# Properties

 Label 1775.1.c.a Level $1775$ Weight $1$ Character orbit 1775.c Analytic conductor $0.886$ Analytic rank $0$ Dimension $6$ Projective image $D_{7}$ CM discriminant -71 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1775 = 5^{2} \cdot 71$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1775.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.885840397424$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.153664.1 Defining polynomial: $$x^{6} + 5 x^{4} + 6 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 71) Projective image: $$D_{7}$$ Projective field: Galois closure of 7.1.357911.1 Artin image: $C_4\times D_7$ Artin field: Galois closure of $$\mathbb{Q}[x]/(x^{28} - \cdots)$$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} -\beta_{3} q^{3} + ( -1 + \beta_{2} ) q^{4} + ( -1 + \beta_{4} ) q^{6} + ( \beta_{1} - \beta_{3} ) q^{8} + ( -1 + \beta_{4} ) q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{2} -\beta_{3} q^{3} + ( -1 + \beta_{2} ) q^{4} + ( -1 + \beta_{4} ) q^{6} + ( \beta_{1} - \beta_{3} ) q^{8} + ( -1 + \beta_{4} ) q^{9} -\beta_{5} q^{12} + \beta_{4} q^{16} + ( \beta_{3} - \beta_{5} ) q^{18} + \beta_{4} q^{19} - q^{24} + ( \beta_{1} - \beta_{5} ) q^{27} + ( 1 - \beta_{2} - \beta_{4} ) q^{29} -\beta_{5} q^{32} -\beta_{4} q^{36} -\beta_{1} q^{37} + ( -\beta_{1} + \beta_{3} - \beta_{5} ) q^{38} + ( -\beta_{1} + \beta_{3} + \beta_{5} ) q^{43} + ( \beta_{1} - \beta_{5} ) q^{48} - q^{49} + ( 2 - \beta_{2} - \beta_{4} ) q^{54} + ( \beta_{1} - \beta_{5} ) q^{57} + ( -\beta_{1} + \beta_{5} ) q^{58} + q^{71} + \beta_{1} q^{72} + ( -\beta_{1} + \beta_{3} + \beta_{5} ) q^{73} + ( -2 + \beta_{2} ) q^{74} + ( -1 + \beta_{2} - \beta_{4} ) q^{76} + \beta_{2} q^{79} + ( 1 - \beta_{2} - \beta_{4} ) q^{81} + \beta_{1} q^{83} + ( -1 + \beta_{2} ) q^{86} + ( -\beta_{1} + 2 \beta_{5} ) q^{87} + ( 1 - \beta_{2} - \beta_{4} ) q^{89} + ( 1 - \beta_{2} - \beta_{4} ) q^{96} + \beta_{1} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 4 q^{4} - 4 q^{6} - 4 q^{9} + O(q^{10})$$ $$6 q - 4 q^{4} - 4 q^{6} - 4 q^{9} + 2 q^{16} + 2 q^{19} - 6 q^{24} + 2 q^{29} - 2 q^{36} - 6 q^{49} + 8 q^{54} + 6 q^{71} - 10 q^{74} - 6 q^{76} + 2 q^{79} + 2 q^{81} - 4 q^{86} + 2 q^{89} + 2 q^{96} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 5 x^{4} + 6 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} + 3 \nu$$ $$\beta_{4}$$ $$=$$ $$\nu^{4} + 3 \nu^{2} + 1$$ $$\beta_{5}$$ $$=$$ $$\nu^{5} + 4 \nu^{3} + 3 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 2$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} - 3 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{4} - 3 \beta_{2} + 5$$ $$\nu^{5}$$ $$=$$ $$\beta_{5} - 4 \beta_{3} + 9 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1775\mathbb{Z}\right)^\times$$.

 $$n$$ $$427$$ $$1001$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1774.1
 1.80194i 1.24698i 0.445042i − 0.445042i − 1.24698i − 1.80194i
1.80194i 0.445042i −2.24698 0 0.801938 0 2.24698i 0.801938 0
1774.2 1.24698i 1.80194i −0.554958 0 −2.24698 0 0.554958i −2.24698 0
1774.3 0.445042i 1.24698i 0.801938 0 −0.554958 0 0.801938i −0.554958 0
1774.4 0.445042i 1.24698i 0.801938 0 −0.554958 0 0.801938i −0.554958 0
1774.5 1.24698i 1.80194i −0.554958 0 −2.24698 0 0.554958i −2.24698 0
1774.6 1.80194i 0.445042i −2.24698 0 0.801938 0 2.24698i 0.801938 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1774.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
71.b odd 2 1 CM by $$\Q(\sqrt{-71})$$
5.b even 2 1 inner
355.c odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1775.1.c.a 6
5.b even 2 1 inner 1775.1.c.a 6
5.c odd 4 1 71.1.b.a 3
5.c odd 4 1 1775.1.d.b 3
15.e even 4 1 639.1.d.a 3
20.e even 4 1 1136.1.h.a 3
35.f even 4 1 3479.1.d.e 3
35.k even 12 2 3479.1.g.d 6
35.l odd 12 2 3479.1.g.e 6
71.b odd 2 1 CM 1775.1.c.a 6
355.c odd 2 1 inner 1775.1.c.a 6
355.e even 4 1 71.1.b.a 3
355.e even 4 1 1775.1.d.b 3
1065.l odd 4 1 639.1.d.a 3
1420.l odd 4 1 1136.1.h.a 3
2485.j odd 4 1 3479.1.d.e 3
2485.be odd 12 2 3479.1.g.d 6
2485.bg even 12 2 3479.1.g.e 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
71.1.b.a 3 5.c odd 4 1
71.1.b.a 3 355.e even 4 1
639.1.d.a 3 15.e even 4 1
639.1.d.a 3 1065.l odd 4 1
1136.1.h.a 3 20.e even 4 1
1136.1.h.a 3 1420.l odd 4 1
1775.1.c.a 6 1.a even 1 1 trivial
1775.1.c.a 6 5.b even 2 1 inner
1775.1.c.a 6 71.b odd 2 1 CM
1775.1.c.a 6 355.c odd 2 1 inner
1775.1.d.b 3 5.c odd 4 1
1775.1.d.b 3 355.e even 4 1
3479.1.d.e 3 35.f even 4 1
3479.1.d.e 3 2485.j odd 4 1
3479.1.g.d 6 35.k even 12 2
3479.1.g.d 6 2485.be odd 12 2
3479.1.g.e 6 35.l odd 12 2
3479.1.g.e 6 2485.bg even 12 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(1775, [\chi])$$.

## Hecke characteristic polynomials

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