# Properties

 Label 2601.2.a.i Level $2601$ Weight $2$ Character orbit 2601.a Self dual yes Analytic conductor $20.769$ Analytic rank $0$ Dimension $1$ 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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 51) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{4} - 4q^{7} - 3q^{8} + O(q^{10})$$ $$q + q^{2} - q^{4} - 4q^{7} - 3q^{8} - 4q^{11} + 2q^{13} - 4q^{14} - q^{16} + 4q^{19} - 4q^{22} + 4q^{23} - 5q^{25} + 2q^{26} + 4q^{28} + 4q^{31} + 5q^{32} - 8q^{37} + 4q^{38} + 8q^{41} + 4q^{43} + 4q^{44} + 4q^{46} + 8q^{47} + 9q^{49} - 5q^{50} - 2q^{52} + 6q^{53} + 12q^{56} - 12q^{59} - 8q^{61} + 4q^{62} + 7q^{64} + 12q^{67} + 12q^{71} - 8q^{74} - 4q^{76} + 16q^{77} - 4q^{79} + 8q^{82} + 12q^{83} + 4q^{86} + 12q^{88} + 10q^{89} - 8q^{91} - 4q^{92} + 8q^{94} - 16q^{97} + 9q^{98} + O(q^{100})$$

## 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
 0
1.00000 0 −1.00000 0 0 −4.00000 −3.00000 0 0
 $$n$$: 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.i 1
3.b odd 2 1 867.2.a.a 1
17.b even 2 1 2601.2.a.j 1
17.c even 4 2 153.2.d.a 2
51.c odd 2 1 867.2.a.b 1
51.f odd 4 2 51.2.d.b 2
51.g odd 8 4 867.2.e.d 4
51.i even 16 8 867.2.h.d 8
68.f odd 4 2 2448.2.c.j 2
204.l even 4 2 816.2.c.c 2
255.i odd 4 2 1275.2.g.a 2
255.k even 4 2 1275.2.d.d 2
255.r even 4 2 1275.2.d.b 2
408.q even 4 2 3264.2.c.d 2
408.t odd 4 2 3264.2.c.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
51.2.d.b 2 51.f odd 4 2
153.2.d.a 2 17.c even 4 2
816.2.c.c 2 204.l even 4 2
867.2.a.a 1 3.b odd 2 1
867.2.a.b 1 51.c odd 2 1
867.2.e.d 4 51.g odd 8 4
867.2.h.d 8 51.i even 16 8
1275.2.d.b 2 255.r even 4 2
1275.2.d.d 2 255.k even 4 2
1275.2.g.a 2 255.i odd 4 2
2448.2.c.j 2 68.f odd 4 2
2601.2.a.i 1 1.a even 1 1 trivial
2601.2.a.j 1 17.b even 2 1
3264.2.c.d 2 408.q even 4 2
3264.2.c.e 2 408.t odd 4 2

## Hecke kernels

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

 $$T_{2} - 1$$ $$T_{5}$$ $$T_{7} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$4 + T$$
$11$ $$4 + T$$
$13$ $$-2 + T$$
$17$ $$T$$
$19$ $$-4 + T$$
$23$ $$-4 + T$$
$29$ $$T$$
$31$ $$-4 + T$$
$37$ $$8 + T$$
$41$ $$-8 + T$$
$43$ $$-4 + T$$
$47$ $$-8 + T$$
$53$ $$-6 + T$$
$59$ $$12 + T$$
$61$ $$8 + T$$
$67$ $$-12 + T$$
$71$ $$-12 + T$$
$73$ $$T$$
$79$ $$4 + T$$
$83$ $$-12 + T$$
$89$ $$-10 + T$$
$97$ $$16 + T$$