Properties

 Label 1200.3.bg.b Level $1200$ Weight $3$ Character orbit 1200.bg Analytic conductor $32.698$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

Related objects

Newspace parameters

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

Newform invariants

 Self dual: no Analytic conductor: $$32.6976317232$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(i, \sqrt{6})$$ Defining polynomial: $$x^{4} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 600) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + ( -4 + 4 \beta_{2} + 2 \beta_{3} ) q^{7} + 3 \beta_{2} q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + ( -4 + 4 \beta_{2} + 2 \beta_{3} ) q^{7} + 3 \beta_{2} q^{9} + ( 2 + 4 \beta_{1} - 4 \beta_{3} ) q^{11} -2 \beta_{1} q^{13} + ( 4 - 4 \beta_{2} + 4 \beta_{3} ) q^{17} + ( 8 \beta_{1} + 14 \beta_{2} + 8 \beta_{3} ) q^{19} + ( -6 - 4 \beta_{1} + 4 \beta_{3} ) q^{21} + ( -12 - 8 \beta_{1} - 12 \beta_{2} ) q^{23} + 3 \beta_{3} q^{27} + ( -8 \beta_{1} + 16 \beta_{2} - 8 \beta_{3} ) q^{29} + ( -14 + 16 \beta_{1} - 16 \beta_{3} ) q^{31} + ( 12 + 2 \beta_{1} + 12 \beta_{2} ) q^{33} + ( 8 - 8 \beta_{2} + 14 \beta_{3} ) q^{37} -6 \beta_{2} q^{39} + ( -2 - 24 \beta_{1} + 24 \beta_{3} ) q^{41} + ( 32 - 8 \beta_{1} + 32 \beta_{2} ) q^{43} + ( 20 - 20 \beta_{2} + 20 \beta_{3} ) q^{47} + ( -16 \beta_{1} + 5 \beta_{2} - 16 \beta_{3} ) q^{49} + ( -12 + 4 \beta_{1} - 4 \beta_{3} ) q^{51} + ( -8 - 36 \beta_{1} - 8 \beta_{2} ) q^{53} + ( -24 + 24 \beta_{2} + 14 \beta_{3} ) q^{57} + ( 20 \beta_{1} + 38 \beta_{2} + 20 \beta_{3} ) q^{59} + ( -30 - 32 \beta_{1} + 32 \beta_{3} ) q^{61} + ( -12 - 6 \beta_{1} - 12 \beta_{2} ) q^{63} + ( -24 + 24 \beta_{2} + 44 \beta_{3} ) q^{67} + ( -12 \beta_{1} - 24 \beta_{2} - 12 \beta_{3} ) q^{69} + ( 8 + 24 \beta_{1} - 24 \beta_{3} ) q^{71} + ( -64 - 12 \beta_{1} - 64 \beta_{2} ) q^{73} + ( -32 + 32 \beta_{2} + 36 \beta_{3} ) q^{77} -66 \beta_{2} q^{79} -9 q^{81} + ( -40 - 36 \beta_{1} - 40 \beta_{2} ) q^{83} + ( 24 - 24 \beta_{2} + 16 \beta_{3} ) q^{87} + ( -8 \beta_{1} + 10 \beta_{2} - 8 \beta_{3} ) q^{89} + ( 12 + 8 \beta_{1} - 8 \beta_{3} ) q^{91} + ( 48 - 14 \beta_{1} + 48 \beta_{2} ) q^{93} + ( 40 - 40 \beta_{2} + 76 \beta_{3} ) q^{97} + ( 12 \beta_{1} + 6 \beta_{2} + 12 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 16 q^{7} + O(q^{10})$$ $$4 q - 16 q^{7} + 8 q^{11} + 16 q^{17} - 24 q^{21} - 48 q^{23} - 56 q^{31} + 48 q^{33} + 32 q^{37} - 8 q^{41} + 128 q^{43} + 80 q^{47} - 48 q^{51} - 32 q^{53} - 96 q^{57} - 120 q^{61} - 48 q^{63} - 96 q^{67} + 32 q^{71} - 256 q^{73} - 128 q^{77} - 36 q^{81} - 160 q^{83} + 96 q^{87} + 48 q^{91} + 192 q^{93} + 160 q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/3$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$3 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$3 \beta_{3}$$

Character values

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

 $$n$$ $$401$$ $$577$$ $$751$$ $$901$$ $$\chi(n)$$ $$1$$ $$-\beta_{2}$$ $$1$$ $$1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
193.1
 −1.22474 − 1.22474i 1.22474 + 1.22474i −1.22474 + 1.22474i 1.22474 − 1.22474i
0 −1.22474 1.22474i 0 0 0 −1.55051 + 1.55051i 0 3.00000i 0
193.2 0 1.22474 + 1.22474i 0 0 0 −6.44949 + 6.44949i 0 3.00000i 0
1057.1 0 −1.22474 + 1.22474i 0 0 0 −1.55051 1.55051i 0 3.00000i 0
1057.2 0 1.22474 1.22474i 0 0 0 −6.44949 6.44949i 0 3.00000i 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
5.c odd 4 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1200.3.bg.b 4
4.b odd 2 1 600.3.u.f yes 4
5.b even 2 1 1200.3.bg.m 4
5.c odd 4 1 inner 1200.3.bg.b 4
5.c odd 4 1 1200.3.bg.m 4
12.b even 2 1 1800.3.v.q 4
20.d odd 2 1 600.3.u.a 4
20.e even 4 1 600.3.u.a 4
20.e even 4 1 600.3.u.f yes 4
60.h even 2 1 1800.3.v.j 4
60.l odd 4 1 1800.3.v.j 4
60.l odd 4 1 1800.3.v.q 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
600.3.u.a 4 20.d odd 2 1
600.3.u.a 4 20.e even 4 1
600.3.u.f yes 4 4.b odd 2 1
600.3.u.f yes 4 20.e even 4 1
1200.3.bg.b 4 1.a even 1 1 trivial
1200.3.bg.b 4 5.c odd 4 1 inner
1200.3.bg.m 4 5.b even 2 1
1200.3.bg.m 4 5.c odd 4 1
1800.3.v.j 4 60.h even 2 1
1800.3.v.j 4 60.l odd 4 1
1800.3.v.q 4 12.b even 2 1
1800.3.v.q 4 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{4} + 16 T_{7}^{3} + 128 T_{7}^{2} + 320 T_{7} + 400$$ acting on $$S_{3}^{\mathrm{new}}(1200, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$9 + T^{4}$$
$5$ $$T^{4}$$
$7$ $$400 + 320 T + 128 T^{2} + 16 T^{3} + T^{4}$$
$11$ $$( -92 - 4 T + T^{2} )^{2}$$
$13$ $$144 + T^{4}$$
$17$ $$256 + 256 T + 128 T^{2} - 16 T^{3} + T^{4}$$
$19$ $$35344 + 1160 T^{2} + T^{4}$$
$23$ $$9216 + 4608 T + 1152 T^{2} + 48 T^{3} + T^{4}$$
$29$ $$16384 + 1280 T^{2} + T^{4}$$
$31$ $$( -1340 + 28 T + T^{2} )^{2}$$
$37$ $$211600 + 14720 T + 512 T^{2} - 32 T^{3} + T^{4}$$
$41$ $$( -3452 + 4 T + T^{2} )^{2}$$
$43$ $$3444736 - 237568 T + 8192 T^{2} - 128 T^{3} + T^{4}$$
$47$ $$160000 + 32000 T + 3200 T^{2} - 80 T^{3} + T^{4}$$
$53$ $$14137600 - 120320 T + 512 T^{2} + 32 T^{3} + T^{4}$$
$59$ $$913936 + 7688 T^{2} + T^{4}$$
$61$ $$( -5244 + 60 T + T^{2} )^{2}$$
$67$ $$21678336 - 446976 T + 4608 T^{2} + 96 T^{3} + T^{4}$$
$71$ $$( -3392 - 16 T + T^{2} )^{2}$$
$73$ $$60217600 + 1986560 T + 32768 T^{2} + 256 T^{3} + T^{4}$$
$79$ $$( 4356 + T^{2} )^{2}$$
$83$ $$473344 - 110080 T + 12800 T^{2} + 160 T^{3} + T^{4}$$
$89$ $$80656 + 968 T^{2} + T^{4}$$
$97$ $$199600384 + 2260480 T + 12800 T^{2} - 160 T^{3} + T^{4}$$