# Properties

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

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## 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 153) 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.

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

## 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
−1.41421 0 0 1.00000 0 1.41421 2.82843 0 −1.41421
1.2 1.41421 0 0 1.00000 0 −1.41421 −2.82843 0 1.41421
 $$n$$: 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
51.c odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2601.2.a.q 2
3.b odd 2 1 2601.2.a.p 2
17.b even 2 1 2601.2.a.p 2
17.d even 8 2 153.2.f.a 4
51.c odd 2 1 inner 2601.2.a.q 2
51.g odd 8 2 153.2.f.a 4
68.g odd 8 2 2448.2.be.r 4
204.p even 8 2 2448.2.be.r 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
153.2.f.a 4 17.d even 8 2
153.2.f.a 4 51.g odd 8 2
2448.2.be.r 4 68.g odd 8 2
2448.2.be.r 4 204.p even 8 2
2601.2.a.p 2 3.b odd 2 1
2601.2.a.p 2 17.b even 2 1
2601.2.a.q 2 1.a even 1 1 trivial
2601.2.a.q 2 51.c odd 2 1 inner

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-2 + T^{2}$$
$3$ $$T^{2}$$
$5$ $$( -1 + T )^{2}$$
$7$ $$-2 + T^{2}$$
$11$ $$( -1 + T )^{2}$$
$13$ $$( -1 + T )^{2}$$
$17$ $$T^{2}$$
$19$ $$( 3 + T )^{2}$$
$23$ $$( 5 + T )^{2}$$
$29$ $$( 8 + T )^{2}$$
$31$ $$-32 + T^{2}$$
$37$ $$-2 + T^{2}$$
$41$ $$( -11 + T )^{2}$$
$43$ $$( 9 + T )^{2}$$
$47$ $$-72 + T^{2}$$
$53$ $$-8 + T^{2}$$
$59$ $$-98 + T^{2}$$
$61$ $$-50 + T^{2}$$
$67$ $$( 10 + T )^{2}$$
$71$ $$( 10 + T )^{2}$$
$73$ $$-128 + T^{2}$$
$79$ $$-50 + T^{2}$$
$83$ $$-200 + T^{2}$$
$89$ $$-18 + T^{2}$$
$97$ $$-200 + T^{2}$$
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