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Properties

Label 783.2.m.a
Level $783$
Weight $2$
Character orbit 783.m
Analytic conductor $6.252$
Analytic rank $0$
Dimension $112$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [783,2,Mod(17,783)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(783, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([10, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("783.17");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 783 = 3^{3} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 783.m (of order \(12\), degree \(4\), not 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: \(6.25228647827\)
Analytic rank: \(0\)
Dimension: \(112\)
Relative dimension: \(28\) over \(\Q(\zeta_{12})\)
Twist minimal: no (minimal twist has level 261)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 112 q + 6 q^{2} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 112 q + 6 q^{2} - 4 q^{7} - 6 q^{11} + 18 q^{14} + 40 q^{16} - 8 q^{19} + 12 q^{20} + 12 q^{23} - 44 q^{25} + 42 q^{29} - 2 q^{31} + 66 q^{32} - 8 q^{37} - 12 q^{40} + 18 q^{41} - 2 q^{43} + 8 q^{46} - 36 q^{49} - 24 q^{50} - 36 q^{52} + 36 q^{55} - 84 q^{56} + 28 q^{58} - 48 q^{59} - 14 q^{61} - 24 q^{65} + 102 q^{68} - 8 q^{73} - 144 q^{74} + 14 q^{76} + 72 q^{77} - 2 q^{79} + 80 q^{82} + 120 q^{83} - 48 q^{85} - 36 q^{88} - 40 q^{94} - 204 q^{95} + 22 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 −0.698882 + 2.60826i 0 −4.58255 − 2.64574i −0.0478262 + 0.0828373i 0 −2.10481 − 3.64565i 6.28468 − 6.28468i 0 −0.182637 − 0.182637i
17.2 −0.583913 + 2.17919i 0 −2.67588 − 1.54492i −1.76353 + 3.05453i 0 1.19891 + 2.07656i 1.73861 − 1.73861i 0 −5.62667 − 5.62667i
17.3 −0.576941 + 2.15317i 0 −2.57124 − 1.48451i 2.00311 − 3.46949i 0 1.25362 + 2.17133i 1.52739 − 1.52739i 0 6.31474 + 6.31474i
17.4 −0.556838 + 2.07815i 0 −2.27657 − 1.31438i −1.18297 + 2.04897i 0 −0.264985 − 0.458967i 0.956542 − 0.956542i 0 −3.59933 − 3.59933i
17.5 −0.533758 + 1.99201i 0 −1.95116 − 1.12650i 0.0869569 − 0.150614i 0 −0.684787 − 1.18609i 0.368943 − 0.368943i 0 0.253610 + 0.253610i
17.6 −0.487125 + 1.81798i 0 −1.33570 − 0.771165i 0.169593 − 0.293744i 0 1.47912 + 2.56191i −0.609091 + 0.609091i 0 0.451407 + 0.451407i
17.7 −0.439342 + 1.63965i 0 −0.763368 − 0.440731i 0.592570 − 1.02636i 0 −2.12656 − 3.68331i −1.34259 + 1.34259i 0 1.42253 + 1.42253i
17.8 −0.306318 + 1.14319i 0 0.518989 + 0.299639i 1.06983 − 1.85299i 0 1.97089 + 3.41367i −2.17527 + 2.17527i 0 1.79062 + 1.79062i
17.9 −0.281409 + 1.05023i 0 0.708253 + 0.408910i −1.55901 + 2.70029i 0 −1.67740 − 2.90535i −2.16641 + 2.16641i 0 −2.39721 − 2.39721i
17.10 −0.263704 + 0.984157i 0 0.833025 + 0.480947i 0.783948 − 1.35784i 0 0.259733 + 0.449871i −2.13391 + 2.13391i 0 1.12960 + 1.12960i
17.11 −0.188476 + 0.703401i 0 1.27280 + 0.734852i −0.0672512 + 0.116482i 0 0.205252 + 0.355506i −1.78664 + 1.78664i 0 −0.0692586 − 0.0692586i
17.12 −0.125672 + 0.469013i 0 1.52787 + 0.882117i 2.09952 − 3.63648i 0 −1.45885 − 2.52681i −1.29242 + 1.29242i 0 1.44170 + 1.44170i
17.13 −0.0590268 + 0.220291i 0 1.68701 + 0.973994i −0.486226 + 0.842168i 0 −2.17228 − 3.76249i −0.636670 + 0.636670i 0 −0.156822 − 0.156822i
17.14 −0.0505829 + 0.188778i 0 1.69897 + 0.980902i −1.89314 + 3.27902i 0 2.26829 + 3.92879i −0.547502 + 0.547502i 0 −0.523246 − 0.523246i
17.15 0.0207936 − 0.0776028i 0 1.72646 + 0.996773i 0.0760639 − 0.131746i 0 0.766676 + 1.32792i 0.226870 − 0.226870i 0 −0.00864225 − 0.00864225i
17.16 0.125113 − 0.466927i 0 1.52968 + 0.883163i −1.57483 + 2.72769i 0 0.454558 + 0.787318i 1.28738 − 1.28738i 0 1.07660 + 1.07660i
17.17 0.194706 − 0.726652i 0 1.24194 + 0.717034i 0.993651 − 1.72105i 0 −0.796821 − 1.38013i 1.82674 − 1.82674i 0 −1.05714 − 1.05714i
17.18 0.263555 − 0.983600i 0 0.834044 + 0.481535i 0.359924 − 0.623407i 0 0.867014 + 1.50171i 2.13354 − 2.13354i 0 −0.518323 − 0.518323i
17.19 0.280495 − 1.04682i 0 0.714895 + 0.412745i 1.56226 − 2.70591i 0 0.431040 + 0.746582i 2.16525 − 2.16525i 0 −2.39439 − 2.39439i
17.20 0.342169 − 1.27699i 0 0.218425 + 0.126108i −1.41847 + 2.45685i 0 −0.670736 − 1.16175i 2.10542 − 2.10542i 0 2.65203 + 2.65203i
See next 80 embeddings (of 112 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner
29.c odd 4 1 inner
261.l even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 783.2.m.a 112
3.b odd 2 1 261.2.l.a ✓ 112
9.c even 3 1 261.2.l.a ✓ 112
9.d odd 6 1 inner 783.2.m.a 112
29.c odd 4 1 inner 783.2.m.a 112
87.f even 4 1 261.2.l.a ✓ 112
261.l even 12 1 inner 783.2.m.a 112
261.m odd 12 1 261.2.l.a ✓ 112
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
261.2.l.a ✓ 112 3.b odd 2 1
261.2.l.a ✓ 112 9.c even 3 1
261.2.l.a ✓ 112 87.f even 4 1
261.2.l.a ✓ 112 261.m odd 12 1
783.2.m.a 112 1.a even 1 1 trivial
783.2.m.a 112 9.d odd 6 1 inner
783.2.m.a 112 29.c odd 4 1 inner
783.2.m.a 112 261.l even 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(783, [\chi])\).