↑

Properties

Label 8034.2.a.r
Level $8034$
Weight $2$
Character orbit 8034.a
Self dual yes
Analytic conductor $64.152$
Analytic rank $1$
Dimension $9$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8034,2,Mod(1,8034)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8034, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("8034.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8034.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: \(64.1518129839\)
Analytic rank: \(1\)
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: \( x^{9} - 4x^{8} - 9x^{7} + 45x^{6} + 7x^{5} - 123x^{4} + 37x^{3} + 87x^{2} - 54x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} - \beta_1 q^{5} - q^{6} - \beta_{3} q^{7} + q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - q^{3} + q^{4} - \beta_1 q^{5} - q^{6} - \beta_{3} q^{7} + q^{8} + q^{9} - \beta_1 q^{10} + ( - \beta_{7} - 1) q^{11} - q^{12} + q^{13} - \beta_{3} q^{14} + \beta_1 q^{15} + q^{16} + (\beta_{6} + \beta_{5} - \beta_{4} + \cdots - 1) q^{17}+ \cdots + ( - \beta_{7} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 9 q^{2} - 9 q^{3} + 9 q^{4} - 4 q^{5} - 9 q^{6} - 4 q^{7} + 9 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + 9 q^{2} - 9 q^{3} + 9 q^{4} - 4 q^{5} - 9 q^{6} - 4 q^{7} + 9 q^{8} + 9 q^{9} - 4 q^{10} - 5 q^{11} - 9 q^{12} + 9 q^{13} - 4 q^{14} + 4 q^{15} + 9 q^{16} - 6 q^{17} + 9 q^{18} - 4 q^{19} - 4 q^{20} + 4 q^{21} - 5 q^{22} - 6 q^{23} - 9 q^{24} - 11 q^{25} + 9 q^{26} - 9 q^{27} - 4 q^{28} - 19 q^{29} + 4 q^{30} - 6 q^{31} + 9 q^{32} + 5 q^{33} - 6 q^{34} + 10 q^{35} + 9 q^{36} - 13 q^{37} - 4 q^{38} - 9 q^{39} - 4 q^{40} - 18 q^{41} + 4 q^{42} - 20 q^{43} - 5 q^{44} - 4 q^{45} - 6 q^{46} + 14 q^{47} - 9 q^{48} - 3 q^{49} - 11 q^{50} + 6 q^{51} + 9 q^{52} - 3 q^{53} - 9 q^{54} - 4 q^{55} - 4 q^{56} + 4 q^{57} - 19 q^{58} - 9 q^{59} + 4 q^{60} - 24 q^{61} - 6 q^{62} - 4 q^{63} + 9 q^{64} - 4 q^{65} + 5 q^{66} - 4 q^{67} - 6 q^{68} + 6 q^{69} + 10 q^{70} - 9 q^{71} + 9 q^{72} - 24 q^{73} - 13 q^{74} + 11 q^{75} - 4 q^{76} + 3 q^{77} - 9 q^{78} - 15 q^{79} - 4 q^{80} + 9 q^{81} - 18 q^{82} + 20 q^{83} + 4 q^{84} - 31 q^{85} - 20 q^{86} + 19 q^{87} - 5 q^{88} + 3 q^{89} - 4 q^{90} - 4 q^{91} - 6 q^{92} + 6 q^{93} + 14 q^{94} - 4 q^{95} - 9 q^{96} - 19 q^{97} - 3 q^{98} - 5 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 4x^{8} - 9x^{7} + 45x^{6} + 7x^{5} - 123x^{4} + 37x^{3} + 87x^{2} - 54x + 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 55\nu^{8} - 272\nu^{7} - 307\nu^{6} + 2869\nu^{5} - 1267\nu^{4} - 6789\nu^{3} + 3935\nu^{2} + 4811\nu - 2412 ) / 634 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 129 \nu^{8} - 442 \nu^{7} - 1331 \nu^{6} + 4781 \nu^{5} + 2815 \nu^{4} - 11785 \nu^{3} - 223 \nu^{2} + \cdots - 3444 ) / 634 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 131 \nu^{8} + 498 \nu^{7} + 1273 \nu^{6} - 5381 \nu^{5} - 2377 \nu^{4} + 12931 \nu^{3} + \cdots + 1964 ) / 634 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 92\nu^{8} - 357\nu^{7} - 819\nu^{6} + 3825\nu^{5} + 774\nu^{4} - 9287\nu^{3} + 1539\nu^{2} + 5938\nu - 1977 ) / 317 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 359 \nu^{8} + 1176 \nu^{7} + 4171 \nu^{6} - 13551 \nu^{5} - 12675 \nu^{4} + 38965 \nu^{3} + \cdots + 5692 ) / 634 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 415 \nu^{8} + 1476 \nu^{7} + 4449 \nu^{6} - 17037 \nu^{5} - 10555 \nu^{4} + 49497 \nu^{3} + \cdots + 10534 ) / 634 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 261 \nu^{8} - 968 \nu^{7} - 2575 \nu^{6} + 10779 \nu^{5} + 4656 \nu^{4} - 28776 \nu^{3} + 28 \nu^{2} + \cdots - 5619 ) / 317 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} + \beta_{3} + \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} + \beta_{6} - 2\beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + 7\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{8} + 2\beta_{7} - \beta_{6} - 11\beta_{5} + \beta_{4} + 8\beta_{3} + 10\beta_{2} + \beta _1 + 21 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 14\beta_{8} + 2\beta_{7} + 10\beta_{6} - 27\beta_{5} - 7\beta_{4} + 12\beta_{3} - 7\beta_{2} + 54\beta _1 + 17 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 34\beta_{8} + 27\beta_{7} - 7\beta_{6} - 114\beta_{5} + 17\beta_{4} + 73\beta_{3} + 86\beta_{2} + 17\beta _1 + 177 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 165 \beta_{8} + 45 \beta_{7} + 86 \beta_{6} - 305 \beta_{5} - 25 \beta_{4} + 141 \beta_{3} - 35 \beta_{2} + \cdots + 213 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 445 \beta_{8} + 315 \beta_{7} - 35 \beta_{6} - 1165 \beta_{5} + 236 \beta_{4} + 715 \beta_{3} + \cdots + 1591 \) 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
3.20969
2.80812
1.90297
0.840786
0.435443
0.254984
−1.25914
−1.44298
−2.74988
1.00000 −1.00000 1.00000 −3.20969 −1.00000 −2.02210 1.00000 1.00000 −3.20969
1.2 1.00000 −1.00000 1.00000 −2.80812 −1.00000 −1.72587 1.00000 1.00000 −2.80812
1.3 1.00000 −1.00000 1.00000 −1.90297 −1.00000 −5.10863 1.00000 1.00000 −1.90297
1.4 1.00000 −1.00000 1.00000 −0.840786 −1.00000 2.87036 1.00000 1.00000 −0.840786
1.5 1.00000 −1.00000 1.00000 −0.435443 −1.00000 1.91967 1.00000 1.00000 −0.435443
1.6 1.00000 −1.00000 1.00000 −0.254984 −1.00000 2.89551 1.00000 1.00000 −0.254984
1.7 1.00000 −1.00000 1.00000 1.25914 −1.00000 −0.797445 1.00000 1.00000 1.25914
1.8 1.00000 −1.00000 1.00000 1.44298 −1.00000 0.367530 1.00000 1.00000 1.44298
1.9 1.00000 −1.00000 1.00000 2.74988 −1.00000 −2.39902 1.00000 1.00000 2.74988
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(13\) \(-1\)
\(103\) \(1\)

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 8034.2.a.r ✓ 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8034.2.a.r ✓ 9 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8034))\):

\( T_{5}^{9} + 4T_{5}^{8} - 9T_{5}^{7} - 45T_{5}^{6} + 7T_{5}^{5} + 123T_{5}^{4} + 37T_{5}^{3} - 87T_{5}^{2} - 54T_{5} - 8 \) Copy content Toggle raw display
\( T_{7}^{9} + 4T_{7}^{8} - 22T_{7}^{7} - 74T_{7}^{6} + 139T_{7}^{5} + 456T_{7}^{4} - 210T_{7}^{3} - 929T_{7}^{2} - 199T_{7} + 200 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{9} \) Copy content Toggle raw display
$3$ \( (T + 1)^{9} \) Copy content Toggle raw display
$5$ \( T^{9} + 4 T^{8} + \cdots - 8 \) Copy content Toggle raw display
$7$ \( T^{9} + 4 T^{8} + \cdots + 200 \) Copy content Toggle raw display
$11$ \( T^{9} + 5 T^{8} + \cdots + 62 \) Copy content Toggle raw display
$13$ \( (T - 1)^{9} \) Copy content Toggle raw display
$17$ \( T^{9} + 6 T^{8} + \cdots - 4562 \) Copy content Toggle raw display
$19$ \( T^{9} + 4 T^{8} + \cdots - 25904 \) Copy content Toggle raw display
$23$ \( T^{9} + 6 T^{8} + \cdots + 32768 \) Copy content Toggle raw display
$29$ \( T^{9} + 19 T^{8} + \cdots + 18820 \) Copy content Toggle raw display
$31$ \( T^{9} + 6 T^{8} + \cdots + 199652 \) Copy content Toggle raw display
$37$ \( T^{9} + 13 T^{8} + \cdots + 370136 \) Copy content Toggle raw display
$41$ \( T^{9} + 18 T^{8} + \cdots + 1900 \) Copy content Toggle raw display
$43$ \( T^{9} + 20 T^{8} + \cdots + 2944376 \) Copy content Toggle raw display
$47$ \( T^{9} - 14 T^{8} + \cdots + 11242472 \) Copy content Toggle raw display
$53$ \( T^{9} + 3 T^{8} + \cdots + 48526 \) Copy content Toggle raw display
$59$ \( T^{9} + 9 T^{8} + \cdots + 6686528 \) Copy content Toggle raw display
$61$ \( T^{9} + 24 T^{8} + \cdots - 8840 \) Copy content Toggle raw display
$67$ \( T^{9} + 4 T^{8} + \cdots - 39059282 \) Copy content Toggle raw display
$71$ \( T^{9} + 9 T^{8} + \cdots - 7122160 \) Copy content Toggle raw display
$73$ \( T^{9} + 24 T^{8} + \cdots + 42267172 \) Copy content Toggle raw display
$79$ \( T^{9} + 15 T^{8} + \cdots + 13472944 \) Copy content Toggle raw display
$83$ \( T^{9} - 20 T^{8} + \cdots - 123824 \) Copy content Toggle raw display
$89$ \( T^{9} - 3 T^{8} + \cdots - 18020120 \) Copy content Toggle raw display
$97$ \( T^{9} + 19 T^{8} + \cdots - 110764 \) Copy content Toggle raw display
show more
show less