# Properties

 Label 3042.2.a.n Level $3042$ Weight $2$ Character orbit 3042.a Self dual yes Analytic conductor $24.290$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3042 = 2 \cdot 3^{2} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3042.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$24.2904922949$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 78) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + 2 q^{5} - 2 q^{7} + q^{8}+O(q^{10})$$ q + q^2 + q^4 + 2 * q^5 - 2 * q^7 + q^8 $$q + q^{2} + q^{4} + 2 q^{5} - 2 q^{7} + q^{8} + 2 q^{10} - 2 q^{14} + q^{16} - 2 q^{17} + 6 q^{19} + 2 q^{20} + 4 q^{23} - q^{25} - 2 q^{28} + 10 q^{29} - 10 q^{31} + q^{32} - 2 q^{34} - 4 q^{35} + 8 q^{37} + 6 q^{38} + 2 q^{40} + 10 q^{41} - 4 q^{43} + 4 q^{46} + 12 q^{47} - 3 q^{49} - q^{50} + 6 q^{53} - 2 q^{56} + 10 q^{58} - 4 q^{59} + 2 q^{61} - 10 q^{62} + q^{64} + 2 q^{67} - 2 q^{68} - 4 q^{70} - 4 q^{73} + 8 q^{74} + 6 q^{76} + 2 q^{80} + 10 q^{82} - 4 q^{83} - 4 q^{85} - 4 q^{86} + 6 q^{89} + 4 q^{92} + 12 q^{94} + 12 q^{95} + 12 q^{97} - 3 q^{98}+O(q^{100})$$ q + q^2 + q^4 + 2 * q^5 - 2 * q^7 + q^8 + 2 * q^10 - 2 * q^14 + q^16 - 2 * q^17 + 6 * q^19 + 2 * q^20 + 4 * q^23 - q^25 - 2 * q^28 + 10 * q^29 - 10 * q^31 + q^32 - 2 * q^34 - 4 * q^35 + 8 * q^37 + 6 * q^38 + 2 * q^40 + 10 * q^41 - 4 * q^43 + 4 * q^46 + 12 * q^47 - 3 * q^49 - q^50 + 6 * q^53 - 2 * q^56 + 10 * q^58 - 4 * q^59 + 2 * q^61 - 10 * q^62 + q^64 + 2 * q^67 - 2 * q^68 - 4 * q^70 - 4 * q^73 + 8 * q^74 + 6 * q^76 + 2 * q^80 + 10 * q^82 - 4 * q^83 - 4 * q^85 - 4 * q^86 + 6 * q^89 + 4 * q^92 + 12 * q^94 + 12 * q^95 + 12 * q^97 - 3 * q^98

## 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 0 1.00000 2.00000 0 −2.00000 1.00000 0 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$13$$ $$-1$$

## 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 3042.2.a.n 1
3.b odd 2 1 1014.2.a.b 1
12.b even 2 1 8112.2.a.g 1
13.b even 2 1 3042.2.a.c 1
13.d odd 4 2 234.2.b.a 2
39.d odd 2 1 1014.2.a.g 1
39.f even 4 2 78.2.b.a 2
39.h odd 6 2 1014.2.e.b 2
39.i odd 6 2 1014.2.e.e 2
39.k even 12 4 1014.2.i.c 4
52.f even 4 2 1872.2.c.b 2
156.h even 2 1 8112.2.a.j 1
156.l odd 4 2 624.2.c.a 2
195.j odd 4 2 1950.2.f.d 2
195.n even 4 2 1950.2.b.c 2
195.u odd 4 2 1950.2.f.g 2
273.o odd 4 2 3822.2.c.d 2
312.w odd 4 2 2496.2.c.m 2
312.y even 4 2 2496.2.c.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.b.a 2 39.f even 4 2
234.2.b.a 2 13.d odd 4 2
624.2.c.a 2 156.l odd 4 2
1014.2.a.b 1 3.b odd 2 1
1014.2.a.g 1 39.d odd 2 1
1014.2.e.b 2 39.h odd 6 2
1014.2.e.e 2 39.i odd 6 2
1014.2.i.c 4 39.k even 12 4
1872.2.c.b 2 52.f even 4 2
1950.2.b.c 2 195.n even 4 2
1950.2.f.d 2 195.j odd 4 2
1950.2.f.g 2 195.u odd 4 2
2496.2.c.f 2 312.y even 4 2
2496.2.c.m 2 312.w odd 4 2
3042.2.a.c 1 13.b even 2 1
3042.2.a.n 1 1.a even 1 1 trivial
3822.2.c.d 2 273.o odd 4 2
8112.2.a.g 1 12.b even 2 1
8112.2.a.j 1 156.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3042))$$:

 $$T_{5} - 2$$ T5 - 2 $$T_{7} + 2$$ T7 + 2 $$T_{11}$$ T11 $$T_{17} + 2$$ T17 + 2

## Hecke characteristic polynomials

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