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Properties

Label 786.2.i.d
Level $786$
Weight $2$
Character orbit 786.i
Analytic conductor $6.276$
Analytic rank $0$
Dimension $72$
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: \(72\)
Relative dimension: \(6\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 72 q + 6 q^{2} - 6 q^{3} - 6 q^{4} + 2 q^{5} + 6 q^{6} - q^{7} + 6 q^{8} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q + 6 q^{2} - 6 q^{3} - 6 q^{4} + 2 q^{5} + 6 q^{6} - q^{7} + 6 q^{8} - 6 q^{9} + 11 q^{10} - 14 q^{11} - 6 q^{12} + 6 q^{13} + q^{14} + 2 q^{15} - 6 q^{16} - q^{17} + 6 q^{18} - 20 q^{19} + 2 q^{20} - q^{21} - 12 q^{22} - 13 q^{23} + 6 q^{24} + 7 q^{26} - 6 q^{27} - q^{28} - 2 q^{30} + 7 q^{31} + 6 q^{32} - q^{33} + q^{34} - 48 q^{35} - 6 q^{36} + 12 q^{37} + 20 q^{38} + 6 q^{39} + 11 q^{40} - 10 q^{41} - 12 q^{42} + 22 q^{43} + 12 q^{44} - 11 q^{45} + 13 q^{46} - 2 q^{47} - 6 q^{48} - 3 q^{49} - q^{51} - 7 q^{52} + 72 q^{53} + 6 q^{54} - 3 q^{55} + q^{56} - 20 q^{57} + 52 q^{58} - 57 q^{59} - 11 q^{60} + 22 q^{61} + 19 q^{62} - q^{63} - 6 q^{64} + 40 q^{65} + 14 q^{66} + 51 q^{67} + 12 q^{68} - 56 q^{70} - 40 q^{71} + 6 q^{72} + 2 q^{73} - 12 q^{74} - 26 q^{75} + 19 q^{76} - 14 q^{77} + 20 q^{78} + 26 q^{79} + 2 q^{80} - 6 q^{81} + 23 q^{82} - 2 q^{83} - q^{84} - 34 q^{85} + 43 q^{86} - 13 q^{87} - 12 q^{88} + 10 q^{89} - 2 q^{90} + 59 q^{91} + 7 q^{93} + 15 q^{94} + 63 q^{95} + 6 q^{96} + 42 q^{97} + 3 q^{98} + 12 q^{99}+O(q^{100}) \) 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 −1.28078 + 3.37713i −0.885456 + 0.464723i 3.06122 + 1.60665i −0.568065 + 0.822984i −0.748511 − 0.663123i −3.19813 + 1.67851i
193.2 0.748511 + 0.663123i −0.354605 + 0.935016i 0.120537 + 0.992709i −0.592195 + 1.56149i −0.885456 + 0.464723i 0.165598 + 0.0869127i −0.568065 + 0.822984i −0.748511 − 0.663123i −1.47872 + 0.776094i
193.3 0.748511 + 0.663123i −0.354605 + 0.935016i 0.120537 + 0.992709i 0.264344 − 0.697017i −0.885456 + 0.464723i −3.81335 − 2.00140i −0.568065 + 0.822984i −0.748511 − 0.663123i 0.660072 − 0.346432i
193.4 0.748511 + 0.663123i −0.354605 + 0.935016i 0.120537 + 0.992709i 0.271405 − 0.715636i −0.885456 + 0.464723i −1.62125 − 0.850898i −0.568065 + 0.822984i −0.748511 − 0.663123i 0.677704 − 0.355686i
193.5 0.748511 + 0.663123i −0.354605 + 0.935016i 0.120537 + 0.992709i 0.620421 − 1.63592i −0.885456 + 0.464723i 2.43530 + 1.27815i −0.568065 + 0.822984i −0.748511 − 0.663123i 1.54921 − 0.813086i
193.6 0.748511 + 0.663123i −0.354605 + 0.935016i 0.120537 + 0.992709i 1.03844 − 2.73815i −0.885456 + 0.464723i −1.27498 − 0.669160i −0.568065 + 0.822984i −0.748511 − 0.663123i 2.59302 − 1.36092i
211.1 0.354605 + 0.935016i 0.568065 − 0.822984i −0.748511 + 0.663123i −2.12984 + 3.08560i 0.970942 + 0.239316i −1.73893 + 0.428607i −0.885456 − 0.464723i −0.354605 − 0.935016i −3.64034 − 0.897264i
211.2 0.354605 + 0.935016i 0.568065 − 0.822984i −0.748511 + 0.663123i −0.663640 + 0.961449i 0.970942 + 0.239316i 2.48314 − 0.612040i −0.885456 − 0.464723i −0.354605 − 0.935016i −1.13430 − 0.279580i
211.3 0.354605 + 0.935016i 0.568065 − 0.822984i −0.748511 + 0.663123i −0.137043 + 0.198541i 0.970942 + 0.239316i 1.36790 − 0.337156i −0.885456 − 0.464723i −0.354605 − 0.935016i −0.234235 − 0.0577337i
211.4 0.354605 + 0.935016i 0.568065 − 0.822984i −0.748511 + 0.663123i 0.651716 − 0.944173i 0.970942 + 0.239316i −4.51866 + 1.11375i −0.885456 − 0.464723i −0.354605 − 0.935016i 1.11392 + 0.274556i
211.5 0.354605 + 0.935016i 0.568065 − 0.822984i −0.748511 + 0.663123i 1.29088 − 1.87016i 0.970942 + 0.239316i −3.02586 + 0.745808i −0.885456 − 0.464723i −0.354605 − 0.935016i 2.20638 + 0.543825i
211.6 0.354605 + 0.935016i 0.568065 − 0.822984i −0.748511 + 0.663123i 2.22118 − 3.21793i 0.970942 + 0.239316i 2.77388 − 0.683699i −0.885456 − 0.464723i −0.354605 − 0.935016i 3.79646 + 0.935744i
301.1 −0.568065 + 0.822984i 0.885456 − 0.464723i −0.354605 − 0.935016i −2.54368 + 1.33503i −0.120537 + 0.992709i 0.145603 + 1.19915i 0.970942 + 0.239316i 0.568065 − 0.822984i 0.346270 − 2.85179i
301.2 −0.568065 + 0.822984i 0.885456 − 0.464723i −0.354605 − 0.935016i −1.06019 + 0.556429i −0.120537 + 0.992709i −0.592991 − 4.88372i 0.970942 + 0.239316i 0.568065 − 0.822984i 0.144323 − 1.18860i
301.3 −0.568065 + 0.822984i 0.885456 − 0.464723i −0.354605 − 0.935016i −0.748561 + 0.392875i −0.120537 + 0.992709i 0.138987 + 1.14466i 0.970942 + 0.239316i 0.568065 − 0.822984i 0.101901 − 0.839232i
301.4 −0.568065 + 0.822984i 0.885456 − 0.464723i −0.354605 − 0.935016i 1.72311 − 0.904355i −0.120537 + 0.992709i 0.552201 + 4.54778i 0.970942 + 0.239316i 0.568065 − 0.822984i −0.234565 + 1.93182i
301.5 −0.568065 + 0.822984i 0.885456 − 0.464723i −0.354605 − 0.935016i 2.39432 − 1.25664i −0.120537 + 0.992709i 0.127325 + 1.04862i 0.970942 + 0.239316i 0.568065 − 0.822984i −0.325938 + 2.68434i
301.6 −0.568065 + 0.822984i 0.885456 − 0.464723i −0.354605 − 0.935016i 3.51191 − 1.84319i −0.120537 + 0.992709i −0.345548 − 2.84585i 0.970942 + 0.239316i 0.568065 − 0.822984i −0.478074 + 3.93729i
307.1 −0.120537 − 0.992709i −0.748511 − 0.663123i −0.970942 + 0.239316i −2.28513 − 2.02445i −0.568065 + 0.822984i −1.49653 − 2.16810i 0.354605 + 0.935016i 0.120537 + 0.992709i −1.73425 + 2.51249i
307.2 −0.120537 − 0.992709i −0.748511 − 0.663123i −0.970942 + 0.239316i −1.82531 − 1.61708i −0.568065 + 0.822984i 1.86455 + 2.70126i 0.354605 + 0.935016i 0.120537 + 0.992709i −1.38528 + 2.00692i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 193.6
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.d ✓ 72
131.e even 13 1 inner 786.2.i.d ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
786.2.i.d ✓ 72 1.a even 1 1 trivial
786.2.i.d ✓ 72 131.e even 13 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{72} - 2 T_{5}^{71} + 17 T_{5}^{70} + 16 T_{5}^{69} + 129 T_{5}^{68} + 663 T_{5}^{67} + \cdots + 12\!\cdots\!21 \) acting on \(S_{2}^{\mathrm{new}}(786, [\chi])\). Copy content Toggle raw display