# Properties

 Label 9072.2.a.cl Level 9072 Weight 2 Character orbit 9072.a Self dual yes Analytic conductor 72.440 Analytic rank 0 Dimension 4 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$9072 = 2^{4} \cdot 3^{4} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9072.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$72.4402847137$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.22545.1 Defining polynomial: $$x^{4} - x^{3} - 6 x^{2} + 5 x + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 504) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta_{1} - \beta_{3} ) q^{5} - q^{7} +O(q^{10})$$ $$q + ( 1 - \beta_{1} - \beta_{3} ) q^{5} - q^{7} + ( 2 - \beta_{1} + \beta_{3} ) q^{11} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{13} + ( 1 + 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{17} + ( \beta_{2} - \beta_{3} ) q^{19} + ( 1 + 2 \beta_{2} + \beta_{3} ) q^{23} + ( 3 - \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{25} + ( -2 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{29} + ( 3 - \beta_{2} ) q^{31} + ( -1 + \beta_{1} + \beta_{3} ) q^{35} + ( 7 - \beta_{2} ) q^{37} + ( 2 - 3 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{41} + ( -3 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{43} + ( -2 + \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{47} + q^{49} + ( 1 + 3 \beta_{1} - 3 \beta_{2} ) q^{53} + ( -1 - 4 \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{55} + ( -2 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{59} + ( 1 + 2 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{61} + ( -1 - 6 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{65} + ( -3 + 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{67} + ( -4 + 3 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{71} + ( 4 - 3 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{73} + ( -2 + \beta_{1} - \beta_{3} ) q^{77} + ( -4 - 5 \beta_{2} + \beta_{3} ) q^{79} + ( 1 + 2 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} ) q^{83} + ( 3 - \beta_{1} - 5 \beta_{2} - 5 \beta_{3} ) q^{85} + ( 4 + \beta_{1} + 3 \beta_{3} ) q^{89} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{91} + ( 6 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{95} + ( 2 - 6 \beta_{1} + \beta_{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{5} - 4q^{7} + O(q^{10})$$ $$4q + 4q^{5} - 4q^{7} + 6q^{11} + 3q^{13} + 8q^{17} + 2q^{19} + 5q^{23} + 14q^{25} + q^{29} + 11q^{31} - 4q^{35} + 27q^{37} + 2q^{41} - 11q^{43} - 7q^{47} + 4q^{49} + 4q^{53} - 6q^{55} - 9q^{59} + 7q^{61} - 9q^{65} - 12q^{67} - 12q^{71} + 13q^{73} - 6q^{77} - 22q^{79} + 6q^{83} + 11q^{85} + 14q^{89} - 3q^{91} + 23q^{95} + q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 6 x^{2} + 5 x + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - 5 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + 5 \beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 2.33866 −0.519120 −2.27060 1.45106
0 0 0 −2.43628 0 −1.00000 0 0 0
1.2 0 0 0 −0.936586 0 −1.00000 0 0 0
1.3 0 0 0 3.62393 0 −1.00000 0 0 0
1.4 0 0 0 3.74893 0 −1.00000 0 0 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
$$2$$ $$1$$
$$3$$ $$-1$$
$$7$$ $$1$$

## 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 9072.2.a.cl 4
3.b odd 2 1 9072.2.a.ce 4
4.b odd 2 1 4536.2.a.ba 4
9.c even 3 2 3024.2.r.l 8
9.d odd 6 2 1008.2.r.m 8
12.b even 2 1 4536.2.a.x 4
36.f odd 6 2 1512.2.r.d 8
36.h even 6 2 504.2.r.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.r.d 8 36.h even 6 2
1008.2.r.m 8 9.d odd 6 2
1512.2.r.d 8 36.f odd 6 2
3024.2.r.l 8 9.c even 3 2
4536.2.a.x 4 12.b even 2 1
4536.2.a.ba 4 4.b odd 2 1
9072.2.a.ce 4 3.b odd 2 1
9072.2.a.cl 4 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9072))$$:

 $$T_{5}^{4} - 4 T_{5}^{3} - 9 T_{5}^{2} + 29 T_{5} + 31$$ $$T_{11}^{4} - 6 T_{11}^{3} - 9 T_{11}^{2} + 81 T_{11} - 54$$ $$T_{13}^{4} - 3 T_{13}^{3} - 27 T_{13}^{2} + 99 T_{13} - 78$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 - 4 T + 11 T^{2} - 31 T^{3} + 91 T^{4} - 155 T^{5} + 275 T^{6} - 500 T^{7} + 625 T^{8}$$
$7$ $$( 1 + T )^{4}$$
$11$ $$1 - 6 T + 35 T^{2} - 117 T^{3} + 474 T^{4} - 1287 T^{5} + 4235 T^{6} - 7986 T^{7} + 14641 T^{8}$$
$13$ $$1 - 3 T + 25 T^{2} - 18 T^{3} + 234 T^{4} - 234 T^{5} + 4225 T^{6} - 6591 T^{7} + 28561 T^{8}$$
$17$ $$1 - 8 T + 35 T^{2} - 167 T^{3} + 886 T^{4} - 2839 T^{5} + 10115 T^{6} - 39304 T^{7} + 83521 T^{8}$$
$19$ $$1 - 2 T + 55 T^{2} - 41 T^{3} + 1309 T^{4} - 779 T^{5} + 19855 T^{6} - 13718 T^{7} + 130321 T^{8}$$
$23$ $$1 - 5 T + 53 T^{2} - 221 T^{3} + 1729 T^{4} - 5083 T^{5} + 28037 T^{6} - 60835 T^{7} + 279841 T^{8}$$
$29$ $$1 - T + 50 T^{2} - 172 T^{3} + 1192 T^{4} - 4988 T^{5} + 42050 T^{6} - 24389 T^{7} + 707281 T^{8}$$
$31$ $$1 - 11 T + 160 T^{2} - 1058 T^{3} + 8008 T^{4} - 32798 T^{5} + 153760 T^{6} - 327701 T^{7} + 923521 T^{8}$$
$37$ $$1 - 27 T + 412 T^{2} - 4104 T^{3} + 29436 T^{4} - 151848 T^{5} + 564028 T^{6} - 1367631 T^{7} + 1874161 T^{8}$$
$41$ $$1 - 2 T + 41 T^{2} - 257 T^{3} + 2008 T^{4} - 10537 T^{5} + 68921 T^{6} - 137842 T^{7} + 2825761 T^{8}$$
$43$ $$1 + 11 T + 148 T^{2} + 1112 T^{3} + 9532 T^{4} + 47816 T^{5} + 273652 T^{6} + 874577 T^{7} + 3418801 T^{8}$$
$47$ $$1 + 7 T + 110 T^{2} + 298 T^{3} + 4906 T^{4} + 14006 T^{5} + 242990 T^{6} + 726761 T^{7} + 4879681 T^{8}$$
$53$ $$1 - 4 T + 83 T^{2} + 197 T^{3} + 2722 T^{4} + 10441 T^{5} + 233147 T^{6} - 595508 T^{7} + 7890481 T^{8}$$
$59$ $$1 + 9 T + 200 T^{2} + 1458 T^{3} + 16542 T^{4} + 86022 T^{5} + 696200 T^{6} + 1848411 T^{7} + 12117361 T^{8}$$
$61$ $$1 - 7 T + 193 T^{2} - 1333 T^{3} + 16135 T^{4} - 81313 T^{5} + 718153 T^{6} - 1588867 T^{7} + 13845841 T^{8}$$
$67$ $$1 + 12 T + 283 T^{2} + 2367 T^{3} + 28956 T^{4} + 158589 T^{5} + 1270387 T^{6} + 3609156 T^{7} + 20151121 T^{8}$$
$71$ $$1 + 12 T + 167 T^{2} + 591 T^{3} + 7587 T^{4} + 41961 T^{5} + 841847 T^{6} + 4294932 T^{7} + 25411681 T^{8}$$
$73$ $$1 - 13 T + 232 T^{2} - 1504 T^{3} + 18820 T^{4} - 109792 T^{5} + 1236328 T^{6} - 5057221 T^{7} + 28398241 T^{8}$$
$79$ $$1 + 22 T + 247 T^{2} + 1909 T^{3} + 16129 T^{4} + 150811 T^{5} + 1541527 T^{6} + 10846858 T^{7} + 38950081 T^{8}$$
$83$ $$1 - 6 T + 149 T^{2} - 1299 T^{3} + 11256 T^{4} - 107817 T^{5} + 1026461 T^{6} - 3430722 T^{7} + 47458321 T^{8}$$
$89$ $$1 - 14 T + 323 T^{2} - 2963 T^{3} + 41152 T^{4} - 263707 T^{5} + 2558483 T^{6} - 9869566 T^{7} + 62742241 T^{8}$$
$97$ $$1 - T + 124 T^{2} - 490 T^{3} + 19024 T^{4} - 47530 T^{5} + 1166716 T^{6} - 912673 T^{7} + 88529281 T^{8}$$