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Properties

Label 85.4.j.a
Level $85$
Weight $4$
Character orbit 85.j
Analytic conductor $5.015$
Analytic rank $0$
Dimension $48$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [85,4,Mod(4,85)] 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("85.4"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(85, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 3])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 85 = 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 85.j (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: \(5.01516235049\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 152 q^{4} + 20 q^{5} - 48 q^{6} - 74 q^{10} + 44 q^{11} - 140 q^{14} + 576 q^{16} + 142 q^{20} + 424 q^{21} - 268 q^{24} - 656 q^{29} - 1288 q^{30} + 388 q^{31} - 1084 q^{34} + 116 q^{35} + 472 q^{39}+ \cdots + 380 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −5.49007 3.49159 + 3.49159i 22.1409 10.9209 + 2.39440i −19.1690 − 19.1690i 9.38887 − 9.38887i −77.6343 − 2.61767i −59.9567 − 13.1454i
4.2 −4.88667 −0.654173 − 0.654173i 15.8796 −11.1284 − 1.07683i 3.19673 + 3.19673i 8.11459 − 8.11459i −38.5048 − 26.1441i 54.3806 + 5.26212i
4.3 −4.42883 −4.71387 − 4.71387i 11.6145 −0.143060 + 11.1794i 20.8769 + 20.8769i −13.3994 + 13.3994i −16.0081 17.4411i 0.633588 − 49.5117i
4.4 −4.05788 3.16864 + 3.16864i 8.46639 −3.10578 − 10.7403i −12.8580 − 12.8580i −16.9033 + 16.9033i −1.89255 − 6.91942i 12.6029 + 43.5829i
4.5 −4.02533 −4.29848 − 4.29848i 8.20326 8.65777 − 7.07411i 17.3028 + 17.3028i 11.1925 − 11.1925i −0.818187 9.95379i −34.8504 + 28.4756i
4.6 −3.35026 4.45603 + 4.45603i 3.22426 1.72766 + 11.0460i −14.9289 − 14.9289i −8.29133 + 8.29133i 16.0000 12.7124i −5.78812 − 37.0072i
4.7 −2.43856 6.23661 + 6.23661i −2.05345 5.63180 − 9.65830i −15.2083 − 15.2083i 23.5847 − 23.5847i 24.5159 50.7905i −13.7335 + 23.5523i
4.8 −1.90094 1.18712 + 1.18712i −4.38641 −9.44148 + 5.98820i −2.25665 − 2.25665i 15.9731 − 15.9731i 23.5459 − 24.1815i 17.9477 − 11.3832i
4.9 −1.79897 −5.09247 − 5.09247i −4.76370 −8.00088 − 7.80935i 9.16121 + 9.16121i −4.33980 + 4.33980i 22.9615 24.8666i 14.3934 + 14.0488i
4.10 −1.76549 −0.0663990 − 0.0663990i −4.88303 10.7808 − 2.96201i 0.117227 + 0.117227i −17.3319 + 17.3319i 22.7449 − 26.9912i −19.0335 + 5.22941i
4.11 −0.730842 −1.09750 − 1.09750i −7.46587 10.1638 + 4.65798i 0.802103 + 0.802103i 4.01572 − 4.01572i 11.3031 − 24.5910i −7.42815 − 3.40425i
4.12 −0.153679 −6.27218 − 6.27218i −7.97638 1.01595 + 11.1341i 0.963902 + 0.963902i 13.3787 − 13.3787i 2.45523 51.6805i −0.156130 − 1.71107i
4.13 0.153679 6.27218 + 6.27218i −7.97638 −11.1341 − 1.01595i 0.963902 + 0.963902i −13.3787 + 13.3787i −2.45523 51.6805i −1.71107 − 0.156130i
4.14 0.730842 1.09750 + 1.09750i −7.46587 −4.65798 − 10.1638i 0.802103 + 0.802103i −4.01572 + 4.01572i −11.3031 − 24.5910i −3.40425 − 7.42815i
4.15 1.76549 0.0663990 + 0.0663990i −4.88303 2.96201 − 10.7808i 0.117227 + 0.117227i 17.3319 − 17.3319i −22.7449 − 26.9912i 5.22941 − 19.0335i
4.16 1.79897 5.09247 + 5.09247i −4.76370 7.80935 + 8.00088i 9.16121 + 9.16121i 4.33980 − 4.33980i −22.9615 24.8666i 14.0488 + 14.3934i
4.17 1.90094 −1.18712 − 1.18712i −4.38641 −5.98820 + 9.44148i −2.25665 − 2.25665i −15.9731 + 15.9731i −23.5459 − 24.1815i −11.3832 + 17.9477i
4.18 2.43856 −6.23661 − 6.23661i −2.05345 9.65830 − 5.63180i −15.2083 − 15.2083i −23.5847 + 23.5847i −24.5159 50.7905i 23.5523 − 13.7335i
4.19 3.35026 −4.45603 − 4.45603i 3.22426 −11.0460 − 1.72766i −14.9289 − 14.9289i 8.29133 − 8.29133i −16.0000 12.7124i −37.0072 − 5.78812i
4.20 4.02533 4.29848 + 4.29848i 8.20326 7.07411 − 8.65777i 17.3028 + 17.3028i −11.1925 + 11.1925i 0.818187 9.95379i 28.4756 − 34.8504i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.24
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.c even 4 1 inner
85.j even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 85.4.j.a ✓ 48
5.b even 2 1 inner 85.4.j.a ✓ 48
17.c even 4 1 inner 85.4.j.a ✓ 48
85.j even 4 1 inner 85.4.j.a ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
85.4.j.a ✓ 48 1.a even 1 1 trivial
85.4.j.a ✓ 48 5.b even 2 1 inner
85.4.j.a ✓ 48 17.c even 4 1 inner
85.4.j.a ✓ 48 85.j even 4 1 inner

Hecke kernels

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