Properties

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

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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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 25)
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.

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

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 0.809017 0.587785i 0 −0.690983 + 2.12663i 0 −0.618034 0 −0.618034 + 1.90211i 0
161.1 0 −0.309017 + 0.951057i 0 −1.80902 1.31433i 0 1.61803 0 1.61803 + 1.17557i 0
241.1 0 −0.309017 0.951057i 0 −1.80902 + 1.31433i 0 1.61803 0 1.61803 1.17557i 0
321.1 0 0.809017 + 0.587785i 0 −0.690983 2.12663i 0 −0.618034 0 −0.618034 1.90211i 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.b 4
4.b odd 2 1 25.2.d.a 4
12.b even 2 1 225.2.h.b 4
20.d odd 2 1 125.2.d.a 4
20.e even 4 2 125.2.e.a 8
25.d even 5 1 inner 400.2.u.b 4
25.d even 5 1 10000.2.a.c 2
25.e even 10 1 10000.2.a.l 2
100.h odd 10 1 125.2.d.a 4
100.h odd 10 1 625.2.a.c 2
100.h odd 10 2 625.2.d.b 4
100.j odd 10 1 25.2.d.a 4
100.j odd 10 1 625.2.a.b 2
100.j odd 10 2 625.2.d.h 4
100.l even 20 2 125.2.e.a 8
100.l even 20 2 625.2.b.a 4
100.l even 20 4 625.2.e.c 8
300.n even 10 1 225.2.h.b 4
300.n even 10 1 5625.2.a.f 2
300.r even 10 1 5625.2.a.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.2.d.a 4 4.b odd 2 1
25.2.d.a 4 100.j odd 10 1
125.2.d.a 4 20.d odd 2 1
125.2.d.a 4 100.h odd 10 1
125.2.e.a 8 20.e even 4 2
125.2.e.a 8 100.l even 20 2
225.2.h.b 4 12.b even 2 1
225.2.h.b 4 300.n even 10 1
400.2.u.b 4 1.a even 1 1 trivial
400.2.u.b 4 25.d even 5 1 inner
625.2.a.b 2 100.j odd 10 1
625.2.a.c 2 100.h odd 10 1
625.2.b.a 4 100.l even 20 2
625.2.d.b 4 100.h odd 10 2
625.2.d.h 4 100.j odd 10 2
625.2.e.c 8 100.l even 20 4
5625.2.a.d 2 300.r even 10 1
5625.2.a.f 2 300.n even 10 1
10000.2.a.c 2 25.d even 5 1
10000.2.a.l 2 25.e even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - T_{3}^{3} + T_{3}^{2} - T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(400, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
$5$ \( 25 + 25 T + 15 T^{2} + 5 T^{3} + T^{4} \)
$7$ \( ( -1 - T + T^{2} )^{2} \)
$11$ \( 16 + 32 T + 24 T^{2} - 2 T^{3} + T^{4} \)
$13$ \( 81 - 54 T + 36 T^{2} - 9 T^{3} + T^{4} \)
$17$ \( 16 + 8 T + 24 T^{2} - 8 T^{3} + T^{4} \)
$19$ \( 25 - 50 T + 40 T^{2} - 5 T^{3} + T^{4} \)
$23$ \( 961 - 31 T + 51 T^{2} - 11 T^{3} + T^{4} \)
$29$ \( 25 + 10 T^{2} - 5 T^{3} + T^{4} \)
$31$ \( 81 + 27 T + 9 T^{2} + 3 T^{3} + T^{4} \)
$37$ \( 1 + 3 T + 19 T^{2} + 7 T^{3} + T^{4} \)
$41$ \( 16 + 8 T + 24 T^{2} - 8 T^{3} + T^{4} \)
$43$ \( ( -9 - 3 T + T^{2} )^{2} \)
$47$ \( 1 - 3 T + 4 T^{2} - 2 T^{3} + T^{4} \)
$53$ \( 361 - 209 T + 61 T^{2} - 9 T^{3} + T^{4} \)
$59$ \( 2025 + 675 T + 90 T^{2} + T^{4} \)
$61$ \( 1681 - 697 T + 139 T^{2} - 13 T^{3} + T^{4} \)
$67$ \( 1936 - 528 T + 64 T^{2} - 2 T^{3} + T^{4} \)
$71$ \( 841 + 87 T + 34 T^{2} + 8 T^{3} + T^{4} \)
$73$ \( 6561 - 729 T + 81 T^{2} - 9 T^{3} + T^{4} \)
$79$ \( 625 + 250 T + 100 T^{2} + 15 T^{3} + T^{4} \)
$83$ \( 121 - 11 T + 31 T^{2} + 9 T^{3} + T^{4} \)
$89$ \( 6400 + 1600 T + 240 T^{2} + 20 T^{3} + T^{4} \)
$97$ \( 121 - 77 T + 34 T^{2} - 8 T^{3} + T^{4} \)
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