# Properties

 Label 3234.2.a.h 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

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{3} + q^{4} - 3 q^{5} - q^{6} - q^{8} + q^{9}+O(q^{10})$$ q - q^2 + q^3 + q^4 - 3 * q^5 - q^6 - q^8 + q^9 $$q - q^{2} + q^{3} + q^{4} - 3 q^{5} - q^{6} - q^{8} + q^{9} + 3 q^{10} - q^{11} + q^{12} + 2 q^{13} - 3 q^{15} + q^{16} + 3 q^{17} - q^{18} + 2 q^{19} - 3 q^{20} + q^{22} + 3 q^{23} - q^{24} + 4 q^{25} - 2 q^{26} + q^{27} - 6 q^{29} + 3 q^{30} - 4 q^{31} - q^{32} - q^{33} - 3 q^{34} + q^{36} + 2 q^{37} - 2 q^{38} + 2 q^{39} + 3 q^{40} - 3 q^{41} + 2 q^{43} - q^{44} - 3 q^{45} - 3 q^{46} - 9 q^{47} + q^{48} - 4 q^{50} + 3 q^{51} + 2 q^{52} + 6 q^{53} - q^{54} + 3 q^{55} + 2 q^{57} + 6 q^{58} - 12 q^{59} - 3 q^{60} + 5 q^{61} + 4 q^{62} + q^{64} - 6 q^{65} + q^{66} + 5 q^{67} + 3 q^{68} + 3 q^{69} + 12 q^{71} - q^{72} - 16 q^{73} - 2 q^{74} + 4 q^{75} + 2 q^{76} - 2 q^{78} + 17 q^{79} - 3 q^{80} + q^{81} + 3 q^{82} + 9 q^{83} - 9 q^{85} - 2 q^{86} - 6 q^{87} + q^{88} - 6 q^{89} + 3 q^{90} + 3 q^{92} - 4 q^{93} + 9 q^{94} - 6 q^{95} - q^{96} + 17 q^{97} - q^{99}+O(q^{100})$$ q - q^2 + q^3 + q^4 - 3 * q^5 - q^6 - q^8 + q^9 + 3 * q^10 - q^11 + q^12 + 2 * q^13 - 3 * q^15 + q^16 + 3 * q^17 - q^18 + 2 * q^19 - 3 * q^20 + q^22 + 3 * q^23 - q^24 + 4 * q^25 - 2 * q^26 + q^27 - 6 * q^29 + 3 * q^30 - 4 * q^31 - q^32 - q^33 - 3 * q^34 + q^36 + 2 * q^37 - 2 * q^38 + 2 * q^39 + 3 * q^40 - 3 * q^41 + 2 * q^43 - q^44 - 3 * q^45 - 3 * q^46 - 9 * q^47 + q^48 - 4 * q^50 + 3 * q^51 + 2 * q^52 + 6 * q^53 - q^54 + 3 * q^55 + 2 * q^57 + 6 * q^58 - 12 * q^59 - 3 * q^60 + 5 * q^61 + 4 * q^62 + q^64 - 6 * q^65 + q^66 + 5 * q^67 + 3 * q^68 + 3 * q^69 + 12 * q^71 - q^72 - 16 * q^73 - 2 * q^74 + 4 * q^75 + 2 * q^76 - 2 * q^78 + 17 * q^79 - 3 * q^80 + q^81 + 3 * q^82 + 9 * q^83 - 9 * q^85 - 2 * q^86 - 6 * q^87 + q^88 - 6 * q^89 + 3 * q^90 + 3 * q^92 - 4 * q^93 + 9 * q^94 - 6 * q^95 - q^96 + 17 * 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 −3.00000 −1.00000 0 −1.00000 1.00000 3.00000
 $$n$$: 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.h 1
3.b odd 2 1 9702.2.a.cf 1
7.b odd 2 1 3234.2.a.g 1
7.c even 3 2 462.2.i.c 2
21.c even 2 1 9702.2.a.bd 1
21.h odd 6 2 1386.2.k.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.i.c 2 7.c even 3 2
1386.2.k.c 2 21.h odd 6 2
3234.2.a.g 1 7.b odd 2 1
3234.2.a.h 1 1.a even 1 1 trivial
9702.2.a.bd 1 21.c even 2 1
9702.2.a.cf 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3234))$$:

 $$T_{5} + 3$$ T5 + 3 $$T_{13} - 2$$ T13 - 2 $$T_{17} - 3$$ T17 - 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T - 1$$
$5$ $$T + 3$$
$7$ $$T$$
$11$ $$T + 1$$
$13$ $$T - 2$$
$17$ $$T - 3$$
$19$ $$T - 2$$
$23$ $$T - 3$$
$29$ $$T + 6$$
$31$ $$T + 4$$
$37$ $$T - 2$$
$41$ $$T + 3$$
$43$ $$T - 2$$
$47$ $$T + 9$$
$53$ $$T - 6$$
$59$ $$T + 12$$
$61$ $$T - 5$$
$67$ $$T - 5$$
$71$ $$T - 12$$
$73$ $$T + 16$$
$79$ $$T - 17$$
$83$ $$T - 9$$
$89$ $$T + 6$$
$97$ $$T - 17$$