Properties

Label 896.1.bm.a
Level $896$
Weight $1$
Character orbit 896.bm
Analytic conductor $0.447$
Analytic rank $0$
Dimension $16$
Projective image $D_{32}$
CM discriminant -7
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 896 = 2^{7} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 896.bm (of order \(32\), degree \(16\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.447162251319\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\Q(\zeta_{32})\)
Defining polynomial: \(x^{16} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{32}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{32} + \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{32}^{13} q^{2} -\zeta_{32}^{10} q^{4} -\zeta_{32}^{9} q^{7} -\zeta_{32}^{7} q^{8} + \zeta_{32}^{15} q^{9} +O(q^{10})\) \( q -\zeta_{32}^{13} q^{2} -\zeta_{32}^{10} q^{4} -\zeta_{32}^{9} q^{7} -\zeta_{32}^{7} q^{8} + \zeta_{32}^{15} q^{9} + ( -\zeta_{32}^{11} + \zeta_{32}^{14} ) q^{11} -\zeta_{32}^{6} q^{14} -\zeta_{32}^{4} q^{16} + \zeta_{32}^{12} q^{18} + ( -\zeta_{32}^{8} + \zeta_{32}^{11} ) q^{22} + ( \zeta_{32}^{2} + \zeta_{32}^{4} ) q^{23} + \zeta_{32}^{5} q^{25} -\zeta_{32}^{3} q^{28} + ( \zeta_{32} + \zeta_{32}^{6} ) q^{29} -\zeta_{32} q^{32} + \zeta_{32}^{9} q^{36} + ( -\zeta_{32}^{3} + \zeta_{32}^{10} ) q^{37} + ( -\zeta_{32}^{5} - \zeta_{32}^{12} ) q^{43} + ( -\zeta_{32}^{5} + \zeta_{32}^{8} ) q^{44} + ( \zeta_{32} - \zeta_{32}^{15} ) q^{46} -\zeta_{32}^{2} q^{49} + \zeta_{32}^{2} q^{50} + ( \zeta_{32}^{12} + \zeta_{32}^{13} ) q^{53} - q^{56} + ( \zeta_{32}^{3} - \zeta_{32}^{14} ) q^{58} + \zeta_{32}^{8} q^{63} + \zeta_{32}^{14} q^{64} + ( \zeta_{32}^{7} - \zeta_{32}^{8} ) q^{67} + ( \zeta_{32}^{3} - \zeta_{32}^{15} ) q^{71} + \zeta_{32}^{6} q^{72} + ( -1 + \zeta_{32}^{7} ) q^{74} + ( -\zeta_{32}^{4} + \zeta_{32}^{7} ) q^{77} + ( -\zeta_{32}^{7} + \zeta_{32}^{13} ) q^{79} -\zeta_{32}^{14} q^{81} + ( -\zeta_{32}^{2} - \zeta_{32}^{9} ) q^{86} + ( -\zeta_{32}^{2} + \zeta_{32}^{5} ) q^{88} + ( -\zeta_{32}^{12} - \zeta_{32}^{14} ) q^{92} + \zeta_{32}^{15} q^{98} + ( \zeta_{32}^{10} - \zeta_{32}^{13} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + O(q^{10}) \) \( 16q - 16q^{56} - 16q^{74} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/896\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(645\)
\(\chi(n)\) \(1\) \(-1\) \(\zeta_{32}^{5}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1
0.831470 + 0.555570i
0.831470 0.555570i
0.195090 + 0.980785i
0.980785 + 0.195090i
−0.555570 + 0.831470i
0.555570 + 0.831470i
−0.980785 + 0.195090i
−0.195090 + 0.980785i
−0.831470 0.555570i
−0.831470 + 0.555570i
−0.195090 0.980785i
−0.980785 0.195090i
0.555570 0.831470i
−0.555570 0.831470i
0.980785 0.195090i
0.195090 0.980785i
−0.195090 0.980785i 0 −0.923880 + 0.382683i 0 0 −0.555570 + 0.831470i 0.555570 + 0.831470i −0.831470 + 0.555570i 0
69.1 −0.195090 + 0.980785i 0 −0.923880 0.382683i 0 0 −0.555570 0.831470i 0.555570 0.831470i −0.831470 0.555570i 0
125.1 −0.555570 + 0.831470i 0 −0.382683 0.923880i 0 0 −0.980785 + 0.195090i 0.980785 + 0.195090i −0.195090 + 0.980785i 0
181.1 0.831470 0.555570i 0 0.382683 0.923880i 0 0 0.195090 0.980785i −0.195090 0.980785i −0.980785 + 0.195090i 0
237.1 0.980785 0.195090i 0 0.923880 0.382683i 0 0 −0.831470 0.555570i 0.831470 0.555570i 0.555570 + 0.831470i 0
293.1 −0.980785 0.195090i 0 0.923880 + 0.382683i 0 0 0.831470 0.555570i −0.831470 0.555570i −0.555570 + 0.831470i 0
349.1 −0.831470 0.555570i 0 0.382683 + 0.923880i 0 0 −0.195090 0.980785i 0.195090 0.980785i 0.980785 + 0.195090i 0
405.1 0.555570 + 0.831470i 0 −0.382683 + 0.923880i 0 0 0.980785 + 0.195090i −0.980785 + 0.195090i 0.195090 + 0.980785i 0
461.1 0.195090 + 0.980785i 0 −0.923880 + 0.382683i 0 0 0.555570 0.831470i −0.555570 0.831470i 0.831470 0.555570i 0
517.1 0.195090 0.980785i 0 −0.923880 0.382683i 0 0 0.555570 + 0.831470i −0.555570 + 0.831470i 0.831470 + 0.555570i 0
573.1 0.555570 0.831470i 0 −0.382683 0.923880i 0 0 0.980785 0.195090i −0.980785 0.195090i 0.195090 0.980785i 0
629.1 −0.831470 + 0.555570i 0 0.382683 0.923880i 0 0 −0.195090 + 0.980785i 0.195090 + 0.980785i 0.980785 0.195090i 0
685.1 −0.980785 + 0.195090i 0 0.923880 0.382683i 0 0 0.831470 + 0.555570i −0.831470 + 0.555570i −0.555570 0.831470i 0
741.1 0.980785 + 0.195090i 0 0.923880 + 0.382683i 0 0 −0.831470 + 0.555570i 0.831470 + 0.555570i 0.555570 0.831470i 0
797.1 0.831470 + 0.555570i 0 0.382683 + 0.923880i 0 0 0.195090 + 0.980785i −0.195090 + 0.980785i −0.980785 0.195090i 0
853.1 −0.555570 0.831470i 0 −0.382683 + 0.923880i 0 0 −0.980785 0.195090i 0.980785 0.195090i −0.195090 0.980785i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 853.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
128.k even 32 1 inner
896.bm odd 32 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 896.1.bm.a 16
4.b odd 2 1 3584.1.bm.a 16
7.b odd 2 1 CM 896.1.bm.a 16
28.d even 2 1 3584.1.bm.a 16
128.k even 32 1 inner 896.1.bm.a 16
128.l odd 32 1 3584.1.bm.a 16
896.bk even 32 1 3584.1.bm.a 16
896.bm odd 32 1 inner 896.1.bm.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
896.1.bm.a 16 1.a even 1 1 trivial
896.1.bm.a 16 7.b odd 2 1 CM
896.1.bm.a 16 128.k even 32 1 inner
896.1.bm.a 16 896.bm odd 32 1 inner
3584.1.bm.a 16 4.b odd 2 1
3584.1.bm.a 16 28.d even 2 1
3584.1.bm.a 16 128.l odd 32 1
3584.1.bm.a 16 896.bk even 32 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(896, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{16} \)
$3$ \( T^{16} \)
$5$ \( T^{16} \)
$7$ \( 1 + T^{16} \)
$11$ \( 2 - 16 T + 24 T^{2} + 32 T^{3} + 148 T^{4} + 176 T^{6} + 2 T^{8} + 16 T^{9} - 32 T^{11} + T^{16} \)
$13$ \( T^{16} \)
$17$ \( T^{16} \)
$19$ \( T^{16} \)
$23$ \( ( 2 + 8 T + 20 T^{2} + 16 T^{3} + 2 T^{4} + T^{8} )^{2} \)
$29$ \( 2 + 16 T + 88 T^{2} + 192 T^{3} + 140 T^{4} + 16 T^{5} + 2 T^{8} - 48 T^{9} + 40 T^{10} + T^{16} \)
$31$ \( T^{16} \)
$37$ \( 2 + 16 T + 40 T^{2} + 140 T^{4} - 48 T^{5} + 192 T^{7} + 2 T^{8} + 88 T^{10} + 16 T^{13} + T^{16} \)
$41$ \( T^{16} \)
$43$ \( 2 + 16 T + 72 T^{2} + 80 T^{3} + 4 T^{4} + 56 T^{6} - 160 T^{7} + 6 T^{8} + 16 T^{11} + 4 T^{12} + T^{16} \)
$47$ \( T^{16} \)
$53$ \( 2 - 16 T + 72 T^{2} - 80 T^{3} + 4 T^{4} + 56 T^{6} + 160 T^{7} + 6 T^{8} - 16 T^{11} + 4 T^{12} + T^{16} \)
$59$ \( T^{16} \)
$61$ \( T^{16} \)
$67$ \( 2 - 16 T + 8 T^{2} + 112 T^{3} + 28 T^{4} - 112 T^{5} + 56 T^{6} + 16 T^{7} + 70 T^{8} + 56 T^{10} + 28 T^{12} + 8 T^{14} + T^{16} \)
$71$ \( 16 + 64 T^{4} + 128 T^{8} - 16 T^{12} + T^{16} \)
$73$ \( T^{16} \)
$79$ \( 4 + 32 T^{2} + 128 T^{4} - 192 T^{6} + 140 T^{8} - 16 T^{10} + T^{16} \)
$83$ \( T^{16} \)
$89$ \( T^{16} \)
$97$ \( T^{16} \)
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