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Properties

Label 700.2.bo.a
Level $700$
Weight $2$
Character orbit 700.bo
Analytic conductor $5.590$
Analytic rank $0$
Dimension $160$
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(9,700)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(700, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([0, 21, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("700.9");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 700.bo (of order \(30\), degree \(8\), 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: \(160\)
Relative dimension: \(20\) over \(\Q(\zeta_{30})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 160 q + q^{5} - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 160 q + q^{5} - 20 q^{9} + 3 q^{11} - 34 q^{15} + 10 q^{17} + 4 q^{19} + 6 q^{21} + 9 q^{25} + 60 q^{27} - 12 q^{29} - 6 q^{31} + 25 q^{33} - 2 q^{39} + 14 q^{41} + 40 q^{45} - 6 q^{49} + 12 q^{51} - 50 q^{53} - 54 q^{55} + 24 q^{59} - 14 q^{61} - 16 q^{65} + 10 q^{69} - 34 q^{71} - 40 q^{73} - 83 q^{75} + 40 q^{77} - 20 q^{79} + 16 q^{81} + 50 q^{83} - 120 q^{85} + 9 q^{89} - 46 q^{91} - q^{95} - 20 q^{97} + 12 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1 0 −2.51715 − 2.26645i 0 2.04497 − 0.904485i 0 −1.86189 − 1.87973i 0 0.885657 + 8.42646i 0
9.2 0 −2.31291 − 2.08255i 0 −2.22416 + 0.230430i 0 2.55305 − 0.694208i 0 0.698934 + 6.64992i 0
9.3 0 −1.78501 − 1.60723i 0 1.21962 − 1.87417i 0 0.892401 + 2.49071i 0 0.289483 + 2.75425i 0
9.4 0 −1.47101 − 1.32450i 0 −0.579612 + 2.15964i 0 0.236972 − 2.63512i 0 0.0959767 + 0.913157i 0
9.5 0 −1.45291 − 1.30820i 0 −0.612599 − 2.15052i 0 2.31328 − 1.28402i 0 0.0859564 + 0.817820i 0
9.6 0 −1.28531 − 1.15730i 0 −1.78619 − 1.34519i 0 −2.55928 + 0.670868i 0 −0.000903994 − 0.00860093i 0
9.7 0 −1.08738 − 0.979079i 0 1.77976 + 1.35368i 0 −2.58311 − 0.572316i 0 −0.0897913 − 0.854307i 0
9.8 0 −0.449004 − 0.404285i 0 1.96759 + 1.06236i 0 2.04908 − 1.67370i 0 −0.275427 − 2.62052i 0
9.9 0 −0.315676 − 0.284236i 0 −1.03939 + 1.97982i 0 2.07002 + 1.64773i 0 −0.294724 − 2.80411i 0
9.10 0 −0.191109 − 0.172075i 0 1.87649 − 1.21606i 0 −0.498434 + 2.59838i 0 −0.306673 − 2.91780i 0
9.11 0 −0.0846702 − 0.0762374i 0 −2.15773 − 0.586689i 0 0.527970 + 2.59254i 0 −0.312228 − 2.97066i 0
9.12 0 0.469755 + 0.422969i 0 −1.20289 + 1.88496i 0 −2.55179 − 0.698839i 0 −0.271819 − 2.58618i 0
9.13 0 0.958999 + 0.863487i 0 0.220195 − 2.22520i 0 −2.19747 − 1.47348i 0 −0.139515 − 1.32740i 0
9.14 0 1.01166 + 0.910898i 0 −2.20805 + 0.352864i 0 0.626527 − 2.57050i 0 −0.119875 − 1.14054i 0
9.15 0 1.09933 + 0.989844i 0 2.09008 − 0.794704i 0 2.62936 − 0.294054i 0 −0.0848430 − 0.807228i 0
9.16 0 1.27456 + 1.14762i 0 1.44518 + 1.70630i 0 −1.85140 + 1.89006i 0 −0.00611204 − 0.0581522i 0
9.17 0 1.60332 + 1.44364i 0 −1.89613 − 1.18520i 0 1.82496 − 1.91560i 0 0.172968 + 1.64568i 0
9.18 0 2.04053 + 1.83730i 0 −0.843393 − 2.07091i 0 −0.517734 + 2.59460i 0 0.474497 + 4.51454i 0
9.19 0 2.13294 + 1.92051i 0 −0.393125 + 2.20124i 0 2.43061 + 1.04506i 0 0.547496 + 5.20908i 0
9.20 0 2.36104 + 2.12589i 0 2.19486 + 0.427322i 0 −1.25664 − 2.32827i 0 0.741513 + 7.05502i 0
See next 80 embeddings (of 160 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
25.e even 10 1 inner
175.t even 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 700.2.bo.a ✓ 160
7.c even 3 1 inner 700.2.bo.a ✓ 160
25.e even 10 1 inner 700.2.bo.a ✓ 160
175.t even 30 1 inner 700.2.bo.a ✓ 160
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
700.2.bo.a ✓ 160 1.a even 1 1 trivial
700.2.bo.a ✓ 160 7.c even 3 1 inner
700.2.bo.a ✓ 160 25.e even 10 1 inner
700.2.bo.a ✓ 160 175.t even 30 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(700, [\chi])\).