# Properties

 Label 3600.3.c.b Level $3600$ Weight $3$ Character orbit 3600.c Analytic conductor $98.093$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3600 = 2^{4} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 3600.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$98.0928951697$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{29}]$$ Coefficient ring index: $$2^{2}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 18) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 4 \zeta_{8}^{2} q^{7} +O(q^{10})$$ $$q + 4 \zeta_{8}^{2} q^{7} + ( -12 \zeta_{8} - 12 \zeta_{8}^{3} ) q^{11} -8 \zeta_{8}^{2} q^{13} + ( 9 \zeta_{8} - 9 \zeta_{8}^{3} ) q^{17} -16 q^{19} + ( 12 \zeta_{8} - 12 \zeta_{8}^{3} ) q^{23} + ( -3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{29} -44 q^{31} -34 \zeta_{8}^{2} q^{37} + ( 33 \zeta_{8} + 33 \zeta_{8}^{3} ) q^{41} -40 \zeta_{8}^{2} q^{43} + ( -60 \zeta_{8} + 60 \zeta_{8}^{3} ) q^{47} + 33 q^{49} + ( 27 \zeta_{8} - 27 \zeta_{8}^{3} ) q^{53} + ( 24 \zeta_{8} + 24 \zeta_{8}^{3} ) q^{59} + 50 q^{61} -8 \zeta_{8}^{2} q^{67} + ( 36 \zeta_{8} + 36 \zeta_{8}^{3} ) q^{71} + 16 \zeta_{8}^{2} q^{73} + ( 48 \zeta_{8} - 48 \zeta_{8}^{3} ) q^{77} -76 q^{79} + ( -84 \zeta_{8} + 84 \zeta_{8}^{3} ) q^{83} + ( -9 \zeta_{8} - 9 \zeta_{8}^{3} ) q^{89} + 32 q^{91} + 176 \zeta_{8}^{2} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q - 64q^{19} - 176q^{31} + 132q^{49} + 200q^{61} - 304q^{79} + 128q^{91} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
449.1
 −0.707107 + 0.707107i 0.707107 − 0.707107i 0.707107 + 0.707107i −0.707107 − 0.707107i
0 0 0 0 0 4.00000i 0 0 0
449.2 0 0 0 0 0 4.00000i 0 0 0
449.3 0 0 0 0 0 4.00000i 0 0 0
449.4 0 0 0 0 0 4.00000i 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
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3600.3.c.b 4
3.b odd 2 1 inner 3600.3.c.b 4
4.b odd 2 1 450.3.b.b 4
5.b even 2 1 inner 3600.3.c.b 4
5.c odd 4 1 144.3.e.b 2
5.c odd 4 1 3600.3.l.d 2
12.b even 2 1 450.3.b.b 4
15.d odd 2 1 inner 3600.3.c.b 4
15.e even 4 1 144.3.e.b 2
15.e even 4 1 3600.3.l.d 2
20.d odd 2 1 450.3.b.b 4
20.e even 4 1 18.3.b.a 2
20.e even 4 1 450.3.d.f 2
40.i odd 4 1 576.3.e.f 2
40.k even 4 1 576.3.e.c 2
45.k odd 12 2 1296.3.q.f 4
45.l even 12 2 1296.3.q.f 4
60.h even 2 1 450.3.b.b 4
60.l odd 4 1 18.3.b.a 2
60.l odd 4 1 450.3.d.f 2
80.i odd 4 1 2304.3.h.c 4
80.j even 4 1 2304.3.h.f 4
80.s even 4 1 2304.3.h.f 4
80.t odd 4 1 2304.3.h.c 4
120.q odd 4 1 576.3.e.c 2
120.w even 4 1 576.3.e.f 2
140.j odd 4 1 882.3.b.a 2
140.w even 12 2 882.3.s.b 4
140.x odd 12 2 882.3.s.d 4
180.v odd 12 2 162.3.d.b 4
180.x even 12 2 162.3.d.b 4
220.i odd 4 1 2178.3.c.d 2
240.z odd 4 1 2304.3.h.f 4
240.bb even 4 1 2304.3.h.c 4
240.bd odd 4 1 2304.3.h.f 4
240.bf even 4 1 2304.3.h.c 4
260.l odd 4 1 3042.3.d.a 4
260.p even 4 1 3042.3.c.e 2
260.s odd 4 1 3042.3.d.a 4
420.w even 4 1 882.3.b.a 2
420.bp odd 12 2 882.3.s.b 4
420.br even 12 2 882.3.s.d 4
660.q even 4 1 2178.3.c.d 2
780.u even 4 1 3042.3.d.a 4
780.w odd 4 1 3042.3.c.e 2
780.bn even 4 1 3042.3.d.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.3.b.a 2 20.e even 4 1
18.3.b.a 2 60.l odd 4 1
144.3.e.b 2 5.c odd 4 1
144.3.e.b 2 15.e even 4 1
162.3.d.b 4 180.v odd 12 2
162.3.d.b 4 180.x even 12 2
450.3.b.b 4 4.b odd 2 1
450.3.b.b 4 12.b even 2 1
450.3.b.b 4 20.d odd 2 1
450.3.b.b 4 60.h even 2 1
450.3.d.f 2 20.e even 4 1
450.3.d.f 2 60.l odd 4 1
576.3.e.c 2 40.k even 4 1
576.3.e.c 2 120.q odd 4 1
576.3.e.f 2 40.i odd 4 1
576.3.e.f 2 120.w even 4 1
882.3.b.a 2 140.j odd 4 1
882.3.b.a 2 420.w even 4 1
882.3.s.b 4 140.w even 12 2
882.3.s.b 4 420.bp odd 12 2
882.3.s.d 4 140.x odd 12 2
882.3.s.d 4 420.br even 12 2
1296.3.q.f 4 45.k odd 12 2
1296.3.q.f 4 45.l even 12 2
2178.3.c.d 2 220.i odd 4 1
2178.3.c.d 2 660.q even 4 1
2304.3.h.c 4 80.i odd 4 1
2304.3.h.c 4 80.t odd 4 1
2304.3.h.c 4 240.bb even 4 1
2304.3.h.c 4 240.bf even 4 1
2304.3.h.f 4 80.j even 4 1
2304.3.h.f 4 80.s even 4 1
2304.3.h.f 4 240.z odd 4 1
2304.3.h.f 4 240.bd odd 4 1
3042.3.c.e 2 260.p even 4 1
3042.3.c.e 2 780.w odd 4 1
3042.3.d.a 4 260.l odd 4 1
3042.3.d.a 4 260.s odd 4 1
3042.3.d.a 4 780.u even 4 1
3042.3.d.a 4 780.bn even 4 1
3600.3.c.b 4 1.a even 1 1 trivial
3600.3.c.b 4 3.b odd 2 1 inner
3600.3.c.b 4 5.b even 2 1 inner
3600.3.c.b 4 15.d odd 2 1 inner
3600.3.l.d 2 5.c odd 4 1
3600.3.l.d 2 15.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(3600, [\chi])$$:

 $$T_{7}^{2} + 16$$ $$T_{13}^{2} + 64$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$( 16 + T^{2} )^{2}$$
$11$ $$( 288 + T^{2} )^{2}$$
$13$ $$( 64 + T^{2} )^{2}$$
$17$ $$( -162 + T^{2} )^{2}$$
$19$ $$( 16 + T )^{4}$$
$23$ $$( -288 + T^{2} )^{2}$$
$29$ $$( 18 + T^{2} )^{2}$$
$31$ $$( 44 + T )^{4}$$
$37$ $$( 1156 + T^{2} )^{2}$$
$41$ $$( 2178 + T^{2} )^{2}$$
$43$ $$( 1600 + T^{2} )^{2}$$
$47$ $$( -7200 + T^{2} )^{2}$$
$53$ $$( -1458 + T^{2} )^{2}$$
$59$ $$( 1152 + T^{2} )^{2}$$
$61$ $$( -50 + T )^{4}$$
$67$ $$( 64 + T^{2} )^{2}$$
$71$ $$( 2592 + T^{2} )^{2}$$
$73$ $$( 256 + T^{2} )^{2}$$
$79$ $$( 76 + T )^{4}$$
$83$ $$( -14112 + T^{2} )^{2}$$
$89$ $$( 162 + T^{2} )^{2}$$
$97$ $$( 30976 + T^{2} )^{2}$$