Properties

Label 700.2.p.e
Level $700$
Weight $2$
Character orbit 700.p
Analytic conductor $5.590$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [700,2,Mod(451,700)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(700, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("700.451");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 700.p (of order \(6\), 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: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 140)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 6 q^{4} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 6 q^{4} - 4 q^{9} - 22 q^{14} + 18 q^{16} - 52 q^{21} + 48 q^{24} - 18 q^{26} - 28 q^{36} + 26 q^{44} - 22 q^{46} - 48 q^{54} - 16 q^{56} + 36 q^{61} - 36 q^{64} - 24 q^{66} - 14 q^{74} + 72 q^{81} - 56 q^{84} + 8 q^{86} + 108 q^{89} - 30 q^{94} + 60 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
451.1 −1.40479 + 0.163027i 0.940044 1.62820i 1.94684 0.458035i 0 −1.05512 + 2.44053i 2.57589 0.603960i −2.66023 + 0.960828i −0.267367 0.463092i 0
451.2 −1.34238 0.444985i −1.29809 + 2.24836i 1.60398 + 1.19468i 0 2.74302 2.44053i 0.603960 + 2.57589i −1.62154 2.31746i −1.87009 3.23909i 0
451.3 −1.24320 0.674125i −0.366453 + 0.634715i 1.09111 + 1.67615i 0 0.883452 0.542044i 0.664037 2.56107i −0.226536 2.81934i 1.23142 + 2.13289i 0
451.4 −1.00031 0.999687i 0.739583 1.28100i 0.00125109 + 2.00000i 0 −2.02041 + 0.542044i −2.56107 0.664037i 1.99812 2.00188i 0.406034 + 0.703271i 0
451.5 −0.843578 + 1.13507i 0.940044 1.62820i −0.576753 1.91503i 0 1.05512 + 2.44053i 2.57589 0.603960i 2.66023 + 0.960828i −0.267367 0.463092i 0
451.6 −0.365598 1.36614i −0.739583 + 1.28100i −1.73268 + 0.998916i 0 2.02041 + 0.542044i 2.56107 + 0.664037i 1.99812 + 2.00188i 0.406034 + 0.703271i 0
451.7 −0.285823 + 1.38503i −1.29809 + 2.24836i −1.83661 0.791746i 0 −2.74302 2.44053i 0.603960 + 2.57589i 1.62154 2.31746i −1.87009 3.23909i 0
451.8 −0.0377920 + 1.41371i −0.366453 + 0.634715i −1.99714 0.106854i 0 −0.883452 0.542044i 0.664037 2.56107i 0.226536 2.81934i 1.23142 + 2.13289i 0
451.9 0.0377920 1.41371i 0.366453 0.634715i −1.99714 0.106854i 0 −0.883452 0.542044i −0.664037 + 2.56107i −0.226536 + 2.81934i 1.23142 + 2.13289i 0
451.10 0.285823 1.38503i 1.29809 2.24836i −1.83661 0.791746i 0 −2.74302 2.44053i −0.603960 2.57589i −1.62154 + 2.31746i −1.87009 3.23909i 0
451.11 0.365598 + 1.36614i 0.739583 1.28100i −1.73268 + 0.998916i 0 2.02041 + 0.542044i −2.56107 0.664037i −1.99812 2.00188i 0.406034 + 0.703271i 0
451.12 0.843578 1.13507i −0.940044 + 1.62820i −0.576753 1.91503i 0 1.05512 + 2.44053i −2.57589 + 0.603960i −2.66023 0.960828i −0.267367 0.463092i 0
451.13 1.00031 + 0.999687i −0.739583 + 1.28100i 0.00125109 + 2.00000i 0 −2.02041 + 0.542044i 2.56107 + 0.664037i −1.99812 + 2.00188i 0.406034 + 0.703271i 0
451.14 1.24320 + 0.674125i 0.366453 0.634715i 1.09111 + 1.67615i 0 0.883452 0.542044i −0.664037 + 2.56107i 0.226536 + 2.81934i 1.23142 + 2.13289i 0
451.15 1.34238 + 0.444985i 1.29809 2.24836i 1.60398 + 1.19468i 0 2.74302 2.44053i −0.603960 2.57589i 1.62154 + 2.31746i −1.87009 3.23909i 0
451.16 1.40479 0.163027i −0.940044 + 1.62820i 1.94684 0.458035i 0 −1.05512 + 2.44053i −2.57589 + 0.603960i 2.66023 0.960828i −0.267367 0.463092i 0
551.1 −1.40479 0.163027i 0.940044 + 1.62820i 1.94684 + 0.458035i 0 −1.05512 2.44053i 2.57589 + 0.603960i −2.66023 0.960828i −0.267367 + 0.463092i 0
551.2 −1.34238 + 0.444985i −1.29809 2.24836i 1.60398 1.19468i 0 2.74302 + 2.44053i 0.603960 2.57589i −1.62154 + 2.31746i −1.87009 + 3.23909i 0
551.3 −1.24320 + 0.674125i −0.366453 0.634715i 1.09111 1.67615i 0 0.883452 + 0.542044i 0.664037 + 2.56107i −0.226536 + 2.81934i 1.23142 2.13289i 0
551.4 −1.00031 + 0.999687i 0.739583 + 1.28100i 0.00125109 2.00000i 0 −2.02041 0.542044i −2.56107 + 0.664037i 1.99812 + 2.00188i 0.406034 0.703271i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 451.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
7.d odd 6 1 inner
20.d odd 2 1 inner
28.f even 6 1 inner
35.i odd 6 1 inner
140.s even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 700.2.p.e 32
4.b odd 2 1 inner 700.2.p.e 32
5.b even 2 1 inner 700.2.p.e 32
5.c odd 4 2 140.2.s.b 32
7.d odd 6 1 inner 700.2.p.e 32
20.d odd 2 1 inner 700.2.p.e 32
20.e even 4 2 140.2.s.b 32
28.f even 6 1 inner 700.2.p.e 32
35.f even 4 2 980.2.s.e 32
35.i odd 6 1 inner 700.2.p.e 32
35.k even 12 2 140.2.s.b 32
35.k even 12 2 980.2.c.d 32
35.l odd 12 2 980.2.c.d 32
35.l odd 12 2 980.2.s.e 32
140.j odd 4 2 980.2.s.e 32
140.s even 6 1 inner 700.2.p.e 32
140.w even 12 2 980.2.c.d 32
140.w even 12 2 980.2.s.e 32
140.x odd 12 2 140.2.s.b 32
140.x odd 12 2 980.2.c.d 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.s.b 32 5.c odd 4 2
140.2.s.b 32 20.e even 4 2
140.2.s.b 32 35.k even 12 2
140.2.s.b 32 140.x odd 12 2
700.2.p.e 32 1.a even 1 1 trivial
700.2.p.e 32 4.b odd 2 1 inner
700.2.p.e 32 5.b even 2 1 inner
700.2.p.e 32 7.d odd 6 1 inner
700.2.p.e 32 20.d odd 2 1 inner
700.2.p.e 32 28.f even 6 1 inner
700.2.p.e 32 35.i odd 6 1 inner
700.2.p.e 32 140.s even 6 1 inner
980.2.c.d 32 35.k even 12 2
980.2.c.d 32 35.l odd 12 2
980.2.c.d 32 140.w even 12 2
980.2.c.d 32 140.x odd 12 2
980.2.s.e 32 35.f even 4 2
980.2.s.e 32 35.l odd 12 2
980.2.s.e 32 140.j odd 4 2
980.2.s.e 32 140.w even 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(700, [\chi])\):

\( T_{3}^{16} + 13T_{3}^{14} + 116T_{3}^{12} + 535T_{3}^{10} + 1780T_{3}^{8} + 3353T_{3}^{6} + 4445T_{3}^{4} + 2156T_{3}^{2} + 784 \) Copy content Toggle raw display
\( T_{17}^{16} - 85 T_{17}^{14} + 5369 T_{17}^{12} - 135448 T_{17}^{10} + 2480860 T_{17}^{8} - 18050816 T_{17}^{6} + 95473040 T_{17}^{4} - 174212096 T_{17}^{2} + 243859456 \) Copy content Toggle raw display