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Properties

Label 891.2.n.h
Level $891$
Weight $2$
Character orbit 891.n
Analytic conductor $7.115$
Analytic rank $0$
Dimension $32$
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [891,2,Mod(136,891)]
 
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([20, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("891.136");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 891 = 3^{4} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 891.n (of order \(15\), 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 \(\Q(\zeta_{15})\)
Twist minimal: no (minimal twist has level 297)
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 10 q^{4} + 24 q^{10} + 10 q^{16} - 4 q^{19} + 36 q^{22} - 32 q^{25} + 84 q^{28} + 26 q^{31} + 48 q^{34} - 48 q^{37} + 20 q^{40} - 24 q^{43} - 32 q^{46} - 24 q^{49} + 40 q^{52} - 32 q^{55} - 106 q^{58}+ \cdots - 72 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
136.1 −2.09957 + 0.934789i 0 2.19611 − 2.43902i −1.55025 − 0.690215i 0 −0.203124 − 0.0431753i −0.910502 + 2.80224i 0 3.90006
136.2 −0.288796 + 0.128580i 0 −1.27139 + 1.41202i 3.70483 + 1.64949i 0 −3.33585 − 0.709056i 0.380991 − 1.17257i 0 −1.28203
136.3 0.288796 − 0.128580i 0 −1.27139 + 1.41202i −3.70483 − 1.64949i 0 −3.33585 − 0.709056i −0.380991 + 1.17257i 0 −1.28203
136.4 2.09957 − 0.934789i 0 2.19611 − 2.43902i 1.55025 + 0.690215i 0 −0.203124 − 0.0431753i 0.910502 − 2.80224i 0 3.90006
190.1 −2.09957 − 0.934789i 0 2.19611 + 2.43902i −1.55025 + 0.690215i 0 −0.203124 + 0.0431753i −0.910502 − 2.80224i 0 3.90006
190.2 −0.288796 − 0.128580i 0 −1.27139 − 1.41202i 3.70483 − 1.64949i 0 −3.33585 + 0.709056i 0.380991 + 1.17257i 0 −1.28203
190.3 0.288796 + 0.128580i 0 −1.27139 − 1.41202i −3.70483 + 1.64949i 0 −3.33585 + 0.709056i −0.380991 − 1.17257i 0 −1.28203
190.4 2.09957 + 0.934789i 0 2.19611 + 2.43902i 1.55025 − 0.690215i 0 −0.203124 + 0.0431753i 0.910502 + 2.80224i 0 3.90006
379.1 −2.40657 − 0.511531i 0 3.70281 + 1.64860i −0.968153 + 0.205787i 0 −0.451792 − 4.29851i −4.08684 − 2.96926i 0 2.43519
379.2 −1.22359 − 0.260081i 0 −0.397568 − 0.177009i 1.60550 − 0.341260i 0 0.307337 + 2.92412i 2.46446 + 1.79053i 0 −2.05323
379.3 1.22359 + 0.260081i 0 −0.397568 − 0.177009i −1.60550 + 0.341260i 0 0.307337 + 2.92412i −2.46446 − 1.79053i 0 −2.05323
379.4 2.40657 + 0.511531i 0 3.70281 + 1.64860i 0.968153 − 0.205787i 0 −0.451792 − 4.29851i 4.08684 + 2.96926i 0 2.43519
433.1 −0.240234 + 2.28568i 0 −3.21031 − 0.682372i −0.177381 − 1.68766i 0 0.138953 − 0.154323i 0.910502 − 2.80224i 0 3.90006
433.2 −0.0330442 + 0.314395i 0 1.85854 + 0.395046i 0.423909 + 4.03322i 0 2.28198 − 2.53440i −0.380991 + 1.17257i 0 −1.28203
433.3 0.0330442 − 0.314395i 0 1.85854 + 0.395046i −0.423909 − 4.03322i 0 2.28198 − 2.53440i 0.380991 − 1.17257i 0 −1.28203
433.4 0.240234 − 2.28568i 0 −3.21031 − 0.682372i 0.177381 + 1.68766i 0 0.138953 − 0.154323i −0.910502 + 2.80224i 0 3.90006
460.1 −1.64628 − 1.82838i 0 −0.423677 + 4.03102i −0.662294 + 0.735552i 0 3.94852 − 1.75799i 4.08684 − 2.96926i 0 2.43519
460.2 −0.837031 − 0.929617i 0 0.0454900 − 0.432808i 1.09829 − 1.21977i 0 −2.68603 + 1.19590i −2.46446 + 1.79053i 0 −2.05323
460.3 0.837031 + 0.929617i 0 0.0454900 − 0.432808i −1.09829 + 1.21977i 0 −2.68603 + 1.19590i 2.46446 − 1.79053i 0 −2.05323
460.4 1.64628 + 1.82838i 0 −0.423677 + 4.03102i 0.662294 − 0.735552i 0 3.94852 − 1.75799i −4.08684 + 2.96926i 0 2.43519
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 136.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner
11.c even 5 1 inner
33.h odd 10 1 inner
99.m even 15 1 inner
99.n odd 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 891.2.n.h 32
3.b odd 2 1 inner 891.2.n.h 32
9.c even 3 1 297.2.f.b ✓ 16
9.c even 3 1 inner 891.2.n.h 32
9.d odd 6 1 297.2.f.b ✓ 16
9.d odd 6 1 inner 891.2.n.h 32
11.c even 5 1 inner 891.2.n.h 32
33.h odd 10 1 inner 891.2.n.h 32
99.m even 15 1 297.2.f.b ✓ 16
99.m even 15 1 inner 891.2.n.h 32
99.m even 15 1 3267.2.a.bj 8
99.n odd 30 1 297.2.f.b ✓ 16
99.n odd 30 1 inner 891.2.n.h 32
99.n odd 30 1 3267.2.a.bj 8
99.o odd 30 1 3267.2.a.bi 8
99.p even 30 1 3267.2.a.bi 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
297.2.f.b ✓ 16 9.c even 3 1
297.2.f.b ✓ 16 9.d odd 6 1
297.2.f.b ✓ 16 99.m even 15 1
297.2.f.b ✓ 16 99.n odd 30 1
891.2.n.h 32 1.a even 1 1 trivial
891.2.n.h 32 3.b odd 2 1 inner
891.2.n.h 32 9.c even 3 1 inner
891.2.n.h 32 9.d odd 6 1 inner
891.2.n.h 32 11.c even 5 1 inner
891.2.n.h 32 33.h odd 10 1 inner
891.2.n.h 32 99.m even 15 1 inner
891.2.n.h 32 99.n odd 30 1 inner
3267.2.a.bi 8 99.o odd 30 1
3267.2.a.bi 8 99.p even 30 1
3267.2.a.bj 8 99.m even 15 1
3267.2.a.bj 8 99.n odd 30 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{32} - 9 T_{2}^{30} + 30 T_{2}^{28} + 39 T_{2}^{26} - 1116 T_{2}^{24} + 10994 T_{2}^{22} + \cdots + 625 \) acting on \(S_{2}^{\mathrm{new}}(891, [\chi])\). Copy content Toggle raw display