# Properties

 Label 3364.1.d.a Level $3364$ Weight $1$ Character orbit 3364.d Analytic conductor $1.679$ Analytic rank $0$ Dimension $6$ Projective image $D_{7}$ CM discriminant -4 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3364 = 2^{2} \cdot 29^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3364.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.67885470250$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.153664.1 Defining polynomial: $$x^{6} + 5 x^{4} + 6 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 116) Projective image: $$D_{7}$$ Projective field: Galois closure of 7.1.38068692544.1

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{5} q^{2} - q^{4} + \beta_{4} q^{5} -\beta_{5} q^{8} - q^{9} +O(q^{10})$$ $$q + \beta_{5} q^{2} - q^{4} + \beta_{4} q^{5} -\beta_{5} q^{8} - q^{9} + \beta_{1} q^{10} + ( 1 - \beta_{2} - \beta_{4} ) q^{13} + q^{16} + ( \beta_{1} - \beta_{3} - \beta_{5} ) q^{17} -\beta_{5} q^{18} -\beta_{4} q^{20} + ( 1 - \beta_{2} ) q^{25} -\beta_{3} q^{26} + \beta_{5} q^{32} + \beta_{2} q^{34} + q^{36} -\beta_{3} q^{37} -\beta_{1} q^{40} -\beta_{3} q^{41} -\beta_{4} q^{45} + q^{49} + ( \beta_{1} - \beta_{3} ) q^{50} + ( -1 + \beta_{2} + \beta_{4} ) q^{52} -\beta_{2} q^{53} + ( \beta_{1} - \beta_{3} - \beta_{5} ) q^{61} - q^{64} + ( -1 + \beta_{4} ) q^{65} + ( -\beta_{1} + \beta_{3} + \beta_{5} ) q^{68} + \beta_{5} q^{72} + ( -\beta_{1} + \beta_{3} + \beta_{5} ) q^{73} + ( -1 + \beta_{2} + \beta_{4} ) q^{74} + \beta_{4} q^{80} + q^{81} + ( -1 + \beta_{2} + \beta_{4} ) q^{82} + ( \beta_{1} - \beta_{3} ) q^{85} -\beta_{1} q^{89} -\beta_{1} q^{90} -\beta_{3} q^{97} + \beta_{5} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 6q^{4} + 2q^{5} - 6q^{9} + O(q^{10})$$ $$6q - 6q^{4} + 2q^{5} - 6q^{9} + 2q^{13} + 6q^{16} - 2q^{20} + 4q^{25} + 2q^{34} + 6q^{36} - 2q^{45} + 6q^{49} - 2q^{52} - 2q^{53} - 6q^{64} - 4q^{65} - 2q^{74} + 2q^{80} + 6q^{81} - 2q^{82} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 5 x^{4} + 6 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} + 3 \nu$$ $$\beta_{4}$$ $$=$$ $$\nu^{4} + 3 \nu^{2} + 1$$ $$\beta_{5}$$ $$=$$ $$\nu^{5} + 4 \nu^{3} + 3 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 2$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} - 3 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{4} - 3 \beta_{2} + 5$$ $$\nu^{5}$$ $$=$$ $$\beta_{5} - 4 \beta_{3} + 9 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3364\mathbb{Z}\right)^\times$$.

 $$n$$ $$1683$$ $$2525$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3363.1
 1.24698i − 0.445042i − 1.80194i − 1.24698i 0.445042i 1.80194i
1.00000i 0 −1.00000 −1.24698 0 0 1.00000i −1.00000 1.24698i
3363.2 1.00000i 0 −1.00000 0.445042 0 0 1.00000i −1.00000 0.445042i
3363.3 1.00000i 0 −1.00000 1.80194 0 0 1.00000i −1.00000 1.80194i
3363.4 1.00000i 0 −1.00000 −1.24698 0 0 1.00000i −1.00000 1.24698i
3363.5 1.00000i 0 −1.00000 0.445042 0 0 1.00000i −1.00000 0.445042i
3363.6 1.00000i 0 −1.00000 1.80194 0 0 1.00000i −1.00000 1.80194i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 3363.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
29.b even 2 1 inner
116.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3364.1.d.a 6
4.b odd 2 1 CM 3364.1.d.a 6
29.b even 2 1 inner 3364.1.d.a 6
29.c odd 4 1 3364.1.b.b 3
29.c odd 4 1 3364.1.b.c 3
29.d even 7 2 3364.1.h.c 12
29.d even 7 2 3364.1.h.d 12
29.d even 7 2 3364.1.h.e 12
29.e even 14 2 3364.1.h.c 12
29.e even 14 2 3364.1.h.d 12
29.e even 14 2 3364.1.h.e 12
29.f odd 28 2 116.1.j.a 6
29.f odd 28 2 3364.1.j.a 6
29.f odd 28 2 3364.1.j.b 6
29.f odd 28 2 3364.1.j.c 6
29.f odd 28 2 3364.1.j.d 6
29.f odd 28 2 3364.1.j.e 6
87.k even 28 2 1044.1.bb.a 6
116.d odd 2 1 inner 3364.1.d.a 6
116.e even 4 1 3364.1.b.b 3
116.e even 4 1 3364.1.b.c 3
116.h odd 14 2 3364.1.h.c 12
116.h odd 14 2 3364.1.h.d 12
116.h odd 14 2 3364.1.h.e 12
116.j odd 14 2 3364.1.h.c 12
116.j odd 14 2 3364.1.h.d 12
116.j odd 14 2 3364.1.h.e 12
116.l even 28 2 116.1.j.a 6
116.l even 28 2 3364.1.j.a 6
116.l even 28 2 3364.1.j.b 6
116.l even 28 2 3364.1.j.c 6
116.l even 28 2 3364.1.j.d 6
116.l even 28 2 3364.1.j.e 6
145.o even 28 2 2900.1.bd.a 12
145.s odd 28 2 2900.1.bj.a 6
145.t even 28 2 2900.1.bd.a 12
232.u odd 28 2 1856.1.bh.a 6
232.v even 28 2 1856.1.bh.a 6
348.v odd 28 2 1044.1.bb.a 6
580.bd odd 28 2 2900.1.bd.a 12
580.be even 28 2 2900.1.bj.a 6
580.bm odd 28 2 2900.1.bd.a 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
116.1.j.a 6 29.f odd 28 2
116.1.j.a 6 116.l even 28 2
1044.1.bb.a 6 87.k even 28 2
1044.1.bb.a 6 348.v odd 28 2
1856.1.bh.a 6 232.u odd 28 2
1856.1.bh.a 6 232.v even 28 2
2900.1.bd.a 12 145.o even 28 2
2900.1.bd.a 12 145.t even 28 2
2900.1.bd.a 12 580.bd odd 28 2
2900.1.bd.a 12 580.bm odd 28 2
2900.1.bj.a 6 145.s odd 28 2
2900.1.bj.a 6 580.be even 28 2
3364.1.b.b 3 29.c odd 4 1
3364.1.b.b 3 116.e even 4 1
3364.1.b.c 3 29.c odd 4 1
3364.1.b.c 3 116.e even 4 1
3364.1.d.a 6 1.a even 1 1 trivial
3364.1.d.a 6 4.b odd 2 1 CM
3364.1.d.a 6 29.b even 2 1 inner
3364.1.d.a 6 116.d odd 2 1 inner
3364.1.h.c 12 29.d even 7 2
3364.1.h.c 12 29.e even 14 2
3364.1.h.c 12 116.h odd 14 2
3364.1.h.c 12 116.j odd 14 2
3364.1.h.d 12 29.d even 7 2
3364.1.h.d 12 29.e even 14 2
3364.1.h.d 12 116.h odd 14 2
3364.1.h.d 12 116.j odd 14 2
3364.1.h.e 12 29.d even 7 2
3364.1.h.e 12 29.e even 14 2
3364.1.h.e 12 116.h odd 14 2
3364.1.h.e 12 116.j odd 14 2
3364.1.j.a 6 29.f odd 28 2
3364.1.j.a 6 116.l even 28 2
3364.1.j.b 6 29.f odd 28 2
3364.1.j.b 6 116.l even 28 2
3364.1.j.c 6 29.f odd 28 2
3364.1.j.c 6 116.l even 28 2
3364.1.j.d 6 29.f odd 28 2
3364.1.j.d 6 116.l even 28 2
3364.1.j.e 6 29.f odd 28 2
3364.1.j.e 6 116.l even 28 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(3364, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{3}$$
$3$ $$T^{6}$$
$5$ $$( 1 - 2 T - T^{2} + T^{3} )^{2}$$
$7$ $$T^{6}$$
$11$ $$T^{6}$$
$13$ $$( 1 - 2 T - T^{2} + T^{3} )^{2}$$
$17$ $$1 + 6 T^{2} + 5 T^{4} + T^{6}$$
$19$ $$T^{6}$$
$23$ $$T^{6}$$
$29$ $$T^{6}$$
$31$ $$T^{6}$$
$37$ $$1 + 6 T^{2} + 5 T^{4} + T^{6}$$
$41$ $$1 + 6 T^{2} + 5 T^{4} + T^{6}$$
$43$ $$T^{6}$$
$47$ $$T^{6}$$
$53$ $$( -1 - 2 T + T^{2} + T^{3} )^{2}$$
$59$ $$T^{6}$$
$61$ $$1 + 6 T^{2} + 5 T^{4} + T^{6}$$
$67$ $$T^{6}$$
$71$ $$T^{6}$$
$73$ $$1 + 6 T^{2} + 5 T^{4} + T^{6}$$
$79$ $$T^{6}$$
$83$ $$T^{6}$$
$89$ $$1 + 6 T^{2} + 5 T^{4} + T^{6}$$
$97$ $$1 + 6 T^{2} + 5 T^{4} + T^{6}$$