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Properties

Label 91.3.x.a
Level $91$
Weight $3$
Character orbit 91.x
Analytic conductor $2.480$
Analytic rank $0$
Dimension $68$
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] = [91,3,Mod(2,91)] 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("91.2"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(91, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([4, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 91.x (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: \(2.47957040568\)
Analytic rank: \(0\)
Dimension: \(68\)
Relative dimension: \(17\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 68 q - 2 q^{2} + 2 q^{3} - 2 q^{5} - 16 q^{6} - 26 q^{7} - 82 q^{9} - 6 q^{10} - 6 q^{11} + 156 q^{12} - 8 q^{13} - 16 q^{14} - 30 q^{15} - 228 q^{16} + 130 q^{18} + 90 q^{19} - 48 q^{20} - 66 q^{21} - 28 q^{22}+ \cdots - 616 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −2.73284 + 2.73284i 0.682403 − 1.18196i − 10.9368i 0.927523 + 3.46156i 1.36520 + 5.09499i 4.93908 − 4.96040i 18.9571 + 18.9571i 3.56865 + 6.18109i −11.9946 − 6.92511i
2.2 −2.48000 + 2.48000i −2.50949 + 4.34656i − 8.30079i −1.61061 − 6.01089i −4.55594 − 17.0030i −1.79680 + 6.76547i 10.6660 + 10.6660i −8.09507 − 14.0211i 18.9013 + 10.9127i
2.3 −1.94732 + 1.94732i 0.380859 − 0.659667i − 3.58414i 0.675147 + 2.51968i 0.542930 + 2.02624i −4.20818 + 5.59385i −0.809822 − 0.809822i 4.20989 + 7.29175i −6.22136 − 3.59191i
2.4 −1.88110 + 1.88110i 2.90151 − 5.02556i − 3.07711i −0.506411 − 1.88995i 3.99556 + 14.9116i −6.50162 − 2.59402i −1.73606 − 1.73606i −12.3375 − 21.3691i 4.50780 + 2.60258i
2.5 −1.57212 + 1.57212i −0.0124943 + 0.0216407i − 0.943116i −2.34948 − 8.76837i −0.0143793 − 0.0536642i 3.71090 − 5.93542i −4.80578 − 4.80578i 4.49969 + 7.79369i 17.4786 + 10.0913i
2.6 −1.54070 + 1.54070i −2.16359 + 3.74744i − 0.747514i 2.05425 + 7.66658i −2.44025 − 9.10712i 6.84154 − 1.48100i −5.01111 − 5.01111i −4.86221 − 8.42160i −14.9769 − 8.64691i
2.7 −0.955045 + 0.955045i 1.18516 − 2.05277i 2.17578i 0.210770 + 0.786604i 0.828598 + 3.09237i 5.06632 + 4.83036i −5.89815 − 5.89815i 1.69077 + 2.92850i −0.952537 − 0.549947i
2.8 −0.542984 + 0.542984i −1.78713 + 3.09539i 3.41034i −0.276444 − 1.03170i −0.710368 − 2.65113i −5.78383 − 3.94300i −4.02369 − 4.02369i −1.88764 − 3.26949i 0.710302 + 0.410093i
2.9 0.00147245 − 0.00147245i 1.25808 − 2.17906i 4.00000i 2.51260 + 9.37716i −0.00135609 − 0.00506101i −5.69803 − 4.06601i 0.0117796 + 0.0117796i 1.33448 + 2.31138i 0.0175071 + 0.0101077i
2.10 0.325516 − 0.325516i −0.796227 + 1.37911i 3.78808i −0.627391 − 2.34145i 0.189736 + 0.708106i −0.963908 + 6.93332i 2.53515 + 2.53515i 3.23204 + 5.59807i −0.966407 − 0.557955i
2.11 0.493820 − 0.493820i 2.48116 − 4.29750i 3.51228i −0.463227 − 1.72879i −0.896943 − 3.34744i 6.93916 − 0.920893i 3.70972 + 3.70972i −7.81233 − 13.5314i −1.08246 − 0.624958i
2.12 1.20446 − 1.20446i −1.14422 + 1.98185i 1.09854i 0.564004 + 2.10489i 1.00889 + 3.76523i 6.07887 − 3.47092i 6.14100 + 6.14100i 1.88152 + 3.25889i 3.21458 + 1.85594i
2.13 1.29783 − 1.29783i 1.10760 − 1.91842i 0.631283i −1.68798 − 6.29962i −1.05231 − 3.92726i −4.85005 − 5.04748i 6.01061 + 6.01061i 2.04643 + 3.54452i −10.3665 − 5.98512i
2.14 1.90832 − 1.90832i −2.78431 + 4.82257i − 3.28338i 1.25431 + 4.68115i 3.88965 + 14.5164i −6.22197 + 3.20734i 1.36753 + 1.36753i −11.0048 − 19.0608i 11.3268 + 6.53952i
2.15 2.05252 − 2.05252i 1.88624 − 3.26706i − 4.42567i 0.850517 + 3.17417i −2.83416 − 10.5772i −3.61983 + 5.99140i −0.873696 − 0.873696i −2.61577 − 4.53064i 8.26075 + 4.76935i
2.16 2.41680 − 2.41680i −0.882981 + 1.52937i − 7.68188i −2.23431 − 8.33855i 1.56219 + 5.83017i 2.62871 + 6.48767i −8.89838 − 8.89838i 2.94069 + 5.09342i −25.5525 − 14.7528i
2.17 2.58534 − 2.58534i −0.168604 + 0.292031i − 9.36797i 1.07275 + 4.00355i 0.319101 + 1.19090i 0.403748 − 6.98835i −13.8780 − 13.8780i 4.44315 + 7.69575i 13.1239 + 7.57711i
32.1 −2.61272 + 2.61272i −0.0106418 − 0.0184322i − 9.65264i 7.99181 + 2.14140i 0.0759624 + 0.0203541i 0.229179 + 6.99625i 14.7688 + 14.7688i 4.49977 − 7.79384i −26.4753 + 15.2855i
32.2 −2.53286 + 2.53286i 2.31370 + 4.00745i − 8.83074i −5.73892 − 1.53774i −16.0106 − 4.29003i −5.46387 − 4.37563i 12.2356 + 12.2356i −6.20645 + 10.7499i 18.4307 − 10.6410i
32.3 −2.08840 + 2.08840i −0.627662 − 1.08714i − 4.72287i −3.32270 − 0.890314i 3.58121 + 0.959581i 6.27635 − 3.09959i 1.50964 + 1.50964i 3.71208 − 6.42951i 8.79847 − 5.07980i
See all 68 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 2.17
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.x odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 91.3.x.a ✓ 68
7.c even 3 1 91.3.bd.a yes 68
13.f odd 12 1 91.3.bd.a yes 68
91.x odd 12 1 inner 91.3.x.a ✓ 68
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.3.x.a ✓ 68 1.a even 1 1 trivial
91.3.x.a ✓ 68 91.x odd 12 1 inner
91.3.bd.a yes 68 7.c even 3 1
91.3.bd.a yes 68 13.f odd 12 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(91, [\chi])\).