# Properties

 Label 4025.2.a.t Level 4025 Weight 2 Character orbit 4025.a Self dual Yes Analytic conductor 32.140 Analytic rank 1 Dimension 8 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: $$1$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + ( -1 - \beta_{6} ) q^{3} + ( 1 + \beta_{2} ) q^{4} + ( \beta_{1} - \beta_{5} ) q^{6} - q^{7} + ( -\beta_{1} - \beta_{3} ) q^{8} + ( 1 - \beta_{2} + \beta_{3} + 2 \beta_{6} ) q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{2} + ( -1 - \beta_{6} ) q^{3} + ( 1 + \beta_{2} ) q^{4} + ( \beta_{1} - \beta_{5} ) q^{6} - q^{7} + ( -\beta_{1} - \beta_{3} ) q^{8} + ( 1 - \beta_{2} + \beta_{3} + 2 \beta_{6} ) q^{9} + ( 1 - \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{11} + ( -1 + \beta_{1} - \beta_{2} + \beta_{4} + \beta_{7} ) q^{12} + ( -1 - \beta_{5} ) q^{13} + \beta_{1} q^{14} + ( \beta_{1} + \beta_{4} ) q^{16} + ( -1 + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} ) q^{17} + ( 1 - \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{18} + ( \beta_{1} + \beta_{4} - \beta_{6} + \beta_{7} ) q^{19} + ( 1 + \beta_{6} ) q^{21} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{6} - \beta_{7} ) q^{22} - q^{23} + ( -1 + \beta_{2} + \beta_{6} - \beta_{7} ) q^{24} + ( 2 \beta_{1} + \beta_{4} - 2 \beta_{6} + \beta_{7} ) q^{26} + ( -3 + 2 \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{6} ) q^{27} + ( -1 - \beta_{2} ) q^{28} + ( -\beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{6} - \beta_{7} ) q^{29} + ( -2 - \beta_{3} - \beta_{4} - \beta_{7} ) q^{31} + ( -2 + 2 \beta_{1} + \beta_{3} - \beta_{5} ) q^{32} + ( -1 + 3 \beta_{1} + \beta_{3} + 3 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} ) q^{33} + ( -1 - 3 \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{34} + ( -2 - 2 \beta_{1} - 3 \beta_{4} + \beta_{5} - 2 \beta_{7} ) q^{36} + ( 1 + \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} + 2 \beta_{6} ) q^{37} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} - 3 \beta_{5} + \beta_{6} - \beta_{7} ) q^{38} + ( 2 - 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} ) q^{39} + ( 2 \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{41} + ( -\beta_{1} + \beta_{5} ) q^{42} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{43} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{44} + \beta_{1} q^{46} + ( -2 + 2 \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{5} + 2 \beta_{7} ) q^{47} + ( 1 - 2 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{48} + q^{49} + ( 1 + 3 \beta_{1} - 2 \beta_{2} - \beta_{3} + 3 \beta_{4} - 2 \beta_{5} + 2 \beta_{7} ) q^{51} + ( -2 - \beta_{1} - \beta_{3} - 2 \beta_{5} + \beta_{6} - \beta_{7} ) q^{52} + ( 2 \beta_{1} - 2 \beta_{2} - \beta_{5} + 3 \beta_{6} ) q^{53} + ( -3 + 2 \beta_{1} + \beta_{2} - \beta_{3} + 3 \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{54} + ( \beta_{1} + \beta_{3} ) q^{56} + ( 3 - \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{57} + ( 2 + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{58} + ( 3 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{59} + ( -2 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{7} ) q^{61} + ( -3 + 4 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{62} + ( -1 + \beta_{2} - \beta_{3} - 2 \beta_{6} ) q^{63} + ( -5 - 3 \beta_{2} - 2 \beta_{4} - 2 \beta_{6} + \beta_{7} ) q^{64} + ( -5 + 2 \beta_{1} - \beta_{2} - 3 \beta_{3} + \beta_{4} - \beta_{5} - 4 \beta_{6} + 2 \beta_{7} ) q^{66} + ( -1 + 2 \beta_{1} + \beta_{2} - 3 \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{67} + ( 1 + 4 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} - \beta_{4} + \beta_{5} + 4 \beta_{6} ) q^{68} + ( 1 + \beta_{6} ) q^{69} + ( 5 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{71} + ( -1 + 3 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} ) q^{72} + ( -4 \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{73} + ( 3 - 5 \beta_{1} + \beta_{2} - 3 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} - \beta_{7} ) q^{74} + ( 1 + 2 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 5 \beta_{6} + 2 \beta_{7} ) q^{76} + ( -1 + \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{77} + ( 6 - 2 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{78} + ( -1 - 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - \beta_{6} - \beta_{7} ) q^{79} + ( 4 - \beta_{1} - \beta_{2} + \beta_{3} + 3 \beta_{4} + 2 \beta_{5} + 4 \beta_{6} + \beta_{7} ) q^{81} + ( -5 - \beta_{1} - 3 \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} ) q^{82} + ( -3 + 2 \beta_{1} - \beta_{2} + \beta_{3} + 3 \beta_{4} - \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{83} + ( 1 - \beta_{1} + \beta_{2} - \beta_{4} - \beta_{7} ) q^{84} + ( -2 - 3 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} + 5 \beta_{6} - 3 \beta_{7} ) q^{86} + ( -2 - \beta_{1} + \beta_{2} - 3 \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{87} + ( 4 - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{88} + ( 1 - 3 \beta_{1} + \beta_{2} - \beta_{3} - 3 \beta_{5} - \beta_{6} + \beta_{7} ) q^{89} + ( 1 + \beta_{5} ) q^{91} + ( -1 - \beta_{2} ) q^{92} + ( 1 - \beta_{1} - \beta_{2} + 2 \beta_{4} + \beta_{5} + 3 \beta_{6} ) q^{93} + ( -5 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} + 6 \beta_{6} - 4 \beta_{7} ) q^{94} + ( 4 - 3 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{96} + ( -2 + 3 \beta_{1} - \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} + 2 \beta_{6} ) q^{97} -\beta_{1} q^{98} + ( -2 - 6 \beta_{1} + 2 \beta_{2} - 5 \beta_{4} + 4 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - q^{2} - 7q^{3} + 7q^{4} + q^{6} - 8q^{7} - 3q^{8} + 9q^{9} + O(q^{10})$$ $$8q - q^{2} - 7q^{3} + 7q^{4} + q^{6} - 8q^{7} - 3q^{8} + 9q^{9} + 6q^{11} - 8q^{12} - 8q^{13} + q^{14} + q^{16} - 4q^{17} + 11q^{18} + 7q^{21} + 8q^{22} - 8q^{23} - 8q^{24} + 2q^{26} - 28q^{27} - 7q^{28} - 3q^{29} - 16q^{31} - 12q^{32} - 5q^{33} - 4q^{34} - 14q^{36} + 7q^{37} - 11q^{38} + 16q^{39} + 7q^{41} - q^{42} + 10q^{43} + 7q^{44} + q^{46} - 19q^{47} + 6q^{48} + 8q^{49} + 7q^{51} - 18q^{52} + q^{53} - 25q^{54} + 3q^{56} + 25q^{57} + 9q^{58} + 21q^{59} - 7q^{61} - 18q^{62} - 9q^{63} - 37q^{64} - 43q^{66} - 11q^{67} + 17q^{68} + 7q^{69} + 8q^{71} - 4q^{72} + 3q^{73} + 12q^{74} + 8q^{76} - 6q^{77} + 50q^{78} - 15q^{79} + 28q^{81} - 41q^{82} - 25q^{83} + 8q^{84} - 12q^{86} - 21q^{87} + 29q^{88} + q^{89} + 8q^{91} - 7q^{92} + 5q^{93} - 36q^{94} + 30q^{96} - 10q^{97} - q^{98} - 18q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{7} - 11 x^{6} + 9 x^{5} + 36 x^{4} - 23 x^{3} - 30 x^{2} + 17 x - 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - 5 \nu$$ $$\beta_{4}$$ $$=$$ $$\nu^{4} - 6 \nu^{2} - \nu + 4$$ $$\beta_{5}$$ $$=$$ $$\nu^{5} - 7 \nu^{3} + 9 \nu - 2$$ $$\beta_{6}$$ $$=$$ $$\nu^{7} - \nu^{6} - 11 \nu^{5} + 8 \nu^{4} + 36 \nu^{3} - 16 \nu^{2} - 30 \nu + 8$$ $$\beta_{7}$$ $$=$$ $$2 \nu^{7} - \nu^{6} - 22 \nu^{5} + 8 \nu^{4} + 72 \nu^{3} - 17 \nu^{2} - 62 \nu + 12$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + 5 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{4} + 6 \beta_{2} + \beta_{1} + 14$$ $$\nu^{5}$$ $$=$$ $$\beta_{5} + 7 \beta_{3} + 26 \beta_{1} + 2$$ $$\nu^{6}$$ $$=$$ $$\beta_{7} - 2 \beta_{6} + 8 \beta_{4} + 33 \beta_{2} + 10 \beta_{1} + 71$$ $$\nu^{7}$$ $$=$$ $$\beta_{7} - \beta_{6} + 11 \beta_{5} + 41 \beta_{3} + \beta_{2} + 138 \beta_{1} + 21$$

## 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
 2.42061 2.28656 1.18794 0.322285 0.181467 −1.17189 −1.94419 −2.28279
−2.42061 0.490228 3.85934 0 −1.18665 −1.00000 −4.50072 −2.75968 0
1.2 −2.28656 −2.13745 3.22836 0 4.88741 −1.00000 −2.80872 1.56870 0
1.3 −1.18794 −1.57054 −0.588791 0 1.86571 −1.00000 3.07534 −0.533418 0
1.4 −0.322285 1.07802 −1.89613 0 −0.347431 −1.00000 1.25566 −1.83786 0
1.5 −0.181467 −3.25071 −1.96707 0 0.589897 −1.00000 0.719892 7.56714 0
1.6 1.17189 1.97939 −0.626674 0 2.31962 −1.00000 −3.07817 0.917966 0
1.7 1.94419 −3.14297 1.77986 0 −6.11052 −1.00000 −0.427999 6.87827 0
1.8 2.28279 −0.445964 3.21111 0 −1.01804 −1.00000 2.76472 −2.80112 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.8 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4025))$$:

 $$T_{2}^{8} + \cdots$$ $$T_{3}^{8} + \cdots$$ $$T_{11}^{8} - \cdots$$