# Properties

 Label 252.2.t.a Level 252 Weight 2 Character orbit 252.t 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.t (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.01223013094$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$3$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{5} + ( 2 - 3 \beta_{1} ) q^{7} +O(q^{10})$$ $$q -\beta_{2} q^{5} + ( 2 - 3 \beta_{1} ) q^{7} + ( -\beta_{2} - \beta_{3} ) q^{11} + ( -1 + 2 \beta_{1} ) q^{13} + ( -2 \beta_{2} + 2 \beta_{3} ) q^{17} + ( 6 - 3 \beta_{1} ) q^{19} + ( -1 + \beta_{1} ) q^{25} + ( 4 \beta_{2} - 2 \beta_{3} ) q^{29} + ( -1 - \beta_{1} ) q^{31} + ( \beta_{2} - 3 \beta_{3} ) q^{35} + 5 \beta_{1} q^{37} + 5 \beta_{3} q^{41} -11 q^{43} -\beta_{2} q^{47} + ( -5 - 3 \beta_{1} ) q^{49} + ( 2 \beta_{2} + 2 \beta_{3} ) q^{53} + ( -6 + 12 \beta_{1} ) q^{55} + ( 2 \beta_{2} - 2 \beta_{3} ) q^{59} + ( -4 + 2 \beta_{1} ) q^{61} + ( -\beta_{2} + 2 \beta_{3} ) q^{65} + ( 7 - 7 \beta_{1} ) q^{67} + ( 6 \beta_{2} - 3 \beta_{3} ) q^{71} + ( 9 + 9 \beta_{1} ) q^{73} + ( 4 \beta_{2} - 5 \beta_{3} ) q^{77} -11 \beta_{1} q^{79} + 5 \beta_{3} q^{83} -12 q^{85} -6 \beta_{2} q^{89} + ( 4 + \beta_{1} ) q^{91} + ( -3 \beta_{2} - 3 \beta_{3} ) q^{95} + ( -2 + 4 \beta_{1} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{7} + O(q^{10})$$ $$4q + 2q^{7} + 18q^{19} - 2q^{25} - 6q^{31} + 10q^{37} - 44q^{43} - 26q^{49} - 12q^{61} + 14q^{67} + 54q^{73} - 22q^{79} - 48q^{85} + 18q^{91} + O(q^{100})$$

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

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

## Character values

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

 $$n$$ $$29$$ $$73$$ $$127$$ $$\chi(n)$$ $$-1$$ $$\beta_{1}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1
 1.22474 + 0.707107i −1.22474 − 0.707107i 1.22474 − 0.707107i −1.22474 + 0.707107i
0 0 0 −1.22474 2.12132i 0 0.500000 2.59808i 0 0 0
17.2 0 0 0 1.22474 + 2.12132i 0 0.500000 2.59808i 0 0 0
89.1 0 0 0 −1.22474 + 2.12132i 0 0.500000 + 2.59808i 0 0 0
89.2 0 0 0 1.22474 2.12132i 0 0.500000 + 2.59808i 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.d odd 6 1 inner
21.g even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.2.t.a 4
3.b odd 2 1 inner 252.2.t.a 4
4.b odd 2 1 1008.2.bt.a 4
5.b even 2 1 6300.2.ch.a 4
5.c odd 4 2 6300.2.dd.a 8
7.b odd 2 1 1764.2.t.a 4
7.c even 3 1 1764.2.f.a 4
7.c even 3 1 1764.2.t.a 4
7.d odd 6 1 inner 252.2.t.a 4
7.d odd 6 1 1764.2.f.a 4
9.c even 3 1 2268.2.w.h 4
9.c even 3 1 2268.2.bm.g 4
9.d odd 6 1 2268.2.w.h 4
9.d odd 6 1 2268.2.bm.g 4
12.b even 2 1 1008.2.bt.a 4
15.d odd 2 1 6300.2.ch.a 4
15.e even 4 2 6300.2.dd.a 8
21.c even 2 1 1764.2.t.a 4
21.g even 6 1 inner 252.2.t.a 4
21.g even 6 1 1764.2.f.a 4
21.h odd 6 1 1764.2.f.a 4
21.h odd 6 1 1764.2.t.a 4
28.f even 6 1 1008.2.bt.a 4
28.f even 6 1 7056.2.k.a 4
28.g odd 6 1 7056.2.k.a 4
35.i odd 6 1 6300.2.ch.a 4
35.k even 12 2 6300.2.dd.a 8
63.i even 6 1 2268.2.bm.g 4
63.k odd 6 1 2268.2.w.h 4
63.s even 6 1 2268.2.w.h 4
63.t odd 6 1 2268.2.bm.g 4
84.j odd 6 1 1008.2.bt.a 4
84.j odd 6 1 7056.2.k.a 4
84.n even 6 1 7056.2.k.a 4
105.p even 6 1 6300.2.ch.a 4
105.w odd 12 2 6300.2.dd.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.t.a 4 1.a even 1 1 trivial
252.2.t.a 4 3.b odd 2 1 inner
252.2.t.a 4 7.d odd 6 1 inner
252.2.t.a 4 21.g even 6 1 inner
1008.2.bt.a 4 4.b odd 2 1
1008.2.bt.a 4 12.b even 2 1
1008.2.bt.a 4 28.f even 6 1
1008.2.bt.a 4 84.j odd 6 1
1764.2.f.a 4 7.c even 3 1
1764.2.f.a 4 7.d odd 6 1
1764.2.f.a 4 21.g even 6 1
1764.2.f.a 4 21.h odd 6 1
1764.2.t.a 4 7.b odd 2 1
1764.2.t.a 4 7.c even 3 1
1764.2.t.a 4 21.c even 2 1
1764.2.t.a 4 21.h odd 6 1
2268.2.w.h 4 9.c even 3 1
2268.2.w.h 4 9.d odd 6 1
2268.2.w.h 4 63.k odd 6 1
2268.2.w.h 4 63.s even 6 1
2268.2.bm.g 4 9.c even 3 1
2268.2.bm.g 4 9.d odd 6 1
2268.2.bm.g 4 63.i even 6 1
2268.2.bm.g 4 63.t odd 6 1
6300.2.ch.a 4 5.b even 2 1
6300.2.ch.a 4 15.d odd 2 1
6300.2.ch.a 4 35.i odd 6 1
6300.2.ch.a 4 105.p even 6 1
6300.2.dd.a 8 5.c odd 4 2
6300.2.dd.a 8 15.e even 4 2
6300.2.dd.a 8 35.k even 12 2
6300.2.dd.a 8 105.w odd 12 2
7056.2.k.a 4 28.f even 6 1
7056.2.k.a 4 28.g odd 6 1
7056.2.k.a 4 84.j odd 6 1
7056.2.k.a 4 84.n even 6 1

## Hecke kernels

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

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 
$3$ 
$5$ $$1 - 4 T^{2} - 9 T^{4} - 100 T^{6} + 625 T^{8}$$
$7$ $$( 1 - T + 7 T^{2} )^{2}$$
$11$ $$1 + 4 T^{2} - 105 T^{4} + 484 T^{6} + 14641 T^{8}$$
$13$ $$( 1 - 7 T + 13 T^{2} )^{2}( 1 + 7 T + 13 T^{2} )^{2}$$
$17$ $$1 - 10 T^{2} - 189 T^{4} - 2890 T^{6} + 83521 T^{8}$$
$19$ $$( 1 - 8 T + 19 T^{2} )^{2}( 1 - T + 19 T^{2} )^{2}$$
$23$ $$( 1 + 23 T^{2} + 529 T^{4} )^{2}$$
$29$ $$( 1 + 14 T^{2} + 841 T^{4} )^{2}$$
$31$ $$( 1 - 4 T + 31 T^{2} )^{2}( 1 + 7 T + 31 T^{2} )^{2}$$
$37$ $$( 1 - 5 T - 12 T^{2} - 185 T^{3} + 1369 T^{4} )^{2}$$
$41$ $$( 1 - 68 T^{2} + 1681 T^{4} )^{2}$$
$43$ $$( 1 + 11 T + 43 T^{2} )^{4}$$
$47$ $$1 - 88 T^{2} + 5535 T^{4} - 194392 T^{6} + 4879681 T^{8}$$
$53$ $$1 + 34 T^{2} - 1653 T^{4} + 95506 T^{6} + 7890481 T^{8}$$
$59$ $$1 - 94 T^{2} + 5355 T^{4} - 327214 T^{6} + 12117361 T^{8}$$
$61$ $$( 1 + 6 T + 73 T^{2} + 366 T^{3} + 3721 T^{4} )^{2}$$
$67$ $$( 1 - 7 T - 18 T^{2} - 469 T^{3} + 4489 T^{4} )^{2}$$
$71$ $$( 1 + 20 T^{2} + 5041 T^{4} )^{2}$$
$73$ $$( 1 - 17 T + 73 T^{2} )^{2}( 1 - 10 T + 73 T^{2} )^{2}$$
$79$ $$( 1 + 11 T + 42 T^{2} + 869 T^{3} + 6241 T^{4} )^{2}$$
$83$ $$( 1 + 16 T^{2} + 6889 T^{4} )^{2}$$
$89$ $$1 + 38 T^{2} - 6477 T^{4} + 300998 T^{6} + 62742241 T^{8}$$
$97$ $$( 1 - 182 T^{2} + 9409 T^{4} )^{2}$$