Properties

Label 13.10.a.b
Level $13$
Weight $10$
Character orbit 13.a
Self dual yes
Analytic conductor $6.695$
Analytic rank $0$
Dimension $5$
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] = [13,10,Mod(1,13)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(13, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("13.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 13 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 13.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: \(6.69546587013\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 1438x^{3} - 4164x^{2} + 396957x - 59580 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 + 3) q^{2} + ( - \beta_{2} + 2 \beta_1 + 32) q^{3} + (\beta_{4} + 10 \beta_1 + 72) q^{4} + ( - 3 \beta_{4} + \beta_{3} + \cdots + 361) q^{5}+ \cdots + ( - 25 \beta_{4} - 29 \beta_{3} + \cdots + 12206) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 3) q^{2} + ( - \beta_{2} + 2 \beta_1 + 32) q^{3} + (\beta_{4} + 10 \beta_1 + 72) q^{4} + ( - 3 \beta_{4} + \beta_{3} + \cdots + 361) q^{5}+ \cdots + (11504 \beta_{4} + 27862 \beta_{3} + \cdots + 602456384) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 15 q^{2} + 161 q^{3} + 361 q^{4} + 1803 q^{5} + 5693 q^{6} + 10099 q^{7} + 23151 q^{8} + 61060 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 15 q^{2} + 161 q^{3} + 361 q^{4} + 1803 q^{5} + 5693 q^{6} + 10099 q^{7} + 23151 q^{8} + 61060 q^{9} + 84505 q^{10} + 121746 q^{11} + 113389 q^{12} + 142805 q^{13} + 8475 q^{14} + 105973 q^{15} - 322463 q^{16} - 495669 q^{17} - 656228 q^{18} - 840738 q^{19} - 1595607 q^{20} - 1599467 q^{21} - 2023594 q^{22} - 592152 q^{23} - 2295657 q^{24} + 1670362 q^{25} + 428415 q^{26} + 6847883 q^{27} + 2587955 q^{28} + 10678182 q^{29} + 5491201 q^{30} + 12885296 q^{31} + 3282927 q^{32} + 17278298 q^{33} - 9934079 q^{34} + 8380731 q^{35} - 20483302 q^{36} + 7171823 q^{37} - 25568814 q^{38} + 4598321 q^{39} - 54359445 q^{40} + 9294012 q^{41} - 69520457 q^{42} + 12831975 q^{43} - 41479074 q^{44} + 26135198 q^{45} - 59319696 q^{46} + 43354215 q^{47} - 86874671 q^{48} + 25249488 q^{49} - 16270770 q^{50} + 16905901 q^{51} + 10310521 q^{52} + 93231780 q^{53} + 58983719 q^{54} + 99448846 q^{55} + 199599225 q^{56} + 90173382 q^{57} + 151020970 q^{58} + 246496182 q^{59} + 90097913 q^{60} - 132232612 q^{61} + 158135724 q^{62} - 416955202 q^{63} + 91019105 q^{64} + 51495483 q^{65} - 323733130 q^{66} - 369388534 q^{67} + 238172073 q^{68} - 579986760 q^{69} - 144857425 q^{70} + 212150457 q^{71} - 415774278 q^{72} - 252729806 q^{73} + 192105957 q^{74} - 752457788 q^{75} - 953775990 q^{76} + 449666118 q^{77} + 162597773 q^{78} - 1247271728 q^{79} + 900649725 q^{80} - 317713115 q^{81} + 169559388 q^{82} + 1696894296 q^{83} + 1247983739 q^{84} - 775363765 q^{85} + 3291621459 q^{86} - 614530466 q^{87} - 220227222 q^{88} - 753854382 q^{89} + 2296265882 q^{90} + 288437539 q^{91} + 13876128 q^{92} - 892784668 q^{93} + 272071215 q^{94} + 1442632962 q^{95} + 930612847 q^{96} + 3824606 q^{97} + 1570614816 q^{98} + 3016199848 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 1438x^{3} - 4164x^{2} + 396957x - 59580 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{4} + 55\nu^{3} + 317\nu^{2} - 39383\nu + 189604 ) / 1088 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 7\nu^{4} - 113\nu^{3} - 6571\nu^{2} + 54545\nu + 495172 ) / 1088 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{2} - 4\nu - 575 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + 4\beta _1 + 575 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 16\beta_{4} + 4\beta_{3} + 28\beta_{2} + 877\beta _1 + 2500 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 1197\beta_{4} + 220\beta_{3} + 452\beta_{2} + 10120\beta _1 + 509379 \) Copy content Toggle raw display

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
−27.7188
−24.3176
0.150341
16.7176
35.1685
−24.7188 194.269 99.0182 −920.299 −4802.09 5359.02 10208.4 18057.4 22748.7
1.2 −21.3176 −195.094 −57.5590 −1277.14 4158.93 −2277.98 12141.6 18378.5 27225.6
1.3 3.15034 −136.532 −502.075 2554.62 −430.124 9399.91 −3194.68 −1041.89 8047.92
1.4 19.7176 250.479 −123.217 1555.58 4938.84 −8329.39 −12524.9 43056.6 30672.3
1.5 38.1685 47.8784 944.833 −109.762 1827.45 5947.44 16520.6 −17390.7 −4189.45
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(13\) \( -1 \)

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 13.10.a.b 5
3.b odd 2 1 117.10.a.e 5
4.b odd 2 1 208.10.a.h 5
5.b even 2 1 325.10.a.b 5
13.b even 2 1 169.10.a.b 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.10.a.b 5 1.a even 1 1 trivial
117.10.a.e 5 3.b odd 2 1
169.10.a.b 5 13.b even 2 1
208.10.a.h 5 4.b odd 2 1
325.10.a.b 5 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{5} - 15T_{2}^{4} - 1348T_{2}^{3} + 8508T_{2}^{2} + 383520T_{2} - 1249344 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(13))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} - 15 T^{4} + \cdots - 1249344 \) Copy content Toggle raw display
$3$ \( T^{5} + \cdots - 62057286864 \) Copy content Toggle raw display
$5$ \( T^{5} + \cdots + 512670311383500 \) Copy content Toggle raw display
$7$ \( T^{5} + \cdots - 56\!\cdots\!88 \) Copy content Toggle raw display
$11$ \( T^{5} + \cdots - 19\!\cdots\!96 \) Copy content Toggle raw display
$13$ \( (T - 28561)^{5} \) Copy content Toggle raw display
$17$ \( T^{5} + \cdots - 39\!\cdots\!92 \) Copy content Toggle raw display
$19$ \( T^{5} + \cdots - 29\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{5} + \cdots - 42\!\cdots\!52 \) Copy content Toggle raw display
$29$ \( T^{5} + \cdots - 44\!\cdots\!20 \) Copy content Toggle raw display
$31$ \( T^{5} + \cdots + 48\!\cdots\!64 \) Copy content Toggle raw display
$37$ \( T^{5} + \cdots - 51\!\cdots\!48 \) Copy content Toggle raw display
$41$ \( T^{5} + \cdots + 49\!\cdots\!12 \) Copy content Toggle raw display
$43$ \( T^{5} + \cdots + 16\!\cdots\!08 \) Copy content Toggle raw display
$47$ \( T^{5} + \cdots - 10\!\cdots\!64 \) Copy content Toggle raw display
$53$ \( T^{5} + \cdots - 36\!\cdots\!12 \) Copy content Toggle raw display
$59$ \( T^{5} + \cdots - 43\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{5} + \cdots + 57\!\cdots\!08 \) Copy content Toggle raw display
$67$ \( T^{5} + \cdots - 85\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( T^{5} + \cdots - 41\!\cdots\!92 \) Copy content Toggle raw display
$73$ \( T^{5} + \cdots + 16\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( T^{5} + \cdots - 41\!\cdots\!40 \) Copy content Toggle raw display
$83$ \( T^{5} + \cdots - 52\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( T^{5} + \cdots - 19\!\cdots\!60 \) Copy content Toggle raw display
$97$ \( T^{5} + \cdots - 14\!\cdots\!32 \) Copy content Toggle raw display
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