# Properties

 Label 252.2.i.a Level 252 Weight 2 Character orbit 252.i Analytic conductor 2.012 Analytic rank 1 Dimension 2 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.i (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.01223013094$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + ( -1 - \zeta_{6} ) q^{3} -2 \zeta_{6} q^{5} + ( -3 + \zeta_{6} ) q^{7} + 3 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( -1 - \zeta_{6} ) q^{3} -2 \zeta_{6} q^{5} + ( -3 + \zeta_{6} ) q^{7} + 3 \zeta_{6} q^{9} + ( -4 + 4 \zeta_{6} ) q^{11} + ( -3 + 3 \zeta_{6} ) q^{13} + ( -2 + 4 \zeta_{6} ) q^{15} -7 \zeta_{6} q^{17} + ( -5 + 5 \zeta_{6} ) q^{19} + ( 4 + \zeta_{6} ) q^{21} -4 \zeta_{6} q^{23} + ( 1 - \zeta_{6} ) q^{25} + ( 3 - 6 \zeta_{6} ) q^{27} + \zeta_{6} q^{29} -3 q^{31} + ( 8 - 4 \zeta_{6} ) q^{33} + ( 2 + 4 \zeta_{6} ) q^{35} + ( -11 + 11 \zeta_{6} ) q^{37} + ( 6 - 3 \zeta_{6} ) q^{39} + ( 9 - 9 \zeta_{6} ) q^{41} -5 \zeta_{6} q^{43} + ( 6 - 6 \zeta_{6} ) q^{45} + 3 q^{47} + ( 8 - 5 \zeta_{6} ) q^{49} + ( -7 + 14 \zeta_{6} ) q^{51} -3 \zeta_{6} q^{53} + 8 q^{55} + ( 10 - 5 \zeta_{6} ) q^{57} -7 q^{59} + 3 q^{61} + ( -3 - 6 \zeta_{6} ) q^{63} + 6 q^{65} + 13 q^{67} + ( -4 + 8 \zeta_{6} ) q^{69} -8 q^{71} -7 \zeta_{6} q^{73} + ( -2 + \zeta_{6} ) q^{75} + ( 8 - 12 \zeta_{6} ) q^{77} -9 q^{79} + ( -9 + 9 \zeta_{6} ) q^{81} -\zeta_{6} q^{83} + ( -14 + 14 \zeta_{6} ) q^{85} + ( 1 - 2 \zeta_{6} ) q^{87} + ( -15 + 15 \zeta_{6} ) q^{89} + ( 6 - 9 \zeta_{6} ) q^{91} + ( 3 + 3 \zeta_{6} ) q^{93} + 10 q^{95} + 17 \zeta_{6} q^{97} -12 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 3q^{3} - 2q^{5} - 5q^{7} + 3q^{9} + O(q^{10})$$ $$2q - 3q^{3} - 2q^{5} - 5q^{7} + 3q^{9} - 4q^{11} - 3q^{13} - 7q^{17} - 5q^{19} + 9q^{21} - 4q^{23} + q^{25} + q^{29} - 6q^{31} + 12q^{33} + 8q^{35} - 11q^{37} + 9q^{39} + 9q^{41} - 5q^{43} + 6q^{45} + 6q^{47} + 11q^{49} - 3q^{53} + 16q^{55} + 15q^{57} - 14q^{59} + 6q^{61} - 12q^{63} + 12q^{65} + 26q^{67} - 16q^{71} - 7q^{73} - 3q^{75} + 4q^{77} - 18q^{79} - 9q^{81} - q^{83} - 14q^{85} - 15q^{89} + 3q^{91} + 9q^{93} + 20q^{95} + 17q^{97} - 24q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$29$$ $$73$$ $$127$$ $$\chi(n)$$ $$-1 + \zeta_{6}$$ $$-1 + \zeta_{6}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
25.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 −1.50000 + 0.866025i 0 −1.00000 + 1.73205i 0 −2.50000 0.866025i 0 1.50000 2.59808i 0
121.1 0 −1.50000 0.866025i 0 −1.00000 1.73205i 0 −2.50000 + 0.866025i 0 1.50000 + 2.59808i 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
63.h even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.2.i.a 2
3.b odd 2 1 756.2.i.a 2
4.b odd 2 1 1008.2.q.f 2
7.b odd 2 1 1764.2.i.b 2
7.c even 3 1 252.2.l.a yes 2
7.c even 3 1 1764.2.j.c 2
7.d odd 6 1 1764.2.j.a 2
7.d odd 6 1 1764.2.l.b 2
9.c even 3 1 252.2.l.a yes 2
9.c even 3 1 2268.2.k.a 2
9.d odd 6 1 756.2.l.a 2
9.d odd 6 1 2268.2.k.b 2
12.b even 2 1 3024.2.q.e 2
21.c even 2 1 5292.2.i.b 2
21.g even 6 1 5292.2.j.b 2
21.g even 6 1 5292.2.l.b 2
21.h odd 6 1 756.2.l.a 2
21.h odd 6 1 5292.2.j.c 2
28.g odd 6 1 1008.2.t.b 2
36.f odd 6 1 1008.2.t.b 2
36.h even 6 1 3024.2.t.b 2
63.g even 3 1 1764.2.j.c 2
63.g even 3 1 2268.2.k.a 2
63.h even 3 1 inner 252.2.i.a 2
63.i even 6 1 5292.2.i.b 2
63.j odd 6 1 756.2.i.a 2
63.k odd 6 1 1764.2.j.a 2
63.l odd 6 1 1764.2.l.b 2
63.n odd 6 1 2268.2.k.b 2
63.n odd 6 1 5292.2.j.c 2
63.o even 6 1 5292.2.l.b 2
63.s even 6 1 5292.2.j.b 2
63.t odd 6 1 1764.2.i.b 2
84.n even 6 1 3024.2.t.b 2
252.u odd 6 1 1008.2.q.f 2
252.bb even 6 1 3024.2.q.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.i.a 2 1.a even 1 1 trivial
252.2.i.a 2 63.h even 3 1 inner
252.2.l.a yes 2 7.c even 3 1
252.2.l.a yes 2 9.c even 3 1
756.2.i.a 2 3.b odd 2 1
756.2.i.a 2 63.j odd 6 1
756.2.l.a 2 9.d odd 6 1
756.2.l.a 2 21.h odd 6 1
1008.2.q.f 2 4.b odd 2 1
1008.2.q.f 2 252.u odd 6 1
1008.2.t.b 2 28.g odd 6 1
1008.2.t.b 2 36.f odd 6 1
1764.2.i.b 2 7.b odd 2 1
1764.2.i.b 2 63.t odd 6 1
1764.2.j.a 2 7.d odd 6 1
1764.2.j.a 2 63.k odd 6 1
1764.2.j.c 2 7.c even 3 1
1764.2.j.c 2 63.g even 3 1
1764.2.l.b 2 7.d odd 6 1
1764.2.l.b 2 63.l odd 6 1
2268.2.k.a 2 9.c even 3 1
2268.2.k.a 2 63.g even 3 1
2268.2.k.b 2 9.d odd 6 1
2268.2.k.b 2 63.n odd 6 1
3024.2.q.e 2 12.b even 2 1
3024.2.q.e 2 252.bb even 6 1
3024.2.t.b 2 36.h even 6 1
3024.2.t.b 2 84.n even 6 1
5292.2.i.b 2 21.c even 2 1
5292.2.i.b 2 63.i even 6 1
5292.2.j.b 2 21.g even 6 1
5292.2.j.b 2 63.s even 6 1
5292.2.j.c 2 21.h odd 6 1
5292.2.j.c 2 63.n odd 6 1
5292.2.l.b 2 21.g even 6 1
5292.2.l.b 2 63.o even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 2 T_{5} + 4$$ acting on $$S_{2}^{\mathrm{new}}(252, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + 3 T + 3 T^{2}$$
$5$ $$1 + 2 T - T^{2} + 10 T^{3} + 25 T^{4}$$
$7$ $$1 + 5 T + 7 T^{2}$$
$11$ $$1 + 4 T + 5 T^{2} + 44 T^{3} + 121 T^{4}$$
$13$ $$1 + 3 T - 4 T^{2} + 39 T^{3} + 169 T^{4}$$
$17$ $$1 + 7 T + 32 T^{2} + 119 T^{3} + 289 T^{4}$$
$19$ $$1 + 5 T + 6 T^{2} + 95 T^{3} + 361 T^{4}$$
$23$ $$1 + 4 T - 7 T^{2} + 92 T^{3} + 529 T^{4}$$
$29$ $$1 - T - 28 T^{2} - 29 T^{3} + 841 T^{4}$$
$31$ $$( 1 + 3 T + 31 T^{2} )^{2}$$
$37$ $$( 1 + T + 37 T^{2} )( 1 + 10 T + 37 T^{2} )$$
$41$ $$1 - 9 T + 40 T^{2} - 369 T^{3} + 1681 T^{4}$$
$43$ $$( 1 - 8 T + 43 T^{2} )( 1 + 13 T + 43 T^{2} )$$
$47$ $$( 1 - 3 T + 47 T^{2} )^{2}$$
$53$ $$1 + 3 T - 44 T^{2} + 159 T^{3} + 2809 T^{4}$$
$59$ $$( 1 + 7 T + 59 T^{2} )^{2}$$
$61$ $$( 1 - 3 T + 61 T^{2} )^{2}$$
$67$ $$( 1 - 13 T + 67 T^{2} )^{2}$$
$71$ $$( 1 + 8 T + 71 T^{2} )^{2}$$
$73$ $$( 1 - 10 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} )$$
$79$ $$( 1 + 9 T + 79 T^{2} )^{2}$$
$83$ $$1 + T - 82 T^{2} + 83 T^{3} + 6889 T^{4}$$
$89$ $$1 + 15 T + 136 T^{2} + 1335 T^{3} + 7921 T^{4}$$
$97$ $$1 - 17 T + 192 T^{2} - 1649 T^{3} + 9409 T^{4}$$