# Properties

 Label 2646.2.e.n Level $2646$ Weight $2$ Character orbit 2646.e Analytic conductor $21.128$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2646 = 2 \cdot 3^{3} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2646.e (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$21.1284163748$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-11})$$ Defining polynomial: $$x^{4} - x^{3} - 2 x^{2} - 3 x + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 126) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + ( \beta_{1} + \beta_{3} ) q^{5} + q^{8} +O(q^{10})$$ $$q + q^{2} + q^{4} + ( \beta_{1} + \beta_{3} ) q^{5} + q^{8} + ( \beta_{1} + \beta_{3} ) q^{10} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{11} + ( -2 + 2 \beta_{1} ) q^{13} + q^{16} + ( -2 \beta_{1} + \beta_{3} ) q^{17} + ( -5 + 5 \beta_{1} ) q^{19} + ( \beta_{1} + \beta_{3} ) q^{20} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{22} + ( -5 \beta_{1} + \beta_{3} ) q^{23} + ( -7 + 4 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{25} + ( -2 + 2 \beta_{1} ) q^{26} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{29} + 2 q^{31} + q^{32} + ( -2 \beta_{1} + \beta_{3} ) q^{34} + ( -2 + 2 \beta_{1} ) q^{37} + ( -5 + 5 \beta_{1} ) q^{38} + ( \beta_{1} + \beta_{3} ) q^{40} + ( 7 - 8 \beta_{1} - \beta_{2} + \beta_{3} ) q^{41} + ( -2 \beta_{1} + 3 \beta_{3} ) q^{43} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{44} + ( -5 \beta_{1} + \beta_{3} ) q^{46} + ( -7 + 4 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{50} + ( -2 + 2 \beta_{1} ) q^{52} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{53} -6 q^{55} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{58} + ( 3 + 3 \beta_{2} ) q^{59} + ( -4 + 3 \beta_{2} ) q^{61} + 2 q^{62} + q^{64} + ( -4 - 2 \beta_{2} ) q^{65} + ( 5 - 3 \beta_{2} ) q^{67} + ( -2 \beta_{1} + \beta_{3} ) q^{68} + 3 \beta_{2} q^{71} + ( -2 \beta_{1} - 3 \beta_{3} ) q^{73} + ( -2 + 2 \beta_{1} ) q^{74} + ( -5 + 5 \beta_{1} ) q^{76} + ( 2 - 3 \beta_{2} ) q^{79} + ( \beta_{1} + \beta_{3} ) q^{80} + ( 7 - 8 \beta_{1} - \beta_{2} + \beta_{3} ) q^{82} + ( 4 \beta_{1} + 4 \beta_{3} ) q^{83} + ( -6 + 6 \beta_{1} ) q^{85} + ( -2 \beta_{1} + 3 \beta_{3} ) q^{86} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{88} + ( 10 - 8 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{89} + ( -5 \beta_{1} + \beta_{3} ) q^{92} + ( -10 - 5 \beta_{2} ) q^{95} + ( -2 \beta_{1} + 3 \beta_{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{2} + 4 q^{4} + 3 q^{5} + 4 q^{8} + O(q^{10})$$ $$4 q + 4 q^{2} + 4 q^{4} + 3 q^{5} + 4 q^{8} + 3 q^{10} + 3 q^{11} - 4 q^{13} + 4 q^{16} - 3 q^{17} - 10 q^{19} + 3 q^{20} + 3 q^{22} - 9 q^{23} - 11 q^{25} - 4 q^{26} - 6 q^{29} + 8 q^{31} + 4 q^{32} - 3 q^{34} - 4 q^{37} - 10 q^{38} + 3 q^{40} + 15 q^{41} - q^{43} + 3 q^{44} - 9 q^{46} - 11 q^{50} - 4 q^{52} - 6 q^{53} - 24 q^{55} - 6 q^{58} + 6 q^{59} - 22 q^{61} + 8 q^{62} + 4 q^{64} - 12 q^{65} + 26 q^{67} - 3 q^{68} - 6 q^{71} - 7 q^{73} - 4 q^{74} - 10 q^{76} + 14 q^{79} + 3 q^{80} + 15 q^{82} + 12 q^{83} - 12 q^{85} - q^{86} + 3 q^{88} + 18 q^{89} - 9 q^{92} - 30 q^{95} - q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 2 x^{2} - 3 x + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 2 \nu^{2} - 2 \nu - 3$$$$)/6$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} + \nu^{2} + 5 \nu$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$($$$$2 \nu^{3} + \nu^{2} + 2 \nu - 9$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} - 2 \beta_{1} + 2$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{3} + 2 \beta_{2} + 8 \beta_{1} + 1$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$4 \beta_{3} - 2 \beta_{2} - 2 \beta_{1} + 11$$$$)/3$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2646\mathbb{Z}\right)^\times$$.

 $$n$$ $$785$$ $$1081$$ $$\chi(n)$$ $$-1 + \beta_{1}$$ $$-1 + \beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1549.1
 −1.18614 − 1.26217i 1.68614 + 0.396143i −1.18614 + 1.26217i 1.68614 − 0.396143i
1.00000 0 1.00000 −0.686141 1.18843i 0 0 1.00000 0 −0.686141 1.18843i
1549.2 1.00000 0 1.00000 2.18614 + 3.78651i 0 0 1.00000 0 2.18614 + 3.78651i
2125.1 1.00000 0 1.00000 −0.686141 + 1.18843i 0 0 1.00000 0 −0.686141 + 1.18843i
2125.2 1.00000 0 1.00000 2.18614 3.78651i 0 0 1.00000 0 2.18614 3.78651i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.h even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2646.2.e.n 4
3.b odd 2 1 882.2.e.l 4
7.b odd 2 1 2646.2.e.m 4
7.c even 3 1 378.2.f.c 4
7.c even 3 1 2646.2.h.k 4
7.d odd 6 1 2646.2.f.j 4
7.d odd 6 1 2646.2.h.l 4
9.c even 3 1 2646.2.h.k 4
9.d odd 6 1 882.2.h.m 4
21.c even 2 1 882.2.e.k 4
21.g even 6 1 882.2.f.k 4
21.g even 6 1 882.2.h.n 4
21.h odd 6 1 126.2.f.d 4
21.h odd 6 1 882.2.h.m 4
28.g odd 6 1 3024.2.r.f 4
63.g even 3 1 378.2.f.c 4
63.h even 3 1 1134.2.a.n 2
63.h even 3 1 inner 2646.2.e.n 4
63.i even 6 1 882.2.e.k 4
63.i even 6 1 7938.2.a.bh 2
63.j odd 6 1 882.2.e.l 4
63.j odd 6 1 1134.2.a.k 2
63.k odd 6 1 2646.2.f.j 4
63.l odd 6 1 2646.2.h.l 4
63.n odd 6 1 126.2.f.d 4
63.o even 6 1 882.2.h.n 4
63.s even 6 1 882.2.f.k 4
63.t odd 6 1 2646.2.e.m 4
63.t odd 6 1 7938.2.a.bs 2
84.n even 6 1 1008.2.r.f 4
252.o even 6 1 1008.2.r.f 4
252.u odd 6 1 9072.2.a.bb 2
252.bb even 6 1 9072.2.a.bm 2
252.bl odd 6 1 3024.2.r.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.f.d 4 21.h odd 6 1
126.2.f.d 4 63.n odd 6 1
378.2.f.c 4 7.c even 3 1
378.2.f.c 4 63.g even 3 1
882.2.e.k 4 21.c even 2 1
882.2.e.k 4 63.i even 6 1
882.2.e.l 4 3.b odd 2 1
882.2.e.l 4 63.j odd 6 1
882.2.f.k 4 21.g even 6 1
882.2.f.k 4 63.s even 6 1
882.2.h.m 4 9.d odd 6 1
882.2.h.m 4 21.h odd 6 1
882.2.h.n 4 21.g even 6 1
882.2.h.n 4 63.o even 6 1
1008.2.r.f 4 84.n even 6 1
1008.2.r.f 4 252.o even 6 1
1134.2.a.k 2 63.j odd 6 1
1134.2.a.n 2 63.h even 3 1
2646.2.e.m 4 7.b odd 2 1
2646.2.e.m 4 63.t odd 6 1
2646.2.e.n 4 1.a even 1 1 trivial
2646.2.e.n 4 63.h even 3 1 inner
2646.2.f.j 4 7.d odd 6 1
2646.2.f.j 4 63.k odd 6 1
2646.2.h.k 4 7.c even 3 1
2646.2.h.k 4 9.c even 3 1
2646.2.h.l 4 7.d odd 6 1
2646.2.h.l 4 63.l odd 6 1
3024.2.r.f 4 28.g odd 6 1
3024.2.r.f 4 252.bl odd 6 1
7938.2.a.bh 2 63.i even 6 1
7938.2.a.bs 2 63.t odd 6 1
9072.2.a.bb 2 252.u odd 6 1
9072.2.a.bm 2 252.bb even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2646, [\chi])$$:

 $$T_{5}^{4} - 3 T_{5}^{3} + 15 T_{5}^{2} + 18 T_{5} + 36$$ $$T_{11}^{4} - 3 T_{11}^{3} + 15 T_{11}^{2} + 18 T_{11} + 36$$ $$T_{13}^{2} + 2 T_{13} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{4}$$
$3$ $$T^{4}$$
$5$ $$36 + 18 T + 15 T^{2} - 3 T^{3} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$36 + 18 T + 15 T^{2} - 3 T^{3} + T^{4}$$
$13$ $$( 4 + 2 T + T^{2} )^{2}$$
$17$ $$36 - 18 T + 15 T^{2} + 3 T^{3} + T^{4}$$
$19$ $$( 25 + 5 T + T^{2} )^{2}$$
$23$ $$144 + 108 T + 69 T^{2} + 9 T^{3} + T^{4}$$
$29$ $$576 - 144 T + 60 T^{2} + 6 T^{3} + T^{4}$$
$31$ $$( -2 + T )^{4}$$
$37$ $$( 4 + 2 T + T^{2} )^{2}$$
$41$ $$2304 - 720 T + 177 T^{2} - 15 T^{3} + T^{4}$$
$43$ $$5476 - 74 T + 75 T^{2} + T^{3} + T^{4}$$
$47$ $$T^{4}$$
$53$ $$576 - 144 T + 60 T^{2} + 6 T^{3} + T^{4}$$
$59$ $$( -72 - 3 T + T^{2} )^{2}$$
$61$ $$( -44 + 11 T + T^{2} )^{2}$$
$67$ $$( -32 - 13 T + T^{2} )^{2}$$
$71$ $$( -72 + 3 T + T^{2} )^{2}$$
$73$ $$3844 - 434 T + 111 T^{2} + 7 T^{3} + T^{4}$$
$79$ $$( -62 - 7 T + T^{2} )^{2}$$
$83$ $$9216 + 1152 T + 240 T^{2} - 12 T^{3} + T^{4}$$
$89$ $$2304 - 864 T + 276 T^{2} - 18 T^{3} + T^{4}$$
$97$ $$5476 - 74 T + 75 T^{2} + T^{3} + T^{4}$$