# Properties

 Label 2601.2.a.ba Level $2601$ Weight $2$ Character orbit 2601.a Self dual yes Analytic conductor $20.769$ Analytic rank $0$ Dimension $4$ CM no 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: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{16})^+$$ Defining polynomial: $$x^{4} - 4 x^{2} + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{4} + ( -\beta_{1} - 2 \beta_{3} ) q^{5} + 2 \beta_{1} q^{7} + 3 q^{8} +O(q^{10})$$ $$q - q^{2} - q^{4} + ( -\beta_{1} - 2 \beta_{3} ) q^{5} + 2 \beta_{1} q^{7} + 3 q^{8} + ( \beta_{1} + 2 \beta_{3} ) q^{10} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{11} -3 \beta_{2} q^{13} -2 \beta_{1} q^{14} - q^{16} + 2 \beta_{2} q^{19} + ( \beta_{1} + 2 \beta_{3} ) q^{20} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{22} + ( 2 \beta_{1} - 4 \beta_{3} ) q^{23} + ( 5 + \beta_{2} ) q^{25} + 3 \beta_{2} q^{26} -2 \beta_{1} q^{28} + ( -2 \beta_{1} - \beta_{3} ) q^{29} -2 \beta_{3} q^{31} -5 q^{32} + ( -4 - 6 \beta_{2} ) q^{35} + ( 3 \beta_{1} - 4 \beta_{3} ) q^{37} -2 \beta_{2} q^{38} + ( -3 \beta_{1} - 6 \beta_{3} ) q^{40} + ( -2 \beta_{1} - \beta_{3} ) q^{41} -4 \beta_{2} q^{43} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{44} + ( -2 \beta_{1} + 4 \beta_{3} ) q^{46} + 2 \beta_{2} q^{47} + ( 1 + 4 \beta_{2} ) q^{49} + ( -5 - \beta_{2} ) q^{50} + 3 \beta_{2} q^{52} + ( -4 + 3 \beta_{2} ) q^{53} + ( 12 + 4 \beta_{2} ) q^{55} + 6 \beta_{1} q^{56} + ( 2 \beta_{1} + \beta_{3} ) q^{58} + ( -4 + 6 \beta_{2} ) q^{59} + ( -8 \beta_{1} + 3 \beta_{3} ) q^{61} + 2 \beta_{3} q^{62} + 7 q^{64} + ( 9 \beta_{1} - 3 \beta_{3} ) q^{65} + ( 4 - 6 \beta_{2} ) q^{67} + ( 4 + 6 \beta_{2} ) q^{70} + ( -2 \beta_{1} + 4 \beta_{3} ) q^{71} + 5 \beta_{1} q^{73} + ( -3 \beta_{1} + 4 \beta_{3} ) q^{74} -2 \beta_{2} q^{76} + ( -8 - 8 \beta_{2} ) q^{77} + ( -6 \beta_{1} + 4 \beta_{3} ) q^{79} + ( \beta_{1} + 2 \beta_{3} ) q^{80} + ( 2 \beta_{1} + \beta_{3} ) q^{82} + ( 4 + 6 \beta_{2} ) q^{83} + 4 \beta_{2} q^{86} + ( -6 \beta_{1} - 6 \beta_{3} ) q^{88} + ( -12 + \beta_{2} ) q^{89} + ( -6 \beta_{1} - 6 \beta_{3} ) q^{91} + ( -2 \beta_{1} + 4 \beta_{3} ) q^{92} -2 \beta_{2} q^{94} + ( -6 \beta_{1} + 2 \beta_{3} ) q^{95} + ( 4 \beta_{1} + 3 \beta_{3} ) q^{97} + ( -1 - 4 \beta_{2} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{2} - 4q^{4} + 12q^{8} + O(q^{10})$$ $$4q - 4q^{2} - 4q^{4} + 12q^{8} - 4q^{16} + 20q^{25} - 20q^{32} - 16q^{35} + 4q^{49} - 20q^{50} - 16q^{53} + 48q^{55} - 16q^{59} + 28q^{64} + 16q^{67} + 16q^{70} - 32q^{77} + 16q^{83} - 48q^{89} - 4q^{98} + O(q^{100})$$

## 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
 1.84776 −0.765367 0.765367 −1.84776
−1.00000 0 −1.00000 −3.37849 0 3.69552 3.00000 0 3.37849
1.2 −1.00000 0 −1.00000 −2.93015 0 −1.53073 3.00000 0 2.93015
1.3 −1.00000 0 −1.00000 2.93015 0 1.53073 3.00000 0 −2.93015
1.4 −1.00000 0 −1.00000 3.37849 0 −3.69552 3.00000 0 −3.37849
 $$n$$: 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
17.b even 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.ba 4
3.b odd 2 1 2601.2.a.bg 4
17.b even 2 1 inner 2601.2.a.ba 4
17.e odd 16 2 153.2.l.b yes 4
51.c odd 2 1 2601.2.a.bg 4
51.i even 16 2 153.2.l.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
153.2.l.a 4 51.i even 16 2
153.2.l.b yes 4 17.e odd 16 2
2601.2.a.ba 4 1.a even 1 1 trivial
2601.2.a.ba 4 17.b even 2 1 inner
2601.2.a.bg 4 3.b odd 2 1
2601.2.a.bg 4 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2601))$$:

 $$T_{2} + 1$$ $$T_{5}^{4} - 20 T_{5}^{2} + 98$$ $$T_{7}^{4} - 16 T_{7}^{2} + 32$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{4}$$
$3$ $$T^{4}$$
$5$ $$98 - 20 T^{2} + T^{4}$$
$7$ $$32 - 16 T^{2} + T^{4}$$
$11$ $$128 - 32 T^{2} + T^{4}$$
$13$ $$( -18 + T^{2} )^{2}$$
$17$ $$T^{4}$$
$19$ $$( -8 + T^{2} )^{2}$$
$23$ $$32 - 80 T^{2} + T^{4}$$
$29$ $$2 - 20 T^{2} + T^{4}$$
$31$ $$32 - 16 T^{2} + T^{4}$$
$37$ $$578 - 100 T^{2} + T^{4}$$
$41$ $$2 - 20 T^{2} + T^{4}$$
$43$ $$( -32 + T^{2} )^{2}$$
$47$ $$( -8 + T^{2} )^{2}$$
$53$ $$( -2 + 8 T + T^{2} )^{2}$$
$59$ $$( -56 + 8 T + T^{2} )^{2}$$
$61$ $$21218 - 292 T^{2} + T^{4}$$
$67$ $$( -56 - 8 T + T^{2} )^{2}$$
$71$ $$32 - 80 T^{2} + T^{4}$$
$73$ $$1250 - 100 T^{2} + T^{4}$$
$79$ $$9248 - 208 T^{2} + T^{4}$$
$83$ $$( -56 - 8 T + T^{2} )^{2}$$
$89$ $$( 142 + 24 T + T^{2} )^{2}$$
$97$ $$578 - 100 T^{2} + T^{4}$$