Properties

Label 2205.1.q.b
Level $2205$
Weight $1$
Character orbit 2205.q
Analytic conductor $1.100$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -35
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2205.q (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.10043835286\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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.

\(f(q)\) \(=\) \( q -\zeta_{6}^{2} q^{3} + q^{4} -\zeta_{6}^{2} q^{5} -\zeta_{6} q^{9} +O(q^{10})\) \( q -\zeta_{6}^{2} q^{3} + q^{4} -\zeta_{6}^{2} q^{5} -\zeta_{6} q^{9} + \zeta_{6} q^{11} -\zeta_{6}^{2} q^{12} -\zeta_{6} q^{13} -\zeta_{6} q^{15} + q^{16} + \zeta_{6}^{2} q^{17} -\zeta_{6}^{2} q^{20} -\zeta_{6} q^{25} - q^{27} + 2 \zeta_{6}^{2} q^{29} + q^{33} -\zeta_{6} q^{36} - q^{39} + \zeta_{6} q^{44} - q^{45} + q^{47} -\zeta_{6}^{2} q^{48} + \zeta_{6} q^{51} -\zeta_{6} q^{52} + q^{55} -\zeta_{6} q^{60} + q^{64} - q^{65} + \zeta_{6}^{2} q^{68} - q^{71} + \zeta_{6}^{2} q^{73} - q^{75} - q^{79} -\zeta_{6}^{2} q^{80} + \zeta_{6}^{2} q^{81} + \zeta_{6}^{2} q^{83} + \zeta_{6} q^{85} + 2 \zeta_{6} q^{87} + \zeta_{6}^{2} q^{97} -\zeta_{6}^{2} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{3} + 2q^{4} + q^{5} - q^{9} + O(q^{10}) \) \( 2q + q^{3} + 2q^{4} + q^{5} - q^{9} + q^{11} + q^{12} - q^{13} - q^{15} + 2q^{16} - q^{17} + q^{20} - q^{25} - 2q^{27} - 2q^{29} + 2q^{33} - q^{36} - 2q^{39} + q^{44} - 2q^{45} + 2q^{47} + q^{48} + q^{51} - q^{52} + 2q^{55} - q^{60} + 2q^{64} - 2q^{65} - q^{68} - 2q^{71} - q^{73} - 2q^{75} - 2q^{79} + q^{80} - q^{81} - q^{83} + q^{85} + 2q^{87} - q^{97} + q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2205\mathbb{Z}\right)^\times\).

\(n\) \(442\) \(1081\) \(1226\)
\(\chi(n)\) \(-1\) \(\zeta_{6}\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
619.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0.500000 0.866025i 1.00000 0.500000 0.866025i 0 0 0 −0.500000 0.866025i 0
1489.1 0 0.500000 + 0.866025i 1.00000 0.500000 + 0.866025i 0 0 0 −0.500000 + 0.866025i 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 \(\Q(\sqrt{-35}) \)
63.h even 3 1 inner
315.q odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2205.1.q.b 2
5.b even 2 1 2205.1.q.a 2
7.b odd 2 1 2205.1.q.a 2
7.c even 3 1 315.1.bg.a 2
7.c even 3 1 2205.1.bn.b 2
7.d odd 6 1 315.1.bg.b yes 2
7.d odd 6 1 2205.1.bn.a 2
9.c even 3 1 2205.1.bn.b 2
21.g even 6 1 945.1.bg.b 2
21.h odd 6 1 945.1.bg.a 2
35.c odd 2 1 CM 2205.1.q.b 2
35.i odd 6 1 315.1.bg.a 2
35.i odd 6 1 2205.1.bn.b 2
35.j even 6 1 315.1.bg.b yes 2
35.j even 6 1 2205.1.bn.a 2
35.k even 12 2 1575.1.y.a 4
35.l odd 12 2 1575.1.y.a 4
45.j even 6 1 2205.1.bn.a 2
63.g even 3 1 315.1.bg.a 2
63.h even 3 1 inner 2205.1.q.b 2
63.h even 3 1 2835.1.e.a 1
63.i even 6 1 2835.1.e.b 1
63.j odd 6 1 2835.1.e.c 1
63.k odd 6 1 315.1.bg.b yes 2
63.l odd 6 1 2205.1.bn.a 2
63.n odd 6 1 945.1.bg.a 2
63.s even 6 1 945.1.bg.b 2
63.t odd 6 1 2205.1.q.a 2
63.t odd 6 1 2835.1.e.d 1
105.o odd 6 1 945.1.bg.b 2
105.p even 6 1 945.1.bg.a 2
315.q odd 6 1 inner 2205.1.q.b 2
315.q odd 6 1 2835.1.e.a 1
315.r even 6 1 2205.1.q.a 2
315.r even 6 1 2835.1.e.d 1
315.u even 6 1 945.1.bg.a 2
315.v odd 6 1 945.1.bg.b 2
315.bg odd 6 1 2205.1.bn.b 2
315.bn odd 6 1 315.1.bg.a 2
315.bo even 6 1 315.1.bg.b yes 2
315.bq even 6 1 2835.1.e.c 1
315.br odd 6 1 2835.1.e.b 1
315.cg even 12 2 1575.1.y.a 4
315.ch 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 7.c even 3 1
315.1.bg.a 2 35.i odd 6 1
315.1.bg.a 2 63.g even 3 1
315.1.bg.a 2 315.bn odd 6 1
315.1.bg.b yes 2 7.d odd 6 1
315.1.bg.b yes 2 35.j even 6 1
315.1.bg.b yes 2 63.k odd 6 1
315.1.bg.b yes 2 315.bo even 6 1
945.1.bg.a 2 21.h odd 6 1
945.1.bg.a 2 63.n odd 6 1
945.1.bg.a 2 105.p even 6 1
945.1.bg.a 2 315.u even 6 1
945.1.bg.b 2 21.g even 6 1
945.1.bg.b 2 63.s even 6 1
945.1.bg.b 2 105.o odd 6 1
945.1.bg.b 2 315.v odd 6 1
1575.1.y.a 4 35.k even 12 2
1575.1.y.a 4 35.l odd 12 2
1575.1.y.a 4 315.cg even 12 2
1575.1.y.a 4 315.ch odd 12 2
2205.1.q.a 2 5.b even 2 1
2205.1.q.a 2 7.b odd 2 1
2205.1.q.a 2 63.t odd 6 1
2205.1.q.a 2 315.r even 6 1
2205.1.q.b 2 1.a even 1 1 trivial
2205.1.q.b 2 35.c odd 2 1 CM
2205.1.q.b 2 63.h even 3 1 inner
2205.1.q.b 2 315.q odd 6 1 inner
2205.1.bn.a 2 7.d odd 6 1
2205.1.bn.a 2 35.j even 6 1
2205.1.bn.a 2 45.j even 6 1
2205.1.bn.a 2 63.l odd 6 1
2205.1.bn.b 2 7.c even 3 1
2205.1.bn.b 2 9.c even 3 1
2205.1.bn.b 2 35.i odd 6 1
2205.1.bn.b 2 315.bg odd 6 1
2835.1.e.a 1 63.h even 3 1
2835.1.e.a 1 315.q odd 6 1
2835.1.e.b 1 63.i even 6 1
2835.1.e.b 1 315.br odd 6 1
2835.1.e.c 1 63.j odd 6 1
2835.1.e.c 1 315.bq even 6 1
2835.1.e.d 1 63.t odd 6 1
2835.1.e.d 1 315.r 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 \(S_{1}^{\mathrm{new}}(2205, [\chi])\).