# Properties

 Label 3600.3.c.k Level $3600$ Weight $3$ Character orbit 3600.c Analytic conductor $98.093$ 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: $$k$$ $$=$$ $$3$$ 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: $$8$$ Coefficient field: 8.0.40960000.1 Defining polynomial: $$x^{8} + 7x^{4} + 1$$ x^8 + 7*x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{29}]$$ Coefficient ring index: $$2^{11}\cdot 3^{6}$$ Twist minimal: no (minimal twist has level 180) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{3} + 2 \beta_1) q^{7}+O(q^{10})$$ q + (-b3 + 2*b1) * q^7 $$q + ( - \beta_{3} + 2 \beta_1) q^{7} + ( - \beta_{5} - \beta_{2}) q^{11} + ( - \beta_{3} - \beta_1) q^{13} + 2 \beta_{4} q^{17} + (\beta_{6} + 8) q^{19} + (\beta_{7} - \beta_{4}) q^{23} + ( - \beta_{5} - 8 \beta_{2}) q^{29} + ( - \beta_{6} - 2) q^{31} + ( - 3 \beta_{3} - 17 \beta_1) q^{37} + ( - 4 \beta_{5} - 3 \beta_{2}) q^{41} + ( - 2 \beta_{3} + 10 \beta_1) q^{43} + ( - 2 \beta_{7} - 7 \beta_{4}) q^{47} + (4 \beta_{6} - 57) q^{49} + (4 \beta_{7} + 5 \beta_{4}) q^{53} + ( - 3 \beta_{5} + 17 \beta_{2}) q^{59} + ( - 3 \beta_{6} - 10) q^{61} + 38 \beta_1 q^{67} + ( - 2 \beta_{5} - 12 \beta_{2}) q^{71} + (6 \beta_{3} - 19 \beta_1) q^{73} + ( - 7 \beta_{7} + 17 \beta_{4}) q^{77} + (3 \beta_{6} + 50) q^{79} + (4 \beta_{7} - \beta_{4}) q^{83} + (2 \beta_{5} - 21 \beta_{2}) q^{89} + (\beta_{6} - 82) q^{91} + (2 \beta_{3} - 53 \beta_1) q^{97}+O(q^{100})$$ q + (-b3 + 2*b1) * q^7 + (-b5 - b2) * q^11 + (-b3 - b1) * q^13 + 2*b4 * q^17 + (b6 + 8) * q^19 + (b7 - b4) * q^23 + (-b5 - 8*b2) * q^29 + (-b6 - 2) * q^31 + (-3*b3 - 17*b1) * q^37 + (-4*b5 - 3*b2) * q^41 + (-2*b3 + 10*b1) * q^43 + (-2*b7 - 7*b4) * q^47 + (4*b6 - 57) * q^49 + (4*b7 + 5*b4) * q^53 + (-3*b5 + 17*b2) * q^59 + (-3*b6 - 10) * q^61 + 38*b1 * q^67 + (-2*b5 - 12*b2) * q^71 + (6*b3 - 19*b1) * q^73 + (-7*b7 + 17*b4) * q^77 + (3*b6 + 50) * q^79 + (4*b7 - b4) * q^83 + (2*b5 - 21*b2) * q^89 + (b6 - 82) * q^91 + (2*b3 - 53*b1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q+O(q^{10})$$ 8 * q $$8 q + 64 q^{19} - 16 q^{31} - 456 q^{49} - 80 q^{61} + 400 q^{79} - 656 q^{91}+O(q^{100})$$ 8 * q + 64 * q^19 - 16 * q^31 - 456 * q^49 - 80 * q^61 + 400 * q^79 - 656 * q^91

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

 $$\beta_{1}$$ $$=$$ $$( 2\nu^{6} + 16\nu^{2} ) / 3$$ (2*v^6 + 16*v^2) / 3 $$\beta_{2}$$ $$=$$ $$2\nu^{7} + \nu^{5} + 13\nu^{3} + 5\nu$$ 2*v^7 + v^5 + 13*v^3 + 5*v $$\beta_{3}$$ $$=$$ $$4\nu^{7} + \nu^{5} + 29\nu^{3} + 11\nu$$ 4*v^7 + v^5 + 29*v^3 + 11*v $$\beta_{4}$$ $$=$$ $$-4\nu^{7} + 2\nu^{5} - 26\nu^{3} + 10\nu$$ -4*v^7 + 2*v^5 - 26*v^3 + 10*v $$\beta_{5}$$ $$=$$ $$-6\nu^{6} - 36\nu^{2}$$ -6*v^6 - 36*v^2 $$\beta_{6}$$ $$=$$ $$-8\nu^{7} + 2\nu^{5} - 58\nu^{3} + 22\nu$$ -8*v^7 + 2*v^5 - 58*v^3 + 22*v $$\beta_{7}$$ $$=$$ $$4\nu^{4} + 14$$ 4*v^4 + 14
 $$\nu$$ $$=$$ $$( \beta_{6} - \beta_{4} + 2\beta_{3} - 2\beta_{2} ) / 24$$ (b6 - b4 + 2*b3 - 2*b2) / 24 $$\nu^{2}$$ $$=$$ $$( \beta_{5} + 9\beta_1 ) / 12$$ (b5 + 9*b1) / 12 $$\nu^{3}$$ $$=$$ $$( -\beta_{6} + 2\beta_{4} + 2\beta_{3} - 4\beta_{2} ) / 12$$ (-b6 + 2*b4 + 2*b3 - 4*b2) / 12 $$\nu^{4}$$ $$=$$ $$( \beta_{7} - 14 ) / 4$$ (b7 - 14) / 4 $$\nu^{5}$$ $$=$$ $$( -5\beta_{6} + 11\beta_{4} - 10\beta_{3} + 22\beta_{2} ) / 24$$ (-5*b6 + 11*b4 - 10*b3 + 22*b2) / 24 $$\nu^{6}$$ $$=$$ $$( -4\beta_{5} - 27\beta_1 ) / 6$$ (-4*b5 - 27*b1) / 6 $$\nu^{7}$$ $$=$$ $$( 13\beta_{6} - 29\beta_{4} - 26\beta_{3} + 58\beta_{2} ) / 24$$ (13*b6 - 29*b4 - 26*b3 + 58*b2) / 24

## 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
 −0.437016 + 0.437016i −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 0 0 0 0 13.4868i 0 0 0
449.2 0 0 0 0 0 13.4868i 0 0 0
449.3 0 0 0 0 0 5.48683i 0 0 0
449.4 0 0 0 0 0 5.48683i 0 0 0
449.5 0 0 0 0 0 5.48683i 0 0 0
449.6 0 0 0 0 0 5.48683i 0 0 0
449.7 0 0 0 0 0 13.4868i 0 0 0
449.8 0 0 0 0 0 13.4868i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 449.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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.k 8
3.b odd 2 1 inner 3600.3.c.k 8
4.b odd 2 1 900.3.b.b 8
5.b even 2 1 inner 3600.3.c.k 8
5.c odd 4 1 720.3.l.c 4
5.c odd 4 1 3600.3.l.n 4
12.b even 2 1 900.3.b.b 8
15.d odd 2 1 inner 3600.3.c.k 8
15.e even 4 1 720.3.l.c 4
15.e even 4 1 3600.3.l.n 4
20.d odd 2 1 900.3.b.b 8
20.e even 4 1 180.3.g.a 4
20.e even 4 1 900.3.g.d 4
40.i odd 4 1 2880.3.l.f 4
40.k even 4 1 2880.3.l.b 4
60.h even 2 1 900.3.b.b 8
60.l odd 4 1 180.3.g.a 4
60.l odd 4 1 900.3.g.d 4
120.q odd 4 1 2880.3.l.b 4
120.w even 4 1 2880.3.l.f 4
180.v odd 12 2 1620.3.o.f 8
180.x even 12 2 1620.3.o.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.3.g.a 4 20.e even 4 1
180.3.g.a 4 60.l odd 4 1
720.3.l.c 4 5.c odd 4 1
720.3.l.c 4 15.e even 4 1
900.3.b.b 8 4.b odd 2 1
900.3.b.b 8 12.b even 2 1
900.3.b.b 8 20.d odd 2 1
900.3.b.b 8 60.h even 2 1
900.3.g.d 4 20.e even 4 1
900.3.g.d 4 60.l odd 4 1
1620.3.o.f 8 180.v odd 12 2
1620.3.o.f 8 180.x even 12 2
2880.3.l.b 4 40.k even 4 1
2880.3.l.b 4 120.q odd 4 1
2880.3.l.f 4 40.i odd 4 1
2880.3.l.f 4 120.w even 4 1
3600.3.c.k 8 1.a even 1 1 trivial
3600.3.c.k 8 3.b odd 2 1 inner
3600.3.c.k 8 5.b even 2 1 inner
3600.3.c.k 8 15.d odd 2 1 inner
3600.3.l.n 4 5.c odd 4 1
3600.3.l.n 4 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 $$S_{3}^{\mathrm{new}}(3600, [\chi])$$:

 $$T_{7}^{4} + 212T_{7}^{2} + 5476$$ T7^4 + 212*T7^2 + 5476 $$T_{13}^{4} + 188T_{13}^{2} + 7396$$ T13^4 + 188*T13^2 + 7396

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$T^{8}$$
$7$ $$(T^{4} + 212 T^{2} + 5476)^{2}$$
$11$ $$(T^{4} + 396 T^{2} + 26244)^{2}$$
$13$ $$(T^{4} + 188 T^{2} + 7396)^{2}$$
$17$ $$(T^{2} - 288)^{4}$$
$19$ $$(T^{2} - 16 T - 296)^{4}$$
$23$ $$(T^{4} - 504 T^{2} + 11664)^{2}$$
$29$ $$(T^{4} + 2664 T^{2} + 944784)^{2}$$
$31$ $$(T^{2} + 4 T - 356)^{4}$$
$37$ $$(T^{4} + 3932 T^{2} + 119716)^{2}$$
$41$ $$(T^{4} + 6084 T^{2} + 7387524)^{2}$$
$43$ $$(T^{4} + 1520 T^{2} + 1600)^{2}$$
$47$ $$(T^{4} - 8496 T^{2} + 7884864)^{2}$$
$53$ $$(T^{4} - 9360 T^{2} + 1166400)^{2}$$
$59$ $$(T^{4} + 13644 T^{2} + 12830724)^{2}$$
$61$ $$(T^{2} + 20 T - 3140)^{4}$$
$67$ $$(T^{2} + 5776)^{4}$$
$71$ $$(T^{4} + 6624 T^{2} + 3504384)^{2}$$
$73$ $$(T^{4} + 9368 T^{2} + 3225616)^{2}$$
$79$ $$(T^{2} - 100 T - 740)^{4}$$
$83$ $$(T^{4} - 5904 T^{2} + 7884864)^{2}$$
$89$ $$(T^{4} + 17316 T^{2} + 52099524)^{2}$$
$97$ $$(T^{4} + 23192 T^{2} + \cdots + 118287376)^{2}$$