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Properties

Label 2013.2.a.e
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: \( x^{13} - 2 x^{12} - 19 x^{11} + 35 x^{10} + 136 x^{9} - 220 x^{8} - 469 x^{7} + 610 x^{6} + 841 x^{5} + \cdots - 47 \) 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.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{13} - 2 x^{12} - 19 x^{11} + 35 x^{10} + 136 x^{9} - 220 x^{8} - 469 x^{7} + 610 x^{6} + 841 x^{5} + \cdots - 47 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 38 \nu^{12} + 47 \nu^{11} + 646 \nu^{10} - 577 \nu^{9} - 3970 \nu^{8} + 1615 \nu^{7} + 11430 \nu^{6} + \cdots - 4830 ) / 1261 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 80 \nu^{12} - 192 \nu^{11} - 1217 \nu^{10} + 2967 \nu^{9} + 5684 \nu^{8} - 15191 \nu^{7} + \cdots - 1964 ) / 1261 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 3 \nu^{12} + 275 \nu^{11} - 352 \nu^{10} - 5029 \nu^{9} + 5943 \nu^{8} + 33414 \nu^{7} + \cdots + 5338 ) / 1261 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 74 \nu^{12} - 448 \nu^{11} - 738 \nu^{10} + 7586 \nu^{9} - 695 \nu^{8} - 44875 \nu^{7} + 24220 \nu^{6} + \cdots - 9293 ) / 1261 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 112 \nu^{12} - 495 \nu^{11} - 1384 \nu^{10} + 8163 \nu^{9} + 3275 \nu^{8} - 46490 \nu^{7} + \cdots - 8246 ) / 1261 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 128 \nu^{12} - 42 \nu^{11} - 2670 \nu^{10} + 699 \nu^{9} + 20498 \nu^{8} - 4257 \nu^{7} - 69946 \nu^{6} + \cdots + 1171 ) / 1261 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 162 \nu^{12} - 524 \nu^{11} - 2507 \nu^{10} + 8823 \nu^{9} + 12632 \nu^{8} - 52242 \nu^{7} + \cdots - 11183 ) / 1261 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 191 \nu^{12} + 448 \nu^{11} + 3260 \nu^{10} - 7625 \nu^{9} - 19377 \nu^{8} + 45850 \nu^{7} + \cdots + 8136 ) / 1261 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 187 \nu^{12} - 883 \nu^{11} - 2451 \nu^{10} + 15172 \nu^{9} + 7541 \nu^{8} - 92220 \nu^{7} + \cdots - 17754 ) / 1261 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 344 \nu^{12} + 953 \nu^{11} + 5640 \nu^{10} - 16038 \nu^{9} - 31573 \nu^{8} + 94362 \nu^{7} + \cdots + 8700 ) / 1261 \) 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
\(\nu^{3}\)\(=\) \( -\beta_{7} + \beta_{6} - \beta_{3} + \beta_{2} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{12} + \beta_{10} - \beta_{9} - \beta_{7} - \beta_{5} + \beta_{4} - \beta_{3} + 9\beta_{2} + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{12} - \beta_{11} - \beta_{8} - 10 \beta_{7} + 9 \beta_{6} - \beta_{5} + \beta_{4} - 9 \beta_{3} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 10 \beta_{12} - 2 \beta_{11} + 11 \beta_{10} - 8 \beta_{9} + \beta_{8} - 11 \beta_{7} + 3 \beta_{6} + \cdots + 79 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 13 \beta_{12} - 16 \beta_{11} - \beta_{10} - 12 \beta_{8} - 79 \beta_{7} + 72 \beta_{6} - 13 \beta_{5} + \cdots - 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 81 \beta_{12} - 33 \beta_{11} + 92 \beta_{10} - 51 \beta_{9} + 13 \beta_{8} - 92 \beta_{7} + \cdots + 488 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 120 \beta_{12} - 176 \beta_{11} - 22 \beta_{10} + \beta_{9} - 105 \beta_{8} - 580 \beta_{7} + \cdots - 29 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 606 \beta_{12} - 375 \beta_{11} + 688 \beta_{10} - 303 \beta_{9} + 127 \beta_{8} - 697 \beta_{7} + \cdots + 3165 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 967 \beta_{12} - 1670 \beta_{11} - 303 \beta_{10} + 23 \beta_{9} - 815 \beta_{8} - 4127 \beta_{7} + \cdots + 124 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 4357 \beta_{12} - 3662 \beta_{11} + 4833 \beta_{10} - 1739 \beta_{9} + 1118 \beta_{8} - 5052 \beta_{7} + \cdots + 21158 \) 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
−2.59890
−2.37960
−1.46794
−1.33092
−0.948254
−0.805107
0.171582
0.822526
1.50067
1.61181
2.14727
2.53773
2.73913
−2.59890 −1.00000 4.75428 −3.50938 2.59890 0.252288 −7.15810 1.00000 9.12053
1.2 −2.37960 −1.00000 3.66252 2.55185 2.37960 1.67712 −3.95613 1.00000 −6.07239
1.3 −1.46794 −1.00000 0.154851 3.84216 1.46794 2.46223 2.70857 1.00000 −5.64006
1.4 −1.33092 −1.00000 −0.228660 −2.36819 1.33092 −2.92169 2.96616 1.00000 3.15187
1.5 −0.948254 −1.00000 −1.10081 −2.25122 0.948254 5.24025 2.94036 1.00000 2.13473
1.6 −0.805107 −1.00000 −1.35180 1.06503 0.805107 −0.203035 2.69856 1.00000 −0.857467
1.7 0.171582 −1.00000 −1.97056 −0.133072 −0.171582 0.615329 −0.681275 1.00000 −0.0228327
1.8 0.822526 −1.00000 −1.32345 2.11599 −0.822526 0.404149 −2.73363 1.00000 1.74046
1.9 1.50067 −1.00000 0.252024 −0.569898 −1.50067 −3.97554 −2.62314 1.00000 −0.855232
1.10 1.61181 −1.00000 0.597924 4.27930 −1.61181 1.98799 −2.25988 1.00000 6.89741
1.11 2.14727 −1.00000 2.61077 −3.62125 −2.14727 2.48742 1.31149 1.00000 −7.77580
1.12 2.53773 −1.00000 4.44007 0.994065 −2.53773 4.88646 6.19224 1.00000 2.52267
1.13 2.73913 −1.00000 5.50285 0.604616 −2.73913 −1.91298 9.59477 1.00000 1.65612
\(n\): 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.e ✓ 13
3.b odd 2 1 6039.2.a.i 13
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2013.2.a.e ✓ 13 1.a even 1 1 trivial
6039.2.a.i 13 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{13} - 2 T_{2}^{12} - 19 T_{2}^{11} + 35 T_{2}^{10} + 136 T_{2}^{9} - 220 T_{2}^{8} - 469 T_{2}^{7} + \cdots - 47 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2013))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{13} - 2 T^{12} + \cdots - 47 \) Copy content Toggle raw display
$3$ \( (T + 1)^{13} \) Copy content Toggle raw display
$5$ \( T^{13} - 3 T^{12} + \cdots - 292 \) Copy content Toggle raw display
$7$ \( T^{13} - 11 T^{12} + \cdots - 148 \) Copy content Toggle raw display
$11$ \( (T - 1)^{13} \) Copy content Toggle raw display
$13$ \( T^{13} - 13 T^{12} + \cdots + 388 \) Copy content Toggle raw display
$17$ \( T^{13} - 17 T^{12} + \cdots + 4744 \) Copy content Toggle raw display
$19$ \( T^{13} - 14 T^{12} + \cdots - 10215742 \) Copy content Toggle raw display
$23$ \( T^{13} - 7 T^{12} + \cdots - 1840672 \) Copy content Toggle raw display
$29$ \( T^{13} + 6 T^{12} + \cdots + 11772848 \) Copy content Toggle raw display
$31$ \( T^{13} - 27 T^{12} + \cdots - 1170458 \) Copy content Toggle raw display
$37$ \( T^{13} - 10 T^{12} + \cdots + 76337468 \) Copy content Toggle raw display
$41$ \( T^{13} - 3 T^{12} + \cdots - 98100918 \) Copy content Toggle raw display
$43$ \( T^{13} + \cdots - 195974572 \) Copy content Toggle raw display
$47$ \( T^{13} - 8 T^{12} + \cdots - 26421804 \) Copy content Toggle raw display
$53$ \( T^{13} + \cdots + 93111825698 \) Copy content Toggle raw display
$59$ \( T^{13} - 13 T^{12} + \cdots + 7051312 \) Copy content Toggle raw display
$61$ \( (T + 1)^{13} \) Copy content Toggle raw display
$67$ \( T^{13} + \cdots - 6958847006 \) Copy content Toggle raw display
$71$ \( T^{13} - 3 T^{12} + \cdots - 48814344 \) Copy content Toggle raw display
$73$ \( T^{13} + \cdots + 28332933844 \) Copy content Toggle raw display
$79$ \( T^{13} + \cdots + 388386798444 \) Copy content Toggle raw display
$83$ \( T^{13} + \cdots + 58130319032 \) Copy content Toggle raw display
$89$ \( T^{13} + \cdots - 120747458 \) Copy content Toggle raw display
$97$ \( T^{13} + \cdots - 87087118744 \) Copy content Toggle raw display
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