# Properties

 Label 400.2.u.c Level $400$ Weight $2$ Character orbit 400.u Analytic conductor $3.194$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$400 = 2^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 400.u (of order $$5$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.19401608085$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 50) 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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{10} - 2 \zeta_{10}^{3} ) q^{3} + ( 2 - 2 \zeta_{10} - \zeta_{10}^{3} ) q^{5} + 3 q^{7} + ( -3 + \zeta_{10} - 3 \zeta_{10}^{2} ) q^{9} +O(q^{10})$$ $$q + ( 1 - \zeta_{10} - 2 \zeta_{10}^{3} ) q^{3} + ( 2 - 2 \zeta_{10} - \zeta_{10}^{3} ) q^{5} + 3 q^{7} + ( -3 + \zeta_{10} - 3 \zeta_{10}^{2} ) q^{9} + ( 1 - 3 \zeta_{10} + 3 \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{11} -\zeta_{10} q^{13} + ( -3 - \zeta_{10} - 3 \zeta_{10}^{2} ) q^{15} + ( 3 \zeta_{10} + 3 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{17} + ( 3 \zeta_{10} - 4 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{19} + ( 3 - 3 \zeta_{10} - 6 \zeta_{10}^{3} ) q^{21} + ( -5 + 3 \zeta_{10} - 3 \zeta_{10}^{2} + 5 \zeta_{10}^{3} ) q^{23} -5 \zeta_{10} q^{25} + ( -1 - \zeta_{10} + \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{27} + ( 4 - 4 \zeta_{10} + 3 \zeta_{10}^{3} ) q^{29} + ( -2 \zeta_{10} - \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{31} + ( \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{33} + ( 6 - 6 \zeta_{10} - 3 \zeta_{10}^{3} ) q^{35} + ( -7 + 4 \zeta_{10} - 7 \zeta_{10}^{2} ) q^{37} + ( -2 + \zeta_{10} - \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{39} + ( 4 + \zeta_{10} + 4 \zeta_{10}^{2} ) q^{41} + ( 3 - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{43} + ( -8 + 7 \zeta_{10} - 7 \zeta_{10}^{2} + 8 \zeta_{10}^{3} ) q^{45} + ( -8 + 8 \zeta_{10} + \zeta_{10}^{3} ) q^{47} + 2 q^{49} + ( 15 + 9 \zeta_{10}^{2} - 9 \zeta_{10}^{3} ) q^{51} + ( -4 + 4 \zeta_{10} - 4 \zeta_{10}^{3} ) q^{53} + ( -4 \zeta_{10} + 7 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{55} + ( 1 + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{57} + ( 4 - 2 \zeta_{10} + 4 \zeta_{10}^{2} ) q^{59} + ( 4 - 7 \zeta_{10} + 7 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{61} + ( -9 + 3 \zeta_{10} - 9 \zeta_{10}^{2} ) q^{63} + ( -1 - \zeta_{10} + \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{65} + ( -2 \zeta_{10} + 9 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{67} + ( 7 \zeta_{10} + 5 \zeta_{10}^{2} + 7 \zeta_{10}^{3} ) q^{69} -3 \zeta_{10}^{3} q^{71} + ( 2 + 4 \zeta_{10} - 4 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{73} + ( -10 + 5 \zeta_{10} - 5 \zeta_{10}^{2} + 10 \zeta_{10}^{3} ) q^{75} + ( 3 - 9 \zeta_{10} + 9 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{77} + ( -2 + 2 \zeta_{10} - 4 \zeta_{10}^{3} ) q^{79} + ( -6 \zeta_{10} + 4 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{81} + ( 4 \zeta_{10} + 7 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{83} + ( 12 + 9 \zeta_{10}^{2} - 9 \zeta_{10}^{3} ) q^{85} + ( -1 + 3 \zeta_{10} - \zeta_{10}^{2} ) q^{87} + ( 2 + 2 \zeta_{10} - 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{89} -3 \zeta_{10} q^{91} + ( -8 - 5 \zeta_{10}^{2} + 5 \zeta_{10}^{3} ) q^{93} + ( 5 - 5 \zeta_{10}^{2} + 5 \zeta_{10}^{3} ) q^{95} + ( -9 + 9 \zeta_{10} + 5 \zeta_{10}^{3} ) q^{97} + ( 4 - 5 \zeta_{10}^{2} + 5 \zeta_{10}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + q^{3} + 5q^{5} + 12q^{7} - 8q^{9} + O(q^{10})$$ $$4q + q^{3} + 5q^{5} + 12q^{7} - 8q^{9} - 3q^{11} - q^{13} - 10q^{15} + 3q^{17} + 10q^{19} + 3q^{21} - 9q^{23} - 5q^{25} - 5q^{27} + 15q^{29} - 3q^{31} + 3q^{33} + 15q^{35} - 17q^{37} - 4q^{39} + 13q^{41} + 16q^{43} - 10q^{45} - 23q^{47} + 8q^{49} + 42q^{51} - 16q^{53} - 15q^{55} + 10q^{59} - 2q^{61} - 24q^{63} - 5q^{65} - 13q^{67} + 9q^{69} - 3q^{71} + 14q^{73} - 20q^{75} - 9q^{77} - 10q^{79} - 16q^{81} + q^{83} + 30q^{85} + 10q^{89} - 3q^{91} - 22q^{93} + 30q^{95} - 22q^{97} + 26q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/400\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$177$$ $$351$$ $$\chi(n)$$ $$1$$ $$-\zeta_{10}^{3}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
81.1
 −0.309017 + 0.951057i 0.809017 + 0.587785i 0.809017 − 0.587785i −0.309017 − 0.951057i
0 −0.309017 + 0.224514i 0 1.80902 1.31433i 0 3.00000 0 −0.881966 + 2.71441i 0
161.1 0 0.809017 2.48990i 0 0.690983 2.12663i 0 3.00000 0 −3.11803 2.26538i 0
241.1 0 0.809017 + 2.48990i 0 0.690983 + 2.12663i 0 3.00000 0 −3.11803 + 2.26538i 0
321.1 0 −0.309017 0.224514i 0 1.80902 + 1.31433i 0 3.00000 0 −0.881966 2.71441i 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
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 400.2.u.c 4
4.b odd 2 1 50.2.d.a 4
12.b even 2 1 450.2.h.a 4
20.d odd 2 1 250.2.d.a 4
20.e even 4 2 250.2.e.b 8
25.d even 5 1 inner 400.2.u.c 4
25.d even 5 1 10000.2.a.n 2
25.e even 10 1 10000.2.a.a 2
100.h odd 10 1 250.2.d.a 4
100.h odd 10 1 1250.2.a.d 2
100.j odd 10 1 50.2.d.a 4
100.j odd 10 1 1250.2.a.a 2
100.l even 20 2 250.2.e.b 8
100.l even 20 2 1250.2.b.b 4
300.n even 10 1 450.2.h.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.2.d.a 4 4.b odd 2 1
50.2.d.a 4 100.j odd 10 1
250.2.d.a 4 20.d odd 2 1
250.2.d.a 4 100.h odd 10 1
250.2.e.b 8 20.e even 4 2
250.2.e.b 8 100.l even 20 2
400.2.u.c 4 1.a even 1 1 trivial
400.2.u.c 4 25.d even 5 1 inner
450.2.h.a 4 12.b even 2 1
450.2.h.a 4 300.n even 10 1
1250.2.a.a 2 100.j odd 10 1
1250.2.a.d 2 100.h odd 10 1
1250.2.b.b 4 100.l even 20 2
10000.2.a.a 2 25.e even 10 1
10000.2.a.n 2 25.d even 5 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 $$S_{2}^{\mathrm{new}}(400, [\chi])$$.

## Hecke characteristic polynomials

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