 q^{31} + 8 q^{32} + 4 q^{34} + 8 q^{38} - 4 q^{41} - 8 q^{46} - 24 q^{47} - 6 q^{49} - 16 q^{52} + 8 q^{56} + 12 q^{58} - 4 q^{62} - 24 q^{71} + 12 q^{73} - 16 q^{74} - 16 q^{76} + 20 q^{79} + 4 q^{82} + 8 q^{86} + 20 q^{89} + 24 q^{94} + 4 q^{97} + 6 q^{98} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
901.1
 1.00000i − 1.00000i
−1.00000 1.00000i 0 2.00000i 0 0 2.00000 2.00000 2.00000i 0 0
901.2 −1.00000 + 1.00000i 0 2.00000i 0 0 2.00000 2.00000 + 2.00000i 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
8.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1800.2.k.a 2
3.b odd 2 1 600.2.k.b 2
4.b odd 2 1 7200.2.k.d 2
5.b even 2 1 72.2.d.b 2
5.c odd 4 1 1800.2.d.b 2
5.c odd 4 1 1800.2.d.i 2
8.b even 2 1 inner 1800.2.k.a 2
8.d odd 2 1 7200.2.k.d 2
12.b even 2 1 2400.2.k.a 2
15.d odd 2 1 24.2.d.a 2
15.e even 4 1 600.2.d.b 2
15.e even 4 1 600.2.d.c 2
20.d odd 2 1 288.2.d.b 2
20.e even 4 1 7200.2.d.d 2
20.e even 4 1 7200.2.d.g 2
24.f even 2 1 2400.2.k.a 2
24.h odd 2 1 600.2.k.b 2
40.e odd 2 1 288.2.d.b 2
40.f even 2 1 72.2.d.b 2
40.i odd 4 1 1800.2.d.b 2
40.i odd 4 1 1800.2.d.i 2
40.k even 4 1 7200.2.d.d 2
40.k even 4 1 7200.2.d.g 2
45.h odd 6 2 648.2.n.k 4
45.j even 6 2 648.2.n.c 4
60.h even 2 1 96.2.d.a 2
60.l odd 4 1 2400.2.d.b 2
60.l odd 4 1 2400.2.d.c 2
80.k odd 4 1 2304.2.a.b 1
80.k odd 4 1 2304.2.a.l 1
80.q even 4 1 2304.2.a.e 1
80.q even 4 1 2304.2.a.o 1
105.g even 2 1 1176.2.c.a 2
120.i odd 2 1 24.2.d.a 2
120.m even 2 1 96.2.d.a 2
120.q odd 4 1 2400.2.d.b 2
120.q odd 4 1 2400.2.d.c 2
120.w even 4 1 600.2.d.b 2
120.w even 4 1 600.2.d.c 2
180.n even 6 2 2592.2.r.f 4
180.p odd 6 2 2592.2.r.g 4
240.t even 4 1 768.2.a.d 1
240.t even 4 1 768.2.a.e 1
240.bm odd 4 1 768.2.a.a 1
240.bm odd 4 1 768.2.a.h 1
360.z odd 6 2 2592.2.r.g 4
360.bd even 6 2 2592.2.r.f 4
360.bh odd 6 2 648.2.n.k 4
360.bk even 6 2 648.2.n.c 4
420.o odd 2 1 4704.2.c.a 2
840.b odd 2 1 4704.2.c.a 2
840.u even 2 1 1176.2.c.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.2.d.a 2 15.d odd 2 1
24.2.d.a 2 120.i odd 2 1
72.2.d.b 2 5.b even 2 1
72.2.d.b 2 40.f even 2 1
96.2.d.a 2 60.h even 2 1
96.2.d.a 2 120.m even 2 1
288.2.d.b 2 20.d odd 2 1
288.2.d.b 2 40.e odd 2 1
600.2.d.b 2 15.e even 4 1
600.2.d.b 2 120.w even 4 1
600.2.d.c 2 15.e even 4 1
600.2.d.c 2 120.w even 4 1
600.2.k.b 2 3.b odd 2 1
600.2.k.b 2 24.h odd 2 1
648.2.n.c 4 45.j even 6 2
648.2.n.c 4 360.bk even 6 2
648.2.n.k 4 45.h odd 6 2
648.2.n.k 4 360.bh odd 6 2
768.2.a.a 1 240.bm odd 4 1
768.2.a.d 1 240.t even 4 1
768.2.a.e 1 240.t even 4 1
768.2.a.h 1 240.bm odd 4 1
1176.2.c.a 2 105.g even 2 1
1176.2.c.a 2 840.u even 2 1
1800.2.d.b 2 5.c odd 4 1
1800.2.d.b 2 40.i odd 4 1
1800.2.d.i 2 5.c odd 4 1
1800.2.d.i 2 40.i odd 4 1
1800.2.k.a 2 1.a even 1 1 trivial
1800.2.k.a 2 8.b even 2 1 inner
2304.2.a.b 1 80.k odd 4 1
2304.2.a.e 1 80.q even 4 1
2304.2.a.l 1 80.k odd 4 1
2304.2.a.o 1 80.q even 4 1
2400.2.d.b 2 60.l odd 4 1
2400.2.d.b 2 120.q odd 4 1
2400.2.d.c 2 60.l odd 4 1
2400.2.d.c 2 120.q odd 4 1
2400.2.k.a 2 12.b even 2 1
2400.2.k.a 2 24.f even 2 1
2592.2.r.f 4 180.n even 6 2
2592.2.r.f 4 360.bd even 6 2
2592.2.r.g 4 180.p odd 6 2
2592.2.r.g 4 360.z odd 6 2
4704.2.c.a 2 420.o odd 2 1
4704.2.c.a 2 840.b odd 2 1
7200.2.d.d 2 20.e even 4 1
7200.2.d.d 2 40.k even 4 1
7200.2.d.g 2 20.e even 4 1
7200.2.d.g 2 40.k even 4 1
7200.2.k.d 2 4.b odd 2 1
7200.2.k.d 2 8.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1800, [\chi])$$:

 $$T_{7} - 2$$ $$T_{11}$$ $$T_{17} + 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$2 + 2 T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$( -2 + T )^{2}$$
$11$ $$T^{2}$$
$13$ $$16 + T^{2}$$
$17$ $$( 2 + T )^{2}$$
$19$ $$16 + T^{2}$$
$23$ $$( -4 + T )^{2}$$
$29$ $$36 + T^{2}$$
$31$ $$( -2 + T )^{2}$$
$37$ $$64 + T^{2}$$
$41$ $$( 2 + T )^{2}$$
$43$ $$16 + T^{2}$$
$47$ $$( 12 + T )^{2}$$
$53$ $$36 + T^{2}$$
$59$ $$16 + T^{2}$$
$61$ $$T^{2}$$
$67$ $$144 + T^{2}$$
$71$ $$( 12 + T )^{2}$$
$73$ $$( -6 + T )^{2}$$
$79$ $$( -10 + T )^{2}$$
$83$ $$256 + T^{2}$$
$89$ $$( -10 + T )^{2}$$
$97$ $$( -2 + T )^{2}$$