# Properties

 Label 2646.2.e.l 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: $$\Q(\sqrt{-2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} - 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ 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.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + ( \beta_{1} - \beta_{2} ) q^{5} - q^{8} +O(q^{10})$$ $$q - q^{2} + q^{4} + ( \beta_{1} - \beta_{2} ) q^{5} - q^{8} + ( -\beta_{1} + \beta_{2} ) q^{10} + ( 2 - 2 \beta_{1} ) q^{11} + ( 2 \beta_{2} - 2 \beta_{3} ) q^{13} + q^{16} + 2 \beta_{1} q^{17} + ( -5 + 5 \beta_{1} - \beta_{2} + \beta_{3} ) q^{19} + ( \beta_{1} - \beta_{2} ) q^{20} + ( -2 + 2 \beta_{1} ) q^{22} -\beta_{1} q^{23} + ( -2 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{25} + ( -2 \beta_{2} + 2 \beta_{3} ) q^{26} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{29} + 6 q^{31} - q^{32} -2 \beta_{1} q^{34} + ( -2 + 2 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{37} + ( 5 - 5 \beta_{1} + \beta_{2} - \beta_{3} ) q^{38} + ( -\beta_{1} + \beta_{2} ) q^{40} + ( -4 \beta_{2} + 4 \beta_{3} ) q^{41} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{43} + ( 2 - 2 \beta_{1} ) q^{44} + \beta_{1} q^{46} -4 \beta_{3} q^{47} + ( 2 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{50} + ( 2 \beta_{2} - 2 \beta_{3} ) q^{52} + ( -6 \beta_{1} - 2 \beta_{2} ) q^{53} + ( 2 - 2 \beta_{3} ) q^{55} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{58} + 2 q^{59} + ( 9 - \beta_{3} ) q^{61} -6 q^{62} + q^{64} + ( 12 - 2 \beta_{3} ) q^{65} + ( -8 - 2 \beta_{3} ) q^{67} + 2 \beta_{1} q^{68} + ( -5 + 2 \beta_{3} ) q^{71} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{73} + ( 2 - 2 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} ) q^{74} + ( -5 + 5 \beta_{1} - \beta_{2} + \beta_{3} ) q^{76} + ( 3 - 2 \beta_{3} ) q^{79} + ( \beta_{1} - \beta_{2} ) q^{80} + ( 4 \beta_{2} - 4 \beta_{3} ) q^{82} + 2 \beta_{1} q^{83} + ( -2 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{85} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{86} + ( -2 + 2 \beta_{1} ) q^{88} + ( -12 + 12 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{89} -\beta_{1} q^{92} + 4 \beta_{3} q^{94} + ( -11 + 6 \beta_{3} ) q^{95} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{2} + 4q^{4} + 2q^{5} - 4q^{8} + O(q^{10})$$ $$4q - 4q^{2} + 4q^{4} + 2q^{5} - 4q^{8} - 2q^{10} + 4q^{11} + 4q^{16} + 4q^{17} - 10q^{19} + 2q^{20} - 4q^{22} - 2q^{23} - 4q^{25} - 4q^{29} + 24q^{31} - 4q^{32} - 4q^{34} - 4q^{37} + 10q^{38} - 2q^{40} - 4q^{43} + 4q^{44} + 2q^{46} + 4q^{50} - 12q^{53} + 8q^{55} + 4q^{58} + 8q^{59} + 36q^{61} - 24q^{62} + 4q^{64} + 48q^{65} - 32q^{67} + 4q^{68} - 20q^{71} + 4q^{73} + 4q^{74} - 10q^{76} + 12q^{79} + 2q^{80} + 4q^{83} - 4q^{85} + 4q^{86} - 4q^{88} - 24q^{89} - 2q^{92} - 44q^{95} + 4q^{97} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 2 \nu$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{3} + 4 \nu$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2}$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$($$$$-2 \beta_{3} + 4 \beta_{2}$$$$)/3$$

## 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
 1.22474 + 0.707107i −1.22474 − 0.707107i 1.22474 − 0.707107i −1.22474 + 0.707107i
−1.00000 0 1.00000 −0.724745 1.25529i 0 0 −1.00000 0 0.724745 + 1.25529i
1549.2 −1.00000 0 1.00000 1.72474 + 2.98735i 0 0 −1.00000 0 −1.72474 2.98735i
2125.1 −1.00000 0 1.00000 −0.724745 + 1.25529i 0 0 −1.00000 0 0.724745 1.25529i
2125.2 −1.00000 0 1.00000 1.72474 2.98735i 0 0 −1.00000 0 −1.72474 + 2.98735i
 $$n$$: 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.l 4
3.b odd 2 1 882.2.e.m 4
7.b odd 2 1 2646.2.e.k 4
7.c even 3 1 378.2.f.d 4
7.c even 3 1 2646.2.h.m 4
7.d odd 6 1 2646.2.f.k 4
7.d odd 6 1 2646.2.h.n 4
9.c even 3 1 2646.2.h.m 4
9.d odd 6 1 882.2.h.k 4
21.c even 2 1 882.2.e.n 4
21.g even 6 1 882.2.f.j 4
21.g even 6 1 882.2.h.l 4
21.h odd 6 1 126.2.f.c 4
21.h odd 6 1 882.2.h.k 4
28.g odd 6 1 3024.2.r.e 4
63.g even 3 1 378.2.f.d 4
63.h even 3 1 1134.2.a.i 2
63.h even 3 1 inner 2646.2.e.l 4
63.i even 6 1 882.2.e.n 4
63.i even 6 1 7938.2.a.bn 2
63.j odd 6 1 882.2.e.m 4
63.j odd 6 1 1134.2.a.p 2
63.k odd 6 1 2646.2.f.k 4
63.l odd 6 1 2646.2.h.n 4
63.n odd 6 1 126.2.f.c 4
63.o even 6 1 882.2.h.l 4
63.s even 6 1 882.2.f.j 4
63.t odd 6 1 2646.2.e.k 4
63.t odd 6 1 7938.2.a.bm 2
84.n even 6 1 1008.2.r.e 4
252.o even 6 1 1008.2.r.e 4
252.u odd 6 1 9072.2.a.bd 2
252.bb even 6 1 9072.2.a.bk 2
252.bl odd 6 1 3024.2.r.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.f.c 4 21.h odd 6 1
126.2.f.c 4 63.n odd 6 1
378.2.f.d 4 7.c even 3 1
378.2.f.d 4 63.g even 3 1
882.2.e.m 4 3.b odd 2 1
882.2.e.m 4 63.j odd 6 1
882.2.e.n 4 21.c even 2 1
882.2.e.n 4 63.i even 6 1
882.2.f.j 4 21.g even 6 1
882.2.f.j 4 63.s even 6 1
882.2.h.k 4 9.d odd 6 1
882.2.h.k 4 21.h odd 6 1
882.2.h.l 4 21.g even 6 1
882.2.h.l 4 63.o even 6 1
1008.2.r.e 4 84.n even 6 1
1008.2.r.e 4 252.o even 6 1
1134.2.a.i 2 63.h even 3 1
1134.2.a.p 2 63.j odd 6 1
2646.2.e.k 4 7.b odd 2 1
2646.2.e.k 4 63.t odd 6 1
2646.2.e.l 4 1.a even 1 1 trivial
2646.2.e.l 4 63.h even 3 1 inner
2646.2.f.k 4 7.d odd 6 1
2646.2.f.k 4 63.k odd 6 1
2646.2.h.m 4 7.c even 3 1
2646.2.h.m 4 9.c even 3 1
2646.2.h.n 4 7.d odd 6 1
2646.2.h.n 4 63.l odd 6 1
3024.2.r.e 4 28.g odd 6 1
3024.2.r.e 4 252.bl odd 6 1
7938.2.a.bm 2 63.t odd 6 1
7938.2.a.bn 2 63.i even 6 1
9072.2.a.bd 2 252.u odd 6 1
9072.2.a.bk 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 $$S_{2}^{\mathrm{new}}(2646, [\chi])$$:

 $$T_{5}^{4} - 2 T_{5}^{3} + 9 T_{5}^{2} + 10 T_{5} + 25$$ $$T_{11}^{2} - 2 T_{11} + 4$$ $$T_{13}^{4} + 24 T_{13}^{2} + 576$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{4}$$
$3$ 1
$5$ $$1 - 2 T - T^{2} + 10 T^{3} - 20 T^{4} + 50 T^{5} - 25 T^{6} - 250 T^{7} + 625 T^{8}$$
$7$ 1
$11$ $$( 1 - 2 T - 7 T^{2} - 22 T^{3} + 121 T^{4} )^{2}$$
$13$ $$1 - 2 T^{2} - 165 T^{4} - 338 T^{6} + 28561 T^{8}$$
$17$ $$( 1 - 2 T - 13 T^{2} - 34 T^{3} + 289 T^{4} )^{2}$$
$19$ $$1 + 10 T + 43 T^{2} + 190 T^{3} + 988 T^{4} + 3610 T^{5} + 15523 T^{6} + 68590 T^{7} + 130321 T^{8}$$
$23$ $$( 1 + T - 22 T^{2} + 23 T^{3} + 529 T^{4} )^{2}$$
$29$ $$1 + 4 T - 22 T^{2} - 80 T^{3} + 139 T^{4} - 2320 T^{5} - 18502 T^{6} + 97556 T^{7} + 707281 T^{8}$$
$31$ $$( 1 - 6 T + 31 T^{2} )^{4}$$
$37$ $$1 + 4 T + 34 T^{2} - 368 T^{3} - 1637 T^{4} - 13616 T^{5} + 46546 T^{6} + 202612 T^{7} + 1874161 T^{8}$$
$41$ $$1 + 14 T^{2} - 1485 T^{4} + 23534 T^{6} + 2825761 T^{8}$$
$43$ $$1 + 4 T - 50 T^{2} - 80 T^{3} + 1819 T^{4} - 3440 T^{5} - 92450 T^{6} + 318028 T^{7} + 3418801 T^{8}$$
$47$ $$( 1 - 2 T^{2} + 2209 T^{4} )^{2}$$
$53$ $$1 + 12 T + 26 T^{2} + 144 T^{3} + 3483 T^{4} + 7632 T^{5} + 73034 T^{6} + 1786524 T^{7} + 7890481 T^{8}$$
$59$ $$( 1 - 2 T + 59 T^{2} )^{4}$$
$61$ $$( 1 - 18 T + 197 T^{2} - 1098 T^{3} + 3721 T^{4} )^{2}$$
$67$ $$( 1 + 16 T + 174 T^{2} + 1072 T^{3} + 4489 T^{4} )^{2}$$
$71$ $$( 1 + 10 T + 143 T^{2} + 710 T^{3} + 5041 T^{4} )^{2}$$
$73$ $$1 - 4 T - 110 T^{2} + 80 T^{3} + 9379 T^{4} + 5840 T^{5} - 586190 T^{6} - 1556068 T^{7} + 28398241 T^{8}$$
$79$ $$( 1 - 6 T + 143 T^{2} - 474 T^{3} + 6241 T^{4} )^{2}$$
$83$ $$( 1 - 2 T - 79 T^{2} - 166 T^{3} + 6889 T^{4} )^{2}$$
$89$ $$1 + 24 T + 278 T^{2} + 2880 T^{3} + 29619 T^{4} + 256320 T^{5} + 2202038 T^{6} + 16919256 T^{7} + 62742241 T^{8}$$
$97$ $$1 - 4 T - 158 T^{2} + 80 T^{3} + 19315 T^{4} + 7760 T^{5} - 1486622 T^{6} - 3650692 T^{7} + 88529281 T^{8}$$