Properties

Label 1575.1.h.c
Level $1575$
Weight $1$
Character orbit 1575.h
Self dual yes
Analytic conductor $0.786$
Analytic rank $0$
Dimension $1$
Projective image $D_{3}$
CM discriminant -7
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1575.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.786027394897\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 175)
Projective image \(D_{3}\)
Projective field Galois closure of 3.1.175.1
Artin image $D_6$
Artin field Galois closure of 6.0.826875.1

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} + q^{7} - q^{8} + O(q^{10}) \) \( q + q^{2} + q^{7} - q^{8} + q^{11} + q^{14} - q^{16} + q^{22} + q^{23} + q^{29} - q^{37} - q^{43} + q^{46} + q^{49} - 2q^{53} - q^{56} + q^{58} + q^{64} - q^{67} + q^{71} - q^{74} + q^{77} - q^{79} - q^{86} - q^{88} + q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1575\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(1226\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1126.1
0
1.00000 0 0 0 0 1.00000 −1.00000 0 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
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1575.1.h.c 1
3.b odd 2 1 175.1.d.a 1
5.b even 2 1 1575.1.h.a 1
5.c odd 4 2 1575.1.e.a 2
7.b odd 2 1 CM 1575.1.h.c 1
12.b even 2 1 2800.1.f.a 1
15.d odd 2 1 175.1.d.b yes 1
15.e even 4 2 175.1.c.a 2
21.c even 2 1 175.1.d.a 1
21.g even 6 2 1225.1.i.b 2
21.h odd 6 2 1225.1.i.b 2
35.c odd 2 1 1575.1.h.a 1
35.f even 4 2 1575.1.e.a 2
60.h even 2 1 2800.1.f.b 1
60.l odd 4 2 2800.1.p.a 2
84.h odd 2 1 2800.1.f.a 1
105.g even 2 1 175.1.d.b yes 1
105.k odd 4 2 175.1.c.a 2
105.o odd 6 2 1225.1.i.a 2
105.p even 6 2 1225.1.i.a 2
105.w odd 12 4 1225.1.j.a 4
105.x even 12 4 1225.1.j.a 4
420.o odd 2 1 2800.1.f.b 1
420.w even 4 2 2800.1.p.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
175.1.c.a 2 15.e even 4 2
175.1.c.a 2 105.k odd 4 2
175.1.d.a 1 3.b odd 2 1
175.1.d.a 1 21.c even 2 1
175.1.d.b yes 1 15.d odd 2 1
175.1.d.b yes 1 105.g even 2 1
1225.1.i.a 2 105.o odd 6 2
1225.1.i.a 2 105.p even 6 2
1225.1.i.b 2 21.g even 6 2
1225.1.i.b 2 21.h odd 6 2
1225.1.j.a 4 105.w odd 12 4
1225.1.j.a 4 105.x even 12 4
1575.1.e.a 2 5.c odd 4 2
1575.1.e.a 2 35.f even 4 2
1575.1.h.a 1 5.b even 2 1
1575.1.h.a 1 35.c odd 2 1
1575.1.h.c 1 1.a even 1 1 trivial
1575.1.h.c 1 7.b odd 2 1 CM
2800.1.f.a 1 12.b even 2 1
2800.1.f.a 1 84.h odd 2 1
2800.1.f.b 1 60.h even 2 1
2800.1.f.b 1 420.o odd 2 1
2800.1.p.a 2 60.l odd 4 2
2800.1.p.a 2 420.w even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(1575, [\chi])\):

\( T_{2} - 1 \)
\( T_{37} + 1 \)