# Properties

 Label 4212.2.i.b 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: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 156) 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.

 $$f(q)$$ $$=$$ $$q -4 \zeta_{6} q^{5} + ( 2 - 2 \zeta_{6} ) q^{7} +O(q^{10})$$ $$q -4 \zeta_{6} q^{5} + ( 2 - 2 \zeta_{6} ) q^{7} + ( -4 + 4 \zeta_{6} ) q^{11} -\zeta_{6} q^{13} -2 q^{17} -2 q^{19} + ( -11 + 11 \zeta_{6} ) q^{25} + ( -6 + 6 \zeta_{6} ) q^{29} + 10 \zeta_{6} q^{31} -8 q^{35} + 10 q^{37} + 8 \zeta_{6} q^{41} + ( -4 + 4 \zeta_{6} ) q^{43} + ( -4 + 4 \zeta_{6} ) q^{47} + 3 \zeta_{6} q^{49} + 10 q^{53} + 16 q^{55} -8 \zeta_{6} q^{59} + ( 14 - 14 \zeta_{6} ) q^{61} + ( -4 + 4 \zeta_{6} ) q^{65} -2 \zeta_{6} q^{67} -16 q^{71} -10 q^{73} + 8 \zeta_{6} q^{77} + ( 16 - 16 \zeta_{6} ) q^{79} + 8 \zeta_{6} q^{85} + 4 q^{89} -2 q^{91} + 8 \zeta_{6} q^{95} + ( 2 - 2 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{5} + 2q^{7} + O(q^{10})$$ $$2q - 4q^{5} + 2q^{7} - 4q^{11} - q^{13} - 4q^{17} - 4q^{19} - 11q^{25} - 6q^{29} + 10q^{31} - 16q^{35} + 20q^{37} + 8q^{41} - 4q^{43} - 4q^{47} + 3q^{49} + 20q^{53} + 32q^{55} - 8q^{59} + 14q^{61} - 4q^{65} - 2q^{67} - 32q^{71} - 20q^{73} + 8q^{77} + 16q^{79} + 8q^{85} + 8q^{89} - 4q^{91} + 8q^{95} + 2q^{97} + O(q^{100})$$

## 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 −2.00000 + 3.46410i 0 1.00000 + 1.73205i 0 0 0
2809.1 0 0 0 −2.00000 3.46410i 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.b 2
3.b odd 2 1 4212.2.i.l 2
9.c even 3 1 468.2.a.d 1
9.c even 3 1 inner 4212.2.i.b 2
9.d odd 6 1 156.2.a.a 1
9.d odd 6 1 4212.2.i.l 2
36.f odd 6 1 1872.2.a.s 1
36.h even 6 1 624.2.a.e 1
45.h odd 6 1 3900.2.a.m 1
45.l even 12 2 3900.2.h.b 2
63.o even 6 1 7644.2.a.k 1
72.j odd 6 1 2496.2.a.bc 1
72.l even 6 1 2496.2.a.o 1
72.n even 6 1 7488.2.a.c 1
72.p odd 6 1 7488.2.a.d 1
117.k odd 6 1 2028.2.i.e 2
117.m odd 6 1 2028.2.i.g 2
117.n odd 6 1 2028.2.a.c 1
117.t even 6 1 6084.2.a.b 1
117.u odd 6 1 2028.2.i.e 2
117.v odd 6 1 2028.2.i.g 2
117.x even 12 2 2028.2.q.h 4
117.y odd 12 2 6084.2.b.j 2
117.z even 12 2 2028.2.b.a 2
117.bc even 12 2 2028.2.q.h 4
468.x even 6 1 8112.2.a.bi 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.2.a.a 1 9.d odd 6 1
468.2.a.d 1 9.c even 3 1
624.2.a.e 1 36.h even 6 1
1872.2.a.s 1 36.f odd 6 1
2028.2.a.c 1 117.n odd 6 1
2028.2.b.a 2 117.z even 12 2
2028.2.i.e 2 117.k odd 6 1
2028.2.i.e 2 117.u odd 6 1
2028.2.i.g 2 117.m odd 6 1
2028.2.i.g 2 117.v odd 6 1
2028.2.q.h 4 117.x even 12 2
2028.2.q.h 4 117.bc even 12 2
2496.2.a.o 1 72.l even 6 1
2496.2.a.bc 1 72.j odd 6 1
3900.2.a.m 1 45.h odd 6 1
3900.2.h.b 2 45.l even 12 2
4212.2.i.b 2 1.a even 1 1 trivial
4212.2.i.b 2 9.c even 3 1 inner
4212.2.i.l 2 3.b odd 2 1
4212.2.i.l 2 9.d odd 6 1
6084.2.a.b 1 117.t even 6 1
6084.2.b.j 2 117.y odd 12 2
7488.2.a.c 1 72.n even 6 1
7488.2.a.d 1 72.p odd 6 1
7644.2.a.k 1 63.o even 6 1
8112.2.a.bi 1 468.x even 6 1

## Hecke kernels

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

 $$T_{5}^{2} + 4 T_{5} + 16$$ $$T_{7}^{2} - 2 T_{7} + 4$$ $$T_{17} + 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$16 + 4 T + T^{2}$$
$7$ $$4 - 2 T + T^{2}$$
$11$ $$16 + 4 T + T^{2}$$
$13$ $$1 + T + T^{2}$$
$17$ $$( 2 + T )^{2}$$
$19$ $$( 2 + T )^{2}$$
$23$ $$T^{2}$$
$29$ $$36 + 6 T + T^{2}$$
$31$ $$100 - 10 T + T^{2}$$
$37$ $$( -10 + T )^{2}$$
$41$ $$64 - 8 T + T^{2}$$
$43$ $$16 + 4 T + T^{2}$$
$47$ $$16 + 4 T + T^{2}$$
$53$ $$( -10 + T )^{2}$$
$59$ $$64 + 8 T + T^{2}$$
$61$ $$196 - 14 T + T^{2}$$
$67$ $$4 + 2 T + T^{2}$$
$71$ $$( 16 + T )^{2}$$
$73$ $$( 10 + T )^{2}$$
$79$ $$256 - 16 T + T^{2}$$
$83$ $$T^{2}$$
$89$ $$( -4 + T )^{2}$$
$97$ $$4 - 2 T + T^{2}$$