# Properties

 Label 3328.2.b.e Level $3328$ Weight $2$ Character orbit 3328.b Analytic conductor $26.574$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$26.5742137927$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 52) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 2 i q^{5} - 2 q^{7} + 3 q^{9} +O(q^{10})$$ q - 2*i * q^5 - 2 * q^7 + 3 * q^9 $$q - 2 i q^{5} - 2 q^{7} + 3 q^{9} - 2 i q^{11} - i q^{13} + 6 q^{17} + 6 i q^{19} + 8 q^{23} + q^{25} + 2 i q^{29} - 10 q^{31} + 4 i q^{35} + 6 i q^{37} + 6 q^{41} + 4 i q^{43} - 6 i q^{45} + 2 q^{47} - 3 q^{49} - 6 i q^{53} - 4 q^{55} - 10 i q^{59} - 2 i q^{61} - 6 q^{63} - 2 q^{65} - 10 i q^{67} + 10 q^{71} - 2 q^{73} + 4 i q^{77} + 4 q^{79} + 9 q^{81} + 6 i q^{83} - 12 i q^{85} + 6 q^{89} + 2 i q^{91} + 12 q^{95} + 2 q^{97} - 6 i q^{99} +O(q^{100})$$ q - 2*i * q^5 - 2 * q^7 + 3 * q^9 - 2*i * q^11 - i * q^13 + 6 * q^17 + 6*i * q^19 + 8 * q^23 + q^25 + 2*i * q^29 - 10 * q^31 + 4*i * q^35 + 6*i * q^37 + 6 * q^41 + 4*i * q^43 - 6*i * q^45 + 2 * q^47 - 3 * q^49 - 6*i * q^53 - 4 * q^55 - 10*i * q^59 - 2*i * q^61 - 6 * q^63 - 2 * q^65 - 10*i * q^67 + 10 * q^71 - 2 * q^73 + 4*i * q^77 + 4 * q^79 + 9 * q^81 + 6*i * q^83 - 12*i * q^85 + 6 * q^89 + 2*i * q^91 + 12 * q^95 + 2 * q^97 - 6*i * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{7} + 6 q^{9}+O(q^{10})$$ 2 * q - 4 * q^7 + 6 * q^9 $$2 q - 4 q^{7} + 6 q^{9} + 12 q^{17} + 16 q^{23} + 2 q^{25} - 20 q^{31} + 12 q^{41} + 4 q^{47} - 6 q^{49} - 8 q^{55} - 12 q^{63} - 4 q^{65} + 20 q^{71} - 4 q^{73} + 8 q^{79} + 18 q^{81} + 12 q^{89} + 24 q^{95} + 4 q^{97}+O(q^{100})$$ 2 * q - 4 * q^7 + 6 * q^9 + 12 * q^17 + 16 * q^23 + 2 * q^25 - 20 * q^31 + 12 * q^41 + 4 * q^47 - 6 * q^49 - 8 * q^55 - 12 * q^63 - 4 * q^65 + 20 * q^71 - 4 * q^73 + 8 * q^79 + 18 * q^81 + 12 * q^89 + 24 * q^95 + 4 * q^97

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1665.1
 1.00000i − 1.00000i
0 0 0 2.00000i 0 −2.00000 0 3.00000 0
1665.2 0 0 0 2.00000i 0 −2.00000 0 3.00000 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.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3328.2.b.e 2
4.b odd 2 1 3328.2.b.q 2
8.b even 2 1 inner 3328.2.b.e 2
8.d odd 2 1 3328.2.b.q 2
16.e even 4 1 208.2.a.c 1
16.e even 4 1 832.2.a.f 1
16.f odd 4 1 52.2.a.a 1
16.f odd 4 1 832.2.a.e 1
48.i odd 4 1 1872.2.a.f 1
48.i odd 4 1 7488.2.a.bw 1
48.k even 4 1 468.2.a.b 1
48.k even 4 1 7488.2.a.bn 1
80.j even 4 1 1300.2.c.c 2
80.k odd 4 1 1300.2.a.d 1
80.q even 4 1 5200.2.a.q 1
80.s even 4 1 1300.2.c.c 2
112.j even 4 1 2548.2.a.e 1
112.u odd 12 2 2548.2.j.e 2
112.v even 12 2 2548.2.j.f 2
144.u even 12 2 4212.2.i.i 2
144.v odd 12 2 4212.2.i.d 2
176.i even 4 1 6292.2.a.g 1
208.l even 4 1 676.2.d.c 2
208.m odd 4 1 2704.2.f.f 2
208.o odd 4 1 676.2.a.c 1
208.p even 4 1 2704.2.a.g 1
208.r odd 4 1 2704.2.f.f 2
208.s even 4 1 676.2.d.c 2
208.bf even 12 2 676.2.h.c 4
208.bg odd 12 2 676.2.e.c 2
208.bi odd 12 2 676.2.e.b 2
208.bk even 12 2 676.2.h.c 4
624.s odd 4 1 6084.2.b.m 2
624.v even 4 1 6084.2.a.m 1
624.bo odd 4 1 6084.2.b.m 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
52.2.a.a 1 16.f odd 4 1
208.2.a.c 1 16.e even 4 1
468.2.a.b 1 48.k even 4 1
676.2.a.c 1 208.o odd 4 1
676.2.d.c 2 208.l even 4 1
676.2.d.c 2 208.s even 4 1
676.2.e.b 2 208.bi odd 12 2
676.2.e.c 2 208.bg odd 12 2
676.2.h.c 4 208.bf even 12 2
676.2.h.c 4 208.bk even 12 2
832.2.a.e 1 16.f odd 4 1
832.2.a.f 1 16.e even 4 1
1300.2.a.d 1 80.k odd 4 1
1300.2.c.c 2 80.j even 4 1
1300.2.c.c 2 80.s even 4 1
1872.2.a.f 1 48.i odd 4 1
2548.2.a.e 1 112.j even 4 1
2548.2.j.e 2 112.u odd 12 2
2548.2.j.f 2 112.v even 12 2
2704.2.a.g 1 208.p even 4 1
2704.2.f.f 2 208.m odd 4 1
2704.2.f.f 2 208.r odd 4 1
3328.2.b.e 2 1.a even 1 1 trivial
3328.2.b.e 2 8.b even 2 1 inner
3328.2.b.q 2 4.b odd 2 1
3328.2.b.q 2 8.d odd 2 1
4212.2.i.d 2 144.v odd 12 2
4212.2.i.i 2 144.u even 12 2
5200.2.a.q 1 80.q even 4 1
6084.2.a.m 1 624.v even 4 1
6084.2.b.m 2 624.s odd 4 1
6084.2.b.m 2 624.bo odd 4 1
6292.2.a.g 1 176.i even 4 1
7488.2.a.bn 1 48.k even 4 1
7488.2.a.bw 1 48.i odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(3328, [\chi])$$:

 $$T_{3}$$ T3 $$T_{5}^{2} + 4$$ T5^2 + 4 $$T_{7} + 2$$ T7 + 2 $$T_{11}^{2} + 4$$ T11^2 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 4$$
$7$ $$(T + 2)^{2}$$
$11$ $$T^{2} + 4$$
$13$ $$T^{2} + 1$$
$17$ $$(T - 6)^{2}$$
$19$ $$T^{2} + 36$$
$23$ $$(T - 8)^{2}$$
$29$ $$T^{2} + 4$$
$31$ $$(T + 10)^{2}$$
$37$ $$T^{2} + 36$$
$41$ $$(T - 6)^{2}$$
$43$ $$T^{2} + 16$$
$47$ $$(T - 2)^{2}$$
$53$ $$T^{2} + 36$$
$59$ $$T^{2} + 100$$
$61$ $$T^{2} + 4$$
$67$ $$T^{2} + 100$$
$71$ $$(T - 10)^{2}$$
$73$ $$(T + 2)^{2}$$
$79$ $$(T - 4)^{2}$$
$83$ $$T^{2} + 36$$
$89$ $$(T - 6)^{2}$$
$97$ $$(T - 2)^{2}$$