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Properties

Label 627.2.r.c
Level $627$
Weight $2$
Character orbit 627.r
Analytic conductor $5.007$
Analytic rank $0$
Dimension $54$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [627,2,Mod(100,627)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(627, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 0, 16]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("627.100");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 627 = 3 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 627.r (of order \(9\), 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: \(5.00662020673\)
Analytic rank: \(0\)
Dimension: \(54\)
Relative dimension: \(9\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 54 q + 3 q^{2} + 3 q^{4} + 3 q^{6} - 18 q^{7} - 12 q^{8} + 30 q^{10} + 27 q^{11} - 33 q^{12} + 9 q^{14} - 21 q^{16} - 24 q^{17} + 6 q^{18} + 3 q^{19} + 30 q^{20} - 3 q^{22} + 12 q^{23} - 39 q^{24} + 30 q^{25}+ \cdots - 54 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
100.1 −2.50055 + 0.910126i 0.173648 − 0.984808i 3.89233 − 3.26605i −1.92507 − 1.61533i 0.462083 + 2.62060i −2.38697 − 4.13436i −4.09942 + 7.10040i −0.939693 − 0.342020i 6.28389 + 2.28715i
100.2 −1.89161 + 0.688491i 0.173648 − 0.984808i 1.57209 − 1.31914i −1.65438 − 1.38819i 0.349556 + 1.98243i 1.57126 + 2.72150i −0.0525637 + 0.0910429i −0.939693 − 0.342020i 4.08520 + 1.48689i
100.3 −1.04904 + 0.381820i 0.173648 − 0.984808i −0.577384 + 0.484483i 3.26470 + 2.73941i 0.193855 + 1.09941i −1.21034 − 2.09636i 1.53708 − 2.66231i −0.939693 − 0.342020i −4.47077 − 1.62723i
100.4 −0.642665 + 0.233911i 0.173648 − 0.984808i −1.17378 + 0.984922i 0.374881 + 0.314563i 0.118760 + 0.673520i −1.38207 − 2.39381i 1.20788 − 2.09210i −0.939693 − 0.342020i −0.314503 − 0.114470i
100.5 −0.529924 + 0.192877i 0.173648 − 0.984808i −1.28847 + 1.08116i −1.59108 − 1.33507i 0.0979261 + 0.555366i 0.832567 + 1.44205i 1.03820 − 1.79821i −0.939693 − 0.342020i 1.10065 + 0.400605i
100.6 1.14605 − 0.417129i 0.173648 − 0.984808i −0.392648 + 0.329471i −1.91969 − 1.61081i −0.211782 − 1.20108i −0.627651 − 1.08712i −1.53217 + 2.65379i −0.939693 − 0.342020i −2.87198 − 1.04531i
100.7 1.41798 − 0.516103i 0.173648 − 0.984808i 0.212218 − 0.178072i 1.54658 + 1.29773i −0.262032 − 1.48606i 1.54228 + 2.67130i −1.29997 + 2.25161i −0.939693 − 0.342020i 2.86278 + 1.04197i
100.8 2.24827 − 0.818302i 0.173648 − 0.984808i 2.85300 − 2.39395i −0.281381 − 0.236107i −0.415463 − 2.35621i 0.491710 + 0.851667i 2.06277 − 3.57283i −0.939693 − 0.342020i −0.825827 − 0.300576i
100.9 2.30150 − 0.837676i 0.173648 − 0.984808i 3.06309 − 2.57024i 2.18544 + 1.83380i −0.425299 − 2.41199i −2.42318 − 4.19708i 2.44746 − 4.23913i −0.939693 − 0.342020i 6.56590 + 2.38979i
199.1 −0.357327 − 2.02650i 0.766044 − 0.642788i −2.09965 + 0.764212i −3.54262 − 1.28941i −1.57634 − 1.32271i −1.09829 + 1.90230i 0.241176 + 0.417728i 0.173648 − 0.984808i −1.34712 + 7.63988i
199.2 −0.312642 − 1.77308i 0.766044 − 0.642788i −1.16668 + 0.424636i 3.18574 + 1.15951i −1.37921 − 1.15729i −0.323948 + 0.561094i −0.682767 − 1.18259i 0.173648 − 0.984808i 1.05991 − 6.01107i
199.3 −0.163057 − 0.924743i 0.766044 − 0.642788i 1.05082 − 0.382468i −1.79310 − 0.652634i −0.719322 − 0.603583i 1.21300 − 2.10097i −1.46404 − 2.53579i 0.173648 − 0.984808i −0.311142 + 1.76457i
199.4 −0.106342 − 0.603094i 0.766044 − 0.642788i 1.52697 − 0.555772i −1.00422 − 0.365505i −0.469124 − 0.393642i 0.304084 − 0.526689i −1.10996 − 1.92251i 0.173648 − 0.984808i −0.113644 + 0.644505i
199.5 0.00745564 + 0.0422831i 0.766044 − 0.642788i 1.87765 − 0.683410i 2.08739 + 0.759748i 0.0328904 + 0.0275983i −2.26349 + 3.92047i 0.0858311 + 0.148664i 0.173648 − 0.984808i −0.0165616 + 0.0939256i
199.6 0.180996 + 1.02648i 0.766044 − 0.642788i 0.858479 − 0.312461i 0.862251 + 0.313834i 0.798461 + 0.669989i 1.34164 − 2.32379i 1.51843 + 2.63001i 0.173648 − 0.984808i −0.166080 + 0.941888i
199.7 0.335406 + 1.90218i 0.766044 − 0.642788i −1.62641 + 0.591963i −2.06412 − 0.751279i 1.47963 + 1.24156i −0.293014 + 0.507515i 0.259998 + 0.450330i 0.173648 − 0.984808i 0.736750 − 4.17831i
199.8 0.441274 + 2.50259i 0.766044 − 0.642788i −4.18885 + 1.52462i 4.09713 + 1.49123i 1.94667 + 1.63345i 1.35128 − 2.34048i −3.12272 − 5.40872i 0.173648 − 0.984808i −1.92399 + 10.9115i
199.9 0.474236 + 2.68953i 0.766044 − 0.642788i −5.12927 + 1.86690i −1.82845 − 0.665503i 2.09208 + 1.75546i −1.52552 + 2.64228i −4.72255 − 8.17969i 0.173648 − 0.984808i 0.922769 − 5.23328i
232.1 −2.50055 − 0.910126i 0.173648 + 0.984808i 3.89233 + 3.26605i −1.92507 + 1.61533i 0.462083 − 2.62060i −2.38697 + 4.13436i −4.09942 − 7.10040i −0.939693 + 0.342020i 6.28389 − 2.28715i
232.2 −1.89161 − 0.688491i 0.173648 + 0.984808i 1.57209 + 1.31914i −1.65438 + 1.38819i 0.349556 − 1.98243i 1.57126 − 2.72150i −0.0525637 − 0.0910429i −0.939693 + 0.342020i 4.08520 − 1.48689i
See all 54 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 100.9
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 627.2.r.c ✓ 54
19.e even 9 1 inner 627.2.r.c ✓ 54
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
627.2.r.c ✓ 54 1.a even 1 1 trivial
627.2.r.c ✓ 54 19.e even 9 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{54} - 3 T_{2}^{53} + 3 T_{2}^{52} + 8 T_{2}^{51} - 12 T_{2}^{50} - 90 T_{2}^{49} + 755 T_{2}^{48} + \cdots + 729 \) acting on \(S_{2}^{\mathrm{new}}(627, [\chi])\). Copy content Toggle raw display