↑

Properties

Label 786.2.m.a
Level $786$
Weight $2$
Character orbit 786.m
Analytic conductor $6.276$
Analytic rank $0$
Dimension $240$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

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: \(240\)
Relative dimension: \(5\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 240 q - 5 q^{2} - 5 q^{3} + 5 q^{4} + 2 q^{5} + 5 q^{6} - 3 q^{7} - 5 q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 240 q - 5 q^{2} - 5 q^{3} + 5 q^{4} + 2 q^{5} + 5 q^{6} - 3 q^{7} - 5 q^{8} + 5 q^{9} - 10 q^{10} - 13 q^{11} - 5 q^{12} - 16 q^{13} + 3 q^{14} + 3 q^{15} + 5 q^{16} - 39 q^{17} + 20 q^{18} - 16 q^{19} + 49 q^{20} + 3 q^{21} - 8 q^{22} + 2 q^{23} - 20 q^{24} - 9 q^{25} + 8 q^{26} - 5 q^{27} - 3 q^{28} + 62 q^{29} + 2 q^{30} - 2 q^{31} + 20 q^{32} + 5 q^{33} + 5 q^{34} - 22 q^{35} + 5 q^{36} + 61 q^{37} + 39 q^{38} - 35 q^{39} + 11 q^{40} - 9 q^{41} + 10 q^{42} - 39 q^{43} + 8 q^{44} + 15 q^{45} - 22 q^{46} - 26 q^{47} - 5 q^{48} - 136 q^{49} + 9 q^{50} + 16 q^{51} + 22 q^{52} + 4 q^{53} + 5 q^{54} + 13 q^{55} + 3 q^{56} - 21 q^{57} - 49 q^{58} + 5 q^{59} + 11 q^{60} + 88 q^{61} - 4 q^{62} + 12 q^{63} + 5 q^{64} - 144 q^{65} + 13 q^{66} + 87 q^{67} - 81 q^{68} - 4 q^{69} + 9 q^{70} + 110 q^{71} - 5 q^{72} - 48 q^{73} - 4 q^{74} + 62 q^{75} + 47 q^{76} - 17 q^{77} + 31 q^{78} + 185 q^{79} + 2 q^{80} + 5 q^{81} - 15 q^{82} - 33 q^{83} - 12 q^{84} + 45 q^{85} + 26 q^{86} + 16 q^{87} + 52 q^{88} - 65 q^{89} + 11 q^{90} - 91 q^{91} + 4 q^{92} + 2 q^{93} - 2 q^{94} + 266 q^{95} + 5 q^{96} - 140 q^{97} + 6 q^{98} + 23 q^{99}+O(q^{100}) \) 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 −4.00032 + 0.387897i 0.958246 − 0.285946i 4.84313 − 1.70407i −0.779674 − 0.626185i 0.485983 + 0.873968i −2.44023 + 3.19347i
7.2 0.681016 − 0.732269i 0.861970 + 0.506960i −0.0724348 − 0.997373i −2.77984 + 0.269552i 0.958246 − 0.285946i −2.25900 + 0.794839i −0.779674 − 0.626185i 0.485983 + 0.873968i −1.69573 + 2.21916i
7.3 0.681016 − 0.732269i 0.861970 + 0.506960i −0.0724348 − 0.997373i −1.51184 + 0.146598i 0.958246 − 0.285946i −2.49824 + 0.879014i −0.779674 − 0.626185i 0.485983 + 0.873968i −0.922236 + 1.20691i
7.4 0.681016 − 0.732269i 0.861970 + 0.506960i −0.0724348 − 0.997373i 1.49010 − 0.144490i 0.958246 − 0.285946i −0.335604 + 0.118084i −0.779674 − 0.626185i 0.485983 + 0.873968i 0.908975 − 1.18955i
7.5 0.681016 − 0.732269i 0.861970 + 0.506960i −0.0724348 − 0.997373i 2.57087 − 0.249289i 0.958246 − 0.285946i 4.13819 − 1.45604i −0.779674 − 0.626185i 0.485983 + 0.873968i 1.56826 − 2.05234i
13.1 0.906874 + 0.421401i −0.995332 + 0.0965139i 0.644842 + 0.764316i −1.10365 − 2.53358i −0.943312 − 0.331908i −1.08940 − 1.42567i 0.262707 + 0.964876i 0.981370 − 0.192127i 0.0667772 − 2.76272i
13.2 0.906874 + 0.421401i −0.995332 + 0.0965139i 0.644842 + 0.764316i −0.0973104 − 0.223388i −0.943312 − 0.331908i 0.124169 + 0.162497i 0.262707 + 0.964876i 0.981370 − 0.192127i 0.00588781 − 0.243592i
13.3 0.906874 + 0.421401i −0.995332 + 0.0965139i 0.644842 + 0.764316i 0.235998 + 0.541764i −0.943312 − 0.331908i −2.41829 − 3.16475i 0.262707 + 0.964876i 0.981370 − 0.192127i −0.0142792 + 0.590762i
13.4 0.906874 + 0.421401i −0.995332 + 0.0965139i 0.644842 + 0.764316i 0.358863 + 0.823817i −0.943312 − 0.331908i 2.96271 + 3.87722i 0.262707 + 0.964876i 0.981370 − 0.192127i −0.0217132 + 0.898324i
13.5 0.906874 + 0.421401i −0.995332 + 0.0965139i 0.644842 + 0.764316i 1.67368 + 3.84214i −0.943312 − 0.331908i 0.713852 + 0.934200i 0.262707 + 0.964876i 0.981370 − 0.192127i −0.101267 + 4.18963i
25.1 0.607163 − 0.794578i 0.989506 − 0.144489i −0.262707 − 0.964876i −0.568434 + 3.32816i 0.485983 − 0.873968i −2.93095 + 0.573803i −0.926175 − 0.377095i 0.958246 − 0.285946i 2.29935 + 2.47240i
25.2 0.607163 − 0.794578i 0.989506 − 0.144489i −0.262707 − 0.964876i −0.329012 + 1.92636i 0.485983 − 0.873968i 0.477356 − 0.0934538i −0.926175 − 0.377095i 0.958246 − 0.285946i 1.33088 + 1.43104i
25.3 0.607163 − 0.794578i 0.989506 − 0.144489i −0.262707 − 0.964876i −0.239958 + 1.40495i 0.485983 − 0.873968i 2.12393 − 0.415811i −0.926175 − 0.377095i 0.958246 − 0.285946i 0.970648 + 1.04370i
25.4 0.607163 − 0.794578i 0.989506 − 0.144489i −0.262707 − 0.964876i 0.234662 − 1.37394i 0.485983 − 0.873968i −3.29183 + 0.644454i −0.926175 − 0.377095i 0.958246 − 0.285946i −0.949223 − 1.02066i
25.5 0.607163 − 0.794578i 0.989506 − 0.144489i −0.262707 − 0.964876i 0.514110 − 3.01010i 0.485983 − 0.873968i 1.32370 − 0.259146i −0.926175 − 0.377095i 0.958246 − 0.285946i −2.07961 − 2.23612i
43.1 0.989506 − 0.144489i 0.527640 − 0.849468i 0.958246 − 0.285946i −2.71993 + 1.10743i 0.399365 − 0.916792i −0.485638 + 2.19755i 0.906874 − 0.421401i −0.443192 − 0.896427i −2.53138 + 1.48881i
43.2 0.989506 − 0.144489i 0.527640 − 0.849468i 0.958246 − 0.285946i −1.40313 + 0.571288i 0.399365 − 0.916792i 0.609461 − 2.75786i 0.906874 − 0.421401i −0.443192 − 0.896427i −1.30586 + 0.768030i
43.3 0.989506 − 0.144489i 0.527640 − 0.849468i 0.958246 − 0.285946i 1.68901 − 0.687686i 0.399365 − 0.916792i −0.831274 + 3.76158i 0.906874 − 0.421401i −0.443192 − 0.896427i 1.57192 − 0.924514i
43.4 0.989506 − 0.144489i 0.527640 − 0.849468i 0.958246 − 0.285946i 2.13755 − 0.870308i 0.399365 − 0.916792i 1.00596 − 4.55203i 0.906874 − 0.421401i −0.443192 − 0.896427i 1.98936 − 1.17003i
43.5 0.989506 − 0.144489i 0.527640 − 0.849468i 0.958246 − 0.285946i 2.81881 − 1.14769i 0.399365 − 0.916792i −0.215743 + 0.976253i 0.906874 − 0.421401i −0.443192 − 0.896427i 2.62340 − 1.54293i
See next 80 embeddings (of 240 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.5
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.a ✓ 240
131.g even 65 1 inner 786.2.m.a ✓ 240
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
786.2.m.a ✓ 240 1.a even 1 1 trivial
786.2.m.a ✓ 240 131.g even 65 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{240} - 2 T_{5}^{239} - 6 T_{5}^{238} + 276 T_{5}^{237} - 712 T_{5}^{236} - 916 T_{5}^{235} + \cdots + 11\!\cdots\!41 \) acting on \(S_{2}^{\mathrm{new}}(786, [\chi])\). Copy content Toggle raw display