# Properties

 Label 3969.1.bz.a Level $3969$ Weight $1$ Character orbit 3969.bz Analytic conductor $1.981$ Analytic rank $0$ Dimension $12$ Projective image $D_{14}$ CM discriminant -3 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3969 = 3^{4} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3969.bz (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: $$x^{12} - x^{11} + x^{9} - x^{8} + x^{6} - x^{4} + x^{3} - x + 1$$ x^12 - x^11 + x^9 - x^8 + x^6 - x^4 + x^3 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 441) Projective image: $$D_{14}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{14} - \cdots)$$

## $q$-expansion

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

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

## Character values

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

 $$n$$ $$2108$$ $$3727$$ $$\chi(n)$$ $$-\zeta_{42}^{7}$$ $$-\zeta_{42}^{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
55.1
 −0.988831 − 0.149042i −0.988831 + 0.149042i 0.955573 − 0.294755i 0.826239 − 0.563320i −0.733052 + 0.680173i 0.365341 − 0.930874i 0.0747301 − 0.997204i 0.0747301 + 0.997204i 0.365341 + 0.930874i −0.733052 − 0.680173i 0.826239 + 0.563320i 0.955573 + 0.294755i
0 0 0.733052 0.680173i 0 0 −0.365341 0.930874i 0 0 0
433.1 0 0 0.733052 + 0.680173i 0 0 −0.365341 + 0.930874i 0 0 0
622.1 0 0 −0.0747301 0.997204i 0 0 0.733052 + 0.680173i 0 0 0
1000.1 0 0 0.988831 0.149042i 0 0 −0.0747301 0.997204i 0 0 0
1189.1 0 0 −0.826239 0.563320i 0 0 −0.955573 0.294755i 0 0 0
1756.1 0 0 −0.955573 + 0.294755i 0 0 0.988831 0.149042i 0 0 0
2134.1 0 0 −0.365341 0.930874i 0 0 −0.826239 0.563320i 0 0 0
2323.1 0 0 −0.365341 + 0.930874i 0 0 −0.826239 + 0.563320i 0 0 0
2701.1 0 0 −0.955573 0.294755i 0 0 0.988831 + 0.149042i 0 0 0
3268.1 0 0 −0.826239 + 0.563320i 0 0 −0.955573 + 0.294755i 0 0 0
3457.1 0 0 0.988831 + 0.149042i 0 0 −0.0747301 + 0.997204i 0 0 0
3835.1 0 0 −0.0747301 + 0.997204i 0 0 0.733052 0.680173i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 3835.1 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 $$\Q(\sqrt{-3})$$
9.c even 3 1 inner
9.d odd 6 1 inner
49.f odd 14 1 inner
147.k even 14 1 inner
441.bh even 42 1 inner
441.bk 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.bz.a 12
3.b odd 2 1 CM 3969.1.bz.a 12
9.c even 3 1 441.1.v.a 6
9.c even 3 1 inner 3969.1.bz.a 12
9.d odd 6 1 441.1.v.a 6
9.d odd 6 1 inner 3969.1.bz.a 12
49.f odd 14 1 inner 3969.1.bz.a 12
63.g even 3 1 3087.1.bj.b 12
63.h even 3 1 3087.1.bj.b 12
63.i even 6 1 3087.1.bj.a 12
63.j odd 6 1 3087.1.bj.b 12
63.k odd 6 1 3087.1.bj.a 12
63.l odd 6 1 3087.1.v.a 6
63.n odd 6 1 3087.1.bj.b 12
63.o even 6 1 3087.1.v.a 6
63.s even 6 1 3087.1.bj.a 12
63.t odd 6 1 3087.1.bj.a 12
147.k even 14 1 inner 3969.1.bz.a 12
441.y even 21 1 3087.1.bj.a 12
441.z even 21 1 3087.1.bj.a 12
441.ba even 21 1 3087.1.v.a 6
441.bc odd 42 1 3087.1.bj.b 12
441.bd even 42 1 3087.1.bj.b 12
441.be odd 42 1 3087.1.v.a 6
441.bh even 42 1 441.1.v.a 6
441.bh even 42 1 inner 3969.1.bz.a 12
441.bi odd 42 1 3087.1.bj.a 12
441.bk odd 42 1 441.1.v.a 6
441.bk odd 42 1 inner 3969.1.bz.a 12
441.bl odd 42 1 3087.1.bj.b 12
441.bm odd 42 1 3087.1.bj.a 12
441.bn even 42 1 3087.1.bj.b 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.1.v.a 6 9.c even 3 1
441.1.v.a 6 9.d odd 6 1
441.1.v.a 6 441.bh even 42 1
441.1.v.a 6 441.bk odd 42 1
3087.1.v.a 6 63.l odd 6 1
3087.1.v.a 6 63.o even 6 1
3087.1.v.a 6 441.ba even 21 1
3087.1.v.a 6 441.be odd 42 1
3087.1.bj.a 12 63.i even 6 1
3087.1.bj.a 12 63.k odd 6 1
3087.1.bj.a 12 63.s even 6 1
3087.1.bj.a 12 63.t odd 6 1
3087.1.bj.a 12 441.y even 21 1
3087.1.bj.a 12 441.z even 21 1
3087.1.bj.a 12 441.bi odd 42 1
3087.1.bj.a 12 441.bm odd 42 1
3087.1.bj.b 12 63.g even 3 1
3087.1.bj.b 12 63.h even 3 1
3087.1.bj.b 12 63.j odd 6 1
3087.1.bj.b 12 63.n odd 6 1
3087.1.bj.b 12 441.bc odd 42 1
3087.1.bj.b 12 441.bd even 42 1
3087.1.bj.b 12 441.bl odd 42 1
3087.1.bj.b 12 441.bn even 42 1
3969.1.bz.a 12 1.a even 1 1 trivial
3969.1.bz.a 12 3.b odd 2 1 CM
3969.1.bz.a 12 9.c even 3 1 inner
3969.1.bz.a 12 9.d odd 6 1 inner
3969.1.bz.a 12 49.f odd 14 1 inner
3969.1.bz.a 12 147.k even 14 1 inner
3969.1.bz.a 12 441.bh even 42 1 inner
3969.1.bz.a 12 441.bk odd 42 1 inner

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$T^{12}$$
$5$ $$T^{12}$$
$7$ $$T^{12} + T^{11} - T^{9} - T^{8} + T^{6} - T^{4} + \cdots + 1$$
$11$ $$T^{12}$$
$13$ $$T^{12} + 14 T^{9} + 7 T^{7} + 63 T^{6} + \cdots + 49$$
$17$ $$T^{12}$$
$19$ $$(T^{6} + 7 T^{4} + 14 T^{2} + 7)^{2}$$
$23$ $$T^{12}$$
$29$ $$T^{12}$$
$31$ $$T^{12} - 7 T^{10} + 35 T^{8} - 84 T^{6} + \cdots + 49$$
$37$ $$(T^{6} + 5 T^{5} + 11 T^{4} + 13 T^{3} + \cdots + 1)^{2}$$
$41$ $$T^{12}$$
$43$ $$T^{12} + 2 T^{11} - 8 T^{9} - 9 T^{8} + \cdots + 1$$
$47$ $$T^{12}$$
$53$ $$T^{12}$$
$59$ $$T^{12}$$
$61$ $$T^{12} - 7 T^{11} + 28 T^{10} - 77 T^{9} + \cdots + 49$$
$67$ $$(T^{6} - T^{5} + 3 T^{4} + 5 T^{2} - 2 T + 1)^{2}$$
$71$ $$T^{12}$$
$73$ $$(T^{6} + 7 T^{2} + 14 T + 7)^{2}$$
$79$ $$(T^{6} - T^{5} + 3 T^{4} + 5 T^{2} - 2 T + 1)^{2}$$
$83$ $$T^{12}$$
$89$ $$T^{12}$$
$97$ $$T^{12} - 7 T^{10} + 35 T^{8} - 84 T^{6} + \cdots + 49$$