# Properties

 Label 2144.1.bu.a Level $2144$ Weight $1$ Character orbit 2144.bu Analytic conductor $1.070$ Analytic rank $0$ Dimension $20$ Projective image $D_{33}$ CM discriminant -8 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2144 = 2^{5} \cdot 67$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2144.bu (of order $$66$$, degree $$20$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.06999538709$$ Analytic rank: $$0$$ Dimension: $$20$$ Coefficient field: $$\Q(\zeta_{33})$$ Defining polynomial: $$x^{20} - x^{19} + x^{17} - x^{16} + x^{14} - x^{13} + x^{11} - x^{10} + x^{9} - x^{7} + x^{6} - x^{4} + x^{3} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 536) Projective image: $$D_{33}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{33} - \cdots)$$

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{66}^{8} + \zeta_{66}^{13} ) q^{3} + ( \zeta_{66}^{16} - \zeta_{66}^{21} + \zeta_{66}^{26} ) q^{9} +O(q^{10})$$ $$q + ( -\zeta_{66}^{8} + \zeta_{66}^{13} ) q^{3} + ( \zeta_{66}^{16} - \zeta_{66}^{21} + \zeta_{66}^{26} ) q^{9} + ( -\zeta_{66}^{4} - \zeta_{66}^{10} ) q^{11} + ( -\zeta_{66}^{15} + \zeta_{66}^{22} ) q^{17} + ( \zeta_{66}^{19} + \zeta_{66}^{27} ) q^{19} -\zeta_{66}^{3} q^{25} + ( \zeta_{66} - \zeta_{66}^{6} - \zeta_{66}^{24} + \zeta_{66}^{29} ) q^{27} + ( \zeta_{66}^{12} - \zeta_{66}^{17} + \zeta_{66}^{18} - \zeta_{66}^{23} ) q^{33} + ( -\zeta_{66} - \zeta_{66}^{25} ) q^{41} + ( \zeta_{66}^{17} + \zeta_{66}^{31} ) q^{43} + \zeta_{66}^{20} q^{49} + ( -\zeta_{66}^{2} + \zeta_{66}^{23} - \zeta_{66}^{28} - \zeta_{66}^{30} ) q^{51} + ( \zeta_{66}^{2} - \zeta_{66}^{7} - \zeta_{66}^{27} + \zeta_{66}^{32} ) q^{57} + ( \zeta_{66}^{5} - \zeta_{66}^{22} ) q^{59} -\zeta_{66}^{18} q^{67} + ( \zeta_{66}^{24} + \zeta_{66}^{28} ) q^{73} + ( \zeta_{66}^{11} - \zeta_{66}^{16} ) q^{75} + ( \zeta_{66}^{4} - \zeta_{66}^{9} + \zeta_{66}^{14} - \zeta_{66}^{19} + \zeta_{66}^{32} ) q^{81} + ( \zeta_{66}^{11} - \zeta_{66}^{14} ) q^{83} + ( -\zeta_{66}^{5} - \zeta_{66}^{7} ) q^{89} + ( -\zeta_{66}^{9} - \zeta_{66}^{13} ) q^{97} + ( \zeta_{66}^{3} - \zeta_{66}^{20} + \zeta_{66}^{25} - \zeta_{66}^{26} - \zeta_{66}^{30} + \zeta_{66}^{31} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20q - 2q^{3} + O(q^{10})$$ $$20q - 2q^{3} - 2q^{11} - 12q^{17} + q^{19} - 2q^{25} + 2q^{27} - 2q^{33} + 2q^{41} - 2q^{43} + q^{49} - q^{51} + q^{57} + 9q^{59} + 2q^{67} - q^{73} + 9q^{75} + 2q^{81} + 9q^{83} + 2q^{89} - q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2144\mathbb{Z}\right)^\times$$.

 $$n$$ $$671$$ $$805$$ $$1409$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-\zeta_{66}^{31}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
47.1
 0.723734 + 0.690079i −0.327068 − 0.945001i 0.235759 + 0.971812i 0.580057 − 0.814576i 0.235759 − 0.971812i 0.928368 − 0.371662i −0.995472 − 0.0950560i 0.580057 + 0.814576i −0.888835 − 0.458227i −0.786053 − 0.618159i −0.888835 + 0.458227i 0.981929 − 0.189251i 0.723734 − 0.690079i 0.928368 + 0.371662i −0.995472 + 0.0950560i −0.786053 + 0.618159i 0.0475819 − 0.998867i 0.0475819 + 0.998867i −0.327068 + 0.945001i 0.981929 + 0.189251i
0 −0.0930932 + 0.647478i 0 0 0 0 0 0.548932 + 0.161181i 0
207.1 0 −0.0395325 + 0.0865641i 0 0 0 0 0 0.648930 + 0.748905i 0
303.1 0 0.279486 + 1.94387i 0 0 0 0 0 −2.74102 + 0.804835i 0
495.1 0 −1.21769 + 0.782560i 0 0 0 0 0 0.454947 0.996196i 0
559.1 0 0.279486 1.94387i 0 0 0 0 0 −2.74102 0.804835i 0
591.1 0 0.759713 0.876756i 0 0 0 0 0 −0.0492216 0.342344i 0
687.1 0 −0.396666 + 0.254922i 0 0 0 0 0 −0.323056 + 0.707394i 0
719.1 0 −1.21769 0.782560i 0 0 0 0 0 0.454947 + 0.996196i 0
1199.1 0 1.78153 + 0.523103i 0 0 0 0 0 2.05894 + 1.32320i 0
1327.1 0 −1.30379 + 1.50465i 0 0 0 0 0 −0.421801 2.93369i 0
1359.1 0 1.78153 0.523103i 0 0 0 0 0 2.05894 1.32320i 0
1423.1 0 0.738471 + 1.61703i 0 0 0 0 0 −1.41457 + 1.63251i 0
1551.1 0 −0.0930932 0.647478i 0 0 0 0 0 0.548932 0.161181i 0
1647.1 0 0.759713 + 0.876756i 0 0 0 0 0 −0.0492216 + 0.342344i 0
1679.1 0 −0.396666 0.254922i 0 0 0 0 0 −0.323056 0.707394i 0
1711.1 0 −1.30379 1.50465i 0 0 0 0 0 −0.421801 + 2.93369i 0
1775.1 0 −1.50842 + 0.442913i 0 0 0 0 0 1.23792 0.795563i 0
1807.1 0 −1.50842 0.442913i 0 0 0 0 0 1.23792 + 0.795563i 0
1999.1 0 −0.0395325 0.0865641i 0 0 0 0 0 0.648930 0.748905i 0
2031.1 0 0.738471 1.61703i 0 0 0 0 0 −1.41457 1.63251i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2031.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$
67.g even 33 1 inner
536.ba odd 66 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2144.1.bu.a 20
4.b odd 2 1 536.1.ba.a 20
8.b even 2 1 536.1.ba.a 20
8.d odd 2 1 CM 2144.1.bu.a 20
67.g even 33 1 inner 2144.1.bu.a 20
268.o odd 66 1 536.1.ba.a 20
536.ba odd 66 1 inner 2144.1.bu.a 20
536.bf even 66 1 536.1.ba.a 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
536.1.ba.a 20 4.b odd 2 1
536.1.ba.a 20 8.b even 2 1
536.1.ba.a 20 268.o odd 66 1
536.1.ba.a 20 536.bf even 66 1
2144.1.bu.a 20 1.a even 1 1 trivial
2144.1.bu.a 20 8.d odd 2 1 CM
2144.1.bu.a 20 67.g even 33 1 inner
2144.1.bu.a 20 536.ba odd 66 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(2144, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{20}$$
$3$ $$1 + 13 T + 157 T^{2} + 565 T^{3} + 1149 T^{4} + 1491 T^{5} + 1613 T^{6} + 767 T^{7} + 768 T^{8} + 1011 T^{9} + 528 T^{10} + 43 T^{11} + 31 T^{12} + 8 T^{13} - 37 T^{14} - 16 T^{15} + 5 T^{16} + 4 T^{17} + 3 T^{18} + 2 T^{19} + T^{20}$$
$5$ $$T^{20}$$
$7$ $$T^{20}$$
$11$ $$1 - 5 T + 11 T^{2} - 62 T^{3} + 178 T^{4} - 77 T^{5} + 49 T^{6} + 19 T^{7} - 154 T^{8} + 130 T^{9} + 164 T^{10} + 49 T^{11} + 11 T^{12} + 40 T^{13} + 42 T^{14} + 11 T^{15} - 5 T^{16} - 8 T^{17} + 2 T^{19} + T^{20}$$
$13$ $$T^{20}$$
$17$ $$1 + 23 T + 198 T^{2} + 615 T^{3} + 967 T^{4} + 1705 T^{5} + 5193 T^{6} + 12860 T^{7} + 22352 T^{8} + 29402 T^{9} + 31107 T^{10} + 27356 T^{11} + 20317 T^{12} + 12816 T^{13} + 6854 T^{14} + 3080 T^{15} + 1143 T^{16} + 340 T^{17} + 77 T^{18} + 12 T^{19} + T^{20}$$
$19$ $$1 - 12 T + 44 T^{2} + 375 T^{3} + 725 T^{4} + 1100 T^{5} + 1563 T^{6} + 604 T^{7} + 11 T^{8} - 10 T^{9} + 120 T^{10} - 109 T^{11} - 33 T^{12} + 98 T^{13} + 12 T^{14} - T^{16} + T^{17} - T^{19} + T^{20}$$
$23$ $$T^{20}$$
$29$ $$T^{20}$$
$31$ $$T^{20}$$
$37$ $$T^{20}$$
$41$ $$1 + 5 T + 22 T^{2} + 29 T^{3} + 24 T^{4} - 231 T^{5} + 71 T^{6} - 8 T^{7} + 198 T^{8} + 24 T^{9} - 243 T^{10} + 116 T^{11} + 121 T^{12} - 95 T^{13} + 20 T^{14} + 22 T^{15} - 16 T^{16} + 8 T^{17} - 2 T^{19} + T^{20}$$
$43$ $$1 - 20 T + 113 T^{2} - 95 T^{3} + 544 T^{4} + 457 T^{5} + 832 T^{6} + 1438 T^{7} + 1802 T^{8} + 1198 T^{9} + 836 T^{10} + 472 T^{11} + 251 T^{12} + 118 T^{13} + 84 T^{14} + 50 T^{15} + 16 T^{16} + 4 T^{17} + 3 T^{18} + 2 T^{19} + T^{20}$$
$47$ $$T^{20}$$
$53$ $$T^{20}$$
$59$ $$1 + 13 T + 36 T^{2} - 381 T^{3} + 742 T^{4} - 874 T^{5} + 1965 T^{6} - 3578 T^{7} + 5069 T^{8} - 6194 T^{9} + 6633 T^{10} - 6194 T^{11} + 5047 T^{12} - 3567 T^{13} + 2174 T^{14} - 1127 T^{15} + 489 T^{16} - 172 T^{17} + 47 T^{18} - 9 T^{19} + T^{20}$$
$61$ $$T^{20}$$
$67$ $$( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2}$$
$71$ $$T^{20}$$
$73$ $$1 - 10 T + 88 T^{2} + 197 T^{3} + 384 T^{4} + 792 T^{5} + 188 T^{6} - 1451 T^{7} - 792 T^{8} + 417 T^{9} + 483 T^{10} + 65 T^{11} + 88 T^{12} + 34 T^{13} - 43 T^{14} - 44 T^{15} - T^{16} - T^{17} + T^{19} + T^{20}$$
$79$ $$T^{20}$$
$83$ $$1 - 5 T + 15 T^{3} + 145 T^{4} - 616 T^{5} + 1116 T^{6} - 1631 T^{7} + 2453 T^{8} - 3302 T^{9} + 3750 T^{10} - 3713 T^{11} + 3223 T^{12} - 2424 T^{13} + 1571 T^{14} - 869 T^{15} + 402 T^{16} - 151 T^{17} + 44 T^{18} - 9 T^{19} + T^{20}$$
$89$ $$1 + 20 T + 113 T^{2} + 95 T^{3} + 544 T^{4} - 457 T^{5} + 832 T^{6} - 1438 T^{7} + 1802 T^{8} - 1198 T^{9} + 836 T^{10} - 472 T^{11} + 251 T^{12} - 118 T^{13} + 84 T^{14} - 50 T^{15} + 16 T^{16} - 4 T^{17} + 3 T^{18} - 2 T^{19} + T^{20}$$
$97$ $$1 + 12 T + 132 T^{2} + 230 T^{3} + 703 T^{4} + 550 T^{5} + 2025 T^{6} + 1431 T^{7} + 2673 T^{8} + 1220 T^{9} + 1935 T^{10} + 714 T^{11} + 968 T^{12} + 254 T^{13} + 320 T^{14} + 66 T^{15} + 76 T^{16} + 10 T^{17} + 11 T^{18} + T^{19} + T^{20}$$