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Properties

Label 153.3.t.a
Level $153$
Weight $3$
Character orbit 153.t
Analytic conductor $4.169$
Analytic rank $0$
Dimension $544$
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] = [153,3,Mod(7,153)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(153, base_ring=CyclotomicField(48)) chi = DirichletCharacter(H, H._module([32, 33])) 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("153.7"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Level: \( N \) \(=\) \( 153 = 3^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 153.t (of order \(48\), degree \(16\), 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: \(544\)
Relative dimension: \(34\) over \(\Q(\zeta_{48})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{48}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 544 q - 8 q^{2} - 16 q^{3} - 8 q^{4} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 32 q^{8} - 16 q^{9} - 32 q^{10} - 8 q^{11} + 176 q^{12} - 8 q^{13} - 8 q^{14} - 40 q^{15} - 32 q^{17} + 32 q^{18} - 32 q^{19} - 8 q^{20}+ \cdots + 448 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −2.34971 − 3.06221i −2.49301 − 1.66880i −2.82067 + 10.5269i 4.51184 − 3.95678i 0.747673 + 11.5553i 7.72900 + 6.77815i 24.5993 − 10.1893i 3.43024 + 8.32066i −22.7180 − 4.51889i
7.2 −2.30425 − 3.00295i 1.06273 + 2.80546i −2.67290 + 9.97539i 1.79172 − 1.57130i 5.97588 − 9.65580i −7.21963 − 6.33144i 22.1266 − 9.16514i −6.74122 + 5.96288i −8.84710 − 1.75980i
7.3 −1.99492 − 2.59982i 2.99475 + 0.177381i −1.74412 + 6.50916i −5.15807 + 4.52351i −5.51312 − 8.13969i 1.60281 + 1.40563i 8.29180 − 3.43458i 8.93707 + 1.06242i 22.0503 + 4.38607i
7.4 −1.98557 − 2.58765i −2.23949 + 1.99617i −1.71816 + 6.41225i −3.06932 + 2.69172i 9.61206 + 1.83148i 1.95779 + 1.71694i 7.95066 − 3.29327i 1.03063 − 8.94079i 13.0596 + 2.59771i
7.5 −1.98320 − 2.58456i 2.10823 − 2.13433i −1.71157 + 6.38767i 3.10070 − 2.71924i −9.69733 − 1.21605i −0.332878 − 0.291926i 7.86457 − 3.25761i −0.110711 − 8.99932i −13.1773 − 2.62114i
7.6 −1.62741 − 2.12088i −0.107726 − 2.99807i −0.814394 + 3.03936i −4.22014 + 3.70097i −6.18322 + 5.10755i 5.62018 + 4.92877i −2.10780 + 0.873079i −8.97679 + 0.645938i 14.7172 + 2.92743i
7.7 −1.60099 − 2.08646i −2.95447 − 0.520684i −0.754842 + 2.81711i 0.0116724 − 0.0102364i 3.64370 + 6.99798i −8.44362 − 7.40486i −2.63265 + 1.09048i 8.45778 + 3.07669i −0.0400452 − 0.00796548i
7.8 −1.45745 − 1.89939i 1.40195 + 2.65227i −0.448230 + 1.67282i 3.48496 − 3.05623i 2.99440 − 6.52840i 7.58002 + 6.64750i −5.01693 + 2.07808i −5.06905 + 7.43671i −10.8841 − 2.16498i
7.9 −1.42645 − 1.85898i −1.02126 − 2.82082i −0.385791 + 1.43979i 1.34400 − 1.17865i −3.78709 + 5.92226i −4.37305 − 3.83506i −5.43247 + 2.25020i −6.91405 + 5.76159i −4.10824 − 0.817180i
7.10 −1.11526 − 1.45343i 2.99911 + 0.0730780i 0.166611 − 0.621802i 5.00581 − 4.38998i −3.23857 − 4.44050i −5.80360 − 5.08962i −7.85979 + 3.25563i 8.98932 + 0.438338i −11.9633 − 2.37965i
7.11 −1.02422 − 1.33479i 1.18397 + 2.75649i 0.302643 − 1.12948i −4.59546 + 4.03011i 2.46668 − 4.40359i −4.37877 − 3.84008i −8.03515 + 3.32827i −6.19644 + 6.52718i 10.0861 + 2.00625i
7.12 −0.729327 − 0.950477i −2.99802 + 0.108885i 0.663787 − 2.47729i 4.84694 − 4.25065i 2.29003 + 2.77014i 3.17511 + 2.78450i −7.26614 + 3.00973i 8.97629 − 0.652881i −7.57515 − 1.50679i
7.13 −0.728437 − 0.949317i −1.51905 + 2.58698i 0.664693 − 2.48067i −1.46882 + 1.28812i 3.56240 − 0.442392i 2.80528 + 2.46016i −7.26114 + 3.00766i −4.38496 − 7.85952i 2.29278 + 0.456063i
7.14 −0.513504 − 0.669211i −2.79121 − 1.09961i 0.851119 − 3.17642i −6.13530 + 5.38051i 0.697425 + 2.43256i 4.80568 + 4.21447i −5.68000 + 2.35273i 6.58171 + 6.13849i 6.75120 + 1.34290i
7.15 −0.434047 − 0.565661i 1.99588 − 2.23974i 0.903701 − 3.37266i −4.79864 + 4.20829i −2.13324 − 0.156838i −7.75304 − 6.79923i −4.93493 + 2.04412i −1.03291 − 8.94053i 4.46330 + 0.887806i
7.16 −0.284452 − 0.370706i 2.74120 − 1.21894i 0.978767 − 3.65281i −0.671178 + 0.588607i −1.23161 − 0.669448i 9.62460 + 8.44054i −3.35931 + 1.39147i 6.02836 − 6.68273i 0.409118 + 0.0813786i
7.17 −0.245388 − 0.319796i 0.0747680 − 2.99907i 0.993222 − 3.70675i 6.39903 − 5.61180i −0.977438 + 0.712026i 3.23973 + 2.84117i −2.91878 + 1.20900i −8.98882 − 0.448469i −3.36488 − 0.669316i
7.18 −0.0485999 − 0.0633367i −0.879538 + 2.86817i 1.03363 − 3.85755i 3.26851 − 2.86641i 0.224406 − 0.0836860i −8.31214 − 7.28955i −0.589587 + 0.244215i −7.45283 − 5.04533i −0.340399 − 0.0677095i
7.19 0.286178 + 0.372954i −1.53721 − 2.57624i 0.978079 − 3.65024i −1.59407 + 1.39796i 0.520903 − 1.31057i −2.48802 − 2.18193i 3.37853 − 1.39943i −4.27399 + 7.92042i −0.977562 − 0.194449i
7.20 0.434420 + 0.566147i 2.71144 + 1.28378i 0.903474 − 3.37181i 1.70774 − 1.49765i 0.451096 + 2.09277i −0.856685 − 0.751292i 4.93860 − 2.04563i 5.70382 + 6.96178i 1.58977 + 0.316224i
See next 80 embeddings (of 544 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.34
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner
17.e odd 16 1 inner
153.t odd 48 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 153.3.t.a ✓ 544
9.c even 3 1 inner 153.3.t.a ✓ 544
17.e odd 16 1 inner 153.3.t.a ✓ 544
153.t odd 48 1 inner 153.3.t.a ✓ 544
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
153.3.t.a ✓ 544 1.a even 1 1 trivial
153.3.t.a ✓ 544 9.c even 3 1 inner
153.3.t.a ✓ 544 17.e odd 16 1 inner
153.3.t.a ✓ 544 153.t odd 48 1 inner

Hecke kernels

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