# Properties

 Label 400.2.u.a Level $400$ Weight $2$ Character orbit 400.u Analytic conductor $3.194$ Analytic rank $1$ 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: $$1$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ x^4 - x^3 + x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 200) 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.

 $$f(q)$$ $$=$$ $$q + (\zeta_{10}^{3} + 2 \zeta_{10} - 2) q^{3} + ( - 2 \zeta_{10}^{2} + \zeta_{10} - 2) q^{5} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - 1) q^{7} - 2 \zeta_{10} q^{9} +O(q^{10})$$ q + (z^3 + 2*z - 2) * q^3 + (-2*z^2 + z - 2) * q^5 + (-z^3 + z^2 - 1) * q^7 - 2*z * q^9 $$q + (\zeta_{10}^{3} + 2 \zeta_{10} - 2) q^{3} + ( - 2 \zeta_{10}^{2} + \zeta_{10} - 2) q^{5} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - 1) q^{7} - 2 \zeta_{10} q^{9} + (2 \zeta_{10}^{3} - 2) q^{11} + ( - \zeta_{10}^{2} - 4 \zeta_{10} - 1) q^{13} + ( - 5 \zeta_{10}^{3} + 5 \zeta_{10}^{2} - 5 \zeta_{10} + 5) q^{15} + ( - 2 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 2 \zeta_{10}) q^{17} + (\zeta_{10}^{3} - 7 \zeta_{10}^{2} + \zeta_{10}) q^{19} + (\zeta_{10}^{3} - 3 \zeta_{10} + 3) q^{21} + (\zeta_{10}^{3} + 3 \zeta_{10}^{2} - 3 \zeta_{10} - 1) q^{23} + 5 \zeta_{10}^{2} q^{25} + (\zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} - 1) q^{27} + ( - 6 \zeta_{10}^{3} - \zeta_{10} + 1) q^{29} + ( - 6 \zeta_{10}^{3} + \zeta_{10}^{2} - 6 \zeta_{10}) q^{31} + ( - 2 \zeta_{10}^{3} - 4 \zeta_{10}^{2} - 2 \zeta_{10}) q^{33} + (3 \zeta_{10}^{2} - 4 \zeta_{10} + 3) q^{35} + ( - 2 \zeta_{10}^{2} + 7 \zeta_{10} - 2) q^{37} + ( - 7 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} + 7) q^{39} + (6 \zeta_{10}^{2} - 2 \zeta_{10} + 6) q^{41} + (\zeta_{10}^{3} - \zeta_{10}^{2} - 6) q^{43} + (4 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 4 \zeta_{10}) q^{45} + (5 \zeta_{10}^{3} + 3 \zeta_{10} - 3) q^{47} + (3 \zeta_{10}^{3} - 3 \zeta_{10}^{2} - 5) q^{49} + (6 \zeta_{10}^{3} - 6 \zeta_{10}^{2} + 2) q^{51} + (5 \zeta_{10}^{3} + 8 \zeta_{10} - 8) q^{53} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} + 6) q^{55} + ( - 13 \zeta_{10}^{3} + 13 \zeta_{10}^{2} + 4) q^{57} + (\zeta_{10}^{2} + 3 \zeta_{10} + 1) q^{59} + (3 \zeta_{10}^{3} + 7 \zeta_{10}^{2} - 7 \zeta_{10} - 3) q^{61} + ( - 2 \zeta_{10}^{2} + 4 \zeta_{10} - 2) q^{63} + (9 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 9 \zeta_{10}) q^{65} + (6 \zeta_{10}^{3} - 10 \zeta_{10}^{2} + 6 \zeta_{10}) q^{67} + (2 \zeta_{10}^{3} - 11 \zeta_{10}^{2} + 2 \zeta_{10}) q^{69} + ( - 7 \zeta_{10}^{3} + \zeta_{10} - 1) q^{71} + (11 \zeta_{10}^{3} - 11 \zeta_{10}^{2} + 11 \zeta_{10} - 11) q^{73} + (10 \zeta_{10}^{3} - 10 \zeta_{10}^{2} - 5) q^{75} + ( - 2 \zeta_{10}^{2} + 2 \zeta_{10}) q^{77} + (6 \zeta_{10}^{3} + 7 \zeta_{10} - 7) q^{79} - 11 \zeta_{10}^{2} q^{81} + (4 \zeta_{10}^{3} + 7 \zeta_{10}^{2} + 4 \zeta_{10}) q^{83} + (2 \zeta_{10}^{3} - 6 \zeta_{10} + 6) q^{85} + (11 \zeta_{10}^{2} - 3 \zeta_{10} + 11) q^{87} + (4 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 4 \zeta_{10} - 4) q^{89} + ( - 3 \zeta_{10}^{2} + 7 \zeta_{10} - 3) q^{91} + ( - 4 \zeta_{10}^{3} + 4 \zeta_{10}^{2} + 17) q^{93} + (4 \zeta_{10}^{3} + 13 \zeta_{10} - 13) q^{95} + ( - 9 \zeta_{10}^{3} - 15 \zeta_{10} + 15) q^{97} + ( - 4 \zeta_{10}^{3} + 4 \zeta_{10}^{2} + 4) q^{99} +O(q^{100})$$ q + (z^3 + 2*z - 2) * q^3 + (-2*z^2 + z - 2) * q^5 + (-z^3 + z^2 - 1) * q^7 - 2*z * q^9 + (2*z^3 - 2) * q^11 + (-z^2 - 4*z - 1) * q^13 + (-5*z^3 + 5*z^2 - 5*z + 5) * q^15 + (-2*z^3 + 4*z^2 - 2*z) * q^17 + (z^3 - 7*z^2 + z) * q^19 + (z^3 - 3*z + 3) * q^21 + (z^3 + 3*z^2 - 3*z - 1) * q^23 + 5*z^2 * q^25 + (z^3 + z^2 - z - 1) * q^27 + (-6*z^3 - z + 1) * q^29 + (-6*z^3 + z^2 - 6*z) * q^31 + (-2*z^3 - 4*z^2 - 2*z) * q^33 + (3*z^2 - 4*z + 3) * q^35 + (-2*z^2 + 7*z - 2) * q^37 + (-7*z^3 - 2*z^2 + 2*z + 7) * q^39 + (6*z^2 - 2*z + 6) * q^41 + (z^3 - z^2 - 6) * q^43 + (4*z^3 - 2*z^2 + 4*z) * q^45 + (5*z^3 + 3*z - 3) * q^47 + (3*z^3 - 3*z^2 - 5) * q^49 + (6*z^3 - 6*z^2 + 2) * q^51 + (5*z^3 + 8*z - 8) * q^53 + (-2*z^3 + 2*z^2 + 6) * q^55 + (-13*z^3 + 13*z^2 + 4) * q^57 + (z^2 + 3*z + 1) * q^59 + (3*z^3 + 7*z^2 - 7*z - 3) * q^61 + (-2*z^2 + 4*z - 2) * q^63 + (9*z^3 - 2*z^2 + 9*z) * q^65 + (6*z^3 - 10*z^2 + 6*z) * q^67 + (2*z^3 - 11*z^2 + 2*z) * q^69 + (-7*z^3 + z - 1) * q^71 + (11*z^3 - 11*z^2 + 11*z - 11) * q^73 + (10*z^3 - 10*z^2 - 5) * q^75 + (-2*z^2 + 2*z) * q^77 + (6*z^3 + 7*z - 7) * q^79 - 11*z^2 * q^81 + (4*z^3 + 7*z^2 + 4*z) * q^83 + (2*z^3 - 6*z + 6) * q^85 + (11*z^2 - 3*z + 11) * q^87 + (4*z^3 - 4*z^2 + 4*z - 4) * q^89 + (-3*z^2 + 7*z - 3) * q^91 + (-4*z^3 + 4*z^2 + 17) * q^93 + (4*z^3 + 13*z - 13) * q^95 + (-9*z^3 - 15*z + 15) * q^97 + (-4*z^3 + 4*z^2 + 4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 5 q^{3} - 5 q^{5} - 6 q^{7} - 2 q^{9}+O(q^{10})$$ 4 * q - 5 * q^3 - 5 * q^5 - 6 * q^7 - 2 * q^9 $$4 q - 5 q^{3} - 5 q^{5} - 6 q^{7} - 2 q^{9} - 6 q^{11} - 7 q^{13} + 5 q^{15} - 8 q^{17} + 9 q^{19} + 10 q^{21} - 9 q^{23} - 5 q^{25} - 5 q^{27} - 3 q^{29} - 13 q^{31} + 5 q^{35} + q^{37} + 25 q^{39} + 16 q^{41} - 22 q^{43} + 10 q^{45} - 4 q^{47} - 14 q^{49} + 20 q^{51} - 19 q^{53} + 20 q^{55} - 10 q^{57} + 6 q^{59} - 23 q^{61} - 2 q^{63} + 20 q^{65} + 22 q^{67} + 15 q^{69} - 10 q^{71} - 11 q^{73} + 4 q^{77} - 15 q^{79} + 11 q^{81} + q^{83} + 20 q^{85} + 30 q^{87} - 4 q^{89} - 2 q^{91} + 60 q^{93} - 35 q^{95} + 36 q^{97} + 8 q^{99}+O(q^{100})$$ 4 * q - 5 * q^3 - 5 * q^5 - 6 * q^7 - 2 * q^9 - 6 * q^11 - 7 * q^13 + 5 * q^15 - 8 * q^17 + 9 * q^19 + 10 * q^21 - 9 * q^23 - 5 * q^25 - 5 * q^27 - 3 * q^29 - 13 * q^31 + 5 * q^35 + q^37 + 25 * q^39 + 16 * q^41 - 22 * q^43 + 10 * q^45 - 4 * q^47 - 14 * q^49 + 20 * q^51 - 19 * q^53 + 20 * q^55 - 10 * q^57 + 6 * q^59 - 23 * q^61 - 2 * q^63 + 20 * q^65 + 22 * q^67 + 15 * q^69 - 10 * q^71 - 11 * q^73 + 4 * q^77 - 15 * q^79 + 11 * q^81 + q^83 + 20 * q^85 + 30 * q^87 - 4 * q^89 - 2 * q^91 + 60 * q^93 - 35 * q^95 + 36 * q^97 + 8 * q^99

## 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 −1.80902 + 1.31433i 0 −0.690983 + 2.12663i 0 −2.61803 0 0.618034 1.90211i 0
161.1 0 −0.690983 + 2.12663i 0 −1.80902 1.31433i 0 −0.381966 0 −1.61803 1.17557i 0
241.1 0 −0.690983 2.12663i 0 −1.80902 + 1.31433i 0 −0.381966 0 −1.61803 + 1.17557i 0
321.1 0 −1.80902 1.31433i 0 −0.690983 2.12663i 0 −2.61803 0 0.618034 + 1.90211i 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.a 4
4.b odd 2 1 200.2.m.a 4
20.d odd 2 1 1000.2.m.a 4
20.e even 4 2 1000.2.q.a 8
25.d even 5 1 inner 400.2.u.a 4
25.d even 5 1 10000.2.a.g 2
25.e even 10 1 10000.2.a.i 2
100.h odd 10 1 1000.2.m.a 4
100.h odd 10 1 5000.2.a.a 2
100.j odd 10 1 200.2.m.a 4
100.j odd 10 1 5000.2.a.c 2
100.l even 20 2 1000.2.q.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.2.m.a 4 4.b odd 2 1
200.2.m.a 4 100.j odd 10 1
400.2.u.a 4 1.a even 1 1 trivial
400.2.u.a 4 25.d even 5 1 inner
1000.2.m.a 4 20.d odd 2 1
1000.2.m.a 4 100.h odd 10 1
1000.2.q.a 8 20.e even 4 2
1000.2.q.a 8 100.l even 20 2
5000.2.a.a 2 100.h odd 10 1
5000.2.a.c 2 100.j odd 10 1
10000.2.a.g 2 25.d even 5 1
10000.2.a.i 2 25.e even 10 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 5T_{3}^{3} + 15T_{3}^{2} + 25T_{3} + 25$$ acting on $$S_{2}^{\mathrm{new}}(400, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 5 T^{3} + 15 T^{2} + 25 T + 25$$
$5$ $$T^{4} + 5 T^{3} + 15 T^{2} + 25 T + 25$$
$7$ $$(T^{2} + 3 T + 1)^{2}$$
$11$ $$T^{4} + 6 T^{3} + 16 T^{2} + 16 T + 16$$
$13$ $$T^{4} + 7 T^{3} + 24 T^{2} + 38 T + 361$$
$17$ $$T^{4} + 8 T^{3} + 24 T^{2} - 8 T + 16$$
$19$ $$T^{4} - 9 T^{3} + 46 T^{2} + \cdots + 1681$$
$23$ $$T^{4} + 9 T^{3} + 61 T^{2} + 209 T + 361$$
$29$ $$T^{4} + 3 T^{3} + 34 T^{2} + 232 T + 841$$
$31$ $$T^{4} + 13 T^{3} + 139 T^{2} + \cdots + 1681$$
$37$ $$T^{4} - T^{3} + 51 T^{2} - 341 T + 961$$
$41$ $$T^{4} - 16 T^{3} + 136 T^{2} + \cdots + 1936$$
$43$ $$(T^{2} + 11 T + 29)^{2}$$
$47$ $$T^{4} + 4 T^{3} + 46 T^{2} - 11 T + 1$$
$53$ $$T^{4} + 19 T^{3} + 241 T^{2} + \cdots + 6241$$
$59$ $$T^{4} - 6 T^{3} + 16 T^{2} - 11 T + 121$$
$61$ $$T^{4} + 23 T^{3} + 379 T^{2} + \cdots + 14641$$
$67$ $$T^{4} - 22 T^{3} + 184 T^{2} + \cdots + 16$$
$71$ $$T^{4} + 10 T^{3} + 60 T^{2} + \cdots + 3025$$
$73$ $$T^{4} + 11 T^{3} + 121 T^{2} + \cdots + 14641$$
$79$ $$T^{4} + 15 T^{3} + 190 T^{2} + \cdots + 3025$$
$83$ $$T^{4} - T^{3} + 141 T^{2} + 1159 T + 3721$$
$89$ $$T^{4} + 4 T^{3} + 16 T^{2} + 64 T + 256$$
$97$ $$T^{4} - 36 T^{3} + 846 T^{2} + \cdots + 77841$$