# Properties

 Label 3234.2.a.j 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 $$q - q^{2} + q^{3} + q^{4} - q^{6} - q^{8} + q^{9} - q^{11} + q^{12} - 4 q^{13} + q^{16} + q^{17} - q^{18} + 3 q^{19} + q^{22} - q^{23} - q^{24} - 5 q^{25} + 4 q^{26} + q^{27} - q^{29} - 6 q^{31} - q^{32} - q^{33} - q^{34} + q^{36} - 3 q^{37} - 3 q^{38} - 4 q^{39} + 6 q^{41} + q^{43} - q^{44} + q^{46} + q^{47} + q^{48} + 5 q^{50} + q^{51} - 4 q^{52} - q^{54} + 3 q^{57} + q^{58} - 7 q^{59} - 6 q^{61} + 6 q^{62} + q^{64} + q^{66} - 4 q^{67} + q^{68} - q^{69} - 15 q^{71} - q^{72} - 12 q^{73} + 3 q^{74} - 5 q^{75} + 3 q^{76} + 4 q^{78} + q^{81} - 6 q^{82} + 16 q^{83} - q^{86} - q^{87} + q^{88} + 8 q^{89} - q^{92} - 6 q^{93} - q^{94} - q^{96} - 7 q^{97} - q^{99}+O(q^{100})$$ q - q^2 + q^3 + q^4 - q^6 - q^8 + q^9 - q^11 + q^12 - 4 * q^13 + q^16 + q^17 - q^18 + 3 * q^19 + q^22 - q^23 - q^24 - 5 * q^25 + 4 * q^26 + q^27 - q^29 - 6 * q^31 - q^32 - q^33 - q^34 + q^36 - 3 * q^37 - 3 * q^38 - 4 * q^39 + 6 * q^41 + q^43 - q^44 + q^46 + q^47 + q^48 + 5 * q^50 + q^51 - 4 * q^52 - q^54 + 3 * q^57 + q^58 - 7 * q^59 - 6 * q^61 + 6 * q^62 + q^64 + q^66 - 4 * q^67 + q^68 - q^69 - 15 * q^71 - q^72 - 12 * q^73 + 3 * q^74 - 5 * q^75 + 3 * q^76 + 4 * q^78 + q^81 - 6 * q^82 + 16 * q^83 - q^86 - q^87 + q^88 + 8 * q^89 - q^92 - 6 * q^93 - q^94 - q^96 - 7 * 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.j 1
3.b odd 2 1 9702.2.a.bq 1
7.b odd 2 1 3234.2.a.c 1
7.d odd 6 2 462.2.i.d 2
21.c even 2 1 9702.2.a.bv 1
21.g even 6 2 1386.2.k.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.i.d 2 7.d odd 6 2
1386.2.k.f 2 21.g even 6 2
3234.2.a.c 1 7.b odd 2 1
3234.2.a.j 1 1.a even 1 1 trivial
9702.2.a.bq 1 3.b odd 2 1
9702.2.a.bv 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 $$T_{17} - 1$$ T17 - 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T - 1$$
$5$ $$T$$
$7$ $$T$$
$11$ $$T + 1$$
$13$ $$T + 4$$
$17$ $$T - 1$$
$19$ $$T - 3$$
$23$ $$T + 1$$
$29$ $$T + 1$$
$31$ $$T + 6$$
$37$ $$T + 3$$
$41$ $$T - 6$$
$43$ $$T - 1$$
$47$ $$T - 1$$
$53$ $$T$$
$59$ $$T + 7$$
$61$ $$T + 6$$
$67$ $$T + 4$$
$71$ $$T + 15$$
$73$ $$T + 12$$
$79$ $$T$$
$83$ $$T - 16$$
$89$ $$T - 8$$
$97$ $$T + 7$$