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Properties

Label 786.2.m.d
Level $786$
Weight $2$
Character orbit 786.m
Analytic conductor $6.276$
Analytic rank $0$
Dimension $288$
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(7,786)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(786, base_ring=CyclotomicField(130))
 
chi = DirichletCharacter(H, H._module([0, 96]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("786.7");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 786 = 2 \cdot 3 \cdot 131 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 786.m (of order \(65\), degree \(48\), 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: \(288\)
Relative dimension: \(6\) over \(\Q(\zeta_{65})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{65}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 288 q + 6 q^{2} - 6 q^{3} + 6 q^{4} - 6 q^{6} + 7 q^{7} + 6 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 288 q + 6 q^{2} - 6 q^{3} + 6 q^{4} - 6 q^{6} + 7 q^{7} + 6 q^{8} + 6 q^{9} + 13 q^{10} - 20 q^{11} - 6 q^{12} + 26 q^{13} - 3 q^{14} + 6 q^{16} + 73 q^{17} - 24 q^{18} - 20 q^{19} + 3 q^{21} - 10 q^{22} + 36 q^{23} + 24 q^{24} - 6 q^{25} - 2 q^{26} - 6 q^{27} - 3 q^{28} - 33 q^{29} + 8 q^{31} - 24 q^{32} - 3 q^{33} + 3 q^{34} + q^{35} + 6 q^{36} + 159 q^{37} - 43 q^{38} + 35 q^{39} - 13 q^{40} - 11 q^{41} + 6 q^{42} - 13 q^{43} - 10 q^{44} + 13 q^{45} + 20 q^{46} + 10 q^{47} - 6 q^{48} + 97 q^{49} - 6 q^{50} + 22 q^{51} - 22 q^{52} + 27 q^{53} - 6 q^{54} + 104 q^{55} + 7 q^{56} - 27 q^{57} + 53 q^{58} - 7 q^{59} + 13 q^{60} - 94 q^{61} + 14 q^{62} - 8 q^{63} + 6 q^{64} + 94 q^{65} - 6 q^{66} - 101 q^{67} + 95 q^{68} + 8 q^{69} + q^{70} + 98 q^{71} + 6 q^{72} + 51 q^{73} - 12 q^{74} + 12 q^{75} - 51 q^{76} + 17 q^{77} - 11 q^{78} + 10 q^{79} + 6 q^{81} + 23 q^{82} + q^{83} + 8 q^{84} + 189 q^{85} - 62 q^{86} + 12 q^{87} + 45 q^{88} - 135 q^{89} + 111 q^{91} - 34 q^{92} - 8 q^{93} - 12 q^{94} - 174 q^{95} - 6 q^{96} + 129 q^{97} + 71 q^{98} - 5 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −0.681016 + 0.732269i 0.861970 + 0.506960i −0.0724348 − 0.997373i −3.72308 + 0.361014i −0.958246 + 0.285946i −2.11795 + 0.745209i 0.779674 + 0.626185i 0.485983 + 0.873968i 2.27112 − 2.97215i
7.2 −0.681016 + 0.732269i 0.861970 + 0.506960i −0.0724348 − 0.997373i −3.07580 + 0.298250i −0.958246 + 0.285946i 2.37618 − 0.836069i 0.779674 + 0.626185i 0.485983 + 0.873968i 1.87627 − 2.45543i
7.3 −0.681016 + 0.732269i 0.861970 + 0.506960i −0.0724348 − 0.997373i −0.975398 + 0.0945810i −0.958246 + 0.285946i −0.701162 + 0.246707i 0.779674 + 0.626185i 0.485983 + 0.873968i 0.595003 − 0.778665i
7.4 −0.681016 + 0.732269i 0.861970 + 0.506960i −0.0724348 − 0.997373i −0.461341 + 0.0447346i −0.958246 + 0.285946i −2.91442 + 1.02545i 0.779674 + 0.626185i 0.485983 + 0.873968i 0.281423 − 0.368290i
7.5 −0.681016 + 0.732269i 0.861970 + 0.506960i −0.0724348 − 0.997373i 1.98690 − 0.192663i −0.958246 + 0.285946i 2.48563 − 0.874577i 0.779674 + 0.626185i 0.485983 + 0.873968i −1.21203 + 1.58615i
7.6 −0.681016 + 0.732269i 0.861970 + 0.506960i −0.0724348 − 0.997373i 2.35284 − 0.228147i −0.958246 + 0.285946i −0.191070 + 0.0672289i 0.779674 + 0.626185i 0.485983 + 0.873968i −1.43525 + 1.87828i
13.1 −0.906874 − 0.421401i −0.995332 + 0.0965139i 0.644842 + 0.764316i −1.07342 − 2.46418i 0.943312 + 0.331908i 1.56977 + 2.05432i −0.262707 − 0.964876i 0.981370 − 0.192127i −0.0649479 + 2.68704i
13.2 −0.906874 − 0.421401i −0.995332 + 0.0965139i 0.644842 + 0.764316i −0.557707 − 1.28029i 0.943312 + 0.331908i 0.614843 + 0.804629i −0.262707 − 0.964876i 0.981370 − 0.192127i −0.0337443 + 1.39608i
13.3 −0.906874 − 0.421401i −0.995332 + 0.0965139i 0.644842 + 0.764316i −0.338856 − 0.777887i 0.943312 + 0.331908i −2.57102 − 3.36462i −0.262707 − 0.964876i 0.981370 − 0.192127i −0.0205026 + 0.848240i
13.4 −0.906874 − 0.421401i −0.995332 + 0.0965139i 0.644842 + 0.764316i 0.780569 + 1.79189i 0.943312 + 0.331908i −2.62328 − 3.43301i −0.262707 − 0.964876i 0.981370 − 0.192127i 0.0472287 − 1.95396i
13.5 −0.906874 − 0.421401i −0.995332 + 0.0965139i 0.644842 + 0.764316i 1.22186 + 2.80493i 0.943312 + 0.331908i 0.287917 + 0.376790i −0.262707 − 0.964876i 0.981370 − 0.192127i 0.0739291 − 3.05861i
13.6 −0.906874 − 0.421401i −0.995332 + 0.0965139i 0.644842 + 0.764316i 1.38912 + 3.18890i 0.943312 + 0.331908i 1.82220 + 2.38467i −0.262707 − 0.964876i 0.981370 − 0.192127i 0.0840494 − 3.47731i
25.1 −0.607163 + 0.794578i 0.989506 − 0.144489i −0.262707 − 0.964876i −0.641628 + 3.75672i −0.485983 + 0.873968i −0.920190 + 0.180149i 0.926175 + 0.377095i 0.958246 − 0.285946i −2.59543 − 2.79076i
25.2 −0.607163 + 0.794578i 0.989506 − 0.144489i −0.262707 − 0.964876i −0.132357 + 0.774948i −0.485983 + 0.873968i 3.63703 − 0.712035i 0.926175 + 0.377095i 0.958246 − 0.285946i −0.535394 − 0.575688i
25.3 −0.607163 + 0.794578i 0.989506 − 0.144489i −0.262707 − 0.964876i −0.115628 + 0.676998i −0.485983 + 0.873968i −4.98517 + 0.975966i 0.926175 + 0.377095i 0.958246 − 0.285946i −0.467722 − 0.502923i
25.4 −0.607163 + 0.794578i 0.989506 − 0.144489i −0.262707 − 0.964876i −0.0656415 + 0.384329i −0.485983 + 0.873968i −0.958358 + 0.187621i 0.926175 + 0.377095i 0.958246 − 0.285946i −0.265524 − 0.285508i
25.5 −0.607163 + 0.794578i 0.989506 − 0.144489i −0.262707 − 0.964876i 0.287906 − 1.68568i −0.485983 + 0.873968i −0.0733138 + 0.0143529i 0.926175 + 0.377095i 0.958246 − 0.285946i 1.16460 + 1.25225i
25.6 −0.607163 + 0.794578i 0.989506 − 0.144489i −0.262707 − 0.964876i 0.613859 − 3.59413i −0.485983 + 0.873968i 4.44290 − 0.869803i 0.926175 + 0.377095i 0.958246 − 0.285946i 2.48310 + 2.66998i
43.1 −0.989506 + 0.144489i 0.527640 − 0.849468i 0.958246 − 0.285946i −3.53310 + 1.43851i −0.399365 + 0.916792i 0.398648 − 1.80391i −0.906874 + 0.421401i −0.443192 − 0.896427i 3.28818 − 1.93391i
43.2 −0.989506 + 0.144489i 0.527640 − 0.849468i 0.958246 − 0.285946i −2.14904 + 0.874988i −0.399365 + 0.916792i −0.362509 + 1.64038i −0.906874 + 0.421401i −0.443192 − 0.896427i 2.00006 − 1.17632i
See next 80 embeddings (of 288 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.6
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 786.2.m.d ✓ 288
131.g even 65 1 inner 786.2.m.d ✓ 288
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
786.2.m.d ✓ 288 1.a even 1 1 trivial
786.2.m.d ✓ 288 131.g even 65 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{288} - 12 T_{5}^{286} + 277 T_{5}^{285} - 133 T_{5}^{284} - 3512 T_{5}^{283} + \cdots + 10\!\cdots\!25 \) acting on \(S_{2}^{\mathrm{new}}(786, [\chi])\). Copy content Toggle raw display