Properties

Label 1014.2.e.i
Level $1014$
Weight $2$
Character orbit 1014.e
Analytic conductor $8.097$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1014,2,Mod(529,1014)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1014, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([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("1014.529");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1014 = 2 \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1014.e (of order \(3\), degree \(2\), 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: \(8.09683076496\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 78)
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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_1 q^{3} + (\beta_1 - 1) q^{4} + ( - \beta_{3} - 2) q^{5} + ( - \beta_1 + 1) q^{6} + ( - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{7} - q^{8} + (\beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_1 q^{3} + (\beta_1 - 1) q^{4} + ( - \beta_{3} - 2) q^{5} + ( - \beta_1 + 1) q^{6} + ( - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{7} - q^{8} + (\beta_1 - 1) q^{9} + ( - \beta_{2} - 2 \beta_1) q^{10} + ( - \beta_{2} + 3 \beta_1) q^{11} + q^{12} + ( - \beta_{3} - 1) q^{14} + (\beta_{2} + 2 \beta_1) q^{15} - \beta_1 q^{16} + (\beta_{3} - \beta_{2} - 4 \beta_1 + 4) q^{17} - q^{18} + (\beta_{3} - \beta_{2} - 3 \beta_1 + 3) q^{19} + (\beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{20} + (\beta_{3} + 1) q^{21} + (\beta_{3} - \beta_{2} + 3 \beta_1 - 3) q^{22} + ( - 3 \beta_{2} + \beta_1) q^{23} + \beta_1 q^{24} + (4 \beta_{3} + 2) q^{25} + q^{27} + ( - \beta_{2} - \beta_1) q^{28} + (2 \beta_{2} + \beta_1) q^{29} + ( - \beta_{3} + \beta_{2} + 2 \beta_1 - 2) q^{30} + ( - 2 \beta_{3} + 2) q^{31} + ( - \beta_1 + 1) q^{32} + ( - \beta_{3} + \beta_{2} - 3 \beta_1 + 3) q^{33} + (\beta_{3} + 4) q^{34} + (3 \beta_{3} - 3 \beta_{2} - 5 \beta_1 + 5) q^{35} - \beta_1 q^{36} + ( - 2 \beta_{2} + 7 \beta_1) q^{37} + (\beta_{3} + 3) q^{38} + (\beta_{3} + 2) q^{40} + ( - 6 \beta_{2} + \beta_1) q^{41} + (\beta_{2} + \beta_1) q^{42} + (5 \beta_{3} - 5 \beta_{2} - \beta_1 + 1) q^{43} + (\beta_{3} - 3) q^{44} + (\beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{45} + (3 \beta_{3} - 3 \beta_{2} + \beta_1 - 1) q^{46} + ( - 3 \beta_{3} + 3) q^{47} + (\beta_1 - 1) q^{48} + ( - 2 \beta_{2} + 3 \beta_1) q^{49} + (4 \beta_{2} + 2 \beta_1) q^{50} + ( - \beta_{3} - 4) q^{51} + ( - 2 \beta_{3} - 3) q^{53} + \beta_1 q^{54} + ( - \beta_{2} - 3 \beta_1) q^{55} + (\beta_{3} - \beta_{2} - \beta_1 + 1) q^{56} + ( - \beta_{3} - 3) q^{57} + ( - 2 \beta_{3} + 2 \beta_{2} + \beta_1 - 1) q^{58} + ( - 8 \beta_1 + 8) q^{59} + ( - \beta_{3} - 2) q^{60} + (3 \beta_{3} - 3 \beta_{2} - 4 \beta_1 + 4) q^{61} + ( - 2 \beta_{2} + 2 \beta_1) q^{62} + ( - \beta_{2} - \beta_1) q^{63} + q^{64} + ( - \beta_{3} + 3) q^{66} + (7 \beta_{2} + \beta_1) q^{67} + (\beta_{2} + 4 \beta_1) q^{68} + ( - 3 \beta_{3} + 3 \beta_{2} - \beta_1 + 1) q^{69} + (3 \beta_{3} + 5) q^{70} + (\beta_{3} - \beta_{2} - 3 \beta_1 + 3) q^{71} + ( - \beta_1 + 1) q^{72} + ( - \beta_{3} + 8) q^{73} + (2 \beta_{3} - 2 \beta_{2} + 7 \beta_1 - 7) q^{74} + ( - 4 \beta_{2} - 2 \beta_1) q^{75} + (\beta_{2} + 3 \beta_1) q^{76} - 2 \beta_{3} q^{77} + (2 \beta_{3} - 6) q^{79} + (\beta_{2} + 2 \beta_1) q^{80} - \beta_1 q^{81} + (6 \beta_{3} - 6 \beta_{2} + \beta_1 - 1) q^{82} + (3 \beta_{3} - 5) q^{83} + ( - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{84} + ( - 6 \beta_{3} + 6 \beta_{2} + 11 \beta_1 - 11) q^{85} + (5 \beta_{3} + 1) q^{86} + (2 \beta_{3} - 2 \beta_{2} - \beta_1 + 1) q^{87} + (\beta_{2} - 3 \beta_1) q^{88} + ( - 2 \beta_{2} - 6 \beta_1) q^{89} + (\beta_{3} + 2) q^{90} + (3 \beta_{3} - 1) q^{92} + (2 \beta_{2} - 2 \beta_1) q^{93} + ( - 3 \beta_{2} + 3 \beta_1) q^{94} + ( - 5 \beta_{3} + 5 \beta_{2} + 9 \beta_1 - 9) q^{95} - q^{96} + (6 \beta_1 - 6) q^{97} + (2 \beta_{3} - 2 \beta_{2} + 3 \beta_1 - 3) q^{98} + (\beta_{3} - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 2 q^{3} - 2 q^{4} - 8 q^{5} + 2 q^{6} - 2 q^{7} - 4 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} - 2 q^{3} - 2 q^{4} - 8 q^{5} + 2 q^{6} - 2 q^{7} - 4 q^{8} - 2 q^{9} - 4 q^{10} + 6 q^{11} + 4 q^{12} - 4 q^{14} + 4 q^{15} - 2 q^{16} + 8 q^{17} - 4 q^{18} + 6 q^{19} + 4 q^{20} + 4 q^{21} - 6 q^{22} + 2 q^{23} + 2 q^{24} + 8 q^{25} + 4 q^{27} - 2 q^{28} + 2 q^{29} - 4 q^{30} + 8 q^{31} + 2 q^{32} + 6 q^{33} + 16 q^{34} + 10 q^{35} - 2 q^{36} + 14 q^{37} + 12 q^{38} + 8 q^{40} + 2 q^{41} + 2 q^{42} + 2 q^{43} - 12 q^{44} + 4 q^{45} - 2 q^{46} + 12 q^{47} - 2 q^{48} + 6 q^{49} + 4 q^{50} - 16 q^{51} - 12 q^{53} + 2 q^{54} - 6 q^{55} + 2 q^{56} - 12 q^{57} - 2 q^{58} + 16 q^{59} - 8 q^{60} + 8 q^{61} + 4 q^{62} - 2 q^{63} + 4 q^{64} + 12 q^{66} + 2 q^{67} + 8 q^{68} + 2 q^{69} + 20 q^{70} + 6 q^{71} + 2 q^{72} + 32 q^{73} - 14 q^{74} - 4 q^{75} + 6 q^{76} - 24 q^{79} + 4 q^{80} - 2 q^{81} - 2 q^{82} - 20 q^{83} - 2 q^{84} - 22 q^{85} + 4 q^{86} + 2 q^{87} - 6 q^{88} - 12 q^{89} + 8 q^{90} - 4 q^{92} - 4 q^{93} + 6 q^{94} - 18 q^{95} - 4 q^{96} - 12 q^{97} - 6 q^{98} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{12}^{3} + \zeta_{12} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( ( -\beta_{3} + 2\beta_{2} ) / 3 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1014\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(847\)
\(\chi(n)\) \(1\) \(-1 + \beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
529.1
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
0.500000 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i −3.73205 0.500000 + 0.866025i −1.36603 2.36603i −1.00000 −0.500000 0.866025i −1.86603 + 3.23205i
529.2 0.500000 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i −0.267949 0.500000 + 0.866025i 0.366025 + 0.633975i −1.00000 −0.500000 0.866025i −0.133975 + 0.232051i
991.1 0.500000 + 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i −3.73205 0.500000 0.866025i −1.36603 + 2.36603i −1.00000 −0.500000 + 0.866025i −1.86603 3.23205i
991.2 0.500000 + 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i −0.267949 0.500000 0.866025i 0.366025 0.633975i −1.00000 −0.500000 + 0.866025i −0.133975 0.232051i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1014.2.e.i 4
13.b even 2 1 1014.2.e.g 4
13.c even 3 1 1014.2.a.i 2
13.c even 3 1 inner 1014.2.e.i 4
13.d odd 4 1 78.2.i.a 4
13.d odd 4 1 1014.2.i.a 4
13.e even 6 1 1014.2.a.k 2
13.e even 6 1 1014.2.e.g 4
13.f odd 12 1 78.2.i.a 4
13.f odd 12 2 1014.2.b.e 4
13.f odd 12 1 1014.2.i.a 4
39.f even 4 1 234.2.l.c 4
39.h odd 6 1 3042.2.a.p 2
39.i odd 6 1 3042.2.a.y 2
39.k even 12 1 234.2.l.c 4
39.k even 12 2 3042.2.b.i 4
52.f even 4 1 624.2.bv.e 4
52.i odd 6 1 8112.2.a.bp 2
52.j odd 6 1 8112.2.a.bj 2
52.l even 12 1 624.2.bv.e 4
65.f even 4 1 1950.2.y.b 4
65.g odd 4 1 1950.2.bc.d 4
65.k even 4 1 1950.2.y.g 4
65.o even 12 1 1950.2.y.g 4
65.s odd 12 1 1950.2.bc.d 4
65.t even 12 1 1950.2.y.b 4
156.l odd 4 1 1872.2.by.h 4
156.v odd 12 1 1872.2.by.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.i.a 4 13.d odd 4 1
78.2.i.a 4 13.f odd 12 1
234.2.l.c 4 39.f even 4 1
234.2.l.c 4 39.k even 12 1
624.2.bv.e 4 52.f even 4 1
624.2.bv.e 4 52.l even 12 1
1014.2.a.i 2 13.c even 3 1
1014.2.a.k 2 13.e even 6 1
1014.2.b.e 4 13.f odd 12 2
1014.2.e.g 4 13.b even 2 1
1014.2.e.g 4 13.e even 6 1
1014.2.e.i 4 1.a even 1 1 trivial
1014.2.e.i 4 13.c even 3 1 inner
1014.2.i.a 4 13.d odd 4 1
1014.2.i.a 4 13.f odd 12 1
1872.2.by.h 4 156.l odd 4 1
1872.2.by.h 4 156.v odd 12 1
1950.2.y.b 4 65.f even 4 1
1950.2.y.b 4 65.t even 12 1
1950.2.y.g 4 65.k even 4 1
1950.2.y.g 4 65.o even 12 1
1950.2.bc.d 4 65.g odd 4 1
1950.2.bc.d 4 65.s odd 12 1
3042.2.a.p 2 39.h odd 6 1
3042.2.a.y 2 39.i odd 6 1
3042.2.b.i 4 39.k even 12 2
8112.2.a.bj 2 52.j odd 6 1
8112.2.a.bp 2 52.i odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1014, [\chi])\):

\( T_{5}^{2} + 4T_{5} + 1 \) Copy content Toggle raw display
\( T_{7}^{4} + 2T_{7}^{3} + 6T_{7}^{2} - 4T_{7} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 4 T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 2 T^{3} + 6 T^{2} - 4 T + 4 \) Copy content Toggle raw display
$11$ \( T^{4} - 6 T^{3} + 30 T^{2} - 36 T + 36 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} - 8 T^{3} + 51 T^{2} - 104 T + 169 \) Copy content Toggle raw display
$19$ \( T^{4} - 6 T^{3} + 30 T^{2} - 36 T + 36 \) Copy content Toggle raw display
$23$ \( T^{4} - 2 T^{3} + 30 T^{2} + 52 T + 676 \) Copy content Toggle raw display
$29$ \( T^{4} - 2 T^{3} + 15 T^{2} + 22 T + 121 \) Copy content Toggle raw display
$31$ \( (T^{2} - 4 T - 8)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 14 T^{3} + 159 T^{2} + \cdots + 1369 \) Copy content Toggle raw display
$41$ \( T^{4} - 2 T^{3} + 111 T^{2} + \cdots + 11449 \) Copy content Toggle raw display
$43$ \( T^{4} - 2 T^{3} + 78 T^{2} + \cdots + 5476 \) Copy content Toggle raw display
$47$ \( (T^{2} - 6 T - 18)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 6 T - 3)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 8 T + 64)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} - 8 T^{3} + 75 T^{2} + 88 T + 121 \) Copy content Toggle raw display
$67$ \( T^{4} - 2 T^{3} + 150 T^{2} + \cdots + 21316 \) Copy content Toggle raw display
$71$ \( T^{4} - 6 T^{3} + 30 T^{2} - 36 T + 36 \) Copy content Toggle raw display
$73$ \( (T^{2} - 16 T + 61)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 12 T + 24)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 10 T - 2)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 12 T^{3} + 120 T^{2} + \cdots + 576 \) Copy content Toggle raw display
$97$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
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