# Properties

 Label 3234.2.a.be Level $3234$ Weight $2$ Character orbit 3234.a Self dual yes Analytic conductor $25.824$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3234.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$25.8236200137$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} + q^{4} + (\beta + 1) q^{5} + q^{6} + q^{8} + q^{9}+O(q^{10})$$ q + q^2 + q^3 + q^4 + (b + 1) * q^5 + q^6 + q^8 + q^9 $$q + q^{2} + q^{3} + q^{4} + (\beta + 1) q^{5} + q^{6} + q^{8} + q^{9} + (\beta + 1) q^{10} - q^{11} + q^{12} + (\beta + 1) q^{15} + q^{16} + ( - \beta + 3) q^{17} + q^{18} + ( - 3 \beta + 1) q^{19} + (\beta + 1) q^{20} - q^{22} + (2 \beta + 2) q^{23} + q^{24} + (2 \beta + 1) q^{25} + q^{27} - 2 \beta q^{29} + (\beta + 1) q^{30} + (\beta + 5) q^{31} + q^{32} - q^{33} + ( - \beta + 3) q^{34} + q^{36} + (4 \beta - 2) q^{37} + ( - 3 \beta + 1) q^{38} + (\beta + 1) q^{40} + ( - \beta + 3) q^{41} + ( - 2 \beta + 2) q^{43} - q^{44} + (\beta + 1) q^{45} + (2 \beta + 2) q^{46} + (3 \beta + 3) q^{47} + q^{48} + (2 \beta + 1) q^{50} + ( - \beta + 3) q^{51} - 6 q^{53} + q^{54} + ( - \beta - 1) q^{55} + ( - 3 \beta + 1) q^{57} - 2 \beta q^{58} + (2 \beta + 6) q^{59} + (\beta + 1) q^{60} + (\beta + 5) q^{62} + q^{64} - q^{66} + ( - 6 \beta + 2) q^{67} + ( - \beta + 3) q^{68} + (2 \beta + 2) q^{69} + (2 \beta + 2) q^{71} + q^{72} + ( - 3 \beta + 9) q^{73} + (4 \beta - 2) q^{74} + (2 \beta + 1) q^{75} + ( - 3 \beta + 1) q^{76} + ( - 6 \beta - 2) q^{79} + (\beta + 1) q^{80} + q^{81} + ( - \beta + 3) q^{82} + (\beta + 1) q^{83} + (2 \beta - 2) q^{85} + ( - 2 \beta + 2) q^{86} - 2 \beta q^{87} - q^{88} + ( - 2 \beta + 2) q^{89} + (\beta + 1) q^{90} + (2 \beta + 2) q^{92} + (\beta + 5) q^{93} + (3 \beta + 3) q^{94} + ( - 2 \beta - 14) q^{95} + q^{96} - q^{99} +O(q^{100})$$ q + q^2 + q^3 + q^4 + (b + 1) * q^5 + q^6 + q^8 + q^9 + (b + 1) * q^10 - q^11 + q^12 + (b + 1) * q^15 + q^16 + (-b + 3) * q^17 + q^18 + (-3*b + 1) * q^19 + (b + 1) * q^20 - q^22 + (2*b + 2) * q^23 + q^24 + (2*b + 1) * q^25 + q^27 - 2*b * q^29 + (b + 1) * q^30 + (b + 5) * q^31 + q^32 - q^33 + (-b + 3) * q^34 + q^36 + (4*b - 2) * q^37 + (-3*b + 1) * q^38 + (b + 1) * q^40 + (-b + 3) * q^41 + (-2*b + 2) * q^43 - q^44 + (b + 1) * q^45 + (2*b + 2) * q^46 + (3*b + 3) * q^47 + q^48 + (2*b + 1) * q^50 + (-b + 3) * q^51 - 6 * q^53 + q^54 + (-b - 1) * q^55 + (-3*b + 1) * q^57 - 2*b * q^58 + (2*b + 6) * q^59 + (b + 1) * q^60 + (b + 5) * q^62 + q^64 - q^66 + (-6*b + 2) * q^67 + (-b + 3) * q^68 + (2*b + 2) * q^69 + (2*b + 2) * q^71 + q^72 + (-3*b + 9) * q^73 + (4*b - 2) * q^74 + (2*b + 1) * q^75 + (-3*b + 1) * q^76 + (-6*b - 2) * q^79 + (b + 1) * q^80 + q^81 + (-b + 3) * q^82 + (b + 1) * q^83 + (2*b - 2) * q^85 + (-2*b + 2) * q^86 - 2*b * q^87 - q^88 + (-2*b + 2) * q^89 + (b + 1) * q^90 + (2*b + 2) * q^92 + (b + 5) * q^93 + (3*b + 3) * q^94 + (-2*b - 14) * q^95 + q^96 - q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} + 2 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^5 + 2 * q^6 + 2 * q^8 + 2 * q^9 $$2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} + 2 q^{8} + 2 q^{9} + 2 q^{10} - 2 q^{11} + 2 q^{12} + 2 q^{15} + 2 q^{16} + 6 q^{17} + 2 q^{18} + 2 q^{19} + 2 q^{20} - 2 q^{22} + 4 q^{23} + 2 q^{24} + 2 q^{25} + 2 q^{27} + 2 q^{30} + 10 q^{31} + 2 q^{32} - 2 q^{33} + 6 q^{34} + 2 q^{36} - 4 q^{37} + 2 q^{38} + 2 q^{40} + 6 q^{41} + 4 q^{43} - 2 q^{44} + 2 q^{45} + 4 q^{46} + 6 q^{47} + 2 q^{48} + 2 q^{50} + 6 q^{51} - 12 q^{53} + 2 q^{54} - 2 q^{55} + 2 q^{57} + 12 q^{59} + 2 q^{60} + 10 q^{62} + 2 q^{64} - 2 q^{66} + 4 q^{67} + 6 q^{68} + 4 q^{69} + 4 q^{71} + 2 q^{72} + 18 q^{73} - 4 q^{74} + 2 q^{75} + 2 q^{76} - 4 q^{79} + 2 q^{80} + 2 q^{81} + 6 q^{82} + 2 q^{83} - 4 q^{85} + 4 q^{86} - 2 q^{88} + 4 q^{89} + 2 q^{90} + 4 q^{92} + 10 q^{93} + 6 q^{94} - 28 q^{95} + 2 q^{96} - 2 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^5 + 2 * q^6 + 2 * q^8 + 2 * q^9 + 2 * q^10 - 2 * q^11 + 2 * q^12 + 2 * q^15 + 2 * q^16 + 6 * q^17 + 2 * q^18 + 2 * q^19 + 2 * q^20 - 2 * q^22 + 4 * q^23 + 2 * q^24 + 2 * q^25 + 2 * q^27 + 2 * q^30 + 10 * q^31 + 2 * q^32 - 2 * q^33 + 6 * q^34 + 2 * q^36 - 4 * q^37 + 2 * q^38 + 2 * q^40 + 6 * q^41 + 4 * q^43 - 2 * q^44 + 2 * q^45 + 4 * q^46 + 6 * q^47 + 2 * q^48 + 2 * q^50 + 6 * q^51 - 12 * q^53 + 2 * q^54 - 2 * q^55 + 2 * q^57 + 12 * q^59 + 2 * q^60 + 10 * q^62 + 2 * q^64 - 2 * q^66 + 4 * q^67 + 6 * q^68 + 4 * q^69 + 4 * q^71 + 2 * q^72 + 18 * q^73 - 4 * q^74 + 2 * q^75 + 2 * q^76 - 4 * q^79 + 2 * q^80 + 2 * q^81 + 6 * q^82 + 2 * q^83 - 4 * q^85 + 4 * q^86 - 2 * q^88 + 4 * q^89 + 2 * q^90 + 4 * q^92 + 10 * q^93 + 6 * q^94 - 28 * q^95 + 2 * q^96 - 2 * q^99

## 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
 −0.618034 1.61803
1.00000 1.00000 1.00000 −1.23607 1.00000 0 1.00000 1.00000 −1.23607
1.2 1.00000 1.00000 1.00000 3.23607 1.00000 0 1.00000 1.00000 3.23607
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$7$$ $$-1$$
$$11$$ $$1$$

## 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 3234.2.a.be yes 2
3.b odd 2 1 9702.2.a.ck 2
7.b odd 2 1 3234.2.a.bb 2
21.c even 2 1 9702.2.a.cw 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3234.2.a.bb 2 7.b odd 2 1
3234.2.a.be yes 2 1.a even 1 1 trivial
9702.2.a.ck 2 3.b odd 2 1
9702.2.a.cw 2 21.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3234))$$:

 $$T_{5}^{2} - 2T_{5} - 4$$ T5^2 - 2*T5 - 4 $$T_{13}$$ T13 $$T_{17}^{2} - 6T_{17} + 4$$ T17^2 - 6*T17 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{2}$$
$3$ $$(T - 1)^{2}$$
$5$ $$T^{2} - 2T - 4$$
$7$ $$T^{2}$$
$11$ $$(T + 1)^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2} - 6T + 4$$
$19$ $$T^{2} - 2T - 44$$
$23$ $$T^{2} - 4T - 16$$
$29$ $$T^{2} - 20$$
$31$ $$T^{2} - 10T + 20$$
$37$ $$T^{2} + 4T - 76$$
$41$ $$T^{2} - 6T + 4$$
$43$ $$T^{2} - 4T - 16$$
$47$ $$T^{2} - 6T - 36$$
$53$ $$(T + 6)^{2}$$
$59$ $$T^{2} - 12T + 16$$
$61$ $$T^{2}$$
$67$ $$T^{2} - 4T - 176$$
$71$ $$T^{2} - 4T - 16$$
$73$ $$T^{2} - 18T + 36$$
$79$ $$T^{2} + 4T - 176$$
$83$ $$T^{2} - 2T - 4$$
$89$ $$T^{2} - 4T - 16$$
$97$ $$T^{2}$$