Properties

Label 1476.1.j.a
Level $1476$
Weight $1$
Character orbit 1476.j
Analytic conductor $0.737$
Analytic rank $0$
Dimension $2$
Projective image $D_{4}$
CM discriminant -4
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.j (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.736619958646\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \( x^{2} + 1 \) 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 4.2.2481156.1

$q$-expansion

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

\(f(q)\) \(=\) \( q + i q^{2} - q^{4} - i q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + i q^{2} - q^{4} - i q^{8} + (i + 1) q^{13} + q^{16} + (i - 1) q^{17} + q^{25} + (i - 1) q^{26} + (i + 1) q^{29} + i q^{32} + ( - i - 1) q^{34} + i q^{41} + i q^{49} + i q^{50} + ( - i - 1) q^{52} + ( - i - 1) q^{53} + (i - 1) q^{58} - q^{64} + ( - i + 1) q^{68} - i q^{73} - q^{82} + ( - i - 1) q^{89} + (i - 1) q^{97} - q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} + 2 q^{13} + 2 q^{16} - 2 q^{17} + 2 q^{25} - 2 q^{26} + 2 q^{29} - 2 q^{34} - 2 q^{52} - 2 q^{53} - 2 q^{58} - 2 q^{64} + 2 q^{68} - 2 q^{82} - 2 q^{89} - 2 q^{97} - 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(739\) \(821\) \(1441\)
\(\chi(n)\) \(-1\) \(1\) \(-i\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
91.1
1.00000i
1.00000i
1.00000i 0 −1.00000 0 0 0 1.00000i 0 0
811.1 1.00000i 0 −1.00000 0 0 0 1.00000i 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
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
41.c even 4 1 inner
164.e 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.j.a 2
3.b odd 2 1 1476.1.j.b yes 2
4.b odd 2 1 CM 1476.1.j.a 2
12.b even 2 1 1476.1.j.b yes 2
41.c even 4 1 inner 1476.1.j.a 2
123.f odd 4 1 1476.1.j.b yes 2
164.e odd 4 1 inner 1476.1.j.a 2
492.l even 4 1 1476.1.j.b yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1476.1.j.a 2 1.a even 1 1 trivial
1476.1.j.a 2 4.b odd 2 1 CM
1476.1.j.a 2 41.c even 4 1 inner
1476.1.j.a 2 164.e odd 4 1 inner
1476.1.j.b yes 2 3.b odd 2 1
1476.1.j.b yes 2 12.b even 2 1
1476.1.j.b yes 2 123.f odd 4 1
1476.1.j.b yes 2 492.l even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{17}^{2} + 2T_{17} + 2 \) acting on \(S_{1}^{\mathrm{new}}(1476, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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