# Properties

 Label 50.2.d.a Level $50$ Weight $2$ Character orbit 50.d Analytic conductor $0.399$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$50 = 2 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 50.d (of order $$5$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.399252010106$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{10}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{2} + ( -1 + \zeta_{10} + 2 \zeta_{10}^{3} ) q^{3} -\zeta_{10}^{3} q^{4} + ( 2 - 2 \zeta_{10} - \zeta_{10}^{3} ) q^{5} + ( \zeta_{10} + \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{6} -3 q^{7} -\zeta_{10}^{2} q^{8} + ( -3 + \zeta_{10} - 3 \zeta_{10}^{2} ) q^{9} +O(q^{10})$$ $$q + ( 1 - \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{2} + ( -1 + \zeta_{10} + 2 \zeta_{10}^{3} ) q^{3} -\zeta_{10}^{3} q^{4} + ( 2 - 2 \zeta_{10} - \zeta_{10}^{3} ) q^{5} + ( \zeta_{10} + \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{6} -3 q^{7} -\zeta_{10}^{2} q^{8} + ( -3 + \zeta_{10} - 3 \zeta_{10}^{2} ) q^{9} + ( -2 \zeta_{10} + \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{10} + ( -1 + 3 \zeta_{10} - 3 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{11} + ( 1 + \zeta_{10} + \zeta_{10}^{2} ) q^{12} -\zeta_{10} q^{13} + ( -3 + 3 \zeta_{10} - 3 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{14} + ( 3 + \zeta_{10} + 3 \zeta_{10}^{2} ) q^{15} -\zeta_{10} q^{16} + ( 3 \zeta_{10} + 3 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{17} + ( -2 - 3 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{18} + ( -3 \zeta_{10} + 4 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{19} + ( -2 + \zeta_{10} - 2 \zeta_{10}^{2} ) q^{20} + ( 3 - 3 \zeta_{10} - 6 \zeta_{10}^{3} ) q^{21} + ( 2 - 2 \zeta_{10} + \zeta_{10}^{3} ) q^{22} + ( 5 - 3 \zeta_{10} + 3 \zeta_{10}^{2} - 5 \zeta_{10}^{3} ) q^{23} + ( 2 + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{24} -5 \zeta_{10} q^{25} - q^{26} + ( 1 + \zeta_{10} - \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{27} + 3 \zeta_{10}^{3} q^{28} + ( 4 - 4 \zeta_{10} + 3 \zeta_{10}^{3} ) q^{29} + ( 4 + 3 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{30} + ( 2 \zeta_{10} + \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{31} - q^{32} + ( \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{33} + ( 3 + 3 \zeta_{10} + 3 \zeta_{10}^{2} ) q^{34} + ( -6 + 6 \zeta_{10} + 3 \zeta_{10}^{3} ) q^{35} + ( -2 - \zeta_{10} + \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{36} + ( -7 + 4 \zeta_{10} - 7 \zeta_{10}^{2} ) q^{37} + ( -3 + 4 \zeta_{10} - 3 \zeta_{10}^{2} ) q^{38} + ( 2 - \zeta_{10} + \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{39} + ( -1 - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{40} + ( 4 + \zeta_{10} + 4 \zeta_{10}^{2} ) q^{41} + ( -3 \zeta_{10} - 3 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{42} + ( -3 + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{43} + ( -2 \zeta_{10} + 3 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{44} + ( -8 + 7 \zeta_{10} - 7 \zeta_{10}^{2} + 8 \zeta_{10}^{3} ) q^{45} + ( 2 - 2 \zeta_{10} - 5 \zeta_{10}^{3} ) q^{46} + ( 8 - 8 \zeta_{10} - \zeta_{10}^{3} ) q^{47} + ( 2 - \zeta_{10} + \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{48} + 2 q^{49} -5 q^{50} + ( -15 - 9 \zeta_{10}^{2} + 9 \zeta_{10}^{3} ) q^{51} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{52} + ( -4 + 4 \zeta_{10} - 4 \zeta_{10}^{3} ) q^{53} + ( 2 - 2 \zeta_{10} - \zeta_{10}^{3} ) q^{54} + ( 4 \zeta_{10} - 7 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{55} + 3 \zeta_{10}^{2} q^{56} + ( 1 + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{57} + ( -4 \zeta_{10} + 7 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{58} + ( -4 + 2 \zeta_{10} - 4 \zeta_{10}^{2} ) q^{59} + ( 4 - \zeta_{10} + \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{60} + ( 4 - 7 \zeta_{10} + 7 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{61} + ( 2 + \zeta_{10} + 2 \zeta_{10}^{2} ) q^{62} + ( 9 - 3 \zeta_{10} + 9 \zeta_{10}^{2} ) q^{63} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{64} + ( -1 - \zeta_{10} + \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{65} + ( 1 - \zeta_{10} + \zeta_{10}^{2} ) q^{66} + ( 2 \zeta_{10} - 9 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{67} + ( 6 + 3 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{68} + ( 7 \zeta_{10} + 5 \zeta_{10}^{2} + 7 \zeta_{10}^{3} ) q^{69} + ( 6 \zeta_{10} - 3 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{70} + 3 \zeta_{10}^{3} q^{71} + ( -3 + 3 \zeta_{10} + 2 \zeta_{10}^{3} ) q^{72} + ( 2 + 4 \zeta_{10} - 4 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{73} + ( -3 - 7 \zeta_{10}^{2} + 7 \zeta_{10}^{3} ) q^{74} + ( 10 - 5 \zeta_{10} + 5 \zeta_{10}^{2} - 10 \zeta_{10}^{3} ) q^{75} + ( 1 - 3 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{76} + ( 3 - 9 \zeta_{10} + 9 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{77} + ( 1 - \zeta_{10} - 2 \zeta_{10}^{3} ) q^{78} + ( 2 - 2 \zeta_{10} + 4 \zeta_{10}^{3} ) q^{79} + ( -1 - \zeta_{10} + \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{80} + ( -6 \zeta_{10} + 4 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{81} + ( 5 + 4 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{82} + ( -4 \zeta_{10} - 7 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{83} + ( -3 - 3 \zeta_{10} - 3 \zeta_{10}^{2} ) q^{84} + ( 12 + 9 \zeta_{10}^{2} - 9 \zeta_{10}^{3} ) q^{85} + ( -3 + 5 \zeta_{10} - 5 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{86} + ( 1 - 3 \zeta_{10} + \zeta_{10}^{2} ) q^{87} + ( -2 + 3 \zeta_{10} - 2 \zeta_{10}^{2} ) q^{88} + ( 2 + 2 \zeta_{10} - 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{89} + ( -1 + \zeta_{10} + 8 \zeta_{10}^{3} ) q^{90} + 3 \zeta_{10} q^{91} + ( -2 \zeta_{10} - 3 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{92} + ( -8 - 5 \zeta_{10}^{2} + 5 \zeta_{10}^{3} ) q^{93} + ( -8 \zeta_{10} + 7 \zeta_{10}^{2} - 8 \zeta_{10}^{3} ) q^{94} + ( -5 + 5 \zeta_{10}^{2} - 5 \zeta_{10}^{3} ) q^{95} + ( 1 - \zeta_{10} - 2 \zeta_{10}^{3} ) q^{96} + ( -9 + 9 \zeta_{10} + 5 \zeta_{10}^{3} ) q^{97} + ( 2 - 2 \zeta_{10} + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{98} + ( -4 + 5 \zeta_{10}^{2} - 5 \zeta_{10}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + q^{2} - q^{3} - q^{4} + 5q^{5} + q^{6} - 12q^{7} + q^{8} - 8q^{9} + O(q^{10})$$ $$4q + q^{2} - q^{3} - q^{4} + 5q^{5} + q^{6} - 12q^{7} + q^{8} - 8q^{9} - 5q^{10} + 3q^{11} + 4q^{12} - q^{13} - 3q^{14} + 10q^{15} - q^{16} + 3q^{17} - 2q^{18} - 10q^{19} - 5q^{20} + 3q^{21} + 7q^{22} + 9q^{23} + 6q^{24} - 5q^{25} - 4q^{26} + 5q^{27} + 3q^{28} + 15q^{29} + 10q^{30} + 3q^{31} - 4q^{32} + 3q^{33} + 12q^{34} - 15q^{35} - 8q^{36} - 17q^{37} - 5q^{38} + 4q^{39} + 13q^{41} - 3q^{42} - 16q^{43} - 7q^{44} - 10q^{45} + q^{46} + 23q^{47} + 4q^{48} + 8q^{49} - 20q^{50} - 42q^{51} - q^{52} - 16q^{53} + 5q^{54} + 15q^{55} - 3q^{56} - 15q^{58} - 10q^{59} + 10q^{60} - 2q^{61} + 7q^{62} + 24q^{63} - q^{64} - 5q^{65} + 2q^{66} + 13q^{67} + 18q^{68} + 9q^{69} + 15q^{70} + 3q^{71} - 7q^{72} + 14q^{73} + 2q^{74} + 20q^{75} + 10q^{76} - 9q^{77} + q^{78} + 10q^{79} - 5q^{80} - 16q^{81} + 12q^{82} - q^{83} - 12q^{84} + 30q^{85} + q^{86} - 3q^{88} + 10q^{89} + 5q^{90} + 3q^{91} - q^{92} - 22q^{93} - 23q^{94} - 30q^{95} + q^{96} - 22q^{97} + 2q^{98} - 26q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/50\mathbb{Z}\right)^\times$$.

 $$n$$ $$27$$ $$\chi(n)$$ $$-\zeta_{10}^{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1
 0.809017 + 0.587785i −0.309017 − 0.951057i −0.309017 + 0.951057i 0.809017 − 0.587785i
0.809017 0.587785i −0.809017 + 2.48990i 0.309017 0.951057i 0.690983 2.12663i 0.809017 + 2.48990i −3.00000 −0.309017 0.951057i −3.11803 2.26538i −0.690983 2.12663i
21.1 −0.309017 + 0.951057i 0.309017 + 0.224514i −0.809017 0.587785i 1.80902 + 1.31433i −0.309017 + 0.224514i −3.00000 0.809017 0.587785i −0.881966 2.71441i −1.80902 + 1.31433i
31.1 −0.309017 0.951057i 0.309017 0.224514i −0.809017 + 0.587785i 1.80902 1.31433i −0.309017 0.224514i −3.00000 0.809017 + 0.587785i −0.881966 + 2.71441i −1.80902 1.31433i
41.1 0.809017 + 0.587785i −0.809017 2.48990i 0.309017 + 0.951057i 0.690983 + 2.12663i 0.809017 2.48990i −3.00000 −0.309017 + 0.951057i −3.11803 + 2.26538i −0.690983 + 2.12663i
 $$n$$: 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
25.d even 5 1 inner

## Twists

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

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.2.d.a 4 1.a even 1 1 trivial
50.2.d.a 4 25.d even 5 1 inner
250.2.d.a 4 5.b even 2 1
250.2.d.a 4 25.e even 10 1
250.2.e.b 8 5.c odd 4 2
250.2.e.b 8 25.f odd 20 2
400.2.u.c 4 4.b odd 2 1
400.2.u.c 4 100.j odd 10 1
450.2.h.a 4 3.b odd 2 1
450.2.h.a 4 75.j odd 10 1
1250.2.a.a 2 25.d even 5 1
1250.2.a.d 2 25.e even 10 1
1250.2.b.b 4 25.f odd 20 2
10000.2.a.a 2 100.h odd 10 1
10000.2.a.n 2 100.j odd 10 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + T_{3}^{3} + 6 T_{3}^{2} - 4 T_{3} + 1$$ acting on $$S_{2}^{\mathrm{new}}(50, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2} - T^{3} + T^{4}$$
$3$ $$1 - 4 T + 6 T^{2} + T^{3} + T^{4}$$
$5$ $$25 - 25 T + 15 T^{2} - 5 T^{3} + T^{4}$$
$7$ $$( 3 + T )^{4}$$
$11$ $$1 - 7 T + 19 T^{2} - 3 T^{3} + T^{4}$$
$13$ $$1 + T + T^{2} + T^{3} + T^{4}$$
$17$ $$81 + 108 T + 54 T^{2} - 3 T^{3} + T^{4}$$
$19$ $$25 + 25 T + 40 T^{2} + 10 T^{3} + T^{4}$$
$23$ $$121 + 11 T + 31 T^{2} - 9 T^{3} + T^{4}$$
$29$ $$25 + 25 T + 85 T^{2} - 15 T^{3} + T^{4}$$
$31$ $$1 - 7 T + 19 T^{2} - 3 T^{3} + T^{4}$$
$37$ $$3721 + 1098 T + 184 T^{2} + 17 T^{3} + T^{4}$$
$41$ $$121 - 77 T + 69 T^{2} - 13 T^{3} + T^{4}$$
$43$ $$( 11 + 8 T + T^{2} )^{2}$$
$47$ $$5041 - 1207 T + 249 T^{2} - 23 T^{3} + T^{4}$$
$53$ $$256 - 64 T + 96 T^{2} + 16 T^{3} + T^{4}$$
$59$ $$400 + 200 T + 60 T^{2} + 10 T^{3} + T^{4}$$
$61$ $$361 - 247 T + 64 T^{2} + 2 T^{3} + T^{4}$$
$67$ $$3481 - 177 T + 79 T^{2} - 13 T^{3} + T^{4}$$
$71$ $$81 - 27 T + 9 T^{2} - 3 T^{3} + T^{4}$$
$73$ $$1936 - 704 T + 136 T^{2} - 14 T^{3} + T^{4}$$
$79$ $$400 + 40 T^{2} - 10 T^{3} + T^{4}$$
$83$ $$3721 - 1159 T + 141 T^{2} + T^{3} + T^{4}$$
$89$ $$400 - 200 T + 60 T^{2} - 10 T^{3} + T^{4}$$
$97$ $$10201 + 2323 T + 304 T^{2} + 22 T^{3} + T^{4}$$