# Properties

 Label 1620.1.bp.b Level $1620$ Weight $1$ Character orbit 1620.bp Analytic conductor $0.808$ Analytic rank $0$ Dimension $18$ Projective image $D_{27}$ CM discriminant -20 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1620 = 2^{2} \cdot 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1620.bp (of order $$54$$, degree $$18$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.808485320465$$ Analytic rank: $$0$$ Dimension: $$18$$ Coefficient field: $$\Q(\zeta_{54})$$ Defining polynomial: $$x^{18} - x^{9} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{27}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{27} - \cdots)$$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\zeta_{54}^{25} q^{2} + \zeta_{54}^{26} q^{3} -\zeta_{54}^{23} q^{4} + \zeta_{54}^{8} q^{5} + \zeta_{54}^{24} q^{6} + ( -\zeta_{54}^{5} + \zeta_{54}^{12} ) q^{7} -\zeta_{54}^{21} q^{8} -\zeta_{54}^{25} q^{9} +O(q^{10})$$ $$q -\zeta_{54}^{25} q^{2} + \zeta_{54}^{26} q^{3} -\zeta_{54}^{23} q^{4} + \zeta_{54}^{8} q^{5} + \zeta_{54}^{24} q^{6} + ( -\zeta_{54}^{5} + \zeta_{54}^{12} ) q^{7} -\zeta_{54}^{21} q^{8} -\zeta_{54}^{25} q^{9} + \zeta_{54}^{6} q^{10} + \zeta_{54}^{22} q^{12} + ( -\zeta_{54}^{3} + \zeta_{54}^{10} ) q^{14} -\zeta_{54}^{7} q^{15} -\zeta_{54}^{19} q^{16} -\zeta_{54}^{23} q^{18} + \zeta_{54}^{4} q^{20} + ( \zeta_{54}^{4} - \zeta_{54}^{11} ) q^{21} + ( 1 + \zeta_{54}^{10} ) q^{23} + \zeta_{54}^{20} q^{24} + \zeta_{54}^{16} q^{25} + \zeta_{54}^{24} q^{27} + ( -\zeta_{54} + \zeta_{54}^{8} ) q^{28} + ( \zeta_{54}^{20} - \zeta_{54}^{21} ) q^{29} -\zeta_{54}^{5} q^{30} -\zeta_{54}^{17} q^{32} + ( -\zeta_{54}^{13} + \zeta_{54}^{20} ) q^{35} -\zeta_{54}^{21} q^{36} + \zeta_{54}^{2} q^{40} + ( -\zeta_{54}^{13} + \zeta_{54}^{18} ) q^{41} + ( \zeta_{54}^{2} - \zeta_{54}^{9} ) q^{42} + ( \zeta_{54}^{4} + \zeta_{54}^{16} ) q^{43} + \zeta_{54}^{6} q^{45} + ( \zeta_{54}^{8} - \zeta_{54}^{25} ) q^{46} + ( -\zeta_{54}^{11} - \zeta_{54}^{15} ) q^{47} + \zeta_{54}^{18} q^{48} + ( \zeta_{54}^{10} - \zeta_{54}^{17} + \zeta_{54}^{24} ) q^{49} + \zeta_{54}^{14} q^{50} + \zeta_{54}^{22} q^{54} + ( \zeta_{54}^{6} + \zeta_{54}^{26} ) q^{56} + ( \zeta_{54}^{18} - \zeta_{54}^{19} ) q^{58} -\zeta_{54}^{3} q^{60} + ( -\zeta_{54}^{3} - \zeta_{54}^{5} ) q^{61} + ( -\zeta_{54}^{3} + \zeta_{54}^{10} ) q^{63} -\zeta_{54}^{15} q^{64} + ( 1 - \zeta_{54}^{13} ) q^{67} + ( -\zeta_{54}^{9} + \zeta_{54}^{26} ) q^{69} + ( -\zeta_{54}^{11} + \zeta_{54}^{18} ) q^{70} -\zeta_{54}^{19} q^{72} -\zeta_{54}^{15} q^{75} + q^{80} -\zeta_{54}^{23} q^{81} + ( -\zeta_{54}^{11} + \zeta_{54}^{16} ) q^{82} + ( -\zeta_{54}^{9} + \zeta_{54}^{14} ) q^{83} + ( 1 - \zeta_{54}^{7} ) q^{84} + ( \zeta_{54}^{2} + \zeta_{54}^{14} ) q^{86} + ( -\zeta_{54}^{19} + \zeta_{54}^{20} ) q^{87} + ( -\zeta_{54}^{19} - \zeta_{54}^{23} ) q^{89} + \zeta_{54}^{4} q^{90} + ( \zeta_{54}^{6} - \zeta_{54}^{23} ) q^{92} + ( -\zeta_{54}^{9} - \zeta_{54}^{13} ) q^{94} + \zeta_{54}^{16} q^{96} + ( \zeta_{54}^{8} - \zeta_{54}^{15} + \zeta_{54}^{22} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$18q + O(q^{10})$$ $$18q + 18q^{23} - 9q^{41} - 9q^{42} - 9q^{48} - 9q^{58} + 18q^{67} - 9q^{69} - 9q^{70} + 18q^{80} - 9q^{83} + 18q^{84} - 9q^{94} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times$$.

 $$n$$ $$811$$ $$1297$$ $$1541$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-\zeta_{54}^{23}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
79.1
 0.686242 − 0.727374i −0.893633 − 0.448799i 0.993238 − 0.116093i 0.993238 + 0.116093i 0.835488 + 0.549509i −0.973045 + 0.230616i 0.286803 + 0.957990i 0.835488 − 0.549509i −0.396080 + 0.918216i −0.597159 + 0.802123i −0.893633 + 0.448799i 0.286803 − 0.957990i −0.973045 − 0.230616i 0.0581448 + 0.998308i −0.597159 − 0.802123i −0.396080 − 0.918216i 0.0581448 − 0.998308i 0.686242 + 0.727374i
−0.0581448 + 0.998308i −0.686242 0.727374i −0.993238 0.116093i 0.973045 0.230616i 0.766044 0.642788i −0.342534 0.460103i 0.173648 0.984808i −0.0581448 + 0.998308i 0.173648 + 0.984808i
139.1 0.597159 0.802123i 0.893633 0.448799i −0.286803 0.957990i −0.835488 0.549509i 0.173648 0.984808i 0.0798028 + 0.0845860i −0.939693 0.342020i 0.597159 0.802123i −0.939693 + 0.342020i
259.1 0.973045 + 0.230616i −0.993238 0.116093i 0.893633 + 0.448799i 0.597159 0.802123i −0.939693 0.342020i −0.661840 0.435299i 0.766044 + 0.642788i 0.973045 + 0.230616i 0.766044 0.642788i
319.1 0.973045 0.230616i −0.993238 + 0.116093i 0.893633 0.448799i 0.597159 + 0.802123i −0.939693 + 0.342020i −0.661840 + 0.435299i 0.766044 0.642788i 0.973045 0.230616i 0.766044 + 0.642788i
439.1 0.396080 0.918216i −0.835488 + 0.549509i −0.686242 0.727374i −0.0581448 0.998308i 0.173648 + 0.984808i 1.73909 + 0.412172i −0.939693 + 0.342020i 0.396080 0.918216i −0.939693 0.342020i
499.1 0.893633 + 0.448799i 0.973045 + 0.230616i 0.597159 + 0.802123i −0.286803 0.957990i 0.766044 + 0.642788i −0.543613 1.26024i 0.173648 + 0.984808i 0.893633 + 0.448799i 0.173648 0.984808i
619.1 −0.835488 0.549509i −0.286803 + 0.957990i 0.396080 + 0.918216i −0.686242 0.727374i 0.766044 0.642788i −1.93293 + 0.225927i 0.173648 0.984808i −0.835488 0.549509i 0.173648 + 0.984808i
679.1 0.396080 + 0.918216i −0.835488 0.549509i −0.686242 + 0.727374i −0.0581448 + 0.998308i 0.173648 0.984808i 1.73909 0.412172i −0.939693 0.342020i 0.396080 + 0.918216i −0.939693 + 0.342020i
799.1 −0.686242 + 0.727374i 0.396080 + 0.918216i −0.0581448 0.998308i −0.993238 0.116093i −0.939693 0.342020i 1.06728 0.536009i 0.766044 + 0.642788i −0.686242 + 0.727374i 0.766044 0.642788i
859.1 −0.286803 + 0.957990i 0.597159 + 0.802123i −0.835488 0.549509i 0.396080 0.918216i −0.939693 + 0.342020i 0.115503 + 1.98312i 0.766044 0.642788i −0.286803 + 0.957990i 0.766044 + 0.642788i
979.1 0.597159 + 0.802123i 0.893633 + 0.448799i −0.286803 + 0.957990i −0.835488 + 0.549509i 0.173648 + 0.984808i 0.0798028 0.0845860i −0.939693 + 0.342020i 0.597159 + 0.802123i −0.939693 0.342020i
1039.1 −0.835488 + 0.549509i −0.286803 0.957990i 0.396080 0.918216i −0.686242 + 0.727374i 0.766044 + 0.642788i −1.93293 0.225927i 0.173648 + 0.984808i −0.835488 + 0.549509i 0.173648 0.984808i
1159.1 0.893633 0.448799i 0.973045 0.230616i 0.597159 0.802123i −0.286803 + 0.957990i 0.766044 0.642788i −0.543613 + 1.26024i 0.173648 0.984808i 0.893633 0.448799i 0.173648 + 0.984808i
1219.1 −0.993238 0.116093i −0.0581448 + 0.998308i 0.973045 + 0.230616i 0.893633 0.448799i 0.173648 0.984808i 0.479241 1.60078i −0.939693 0.342020i −0.993238 0.116093i −0.939693 + 0.342020i
1339.1 −0.286803 0.957990i 0.597159 0.802123i −0.835488 + 0.549509i 0.396080 + 0.918216i −0.939693 0.342020i 0.115503 1.98312i 0.766044 + 0.642788i −0.286803 0.957990i 0.766044 0.642788i
1399.1 −0.686242 0.727374i 0.396080 0.918216i −0.0581448 + 0.998308i −0.993238 + 0.116093i −0.939693 + 0.342020i 1.06728 + 0.536009i 0.766044 0.642788i −0.686242 0.727374i 0.766044 + 0.642788i
1519.1 −0.993238 + 0.116093i −0.0581448 0.998308i 0.973045 0.230616i 0.893633 + 0.448799i 0.173648 + 0.984808i 0.479241 + 1.60078i −0.939693 + 0.342020i −0.993238 + 0.116093i −0.939693 0.342020i
1579.1 −0.0581448 0.998308i −0.686242 + 0.727374i −0.993238 + 0.116093i 0.973045 + 0.230616i 0.766044 + 0.642788i −0.342534 + 0.460103i 0.173648 + 0.984808i −0.0581448 0.998308i 0.173648 0.984808i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1579.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 CM by $$\Q(\sqrt{-5})$$
81.g even 27 1 inner
1620.bp odd 54 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1620.1.bp.b yes 18
4.b odd 2 1 1620.1.bp.a 18
5.b even 2 1 1620.1.bp.a 18
20.d odd 2 1 CM 1620.1.bp.b yes 18
81.g even 27 1 inner 1620.1.bp.b yes 18
324.n odd 54 1 1620.1.bp.a 18
405.t even 54 1 1620.1.bp.a 18
1620.bp odd 54 1 inner 1620.1.bp.b yes 18

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1620.1.bp.a 18 4.b odd 2 1
1620.1.bp.a 18 5.b even 2 1
1620.1.bp.a 18 324.n odd 54 1
1620.1.bp.a 18 405.t even 54 1
1620.1.bp.b yes 18 1.a even 1 1 trivial
1620.1.bp.b yes 18 20.d odd 2 1 CM
1620.1.bp.b yes 18 81.g even 27 1 inner
1620.1.bp.b yes 18 1620.bp odd 54 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{18} + \cdots$$ acting on $$S_{1}^{\mathrm{new}}(1620, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{9} + T^{18}$$
$3$ $$1 + T^{9} + T^{18}$$
$5$ $$1 + T^{9} + T^{18}$$
$7$ $$1 - 9 T + 45 T^{2} + 159 T^{3} + 297 T^{4} + 63 T^{5} - 159 T^{6} - 135 T^{7} + 135 T^{8} + 8 T^{9} + 99 T^{10} - 9 T^{11} + 6 T^{12} - 18 T^{14} + 3 T^{15} + T^{18}$$
$11$ $$T^{18}$$
$13$ $$T^{18}$$
$17$ $$T^{18}$$
$19$ $$T^{18}$$
$23$ $$1 - 9 T + 117 T^{2} - 732 T^{3} + 2934 T^{4} - 8442 T^{5} + 18480 T^{6} - 31788 T^{7} + 43749 T^{8} - 48619 T^{9} + 43758 T^{10} - 31824 T^{11} + 18564 T^{12} - 8568 T^{13} + 3060 T^{14} - 816 T^{15} + 153 T^{16} - 18 T^{17} + T^{18}$$
$29$ $$1 - 9 T + 45 T^{2} + 159 T^{3} + 297 T^{4} + 63 T^{5} - 159 T^{6} - 135 T^{7} + 135 T^{8} + 8 T^{9} + 99 T^{10} - 9 T^{11} + 6 T^{12} - 18 T^{14} + 3 T^{15} + T^{18}$$
$31$ $$T^{18}$$
$37$ $$T^{18}$$
$41$ $$1 - 9 T + 9 T^{2} + 240 T^{3} + 666 T^{4} + 1008 T^{5} + 1470 T^{6} + 2232 T^{7} + 2898 T^{8} + 3140 T^{9} + 2907 T^{10} + 2304 T^{11} + 1554 T^{12} + 882 T^{13} + 414 T^{14} + 156 T^{15} + 45 T^{16} + 9 T^{17} + T^{18}$$
$43$ $$1 - 36 T^{3} + 549 T^{6} + 323 T^{9} + 72 T^{12} + 9 T^{15} + T^{18}$$
$47$ $$1 + 9 T + 90 T^{2} + 168 T^{3} + 72 T^{4} - 135 T^{5} - 150 T^{6} - 135 T^{7} + 297 T^{8} + 8 T^{9} - 72 T^{10} - 18 T^{11} + 6 T^{12} + 18 T^{13} + 3 T^{15} + T^{18}$$
$53$ $$T^{18}$$
$59$ $$T^{18}$$
$61$ $$1 + 18 T + 99 T^{2} + 159 T^{3} + 297 T^{4} + 63 T^{5} + 84 T^{6} - 216 T^{7} - 108 T^{8} + 8 T^{9} + 72 T^{10} - 63 T^{11} + 6 T^{12} + 9 T^{14} + 3 T^{15} + T^{18}$$
$67$ $$1 - 9 T + 117 T^{2} - 732 T^{3} + 2934 T^{4} - 8442 T^{5} + 18480 T^{6} - 31788 T^{7} + 43749 T^{8} - 48619 T^{9} + 43758 T^{10} - 31824 T^{11} + 18564 T^{12} - 8568 T^{13} + 3060 T^{14} - 816 T^{15} + 153 T^{16} - 18 T^{17} + T^{18}$$
$71$ $$T^{18}$$
$73$ $$T^{18}$$
$79$ $$T^{18}$$
$83$ $$1 - 9 T + 9 T^{2} + 240 T^{3} + 666 T^{4} + 1008 T^{5} + 1470 T^{6} + 2232 T^{7} + 2898 T^{8} + 3140 T^{9} + 2907 T^{10} + 2304 T^{11} + 1554 T^{12} + 882 T^{13} + 414 T^{14} + 156 T^{15} + 45 T^{16} + 9 T^{17} + T^{18}$$
$89$ $$1 + 9 T + 108 T^{2} + 516 T^{3} + 1278 T^{4} + 1782 T^{5} + 1386 T^{6} + 540 T^{7} + 81 T^{8} + 2 T^{9} + 9 T^{10} + 27 T^{11} + 30 T^{12} + 9 T^{13} + T^{18}$$
$97$ $$T^{18}$$