# Properties

 Label 3234.2.a.t 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

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} + q^{4} - 2 q^{5} + q^{6} + q^{8} + q^{9}+O(q^{10})$$ q + q^2 + q^3 + q^4 - 2 * q^5 + q^6 + q^8 + q^9 $$q + q^{2} + q^{3} + q^{4} - 2 q^{5} + q^{6} + q^{8} + q^{9} - 2 q^{10} - q^{11} + q^{12} + 6 q^{13} - 2 q^{15} + q^{16} - 2 q^{17} + q^{18} - 4 q^{19} - 2 q^{20} - q^{22} + 4 q^{23} + q^{24} - q^{25} + 6 q^{26} + q^{27} + 6 q^{29} - 2 q^{30} + q^{32} - q^{33} - 2 q^{34} + q^{36} + 6 q^{37} - 4 q^{38} + 6 q^{39} - 2 q^{40} + 6 q^{41} + 4 q^{43} - q^{44} - 2 q^{45} + 4 q^{46} + 12 q^{47} + q^{48} - q^{50} - 2 q^{51} + 6 q^{52} + 2 q^{53} + q^{54} + 2 q^{55} - 4 q^{57} + 6 q^{58} - 12 q^{59} - 2 q^{60} + 14 q^{61} + q^{64} - 12 q^{65} - q^{66} + 4 q^{67} - 2 q^{68} + 4 q^{69} - 12 q^{71} + q^{72} + 6 q^{73} + 6 q^{74} - q^{75} - 4 q^{76} + 6 q^{78} - 4 q^{79} - 2 q^{80} + q^{81} + 6 q^{82} - 4 q^{83} + 4 q^{85} + 4 q^{86} + 6 q^{87} - q^{88} - 10 q^{89} - 2 q^{90} + 4 q^{92} + 12 q^{94} + 8 q^{95} + q^{96} + 14 q^{97} - q^{99}+O(q^{100})$$ q + q^2 + q^3 + q^4 - 2 * q^5 + q^6 + q^8 + q^9 - 2 * q^10 - q^11 + q^12 + 6 * q^13 - 2 * q^15 + q^16 - 2 * q^17 + q^18 - 4 * q^19 - 2 * q^20 - q^22 + 4 * q^23 + q^24 - q^25 + 6 * q^26 + q^27 + 6 * q^29 - 2 * q^30 + q^32 - q^33 - 2 * q^34 + q^36 + 6 * q^37 - 4 * q^38 + 6 * q^39 - 2 * q^40 + 6 * q^41 + 4 * q^43 - q^44 - 2 * q^45 + 4 * q^46 + 12 * q^47 + q^48 - q^50 - 2 * q^51 + 6 * q^52 + 2 * q^53 + q^54 + 2 * q^55 - 4 * q^57 + 6 * q^58 - 12 * q^59 - 2 * q^60 + 14 * q^61 + q^64 - 12 * q^65 - q^66 + 4 * q^67 - 2 * q^68 + 4 * q^69 - 12 * q^71 + q^72 + 6 * q^73 + 6 * q^74 - q^75 - 4 * q^76 + 6 * q^78 - 4 * q^79 - 2 * q^80 + q^81 + 6 * q^82 - 4 * q^83 + 4 * q^85 + 4 * q^86 + 6 * q^87 - q^88 - 10 * q^89 - 2 * q^90 + 4 * q^92 + 12 * q^94 + 8 * q^95 + q^96 + 14 * 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 −2.00000 1.00000 0 1.00000 1.00000 −2.00000
 $$n$$: 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.t 1
3.b odd 2 1 9702.2.a.x 1
7.b odd 2 1 66.2.a.b 1
21.c even 2 1 198.2.a.a 1
28.d even 2 1 528.2.a.j 1
35.c odd 2 1 1650.2.a.k 1
35.f even 4 2 1650.2.c.e 2
56.e even 2 1 2112.2.a.e 1
56.h odd 2 1 2112.2.a.r 1
63.l odd 6 2 1782.2.e.e 2
63.o even 6 2 1782.2.e.v 2
77.b even 2 1 726.2.a.c 1
77.j odd 10 4 726.2.e.g 4
77.l even 10 4 726.2.e.o 4
84.h odd 2 1 1584.2.a.f 1
105.g even 2 1 4950.2.a.bu 1
105.k odd 4 2 4950.2.c.p 2
168.e odd 2 1 6336.2.a.cj 1
168.i even 2 1 6336.2.a.bw 1
231.h odd 2 1 2178.2.a.g 1
308.g odd 2 1 5808.2.a.bc 1

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

 $$T_{5} + 2$$ T5 + 2 $$T_{13} - 6$$ T13 - 6 $$T_{17} + 2$$ T17 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T - 1$$
$5$ $$T + 2$$
$7$ $$T$$
$11$ $$T + 1$$
$13$ $$T - 6$$
$17$ $$T + 2$$
$19$ $$T + 4$$
$23$ $$T - 4$$
$29$ $$T - 6$$
$31$ $$T$$
$37$ $$T - 6$$
$41$ $$T - 6$$
$43$ $$T - 4$$
$47$ $$T - 12$$
$53$ $$T - 2$$
$59$ $$T + 12$$
$61$ $$T - 14$$
$67$ $$T - 4$$
$71$ $$T + 12$$
$73$ $$T - 6$$
$79$ $$T + 4$$
$83$ $$T + 4$$
$89$ $$T + 10$$
$97$ $$T - 14$$