# Properties

 Label 288.3.h.a Level $288$ Weight $3$ Character orbit 288.h Analytic conductor $7.847$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

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## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$7.84743161358$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.33808912384.2 Defining polynomial: $$x^{8} - 2 x^{7} + 11 x^{6} - 18 x^{5} + 47 x^{4} - 28 x^{3} - 44 x^{2} + 48 x + 36$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{14}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 72) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{5} + \beta_{4} q^{7} +O(q^{10})$$ $$q + \beta_{2} q^{5} + \beta_{4} q^{7} + ( -\beta_{2} - \beta_{3} ) q^{11} + \beta_{7} q^{13} + ( \beta_{5} + 2 \beta_{6} ) q^{17} + ( \beta_{1} + \beta_{7} ) q^{19} + ( 3 \beta_{5} + \beta_{6} ) q^{23} + ( 5 + 4 \beta_{4} ) q^{25} + ( \beta_{2} - 2 \beta_{3} ) q^{29} + ( 16 + 3 \beta_{4} ) q^{31} + ( 7 \beta_{2} - \beta_{3} ) q^{35} + ( \beta_{1} + \beta_{7} ) q^{37} + ( \beta_{5} - 2 \beta_{6} ) q^{41} + ( -2 \beta_{1} + 2 \beta_{7} ) q^{43} + ( -7 \beta_{5} + 3 \beta_{6} ) q^{47} + 3 q^{49} + ( -5 \beta_{2} - 2 \beta_{3} ) q^{53} + ( -32 - 2 \beta_{4} ) q^{55} + ( -4 \beta_{2} + 4 \beta_{3} ) q^{59} + ( \beta_{1} - 3 \beta_{7} ) q^{61} -2 \beta_{6} q^{65} + ( \beta_{1} - 3 \beta_{7} ) q^{67} + ( 15 \beta_{5} - 3 \beta_{6} ) q^{71} + ( -20 - 8 \beta_{4} ) q^{73} + ( -4 \beta_{2} + 8 \beta_{3} ) q^{77} + ( 48 - 11 \beta_{4} ) q^{79} + ( -7 \beta_{2} + \beta_{3} ) q^{83} + ( -5 \beta_{1} - 2 \beta_{7} ) q^{85} + ( -7 \beta_{5} + 4 \beta_{6} ) q^{89} + ( \beta_{1} - 7 \beta_{7} ) q^{91} + ( -34 \beta_{5} - 14 \beta_{6} ) q^{95} + ( -24 - 4 \beta_{4} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q + 40q^{25} + 128q^{31} + 24q^{49} - 256q^{55} - 160q^{73} + 384q^{79} - 192q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2 x^{7} + 11 x^{6} - 18 x^{5} + 47 x^{4} - 28 x^{3} - 44 x^{2} + 48 x + 36$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-449 \nu^{7} - 3617 \nu^{6} - 2407 \nu^{5} - 40233 \nu^{4} + 8129 \nu^{3} - 164911 \nu^{2} + 13066 \nu + 80772$$$$)/9753$$ $$\beta_{2}$$ $$=$$ $$($$$$432 \nu^{7} - 495 \nu^{6} + 4126 \nu^{5} - 4929 \nu^{4} + 13046 \nu^{3} - 4339 \nu^{2} - 30774 \nu - 7582$$$$)/6502$$ $$\beta_{3}$$ $$=$$ $$($$$$574 \nu^{7} - 1335 \nu^{6} + 6596 \nu^{5} - 10929 \nu^{4} + 24348 \nu^{3} - 4215 \nu^{2} - 53442 \nu + 73338$$$$)/6502$$ $$\beta_{4}$$ $$=$$ $$($$$$-321 \nu^{7} + 571 \nu^{6} - 3111 \nu^{5} + 4543 \nu^{4} - 12087 \nu^{3} + 3337 \nu^{2} + 27066 \nu - 10576$$$$)/3251$$ $$\beta_{5}$$ $$=$$ $$($$$$-3067 \nu^{7} + 6833 \nu^{6} - 36773 \nu^{5} + 65991 \nu^{4} - 176981 \nu^{3} + 149203 \nu^{2} - 2632 \nu - 110994$$$$)/19506$$ $$\beta_{6}$$ $$=$$ $$($$$$5357 \nu^{7} - 17449 \nu^{6} + 72943 \nu^{5} - 171543 \nu^{4} + 379759 \nu^{3} - 515711 \nu^{2} + 31220 \nu + 336090$$$$)/19506$$ $$\beta_{7}$$ $$=$$ $$($$$$-6103 \nu^{7} + 16001 \nu^{6} - 78593 \nu^{5} + 162129 \nu^{4} - 402137 \nu^{3} + 432823 \nu^{2} - 17722 \nu - 298464$$$$)/9753$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$3 \beta_{7} - 16 \beta_{5} + 6 \beta_{4} + \beta_{1} + 12$$$$)/48$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{7} - 4 \beta_{6} - 4 \beta_{5} + 2 \beta_{4} + 4 \beta_{3} - 4 \beta_{2} + \beta_{1} - 36$$$$)/16$$ $$\nu^{3}$$ $$=$$ $$($$$$-9 \beta_{7} - 9 \beta_{6} + 19 \beta_{5} - 24 \beta_{4} - 9 \beta_{3} - 27 \beta_{2} - \beta_{1} - 12$$$$)/24$$ $$\nu^{4}$$ $$=$$ $$($$$$15 \beta_{7} + 32 \beta_{6} - 26 \beta_{4} - 16 \beta_{3} - 16 \beta_{2} - 11 \beta_{1} + 76$$$$)/16$$ $$\nu^{5}$$ $$=$$ $$($$$$51 \beta_{7} + 150 \beta_{6} + 86 \beta_{5} + 354 \beta_{4} + 210 \beta_{3} + 270 \beta_{2} - 59 \beta_{1} - 948$$$$)/48$$ $$\nu^{6}$$ $$=$$ $$($$$$-38 \beta_{7} - 49 \beta_{6} + 67 \beta_{5} + 41 \beta_{4} + 3 \beta_{3} + 65 \beta_{2} + 7 \beta_{1} + 180$$$$)/4$$ $$\nu^{7}$$ $$=$$ $$($$$$231 \beta_{7} - 588 \beta_{6} - 2308 \beta_{5} - 1770 \beta_{4} - 1848 \beta_{3} + 445 \beta_{1} + 15468$$$$)/48$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1
 1.15139 + 0.593052i 1.15139 − 0.593052i −0.651388 − 2.66948i −0.651388 + 2.66948i −0.651388 + 0.158947i −0.651388 − 0.158947i 1.15139 − 2.23537i 1.15139 + 2.23537i
0 0 0 −7.67101 0 7.21110 0 0 0
17.2 0 0 0 −7.67101 0 7.21110 0 0 0
17.3 0 0 0 −1.07498 0 −7.21110 0 0 0
17.4 0 0 0 −1.07498 0 −7.21110 0 0 0
17.5 0 0 0 1.07498 0 −7.21110 0 0 0
17.6 0 0 0 1.07498 0 −7.21110 0 0 0
17.7 0 0 0 7.67101 0 7.21110 0 0 0
17.8 0 0 0 7.67101 0 7.21110 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 17.8 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
8.b even 2 1 inner
24.h odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 288.3.h.a 8
3.b odd 2 1 inner 288.3.h.a 8
4.b odd 2 1 72.3.h.a 8
8.b even 2 1 inner 288.3.h.a 8
8.d odd 2 1 72.3.h.a 8
12.b even 2 1 72.3.h.a 8
16.e even 4 2 2304.3.e.o 8
16.f odd 4 2 2304.3.e.n 8
24.f even 2 1 72.3.h.a 8
24.h odd 2 1 inner 288.3.h.a 8
48.i odd 4 2 2304.3.e.o 8
48.k even 4 2 2304.3.e.n 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.3.h.a 8 4.b odd 2 1
72.3.h.a 8 8.d odd 2 1
72.3.h.a 8 12.b even 2 1
72.3.h.a 8 24.f even 2 1
288.3.h.a 8 1.a even 1 1 trivial
288.3.h.a 8 3.b odd 2 1 inner
288.3.h.a 8 8.b even 2 1 inner
288.3.h.a 8 24.h odd 2 1 inner
2304.3.e.n 8 16.f odd 4 2
2304.3.e.n 8 48.k even 4 2
2304.3.e.o 8 16.e even 4 2
2304.3.e.o 8 48.i odd 4 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(288, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$( 68 - 60 T^{2} + T^{4} )^{2}$$
$7$ $$( -52 + T^{2} )^{4}$$
$11$ $$( 9792 - 304 T^{2} + T^{4} )^{2}$$
$13$ $$( 2448 + 472 T^{2} + T^{4} )^{2}$$
$17$ $$( 171396 + 836 T^{2} + T^{4} )^{2}$$
$19$ $$( 352512 + 1504 T^{2} + T^{4} )^{2}$$
$23$ $$( 576 + 464 T^{2} + T^{4} )^{2}$$
$29$ $$( 103428 - 988 T^{2} + T^{4} )^{2}$$
$31$ $$( -212 - 32 T + T^{2} )^{4}$$
$37$ $$( 352512 + 1504 T^{2} + T^{4} )^{2}$$
$41$ $$( 133956 + 932 T^{2} + T^{4} )^{2}$$
$43$ $$( 10027008 + 6400 T^{2} + T^{4} )^{2}$$
$47$ $$( 46656 + 4176 T^{2} + T^{4} )^{2}$$
$53$ $$( 1591812 - 2524 T^{2} + T^{4} )^{2}$$
$59$ $$( 4456448 - 4608 T^{2} + T^{4} )^{2}$$
$61$ $$( 3172608 + 5472 T^{2} + T^{4} )^{2}$$
$67$ $$( 3172608 + 5472 T^{2} + T^{4} )^{2}$$
$71$ $$( 13483584 + 11088 T^{2} + T^{4} )^{2}$$
$73$ $$( -2928 + 40 T + T^{2} )^{4}$$
$79$ $$( -3988 - 96 T + T^{2} )^{4}$$
$83$ $$( 183872 - 3120 T^{2} + T^{4} )^{2}$$
$89$ $$( 171396 + 5828 T^{2} + T^{4} )^{2}$$
$97$ $$( -256 + 48 T + T^{2} )^{4}$$
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