# Properties

 Label 648.4.i.o 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: $$\Q(\sqrt{-3}, \sqrt{5})$$ Defining polynomial: $$x^{4} - x^{3} + 2 x^{2} + x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}\cdot 3^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q + ( -2 \beta_{1} - \beta_{2} ) q^{5} + ( 3 - 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{7} +O(q^{10})$$ $$q + ( -2 \beta_{1} - \beta_{2} ) q^{5} + ( 3 - 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{7} + ( -26 + 26 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{11} + ( -13 \beta_{1} + 2 \beta_{2} ) q^{13} + ( 94 + \beta_{3} ) q^{17} + ( -37 + 8 \beta_{3} ) q^{19} + ( -74 \beta_{1} - 5 \beta_{2} ) q^{23} + ( -59 + 59 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} ) q^{25} + ( -144 + 144 \beta_{1} + 8 \beta_{2} + 8 \beta_{3} ) q^{29} + ( 124 \beta_{1} - 8 \beta_{2} ) q^{31} + ( 354 - \beta_{3} ) q^{35} + ( 171 - 10 \beta_{3} ) q^{37} + 32 \beta_{2} q^{41} + ( 128 - 128 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} ) q^{43} + ( -66 + 66 \beta_{1} + \beta_{2} + \beta_{3} ) q^{47} + ( -386 \beta_{1} + 12 \beta_{2} ) q^{49} + ( 476 - 14 \beta_{3} ) q^{53} + ( -488 - 20 \beta_{3} ) q^{55} + ( 502 \beta_{1} - 21 \beta_{2} ) q^{59} + ( 17 - 17 \beta_{1} - 34 \beta_{2} - 34 \beta_{3} ) q^{61} + ( 334 - 334 \beta_{1} + 9 \beta_{2} + 9 \beta_{3} ) q^{65} + ( 433 \beta_{1} + 16 \beta_{2} ) q^{67} + ( 388 + 30 \beta_{3} ) q^{71} + ( 937 + 12 \beta_{3} ) q^{73} + ( 1158 \beta_{1} - 61 \beta_{2} ) q^{77} + ( -91 + 91 \beta_{1} - 26 \beta_{2} - 26 \beta_{3} ) q^{79} + ( 668 - 668 \beta_{1} - 46 \beta_{2} - 46 \beta_{3} ) q^{83} + ( -8 \beta_{1} - 92 \beta_{2} ) q^{85} + ( 438 + 21 \beta_{3} ) q^{89} + ( -759 - 32 \beta_{3} ) q^{91} + ( 1514 \beta_{1} + 53 \beta_{2} ) q^{95} + ( 19 - 19 \beta_{1} + 88 \beta_{2} + 88 \beta_{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{5} + 6q^{7} + O(q^{10})$$ $$4q - 4q^{5} + 6q^{7} - 52q^{11} - 26q^{13} + 376q^{17} - 148q^{19} - 148q^{23} - 118q^{25} - 288q^{29} + 248q^{31} + 1416q^{35} + 684q^{37} + 256q^{43} - 132q^{47} - 772q^{49} + 1904q^{53} - 1952q^{55} + 1004q^{59} + 34q^{61} + 668q^{65} + 866q^{67} + 1552q^{71} + 3748q^{73} + 2316q^{77} - 182q^{79} + 1336q^{83} - 16q^{85} + 1752q^{89} - 3036q^{91} + 3028q^{95} + 38q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 2 x^{2} + x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{3} + 2 \nu^{2} - 2 \nu + 1$$$$)/2$$ $$\beta_{2}$$ $$=$$ $$3 \nu^{3} - 6 \nu^{2} + 18 \nu - 3$$ $$\beta_{3}$$ $$=$$ $$6 \nu^{3} + 12$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + 6 \beta_{1}$$$$)/12$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} + 18 \beta_{1} - 18$$$$)/12$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{3} - 12$$$$)/6$$

## Character values

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

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

## 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
 0.809017 + 1.40126i −0.309017 − 0.535233i 0.809017 − 1.40126i −0.309017 + 0.535233i
0 0 0 −7.70820 13.3510i 0 −11.9164 + 20.6398i 0 0 0
217.2 0 0 0 5.70820 + 9.88690i 0 14.9164 25.8360i 0 0 0
433.1 0 0 0 −7.70820 + 13.3510i 0 −11.9164 20.6398i 0 0 0
433.2 0 0 0 5.70820 9.88690i 0 14.9164 + 25.8360i 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.o 4
3.b odd 2 1 648.4.i.r 4
9.c even 3 1 216.4.a.g yes 2
9.c even 3 1 inner 648.4.i.o 4
9.d odd 6 1 216.4.a.f 2
9.d odd 6 1 648.4.i.r 4
36.f odd 6 1 432.4.a.r 2
36.h even 6 1 432.4.a.p 2
72.j odd 6 1 1728.4.a.bq 2
72.l even 6 1 1728.4.a.br 2
72.n even 6 1 1728.4.a.bi 2
72.p odd 6 1 1728.4.a.bj 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.4.a.f 2 9.d odd 6 1
216.4.a.g yes 2 9.c even 3 1
432.4.a.p 2 36.h even 6 1
432.4.a.r 2 36.f odd 6 1
648.4.i.o 4 1.a even 1 1 trivial
648.4.i.o 4 9.c even 3 1 inner
648.4.i.r 4 3.b odd 2 1
648.4.i.r 4 9.d odd 6 1
1728.4.a.bi 2 72.n even 6 1
1728.4.a.bj 2 72.p odd 6 1
1728.4.a.bq 2 72.j odd 6 1
1728.4.a.br 2 72.l even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 4 T_{5}^{3} + 192 T_{5}^{2} - 704 T_{5} + 30976$$ acting on $$S_{4}^{\mathrm{new}}(648, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$30976 - 704 T + 192 T^{2} + 4 T^{3} + T^{4}$$
$7$ $$505521 + 4266 T + 747 T^{2} - 6 T^{3} + T^{4}$$
$11$ $$891136 - 49088 T + 3648 T^{2} + 52 T^{3} + T^{4}$$
$13$ $$303601 - 14326 T + 1227 T^{2} + 26 T^{3} + T^{4}$$
$17$ $$( 8656 - 188 T + T^{2} )^{2}$$
$19$ $$( -10151 + 74 T + T^{2} )^{2}$$
$23$ $$952576 + 144448 T + 20928 T^{2} + 148 T^{3} + T^{4}$$
$29$ $$84934656 + 2654208 T + 73728 T^{2} + 288 T^{3} + T^{4}$$
$31$ $$14868736 - 956288 T + 57648 T^{2} - 248 T^{3} + T^{4}$$
$37$ $$( 11241 - 342 T + T^{2} )^{2}$$
$41$ $$33973862400 + 184320 T^{2} + T^{4}$$
$43$ $$182358016 - 3457024 T + 52032 T^{2} - 256 T^{3} + T^{4}$$
$47$ $$17438976 + 551232 T + 13248 T^{2} + 132 T^{3} + T^{4}$$
$53$ $$( 191296 - 952 T + T^{2} )^{2}$$
$59$ $$29799045376 - 173314496 T + 835392 T^{2} - 1004 T^{3} + T^{4}$$
$61$ $$43177099681 + 7064894 T + 208947 T^{2} - 34 T^{3} + T^{4}$$
$67$ $$19996505281 - 122460194 T + 608547 T^{2} - 866 T^{3} + T^{4}$$
$71$ $$( -11456 - 776 T + T^{2} )^{2}$$
$73$ $$( 852049 - 1874 T + T^{2} )^{2}$$
$79$ $$12859333201 - 20638618 T + 146523 T^{2} + 182 T^{3} + T^{4}$$
$83$ $$4269838336 - 87299584 T + 1719552 T^{2} - 1336 T^{3} + T^{4}$$
$89$ $$( 112464 - 876 T + T^{2} )^{2}$$
$97$ $$1942006686481 + 52955242 T + 1395003 T^{2} - 38 T^{3} + T^{4}$$