# Properties

 Label 252.2.f.a Level 252 Weight 2 Character orbit 252.f Analytic conductor 2.012 Analytic rank 0 Dimension 4 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$252 = 2^{2} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 252.f (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.01223013094$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{3})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}\cdot 3$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{5} + ( -1 - \beta_{1} ) q^{7} +O(q^{10})$$ $$q + \beta_{3} q^{5} + ( -1 - \beta_{1} ) q^{7} -\beta_{2} q^{11} -2 \beta_{1} q^{13} + \beta_{3} q^{17} + 2 \beta_{1} q^{19} + \beta_{2} q^{23} + 7 q^{25} -\beta_{2} q^{29} + ( 2 \beta_{2} - \beta_{3} ) q^{35} -8 q^{37} -\beta_{3} q^{41} -2 q^{43} -2 \beta_{3} q^{47} + ( -5 + 2 \beta_{1} ) q^{49} -3 \beta_{2} q^{53} + 6 \beta_{1} q^{55} -4 \beta_{3} q^{59} -4 \beta_{1} q^{61} + 4 \beta_{2} q^{65} + 8 q^{67} + \beta_{2} q^{71} + 2 \beta_{1} q^{73} + ( \beta_{2} + 3 \beta_{3} ) q^{77} -4 q^{79} + 2 \beta_{3} q^{83} + 12 q^{85} -3 \beta_{3} q^{89} + ( -12 + 2 \beta_{1} ) q^{91} -4 \beta_{2} q^{95} -2 \beta_{1} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{7} + O(q^{10})$$ $$4q - 4q^{7} + 28q^{25} - 32q^{37} - 8q^{43} - 20q^{49} + 32q^{67} - 16q^{79} + 48q^{85} - 48q^{91} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 4 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{3} + 5 \nu$$ $$\beta_{2}$$ $$=$$ $$-3 \nu^{3} - 9 \nu$$ $$\beta_{3}$$ $$=$$ $$2 \nu^{2} + 4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + 3 \beta_{1}$$$$)/6$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} - 4$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-5 \beta_{2} - 9 \beta_{1}$$$$)/6$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/252\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$73$$ $$127$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
125.1
 1.93185i − 1.93185i 0.517638i − 0.517638i
0 0 0 −3.46410 0 −1.00000 2.44949i 0 0 0
125.2 0 0 0 −3.46410 0 −1.00000 + 2.44949i 0 0 0
125.3 0 0 0 3.46410 0 −1.00000 2.44949i 0 0 0
125.4 0 0 0 3.46410 0 −1.00000 + 2.44949i 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
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.2.f.a 4
3.b odd 2 1 inner 252.2.f.a 4
4.b odd 2 1 1008.2.k.b 4
5.b even 2 1 6300.2.d.c 4
5.c odd 4 2 6300.2.f.b 8
7.b odd 2 1 inner 252.2.f.a 4
7.c even 3 2 1764.2.t.b 8
7.d odd 6 2 1764.2.t.b 8
8.b even 2 1 4032.2.k.a 4
8.d odd 2 1 4032.2.k.d 4
9.c even 3 2 2268.2.x.i 8
9.d odd 6 2 2268.2.x.i 8
12.b even 2 1 1008.2.k.b 4
15.d odd 2 1 6300.2.d.c 4
15.e even 4 2 6300.2.f.b 8
21.c even 2 1 inner 252.2.f.a 4
21.g even 6 2 1764.2.t.b 8
21.h odd 6 2 1764.2.t.b 8
24.f even 2 1 4032.2.k.d 4
24.h odd 2 1 4032.2.k.a 4
28.d even 2 1 1008.2.k.b 4
35.c odd 2 1 6300.2.d.c 4
35.f even 4 2 6300.2.f.b 8
56.e even 2 1 4032.2.k.d 4
56.h odd 2 1 4032.2.k.a 4
63.l odd 6 2 2268.2.x.i 8
63.o even 6 2 2268.2.x.i 8
84.h odd 2 1 1008.2.k.b 4
105.g even 2 1 6300.2.d.c 4
105.k odd 4 2 6300.2.f.b 8
168.e odd 2 1 4032.2.k.d 4
168.i even 2 1 4032.2.k.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.f.a 4 1.a even 1 1 trivial
252.2.f.a 4 3.b odd 2 1 inner
252.2.f.a 4 7.b odd 2 1 inner
252.2.f.a 4 21.c even 2 1 inner
1008.2.k.b 4 4.b odd 2 1
1008.2.k.b 4 12.b even 2 1
1008.2.k.b 4 28.d even 2 1
1008.2.k.b 4 84.h odd 2 1
1764.2.t.b 8 7.c even 3 2
1764.2.t.b 8 7.d odd 6 2
1764.2.t.b 8 21.g even 6 2
1764.2.t.b 8 21.h odd 6 2
2268.2.x.i 8 9.c even 3 2
2268.2.x.i 8 9.d odd 6 2
2268.2.x.i 8 63.l odd 6 2
2268.2.x.i 8 63.o even 6 2
4032.2.k.a 4 8.b even 2 1
4032.2.k.a 4 24.h odd 2 1
4032.2.k.a 4 56.h odd 2 1
4032.2.k.a 4 168.i even 2 1
4032.2.k.d 4 8.d odd 2 1
4032.2.k.d 4 24.f even 2 1
4032.2.k.d 4 56.e even 2 1
4032.2.k.d 4 168.e odd 2 1
6300.2.d.c 4 5.b even 2 1
6300.2.d.c 4 15.d odd 2 1
6300.2.d.c 4 35.c odd 2 1
6300.2.d.c 4 105.g even 2 1
6300.2.f.b 8 5.c odd 4 2
6300.2.f.b 8 15.e even 4 2
6300.2.f.b 8 35.f even 4 2
6300.2.f.b 8 105.k odd 4 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(252, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 
$3$ 
$5$ $$( 1 - 2 T^{2} + 25 T^{4} )^{2}$$
$7$ $$( 1 + 2 T + 7 T^{2} )^{2}$$
$11$ $$( 1 - 4 T^{2} + 121 T^{4} )^{2}$$
$13$ $$( 1 - 2 T^{2} + 169 T^{4} )^{2}$$
$17$ $$( 1 + 22 T^{2} + 289 T^{4} )^{2}$$
$19$ $$( 1 - 14 T^{2} + 361 T^{4} )^{2}$$
$23$ $$( 1 - 28 T^{2} + 529 T^{4} )^{2}$$
$29$ $$( 1 - 40 T^{2} + 841 T^{4} )^{2}$$
$31$ $$( 1 - 31 T^{2} )^{4}$$
$37$ $$( 1 + 8 T + 37 T^{2} )^{4}$$
$41$ $$( 1 + 70 T^{2} + 1681 T^{4} )^{2}$$
$43$ $$( 1 + 2 T + 43 T^{2} )^{4}$$
$47$ $$( 1 + 46 T^{2} + 2209 T^{4} )^{2}$$
$53$ $$( 1 + 56 T^{2} + 2809 T^{4} )^{2}$$
$59$ $$( 1 - 74 T^{2} + 3481 T^{4} )^{2}$$
$61$ $$( 1 - 26 T^{2} + 3721 T^{4} )^{2}$$
$67$ $$( 1 - 8 T + 67 T^{2} )^{4}$$
$71$ $$( 1 - 124 T^{2} + 5041 T^{4} )^{2}$$
$73$ $$( 1 - 122 T^{2} + 5329 T^{4} )^{2}$$
$79$ $$( 1 + 4 T + 79 T^{2} )^{4}$$
$83$ $$( 1 + 118 T^{2} + 6889 T^{4} )^{2}$$
$89$ $$( 1 + 70 T^{2} + 7921 T^{4} )^{2}$$
$97$ $$( 1 - 170 T^{2} + 9409 T^{4} )^{2}$$