Properties

Label 891.2.n.c
Level $891$
Weight $2$
Character orbit 891.n
Analytic conductor $7.115$
Analytic rank $0$
Dimension $8$
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: \(8\)
Coefficient field: \(\Q(\zeta_{15})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{15}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (2 \zeta_{15}^{6} - \zeta_{15}^{5} - \zeta_{15}^{2} + 2 \zeta_{15}) q^{2} + (3 \zeta_{15}^{5} - 3 \zeta_{15}^{4} + 3) q^{4} + (\zeta_{15}^{7} + \zeta_{15}^{4} + \zeta_{15}) q^{5} - \zeta_{15}^{2} q^{7} + (4 \zeta_{15}^{6} - \zeta_{15}^{3} + 4) q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + (2 \zeta_{15}^{6} - \zeta_{15}^{5} - \zeta_{15}^{2} + 2 \zeta_{15}) q^{2} + (3 \zeta_{15}^{5} - 3 \zeta_{15}^{4} + 3) q^{4} + (\zeta_{15}^{7} + \zeta_{15}^{4} + \zeta_{15}) q^{5} - \zeta_{15}^{2} q^{7} + (4 \zeta_{15}^{6} - \zeta_{15}^{3} + 4) q^{8} + (\zeta_{15}^{7} - \zeta_{15}^{3} + \zeta_{15}^{2}) q^{10} + ( - \zeta_{15}^{7} + \zeta_{15}^{6} - 2 \zeta_{15}^{5} - \zeta_{15}^{4} + \zeta_{15}^{3} + \zeta_{15}^{2} + 1) q^{11} + (2 \zeta_{15}^{7} - 2 \zeta_{15}^{5} - \zeta_{15} - 2) q^{13} + ( - \zeta_{15}^{7} + 2 \zeta_{15}^{5} - \zeta_{15}^{4} - 2 \zeta_{15} + 2) q^{14} + ( - 5 \zeta_{15}^{7} + 8 \zeta_{15}^{6} - 5 \zeta_{15}^{4} + 5 \zeta_{15}^{3} - 8 \zeta_{15}^{2} + 3 \zeta_{15} + 5) q^{16} + ( - 3 \zeta_{15}^{7} + 3 \zeta_{15}^{6} + 6 \zeta_{15}^{3} - 3 \zeta_{15}^{2} + 6) q^{17} + ( - 3 \zeta_{15}^{6} + \zeta_{15}^{3} - 3) q^{19} + (3 \zeta_{15}^{6} + 3 \zeta_{15}) q^{20} + (4 \zeta_{15}^{7} - 4 \zeta_{15}^{5} + \zeta_{15}^{4} + 7 \zeta_{15} - 4) q^{22} + (\zeta_{15}^{5} + 2 \zeta_{15}^{4} + 2 \zeta_{15} + 1) q^{23} + ( - 3 \zeta_{15}^{7} - \zeta_{15}^{6} + 4 \zeta_{15}^{5} - 3 \zeta_{15}^{4} + 3 \zeta_{15}^{3} - 4 \zeta_{15} + 3) q^{25} + ( - \zeta_{15}^{6} - \zeta_{15}^{3}) q^{26} + ( - 3 \zeta_{15}^{7} + 3 \zeta_{15}^{6} - 3 \zeta_{15}^{2}) q^{28} - 6 \zeta_{15}^{2} q^{29} + ( - 5 \zeta_{15}^{7} + 5 \zeta_{15}^{5} - 3 \zeta_{15} + 5) q^{31} + (9 \zeta_{15}^{5} - 3 \zeta_{15}^{4} - 3 \zeta_{15} + 9) q^{32} + 3 \zeta_{15}^{5} q^{34} + ( - \zeta_{15}^{7} - \zeta_{15}^{2} + 1) q^{35} + (5 \zeta_{15}^{7} - 2 \zeta_{15}^{6} - 2 \zeta_{15}^{3} + 5 \zeta_{15}^{2}) q^{37} + (4 \zeta_{15}^{7} - 11 \zeta_{15}^{6} + 4 \zeta_{15}^{4} - 4 \zeta_{15}^{3} + 11 \zeta_{15}^{2} - 7 \zeta_{15} - 4) q^{38} + (3 \zeta_{15}^{7} + \zeta_{15}^{5} - \zeta_{15}^{4} + 1) q^{40} + (3 \zeta_{15}^{7} - \zeta_{15}^{5} + 3 \zeta_{15}^{4} + \zeta_{15} - 1) q^{41} + (6 \zeta_{15}^{7} - 3 \zeta_{15}^{5} + 6 \zeta_{15}^{4} - 6 \zeta_{15}^{3} + 6 \zeta_{15}^{2} + 6 \zeta_{15} - 6) q^{43} + (9 \zeta_{15}^{7} - 9 \zeta_{15}^{6} - 6 \zeta_{15}^{3} + 9 \zeta_{15}^{2} + 3) q^{44} + (\zeta_{15}^{7} + \zeta_{15}^{2} - 1) q^{46} + ( - 2 \zeta_{15}^{7} - 5 \zeta_{15}^{6} + 7 \zeta_{15}^{5} - 2 \zeta_{15}^{4} + 2 \zeta_{15}^{3} - 7 \zeta_{15} + 2) q^{47} - 6 \zeta_{15}^{4} q^{49} + ( - 5 \zeta_{15}^{7} + 4 \zeta_{15}^{4} - 5 \zeta_{15}) q^{50} + (6 \zeta_{15}^{7} - 3 \zeta_{15}^{6} - 3 \zeta_{15}^{5} + 6 \zeta_{15}^{4} - 6 \zeta_{15}^{3} + 3 \zeta_{15} - 6) q^{52} + ( - \zeta_{15}^{7} - \zeta_{15}^{6} - \zeta_{15}^{2} + 1) q^{53} + (2 \zeta_{15}^{7} + \zeta_{15}^{6} + 3 \zeta_{15}^{3} + 2 \zeta_{15}^{2} + 1) q^{55} + ( - 4 \zeta_{15}^{7} + 5 \zeta_{15}^{5} - 4 \zeta_{15}^{4} + 4 \zeta_{15}^{3} - 4 \zeta_{15}^{2} - 4 \zeta_{15} + 4) q^{56} + ( - 6 \zeta_{15}^{7} + 12 \zeta_{15}^{5} - 6 \zeta_{15}^{4} - 12 \zeta_{15} + 12) q^{58} + (9 \zeta_{15}^{7} - \zeta_{15}^{5} + \zeta_{15}^{4} - 1) q^{59} + ( - 3 \zeta_{15}^{7} - 6 \zeta_{15}^{6} - 3 \zeta_{15}^{4} + 3 \zeta_{15}^{3} + 6 \zeta_{15}^{2} - 9 \zeta_{15} + 3) q^{61} + ( - 11 \zeta_{15}^{7} + 8 \zeta_{15}^{6} + 8 \zeta_{15}^{3} - 11 \zeta_{15}^{2}) q^{62} + ( - 6 \zeta_{15}^{7} + 5 \zeta_{15}^{6} - 6 \zeta_{15}^{2} + 6) q^{64} + ( - 3 \zeta_{15}^{7} - 2 \zeta_{15}^{5} - 3 \zeta_{15}^{4} + 3 \zeta_{15}^{3} - 3 \zeta_{15}^{2} - 3 \zeta_{15} + 3) q^{65} + (3 \zeta_{15}^{4} + 3 \zeta_{15}) q^{67} + ( - 9 \zeta_{15}^{7} + 9 \zeta_{15}^{5} + 9) q^{68} + ( - \zeta_{15}^{7} + \zeta_{15}^{6} + \zeta_{15}^{5} - \zeta_{15}^{4} + \zeta_{15}^{3} - \zeta_{15}^{2} + 1) q^{70} + (6 \zeta_{15}^{7} - 6 \zeta_{15}^{6} + \zeta_{15}^{3} + 6 \zeta_{15}^{2} + 1) q^{71} + (2 \zeta_{15}^{7} - 6 \zeta_{15}^{6} - 6 \zeta_{15}^{3} + 2 \zeta_{15}^{2}) q^{73} + (5 \zeta_{15}^{7} + 7 \zeta_{15}^{6} - 12 \zeta_{15}^{5} + 5 \zeta_{15}^{4} - 5 \zeta_{15}^{3} + 12 \zeta_{15} - 5) q^{74} + ( - 21 \zeta_{15}^{5} + 12 \zeta_{15}^{4} + 12 \zeta_{15} - 21) q^{76} + (2 \zeta_{15}^{7} - 2 \zeta_{15}^{4} - \zeta_{15}) q^{77} + ( - 11 \zeta_{15}^{6} - 11 \zeta_{15}) q^{79} + (5 \zeta_{15}^{6} + 2 \zeta_{15}^{3} + 5) q^{80} + ( - \zeta_{15}^{3} - 1) q^{82} + (5 \zeta_{15}^{7} - \zeta_{15}^{6} + 5 \zeta_{15}^{4} - 5 \zeta_{15}^{3} + \zeta_{15}^{2} + 4 \zeta_{15} - 5) q^{83} + (9 \zeta_{15}^{7} - 3 \zeta_{15}^{5} + 9 \zeta_{15}^{4} + 3 \zeta_{15} - 3) q^{85} + (9 \zeta_{15}^{7} - 9 \zeta_{15}^{5} + 12 \zeta_{15} - 9) q^{86} + (6 \zeta_{15}^{7} + 12 \zeta_{15}^{6} - 16 \zeta_{15}^{5} + 6 \zeta_{15}^{4} - 6 \zeta_{15}^{3} + 3 \zeta_{15}^{2} + \cdots - 6) q^{88} + \cdots + (6 \zeta_{15}^{7} - 6 \zeta_{15}^{3} + 6 \zeta_{15}^{2} + 6) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{2} + 9 q^{4} + 3 q^{5} - q^{7} + 26 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + q^{2} + 9 q^{4} + 3 q^{5} - q^{7} + 26 q^{8} + 4 q^{10} + 11 q^{11} - 7 q^{13} + 4 q^{14} - q^{16} + 24 q^{17} - 20 q^{19} - 3 q^{20} - 4 q^{22} + 8 q^{23} - 6 q^{25} + 4 q^{26} - 12 q^{28} - 6 q^{29} + 12 q^{31} + 30 q^{32} - 12 q^{34} + 6 q^{35} + 18 q^{37} + 10 q^{38} + 6 q^{40} + 3 q^{41} + 72 q^{44} - 6 q^{46} - 17 q^{47} - 6 q^{49} - 6 q^{50} - 3 q^{52} + 8 q^{53} + 4 q^{55} - 12 q^{56} + 24 q^{58} + 6 q^{59} + 21 q^{61} - 54 q^{62} + 26 q^{64} + 14 q^{65} + 6 q^{67} + 27 q^{68} - 3 q^{70} + 30 q^{71} + 28 q^{73} + 26 q^{74} - 60 q^{76} - q^{77} + 11 q^{79} + 26 q^{80} - 6 q^{82} - 13 q^{83} + 9 q^{85} - 15 q^{86} + 37 q^{88} - 48 q^{89} + 6 q^{91} - 3 q^{92} - 17 q^{94} - 5 q^{95} - 3 q^{97} + 72 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/891\mathbb{Z}\right)^\times\).

\(n\) \(244\) \(650\)
\(\chi(n)\) \(-\zeta_{15}^{2} - \zeta_{15}^{7}\) \(-1 - \zeta_{15}^{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
136.1
−0.104528 + 0.994522i
−0.104528 0.994522i
0.669131 0.743145i
0.913545 0.406737i
−0.978148 + 0.207912i
−0.978148 0.207912i
0.669131 + 0.743145i
0.913545 + 0.406737i
−0.348943 + 0.155360i 0 −1.24064 + 1.37787i 1.47815 + 0.658114i 0 0.978148 + 0.207912i 0.454915 1.40008i 0 −0.618034
190.1 −0.348943 0.155360i 0 −1.24064 1.37787i 1.47815 0.658114i 0 0.978148 0.207912i 0.454915 + 1.40008i 0 −0.618034
379.1 2.56082 + 0.544320i 0 4.43444 + 1.97434i 0.604528 0.128496i 0 0.104528 + 0.994522i 6.04508 + 4.39201i 0 1.61803
433.1 0.0399263 0.379874i 0 1.81359 + 0.385489i −0.169131 1.60917i 0 −0.669131 + 0.743145i 0.454915 1.40008i 0 −0.618034
460.1 −1.75181 1.94558i 0 −0.507392 + 4.82751i −0.413545 + 0.459289i 0 −0.913545 + 0.406737i 6.04508 4.39201i 0 1.61803
676.1 −1.75181 + 1.94558i 0 −0.507392 4.82751i −0.413545 0.459289i 0 −0.913545 0.406737i 6.04508 + 4.39201i 0 1.61803
757.1 2.56082 0.544320i 0 4.43444 1.97434i 0.604528 + 0.128496i 0 0.104528 0.994522i 6.04508 4.39201i 0 1.61803
784.1 0.0399263 + 0.379874i 0 1.81359 0.385489i −0.169131 + 1.60917i 0 −0.669131 0.743145i 0.454915 + 1.40008i 0 −0.618034
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 136.1
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.c 8
3.b odd 2 1 891.2.n.b 8
9.c even 3 1 33.2.e.b 4
9.c even 3 1 inner 891.2.n.c 8
9.d odd 6 1 99.2.f.a 4
9.d odd 6 1 891.2.n.b 8
11.c even 5 1 inner 891.2.n.c 8
33.h odd 10 1 891.2.n.b 8
36.f odd 6 1 528.2.y.b 4
45.j even 6 1 825.2.n.c 4
45.k odd 12 2 825.2.bx.d 8
99.h odd 6 1 363.2.e.f 4
99.m even 15 1 33.2.e.b 4
99.m even 15 1 363.2.a.d 2
99.m even 15 2 363.2.e.k 4
99.m even 15 1 inner 891.2.n.c 8
99.n odd 30 1 99.2.f.a 4
99.n odd 30 1 891.2.n.b 8
99.n odd 30 1 1089.2.a.t 2
99.o odd 30 1 363.2.a.i 2
99.o odd 30 2 363.2.e.b 4
99.o odd 30 1 363.2.e.f 4
99.p even 30 1 1089.2.a.l 2
396.be odd 30 1 528.2.y.b 4
396.be odd 30 1 5808.2.a.cj 2
396.bf even 30 1 5808.2.a.ci 2
495.bl even 30 1 825.2.n.c 4
495.bl even 30 1 9075.2.a.cb 2
495.br odd 30 1 9075.2.a.u 2
495.bt odd 60 2 825.2.bx.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.2.e.b 4 9.c even 3 1
33.2.e.b 4 99.m even 15 1
99.2.f.a 4 9.d odd 6 1
99.2.f.a 4 99.n odd 30 1
363.2.a.d 2 99.m even 15 1
363.2.a.i 2 99.o odd 30 1
363.2.e.b 4 99.o odd 30 2
363.2.e.f 4 99.h odd 6 1
363.2.e.f 4 99.o odd 30 1
363.2.e.k 4 99.m even 15 2
528.2.y.b 4 36.f odd 6 1
528.2.y.b 4 396.be odd 30 1
825.2.n.c 4 45.j even 6 1
825.2.n.c 4 495.bl even 30 1
825.2.bx.d 8 45.k odd 12 2
825.2.bx.d 8 495.bt odd 60 2
891.2.n.b 8 3.b odd 2 1
891.2.n.b 8 9.d odd 6 1
891.2.n.b 8 33.h odd 10 1
891.2.n.b 8 99.n odd 30 1
891.2.n.c 8 1.a even 1 1 trivial
891.2.n.c 8 9.c even 3 1 inner
891.2.n.c 8 11.c even 5 1 inner
891.2.n.c 8 99.m even 15 1 inner
1089.2.a.l 2 99.p even 30 1
1089.2.a.t 2 99.n odd 30 1
5808.2.a.ci 2 396.bf even 30 1
5808.2.a.cj 2 396.be odd 30 1
9075.2.a.u 2 495.br odd 30 1
9075.2.a.cb 2 495.bl even 30 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - T_{2}^{7} - 5T_{2}^{6} - 14T_{2}^{5} + 39T_{2}^{4} + 26T_{2}^{3} + 10T_{2}^{2} + 4T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(891, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - T^{7} - 5 T^{6} - 14 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 3 T^{7} + 5 T^{6} - 8 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{8} + T^{7} - T^{5} - T^{4} - T^{3} + T + 1 \) Copy content Toggle raw display
$11$ \( T^{8} - 11 T^{7} + 70 T^{6} + \cdots + 14641 \) Copy content Toggle raw display
$13$ \( T^{8} + 7 T^{7} + 30 T^{6} + 127 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( (T^{4} - 12 T^{3} + 54 T^{2} + 27 T + 81)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 10 T^{3} + 40 T^{2} + 25 T + 25)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 4 T^{3} + 17 T^{2} + 4 T + 1)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + 6 T^{7} - 216 T^{5} + \cdots + 1679616 \) Copy content Toggle raw display
$31$ \( T^{8} - 12 T^{7} + 50 T^{6} + \cdots + 923521 \) Copy content Toggle raw display
$37$ \( (T^{4} - 9 T^{3} + 31 T^{2} + 11 T + 121)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} - 3 T^{7} - 10 T^{6} - 43 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$43$ \( (T^{4} + 45 T^{2} + 2025)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 17 T^{7} + 175 T^{6} + \cdots + 14641 \) Copy content Toggle raw display
$53$ \( (T^{4} - 4 T^{3} + 6 T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} - 6 T^{7} - 40 T^{6} + \cdots + 25411681 \) Copy content Toggle raw display
$61$ \( T^{8} - 21 T^{7} + 135 T^{6} + \cdots + 96059601 \) Copy content Toggle raw display
$67$ \( (T^{4} - 3 T^{3} + 18 T^{2} + 27 T + 81)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 15 T^{3} + 190 T^{2} - 1100 T + 3025)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 14 T^{3} + 136 T^{2} - 704 T + 1936)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} - 11 T^{7} + \cdots + 214358881 \) Copy content Toggle raw display
$83$ \( T^{8} + 13 T^{7} + 100 T^{6} + \cdots + 14641 \) Copy content Toggle raw display
$89$ \( (T^{2} + 12 T + 31)^{4} \) Copy content Toggle raw display
$97$ \( T^{8} + 3 T^{7} - 45 T^{6} + \cdots + 6561 \) Copy content Toggle raw display
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