Properties

Label 31.1.b.a
Level 31
Weight 1
Character orbit 31.b
Self dual yes
Analytic conductor 0.015
Analytic rank 0
Dimension 1
Projective image \(D_{3}\)
CM discriminant -31
Inner twists 2

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(0.0154710153916\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{3}\)
Projective field Galois closure of 3.1.31.1
Artin image $S_3$
Artin field Galois closure of 3.1.31.1

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} - q^{5} - q^{7} + q^{8} + q^{9} + O(q^{10}) \) \( q - q^{2} - q^{5} - q^{7} + q^{8} + q^{9} + q^{10} + q^{14} - q^{16} - q^{18} - q^{19} + q^{31} + q^{35} + q^{38} - q^{40} - q^{41} - q^{45} + 2q^{47} - q^{56} - q^{59} - q^{62} - q^{63} + q^{64} + 2q^{67} - q^{70} - q^{71} + q^{72} + q^{80} + q^{81} + q^{82} + q^{90} - 2q^{94} + q^{95} - q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/31\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
30.1
0
−1.00000 0 0 −1.00000 0 −1.00000 1.00000 1.00000 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.b odd 2 1 CM by \(\Q(\sqrt{-31}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 31.1.b.a 1
3.b odd 2 1 279.1.d.b 1
4.b odd 2 1 496.1.e.a 1
5.b even 2 1 775.1.d.b 1
5.c odd 4 2 775.1.c.a 2
7.b odd 2 1 1519.1.c.a 1
7.c even 3 2 1519.1.n.b 2
7.d odd 6 2 1519.1.n.a 2
8.b even 2 1 1984.1.e.a 1
8.d odd 2 1 1984.1.e.b 1
9.c even 3 2 2511.1.m.e 2
9.d odd 6 2 2511.1.m.a 2
11.b odd 2 1 3751.1.d.b 1
11.c even 5 4 3751.1.t.c 4
11.d odd 10 4 3751.1.t.a 4
31.b odd 2 1 CM 31.1.b.a 1
31.c even 3 2 961.1.e.a 2
31.d even 5 4 961.1.f.a 4
31.e odd 6 2 961.1.e.a 2
31.f odd 10 4 961.1.f.a 4
31.g even 15 8 961.1.h.a 8
31.h odd 30 8 961.1.h.a 8
93.c even 2 1 279.1.d.b 1
124.d even 2 1 496.1.e.a 1
155.c odd 2 1 775.1.d.b 1
155.f even 4 2 775.1.c.a 2
217.d even 2 1 1519.1.c.a 1
217.m even 6 2 1519.1.n.a 2
217.n odd 6 2 1519.1.n.b 2
248.b even 2 1 1984.1.e.b 1
248.g odd 2 1 1984.1.e.a 1
279.m odd 6 2 2511.1.m.e 2
279.s even 6 2 2511.1.m.a 2
341.b even 2 1 3751.1.d.b 1
341.t odd 10 4 3751.1.t.c 4
341.ba even 10 4 3751.1.t.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.1.b.a 1 1.a even 1 1 trivial
31.1.b.a 1 31.b odd 2 1 CM
279.1.d.b 1 3.b odd 2 1
279.1.d.b 1 93.c even 2 1
496.1.e.a 1 4.b odd 2 1
496.1.e.a 1 124.d even 2 1
775.1.c.a 2 5.c odd 4 2
775.1.c.a 2 155.f even 4 2
775.1.d.b 1 5.b even 2 1
775.1.d.b 1 155.c odd 2 1
961.1.e.a 2 31.c even 3 2
961.1.e.a 2 31.e odd 6 2
961.1.f.a 4 31.d even 5 4
961.1.f.a 4 31.f odd 10 4
961.1.h.a 8 31.g even 15 8
961.1.h.a 8 31.h odd 30 8
1519.1.c.a 1 7.b odd 2 1
1519.1.c.a 1 217.d even 2 1
1519.1.n.a 2 7.d odd 6 2
1519.1.n.a 2 217.m even 6 2
1519.1.n.b 2 7.c even 3 2
1519.1.n.b 2 217.n odd 6 2
1984.1.e.a 1 8.b even 2 1
1984.1.e.a 1 248.g odd 2 1
1984.1.e.b 1 8.d odd 2 1
1984.1.e.b 1 248.b even 2 1
2511.1.m.a 2 9.d odd 6 2
2511.1.m.a 2 279.s even 6 2
2511.1.m.e 2 9.c even 3 2
2511.1.m.e 2 279.m odd 6 2
3751.1.d.b 1 11.b odd 2 1
3751.1.d.b 1 341.b even 2 1
3751.1.t.a 4 11.d odd 10 4
3751.1.t.a 4 341.ba even 10 4
3751.1.t.c 4 11.c even 5 4
3751.1.t.c 4 341.t odd 10 4

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(31, [\chi])\).

Hecke characteristic polynomials

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