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Properties

Label 700.2.n.d
Level $700$
Weight $2$
Character orbit 700.n
Analytic conductor $5.590$
Analytic rank $0$
Dimension $28$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [700,2,Mod(141,700)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("700.141"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(700, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 2, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 700.n (of order \(5\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [28,0,-4] 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: \(5.58952814149\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(7\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 28 q - 4 q^{3} + 6 q^{5} - 28 q^{7} - q^{9} - 4 q^{11} - 7 q^{15} + 2 q^{17} + 4 q^{19} + 4 q^{21} - 3 q^{23} + 4 q^{25} + 5 q^{27} + 15 q^{29} - 25 q^{31} + 21 q^{33} - 6 q^{35} - 15 q^{37} - 22 q^{39}+ \cdots - 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
141.1 0 −0.823863 − 2.53559i 0 2.21192 − 0.327720i 0 −1.00000 0 −3.32341 + 2.41460i 0
141.2 0 −0.453512 − 1.39577i 0 −2.08808 − 0.799951i 0 −1.00000 0 0.684559 − 0.497362i 0
141.3 0 −0.445089 − 1.36984i 0 −1.51970 + 1.64028i 0 −1.00000 0 0.748683 − 0.543950i 0
141.4 0 0.158002 + 0.486281i 0 1.04138 + 1.97877i 0 −1.00000 0 2.21555 − 1.60969i 0
141.5 0 0.328972 + 1.01247i 0 0.590587 − 2.15667i 0 −1.00000 0 1.51017 − 1.09721i 0
141.6 0 0.451984 + 1.39106i 0 −1.97011 − 1.05767i 0 −1.00000 0 0.696282 − 0.505879i 0
141.7 0 0.901540 + 2.77465i 0 2.11597 + 0.722957i 0 −1.00000 0 −4.45888 + 3.23957i 0
281.1 0 −2.73964 + 1.99046i 0 0.753315 + 2.10535i 0 −1.00000 0 2.61662 − 8.05312i 0
281.2 0 −1.74108 + 1.26497i 0 0.853584 − 2.06674i 0 −1.00000 0 0.504169 − 1.55167i 0
281.3 0 −0.743793 + 0.540397i 0 2.18245 + 0.486729i 0 −1.00000 0 −0.665852 + 2.04928i 0
281.4 0 −0.238779 + 0.173483i 0 −1.65249 − 1.50641i 0 −1.00000 0 −0.900132 + 2.77032i 0
281.5 0 0.256610 − 0.186438i 0 −1.76490 + 1.37300i 0 −1.00000 0 −0.895962 + 2.75749i 0
281.6 0 0.918117 − 0.667051i 0 1.45372 + 1.69903i 0 −1.00000 0 −0.529069 + 1.62831i 0
281.7 0 2.17053 − 1.57698i 0 0.792358 − 2.09097i 0 −1.00000 0 1.29728 − 3.99262i 0
421.1 0 −2.73964 − 1.99046i 0 0.753315 − 2.10535i 0 −1.00000 0 2.61662 + 8.05312i 0
421.2 0 −1.74108 − 1.26497i 0 0.853584 + 2.06674i 0 −1.00000 0 0.504169 + 1.55167i 0
421.3 0 −0.743793 − 0.540397i 0 2.18245 − 0.486729i 0 −1.00000 0 −0.665852 − 2.04928i 0
421.4 0 −0.238779 − 0.173483i 0 −1.65249 + 1.50641i 0 −1.00000 0 −0.900132 − 2.77032i 0
421.5 0 0.256610 + 0.186438i 0 −1.76490 − 1.37300i 0 −1.00000 0 −0.895962 − 2.75749i 0
421.6 0 0.918117 + 0.667051i 0 1.45372 − 1.69903i 0 −1.00000 0 −0.529069 − 1.62831i 0
See all 28 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 141.7
Significant digits:
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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 700.2.n.d ✓ 28
25.d even 5 1 inner 700.2.n.d ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
700.2.n.d ✓ 28 1.a even 1 1 trivial
700.2.n.d ✓ 28 25.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{28} + 4 T_{3}^{27} + 19 T_{3}^{26} + 51 T_{3}^{25} + 207 T_{3}^{24} + 379 T_{3}^{23} + 1577 T_{3}^{22} + \cdots + 625 \) acting on \(S_{2}^{\mathrm{new}}(700, [\chi])\). Copy content Toggle raw display