# Properties

 Label 250.2.e.b Level $250$ Weight $2$ Character orbit 250.e Analytic conductor $1.996$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$250 = 2 \cdot 5^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 250.e (of order $$10$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.99626005053$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{10})$$ Coefficient field: $$\Q(\zeta_{20})$$ Defining polynomial: $$x^{8} - x^{6} + x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 50) Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{20}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/250\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$\chi(n)$$ $$\zeta_{20}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 −0.587785 + 0.809017i 0.587785 − 0.809017i −0.951057 + 0.309017i 0.951057 − 0.309017i −0.951057 − 0.309017i 0.951057 + 0.309017i −0.587785 − 0.809017i 0.587785 + 0.809017i
−0.587785 + 0.809017i −2.48990 + 0.809017i −0.309017 0.951057i 0 0.809017 2.48990i 3.00000i 0.951057 + 0.309017i 3.11803 2.26538i 0
49.2 0.587785 0.809017i 2.48990 0.809017i −0.309017 0.951057i 0 0.809017 2.48990i 3.00000i −0.951057 0.309017i 3.11803 2.26538i 0
99.1 −0.951057 + 0.309017i 0.224514 + 0.309017i 0.809017 0.587785i 0 −0.309017 0.224514i 3.00000i −0.587785 + 0.809017i 0.881966 2.71441i 0
99.2 0.951057 0.309017i −0.224514 0.309017i 0.809017 0.587785i 0 −0.309017 0.224514i 3.00000i 0.587785 0.809017i 0.881966 2.71441i 0
149.1 −0.951057 0.309017i 0.224514 0.309017i 0.809017 + 0.587785i 0 −0.309017 + 0.224514i 3.00000i −0.587785 0.809017i 0.881966 + 2.71441i 0
149.2 0.951057 + 0.309017i −0.224514 + 0.309017i 0.809017 + 0.587785i 0 −0.309017 + 0.224514i 3.00000i 0.587785 + 0.809017i 0.881966 + 2.71441i 0
199.1 −0.587785 0.809017i −2.48990 0.809017i −0.309017 + 0.951057i 0 0.809017 + 2.48990i 3.00000i 0.951057 0.309017i 3.11803 + 2.26538i 0
199.2 0.587785 + 0.809017i 2.48990 + 0.809017i −0.309017 + 0.951057i 0 0.809017 + 2.48990i 3.00000i −0.951057 + 0.309017i 3.11803 + 2.26538i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 199.2 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
25.d even 5 1 inner
25.e even 10 1 inner

## Twists

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

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.2.d.a 4 5.c odd 4 1
50.2.d.a 4 25.f odd 20 1
250.2.d.a 4 5.c odd 4 1
250.2.d.a 4 25.f odd 20 1
250.2.e.b 8 1.a even 1 1 trivial
250.2.e.b 8 5.b even 2 1 inner
250.2.e.b 8 25.d even 5 1 inner
250.2.e.b 8 25.e even 10 1 inner
400.2.u.c 4 20.e even 4 1
400.2.u.c 4 100.l even 20 1
450.2.h.a 4 15.e even 4 1
450.2.h.a 4 75.l even 20 1
1250.2.a.a 2 25.f odd 20 1
1250.2.a.d 2 25.f odd 20 1
1250.2.b.b 4 25.d even 5 1
1250.2.b.b 4 25.e even 10 1
10000.2.a.a 2 100.l even 20 1
10000.2.a.n 2 100.l even 20 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{8} - 11 T_{3}^{6} + 46 T_{3}^{4} + 4 T_{3}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(250, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + T^{4} - T^{6} + T^{8}$$
$3$ $$1 + 4 T^{2} + 46 T^{4} - 11 T^{6} + T^{8}$$
$5$ $$T^{8}$$
$7$ $$( 9 + T^{2} )^{4}$$
$11$ $$( 1 - 7 T + 19 T^{2} - 3 T^{3} + T^{4} )^{2}$$
$13$ $$1 - T^{2} + T^{4} - T^{6} + T^{8}$$
$17$ $$6561 + 2916 T^{2} + 3726 T^{4} - 99 T^{6} + T^{8}$$
$19$ $$( 25 - 25 T + 40 T^{2} - 10 T^{3} + T^{4} )^{2}$$
$23$ $$14641 - 7381 T^{2} + 1401 T^{4} + 19 T^{6} + T^{8}$$
$29$ $$( 25 - 25 T + 85 T^{2} + 15 T^{3} + T^{4} )^{2}$$
$31$ $$( 1 - 7 T + 19 T^{2} - 3 T^{3} + T^{4} )^{2}$$
$37$ $$13845841 - 163724 T^{2} + 3966 T^{4} - 79 T^{6} + T^{8}$$
$41$ $$( 121 - 77 T + 69 T^{2} - 13 T^{3} + T^{4} )^{2}$$
$43$ $$( 121 + 42 T^{2} + T^{4} )^{2}$$
$47$ $$25411681 - 1053569 T^{2} + 16561 T^{4} + 31 T^{6} + T^{8}$$
$53$ $$65536 - 45056 T^{2} + 11776 T^{4} + 64 T^{6} + T^{8}$$
$59$ $$( 400 - 200 T + 60 T^{2} - 10 T^{3} + T^{4} )^{2}$$
$61$ $$( 361 - 247 T + 64 T^{2} + 2 T^{3} + T^{4} )^{2}$$
$67$ $$12117361 - 518669 T^{2} + 8601 T^{4} + 11 T^{6} + T^{8}$$
$71$ $$( 81 - 27 T + 9 T^{2} - 3 T^{3} + T^{4} )^{2}$$
$73$ $$3748096 - 30976 T^{2} + 2656 T^{4} - 76 T^{6} + T^{8}$$
$79$ $$( 400 + 40 T^{2} + 10 T^{3} + T^{4} )^{2}$$
$83$ $$13845841 + 293959 T^{2} + 29641 T^{4} - 281 T^{6} + T^{8}$$
$89$ $$( 400 + 200 T + 60 T^{2} + 10 T^{3} + T^{4} )^{2}$$
$97$ $$104060401 - 805879 T^{2} + 10606 T^{4} - 124 T^{6} + T^{8}$$