Properties

Label 1815.1.g.g
Level $1815$
Weight $1$
Character orbit 1815.g
Self dual yes
Analytic conductor $0.906$
Analytic rank $0$
Dimension $2$
Projective image $D_{6}$
CM discriminant -15
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1815 = 3 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1815.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.905802997929\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.0.36236475.1

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta q^{2} - q^{3} + 2 q^{4} + q^{5} + \beta q^{6} -\beta q^{8} + q^{9} +O(q^{10})\) \( q -\beta q^{2} - q^{3} + 2 q^{4} + q^{5} + \beta q^{6} -\beta q^{8} + q^{9} -\beta q^{10} -2 q^{12} - q^{15} + q^{16} + \beta q^{17} -\beta q^{18} + 2 q^{20} + q^{23} + \beta q^{24} + q^{25} - q^{27} + \beta q^{30} - q^{31} -3 q^{34} + 2 q^{36} -\beta q^{40} + q^{45} -\beta q^{46} - q^{47} - q^{48} + q^{49} -\beta q^{50} -\beta q^{51} - q^{53} + \beta q^{54} -2 q^{60} -\beta q^{61} + \beta q^{62} - q^{64} + 2 \beta q^{68} - q^{69} -\beta q^{72} - q^{75} + \beta q^{79} + q^{80} + q^{81} + \beta q^{85} -\beta q^{90} + 2 q^{92} + q^{93} + \beta q^{94} -\beta q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 4q^{4} + 2q^{5} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} + 4q^{4} + 2q^{5} + 2q^{9} - 4q^{12} - 2q^{15} + 2q^{16} + 4q^{20} + 2q^{23} + 2q^{25} - 2q^{27} - 2q^{31} - 6q^{34} + 4q^{36} + 2q^{45} - 2q^{47} - 2q^{48} + 2q^{49} - 2q^{53} - 4q^{60} - 2q^{64} - 2q^{69} - 2q^{75} + 2q^{80} + 2q^{81} + 4q^{92} + 2q^{93} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1815\mathbb{Z}\right)^\times\).

\(n\) \(727\) \(1211\) \(1696\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1574.1
1.73205
−1.73205
−1.73205 −1.00000 2.00000 1.00000 1.73205 0 −1.73205 1.00000 −1.73205
1574.2 1.73205 −1.00000 2.00000 1.00000 −1.73205 0 1.73205 1.00000 1.73205
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 CM by \(\Q(\sqrt{-15}) \)
11.b odd 2 1 inner
165.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1815.1.g.g 2
3.b odd 2 1 1815.1.g.h yes 2
5.b even 2 1 1815.1.g.h yes 2
11.b odd 2 1 inner 1815.1.g.g 2
11.c even 5 4 1815.1.o.h 8
11.d odd 10 4 1815.1.o.h 8
15.d odd 2 1 CM 1815.1.g.g 2
33.d even 2 1 1815.1.g.h yes 2
33.f even 10 4 1815.1.o.g 8
33.h odd 10 4 1815.1.o.g 8
55.d odd 2 1 1815.1.g.h yes 2
55.h odd 10 4 1815.1.o.g 8
55.j even 10 4 1815.1.o.g 8
165.d even 2 1 inner 1815.1.g.g 2
165.o odd 10 4 1815.1.o.h 8
165.r even 10 4 1815.1.o.h 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1815.1.g.g 2 1.a even 1 1 trivial
1815.1.g.g 2 11.b odd 2 1 inner
1815.1.g.g 2 15.d odd 2 1 CM
1815.1.g.g 2 165.d even 2 1 inner
1815.1.g.h yes 2 3.b odd 2 1
1815.1.g.h yes 2 5.b even 2 1
1815.1.g.h yes 2 33.d even 2 1
1815.1.g.h yes 2 55.d odd 2 1
1815.1.o.g 8 33.f even 10 4
1815.1.o.g 8 33.h odd 10 4
1815.1.o.g 8 55.h odd 10 4
1815.1.o.g 8 55.j even 10 4
1815.1.o.h 8 11.c even 5 4
1815.1.o.h 8 11.d odd 10 4
1815.1.o.h 8 165.o odd 10 4
1815.1.o.h 8 165.r even 10 4

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(1815, [\chi])\):

\( T_{2}^{2} - 3 \)
\( T_{19} \)
\( T_{23} - 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -3 + T^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( T^{2} \)
$11$ \( T^{2} \)
$13$ \( T^{2} \)
$17$ \( -3 + T^{2} \)
$19$ \( T^{2} \)
$23$ \( ( -1 + T )^{2} \)
$29$ \( T^{2} \)
$31$ \( ( 1 + T )^{2} \)
$37$ \( T^{2} \)
$41$ \( T^{2} \)
$43$ \( T^{2} \)
$47$ \( ( 1 + T )^{2} \)
$53$ \( ( 1 + T )^{2} \)
$59$ \( T^{2} \)
$61$ \( -3 + T^{2} \)
$67$ \( T^{2} \)
$71$ \( T^{2} \)
$73$ \( T^{2} \)
$79$ \( -3 + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( T^{2} \)
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