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Properties

Label 576.4.f.b
Level $576$
Weight $4$
Character orbit 576.f
Analytic conductor $33.985$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [576,4,Mod(287,576)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(576, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("576.287");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 576.f (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: \(33.9851001633\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: 16.0.196571825135013064605696.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 49x^{12} + 2145x^{8} - 12544x^{4} + 65536 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{48}\cdot 3^{12} \)
Twist minimal: yes
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 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.

\(f(q)\) \(=\) \( q + \beta_{9} q^{5} + (\beta_{12} + \beta_{11}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{9} q^{5} + (\beta_{12} + \beta_{11}) q^{7} - \beta_{14} q^{11} + (\beta_{3} - 4 \beta_1) q^{13} + ( - \beta_{7} + 4 \beta_{5}) q^{17} - 5 \beta_{6} q^{19} + \beta_{8} q^{23} + (\beta_{2} + 1) q^{25} + ( - 5 \beta_{10} - 4 \beta_{9}) q^{29} + ( - \beta_{12} - 4 \beta_{11}) q^{31} + (5 \beta_{14} + 9 \beta_{13}) q^{35} + ( - 4 \beta_{3} - 17 \beta_1) q^{37} + (5 \beta_{7} + 2 \beta_{5}) q^{41} + ( - \beta_{15} - 42 \beta_{6}) q^{43} + (\beta_{8} - 2 \beta_{4}) q^{47} + ( - 4 \beta_{2} - 333) q^{49} + ( - 17 \beta_{10} + 30 \beta_{9}) q^{53} + ( - 10 \beta_{12} - 32 \beta_{11}) q^{55} + ( - 6 \beta_{14} + 19 \beta_{13}) q^{59} - 61 \beta_1 q^{61} + ( - 7 \beta_{7} + 47 \beta_{5}) q^{65} + (5 \beta_{15} - 25 \beta_{6}) q^{67} + (2 \beta_{8} - 13 \beta_{4}) q^{71} + (2 \beta_{2} - 332) q^{73} + ( - 28 \beta_{10} + 72 \beta_{9}) q^{77} + (9 \beta_{12} + 4 \beta_{11}) q^{79} + ( - 5 \beta_{14} + 31 \beta_{13}) q^{83} + (19 \beta_{3} - 135 \beta_1) q^{85} + ( - 2 \beta_{7} + 49 \beta_{5}) q^{89} + ( - 6 \beta_{15} - 185 \beta_{6}) q^{91} + (5 \beta_{8} + 5 \beta_{4}) q^{95} + (7 \beta_{2} + 8) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 16 q^{25} - 5328 q^{49} - 5312 q^{73} + 128 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 49x^{12} + 2145x^{8} - 12544x^{4} + 65536 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -49\nu^{12} + 2145\nu^{8} - 105105\nu^{4} + 340096 ) / 68640 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 16\nu^{12} + 640136 ) / 6435 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{12} - 49\nu^{8} + 1889\nu^{4} - 6272 ) / 192 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -112\nu^{15} - 33\nu^{13} - 32175\nu^{9} - 70785\nu^{5} + 615568\nu^{3} + 413952\nu ) / 3294720 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 16\nu^{15} + 233\nu^{13} - 12441\nu^{9} + 499785\nu^{5} + 1041680\nu^{3} - 2922752\nu ) / 439296 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 81\nu^{14} - 4225\nu^{10} + 190385\nu^{6} - 2097664\nu^{2} ) / 599040 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -752\nu^{15} - 17607\nu^{13} + 804375\nu^{9} - 37767015\nu^{5} - 80588272\nu^{3} + 220862208\nu ) / 6589440 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 848\nu^{15} - 14013\nu^{13} + 714285\nu^{9} - 30057885\nu^{5} + 63116368\nu^{3} + 175779072\nu ) / 3294720 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 16381 \nu^{15} + 22824 \nu^{13} + 760045 \nu^{11} - 909480 \nu^{9} - 33032285 \nu^{7} + \cdots + 179349504 \nu ) / 52715520 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 3307 \nu^{15} - 6168 \nu^{13} - 158587 \nu^{11} + 216216 \nu^{9} + 6416267 \nu^{7} + \cdots - 33331200 \nu ) / 10543104 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -49\nu^{14} + 2145\nu^{10} - 96657\nu^{6} + 65536\nu^{2} ) / 76032 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( -30919\nu^{14} + 1353495\nu^{10} - 53691495\nu^{6} + 41353216\nu^{2} ) / 39536640 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 211 \nu^{15} - 344 \nu^{13} + 9955 \nu^{11} + 12760 \nu^{9} - 417395 \nu^{7} + \cdots - 2217984 \nu ) / 337920 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 24533 \nu^{15} - 32232 \nu^{13} + 1131845 \nu^{11} + 1321320 \nu^{9} - 49786165 \nu^{7} + \cdots - 272197632 \nu ) / 26357760 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 17\nu^{14} - 1089\nu^{10} + 44913\nu^{6} - 492032\nu^{2} ) / 5632 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -6\beta_{14} + 3\beta_{13} + 6\beta_{10} + 18\beta_{9} + \beta_{8} + 2\beta_{7} + 2\beta_{5} + 3\beta_{4} ) / 192 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{15} + 8\beta_{12} - 23\beta_{11} - 108\beta_{6} ) / 192 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 7\beta_{8} - 30\beta_{7} - 94\beta_{5} + 53\beta_{4} ) / 288 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -12\beta_{3} - 3\beta_{2} - 98\beta _1 + 392 ) / 32 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 174 \beta_{14} + 231 \beta_{13} + 318 \beta_{10} + 378 \beta_{9} - 5 \beta_{8} + \cdots - 111 \beta_{4} ) / 192 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 520\beta_{12} - 631\beta_{11} ) / 96 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 3258 \beta_{14} + 4749 \beta_{13} - 6378 \beta_{10} - 6654 \beta_{9} + 23 \beta_{8} + \cdots + 2149 \beta_{4} ) / 576 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -588\beta_{3} + 147\beta_{2} - 3778\beta _1 - 15112 ) / 32 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 11\beta_{8} - 2330\beta_{7} - 11738\beta_{5} - 4671\beta_{4} ) / 96 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -5858\beta_{15} + 23432\beta_{12} - 25031\beta_{11} + 146988\beta_{6} ) / 192 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 136602 \beta_{14} + 208893 \beta_{13} - 277194 \beta_{10} - 269214 \beta_{9} + \cdots - 91733 \beta_{4} ) / 576 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 6435\beta_{2} - 640136 ) / 16 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 297966 \beta_{14} - 457863 \beta_{13} - 606846 \beta_{10} - 585018 \beta_{9} + 1819 \beta_{8} + \cdots - 200463 \beta_{4} ) / 192 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( -253762\beta_{15} - 1015048\beta_{12} + 1064983\beta_{11} + 6290028\beta_{6} ) / 192 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 38473\beta_{8} + 1953150\beta_{7} + 10073534\beta_{5} - 3944773\beta_{4} ) / 288 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(65\) \(127\) \(325\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
287.1
−0.404160 − 1.50834i
−1.50834 − 0.404160i
−1.50834 + 0.404160i
−0.404160 + 1.50834i
−0.662979 + 2.47427i
−2.47427 + 0.662979i
−2.47427 − 0.662979i
−0.662979 − 2.47427i
0.662979 − 2.47427i
2.47427 − 0.662979i
2.47427 + 0.662979i
0.662979 + 2.47427i
0.404160 + 1.50834i
1.50834 + 0.404160i
1.50834 − 0.404160i
0.404160 − 1.50834i
0 0 0 −14.9985 0 − 32.7386i 0 0 0
287.2 0 0 0 −14.9985 0 − 32.7386i 0 0 0
287.3 0 0 0 −14.9985 0 32.7386i 0 0 0
287.4 0 0 0 −14.9985 0 32.7386i 0 0 0
287.5 0 0 0 −5.20053 0 − 16.7386i 0 0 0
287.6 0 0 0 −5.20053 0 − 16.7386i 0 0 0
287.7 0 0 0 −5.20053 0 16.7386i 0 0 0
287.8 0 0 0 −5.20053 0 16.7386i 0 0 0
287.9 0 0 0 5.20053 0 − 16.7386i 0 0 0
287.10 0 0 0 5.20053 0 − 16.7386i 0 0 0
287.11 0 0 0 5.20053 0 16.7386i 0 0 0
287.12 0 0 0 5.20053 0 16.7386i 0 0 0
287.13 0 0 0 14.9985 0 − 32.7386i 0 0 0
287.14 0 0 0 14.9985 0 − 32.7386i 0 0 0
287.15 0 0 0 14.9985 0 32.7386i 0 0 0
287.16 0 0 0 14.9985 0 32.7386i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 287.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
12.b even 2 1 inner
24.f even 2 1 inner
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 576.4.f.b ✓ 16
3.b odd 2 1 inner 576.4.f.b ✓ 16
4.b odd 2 1 inner 576.4.f.b ✓ 16
8.b even 2 1 inner 576.4.f.b ✓ 16
8.d odd 2 1 inner 576.4.f.b ✓ 16
12.b even 2 1 inner 576.4.f.b ✓ 16
16.e even 4 2 2304.4.c.m 16
16.f odd 4 2 2304.4.c.m 16
24.f even 2 1 inner 576.4.f.b ✓ 16
24.h odd 2 1 inner 576.4.f.b ✓ 16
48.i odd 4 2 2304.4.c.m 16
48.k even 4 2 2304.4.c.m 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
576.4.f.b ✓ 16 1.a even 1 1 trivial
576.4.f.b ✓ 16 3.b odd 2 1 inner
576.4.f.b ✓ 16 4.b odd 2 1 inner
576.4.f.b ✓ 16 8.b even 2 1 inner
576.4.f.b ✓ 16 8.d odd 2 1 inner
576.4.f.b ✓ 16 12.b even 2 1 inner
576.4.f.b ✓ 16 24.f even 2 1 inner
576.4.f.b ✓ 16 24.h odd 2 1 inner
2304.4.c.m 16 16.e even 4 2
2304.4.c.m 16 16.f odd 4 2
2304.4.c.m 16 48.i odd 4 2
2304.4.c.m 16 48.k even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 252T_{5}^{2} + 6084 \) acting on \(S_{4}^{\mathrm{new}}(576, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{4} - 252 T^{2} + 6084)^{4} \) Copy content Toggle raw display
$7$ \( (T^{4} + 1352 T^{2} + 300304)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} + 3312 T^{2} + 2585664)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} + 5208 T^{2} + 1140624)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} + 14148 T^{2} + 7387524)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} - 1200)^{8} \) Copy content Toggle raw display
$23$ \( (T^{4} - 11664 T^{2} + 11451456)^{4} \) Copy content Toggle raw display
$29$ \( (T^{4} - 37692 T^{2} + 5391684)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 7496 T^{2} + 6370576)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + 86496 T^{2} + 240374016)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + 249156 T^{2} + 14453329284)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} - 228096 T^{2} + 3057647616)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} - 39312 T^{2} + 296115264)^{4} \) Copy content Toggle raw display
$53$ \( (T^{4} - 669276 T^{2} + 90097226244)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} + 677376 T^{2} + 48922361856)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 178608)^{8} \) Copy content Toggle raw display
$67$ \( (T^{4} - 1528800 T^{2} + 496179360000)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} - 962064 T^{2} + 34159910976)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 664 T + 71056)^{8} \) Copy content Toggle raw display
$79$ \( (T^{4} + 99272 T^{2} + 2451042064)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 1654128 T^{2} + 555716575296)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} + 796068 T^{2} + 128784805956)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} - 16 T - 479744)^{8} \) Copy content Toggle raw display
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