# Properties

 Label 3328.2.b.j 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 26) 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.

 $$f(q)$$ $$=$$ $$q + i q^{3} - 3 i q^{5} - q^{7} + 2 q^{9} +O(q^{10})$$ q + i * q^3 - 3*i * q^5 - q^7 + 2 * q^9 $$q + i q^{3} - 3 i q^{5} - q^{7} + 2 q^{9} - 6 i q^{11} - i q^{13} + 3 q^{15} - 3 q^{17} + 2 i q^{19} - i q^{21} - 4 q^{25} + 5 i q^{27} - 6 i q^{29} + 4 q^{31} + 6 q^{33} + 3 i q^{35} - 7 i q^{37} + q^{39} + i q^{43} - 6 i q^{45} - 3 q^{47} - 6 q^{49} - 3 i q^{51} - 18 q^{55} - 2 q^{57} + 6 i q^{59} - 8 i q^{61} - 2 q^{63} - 3 q^{65} + 14 i q^{67} - 3 q^{71} - 2 q^{73} - 4 i q^{75} + 6 i q^{77} - 8 q^{79} + q^{81} + 12 i q^{83} + 9 i q^{85} + 6 q^{87} + 6 q^{89} + i q^{91} + 4 i q^{93} + 6 q^{95} - 10 q^{97} - 12 i q^{99} +O(q^{100})$$ q + i * q^3 - 3*i * q^5 - q^7 + 2 * q^9 - 6*i * q^11 - i * q^13 + 3 * q^15 - 3 * q^17 + 2*i * q^19 - i * q^21 - 4 * q^25 + 5*i * q^27 - 6*i * q^29 + 4 * q^31 + 6 * q^33 + 3*i * q^35 - 7*i * q^37 + q^39 + i * q^43 - 6*i * q^45 - 3 * q^47 - 6 * q^49 - 3*i * q^51 - 18 * q^55 - 2 * q^57 + 6*i * q^59 - 8*i * q^61 - 2 * q^63 - 3 * q^65 + 14*i * q^67 - 3 * q^71 - 2 * q^73 - 4*i * q^75 + 6*i * q^77 - 8 * q^79 + q^81 + 12*i * q^83 + 9*i * q^85 + 6 * q^87 + 6 * q^89 + i * q^91 + 4*i * q^93 + 6 * q^95 - 10 * q^97 - 12*i * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{7} + 4 q^{9}+O(q^{10})$$ 2 * q - 2 * q^7 + 4 * q^9 $$2 q - 2 q^{7} + 4 q^{9} + 6 q^{15} - 6 q^{17} - 8 q^{25} + 8 q^{31} + 12 q^{33} + 2 q^{39} - 6 q^{47} - 12 q^{49} - 36 q^{55} - 4 q^{57} - 4 q^{63} - 6 q^{65} - 6 q^{71} - 4 q^{73} - 16 q^{79} + 2 q^{81} + 12 q^{87} + 12 q^{89} + 12 q^{95} - 20 q^{97}+O(q^{100})$$ 2 * q - 2 * q^7 + 4 * q^9 + 6 * q^15 - 6 * q^17 - 8 * q^25 + 8 * q^31 + 12 * q^33 + 2 * q^39 - 6 * q^47 - 12 * q^49 - 36 * q^55 - 4 * q^57 - 4 * q^63 - 6 * q^65 - 6 * q^71 - 4 * q^73 - 16 * q^79 + 2 * q^81 + 12 * q^87 + 12 * q^89 + 12 * q^95 - 20 * 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 1.00000i 0 3.00000i 0 −1.00000 0 2.00000 0
1665.2 0 1.00000i 0 3.00000i 0 −1.00000 0 2.00000 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.j 2
4.b odd 2 1 3328.2.b.m 2
8.b even 2 1 inner 3328.2.b.j 2
8.d odd 2 1 3328.2.b.m 2
16.e even 4 1 208.2.a.a 1
16.e even 4 1 832.2.a.i 1
16.f odd 4 1 26.2.a.a 1
16.f odd 4 1 832.2.a.d 1
48.i odd 4 1 1872.2.a.q 1
48.i odd 4 1 7488.2.a.h 1
48.k even 4 1 234.2.a.e 1
48.k even 4 1 7488.2.a.g 1
80.j even 4 1 650.2.b.d 2
80.k odd 4 1 650.2.a.j 1
80.q even 4 1 5200.2.a.x 1
80.s even 4 1 650.2.b.d 2
112.j even 4 1 1274.2.a.d 1
112.u odd 12 2 1274.2.f.p 2
112.v even 12 2 1274.2.f.r 2
144.u even 12 2 2106.2.e.b 2
144.v odd 12 2 2106.2.e.ba 2
176.i even 4 1 3146.2.a.n 1
208.l even 4 1 338.2.b.c 2
208.m odd 4 1 2704.2.f.d 2
208.o odd 4 1 338.2.a.f 1
208.p even 4 1 2704.2.a.f 1
208.r odd 4 1 2704.2.f.d 2
208.s even 4 1 338.2.b.c 2
208.bf even 12 2 338.2.e.a 4
208.bg odd 12 2 338.2.c.d 2
208.bi odd 12 2 338.2.c.a 2
208.bk even 12 2 338.2.e.a 4
240.t even 4 1 5850.2.a.p 1
240.z odd 4 1 5850.2.e.a 2
240.bd odd 4 1 5850.2.e.a 2
272.k odd 4 1 7514.2.a.c 1
304.m even 4 1 9386.2.a.j 1
624.s odd 4 1 3042.2.b.a 2
624.v even 4 1 3042.2.a.a 1
624.bo odd 4 1 3042.2.b.a 2
1040.cb odd 4 1 8450.2.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.2.a.a 1 16.f odd 4 1
208.2.a.a 1 16.e even 4 1
234.2.a.e 1 48.k even 4 1
338.2.a.f 1 208.o odd 4 1
338.2.b.c 2 208.l even 4 1
338.2.b.c 2 208.s even 4 1
338.2.c.a 2 208.bi odd 12 2
338.2.c.d 2 208.bg odd 12 2
338.2.e.a 4 208.bf even 12 2
338.2.e.a 4 208.bk even 12 2
650.2.a.j 1 80.k odd 4 1
650.2.b.d 2 80.j even 4 1
650.2.b.d 2 80.s even 4 1
832.2.a.d 1 16.f odd 4 1
832.2.a.i 1 16.e even 4 1
1274.2.a.d 1 112.j even 4 1
1274.2.f.p 2 112.u odd 12 2
1274.2.f.r 2 112.v even 12 2
1872.2.a.q 1 48.i odd 4 1
2106.2.e.b 2 144.u even 12 2
2106.2.e.ba 2 144.v odd 12 2
2704.2.a.f 1 208.p even 4 1
2704.2.f.d 2 208.m odd 4 1
2704.2.f.d 2 208.r odd 4 1
3042.2.a.a 1 624.v even 4 1
3042.2.b.a 2 624.s odd 4 1
3042.2.b.a 2 624.bo odd 4 1
3146.2.a.n 1 176.i even 4 1
3328.2.b.j 2 1.a even 1 1 trivial
3328.2.b.j 2 8.b even 2 1 inner
3328.2.b.m 2 4.b odd 2 1
3328.2.b.m 2 8.d odd 2 1
5200.2.a.x 1 80.q even 4 1
5850.2.a.p 1 240.t even 4 1
5850.2.e.a 2 240.z odd 4 1
5850.2.e.a 2 240.bd odd 4 1
7488.2.a.g 1 48.k even 4 1
7488.2.a.h 1 48.i odd 4 1
7514.2.a.c 1 272.k odd 4 1
8450.2.a.c 1 1040.cb odd 4 1
9386.2.a.j 1 304.m even 4 1

## Hecke kernels

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

 $$T_{3}^{2} + 1$$ T3^2 + 1 $$T_{5}^{2} + 9$$ T5^2 + 9 $$T_{7} + 1$$ T7 + 1 $$T_{11}^{2} + 36$$ T11^2 + 36

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2} + 9$$
$7$ $$(T + 1)^{2}$$
$11$ $$T^{2} + 36$$
$13$ $$T^{2} + 1$$
$17$ $$(T + 3)^{2}$$
$19$ $$T^{2} + 4$$
$23$ $$T^{2}$$
$29$ $$T^{2} + 36$$
$31$ $$(T - 4)^{2}$$
$37$ $$T^{2} + 49$$
$41$ $$T^{2}$$
$43$ $$T^{2} + 1$$
$47$ $$(T + 3)^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2} + 36$$
$61$ $$T^{2} + 64$$
$67$ $$T^{2} + 196$$
$71$ $$(T + 3)^{2}$$
$73$ $$(T + 2)^{2}$$
$79$ $$(T + 8)^{2}$$
$83$ $$T^{2} + 144$$
$89$ $$(T - 6)^{2}$$
$97$ $$(T + 10)^{2}$$