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Properties

Label 2527.1.y.e
Level $2527$
Weight $1$
Character orbit 2527.y
Analytic conductor $1.261$
Analytic rank $0$
Dimension $12$
Projective image $A_{4}$
CM/RM no
Inner twists $12$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2527,1,Mod(62,2527)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2527, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([9, 16]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2527.62");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2527 = 7 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2527.y (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: \(1.26113728692\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 133)
Projective image: \(A_{4}\)
Projective field: Galois closure of 4.0.17689.1

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{36}^{16} q^{2} - \zeta_{36} q^{3} + \zeta_{36}^{13} q^{5} - \zeta_{36}^{17} q^{6} + \zeta_{36}^{15} q^{7} - \zeta_{36}^{12} q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{36}^{16} q^{2} - \zeta_{36} q^{3} + \zeta_{36}^{13} q^{5} - \zeta_{36}^{17} q^{6} + \zeta_{36}^{15} q^{7} - \zeta_{36}^{12} q^{8} - \zeta_{36}^{11} q^{10} - \zeta_{36}^{17} q^{13} - \zeta_{36}^{13} q^{14} - \zeta_{36}^{14} q^{15} + \zeta_{36}^{10} q^{16} + \zeta_{36}^{7} q^{17} - \zeta_{36}^{16} q^{21} - \zeta_{36}^{14} q^{23} + \zeta_{36}^{13} q^{24} + \zeta_{36}^{15} q^{26} + \zeta_{36}^{3} q^{27} + \zeta_{36}^{2} q^{29} + \zeta_{36}^{12} q^{30} + \zeta_{36}^{8} q^{32} - \zeta_{36}^{5} q^{34} - \zeta_{36}^{10} q^{35} - q^{39} + \zeta_{36}^{7} q^{40} - \zeta_{36} q^{41} + \zeta_{36}^{14} q^{42} + \zeta_{36}^{4} q^{43} + \zeta_{36}^{12} q^{46} + \zeta_{36}^{11} q^{47} - \zeta_{36}^{11} q^{48} - \zeta_{36}^{12} q^{49} - \zeta_{36}^{8} q^{51} + \zeta_{36}^{14} q^{53} - \zeta_{36} q^{54} + \zeta_{36}^{9} q^{56} - q^{58} - \zeta_{36}^{7} q^{59} + \zeta_{36}^{5} q^{61} - \zeta_{36}^{6} q^{64} + \zeta_{36}^{12} q^{65} - \zeta_{36}^{2} q^{67} + \zeta_{36}^{15} q^{69} + \zeta_{36}^{8} q^{70} - \zeta_{36}^{4} q^{71} + \zeta_{36} q^{73} - \zeta_{36}^{16} q^{78} - \zeta_{36}^{10} q^{79} - \zeta_{36}^{5} q^{80} - \zeta_{36}^{4} q^{81} - \zeta_{36}^{17} q^{82} - \zeta_{36}^{2} q^{85} - \zeta_{36}^{2} q^{86} - \zeta_{36}^{3} q^{87} + \zeta_{36}^{17} q^{89} + \zeta_{36}^{14} q^{91} - \zeta_{36}^{9} q^{94} + \zeta_{36}^{7} q^{97} + \zeta_{36}^{10} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 6 q^{8} - 6 q^{30} - 12 q^{39} - 6 q^{46} + 6 q^{49} - 12 q^{58} - 6 q^{64} - 6 q^{65}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2527\mathbb{Z}\right)^\times\).

\(n\) \(1445\) \(1807\)
\(\chi(n)\) \(-1\) \(-\zeta_{36}^{14}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
62.1
0.984808 + 0.173648i
−0.984808 − 0.173648i
0.342020 + 0.939693i
−0.342020 − 0.939693i
0.984808 − 0.173648i
−0.984808 + 0.173648i
0.642788 + 0.766044i
−0.642788 − 0.766044i
0.642788 − 0.766044i
−0.642788 + 0.766044i
0.342020 − 0.939693i
−0.342020 + 0.939693i
−0.939693 + 0.342020i −0.984808 − 0.173648i 0 −0.642788 + 0.766044i 0.984808 − 0.173648i −0.866025 + 0.500000i 0.500000 − 0.866025i 0 0.342020 − 0.939693i
62.2 −0.939693 + 0.342020i 0.984808 + 0.173648i 0 0.642788 − 0.766044i −0.984808 + 0.173648i 0.866025 − 0.500000i 0.500000 − 0.866025i 0 −0.342020 + 0.939693i
776.1 0.766044 + 0.642788i −0.342020 − 0.939693i 0 −0.984808 − 0.173648i 0.342020 − 0.939693i 0.866025 − 0.500000i 0.500000 − 0.866025i 0 −0.642788 − 0.766044i
776.2 0.766044 + 0.642788i 0.342020 + 0.939693i 0 0.984808 + 0.173648i −0.342020 + 0.939693i −0.866025 + 0.500000i 0.500000 − 0.866025i 0 0.642788 + 0.766044i
1182.1 −0.939693 − 0.342020i −0.984808 + 0.173648i 0 −0.642788 − 0.766044i 0.984808 + 0.173648i −0.866025 − 0.500000i 0.500000 + 0.866025i 0 0.342020 + 0.939693i
1182.2 −0.939693 − 0.342020i 0.984808 − 0.173648i 0 0.642788 + 0.766044i −0.984808 − 0.173648i 0.866025 + 0.500000i 0.500000 + 0.866025i 0 −0.342020 − 0.939693i
1833.1 0.173648 + 0.984808i −0.642788 − 0.766044i 0 0.342020 − 0.939693i 0.642788 − 0.766044i 0.866025 + 0.500000i 0.500000 + 0.866025i 0 0.984808 + 0.173648i
1833.2 0.173648 + 0.984808i 0.642788 + 0.766044i 0 −0.342020 + 0.939693i −0.642788 + 0.766044i −0.866025 − 0.500000i 0.500000 + 0.866025i 0 −0.984808 − 0.173648i
2050.1 0.173648 − 0.984808i −0.642788 + 0.766044i 0 0.342020 + 0.939693i 0.642788 + 0.766044i 0.866025 − 0.500000i 0.500000 − 0.866025i 0 0.984808 − 0.173648i
2050.2 0.173648 − 0.984808i 0.642788 − 0.766044i 0 −0.342020 − 0.939693i −0.642788 − 0.766044i −0.866025 + 0.500000i 0.500000 − 0.866025i 0 −0.984808 + 0.173648i
2400.1 0.766044 − 0.642788i −0.342020 + 0.939693i 0 −0.984808 + 0.173648i 0.342020 + 0.939693i 0.866025 + 0.500000i 0.500000 + 0.866025i 0 −0.642788 + 0.766044i
2400.2 0.766044 − 0.642788i 0.342020 − 0.939693i 0 0.984808 − 0.173648i −0.342020 − 0.939693i −0.866025 − 0.500000i 0.500000 + 0.866025i 0 0.642788 − 0.766044i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 62.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
19.c even 3 2 inner
19.e even 9 3 inner
133.m odd 6 2 inner
133.y odd 18 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2527.1.y.e 12
7.b odd 2 1 inner 2527.1.y.e 12
19.b odd 2 1 2527.1.y.d 12
19.c even 3 2 inner 2527.1.y.e 12
19.d odd 6 2 2527.1.y.d 12
19.e even 9 2 133.1.m.a ✓ 4
19.e even 9 1 2527.1.d.e 2
19.e even 9 3 inner 2527.1.y.e 12
19.f odd 18 1 2527.1.d.b 2
19.f odd 18 2 2527.1.m.d 4
19.f odd 18 3 2527.1.y.d 12
57.l odd 18 2 1197.1.ci.c 4
76.l odd 18 2 2128.1.cy.a 4
95.p even 18 2 3325.1.bc.a 4
95.q odd 36 2 3325.1.bi.a 4
95.q odd 36 2 3325.1.bi.b 4
133.c even 2 1 2527.1.y.d 12
133.m odd 6 2 inner 2527.1.y.e 12
133.p even 6 2 2527.1.y.d 12
133.u even 9 1 931.1.k.a 4
133.u even 9 1 931.1.t.a 4
133.w even 9 1 931.1.k.a 4
133.w even 9 1 931.1.t.a 4
133.x odd 18 1 931.1.k.a 4
133.x odd 18 1 931.1.t.a 4
133.y odd 18 2 133.1.m.a ✓ 4
133.y odd 18 1 2527.1.d.e 2
133.y odd 18 3 inner 2527.1.y.e 12
133.z odd 18 1 931.1.k.a 4
133.z odd 18 1 931.1.t.a 4
133.ba even 18 1 2527.1.d.b 2
133.ba even 18 2 2527.1.m.d 4
133.ba even 18 3 2527.1.y.d 12
399.cj even 18 2 1197.1.ci.c 4
532.cc even 18 2 2128.1.cy.a 4
665.cv odd 18 2 3325.1.bc.a 4
665.dm even 36 2 3325.1.bi.a 4
665.dm even 36 2 3325.1.bi.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
133.1.m.a ✓ 4 19.e even 9 2
133.1.m.a ✓ 4 133.y odd 18 2
931.1.k.a 4 133.u even 9 1
931.1.k.a 4 133.w even 9 1
931.1.k.a 4 133.x odd 18 1
931.1.k.a 4 133.z odd 18 1
931.1.t.a 4 133.u even 9 1
931.1.t.a 4 133.w even 9 1
931.1.t.a 4 133.x odd 18 1
931.1.t.a 4 133.z odd 18 1
1197.1.ci.c 4 57.l odd 18 2
1197.1.ci.c 4 399.cj even 18 2
2128.1.cy.a 4 76.l odd 18 2
2128.1.cy.a 4 532.cc even 18 2
2527.1.d.b 2 19.f odd 18 1
2527.1.d.b 2 133.ba even 18 1
2527.1.d.e 2 19.e even 9 1
2527.1.d.e 2 133.y odd 18 1
2527.1.m.d 4 19.f odd 18 2
2527.1.m.d 4 133.ba even 18 2
2527.1.y.d 12 19.b odd 2 1
2527.1.y.d 12 19.d odd 6 2
2527.1.y.d 12 19.f odd 18 3
2527.1.y.d 12 133.c even 2 1
2527.1.y.d 12 133.p even 6 2
2527.1.y.d 12 133.ba even 18 3
2527.1.y.e 12 1.a even 1 1 trivial
2527.1.y.e 12 7.b odd 2 1 inner
2527.1.y.e 12 19.c even 3 2 inner
2527.1.y.e 12 19.e even 9 3 inner
2527.1.y.e 12 133.m odd 6 2 inner
2527.1.y.e 12 133.y odd 18 3 inner
3325.1.bc.a 4 95.p even 18 2
3325.1.bc.a 4 665.cv odd 18 2
3325.1.bi.a 4 95.q odd 36 2
3325.1.bi.a 4 665.dm even 36 2
3325.1.bi.b 4 95.q odd 36 2
3325.1.bi.b 4 665.dm even 36 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + T_{2}^{3} + 1 \) acting on \(S_{1}^{\mathrm{new}}(2527, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} + T^{3} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} - T^{6} + 1 \) Copy content Toggle raw display
$5$ \( T^{12} - T^{6} + 1 \) Copy content Toggle raw display
$7$ \( (T^{4} - T^{2} + 1)^{3} \) Copy content Toggle raw display
$11$ \( T^{12} \) 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
$29$ \( (T^{6} - T^{3} + 1)^{2} \) 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
$43$ \( (T^{6} + T^{3} + 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} - T^{6} + 1 \) Copy content Toggle raw display
$53$ \( (T^{6} - T^{3} + 1)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} - T^{6} + 1 \) 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
$79$ \( (T^{6} + T^{3} + 1)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} \) Copy content Toggle raw display
$89$ \( T^{12} - T^{6} + 1 \) Copy content Toggle raw display
$97$ \( T^{12} - T^{6} + 1 \) Copy content Toggle raw display
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