# Properties

 Label 1872.4.a.k Level $1872$ Weight $4$ Character orbit 1872.a Self dual yes Analytic conductor $110.452$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: yes Analytic conductor: $$110.451575531$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 13) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 7 q^{5} + 13 q^{7} + O(q^{10})$$ $$q + 7 q^{5} + 13 q^{7} - 26 q^{11} + 13 q^{13} - 77 q^{17} + 126 q^{19} - 96 q^{23} - 76 q^{25} + 82 q^{29} - 196 q^{31} + 91 q^{35} - 131 q^{37} - 336 q^{41} + 201 q^{43} - 105 q^{47} - 174 q^{49} + 432 q^{53} - 182 q^{55} - 294 q^{59} - 56 q^{61} + 91 q^{65} - 478 q^{67} + 9 q^{71} + 98 q^{73} - 338 q^{77} - 1304 q^{79} - 308 q^{83} - 539 q^{85} + 1190 q^{89} + 169 q^{91} + 882 q^{95} + 70 q^{97} + O(q^{100})$$

## 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 7.00000 0 13.0000 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 1872.4.a.k 1
3.b odd 2 1 208.4.a.g 1
4.b odd 2 1 117.4.a.b 1
12.b even 2 1 13.4.a.a 1
24.f even 2 1 832.4.a.r 1
24.h odd 2 1 832.4.a.a 1
52.b odd 2 1 1521.4.a.a 1
60.h even 2 1 325.4.a.d 1
60.l odd 4 2 325.4.b.b 2
84.h odd 2 1 637.4.a.a 1
132.d odd 2 1 1573.4.a.a 1
156.h even 2 1 169.4.a.e 1
156.l odd 4 2 169.4.b.a 2
156.p even 6 2 169.4.c.e 2
156.r even 6 2 169.4.c.a 2
156.v odd 12 4 169.4.e.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.a.a 1 12.b even 2 1
117.4.a.b 1 4.b odd 2 1
169.4.a.e 1 156.h even 2 1
169.4.b.a 2 156.l odd 4 2
169.4.c.a 2 156.r even 6 2
169.4.c.e 2 156.p even 6 2
169.4.e.e 4 156.v odd 12 4
208.4.a.g 1 3.b odd 2 1
325.4.a.d 1 60.h even 2 1
325.4.b.b 2 60.l odd 4 2
637.4.a.a 1 84.h odd 2 1
832.4.a.a 1 24.h odd 2 1
832.4.a.r 1 24.f even 2 1
1521.4.a.a 1 52.b odd 2 1
1573.4.a.a 1 132.d odd 2 1
1872.4.a.k 1 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1872))$$:

 $$T_{5} - 7$$ $$T_{7} - 13$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$-7 + T$$
$7$ $$-13 + T$$
$11$ $$26 + T$$
$13$ $$-13 + T$$
$17$ $$77 + T$$
$19$ $$-126 + T$$
$23$ $$96 + T$$
$29$ $$-82 + T$$
$31$ $$196 + T$$
$37$ $$131 + T$$
$41$ $$336 + T$$
$43$ $$-201 + T$$
$47$ $$105 + T$$
$53$ $$-432 + T$$
$59$ $$294 + T$$
$61$ $$56 + T$$
$67$ $$478 + T$$
$71$ $$-9 + T$$
$73$ $$-98 + T$$
$79$ $$1304 + T$$
$83$ $$308 + T$$
$89$ $$-1190 + T$$
$97$ $$-70 + T$$