Properties

 Label 625.2.b.a Level $625$ Weight $2$ Character orbit 625.b Analytic conductor $4.991$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$625 = 5^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 625.b (of order $$2$$, degree $$1$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$4.99065012633$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ Defining polynomial: $$x^{4} + 3 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 25) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 3 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 1$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} + 2 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 1$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} - 2 \beta_{1}$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/625\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
624.1
 − 1.61803i − 0.618034i 0.618034i 1.61803i
1.61803i 1.00000i −0.618034 0 −1.61803 0.618034i 2.23607i 2.00000 0
624.2 0.618034i 1.00000i 1.61803 0 0.618034 1.61803i 2.23607i 2.00000 0
624.3 0.618034i 1.00000i 1.61803 0 0.618034 1.61803i 2.23607i 2.00000 0
624.4 1.61803i 1.00000i −0.618034 0 −1.61803 0.618034i 2.23607i 2.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

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 625.2.b.a 4
5.b even 2 1 inner 625.2.b.a 4
5.c odd 4 1 625.2.a.b 2
5.c odd 4 1 625.2.a.c 2
15.e even 4 1 5625.2.a.d 2
15.e even 4 1 5625.2.a.f 2
20.e even 4 1 10000.2.a.c 2
20.e even 4 1 10000.2.a.l 2
25.d even 5 2 125.2.e.a 8
25.d even 5 2 625.2.e.c 8
25.e even 10 2 125.2.e.a 8
25.e even 10 2 625.2.e.c 8
25.f odd 20 2 25.2.d.a 4
25.f odd 20 2 125.2.d.a 4
25.f odd 20 2 625.2.d.b 4
25.f odd 20 2 625.2.d.h 4
75.l even 20 2 225.2.h.b 4
100.l even 20 2 400.2.u.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.2.d.a 4 25.f odd 20 2
125.2.d.a 4 25.f odd 20 2
125.2.e.a 8 25.d even 5 2
125.2.e.a 8 25.e even 10 2
225.2.h.b 4 75.l even 20 2
400.2.u.b 4 100.l even 20 2
625.2.a.b 2 5.c odd 4 1
625.2.a.c 2 5.c odd 4 1
625.2.b.a 4 1.a even 1 1 trivial
625.2.b.a 4 5.b even 2 1 inner
625.2.d.b 4 25.f odd 20 2
625.2.d.h 4 25.f odd 20 2
625.2.e.c 8 25.d even 5 2
625.2.e.c 8 25.e even 10 2
5625.2.a.d 2 15.e even 4 1
5625.2.a.f 2 15.e even 4 1
10000.2.a.c 2 20.e even 4 1
10000.2.a.l 2 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(625, [\chi])$$:

 $$T_{2}^{4} + 3 T_{2}^{2} + 1$$ $$T_{3}^{2} + 1$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 3 T^{2} + T^{4}$$
$3$ $$( 1 + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$1 + 3 T^{2} + T^{4}$$
$11$ $$( 4 + 6 T + T^{2} )^{2}$$
$13$ $$81 + 27 T^{2} + T^{4}$$
$17$ $$16 + 28 T^{2} + T^{4}$$
$19$ $$( -5 - 5 T + T^{2} )^{2}$$
$23$ $$961 + 82 T^{2} + T^{4}$$
$29$ $$( 5 - 5 T + T^{2} )^{2}$$
$31$ $$( 3 + T )^{4}$$
$37$ $$1 + 18 T^{2} + T^{4}$$
$41$ $$( 4 + 6 T + T^{2} )^{2}$$
$43$ $$81 + 27 T^{2} + T^{4}$$
$47$ $$1 + 3 T^{2} + T^{4}$$
$53$ $$361 + 42 T^{2} + T^{4}$$
$59$ $$( 45 - 15 T + T^{2} )^{2}$$
$61$ $$( -41 - 4 T + T^{2} )^{2}$$
$67$ $$1936 + 108 T^{2} + T^{4}$$
$71$ $$( 29 + 11 T + T^{2} )^{2}$$
$73$ $$( 81 + T^{2} )^{2}$$
$79$ $$( -25 - 5 T + T^{2} )^{2}$$
$83$ $$121 + 42 T^{2} + T^{4}$$
$89$ $$( -80 + T^{2} )^{2}$$
$97$ $$121 + 23 T^{2} + T^{4}$$