Properties

Label 945.1.bg.a
Level $945$
Weight $1$
Character orbit 945.bg
Analytic conductor $0.472$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -35
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 945.bg (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.471616436938\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
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 $C_6\times S_3$
Artin field Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{6} q^{4} -\zeta_{6} q^{5} -\zeta_{6}^{2} q^{7} +O(q^{10})\) \( q -\zeta_{6} q^{4} -\zeta_{6} q^{5} -\zeta_{6}^{2} q^{7} + \zeta_{6}^{2} q^{11} -\zeta_{6} q^{13} + \zeta_{6}^{2} q^{16} - q^{17} + \zeta_{6}^{2} q^{20} + \zeta_{6}^{2} q^{25} - q^{28} -2 \zeta_{6}^{2} q^{29} - q^{35} + q^{44} -\zeta_{6}^{2} q^{47} -\zeta_{6} q^{49} + \zeta_{6}^{2} q^{52} + q^{55} + q^{64} + \zeta_{6}^{2} q^{65} + \zeta_{6} q^{68} + q^{71} + q^{73} + \zeta_{6} q^{77} -\zeta_{6}^{2} q^{79} + q^{80} -\zeta_{6}^{2} q^{83} + \zeta_{6} q^{85} - q^{91} + \zeta_{6}^{2} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{4} - q^{5} + q^{7} + O(q^{10}) \) \( 2q - q^{4} - q^{5} + q^{7} - q^{11} - q^{13} - q^{16} - 2q^{17} - q^{20} - q^{25} - 2q^{28} + 2q^{29} - 2q^{35} + 2q^{44} + q^{47} - q^{49} - q^{52} + 2q^{55} + 2q^{64} - q^{65} + q^{68} + 2q^{71} + 2q^{73} + q^{77} + q^{79} + 2q^{80} + q^{83} + q^{85} - 2q^{91} - q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/945\mathbb{Z}\right)^\times\).

\(n\) \(136\) \(596\) \(757\)
\(\chi(n)\) \(-1\) \(-\zeta_{6}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
559.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 −0.500000 0.866025i −0.500000 0.866025i 0 0.500000 0.866025i 0 0 0
874.1 0 0 −0.500000 + 0.866025i −0.500000 + 0.866025i 0 0.500000 + 0.866025i 0 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
35.c odd 2 1 CM by \(\Q(\sqrt{-35}) \)
9.c even 3 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 945.1.bg.a 2
3.b odd 2 1 315.1.bg.a 2
5.b even 2 1 945.1.bg.b 2
7.b odd 2 1 945.1.bg.b 2
9.c even 3 1 inner 945.1.bg.a 2
9.c even 3 1 2835.1.e.c 1
9.d odd 6 1 315.1.bg.a 2
9.d odd 6 1 2835.1.e.a 1
15.d odd 2 1 315.1.bg.b yes 2
15.e even 4 2 1575.1.y.a 4
21.c even 2 1 315.1.bg.b yes 2
21.g even 6 1 2205.1.q.a 2
21.g even 6 1 2205.1.bn.a 2
21.h odd 6 1 2205.1.q.b 2
21.h odd 6 1 2205.1.bn.b 2
35.c odd 2 1 CM 945.1.bg.a 2
45.h odd 6 1 315.1.bg.b yes 2
45.h odd 6 1 2835.1.e.d 1
45.j even 6 1 945.1.bg.b 2
45.j even 6 1 2835.1.e.b 1
45.l even 12 2 1575.1.y.a 4
63.i even 6 1 2205.1.bn.a 2
63.j odd 6 1 2205.1.bn.b 2
63.l odd 6 1 945.1.bg.b 2
63.l odd 6 1 2835.1.e.b 1
63.n odd 6 1 2205.1.q.b 2
63.o even 6 1 315.1.bg.b yes 2
63.o even 6 1 2835.1.e.d 1
63.s even 6 1 2205.1.q.a 2
105.g even 2 1 315.1.bg.a 2
105.k odd 4 2 1575.1.y.a 4
105.o odd 6 1 2205.1.q.a 2
105.o odd 6 1 2205.1.bn.a 2
105.p even 6 1 2205.1.q.b 2
105.p even 6 1 2205.1.bn.b 2
315.u even 6 1 2205.1.q.b 2
315.v odd 6 1 2205.1.q.a 2
315.z even 6 1 315.1.bg.a 2
315.z even 6 1 2835.1.e.a 1
315.bg odd 6 1 inner 945.1.bg.a 2
315.bg odd 6 1 2835.1.e.c 1
315.bq even 6 1 2205.1.bn.b 2
315.br odd 6 1 2205.1.bn.a 2
315.cf odd 12 2 1575.1.y.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.1.bg.a 2 3.b odd 2 1
315.1.bg.a 2 9.d odd 6 1
315.1.bg.a 2 105.g even 2 1
315.1.bg.a 2 315.z even 6 1
315.1.bg.b yes 2 15.d odd 2 1
315.1.bg.b yes 2 21.c even 2 1
315.1.bg.b yes 2 45.h odd 6 1
315.1.bg.b yes 2 63.o even 6 1
945.1.bg.a 2 1.a even 1 1 trivial
945.1.bg.a 2 9.c even 3 1 inner
945.1.bg.a 2 35.c odd 2 1 CM
945.1.bg.a 2 315.bg odd 6 1 inner
945.1.bg.b 2 5.b even 2 1
945.1.bg.b 2 7.b odd 2 1
945.1.bg.b 2 45.j even 6 1
945.1.bg.b 2 63.l odd 6 1
1575.1.y.a 4 15.e even 4 2
1575.1.y.a 4 45.l even 12 2
1575.1.y.a 4 105.k odd 4 2
1575.1.y.a 4 315.cf odd 12 2
2205.1.q.a 2 21.g even 6 1
2205.1.q.a 2 63.s even 6 1
2205.1.q.a 2 105.o odd 6 1
2205.1.q.a 2 315.v odd 6 1
2205.1.q.b 2 21.h odd 6 1
2205.1.q.b 2 63.n odd 6 1
2205.1.q.b 2 105.p even 6 1
2205.1.q.b 2 315.u even 6 1
2205.1.bn.a 2 21.g even 6 1
2205.1.bn.a 2 63.i even 6 1
2205.1.bn.a 2 105.o odd 6 1
2205.1.bn.a 2 315.br odd 6 1
2205.1.bn.b 2 21.h odd 6 1
2205.1.bn.b 2 63.j odd 6 1
2205.1.bn.b 2 105.p even 6 1
2205.1.bn.b 2 315.bq even 6 1
2835.1.e.a 1 9.d odd 6 1
2835.1.e.a 1 315.z even 6 1
2835.1.e.b 1 45.j even 6 1
2835.1.e.b 1 63.l odd 6 1
2835.1.e.c 1 9.c even 3 1
2835.1.e.c 1 315.bg odd 6 1
2835.1.e.d 1 45.h odd 6 1
2835.1.e.d 1 63.o even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13}^{2} + T_{13} + 1 \) acting on \(S_{1}^{\mathrm{new}}(945, [\chi])\).