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Properties

Label 2006.2.a.w
Level $2006$
Weight $2$
Character orbit 2006.a
Self dual yes
Analytic conductor $16.018$
Analytic rank $0$
Dimension $13$
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] = [2006,2,Mod(1,2006)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2006, 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("2006.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2006 = 2 \cdot 17 \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2006.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: \(16.0179906455\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 6 x^{12} - 9 x^{11} + 91 x^{10} + 14 x^{9} - 517 x^{8} + 70 x^{7} + 1296 x^{6} - 300 x^{5} + \cdots + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{7} \)
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 \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{13} - 6 x^{12} - 9 x^{11} + 91 x^{10} + 14 x^{9} - 517 x^{8} + 70 x^{7} + 1296 x^{6} - 300 x^{5} + \cdots + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2\nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{12} - 26 \nu^{11} + 141 \nu^{10} + 11 \nu^{9} - 1432 \nu^{8} + 1301 \nu^{7} + 5042 \nu^{6} + \cdots - 192 ) / 64 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2 \nu^{12} - 19 \nu^{11} + 36 \nu^{10} + 151 \nu^{9} - 545 \nu^{8} - 158 \nu^{7} + 2165 \nu^{6} + \cdots + 216 ) / 32 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3 \nu^{12} - 28 \nu^{11} + 47 \nu^{10} + 247 \nu^{9} - 742 \nu^{8} - 609 \nu^{7} + 3092 \nu^{6} + \cdots + 64 ) / 32 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 13 \nu^{12} + 82 \nu^{11} + 63 \nu^{10} - 991 \nu^{9} + 48 \nu^{8} + 4551 \nu^{7} + 70 \nu^{6} + \cdots - 64 ) / 64 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 9 \nu^{12} - 78 \nu^{11} + 117 \nu^{10} + 607 \nu^{9} - 1692 \nu^{8} - 955 \nu^{7} + 5950 \nu^{6} + \cdots + 672 ) / 64 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 15 \nu^{12} - 110 \nu^{11} + 35 \nu^{10} + 1141 \nu^{9} - 1216 \nu^{8} - 4333 \nu^{7} + 4398 \nu^{6} + \cdots + 32 ) / 64 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 3 \nu^{12} + 6 \nu^{11} - 201 \nu^{10} + 301 \nu^{9} + 1996 \nu^{8} - 3601 \nu^{7} - 7534 \nu^{6} + \cdots - 704 ) / 64 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 7 \nu^{12} - 72 \nu^{11} + 167 \nu^{10} + 507 \nu^{9} - 2294 \nu^{8} - 93 \nu^{7} + 8876 \nu^{6} + \cdots + 528 ) / 64 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 6 \nu^{12} - 53 \nu^{11} + 80 \nu^{10} + 441 \nu^{9} - 1219 \nu^{8} - 926 \nu^{7} + 4659 \nu^{6} + \cdots + 40 ) / 32 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 27 \nu^{12} + 182 \nu^{11} + 73 \nu^{10} - 2249 \nu^{9} + 1192 \nu^{8} + 10265 \nu^{7} - 6038 \nu^{6} + \cdots + 800 ) / 64 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{11} - \beta_{9} - \beta_{6} + \beta_{5} - \beta_{4} + 3\beta_{2} + 11\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{12} - 2 \beta_{11} - 5 \beta_{9} + \beta_{8} - 2 \beta_{7} - 5 \beta_{6} + 5 \beta_{5} + \cdots + 23 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4 \beta_{12} - 12 \beta_{11} - \beta_{10} - 29 \beta_{9} + 6 \beta_{8} - 10 \beta_{7} - 26 \beta_{6} + \cdots + 59 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 24 \beta_{12} - 34 \beta_{11} - 9 \beta_{10} - 136 \beta_{9} + 36 \beta_{8} - 60 \beta_{7} - 118 \beta_{6} + \cdots + 276 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 103 \beta_{12} - 144 \beta_{11} - 54 \beta_{10} - 635 \beta_{9} + 180 \beta_{8} - 277 \beta_{7} + \cdots + 978 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 478 \beta_{12} - 479 \beta_{11} - 291 \beta_{10} - 2809 \beta_{9} + 862 \beta_{8} - 1299 \beta_{7} + \cdots + 4102 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2040 \beta_{12} - 1852 \beta_{11} - 1415 \beta_{10} - 12274 \beta_{9} + 3921 \beta_{8} - 5693 \beta_{7} + \cdots + 16117 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 8819 \beta_{12} - 6679 \beta_{11} - 6610 \beta_{10} - 52619 \beta_{9} + 17412 \beta_{8} - 24812 \beta_{7} + \cdots + 66343 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 37190 \beta_{12} - 25567 \beta_{11} - 29721 \beta_{10} - 223769 \beta_{9} + 75731 \beta_{8} + \cdots + 269419 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 156858 \beta_{12} - 96703 \beta_{11} - 130973 \beta_{10} - 943924 \beta_{9} + 325271 \beta_{8} + \cdots + 1108948 \) 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
4.14219
3.13025
2.78728
1.87304
0.958065
0.714077
0.333226
0.129950
−0.723365
−1.58918
−1.63561
−2.05797
−2.06195
1.00000 −3.14219 1.00000 1.20202 −3.14219 2.33770 1.00000 6.87337 1.20202
1.2 1.00000 −2.13025 1.00000 −0.954710 −2.13025 −2.87838 1.00000 1.53797 −0.954710
1.3 1.00000 −1.78728 1.00000 0.747937 −1.78728 2.14313 1.00000 0.194354 0.747937
1.4 1.00000 −0.873044 1.00000 3.87247 −0.873044 2.67335 1.00000 −2.23779 3.87247
1.5 1.00000 0.0419348 1.00000 −2.69413 0.0419348 −1.02701 1.00000 −2.99824 −2.69413
1.6 1.00000 0.285923 1.00000 −4.07831 0.285923 3.91932 1.00000 −2.91825 −4.07831
1.7 1.00000 0.666774 1.00000 2.46083 0.666774 −4.80169 1.00000 −2.55541 2.46083
1.8 1.00000 0.870050 1.00000 1.72202 0.870050 2.12442 1.00000 −2.24301 1.72202
1.9 1.00000 1.72336 1.00000 −0.202404 1.72336 3.37958 1.00000 −0.0300134 −0.202404
1.10 1.00000 2.58918 1.00000 2.47054 2.58918 −1.02544 1.00000 3.70384 2.47054
1.11 1.00000 2.63561 1.00000 −0.925008 2.63561 4.83912 1.00000 3.94645 −0.925008
1.12 1.00000 3.05797 1.00000 −4.28893 3.05797 −2.80349 1.00000 6.35118 −4.28893
1.13 1.00000 3.06195 1.00000 1.66767 3.06195 −0.880611 1.00000 6.37557 1.66767
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.13
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(17\) \(-1\)
\(59\) \(-1\)

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 2006.2.a.w ✓ 13
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2006.2.a.w ✓ 13 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2006))\):

\( T_{3}^{13} - 7 T_{3}^{12} - 3 T_{3}^{11} + 118 T_{3}^{10} - 176 T_{3}^{9} - 536 T_{3}^{8} + 1352 T_{3}^{7} + \cdots - 8 \) Copy content Toggle raw display
\( T_{5}^{13} - T_{5}^{12} - 39 T_{5}^{11} + 62 T_{5}^{10} + 491 T_{5}^{9} - 1083 T_{5}^{8} - 1953 T_{5}^{7} + \cdots - 512 \) Copy content Toggle raw display
\( T_{11}^{13} - 13 T_{11}^{12} + T_{11}^{11} + 598 T_{11}^{10} - 1682 T_{11}^{9} - 8876 T_{11}^{8} + \cdots + 1642496 \) Copy content Toggle raw display
\( T_{31}^{13} - 32 T_{31}^{12} + 263 T_{31}^{11} + 2340 T_{31}^{10} - 56231 T_{31}^{9} + 327892 T_{31}^{8} + \cdots + 550915744 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{13} \) Copy content Toggle raw display
$3$ \( T^{13} - 7 T^{12} + \cdots - 8 \) Copy content Toggle raw display
$5$ \( T^{13} - T^{12} + \cdots - 512 \) Copy content Toggle raw display
$7$ \( T^{13} - 8 T^{12} + \cdots - 65536 \) Copy content Toggle raw display
$11$ \( T^{13} - 13 T^{12} + \cdots + 1642496 \) Copy content Toggle raw display
$13$ \( T^{13} - 9 T^{12} + \cdots + 5536 \) Copy content Toggle raw display
$17$ \( (T - 1)^{13} \) Copy content Toggle raw display
$19$ \( T^{13} - 16 T^{12} + \cdots + 4096 \) Copy content Toggle raw display
$23$ \( T^{13} - 18 T^{12} + \cdots - 7056512 \) Copy content Toggle raw display
$29$ \( T^{13} - 8 T^{12} + \cdots + 373376 \) Copy content Toggle raw display
$31$ \( T^{13} + \cdots + 550915744 \) Copy content Toggle raw display
$37$ \( T^{13} + \cdots + 238944256 \) Copy content Toggle raw display
$41$ \( T^{13} + \cdots + 243425792 \) Copy content Toggle raw display
$43$ \( T^{13} + \cdots - 1060831232 \) Copy content Toggle raw display
$47$ \( T^{13} + \cdots - 9668263936 \) Copy content Toggle raw display
$53$ \( T^{13} + \cdots - 3226658816 \) Copy content Toggle raw display
$59$ \( (T - 1)^{13} \) Copy content Toggle raw display
$61$ \( T^{13} + \cdots - 426975232 \) Copy content Toggle raw display
$67$ \( T^{13} + \cdots + 2954987264 \) Copy content Toggle raw display
$71$ \( T^{13} + \cdots + 15233368064 \) Copy content Toggle raw display
$73$ \( T^{13} + \cdots - 1130571776 \) Copy content Toggle raw display
$79$ \( T^{13} + \cdots + 393997186048 \) Copy content Toggle raw display
$83$ \( T^{13} + \cdots - 1935106816 \) Copy content Toggle raw display
$89$ \( T^{13} + \cdots - 172095488 \) Copy content Toggle raw display
$97$ \( T^{13} + \cdots + 1209382432 \) Copy content Toggle raw display
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