# Properties

 Label 5625.2.a.e Level $5625$ Weight $2$ Character orbit 5625.a Self dual yes Analytic conductor $44.916$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5625 = 3^{2} \cdot 5^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5625.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$44.9158511370$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 625) 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.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + (\beta - 1) q^{4} + ( - 3 \beta + 1) q^{7} + ( - 2 \beta + 1) q^{8}+O(q^{10})$$ q + b * q^2 + (b - 1) * q^4 + (-3*b + 1) * q^7 + (-2*b + 1) * q^8 $$q + \beta q^{2} + (\beta - 1) q^{4} + ( - 3 \beta + 1) q^{7} + ( - 2 \beta + 1) q^{8} + ( - \beta + 1) q^{11} + (4 \beta - 1) q^{13} + ( - 2 \beta - 3) q^{14} - 3 \beta q^{16} + ( - 4 \beta + 5) q^{17} + ( - 3 \beta + 4) q^{19} - q^{22} + ( - 3 \beta + 3) q^{23} + (3 \beta + 4) q^{26} + (\beta - 4) q^{28} + ( - 2 \beta + 6) q^{29} + 2 q^{31} + (\beta - 5) q^{32} + (\beta - 4) q^{34} - 3 q^{37} + (\beta - 3) q^{38} - 4 \beta q^{41} + (4 \beta - 6) q^{43} + (\beta - 2) q^{44} - 3 q^{46} + (\beta + 10) q^{47} + (3 \beta + 3) q^{49} + ( - \beta + 5) q^{52} + (4 \beta + 2) q^{53} + (\beta + 7) q^{56} + (4 \beta - 2) q^{58} + (8 \beta + 1) q^{59} + ( - 3 \beta - 4) q^{61} + 2 \beta q^{62} + (2 \beta + 1) q^{64} + (4 \beta + 5) q^{67} + (5 \beta - 9) q^{68} + (7 \beta - 8) q^{71} + ( - 3 \beta + 5) q^{73} - 3 \beta q^{74} + (4 \beta - 7) q^{76} + ( - \beta + 4) q^{77} + ( - 6 \beta + 3) q^{79} + ( - 4 \beta - 4) q^{82} + 8 \beta q^{83} + ( - 2 \beta + 4) q^{86} + ( - \beta + 3) q^{88} + ( - \beta - 2) q^{89} + ( - 5 \beta - 13) q^{91} + (3 \beta - 6) q^{92} + (11 \beta + 1) q^{94} + (\beta + 9) q^{97} + (6 \beta + 3) q^{98} +O(q^{100})$$ q + b * q^2 + (b - 1) * q^4 + (-3*b + 1) * q^7 + (-2*b + 1) * q^8 + (-b + 1) * q^11 + (4*b - 1) * q^13 + (-2*b - 3) * q^14 - 3*b * q^16 + (-4*b + 5) * q^17 + (-3*b + 4) * q^19 - q^22 + (-3*b + 3) * q^23 + (3*b + 4) * q^26 + (b - 4) * q^28 + (-2*b + 6) * q^29 + 2 * q^31 + (b - 5) * q^32 + (b - 4) * q^34 - 3 * q^37 + (b - 3) * q^38 - 4*b * q^41 + (4*b - 6) * q^43 + (b - 2) * q^44 - 3 * q^46 + (b + 10) * q^47 + (3*b + 3) * q^49 + (-b + 5) * q^52 + (4*b + 2) * q^53 + (b + 7) * q^56 + (4*b - 2) * q^58 + (8*b + 1) * q^59 + (-3*b - 4) * q^61 + 2*b * q^62 + (2*b + 1) * q^64 + (4*b + 5) * q^67 + (5*b - 9) * q^68 + (7*b - 8) * q^71 + (-3*b + 5) * q^73 - 3*b * q^74 + (4*b - 7) * q^76 + (-b + 4) * q^77 + (-6*b + 3) * q^79 + (-4*b - 4) * q^82 + 8*b * q^83 + (-2*b + 4) * q^86 + (-b + 3) * q^88 + (-b - 2) * q^89 + (-5*b - 13) * q^91 + (3*b - 6) * q^92 + (11*b + 1) * q^94 + (b + 9) * q^97 + (6*b + 3) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} - q^{4} - q^{7}+O(q^{10})$$ 2 * q + q^2 - q^4 - q^7 $$2 q + q^{2} - q^{4} - q^{7} + q^{11} + 2 q^{13} - 8 q^{14} - 3 q^{16} + 6 q^{17} + 5 q^{19} - 2 q^{22} + 3 q^{23} + 11 q^{26} - 7 q^{28} + 10 q^{29} + 4 q^{31} - 9 q^{32} - 7 q^{34} - 6 q^{37} - 5 q^{38} - 4 q^{41} - 8 q^{43} - 3 q^{44} - 6 q^{46} + 21 q^{47} + 9 q^{49} + 9 q^{52} + 8 q^{53} + 15 q^{56} + 10 q^{59} - 11 q^{61} + 2 q^{62} + 4 q^{64} + 14 q^{67} - 13 q^{68} - 9 q^{71} + 7 q^{73} - 3 q^{74} - 10 q^{76} + 7 q^{77} - 12 q^{82} + 8 q^{83} + 6 q^{86} + 5 q^{88} - 5 q^{89} - 31 q^{91} - 9 q^{92} + 13 q^{94} + 19 q^{97} + 12 q^{98}+O(q^{100})$$ 2 * q + q^2 - q^4 - q^7 + q^11 + 2 * q^13 - 8 * q^14 - 3 * q^16 + 6 * q^17 + 5 * q^19 - 2 * q^22 + 3 * q^23 + 11 * q^26 - 7 * q^28 + 10 * q^29 + 4 * q^31 - 9 * q^32 - 7 * q^34 - 6 * q^37 - 5 * q^38 - 4 * q^41 - 8 * q^43 - 3 * q^44 - 6 * q^46 + 21 * q^47 + 9 * q^49 + 9 * q^52 + 8 * q^53 + 15 * q^56 + 10 * q^59 - 11 * q^61 + 2 * q^62 + 4 * q^64 + 14 * q^67 - 13 * q^68 - 9 * q^71 + 7 * q^73 - 3 * q^74 - 10 * q^76 + 7 * q^77 - 12 * q^82 + 8 * q^83 + 6 * q^86 + 5 * q^88 - 5 * q^89 - 31 * q^91 - 9 * q^92 + 13 * q^94 + 19 * q^97 + 12 * q^98

## 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 0 −1.61803 0 0 2.85410 2.23607 0 0
1.2 1.61803 0 0.618034 0 0 −3.85410 −2.23607 0 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
$$3$$ $$-1$$
$$5$$ $$1$$

## 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 5625.2.a.e 2
3.b odd 2 1 625.2.a.a 2
5.b even 2 1 5625.2.a.c 2
12.b even 2 1 10000.2.a.m 2
15.d odd 2 1 625.2.a.d yes 2
15.e even 4 2 625.2.b.b 4
60.h even 2 1 10000.2.a.b 2
75.h odd 10 2 625.2.d.c 4
75.h odd 10 2 625.2.d.f 4
75.j odd 10 2 625.2.d.e 4
75.j odd 10 2 625.2.d.i 4
75.l even 20 4 625.2.e.e 8
75.l even 20 4 625.2.e.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
625.2.a.a 2 3.b odd 2 1
625.2.a.d yes 2 15.d odd 2 1
625.2.b.b 4 15.e even 4 2
625.2.d.c 4 75.h odd 10 2
625.2.d.e 4 75.j odd 10 2
625.2.d.f 4 75.h odd 10 2
625.2.d.i 4 75.j odd 10 2
625.2.e.e 8 75.l even 20 4
625.2.e.f 8 75.l even 20 4
5625.2.a.c 2 5.b even 2 1
5625.2.a.e 2 1.a even 1 1 trivial
10000.2.a.b 2 60.h even 2 1
10000.2.a.m 2 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(5625))$$:

 $$T_{2}^{2} - T_{2} - 1$$ T2^2 - T2 - 1 $$T_{7}^{2} + T_{7} - 11$$ T7^2 + T7 - 11

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T - 1$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + T - 11$$
$11$ $$T^{2} - T - 1$$
$13$ $$T^{2} - 2T - 19$$
$17$ $$T^{2} - 6T - 11$$
$19$ $$T^{2} - 5T - 5$$
$23$ $$T^{2} - 3T - 9$$
$29$ $$T^{2} - 10T + 20$$
$31$ $$(T - 2)^{2}$$
$37$ $$(T + 3)^{2}$$
$41$ $$T^{2} + 4T - 16$$
$43$ $$T^{2} + 8T - 4$$
$47$ $$T^{2} - 21T + 109$$
$53$ $$T^{2} - 8T - 4$$
$59$ $$T^{2} - 10T - 55$$
$61$ $$T^{2} + 11T + 19$$
$67$ $$T^{2} - 14T + 29$$
$71$ $$T^{2} + 9T - 41$$
$73$ $$T^{2} - 7T + 1$$
$79$ $$T^{2} - 45$$
$83$ $$T^{2} - 8T - 64$$
$89$ $$T^{2} + 5T + 5$$
$97$ $$T^{2} - 19T + 89$$