↑

Properties

Label 1755.2.i.h
Level $1755$
Weight $2$
Character orbit 1755.i
Analytic conductor $14.014$
Analytic rank $0$
Dimension $30$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1755,2,Mod(586,1755)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1755, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1755.586");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1755 = 3^{3} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1755.i (of order \(3\), degree \(2\), not 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: \(14.0137455547\)
Analytic rank: \(0\)
Dimension: \(30\)
Relative dimension: \(15\) over \(\Q(\zeta_{3})\)
Twist minimal: no (minimal twist has level 585)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 30 q - q^{2} - 21 q^{4} - 15 q^{5} - 10 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q - q^{2} - 21 q^{4} - 15 q^{5} - 10 q^{7} + 2 q^{10} - 9 q^{11} - 15 q^{13} - 3 q^{14} - 33 q^{16} - 6 q^{17} + 30 q^{19} - 21 q^{20} - 10 q^{22} + 6 q^{23} - 15 q^{25} + 2 q^{26} + 70 q^{28} - 8 q^{29} - 22 q^{31} - 21 q^{32} - 9 q^{34} + 20 q^{35} + 8 q^{37} + 14 q^{38} - 13 q^{41} - 24 q^{43} - 10 q^{44} - 6 q^{46} + q^{47} - 37 q^{49} - q^{50} - 21 q^{52} - 14 q^{53} + 18 q^{55} - 17 q^{56} - 22 q^{58} - 19 q^{59} - 16 q^{61} - 26 q^{62} + 72 q^{64} - 15 q^{65} - 11 q^{67} + 28 q^{68} - 3 q^{70} + 56 q^{71} + 52 q^{73} - 8 q^{74} - 18 q^{76} + 24 q^{77} - 44 q^{79} + 66 q^{80} + 70 q^{82} + 3 q^{83} + 3 q^{85} - 40 q^{86} - 37 q^{88} + 8 q^{89} + 20 q^{91} + 74 q^{92} - 2 q^{94} - 15 q^{95} - 33 q^{97} - 6 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
586.1 −1.38695 + 2.40227i 0 −2.84726 − 4.93159i −0.500000 − 0.866025i 0 −2.15445 + 3.73162i 10.2482 0 2.77390
586.2 −1.26258 + 2.18685i 0 −2.18820 − 3.79007i −0.500000 − 0.866025i 0 0.799357 − 1.38453i 6.00077 0 2.52515
586.3 −1.17680 + 2.03828i 0 −1.76973 − 3.06526i −0.500000 − 0.866025i 0 0.0389983 − 0.0675470i 3.62327 0 2.35360
586.4 −0.909537 + 1.57536i 0 −0.654515 − 1.13365i −0.500000 − 0.866025i 0 −1.45386 + 2.51816i −1.25693 0 1.81907
586.5 −0.628641 + 1.08884i 0 0.209620 + 0.363073i −0.500000 − 0.866025i 0 −2.28445 + 3.95678i −3.04167 0 1.25728
586.6 −0.562020 + 0.973448i 0 0.368266 + 0.637856i −0.500000 − 0.866025i 0 1.91433 − 3.31571i −3.07597 0 1.12404
586.7 −0.300757 + 0.520926i 0 0.819091 + 1.41871i −0.500000 − 0.866025i 0 1.29950 − 2.25080i −2.18842 0 0.601514
586.8 −0.210096 + 0.363896i 0 0.911720 + 1.57914i −0.500000 − 0.866025i 0 0.421896 − 0.730745i −1.60658 0 0.420191
586.9 0.264400 − 0.457954i 0 0.860185 + 1.48988i −0.500000 − 0.866025i 0 −1.64961 + 2.85721i 1.96733 0 −0.528800
586.10 0.332241 − 0.575458i 0 0.779232 + 1.34967i −0.500000 − 0.866025i 0 −1.40460 + 2.43283i 2.36453 0 −0.664482
586.11 0.644173 − 1.11574i 0 0.170081 + 0.294590i −0.500000 − 0.866025i 0 1.40888 − 2.44026i 3.01494 0 −1.28835
586.12 1.01209 − 1.75298i 0 −1.04864 − 1.81629i −0.500000 − 0.866025i 0 −1.83320 + 3.17519i −0.196897 0 −2.02417
586.13 1.09333 − 1.89370i 0 −1.39073 − 2.40882i −0.500000 − 0.866025i 0 2.12668 − 3.68352i −1.70880 0 −2.18666
586.14 1.26000 − 2.18239i 0 −2.17520 − 3.76756i −0.500000 − 0.866025i 0 −1.50610 + 2.60865i −5.92303 0 −2.52000
586.15 1.33115 − 2.30562i 0 −2.54392 − 4.40620i −0.500000 − 0.866025i 0 −0.723369 + 1.25291i −8.22077 0 −2.66230
1171.1 −1.38695 − 2.40227i 0 −2.84726 + 4.93159i −0.500000 + 0.866025i 0 −2.15445 − 3.73162i 10.2482 0 2.77390
1171.2 −1.26258 − 2.18685i 0 −2.18820 + 3.79007i −0.500000 + 0.866025i 0 0.799357 + 1.38453i 6.00077 0 2.52515
1171.3 −1.17680 − 2.03828i 0 −1.76973 + 3.06526i −0.500000 + 0.866025i 0 0.0389983 + 0.0675470i 3.62327 0 2.35360
1171.4 −0.909537 − 1.57536i 0 −0.654515 + 1.13365i −0.500000 + 0.866025i 0 −1.45386 − 2.51816i −1.25693 0 1.81907
1171.5 −0.628641 − 1.08884i 0 0.209620 − 0.363073i −0.500000 + 0.866025i 0 −2.28445 − 3.95678i −3.04167 0 1.25728
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 586.15
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1755.2.i.h 30
3.b odd 2 1 585.2.i.h ✓ 30
9.c even 3 1 inner 1755.2.i.h 30
9.c even 3 1 5265.2.a.bl 15
9.d odd 6 1 585.2.i.h ✓ 30
9.d odd 6 1 5265.2.a.bk 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
585.2.i.h ✓ 30 3.b odd 2 1
585.2.i.h ✓ 30 9.d odd 6 1
1755.2.i.h 30 1.a even 1 1 trivial
1755.2.i.h 30 9.c even 3 1 inner
5265.2.a.bk 15 9.d odd 6 1
5265.2.a.bl 15 9.c even 3 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1755, [\chi])\):

\( T_{2}^{30} + T_{2}^{29} + 26 T_{2}^{28} + 23 T_{2}^{27} + 405 T_{2}^{26} + 338 T_{2}^{25} + 4110 T_{2}^{24} + \cdots + 20736 \) Copy content Toggle raw display
\( T_{7}^{30} + 10 T_{7}^{29} + 121 T_{7}^{28} + 770 T_{7}^{27} + 5910 T_{7}^{26} + 30389 T_{7}^{25} + \cdots + 11314151424 \) Copy content Toggle raw display