Properties

Label 700.2.bg.a
Level $700$
Weight $2$
Character orbit 700.bg
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(81,700)]
 
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, 12, 20]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("700.81");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 700.bg (of order \(15\), 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 \(\Q(\zeta_{15})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 160 q + q^{5} + 2 q^{7} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 160 q + q^{5} + 2 q^{7} + 20 q^{9} - 3 q^{11} + 30 q^{15} + 12 q^{17} + 4 q^{19} - 6 q^{21} + 20 q^{23} + q^{25} - 12 q^{27} - 12 q^{29} + 6 q^{31} - 19 q^{33} - 10 q^{35} + 14 q^{37} - 2 q^{39} - 14 q^{41} - 76 q^{43} + 30 q^{45} - 10 q^{47} - 6 q^{49} - 12 q^{51} - 8 q^{53} + 26 q^{55} - 172 q^{57} + 24 q^{59} + 14 q^{61} + 38 q^{63} + 10 q^{65} - 24 q^{67} + 10 q^{69} + 34 q^{71} + 42 q^{73} + 27 q^{75} + 30 q^{77} - 20 q^{79} + 24 q^{81} - 2 q^{83} + 104 q^{85} - 18 q^{87} + 9 q^{89} + 46 q^{91} - 58 q^{93} - 21 q^{95} + 60 q^{97} + 12 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
81.1 0 −0.297443 2.82998i 0 0.228995 + 2.22431i 0 −1.76280 1.97295i 0 −4.98590 + 1.05978i 0
81.2 0 −0.296036 2.81659i 0 −1.05498 1.97155i 0 −1.22751 2.34376i 0 −4.91111 + 1.04389i 0
81.3 0 −0.272347 2.59120i 0 1.41505 + 1.73137i 0 −0.381973 + 2.61803i 0 −3.70573 + 0.787676i 0
81.4 0 −0.260227 2.47589i 0 −1.74304 + 1.40065i 0 2.53743 0.749316i 0 −3.12789 + 0.664854i 0
81.5 0 −0.221467 2.10712i 0 −2.22616 0.210296i 0 1.31775 + 2.29424i 0 −1.45644 + 0.309577i 0
81.6 0 −0.164810 1.56806i 0 1.06004 1.96884i 0 2.59162 0.532438i 0 0.502795 0.106872i 0
81.7 0 −0.115689 1.10070i 0 −0.0767094 2.23475i 0 −0.987418 + 2.45459i 0 1.73628 0.369057i 0
81.8 0 −0.0806619 0.767447i 0 2.11029 0.739383i 0 0.187493 2.63910i 0 2.35197 0.499928i 0
81.9 0 −0.0610054 0.580427i 0 2.23452 + 0.0832050i 0 −2.51126 0.832804i 0 2.60127 0.552917i 0
81.10 0 −0.0323219 0.307522i 0 −2.23362 0.104508i 0 −2.61097 + 0.427582i 0 2.84092 0.603856i 0
81.11 0 −0.0229598 0.218448i 0 1.54824 + 1.61337i 0 2.37030 + 1.17545i 0 2.88725 0.613704i 0
81.12 0 0.0118737 + 0.112971i 0 −1.67957 + 1.47616i 0 1.14088 2.38713i 0 2.92182 0.621052i 0
81.13 0 0.160475 + 1.52682i 0 −2.09374 0.785010i 0 1.96337 + 1.77347i 0 0.629027 0.133704i 0
81.14 0 0.162396 + 1.54509i 0 0.272993 + 2.21934i 0 −0.0456598 2.64536i 0 0.573508 0.121903i 0
81.15 0 0.173464 + 1.65040i 0 0.0779163 2.23471i 0 2.53599 0.754166i 0 0.240704 0.0511632i 0
81.16 0 0.180708 + 1.71932i 0 −0.0662493 2.23509i 0 −2.63897 0.189349i 0 0.0110297 0.00234443i 0
81.17 0 0.212873 + 2.02535i 0 2.17143 0.533747i 0 −0.472995 + 2.60313i 0 −1.12230 + 0.238553i 0
81.18 0 0.225525 + 2.14572i 0 −0.830377 + 2.07617i 0 −0.864256 + 2.50061i 0 −1.61882 + 0.344092i 0
81.19 0 0.340003 + 3.23492i 0 −2.21544 0.302996i 0 −1.66018 2.06005i 0 −7.41464 + 1.57603i 0
81.20 0 0.357649 + 3.40280i 0 2.12228 + 0.704219i 0 2.44622 1.00797i 0 −8.51670 + 1.81028i 0
See next 80 embeddings (of 160 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 81.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
25.d even 5 1 inner
175.q even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 700.2.bg.a 160
7.c even 3 1 inner 700.2.bg.a 160
25.d even 5 1 inner 700.2.bg.a 160
175.q even 15 1 inner 700.2.bg.a 160
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
700.2.bg.a 160 1.a even 1 1 trivial
700.2.bg.a 160 7.c even 3 1 inner
700.2.bg.a 160 25.d even 5 1 inner
700.2.bg.a 160 175.q even 15 1 inner

Hecke kernels

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