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

Downloads

Learn more about

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:

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

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