# Properties

 Label 1200.4.a.bn Level $1200$ Weight $4$ Character orbit 1200.a Self dual yes Analytic conductor $70.802$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1200 = 2^{4} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1200.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$70.8022920069$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{41})$$ Defining polynomial: $$x^{2} - x - 10$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 15) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{41}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -3 q^{3} + ( -3 - 3 \beta ) q^{7} + 9 q^{9} +O(q^{10})$$ $$q -3 q^{3} + ( -3 - 3 \beta ) q^{7} + 9 q^{9} + ( 21 - 3 \beta ) q^{11} + ( 39 + 3 \beta ) q^{13} + ( 51 - 5 \beta ) q^{17} + ( -28 - 12 \beta ) q^{19} + ( 9 + 9 \beta ) q^{21} + ( 24 + 4 \beta ) q^{23} -27 q^{27} + ( -159 - 21 \beta ) q^{29} + ( -26 - 6 \beta ) q^{31} + ( -63 + 9 \beta ) q^{33} + ( 153 - 27 \beta ) q^{37} + ( -117 - 9 \beta ) q^{39} + ( -204 + 6 \beta ) q^{41} + ( -60 + 48 \beta ) q^{43} + ( -90 - 46 \beta ) q^{47} + ( 35 + 18 \beta ) q^{49} + ( -153 + 15 \beta ) q^{51} + ( 201 + 41 \beta ) q^{53} + ( 84 + 36 \beta ) q^{57} + ( 93 - 3 \beta ) q^{59} + ( 170 - 48 \beta ) q^{61} + ( -27 - 27 \beta ) q^{63} + ( -366 + 30 \beta ) q^{67} + ( -72 - 12 \beta ) q^{69} + ( 18 + 90 \beta ) q^{71} + ( 666 - 54 \beta ) q^{73} + ( 306 - 54 \beta ) q^{77} + ( -190 + 150 \beta ) q^{79} + 81 q^{81} + ( 492 - 104 \beta ) q^{83} + ( 477 + 63 \beta ) q^{87} + ( 558 + 72 \beta ) q^{89} + ( -486 - 126 \beta ) q^{91} + ( 78 + 18 \beta ) q^{93} + ( -384 + 120 \beta ) q^{97} + ( 189 - 27 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 6 q^{3} - 6 q^{7} + 18 q^{9} + O(q^{10})$$ $$2 q - 6 q^{3} - 6 q^{7} + 18 q^{9} + 42 q^{11} + 78 q^{13} + 102 q^{17} - 56 q^{19} + 18 q^{21} + 48 q^{23} - 54 q^{27} - 318 q^{29} - 52 q^{31} - 126 q^{33} + 306 q^{37} - 234 q^{39} - 408 q^{41} - 120 q^{43} - 180 q^{47} + 70 q^{49} - 306 q^{51} + 402 q^{53} + 168 q^{57} + 186 q^{59} + 340 q^{61} - 54 q^{63} - 732 q^{67} - 144 q^{69} + 36 q^{71} + 1332 q^{73} + 612 q^{77} - 380 q^{79} + 162 q^{81} + 984 q^{83} + 954 q^{87} + 1116 q^{89} - 972 q^{91} + 156 q^{93} - 768 q^{97} + 378 q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 3.70156 −2.70156
0 −3.00000 0 0 0 −22.2094 0 9.00000 0
1.2 0 −3.00000 0 0 0 16.2094 0 9.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$
$$5$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1200.4.a.bn 2
4.b odd 2 1 75.4.a.f 2
5.b even 2 1 1200.4.a.bt 2
5.c odd 4 2 240.4.f.f 4
12.b even 2 1 225.4.a.i 2
15.e even 4 2 720.4.f.j 4
20.d odd 2 1 75.4.a.c 2
20.e even 4 2 15.4.b.a 4
40.i odd 4 2 960.4.f.p 4
40.k even 4 2 960.4.f.q 4
60.h even 2 1 225.4.a.o 2
60.l odd 4 2 45.4.b.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.4.b.a 4 20.e even 4 2
45.4.b.b 4 60.l odd 4 2
75.4.a.c 2 20.d odd 2 1
75.4.a.f 2 4.b odd 2 1
225.4.a.i 2 12.b even 2 1
225.4.a.o 2 60.h even 2 1
240.4.f.f 4 5.c odd 4 2
720.4.f.j 4 15.e even 4 2
960.4.f.p 4 40.i odd 4 2
960.4.f.q 4 40.k even 4 2
1200.4.a.bn 2 1.a even 1 1 trivial
1200.4.a.bt 2 5.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1200))$$:

 $$T_{7}^{2} + 6 T_{7} - 360$$ $$T_{11}^{2} - 42 T_{11} + 72$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( 3 + T )^{2}$$
$5$ $$T^{2}$$
$7$ $$-360 + 6 T + T^{2}$$
$11$ $$72 - 42 T + T^{2}$$
$13$ $$1152 - 78 T + T^{2}$$
$17$ $$1576 - 102 T + T^{2}$$
$19$ $$-5120 + 56 T + T^{2}$$
$23$ $$-80 - 48 T + T^{2}$$
$29$ $$7200 + 318 T + T^{2}$$
$31$ $$-800 + 52 T + T^{2}$$
$37$ $$-6480 - 306 T + T^{2}$$
$41$ $$40140 + 408 T + T^{2}$$
$43$ $$-90864 + 120 T + T^{2}$$
$47$ $$-78656 + 180 T + T^{2}$$
$53$ $$-28520 - 402 T + T^{2}$$
$59$ $$8280 - 186 T + T^{2}$$
$61$ $$-65564 - 340 T + T^{2}$$
$67$ $$97056 + 732 T + T^{2}$$
$71$ $$-331776 - 36 T + T^{2}$$
$73$ $$324000 - 1332 T + T^{2}$$
$79$ $$-886400 + 380 T + T^{2}$$
$83$ $$-201392 - 984 T + T^{2}$$
$89$ $$98820 - 1116 T + T^{2}$$
$97$ $$-442944 + 768 T + T^{2}$$