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Properties

Label 1029.1.n.a
Level $1029$
Weight $1$
Character orbit 1029.n
Analytic conductor $0.514$
Analytic rank $0$
Dimension $12$
Projective image $D_{7}$
CM discriminant -3
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1029,1,Mod(116,1029)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1029, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([21, 4]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1029.116");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1029 = 3 \cdot 7^{3} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1029.n (of order \(42\), degree \(12\), 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: \(0.513537897999\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\Q(\zeta_{21})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} + x^{9} - x^{8} + x^{6} - x^{4} + x^{3} - x + 1 \) 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 147)
Projective image: \(D_{7}\)
Projective field: Galois closure of 7.1.373714754427.1

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{42}^{13} q^{3} + \zeta_{42}^{4} q^{4} - \zeta_{42}^{5} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{42}^{13} q^{3} + \zeta_{42}^{4} q^{4} - \zeta_{42}^{5} q^{9} + \zeta_{42}^{17} q^{12} + (\zeta_{42}^{15} + \zeta_{42}^{3}) q^{13} + \zeta_{42}^{8} q^{16} + (\zeta_{42}^{5} - \zeta_{42}^{2}) q^{19} - \zeta_{42}^{19} q^{25} - \zeta_{42}^{18} q^{27} + ( - \zeta_{42}^{10} - \zeta_{42}^{4}) q^{31} - \zeta_{42}^{9} q^{36} + (\zeta_{42}^{20} + \zeta_{42}^{14}) q^{37} + (\zeta_{42}^{16} - \zeta_{42}^{7}) q^{39} + (\zeta_{42}^{6} - \zeta_{42}^{3}) q^{43} - q^{48} + (\zeta_{42}^{19} + \zeta_{42}^{7}) q^{52} + (\zeta_{42}^{18} - \zeta_{42}^{15}) q^{57} + ( - \zeta_{42}^{20} - \zeta_{42}^{14}) q^{61} + \zeta_{42}^{12} q^{64} + ( - \zeta_{42}^{13} - \zeta_{42}) q^{67} + ( - \zeta_{42}^{16} + \zeta_{42}) q^{73} + \zeta_{42}^{11} q^{75} + (\zeta_{42}^{9} - \zeta_{42}^{6}) q^{76} + ( - \zeta_{42}^{17} - \zeta_{42}^{11}) q^{79} + \zeta_{42}^{10} q^{81} + ( - \zeta_{42}^{17} + \zeta_{42}^{2}) q^{93} + ( - \zeta_{42}^{12} + \zeta_{42}^{9}) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - q^{3} + q^{4} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - q^{3} + q^{4} + q^{9} - q^{12} + 4 q^{13} + q^{16} - 2 q^{19} + q^{25} + 2 q^{27} - 2 q^{31} - 2 q^{36} - 5 q^{37} - 5 q^{39} - 4 q^{43} - 12 q^{48} + 5 q^{52} - 4 q^{57} + 5 q^{61} - 2 q^{64} + 2 q^{67} - 2 q^{73} - q^{75} + 4 q^{76} + 2 q^{79} + q^{81} + 2 q^{93} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1029\mathbb{Z}\right)^\times\).

\(n\) \(344\) \(346\)
\(\chi(n)\) \(-1\) \(-\zeta_{42}^{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
116.1
−0.733052 − 0.680173i
−0.988831 − 0.149042i
0.955573 + 0.294755i
−0.733052 + 0.680173i
−0.988831 + 0.149042i
0.0747301 + 0.997204i
0.826239 − 0.563320i
0.826239 + 0.563320i
0.955573 − 0.294755i
0.0747301 − 0.997204i
0.365341 − 0.930874i
0.365341 + 0.930874i
0 −0.955573 − 0.294755i −0.988831 + 0.149042i 0 0 0 0 0.826239 + 0.563320i 0
128.1 0 −0.365341 + 0.930874i 0.826239 + 0.563320i 0 0 0 0 −0.733052 − 0.680173i 0
263.1 0 0.733052 + 0.680173i 0.365341 + 0.930874i 0 0 0 0 0.0747301 + 0.997204i 0
275.1 0 −0.955573 + 0.294755i −0.988831 − 0.149042i 0 0 0 0 0.826239 − 0.563320i 0
410.1 0 −0.365341 − 0.930874i 0.826239 − 0.563320i 0 0 0 0 −0.733052 + 0.680173i 0
422.1 0 −0.826239 − 0.563320i 0.955573 − 0.294755i 0 0 0 0 0.365341 + 0.930874i 0
557.1 0 −0.0747301 + 0.997204i −0.733052 − 0.680173i 0 0 0 0 −0.988831 − 0.149042i 0
569.1 0 −0.0747301 − 0.997204i −0.733052 + 0.680173i 0 0 0 0 −0.988831 + 0.149042i 0
716.1 0 0.733052 − 0.680173i 0.365341 − 0.930874i 0 0 0 0 0.0747301 − 0.997204i 0
851.1 0 −0.826239 + 0.563320i 0.955573 + 0.294755i 0 0 0 0 0.365341 − 0.930874i 0
863.1 0 0.988831 + 0.149042i 0.0747301 + 0.997204i 0 0 0 0 0.955573 + 0.294755i 0
998.1 0 0.988831 − 0.149042i 0.0747301 − 0.997204i 0 0 0 0 0.955573 − 0.294755i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 116.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
7.c even 3 1 inner
21.h odd 6 1 inner
49.e even 7 1 inner
49.g even 21 1 inner
147.l odd 14 1 inner
147.n odd 42 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1029.1.n.a 12
3.b odd 2 1 CM 1029.1.n.a 12
7.b odd 2 1 1029.1.n.b 12
7.c even 3 1 1029.1.l.a 6
7.c even 3 1 inner 1029.1.n.a 12
7.d odd 6 1 147.1.l.a ✓ 6
7.d odd 6 1 1029.1.n.b 12
21.c even 2 1 1029.1.n.b 12
21.g even 6 1 147.1.l.a ✓ 6
21.g even 6 1 1029.1.n.b 12
21.h odd 6 1 1029.1.l.a 6
21.h odd 6 1 inner 1029.1.n.a 12
28.f even 6 1 2352.1.cj.a 6
35.i odd 6 1 3675.1.bm.a 6
35.k even 12 2 3675.1.bj.a 12
49.e even 7 1 inner 1029.1.n.a 12
49.f odd 14 1 1029.1.n.b 12
49.g even 21 1 1029.1.l.a 6
49.g even 21 1 inner 1029.1.n.a 12
49.h odd 42 1 147.1.l.a ✓ 6
49.h odd 42 1 1029.1.n.b 12
63.i even 6 1 3969.1.bt.a 12
63.k odd 6 1 3969.1.bt.a 12
63.s even 6 1 3969.1.bt.a 12
63.t odd 6 1 3969.1.bt.a 12
84.j odd 6 1 2352.1.cj.a 6
105.p even 6 1 3675.1.bm.a 6
105.w odd 12 2 3675.1.bj.a 12
147.k even 14 1 1029.1.n.b 12
147.l odd 14 1 inner 1029.1.n.a 12
147.n odd 42 1 1029.1.l.a 6
147.n odd 42 1 inner 1029.1.n.a 12
147.o even 42 1 147.1.l.a ✓ 6
147.o even 42 1 1029.1.n.b 12
196.p even 42 1 2352.1.cj.a 6
245.u odd 42 1 3675.1.bm.a 6
245.x even 84 2 3675.1.bj.a 12
441.bc odd 42 1 3969.1.bt.a 12
441.bd even 42 1 3969.1.bt.a 12
441.bl odd 42 1 3969.1.bt.a 12
441.bn even 42 1 3969.1.bt.a 12
588.bf odd 42 1 2352.1.cj.a 6
735.bp even 42 1 3675.1.bm.a 6
735.bs odd 84 2 3675.1.bj.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.1.l.a ✓ 6 7.d odd 6 1
147.1.l.a ✓ 6 21.g even 6 1
147.1.l.a ✓ 6 49.h odd 42 1
147.1.l.a ✓ 6 147.o even 42 1
1029.1.l.a 6 7.c even 3 1
1029.1.l.a 6 21.h odd 6 1
1029.1.l.a 6 49.g even 21 1
1029.1.l.a 6 147.n odd 42 1
1029.1.n.a 12 1.a even 1 1 trivial
1029.1.n.a 12 3.b odd 2 1 CM
1029.1.n.a 12 7.c even 3 1 inner
1029.1.n.a 12 21.h odd 6 1 inner
1029.1.n.a 12 49.e even 7 1 inner
1029.1.n.a 12 49.g even 21 1 inner
1029.1.n.a 12 147.l odd 14 1 inner
1029.1.n.a 12 147.n odd 42 1 inner
1029.1.n.b 12 7.b odd 2 1
1029.1.n.b 12 7.d odd 6 1
1029.1.n.b 12 21.c even 2 1
1029.1.n.b 12 21.g even 6 1
1029.1.n.b 12 49.f odd 14 1
1029.1.n.b 12 49.h odd 42 1
1029.1.n.b 12 147.k even 14 1
1029.1.n.b 12 147.o even 42 1
2352.1.cj.a 6 28.f even 6 1
2352.1.cj.a 6 84.j odd 6 1
2352.1.cj.a 6 196.p even 42 1
2352.1.cj.a 6 588.bf odd 42 1
3675.1.bj.a 12 35.k even 12 2
3675.1.bj.a 12 105.w odd 12 2
3675.1.bj.a 12 245.x even 84 2
3675.1.bj.a 12 735.bs odd 84 2
3675.1.bm.a 6 35.i odd 6 1
3675.1.bm.a 6 105.p even 6 1
3675.1.bm.a 6 245.u odd 42 1
3675.1.bm.a 6 735.bp even 42 1
3969.1.bt.a 12 63.i even 6 1
3969.1.bt.a 12 63.k odd 6 1
3969.1.bt.a 12 63.s even 6 1
3969.1.bt.a 12 63.t odd 6 1
3969.1.bt.a 12 441.bc odd 42 1
3969.1.bt.a 12 441.bd even 42 1
3969.1.bt.a 12 441.bl odd 42 1
3969.1.bt.a 12 441.bn even 42 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13}^{6} - 2T_{13}^{5} + 4T_{13}^{4} - T_{13}^{3} + 2T_{13}^{2} + 3T_{13} + 1 \) acting on \(S_{1}^{\mathrm{new}}(1029, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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