# Properties

 Label 648.4.i.n 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: $$\Q(\sqrt{-3}, \sqrt{-67})$$ Defining polynomial: $$x^{4} - x^{3} - 16 x^{2} - 17 x + 289$$ 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.

 $$f(q)$$ $$=$$ $$q + ( -4 - 4 \beta_{1} + \beta_{3} ) q^{5} + ( -15 \beta_{1} + \beta_{2} - \beta_{3} ) q^{7} +O(q^{10})$$ $$q + ( -4 - 4 \beta_{1} + \beta_{3} ) q^{5} + ( -15 \beta_{1} + \beta_{2} - \beta_{3} ) q^{7} + ( -23 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{11} + ( -46 - 46 \beta_{1} - \beta_{3} ) q^{13} + ( 11 - 8 \beta_{2} ) q^{17} + ( -43 - 5 \beta_{2} ) q^{19} + ( 29 + 29 \beta_{1} - \beta_{3} ) q^{23} + ( 92 \beta_{1} + 8 \beta_{2} - 8 \beta_{3} ) q^{25} + ( 54 \beta_{1} - \beta_{2} + \beta_{3} ) q^{29} + ( -68 - 68 \beta_{1} + 16 \beta_{3} ) q^{31} + ( 141 + 11 \beta_{2} ) q^{35} + ( -90 + \beta_{2} ) q^{37} + ( -336 - 336 \beta_{1} - 2 \beta_{3} ) q^{41} + ( -35 \beta_{1} - 23 \beta_{2} + 23 \beta_{3} ) q^{43} + ( 426 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{47} + ( -83 - 83 \beta_{1} - 30 \beta_{3} ) q^{49} + ( 334 + 16 \beta_{2} ) q^{53} + ( -695 + 35 \beta_{2} ) q^{55} + ( -274 - 274 \beta_{1} - 6 \beta_{3} ) q^{59} + ( 142 \beta_{1} - 17 \beta_{2} + 17 \beta_{3} ) q^{61} + ( -17 \beta_{1} + 42 \beta_{2} - 42 \beta_{3} ) q^{65} + ( -581 - 581 \beta_{1} - 11 \beta_{3} ) q^{67} + ( -403 + 3 \beta_{2} ) q^{71} + ( -755 - 18 \beta_{2} ) q^{73} + ( 258 + 258 \beta_{1} + 22 \beta_{3} ) q^{77} + ( -11 \beta_{1} + 41 \beta_{2} - 41 \beta_{3} ) q^{79} + ( 770 \beta_{1} + 38 \beta_{2} - 38 \beta_{3} ) q^{83} + ( -1652 - 1652 \beta_{1} + 43 \beta_{3} ) q^{85} + 1323 q^{89} + ( -891 - 61 \beta_{2} ) q^{91} + ( -833 - 833 \beta_{1} - 23 \beta_{3} ) q^{95} + ( 236 \beta_{1} + 50 \beta_{2} - 50 \beta_{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 8q^{5} + 30q^{7} + O(q^{10})$$ $$4q - 8q^{5} + 30q^{7} + 46q^{11} - 92q^{13} + 44q^{17} - 172q^{19} + 58q^{23} - 184q^{25} - 108q^{29} - 136q^{31} + 564q^{35} - 360q^{37} - 672q^{41} + 70q^{43} - 852q^{47} - 166q^{49} + 1336q^{53} - 2780q^{55} - 548q^{59} - 284q^{61} + 34q^{65} - 1162q^{67} - 1612q^{71} - 3020q^{73} + 516q^{77} + 22q^{79} - 1540q^{83} - 3304q^{85} + 5292q^{89} - 3564q^{91} - 1666q^{95} - 472q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 16 x^{2} - 17 x + 289$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 16 \nu^{2} - 16 \nu - 289$$$$)/272$$ $$\beta_{2}$$ $$=$$ $$($$$$-2 \nu^{3} + 2 \nu^{2} + 66 \nu + 17$$$$)/17$$ $$\beta_{3}$$ $$=$$ $$($$$$35 \nu^{3} + 16 \nu^{2} + 528 \nu - 1139$$$$)/272$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} - 3 \beta_{1}$$$$)/6$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{3} + 2 \beta_{2} + 99 \beta_{1} + 99$$$$)/6$$ $$\nu^{3}$$ $$=$$ $$($$$$16 \beta_{3} - 8 \beta_{2} + 75$$$$)/3$$

## 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
 −3.29436 − 2.47935i 3.79436 + 1.61333i −3.29436 + 2.47935i 3.79436 − 1.61333i
0 0 0 −9.08872 15.7421i 0 0.411277 0.712352i 0 0 0
217.2 0 0 0 5.08872 + 8.81393i 0 14.5887 25.2684i 0 0 0
433.1 0 0 0 −9.08872 + 15.7421i 0 0.411277 + 0.712352i 0 0 0
433.2 0 0 0 5.08872 8.81393i 0 14.5887 + 25.2684i 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.n 4
3.b odd 2 1 648.4.i.t 4
9.c even 3 1 648.4.a.f yes 2
9.c even 3 1 inner 648.4.i.n 4
9.d odd 6 1 648.4.a.c 2
9.d odd 6 1 648.4.i.t 4
36.f odd 6 1 1296.4.a.q 2
36.h even 6 1 1296.4.a.m 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 8 T_{5}^{3} + 249 T_{5}^{2} - 1480 T_{5} + 34225$$ acting on $$S_{4}^{\mathrm{new}}(648, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$34225 - 1480 T + 249 T^{2} + 8 T^{3} + T^{4}$$
$7$ $$576 - 720 T + 876 T^{2} - 30 T^{3} + T^{4}$$
$11$ $$1638400 + 58880 T + 3396 T^{2} - 46 T^{3} + T^{4}$$
$13$ $$3667225 + 176180 T + 6549 T^{2} + 92 T^{3} + T^{4}$$
$17$ $$( -12743 - 22 T + T^{2} )^{2}$$
$19$ $$( -3176 + 86 T + T^{2} )^{2}$$
$23$ $$409600 - 37120 T + 2724 T^{2} - 58 T^{3} + T^{4}$$
$29$ $$7371225 + 293220 T + 8949 T^{2} + 108 T^{3} + T^{4}$$
$31$ $$2193236224 - 6369152 T + 65328 T^{2} + 136 T^{3} + T^{4}$$
$37$ $$( 7899 + 180 T + T^{2} )^{2}$$
$41$ $$12564616464 + 75325824 T + 339492 T^{2} + 672 T^{3} + T^{4}$$
$43$ $$11046850816 + 7357280 T + 110004 T^{2} - 70 T^{3} + T^{4}$$
$47$ $$32642371584 + 153932544 T + 545232 T^{2} + 852 T^{3} + T^{4}$$
$53$ $$( 60100 - 668 T + T^{2} )^{2}$$
$59$ $$4602265600 + 37176320 T + 232464 T^{2} + 548 T^{3} + T^{4}$$
$61$ $$1438305625 - 10770700 T + 118581 T^{2} + 284 T^{3} + T^{4}$$
$67$ $$98119297600 + 363984880 T + 1037004 T^{2} + 1162 T^{3} + T^{4}$$
$71$ $$( 160600 + 806 T + T^{2} )^{2}$$
$73$ $$( 504901 + 1510 T + T^{2} )^{2}$$
$79$ $$114081817600 + 7430720 T + 338244 T^{2} - 22 T^{3} + T^{4}$$
$83$ $$91600654336 + 466090240 T + 2068944 T^{2} + 1540 T^{3} + T^{4}$$
$89$ $$( -1323 + T )^{4}$$
$97$ $$199633814416 - 210891488 T + 669588 T^{2} + 472 T^{3} + T^{4}$$