Properties

Label 572.2.bb.a
Level $572$
Weight $2$
Character orbit 572.bb
Analytic conductor $4.567$
Analytic rank $0$
Dimension $16$
CM discriminant -52
Inner twists $8$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 572 = 2^{2} \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 572.bb (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.56744299562\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{10})\)
Coefficient field: 16.0.855355656503296000000000000.9
Defining polynomial: \(x^{16} + 12 x^{14} + 95 x^{12} + 552 x^{10} + 1969 x^{8} + 27048 x^{6} + 228095 x^{4} + 1411788 x^{2} + 5764801\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{10}]$

$q$-expansion

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.

\(f(q)\) \(=\) \( q -\beta_{8} q^{2} + ( 2 + 2 \beta_{2} - 2 \beta_{7} - 2 \beta_{9} ) q^{4} + ( \beta_{3} - \beta_{8} - \beta_{10} - \beta_{12} + \beta_{13} - \beta_{14} ) q^{7} + ( 2 \beta_{3} - 2 \beta_{8} - 2 \beta_{10} + 2 \beta_{13} ) q^{8} -3 \beta_{7} q^{9} +O(q^{10})\) \( q -\beta_{8} q^{2} + ( 2 + 2 \beta_{2} - 2 \beta_{7} - 2 \beta_{9} ) q^{4} + ( \beta_{3} - \beta_{8} - \beta_{10} - \beta_{12} + \beta_{13} - \beta_{14} ) q^{7} + ( 2 \beta_{3} - 2 \beta_{8} - 2 \beta_{10} + 2 \beta_{13} ) q^{8} -3 \beta_{7} q^{9} + ( \beta_{1} + \beta_{3} ) q^{11} + ( -\beta_{4} + \beta_{5} + \beta_{6} - \beta_{11} ) q^{13} + ( -1 + \beta_{5} - \beta_{6} + \beta_{7} + \beta_{9} ) q^{14} + 4 \beta_{2} q^{16} + ( -2 + \beta_{4} - 2 \beta_{9} + \beta_{11} ) q^{17} + 3 \beta_{3} q^{18} + ( -2 \beta_{8} - 2 \beta_{10} + \beta_{12} - \beta_{15} ) q^{19} + ( 3 \beta_{9} - \beta_{11} ) q^{22} -5 \beta_{9} q^{25} + ( 2 \beta_{1} - \beta_{3} ) q^{26} + ( -2 \beta_{14} - 2 \beta_{15} ) q^{28} + ( 4 - 4 \beta_{2} - \beta_{4} + \beta_{6} ) q^{29} + ( -\beta_{1} + 4 \beta_{3} - 3 \beta_{8} - 4 \beta_{10} + \beta_{12} + 7 \beta_{13} + \beta_{15} ) q^{31} -4 \beta_{10} q^{32} + ( -2 \beta_{1} + 2 \beta_{3} + \beta_{8} - 2 \beta_{10} + 3 \beta_{13} - 2 \beta_{14} + 2 \beta_{15} ) q^{34} + 6 \beta_{9} q^{36} + ( 5 + 5 \beta_{2} + \beta_{4} - 2 \beta_{5} - \beta_{6} - 5 \beta_{9} + \beta_{11} ) q^{38} + ( 2 \beta_{1} - 2 \beta_{3} + 2 \beta_{10} - 2 \beta_{12} - 4 \beta_{13} + 2 \beta_{14} - 2 \beta_{15} ) q^{44} + ( 4 \beta_{3} + \beta_{10} - \beta_{12} - 5 \beta_{13} + \beta_{14} - \beta_{15} ) q^{47} + ( -6 - 13 \beta_{2} + \beta_{4} - 2 \beta_{5} - \beta_{6} + 6 \beta_{9} + \beta_{11} ) q^{49} + 5 \beta_{13} q^{50} -2 \beta_{11} q^{52} + ( -\beta_{2} + 2 \beta_{6} + \beta_{9} + 2 \beta_{11} ) q^{53} + ( -2 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} - 4 \beta_{6} + 2 \beta_{7} + 2 \beta_{11} ) q^{56} + ( -3 \beta_{8} + 3 \beta_{10} + 2 \beta_{12} + 2 \beta_{15} ) q^{58} + ( -3 \beta_{1} + 3 \beta_{3} - \beta_{10} + 3 \beta_{12} + 2 \beta_{13} - 3 \beta_{14} ) q^{59} + ( -3 - 3 \beta_{2} - 2 \beta_{4} + 2 \beta_{6} + 6 \beta_{7} + 3 \beta_{9} - 2 \beta_{11} ) q^{61} + ( -7 + 7 \beta_{2} - \beta_{4} + \beta_{6} ) q^{62} + ( 6 \beta_{1} - 6 \beta_{3} + 3 \beta_{10} - 3 \beta_{12} - 3 \beta_{13} + 3 \beta_{14} - 3 \beta_{15} ) q^{63} + 8 \beta_{7} q^{64} + ( -\beta_{10} + 2 \beta_{15} ) q^{67} + ( -8 - 4 \beta_{2} - 2 \beta_{4} + 2 \beta_{5} + 4 \beta_{7} + 4 \beta_{9} ) q^{68} + ( -\beta_{3} + \beta_{8} + 2 \beta_{10} - \beta_{13} - 3 \beta_{14} + 3 \beta_{15} ) q^{71} -6 \beta_{13} q^{72} + ( -2 \beta_{1} - 4 \beta_{8} - 4 \beta_{10} + 4 \beta_{13} + 2 \beta_{14} ) q^{76} + ( -5 + 2 \beta_{4} - 8 \beta_{9} - \beta_{11} ) q^{77} + ( -9 - 9 \beta_{2} + 9 \beta_{7} + 9 \beta_{9} ) q^{81} + ( -3 \beta_{1} - 2 \beta_{3} + 2 \beta_{8} - 3 \beta_{12} ) q^{83} + ( 6 + 2 \beta_{4} ) q^{88} + ( -7 \beta_{3} + \beta_{10} - \beta_{12} + 6 \beta_{13} + \beta_{14} - \beta_{15} ) q^{91} + ( 9 + \beta_{4} + 9 \beta_{9} + \beta_{11} ) q^{94} + ( -2 \beta_{1} + 7 \beta_{8} + 14 \beta_{10} - 7 \beta_{13} + 2 \beta_{14} ) q^{98} + ( -6 \beta_{3} + 6 \beta_{8} + 6 \beta_{10} - 6 \beta_{13} + 3 \beta_{14} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 8q^{4} - 12q^{9} + O(q^{10}) \) \( 16q + 8q^{4} - 12q^{9} - 8q^{14} - 16q^{16} - 40q^{17} + 12q^{22} - 20q^{25} + 80q^{29} + 24q^{36} + 40q^{38} - 20q^{49} + 8q^{53} + 16q^{56} - 140q^{62} + 32q^{64} - 80q^{68} - 112q^{77} - 36q^{81} + 96q^{88} + 180q^{94} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} + 12 x^{14} + 95 x^{12} + 552 x^{10} + 1969 x^{8} + 27048 x^{6} + 228095 x^{4} + 1411788 x^{2} + 5764801\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{12} - 3420 \nu^{2} \)\()/96481\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{11} + 17203 \nu \)\()/13783\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{10} - 15234 \)\()/1969\)
\(\beta_{5}\)\(=\)\((\)\( 216 \nu^{14} + 1710 \nu^{12} + 9936 \nu^{10} + 35442 \nu^{8} + 2644367 \nu^{6} + 4105710 \nu^{4} + 25412184 \nu^{2} + 103766418 \)\()/ 231650881 \)
\(\beta_{6}\)\(=\)\((\)\( 6 \nu^{12} + 117001 \nu^{2} \)\()/96481\)
\(\beta_{7}\)\(=\)\((\)\( 552 \nu^{14} + 6624 \nu^{12} + 52440 \nu^{10} + 187055 \nu^{8} + 1086888 \nu^{6} + 14930496 \nu^{4} + 125908440 \nu^{2} + 779306976 \)\()/ 231650881 \)
\(\beta_{8}\)\(=\)\((\)\( -5 \nu^{13} - 113581 \nu^{3} \)\()/675367\)
\(\beta_{9}\)\(=\)\((\)\( 12 \nu^{14} + 137521 \nu^{4} \)\()/4727569\)
\(\beta_{10}\)\(=\)\((\)\( -\nu^{15} - 47275 \nu^{5} \)\()/3008453\)
\(\beta_{11}\)\(=\)\((\)\( -23 \nu^{14} - 657546 \nu^{4} \)\()/4727569\)
\(\beta_{12}\)\(=\)\((\)\( 12 \nu^{13} + 137521 \nu^{3} \)\()/675367\)
\(\beta_{13}\)\(=\)\((\)\( 1356 \nu^{15} + 10735 \nu^{13} + 62376 \nu^{11} + 222497 \nu^{9} + 3731255 \nu^{7} + 25774735 \nu^{5} + 159532044 \nu^{3} + 651422513 \nu \)\()/ 1621556167 \)
\(\beta_{14}\)\(=\)\((\)\( -\nu^{15} - 12 \nu^{13} - 95 \nu^{11} - 552 \nu^{9} - 1969 \nu^{7} - 27048 \nu^{5} - 228095 \nu^{3} - 1411788 \nu \)\()/823543\)
\(\beta_{15}\)\(=\)\((\)\( 12 \nu^{15} + 137521 \nu^{5} \)\()/4727569\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{6} + 6 \beta_{2}\)
\(\nu^{3}\)\(=\)\(-5 \beta_{12} - 12 \beta_{8}\)
\(\nu^{4}\)\(=\)\(-12 \beta_{11} - 23 \beta_{9}\)
\(\nu^{5}\)\(=\)\(-11 \beta_{15} - 84 \beta_{10}\)
\(\nu^{6}\)\(=\)\(-18 \beta_{9} - 18 \beta_{7} + 95 \beta_{5} + 18 \beta_{2} + 18\)
\(\nu^{7}\)\(=\)\(-113 \beta_{15} + 113 \beta_{14} + 552 \beta_{13} - 113 \beta_{12} + 113 \beta_{10} - 113 \beta_{3} + 113 \beta_{1}\)
\(\nu^{8}\)\(=\)\(552 \beta_{11} + 1343 \beta_{7} - 552 \beta_{6} - 552 \beta_{5} + 552 \beta_{4}\)
\(\nu^{9}\)\(=\)\(-1895 \beta_{14} - 1969 \beta_{13} + 1969 \beta_{10} + 1969 \beta_{8} - 1969 \beta_{3}\)
\(\nu^{10}\)\(=\)\(-1969 \beta_{4} - 15234\)
\(\nu^{11}\)\(=\)\(13783 \beta_{3} - 17203 \beta_{1}\)
\(\nu^{12}\)\(=\)\(-3420 \beta_{6} - 117001 \beta_{2}\)
\(\nu^{13}\)\(=\)\(113581 \beta_{12} + 137521 \beta_{8}\)
\(\nu^{14}\)\(=\)\(137521 \beta_{11} + 657546 \beta_{9}\)
\(\nu^{15}\)\(=\)\(520025 \beta_{15} + 962647 \beta_{10}\)

Character values

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

\(n\) \(287\) \(353\) \(365\)
\(\chi(n)\) \(-1\) \(-1\) \(1 + \beta_{2} - \beta_{7} - \beta_{9}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
51.1
1.46034 + 2.20622i
−0.115343 2.64324i
0.115343 + 2.64324i
−1.46034 2.20622i
1.64697 2.07063i
−2.47822 + 0.926503i
2.47822 0.926503i
−1.64697 + 2.07063i
1.46034 2.20622i
−0.115343 + 2.64324i
0.115343 2.64324i
−1.46034 + 2.20622i
1.64697 + 2.07063i
−2.47822 0.926503i
2.47822 + 0.926503i
−1.64697 2.07063i
−0.831254 + 1.14412i 0 −0.618034 1.90211i 0 0 −1.76231 + 0.572610i 2.68999 + 0.874032i −2.42705 1.76336i 0
51.2 −0.831254 + 1.14412i 0 −0.618034 1.90211i 0 0 3.93856 1.27972i 2.68999 + 0.874032i −2.42705 1.76336i 0
51.3 0.831254 1.14412i 0 −0.618034 1.90211i 0 0 −3.93856 + 1.27972i −2.68999 0.874032i −2.42705 1.76336i 0
51.4 0.831254 1.14412i 0 −0.618034 1.90211i 0 0 1.76231 0.572610i −2.68999 0.874032i −2.42705 1.76336i 0
259.1 −1.34500 + 0.437016i 0 1.61803 1.17557i 0 0 −2.59357 3.56974i −1.66251 + 2.28825i 0.927051 + 2.85317i 0
259.2 −1.34500 + 0.437016i 0 1.61803 1.17557i 0 0 3.10731 + 4.27685i −1.66251 + 2.28825i 0.927051 + 2.85317i 0
259.3 1.34500 0.437016i 0 1.61803 1.17557i 0 0 −3.10731 4.27685i 1.66251 2.28825i 0.927051 + 2.85317i 0
259.4 1.34500 0.437016i 0 1.61803 1.17557i 0 0 2.59357 + 3.56974i 1.66251 2.28825i 0.927051 + 2.85317i 0
415.1 −0.831254 1.14412i 0 −0.618034 + 1.90211i 0 0 −1.76231 0.572610i 2.68999 0.874032i −2.42705 + 1.76336i 0
415.2 −0.831254 1.14412i 0 −0.618034 + 1.90211i 0 0 3.93856 + 1.27972i 2.68999 0.874032i −2.42705 + 1.76336i 0
415.3 0.831254 + 1.14412i 0 −0.618034 + 1.90211i 0 0 −3.93856 1.27972i −2.68999 + 0.874032i −2.42705 + 1.76336i 0
415.4 0.831254 + 1.14412i 0 −0.618034 + 1.90211i 0 0 1.76231 + 0.572610i −2.68999 + 0.874032i −2.42705 + 1.76336i 0
519.1 −1.34500 0.437016i 0 1.61803 + 1.17557i 0 0 −2.59357 + 3.56974i −1.66251 2.28825i 0.927051 2.85317i 0
519.2 −1.34500 0.437016i 0 1.61803 + 1.17557i 0 0 3.10731 4.27685i −1.66251 2.28825i 0.927051 2.85317i 0
519.3 1.34500 + 0.437016i 0 1.61803 + 1.17557i 0 0 −3.10731 + 4.27685i 1.66251 + 2.28825i 0.927051 2.85317i 0
519.4 1.34500 + 0.437016i 0 1.61803 + 1.17557i 0 0 2.59357 3.56974i 1.66251 + 2.28825i 0.927051 2.85317i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 519.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
52.b odd 2 1 CM by \(\Q(\sqrt{-13}) \)
4.b odd 2 1 inner
11.d odd 10 1 inner
13.b even 2 1 inner
44.g even 10 1 inner
143.l odd 10 1 inner
572.bb even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 572.2.bb.a 16
4.b odd 2 1 inner 572.2.bb.a 16
11.d odd 10 1 inner 572.2.bb.a 16
13.b even 2 1 inner 572.2.bb.a 16
44.g even 10 1 inner 572.2.bb.a 16
52.b odd 2 1 CM 572.2.bb.a 16
143.l odd 10 1 inner 572.2.bb.a 16
572.bb even 10 1 inner 572.2.bb.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
572.2.bb.a 16 1.a even 1 1 trivial
572.2.bb.a 16 4.b odd 2 1 inner
572.2.bb.a 16 11.d odd 10 1 inner
572.2.bb.a 16 13.b even 2 1 inner
572.2.bb.a 16 44.g even 10 1 inner
572.2.bb.a 16 52.b odd 2 1 CM
572.2.bb.a 16 143.l odd 10 1 inner
572.2.bb.a 16 572.bb even 10 1 inner

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 16 - 8 T^{2} + 4 T^{4} - 2 T^{6} + T^{8} )^{2} \)
$3$ \( T^{16} \)
$5$ \( T^{16} \)
$7$ \( 1026625681 - 525344236 T^{2} + 109681897 T^{4} - 5105368 T^{6} + 340350 T^{8} - 18092 T^{10} + 852 T^{12} - 4 T^{14} + T^{16} \)
$11$ \( 214358881 + 7086244 T^{2} - 1537305 T^{4} - 109384 T^{6} + 9089 T^{8} - 904 T^{10} - 105 T^{12} + 4 T^{14} + T^{16} \)
$13$ \( ( 28561 - 2197 T^{2} + 169 T^{4} - 13 T^{6} + T^{8} )^{2} \)
$17$ \( ( 73441 + 97560 T + 41452 T^{2} + 6180 T^{3} + 2294 T^{4} + 960 T^{5} + 193 T^{6} + 20 T^{7} + T^{8} )^{2} \)
$19$ \( 39213900625 - 6633837500 T^{2} + 897840500 T^{4} - 117831000 T^{6} + 16793150 T^{8} - 380400 T^{10} + 4745 T^{12} - 40 T^{14} + T^{16} \)
$23$ \( T^{16} \)
$29$ \( ( 398161 + 151440 T + 111532 T^{2} - 140040 T^{3} + 46454 T^{4} - 7680 T^{5} + 733 T^{6} - 40 T^{7} + T^{8} )^{2} \)
$31$ \( 4432713370801 + 2505873535012 T^{2} + 542853268873 T^{4} + 1250230024 T^{6} + 135509550 T^{8} + 1515524 T^{10} + 13188 T^{12} + 52 T^{14} + T^{16} \)
$37$ \( T^{16} \)
$41$ \( T^{16} \)
$43$ \( T^{16} \)
$47$ \( 991199501189041 + 145497397469252 T^{2} + 8335584423913 T^{4} + 36044161864 T^{6} + 693153390 T^{8} + 5487524 T^{10} + 33988 T^{12} + 52 T^{14} + T^{16} \)
$53$ \( ( 169026001 + 6682514 T + 3821757 T^{2} - 49948 T^{3} + 24990 T^{4} + 498 T^{5} + 12 T^{6} - 4 T^{7} + T^{8} )^{2} \)
$59$ \( 96101965591921 + 17025620260428 T^{2} + 1266251168228 T^{4} + 32614648736 T^{6} + 336772590 T^{8} - 420544 T^{10} + 44793 T^{12} - 112 T^{14} + T^{16} \)
$61$ \( ( 2468041 - 2874930 T + 2131453 T^{2} - 380640 T^{3} + 17034 T^{4} + 1830 T^{5} - 208 T^{6} + T^{8} )^{2} \)
$67$ \( ( -26 + T^{2} )^{8} \)
$71$ \( 93429255506161 + 37957798696428 T^{2} + 5914585455188 T^{4} + 12070779896 T^{6} + 1070774430 T^{8} + 11182256 T^{10} + 58833 T^{12} + 8 T^{14} + T^{16} \)
$73$ \( T^{16} \)
$79$ \( T^{16} \)
$83$ \( 130362526004881 - 218360146267364 T^{2} + 139658310844897 T^{4} + 139318106168 T^{6} + 7277184750 T^{8} - 26538308 T^{10} + 85252 T^{12} - 196 T^{14} + T^{16} \)
$89$ \( T^{16} \)
$97$ \( T^{16} \)
show more
show less