# Properties

 Label 6025.2.a.a 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.

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

## 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 −0.618034 −1.00000 0 0.618034 −3.23607 3.00000 −2.61803 0
1.2 −1.00000 1.61803 −1.00000 0 −1.61803 1.23607 3.00000 −0.381966 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.a 2
5.b even 2 1 6025.2.a.d 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 1.a even 1 1 trivial
6025.2.a.d 2 5.b even 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$-1$$
$$241$$ $$-1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6025))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T + 2 T^{2} )^{2}$$
$3$ $$1 - T + 5 T^{2} - 3 T^{3} + 9 T^{4}$$
$5$ 1
$7$ $$1 + 2 T + 10 T^{2} + 14 T^{3} + 49 T^{4}$$
$11$ $$1 - 5 T + 27 T^{2} - 55 T^{3} + 121 T^{4}$$
$13$ $$1 - 3 T + 27 T^{2} - 39 T^{3} + 169 T^{4}$$
$17$ $$1 - T + 23 T^{2} - 17 T^{3} + 289 T^{4}$$
$19$ $$1 + T + 37 T^{2} + 19 T^{3} + 361 T^{4}$$
$23$ $$1 + 8 T + 42 T^{2} + 184 T^{3} + 529 T^{4}$$
$29$ $$1 + 3 T + 29 T^{2} + 87 T^{3} + 841 T^{4}$$
$31$ $$1 + 12 T + 78 T^{2} + 372 T^{3} + 961 T^{4}$$
$37$ $$1 + 54 T^{2} + 1369 T^{4}$$
$41$ $$1 - 3 T + 53 T^{2} - 123 T^{3} + 1681 T^{4}$$
$43$ $$1 + 4 T + 70 T^{2} + 172 T^{3} + 1849 T^{4}$$
$47$ $$1 + 9 T + 103 T^{2} + 423 T^{3} + 2209 T^{4}$$
$53$ $$( 1 - 4 T + 53 T^{2} )^{2}$$
$59$ $$1 + 10 T + 138 T^{2} + 590 T^{3} + 3481 T^{4}$$
$61$ $$1 - 3 T + 113 T^{2} - 183 T^{3} + 3721 T^{4}$$
$67$ $$1 + T - 17 T^{2} + 67 T^{3} + 4489 T^{4}$$
$71$ $$1 + 5 T - 3 T^{2} + 355 T^{3} + 5041 T^{4}$$
$73$ $$1 + 9 T + 135 T^{2} + 657 T^{3} + 5329 T^{4}$$
$79$ $$1 + 2 T - 86 T^{2} + 158 T^{3} + 6241 T^{4}$$
$83$ $$1 - 29 T + 375 T^{2} - 2407 T^{3} + 6889 T^{4}$$
$89$ $$1 - 24 T + 302 T^{2} - 2136 T^{3} + 7921 T^{4}$$
$97$ $$1 - 30 T + 414 T^{2} - 2910 T^{3} + 9409 T^{4}$$