# Properties

 Label 6171.2.a.p Level $6171$ Weight $2$ Character orbit 6171.a Self dual yes Analytic conductor $49.276$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$6171 = 3 \cdot 11^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6171.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$49.2756830873$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ Coefficient ring: $$\Z[a_1, a_2]$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} - q^{3} + ( 2 + \beta ) q^{4} + ( 1 + \beta ) q^{5} -\beta q^{6} + ( 4 + \beta ) q^{8} + q^{9} +O(q^{10})$$ $$q + \beta q^{2} - q^{3} + ( 2 + \beta ) q^{4} + ( 1 + \beta ) q^{5} -\beta q^{6} + ( 4 + \beta ) q^{8} + q^{9} + ( 4 + 2 \beta ) q^{10} + ( -2 - \beta ) q^{12} + ( -3 + \beta ) q^{13} + ( -1 - \beta ) q^{15} + 3 \beta q^{16} - q^{17} + \beta q^{18} + ( -3 + 3 \beta ) q^{19} + ( 6 + 4 \beta ) q^{20} + ( -5 + \beta ) q^{23} + ( -4 - \beta ) q^{24} + 3 \beta q^{25} + ( 4 - 2 \beta ) q^{26} - q^{27} + ( -2 + 4 \beta ) q^{29} + ( -4 - 2 \beta ) q^{30} + ( -2 + 2 \beta ) q^{31} + ( 4 + \beta ) q^{32} -\beta q^{34} + ( 2 + \beta ) q^{36} -2 \beta q^{37} + 12 q^{38} + ( 3 - \beta ) q^{39} + ( 8 + 6 \beta ) q^{40} + ( 1 + \beta ) q^{41} + ( 3 - 3 \beta ) q^{43} + ( 1 + \beta ) q^{45} + ( 4 - 4 \beta ) q^{46} + ( -6 - 2 \beta ) q^{47} -3 \beta q^{48} -7 q^{49} + ( 12 + 3 \beta ) q^{50} + q^{51} -2 q^{52} + ( 2 + 4 \beta ) q^{53} -\beta q^{54} + ( 3 - 3 \beta ) q^{57} + ( 16 + 2 \beta ) q^{58} + ( 2 + 2 \beta ) q^{59} + ( -6 - 4 \beta ) q^{60} + ( -4 - 2 \beta ) q^{61} + 8 q^{62} + ( 4 - \beta ) q^{64} + ( 1 - \beta ) q^{65} + 4 q^{67} + ( -2 - \beta ) q^{68} + ( 5 - \beta ) q^{69} + ( 4 - 4 \beta ) q^{71} + ( 4 + \beta ) q^{72} + ( 2 + 4 \beta ) q^{73} + ( -8 - 2 \beta ) q^{74} -3 \beta q^{75} + ( 6 + 6 \beta ) q^{76} + ( -4 + 2 \beta ) q^{78} + ( -6 + 6 \beta ) q^{79} + ( 12 + 6 \beta ) q^{80} + q^{81} + ( 4 + 2 \beta ) q^{82} + ( 6 - 2 \beta ) q^{83} + ( -1 - \beta ) q^{85} -12 q^{86} + ( 2 - 4 \beta ) q^{87} + ( 4 - 2 \beta ) q^{89} + ( 4 + 2 \beta ) q^{90} + ( -6 - 2 \beta ) q^{92} + ( 2 - 2 \beta ) q^{93} + ( -8 - 8 \beta ) q^{94} + ( 9 + 3 \beta ) q^{95} + ( -4 - \beta ) q^{96} + ( -8 + 2 \beta ) q^{97} -7 \beta q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{2} - 2q^{3} + 5q^{4} + 3q^{5} - q^{6} + 9q^{8} + 2q^{9} + O(q^{10})$$ $$2q + q^{2} - 2q^{3} + 5q^{4} + 3q^{5} - q^{6} + 9q^{8} + 2q^{9} + 10q^{10} - 5q^{12} - 5q^{13} - 3q^{15} + 3q^{16} - 2q^{17} + q^{18} - 3q^{19} + 16q^{20} - 9q^{23} - 9q^{24} + 3q^{25} + 6q^{26} - 2q^{27} - 10q^{30} - 2q^{31} + 9q^{32} - q^{34} + 5q^{36} - 2q^{37} + 24q^{38} + 5q^{39} + 22q^{40} + 3q^{41} + 3q^{43} + 3q^{45} + 4q^{46} - 14q^{47} - 3q^{48} - 14q^{49} + 27q^{50} + 2q^{51} - 4q^{52} + 8q^{53} - q^{54} + 3q^{57} + 34q^{58} + 6q^{59} - 16q^{60} - 10q^{61} + 16q^{62} + 7q^{64} + q^{65} + 8q^{67} - 5q^{68} + 9q^{69} + 4q^{71} + 9q^{72} + 8q^{73} - 18q^{74} - 3q^{75} + 18q^{76} - 6q^{78} - 6q^{79} + 30q^{80} + 2q^{81} + 10q^{82} + 10q^{83} - 3q^{85} - 24q^{86} + 6q^{89} + 10q^{90} - 14q^{92} + 2q^{93} - 24q^{94} + 21q^{95} - 9q^{96} - 14q^{97} - 7q^{98} + O(q^{100})$$

## 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
 −1.56155 2.56155
−1.56155 −1.00000 0.438447 −0.561553 1.56155 0 2.43845 1.00000 0.876894
1.2 2.56155 −1.00000 4.56155 3.56155 −2.56155 0 6.56155 1.00000 9.12311
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$11$$ $$-1$$
$$17$$ $$1$$

## 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 6171.2.a.p 2
11.b odd 2 1 51.2.a.b 2
33.d even 2 1 153.2.a.e 2
44.c even 2 1 816.2.a.m 2
55.d odd 2 1 1275.2.a.n 2
55.e even 4 2 1275.2.b.d 4
77.b even 2 1 2499.2.a.o 2
88.b odd 2 1 3264.2.a.bl 2
88.g even 2 1 3264.2.a.bg 2
132.d odd 2 1 2448.2.a.v 2
143.d odd 2 1 8619.2.a.q 2
165.d even 2 1 3825.2.a.s 2
187.b odd 2 1 867.2.a.f 2
187.f odd 4 2 867.2.d.c 4
187.i odd 8 4 867.2.e.f 8
187.m even 16 8 867.2.h.j 16
231.h odd 2 1 7497.2.a.v 2
264.m even 2 1 9792.2.a.cy 2
264.p odd 2 1 9792.2.a.cz 2
561.h even 2 1 2601.2.a.t 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
51.2.a.b 2 11.b odd 2 1
153.2.a.e 2 33.d even 2 1
816.2.a.m 2 44.c even 2 1
867.2.a.f 2 187.b odd 2 1
867.2.d.c 4 187.f odd 4 2
867.2.e.f 8 187.i odd 8 4
867.2.h.j 16 187.m even 16 8
1275.2.a.n 2 55.d odd 2 1
1275.2.b.d 4 55.e even 4 2
2448.2.a.v 2 132.d odd 2 1
2499.2.a.o 2 77.b even 2 1
2601.2.a.t 2 561.h even 2 1
3264.2.a.bg 2 88.g even 2 1
3264.2.a.bl 2 88.b odd 2 1
3825.2.a.s 2 165.d even 2 1
6171.2.a.p 2 1.a even 1 1 trivial
7497.2.a.v 2 231.h odd 2 1
8619.2.a.q 2 143.d odd 2 1
9792.2.a.cy 2 264.m even 2 1
9792.2.a.cz 2 264.p odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6171))$$:

 $$T_{2}^{2} - T_{2} - 4$$ $$T_{5}^{2} - 3 T_{5} - 2$$ $$T_{7}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-4 - T + T^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$-2 - 3 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$2 + 5 T + T^{2}$$
$17$ $$( 1 + T )^{2}$$
$19$ $$-36 + 3 T + T^{2}$$
$23$ $$16 + 9 T + T^{2}$$
$29$ $$-68 + T^{2}$$
$31$ $$-16 + 2 T + T^{2}$$
$37$ $$-16 + 2 T + T^{2}$$
$41$ $$-2 - 3 T + T^{2}$$
$43$ $$-36 - 3 T + T^{2}$$
$47$ $$32 + 14 T + T^{2}$$
$53$ $$-52 - 8 T + T^{2}$$
$59$ $$-8 - 6 T + T^{2}$$
$61$ $$8 + 10 T + T^{2}$$
$67$ $$( -4 + T )^{2}$$
$71$ $$-64 - 4 T + T^{2}$$
$73$ $$-52 - 8 T + T^{2}$$
$79$ $$-144 + 6 T + T^{2}$$
$83$ $$8 - 10 T + T^{2}$$
$89$ $$-8 - 6 T + T^{2}$$
$97$ $$32 + 14 T + T^{2}$$