# Properties

 Label 9200.2.a.de Level $9200$ Weight $2$ Character orbit 9200.a Self dual yes Analytic conductor $73.462$ Analytic rank $0$ Dimension $8$ 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: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - 3 x^{7} - 13 x^{6} + 38 x^{5} + 41 x^{4} - 123 x^{3} + 15 x^{2} + 32 x - 8$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 920) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + ( 1 + \beta_{7} ) q^{7} + ( 2 + \beta_{3} + \beta_{5} + \beta_{7} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + ( 1 + \beta_{7} ) q^{7} + ( 2 + \beta_{3} + \beta_{5} + \beta_{7} ) q^{9} + ( -1 - \beta_{3} ) q^{11} + ( 1 + \beta_{2} ) q^{13} + ( -1 + \beta_{4} + \beta_{6} ) q^{17} + ( -1 + \beta_{4} + \beta_{7} ) q^{19} + ( -2 + 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{21} - q^{23} + ( 1 + \beta_{1} + \beta_{3} + \beta_{6} + \beta_{7} ) q^{27} + ( 3 - \beta_{1} + \beta_{4} + \beta_{6} ) q^{29} + ( -2 + \beta_{1} - \beta_{3} - \beta_{4} - \beta_{7} ) q^{31} + ( -1 - 2 \beta_{1} - 2 \beta_{4} - \beta_{7} ) q^{33} + ( 2 - \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{37} + ( -2 + 2 \beta_{1} + \beta_{2} + \beta_{4} - 2 \beta_{5} + \beta_{6} ) q^{39} + ( 2 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{41} + ( 3 + \beta_{1} - \beta_{3} + \beta_{5} - \beta_{6} ) q^{43} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{47} + ( 5 + \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{6} + \beta_{7} ) q^{49} + ( 2 - \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{6} ) q^{51} + ( 3 - 3 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{5} + \beta_{7} ) q^{53} + ( -2 + 2 \beta_{2} + 2 \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{7} ) q^{57} + ( -5 + \beta_{1} - \beta_{4} - \beta_{5} ) q^{59} + ( 3 - \beta_{2} + \beta_{5} + \beta_{7} ) q^{61} + ( 6 + \beta_{2} + 2 \beta_{3} + 3 \beta_{4} + \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{63} + ( -2 + 2 \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{67} -\beta_{1} q^{69} + ( -1 + 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{71} + ( 2 - 2 \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{6} + \beta_{7} ) q^{73} + ( 4 - \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} - 4 \beta_{6} ) q^{77} + ( 2 \beta_{1} + \beta_{2} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{79} + ( 3 \beta_{1} + \beta_{2} - 2 \beta_{5} + 2 \beta_{6} ) q^{81} + ( 4 + \beta_{4} + 2 \beta_{6} ) q^{83} + ( -3 + 3 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{87} + ( 4 + \beta_{2} - 2 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{89} + ( 4 + 3 \beta_{1} + \beta_{2} - 2 \beta_{6} + 3 \beta_{7} ) q^{91} + ( 6 - 4 \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} + 3 \beta_{5} - \beta_{6} + \beta_{7} ) q^{93} + ( 1 - \beta_{2} - \beta_{4} + \beta_{5} + \beta_{7} ) q^{97} + ( -5 - 2 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 3 q^{3} + 7 q^{7} + 11 q^{9} + O(q^{10})$$ $$8 q + 3 q^{3} + 7 q^{7} + 11 q^{9} - 7 q^{11} + 11 q^{13} - 7 q^{17} - 11 q^{19} - 8 q^{23} + 12 q^{27} + 22 q^{29} - 9 q^{31} - 9 q^{33} + 4 q^{37} + 7 q^{41} + 22 q^{43} + 4 q^{47} + 39 q^{49} + 19 q^{51} + 4 q^{53} - 32 q^{59} + 17 q^{61} + 44 q^{63} - 4 q^{67} - 3 q^{69} - 15 q^{71} + 6 q^{73} + 18 q^{77} + 2 q^{79} + 24 q^{81} + 36 q^{83} - 4 q^{87} + 46 q^{89} + 35 q^{91} + 20 q^{93} + 3 q^{97} - 61 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 3 x^{7} - 13 x^{6} + 38 x^{5} + 41 x^{4} - 123 x^{3} + 15 x^{2} + 32 x - 8$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{4} - 2 \nu^{3} - 7 \nu^{2} + 11 \nu + 1$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{6} - 4 \nu^{5} - 5 \nu^{4} + 29 \nu^{3} - 8 \nu^{2} - 21 \nu - 2$$$$)/2$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{7} - 3 \nu^{6} - 11 \nu^{5} + 30 \nu^{4} + 33 \nu^{3} - 69 \nu^{2} - 17 \nu + 12$$$$)/2$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{7} - 3 \nu^{6} - 13 \nu^{5} + 36 \nu^{4} + 45 \nu^{3} - 107 \nu^{2} - 9 \nu + 18$$$$)/2$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{7} - 3 \nu^{6} - 13 \nu^{5} + 36 \nu^{4} + 47 \nu^{3} - 109 \nu^{2} - 23 \nu + 26$$$$)/2$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{7} + 2 \nu^{6} + 17 \nu^{5} - 31 \nu^{4} - 74 \nu^{3} + 117 \nu^{2} + 30 \nu - 26$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{7} + \beta_{5} + \beta_{3} + 5$$ $$\nu^{3}$$ $$=$$ $$\beta_{7} + \beta_{6} + \beta_{3} + 7 \beta_{1} + 1$$ $$\nu^{4}$$ $$=$$ $$9 \beta_{7} + 2 \beta_{6} + 7 \beta_{5} + 9 \beta_{3} + \beta_{2} + 3 \beta_{1} + 36$$ $$\nu^{5}$$ $$=$$ $$14 \beta_{7} + 12 \beta_{6} + \beta_{5} + \beta_{4} + 14 \beta_{3} + 3 \beta_{2} + 55 \beta_{1} + 22$$ $$\nu^{6}$$ $$=$$ $$80 \beta_{7} + 29 \beta_{6} + 47 \beta_{5} + 4 \beta_{4} + 82 \beta_{3} + 17 \beta_{2} + 53 \beta_{1} + 281$$ $$\nu^{7}$$ $$=$$ $$160 \beta_{7} + 126 \beta_{6} + 11 \beta_{5} + 25 \beta_{4} + 166 \beta_{3} + 54 \beta_{2} + 460 \beta_{1} + 305$$

## 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
 −2.54092 −2.51561 −0.540724 0.296848 0.493532 1.69755 2.89996 3.20935
0 −2.54092 0 0 0 −0.780573 0 3.45626 0
1.2 0 −2.51561 0 0 0 4.64022 0 3.32827 0
1.3 0 −0.540724 0 0 0 1.15693 0 −2.70762 0
1.4 0 0.296848 0 0 0 −3.46037 0 −2.91188 0
1.5 0 0.493532 0 0 0 4.54439 0 −2.75643 0
1.6 0 1.69755 0 0 0 −4.22860 0 −0.118308 0
1.7 0 2.89996 0 0 0 0.580879 0 5.40975 0
1.8 0 3.20935 0 0 0 4.54713 0 7.29996 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.8 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.de 8
4.b odd 2 1 4600.2.a.bj 8
5.b even 2 1 9200.2.a.dd 8
5.c odd 4 2 1840.2.e.h 16
20.d odd 2 1 4600.2.a.bk 8
20.e even 4 2 920.2.e.c 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.2.e.c 16 20.e even 4 2
1840.2.e.h 16 5.c odd 4 2
4600.2.a.bj 8 4.b odd 2 1
4600.2.a.bk 8 20.d odd 2 1
9200.2.a.dd 8 5.b even 2 1
9200.2.a.de 8 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(9200))$$:

 $$T_{3}^{8} - \cdots$$ $$T_{7}^{8} - \cdots$$ $$T_{11}^{8} + \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$-8 + 32 T + 15 T^{2} - 123 T^{3} + 41 T^{4} + 38 T^{5} - 13 T^{6} - 3 T^{7} + T^{8}$$
$5$ $$T^{8}$$
$7$ $$-736 + 1056 T + 1316 T^{2} - 1720 T^{3} + 20 T^{4} + 218 T^{5} - 23 T^{6} - 7 T^{7} + T^{8}$$
$11$ $$440 - 5456 T + 28 T^{2} + 2364 T^{3} + 162 T^{4} - 248 T^{5} - 29 T^{6} + 7 T^{7} + T^{8}$$
$13$ $$-14020 - 1696 T + 11569 T^{2} - 2883 T^{3} - 1115 T^{4} + 438 T^{5} - 7 T^{6} - 11 T^{7} + T^{8}$$
$17$ $$176 - 64 T - 2700 T^{2} + 2292 T^{3} + 706 T^{4} - 296 T^{5} - 49 T^{6} + 7 T^{7} + T^{8}$$
$19$ $$11192 + 23600 T + 16488 T^{2} + 3044 T^{3} - 1116 T^{4} - 432 T^{5} - 7 T^{6} + 11 T^{7} + T^{8}$$
$23$ $$( 1 + T )^{8}$$
$29$ $$-400 - 4632 T - 344 T^{2} + 5614 T^{3} - 2107 T^{4} - 96 T^{5} + 146 T^{6} - 22 T^{7} + T^{8}$$
$31$ $$-287276 - 183760 T + 5147 T^{2} + 20781 T^{3} + 1671 T^{4} - 766 T^{5} - 79 T^{6} + 9 T^{7} + T^{8}$$
$37$ $$16096 - 3792 T - 22120 T^{2} - 112 T^{3} + 3340 T^{4} + 212 T^{5} - 106 T^{6} - 4 T^{7} + T^{8}$$
$41$ $$1197584 + 601624 T - 181313 T^{2} - 44475 T^{3} + 8683 T^{4} + 1026 T^{5} - 163 T^{6} - 7 T^{7} + T^{8}$$
$43$ $$-244448 - 74096 T + 72296 T^{2} + 4616 T^{3} - 6976 T^{4} + 716 T^{5} + 108 T^{6} - 22 T^{7} + T^{8}$$
$47$ $$-73520 - 138952 T - 20048 T^{2} + 27562 T^{3} + 8173 T^{4} - 34 T^{5} - 166 T^{6} - 4 T^{7} + T^{8}$$
$53$ $$202880 - 254336 T - 228000 T^{2} - 13376 T^{3} + 12488 T^{4} + 568 T^{5} - 206 T^{6} - 4 T^{7} + T^{8}$$
$59$ $$29696 + 1536 T - 24896 T^{2} - 9024 T^{3} + 2464 T^{4} + 1872 T^{5} + 376 T^{6} + 32 T^{7} + T^{8}$$
$61$ $$200 + 320 T - 592 T^{2} - 1276 T^{3} - 400 T^{4} + 208 T^{5} + 49 T^{6} - 17 T^{7} + T^{8}$$
$67$ $$680608 - 115216 T - 135192 T^{2} + 13376 T^{3} + 8428 T^{4} - 388 T^{5} - 174 T^{6} + 4 T^{7} + T^{8}$$
$71$ $$-8900000 - 4219360 T + 126317 T^{2} + 242207 T^{3} + 10847 T^{4} - 3982 T^{5} - 253 T^{6} + 15 T^{7} + T^{8}$$
$73$ $$-557680 - 780576 T - 245128 T^{2} + 28224 T^{3} + 16801 T^{4} + 298 T^{5} - 258 T^{6} - 6 T^{7} + T^{8}$$
$79$ $$-5248 + 34496 T - 13136 T^{2} - 11992 T^{3} + 4108 T^{4} + 984 T^{5} - 206 T^{6} - 2 T^{7} + T^{8}$$
$83$ $$-4183520 - 553168 T + 505448 T^{2} + 25120 T^{3} - 22872 T^{4} + 708 T^{5} + 352 T^{6} - 36 T^{7} + T^{8}$$
$89$ $$12804160 - 4496608 T - 440608 T^{2} + 349256 T^{3} - 36348 T^{4} - 2600 T^{5} + 710 T^{6} - 46 T^{7} + T^{8}$$
$97$ $$2960 + 3344 T - 8172 T^{2} + 1216 T^{3} + 1712 T^{4} - 174 T^{5} - 123 T^{6} - 3 T^{7} + T^{8}$$