# Properties

 Label 9200.2.a.cx Level $9200$ Weight $2$ Character orbit 9200.a Self dual yes Analytic conductor $73.462$ Analytic rank $1$ Dimension $6$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$9200 = 2^{4} \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9200.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$73.4623698596$$ Analytic rank: $$1$$ Dimension: $$6$$ Coefficient field: 6.6.143376304.1 Defining polynomial: $$x^{6} - 12 x^{4} + 22 x^{2} - 6 x - 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 460) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 - \beta_{2} ) q^{3} + ( -2 + \beta_{3} ) q^{7} + ( 2 + \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})$$ $$q + ( -1 - \beta_{2} ) q^{3} + ( -2 + \beta_{3} ) q^{7} + ( 2 + \beta_{1} + \beta_{2} ) q^{9} + ( -1 + \beta_{2} - \beta_{4} + \beta_{5} ) q^{11} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} ) q^{13} + ( 1 - \beta_{2} - \beta_{5} ) q^{17} + ( -1 - \beta_{2} + \beta_{3} - \beta_{5} ) q^{19} + ( 1 - 2 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{21} - q^{23} + ( -3 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} ) q^{27} + ( 2 + 2 \beta_{2} - \beta_{5} ) q^{29} + ( -\beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} ) q^{31} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{33} + ( 2 - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{37} + ( 1 - \beta_{2} - 3 \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{39} + ( -\beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{41} + ( -1 - \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{43} + ( -3 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} ) q^{47} + ( 4 + \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{5} ) q^{49} + ( 2 - \beta_{3} - \beta_{4} ) q^{51} + ( 1 - \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{5} ) q^{53} + ( 3 - 2 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{57} + ( 3 + 2 \beta_{1} - \beta_{2} + \beta_{3} + 3 \beta_{4} - \beta_{5} ) q^{59} + ( 1 - \beta_{2} + \beta_{3} + 2 \beta_{4} - 3 \beta_{5} ) q^{61} + ( -3 + \beta_{1} - \beta_{2} + 4 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} ) q^{63} + ( -4 + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{67} + ( 1 + \beta_{2} ) q^{69} + ( 4 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{5} ) q^{71} + ( -3 - 2 \beta_{1} + \beta_{2} + 2 \beta_{5} ) q^{73} + ( 4 - 4 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} ) q^{77} + ( -1 - \beta_{2} - \beta_{3} - \beta_{5} ) q^{79} + ( 1 + \beta_{1} + 2 \beta_{2} + 3 \beta_{3} - \beta_{4} ) q^{81} + ( -3 - 2 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} + 3 \beta_{5} ) q^{83} + ( -11 - 3 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} ) q^{87} + ( -5 - 2 \beta_{1} - 3 \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{89} + ( 1 - 2 \beta_{1} + 7 \beta_{2} - 3 \beta_{4} + \beta_{5} ) q^{91} + ( -1 + 2 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} ) q^{93} + ( 1 - 3 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{97} + ( 4 + 4 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 4 q^{3} - 9 q^{7} + 10 q^{9} + O(q^{10})$$ $$6 q - 4 q^{3} - 9 q^{7} + 10 q^{9} - 2 q^{11} + 8 q^{13} + 5 q^{17} - 4 q^{19} - 6 q^{23} - 22 q^{27} + 5 q^{29} - 9 q^{31} - 10 q^{33} + 21 q^{37} + 8 q^{39} - q^{41} - 16 q^{43} - 16 q^{47} + 19 q^{49} + 12 q^{51} - q^{53} + 12 q^{57} + 11 q^{59} - 4 q^{61} - 19 q^{63} - 25 q^{67} + 4 q^{69} + 17 q^{71} - 14 q^{73} + 20 q^{77} - 10 q^{79} + 14 q^{81} - 21 q^{83} - 64 q^{87} - 24 q^{89} + 4 q^{91} + 4 q^{97} + 16 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 12 x^{4} + 22 x^{2} - 6 x - 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{5} - 3 \nu^{4} + 11 \nu^{3} + 33 \nu^{2} - 3 \nu - 27$$$$)/8$$ $$\beta_{2}$$ $$=$$ $$($$$$3 \nu^{5} + \nu^{4} - 33 \nu^{3} - 11 \nu^{2} + 41 \nu - 7$$$$)/8$$ $$\beta_{3}$$ $$=$$ $$($$$$-7 \nu^{5} + 3 \nu^{4} + 85 \nu^{3} - 25 \nu^{2} - 157 \nu + 43$$$$)/16$$ $$\beta_{4}$$ $$=$$ $$($$$$9 \nu^{5} + 3 \nu^{4} - 107 \nu^{3} - 41 \nu^{2} + 195 \nu + 11$$$$)/16$$ $$\beta_{5}$$ $$=$$ $$($$$$-9 \nu^{5} - 3 \nu^{4} + 107 \nu^{3} + 25 \nu^{2} - 179 \nu + 53$$$$)/16$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{4} + \beta_{3} + \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-2 \beta_{5} - \beta_{4} + \beta_{3} + \beta_{1} + 8$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$\beta_{5} + 3 \beta_{4} + 4 \beta_{3} + 3 \beta_{2} + 4 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$($$$$-22 \beta_{5} - 7 \beta_{4} + 15 \beta_{3} - 2 \beta_{2} + 9 \beta_{1} + 66$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$22 \beta_{5} + 51 \beta_{4} + 73 \beta_{3} + 72 \beta_{2} + 75 \beta_{1} + 12$$$$)/2$$

## 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
 3.16223 −1.65047 0.420790 −0.116918 −3.08006 1.26443
0 −3.21923 0 0 0 2.43185 0 7.36343 0
1.2 0 −2.80150 0 0 0 −4.50896 0 4.84843 0
1.3 0 −1.73961 0 0 0 −3.32224 0 0.0262434 0
1.4 0 0.486391 0 0 0 1.80495 0 −2.76342 0
1.5 0 0.873449 0 0 0 −0.992530 0 −2.23709 0
1.6 0 2.40050 0 0 0 −4.41307 0 2.76241 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$-1$$
$$23$$ $$1$$

## 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 9200.2.a.cx 6
4.b odd 2 1 2300.2.a.o 6
5.b even 2 1 9200.2.a.cy 6
5.c odd 4 2 1840.2.e.f 12
20.d odd 2 1 2300.2.a.n 6
20.e even 4 2 460.2.c.a 12
60.l odd 4 2 4140.2.f.b 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
460.2.c.a 12 20.e even 4 2
1840.2.e.f 12 5.c odd 4 2
2300.2.a.n 6 20.d odd 2 1
2300.2.a.o 6 4.b odd 2 1
4140.2.f.b 12 60.l odd 4 2
9200.2.a.cx 6 1.a even 1 1 trivial
9200.2.a.cy 6 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9200))$$:

 $$T_{3}^{6} + 4 T_{3}^{5} - 6 T_{3}^{4} - 30 T_{3}^{3} + 5 T_{3}^{2} + 38 T_{3} - 16$$ $$T_{7}^{6} + 9 T_{7}^{5} + 10 T_{7}^{4} - 88 T_{7}^{3} - 152 T_{7}^{2} + 228 T_{7} + 288$$ $$T_{11}^{6} + 2 T_{11}^{5} - 32 T_{11}^{4} - 28 T_{11}^{3} + 212 T_{11}^{2} + 144 T_{11} - 256$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$-16 + 38 T + 5 T^{2} - 30 T^{3} - 6 T^{4} + 4 T^{5} + T^{6}$$
$5$ $$T^{6}$$
$7$ $$288 + 228 T - 152 T^{2} - 88 T^{3} + 10 T^{4} + 9 T^{5} + T^{6}$$
$11$ $$-256 + 144 T + 212 T^{2} - 28 T^{3} - 32 T^{4} + 2 T^{5} + T^{6}$$
$13$ $$1184 - 1354 T + 45 T^{2} + 222 T^{3} - 22 T^{4} - 8 T^{5} + T^{6}$$
$17$ $$16 - 60 T + 36 T^{2} + 30 T^{3} - 16 T^{4} - 5 T^{5} + T^{6}$$
$19$ $$256 + 848 T + 236 T^{2} - 140 T^{3} - 42 T^{4} + 4 T^{5} + T^{6}$$
$23$ $$( 1 + T )^{6}$$
$29$ $$11862 - 9453 T + 1229 T^{2} + 450 T^{3} - 84 T^{4} - 5 T^{5} + T^{6}$$
$31$ $$916 - 1987 T - 2651 T^{2} - 838 T^{3} - 58 T^{4} + 9 T^{5} + T^{6}$$
$37$ $$63216 - 12012 T - 7096 T^{2} + 1444 T^{3} + 42 T^{4} - 21 T^{5} + T^{6}$$
$41$ $$-2 + 477 T + 257 T^{2} - 158 T^{3} - 64 T^{4} + T^{5} + T^{6}$$
$43$ $$91648 + 17072 T - 6612 T^{2} - 1764 T^{3} - 42 T^{4} + 16 T^{5} + T^{6}$$
$47$ $$-464 + 2178 T - 1959 T^{2} - 678 T^{3} + 14 T^{4} + 16 T^{5} + T^{6}$$
$53$ $$12224 - 13552 T + 3648 T^{2} + 280 T^{3} - 136 T^{4} + T^{5} + T^{6}$$
$59$ $$-360576 - 63024 T + 12832 T^{2} + 1692 T^{3} - 188 T^{4} - 11 T^{5} + T^{6}$$
$61$ $$-171088 + 10976 T + 10996 T^{2} - 532 T^{3} - 198 T^{4} + 4 T^{5} + T^{6}$$
$67$ $$7424 - 2772 T - 6472 T^{2} - 692 T^{3} + 142 T^{4} + 25 T^{5} + T^{6}$$
$71$ $$-16108 - 20769 T + 569 T^{2} + 1570 T^{3} - 62 T^{4} - 17 T^{5} + T^{6}$$
$73$ $$5504 + 13624 T - 4847 T^{2} - 1914 T^{3} - 90 T^{4} + 14 T^{5} + T^{6}$$
$79$ $$-128 + 584 T + 84 T^{2} - 208 T^{3} - 10 T^{4} + 10 T^{5} + T^{6}$$
$83$ $$-633344 + 145852 T + 15572 T^{2} - 3726 T^{3} - 168 T^{4} + 21 T^{5} + T^{6}$$
$89$ $$180432 + 167568 T - 3212 T^{2} - 5172 T^{3} - 142 T^{4} + 24 T^{5} + T^{6}$$
$97$ $$45728 - 43992 T + 10148 T^{2} + 932 T^{3} - 344 T^{4} - 4 T^{5} + T^{6}$$