Properties

 Label 648.4.i.p Level $648$ Weight $4$ Character orbit 648.i Analytic conductor $38.233$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

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Newspace parameters

 Level: $$N$$ $$=$$ $$648 = 2^{3} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 648.i (of order $$3$$, degree $$2$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$38.2332376837$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-43})$$ Defining polynomial: $$x^{4} - x^{3} - 10 x^{2} - 11 x + 121$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}\cdot 3$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

 $$f(q)$$ $$=$$ $$q + ( -2 - 2 \beta_{1} + \beta_{3} ) q^{5} + ( -3 \beta_{1} - \beta_{2} + \beta_{3} ) q^{7} +O(q^{10})$$ $$q + ( -2 - 2 \beta_{1} + \beta_{3} ) q^{5} + ( -3 \beta_{1} - \beta_{2} + \beta_{3} ) q^{7} + ( 5 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{11} + ( 20 + 20 \beta_{1} - 5 \beta_{3} ) q^{13} + ( -17 + 4 \beta_{2} ) q^{17} + ( 17 + 5 \beta_{2} ) q^{19} + ( 49 + 49 \beta_{1} - 7 \beta_{3} ) q^{23} + ( 8 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{25} + ( -60 \beta_{1} + 11 \beta_{2} - 11 \beta_{3} ) q^{29} + ( 100 + 100 \beta_{1} + 8 \beta_{3} ) q^{31} + ( -135 + 5 \beta_{2} ) q^{35} + ( 240 - 7 \beta_{2} ) q^{37} + ( 96 + 96 \beta_{1} + 10 \beta_{3} ) q^{41} + ( -167 \beta_{1} - 25 \beta_{2} + 25 \beta_{3} ) q^{43} + ( -150 \beta_{1} - 38 \beta_{2} + 38 \beta_{3} ) q^{47} + ( 205 + 205 \beta_{1} + 6 \beta_{3} ) q^{49} + ( -34 - 32 \beta_{2} ) q^{53} + ( 397 - 11 \beta_{2} ) q^{55} + ( 310 + 310 \beta_{1} + 6 \beta_{3} ) q^{59} + ( 100 \beta_{1} + 35 \beta_{2} - 35 \beta_{3} ) q^{61} + ( -685 \beta_{1} - 30 \beta_{2} + 30 \beta_{3} ) q^{65} + ( 703 + 703 \beta_{1} + 11 \beta_{3} ) q^{67} + ( -695 - 3 \beta_{2} ) q^{71} + ( 901 + 18 \beta_{2} ) q^{73} + ( 402 + 402 \beta_{1} - 14 \beta_{3} ) q^{77} + ( -167 \beta_{1} + 79 \beta_{2} - 79 \beta_{3} ) q^{79} + ( 250 \beta_{1} + 74 \beta_{2} - 74 \beta_{3} ) q^{83} + ( 550 + 550 \beta_{1} - 25 \beta_{3} ) q^{85} + ( -45 + 24 \beta_{2} ) q^{89} + ( 705 - 35 \beta_{2} ) q^{91} + ( 611 + 611 \beta_{1} + 7 \beta_{3} ) q^{95} + ( -100 \beta_{1} - 122 \beta_{2} + 122 \beta_{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{5} + 6q^{7} + O(q^{10})$$ $$4q - 4q^{5} + 6q^{7} - 10q^{11} + 40q^{13} - 68q^{17} + 68q^{19} + 98q^{23} - 16q^{25} + 120q^{29} + 200q^{31} - 540q^{35} + 960q^{37} + 192q^{41} + 334q^{43} + 300q^{47} + 410q^{49} - 136q^{53} + 1588q^{55} + 620q^{59} - 200q^{61} + 1370q^{65} + 1406q^{67} - 2780q^{71} + 3604q^{73} + 804q^{77} + 334q^{79} - 500q^{83} + 1100q^{85} - 180q^{89} + 2820q^{91} + 1222q^{95} + 200q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 10 x^{2} - 11 x + 121$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 10 \nu^{2} - 10 \nu - 121$$$$)/110$$ $$\beta_{2}$$ $$=$$ $$($$$$-2 \nu^{3} + 2 \nu^{2} + 42 \nu + 11$$$$)/11$$ $$\beta_{3}$$ $$=$$ $$($$$$23 \nu^{3} + 10 \nu^{2} + 210 \nu - 473$$$$)/110$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} - 3 \beta_{1}$$$$)/6$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{3} + 2 \beta_{2} + 63 \beta_{1} + 63$$$$)/6$$ $$\nu^{3}$$ $$=$$ $$($$$$10 \beta_{3} - 5 \beta_{2} + 48$$$$)/3$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/648\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$487$$ $$569$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1 - \beta_{1}$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
217.1
 −2.58945 − 2.07237i 3.08945 + 1.20635i −2.58945 + 2.07237i 3.08945 − 1.20635i
0 0 0 −6.67891 11.5682i 0 7.17891 12.4342i 0 0 0
217.2 0 0 0 4.67891 + 8.10411i 0 −4.17891 + 7.23808i 0 0 0
433.1 0 0 0 −6.67891 + 11.5682i 0 7.17891 + 12.4342i 0 0 0
433.2 0 0 0 4.67891 8.10411i 0 −4.17891 7.23808i 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
9.c even 3 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 648.4.i.p 4
3.b odd 2 1 648.4.i.q 4
9.c even 3 1 648.4.a.e yes 2
9.c even 3 1 inner 648.4.i.p 4
9.d odd 6 1 648.4.a.d 2
9.d odd 6 1 648.4.i.q 4
36.f odd 6 1 1296.4.a.p 2
36.h even 6 1 1296.4.a.n 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
648.4.a.d 2 9.d odd 6 1
648.4.a.e yes 2 9.c even 3 1
648.4.i.p 4 1.a even 1 1 trivial
648.4.i.p 4 9.c even 3 1 inner
648.4.i.q 4 3.b odd 2 1
648.4.i.q 4 9.d odd 6 1
1296.4.a.n 2 36.h even 6 1
1296.4.a.p 2 36.f odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 4 T_{5}^{3} + 141 T_{5}^{2} - 500 T_{5} + 15625$$ acting on $$S_{4}^{\mathrm{new}}(648, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$15625 - 500 T + 141 T^{2} + 4 T^{3} + T^{4}$$
$7$ $$14400 + 720 T + 156 T^{2} - 6 T^{3} + T^{4}$$
$11$ $$1290496 - 11360 T + 1236 T^{2} + 10 T^{3} + T^{4}$$
$13$ $$7980625 + 113000 T + 4425 T^{2} - 40 T^{3} + T^{4}$$
$17$ $$( -1775 + 34 T + T^{2} )^{2}$$
$19$ $$( -2936 - 34 T + T^{2} )^{2}$$
$23$ $$15366400 + 384160 T + 13524 T^{2} - 98 T^{3} + T^{4}$$
$29$ $$144216081 + 1441080 T + 26409 T^{2} - 120 T^{3} + T^{4}$$
$31$ $$3041536 - 348800 T + 38256 T^{2} - 200 T^{3} + T^{4}$$
$37$ $$( 51279 - 480 T + T^{2} )^{2}$$
$41$ $$13571856 + 707328 T + 40548 T^{2} - 192 T^{3} + T^{4}$$
$43$ $$2781085696 + 17613824 T + 164292 T^{2} - 334 T^{3} + T^{4}$$
$47$ $$26822578176 + 49132800 T + 253776 T^{2} - 300 T^{3} + T^{4}$$
$53$ $$( -130940 + 68 T + T^{2} )^{2}$$
$59$ $$8364199936 - 56702720 T + 292944 T^{2} - 620 T^{3} + T^{4}$$
$61$ $$21911400625 - 29605000 T + 188025 T^{2} + 200 T^{3} + T^{4}$$
$67$ $$229057960000 - 672911600 T + 1498236 T^{2} - 1406 T^{3} + T^{4}$$
$71$ $$( 481864 + 1390 T + T^{2} )^{2}$$
$73$ $$( 770005 - 1802 T + T^{2} )^{2}$$
$79$ $$604039840000 + 259584800 T + 888756 T^{2} - 334 T^{3} + T^{4}$$
$83$ $$414612361216 - 321952000 T + 893904 T^{2} + 500 T^{3} + T^{4}$$
$89$ $$( -72279 + 90 T + T^{2} )^{2}$$
$97$ $$3648237521296 + 382007200 T + 1950036 T^{2} - 200 T^{3} + T^{4}$$
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