# Properties

 Label 3776.1.h.a Level $3776$ Weight $1$ Character orbit 3776.h Self dual yes Analytic conductor $1.884$ Analytic rank $0$ Dimension $1$ Projective image $D_{3}$ CM discriminant -59 Inner twists $2$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$3776 = 2^{6} \cdot 59$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3776.h (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$1.88446948770$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 59) Projective image $$D_{3}$$ Projective field Galois closure of 3.1.59.1 Artin image $D_6$ Artin field Galois closure of 6.0.1782272.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{3} + q^{5} + q^{7} + O(q^{10})$$ $$q - q^{3} + q^{5} + q^{7} - q^{15} + 2q^{17} - q^{19} - q^{21} + q^{27} + q^{29} + q^{35} - q^{41} - 2q^{51} + q^{53} + q^{57} + q^{59} - 2q^{71} + q^{79} - q^{81} + 2q^{85} - q^{87} - q^{95} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3776\mathbb{Z}\right)^\times$$.

 $$n$$ $$709$$ $$769$$ $$1535$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2241.1
 0
0 −1.00000 0 1.00000 0 1.00000 0 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
59.b odd 2 1 CM by $$\Q(\sqrt{-59})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3776.1.h.a 1
4.b odd 2 1 3776.1.h.b 1
8.b even 2 1 944.1.h.a 1
8.d odd 2 1 59.1.b.a 1
24.f even 2 1 531.1.c.a 1
40.e odd 2 1 1475.1.c.b 1
40.k even 4 2 1475.1.d.a 2
56.e even 2 1 2891.1.c.e 1
56.k odd 6 2 2891.1.g.d 2
56.m even 6 2 2891.1.g.b 2
59.b odd 2 1 CM 3776.1.h.a 1
236.c even 2 1 3776.1.h.b 1
472.c odd 2 1 944.1.h.a 1
472.f even 2 1 59.1.b.a 1
472.k odd 58 28 3481.1.d.a 28
472.l even 58 28 3481.1.d.a 28
1416.m odd 2 1 531.1.c.a 1
2360.f even 2 1 1475.1.c.b 1
2360.q odd 4 2 1475.1.d.a 2
3304.f odd 2 1 2891.1.c.e 1
3304.x even 6 2 2891.1.g.d 2
3304.ba odd 6 2 2891.1.g.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
59.1.b.a 1 8.d odd 2 1
59.1.b.a 1 472.f even 2 1
531.1.c.a 1 24.f even 2 1
531.1.c.a 1 1416.m odd 2 1
944.1.h.a 1 8.b even 2 1
944.1.h.a 1 472.c odd 2 1
1475.1.c.b 1 40.e odd 2 1
1475.1.c.b 1 2360.f even 2 1
1475.1.d.a 2 40.k even 4 2
1475.1.d.a 2 2360.q odd 4 2
2891.1.c.e 1 56.e even 2 1
2891.1.c.e 1 3304.f odd 2 1
2891.1.g.b 2 56.m even 6 2
2891.1.g.b 2 3304.ba odd 6 2
2891.1.g.d 2 56.k odd 6 2
2891.1.g.d 2 3304.x even 6 2
3481.1.d.a 28 472.k odd 58 28
3481.1.d.a 28 472.l even 58 28
3776.1.h.a 1 1.a even 1 1 trivial
3776.1.h.a 1 59.b odd 2 1 CM
3776.1.h.b 1 4.b odd 2 1
3776.1.h.b 1 236.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(3776, [\chi])$$:

 $$T_{3} + 1$$ $$T_{7} - 1$$

## Hecke characteristic polynomials

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