# Properties

 Label 9072.2.a.cg Level $9072$ Weight $2$ Character orbit 9072.a Self dual yes Analytic conductor $72.440$ Analytic rank $1$ 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: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.45729.1 Defining polynomial: $$x^{4} - x^{3} - 11 x^{2} + 12 x - 3$$ 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{1} ) q^{5} + q^{7} +O(q^{10})$$ $$q + ( -1 + \beta_{1} ) q^{5} + q^{7} + ( -2 + \beta_{1} ) q^{11} + ( -1 - \beta_{2} ) q^{13} + ( -\beta_{1} + \beta_{2} - \beta_{3} ) q^{17} + ( 1 - \beta_{1} + \beta_{3} ) q^{19} + ( \beta_{2} - \beta_{3} ) q^{23} + ( 2 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{25} + ( -3 - 2 \beta_{2} + \beta_{3} ) q^{29} + ( 1 - \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{31} + ( -1 + \beta_{1} ) q^{35} + ( 1 - \beta_{1} ) q^{37} + ( 2 + \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{41} + ( 1 + \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{43} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{47} + q^{49} + ( -4 \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{53} + ( 8 - 3 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{55} + ( -3 + \beta_{2} + 3 \beta_{3} ) q^{59} + ( -6 - 3 \beta_{2} + \beta_{3} ) q^{61} + ( 1 - 3 \beta_{1} ) q^{65} + ( 4 - \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{67} + ( -1 + \beta_{3} ) q^{71} + ( -5 - \beta_{1} - 3 \beta_{3} ) q^{73} + ( -2 + \beta_{1} ) q^{77} + ( 5 + \beta_{1} + 3 \beta_{2} + 3 \beta_{3} ) q^{79} + ( -2 - \beta_{1} + \beta_{3} ) q^{83} + ( -3 + 2 \beta_{1} - \beta_{2} ) q^{85} + ( -3 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{89} + ( -1 - \beta_{2} ) q^{91} + ( -10 + 3 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} ) q^{95} + ( -3 - 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 3q^{5} + 4q^{7} + O(q^{10})$$ $$4q - 3q^{5} + 4q^{7} - 7q^{11} - 3q^{13} - 3q^{17} + 4q^{19} - 2q^{23} + 5q^{25} - 9q^{29} + 3q^{31} - 3q^{35} + 3q^{37} + 9q^{41} + 8q^{43} - 3q^{47} + 4q^{49} - 6q^{53} + 28q^{55} - 10q^{59} - 20q^{61} + q^{65} + 11q^{67} - 3q^{71} - 24q^{73} - 7q^{77} + 21q^{79} - 8q^{83} - 9q^{85} - 6q^{89} - 3q^{91} - 36q^{95} - 16q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 11 x^{2} + 12 x - 3$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{3} - 11 \nu + 3$$ $$\beta_{3}$$ $$=$$ $$-2 \nu^{3} + \nu^{2} + 22 \nu - 12$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + 2 \beta_{2} + 6$$ $$\nu^{3}$$ $$=$$ $$\beta_{2} + 11 \beta_{1} - 3$$

## 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
 −3.38095 0.399463 0.670984 3.31050
0 0 0 −4.38095 0 1.00000 0 0 0
1.2 0 0 0 −0.600537 0 1.00000 0 0 0
1.3 0 0 0 −0.329016 0 1.00000 0 0 0
1.4 0 0 0 2.31050 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.cg 4
3.b odd 2 1 9072.2.a.cj 4
4.b odd 2 1 4536.2.a.y 4
9.c even 3 2 3024.2.r.m 8
9.d odd 6 2 1008.2.r.l 8
12.b even 2 1 4536.2.a.z 4
36.f odd 6 2 1512.2.r.e 8
36.h even 6 2 504.2.r.e 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.r.e 8 36.h even 6 2
1008.2.r.l 8 9.d odd 6 2
1512.2.r.e 8 36.f odd 6 2
3024.2.r.m 8 9.c even 3 2
4536.2.a.y 4 4.b odd 2 1
4536.2.a.z 4 12.b even 2 1
9072.2.a.cg 4 1.a even 1 1 trivial
9072.2.a.cj 4 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(9072))$$:

 $$T_{5}^{4} + 3 T_{5}^{3} - 8 T_{5}^{2} - 9 T_{5} - 2$$ $$T_{11}^{4} + 7 T_{11}^{3} + 7 T_{11}^{2} - 12 T_{11} - 15$$ $$T_{13}^{4} + 3 T_{13}^{3} - 11 T_{13}^{2} - 27 T_{13} + 10$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 + 3 T + 12 T^{2} + 36 T^{3} + 68 T^{4} + 180 T^{5} + 300 T^{6} + 375 T^{7} + 625 T^{8}$$
$7$ $$( 1 - T )^{4}$$
$11$ $$1 + 7 T + 51 T^{2} + 219 T^{3} + 865 T^{4} + 2409 T^{5} + 6171 T^{6} + 9317 T^{7} + 14641 T^{8}$$
$13$ $$1 + 3 T + 41 T^{2} + 90 T^{3} + 738 T^{4} + 1170 T^{5} + 6929 T^{6} + 6591 T^{7} + 28561 T^{8}$$
$17$ $$1 + 3 T + 41 T^{2} + 189 T^{3} + 807 T^{4} + 3213 T^{5} + 11849 T^{6} + 14739 T^{7} + 83521 T^{8}$$
$19$ $$1 - 4 T + 49 T^{2} - 193 T^{3} + 1297 T^{4} - 3667 T^{5} + 17689 T^{6} - 27436 T^{7} + 130321 T^{8}$$
$23$ $$1 + 2 T + 63 T^{2} + 159 T^{3} + 1876 T^{4} + 3657 T^{5} + 33327 T^{6} + 24334 T^{7} + 279841 T^{8}$$
$29$ $$1 + 9 T + 68 T^{2} + 270 T^{3} + 1290 T^{4} + 7830 T^{5} + 57188 T^{6} + 219501 T^{7} + 707281 T^{8}$$
$31$ $$1 - 3 T + 50 T^{2} - 252 T^{3} + 1872 T^{4} - 7812 T^{5} + 48050 T^{6} - 89373 T^{7} + 923521 T^{8}$$
$37$ $$1 - 3 T + 140 T^{2} - 324 T^{3} + 7620 T^{4} - 11988 T^{5} + 191660 T^{6} - 151959 T^{7} + 1874161 T^{8}$$
$41$ $$1 - 9 T + 117 T^{2} - 717 T^{3} + 6341 T^{4} - 29397 T^{5} + 196677 T^{6} - 620289 T^{7} + 2825761 T^{8}$$
$43$ $$1 - 8 T + 137 T^{2} - 1005 T^{3} + 8087 T^{4} - 43215 T^{5} + 253313 T^{6} - 636056 T^{7} + 3418801 T^{8}$$
$47$ $$1 + 3 T + 150 T^{2} + 468 T^{3} + 9674 T^{4} + 21996 T^{5} + 331350 T^{6} + 311469 T^{7} + 4879681 T^{8}$$
$53$ $$1 + 6 T + 59 T^{2} + 621 T^{3} + 4704 T^{4} + 32913 T^{5} + 165731 T^{6} + 893262 T^{7} + 7890481 T^{8}$$
$59$ $$1 + 10 T + 171 T^{2} + 1623 T^{3} + 13357 T^{4} + 95757 T^{5} + 595251 T^{6} + 2053790 T^{7} + 12117361 T^{8}$$
$61$ $$1 + 20 T + 239 T^{2} + 2109 T^{3} + 15872 T^{4} + 128649 T^{5} + 889319 T^{6} + 4539620 T^{7} + 13845841 T^{8}$$
$67$ $$1 - 11 T + 203 T^{2} - 1461 T^{3} + 17099 T^{4} - 97887 T^{5} + 911267 T^{6} - 3308393 T^{7} + 20151121 T^{8}$$
$71$ $$1 + 3 T + 276 T^{2} + 630 T^{3} + 29126 T^{4} + 44730 T^{5} + 1391316 T^{6} + 1073733 T^{7} + 25411681 T^{8}$$
$73$ $$1 + 24 T + 425 T^{2} + 4923 T^{3} + 48495 T^{4} + 359379 T^{5} + 2264825 T^{6} + 9336408 T^{7} + 28398241 T^{8}$$
$79$ $$1 - 21 T + 308 T^{2} - 3360 T^{3} + 34200 T^{4} - 265440 T^{5} + 1922228 T^{6} - 10353819 T^{7} + 38950081 T^{8}$$
$83$ $$1 + 8 T + 323 T^{2} + 1865 T^{3} + 39832 T^{4} + 154795 T^{5} + 2225147 T^{6} + 4574296 T^{7} + 47458321 T^{8}$$
$89$ $$1 + 6 T + 263 T^{2} + 1377 T^{3} + 32646 T^{4} + 122553 T^{5} + 2083223 T^{6} + 4229814 T^{7} + 62742241 T^{8}$$
$97$ $$1 + 16 T + 431 T^{2} + 4491 T^{3} + 64415 T^{4} + 435627 T^{5} + 4055279 T^{6} + 14602768 T^{7} + 88529281 T^{8}$$