# Properties

 Label 3724.1.dj.a Level $3724$ Weight $1$ Character orbit 3724.dj Analytic conductor $1.859$ Analytic rank $0$ Dimension $12$ Projective image $D_{21}$ CM discriminant -19 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3724 = 2^{2} \cdot 7^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3724.dj (of order $$42$$, degree $$12$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.85851810705$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\Q(\zeta_{21})$$ Defining polynomial: $$x^{12} - x^{11} + x^{9} - x^{8} + x^{6} - x^{4} + x^{3} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{21}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{21} - \cdots)$$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{42} - \zeta_{42}^{3} ) q^{5} -\zeta_{42}^{9} q^{7} -\zeta_{42}^{11} q^{9} +O(q^{10})$$ $$q + ( -\zeta_{42} - \zeta_{42}^{3} ) q^{5} -\zeta_{42}^{9} q^{7} -\zeta_{42}^{11} q^{9} + ( -\zeta_{42}^{7} - \zeta_{42}^{13} ) q^{11} + ( -\zeta_{42}^{9} + \zeta_{42}^{14} ) q^{17} + \zeta_{42}^{14} q^{19} + ( -\zeta_{42}^{9} + \zeta_{42}^{10} ) q^{23} + ( \zeta_{42}^{2} + \zeta_{42}^{4} + \zeta_{42}^{6} ) q^{25} + ( \zeta_{42}^{10} + \zeta_{42}^{12} ) q^{35} + ( \zeta_{42}^{4} + \zeta_{42}^{20} ) q^{43} + ( \zeta_{42}^{12} + \zeta_{42}^{14} ) q^{45} + ( -\zeta_{42}^{5} + \zeta_{42}^{8} ) q^{47} + \zeta_{42}^{18} q^{49} + ( \zeta_{42}^{8} + \zeta_{42}^{10} + \zeta_{42}^{14} + \zeta_{42}^{16} ) q^{55} + ( -\zeta_{42}^{5} - \zeta_{42}^{11} ) q^{61} + \zeta_{42}^{20} q^{63} + 2 \zeta_{42}^{4} q^{73} + ( -\zeta_{42} + \zeta_{42}^{16} ) q^{77} -\zeta_{42} q^{81} + ( -\zeta_{42}^{17} - \zeta_{42}^{19} ) q^{83} + ( \zeta_{42}^{10} + \zeta_{42}^{12} - \zeta_{42}^{15} - \zeta_{42}^{17} ) q^{85} + ( -\zeta_{42}^{15} - \zeta_{42}^{17} ) q^{95} + ( -\zeta_{42}^{3} + \zeta_{42}^{18} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q - q^{5} - 2q^{7} + q^{9} + O(q^{10})$$ $$12q - q^{5} - 2q^{7} + q^{9} - 5q^{11} - 8q^{17} - 6q^{19} - q^{23} - q^{35} + 2q^{43} - 8q^{45} + 2q^{47} - 2q^{49} - 3q^{55} + 2q^{61} + q^{63} + 2q^{73} + 2q^{77} + q^{81} + 2q^{83} - 2q^{85} - q^{95} - 4q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$1863$$ $$3041$$ $$3137$$ $$\chi(n)$$ $$1$$ $$-\zeta_{42}^{11}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1
 −0.988831 − 0.149042i 0.0747301 − 0.997204i 0.365341 − 0.930874i 0.826239 − 0.563320i 0.955573 − 0.294755i −0.988831 + 0.149042i 0.826239 + 0.563320i 0.955573 + 0.294755i 0.0747301 + 0.997204i −0.733052 − 0.680173i −0.733052 + 0.680173i 0.365341 + 0.930874i
0 0 0 −1.88980 0.582926i 0 −0.222521 0.974928i 0 0.0747301 0.997204i 0
417.1 0 0 0 −0.147791 0.0222759i 0 0.623490 0.781831i 0 −0.733052 + 0.680173i 0
1101.1 0 0 0 −0.535628 0.496990i 0 −0.222521 + 0.974928i 0 0.826239 0.563320i 0
1481.1 0 0 0 0.603718 1.53825i 0 0.623490 + 0.781831i 0 0.955573 0.294755i 0
1633.1 0 0 0 1.57906 1.07659i 0 −0.900969 0.433884i 0 −0.988831 + 0.149042i 0
2013.1 0 0 0 −1.88980 + 0.582926i 0 −0.222521 + 0.974928i 0 0.0747301 + 0.997204i 0
2165.1 0 0 0 0.603718 + 1.53825i 0 0.623490 0.781831i 0 0.955573 + 0.294755i 0
2545.1 0 0 0 1.57906 + 1.07659i 0 −0.900969 + 0.433884i 0 −0.988831 0.149042i 0
2697.1 0 0 0 −0.147791 + 0.0222759i 0 0.623490 + 0.781831i 0 −0.733052 0.680173i 0
3077.1 0 0 0 −0.109562 1.46200i 0 −0.900969 0.433884i 0 0.365341 0.930874i 0
3229.1 0 0 0 −0.109562 + 1.46200i 0 −0.900969 + 0.433884i 0 0.365341 + 0.930874i 0
3609.1 0 0 0 −0.535628 + 0.496990i 0 −0.222521 0.974928i 0 0.826239 + 0.563320i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 3609.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.b odd 2 1 CM by $$\Q(\sqrt{-19})$$
49.g even 21 1 inner
931.bq odd 42 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3724.1.dj.a 12
19.b odd 2 1 CM 3724.1.dj.a 12
49.g even 21 1 inner 3724.1.dj.a 12
931.bq odd 42 1 inner 3724.1.dj.a 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3724.1.dj.a 12 1.a even 1 1 trivial
3724.1.dj.a 12 19.b odd 2 1 CM
3724.1.dj.a 12 49.g even 21 1 inner
3724.1.dj.a 12 931.bq odd 42 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{12} + \cdots$$ acting on $$S_{1}^{\mathrm{new}}(3724, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$T^{12}$$
$5$ $$1 + 15 T + 70 T^{2} + 104 T^{3} + 90 T^{4} + 35 T^{5} + 43 T^{6} + 7 T^{7} + 6 T^{8} + 6 T^{9} + T^{11} + T^{12}$$
$7$ $$( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}$$
$11$ $$1 + 3 T + T^{3} + 31 T^{4} + 56 T^{5} + 57 T^{6} + 56 T^{7} + 47 T^{8} + 29 T^{9} + 14 T^{10} + 5 T^{11} + T^{12}$$
$13$ $$T^{12}$$
$17$ $$1 - 6 T + 118 T^{3} + 349 T^{4} + 518 T^{5} + 519 T^{6} + 392 T^{7} + 230 T^{8} + 104 T^{9} + 35 T^{10} + 8 T^{11} + T^{12}$$
$19$ $$( 1 + T + T^{2} )^{6}$$
$23$ $$1 + 15 T + 70 T^{2} + 104 T^{3} + 90 T^{4} + 35 T^{5} + 43 T^{6} + 7 T^{7} + 6 T^{8} + 6 T^{9} + T^{11} + T^{12}$$
$29$ $$T^{12}$$
$31$ $$T^{12}$$
$37$ $$T^{12}$$
$41$ $$T^{12}$$
$43$ $$1 - 9 T + 52 T^{2} - 46 T^{3} + 96 T^{4} - 97 T^{5} + 49 T^{6} + T^{7} - 9 T^{8} + 3 T^{9} + 3 T^{10} - 2 T^{11} + T^{12}$$
$47$ $$1 + 3 T + 7 T^{2} + 8 T^{3} + 3 T^{4} - 28 T^{5} + T^{6} + 7 T^{7} + 12 T^{8} - 6 T^{9} - 2 T^{11} + T^{12}$$
$53$ $$T^{12}$$
$59$ $$T^{12}$$
$61$ $$1 + 3 T + 7 T^{2} + 8 T^{3} + 3 T^{4} - 28 T^{5} + T^{6} + 7 T^{7} + 12 T^{8} - 6 T^{9} - 2 T^{11} + T^{12}$$
$67$ $$T^{12}$$
$71$ $$T^{12}$$
$73$ $$4096 - 2048 T + 512 T^{3} - 256 T^{4} + 64 T^{6} - 16 T^{8} + 8 T^{9} - 2 T^{11} + T^{12}$$
$79$ $$T^{12}$$
$83$ $$1 + 5 T + 52 T^{2} + 94 T^{3} + 54 T^{4} - 6 T^{5} + 7 T^{6} - 6 T^{7} + 12 T^{8} - 4 T^{9} + 3 T^{10} - 2 T^{11} + T^{12}$$
$89$ $$T^{12}$$
$97$ $$T^{12}$$