# Properties

 Label 3330.2.e.b Level $3330$ Weight $2$ Character orbit 3330.e Analytic conductor $26.590$ Analytic rank $1$ 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.e (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$26.5901838731$$ Analytic rank: $$1$$ 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 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.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + ( -2 + i ) q^{5} + 3 i q^{7} + q^{8} +O(q^{10})$$ $$q + q^{2} + q^{4} + ( -2 + i ) q^{5} + 3 i q^{7} + q^{8} + ( -2 + i ) q^{10} + 3 q^{11} -4 q^{13} + 3 i q^{14} + q^{16} -7 q^{17} -4 i q^{19} + ( -2 + i ) q^{20} + 3 q^{22} -6 q^{23} + ( 3 - 4 i ) q^{25} -4 q^{26} + 3 i q^{28} + 9 i q^{29} -5 i q^{31} + q^{32} -7 q^{34} + ( -3 - 6 i ) q^{35} + ( 6 - i ) q^{37} -4 i q^{38} + ( -2 + i ) q^{40} -7 q^{41} + q^{43} + 3 q^{44} -6 q^{46} -8 i q^{47} -2 q^{49} + ( 3 - 4 i ) q^{50} -4 q^{52} -i q^{53} + ( -6 + 3 i ) q^{55} + 3 i q^{56} + 9 i q^{58} -6 i q^{59} -5 i q^{61} -5 i q^{62} + q^{64} + ( 8 - 4 i ) q^{65} -2 i q^{67} -7 q^{68} + ( -3 - 6 i ) q^{70} -12 q^{71} -4 i q^{73} + ( 6 - i ) q^{74} -4 i q^{76} + 9 i q^{77} -4 i q^{79} + ( -2 + i ) q^{80} -7 q^{82} + 14 i q^{83} + ( 14 - 7 i ) q^{85} + q^{86} + 3 q^{88} -6 i q^{89} -12 i q^{91} -6 q^{92} -8 i q^{94} + ( 4 + 8 i ) q^{95} + 7 q^{97} -2 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} + 2q^{4} - 4q^{5} + 2q^{8} + O(q^{10})$$ $$2q + 2q^{2} + 2q^{4} - 4q^{5} + 2q^{8} - 4q^{10} + 6q^{11} - 8q^{13} + 2q^{16} - 14q^{17} - 4q^{20} + 6q^{22} - 12q^{23} + 6q^{25} - 8q^{26} + 2q^{32} - 14q^{34} - 6q^{35} + 12q^{37} - 4q^{40} - 14q^{41} + 2q^{43} + 6q^{44} - 12q^{46} - 4q^{49} + 6q^{50} - 8q^{52} - 12q^{55} + 2q^{64} + 16q^{65} - 14q^{68} - 6q^{70} - 24q^{71} + 12q^{74} - 4q^{80} - 14q^{82} + 28q^{85} + 2q^{86} + 6q^{88} - 12q^{92} + 8q^{95} + 14q^{97} - 4q^{98} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
739.1
 − 1.00000i 1.00000i
1.00000 0 1.00000 −2.00000 1.00000i 0 3.00000i 1.00000 0 −2.00000 1.00000i
739.2 1.00000 0 1.00000 −2.00000 + 1.00000i 0 3.00000i 1.00000 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
185.d 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.e.b 2
3.b odd 2 1 1110.2.e.a 2
5.b even 2 1 3330.2.e.a 2
15.d odd 2 1 1110.2.e.b yes 2
37.b even 2 1 3330.2.e.a 2
111.d odd 2 1 1110.2.e.b yes 2
185.d even 2 1 inner 3330.2.e.b 2
555.b odd 2 1 1110.2.e.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.e.a 2 3.b odd 2 1
1110.2.e.a 2 555.b odd 2 1
1110.2.e.b yes 2 15.d odd 2 1
1110.2.e.b yes 2 111.d odd 2 1
3330.2.e.a 2 5.b even 2 1
3330.2.e.a 2 37.b even 2 1
3330.2.e.b 2 1.a even 1 1 trivial
3330.2.e.b 2 185.d 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 $$S_{2}^{\mathrm{new}}(3330, [\chi])$$:

 $$T_{7}^{2} + 9$$ $$T_{13} + 4$$ $$T_{17} + 7$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{2}$$
$3$ $$T^{2}$$
$5$ $$5 + 4 T + T^{2}$$
$7$ $$9 + T^{2}$$
$11$ $$( -3 + T )^{2}$$
$13$ $$( 4 + T )^{2}$$
$17$ $$( 7 + T )^{2}$$
$19$ $$16 + T^{2}$$
$23$ $$( 6 + T )^{2}$$
$29$ $$81 + T^{2}$$
$31$ $$25 + T^{2}$$
$37$ $$37 - 12 T + T^{2}$$
$41$ $$( 7 + T )^{2}$$
$43$ $$( -1 + T )^{2}$$
$47$ $$64 + T^{2}$$
$53$ $$1 + T^{2}$$
$59$ $$36 + T^{2}$$
$61$ $$25 + T^{2}$$
$67$ $$4 + T^{2}$$
$71$ $$( 12 + T )^{2}$$
$73$ $$16 + T^{2}$$
$79$ $$16 + T^{2}$$
$83$ $$196 + T^{2}$$
$89$ $$36 + T^{2}$$
$97$ $$( -7 + T )^{2}$$