# Properties

 Label 288.4.d.a Level $288$ Weight $4$ Character orbit 288.d Analytic conductor $16.993$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$288 = 2^{5} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 288.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$16.9925500817$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-7})$$ Defining polynomial: $$x^{2} - x + 2$$ x^2 - x + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 8) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{-7}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 2 \beta q^{5} + 8 q^{7} +O(q^{10})$$ q - 2*b * q^5 + 8 * q^7 $$q - 2 \beta q^{5} + 8 q^{7} + 3 \beta q^{11} - 10 \beta q^{13} + 14 q^{17} - 7 \beta q^{19} - 152 q^{23} + 13 q^{25} - 30 \beta q^{29} - 224 q^{31} - 16 \beta q^{35} - 46 \beta q^{37} + 70 q^{41} - 83 \beta q^{43} + 336 q^{47} - 279 q^{49} + 6 \beta q^{53} + 168 q^{55} - 101 \beta q^{59} - 18 \beta q^{61} - 560 q^{65} + 33 \beta q^{67} - 72 q^{71} - 294 q^{73} + 24 \beta q^{77} + 464 q^{79} + 103 \beta q^{83} - 28 \beta q^{85} - 266 q^{89} - 80 \beta q^{91} - 392 q^{95} + 994 q^{97} +O(q^{100})$$ q - 2*b * q^5 + 8 * q^7 + 3*b * q^11 - 10*b * q^13 + 14 * q^17 - 7*b * q^19 - 152 * q^23 + 13 * q^25 - 30*b * q^29 - 224 * q^31 - 16*b * q^35 - 46*b * q^37 + 70 * q^41 - 83*b * q^43 + 336 * q^47 - 279 * q^49 + 6*b * q^53 + 168 * q^55 - 101*b * q^59 - 18*b * q^61 - 560 * q^65 + 33*b * q^67 - 72 * q^71 - 294 * q^73 + 24*b * q^77 + 464 * q^79 + 103*b * q^83 - 28*b * q^85 - 266 * q^89 - 80*b * q^91 - 392 * q^95 + 994 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 16 q^{7}+O(q^{10})$$ 2 * q + 16 * q^7 $$2 q + 16 q^{7} + 28 q^{17} - 304 q^{23} + 26 q^{25} - 448 q^{31} + 140 q^{41} + 672 q^{47} - 558 q^{49} + 336 q^{55} - 1120 q^{65} - 144 q^{71} - 588 q^{73} + 928 q^{79} - 532 q^{89} - 784 q^{95} + 1988 q^{97}+O(q^{100})$$ 2 * q + 16 * q^7 + 28 * q^17 - 304 * q^23 + 26 * q^25 - 448 * q^31 + 140 * q^41 + 672 * q^47 - 558 * q^49 + 336 * q^55 - 1120 * q^65 - 144 * q^71 - 588 * q^73 + 928 * q^79 - 532 * q^89 - 784 * q^95 + 1988 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/288\mathbb{Z}\right)^\times$$.

 $$n$$ $$37$$ $$65$$ $$127$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
145.1
 0.5 + 1.32288i 0.5 − 1.32288i
0 0 0 10.5830i 0 8.00000 0 0 0
145.2 0 0 0 10.5830i 0 8.00000 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
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 288.4.d.a 2
3.b odd 2 1 32.4.b.a 2
4.b odd 2 1 72.4.d.b 2
8.b even 2 1 inner 288.4.d.a 2
8.d odd 2 1 72.4.d.b 2
12.b even 2 1 8.4.b.a 2
15.d odd 2 1 800.4.d.a 2
15.e even 4 2 800.4.f.a 4
16.e even 4 2 2304.4.a.v 2
16.f odd 4 2 2304.4.a.bn 2
24.f even 2 1 8.4.b.a 2
24.h odd 2 1 32.4.b.a 2
48.i odd 4 2 256.4.a.j 2
48.k even 4 2 256.4.a.l 2
60.h even 2 1 200.4.d.a 2
60.l odd 4 2 200.4.f.a 4
120.i odd 2 1 800.4.d.a 2
120.m even 2 1 200.4.d.a 2
120.q odd 4 2 200.4.f.a 4
120.w even 4 2 800.4.f.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.4.b.a 2 12.b even 2 1
8.4.b.a 2 24.f even 2 1
32.4.b.a 2 3.b odd 2 1
32.4.b.a 2 24.h odd 2 1
72.4.d.b 2 4.b odd 2 1
72.4.d.b 2 8.d odd 2 1
200.4.d.a 2 60.h even 2 1
200.4.d.a 2 120.m even 2 1
200.4.f.a 4 60.l odd 4 2
200.4.f.a 4 120.q odd 4 2
256.4.a.j 2 48.i odd 4 2
256.4.a.l 2 48.k even 4 2
288.4.d.a 2 1.a even 1 1 trivial
288.4.d.a 2 8.b even 2 1 inner
800.4.d.a 2 15.d odd 2 1
800.4.d.a 2 120.i odd 2 1
800.4.f.a 4 15.e even 4 2
800.4.f.a 4 120.w even 4 2
2304.4.a.v 2 16.e even 4 2
2304.4.a.bn 2 16.f odd 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 112$$ acting on $$S_{4}^{\mathrm{new}}(288, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 112$$
$7$ $$(T - 8)^{2}$$
$11$ $$T^{2} + 252$$
$13$ $$T^{2} + 2800$$
$17$ $$(T - 14)^{2}$$
$19$ $$T^{2} + 1372$$
$23$ $$(T + 152)^{2}$$
$29$ $$T^{2} + 25200$$
$31$ $$(T + 224)^{2}$$
$37$ $$T^{2} + 59248$$
$41$ $$(T - 70)^{2}$$
$43$ $$T^{2} + 192892$$
$47$ $$(T - 336)^{2}$$
$53$ $$T^{2} + 1008$$
$59$ $$T^{2} + 285628$$
$61$ $$T^{2} + 9072$$
$67$ $$T^{2} + 30492$$
$71$ $$(T + 72)^{2}$$
$73$ $$(T + 294)^{2}$$
$79$ $$(T - 464)^{2}$$
$83$ $$T^{2} + 297052$$
$89$ $$(T + 266)^{2}$$
$97$ $$(T - 994)^{2}$$
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