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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [85,4,Mod(84,85)] 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("85.84"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(85, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 85 = 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 85.c (of order \(2\), degree \(1\), 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: \(5.01516235049\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 72 q^{4} + 172 q^{9} + 40 q^{15} - 64 q^{16} - 156 q^{19} - 120 q^{21} - 116 q^{25} + 524 q^{26} + 44 q^{30} - 132 q^{34} - 516 q^{35} - 196 q^{36} + 160 q^{49} + 1372 q^{50} + 36 q^{51} + 1128 q^{55}+ \cdots - 1564 q^{94}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
84.1 − 5.07030i −3.15076 −17.7079 2.26997 + 10.9475i 15.9753i −2.84475 49.2221i −17.0727 55.5070 − 11.5094i
84.2 − 5.07030i 3.15076 −17.7079 −2.26997 − 10.9475i − 15.9753i 2.84475 49.2221i −17.0727 −55.5070 + 11.5094i
84.3 − 4.28591i −8.78504 −10.3690 −9.76384 − 5.44678i 37.6519i 13.5594 10.1535i 50.1770 −23.3444 + 41.8470i
84.4 − 4.28591i 8.78504 −10.3690 9.76384 + 5.44678i − 37.6519i −13.5594 10.1535i 50.1770 23.3444 − 41.8470i
84.5 − 3.30829i −0.644147 −2.94478 10.5296 − 3.75871i 2.13103i 24.9231 − 16.7241i −26.5851 −12.4349 − 34.8349i
84.6 − 3.30829i 0.644147 −2.94478 −10.5296 + 3.75871i − 2.13103i −24.9231 − 16.7241i −26.5851 12.4349 + 34.8349i
84.7 − 2.69457i −5.79684 0.739315 8.08691 − 7.72022i 15.6200i −34.0883 − 23.5487i 6.60337 −20.8027 − 21.7907i
84.8 − 2.69457i 5.79684 0.739315 −8.08691 + 7.72022i − 15.6200i 34.0883 − 23.5487i 6.60337 20.8027 + 21.7907i
84.9 − 1.90493i −8.33875 4.37124 5.21088 + 9.89175i 15.8847i 10.3098 − 23.5663i 42.5348 18.8431 − 9.92636i
84.10 − 1.90493i 8.33875 4.37124 −5.21088 − 9.89175i − 15.8847i −10.3098 − 23.5663i 42.5348 −18.8431 + 9.92636i
84.11 − 0.298039i −3.78716 7.91117 −7.52271 − 8.27097i 1.12872i 4.07231 − 4.74215i −12.6574 −2.46507 + 2.24206i
84.12 − 0.298039i 3.78716 7.91117 7.52271 + 8.27097i − 1.12872i −4.07231 − 4.74215i −12.6574 2.46507 − 2.24206i
84.13 0.298039i −3.78716 7.91117 −7.52271 + 8.27097i − 1.12872i 4.07231 4.74215i −12.6574 −2.46507 − 2.24206i
84.14 0.298039i 3.78716 7.91117 7.52271 − 8.27097i 1.12872i −4.07231 4.74215i −12.6574 2.46507 + 2.24206i
84.15 1.90493i −8.33875 4.37124 5.21088 − 9.89175i − 15.8847i 10.3098 23.5663i 42.5348 18.8431 + 9.92636i
84.16 1.90493i 8.33875 4.37124 −5.21088 + 9.89175i 15.8847i −10.3098 23.5663i 42.5348 −18.8431 − 9.92636i
84.17 2.69457i −5.79684 0.739315 8.08691 + 7.72022i − 15.6200i −34.0883 23.5487i 6.60337 −20.8027 + 21.7907i
84.18 2.69457i 5.79684 0.739315 −8.08691 − 7.72022i 15.6200i 34.0883 23.5487i 6.60337 20.8027 − 21.7907i
84.19 3.30829i −0.644147 −2.94478 10.5296 + 3.75871i − 2.13103i 24.9231 16.7241i −26.5851 −12.4349 + 34.8349i
84.20 3.30829i 0.644147 −2.94478 −10.5296 − 3.75871i 2.13103i −24.9231 16.7241i −26.5851 12.4349 − 34.8349i
See all 24 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 84.24
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
17.b even 2 1 inner
85.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 85.4.c.a ✓ 24
5.b even 2 1 inner 85.4.c.a ✓ 24
5.c odd 4 2 425.4.d.h 24
17.b even 2 1 inner 85.4.c.a ✓ 24
85.c even 2 1 inner 85.4.c.a ✓ 24
85.g odd 4 2 425.4.d.h 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
85.4.c.a ✓ 24 1.a even 1 1 trivial
85.4.c.a ✓ 24 5.b even 2 1 inner
85.4.c.a ✓ 24 17.b even 2 1 inner
85.4.c.a ✓ 24 85.c even 2 1 inner
425.4.d.h 24 5.c odd 4 2
425.4.d.h 24 85.g odd 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(85, [\chi])\).