# Properties

 Label 5625.2.a.x Level $5625$ Weight $2$ Character orbit 5625.a Self dual yes Analytic conductor $44.916$ Analytic rank $1$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5625 = 3^{2} \cdot 5^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5625.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$44.9158511370$$ Analytic rank: $$1$$ Dimension: $$8$$ Coefficient field: 8.8.14884000000.2 Defining polynomial: $$x^{8} - 11 x^{6} + 36 x^{4} - 31 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 25) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + ( -\beta_{1} - \beta_{6} + \beta_{7} ) q^{7} + ( \beta_{6} + \beta_{7} ) q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + ( -\beta_{1} - \beta_{6} + \beta_{7} ) q^{7} + ( \beta_{6} + \beta_{7} ) q^{8} -2 q^{11} + \beta_{5} q^{13} + ( -3 - \beta_{3} - 2 \beta_{4} ) q^{14} + ( -\beta_{2} + \beta_{3} ) q^{16} + ( \beta_{1} + \beta_{6} - 2 \beta_{7} ) q^{17} + ( 2 - \beta_{2} + 2 \beta_{4} ) q^{19} -2 \beta_{1} q^{22} + ( -\beta_{5} - \beta_{6} ) q^{23} + ( 1 - \beta_{2} + \beta_{3} + 3 \beta_{4} ) q^{26} + ( -\beta_{1} - 3 \beta_{5} - 3 \beta_{7} ) q^{28} + ( -4 + 2 \beta_{2} - \beta_{3} - 3 \beta_{4} ) q^{29} + ( 1 - 2 \beta_{2} - \beta_{3} ) q^{31} + ( -\beta_{1} + \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{32} + ( 2 - \beta_{2} + \beta_{3} + 3 \beta_{4} ) q^{34} + ( \beta_{1} - \beta_{5} - 4 \beta_{7} ) q^{37} + ( \beta_{1} + 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{38} + ( -2 - \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{41} + ( -\beta_{1} - \beta_{5} + 2 \beta_{6} + 3 \beta_{7} ) q^{43} + ( -2 - 2 \beta_{2} ) q^{44} + ( -2 + \beta_{2} - 2 \beta_{3} - 4 \beta_{4} ) q^{46} + ( -3 \beta_{1} + 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{47} + ( -1 - \beta_{2} + 2 \beta_{3} ) q^{49} + ( 2 \beta_{5} + \beta_{6} ) q^{52} + ( -3 \beta_{1} - 2 \beta_{5} + 2 \beta_{6} - 3 \beta_{7} ) q^{53} + ( -3 - \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{56} + ( -2 \beta_{1} - 4 \beta_{5} + \beta_{7} ) q^{58} + ( -4 - \beta_{2} - 2 \beta_{3} + 2 \beta_{4} ) q^{59} + ( 3 + \beta_{2} + 3 \beta_{3} + \beta_{4} ) q^{61} + ( -\beta_{1} - \beta_{5} - 4 \beta_{6} - 3 \beta_{7} ) q^{62} + ( -5 - 2 \beta_{2} - 2 \beta_{3} + 4 \beta_{4} ) q^{64} + ( 4 \beta_{1} + 2 \beta_{6} - 2 \beta_{7} ) q^{67} + ( -\beta_{1} + 4 \beta_{5} - \beta_{6} + 4 \beta_{7} ) q^{68} + ( -7 - 3 \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{71} + ( \beta_{1} - 2 \beta_{5} + 2 \beta_{6} - 3 \beta_{7} ) q^{73} + ( -2 - 2 \beta_{2} - \beta_{3} + \beta_{4} ) q^{74} + ( -1 + \beta_{3} + 2 \beta_{4} ) q^{76} + ( 2 \beta_{1} + 2 \beta_{6} - 2 \beta_{7} ) q^{77} + ( -2 + 3 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} ) q^{79} + ( -3 \beta_{1} + 3 \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{82} + ( -4 \beta_{1} + \beta_{5} - 5 \beta_{6} + 4 \beta_{7} ) q^{83} + ( 1 + 3 \beta_{2} + \beta_{3} - 4 \beta_{4} ) q^{86} + ( -2 \beta_{6} - 2 \beta_{7} ) q^{88} + ( -\beta_{2} + \beta_{3} + 7 \beta_{4} ) q^{89} + ( -2 - \beta_{2} ) q^{91} + ( -\beta_{1} - 4 \beta_{5} - \beta_{6} - \beta_{7} ) q^{92} + ( -7 - 4 \beta_{2} + \beta_{3} + 4 \beta_{4} ) q^{94} + ( -4 \beta_{1} + 4 \beta_{5} + 4 \beta_{6} + 5 \beta_{7} ) q^{97} + ( -2 \beta_{1} + 2 \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 6q^{4} + O(q^{10})$$ $$8q + 6q^{4} - 16q^{11} - 12q^{14} - 2q^{16} + 10q^{19} - 6q^{26} - 20q^{29} + 16q^{31} + 2q^{34} - 26q^{41} - 12q^{44} + 6q^{46} - 14q^{49} - 10q^{56} - 30q^{59} + 6q^{61} - 44q^{64} - 46q^{71} - 12q^{74} - 20q^{76} + 10q^{79} + 14q^{86} - 30q^{89} - 14q^{91} - 68q^{94} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 11 x^{6} + 36 x^{4} - 31 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ $$\beta_{3}$$ $$=$$ $$\nu^{4} - 5 \nu^{2} + 1$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{6} - 8 \nu^{4} + 16 \nu^{2} - 7$$$$)/4$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{7} - 8 \nu^{5} + 16 \nu^{3} - 7 \nu$$$$)/4$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{7} + 12 \nu^{5} - 40 \nu^{3} + 27 \nu$$$$)/4$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{7} - 12 \nu^{5} + 44 \nu^{3} - 43 \nu$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{7} + \beta_{6} + 4 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{3} + 5 \beta_{2} + 14$$ $$\nu^{5}$$ $$=$$ $$6 \beta_{7} + 7 \beta_{6} + \beta_{5} + 19 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$4 \beta_{4} + 8 \beta_{3} + 24 \beta_{2} + 71$$ $$\nu^{7}$$ $$=$$ $$32 \beta_{7} + 40 \beta_{6} + 12 \beta_{5} + 95 \beta_{1}$$

## 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
 −2.30927 −2.08529 −1.13370 −0.183172 0.183172 1.13370 2.08529 2.30927
−2.30927 0 3.33275 0 0 3.03582 −3.07768 0 0
1.2 −2.08529 0 2.34841 0 0 −0.992398 −0.726543 0 0
1.3 −1.13370 0 −0.714715 0 0 0.407162 3.07768 0 0
1.4 −0.183172 0 −1.96645 0 0 3.26086 0.726543 0 0
1.5 0.183172 0 −1.96645 0 0 −3.26086 −0.726543 0 0
1.6 1.13370 0 −0.714715 0 0 −0.407162 −3.07768 0 0
1.7 2.08529 0 2.34841 0 0 0.992398 0.726543 0 0
1.8 2.30927 0 3.33275 0 0 −3.03582 3.07768 0 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

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$5$$ $$-1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

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

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.2.e.a 8 75.l even 20 2
125.2.d.b 16 75.h odd 10 2
125.2.d.b 16 75.j odd 10 2
125.2.e.b 8 75.l even 20 2
225.2.m.a 8 25.f odd 20 2
400.2.y.c 8 300.u odd 20 2
625.2.a.f 8 3.b odd 2 1
625.2.a.f 8 15.d odd 2 1
625.2.b.c 8 15.e even 4 2
625.2.d.o 16 75.h odd 10 2
625.2.d.o 16 75.j odd 10 2
625.2.e.a 8 75.l even 20 2
625.2.e.i 8 75.l even 20 2
5625.2.a.x 8 1.a even 1 1 trivial
5625.2.a.x 8 5.b even 2 1 inner
10000.2.a.bj 8 12.b even 2 1
10000.2.a.bj 8 60.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(5625))$$:

 $$T_{2}^{8} - 11 T_{2}^{6} + 36 T_{2}^{4} - 31 T_{2}^{2} + 1$$ $$T_{7}^{8} - 21 T_{7}^{6} + 121 T_{7}^{4} - 116 T_{7}^{2} + 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 31 T^{2} + 36 T^{4} - 11 T^{6} + T^{8}$$
$3$ $$T^{8}$$
$5$ $$T^{8}$$
$7$ $$16 - 116 T^{2} + 121 T^{4} - 21 T^{6} + T^{8}$$
$11$ $$( 2 + T )^{8}$$
$13$ $$1 - 14 T^{2} + 31 T^{4} - 14 T^{6} + T^{8}$$
$17$ $$1936 - 1636 T^{2} + 441 T^{4} - 41 T^{6} + T^{8}$$
$19$ $$( -20 + 30 T - 5 T^{2} - 5 T^{3} + T^{4} )^{2}$$
$23$ $$256 - 544 T^{2} + 301 T^{4} - 34 T^{6} + T^{8}$$
$29$ $$( -695 - 290 T - 5 T^{2} + 10 T^{3} + T^{4} )^{2}$$
$31$ $$( -44 + 328 T - 41 T^{2} - 8 T^{3} + T^{4} )^{2}$$
$37$ $$116281 - 39331 T^{2} + 3556 T^{4} - 111 T^{6} + T^{8}$$
$41$ $$( 116 - 148 T + 19 T^{2} + 13 T^{3} + T^{4} )^{2}$$
$43$ $$246016 - 56784 T^{2} + 4421 T^{4} - 129 T^{6} + T^{8}$$
$47$ $$65536 - 47616 T^{2} + 4661 T^{4} - 141 T^{6} + T^{8}$$
$53$ $$8755681 - 722619 T^{2} + 20356 T^{4} - 239 T^{6} + T^{8}$$
$59$ $$( -2020 - 630 T + 5 T^{2} + 15 T^{3} + T^{4} )^{2}$$
$61$ $$( 341 - 237 T - 146 T^{2} - 3 T^{3} + T^{4} )^{2}$$
$67$ $$246016 - 84736 T^{2} + 7776 T^{4} - 176 T^{6} + T^{8}$$
$71$ $$( -4924 - 798 T + 99 T^{2} + 23 T^{3} + T^{4} )^{2}$$
$73$ $$1 - 19 T^{2} + 76 T^{4} - 79 T^{6} + T^{8}$$
$79$ $$( 5780 - 195 T^{2} - 5 T^{3} + T^{4} )^{2}$$
$83$ $$99856 - 138164 T^{2} + 35061 T^{4} - 374 T^{6} + T^{8}$$
$89$ $$( 1180 - 530 T - 35 T^{2} + 15 T^{3} + T^{4} )^{2}$$
$97$ $$301334881 - 12038986 T^{2} + 146971 T^{4} - 666 T^{6} + T^{8}$$