# Properties

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

# Related objects

## 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: $$\Q(\sqrt{-3}, \sqrt{-11})$$ Defining polynomial: $$x^{4} - x^{3} - 2 x^{2} - 3 x + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}\cdot 3^{3}$$ Twist minimal: no (minimal twist has level 216) 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.

 $$f(q)$$ $$=$$ $$q + ( -4 - 4 \beta_{1} - \beta_{3} ) q^{5} + ( 12 \beta_{1} - \beta_{2} + \beta_{3} ) q^{7} +O(q^{10})$$ $$q + ( -4 - 4 \beta_{1} - \beta_{3} ) q^{5} + ( 12 \beta_{1} - \beta_{2} + \beta_{3} ) q^{7} + \beta_{1} q^{11} + ( -16 - 16 \beta_{1} + 4 \beta_{3} ) q^{13} + ( -28 - 4 \beta_{2} ) q^{17} + ( 92 + 2 \beta_{2} ) q^{19} + ( -46 - 46 \beta_{1} + 4 \beta_{3} ) q^{23} + ( 188 \beta_{1} - 8 \beta_{2} + 8 \beta_{3} ) q^{25} + ( -168 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{29} + ( -188 - 188 \beta_{1} + 5 \beta_{3} ) q^{31} + ( 345 + 16 \beta_{2} ) q^{35} + ( 174 - 4 \beta_{2} ) q^{37} + ( 156 + 156 \beta_{1} - 10 \beta_{3} ) q^{41} + ( 40 \beta_{1} + 14 \beta_{2} - 14 \beta_{3} ) q^{43} + ( -114 \beta_{1} + 8 \beta_{2} - 8 \beta_{3} ) q^{47} + ( -98 - 98 \beta_{1} - 24 \beta_{3} ) q^{49} + ( 76 - 13 \beta_{2} ) q^{53} + ( 4 + \beta_{2} ) q^{55} + ( -340 - 340 \beta_{1} + 24 \beta_{3} ) q^{59} + ( -56 \beta_{1} + 32 \beta_{2} - 32 \beta_{3} ) q^{61} -1124 \beta_{1} q^{65} + ( -176 - 176 \beta_{1} - 34 \beta_{3} ) q^{67} + ( 908 - 12 \beta_{2} ) q^{71} -287 q^{73} + ( -12 - 12 \beta_{1} - \beta_{3} ) q^{77} + ( -680 \beta_{1} - 32 \beta_{2} + 32 \beta_{3} ) q^{79} + ( -391 \beta_{1} - 32 \beta_{2} + 32 \beta_{3} ) q^{83} + ( 1300 + 1300 \beta_{1} + 44 \beta_{3} ) q^{85} + ( 120 + 18 \beta_{2} ) q^{89} + ( -996 - 32 \beta_{2} ) q^{91} + ( -962 - 962 \beta_{1} - 100 \beta_{3} ) q^{95} + ( -169 \beta_{1} - 8 \beta_{2} + 8 \beta_{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 8q^{5} - 24q^{7} + O(q^{10})$$ $$4q - 8q^{5} - 24q^{7} - 2q^{11} - 32q^{13} - 112q^{17} + 368q^{19} - 92q^{23} - 376q^{25} + 336q^{29} - 376q^{31} + 1380q^{35} + 696q^{37} + 312q^{41} - 80q^{43} + 228q^{47} - 196q^{49} + 304q^{53} + 16q^{55} - 680q^{59} + 112q^{61} + 2248q^{65} - 352q^{67} + 3632q^{71} - 1148q^{73} - 24q^{77} + 1360q^{79} + 782q^{83} + 2600q^{85} + 480q^{89} - 3984q^{91} - 1924q^{95} + 338q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 2 x^{2} - 3 x + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 2 \nu^{2} - 2 \nu - 9$$$$)/6$$ $$\beta_{2}$$ $$=$$ $$-2 \nu^{3} + 2 \nu^{2} + 10 \nu + 3$$ $$\beta_{3}$$ $$=$$ $$($$$$7 \nu^{3} + 2 \nu^{2} + 10 \nu - 33$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} - 9 \beta_{1}$$$$)/18$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{3} + 2 \beta_{2} + 45 \beta_{1} + 45$$$$)/18$$ $$\nu^{3}$$ $$=$$ $$($$$$2 \beta_{3} - \beta_{2} + 36$$$$)/9$$

## 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
 1.68614 + 0.396143i −1.18614 − 1.26217i 1.68614 − 0.396143i −1.18614 + 1.26217i
0 0 0 −10.6168 18.3889i 0 −14.6168 + 25.3171i 0 0 0
217.2 0 0 0 6.61684 + 11.4607i 0 2.61684 4.53251i 0 0 0
433.1 0 0 0 −10.6168 + 18.3889i 0 −14.6168 25.3171i 0 0 0
433.2 0 0 0 6.61684 11.4607i 0 2.61684 + 4.53251i 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.m 4
3.b odd 2 1 648.4.i.s 4
9.c even 3 1 216.4.a.h yes 2
9.c even 3 1 inner 648.4.i.m 4
9.d odd 6 1 216.4.a.e 2
9.d odd 6 1 648.4.i.s 4
36.f odd 6 1 432.4.a.s 2
36.h even 6 1 432.4.a.o 2
72.j odd 6 1 1728.4.a.bt 2
72.l even 6 1 1728.4.a.bs 2
72.n even 6 1 1728.4.a.bh 2
72.p odd 6 1 1728.4.a.bg 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.4.a.e 2 9.d odd 6 1
216.4.a.h yes 2 9.c even 3 1
432.4.a.o 2 36.h even 6 1
432.4.a.s 2 36.f odd 6 1
648.4.i.m 4 1.a even 1 1 trivial
648.4.i.m 4 9.c even 3 1 inner
648.4.i.s 4 3.b odd 2 1
648.4.i.s 4 9.d odd 6 1
1728.4.a.bg 2 72.p odd 6 1
1728.4.a.bh 2 72.n even 6 1
1728.4.a.bs 2 72.l even 6 1
1728.4.a.bt 2 72.j odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 8 T_{5}^{3} + 345 T_{5}^{2} - 2248 T_{5} + 78961$$ acting on $$S_{4}^{\mathrm{new}}(648, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$78961 - 2248 T + 345 T^{2} + 8 T^{3} + T^{4}$$
$7$ $$23409 - 3672 T + 729 T^{2} + 24 T^{3} + T^{4}$$
$11$ $$( 1 + T + T^{2} )^{2}$$
$13$ $$20214016 - 143872 T + 5520 T^{2} + 32 T^{3} + T^{4}$$
$17$ $$( -3968 + 56 T + T^{2} )^{2}$$
$19$ $$( 7276 - 184 T + T^{2} )^{2}$$
$23$ $$6948496 - 242512 T + 11100 T^{2} + 92 T^{3} + T^{4}$$
$29$ $$730945296 - 9084096 T + 85860 T^{2} - 336 T^{3} + T^{4}$$
$31$ $$779470561 + 10497544 T + 113457 T^{2} + 376 T^{3} + T^{4}$$
$37$ $$( 25524 - 348 T + T^{2} )^{2}$$
$41$ $$28772496 + 1673568 T + 102708 T^{2} - 312 T^{3} + T^{4}$$
$43$ $$3204918544 - 4528960 T + 63012 T^{2} + 80 T^{3} + T^{4}$$
$47$ $$36144144 + 1370736 T + 57996 T^{2} - 228 T^{3} + T^{4}$$
$53$ $$( -44417 - 152 T + T^{2} )^{2}$$
$59$ $$3077142784 - 37720960 T + 517872 T^{2} + 680 T^{3} + T^{4}$$
$61$ $$90596184064 + 33711104 T + 313536 T^{2} - 112 T^{3} + T^{4}$$
$67$ $$97566270736 - 109949312 T + 436260 T^{2} + 352 T^{3} + T^{4}$$
$71$ $$( 781696 - 1816 T + T^{2} )^{2}$$
$73$ $$( 287 + T )^{4}$$
$79$ $$25050025984 - 215249920 T + 1691328 T^{2} - 1360 T^{3} + T^{4}$$
$83$ $$22875655009 + 118275154 T + 762771 T^{2} - 782 T^{3} + T^{4}$$
$89$ $$( -81828 - 240 T + T^{2} )^{2}$$
$97$ $$91259809 - 3228914 T + 104691 T^{2} - 338 T^{3} + T^{4}$$