# Properties

 Label 3234.2.a.n Level $3234$ Weight $2$ Character orbit 3234.a Self dual yes Analytic conductor $25.824$ Analytic rank $0$ 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: $$0$$ 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

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{3} + q^{4} + 2 q^{5} - q^{6} - q^{8} + q^{9}+O(q^{10})$$ q - q^2 + q^3 + q^4 + 2 * q^5 - q^6 - q^8 + q^9 $$q - q^{2} + q^{3} + q^{4} + 2 q^{5} - q^{6} - q^{8} + q^{9} - 2 q^{10} + q^{11} + q^{12} - 2 q^{13} + 2 q^{15} + q^{16} + 6 q^{17} - q^{18} + 4 q^{19} + 2 q^{20} - q^{22} - 4 q^{23} - q^{24} - q^{25} + 2 q^{26} + q^{27} + 2 q^{29} - 2 q^{30} + 4 q^{31} - q^{32} + q^{33} - 6 q^{34} + q^{36} - 2 q^{37} - 4 q^{38} - 2 q^{39} - 2 q^{40} + 6 q^{41} + q^{44} + 2 q^{45} + 4 q^{46} + 8 q^{47} + q^{48} + q^{50} + 6 q^{51} - 2 q^{52} - 14 q^{53} - q^{54} + 2 q^{55} + 4 q^{57} - 2 q^{58} - 12 q^{59} + 2 q^{60} + 14 q^{61} - 4 q^{62} + q^{64} - 4 q^{65} - q^{66} + 4 q^{67} + 6 q^{68} - 4 q^{69} + 12 q^{71} - q^{72} - 6 q^{73} + 2 q^{74} - q^{75} + 4 q^{76} + 2 q^{78} + 2 q^{80} + q^{81} - 6 q^{82} + 12 q^{85} + 2 q^{87} - q^{88} + 6 q^{89} - 2 q^{90} - 4 q^{92} + 4 q^{93} - 8 q^{94} + 8 q^{95} - q^{96} + 14 q^{97} + q^{99}+O(q^{100})$$ q - q^2 + q^3 + q^4 + 2 * q^5 - q^6 - q^8 + q^9 - 2 * q^10 + q^11 + q^12 - 2 * q^13 + 2 * q^15 + q^16 + 6 * q^17 - q^18 + 4 * q^19 + 2 * q^20 - q^22 - 4 * q^23 - q^24 - q^25 + 2 * q^26 + q^27 + 2 * q^29 - 2 * q^30 + 4 * q^31 - q^32 + q^33 - 6 * q^34 + q^36 - 2 * q^37 - 4 * q^38 - 2 * q^39 - 2 * q^40 + 6 * q^41 + q^44 + 2 * q^45 + 4 * q^46 + 8 * q^47 + q^48 + q^50 + 6 * q^51 - 2 * q^52 - 14 * q^53 - q^54 + 2 * q^55 + 4 * q^57 - 2 * q^58 - 12 * q^59 + 2 * q^60 + 14 * q^61 - 4 * q^62 + q^64 - 4 * q^65 - q^66 + 4 * q^67 + 6 * q^68 - 4 * q^69 + 12 * q^71 - q^72 - 6 * q^73 + 2 * q^74 - q^75 + 4 * q^76 + 2 * q^78 + 2 * q^80 + q^81 - 6 * q^82 + 12 * q^85 + 2 * q^87 - q^88 + 6 * q^89 - 2 * q^90 - 4 * q^92 + 4 * q^93 - 8 * q^94 + 8 * q^95 - q^96 + 14 * 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 2.00000 −1.00000 0 −1.00000 1.00000 −2.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

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

## 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.n 1
3.b odd 2 1 9702.2.a.bf 1
7.b odd 2 1 462.2.a.a 1
21.c even 2 1 1386.2.a.k 1
28.d even 2 1 3696.2.a.s 1
77.b even 2 1 5082.2.a.q 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.a.a 1 7.b odd 2 1
1386.2.a.k 1 21.c even 2 1
3234.2.a.n 1 1.a even 1 1 trivial
3696.2.a.s 1 28.d even 2 1
5082.2.a.q 1 77.b even 2 1
9702.2.a.bf 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3234))$$:

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

## Hecke characteristic polynomials

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