Properties

Label 648.1.bf.a
Level $648$
Weight $1$
Character orbit 648.bf
Analytic conductor $0.323$
Analytic rank $0$
Dimension $18$
Projective image $D_{27}$
CM discriminant -8
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 648 = 2^{3} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 648.bf (of order \(54\), degree \(18\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.323394128186\)
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.

\(f(q)\) \(=\) \( q -\zeta_{54}^{7} q^{2} -\zeta_{54}^{19} q^{3} + \zeta_{54}^{14} q^{4} + \zeta_{54}^{26} q^{6} -\zeta_{54}^{21} q^{8} -\zeta_{54}^{11} q^{9} +O(q^{10})\) \( q -\zeta_{54}^{7} q^{2} -\zeta_{54}^{19} q^{3} + \zeta_{54}^{14} q^{4} + \zeta_{54}^{26} q^{6} -\zeta_{54}^{21} q^{8} -\zeta_{54}^{11} q^{9} + ( -\zeta_{54}^{23} + \zeta_{54}^{24} ) q^{11} + \zeta_{54}^{6} q^{12} -\zeta_{54} q^{16} + ( -\zeta_{54}^{13} - \zeta_{54}^{17} ) q^{17} + \zeta_{54}^{18} q^{18} + ( \zeta_{54}^{2} + \zeta_{54}^{22} ) q^{19} + ( -\zeta_{54}^{3} + \zeta_{54}^{4} ) q^{22} -\zeta_{54}^{13} q^{24} -\zeta_{54}^{25} q^{25} -\zeta_{54}^{3} q^{27} + \zeta_{54}^{8} q^{32} + ( -\zeta_{54}^{15} + \zeta_{54}^{16} ) q^{33} + ( \zeta_{54}^{20} + \zeta_{54}^{24} ) q^{34} -\zeta_{54}^{25} q^{36} + ( \zeta_{54}^{2} - \zeta_{54}^{9} ) q^{38} + ( -\zeta_{54} + \zeta_{54}^{12} ) q^{41} + ( \zeta_{54}^{12} + \zeta_{54}^{26} ) q^{43} + ( \zeta_{54}^{10} - \zeta_{54}^{11} ) q^{44} + \zeta_{54}^{20} q^{48} + \zeta_{54}^{8} q^{49} -\zeta_{54}^{5} q^{50} + ( -\zeta_{54}^{5} - \zeta_{54}^{9} ) q^{51} + \zeta_{54}^{10} q^{54} + ( \zeta_{54}^{14} - \zeta_{54}^{21} ) q^{57} + ( \zeta_{54}^{16} + \zeta_{54}^{18} ) q^{59} -\zeta_{54}^{15} q^{64} + ( \zeta_{54}^{22} - \zeta_{54}^{23} ) q^{66} + ( -\zeta_{54}^{7} - \zeta_{54}^{15} ) q^{67} + ( 1 + \zeta_{54}^{4} ) q^{68} -\zeta_{54}^{5} q^{72} + ( -\zeta_{54}^{5} + \zeta_{54}^{10} ) q^{73} -\zeta_{54}^{17} q^{75} + ( -\zeta_{54}^{9} + \zeta_{54}^{16} ) q^{76} + \zeta_{54}^{22} q^{81} + ( \zeta_{54}^{8} - \zeta_{54}^{19} ) q^{82} + ( \zeta_{54}^{4} + \zeta_{54}^{10} ) q^{83} + ( \zeta_{54}^{6} - \zeta_{54}^{19} ) q^{86} + ( -\zeta_{54}^{17} + \zeta_{54}^{18} ) q^{88} + ( \zeta_{54}^{6} - \zeta_{54}^{9} ) q^{89} + q^{96} + ( \zeta_{54}^{6} - \zeta_{54}^{23} ) q^{97} -\zeta_{54}^{15} q^{98} + ( -\zeta_{54}^{7} + \zeta_{54}^{8} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18q + O(q^{10}) \) \( 18q - 9q^{18} - 9q^{38} - 9q^{51} - 9q^{59} + 18q^{68} - 9q^{76} - 9q^{88} - 9q^{89} + 18q^{96} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/648\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(487\) \(569\)
\(\chi(n)\) \(-1\) \(-1\) \(\zeta_{54}^{14}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1
−0.396080 + 0.918216i
0.686242 + 0.727374i
0.0581448 0.998308i
0.835488 0.549509i
0.286803 + 0.957990i
−0.396080 0.918216i
−0.597159 0.802123i
−0.973045 + 0.230616i
0.835488 + 0.549509i
0.0581448 + 0.998308i
−0.973045 0.230616i
0.993238 + 0.116093i
0.993238 0.116093i
0.286803 0.957990i
−0.893633 + 0.448799i
−0.893633 0.448799i
0.686242 0.727374i
−0.597159 + 0.802123i
−0.286803 0.957990i −0.993238 + 0.116093i −0.835488 + 0.549509i 0 0.396080 + 0.918216i 0 0.766044 + 0.642788i 0.973045 0.230616i 0
67.1 −0.835488 + 0.549509i 0.973045 0.230616i 0.396080 0.918216i 0 −0.686242 + 0.727374i 0 0.173648 + 0.984808i 0.893633 0.448799i 0
115.1 0.396080 0.918216i 0.893633 0.448799i −0.686242 0.727374i 0 −0.0581448 0.998308i 0 −0.939693 + 0.342020i 0.597159 0.802123i 0
139.1 0.597159 0.802123i −0.0581448 0.998308i −0.286803 0.957990i 0 −0.835488 0.549509i 0 −0.939693 0.342020i −0.993238 + 0.116093i 0
187.1 0.893633 0.448799i −0.686242 + 0.727374i 0.597159 0.802123i 0 −0.286803 + 0.957990i 0 0.173648 0.984808i −0.0581448 0.998308i 0
211.1 −0.286803 + 0.957990i −0.993238 0.116093i −0.835488 0.549509i 0 0.396080 0.918216i 0 0.766044 0.642788i 0.973045 + 0.230616i 0
259.1 0.973045 + 0.230616i 0.396080 0.918216i 0.893633 + 0.448799i 0 0.597159 0.802123i 0 0.766044 + 0.642788i −0.686242 0.727374i 0
283.1 −0.0581448 0.998308i −0.286803 + 0.957990i −0.993238 + 0.116093i 0 0.973045 + 0.230616i 0 0.173648 + 0.984808i −0.835488 0.549509i 0
331.1 0.597159 + 0.802123i −0.0581448 + 0.998308i −0.286803 + 0.957990i 0 −0.835488 + 0.549509i 0 −0.939693 + 0.342020i −0.993238 0.116093i 0
355.1 0.396080 + 0.918216i 0.893633 + 0.448799i −0.686242 + 0.727374i 0 −0.0581448 + 0.998308i 0 −0.939693 0.342020i 0.597159 + 0.802123i 0
403.1 −0.0581448 + 0.998308i −0.286803 0.957990i −0.993238 0.116093i 0 0.973045 0.230616i 0 0.173648 0.984808i −0.835488 + 0.549509i 0
427.1 −0.686242 0.727374i 0.597159 0.802123i −0.0581448 + 0.998308i 0 −0.993238 + 0.116093i 0 0.766044 0.642788i −0.286803 0.957990i 0
475.1 −0.686242 + 0.727374i 0.597159 + 0.802123i −0.0581448 0.998308i 0 −0.993238 0.116093i 0 0.766044 + 0.642788i −0.286803 + 0.957990i 0
499.1 0.893633 + 0.448799i −0.686242 0.727374i 0.597159 + 0.802123i 0 −0.286803 0.957990i 0 0.173648 + 0.984808i −0.0581448 + 0.998308i 0
547.1 −0.993238 + 0.116093i −0.835488 0.549509i 0.973045 0.230616i 0 0.893633 + 0.448799i 0 −0.939693 + 0.342020i 0.396080 + 0.918216i 0
571.1 −0.993238 0.116093i −0.835488 + 0.549509i 0.973045 + 0.230616i 0 0.893633 0.448799i 0 −0.939693 0.342020i 0.396080 0.918216i 0
619.1 −0.835488 0.549509i 0.973045 + 0.230616i 0.396080 + 0.918216i 0 −0.686242 0.727374i 0 0.173648 0.984808i 0.893633 + 0.448799i 0
643.1 0.973045 0.230616i 0.396080 + 0.918216i 0.893633 0.448799i 0 0.597159 + 0.802123i 0 0.766044 0.642788i −0.686242 + 0.727374i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 643.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
81.g even 27 1 inner
648.bf odd 54 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 648.1.bf.a 18
3.b odd 2 1 1944.1.bf.a 18
4.b odd 2 1 2592.1.bz.a 18
8.b even 2 1 2592.1.bz.a 18
8.d odd 2 1 CM 648.1.bf.a 18
24.f even 2 1 1944.1.bf.a 18
81.g even 27 1 inner 648.1.bf.a 18
81.h odd 54 1 1944.1.bf.a 18
324.n odd 54 1 2592.1.bz.a 18
648.bb even 54 1 1944.1.bf.a 18
648.bd even 54 1 2592.1.bz.a 18
648.bf odd 54 1 inner 648.1.bf.a 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
648.1.bf.a 18 1.a even 1 1 trivial
648.1.bf.a 18 8.d odd 2 1 CM
648.1.bf.a 18 81.g even 27 1 inner
648.1.bf.a 18 648.bf odd 54 1 inner
1944.1.bf.a 18 3.b odd 2 1
1944.1.bf.a 18 24.f even 2 1
1944.1.bf.a 18 81.h odd 54 1
1944.1.bf.a 18 648.bb even 54 1
2592.1.bz.a 18 4.b odd 2 1
2592.1.bz.a 18 8.b even 2 1
2592.1.bz.a 18 324.n odd 54 1
2592.1.bz.a 18 648.bd even 54 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(648, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{9} + T^{18} \)
$3$ \( 1 + T^{9} + T^{18} \)
$5$ \( T^{18} \)
$7$ \( T^{18} \)
$11$ \( 1 + 9 T + 90 T^{2} + 168 T^{3} + 72 T^{4} - 135 T^{5} - 150 T^{6} - 135 T^{7} + 297 T^{8} + 8 T^{9} - 72 T^{10} - 18 T^{11} + 6 T^{12} + 18 T^{13} + 3 T^{15} + T^{18} \)
$13$ \( T^{18} \)
$17$ \( 1 - 18 T + 108 T^{2} - 213 T^{3} + 441 T^{4} - 648 T^{5} + 657 T^{6} - 270 T^{7} + 81 T^{8} + 2 T^{9} - 18 T^{10} + 27 T^{11} + 30 T^{12} - 18 T^{13} + T^{18} \)
$19$ \( 1 + 9 T + 27 T^{2} - 213 T^{3} + 468 T^{4} - 648 T^{5} + 657 T^{6} - 270 T^{7} + 81 T^{8} + 2 T^{9} + 9 T^{10} - 54 T^{11} + 30 T^{12} + 9 T^{13} + T^{18} \)
$23$ \( T^{18} \)
$29$ \( T^{18} \)
$31$ \( T^{18} \)
$37$ \( T^{18} \)
$41$ \( 1 + 9 T + 63 T^{2} - 75 T^{3} + 126 T^{4} - 216 T^{5} - 150 T^{6} + 108 T^{7} + 297 T^{8} + 8 T^{9} - 45 T^{10} + 9 T^{11} + 6 T^{12} - 36 T^{13} + 3 T^{15} + T^{18} \)
$43$ \( 1 - 9 T + 99 T^{2} - 327 T^{3} + 378 T^{4} - 126 T^{5} + 84 T^{6} - 135 T^{7} - 27 T^{8} + 8 T^{9} + 72 T^{10} + 72 T^{11} + 6 T^{12} + 9 T^{14} + 3 T^{15} + T^{18} \)
$47$ \( T^{18} \)
$53$ \( T^{18} \)
$59$ \( 1 - 9 T + 9 T^{2} + 240 T^{3} + 666 T^{4} + 1008 T^{5} + 1470 T^{6} + 2232 T^{7} + 2898 T^{8} + 3140 T^{9} + 2907 T^{10} + 2304 T^{11} + 1554 T^{12} + 882 T^{13} + 414 T^{14} + 156 T^{15} + 45 T^{16} + 9 T^{17} + T^{18} \)
$61$ \( T^{18} \)
$67$ \( 1 + 18 T + 99 T^{2} + 159 T^{3} + 297 T^{4} + 63 T^{5} + 84 T^{6} - 216 T^{7} - 108 T^{8} + 8 T^{9} + 72 T^{10} - 63 T^{11} + 6 T^{12} + 9 T^{14} + 3 T^{15} + T^{18} \)
$71$ \( T^{18} \)
$73$ \( 1 + 9 T + 108 T^{2} + 516 T^{3} + 1278 T^{4} + 1782 T^{5} + 1386 T^{6} + 540 T^{7} + 81 T^{8} + 2 T^{9} + 9 T^{10} + 27 T^{11} + 30 T^{12} + 9 T^{13} + T^{18} \)
$79$ \( T^{18} \)
$83$ \( 1 + 45 T^{3} + 576 T^{6} + 80 T^{9} + 45 T^{12} + 9 T^{15} + T^{18} \)
$89$ \( ( 1 - 3 T + 3 T^{2} + 8 T^{3} + 6 T^{4} + 3 T^{5} + T^{6} )^{3} \)
$97$ \( 1 + 9 T + 63 T^{2} - 75 T^{3} + 126 T^{4} - 216 T^{5} - 150 T^{6} + 108 T^{7} + 297 T^{8} + 8 T^{9} - 45 T^{10} + 9 T^{11} + 6 T^{12} - 36 T^{13} + 3 T^{15} + T^{18} \)
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