# Properties

 Label 6025.2.a.d Level $6025$ Weight $2$ Character orbit 6025.a Self dual yes Analytic conductor $48.110$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$6025 = 5^{2} \cdot 241$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6025.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$48.1098672178$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1205) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + ( -1 + \beta ) q^{3} - q^{4} + ( -1 + \beta ) q^{6} + 2 \beta q^{7} -3 q^{8} + ( -1 - \beta ) q^{9} +O(q^{10})$$ $$q + q^{2} + ( -1 + \beta ) q^{3} - q^{4} + ( -1 + \beta ) q^{6} + 2 \beta q^{7} -3 q^{8} + ( -1 - \beta ) q^{9} + ( 3 - \beta ) q^{11} + ( 1 - \beta ) q^{12} + ( -1 - \beta ) q^{13} + 2 \beta q^{14} - q^{16} + ( -2 + 3 \beta ) q^{17} + ( -1 - \beta ) q^{18} -\beta q^{19} + 2 q^{21} + ( 3 - \beta ) q^{22} + ( 6 - 4 \beta ) q^{23} + ( 3 - 3 \beta ) q^{24} + ( -1 - \beta ) q^{26} + ( 3 - 4 \beta ) q^{27} -2 \beta q^{28} + ( -4 + 5 \beta ) q^{29} + ( -4 - 4 \beta ) q^{31} + 5 q^{32} + ( -4 + 3 \beta ) q^{33} + ( -2 + 3 \beta ) q^{34} + ( 1 + \beta ) q^{36} + ( 2 - 4 \beta ) q^{37} -\beta q^{38} -\beta q^{39} + ( -1 + 5 \beta ) q^{41} + 2 q^{42} + ( 4 - 4 \beta ) q^{43} + ( -3 + \beta ) q^{44} + ( 6 - 4 \beta ) q^{46} + ( 6 - 3 \beta ) q^{47} + ( 1 - \beta ) q^{48} + ( -3 + 4 \beta ) q^{49} + ( 5 - 2 \beta ) q^{51} + ( 1 + \beta ) q^{52} -4 q^{53} + ( 3 - 4 \beta ) q^{54} -6 \beta q^{56} - q^{57} + ( -4 + 5 \beta ) q^{58} + ( -4 - 2 \beta ) q^{59} + ( 3 - 3 \beta ) q^{61} + ( -4 - 4 \beta ) q^{62} + ( -2 - 4 \beta ) q^{63} + 7 q^{64} + ( -4 + 3 \beta ) q^{66} + ( 6 - 11 \beta ) q^{67} + ( 2 - 3 \beta ) q^{68} + ( -10 + 6 \beta ) q^{69} + ( -8 + 11 \beta ) q^{71} + ( 3 + 3 \beta ) q^{72} + ( 2 + 5 \beta ) q^{73} + ( 2 - 4 \beta ) q^{74} + \beta q^{76} + ( -2 + 4 \beta ) q^{77} -\beta q^{78} + ( 6 - 14 \beta ) q^{79} + ( -4 + 6 \beta ) q^{81} + ( -1 + 5 \beta ) q^{82} + ( -14 - \beta ) q^{83} -2 q^{84} + ( 4 - 4 \beta ) q^{86} + ( 9 - 4 \beta ) q^{87} + ( -9 + 3 \beta ) q^{88} + ( 14 - 4 \beta ) q^{89} + ( -2 - 4 \beta ) q^{91} + ( -6 + 4 \beta ) q^{92} -4 \beta q^{93} + ( 6 - 3 \beta ) q^{94} + ( -5 + 5 \beta ) q^{96} + ( -14 - 2 \beta ) q^{97} + ( -3 + 4 \beta ) q^{98} + ( -2 - \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} - q^{3} - 2q^{4} - q^{6} + 2q^{7} - 6q^{8} - 3q^{9} + O(q^{10})$$ $$2q + 2q^{2} - q^{3} - 2q^{4} - q^{6} + 2q^{7} - 6q^{8} - 3q^{9} + 5q^{11} + q^{12} - 3q^{13} + 2q^{14} - 2q^{16} - q^{17} - 3q^{18} - q^{19} + 4q^{21} + 5q^{22} + 8q^{23} + 3q^{24} - 3q^{26} + 2q^{27} - 2q^{28} - 3q^{29} - 12q^{31} + 10q^{32} - 5q^{33} - q^{34} + 3q^{36} - q^{38} - q^{39} + 3q^{41} + 4q^{42} + 4q^{43} - 5q^{44} + 8q^{46} + 9q^{47} + q^{48} - 2q^{49} + 8q^{51} + 3q^{52} - 8q^{53} + 2q^{54} - 6q^{56} - 2q^{57} - 3q^{58} - 10q^{59} + 3q^{61} - 12q^{62} - 8q^{63} + 14q^{64} - 5q^{66} + q^{67} + q^{68} - 14q^{69} - 5q^{71} + 9q^{72} + 9q^{73} + q^{76} - q^{78} - 2q^{79} - 2q^{81} + 3q^{82} - 29q^{83} - 4q^{84} + 4q^{86} + 14q^{87} - 15q^{88} + 24q^{89} - 8q^{91} - 8q^{92} - 4q^{93} + 9q^{94} - 5q^{96} - 30q^{97} - 2q^{98} - 5q^{99} + O(q^{100})$$

## 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
 −0.618034 1.61803
1.00000 −1.61803 −1.00000 0 −1.61803 −1.23607 −3.00000 −0.381966 0
1.2 1.00000 0.618034 −1.00000 0 0.618034 3.23607 −3.00000 −2.61803 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
$$5$$ $$-1$$
$$241$$ $$-1$$

## 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 6025.2.a.d 2
5.b even 2 1 6025.2.a.a 2
5.c odd 4 2 1205.2.b.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1205.2.b.b 4 5.c odd 4 2
6025.2.a.a 2 5.b even 2 1
6025.2.a.d 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(6025))$$:

 $$T_{2} - 1$$ $$T_{3}^{2} + T_{3} - 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{2}$$
$3$ $$-1 + T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$-4 - 2 T + T^{2}$$
$11$ $$5 - 5 T + T^{2}$$
$13$ $$1 + 3 T + T^{2}$$
$17$ $$-11 + T + T^{2}$$
$19$ $$-1 + T + T^{2}$$
$23$ $$-4 - 8 T + T^{2}$$
$29$ $$-29 + 3 T + T^{2}$$
$31$ $$16 + 12 T + T^{2}$$
$37$ $$-20 + T^{2}$$
$41$ $$-29 - 3 T + T^{2}$$
$43$ $$-16 - 4 T + T^{2}$$
$47$ $$9 - 9 T + T^{2}$$
$53$ $$( 4 + T )^{2}$$
$59$ $$20 + 10 T + T^{2}$$
$61$ $$-9 - 3 T + T^{2}$$
$67$ $$-151 - T + T^{2}$$
$71$ $$-145 + 5 T + T^{2}$$
$73$ $$-11 - 9 T + T^{2}$$
$79$ $$-244 + 2 T + T^{2}$$
$83$ $$209 + 29 T + T^{2}$$
$89$ $$124 - 24 T + T^{2}$$
$97$ $$220 + 30 T + T^{2}$$