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Properties

Label 891.2.n.f
Level $891$
Weight $2$
Character orbit 891.n
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(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 dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 2 q^{2} + 4 q^{4} - q^{5} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 2 q^{2} + 4 q^{4} - q^{5} + 2 q^{7} + 12 q^{10} - 13 q^{11} + 2 q^{13} + 22 q^{14} + 24 q^{16} - 4 q^{17} - 4 q^{19} - 15 q^{22} - 14 q^{23} + 19 q^{25} + 42 q^{26} + 30 q^{28} - q^{29} - 14 q^{31} + 48 q^{32} - 10 q^{34} - 36 q^{35} + 18 q^{37} - 11 q^{38} - 33 q^{40} - 25 q^{41} - 14 q^{43} + 28 q^{44} + 8 q^{46} + 28 q^{47} + 4 q^{49} + 63 q^{50} - 10 q^{52} + 2 q^{53} - 80 q^{55} - 96 q^{56} + 20 q^{58} - 41 q^{59} - 10 q^{62} - 184 q^{64} + 60 q^{65} + 48 q^{67} - 25 q^{68} + 31 q^{70} + 6 q^{71} - 26 q^{73} - 29 q^{74} + 58 q^{76} + 2 q^{77} - 166 q^{80} + 82 q^{82} + 14 q^{83} + 10 q^{85} + 56 q^{86} - 86 q^{88} + 164 q^{89} + 28 q^{91} - 74 q^{92} + 2 q^{94} + 56 q^{95} - 12 q^{97} + 52 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
136.1 −2.53854 + 1.13023i 0 3.82850 − 4.25198i −0.924300 − 0.411525i 0 1.86094 + 0.395555i −3.19570 + 9.83534i 0 2.81149
136.2 −1.11141 + 0.494834i 0 −0.347879 + 0.386359i −0.271292 − 0.120787i 0 −4.20378 − 0.893541i 0.947351 − 2.91565i 0 0.361288
136.3 0.330935 − 0.147342i 0 −1.25045 + 1.38877i −2.45353 − 1.09238i 0 2.43963 + 0.518560i −0.433081 + 1.33288i 0 −0.972914
136.4 1.49193 − 0.664249i 0 0.446358 − 0.495731i 2.73558 + 1.21796i 0 1.48588 + 0.315834i −0.672677 + 2.07029i 0 4.89031
190.1 −2.53854 − 1.13023i 0 3.82850 + 4.25198i −0.924300 + 0.411525i 0 1.86094 − 0.395555i −3.19570 − 9.83534i 0 2.81149
190.2 −1.11141 − 0.494834i 0 −0.347879 − 0.386359i −0.271292 + 0.120787i 0 −4.20378 + 0.893541i 0.947351 + 2.91565i 0 0.361288
190.3 0.330935 + 0.147342i 0 −1.25045 − 1.38877i −2.45353 + 1.09238i 0 2.43963 − 0.518560i −0.433081 − 1.33288i 0 −0.972914
190.4 1.49193 + 0.664249i 0 0.446358 + 0.495731i 2.73558 − 1.21796i 0 1.48588 − 0.315834i −0.672677 − 2.07029i 0 4.89031
379.1 −1.79145 − 0.380785i 0 1.23721 + 0.550844i 1.63692 − 0.347939i 0 −0.432236 − 4.11245i 0.956730 + 0.695105i 0 −3.06496
379.2 −0.260096 − 0.0552850i 0 −1.76250 − 0.784714i −1.07651 + 0.228819i 0 0.161379 + 1.53541i 0.845280 + 0.614132i 0 0.292645
379.3 1.70989 + 0.363447i 0 0.964524 + 0.429434i 3.77644 − 0.802707i 0 −0.0917633 − 0.873070i −1.33531 − 0.970162i 0 6.74902
379.4 2.29796 + 0.488446i 0 3.21494 + 1.43138i −3.35871 + 0.713915i 0 0.298019 + 2.83546i 2.88741 + 2.09782i 0 −8.06687
433.1 −0.170707 + 1.62417i 0 −0.652494 − 0.138692i −0.313007 − 2.97806i 0 −1.01646 + 1.12890i −0.672677 + 2.07029i 0 4.89031
433.2 −0.0378659 + 0.360270i 0 1.82793 + 0.388540i 0.280735 + 2.67101i 0 −1.66890 + 1.85350i −0.433081 + 1.33288i 0 −0.972914
433.3 0.127169 − 1.20993i 0 0.508536 + 0.108093i 0.0310414 + 0.295339i 0 2.87572 − 3.19381i 0.947351 − 2.91565i 0 0.361288
433.4 0.290461 − 2.76355i 0 −5.59657 − 1.18959i 0.105759 + 1.00623i 0 −1.27303 + 1.41384i −3.19570 + 9.83534i 0 2.81149
460.1 −1.57199 − 1.74587i 0 −0.367856 + 3.49991i 2.29762 − 2.55177i 0 −2.60459 + 1.15964i 2.88741 − 2.09782i 0 −8.06687
460.2 −1.16970 − 1.29908i 0 −0.110361 + 1.05002i −2.58338 + 2.86914i 0 0.801982 − 0.357065i −1.33531 + 0.970162i 0 6.74902
460.3 0.177926 + 0.197607i 0 0.201666 − 1.91872i 0.736416 − 0.817873i 0 −1.41040 + 0.627949i 0.845280 − 0.614132i 0 0.292645
460.4 1.22550 + 1.36105i 0 −0.141563 + 1.34688i −1.11978 + 1.24365i 0 3.77761 − 1.68190i 0.956730 − 0.695105i 0 −3.06496
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
9.c even 3 1 inner
11.c even 5 1 inner
99.m even 15 1 inner

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{32} + 2 T_{2}^{31} - 4 T_{2}^{30} - 16 T_{2}^{29} - 24 T_{2}^{28} - 80 T_{2}^{27} + 180 T_{2}^{26} + \cdots + 256 \) acting on \(S_{2}^{\mathrm{new}}(891, [\chi])\). Copy content Toggle raw display