# Properties

 Label 1250.2.b.e Level $1250$ Weight $2$ Character orbit 1250.b Analytic conductor $9.981$ Analytic rank $0$ Dimension $8$ 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: $$8$$ Coefficient field: 8.0.14884000000.15 Defining polynomial: $$x^{8} + 19 x^{6} + 121 x^{4} + 304 x^{2} + 256$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 19 x^{6} + 121 x^{4} + 304 x^{2} + 256$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{7} + 19 \nu^{5} + 121 \nu^{3} + 240 \nu$$$$)/64$$ $$\beta_{3}$$ $$=$$ $$($$$$-5 \nu^{7} - 79 \nu^{5} - 365 \nu^{3} - 480 \nu$$$$)/64$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{6} + 19 \nu^{4} + 105 \nu^{2} + 144$$$$)/16$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{7} + 19 \nu^{5} + 105 \nu^{3} + 144 \nu$$$$)/16$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{6} - 15 \nu^{4} - 61 \nu^{2} - 64$$$$)/4$$ $$\beta_{7}$$ $$=$$ $$($$$$5 \nu^{6} + 79 \nu^{4} + 365 \nu^{2} + 480$$$$)/16$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{7} + \beta_{6} - \beta_{4} - 5$$ $$\nu^{3}$$ $$=$$ $$-\beta_{5} + 4 \beta_{2} - 6 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$-11 \beta_{7} - 10 \beta_{6} + 15 \beta_{4} + 35$$ $$\nu^{5}$$ $$=$$ $$15 \beta_{5} + 4 \beta_{3} - 40 \beta_{2} + 45 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$104 \beta_{7} + 85 \beta_{6} - 164 \beta_{4} - 284$$ $$\nu^{7}$$ $$=$$ $$-164 \beta_{5} - 76 \beta_{3} + 340 \beta_{2} - 369 \beta_{1}$$

## 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
 − 2.33275i − 1.34841i 1.71472i 2.96645i − 2.96645i − 1.71472i 1.34841i 2.33275i
1.00000i 2.33275i −1.00000 0 −2.33275 3.77447i 1.00000i −2.44172 0
1249.2 1.00000i 1.34841i −1.00000 0 −1.34841 0.833366i 1.00000i 1.18178 0
1249.3 1.00000i 1.71472i −1.00000 0 1.71472 2.77447i 1.00000i 0.0597522 0
1249.4 1.00000i 2.96645i −1.00000 0 2.96645 1.83337i 1.00000i −5.79981 0
1249.5 1.00000i 2.96645i −1.00000 0 2.96645 1.83337i 1.00000i −5.79981 0
1249.6 1.00000i 1.71472i −1.00000 0 1.71472 2.77447i 1.00000i 0.0597522 0
1249.7 1.00000i 1.34841i −1.00000 0 −1.34841 0.833366i 1.00000i 1.18178 0
1249.8 1.00000i 2.33275i −1.00000 0 −2.33275 3.77447i 1.00000i −2.44172 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1249.8 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.e 8
5.b even 2 1 inner 1250.2.b.e 8
5.c odd 4 1 1250.2.a.f 4
5.c odd 4 1 1250.2.a.l 4
20.e even 4 1 10000.2.a.t 4
20.e even 4 1 10000.2.a.x 4
25.d even 5 2 250.2.e.c 16
25.e even 10 2 250.2.e.c 16
25.f odd 20 2 50.2.d.b 8
25.f odd 20 2 250.2.d.d 8
75.l even 20 2 450.2.h.e 8
100.l even 20 2 400.2.u.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.2.d.b 8 25.f odd 20 2
250.2.d.d 8 25.f odd 20 2
250.2.e.c 16 25.d even 5 2
250.2.e.c 16 25.e even 10 2
400.2.u.d 8 100.l even 20 2
450.2.h.e 8 75.l even 20 2
1250.2.a.f 4 5.c odd 4 1
1250.2.a.l 4 5.c odd 4 1
1250.2.b.e 8 1.a even 1 1 trivial
1250.2.b.e 8 5.b even 2 1 inner
10000.2.a.t 4 20.e even 4 1
10000.2.a.x 4 20.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{8} + 19 T_{3}^{6} + 121 T_{3}^{4} + 304 T_{3}^{2} + 256$$ acting on $$S_{2}^{\mathrm{new}}(1250, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{4}$$
$3$ $$256 + 304 T^{2} + 121 T^{4} + 19 T^{6} + T^{8}$$
$5$ $$T^{8}$$
$7$ $$256 + 496 T^{2} + 201 T^{4} + 26 T^{6} + T^{8}$$
$11$ $$( 16 - 12 T - 11 T^{2} + 2 T^{3} + T^{4} )^{2}$$
$13$ $$39601 + 13829 T^{2} + 1576 T^{4} + 69 T^{6} + T^{8}$$
$17$ $$11881 + 8506 T^{2} + 1351 T^{4} + 66 T^{6} + T^{8}$$
$19$ $$( 80 + 20 T - 35 T^{2} - 5 T^{3} + T^{4} )^{2}$$
$23$ $$256 + 944 T^{2} + 841 T^{4} + 74 T^{6} + T^{8}$$
$29$ $$( 5 + 105 T + 70 T^{2} + 15 T^{3} + T^{4} )^{2}$$
$31$ $$( -1264 - 432 T - T^{2} + 12 T^{3} + T^{4} )^{2}$$
$37$ $$5041 + 7706 T^{2} + 2951 T^{4} + 106 T^{6} + T^{8}$$
$41$ $$( -89 + 23 T + 34 T^{2} - 13 T^{3} + T^{4} )^{2}$$
$43$ $$30976 + 17824 T^{2} + 2641 T^{4} + 114 T^{6} + T^{8}$$
$47$ $$256 + 496 T^{2} + 201 T^{4} + 26 T^{6} + T^{8}$$
$53$ $$65536 + 240384 T^{2} + 13121 T^{4} + 219 T^{6} + T^{8}$$
$59$ $$( -320 + 560 T - 140 T^{2} + T^{4} )^{2}$$
$61$ $$( -1709 + 958 T - 101 T^{2} - 8 T^{3} + T^{4} )^{2}$$
$67$ $$891136 + 457456 T^{2} + 28041 T^{4} + 346 T^{6} + T^{8}$$
$71$ $$( -64 + 168 T + 129 T^{2} + 22 T^{3} + T^{4} )^{2}$$
$73$ $$1175056 + 221404 T^{2} + 13381 T^{4} + 279 T^{6} + T^{8}$$
$79$ $$( -320 + 240 T - 20 T^{2} - 10 T^{3} + T^{4} )^{2}$$
$83$ $$13424896 + 1167904 T^{2} + 31721 T^{4} + 314 T^{6} + T^{8}$$
$89$ $$( -3100 + 1500 T - 165 T^{2} - 5 T^{3} + T^{4} )^{2}$$
$97$ $$( 1 + 123 T^{2} + T^{4} )^{2}$$