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Properties

Label 786.2.m.b
Level $786$
Weight $2$
Character orbit 786.m
Analytic conductor $6.276$
Analytic rank $0$
Dimension $240$
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: \(240\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 240 q + 5 q^{2} + 5 q^{3} + 5 q^{4} - 8 q^{5} + 5 q^{6} + 3 q^{7} + 5 q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 240 q + 5 q^{2} + 5 q^{3} + 5 q^{4} - 8 q^{5} + 5 q^{6} + 3 q^{7} + 5 q^{8} + 5 q^{9} + 20 q^{10} + 13 q^{11} + 5 q^{12} + 8 q^{13} - 7 q^{14} + 7 q^{15} + 5 q^{16} - 39 q^{17} - 20 q^{18} + 28 q^{19} - 45 q^{20} - 7 q^{21} + 8 q^{22} + 34 q^{23} - 20 q^{24} + 7 q^{25} + 5 q^{27} - 7 q^{28} - 36 q^{29} - 8 q^{30} + 10 q^{31} - 20 q^{32} - 5 q^{33} - 5 q^{34} + 48 q^{35} + 5 q^{36} - 69 q^{37} - 27 q^{38} + 23 q^{39} + 5 q^{40} - 9 q^{41} + 16 q^{42} + 37 q^{43} + 8 q^{44} + 15 q^{45} + 18 q^{46} + 2 q^{47} + 5 q^{48} - 44 q^{49} + 7 q^{50} + 36 q^{51} + 10 q^{52} + 54 q^{53} + 5 q^{54} + 65 q^{55} + 3 q^{56} + 13 q^{57} + 55 q^{58} + 75 q^{59} + 15 q^{60} + 32 q^{61} + 16 q^{62} + 8 q^{63} + 5 q^{64} - 50 q^{65} + 13 q^{66} - 3 q^{67} - 81 q^{68} + 35 q^{70} - 118 q^{71} + 5 q^{72} + 92 q^{73} + 14 q^{74} - 14 q^{75} - 65 q^{76} + 7 q^{77} + 39 q^{78} + 49 q^{79} + 2 q^{80} + 5 q^{81} + 49 q^{82} + q^{83} + 8 q^{84} + 161 q^{85} - 30 q^{86} + 16 q^{87} - 52 q^{88} + 195 q^{89} + 5 q^{90} + 117 q^{91} + 26 q^{92} + 10 q^{93} - 10 q^{94} - 60 q^{95} + 5 q^{96} - 64 q^{97} + 86 q^{98} + 23 q^{99}+O(q^{100}) \) 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 −3.26465 + 0.316562i 0.958246 − 0.285946i −4.33290 + 1.52455i 0.779674 + 0.626185i 0.485983 + 0.873968i 1.99147 − 2.60619i
7.2 −0.681016 + 0.732269i −0.861970 − 0.506960i −0.0724348 − 0.997373i −1.10143 + 0.106802i 0.958246 − 0.285946i −1.14767 + 0.403812i 0.779674 + 0.626185i 0.485983 + 0.873968i 0.671886 − 0.879280i
7.3 −0.681016 + 0.732269i −0.861970 − 0.506960i −0.0724348 − 0.997373i −0.865652 + 0.0839393i 0.958246 − 0.285946i −0.321753 + 0.113210i 0.779674 + 0.626185i 0.485983 + 0.873968i 0.528056 − 0.691054i
7.4 −0.681016 + 0.732269i −0.861970 − 0.506960i −0.0724348 − 0.997373i −0.00428502 0.000415504i 0.958246 − 0.285946i 4.40531 − 1.55002i 0.779674 + 0.626185i 0.485983 + 0.873968i 0.00261391 − 0.00342075i
7.5 −0.681016 + 0.732269i −0.861970 − 0.506960i −0.0724348 − 0.997373i 3.66130 − 0.355024i 0.958246 − 0.285946i 1.55184 − 0.546019i 0.779674 + 0.626185i 0.485983 + 0.873968i −2.23343 + 2.92283i
13.1 −0.906874 − 0.421401i 0.995332 − 0.0965139i 0.644842 + 0.764316i −1.47355 − 3.38273i −0.943312 − 0.331908i −1.88046 − 2.46090i −0.262707 − 0.964876i 0.981370 − 0.192127i −0.0891579 + 3.68866i
13.2 −0.906874 − 0.421401i 0.995332 − 0.0965139i 0.644842 + 0.764316i −0.811703 − 1.86337i −0.943312 − 0.331908i 0.867413 + 1.13516i −0.262707 − 0.964876i 0.981370 − 0.192127i −0.0491125 + 2.03189i
13.3 −0.906874 − 0.421401i 0.995332 − 0.0965139i 0.644842 + 0.764316i 0.367463 + 0.843557i −0.943312 − 0.331908i −1.13035 − 1.47926i −0.262707 − 0.964876i 0.981370 − 0.192127i 0.0222335 − 0.919849i
13.4 −0.906874 − 0.421401i 0.995332 − 0.0965139i 0.644842 + 0.764316i 0.480488 + 1.10302i −0.943312 − 0.331908i −0.677281 − 0.886340i −0.262707 − 0.964876i 0.981370 − 0.192127i 0.0290721 − 1.20278i
13.5 −0.906874 − 0.421401i 0.995332 − 0.0965139i 0.644842 + 0.764316i 1.07790 + 2.47446i −0.943312 − 0.331908i 2.70483 + 3.53974i −0.262707 − 0.964876i 0.981370 − 0.192127i 0.0652190 − 2.69826i
25.1 −0.607163 + 0.794578i −0.989506 + 0.144489i −0.262707 − 0.964876i −0.389448 + 2.28021i 0.485983 − 0.873968i 2.50054 − 0.489541i 0.926175 + 0.377095i 0.958246 − 0.285946i −1.57534 − 1.69390i
25.2 −0.607163 + 0.794578i −0.989506 + 0.144489i −0.262707 − 0.964876i −0.0724979 + 0.424474i 0.485983 − 0.873968i −0.423172 + 0.0828461i 0.926175 + 0.377095i 0.958246 − 0.285946i −0.293259 − 0.315330i
25.3 −0.607163 + 0.794578i −0.989506 + 0.144489i −0.262707 − 0.964876i 0.296293 − 1.73479i 0.485983 − 0.873968i −1.45604 + 0.285055i 0.926175 + 0.377095i 0.958246 − 0.285946i 1.19853 + 1.28873i
25.4 −0.607163 + 0.794578i −0.989506 + 0.144489i −0.262707 − 0.964876i 0.592097 − 3.46671i 0.485983 − 0.873968i 3.23589 − 0.633503i 0.926175 + 0.377095i 0.958246 − 0.285946i 2.39507 + 2.57533i
25.5 −0.607163 + 0.794578i −0.989506 + 0.144489i −0.262707 − 0.964876i 0.626177 − 3.66625i 0.485983 − 0.873968i −2.77136 + 0.542560i 0.926175 + 0.377095i 0.958246 − 0.285946i 2.53293 + 2.72355i
43.1 −0.989506 + 0.144489i −0.527640 + 0.849468i 0.958246 − 0.285946i −3.15360 + 1.28400i 0.399365 − 0.916792i −1.09998 + 4.97748i −0.906874 + 0.421401i −0.443192 − 0.896427i 2.93498 − 1.72618i
43.2 −0.989506 + 0.144489i −0.527640 + 0.849468i 0.958246 − 0.285946i −2.35932 + 0.960606i 0.399365 − 0.916792i −0.551677 + 2.49638i −0.906874 + 0.421401i −0.443192 − 0.896427i 2.19577 − 1.29142i
43.3 −0.989506 + 0.144489i −0.527640 + 0.849468i 0.958246 − 0.285946i −2.04245 + 0.831591i 0.399365 − 0.916792i 1.03568 − 4.68653i −0.906874 + 0.421401i −0.443192 − 0.896427i 1.90086 − 1.11798i
43.4 −0.989506 + 0.144489i −0.527640 + 0.849468i 0.958246 − 0.285946i 0.188590 − 0.0767850i 0.399365 − 0.916792i −0.0188213 + 0.0851680i −0.906874 + 0.421401i −0.443192 − 0.896427i −0.175516 + 0.103228i
43.5 −0.989506 + 0.144489i −0.527640 + 0.849468i 0.958246 − 0.285946i 1.51260 − 0.615860i 0.399365 − 0.916792i −0.00443006 + 0.0200464i −0.906874 + 0.421401i −0.443192 − 0.896427i −1.40774 + 0.827951i
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.b ✓ 240
131.g even 65 1 inner 786.2.m.b ✓ 240
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
786.2.m.b ✓ 240 1.a even 1 1 trivial
786.2.m.b ✓ 240 131.g even 65 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{240} + 8 T_{5}^{239} + 16 T_{5}^{238} + 128 T_{5}^{237} + 1018 T_{5}^{236} + \cdots + 15\!\cdots\!61 \) acting on \(S_{2}^{\mathrm{new}}(786, [\chi])\). Copy content Toggle raw display