# Properties

 Label 3969.1.bt.a Level $3969$ Weight $1$ Character orbit 3969.bt Analytic conductor $1.981$ Analytic rank $0$ Dimension $12$ Projective image $D_{7}$ 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.bt (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: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 147) Projective image: $$D_{7}$$ Projective field: Galois closure of 7.1.373714754427.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\zeta_{42}^{11} q^{4} + \zeta_{42}^{16} q^{7} +O(q^{10})$$ $$q -\zeta_{42}^{11} q^{4} + \zeta_{42}^{16} q^{7} + ( \zeta_{42}^{8} - \zeta_{42}^{17} ) q^{13} -\zeta_{42} q^{16} + ( -\zeta_{42}^{9} + \zeta_{42}^{12} ) q^{19} -\zeta_{42}^{19} q^{25} + \zeta_{42}^{6} q^{28} + ( -\zeta_{42}^{11} - \zeta_{42}^{17} ) q^{31} + ( 1 + \zeta_{42}^{6} ) q^{37} + ( \zeta_{42}^{10} - \zeta_{42}^{13} ) q^{43} -\zeta_{42}^{11} q^{49} + ( -\zeta_{42}^{7} - \zeta_{42}^{19} ) q^{52} + ( -\zeta_{42}^{7} - \zeta_{42}^{13} ) q^{61} + \zeta_{42}^{12} q^{64} + ( \zeta_{42}^{8} + \zeta_{42}^{20} ) q^{67} + ( -\zeta_{42}^{9} - \zeta_{42}^{15} ) q^{73} + ( \zeta_{42}^{2} + \zeta_{42}^{20} ) q^{76} + ( \zeta_{42}^{4} + \zeta_{42}^{10} ) q^{79} + ( -\zeta_{42}^{3} + \zeta_{42}^{12} ) q^{91} + ( \zeta_{42}^{16} - \zeta_{42}^{19} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + q^{4} + q^{7} + O(q^{10})$$ $$12 q + q^{4} + q^{7} + 2 q^{13} + q^{16} - 4 q^{19} + q^{25} - 2 q^{28} + 2 q^{31} + 10 q^{37} + 2 q^{43} + q^{49} - 5 q^{52} - 5 q^{61} - 2 q^{64} + 2 q^{67} - 4 q^{73} + 2 q^{76} + 2 q^{79} - 4 q^{91} + 2 q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$2108$$ $$3727$$ $$\chi(n)$$ $$-\zeta_{42}^{14}$$ $$\zeta_{42}^{12}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
134.1
 −0.988831 − 0.149042i 0.955573 + 0.294755i 0.365341 − 0.930874i 0.826239 + 0.563320i −0.733052 − 0.680173i −0.733052 + 0.680173i 0.826239 − 0.563320i 0.365341 + 0.930874i 0.955573 − 0.294755i −0.988831 + 0.149042i 0.0747301 + 0.997204i 0.0747301 − 0.997204i
0 0 0.0747301 0.997204i 0 0 −0.733052 + 0.680173i 0 0 0
512.1 0 0 −0.988831 0.149042i 0 0 0.0747301 0.997204i 0 0 0
701.1 0 0 0.826239 0.563320i 0 0 0.955573 0.294755i 0 0 0
1268.1 0 0 0.955573 + 0.294755i 0 0 −0.988831 0.149042i 0 0 0
1646.1 0 0 0.365341 0.930874i 0 0 0.826239 0.563320i 0 0 0
1835.1 0 0 0.365341 + 0.930874i 0 0 0.826239 + 0.563320i 0 0 0
2213.1 0 0 0.955573 0.294755i 0 0 −0.988831 + 0.149042i 0 0 0
2780.1 0 0 0.826239 + 0.563320i 0 0 0.955573 + 0.294755i 0 0 0
2969.1 0 0 −0.988831 + 0.149042i 0 0 0.0747301 + 0.997204i 0 0 0
3347.1 0 0 0.0747301 + 0.997204i 0 0 −0.733052 0.680173i 0 0 0
3536.1 0 0 −0.733052 0.680173i 0 0 0.365341 0.930874i 0 0 0
3914.1 0 0 −0.733052 + 0.680173i 0 0 0.365341 + 0.930874i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 3914.1 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.e even 7 1 inner
147.l odd 14 1 inner
441.ba even 21 1 inner
441.be 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.bt.a 12
3.b odd 2 1 CM 3969.1.bt.a 12
9.c even 3 1 147.1.l.a 6
9.c even 3 1 inner 3969.1.bt.a 12
9.d odd 6 1 147.1.l.a 6
9.d odd 6 1 inner 3969.1.bt.a 12
36.f odd 6 1 2352.1.cj.a 6
36.h even 6 1 2352.1.cj.a 6
45.h odd 6 1 3675.1.bm.a 6
45.j even 6 1 3675.1.bm.a 6
45.k odd 12 2 3675.1.bj.a 12
45.l even 12 2 3675.1.bj.a 12
49.e even 7 1 inner 3969.1.bt.a 12
63.g even 3 1 1029.1.n.b 12
63.h even 3 1 1029.1.n.b 12
63.i even 6 1 1029.1.n.a 12
63.j odd 6 1 1029.1.n.b 12
63.k odd 6 1 1029.1.n.a 12
63.l odd 6 1 1029.1.l.a 6
63.n odd 6 1 1029.1.n.b 12
63.o even 6 1 1029.1.l.a 6
63.s even 6 1 1029.1.n.a 12
63.t odd 6 1 1029.1.n.a 12
147.l odd 14 1 inner 3969.1.bt.a 12
441.y even 21 1 1029.1.n.b 12
441.z even 21 1 1029.1.n.b 12
441.ba even 21 1 147.1.l.a 6
441.ba even 21 1 inner 3969.1.bt.a 12
441.bc odd 42 1 1029.1.n.a 12
441.bd even 42 1 1029.1.n.a 12
441.be odd 42 1 147.1.l.a 6
441.be odd 42 1 inner 3969.1.bt.a 12
441.bh even 42 1 1029.1.l.a 6
441.bi odd 42 1 1029.1.n.b 12
441.bk odd 42 1 1029.1.l.a 6
441.bl odd 42 1 1029.1.n.a 12
441.bm odd 42 1 1029.1.n.b 12
441.bn even 42 1 1029.1.n.a 12
1764.cn even 42 1 2352.1.cj.a 6
1764.cs odd 42 1 2352.1.cj.a 6
2205.di even 42 1 3675.1.bm.a 6
2205.dp odd 42 1 3675.1.bm.a 6
2205.ec even 84 2 3675.1.bj.a 12
2205.eg odd 84 2 3675.1.bj.a 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.1.l.a 6 9.c even 3 1
147.1.l.a 6 9.d odd 6 1
147.1.l.a 6 441.ba even 21 1
147.1.l.a 6 441.be odd 42 1
1029.1.l.a 6 63.l odd 6 1
1029.1.l.a 6 63.o even 6 1
1029.1.l.a 6 441.bh even 42 1
1029.1.l.a 6 441.bk odd 42 1
1029.1.n.a 12 63.i even 6 1
1029.1.n.a 12 63.k odd 6 1
1029.1.n.a 12 63.s even 6 1
1029.1.n.a 12 63.t odd 6 1
1029.1.n.a 12 441.bc odd 42 1
1029.1.n.a 12 441.bd even 42 1
1029.1.n.a 12 441.bl odd 42 1
1029.1.n.a 12 441.bn even 42 1
1029.1.n.b 12 63.g even 3 1
1029.1.n.b 12 63.h even 3 1
1029.1.n.b 12 63.j odd 6 1
1029.1.n.b 12 63.n odd 6 1
1029.1.n.b 12 441.y even 21 1
1029.1.n.b 12 441.z even 21 1
1029.1.n.b 12 441.bi odd 42 1
1029.1.n.b 12 441.bm odd 42 1
2352.1.cj.a 6 36.f odd 6 1
2352.1.cj.a 6 36.h even 6 1
2352.1.cj.a 6 1764.cn even 42 1
2352.1.cj.a 6 1764.cs odd 42 1
3675.1.bj.a 12 45.k odd 12 2
3675.1.bj.a 12 45.l even 12 2
3675.1.bj.a 12 2205.ec even 84 2
3675.1.bj.a 12 2205.eg odd 84 2
3675.1.bm.a 6 45.h odd 6 1
3675.1.bm.a 6 45.j even 6 1
3675.1.bm.a 6 2205.di even 42 1
3675.1.bm.a 6 2205.dp odd 42 1
3969.1.bt.a 12 1.a even 1 1 trivial
3969.1.bt.a 12 3.b odd 2 1 CM
3969.1.bt.a 12 9.c even 3 1 inner
3969.1.bt.a 12 9.d odd 6 1 inner
3969.1.bt.a 12 49.e even 7 1 inner
3969.1.bt.a 12 147.l odd 14 1 inner
3969.1.bt.a 12 441.ba even 21 1 inner
3969.1.bt.a 12 441.be odd 42 1 inner

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$T^{12}$$
$5$ $$T^{12}$$
$7$ $$1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12}$$
$11$ $$T^{12}$$
$13$ $$1 + 3 T + 7 T^{2} + 8 T^{3} + 3 T^{4} - 28 T^{5} + T^{6} + 7 T^{7} + 12 T^{8} - 6 T^{9} - 2 T^{11} + T^{12}$$
$17$ $$T^{12}$$
$19$ $$( -1 - 2 T + T^{2} + T^{3} )^{4}$$
$23$ $$T^{12}$$
$29$ $$T^{12}$$
$31$ $$( 1 - 2 T + 5 T^{2} + 3 T^{4} - T^{5} + T^{6} )^{2}$$
$37$ $$( 1 - 3 T + 9 T^{2} - 13 T^{3} + 11 T^{4} - 5 T^{5} + T^{6} )^{2}$$
$41$ $$T^{12}$$
$43$ $$1 - 4 T + 7 T^{2} - 20 T^{3} + 45 T^{4} - 42 T^{5} + 22 T^{6} - 9 T^{8} + 8 T^{9} - 2 T^{11} + T^{12}$$
$47$ $$T^{12}$$
$53$ $$T^{12}$$
$59$ $$T^{12}$$
$61$ $$1 + 3 T + T^{3} + 31 T^{4} + 56 T^{5} + 57 T^{6} + 56 T^{7} + 47 T^{8} + 29 T^{9} + 14 T^{10} + 5 T^{11} + T^{12}$$
$67$ $$( 1 - 2 T + 5 T^{2} + 3 T^{4} - T^{5} + T^{6} )^{2}$$
$71$ $$T^{12}$$
$73$ $$( 1 + 4 T + 9 T^{2} + 8 T^{3} + 4 T^{4} + 2 T^{5} + T^{6} )^{2}$$
$79$ $$( 1 - 2 T + 5 T^{2} + 3 T^{4} - T^{5} + T^{6} )^{2}$$
$83$ $$T^{12}$$
$89$ $$T^{12}$$
$97$ $$( 1 - 2 T + 5 T^{2} + 3 T^{4} - T^{5} + T^{6} )^{2}$$