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Properties

Label 400.8.c.a
Level $400$
Weight $8$
Character orbit 400.c
Analytic conductor $124.954$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [400,8,Mod(49,400)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 8, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("400.49"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 400.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,0,0,-10764] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(124.954010194\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 50)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

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

\(f(q)\) \(=\) \( q + 87 i q^{3} - 1366 i q^{7} - 5382 q^{9} + 1083 q^{11} + 5468 i q^{13} - 25269 i q^{17} + 33485 q^{19} + 118842 q^{21} - 5838 i q^{23} - 277965 i q^{27} - 125280 q^{29} + 73798 q^{31} + 94221 i q^{33} + \cdots - 5828706 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 10764 q^{9} + 2166 q^{11} + 66970 q^{19} + 237684 q^{21} - 250560 q^{29} + 147596 q^{31} - 951432 q^{39} - 45366 q^{41} - 2084826 q^{49} + 4396806 q^{51} + 2196720 q^{59} - 845996 q^{61} + 1015812 q^{69}+ \cdots - 11657412 q^{99}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
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 − 87.0000i 0 0 0 1366.00i 0 −5382.00 0
49.2 0 87.0000i 0 0 0 − 1366.00i 0 −5382.00 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 400.8.c.a 2
4.b odd 2 1 50.8.b.a 2
5.b even 2 1 inner 400.8.c.a 2
5.c odd 4 1 400.8.a.a 1
5.c odd 4 1 400.8.a.s 1
12.b even 2 1 450.8.c.l 2
20.d odd 2 1 50.8.b.a 2
20.e even 4 1 50.8.a.d ✓ 1
20.e even 4 1 50.8.a.e yes 1
60.h even 2 1 450.8.c.l 2
60.l odd 4 1 450.8.a.a 1
60.l odd 4 1 450.8.a.z 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.8.a.d ✓ 1 20.e even 4 1
50.8.a.e yes 1 20.e even 4 1
50.8.b.a 2 4.b odd 2 1
50.8.b.a 2 20.d odd 2 1
400.8.a.a 1 5.c odd 4 1
400.8.a.s 1 5.c odd 4 1
400.8.c.a 2 1.a even 1 1 trivial
400.8.c.a 2 5.b even 2 1 inner
450.8.a.a 1 60.l odd 4 1
450.8.a.z 1 60.l odd 4 1
450.8.c.l 2 12.b even 2 1
450.8.c.l 2 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 7569 \) acting on \(S_{8}^{\mathrm{new}}(400, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 7569 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 1865956 \) Copy content Toggle raw display
$11$ \( (T - 1083)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 29899024 \) Copy content Toggle raw display
$17$ \( T^{2} + 638522361 \) Copy content Toggle raw display
$19$ \( (T - 33485)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 34082244 \) Copy content Toggle raw display
$29$ \( (T + 125280)^{2} \) Copy content Toggle raw display
$31$ \( (T - 73798)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 156757397476 \) Copy content Toggle raw display
$41$ \( (T + 22683)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 10029621904 \) Copy content Toggle raw display
$47$ \( T^{2} + 1311583819536 \) Copy content Toggle raw display
$53$ \( T^{2} + 125941233924 \) Copy content Toggle raw display
$59$ \( (T - 1098360)^{2} \) Copy content Toggle raw display
$61$ \( (T + 422998)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 6546326499241 \) Copy content Toggle raw display
$71$ \( (T - 2287428)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 40608029788249 \) Copy content Toggle raw display
$79$ \( (T + 2019250)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 63568457918289 \) Copy content Toggle raw display
$89$ \( (T + 2185935)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 33914852733316 \) Copy content Toggle raw display
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