# Properties

 Label 648.4.a.i Level $648$ Weight $4$ Character orbit 648.a Self dual yes Analytic conductor $38.233$ Analytic rank $1$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$648 = 2^{3} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 648.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$38.2332376837$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.72153.1 Defining polynomial: $$x^{4} - x^{3} - 8 x^{2} + 3 x + 12$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{3}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 72) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta_{2} ) q^{5} + ( -1 - \beta_{3} ) q^{7} +O(q^{10})$$ $$q + ( 1 + \beta_{2} ) q^{5} + ( -1 - \beta_{3} ) q^{7} + ( -4 + \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{11} + ( -6 - \beta_{1} + 4 \beta_{3} ) q^{13} + ( 1 - 5 \beta_{1} - 10 \beta_{2} + 6 \beta_{3} ) q^{17} + ( -28 + 6 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{19} + ( -10 - \beta_{1} - 3 \beta_{2} - 7 \beta_{3} ) q^{23} + ( -26 - 4 \beta_{1} + 9 \beta_{2} - 2 \beta_{3} ) q^{25} + ( -4 + 17 \beta_{1} + 2 \beta_{2} ) q^{29} + ( -70 - 17 \beta_{1} - 25 \beta_{2} + 9 \beta_{3} ) q^{31} + ( -41 - 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} ) q^{35} + ( -103 + 13 \beta_{1} + 17 \beta_{2} - 16 \beta_{3} ) q^{37} + ( 86 - 7 \beta_{1} + 21 \beta_{2} - 20 \beta_{3} ) q^{41} + ( -141 + 10 \beta_{1} + 19 \beta_{2} - 29 \beta_{3} ) q^{43} + ( -43 - 6 \beta_{1} + 10 \beta_{2} + 3 \beta_{3} ) q^{47} + ( -211 + 11 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{49} + ( 119 - 25 \beta_{1} - 9 \beta_{2} + 32 \beta_{3} ) q^{53} + ( -280 + 5 \beta_{1} - 35 \beta_{2} + 13 \beta_{3} ) q^{55} + ( -149 + 8 \beta_{1} + 13 \beta_{2} + 41 \beta_{3} ) q^{59} + ( -308 - 27 \beta_{1} - 44 \beta_{2} + 28 \beta_{3} ) q^{61} + ( 194 + 9 \beta_{1} + 12 \beta_{2} - 18 \beta_{3} ) q^{65} + ( -300 - 9 \beta_{1} + 56 \beta_{2} + 9 \beta_{3} ) q^{67} + ( -121 + 37 \beta_{1} + 47 \beta_{2} - 56 \beta_{3} ) q^{71} + ( -321 - 39 \beta_{1} + 26 \beta_{2} + 10 \beta_{3} ) q^{73} + ( 175 + 12 \beta_{1} + 9 \beta_{2} - 20 \beta_{3} ) q^{77} + ( -470 + 29 \beta_{1} + 13 \beta_{2} + 15 \beta_{3} ) q^{79} + ( -333 - 60 \beta_{1} - 24 \beta_{2} + 5 \beta_{3} ) q^{83} + ( -539 + 57 \beta_{1} - 3 \beta_{2} - 28 \beta_{3} ) q^{85} + ( 181 + 41 \beta_{1} + 15 \beta_{2} - 36 \beta_{3} ) q^{89} + ( -478 - 41 \beta_{1} - 11 \beta_{2} + 31 \beta_{3} ) q^{91} + ( -286 + 2 \beta_{1} - 118 \beta_{2} + 32 \beta_{3} ) q^{95} + ( -606 + 21 \beta_{1} - 39 \beta_{2} + 64 \beta_{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 5q^{5} - 3q^{7} + O(q^{10})$$ $$4q + 5q^{5} - 3q^{7} - 16q^{11} - 29q^{13} - 17q^{17} - 109q^{19} - 37q^{23} - 97q^{25} + 3q^{29} - 331q^{31} - 171q^{35} - 366q^{37} + 378q^{41} - 506q^{43} - 171q^{47} - 829q^{49} + 410q^{53} - 1163q^{55} - 616q^{59} - 1331q^{61} + 815q^{65} - 1162q^{67} - 344q^{71} - 1307q^{73} + 741q^{77} - 1853q^{79} - 1421q^{83} - 2074q^{85} + 816q^{89} - 1995q^{91} - 1292q^{95} - 2506q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 8 x^{2} + 3 x + 12$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$3 \nu^{3} - 3 \nu^{2} - 12 \nu + 4$$ $$\beta_{2}$$ $$=$$ $$-\nu^{3} + 5 \nu^{2} - 17$$ $$\beta_{3}$$ $$=$$ $$-\nu^{3} + 5 \nu^{2} + 6 \nu - 19$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} - \beta_{2} + 2$$$$)/6$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{3} + \beta_{2} + \beta_{1} + 51$$$$)/12$$ $$\nu^{3}$$ $$=$$ $$($$$$10 \beta_{3} - 7 \beta_{2} + 5 \beta_{1} + 51$$$$)/12$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 1.51317 −1.26386 2.90825 −2.15756
0 0 0 −8.01626 0 0.937231 0 0 0
1.2 0 0 0 −5.99447 0 15.5776 0 0 0
1.3 0 0 0 1.69184 0 −17.1413 0 0 0
1.4 0 0 0 17.3189 0 −2.37353 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 648.4.a.i 4
3.b odd 2 1 648.4.a.h 4
4.b odd 2 1 1296.4.a.ba 4
9.c even 3 2 72.4.i.a 8
9.d odd 6 2 216.4.i.a 8
12.b even 2 1 1296.4.a.y 4
36.f odd 6 2 144.4.i.e 8
36.h even 6 2 432.4.i.e 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.4.i.a 8 9.c even 3 2
144.4.i.e 8 36.f odd 6 2
216.4.i.a 8 9.d odd 6 2
432.4.i.e 8 36.h even 6 2
648.4.a.h 4 3.b odd 2 1
648.4.a.i 4 1.a even 1 1 trivial
1296.4.a.y 4 12.b even 2 1
1296.4.a.ba 4 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} - 5 T_{5}^{3} - 189 T_{5}^{2} - 503 T_{5} + 1408$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(648))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$1408 - 503 T - 189 T^{2} - 5 T^{3} + T^{4}$$
$7$ $$594 - 387 T - 267 T^{2} + 3 T^{3} + T^{4}$$
$11$ $$38929 + 2056 T - 1866 T^{2} + 16 T^{3} + T^{4}$$
$13$ $$141262 + 33335 T - 3771 T^{2} + 29 T^{3} + T^{4}$$
$17$ $$2567224 - 723268 T - 14562 T^{2} + 17 T^{3} + T^{4}$$
$19$ $$-5081408 - 1246544 T - 11772 T^{2} + 109 T^{3} + T^{4}$$
$23$ $$37951678 - 275669 T - 18855 T^{2} + 37 T^{3} + T^{4}$$
$29$ $$1572741522 - 2926449 T - 94407 T^{2} - 3 T^{3} + T^{4}$$
$31$ $$-3182697596 - 34618859 T - 67149 T^{2} + 331 T^{3} + T^{4}$$
$37$ $$-215981856 - 7645176 T - 18492 T^{2} + 366 T^{3} + T^{4}$$
$41$ $$-360819333 + 38061882 T - 135996 T^{2} - 378 T^{3} + T^{4}$$
$43$ $$-567289217 - 45189154 T - 68328 T^{2} + 506 T^{3} + T^{4}$$
$47$ $$-310694022 - 7460379 T - 37059 T^{2} + 171 T^{3} + T^{4}$$
$53$ $$-15963093536 + 137282536 T - 239580 T^{2} - 410 T^{3} + T^{4}$$
$59$ $$1415131633 - 240406592 T - 485994 T^{2} + 616 T^{3} + T^{4}$$
$61$ $$-48115815704 - 133690159 T + 349659 T^{2} + 1331 T^{3} + T^{4}$$
$67$ $$-137320199621 - 658601234 T - 329940 T^{2} + 1162 T^{3} + T^{4}$$
$71$ $$9762389248 - 196817728 T - 654384 T^{2} + 344 T^{3} + T^{4}$$
$73$ $$-88005243128 - 613828588 T - 217110 T^{2} + 1307 T^{3} + T^{4}$$
$79$ $$-98670843044 - 91124605 T + 875259 T^{2} + 1853 T^{3} + T^{4}$$
$83$ $$67888747828 - 495031429 T - 280773 T^{2} + 1421 T^{3} + T^{4}$$
$89$ $$22003976592 + 86780160 T - 307656 T^{2} - 816 T^{3} + T^{4}$$
$97$ $$-818649939857 - 1529506394 T + 865152 T^{2} + 2506 T^{3} + T^{4}$$