Properties

Label 126.16.a.n
Level $126$
Weight $16$
Character orbit 126.a
Self dual yes
Analytic conductor $179.794$
Analytic rank $1$
Dimension $4$
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] = [126,16,Mod(1,126)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(126, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 16, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("126.1");
 
S:= CuspForms(chi, 16);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 126.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: \(179.793816426\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 201842416x^{2} + 1462712685511x - 2792107035723495 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{8}\cdot 5^{2}\cdot 7^{2} \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 128 q^{2} + 16384 q^{4} + ( - \beta_1 - 72828) q^{5} + 823543 q^{7} - 2097152 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 128 q^{2} + 16384 q^{4} + ( - \beta_1 - 72828) q^{5} + 823543 q^{7} - 2097152 q^{8} + (128 \beta_1 + 9321984) q^{10} + (\beta_{3} + 36 \beta_1 - 30286188) q^{11} + ( - \beta_{3} + \beta_{2} + 198 \beta_1 + 42541310) q^{13} - 105413504 q^{14} + 268435456 q^{16} + ( - 8 \beta_{3} - 8 \beta_{2} - 171 \beta_1 - 86421300) q^{17} + ( - 13 \beta_{3} - 3 \beta_{2} - 12002 \beta_1 + 1184698424) q^{19} + ( - 16384 \beta_1 - 1193213952) q^{20} + ( - 128 \beta_{3} - 4608 \beta_1 + 3876632064) q^{22} + (96 \beta_{3} + 8 \beta_{2} - 76395 \beta_1 - 4255956756) q^{23} + ( - 281 \beta_{3} - 39 \beta_{2} + 125846 \beta_1 + 9581918131) q^{25} + (128 \beta_{3} - 128 \beta_{2} - 25344 \beta_1 - 5445287680) q^{26} + 13492928512 q^{28} + ( - 785 \beta_{3} + 464 \beta_{2} + 252779 \beta_1 - 24060598032) q^{29} + (1343 \beta_{3} - 95 \beta_{2} - 46746 \beta_1 + 16416332408) q^{31} - 34359738368 q^{32} + (1024 \beta_{3} + 1024 \beta_{2} + 21888 \beta_1 + 11061926400) q^{34} + ( - 823543 \beta_1 - 59976989604) q^{35} + (1971 \beta_{3} - 291 \beta_{2} + 2562558 \beta_1 + 69912624062) q^{37} + (1664 \beta_{3} + 384 \beta_{2} + 1536256 \beta_1 - 151641398272) q^{38} + (2097152 \beta_1 + 152731385856) q^{40} + ( - 1922 \beta_{3} - 3720 \beta_{2} + 567765 \beta_1 - 68626951260) q^{41} + (1620 \beta_{3} - 5188 \beta_{2} + 4712440 \beta_1 + 387983163620) q^{43} + (16384 \beta_{3} + 589824 \beta_1 - 496208904192) q^{44} + ( - 12288 \beta_{3} - 1024 \beta_{2} + \cdots + 544762464768) q^{46}+ \cdots - 86812553324672 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 512 q^{2} + 65536 q^{4} - 291312 q^{5} + 3294172 q^{7} - 8388608 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 512 q^{2} + 65536 q^{4} - 291312 q^{5} + 3294172 q^{7} - 8388608 q^{8} + 37287936 q^{10} - 121144752 q^{11} + 170165240 q^{13} - 421654016 q^{14} + 1073741824 q^{16} - 345685200 q^{17} + 4738793696 q^{19} - 4772855808 q^{20} + 15506528256 q^{22} - 17023827024 q^{23} + 38327672524 q^{25} - 21781150720 q^{26} + 53971714048 q^{28} - 96242392128 q^{29} + 65665329632 q^{31} - 137438953472 q^{32} + 44247705600 q^{34} - 239907958416 q^{35} + 279650496248 q^{37} - 606565593088 q^{38} + 610925543424 q^{40} - 274507805040 q^{41} + 1551932654480 q^{43} - 1984835616768 q^{44} + 2179049859072 q^{46} - 686050223712 q^{47} + 2712892291396 q^{49} - 4905942083072 q^{50} + 2787987292160 q^{52} - 4929897017760 q^{53} + 3835130598432 q^{55} - 6908379398144 q^{56} + 12319026192384 q^{58} - 15661604798112 q^{59} + 7270740102248 q^{61} - 8405162192896 q^{62} + 17592186044416 q^{64} - 39986092580832 q^{65} + 2642113680560 q^{67} - 5663706316800 q^{68} + 30708218677248 q^{70} - 62559334721520 q^{71} - 53486860270312 q^{73} - 35795263519744 q^{74} + 77640395915264 q^{76} - 99767912496336 q^{77} + 77560377412544 q^{79} - 78198469558272 q^{80} + 35136999045120 q^{82} + 58088393760768 q^{83} + 48889916091648 q^{85} - 198647379773440 q^{86} + 254058958946304 q^{88} + 296914762526928 q^{89} + 140138392245320 q^{91} - 278918381961216 q^{92} + 87814428635136 q^{94} + 13\!\cdots\!48 q^{95}+ \cdots - 347250213298688 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 201842416x^{2} + 1462712685511x - 2792107035723495 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -79\nu^{3} - 432876\nu^{2} + 13965956644\nu - 42970890000093 ) / 34475428 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -97559\nu^{3} - 1665712140\nu^{2} + 5301081396164\nu + 61093534989039675 ) / 172377140 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 363991\nu^{3} + 2277253740\nu^{2} - 58103498526436\nu + 169446915158653365 ) / 172377140 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 4\beta_{3} + \beta_{2} + 3439\beta _1 + 18900 ) / 75600 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -25346\beta_{3} - 13249\beta_{2} - 20083941\beta _1 + 4577786006220 ) / 45360 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 16760839\beta_{3} + 5317491\beta_{2} + 13542572924\beta _1 - 1480792228633350 ) / 1350 \) 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
6146.66
7587.42
3478.41
−17211.5
−128.000 0 16384.0 −309875. 0 823543. −2.09715e6 0 3.96640e7
1.2 −128.000 0 16384.0 −176302. 0 823543. −2.09715e6 0 2.25667e7
1.3 −128.000 0 16384.0 12853.6 0 823543. −2.09715e6 0 −1.64526e6
1.4 −128.000 0 16384.0 182011. 0 823543. −2.09715e6 0 −2.32974e7
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)
\(7\) \(-1\)

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 126.16.a.n 4
3.b odd 2 1 126.16.a.q yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.16.a.n 4 1.a even 1 1 trivial
126.16.a.q yes 4 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 291312T_{5}^{3} - 37767651840T_{5}^{2} - 9508350109406400T_{5} + 127810130166316422000 \) acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(126))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 128)^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 291312 T^{3} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T - 823543)^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 121144752 T^{3} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{4} - 170165240 T^{3} + \cdots + 16\!\cdots\!12 \) Copy content Toggle raw display
$17$ \( T^{4} + 345685200 T^{3} + \cdots - 11\!\cdots\!48 \) Copy content Toggle raw display
$19$ \( T^{4} - 4738793696 T^{3} + \cdots + 11\!\cdots\!56 \) Copy content Toggle raw display
$23$ \( T^{4} + 17023827024 T^{3} + \cdots + 86\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{4} + 96242392128 T^{3} + \cdots + 14\!\cdots\!72 \) Copy content Toggle raw display
$31$ \( T^{4} - 65665329632 T^{3} + \cdots + 80\!\cdots\!48 \) Copy content Toggle raw display
$37$ \( T^{4} - 279650496248 T^{3} + \cdots + 24\!\cdots\!20 \) Copy content Toggle raw display
$41$ \( T^{4} + 274507805040 T^{3} + \cdots - 33\!\cdots\!60 \) Copy content Toggle raw display
$43$ \( T^{4} - 1551932654480 T^{3} + \cdots + 51\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( T^{4} + 686050223712 T^{3} + \cdots - 52\!\cdots\!20 \) Copy content Toggle raw display
$53$ \( T^{4} + 4929897017760 T^{3} + \cdots + 47\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( T^{4} + 15661604798112 T^{3} + \cdots + 82\!\cdots\!04 \) Copy content Toggle raw display
$61$ \( T^{4} - 7270740102248 T^{3} + \cdots + 61\!\cdots\!36 \) Copy content Toggle raw display
$67$ \( T^{4} - 2642113680560 T^{3} + \cdots + 17\!\cdots\!20 \) Copy content Toggle raw display
$71$ \( T^{4} + 62559334721520 T^{3} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{4} + 53486860270312 T^{3} + \cdots + 84\!\cdots\!20 \) Copy content Toggle raw display
$79$ \( T^{4} - 77560377412544 T^{3} + \cdots - 35\!\cdots\!44 \) Copy content Toggle raw display
$83$ \( T^{4} - 58088393760768 T^{3} + \cdots - 14\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{4} - 296914762526928 T^{3} + \cdots - 76\!\cdots\!32 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots - 66\!\cdots\!76 \) Copy content Toggle raw display
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