# Properties

 Label 3330.2.d.d Level $3330$ Weight $2$ Character orbit 3330.d Analytic conductor $26.590$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3330.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$26.5901838731$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1110) 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.

 $$f(q)$$ $$=$$ $$q -i q^{2} - q^{4} + ( -1 + 2 i ) q^{5} + i q^{7} + i q^{8} +O(q^{10})$$ $$q -i q^{2} - q^{4} + ( -1 + 2 i ) q^{5} + i q^{7} + i q^{8} + ( 2 + i ) q^{10} -5 q^{11} + q^{14} + q^{16} -i q^{17} + ( 1 - 2 i ) q^{20} + 5 i q^{22} + 4 i q^{23} + ( -3 - 4 i ) q^{25} -i q^{28} -3 q^{29} + q^{31} -i q^{32} - q^{34} + ( -2 - i ) q^{35} -i q^{37} + ( -2 - i ) q^{40} + q^{41} -7 i q^{43} + 5 q^{44} + 4 q^{46} + 4 i q^{47} + 6 q^{49} + ( -4 + 3 i ) q^{50} + 3 i q^{53} + ( 5 - 10 i ) q^{55} - q^{56} + 3 i q^{58} -8 q^{59} + 5 q^{61} -i q^{62} - q^{64} -4 i q^{67} + i q^{68} + ( -1 + 2 i ) q^{70} + 6 q^{71} -10 i q^{73} - q^{74} -5 i q^{77} + ( -1 + 2 i ) q^{80} -i q^{82} -8 i q^{83} + ( 2 + i ) q^{85} -7 q^{86} -5 i q^{88} + 8 q^{89} -4 i q^{92} + 4 q^{94} -i q^{97} -6 i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} - 2q^{5} + O(q^{10})$$ $$2q - 2q^{4} - 2q^{5} + 4q^{10} - 10q^{11} + 2q^{14} + 2q^{16} + 2q^{20} - 6q^{25} - 6q^{29} + 2q^{31} - 2q^{34} - 4q^{35} - 4q^{40} + 2q^{41} + 10q^{44} + 8q^{46} + 12q^{49} - 8q^{50} + 10q^{55} - 2q^{56} - 16q^{59} + 10q^{61} - 2q^{64} - 2q^{70} + 12q^{71} - 2q^{74} - 2q^{80} + 4q^{85} - 14q^{86} + 16q^{89} + 8q^{94} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3330\mathbb{Z}\right)^\times$$.

 $$n$$ $$371$$ $$631$$ $$667$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1999.1
 1.00000i − 1.00000i
1.00000i 0 −1.00000 −1.00000 + 2.00000i 0 1.00000i 1.00000i 0 2.00000 + 1.00000i
1999.2 1.00000i 0 −1.00000 −1.00000 2.00000i 0 1.00000i 1.00000i 0 2.00000 1.00000i
 $$n$$: 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
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3330.2.d.d 2
3.b odd 2 1 1110.2.d.c 2
5.b even 2 1 inner 3330.2.d.d 2
15.d odd 2 1 1110.2.d.c 2
15.e even 4 1 5550.2.a.d 1
15.e even 4 1 5550.2.a.bo 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.d.c 2 3.b odd 2 1
1110.2.d.c 2 15.d odd 2 1
3330.2.d.d 2 1.a even 1 1 trivial
3330.2.d.d 2 5.b even 2 1 inner
5550.2.a.d 1 15.e even 4 1
5550.2.a.bo 1 15.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(3330, [\chi])$$:

 $$T_{7}^{2} + 1$$ $$T_{11} + 5$$ $$T_{17}^{2} + 1$$ $$T_{29} + 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ $$T^{2}$$
$5$ $$5 + 2 T + T^{2}$$
$7$ $$1 + T^{2}$$
$11$ $$( 5 + T )^{2}$$
$13$ $$T^{2}$$
$17$ $$1 + T^{2}$$
$19$ $$T^{2}$$
$23$ $$16 + T^{2}$$
$29$ $$( 3 + T )^{2}$$
$31$ $$( -1 + T )^{2}$$
$37$ $$1 + T^{2}$$
$41$ $$( -1 + T )^{2}$$
$43$ $$49 + T^{2}$$
$47$ $$16 + T^{2}$$
$53$ $$9 + T^{2}$$
$59$ $$( 8 + T )^{2}$$
$61$ $$( -5 + T )^{2}$$
$67$ $$16 + T^{2}$$
$71$ $$( -6 + T )^{2}$$
$73$ $$100 + T^{2}$$
$79$ $$T^{2}$$
$83$ $$64 + T^{2}$$
$89$ $$( -8 + T )^{2}$$
$97$ $$1 + T^{2}$$