# Properties

 Label 1872.2.a.q Level $1872$ Weight $2$ Character orbit 1872.a Self dual yes Analytic conductor $14.948$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1872 = 2^{4} \cdot 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1872.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$14.9479952584$$ 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

 $$f(q)$$ $$=$$ $$q + 3 q^{5} + q^{7}+O(q^{10})$$ q + 3 * q^5 + q^7 $$q + 3 q^{5} + q^{7} + 6 q^{11} + q^{13} + 3 q^{17} - 2 q^{19} + 4 q^{25} - 6 q^{29} + 4 q^{31} + 3 q^{35} - 7 q^{37} + q^{43} + 3 q^{47} - 6 q^{49} + 18 q^{55} - 6 q^{59} + 8 q^{61} + 3 q^{65} - 14 q^{67} - 3 q^{71} + 2 q^{73} + 6 q^{77} - 8 q^{79} + 12 q^{83} + 9 q^{85} + 6 q^{89} + q^{91} - 6 q^{95} - 10 q^{97}+O(q^{100})$$ q + 3 * q^5 + q^7 + 6 * q^11 + q^13 + 3 * q^17 - 2 * q^19 + 4 * q^25 - 6 * q^29 + 4 * q^31 + 3 * q^35 - 7 * q^37 + q^43 + 3 * q^47 - 6 * q^49 + 18 * q^55 - 6 * q^59 + 8 * q^61 + 3 * q^65 - 14 * q^67 - 3 * q^71 + 2 * q^73 + 6 * q^77 - 8 * q^79 + 12 * q^83 + 9 * q^85 + 6 * q^89 + q^91 - 6 * q^95 - 10 * q^97

## 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
0 0 0 3.00000 0 1.00000 0 0 0
 $$n$$: 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 1872.2.a.q 1
3.b odd 2 1 208.2.a.a 1
4.b odd 2 1 234.2.a.e 1
8.b even 2 1 7488.2.a.h 1
8.d odd 2 1 7488.2.a.g 1
12.b even 2 1 26.2.a.a 1
15.d odd 2 1 5200.2.a.x 1
20.d odd 2 1 5850.2.a.p 1
20.e even 4 2 5850.2.e.a 2
24.f even 2 1 832.2.a.d 1
24.h odd 2 1 832.2.a.i 1
36.f odd 6 2 2106.2.e.b 2
36.h even 6 2 2106.2.e.ba 2
39.d odd 2 1 2704.2.a.f 1
39.f even 4 2 2704.2.f.d 2
48.i odd 4 2 3328.2.b.j 2
48.k even 4 2 3328.2.b.m 2
52.b odd 2 1 3042.2.a.a 1
52.f even 4 2 3042.2.b.a 2
60.h even 2 1 650.2.a.j 1
60.l odd 4 2 650.2.b.d 2
84.h odd 2 1 1274.2.a.d 1
84.j odd 6 2 1274.2.f.r 2
84.n even 6 2 1274.2.f.p 2
132.d odd 2 1 3146.2.a.n 1
156.h even 2 1 338.2.a.f 1
156.l odd 4 2 338.2.b.c 2
156.p even 6 2 338.2.c.d 2
156.r even 6 2 338.2.c.a 2
156.v odd 12 4 338.2.e.a 4
204.h even 2 1 7514.2.a.c 1
228.b odd 2 1 9386.2.a.j 1
780.d even 2 1 8450.2.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.2.a.a 1 12.b even 2 1
208.2.a.a 1 3.b odd 2 1
234.2.a.e 1 4.b odd 2 1
338.2.a.f 1 156.h even 2 1
338.2.b.c 2 156.l odd 4 2
338.2.c.a 2 156.r even 6 2
338.2.c.d 2 156.p even 6 2
338.2.e.a 4 156.v odd 12 4
650.2.a.j 1 60.h even 2 1
650.2.b.d 2 60.l odd 4 2
832.2.a.d 1 24.f even 2 1
832.2.a.i 1 24.h odd 2 1
1274.2.a.d 1 84.h odd 2 1
1274.2.f.p 2 84.n even 6 2
1274.2.f.r 2 84.j odd 6 2
1872.2.a.q 1 1.a even 1 1 trivial
2106.2.e.b 2 36.f odd 6 2
2106.2.e.ba 2 36.h even 6 2
2704.2.a.f 1 39.d odd 2 1
2704.2.f.d 2 39.f even 4 2
3042.2.a.a 1 52.b odd 2 1
3042.2.b.a 2 52.f even 4 2
3146.2.a.n 1 132.d odd 2 1
3328.2.b.j 2 48.i odd 4 2
3328.2.b.m 2 48.k even 4 2
5200.2.a.x 1 15.d odd 2 1
5850.2.a.p 1 20.d odd 2 1
5850.2.e.a 2 20.e even 4 2
7488.2.a.g 1 8.d odd 2 1
7488.2.a.h 1 8.b even 2 1
7514.2.a.c 1 204.h even 2 1
8450.2.a.c 1 780.d even 2 1
9386.2.a.j 1 228.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1872))$$:

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

## Hecke characteristic polynomials

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