# Properties

 Label 72.7.e.b Level $72$ Weight $7$ Character orbit 72.e Analytic conductor $16.564$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$72 = 2^{3} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 72.e (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$16.5638940206$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{145})$$ Defining polynomial: $$x^{4} - 2x^{3} - 67x^{2} + 68x + 1446$$ x^4 - 2*x^3 - 67*x^2 + 68*x + 1446 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{7}\cdot 3^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{3} - 25 \beta_1) q^{5} + ( - \beta_{2} - 108) q^{7}+O(q^{10})$$ q + (-b3 - 25*b1) * q^5 + (-b2 - 108) * q^7 $$q + ( - \beta_{3} - 25 \beta_1) q^{5} + ( - \beta_{2} - 108) q^{7} + (6 \beta_{3} + 524 \beta_1) q^{11} + (7 \beta_{2} - 1088) q^{13} + ( - 2 \beta_{3} + 1633 \beta_1) q^{17} + ( - 8 \beta_{2} - 6736) q^{19} + ( - 50 \beta_{3} + 9188 \beta_1) q^{23} + ( - 50 \beta_{2} - 27385) q^{25} + (67 \beta_{3} + 7761 \beta_1) q^{29} + (131 \beta_{2} - 14044) q^{31} + (158 \beta_{3} + 44460 \beta_1) q^{35} + (76 \beta_{2} - 59250) q^{37} + ( - 250 \beta_{3} - 34905 \beta_1) q^{41} + ( - 514 \beta_{2} + 712) q^{43} + ( - 298 \beta_{3} + 50292 \beta_1) q^{47} + (216 \beta_{2} - 22465) q^{49} + (175 \beta_{3} - 127975 \beta_1) q^{53} + (674 \beta_{2} + 276760) q^{55} + (708 \beta_{3} + 32504 \beta_1) q^{59} + ( - 556 \beta_{2} + 36706) q^{61} + (738 \beta_{3} - 265120 \beta_1) q^{65} + (302 \beta_{2} + 386552) q^{67} + ( - 1374 \beta_{3} + 76300 \beta_1) q^{71} + ( - 564 \beta_{2} + 70480) q^{73} + ( - 1696 \beta_{3} - 307152 \beta_1) q^{77} + ( - 1067 \beta_{2} + 314620) q^{79} + ( - 686 \beta_{3} - 126668 \beta_1) q^{83} + (1583 \beta_{2} - 1870) q^{85} + (1716 \beta_{3} + 275415 \beta_1) q^{89} + (332 \beta_{2} - 467136) q^{91} + (7136 \beta_{3} + 502480 \beta_1) q^{95} + (2350 \beta_{2} - 330592) q^{97}+O(q^{100})$$ q + (-b3 - 25*b1) * q^5 + (-b2 - 108) * q^7 + (6*b3 + 524*b1) * q^11 + (7*b2 - 1088) * q^13 + (-2*b3 + 1633*b1) * q^17 + (-8*b2 - 6736) * q^19 + (-50*b3 + 9188*b1) * q^23 + (-50*b2 - 27385) * q^25 + (67*b3 + 7761*b1) * q^29 + (131*b2 - 14044) * q^31 + (158*b3 + 44460*b1) * q^35 + (76*b2 - 59250) * q^37 + (-250*b3 - 34905*b1) * q^41 + (-514*b2 + 712) * q^43 + (-298*b3 + 50292*b1) * q^47 + (216*b2 - 22465) * q^49 + (175*b3 - 127975*b1) * q^53 + (674*b2 + 276760) * q^55 + (708*b3 + 32504*b1) * q^59 + (-556*b2 + 36706) * q^61 + (738*b3 - 265120*b1) * q^65 + (302*b2 + 386552) * q^67 + (-1374*b3 + 76300*b1) * q^71 + (-564*b2 + 70480) * q^73 + (-1696*b3 - 307152*b1) * q^77 + (-1067*b2 + 314620) * q^79 + (-686*b3 - 126668*b1) * q^83 + (1583*b2 - 1870) * q^85 + (1716*b3 + 275415*b1) * q^89 + (332*b2 - 467136) * q^91 + (7136*b3 + 502480*b1) * q^95 + (2350*b2 - 330592) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 432 q^{7}+O(q^{10})$$ 4 * q - 432 * q^7 $$4 q - 432 q^{7} - 4352 q^{13} - 26944 q^{19} - 109540 q^{25} - 56176 q^{31} - 237000 q^{37} + 2848 q^{43} - 89860 q^{49} + 1107040 q^{55} + 146824 q^{61} + 1546208 q^{67} + 281920 q^{73} + 1258480 q^{79} - 7480 q^{85} - 1868544 q^{91} - 1322368 q^{97}+O(q^{100})$$ 4 * q - 432 * q^7 - 4352 * q^13 - 26944 * q^19 - 109540 * q^25 - 56176 * q^31 - 237000 * q^37 + 2848 * q^43 - 89860 * q^49 + 1107040 * q^55 + 146824 * q^61 + 1546208 * q^67 + 281920 * q^73 + 1258480 * q^79 - 7480 * q^85 - 1868544 * q^91 - 1322368 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2x^{3} - 67x^{2} + 68x + 1446$$ :

 $$\beta_{1}$$ $$=$$ $$( 2\nu^{3} - 3\nu^{2} - 59\nu + 30 ) / 153$$ (2*v^3 - 3*v^2 - 59*v + 30) / 153 $$\beta_{2}$$ $$=$$ $$( -32\nu^{3} + 48\nu^{2} + 3392\nu - 1704 ) / 51$$ (-32*v^3 + 48*v^2 + 3392*v - 1704) / 51 $$\beta_{3}$$ $$=$$ $$12\nu^{2} - 12\nu - 408$$ 12*v^2 - 12*v - 408
 $$\nu$$ $$=$$ $$( \beta_{2} + 48\beta _1 + 24 ) / 48$$ (b2 + 48*b1 + 24) / 48 $$\nu^{2}$$ $$=$$ $$( 4\beta_{3} + \beta_{2} + 48\beta _1 + 1656 ) / 48$$ (4*b3 + b2 + 48*b1 + 1656) / 48 $$\nu^{3}$$ $$=$$ $$( 6\beta_{3} + 31\beta_{2} + 5160\beta _1 + 2472 ) / 48$$ (6*b3 + 31*b2 + 5160*b1 + 2472) / 48

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/72\mathbb{Z}\right)^\times$$.

 $$n$$ $$37$$ $$55$$ $$65$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1
 6.52080 + 1.41421i −5.52080 − 1.41421i −5.52080 + 1.41421i 6.52080 − 1.41421i
0 0 0 239.708i 0 −396.998 0 0 0
17.2 0 0 0 168.997i 0 180.998 0 0 0
17.3 0 0 0 168.997i 0 180.998 0 0 0
17.4 0 0 0 239.708i 0 −396.998 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 72.7.e.b 4
3.b odd 2 1 inner 72.7.e.b 4
4.b odd 2 1 144.7.e.e 4
8.b even 2 1 576.7.e.m 4
8.d odd 2 1 576.7.e.r 4
12.b even 2 1 144.7.e.e 4
24.f even 2 1 576.7.e.r 4
24.h odd 2 1 576.7.e.m 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.7.e.b 4 1.a even 1 1 trivial
72.7.e.b 4 3.b odd 2 1 inner
144.7.e.e 4 4.b odd 2 1
144.7.e.e 4 12.b even 2 1
576.7.e.m 4 8.b even 2 1
576.7.e.m 4 24.h odd 2 1
576.7.e.r 4 8.d odd 2 1
576.7.e.r 4 24.f even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 86020T_{5}^{2} + 1641060100$$ acting on $$S_{7}^{\mathrm{new}}(72, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 86020 T^{2} + \cdots + 1641060100$$
$7$ $$(T^{2} + 216 T - 71856)^{2}$$
$11$ $$T^{4} + 4105024 T^{2} + \cdots + 910512907264$$
$13$ $$(T^{2} + 2176 T - 2908736)^{2}$$
$17$ $$T^{4} + 11000836 T^{2} + \cdots + 26691048330244$$
$19$ $$(T^{2} + 13472 T + 40028416)^{2}$$
$23$ $$T^{4} + 546477376 T^{2} + \cdots + 41\!\cdots\!44$$
$29$ $$T^{4} + 615853764 T^{2} + \cdots + 44\!\cdots\!04$$
$31$ $$(T^{2} + 28088 T - 1236052784)^{2}$$
$37$ $$(T^{2} + 118500 T + 3028150980)^{2}$$
$41$ $$T^{4} + 10093436100 T^{2} + \cdots + 30\!\cdots\!00$$
$43$ $$(T^{2} - 1424 T - 22065142976)^{2}$$
$47$ $$T^{4} + 17534051136 T^{2} + \cdots + 18\!\cdots\!44$$
$53$ $$T^{4} + 68068202500 T^{2} + \cdots + 99\!\cdots\!00$$
$59$ $$T^{4} + 46091609344 T^{2} + \cdots + 35\!\cdots\!64$$
$61$ $$(T^{2} - 73412 T - 24471708284)^{2}$$
$67$ $$(T^{2} - 773104 T + 141805090624)^{2}$$
$71$ $$T^{4} + 180962163520 T^{2} + \cdots + 45\!\cdots\!00$$
$73$ $$(T^{2} - 140960 T - 21599947520)^{2}$$
$79$ $$(T^{2} - 629240 T + 3899143120)^{2}$$
$83$ $$T^{4} + 103483306816 T^{2} + \cdots + 15\!\cdots\!44$$
$89$ $$T^{4} + 549351358020 T^{2} + \cdots + 82\!\cdots\!00$$
$97$ $$(T^{2} + 661184 T - 351948129536)^{2}$$