# Properties

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

# 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: $$2$$ Coefficient field: $$\Q(\sqrt{-7})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 28) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{-7})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + ( -2 + \beta ) q^{4} + ( -1 + 2 \beta ) q^{7} + ( -2 - \beta ) q^{8} +O(q^{10})$$ $$q + \beta q^{2} + ( -2 + \beta ) q^{4} + ( -1 + 2 \beta ) q^{7} + ( -2 - \beta ) q^{8} + ( -2 + 4 \beta ) q^{11} + ( -4 + \beta ) q^{14} + ( 2 - 3 \beta ) q^{16} + ( -8 + 2 \beta ) q^{22} + ( 2 - 4 \beta ) q^{23} + 5 q^{25} + ( -2 - 3 \beta ) q^{28} + 2 q^{29} + ( 6 - \beta ) q^{32} + 6 q^{37} + ( 2 - 4 \beta ) q^{43} + ( -4 - 6 \beta ) q^{44} + ( 8 - 2 \beta ) q^{46} -7 q^{49} + 5 \beta q^{50} + 10 q^{53} + ( 6 - 5 \beta ) q^{56} + 2 \beta q^{58} + ( 2 + 5 \beta ) q^{64} + ( -6 + 12 \beta ) q^{67} + ( 2 - 4 \beta ) q^{71} + 6 \beta q^{74} -14 q^{77} + ( 6 - 12 \beta ) q^{79} + ( 8 - 2 \beta ) q^{86} + ( 12 - 10 \beta ) q^{88} + ( 4 + 6 \beta ) q^{92} -7 \beta q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{2} - 3q^{4} - 5q^{8} + O(q^{10})$$ $$2q + q^{2} - 3q^{4} - 5q^{8} - 7q^{14} + q^{16} - 14q^{22} + 10q^{25} - 7q^{28} + 4q^{29} + 11q^{32} + 12q^{37} - 14q^{44} + 14q^{46} - 14q^{49} + 5q^{50} + 20q^{53} + 7q^{56} + 2q^{58} + 9q^{64} + 6q^{74} - 28q^{77} + 14q^{86} + 14q^{88} + 14q^{92} - 7q^{98} + O(q^{100})$$

## Character values

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

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

## 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
 0.5 − 1.32288i 0.5 + 1.32288i
0.500000 1.32288i 0 −1.50000 1.32288i 0 0 2.64575i −2.50000 + 1.32288i 0 0
55.2 0.500000 + 1.32288i 0 −1.50000 + 1.32288i 0 0 2.64575i −2.50000 1.32288i 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 $$\Q(\sqrt{-7})$$
4.b odd 2 1 inner
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.a 2
3.b odd 2 1 28.2.d.a 2
4.b odd 2 1 inner 252.2.b.a 2
7.b odd 2 1 CM 252.2.b.a 2
8.b even 2 1 4032.2.b.e 2
8.d odd 2 1 4032.2.b.e 2
12.b even 2 1 28.2.d.a 2
15.d odd 2 1 700.2.g.a 2
15.e even 4 2 700.2.c.d 4
21.c even 2 1 28.2.d.a 2
21.g even 6 2 196.2.f.b 4
21.h odd 6 2 196.2.f.b 4
24.f even 2 1 448.2.f.b 2
24.h odd 2 1 448.2.f.b 2
28.d even 2 1 inner 252.2.b.a 2
48.i odd 4 2 1792.2.e.b 4
48.k even 4 2 1792.2.e.b 4
56.e even 2 1 4032.2.b.e 2
56.h odd 2 1 4032.2.b.e 2
60.h even 2 1 700.2.g.a 2
60.l odd 4 2 700.2.c.d 4
84.h odd 2 1 28.2.d.a 2
84.j odd 6 2 196.2.f.b 4
84.n even 6 2 196.2.f.b 4
105.g even 2 1 700.2.g.a 2
105.k odd 4 2 700.2.c.d 4
168.e odd 2 1 448.2.f.b 2
168.i even 2 1 448.2.f.b 2
336.v odd 4 2 1792.2.e.b 4
336.y even 4 2 1792.2.e.b 4
420.o odd 2 1 700.2.g.a 2
420.w even 4 2 700.2.c.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.2.d.a 2 3.b odd 2 1
28.2.d.a 2 12.b even 2 1
28.2.d.a 2 21.c even 2 1
28.2.d.a 2 84.h odd 2 1
196.2.f.b 4 21.g even 6 2
196.2.f.b 4 21.h odd 6 2
196.2.f.b 4 84.j odd 6 2
196.2.f.b 4 84.n even 6 2
252.2.b.a 2 1.a even 1 1 trivial
252.2.b.a 2 4.b odd 2 1 inner
252.2.b.a 2 7.b odd 2 1 CM
252.2.b.a 2 28.d even 2 1 inner
448.2.f.b 2 24.f even 2 1
448.2.f.b 2 24.h odd 2 1
448.2.f.b 2 168.e odd 2 1
448.2.f.b 2 168.i even 2 1
700.2.c.d 4 15.e even 4 2
700.2.c.d 4 60.l odd 4 2
700.2.c.d 4 105.k odd 4 2
700.2.c.d 4 420.w even 4 2
700.2.g.a 2 15.d odd 2 1
700.2.g.a 2 60.h even 2 1
700.2.g.a 2 105.g even 2 1
700.2.g.a 2 420.o odd 2 1
1792.2.e.b 4 48.i odd 4 2
1792.2.e.b 4 48.k even 4 2
1792.2.e.b 4 336.v odd 4 2
1792.2.e.b 4 336.y even 4 2
4032.2.b.e 2 8.b even 2 1
4032.2.b.e 2 8.d odd 2 1
4032.2.b.e 2 56.e even 2 1
4032.2.b.e 2 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 $$S_{2}^{\mathrm{new}}(252, [\chi])$$:

 $$T_{5}$$ $$T_{11}^{2} + 28$$ $$T_{19}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - T + 2 T^{2}$$
$3$ 
$5$ $$( 1 - 5 T^{2} )^{2}$$
$7$ $$1 + 7 T^{2}$$
$11$ $$( 1 - 4 T + 11 T^{2} )( 1 + 4 T + 11 T^{2} )$$
$13$ $$( 1 - 13 T^{2} )^{2}$$
$17$ $$( 1 - 17 T^{2} )^{2}$$
$19$ $$( 1 + 19 T^{2} )^{2}$$
$23$ $$( 1 - 8 T + 23 T^{2} )( 1 + 8 T + 23 T^{2} )$$
$29$ $$( 1 - 2 T + 29 T^{2} )^{2}$$
$31$ $$( 1 + 31 T^{2} )^{2}$$
$37$ $$( 1 - 6 T + 37 T^{2} )^{2}$$
$41$ $$( 1 - 41 T^{2} )^{2}$$
$43$ $$( 1 - 12 T + 43 T^{2} )( 1 + 12 T + 43 T^{2} )$$
$47$ $$( 1 + 47 T^{2} )^{2}$$
$53$ $$( 1 - 10 T + 53 T^{2} )^{2}$$
$59$ $$( 1 + 59 T^{2} )^{2}$$
$61$ $$( 1 - 61 T^{2} )^{2}$$
$67$ $$( 1 - 4 T + 67 T^{2} )( 1 + 4 T + 67 T^{2} )$$
$71$ $$( 1 - 16 T + 71 T^{2} )( 1 + 16 T + 71 T^{2} )$$
$73$ $$( 1 - 73 T^{2} )^{2}$$
$79$ $$( 1 - 8 T + 79 T^{2} )( 1 + 8 T + 79 T^{2} )$$
$83$ $$( 1 + 83 T^{2} )^{2}$$
$89$ $$( 1 - 89 T^{2} )^{2}$$
$97$ $$( 1 - 97 T^{2} )^{2}$$