# Properties

 Label 3234.2.a.s 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 66) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{3} + q^{4} + 4 q^{5} - q^{6} + q^{8} + q^{9}+O(q^{10})$$ q + q^2 - q^3 + q^4 + 4 * q^5 - q^6 + q^8 + q^9 $$q + q^{2} - q^{3} + q^{4} + 4 q^{5} - q^{6} + q^{8} + q^{9} + 4 q^{10} + q^{11} - q^{12} - 4 q^{13} - 4 q^{15} + q^{16} + 2 q^{17} + q^{18} + 4 q^{20} + q^{22} - 6 q^{23} - q^{24} + 11 q^{25} - 4 q^{26} - q^{27} + 10 q^{29} - 4 q^{30} + 8 q^{31} + q^{32} - q^{33} + 2 q^{34} + q^{36} - 2 q^{37} + 4 q^{39} + 4 q^{40} - 2 q^{41} + 4 q^{43} + q^{44} + 4 q^{45} - 6 q^{46} + 2 q^{47} - q^{48} + 11 q^{50} - 2 q^{51} - 4 q^{52} + 4 q^{53} - q^{54} + 4 q^{55} + 10 q^{58} - 4 q^{60} + 8 q^{61} + 8 q^{62} + q^{64} - 16 q^{65} - q^{66} - 12 q^{67} + 2 q^{68} + 6 q^{69} + 2 q^{71} + q^{72} + 6 q^{73} - 2 q^{74} - 11 q^{75} + 4 q^{78} + 10 q^{79} + 4 q^{80} + q^{81} - 2 q^{82} - 4 q^{83} + 8 q^{85} + 4 q^{86} - 10 q^{87} + q^{88} - 10 q^{89} + 4 q^{90} - 6 q^{92} - 8 q^{93} + 2 q^{94} - q^{96} + 2 q^{97} + q^{99}+O(q^{100})$$ q + q^2 - q^3 + q^4 + 4 * q^5 - q^6 + q^8 + q^9 + 4 * q^10 + q^11 - q^12 - 4 * q^13 - 4 * q^15 + q^16 + 2 * q^17 + q^18 + 4 * q^20 + q^22 - 6 * q^23 - q^24 + 11 * q^25 - 4 * q^26 - q^27 + 10 * q^29 - 4 * q^30 + 8 * q^31 + q^32 - q^33 + 2 * q^34 + q^36 - 2 * q^37 + 4 * q^39 + 4 * q^40 - 2 * q^41 + 4 * q^43 + q^44 + 4 * q^45 - 6 * q^46 + 2 * q^47 - q^48 + 11 * q^50 - 2 * q^51 - 4 * q^52 + 4 * q^53 - q^54 + 4 * q^55 + 10 * q^58 - 4 * q^60 + 8 * q^61 + 8 * q^62 + q^64 - 16 * q^65 - q^66 - 12 * q^67 + 2 * q^68 + 6 * q^69 + 2 * q^71 + q^72 + 6 * q^73 - 2 * q^74 - 11 * q^75 + 4 * q^78 + 10 * q^79 + 4 * q^80 + q^81 - 2 * q^82 - 4 * q^83 + 8 * q^85 + 4 * q^86 - 10 * q^87 + q^88 - 10 * q^89 + 4 * q^90 - 6 * q^92 - 8 * q^93 + 2 * q^94 - q^96 + 2 * 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 4.00000 −1.00000 0 1.00000 1.00000 4.00000
 $$n$$: 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.s 1
3.b odd 2 1 9702.2.a.a 1
7.b odd 2 1 66.2.a.c 1
21.c even 2 1 198.2.a.c 1
28.d even 2 1 528.2.a.a 1
35.c odd 2 1 1650.2.a.c 1
35.f even 4 2 1650.2.c.m 2
56.e even 2 1 2112.2.a.bd 1
56.h odd 2 1 2112.2.a.n 1
63.l odd 6 2 1782.2.e.l 2
63.o even 6 2 1782.2.e.n 2
77.b even 2 1 726.2.a.d 1
77.j odd 10 4 726.2.e.e 4
77.l even 10 4 726.2.e.m 4
84.h odd 2 1 1584.2.a.s 1
105.g even 2 1 4950.2.a.bo 1
105.k odd 4 2 4950.2.c.d 2
168.e odd 2 1 6336.2.a.d 1
168.i even 2 1 6336.2.a.c 1
231.h odd 2 1 2178.2.a.m 1
308.g odd 2 1 5808.2.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
66.2.a.c 1 7.b odd 2 1
198.2.a.c 1 21.c even 2 1
528.2.a.a 1 28.d even 2 1
726.2.a.d 1 77.b even 2 1
726.2.e.e 4 77.j odd 10 4
726.2.e.m 4 77.l even 10 4
1584.2.a.s 1 84.h odd 2 1
1650.2.a.c 1 35.c odd 2 1
1650.2.c.m 2 35.f even 4 2
1782.2.e.l 2 63.l odd 6 2
1782.2.e.n 2 63.o even 6 2
2112.2.a.n 1 56.h odd 2 1
2112.2.a.bd 1 56.e even 2 1
2178.2.a.m 1 231.h odd 2 1
3234.2.a.s 1 1.a even 1 1 trivial
4950.2.a.bo 1 105.g even 2 1
4950.2.c.d 2 105.k odd 4 2
5808.2.a.b 1 308.g odd 2 1
6336.2.a.c 1 168.i even 2 1
6336.2.a.d 1 168.e odd 2 1
9702.2.a.a 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3234))$$:

 $$T_{5} - 4$$ T5 - 4 $$T_{13} + 4$$ T13 + 4 $$T_{17} - 2$$ T17 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T + 1$$
$5$ $$T - 4$$
$7$ $$T$$
$11$ $$T - 1$$
$13$ $$T + 4$$
$17$ $$T - 2$$
$19$ $$T$$
$23$ $$T + 6$$
$29$ $$T - 10$$
$31$ $$T - 8$$
$37$ $$T + 2$$
$41$ $$T + 2$$
$43$ $$T - 4$$
$47$ $$T - 2$$
$53$ $$T - 4$$
$59$ $$T$$
$61$ $$T - 8$$
$67$ $$T + 12$$
$71$ $$T - 2$$
$73$ $$T - 6$$
$79$ $$T - 10$$
$83$ $$T + 4$$
$89$ $$T + 10$$
$97$ $$T - 2$$