# Properties

 Label 3328.1.c.e 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: $$x^{2} + 1$$ 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 12.4.981348487528448.4

## $q$-expansion

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

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

## 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.e 2
4.b odd 2 1 3328.1.c.a 2
8.b even 2 1 inner 3328.1.c.e 2
8.d odd 2 1 3328.1.c.a 2
13.b even 2 1 3328.1.c.a 2
16.e even 4 1 104.1.h.b yes 1
16.e even 4 1 416.1.h.b 1
16.f odd 4 1 104.1.h.a 1
16.f odd 4 1 416.1.h.a 1
48.i odd 4 1 936.1.o.a 1
48.i odd 4 1 3744.1.o.a 1
48.k even 4 1 936.1.o.b 1
48.k even 4 1 3744.1.o.b 1
52.b odd 2 1 inner 3328.1.c.e 2
80.i odd 4 1 2600.1.b.a 2
80.j even 4 1 2600.1.b.b 2
80.k odd 4 1 2600.1.o.d 1
80.q even 4 1 2600.1.o.b 1
80.s even 4 1 2600.1.b.b 2
80.t odd 4 1 2600.1.b.a 2
104.e even 2 1 3328.1.c.a 2
104.h odd 2 1 CM 3328.1.c.e 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.b yes 1
208.o odd 4 1 416.1.h.b 1
208.p even 4 1 104.1.h.a 1
208.p even 4 1 416.1.h.a 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.b 2
208.bh even 12 2 1352.1.p.b 2
208.bi odd 12 2 1352.1.p.a 2
208.bj even 12 2 1352.1.p.a 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.a 1
624.v even 4 1 3744.1.o.a 1
624.bi odd 4 1 936.1.o.b 1
624.bi odd 4 1 3744.1.o.b 1
1040.w odd 4 1 2600.1.b.b 2
1040.y even 4 1 2600.1.b.a 2
1040.be even 4 1 2600.1.o.d 1
1040.cb odd 4 1 2600.1.o.b 1
1040.co odd 4 1 2600.1.b.b 2
1040.cq even 4 1 2600.1.b.a 2

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

## Hecke kernels

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

 $$T_{3}^{2} + 1$$ $$T_{5}^{2} + 1$$ $$T_{7} - 1$$ $$T_{29}$$

## Hecke characteristic polynomials

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