# Properties

 Label 2601.2.a.bf Level $2601$ Weight $2$ Character orbit 2601.a Self dual yes Analytic conductor $20.769$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: 4.4.7232.1 Defining polynomial: $$x^{4} - 2 x^{3} - 5 x^{2} + 4 x + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 51) 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.

 $$f(q)$$ $$=$$ $$q + ( \beta_{1} - \beta_{3} ) q^{2} + ( 1 + \beta_{1} - \beta_{2} ) q^{4} + ( 1 + \beta_{1} ) q^{5} + ( -1 + \beta_{2} - \beta_{3} ) q^{7} + ( 1 + \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{8} +O(q^{10})$$ $$q + ( \beta_{1} - \beta_{3} ) q^{2} + ( 1 + \beta_{1} - \beta_{2} ) q^{4} + ( 1 + \beta_{1} ) q^{5} + ( -1 + \beta_{2} - \beta_{3} ) q^{7} + ( 1 + \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{8} + ( 2 + 2 \beta_{1} - \beta_{3} ) q^{10} + ( 2 - \beta_{1} + \beta_{2} ) q^{11} + ( \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{13} + ( 2 - 2 \beta_{1} + 4 \beta_{3} ) q^{14} + ( 1 + \beta_{1} - \beta_{2} - 4 \beta_{3} ) q^{16} + ( 2 + \beta_{1} - \beta_{2} ) q^{19} + ( 3 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{20} + ( -1 + \beta_{2} + \beta_{3} ) q^{22} + ( -2 + \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{23} + ( -1 + 3 \beta_{1} + \beta_{2} ) q^{25} + ( 5 - \beta_{2} + 3 \beta_{3} ) q^{26} + ( -6 + 2 \beta_{2} ) q^{28} + ( 5 - 2 \beta_{1} + \beta_{2} ) q^{29} + ( -2 + 2 \beta_{1} ) q^{31} + ( 3 + \beta_{1} - 3 \beta_{2} ) q^{32} + ( -1 + \beta_{2} + \beta_{3} ) q^{35} + ( -4 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{37} + ( 1 + 4 \beta_{1} - \beta_{2} - 5 \beta_{3} ) q^{38} + ( 1 + 2 \beta_{1} - 3 \beta_{2} - 4 \beta_{3} ) q^{40} + ( -3 - \beta_{1} + 2 \beta_{2} - 4 \beta_{3} ) q^{41} + ( 4 - \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{43} + ( -4 + 4 \beta_{3} ) q^{44} + ( 3 - 3 \beta_{2} - \beta_{3} ) q^{46} + ( -2 + 2 \beta_{1} + 2 \beta_{3} ) q^{47} + ( 1 - 2 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{49} + ( 7 + \beta_{1} + \beta_{2} + 4 \beta_{3} ) q^{50} + ( -4 + 4 \beta_{1} - 4 \beta_{3} ) q^{52} + ( 6 - 2 \beta_{1} + 2 \beta_{3} ) q^{53} + ( \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{55} + ( -2 - 4 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} ) q^{56} + ( -3 + 2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{58} + ( 5 + 2 \beta_{1} + \beta_{2} - 5 \beta_{3} ) q^{59} + ( -2 - 2 \beta_{1} + 3 \beta_{3} ) q^{61} + ( 4 + 2 \beta_{3} ) q^{62} + ( -3 + 5 \beta_{1} - \beta_{2} - 4 \beta_{3} ) q^{64} + ( 2 + 3 \beta_{1} + \beta_{2} ) q^{65} + ( 2 - 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{67} + ( -2 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} ) q^{70} + ( 2 - 2 \beta_{1} + 4 \beta_{3} ) q^{71} + ( -5 + \beta_{2} + 2 \beta_{3} ) q^{73} + ( 3 - 3 \beta_{2} - 2 \beta_{3} ) q^{74} + ( 8 + 4 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} ) q^{76} + ( 3 + \beta_{2} - 3 \beta_{3} ) q^{77} + ( -1 - 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{79} + ( -1 + 2 \beta_{1} - 5 \beta_{2} - 6 \beta_{3} ) q^{80} + ( 4 - 6 \beta_{1} - 2 \beta_{2} + 9 \beta_{3} ) q^{82} + ( 2 - 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{83} + ( -1 + 4 \beta_{1} - 3 \beta_{2} - 7 \beta_{3} ) q^{86} + ( -2 - 4 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{88} + ( 1 - 4 \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{89} + ( 9 - 4 \beta_{1} - \beta_{2} + 7 \beta_{3} ) q^{91} + ( 2 + 4 \beta_{1} - 2 \beta_{2} - 8 \beta_{3} ) q^{92} + ( 2 + 2 \beta_{2} + 2 \beta_{3} ) q^{94} + ( 4 + 3 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{95} + ( -5 - 2 \beta_{1} - \beta_{2} ) q^{97} + ( -10 + \beta_{1} + 2 \beta_{2} - 7 \beta_{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{2} + 6q^{4} + 6q^{5} - 4q^{7} + 6q^{8} + O(q^{10})$$ $$4q + 2q^{2} + 6q^{4} + 6q^{5} - 4q^{7} + 6q^{8} + 12q^{10} + 6q^{11} + 2q^{13} + 4q^{14} + 6q^{16} + 10q^{19} + 16q^{20} - 4q^{22} - 6q^{23} + 2q^{25} + 20q^{26} - 24q^{28} + 16q^{29} - 4q^{31} + 14q^{32} - 4q^{35} - 12q^{37} + 12q^{38} + 8q^{40} - 14q^{41} + 14q^{43} - 16q^{44} + 12q^{46} - 4q^{47} + 30q^{50} - 8q^{52} + 20q^{53} + 2q^{55} - 16q^{56} - 8q^{58} + 24q^{59} - 12q^{61} + 16q^{62} - 2q^{64} + 14q^{65} + 4q^{67} - 4q^{70} + 4q^{71} - 20q^{73} + 12q^{74} + 40q^{76} + 12q^{77} - 8q^{79} + 4q^{82} + 4q^{83} + 4q^{86} - 16q^{88} - 4q^{89} + 28q^{91} + 16q^{92} + 8q^{94} + 22q^{95} - 24q^{97} - 38q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2 x^{3} - 5 x^{2} + 4 x + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} - 2 \nu^{2} - 3 \nu + 2$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 3$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{3} + 2 \beta_{2} + 5 \beta_{1} + 4$$

## 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
 −0.652223 −1.63640 3.06644 1.22219
−2.06644 0 2.27016 0.347777 0 −4.33660 −0.558268 0 −0.718659
1.2 −0.222191 0 −1.95063 −0.636405 0 1.72844 0.877796 0 0.141404
1.3 1.65222 0 0.729840 4.06644 0 0.922382 −2.09859 0 6.71866
1.4 2.63640 0 4.95063 2.22219 0 −2.31423 7.77906 0 5.85860
 $$n$$: 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.bf 4
3.b odd 2 1 867.2.a.k 4
17.b even 2 1 2601.2.a.be 4
17.d even 8 2 153.2.f.b 8
51.c odd 2 1 867.2.a.l 4
51.f odd 4 2 867.2.d.f 8
51.g odd 8 2 51.2.e.a 8
51.g odd 8 2 867.2.e.g 8
51.i even 16 4 867.2.h.i 16
51.i even 16 4 867.2.h.k 16
68.g odd 8 2 2448.2.be.x 8
204.p even 8 2 816.2.bd.e 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
51.2.e.a 8 51.g odd 8 2
153.2.f.b 8 17.d even 8 2
816.2.bd.e 8 204.p even 8 2
867.2.a.k 4 3.b odd 2 1
867.2.a.l 4 51.c odd 2 1
867.2.d.f 8 51.f odd 4 2
867.2.e.g 8 51.g odd 8 2
867.2.h.i 16 51.i even 16 4
867.2.h.k 16 51.i even 16 4
2448.2.be.x 8 68.g odd 8 2
2601.2.a.be 4 17.b even 2 1
2601.2.a.bf 4 1.a even 1 1 trivial

## Hecke kernels

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

 $$T_{2}^{4} - 2 T_{2}^{3} - 5 T_{2}^{2} + 8 T_{2} + 2$$ $$T_{5}^{4} - 6 T_{5}^{3} + 7 T_{5}^{2} + 4 T_{5} - 2$$ $$T_{7}^{4} + 4 T_{7}^{3} - 6 T_{7}^{2} - 16 T_{7} + 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$2 + 8 T - 5 T^{2} - 2 T^{3} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$-2 + 4 T + 7 T^{2} - 6 T^{3} + T^{4}$$
$7$ $$16 - 16 T - 6 T^{2} + 4 T^{3} + T^{4}$$
$11$ $$-16 + 24 T + T^{2} - 6 T^{3} + T^{4}$$
$13$ $$-64 + 80 T - 23 T^{2} - 2 T^{3} + T^{4}$$
$17$ $$T^{4}$$
$19$ $$-32 + 25 T^{2} - 10 T^{3} + T^{4}$$
$23$ $$-184 - 136 T - 15 T^{2} + 6 T^{3} + T^{4}$$
$29$ $$-16 - 80 T + 70 T^{2} - 16 T^{3} + T^{4}$$
$31$ $$32 - 64 T - 20 T^{2} + 4 T^{3} + T^{4}$$
$37$ $$-772 - 344 T + 12 T^{3} + T^{4}$$
$41$ $$-1666 - 644 T - 17 T^{2} + 14 T^{3} + T^{4}$$
$43$ $$-1168 + 392 T + 17 T^{2} - 14 T^{3} + T^{4}$$
$47$ $$-64 - 160 T - 52 T^{2} + 4 T^{3} + T^{4}$$
$53$ $$128 - 256 T + 124 T^{2} - 20 T^{3} + T^{4}$$
$59$ $$-4672 + 608 T + 126 T^{2} - 24 T^{3} + T^{4}$$
$61$ $$-356 - 152 T + 16 T^{2} + 12 T^{3} + T^{4}$$
$67$ $$2048 + 1024 T - 196 T^{2} - 4 T^{3} + T^{4}$$
$71$ $$32 + 64 T - 52 T^{2} - 4 T^{3} + T^{4}$$
$73$ $$-328 + 120 T + 114 T^{2} + 20 T^{3} + T^{4}$$
$79$ $$-16 + 48 T - 42 T^{2} + 8 T^{3} + T^{4}$$
$83$ $$5248 + 128 T - 164 T^{2} - 4 T^{3} + T^{4}$$
$89$ $$1088 - 352 T - 110 T^{2} + 4 T^{3} + T^{4}$$
$97$ $$368 + 432 T + 166 T^{2} + 24 T^{3} + T^{4}$$