# Properties

 Label 648.4.a.h 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
 −2.15756 2.90825 −1.26386 1.51317
0 0 0 −17.3189 0 −2.37353 0 0 0
1.2 0 0 0 −1.69184 0 −17.1413 0 0 0
1.3 0 0 0 5.99447 0 15.5776 0 0 0
1.4 0 0 0 8.01626 0 0.937231 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.h 4
3.b odd 2 1 648.4.a.i 4
4.b odd 2 1 1296.4.a.y 4
9.c even 3 2 216.4.i.a 8
9.d odd 6 2 72.4.i.a 8
12.b even 2 1 1296.4.a.ba 4
36.f odd 6 2 432.4.i.e 8
36.h even 6 2 144.4.i.e 8

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

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