# Properties

 Label 8025.2.a.r Level 8025 Weight 2 Character orbit 8025.a Self dual yes Analytic conductor 64.080 Analytic rank 1 Dimension 2 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$8025 = 3 \cdot 5^{2} \cdot 107$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8025.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$64.0799476221$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 321) 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.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} - q^{3} + ( -1 + \beta ) q^{4} -\beta q^{6} + ( 2 - 2 \beta ) q^{7} + ( 1 - 2 \beta ) q^{8} + q^{9} +O(q^{10})$$ $$q + \beta q^{2} - q^{3} + ( -1 + \beta ) q^{4} -\beta q^{6} + ( 2 - 2 \beta ) q^{7} + ( 1 - 2 \beta ) q^{8} + q^{9} -2 q^{11} + ( 1 - \beta ) q^{12} + q^{13} -2 q^{14} -3 \beta q^{16} + ( 1 + 4 \beta ) q^{17} + \beta q^{18} + ( 1 - 2 \beta ) q^{19} + ( -2 + 2 \beta ) q^{21} -2 \beta q^{22} + 4 q^{23} + ( -1 + 2 \beta ) q^{24} + \beta q^{26} - q^{27} + ( -4 + 2 \beta ) q^{28} + ( -2 + 2 \beta ) q^{29} -2 q^{31} + ( -5 + \beta ) q^{32} + 2 q^{33} + ( 4 + 5 \beta ) q^{34} + ( -1 + \beta ) q^{36} + ( -5 + 8 \beta ) q^{37} + ( -2 - \beta ) q^{38} - q^{39} + ( -4 - 2 \beta ) q^{41} + 2 q^{42} + ( 2 - 4 \beta ) q^{43} + ( 2 - 2 \beta ) q^{44} + 4 \beta q^{46} + ( 2 - 6 \beta ) q^{47} + 3 \beta q^{48} + ( 1 - 4 \beta ) q^{49} + ( -1 - 4 \beta ) q^{51} + ( -1 + \beta ) q^{52} + ( 10 - 4 \beta ) q^{53} -\beta q^{54} + ( 6 - 2 \beta ) q^{56} + ( -1 + 2 \beta ) q^{57} + 2 q^{58} + ( -4 + 8 \beta ) q^{59} + ( -5 + 8 \beta ) q^{61} -2 \beta q^{62} + ( 2 - 2 \beta ) q^{63} + ( 1 + 2 \beta ) q^{64} + 2 \beta q^{66} + ( -8 + 4 \beta ) q^{67} + ( 3 + \beta ) q^{68} -4 q^{69} + ( 1 - 6 \beta ) q^{71} + ( 1 - 2 \beta ) q^{72} + ( -6 + 6 \beta ) q^{73} + ( 8 + 3 \beta ) q^{74} + ( -3 + \beta ) q^{76} + ( -4 + 4 \beta ) q^{77} -\beta q^{78} -8 q^{79} + q^{81} + ( -2 - 6 \beta ) q^{82} + ( -4 - 6 \beta ) q^{83} + ( 4 - 2 \beta ) q^{84} + ( -4 - 2 \beta ) q^{86} + ( 2 - 2 \beta ) q^{87} + ( -2 + 4 \beta ) q^{88} + ( -4 + 12 \beta ) q^{89} + ( 2 - 2 \beta ) q^{91} + ( -4 + 4 \beta ) q^{92} + 2 q^{93} + ( -6 - 4 \beta ) q^{94} + ( 5 - \beta ) q^{96} -10 q^{97} + ( -4 - 3 \beta ) q^{98} -2 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{2} - 2q^{3} - q^{4} - q^{6} + 2q^{7} + 2q^{9} + O(q^{10})$$ $$2q + q^{2} - 2q^{3} - q^{4} - q^{6} + 2q^{7} + 2q^{9} - 4q^{11} + q^{12} + 2q^{13} - 4q^{14} - 3q^{16} + 6q^{17} + q^{18} - 2q^{21} - 2q^{22} + 8q^{23} + q^{26} - 2q^{27} - 6q^{28} - 2q^{29} - 4q^{31} - 9q^{32} + 4q^{33} + 13q^{34} - q^{36} - 2q^{37} - 5q^{38} - 2q^{39} - 10q^{41} + 4q^{42} + 2q^{44} + 4q^{46} - 2q^{47} + 3q^{48} - 2q^{49} - 6q^{51} - q^{52} + 16q^{53} - q^{54} + 10q^{56} + 4q^{58} - 2q^{61} - 2q^{62} + 2q^{63} + 4q^{64} + 2q^{66} - 12q^{67} + 7q^{68} - 8q^{69} - 4q^{71} - 6q^{73} + 19q^{74} - 5q^{76} - 4q^{77} - q^{78} - 16q^{79} + 2q^{81} - 10q^{82} - 14q^{83} + 6q^{84} - 10q^{86} + 2q^{87} + 4q^{89} + 2q^{91} - 4q^{92} + 4q^{93} - 16q^{94} + 9q^{96} - 20q^{97} - 11q^{98} - 4q^{99} + O(q^{100})$$

## 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
−0.618034 −1.00000 −1.61803 0 0.618034 3.23607 2.23607 1.00000 0
1.2 1.61803 −1.00000 0.618034 0 −1.61803 −1.23607 −2.23607 1.00000 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
$$3$$ $$1$$
$$5$$ $$1$$
$$107$$ $$1$$

## 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 8025.2.a.r 2
5.b even 2 1 321.2.a.b 2
15.d odd 2 1 963.2.a.c 2
20.d odd 2 1 5136.2.a.t 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
321.2.a.b 2 5.b even 2 1
963.2.a.c 2 15.d odd 2 1
5136.2.a.t 2 20.d odd 2 1
8025.2.a.r 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8025))$$:

 $$T_{2}^{2} - T_{2} - 1$$ $$T_{7}^{2} - 2 T_{7} - 4$$ $$T_{11} + 2$$ $$T_{13} - 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + 3 T^{2} - 2 T^{3} + 4 T^{4}$$
$3$ $$( 1 + T )^{2}$$
$5$ 1
$7$ $$1 - 2 T + 10 T^{2} - 14 T^{3} + 49 T^{4}$$
$11$ $$( 1 + 2 T + 11 T^{2} )^{2}$$
$13$ $$( 1 - T + 13 T^{2} )^{2}$$
$17$ $$1 - 6 T + 23 T^{2} - 102 T^{3} + 289 T^{4}$$
$19$ $$1 + 33 T^{2} + 361 T^{4}$$
$23$ $$( 1 - 4 T + 23 T^{2} )^{2}$$
$29$ $$1 + 2 T + 54 T^{2} + 58 T^{3} + 841 T^{4}$$
$31$ $$( 1 + 2 T + 31 T^{2} )^{2}$$
$37$ $$1 + 2 T - 5 T^{2} + 74 T^{3} + 1369 T^{4}$$
$41$ $$1 + 10 T + 102 T^{2} + 410 T^{3} + 1681 T^{4}$$
$43$ $$1 + 66 T^{2} + 1849 T^{4}$$
$47$ $$1 + 2 T + 50 T^{2} + 94 T^{3} + 2209 T^{4}$$
$53$ $$1 - 16 T + 150 T^{2} - 848 T^{3} + 2809 T^{4}$$
$59$ $$1 + 38 T^{2} + 3481 T^{4}$$
$61$ $$1 + 2 T + 43 T^{2} + 122 T^{3} + 3721 T^{4}$$
$67$ $$1 + 12 T + 150 T^{2} + 804 T^{3} + 4489 T^{4}$$
$71$ $$1 + 4 T + 101 T^{2} + 284 T^{3} + 5041 T^{4}$$
$73$ $$1 + 6 T + 110 T^{2} + 438 T^{3} + 5329 T^{4}$$
$79$ $$( 1 + 8 T + 79 T^{2} )^{2}$$
$83$ $$1 + 14 T + 170 T^{2} + 1162 T^{3} + 6889 T^{4}$$
$89$ $$1 - 4 T + 2 T^{2} - 356 T^{3} + 7921 T^{4}$$
$97$ $$( 1 + 10 T + 97 T^{2} )^{2}$$