# Properties

 Label 3600.2.w.j Level $3600$ Weight $2$ Character orbit 3600.w Analytic conductor $28.746$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3600 = 2^{4} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3600.w (of order $$4$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$28.7461447277$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: 8.0.40960000.1 Defining polynomial: $$x^{8} + 7 x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{29}]$$ Coefficient ring index: $$2^{7}$$ Twist minimal: no (minimal twist has level 360) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{1} + \beta_{4} ) q^{7} +O(q^{10})$$ $$q + ( -1 + \beta_{1} + \beta_{4} ) q^{7} + ( -\beta_{3} - \beta_{6} ) q^{11} + ( 2 + 2 \beta_{1} + \beta_{5} ) q^{13} + ( \beta_{2} + \beta_{3} - \beta_{6} - \beta_{7} ) q^{17} + ( 2 \beta_{1} - \beta_{4} - \beta_{5} ) q^{19} + ( 2 \beta_{2} - 2 \beta_{3} ) q^{23} + ( 2 \beta_{2} + \beta_{7} ) q^{29} + ( -6 - \beta_{4} + \beta_{5} ) q^{31} + \beta_{4} q^{37} + 5 \beta_{3} q^{41} + ( -2 - 2 \beta_{1} + 2 \beta_{5} ) q^{43} + ( \beta_{2} + \beta_{3} - \beta_{6} - \beta_{7} ) q^{47} + ( -5 \beta_{1} - 2 \beta_{4} - 2 \beta_{5} ) q^{49} + ( -3 \beta_{2} + \beta_{7} ) q^{59} + ( 6 - \beta_{4} + \beta_{5} ) q^{61} + ( -6 + 6 \beta_{1} - 2 \beta_{4} ) q^{67} + 4 \beta_{6} q^{71} + ( 5 + 5 \beta_{1} - 2 \beta_{5} ) q^{73} + ( 6 \beta_{2} + 6 \beta_{3} + 2 \beta_{6} + 2 \beta_{7} ) q^{77} + ( 10 \beta_{1} + \beta_{4} + \beta_{5} ) q^{79} + ( 3 \beta_{2} - 3 \beta_{3} + \beta_{6} - \beta_{7} ) q^{83} + ( 5 \beta_{2} - 2 \beta_{7} ) q^{89} + ( -14 + 3 \beta_{4} - 3 \beta_{5} ) q^{91} + ( -1 + \beta_{1} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 8q^{7} + O(q^{10})$$ $$8q - 8q^{7} + 16q^{13} - 48q^{31} - 16q^{43} + 48q^{61} - 48q^{67} + 40q^{73} - 112q^{91} - 8q^{97} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{6} + 8 \nu^{2}$$$$)/3$$ $$\beta_{2}$$ $$=$$ $$($$$$2 \nu^{7} - \nu^{5} + 13 \nu^{3} - 5 \nu$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$($$$$-2 \nu^{7} - \nu^{5} - 13 \nu^{3} - 5 \nu$$$$)/3$$ $$\beta_{4}$$ $$=$$ $$($$$$-3 \nu^{6} + 2 \nu^{4} - 18 \nu^{2} + 7$$$$)/3$$ $$\beta_{5}$$ $$=$$ $$($$$$-3 \nu^{6} - 2 \nu^{4} - 18 \nu^{2} - 7$$$$)/3$$ $$\beta_{6}$$ $$=$$ $$($$$$4 \nu^{7} + \nu^{5} + 29 \nu^{3} + 11 \nu$$$$)/3$$ $$\beta_{7}$$ $$=$$ $$($$$$-4 \nu^{7} + \nu^{5} - 29 \nu^{3} + 11 \nu$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7} + \beta_{6} + \beta_{3} + \beta_{2}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{5} + \beta_{4} + 6 \beta_{1}$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{7} + \beta_{6} + 2 \beta_{3} - 2 \beta_{2}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-3 \beta_{5} + 3 \beta_{4} - 14$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$-5 \beta_{7} - 5 \beta_{6} - 11 \beta_{3} - 11 \beta_{2}$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$-2 \beta_{5} - 2 \beta_{4} - 9 \beta_{1}$$ $$\nu^{7}$$ $$=$$ $$($$$$13 \beta_{7} - 13 \beta_{6} - 29 \beta_{3} + 29 \beta_{2}$$$$)/4$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3600\mathbb{Z}\right)^\times$$.

 $$n$$ $$577$$ $$901$$ $$2801$$ $$3151$$ $$\chi(n)$$ $$-\beta_{1}$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
593.1
 1.14412 + 1.14412i −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
0 0 0 0 0 −3.23607 + 3.23607i 0 0 0
593.2 0 0 0 0 0 −3.23607 + 3.23607i 0 0 0
593.3 0 0 0 0 0 1.23607 1.23607i 0 0 0
593.4 0 0 0 0 0 1.23607 1.23607i 0 0 0
1457.1 0 0 0 0 0 −3.23607 3.23607i 0 0 0
1457.2 0 0 0 0 0 −3.23607 3.23607i 0 0 0
1457.3 0 0 0 0 0 1.23607 + 1.23607i 0 0 0
1457.4 0 0 0 0 0 1.23607 + 1.23607i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1457.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3600.2.w.j 8
3.b odd 2 1 inner 3600.2.w.j 8
4.b odd 2 1 1800.2.s.e 8
5.b even 2 1 720.2.w.e 8
5.c odd 4 1 720.2.w.e 8
5.c odd 4 1 inner 3600.2.w.j 8
12.b even 2 1 1800.2.s.e 8
15.d odd 2 1 720.2.w.e 8
15.e even 4 1 720.2.w.e 8
15.e even 4 1 inner 3600.2.w.j 8
20.d odd 2 1 360.2.s.b 8
20.e even 4 1 360.2.s.b 8
20.e even 4 1 1800.2.s.e 8
40.e odd 2 1 2880.2.w.m 8
40.f even 2 1 2880.2.w.o 8
40.i odd 4 1 2880.2.w.o 8
40.k even 4 1 2880.2.w.m 8
60.h even 2 1 360.2.s.b 8
60.l odd 4 1 360.2.s.b 8
60.l odd 4 1 1800.2.s.e 8
120.i odd 2 1 2880.2.w.o 8
120.m even 2 1 2880.2.w.m 8
120.q odd 4 1 2880.2.w.m 8
120.w even 4 1 2880.2.w.o 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.s.b 8 20.d odd 2 1
360.2.s.b 8 20.e even 4 1
360.2.s.b 8 60.h even 2 1
360.2.s.b 8 60.l odd 4 1
720.2.w.e 8 5.b even 2 1
720.2.w.e 8 5.c odd 4 1
720.2.w.e 8 15.d odd 2 1
720.2.w.e 8 15.e even 4 1
1800.2.s.e 8 4.b odd 2 1
1800.2.s.e 8 12.b even 2 1
1800.2.s.e 8 20.e even 4 1
1800.2.s.e 8 60.l odd 4 1
2880.2.w.m 8 40.e odd 2 1
2880.2.w.m 8 40.k even 4 1
2880.2.w.m 8 120.m even 2 1
2880.2.w.m 8 120.q odd 4 1
2880.2.w.o 8 40.f even 2 1
2880.2.w.o 8 40.i odd 4 1
2880.2.w.o 8 120.i odd 2 1
2880.2.w.o 8 120.w even 4 1
3600.2.w.j 8 1.a even 1 1 trivial
3600.2.w.j 8 3.b odd 2 1 inner
3600.2.w.j 8 5.c odd 4 1 inner
3600.2.w.j 8 15.e even 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(3600, [\chi])$$:

 $$T_{7}^{4} + 4 T_{7}^{3} + 8 T_{7}^{2} - 32 T_{7} + 64$$ $$T_{13}^{4} - 8 T_{13}^{3} + 32 T_{13}^{2} + 16 T_{13} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$T^{8}$$
$7$ $$( 64 - 32 T + 8 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$11$ $$( 64 + 24 T^{2} + T^{4} )^{2}$$
$13$ $$( 4 + 16 T + 32 T^{2} - 8 T^{3} + T^{4} )^{2}$$
$17$ $$65536 + 1792 T^{4} + T^{8}$$
$19$ $$( 256 + 48 T^{2} + T^{4} )^{2}$$
$23$ $$( 256 + T^{4} )^{2}$$
$29$ $$( 4 - 36 T^{2} + T^{4} )^{2}$$
$31$ $$( 16 + 12 T + T^{2} )^{4}$$
$37$ $$( 100 + T^{4} )^{2}$$
$41$ $$( 50 + T^{2} )^{4}$$
$43$ $$( 1024 - 256 T + 32 T^{2} + 8 T^{3} + T^{4} )^{2}$$
$47$ $$65536 + 1792 T^{4} + T^{8}$$
$53$ $$T^{8}$$
$59$ $$( 64 - 56 T^{2} + T^{4} )^{2}$$
$61$ $$( 16 - 12 T + T^{2} )^{4}$$
$67$ $$( 1024 + 768 T + 288 T^{2} + 24 T^{3} + T^{4} )^{2}$$
$71$ $$( 160 + T^{2} )^{4}$$
$73$ $$( 100 - 200 T + 200 T^{2} - 20 T^{3} + T^{4} )^{2}$$
$79$ $$( 6400 + 240 T^{2} + T^{4} )^{2}$$
$83$ $$65536 + 12032 T^{4} + T^{8}$$
$89$ $$( 100 - 180 T^{2} + T^{4} )^{2}$$
$97$ $$( 2 + 2 T + T^{2} )^{4}$$