# Properties

 Label 1800.2.b.f Level $1800$ Weight $2$ Character orbit 1800.b Analytic conductor $14.373$ Analytic rank $0$ Dimension $8$ CM discriminant -40 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1800 = 2^{3} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1800.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.3730723638$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.40960000.1 Defining polynomial: $$x^{8} + 7 x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{10}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 360) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + 2 q^{4} -\beta_{5} q^{7} + 2 \beta_{2} q^{8} +O(q^{10})$$ $$q + \beta_{2} q^{2} + 2 q^{4} -\beta_{5} q^{7} + 2 \beta_{2} q^{8} -\beta_{3} q^{11} -\beta_{4} q^{13} + ( \beta_{3} + \beta_{6} ) q^{14} + 4 q^{16} + \beta_{7} q^{19} + ( -\beta_{4} + \beta_{5} ) q^{22} + \beta_{1} q^{23} + ( -\beta_{3} + \beta_{6} ) q^{26} -2 \beta_{5} q^{28} + 4 \beta_{2} q^{32} + ( -\beta_{4} - 2 \beta_{5} ) q^{37} -2 \beta_{1} q^{38} + ( 2 \beta_{3} + \beta_{6} ) q^{41} -2 \beta_{3} q^{44} -\beta_{7} q^{46} + 2 \beta_{2} q^{47} + ( -7 - 2 \beta_{7} ) q^{49} -2 \beta_{4} q^{52} + 4 \beta_{2} q^{53} + ( 2 \beta_{3} + 2 \beta_{6} ) q^{56} + ( \beta_{3} - 2 \beta_{6} ) q^{59} + 8 q^{64} + ( \beta_{3} + 3 \beta_{6} ) q^{74} + 2 \beta_{7} q^{76} + ( -\beta_{1} + 8 \beta_{2} ) q^{77} + ( \beta_{4} - 3 \beta_{5} ) q^{82} + ( -2 \beta_{4} + 2 \beta_{5} ) q^{88} + ( -2 \beta_{3} - 3 \beta_{6} ) q^{89} + ( 2 - \beta_{7} ) q^{91} + 2 \beta_{1} q^{92} + 4 q^{94} + ( 4 \beta_{1} - 7 \beta_{2} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 16q^{4} + O(q^{10})$$ $$8q + 16q^{4} + 32q^{16} - 56q^{49} + 64q^{64} + 16q^{91} + 32q^{94} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 7 x^{4} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$4 \nu^{4} + 14$$$$)/3$$ $$\beta_{2}$$ $$=$$ $$($$$$2 \nu^{7} - \nu^{5} + 13 \nu^{3} - 5 \nu$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$($$$$2 \nu^{7} - 6 \nu^{6} + \nu^{5} + 13 \nu^{3} - 36 \nu^{2} + 5 \nu$$$$)/3$$ $$\beta_{4}$$ $$=$$ $$($$$$4 \nu^{7} - 4 \nu^{6} + \nu^{5} + 29 \nu^{3} - 32 \nu^{2} + 11 \nu$$$$)/3$$ $$\beta_{5}$$ $$=$$ $$($$$$-4 \nu^{7} - 2 \nu^{6} - \nu^{5} - 29 \nu^{3} - 16 \nu^{2} - 11 \nu$$$$)/3$$ $$\beta_{6}$$ $$=$$ $$-2 \nu^{7} - \nu^{5} - 13 \nu^{3} - 5 \nu$$ $$\beta_{7}$$ $$=$$ $$($$$$-8 \nu^{7} + 2 \nu^{5} - 58 \nu^{3} + 22 \nu$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$3 \beta_{7} + 2 \beta_{6} - 4 \beta_{5} + 2 \beta_{4} + 6 \beta_{2}$$$$)/24$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{6} - 3 \beta_{5} - 3 \beta_{4} + 3 \beta_{3}$$$$)/12$$ $$\nu^{3}$$ $$=$$ $$($$$$-3 \beta_{7} + 4 \beta_{6} - 4 \beta_{5} + 2 \beta_{4} - 12 \beta_{2}$$$$)/12$$ $$\nu^{4}$$ $$=$$ $$($$$$3 \beta_{1} - 14$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$-15 \beta_{7} - 22 \beta_{6} + 20 \beta_{5} - 10 \beta_{4} - 66 \beta_{2}$$$$)/24$$ $$\nu^{6}$$ $$=$$ $$($$$$-4 \beta_{6} + 9 \beta_{5} + 9 \beta_{4} - 12 \beta_{3}$$$$)/6$$ $$\nu^{7}$$ $$=$$ $$($$$$39 \beta_{7} - 58 \beta_{6} + 52 \beta_{5} - 26 \beta_{4} + 174 \beta_{2}$$$$)/24$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
251.1
 0.437016 − 0.437016i −1.14412 − 1.14412i −1.14412 + 1.14412i 0.437016 + 0.437016i 1.14412 − 1.14412i −0.437016 − 0.437016i −0.437016 + 0.437016i 1.14412 + 1.14412i
−1.41421 0 2.00000 0 0 5.16228i −2.82843 0 0
251.2 −1.41421 0 2.00000 0 0 1.16228i −2.82843 0 0
251.3 −1.41421 0 2.00000 0 0 1.16228i −2.82843 0 0
251.4 −1.41421 0 2.00000 0 0 5.16228i −2.82843 0 0
251.5 1.41421 0 2.00000 0 0 5.16228i 2.82843 0 0
251.6 1.41421 0 2.00000 0 0 1.16228i 2.82843 0 0
251.7 1.41421 0 2.00000 0 0 1.16228i 2.82843 0 0
251.8 1.41421 0 2.00000 0 0 5.16228i 2.82843 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 251.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
40.e odd 2 1 CM by $$\Q(\sqrt{-10})$$
3.b odd 2 1 inner
5.b even 2 1 inner
8.d odd 2 1 inner
15.d odd 2 1 inner
24.f even 2 1 inner
120.m 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.b.f 8
3.b odd 2 1 inner 1800.2.b.f 8
4.b odd 2 1 7200.2.b.f 8
5.b even 2 1 inner 1800.2.b.f 8
5.c odd 4 1 360.2.m.a 4
5.c odd 4 1 360.2.m.b yes 4
8.b even 2 1 7200.2.b.f 8
8.d odd 2 1 inner 1800.2.b.f 8
12.b even 2 1 7200.2.b.f 8
15.d odd 2 1 inner 1800.2.b.f 8
15.e even 4 1 360.2.m.a 4
15.e even 4 1 360.2.m.b yes 4
20.d odd 2 1 7200.2.b.f 8
20.e even 4 1 1440.2.m.a 4
20.e even 4 1 1440.2.m.b 4
24.f even 2 1 inner 1800.2.b.f 8
24.h odd 2 1 7200.2.b.f 8
40.e odd 2 1 CM 1800.2.b.f 8
40.f even 2 1 7200.2.b.f 8
40.i odd 4 1 1440.2.m.a 4
40.i odd 4 1 1440.2.m.b 4
40.k even 4 1 360.2.m.a 4
40.k even 4 1 360.2.m.b yes 4
60.h even 2 1 7200.2.b.f 8
60.l odd 4 1 1440.2.m.a 4
60.l odd 4 1 1440.2.m.b 4
120.i odd 2 1 7200.2.b.f 8
120.m even 2 1 inner 1800.2.b.f 8
120.q odd 4 1 360.2.m.a 4
120.q odd 4 1 360.2.m.b yes 4
120.w even 4 1 1440.2.m.a 4
120.w even 4 1 1440.2.m.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.m.a 4 5.c odd 4 1
360.2.m.a 4 15.e even 4 1
360.2.m.a 4 40.k even 4 1
360.2.m.a 4 120.q odd 4 1
360.2.m.b yes 4 5.c odd 4 1
360.2.m.b yes 4 15.e even 4 1
360.2.m.b yes 4 40.k even 4 1
360.2.m.b yes 4 120.q odd 4 1
1440.2.m.a 4 20.e even 4 1
1440.2.m.a 4 40.i odd 4 1
1440.2.m.a 4 60.l odd 4 1
1440.2.m.a 4 120.w even 4 1
1440.2.m.b 4 20.e even 4 1
1440.2.m.b 4 40.i odd 4 1
1440.2.m.b 4 60.l odd 4 1
1440.2.m.b 4 120.w even 4 1
1800.2.b.f 8 1.a even 1 1 trivial
1800.2.b.f 8 3.b odd 2 1 inner
1800.2.b.f 8 5.b even 2 1 inner
1800.2.b.f 8 8.d odd 2 1 inner
1800.2.b.f 8 15.d odd 2 1 inner
1800.2.b.f 8 24.f even 2 1 inner
1800.2.b.f 8 40.e odd 2 1 CM
1800.2.b.f 8 120.m even 2 1 inner
7200.2.b.f 8 4.b odd 2 1
7200.2.b.f 8 8.b even 2 1
7200.2.b.f 8 12.b even 2 1
7200.2.b.f 8 20.d odd 2 1
7200.2.b.f 8 24.h odd 2 1
7200.2.b.f 8 40.f even 2 1
7200.2.b.f 8 60.h even 2 1
7200.2.b.f 8 120.i odd 2 1

## Hecke kernels

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

 $$T_{7}^{4} + 28 T_{7}^{2} + 36$$ $$T_{23}^{2} - 20$$ $$T_{43}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -2 + T^{2} )^{4}$$
$3$ $$T^{8}$$
$5$ $$T^{8}$$
$7$ $$( 36 + 28 T^{2} + T^{4} )^{2}$$
$11$ $$( 324 + 44 T^{2} + T^{4} )^{2}$$
$13$ $$( 36 + 52 T^{2} + T^{4} )^{2}$$
$17$ $$T^{8}$$
$19$ $$( -40 + T^{2} )^{4}$$
$23$ $$( -20 + T^{2} )^{4}$$
$29$ $$T^{8}$$
$31$ $$T^{8}$$
$37$ $$( 2916 + 148 T^{2} + T^{4} )^{2}$$
$41$ $$( 6084 + 164 T^{2} + T^{4} )^{2}$$
$43$ $$T^{8}$$
$47$ $$( -8 + T^{2} )^{4}$$
$53$ $$( -32 + T^{2} )^{4}$$
$59$ $$( 6084 + 236 T^{2} + T^{4} )^{2}$$
$61$ $$T^{8}$$
$67$ $$T^{8}$$
$71$ $$T^{8}$$
$73$ $$T^{8}$$
$79$ $$T^{8}$$
$83$ $$T^{8}$$
$89$ $$( 324 + 356 T^{2} + T^{4} )^{2}$$
$97$ $$T^{8}$$