Properties

Label 403.3.b.a
Level 403
Weight 3
Character orbit 403.b
Self dual yes
Analytic conductor 10.981
Analytic rank 0
Dimension 1
CM discriminant -403
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 403 = 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 403.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(10.9809546537\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\) \(=\) \( q + 4q^{4} + 9q^{9} + O(q^{10}) \) \( q + 4q^{4} + 9q^{9} - 9q^{11} + 13q^{13} + 16q^{16} + 25q^{25} - 31q^{31} + 36q^{36} + 43q^{37} - 36q^{44} + 49q^{49} + 52q^{52} + 64q^{64} - 133q^{73} + 81q^{81} + 42q^{83} + 147q^{89} - 81q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/403\mathbb{Z}\right)^\times\).

\(n\) \(249\) \(313\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
402.1
0
0 0 4.00000 0 0 0 0 9.00000 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
403.b odd 2 1 CM by \(\Q(\sqrt{-403}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 403.3.b.a 1
13.b even 2 1 403.3.b.b yes 1
31.b odd 2 1 403.3.b.b yes 1
403.b odd 2 1 CM 403.3.b.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
403.3.b.a 1 1.a even 1 1 trivial
403.3.b.a 1 403.b odd 2 1 CM
403.3.b.b yes 1 13.b even 2 1
403.3.b.b yes 1 31.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(403, [\chi])\):

\( T_{2} \)
\( T_{11} + 9 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 2 T )( 1 + 2 T ) \)
$3$ \( ( 1 - 3 T )( 1 + 3 T ) \)
$5$ \( ( 1 - 5 T )( 1 + 5 T ) \)
$7$ \( ( 1 - 7 T )( 1 + 7 T ) \)
$11$ \( 1 + 9 T + 121 T^{2} \)
$13$ \( 1 - 13 T \)
$17$ \( ( 1 - 17 T )( 1 + 17 T ) \)
$19$ \( ( 1 - 19 T )( 1 + 19 T ) \)
$23$ \( ( 1 - 23 T )( 1 + 23 T ) \)
$29$ \( ( 1 - 29 T )( 1 + 29 T ) \)
$31$ \( 1 + 31 T \)
$37$ \( 1 - 43 T + 1369 T^{2} \)
$41$ \( ( 1 - 41 T )( 1 + 41 T ) \)
$43$ \( ( 1 - 43 T )( 1 + 43 T ) \)
$47$ \( ( 1 - 47 T )( 1 + 47 T ) \)
$53$ \( ( 1 - 53 T )( 1 + 53 T ) \)
$59$ \( ( 1 - 59 T )( 1 + 59 T ) \)
$61$ \( ( 1 - 61 T )( 1 + 61 T ) \)
$67$ \( ( 1 - 67 T )( 1 + 67 T ) \)
$71$ \( ( 1 - 71 T )( 1 + 71 T ) \)
$73$ \( 1 + 133 T + 5329 T^{2} \)
$79$ \( ( 1 - 79 T )( 1 + 79 T ) \)
$83$ \( 1 - 42 T + 6889 T^{2} \)
$89$ \( 1 - 147 T + 7921 T^{2} \)
$97$ \( ( 1 - 97 T )( 1 + 97 T ) \)
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