# 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 $4$

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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

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