Properties

Label 786.2.o.a
Level $786$
Weight $2$
Character orbit 786.o
Analytic conductor $6.276$
Analytic rank $0$
Dimension $1056$
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(17,786)]
 
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([65, 43]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("786.17");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 786 = 2 \cdot 3 \cdot 131 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 786.o (of order \(130\), 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: \(1056\)
Relative dimension: \(22\) over \(\Q(\zeta_{130})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{130}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1056 q - 22 q^{2} + q^{3} + 22 q^{4} - q^{6} - 2 q^{7} - 22 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1056 q - 22 q^{2} + q^{3} + 22 q^{4} - q^{6} - 2 q^{7} - 22 q^{8} + q^{9} - 5 q^{10} + 5 q^{11} + q^{12} + 6 q^{13} + 2 q^{14} + 43 q^{15} + 22 q^{16} - 39 q^{17} + 4 q^{18} + 47 q^{20} - 3 q^{21} - 15 q^{22} - 2 q^{23} + 4 q^{24} - 16 q^{25} + 4 q^{26} + 28 q^{27} - 2 q^{28} - 81 q^{29} + 31 q^{30} - 10 q^{31} + 88 q^{32} - 39 q^{33} + 5 q^{34} + 71 q^{35} + q^{36} - 10 q^{38} + 80 q^{39} + 13 q^{42} + 40 q^{43} - 15 q^{44} + 46 q^{45} + 2 q^{46} + 32 q^{47} + q^{48} + 48 q^{49} + 16 q^{50} - 52 q^{51} - 4 q^{52} - 15 q^{53} + 11 q^{54} + 121 q^{55} + 2 q^{56} - 298 q^{57} + 12 q^{58} - 10 q^{59} - 22 q^{60} + 12 q^{61} - 26 q^{62} + 31 q^{63} + 22 q^{64} + 30 q^{65} + 124 q^{66} - 16 q^{67} + 10 q^{68} + 64 q^{69} - 20 q^{70} - 16 q^{71} + 25 q^{72} - 45 q^{73} - 60 q^{75} + 15 q^{76} + 29 q^{78} + 91 q^{79} + 25 q^{81} + 5 q^{82} - 41 q^{83} + 2 q^{84} + 64 q^{85} - 130 q^{86} + 74 q^{87} + 5 q^{88} + 45 q^{89} + 44 q^{90} - 42 q^{91} + 8 q^{92} + 40 q^{93} - 12 q^{94} + 164 q^{95} - q^{96} + q^{97} - 48 q^{98} - 146 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 0.861970 0.506960i −1.73196 + 0.0180215i 0.485983 0.873968i −0.217210 + 0.270452i −1.48376 + 0.893567i −1.62737 + 1.58851i −0.0241637 0.999708i 2.99935 0.0624248i −0.0501201 + 0.343238i
17.2 0.861970 0.506960i −1.73181 0.0289032i 0.485983 0.873968i −2.53121 + 3.15166i −1.50742 + 0.853044i 1.00267 0.978730i −0.0241637 0.999708i 2.99833 + 0.100110i −0.584064 + 3.99986i
17.3 0.861970 0.506960i −1.73173 0.0335448i 0.485983 0.873968i 2.64422 3.29236i −1.50970 + 0.849001i 3.08442 3.01077i −0.0241637 0.999708i 2.99775 + 0.116181i 0.610139 4.17842i
17.4 0.861970 0.506960i −1.54412 + 0.784663i 0.485983 0.873968i −1.48246 + 1.84583i −0.933191 + 1.45916i −0.0328248 + 0.0320410i −0.0241637 0.999708i 1.76861 2.42323i −0.342069 + 2.34260i
17.5 0.861970 0.506960i −1.51992 0.830568i 0.485983 0.873968i 1.07056 1.33297i −1.73119 + 0.0546139i −2.21239 + 2.15956i −0.0241637 0.999708i 1.62031 + 2.52479i 0.247026 1.69171i
17.6 0.861970 0.506960i −1.46706 + 0.920726i 0.485983 0.873968i 1.52227 1.89541i −0.797790 + 1.53738i −2.35040 + 2.29427i −0.0241637 0.999708i 1.30453 2.70152i 0.351257 2.40552i
17.7 0.861970 0.506960i −1.46539 0.923374i 0.485983 0.873968i −0.538075 + 0.669967i −1.73124 0.0530244i 2.12214 2.07147i −0.0241637 0.999708i 1.29476 + 2.70621i −0.124158 + 0.850273i
17.8 0.861970 0.506960i −0.852293 1.50785i 0.485983 0.873968i −1.30402 + 1.62366i −1.49907 0.867639i −2.81166 + 2.74452i −0.0241637 0.999708i −1.54719 + 2.57025i −0.300896 + 2.06063i
17.9 0.861970 0.506960i −0.754138 + 1.55926i 0.485983 0.873968i −0.859114 + 1.06970i 0.140436 + 1.72635i 2.42971 2.37170i −0.0241637 0.999708i −1.86255 2.35179i −0.198236 + 1.35758i
17.10 0.861970 0.506960i −0.369930 1.69208i 0.485983 0.873968i 1.10905 1.38089i −1.17669 1.27099i 1.51579 1.47960i −0.0241637 0.999708i −2.72630 + 1.25191i 0.255907 1.75253i
17.11 0.861970 0.506960i 0.0550939 + 1.73117i 0.485983 0.873968i 2.53597 3.15759i 0.925125 + 1.46429i 0.530869 0.518192i −0.0241637 0.999708i −2.99393 + 0.190754i 0.585163 4.00738i
17.12 0.861970 0.506960i 0.115640 + 1.72819i 0.485983 0.873968i −1.12802 + 1.40452i 0.975799 + 1.43102i −1.22155 + 1.19238i −0.0241637 0.999708i −2.97325 + 0.399696i −0.260285 + 1.78252i
17.13 0.861970 0.506960i 0.352064 1.69589i 0.485983 0.873968i −1.33349 + 1.66036i −0.556281 1.64029i −2.27735 + 2.22297i −0.0241637 0.999708i −2.75210 1.19412i −0.307697 + 2.10720i
17.14 0.861970 0.506960i 0.484131 1.66301i 0.485983 0.873968i 1.44498 1.79917i −0.425775 1.67890i 0.195393 0.190727i −0.0241637 0.999708i −2.53123 1.61024i 0.333422 2.28338i
17.15 0.861970 0.506960i 0.654747 + 1.60353i 0.485983 0.873968i 0.245999 0.306298i 1.37730 + 1.05026i −3.12970 + 3.05497i −0.0241637 0.999708i −2.14261 + 2.09981i 0.0567630 0.388731i
17.16 0.861970 0.506960i 1.08971 + 1.34631i 0.485983 0.873968i 0.430186 0.535632i 1.62182 + 0.608037i 2.60062 2.53852i −0.0241637 0.999708i −0.625078 + 2.93416i 0.0992632 0.679786i
17.17 0.861970 0.506960i 1.39373 1.02835i 0.485983 0.873968i −2.14360 + 2.66904i 0.680023 1.59297i 0.570936 0.557303i −0.0241637 0.999708i 0.884987 2.86650i −0.494625 + 3.38735i
17.18 0.861970 0.506960i 1.51545 0.838687i 0.485983 0.873968i 0.0454281 0.0565633i 0.881095 1.49120i 0.0476998 0.0465607i −0.0241637 0.999708i 1.59321 2.54199i 0.0104823 0.0717861i
17.19 0.861970 0.506960i 1.59457 + 0.676284i 0.485983 0.873968i 0.775130 0.965128i 1.71732 0.225445i −0.968994 + 0.945856i −0.0241637 0.999708i 2.08528 + 2.15676i 0.178857 1.22487i
17.20 0.861970 0.506960i 1.62099 0.610234i 0.485983 0.873968i −0.126884 + 0.157986i 1.08788 1.34778i 3.51655 3.43258i −0.0241637 0.999708i 2.25523 1.97837i −0.0292779 + 0.200504i
See next 80 embeddings (of 1056 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
393.p even 130 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 786.2.o.a 1056
3.b odd 2 1 786.2.o.b yes 1056
131.h odd 130 1 786.2.o.b yes 1056
393.p even 130 1 inner 786.2.o.a 1056
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
786.2.o.a 1056 1.a even 1 1 trivial
786.2.o.a 1056 393.p even 130 1 inner
786.2.o.b yes 1056 3.b odd 2 1
786.2.o.b yes 1056 131.h odd 130 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{1056} + 63 T_{5}^{1054} - 75 T_{5}^{1053} + 1696 T_{5}^{1052} - 3711 T_{5}^{1051} + 30367 T_{5}^{1050} - 73873 T_{5}^{1049} + 482702 T_{5}^{1048} - 957509 T_{5}^{1047} + 4965134 T_{5}^{1046} - 12955898 T_{5}^{1045} + \cdots + 12\!\cdots\!76 \) acting on \(S_{2}^{\mathrm{new}}(786, [\chi])\). Copy content Toggle raw display