# Properties

 Label 4425.2.a.w Level $4425$ Weight $2$ Character orbit 4425.a Self dual yes Analytic conductor $35.334$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4425 = 3 \cdot 5^{2} \cdot 59$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4425.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$35.3338028944$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.229.1 Defining polynomial: $$x^{3} - 4 x - 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 177) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 4 x - 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$

## 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
 2.11491 −0.254102 −1.86081
−2.11491 1.00000 2.47283 0 −2.11491 −5.11491 −1.00000 1.00000 0
1.2 0.254102 1.00000 −1.93543 0 0.254102 −2.74590 −1.00000 1.00000 0
1.3 1.86081 1.00000 1.46260 0 1.86081 −1.13919 −1.00000 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
$$3$$ $$-1$$
$$5$$ $$1$$
$$59$$ $$-1$$

## 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 4425.2.a.w 3
5.b even 2 1 177.2.a.d 3
15.d odd 2 1 531.2.a.d 3
20.d odd 2 1 2832.2.a.t 3
35.c odd 2 1 8673.2.a.s 3
60.h even 2 1 8496.2.a.bl 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.2.a.d 3 5.b even 2 1
531.2.a.d 3 15.d odd 2 1
2832.2.a.t 3 20.d odd 2 1
4425.2.a.w 3 1.a even 1 1 trivial
8496.2.a.bl 3 60.h even 2 1
8673.2.a.s 3 35.c odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4425))$$:

 $$T_{2}^{3} - 4 T_{2} + 1$$ $$T_{7}^{3} + 9 T_{7}^{2} + 23 T_{7} + 16$$ $$T_{11}^{3} + 2 T_{11}^{2} - 11 T_{11} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 4 T + T^{3}$$
$3$ $$( -1 + T )^{3}$$
$5$ $$T^{3}$$
$7$ $$16 + 23 T + 9 T^{2} + T^{3}$$
$11$ $$4 - 11 T + 2 T^{2} + T^{3}$$
$13$ $$-26 - 7 T + 4 T^{2} + T^{3}$$
$17$ $$-98 - 43 T + 3 T^{2} + T^{3}$$
$19$ $$-4 + 11 T - 7 T^{2} + T^{3}$$
$23$ $$-64 - 27 T + T^{2} + T^{3}$$
$29$ $$-74 + 9 T + 11 T^{2} + T^{3}$$
$31$ $$28 + 37 T - 13 T^{2} + T^{3}$$
$37$ $$-14 - 19 T - 5 T^{2} + T^{3}$$
$41$ $$74 - 39 T + T^{2} + T^{3}$$
$43$ $$-592 - 91 T + 6 T^{2} + T^{3}$$
$47$ $$-496 - 37 T + 11 T^{2} + T^{3}$$
$53$ $$58 - 89 T + 2 T^{2} + T^{3}$$
$59$ $$( -1 + T )^{3}$$
$61$ $$98 - 101 T + T^{2} + T^{3}$$
$67$ $$-784 - 119 T + 10 T^{2} + T^{3}$$
$71$ $$-424 + 193 T - 26 T^{2} + T^{3}$$
$73$ $$-718 - 141 T + 7 T^{2} + T^{3}$$
$79$ $$-32 - 31 T - 2 T^{2} + T^{3}$$
$83$ $$148 - 199 T - 3 T^{2} + T^{3}$$
$89$ $$-278 + 91 T + 23 T^{2} + T^{3}$$
$97$ $$-202 - 25 T + 14 T^{2} + T^{3}$$