Properties

Label 882.2.u.c
Level $882$
Weight $2$
Character orbit 882.u
Analytic conductor $7.043$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [882,2,Mod(127,882)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(882, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([0, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("882.127");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 882.u (of order \(7\), degree \(6\), 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: \(7.04280545828\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{14})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-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 \(\zeta_{14}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/882\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(785\)
\(\chi(n)\) \(-\zeta_{14}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
0.900969 0.433884i
−0.623490 + 0.781831i
0.222521 0.974928i
0.222521 + 0.974928i
−0.623490 0.781831i
0.900969 + 0.433884i
0.900969 + 0.433884i 0 0.623490 + 0.781831i 0.801938 + 3.51352i 0 1.14795 2.38374i 0.222521 + 0.974928i 0 −0.801938 + 3.51352i
253.1 −0.623490 0.781831i 0 −0.222521 + 0.974928i −2.24698 + 1.08209i 0 −2.06853 + 1.64960i 0.900969 0.433884i 0 2.24698 + 1.08209i
379.1 0.222521 + 0.974928i 0 −0.900969 + 0.433884i −0.554958 0.695895i 0 −2.57942 + 0.588735i −0.623490 0.781831i 0 0.554958 0.695895i
505.1 0.222521 0.974928i 0 −0.900969 0.433884i −0.554958 + 0.695895i 0 −2.57942 0.588735i −0.623490 + 0.781831i 0 0.554958 + 0.695895i
631.1 −0.623490 + 0.781831i 0 −0.222521 0.974928i −2.24698 1.08209i 0 −2.06853 1.64960i 0.900969 + 0.433884i 0 2.24698 1.08209i
757.1 0.900969 0.433884i 0 0.623490 0.781831i 0.801938 3.51352i 0 1.14795 + 2.38374i 0.222521 0.974928i 0 −0.801938 3.51352i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 127.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
49.e even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 882.2.u.c yes 6
3.b odd 2 1 882.2.u.a 6
49.e even 7 1 inner 882.2.u.c yes 6
147.l odd 14 1 882.2.u.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
882.2.u.a 6 3.b odd 2 1
882.2.u.a 6 147.l odd 14 1
882.2.u.c yes 6 1.a even 1 1 trivial
882.2.u.c yes 6 49.e even 7 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + 4T_{5}^{5} + 16T_{5}^{4} + 64T_{5}^{3} + 144T_{5}^{2} + 128T_{5} + 64 \) acting on \(S_{2}^{\mathrm{new}}(882, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - T^{5} + T^{4} - T^{3} + T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + 4 T^{5} + 16 T^{4} + 64 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$7$ \( T^{6} + 7 T^{5} + 21 T^{4} + 49 T^{3} + \cdots + 343 \) Copy content Toggle raw display
$11$ \( T^{6} + 10 T^{5} + 44 T^{4} + 104 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$13$ \( T^{6} - 12 T^{5} + 60 T^{4} + \cdots + 1681 \) Copy content Toggle raw display
$17$ \( T^{6} - 6 T^{5} + 8 T^{4} + 8 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$19$ \( (T^{3} - 13 T^{2} + 54 T - 71)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} - 22 T^{5} + 204 T^{4} + \cdots + 10816 \) Copy content Toggle raw display
$29$ \( T^{6} - 2 T^{5} + 4 T^{4} + \cdots + 10816 \) Copy content Toggle raw display
$31$ \( (T^{3} + 7 T^{2} - 56 T - 301)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 25 T^{5} + 331 T^{4} + \cdots + 19321 \) Copy content Toggle raw display
$41$ \( T^{6} - 280 T^{3} + 1568 T^{2} + \cdots + 3136 \) Copy content Toggle raw display
$43$ \( T^{6} - 18 T^{5} + 156 T^{4} + \cdots + 9409 \) Copy content Toggle raw display
$47$ \( T^{6} + 30 T^{5} + 452 T^{4} + \cdots + 322624 \) Copy content Toggle raw display
$53$ \( T^{6} - 32 T^{5} + 464 T^{4} + \cdots + 118336 \) Copy content Toggle raw display
$59$ \( T^{6} - 32 T^{5} + 576 T^{4} + \cdots + 692224 \) Copy content Toggle raw display
$61$ \( T^{6} + 17 T^{5} + 79 T^{4} + \cdots + 121801 \) Copy content Toggle raw display
$67$ \( (T^{3} - 13 T^{2} + 40 T + 13)^{2} \) Copy content Toggle raw display
$71$ \( T^{6} + 28 T^{5} + 280 T^{4} + \cdots + 3136 \) Copy content Toggle raw display
$73$ \( T^{6} + 2 T^{5} + 32 T^{4} + \cdots + 5041 \) Copy content Toggle raw display
$79$ \( (T^{3} + 13 T^{2} - 16 T - 167)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} - 16 T^{5} + 144 T^{4} + \cdots + 4096 \) Copy content Toggle raw display
$89$ \( T^{6} - 22 T^{5} + 260 T^{4} + \cdots + 3655744 \) Copy content Toggle raw display
$97$ \( (T^{3} + 11 T^{2} - 74 T + 13)^{2} \) Copy content Toggle raw display
show more
show less