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Properties

Label 663.2.cd.a
Level $663$
Weight $2$
Character orbit 663.cd
Analytic conductor $5.294$
Analytic rank $0$
Dimension $336$
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] = [663,2,Mod(73,663)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(663, base_ring=CyclotomicField(16))
 
chi = DirichletCharacter(H, H._module([0, 4, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("663.73");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 663 = 3 \cdot 13 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 663.cd (of order \(16\), degree \(8\), 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: \(5.29408165401\)
Analytic rank: \(0\)
Dimension: \(336\)
Relative dimension: \(42\) over \(\Q(\zeta_{16})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 336 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 336 q - 16 q^{13} - 32 q^{14} + 32 q^{17} + 80 q^{20} + 32 q^{22} + 32 q^{26} + 48 q^{28} - 80 q^{29} + 32 q^{31} - 160 q^{32} - 16 q^{33} + 32 q^{34} - 160 q^{38} - 16 q^{41} - 96 q^{42} - 64 q^{43} + 128 q^{44} + 16 q^{45} - 32 q^{46} + 32 q^{49} + 48 q^{53} + 32 q^{55} - 96 q^{58} - 64 q^{59} - 96 q^{61} - 96 q^{64} + 64 q^{67} + 64 q^{70} + 128 q^{71} - 128 q^{73} - 240 q^{74} + 64 q^{76} + 48 q^{78} + 128 q^{79} + 128 q^{80} + 16 q^{85} - 96 q^{87} - 16 q^{90} + 64 q^{91} + 256 q^{95} - 96 q^{96} + 48 q^{97} + 144 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
73.1 −1.04402 + 2.52048i −0.555570 − 0.831470i −3.84865 − 3.84865i −2.73654 + 1.82850i 2.67573 − 0.532236i −0.452682 − 0.302472i 8.67754 − 3.59436i −0.382683 + 0.923880i −1.75170 − 8.80640i
73.2 −0.989322 + 2.38843i −0.555570 − 0.831470i −3.31165 − 3.31165i 3.25198 − 2.17290i 2.53555 − 0.504352i −3.01105 − 2.01192i 6.40907 − 2.65472i −0.382683 + 0.923880i 1.97258 + 9.91683i
73.3 −0.986599 + 2.38186i 0.555570 + 0.831470i −3.28567 − 3.28567i 1.55439 − 1.03861i −2.52857 + 0.502964i −2.47879 − 1.65627i 6.30393 − 2.61117i −0.382683 + 0.923880i 0.940264 + 4.72703i
73.4 −0.941279 + 2.27245i 0.555570 + 0.831470i −2.86380 − 2.86380i 0.308269 − 0.205978i −2.41242 + 0.479860i 2.30272 + 1.53863i 4.65857 − 1.92964i −0.382683 + 0.923880i 0.177909 + 0.894407i
73.5 −0.916621 + 2.21292i 0.555570 + 0.831470i −2.64260 − 2.64260i −1.28577 + 0.859121i −2.34922 + 0.467289i 0.534134 + 0.356897i 3.84430 − 1.59236i −0.382683 + 0.923880i −0.722606 − 3.63278i
73.6 −0.840664 + 2.02954i −0.555570 − 0.831470i −1.99812 − 1.99812i −1.47786 + 0.987477i 2.15455 − 0.428567i −3.41094 − 2.27912i 1.67592 − 0.694189i −0.382683 + 0.923880i −0.761740 − 3.82952i
73.7 −0.821822 + 1.98405i −0.555570 − 0.831470i −1.84687 − 1.84687i −1.85663 + 1.24056i 2.10626 − 0.418961i 3.34569 + 2.23552i 1.21397 − 0.502843i −0.382683 + 0.923880i −0.935521 − 4.70318i
73.8 −0.755685 + 1.82439i 0.555570 + 0.831470i −1.34311 − 1.34311i −3.18823 + 2.13031i −1.93676 + 0.385245i −1.74102 − 1.16331i −0.183450 + 0.0759876i −0.382683 + 0.923880i −1.47721 − 7.42641i
73.9 −0.692581 + 1.67204i −0.555570 − 0.831470i −0.901831 − 0.901831i 1.77905 − 1.18872i 1.77503 − 0.353075i −0.0241506 − 0.0161369i −1.21159 + 0.501857i −0.382683 + 0.923880i 0.755454 + 3.79793i
73.10 −0.529736 + 1.27890i 0.555570 + 0.831470i 0.0592595 + 0.0592595i 0.367010 − 0.245228i −1.35767 + 0.270057i 3.46921 + 2.31805i −2.66497 + 1.10387i −0.382683 + 0.923880i 0.119203 + 0.599273i
73.11 −0.522423 + 1.26124i 0.555570 + 0.831470i 0.0964111 + 0.0964111i 0.505483 − 0.337753i −1.33893 + 0.266329i −4.22805 − 2.82510i −2.69445 + 1.11608i −0.382683 + 0.923880i 0.161912 + 0.813986i
73.12 −0.519761 + 1.25481i 0.555570 + 0.831470i 0.109806 + 0.109806i 2.76974 − 1.85068i −1.33210 + 0.264972i 3.05841 + 2.04356i −2.70449 + 1.12024i −0.382683 + 0.923880i 0.882657 + 4.43742i
73.13 −0.482610 + 1.16512i −0.555570 − 0.831470i 0.289613 + 0.289613i −0.357538 + 0.238899i 1.23689 − 0.246032i 0.113379 + 0.0757572i −2.80745 + 1.16288i −0.382683 + 0.923880i −0.105796 − 0.531870i
73.14 −0.401022 + 0.968154i 0.555570 + 0.831470i 0.637711 + 0.637711i −0.0650574 + 0.0434700i −1.02779 + 0.204439i −2.02096 − 1.35036i −2.80945 + 1.16371i −0.382683 + 0.923880i −0.0159961 − 0.0804180i
73.15 −0.381647 + 0.921377i −0.555570 − 0.831470i 0.710932 + 0.710932i −3.51517 + 2.34876i 0.978129 − 0.194562i −0.167056 − 0.111623i −2.76912 + 1.14701i −0.382683 + 0.923880i −0.822542 − 4.13520i
73.16 −0.276671 + 0.667944i −0.555570 − 0.831470i 1.04461 + 1.04461i 1.96855 − 1.31534i 0.709085 − 0.141046i −3.35790 − 2.24368i −2.32264 + 0.962071i −0.382683 + 0.923880i 0.333934 + 1.67880i
73.17 −0.207049 + 0.499861i −0.555570 − 0.831470i 1.20722 + 1.20722i −0.0184735 + 0.0123436i 0.530649 − 0.105553i 2.13460 + 1.42629i −1.85312 + 0.767587i −0.382683 + 0.923880i −0.00234516 − 0.0117899i
73.18 −0.199843 + 0.482463i −0.555570 − 0.831470i 1.22138 + 1.22138i 3.26687 − 2.18285i 0.512180 − 0.101879i 3.65330 + 2.44106i −1.79828 + 0.744873i −0.382683 + 0.923880i 0.400286 + 2.01237i
73.19 −0.174920 + 0.422294i 0.555570 + 0.831470i 1.26648 + 1.26648i 0.0830991 − 0.0555250i −0.448305 + 0.0891734i 0.591370 + 0.395141i −1.60095 + 0.663134i −0.382683 + 0.923880i 0.00891220 + 0.0448046i
73.20 −0.0684682 + 0.165297i 0.555570 + 0.831470i 1.39158 + 1.39158i 1.93898 − 1.29558i −0.175478 + 0.0349048i −0.922019 − 0.616073i −0.655896 + 0.271681i −0.382683 + 0.923880i 0.0813975 + 0.409213i
See next 80 embeddings (of 336 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 73.42
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
221.z even 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 663.2.cd.a yes 336
13.d odd 4 1 663.2.ca.a ✓ 336
17.e odd 16 1 663.2.ca.a ✓ 336
221.z even 16 1 inner 663.2.cd.a yes 336
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
663.2.ca.a ✓ 336 13.d odd 4 1
663.2.ca.a ✓ 336 17.e odd 16 1
663.2.cd.a yes 336 1.a even 1 1 trivial
663.2.cd.a yes 336 221.z even 16 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(663, [\chi])\).