# Properties

 Label 3328.1.c.a Level $3328$ Weight $1$ Character orbit 3328.c Analytic conductor $1.661$ Analytic rank $0$ Dimension $2$ Projective image $D_{3}$ CM discriminant -104 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3328 = 2^{8} \cdot 13$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3328.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.66088836204$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 104) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.104.1 Artin image: $C_4\times S_3$ Artin field: Galois closure of 12.0.981348487528448.5

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q - i q^{3} - i q^{5} - q^{7} +O(q^{10})$$ q - z * q^3 - z * q^5 - q^7 $$q - i q^{3} - i q^{5} - q^{7} - i q^{13} - q^{15} - q^{17} + i q^{21} - i q^{27} - q^{31} + i q^{35} - i q^{37} - q^{39} + i q^{43} + q^{47} + i q^{51} - q^{65} - q^{71} - q^{81} + i q^{85} + i q^{91} + 2 i q^{93} +O(q^{100})$$ q - z * q^3 - z * q^5 - q^7 - z * q^13 - q^15 - q^17 + z * q^21 - z * q^27 - q^31 + z * q^35 - z * q^37 - q^39 + z * q^43 + q^47 + z * q^51 - q^65 - q^71 - q^81 + z * q^85 + z * q^91 + 2*z * q^93 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{7}+O(q^{10})$$ 2 * q - 2 * q^7 $$2 q - 2 q^{7} - 2 q^{15} - 2 q^{17} - 4 q^{31} - 2 q^{39} + 2 q^{47} - 2 q^{65} - 2 q^{71} - 2 q^{81}+O(q^{100})$$ 2 * q - 2 * q^7 - 2 * q^15 - 2 * q^17 - 4 * q^31 - 2 * q^39 + 2 * q^47 - 2 * q^65 - 2 * q^71 - 2 * q^81

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3328\mathbb{Z}\right)^\times$$.

 $$n$$ $$261$$ $$769$$ $$1535$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3327.1
 1.00000i − 1.00000i
0 1.00000i 0 1.00000i 0 −1.00000 0 0 0
3327.2 0 1.00000i 0 1.00000i 0 −1.00000 0 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
104.h odd 2 1 CM by $$\Q(\sqrt{-26})$$
8.b even 2 1 inner
52.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3328.1.c.a 2
4.b odd 2 1 3328.1.c.e 2
8.b even 2 1 inner 3328.1.c.a 2
8.d odd 2 1 3328.1.c.e 2
13.b even 2 1 3328.1.c.e 2
16.e even 4 1 104.1.h.a 1
16.e even 4 1 416.1.h.a 1
16.f odd 4 1 104.1.h.b yes 1
16.f odd 4 1 416.1.h.b 1
48.i odd 4 1 936.1.o.b 1
48.i odd 4 1 3744.1.o.b 1
48.k even 4 1 936.1.o.a 1
48.k even 4 1 3744.1.o.a 1
52.b odd 2 1 inner 3328.1.c.a 2
80.i odd 4 1 2600.1.b.b 2
80.j even 4 1 2600.1.b.a 2
80.k odd 4 1 2600.1.o.b 1
80.q even 4 1 2600.1.o.d 1
80.s even 4 1 2600.1.b.a 2
80.t odd 4 1 2600.1.b.b 2
104.e even 2 1 3328.1.c.e 2
104.h odd 2 1 CM 3328.1.c.a 2
208.l even 4 1 1352.1.g.a 2
208.m odd 4 1 1352.1.g.a 2
208.o odd 4 1 104.1.h.a 1
208.o odd 4 1 416.1.h.a 1
208.p even 4 1 104.1.h.b yes 1
208.p even 4 1 416.1.h.b 1
208.r odd 4 1 1352.1.g.a 2
208.s even 4 1 1352.1.g.a 2
208.be odd 12 2 1352.1.n.a 4
208.bf even 12 2 1352.1.n.a 4
208.bg odd 12 2 1352.1.p.a 2
208.bh even 12 2 1352.1.p.a 2
208.bi odd 12 2 1352.1.p.b 2
208.bj even 12 2 1352.1.p.b 2
208.bk even 12 2 1352.1.n.a 4
208.bl odd 12 2 1352.1.n.a 4
624.v even 4 1 936.1.o.b 1
624.v even 4 1 3744.1.o.b 1
624.bi odd 4 1 936.1.o.a 1
624.bi odd 4 1 3744.1.o.a 1
1040.w odd 4 1 2600.1.b.a 2
1040.y even 4 1 2600.1.b.b 2
1040.be even 4 1 2600.1.o.b 1
1040.cb odd 4 1 2600.1.o.d 1
1040.co odd 4 1 2600.1.b.a 2
1040.cq even 4 1 2600.1.b.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
104.1.h.a 1 16.e even 4 1
104.1.h.a 1 208.o odd 4 1
104.1.h.b yes 1 16.f odd 4 1
104.1.h.b yes 1 208.p even 4 1
416.1.h.a 1 16.e even 4 1
416.1.h.a 1 208.o odd 4 1
416.1.h.b 1 16.f odd 4 1
416.1.h.b 1 208.p even 4 1
936.1.o.a 1 48.k even 4 1
936.1.o.a 1 624.bi odd 4 1
936.1.o.b 1 48.i odd 4 1
936.1.o.b 1 624.v even 4 1
1352.1.g.a 2 208.l even 4 1
1352.1.g.a 2 208.m odd 4 1
1352.1.g.a 2 208.r odd 4 1
1352.1.g.a 2 208.s even 4 1
1352.1.n.a 4 208.be odd 12 2
1352.1.n.a 4 208.bf even 12 2
1352.1.n.a 4 208.bk even 12 2
1352.1.n.a 4 208.bl odd 12 2
1352.1.p.a 2 208.bg odd 12 2
1352.1.p.a 2 208.bh even 12 2
1352.1.p.b 2 208.bi odd 12 2
1352.1.p.b 2 208.bj even 12 2
2600.1.b.a 2 80.j even 4 1
2600.1.b.a 2 80.s even 4 1
2600.1.b.a 2 1040.w odd 4 1
2600.1.b.a 2 1040.co odd 4 1
2600.1.b.b 2 80.i odd 4 1
2600.1.b.b 2 80.t odd 4 1
2600.1.b.b 2 1040.y even 4 1
2600.1.b.b 2 1040.cq even 4 1
2600.1.o.b 1 80.k odd 4 1
2600.1.o.b 1 1040.be even 4 1
2600.1.o.d 1 80.q even 4 1
2600.1.o.d 1 1040.cb odd 4 1
3328.1.c.a 2 1.a even 1 1 trivial
3328.1.c.a 2 8.b even 2 1 inner
3328.1.c.a 2 52.b odd 2 1 inner
3328.1.c.a 2 104.h odd 2 1 CM
3328.1.c.e 2 4.b odd 2 1
3328.1.c.e 2 8.d odd 2 1
3328.1.c.e 2 13.b even 2 1
3328.1.c.e 2 104.e even 2 1
3744.1.o.a 1 48.k even 4 1
3744.1.o.a 1 624.bi odd 4 1
3744.1.o.b 1 48.i odd 4 1
3744.1.o.b 1 624.v even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(3328, [\chi])$$:

 $$T_{3}^{2} + 1$$ T3^2 + 1 $$T_{5}^{2} + 1$$ T5^2 + 1 $$T_{7} + 1$$ T7 + 1 $$T_{29}$$ T29

## Hecke characteristic polynomials

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