# Properties

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

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## 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 26) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} - 3 q^{5} + q^{7} - q^{8}+O(q^{10})$$ q - q^2 + q^4 - 3 * q^5 + q^7 - q^8 $$q - q^{2} + q^{4} - 3 q^{5} + q^{7} - q^{8} + 3 q^{10} + 6 q^{11} - q^{14} + q^{16} + 3 q^{17} - 2 q^{19} - 3 q^{20} - 6 q^{22} + 4 q^{25} + q^{28} - 6 q^{29} + 4 q^{31} - q^{32} - 3 q^{34} - 3 q^{35} + 7 q^{37} + 2 q^{38} + 3 q^{40} - q^{43} + 6 q^{44} + 3 q^{47} - 6 q^{49} - 4 q^{50} - 18 q^{55} - q^{56} + 6 q^{58} - 6 q^{59} + 8 q^{61} - 4 q^{62} + q^{64} - 14 q^{67} + 3 q^{68} + 3 q^{70} - 3 q^{71} - 2 q^{73} - 7 q^{74} - 2 q^{76} + 6 q^{77} + 8 q^{79} - 3 q^{80} + 12 q^{83} - 9 q^{85} + q^{86} - 6 q^{88} - 6 q^{89} - 3 q^{94} + 6 q^{95} + 10 q^{97} + 6 q^{98}+O(q^{100})$$ q - q^2 + q^4 - 3 * q^5 + q^7 - q^8 + 3 * q^10 + 6 * q^11 - q^14 + q^16 + 3 * q^17 - 2 * q^19 - 3 * q^20 - 6 * q^22 + 4 * q^25 + q^28 - 6 * q^29 + 4 * q^31 - q^32 - 3 * q^34 - 3 * q^35 + 7 * q^37 + 2 * q^38 + 3 * q^40 - q^43 + 6 * q^44 + 3 * q^47 - 6 * q^49 - 4 * q^50 - 18 * q^55 - q^56 + 6 * q^58 - 6 * q^59 + 8 * q^61 - 4 * q^62 + q^64 - 14 * q^67 + 3 * q^68 + 3 * q^70 - 3 * q^71 - 2 * q^73 - 7 * q^74 - 2 * q^76 + 6 * q^77 + 8 * q^79 - 3 * q^80 + 12 * q^83 - 9 * q^85 + q^86 - 6 * q^88 - 6 * q^89 - 3 * q^94 + 6 * q^95 + 10 * q^97 + 6 * 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 −3.00000 0 1.00000 −1.00000 0 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
$$2$$ $$1$$
$$3$$ $$-1$$
$$13$$ $$1$$

## 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.a 1
3.b odd 2 1 338.2.a.f 1
12.b even 2 1 2704.2.a.f 1
13.b even 2 1 234.2.a.e 1
13.d odd 4 2 3042.2.b.a 2
15.d odd 2 1 8450.2.a.c 1
39.d odd 2 1 26.2.a.a 1
39.f even 4 2 338.2.b.c 2
39.h odd 6 2 338.2.c.d 2
39.i odd 6 2 338.2.c.a 2
39.k even 12 4 338.2.e.a 4
52.b odd 2 1 1872.2.a.q 1
65.d even 2 1 5850.2.a.p 1
65.h odd 4 2 5850.2.e.a 2
104.e even 2 1 7488.2.a.g 1
104.h odd 2 1 7488.2.a.h 1
117.n odd 6 2 2106.2.e.ba 2
117.t even 6 2 2106.2.e.b 2
156.h even 2 1 208.2.a.a 1
156.l odd 4 2 2704.2.f.d 2
195.e odd 2 1 650.2.a.j 1
195.s even 4 2 650.2.b.d 2
273.g even 2 1 1274.2.a.d 1
273.w odd 6 2 1274.2.f.p 2
273.ba even 6 2 1274.2.f.r 2
312.b odd 2 1 832.2.a.d 1
312.h even 2 1 832.2.a.i 1
429.e even 2 1 3146.2.a.n 1
624.v even 4 2 3328.2.b.j 2
624.bi odd 4 2 3328.2.b.m 2
663.g odd 2 1 7514.2.a.c 1
741.d even 2 1 9386.2.a.j 1
780.d even 2 1 5200.2.a.x 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.2.a.a 1 39.d odd 2 1
208.2.a.a 1 156.h even 2 1
234.2.a.e 1 13.b even 2 1
338.2.a.f 1 3.b odd 2 1
338.2.b.c 2 39.f even 4 2
338.2.c.a 2 39.i odd 6 2
338.2.c.d 2 39.h odd 6 2
338.2.e.a 4 39.k even 12 4
650.2.a.j 1 195.e odd 2 1
650.2.b.d 2 195.s even 4 2
832.2.a.d 1 312.b odd 2 1
832.2.a.i 1 312.h even 2 1
1274.2.a.d 1 273.g even 2 1
1274.2.f.p 2 273.w odd 6 2
1274.2.f.r 2 273.ba even 6 2
1872.2.a.q 1 52.b odd 2 1
2106.2.e.b 2 117.t even 6 2
2106.2.e.ba 2 117.n odd 6 2
2704.2.a.f 1 12.b even 2 1
2704.2.f.d 2 156.l odd 4 2
3042.2.a.a 1 1.a even 1 1 trivial
3042.2.b.a 2 13.d odd 4 2
3146.2.a.n 1 429.e even 2 1
3328.2.b.j 2 624.v even 4 2
3328.2.b.m 2 624.bi odd 4 2
5200.2.a.x 1 780.d even 2 1
5850.2.a.p 1 65.d even 2 1
5850.2.e.a 2 65.h odd 4 2
7488.2.a.g 1 104.e even 2 1
7488.2.a.h 1 104.h odd 2 1
7514.2.a.c 1 663.g odd 2 1
8450.2.a.c 1 15.d odd 2 1
9386.2.a.j 1 741.d even 2 1

## Hecke kernels

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

 $$T_{5} + 3$$ T5 + 3 $$T_{7} - 1$$ T7 - 1 $$T_{11} - 6$$ T11 - 6 $$T_{17} - 3$$ T17 - 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T$$
$5$ $$T + 3$$
$7$ $$T - 1$$
$11$ $$T - 6$$
$13$ $$T$$
$17$ $$T - 3$$
$19$ $$T + 2$$
$23$ $$T$$
$29$ $$T + 6$$
$31$ $$T - 4$$
$37$ $$T - 7$$
$41$ $$T$$
$43$ $$T + 1$$
$47$ $$T - 3$$
$53$ $$T$$
$59$ $$T + 6$$
$61$ $$T - 8$$
$67$ $$T + 14$$
$71$ $$T + 3$$
$73$ $$T + 2$$
$79$ $$T - 8$$
$83$ $$T - 12$$
$89$ $$T + 6$$
$97$ $$T - 10$$
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