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 $2$

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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: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
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.

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

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:

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 \(S_{2}^{\mathrm{new}}(400, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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