# Properties

 Label 72.8.d.a Level $72$ Weight $8$ Character orbit 72.d Analytic conductor $22.492$ Analytic rank $0$ Dimension $2$ CM discriminant -24 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$72 = 2^{3} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 72.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$22.4917218349$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ Defining polynomial: $$x^{2} + 2$$ x^2 + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 4 \beta q^{2} - 128 q^{4} + 197 \beta q^{5} - 754 q^{7} - 512 \beta q^{8} +O(q^{10})$$ q + 4*b * q^2 - 128 * q^4 + 197*b * q^5 - 754 * q^7 - 512*b * q^8 $$q + 4 \beta q^{2} - 128 q^{4} + 197 \beta q^{5} - 754 q^{7} - 512 \beta q^{8} - 6304 q^{10} + 2378 \beta q^{11} - 3016 \beta q^{14} + 16384 q^{16} - 25216 \beta q^{20} - 76096 q^{22} - 232347 q^{25} + 96512 q^{28} - 89011 \beta q^{29} + 331370 q^{31} + 65536 \beta q^{32} - 148538 \beta q^{35} + 806912 q^{40} - 304384 \beta q^{44} - 255027 q^{49} - 929388 \beta q^{50} + 84695 \beta q^{53} - 3747728 q^{55} + 386048 \beta q^{56} + 2848352 q^{58} + 523436 \beta q^{59} + 1325480 \beta q^{62} - 2097152 q^{64} + 4753216 q^{70} - 2819362 q^{73} - 1793012 \beta q^{77} - 7561270 q^{79} + 3227648 \beta q^{80} + 2956826 \beta q^{83} + 9740288 q^{88} - 11745214 q^{97} - 1020108 \beta q^{98} +O(q^{100})$$ q + 4*b * q^2 - 128 * q^4 + 197*b * q^5 - 754 * q^7 - 512*b * q^8 - 6304 * q^10 + 2378*b * q^11 - 3016*b * q^14 + 16384 * q^16 - 25216*b * q^20 - 76096 * q^22 - 232347 * q^25 + 96512 * q^28 - 89011*b * q^29 + 331370 * q^31 + 65536*b * q^32 - 148538*b * q^35 + 806912 * q^40 - 304384*b * q^44 - 255027 * q^49 - 929388*b * q^50 + 84695*b * q^53 - 3747728 * q^55 + 386048*b * q^56 + 2848352 * q^58 + 523436*b * q^59 + 1325480*b * q^62 - 2097152 * q^64 + 4753216 * q^70 - 2819362 * q^73 - 1793012*b * q^77 - 7561270 * q^79 + 3227648*b * q^80 + 2956826*b * q^83 + 9740288 * q^88 - 11745214 * q^97 - 1020108*b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 256 q^{4} - 1508 q^{7}+O(q^{10})$$ 2 * q - 256 * q^4 - 1508 * q^7 $$2 q - 256 q^{4} - 1508 q^{7} - 12608 q^{10} + 32768 q^{16} - 152192 q^{22} - 464694 q^{25} + 193024 q^{28} + 662740 q^{31} + 1613824 q^{40} - 510054 q^{49} - 7495456 q^{55} + 5696704 q^{58} - 4194304 q^{64} + 9506432 q^{70} - 5638724 q^{73} - 15122540 q^{79} + 19480576 q^{88} - 23490428 q^{97}+O(q^{100})$$ 2 * q - 256 * q^4 - 1508 * q^7 - 12608 * q^10 + 32768 * q^16 - 152192 * q^22 - 464694 * q^25 + 193024 * q^28 + 662740 * q^31 + 1613824 * q^40 - 510054 * q^49 - 7495456 * q^55 + 5696704 * q^58 - 4194304 * q^64 + 9506432 * q^70 - 5638724 * q^73 - 15122540 * q^79 + 19480576 * q^88 - 23490428 * q^97

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1
 − 1.41421i 1.41421i
11.3137i 0 −128.000 557.200i 0 −754.000 1448.15i 0 −6304.00
37.2 11.3137i 0 −128.000 557.200i 0 −754.000 1448.15i 0 −6304.00
 $$n$$: 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
24.h odd 2 1 CM by $$\Q(\sqrt{-6})$$
3.b odd 2 1 inner
8.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 72.8.d.a 2
3.b odd 2 1 inner 72.8.d.a 2
4.b odd 2 1 288.8.d.a 2
8.b even 2 1 inner 72.8.d.a 2
8.d odd 2 1 288.8.d.a 2
12.b even 2 1 288.8.d.a 2
24.f even 2 1 288.8.d.a 2
24.h odd 2 1 CM 72.8.d.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.8.d.a 2 1.a even 1 1 trivial
72.8.d.a 2 3.b odd 2 1 inner
72.8.d.a 2 8.b even 2 1 inner
72.8.d.a 2 24.h odd 2 1 CM
288.8.d.a 2 4.b odd 2 1
288.8.d.a 2 8.d odd 2 1
288.8.d.a 2 12.b even 2 1
288.8.d.a 2 24.f even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 310472$$ acting on $$S_{8}^{\mathrm{new}}(72, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 128$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 310472$$
$7$ $$(T + 754)^{2}$$
$11$ $$T^{2} + 45239072$$
$13$ $$T^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2} + 63383664968$$
$31$ $$(T - 331370)^{2}$$
$37$ $$T^{2}$$
$41$ $$T^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 57385944200$$
$59$ $$T^{2} + 2191881968768$$
$61$ $$T^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$(T + 2819362)^{2}$$
$79$ $$(T + 7561270)^{2}$$
$83$ $$T^{2} + 69942559954208$$
$89$ $$T^{2}$$
$97$ $$(T + 11745214)^{2}$$