Properties

Label 775.1.d.b
Level $775$
Weight $1$
Character orbit 775.d
Self dual yes
Analytic conductor $0.387$
Analytic rank $0$
Dimension $1$
Projective image $D_{3}$
CM discriminant -31
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 775 = 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 775.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.386775384791\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 31)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.31.1
Artin image: $D_6$
Artin field: Galois closure of 6.2.120125.1

$q$-expansion

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/775\mathbb{Z}\right)^\times\).

\(n\) \(251\) \(652\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
526.1
0
1.00000 0 0 0 0 1.00000 −1.00000 1.00000 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
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 775.1.d.b 1
5.b even 2 1 31.1.b.a 1
5.c odd 4 2 775.1.c.a 2
15.d odd 2 1 279.1.d.b 1
20.d odd 2 1 496.1.e.a 1
31.b odd 2 1 CM 775.1.d.b 1
35.c odd 2 1 1519.1.c.a 1
35.i odd 6 2 1519.1.n.a 2
35.j even 6 2 1519.1.n.b 2
40.e odd 2 1 1984.1.e.b 1
40.f even 2 1 1984.1.e.a 1
45.h odd 6 2 2511.1.m.a 2
45.j even 6 2 2511.1.m.e 2
55.d odd 2 1 3751.1.d.b 1
55.h odd 10 4 3751.1.t.a 4
55.j even 10 4 3751.1.t.c 4
155.c odd 2 1 31.1.b.a 1
155.f even 4 2 775.1.c.a 2
155.i odd 6 2 961.1.e.a 2
155.j even 6 2 961.1.e.a 2
155.m odd 10 4 961.1.f.a 4
155.n even 10 4 961.1.f.a 4
155.u even 30 8 961.1.h.a 8
155.v odd 30 8 961.1.h.a 8
465.g even 2 1 279.1.d.b 1
620.e even 2 1 496.1.e.a 1
1085.b even 2 1 1519.1.c.a 1
1085.bh odd 6 2 1519.1.n.b 2
1085.bn even 6 2 1519.1.n.a 2
1240.e odd 2 1 1984.1.e.a 1
1240.o even 2 1 1984.1.e.b 1
1395.ba even 6 2 2511.1.m.a 2
1395.bm odd 6 2 2511.1.m.e 2
1705.h even 2 1 3751.1.d.b 1
1705.bi even 10 4 3751.1.t.a 4
1705.cb odd 10 4 3751.1.t.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.1.b.a 1 5.b even 2 1
31.1.b.a 1 155.c odd 2 1
279.1.d.b 1 15.d odd 2 1
279.1.d.b 1 465.g even 2 1
496.1.e.a 1 20.d odd 2 1
496.1.e.a 1 620.e 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 1.a even 1 1 trivial
775.1.d.b 1 31.b odd 2 1 CM
961.1.e.a 2 155.i odd 6 2
961.1.e.a 2 155.j even 6 2
961.1.f.a 4 155.m odd 10 4
961.1.f.a 4 155.n even 10 4
961.1.h.a 8 155.u even 30 8
961.1.h.a 8 155.v odd 30 8
1519.1.c.a 1 35.c odd 2 1
1519.1.c.a 1 1085.b even 2 1
1519.1.n.a 2 35.i odd 6 2
1519.1.n.a 2 1085.bn even 6 2
1519.1.n.b 2 35.j even 6 2
1519.1.n.b 2 1085.bh odd 6 2
1984.1.e.a 1 40.f even 2 1
1984.1.e.a 1 1240.e odd 2 1
1984.1.e.b 1 40.e odd 2 1
1984.1.e.b 1 1240.o even 2 1
2511.1.m.a 2 45.h odd 6 2
2511.1.m.a 2 1395.ba even 6 2
2511.1.m.e 2 45.j even 6 2
2511.1.m.e 2 1395.bm odd 6 2
3751.1.d.b 1 55.d odd 2 1
3751.1.d.b 1 1705.h even 2 1
3751.1.t.a 4 55.h odd 10 4
3751.1.t.a 4 1705.bi even 10 4
3751.1.t.c 4 55.j even 10 4
3751.1.t.c 4 1705.cb odd 10 4

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 1 \) acting on \(S_{1}^{\mathrm{new}}(775, [\chi])\).

Hecke characteristic polynomials

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