Properties

Label 76.1.c.a
Level 76
Weight 1
Character orbit 76.c
Self dual yes
Analytic conductor 0.038
Analytic rank 0
Dimension 1
Projective image \(D_{3}\)
CM discriminant -19
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 76.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.0379289409601\)
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 3.1.76.1
Artin image $S_3$
Artin field Galois closure of 3.1.76.1

$q$-expansion

\(f(q)\) \(=\) \( q - q^{5} - q^{7} + q^{9} + O(q^{10}) \) \( q - q^{5} - q^{7} + q^{9} - q^{11} - q^{17} + q^{19} + 2q^{23} + q^{35} - q^{43} - q^{45} - q^{47} + q^{55} - q^{61} - q^{63} - q^{73} + q^{77} + q^{81} + 2q^{83} + q^{85} - q^{95} - q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/76\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(39\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
0
0 0 0 −1.00000 0 −1.00000 0 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
19.b odd 2 1 CM by \(\Q(\sqrt{-19}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 76.1.c.a 1
3.b odd 2 1 684.1.h.a 1
4.b odd 2 1 304.1.e.a 1
5.b even 2 1 1900.1.e.a 1
5.c odd 4 2 1900.1.g.a 2
7.b odd 2 1 3724.1.e.c 1
7.c even 3 2 3724.1.bc.c 2
7.d odd 6 2 3724.1.bc.b 2
8.b even 2 1 1216.1.e.a 1
8.d odd 2 1 1216.1.e.b 1
12.b even 2 1 2736.1.o.b 1
19.b odd 2 1 CM 76.1.c.a 1
19.c even 3 2 1444.1.h.a 2
19.d odd 6 2 1444.1.h.a 2
19.e even 9 6 1444.1.j.a 6
19.f odd 18 6 1444.1.j.a 6
57.d even 2 1 684.1.h.a 1
76.d even 2 1 304.1.e.a 1
95.d odd 2 1 1900.1.e.a 1
95.g even 4 2 1900.1.g.a 2
133.c even 2 1 3724.1.e.c 1
133.o even 6 2 3724.1.bc.b 2
133.r odd 6 2 3724.1.bc.c 2
152.b even 2 1 1216.1.e.b 1
152.g odd 2 1 1216.1.e.a 1
228.b odd 2 1 2736.1.o.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.1.c.a 1 1.a even 1 1 trivial
76.1.c.a 1 19.b odd 2 1 CM
304.1.e.a 1 4.b odd 2 1
304.1.e.a 1 76.d even 2 1
684.1.h.a 1 3.b odd 2 1
684.1.h.a 1 57.d even 2 1
1216.1.e.a 1 8.b even 2 1
1216.1.e.a 1 152.g odd 2 1
1216.1.e.b 1 8.d odd 2 1
1216.1.e.b 1 152.b even 2 1
1444.1.h.a 2 19.c even 3 2
1444.1.h.a 2 19.d odd 6 2
1444.1.j.a 6 19.e even 9 6
1444.1.j.a 6 19.f odd 18 6
1900.1.e.a 1 5.b even 2 1
1900.1.e.a 1 95.d odd 2 1
1900.1.g.a 2 5.c odd 4 2
1900.1.g.a 2 95.g even 4 2
2736.1.o.b 1 12.b even 2 1
2736.1.o.b 1 228.b odd 2 1
3724.1.e.c 1 7.b odd 2 1
3724.1.e.c 1 133.c even 2 1
3724.1.bc.b 2 7.d odd 6 2
3724.1.bc.b 2 133.o even 6 2
3724.1.bc.c 2 7.c even 3 2
3724.1.bc.c 2 133.r odd 6 2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(76, [\chi])\).

Hecke characteristic polynomials

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