Properties

Label 144.9.g.g
Level $144$
Weight $9$
Character orbit 144.g
Analytic conductor $58.663$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [144,9,Mod(127,144)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(144, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("144.127");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 144.g (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: \(58.6625198488\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-35}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{3} \)
Twist minimal: no (minimal twist has level 16)
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 \(\beta = 216\sqrt{-35}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 510 q^{5} - 2 \beta q^{7} +O(q^{10}) \) Copy content Toggle raw display \( q + 510 q^{5} - 2 \beta q^{7} - 15 \beta q^{11} - 27710 q^{13} - 50370 q^{17} + 85 \beta q^{19} - 138 \beta q^{23} - 130525 q^{25} - 54978 q^{29} + 920 \beta q^{31} - 1020 \beta q^{35} + 793730 q^{37} + 75582 q^{41} + 391 \beta q^{43} - 2244 \beta q^{47} - 767039 q^{49} - 11166210 q^{53} - 7650 \beta q^{55} + 17085 \beta q^{59} - 23826622 q^{61} - 14132100 q^{65} + 5865 \beta q^{67} + 7890 \beta q^{71} + 6516610 q^{73} - 48988800 q^{77} - 38180 \beta q^{79} - 57477 \beta q^{83} - 25688700 q^{85} - 86795778 q^{89} + 55420 \beta q^{91} + 43350 \beta q^{95} - 46670270 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 1020 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 1020 q^{5} - 55420 q^{13} - 100740 q^{17} - 261050 q^{25} - 109956 q^{29} + 1587460 q^{37} + 151164 q^{41} - 1534078 q^{49} - 22332420 q^{53} - 47653244 q^{61} - 28264200 q^{65} + 13033220 q^{73} - 97977600 q^{77} - 51377400 q^{85} - 173591556 q^{89} - 93340540 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/144\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
0.500000 + 2.95804i
0.500000 2.95804i
0 0 0 510.000 0 2555.75i 0 0 0
127.2 0 0 0 510.000 0 2555.75i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 144.9.g.g 2
3.b odd 2 1 16.9.c.a 2
4.b odd 2 1 inner 144.9.g.g 2
12.b even 2 1 16.9.c.a 2
15.d odd 2 1 400.9.b.c 2
15.e even 4 2 400.9.h.b 4
24.f even 2 1 64.9.c.d 2
24.h odd 2 1 64.9.c.d 2
48.i odd 4 2 256.9.d.f 4
48.k even 4 2 256.9.d.f 4
60.h even 2 1 400.9.b.c 2
60.l odd 4 2 400.9.h.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.9.c.a 2 3.b odd 2 1
16.9.c.a 2 12.b even 2 1
64.9.c.d 2 24.f even 2 1
64.9.c.d 2 24.h odd 2 1
144.9.g.g 2 1.a even 1 1 trivial
144.9.g.g 2 4.b odd 2 1 inner
256.9.d.f 4 48.i odd 4 2
256.9.d.f 4 48.k even 4 2
400.9.b.c 2 15.d odd 2 1
400.9.b.c 2 60.h even 2 1
400.9.h.b 4 15.e even 4 2
400.9.h.b 4 60.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 510 \) acting on \(S_{9}^{\mathrm{new}}(144, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T - 510)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 6531840 \) Copy content Toggle raw display
$11$ \( T^{2} + 367416000 \) Copy content Toggle raw display
$13$ \( (T + 27710)^{2} \) Copy content Toggle raw display
$17$ \( (T + 50370)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 11798136000 \) Copy content Toggle raw display
$23$ \( T^{2} + 31098090240 \) Copy content Toggle raw display
$29$ \( (T + 54978)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 1382137344000 \) Copy content Toggle raw display
$37$ \( (T - 793730)^{2} \) Copy content Toggle raw display
$41$ \( (T - 75582)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 249648557760 \) Copy content Toggle raw display
$47$ \( T^{2} + 8222828866560 \) Copy content Toggle raw display
$53$ \( (T + 11166210)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots + 476656492536000 \) Copy content Toggle raw display
$61$ \( (T + 23826622)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 56170925496000 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots + 101655189216000 \) Copy content Toggle raw display
$73$ \( (T - 6516610)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 23\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{2} + 53\!\cdots\!40 \) Copy content Toggle raw display
$89$ \( (T + 86795778)^{2} \) Copy content Toggle raw display
$97$ \( (T + 46670270)^{2} \) Copy content Toggle raw display
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