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Properties

Label 763.2.k.a
Level $763$
Weight $2$
Character orbit 763.k
Analytic conductor $6.093$
Analytic rank $0$
Dimension $52$
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(64,763)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(763, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("763.64");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 763 = 7 \cdot 109 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 763.k (of order \(6\), degree \(2\), 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: \(52\)
Relative dimension: \(26\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 52 q + 3 q^{3} - 48 q^{4} + 6 q^{5} + 6 q^{6} - 26 q^{7} - 23 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 52 q + 3 q^{3} - 48 q^{4} + 6 q^{5} + 6 q^{6} - 26 q^{7} - 23 q^{9} - 12 q^{10} - 10 q^{12} + 6 q^{13} + 7 q^{15} + 32 q^{16} + 21 q^{18} + 7 q^{20} + 3 q^{21} + 7 q^{22} - 45 q^{24} - 18 q^{25} - 3 q^{26} + 18 q^{27} + 24 q^{28} - 8 q^{29} + 15 q^{30} - 14 q^{31} - 58 q^{34} + 6 q^{35} - 12 q^{36} - 21 q^{37} + 32 q^{38} + 12 q^{39} + 3 q^{40} - 6 q^{42} - 14 q^{43} + 78 q^{44} - 88 q^{45} - 12 q^{46} + 12 q^{47} + 15 q^{48} - 26 q^{49} + 105 q^{50} - 27 q^{51} - 30 q^{53} - 6 q^{57} - 3 q^{58} + 15 q^{59} - 69 q^{60} - 26 q^{61} + 33 q^{62} + 46 q^{63} - 32 q^{64} + 60 q^{65} + 76 q^{66} - 18 q^{67} - 12 q^{69} + 12 q^{70} + 42 q^{71} - 39 q^{72} + 31 q^{73} - 64 q^{74} - 32 q^{75} + 16 q^{78} - 9 q^{79} - 19 q^{80} - 6 q^{81} + 16 q^{82} - 59 q^{83} - 10 q^{84} - 36 q^{85} - 30 q^{87} - 26 q^{88} - 3 q^{89} - 6 q^{91} + 34 q^{93} - 38 q^{94} + 57 q^{95} + 93 q^{96} + 21 q^{97} - 45 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
64.1 − 2.73066i 0.718523 − 1.24452i −5.45648 −0.435138 + 0.753681i −3.39835 − 1.96204i −0.500000 + 0.866025i 9.43846i 0.467450 + 0.809646i 2.05804 + 1.18821i
64.2 − 2.65435i −0.773143 + 1.33912i −5.04559 1.18193 − 2.04716i 3.55451 + 2.05220i −0.500000 + 0.866025i 8.08407i 0.304499 + 0.527407i −5.43389 − 3.13726i
64.3 − 2.16638i −0.864864 + 1.49799i −2.69322 −0.207728 + 0.359795i 3.24522 + 1.87363i −0.500000 + 0.866025i 1.50178i 0.00402078 + 0.00696419i 0.779455 + 0.450018i
64.4 − 2.09847i 0.110463 − 0.191328i −2.40356 −0.604518 + 1.04706i −0.401495 − 0.231803i −0.500000 + 0.866025i 0.846859i 1.47560 + 2.55581i 2.19721 + 1.26856i
64.5 − 1.89699i 0.573885 − 0.993998i −1.59859 −1.68553 + 2.91942i −1.88561 − 1.08866i −0.500000 + 0.866025i − 0.761474i 0.841313 + 1.45720i 5.53812 + 3.19743i
64.6 − 1.76393i −1.39560 + 2.41726i −1.11144 1.01161 − 1.75216i 4.26386 + 2.46174i −0.500000 + 0.866025i − 1.56736i −2.39542 − 4.14898i −3.09067 − 1.78440i
64.7 − 1.71061i 1.42320 − 2.46505i −0.926179 0.997584 − 1.72787i −4.21674 − 2.43454i −0.500000 + 0.866025i − 1.83689i −2.55099 − 4.41845i −2.95570 − 1.70648i
64.8 − 1.28908i 1.42302 − 2.46474i 0.338280 −0.447512 + 0.775114i −3.17723 − 1.83438i −0.500000 + 0.866025i − 3.01422i −2.54995 − 4.41664i 0.999182 + 0.576878i
64.9 − 1.24040i 0.715320 − 1.23897i 0.461398 1.64671 − 2.85218i −1.53682 − 0.887286i −0.500000 + 0.866025i − 3.05313i 0.476635 + 0.825556i −3.53785 − 2.04258i
64.10 − 0.946968i −0.781822 + 1.35416i 1.10325 −0.316827 + 0.548760i 1.28234 + 0.740360i −0.500000 + 0.866025i − 2.93868i 0.277508 + 0.480658i 0.519658 + 0.300024i
64.11 − 0.866985i −0.355190 + 0.615206i 1.24834 1.77854 − 3.08052i 0.533375 + 0.307944i −0.500000 + 0.866025i − 2.81626i 1.24768 + 2.16105i −2.67077 − 1.54197i
64.12 − 0.185679i 0.475274 − 0.823199i 1.96552 −0.261510 + 0.452948i −0.152850 − 0.0882483i −0.500000 + 0.866025i − 0.736313i 1.04823 + 1.81559i 0.0841027 + 0.0485567i
64.13 0.0364097i 0.879894 − 1.52402i 1.99867 0.0834930 − 0.144614i 0.0554891 + 0.0320366i −0.500000 + 0.866025i 0.145590i −0.0484255 − 0.0838755i 0.00526535 + 0.00303995i
64.14 0.178263i −1.71925 + 2.97783i 1.96822 1.31766 − 2.28226i −0.530838 − 0.306480i −0.500000 + 0.866025i 0.707388i −4.41166 − 7.64123i 0.406843 + 0.234891i
64.15 0.390319i −0.736468 + 1.27560i 1.84765 −1.82805 + 3.16628i −0.497891 − 0.287457i −0.500000 + 0.866025i 1.50181i 0.415229 + 0.719197i −1.23586 − 0.713522i
64.16 0.549945i −1.13933 + 1.97337i 1.69756 −0.891896 + 1.54481i −1.08525 − 0.626567i −0.500000 + 0.866025i 2.03345i −1.09613 − 1.89856i −0.849560 − 0.490494i
64.17 0.813380i 1.53325 − 2.65566i 1.33841 1.90521 − 3.29992i 2.16006 + 1.24711i −0.500000 + 0.866025i 2.71540i −3.20170 − 5.54550i 2.68409 + 1.54966i
64.18 1.02005i 0.236751 − 0.410064i 0.959490 0.827708 − 1.43363i 0.418288 + 0.241499i −0.500000 + 0.866025i 3.01884i 1.38790 + 2.40391i 1.46238 + 0.844307i
64.19 1.36974i 0.341376 − 0.591280i 0.123811 −0.0792482 + 0.137262i 0.809900 + 0.467596i −0.500000 + 0.866025i 2.90907i 1.26693 + 2.19438i −0.188013 − 0.108549i
64.20 1.53291i 1.39624 − 2.41835i −0.349826 −1.42765 + 2.47276i 3.70713 + 2.14031i −0.500000 + 0.866025i 2.52958i −2.39895 − 4.15511i −3.79053 − 2.18846i
See all 52 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 64.26
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
109.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 763.2.k.a ✓ 52
109.e even 6 1 inner 763.2.k.a ✓ 52
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
763.2.k.a ✓ 52 1.a even 1 1 trivial
763.2.k.a ✓ 52 109.e even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{52} + 76 T_{2}^{50} + 2704 T_{2}^{48} + 59872 T_{2}^{46} + 925128 T_{2}^{44} + 10602663 T_{2}^{42} + \cdots + 225 \) acting on \(S_{2}^{\mathrm{new}}(763, [\chi])\). Copy content Toggle raw display