# Properties

 Label 252.2.b.c Level $252$ Weight $2$ Character orbit 252.b Analytic conductor $2.012$ Analytic rank $0$ Dimension $4$ CM discriminant -7 Inner twists $8$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$2.01223013094$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{7})$$ Defining polynomial: $$x^{4} - 3 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + ( -1 + 2 \beta_{2} ) q^{7} + ( \beta_{1} + \beta_{3} ) q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + ( -1 + 2 \beta_{2} ) q^{7} + ( \beta_{1} + \beta_{3} ) q^{8} -2 \beta_{3} q^{11} + ( -\beta_{1} + 2 \beta_{3} ) q^{14} + ( -1 + 3 \beta_{2} ) q^{16} + ( 4 - 4 \beta_{2} ) q^{22} -4 \beta_{3} q^{23} + 5 q^{25} + ( -5 + 3 \beta_{2} ) q^{28} + ( -8 \beta_{1} + 2 \beta_{3} ) q^{29} + ( -\beta_{1} + 3 \beta_{3} ) q^{32} -6 q^{37} + ( 2 - 4 \beta_{2} ) q^{43} + ( 4 \beta_{1} - 4 \beta_{3} ) q^{44} + ( 8 - 8 \beta_{2} ) q^{46} -7 q^{49} + 5 \beta_{1} q^{50} + ( 8 \beta_{1} - 2 \beta_{3} ) q^{53} + ( -5 \beta_{1} + 3 \beta_{3} ) q^{56} + ( -12 - 4 \beta_{2} ) q^{58} + ( -7 + 5 \beta_{2} ) q^{64} + ( 6 - 12 \beta_{2} ) q^{67} + 8 \beta_{3} q^{71} -6 \beta_{1} q^{74} + ( 8 \beta_{1} - 2 \beta_{3} ) q^{77} + ( -6 + 12 \beta_{2} ) q^{79} + ( 2 \beta_{1} - 4 \beta_{3} ) q^{86} + ( 12 - 4 \beta_{2} ) q^{88} + ( 8 \beta_{1} - 8 \beta_{3} ) q^{92} -7 \beta_{1} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 6 q^{4} + O(q^{10})$$ $$4 q + 6 q^{4} + 2 q^{16} + 8 q^{22} + 20 q^{25} - 14 q^{28} - 24 q^{37} + 16 q^{46} - 28 q^{49} - 56 q^{58} - 18 q^{64} + 40 q^{88} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 3 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 1$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 1$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + \beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
55.1
 −1.32288 − 0.500000i −1.32288 + 0.500000i 1.32288 − 0.500000i 1.32288 + 0.500000i
−1.32288 0.500000i 0 1.50000 + 1.32288i 0 0 2.64575i −1.32288 2.50000i 0 0
55.2 −1.32288 + 0.500000i 0 1.50000 1.32288i 0 0 2.64575i −1.32288 + 2.50000i 0 0
55.3 1.32288 0.500000i 0 1.50000 1.32288i 0 0 2.64575i 1.32288 2.50000i 0 0
55.4 1.32288 + 0.500000i 0 1.50000 + 1.32288i 0 0 2.64575i 1.32288 + 2.50000i 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
7.b odd 2 1 CM by $$\Q(\sqrt{-7})$$
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner
21.c even 2 1 inner
28.d even 2 1 inner
84.h odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.2.b.c 4
3.b odd 2 1 inner 252.2.b.c 4
4.b odd 2 1 inner 252.2.b.c 4
7.b odd 2 1 CM 252.2.b.c 4
8.b even 2 1 4032.2.b.m 4
8.d odd 2 1 4032.2.b.m 4
12.b even 2 1 inner 252.2.b.c 4
21.c even 2 1 inner 252.2.b.c 4
24.f even 2 1 4032.2.b.m 4
24.h odd 2 1 4032.2.b.m 4
28.d even 2 1 inner 252.2.b.c 4
56.e even 2 1 4032.2.b.m 4
56.h odd 2 1 4032.2.b.m 4
84.h odd 2 1 inner 252.2.b.c 4
168.e odd 2 1 4032.2.b.m 4
168.i even 2 1 4032.2.b.m 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.b.c 4 1.a even 1 1 trivial
252.2.b.c 4 3.b odd 2 1 inner
252.2.b.c 4 4.b odd 2 1 inner
252.2.b.c 4 7.b odd 2 1 CM
252.2.b.c 4 12.b even 2 1 inner
252.2.b.c 4 21.c even 2 1 inner
252.2.b.c 4 28.d even 2 1 inner
252.2.b.c 4 84.h odd 2 1 inner
4032.2.b.m 4 8.b even 2 1
4032.2.b.m 4 8.d odd 2 1
4032.2.b.m 4 24.f even 2 1
4032.2.b.m 4 24.h odd 2 1
4032.2.b.m 4 56.e even 2 1
4032.2.b.m 4 56.h odd 2 1
4032.2.b.m 4 168.e odd 2 1
4032.2.b.m 4 168.i even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(252, [\chi])$$:

 $$T_{5}$$ $$T_{11}^{2} + 16$$ $$T_{19}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 - 3 T^{2} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$( 7 + T^{2} )^{2}$$
$11$ $$( 16 + T^{2} )^{2}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$( 64 + T^{2} )^{2}$$
$29$ $$( -112 + T^{2} )^{2}$$
$31$ $$T^{4}$$
$37$ $$( 6 + T )^{4}$$
$41$ $$T^{4}$$
$43$ $$( 28 + T^{2} )^{2}$$
$47$ $$T^{4}$$
$53$ $$( -112 + T^{2} )^{2}$$
$59$ $$T^{4}$$
$61$ $$T^{4}$$
$67$ $$( 252 + T^{2} )^{2}$$
$71$ $$( 256 + T^{2} )^{2}$$
$73$ $$T^{4}$$
$79$ $$( 252 + T^{2} )^{2}$$
$83$ $$T^{4}$$
$89$ $$T^{4}$$
$97$ $$T^{4}$$