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Properties

Label 2013.2.a.h
Level $2013$
Weight $2$
Character orbit 2013.a
Self dual yes
Analytic conductor $16.074$
Analytic rank $0$
Dimension $14$
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] = [2013,2,Mod(1,2013)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2013, 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("2013.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2013 = 3 \cdot 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2013.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: \(16.0738859269\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - x^{13} - 21 x^{12} + 20 x^{11} + 167 x^{10} - 148 x^{9} - 627 x^{8} + 497 x^{7} + 1123 x^{6} + \cdots - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - x^{13} - 21 x^{12} + 20 x^{11} + 167 x^{10} - 148 x^{9} - 627 x^{8} + 497 x^{7} + 1123 x^{6} + \cdots - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 5\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - \nu^{13} - 71 \nu^{12} - 586 \nu^{11} + 2838 \nu^{10} + 10454 \nu^{9} - 35539 \nu^{8} + \cdots - 13953 ) / 4505 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 82 \nu^{13} - 1317 \nu^{12} + 1503 \nu^{11} + 25486 \nu^{10} - 7732 \nu^{9} - 179663 \nu^{8} + \cdots - 22401 ) / 4505 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 311 \nu^{13} + 444 \nu^{12} + 6964 \nu^{11} - 9372 \nu^{10} - 59981 \nu^{9} + 74721 \nu^{8} + \cdots - 1068 ) / 4505 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 349 \nu^{13} - 2254 \nu^{12} + 7221 \nu^{11} + 44412 \nu^{10} - 54664 \nu^{9} - 320701 \nu^{8} + \cdots - 17712 ) / 4505 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 251 \nu^{13} + 199 \nu^{12} + 5183 \nu^{11} - 3957 \nu^{10} - 40303 \nu^{9} + 29344 \nu^{8} + \cdots + 885 ) / 901 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1423 \nu^{13} - 2582 \nu^{12} - 31082 \nu^{11} + 52066 \nu^{10} + 260758 \nu^{9} - 388563 \nu^{8} + \cdots - 20941 ) / 4505 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 1433 \nu^{13} - 1872 \nu^{12} - 29727 \nu^{11} + 37201 \nu^{10} + 232803 \nu^{9} - 271938 \nu^{8} + \cdots - 12056 ) / 4505 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 303 \nu^{13} + 111 \nu^{12} + 6246 \nu^{11} - 2343 \nu^{10} - 48107 \nu^{9} + 18455 \nu^{8} + \cdots + 3337 ) / 901 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 1576 \nu^{13} + 729 \nu^{12} + 31524 \nu^{11} - 14292 \nu^{10} - 233541 \nu^{9} + 104816 \nu^{8} + \cdots + 21502 ) / 4505 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 3393 \nu^{13} + 2367 \nu^{12} + 70487 \nu^{11} - 47406 \nu^{10} - 551558 \nu^{9} + 352898 \nu^{8} + \cdots + 36556 ) / 4505 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} + \beta_{10} - \beta_{5} + 7\beta_{2} + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{12} + \beta_{11} + \beta_{9} + \beta_{8} + \beta_{6} - \beta_{5} + 9\beta_{3} + 28\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{13} + 9 \beta_{11} + 9 \beta_{10} + \beta_{9} - 2 \beta_{8} + 2 \beta_{6} - 10 \beta_{5} + \cdots + 84 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2 \beta_{13} - 12 \beta_{12} + 10 \beta_{11} - \beta_{10} + 13 \beta_{9} + 9 \beta_{8} + 12 \beta_{6} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 16 \beta_{13} - 2 \beta_{12} + 64 \beta_{11} + 66 \beta_{10} + 16 \beta_{9} - 26 \beta_{8} - \beta_{7} + \cdots + 504 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 35 \beta_{13} - 109 \beta_{12} + 79 \beta_{11} - 10 \beta_{10} + 125 \beta_{9} + 57 \beta_{8} + \cdots - 12 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 176 \beta_{13} - 37 \beta_{12} + 426 \beta_{11} + 464 \beta_{10} + 174 \beta_{9} - 243 \beta_{8} + \cdots + 3147 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 405 \beta_{13} - 898 \beta_{12} + 586 \beta_{11} - 56 \beta_{10} + 1066 \beta_{9} + 302 \beta_{8} + \cdots - 89 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1652 \beta_{13} - 443 \beta_{12} + 2777 \beta_{11} + 3245 \beta_{10} + 1609 \beta_{9} - 2010 \beta_{8} + \cdots + 20093 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 3916 \beta_{13} - 7055 \beta_{12} + 4261 \beta_{11} - 128 \beta_{10} + 8540 \beta_{9} + 1339 \beta_{8} + \cdots - 465 \) 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
2.63401
2.45909
1.93923
1.67203
1.61392
0.546298
0.179763
−0.0561655
−0.231279
−1.18140
−1.69494
−1.76637
−2.54331
−2.57087
−2.63401 −1.00000 4.93799 2.62502 2.63401 5.18852 −7.73867 1.00000 −6.91432
1.2 −2.45909 −1.00000 4.04715 −1.85442 2.45909 2.32343 −5.03413 1.00000 4.56020
1.3 −1.93923 −1.00000 1.76061 3.07794 1.93923 −3.15900 0.464237 1.00000 −5.96883
1.4 −1.67203 −1.00000 0.795679 −1.45953 1.67203 −0.113904 2.01366 1.00000 2.44037
1.5 −1.61392 −1.00000 0.604750 −3.65776 1.61392 0.985739 2.25183 1.00000 5.90335
1.6 −0.546298 −1.00000 −1.70156 −0.842631 0.546298 −4.19208 2.02216 1.00000 0.460328
1.7 −0.179763 −1.00000 −1.96769 2.20477 0.179763 3.17177 0.713244 1.00000 −0.396336
1.8 0.0561655 −1.00000 −1.99685 −2.87016 −0.0561655 3.53988 −0.224485 1.00000 −0.161204
1.9 0.231279 −1.00000 −1.94651 3.70694 −0.231279 −0.911108 −0.912746 1.00000 0.857338
1.10 1.18140 −1.00000 −0.604292 −2.90081 −1.18140 −0.528840 −3.07671 1.00000 −3.42702
1.11 1.69494 −1.00000 0.872835 1.13650 −1.69494 4.14659 −1.91048 1.00000 1.92630
1.12 1.76637 −1.00000 1.12006 −2.50284 −1.76637 −4.08919 −1.55430 1.00000 −4.42094
1.13 2.54331 −1.00000 4.46845 0.329781 −2.54331 0.595463 6.27804 1.00000 0.838736
1.14 2.57087 −1.00000 4.60938 4.00721 −2.57087 2.04273 6.70837 1.00000 10.3020
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(11\) \(1\)
\(61\) \(-1\)

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 2013.2.a.h ✓ 14
3.b odd 2 1 6039.2.a.j 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2013.2.a.h ✓ 14 1.a even 1 1 trivial
6039.2.a.j 14 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{14} + T_{2}^{13} - 21 T_{2}^{12} - 20 T_{2}^{11} + 167 T_{2}^{10} + 148 T_{2}^{9} - 627 T_{2}^{8} + \cdots - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2013))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + T^{13} + \cdots - 1 \) Copy content Toggle raw display
$3$ \( (T + 1)^{14} \) Copy content Toggle raw display
$5$ \( T^{14} - T^{13} + \cdots - 17240 \) Copy content Toggle raw display
$7$ \( T^{14} - 9 T^{13} + \cdots + 2000 \) Copy content Toggle raw display
$11$ \( (T + 1)^{14} \) Copy content Toggle raw display
$13$ \( T^{14} - T^{13} + \cdots - 30904 \) Copy content Toggle raw display
$17$ \( T^{14} + 9 T^{13} + \cdots - 2464 \) Copy content Toggle raw display
$19$ \( T^{14} - 22 T^{13} + \cdots - 372904 \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots - 101900800 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 4669602512 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 302065736 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots - 207834696 \) Copy content Toggle raw display
$41$ \( T^{14} + 25 T^{13} + \cdots + 7101316 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 6851179168 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots - 2272971239168 \) Copy content Toggle raw display
$53$ \( T^{14} - 340 T^{12} + \cdots - 50664924 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 10127082816 \) Copy content Toggle raw display
$61$ \( (T - 1)^{14} \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots - 11152814800 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 6346448960 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 457179272 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots - 33860837584 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 1517677024 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 12934626280804 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 39289496832 \) Copy content Toggle raw display
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