# Properties

 Label 3267.1.w.b Level $3267$ Weight $1$ Character orbit 3267.w Analytic conductor $1.630$ Analytic rank $0$ Dimension $16$ Projective image $S_{4}$ CM/RM no Inner twists $16$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3267 = 3^{3} \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3267.w (of order $$30$$, degree $$8$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.63044539627$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$2$$ over $$\Q(\zeta_{30})$$ Coefficient field: 16.0.26873856000000000000.1 Defining polynomial: $$x^{16} + 2 x^{14} - 8 x^{10} - 16 x^{8} - 32 x^{6} + 128 x^{2} + 256$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1089) Projective image: $$S_{4}$$ Projective field: Galois closure of 4.2.107811.1

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + \beta_{2} q^{4} + ( 1 + \beta_{2} - \beta_{8} - \beta_{10} - \beta_{15} ) q^{5} + \beta_{7} q^{7} +O(q^{10})$$ $$q -\beta_{1} q^{2} + \beta_{2} q^{4} + ( 1 + \beta_{2} - \beta_{8} - \beta_{10} - \beta_{15} ) q^{5} + \beta_{7} q^{7} + \beta_{13} q^{10} -2 \beta_{8} q^{14} -\beta_{4} q^{16} + ( -\beta_{6} + \beta_{15} ) q^{20} + \beta_{9} q^{28} + \beta_{7} q^{29} + \beta_{15} q^{31} + \beta_{5} q^{32} + ( \beta_{1} - \beta_{11} ) q^{35} -\beta_{12} q^{37} + ( -\beta_{2} - \beta_{4} - \beta_{6} + \beta_{10} + \beta_{12} + \beta_{15} ) q^{47} + ( -1 - \beta_{2} + \beta_{8} + \beta_{10} + \beta_{15} ) q^{49} + \beta_{6} q^{53} -2 \beta_{8} q^{58} + \beta_{2} q^{59} + ( \beta_{1} + \beta_{3} - \beta_{5} - \beta_{7} + \beta_{13} + \beta_{14} ) q^{61} + ( -\beta_{1} - \beta_{3} + \beta_{9} + \beta_{11} - \beta_{13} ) q^{62} -\beta_{6} q^{64} + ( 1 - \beta_{10} ) q^{67} + ( -2 \beta_{2} + 2 \beta_{12} ) q^{70} + ( -1 - \beta_{2} - \beta_{4} + \beta_{8} + \beta_{10} + \beta_{15} ) q^{71} + ( -\beta_{1} - \beta_{3} + \beta_{9} + \beta_{11} - \beta_{13} ) q^{73} + ( \beta_{3} + \beta_{14} ) q^{74} + ( -\beta_{2} - \beta_{4} - \beta_{6} + \beta_{8} + \beta_{10} + \beta_{12} + \beta_{15} ) q^{80} + ( -\beta_{1} - \beta_{3} + \beta_{5} + \beta_{7} - \beta_{13} - \beta_{14} ) q^{83} + ( -\beta_{1} - \beta_{3} + \beta_{5} + \beta_{7} + \beta_{9} - \beta_{13} - \beta_{14} ) q^{94} + ( \beta_{6} - \beta_{15} ) q^{97} -\beta_{13} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q - 2 q^{4} + 2 q^{5} + O(q^{10})$$ $$16 q - 2 q^{4} + 2 q^{5} - 4 q^{14} - 2 q^{16} - 2 q^{20} + 2 q^{31} + 4 q^{37} + 2 q^{47} - 2 q^{49} + 4 q^{53} - 4 q^{58} - 2 q^{59} - 4 q^{64} + 8 q^{67} - 4 q^{70} - 4 q^{71} + 4 q^{80} + 2 q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 2 x^{14} - 8 x^{10} - 16 x^{8} - 32 x^{6} + 128 x^{2} + 256$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/2$$ $$\beta_{4}$$ $$=$$ $$\nu^{4}$$$$/4$$ $$\beta_{5}$$ $$=$$ $$\nu^{5}$$$$/4$$ $$\beta_{6}$$ $$=$$ $$\nu^{6}$$$$/8$$ $$\beta_{7}$$ $$=$$ $$\nu^{7}$$$$/8$$ $$\beta_{8}$$ $$=$$ $$\nu^{8}$$$$/16$$ $$\beta_{9}$$ $$=$$ $$\nu^{9}$$$$/16$$ $$\beta_{10}$$ $$=$$ $$\nu^{10}$$$$/32$$ $$\beta_{11}$$ $$=$$ $$\nu^{11}$$$$/32$$ $$\beta_{12}$$ $$=$$ $$\nu^{12}$$$$/64$$ $$\beta_{13}$$ $$=$$ $$\nu^{15}$$$$/128$$ $$\beta_{14}$$ $$=$$ $$($$$$\nu^{13} - 32 \nu^{3}$$$$)/64$$ $$\beta_{15}$$ $$=$$ $$($$$$\nu^{14} - 4 \nu^{10} - 8 \nu^{8} + 64 \nu^{2} + 128$$$$)/128$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{3}$$ $$\nu^{4}$$ $$=$$ $$4 \beta_{4}$$ $$\nu^{5}$$ $$=$$ $$4 \beta_{5}$$ $$\nu^{6}$$ $$=$$ $$8 \beta_{6}$$ $$\nu^{7}$$ $$=$$ $$8 \beta_{7}$$ $$\nu^{8}$$ $$=$$ $$16 \beta_{8}$$ $$\nu^{9}$$ $$=$$ $$16 \beta_{9}$$ $$\nu^{10}$$ $$=$$ $$32 \beta_{10}$$ $$\nu^{11}$$ $$=$$ $$32 \beta_{11}$$ $$\nu^{12}$$ $$=$$ $$64 \beta_{12}$$ $$\nu^{13}$$ $$=$$ $$64 \beta_{14} + 64 \beta_{3}$$ $$\nu^{14}$$ $$=$$ $$128 \beta_{15} + 128 \beta_{10} + 128 \beta_{8} - 128 \beta_{2} - 128$$ $$\nu^{15}$$ $$=$$ $$128 \beta_{13}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3267\mathbb{Z}\right)^\times$$.

 $$n$$ $$244$$ $$3026$$ $$\chi(n)$$ $$-\beta_{12}$$ $$-1 + \beta_{10}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
118.1
 0.575212 − 1.29195i −0.575212 + 1.29195i 1.40647 + 0.147826i −1.40647 − 0.147826i 1.05097 + 0.946294i −1.05097 − 0.946294i 1.40647 − 0.147826i −1.40647 + 0.147826i 0.575212 + 1.29195i −0.575212 − 1.29195i 0.294032 − 1.38331i −0.294032 + 1.38331i 0.294032 + 1.38331i −0.294032 − 1.38331i 1.05097 − 0.946294i −1.05097 + 0.946294i
−0.575212 + 1.29195i 0 −0.669131 0.743145i 0.913545 0.406737i 0 −0.294032 1.38331i 0 0 1.41421i
118.2 0.575212 1.29195i 0 −0.669131 0.743145i 0.913545 0.406737i 0 0.294032 + 1.38331i 0 0 1.41421i
766.1 −1.40647 0.147826i 0 0.978148 + 0.207912i −0.104528 0.994522i 0 1.05097 + 0.946294i 0 0 1.41421i
766.2 1.40647 + 0.147826i 0 0.978148 + 0.207912i −0.104528 0.994522i 0 −1.05097 0.946294i 0 0 1.41421i
820.1 −1.05097 0.946294i 0 0.104528 + 0.994522i 0.669131 + 0.743145i 0 0.575212 1.29195i 0 0 1.41421i
820.2 1.05097 + 0.946294i 0 0.104528 + 0.994522i 0.669131 + 0.743145i 0 −0.575212 + 1.29195i 0 0 1.41421i
1207.1 −1.40647 + 0.147826i 0 0.978148 0.207912i −0.104528 + 0.994522i 0 1.05097 0.946294i 0 0 1.41421i
1207.2 1.40647 0.147826i 0 0.978148 0.207912i −0.104528 + 0.994522i 0 −1.05097 + 0.946294i 0 0 1.41421i
1855.1 −0.575212 1.29195i 0 −0.669131 + 0.743145i 0.913545 + 0.406737i 0 −0.294032 + 1.38331i 0 0 1.41421i
1855.2 0.575212 + 1.29195i 0 −0.669131 + 0.743145i 0.913545 + 0.406737i 0 0.294032 1.38331i 0 0 1.41421i
1909.1 −0.294032 + 1.38331i 0 −0.913545 0.406737i −0.978148 + 0.207912i 0 −1.40647 + 0.147826i 0 0 1.41421i
1909.2 0.294032 1.38331i 0 −0.913545 0.406737i −0.978148 + 0.207912i 0 1.40647 0.147826i 0 0 1.41421i
1927.1 −0.294032 1.38331i 0 −0.913545 + 0.406737i −0.978148 0.207912i 0 −1.40647 0.147826i 0 0 1.41421i
1927.2 0.294032 + 1.38331i 0 −0.913545 + 0.406737i −0.978148 0.207912i 0 1.40647 + 0.147826i 0 0 1.41421i
3016.1 −1.05097 + 0.946294i 0 0.104528 0.994522i 0.669131 0.743145i 0 0.575212 + 1.29195i 0 0 1.41421i
3016.2 1.05097 0.946294i 0 0.104528 0.994522i 0.669131 0.743145i 0 −0.575212 1.29195i 0 0 1.41421i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 3016.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner
11.b odd 2 1 inner
11.c even 5 3 inner
11.d odd 10 3 inner
99.h odd 6 1 inner
99.m even 15 3 inner
99.o odd 30 3 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3267.1.w.b 16
3.b odd 2 1 1089.1.s.b 16
9.c even 3 1 inner 3267.1.w.b 16
9.d odd 6 1 1089.1.s.b 16
11.b odd 2 1 inner 3267.1.w.b 16
11.c even 5 1 3267.1.h.a 4
11.c even 5 3 inner 3267.1.w.b 16
11.d odd 10 1 3267.1.h.a 4
11.d odd 10 3 inner 3267.1.w.b 16
33.d even 2 1 1089.1.s.b 16
33.f even 10 1 1089.1.h.a 4
33.f even 10 3 1089.1.s.b 16
33.h odd 10 1 1089.1.h.a 4
33.h odd 10 3 1089.1.s.b 16
99.g even 6 1 1089.1.s.b 16
99.h odd 6 1 inner 3267.1.w.b 16
99.m even 15 1 3267.1.h.a 4
99.m even 15 3 inner 3267.1.w.b 16
99.n odd 30 1 1089.1.h.a 4
99.n odd 30 3 1089.1.s.b 16
99.o odd 30 1 3267.1.h.a 4
99.o odd 30 3 inner 3267.1.w.b 16
99.p even 30 1 1089.1.h.a 4
99.p even 30 3 1089.1.s.b 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1089.1.h.a 4 33.f even 10 1
1089.1.h.a 4 33.h odd 10 1
1089.1.h.a 4 99.n odd 30 1
1089.1.h.a 4 99.p even 30 1
1089.1.s.b 16 3.b odd 2 1
1089.1.s.b 16 9.d odd 6 1
1089.1.s.b 16 33.d even 2 1
1089.1.s.b 16 33.f even 10 3
1089.1.s.b 16 33.h odd 10 3
1089.1.s.b 16 99.g even 6 1
1089.1.s.b 16 99.n odd 30 3
1089.1.s.b 16 99.p even 30 3
3267.1.h.a 4 11.c even 5 1
3267.1.h.a 4 11.d odd 10 1
3267.1.h.a 4 99.m even 15 1
3267.1.h.a 4 99.o odd 30 1
3267.1.w.b 16 1.a even 1 1 trivial
3267.1.w.b 16 9.c even 3 1 inner
3267.1.w.b 16 11.b odd 2 1 inner
3267.1.w.b 16 11.c even 5 3 inner
3267.1.w.b 16 11.d odd 10 3 inner
3267.1.w.b 16 99.h odd 6 1 inner
3267.1.w.b 16 99.m even 15 3 inner
3267.1.w.b 16 99.o odd 30 3 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{16} + 2 T_{2}^{14} - 8 T_{2}^{10} - 16 T_{2}^{8} - 32 T_{2}^{6} + 128 T_{2}^{2} + 256$$ acting on $$S_{1}^{\mathrm{new}}(3267, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$256 + 128 T^{2} - 32 T^{6} - 16 T^{8} - 8 T^{10} + 2 T^{14} + T^{16}$$
$3$ $$T^{16}$$
$5$ $$( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2}$$
$7$ $$256 + 128 T^{2} - 32 T^{6} - 16 T^{8} - 8 T^{10} + 2 T^{14} + T^{16}$$
$11$ $$T^{16}$$
$13$ $$T^{16}$$
$17$ $$T^{16}$$
$19$ $$T^{16}$$
$23$ $$T^{16}$$
$29$ $$256 + 128 T^{2} - 32 T^{6} - 16 T^{8} - 8 T^{10} + 2 T^{14} + T^{16}$$
$31$ $$( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2}$$
$37$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{4}$$
$41$ $$T^{16}$$
$43$ $$T^{16}$$
$47$ $$( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2}$$
$53$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{4}$$
$59$ $$( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2}$$
$61$ $$256 + 128 T^{2} - 32 T^{6} - 16 T^{8} - 8 T^{10} + 2 T^{14} + T^{16}$$
$67$ $$( 1 - T + T^{2} )^{8}$$
$71$ $$( 1 + T + T^{2} + T^{3} + T^{4} )^{4}$$
$73$ $$( 16 - 8 T^{2} + 4 T^{4} - 2 T^{6} + T^{8} )^{2}$$
$79$ $$T^{16}$$
$83$ $$256 + 128 T^{2} - 32 T^{6} - 16 T^{8} - 8 T^{10} + 2 T^{14} + T^{16}$$
$89$ $$T^{16}$$
$97$ $$( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2}$$