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Properties

Label 763.2.bn.b
Level $763$
Weight $2$
Character orbit 763.bn
Analytic conductor $6.093$
Analytic rank $0$
Dimension $150$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [763,2,Mod(43,763)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(763, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("763.43");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 763 = 7 \cdot 109 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 763.bn (of order \(18\), degree \(6\), 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: \(6.09258567422\)
Analytic rank: \(0\)
Dimension: \(150\)
Relative dimension: \(25\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 150 q - 9 q^{3} + 72 q^{4} + 3 q^{5} - 12 q^{6} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 150 q - 9 q^{3} + 72 q^{4} + 3 q^{5} - 12 q^{6} - 9 q^{9} - 15 q^{10} + 6 q^{12} - 24 q^{13} - 3 q^{14} - 66 q^{16} + 45 q^{18} + 39 q^{20} - 9 q^{21} + 9 q^{22} - 45 q^{23} - 51 q^{24} + 21 q^{25} + 36 q^{26} + 51 q^{27} + 21 q^{28} + 24 q^{29} - 24 q^{30} - 15 q^{31} - 81 q^{32} - 63 q^{34} - 6 q^{35} - 30 q^{36} + 9 q^{37} + 42 q^{38} - 21 q^{39} + 42 q^{40} - 15 q^{42} - 33 q^{43} + 39 q^{44} + 30 q^{45} + 66 q^{46} + 54 q^{47} - 3 q^{48} - 102 q^{50} - 21 q^{51} + 81 q^{52} - 6 q^{53} - 90 q^{54} - 36 q^{55} - 3 q^{56} + 18 q^{57} + 66 q^{58} + 9 q^{59} - 87 q^{60} - 45 q^{61} - 75 q^{62} + 114 q^{63} - 84 q^{64} + 66 q^{65} - 39 q^{66} - 45 q^{67} - 72 q^{69} + 15 q^{70} - 63 q^{71} + 42 q^{72} - 9 q^{73} + 57 q^{74} + 84 q^{75} - 195 q^{78} + 15 q^{79} - 78 q^{80} - 57 q^{81} - 99 q^{82} - 39 q^{83} - 21 q^{84} + 90 q^{86} + 45 q^{87} + 144 q^{88} + 51 q^{89} - 162 q^{90} - 3 q^{91} + 117 q^{92} - 108 q^{93} + 30 q^{94} + 6 q^{95} + 60 q^{96} - 27 q^{97} + 3 q^{98} + 144 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1 −2.20946 + 1.27563i 0.0179788 + 0.101963i 2.25447 − 3.90485i 2.94549 + 1.07207i −0.169790 − 0.202348i −0.939693 − 0.342020i 6.40093i 2.80900 − 1.02239i −7.87549 + 1.38866i
43.2 −2.07687 + 1.19908i 0.437476 + 2.48105i 1.87559 − 3.24862i 0.323841 + 0.117868i −3.88357 − 4.62825i −0.939693 − 0.342020i 4.19963i −3.14515 + 1.14474i −0.813910 + 0.143514i
43.3 −1.88239 + 1.08680i −0.456225 − 2.58738i 1.36225 − 2.35948i −2.52011 − 0.917245i 3.67074 + 4.37462i −0.939693 − 0.342020i 1.57476i −3.66731 + 1.33479i 5.74067 − 1.01224i
43.4 −1.80496 + 1.04209i 0.333622 + 1.89206i 1.17192 − 2.02982i −2.74307 − 0.998398i −2.57388 − 3.06743i −0.939693 − 0.342020i 0.716616i −0.649522 + 0.236407i 5.99156 − 1.05647i
43.5 −1.77645 + 1.02563i −0.451400 − 2.56002i 1.10384 − 1.91191i 2.74028 + 0.997382i 3.42752 + 4.08476i −0.939693 − 0.342020i 0.426010i −3.53085 + 1.28513i −5.89091 + 1.03873i
43.6 −1.53464 + 0.886027i 0.0834004 + 0.472987i 0.570086 − 0.987418i 2.83728 + 1.03269i −0.547069 − 0.651972i −0.939693 − 0.342020i − 1.52366i 2.60232 − 0.947166i −5.26920 + 0.929103i
43.7 −1.17384 + 0.677717i −0.251169 − 1.42445i −0.0813980 + 0.140986i −1.50107 − 0.546345i 1.26021 + 1.50186i −0.939693 − 0.342020i − 2.93153i 0.853099 − 0.310503i 2.13228 − 0.375979i
43.8 −0.784259 + 0.452792i 0.0835322 + 0.473734i −0.589959 + 1.02184i 2.35341 + 0.856573i −0.280014 − 0.333708i −0.939693 − 0.342020i − 2.87968i 2.60163 − 0.946916i −2.23354 + 0.393832i
43.9 −0.693911 + 0.400630i 0.519668 + 2.94718i −0.678992 + 1.17605i 0.255391 + 0.0929547i −1.54133 − 1.83689i −0.939693 − 0.342020i − 2.69062i −5.59675 + 2.03705i −0.214459 + 0.0378149i
43.10 −0.499985 + 0.288667i −0.451848 − 2.56256i −0.833343 + 1.44339i 2.47886 + 0.902231i 0.965642 + 1.15081i −0.939693 − 0.342020i − 2.11690i −3.54346 + 1.28971i −1.49984 + 0.264462i
43.11 −0.182353 + 0.105281i 0.274413 + 1.55627i −0.977832 + 1.69365i −2.80530 − 1.02104i −0.213887 − 0.254900i −0.939693 − 0.342020i − 0.832916i 0.472391 − 0.171936i 0.619051 − 0.109155i
43.12 −0.115843 + 0.0668817i −0.538673 − 3.05497i −0.991054 + 1.71656i −3.78304 − 1.37691i 0.266723 + 0.317868i −0.939693 − 0.342020i − 0.532660i −6.22357 + 2.26520i 0.530327 − 0.0935110i
43.13 0.117024 − 0.0675636i −0.167296 − 0.948783i −0.990870 + 1.71624i −0.109871 − 0.0399897i −0.0836808 − 0.0997270i −0.939693 − 0.342020i 0.538042i 1.94688 − 0.708605i −0.0155593 + 0.00274353i
43.14 0.269225 − 0.155437i 0.451384 + 2.55993i −0.951678 + 1.64836i 3.08882 + 1.12424i 0.519433 + 0.619036i −0.939693 − 0.342020i 1.21345i −3.53041 + 1.28496i 1.00634 − 0.177445i
43.15 0.284368 − 0.164180i −0.409525 − 2.32253i −0.946090 + 1.63868i 1.09750 + 0.399456i −0.497768 − 0.593217i −0.939693 − 0.342020i 1.27803i −2.40736 + 0.876208i 0.377676 − 0.0665944i
43.16 0.515993 − 0.297909i 0.201200 + 1.14106i −0.822501 + 1.42461i −0.0596727 − 0.0217191i 0.443749 + 0.528840i −0.939693 − 0.342020i 2.17175i 1.55754 − 0.566899i −0.0372610 + 0.00657012i
43.17 0.713717 − 0.412065i −0.141117 − 0.800314i −0.660406 + 1.14386i −1.55450 − 0.565791i −0.430498 − 0.513048i −0.939693 − 0.342020i 2.73678i 2.19849 − 0.800185i −1.34261 + 0.236739i
43.18 1.35632 − 0.783073i 0.513445 + 2.91189i 0.226405 − 0.392146i −1.03380 − 0.376274i 2.97662 + 3.54739i −0.939693 − 0.342020i 2.42312i −5.39639 + 1.96413i −1.69682 + 0.299195i
43.19 1.37707 − 0.795054i 0.0266975 + 0.151409i 0.264220 − 0.457643i −3.70858 − 1.34981i 0.157143 + 0.187275i −0.939693 − 0.342020i 2.33994i 2.79687 − 1.01798i −6.18017 + 1.08973i
43.20 1.39778 − 0.807008i 0.299605 + 1.69914i 0.302525 − 0.523988i 3.23492 + 1.17741i 1.79000 + 2.13324i −0.939693 − 0.342020i 2.25147i 0.0217531 − 0.00791749i 5.47189 − 0.964841i
See next 80 embeddings (of 150 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.25
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
109.h even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 763.2.bn.b ✓ 150
109.h even 18 1 inner 763.2.bn.b ✓ 150
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
763.2.bn.b ✓ 150 1.a even 1 1 trivial
763.2.bn.b ✓ 150 109.h even 18 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{150} - 111 T_{2}^{148} + 6435 T_{2}^{146} + 45 T_{2}^{145} - 256310 T_{2}^{144} - 4698 T_{2}^{143} + \cdots + 15417867 \) acting on \(S_{2}^{\mathrm{new}}(763, [\chi])\). Copy content Toggle raw display