# Properties

 Label 4025.2.a.p Level 4025 Weight 2 Character orbit 4025.a Self dual Yes Analytic conductor 32.140 Analytic rank 0 Dimension 5 CM No Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$4025 = 5^{2} \cdot 7 \cdot 23$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 4025.a (trivial)

## Newform invariants

 Self dual: Yes Analytic conductor: $$32.1397868136$$ Analytic rank: $$0$$ Dimension: $$5$$ Coefficient field: 5.5.2147108.1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} -\beta_{3} q^{3} + ( 2 + \beta_{2} ) q^{4} + ( -\beta_{2} + \beta_{4} ) q^{6} - q^{7} + ( -\beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} ) q^{8} + ( 3 - \beta_{1} - \beta_{2} ) q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{2} -\beta_{3} q^{3} + ( 2 + \beta_{2} ) q^{4} + ( -\beta_{2} + \beta_{4} ) q^{6} - q^{7} + ( -\beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} ) q^{8} + ( 3 - \beta_{1} - \beta_{2} ) q^{9} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} ) q^{11} + ( -1 + 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{12} + ( 1 - \beta_{3} - \beta_{4} ) q^{13} + \beta_{1} q^{14} + ( 1 + 3 \beta_{2} + \beta_{3} + \beta_{4} ) q^{16} + ( 3 - 2 \beta_{2} - \beta_{4} ) q^{17} + ( 4 - 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} ) q^{18} + ( 2 - 2 \beta_{1} ) q^{19} + \beta_{3} q^{21} + ( -3 + \beta_{1} + \beta_{2} + \beta_{3} + 3 \beta_{4} ) q^{22} + q^{23} + ( -8 + 2 \beta_{1} - \beta_{3} ) q^{24} + ( 1 - 2 \beta_{1} + \beta_{2} + 2 \beta_{3} + 2 \beta_{4} ) q^{26} + ( 1 - 2 \beta_{1} - \beta_{3} + \beta_{4} ) q^{27} + ( -2 - \beta_{2} ) q^{28} + ( \beta_{1} - 3 \beta_{2} ) q^{29} + ( 6 - \beta_{3} ) q^{31} + ( -1 - \beta_{1} - 2 \beta_{2} - \beta_{3} - 3 \beta_{4} ) q^{32} + ( 4 - 2 \beta_{4} ) q^{33} + ( 1 - 2 \beta_{1} + 4 \beta_{2} + 3 \beta_{4} ) q^{34} + ( 1 - 3 \beta_{1} - \beta_{2} - 2 \beta_{4} ) q^{36} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{37} + ( 8 - 2 \beta_{1} + 2 \beta_{2} ) q^{38} + ( 3 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} ) q^{39} + ( 1 + \beta_{3} - \beta_{4} ) q^{41} + ( \beta_{2} - \beta_{4} ) q^{42} + ( 2 - 2 \beta_{4} ) q^{43} + ( -5 + 3 \beta_{1} - 5 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} ) q^{44} -\beta_{1} q^{46} + ( -2 + 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} ) q^{47} + ( -6 + 4 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{48} + q^{49} + ( -1 - \beta_{1} + \beta_{2} - 5 \beta_{3} - \beta_{4} ) q^{51} + ( 4 - \beta_{2} - \beta_{3} - 3 \beta_{4} ) q^{52} + ( -4 + 2 \beta_{2} ) q^{53} + ( 7 - \beta_{2} - 2 \beta_{3} ) q^{54} + ( \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{56} + ( -2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} ) q^{57} + ( -4 + 3 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} ) q^{58} + ( 5 - 2 \beta_{1} - \beta_{4} ) q^{59} + ( -3 - 2 \beta_{1} - 2 \beta_{3} - \beta_{4} ) q^{61} + ( -6 \beta_{1} - \beta_{2} + \beta_{4} ) q^{62} + ( -3 + \beta_{1} + \beta_{2} ) q^{63} + ( 5 + 2 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} ) q^{64} + ( 2 - 6 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} ) q^{66} + ( 3 - 5 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} ) q^{67} + ( -1 - 2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} - 5 \beta_{4} ) q^{68} -\beta_{3} q^{69} + ( 1 + 3 \beta_{3} + \beta_{4} ) q^{71} + ( 6 + 2 \beta_{1} + 4 \beta_{2} + 5 \beta_{3} + \beta_{4} ) q^{72} + ( -1 + 4 \beta_{2} + \beta_{3} + \beta_{4} ) q^{73} + ( 8 + 2 \beta_{4} ) q^{74} + ( 4 - 6 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} ) q^{76} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} ) q^{77} + ( -7 - 2 \beta_{1} + \beta_{2} + 4 \beta_{4} ) q^{78} + ( 7 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} ) q^{79} + ( -\beta_{1} + \beta_{2} - \beta_{3} + 3 \beta_{4} ) q^{81} + ( 1 - 2 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{82} + ( -2 - 2 \beta_{4} ) q^{83} + ( 1 - 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{84} + ( 2 - 4 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} ) q^{86} + ( 3 - 6 \beta_{1} + 4 \beta_{2} - 3 \beta_{3} - \beta_{4} ) q^{87} + ( -3 + 5 \beta_{1} + 3 \beta_{2} - \beta_{3} + 5 \beta_{4} ) q^{88} + ( -5 + 2 \beta_{1} - 2 \beta_{3} - \beta_{4} ) q^{89} + ( -1 + \beta_{3} + \beta_{4} ) q^{91} + ( 2 + \beta_{2} ) q^{92} + ( 6 - \beta_{1} - \beta_{2} - 6 \beta_{3} ) q^{93} + ( -8 + 4 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{94} + ( -1 + 4 \beta_{1} - 3 \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{96} + ( 3 - 2 \beta_{1} + 2 \beta_{2} + 3 \beta_{4} ) q^{97} -\beta_{1} q^{98} + ( -3 + 3 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5q - 2q^{2} + 12q^{4} - 3q^{6} - 5q^{7} - 3q^{8} + 11q^{9} + O(q^{10})$$ $$5q - 2q^{2} + 12q^{4} - 3q^{6} - 5q^{7} - 3q^{8} + 11q^{9} - 4q^{11} - 3q^{12} + 6q^{13} + 2q^{14} + 10q^{16} + 12q^{17} + 19q^{18} + 6q^{19} - 14q^{22} + 5q^{23} - 36q^{24} + q^{26} - 12q^{28} - 4q^{29} + 30q^{31} - 8q^{32} + 22q^{33} + 6q^{34} - q^{36} - 4q^{37} + 40q^{38} + 16q^{39} + 6q^{41} + 3q^{42} + 12q^{43} - 26q^{44} - 2q^{46} - 10q^{47} - 25q^{48} + 5q^{49} - 4q^{51} + 21q^{52} - 16q^{53} + 33q^{54} + 3q^{56} - 6q^{57} - 13q^{58} + 22q^{59} - 18q^{61} - 15q^{62} - 11q^{63} + 25q^{64} + 4q^{66} + 2q^{67} - 12q^{68} + 4q^{71} + 41q^{72} + 2q^{73} + 38q^{74} + 10q^{76} + 4q^{77} - 41q^{78} + 30q^{79} - 3q^{81} + 7q^{82} - 8q^{83} + 3q^{84} + 8q^{86} + 12q^{87} - 4q^{88} - 20q^{89} - 6q^{91} + 12q^{92} + 26q^{93} - 25q^{94} - q^{96} + 12q^{97} - 2q^{98} - 8q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - 2 x^{4} - 9 x^{3} + 17 x^{2} + 16 x - 27$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 4$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{4} - \nu^{3} - 8 \nu^{2} + 5 \nu + 11$$$$)/2$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{4} + \nu^{3} - 10 \nu^{2} - 5 \nu + 19$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 4$$ $$\nu^{3}$$ $$=$$ $$\beta_{4} - \beta_{3} + \beta_{2} + 5 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{4} + \beta_{3} + 9 \beta_{2} + 21$$

## 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
 2.69017 2.11948 1.23828 −1.50216 −2.54577
−2.69017 0.269842 5.23702 0 −0.725921 −1.00000 −8.70812 −2.92719 0
1.2 −2.11948 1.84074 2.49221 0 −3.90141 −1.00000 −1.04322 0.388311 0
1.3 −1.23828 −2.68857 −0.466664 0 3.32920 −1.00000 3.05442 4.22838 0
1.4 1.50216 3.04067 0.256481 0 4.56757 −1.00000 −2.61904 6.24568 0
1.5 2.54577 −2.46268 4.48096 0 −6.26943 −1.00000 6.31597 3.06481 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$1$$
$$7$$ $$1$$
$$23$$ $$-1$$

## Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4025))$$:

 $$T_{2}^{5} + 2 T_{2}^{4} - 9 T_{2}^{3} - 17 T_{2}^{2} + 16 T_{2} + 27$$ $$T_{3}^{5} - 13 T_{3}^{3} + 38 T_{3} - 10$$ $$T_{11}^{5} + 4 T_{11}^{4} - 28 T_{11}^{3} - 148 T_{11}^{2} - 160 T_{11} - 48$$