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Properties

Label 624.2.cz.a
Level $624$
Weight $2$
Character orbit 624.cz
Analytic conductor $4.983$
Analytic rank $0$
Dimension $224$
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] = [624,2,Mod(115,624)] 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("624.115"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(624, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([6, 9, 0, 7])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 624.cz (of order \(12\), 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: \(4.98266508613\)
Analytic rank: \(0\)
Dimension: \(224\)
Relative dimension: \(56\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 224 q + 8 q^{10} - 32 q^{14} + 48 q^{16} - 52 q^{20} + 8 q^{22} - 12 q^{24} + 224 q^{25} + 16 q^{26} - 52 q^{28} - 24 q^{34} + 40 q^{38} - 40 q^{40} + 16 q^{43} - 56 q^{44} + 56 q^{46} + 56 q^{50} - 56 q^{52}+ \cdots - 88 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
115.1 −1.41421 + 0.00410420i −0.965926 − 0.258819i 1.99997 − 0.0116084i 1.50565 1.36708 + 0.362060i −0.0849297 − 0.316962i −2.82832 + 0.0246249i 0.866025 + 0.500000i −2.12930 + 0.00617946i
115.2 −1.41305 − 0.0572703i 0.965926 + 0.258819i 1.99344 + 0.161852i −3.89484 −1.35008 − 0.421044i −0.475163 − 1.77333i −2.80757 − 0.342870i 0.866025 + 0.500000i 5.50362 + 0.223059i
115.3 −1.40660 − 0.146507i 0.965926 + 0.258819i 1.95707 + 0.412156i −0.365905 −1.32076 − 0.505571i 0.871571 + 3.25275i −2.69244 − 0.866465i 0.866025 + 0.500000i 0.514683 + 0.0536077i
115.4 −1.38755 − 0.273346i 0.965926 + 0.258819i 1.85056 + 0.758560i 3.57778 −1.26952 − 0.623155i 0.0295498 + 0.110281i −2.36039 − 1.55838i 0.866025 + 0.500000i −4.96433 − 0.977972i
115.5 −1.37476 + 0.331717i 0.965926 + 0.258819i 1.77993 − 0.912062i 0.774097 −1.41377 − 0.0353999i −0.582382 − 2.17348i −2.14443 + 1.84430i 0.866025 + 0.500000i −1.06420 + 0.256781i
115.6 −1.36730 + 0.361218i −0.965926 − 0.258819i 1.73904 − 0.987790i −1.10145 1.41420 + 0.00497472i 1.00137 + 3.73716i −2.02099 + 1.97878i 0.866025 + 0.500000i 1.50602 − 0.397863i
115.7 −1.36005 − 0.387632i −0.965926 − 0.258819i 1.69948 + 1.05440i −0.776589 1.21338 + 0.726431i −1.25064 − 4.66744i −1.90266 − 2.09281i 0.866025 + 0.500000i 1.05620 + 0.301031i
115.8 −1.30591 + 0.542774i −0.965926 − 0.258819i 1.41079 − 1.41763i −3.89459 1.40189 − 0.186286i −0.592612 − 2.21166i −1.07291 + 2.61703i 0.866025 + 0.500000i 5.08598 − 2.11389i
115.9 −1.19769 + 0.752027i 0.965926 + 0.258819i 0.868912 − 1.80139i 2.40258 −1.35152 + 0.416418i 0.525989 + 1.96302i 0.314006 + 2.81094i 0.866025 + 0.500000i −2.87755 + 1.80681i
115.10 −1.17759 − 0.783126i 0.965926 + 0.258819i 0.773429 + 1.84440i −1.72926 −0.934775 − 1.06122i 1.28760 + 4.80539i 0.533616 − 2.77763i 0.866025 + 0.500000i 2.03635 + 1.35423i
115.11 −1.13064 + 0.849506i −0.965926 − 0.258819i 0.556678 − 1.92097i 3.27890 1.31198 − 0.527930i 1.22700 + 4.57923i 1.00247 + 2.64482i 0.866025 + 0.500000i −3.70724 + 2.78545i
115.12 −1.07992 − 0.913109i −0.965926 − 0.258819i 0.332465 + 1.97217i −3.21365 0.806795 + 1.16150i 0.196353 + 0.732798i 1.44177 − 2.43337i 0.866025 + 0.500000i 3.47050 + 2.93442i
115.13 −1.07567 − 0.918114i −0.965926 − 0.258819i 0.314134 + 1.97518i 2.72029 0.801393 + 1.16523i 0.494259 + 1.84460i 1.47553 − 2.41305i 0.866025 + 0.500000i −2.92613 − 2.49753i
115.14 −1.06469 + 0.930825i −0.965926 − 0.258819i 0.267129 − 1.98208i 0.770103 1.26933 − 0.623546i −0.864796 − 3.22746i 1.56056 + 2.35895i 0.866025 + 0.500000i −0.819921 + 0.716831i
115.15 −1.01187 − 0.987984i 0.965926 + 0.258819i 0.0477771 + 1.99943i −0.533822 −0.721686 − 1.21621i −0.607709 − 2.26800i 1.92706 − 2.07037i 0.866025 + 0.500000i 0.540160 + 0.527407i
115.16 −0.905962 + 1.08592i −0.965926 − 0.258819i −0.358465 − 1.96761i 2.01134 1.15615 − 0.814442i −0.741068 − 2.76570i 2.46144 + 1.39332i 0.866025 + 0.500000i −1.82219 + 2.18416i
115.17 −0.903662 + 1.08784i 0.965926 + 0.258819i −0.366791 − 1.96608i −2.92410 −1.15442 + 0.816888i 0.425945 + 1.58965i 2.47023 + 1.37766i 0.866025 + 0.500000i 2.64239 − 3.18095i
115.18 −0.793687 − 1.17050i −0.965926 − 0.258819i −0.740122 + 1.85801i −0.404954 0.463696 + 1.33603i −0.266540 − 0.994741i 2.76222 − 0.608372i 0.866025 + 0.500000i 0.321407 + 0.473997i
115.19 −0.759205 − 1.19315i 0.965926 + 0.258819i −0.847214 + 1.81169i 4.27366 −0.424526 − 1.34899i 0.315701 + 1.17821i 2.80483 − 0.364593i 0.866025 + 0.500000i −3.24458 − 5.09911i
115.20 −0.758340 + 1.19370i 0.965926 + 0.258819i −0.849842 − 1.81046i −2.13330 −1.04145 + 0.956754i −0.202379 − 0.755287i 2.80562 + 0.358487i 0.866025 + 0.500000i 1.61776 − 2.54652i
See next 80 embeddings (of 224 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 115.56
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
208.bk even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 624.2.cz.a yes 224
13.f odd 12 1 624.2.ch.a ✓ 224
16.f odd 4 1 624.2.ch.a ✓ 224
208.bk even 12 1 inner 624.2.cz.a yes 224
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
624.2.ch.a ✓ 224 13.f odd 12 1
624.2.ch.a ✓ 224 16.f odd 4 1
624.2.cz.a yes 224 1.a even 1 1 trivial
624.2.cz.a yes 224 208.bk even 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(624, [\chi])\).