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Properties

Label 1029.2.a.h
Level $1029$
Weight $2$
Character orbit 1029.a
Self dual yes
Analytic conductor $8.217$
Analytic rank $0$
Dimension $9$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1029,2,Mod(1,1029)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1029, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1029.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1029 = 3 \cdot 7^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1029.a (trivial)

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: yes
Analytic conductor: \(8.21660636799\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 4x^{8} - 9x^{7} + 35x^{6} + 40x^{5} - 92x^{4} - 88x^{3} + 52x^{2} + 32x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} + q^{3} + ( - \beta_{6} + \beta_{5} + 2) q^{4} - \beta_{7} q^{5} + ( - \beta_1 + 1) q^{6} + (\beta_{5} + \beta_{3} - \beta_{2} + \cdots + 3) q^{8}+ \cdots + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{2} + q^{3} + ( - \beta_{6} + \beta_{5} + 2) q^{4} - \beta_{7} q^{5} + ( - \beta_1 + 1) q^{6} + (\beta_{5} + \beta_{3} - \beta_{2} + \cdots + 3) q^{8}+ \cdots + ( - \beta_{8} - \beta_{4} + \beta_{2} + \cdots + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 5 q^{2} + 9 q^{3} + 17 q^{4} - q^{5} + 5 q^{6} + 12 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + 5 q^{2} + 9 q^{3} + 17 q^{4} - q^{5} + 5 q^{6} + 12 q^{8} + 9 q^{9} - q^{10} + 21 q^{11} + 17 q^{12} + 11 q^{13} - q^{15} + 29 q^{16} - 6 q^{17} + 5 q^{18} - 2 q^{19} - 20 q^{20} + 9 q^{23} + 12 q^{24} + 10 q^{25} - 23 q^{26} + 9 q^{27} + 12 q^{29} - q^{30} - 3 q^{31} + 24 q^{32} + 21 q^{33} + 12 q^{34} + 17 q^{36} + 8 q^{37} - 15 q^{38} + 11 q^{39} + 7 q^{40} - 17 q^{41} + 16 q^{43} + 46 q^{44} - q^{45} + 13 q^{46} - 7 q^{47} + 29 q^{48} + 3 q^{50} - 6 q^{51} + 26 q^{52} - 8 q^{53} + 5 q^{54} - 6 q^{55} - 2 q^{57} - 22 q^{58} - 30 q^{59} - 20 q^{60} + 7 q^{61} - 43 q^{62} + 42 q^{64} - 15 q^{65} + 33 q^{67} - 66 q^{68} + 9 q^{69} + 30 q^{71} + 12 q^{72} + 48 q^{73} - 8 q^{74} + 10 q^{75} - 34 q^{76} - 23 q^{78} + 7 q^{79} - 98 q^{80} + 9 q^{81} - 10 q^{82} - 11 q^{83} - 18 q^{85} - 3 q^{86} + 12 q^{87} - 47 q^{88} - 25 q^{89} - q^{90} - 21 q^{92} - 3 q^{93} + 11 q^{94} - 13 q^{95} + 24 q^{96} + 50 q^{97} + 21 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 4x^{8} - 9x^{7} + 35x^{6} + 40x^{5} - 92x^{4} - 88x^{3} + 52x^{2} + 32x - 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{8} - 5\nu^{7} - 3\nu^{6} + 34\nu^{5} - \nu^{4} - 60\nu^{3} - 10\nu^{2} - 4\nu + 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{8} + 11\nu^{7} + 2\nu^{6} - 75\nu^{5} + 29\nu^{4} + 154\nu^{3} - 42\nu^{2} - 64\nu + 12 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{8} - 17\nu^{7} + \nu^{6} + 106\nu^{5} - 59\nu^{4} - 196\nu^{3} + 80\nu^{2} + 64\nu - 20 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{8} - 16\nu^{7} - 5\nu^{6} + 109\nu^{5} - 30\nu^{4} - 218\nu^{3} + 44\nu^{2} + 72\nu - 28 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3\nu^{8} - 16\nu^{7} - 5\nu^{6} + 109\nu^{5} - 30\nu^{4} - 218\nu^{3} + 40\nu^{2} + 80\nu - 16 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 4\nu^{8} - 21\nu^{7} - 8\nu^{6} + 141\nu^{5} - 19\nu^{4} - 288\nu^{3} - 4\nu^{2} + 108\nu - 4 ) / 4 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -5\nu^{8} + 28\nu^{7} + \nu^{6} - 179\nu^{5} + 80\nu^{4} + 344\nu^{3} - 92\nu^{2} - 116\nu + 24 ) / 4 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{6} + \beta_{5} + 2\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -3\beta_{6} + 2\beta_{5} - \beta_{3} + \beta_{2} + 9\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{8} + \beta_{7} - 14\beta_{6} + 9\beta_{5} + \beta_{4} - 5\beta_{3} + 3\beta_{2} + 27\beta _1 + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 6\beta_{8} + 4\beta_{7} - 50\beta_{6} + 27\beta_{5} + 6\beta_{4} - 25\beta_{3} + 15\beta_{2} + 99\beta _1 + 32 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 31\beta_{8} + 21\beta_{7} - 193\beta_{6} + 99\beta_{5} + 33\beta_{4} - 100\beta_{3} + 54\beta_{2} + 336\beta _1 + 113 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 139 \beta_{8} + 85 \beta_{7} - 704 \beta_{6} + 336 \beta_{5} + 147 \beta_{4} - 402 \beta_{3} + \cdots + 300 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 585 \beta_{8} + 353 \beta_{7} - 2603 \beta_{6} + 1198 \beta_{5} + 631 \beta_{4} - 1525 \beta_{3} + \cdots + 969 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
3.63032
2.77214
2.15127
0.664823
0.209021
−0.671292
−1.49320
−1.53612
−1.72696
−2.63032 1.00000 4.91857 −2.91453 −2.63032 0 −7.67677 1.00000 7.66614
1.2 −1.77214 1.00000 1.14049 2.38502 −1.77214 0 1.52317 1.00000 −4.22660
1.3 −1.15127 1.00000 −0.674571 0.942951 −1.15127 0 3.07916 1.00000 −1.08559
1.4 0.335177 1.00000 −1.88766 1.43430 0.335177 0 −1.30305 1.00000 0.480743
1.5 0.790979 1.00000 −1.37435 −3.94710 0.790979 0 −2.66904 1.00000 −3.12208
1.6 1.67129 1.00000 0.793217 3.38417 1.67129 0 −2.01689 1.00000 5.65594
1.7 2.49320 1.00000 4.21606 −0.321703 2.49320 0 5.52510 1.00000 −0.802071
1.8 2.53612 1.00000 4.43192 1.11704 2.53612 0 6.16765 1.00000 2.83295
1.9 2.72696 1.00000 5.43631 −3.08015 2.72696 0 9.37069 1.00000 −8.39943
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)
\(7\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1029.2.a.h yes 9
3.b odd 2 1 3087.2.a.m 9
7.b odd 2 1 1029.2.a.g ✓ 9
7.c even 3 2 1029.2.e.e 18
7.d odd 6 2 1029.2.e.f 18
21.c even 2 1 3087.2.a.l 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1029.2.a.g ✓ 9 7.b odd 2 1
1029.2.a.h yes 9 1.a even 1 1 trivial
1029.2.e.e 18 7.c even 3 2
1029.2.e.f 18 7.d odd 6 2
3087.2.a.l 9 21.c even 2 1
3087.2.a.m 9 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1029))\):

\( T_{2}^{9} - 5T_{2}^{8} - 5T_{2}^{7} + 56T_{2}^{6} - 37T_{2}^{5} - 164T_{2}^{4} + 189T_{2}^{3} + 104T_{2}^{2} - 172T_{2} + 41 \) Copy content Toggle raw display
\( T_{5}^{9} + T_{5}^{8} - 27T_{5}^{7} - 8T_{5}^{6} + 240T_{5}^{5} - 109T_{5}^{4} - 713T_{5}^{3} + 847T_{5}^{2} - 92T_{5} - 139 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} - 5 T^{8} + \cdots + 41 \) Copy content Toggle raw display
$3$ \( (T - 1)^{9} \) Copy content Toggle raw display
$5$ \( T^{9} + T^{8} + \cdots - 139 \) Copy content Toggle raw display
$7$ \( T^{9} \) Copy content Toggle raw display
$11$ \( T^{9} - 21 T^{8} + \cdots + 3569 \) Copy content Toggle raw display
$13$ \( T^{9} - 11 T^{8} + \cdots - 3136 \) Copy content Toggle raw display
$17$ \( T^{9} + 6 T^{8} + \cdots - 56 \) Copy content Toggle raw display
$19$ \( T^{9} + 2 T^{8} + \cdots + 953 \) Copy content Toggle raw display
$23$ \( T^{9} - 9 T^{8} + \cdots + 14063 \) Copy content Toggle raw display
$29$ \( T^{9} - 12 T^{8} + \cdots + 1857472 \) Copy content Toggle raw display
$31$ \( T^{9} + 3 T^{8} + \cdots + 8827 \) Copy content Toggle raw display
$37$ \( T^{9} - 8 T^{8} + \cdots + 3704 \) Copy content Toggle raw display
$41$ \( T^{9} + 17 T^{8} + \cdots - 216881 \) Copy content Toggle raw display
$43$ \( T^{9} - 16 T^{8} + \cdots + 799168 \) Copy content Toggle raw display
$47$ \( T^{9} + 7 T^{8} + \cdots + 1320256 \) Copy content Toggle raw display
$53$ \( T^{9} + 8 T^{8} + \cdots + 2542016 \) Copy content Toggle raw display
$59$ \( T^{9} + 30 T^{8} + \cdots - 18752 \) Copy content Toggle raw display
$61$ \( T^{9} - 7 T^{8} + \cdots - 4126912 \) Copy content Toggle raw display
$67$ \( T^{9} - 33 T^{8} + \cdots + 860672 \) Copy content Toggle raw display
$71$ \( T^{9} - 30 T^{8} + \cdots + 1412957 \) Copy content Toggle raw display
$73$ \( T^{9} - 48 T^{8} + \cdots + 258637504 \) Copy content Toggle raw display
$79$ \( T^{9} - 7 T^{8} + \cdots - 2126144 \) Copy content Toggle raw display
$83$ \( T^{9} + 11 T^{8} + \cdots + 4798528 \) Copy content Toggle raw display
$89$ \( T^{9} + 25 T^{8} + \cdots - 14421673 \) Copy content Toggle raw display
$97$ \( T^{9} - 50 T^{8} + \cdots + 14149184 \) Copy content Toggle raw display
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