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Properties

Label 131.2.g.a
Level $131$
Weight $2$
Character orbit 131.g
Analytic conductor $1.046$
Analytic rank $0$
Dimension $480$
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] = [131,2,Mod(3,131)] 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("131.3"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(131, base_ring=CyclotomicField(130)) chi = DirichletCharacter(H, H._module([72])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 131 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 131.g (of order \(65\), degree \(48\), 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: \(1.04604026648\)
Analytic rank: \(0\)
Dimension: \(480\)
Relative dimension: \(10\) over \(\Q(\zeta_{65})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{65}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 480 q - 50 q^{2} - 51 q^{3} - 40 q^{4} - 51 q^{5} - 63 q^{6} - 50 q^{7} + q^{8} - 47 q^{9} - 40 q^{10} - 57 q^{11} - 2 q^{12} - 43 q^{13} - 23 q^{14} - 51 q^{15} - 46 q^{16} - 6 q^{17} + 15 q^{18} - 34 q^{19}+ \cdots - 98 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −0.417542 + 2.44470i −2.93863 + 1.19647i −3.91558 − 1.37771i 0.0646001 + 0.889495i −1.69801 − 7.68365i 3.01288 + 1.77200i 2.59244 − 4.66211i 5.05721 − 4.93645i −2.20152 − 0.213474i
3.2 −0.383228 + 2.24379i 0.905506 − 0.368680i −3.00112 − 1.05595i 0.162357 + 2.23554i 0.480225 + 2.17306i −0.544265 − 0.320104i 1.30698 − 2.35041i −1.46278 + 1.42785i −5.07830 − 0.492426i
3.3 −0.320177 + 1.87463i 2.01006 − 0.818401i −1.52510 − 0.536611i −0.270841 − 3.72928i 0.890624 + 4.03015i 2.70263 + 1.58953i −0.354213 + 0.636999i 1.22376 − 1.19454i 7.07773 + 0.686303i
3.4 −0.0849272 + 0.497247i −1.07782 + 0.438839i 1.64658 + 0.579356i −0.0828582 − 1.14090i −0.126675 − 0.573213i 2.74375 + 1.61371i −0.918229 + 1.65130i −1.17767 + 1.14955i 0.574343 + 0.0556921i
3.5 −0.0582900 + 0.341286i 1.74884 − 0.712045i 1.77355 + 0.624029i 0.0136268 + 0.187631i 0.141071 + 0.638360i −2.92116 − 1.71806i −0.652875 + 1.17410i 0.404635 − 0.394973i −0.0648303 − 0.00628638i
3.6 0.162150 − 0.949383i −3.01631 + 1.22810i 1.01159 + 0.355931i −0.248835 − 3.42627i 0.676843 + 3.06277i −3.10599 − 1.82676i 1.43807 − 2.58616i 5.44310 − 5.31313i −3.29319 − 0.319330i
3.7 0.184536 − 1.08045i −1.64449 + 0.669559i 0.753296 + 0.265050i 0.315521 + 4.34449i 0.419960 + 1.90035i 0.143448 + 0.0843677i 1.49076 − 2.68090i 0.109242 − 0.106633i 4.75224 + 0.460809i
3.8 0.212235 − 1.24263i 0.522363 − 0.212682i 0.387543 + 0.136358i −0.0409240 − 0.563492i −0.153421 − 0.694241i −0.403904 − 0.237552i 1.47698 − 2.65612i −1.91917 + 1.87334i −0.708897 − 0.0687393i
3.9 0.376358 − 2.20356i 0.103859 − 0.0422867i −2.82743 − 0.994842i −0.157578 − 2.16973i −0.0540932 − 0.244776i 1.63015 + 0.958759i −1.08352 + 1.94854i −2.13780 + 2.08675i −4.84044 − 0.469361i
3.10 0.440556 − 2.57944i 2.32788 − 0.947804i −4.57281 − 1.60896i 0.265698 + 3.65846i −1.41924 − 6.42219i −2.67596 − 1.57384i −3.62136 + 6.51247i 2.37390 − 2.31722i 9.55384 + 0.926403i
4.1 −2.63776 − 0.127588i 1.44222 + 0.507450i 4.95082 + 0.480065i 0.641000 − 0.838859i −3.73948 − 1.52254i 0.110826 + 1.52599i −7.77156 − 1.13481i −0.516534 − 0.414847i −1.79783 + 2.13092i
4.2 −1.90113 − 0.0919573i −0.302711 − 0.106510i 1.61517 + 0.156617i −0.760613 + 0.995395i 0.565697 + 0.230325i −0.218497 − 3.00854i 0.710516 + 0.103750i −2.25873 − 1.81407i 1.53756 − 1.82243i
4.3 −1.48472 − 0.0718158i −2.21695 − 0.780044i 0.208576 + 0.0202249i −0.0611959 + 0.0800854i 3.23554 + 1.31736i 0.138086 + 1.90134i 2.63349 + 0.384546i 1.96740 + 1.58009i 0.0966102 − 0.114510i
4.4 −1.06803 − 0.0516606i 2.81832 + 0.991638i −0.852637 − 0.0826773i 1.39031 − 1.81946i −2.95884 − 1.20470i −0.157856 − 2.17356i 3.02250 + 0.441349i 4.62058 + 3.71096i −1.57889 + 1.87142i
4.5 −0.713077 − 0.0344914i 1.84003 + 0.647423i −1.48337 − 0.143838i −2.00796 + 2.62776i −1.28975 − 0.525127i 0.293226 + 4.03750i 2.46564 + 0.360036i 0.627542 + 0.504002i 1.52246 − 1.80454i
4.6 0.144335 + 0.00698147i −1.12734 − 0.396657i −1.96988 − 0.191012i 1.78680 − 2.33834i −0.159945 − 0.0651221i −0.124981 − 1.72090i −0.568965 − 0.0830810i −1.22547 − 0.984223i 0.274223 − 0.325030i
4.7 0.997302 + 0.0482394i −2.80109 − 0.985575i −0.998378 − 0.0968093i −2.35478 + 3.08163i −2.74599 − 1.11804i −0.0816750 − 1.12460i −2.96700 − 0.433245i 4.53573 + 3.64281i −2.49708 + 2.95973i
4.8 1.16502 + 0.0563520i 1.56924 + 0.552141i −0.636561 − 0.0617251i 0.648155 − 0.848224i 1.79708 + 0.731687i 0.101524 + 1.39791i −3.04642 − 0.444842i −0.181383 − 0.145676i 0.802915 − 0.951675i
4.9 2.16151 + 0.104552i −0.0144134 − 0.00507141i 2.67055 + 0.258954i −0.996208 + 1.30371i −0.0306245 − 0.0124689i −0.0912295 − 1.25616i 1.46270 + 0.213585i −2.33884 − 1.87841i −2.28962 + 2.71384i
4.10 2.34004 + 0.113188i −2.75957 − 0.970966i 3.47232 + 0.336699i 2.10187 − 2.75066i −6.34761 − 2.58445i 0.151762 + 2.08964i 3.45089 + 0.503903i 4.33344 + 3.48034i 5.22980 − 6.19876i
See next 80 embeddings (of 480 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
131.g even 65 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 131.2.g.a ✓ 480
131.g even 65 1 inner 131.2.g.a ✓ 480
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
131.2.g.a ✓ 480 1.a even 1 1 trivial
131.2.g.a ✓ 480 131.g even 65 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(131, [\chi])\).