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Properties

Label 409.2.d.a
Level $409$
Weight $2$
Character orbit 409.d
Analytic conductor $3.266$
Analytic rank $0$
Dimension $66$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [409,2,Mod(143,409)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(409, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("409.143"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 409 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 409.d (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.26588144267\)
Analytic rank: \(0\)
Dimension: \(66\)
Relative dimension: \(33\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 66 q - 58 q^{4} + 16 q^{6} - 86 q^{9} - 4 q^{11} + 14 q^{13} - 6 q^{14} + 34 q^{16} + 16 q^{17} + 6 q^{19} - 24 q^{20} + 6 q^{21} + 4 q^{22} - 36 q^{24} + 38 q^{25} + 6 q^{26} - 24 q^{28} + 8 q^{29} - 24 q^{30}+ \cdots + 18 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
143.1 − 2.80746i 2.16504i −5.88184 0.263151 6.07827 0.911633 − 0.911633i 10.8981i −1.68741 − 0.738786i
143.2 − 2.43543i − 2.99595i −3.93133 0.519934 −7.29644 −1.17659 + 1.17659i 4.70363i −5.97574 − 1.26626i
143.3 − 2.43426i − 0.775144i −3.92563 2.74879 −1.88690 1.18782 − 1.18782i 4.68749i 2.39915 − 6.69129i
143.4 − 2.09667i 1.97910i −2.39602 −2.60133 4.14952 0.312632 − 0.312632i 0.830330i −0.916830 5.45413i
143.5 − 2.05898i − 2.54764i −2.23939 −1.88272 −5.24554 3.39644 − 3.39644i 0.492891i −3.49048 3.87647i
143.6 − 2.05379i 1.19943i −2.21804 −0.896544 2.46336 −3.72745 + 3.72745i 0.447809i 1.56138 1.84131i
143.7 − 1.96022i − 0.0523983i −1.84245 1.99566 −0.102712 0.623221 − 0.623221i − 0.308840i 2.99725 − 3.91192i
143.8 − 1.65580i 2.89110i −0.741661 2.89126 4.78707 1.70816 − 1.70816i − 2.08355i −5.35844 − 4.78733i
143.9 − 1.34891i − 0.276022i 0.180448 −0.908760 −0.372328 0.0810166 − 0.0810166i − 2.94122i 2.92381 1.22583i
143.10 − 1.28373i − 2.13946i 0.352045 −3.25886 −2.74648 −2.13840 + 2.13840i − 3.01938i −1.57728 4.18348i
143.11 − 1.14090i − 2.12403i 0.698354 3.71349 −2.42330 −2.57191 + 2.57191i − 3.07854i −1.51151 − 4.23671i
143.12 − 0.991585i 2.21084i 1.01676 −3.39717 2.19224 3.18091 − 3.18091i − 2.99137i −1.88782 3.36858i
143.13 − 0.626528i 3.21664i 1.60746 −0.967528 2.01532 −2.91712 + 2.91712i − 2.26018i −7.34680 0.606183i
143.14 − 0.625605i 1.14546i 1.60862 2.66443 0.716604 −1.71683 + 1.71683i − 2.25757i 1.68793 − 1.66688i
143.15 − 0.617267i − 3.01639i 1.61898 0.890763 −1.86192 1.10470 − 1.10470i − 2.23388i −6.09859 − 0.549839i
143.16 − 0.0897084i 1.11914i 1.99195 0.205533 0.100396 3.05640 − 3.05640i − 0.358112i 1.74753 − 0.0184380i
143.17 0.0671373i − 1.35085i 1.99549 0.275246 0.0906926 0.249444 − 0.249444i 0.268247i 1.17520 0.0184793i
143.18 0.0702259i 0.0945115i 1.99507 −2.90843 −0.00663715 −0.764853 + 0.764853i 0.280557i 2.99107 − 0.204247i
143.19 0.417704i 2.31344i 1.82552 1.64451 −0.966331 0.111455 − 0.111455i 1.59793i −2.35200 0.686917i
143.20 0.822130i − 2.29851i 1.32410 −3.97213 1.88967 2.86832 − 2.86832i 2.73284i −2.28316 − 3.26560i
See all 66 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 143.33
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
409.d even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 409.2.d.a ✓ 66
409.d even 4 1 inner 409.2.d.a ✓ 66
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
409.2.d.a ✓ 66 1.a even 1 1 trivial
409.2.d.a ✓ 66 409.d even 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(409, [\chi])\).