# Properties

 Label 1805.1.o.b Level $1805$ Weight $1$ Character orbit 1805.o Analytic conductor $0.901$ Analytic rank $0$ Dimension $12$ Projective image $D_{4}$ CM discriminant -95 Inner twists $24$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1805 = 5 \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1805.o (of order $$18$$, degree $$6$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.900812347803$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$2$$ over $$\Q(\zeta_{18})$$ Coefficient field: 12.0.101559956668416.2 Defining polynomial: $$x^{12} + 8 x^{6} + 64$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 95) Projective image: $$D_{4}$$ Projective field: Galois closure of 4.2.475.1 Artin image: $C_9\times D_8$ Artin field: Galois closure of $$\mathbb{Q}[x]/(x^{72} - \cdots)$$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \beta_{5} + \beta_{11} ) q^{2} + \beta_{5} q^{3} + ( -\beta_{4} - \beta_{10} ) q^{4} -\beta_{2} q^{5} -2 \beta_{4} q^{6} + \beta_{10} q^{9} +O(q^{10})$$ $$q + ( \beta_{5} + \beta_{11} ) q^{2} + \beta_{5} q^{3} + ( -\beta_{4} - \beta_{10} ) q^{4} -\beta_{2} q^{5} -2 \beta_{4} q^{6} + \beta_{10} q^{9} + \beta_{1} q^{10} + \beta_{3} q^{12} + ( \beta_{1} + \beta_{7} ) q^{13} -\beta_{7} q^{15} + ( \beta_{2} + \beta_{8} ) q^{16} -\beta_{9} q^{18} - q^{20} + \beta_{4} q^{25} + ( -2 - 2 \beta_{6} ) q^{26} + 2 \beta_{6} q^{30} + ( -\beta_{1} - \beta_{7} ) q^{32} + \beta_{8} q^{36} + \beta_{9} q^{37} -2 q^{39} + ( 1 + \beta_{6} ) q^{45} -\beta_{1} q^{48} + \beta_{6} q^{49} -\beta_{3} q^{50} -\beta_{11} q^{52} + \beta_{7} q^{53} -\beta_{5} q^{60} + ( 1 + \beta_{6} ) q^{64} + ( -\beta_{3} - \beta_{9} ) q^{65} -\beta_{1} q^{67} -2 \beta_{8} q^{74} + \beta_{9} q^{75} + ( -2 \beta_{5} - 2 \beta_{11} ) q^{78} + ( -\beta_{4} - \beta_{10} ) q^{80} -\beta_{2} q^{81} + \beta_{11} q^{90} + 2 q^{96} + ( -\beta_{5} - \beta_{11} ) q^{97} -\beta_{5} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + O(q^{10})$$ $$12 q - 12 q^{20} - 12 q^{26} - 12 q^{30} - 24 q^{39} + 6 q^{45} - 6 q^{49} + 6 q^{64} + 24 q^{96} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + 8 x^{6} + 64$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/2$$ $$\beta_{4}$$ $$=$$ $$\nu^{4}$$$$/4$$ $$\beta_{5}$$ $$=$$ $$\nu^{5}$$$$/4$$ $$\beta_{6}$$ $$=$$ $$\nu^{6}$$$$/8$$ $$\beta_{7}$$ $$=$$ $$\nu^{7}$$$$/8$$ $$\beta_{8}$$ $$=$$ $$\nu^{8}$$$$/16$$ $$\beta_{9}$$ $$=$$ $$\nu^{9}$$$$/16$$ $$\beta_{10}$$ $$=$$ $$\nu^{10}$$$$/32$$ $$\beta_{11}$$ $$=$$ $$\nu^{11}$$$$/32$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{3}$$ $$\nu^{4}$$ $$=$$ $$4 \beta_{4}$$ $$\nu^{5}$$ $$=$$ $$4 \beta_{5}$$ $$\nu^{6}$$ $$=$$ $$8 \beta_{6}$$ $$\nu^{7}$$ $$=$$ $$8 \beta_{7}$$ $$\nu^{8}$$ $$=$$ $$16 \beta_{8}$$ $$\nu^{9}$$ $$=$$ $$16 \beta_{9}$$ $$\nu^{10}$$ $$=$$ $$32 \beta_{10}$$ $$\nu^{11}$$ $$=$$ $$32 \beta_{11}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1805\mathbb{Z}\right)^\times$$.

 $$n$$ $$362$$ $$1446$$ $$\chi(n)$$ $$-1$$ $$\beta_{4} + \beta_{10}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
299.1
 1.32893 + 0.483690i −1.32893 − 0.483690i 0.245576 − 1.39273i −0.245576 + 1.39273i 1.08335 + 0.909039i −1.08335 − 0.909039i 1.32893 − 0.483690i −1.32893 + 0.483690i 1.08335 − 0.909039i −1.08335 + 0.909039i 0.245576 + 1.39273i −0.245576 − 1.39273i
−1.32893 + 0.483690i −0.245576 + 1.39273i 0.766044 0.642788i −0.766044 0.642788i −0.347296 1.96962i 0 0 −0.939693 0.342020i 1.32893 + 0.483690i
299.2 1.32893 0.483690i 0.245576 1.39273i 0.766044 0.642788i −0.766044 0.642788i −0.347296 1.96962i 0 0 −0.939693 0.342020i −1.32893 0.483690i
694.1 −0.245576 1.39273i 1.08335 0.909039i −0.939693 + 0.342020i 0.939693 + 0.342020i −1.53209 1.28558i 0 0 0.173648 0.984808i 0.245576 1.39273i
694.2 0.245576 + 1.39273i −1.08335 + 0.909039i −0.939693 + 0.342020i 0.939693 + 0.342020i −1.53209 1.28558i 0 0 0.173648 0.984808i −0.245576 + 1.39273i
849.1 −1.08335 + 0.909039i −1.32893 0.483690i 0.173648 0.984808i −0.173648 0.984808i 1.87939 0.684040i 0 0 0.766044 + 0.642788i 1.08335 + 0.909039i
849.2 1.08335 0.909039i 1.32893 + 0.483690i 0.173648 0.984808i −0.173648 0.984808i 1.87939 0.684040i 0 0 0.766044 + 0.642788i −1.08335 0.909039i
984.1 −1.32893 0.483690i −0.245576 1.39273i 0.766044 + 0.642788i −0.766044 + 0.642788i −0.347296 + 1.96962i 0 0 −0.939693 + 0.342020i 1.32893 0.483690i
984.2 1.32893 + 0.483690i 0.245576 + 1.39273i 0.766044 + 0.642788i −0.766044 + 0.642788i −0.347296 + 1.96962i 0 0 −0.939693 + 0.342020i −1.32893 + 0.483690i
1029.1 −1.08335 0.909039i −1.32893 + 0.483690i 0.173648 + 0.984808i −0.173648 + 0.984808i 1.87939 + 0.684040i 0 0 0.766044 0.642788i 1.08335 0.909039i
1029.2 1.08335 + 0.909039i 1.32893 0.483690i 0.173648 + 0.984808i −0.173648 + 0.984808i 1.87939 + 0.684040i 0 0 0.766044 0.642788i −1.08335 + 0.909039i
1199.1 −0.245576 + 1.39273i 1.08335 + 0.909039i −0.939693 0.342020i 0.939693 0.342020i −1.53209 + 1.28558i 0 0 0.173648 + 0.984808i 0.245576 + 1.39273i
1199.2 0.245576 1.39273i −1.08335 0.909039i −0.939693 0.342020i 0.939693 0.342020i −1.53209 + 1.28558i 0 0 0.173648 + 0.984808i −0.245576 1.39273i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1199.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
95.d odd 2 1 CM by $$\Q(\sqrt{-95})$$
5.b even 2 1 inner
19.b odd 2 1 inner
19.c even 3 2 inner
19.d odd 6 2 inner
19.e even 9 3 inner
19.f odd 18 3 inner
95.h odd 6 2 inner
95.i even 6 2 inner
95.o odd 18 3 inner
95.p even 18 3 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1805.1.o.b 12
5.b even 2 1 inner 1805.1.o.b 12
19.b odd 2 1 inner 1805.1.o.b 12
19.c even 3 2 inner 1805.1.o.b 12
19.d odd 6 2 inner 1805.1.o.b 12
19.e even 9 1 95.1.d.b 2
19.e even 9 2 1805.1.h.b 4
19.e even 9 3 inner 1805.1.o.b 12
19.f odd 18 1 95.1.d.b 2
19.f odd 18 2 1805.1.h.b 4
19.f odd 18 3 inner 1805.1.o.b 12
57.j even 18 1 855.1.g.c 2
57.l odd 18 1 855.1.g.c 2
76.k even 18 1 1520.1.m.b 2
76.l odd 18 1 1520.1.m.b 2
95.d odd 2 1 CM 1805.1.o.b 12
95.h odd 6 2 inner 1805.1.o.b 12
95.i even 6 2 inner 1805.1.o.b 12
95.o odd 18 1 95.1.d.b 2
95.o odd 18 2 1805.1.h.b 4
95.o odd 18 3 inner 1805.1.o.b 12
95.p even 18 1 95.1.d.b 2
95.p even 18 2 1805.1.h.b 4
95.p even 18 3 inner 1805.1.o.b 12
95.q odd 36 2 475.1.c.b 2
95.r even 36 2 475.1.c.b 2
285.bd odd 18 1 855.1.g.c 2
285.bf even 18 1 855.1.g.c 2
380.ba odd 18 1 1520.1.m.b 2
380.bb even 18 1 1520.1.m.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.1.d.b 2 19.e even 9 1
95.1.d.b 2 19.f odd 18 1
95.1.d.b 2 95.o odd 18 1
95.1.d.b 2 95.p even 18 1
475.1.c.b 2 95.q odd 36 2
475.1.c.b 2 95.r even 36 2
855.1.g.c 2 57.j even 18 1
855.1.g.c 2 57.l odd 18 1
855.1.g.c 2 285.bd odd 18 1
855.1.g.c 2 285.bf even 18 1
1520.1.m.b 2 76.k even 18 1
1520.1.m.b 2 76.l odd 18 1
1520.1.m.b 2 380.ba odd 18 1
1520.1.m.b 2 380.bb even 18 1
1805.1.h.b 4 19.e even 9 2
1805.1.h.b 4 19.f odd 18 2
1805.1.h.b 4 95.o odd 18 2
1805.1.h.b 4 95.p even 18 2
1805.1.o.b 12 1.a even 1 1 trivial
1805.1.o.b 12 5.b even 2 1 inner
1805.1.o.b 12 19.b odd 2 1 inner
1805.1.o.b 12 19.c even 3 2 inner
1805.1.o.b 12 19.d odd 6 2 inner
1805.1.o.b 12 19.e even 9 3 inner
1805.1.o.b 12 19.f odd 18 3 inner
1805.1.o.b 12 95.d odd 2 1 CM
1805.1.o.b 12 95.h odd 6 2 inner
1805.1.o.b 12 95.i even 6 2 inner
1805.1.o.b 12 95.o odd 18 3 inner
1805.1.o.b 12 95.p even 18 3 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{12} + 8 T_{2}^{6} + 64$$ acting on $$S_{1}^{\mathrm{new}}(1805, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$64 + 8 T^{6} + T^{12}$$
$3$ $$64 + 8 T^{6} + T^{12}$$
$5$ $$( 1 - T^{3} + T^{6} )^{2}$$
$7$ $$T^{12}$$
$11$ $$T^{12}$$
$13$ $$64 + 8 T^{6} + T^{12}$$
$17$ $$T^{12}$$
$19$ $$T^{12}$$
$23$ $$T^{12}$$
$29$ $$T^{12}$$
$31$ $$T^{12}$$
$37$ $$( -2 + T^{2} )^{6}$$
$41$ $$T^{12}$$
$43$ $$T^{12}$$
$47$ $$T^{12}$$
$53$ $$64 + 8 T^{6} + T^{12}$$
$59$ $$T^{12}$$
$61$ $$T^{12}$$
$67$ $$64 + 8 T^{6} + T^{12}$$
$71$ $$T^{12}$$
$73$ $$T^{12}$$
$79$ $$T^{12}$$
$83$ $$T^{12}$$
$89$ $$T^{12}$$
$97$ $$64 + 8 T^{6} + T^{12}$$