# 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

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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

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