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Properties

Label 763.2.bi.a
Level $763$
Weight $2$
Character orbit 763.bi
Analytic conductor $6.093$
Analytic rank $0$
Dimension $432$
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(4,763)]
 
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([12, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("763.4");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 763 = 7 \cdot 109 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 763.bi (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: \(432\)
Relative dimension: \(72\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 432 q - 3 q^{3} - 438 q^{4} - 3 q^{5} - 12 q^{6} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 432 q - 3 q^{3} - 438 q^{4} - 3 q^{5} - 12 q^{6} - 9 q^{9} + 39 q^{10} - 9 q^{11} - 33 q^{12} - 12 q^{13} + 24 q^{14} - 18 q^{15} + 438 q^{16} - 9 q^{17} + 51 q^{18} - 18 q^{20} - 15 q^{21} - 54 q^{22} + 9 q^{23} + 72 q^{24} - 15 q^{25} + 3 q^{26} - 12 q^{27} - 18 q^{28} - 6 q^{29} + 21 q^{30} - 3 q^{31} - 6 q^{34} + 63 q^{35} + 27 q^{36} - 3 q^{37} + 114 q^{38} + 42 q^{39} - 66 q^{40} - 54 q^{42} - 6 q^{43} - 57 q^{44} - 42 q^{45} + 6 q^{46} + 9 q^{47} - 15 q^{48} + 30 q^{50} - 45 q^{51} + 9 q^{52} + 21 q^{53} + 27 q^{54} - 18 q^{55} + 51 q^{56} - 12 q^{57} + 42 q^{58} + 21 q^{59} + 117 q^{60} + 33 q^{61} + 15 q^{62} - 12 q^{63} - 426 q^{64} - 21 q^{65} - 63 q^{66} - 27 q^{67} + 72 q^{68} - 27 q^{69} - 27 q^{70} - 30 q^{71} - 126 q^{72} - 6 q^{73} - 3 q^{74} - 54 q^{75} - 90 q^{77} - 24 q^{78} + 27 q^{79} + 39 q^{80} - 57 q^{81} - 114 q^{82} + 12 q^{83} - 69 q^{84} + 51 q^{85} + 54 q^{86} - 66 q^{87} + 129 q^{88} + 3 q^{89} - 108 q^{90} + 54 q^{91} - 36 q^{92} - 96 q^{93} + 75 q^{94} + 33 q^{95} - 117 q^{96} + 33 q^{97} - 36 q^{98} - 105 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 − 2.76864i 0.330183 − 1.87256i −5.66538 0.733974 − 4.16257i −5.18445 − 0.914158i 1.68123 − 2.04291i 10.1481i −0.578383 − 0.210514i −11.5247 − 2.03211i
4.2 − 2.72909i 0.531427 − 3.01387i −5.44794 −0.479533 + 2.71956i −8.22514 − 1.45031i 0.827609 + 2.51298i 9.40974i −5.98194 − 2.17725i 7.42194 + 1.30869i
4.3 − 2.70816i −0.381253 + 2.16219i −5.33415 −0.0304949 + 0.172945i 5.85558 + 1.03250i −0.211166 − 2.63731i 9.02943i −1.71065 − 0.622626i 0.468364 + 0.0825852i
4.4 − 2.66483i −0.241076 + 1.36721i −5.10133 0.380597 − 2.15847i 3.64339 + 0.642428i 0.970470 + 2.46134i 8.26452i 1.00793 + 0.366855i −5.75196 − 1.01423i
4.5 − 2.65043i 0.108787 − 0.616960i −5.02480 −0.320926 + 1.82006i −1.63521 − 0.288332i −2.64291 − 0.122616i 8.01702i 2.45027 + 0.891826i 4.82395 + 0.850592i
4.6 − 2.40361i 0.0608550 − 0.345126i −3.77732 −0.712532 + 4.04097i −0.829547 − 0.146271i 0.813012 − 2.51774i 4.27197i 2.70367 + 0.984055i 9.71290 + 1.71265i
4.7 − 2.34105i −0.0641756 + 0.363958i −3.48052 0.0799039 − 0.453157i 0.852044 + 0.150238i 1.23151 + 2.34166i 3.46597i 2.69073 + 0.979346i −1.06086 − 0.187059i
4.8 − 2.33802i −0.496297 + 2.81464i −3.46635 −0.542802 + 3.07839i 6.58069 + 1.16035i 2.45708 + 0.981197i 3.42836i −4.85680 − 1.76773i 7.19733 + 1.26908i
4.9 − 2.26767i 0.359541 − 2.03906i −3.14233 0.0440259 − 0.249683i −4.62391 − 0.815321i 2.51018 − 0.836073i 2.59043i −1.20941 − 0.440190i −0.566199 − 0.0998362i
4.10 − 2.26713i −0.491256 + 2.78605i −3.13988 0.587071 − 3.32945i 6.31633 + 1.11374i −2.35242 + 1.21084i 2.58425i −4.70166 − 1.71126i −7.54828 − 1.33097i
4.11 − 2.18272i 0.0353123 − 0.200266i −2.76425 0.170402 − 0.966399i −0.437124 − 0.0770768i −2.24709 − 1.39663i 1.66814i 2.78022 + 1.01192i −2.10937 − 0.371939i
4.12 − 2.17241i 0.400945 − 2.27387i −2.71938 0.512853 − 2.90853i −4.93979 − 0.871018i −2.20141 + 1.46759i 1.56280i −2.19066 − 0.797334i −6.31854 − 1.11413i
4.13 − 2.11450i 0.544200 − 3.08631i −2.47112 −0.0642217 + 0.364220i −6.52601 − 1.15071i −1.20060 − 2.35766i 0.996180i −6.41009 − 2.33308i 0.770143 + 0.135797i
4.14 − 2.02222i −0.509666 + 2.89046i −2.08936 −0.161985 + 0.918663i 5.84513 + 1.03065i −2.50383 − 0.854897i 0.180696i −5.27591 − 1.92027i 1.85773 + 0.327569i
4.15 − 1.98186i −0.190964 + 1.08301i −1.92777 −0.538105 + 3.05175i 2.14638 + 0.378464i −1.07643 + 2.41688i − 0.143157i 1.68263 + 0.612428i 6.04813 + 1.06645i
4.16 − 1.83054i −0.100200 + 0.568261i −1.35086 0.414290 − 2.34955i 1.04022 + 0.183419i 0.636579 − 2.56803i − 1.18827i 2.50620 + 0.912181i −4.30094 − 0.758372i
4.17 − 1.71684i 0.312100 − 1.77001i −0.947543 −0.500644 + 2.83929i −3.03882 − 0.535826i 0.476616 + 2.60247i − 1.80690i −0.216444 − 0.0787790i 4.87461 + 0.859525i
4.18 − 1.69851i −0.425056 + 2.41061i −0.884948 0.601460 − 3.41105i 4.09446 + 0.721963i 2.60485 − 0.463430i − 1.89393i −2.81130 − 1.02323i −5.79371 − 1.02159i
4.19 − 1.63786i 0.286531 − 1.62500i −0.682569 0.246279 − 1.39672i −2.66151 − 0.469297i 2.49620 + 0.876925i − 2.15776i 0.260553 + 0.0948337i −2.28762 − 0.403370i
4.20 − 1.54319i −0.201900 + 1.14503i −0.381427 −0.275241 + 1.56097i 1.76700 + 0.311569i 2.37169 − 1.17264i − 2.49776i 1.54875 + 0.563698i 2.40887 + 0.424749i
See next 80 embeddings (of 432 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.72
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
763.bi 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.bi.a ✓ 432
7.c even 3 1 763.2.bl.a yes 432
109.h even 18 1 763.2.bl.a yes 432
763.bi even 18 1 inner 763.2.bi.a ✓ 432
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
763.2.bi.a ✓ 432 1.a even 1 1 trivial
763.2.bi.a ✓ 432 763.bi even 18 1 inner
763.2.bl.a yes 432 7.c even 3 1
763.2.bl.a yes 432 109.h even 18 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(763, [\chi])\).