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Properties

Label 84.9.d.a
Level $84$
Weight $9$
Character orbit 84.d
Analytic conductor $34.220$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [84,9,Mod(13,84)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(84, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("84.13");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 84.d (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(34.2198032451\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} + 14312 x^{8} - 30343 x^{7} + 170123918 x^{6} - 875537263 x^{5} + 496509566533 x^{4} + \cdots + 39\!\cdots\!76 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{15}\cdot 3^{15}\cdot 7^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + (\beta_{5} + 3 \beta_1) q^{5} + (\beta_{6} + 7 \beta_1 - 234) q^{7} - 2187 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + (\beta_{5} + 3 \beta_1) q^{5} + (\beta_{6} + 7 \beta_1 - 234) q^{7} - 2187 q^{9} + (\beta_{7} + \beta_{6} - \beta_{4} + \cdots - 3760) q^{11}+ \cdots + ( - 2187 \beta_{7} - 2187 \beta_{6} + \cdots + 8223120) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 2338 q^{7} - 21870 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 2338 q^{7} - 21870 q^{9} - 37596 q^{11} + 59616 q^{15} + 142884 q^{21} - 22380 q^{23} - 303998 q^{25} - 308892 q^{29} - 1480584 q^{35} - 5471108 q^{37} + 655128 q^{39} + 1177324 q^{43} - 6064142 q^{49} - 6353640 q^{51} - 129132 q^{53} - 7286112 q^{57} + 5113206 q^{63} - 106801008 q^{65} - 5722372 q^{67} + 26985540 q^{71} + 48770148 q^{77} - 181197556 q^{79} + 47829690 q^{81} + 337759224 q^{85} + 67638816 q^{91} - 120586320 q^{93} + 103302096 q^{95} + 82222452 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} + 14312 x^{8} - 30343 x^{7} + 170123918 x^{6} - 875537263 x^{5} + 496509566533 x^{4} + \cdots + 39\!\cdots\!76 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 56\!\cdots\!23 \nu^{9} + \cdots + 63\!\cdots\!33 ) / 22\!\cdots\!97 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 14\!\cdots\!45 \nu^{9} + \cdots + 90\!\cdots\!92 ) / 58\!\cdots\!54 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 51\!\cdots\!06 \nu^{9} + \cdots + 72\!\cdots\!44 ) / 16\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 51\!\cdots\!36 \nu^{9} + \cdots - 14\!\cdots\!24 ) / 81\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 41\!\cdots\!64 \nu^{9} + \cdots - 27\!\cdots\!64 ) / 51\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 17\!\cdots\!32 \nu^{9} + \cdots + 34\!\cdots\!04 ) / 20\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 13\!\cdots\!88 \nu^{9} + \cdots - 13\!\cdots\!44 ) / 14\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 35\!\cdots\!86 \nu^{9} + \cdots + 21\!\cdots\!72 ) / 10\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 21\!\cdots\!52 \nu^{9} + \cdots + 16\!\cdots\!24 ) / 25\!\cdots\!28 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( -6\beta_{9} + 6\beta_{8} + 9\beta_{7} + 9\beta_{6} - 36\beta_{5} - 3\beta_{3} - 14\beta _1 + 69 ) / 756 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 129 \beta_{9} + 312 \beta_{8} - 573 \beta_{7} + 687 \beta_{6} + 1656 \beta_{5} + 63 \beta_{4} + \cdots - 2164107 ) / 756 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -4177\beta_{7} - 4177\beta_{6} + 236\beta_{4} + 424\beta_{3} + 46\beta_{2} + 88296 ) / 18 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 942285 \beta_{9} - 4720596 \beta_{8} + 7075857 \beta_{7} - 7021779 \beta_{6} - 23313996 \beta_{5} + \cdots - 20467963221 ) / 756 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 769843806 \beta_{9} - 576325950 \beta_{8} + 1591774743 \beta_{7} + 296696091 \beta_{6} + \cdots + 179819092209 ) / 756 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 295509413 \beta_{7} + 295509413 \beta_{6} - 552613029 \beta_{4} + 1136452554 \beta_{3} + \cdots + 10386708923577 ) / 18 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 8584553756148 \beta_{9} + 6489457288026 \beta_{8} + 2445099616983 \beta_{7} + 18435018286659 \beta_{6} + \cdots + 37\!\cdots\!71 ) / 756 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 19237277705565 \beta_{9} + 720420180394380 \beta_{8} + \cdots - 24\!\cdots\!57 ) / 756 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 55\!\cdots\!53 \beta_{7} + \cdots - 27\!\cdots\!86 ) / 18 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/84\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(43\) \(73\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1
−51.3922 + 89.0139i
10.9569 − 18.9779i
54.2016 − 93.8799i
−32.7935 + 56.7999i
19.5272 − 33.8221i
19.5272 + 33.8221i
−32.7935 − 56.7999i
54.2016 + 93.8799i
10.9569 + 18.9779i
−51.3922 − 89.0139i
0 − 46.7654i 0 − 803.413i 0 −2396.47 − 147.366i 0 −2187.00 0
13.2 0 − 46.7654i 0 − 307.159i 0 222.189 + 2390.70i 0 −2187.00 0
13.3 0 − 46.7654i 0 264.232i 0 2038.47 − 1268.64i 0 −2187.00 0
13.4 0 − 46.7654i 0 429.816i 0 −1615.33 − 1776.38i 0 −2187.00 0
13.5 0 − 46.7654i 0 1053.92i 0 582.142 + 2329.36i 0 −2187.00 0
13.6 0 46.7654i 0 − 1053.92i 0 582.142 − 2329.36i 0 −2187.00 0
13.7 0 46.7654i 0 − 429.816i 0 −1615.33 + 1776.38i 0 −2187.00 0
13.8 0 46.7654i 0 − 264.232i 0 2038.47 + 1268.64i 0 −2187.00 0
13.9 0 46.7654i 0 307.159i 0 222.189 − 2390.70i 0 −2187.00 0
13.10 0 46.7654i 0 803.413i 0 −2396.47 + 147.366i 0 −2187.00 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 84.9.d.a ✓ 10
3.b odd 2 1 252.9.d.d 10
4.b odd 2 1 336.9.f.a 10
7.b odd 2 1 inner 84.9.d.a ✓ 10
21.c even 2 1 252.9.d.d 10
28.d even 2 1 336.9.f.a 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.9.d.a ✓ 10 1.a even 1 1 trivial
84.9.d.a ✓ 10 7.b odd 2 1 inner
252.9.d.d 10 3.b odd 2 1
252.9.d.d 10 21.c even 2 1
336.9.f.a 10 4.b odd 2 1
336.9.f.a 10 28.d even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(84, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( (T^{2} + 2187)^{5} \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 87\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 63\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( (T^{5} + \cdots - 18\!\cdots\!28)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 16\!\cdots\!88 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 52\!\cdots\!68 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 17\!\cdots\!52 \) Copy content Toggle raw display
$23$ \( (T^{5} + \cdots + 99\!\cdots\!48)^{2} \) Copy content Toggle raw display
$29$ \( (T^{5} + \cdots - 76\!\cdots\!32)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots + 21\!\cdots\!92 \) Copy content Toggle raw display
$37$ \( (T^{5} + \cdots - 88\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{5} + \cdots - 15\!\cdots\!52)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 76\!\cdots\!88 \) Copy content Toggle raw display
$53$ \( (T^{5} + \cdots - 68\!\cdots\!40)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 12\!\cdots\!08 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 62\!\cdots\!32 \) Copy content Toggle raw display
$67$ \( (T^{5} + \cdots - 43\!\cdots\!52)^{2} \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots - 22\!\cdots\!52)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 86\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots - 10\!\cdots\!96)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 84\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 90\!\cdots\!72 \) Copy content Toggle raw display
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