# Properties

 Label 3724.1.n.a Level $3724$ Weight $1$ Character orbit 3724.n Analytic conductor $1.859$ Analytic rank $0$ Dimension $4$ Projective image $A_{4}$ CM/RM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3724 = 2^{2} \cdot 7^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3724.n (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.85851810705$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 532) Projective image: $$A_{4}$$ Projective field: Galois closure of 4.0.283024.1

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + \zeta_{12} q^{2} + \zeta_{12}^{3} q^{3} + \zeta_{12}^{2} q^{4} + \zeta_{12}^{4} q^{6} + \zeta_{12}^{3} q^{8} +O(q^{10})$$ $$q + \zeta_{12} q^{2} + \zeta_{12}^{3} q^{3} + \zeta_{12}^{2} q^{4} + \zeta_{12}^{4} q^{6} + \zeta_{12}^{3} q^{8} -\zeta_{12} q^{11} + \zeta_{12}^{5} q^{12} + \zeta_{12}^{4} q^{13} + \zeta_{12}^{4} q^{16} + q^{17} + \zeta_{12} q^{19} -\zeta_{12}^{2} q^{22} -\zeta_{12}^{3} q^{23} - q^{24} + \zeta_{12}^{2} q^{25} + \zeta_{12}^{5} q^{26} + \zeta_{12}^{3} q^{27} + \zeta_{12}^{4} q^{29} -\zeta_{12} q^{31} + \zeta_{12}^{5} q^{32} -\zeta_{12}^{4} q^{33} + \zeta_{12} q^{34} -\zeta_{12}^{2} q^{37} + \zeta_{12}^{2} q^{38} -\zeta_{12} q^{39} + \zeta_{12}^{2} q^{41} + \zeta_{12}^{5} q^{43} -\zeta_{12}^{3} q^{44} -\zeta_{12}^{4} q^{46} -\zeta_{12}^{3} q^{47} -\zeta_{12} q^{48} + \zeta_{12}^{3} q^{50} + \zeta_{12}^{3} q^{51} - q^{52} + \zeta_{12}^{4} q^{54} + \zeta_{12}^{4} q^{57} + \zeta_{12}^{5} q^{58} -\zeta_{12}^{3} q^{59} - q^{61} -\zeta_{12}^{2} q^{62} - q^{64} -\zeta_{12}^{5} q^{66} + \zeta_{12}^{2} q^{68} + q^{69} -\zeta_{12}^{5} q^{71} + q^{73} -\zeta_{12}^{3} q^{74} + \zeta_{12}^{5} q^{75} + \zeta_{12}^{3} q^{76} -\zeta_{12}^{2} q^{78} - q^{81} + \zeta_{12}^{3} q^{82} - q^{86} -\zeta_{12} q^{87} -\zeta_{12}^{4} q^{88} + q^{89} -\zeta_{12}^{5} q^{92} -\zeta_{12}^{4} q^{93} -\zeta_{12}^{4} q^{94} -\zeta_{12}^{2} q^{96} -\zeta_{12}^{2} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{4} - 2q^{6} + O(q^{10})$$ $$4q + 2q^{4} - 2q^{6} - 2q^{13} - 2q^{16} + 4q^{17} - 2q^{22} - 4q^{24} + 2q^{25} - 2q^{29} + 2q^{33} - 2q^{37} + 2q^{38} + 2q^{41} + 2q^{46} - 4q^{52} - 2q^{54} - 2q^{57} - 4q^{61} - 2q^{62} - 4q^{64} + 2q^{68} + 4q^{69} + 4q^{73} - 2q^{78} - 4q^{81} - 4q^{86} + 2q^{88} + 4q^{89} + 2q^{93} + 2q^{94} - 2q^{96} - 2q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3724\mathbb{Z}\right)^\times$$.

 $$n$$ $$1863$$ $$3041$$ $$3137$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{12}^{2}$$ $$\zeta_{12}^{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1831.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
−0.866025 0.500000i 1.00000i 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0 1.00000i 0 0
1831.2 0.866025 + 0.500000i 1.00000i 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0 1.00000i 0 0
2823.1 −0.866025 + 0.500000i 1.00000i 0.500000 0.866025i 0 −0.500000 0.866025i 0 1.00000i 0 0
2823.2 0.866025 0.500000i 1.00000i 0.500000 0.866025i 0 −0.500000 0.866025i 0 1.00000i 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
4.b odd 2 1 inner
133.g even 3 1 inner
532.n odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3724.1.n.a 4
4.b odd 2 1 inner 3724.1.n.a 4
7.b odd 2 1 532.1.n.a 4
7.c even 3 1 3724.1.bb.a 4
7.c even 3 1 3724.1.bk.a 4
7.d odd 6 1 532.1.bk.a yes 4
7.d odd 6 1 3724.1.bb.b 4
19.c even 3 1 3724.1.bk.a 4
28.d even 2 1 532.1.n.a 4
28.f even 6 1 532.1.bk.a yes 4
28.f even 6 1 3724.1.bb.b 4
28.g odd 6 1 3724.1.bb.a 4
28.g odd 6 1 3724.1.bk.a 4
76.g odd 6 1 3724.1.bk.a 4
133.g even 3 1 inner 3724.1.n.a 4
133.h even 3 1 3724.1.bb.a 4
133.k odd 6 1 532.1.n.a 4
133.m odd 6 1 532.1.bk.a yes 4
133.t odd 6 1 3724.1.bb.b 4
532.n odd 6 1 inner 3724.1.n.a 4
532.r even 6 1 3724.1.bb.b 4
532.u even 6 1 532.1.bk.a yes 4
532.bk odd 6 1 3724.1.bb.a 4
532.bn even 6 1 532.1.n.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
532.1.n.a 4 7.b odd 2 1
532.1.n.a 4 28.d even 2 1
532.1.n.a 4 133.k odd 6 1
532.1.n.a 4 532.bn even 6 1
532.1.bk.a yes 4 7.d odd 6 1
532.1.bk.a yes 4 28.f even 6 1
532.1.bk.a yes 4 133.m odd 6 1
532.1.bk.a yes 4 532.u even 6 1
3724.1.n.a 4 1.a even 1 1 trivial
3724.1.n.a 4 4.b odd 2 1 inner
3724.1.n.a 4 133.g even 3 1 inner
3724.1.n.a 4 532.n odd 6 1 inner
3724.1.bb.a 4 7.c even 3 1
3724.1.bb.a 4 28.g odd 6 1
3724.1.bb.a 4 133.h even 3 1
3724.1.bb.a 4 532.bk odd 6 1
3724.1.bb.b 4 7.d odd 6 1
3724.1.bb.b 4 28.f even 6 1
3724.1.bb.b 4 133.t odd 6 1
3724.1.bb.b 4 532.r even 6 1
3724.1.bk.a 4 7.c even 3 1
3724.1.bk.a 4 19.c even 3 1
3724.1.bk.a 4 28.g odd 6 1
3724.1.bk.a 4 76.g odd 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(3724, [\chi])$$.

## Hecke characteristic polynomials

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