# Properties

 Label 4025.2.a.q Level 4025 Weight 2 Character orbit 4025.a Self dual Yes Analytic conductor 32.140 Analytic rank 1 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: $$1$$ Dimension: $$5$$ Coefficient field: 5.5.255877.1 Coefficient ring: $$\Z[a_1, a_2]$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5}\mathstrut -\mathstrut$$ $$x^{4}\mathstrut -\mathstrut$$ $$6$$ $$x^{3}\mathstrut +\mathstrut$$ $$6$$ $$x^{2}\mathstrut +\mathstrut$$ $$6$$ $$x\mathstrut -\mathstrut$$ $$5$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - 4 \nu + 1$$ $$\beta_{4}$$ $$=$$ $$\nu^{4} - 5 \nu^{2} + 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$3$$ $$\nu^{3}$$ $$=$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$4$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$1$$ $$\nu^{4}$$ $$=$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$5$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$12$$

## 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
 −2.19548 −1.06459 0.698160 1.45275 2.10917
−2.19548 3.13309 2.82015 0 −6.87865 −1.00000 −1.80062 6.81626 0
1.2 −1.06459 −0.382286 −0.866643 0 0.406979 −1.00000 3.05181 −2.85386 0
1.3 0.698160 1.80045 −1.51257 0 1.25700 −1.00000 −2.45234 0.241606 0
1.4 1.45275 −2.09827 0.110473 0 −3.04825 −1.00000 −2.74500 1.40273 0
1.5 2.10917 1.54702 2.44859 0 3.26292 −1.00000 0.946160 −0.606735 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4025))$$:

 $$T_{2}^{5}$$ $$\mathstrut -\mathstrut T_{2}^{4}$$ $$\mathstrut -\mathstrut 6 T_{2}^{3}$$ $$\mathstrut +\mathstrut 6 T_{2}^{2}$$ $$\mathstrut +\mathstrut 6 T_{2}$$ $$\mathstrut -\mathstrut 5$$ $$T_{3}^{5}$$ $$\mathstrut -\mathstrut 4 T_{3}^{4}$$ $$\mathstrut -\mathstrut 2 T_{3}^{3}$$ $$\mathstrut +\mathstrut 19 T_{3}^{2}$$ $$\mathstrut -\mathstrut 11 T_{3}$$ $$\mathstrut -\mathstrut 7$$ $$T_{11}^{5}$$ $$\mathstrut +\mathstrut 7 T_{11}^{4}$$ $$\mathstrut -\mathstrut 4 T_{11}^{3}$$ $$\mathstrut -\mathstrut 107 T_{11}^{2}$$ $$\mathstrut -\mathstrut 156 T_{11}$$ $$\mathstrut +\mathstrut 68$$