Properties

Label 400.6.c.d
Level $400$
Weight $6$
Character orbit 400.c
Analytic conductor $64.154$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 400.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(64.1535279252\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 8)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 10 \beta q^{3} + 12 \beta q^{7} - 157 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 10 \beta q^{3} + 12 \beta q^{7} - 157 q^{9} - 124 q^{11} - 239 \beta q^{13} - 599 \beta q^{17} + 3044 q^{19} - 480 q^{21} + 92 \beta q^{23} + 860 \beta q^{27} + 3282 q^{29} + 5728 q^{31} - 1240 \beta q^{33} + 5163 \beta q^{37} + 9560 q^{39} - 8886 q^{41} - 4594 \beta q^{43} - 11832 \beta q^{47} + 16231 q^{49} + 23960 q^{51} - 5843 \beta q^{53} + 30440 \beta q^{57} + 16876 q^{59} - 18482 q^{61} - 1884 \beta q^{63} + 7766 \beta q^{67} - 3680 q^{69} + 31960 q^{71} + 2443 \beta q^{73} - 1488 \beta q^{77} + 44560 q^{79} - 72551 q^{81} + 33682 \beta q^{83} + 32820 \beta q^{87} - 71994 q^{89} + 11472 q^{91} + 57280 \beta q^{93} + 24433 \beta q^{97} + 19468 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 314 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 314 q^{9} - 248 q^{11} + 6088 q^{19} - 960 q^{21} + 6564 q^{29} + 11456 q^{31} + 19120 q^{39} - 17772 q^{41} + 32462 q^{49} + 47920 q^{51} + 33752 q^{59} - 36964 q^{61} - 7360 q^{69} + 63920 q^{71} + 89120 q^{79} - 145102 q^{81} - 143988 q^{89} + 22944 q^{91} + 38936 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(177\) \(351\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
1.00000i
1.00000i
0 20.0000i 0 0 0 24.0000i 0 −157.000 0
49.2 0 20.0000i 0 0 0 24.0000i 0 −157.000 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
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 400.6.c.d 2
4.b odd 2 1 200.6.c.a 2
5.b even 2 1 inner 400.6.c.d 2
5.c odd 4 1 16.6.a.a 1
5.c odd 4 1 400.6.a.l 1
15.e even 4 1 144.6.a.k 1
20.d odd 2 1 200.6.c.a 2
20.e even 4 1 8.6.a.a 1
20.e even 4 1 200.6.a.a 1
35.f even 4 1 784.6.a.l 1
40.i odd 4 1 64.6.a.g 1
40.k even 4 1 64.6.a.a 1
60.l odd 4 1 72.6.a.f 1
80.i odd 4 1 256.6.b.d 2
80.j even 4 1 256.6.b.f 2
80.s even 4 1 256.6.b.f 2
80.t odd 4 1 256.6.b.d 2
120.q odd 4 1 576.6.a.g 1
120.w even 4 1 576.6.a.h 1
140.j odd 4 1 392.6.a.b 1
140.w even 12 2 392.6.i.b 2
140.x odd 12 2 392.6.i.e 2
220.i odd 4 1 968.6.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.6.a.a 1 20.e even 4 1
16.6.a.a 1 5.c odd 4 1
64.6.a.a 1 40.k even 4 1
64.6.a.g 1 40.i odd 4 1
72.6.a.f 1 60.l odd 4 1
144.6.a.k 1 15.e even 4 1
200.6.a.a 1 20.e even 4 1
200.6.c.a 2 4.b odd 2 1
200.6.c.a 2 20.d odd 2 1
256.6.b.d 2 80.i odd 4 1
256.6.b.d 2 80.t odd 4 1
256.6.b.f 2 80.j even 4 1
256.6.b.f 2 80.s even 4 1
392.6.a.b 1 140.j odd 4 1
392.6.i.b 2 140.w even 12 2
392.6.i.e 2 140.x odd 12 2
400.6.a.l 1 5.c odd 4 1
400.6.c.d 2 1.a even 1 1 trivial
400.6.c.d 2 5.b even 2 1 inner
576.6.a.g 1 120.q odd 4 1
576.6.a.h 1 120.w even 4 1
784.6.a.l 1 35.f even 4 1
968.6.a.a 1 220.i odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 400 \) acting on \(S_{6}^{\mathrm{new}}(400, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 400 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 576 \) Copy content Toggle raw display
$11$ \( (T + 124)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 228484 \) Copy content Toggle raw display
$17$ \( T^{2} + 1435204 \) Copy content Toggle raw display
$19$ \( (T - 3044)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 33856 \) Copy content Toggle raw display
$29$ \( (T - 3282)^{2} \) Copy content Toggle raw display
$31$ \( (T - 5728)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 106626276 \) Copy content Toggle raw display
$41$ \( (T + 8886)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 84419344 \) Copy content Toggle raw display
$47$ \( T^{2} + 559984896 \) Copy content Toggle raw display
$53$ \( T^{2} + 136562596 \) Copy content Toggle raw display
$59$ \( (T - 16876)^{2} \) Copy content Toggle raw display
$61$ \( (T + 18482)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 241243024 \) Copy content Toggle raw display
$71$ \( (T - 31960)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 23872996 \) Copy content Toggle raw display
$79$ \( (T - 44560)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 4537908496 \) Copy content Toggle raw display
$89$ \( (T + 71994)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 2387885956 \) Copy content Toggle raw display
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