Properties

Label 2312.1.f.b
Level $2312$
Weight $1$
Character orbit 2312.f
Self dual yes
Analytic conductor $1.154$
Analytic rank $0$
Dimension $2$
Projective image $D_{4}$
CM discriminant -8
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2312 = 2^{3} \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2312.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(1.15383830921\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 136)
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.0.314432.1
Artin image: $D_8$
Artin field: Galois closure of 8.2.210093400576.1

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} -\beta q^{3} + q^{4} -\beta q^{6} + q^{8} + q^{9} +O(q^{10})\) \( q + q^{2} -\beta q^{3} + q^{4} -\beta q^{6} + q^{8} + q^{9} + \beta q^{11} -\beta q^{12} + q^{16} + q^{18} -2 q^{19} + \beta q^{22} -\beta q^{24} + q^{25} + q^{32} -2 q^{33} + q^{36} -2 q^{38} + \beta q^{41} + \beta q^{44} -\beta q^{48} + q^{49} + q^{50} + 2 \beta q^{57} + q^{64} -2 q^{66} + q^{72} + \beta q^{73} -\beta q^{75} -2 q^{76} - q^{81} + \beta q^{82} + \beta q^{88} -\beta q^{96} -\beta q^{97} + q^{98} + \beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{4} + 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{4} + 2q^{8} + 2q^{9} + 2q^{16} + 2q^{18} - 4q^{19} + 2q^{25} + 2q^{32} - 4q^{33} + 2q^{36} - 4q^{38} + 2q^{49} + 2q^{50} + 2q^{64} - 4q^{66} + 2q^{72} - 4q^{76} - 2q^{81} + 2q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2312\mathbb{Z}\right)^\times\).

\(n\) \(1157\) \(1735\) \(1737\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
579.1
1.41421
−1.41421
1.00000 −1.41421 1.00000 0 −1.41421 0 1.00000 1.00000 0
579.2 1.00000 1.41421 1.00000 0 1.41421 0 1.00000 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
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
17.b even 2 1 inner
136.e odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2312.1.f.b 2
8.d odd 2 1 CM 2312.1.f.b 2
17.b even 2 1 inner 2312.1.f.b 2
17.c even 4 2 2312.1.e.a 2
17.d even 8 2 136.1.j.a 2
17.d even 8 2 2312.1.j.b 2
17.e odd 16 8 2312.1.p.e 8
51.g odd 8 2 1224.1.s.a 2
68.g odd 8 2 544.1.n.a 2
85.k odd 8 2 3400.1.bc.a 2
85.m even 8 2 3400.1.y.a 2
85.n odd 8 2 3400.1.bc.b 2
136.e odd 2 1 inner 2312.1.f.b 2
136.j odd 4 2 2312.1.e.a 2
136.o even 8 2 544.1.n.a 2
136.p odd 8 2 136.1.j.a 2
136.p odd 8 2 2312.1.j.b 2
136.s even 16 8 2312.1.p.e 8
408.bd even 8 2 1224.1.s.a 2
680.bq odd 8 2 3400.1.y.a 2
680.bw even 8 2 3400.1.bc.a 2
680.bz even 8 2 3400.1.bc.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.1.j.a 2 17.d even 8 2
136.1.j.a 2 136.p odd 8 2
544.1.n.a 2 68.g odd 8 2
544.1.n.a 2 136.o even 8 2
1224.1.s.a 2 51.g odd 8 2
1224.1.s.a 2 408.bd even 8 2
2312.1.e.a 2 17.c even 4 2
2312.1.e.a 2 136.j odd 4 2
2312.1.f.b 2 1.a even 1 1 trivial
2312.1.f.b 2 8.d odd 2 1 CM
2312.1.f.b 2 17.b even 2 1 inner
2312.1.f.b 2 136.e odd 2 1 inner
2312.1.j.b 2 17.d even 8 2
2312.1.j.b 2 136.p odd 8 2
2312.1.p.e 8 17.e odd 16 8
2312.1.p.e 8 136.s even 16 8
3400.1.y.a 2 85.m even 8 2
3400.1.y.a 2 680.bq odd 8 2
3400.1.bc.a 2 85.k odd 8 2
3400.1.bc.a 2 680.bw even 8 2
3400.1.bc.b 2 85.n odd 8 2
3400.1.bc.b 2 680.bz even 8 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 2 \) acting on \(S_{1}^{\mathrm{new}}(2312, [\chi])\).

Hecke characteristic polynomials

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