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Properties

Label 598.2.k.f
Level $598$
Weight $2$
Character orbit 598.k
Analytic conductor $4.775$
Analytic rank $0$
Dimension $60$
Inner twists $2$

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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] = [598,2,Mod(27,598)] 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("598.27"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(598, base_ring=CyclotomicField(22)) chi = DirichletCharacter(H, H._module([0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 598 = 2 \cdot 13 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 598.k (of order \(11\), degree \(10\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [60,6,0,-6,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.77505404087\)
Analytic rank: \(0\)
Dimension: \(60\)
Relative dimension: \(6\) over \(\Q(\zeta_{11})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 60 q + 6 q^{2} - 6 q^{4} - 3 q^{5} + 13 q^{7} + 6 q^{8} - 2 q^{9} - 8 q^{10} - 4 q^{11} + 11 q^{12} - 6 q^{13} + 9 q^{14} + 21 q^{15} - 6 q^{16} + 18 q^{17} + 2 q^{18} - 11 q^{19} - 3 q^{20} - 29 q^{21}+ \cdots + 36 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
27.1 −0.415415 − 0.909632i −3.13267 + 0.919834i −0.654861 + 0.755750i −0.985274 − 0.633197i 2.13807 + 2.46746i −0.121161 − 0.842695i 0.959493 + 0.281733i 6.44375 − 4.14115i −0.166679 + 1.15928i
27.2 −0.415415 − 0.909632i −1.65011 + 0.484516i −0.654861 + 0.755750i −0.986982 − 0.634295i 1.12621 + 1.29972i 0.0179750 + 0.125019i 0.959493 + 0.281733i −0.0356585 + 0.0229163i −0.166968 + 1.16129i
27.3 −0.415415 − 0.909632i −0.263871 + 0.0774796i −0.654861 + 0.755750i 2.41093 + 1.54941i 0.180094 + 0.207839i 0.284581 + 1.97930i 0.959493 + 0.281733i −2.46014 + 1.58103i 0.407858 − 2.83671i
27.4 −0.415415 − 0.909632i 0.310477 − 0.0911643i −0.654861 + 0.755750i −1.40637 − 0.903821i −0.211903 − 0.244549i −0.751602 − 5.22750i 0.959493 + 0.281733i −2.43568 + 1.56531i −0.237916 + 1.65474i
27.5 −0.415415 − 0.909632i 0.984709 − 0.289137i −0.654861 + 0.755750i −3.21203 − 2.06425i −0.672071 − 0.775611i 0.658581 + 4.58053i 0.959493 + 0.281733i −1.63771 + 1.05249i −0.543379 + 3.77928i
27.6 −0.415415 − 0.909632i 2.68119 − 0.787267i −0.654861 + 0.755750i 1.99722 + 1.28353i −1.82993 − 2.11185i −0.465836 − 3.23996i 0.959493 + 0.281733i 4.04521 − 2.59970i 0.337869 − 2.34993i
105.1 0.654861 + 0.755750i −1.54754 − 0.994541i −0.142315 + 0.989821i −1.83517 + 4.01847i −0.261797 − 1.82084i −0.992517 − 0.291429i −0.841254 + 0.540641i 0.159512 + 0.349283i −4.23874 + 1.24461i
105.2 0.654861 + 0.755750i −1.49766 − 0.962490i −0.142315 + 0.989821i −0.500058 + 1.09497i −0.253360 − 1.76215i 4.15445 + 1.21986i −0.841254 + 0.540641i 0.0703631 + 0.154074i −1.15500 + 0.339137i
105.3 0.654861 + 0.755750i −0.213281 − 0.137067i −0.142315 + 0.989821i 1.54954 − 3.39303i −0.0360808 − 0.250947i −3.09616 − 0.909114i −0.841254 + 0.540641i −1.21954 − 2.67043i 3.57901 − 1.05089i
105.4 0.654861 + 0.755750i 0.483269 + 0.310578i −0.142315 + 0.989821i 0.896658 − 1.96341i 0.0817547 + 0.568616i 2.62306 + 0.770200i −0.841254 + 0.540641i −1.10915 − 2.42871i 2.07103 − 0.608110i
105.5 0.654861 + 0.755750i 1.39176 + 0.894432i −0.142315 + 0.989821i −1.10621 + 2.42226i 0.235445 + 1.63755i −4.07399 − 1.19623i −0.841254 + 0.540641i −0.109248 − 0.239220i −2.55503 + 0.750226i
105.6 0.654861 + 0.755750i 1.89599 + 1.21848i −0.142315 + 0.989821i −0.335592 + 0.734843i 0.320745 + 2.23083i 1.55839 + 0.457585i −0.841254 + 0.540641i 0.863851 + 1.89157i −0.775123 + 0.227597i
131.1 0.654861 − 0.755750i −1.54754 + 0.994541i −0.142315 − 0.989821i −1.83517 − 4.01847i −0.261797 + 1.82084i −0.992517 + 0.291429i −0.841254 − 0.540641i 0.159512 − 0.349283i −4.23874 − 1.24461i
131.2 0.654861 − 0.755750i −1.49766 + 0.962490i −0.142315 − 0.989821i −0.500058 − 1.09497i −0.253360 + 1.76215i 4.15445 − 1.21986i −0.841254 − 0.540641i 0.0703631 − 0.154074i −1.15500 − 0.339137i
131.3 0.654861 − 0.755750i −0.213281 + 0.137067i −0.142315 − 0.989821i 1.54954 + 3.39303i −0.0360808 + 0.250947i −3.09616 + 0.909114i −0.841254 − 0.540641i −1.21954 + 2.67043i 3.57901 + 1.05089i
131.4 0.654861 − 0.755750i 0.483269 − 0.310578i −0.142315 − 0.989821i 0.896658 + 1.96341i 0.0817547 − 0.568616i 2.62306 − 0.770200i −0.841254 − 0.540641i −1.10915 + 2.42871i 2.07103 + 0.608110i
131.5 0.654861 − 0.755750i 1.39176 − 0.894432i −0.142315 − 0.989821i −1.10621 − 2.42226i 0.235445 − 1.63755i −4.07399 + 1.19623i −0.841254 − 0.540641i −0.109248 + 0.239220i −2.55503 − 0.750226i
131.6 0.654861 − 0.755750i 1.89599 − 1.21848i −0.142315 − 0.989821i −0.335592 − 0.734843i 0.320745 − 2.23083i 1.55839 − 0.457585i −0.841254 − 0.540641i 0.863851 − 1.89157i −0.775123 − 0.227597i
209.1 −0.841254 − 0.540641i −0.339420 − 2.36072i 0.415415 + 0.909632i −3.64185 − 1.06934i −0.990764 + 2.16947i 1.23762 − 1.42829i 0.142315 − 0.989821i −2.57932 + 0.757356i 2.48559 + 2.86852i
209.2 −0.841254 − 0.540641i −0.317967 − 2.21151i 0.415415 + 0.909632i 3.91448 + 1.14940i −0.928140 + 2.03234i −1.81293 + 2.09223i 0.142315 − 0.989821i −1.91118 + 0.561172i −2.67166 − 3.08326i
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 27.6
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.c even 11 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 598.2.k.f ✓ 60
23.c even 11 1 inner 598.2.k.f ✓ 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
598.2.k.f ✓ 60 1.a even 1 1 trivial
598.2.k.f ✓ 60 23.c even 11 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{60} + 10 T_{3}^{58} + 7 T_{3}^{57} + 72 T_{3}^{56} + 196 T_{3}^{55} + 93 T_{3}^{54} + 804 T_{3}^{53} + \cdots + 121 \) acting on \(S_{2}^{\mathrm{new}}(598, [\chi])\). Copy content Toggle raw display