Properties

Label 775.1.d.c
Level $775$
Weight $1$
Character orbit 775.d
Self dual yes
Analytic conductor $0.387$
Analytic rank $0$
Dimension $2$
Projective image $D_{6}$
CM discriminant -31
Inner twists $4$

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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: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 155)
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.2.120125.1

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta q^{2} + 2 q^{4} + \beta q^{7} -\beta q^{8} + q^{9} +O(q^{10})\) \( q -\beta q^{2} + 2 q^{4} + \beta q^{7} -\beta q^{8} + q^{9} -3 q^{14} + q^{16} -\beta q^{18} - q^{19} + 2 \beta q^{28} - q^{31} + 2 q^{36} + \beta q^{38} - q^{41} + 2 q^{49} -3 q^{56} + q^{59} + \beta q^{62} + \beta q^{63} - q^{64} + q^{71} -\beta q^{72} -2 q^{76} + q^{81} + \beta q^{82} -\beta q^{97} -2 \beta q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{4} + 2 q^{9} + O(q^{10}) \) \( 2 q + 4 q^{4} + 2 q^{9} - 6 q^{14} + 2 q^{16} - 2 q^{19} - 2 q^{31} + 4 q^{36} - 2 q^{41} + 4 q^{49} - 6 q^{56} + 2 q^{59} - 2 q^{64} + 2 q^{71} - 4 q^{76} + 2 q^{81} + O(q^{100}) \)

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
1.73205
−1.73205
−1.73205 0 2.00000 0 0 1.73205 −1.73205 1.00000 0
526.2 1.73205 0 2.00000 0 0 −1.73205 1.73205 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 \(\Q(\sqrt{-31}) \)
5.b even 2 1 inner
155.c odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 775.1.d.c 2
5.b even 2 1 inner 775.1.d.c 2
5.c odd 4 2 155.1.c.b 2
15.e even 4 2 1395.1.b.c 2
20.e even 4 2 2480.1.k.e 2
31.b odd 2 1 CM 775.1.d.c 2
155.c odd 2 1 inner 775.1.d.c 2
155.f even 4 2 155.1.c.b 2
465.m odd 4 2 1395.1.b.c 2
620.m odd 4 2 2480.1.k.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
155.1.c.b 2 5.c odd 4 2
155.1.c.b 2 155.f even 4 2
775.1.d.c 2 1.a even 1 1 trivial
775.1.d.c 2 5.b even 2 1 inner
775.1.d.c 2 31.b odd 2 1 CM
775.1.d.c 2 155.c odd 2 1 inner
1395.1.b.c 2 15.e even 4 2
1395.1.b.c 2 465.m odd 4 2
2480.1.k.e 2 20.e even 4 2
2480.1.k.e 2 620.m odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 3 \) acting on \(S_{1}^{\mathrm{new}}(775, [\chi])\).

Hecke characteristic polynomials

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