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Properties

Label 153.3.j.a
Level $153$
Weight $3$
Character orbit 153.j
Analytic conductor $4.169$
Analytic rank $0$
Dimension $64$
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] = [153,3,Mod(86,153)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(153, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([5, 0])) N = Newforms(chi, 3, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("153.86"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Level: \( N \) \(=\) \( 153 = 3^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 153.j (of order \(6\), 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: \(4.16894804471\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(32\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 64 q - 2 q^{3} + 64 q^{4} - 18 q^{5} - 22 q^{6} + 2 q^{7} - 6 q^{9} - 44 q^{12} - 10 q^{13} + 72 q^{14} - 36 q^{15} - 128 q^{16} - 38 q^{18} - 28 q^{19} - 18 q^{20} + 88 q^{21} + 144 q^{23} - 42 q^{24}+ \cdots + 58 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
86.1 −3.46043 + 1.99788i −0.0213833 + 2.99992i 5.98304 − 10.3629i 2.66319 + 1.53759i −5.91949 − 10.4237i −0.00954207 − 0.0165273i 31.8305i −8.99909 − 0.128297i −12.2877
86.2 −3.17876 + 1.83526i −2.03325 − 2.20588i 4.73636 − 8.20362i −7.30066 − 4.21504i 10.5116 + 3.28044i −6.34017 − 10.9815i 20.0878i −0.731824 + 8.97020i 30.9428
86.3 −3.03806 + 1.75403i 1.37044 − 2.66869i 4.15322 − 7.19358i 5.76358 + 3.32760i 0.517471 + 10.5114i −1.13321 − 1.96277i 15.1072i −5.24380 − 7.31455i −23.3468
86.4 −2.91390 + 1.68234i 2.99174 + 0.222513i 3.66053 − 6.34022i −4.65913 − 2.68995i −9.09195 + 4.38474i 2.01660 + 3.49286i 11.1743i 8.90098 + 1.33140i 18.1016
86.5 −2.78989 + 1.61074i −2.97400 − 0.394134i 3.18897 − 5.52346i 0.946468 + 0.546443i 8.93196 − 3.69075i 5.45492 + 9.44819i 7.66050i 8.68932 + 2.34431i −3.52072
86.6 −2.39630 + 1.38350i −2.93584 + 0.617116i 1.82816 − 3.16647i 5.03797 + 2.90867i 6.18137 − 5.54054i −4.07760 − 7.06261i − 0.950959i 8.23834 − 3.62351i −16.0966
86.7 −2.19452 + 1.26701i 1.16546 − 2.76436i 1.21061 − 2.09685i −3.87953 − 2.23985i 0.944846 + 7.54310i 2.72279 + 4.71600i − 4.00063i −6.28341 − 6.44350i 11.3516
86.8 −2.16313 + 1.24888i −1.46489 + 2.61804i 1.11942 − 1.93890i −4.93053 − 2.84664i −0.100885 − 7.49263i 1.17060 + 2.02753i − 4.39895i −4.70822 − 7.67024i 14.2205
86.9 −2.08063 + 1.20125i 2.00274 + 2.23361i 0.886009 − 1.53461i 4.81128 + 2.77779i −6.85009 − 2.24153i 3.83640 + 6.64484i − 5.35273i −0.978063 + 8.94670i −13.3473
86.10 −1.99799 + 1.15354i 1.41704 + 2.64424i 0.661304 − 1.14541i −1.02022 − 0.589023i −5.88146 − 3.64854i −6.78276 − 11.7481i − 6.17695i −4.98399 + 7.49399i 2.71784
86.11 −1.07474 + 0.620503i 2.92558 − 0.664084i −1.22995 + 2.13034i 6.36425 + 3.67440i −2.73217 + 2.52905i 0.856563 + 1.48361i − 8.01678i 8.11798 − 3.88566i −9.11990
86.12 −0.910309 + 0.525567i 2.55415 − 1.57363i −1.44756 + 2.50724i −2.56865 − 1.48301i −1.49802 + 2.77487i −4.96653 − 8.60228i − 7.24770i 4.04736 − 8.03859i 3.11769
86.13 −0.759776 + 0.438657i −2.72733 − 1.24967i −1.61516 + 2.79754i −6.70632 − 3.87189i 2.62034 − 0.246889i 4.09746 + 7.09700i − 6.34326i 5.87663 + 6.81654i 6.79373
86.14 −0.510714 + 0.294861i −1.38625 + 2.66051i −1.82611 + 3.16292i 4.01198 + 2.31632i −0.0765013 − 1.76751i 2.98103 + 5.16330i − 4.51269i −5.15661 − 7.37627i −2.73197
86.15 −0.393387 + 0.227122i 2.39344 + 1.80872i −1.89683 + 3.28541i −5.60074 − 3.23359i −1.35235 − 0.167922i 2.18449 + 3.78365i − 3.54022i 2.45709 + 8.65810i 2.93768
86.16 −0.0918678 + 0.0530399i −2.79064 + 1.10106i −1.99437 + 3.45436i −0.368715 − 0.212878i 0.197970 − 0.249167i −3.49077 − 6.04619i − 0.847445i 6.57534 − 6.14532i 0.0451640
86.17 0.424981 − 0.245363i 0.705860 − 2.91578i −1.87959 + 3.25555i 3.47947 + 2.00887i −0.415447 − 1.41234i 6.36001 + 11.0159i 3.80764i −8.00352 − 4.11626i 1.97161
86.18 0.656590 − 0.379082i −2.05010 − 2.19023i −1.71259 + 2.96630i −0.533620 − 0.308086i −2.17635 − 0.660923i −2.18450 − 3.78366i 5.62951i −0.594178 + 8.98036i −0.467159
86.19 0.700836 − 0.404628i 0.807723 − 2.88922i −1.67255 + 2.89695i −6.73682 − 3.88950i −0.602977 − 2.35170i −1.30137 − 2.25403i 5.94407i −7.69517 − 4.66738i −6.29521
86.20 0.784435 − 0.452894i 1.03612 + 2.81540i −1.58977 + 2.75357i 6.71259 + 3.87552i 2.08785 + 1.73924i −5.18495 − 8.98060i 6.50315i −6.85291 + 5.83418i 7.02079
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 86.32
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 153.3.j.a ✓ 64
3.b odd 2 1 459.3.j.a 64
9.c even 3 1 459.3.j.a 64
9.d odd 6 1 inner 153.3.j.a ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
153.3.j.a ✓ 64 1.a even 1 1 trivial
153.3.j.a ✓ 64 9.d odd 6 1 inner
459.3.j.a 64 3.b odd 2 1
459.3.j.a 64 9.c even 3 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(153, [\chi])\).