# Properties

 Label 3234.2.a.k Level $3234$ Weight $2$ Character orbit 3234.a Self dual yes Analytic conductor $25.824$ Analytic rank $1$ Dimension $1$ 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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 462) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{3} + q^{4} - q^{6} - q^{8} + q^{9}+O(q^{10})$$ q - q^2 + q^3 + q^4 - q^6 - q^8 + q^9 $$q - q^{2} + q^{3} + q^{4} - q^{6} - q^{8} + q^{9} - q^{11} + q^{12} + 2 q^{13} + q^{16} + 4 q^{17} - q^{18} - 6 q^{19} + q^{22} - 4 q^{23} - q^{24} - 5 q^{25} - 2 q^{26} + q^{27} - 10 q^{29} - 6 q^{31} - q^{32} - q^{33} - 4 q^{34} + q^{36} - 6 q^{37} + 6 q^{38} + 2 q^{39} + 12 q^{41} - 8 q^{43} - q^{44} + 4 q^{46} - 2 q^{47} + q^{48} + 5 q^{50} + 4 q^{51} + 2 q^{52} + 6 q^{53} - q^{54} - 6 q^{57} + 10 q^{58} + 8 q^{59} - 6 q^{61} + 6 q^{62} + q^{64} + q^{66} - 4 q^{67} + 4 q^{68} - 4 q^{69} - q^{72} + 12 q^{73} + 6 q^{74} - 5 q^{75} - 6 q^{76} - 2 q^{78} + q^{81} - 12 q^{82} - 14 q^{83} + 8 q^{86} - 10 q^{87} + q^{88} - 10 q^{89} - 4 q^{92} - 6 q^{93} + 2 q^{94} - q^{96} - 10 q^{97} - q^{99}+O(q^{100})$$ q - q^2 + q^3 + q^4 - q^6 - q^8 + q^9 - q^11 + q^12 + 2 * q^13 + q^16 + 4 * q^17 - q^18 - 6 * q^19 + q^22 - 4 * q^23 - q^24 - 5 * q^25 - 2 * q^26 + q^27 - 10 * q^29 - 6 * q^31 - q^32 - q^33 - 4 * q^34 + q^36 - 6 * q^37 + 6 * q^38 + 2 * q^39 + 12 * q^41 - 8 * q^43 - q^44 + 4 * q^46 - 2 * q^47 + q^48 + 5 * q^50 + 4 * q^51 + 2 * q^52 + 6 * q^53 - q^54 - 6 * q^57 + 10 * q^58 + 8 * q^59 - 6 * q^61 + 6 * q^62 + q^64 + q^66 - 4 * q^67 + 4 * q^68 - 4 * q^69 - q^72 + 12 * q^73 + 6 * q^74 - 5 * q^75 - 6 * q^76 - 2 * q^78 + q^81 - 12 * q^82 - 14 * q^83 + 8 * q^86 - 10 * q^87 + q^88 - 10 * q^89 - 4 * q^92 - 6 * q^93 + 2 * q^94 - q^96 - 10 * q^97 - 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
−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.k 1
3.b odd 2 1 9702.2.a.bt 1
7.b odd 2 1 462.2.a.b 1
21.c even 2 1 1386.2.a.i 1
28.d even 2 1 3696.2.a.y 1
77.b even 2 1 5082.2.a.s 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.a.b 1 7.b odd 2 1
1386.2.a.i 1 21.c even 2 1
3234.2.a.k 1 1.a even 1 1 trivial
3696.2.a.y 1 28.d even 2 1
5082.2.a.s 1 77.b even 2 1
9702.2.a.bt 1 3.b odd 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 $$T_{13} - 2$$ T13 - 2 $$T_{17} - 4$$ T17 - 4

## Hecke characteristic polynomials

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