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Properties

Label 1001.2.d.c
Level $1001$
Weight $2$
Character orbit 1001.d
Analytic conductor $7.993$
Analytic rank $0$
Dimension $40$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1001,2,Mod(155,1001)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1001, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1001.155");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1001 = 7 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1001.d (of order \(2\), degree \(1\), 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: \(7.99302524233\)
Analytic rank: \(0\)
Dimension: \(40\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 12 q^{3} - 50 q^{4} + 52 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q + 12 q^{3} - 50 q^{4} + 52 q^{9} - 24 q^{10} - 12 q^{12} - 2 q^{13} + 6 q^{14} + 70 q^{16} + 6 q^{22} - 22 q^{23} - 62 q^{25} + 20 q^{26} + 72 q^{27} - 2 q^{29} - 96 q^{30} - 2 q^{35} - 98 q^{36} + 44 q^{38} + 12 q^{39} + 72 q^{40} + 4 q^{42} - 78 q^{43} + 88 q^{48} - 40 q^{49} + 4 q^{51} + 10 q^{52} + 30 q^{53} - 2 q^{55} - 30 q^{56} + 4 q^{62} - 74 q^{64} + 34 q^{65} + 4 q^{66} - 100 q^{68} - 60 q^{69} + 16 q^{74} - 80 q^{75} - 40 q^{77} + 76 q^{78} + 46 q^{79} + 112 q^{81} - 24 q^{82} + 4 q^{87} - 30 q^{88} + 80 q^{90} - 6 q^{91} + 60 q^{92} + 24 q^{94} - 54 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
155.1 − 2.82091i −0.805070 −5.95752 − 3.58599i 2.27103i 1.00000i 11.1638i −2.35186 −10.1157
155.2 − 2.71355i 3.00445 −5.36334 − 3.32321i − 8.15272i − 1.00000i 9.12658i 6.02672 −9.01769
155.3 − 2.62771i 3.15598 −4.90488 2.25021i − 8.29302i 1.00000i 7.63321i 6.96022 5.91291
155.4 − 2.40283i −2.60261 −3.77359 3.37332i 6.25363i 1.00000i 4.26164i 3.77357 8.10552
155.5 − 2.32366i −0.856155 −3.39939 − 3.23368i 1.98941i − 1.00000i 3.25171i −2.26700 −7.51397
155.6 − 2.28880i −1.09659 −3.23859 1.64284i 2.50988i − 1.00000i 2.83488i −1.79749 3.76012
155.7 − 2.25762i 1.73265 −3.09686 − 2.19658i − 3.91167i 1.00000i 2.47630i 0.00207277 −4.95904
155.8 − 2.07366i −1.87645 −2.30008 0.0660844i 3.89113i 1.00000i 0.622272i 0.521074 0.137037
155.9 − 2.05335i −0.142660 −2.21626 1.42214i 0.292931i 1.00000i 0.444061i −2.97965 2.92015
155.10 − 1.86279i −2.59577 −1.46999 2.75718i 4.83538i − 1.00000i − 0.987300i 3.73803 5.13605
155.11 − 1.64253i 2.84243 −0.697905 − 0.421993i − 4.66879i − 1.00000i − 2.13873i 5.07944 −0.693136
155.12 − 1.39136i 0.232646 0.0641059 − 0.592388i − 0.323695i − 1.00000i − 2.87192i −2.94588 −0.824228
155.13 − 1.25823i 3.20372 0.416855 − 3.76829i − 4.03102i 1.00000i − 3.04096i 7.26383 −4.74138
155.14 − 0.932838i 2.05268 1.12981 3.23603i − 1.91481i − 1.00000i − 2.91961i 1.21348 3.01869
155.15 − 0.907716i 1.17354 1.17605 − 0.404680i − 1.06524i 1.00000i − 2.88295i −1.62281 −0.367334
155.16 − 0.801051i −3.20290 1.35832 − 0.640436i 2.56569i − 1.00000i − 2.69018i 7.25855 −0.513022
155.17 − 0.560617i 2.73380 1.68571 2.40028i − 1.53261i 1.00000i − 2.06627i 4.47364 1.34564
155.18 − 0.476896i 0.895425 1.77257 − 4.36314i − 0.427024i − 1.00000i − 1.79912i −2.19821 −2.08076
155.19 − 0.427983i −1.19164 1.81683 − 3.53133i 0.510001i 1.00000i − 1.63354i −1.58000 −1.51135
155.20 − 0.0428332i −0.657470 1.99817 0.0360367i 0.0281616i 1.00000i − 0.171254i −2.56773 0.00154357
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 155.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1001.2.d.c ✓ 40
13.b even 2 1 inner 1001.2.d.c ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1001.2.d.c ✓ 40 1.a even 1 1 trivial
1001.2.d.c ✓ 40 13.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{40} + 65 T_{2}^{38} + 1940 T_{2}^{36} + 35254 T_{2}^{34} + 436094 T_{2}^{32} + 3889320 T_{2}^{30} + \cdots + 1936 \) acting on \(S_{2}^{\mathrm{new}}(1001, [\chi])\). Copy content Toggle raw display