# Properties

 Label 28.2.d.a Level $28$ Weight $2$ Character orbit 28.d Analytic conductor $0.224$ Analytic rank $0$ Dimension $2$ CM discriminant -7 Inner twists $4$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$28 = 2^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 28.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.223581125660$$ 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: yes 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.

 $$f(q)$$ $$=$$ $$q -\beta q^{2} + ( -2 + \beta ) q^{4} + ( -1 + 2 \beta ) q^{7} + ( 2 + \beta ) q^{8} -3 q^{9} +O(q^{10})$$ $$q -\beta q^{2} + ( -2 + \beta ) q^{4} + ( -1 + 2 \beta ) q^{7} + ( 2 + \beta ) q^{8} -3 q^{9} + ( 2 - 4 \beta ) q^{11} + ( 4 - \beta ) q^{14} + ( 2 - 3 \beta ) q^{16} + 3 \beta q^{18} + ( -8 + 2 \beta ) q^{22} + ( -2 + 4 \beta ) q^{23} + 5 q^{25} + ( -2 - 3 \beta ) q^{28} -2 q^{29} + ( -6 + \beta ) q^{32} + ( 6 - 3 \beta ) q^{36} + 6 q^{37} + ( 2 - 4 \beta ) q^{43} + ( 4 + 6 \beta ) q^{44} + ( 8 - 2 \beta ) q^{46} -7 q^{49} -5 \beta q^{50} -10 q^{53} + ( -6 + 5 \beta ) q^{56} + 2 \beta q^{58} + ( 3 - 6 \beta ) q^{63} + ( 2 + 5 \beta ) q^{64} + ( -6 + 12 \beta ) q^{67} + ( -2 + 4 \beta ) q^{71} + ( -6 - 3 \beta ) q^{72} -6 \beta q^{74} + 14 q^{77} + ( 6 - 12 \beta ) q^{79} + 9 q^{81} + ( -8 + 2 \beta ) q^{86} + ( 12 - 10 \beta ) q^{88} + ( -4 - 6 \beta ) q^{92} + 7 \beta q^{98} + ( -6 + 12 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} - 3q^{4} + 5q^{8} - 6q^{9} + O(q^{10})$$ $$2q - q^{2} - 3q^{4} + 5q^{8} - 6q^{9} + 7q^{14} + q^{16} + 3q^{18} - 14q^{22} + 10q^{25} - 7q^{28} - 4q^{29} - 11q^{32} + 9q^{36} + 12q^{37} + 14q^{44} + 14q^{46} - 14q^{49} - 5q^{50} - 20q^{53} - 7q^{56} + 2q^{58} + 9q^{64} - 15q^{72} - 6q^{74} + 28q^{77} + 18q^{81} - 14q^{86} + 14q^{88} - 14q^{92} + 7q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/28\mathbb{Z}\right)^\times$$.

 $$n$$ $$15$$ $$17$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
27.1
 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 −3.00000 0
27.2 −0.500000 + 1.32288i 0 −1.50000 1.32288i 0 0 2.64575i 2.50000 1.32288i −3.00000 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 28.2.d.a 2
3.b odd 2 1 252.2.b.a 2
4.b odd 2 1 inner 28.2.d.a 2
5.b even 2 1 700.2.g.a 2
5.c odd 4 2 700.2.c.d 4
7.b odd 2 1 CM 28.2.d.a 2
7.c even 3 2 196.2.f.b 4
7.d odd 6 2 196.2.f.b 4
8.b even 2 1 448.2.f.b 2
8.d odd 2 1 448.2.f.b 2
12.b even 2 1 252.2.b.a 2
16.e even 4 2 1792.2.e.b 4
16.f odd 4 2 1792.2.e.b 4
20.d odd 2 1 700.2.g.a 2
20.e even 4 2 700.2.c.d 4
21.c even 2 1 252.2.b.a 2
24.f even 2 1 4032.2.b.e 2
24.h odd 2 1 4032.2.b.e 2
28.d even 2 1 inner 28.2.d.a 2
28.f even 6 2 196.2.f.b 4
28.g odd 6 2 196.2.f.b 4
35.c odd 2 1 700.2.g.a 2
35.f even 4 2 700.2.c.d 4
56.e even 2 1 448.2.f.b 2
56.h odd 2 1 448.2.f.b 2
84.h odd 2 1 252.2.b.a 2
112.j even 4 2 1792.2.e.b 4
112.l odd 4 2 1792.2.e.b 4
140.c even 2 1 700.2.g.a 2
140.j odd 4 2 700.2.c.d 4
168.e odd 2 1 4032.2.b.e 2
168.i even 2 1 4032.2.b.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.2.d.a 2 1.a even 1 1 trivial
28.2.d.a 2 4.b odd 2 1 inner
28.2.d.a 2 7.b odd 2 1 CM
28.2.d.a 2 28.d even 2 1 inner
196.2.f.b 4 7.c even 3 2
196.2.f.b 4 7.d odd 6 2
196.2.f.b 4 28.f even 6 2
196.2.f.b 4 28.g odd 6 2
252.2.b.a 2 3.b odd 2 1
252.2.b.a 2 12.b even 2 1
252.2.b.a 2 21.c even 2 1
252.2.b.a 2 84.h odd 2 1
448.2.f.b 2 8.b even 2 1
448.2.f.b 2 8.d odd 2 1
448.2.f.b 2 56.e even 2 1
448.2.f.b 2 56.h odd 2 1
700.2.c.d 4 5.c odd 4 2
700.2.c.d 4 20.e even 4 2
700.2.c.d 4 35.f even 4 2
700.2.c.d 4 140.j odd 4 2
700.2.g.a 2 5.b even 2 1
700.2.g.a 2 20.d odd 2 1
700.2.g.a 2 35.c odd 2 1
700.2.g.a 2 140.c even 2 1
1792.2.e.b 4 16.e even 4 2
1792.2.e.b 4 16.f odd 4 2
1792.2.e.b 4 112.j even 4 2
1792.2.e.b 4 112.l odd 4 2
4032.2.b.e 2 24.f even 2 1
4032.2.b.e 2 24.h odd 2 1
4032.2.b.e 2 168.e odd 2 1
4032.2.b.e 2 168.i even 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(28, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$2 + T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$7 + T^{2}$$
$11$ $$28 + T^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2}$$
$23$ $$28 + T^{2}$$
$29$ $$( 2 + T )^{2}$$
$31$ $$T^{2}$$
$37$ $$( -6 + T )^{2}$$
$41$ $$T^{2}$$
$43$ $$28 + T^{2}$$
$47$ $$T^{2}$$
$53$ $$( 10 + T )^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2}$$
$67$ $$252 + T^{2}$$
$71$ $$28 + T^{2}$$
$73$ $$T^{2}$$
$79$ $$252 + T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2}$$
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