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Properties

Label 2010.2.e.j
Level $2010$
Weight $2$
Character orbit 2010.e
Analytic conductor $16.050$
Analytic rank $0$
Dimension $18$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2010,2,Mod(1609,2010)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2010, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2010.1609");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2010 = 2 \cdot 3 \cdot 5 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2010.e (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(16.0499308063\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 4 x^{17} + 10 x^{16} - 16 x^{15} - x^{14} + 104 x^{13} - 279 x^{12} + 160 x^{11} + \cdots + 1953125 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{7} q^{2} + \beta_{7} q^{3} - q^{4} - \beta_{5} q^{5} + q^{6} + ( - \beta_{8} - \beta_{7}) q^{7} + \beta_{7} q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{7} q^{2} + \beta_{7} q^{3} - q^{4} - \beta_{5} q^{5} + q^{6} + ( - \beta_{8} - \beta_{7}) q^{7} + \beta_{7} q^{8} - q^{9} + \beta_{11} q^{10} + (\beta_{17} + \beta_{15} + \beta_{14} + \cdots + 1) q^{11}+ \cdots + ( - \beta_{17} - \beta_{15} - \beta_{14} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 18 q^{4} + 4 q^{5} + 18 q^{6} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 18 q^{4} + 4 q^{5} + 18 q^{6} - 18 q^{9} + 14 q^{11} - 18 q^{14} + 18 q^{16} - 4 q^{19} - 4 q^{20} + 18 q^{21} - 18 q^{24} - 4 q^{25} + 16 q^{26} - 28 q^{29} + 4 q^{30} + 8 q^{31} - 16 q^{34} - 10 q^{35} + 18 q^{36} - 16 q^{39} + 48 q^{41} - 14 q^{44} - 4 q^{45} + 16 q^{46} - 40 q^{49} + 8 q^{50} + 16 q^{51} - 18 q^{54} - 10 q^{55} + 18 q^{56} - 20 q^{59} + 10 q^{61} - 18 q^{64} - 28 q^{65} + 14 q^{66} - 16 q^{69} + 22 q^{70} - 10 q^{71} - 30 q^{74} - 8 q^{75} + 4 q^{76} + 16 q^{79} + 4 q^{80} + 18 q^{81} - 18 q^{84} + 24 q^{86} - 78 q^{89} - 36 q^{91} - 16 q^{94} + 4 q^{95} + 18 q^{96} - 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 4 x^{17} + 10 x^{16} - 16 x^{15} - x^{14} + 104 x^{13} - 279 x^{12} + 160 x^{11} + \cdots + 1953125 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 15056 \nu^{17} - 337219 \nu^{16} + 555040 \nu^{15} + 62529 \nu^{14} - 3750636 \nu^{13} + \cdots + 33748828125 ) / 11725000000 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 19536 \nu^{17} + 161309 \nu^{16} - 762520 \nu^{15} - 772399 \nu^{14} + 4217396 \nu^{13} + \cdots - 77613671875 ) / 11725000000 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 14968 \nu^{17} - 9217 \nu^{16} + 111210 \nu^{15} - 595913 \nu^{14} + 939052 \nu^{13} + \cdots - 52581640625 ) / 5862500000 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{17} - 4 \nu^{16} + 10 \nu^{15} - 16 \nu^{14} - \nu^{13} + 104 \nu^{12} - 279 \nu^{11} + \cdots - 1562500 ) / 390625 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1054659 \nu^{17} + 5761219 \nu^{16} - 33944455 \nu^{15} + 64063381 \nu^{14} + \cdots + 3173994140625 ) / 340025000000 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 295237 \nu^{17} + 165451 \nu^{16} - 2100221 \nu^{15} + 4347453 \nu^{14} - 9764546 \nu^{13} + \cdots + 361752890625 ) / 68005000000 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 148055 \nu^{17} + 892093 \nu^{16} - 2096777 \nu^{15} + 1751475 \nu^{14} - 5065138 \nu^{13} + \cdots + 306865234375 ) / 34002500000 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 2144244 \nu^{17} - 7264521 \nu^{16} - 3078830 \nu^{15} + 43423321 \nu^{14} + \cdots - 1080715234375 ) / 340025000000 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1108357 \nu^{17} + 5274668 \nu^{16} - 5671855 \nu^{15} + 10585162 \nu^{14} + \cdots + 1784652343750 ) / 170012500000 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 4630437 \nu^{17} - 25902673 \nu^{16} + 42168095 \nu^{15} - 21581467 \nu^{14} + \cdots - 4148374609375 ) / 340025000000 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 4656697 \nu^{17} + 271052 \nu^{16} - 7996815 \nu^{15} + 13321948 \nu^{14} + \cdots + 3092468750000 ) / 340025000000 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 533999 \nu^{17} - 973338 \nu^{16} + 540553 \nu^{15} - 2825484 \nu^{14} - 4777502 \nu^{13} + \cdots - 328570625000 ) / 34002500000 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 2944652 \nu^{17} - 9700148 \nu^{16} + 16806855 \nu^{15} - 29318407 \nu^{14} + \cdots - 2547061328125 ) / 170012500000 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 3330252 \nu^{17} - 15355968 \nu^{16} + 20606335 \nu^{15} - 4217657 \nu^{14} + \cdots - 1120940234375 ) / 170012500000 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 1346399 \nu^{17} - 5052591 \nu^{16} + 9071245 \nu^{15} - 9469309 \nu^{14} + \cdots - 576634765625 ) / 68005000000 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 118016 \nu^{17} + 401279 \nu^{16} - 287320 \nu^{15} + 877731 \nu^{14} + 1003076 \nu^{13} + \cdots + 97987109375 ) / 5862500000 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( - \beta_{17} - \beta_{15} - \beta_{13} + \beta_{12} + \beta_{11} + \beta_{10} - \beta_{8} + \cdots - \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2 \beta_{17} + \beta_{16} + \beta_{15} - 2 \beta_{14} + 2 \beta_{13} + \beta_{12} + \beta_{11} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 5 \beta_{17} - \beta_{16} + \beta_{15} + 6 \beta_{14} - 2 \beta_{13} - 2 \beta_{11} - 4 \beta_{10} + \cdots + 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 10 \beta_{17} + 6 \beta_{16} - 3 \beta_{15} - 15 \beta_{14} - 4 \beta_{13} - 4 \beta_{12} + 2 \beta_{11} + \cdots - 17 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 20 \beta_{17} + 10 \beta_{16} + 13 \beta_{15} - 7 \beta_{14} - 33 \beta_{13} + 9 \beta_{12} + \cdots - 49 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 54 \beta_{16} - 13 \beta_{15} - 17 \beta_{14} + 46 \beta_{13} + 16 \beta_{12} + 54 \beta_{11} + \cdots + 105 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 15 \beta_{17} - 32 \beta_{16} + 54 \beta_{15} + 49 \beta_{14} - 2 \beta_{13} - 6 \beta_{12} + \cdots + 44 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 192 \beta_{17} - 383 \beta_{16} - 40 \beta_{15} - 17 \beta_{14} - 14 \beta_{13} + 255 \beta_{12} + \cdots - 792 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 199 \beta_{17} + 691 \beta_{16} + 28 \beta_{15} - 613 \beta_{14} - 83 \beta_{13} + 289 \beta_{12} + \cdots - 1253 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 2058 \beta_{17} - 1260 \beta_{16} + 1638 \beta_{15} + 1182 \beta_{14} + 1150 \beta_{13} - 964 \beta_{12} + \cdots - 638 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 1012 \beta_{17} + 662 \beta_{16} - 1208 \beta_{15} + 520 \beta_{14} - 210 \beta_{13} - 2126 \beta_{12} + \cdots + 297 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 4906 \beta_{17} - 1688 \beta_{16} + 9378 \beta_{15} - 4670 \beta_{14} - 3690 \beta_{13} + \cdots - 15138 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 9431 \beta_{17} - 18 \beta_{16} - 7949 \beta_{15} + 6332 \beta_{14} + 2545 \beta_{13} + 2131 \beta_{12} + \cdots + 17316 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 17756 \beta_{17} - 17703 \beta_{16} + 15615 \beta_{15} + 23260 \beta_{14} + 30532 \beta_{13} + \cdots + 80441 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 27345 \beta_{17} - 55871 \beta_{16} - 19055 \beta_{15} + 89814 \beta_{14} + 39104 \beta_{13} + \cdots - 87695 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 17284 \beta_{17} + 142610 \beta_{16} - 23961 \beta_{15} - 92029 \beta_{14} - 66954 \beta_{13} + \cdots + 7773 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2010\mathbb{Z}\right)^\times\).

\(n\) \(671\) \(1141\) \(1207\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1609.1
−2.20235 + 0.386880i
−1.90147 − 1.17660i
−0.442082 + 2.19193i
−0.325759 − 2.21221i
−0.0990354 + 2.23387i
1.34528 − 1.78612i
1.38463 + 1.75579i
2.07801 − 0.825764i
2.16278 − 0.567780i
−2.20235 − 0.386880i
−1.90147 + 1.17660i
−0.442082 − 2.19193i
−0.325759 + 2.21221i
−0.0990354 − 2.23387i
1.34528 + 1.78612i
1.38463 − 1.75579i
2.07801 + 0.825764i
2.16278 + 0.567780i
− 1.00000i 1.00000i −1.00000 −2.20235 − 0.386880i 1.00000 − 2.02241i 1.00000i −1.00000 −0.386880 + 2.20235i
1609.2 − 1.00000i 1.00000i −1.00000 −1.90147 + 1.17660i 1.00000 − 1.53330i 1.00000i −1.00000 1.17660 + 1.90147i
1609.3 − 1.00000i 1.00000i −1.00000 −0.442082 − 2.19193i 1.00000 − 5.06658i 1.00000i −1.00000 −2.19193 + 0.442082i
1609.4 − 1.00000i 1.00000i −1.00000 −0.325759 + 2.21221i 1.00000 − 0.443432i 1.00000i −1.00000 2.21221 + 0.325759i
1609.5 − 1.00000i 1.00000i −1.00000 −0.0990354 − 2.23387i 1.00000 2.04713i 1.00000i −1.00000 −2.23387 + 0.0990354i
1609.6 − 1.00000i 1.00000i −1.00000 1.34528 + 1.78612i 1.00000 − 4.39014i 1.00000i −1.00000 1.78612 − 1.34528i
1609.7 − 1.00000i 1.00000i −1.00000 1.38463 − 1.75579i 1.00000 − 2.87153i 1.00000i −1.00000 −1.75579 − 1.38463i
1609.8 − 1.00000i 1.00000i −1.00000 2.07801 + 0.825764i 1.00000 1.05244i 1.00000i −1.00000 0.825764 − 2.07801i
1609.9 − 1.00000i 1.00000i −1.00000 2.16278 + 0.567780i 1.00000 4.22783i 1.00000i −1.00000 0.567780 − 2.16278i
1609.10 1.00000i − 1.00000i −1.00000 −2.20235 + 0.386880i 1.00000 2.02241i − 1.00000i −1.00000 −0.386880 − 2.20235i
1609.11 1.00000i − 1.00000i −1.00000 −1.90147 − 1.17660i 1.00000 1.53330i − 1.00000i −1.00000 1.17660 − 1.90147i
1609.12 1.00000i − 1.00000i −1.00000 −0.442082 + 2.19193i 1.00000 5.06658i − 1.00000i −1.00000 −2.19193 − 0.442082i
1609.13 1.00000i − 1.00000i −1.00000 −0.325759 − 2.21221i 1.00000 0.443432i − 1.00000i −1.00000 2.21221 − 0.325759i
1609.14 1.00000i − 1.00000i −1.00000 −0.0990354 + 2.23387i 1.00000 − 2.04713i − 1.00000i −1.00000 −2.23387 − 0.0990354i
1609.15 1.00000i − 1.00000i −1.00000 1.34528 − 1.78612i 1.00000 4.39014i − 1.00000i −1.00000 1.78612 + 1.34528i
1609.16 1.00000i − 1.00000i −1.00000 1.38463 + 1.75579i 1.00000 2.87153i − 1.00000i −1.00000 −1.75579 + 1.38463i
1609.17 1.00000i − 1.00000i −1.00000 2.07801 − 0.825764i 1.00000 − 1.05244i − 1.00000i −1.00000 0.825764 + 2.07801i
1609.18 1.00000i − 1.00000i −1.00000 2.16278 − 0.567780i 1.00000 − 4.22783i − 1.00000i −1.00000 0.567780 + 2.16278i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1609.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2010.2.e.j ✓ 18
5.b even 2 1 inner 2010.2.e.j ✓ 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2010.2.e.j ✓ 18 1.a even 1 1 trivial
2010.2.e.j ✓ 18 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2010, [\chi])\):

\( T_{7}^{18} + 83 T_{7}^{16} + 2715 T_{7}^{14} + 44917 T_{7}^{12} + 404784 T_{7}^{10} + 2029720 T_{7}^{8} + \cdots + 640000 \) Copy content Toggle raw display
\( T_{11}^{9} - 7 T_{11}^{8} - 24 T_{11}^{7} + 250 T_{11}^{6} - 136 T_{11}^{5} - 2100 T_{11}^{4} + \cdots + 320 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{9} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{9} \) Copy content Toggle raw display
$5$ \( T^{18} - 4 T^{17} + \cdots + 1953125 \) Copy content Toggle raw display
$7$ \( T^{18} + 83 T^{16} + \cdots + 640000 \) Copy content Toggle raw display
$11$ \( (T^{9} - 7 T^{8} + \cdots + 320)^{2} \) Copy content Toggle raw display
$13$ \( T^{18} + 88 T^{16} + \cdots + 16777216 \) Copy content Toggle raw display
$17$ \( T^{18} + 108 T^{16} + \cdots + 16384 \) Copy content Toggle raw display
$19$ \( (T^{9} + 2 T^{8} + \cdots + 24320)^{2} \) Copy content Toggle raw display
$23$ \( T^{18} + 170 T^{16} + \cdots + 1081600 \) Copy content Toggle raw display
$29$ \( (T^{9} + 14 T^{8} + \cdots + 194560)^{2} \) Copy content Toggle raw display
$31$ \( (T^{9} - 4 T^{8} + \cdots - 3560)^{2} \) Copy content Toggle raw display
$37$ \( T^{18} + 243 T^{16} + \cdots + 640000 \) Copy content Toggle raw display
$41$ \( (T^{9} - 24 T^{8} + \cdots + 605120)^{2} \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 961551631138816 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 252148596736 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 161539686400 \) Copy content Toggle raw display
$59$ \( (T^{9} + 10 T^{8} + \cdots - 7532416)^{2} \) Copy content Toggle raw display
$61$ \( (T^{9} - 5 T^{8} + \cdots + 5533744)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 1)^{9} \) Copy content Toggle raw display
$71$ \( (T^{9} + 5 T^{8} + \cdots + 3265664)^{2} \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 75\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{9} - 8 T^{8} + \cdots + 53276000)^{2} \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 15424659275776 \) Copy content Toggle raw display
$89$ \( (T^{9} + 39 T^{8} + \cdots - 44257280)^{2} \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
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