# Properties

 Label 1200.3.c.k Level $1200$ Weight $3$ Character orbit 1200.c Analytic conductor $32.698$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1200 = 2^{4} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1200.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$32.6976317232$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.40960000.1 Defining polynomial: $$x^{8} + 7 x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{8}\cdot 3^{4}$$ Twist minimal: no (minimal twist has level 30) 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.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{3} + ( -2 \beta_{1} - 2 \beta_{2} + \beta_{7} ) q^{7} + ( 2 - \beta_{5} + \beta_{6} ) q^{9} +O(q^{10})$$ $$q -\beta_{2} q^{3} + ( -2 \beta_{1} - 2 \beta_{2} + \beta_{7} ) q^{7} + ( 2 - \beta_{5} + \beta_{6} ) q^{9} + ( -\beta_{4} - \beta_{5} ) q^{11} + 5 \beta_{1} q^{13} + ( 2 \beta_{1} + 4 \beta_{2} + 5 \beta_{3} + \beta_{7} ) q^{17} + ( 8 - 3 \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{19} + ( -13 - 3 \beta_{5} - \beta_{6} ) q^{21} + ( -2 \beta_{1} - 4 \beta_{2} - \beta_{7} ) q^{23} + ( -4 \beta_{1} - \beta_{2} + 6 \beta_{3} + \beta_{7} ) q^{27} + ( 4 \beta_{4} - 4 \beta_{6} ) q^{29} -8 q^{31} + ( -3 \beta_{1} + 3 \beta_{3} + 3 \beta_{7} ) q^{33} + ( -15 \beta_{1} - 8 \beta_{2} + 4 \beta_{7} ) q^{37} + ( -10 + 5 \beta_{6} ) q^{39} + ( 6 \beta_{4} + 4 \beta_{5} - 2 \beta_{6} ) q^{41} + ( 10 \beta_{1} + 6 \beta_{2} - 3 \beta_{7} ) q^{43} + ( 10 \beta_{1} + 20 \beta_{2} + 5 \beta_{7} ) q^{47} + ( -45 - 6 \beta_{4} + 2 \beta_{5} - 4 \beta_{6} ) q^{49} + ( -18 + 15 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} ) q^{51} + ( -2 \beta_{1} - 4 \beta_{2} - 5 \beta_{3} - \beta_{7} ) q^{53} + ( 15 \beta_{1} - 8 \beta_{2} - 9 \beta_{3} + 3 \beta_{7} ) q^{57} + ( -2 \beta_{4} - 6 \beta_{5} - 4 \beta_{6} ) q^{59} + ( -16 + 6 \beta_{4} - 2 \beta_{5} + 4 \beta_{6} ) q^{61} + ( -28 \beta_{1} + 8 \beta_{2} + 6 \beta_{3} + 7 \beta_{7} ) q^{63} + ( 38 \beta_{1} - 6 \beta_{2} + 3 \beta_{7} ) q^{67} + ( 33 - 3 \beta_{5} + 3 \beta_{6} ) q^{69} + ( -\beta_{4} - 5 \beta_{5} - 4 \beta_{6} ) q^{71} + ( -21 \beta_{1} + 8 \beta_{2} - 4 \beta_{7} ) q^{73} + ( 12 \beta_{1} + 24 \beta_{2} + 2 \beta_{3} + 6 \beta_{7} ) q^{77} + ( -28 + 3 \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{79} + ( 7 + 18 \beta_{4} - 2 \beta_{5} - 4 \beta_{6} ) q^{81} + ( -2 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} - \beta_{7} ) q^{83} + ( -36 \beta_{1} - 12 \beta_{2} - 12 \beta_{3} ) q^{87} + ( -8 \beta_{4} - 4 \beta_{5} + 4 \beta_{6} ) q^{89} + ( 20 + 15 \beta_{4} - 5 \beta_{5} + 10 \beta_{6} ) q^{91} + 8 \beta_{2} q^{93} + ( 41 \beta_{1} + 8 \beta_{2} - 4 \beta_{7} ) q^{97} + ( -48 + 9 \beta_{4} - 3 \beta_{5} - 6 \beta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 16q^{9} + O(q^{10})$$ $$8q + 16q^{9} + 64q^{19} - 104q^{21} - 64q^{31} - 80q^{39} - 360q^{49} - 144q^{51} - 128q^{61} + 264q^{69} - 224q^{79} + 56q^{81} + 160q^{91} - 384q^{99} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-2 \nu^{6} - 16 \nu^{2}$$$$)/3$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{7} + \nu^{6} - \nu^{5} - 2 \nu^{4} - 8 \nu^{3} + 8 \nu^{2} - 8 \nu - 7$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$-2 \nu^{7} + \nu^{5} - 13 \nu^{3} + 5 \nu$$ $$\beta_{4}$$ $$=$$ $$-2 \nu^{7} + \nu^{6} - 14 \nu^{3} + 6 \nu^{2} + 2 \nu$$ $$\beta_{5}$$ $$=$$ $$-2 \nu^{7} - \nu^{6} - 2 \nu^{5} - 12 \nu^{3} - 6 \nu^{2} - 12 \nu$$ $$\beta_{6}$$ $$=$$ $$-2 \nu^{7} - 2 \nu^{6} - 14 \nu^{3} - 12 \nu^{2} + 2 \nu$$ $$\beta_{7}$$ $$=$$ $$($$$$10 \nu^{7} + \nu^{5} - 4 \nu^{4} + 71 \nu^{3} + 17 \nu - 14$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7} + \beta_{6} + 2 \beta_{4} - \beta_{3} - 2 \beta_{2} - \beta_{1}$$$$)/12$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{6} - 2 \beta_{4} - 9 \beta_{1}$$$$)/12$$ $$\nu^{3}$$ $$=$$ $$($$$$2 \beta_{7} - 2 \beta_{6} + 3 \beta_{5} - \beta_{4} + 4 \beta_{3} - 4 \beta_{2} - 2 \beta_{1}$$$$)/12$$ $$\nu^{4}$$ $$=$$ $$($$$$-\beta_{7} - \beta_{3} - 4 \beta_{2} - 2 \beta_{1} - 14$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$-5 \beta_{7} - 5 \beta_{6} - 3 \beta_{5} - 13 \beta_{4} + 11 \beta_{3} + 10 \beta_{2} + 5 \beta_{1}$$$$)/12$$ $$\nu^{6}$$ $$=$$ $$($$$$-8 \beta_{6} + 8 \beta_{4} + 27 \beta_{1}$$$$)/6$$ $$\nu^{7}$$ $$=$$ $$($$$$-13 \beta_{7} + 13 \beta_{6} - 21 \beta_{5} + 5 \beta_{4} - 29 \beta_{3} + 26 \beta_{2} + 13 \beta_{1}$$$$)/12$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times$$.

 $$n$$ $$401$$ $$577$$ $$751$$ $$901$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$ $$1$$

## 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
 1.14412 − 1.14412i 1.14412 + 1.14412i −1.14412 − 1.14412i −1.14412 + 1.14412i −0.437016 − 0.437016i −0.437016 + 0.437016i 0.437016 − 0.437016i 0.437016 + 0.437016i
0 −2.94317 0.581139i 0 0 0 11.4868i 0 8.32456 + 3.42079i 0
449.2 0 −2.94317 + 0.581139i 0 0 0 11.4868i 0 8.32456 3.42079i 0
449.3 0 −1.52896 2.58114i 0 0 0 7.48683i 0 −4.32456 + 7.89292i 0
449.4 0 −1.52896 + 2.58114i 0 0 0 7.48683i 0 −4.32456 7.89292i 0
449.5 0 1.52896 2.58114i 0 0 0 7.48683i 0 −4.32456 7.89292i 0
449.6 0 1.52896 + 2.58114i 0 0 0 7.48683i 0 −4.32456 + 7.89292i 0
449.7 0 2.94317 0.581139i 0 0 0 11.4868i 0 8.32456 3.42079i 0
449.8 0 2.94317 + 0.581139i 0 0 0 11.4868i 0 8.32456 + 3.42079i 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 1200.3.c.k 8
3.b odd 2 1 inner 1200.3.c.k 8
4.b odd 2 1 150.3.b.b 8
5.b even 2 1 inner 1200.3.c.k 8
5.c odd 4 1 240.3.l.c 4
5.c odd 4 1 1200.3.l.u 4
12.b even 2 1 150.3.b.b 8
15.d odd 2 1 inner 1200.3.c.k 8
15.e even 4 1 240.3.l.c 4
15.e even 4 1 1200.3.l.u 4
20.d odd 2 1 150.3.b.b 8
20.e even 4 1 30.3.d.a 4
20.e even 4 1 150.3.d.c 4
40.i odd 4 1 960.3.l.f 4
40.k even 4 1 960.3.l.e 4
60.h even 2 1 150.3.b.b 8
60.l odd 4 1 30.3.d.a 4
60.l odd 4 1 150.3.d.c 4
120.q odd 4 1 960.3.l.e 4
120.w even 4 1 960.3.l.f 4
180.v odd 12 2 810.3.h.a 8
180.x even 12 2 810.3.h.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.3.d.a 4 20.e even 4 1
30.3.d.a 4 60.l odd 4 1
150.3.b.b 8 4.b odd 2 1
150.3.b.b 8 12.b even 2 1
150.3.b.b 8 20.d odd 2 1
150.3.b.b 8 60.h even 2 1
150.3.d.c 4 20.e even 4 1
150.3.d.c 4 60.l odd 4 1
240.3.l.c 4 5.c odd 4 1
240.3.l.c 4 15.e even 4 1
810.3.h.a 8 180.v odd 12 2
810.3.h.a 8 180.x even 12 2
960.3.l.e 4 40.k even 4 1
960.3.l.e 4 120.q odd 4 1
960.3.l.f 4 40.i odd 4 1
960.3.l.f 4 120.w even 4 1
1200.3.c.k 8 1.a even 1 1 trivial
1200.3.c.k 8 3.b odd 2 1 inner
1200.3.c.k 8 5.b even 2 1 inner
1200.3.c.k 8 15.d odd 2 1 inner
1200.3.l.u 4 5.c odd 4 1
1200.3.l.u 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 $$S_{3}^{\mathrm{new}}(1200, [\chi])$$:

 $$T_{7}^{4} + 188 T_{7}^{2} + 7396$$ $$T_{11}^{2} + 72$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$6561 - 648 T^{2} + 18 T^{4} - 8 T^{6} + T^{8}$$
$5$ $$T^{8}$$
$7$ $$( 7396 + 188 T^{2} + T^{4} )^{2}$$
$11$ $$( 72 + T^{2} )^{4}$$
$13$ $$( 100 + T^{2} )^{4}$$
$17$ $$( 11664 - 936 T^{2} + T^{4} )^{2}$$
$19$ $$( -296 - 16 T + T^{2} )^{4}$$
$23$ $$( 26244 - 396 T^{2} + T^{4} )^{2}$$
$29$ $$( 720 + T^{2} )^{4}$$
$31$ $$( 8 + T )^{8}$$
$37$ $$( 913936 + 3848 T^{2} + T^{4} )^{2}$$
$41$ $$( 944784 + 2664 T^{2} + T^{4} )^{2}$$
$43$ $$( 376996 + 2012 T^{2} + T^{4} )^{2}$$
$47$ $$( 16402500 - 9900 T^{2} + T^{4} )^{2}$$
$53$ $$( 11664 - 936 T^{2} + T^{4} )^{2}$$
$59$ $$( 3504384 + 6624 T^{2} + T^{4} )^{2}$$
$61$ $$( -1184 + 32 T + T^{2} )^{4}$$
$67$ $$( 34975396 + 15068 T^{2} + T^{4} )^{2}$$
$71$ $$( 1166400 + 5040 T^{2} + T^{4} )^{2}$$
$73$ $$( 1123600 + 7880 T^{2} + T^{4} )^{2}$$
$79$ $$( 424 + 56 T + T^{2} )^{4}$$
$83$ $$( 324 - 684 T^{2} + T^{4} )^{2}$$
$89$ $$( 186624 + 3744 T^{2} + T^{4} )^{2}$$
$97$ $$( 16289296 + 13832 T^{2} + T^{4} )^{2}$$