# Properties

 Label 2601.2.a.m Level $2601$ Weight $2$ Character orbit 2601.a Self dual yes Analytic conductor $20.769$ Analytic rank $1$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 867) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta ) q^{2} + ( 1 - 2 \beta ) q^{4} + 2 \beta q^{5} -3 q^{7} + ( -3 + \beta ) q^{8} +O(q^{10})$$ $$q + ( -1 + \beta ) q^{2} + ( 1 - 2 \beta ) q^{4} + 2 \beta q^{5} -3 q^{7} + ( -3 + \beta ) q^{8} + ( 4 - 2 \beta ) q^{10} + ( 2 + 2 \beta ) q^{11} + ( 1 - 4 \beta ) q^{13} + ( 3 - 3 \beta ) q^{14} + 3 q^{16} -3 q^{19} + ( -8 + 2 \beta ) q^{20} + 2 q^{22} + 2 \beta q^{23} + 3 q^{25} + ( -9 + 5 \beta ) q^{26} + ( -3 + 6 \beta ) q^{28} + ( -6 - 2 \beta ) q^{29} + ( -1 - 4 \beta ) q^{31} + ( 3 + \beta ) q^{32} -6 \beta q^{35} + ( -1 + 4 \beta ) q^{37} + ( 3 - 3 \beta ) q^{38} + ( 4 - 6 \beta ) q^{40} + ( 6 - 2 \beta ) q^{41} + 3 q^{43} + ( -6 - 2 \beta ) q^{44} + ( 4 - 2 \beta ) q^{46} + ( 4 - 4 \beta ) q^{47} + 2 q^{49} + ( -3 + 3 \beta ) q^{50} + ( 17 - 6 \beta ) q^{52} + ( -4 - 4 \beta ) q^{53} + ( 8 + 4 \beta ) q^{55} + ( 9 - 3 \beta ) q^{56} + ( 2 - 4 \beta ) q^{58} + ( 6 - 2 \beta ) q^{59} + ( -1 - 4 \beta ) q^{61} + ( -7 + 3 \beta ) q^{62} + ( -7 + 2 \beta ) q^{64} + ( -16 + 2 \beta ) q^{65} + q^{67} + ( -12 + 6 \beta ) q^{70} + ( -8 - 2 \beta ) q^{71} + ( 2 - 4 \beta ) q^{73} + ( 9 - 5 \beta ) q^{74} + ( -3 + 6 \beta ) q^{76} + ( -6 - 6 \beta ) q^{77} -12 q^{79} + 6 \beta q^{80} + ( -10 + 8 \beta ) q^{82} + ( -4 - 2 \beta ) q^{83} + ( -3 + 3 \beta ) q^{86} + ( -2 - 4 \beta ) q^{88} + ( 6 + 8 \beta ) q^{89} + ( -3 + 12 \beta ) q^{91} + ( -8 + 2 \beta ) q^{92} + ( -12 + 8 \beta ) q^{94} -6 \beta q^{95} + ( 5 - 4 \beta ) q^{97} + ( -2 + 2 \beta ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} + 2q^{4} - 6q^{7} - 6q^{8} + O(q^{10})$$ $$2q - 2q^{2} + 2q^{4} - 6q^{7} - 6q^{8} + 8q^{10} + 4q^{11} + 2q^{13} + 6q^{14} + 6q^{16} - 6q^{19} - 16q^{20} + 4q^{22} + 6q^{25} - 18q^{26} - 6q^{28} - 12q^{29} - 2q^{31} + 6q^{32} - 2q^{37} + 6q^{38} + 8q^{40} + 12q^{41} + 6q^{43} - 12q^{44} + 8q^{46} + 8q^{47} + 4q^{49} - 6q^{50} + 34q^{52} - 8q^{53} + 16q^{55} + 18q^{56} + 4q^{58} + 12q^{59} - 2q^{61} - 14q^{62} - 14q^{64} - 32q^{65} + 2q^{67} - 24q^{70} - 16q^{71} + 4q^{73} + 18q^{74} - 6q^{76} - 12q^{77} - 24q^{79} - 20q^{82} - 8q^{83} - 6q^{86} - 4q^{88} + 12q^{89} - 6q^{91} - 16q^{92} - 24q^{94} + 10q^{97} - 4q^{98} + O(q^{100})$$

## 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
 −1.41421 1.41421
−2.41421 0 3.82843 −2.82843 0 −3.00000 −4.41421 0 6.82843
1.2 0.414214 0 −1.82843 2.82843 0 −3.00000 −1.58579 0 1.17157
 $$n$$: 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.m 2
3.b odd 2 1 867.2.a.g 2
17.b even 2 1 2601.2.a.n 2
51.c odd 2 1 867.2.a.h yes 2
51.f odd 4 2 867.2.d.b 4
51.g odd 8 2 867.2.e.b 4
51.g odd 8 2 867.2.e.c 4
51.i even 16 4 867.2.h.a 8
51.i even 16 4 867.2.h.h 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
867.2.a.g 2 3.b odd 2 1
867.2.a.h yes 2 51.c odd 2 1
867.2.d.b 4 51.f odd 4 2
867.2.e.b 4 51.g odd 8 2
867.2.e.c 4 51.g odd 8 2
867.2.h.a 8 51.i even 16 4
867.2.h.h 8 51.i even 16 4
2601.2.a.m 2 1.a even 1 1 trivial
2601.2.a.n 2 17.b even 2 1

## Hecke kernels

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

 $$T_{2}^{2} + 2 T_{2} - 1$$ $$T_{5}^{2} - 8$$ $$T_{7} + 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + 2 T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$-8 + T^{2}$$
$7$ $$( 3 + T )^{2}$$
$11$ $$-4 - 4 T + T^{2}$$
$13$ $$-31 - 2 T + T^{2}$$
$17$ $$T^{2}$$
$19$ $$( 3 + T )^{2}$$
$23$ $$-8 + T^{2}$$
$29$ $$28 + 12 T + T^{2}$$
$31$ $$-31 + 2 T + T^{2}$$
$37$ $$-31 + 2 T + T^{2}$$
$41$ $$28 - 12 T + T^{2}$$
$43$ $$( -3 + T )^{2}$$
$47$ $$-16 - 8 T + T^{2}$$
$53$ $$-16 + 8 T + T^{2}$$
$59$ $$28 - 12 T + T^{2}$$
$61$ $$-31 + 2 T + T^{2}$$
$67$ $$( -1 + T )^{2}$$
$71$ $$56 + 16 T + T^{2}$$
$73$ $$-28 - 4 T + T^{2}$$
$79$ $$( 12 + T )^{2}$$
$83$ $$8 + 8 T + T^{2}$$
$89$ $$-92 - 12 T + T^{2}$$
$97$ $$-7 - 10 T + T^{2}$$