# Properties

 Label 3969.1.b.a Level $3969$ Weight $1$ Character orbit 3969.b Analytic conductor $1.981$ Analytic rank $0$ Dimension $4$ Projective image $D_{12}$ CM discriminant -7 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3969 = 3^{4} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3969.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.98078903514$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{3})$$ Defining polynomial: $$x^{4} + 4 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{12}$$ Projective field: Galois closure of 12.2.136738899331083.1

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + ( -1 + \beta_{2} ) q^{4} + ( 2 \beta_{1} - \beta_{3} ) q^{8} +O(q^{10})$$ $$q -\beta_{1} q^{2} + ( -1 + \beta_{2} ) q^{4} + ( 2 \beta_{1} - \beta_{3} ) q^{8} + \beta_{3} q^{11} + ( 2 - \beta_{2} ) q^{16} + q^{22} + ( -\beta_{1} + \beta_{3} ) q^{23} + q^{25} + ( -\beta_{1} + \beta_{3} ) q^{29} -2 \beta_{1} q^{32} -\beta_{2} q^{37} + \beta_{2} q^{43} + ( -\beta_{1} + \beta_{3} ) q^{44} + ( -1 + \beta_{2} ) q^{46} -\beta_{1} q^{50} -\beta_{3} q^{53} + ( -1 + \beta_{2} ) q^{58} + ( -2 + \beta_{2} ) q^{64} + q^{67} -\beta_{1} q^{71} + ( -2 \beta_{1} + \beta_{3} ) q^{74} - q^{79} + ( 2 \beta_{1} - \beta_{3} ) q^{86} + \beta_{2} q^{88} + 2 \beta_{1} q^{92} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4} + O(q^{10})$$ $$4 q - 4 q^{4} + 8 q^{16} + 4 q^{22} + 4 q^{25} - 4 q^{46} - 4 q^{58} - 8 q^{64} + 4 q^{67} - 4 q^{79} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 4 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} + 4 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 2$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} - 4 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3969\mathbb{Z}\right)^\times$$.

 $$n$$ $$2108$$ $$3727$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3725.1
 1.93185i 0.517638i − 0.517638i − 1.93185i
1.93185i 0 −2.73205 0 0 0 3.34607i 0 0
3725.2 0.517638i 0 0.732051 0 0 0 0.896575i 0 0
3725.3 0.517638i 0 0.732051 0 0 0 0.896575i 0 0
3725.4 1.93185i 0 −2.73205 0 0 0 3.34607i 0 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
7.b odd 2 1 CM by $$\Q(\sqrt{-7})$$
3.b odd 2 1 inner
21.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3969.1.b.a 4
3.b odd 2 1 inner 3969.1.b.a 4
7.b odd 2 1 CM 3969.1.b.a 4
7.c even 3 2 3969.1.q.a 8
7.d odd 6 2 3969.1.q.a 8
9.c even 3 2 3969.1.r.d 8
9.d odd 6 2 3969.1.r.d 8
21.c even 2 1 inner 3969.1.b.a 4
21.g even 6 2 3969.1.q.a 8
21.h odd 6 2 3969.1.q.a 8
63.g even 3 2 3969.1.n.c 8
63.h even 3 2 3969.1.j.c 8
63.i even 6 2 3969.1.j.c 8
63.j odd 6 2 3969.1.j.c 8
63.k odd 6 2 3969.1.n.c 8
63.l odd 6 2 3969.1.r.d 8
63.n odd 6 2 3969.1.n.c 8
63.o even 6 2 3969.1.r.d 8
63.s even 6 2 3969.1.n.c 8
63.t odd 6 2 3969.1.j.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3969.1.b.a 4 1.a even 1 1 trivial
3969.1.b.a 4 3.b odd 2 1 inner
3969.1.b.a 4 7.b odd 2 1 CM
3969.1.b.a 4 21.c even 2 1 inner
3969.1.j.c 8 63.h even 3 2
3969.1.j.c 8 63.i even 6 2
3969.1.j.c 8 63.j odd 6 2
3969.1.j.c 8 63.t odd 6 2
3969.1.n.c 8 63.g even 3 2
3969.1.n.c 8 63.k odd 6 2
3969.1.n.c 8 63.n odd 6 2
3969.1.n.c 8 63.s even 6 2
3969.1.q.a 8 7.c even 3 2
3969.1.q.a 8 7.d odd 6 2
3969.1.q.a 8 21.g even 6 2
3969.1.q.a 8 21.h odd 6 2
3969.1.r.d 8 9.c even 3 2
3969.1.r.d 8 9.d odd 6 2
3969.1.r.d 8 63.l odd 6 2
3969.1.r.d 8 63.o even 6 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(3969, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 4 T^{2} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$1 + 4 T^{2} + T^{4}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$( 2 + T^{2} )^{2}$$
$29$ $$( 2 + T^{2} )^{2}$$
$31$ $$T^{4}$$
$37$ $$( -3 + T^{2} )^{2}$$
$41$ $$T^{4}$$
$43$ $$( -3 + T^{2} )^{2}$$
$47$ $$T^{4}$$
$53$ $$1 + 4 T^{2} + T^{4}$$
$59$ $$T^{4}$$
$61$ $$T^{4}$$
$67$ $$( -1 + T )^{4}$$
$71$ $$1 + 4 T^{2} + T^{4}$$
$73$ $$T^{4}$$
$79$ $$( 1 + T )^{4}$$
$83$ $$T^{4}$$
$89$ $$T^{4}$$
$97$ $$T^{4}$$