# Properties

 Label 3969.1.cb.a Level $3969$ Weight $1$ Character orbit 3969.cb Analytic conductor $1.981$ Analytic rank $0$ Dimension $12$ Projective image $D_{21}$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3969 = 3^{4} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3969.cb (of order $$42$$, degree $$12$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.98078903514$$ 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: no (minimal twist has level 1323) 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{42}^{18} q^{4} + \zeta_{42}^{16} q^{7} +O(q^{10})$$ $$q + \zeta_{42}^{18} q^{4} + \zeta_{42}^{16} q^{7} + ( -\zeta_{42} - \zeta_{42}^{3} ) q^{13} -\zeta_{42}^{15} q^{16} + ( \zeta_{42}^{2} - \zeta_{42}^{5} ) q^{19} -\zeta_{42}^{5} q^{25} -\zeta_{42}^{13} q^{28} + ( \zeta_{42}^{4} - \zeta_{42}^{17} ) q^{31} + ( 1 - \zeta_{42}^{13} ) q^{37} + ( \zeta_{42}^{6} - \zeta_{42}^{17} ) q^{43} -\zeta_{42}^{11} q^{49} + ( 1 - \zeta_{42}^{19} ) q^{52} + ( -\zeta_{42}^{13} + \zeta_{42}^{14} ) q^{61} + \zeta_{42}^{12} q^{64} + ( \zeta_{42}^{8} - \zeta_{42}^{13} ) q^{67} + ( -\zeta_{42} + \zeta_{42}^{16} ) q^{73} + ( \zeta_{42}^{2} + \zeta_{42}^{20} ) q^{76} + ( \zeta_{42}^{10} - \zeta_{42}^{11} ) q^{79} + ( -\zeta_{42}^{17} - \zeta_{42}^{19} ) q^{91} + ( \zeta_{42}^{2} + \zeta_{42}^{12} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 2 q^{4} + q^{7} + O(q^{10})$$ $$12 q - 2 q^{4} + q^{7} - q^{13} - 2 q^{16} + 2 q^{19} + q^{25} + q^{28} + 2 q^{31} + 13 q^{37} - q^{43} + q^{49} + 13 q^{52} - 5 q^{61} - 2 q^{64} + 2 q^{67} + 2 q^{73} + 2 q^{76} + 2 q^{79} + 2 q^{91} - q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3969\mathbb{Z}\right)^\times$$.

 $$n$$ $$2108$$ $$3727$$ $$\chi(n)$$ $$-\zeta_{42}^{14}$$ $$-\zeta_{42}^{5}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
296.1
 −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 −0.900969 + 0.433884i 0 0 −0.733052 + 0.680173i 0 0 0
431.1 0 0 −0.222521 0.974928i 0 0 0.365341 + 0.930874i 0 0 0
1430.1 0 0 −0.900969 0.433884i 0 0 0.955573 0.294755i 0 0 0
1565.1 0 0 −0.222521 + 0.974928i 0 0 −0.988831 + 0.149042i 0 0 0
1997.1 0 0 0.623490 + 0.781831i 0 0 0.0747301 + 0.997204i 0 0 0
2132.1 0 0 −0.900969 0.433884i 0 0 −0.733052 0.680173i 0 0 0
2564.1 0 0 −0.222521 0.974928i 0 0 −0.988831 0.149042i 0 0 0
2699.1 0 0 0.623490 0.781831i 0 0 0.0747301 0.997204i 0 0 0
3131.1 0 0 −0.222521 + 0.974928i 0 0 0.365341 0.930874i 0 0 0
3266.1 0 0 0.623490 + 0.781831i 0 0 0.826239 0.563320i 0 0 0
3698.1 0 0 0.623490 0.781831i 0 0 0.826239 + 0.563320i 0 0 0
3833.1 0 0 −0.900969 + 0.433884i 0 0 0.955573 + 0.294755i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 3833.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
441.y even 21 1 inner
441.bm odd 42 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3969.1.cb.a 12
3.b odd 2 1 CM 3969.1.cb.a 12
9.c even 3 1 1323.1.br.a 12
9.c even 3 1 3969.1.bx.a 12
9.d odd 6 1 1323.1.br.a 12
9.d odd 6 1 3969.1.bx.a 12
49.g even 21 1 3969.1.bx.a 12
147.n odd 42 1 3969.1.bx.a 12
441.y even 21 1 inner 3969.1.cb.a 12
441.z even 21 1 1323.1.br.a 12
441.bi odd 42 1 1323.1.br.a 12
441.bm odd 42 1 inner 3969.1.cb.a 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1323.1.br.a 12 9.c even 3 1
1323.1.br.a 12 9.d odd 6 1
1323.1.br.a 12 441.z even 21 1
1323.1.br.a 12 441.bi odd 42 1
3969.1.bx.a 12 9.c even 3 1
3969.1.bx.a 12 9.d odd 6 1
3969.1.bx.a 12 49.g even 21 1
3969.1.bx.a 12 147.n odd 42 1
3969.1.cb.a 12 1.a even 1 1 trivial
3969.1.cb.a 12 3.b odd 2 1 CM
3969.1.cb.a 12 441.y even 21 1 inner
3969.1.cb.a 12 441.bm odd 42 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(3969, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$T^{12}$$
$5$ $$T^{12}$$
$7$ $$1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12}$$
$11$ $$T^{12}$$
$13$ $$1 + 15 T + 70 T^{2} + 104 T^{3} + 90 T^{4} + 35 T^{5} + 43 T^{6} + 7 T^{7} + 6 T^{8} + 6 T^{9} + T^{11} + T^{12}$$
$17$ $$T^{12}$$
$19$ $$( 1 - 2 T + 5 T^{2} + 3 T^{4} - T^{5} + T^{6} )^{2}$$
$23$ $$T^{12}$$
$29$ $$T^{12}$$
$31$ $$( 1 - 8 T + 8 T^{2} + 6 T^{3} - 6 T^{4} - T^{5} + T^{6} )^{2}$$
$37$ $$1 - 6 T + 63 T^{2} - 260 T^{3} + 643 T^{4} - 1078 T^{5} + 1275 T^{6} - 1078 T^{7} + 650 T^{8} - 274 T^{9} + 77 T^{10} - 13 T^{11} + T^{12}$$
$41$ $$T^{12}$$
$43$ $$1 - 13 T + 49 T^{2} - 29 T^{3} + 69 T^{4} - 20 T^{6} - 21 T^{7} + 6 T^{8} - T^{9} + T^{11} + T^{12}$$
$47$ $$T^{12}$$
$53$ $$T^{12}$$
$59$ $$T^{12}$$
$61$ $$1 + 12 T + 45 T^{2} + 10 T^{3} + 61 T^{4} + 92 T^{5} + 105 T^{6} + 92 T^{7} + 68 T^{8} + 38 T^{9} + 17 T^{10} + 5 T^{11} + T^{12}$$
$67$ $$( 1 - 8 T + 8 T^{2} + 6 T^{3} - 6 T^{4} - T^{5} + T^{6} )^{2}$$
$71$ $$T^{12}$$
$73$ $$1 - 4 T + 7 T^{2} - 20 T^{3} + 45 T^{4} - 42 T^{5} + 22 T^{6} - 9 T^{8} + 8 T^{9} - 2 T^{11} + T^{12}$$
$79$ $$( 1 - 8 T + 8 T^{2} + 6 T^{3} - 6 T^{4} - T^{5} + T^{6} )^{2}$$
$83$ $$T^{12}$$
$89$ $$T^{12}$$
$97$ $$1 + 8 T + 56 T^{2} + 76 T^{3} + 118 T^{4} + 49 T^{5} + 78 T^{6} + 28 T^{7} + 34 T^{8} + 6 T^{9} + 7 T^{10} + T^{11} + T^{12}$$