Properties

Label 936.1.o.c
Level $936$
Weight $1$
Character orbit 936.o
Analytic conductor $0.467$
Analytic rank $0$
Dimension $4$
Projective image $D_{4}$
CM discriminant -39
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 936 = 2^{3} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 936.o (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.467124851824\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.32448.1

$q$-expansion

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

\(f(q)\) \(=\) \( q - \zeta_{8} q^{2} + \zeta_{8}^{2} q^{4} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{5} - \zeta_{8}^{3} q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{8} q^{2} + \zeta_{8}^{2} q^{4} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{5} - \zeta_{8}^{3} q^{8} + ( - \zeta_{8}^{2} - 1) q^{10} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{11} + \zeta_{8}^{2} q^{13} - q^{16} + (\zeta_{8}^{3} + \zeta_{8}) q^{20} + (\zeta_{8}^{2} - 1) q^{22} + q^{25} - \zeta_{8}^{3} q^{26} + \zeta_{8} q^{32} + ( - \zeta_{8}^{2} + 1) q^{40} + (\zeta_{8}^{3} + \zeta_{8}) q^{41} + q^{43} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{44} + (\zeta_{8}^{3} - \zeta_{8}) q^{47} - q^{49} - \zeta_{8} q^{50} - q^{52} - \zeta_{8}^{2} q^{55} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{59} - \zeta_{8}^{2} q^{61} - \zeta_{8}^{2} q^{64} + (\zeta_{8}^{3} + \zeta_{8}) q^{65} + (\zeta_{8}^{3} - \zeta_{8}) q^{71} + \zeta_{8}^{2} q^{79} + (\zeta_{8}^{3} - \zeta_{8}) q^{80} + ( - \zeta_{8}^{2} + 1) q^{82} + (\zeta_{8}^{3} + \zeta_{8}) q^{83} - 2 \zeta_{8} q^{86} + ( - \zeta_{8}^{2} - 1) q^{88} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{89} + (\zeta_{8}^{2} + 1) q^{94} + \zeta_{8} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{10} - 4 q^{16} - 4 q^{22} + 4 q^{25} + 4 q^{40} + 8 q^{43} - 4 q^{49} - 4 q^{52} + 4 q^{82} - 4 q^{88} + 4 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/936\mathbb{Z}\right)^\times\).

\(n\) \(145\) \(209\) \(469\) \(703\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
883.1
0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 + 0.707107i
−0.707107 0.707107i
−0.707107 0.707107i 0 1.00000i 1.41421 0 0 0.707107 0.707107i 0 −1.00000 1.00000i
883.2 −0.707107 + 0.707107i 0 1.00000i 1.41421 0 0 0.707107 + 0.707107i 0 −1.00000 + 1.00000i
883.3 0.707107 0.707107i 0 1.00000i −1.41421 0 0 −0.707107 0.707107i 0 −1.00000 + 1.00000i
883.4 0.707107 + 0.707107i 0 1.00000i −1.41421 0 0 −0.707107 + 0.707107i 0 −1.00000 1.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
39.d odd 2 1 CM by \(\Q(\sqrt{-39}) \)
3.b odd 2 1 inner
8.d odd 2 1 inner
13.b even 2 1 inner
24.f even 2 1 inner
104.h odd 2 1 inner
312.h even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 936.1.o.c 4
3.b odd 2 1 inner 936.1.o.c 4
4.b odd 2 1 3744.1.o.c 4
8.b even 2 1 3744.1.o.c 4
8.d odd 2 1 inner 936.1.o.c 4
12.b even 2 1 3744.1.o.c 4
13.b even 2 1 inner 936.1.o.c 4
24.f even 2 1 inner 936.1.o.c 4
24.h odd 2 1 3744.1.o.c 4
39.d odd 2 1 CM 936.1.o.c 4
52.b odd 2 1 3744.1.o.c 4
104.e even 2 1 3744.1.o.c 4
104.h odd 2 1 inner 936.1.o.c 4
156.h even 2 1 3744.1.o.c 4
312.b odd 2 1 3744.1.o.c 4
312.h even 2 1 inner 936.1.o.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
936.1.o.c 4 1.a even 1 1 trivial
936.1.o.c 4 3.b odd 2 1 inner
936.1.o.c 4 8.d odd 2 1 inner
936.1.o.c 4 13.b even 2 1 inner
936.1.o.c 4 24.f even 2 1 inner
936.1.o.c 4 39.d odd 2 1 CM
936.1.o.c 4 104.h odd 2 1 inner
936.1.o.c 4 312.h even 2 1 inner
3744.1.o.c 4 4.b odd 2 1
3744.1.o.c 4 8.b even 2 1
3744.1.o.c 4 12.b even 2 1
3744.1.o.c 4 24.h odd 2 1
3744.1.o.c 4 52.b odd 2 1
3744.1.o.c 4 104.e even 2 1
3744.1.o.c 4 156.h even 2 1
3744.1.o.c 4 312.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 2 \) acting on \(S_{1}^{\mathrm{new}}(936, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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