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Properties

Label 952.2.s.a
Level $952$
Weight $2$
Character orbit 952.s
Analytic conductor $7.602$
Analytic rank $0$
Dimension $216$
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] = [952,2,Mod(421,952)] 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("952.421"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(952, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 2, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 952 = 2^{3} \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 952.s (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: \(7.60175827243\)
Analytic rank: \(0\)
Dimension: \(216\)
Relative dimension: \(108\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 216 q - 8 q^{6} - 8 q^{10} - 16 q^{16} + 8 q^{17} + 8 q^{24} - 20 q^{30} + 32 q^{31} - 32 q^{34} - 40 q^{38} + 4 q^{40} + 24 q^{41} - 8 q^{44} - 8 q^{46} + 84 q^{48} + 36 q^{50} + 84 q^{54} - 16 q^{57}+ \cdots + 8 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
421.1 −1.41171 − 0.0840655i 0.875373 − 0.875373i 1.98587 + 0.237353i 2.76924 − 2.76924i −1.30936 + 1.16219i 0.707107 − 0.707107i −2.78352 − 0.502017i 1.46744i −4.14218 + 3.67658i
421.2 −1.40876 + 0.124097i −1.34257 + 1.34257i 1.96920 − 0.349646i 2.73870 − 2.73870i 1.72475 − 2.05797i −0.707107 + 0.707107i −2.73074 + 0.736939i − 0.604996i −3.51830 + 4.19802i
421.3 −1.40841 − 0.127997i 0.741274 − 0.741274i 1.96723 + 0.360544i −2.18213 + 2.18213i −1.13890 + 0.949137i −0.707107 + 0.707107i −2.72452 − 0.759593i 1.90102i 3.35263 − 2.79402i
421.4 −1.40220 + 0.183965i −0.228586 + 0.228586i 1.93231 − 0.515911i −1.81956 + 1.81956i 0.278471 − 0.362574i 0.707107 − 0.707107i −2.61457 + 1.07889i 2.89550i 2.21664 − 2.88611i
421.5 −1.39987 − 0.200922i 2.09682 − 2.09682i 1.91926 + 0.562527i 0.164543 − 0.164543i −3.35657 + 2.51398i 0.707107 − 0.707107i −2.57369 − 1.17308i − 5.79332i −0.263399 + 0.197278i
421.6 −1.39898 + 0.207009i 0.660275 − 0.660275i 1.91429 − 0.579204i 0.0855425 − 0.0855425i −0.787029 + 1.06040i −0.707107 + 0.707107i −2.55816 + 1.20657i 2.12807i −0.101964 + 0.137380i
421.7 −1.39498 + 0.232457i −1.18147 + 1.18147i 1.89193 − 0.648545i −2.00232 + 2.00232i 1.37348 − 1.92276i −0.707107 + 0.707107i −2.48844 + 1.34450i 0.208274i 2.32774 − 3.25864i
421.8 −1.38700 + 0.276112i −2.14609 + 2.14609i 1.84752 − 0.765935i 0.212977 − 0.212977i 2.38406 − 3.56918i 0.707107 − 0.707107i −2.35103 + 1.57247i − 6.21139i −0.236593 + 0.354204i
421.9 −1.36959 − 0.352437i 1.19854 − 1.19854i 1.75158 + 0.965391i −0.621263 + 0.621263i −2.06392 + 1.21910i 0.707107 − 0.707107i −2.05871 − 1.93951i 0.127012i 1.06983 − 0.631922i
421.10 −1.36581 − 0.366847i −1.09207 + 1.09207i 1.73085 + 1.00208i 0.248870 − 0.248870i 1.89218 − 1.09093i 0.707107 − 0.707107i −1.99639 − 2.00360i 0.614762i −0.431206 + 0.248611i
421.11 −1.33788 − 0.458333i 2.21425 − 2.21425i 1.57986 + 1.22639i −1.94010 + 1.94010i −3.97728 + 1.94755i −0.707107 + 0.707107i −1.55158 − 2.36487i − 6.80584i 3.48485 − 1.70642i
421.12 −1.31525 − 0.519727i 0.229425 − 0.229425i 1.45977 + 1.36714i 0.757209 − 0.757209i −0.420991 + 0.182513i −0.707107 + 0.707107i −1.20942 − 2.55681i 2.89473i −1.38946 + 0.602378i
421.13 −1.30571 + 0.543253i 0.265770 − 0.265770i 1.40975 − 1.41866i 1.57608 − 1.57608i −0.202638 + 0.491399i −0.707107 + 0.707107i −1.07003 + 2.61821i 2.85873i −1.20169 + 2.91411i
421.14 −1.29470 − 0.568988i −1.37745 + 1.37745i 1.35251 + 1.47334i −1.95944 + 1.95944i 2.56714 − 0.999634i −0.707107 + 0.707107i −0.912778 − 2.67709i − 0.794737i 3.65179 − 1.42199i
421.15 −1.27820 + 0.605141i −0.176712 + 0.176712i 1.26761 − 1.54699i 0.338110 − 0.338110i 0.118938 − 0.332809i 0.707107 − 0.707107i −0.684114 + 2.74445i 2.93755i −0.227569 + 0.636778i
421.16 −1.26155 + 0.639135i −1.75704 + 1.75704i 1.18301 − 1.61260i 2.35713 − 2.35713i 1.09361 − 3.33959i 0.707107 − 0.707107i −0.461762 + 2.79048i − 3.17441i −1.46711 + 4.48016i
421.17 −1.22275 + 0.710547i 1.87007 − 1.87007i 0.990245 − 1.73765i −1.87226 + 1.87226i −0.957859 + 3.61541i −0.707107 + 0.707107i 0.0238547 + 2.82833i − 3.99433i 0.958982 − 3.61964i
421.18 −1.21076 − 0.730799i 1.63412 − 1.63412i 0.931865 + 1.76964i 2.89515 − 2.89515i −3.17274 + 0.784310i −0.707107 + 0.707107i 0.164990 − 2.82361i − 2.34072i −5.62109 + 1.38955i
421.19 −1.20590 − 0.738788i 1.23543 − 1.23543i 0.908383 + 1.78181i −3.04729 + 3.04729i −2.40252 + 0.577081i 0.707107 − 0.707107i 0.220961 − 2.81978i − 0.0525658i 5.92602 − 1.42342i
421.20 −1.19990 + 0.748495i −1.76490 + 1.76490i 0.879511 − 1.79623i −1.06907 + 1.06907i 0.796682 − 3.43872i −0.707107 + 0.707107i 0.289148 + 2.81361i − 3.22974i 0.482581 − 2.08296i
See next 80 embeddings (of 216 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 421.108
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
17.c even 4 1 inner
136.i even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 952.2.s.a ✓ 216
8.b even 2 1 inner 952.2.s.a ✓ 216
17.c even 4 1 inner 952.2.s.a ✓ 216
136.i even 4 1 inner 952.2.s.a ✓ 216
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
952.2.s.a ✓ 216 1.a even 1 1 trivial
952.2.s.a ✓ 216 8.b even 2 1 inner
952.2.s.a ✓ 216 17.c even 4 1 inner
952.2.s.a ✓ 216 136.i even 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(952, [\chi])\).