# Properties

 Label 3330.2.a.bd Level $3330$ Weight $2$ Character orbit 3330.a Self dual yes Analytic conductor $26.590$ Analytic rank $0$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ x^2 - 3 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 $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} - q^{5} + (\beta - 3) q^{7} + q^{8}+O(q^{10})$$ q + q^2 + q^4 - q^5 + (b - 3) * q^7 + q^8 $$q + q^{2} + q^{4} - q^{5} + (\beta - 3) q^{7} + q^{8} - q^{10} + ( - 2 \beta + 2) q^{11} + (2 \beta - 2) q^{13} + (\beta - 3) q^{14} + q^{16} + (2 \beta - 2) q^{17} + ( - 3 \beta + 1) q^{19} - q^{20} + ( - 2 \beta + 2) q^{22} + 8 q^{23} + q^{25} + (2 \beta - 2) q^{26} + (\beta - 3) q^{28} + (4 \beta + 2) q^{29} + ( - \beta - 1) q^{31} + q^{32} + (2 \beta - 2) q^{34} + ( - \beta + 3) q^{35} + q^{37} + ( - 3 \beta + 1) q^{38} - q^{40} + 2 q^{41} - 4 \beta q^{43} + ( - 2 \beta + 2) q^{44} + 8 q^{46} + ( - \beta + 3) q^{47} + ( - 6 \beta + 5) q^{49} + q^{50} + (2 \beta - 2) q^{52} + 6 q^{53} + (2 \beta - 2) q^{55} + (\beta - 3) q^{56} + (4 \beta + 2) q^{58} + ( - 3 \beta + 5) q^{59} + (4 \beta + 2) q^{61} + ( - \beta - 1) q^{62} + q^{64} + ( - 2 \beta + 2) q^{65} + (5 \beta + 5) q^{67} + (2 \beta - 2) q^{68} + ( - \beta + 3) q^{70} + (4 \beta + 4) q^{71} + (4 \beta + 6) q^{73} + q^{74} + ( - 3 \beta + 1) q^{76} + (8 \beta - 12) q^{77} + ( - \beta + 7) q^{79} - q^{80} + 2 q^{82} + ( - \beta + 7) q^{83} + ( - 2 \beta + 2) q^{85} - 4 \beta q^{86} + ( - 2 \beta + 2) q^{88} + 2 q^{89} + ( - 8 \beta + 12) q^{91} + 8 q^{92} + ( - \beta + 3) q^{94} + (3 \beta - 1) q^{95} - 2 q^{97} + ( - 6 \beta + 5) q^{98} +O(q^{100})$$ q + q^2 + q^4 - q^5 + (b - 3) * q^7 + q^8 - q^10 + (-2*b + 2) * q^11 + (2*b - 2) * q^13 + (b - 3) * q^14 + q^16 + (2*b - 2) * q^17 + (-3*b + 1) * q^19 - q^20 + (-2*b + 2) * q^22 + 8 * q^23 + q^25 + (2*b - 2) * q^26 + (b - 3) * q^28 + (4*b + 2) * q^29 + (-b - 1) * q^31 + q^32 + (2*b - 2) * q^34 + (-b + 3) * q^35 + q^37 + (-3*b + 1) * q^38 - q^40 + 2 * q^41 - 4*b * q^43 + (-2*b + 2) * q^44 + 8 * q^46 + (-b + 3) * q^47 + (-6*b + 5) * q^49 + q^50 + (2*b - 2) * q^52 + 6 * q^53 + (2*b - 2) * q^55 + (b - 3) * q^56 + (4*b + 2) * q^58 + (-3*b + 5) * q^59 + (4*b + 2) * q^61 + (-b - 1) * q^62 + q^64 + (-2*b + 2) * q^65 + (5*b + 5) * q^67 + (2*b - 2) * q^68 + (-b + 3) * q^70 + (4*b + 4) * q^71 + (4*b + 6) * q^73 + q^74 + (-3*b + 1) * q^76 + (8*b - 12) * q^77 + (-b + 7) * q^79 - q^80 + 2 * q^82 + (-b + 7) * q^83 + (-2*b + 2) * q^85 - 4*b * q^86 + (-2*b + 2) * q^88 + 2 * q^89 + (-8*b + 12) * q^91 + 8 * q^92 + (-b + 3) * q^94 + (3*b - 1) * q^95 - 2 * q^97 + (-6*b + 5) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 6 q^{7} + 2 q^{8}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 - 6 * q^7 + 2 * q^8 $$2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 6 q^{7} + 2 q^{8} - 2 q^{10} + 4 q^{11} - 4 q^{13} - 6 q^{14} + 2 q^{16} - 4 q^{17} + 2 q^{19} - 2 q^{20} + 4 q^{22} + 16 q^{23} + 2 q^{25} - 4 q^{26} - 6 q^{28} + 4 q^{29} - 2 q^{31} + 2 q^{32} - 4 q^{34} + 6 q^{35} + 2 q^{37} + 2 q^{38} - 2 q^{40} + 4 q^{41} + 4 q^{44} + 16 q^{46} + 6 q^{47} + 10 q^{49} + 2 q^{50} - 4 q^{52} + 12 q^{53} - 4 q^{55} - 6 q^{56} + 4 q^{58} + 10 q^{59} + 4 q^{61} - 2 q^{62} + 2 q^{64} + 4 q^{65} + 10 q^{67} - 4 q^{68} + 6 q^{70} + 8 q^{71} + 12 q^{73} + 2 q^{74} + 2 q^{76} - 24 q^{77} + 14 q^{79} - 2 q^{80} + 4 q^{82} + 14 q^{83} + 4 q^{85} + 4 q^{88} + 4 q^{89} + 24 q^{91} + 16 q^{92} + 6 q^{94} - 2 q^{95} - 4 q^{97} + 10 q^{98}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 - 6 * q^7 + 2 * q^8 - 2 * q^10 + 4 * q^11 - 4 * q^13 - 6 * q^14 + 2 * q^16 - 4 * q^17 + 2 * q^19 - 2 * q^20 + 4 * q^22 + 16 * q^23 + 2 * q^25 - 4 * q^26 - 6 * q^28 + 4 * q^29 - 2 * q^31 + 2 * q^32 - 4 * q^34 + 6 * q^35 + 2 * q^37 + 2 * q^38 - 2 * q^40 + 4 * q^41 + 4 * q^44 + 16 * q^46 + 6 * q^47 + 10 * q^49 + 2 * q^50 - 4 * q^52 + 12 * q^53 - 4 * q^55 - 6 * q^56 + 4 * q^58 + 10 * q^59 + 4 * q^61 - 2 * q^62 + 2 * q^64 + 4 * q^65 + 10 * q^67 - 4 * q^68 + 6 * q^70 + 8 * q^71 + 12 * q^73 + 2 * q^74 + 2 * q^76 - 24 * q^77 + 14 * q^79 - 2 * q^80 + 4 * q^82 + 14 * q^83 + 4 * q^85 + 4 * q^88 + 4 * q^89 + 24 * q^91 + 16 * q^92 + 6 * q^94 - 2 * q^95 - 4 * q^97 + 10 * q^98

## 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
 −1.73205 1.73205
1.00000 0 1.00000 −1.00000 0 −4.73205 1.00000 0 −1.00000
1.2 1.00000 0 1.00000 −1.00000 0 −1.26795 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.bd 2
3.b odd 2 1 370.2.a.e 2
12.b even 2 1 2960.2.a.q 2
15.d odd 2 1 1850.2.a.x 2
15.e even 4 2 1850.2.b.l 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.a.e 2 3.b odd 2 1
1850.2.a.x 2 15.d odd 2 1
1850.2.b.l 4 15.e even 4 2
2960.2.a.q 2 12.b even 2 1
3330.2.a.bd 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3330))$$:

 $$T_{7}^{2} + 6T_{7} + 6$$ T7^2 + 6*T7 + 6 $$T_{11}^{2} - 4T_{11} - 8$$ T11^2 - 4*T11 - 8 $$T_{13}^{2} + 4T_{13} - 8$$ T13^2 + 4*T13 - 8 $$T_{17}^{2} + 4T_{17} - 8$$ T17^2 + 4*T17 - 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{2}$$
$3$ $$T^{2}$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2} + 6T + 6$$
$11$ $$T^{2} - 4T - 8$$
$13$ $$T^{2} + 4T - 8$$
$17$ $$T^{2} + 4T - 8$$
$19$ $$T^{2} - 2T - 26$$
$23$ $$(T - 8)^{2}$$
$29$ $$T^{2} - 4T - 44$$
$31$ $$T^{2} + 2T - 2$$
$37$ $$(T - 1)^{2}$$
$41$ $$(T - 2)^{2}$$
$43$ $$T^{2} - 48$$
$47$ $$T^{2} - 6T + 6$$
$53$ $$(T - 6)^{2}$$
$59$ $$T^{2} - 10T - 2$$
$61$ $$T^{2} - 4T - 44$$
$67$ $$T^{2} - 10T - 50$$
$71$ $$T^{2} - 8T - 32$$
$73$ $$T^{2} - 12T - 12$$
$79$ $$T^{2} - 14T + 46$$
$83$ $$T^{2} - 14T + 46$$
$89$ $$(T - 2)^{2}$$
$97$ $$(T + 2)^{2}$$