Properties

 Label 2601.2.a.u Level $2601$ Weight $2$ Character orbit 2601.a Self dual yes Analytic conductor $20.769$ Analytic rank $1$ Dimension $3$ CM discriminant -3 Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$2601 = 3^{2} \cdot 17^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2601.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$20.7690895657$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{18})^+$$ Defining polynomial: $$x^{3} - 3 x - 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $N(\mathrm{U}(1))$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -2 q^{4} + ( 3 \beta_{1} - \beta_{2} ) q^{7} +O(q^{10})$$ $$q -2 q^{4} + ( 3 \beta_{1} - \beta_{2} ) q^{7} + ( -4 \beta_{1} + 3 \beta_{2} ) q^{13} + 4 q^{16} + ( -3 \beta_{1} - 2 \beta_{2} ) q^{19} -5 q^{25} + ( -6 \beta_{1} + 2 \beta_{2} ) q^{28} + ( \beta_{1} - 6 \beta_{2} ) q^{31} + ( -4 \beta_{1} + 7 \beta_{2} ) q^{37} + ( 7 \beta_{1} - \beta_{2} ) q^{43} + ( 7 - 5 \beta_{1} + 8 \beta_{2} ) q^{49} + ( 8 \beta_{1} - 6 \beta_{2} ) q^{52} + ( 5 \beta_{1} + 4 \beta_{2} ) q^{61} -8 q^{64} + ( -2 \beta_{1} - 7 \beta_{2} ) q^{67} -17 q^{73} + ( 6 \beta_{1} + 4 \beta_{2} ) q^{76} -17 q^{79} + ( -17 + 10 \beta_{1} - 9 \beta_{2} ) q^{91} + ( -8 \beta_{1} - 3 \beta_{2} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - 6q^{4} + O(q^{10})$$ $$3q - 6q^{4} + 12q^{16} - 15q^{25} + 21q^{49} - 24q^{64} - 51q^{73} - 51q^{79} - 51q^{91} + O(q^{100})$$

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
 −1.53209 −0.347296 1.87939
0 0 −2.00000 0 0 −4.94356 0 0 0
1.2 0 0 −2.00000 0 0 0.837496 0 0 0
1.3 0 0 −2.00000 0 0 4.10607 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
$$3$$ $$1$$
$$17$$ $$1$$

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2601.2.a.u 3
3.b odd 2 1 CM 2601.2.a.u 3
17.b even 2 1 2601.2.a.v yes 3
51.c odd 2 1 2601.2.a.v yes 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2601.2.a.u 3 1.a even 1 1 trivial
2601.2.a.u 3 3.b odd 2 1 CM
2601.2.a.v yes 3 17.b even 2 1
2601.2.a.v yes 3 51.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2601))$$:

 $$T_{2}$$ $$T_{5}$$ $$T_{7}^{3} - 21 T_{7} + 17$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3}$$
$5$ $$T^{3}$$
$7$ $$17 - 21 T + T^{3}$$
$11$ $$T^{3}$$
$13$ $$-89 - 39 T + T^{3}$$
$17$ $$T^{3}$$
$19$ $$163 - 57 T + T^{3}$$
$23$ $$T^{3}$$
$29$ $$T^{3}$$
$31$ $$-289 - 93 T + T^{3}$$
$37$ $$323 - 111 T + T^{3}$$
$41$ $$T^{3}$$
$43$ $$-71 - 129 T + T^{3}$$
$47$ $$T^{3}$$
$53$ $$T^{3}$$
$59$ $$T^{3}$$
$61$ $$-901 - 183 T + T^{3}$$
$67$ $$127 - 201 T + T^{3}$$
$71$ $$T^{3}$$
$73$ $$( 17 + T )^{3}$$
$79$ $$( 17 + T )^{3}$$
$83$ $$T^{3}$$
$89$ $$T^{3}$$
$97$ $$1853 - 291 T + T^{3}$$