# Properties

 Label 7200.2.b.f Level $7200$ Weight $2$ Character orbit 7200.b Analytic conductor $57.492$ Analytic rank $0$ Dimension $8$ CM discriminant -40 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$7200 = 2^{5} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7200.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$57.4922894553$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.40960000.1 Defining polynomial: $$x^{8} + 7 x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{41}]$$ Coefficient ring index: $$2^{11}\cdot 3^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q -\beta_{4} q^{7} +O(q^{10})$$ $$q -\beta_{4} q^{7} + \beta_{2} q^{11} + \beta_{3} q^{13} + \beta_{7} q^{19} + \beta_{1} q^{23} + ( \beta_{3} + 2 \beta_{4} ) q^{37} + ( 2 \beta_{2} - \beta_{6} ) q^{41} + \beta_{5} q^{47} + ( -7 + 2 \beta_{7} ) q^{49} -2 \beta_{5} q^{53} + ( -\beta_{2} - 2 \beta_{6} ) q^{59} + ( \beta_{1} - 4 \beta_{5} ) q^{77} + ( -2 \beta_{2} + 3 \beta_{6} ) q^{89} + ( -2 - \beta_{7} ) q^{91} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q - 56q^{49} - 16q^{91} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$4 \nu^{4} + 14$$$$)/3$$ $$\beta_{2}$$ $$=$$ $$($$$$2 \nu^{7} - 6 \nu^{6} + \nu^{5} + 13 \nu^{3} - 36 \nu^{2} + 5 \nu$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$($$$$4 \nu^{7} - 4 \nu^{6} + \nu^{5} + 29 \nu^{3} - 32 \nu^{2} + 11 \nu$$$$)/3$$ $$\beta_{4}$$ $$=$$ $$($$$$-4 \nu^{7} - 2 \nu^{6} - \nu^{5} - 29 \nu^{3} - 16 \nu^{2} - 11 \nu$$$$)/3$$ $$\beta_{5}$$ $$=$$ $$($$$$4 \nu^{7} - 2 \nu^{5} + 26 \nu^{3} - 10 \nu$$$$)/3$$ $$\beta_{6}$$ $$=$$ $$2 \nu^{7} + \nu^{5} + 13 \nu^{3} + 5 \nu$$ $$\beta_{7}$$ $$=$$ $$($$$$8 \nu^{7} - 2 \nu^{5} + 58 \nu^{3} - 22 \nu$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-3 \beta_{7} - 2 \beta_{6} + 3 \beta_{5} - 4 \beta_{4} + 2 \beta_{3}$$$$)/24$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{6} - 3 \beta_{4} - 3 \beta_{3} + 3 \beta_{2}$$$$)/12$$ $$\nu^{3}$$ $$=$$ $$($$$$3 \beta_{7} - 4 \beta_{6} - 6 \beta_{5} - 4 \beta_{4} + 2 \beta_{3}$$$$)/12$$ $$\nu^{4}$$ $$=$$ $$($$$$3 \beta_{1} - 14$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$15 \beta_{7} + 22 \beta_{6} - 33 \beta_{5} + 20 \beta_{4} - 10 \beta_{3}$$$$)/24$$ $$\nu^{6}$$ $$=$$ $$($$$$4 \beta_{6} + 9 \beta_{4} + 9 \beta_{3} - 12 \beta_{2}$$$$)/6$$ $$\nu^{7}$$ $$=$$ $$($$$$-39 \beta_{7} + 58 \beta_{6} + 87 \beta_{5} + 52 \beta_{4} - 26 \beta_{3}$$$$)/24$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/7200\mathbb{Z}\right)^\times$$.

 $$n$$ $$577$$ $$901$$ $$6401$$ $$6751$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4751.1
 1.14412 − 1.14412i 0.437016 − 0.437016i −0.437016 − 0.437016i −1.14412 − 1.14412i −1.14412 + 1.14412i −0.437016 + 0.437016i 0.437016 + 0.437016i 1.14412 + 1.14412i
0 0 0 0 0 5.16228i 0 0 0
4751.2 0 0 0 0 0 5.16228i 0 0 0
4751.3 0 0 0 0 0 1.16228i 0 0 0
4751.4 0 0 0 0 0 1.16228i 0 0 0
4751.5 0 0 0 0 0 1.16228i 0 0 0
4751.6 0 0 0 0 0 1.16228i 0 0 0
4751.7 0 0 0 0 0 5.16228i 0 0 0
4751.8 0 0 0 0 0 5.16228i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 4751.8 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 7200.2.b.f 8
3.b odd 2 1 inner 7200.2.b.f 8
4.b odd 2 1 1800.2.b.f 8
5.b even 2 1 inner 7200.2.b.f 8
5.c odd 4 1 1440.2.m.a 4
5.c odd 4 1 1440.2.m.b 4
8.b even 2 1 1800.2.b.f 8
8.d odd 2 1 inner 7200.2.b.f 8
12.b even 2 1 1800.2.b.f 8
15.d odd 2 1 inner 7200.2.b.f 8
15.e even 4 1 1440.2.m.a 4
15.e even 4 1 1440.2.m.b 4
20.d odd 2 1 1800.2.b.f 8
20.e even 4 1 360.2.m.a 4
20.e even 4 1 360.2.m.b yes 4
24.f even 2 1 inner 7200.2.b.f 8
24.h odd 2 1 1800.2.b.f 8
40.e odd 2 1 CM 7200.2.b.f 8
40.f even 2 1 1800.2.b.f 8
40.i odd 4 1 360.2.m.a 4
40.i odd 4 1 360.2.m.b yes 4
40.k even 4 1 1440.2.m.a 4
40.k even 4 1 1440.2.m.b 4
60.h even 2 1 1800.2.b.f 8
60.l odd 4 1 360.2.m.a 4
60.l odd 4 1 360.2.m.b yes 4
120.i odd 2 1 1800.2.b.f 8
120.m even 2 1 inner 7200.2.b.f 8
120.q odd 4 1 1440.2.m.a 4
120.q odd 4 1 1440.2.m.b 4
120.w even 4 1 360.2.m.a 4
120.w even 4 1 360.2.m.b yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.m.a 4 20.e even 4 1
360.2.m.a 4 40.i odd 4 1
360.2.m.a 4 60.l odd 4 1
360.2.m.a 4 120.w even 4 1
360.2.m.b yes 4 20.e even 4 1
360.2.m.b yes 4 40.i odd 4 1
360.2.m.b yes 4 60.l odd 4 1
360.2.m.b yes 4 120.w even 4 1
1440.2.m.a 4 5.c odd 4 1
1440.2.m.a 4 15.e even 4 1
1440.2.m.a 4 40.k even 4 1
1440.2.m.a 4 120.q odd 4 1
1440.2.m.b 4 5.c odd 4 1
1440.2.m.b 4 15.e even 4 1
1440.2.m.b 4 40.k even 4 1
1440.2.m.b 4 120.q odd 4 1
1800.2.b.f 8 4.b odd 2 1
1800.2.b.f 8 8.b even 2 1
1800.2.b.f 8 12.b even 2 1
1800.2.b.f 8 20.d odd 2 1
1800.2.b.f 8 24.h odd 2 1
1800.2.b.f 8 40.f even 2 1
1800.2.b.f 8 60.h even 2 1
1800.2.b.f 8 120.i odd 2 1
7200.2.b.f 8 1.a even 1 1 trivial
7200.2.b.f 8 3.b odd 2 1 inner
7200.2.b.f 8 5.b even 2 1 inner
7200.2.b.f 8 8.d odd 2 1 inner
7200.2.b.f 8 15.d odd 2 1 inner
7200.2.b.f 8 24.f even 2 1 inner
7200.2.b.f 8 40.e odd 2 1 CM
7200.2.b.f 8 120.m 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 $$S_{2}^{\mathrm{new}}(7200, [\chi])$$:

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

## Hecke characteristic polynomials

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