Properties

Label 9.16.a.c
Level $9$
Weight $16$
Character orbit 9.a
Self dual yes
Analytic conductor $12.842$
Analytic rank $1$
Dimension $1$
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] = [9,16,Mod(1,9)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9, 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("9.1");
 
S:= CuspForms(chi, 16);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 9.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: \(12.8424154590\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3)
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)
 
\(f(q)\) \(=\) \( q + 72 q^{2} - 27584 q^{4} + 221490 q^{5} - 2149000 q^{7} - 4345344 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 72 q^{2} - 27584 q^{4} + 221490 q^{5} - 2149000 q^{7} - 4345344 q^{8} + 15947280 q^{10} - 37169316 q^{11} - 279974266 q^{13} - 154728000 q^{14} + 591007744 q^{16} - 2492912754 q^{17} - 4669782244 q^{19} - 6109580160 q^{20} - 2676190752 q^{22} + 18467933400 q^{23} + 18540241975 q^{25} - 20158147152 q^{26} + 59278016000 q^{28} + 115953449418 q^{29} - 56187023200 q^{31} + 184940789760 q^{32} - 179489718288 q^{34} - 475982010000 q^{35} + 614764926830 q^{37} - 336224321568 q^{38} - 962450242560 q^{40} - 549859792410 q^{41} - 982884444028 q^{43} + 1025278412544 q^{44} + 1329691204800 q^{46} - 2076144322896 q^{47} - 129360509943 q^{49} + 1334897422200 q^{50} + 7722810153344 q^{52} + 12048378188130 q^{53} - 8232631800840 q^{55} + 9338144256000 q^{56} + 8348648358096 q^{58} - 23087905758324 q^{59} - 8505809142442 q^{61} - 4045465670400 q^{62} - 6050404892672 q^{64} - 62011500176340 q^{65} - 12331010771476 q^{67} + 68764505406336 q^{68} - 34270704720000 q^{70} - 58989192692472 q^{71} - 5609828808070 q^{73} + 44263074731760 q^{74} + 128811273418496 q^{76} + 79876860084000 q^{77} + 159918683826800 q^{79} + 130902305218560 q^{80} - 39589905053520 q^{82} - 57675894342876 q^{83} - 552155245883460 q^{85} - 70767679970016 q^{86} + 161513464264704 q^{88} + 362287610413974 q^{89} + 601664697634000 q^{91} - 509419474905600 q^{92} - 149482391248512 q^{94} - 10\!\cdots\!60 q^{95}+ \cdots - 9313956715896 q^{98}+O(q^{100}) \) 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
0
72.0000 0 −27584.0 221490. 0 −2.14900e6 −4.34534e6 0 1.59473e7
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 9.16.a.c 1
3.b odd 2 1 3.16.a.b 1
4.b odd 2 1 144.16.a.l 1
12.b even 2 1 48.16.a.a 1
15.d odd 2 1 75.16.a.a 1
15.e even 4 2 75.16.b.b 2
21.c even 2 1 147.16.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.16.a.b 1 3.b odd 2 1
9.16.a.c 1 1.a even 1 1 trivial
48.16.a.a 1 12.b even 2 1
75.16.a.a 1 15.d odd 2 1
75.16.b.b 2 15.e even 4 2
144.16.a.l 1 4.b odd 2 1
147.16.a.b 1 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 72 \) acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(9))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 72 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 221490 \) Copy content Toggle raw display
$7$ \( T + 2149000 \) Copy content Toggle raw display
$11$ \( T + 37169316 \) Copy content Toggle raw display
$13$ \( T + 279974266 \) Copy content Toggle raw display
$17$ \( T + 2492912754 \) Copy content Toggle raw display
$19$ \( T + 4669782244 \) Copy content Toggle raw display
$23$ \( T - 18467933400 \) Copy content Toggle raw display
$29$ \( T - 115953449418 \) Copy content Toggle raw display
$31$ \( T + 56187023200 \) Copy content Toggle raw display
$37$ \( T - 614764926830 \) Copy content Toggle raw display
$41$ \( T + 549859792410 \) Copy content Toggle raw display
$43$ \( T + 982884444028 \) Copy content Toggle raw display
$47$ \( T + 2076144322896 \) Copy content Toggle raw display
$53$ \( T - 12048378188130 \) Copy content Toggle raw display
$59$ \( T + 23087905758324 \) Copy content Toggle raw display
$61$ \( T + 8505809142442 \) Copy content Toggle raw display
$67$ \( T + 12331010771476 \) Copy content Toggle raw display
$71$ \( T + 58989192692472 \) Copy content Toggle raw display
$73$ \( T + 5609828808070 \) Copy content Toggle raw display
$79$ \( T - 159918683826800 \) Copy content Toggle raw display
$83$ \( T + 57675894342876 \) Copy content Toggle raw display
$89$ \( T - 362287610413974 \) Copy content Toggle raw display
$97$ \( T + 539786645144926 \) Copy content Toggle raw display
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