Properties

Label 84.2.i.a
Level $84$
Weight $2$
Character orbit 84.i
Analytic conductor $0.671$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [84,2,Mod(25,84)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(84, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("84.25");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 84.i (of order \(3\), degree \(2\), 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: \(0.670743376979\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{6} q^{3} + ( - 2 \zeta_{6} + 2) q^{5} + (3 \zeta_{6} - 1) q^{7} + (\zeta_{6} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{6} q^{3} + ( - 2 \zeta_{6} + 2) q^{5} + (3 \zeta_{6} - 1) q^{7} + (\zeta_{6} - 1) q^{9} - 2 \zeta_{6} q^{11} - 3 q^{13} + 2 q^{15} - 8 \zeta_{6} q^{17} + ( - \zeta_{6} + 1) q^{19} + (2 \zeta_{6} - 3) q^{21} + (8 \zeta_{6} - 8) q^{23} + \zeta_{6} q^{25} - q^{27} + 4 q^{29} - 3 \zeta_{6} q^{31} + ( - 2 \zeta_{6} + 2) q^{33} + (2 \zeta_{6} + 4) q^{35} + ( - \zeta_{6} + 1) q^{37} - 3 \zeta_{6} q^{39} + 6 q^{41} + 11 q^{43} + 2 \zeta_{6} q^{45} + (6 \zeta_{6} - 6) q^{47} + (3 \zeta_{6} - 8) q^{49} + ( - 8 \zeta_{6} + 8) q^{51} + 12 \zeta_{6} q^{53} - 4 q^{55} + q^{57} - 4 \zeta_{6} q^{59} + ( - 6 \zeta_{6} + 6) q^{61} + ( - \zeta_{6} - 2) q^{63} + (6 \zeta_{6} - 6) q^{65} - 13 \zeta_{6} q^{67} - 8 q^{69} - 10 q^{71} + 11 \zeta_{6} q^{73} + (\zeta_{6} - 1) q^{75} + ( - 4 \zeta_{6} + 6) q^{77} + ( - 3 \zeta_{6} + 3) q^{79} - \zeta_{6} q^{81} + 2 q^{83} - 16 q^{85} + 4 \zeta_{6} q^{87} + ( - 9 \zeta_{6} + 3) q^{91} + ( - 3 \zeta_{6} + 3) q^{93} - 2 \zeta_{6} q^{95} + 10 q^{97} + 2 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} + 2 q^{5} + q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{3} + 2 q^{5} + q^{7} - q^{9} - 2 q^{11} - 6 q^{13} + 4 q^{15} - 8 q^{17} + q^{19} - 4 q^{21} - 8 q^{23} + q^{25} - 2 q^{27} + 8 q^{29} - 3 q^{31} + 2 q^{33} + 10 q^{35} + q^{37} - 3 q^{39} + 12 q^{41} + 22 q^{43} + 2 q^{45} - 6 q^{47} - 13 q^{49} + 8 q^{51} + 12 q^{53} - 8 q^{55} + 2 q^{57} - 4 q^{59} + 6 q^{61} - 5 q^{63} - 6 q^{65} - 13 q^{67} - 16 q^{69} - 20 q^{71} + 11 q^{73} - q^{75} + 8 q^{77} + 3 q^{79} - q^{81} + 4 q^{83} - 32 q^{85} + 4 q^{87} - 3 q^{91} + 3 q^{93} - 2 q^{95} + 20 q^{97} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/84\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(43\) \(73\)
\(\chi(n)\) \(1\) \(1\) \(-1 + \zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0.500000 0.866025i 0 1.00000 + 1.73205i 0 0.500000 2.59808i 0 −0.500000 0.866025i 0
37.1 0 0.500000 + 0.866025i 0 1.00000 1.73205i 0 0.500000 + 2.59808i 0 −0.500000 + 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 84.2.i.a 2
3.b odd 2 1 252.2.k.a 2
4.b odd 2 1 336.2.q.c 2
5.b even 2 1 2100.2.q.b 2
5.c odd 4 2 2100.2.bc.a 4
7.b odd 2 1 588.2.i.b 2
7.c even 3 1 inner 84.2.i.a 2
7.c even 3 1 588.2.a.a 1
7.d odd 6 1 588.2.a.f 1
7.d odd 6 1 588.2.i.b 2
8.b even 2 1 1344.2.q.b 2
8.d odd 2 1 1344.2.q.n 2
9.c even 3 1 2268.2.i.g 2
9.c even 3 1 2268.2.l.b 2
9.d odd 6 1 2268.2.i.b 2
9.d odd 6 1 2268.2.l.g 2
12.b even 2 1 1008.2.s.c 2
21.c even 2 1 1764.2.k.j 2
21.g even 6 1 1764.2.a.c 1
21.g even 6 1 1764.2.k.j 2
21.h odd 6 1 252.2.k.a 2
21.h odd 6 1 1764.2.a.h 1
28.d even 2 1 2352.2.q.q 2
28.f even 6 1 2352.2.a.k 1
28.f even 6 1 2352.2.q.q 2
28.g odd 6 1 336.2.q.c 2
28.g odd 6 1 2352.2.a.o 1
35.j even 6 1 2100.2.q.b 2
35.l odd 12 2 2100.2.bc.a 4
56.j odd 6 1 9408.2.a.i 1
56.k odd 6 1 1344.2.q.n 2
56.k odd 6 1 9408.2.a.bi 1
56.m even 6 1 9408.2.a.bx 1
56.p even 6 1 1344.2.q.b 2
56.p even 6 1 9408.2.a.cx 1
63.g even 3 1 2268.2.i.g 2
63.h even 3 1 2268.2.l.b 2
63.j odd 6 1 2268.2.l.g 2
63.n odd 6 1 2268.2.i.b 2
84.j odd 6 1 7056.2.a.o 1
84.n even 6 1 1008.2.s.c 2
84.n even 6 1 7056.2.a.bs 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.2.i.a 2 1.a even 1 1 trivial
84.2.i.a 2 7.c even 3 1 inner
252.2.k.a 2 3.b odd 2 1
252.2.k.a 2 21.h odd 6 1
336.2.q.c 2 4.b odd 2 1
336.2.q.c 2 28.g odd 6 1
588.2.a.a 1 7.c even 3 1
588.2.a.f 1 7.d odd 6 1
588.2.i.b 2 7.b odd 2 1
588.2.i.b 2 7.d odd 6 1
1008.2.s.c 2 12.b even 2 1
1008.2.s.c 2 84.n even 6 1
1344.2.q.b 2 8.b even 2 1
1344.2.q.b 2 56.p even 6 1
1344.2.q.n 2 8.d odd 2 1
1344.2.q.n 2 56.k odd 6 1
1764.2.a.c 1 21.g even 6 1
1764.2.a.h 1 21.h odd 6 1
1764.2.k.j 2 21.c even 2 1
1764.2.k.j 2 21.g even 6 1
2100.2.q.b 2 5.b even 2 1
2100.2.q.b 2 35.j even 6 1
2100.2.bc.a 4 5.c odd 4 2
2100.2.bc.a 4 35.l odd 12 2
2268.2.i.b 2 9.d odd 6 1
2268.2.i.b 2 63.n odd 6 1
2268.2.i.g 2 9.c even 3 1
2268.2.i.g 2 63.g even 3 1
2268.2.l.b 2 9.c even 3 1
2268.2.l.b 2 63.h even 3 1
2268.2.l.g 2 9.d odd 6 1
2268.2.l.g 2 63.j odd 6 1
2352.2.a.k 1 28.f even 6 1
2352.2.a.o 1 28.g odd 6 1
2352.2.q.q 2 28.d even 2 1
2352.2.q.q 2 28.f even 6 1
7056.2.a.o 1 84.j odd 6 1
7056.2.a.bs 1 84.n even 6 1
9408.2.a.i 1 56.j odd 6 1
9408.2.a.bi 1 56.k odd 6 1
9408.2.a.bx 1 56.m even 6 1
9408.2.a.cx 1 56.p even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(84, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$7$ \( T^{2} - T + 7 \) Copy content Toggle raw display
$11$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$13$ \( (T + 3)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$19$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$23$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$29$ \( (T - 4)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$37$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$41$ \( (T - 6)^{2} \) Copy content Toggle raw display
$43$ \( (T - 11)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$53$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$59$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$61$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$67$ \( T^{2} + 13T + 169 \) Copy content Toggle raw display
$71$ \( (T + 10)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 11T + 121 \) Copy content Toggle raw display
$79$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$83$ \( (T - 2)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( (T - 10)^{2} \) Copy content Toggle raw display
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