Properties

Label 13.14.a.a
Level $13$
Weight $14$
Character orbit 13.a
Self dual yes
Analytic conductor $13.940$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [13,14,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, 14, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("13.1");
 
S:= CuspForms(chi, 14);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 13 \)
Weight: \( k \) \(=\) \( 14 \)
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: \(13.9400207637\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 34578x^{4} - 245250x^{3} + 301528653x^{2} + 1141403491x - 761768771036 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4} \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 11) q^{2} + (\beta_{2} + 2 \beta_1 - 365) q^{3} + (\beta_{3} - 3 \beta_{2} - 10 \beta_1 + 3453) q^{4} + ( - \beta_{4} - 5 \beta_{3} + \cdots - 12794) q^{5}+ \cdots + (39 \beta_{4} + 123 \beta_{3} + \cdots + 415573) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 11) q^{2} + (\beta_{2} + 2 \beta_1 - 365) q^{3} + (\beta_{3} - 3 \beta_{2} - 10 \beta_1 + 3453) q^{4} + ( - \beta_{4} - 5 \beta_{3} + \cdots - 12794) q^{5}+ \cdots + (341352396 \beta_{5} + \cdots + 562146370654) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 65 q^{2} - 2188 q^{3} + 20709 q^{4} - 76752 q^{5} + 155623 q^{6} + 271440 q^{7} - 385359 q^{8} + 2482058 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 65 q^{2} - 2188 q^{3} + 20709 q^{4} - 76752 q^{5} + 155623 q^{6} + 271440 q^{7} - 385359 q^{8} + 2482058 q^{9} + 1958265 q^{10} - 10381540 q^{11} - 45610903 q^{12} - 28960854 q^{13} - 90325765 q^{14} + 44277948 q^{15} - 114616719 q^{16} - 158168768 q^{17} - 817311638 q^{18} - 344467812 q^{19} - 1606802717 q^{20} - 1362000276 q^{21} - 994063974 q^{22} - 819335384 q^{23} + 811229133 q^{24} + 1191461742 q^{25} + 313742585 q^{26} - 2066222572 q^{27} + 6336076773 q^{28} + 3016871284 q^{29} - 4832579039 q^{30} + 10160650812 q^{31} + 10452982137 q^{32} + 20382532352 q^{33} + 33491602131 q^{34} - 3217526044 q^{35} + 58566370862 q^{36} + 11333857080 q^{37} - 35164839374 q^{38} + 10561058092 q^{39} + 25796611575 q^{40} - 29997423092 q^{41} + 87796840043 q^{42} - 60780215652 q^{43} - 127689276442 q^{44} - 248661587772 q^{45} - 30575990796 q^{46} - 103938364920 q^{47} - 163563054627 q^{48} - 213701720838 q^{49} - 194717671980 q^{50} - 290589694748 q^{51} - 99958387581 q^{52} - 398877212524 q^{53} + 883021761349 q^{54} + 203094660456 q^{55} - 41234376295 q^{56} - 408224238552 q^{57} - 109548099570 q^{58} + 464482663996 q^{59} + 2610980312863 q^{60} - 64329382668 q^{61} + 621304219564 q^{62} + 1241788189356 q^{63} + 106114141137 q^{64} + 370467244368 q^{65} + 1052192441050 q^{66} - 908043289140 q^{67} + 2134057181201 q^{68} - 912381570632 q^{69} + 3777680409165 q^{70} - 2885058582224 q^{71} - 2138988475338 q^{72} - 848522826060 q^{73} + 4049561594517 q^{74} - 6736509337008 q^{75} + 1861241597358 q^{76} - 4890475722168 q^{77} - 751162497007 q^{78} - 5609407048632 q^{79} - 1964951726985 q^{80} - 2724107904754 q^{81} - 1610417512992 q^{82} - 6954788539660 q^{83} - 8688526273063 q^{84} - 6931231220940 q^{85} + 3474340081329 q^{86} - 9000693405584 q^{87} + 3051890126718 q^{88} - 7286778533892 q^{89} + 33445975464982 q^{90} - 1310189034960 q^{91} + 6008043339532 q^{92} - 7081056298472 q^{93} + 24204370752975 q^{94} - 6374489271248 q^{95} + 12194866385637 q^{96} + 28132662431796 q^{97} - 1411077811754 q^{98} + 3415702098380 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} - 34578x^{4} - 245250x^{3} + 301528653x^{2} + 1141403491x - 761768771036 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 455\nu^{5} + 20106\nu^{4} - 13155416\nu^{3} - 753334486\nu^{2} + 58401824033\nu + 3416956810460 ) / 187891968 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 455\nu^{5} + 20106\nu^{4} - 13155416\nu^{3} - 690703830\nu^{2} + 57650256161\nu + 2695201130716 ) / 62630656 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 133\nu^{5} + 9641\nu^{4} - 3962109\nu^{3} - 317091025\nu^{2} + 18008741056\nu + 1387523313920 ) / 3914416 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 5889 \nu^{5} - 483910 \nu^{4} + 172729160 \nu^{3} + 15794382618 \nu^{2} + \cdots - 70012689899972 ) / 187891968 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 3\beta_{2} + 12\beta _1 + 11524 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -42\beta_{5} - 52\beta_{4} + 64\beta_{3} - 6\beta_{2} + 17173\beta _1 + 137038 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -1302\beta_{5} - 572\beta_{4} + 27725\beta_{3} - 92001\beta_{2} + 592192\beta _1 + 197610402 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -1156812\beta_{5} - 1478200\beta_{4} + 2280972\beta_{3} - 662136\beta_{2} + 361867113\beta _1 + 6800235080 \) 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
−140.869
−93.5755
−68.8369
71.8052
73.5789
158.897
−151.869 −2279.81 14872.1 −53923.4 346232. 358132. −1.01450e6 3.60323e6 8.18928e6
1.2 −104.576 1639.96 2744.04 −27927.9 −171499. 85412.5 569723. 1.09513e6 2.92058e6
1.3 −79.8369 −1213.97 −1818.06 39770.7 96919.6 108654. 799173. −120602. −3.17517e6
1.4 60.8052 1156.50 −4494.73 −18981.4 70321.3 −435876. −771419. −256825. −1.15417e6
1.5 62.5789 −399.859 −4275.88 29322.8 −25022.8 173095. −780226. −1.43444e6 1.83499e6
1.6 147.897 −1090.82 13681.5 −45012.7 −161328. −17976.6 811887. −404445. −6.65725e6
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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 13.14.a.a 6
3.b odd 2 1 117.14.a.c 6
13.b even 2 1 169.14.a.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.14.a.a 6 1.a even 1 1 trivial
117.14.a.c 6 3.b odd 2 1
169.14.a.a 6 13.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 65T_{2}^{5} - 32818T_{2}^{4} - 1741272T_{2}^{3} + 268538080T_{2}^{2} + 7502807936T_{2} - 713559439360 \) acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(13))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + \cdots - 713559439360 \) Copy content Toggle raw display
$3$ \( T^{6} + \cdots + 22\!\cdots\!44 \) Copy content Toggle raw display
$5$ \( T^{6} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{6} + \cdots + 45\!\cdots\!08 \) Copy content Toggle raw display
$11$ \( T^{6} + \cdots + 11\!\cdots\!04 \) Copy content Toggle raw display
$13$ \( (T + 4826809)^{6} \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots + 17\!\cdots\!32 \) Copy content Toggle raw display
$19$ \( T^{6} + \cdots + 51\!\cdots\!20 \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots + 10\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( T^{6} + \cdots - 13\!\cdots\!60 \) Copy content Toggle raw display
$31$ \( T^{6} + \cdots + 13\!\cdots\!16 \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots - 22\!\cdots\!44 \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots - 56\!\cdots\!92 \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots - 83\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 14\!\cdots\!80 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots - 18\!\cdots\!08 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots - 48\!\cdots\!20 \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots - 28\!\cdots\!32 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 24\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots - 17\!\cdots\!72 \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots - 26\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots - 12\!\cdots\!80 \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots - 46\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots - 38\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots + 33\!\cdots\!80 \) Copy content Toggle raw display
show more
show less