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Properties

Label 2013.2.a.f
Level $2013$
Weight $2$
Character orbit 2013.a
Self dual yes
Analytic conductor $16.074$
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] = [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: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 4 x^{12} - 11 x^{11} + 55 x^{10} + 32 x^{9} - 266 x^{8} + 13 x^{7} + 534 x^{6} - 141 x^{5} + \cdots - 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{13} - 4 x^{12} - 11 x^{11} + 55 x^{10} + 32 x^{9} - 266 x^{8} + 13 x^{7} + 534 x^{6} - 141 x^{5} + \cdots - 11 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 4\nu + 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{12} - 5 \nu^{11} - 11 \nu^{10} + 73 \nu^{9} + 29 \nu^{8} - 374 \nu^{7} + 46 \nu^{6} + 770 \nu^{5} + \cdots - 47 ) / 17 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - \nu^{12} + \nu^{11} + 20 \nu^{10} - 17 \nu^{9} - 150 \nu^{8} + 108 \nu^{7} + 506 \nu^{6} - 320 \nu^{5} + \cdots + 8 ) / 17 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 2 \nu^{12} + 8 \nu^{11} + 18 \nu^{10} - 101 \nu^{9} - 8 \nu^{8} + 411 \nu^{7} - 292 \nu^{6} + \cdots + 66 ) / 17 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2 \nu^{12} - 7 \nu^{11} - 33 \nu^{10} + 104 \nu^{9} + 221 \nu^{8} - 540 \nu^{7} - 764 \nu^{6} + \cdots + 186 ) / 17 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 5 \nu^{12} + 14 \nu^{11} + 67 \nu^{10} - 194 \nu^{9} - 312 \nu^{8} + 960 \nu^{7} + 591 \nu^{6} + \cdots - 21 ) / 17 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - \nu^{12} + 11 \nu^{11} - 11 \nu^{10} - 140 \nu^{9} + 263 \nu^{8} + 586 \nu^{7} - 1384 \nu^{6} + \cdots + 267 ) / 17 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 6 \nu^{12} + 26 \nu^{11} + 58 \nu^{10} - 348 \nu^{9} - 91 \nu^{8} + 1604 \nu^{7} - 489 \nu^{6} + \cdots - 12 ) / 17 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 6 \nu^{12} - 26 \nu^{11} - 58 \nu^{10} + 348 \nu^{9} + 91 \nu^{8} - 1604 \nu^{7} + 506 \nu^{6} + \cdots - 141 ) / 17 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 7 \nu^{12} - 33 \nu^{11} - 56 \nu^{10} + 432 \nu^{9} - 51 \nu^{8} - 1924 \nu^{7} + 1321 \nu^{6} + \cdots - 233 ) / 17 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{10} + \beta_{9} + \beta_{8} - \beta_{7} - \beta_{5} + \beta_{4} + \beta_{3} + 6\beta_{2} + 8\beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{12} - \beta_{10} - \beta_{9} - \beta_{5} - \beta_{4} + 8\beta_{3} + 9\beta_{2} + 29\beta _1 + 10 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - \beta_{12} + \beta_{11} - 10 \beta_{10} + 9 \beta_{9} + 10 \beta_{8} - 10 \beta_{7} - 11 \beta_{5} + \cdots + 84 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 13 \beta_{12} + 2 \beta_{11} - 13 \beta_{10} - 10 \beta_{9} + 2 \beta_{8} - 2 \beta_{7} - \beta_{6} + \cdots + 83 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 17 \beta_{12} + 15 \beta_{11} - 79 \beta_{10} + 61 \beta_{9} + 76 \beta_{8} - 76 \beta_{7} + \cdots + 498 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 123 \beta_{12} + 32 \beta_{11} - 124 \beta_{10} - 75 \beta_{9} + 31 \beta_{8} - 33 \beta_{7} + \cdots + 636 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 193 \beta_{12} + 155 \beta_{11} - 581 \beta_{10} + 369 \beta_{9} + 526 \beta_{8} - 530 \beta_{7} + \cdots + 3068 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 1028 \beta_{12} + 348 \beta_{11} - 1044 \beta_{10} - 508 \beta_{9} + 328 \beta_{8} - 365 \beta_{7} + \cdots + 4676 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 1837 \beta_{12} + 1376 \beta_{11} - 4155 \beta_{10} + 2096 \beta_{9} + 3502 \beta_{8} - 3585 \beta_{7} + \cdots + 19419 \) 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
−2.33909
−1.96783
−1.74673
−0.762797
−0.534142
−0.329361
0.468970
0.684638
1.03857
1.88022
2.38255
2.60074
2.62425
−2.33909 1.00000 3.47134 −0.256721 −2.33909 −3.14990 −3.44160 1.00000 0.600493
1.2 −1.96783 1.00000 1.87234 3.39588 −1.96783 0.824946 0.251204 1.00000 −6.68250
1.3 −1.74673 1.00000 1.05105 −2.68410 −1.74673 −0.749083 1.65755 1.00000 4.68839
1.4 −0.762797 1.00000 −1.41814 −1.72725 −0.762797 1.91723 2.60735 1.00000 1.31754
1.5 −0.534142 1.00000 −1.71469 3.62612 −0.534142 4.31243 1.98417 1.00000 −1.93686
1.6 −0.329361 1.00000 −1.89152 2.08793 −0.329361 −4.32416 1.28171 1.00000 −0.687681
1.7 0.468970 1.00000 −1.78007 −3.25579 0.468970 −3.43403 −1.77274 1.00000 −1.52687
1.8 0.684638 1.00000 −1.53127 −1.50791 0.684638 2.85777 −2.41764 1.00000 −1.03237
1.9 1.03857 1.00000 −0.921370 2.59019 1.03857 2.45800 −3.03405 1.00000 2.69009
1.10 1.88022 1.00000 1.53524 1.27827 1.88022 0.0298223 −0.873847 1.00000 2.40344
1.11 2.38255 1.00000 3.67655 3.76064 2.38255 −1.09133 3.99445 1.00000 8.95992
1.12 2.60074 1.00000 4.76385 −0.540623 2.60074 4.31275 7.18805 1.00000 −1.40602
1.13 2.62425 1.00000 4.88668 0.233370 2.62425 1.03556 7.57538 1.00000 0.612421
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.13
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.f ✓ 13
3.b odd 2 1 6039.2.a.g 13
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2013.2.a.f ✓ 13 1.a even 1 1 trivial
6039.2.a.g 13 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{13} - 4 T_{2}^{12} - 11 T_{2}^{11} + 55 T_{2}^{10} + 32 T_{2}^{9} - 266 T_{2}^{8} + 13 T_{2}^{7} + \cdots - 11 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2013))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{13} - 4 T^{12} + \cdots - 11 \) Copy content Toggle raw display
$3$ \( (T - 1)^{13} \) Copy content Toggle raw display
$5$ \( T^{13} - 7 T^{12} + \cdots - 236 \) Copy content Toggle raw display
$7$ \( T^{13} - 5 T^{12} + \cdots + 244 \) Copy content Toggle raw display
$11$ \( (T + 1)^{13} \) Copy content Toggle raw display
$13$ \( T^{13} + 9 T^{12} + \cdots - 7972 \) Copy content Toggle raw display
$17$ \( T^{13} - 7 T^{12} + \cdots - 20968 \) Copy content Toggle raw display
$19$ \( T^{13} - 2 T^{12} + \cdots + 69050 \) Copy content Toggle raw display
$23$ \( T^{13} - 23 T^{12} + \cdots + 50081152 \) Copy content Toggle raw display
$29$ \( T^{13} - 16 T^{12} + \cdots + 704192 \) Copy content Toggle raw display
$31$ \( T^{13} - 9 T^{12} + \cdots + 1383602 \) Copy content Toggle raw display
$37$ \( T^{13} - 14 T^{12} + \cdots + 288220 \) Copy content Toggle raw display
$41$ \( T^{13} - 19 T^{12} + \cdots + 1984822 \) Copy content Toggle raw display
$43$ \( T^{13} - 7 T^{12} + \cdots + 92060 \) Copy content Toggle raw display
$47$ \( T^{13} - 26 T^{12} + \cdots + 232556 \) Copy content Toggle raw display
$53$ \( T^{13} + \cdots - 2168121694 \) Copy content Toggle raw display
$59$ \( T^{13} + \cdots - 306402637808 \) Copy content Toggle raw display
$61$ \( (T + 1)^{13} \) Copy content Toggle raw display
$67$ \( T^{13} + \cdots + 22880121982 \) Copy content Toggle raw display
$71$ \( T^{13} + \cdots - 3379107441064 \) Copy content Toggle raw display
$73$ \( T^{13} + 16 T^{12} + \cdots - 41176692 \) Copy content Toggle raw display
$79$ \( T^{13} + \cdots + 381289224964 \) Copy content Toggle raw display
$83$ \( T^{13} + \cdots + 49887527848 \) Copy content Toggle raw display
$89$ \( T^{13} + \cdots - 820098746 \) Copy content Toggle raw display
$97$ \( T^{13} + \cdots - 3579851464 \) Copy content Toggle raw display
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