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Properties

Label 207.4.i.d
Level $207$
Weight $4$
Character orbit 207.i
Analytic conductor $12.213$
Analytic rank $0$
Dimension $120$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [207,4,Mod(55,207)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(207, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([0, 10]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("207.55");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 207 = 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 207.i (of order \(11\), 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: \(12.2133953712\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(12\) over \(\Q(\zeta_{11})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 120 q - 16 q^{4} - 20 q^{7} - 100 q^{10} + 32 q^{13} - 888 q^{16} + 164 q^{19} + 988 q^{22} + 476 q^{25} - 832 q^{28} - 996 q^{31} + 1954 q^{34} + 1152 q^{37} + 226 q^{40} - 892 q^{43} - 3776 q^{46} - 412 q^{49}+ \cdots + 5476 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1 −5.10814 + 1.49988i 0 17.1134 − 10.9981i 0.282807 + 1.96697i 0 0.431353 − 0.944532i −43.0309 + 49.6603i 0 −4.39484 − 9.62336i
55.2 −3.54609 + 1.04123i 0 4.76056 − 3.05942i 2.17870 + 15.1532i 0 5.20421 − 11.3956i 5.66604 − 6.53896i 0 −23.5037 − 51.4660i
55.3 −3.45688 + 1.01503i 0 4.18971 − 2.69257i −1.06996 − 7.44170i 0 −11.8480 + 25.9435i 7.12446 − 8.22207i 0 11.2523 + 24.6390i
55.4 −3.23376 + 0.949517i 0 2.82558 − 1.81589i −2.31518 − 16.1024i 0 13.7180 − 30.0382i 10.2435 − 11.8216i 0 22.7762 + 49.8730i
55.5 −1.25127 + 0.367407i 0 −5.29933 + 3.40567i 0.179870 + 1.25102i 0 4.43639 − 9.71434i 12.2117 − 14.0930i 0 −0.684701 − 1.49929i
55.6 −0.778533 + 0.228598i 0 −6.17617 + 3.96918i 2.23693 + 15.5582i 0 −5.88963 + 12.8965i 8.15184 − 9.40772i 0 −5.29808 − 11.6012i
55.7 0.778533 − 0.228598i 0 −6.17617 + 3.96918i −2.23693 − 15.5582i 0 −5.88963 + 12.8965i −8.15184 + 9.40772i 0 −5.29808 − 11.6012i
55.8 1.25127 − 0.367407i 0 −5.29933 + 3.40567i −0.179870 − 1.25102i 0 4.43639 − 9.71434i −12.2117 + 14.0930i 0 −0.684701 − 1.49929i
55.9 3.23376 − 0.949517i 0 2.82558 − 1.81589i 2.31518 + 16.1024i 0 13.7180 − 30.0382i −10.2435 + 11.8216i 0 22.7762 + 49.8730i
55.10 3.45688 − 1.01503i 0 4.18971 − 2.69257i 1.06996 + 7.44170i 0 −11.8480 + 25.9435i −7.12446 + 8.22207i 0 11.2523 + 24.6390i
55.11 3.54609 − 1.04123i 0 4.76056 − 3.05942i −2.17870 − 15.1532i 0 5.20421 − 11.3956i −5.66604 + 6.53896i 0 −23.5037 − 51.4660i
55.12 5.10814 − 1.49988i 0 17.1134 − 10.9981i −0.282807 − 1.96697i 0 0.431353 − 0.944532i 43.0309 − 49.6603i 0 −4.39484 − 9.62336i
64.1 −5.10814 − 1.49988i 0 17.1134 + 10.9981i 0.282807 − 1.96697i 0 0.431353 + 0.944532i −43.0309 − 49.6603i 0 −4.39484 + 9.62336i
64.2 −3.54609 − 1.04123i 0 4.76056 + 3.05942i 2.17870 − 15.1532i 0 5.20421 + 11.3956i 5.66604 + 6.53896i 0 −23.5037 + 51.4660i
64.3 −3.45688 − 1.01503i 0 4.18971 + 2.69257i −1.06996 + 7.44170i 0 −11.8480 − 25.9435i 7.12446 + 8.22207i 0 11.2523 − 24.6390i
64.4 −3.23376 − 0.949517i 0 2.82558 + 1.81589i −2.31518 + 16.1024i 0 13.7180 + 30.0382i 10.2435 + 11.8216i 0 22.7762 − 49.8730i
64.5 −1.25127 − 0.367407i 0 −5.29933 − 3.40567i 0.179870 − 1.25102i 0 4.43639 + 9.71434i 12.2117 + 14.0930i 0 −0.684701 + 1.49929i
64.6 −0.778533 − 0.228598i 0 −6.17617 − 3.96918i 2.23693 − 15.5582i 0 −5.88963 − 12.8965i 8.15184 + 9.40772i 0 −5.29808 + 11.6012i
64.7 0.778533 + 0.228598i 0 −6.17617 − 3.96918i −2.23693 + 15.5582i 0 −5.88963 − 12.8965i −8.15184 − 9.40772i 0 −5.29808 + 11.6012i
64.8 1.25127 + 0.367407i 0 −5.29933 − 3.40567i −0.179870 + 1.25102i 0 4.43639 + 9.71434i −12.2117 − 14.0930i 0 −0.684701 + 1.49929i
See next 80 embeddings (of 120 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 55.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
23.c even 11 1 inner
69.h odd 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 207.4.i.d ✓ 120
3.b odd 2 1 inner 207.4.i.d ✓ 120
23.c even 11 1 inner 207.4.i.d ✓ 120
69.h odd 22 1 inner 207.4.i.d ✓ 120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
207.4.i.d ✓ 120 1.a even 1 1 trivial
207.4.i.d ✓ 120 3.b odd 2 1 inner
207.4.i.d ✓ 120 23.c even 11 1 inner
207.4.i.d ✓ 120 69.h odd 22 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{120} + 56 T_{2}^{118} + 2930 T_{2}^{116} + 122872 T_{2}^{114} + 4643635 T_{2}^{112} + \cdots + 29\!\cdots\!04 \) acting on \(S_{4}^{\mathrm{new}}(207, [\chi])\). Copy content Toggle raw display