# Properties

 Label 4056.2.a.i Level $4056$ Weight $2$ Character orbit 4056.a Self dual yes Analytic conductor $32.387$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4056 = 2^{3} \cdot 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4056.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$32.3873230598$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 24) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{3} + 2 q^{5} + q^{9}+O(q^{10})$$ q - q^3 + 2 * q^5 + q^9 $$q - q^{3} + 2 q^{5} + q^{9} - 4 q^{11} - 2 q^{15} + 2 q^{17} + 4 q^{19} - 8 q^{23} - q^{25} - q^{27} + 6 q^{29} - 8 q^{31} + 4 q^{33} - 6 q^{37} + 6 q^{41} + 4 q^{43} + 2 q^{45} - 7 q^{49} - 2 q^{51} - 2 q^{53} - 8 q^{55} - 4 q^{57} - 4 q^{59} - 2 q^{61} + 4 q^{67} + 8 q^{69} - 8 q^{71} - 10 q^{73} + q^{75} - 8 q^{79} + q^{81} + 4 q^{83} + 4 q^{85} - 6 q^{87} + 6 q^{89} + 8 q^{93} + 8 q^{95} - 2 q^{97} - 4 q^{99}+O(q^{100})$$ q - q^3 + 2 * q^5 + q^9 - 4 * q^11 - 2 * q^15 + 2 * q^17 + 4 * q^19 - 8 * q^23 - q^25 - q^27 + 6 * q^29 - 8 * q^31 + 4 * q^33 - 6 * q^37 + 6 * q^41 + 4 * q^43 + 2 * q^45 - 7 * q^49 - 2 * q^51 - 2 * q^53 - 8 * q^55 - 4 * q^57 - 4 * q^59 - 2 * q^61 + 4 * q^67 + 8 * q^69 - 8 * q^71 - 10 * q^73 + q^75 - 8 * q^79 + q^81 + 4 * q^83 + 4 * q^85 - 6 * q^87 + 6 * q^89 + 8 * q^93 + 8 * q^95 - 2 * q^97 - 4 * 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
0 −1.00000 0 2.00000 0 0 0 1.00000 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 4056.2.a.i 1
4.b odd 2 1 8112.2.a.be 1
13.b even 2 1 24.2.a.a 1
13.d odd 4 2 4056.2.c.e 2
39.d odd 2 1 72.2.a.a 1
52.b odd 2 1 48.2.a.a 1
65.d even 2 1 600.2.a.h 1
65.h odd 4 2 600.2.f.e 2
91.b odd 2 1 1176.2.a.i 1
91.r even 6 2 1176.2.q.i 2
91.s odd 6 2 1176.2.q.a 2
104.e even 2 1 192.2.a.d 1
104.h odd 2 1 192.2.a.b 1
117.n odd 6 2 648.2.i.b 2
117.t even 6 2 648.2.i.g 2
143.d odd 2 1 2904.2.a.c 1
156.h even 2 1 144.2.a.b 1
195.e odd 2 1 1800.2.a.m 1
195.s even 4 2 1800.2.f.c 2
208.o odd 4 2 768.2.d.d 2
208.p even 4 2 768.2.d.e 2
221.b even 2 1 6936.2.a.p 1
247.d odd 2 1 8664.2.a.j 1
260.g odd 2 1 1200.2.a.d 1
260.p even 4 2 1200.2.f.b 2
273.g even 2 1 3528.2.a.d 1
273.w odd 6 2 3528.2.s.j 2
273.ba even 6 2 3528.2.s.y 2
312.b odd 2 1 576.2.a.d 1
312.h even 2 1 576.2.a.b 1
364.h even 2 1 2352.2.a.i 1
364.x even 6 2 2352.2.q.r 2
364.bl odd 6 2 2352.2.q.l 2
429.e even 2 1 8712.2.a.u 1
468.x even 6 2 1296.2.i.e 2
468.bg odd 6 2 1296.2.i.m 2
520.b odd 2 1 4800.2.a.cc 1
520.p even 2 1 4800.2.a.q 1
520.bc even 4 2 4800.2.f.bg 2
520.bg odd 4 2 4800.2.f.d 2
572.b even 2 1 5808.2.a.s 1
624.v even 4 2 2304.2.d.k 2
624.bi odd 4 2 2304.2.d.i 2
728.b even 2 1 9408.2.a.cc 1
728.l odd 2 1 9408.2.a.h 1
780.d even 2 1 3600.2.a.v 1
780.w odd 4 2 3600.2.f.r 2
1092.d odd 2 1 7056.2.a.q 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.2.a.a 1 13.b even 2 1
48.2.a.a 1 52.b odd 2 1
72.2.a.a 1 39.d odd 2 1
144.2.a.b 1 156.h even 2 1
192.2.a.b 1 104.h odd 2 1
192.2.a.d 1 104.e even 2 1
576.2.a.b 1 312.h even 2 1
576.2.a.d 1 312.b odd 2 1
600.2.a.h 1 65.d even 2 1
600.2.f.e 2 65.h odd 4 2
648.2.i.b 2 117.n odd 6 2
648.2.i.g 2 117.t even 6 2
768.2.d.d 2 208.o odd 4 2
768.2.d.e 2 208.p even 4 2
1176.2.a.i 1 91.b odd 2 1
1176.2.q.a 2 91.s odd 6 2
1176.2.q.i 2 91.r even 6 2
1200.2.a.d 1 260.g odd 2 1
1200.2.f.b 2 260.p even 4 2
1296.2.i.e 2 468.x even 6 2
1296.2.i.m 2 468.bg odd 6 2
1800.2.a.m 1 195.e odd 2 1
1800.2.f.c 2 195.s even 4 2
2304.2.d.i 2 624.bi odd 4 2
2304.2.d.k 2 624.v even 4 2
2352.2.a.i 1 364.h even 2 1
2352.2.q.l 2 364.bl odd 6 2
2352.2.q.r 2 364.x even 6 2
2904.2.a.c 1 143.d odd 2 1
3528.2.a.d 1 273.g even 2 1
3528.2.s.j 2 273.w odd 6 2
3528.2.s.y 2 273.ba even 6 2
3600.2.a.v 1 780.d even 2 1
3600.2.f.r 2 780.w odd 4 2
4056.2.a.i 1 1.a even 1 1 trivial
4056.2.c.e 2 13.d odd 4 2
4800.2.a.q 1 520.p even 2 1
4800.2.a.cc 1 520.b odd 2 1
4800.2.f.d 2 520.bg odd 4 2
4800.2.f.bg 2 520.bc even 4 2
5808.2.a.s 1 572.b even 2 1
6936.2.a.p 1 221.b even 2 1
7056.2.a.q 1 1092.d odd 2 1
8112.2.a.be 1 4.b odd 2 1
8664.2.a.j 1 247.d odd 2 1
8712.2.a.u 1 429.e even 2 1
9408.2.a.h 1 728.l odd 2 1
9408.2.a.cc 1 728.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4056))$$:

 $$T_{5} - 2$$ T5 - 2 $$T_{7}$$ T7

## Hecke characteristic polynomials

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