# Properties

 Label 3234.2.a.c 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

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{3} + q^{4} + q^{6} - q^{8} + q^{9} + O(q^{10})$$ $$q - q^{2} - q^{3} + q^{4} + q^{6} - q^{8} + q^{9} - q^{11} - q^{12} + 4q^{13} + q^{16} - q^{17} - q^{18} - 3q^{19} + q^{22} - q^{23} + q^{24} - 5q^{25} - 4q^{26} - q^{27} - q^{29} + 6q^{31} - q^{32} + q^{33} + q^{34} + q^{36} - 3q^{37} + 3q^{38} - 4q^{39} - 6q^{41} + q^{43} - q^{44} + q^{46} - q^{47} - q^{48} + 5q^{50} + q^{51} + 4q^{52} + q^{54} + 3q^{57} + q^{58} + 7q^{59} + 6q^{61} - 6q^{62} + q^{64} - q^{66} - 4q^{67} - q^{68} + q^{69} - 15q^{71} - q^{72} + 12q^{73} + 3q^{74} + 5q^{75} - 3q^{76} + 4q^{78} + q^{81} + 6q^{82} - 16q^{83} - q^{86} + q^{87} + q^{88} - 8q^{89} - q^{92} - 6q^{93} + q^{94} + q^{96} + 7q^{97} - q^{99} + O(q^{100})$$

## 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.c 1
3.b odd 2 1 9702.2.a.bv 1
7.b odd 2 1 3234.2.a.j 1
7.c even 3 2 462.2.i.d 2
21.c even 2 1 9702.2.a.bq 1
21.h odd 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.c even 3 2
1386.2.k.f 2 21.h odd 6 2
3234.2.a.c 1 1.a even 1 1 trivial
3234.2.a.j 1 7.b odd 2 1
9702.2.a.bq 1 21.c even 2 1
9702.2.a.bv 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3234))$$:

 $$T_{5}$$ $$T_{13} - 4$$ $$T_{17} + 1$$

## Hecke characteristic polynomials

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