# Properties

 Label 252.4.b.a Level $252$ Weight $4$ Character orbit 252.b Analytic conductor $14.868$ Analytic rank $0$ Dimension $2$ CM discriminant -7 Inner twists $4$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$252 = 2^{2} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 252.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.8684813214$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-7})$$ Defining polynomial: $$x^{2} - x + 2$$ 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.

 $$f(q)$$ $$=$$ $$q + ( -3 + \beta ) q^{2} + ( 7 - 5 \beta ) q^{4} + ( -7 + 14 \beta ) q^{7} + ( -11 + 17 \beta ) q^{8} +O(q^{10})$$ $$q + ( -3 + \beta ) q^{2} + ( 7 - 5 \beta ) q^{4} + ( -7 + 14 \beta ) q^{7} + ( -11 + 17 \beta ) q^{8} + ( 10 - 20 \beta ) q^{11} + ( -7 - 35 \beta ) q^{14} + ( -1 - 45 \beta ) q^{16} + ( 10 + 50 \beta ) q^{22} + ( -82 + 164 \beta ) q^{23} + 125 q^{25} + ( 91 + 63 \beta ) q^{28} -166 q^{29} + ( 93 + 89 \beta ) q^{32} -450 q^{37} + ( -202 + 404 \beta ) q^{43} + ( -130 - 90 \beta ) q^{44} + ( -82 - 410 \beta ) q^{46} -343 q^{49} + ( -375 + 125 \beta ) q^{50} -590 q^{53} + ( -399 - 35 \beta ) q^{56} + ( 498 - 166 \beta ) q^{58} + ( -457 - 85 \beta ) q^{64} + ( -306 + 612 \beta ) q^{67} + ( -370 + 740 \beta ) q^{71} + ( 1350 - 450 \beta ) q^{74} + 490 q^{77} + ( 90 - 180 \beta ) q^{79} + ( -202 - 1010 \beta ) q^{86} + ( 570 + 50 \beta ) q^{88} + ( 1066 + 738 \beta ) q^{92} + ( 1029 - 343 \beta ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 5q^{2} + 9q^{4} - 5q^{8} + O(q^{10})$$ $$2q - 5q^{2} + 9q^{4} - 5q^{8} - 49q^{14} - 47q^{16} + 70q^{22} + 250q^{25} + 245q^{28} - 332q^{29} + 275q^{32} - 900q^{37} - 350q^{44} - 574q^{46} - 686q^{49} - 625q^{50} - 1180q^{53} - 833q^{56} + 830q^{58} - 999q^{64} + 2250q^{74} + 980q^{77} - 1414q^{86} + 1190q^{88} + 2870q^{92} + 1715q^{98} + O(q^{100})$$

## 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
−2.50000 1.32288i 0 4.50000 + 6.61438i 0 0 18.5203i −2.50000 22.4889i 0 0
55.2 −2.50000 + 1.32288i 0 4.50000 6.61438i 0 0 18.5203i −2.50000 + 22.4889i 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 252.4.b.a 2
3.b odd 2 1 28.4.d.a 2
4.b odd 2 1 inner 252.4.b.a 2
7.b odd 2 1 CM 252.4.b.a 2
12.b even 2 1 28.4.d.a 2
21.c even 2 1 28.4.d.a 2
21.g even 6 2 196.4.f.a 4
21.h odd 6 2 196.4.f.a 4
24.f even 2 1 448.4.f.a 2
24.h odd 2 1 448.4.f.a 2
28.d even 2 1 inner 252.4.b.a 2
84.h odd 2 1 28.4.d.a 2
84.j odd 6 2 196.4.f.a 4
84.n even 6 2 196.4.f.a 4
168.e odd 2 1 448.4.f.a 2
168.i even 2 1 448.4.f.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.4.d.a 2 3.b odd 2 1
28.4.d.a 2 12.b even 2 1
28.4.d.a 2 21.c even 2 1
28.4.d.a 2 84.h odd 2 1
196.4.f.a 4 21.g even 6 2
196.4.f.a 4 21.h odd 6 2
196.4.f.a 4 84.j odd 6 2
196.4.f.a 4 84.n even 6 2
252.4.b.a 2 1.a even 1 1 trivial
252.4.b.a 2 4.b odd 2 1 inner
252.4.b.a 2 7.b odd 2 1 CM
252.4.b.a 2 28.d even 2 1 inner
448.4.f.a 2 24.f even 2 1
448.4.f.a 2 24.h odd 2 1
448.4.f.a 2 168.e odd 2 1
448.4.f.a 2 168.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 $$S_{4}^{\mathrm{new}}(252, [\chi])$$:

 $$T_{5}$$ $$T_{11}^{2} + 700$$ $$T_{19}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$8 + 5 T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$343 + T^{2}$$
$11$ $$700 + T^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2}$$
$23$ $$47068 + T^{2}$$
$29$ $$( 166 + T )^{2}$$
$31$ $$T^{2}$$
$37$ $$( 450 + T )^{2}$$
$41$ $$T^{2}$$
$43$ $$285628 + T^{2}$$
$47$ $$T^{2}$$
$53$ $$( 590 + T )^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2}$$
$67$ $$655452 + T^{2}$$
$71$ $$958300 + T^{2}$$
$73$ $$T^{2}$$
$79$ $$56700 + T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2}$$
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