Properties

Label 1521.1.j.c
Level $1521$
Weight $1$
Character orbit 1521.j
Analytic conductor $0.759$
Analytic rank $0$
Dimension $2$
Projective image $D_{4}$
CM discriminant -39
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1521.j (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.759077884215\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.507.1

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( 1 + i ) q^{2} + i q^{4} + ( -1 - i ) q^{5} +O(q^{10})\) \( q + ( 1 + i ) q^{2} + i q^{4} + ( -1 - i ) q^{5} -2 i q^{10} + ( 1 - i ) q^{11} + q^{16} + ( 1 - i ) q^{20} + 2 q^{22} + i q^{25} + ( 1 + i ) q^{32} + ( -1 - i ) q^{41} + 2 i q^{43} + ( 1 + i ) q^{44} + ( 1 - i ) q^{47} + i q^{49} + ( -1 + i ) q^{50} -2 q^{55} + ( -1 + i ) q^{59} + i q^{64} + ( -1 - i ) q^{71} + ( -1 - i ) q^{80} -2 i q^{82} + ( 1 + i ) q^{83} + ( -2 + 2 i ) q^{86} + ( -1 + i ) q^{89} + 2 q^{94} + ( -1 + i ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 2 q^{5} + O(q^{10}) \) \( 2 q + 2 q^{2} - 2 q^{5} + 2 q^{11} + 2 q^{16} + 2 q^{20} + 4 q^{22} + 2 q^{32} - 2 q^{41} + 2 q^{44} + 2 q^{47} - 2 q^{50} - 4 q^{55} - 2 q^{59} - 2 q^{71} - 2 q^{80} + 2 q^{83} - 4 q^{86} - 2 q^{89} + 4 q^{94} - 2 q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1521\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(847\)
\(\chi(n)\) \(1\) \(i\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
577.1
1.00000i
1.00000i
1.00000 1.00000i 0 1.00000i −1.00000 + 1.00000i 0 0 0 0 2.00000i
775.1 1.00000 + 1.00000i 0 1.00000i −1.00000 1.00000i 0 0 0 0 2.00000i
\(n\): 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 \(\Q(\sqrt{-39}) \)
13.d odd 4 1 inner
39.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1521.1.j.c yes 2
3.b odd 2 1 1521.1.j.a 2
13.b even 2 1 1521.1.j.a 2
13.c even 3 2 1521.1.bd.a 4
13.d odd 4 1 1521.1.j.a 2
13.d odd 4 1 inner 1521.1.j.c yes 2
13.e even 6 2 1521.1.bd.d 4
13.f odd 12 2 1521.1.bd.a 4
13.f odd 12 2 1521.1.bd.d 4
39.d odd 2 1 CM 1521.1.j.c yes 2
39.f even 4 1 1521.1.j.a 2
39.f even 4 1 inner 1521.1.j.c yes 2
39.h odd 6 2 1521.1.bd.a 4
39.i odd 6 2 1521.1.bd.d 4
39.k even 12 2 1521.1.bd.a 4
39.k even 12 2 1521.1.bd.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1521.1.j.a 2 3.b odd 2 1
1521.1.j.a 2 13.b even 2 1
1521.1.j.a 2 13.d odd 4 1
1521.1.j.a 2 39.f even 4 1
1521.1.j.c yes 2 1.a even 1 1 trivial
1521.1.j.c yes 2 13.d odd 4 1 inner
1521.1.j.c yes 2 39.d odd 2 1 CM
1521.1.j.c yes 2 39.f even 4 1 inner
1521.1.bd.a 4 13.c even 3 2
1521.1.bd.a 4 13.f odd 12 2
1521.1.bd.a 4 39.h odd 6 2
1521.1.bd.a 4 39.k even 12 2
1521.1.bd.d 4 13.e even 6 2
1521.1.bd.d 4 13.f odd 12 2
1521.1.bd.d 4 39.i odd 6 2
1521.1.bd.d 4 39.k even 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 2 T_{2} + 2 \) acting on \(S_{1}^{\mathrm{new}}(1521, [\chi])\).

Hecke characteristic polynomials

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