Properties

Label 1476.1.l.a
Level $1476$
Weight $1$
Character orbit 1476.l
Analytic conductor $0.737$
Analytic rank $0$
Dimension $4$
Projective image $S_{4}$
CM/RM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1476 = 2^{2} \cdot 3^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1476.l (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.736619958646\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(S_{4}\)
Projective field: Galois closure of 4.2.7443468.1

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( \zeta_{8} - \zeta_{8}^{3} ) q^{5} +O(q^{10})\) \( q + ( \zeta_{8} - \zeta_{8}^{3} ) q^{5} + \zeta_{8}^{3} q^{11} + \zeta_{8} q^{17} + ( 1 - \zeta_{8}^{2} ) q^{19} + q^{25} + \zeta_{8}^{3} q^{29} - q^{31} + q^{37} + \zeta_{8}^{3} q^{41} -\zeta_{8}^{2} q^{43} -\zeta_{8} q^{47} -\zeta_{8}^{2} q^{49} + ( -1 + \zeta_{8}^{2} ) q^{55} -\zeta_{8}^{2} q^{61} + ( -1 + \zeta_{8}^{2} ) q^{67} -\zeta_{8}^{3} q^{71} -\zeta_{8}^{2} q^{73} + ( 1 + \zeta_{8}^{2} ) q^{79} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{83} + ( 1 + \zeta_{8}^{2} ) q^{85} -2 \zeta_{8}^{3} q^{95} + ( -1 + \zeta_{8}^{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + O(q^{10}) \) \( 4 q + 4 q^{19} + 4 q^{25} - 4 q^{31} + 4 q^{37} - 4 q^{55} - 4 q^{67} + 4 q^{79} + 4 q^{85} - 4 q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1476\mathbb{Z}\right)^\times\).

\(n\) \(739\) \(821\) \(1441\)
\(\chi(n)\) \(1\) \(-1\) \(-\zeta_{8}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
665.1
−0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 + 0.707107i
0.707107 0.707107i
0 0 0 −1.41421 0 0 0 0 0
665.2 0 0 0 1.41421 0 0 0 0 0
1385.1 0 0 0 −1.41421 0 0 0 0 0
1385.2 0 0 0 1.41421 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
41.c even 4 1 inner
123.f odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1476.1.l.a 4
3.b odd 2 1 inner 1476.1.l.a 4
41.c even 4 1 inner 1476.1.l.a 4
123.f odd 4 1 inner 1476.1.l.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1476.1.l.a 4 1.a even 1 1 trivial
1476.1.l.a 4 3.b odd 2 1 inner
1476.1.l.a 4 41.c even 4 1 inner
1476.1.l.a 4 123.f odd 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1476, [\chi])\).

Hecke characteristic polynomials

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