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Properties

Label 576.3.o.g
Level $576$
Weight $3$
Character orbit 576.o
Analytic conductor $15.695$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [576,3,Mod(319,576)]
 
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([3, 0, 2]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("576.319");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 576.o (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: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 3 x^{15} + 7 x^{14} - 30 x^{13} + 76 x^{12} - 144 x^{11} + 424 x^{10} - 912 x^{9} + \cdots + 65536 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{16}\cdot 3^{3} \)
Twist minimal: no (minimal twist has level 36)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{6} + \beta_{3}) q^{3} + (\beta_{7} - \beta_1) q^{5} + ( - \beta_{11} + \beta_{9} + \beta_{3}) q^{7} + ( - \beta_{8} + \beta_{7} + \beta_{4} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{6} + \beta_{3}) q^{3} + (\beta_{7} - \beta_1) q^{5} + ( - \beta_{11} + \beta_{9} + \beta_{3}) q^{7} + ( - \beta_{8} + \beta_{7} + \beta_{4} + \cdots - 1) q^{9}+ \cdots + ( - 11 \beta_{13} + \cdots + 34 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 6 q^{5} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 6 q^{5} + 18 q^{9} + 46 q^{13} + 12 q^{17} + 66 q^{21} - 30 q^{25} - 42 q^{29} - 168 q^{33} - 56 q^{37} + 84 q^{41} - 174 q^{45} + 58 q^{49} + 72 q^{53} + 366 q^{57} + 34 q^{61} - 30 q^{65} + 54 q^{69} + 116 q^{73} + 330 q^{77} - 102 q^{81} + 140 q^{85} - 384 q^{89} + 486 q^{93} - 148 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 3 x^{15} + 7 x^{14} - 30 x^{13} + 76 x^{12} - 144 x^{11} + 424 x^{10} - 912 x^{9} + \cdots + 65536 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - \nu^{15} - 17 \nu^{14} - 83 \nu^{13} + 394 \nu^{12} - 204 \nu^{11} + 2224 \nu^{10} + \cdots + 1769472 ) / 540672 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - \nu^{15} + 3 \nu^{14} + 5 \nu^{13} - 22 \nu^{12} - 8 \nu^{11} - 72 \nu^{10} + 328 \nu^{9} + \cdots - 122880 ) / 24576 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - \nu^{15} + 5 \nu^{14} + 15 \nu^{13} - 8 \nu^{12} - 36 \nu^{11} - 128 \nu^{10} + 136 \nu^{9} + \cdots - 40960 ) / 24576 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 27 \nu^{15} + 333 \nu^{14} - 1273 \nu^{13} + 2630 \nu^{12} - 9380 \nu^{11} + 23440 \nu^{10} + \cdots + 4341760 ) / 540672 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 17 \nu^{15} - 179 \nu^{14} + 371 \nu^{13} - 628 \nu^{12} + 3440 \nu^{11} - 6984 \nu^{10} + \cdots - 286720 ) / 270336 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 41 \nu^{15} - 37 \nu^{14} - 279 \nu^{13} + 534 \nu^{12} - 180 \nu^{11} + 1600 \nu^{10} + \cdots - 622592 ) / 540672 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 49 \nu^{15} + 135 \nu^{14} - 723 \nu^{13} + 2762 \nu^{12} - 4100 \nu^{11} + 12352 \nu^{10} + \cdots + 7045120 ) / 540672 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 3 \nu^{15} + 27 \nu^{14} - 51 \nu^{13} + 176 \nu^{12} - 504 \nu^{11} + 984 \nu^{10} + \cdots + 131072 ) / 24576 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 71 \nu^{15} + 283 \nu^{14} - 223 \nu^{13} - 1090 \nu^{12} - 652 \nu^{11} + 2256 \nu^{10} + \cdots - 2179072 ) / 540672 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 53 \nu^{15} + \nu^{14} + 23 \nu^{13} + 708 \nu^{12} - 120 \nu^{11} - 488 \nu^{10} + \cdots + 786432 ) / 270336 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 113 \nu^{15} - 147 \nu^{14} + 535 \nu^{13} - 2590 \nu^{12} + 3692 \nu^{11} - 7728 \nu^{10} + \cdots - 3866624 ) / 540672 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 149 \nu^{15} + 459 \nu^{14} - 1103 \nu^{13} + 3442 \nu^{12} - 8044 \nu^{11} + 17744 \nu^{10} + \cdots + 524288 ) / 540672 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 75 \nu^{15} + 221 \nu^{14} - 593 \nu^{13} + 1918 \nu^{12} - 4124 \nu^{11} + 9984 \nu^{10} + \cdots + 1146880 ) / 270336 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 193 \nu^{15} + 459 \nu^{14} - 1455 \nu^{13} + 4102 \nu^{12} - 10156 \nu^{11} + 24080 \nu^{10} + \cdots + 2146304 ) / 540672 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 59 \nu^{15} - 163 \nu^{14} + 277 \nu^{13} - 1510 \nu^{12} + 3126 \nu^{11} - 5420 \nu^{10} + \cdots - 1040384 ) / 135168 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{13} + \beta_{12} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{15} + \beta_{10} + \beta_{8} + \beta_{6} + \beta_{3} + \beta _1 - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{14} + \beta_{13} - \beta_{12} + 4 \beta_{11} + \beta_{10} + \beta_{7} + 4 \beta_{6} - \beta_{5} + \cdots + 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2 \beta_{15} + \beta_{13} + 3 \beta_{12} + 4 \beta_{11} - \beta_{9} + 3 \beta_{7} + 2 \beta_{6} + \cdots - 7 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 6 \beta_{15} - 3 \beta_{14} - 10 \beta_{11} - 3 \beta_{10} - \beta_{9} + 6 \beta_{8} + 8 \beta_{6} + \cdots - 19 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 9 \beta_{14} + \beta_{13} - 9 \beta_{12} + 2 \beta_{11} + \beta_{10} - 6 \beta_{8} + 9 \beta_{7} + \cdots - 31 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 6 \beta_{15} + 17 \beta_{13} - \beta_{12} + 20 \beta_{11} - 11 \beta_{9} + 33 \beta_{7} + \cdots - 103 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 26 \beta_{15} - 57 \beta_{14} - 46 \beta_{11} - 33 \beta_{10} + 13 \beta_{9} - 26 \beta_{8} + \cdots - 121 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 65 \beta_{14} + 39 \beta_{13} + 41 \beta_{12} - 114 \beta_{11} + 39 \beta_{10} - 90 \beta_{8} + \cdots - 209 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 22 \beta_{15} + 111 \beta_{13} + 33 \beta_{12} + 108 \beta_{11} - 77 \beta_{9} - 57 \beta_{7} + \cdots + 407 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 294 \beta_{15} - 111 \beta_{14} + 190 \beta_{11} + 73 \beta_{10} + 379 \beta_{9} - 294 \beta_{8} + \cdots - 1567 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 855 \beta_{14} + 929 \beta_{13} + 207 \beta_{12} + 418 \beta_{11} + 929 \beta_{10} + 138 \beta_{8} + \cdots - 1543 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 762 \beta_{15} + 1401 \beta_{13} - 265 \beta_{12} + 2548 \beta_{11} - 523 \beta_{9} + \cdots + 7313 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 22 \beta_{15} + 1767 \beta_{14} - 2222 \beta_{11} - 1297 \beta_{10} + 3245 \beta_{9} + 22 \beta_{8} + \cdots - 3433 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 2609 \beta_{14} - 1129 \beta_{13} - 3079 \beta_{12} - 1330 \beta_{11} - 1129 \beta_{10} + \cdots - 20225 ) / 2 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
319.1
−0.523926 − 1.93016i
1.63139 − 1.15696i
−0.710719 + 1.86946i
1.84233 + 0.778342i
−1.59523 − 1.20633i
−1.26364 + 1.55023i
0.186266 − 1.99131i
1.93353 − 0.511345i
−0.523926 + 1.93016i
1.63139 + 1.15696i
−0.710719 − 1.86946i
1.84233 − 0.778342i
−1.59523 + 1.20633i
−1.26364 − 1.55023i
0.186266 + 1.99131i
1.93353 + 0.511345i
0 −2.76570 + 1.16229i 0 4.03104 − 6.98197i 0 3.90254 − 2.25313i 0 6.29815 − 6.42910i 0
319.2 0 −2.67178 − 1.36441i 0 −3.07403 + 5.32438i 0 −0.511543 + 0.295340i 0 5.27677 + 7.29079i 0
319.3 0 −2.32245 + 1.89900i 0 −1.35609 + 2.34881i 0 −10.0431 + 5.79837i 0 1.78756 − 8.82069i 0
319.4 0 −0.262217 + 2.98852i 0 −1.10093 + 1.90686i 0 7.23844 − 4.17912i 0 −8.86248 − 1.56728i 0
319.5 0 0.262217 − 2.98852i 0 −1.10093 + 1.90686i 0 −7.23844 + 4.17912i 0 −8.86248 − 1.56728i 0
319.6 0 2.32245 − 1.89900i 0 −1.35609 + 2.34881i 0 10.0431 − 5.79837i 0 1.78756 − 8.82069i 0
319.7 0 2.67178 + 1.36441i 0 −3.07403 + 5.32438i 0 0.511543 − 0.295340i 0 5.27677 + 7.29079i 0
319.8 0 2.76570 − 1.16229i 0 4.03104 − 6.98197i 0 −3.90254 + 2.25313i 0 6.29815 − 6.42910i 0
511.1 0 −2.76570 − 1.16229i 0 4.03104 + 6.98197i 0 3.90254 + 2.25313i 0 6.29815 + 6.42910i 0
511.2 0 −2.67178 + 1.36441i 0 −3.07403 − 5.32438i 0 −0.511543 − 0.295340i 0 5.27677 − 7.29079i 0
511.3 0 −2.32245 − 1.89900i 0 −1.35609 − 2.34881i 0 −10.0431 − 5.79837i 0 1.78756 + 8.82069i 0
511.4 0 −0.262217 − 2.98852i 0 −1.10093 − 1.90686i 0 7.23844 + 4.17912i 0 −8.86248 + 1.56728i 0
511.5 0 0.262217 + 2.98852i 0 −1.10093 − 1.90686i 0 −7.23844 − 4.17912i 0 −8.86248 + 1.56728i 0
511.6 0 2.32245 + 1.89900i 0 −1.35609 − 2.34881i 0 10.0431 + 5.79837i 0 1.78756 + 8.82069i 0
511.7 0 2.67178 − 1.36441i 0 −3.07403 − 5.32438i 0 0.511543 + 0.295340i 0 5.27677 − 7.29079i 0
511.8 0 2.76570 + 1.16229i 0 4.03104 + 6.98197i 0 −3.90254 − 2.25313i 0 6.29815 + 6.42910i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 319.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
9.c even 3 1 inner
36.f 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.o.g 16
3.b odd 2 1 1728.3.o.g 16
4.b odd 2 1 inner 576.3.o.g 16
8.b even 2 1 36.3.f.c ✓ 16
8.d odd 2 1 36.3.f.c ✓ 16
9.c even 3 1 inner 576.3.o.g 16
9.d odd 6 1 1728.3.o.g 16
12.b even 2 1 1728.3.o.g 16
24.f even 2 1 108.3.f.c 16
24.h odd 2 1 108.3.f.c 16
36.f odd 6 1 inner 576.3.o.g 16
36.h even 6 1 1728.3.o.g 16
72.j odd 6 1 108.3.f.c 16
72.j odd 6 1 324.3.d.g 8
72.l even 6 1 108.3.f.c 16
72.l even 6 1 324.3.d.g 8
72.n even 6 1 36.3.f.c ✓ 16
72.n even 6 1 324.3.d.i 8
72.p odd 6 1 36.3.f.c ✓ 16
72.p odd 6 1 324.3.d.i 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.3.f.c ✓ 16 8.b even 2 1
36.3.f.c ✓ 16 8.d odd 2 1
36.3.f.c ✓ 16 72.n even 6 1
36.3.f.c ✓ 16 72.p odd 6 1
108.3.f.c 16 24.f even 2 1
108.3.f.c 16 24.h odd 2 1
108.3.f.c 16 72.j odd 6 1
108.3.f.c 16 72.l even 6 1
324.3.d.g 8 72.j odd 6 1
324.3.d.g 8 72.l even 6 1
324.3.d.i 8 72.n even 6 1
324.3.d.i 8 72.p odd 6 1
576.3.o.g 16 1.a even 1 1 trivial
576.3.o.g 16 4.b odd 2 1 inner
576.3.o.g 16 9.c even 3 1 inner
576.3.o.g 16 36.f odd 6 1 inner
1728.3.o.g 16 3.b odd 2 1
1728.3.o.g 16 9.d odd 6 1
1728.3.o.g 16 12.b even 2 1
1728.3.o.g 16 36.h even 6 1

Hecke kernels

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

\( T_{5}^{8} + 3T_{5}^{7} + 62T_{5}^{6} + 351T_{5}^{5} + 3870T_{5}^{4} + 15291T_{5}^{3} + 49337T_{5}^{2} + 75480T_{5} + 87616 \) Copy content Toggle raw display
\( T_{7}^{16} - 225 T_{7}^{14} + 37002 T_{7}^{12} - 2674161 T_{7}^{10} + 141530490 T_{7}^{8} + \cdots + 4430766096 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} - 9 T^{14} + \cdots + 43046721 \) Copy content Toggle raw display
$5$ \( (T^{8} + 3 T^{7} + \cdots + 87616)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 4430766096 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 134421415700625 \) Copy content Toggle raw display
$13$ \( (T^{8} - 23 T^{7} + \cdots + 12100)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 3 T^{3} + \cdots + 2200)^{4} \) Copy content Toggle raw display
$19$ \( (T^{8} + 1215 T^{6} + \cdots + 7464960000)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 14\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( (T^{8} + 21 T^{7} + \cdots + 3658072324)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 86\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{4} + 14 T^{3} + \cdots + 5920)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} - 42 T^{7} + \cdots + 12391919761)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 14\!\cdots\!41 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 17\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( (T^{4} - 18 T^{3} + \cdots - 16160)^{4} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 23\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 933144135518464)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 15\!\cdots\!81 \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 29 T^{3} + \cdots + 20112040)^{4} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 36\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( (T^{4} + 96 T^{3} + \cdots - 4957424)^{4} \) Copy content Toggle raw display
$97$ \( (T^{8} + 74 T^{7} + \cdots + 4182855625)^{2} \) Copy content Toggle raw display
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