# Properties

 Label 3300.2.a.w Level $3300$ Weight $2$ Character orbit 3300.a Self dual yes Analytic conductor $26.351$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3300 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3300.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$26.3506326670$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ Defining polynomial: $$x^{2} - x - 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 660) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{13}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{3} + ( -1 - \beta ) q^{7} + q^{9} +O(q^{10})$$ $$q + q^{3} + ( -1 - \beta ) q^{7} + q^{9} + q^{11} + ( 1 + \beta ) q^{13} + ( -3 - \beta ) q^{17} -2 \beta q^{19} + ( -1 - \beta ) q^{21} + q^{27} + 8 q^{29} + ( 2 + 2 \beta ) q^{31} + q^{33} + ( -4 + 2 \beta ) q^{37} + ( 1 + \beta ) q^{39} + 8 q^{41} + ( 7 - \beta ) q^{43} + ( -2 + 2 \beta ) q^{47} + ( 7 + 2 \beta ) q^{49} + ( -3 - \beta ) q^{51} -2 q^{53} -2 \beta q^{57} + 8 q^{59} + 2 \beta q^{61} + ( -1 - \beta ) q^{63} + 4 q^{67} -4 \beta q^{71} + ( 3 - \beta ) q^{73} + ( -1 - \beta ) q^{77} + ( -4 - 2 \beta ) q^{79} + q^{81} + ( 7 + \beta ) q^{83} + 8 q^{87} + 6 q^{89} + ( -14 - 2 \beta ) q^{91} + ( 2 + 2 \beta ) q^{93} + ( -2 + 4 \beta ) q^{97} + q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} - 2 q^{7} + 2 q^{9} + O(q^{10})$$ $$2 q + 2 q^{3} - 2 q^{7} + 2 q^{9} + 2 q^{11} + 2 q^{13} - 6 q^{17} - 2 q^{21} + 2 q^{27} + 16 q^{29} + 4 q^{31} + 2 q^{33} - 8 q^{37} + 2 q^{39} + 16 q^{41} + 14 q^{43} - 4 q^{47} + 14 q^{49} - 6 q^{51} - 4 q^{53} + 16 q^{59} - 2 q^{63} + 8 q^{67} + 6 q^{73} - 2 q^{77} - 8 q^{79} + 2 q^{81} + 14 q^{83} + 16 q^{87} + 12 q^{89} - 28 q^{91} + 4 q^{93} - 4 q^{97} + 2 q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 2.30278 −1.30278
0 1.00000 0 0 0 −4.60555 0 1.00000 0
1.2 0 1.00000 0 0 0 2.60555 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$1$$
$$11$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3300.2.a.w 2
3.b odd 2 1 9900.2.a.bl 2
5.b even 2 1 660.2.a.e 2
5.c odd 4 2 3300.2.c.l 4
15.d odd 2 1 1980.2.a.h 2
15.e even 4 2 9900.2.c.q 4
20.d odd 2 1 2640.2.a.bc 2
55.d odd 2 1 7260.2.a.w 2
60.h even 2 1 7920.2.a.bo 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
660.2.a.e 2 5.b even 2 1
1980.2.a.h 2 15.d odd 2 1
2640.2.a.bc 2 20.d odd 2 1
3300.2.a.w 2 1.a even 1 1 trivial
3300.2.c.l 4 5.c odd 4 2
7260.2.a.w 2 55.d odd 2 1
7920.2.a.bo 2 60.h even 2 1
9900.2.a.bl 2 3.b odd 2 1
9900.2.c.q 4 15.e even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3300))$$:

 $$T_{7}^{2} + 2 T_{7} - 12$$ $$T_{13}^{2} - 2 T_{13} - 12$$ $$T_{17}^{2} + 6 T_{17} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$T^{2}$$
$7$ $$-12 + 2 T + T^{2}$$
$11$ $$( -1 + T )^{2}$$
$13$ $$-12 - 2 T + T^{2}$$
$17$ $$-4 + 6 T + T^{2}$$
$19$ $$-52 + T^{2}$$
$23$ $$T^{2}$$
$29$ $$( -8 + T )^{2}$$
$31$ $$-48 - 4 T + T^{2}$$
$37$ $$-36 + 8 T + T^{2}$$
$41$ $$( -8 + T )^{2}$$
$43$ $$36 - 14 T + T^{2}$$
$47$ $$-48 + 4 T + T^{2}$$
$53$ $$( 2 + T )^{2}$$
$59$ $$( -8 + T )^{2}$$
$61$ $$-52 + T^{2}$$
$67$ $$( -4 + T )^{2}$$
$71$ $$-208 + T^{2}$$
$73$ $$-4 - 6 T + T^{2}$$
$79$ $$-36 + 8 T + T^{2}$$
$83$ $$36 - 14 T + T^{2}$$
$89$ $$( -6 + T )^{2}$$
$97$ $$-204 + 4 T + T^{2}$$