Properties

Label 1575.1.y.a
Level $1575$
Weight $1$
Character orbit 1575.y
Analytic conductor $0.786$
Analytic rank $0$
Dimension $4$
Projective image $D_{3}$
CM discriminant -35
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1575.y (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.786027394897\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 315)
Projective image \(D_{3}\)
Projective field Galois closure of 3.1.2835.1
Artin image $C_{12}\times S_3$
Artin field Galois closure of \(\mathbb{Q}[x]/(x^{24} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{12}^{3} q^{3} + \zeta_{12}^{2} q^{4} -\zeta_{12} q^{7} - q^{9} +O(q^{10})\) \( q -\zeta_{12}^{3} q^{3} + \zeta_{12}^{2} q^{4} -\zeta_{12} q^{7} - q^{9} -\zeta_{12}^{4} q^{11} -\zeta_{12}^{5} q^{12} -\zeta_{12}^{5} q^{13} + \zeta_{12}^{4} q^{16} -\zeta_{12}^{3} q^{17} + \zeta_{12}^{4} q^{21} + \zeta_{12}^{3} q^{27} -\zeta_{12}^{3} q^{28} -2 \zeta_{12}^{4} q^{29} -\zeta_{12} q^{33} -\zeta_{12}^{2} q^{36} -\zeta_{12}^{2} q^{39} + q^{44} + \zeta_{12} q^{47} + \zeta_{12} q^{48} + \zeta_{12}^{2} q^{49} - q^{51} + \zeta_{12} q^{52} + \zeta_{12} q^{63} - q^{64} -\zeta_{12}^{5} q^{68} - q^{71} + \zeta_{12}^{3} q^{73} + \zeta_{12}^{5} q^{77} + \zeta_{12}^{4} q^{79} + q^{81} -\zeta_{12} q^{83} - q^{84} -2 \zeta_{12} q^{87} - q^{91} + \zeta_{12} q^{97} + \zeta_{12}^{4} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{4} - 4q^{9} + O(q^{10}) \) \( 4q + 2q^{4} - 4q^{9} + 2q^{11} - 2q^{16} - 2q^{21} + 4q^{29} - 2q^{36} - 2q^{39} + 4q^{44} + 2q^{49} - 4q^{51} - 4q^{64} - 4q^{71} - 2q^{79} + 4q^{81} - 4q^{84} - 4q^{91} - 2q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(127\) \(451\) \(1226\)
\(\chi(n)\) \(1\) \(-1\) \(-\zeta_{12}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
76.1
−0.866025 + 0.500000i
0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
0 1.00000i 0.500000 0.866025i 0 0 0.866025 0.500000i 0 −1.00000 0
76.2 0 1.00000i 0.500000 0.866025i 0 0 −0.866025 + 0.500000i 0 −1.00000 0
601.1 0 1.00000i 0.500000 + 0.866025i 0 0 −0.866025 0.500000i 0 −1.00000 0
601.2 0 1.00000i 0.500000 + 0.866025i 0 0 0.866025 + 0.500000i 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
35.c odd 2 1 CM by \(\Q(\sqrt{-35}) \)
5.b even 2 1 inner
7.b odd 2 1 inner
9.c even 3 1 inner
45.j even 6 1 inner
63.l odd 6 1 inner
315.bg odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1575.1.y.a 4
5.b even 2 1 inner 1575.1.y.a 4
5.c odd 4 1 315.1.bg.a 2
5.c odd 4 1 315.1.bg.b yes 2
7.b odd 2 1 inner 1575.1.y.a 4
9.c even 3 1 inner 1575.1.y.a 4
15.e even 4 1 945.1.bg.a 2
15.e even 4 1 945.1.bg.b 2
35.c odd 2 1 CM 1575.1.y.a 4
35.f even 4 1 315.1.bg.a 2
35.f even 4 1 315.1.bg.b yes 2
35.k even 12 1 2205.1.q.a 2
35.k even 12 1 2205.1.q.b 2
35.k even 12 1 2205.1.bn.a 2
35.k even 12 1 2205.1.bn.b 2
35.l odd 12 1 2205.1.q.a 2
35.l odd 12 1 2205.1.q.b 2
35.l odd 12 1 2205.1.bn.a 2
35.l odd 12 1 2205.1.bn.b 2
45.j even 6 1 inner 1575.1.y.a 4
45.k odd 12 1 315.1.bg.a 2
45.k odd 12 1 315.1.bg.b yes 2
45.k odd 12 1 2835.1.e.a 1
45.k odd 12 1 2835.1.e.d 1
45.l even 12 1 945.1.bg.a 2
45.l even 12 1 945.1.bg.b 2
45.l even 12 1 2835.1.e.b 1
45.l even 12 1 2835.1.e.c 1
63.l odd 6 1 inner 1575.1.y.a 4
105.k odd 4 1 945.1.bg.a 2
105.k odd 4 1 945.1.bg.b 2
315.bg odd 6 1 inner 1575.1.y.a 4
315.bs even 12 1 2205.1.bn.a 2
315.bs even 12 1 2205.1.bn.b 2
315.bt odd 12 1 2205.1.bn.a 2
315.bt odd 12 1 2205.1.bn.b 2
315.cb even 12 1 315.1.bg.a 2
315.cb even 12 1 315.1.bg.b yes 2
315.cb even 12 1 2835.1.e.a 1
315.cb even 12 1 2835.1.e.d 1
315.cf odd 12 1 945.1.bg.a 2
315.cf odd 12 1 945.1.bg.b 2
315.cf odd 12 1 2835.1.e.b 1
315.cf odd 12 1 2835.1.e.c 1
315.cg even 12 1 2205.1.q.a 2
315.cg even 12 1 2205.1.q.b 2
315.ch odd 12 1 2205.1.q.a 2
315.ch odd 12 1 2205.1.q.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.1.bg.a 2 5.c odd 4 1
315.1.bg.a 2 35.f even 4 1
315.1.bg.a 2 45.k odd 12 1
315.1.bg.a 2 315.cb even 12 1
315.1.bg.b yes 2 5.c odd 4 1
315.1.bg.b yes 2 35.f even 4 1
315.1.bg.b yes 2 45.k odd 12 1
315.1.bg.b yes 2 315.cb even 12 1
945.1.bg.a 2 15.e even 4 1
945.1.bg.a 2 45.l even 12 1
945.1.bg.a 2 105.k odd 4 1
945.1.bg.a 2 315.cf odd 12 1
945.1.bg.b 2 15.e even 4 1
945.1.bg.b 2 45.l even 12 1
945.1.bg.b 2 105.k odd 4 1
945.1.bg.b 2 315.cf odd 12 1
1575.1.y.a 4 1.a even 1 1 trivial
1575.1.y.a 4 5.b even 2 1 inner
1575.1.y.a 4 7.b odd 2 1 inner
1575.1.y.a 4 9.c even 3 1 inner
1575.1.y.a 4 35.c odd 2 1 CM
1575.1.y.a 4 45.j even 6 1 inner
1575.1.y.a 4 63.l odd 6 1 inner
1575.1.y.a 4 315.bg odd 6 1 inner
2205.1.q.a 2 35.k even 12 1
2205.1.q.a 2 35.l odd 12 1
2205.1.q.a 2 315.cg even 12 1
2205.1.q.a 2 315.ch odd 12 1
2205.1.q.b 2 35.k even 12 1
2205.1.q.b 2 35.l odd 12 1
2205.1.q.b 2 315.cg even 12 1
2205.1.q.b 2 315.ch odd 12 1
2205.1.bn.a 2 35.k even 12 1
2205.1.bn.a 2 35.l odd 12 1
2205.1.bn.a 2 315.bs even 12 1
2205.1.bn.a 2 315.bt odd 12 1
2205.1.bn.b 2 35.k even 12 1
2205.1.bn.b 2 35.l odd 12 1
2205.1.bn.b 2 315.bs even 12 1
2205.1.bn.b 2 315.bt odd 12 1
2835.1.e.a 1 45.k odd 12 1
2835.1.e.a 1 315.cb even 12 1
2835.1.e.b 1 45.l even 12 1
2835.1.e.b 1 315.cf odd 12 1
2835.1.e.c 1 45.l even 12 1
2835.1.e.c 1 315.cf odd 12 1
2835.1.e.d 1 45.k odd 12 1
2835.1.e.d 1 315.cb even 12 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1575, [\chi])\).