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Properties

Label 9.9.d.a
Level $9$
Weight $9$
Character orbit 9.d
Analytic conductor $3.666$
Analytic rank $0$
Dimension $14$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [9,9,Mod(2,9)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9.2");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 9.d (of order \(6\), degree \(2\), 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: \(3.66640749055\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - x^{13} + 427 x^{12} - 1362 x^{11} + 135762 x^{10} - 371244 x^{9} + 18261508 x^{8} + \cdots + 872385888256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{21} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{6} + \beta_{4} - 3 \beta_{3} - 5) q^{3} + (\beta_{10} - \beta_{4} - 110 \beta_{3} + \cdots + 110) q^{4} + (\beta_{12} - 2 \beta_{6} - 2 \beta_{4} + \cdots + 20) q^{5} + (\beta_{12} + \beta_{10} - \beta_{8} + \cdots - 136) q^{6}+ \cdots + ( - 53970 \beta_{13} - 21477 \beta_{12} + \cdots - 45183069) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 3 q^{2} - 93 q^{3} + 767 q^{4} + 438 q^{5} - 2259 q^{6} + 922 q^{7} + 17973 q^{9} - 516 q^{10} - 28677 q^{11} - 55110 q^{12} + 1684 q^{13} + 120966 q^{14} - 75276 q^{15} - 65281 q^{16} + 243324 q^{18}+ \cdots - 511060752 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - x^{13} + 427 x^{12} - 1362 x^{11} + 135762 x^{10} - 371244 x^{9} + 18261508 x^{8} + \cdots + 872385888256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 44\!\cdots\!83 \nu^{13} + \cdots - 86\!\cdots\!64 ) / 10\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 44\!\cdots\!83 \nu^{13} + \cdots - 86\!\cdots\!64 ) / 10\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 14\!\cdots\!61 \nu^{13} + \cdots + 16\!\cdots\!48 ) / 15\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 13\!\cdots\!63 \nu^{13} + \cdots - 10\!\cdots\!00 ) / 13\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 12\!\cdots\!89 \nu^{13} + \cdots + 66\!\cdots\!16 ) / 65\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 28\!\cdots\!05 \nu^{13} + \cdots - 95\!\cdots\!88 ) / 31\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 12\!\cdots\!61 \nu^{13} + \cdots + 59\!\cdots\!48 ) / 13\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 57\!\cdots\!61 \nu^{13} + \cdots - 64\!\cdots\!52 ) / 47\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 73\!\cdots\!67 \nu^{13} + \cdots - 67\!\cdots\!04 ) / 32\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 14\!\cdots\!07 \nu^{13} + \cdots - 25\!\cdots\!12 ) / 43\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 75\!\cdots\!37 \nu^{13} + \cdots + 77\!\cdots\!56 ) / 13\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 80\!\cdots\!19 \nu^{13} + \cdots - 14\!\cdots\!20 ) / 13\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 15\!\cdots\!33 \nu^{13} + \cdots - 12\!\cdots\!72 ) / 15\!\cdots\!08 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{2} + \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{10} + \beta_{7} + \beta_{6} - 366\beta_{3} + 3\beta_{2} + 2\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 3 \beta_{13} - \beta_{12} + \beta_{11} - 3 \beta_{10} + 3 \beta_{9} + 3 \beta_{8} - 18 \beta_{7} + \cdots + 2060 ) / 9 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 62 \beta_{13} - 5 \beta_{12} - 10 \beta_{11} - 833 \beta_{10} + 13 \beta_{9} - 26 \beta_{8} + \cdots - 225385 ) / 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 775 \beta_{13} + 890 \beta_{12} + 445 \beta_{11} + 7254 \beta_{10} - 2258 \beta_{9} + 1129 \beta_{8} + \cdots + 895 ) / 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 5595 \beta_{13} - 353 \beta_{12} + 353 \beta_{11} - 2227 \beta_{10} + 2449 \beta_{9} + 2449 \beta_{8} + \cdots + 18161930 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 40474 \beta_{13} - 48791 \beta_{12} - 97582 \beta_{11} - 775309 \beta_{10} + 115135 \beta_{9} + \cdots - 140887695 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 797311 \beta_{13} + 257290 \beta_{12} + 128645 \beta_{11} + 22245998 \beta_{10} - 1957994 \beta_{9} + \cdots - 925069 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 3055185 \beta_{13} - 42911629 \beta_{12} + 42911629 \beta_{11} - 53614551 \beta_{10} + \cdots + 145236939578 ) / 9 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 2074068722 \beta_{13} - 170625935 \beta_{12} - 341251870 \beta_{11} - 18456045845 \beta_{10} + \cdots - 3978594065863 ) / 9 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 15336140671 \beta_{13} + 24106321802 \beta_{12} + 12053160901 \beta_{11} + 263284992822 \beta_{10} + \cdots + 178934899 ) / 9 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 116297982111 \beta_{13} - 23192247469 \beta_{12} + 23192247469 \beta_{11} - 81279180215 \beta_{10} + \cdots + 378122825249866 ) / 3 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 2093005439570 \beta_{13} - 1117773709951 \beta_{12} - 2235547419902 \beta_{11} + \cdots - 50\!\cdots\!27 ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/9\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(1 - \beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
7.38374 + 12.7890i
5.69757 + 9.86849i
4.05115 + 7.01679i
−0.447645 − 0.775344i
−2.00397 − 3.47098i
−5.49482 − 9.51731i
−8.68602 − 15.0446i
7.38374 − 12.7890i
5.69757 − 9.86849i
4.05115 − 7.01679i
−0.447645 + 0.775344i
−2.00397 + 3.47098i
−5.49482 + 9.51731i
−8.68602 + 15.0446i
−22.1512 − 12.7890i −80.9424 − 3.05302i 199.117 + 344.881i 570.352 − 329.293i 1753.93 + 1102.80i −1512.41 + 2619.58i − 3638.08i 6542.36 + 494.238i −16845.3
2.2 −17.0927 − 9.86849i 51.0315 − 62.9030i 66.7741 + 115.656i −896.557 + 517.627i −1493.02 + 571.581i −21.9132 + 37.9547i 2416.83i −1352.58 − 6420.07i 20432.8
2.3 −12.1534 − 7.01679i 54.3339 + 60.0735i −29.5292 − 51.1461i 676.216 − 390.413i −238.821 − 1111.35i 2168.61 − 3756.14i 4421.40i −656.650 + 6528.06i −10957.8
2.4 1.34294 + 0.775344i −44.1317 + 67.9220i −126.798 − 219.620i −604.549 + 349.037i −111.929 + 56.9976i −1124.45 + 1947.61i − 790.223i −2665.79 − 5995.02i −1082.49
2.5 6.01192 + 3.47098i −29.7082 − 75.3553i −103.905 − 179.968i 331.396 − 191.331i 82.9534 − 556.147i 467.516 − 809.762i − 3219.75i −4795.84 + 4477.35i 2656.43
2.6 16.4845 + 9.51731i 80.5605 + 8.42672i 53.1583 + 92.0729i −46.9888 + 27.1290i 1247.80 + 905.629i −921.012 + 1595.24i − 2849.17i 6418.98 + 1357.72i −1032.78
2.7 26.0581 + 15.0446i −77.6435 + 23.0757i 324.682 + 562.365i 189.131 − 109.195i −2370.40 − 566.810i 1404.67 − 2432.96i 11836.0i 5496.03 − 3583.35i 6571.17
5.1 −22.1512 + 12.7890i −80.9424 + 3.05302i 199.117 − 344.881i 570.352 + 329.293i 1753.93 − 1102.80i −1512.41 − 2619.58i 3638.08i 6542.36 − 494.238i −16845.3
5.2 −17.0927 + 9.86849i 51.0315 + 62.9030i 66.7741 − 115.656i −896.557 − 517.627i −1493.02 − 571.581i −21.9132 − 37.9547i − 2416.83i −1352.58 + 6420.07i 20432.8
5.3 −12.1534 + 7.01679i 54.3339 − 60.0735i −29.5292 + 51.1461i 676.216 + 390.413i −238.821 + 1111.35i 2168.61 + 3756.14i − 4421.40i −656.650 − 6528.06i −10957.8
5.4 1.34294 − 0.775344i −44.1317 − 67.9220i −126.798 + 219.620i −604.549 − 349.037i −111.929 − 56.9976i −1124.45 − 1947.61i 790.223i −2665.79 + 5995.02i −1082.49
5.5 6.01192 − 3.47098i −29.7082 + 75.3553i −103.905 + 179.968i 331.396 + 191.331i 82.9534 + 556.147i 467.516 + 809.762i 3219.75i −4795.84 − 4477.35i 2656.43
5.6 16.4845 − 9.51731i 80.5605 − 8.42672i 53.1583 − 92.0729i −46.9888 − 27.1290i 1247.80 − 905.629i −921.012 − 1595.24i 2849.17i 6418.98 − 1357.72i −1032.78
5.7 26.0581 − 15.0446i −77.6435 − 23.0757i 324.682 − 562.365i 189.131 + 109.195i −2370.40 + 566.810i 1404.67 + 2432.96i − 11836.0i 5496.03 + 3583.35i 6571.17
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9.9.d.a ✓ 14
3.b odd 2 1 27.9.d.a 14
4.b odd 2 1 144.9.q.a 14
9.c even 3 1 27.9.d.a 14
9.c even 3 1 81.9.b.a 14
9.d odd 6 1 inner 9.9.d.a ✓ 14
9.d odd 6 1 81.9.b.a 14
12.b even 2 1 432.9.q.a 14
36.f odd 6 1 432.9.q.a 14
36.h even 6 1 144.9.q.a 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.9.d.a ✓ 14 1.a even 1 1 trivial
9.9.d.a ✓ 14 9.d odd 6 1 inner
27.9.d.a 14 3.b odd 2 1
27.9.d.a 14 9.c even 3 1
81.9.b.a 14 9.c even 3 1
81.9.b.a 14 9.d odd 6 1
144.9.q.a 14 4.b odd 2 1
144.9.q.a 14 36.h even 6 1
432.9.q.a 14 12.b even 2 1
432.9.q.a 14 36.f odd 6 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + \cdots + 19\!\cdots\!72 \) Copy content Toggle raw display
$3$ \( T^{14} + \cdots + 52\!\cdots\!21 \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots + 39\!\cdots\!36 \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 15\!\cdots\!43 \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 41\!\cdots\!96 \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots + 19\!\cdots\!68 \) Copy content Toggle raw display
$19$ \( (T^{7} + \cdots + 18\!\cdots\!88)^{2} \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 83\!\cdots\!52 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 26\!\cdots\!88 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 25\!\cdots\!84 \) Copy content Toggle raw display
$37$ \( (T^{7} + \cdots + 33\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 20\!\cdots\!23 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 11\!\cdots\!69 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 35\!\cdots\!72 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 10\!\cdots\!67 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 26\!\cdots\!04 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 48\!\cdots\!49 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{7} + \cdots + 40\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 49\!\cdots\!84 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 55\!\cdots\!52 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 78\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 18\!\cdots\!25 \) Copy content Toggle raw display
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