Properties

 Label 3330.2.a.bb Level $3330$ Weight $2$ Character orbit 3330.a Self dual yes Analytic conductor $26.590$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3330.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$26.5901838731$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{33})$$ Defining polynomial: $$x^{2} - x - 8$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 370) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{33})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} - q^{5} + ( -1 - \beta ) q^{7} - q^{8} +O(q^{10})$$ $$q - q^{2} + q^{4} - q^{5} + ( -1 - \beta ) q^{7} - q^{8} + q^{10} + ( 1 - \beta ) q^{11} + 2 \beta q^{13} + ( 1 + \beta ) q^{14} + q^{16} + ( 3 - \beta ) q^{17} -2 q^{19} - q^{20} + ( -1 + \beta ) q^{22} + ( -2 + 2 \beta ) q^{23} + q^{25} -2 \beta q^{26} + ( -1 - \beta ) q^{28} + ( -1 + 3 \beta ) q^{29} + ( -5 - \beta ) q^{31} - q^{32} + ( -3 + \beta ) q^{34} + ( 1 + \beta ) q^{35} + q^{37} + 2 q^{38} + q^{40} + ( -3 + \beta ) q^{41} + ( 5 - \beta ) q^{43} + ( 1 - \beta ) q^{44} + ( 2 - 2 \beta ) q^{46} + ( 4 - 2 \beta ) q^{47} + ( 2 + 3 \beta ) q^{49} - q^{50} + 2 \beta q^{52} + ( 1 + \beta ) q^{53} + ( -1 + \beta ) q^{55} + ( 1 + \beta ) q^{56} + ( 1 - 3 \beta ) q^{58} + ( -8 + 2 \beta ) q^{59} + ( -3 + \beta ) q^{61} + ( 5 + \beta ) q^{62} + q^{64} -2 \beta q^{65} -2 \beta q^{67} + ( 3 - \beta ) q^{68} + ( -1 - \beta ) q^{70} + ( 2 - 2 \beta ) q^{71} + ( 4 - 2 \beta ) q^{73} - q^{74} -2 q^{76} + ( 7 + \beta ) q^{77} + 2 \beta q^{79} - q^{80} + ( 3 - \beta ) q^{82} + ( -4 - 2 \beta ) q^{83} + ( -3 + \beta ) q^{85} + ( -5 + \beta ) q^{86} + ( -1 + \beta ) q^{88} -10 q^{89} + ( -16 - 4 \beta ) q^{91} + ( -2 + 2 \beta ) q^{92} + ( -4 + 2 \beta ) q^{94} + 2 q^{95} + ( 7 + 3 \beta ) q^{97} + ( -2 - 3 \beta ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} + 2q^{4} - 2q^{5} - 3q^{7} - 2q^{8} + O(q^{10})$$ $$2q - 2q^{2} + 2q^{4} - 2q^{5} - 3q^{7} - 2q^{8} + 2q^{10} + q^{11} + 2q^{13} + 3q^{14} + 2q^{16} + 5q^{17} - 4q^{19} - 2q^{20} - q^{22} - 2q^{23} + 2q^{25} - 2q^{26} - 3q^{28} + q^{29} - 11q^{31} - 2q^{32} - 5q^{34} + 3q^{35} + 2q^{37} + 4q^{38} + 2q^{40} - 5q^{41} + 9q^{43} + q^{44} + 2q^{46} + 6q^{47} + 7q^{49} - 2q^{50} + 2q^{52} + 3q^{53} - q^{55} + 3q^{56} - q^{58} - 14q^{59} - 5q^{61} + 11q^{62} + 2q^{64} - 2q^{65} - 2q^{67} + 5q^{68} - 3q^{70} + 2q^{71} + 6q^{73} - 2q^{74} - 4q^{76} + 15q^{77} + 2q^{79} - 2q^{80} + 5q^{82} - 10q^{83} - 5q^{85} - 9q^{86} - q^{88} - 20q^{89} - 36q^{91} - 2q^{92} - 6q^{94} + 4q^{95} + 17q^{97} - 7q^{98} + O(q^{100})$$

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
 3.37228 −2.37228
−1.00000 0 1.00000 −1.00000 0 −4.37228 −1.00000 0 1.00000
1.2 −1.00000 0 1.00000 −1.00000 0 1.37228 −1.00000 0 1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$-1$$
$$5$$ $$1$$
$$37$$ $$-1$$

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 3330.2.a.bb 2
3.b odd 2 1 370.2.a.f 2
12.b even 2 1 2960.2.a.o 2
15.d odd 2 1 1850.2.a.q 2
15.e even 4 2 1850.2.b.m 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.a.f 2 3.b odd 2 1
1850.2.a.q 2 15.d odd 2 1
1850.2.b.m 4 15.e even 4 2
2960.2.a.o 2 12.b even 2 1
3330.2.a.bb 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3330))$$:

 $$T_{7}^{2} + 3 T_{7} - 6$$ $$T_{11}^{2} - T_{11} - 8$$ $$T_{13}^{2} - 2 T_{13} - 32$$ $$T_{17}^{2} - 5 T_{17} - 2$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$T^{2}$$
$5$ $$( 1 + T )^{2}$$
$7$ $$-6 + 3 T + T^{2}$$
$11$ $$-8 - T + T^{2}$$
$13$ $$-32 - 2 T + T^{2}$$
$17$ $$-2 - 5 T + T^{2}$$
$19$ $$( 2 + T )^{2}$$
$23$ $$-32 + 2 T + T^{2}$$
$29$ $$-74 - T + T^{2}$$
$31$ $$22 + 11 T + T^{2}$$
$37$ $$( -1 + T )^{2}$$
$41$ $$-2 + 5 T + T^{2}$$
$43$ $$12 - 9 T + T^{2}$$
$47$ $$-24 - 6 T + T^{2}$$
$53$ $$-6 - 3 T + T^{2}$$
$59$ $$16 + 14 T + T^{2}$$
$61$ $$-2 + 5 T + T^{2}$$
$67$ $$-32 + 2 T + T^{2}$$
$71$ $$-32 - 2 T + T^{2}$$
$73$ $$-24 - 6 T + T^{2}$$
$79$ $$-32 - 2 T + T^{2}$$
$83$ $$-8 + 10 T + T^{2}$$
$89$ $$( 10 + T )^{2}$$
$97$ $$-2 - 17 T + T^{2}$$