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Properties

Label 786.2.m.c
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.

\(\operatorname{Tr}(f)(q) = \) \( 288 q - 6 q^{2} + 6 q^{3} + 6 q^{4} - 2 q^{5} - 6 q^{6} + q^{7} - 6 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 288 q - 6 q^{2} + 6 q^{3} + 6 q^{4} - 2 q^{5} - 6 q^{6} + q^{7} - 6 q^{8} + 6 q^{9} - 21 q^{10} + 4 q^{11} + 6 q^{12} - 6 q^{13} - q^{14} + 8 q^{15} + 6 q^{16} + 61 q^{17} + 24 q^{18} + 40 q^{19} + 8 q^{20} + q^{21} + 2 q^{22} + 8 q^{23} + 24 q^{24} + 10 q^{25} - 2 q^{26} + 6 q^{27} + q^{28} + 75 q^{29} + 2 q^{30} + 8 q^{31} + 24 q^{32} + 11 q^{33} - 11 q^{34} + 33 q^{35} + 6 q^{36} - 127 q^{37} + 35 q^{38} - 31 q^{39} - 11 q^{40} + 5 q^{41} + 12 q^{42} - 17 q^{43} - 2 q^{44} + q^{45} - 8 q^{46} - 18 q^{47} + 6 q^{48} + 133 q^{49} - 10 q^{50} - 14 q^{51} - 18 q^{52} - 57 q^{53} - 6 q^{54} + 28 q^{55} - q^{56} + 15 q^{57} - 47 q^{58} + 47 q^{59} + q^{60} - 62 q^{61} - 34 q^{62} - 4 q^{63} + 6 q^{64} + 160 q^{65} - 4 q^{66} + 69 q^{67} + 103 q^{68} + 20 q^{69} + 71 q^{70} - 90 q^{71} - 6 q^{72} - 17 q^{73} + 22 q^{74} + 36 q^{75} + 41 q^{76} - 31 q^{77} - 15 q^{78} - 126 q^{79} - 12 q^{80} + 6 q^{81} - 33 q^{82} - 33 q^{83} - 4 q^{84} + 89 q^{85} + 2 q^{86} + 8 q^{87} - 43 q^{88} - 35 q^{89} + 2 q^{90} - 109 q^{91} + 20 q^{92} + 8 q^{93} + 192 q^{95} - 6 q^{96} + 93 q^{97} - 3 q^{98} - 17 q^{99}+O(q^{100}) \) 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 −3.00802 + 0.291678i −0.958246 + 0.285946i 2.64463 − 0.930524i −0.779674 − 0.626185i 0.485983 + 0.873968i −1.83492 + 2.40132i
7.2 0.681016 − 0.732269i −0.861970 − 0.506960i −0.0724348 − 0.997373i −0.600196 + 0.0581989i −0.958246 + 0.285946i −2.69233 + 0.947307i −0.779674 − 0.626185i 0.485983 + 0.873968i −0.366125 + 0.479139i
7.3 0.681016 − 0.732269i −0.861970 − 0.506960i −0.0724348 − 0.997373i −0.381220 + 0.0369656i −0.958246 + 0.285946i 2.87168 − 1.01041i −0.779674 − 0.626185i 0.485983 + 0.873968i −0.232548 + 0.304330i
7.4 0.681016 − 0.732269i −0.861970 − 0.506960i −0.0724348 − 0.997373i 0.773859 − 0.0750385i −0.958246 + 0.285946i −3.78665 + 1.33235i −0.779674 − 0.626185i 0.485983 + 0.873968i 0.472062 − 0.617775i
7.5 0.681016 − 0.732269i −0.861970 − 0.506960i −0.0724348 − 0.997373i 1.92909 − 0.187057i −0.958246 + 0.285946i 3.57055 − 1.25631i −0.779674 − 0.626185i 0.485983 + 0.873968i 1.17676 − 1.54000i
7.6 0.681016 − 0.732269i −0.861970 − 0.506960i −0.0724348 − 0.997373i 3.35795 − 0.325609i −0.958246 + 0.285946i 0.0629706 − 0.0221565i −0.779674 − 0.626185i 0.485983 + 0.873968i 2.04838 − 2.68066i
13.1 0.906874 + 0.421401i 0.995332 − 0.0965139i 0.644842 + 0.764316i −1.14848 − 2.63648i 0.943312 + 0.331908i −0.366867 − 0.480109i 0.262707 + 0.964876i 0.981370 − 0.192127i 0.0694893 − 2.87493i
13.2 0.906874 + 0.421401i 0.995332 − 0.0965139i 0.644842 + 0.764316i −0.631716 − 1.45018i 0.943312 + 0.331908i −2.51798 − 3.29521i 0.262707 + 0.964876i 0.981370 − 0.192127i 0.0382223 − 1.58134i
13.3 0.906874 + 0.421401i 0.995332 − 0.0965139i 0.644842 + 0.764316i −0.439597 − 1.00915i 0.943312 + 0.331908i 2.03543 + 2.66371i 0.262707 + 0.964876i 0.981370 − 0.192127i 0.0265980 − 1.10042i
13.4 0.906874 + 0.421401i 0.995332 − 0.0965139i 0.644842 + 0.764316i 0.429049 + 0.984937i 0.943312 + 0.331908i 1.94186 + 2.54126i 0.262707 + 0.964876i 0.981370 − 0.192127i −0.0259598 + 1.07402i
13.5 0.906874 + 0.421401i 0.995332 − 0.0965139i 0.644842 + 0.764316i 1.13484 + 2.60517i 0.943312 + 0.331908i −1.83038 − 2.39538i 0.262707 + 0.964876i 0.981370 − 0.192127i −0.0686641 + 2.84079i
13.6 0.906874 + 0.421401i 0.995332 − 0.0965139i 0.644842 + 0.764316i 1.53997 + 3.53519i 0.943312 + 0.331908i 0.247263 + 0.323586i 0.262707 + 0.964876i 0.981370 − 0.192127i −0.0931765 + 3.85492i
25.1 0.607163 − 0.794578i −0.989506 + 0.144489i −0.262707 − 0.964876i −0.570596 + 3.34082i −0.485983 + 0.873968i −3.35405 + 0.656636i −0.926175 − 0.377095i 0.958246 − 0.285946i 2.30810 + 2.48181i
25.2 0.607163 − 0.794578i −0.989506 + 0.144489i −0.262707 − 0.964876i −0.466535 + 2.73155i −0.485983 + 0.873968i 2.28822 − 0.447974i −0.926175 − 0.377095i 0.958246 − 0.285946i 1.88717 + 2.02919i
25.3 0.607163 − 0.794578i −0.989506 + 0.144489i −0.262707 − 0.964876i 0.00652668 − 0.0382135i −0.485983 + 0.873968i 1.42164 − 0.278320i −0.926175 − 0.377095i 0.958246 − 0.285946i −0.0264009 − 0.0283878i
25.4 0.607163 − 0.794578i −0.989506 + 0.144489i −0.262707 − 0.964876i 0.0434492 − 0.254394i −0.485983 + 0.873968i −1.44418 + 0.282732i −0.926175 − 0.377095i 0.958246 − 0.285946i −0.175755 − 0.188982i
25.5 0.607163 − 0.794578i −0.989506 + 0.144489i −0.262707 − 0.964876i 0.449237 − 2.63027i −0.485983 + 0.873968i 3.13940 − 0.614613i −0.926175 − 0.377095i 0.958246 − 0.285946i −1.81719 − 1.95395i
25.6 0.607163 − 0.794578i −0.989506 + 0.144489i −0.262707 − 0.964876i 0.581969 − 3.40741i −0.485983 + 0.873968i −4.29179 + 0.840220i −0.926175 − 0.377095i 0.958246 − 0.285946i −2.35411 − 2.53127i
43.1 0.989506 − 0.144489i −0.527640 + 0.849468i 0.958246 − 0.285946i −3.71244 + 1.51153i −0.399365 + 0.916792i −0.108120 + 0.489250i 0.906874 − 0.421401i −0.443192 − 0.896427i −3.45508 + 2.03207i
43.2 0.989506 − 0.144489i −0.527640 + 0.849468i 0.958246 − 0.285946i −1.57400 + 0.640857i −0.399365 + 0.916792i −0.210083 + 0.950641i 0.906874 − 0.421401i −0.443192 − 0.896427i −1.46488 + 0.861557i
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.c ✓ 288
131.g even 65 1 inner 786.2.m.c ✓ 288
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
786.2.m.c ✓ 288 1.a even 1 1 trivial
786.2.m.c ✓ 288 131.g even 65 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{288} + 2 T_{5}^{287} - 18 T_{5}^{286} + 219 T_{5}^{285} + 723 T_{5}^{284} - 3832 T_{5}^{283} + \cdots + 16\!\cdots\!01 \) acting on \(S_{2}^{\mathrm{new}}(786, [\chi])\). Copy content Toggle raw display