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Properties

Label 861.2.d.c
Level $861$
Weight $2$
Character orbit 861.d
Analytic conductor $6.875$
Analytic rank $0$
Dimension $50$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [861,2,Mod(83,861)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(861, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("861.83");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 861 = 3 \cdot 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 861.d (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.87511961403\)
Analytic rank: \(0\)
Dimension: \(50\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 50 q - 4 q^{3} - 56 q^{4} + 4 q^{6} - 6 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 50 q - 4 q^{3} - 56 q^{4} + 4 q^{6} - 6 q^{7} + 6 q^{9} + 7 q^{12} + 2 q^{14} - 2 q^{15} + 56 q^{16} + 12 q^{17} + 10 q^{18} + 24 q^{20} + 6 q^{21} - 28 q^{22} - 4 q^{24} + 54 q^{25} - 54 q^{26} + 2 q^{27} - 28 q^{30} + 6 q^{33} - 24 q^{35} - 8 q^{36} - 24 q^{37} + 12 q^{38} + 26 q^{39} + 50 q^{41} + 38 q^{42} - 52 q^{43} + 24 q^{45} + 16 q^{46} - 4 q^{47} - 108 q^{48} + 30 q^{49} - 6 q^{51} + 53 q^{54} + 12 q^{56} + 10 q^{57} + 16 q^{58} - 44 q^{59} - 8 q^{60} + 176 q^{62} - 2 q^{63} - 80 q^{64} + 24 q^{66} + 12 q^{67} - 48 q^{68} - 92 q^{69} + 16 q^{70} + 5 q^{72} + 24 q^{75} + 34 q^{77} - 39 q^{78} + 8 q^{79} - 96 q^{80} - 6 q^{81} + 56 q^{83} + 35 q^{84} - 44 q^{85} + 44 q^{87} + 24 q^{88} + 8 q^{89} - 236 q^{90} - 40 q^{91} + 24 q^{93} + 69 q^{96} - 86 q^{98} - 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
83.1 − 2.76236i −1.63382 + 0.575020i −5.63062 −1.44209 1.58841 + 4.51318i 2.41007 − 1.09159i 10.0291i 2.33870 − 1.87895i 3.98356i
83.2 − 2.70025i −1.17447 + 1.27303i −5.29136 2.61972 3.43751 + 3.17138i −2.40496 + 1.10280i 8.88749i −0.241220 − 2.99029i − 7.07390i
83.3 − 2.57987i −1.40722 − 1.00981i −4.65573 −1.65674 −2.60518 + 3.63046i −2.40049 − 1.11250i 6.85144i 0.960564 + 2.84206i 4.27416i
83.4 − 2.56842i 1.31236 + 1.13036i −4.59681 −3.90592 2.90324 − 3.37070i 0.894853 − 2.48983i 6.66970i 0.444580 + 2.96688i 10.0321i
83.5 − 2.42487i 0.214566 − 1.71871i −3.88001 −1.01003 −4.16765 − 0.520296i 0.00714389 + 2.64574i 4.55878i −2.90792 − 0.737554i 2.44919i
83.6 − 2.40125i 1.06878 + 1.36298i −3.76598 −1.47773 3.27284 − 2.56641i −0.566624 + 2.58436i 4.24056i −0.715412 + 2.91345i 3.54840i
83.7 − 2.32666i −0.0978850 − 1.72928i −3.41333 1.17011 −4.02345 + 0.227745i 2.28747 − 1.32947i 3.28832i −2.98084 + 0.338542i − 2.72244i
83.8 − 2.27654i 1.23228 − 1.21716i −3.18261 4.21613 −2.77092 − 2.80533i −1.74132 − 1.99194i 2.69227i 0.0370191 − 2.99977i − 9.59816i
83.9 − 2.05867i −1.73138 − 0.0483552i −2.23814 −0.395190 −0.0995476 + 3.56434i 1.76294 + 1.97283i 0.490251i 2.99532 + 0.167442i 0.813566i
83.10 − 1.76680i 1.73178 − 0.0308681i −1.12157 2.85036 −0.0545378 − 3.05970i −0.0791274 + 2.64457i − 1.55200i 2.99809 − 0.106913i − 5.03601i
83.11 − 1.74668i 1.72461 + 0.160414i −1.05089 −2.41543 0.280193 − 3.01234i −2.57144 − 0.622651i − 1.65779i 2.94853 + 0.553303i 4.21899i
83.12 − 1.68620i 1.38484 + 1.04030i −0.843260 2.29271 1.75415 − 2.33511i 2.61066 + 0.429468i − 1.95049i 0.835547 + 2.88130i − 3.86596i
83.13 − 1.65427i −1.04297 − 1.38283i −0.736612 −3.11018 −2.28757 + 1.72536i −1.81778 + 1.92241i − 2.08999i −0.824421 + 2.88450i 5.14508i
83.14 − 1.63538i 0.280357 + 1.70921i −0.674475 −1.11581 2.79521 − 0.458491i −2.50636 − 0.847446i − 2.16774i −2.84280 + 0.958378i 1.82478i
83.15 − 1.58243i −1.37768 + 1.04976i −0.504078 3.55190 1.66117 + 2.18008i 1.85221 − 1.88926i − 2.36719i 0.795999 − 2.89247i − 5.62062i
83.16 − 1.19100i 0.0128999 + 1.73200i 0.581527 −4.05307 2.06281 − 0.0153637i 2.43680 − 1.03053i − 3.07459i −2.99967 + 0.0446852i 4.82720i
83.17 − 1.13331i 1.48427 − 0.892712i 0.715618 1.62699 −1.01172 − 1.68214i −0.803122 + 2.52091i − 3.07763i 1.40613 − 2.65006i − 1.84387i
83.18 − 1.12299i −0.649768 + 1.60555i 0.738904 −0.0655748 1.80301 + 0.729680i 2.51092 − 0.833845i − 3.07575i −2.15560 − 2.08648i 0.0736396i
83.19 − 0.906441i −1.42479 − 0.984878i 1.17837 2.73739 −0.892734 + 1.29148i −0.111001 − 2.64342i − 2.88100i 1.06003 + 2.80648i − 2.48128i
83.20 − 0.815001i −1.70768 − 0.289505i 1.33577 1.47028 −0.235947 + 1.39177i −1.24747 + 2.33320i − 2.71866i 2.83237 + 0.988766i − 1.19828i
See all 50 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 83.50
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 861.2.d.c ✓ 50
3.b odd 2 1 861.2.d.d yes 50
7.b odd 2 1 861.2.d.d yes 50
21.c even 2 1 inner 861.2.d.c ✓ 50
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
861.2.d.c ✓ 50 1.a even 1 1 trivial
861.2.d.c ✓ 50 21.c even 2 1 inner
861.2.d.d yes 50 3.b odd 2 1
861.2.d.d yes 50 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(861, [\chi])\):

\( T_{2}^{50} + 78 T_{2}^{48} + 2844 T_{2}^{46} + 64424 T_{2}^{44} + 1016438 T_{2}^{42} + 11868478 T_{2}^{40} + \cdots + 22188 \) Copy content Toggle raw display
\( T_{5}^{25} - 76 T_{5}^{23} - 20 T_{5}^{22} + 2461 T_{5}^{21} + 1296 T_{5}^{20} - 44483 T_{5}^{19} + \cdots - 172032 \) Copy content Toggle raw display
\( T_{17}^{25} - 6 T_{17}^{24} - 221 T_{17}^{23} + 1246 T_{17}^{22} + 20674 T_{17}^{21} + \cdots - 71166448512 \) Copy content Toggle raw display