# Properties

 Label 4212.2.i.d Level $4212$ Weight $2$ Character orbit 4212.i Analytic conductor $33.633$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4212 = 2^{2} \cdot 3^{4} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4212.i (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$33.6329893314$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 52) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 2 \zeta_{6} q^{5} + ( - 2 \zeta_{6} + 2) q^{7} +O(q^{10})$$ q - 2*z * q^5 + (-2*z + 2) * q^7 $$q - 2 \zeta_{6} q^{5} + ( - 2 \zeta_{6} + 2) q^{7} + ( - 2 \zeta_{6} + 2) q^{11} + \zeta_{6} q^{13} + 6 q^{17} - 6 q^{19} - 8 \zeta_{6} q^{23} + ( - \zeta_{6} + 1) q^{25} + (2 \zeta_{6} - 2) q^{29} - 10 \zeta_{6} q^{31} - 4 q^{35} - 6 q^{37} + 6 \zeta_{6} q^{41} + (4 \zeta_{6} - 4) q^{43} + ( - 2 \zeta_{6} + 2) q^{47} + 3 \zeta_{6} q^{49} + 6 q^{53} - 4 q^{55} + 10 \zeta_{6} q^{59} + ( - 2 \zeta_{6} + 2) q^{61} + ( - 2 \zeta_{6} + 2) q^{65} - 10 \zeta_{6} q^{67} + 10 q^{71} + 2 q^{73} - 4 \zeta_{6} q^{77} + ( - 4 \zeta_{6} + 4) q^{79} + ( - 6 \zeta_{6} + 6) q^{83} - 12 \zeta_{6} q^{85} - 6 q^{89} + 2 q^{91} + 12 \zeta_{6} q^{95} + (2 \zeta_{6} - 2) q^{97} +O(q^{100})$$ q - 2*z * q^5 + (-2*z + 2) * q^7 + (-2*z + 2) * q^11 + z * q^13 + 6 * q^17 - 6 * q^19 - 8*z * q^23 + (-z + 1) * q^25 + (2*z - 2) * q^29 - 10*z * q^31 - 4 * q^35 - 6 * q^37 + 6*z * q^41 + (4*z - 4) * q^43 + (-2*z + 2) * q^47 + 3*z * q^49 + 6 * q^53 - 4 * q^55 + 10*z * q^59 + (-2*z + 2) * q^61 + (-2*z + 2) * q^65 - 10*z * q^67 + 10 * q^71 + 2 * q^73 - 4*z * q^77 + (-4*z + 4) * q^79 + (-6*z + 6) * q^83 - 12*z * q^85 - 6 * q^89 + 2 * q^91 + 12*z * q^95 + (2*z - 2) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{5} + 2 q^{7}+O(q^{10})$$ 2 * q - 2 * q^5 + 2 * q^7 $$2 q - 2 q^{5} + 2 q^{7} + 2 q^{11} + q^{13} + 12 q^{17} - 12 q^{19} - 8 q^{23} + q^{25} - 2 q^{29} - 10 q^{31} - 8 q^{35} - 12 q^{37} + 6 q^{41} - 4 q^{43} + 2 q^{47} + 3 q^{49} + 12 q^{53} - 8 q^{55} + 10 q^{59} + 2 q^{61} + 2 q^{65} - 10 q^{67} + 20 q^{71} + 4 q^{73} - 4 q^{77} + 4 q^{79} + 6 q^{83} - 12 q^{85} - 12 q^{89} + 4 q^{91} + 12 q^{95} - 2 q^{97}+O(q^{100})$$ 2 * q - 2 * q^5 + 2 * q^7 + 2 * q^11 + q^13 + 12 * q^17 - 12 * q^19 - 8 * q^23 + q^25 - 2 * q^29 - 10 * q^31 - 8 * q^35 - 12 * q^37 + 6 * q^41 - 4 * q^43 + 2 * q^47 + 3 * q^49 + 12 * q^53 - 8 * q^55 + 10 * q^59 + 2 * q^61 + 2 * q^65 - 10 * q^67 + 20 * q^71 + 4 * q^73 - 4 * q^77 + 4 * q^79 + 6 * q^83 - 12 * q^85 - 12 * q^89 + 4 * q^91 + 12 * q^95 - 2 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4212\mathbb{Z}\right)^\times$$.

 $$n$$ $$2107$$ $$3485$$ $$3889$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1405.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 0 −1.00000 + 1.73205i 0 1.00000 + 1.73205i 0 0 0
2809.1 0 0 0 −1.00000 1.73205i 0 1.00000 1.73205i 0 0 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
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4212.2.i.d 2
3.b odd 2 1 4212.2.i.i 2
9.c even 3 1 52.2.a.a 1
9.c even 3 1 inner 4212.2.i.d 2
9.d odd 6 1 468.2.a.b 1
9.d odd 6 1 4212.2.i.i 2
36.f odd 6 1 208.2.a.c 1
36.h even 6 1 1872.2.a.f 1
45.j even 6 1 1300.2.a.d 1
45.k odd 12 2 1300.2.c.c 2
63.g even 3 1 2548.2.j.e 2
63.h even 3 1 2548.2.j.e 2
63.k odd 6 1 2548.2.j.f 2
63.l odd 6 1 2548.2.a.e 1
63.t odd 6 1 2548.2.j.f 2
72.j odd 6 1 7488.2.a.bn 1
72.l even 6 1 7488.2.a.bw 1
72.n even 6 1 832.2.a.e 1
72.p odd 6 1 832.2.a.f 1
99.h odd 6 1 6292.2.a.g 1
117.f even 3 1 676.2.e.c 2
117.h even 3 1 676.2.e.c 2
117.l even 6 1 676.2.e.b 2
117.n odd 6 1 6084.2.a.m 1
117.r even 6 1 676.2.e.b 2
117.t even 6 1 676.2.a.c 1
117.w odd 12 2 676.2.h.c 4
117.y odd 12 2 676.2.d.c 2
117.z even 12 2 6084.2.b.m 2
117.bb odd 12 2 676.2.h.c 4
144.v odd 12 2 3328.2.b.e 2
144.x even 12 2 3328.2.b.q 2
180.p odd 6 1 5200.2.a.q 1
468.bg odd 6 1 2704.2.a.g 1
468.bs even 12 2 2704.2.f.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
52.2.a.a 1 9.c even 3 1
208.2.a.c 1 36.f odd 6 1
468.2.a.b 1 9.d odd 6 1
676.2.a.c 1 117.t even 6 1
676.2.d.c 2 117.y odd 12 2
676.2.e.b 2 117.l even 6 1
676.2.e.b 2 117.r even 6 1
676.2.e.c 2 117.f even 3 1
676.2.e.c 2 117.h even 3 1
676.2.h.c 4 117.w odd 12 2
676.2.h.c 4 117.bb odd 12 2
832.2.a.e 1 72.n even 6 1
832.2.a.f 1 72.p odd 6 1
1300.2.a.d 1 45.j even 6 1
1300.2.c.c 2 45.k odd 12 2
1872.2.a.f 1 36.h even 6 1
2548.2.a.e 1 63.l odd 6 1
2548.2.j.e 2 63.g even 3 1
2548.2.j.e 2 63.h even 3 1
2548.2.j.f 2 63.k odd 6 1
2548.2.j.f 2 63.t odd 6 1
2704.2.a.g 1 468.bg odd 6 1
2704.2.f.f 2 468.bs even 12 2
3328.2.b.e 2 144.v odd 12 2
3328.2.b.q 2 144.x even 12 2
4212.2.i.d 2 1.a even 1 1 trivial
4212.2.i.d 2 9.c even 3 1 inner
4212.2.i.i 2 3.b odd 2 1
4212.2.i.i 2 9.d odd 6 1
5200.2.a.q 1 180.p odd 6 1
6084.2.a.m 1 117.n odd 6 1
6084.2.b.m 2 117.z even 12 2
6292.2.a.g 1 99.h odd 6 1
7488.2.a.bn 1 72.j odd 6 1
7488.2.a.bw 1 72.l even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(4212, [\chi])$$:

 $$T_{5}^{2} + 2T_{5} + 4$$ T5^2 + 2*T5 + 4 $$T_{7}^{2} - 2T_{7} + 4$$ T7^2 - 2*T7 + 4 $$T_{17} - 6$$ T17 - 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 2T + 4$$
$7$ $$T^{2} - 2T + 4$$
$11$ $$T^{2} - 2T + 4$$
$13$ $$T^{2} - T + 1$$
$17$ $$(T - 6)^{2}$$
$19$ $$(T + 6)^{2}$$
$23$ $$T^{2} + 8T + 64$$
$29$ $$T^{2} + 2T + 4$$
$31$ $$T^{2} + 10T + 100$$
$37$ $$(T + 6)^{2}$$
$41$ $$T^{2} - 6T + 36$$
$43$ $$T^{2} + 4T + 16$$
$47$ $$T^{2} - 2T + 4$$
$53$ $$(T - 6)^{2}$$
$59$ $$T^{2} - 10T + 100$$
$61$ $$T^{2} - 2T + 4$$
$67$ $$T^{2} + 10T + 100$$
$71$ $$(T - 10)^{2}$$
$73$ $$(T - 2)^{2}$$
$79$ $$T^{2} - 4T + 16$$
$83$ $$T^{2} - 6T + 36$$
$89$ $$(T + 6)^{2}$$
$97$ $$T^{2} + 2T + 4$$