Properties

Label 756.1.h.c
Level $756$
Weight $1$
Character orbit 756.h
Self dual yes
Analytic conductor $0.377$
Analytic rank $0$
Dimension $1$
Projective image $D_{3}$
CM discriminant -84
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 756.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.377293149551\)
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 3.1.756.1
Artin image: $S_3$
Artin field: Galois closure of 3.1.756.1

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} + q^{4} - q^{5} + q^{7} + q^{8} + O(q^{10}) \) \( q + q^{2} + q^{4} - q^{5} + q^{7} + q^{8} - q^{10} - q^{11} + q^{14} + q^{16} + 2 q^{17} - q^{19} - q^{20} - q^{22} - q^{23} + q^{28} - q^{31} + q^{32} + 2 q^{34} - q^{35} - q^{37} - q^{38} - q^{40} - q^{41} - q^{44} - q^{46} + q^{49} + q^{55} + q^{56} - q^{62} + q^{64} + 2 q^{68} - q^{70} - q^{71} - q^{74} - q^{76} - q^{77} - q^{80} - q^{82} - 2 q^{85} - q^{88} - q^{89} - q^{92} + q^{95} + q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/756\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(325\) \(379\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
755.1
0
1.00000 0 1.00000 −1.00000 0 1.00000 1.00000 0 −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
84.h odd 2 1 CM by \(\Q(\sqrt{-21}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 756.1.h.c yes 1
3.b odd 2 1 756.1.h.b yes 1
4.b odd 2 1 756.1.h.a 1
7.b odd 2 1 756.1.h.d yes 1
9.c even 3 2 2268.1.s.b 2
9.d odd 6 2 2268.1.s.c 2
12.b even 2 1 756.1.h.d yes 1
21.c even 2 1 756.1.h.a 1
28.d even 2 1 756.1.h.b yes 1
36.f odd 6 2 2268.1.s.d 2
36.h even 6 2 2268.1.s.a 2
63.l odd 6 2 2268.1.s.a 2
63.o even 6 2 2268.1.s.d 2
84.h odd 2 1 CM 756.1.h.c yes 1
252.s odd 6 2 2268.1.s.b 2
252.bi even 6 2 2268.1.s.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.1.h.a 1 4.b odd 2 1
756.1.h.a 1 21.c even 2 1
756.1.h.b yes 1 3.b odd 2 1
756.1.h.b yes 1 28.d even 2 1
756.1.h.c yes 1 1.a even 1 1 trivial
756.1.h.c yes 1 84.h odd 2 1 CM
756.1.h.d yes 1 7.b odd 2 1
756.1.h.d yes 1 12.b even 2 1
2268.1.s.a 2 36.h even 6 2
2268.1.s.a 2 63.l odd 6 2
2268.1.s.b 2 9.c even 3 2
2268.1.s.b 2 252.s odd 6 2
2268.1.s.c 2 9.d odd 6 2
2268.1.s.c 2 252.bi even 6 2
2268.1.s.d 2 36.f odd 6 2
2268.1.s.d 2 63.o even 6 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(756, [\chi])\):

\( T_{5} + 1 \)
\( T_{11} + 1 \)

Hecke characteristic polynomials

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