# Properties

 Label 25.2.e.a Level 25 Weight 2 Character orbit 25.e Analytic conductor 0.200 Analytic rank 0 Dimension 8 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$25 = 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 25.e (of order $$10$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.199626005053$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{10})$$ Coefficient field: 8.0.58140625.2 Defining polynomial: $$x^{8} - 3 x^{7} + 4 x^{6} - 7 x^{5} + 11 x^{4} + 5 x^{3} - 10 x^{2} - 25 x + 25$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{1} ) q^{2} + ( -1 + \beta_{2} - \beta_{3} + \beta_{7} ) q^{3} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} ) q^{4} + ( 1 - \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} ) q^{5} + ( -2 - \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{6} + ( -1 + 2 \beta_{2} + \beta_{4} + \beta_{6} ) q^{7} + ( 2 - 2 \beta_{2} + \beta_{3} ) q^{8} + ( \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{9} +O(q^{10})$$ $$q + ( -1 + \beta_{1} ) q^{2} + ( -1 + \beta_{2} - \beta_{3} + \beta_{7} ) q^{3} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} ) q^{4} + ( 1 - \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} ) q^{5} + ( -2 - \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{6} + ( -1 + 2 \beta_{2} + \beta_{4} + \beta_{6} ) q^{7} + ( 2 - 2 \beta_{2} + \beta_{3} ) q^{8} + ( \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{9} + ( -1 + \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{10} -2 \beta_{4} q^{11} + ( 2 + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} + \beta_{7} ) q^{12} + ( -1 + \beta_{1} + \beta_{4} + \beta_{5} ) q^{13} + ( 2 + \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{14} + ( -\beta_{1} + \beta_{2} - 2 \beta_{3} + 3 \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{15} + ( -1 + 2 \beta_{1} + 2 \beta_{2} + \beta_{6} + \beta_{7} ) q^{16} + ( 1 - \beta_{1} - 4 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} - \beta_{5} - \beta_{7} ) q^{17} + ( 1 - \beta_{1} + \beta_{2} + 3 \beta_{3} - \beta_{4} - \beta_{7} ) q^{18} + ( -2 - 3 \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{19} + ( -2 + \beta_{2} + 3 \beta_{3} + 2 \beta_{6} - \beta_{7} ) q^{20} + ( 2 - 2 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{6} ) q^{21} + ( 2 \beta_{4} + 2 \beta_{5} ) q^{22} + ( -\beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} ) q^{23} + ( -4 + \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{24} + ( -3 \beta_{2} + \beta_{5} + 2 \beta_{7} ) q^{25} + ( 1 - 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{26} + ( 2 - 3 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} ) q^{27} + ( -3 - 3 \beta_{3} - \beta_{4} - \beta_{5} - 3 \beta_{7} ) q^{28} + ( -3 + 4 \beta_{1} + 5 \beta_{2} - \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{29} + ( 3 + 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} - 5 \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{30} + ( \beta_{1} + 4 \beta_{3} - 2 \beta_{5} - 3 \beta_{6} + 2 \beta_{7} ) q^{31} + ( -2 - \beta_{1} + \beta_{2} - 3 \beta_{3} + 6 \beta_{4} - \beta_{6} - \beta_{7} ) q^{32} + ( 2 \beta_{1} + 2 \beta_{6} ) q^{33} + ( 3 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{6} - \beta_{7} ) q^{34} + ( 2 \beta_{1} + 3 \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} + 3 \beta_{6} - \beta_{7} ) q^{35} + ( 4 + \beta_{1} - 3 \beta_{2} + \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{36} + ( 2 + \beta_{1} - 4 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{37} + ( 2 - 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{7} ) q^{38} + ( -1 - \beta_{1} + 3 \beta_{4} - 2 \beta_{7} ) q^{39} + ( 3 + \beta_{1} + 3 \beta_{3} + \beta_{5} - 3 \beta_{6} + 3 \beta_{7} ) q^{40} + ( -1 - 3 \beta_{1} + \beta_{2} - 3 \beta_{3} + 4 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{41} + ( -2 + \beta_{1} - 2 \beta_{4} + \beta_{6} ) q^{42} + ( 3 + \beta_{1} - \beta_{2} + 5 \beta_{3} - 6 \beta_{4} - \beta_{6} + \beta_{7} ) q^{43} + ( -2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{44} + ( -3 + 4 \beta_{2} - 3 \beta_{3} - 2 \beta_{6} + \beta_{7} ) q^{45} + ( -3 + 2 \beta_{1} + \beta_{3} - 3 \beta_{4} + \beta_{5} + 4 \beta_{6} - \beta_{7} ) q^{46} + ( \beta_{2} - 3 \beta_{4} - 3 \beta_{5} + 2 \beta_{7} ) q^{47} + ( -3 - 3 \beta_{2} + \beta_{5} + \beta_{6} ) q^{48} + ( 4 - 3 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} - \beta_{6} - \beta_{7} ) q^{49} + ( -4 - 3 \beta_{1} - 4 \beta_{3} + 5 \beta_{4} - 3 \beta_{5} - \beta_{6} - 4 \beta_{7} ) q^{50} + ( -4 + 4 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} + 2 \beta_{6} ) q^{51} + ( -1 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} + 2 \beta_{5} + 2 \beta_{6} ) q^{52} + ( -2 + 2 \beta_{2} + \beta_{3} + 6 \beta_{4} + 3 \beta_{5} + 2 \beta_{7} ) q^{53} + ( -1 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} ) q^{54} + ( -4 \beta_{1} - 2 \beta_{2} - 2 \beta_{5} - 2 \beta_{7} ) q^{55} + ( 2 - \beta_{1} + \beta_{3} - 2 \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{56} + ( 1 + \beta_{1} + 3 \beta_{2} + 5 \beta_{3} - 3 \beta_{6} + \beta_{7} ) q^{57} + ( 3 - 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + 5 \beta_{4} + 2 \beta_{5} - 4 \beta_{6} + 2 \beta_{7} ) q^{58} + ( 3 - \beta_{1} - 4 \beta_{2} + \beta_{3} - 3 \beta_{4} + 3 \beta_{5} + \beta_{6} + \beta_{7} ) q^{59} + ( -3 \beta_{1} - 4 \beta_{2} - \beta_{3} + 4 \beta_{4} + \beta_{5} - 2 \beta_{6} - 3 \beta_{7} ) q^{60} + ( -2 - 3 \beta_{1} - \beta_{2} - 2 \beta_{4} - 4 \beta_{5} + 4 \beta_{6} - 6 \beta_{7} ) q^{61} + ( -3 - \beta_{1} + 4 \beta_{2} + 3 \beta_{3} - 5 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{62} + ( 2 - 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{6} + 2 \beta_{7} ) q^{63} + ( 2 - 2 \beta_{1} - 4 \beta_{2} - 9 \beta_{4} - 4 \beta_{5} + 4 \beta_{6} - 4 \beta_{7} ) q^{64} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{7} ) q^{65} + ( 2 - 2 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{66} + ( 4 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} + 4 \beta_{7} ) q^{67} + ( -4 + 4 \beta_{1} + 5 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} + \beta_{6} + 4 \beta_{7} ) q^{68} + ( 4 - \beta_{1} + 2 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{69} + ( 4 + \beta_{1} - \beta_{2} - \beta_{3} + 5 \beta_{4} + 2 \beta_{5} - 4 \beta_{6} + 3 \beta_{7} ) q^{70} + ( -4 + 6 \beta_{1} + \beta_{2} - 6 \beta_{3} + 5 \beta_{4} + 3 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{71} + ( 1 - 2 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} - 3 \beta_{7} ) q^{72} + ( -2 + 3 \beta_{1} + 5 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} ) q^{73} + ( -3 + \beta_{1} - \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 3 \beta_{7} ) q^{74} + ( 5 \beta_{1} - \beta_{2} + 5 \beta_{3} - 5 \beta_{4} + 2 \beta_{5} - \beta_{7} ) q^{75} + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{76} + ( -2 + 2 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} ) q^{77} + ( 5 - \beta_{2} + \beta_{3} - 6 \beta_{4} - 2 \beta_{5} + 3 \beta_{7} ) q^{78} + ( 7 - 6 \beta_{1} - 4 \beta_{2} + 3 \beta_{3} + \beta_{4} - 3 \beta_{5} - 8 \beta_{6} + \beta_{7} ) q^{79} + ( 1 + 4 \beta_{2} - 4 \beta_{3} + \beta_{5} - \beta_{6} ) q^{80} + ( 4 - 2 \beta_{1} - 3 \beta_{3} - 4 \beta_{4} + 2 \beta_{6} ) q^{81} + ( -1 + 3 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} + 3 \beta_{6} + 3 \beta_{7} ) q^{82} + ( -5 \beta_{1} - 8 \beta_{2} + 4 \beta_{3} - 8 \beta_{4} - 4 \beta_{5} - \beta_{6} - 4 \beta_{7} ) q^{83} + ( -1 + \beta_{1} + 6 \beta_{2} - \beta_{3} + 5 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{84} + ( -3 \beta_{1} + \beta_{2} + 4 \beta_{3} - \beta_{4} - 4 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{85} + ( -1 + \beta_{1} + 2 \beta_{2} + 5 \beta_{4} + 4 \beta_{5} - 4 \beta_{6} + 2 \beta_{7} ) q^{86} + ( -3 - \beta_{1} - 6 \beta_{2} + \beta_{3} + 5 \beta_{4} - \beta_{5} + 3 \beta_{6} - 3 \beta_{7} ) q^{87} + ( -2 + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} ) q^{88} + ( -6 - \beta_{1} + 5 \beta_{2} + 7 \beta_{4} - 2 \beta_{7} ) q^{89} + ( -4 + \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} + 4 \beta_{5} + 5 \beta_{6} ) q^{90} + ( 1 + \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{91} + ( 6 - 3 \beta_{1} - 2 \beta_{2} + \beta_{3} + 4 \beta_{4} + \beta_{5} - 4 \beta_{6} + \beta_{7} ) q^{92} + ( -5 \beta_{1} - 7 \beta_{2} - 7 \beta_{3} + \beta_{4} + 4 \beta_{6} - 5 \beta_{7} ) q^{93} + ( -4 + \beta_{1} + 4 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} + 3 \beta_{6} - 4 \beta_{7} ) q^{94} + ( 5 - 6 \beta_{2} + 5 \beta_{3} - 5 \beta_{4} - 3 \beta_{5} - \beta_{7} ) q^{95} + ( 6 - 6 \beta_{1} + \beta_{2} + 4 \beta_{3} - \beta_{4} - 3 \beta_{5} - 6 \beta_{6} ) q^{96} + ( -12 + 4 \beta_{2} - 9 \beta_{3} + 7 \beta_{4} + 4 \beta_{5} - 4 \beta_{7} ) q^{97} + ( 4 \beta_{1} - 2 \beta_{2} + 6 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} ) q^{98} + ( -2 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} + 2 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 5q^{2} - 5q^{3} - q^{4} - 9q^{6} + 10q^{8} + q^{9} + O(q^{10})$$ $$8q - 5q^{2} - 5q^{3} - q^{4} - 9q^{6} + 10q^{8} + q^{9} - 5q^{10} - 4q^{11} + 15q^{12} - 5q^{13} + 13q^{14} + 15q^{15} + 3q^{16} - 10q^{17} - 5q^{19} - 15q^{20} - 4q^{21} + 5q^{23} - 20q^{24} - 10q^{25} + 6q^{26} - 5q^{27} - 15q^{28} - 5q^{29} + 15q^{30} - 9q^{31} + 10q^{33} + 13q^{34} + 15q^{35} + 23q^{36} + 30q^{37} + 15q^{38} - 3q^{39} + 10q^{40} - 4q^{41} - 15q^{42} - 2q^{44} - 15q^{45} - 19q^{46} - 30q^{48} + 14q^{49} - 15q^{50} - 4q^{51} - 10q^{52} - 10q^{53} - 5q^{54} - 10q^{55} + 10q^{56} + 20q^{58} - 10q^{60} - 9q^{61} - 30q^{62} + 10q^{63} + 4q^{64} + 5q^{65} + 12q^{66} + 20q^{67} + 17q^{69} + 30q^{70} + 6q^{71} + 5q^{72} + 15q^{73} - 12q^{74} - 10q^{75} - 20q^{76} + 10q^{77} + 25q^{78} + 15q^{79} + 20q^{80} + 28q^{81} - 45q^{83} + 18q^{84} - 15q^{85} - 9q^{86} - 20q^{87} - 20q^{88} - 25q^{89} - 25q^{90} + 6q^{91} + 30q^{92} - 27q^{94} + 15q^{95} + 16q^{96} - 60q^{97} - 10q^{98} - 8q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 3 x^{7} + 4 x^{6} - 7 x^{5} + 11 x^{4} + 5 x^{3} - 10 x^{2} - 25 x + 25$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$406 \nu^{7} - 714 \nu^{6} + 747 \nu^{5} - 1896 \nu^{4} + 2103 \nu^{3} + 4949 \nu^{2} + 1065 \nu - 7800$$$$)/1355$$ $$\beta_{3}$$ $$=$$ $$($$$$420 \nu^{7} - 776 \nu^{6} + 698 \nu^{5} - 1924 \nu^{4} + 2297 \nu^{3} + 5129 \nu^{2} + 1055 \nu - 10265$$$$)/1355$$ $$\beta_{4}$$ $$=$$ $$($$$$728 \nu^{7} - 1327 \nu^{6} + 1246 \nu^{5} - 3353 \nu^{4} + 3584 \nu^{3} + 8547 \nu^{2} + 2190 \nu - 15715$$$$)/1355$$ $$\beta_{5}$$ $$=$$ $$($$$$-857 \nu^{7} + 1666 \nu^{6} - 1743 \nu^{5} + 4424 \nu^{4} - 4907 \nu^{3} - 9470 \nu^{2} - 2485 \nu + 18200$$$$)/1355$$ $$\beta_{6}$$ $$=$$ $$($$$$891 \nu^{7} - 1623 \nu^{6} + 1624 \nu^{5} - 4492 \nu^{4} + 4991 \nu^{3} + 9520 \nu^{2} + 3235 \nu - 18960$$$$)/1355$$ $$\beta_{7}$$ $$=$$ $$($$$$955 \nu^{7} - 1829 \nu^{6} + 1942 \nu^{5} - 4891 \nu^{4} + 5723 \nu^{3} + 9646 \nu^{2} + 2415 \nu - 20550$$$$)/1355$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{7} - \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$\beta_{7} + \beta_{5} - 3 \beta_{4} + 4 \beta_{3} + \beta_{2} + \beta_{1} + 3$$ $$\nu^{4}$$ $$=$$ $$5 \beta_{7} - 5 \beta_{6} + 4 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 4 \beta_{1} + 2$$ $$\nu^{5}$$ $$=$$ $$4 \beta_{7} - 6 \beta_{6} - 7 \beta_{3} + 11 \beta_{2} + 4 \beta_{1} - 13$$ $$\nu^{6}$$ $$=$$ $$7 \beta_{6} + 7 \beta_{5} - 8 \beta_{4} - 7 \beta_{3} + 21 \beta_{2} - 2 \beta_{1} - 21$$ $$\nu^{7}$$ $$=$$ $$23 \beta_{7} + 38 \beta_{5} + 23 \beta_{4} - 23 \beta_{3} + 12 \beta_{2}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/25\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1
 −0.983224 − 0.644389i 1.17421 + 0.0566033i −0.357358 + 1.86824i 1.66637 − 0.917186i −0.357358 − 1.86824i 1.66637 + 0.917186i −0.983224 + 0.644389i 1.17421 − 0.0566033i
−1.98322 0.644389i 1.29224 1.77862i 1.89991 + 1.38036i −1.22570 + 1.87020i −3.70892 + 2.69469i 0.992398i −0.427051 0.587785i −0.566541 1.74363i 3.63597 2.91920i
4.2 0.174207 + 0.0566033i −0.865190 + 1.19083i −1.59089 1.15585i 0.107666 2.23347i −0.218127 + 0.158479i 3.26086i −0.427051 0.587785i 0.257524 + 0.792578i 0.145178 0.382993i
9.1 −1.35736 + 1.86824i −0.451659 + 0.146753i −1.02988 3.16963i 2.19625 + 0.420099i 0.338893 1.04301i 3.03582i 2.92705 + 0.951057i −2.24459 + 1.63079i −3.76594 + 3.53290i
9.2 0.666375 0.917186i −2.47539 + 0.804303i 0.220859 + 0.679734i −1.07822 1.95894i −0.911842 + 2.80636i 0.407162i 2.92705 + 0.951057i 3.05361 2.21858i −2.51521 0.316463i
14.1 −1.35736 1.86824i −0.451659 0.146753i −1.02988 + 3.16963i 2.19625 0.420099i 0.338893 + 1.04301i 3.03582i 2.92705 0.951057i −2.24459 1.63079i −3.76594 3.53290i
14.2 0.666375 + 0.917186i −2.47539 0.804303i 0.220859 0.679734i −1.07822 + 1.95894i −0.911842 2.80636i 0.407162i 2.92705 0.951057i 3.05361 + 2.21858i −2.51521 + 0.316463i
19.1 −1.98322 + 0.644389i 1.29224 + 1.77862i 1.89991 1.38036i −1.22570 1.87020i −3.70892 2.69469i 0.992398i −0.427051 + 0.587785i −0.566541 + 1.74363i 3.63597 + 2.91920i
19.2 0.174207 0.0566033i −0.865190 1.19083i −1.59089 + 1.15585i 0.107666 + 2.23347i −0.218127 0.158479i 3.26086i −0.427051 + 0.587785i 0.257524 0.792578i 0.145178 + 0.382993i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 19.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.e even 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 25.2.e.a 8
3.b odd 2 1 225.2.m.a 8
4.b odd 2 1 400.2.y.c 8
5.b even 2 1 125.2.e.b 8
5.c odd 4 2 125.2.d.b 16
25.d even 5 1 125.2.e.b 8
25.d even 5 1 625.2.b.c 8
25.d even 5 1 625.2.e.a 8
25.d even 5 1 625.2.e.i 8
25.e even 10 1 inner 25.2.e.a 8
25.e even 10 1 625.2.b.c 8
25.e even 10 1 625.2.e.a 8
25.e even 10 1 625.2.e.i 8
25.f odd 20 2 125.2.d.b 16
25.f odd 20 2 625.2.a.f 8
25.f odd 20 4 625.2.d.o 16
75.h odd 10 1 225.2.m.a 8
75.l even 20 2 5625.2.a.x 8
100.h odd 10 1 400.2.y.c 8
100.l even 20 2 10000.2.a.bj 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.2.e.a 8 1.a even 1 1 trivial
25.2.e.a 8 25.e even 10 1 inner
125.2.d.b 16 5.c odd 4 2
125.2.d.b 16 25.f odd 20 2
125.2.e.b 8 5.b even 2 1
125.2.e.b 8 25.d even 5 1
225.2.m.a 8 3.b odd 2 1
225.2.m.a 8 75.h odd 10 1
400.2.y.c 8 4.b odd 2 1
400.2.y.c 8 100.h odd 10 1
625.2.a.f 8 25.f odd 20 2
625.2.b.c 8 25.d even 5 1
625.2.b.c 8 25.e even 10 1
625.2.d.o 16 25.f odd 20 4
625.2.e.a 8 25.d even 5 1
625.2.e.a 8 25.e even 10 1
625.2.e.i 8 25.d even 5 1
625.2.e.i 8 25.e even 10 1
5625.2.a.x 8 75.l even 20 2
10000.2.a.bj 8 100.l even 20 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(25, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 5 T + 15 T^{2} + 30 T^{3} + 41 T^{4} + 30 T^{5} - 20 T^{6} - 110 T^{7} - 199 T^{8} - 220 T^{9} - 80 T^{10} + 240 T^{11} + 656 T^{12} + 960 T^{13} + 960 T^{14} + 640 T^{15} + 256 T^{16}$$
$3$ $$1 + 5 T + 15 T^{2} + 30 T^{3} + 36 T^{4} + 5 T^{5} - 120 T^{6} - 400 T^{7} - 809 T^{8} - 1200 T^{9} - 1080 T^{10} + 135 T^{11} + 2916 T^{12} + 7290 T^{13} + 10935 T^{14} + 10935 T^{15} + 6561 T^{16}$$
$5$ $$1 + 5 T^{2} - 20 T^{3} + 5 T^{4} - 100 T^{5} + 125 T^{6} + 625 T^{8}$$
$7$ $$1 - 35 T^{2} + 611 T^{4} - 7045 T^{6} + 57976 T^{8} - 345205 T^{10} + 1467011 T^{12} - 4117715 T^{14} + 5764801 T^{16}$$
$11$ $$( 1 + 2 T - 7 T^{2} - 36 T^{3} + 5 T^{4} - 396 T^{5} - 847 T^{6} + 2662 T^{7} + 14641 T^{8} )^{2}$$
$13$ $$1 + 5 T + 30 T^{2} + 60 T^{3} + 346 T^{4} + 655 T^{5} + 6335 T^{6} + 13320 T^{7} + 88856 T^{8} + 173160 T^{9} + 1070615 T^{10} + 1439035 T^{11} + 9882106 T^{12} + 22277580 T^{13} + 144804270 T^{14} + 313742585 T^{15} + 815730721 T^{16}$$
$17$ $$1 + 10 T + 90 T^{2} + 720 T^{3} + 4451 T^{4} + 26310 T^{5} + 136780 T^{6} + 638720 T^{7} + 2802941 T^{8} + 10858240 T^{9} + 39529420 T^{10} + 129261030 T^{11} + 371751971 T^{12} + 1022297040 T^{13} + 2172381210 T^{14} + 4103386730 T^{15} + 6975757441 T^{16}$$
$19$ $$1 + 5 T - 8 T^{2} + 40 T^{3} + 878 T^{4} + 1705 T^{5} - 1861 T^{6} + 31550 T^{7} + 293380 T^{8} + 599450 T^{9} - 671821 T^{10} + 11694595 T^{11} + 114421838 T^{12} + 99043960 T^{13} - 376367048 T^{14} + 4469358695 T^{15} + 16983563041 T^{16}$$
$23$ $$1 - 5 T + 45 T^{2} + 100 T^{3} - 104 T^{4} + 7645 T^{5} + 18890 T^{6} + 8420 T^{7} + 1238691 T^{8} + 193660 T^{9} + 9992810 T^{10} + 93016715 T^{11} - 29103464 T^{12} + 643634300 T^{13} + 6661615005 T^{14} - 17024127235 T^{15} + 78310985281 T^{16}$$
$29$ $$1 + 5 T - 28 T^{2} + 140 T^{3} + 1268 T^{4} - 5045 T^{5} + 48219 T^{6} + 207000 T^{7} - 1738520 T^{8} + 6003000 T^{9} + 40552179 T^{10} - 123042505 T^{11} + 896832308 T^{12} + 2871560860 T^{13} - 16655052988 T^{14} + 86249381545 T^{15} + 500246412961 T^{16}$$
$31$ $$1 + 9 T + 55 T^{2} + 390 T^{3} + 2980 T^{4} + 20297 T^{5} + 114748 T^{6} + 589990 T^{7} + 3582095 T^{8} + 18289690 T^{9} + 110272828 T^{10} + 604667927 T^{11} + 2752092580 T^{12} + 11165368890 T^{13} + 48812702455 T^{14} + 247613526999 T^{15} + 852891037441 T^{16}$$
$37$ $$1 - 30 T + 480 T^{2} - 5675 T^{3} + 56171 T^{4} - 489630 T^{5} + 3826535 T^{6} - 26904445 T^{7} + 171416106 T^{8} - 995464465 T^{9} + 5238526415 T^{10} - 24801228390 T^{11} + 105273497531 T^{12} - 393526955975 T^{13} + 1231548676320 T^{14} - 2847956313990 T^{15} + 3512479453921 T^{16}$$
$41$ $$1 + 4 T - 30 T^{2} - 240 T^{3} + 195 T^{4} + 18372 T^{5} + 77388 T^{6} - 230240 T^{7} - 1438395 T^{8} - 9439840 T^{9} + 130089228 T^{10} + 1266216612 T^{11} + 551023395 T^{12} - 27805488240 T^{13} - 142503127230 T^{14} + 779017095524 T^{15} + 7984925229121 T^{16}$$
$43$ $$1 - 215 T^{2} + 22911 T^{4} - 1578205 T^{6} + 78597176 T^{8} - 2918101045 T^{10} + 78328149711 T^{12} - 1359093055535 T^{14} + 11688200277601 T^{16}$$
$47$ $$1 + 110 T^{2} - 90 T^{3} + 4101 T^{4} - 9900 T^{5} - 3830 T^{6} - 910260 T^{7} - 5610889 T^{8} - 42782220 T^{9} - 8460470 T^{10} - 1027847700 T^{11} + 20011571781 T^{12} - 20641050630 T^{13} + 1185713686190 T^{14} + 23811286661761 T^{16}$$
$53$ $$1 + 10 T + 100 T^{2} + 1625 T^{3} + 11531 T^{4} + 95310 T^{5} + 857875 T^{6} + 5635035 T^{7} + 42472426 T^{8} + 298656855 T^{9} + 2409770875 T^{10} + 14189466870 T^{11} + 90985136411 T^{12} + 679567676125 T^{13} + 2216436112900 T^{14} + 11747111398370 T^{15} + 62259690411361 T^{16}$$
$59$ $$1 - 118 T^{2} + 900 T^{3} + 3393 T^{4} - 96300 T^{5} + 615034 T^{6} + 2943900 T^{7} - 65890945 T^{8} + 173690100 T^{9} + 2140933354 T^{10} - 19777997700 T^{11} + 41114205873 T^{12} + 643431869100 T^{13} - 4977302969638 T^{14} + 146830437604321 T^{16}$$
$61$ $$1 + 9 T - 165 T^{2} - 1800 T^{3} + 7560 T^{4} + 154437 T^{5} + 526138 T^{6} - 4559670 T^{7} - 69838275 T^{8} - 278139870 T^{9} + 1957759498 T^{10} + 35054264697 T^{11} + 104674557960 T^{12} - 1520273341800 T^{13} - 8500861769565 T^{14} + 28284685524189 T^{15} + 191707312997281 T^{16}$$
$67$ $$1 - 20 T + 250 T^{2} - 2600 T^{3} + 26091 T^{4} - 241820 T^{5} + 2034700 T^{6} - 15361680 T^{7} + 117317461 T^{8} - 1029232560 T^{9} + 9133768300 T^{10} - 72730508660 T^{11} + 525762898011 T^{12} - 3510325278200 T^{13} + 22614595542250 T^{14} - 121214232106460 T^{15} + 406067677556641 T^{16}$$
$71$ $$1 - 6 T - 910 T^{3} + 4875 T^{4} + 34402 T^{5} + 474398 T^{6} - 3835500 T^{7} - 25760305 T^{8} - 272320500 T^{9} + 2391440318 T^{10} + 12312854222 T^{11} + 123881944875 T^{12} - 1641848709410 T^{13} - 54570720950346 T^{15} + 645753531245761 T^{16}$$
$73$ $$1 - 15 T + 195 T^{2} - 1340 T^{3} + 15786 T^{4} - 132465 T^{5} + 1900340 T^{6} - 14658400 T^{7} + 154250461 T^{8} - 1070063200 T^{9} + 10126911860 T^{10} - 51531136905 T^{11} + 448294632426 T^{12} - 2777915934620 T^{13} + 29510174126355 T^{14} - 165710977786455 T^{15} + 806460091894081 T^{16}$$
$79$ $$1 - 15 T - 58 T^{2} + 2180 T^{3} - 7802 T^{4} - 148865 T^{5} + 1559169 T^{6} + 5584500 T^{7} - 169067020 T^{8} + 441175500 T^{9} + 9730773729 T^{10} - 73396250735 T^{11} - 303888531962 T^{12} + 6707982949820 T^{13} - 14099072420218 T^{14} - 288058634792385 T^{15} + 1517108809906561 T^{16}$$
$83$ $$1 + 45 T + 1115 T^{2} + 19890 T^{3} + 284376 T^{4} + 3450645 T^{5} + 37197280 T^{6} + 367888080 T^{7} + 3431240591 T^{8} + 30534710640 T^{9} + 256252061920 T^{10} + 1973033952615 T^{11} + 13496007492696 T^{12} + 78347518389270 T^{13} + 364538516306435 T^{14} + 1221122294533215 T^{15} + 2252292232139041 T^{16}$$
$89$ $$1 + 25 T + 342 T^{2} + 5000 T^{3} + 78868 T^{4} + 928525 T^{5} + 9098049 T^{6} + 101226750 T^{7} + 1076434080 T^{8} + 9009180750 T^{9} + 72065646129 T^{10} + 654581340725 T^{11} + 4948355063188 T^{12} + 27920297245000 T^{13} + 169967601508662 T^{14} + 1105783372388225 T^{15} + 3936588805702081 T^{16}$$
$97$ $$1 + 60 T + 1830 T^{2} + 35660 T^{3} + 467711 T^{4} + 3770460 T^{5} + 6583460 T^{6} - 292271720 T^{7} - 4451764059 T^{8} - 28350356840 T^{9} + 61943775140 T^{10} + 3441197039580 T^{11} + 41406118545791 T^{12} + 306224553564620 T^{13} + 1524338769020070 T^{14} + 4847897068686780 T^{15} + 7837433594376961 T^{16}$$