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Properties

Label 3400.1.dh.c
Level $3400$
Weight $1$
Character orbit 3400.dh
Analytic conductor $1.697$
Analytic rank $0$
Dimension $16$
Projective image $D_{48}$
CM discriminant -8
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3400,1,Mod(243,3400)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3400, base_ring=CyclotomicField(16)) chi = DirichletCharacter(H, H._module([8, 8, 12, 5])) N = Newforms(chi, 1, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3400.243"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Level: \( N \) \(=\) \( 3400 = 2^{3} \cdot 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3400.dh (of order \(16\), degree \(8\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.69682104295\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(2\) over \(\Q(\zeta_{16})\)
Coefficient field: \(\Q(\zeta_{48})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{48}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{48} - \cdots)\)

$q$-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)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + \zeta_{48}^{9} q^{2} + (\zeta_{48}^{17} + \zeta_{48}^{16}) q^{3} + \zeta_{48}^{18} q^{4} + ( - \zeta_{48}^{2} - \zeta_{48}) q^{6} - \zeta_{48}^{3} q^{8} + ( - \zeta_{48}^{10} + \cdots - \zeta_{48}^{8}) q^{9} + \cdots + (\zeta_{48}^{17} + \cdots - \zeta_{48}^{4}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 8 q^{3} - 8 q^{9} + 8 q^{22} + 16 q^{27} + 16 q^{33} - 16 q^{43} + 16 q^{59} + 8 q^{66} + 8 q^{73} - 8 q^{81} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3400\mathbb{Z}\right)^\times\).

\(n\) \(1601\) \(1701\) \(2177\) \(2551\)
\(\chi(n)\) \(-\zeta_{48}^{21}\) \(-1\) \(\zeta_{48}^{12}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
243.1
0.793353 − 0.608761i
0.130526 + 0.991445i
0.608761 + 0.793353i
−0.991445 + 0.130526i
−0.130526 + 0.991445i
−0.793353 − 0.608761i
0.991445 + 0.130526i
−0.608761 + 0.793353i
0.608761 − 0.793353i
−0.991445 − 0.130526i
0.793353 + 0.608761i
0.130526 − 0.991445i
0.991445 − 0.130526i
−0.608761 − 0.793353i
−0.130526 − 0.991445i
−0.793353 + 0.608761i
0.923880 + 0.382683i −0.369474 + 1.85747i 0.707107 + 0.707107i 0 −1.05217 + 1.57469i 0 0.382683 + 0.923880i −2.38981 − 0.989890i 0
243.2 0.923880 + 0.382683i 0.293353 − 1.47479i 0.707107 + 0.707107i 0 0.835400 − 1.25026i 0 0.382683 + 0.923880i −1.16506 − 0.482584i 0
643.1 −0.382683 + 0.923880i −1.49144 + 0.996552i −0.707107 − 0.707107i 0 −0.349942 − 1.75928i 0 0.923880 − 0.382683i 0.848609 − 2.04872i 0
643.2 −0.382683 + 0.923880i 0.108761 − 0.0726721i −0.707107 − 0.707107i 0 0.0255190 + 0.128293i 0 0.923880 − 0.382683i −0.376136 + 0.908072i 0
707.1 −0.923880 + 0.382683i −1.29335 + 0.257264i 0.707107 − 0.707107i 0 1.09645 − 0.732626i 0 −0.382683 + 0.923880i 0.682699 − 0.282783i 0
707.2 −0.923880 + 0.382683i −0.630526 + 0.125419i 0.707107 − 0.707107i 0 0.534534 − 0.357164i 0 −0.382683 + 0.923880i −0.542046 + 0.224523i 0
907.1 0.382683 + 0.923880i −1.10876 + 1.65938i −0.707107 + 0.707107i 0 −1.95737 − 0.389345i 0 −0.923880 − 0.382683i −1.14150 − 2.75583i 0
907.2 0.382683 + 0.923880i 0.491445 − 0.735499i −0.707107 + 0.707107i 0 0.867580 + 0.172572i 0 −0.923880 − 0.382683i 0.0832424 + 0.200965i 0
1507.1 −0.382683 − 0.923880i −1.49144 − 0.996552i −0.707107 + 0.707107i 0 −0.349942 + 1.75928i 0 0.923880 + 0.382683i 0.848609 + 2.04872i 0
1507.2 −0.382683 − 0.923880i 0.108761 + 0.0726721i −0.707107 + 0.707107i 0 0.0255190 − 0.128293i 0 0.923880 + 0.382683i −0.376136 − 0.908072i 0
1707.1 0.923880 − 0.382683i −0.369474 − 1.85747i 0.707107 − 0.707107i 0 −1.05217 − 1.57469i 0 0.382683 − 0.923880i −2.38981 + 0.989890i 0
1707.2 0.923880 − 0.382683i 0.293353 + 1.47479i 0.707107 − 0.707107i 0 0.835400 + 1.25026i 0 0.382683 − 0.923880i −1.16506 + 0.482584i 0
2043.1 0.382683 − 0.923880i −1.10876 − 1.65938i −0.707107 − 0.707107i 0 −1.95737 + 0.389345i 0 −0.923880 + 0.382683i −1.14150 + 2.75583i 0
2043.2 0.382683 − 0.923880i 0.491445 + 0.735499i −0.707107 − 0.707107i 0 0.867580 − 0.172572i 0 −0.923880 + 0.382683i 0.0832424 − 0.200965i 0
2443.1 −0.923880 − 0.382683i −1.29335 − 0.257264i 0.707107 + 0.707107i 0 1.09645 + 0.732626i 0 −0.382683 − 0.923880i 0.682699 + 0.282783i 0
2443.2 −0.923880 − 0.382683i −0.630526 − 0.125419i 0.707107 + 0.707107i 0 0.534534 + 0.357164i 0 −0.382683 − 0.923880i −0.542046 − 0.224523i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 243.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
85.o even 16 1 inner
680.cr odd 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3400.1.dh.c yes 16
5.b even 2 1 3400.1.dh.d yes 16
5.c odd 4 1 3400.1.cx.c ✓ 16
5.c odd 4 1 3400.1.cx.d yes 16
8.d odd 2 1 CM 3400.1.dh.c yes 16
17.e odd 16 1 3400.1.cx.c ✓ 16
40.e odd 2 1 3400.1.dh.d yes 16
40.k even 4 1 3400.1.cx.c ✓ 16
40.k even 4 1 3400.1.cx.d yes 16
85.o even 16 1 inner 3400.1.dh.c yes 16
85.p odd 16 1 3400.1.cx.d yes 16
85.r even 16 1 3400.1.dh.d yes 16
136.s even 16 1 3400.1.cx.c ✓ 16
680.ch odd 16 1 3400.1.dh.d yes 16
680.co even 16 1 3400.1.cx.d yes 16
680.cr odd 16 1 inner 3400.1.dh.c yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3400.1.cx.c ✓ 16 5.c odd 4 1
3400.1.cx.c ✓ 16 17.e odd 16 1
3400.1.cx.c ✓ 16 40.k even 4 1
3400.1.cx.c ✓ 16 136.s even 16 1
3400.1.cx.d yes 16 5.c odd 4 1
3400.1.cx.d yes 16 40.k even 4 1
3400.1.cx.d yes 16 85.p odd 16 1
3400.1.cx.d yes 16 680.co even 16 1
3400.1.dh.c yes 16 1.a even 1 1 trivial
3400.1.dh.c yes 16 8.d odd 2 1 CM
3400.1.dh.c yes 16 85.o even 16 1 inner
3400.1.dh.c yes 16 680.cr odd 16 1 inner
3400.1.dh.d yes 16 5.b even 2 1
3400.1.dh.d yes 16 40.e odd 2 1
3400.1.dh.d yes 16 85.r even 16 1
3400.1.dh.d yes 16 680.ch odd 16 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} + 8 T_{3}^{15} + 36 T_{3}^{14} + 112 T_{3}^{13} + 266 T_{3}^{12} + 504 T_{3}^{11} + 784 T_{3}^{10} + \cdots + 1 \) acting on \(S_{1}^{\mathrm{new}}(3400, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} + 8 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( T^{16} - 4 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{16} \) Copy content Toggle raw display
$17$ \( T^{16} - T^{8} + 1 \) Copy content Toggle raw display
$19$ \( T^{16} + 194T^{8} + 1 \) Copy content Toggle raw display
$23$ \( T^{16} \) Copy content Toggle raw display
$29$ \( T^{16} \) Copy content Toggle raw display
$31$ \( T^{16} \) Copy content Toggle raw display
$37$ \( T^{16} \) Copy content Toggle raw display
$41$ \( T^{16} - 2 T^{12} + \cdots + 1 \) Copy content Toggle raw display
$43$ \( (T^{4} + 4 T^{3} + 6 T^{2} + \cdots + 2)^{4} \) Copy content Toggle raw display
$47$ \( T^{16} \) Copy content Toggle raw display
$53$ \( T^{16} \) Copy content Toggle raw display
$59$ \( (T^{4} - 4 T^{3} + 6 T^{2} + \cdots + 2)^{4} \) Copy content Toggle raw display
$61$ \( T^{16} \) Copy content Toggle raw display
$67$ \( T^{16} + 24 T^{12} + \cdots + 1 \) Copy content Toggle raw display
$71$ \( T^{16} \) Copy content Toggle raw display
$73$ \( T^{16} - 8 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( T^{16} \) Copy content Toggle raw display
$83$ \( (T^{8} - 2 T^{6} - 4 T^{5} + \cdots + 1)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + 24 T^{12} + \cdots + 1 \) Copy content Toggle raw display
$97$ \( (T^{8} + 2 T^{4} + 16 T^{3} + \cdots + 2)^{2} \) Copy content Toggle raw display
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