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Properties

Label 672.2.cj.a
Level $672$
Weight $2$
Character orbit 672.cj
Analytic conductor $5.366$
Analytic rank $0$
Dimension $512$
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] = [672,2,Mod(37,672)] 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("672.37"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(672, base_ring=CyclotomicField(24)) chi = DirichletCharacter(H, H._module([0, 3, 0, 8])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 672.cj (of order \(24\), degree \(8\), 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: \(5.36594701583\)
Analytic rank: \(0\)
Dimension: \(512\)
Relative dimension: \(64\) over \(\Q(\zeta_{24})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{24}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 512 q - 64 q^{14} + 8 q^{16} + 8 q^{18} + 64 q^{20} + 16 q^{22} + 16 q^{23} - 40 q^{28} + 96 q^{31} + 48 q^{35} - 80 q^{38} + 64 q^{40} - 32 q^{43} - 8 q^{44} - 48 q^{50} + 24 q^{52} + 32 q^{53} - 32 q^{58}+ \cdots - 88 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1 −1.41272 + 0.0650669i 0.608761 − 0.793353i 1.99153 − 0.183842i 0.793153 − 0.608607i −0.808386 + 1.16039i 2.15682 + 1.53235i −2.80151 + 0.389300i −0.258819 − 0.965926i −1.08090 + 0.911397i
37.2 −1.41057 + 0.101443i −0.608761 + 0.793353i 1.97942 − 0.286185i 2.37132 − 1.81958i 0.778221 − 1.18084i 0.892352 + 2.49072i −2.76308 + 0.604481i −0.258819 − 0.965926i −3.16033 + 2.80720i
37.3 −1.40602 + 0.152013i −0.608761 + 0.793353i 1.95378 − 0.427466i −3.05794 + 2.34644i 0.735331 − 1.20801i −1.56959 − 2.12988i −2.68208 + 0.898026i −0.258819 − 0.965926i 3.94284 − 3.76399i
37.4 −1.40441 − 0.166200i 0.608761 − 0.793353i 1.94476 + 0.466826i −3.53997 + 2.71631i −0.986808 + 1.01302i −1.01327 + 2.44403i −2.65365 − 0.978835i −0.258819 − 0.965926i 5.42303 − 3.22649i
37.5 −1.39097 + 0.255335i 0.608761 − 0.793353i 1.86961 − 0.710329i 1.77094 − 1.35889i −0.644199 + 1.25897i −2.25409 + 1.38531i −2.41920 + 1.46542i −0.258819 − 0.965926i −2.11636 + 2.34236i
37.6 −1.38887 + 0.266508i 0.608761 − 0.793353i 1.85795 − 0.740294i −0.342802 + 0.263041i −0.634058 + 1.26411i −0.219164 − 2.63666i −2.38316 + 1.52333i −0.258819 − 0.965926i 0.406006 − 0.456691i
37.7 −1.38600 − 0.281097i −0.608761 + 0.793353i 1.84197 + 0.779198i −1.29803 + 0.996014i 1.06675 − 0.928464i 1.75011 + 1.98422i −2.33393 − 1.59774i −0.258819 − 0.965926i 2.07904 − 1.01560i
37.8 −1.33766 − 0.458976i 0.608761 − 0.793353i 1.57868 + 1.22791i 1.59131 − 1.22106i −1.17845 + 0.781832i 2.60325 − 0.472310i −1.54816 − 2.36711i −0.258819 − 0.965926i −2.68908 + 0.902988i
37.9 −1.30025 + 0.556198i −0.608761 + 0.793353i 1.38129 − 1.44639i 0.874472 − 0.671006i 0.350279 − 1.37015i 2.18520 − 1.49161i −0.991534 + 2.64894i −0.258819 − 0.965926i −0.763817 + 1.35885i
37.10 −1.27720 + 0.607254i −0.608761 + 0.793353i 1.26249 − 1.55117i 0.101604 − 0.0779635i 0.295744 − 1.38294i −2.55596 + 0.683426i −0.670493 + 2.74781i −0.258819 − 0.965926i −0.0824251 + 0.161274i
37.11 −1.22621 − 0.704559i −0.608761 + 0.793353i 1.00719 + 1.72788i −0.601499 + 0.461547i 1.30543 − 0.543912i −2.34514 + 1.22487i −0.0176438 − 2.82837i −0.258819 − 0.965926i 1.06275 − 0.142163i
37.12 −1.19253 − 0.760181i −0.608761 + 0.793353i 0.844250 + 1.81308i 1.70446 − 1.30788i 1.32906 − 0.483328i −1.89849 − 1.84275i 0.371472 − 2.80393i −0.258819 − 0.965926i −3.02685 + 0.263985i
37.13 −1.14852 + 0.825172i 0.608761 − 0.793353i 0.638182 − 1.89545i −1.92032 + 1.47351i −0.0445198 + 1.41351i 2.30408 + 1.30046i 0.831107 + 2.70356i −0.258819 − 0.965926i 0.989617 − 3.27695i
37.14 −1.14821 − 0.825600i 0.608761 − 0.793353i 0.636770 + 1.89592i −2.87168 + 2.20351i −1.35398 + 0.408343i 1.27376 − 2.31895i 0.834128 − 2.70263i −0.258819 − 0.965926i 5.11651 − 0.159241i
37.15 −1.09201 + 0.898623i 0.608761 − 0.793353i 0.384954 − 1.96260i 3.01788 − 2.31570i 0.0481542 + 1.41339i −0.423858 − 2.61158i 1.34327 + 2.48910i −0.258819 − 0.965926i −1.21460 + 5.24069i
37.16 −1.05500 + 0.941796i 0.608761 − 0.793353i 0.226042 − 1.98719i −1.87609 + 1.43957i 0.104934 + 1.41032i −2.64497 − 0.0642908i 1.63305 + 2.30936i −0.258819 − 0.965926i 0.623485 − 3.28563i
37.17 −0.976671 − 1.02280i 0.608761 − 0.793353i −0.0922281 + 1.99787i −0.440961 + 0.338361i −1.40600 + 0.152206i −0.156173 + 2.64114i 2.13349 − 1.85693i −0.258819 − 0.965926i 0.776749 + 0.120546i
37.18 −0.961805 + 1.03679i −0.608761 + 0.793353i −0.149861 − 1.99438i −1.59412 + 1.22321i −0.237030 − 1.39421i −1.44227 + 2.21808i 2.21188 + 1.76283i −0.258819 − 0.965926i 0.265022 − 2.82925i
37.19 −0.958402 − 1.03994i −0.608761 + 0.793353i −0.162930 + 1.99335i 2.79050 − 2.14123i 1.40847 − 0.127279i 2.64551 − 0.0353797i 2.22911 − 1.74100i −0.258819 − 0.965926i −4.90116 − 0.849783i
37.20 −0.937240 − 1.05905i −0.608761 + 0.793353i −0.243161 + 1.98516i −1.89948 + 1.45752i 1.41075 − 0.0988559i −0.272298 − 2.63170i 2.33028 − 1.60306i −0.258819 − 0.965926i 3.32386 + 0.645590i
See next 80 embeddings (of 512 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.64
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
32.g even 8 1 inner
224.bd even 24 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 672.2.cj.a ✓ 512
7.c even 3 1 inner 672.2.cj.a ✓ 512
32.g even 8 1 inner 672.2.cj.a ✓ 512
224.bd even 24 1 inner 672.2.cj.a ✓ 512
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.2.cj.a ✓ 512 1.a even 1 1 trivial
672.2.cj.a ✓ 512 7.c even 3 1 inner
672.2.cj.a ✓ 512 32.g even 8 1 inner
672.2.cj.a ✓ 512 224.bd even 24 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(672, [\chi])\).