Properties

Label 1575.1.e.a
Level $1575$
Weight $1$
Character orbit 1575.e
Analytic conductor $0.786$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -7
Inner twists $4$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1575.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.786027394897\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
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

$q$-expansion

The \(q\)-expansion and trace form are shown below.

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
874.1
1.00000i
1.00000i
1.00000i 0 0 0 0 1.00000i 1.00000i 0 0
874.2 1.00000i 0 0 0 0 1.00000i 1.00000i 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 \(\Q(\sqrt{-7}) \)
5.b even 2 1 inner
35.c odd 2 1 inner

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 1 \) acting on \(S_{1}^{\mathrm{new}}(1575, [\chi])\).