# Properties

 Label 648.2.f.a Level 648 Weight 2 Character orbit 648.f Analytic conductor 5.174 Analytic rank 0 Dimension 4 CM discriminant -8 Inner twists 4

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## Newspace parameters

 Level: $$N$$ = $$648 = 2^{3} \cdot 3^{4}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 648.f (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.17430605098$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 72) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} -2 q^{4} -2 \beta_{1} q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} -2 q^{4} -2 \beta_{1} q^{8} + ( -\beta_{1} - \beta_{2} ) q^{11} + 4 q^{16} + ( 2 \beta_{1} - \beta_{2} ) q^{17} + ( -1 + \beta_{3} ) q^{19} + ( 2 + \beta_{3} ) q^{22} -5 q^{25} + 4 \beta_{1} q^{32} + ( -4 + \beta_{3} ) q^{34} + ( -\beta_{1} + 2 \beta_{2} ) q^{38} + ( -4 \beta_{1} - \beta_{2} ) q^{41} + ( 5 + \beta_{3} ) q^{43} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{44} + 7 q^{49} -5 \beta_{1} q^{50} + ( 5 \beta_{1} - \beta_{2} ) q^{59} -8 q^{64} + ( -7 + \beta_{3} ) q^{67} + ( -4 \beta_{1} + 2 \beta_{2} ) q^{68} + ( -1 - 2 \beta_{3} ) q^{73} + ( 2 - 2 \beta_{3} ) q^{76} + ( 8 + \beta_{3} ) q^{82} + 2 \beta_{1} q^{83} + ( 5 \beta_{1} + 2 \beta_{2} ) q^{86} + ( -4 - 2 \beta_{3} ) q^{88} -4 \beta_{1} q^{89} + ( 5 - 2 \beta_{3} ) q^{97} + 7 \beta_{1} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 8q^{4} + O(q^{10})$$ $$4q - 8q^{4} + 16q^{16} - 4q^{19} + 8q^{22} - 20q^{25} - 16q^{34} + 20q^{43} + 28q^{49} - 32q^{64} - 28q^{67} - 4q^{73} + 8q^{76} + 32q^{82} - 16q^{88} + 20q^{97} + O(q^{100})$$

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

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

## Character values

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

 $$n$$ $$325$$ $$487$$ $$569$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
323.1
 −1.22474 − 0.707107i 1.22474 − 0.707107i 1.22474 + 0.707107i −1.22474 + 0.707107i
1.41421i 0 −2.00000 0 0 0 2.82843i 0 0
323.2 1.41421i 0 −2.00000 0 0 0 2.82843i 0 0
323.3 1.41421i 0 −2.00000 0 0 0 2.82843i 0 0
323.4 1.41421i 0 −2.00000 0 0 0 2.82843i 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
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$
3.b odd 2 1 inner
24.f even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 648.2.f.a 4
3.b odd 2 1 inner 648.2.f.a 4
4.b odd 2 1 2592.2.f.a 4
8.b even 2 1 2592.2.f.a 4
8.d odd 2 1 CM 648.2.f.a 4
9.c even 3 1 72.2.l.a 4
9.c even 3 1 216.2.l.a 4
9.d odd 6 1 72.2.l.a 4
9.d odd 6 1 216.2.l.a 4
12.b even 2 1 2592.2.f.a 4
24.f even 2 1 inner 648.2.f.a 4
24.h odd 2 1 2592.2.f.a 4
36.f odd 6 1 288.2.p.a 4
36.f odd 6 1 864.2.p.a 4
36.h even 6 1 288.2.p.a 4
36.h even 6 1 864.2.p.a 4
72.j odd 6 1 288.2.p.a 4
72.j odd 6 1 864.2.p.a 4
72.l even 6 1 72.2.l.a 4
72.l even 6 1 216.2.l.a 4
72.n even 6 1 288.2.p.a 4
72.n even 6 1 864.2.p.a 4
72.p odd 6 1 72.2.l.a 4
72.p odd 6 1 216.2.l.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.2.l.a 4 9.c even 3 1
72.2.l.a 4 9.d odd 6 1
72.2.l.a 4 72.l even 6 1
72.2.l.a 4 72.p odd 6 1
216.2.l.a 4 9.c even 3 1
216.2.l.a 4 9.d odd 6 1
216.2.l.a 4 72.l even 6 1
216.2.l.a 4 72.p odd 6 1
288.2.p.a 4 36.f odd 6 1
288.2.p.a 4 36.h even 6 1
288.2.p.a 4 72.j odd 6 1
288.2.p.a 4 72.n even 6 1
648.2.f.a 4 1.a even 1 1 trivial
648.2.f.a 4 3.b odd 2 1 inner
648.2.f.a 4 8.d odd 2 1 CM
648.2.f.a 4 24.f even 2 1 inner
864.2.p.a 4 36.f odd 6 1
864.2.p.a 4 36.h even 6 1
864.2.p.a 4 72.j odd 6 1
864.2.p.a 4 72.n even 6 1
2592.2.f.a 4 4.b odd 2 1
2592.2.f.a 4 8.b even 2 1
2592.2.f.a 4 12.b even 2 1
2592.2.f.a 4 24.h odd 2 1

## Hecke kernels

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

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + 2 T^{2} )^{2}$$
$3$ 1
$5$ $$( 1 + 5 T^{2} )^{4}$$
$7$ $$( 1 - 7 T^{2} )^{4}$$
$11$ $$( 1 - 6 T + 25 T^{2} - 66 T^{3} + 121 T^{4} )( 1 + 6 T + 25 T^{2} + 66 T^{3} + 121 T^{4} )$$
$13$ $$( 1 - 13 T^{2} )^{4}$$
$17$ $$( 1 - 6 T + 19 T^{2} - 102 T^{3} + 289 T^{4} )( 1 + 6 T + 19 T^{2} + 102 T^{3} + 289 T^{4} )$$
$19$ $$( 1 + 2 T - 15 T^{2} + 38 T^{3} + 361 T^{4} )^{2}$$
$23$ $$( 1 + 23 T^{2} )^{4}$$
$29$ $$( 1 + 29 T^{2} )^{4}$$
$31$ $$( 1 - 31 T^{2} )^{4}$$
$37$ $$( 1 - 37 T^{2} )^{4}$$
$41$ $$( 1 - 6 T - 5 T^{2} - 246 T^{3} + 1681 T^{4} )( 1 + 6 T - 5 T^{2} + 246 T^{3} + 1681 T^{4} )$$
$43$ $$( 1 - 10 T + 57 T^{2} - 430 T^{3} + 1849 T^{4} )^{2}$$
$47$ $$( 1 + 47 T^{2} )^{4}$$
$53$ $$( 1 + 53 T^{2} )^{4}$$
$59$ $$( 1 - 6 T - 23 T^{2} - 354 T^{3} + 3481 T^{4} )( 1 + 6 T - 23 T^{2} + 354 T^{3} + 3481 T^{4} )$$
$61$ $$( 1 - 61 T^{2} )^{4}$$
$67$ $$( 1 + 14 T + 129 T^{2} + 938 T^{3} + 4489 T^{4} )^{2}$$
$71$ $$( 1 + 71 T^{2} )^{4}$$
$73$ $$( 1 + 2 T - 69 T^{2} + 146 T^{3} + 5329 T^{4} )^{2}$$
$79$ $$( 1 - 79 T^{2} )^{4}$$
$83$ $$( 1 - 18 T + 83 T^{2} )^{2}( 1 + 18 T + 83 T^{2} )^{2}$$
$89$ $$( 1 - 18 T + 89 T^{2} )^{2}( 1 + 18 T + 89 T^{2} )^{2}$$
$97$ $$( 1 - 10 T + 3 T^{2} - 970 T^{3} + 9409 T^{4} )^{2}$$
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