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Properties

Label 725.2.q.e
Level $725$
Weight $2$
Character orbit 725.q
Analytic conductor $5.789$
Analytic rank $0$
Dimension $60$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [725,2,Mod(51,725)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(725, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([0, 13]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("725.51");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 725 = 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 725.q (of order \(14\), degree \(6\), 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: \(5.78915414654\)
Analytic rank: \(0\)
Dimension: \(60\)
Relative dimension: \(10\) over \(\Q(\zeta_{14})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 60 q + 12 q^{4} - 16 q^{6} + 4 q^{7} + 21 q^{8} + 10 q^{9} + 14 q^{11} - 4 q^{13} + 10 q^{16} - 35 q^{22} + 37 q^{23} + 48 q^{24} - 21 q^{27} - 44 q^{28} - 4 q^{29} + 14 q^{31} - 98 q^{32} + 41 q^{33} + 10 q^{34}+ \cdots - 91 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
51.1 −2.52619 + 0.576587i −0.752390 + 1.56235i 4.24725 − 2.04537i 0 0.999849 − 4.38063i 2.08151 + 1.00240i −5.49834 + 4.38478i −0.00439215 − 0.00550758i 0
51.2 −2.36797 + 0.540475i 1.05066 − 2.18173i 3.51325 − 1.69189i 0 −1.30878 + 5.73413i −0.976330 − 0.470176i −3.60693 + 2.87643i −1.78556 − 2.23903i 0
51.3 −1.51798 + 0.346469i −0.366652 + 0.761361i 0.382281 − 0.184097i 0 0.292782 − 1.28276i −3.29425 − 1.58643i 1.91814 − 1.52966i 1.42523 + 1.78719i 0
51.4 −1.11356 + 0.254163i 0.486219 − 1.00964i −0.626516 + 0.301714i 0 −0.284821 + 1.24788i 3.58912 + 1.72843i 2.40699 − 1.91951i 1.08750 + 1.36368i 0
51.5 −0.334311 + 0.0763043i −1.46374 + 3.03949i −1.69600 + 0.816749i 0 0.257418 − 1.12782i 0.936929 + 0.451201i 1.04086 − 0.830059i −5.22548 − 6.55255i 0
51.6 0.00717851 − 0.00163845i −0.170304 + 0.353640i −1.80189 + 0.867744i 0 −0.000643109 0.00281765i −1.03412 − 0.498007i −0.0230266 + 0.0183631i 1.77441 + 2.22504i 0
51.7 0.714081 − 0.162984i 0.656356 − 1.36294i −1.31859 + 0.635000i 0 0.246554 − 1.08022i 0.573111 + 0.275996i −1.98338 + 1.58169i 0.443675 + 0.556351i 0
51.8 1.68193 − 0.383889i −0.800342 + 1.66193i 0.879573 − 0.423580i 0 −0.708122 + 3.10248i 0.672793 + 0.324000i −1.38083 + 1.10118i −0.250983 − 0.314723i 0
51.9 2.09792 − 0.478836i 1.31670 − 2.73416i 2.37003 − 1.14135i 0 1.45312 − 6.36653i 3.53937 + 1.70447i 1.06081 − 0.845968i −3.87148 − 4.85468i 0
51.10 2.51290 − 0.573554i 0.0434857 − 0.0902990i 4.18378 − 2.01480i 0 0.0574840 − 0.251854i −2.48426 − 1.19636i 5.32746 − 4.24851i 1.86421 + 2.33764i 0
151.1 −1.84674 − 1.47273i −1.74149 − 0.397484i 0.796488 + 3.48964i 0 2.63070 + 3.29879i 0.432997 − 1.89709i 1.61865 − 3.36117i 0.171885 + 0.0827756i 0
151.2 −1.67511 − 1.33586i 1.23313 + 0.281454i 0.576441 + 2.52555i 0 −1.68965 − 2.11875i −0.897887 + 3.93390i 0.548942 − 1.13989i −1.26151 − 0.607511i 0
151.3 −1.18203 − 0.942638i 1.32937 + 0.303420i 0.0635887 + 0.278601i 0 −1.28534 − 1.61177i 0.435729 − 1.90905i −1.12450 + 2.33505i −1.02775 − 0.494937i 0
151.4 −0.187413 − 0.149457i −0.702150 − 0.160261i −0.432256 − 1.89384i 0 0.107640 + 0.134976i −0.331102 + 1.45065i −0.410048 + 0.851474i −2.23558 − 1.07660i 0
151.5 −0.0774561 − 0.0617692i 2.30681 + 0.526514i −0.442858 − 1.94029i 0 −0.146154 − 0.183271i 0.506653 − 2.21979i −0.171518 + 0.356160i 2.34124 + 1.12748i 0
151.6 0.584832 + 0.466388i −3.00267 − 0.685340i −0.320531 − 1.40434i 0 −1.43643 − 1.80122i 1.10112 − 4.82431i 1.11662 − 2.31870i 5.84344 + 2.81405i 0
151.7 1.17272 + 0.935212i −1.17572 − 0.268351i 0.0556064 + 0.243627i 0 −1.12783 − 1.41425i −0.494498 + 2.16654i 1.13899 − 2.36513i −1.39260 − 0.670639i 0
151.8 1.19581 + 0.953623i 2.47243 + 0.564317i 0.0755119 + 0.330839i 0 2.41840 + 3.03258i −0.919048 + 4.02661i 1.10205 − 2.28842i 3.09157 + 1.48882i 0
151.9 1.47919 + 1.17961i 1.35735 + 0.309807i 0.351469 + 1.53989i 0 1.64233 + 2.05942i 1.09691 − 4.80589i 0.345193 − 0.716800i −0.956476 − 0.460614i 0
151.10 2.06066 + 1.64333i −2.07706 − 0.474076i 1.10078 + 4.82283i 0 −3.50107 − 4.39020i −0.0407903 + 0.178714i −3.36998 + 6.99783i 1.38654 + 0.667721i 0
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 51.10
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.e even 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 725.2.q.e yes 60
5.b even 2 1 725.2.q.d ✓ 60
5.c odd 4 2 725.2.p.d 120
29.e even 14 1 inner 725.2.q.e yes 60
145.l even 14 1 725.2.q.d ✓ 60
145.q odd 28 2 725.2.p.d 120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
725.2.p.d 120 5.c odd 4 2
725.2.p.d 120 145.q odd 28 2
725.2.q.d ✓ 60 5.b even 2 1
725.2.q.d ✓ 60 145.l even 14 1
725.2.q.e yes 60 1.a even 1 1 trivial
725.2.q.e yes 60 29.e even 14 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{60} - 16 T_{2}^{58} - 7 T_{2}^{57} + 153 T_{2}^{56} + 182 T_{2}^{55} - 1268 T_{2}^{54} - 2191 T_{2}^{53} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(725, [\chi])\). Copy content Toggle raw display