# Properties

 Label 1250.2.b.b Level $1250$ Weight $2$ Character orbit 1250.b Analytic conductor $9.981$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1250 = 2 \cdot 5^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1250.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.98130025266$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ Defining polynomial: $$x^{4} + 3 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 50) 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.

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

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 $$\left(\mathbb{Z}/1250\mathbb{Z}\right)^\times$$.

 $$n$$ $$627$$ $$\chi(n)$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1249.1
 − 0.618034i 1.61803i − 1.61803i 0.618034i
1.00000i 0.381966i −1.00000 0 0.381966 3.00000i 1.00000i 2.85410 0
1249.2 1.00000i 2.61803i −1.00000 0 2.61803 3.00000i 1.00000i −3.85410 0
1249.3 1.00000i 2.61803i −1.00000 0 2.61803 3.00000i 1.00000i −3.85410 0
1249.4 1.00000i 0.381966i −1.00000 0 0.381966 3.00000i 1.00000i 2.85410 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 1250.2.b.b 4
5.b even 2 1 inner 1250.2.b.b 4
5.c odd 4 1 1250.2.a.a 2
5.c odd 4 1 1250.2.a.d 2
20.e even 4 1 10000.2.a.a 2
20.e even 4 1 10000.2.a.n 2
25.d even 5 2 250.2.e.b 8
25.e even 10 2 250.2.e.b 8
25.f odd 20 2 50.2.d.a 4
25.f odd 20 2 250.2.d.a 4
75.l even 20 2 450.2.h.a 4
100.l even 20 2 400.2.u.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.2.d.a 4 25.f odd 20 2
250.2.d.a 4 25.f odd 20 2
250.2.e.b 8 25.d even 5 2
250.2.e.b 8 25.e even 10 2
400.2.u.c 4 100.l even 20 2
450.2.h.a 4 75.l even 20 2
1250.2.a.a 2 5.c odd 4 1
1250.2.a.d 2 5.c odd 4 1
1250.2.b.b 4 1.a even 1 1 trivial
1250.2.b.b 4 5.b even 2 1 inner
10000.2.a.a 2 20.e even 4 1
10000.2.a.n 2 20.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 7 T_{3}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(1250, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{2}$$
$3$ $$1 + 7 T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$( 9 + T^{2} )^{2}$$
$11$ $$( -1 - 4 T + T^{2} )^{2}$$
$13$ $$( 1 + T^{2} )^{2}$$
$17$ $$81 + 63 T^{2} + T^{4}$$
$19$ $$( -5 + 5 T + T^{2} )^{2}$$
$23$ $$121 + 42 T^{2} + T^{4}$$
$29$ $$( 5 - 10 T + T^{2} )^{2}$$
$31$ $$( -1 - 4 T + T^{2} )^{2}$$
$37$ $$3721 + 123 T^{2} + T^{4}$$
$41$ $$( -11 + 6 T + T^{2} )^{2}$$
$43$ $$121 + 42 T^{2} + T^{4}$$
$47$ $$5041 + 178 T^{2} + T^{4}$$
$53$ $$256 + 112 T^{2} + T^{4}$$
$59$ $$( -20 + T^{2} )^{2}$$
$61$ $$( 19 + 11 T + T^{2} )^{2}$$
$67$ $$3481 + 138 T^{2} + T^{4}$$
$71$ $$( 3 + T )^{4}$$
$73$ $$1936 + 92 T^{2} + T^{4}$$
$79$ $$( 20 - 10 T + T^{2} )^{2}$$
$83$ $$3721 + 202 T^{2} + T^{4}$$
$89$ $$( -20 + T^{2} )^{2}$$
$97$ $$10201 + 203 T^{2} + T^{4}$$