Properties

Label 8016.2.a.ba
Level $8016$
Weight $2$
Character orbit 8016.a
Self dual yes
Analytic conductor $64.008$
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] = [8016,2,Mod(1,8016)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8016, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("8016.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8016.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: \(64.0080822603\)
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: \( x^{9} - 3x^{8} - 16x^{7} + 45x^{6} + 67x^{5} - 166x^{4} - 83x^{3} + 152x^{2} + 51x - 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 4008)
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.

\(f(q)\) \(=\) \( q - q^{3} + (\beta_1 - 1) q^{5} + ( - \beta_{7} - \beta_{5} + 2) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} + (\beta_1 - 1) q^{5} + ( - \beta_{7} - \beta_{5} + 2) q^{7} + q^{9} + (\beta_{8} - \beta_{6} + \beta_{5} + \beta_{4} + \beta_1 - 2) q^{11} + (\beta_{8} - \beta_{7} + \beta_{4} + \beta_{3} - \beta_{2} - \beta_1) q^{13} + ( - \beta_1 + 1) q^{15} + ( - \beta_{7} - \beta_{6} - \beta_{2} - 1) q^{17} + (\beta_{8} + \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} - 3 \beta_{2} - \beta_1 + 1) q^{19} + (\beta_{7} + \beta_{5} - 2) q^{21} + ( - \beta_{8} + \beta_{7} - \beta_{6} + \beta_1) q^{23} + ( - \beta_{7} - \beta_{5} + \beta_{4} - \beta_{2} - 2 \beta_1 + 1) q^{25} - q^{27} + (2 \beta_{8} + \beta_{7} - 2 \beta_{6} + \beta_{5} + 3 \beta_{4} - \beta_{3} + \beta_{2} - 5) q^{29} + ( - \beta_{7} + \beta_{5} + \beta_{4} + \beta_{3} + 3) q^{31} + ( - \beta_{8} + \beta_{6} - \beta_{5} - \beta_{4} - \beta_1 + 2) q^{33} + (\beta_{6} - \beta_{4} + \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{35} + (\beta_{8} - 3 \beta_{7} + \beta_{6} - \beta_{5} - \beta_{2} + 1) q^{37} + ( - \beta_{8} + \beta_{7} - \beta_{4} - \beta_{3} + \beta_{2} + \beta_1) q^{39} + (\beta_{8} + \beta_{7} - 2 \beta_{6} + \beta_{4} - \beta_{3} + 2 \beta_{2} - 3) q^{41} + ( - \beta_{7} + 2 \beta_{3} + 4) q^{43} + (\beta_1 - 1) q^{45} + ( - \beta_{7} - \beta_{6} - \beta_{3} + \beta_1 + 1) q^{47} + ( - \beta_{8} - 2 \beta_{7} - \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{49} + (\beta_{7} + \beta_{6} + \beta_{2} + 1) q^{51} + ( - \beta_{8} + \beta_{6} + \beta_{5} - 3 \beta_{4} + 3 \beta_{3} - 3 \beta_{2}) q^{53} + (\beta_{7} - \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - 2 \beta_1 + 3) q^{55} + ( - \beta_{8} - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + 3 \beta_{2} + \beta_1 - 1) q^{57} + (2 \beta_{8} - 2 \beta_{3} + \beta_{2} + 2 \beta_1 - 2) q^{59} + ( - \beta_{8} + \beta_{7} + \beta_{6} - 2 \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 - 1) q^{61} + ( - \beta_{7} - \beta_{5} + 2) q^{63} + ( - \beta_{8} + 2 \beta_{6} - 2 \beta_{4} + 2 \beta_{3} + 2) q^{65} + (2 \beta_{8} - 2 \beta_{6} - \beta_{5} + \beta_{4} - 3 \beta_{3} + 4 \beta_{2} + 2 \beta_1 - 1) q^{67} + (\beta_{8} - \beta_{7} + \beta_{6} - \beta_1) q^{69} + (2 \beta_{8} - \beta_{5} + \beta_{4} - \beta_{3} - 2 \beta_{2} + \beta_1) q^{71} + ( - \beta_{7} - 2 \beta_{5} - 3 \beta_{4} + 3 \beta_{3} - \beta_{2} + \beta_1 + 2) q^{73} + (\beta_{7} + \beta_{5} - \beta_{4} + \beta_{2} + 2 \beta_1 - 1) q^{75} + (2 \beta_{8} - \beta_{7} - \beta_{2} + \beta_1 - 3) q^{77} + (2 \beta_{8} - \beta_{7} - 2 \beta_{4} - 2 \beta_{2} + 4) q^{79} + q^{81} + ( - 3 \beta_{8} + \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + 3 \beta_{2} + 2) q^{83} + ( - \beta_{8} + 2 \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + \cdots - 2) q^{85}+ \cdots + (\beta_{8} - \beta_{6} + \beta_{5} + \beta_{4} + \beta_1 - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 9 q^{3} - 6 q^{5} + 11 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - 9 q^{3} - 6 q^{5} + 11 q^{7} + 9 q^{9} - q^{11} - 4 q^{13} + 6 q^{15} - 9 q^{17} + 8 q^{19} - 11 q^{21} + 7 q^{23} - q^{25} - 9 q^{27} - 9 q^{29} + 25 q^{31} + q^{33} - 5 q^{35} - 6 q^{37} + 4 q^{39} - 4 q^{41} + 24 q^{43} - 6 q^{45} + 16 q^{47} + 4 q^{49} + 9 q^{51} - 26 q^{53} + 29 q^{55} - 8 q^{57} + 4 q^{59} - 20 q^{61} + 11 q^{63} - 8 q^{65} + 25 q^{67} - 7 q^{69} + 15 q^{71} - 10 q^{73} + q^{75} - 20 q^{77} + 34 q^{79} + 9 q^{81} + 4 q^{83} - 13 q^{85} + 9 q^{87} + 13 q^{89} + 21 q^{91} - 25 q^{93} + 7 q^{95} - 4 q^{97} - q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 3x^{8} - 16x^{7} + 45x^{6} + 67x^{5} - 166x^{4} - 83x^{3} + 152x^{2} + 51x - 10 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 253 \nu^{8} + 10903 \nu^{7} - 17171 \nu^{6} - 175482 \nu^{5} + 247955 \nu^{4} + 716568 \nu^{3} - 563660 \nu^{2} - 635601 \nu + 68945 ) / 102665 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 428 \nu^{8} + 9923 \nu^{7} - 2266 \nu^{6} - 178372 \nu^{5} + 67645 \nu^{4} + 878658 \nu^{3} - 38080 \nu^{2} - 988001 \nu - 301330 ) / 102665 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 3031 \nu^{8} + 7666 \nu^{7} + 48718 \nu^{6} - 99334 \nu^{5} - 199070 \nu^{4} + 207921 \nu^{3} + 208150 \nu^{2} + 200063 \nu - 130065 ) / 102665 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3811 \nu^{8} - 23831 \nu^{7} - 35953 \nu^{6} + 387944 \nu^{5} - 79645 \nu^{4} - 1646101 \nu^{3} + 520415 \nu^{2} + 1605297 \nu + 255125 ) / 102665 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 5684 \nu^{8} - 5129 \nu^{7} - 118627 \nu^{6} + 65121 \nu^{5} + 751965 \nu^{4} - 184819 \nu^{3} - 1469865 \nu^{2} + 296533 \nu + 445920 ) / 102665 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 6589 \nu^{8} + 20594 \nu^{7} + 101842 \nu^{6} - 311796 \nu^{5} - 367380 \nu^{4} + 1137454 \nu^{3} + 148730 \nu^{2} - 769633 \nu + 59190 ) / 102665 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 16128 \nu^{8} - 24668 \nu^{7} - 293609 \nu^{6} + 290982 \nu^{5} + 1500975 \nu^{4} - 446023 \nu^{3} - 2000020 \nu^{2} - 396149 \nu + 286160 ) / 102665 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{7} - \beta_{5} + \beta_{4} - \beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{8} - 3\beta_{7} + 2\beta_{6} - 2\beta_{5} - 3\beta_{4} + 2\beta_{3} - 2\beta_{2} + 8\beta _1 + 7 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -3\beta_{8} - 12\beta_{7} + 6\beta_{6} - 12\beta_{5} + 7\beta_{4} + 5\beta_{3} - 17\beta_{2} - 2\beta _1 + 54 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -30\beta_{8} - 45\beta_{7} + 33\beta_{6} - 27\beta_{5} - 36\beta_{4} + 34\beta_{3} - 32\beta_{2} + 73\beta _1 + 109 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 56 \beta_{8} - 145 \beta_{7} + 106 \beta_{6} - 132 \beta_{5} + 54 \beta_{4} + 105 \beta_{3} - 225 \beta_{2} - 33 \beta _1 + 635 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 391 \beta_{8} - 576 \beta_{7} + 466 \beta_{6} - 324 \beta_{5} - 395 \beta_{4} + 506 \beta_{3} - 459 \beta_{2} + 691 \beta _1 + 1496 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 846 \beta_{8} - 1799 \beta_{7} + 1544 \beta_{6} - 1474 \beta_{5} + 418 \beta_{4} + 1662 \beta_{3} - 2818 \beta_{2} - 429 \beta _1 + 7652 \) Copy content Toggle raw display

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
−3.17025
−1.99382
−0.985991
−0.482381
0.141931
1.41388
1.58856
2.94484
3.54323
0 −1.00000 0 −4.17025 0 1.78655 0 1.00000 0
1.2 0 −1.00000 0 −2.99382 0 4.65862 0 1.00000 0
1.3 0 −1.00000 0 −1.98599 0 0.0909950 0 1.00000 0
1.4 0 −1.00000 0 −1.48238 0 1.03908 0 1.00000 0
1.5 0 −1.00000 0 −0.858069 0 −2.33225 0 1.00000 0
1.6 0 −1.00000 0 0.413878 0 −2.72537 0 1.00000 0
1.7 0 −1.00000 0 0.588564 0 1.23518 0 1.00000 0
1.8 0 −1.00000 0 1.94484 0 3.19705 0 1.00000 0
1.9 0 −1.00000 0 2.54323 0 4.05015 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)
\(167\) \(-1\)

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 8016.2.a.ba 9
4.b odd 2 1 4008.2.a.i 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4008.2.a.i 9 4.b odd 2 1
8016.2.a.ba 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8016))\):

\( T_{5}^{9} + 6T_{5}^{8} - 4T_{5}^{7} - 67T_{5}^{6} - 41T_{5}^{5} + 200T_{5}^{4} + 179T_{5}^{3} - 132T_{5}^{2} - 80T_{5} + 38 \) Copy content Toggle raw display
\( T_{7}^{9} - 11T_{7}^{8} + 27T_{7}^{7} + 88T_{7}^{6} - 446T_{7}^{5} + 242T_{7}^{4} + 1265T_{7}^{3} - 2148T_{7}^{2} + 1064T_{7} - 80 \) Copy content Toggle raw display
\( T_{11}^{9} + T_{11}^{8} - 47T_{11}^{7} - 18T_{11}^{6} + 595T_{11}^{5} - 71T_{11}^{4} - 1612T_{11}^{3} - 600T_{11}^{2} + 780T_{11} + 400 \) Copy content Toggle raw display
\( T_{13}^{9} + 4 T_{13}^{8} - 43 T_{13}^{7} - 152 T_{13}^{6} + 612 T_{13}^{5} + 1539 T_{13}^{4} - 4140 T_{13}^{3} - 4048 T_{13}^{2} + 11568 T_{13} - 5360 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} \) Copy content Toggle raw display
$3$ \( (T + 1)^{9} \) Copy content Toggle raw display
$5$ \( T^{9} + 6 T^{8} - 4 T^{7} - 67 T^{6} + \cdots + 38 \) Copy content Toggle raw display
$7$ \( T^{9} - 11 T^{8} + 27 T^{7} + 88 T^{6} + \cdots - 80 \) Copy content Toggle raw display
$11$ \( T^{9} + T^{8} - 47 T^{7} - 18 T^{6} + \cdots + 400 \) Copy content Toggle raw display
$13$ \( T^{9} + 4 T^{8} - 43 T^{7} + \cdots - 5360 \) Copy content Toggle raw display
$17$ \( T^{9} + 9 T^{8} - 48 T^{7} + \cdots + 2620 \) Copy content Toggle raw display
$19$ \( T^{9} - 8 T^{8} - 97 T^{7} + \cdots - 8000 \) Copy content Toggle raw display
$23$ \( T^{9} - 7 T^{8} - 85 T^{7} + \cdots + 239104 \) Copy content Toggle raw display
$29$ \( T^{9} + 9 T^{8} - 101 T^{7} + \cdots - 539584 \) Copy content Toggle raw display
$31$ \( T^{9} - 25 T^{8} + 152 T^{7} + \cdots + 2058080 \) Copy content Toggle raw display
$37$ \( T^{9} + 6 T^{8} - 168 T^{7} + \cdots - 40760 \) Copy content Toggle raw display
$41$ \( T^{9} + 4 T^{8} - 112 T^{7} + \cdots - 380 \) Copy content Toggle raw display
$43$ \( T^{9} - 24 T^{8} + 128 T^{7} + \cdots + 1370216 \) Copy content Toggle raw display
$47$ \( T^{9} - 16 T^{8} + 20 T^{7} + \cdots - 10204 \) Copy content Toggle raw display
$53$ \( T^{9} + 26 T^{8} + 50 T^{7} + \cdots + 270098 \) Copy content Toggle raw display
$59$ \( T^{9} - 4 T^{8} - 206 T^{7} + \cdots + 10720 \) Copy content Toggle raw display
$61$ \( T^{9} + 20 T^{8} + 53 T^{7} + \cdots + 160 \) Copy content Toggle raw display
$67$ \( T^{9} - 25 T^{8} - 35 T^{7} + \cdots + 1990 \) Copy content Toggle raw display
$71$ \( T^{9} - 15 T^{8} - 190 T^{7} + \cdots - 5162848 \) Copy content Toggle raw display
$73$ \( T^{9} + 10 T^{8} + \cdots - 187912544 \) Copy content Toggle raw display
$79$ \( T^{9} - 34 T^{8} + 230 T^{7} + \cdots + 237200 \) Copy content Toggle raw display
$83$ \( T^{9} - 4 T^{8} - 296 T^{7} + \cdots + 5840 \) Copy content Toggle raw display
$89$ \( T^{9} - 13 T^{8} - 321 T^{7} + \cdots + 69802664 \) Copy content Toggle raw display
$97$ \( T^{9} + 4 T^{8} - 567 T^{7} + \cdots - 10035400 \) Copy content Toggle raw display
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