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Properties

Label 600.2.bg.a
Level $600$
Weight $2$
Character orbit 600.bg
Analytic conductor $4.791$
Analytic rank $0$
Dimension $464$
Inner twists $8$

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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] = [600,2,Mod(11,600)] 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("600.11"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(600, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([5, 5, 5, 8])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 600.bg (of order \(10\), degree \(4\), 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: \(4.79102412128\)
Analytic rank: \(0\)
Dimension: \(464\)
Relative dimension: \(116\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 464 q - 6 q^{3} - 6 q^{4} - 7 q^{6} - 6 q^{9} - 8 q^{10} + 3 q^{12} + 6 q^{16} + 2 q^{18} - 12 q^{19} - 22 q^{22} - 16 q^{25} - 6 q^{27} + 34 q^{28} + 3 q^{30} - 18 q^{33} - 30 q^{34} + 35 q^{36} + 4 q^{40}+ \cdots - 52 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −1.41195 − 0.0799117i 0.555376 − 1.64060i 1.98723 + 0.225663i −0.650521 + 2.13935i −0.915268 + 2.27207i − 0.00558951i −2.78784 − 0.477429i −2.38311 − 1.82230i 1.08946 − 2.96868i
11.2 −1.41061 − 0.100834i −1.43639 + 0.967882i 1.97967 + 0.284475i 1.59829 − 1.56380i 2.12378 − 1.22047i − 3.56157i −2.76386 − 0.600901i 1.12641 − 2.78050i −2.41225 + 2.04476i
11.3 −1.40993 − 0.110049i 0.774089 + 1.54945i 1.97578 + 0.310323i 0.0342163 − 2.23581i −0.920892 − 2.26979i 0.444006i −2.75155 − 0.654965i −1.80157 + 2.39882i −0.294292 + 3.14855i
11.4 −1.40671 − 0.145467i 1.73000 + 0.0842968i 1.95768 + 0.409261i 1.75536 − 1.38518i −2.42135 − 0.370239i − 3.20070i −2.69436 − 0.860490i 2.98579 + 0.291667i −2.67078 + 1.69320i
11.5 −1.40552 + 0.156556i −0.229611 + 1.71676i 1.95098 − 0.440084i −1.91555 + 1.15354i 0.0539541 − 2.44890i 3.08290i −2.67325 + 0.923985i −2.89456 − 0.788375i 2.51176 − 1.92121i
11.6 −1.39814 + 0.212628i −1.71398 + 0.249524i 1.90958 − 0.594567i 2.23545 + 0.0525986i 2.34333 − 0.713310i 5.07150i −2.54343 + 1.23732i 2.87548 − 0.855360i −3.13665 + 0.401779i
11.7 −1.39616 + 0.225221i 1.71428 + 0.247508i 1.89855 − 0.628892i −2.15684 + 0.589963i −2.44915 + 0.0405298i − 2.80239i −2.50905 + 1.30563i 2.87748 + 0.848593i 2.87843 − 1.30945i
11.8 −1.39087 + 0.255890i −1.52542 − 0.820417i 1.86904 − 0.711820i −1.32561 − 1.80077i 2.33160 + 0.750753i − 1.33729i −2.41745 + 1.46832i 1.65383 + 2.50297i 2.30455 + 2.16542i
11.9 −1.38135 − 0.303105i −1.08687 − 1.34860i 1.81625 + 0.837389i 1.78652 + 1.34475i 1.09258 + 2.19232i − 1.71139i −2.25506 − 1.70724i −0.637420 + 2.93150i −2.06020 − 2.39908i
11.10 −1.37376 + 0.335826i −0.521934 + 1.65154i 1.77444 − 0.922689i 1.55431 + 1.60752i 0.162383 − 2.44410i − 2.63515i −2.12780 + 1.86346i −2.45517 − 1.72399i −2.67510 − 1.68637i
11.11 −1.37372 + 0.335982i −0.117705 − 1.72805i 1.77423 − 0.923092i 0.424365 − 2.19543i 0.742286 + 2.33431i 2.21911i −2.12716 + 1.86418i −2.97229 + 0.406799i 0.154664 + 3.15849i
11.12 −1.35704 − 0.398052i 1.73190 + 0.0228378i 1.68311 + 1.08034i 0.581184 + 2.15922i −2.34117 − 0.720378i 4.62248i −1.85401 − 2.13603i 2.99896 + 0.0791057i 0.0707914 − 3.16149i
11.13 −1.31367 − 0.523711i −1.69362 + 0.362841i 1.45145 + 1.37597i −1.56947 + 1.59273i 2.41488 + 0.410313i − 0.701288i −1.18612 − 2.56771i 2.73669 − 1.22903i 2.89589 − 1.27037i
11.14 −1.29339 − 0.571955i 0.748811 + 1.56182i 1.34573 + 1.47953i 2.17429 + 0.521965i −0.0752160 − 2.44833i 1.85317i −0.894342 − 2.68331i −1.87856 + 2.33902i −2.51368 − 1.91870i
11.15 −1.29234 − 0.574328i 0.528605 − 1.64942i 1.34029 + 1.48446i −1.48979 − 1.66749i −1.63044 + 1.82802i − 4.77372i −0.879555 − 2.68819i −2.44115 − 1.74378i 0.967627 + 3.01060i
11.16 −1.28564 + 0.589172i 1.31833 − 1.12339i 1.30575 − 1.51493i 1.92718 + 1.13402i −1.03304 + 2.22100i − 1.75442i −0.786177 + 2.71697i 0.476004 − 2.96200i −3.14579 − 0.322500i
11.17 −1.27365 − 0.614667i −0.999725 + 1.41441i 1.24437 + 1.56574i −0.851239 − 2.06770i 2.14269 − 1.18696i 1.35122i −0.622484 − 2.75908i −1.00110 − 2.82804i −0.186766 + 3.15676i
11.18 −1.26700 + 0.628261i 1.28529 − 1.16105i 1.21058 − 1.59201i −2.23275 − 0.121818i −0.899016 + 2.27855i 2.37421i −0.533598 + 2.77764i 0.303928 − 2.98456i 2.90542 − 1.24840i
11.19 −1.22532 + 0.706109i 1.70092 + 0.326889i 1.00282 − 1.73042i 1.59818 − 1.56391i −2.31500 + 0.800494i 3.30892i −0.00691212 + 2.82842i 2.78629 + 1.11203i −0.853993 + 3.04478i
11.20 −1.21408 − 0.725267i 0.386842 + 1.68830i 0.947976 + 1.76106i −0.265182 + 2.22029i 0.754810 − 2.33029i − 4.86011i 0.126322 − 2.82560i −2.70071 + 1.30621i 1.93225 − 2.50328i
See next 80 embeddings (of 464 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.116
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.d odd 2 1 inner
24.f even 2 1 inner
25.d even 5 1 inner
75.j odd 10 1 inner
200.n odd 10 1 inner
600.bg even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 600.2.bg.a ✓ 464
3.b odd 2 1 inner 600.2.bg.a ✓ 464
8.d odd 2 1 inner 600.2.bg.a ✓ 464
24.f even 2 1 inner 600.2.bg.a ✓ 464
25.d even 5 1 inner 600.2.bg.a ✓ 464
75.j odd 10 1 inner 600.2.bg.a ✓ 464
200.n odd 10 1 inner 600.2.bg.a ✓ 464
600.bg even 10 1 inner 600.2.bg.a ✓ 464
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
600.2.bg.a ✓ 464 1.a even 1 1 trivial
600.2.bg.a ✓ 464 3.b odd 2 1 inner
600.2.bg.a ✓ 464 8.d odd 2 1 inner
600.2.bg.a ✓ 464 24.f even 2 1 inner
600.2.bg.a ✓ 464 25.d even 5 1 inner
600.2.bg.a ✓ 464 75.j odd 10 1 inner
600.2.bg.a ✓ 464 200.n odd 10 1 inner
600.2.bg.a ✓ 464 600.bg even 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(600, [\chi])\).