Properties

Label 13.16.a.a
Level $13$
Weight $16$
Character orbit 13.a
Self dual yes
Analytic conductor $18.550$
Analytic rank $1$
Dimension $7$
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] = [13,16,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, 16, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("13.1");
 
S:= CuspForms(chi, 16);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 13 \)
Weight: \( k \) \(=\) \( 16 \)
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: \(18.5501556630\)
Analytic rank: \(1\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - 2 x^{6} - 172505 x^{5} + 9594016 x^{4} + 7133274223 x^{3} - 680104282298 x^{2} + \cdots + 23\!\cdots\!32 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{8}\cdot 3^{2}\cdot 5 \)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 49) q^{2} + (\beta_{4} + 7 \beta_1 - 149) q^{3} + (\beta_{4} + \beta_{3} + 17 \beta_1 + 18944) q^{4} + (\beta_{6} - \beta_{5} - 13 \beta_{4} + \cdots - 24149) q^{5}+ \cdots + (71 \beta_{6} - 453 \beta_{5} + \cdots + 6742948) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 49) q^{2} + (\beta_{4} + 7 \beta_1 - 149) q^{3} + (\beta_{4} + \beta_{3} + 17 \beta_1 + 18944) q^{4} + (\beta_{6} - \beta_{5} - 13 \beta_{4} + \cdots - 24149) q^{5}+ \cdots + ( - 1433702926 \beta_{6} + \cdots - 618683833154858) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - 345 q^{2} - 1027 q^{3} + 132641 q^{4} - 169503 q^{5} - 2487127 q^{6} - 3685093 q^{7} - 1198077 q^{8} + 47218000 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q - 345 q^{2} - 1027 q^{3} + 132641 q^{4} - 169503 q^{5} - 2487127 q^{6} - 3685093 q^{7} - 1198077 q^{8} + 47218000 q^{9} + 84988937 q^{10} - 87652446 q^{11} + 209621293 q^{12} + 439239619 q^{13} - 1714149813 q^{14} - 2295843203 q^{15} + 2410487249 q^{16} - 2022637899 q^{17} - 5825323712 q^{18} - 8397579954 q^{19} - 30242985531 q^{20} - 7399278761 q^{21} - 43472740778 q^{22} - 58438202280 q^{23} - 191099405565 q^{24} - 40302720622 q^{25} - 21648238365 q^{26} - 185953338217 q^{27} - 185131242041 q^{28} + 37766024250 q^{29} + 207533912593 q^{30} - 199753323272 q^{31} - 656671555461 q^{32} + 677544262406 q^{33} - 943686679699 q^{34} - 613877280513 q^{35} + 1393279187810 q^{36} + 473401973917 q^{37} + 2141758355874 q^{38} - 64442726959 q^{39} + 8026383370011 q^{40} + 520296901884 q^{41} + 7804861824823 q^{42} - 1230378630045 q^{43} + 1307736426078 q^{44} + 4729221421730 q^{45} + 2133222620448 q^{46} - 461360462985 q^{47} + 7051961635657 q^{48} + 8535097100916 q^{49} - 18345202573494 q^{50} + 4885344111721 q^{51} + 8323026043397 q^{52} - 21385360950924 q^{53} - 2718520238389 q^{54} - 34124449164658 q^{55} - 79106992661847 q^{56} - 46362964354086 q^{57} + 25840422614546 q^{58} - 81065343380394 q^{59} - 124743933102355 q^{60} - 57955864231844 q^{61} + 85181132767788 q^{62} - 43003539964762 q^{63} + 118171820808937 q^{64} - 10636061877051 q^{65} + 89158090393838 q^{66} - 12951133598774 q^{67} - 74477982425439 q^{68} + 89528090266392 q^{69} + 253524584445475 q^{70} + 41185199555217 q^{71} + 248414735410218 q^{72} - 115509235801498 q^{73} + 801044021675181 q^{74} + 351010548322804 q^{75} - 110199520254870 q^{76} + 100395018485442 q^{77} - 156063530840659 q^{78} + 197526004984352 q^{79} - 663670784889039 q^{80} - 418232128495097 q^{81} + 766696174770372 q^{82} - 153490882935360 q^{83} + 20\!\cdots\!75 q^{84}+ \cdots - 43\!\cdots\!32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - 2 x^{6} - 172505 x^{5} + 9594016 x^{4} + 7133274223 x^{3} - 680104282298 x^{2} + \cdots + 23\!\cdots\!32 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 41041543 \nu^{6} + 14123740235 \nu^{5} - 5785749609114 \nu^{4} + \cdots - 14\!\cdots\!64 ) / 41\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 25853207 \nu^{6} - 3848269423 \nu^{5} + 3853554298314 \nu^{4} + 319114537852750 \nu^{3} + \cdots - 57\!\cdots\!20 ) / 20\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 25853207 \nu^{6} + 3848269423 \nu^{5} - 3853554298314 \nu^{4} - 319114537852750 \nu^{3} + \cdots - 45\!\cdots\!20 ) / 20\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 23661403 \nu^{6} - 5201947975 \nu^{5} + 3187138376114 \nu^{4} + 513969319377406 \nu^{3} + \cdots + 15\!\cdots\!64 ) / 92\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 463704677 \nu^{6} + 73231607569 \nu^{5} - 67845749579310 \nu^{4} + \cdots - 65\!\cdots\!08 ) / 13\!\cdots\!60 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{3} - 81\beta _1 + 49311 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -48\beta_{6} - 130\beta_{5} + 964\beta_{4} - 97\beta_{3} - 47\beta_{2} + 89264\beta _1 - 3989668 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7520 \beta_{6} - 13742 \beta_{5} - 88231 \beta_{4} + 122694 \beta_{3} - 24809 \beta_{2} + \cdots + 4419484419 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 7861216 \beta_{6} - 19879132 \beta_{5} + 152277092 \beta_{4} - 19231490 \beta_{3} + \cdots - 666852880464 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 1698563264 \beta_{6} - 693924636 \beta_{5} - 28227325375 \beta_{4} + 14843004603 \beta_{3} + \cdots + 474171031456999 \) 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
299.737
170.106
159.673
49.0235
−63.8141
−249.146
−363.580
−348.737 4816.93 88849.8 −293140. −1.67985e6 2.09132e6 −1.95578e7 8.85394e6 1.02229e8
1.2 −219.106 −3377.01 15239.6 −113774. 739926. 714392. 3.84058e6 −2.94468e6 2.49287e7
1.3 −208.673 5348.59 10776.4 82594.0 −1.11610e6 −3.96367e6 4.58906e6 1.42585e7 −1.72351e7
1.4 −98.0235 −6818.66 −23159.4 151115. 668388. 1.60747e6 5.48220e6 3.21452e7 −1.48128e7
1.5 14.8141 1585.00 −32548.5 102776. 23480.3 1.31907e6 −967607. −1.18367e7 1.52254e6
1.6 200.146 2715.74 7290.37 −170608. 543544. −1.73233e6 −5.09924e6 −6.97367e6 −3.41464e7
1.7 314.580 −5297.58 66192.8 71533.9 −1.66652e6 −3.72135e6 1.05148e7 1.37155e7 2.25031e7
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
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 13.16.a.a 7
3.b odd 2 1 117.16.a.c 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.16.a.a 7 1.a even 1 1 trivial
117.16.a.c 7 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{7} + 345 T_{2}^{6} - 121496 T_{2}^{5} - 47667996 T_{2}^{4} + 1317476032 T_{2}^{3} + \cdots - 14\!\cdots\!00 \) acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(13))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} + \cdots - 14\!\cdots\!00 \) Copy content Toggle raw display
$3$ \( T^{7} + \cdots + 13\!\cdots\!24 \) Copy content Toggle raw display
$5$ \( T^{7} + \cdots + 52\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{7} + \cdots + 80\!\cdots\!68 \) Copy content Toggle raw display
$11$ \( T^{7} + \cdots - 10\!\cdots\!52 \) Copy content Toggle raw display
$13$ \( (T - 62748517)^{7} \) Copy content Toggle raw display
$17$ \( T^{7} + \cdots - 23\!\cdots\!28 \) Copy content Toggle raw display
$19$ \( T^{7} + \cdots - 13\!\cdots\!04 \) Copy content Toggle raw display
$23$ \( T^{7} + \cdots + 40\!\cdots\!72 \) Copy content Toggle raw display
$29$ \( T^{7} + \cdots - 46\!\cdots\!76 \) Copy content Toggle raw display
$31$ \( T^{7} + \cdots - 10\!\cdots\!20 \) Copy content Toggle raw display
$37$ \( T^{7} + \cdots + 99\!\cdots\!52 \) Copy content Toggle raw display
$41$ \( T^{7} + \cdots + 82\!\cdots\!64 \) Copy content Toggle raw display
$43$ \( T^{7} + \cdots + 55\!\cdots\!20 \) Copy content Toggle raw display
$47$ \( T^{7} + \cdots - 12\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{7} + \cdots + 11\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{7} + \cdots + 32\!\cdots\!04 \) Copy content Toggle raw display
$61$ \( T^{7} + \cdots - 67\!\cdots\!88 \) Copy content Toggle raw display
$67$ \( T^{7} + \cdots - 20\!\cdots\!92 \) Copy content Toggle raw display
$71$ \( T^{7} + \cdots + 63\!\cdots\!64 \) Copy content Toggle raw display
$73$ \( T^{7} + \cdots + 36\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( T^{7} + \cdots - 91\!\cdots\!76 \) Copy content Toggle raw display
$83$ \( T^{7} + \cdots + 10\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( T^{7} + \cdots + 90\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{7} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
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