# Properties

 Label 2601.2.a.s Level $2601$ Weight $2$ Character orbit 2601.a Self dual yes Analytic conductor $20.769$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ Defining polynomial: $$x^{2} - x - 3$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{13})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + ( 1 + \beta ) q^{4} + \beta q^{5} + ( -2 + \beta ) q^{7} + 3 q^{8} +O(q^{10})$$ $$q + \beta q^{2} + ( 1 + \beta ) q^{4} + \beta q^{5} + ( -2 + \beta ) q^{7} + 3 q^{8} + ( 3 + \beta ) q^{10} + 3 q^{11} + ( -1 - \beta ) q^{13} + ( 3 - \beta ) q^{14} + ( -2 + \beta ) q^{16} + ( -1 + 3 \beta ) q^{19} + ( 3 + 2 \beta ) q^{20} + 3 \beta q^{22} + \beta q^{23} + ( -2 + \beta ) q^{25} + ( -3 - 2 \beta ) q^{26} + q^{28} + ( 3 + 3 \beta ) q^{29} + ( 1 - 2 \beta ) q^{31} + ( -3 - \beta ) q^{32} + ( 3 - \beta ) q^{35} + ( 4 - 2 \beta ) q^{37} + ( 9 + 2 \beta ) q^{38} + 3 \beta q^{40} + 6 q^{41} + ( -7 + 2 \beta ) q^{43} + ( 3 + 3 \beta ) q^{44} + ( 3 + \beta ) q^{46} -3 q^{47} -3 \beta q^{49} + ( 3 - \beta ) q^{50} + ( -4 - 3 \beta ) q^{52} + ( -9 + 3 \beta ) q^{53} + 3 \beta q^{55} + ( -6 + 3 \beta ) q^{56} + ( 9 + 6 \beta ) q^{58} -6 q^{59} + ( -5 + 6 \beta ) q^{61} + ( -6 - \beta ) q^{62} + ( 1 - 6 \beta ) q^{64} + ( -3 - 2 \beta ) q^{65} + ( 8 + 2 \beta ) q^{67} + ( -3 + 2 \beta ) q^{70} + ( 6 - 4 \beta ) q^{71} + ( -5 + 2 \beta ) q^{73} + ( -6 + 2 \beta ) q^{74} + ( 8 + 5 \beta ) q^{76} + ( -6 + 3 \beta ) q^{77} + ( -2 - 4 \beta ) q^{79} + ( 3 - \beta ) q^{80} + 6 \beta q^{82} + ( 9 - 5 \beta ) q^{83} + ( 6 - 5 \beta ) q^{86} + 9 q^{88} + ( 6 - 4 \beta ) q^{89} - q^{91} + ( 3 + 2 \beta ) q^{92} -3 \beta q^{94} + ( 9 + 2 \beta ) q^{95} + ( 4 + 3 \beta ) q^{97} + ( -9 - 3 \beta ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{2} + 3q^{4} + q^{5} - 3q^{7} + 6q^{8} + O(q^{10})$$ $$2q + q^{2} + 3q^{4} + q^{5} - 3q^{7} + 6q^{8} + 7q^{10} + 6q^{11} - 3q^{13} + 5q^{14} - 3q^{16} + q^{19} + 8q^{20} + 3q^{22} + q^{23} - 3q^{25} - 8q^{26} + 2q^{28} + 9q^{29} - 7q^{32} + 5q^{35} + 6q^{37} + 20q^{38} + 3q^{40} + 12q^{41} - 12q^{43} + 9q^{44} + 7q^{46} - 6q^{47} - 3q^{49} + 5q^{50} - 11q^{52} - 15q^{53} + 3q^{55} - 9q^{56} + 24q^{58} - 12q^{59} - 4q^{61} - 13q^{62} - 4q^{64} - 8q^{65} + 18q^{67} - 4q^{70} + 8q^{71} - 8q^{73} - 10q^{74} + 21q^{76} - 9q^{77} - 8q^{79} + 5q^{80} + 6q^{82} + 13q^{83} + 7q^{86} + 18q^{88} + 8q^{89} - 2q^{91} + 8q^{92} - 3q^{94} + 20q^{95} + 11q^{97} - 21q^{98} + O(q^{100})$$

## 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
 −1.30278 2.30278
−1.30278 0 −0.302776 −1.30278 0 −3.30278 3.00000 0 1.69722
1.2 2.30278 0 3.30278 2.30278 0 0.302776 3.00000 0 5.30278
 $$n$$: 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.s 2
3.b odd 2 1 289.2.a.b 2
12.b even 2 1 4624.2.a.v 2
15.d odd 2 1 7225.2.a.n 2
17.b even 2 1 2601.2.a.r 2
51.c odd 2 1 289.2.a.c yes 2
51.f odd 4 2 289.2.b.c 4
51.g odd 8 4 289.2.c.b 8
51.i even 16 8 289.2.d.e 16
204.h even 2 1 4624.2.a.j 2
255.h odd 2 1 7225.2.a.m 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
289.2.a.b 2 3.b odd 2 1
289.2.a.c yes 2 51.c odd 2 1
289.2.b.c 4 51.f odd 4 2
289.2.c.b 8 51.g odd 8 4
289.2.d.e 16 51.i even 16 8
2601.2.a.r 2 17.b even 2 1
2601.2.a.s 2 1.a even 1 1 trivial
4624.2.a.j 2 204.h even 2 1
4624.2.a.v 2 12.b even 2 1
7225.2.a.m 2 255.h odd 2 1
7225.2.a.n 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2601))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-3 - T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$-3 - T + T^{2}$$
$7$ $$-1 + 3 T + T^{2}$$
$11$ $$( -3 + T )^{2}$$
$13$ $$-1 + 3 T + T^{2}$$
$17$ $$T^{2}$$
$19$ $$-29 - T + T^{2}$$
$23$ $$-3 - T + T^{2}$$
$29$ $$-9 - 9 T + T^{2}$$
$31$ $$-13 + T^{2}$$
$37$ $$-4 - 6 T + T^{2}$$
$41$ $$( -6 + T )^{2}$$
$43$ $$23 + 12 T + T^{2}$$
$47$ $$( 3 + T )^{2}$$
$53$ $$27 + 15 T + T^{2}$$
$59$ $$( 6 + T )^{2}$$
$61$ $$-113 + 4 T + T^{2}$$
$67$ $$68 - 18 T + T^{2}$$
$71$ $$-36 - 8 T + T^{2}$$
$73$ $$3 + 8 T + T^{2}$$
$79$ $$-36 + 8 T + T^{2}$$
$83$ $$-39 - 13 T + T^{2}$$
$89$ $$-36 - 8 T + T^{2}$$
$97$ $$1 - 11 T + T^{2}$$