Properties

Label 572.2.f.c
Level $572$
Weight $2$
Character orbit 572.f
Analytic conductor $4.567$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 572 = 2^{2} \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 572.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.56744299562\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Defining polynomial: \(x^{8} + 21 x^{6} + 136 x^{4} + 309 x^{2} + 225\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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.

\(f(q)\) \(=\) \( q -\beta_{2} q^{3} + ( \beta_{1} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{5} + ( \beta_{1} + \beta_{4} ) q^{7} + ( 2 + \beta_{3} - \beta_{5} + \beta_{6} ) q^{9} +O(q^{10})\) \( q -\beta_{2} q^{3} + ( \beta_{1} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{5} + ( \beta_{1} + \beta_{4} ) q^{7} + ( 2 + \beta_{3} - \beta_{5} + \beta_{6} ) q^{9} + \beta_{4} q^{11} + ( -1 - \beta_{1} - \beta_{3} - \beta_{6} ) q^{13} + ( \beta_{1} - 2 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{15} + ( 2 - 2 \beta_{2} - \beta_{5} + \beta_{6} ) q^{17} + ( -2 \beta_{1} - \beta_{7} ) q^{19} + ( -\beta_{1} - 5 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{21} + ( 1 + 2 \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} ) q^{23} + ( -7 - \beta_{2} - \beta_{3} - \beta_{5} + \beta_{6} ) q^{25} + ( -3 + 2 \beta_{5} - 2 \beta_{6} ) q^{27} + ( -2 \beta_{2} + 2 \beta_{3} - \beta_{5} + \beta_{6} ) q^{29} + ( \beta_{1} - 6 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{31} -\beta_{1} q^{33} + ( -2 - 2 \beta_{5} + 2 \beta_{6} ) q^{35} + ( -\beta_{1} + 2 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{37} + ( 1 + 2 \beta_{2} - \beta_{3} + 3 \beta_{4} + \beta_{6} ) q^{39} + ( \beta_{1} + \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{41} + ( -2 - 2 \beta_{2} + 2 \beta_{3} - \beta_{5} + \beta_{6} ) q^{43} + ( -10 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{45} + ( -2 \beta_{1} + 4 \beta_{4} ) q^{47} + ( 1 - 2 \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} ) q^{49} + ( 6 - 2 \beta_{2} + \beta_{5} - \beta_{6} ) q^{51} + ( 4 - \beta_{3} - \beta_{5} + \beta_{6} ) q^{53} + ( -\beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} ) q^{55} + ( \beta_{1} + 9 \beta_{4} - \beta_{5} - \beta_{6} ) q^{57} + ( -\beta_{1} - 2 \beta_{4} - \beta_{5} - \beta_{6} + 3 \beta_{7} ) q^{59} + ( -4 - 2 \beta_{2} + 2 \beta_{3} - \beta_{5} + \beta_{6} ) q^{61} + ( 3 \beta_{1} + 5 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} + \beta_{7} ) q^{63} + ( 5 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} + 7 \beta_{4} - 3 \beta_{6} + \beta_{7} ) q^{65} + ( 3 \beta_{1} - 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{67} + ( -7 - 2 \beta_{3} ) q^{69} + ( -\beta_{1} - 4 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{71} + ( -2 \beta_{1} + 6 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{73} + ( 8 \beta_{2} - 3 \beta_{3} + 3 \beta_{5} - 3 \beta_{6} ) q^{75} + ( -1 - \beta_{2} ) q^{77} + ( 4 - 2 \beta_{3} - \beta_{5} + \beta_{6} ) q^{79} + ( 2 + 3 \beta_{2} + \beta_{3} - 3 \beta_{5} + 3 \beta_{6} ) q^{81} + ( \beta_{1} + 3 \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{83} + ( -2 \beta_{1} - 10 \beta_{4} - \beta_{5} - \beta_{6} - 4 \beta_{7} ) q^{85} + ( 8 - 2 \beta_{2} + 4 \beta_{3} - \beta_{5} + \beta_{6} ) q^{87} + ( 5 \beta_{1} - 4 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{89} + ( 3 - 2 \beta_{1} + \beta_{2} - 2 \beta_{4} + 2 \beta_{5} ) q^{91} + ( 7 \beta_{1} - 2 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{93} + ( 2 \beta_{2} + 2 \beta_{3} + 3 \beta_{5} - 3 \beta_{6} ) q^{95} + ( -\beta_{1} - 8 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{97} + ( 2 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 2q^{3} + 18q^{9} + O(q^{10}) \) \( 8q - 2q^{3} + 18q^{9} - 8q^{13} + 16q^{17} + 10q^{23} - 52q^{25} - 32q^{27} - 4q^{29} - 8q^{35} + 16q^{39} - 20q^{43} + 2q^{49} + 40q^{51} + 38q^{53} - 36q^{61} + 36q^{65} - 52q^{69} + 10q^{75} - 10q^{77} + 40q^{79} + 32q^{81} + 56q^{87} + 22q^{91} - 12q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 21 x^{6} + 136 x^{4} + 309 x^{2} + 225\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{6} - 18 \nu^{4} - 88 \nu^{2} - 105 \)\()/6\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{6} - 20 \nu^{4} - 112 \nu^{2} - 149 \)\()/4\)
\(\beta_{4}\)\(=\)\((\)\( -7 \nu^{7} - 132 \nu^{5} - 682 \nu^{3} - 843 \nu \)\()/90\)
\(\beta_{5}\)\(=\)\((\)\( -7 \nu^{7} - 15 \nu^{6} - 132 \nu^{5} - 300 \nu^{4} - 652 \nu^{3} - 1620 \nu^{2} - 663 \nu - 1935 \)\()/120\)
\(\beta_{6}\)\(=\)\((\)\( -7 \nu^{7} + 15 \nu^{6} - 132 \nu^{5} + 300 \nu^{4} - 652 \nu^{3} + 1620 \nu^{2} - 663 \nu + 1935 \)\()/120\)
\(\beta_{7}\)\(=\)\((\)\( 19 \nu^{7} + 384 \nu^{5} + 2224 \nu^{3} + 3291 \nu \)\()/180\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{6} + \beta_{5} - \beta_{3} - 5\)
\(\nu^{3}\)\(=\)\(2 \beta_{6} + 2 \beta_{5} - 3 \beta_{4} - 6 \beta_{1}\)
\(\nu^{4}\)\(=\)\(12 \beta_{6} - 12 \beta_{5} + 10 \beta_{3} + 3 \beta_{2} + 38\)
\(\nu^{5}\)\(=\)\(7 \beta_{7} - 29 \beta_{6} - 29 \beta_{5} + 53 \beta_{4} + 48 \beta_{1}\)
\(\nu^{6}\)\(=\)\(-128 \beta_{6} + 128 \beta_{5} - 92 \beta_{3} - 60 \beta_{2} - 349\)
\(\nu^{7}\)\(=\)\(-132 \beta_{7} + 352 \beta_{6} + 352 \beta_{5} - 720 \beta_{4} - 441 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/572\mathbb{Z}\right)^\times\).

\(n\) \(287\) \(353\) \(365\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
441.1
3.32727i
3.32727i
1.43491i
1.43491i
1.25126i
1.25126i
2.51091i
2.51091i
0 −3.32727 0 2.74343i 0 4.32727i 0 8.07070 0
441.2 0 −3.32727 0 2.74343i 0 4.32727i 0 8.07070 0
441.3 0 −1.43491 0 4.37595i 0 2.43491i 0 −0.941037 0
441.4 0 −1.43491 0 4.37595i 0 2.43491i 0 −0.941037 0
441.5 0 1.25126 0 2.18309i 0 0.251260i 0 −1.43435 0
441.6 0 1.25126 0 2.18309i 0 0.251260i 0 −1.43435 0
441.7 0 2.51091 0 3.81560i 0 1.51091i 0 3.30469 0
441.8 0 2.51091 0 3.81560i 0 1.51091i 0 3.30469 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 441.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 572.2.f.c 8
3.b odd 2 1 5148.2.e.c 8
4.b odd 2 1 2288.2.j.i 8
13.b even 2 1 inner 572.2.f.c 8
13.d odd 4 1 7436.2.a.o 4
13.d odd 4 1 7436.2.a.p 4
39.d odd 2 1 5148.2.e.c 8
52.b odd 2 1 2288.2.j.i 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
572.2.f.c 8 1.a even 1 1 trivial
572.2.f.c 8 13.b even 2 1 inner
2288.2.j.i 8 4.b odd 2 1
2288.2.j.i 8 52.b odd 2 1
5148.2.e.c 8 3.b odd 2 1
5148.2.e.c 8 39.d odd 2 1
7436.2.a.o 4 13.d odd 4 1
7436.2.a.p 4 13.d odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + T_{3}^{3} - 10 T_{3}^{2} - 3 T_{3} + 15 \) acting on \(S_{2}^{\mathrm{new}}(572, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( 15 - 3 T - 10 T^{2} + T^{3} + T^{4} )^{2} \)
$5$ \( 10000 + 4636 T^{2} + 729 T^{4} + 46 T^{6} + T^{8} \)
$7$ \( 16 + 264 T^{2} + 169 T^{4} + 27 T^{6} + T^{8} \)
$11$ \( ( 1 + T^{2} )^{4} \)
$13$ \( 28561 + 17576 T + 3042 T^{2} + 104 T^{3} + 2 T^{4} + 8 T^{5} + 18 T^{6} + 8 T^{7} + T^{8} \)
$17$ \( ( -352 + 216 T - 14 T^{2} - 8 T^{3} + T^{4} )^{2} \)
$19$ \( 106276 + 42656 T^{2} + 3641 T^{4} + 107 T^{6} + T^{8} \)
$23$ \( ( -111 + 123 T - 28 T^{2} - 5 T^{3} + T^{4} )^{2} \)
$29$ \( ( -344 - 340 T - 70 T^{2} + 2 T^{3} + T^{4} )^{2} \)
$31$ \( 283024 + 133372 T^{2} + 10377 T^{4} + 190 T^{6} + T^{8} \)
$37$ \( 309136 + 67784 T^{2} + 4817 T^{4} + 122 T^{6} + T^{8} \)
$41$ \( 9326916 + 880896 T^{2} + 25297 T^{4} + 275 T^{6} + T^{8} \)
$43$ \( ( -1272 - 564 T - 34 T^{2} + 10 T^{3} + T^{4} )^{2} \)
$47$ \( 30976 + 29760 T^{2} + 4480 T^{4} + 132 T^{6} + T^{8} \)
$53$ \( ( 216 + 288 T + 59 T^{2} - 19 T^{3} + T^{4} )^{2} \)
$59$ \( 156816 + 176328 T^{2} + 15937 T^{4} + 266 T^{6} + T^{8} \)
$61$ \( ( -2440 - 548 T + 50 T^{2} + 18 T^{3} + T^{4} )^{2} \)
$67$ \( 5776 + 176844 T^{2} + 12889 T^{4} + 222 T^{6} + T^{8} \)
$71$ \( 17424 + 225372 T^{2} + 21385 T^{4} + 318 T^{6} + T^{8} \)
$73$ \( 22429696 + 1499392 T^{2} + 34569 T^{4} + 319 T^{6} + T^{8} \)
$79$ \( ( 2456 + 1008 T - 2 T^{2} - 20 T^{3} + T^{4} )^{2} \)
$83$ \( 1119364 + 215832 T^{2} + 11401 T^{4} + 207 T^{6} + T^{8} \)
$89$ \( 23931664 + 1784300 T^{2} + 45881 T^{4} + 446 T^{6} + T^{8} \)
$97$ \( 7683984 + 841212 T^{2} + 27961 T^{4} + 302 T^{6} + T^{8} \)
show more
show less