# Properties

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

# 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: 4.0.2312.1 Defining polynomial: $$x^{4} - x^{3} - 2 x + 4$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 84) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + \beta_{2} q^{4} + ( -\beta_{1} - \beta_{3} ) q^{5} + ( -1 + \beta_{1} - \beta_{3} ) q^{7} + ( 2 + \beta_{3} ) q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + \beta_{2} q^{4} + ( -\beta_{1} - \beta_{3} ) q^{5} + ( -1 + \beta_{1} - \beta_{3} ) q^{7} + ( 2 + \beta_{3} ) q^{8} + ( 2 - \beta_{2} - \beta_{3} ) q^{10} + ( -\beta_{1} + \beta_{2} ) q^{11} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{13} + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{14} + ( -2 + 2 \beta_{1} + \beta_{3} ) q^{16} + ( -\beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{17} + ( -2 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{19} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{20} + ( 2 - \beta_{2} + \beta_{3} ) q^{22} + ( \beta_{1} - \beta_{2} ) q^{23} + ( -1 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{25} + ( 4 - 2 \beta_{2} + 2 \beta_{3} ) q^{26} + ( 4 + 2 \beta_{1} - \beta_{2} ) q^{28} + 2 q^{29} + ( -2 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{32} + ( 2 - \beta_{2} + 3 \beta_{3} ) q^{34} + ( -2 - \beta_{1} - 3 \beta_{2} ) q^{35} + ( -4 + 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{37} + ( -4 - 2 \beta_{1} - 2 \beta_{2} ) q^{38} + ( 4 + 2 \beta_{2} - 2 \beta_{3} ) q^{40} + ( 3 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{41} + ( -4 \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{43} + ( -4 + 2 \beta_{1} ) q^{44} + ( -2 + \beta_{2} - \beta_{3} ) q^{46} + ( -4 + 4 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{47} + ( 3 - 4 \beta_{1} - \beta_{2} - \beta_{3} ) q^{49} + ( -4 - \beta_{1} - 2 \beta_{2} ) q^{50} + ( -8 + 4 \beta_{1} ) q^{52} + ( -2 - 4 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{53} + ( -2 + 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{55} + ( -2 + 4 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{56} + 2 \beta_{1} q^{58} + 4 q^{59} + ( 4 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{61} + ( 2 - 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} ) q^{64} + ( -4 + 4 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{65} + ( 2 \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{67} + ( -8 + 2 \beta_{1} + 2 \beta_{3} ) q^{68} + ( -6 - 2 \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{70} + ( -5 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} ) q^{71} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{73} + ( 4 - 4 \beta_{1} + 2 \beta_{2} ) q^{74} + ( -4 - 4 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{76} + ( 2 + 3 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{77} + ( 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{79} + ( 8 + 4 \beta_{1} ) q^{80} + ( -6 + 3 \beta_{2} - \beta_{3} ) q^{82} + ( -4 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{83} + ( 2 + 6 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{85} + ( 8 - 4 \beta_{2} - 2 \beta_{3} ) q^{86} + ( -4 \beta_{1} + 2 \beta_{2} ) q^{88} + ( -3 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{89} + ( 4 + 6 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{91} + ( 4 - 2 \beta_{1} ) q^{92} + ( 8 - 4 \beta_{1} + 4 \beta_{2} ) q^{94} + ( 2 \beta_{1} + 4 \beta_{2} + 6 \beta_{3} ) q^{95} + ( 2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{97} + ( 3 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + q^{2} + q^{4} - 2 q^{7} + 7 q^{8} + O(q^{10})$$ $$4 q + q^{2} + q^{4} - 2 q^{7} + 7 q^{8} + 8 q^{10} + 9 q^{14} - 7 q^{16} - 12 q^{19} + 4 q^{20} + 6 q^{22} - 8 q^{25} + 12 q^{26} + 17 q^{28} + 8 q^{29} - 9 q^{32} + 4 q^{34} - 12 q^{35} - 12 q^{37} - 20 q^{38} + 20 q^{40} - 14 q^{44} - 6 q^{46} - 8 q^{47} + 8 q^{49} - 19 q^{50} - 28 q^{52} - 16 q^{53} - 4 q^{55} - q^{56} + 2 q^{58} + 16 q^{59} + q^{64} - 8 q^{65} - 32 q^{68} - 24 q^{70} + 14 q^{74} - 20 q^{76} + 8 q^{77} + 36 q^{80} - 20 q^{82} - 8 q^{83} + 20 q^{85} + 30 q^{86} - 2 q^{88} + 16 q^{91} + 14 q^{92} + 32 q^{94} + q^{98} + O(q^{100})$$

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

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

## 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
 −0.780776 − 1.17915i −0.780776 + 1.17915i 1.28078 − 0.599676i 1.28078 + 0.599676i
−0.780776 1.17915i 0 −0.780776 + 1.84130i 1.69614i 0 −2.56155 0.662153i 2.78078 0.516994i 0 2.00000 1.32431i
55.2 −0.780776 + 1.17915i 0 −0.780776 1.84130i 1.69614i 0 −2.56155 + 0.662153i 2.78078 + 0.516994i 0 2.00000 + 1.32431i
55.3 1.28078 0.599676i 0 1.28078 1.53610i 3.33513i 0 1.56155 + 2.13578i 0.719224 2.73546i 0 2.00000 + 4.27156i
55.4 1.28078 + 0.599676i 0 1.28078 + 1.53610i 3.33513i 0 1.56155 2.13578i 0.719224 + 2.73546i 0 2.00000 4.27156i
 $$n$$: 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
28.d 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.b.d 4
3.b odd 2 1 84.2.b.b yes 4
4.b odd 2 1 252.2.b.e 4
7.b odd 2 1 252.2.b.e 4
8.b even 2 1 4032.2.b.j 4
8.d odd 2 1 4032.2.b.n 4
12.b even 2 1 84.2.b.a 4
21.c even 2 1 84.2.b.a 4
21.g even 6 2 588.2.o.c 8
21.h odd 6 2 588.2.o.a 8
24.f even 2 1 1344.2.b.f 4
24.h odd 2 1 1344.2.b.e 4
28.d even 2 1 inner 252.2.b.d 4
56.e even 2 1 4032.2.b.j 4
56.h odd 2 1 4032.2.b.n 4
84.h odd 2 1 84.2.b.b yes 4
84.j odd 6 2 588.2.o.a 8
84.n even 6 2 588.2.o.c 8
168.e odd 2 1 1344.2.b.e 4
168.i even 2 1 1344.2.b.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.2.b.a 4 12.b even 2 1
84.2.b.a 4 21.c even 2 1
84.2.b.b yes 4 3.b odd 2 1
84.2.b.b yes 4 84.h odd 2 1
252.2.b.d 4 1.a even 1 1 trivial
252.2.b.d 4 28.d even 2 1 inner
252.2.b.e 4 4.b odd 2 1
252.2.b.e 4 7.b odd 2 1
588.2.o.a 8 21.h odd 6 2
588.2.o.a 8 84.j odd 6 2
588.2.o.c 8 21.g even 6 2
588.2.o.c 8 84.n even 6 2
1344.2.b.e 4 24.h odd 2 1
1344.2.b.e 4 168.e odd 2 1
1344.2.b.f 4 24.f even 2 1
1344.2.b.f 4 168.i even 2 1
4032.2.b.j 4 8.b even 2 1
4032.2.b.j 4 56.e even 2 1
4032.2.b.n 4 8.d odd 2 1
4032.2.b.n 4 56.h odd 2 1

## Hecke kernels

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

 $$T_{5}^{4} + 14 T_{5}^{2} + 32$$ $$T_{11}^{4} + 10 T_{11}^{2} + 8$$ $$T_{19}^{2} + 6 T_{19} - 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 - 2 T - T^{3} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$32 + 14 T^{2} + T^{4}$$
$7$ $$49 + 14 T - 2 T^{2} + 2 T^{3} + T^{4}$$
$11$ $$8 + 10 T^{2} + T^{4}$$
$13$ $$128 + 40 T^{2} + T^{4}$$
$17$ $$512 + 46 T^{2} + T^{4}$$
$19$ $$( -8 + 6 T + T^{2} )^{2}$$
$23$ $$8 + 10 T^{2} + T^{4}$$
$29$ $$( -2 + T )^{4}$$
$31$ $$T^{4}$$
$37$ $$( -8 + 6 T + T^{2} )^{2}$$
$41$ $$128 + 62 T^{2} + T^{4}$$
$43$ $$5408 + 148 T^{2} + T^{4}$$
$47$ $$( -64 + 4 T + T^{2} )^{2}$$
$53$ $$( -52 + 8 T + T^{2} )^{2}$$
$59$ $$( -4 + T )^{4}$$
$61$ $$2048 + 112 T^{2} + T^{4}$$
$67$ $$512 + 124 T^{2} + T^{4}$$
$71$ $$2312 + 170 T^{2} + T^{4}$$
$73$ $$512 + 56 T^{2} + T^{4}$$
$79$ $$128 + 28 T^{2} + T^{4}$$
$83$ $$( -64 + 4 T + T^{2} )^{2}$$
$89$ $$128 + 62 T^{2} + T^{4}$$
$97$ $$8192 + 184 T^{2} + T^{4}$$