# Properties

 Label 3234.2.a.l 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 $$q - q^{2} + q^{3} + q^{4} - q^{6} - q^{8} + q^{9} + q^{11} + q^{12} - 4 q^{13} + q^{16} + 3 q^{17} - q^{18} - q^{19} - q^{22} - 3 q^{23} - q^{24} - 5 q^{25} + 4 q^{26} + q^{27} - 9 q^{29} + 2 q^{31} - q^{32} + q^{33} - 3 q^{34} + q^{36} - 7 q^{37} + q^{38} - 4 q^{39} - 6 q^{41} + 11 q^{43} + q^{44} + 3 q^{46} - 3 q^{47} + q^{48} + 5 q^{50} + 3 q^{51} - 4 q^{52} - q^{54} - q^{57} + 9 q^{58} + 9 q^{59} - 10 q^{61} - 2 q^{62} + q^{64} - q^{66} - 4 q^{67} + 3 q^{68} - 3 q^{69} + 3 q^{71} - q^{72} - 4 q^{73} + 7 q^{74} - 5 q^{75} - q^{76} + 4 q^{78} - 16 q^{79} + q^{81} + 6 q^{82} - 11 q^{86} - 9 q^{87} - q^{88} - 3 q^{92} + 2 q^{93} + 3 q^{94} - q^{96} - q^{97} + q^{99}+O(q^{100})$$ q - q^2 + q^3 + q^4 - q^6 - q^8 + q^9 + q^11 + q^12 - 4 * q^13 + q^16 + 3 * q^17 - q^18 - q^19 - q^22 - 3 * q^23 - q^24 - 5 * q^25 + 4 * q^26 + q^27 - 9 * q^29 + 2 * q^31 - q^32 + q^33 - 3 * q^34 + q^36 - 7 * q^37 + q^38 - 4 * q^39 - 6 * q^41 + 11 * q^43 + q^44 + 3 * q^46 - 3 * q^47 + q^48 + 5 * q^50 + 3 * q^51 - 4 * q^52 - q^54 - q^57 + 9 * q^58 + 9 * q^59 - 10 * q^61 - 2 * q^62 + q^64 - q^66 - 4 * q^67 + 3 * q^68 - 3 * q^69 + 3 * q^71 - q^72 - 4 * q^73 + 7 * q^74 - 5 * q^75 - q^76 + 4 * q^78 - 16 * q^79 + q^81 + 6 * q^82 - 11 * q^86 - 9 * q^87 - q^88 - 3 * q^92 + 2 * q^93 + 3 * q^94 - q^96 - 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.l 1
3.b odd 2 1 9702.2.a.bl 1
7.b odd 2 1 3234.2.a.f 1
7.c even 3 2 462.2.i.b 2
21.c even 2 1 9702.2.a.bn 1
21.h odd 6 2 1386.2.k.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.i.b 2 7.c even 3 2
1386.2.k.d 2 21.h odd 6 2
3234.2.a.f 1 7.b odd 2 1
3234.2.a.l 1 1.a even 1 1 trivial
9702.2.a.bl 1 3.b odd 2 1
9702.2.a.bn 1 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 $$T_{13} + 4$$ T13 + 4 $$T_{17} - 3$$ T17 - 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T - 1$$
$5$ $$T$$
$7$ $$T$$
$11$ $$T - 1$$
$13$ $$T + 4$$
$17$ $$T - 3$$
$19$ $$T + 1$$
$23$ $$T + 3$$
$29$ $$T + 9$$
$31$ $$T - 2$$
$37$ $$T + 7$$
$41$ $$T + 6$$
$43$ $$T - 11$$
$47$ $$T + 3$$
$53$ $$T$$
$59$ $$T - 9$$
$61$ $$T + 10$$
$67$ $$T + 4$$
$71$ $$T - 3$$
$73$ $$T + 4$$
$79$ $$T + 16$$
$83$ $$T$$
$89$ $$T$$
$97$ $$T + 1$$