# Properties

 Label 72.7.e.a Level $72$ Weight $7$ Character orbit 72.e Analytic conductor $16.564$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ Defining polynomial: $$x^{2} + 2$$ x^2 + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + 5 \beta q^{5} + 60 q^{7}+O(q^{10})$$ q + 5*b * q^5 + 60 * q^7 $$q + 5 \beta q^{5} + 60 q^{7} + 236 \beta q^{11} + 1192 q^{13} + 2935 \beta q^{17} + 8432 q^{19} + 4532 \beta q^{23} + 15575 q^{25} + 23787 \beta q^{29} + 46892 q^{31} + 300 \beta q^{35} + 10926 q^{37} + 51921 \beta q^{41} + 59416 q^{43} - 83004 \beta q^{47} - 114049 q^{49} + 58259 \beta q^{53} - 2360 q^{55} - 199336 \beta q^{59} - 339902 q^{61} + 5960 \beta q^{65} - 148024 q^{67} - 291044 \beta q^{71} - 401552 q^{73} + 14160 \beta q^{77} + 79156 q^{79} - 70988 \beta q^{83} - 29350 q^{85} + 265785 \beta q^{89} + 71520 q^{91} + 42160 \beta q^{95} + 663920 q^{97} +O(q^{100})$$ q + 5*b * q^5 + 60 * q^7 + 236*b * q^11 + 1192 * q^13 + 2935*b * q^17 + 8432 * q^19 + 4532*b * q^23 + 15575 * q^25 + 23787*b * q^29 + 46892 * q^31 + 300*b * q^35 + 10926 * q^37 + 51921*b * q^41 + 59416 * q^43 - 83004*b * q^47 - 114049 * q^49 + 58259*b * q^53 - 2360 * q^55 - 199336*b * q^59 - 339902 * q^61 + 5960*b * q^65 - 148024 * q^67 - 291044*b * q^71 - 401552 * q^73 + 14160*b * q^77 + 79156 * q^79 - 70988*b * q^83 - 29350 * q^85 + 265785*b * q^89 + 71520 * q^91 + 42160*b * q^95 + 663920 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 120 q^{7}+O(q^{10})$$ 2 * q + 120 * q^7 $$2 q + 120 q^{7} + 2384 q^{13} + 16864 q^{19} + 31150 q^{25} + 93784 q^{31} + 21852 q^{37} + 118832 q^{43} - 228098 q^{49} - 4720 q^{55} - 679804 q^{61} - 296048 q^{67} - 803104 q^{73} + 158312 q^{79} - 58700 q^{85} + 143040 q^{91} + 1327840 q^{97}+O(q^{100})$$ 2 * q + 120 * q^7 + 2384 * q^13 + 16864 * q^19 + 31150 * q^25 + 93784 * q^31 + 21852 * q^37 + 118832 * q^43 - 228098 * q^49 - 4720 * q^55 - 679804 * q^61 - 296048 * q^67 - 803104 * q^73 + 158312 * q^79 - 58700 * q^85 + 143040 * q^91 + 1327840 * 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1
 − 1.41421i 1.41421i
0 0 0 7.07107i 0 60.0000 0 0 0
17.2 0 0 0 7.07107i 0 60.0000 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.a 2
3.b odd 2 1 inner 72.7.e.a 2
4.b odd 2 1 144.7.e.b 2
8.b even 2 1 576.7.e.h 2
8.d odd 2 1 576.7.e.e 2
12.b even 2 1 144.7.e.b 2
24.f even 2 1 576.7.e.e 2
24.h odd 2 1 576.7.e.h 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 50$$ acting on $$S_{7}^{\mathrm{new}}(72, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 50$$
$7$ $$(T - 60)^{2}$$
$11$ $$T^{2} + 111392$$
$13$ $$(T - 1192)^{2}$$
$17$ $$T^{2} + 17228450$$
$19$ $$(T - 8432)^{2}$$
$23$ $$T^{2} + 41078048$$
$29$ $$T^{2} + 1131642738$$
$31$ $$(T - 46892)^{2}$$
$37$ $$(T - 10926)^{2}$$
$41$ $$T^{2} + 5391580482$$
$43$ $$(T - 59416)^{2}$$
$47$ $$T^{2} + 13779328032$$
$53$ $$T^{2} + 6788222162$$
$59$ $$T^{2} + 79469681792$$
$61$ $$(T + 339902)^{2}$$
$67$ $$(T + 148024)^{2}$$
$71$ $$T^{2} + 169413219872$$
$73$ $$(T + 401552)^{2}$$
$79$ $$(T - 79156)^{2}$$
$83$ $$T^{2} + 10078592288$$
$89$ $$T^{2} + 141283332450$$
$97$ $$(T - 663920)^{2}$$