# Properties

 Label 3234.2.a.y 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: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{3} + q^{4} + q^{6} - q^{8} + q^{9}+O(q^{10})$$ q - q^2 - q^3 + q^4 + q^6 - q^8 + q^9 $$q - q^{2} - q^{3} + q^{4} + q^{6} - q^{8} + q^{9} + q^{11} - q^{12} - \beta q^{13} + q^{16} + 4 q^{17} - q^{18} + ( - \beta + 4) q^{19} - q^{22} + ( - 4 \beta + 2) q^{23} + q^{24} - 5 q^{25} + \beta q^{26} - q^{27} + 4 \beta q^{29} + (5 \beta + 4) q^{31} - q^{32} - q^{33} - 4 q^{34} + q^{36} + ( - 4 \beta - 2) q^{37} + (\beta - 4) q^{38} + \beta q^{39} + ( - 4 \beta + 4) q^{41} - 10 q^{43} + q^{44} + (4 \beta - 2) q^{46} + \beta q^{47} - q^{48} + 5 q^{50} - 4 q^{51} - \beta q^{52} + (4 \beta + 2) q^{53} + q^{54} + (\beta - 4) q^{57} - 4 \beta q^{58} + ( - 2 \beta + 4) q^{59} + \beta q^{61} + ( - 5 \beta - 4) q^{62} + q^{64} + q^{66} + 8 \beta q^{67} + 4 q^{68} + (4 \beta - 2) q^{69} + ( - 4 \beta + 8) q^{71} - q^{72} + 4 q^{73} + (4 \beta + 2) q^{74} + 5 q^{75} + ( - \beta + 4) q^{76} - \beta q^{78} + ( - 4 \beta - 4) q^{79} + q^{81} + (4 \beta - 4) q^{82} + (5 \beta - 8) q^{83} + 10 q^{86} - 4 \beta q^{87} - q^{88} + (\beta + 4) q^{89} + ( - 4 \beta + 2) q^{92} + ( - 5 \beta - 4) q^{93} - \beta q^{94} + q^{96} - 9 \beta q^{97} + q^{99} +O(q^{100})$$ q - q^2 - q^3 + q^4 + q^6 - q^8 + q^9 + q^11 - q^12 - b * q^13 + q^16 + 4 * q^17 - q^18 + (-b + 4) * q^19 - q^22 + (-4*b + 2) * q^23 + q^24 - 5 * q^25 + b * q^26 - q^27 + 4*b * q^29 + (5*b + 4) * q^31 - q^32 - q^33 - 4 * q^34 + q^36 + (-4*b - 2) * q^37 + (b - 4) * q^38 + b * q^39 + (-4*b + 4) * q^41 - 10 * q^43 + q^44 + (4*b - 2) * q^46 + b * q^47 - q^48 + 5 * q^50 - 4 * q^51 - b * q^52 + (4*b + 2) * q^53 + q^54 + (b - 4) * q^57 - 4*b * q^58 + (-2*b + 4) * q^59 + b * q^61 + (-5*b - 4) * q^62 + q^64 + q^66 + 8*b * q^67 + 4 * q^68 + (4*b - 2) * q^69 + (-4*b + 8) * q^71 - q^72 + 4 * q^73 + (4*b + 2) * q^74 + 5 * q^75 + (-b + 4) * q^76 - b * q^78 + (-4*b - 4) * q^79 + q^81 + (4*b - 4) * q^82 + (5*b - 8) * q^83 + 10 * q^86 - 4*b * q^87 - q^88 + (b + 4) * q^89 + (-4*b + 2) * q^92 + (-5*b - 4) * q^93 - b * q^94 + q^96 - 9*b * q^97 + q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 2 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 - 2 * q^8 + 2 * q^9 $$2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 2 q^{8} + 2 q^{9} + 2 q^{11} - 2 q^{12} + 2 q^{16} + 8 q^{17} - 2 q^{18} + 8 q^{19} - 2 q^{22} + 4 q^{23} + 2 q^{24} - 10 q^{25} - 2 q^{27} + 8 q^{31} - 2 q^{32} - 2 q^{33} - 8 q^{34} + 2 q^{36} - 4 q^{37} - 8 q^{38} + 8 q^{41} - 20 q^{43} + 2 q^{44} - 4 q^{46} - 2 q^{48} + 10 q^{50} - 8 q^{51} + 4 q^{53} + 2 q^{54} - 8 q^{57} + 8 q^{59} - 8 q^{62} + 2 q^{64} + 2 q^{66} + 8 q^{68} - 4 q^{69} + 16 q^{71} - 2 q^{72} + 8 q^{73} + 4 q^{74} + 10 q^{75} + 8 q^{76} - 8 q^{79} + 2 q^{81} - 8 q^{82} - 16 q^{83} + 20 q^{86} - 2 q^{88} + 8 q^{89} + 4 q^{92} - 8 q^{93} + 2 q^{96} + 2 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 - 2 * q^8 + 2 * q^9 + 2 * q^11 - 2 * q^12 + 2 * q^16 + 8 * q^17 - 2 * q^18 + 8 * q^19 - 2 * q^22 + 4 * q^23 + 2 * q^24 - 10 * q^25 - 2 * q^27 + 8 * q^31 - 2 * q^32 - 2 * q^33 - 8 * q^34 + 2 * q^36 - 4 * q^37 - 8 * q^38 + 8 * q^41 - 20 * q^43 + 2 * q^44 - 4 * q^46 - 2 * q^48 + 10 * q^50 - 8 * q^51 + 4 * q^53 + 2 * q^54 - 8 * q^57 + 8 * q^59 - 8 * q^62 + 2 * q^64 + 2 * q^66 + 8 * q^68 - 4 * q^69 + 16 * q^71 - 2 * q^72 + 8 * q^73 + 4 * q^74 + 10 * q^75 + 8 * q^76 - 8 * q^79 + 2 * q^81 - 8 * q^82 - 16 * q^83 + 20 * q^86 - 2 * q^88 + 8 * q^89 + 4 * q^92 - 8 * q^93 + 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
 1.41421 −1.41421
−1.00000 −1.00000 1.00000 0 1.00000 0 −1.00000 1.00000 0
1.2 −1.00000 −1.00000 1.00000 0 1.00000 0 −1.00000 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
$$2$$ $$1$$
$$3$$ $$1$$
$$7$$ $$1$$
$$11$$ $$-1$$

## 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.y 2
3.b odd 2 1 9702.2.a.de 2
7.b odd 2 1 3234.2.a.z yes 2
21.c even 2 1 9702.2.a.dl 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3234.2.a.y 2 1.a even 1 1 trivial
3234.2.a.z yes 2 7.b odd 2 1
9702.2.a.de 2 3.b odd 2 1
9702.2.a.dl 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3234))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$(T + 1)^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$(T - 1)^{2}$$
$13$ $$T^{2} - 2$$
$17$ $$(T - 4)^{2}$$
$19$ $$T^{2} - 8T + 14$$
$23$ $$T^{2} - 4T - 28$$
$29$ $$T^{2} - 32$$
$31$ $$T^{2} - 8T - 34$$
$37$ $$T^{2} + 4T - 28$$
$41$ $$T^{2} - 8T - 16$$
$43$ $$(T + 10)^{2}$$
$47$ $$T^{2} - 2$$
$53$ $$T^{2} - 4T - 28$$
$59$ $$T^{2} - 8T + 8$$
$61$ $$T^{2} - 2$$
$67$ $$T^{2} - 128$$
$71$ $$T^{2} - 16T + 32$$
$73$ $$(T - 4)^{2}$$
$79$ $$T^{2} + 8T - 16$$
$83$ $$T^{2} + 16T + 14$$
$89$ $$T^{2} - 8T + 14$$
$97$ $$T^{2} - 162$$