Properties

Label 712.2.u.a
Level $712$
Weight $2$
Character orbit 712.u
Analytic conductor $5.685$
Analytic rank $0$
Dimension $100$
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] = [712,2,Mod(25,712)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(712, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([0, 0, 13]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("712.25");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 712 = 2^{3} \cdot 89 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 712.u (of order \(22\), degree \(10\), 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: \(5.68534862392\)
Analytic rank: \(0\)
Dimension: \(100\)
Relative dimension: \(10\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 100 q + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 100 q + 10 q^{9} + 14 q^{11} + 22 q^{13} - 8 q^{17} - 11 q^{19} - 34 q^{21} + 33 q^{23} + 24 q^{25} - 66 q^{27} - 22 q^{33} + 33 q^{35} - 36 q^{39} + 55 q^{41} + 8 q^{45} - 40 q^{47} - 30 q^{49} + 22 q^{51} - 38 q^{53} + 42 q^{55} + 54 q^{57} + 22 q^{59} + 11 q^{61} + 11 q^{63} - 33 q^{65} + 2 q^{67} + 17 q^{69} - 13 q^{71} + 20 q^{73} - 12 q^{79} - 38 q^{81} + 11 q^{83} + 35 q^{85} - 50 q^{87} + 47 q^{89} + 26 q^{91} - 2 q^{93} + 44 q^{95} + 36 q^{97} - 38 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1 0 −0.772544 + 2.63104i 0 −0.895068 1.95993i 0 −2.95434 + 1.34920i 0 −3.80180 2.44327i 0
25.2 0 −0.619520 + 2.10989i 0 0.619352 + 1.35619i 0 −0.0646412 + 0.0295207i 0 −1.54407 0.992316i 0
25.3 0 −0.387014 + 1.31805i 0 −0.196511 0.430300i 0 1.23694 0.564893i 0 0.936289 + 0.601717i 0
25.4 0 −0.379489 + 1.29242i 0 1.36747 + 2.99434i 0 1.93588 0.884085i 0 0.997424 + 0.641005i 0
25.5 0 −0.0328548 + 0.111893i 0 1.07829 + 2.36111i 0 −3.17713 + 1.45095i 0 2.51232 + 1.61457i 0
25.6 0 0.0384357 0.130900i 0 −1.08099 2.36703i 0 3.22958 1.47490i 0 2.50810 + 1.61186i 0
25.7 0 0.264940 0.902302i 0 −0.688724 1.50810i 0 −3.19771 + 1.46035i 0 1.77980 + 1.14381i 0
25.8 0 0.433312 1.47573i 0 0.425320 + 0.931322i 0 4.36812 1.99486i 0 0.533753 + 0.343022i 0
25.9 0 0.664476 2.26300i 0 0.831055 + 1.81976i 0 −0.758784 + 0.346525i 0 −2.15587 1.38549i 0
25.10 0 0.703549 2.39607i 0 −1.32830 2.90857i 0 −0.192082 + 0.0877207i 0 −2.72241 1.74958i 0
57.1 0 −0.772544 2.63104i 0 −0.895068 + 1.95993i 0 −2.95434 1.34920i 0 −3.80180 + 2.44327i 0
57.2 0 −0.619520 2.10989i 0 0.619352 1.35619i 0 −0.0646412 0.0295207i 0 −1.54407 + 0.992316i 0
57.3 0 −0.387014 1.31805i 0 −0.196511 + 0.430300i 0 1.23694 + 0.564893i 0 0.936289 0.601717i 0
57.4 0 −0.379489 1.29242i 0 1.36747 2.99434i 0 1.93588 + 0.884085i 0 0.997424 0.641005i 0
57.5 0 −0.0328548 0.111893i 0 1.07829 2.36111i 0 −3.17713 1.45095i 0 2.51232 1.61457i 0
57.6 0 0.0384357 + 0.130900i 0 −1.08099 + 2.36703i 0 3.22958 + 1.47490i 0 2.50810 1.61186i 0
57.7 0 0.264940 + 0.902302i 0 −0.688724 + 1.50810i 0 −3.19771 1.46035i 0 1.77980 1.14381i 0
57.8 0 0.433312 + 1.47573i 0 0.425320 0.931322i 0 4.36812 + 1.99486i 0 0.533753 0.343022i 0
57.9 0 0.664476 + 2.26300i 0 0.831055 1.81976i 0 −0.758784 0.346525i 0 −2.15587 + 1.38549i 0
57.10 0 0.703549 + 2.39607i 0 −1.32830 + 2.90857i 0 −0.192082 0.0877207i 0 −2.72241 + 1.74958i 0
See all 100 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
89.f even 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 712.2.u.a 100
89.f even 22 1 inner 712.2.u.a 100
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
712.2.u.a 100 1.a even 1 1 trivial
712.2.u.a 100 89.f even 22 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{100} - 20 T_{3}^{98} + 44 T_{3}^{97} + 277 T_{3}^{96} - 880 T_{3}^{95} - 2439 T_{3}^{94} + 10659 T_{3}^{93} + 18969 T_{3}^{92} - 94314 T_{3}^{91} - 153704 T_{3}^{90} + 762663 T_{3}^{89} + 1344841 T_{3}^{88} + \cdots + 2111209 \) acting on \(S_{2}^{\mathrm{new}}(712, [\chi])\). Copy content Toggle raw display