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Properties

Label 891.2.u.d
Level $891$
Weight $2$
Character orbit 891.u
Analytic conductor $7.115$
Analytic rank $0$
Dimension $32$
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] = [891,2,Mod(107,891)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(891, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([5, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("891.107");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 891 = 3^{4} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 891.u (of order \(30\), degree \(8\), 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: \(7.11467082010\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(4\) over \(\Q(\zeta_{30})\)
Twist minimal: no (minimal twist has level 297)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 4 q^{4} + 15 q^{5}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 4 q^{4} + 15 q^{5} + 33 q^{11} - 12 q^{14} - 16 q^{16} - 36 q^{20} + 15 q^{22} + 18 q^{23} - 13 q^{25} - 10 q^{28} + 15 q^{29} + 8 q^{31} - 22 q^{34} + 18 q^{37} - 105 q^{38} - 15 q^{40} - 75 q^{41} + 40 q^{46} + 36 q^{47} + 12 q^{49} + 15 q^{50} + 40 q^{52} - 16 q^{55} + 60 q^{56} + 24 q^{58} - 39 q^{59} + 30 q^{61} + 48 q^{67} + 165 q^{68} + 41 q^{70} - 70 q^{73} - 15 q^{74} - 42 q^{77} + 60 q^{79} - 66 q^{82} - 60 q^{83} - 80 q^{85} + 6 q^{86} - 24 q^{88} - 60 q^{91} + 42 q^{92} - 100 q^{94} - 60 q^{95} - 24 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1 −0.189255 − 1.80064i 0 −1.25021 + 0.265739i 4.27136 + 0.448938i 0 2.14141 − 1.92813i −0.403879 − 1.24301i 0 − 7.77615i
107.2 −0.151369 − 1.44018i 0 −0.0949044 + 0.0201726i −2.18492 − 0.229644i 0 −2.71309 + 2.44288i −0.851564 − 2.62084i 0 3.18143i
107.3 0.0855419 + 0.813877i 0 1.30122 − 0.276582i 1.55568 + 0.163508i 0 −0.725520 + 0.653261i 0.842187 + 2.59198i 0 1.28012i
107.4 0.255082 + 2.42694i 0 −3.86870 + 0.822317i 0.209653 + 0.0220354i 0 0.423586 − 0.381399i −1.47436 − 4.53760i 0 0.514437i
134.1 −2.00789 + 0.426789i 0 2.02236 − 0.900414i −0.370816 + 1.74455i 0 0.305114 + 0.0320688i −0.354978 + 0.257906i 0 − 3.66112i
134.2 −0.775410 + 0.164818i 0 −1.25300 + 0.557869i 0.689350 − 3.24314i 0 4.74370 + 0.498583i 2.16231 − 1.57101i 0 2.62838i
134.3 0.172007 − 0.0365613i 0 −1.79884 + 0.800896i −0.384570 + 1.80926i 0 −3.33045 − 0.350045i −0.564664 + 0.410252i 0 0.325266i
134.4 2.61129 − 0.555046i 0 4.68366 − 2.08530i −0.739203 + 3.47767i 0 0.173335 + 0.0182182i 6.75339 − 4.90663i 0 9.49150i
215.1 −1.37355 + 1.52548i 0 −0.231400 − 2.20162i 1.32542 − 1.19341i 0 −0.124785 + 0.280271i 0.354978 + 0.257906i 0 3.66112i
215.2 −0.530442 + 0.589116i 0 0.143369 + 1.36406i −2.46396 + 2.21856i 0 −1.94006 + 4.35745i −2.16231 − 1.57101i 0 − 2.62838i
215.3 0.117667 − 0.130682i 0 0.205825 + 1.95829i 1.37458 − 1.23768i 0 1.36208 − 3.05928i 0.564664 + 0.410252i 0 − 0.325266i
215.4 1.78633 − 1.98392i 0 −0.535907 − 5.09881i 2.64215 − 2.37901i 0 −0.0708899 + 0.159221i −6.75339 − 4.90663i 0 − 9.49150i
431.1 −1.37355 − 1.52548i 0 −0.231400 + 2.20162i 1.32542 + 1.19341i 0 −0.124785 − 0.280271i 0.354978 − 0.257906i 0 − 3.66112i
431.2 −0.530442 − 0.589116i 0 0.143369 − 1.36406i −2.46396 − 2.21856i 0 −1.94006 − 4.35745i −2.16231 + 1.57101i 0 2.62838i
431.3 0.117667 + 0.130682i 0 0.205825 − 1.95829i 1.37458 + 1.23768i 0 1.36208 + 3.05928i 0.564664 − 0.410252i 0 0.325266i
431.4 1.78633 + 1.98392i 0 −0.535907 + 5.09881i 2.64215 + 2.37901i 0 −0.0708899 − 0.159221i −6.75339 + 4.90663i 0 9.49150i
458.1 −0.189255 + 1.80064i 0 −1.25021 − 0.265739i 4.27136 − 0.448938i 0 2.14141 + 1.92813i −0.403879 + 1.24301i 0 7.77615i
458.2 −0.151369 + 1.44018i 0 −0.0949044 − 0.0201726i −2.18492 + 0.229644i 0 −2.71309 − 2.44288i −0.851564 + 2.62084i 0 − 3.18143i
458.3 0.0855419 − 0.813877i 0 1.30122 + 0.276582i 1.55568 − 0.163508i 0 −0.725520 − 0.653261i 0.842187 − 2.59198i 0 − 1.28012i
458.4 0.255082 − 2.42694i 0 −3.86870 − 0.822317i 0.209653 − 0.0220354i 0 0.423586 + 0.381399i −1.47436 + 4.53760i 0 − 0.514437i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 107.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner
11.d odd 10 1 inner
99.p even 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 891.2.u.d 32
3.b odd 2 1 891.2.u.b 32
9.c even 3 1 297.2.k.b ✓ 32
9.c even 3 1 891.2.u.b 32
9.d odd 6 1 297.2.k.b ✓ 32
9.d odd 6 1 inner 891.2.u.d 32
11.d odd 10 1 inner 891.2.u.d 32
33.f even 10 1 891.2.u.b 32
99.o odd 30 1 297.2.k.b ✓ 32
99.o odd 30 1 891.2.u.b 32
99.p even 30 1 297.2.k.b ✓ 32
99.p even 30 1 inner 891.2.u.d 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
297.2.k.b ✓ 32 9.c even 3 1
297.2.k.b ✓ 32 9.d odd 6 1
297.2.k.b ✓ 32 99.o odd 30 1
297.2.k.b ✓ 32 99.p even 30 1
891.2.u.b 32 3.b odd 2 1
891.2.u.b 32 9.c even 3 1
891.2.u.b 32 33.f even 10 1
891.2.u.b 32 99.o odd 30 1
891.2.u.d 32 1.a even 1 1 trivial
891.2.u.d 32 9.d odd 6 1 inner
891.2.u.d 32 11.d odd 10 1 inner
891.2.u.d 32 99.p even 30 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{32} - 6 T_{2}^{30} + 8 T_{2}^{28} + 72 T_{2}^{26} - 30 T_{2}^{25} - 842 T_{2}^{24} + 180 T_{2}^{23} + \cdots + 256 \) acting on \(S_{2}^{\mathrm{new}}(891, [\chi])\). Copy content Toggle raw display