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Properties

Label 700.2.k.b
Level $700$
Weight $2$
Character orbit 700.k
Analytic conductor $5.590$
Analytic rank $0$
Dimension $36$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [700,2,Mod(43,700)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(700, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("700.43");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 700.k (of order \(4\), degree \(2\), 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: \(5.58952814149\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(18\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 140)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 8 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q - 8 q^{6} + 16 q^{12} + 4 q^{13} - 8 q^{16} + 20 q^{17} - 28 q^{18} - 4 q^{22} - 32 q^{26} - 20 q^{37} + 20 q^{42} + 16 q^{46} + 24 q^{48} - 16 q^{52} + 44 q^{53} - 24 q^{56} + 16 q^{57} + 4 q^{58} - 64 q^{61} - 40 q^{62} + 32 q^{66} - 80 q^{68} - 80 q^{72} - 52 q^{73} + 8 q^{76} + 76 q^{78} - 36 q^{81} - 56 q^{82} + 56 q^{86} + 40 q^{88} + 56 q^{92} - 32 q^{93} + 120 q^{96} - 20 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1 −1.35755 − 0.396294i −0.945787 + 0.945787i 1.68590 + 1.07598i 0 1.65877 − 0.909146i 0.707107 + 0.707107i −1.86230 − 2.12881i 1.21097i 0
43.2 −1.34141 + 0.447909i 0.396892 − 0.396892i 1.59875 − 1.20166i 0 −0.354623 + 0.710167i 0.707107 + 0.707107i −1.60635 + 2.32801i 2.68495i 0
43.3 −1.29055 + 0.578354i 1.27396 − 1.27396i 1.33101 − 1.49278i 0 −0.907305 + 2.38090i −0.707107 − 0.707107i −0.854378 + 2.69630i − 0.245954i 0
43.4 −1.26992 − 0.622342i 2.09607 − 2.09607i 1.22538 + 1.58065i 0 −3.96631 + 1.35737i −0.707107 − 0.707107i −0.572433 − 2.76990i − 5.78704i 0
43.5 −0.903055 − 1.08834i 1.00798 − 1.00798i −0.368985 + 1.96567i 0 −2.00729 − 0.186768i 0.707107 + 0.707107i 2.47254 − 1.37352i 0.967954i 0
43.6 −0.802007 − 1.16481i −1.75731 + 1.75731i −0.713568 + 1.86837i 0 3.45630 + 0.637557i −0.707107 − 0.707107i 2.74859 − 0.667278i − 3.17626i 0
43.7 −0.578354 + 1.29055i −1.27396 + 1.27396i −1.33101 − 1.49278i 0 −0.907305 − 2.38090i 0.707107 + 0.707107i 2.69630 − 0.854378i − 0.245954i 0
43.8 −0.447909 + 1.34141i −0.396892 + 0.396892i −1.59875 − 1.20166i 0 −0.354623 − 0.710167i −0.707107 − 0.707107i 2.32801 − 1.60635i 2.68495i 0
43.9 −0.297828 − 1.38250i −0.137886 + 0.137886i −1.82260 + 0.823494i 0 0.231693 + 0.149560i −0.707107 − 0.707107i 1.68130 + 2.27447i 2.96198i 0
43.10 0.361308 − 1.36728i −1.26588 + 1.26588i −1.73891 − 0.988019i 0 1.27344 + 2.18818i 0.707107 + 0.707107i −1.97918 + 2.02060i − 0.204893i 0
43.11 0.396294 + 1.35755i 0.945787 − 0.945787i −1.68590 + 1.07598i 0 1.65877 + 0.909146i −0.707107 − 0.707107i −2.12881 − 1.86230i 1.21097i 0
43.12 0.622342 + 1.26992i −2.09607 + 2.09607i −1.22538 + 1.58065i 0 −3.96631 − 1.35737i 0.707107 + 0.707107i −2.76990 − 0.572433i − 5.78704i 0
43.13 0.649412 − 1.25629i 2.28163 − 2.28163i −1.15653 − 1.63170i 0 −1.38467 − 4.34811i 0.707107 + 0.707107i −2.80095 + 0.393286i − 7.41170i 0
43.14 1.08834 + 0.903055i −1.00798 + 1.00798i 0.368985 + 1.96567i 0 −2.00729 + 0.186768i −0.707107 − 0.707107i −1.37352 + 2.47254i 0.967954i 0
43.15 1.16481 + 0.802007i 1.75731 − 1.75731i 0.713568 + 1.86837i 0 3.45630 − 0.637557i 0.707107 + 0.707107i −0.667278 + 2.74859i − 3.17626i 0
43.16 1.25629 − 0.649412i −2.28163 + 2.28163i 1.15653 − 1.63170i 0 −1.38467 + 4.34811i −0.707107 − 0.707107i 0.393286 − 2.80095i − 7.41170i 0
43.17 1.36728 − 0.361308i 1.26588 − 1.26588i 1.73891 − 0.988019i 0 1.27344 − 2.18818i −0.707107 − 0.707107i 2.02060 − 1.97918i − 0.204893i 0
43.18 1.38250 + 0.297828i 0.137886 − 0.137886i 1.82260 + 0.823494i 0 0.231693 − 0.149560i 0.707107 + 0.707107i 2.27447 + 1.68130i 2.96198i 0
407.1 −1.35755 + 0.396294i −0.945787 − 0.945787i 1.68590 − 1.07598i 0 1.65877 + 0.909146i 0.707107 − 0.707107i −1.86230 + 2.12881i − 1.21097i 0
407.2 −1.34141 − 0.447909i 0.396892 + 0.396892i 1.59875 + 1.20166i 0 −0.354623 − 0.710167i 0.707107 − 0.707107i −1.60635 − 2.32801i − 2.68495i 0
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.c odd 4 1 inner
20.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 700.2.k.b 36
4.b odd 2 1 inner 700.2.k.b 36
5.b even 2 1 140.2.k.a ✓ 36
5.c odd 4 1 140.2.k.a ✓ 36
5.c odd 4 1 inner 700.2.k.b 36
20.d odd 2 1 140.2.k.a ✓ 36
20.e even 4 1 140.2.k.a ✓ 36
20.e even 4 1 inner 700.2.k.b 36
35.c odd 2 1 980.2.k.l 36
35.f even 4 1 980.2.k.l 36
35.i odd 6 2 980.2.x.l 72
35.j even 6 2 980.2.x.k 72
35.k even 12 2 980.2.x.l 72
35.l odd 12 2 980.2.x.k 72
140.c even 2 1 980.2.k.l 36
140.j odd 4 1 980.2.k.l 36
140.p odd 6 2 980.2.x.k 72
140.s even 6 2 980.2.x.l 72
140.w even 12 2 980.2.x.k 72
140.x odd 12 2 980.2.x.l 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.k.a ✓ 36 5.b even 2 1
140.2.k.a ✓ 36 5.c odd 4 1
140.2.k.a ✓ 36 20.d odd 2 1
140.2.k.a ✓ 36 20.e even 4 1
700.2.k.b 36 1.a even 1 1 trivial
700.2.k.b 36 4.b odd 2 1 inner
700.2.k.b 36 5.c odd 4 1 inner
700.2.k.b 36 20.e even 4 1 inner
980.2.k.l 36 35.c odd 2 1
980.2.k.l 36 35.f even 4 1
980.2.k.l 36 140.c even 2 1
980.2.k.l 36 140.j odd 4 1
980.2.x.k 72 35.j even 6 2
980.2.x.k 72 35.l odd 12 2
980.2.x.k 72 140.p odd 6 2
980.2.x.k 72 140.w even 12 2
980.2.x.l 72 35.i odd 6 2
980.2.x.l 72 35.k even 12 2
980.2.x.l 72 140.s even 6 2
980.2.x.l 72 140.x odd 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{36} + 252 T_{3}^{32} + 22046 T_{3}^{28} + 818612 T_{3}^{24} + 13539297 T_{3}^{20} + 105649592 T_{3}^{16} + \cdots + 65536 \) acting on \(S_{2}^{\mathrm{new}}(700, [\chi])\). Copy content Toggle raw display