# Properties

 Label 6025.2.a.c 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

# Learn more about

## 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.

 $$f(q)$$ $$=$$ $$q + ( 1 - 2 \beta ) q^{2} + ( 1 + \beta ) q^{3} + 3 q^{4} + ( -1 - 3 \beta ) q^{6} -2 \beta q^{7} + ( 1 - 2 \beta ) q^{8} + ( -1 + 3 \beta ) q^{9} +O(q^{10})$$ $$q + ( 1 - 2 \beta ) q^{2} + ( 1 + \beta ) q^{3} + 3 q^{4} + ( -1 - 3 \beta ) q^{6} -2 \beta q^{7} + ( 1 - 2 \beta ) q^{8} + ( -1 + 3 \beta ) q^{9} + ( -1 + \beta ) q^{11} + ( 3 + 3 \beta ) q^{12} + ( 1 - 3 \beta ) q^{13} + ( 4 + 2 \beta ) q^{14} - q^{16} + ( -2 + 5 \beta ) q^{17} + ( -7 - \beta ) q^{18} + ( -6 - \beta ) q^{19} + ( -2 - 4 \beta ) q^{21} + ( -3 + \beta ) q^{22} + ( -2 + 4 \beta ) q^{23} + ( -1 - 3 \beta ) q^{24} + ( 7 + \beta ) q^{26} + ( -1 + 2 \beta ) q^{27} -6 \beta q^{28} + ( -4 + 5 \beta ) q^{29} -4 \beta q^{31} + ( -3 + 6 \beta ) q^{32} + \beta q^{33} + ( -12 - \beta ) q^{34} + ( -3 + 9 \beta ) q^{36} + ( -2 + 4 \beta ) q^{37} + ( -4 + 13 \beta ) q^{38} + ( -2 - 5 \beta ) q^{39} + ( 3 + \beta ) q^{41} + ( 6 + 8 \beta ) q^{42} + ( 4 + 4 \beta ) q^{43} + ( -3 + 3 \beta ) q^{44} -10 q^{46} + ( -4 - \beta ) q^{47} + ( -1 - \beta ) q^{48} + ( -3 + 4 \beta ) q^{49} + ( 3 + 8 \beta ) q^{51} + ( 3 - 9 \beta ) q^{52} + ( -8 + 4 \beta ) q^{53} -5 q^{54} + ( 4 + 2 \beta ) q^{56} + ( -7 - 8 \beta ) q^{57} + ( -14 + 3 \beta ) q^{58} + ( 4 - 6 \beta ) q^{59} + ( 11 - 3 \beta ) q^{61} + ( 8 + 4 \beta ) q^{62} + ( -6 - 4 \beta ) q^{63} -13 q^{64} + ( -2 - \beta ) q^{66} + ( 4 - 5 \beta ) q^{67} + ( -6 + 15 \beta ) q^{68} + ( 2 + 6 \beta ) q^{69} + ( 2 - 9 \beta ) q^{71} + ( -7 - \beta ) q^{72} + ( -6 - \beta ) q^{73} -10 q^{74} + ( -18 - 3 \beta ) q^{76} -2 q^{77} + ( 8 + 9 \beta ) q^{78} + ( 10 - 10 \beta ) q^{79} + ( 4 - 6 \beta ) q^{81} + ( 1 - 7 \beta ) q^{82} + ( 4 + 5 \beta ) q^{83} + ( -6 - 12 \beta ) q^{84} + ( -4 - 12 \beta ) q^{86} + ( 1 + 6 \beta ) q^{87} + ( -3 + \beta ) q^{88} + ( -10 + 8 \beta ) q^{89} + ( 6 + 4 \beta ) q^{91} + ( -6 + 12 \beta ) q^{92} + ( -4 - 8 \beta ) q^{93} + ( -2 + 9 \beta ) q^{94} + ( 3 + 9 \beta ) q^{96} + ( 6 - 2 \beta ) q^{97} + ( -11 + 2 \beta ) q^{98} + ( 4 - \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 3q^{3} + 6q^{4} - 5q^{6} - 2q^{7} + q^{9} + O(q^{10})$$ $$2q + 3q^{3} + 6q^{4} - 5q^{6} - 2q^{7} + q^{9} - q^{11} + 9q^{12} - q^{13} + 10q^{14} - 2q^{16} + q^{17} - 15q^{18} - 13q^{19} - 8q^{21} - 5q^{22} - 5q^{24} + 15q^{26} - 6q^{28} - 3q^{29} - 4q^{31} + q^{33} - 25q^{34} + 3q^{36} + 5q^{38} - 9q^{39} + 7q^{41} + 20q^{42} + 12q^{43} - 3q^{44} - 20q^{46} - 9q^{47} - 3q^{48} - 2q^{49} + 14q^{51} - 3q^{52} - 12q^{53} - 10q^{54} + 10q^{56} - 22q^{57} - 25q^{58} + 2q^{59} + 19q^{61} + 20q^{62} - 16q^{63} - 26q^{64} - 5q^{66} + 3q^{67} + 3q^{68} + 10q^{69} - 5q^{71} - 15q^{72} - 13q^{73} - 20q^{74} - 39q^{76} - 4q^{77} + 25q^{78} + 10q^{79} + 2q^{81} - 5q^{82} + 13q^{83} - 24q^{84} - 20q^{86} + 8q^{87} - 5q^{88} - 12q^{89} + 16q^{91} - 16q^{93} + 5q^{94} + 15q^{96} + 10q^{97} - 20q^{98} + 7q^{99} + O(q^{100})$$

## 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
 1.61803 −0.618034
−2.23607 2.61803 3.00000 0 −5.85410 −3.23607 −2.23607 3.85410 0
1.2 2.23607 0.381966 3.00000 0 0.854102 1.23607 2.23607 −2.85410 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.c 2
5.b even 2 1 6025.2.a.b 2
5.c odd 4 2 1205.2.b.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1205.2.b.a 4 5.c odd 4 2
6025.2.a.b 2 5.b even 2 1
6025.2.a.c 2 1.a even 1 1 trivial

## 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(6025))$$:

 $$T_{2}^{2} - 5$$ $$T_{3}^{2} - 3 T_{3} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + 4 T^{4}$$
$3$ $$1 - 3 T + 7 T^{2} - 9 T^{3} + 9 T^{4}$$
$5$ 1
$7$ $$1 + 2 T + 10 T^{2} + 14 T^{3} + 49 T^{4}$$
$11$ $$1 + T + 21 T^{2} + 11 T^{3} + 121 T^{4}$$
$13$ $$1 + T + 15 T^{2} + 13 T^{3} + 169 T^{4}$$
$17$ $$1 - T + 3 T^{2} - 17 T^{3} + 289 T^{4}$$
$19$ $$1 + 13 T + 79 T^{2} + 247 T^{3} + 361 T^{4}$$
$23$ $$1 + 26 T^{2} + 529 T^{4}$$
$29$ $$1 + 3 T + 29 T^{2} + 87 T^{3} + 841 T^{4}$$
$31$ $$1 + 4 T + 46 T^{2} + 124 T^{3} + 961 T^{4}$$
$37$ $$1 + 54 T^{2} + 1369 T^{4}$$
$41$ $$1 - 7 T + 93 T^{2} - 287 T^{3} + 1681 T^{4}$$
$43$ $$1 - 12 T + 102 T^{2} - 516 T^{3} + 1849 T^{4}$$
$47$ $$1 + 9 T + 113 T^{2} + 423 T^{3} + 2209 T^{4}$$
$53$ $$1 + 12 T + 122 T^{2} + 636 T^{3} + 2809 T^{4}$$
$59$ $$1 - 2 T + 74 T^{2} - 118 T^{3} + 3481 T^{4}$$
$61$ $$1 - 19 T + 201 T^{2} - 1159 T^{3} + 3721 T^{4}$$
$67$ $$1 - 3 T + 105 T^{2} - 201 T^{3} + 4489 T^{4}$$
$71$ $$1 + 5 T + 47 T^{2} + 355 T^{3} + 5041 T^{4}$$
$73$ $$1 + 13 T + 187 T^{2} + 949 T^{3} + 5329 T^{4}$$
$79$ $$1 - 10 T + 58 T^{2} - 790 T^{3} + 6241 T^{4}$$
$83$ $$1 - 13 T + 177 T^{2} - 1079 T^{3} + 6889 T^{4}$$
$89$ $$1 + 12 T + 134 T^{2} + 1068 T^{3} + 7921 T^{4}$$
$97$ $$1 - 10 T + 214 T^{2} - 970 T^{3} + 9409 T^{4}$$
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