Properties

Label 3200.1.m.e
Level $3200$
Weight $1$
Character orbit 3200.m
Analytic conductor $1.597$
Analytic rank $0$
Dimension $8$
Projective image $D_{6}$
CM discriminant -8
Inner twists $16$

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Newspace parameters

Level: \( N \) \(=\) \( 3200 = 2^{7} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3200.m (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.59700804043\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.2.6400000.1

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{24}^{3} q^{3}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{24}^{3} q^{3} + ( - \zeta_{24}^{8} - \zeta_{24}^{4}) q^{11} + ( - \zeta_{24}^{5} - \zeta_{24}) q^{17} + ( - \zeta_{24}^{10} + \zeta_{24}^{2}) q^{19} - \zeta_{24}^{9} q^{27} + ( - \zeta_{24}^{11} - \zeta_{24}^{7}) q^{33} - q^{41} + \zeta_{24}^{3} q^{43} - \zeta_{24}^{6} q^{49} + ( - \zeta_{24}^{8} - \zeta_{24}^{4}) q^{51} + (\zeta_{24}^{5} + \zeta_{24}) q^{57} + \zeta_{24}^{9} q^{67} + (\zeta_{24}^{11} + \zeta_{24}^{7}) q^{73} + q^{81} + \zeta_{24}^{3} q^{83} + \zeta_{24}^{6} q^{89} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{41} + 8 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3200\mathbb{Z}\right)^\times\).

\(n\) \(901\) \(1151\) \(2177\)
\(\chi(n)\) \(-1\) \(1\) \(\zeta_{24}^{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
193.1
0.258819 0.965926i
−0.965926 + 0.258819i
−0.258819 + 0.965926i
0.965926 0.258819i
−0.965926 0.258819i
0.258819 + 0.965926i
0.965926 + 0.258819i
−0.258819 0.965926i
0 −0.707107 + 0.707107i 0 0 0 0 0 0 0
193.2 0 −0.707107 + 0.707107i 0 0 0 0 0 0 0
193.3 0 0.707107 0.707107i 0 0 0 0 0 0 0
193.4 0 0.707107 0.707107i 0 0 0 0 0 0 0
1857.1 0 −0.707107 0.707107i 0 0 0 0 0 0 0
1857.2 0 −0.707107 0.707107i 0 0 0 0 0 0 0
1857.3 0 0.707107 + 0.707107i 0 0 0 0 0 0 0
1857.4 0 0.707107 + 0.707107i 0 0 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1857.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
4.b odd 2 1 inner
5.b even 2 1 inner
5.c odd 4 2 inner
8.b even 2 1 inner
20.d odd 2 1 inner
20.e even 4 2 inner
40.e odd 2 1 inner
40.f even 2 1 inner
40.i odd 4 2 inner
40.k even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3200.1.m.e 8
4.b odd 2 1 inner 3200.1.m.e 8
5.b even 2 1 inner 3200.1.m.e 8
5.c odd 4 2 inner 3200.1.m.e 8
8.b even 2 1 inner 3200.1.m.e 8
8.d odd 2 1 CM 3200.1.m.e 8
20.d odd 2 1 inner 3200.1.m.e 8
20.e even 4 2 inner 3200.1.m.e 8
40.e odd 2 1 inner 3200.1.m.e 8
40.f even 2 1 inner 3200.1.m.e 8
40.i odd 4 2 inner 3200.1.m.e 8
40.k even 4 2 inner 3200.1.m.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3200.1.m.e 8 1.a even 1 1 trivial
3200.1.m.e 8 4.b odd 2 1 inner
3200.1.m.e 8 5.b even 2 1 inner
3200.1.m.e 8 5.c odd 4 2 inner
3200.1.m.e 8 8.b even 2 1 inner
3200.1.m.e 8 8.d odd 2 1 CM
3200.1.m.e 8 20.d odd 2 1 inner
3200.1.m.e 8 20.e even 4 2 inner
3200.1.m.e 8 40.e odd 2 1 inner
3200.1.m.e 8 40.f even 2 1 inner
3200.1.m.e 8 40.i odd 4 2 inner
3200.1.m.e 8 40.k even 4 2 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3200, [\chi])\):

\( T_{3}^{4} + 1 \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{2} + 3)^{4} \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( (T^{4} + 9)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 3)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( (T + 1)^{8} \) Copy content Toggle raw display
$43$ \( (T^{4} + 16)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( (T^{4} + 9)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$97$ \( T^{8} \) Copy content Toggle raw display
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