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Properties

Label 144.3.m.a
Level $144$
Weight $3$
Character orbit 144.m
Analytic conductor $3.924$
Analytic rank $0$
Dimension $6$
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] = [144,3,Mod(19,144)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(144, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("144.19");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 144.m (of order \(4\), 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: \(3.92371580679\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(i)\)
Coefficient field: 6.0.399424.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 16)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + ( - \beta_{5} + \beta_{4} - \beta_1 - 1) q^{4} + (\beta_{5} + \beta_{4} + \cdots + \beta_{2}) q^{5}+ \cdots + ( - 2 \beta_{5} - 2 \beta_{3} + \cdots + 6 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} + ( - \beta_{5} + \beta_{4} - \beta_1 - 1) q^{4} + (\beta_{5} + \beta_{4} + \cdots + \beta_{2}) q^{5}+ \cdots + ( - 8 \beta_{5} - 16 \beta_{4} + \cdots - 56 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{2} - 8 q^{4} + 2 q^{5} - 4 q^{7} - 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 2 q^{2} - 8 q^{4} + 2 q^{5} - 4 q^{7} - 4 q^{8} + 36 q^{10} + 18 q^{11} - 2 q^{13} - 12 q^{14} - 40 q^{16} + 4 q^{17} + 30 q^{19} + 84 q^{20} - 52 q^{22} - 60 q^{23} - 96 q^{26} + 56 q^{28} + 18 q^{29} - 8 q^{32} - 76 q^{34} + 100 q^{35} + 46 q^{37} - 40 q^{38} + 40 q^{40} - 114 q^{43} - 20 q^{44} + 28 q^{46} - 46 q^{49} - 46 q^{50} + 100 q^{52} - 78 q^{53} + 252 q^{55} + 168 q^{56} - 176 q^{58} - 206 q^{59} + 30 q^{61} + 144 q^{62} + 64 q^{64} - 12 q^{65} - 226 q^{67} - 112 q^{68} - 16 q^{70} + 260 q^{71} + 92 q^{74} - 188 q^{76} + 212 q^{77} - 232 q^{80} + 304 q^{82} - 318 q^{83} - 212 q^{85} - 268 q^{86} - 8 q^{88} + 188 q^{91} + 168 q^{92} + 48 q^{94} - 4 q^{97} - 10 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{5} - 3\nu^{3} + 4\nu^{2} - 2\nu + 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} + \nu^{3} - 2\nu^{2} + 2\nu - 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{5} - \nu^{3} + 2\nu^{2} + 2\nu + 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\nu^{5} + \nu^{4} - 2\nu^{3} + 4\nu^{2} - 2\nu + 5 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 5\nu^{5} - 4\nu^{4} + 11\nu^{3} - 12\nu^{2} + 10\nu - 24 ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + \beta_{4} + \beta _1 - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{5} + \beta_{4} - 2\beta_{2} - 3\beta _1 + 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{5} + \beta_{4} + 2\beta_{2} - 5\beta _1 + 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( \beta_{5} + \beta_{4} - 2\beta_{3} + 4\beta_{2} + 5\beta _1 + 3 ) / 2 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(-\beta_{1}\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
0.264658 + 1.38923i
−0.671462 − 1.24464i
1.40680 − 0.144584i
0.264658 − 1.38923i
−0.671462 + 1.24464i
1.40680 + 0.144584i
−1.12457 + 1.65389i 0 −1.47068 − 3.71982i 0.0586332 − 0.0586332i 0 4.61555 7.80605 + 1.75086i 0 0.0310355 + 0.162910i
19.2 0.573183 − 1.91611i 0 −3.34292 − 2.19656i −3.68585 + 3.68585i 0 −9.66442 −6.12494 + 5.14637i 0 4.94981 + 9.17513i
19.3 1.55139 + 1.26222i 0 0.813607 + 3.91638i 4.62721 − 4.62721i 0 3.04888 −3.68111 + 7.10278i 0 13.0192 − 1.33804i
91.1 −1.12457 − 1.65389i 0 −1.47068 + 3.71982i 0.0586332 + 0.0586332i 0 4.61555 7.80605 − 1.75086i 0 0.0310355 − 0.162910i
91.2 0.573183 + 1.91611i 0 −3.34292 + 2.19656i −3.68585 − 3.68585i 0 −9.66442 −6.12494 − 5.14637i 0 4.94981 − 9.17513i
91.3 1.55139 − 1.26222i 0 0.813607 − 3.91638i 4.62721 + 4.62721i 0 3.04888 −3.68111 − 7.10278i 0 13.0192 + 1.33804i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.f odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 144.3.m.a 6
3.b odd 2 1 16.3.f.a ✓ 6
4.b odd 2 1 576.3.m.a 6
8.b even 2 1 1152.3.m.a 6
8.d odd 2 1 1152.3.m.b 6
12.b even 2 1 64.3.f.a 6
15.d odd 2 1 400.3.r.c 6
15.e even 4 1 400.3.k.c 6
15.e even 4 1 400.3.k.d 6
16.e even 4 1 576.3.m.a 6
16.e even 4 1 1152.3.m.b 6
16.f odd 4 1 inner 144.3.m.a 6
16.f odd 4 1 1152.3.m.a 6
24.f even 2 1 128.3.f.a 6
24.h odd 2 1 128.3.f.b 6
48.i odd 4 1 64.3.f.a 6
48.i odd 4 1 128.3.f.a 6
48.k even 4 1 16.3.f.a ✓ 6
48.k even 4 1 128.3.f.b 6
96.o even 8 2 1024.3.c.j 12
96.o even 8 2 1024.3.d.k 12
96.p odd 8 2 1024.3.c.j 12
96.p odd 8 2 1024.3.d.k 12
240.t even 4 1 400.3.r.c 6
240.z odd 4 1 400.3.k.c 6
240.bd odd 4 1 400.3.k.d 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.3.f.a ✓ 6 3.b odd 2 1
16.3.f.a ✓ 6 48.k even 4 1
64.3.f.a 6 12.b even 2 1
64.3.f.a 6 48.i odd 4 1
128.3.f.a 6 24.f even 2 1
128.3.f.a 6 48.i odd 4 1
128.3.f.b 6 24.h odd 2 1
128.3.f.b 6 48.k even 4 1
144.3.m.a 6 1.a even 1 1 trivial
144.3.m.a 6 16.f odd 4 1 inner
400.3.k.c 6 15.e even 4 1
400.3.k.c 6 240.z odd 4 1
400.3.k.d 6 15.e even 4 1
400.3.k.d 6 240.bd odd 4 1
400.3.r.c 6 15.d odd 2 1
400.3.r.c 6 240.t even 4 1
576.3.m.a 6 4.b odd 2 1
576.3.m.a 6 16.e even 4 1
1024.3.c.j 12 96.o even 8 2
1024.3.c.j 12 96.p odd 8 2
1024.3.d.k 12 96.o even 8 2
1024.3.d.k 12 96.p odd 8 2
1152.3.m.a 6 8.b even 2 1
1152.3.m.a 6 16.f odd 4 1
1152.3.m.b 6 8.d odd 2 1
1152.3.m.b 6 16.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} - 2T_{5}^{5} + 2T_{5}^{4} + 64T_{5}^{3} + 1156T_{5}^{2} - 136T_{5} + 8 \) acting on \(S_{3}^{\mathrm{new}}(144, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - 2 T^{5} + \cdots + 64 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} - 2 T^{5} + \cdots + 8 \) Copy content Toggle raw display
$7$ \( (T^{3} + 2 T^{2} + \cdots + 136)^{2} \) Copy content Toggle raw display
$11$ \( T^{6} - 18 T^{5} + \cdots + 587528 \) Copy content Toggle raw display
$13$ \( T^{6} + 2 T^{5} + \cdots + 1286408 \) Copy content Toggle raw display
$17$ \( (T^{3} - 2 T^{2} + \cdots + 1544)^{2} \) Copy content Toggle raw display
$19$ \( T^{6} - 30 T^{5} + \cdots + 13448 \) Copy content Toggle raw display
$23$ \( (T^{3} + 30 T^{2} + \cdots - 968)^{2} \) Copy content Toggle raw display
$29$ \( T^{6} - 18 T^{5} + \cdots + 19046792 \) Copy content Toggle raw display
$31$ \( T^{6} + 1920 T^{4} + \cdots + 16777216 \) Copy content Toggle raw display
$37$ \( T^{6} - 46 T^{5} + \cdots + 42632 \) Copy content Toggle raw display
$41$ \( T^{6} + 4992 T^{4} + \cdots + 67108864 \) Copy content Toggle raw display
$43$ \( T^{6} + 114 T^{5} + \cdots + 42632 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 6056574976 \) Copy content Toggle raw display
$53$ \( T^{6} + 78 T^{5} + \cdots + 783752 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 8410007432 \) Copy content Toggle raw display
$61$ \( T^{6} - 30 T^{5} + \cdots + 151449608 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 87233303432 \) Copy content Toggle raw display
$71$ \( (T^{3} - 130 T^{2} + \cdots + 391864)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 7310934016 \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots + 1550483193856 \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 105636303368 \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots + 25681985536 \) Copy content Toggle raw display
$97$ \( (T^{3} + 2 T^{2} + \cdots + 519928)^{2} \) Copy content Toggle raw display
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