Properties

Label 8016.2.a.bf
Level $8016$
Weight $2$
Character orbit 8016.a
Self dual yes
Analytic conductor $64.008$
Analytic rank $1$
Dimension $12$
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: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} - 31 x^{10} + 131 x^{9} + 309 x^{8} - 1453 x^{7} - 1072 x^{6} + 6350 x^{5} + 1411 x^{4} - 11022 x^{3} - 2450 x^{2} + 6960 x + 3008 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + \beta_1 q^{5} + (\beta_{9} - 1) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} + \beta_1 q^{5} + (\beta_{9} - 1) q^{7} + q^{9} + (\beta_{10} - \beta_1) q^{11} + ( - \beta_{5} + 1) q^{13} - \beta_1 q^{15} + ( - \beta_{7} + \beta_{6} + \beta_{5}) q^{17} + (\beta_{11} - 1) q^{19} + ( - \beta_{9} + 1) q^{21} + ( - \beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} - \beta_{5}) q^{23} + ( - \beta_{9} + \beta_{7} - \beta_{6} + \beta_{4} + 2) q^{25} - q^{27} + ( - \beta_{10} + \beta_{7} + \beta_1) q^{29} + ( - \beta_{11} - \beta_{10} - \beta_{8} - \beta_{7} + \beta_{6} + \beta_{5} + \beta_{3} + \beta_1 - 3) q^{31} + ( - \beta_{10} + \beta_1) q^{33} + ( - \beta_{11} - \beta_{10} + \beta_{9} - \beta_{8} - \beta_{7} + \beta_{5} - \beta_{3} - 2 \beta_1 - 1) q^{35} + ( - \beta_{11} - \beta_{7} + \beta_{5} - \beta_{4} - \beta_{2} - \beta_1) q^{37} + (\beta_{5} - 1) q^{39} + (\beta_{9} - \beta_{8} - \beta_{7} - 2 \beta_{4} + \beta_{2} - 1) q^{41} + ( - \beta_{9} + \beta_{8} + \beta_{7} + \beta_{5} - \beta_{3} - 2) q^{43} + \beta_1 q^{45} + ( - \beta_{10} - \beta_{8} + \beta_{7} - \beta_{5} - \beta_{4} + \beta_{3} + \beta_1 - 2) q^{47} + (\beta_{11} + \beta_{10} - 2 \beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} - \beta_{5} - \beta_{3} + \beta_{2} - \beta_1 + 3) q^{49} + (\beta_{7} - \beta_{6} - \beta_{5}) q^{51} + ( - \beta_{10} - \beta_{6} + \beta_{4} + \beta_{3} + 3) q^{53} + ( - \beta_{11} - \beta_{7} + \beta_{6} - \beta_{4} - \beta_{3} - 4) q^{55} + ( - \beta_{11} + 1) q^{57} + ( - \beta_{9} + \beta_{8} + 2 \beta_{4} - \beta_{3} - \beta_1 + 1) q^{59} + ( - \beta_{11} - \beta_{10} + \beta_{7} + \beta_{6} - \beta_{4} + \beta_{3} + \beta_1) q^{61} + (\beta_{9} - 1) q^{63} + (\beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} + 2 \beta_{7} - \beta_{6} - \beta_{5} + 2 \beta_{4} - \beta_{3} + \cdots - \beta_1) q^{65}+ \cdots + (\beta_{10} - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{3} + 4 q^{5} - 11 q^{7} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{3} + 4 q^{5} - 11 q^{7} + 12 q^{9} - q^{11} + 8 q^{13} - 4 q^{15} + 3 q^{17} - 12 q^{19} + 11 q^{21} - 7 q^{23} + 18 q^{25} - 12 q^{27} + 5 q^{29} - 33 q^{31} + q^{33} - 15 q^{35} + 8 q^{37} - 8 q^{39} - 6 q^{41} - 16 q^{43} + 4 q^{45} - 18 q^{47} + 25 q^{49} - 3 q^{51} + 20 q^{53} - 39 q^{55} + 12 q^{57} - 4 q^{59} + 10 q^{61} - 11 q^{63} - 9 q^{67} + 7 q^{69} - 11 q^{71} + 22 q^{73} - 18 q^{75} + 24 q^{77} - 56 q^{79} + 12 q^{81} - 26 q^{83} + 15 q^{85} - 5 q^{87} - 15 q^{89} - 11 q^{91} + 33 q^{93} - 3 q^{95} + 8 q^{97} - q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} - 31 x^{10} + 131 x^{9} + 309 x^{8} - 1453 x^{7} - 1072 x^{6} + 6350 x^{5} + 1411 x^{4} - 11022 x^{3} - 2450 x^{2} + 6960 x + 3008 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 2612041 \nu^{11} + 172266458 \nu^{10} - 431149861 \nu^{9} - 5604811029 \nu^{8} + 15598783254 \nu^{7} + 61532588360 \nu^{6} + \cdots - 506734859797 ) / 6887737669 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 48270257 \nu^{11} - 393971708 \nu^{10} - 803884765 \nu^{9} + 12731441603 \nu^{8} - 7245605191 \nu^{7} - 136791943933 \nu^{6} + \cdots + 623931970190 ) / 13775475338 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 433554181 \nu^{11} - 1027661208 \nu^{10} - 15151502491 \nu^{9} + 32370295155 \nu^{8} + 187572944809 \nu^{7} - 331978409981 \nu^{6} + \cdots - 810005934996 ) / 27550950676 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 924523625 \nu^{11} + 3335040372 \nu^{10} + 28727151823 \nu^{9} - 105506939067 \nu^{8} - 286439821885 \nu^{7} + \cdots - 1194286516536 ) / 55101901352 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 497566513 \nu^{11} - 583717328 \nu^{10} - 19421435387 \nu^{9} + 18043299379 \nu^{8} + 279234229797 \nu^{7} - 177472297861 \nu^{6} + \cdots - 2291097916892 ) / 27550950676 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 608147925 \nu^{11} + 2540267068 \nu^{10} + 17866911943 \nu^{9} - 80927081243 \nu^{8} - 156734313797 \nu^{7} + \cdots - 1570311572784 ) / 27550950676 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1276718751 \nu^{11} - 4447291652 \nu^{10} - 40378997177 \nu^{9} + 141144188077 \nu^{8} + 420342586963 \nu^{7} + \cdots + 1003995641592 ) / 55101901352 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 672160257 \nu^{11} + 2096323188 \nu^{10} + 22136844839 \nu^{9} - 66600085467 \nu^{8} - 248395598785 \nu^{7} + \cdots + 103637063844 ) / 27550950676 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 825574067 \nu^{11} - 1232246264 \nu^{10} - 31068710789 \nu^{9} + 37574547953 \nu^{8} + 427577508547 \nu^{7} - 358636986823 \nu^{6} + \cdots - 3936944027888 ) / 27550950676 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 1494430261 \nu^{11} + 4167534940 \nu^{10} + 50046579515 \nu^{9} - 131337919583 \nu^{8} - 578035323361 \nu^{7} + \cdots + 1214478641340 ) / 27550950676 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{9} + \beta_{7} - \beta_{6} + \beta_{4} + 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{10} + \beta_{7} - 2\beta_{5} + \beta_{4} + \beta_{3} + 2\beta_{2} + 12\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} - 13 \beta_{9} + 2 \beta_{8} + 15 \beta_{7} - 15 \beta_{6} + 19 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 2 \beta _1 + 78 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{11} - 15 \beta_{10} + \beta_{9} - 3 \beta_{8} + 11 \beta_{7} - \beta_{6} - 31 \beta_{5} + 23 \beta_{4} + 12 \beta_{3} + 30 \beta_{2} + 144 \beta _1 + 31 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 26 \beta_{11} + 6 \beta_{10} - 157 \beta_{9} + 34 \beta_{8} + 188 \beta_{7} - 202 \beta_{6} + 4 \beta_{5} + 292 \beta_{4} - 38 \beta_{3} - 41 \beta_{2} - 46 \beta _1 + 976 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 49 \beta_{11} - 199 \beta_{10} + 29 \beta_{9} - 57 \beta_{8} + 96 \beta_{7} - 32 \beta_{6} - 401 \beta_{5} + 406 \beta_{4} + 108 \beta_{3} + 394 \beta_{2} + 1764 \beta _1 + 484 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 465 \beta_{11} + 136 \beta_{10} - 1897 \beta_{9} + 446 \beta_{8} + 2264 \beta_{7} - 2645 \beta_{6} + 150 \beta_{5} + 4221 \beta_{4} - 533 \beta_{3} - 623 \beta_{2} - 730 \beta _1 + 12688 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 810 \beta_{11} - 2629 \beta_{10} + 581 \beta_{9} - 879 \beta_{8} + 663 \beta_{7} - 715 \beta_{6} - 4895 \beta_{5} + 6479 \beta_{4} + 759 \beta_{3} + 5122 \beta_{2} + 22012 \beta _1 + 7620 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 7141 \beta_{11} + 2177 \beta_{10} - 22913 \beta_{9} + 5288 \beta_{8} + 26926 \beta_{7} - 34360 \beta_{6} + 3630 \beta_{5} + 59518 \beta_{4} - 6696 \beta_{3} - 8532 \beta_{2} - 9985 \beta _1 + 167606 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 11158 \beta_{11} - 35307 \beta_{10} + 10456 \beta_{9} - 13224 \beta_{8} + 1478 \beta_{7} - 13396 \beta_{6} - 57692 \beta_{5} + 98497 \beta_{4} + 2215 \beta_{3} + 67138 \beta_{2} + 278574 \beta _1 + 119224 \) 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
−3.64003
−3.49351
−2.09997
−1.14709
−0.716151
−0.619387
1.53322
1.56937
2.31305
2.93317
3.59055
3.77676
0 −1.00000 0 −3.64003 0 1.45249 0 1.00000 0
1.2 0 −1.00000 0 −3.49351 0 −4.58171 0 1.00000 0
1.3 0 −1.00000 0 −2.09997 0 −2.97416 0 1.00000 0
1.4 0 −1.00000 0 −1.14709 0 3.32442 0 1.00000 0
1.5 0 −1.00000 0 −0.716151 0 3.33002 0 1.00000 0
1.6 0 −1.00000 0 −0.619387 0 −0.954126 0 1.00000 0
1.7 0 −1.00000 0 1.53322 0 0.915565 0 1.00000 0
1.8 0 −1.00000 0 1.56937 0 −3.21458 0 1.00000 0
1.9 0 −1.00000 0 2.31305 0 −4.58648 0 1.00000 0
1.10 0 −1.00000 0 2.93317 0 −1.35675 0 1.00000 0
1.11 0 −1.00000 0 3.59055 0 −4.10828 0 1.00000 0
1.12 0 −1.00000 0 3.77676 0 1.75359 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
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.bf 12
4.b odd 2 1 4008.2.a.l 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4008.2.a.l 12 4.b odd 2 1
8016.2.a.bf 12 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8016))\):

\( T_{5}^{12} - 4 T_{5}^{11} - 31 T_{5}^{10} + 131 T_{5}^{9} + 309 T_{5}^{8} - 1453 T_{5}^{7} - 1072 T_{5}^{6} + 6350 T_{5}^{5} + 1411 T_{5}^{4} - 11022 T_{5}^{3} - 2450 T_{5}^{2} + 6960 T_{5} + 3008 \) Copy content Toggle raw display
\( T_{7}^{12} + 11 T_{7}^{11} + 6 T_{7}^{10} - 297 T_{7}^{9} - 719 T_{7}^{8} + 2618 T_{7}^{7} + 8707 T_{7}^{6} - 8480 T_{7}^{5} - 36387 T_{7}^{4} + 11176 T_{7}^{3} + 57280 T_{7}^{2} - 4416 T_{7} - 27584 \) Copy content Toggle raw display
\( T_{11}^{12} + T_{11}^{11} - 75 T_{11}^{10} - 130 T_{11}^{9} + 1567 T_{11}^{8} + 4085 T_{11}^{7} - 6648 T_{11}^{6} - 22660 T_{11}^{5} + 3892 T_{11}^{4} + 32848 T_{11}^{3} + 7728 T_{11}^{2} - 2624 T_{11} + 128 \) Copy content Toggle raw display
\( T_{13}^{12} - 8 T_{13}^{11} - 53 T_{13}^{10} + 520 T_{13}^{9} + 554 T_{13}^{8} - 10279 T_{13}^{7} + 3618 T_{13}^{6} + 71278 T_{13}^{5} - 32028 T_{13}^{4} - 206112 T_{13}^{3} + 19496 T_{13}^{2} + 196480 T_{13} + 55424 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T + 1)^{12} \) Copy content Toggle raw display
$5$ \( T^{12} - 4 T^{11} - 31 T^{10} + \cdots + 3008 \) Copy content Toggle raw display
$7$ \( T^{12} + 11 T^{11} + 6 T^{10} + \cdots - 27584 \) Copy content Toggle raw display
$11$ \( T^{12} + T^{11} - 75 T^{10} - 130 T^{9} + \cdots + 128 \) Copy content Toggle raw display
$13$ \( T^{12} - 8 T^{11} - 53 T^{10} + \cdots + 55424 \) Copy content Toggle raw display
$17$ \( T^{12} - 3 T^{11} - 86 T^{10} + \cdots + 81616 \) Copy content Toggle raw display
$19$ \( T^{12} + 12 T^{11} - 29 T^{10} + \cdots - 93568 \) Copy content Toggle raw display
$23$ \( T^{12} + 7 T^{11} - 109 T^{10} + \cdots + 1079296 \) Copy content Toggle raw display
$29$ \( T^{12} - 5 T^{11} - 83 T^{10} + \cdots - 2048 \) Copy content Toggle raw display
$31$ \( T^{12} + 33 T^{11} + \cdots + 160663936 \) Copy content Toggle raw display
$37$ \( T^{12} - 8 T^{11} + \cdots + 1249349168 \) Copy content Toggle raw display
$41$ \( T^{12} + 6 T^{11} - 246 T^{10} + \cdots + 59055712 \) Copy content Toggle raw display
$43$ \( T^{12} + 16 T^{11} + \cdots - 142411744 \) Copy content Toggle raw display
$47$ \( T^{12} + 18 T^{11} + \cdots - 1855943024 \) Copy content Toggle raw display
$53$ \( T^{12} - 20 T^{11} - 87 T^{10} + \cdots + 7459216 \) Copy content Toggle raw display
$59$ \( T^{12} + 4 T^{11} - 331 T^{10} + \cdots - 176553400 \) Copy content Toggle raw display
$61$ \( T^{12} - 10 T^{11} + \cdots - 116056064 \) Copy content Toggle raw display
$67$ \( T^{12} + 9 T^{11} + \cdots + 1073105416 \) Copy content Toggle raw display
$71$ \( T^{12} + 11 T^{11} + \cdots + 23931793408 \) Copy content Toggle raw display
$73$ \( T^{12} - 22 T^{11} - 243 T^{10} + \cdots + 63085376 \) Copy content Toggle raw display
$79$ \( T^{12} + 56 T^{11} + \cdots - 3173513344 \) Copy content Toggle raw display
$83$ \( T^{12} + 26 T^{11} - 23 T^{10} + \cdots - 56792 \) Copy content Toggle raw display
$89$ \( T^{12} + 15 T^{11} + \cdots + 40701142672 \) Copy content Toggle raw display
$97$ \( T^{12} - 8 T^{11} + \cdots - 1191476768 \) Copy content Toggle raw display
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