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Properties

Label 576.3.q.i
Level $576$
Weight $3$
Character orbit 576.q
Analytic conductor $15.695$
Analytic rank $0$
Dimension $8$
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] = [576,3,Mod(65,576)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(576, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("576.65");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 576.q (of order \(6\), 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: \(15.6948632272\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.19269881856.9
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 15x^{6} - 2x^{5} + 133x^{4} - 84x^{3} + 276x^{2} + 144x + 144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: no (minimal twist has level 72)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{7} + \beta_{2} - 1) q^{3} + (\beta_{7} + \beta_{5} + \beta_{3} + \cdots + 1) q^{5}+ \cdots + ( - 2 \beta_{7} + \beta_{6} - \beta_{4} + \cdots - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{7} + \beta_{2} - 1) q^{3} + (\beta_{7} + \beta_{5} + \beta_{3} + \cdots + 1) q^{5}+ \cdots + ( - 7 \beta_{7} - 22 \beta_{6} + \cdots + 64) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 10 q^{3} + 6 q^{5} + 6 q^{7} - 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 10 q^{3} + 6 q^{5} + 6 q^{7} - 22 q^{9} - 36 q^{11} - 14 q^{13} + 10 q^{15} - 4 q^{19} + 54 q^{21} - 102 q^{23} + 10 q^{25} + 20 q^{27} + 114 q^{29} - 50 q^{31} - 104 q^{33} - 120 q^{37} + 82 q^{39} + 264 q^{41} + 28 q^{43} - 206 q^{45} + 150 q^{47} + 94 q^{49} - 170 q^{51} - 244 q^{55} - 178 q^{57} + 108 q^{59} - 14 q^{61} - 210 q^{63} - 198 q^{65} + 20 q^{67} + 14 q^{69} - 76 q^{73} - 326 q^{75} - 66 q^{77} + 26 q^{79} + 194 q^{81} - 246 q^{83} + 224 q^{85} - 18 q^{87} - 108 q^{91} + 130 q^{93} - 456 q^{95} - 236 q^{97} + 634 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 15x^{6} - 2x^{5} + 133x^{4} - 84x^{3} + 276x^{2} + 144x + 144 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 449 \nu^{7} - 2076 \nu^{6} + 4389 \nu^{5} - 53288 \nu^{4} + 35895 \nu^{3} - 330114 \nu^{2} + \cdots - 586944 ) / 318600 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 677 \nu^{7} + 1827 \nu^{6} - 10353 \nu^{5} + 6901 \nu^{4} - 82215 \nu^{3} + 132153 \nu^{2} + \cdots - 78912 ) / 159300 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1013 \nu^{7} - 2688 \nu^{6} + 16707 \nu^{5} - 19444 \nu^{4} + 143085 \nu^{3} - 232782 \nu^{2} + \cdots - 401472 ) / 159300 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2341 \nu^{7} + 1284 \nu^{6} + 17799 \nu^{5} + 84592 \nu^{4} + 265245 \nu^{3} + 449826 \nu^{2} + \cdots + 1295496 ) / 318600 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1439 \nu^{7} + 3964 \nu^{6} - 24921 \nu^{5} + 18832 \nu^{4} - 197555 \nu^{3} + 226746 \nu^{2} + \cdots + 9216 ) / 106200 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 13\nu^{7} - 38\nu^{6} + 207\nu^{5} - 194\nu^{4} + 1585\nu^{3} - 2532\nu^{2} + 2856\nu - 72 ) / 900 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5263 \nu^{7} - 12288 \nu^{6} + 85857 \nu^{5} - 40844 \nu^{4} + 743235 \nu^{3} - 620382 \nu^{2} + \cdots + 485928 ) / 318600 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -2\beta_{7} - 2\beta_{5} + \beta_{4} + \beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{7} + 6\beta_{6} - \beta_{5} + 2\beta_{4} - 3\beta_{3} + 18\beta_{2} + 2\beta _1 - 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 7\beta_{7} + 6\beta_{6} + 13\beta_{5} + 7\beta_{4} - 12\beta_{3} - 6\beta_{2} + 19\beta _1 - 38 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 50\beta_{7} - 45\beta_{6} + 38\beta_{5} - 25\beta_{4} - 45\beta_{3} - 186\beta_{2} - 13\beta _1 - 211 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 259\beta_{7} - 240\beta_{6} + 79\beta_{5} - 248\beta_{4} + 120\beta_{3} - 198\beta_{2} - 338\beta _1 + 169 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -205\beta_{7} - 627\beta_{6} - 445\beta_{5} - 205\beta_{4} + 1254\beta_{3} + 240\beta_{2} - 685\beta _1 + 3152 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 4586 \beta_{7} + 1962 \beta_{6} - 3332 \beta_{5} + 2293 \beta_{4} + 1962 \beta_{3} + 5316 \beta_{2} + \cdots + 7609 ) / 3 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(65\) \(127\) \(325\)
\(\chi(n)\) \(-\beta_{2}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
1.91950 − 3.32468i
0.831167 − 1.43962i
−0.331167 + 0.573598i
−1.41950 + 2.45865i
1.91950 + 3.32468i
0.831167 + 1.43962i
−0.331167 − 0.573598i
−1.41950 − 2.45865i
0 −2.91950 + 0.690286i 0 −1.80902 + 1.04444i 0 −0.781452 + 1.35351i 0 8.04701 − 4.03058i 0
65.2 0 −1.83117 − 2.37631i 0 −3.44299 + 1.98781i 0 −1.80469 + 3.12582i 0 −2.29365 + 8.70282i 0
65.3 0 −0.668833 + 2.92449i 0 0.0440114 − 0.0254100i 0 4.52944 − 7.84521i 0 −8.10532 − 3.91200i 0
65.4 0 0.419504 − 2.97052i 0 8.20800 − 4.73889i 0 1.05671 − 1.83027i 0 −8.64803 − 2.49230i 0
257.1 0 −2.91950 − 0.690286i 0 −1.80902 − 1.04444i 0 −0.781452 − 1.35351i 0 8.04701 + 4.03058i 0
257.2 0 −1.83117 + 2.37631i 0 −3.44299 − 1.98781i 0 −1.80469 − 3.12582i 0 −2.29365 − 8.70282i 0
257.3 0 −0.668833 − 2.92449i 0 0.0440114 + 0.0254100i 0 4.52944 + 7.84521i 0 −8.10532 + 3.91200i 0
257.4 0 0.419504 + 2.97052i 0 8.20800 + 4.73889i 0 1.05671 + 1.83027i 0 −8.64803 + 2.49230i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 65.4
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 576.3.q.i 8
3.b odd 2 1 1728.3.q.j 8
4.b odd 2 1 576.3.q.j 8
8.b even 2 1 72.3.m.b ✓ 8
8.d odd 2 1 144.3.q.e 8
9.c even 3 1 1728.3.q.j 8
9.d odd 6 1 inner 576.3.q.i 8
12.b even 2 1 1728.3.q.i 8
24.f even 2 1 432.3.q.e 8
24.h odd 2 1 216.3.m.b 8
36.f odd 6 1 1728.3.q.i 8
36.h even 6 1 576.3.q.j 8
72.j odd 6 1 72.3.m.b ✓ 8
72.j odd 6 1 648.3.e.c 8
72.l even 6 1 144.3.q.e 8
72.l even 6 1 1296.3.e.i 8
72.n even 6 1 216.3.m.b 8
72.n even 6 1 648.3.e.c 8
72.p odd 6 1 432.3.q.e 8
72.p odd 6 1 1296.3.e.i 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.3.m.b ✓ 8 8.b even 2 1
72.3.m.b ✓ 8 72.j odd 6 1
144.3.q.e 8 8.d odd 2 1
144.3.q.e 8 72.l even 6 1
216.3.m.b 8 24.h odd 2 1
216.3.m.b 8 72.n even 6 1
432.3.q.e 8 24.f even 2 1
432.3.q.e 8 72.p odd 6 1
576.3.q.i 8 1.a even 1 1 trivial
576.3.q.i 8 9.d odd 6 1 inner
576.3.q.j 8 4.b odd 2 1
576.3.q.j 8 36.h even 6 1
648.3.e.c 8 72.j odd 6 1
648.3.e.c 8 72.n even 6 1
1296.3.e.i 8 72.l even 6 1
1296.3.e.i 8 72.p odd 6 1
1728.3.q.i 8 12.b even 2 1
1728.3.q.i 8 36.f odd 6 1
1728.3.q.j 8 3.b odd 2 1
1728.3.q.j 8 9.c even 3 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(576, [\chi])\):

\( T_{5}^{8} - 6T_{5}^{7} - 37T_{5}^{6} + 294T_{5}^{5} + 2661T_{5}^{4} + 6468T_{5}^{3} + 5612T_{5}^{2} - 528T_{5} + 16 \) Copy content Toggle raw display
\( T_{7}^{8} - 6T_{7}^{7} + 69T_{7}^{6} + 126T_{7}^{5} + 1197T_{7}^{4} + 108T_{7}^{3} + 4860T_{7}^{2} + 3888T_{7} + 11664 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 10 T^{7} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( T^{8} - 6 T^{7} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( T^{8} - 6 T^{7} + \cdots + 11664 \) Copy content Toggle raw display
$11$ \( T^{8} + 36 T^{7} + \cdots + 105616729 \) Copy content Toggle raw display
$13$ \( T^{8} + 14 T^{7} + \cdots + 2611456 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 7020428944 \) Copy content Toggle raw display
$19$ \( (T^{4} + 2 T^{3} + \cdots + 226348)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 11198718976 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 106450807824 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 152712134656 \) Copy content Toggle raw display
$37$ \( (T^{4} + 60 T^{3} + \cdots + 206496)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 1919025613521 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 1352729498761 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 4615347568896 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 78435844096 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 127589696809 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 133593174016 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 17391015625 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 114698616545536 \) Copy content Toggle raw display
$73$ \( (T^{4} + 38 T^{3} + \cdots + 2961976)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 103529078405776 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 1085363908864 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 309931236458496 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 3435006304129 \) Copy content Toggle raw display
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