# Properties

 Label 3380.2.f.b Level $3380$ Weight $2$ Character orbit 3380.f Analytic conductor $26.989$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3380 = 2^{2} \cdot 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3380.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$26.9894358832$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 20) 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.

 $$f(q)$$ $$=$$ $$q -2 q^{3} + i q^{5} + 2 i q^{7} + q^{9} +O(q^{10})$$ $$q -2 q^{3} + i q^{5} + 2 i q^{7} + q^{9} -2 i q^{15} + 6 q^{17} + 4 i q^{19} -4 i q^{21} -6 q^{23} - q^{25} + 4 q^{27} + 6 q^{29} + 4 i q^{31} -2 q^{35} + 2 i q^{37} -6 i q^{41} + 10 q^{43} + i q^{45} -6 i q^{47} + 3 q^{49} -12 q^{51} -6 q^{53} -8 i q^{57} + 12 i q^{59} + 2 q^{61} + 2 i q^{63} -2 i q^{67} + 12 q^{69} + 12 i q^{71} + 2 i q^{73} + 2 q^{75} + 8 q^{79} -11 q^{81} -6 i q^{83} + 6 i q^{85} -12 q^{87} -6 i q^{89} -8 i q^{93} -4 q^{95} -2 i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{3} + 2q^{9} + O(q^{10})$$ $$2q - 4q^{3} + 2q^{9} + 12q^{17} - 12q^{23} - 2q^{25} + 8q^{27} + 12q^{29} - 4q^{35} + 20q^{43} + 6q^{49} - 24q^{51} - 12q^{53} + 4q^{61} + 24q^{69} + 4q^{75} + 16q^{79} - 22q^{81} - 24q^{87} - 8q^{95} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3380\mathbb{Z}\right)^\times$$.

 $$n$$ $$677$$ $$1691$$ $$1861$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3041.1
 − 1.00000i 1.00000i
0 −2.00000 0 1.00000i 0 2.00000i 0 1.00000 0
3041.2 0 −2.00000 0 1.00000i 0 2.00000i 0 1.00000 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
13.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3380.2.f.b 2
13.b even 2 1 inner 3380.2.f.b 2
13.d odd 4 1 20.2.a.a 1
13.d odd 4 1 3380.2.a.c 1
39.f even 4 1 180.2.a.a 1
52.f even 4 1 80.2.a.b 1
65.f even 4 1 100.2.c.a 2
65.g odd 4 1 100.2.a.a 1
65.k even 4 1 100.2.c.a 2
91.i even 4 1 980.2.a.h 1
91.z odd 12 2 980.2.i.i 2
91.bb even 12 2 980.2.i.c 2
104.j odd 4 1 320.2.a.f 1
104.m even 4 1 320.2.a.a 1
117.y odd 12 2 1620.2.i.h 2
117.z even 12 2 1620.2.i.b 2
143.g even 4 1 2420.2.a.a 1
156.l odd 4 1 720.2.a.h 1
195.j odd 4 1 900.2.d.c 2
195.n even 4 1 900.2.a.b 1
195.u odd 4 1 900.2.d.c 2
208.l even 4 1 1280.2.d.g 2
208.m odd 4 1 1280.2.d.c 2
208.r odd 4 1 1280.2.d.c 2
208.s even 4 1 1280.2.d.g 2
221.g odd 4 1 5780.2.a.f 1
221.h odd 4 1 5780.2.c.a 2
221.i odd 4 1 5780.2.c.a 2
247.i even 4 1 7220.2.a.f 1
260.l odd 4 1 400.2.c.b 2
260.s odd 4 1 400.2.c.b 2
260.u even 4 1 400.2.a.c 1
273.o odd 4 1 8820.2.a.g 1
312.w odd 4 1 2880.2.a.f 1
312.y even 4 1 2880.2.a.m 1
364.p odd 4 1 3920.2.a.h 1
455.n odd 4 1 4900.2.e.f 2
455.u even 4 1 4900.2.a.e 1
455.w odd 4 1 4900.2.e.f 2
520.t even 4 1 1600.2.a.w 1
520.x odd 4 1 1600.2.c.e 2
520.y even 4 1 1600.2.c.d 2
520.bj even 4 1 1600.2.c.d 2
520.bk odd 4 1 1600.2.c.e 2
520.bo odd 4 1 1600.2.a.c 1
572.k odd 4 1 9680.2.a.ba 1
780.u even 4 1 3600.2.f.j 2
780.bb odd 4 1 3600.2.a.be 1
780.bn even 4 1 3600.2.f.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.2.a.a 1 13.d odd 4 1
80.2.a.b 1 52.f even 4 1
100.2.a.a 1 65.g odd 4 1
100.2.c.a 2 65.f even 4 1
100.2.c.a 2 65.k even 4 1
180.2.a.a 1 39.f even 4 1
320.2.a.a 1 104.m even 4 1
320.2.a.f 1 104.j odd 4 1
400.2.a.c 1 260.u even 4 1
400.2.c.b 2 260.l odd 4 1
400.2.c.b 2 260.s odd 4 1
720.2.a.h 1 156.l odd 4 1
900.2.a.b 1 195.n even 4 1
900.2.d.c 2 195.j odd 4 1
900.2.d.c 2 195.u odd 4 1
980.2.a.h 1 91.i even 4 1
980.2.i.c 2 91.bb even 12 2
980.2.i.i 2 91.z odd 12 2
1280.2.d.c 2 208.m odd 4 1
1280.2.d.c 2 208.r odd 4 1
1280.2.d.g 2 208.l even 4 1
1280.2.d.g 2 208.s even 4 1
1600.2.a.c 1 520.bo odd 4 1
1600.2.a.w 1 520.t even 4 1
1600.2.c.d 2 520.y even 4 1
1600.2.c.d 2 520.bj even 4 1
1600.2.c.e 2 520.x odd 4 1
1600.2.c.e 2 520.bk odd 4 1
1620.2.i.b 2 117.z even 12 2
1620.2.i.h 2 117.y odd 12 2
2420.2.a.a 1 143.g even 4 1
2880.2.a.f 1 312.w odd 4 1
2880.2.a.m 1 312.y even 4 1
3380.2.a.c 1 13.d odd 4 1
3380.2.f.b 2 1.a even 1 1 trivial
3380.2.f.b 2 13.b even 2 1 inner
3600.2.a.be 1 780.bb odd 4 1
3600.2.f.j 2 780.u even 4 1
3600.2.f.j 2 780.bn even 4 1
3920.2.a.h 1 364.p odd 4 1
4900.2.a.e 1 455.u even 4 1
4900.2.e.f 2 455.n odd 4 1
4900.2.e.f 2 455.w odd 4 1
5780.2.a.f 1 221.g odd 4 1
5780.2.c.a 2 221.h odd 4 1
5780.2.c.a 2 221.i odd 4 1
7220.2.a.f 1 247.i even 4 1
8820.2.a.g 1 273.o odd 4 1
9680.2.a.ba 1 572.k odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(3380, [\chi])$$:

 $$T_{3} + 2$$ $$T_{19}^{2} + 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( 2 + T )^{2}$$
$5$ $$1 + T^{2}$$
$7$ $$4 + T^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2}$$
$17$ $$( -6 + T )^{2}$$
$19$ $$16 + T^{2}$$
$23$ $$( 6 + T )^{2}$$
$29$ $$( -6 + T )^{2}$$
$31$ $$16 + T^{2}$$
$37$ $$4 + T^{2}$$
$41$ $$36 + T^{2}$$
$43$ $$( -10 + T )^{2}$$
$47$ $$36 + T^{2}$$
$53$ $$( 6 + T )^{2}$$
$59$ $$144 + T^{2}$$
$61$ $$( -2 + T )^{2}$$
$67$ $$4 + T^{2}$$
$71$ $$144 + T^{2}$$
$73$ $$4 + T^{2}$$
$79$ $$( -8 + T )^{2}$$
$83$ $$36 + T^{2}$$
$89$ $$36 + T^{2}$$
$97$ $$4 + T^{2}$$