Properties

Label 16.16.a.f
Level $16$
Weight $16$
Character orbit 16.a
Self dual yes
Analytic conductor $22.831$
Analytic rank $0$
Dimension $2$
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] = [16,16,Mod(1,16)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(16, 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("16.1");
 
S:= CuspForms(chi, 16);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 16 = 2^{4} \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 16.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: \(22.8309608160\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{58}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 58 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{7}\cdot 5 \)
Twist minimal: no (minimal twist has level 8)
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 \(\beta = 640\sqrt{58}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 2036) q^{3} + ( - 36 \beta - 70130) q^{5} + (90 \beta - 63096) q^{7} + (4072 \beta + 13553189) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 2036) q^{3} + ( - 36 \beta - 70130) q^{5} + (90 \beta - 63096) q^{7} + (4072 \beta + 13553189) q^{9} + (17307 \beta - 10341316) q^{11} + ( - 2340 \beta + 249903238) q^{13} + ( - 143426 \beta - 998029480) q^{15} + ( - 175896 \beta + 1569758450) q^{17} + (285813 \beta + 237334276) q^{19} + (120144 \beta + 2009648544) q^{21} + ( - 407250 \beta + 20388001304) q^{23} + (5049360 \beta + 5189451575) q^{25} + (7494874 \beta + 95117607752) q^{27} + ( - 19501524 \beta + 17626578678) q^{29} + ( - 30813336 \beta + 17194596640) q^{31} + (24895736 \beta + 390104018224) q^{33} + ( - 4040244 \beta - 72547109520) q^{35} + ( - 65307348 \beta + 435114282222) q^{37} + (245138998 \beta + 453212080568) q^{39} + (384323472 \beta + 450042726042) q^{41} + ( - 376557597 \beta - 250353999268) q^{43} + ( - 773484164 \beta - 4433041970170) q^{45} + ( - 356149188 \beta - 604029559632) q^{47} + ( - 11357280 \beta - 4551150324727) q^{49} + (1211634194 \beta - 982697888600) q^{51} + (1167361452 \beta + 618367101022) q^{53} + ( - 841452534 \beta - 14076485262520) q^{55} + (819249544 \beta + 7273214864336) q^{57} + (884641527 \beta - 7220987952532) q^{59} + ( - 1334372868 \beta + 9168151708630) q^{61} + (962860098 \beta + 7851240050856) q^{63} + ( - 8832412368 \beta - 15524441248940) q^{65} + ( - 8202370383 \beta + 38300710757428) q^{67} + (19558840304 \beta + 31835013854944) q^{69} + (1679200362 \beta + 72938586943432) q^{71} + ( - 17753485176 \beta - 13208634962054) q^{73} + (15469948535 \beta + 130522359054700) q^{75} + ( - 2022720912 \beta + 37656800058336) q^{77} + (12009529764 \beta - 128953916694064) q^{79} + (51948421912 \beta + 177240223511849) q^{81} + ( - 35231817627 \beta - 127756403291324) q^{83} + ( - 44175717720 \beta + 40346979242300) q^{85} + ( - 22078524186 \beta - 427406091174792) q^{87} + ( - 61727093496 \beta - 359897305856406) q^{89} + (22638936060 \beta - 20771076784848) q^{91} + ( - 45541355456 \beta - 697018061925760) q^{93} + ( - 28588099626 \beta - 261084334798280) q^{95} + (259618417416 \beta + 203817795209378) q^{97} + (192455203271 \beta + 15\!\cdots\!76) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4072 q^{3} - 140260 q^{5} - 126192 q^{7} + 27106378 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4072 q^{3} - 140260 q^{5} - 126192 q^{7} + 27106378 q^{9} - 20682632 q^{11} + 499806476 q^{13} - 1996058960 q^{15} + 3139516900 q^{17} + 474668552 q^{19} + 4019297088 q^{21} + 40776002608 q^{23} + 10378903150 q^{25} + 190235215504 q^{27} + 35253157356 q^{29} + 34389193280 q^{31} + 780208036448 q^{33} - 145094219040 q^{35} + 870228564444 q^{37} + 906424161136 q^{39} + 900085452084 q^{41} - 500707998536 q^{43} - 8866083940340 q^{45} - 1208059119264 q^{47} - 9102300649454 q^{49} - 1965395777200 q^{51} + 1236734202044 q^{53} - 28152970525040 q^{55} + 14546429728672 q^{57} - 14441975905064 q^{59} + 18336303417260 q^{61} + 15702480101712 q^{63} - 31048882497880 q^{65} + 76601421514856 q^{67} + 63670027709888 q^{69} + 145877173886864 q^{71} - 26417269924108 q^{73} + 261044718109400 q^{75} + 75313600116672 q^{77} - 257907833388128 q^{79} + 354480447023698 q^{81} - 255512806582648 q^{83} + 80693958484600 q^{85} - 854812182349584 q^{87} - 719794611712812 q^{89} - 41542153569696 q^{91} - 13\!\cdots\!20 q^{93}+ \cdots + 30\!\cdots\!52 q^{99}+O(q^{100}) \) 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
−7.61577
7.61577
0 −2838.09 0 105337. 0 −501765. 0 −6.29412e6 0
1.2 0 6910.09 0 −245597. 0 375573. 0 3.34005e7 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)

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 16.16.a.f 2
3.b odd 2 1 144.16.a.v 2
4.b odd 2 1 8.16.a.c 2
8.b even 2 1 64.16.a.l 2
8.d odd 2 1 64.16.a.n 2
12.b even 2 1 72.16.a.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.16.a.c 2 4.b odd 2 1
16.16.a.f 2 1.a even 1 1 trivial
64.16.a.l 2 8.b even 2 1
64.16.a.n 2 8.d odd 2 1
72.16.a.g 2 12.b even 2 1
144.16.a.v 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 4072T_{3} - 19611504 \) acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(16))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 4072 T - 19611504 \) Copy content Toggle raw display
$5$ \( T^{2} + 140260 T - 25870595900 \) Copy content Toggle raw display
$7$ \( T^{2} + 126192 T - 188448974784 \) Copy content Toggle raw display
$11$ \( T^{2} + 20682632 T - 70\!\cdots\!44 \) Copy content Toggle raw display
$13$ \( T^{2} - 499806476 T + 62\!\cdots\!44 \) Copy content Toggle raw display
$17$ \( T^{2} - 3139516900 T + 17\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{2} - 474668552 T - 18\!\cdots\!24 \) Copy content Toggle raw display
$23$ \( T^{2} - 40776002608 T + 41\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( T^{2} - 35253157356 T - 87\!\cdots\!16 \) Copy content Toggle raw display
$31$ \( T^{2} - 34389193280 T - 22\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{2} - 870228564444 T + 88\!\cdots\!84 \) Copy content Toggle raw display
$41$ \( T^{2} - 900085452084 T - 33\!\cdots\!36 \) Copy content Toggle raw display
$43$ \( T^{2} + 500707998536 T - 33\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( T^{2} + 1208059119264 T - 26\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{2} - 1236734202044 T - 31\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( T^{2} + 14441975905064 T + 33\!\cdots\!24 \) Copy content Toggle raw display
$61$ \( T^{2} - 18336303417260 T + 41\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{2} - 76601421514856 T - 13\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( T^{2} - 145877173886864 T + 52\!\cdots\!24 \) Copy content Toggle raw display
$73$ \( T^{2} + 26417269924108 T - 73\!\cdots\!84 \) Copy content Toggle raw display
$79$ \( T^{2} + 257907833388128 T + 13\!\cdots\!96 \) Copy content Toggle raw display
$83$ \( T^{2} + 255512806582648 T - 13\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( T^{2} + 719794611712812 T + 39\!\cdots\!36 \) Copy content Toggle raw display
$97$ \( T^{2} - 407635590418756 T - 15\!\cdots\!16 \) Copy content Toggle raw display
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