Properties

 Label 4056.2.c.e Level $4056$ Weight $2$ Character orbit 4056.c Analytic conductor $32.387$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$4056 = 2^{3} \cdot 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4056.c (of order $$2$$, degree $$1$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$32.3873230598$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 24) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{3} + 2 i q^{5} + q^{9} +O(q^{10})$$ $$q - q^{3} + 2 i q^{5} + q^{9} + 4 i q^{11} -2 i q^{15} -2 q^{17} + 4 i q^{19} + 8 q^{23} + q^{25} - q^{27} + 6 q^{29} -8 i q^{31} -4 i q^{33} + 6 i q^{37} + 6 i q^{41} -4 q^{43} + 2 i q^{45} + 7 q^{49} + 2 q^{51} -2 q^{53} -8 q^{55} -4 i q^{57} + 4 i q^{59} -2 q^{61} + 4 i q^{67} -8 q^{69} -8 i q^{71} + 10 i q^{73} - q^{75} -8 q^{79} + q^{81} + 4 i q^{83} -4 i q^{85} -6 q^{87} -6 i q^{89} + 8 i q^{93} -8 q^{95} -2 i q^{97} + 4 i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + 2 q^{9} + O(q^{10})$$ $$2 q - 2 q^{3} + 2 q^{9} - 4 q^{17} + 16 q^{23} + 2 q^{25} - 2 q^{27} + 12 q^{29} - 8 q^{43} + 14 q^{49} + 4 q^{51} - 4 q^{53} - 16 q^{55} - 4 q^{61} - 16 q^{69} - 2 q^{75} - 16 q^{79} + 2 q^{81} - 12 q^{87} - 16 q^{95} + O(q^{100})$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4056\mathbb{Z}\right)^\times$$.

 $$n$$ $$1015$$ $$2029$$ $$2705$$ $$3889$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
337.1
 − 1.00000i 1.00000i
0 −1.00000 0 2.00000i 0 0 0 1.00000 0
337.2 0 −1.00000 0 2.00000i 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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4056.2.c.e 2
13.b even 2 1 inner 4056.2.c.e 2
13.d odd 4 1 24.2.a.a 1
13.d odd 4 1 4056.2.a.i 1
39.f even 4 1 72.2.a.a 1
52.f even 4 1 48.2.a.a 1
52.f even 4 1 8112.2.a.be 1
65.f even 4 1 600.2.f.e 2
65.g odd 4 1 600.2.a.h 1
65.k even 4 1 600.2.f.e 2
91.i even 4 1 1176.2.a.i 1
91.z odd 12 2 1176.2.q.i 2
91.bb even 12 2 1176.2.q.a 2
104.j odd 4 1 192.2.a.d 1
104.m even 4 1 192.2.a.b 1
117.y odd 12 2 648.2.i.g 2
117.z even 12 2 648.2.i.b 2
143.g even 4 1 2904.2.a.c 1
156.l odd 4 1 144.2.a.b 1
195.j odd 4 1 1800.2.f.c 2
195.n even 4 1 1800.2.a.m 1
195.u odd 4 1 1800.2.f.c 2
208.l even 4 1 768.2.d.d 2
208.m odd 4 1 768.2.d.e 2
208.r odd 4 1 768.2.d.e 2
208.s even 4 1 768.2.d.d 2
221.g odd 4 1 6936.2.a.p 1
247.i even 4 1 8664.2.a.j 1
260.l odd 4 1 1200.2.f.b 2
260.s odd 4 1 1200.2.f.b 2
260.u even 4 1 1200.2.a.d 1
273.o odd 4 1 3528.2.a.d 1
273.cb odd 12 2 3528.2.s.y 2
273.cd even 12 2 3528.2.s.j 2
312.w odd 4 1 576.2.a.b 1
312.y even 4 1 576.2.a.d 1
364.p odd 4 1 2352.2.a.i 1
364.bw odd 12 2 2352.2.q.r 2
364.ce even 12 2 2352.2.q.l 2
429.l odd 4 1 8712.2.a.u 1
468.bs even 12 2 1296.2.i.m 2
468.ch odd 12 2 1296.2.i.e 2
520.t even 4 1 4800.2.a.cc 1
520.x odd 4 1 4800.2.f.bg 2
520.y even 4 1 4800.2.f.d 2
520.bj even 4 1 4800.2.f.d 2
520.bk odd 4 1 4800.2.f.bg 2
520.bo odd 4 1 4800.2.a.q 1
572.k odd 4 1 5808.2.a.s 1
624.s odd 4 1 2304.2.d.k 2
624.u even 4 1 2304.2.d.i 2
624.bm even 4 1 2304.2.d.i 2
624.bo odd 4 1 2304.2.d.k 2
728.x odd 4 1 9408.2.a.cc 1
728.ba even 4 1 9408.2.a.h 1
780.u even 4 1 3600.2.f.r 2
780.bb odd 4 1 3600.2.a.v 1
780.bn even 4 1 3600.2.f.r 2
1092.u even 4 1 7056.2.a.q 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.2.a.a 1 13.d odd 4 1
48.2.a.a 1 52.f even 4 1
72.2.a.a 1 39.f even 4 1
144.2.a.b 1 156.l odd 4 1
192.2.a.b 1 104.m even 4 1
192.2.a.d 1 104.j odd 4 1
576.2.a.b 1 312.w odd 4 1
576.2.a.d 1 312.y even 4 1
600.2.a.h 1 65.g odd 4 1
600.2.f.e 2 65.f even 4 1
600.2.f.e 2 65.k even 4 1
648.2.i.b 2 117.z even 12 2
648.2.i.g 2 117.y odd 12 2
768.2.d.d 2 208.l even 4 1
768.2.d.d 2 208.s even 4 1
768.2.d.e 2 208.m odd 4 1
768.2.d.e 2 208.r odd 4 1
1176.2.a.i 1 91.i even 4 1
1176.2.q.a 2 91.bb even 12 2
1176.2.q.i 2 91.z odd 12 2
1200.2.a.d 1 260.u even 4 1
1200.2.f.b 2 260.l odd 4 1
1200.2.f.b 2 260.s odd 4 1
1296.2.i.e 2 468.ch odd 12 2
1296.2.i.m 2 468.bs even 12 2
1800.2.a.m 1 195.n even 4 1
1800.2.f.c 2 195.j odd 4 1
1800.2.f.c 2 195.u odd 4 1
2304.2.d.i 2 624.u even 4 1
2304.2.d.i 2 624.bm even 4 1
2304.2.d.k 2 624.s odd 4 1
2304.2.d.k 2 624.bo odd 4 1
2352.2.a.i 1 364.p odd 4 1
2352.2.q.l 2 364.ce even 12 2
2352.2.q.r 2 364.bw odd 12 2
2904.2.a.c 1 143.g even 4 1
3528.2.a.d 1 273.o odd 4 1
3528.2.s.j 2 273.cd even 12 2
3528.2.s.y 2 273.cb odd 12 2
3600.2.a.v 1 780.bb odd 4 1
3600.2.f.r 2 780.u even 4 1
3600.2.f.r 2 780.bn even 4 1
4056.2.a.i 1 13.d odd 4 1
4056.2.c.e 2 1.a even 1 1 trivial
4056.2.c.e 2 13.b even 2 1 inner
4800.2.a.q 1 520.bo odd 4 1
4800.2.a.cc 1 520.t even 4 1
4800.2.f.d 2 520.y even 4 1
4800.2.f.d 2 520.bj even 4 1
4800.2.f.bg 2 520.x odd 4 1
4800.2.f.bg 2 520.bk odd 4 1
5808.2.a.s 1 572.k odd 4 1
6936.2.a.p 1 221.g odd 4 1
7056.2.a.q 1 1092.u even 4 1
8112.2.a.be 1 52.f even 4 1
8664.2.a.j 1 247.i even 4 1
8712.2.a.u 1 429.l odd 4 1
9408.2.a.h 1 728.ba even 4 1
9408.2.a.cc 1 728.x odd 4 1

Hecke kernels

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

 $$T_{5}^{2} + 4$$ $$T_{7}$$ $$T_{11}^{2} + 16$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$4 + T^{2}$$
$7$ $$T^{2}$$
$11$ $$16 + T^{2}$$
$13$ $$T^{2}$$
$17$ $$( 2 + T )^{2}$$
$19$ $$16 + T^{2}$$
$23$ $$( -8 + T )^{2}$$
$29$ $$( -6 + T )^{2}$$
$31$ $$64 + T^{2}$$
$37$ $$36 + T^{2}$$
$41$ $$36 + T^{2}$$
$43$ $$( 4 + T )^{2}$$
$47$ $$T^{2}$$
$53$ $$( 2 + T )^{2}$$
$59$ $$16 + T^{2}$$
$61$ $$( 2 + T )^{2}$$
$67$ $$16 + T^{2}$$
$71$ $$64 + T^{2}$$
$73$ $$100 + T^{2}$$
$79$ $$( 8 + T )^{2}$$
$83$ $$16 + T^{2}$$
$89$ $$36 + T^{2}$$
$97$ $$4 + T^{2}$$