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Properties

Label 576.6.k.a
Level $576$
Weight $6$
Character orbit 576.k
Analytic conductor $92.381$
Analytic rank $0$
Dimension $18$
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] = [576,6,Mod(145,576)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(576, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("576.145");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 576.k (of order \(4\), degree \(2\), not 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: \(92.3810802123\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(9\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 8 x^{17} - 3867 x^{16} + 20528 x^{15} + 5993890 x^{14} - 12125584 x^{13} + \cdots + 93\!\cdots\!16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{72}\cdot 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{5} + ( - \beta_{2} - 11 \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{5} + ( - \beta_{2} - 11 \beta_1) q^{7} + (\beta_{11} + \beta_{4} + 34 \beta_1 - 34) q^{11} + (\beta_{16} + \beta_{8} + \cdots + \beta_{2}) q^{13}+ \cdots + (17 \beta_{16} + 51 \beta_{15} + \cdots - 397) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 2 q^{5} - 606 q^{11} - 2 q^{13} + 4 q^{17} + 2362 q^{19} - 4070 q^{29} + 11536 q^{31} + 8636 q^{35} - 10650 q^{37} + 15382 q^{43} + 44176 q^{47} - 14410 q^{49} - 24726 q^{53} - 29734 q^{59} - 48082 q^{61} - 27684 q^{65} + 75210 q^{67} - 41060 q^{77} + 52864 q^{79} + 227838 q^{83} - 138652 q^{85} + 231164 q^{91} - 250380 q^{95} - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 8 x^{17} - 3867 x^{16} + 20528 x^{15} + 5993890 x^{14} - 12125584 x^{13} + \cdots + 93\!\cdots\!16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 65\!\cdots\!89 \nu^{17} + \cdots - 48\!\cdots\!96 ) / 76\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 88\!\cdots\!03 \nu^{17} + \cdots + 43\!\cdots\!20 ) / 65\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 16\!\cdots\!39 \nu^{17} + \cdots - 11\!\cdots\!44 ) / 76\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 19\!\cdots\!95 \nu^{17} + \cdots - 13\!\cdots\!00 ) / 76\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 35\!\cdots\!25 \nu^{17} + \cdots - 68\!\cdots\!16 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 24\!\cdots\!38 \nu^{17} + \cdots - 20\!\cdots\!48 ) / 72\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 57\!\cdots\!34 \nu^{17} + \cdots - 39\!\cdots\!68 ) / 72\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 36\!\cdots\!56 \nu^{17} + \cdots - 26\!\cdots\!36 ) / 36\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 70\!\cdots\!59 \nu^{17} + \cdots - 47\!\cdots\!40 ) / 54\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 27\!\cdots\!62 \nu^{17} + \cdots + 19\!\cdots\!72 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 16\!\cdots\!09 \nu^{17} + \cdots - 91\!\cdots\!76 ) / 65\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 15\!\cdots\!56 \nu^{17} + \cdots + 10\!\cdots\!60 ) / 54\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 32\!\cdots\!76 \nu^{17} + \cdots + 23\!\cdots\!72 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 42\!\cdots\!15 \nu^{17} + \cdots - 27\!\cdots\!68 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 37\!\cdots\!28 \nu^{17} + \cdots + 26\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 28\!\cdots\!67 \nu^{17} + \cdots - 22\!\cdots\!76 ) / 65\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 33\!\cdots\!95 \nu^{17} + \cdots - 23\!\cdots\!60 ) / 65\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} - \beta_{3} - 4\beta _1 + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 3 \beta_{16} - \beta_{15} - \beta_{14} + 3 \beta_{13} - 3 \beta_{11} - 3 \beta_{10} + 10 \beta_{8} + \cdots + 3477 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 6 \beta_{17} + 87 \beta_{16} - 2 \beta_{15} + 7 \beta_{14} + 16 \beta_{13} - 18 \beta_{12} + \cdots + 29350 ) / 16 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 2 \beta_{17} - 1426 \beta_{16} - 319 \beta_{15} - 1026 \beta_{14} + 2092 \beta_{13} + 374 \beta_{12} + \cdots + 1400265 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 6420 \beta_{17} + 120583 \beta_{16} + 17666 \beta_{15} - 28389 \beta_{14} + 25882 \beta_{13} + \cdots + 24029366 ) / 16 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 52958 \beta_{17} - 2880991 \beta_{16} - 399789 \beta_{15} - 2740391 \beta_{14} + 4899542 \beta_{13} + \cdots + 2716235751 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 5687038 \beta_{17} + 150019351 \beta_{16} + 36050354 \beta_{15} - 62431769 \beta_{14} + \cdots + 18626263034 ) / 16 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 72594908 \beta_{17} - 1544478107 \beta_{16} - 113010286 \beta_{15} - 1645419463 \beta_{14} + \cdots + 1410084220398 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 4205291400 \beta_{17} + 178731101515 \beta_{16} + 50746701130 \beta_{15} - 95089531625 \beta_{14} + \cdots + 16173078679574 ) / 16 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 258095130146 \beta_{17} - 3408822562557 \beta_{16} - 100776546823 \beta_{15} - 3793815870757 \beta_{14} + \cdots + 30\!\cdots\!01 ) / 8 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 2372283494922 \beta_{17} + 207897422633891 \beta_{16} + 62590889494650 \beta_{15} + \cdots + 16\!\cdots\!02 ) / 16 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 193469963676858 \beta_{17} + \cdots + 16\!\cdots\!03 ) / 4 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 419123281090956 \beta_{17} + \cdots + 17\!\cdots\!02 ) / 16 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 53\!\cdots\!02 \beta_{17} + \cdots + 34\!\cdots\!83 ) / 8 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 15\!\cdots\!62 \beta_{17} + \cdots + 20\!\cdots\!38 ) / 16 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 34\!\cdots\!60 \beta_{17} + \cdots + 18\!\cdots\!56 ) / 4 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 37\!\cdots\!16 \beta_{17} + \cdots + 23\!\cdots\!62 ) / 16 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/576\mathbb{Z}\right)^\times\).

\(n\) \(65\) \(127\) \(325\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
145.1
33.5024 − 1.00000i
24.7991 − 1.00000i
20.4066 − 1.00000i
4.86757 − 1.00000i
−3.70323 − 1.00000i
−12.5478 − 1.00000i
−13.6637 − 1.00000i
−16.4787 − 1.00000i
−33.1823 − 1.00000i
33.5024 + 1.00000i
24.7991 + 1.00000i
20.4066 + 1.00000i
4.86757 + 1.00000i
−3.70323 + 1.00000i
−12.5478 + 1.00000i
−13.6637 + 1.00000i
−16.4787 + 1.00000i
−33.1823 + 1.00000i
0 0 0 −66.0049 + 66.0049i 0 − 75.3048i 0 0 0
145.2 0 0 0 −48.5981 + 48.5981i 0 106.338i 0 0 0
145.3 0 0 0 −39.8132 + 39.8132i 0 − 248.565i 0 0 0
145.4 0 0 0 −8.73514 + 8.73514i 0 28.0117i 0 0 0
145.5 0 0 0 8.40645 − 8.40645i 0 149.265i 0 0 0
145.6 0 0 0 26.0955 − 26.0955i 0 − 106.802i 0 0 0
145.7 0 0 0 28.3274 − 28.3274i 0 55.5494i 0 0 0
145.8 0 0 0 33.9573 − 33.9573i 0 141.886i 0 0 0
145.9 0 0 0 67.3647 − 67.3647i 0 − 148.379i 0 0 0
433.1 0 0 0 −66.0049 − 66.0049i 0 75.3048i 0 0 0
433.2 0 0 0 −48.5981 − 48.5981i 0 − 106.338i 0 0 0
433.3 0 0 0 −39.8132 − 39.8132i 0 248.565i 0 0 0
433.4 0 0 0 −8.73514 − 8.73514i 0 − 28.0117i 0 0 0
433.5 0 0 0 8.40645 + 8.40645i 0 − 149.265i 0 0 0
433.6 0 0 0 26.0955 + 26.0955i 0 106.802i 0 0 0
433.7 0 0 0 28.3274 + 28.3274i 0 − 55.5494i 0 0 0
433.8 0 0 0 33.9573 + 33.9573i 0 − 141.886i 0 0 0
433.9 0 0 0 67.3647 + 67.3647i 0 148.379i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 145.9
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 576.6.k.a 18
3.b odd 2 1 64.6.e.a 18
4.b odd 2 1 144.6.k.a 18
12.b even 2 1 16.6.e.a ✓ 18
16.e even 4 1 inner 576.6.k.a 18
16.f odd 4 1 144.6.k.a 18
24.f even 2 1 128.6.e.b 18
24.h odd 2 1 128.6.e.a 18
48.i odd 4 1 64.6.e.a 18
48.i odd 4 1 128.6.e.a 18
48.k even 4 1 16.6.e.a ✓ 18
48.k even 4 1 128.6.e.b 18
96.o even 8 2 1024.6.a.k 18
96.p odd 8 2 1024.6.a.l 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.6.e.a ✓ 18 12.b even 2 1
16.6.e.a ✓ 18 48.k even 4 1
64.6.e.a 18 3.b odd 2 1
64.6.e.a 18 48.i odd 4 1
128.6.e.a 18 24.h odd 2 1
128.6.e.a 18 48.i odd 4 1
128.6.e.b 18 24.f even 2 1
128.6.e.b 18 48.k even 4 1
144.6.k.a 18 4.b odd 2 1
144.6.k.a 18 16.f odd 4 1
576.6.k.a 18 1.a even 1 1 trivial
576.6.k.a 18 16.e even 4 1 inner
1024.6.a.k 18 96.o even 8 2
1024.6.a.l 18 96.p odd 8 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{18} - 2 T_{5}^{17} + 2 T_{5}^{16} - 106880 T_{5}^{15} + 100438544 T_{5}^{14} + \cdots + 12\!\cdots\!92 \) acting on \(S_{6}^{\mathrm{new}}(576, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} \) Copy content Toggle raw display
$3$ \( T^{18} \) Copy content Toggle raw display
$5$ \( T^{18} + \cdots + 12\!\cdots\!92 \) Copy content Toggle raw display
$7$ \( T^{18} + \cdots + 10\!\cdots\!56 \) Copy content Toggle raw display
$11$ \( T^{18} + \cdots + 18\!\cdots\!52 \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots + 88\!\cdots\!32 \) Copy content Toggle raw display
$17$ \( (T^{9} + \cdots - 13\!\cdots\!16)^{2} \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots + 33\!\cdots\!28 \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 32\!\cdots\!36 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 99\!\cdots\!52 \) Copy content Toggle raw display
$31$ \( (T^{9} + \cdots - 68\!\cdots\!08)^{2} \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 78\!\cdots\!32 \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 41\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( (T^{9} + \cdots - 16\!\cdots\!92)^{2} \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 46\!\cdots\!48 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 14\!\cdots\!48 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 17\!\cdots\!84 \) Copy content Toggle raw display
$79$ \( (T^{9} + \cdots - 28\!\cdots\!60)^{2} \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 33\!\cdots\!92 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 63\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( (T^{9} + \cdots + 84\!\cdots\!16)^{2} \) Copy content Toggle raw display
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