# Properties

 Label 7200.2.o.j Level $7200$ Weight $2$ Character orbit 7200.o Analytic conductor $57.492$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$57.4922894553$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 1440) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{3} + 2) q^{7}+O(q^{10})$$ q + (b3 + 2) * q^7 $$q + (\beta_{3} + 2) q^{7} + (\beta_{3} - 4) q^{11} + ( - \beta_{2} - \beta_1) q^{13} + (2 \beta_{3} - 4) q^{17} + ( - 2 \beta_{2} - \beta_1) q^{23} + 3 \beta_1 q^{29} + (6 \beta_{2} - \beta_1) q^{31} + (5 \beta_{2} + \beta_1) q^{37} + ( - 5 \beta_{2} - 2 \beta_1) q^{41} + (2 \beta_{3} + 4) q^{43} - 4 \beta_{2} q^{47} + (4 \beta_{3} - 1) q^{49} + ( - 2 \beta_{3} + 4) q^{53} + ( - \beta_{3} + 8) q^{59} + (2 \beta_{3} + 10) q^{61} + 8 q^{67} - 4 \beta_{3} q^{71} + ( - 6 \beta_{2} - \beta_1) q^{73} + ( - 2 \beta_{3} - 6) q^{77} + ( - 6 \beta_{2} - 3 \beta_1) q^{79} + (2 \beta_{2} - 6 \beta_1) q^{83} - 3 \beta_{2} q^{89} + ( - 4 \beta_{2} - 3 \beta_1) q^{91} + ( - 6 \beta_{2} + 3 \beta_1) q^{97}+O(q^{100})$$ q + (b3 + 2) * q^7 + (b3 - 4) * q^11 + (-b2 - b1) * q^13 + (2*b3 - 4) * q^17 + (-2*b2 - b1) * q^23 + 3*b1 * q^29 + (6*b2 - b1) * q^31 + (5*b2 + b1) * q^37 + (-5*b2 - 2*b1) * q^41 + (2*b3 + 4) * q^43 - 4*b2 * q^47 + (4*b3 - 1) * q^49 + (-2*b3 + 4) * q^53 + (-b3 + 8) * q^59 + (2*b3 + 10) * q^61 + 8 * q^67 - 4*b3 * q^71 + (-6*b2 - b1) * q^73 + (-2*b3 - 6) * q^77 + (-6*b2 - 3*b1) * q^79 + (2*b2 - 6*b1) * q^83 - 3*b2 * q^89 + (-4*b2 - 3*b1) * q^91 + (-6*b2 + 3*b1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 8 q^{7}+O(q^{10})$$ 4 * q + 8 * q^7 $$4 q + 8 q^{7} - 16 q^{11} - 16 q^{17} + 16 q^{43} - 4 q^{49} + 16 q^{53} + 32 q^{59} + 40 q^{61} + 32 q^{67} - 24 q^{77}+O(q^{100})$$ 4 * q + 8 * q^7 - 16 * q^11 - 16 * q^17 + 16 * q^43 - 4 * q^49 + 16 * q^53 + 32 * q^59 + 40 * q^61 + 32 * q^67 - 24 * q^77

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$2\zeta_{8}^{2}$$ 2*v^2 $$\beta_{2}$$ $$=$$ $$\zeta_{8}^{3} + \zeta_{8}$$ v^3 + v $$\beta_{3}$$ $$=$$ $$-\zeta_{8}^{3} + \zeta_{8}$$ -v^3 + v
 $$\zeta_{8}$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 2$$ (b3 + b2) / 2 $$\zeta_{8}^{2}$$ $$=$$ $$( \beta_1 ) / 2$$ (b1) / 2 $$\zeta_{8}^{3}$$ $$=$$ $$( -\beta_{3} + \beta_{2} ) / 2$$ (-b3 + b2) / 2

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7199.1
 −0.707107 − 0.707107i −0.707107 + 0.707107i 0.707107 + 0.707107i 0.707107 − 0.707107i
0 0 0 0 0 0.585786 0 0 0
7199.2 0 0 0 0 0 0.585786 0 0 0
7199.3 0 0 0 0 0 3.41421 0 0 0
7199.4 0 0 0 0 0 3.41421 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
60.h 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.o.j 4
3.b odd 2 1 7200.2.o.m 4
4.b odd 2 1 7200.2.o.e 4
5.b even 2 1 7200.2.o.b 4
5.c odd 4 1 1440.2.h.a 4
5.c odd 4 1 7200.2.h.c 4
12.b even 2 1 7200.2.o.b 4
15.d odd 2 1 7200.2.o.e 4
15.e even 4 1 1440.2.h.d yes 4
15.e even 4 1 7200.2.h.i 4
20.d odd 2 1 7200.2.o.m 4
20.e even 4 1 1440.2.h.d yes 4
20.e even 4 1 7200.2.h.i 4
40.i odd 4 1 2880.2.h.d 4
40.k even 4 1 2880.2.h.a 4
60.h even 2 1 inner 7200.2.o.j 4
60.l odd 4 1 1440.2.h.a 4
60.l odd 4 1 7200.2.h.c 4
120.q odd 4 1 2880.2.h.d 4
120.w even 4 1 2880.2.h.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1440.2.h.a 4 5.c odd 4 1
1440.2.h.a 4 60.l odd 4 1
1440.2.h.d yes 4 15.e even 4 1
1440.2.h.d yes 4 20.e even 4 1
2880.2.h.a 4 40.k even 4 1
2880.2.h.a 4 120.w even 4 1
2880.2.h.d 4 40.i odd 4 1
2880.2.h.d 4 120.q odd 4 1
7200.2.h.c 4 5.c odd 4 1
7200.2.h.c 4 60.l odd 4 1
7200.2.h.i 4 15.e even 4 1
7200.2.h.i 4 20.e even 4 1
7200.2.o.b 4 5.b even 2 1
7200.2.o.b 4 12.b even 2 1
7200.2.o.e 4 4.b odd 2 1
7200.2.o.e 4 15.d odd 2 1
7200.2.o.j 4 1.a even 1 1 trivial
7200.2.o.j 4 60.h even 2 1 inner
7200.2.o.m 4 3.b odd 2 1
7200.2.o.m 4 20.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 $$S_{2}^{\mathrm{new}}(7200, [\chi])$$:

 $$T_{7}^{2} - 4T_{7} + 2$$ T7^2 - 4*T7 + 2 $$T_{11}^{2} + 8T_{11} + 14$$ T11^2 + 8*T11 + 14 $$T_{17}^{2} + 8T_{17} + 8$$ T17^2 + 8*T17 + 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$(T^{2} - 4 T + 2)^{2}$$
$11$ $$(T^{2} + 8 T + 14)^{2}$$
$13$ $$T^{4} + 12T^{2} + 4$$
$17$ $$(T^{2} + 8 T + 8)^{2}$$
$19$ $$T^{4}$$
$23$ $$T^{4} + 24T^{2} + 16$$
$29$ $$(T^{2} + 36)^{2}$$
$31$ $$T^{4} + 152T^{2} + 4624$$
$37$ $$T^{4} + 108T^{2} + 2116$$
$41$ $$T^{4} + 132T^{2} + 1156$$
$43$ $$(T^{2} - 8 T + 8)^{2}$$
$47$ $$(T^{2} + 32)^{2}$$
$53$ $$(T^{2} - 8 T + 8)^{2}$$
$59$ $$(T^{2} - 16 T + 62)^{2}$$
$61$ $$(T^{2} - 20 T + 92)^{2}$$
$67$ $$(T - 8)^{4}$$
$71$ $$(T^{2} - 32)^{2}$$
$73$ $$T^{4} + 152T^{2} + 4624$$
$79$ $$T^{4} + 216T^{2} + 1296$$
$83$ $$T^{4} + 304 T^{2} + 18496$$
$89$ $$(T^{2} + 18)^{2}$$
$97$ $$T^{4} + 216T^{2} + 1296$$