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Properties

Label 615.2.k.a
Level $615$
Weight $2$
Character orbit 615.k
Analytic conductor $4.911$
Analytic rank $0$
Dimension $160$
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] = [615,2,Mod(278,615)] 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("615.278"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(615, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 3, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 615 = 3 \cdot 5 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 615.k (of order \(4\), 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: \(4.91079972431\)
Analytic rank: \(0\)
Dimension: \(160\)
Relative dimension: \(80\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 160 q - 4 q^{3} - 8 q^{6} - 24 q^{10} - 2 q^{15} - 152 q^{16} + 4 q^{18} + 8 q^{22} - 8 q^{24} - 8 q^{25} - 16 q^{27} + 40 q^{28} + 6 q^{30} - 16 q^{31} + 12 q^{33} + 16 q^{34} - 8 q^{37} - 24 q^{39} + 24 q^{40}+ \cdots - 24 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
278.1 −1.93141 + 1.93141i 0.0901717 + 1.72970i − 5.46066i −2.22866 − 0.181832i −3.51491 − 3.16660i 2.36236i 6.68393 + 6.68393i −2.98374 + 0.311940i 4.65564 − 3.95326i
278.2 −1.90761 + 1.90761i 1.66224 − 0.486774i − 5.27794i 0.0903349 − 2.23424i −2.24234 + 4.09948i − 2.32577i 6.25303 + 6.25303i 2.52610 − 1.61827i 4.08974 + 4.43438i
278.3 −1.88681 + 1.88681i −1.45763 + 0.935589i − 5.12009i 1.55532 + 1.60654i 0.984982 − 4.51554i − 2.86502i 5.88701 + 5.88701i 1.24934 − 2.72748i −5.96583 − 0.0966353i
278.4 −1.85429 + 1.85429i −0.587012 − 1.62955i − 4.87676i 2.23239 − 0.128162i 4.11013 + 1.93316i − 0.659233i 5.33433 + 5.33433i −2.31083 + 1.91312i −3.90185 + 4.37714i
278.5 −1.84151 + 1.84151i 1.46911 + 0.917450i − 4.78231i 2.06963 + 0.846535i −4.39487 + 1.01589i 2.38837i 5.12365 + 5.12365i 1.31657 + 2.69567i −5.37015 + 2.25234i
278.6 −1.81403 + 1.81403i 1.00800 − 1.40852i − 4.58142i −0.973951 + 2.01281i 0.726548 + 4.38365i − 0.972545i 4.68279 + 4.68279i −0.967855 − 2.83959i −1.88453 − 5.41809i
278.7 −1.75032 + 1.75032i −1.73205 − 0.00348917i − 4.12722i 0.957415 − 2.02073i 3.03774 − 3.02552i 4.74925i 3.72330 + 3.72330i 2.99998 + 0.0120868i 1.86114 + 5.21270i
278.8 −1.70813 + 1.70813i −1.73183 − 0.0275900i − 3.83543i −2.18111 + 0.492700i 3.00532 − 2.91107i − 0.802352i 3.13516 + 3.13516i 2.99848 + 0.0955623i 2.88403 − 4.56722i
278.9 −1.58028 + 1.58028i 0.375547 − 1.69085i − 2.99458i −1.64966 − 1.50951i 2.07854 + 3.26548i 2.25110i 1.57171 + 1.57171i −2.71793 − 1.26999i 4.99238 − 0.221462i
278.10 −1.53653 + 1.53653i 0.382539 + 1.68928i − 2.72185i 2.01492 − 0.969582i −3.18341 − 2.00785i − 3.61671i 1.10915 + 1.10915i −2.70733 + 1.29243i −1.60620 + 4.58578i
278.11 −1.51359 + 1.51359i −1.39249 − 1.03003i − 2.58193i −0.0650452 − 2.23512i 3.66672 − 0.548616i − 3.82040i 0.880802 + 0.880802i 0.878066 + 2.86862i 3.48152 + 3.28461i
278.12 −1.51198 + 1.51198i −0.958557 − 1.44263i − 2.57215i −0.441633 + 2.19202i 3.63053 + 0.731902i 4.88662i 0.865083 + 0.865083i −1.16234 + 2.76568i −2.64655 − 3.98203i
278.13 −1.50851 + 1.50851i −0.903314 + 1.47784i − 2.55120i 0.936795 + 2.03037i −0.866683 − 3.59200i 2.95622i 0.831486 + 0.831486i −1.36805 − 2.66992i −4.47600 − 1.64967i
278.14 −1.50520 + 1.50520i 1.16925 + 1.27783i − 2.53128i −1.05186 − 1.97322i −3.68336 − 0.163428i − 0.392544i 0.799687 + 0.799687i −0.265692 + 2.98821i 4.55336 + 1.38683i
278.15 −1.45876 + 1.45876i 1.65864 + 0.498902i − 2.25597i −0.723075 + 2.11593i −3.14735 + 1.69179i 1.48908i 0.373406 + 0.373406i 2.50219 + 1.65500i −2.03185 − 4.14143i
278.16 −1.29733 + 1.29733i −0.252374 + 1.71357i − 1.36614i −1.49068 + 1.66670i −1.89565 − 2.55048i − 3.24596i −0.822328 − 0.822328i −2.87261 − 0.864918i −0.228361 − 4.09616i
278.17 −1.29283 + 1.29283i 1.57218 − 0.726807i − 1.34283i 1.82412 − 1.29328i −1.09293 + 2.97221i 3.45365i −0.849605 − 0.849605i 1.94350 − 2.28534i −0.686282 + 4.03028i
278.18 −1.22901 + 1.22901i 1.35174 − 1.08296i − 1.02094i 1.33308 + 1.79524i −0.330327 + 2.99227i − 2.11054i −1.20328 − 1.20328i 0.654385 − 2.92776i −3.84474 − 0.567996i
278.19 −1.20133 + 1.20133i −1.03181 + 1.39118i − 0.886366i −1.33262 − 1.79558i −0.431722 − 2.91079i 0.826459i −1.33784 − 1.33784i −0.870750 − 2.87085i 3.75799 + 0.556170i
278.20 −1.15430 + 1.15430i 0.0985233 − 1.72925i − 0.664797i 2.22385 + 0.233461i 1.88234 + 2.10979i − 0.533046i −1.54122 − 1.54122i −2.98059 − 0.340742i −2.83646 + 2.29749i
See next 80 embeddings (of 160 total)
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 278.80
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
205.i odd 4 1 inner
615.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 615.2.k.a ✓ 160
3.b odd 2 1 inner 615.2.k.a ✓ 160
5.c odd 4 1 615.2.r.a yes 160
15.e even 4 1 615.2.r.a yes 160
41.c even 4 1 615.2.r.a yes 160
123.f odd 4 1 615.2.r.a yes 160
205.i odd 4 1 inner 615.2.k.a ✓ 160
615.k even 4 1 inner 615.2.k.a ✓ 160
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
615.2.k.a ✓ 160 1.a even 1 1 trivial
615.2.k.a ✓ 160 3.b odd 2 1 inner
615.2.k.a ✓ 160 205.i odd 4 1 inner
615.2.k.a ✓ 160 615.k even 4 1 inner
615.2.r.a yes 160 5.c odd 4 1
615.2.r.a yes 160 15.e even 4 1
615.2.r.a yes 160 41.c even 4 1
615.2.r.a yes 160 123.f odd 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(615, [\chi])\).