# Properties

 Label 1800.2.k.q Level $1800$ Weight $2$ Character orbit 1800.k Analytic conductor $14.373$ Analytic rank $0$ Dimension $8$ 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: $$8$$ Coefficient field: 8.0.214798336.3 Defining polynomial: $$x^{8} - 2 x^{7} - 2 x^{5} + 9 x^{4} - 4 x^{3} - 16 x + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 600) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{2} + ( -\beta_{4} - \beta_{5} + \beta_{6} ) q^{4} + ( -1 - \beta_{1} + \beta_{3} + \beta_{6} ) q^{7} + ( 1 + \beta_{2} - \beta_{6} + \beta_{7} ) q^{8} +O(q^{10})$$ $$q + \beta_{3} q^{2} + ( -\beta_{4} - \beta_{5} + \beta_{6} ) q^{4} + ( -1 - \beta_{1} + \beta_{3} + \beta_{6} ) q^{7} + ( 1 + \beta_{2} - \beta_{6} + \beta_{7} ) q^{8} + ( 1 + \beta_{3} - \beta_{4} + \beta_{5} ) q^{11} + ( 1 - \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{13} + ( \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{5} - \beta_{6} ) q^{14} + ( -\beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{16} + ( \beta_{1} + \beta_{4} - 2 \beta_{6} - \beta_{7} ) q^{17} + ( -1 - 2 \beta_{1} - 3 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{7} ) q^{19} + ( -2 - \beta_{1} - 3 \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{22} + ( 1 - 2 \beta_{1} + 3 \beta_{3} + 3 \beta_{4} + \beta_{5} - 2 \beta_{7} ) q^{23} + ( -2 \beta_{1} - 4 \beta_{2} + \beta_{3} - 2 \beta_{7} ) q^{26} + ( 2 - \beta_{1} + 3 \beta_{2} + 2 \beta_{3} + 2 \beta_{5} ) q^{28} + ( \beta_{1} + 4 \beta_{2} + 2 \beta_{3} - \beta_{4} + 2 \beta_{6} - \beta_{7} ) q^{29} + ( 1 + \beta_{1} + \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{31} + ( 4 + 2 \beta_{1} + 2 \beta_{7} ) q^{32} + ( 2 - \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} + 3 \beta_{6} ) q^{34} + ( 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} - 2 \beta_{7} ) q^{37} + ( 6 - \beta_{1} - 3 \beta_{2} + 3 \beta_{5} - 3 \beta_{6} ) q^{38} + ( -\beta_{1} + 2 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{41} + ( -1 - \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{43} + ( 2 - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{6} ) q^{44} + ( 2 + 3 \beta_{1} - 3 \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{46} + ( 4 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{47} + ( 4 \beta_{1} - 6 \beta_{3} - 2 \beta_{4} - 2 \beta_{6} + 2 \beta_{7} ) q^{49} + ( 4 - 4 \beta_{2} - 3 \beta_{4} + \beta_{5} - \beta_{6} ) q^{52} + ( 3 \beta_{1} + 2 \beta_{2} + 3 \beta_{4} + 3 \beta_{7} ) q^{53} + ( -3 - 5 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{7} ) q^{56} + ( -2 + \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + 4 \beta_{7} ) q^{58} + ( -4 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{4} - 4 \beta_{5} + 4 \beta_{6} - 2 \beta_{7} ) q^{59} + ( 1 - \beta_{1} - 5 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{61} + ( 4 - 3 \beta_{1} + 3 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} - \beta_{5} + 3 \beta_{6} ) q^{62} + ( -4 + 4 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} ) q^{64} + ( -3 + 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - 3 \beta_{5} + 4 \beta_{6} - 2 \beta_{7} ) q^{67} + ( -4 + 2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{68} + ( 5 - 2 \beta_{1} + 3 \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{71} + ( 2 - 2 \beta_{3} + 6 \beta_{4} + 4 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{73} + ( -4 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 4 \beta_{7} ) q^{74} + ( -2 - 3 \beta_{1} - 7 \beta_{2} + 6 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} - 4 \beta_{7} ) q^{76} + ( -4 - \beta_{1} - 4 \beta_{2} - 2 \beta_{3} + \beta_{4} - 4 \beta_{5} + 2 \beta_{6} - 3 \beta_{7} ) q^{77} + ( -2 - 2 \beta_{1} + 6 \beta_{3} + 4 \beta_{4} - 2 \beta_{6} - 4 \beta_{7} ) q^{79} + ( 2 - \beta_{1} + \beta_{2} - \beta_{4} - 3 \beta_{5} - \beta_{6} ) q^{82} + ( 2 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} - 2 \beta_{7} ) q^{83} + ( 2 + \beta_{1} + 7 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} - 4 \beta_{7} ) q^{86} + ( 2 - 4 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{7} ) q^{88} + ( -2 \beta_{1} + 6 \beta_{4} + 4 \beta_{5} - 2 \beta_{7} ) q^{89} + ( -3 - 5 \beta_{2} - \beta_{3} + \beta_{4} - 3 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{91} + ( -2 + 4 \beta_{2} + 2 \beta_{3} - 6 \beta_{4} + 2 \beta_{6} + 4 \beta_{7} ) q^{92} + ( 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{5} + 2 \beta_{6} ) q^{94} + ( 1 - 2 \beta_{1} + 6 \beta_{3} + 4 \beta_{4} - 2 \beta_{6} - 4 \beta_{7} ) q^{97} + ( -4 - 4 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 2 q^{2} + 4 q^{4} - 8 q^{7} + 4 q^{8} + O(q^{10})$$ $$8 q - 2 q^{2} + 4 q^{4} - 8 q^{7} + 4 q^{8} + 6 q^{14} + 8 q^{16} - 12 q^{22} + 8 q^{23} + 2 q^{26} + 4 q^{28} + 8 q^{31} + 28 q^{32} + 12 q^{34} + 30 q^{38} + 12 q^{44} + 20 q^{46} + 20 q^{52} - 8 q^{56} - 12 q^{58} + 30 q^{62} - 32 q^{64} - 28 q^{68} + 40 q^{71} + 16 q^{73} - 8 q^{74} - 20 q^{76} - 16 q^{79} + 24 q^{82} + 18 q^{86} + 8 q^{88} - 36 q^{92} - 4 q^{94} + 8 q^{97} - 48 q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2 x^{7} - 2 x^{5} + 9 x^{4} - 4 x^{3} - 16 x + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{7} + 2 \nu^{4} + 3 \nu^{3} - 6 \nu^{2} - 4 \nu$$$$)/8$$ $$\beta_{2}$$ $$=$$ $$($$$$-5 \nu^{7} + 2 \nu^{6} + 4 \nu^{5} + 18 \nu^{4} - 21 \nu^{3} - 12 \nu^{2} - 20 \nu + 56$$$$)/8$$ $$\beta_{3}$$ $$=$$ $$($$$$5 \nu^{7} - 4 \nu^{6} - 4 \nu^{5} - 18 \nu^{4} + 25 \nu^{3} + 10 \nu^{2} + 24 \nu - 64$$$$)/8$$ $$\beta_{4}$$ $$=$$ $$($$$$3 \nu^{7} - 2 \nu^{6} - 2 \nu^{5} - 10 \nu^{4} + 15 \nu^{3} + 8 \nu^{2} + 10 \nu - 36$$$$)/4$$ $$\beta_{5}$$ $$=$$ $$($$$$-3 \nu^{7} + 2 \nu^{6} + 4 \nu^{5} + 10 \nu^{4} - 15 \nu^{3} - 12 \nu^{2} - 8 \nu + 36$$$$)/4$$ $$\beta_{6}$$ $$=$$ $$($$$$-4 \nu^{7} + 3 \nu^{6} + 4 \nu^{5} + 12 \nu^{4} - 18 \nu^{3} - 9 \nu^{2} - 10 \nu + 44$$$$)/4$$ $$\beta_{7}$$ $$=$$ $$($$$$7 \nu^{7} - 4 \nu^{6} - 6 \nu^{5} - 22 \nu^{4} + 31 \nu^{3} + 18 \nu^{2} + 26 \nu - 80$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7} + \beta_{6} - \beta_{4} + \beta_{3} + \beta_{2} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{6} - \beta_{5} + \beta_{4} + \beta_{2} - \beta_{1}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{7} + \beta_{6} + \beta_{4} - \beta_{3} + \beta_{2} + 2 \beta_{1} + 3$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$4 \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} - 2 \beta_{3} + 5 \beta_{2} + \beta_{1}$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-\beta_{7} + \beta_{6} + 2 \beta_{5} + 7 \beta_{4} - \beta_{3} + \beta_{2} - 2 \beta_{1} - 1$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$4 \beta_{7} + 3 \beta_{6} + \beta_{5} - \beta_{4} - 8 \beta_{3} - 5 \beta_{2} + 5 \beta_{1}$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$7 \beta_{7} - 5 \beta_{6} + 8 \beta_{5} - \beta_{4} - 11 \beta_{3} + 3 \beta_{2} - 2 \beta_{1} + 5$$$$)/2$$

## 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
 1.41216 − 0.0762223i 1.41216 + 0.0762223i −0.565036 − 1.29643i −0.565036 + 1.29643i −1.08003 − 0.912978i −1.08003 + 0.912978i 1.23291 − 0.692769i 1.23291 + 0.692769i
−1.29150 0.576222i 0 1.33594 + 1.48838i 0 0 1.97676 −0.867721 2.69204i 0 0
901.2 −1.29150 + 0.576222i 0 1.33594 1.48838i 0 0 1.97676 −0.867721 + 2.69204i 0 0
901.3 −1.16863 0.796431i 0 0.731395 + 1.86147i 0 0 −4.72294 0.627801 2.75787i 0 0
901.4 −1.16863 + 0.796431i 0 0.731395 1.86147i 0 0 −4.72294 0.627801 + 2.75787i 0 0
901.5 0.0591148 1.41298i 0 −1.99301 0.167056i 0 0 −1.33411 −0.353863 + 2.80620i 0 0
901.6 0.0591148 + 1.41298i 0 −1.99301 + 0.167056i 0 0 −1.33411 −0.353863 2.80620i 0 0
901.7 1.40101 0.192769i 0 1.92568 0.540143i 0 0 0.0802864 2.59378 1.12796i 0 0
901.8 1.40101 + 0.192769i 0 1.92568 + 0.540143i 0 0 0.0802864 2.59378 + 1.12796i 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 901.8 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.q 8
3.b odd 2 1 600.2.k.e yes 8
4.b odd 2 1 7200.2.k.s 8
5.b even 2 1 1800.2.k.t 8
5.c odd 4 1 1800.2.d.s 8
5.c odd 4 1 1800.2.d.t 8
8.b even 2 1 inner 1800.2.k.q 8
8.d odd 2 1 7200.2.k.s 8
12.b even 2 1 2400.2.k.e 8
15.d odd 2 1 600.2.k.d 8
15.e even 4 1 600.2.d.g 8
15.e even 4 1 600.2.d.h 8
20.d odd 2 1 7200.2.k.r 8
20.e even 4 1 7200.2.d.s 8
20.e even 4 1 7200.2.d.t 8
24.f even 2 1 2400.2.k.e 8
24.h odd 2 1 600.2.k.e yes 8
40.e odd 2 1 7200.2.k.r 8
40.f even 2 1 1800.2.k.t 8
40.i odd 4 1 1800.2.d.s 8
40.i odd 4 1 1800.2.d.t 8
40.k even 4 1 7200.2.d.s 8
40.k even 4 1 7200.2.d.t 8
60.h even 2 1 2400.2.k.d 8
60.l odd 4 1 2400.2.d.g 8
60.l odd 4 1 2400.2.d.h 8
120.i odd 2 1 600.2.k.d 8
120.m even 2 1 2400.2.k.d 8
120.q odd 4 1 2400.2.d.g 8
120.q odd 4 1 2400.2.d.h 8
120.w even 4 1 600.2.d.g 8
120.w even 4 1 600.2.d.h 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
600.2.d.g 8 15.e even 4 1
600.2.d.g 8 120.w even 4 1
600.2.d.h 8 15.e even 4 1
600.2.d.h 8 120.w even 4 1
600.2.k.d 8 15.d odd 2 1
600.2.k.d 8 120.i odd 2 1
600.2.k.e yes 8 3.b odd 2 1
600.2.k.e yes 8 24.h odd 2 1
1800.2.d.s 8 5.c odd 4 1
1800.2.d.s 8 40.i odd 4 1
1800.2.d.t 8 5.c odd 4 1
1800.2.d.t 8 40.i odd 4 1
1800.2.k.q 8 1.a even 1 1 trivial
1800.2.k.q 8 8.b even 2 1 inner
1800.2.k.t 8 5.b even 2 1
1800.2.k.t 8 40.f even 2 1
2400.2.d.g 8 60.l odd 4 1
2400.2.d.g 8 120.q odd 4 1
2400.2.d.h 8 60.l odd 4 1
2400.2.d.h 8 120.q odd 4 1
2400.2.k.d 8 60.h even 2 1
2400.2.k.d 8 120.m even 2 1
2400.2.k.e 8 12.b even 2 1
2400.2.k.e 8 24.f even 2 1
7200.2.d.s 8 20.e even 4 1
7200.2.d.s 8 40.k even 4 1
7200.2.d.t 8 20.e even 4 1
7200.2.d.t 8 40.k even 4 1
7200.2.k.r 8 20.d odd 2 1
7200.2.k.r 8 40.e odd 2 1
7200.2.k.s 8 4.b odd 2 1
7200.2.k.s 8 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 $$S_{2}^{\mathrm{new}}(1800, [\chi])$$:

 $$T_{7}^{4} + 4 T_{7}^{3} - 6 T_{7}^{2} - 12 T_{7} + 1$$ $$T_{11}^{8} + 32 T_{11}^{6} + 336 T_{11}^{4} + 1344 T_{11}^{2} + 1600$$ $$T_{17}^{4} - 40 T_{17}^{2} + 104 T_{17} - 24$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$16 + 16 T - 8 T^{3} - 6 T^{4} - 4 T^{5} + 2 T^{7} + T^{8}$$
$3$ $$T^{8}$$
$5$ $$T^{8}$$
$7$ $$( 1 - 12 T - 6 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$11$ $$1600 + 1344 T^{2} + 336 T^{4} + 32 T^{6} + T^{8}$$
$13$ $$81 + 1420 T^{2} + 502 T^{4} + 44 T^{6} + T^{8}$$
$17$ $$( -24 + 104 T - 40 T^{2} + T^{4} )^{2}$$
$19$ $$380689 + 75156 T^{2} + 4662 T^{4} + 116 T^{6} + T^{8}$$
$23$ $$( -88 + 152 T - 56 T^{2} - 4 T^{3} + T^{4} )^{2}$$
$29$ $$627264 + 107584 T^{2} + 6288 T^{4} + 144 T^{6} + T^{8}$$
$31$ $$( 673 + 204 T - 70 T^{2} - 4 T^{3} + T^{4} )^{2}$$
$37$ $$4096 + 24576 T^{2} + 4224 T^{4} + 128 T^{6} + T^{8}$$
$41$ $$( 328 - 56 T - 64 T^{2} + T^{4} )^{2}$$
$43$ $$4363921 + 508692 T^{2} + 18614 T^{4} + 244 T^{6} + T^{8}$$
$47$ $$( -176 - 256 T - 72 T^{2} + T^{4} )^{2}$$
$53$ $$23104 + 425920 T^{2} + 19536 T^{4} + 256 T^{6} + T^{8}$$
$59$ $$31181056 + 2477824 T^{2} + 57824 T^{4} + 432 T^{6} + T^{8}$$
$61$ $$3025 + 216396 T^{2} + 14838 T^{4} + 236 T^{6} + T^{8}$$
$67$ $$25979409 + 1957780 T^{2} + 44598 T^{4} + 372 T^{6} + T^{8}$$
$71$ $$( -536 + 72 T + 96 T^{2} - 20 T^{3} + T^{4} )^{2}$$
$73$ $$( -432 + 864 T - 168 T^{2} - 8 T^{3} + T^{4} )^{2}$$
$79$ $$( 8080 - 864 T - 184 T^{2} + 8 T^{3} + T^{4} )^{2}$$
$83$ $$3873024 + 1762048 T^{2} + 44256 T^{4} + 368 T^{6} + T^{8}$$
$89$ $$( 10880 - 64 T - 224 T^{2} + T^{4} )^{2}$$
$97$ $$( 8881 + 348 T - 202 T^{2} - 4 T^{3} + T^{4} )^{2}$$