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Properties

Label 786.2.i.a
Level $786$
Weight $2$
Character orbit 786.i
Analytic conductor $6.276$
Analytic rank $0$
Dimension $60$
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] = [786,2,Mod(193,786)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(786, base_ring=CyclotomicField(26))
 
chi = DirichletCharacter(H, H._module([0, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("786.193");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 786 = 2 \cdot 3 \cdot 131 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 786.i (of order \(13\), degree \(12\), 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: \(6.27624159887\)
Analytic rank: \(0\)
Dimension: \(60\)
Relative dimension: \(5\) over \(\Q(\zeta_{13})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{13}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 60 q - 5 q^{2} - 5 q^{3} - 5 q^{4} + 8 q^{5} - 5 q^{6} + 7 q^{7} - 5 q^{8} - 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 60 q - 5 q^{2} - 5 q^{3} - 5 q^{4} + 8 q^{5} - 5 q^{6} + 7 q^{7} - 5 q^{8} - 5 q^{9} - 5 q^{10} + 2 q^{11} - 5 q^{12} + 2 q^{13} + 7 q^{14} + 8 q^{15} - 5 q^{16} - 11 q^{17} - 5 q^{18} - 8 q^{19} - 5 q^{20} + 7 q^{21} + 2 q^{22} - 19 q^{23} - 5 q^{24} + 23 q^{25} + 15 q^{26} - 5 q^{27} + 7 q^{28} - 4 q^{29} + 8 q^{30} + 25 q^{31} - 5 q^{32} + 15 q^{33} + 15 q^{34} - 13 q^{35} - 5 q^{36} - 26 q^{37} - 8 q^{38} + 2 q^{39} - 5 q^{40} + 44 q^{41} - 6 q^{42} + 18 q^{43} + 2 q^{44} - 5 q^{45} + 7 q^{46} + 58 q^{47} - 5 q^{48} - 26 q^{49} + 23 q^{50} - 11 q^{51} + 15 q^{52} - 14 q^{53} - 5 q^{54} - 25 q^{55} + 7 q^{56} - 8 q^{57} - 30 q^{58} + 15 q^{59} - 5 q^{60} + 38 q^{61} - q^{62} + 7 q^{63} - 5 q^{64} - 50 q^{65} + 2 q^{66} - 57 q^{67} - 24 q^{68} + 20 q^{69} + 48 q^{71} - 5 q^{72} - 32 q^{73} + 26 q^{74} - 16 q^{75} + 5 q^{76} + 98 q^{77} - 24 q^{78} + q^{79} + 8 q^{80} - 5 q^{81} + 31 q^{82} + 64 q^{83} + 7 q^{84} - 76 q^{85} + 5 q^{86} + 9 q^{87} + 2 q^{88} - 150 q^{89} - 5 q^{90} + 33 q^{91} - 6 q^{92} + 25 q^{93} + 45 q^{94} - 65 q^{95} - 5 q^{96} - 16 q^{97} - 26 q^{98} + 2 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
193.1 −0.748511 − 0.663123i −0.354605 + 0.935016i 0.120537 + 0.992709i −0.624633 + 1.64702i 0.885456 − 0.464723i −0.883439 − 0.463665i 0.568065 − 0.822984i −0.748511 − 0.663123i 1.55972 − 0.818605i
193.2 −0.748511 − 0.663123i −0.354605 + 0.935016i 0.120537 + 0.992709i 0.0529848 − 0.139709i 0.885456 − 0.464723i −3.79561 − 1.99209i 0.568065 − 0.822984i −0.748511 − 0.663123i −0.132304 + 0.0694386i
193.3 −0.748511 − 0.663123i −0.354605 + 0.935016i 0.120537 + 0.992709i 0.133945 − 0.353184i 0.885456 − 0.464723i 3.98294 + 2.09041i 0.568065 − 0.822984i −0.748511 − 0.663123i −0.334464 + 0.175540i
193.4 −0.748511 − 0.663123i −0.354605 + 0.935016i 0.120537 + 0.992709i 0.306783 − 0.808921i 0.885456 − 0.464723i 1.93940 + 1.01787i 0.568065 − 0.822984i −0.748511 − 0.663123i −0.766045 + 0.402051i
193.5 −0.748511 − 0.663123i −0.354605 + 0.935016i 0.120537 + 0.992709i 1.18238 − 3.11768i 0.885456 − 0.464723i −1.62874 − 0.854831i 0.568065 − 0.822984i −0.748511 − 0.663123i −2.95243 + 1.54955i
211.1 −0.354605 − 0.935016i 0.568065 − 0.822984i −0.748511 + 0.663123i −1.80124 + 2.60955i −0.970942 − 0.239316i 4.26907 − 1.05223i 0.885456 + 0.464723i −0.354605 − 0.935016i 3.07870 + 0.758832i
211.2 −0.354605 − 0.935016i 0.568065 − 0.822984i −0.748511 + 0.663123i −0.564244 + 0.817448i −0.970942 − 0.239316i −1.11254 + 0.274216i 0.885456 + 0.464723i −0.354605 − 0.935016i 0.964411 + 0.237706i
211.3 −0.354605 − 0.935016i 0.568065 − 0.822984i −0.748511 + 0.663123i −0.190992 + 0.276700i −0.970942 − 0.239316i 1.33374 − 0.328737i 0.885456 + 0.464723i −0.354605 − 0.935016i 0.326446 + 0.0804617i
211.4 −0.354605 − 0.935016i 0.568065 − 0.822984i −0.748511 + 0.663123i −0.0564571 + 0.0817922i −0.970942 − 0.239316i −3.63389 + 0.895673i 0.885456 + 0.464723i −0.354605 − 0.935016i 0.0964970 + 0.0237844i
211.5 −0.354605 − 0.935016i 0.568065 − 0.822984i −0.748511 + 0.663123i 1.43698 − 2.08183i −0.970942 − 0.239316i 0.614556 − 0.151474i 0.885456 + 0.464723i −0.354605 − 0.935016i −2.45611 − 0.605376i
301.1 0.568065 − 0.822984i 0.885456 − 0.464723i −0.354605 − 0.935016i −2.64920 + 1.39041i 0.120537 − 0.992709i 0.295594 + 2.43443i −0.970942 − 0.239316i 0.568065 − 0.822984i −0.360634 + 2.97009i
301.2 0.568065 − 0.822984i 0.885456 − 0.464723i −0.354605 − 0.935016i −1.83329 + 0.962187i 0.120537 − 0.992709i −0.227040 − 1.86984i −0.970942 − 0.239316i 0.568065 − 0.822984i −0.249565 + 2.05536i
301.3 0.568065 − 0.822984i 0.885456 − 0.464723i −0.354605 − 0.935016i 1.02853 − 0.539815i 0.120537 − 0.992709i 0.610873 + 5.03099i −0.970942 − 0.239316i 0.568065 − 0.822984i 0.140014 − 1.15312i
301.4 0.568065 − 0.822984i 0.885456 − 0.464723i −0.354605 − 0.935016i 2.07146 − 1.08719i 0.120537 − 0.992709i −0.415147 − 3.41905i −0.970942 − 0.239316i 0.568065 − 0.822984i 0.281987 − 2.32237i
301.5 0.568065 − 0.822984i 0.885456 − 0.464723i −0.354605 − 0.935016i 2.44312 − 1.28225i 0.120537 − 0.992709i 0.115184 + 0.948629i −0.970942 − 0.239316i 0.568065 − 0.822984i 0.332581 − 2.73905i
307.1 0.120537 + 0.992709i −0.748511 − 0.663123i −0.970942 + 0.239316i −2.09108 − 1.85254i 0.568065 − 0.822984i −1.80465 − 2.61449i −0.354605 − 0.935016i 0.120537 + 0.992709i 1.58698 − 2.29913i
307.2 0.120537 + 0.992709i −0.748511 − 0.663123i −0.970942 + 0.239316i −0.0689369 − 0.0610728i 0.568065 − 0.822984i −0.0119534 − 0.0173175i −0.354605 − 0.935016i 0.120537 + 0.992709i 0.0523181 − 0.0757958i
307.3 0.120537 + 0.992709i −0.748511 − 0.663123i −0.970942 + 0.239316i 1.08579 + 0.961923i 0.568065 − 0.822984i 1.17583 + 1.70348i −0.354605 − 0.935016i 0.120537 + 0.992709i −0.824032 + 1.19382i
307.4 0.120537 + 0.992709i −0.748511 − 0.663123i −0.970942 + 0.239316i 1.21524 + 1.07661i 0.568065 − 0.822984i −1.95282 − 2.82915i −0.354605 − 0.935016i 0.120537 + 0.992709i −0.922281 + 1.33616i
307.5 0.120537 + 0.992709i −0.748511 − 0.663123i −0.970942 + 0.239316i 3.00111 + 2.65875i 0.568065 − 0.822984i 2.52554 + 3.65887i −0.354605 − 0.935016i 0.120537 + 0.992709i −2.27762 + 3.29970i
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 193.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
131.e even 13 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 786.2.i.a ✓ 60
131.e even 13 1 inner 786.2.i.a ✓ 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
786.2.i.a ✓ 60 1.a even 1 1 trivial
786.2.i.a ✓ 60 131.e even 13 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{60} - 8 T_{5}^{59} + 33 T_{5}^{58} - 74 T_{5}^{57} + 35 T_{5}^{56} + 557 T_{5}^{55} + \cdots + 7733961 \) acting on \(S_{2}^{\mathrm{new}}(786, [\chi])\). Copy content Toggle raw display