Properties

Label 964.1.bh.a
Level $964$
Weight $1$
Character orbit 964.bh
Analytic conductor $0.481$
Analytic rank $0$
Dimension $16$
Projective image $D_{60}$
CM discriminant -4
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 964 = 2^{2} \cdot 241 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 964.bh (of order \(60\), degree \(16\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.481098672178\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\Q(\zeta_{60})\)
Defining polynomial: \(x^{16} + x^{14} - x^{10} - x^{8} - x^{6} + x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{60}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{60} - \cdots)\)

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q -\zeta_{60}^{5} q^{2} + \zeta_{60}^{10} q^{4} + ( -\zeta_{60} + \zeta_{60}^{23} ) q^{5} -\zeta_{60}^{15} q^{8} + \zeta_{60}^{26} q^{9} +O(q^{10})\) \( q -\zeta_{60}^{5} q^{2} + \zeta_{60}^{10} q^{4} + ( -\zeta_{60} + \zeta_{60}^{23} ) q^{5} -\zeta_{60}^{15} q^{8} + \zeta_{60}^{26} q^{9} + ( \zeta_{60}^{6} - \zeta_{60}^{28} ) q^{10} + ( -\zeta_{60}^{12} - \zeta_{60}^{29} ) q^{13} + \zeta_{60}^{20} q^{16} + ( -\zeta_{60}^{10} - \zeta_{60}^{23} ) q^{17} + \zeta_{60} q^{18} + ( -\zeta_{60}^{3} - \zeta_{60}^{11} ) q^{20} + ( \zeta_{60}^{2} - \zeta_{60}^{16} - \zeta_{60}^{24} ) q^{25} + ( -\zeta_{60}^{4} + \zeta_{60}^{17} ) q^{26} + ( -\zeta_{60}^{8} - \zeta_{60}^{14} ) q^{29} -\zeta_{60}^{25} q^{32} + ( \zeta_{60}^{15} + \zeta_{60}^{28} ) q^{34} -\zeta_{60}^{6} q^{36} + ( -\zeta_{60}^{2} - \zeta_{60}^{17} ) q^{37} + ( \zeta_{60}^{8} + \zeta_{60}^{16} ) q^{40} + ( \zeta_{60}^{4} + \zeta_{60}^{14} ) q^{41} + ( -\zeta_{60}^{19} - \zeta_{60}^{27} ) q^{45} -\zeta_{60}^{13} q^{49} + ( -\zeta_{60}^{7} + \zeta_{60}^{21} + \zeta_{60}^{29} ) q^{50} + ( \zeta_{60}^{9} - \zeta_{60}^{22} ) q^{52} + ( \zeta_{60}^{12} - \zeta_{60}^{20} ) q^{53} + ( \zeta_{60}^{13} + \zeta_{60}^{19} ) q^{58} + ( -\zeta_{60}^{9} + \zeta_{60}^{27} ) q^{61} - q^{64} + ( -1 + \zeta_{60}^{5} + \zeta_{60}^{13} + \zeta_{60}^{22} ) q^{65} + ( \zeta_{60}^{3} - \zeta_{60}^{20} ) q^{68} + \zeta_{60}^{11} q^{72} + ( \zeta_{60}^{18} + \zeta_{60}^{21} ) q^{73} + ( \zeta_{60}^{7} + \zeta_{60}^{22} ) q^{74} + ( -\zeta_{60}^{13} - \zeta_{60}^{21} ) q^{80} -\zeta_{60}^{22} q^{81} + ( -\zeta_{60}^{9} - \zeta_{60}^{19} ) q^{82} + ( \zeta_{60}^{3} + \zeta_{60}^{11} + \zeta_{60}^{16} + \zeta_{60}^{24} ) q^{85} + ( -\zeta_{60}^{7} - \zeta_{60}^{18} ) q^{89} + ( -\zeta_{60}^{2} + \zeta_{60}^{24} ) q^{90} -\zeta_{60}^{16} q^{97} + \zeta_{60}^{18} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 8q^{4} - 2q^{9} + O(q^{10}) \) \( 16q + 8q^{4} - 2q^{9} + 2q^{10} + 4q^{13} - 8q^{16} - 8q^{17} - 2q^{26} + 2q^{34} - 4q^{36} + 2q^{37} + 4q^{40} + 2q^{52} + 4q^{53} - 16q^{64} - 18q^{65} + 8q^{68} + 4q^{73} - 2q^{74} + 2q^{81} - 2q^{85} - 4q^{89} - 2q^{90} - 2q^{97} + 4q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/964\mathbb{Z}\right)^\times\).

\(n\) \(483\) \(489\)
\(\chi(n)\) \(-1\) \(-\zeta_{60}^{13}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
83.1
−0.406737 + 0.913545i
0.743145 0.669131i
−0.994522 + 0.104528i
−0.406737 0.913545i
0.207912 + 0.978148i
−0.207912 0.978148i
0.406737 + 0.913545i
0.994522 0.104528i
−0.743145 + 0.669131i
0.406737 0.913545i
−0.743145 0.669131i
−0.207912 + 0.978148i
−0.994522 0.104528i
0.994522 + 0.104528i
0.207912 0.978148i
0.743145 + 0.669131i
0.866025 + 0.500000i 0 0.500000 + 0.866025i 0.198825 + 0.0646021i 0 0 1.00000i 0.104528 + 0.994522i 0.139886 + 0.155360i
107.1 0.866025 0.500000i 0 0.500000 0.866025i −1.14988 + 1.58268i 0 0 1.00000i 0.978148 0.207912i −0.204489 + 1.94558i
123.1 0.866025 0.500000i 0 0.500000 0.866025i 1.73767 + 0.564602i 0 0 1.00000i −0.913545 0.406737i 1.78716 0.379874i
151.1 0.866025 0.500000i 0 0.500000 0.866025i 0.198825 0.0646021i 0 0 1.00000i 0.104528 0.994522i 0.139886 0.155360i
159.1 −0.866025 0.500000i 0 0.500000 + 0.866025i 0.786610 1.08268i 0 0 1.00000i −0.669131 0.743145i −1.22256 + 0.544320i
323.1 0.866025 + 0.500000i 0 0.500000 + 0.866025i −0.786610 + 1.08268i 0 0 1.00000i −0.669131 0.743145i −1.22256 + 0.544320i
331.1 −0.866025 + 0.500000i 0 0.500000 0.866025i −0.198825 + 0.0646021i 0 0 1.00000i 0.104528 0.994522i 0.139886 0.155360i
359.1 −0.866025 + 0.500000i 0 0.500000 0.866025i −1.73767 0.564602i 0 0 1.00000i −0.913545 0.406737i 1.78716 0.379874i
375.1 −0.866025 + 0.500000i 0 0.500000 0.866025i 1.14988 1.58268i 0 0 1.00000i 0.978148 0.207912i −0.204489 + 1.94558i
399.1 −0.866025 0.500000i 0 0.500000 + 0.866025i −0.198825 0.0646021i 0 0 1.00000i 0.104528 + 0.994522i 0.139886 + 0.155360i
491.1 −0.866025 0.500000i 0 0.500000 + 0.866025i 1.14988 + 1.58268i 0 0 1.00000i 0.978148 + 0.207912i −0.204489 1.94558i
579.1 0.866025 0.500000i 0 0.500000 0.866025i −0.786610 1.08268i 0 0 1.00000i −0.669131 + 0.743145i −1.22256 0.544320i
627.1 0.866025 + 0.500000i 0 0.500000 + 0.866025i 1.73767 0.564602i 0 0 1.00000i −0.913545 + 0.406737i 1.78716 + 0.379874i
819.1 −0.866025 0.500000i 0 0.500000 + 0.866025i −1.73767 + 0.564602i 0 0 1.00000i −0.913545 + 0.406737i 1.78716 + 0.379874i
867.1 −0.866025 + 0.500000i 0 0.500000 0.866025i 0.786610 + 1.08268i 0 0 1.00000i −0.669131 + 0.743145i −1.22256 0.544320i
955.1 0.866025 + 0.500000i 0 0.500000 + 0.866025i −1.14988 1.58268i 0 0 1.00000i 0.978148 + 0.207912i −0.204489 1.94558i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 955.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
241.q even 60 1 inner
964.bh odd 60 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 964.1.bh.a 16
4.b odd 2 1 CM 964.1.bh.a 16
241.q even 60 1 inner 964.1.bh.a 16
964.bh odd 60 1 inner 964.1.bh.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
964.1.bh.a 16 1.a even 1 1 trivial
964.1.bh.a 16 4.b odd 2 1 CM
964.1.bh.a 16 241.q even 60 1 inner
964.1.bh.a 16 964.bh odd 60 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(964, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T^{2} + T^{4} )^{4} \)
$3$ \( T^{16} \)
$5$ \( 1 - 37 T^{2} + 523 T^{4} + T^{6} + 150 T^{8} - 49 T^{10} + 13 T^{12} - 2 T^{14} + T^{16} \)
$7$ \( T^{16} \)
$11$ \( T^{16} \)
$13$ \( 1 - 12 T + 89 T^{2} + 28 T^{3} - 204 T^{4} + 176 T^{5} + 52 T^{6} - 206 T^{7} + 177 T^{8} - 58 T^{9} + 8 T^{10} + 2 T^{11} + 16 T^{12} - 16 T^{13} + 11 T^{14} - 4 T^{15} + T^{16} \)
$17$ \( 1 + 22 T + 187 T^{2} + 658 T^{3} + 1302 T^{4} + 1840 T^{5} + 2213 T^{6} + 2344 T^{7} + 2135 T^{8} + 1676 T^{9} + 1133 T^{10} + 650 T^{11} + 312 T^{12} + 122 T^{13} + 37 T^{14} + 8 T^{15} + T^{16} \)
$19$ \( T^{16} \)
$23$ \( T^{16} \)
$29$ \( ( 25 + 25 T + 25 T^{2} - 5 T^{4} - 10 T^{5} + T^{8} )^{2} \)
$31$ \( T^{16} \)
$37$ \( 256 - 256 T + 128 T^{2} - 128 T^{3} + 128 T^{4} - 128 T^{5} + 96 T^{6} - 64 T^{7} + 48 T^{8} - 32 T^{9} + 24 T^{10} - 16 T^{11} + 8 T^{12} - 4 T^{13} + 2 T^{14} - 2 T^{15} + T^{16} \)
$41$ \( ( 81 - 27 T^{2} + 9 T^{4} - 3 T^{6} + T^{8} )^{2} \)
$43$ \( T^{16} \)
$47$ \( T^{16} \)
$53$ \( ( 1 + T - T^{2} - 8 T^{3} + 9 T^{4} - 6 T^{5} + 6 T^{6} - 2 T^{7} + T^{8} )^{2} \)
$59$ \( T^{16} \)
$61$ \( ( 1 + T^{2} + 6 T^{4} - 4 T^{6} + T^{8} )^{2} \)
$67$ \( T^{16} \)
$71$ \( T^{16} \)
$73$ \( ( 1 - 6 T + 13 T^{2} - 10 T^{3} + 16 T^{4} - 10 T^{5} + 2 T^{6} - 2 T^{7} + T^{8} )^{2} \)
$79$ \( T^{16} \)
$83$ \( T^{16} \)
$89$ \( 1 - 18 T + 84 T^{2} + 22 T^{3} + 76 T^{4} + 144 T^{5} + 2 T^{6} - 84 T^{7} - 93 T^{8} - 52 T^{9} + 28 T^{10} + 48 T^{11} + 41 T^{12} + 26 T^{13} + 11 T^{14} + 4 T^{15} + T^{16} \)
$97$ \( ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \)
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