# Properties

 Label 2601.2.a.x Level $2601$ Weight $2$ Character orbit 2601.a Self dual yes Analytic conductor $20.769$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# 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: $$0$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{18})^+$$ Defining polynomial: $$x^{3} - 3 x - 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 289) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + \beta_{2} q^{4} + ( 2 - \beta_{1} ) q^{5} + \beta_{2} q^{7} + ( 1 - \beta_{1} ) q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + \beta_{2} q^{4} + ( 2 - \beta_{1} ) q^{5} + \beta_{2} q^{7} + ( 1 - \beta_{1} ) q^{8} + ( -2 + 2 \beta_{1} - \beta_{2} ) q^{10} + ( 2 + 2 \beta_{1} - 2 \beta_{2} ) q^{11} + ( 2 + 3 \beta_{1} - 2 \beta_{2} ) q^{13} + ( 1 + \beta_{1} ) q^{14} + ( -2 + \beta_{1} - 3 \beta_{2} ) q^{16} -\beta_{1} q^{19} + ( -1 - \beta_{1} + 2 \beta_{2} ) q^{20} + ( 2 + 2 \beta_{2} ) q^{22} + ( 4 + \beta_{1} + \beta_{2} ) q^{23} + ( 1 - 4 \beta_{1} + \beta_{2} ) q^{25} + ( 4 + 3 \beta_{2} ) q^{26} + ( 2 + \beta_{1} - \beta_{2} ) q^{28} + ( \beta_{1} + \beta_{2} ) q^{29} + ( -3 + 2 \beta_{1} - 3 \beta_{2} ) q^{31} + ( -3 - 3 \beta_{1} + \beta_{2} ) q^{32} + ( -1 - \beta_{1} + 2 \beta_{2} ) q^{35} + ( 1 - \beta_{1} + 4 \beta_{2} ) q^{37} + ( -2 - \beta_{2} ) q^{38} + ( 4 - 3 \beta_{1} + \beta_{2} ) q^{40} + ( 2 - \beta_{1} + 4 \beta_{2} ) q^{41} + ( 5 - 3 \beta_{1} + 4 \beta_{2} ) q^{43} + ( -2 + 4 \beta_{2} ) q^{44} + ( 3 + 5 \beta_{1} + \beta_{2} ) q^{46} + ( 7 + \beta_{1} - \beta_{2} ) q^{47} + ( -5 + \beta_{1} - \beta_{2} ) q^{49} + ( -7 + 2 \beta_{1} - 4 \beta_{2} ) q^{50} + ( -1 + \beta_{1} + 4 \beta_{2} ) q^{52} + ( 6 - 2 \beta_{1} - 2 \beta_{2} ) q^{53} + ( 2 + 4 \beta_{1} - 6 \beta_{2} ) q^{55} + ( -1 - \beta_{1} + \beta_{2} ) q^{56} + ( 3 + \beta_{1} + \beta_{2} ) q^{58} + ( 3 - 4 \beta_{1} + 5 \beta_{2} ) q^{59} + ( 1 - 2 \beta_{1} + \beta_{2} ) q^{61} + ( 1 - 6 \beta_{1} + 2 \beta_{2} ) q^{62} + ( -1 - 4 \beta_{1} + 3 \beta_{2} ) q^{64} + ( 6 \beta_{1} - 7 \beta_{2} ) q^{65} + ( -3 - 7 \beta_{1} + \beta_{2} ) q^{67} + ( \beta_{1} - \beta_{2} ) q^{70} + ( 7 - 3 \beta_{1} - \beta_{2} ) q^{71} + ( 7 + 5 \beta_{1} - 3 \beta_{2} ) q^{73} + ( 2 + 5 \beta_{1} - \beta_{2} ) q^{74} + ( -1 - \beta_{1} ) q^{76} + ( -2 + 4 \beta_{2} ) q^{77} + ( -1 + 6 \beta_{1} - 4 \beta_{2} ) q^{79} + ( -3 + 7 \beta_{1} - 7 \beta_{2} ) q^{80} + ( 2 + 6 \beta_{1} - \beta_{2} ) q^{82} + ( -3 - 2 \beta_{1} + 6 \beta_{2} ) q^{83} + ( -2 + 9 \beta_{1} - 3 \beta_{2} ) q^{86} + ( 2 \beta_{1} - 4 \beta_{2} ) q^{88} + ( 5 + 7 \beta_{1} - 2 \beta_{2} ) q^{89} + ( -1 + \beta_{1} + 4 \beta_{2} ) q^{91} + ( 3 + 2 \beta_{1} + 3 \beta_{2} ) q^{92} + ( 1 + 6 \beta_{1} + \beta_{2} ) q^{94} + ( 2 - 2 \beta_{1} + \beta_{2} ) q^{95} + ( -2 - 6 \beta_{2} ) q^{97} + ( 1 - 6 \beta_{1} + \beta_{2} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + 6q^{5} + 3q^{8} + O(q^{10})$$ $$3q + 6q^{5} + 3q^{8} - 6q^{10} + 6q^{11} + 6q^{13} + 3q^{14} - 6q^{16} - 3q^{20} + 6q^{22} + 12q^{23} + 3q^{25} + 12q^{26} + 6q^{28} - 9q^{31} - 9q^{32} - 3q^{35} + 3q^{37} - 6q^{38} + 12q^{40} + 6q^{41} + 15q^{43} - 6q^{44} + 9q^{46} + 21q^{47} - 15q^{49} - 21q^{50} - 3q^{52} + 18q^{53} + 6q^{55} - 3q^{56} + 9q^{58} + 9q^{59} + 3q^{61} + 3q^{62} - 3q^{64} - 9q^{67} + 21q^{71} + 21q^{73} + 6q^{74} - 3q^{76} - 6q^{77} - 3q^{79} - 9q^{80} + 6q^{82} - 9q^{83} - 6q^{86} + 15q^{89} - 3q^{91} + 9q^{92} + 3q^{94} + 6q^{95} - 6q^{97} + 3q^{98} + O(q^{100})$$

## 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
−1.53209 0 0.347296 3.53209 0 0.347296 2.53209 0 −5.41147
1.2 −0.347296 0 −1.87939 2.34730 0 −1.87939 1.34730 0 −0.815207
1.3 1.87939 0 1.53209 0.120615 0 1.53209 −0.879385 0 0.226682
 $$n$$: 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

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 2601.2.a.x 3
3.b odd 2 1 289.2.a.d 3
12.b even 2 1 4624.2.a.bg 3
15.d odd 2 1 7225.2.a.t 3
17.b even 2 1 2601.2.a.w 3
51.c odd 2 1 289.2.a.e yes 3
51.f odd 4 2 289.2.b.d 6
51.g odd 8 4 289.2.c.d 12
51.i even 16 8 289.2.d.f 24
204.h even 2 1 4624.2.a.bd 3
255.h odd 2 1 7225.2.a.s 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
289.2.a.d 3 3.b odd 2 1
289.2.a.e yes 3 51.c odd 2 1
289.2.b.d 6 51.f odd 4 2
289.2.c.d 12 51.g odd 8 4
289.2.d.f 24 51.i even 16 8
2601.2.a.w 3 17.b even 2 1
2601.2.a.x 3 1.a even 1 1 trivial
4624.2.a.bd 3 204.h even 2 1
4624.2.a.bg 3 12.b even 2 1
7225.2.a.s 3 255.h odd 2 1
7225.2.a.t 3 15.d odd 2 1

## Hecke kernels

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

 $$T_{2}^{3} - 3 T_{2} - 1$$ $$T_{5}^{3} - 6 T_{5}^{2} + 9 T_{5} - 1$$ $$T_{7}^{3} - 3 T_{7} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 - 3 T + T^{3}$$
$3$ $$T^{3}$$
$5$ $$-1 + 9 T - 6 T^{2} + T^{3}$$
$7$ $$1 - 3 T + T^{3}$$
$11$ $$24 - 6 T^{2} + T^{3}$$
$13$ $$71 - 9 T - 6 T^{2} + T^{3}$$
$17$ $$T^{3}$$
$19$ $$1 - 3 T + T^{3}$$
$23$ $$-37 + 39 T - 12 T^{2} + T^{3}$$
$29$ $$-9 - 9 T + T^{3}$$
$31$ $$-53 + 6 T + 9 T^{2} + T^{3}$$
$37$ $$127 - 36 T - 3 T^{2} + T^{3}$$
$41$ $$159 - 27 T - 6 T^{2} + T^{3}$$
$43$ $$89 + 36 T - 15 T^{2} + T^{3}$$
$47$ $$-321 + 144 T - 21 T^{2} + T^{3}$$
$53$ $$72 + 72 T - 18 T^{2} + T^{3}$$
$59$ $$171 - 36 T - 9 T^{2} + T^{3}$$
$61$ $$-1 - 6 T - 3 T^{2} + T^{3}$$
$67$ $$-289 - 102 T + 9 T^{2} + T^{3}$$
$71$ $$19 + 108 T - 21 T^{2} + T^{3}$$
$73$ $$219 + 90 T - 21 T^{2} + T^{3}$$
$79$ $$213 - 81 T + 3 T^{2} + T^{3}$$
$83$ $$71 - 57 T + 9 T^{2} + T^{3}$$
$89$ $$613 - 42 T - 15 T^{2} + T^{3}$$
$97$ $$-424 - 96 T + 6 T^{2} + T^{3}$$