# Properties

 Label 3380.2.f.a 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 260) 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.

 $$f(q)$$ $$=$$ $$q -3 q^{3} -i q^{5} + 3 i q^{7} + 6 q^{9} +O(q^{10})$$ $$q -3 q^{3} -i q^{5} + 3 i q^{7} + 6 q^{9} + 3 i q^{11} + 3 i q^{15} + 7 q^{17} -i q^{19} -9 i q^{21} + 7 q^{23} - q^{25} -9 q^{27} -5 q^{29} + 4 i q^{31} -9 i q^{33} + 3 q^{35} -3 i q^{37} -7 i q^{41} + 9 q^{43} -6 i q^{45} + 8 i q^{47} -2 q^{49} -21 q^{51} -6 q^{53} + 3 q^{55} + 3 i q^{57} + 5 i q^{59} -5 q^{61} + 18 i q^{63} -13 i q^{67} -21 q^{69} + 3 i q^{71} -14 i q^{73} + 3 q^{75} -9 q^{77} -8 q^{79} + 9 q^{81} -12 i q^{83} -7 i q^{85} + 15 q^{87} + 7 i q^{89} -12 i q^{93} - q^{95} + 11 i q^{97} + 18 i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 6 q^{3} + 12 q^{9} + O(q^{10})$$ $$2 q - 6 q^{3} + 12 q^{9} + 14 q^{17} + 14 q^{23} - 2 q^{25} - 18 q^{27} - 10 q^{29} + 6 q^{35} + 18 q^{43} - 4 q^{49} - 42 q^{51} - 12 q^{53} + 6 q^{55} - 10 q^{61} - 42 q^{69} + 6 q^{75} - 18 q^{77} - 16 q^{79} + 18 q^{81} + 30 q^{87} - 2 q^{95} + O(q^{100})$$

## 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 −3.00000 0 1.00000i 0 3.00000i 0 6.00000 0
3041.2 0 −3.00000 0 1.00000i 0 3.00000i 0 6.00000 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.a 2
13.b even 2 1 inner 3380.2.f.a 2
13.d odd 4 1 3380.2.a.a 1
13.d odd 4 1 3380.2.a.b 1
13.f odd 12 2 260.2.i.d 2
39.k even 12 2 2340.2.q.a 2
52.l even 12 2 1040.2.q.b 2
65.o even 12 2 1300.2.bb.e 4
65.s odd 12 2 1300.2.i.a 2
65.t even 12 2 1300.2.bb.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.i.d 2 13.f odd 12 2
1040.2.q.b 2 52.l even 12 2
1300.2.i.a 2 65.s odd 12 2
1300.2.bb.e 4 65.o even 12 2
1300.2.bb.e 4 65.t even 12 2
2340.2.q.a 2 39.k even 12 2
3380.2.a.a 1 13.d odd 4 1
3380.2.a.b 1 13.d odd 4 1
3380.2.f.a 2 1.a even 1 1 trivial
3380.2.f.a 2 13.b even 2 1 inner

## Hecke kernels

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

 $$T_{3} + 3$$ $$T_{19}^{2} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( 3 + T )^{2}$$
$5$ $$1 + T^{2}$$
$7$ $$9 + T^{2}$$
$11$ $$9 + T^{2}$$
$13$ $$T^{2}$$
$17$ $$( -7 + T )^{2}$$
$19$ $$1 + T^{2}$$
$23$ $$( -7 + T )^{2}$$
$29$ $$( 5 + T )^{2}$$
$31$ $$16 + T^{2}$$
$37$ $$9 + T^{2}$$
$41$ $$49 + T^{2}$$
$43$ $$( -9 + T )^{2}$$
$47$ $$64 + T^{2}$$
$53$ $$( 6 + T )^{2}$$
$59$ $$25 + T^{2}$$
$61$ $$( 5 + T )^{2}$$
$67$ $$169 + T^{2}$$
$71$ $$9 + T^{2}$$
$73$ $$196 + T^{2}$$
$79$ $$( 8 + T )^{2}$$
$83$ $$144 + T^{2}$$
$89$ $$49 + T^{2}$$
$97$ $$121 + T^{2}$$