↑

Properties

Label 3971.1.q.d
Level $3971$
Weight $1$
Character orbit 3971.q
Analytic conductor $1.982$
Analytic rank $0$
Dimension $12$
Projective image $A_{4}$
CM/RM no
Inner twists $12$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3971,1,Mod(54,3971)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3971, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([9, 4]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3971.54");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3971 = 11 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3971.q (of order \(18\), degree \(6\), 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: \(1.98178716517\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{18})\)
Coefficient field: \(\Q(\zeta_{36})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{6} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 209)
Projective image: \(A_{4}\)
Projective field: Galois closure of 4.0.43681.1

$q$-expansion

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

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + \zeta_{36}^{11} q^{2} - \zeta_{36}^{8} q^{3} - \zeta_{36}^{14} q^{5} + \zeta_{36} q^{6} + \zeta_{36}^{15} q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{36}^{11} q^{2} - \zeta_{36}^{8} q^{3} - \zeta_{36}^{14} q^{5} + \zeta_{36} q^{6} + \zeta_{36}^{15} q^{8} + \zeta_{36}^{7} q^{10} - \zeta_{36}^{15} q^{11} + \zeta_{36} q^{13} - \zeta_{36}^{4} q^{15} - \zeta_{36}^{8} q^{16} + \zeta_{36}^{11} q^{17} + \zeta_{36}^{8} q^{22} + \zeta_{36}^{4} q^{23} + \zeta_{36}^{5} q^{24} + \zeta_{36}^{12} q^{26} - \zeta_{36}^{6} q^{27} + \zeta_{36}^{7} q^{29} - \zeta_{36}^{15} q^{30} - \zeta_{36} q^{32} - \zeta_{36}^{5} q^{33} - \zeta_{36}^{4} q^{34} - \zeta_{36}^{9} q^{39} + \zeta_{36}^{11} q^{40} - \zeta_{36}^{17} q^{41} - \zeta_{36}^{5} q^{43} + \zeta_{36}^{15} q^{46} + \zeta_{36}^{16} q^{47} + \zeta_{36}^{16} q^{48} - \zeta_{36}^{6} q^{49} + \zeta_{36} q^{51} + \zeta_{36}^{4} q^{53} - \zeta_{36}^{17} q^{54} - \zeta_{36}^{11} q^{55} - q^{58} + \zeta_{36}^{2} q^{59} + \zeta_{36}^{13} q^{61} - \zeta_{36}^{12} q^{64} - \zeta_{36}^{15} q^{65} - \zeta_{36}^{16} q^{66} + \zeta_{36}^{16} q^{67} - \zeta_{36}^{12} q^{69} - \zeta_{36}^{14} q^{71} - \zeta_{36}^{17} q^{73} + \zeta_{36}^{2} q^{78} + \zeta_{36}^{17} q^{79} - \zeta_{36}^{4} q^{80} + \zeta_{36}^{14} q^{81} + \zeta_{36}^{10} q^{82} + \zeta_{36}^{7} q^{85} - \zeta_{36}^{16} q^{86} - \zeta_{36}^{15} q^{87} + \zeta_{36}^{12} q^{88} - \zeta_{36}^{10} q^{89} - \zeta_{36}^{9} q^{94} - \zeta_{36}^{2} q^{97} - \zeta_{36}^{17} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 6 q^{26} - 6 q^{27} - 6 q^{49} - 12 q^{58} + 6 q^{64} + 6 q^{69} - 6 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3971\mathbb{Z}\right)^\times\).

\(n\) \(1806\) \(2168\)
\(\chi(n)\) \(-1\) \(\zeta_{36}^{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
54.1
−0.342020 + 0.939693i
0.342020 − 0.939693i
−0.342020 − 0.939693i
0.342020 + 0.939693i
0.642788 + 0.766044i
−0.642788 − 0.766044i
0.984808 − 0.173648i
−0.984808 + 0.173648i
0.984808 + 0.173648i
−0.984808 − 0.173648i
0.642788 − 0.766044i
−0.642788 + 0.766044i
−0.642788 + 0.766044i 0.939693 − 0.342020i 0 0.173648 − 0.984808i −0.342020 + 0.939693i 0 −0.866025 − 0.500000i 0 0.642788 + 0.766044i
54.2 0.642788 − 0.766044i 0.939693 − 0.342020i 0 0.173648 − 0.984808i 0.342020 − 0.939693i 0 0.866025 + 0.500000i 0 −0.642788 − 0.766044i
956.1 −0.642788 − 0.766044i 0.939693 + 0.342020i 0 0.173648 + 0.984808i −0.342020 − 0.939693i 0 −0.866025 + 0.500000i 0 0.642788 − 0.766044i
956.2 0.642788 + 0.766044i 0.939693 + 0.342020i 0 0.173648 + 0.984808i 0.342020 + 0.939693i 0 0.866025 − 0.500000i 0 −0.642788 + 0.766044i
967.1 −0.984808 − 0.173648i −0.766044 − 0.642788i 0 −0.939693 + 0.342020i 0.642788 + 0.766044i 0 0.866025 + 0.500000i 0 0.984808 − 0.173648i
967.2 0.984808 + 0.173648i −0.766044 − 0.642788i 0 −0.939693 + 0.342020i −0.642788 − 0.766044i 0 −0.866025 − 0.500000i 0 −0.984808 + 0.173648i
1506.1 −0.342020 − 0.939693i −0.173648 + 0.984808i 0 0.766044 + 0.642788i 0.984808 − 0.173648i 0 −0.866025 − 0.500000i 0 0.342020 − 0.939693i
1506.2 0.342020 + 0.939693i −0.173648 + 0.984808i 0 0.766044 + 0.642788i −0.984808 + 0.173648i 0 0.866025 + 0.500000i 0 −0.342020 + 0.939693i
2265.1 −0.342020 + 0.939693i −0.173648 − 0.984808i 0 0.766044 − 0.642788i 0.984808 + 0.173648i 0 −0.866025 + 0.500000i 0 0.342020 + 0.939693i
2265.2 0.342020 − 0.939693i −0.173648 − 0.984808i 0 0.766044 − 0.642788i −0.984808 − 0.173648i 0 0.866025 − 0.500000i 0 −0.342020 − 0.939693i
3277.1 −0.984808 + 0.173648i −0.766044 + 0.642788i 0 −0.939693 − 0.342020i 0.642788 − 0.766044i 0 0.866025 − 0.500000i 0 0.984808 + 0.173648i
3277.2 0.984808 − 0.173648i −0.766044 + 0.642788i 0 −0.939693 − 0.342020i −0.642788 + 0.766044i 0 −0.866025 + 0.500000i 0 −0.984808 − 0.173648i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 54.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner
19.c even 3 2 inner
19.e even 9 3 inner
209.h odd 6 2 inner
209.q odd 18 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3971.1.q.d 12
11.b odd 2 1 inner 3971.1.q.d 12
19.b odd 2 1 3971.1.q.e 12
19.c even 3 2 inner 3971.1.q.d 12
19.d odd 6 2 3971.1.q.e 12
19.e even 9 1 3971.1.c.b 2
19.e even 9 2 3971.1.h.d 4
19.e even 9 3 inner 3971.1.q.d 12
19.f odd 18 2 209.1.h.a ✓ 4
19.f odd 18 1 3971.1.c.e 2
19.f odd 18 3 3971.1.q.e 12
57.j even 18 2 1881.1.bc.a 4
76.k even 18 2 3344.1.bb.a 4
209.d even 2 1 3971.1.q.e 12
209.g even 6 2 3971.1.q.e 12
209.h odd 6 2 inner 3971.1.q.d 12
209.p even 18 2 209.1.h.a ✓ 4
209.p even 18 1 3971.1.c.e 2
209.p even 18 3 3971.1.q.e 12
209.q odd 18 1 3971.1.c.b 2
209.q odd 18 2 3971.1.h.d 4
209.q odd 18 3 inner 3971.1.q.d 12
209.w even 90 8 2299.1.w.a 16
209.x odd 90 8 2299.1.w.a 16
627.bf odd 18 2 1881.1.bc.a 4
836.ba odd 18 2 3344.1.bb.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
209.1.h.a ✓ 4 19.f odd 18 2
209.1.h.a ✓ 4 209.p even 18 2
1881.1.bc.a 4 57.j even 18 2
1881.1.bc.a 4 627.bf odd 18 2
2299.1.w.a 16 209.w even 90 8
2299.1.w.a 16 209.x odd 90 8
3344.1.bb.a 4 76.k even 18 2
3344.1.bb.a 4 836.ba odd 18 2
3971.1.c.b 2 19.e even 9 1
3971.1.c.b 2 209.q odd 18 1
3971.1.c.e 2 19.f odd 18 1
3971.1.c.e 2 209.p even 18 1
3971.1.h.d 4 19.e even 9 2
3971.1.h.d 4 209.q odd 18 2
3971.1.q.d 12 1.a even 1 1 trivial
3971.1.q.d 12 11.b odd 2 1 inner
3971.1.q.d 12 19.c even 3 2 inner
3971.1.q.d 12 19.e even 9 3 inner
3971.1.q.d 12 209.h odd 6 2 inner
3971.1.q.d 12 209.q odd 18 3 inner
3971.1.q.e 12 19.b odd 2 1
3971.1.q.e 12 19.d odd 6 2
3971.1.q.e 12 19.f odd 18 3
3971.1.q.e 12 209.d even 2 1
3971.1.q.e 12 209.g even 6 2
3971.1.q.e 12 209.p even 18 3

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3971, [\chi])\):

\( T_{2}^{12} - T_{2}^{6} + 1 \) Copy content Toggle raw display
\( T_{3}^{6} - T_{3}^{3} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - T^{6} + 1 \) Copy content Toggle raw display
$3$ \( (T^{6} - T^{3} + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{6} + T^{3} + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( (T^{4} - T^{2} + 1)^{3} \) Copy content Toggle raw display
$13$ \( T^{12} - T^{6} + 1 \) Copy content Toggle raw display
$17$ \( T^{12} - T^{6} + 1 \) Copy content Toggle raw display
$19$ \( T^{12} \) Copy content Toggle raw display
$23$ \( (T^{6} + T^{3} + 1)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} - T^{6} + 1 \) Copy content Toggle raw display
$31$ \( T^{12} \) Copy content Toggle raw display
$37$ \( T^{12} \) Copy content Toggle raw display
$41$ \( T^{12} - T^{6} + 1 \) Copy content Toggle raw display
$43$ \( T^{12} - T^{6} + 1 \) Copy content Toggle raw display
$47$ \( (T^{6} + T^{3} + 1)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + T^{3} + 1)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} - T^{3} + 1)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} - T^{6} + 1 \) Copy content Toggle raw display
$67$ \( (T^{6} + T^{3} + 1)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + T^{3} + 1)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} - T^{6} + 1 \) Copy content Toggle raw display
$79$ \( T^{12} - T^{6} + 1 \) Copy content Toggle raw display
$83$ \( T^{12} \) Copy content Toggle raw display
$89$ \( (T^{6} + T^{3} + 1)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + T^{3} + 1)^{2} \) Copy content Toggle raw display
show more
show less