# Properties

 Label 2646.2.h.r Level $2646$ Weight $2$ Character orbit 2646.h Analytic conductor $21.128$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$21.1284163748$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: 8.0.31116960000.2 Defining polynomial: $$x^{8} + x^{6} - 8x^{4} + 9x^{2} + 81$$ x^8 + x^6 - 8*x^4 + 9*x^2 + 81 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: no (minimal twist has level 882) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} - 1) q^{2} + \beta_{2} q^{4} + (\beta_{4} - \beta_{3}) q^{5} + q^{8}+O(q^{10})$$ q + (-b2 - 1) * q^2 + b2 * q^4 + (b4 - b3) * q^5 + q^8 $$q + ( - \beta_{2} - 1) q^{2} + \beta_{2} q^{4} + (\beta_{4} - \beta_{3}) q^{5} + q^{8} + \beta_{3} q^{10} + \beta_{5} q^{11} + ( - 2 \beta_{7} + 2 \beta_1) q^{13} + ( - \beta_{2} - 1) q^{16} + ( - \beta_{7} + 2 \beta_{3} + \beta_1) q^{17} + ( - \beta_{4} - \beta_1) q^{19} - \beta_{4} q^{20} + \beta_{6} q^{22} + (\beta_{5} - 1) q^{23} - \beta_{5} q^{25} - 2 \beta_1 q^{26} + 4 \beta_{2} q^{29} + 2 \beta_1 q^{31} + \beta_{2} q^{32} + ( - 2 \beta_{4} - \beta_1) q^{34} + 6 \beta_{2} q^{37} + (\beta_{7} + \beta_{4} - \beta_{3}) q^{38} + (\beta_{4} - \beta_{3}) q^{40} + ( - \beta_{7} + 2 \beta_{3} + \beta_1) q^{41} + (\beta_{6} + \beta_{5} + 2 \beta_{2}) q^{43} + ( - \beta_{6} - \beta_{5}) q^{44} + (\beta_{6} + \beta_{2} + 1) q^{46} + (4 \beta_{7} - 4 \beta_{3} - 4 \beta_1) q^{47} - \beta_{6} q^{50} + 2 \beta_{7} q^{52} + ( - 4 \beta_{2} - 4) q^{53} + ( - 2 \beta_{7} - 6 \beta_{4} + 6 \beta_{3}) q^{55} + 4 q^{58} + ( - 4 \beta_{4} + 3 \beta_1) q^{59} + ( - 2 \beta_{7} - \beta_{3} + 2 \beta_1) q^{61} - 2 \beta_{7} q^{62} + q^{64} + (4 \beta_{2} + 4) q^{65} + ( - \beta_{6} - \beta_{5}) q^{67} + (\beta_{7} + 2 \beta_{4} - 2 \beta_{3}) q^{68} + ( - \beta_{5} + 5) q^{71} + ( - 3 \beta_{7} + 4 \beta_{3} + 3 \beta_1) q^{73} + 6 q^{74} + ( - \beta_{7} + \beta_{3} + \beta_1) q^{76} + (\beta_{6} - 9 \beta_{2} - 9) q^{79} + \beta_{3} q^{80} + ( - 2 \beta_{4} - \beta_1) q^{82} + ( - 2 \beta_{4} + 2 \beta_1) q^{83} + ( - 2 \beta_{6} - 8 \beta_{2} - 8) q^{85} + ( - \beta_{5} + 2) q^{86} + \beta_{5} q^{88} + (2 \beta_{4} - 4 \beta_1) q^{89} + ( - \beta_{6} - \beta_{5} - \beta_{2}) q^{92} + (4 \beta_{4} + 4 \beta_1) q^{94} + (\beta_{6} + \beta_{5} + 3 \beta_{2}) q^{95} + 5 \beta_1 q^{97}+O(q^{100})$$ q + (-b2 - 1) * q^2 + b2 * q^4 + (b4 - b3) * q^5 + q^8 + b3 * q^10 + b5 * q^11 + (-2*b7 + 2*b1) * q^13 + (-b2 - 1) * q^16 + (-b7 + 2*b3 + b1) * q^17 + (-b4 - b1) * q^19 - b4 * q^20 + b6 * q^22 + (b5 - 1) * q^23 - b5 * q^25 - 2*b1 * q^26 + 4*b2 * q^29 + 2*b1 * q^31 + b2 * q^32 + (-2*b4 - b1) * q^34 + 6*b2 * q^37 + (b7 + b4 - b3) * q^38 + (b4 - b3) * q^40 + (-b7 + 2*b3 + b1) * q^41 + (b6 + b5 + 2*b2) * q^43 + (-b6 - b5) * q^44 + (b6 + b2 + 1) * q^46 + (4*b7 - 4*b3 - 4*b1) * q^47 - b6 * q^50 + 2*b7 * q^52 + (-4*b2 - 4) * q^53 + (-2*b7 - 6*b4 + 6*b3) * q^55 + 4 * q^58 + (-4*b4 + 3*b1) * q^59 + (-2*b7 - b3 + 2*b1) * q^61 - 2*b7 * q^62 + q^64 + (4*b2 + 4) * q^65 + (-b6 - b5) * q^67 + (b7 + 2*b4 - 2*b3) * q^68 + (-b5 + 5) * q^71 + (-3*b7 + 4*b3 + 3*b1) * q^73 + 6 * q^74 + (-b7 + b3 + b1) * q^76 + (b6 - 9*b2 - 9) * q^79 + b3 * q^80 + (-2*b4 - b1) * q^82 + (-2*b4 + 2*b1) * q^83 + (-2*b6 - 8*b2 - 8) * q^85 + (-b5 + 2) * q^86 + b5 * q^88 + (2*b4 - 4*b1) * q^89 + (-b6 - b5 - b2) * q^92 + (4*b4 + 4*b1) * q^94 + (b6 + b5 + 3*b2) * q^95 + 5*b1 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 4 q^{2} - 4 q^{4} + 8 q^{8}+O(q^{10})$$ 8 * q - 4 * q^2 - 4 * q^4 + 8 * q^8 $$8 q - 4 q^{2} - 4 q^{4} + 8 q^{8} - 4 q^{11} - 4 q^{16} + 2 q^{22} - 12 q^{23} + 4 q^{25} - 16 q^{29} - 4 q^{32} - 24 q^{37} - 10 q^{43} + 2 q^{44} + 6 q^{46} - 2 q^{50} - 16 q^{53} + 32 q^{58} + 8 q^{64} + 16 q^{65} + 2 q^{67} + 44 q^{71} + 48 q^{74} - 34 q^{79} - 36 q^{85} + 20 q^{86} - 4 q^{88} + 6 q^{92} - 14 q^{95}+O(q^{100})$$ 8 * q - 4 * q^2 - 4 * q^4 + 8 * q^8 - 4 * q^11 - 4 * q^16 + 2 * q^22 - 12 * q^23 + 4 * q^25 - 16 * q^29 - 4 * q^32 - 24 * q^37 - 10 * q^43 + 2 * q^44 + 6 * q^46 - 2 * q^50 - 16 * q^53 + 32 * q^58 + 8 * q^64 + 16 * q^65 + 2 * q^67 + 44 * q^71 + 48 * q^74 - 34 * q^79 - 36 * q^85 + 20 * q^86 - 4 * q^88 + 6 * q^92 - 14 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + x^{6} - 8x^{4} + 9x^{2} + 81$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{7} + 8\nu^{5} - 64\nu^{3} + 135\nu ) / 216$$ (-v^7 + 8*v^5 - 64*v^3 + 135*v) / 216 $$\beta_{2}$$ $$=$$ $$( -\nu^{6} + 8\nu^{4} + 8\nu^{2} - 81 ) / 72$$ (-v^6 + 8*v^4 + 8*v^2 - 81) / 72 $$\beta_{3}$$ $$=$$ $$( -\nu^{7} + 8\nu^{5} + 44\nu^{3} + 27\nu ) / 108$$ (-v^7 + 8*v^5 + 44*v^3 + 27*v) / 108 $$\beta_{4}$$ $$=$$ $$( \nu^{7} + 4\nu^{5} + 4\nu^{3} - 15\nu ) / 36$$ (v^7 + 4*v^5 + 4*v^3 - 15*v) / 36 $$\beta_{5}$$ $$=$$ $$( \nu^{6} + \nu^{4} - 17\nu^{2} ) / 9$$ (v^6 + v^4 - 17*v^2) / 9 $$\beta_{6}$$ $$=$$ $$( \nu^{6} + 8\nu^{2} + 17 ) / 8$$ (v^6 + 8*v^2 + 17) / 8 $$\beta_{7}$$ $$=$$ $$( -\nu^{7} + 2\nu^{5} + 2\nu^{3} - 15\nu ) / 18$$ (-v^7 + 2*v^5 + 2*v^3 - 15*v) / 18
 $$\nu$$ $$=$$ $$( -\beta_{7} - \beta_{4} + 2\beta_{3} + 2\beta_1 ) / 3$$ (-b7 - b4 + 2*b3 + 2*b1) / 3 $$\nu^{2}$$ $$=$$ $$( \beta_{6} - \beta_{5} + \beta_{2} - 1 ) / 3$$ (b6 - b5 + b2 - 1) / 3 $$\nu^{3}$$ $$=$$ $$( -\beta_{7} - \beta_{4} + 5\beta_{3} - 4\beta_1 ) / 3$$ (-b7 - b4 + 5*b3 - 4*b1) / 3 $$\nu^{4}$$ $$=$$ $$( \beta_{6} + 2\beta_{5} + 25\beta_{2} + 26 ) / 3$$ (b6 + 2*b5 + 25*b2 + 26) / 3 $$\nu^{5}$$ $$=$$ $$( 5\beta_{7} + 14\beta_{4} + 5\beta_{3} + 14\beta_1 ) / 3$$ (5*b7 + 14*b4 + 5*b3 + 14*b1) / 3 $$\nu^{6}$$ $$=$$ $$( 16\beta_{6} + 8\beta_{5} - 8\beta_{2} - 43 ) / 3$$ (16*b6 + 8*b5 - 8*b2 - 43) / 3 $$\nu^{7}$$ $$=$$ $$( -31\beta_{7} + 41\beta_{4} - 10\beta_{3} - 10\beta_1 ) / 3$$ (-31*b7 + 41*b4 - 10*b3 - 10*b1) / 3

## Character values

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

 $$n$$ $$785$$ $$1081$$ $$\chi(n)$$ $$-1 - \beta_{2}$$ $$\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
361.1
 1.62968 − 0.586627i 0.306808 + 1.70466i −0.306808 − 1.70466i −1.62968 + 0.586627i 1.62968 + 0.586627i 0.306808 − 1.70466i −0.306808 + 1.70466i −1.62968 − 0.586627i
−0.500000 + 0.866025i 0 −0.500000 0.866025i −3.25937 0 0 1.00000 0 1.62968 2.82269i
361.2 −0.500000 + 0.866025i 0 −0.500000 0.866025i −0.613616 0 0 1.00000 0 0.306808 0.531407i
361.3 −0.500000 + 0.866025i 0 −0.500000 0.866025i 0.613616 0 0 1.00000 0 −0.306808 + 0.531407i
361.4 −0.500000 + 0.866025i 0 −0.500000 0.866025i 3.25937 0 0 1.00000 0 −1.62968 + 2.82269i
667.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i −3.25937 0 0 1.00000 0 1.62968 + 2.82269i
667.2 −0.500000 0.866025i 0 −0.500000 + 0.866025i −0.613616 0 0 1.00000 0 0.306808 + 0.531407i
667.3 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0.613616 0 0 1.00000 0 −0.306808 0.531407i
667.4 −0.500000 0.866025i 0 −0.500000 + 0.866025i 3.25937 0 0 1.00000 0 −1.62968 2.82269i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 667.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
63.g even 3 1 inner
63.k odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2646.2.h.r 8
3.b odd 2 1 882.2.h.s 8
7.b odd 2 1 inner 2646.2.h.r 8
7.c even 3 1 2646.2.e.s 8
7.c even 3 1 2646.2.f.p 8
7.d odd 6 1 2646.2.e.s 8
7.d odd 6 1 2646.2.f.p 8
9.c even 3 1 2646.2.e.s 8
9.d odd 6 1 882.2.e.r 8
21.c even 2 1 882.2.h.s 8
21.g even 6 1 882.2.e.r 8
21.g even 6 1 882.2.f.r 8
21.h odd 6 1 882.2.e.r 8
21.h odd 6 1 882.2.f.r 8
63.g even 3 1 inner 2646.2.h.r 8
63.g even 3 1 7938.2.a.cq 4
63.h even 3 1 2646.2.f.p 8
63.i even 6 1 882.2.f.r 8
63.j odd 6 1 882.2.f.r 8
63.k odd 6 1 inner 2646.2.h.r 8
63.k odd 6 1 7938.2.a.cq 4
63.l odd 6 1 2646.2.e.s 8
63.n odd 6 1 882.2.h.s 8
63.n odd 6 1 7938.2.a.ch 4
63.o even 6 1 882.2.e.r 8
63.s even 6 1 882.2.h.s 8
63.s even 6 1 7938.2.a.ch 4
63.t odd 6 1 2646.2.f.p 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
882.2.e.r 8 9.d odd 6 1
882.2.e.r 8 21.g even 6 1
882.2.e.r 8 21.h odd 6 1
882.2.e.r 8 63.o even 6 1
882.2.f.r 8 21.g even 6 1
882.2.f.r 8 21.h odd 6 1
882.2.f.r 8 63.i even 6 1
882.2.f.r 8 63.j odd 6 1
882.2.h.s 8 3.b odd 2 1
882.2.h.s 8 21.c even 2 1
882.2.h.s 8 63.n odd 6 1
882.2.h.s 8 63.s even 6 1
2646.2.e.s 8 7.c even 3 1
2646.2.e.s 8 7.d odd 6 1
2646.2.e.s 8 9.c even 3 1
2646.2.e.s 8 63.l odd 6 1
2646.2.f.p 8 7.c even 3 1
2646.2.f.p 8 7.d odd 6 1
2646.2.f.p 8 63.h even 3 1
2646.2.f.p 8 63.t odd 6 1
2646.2.h.r 8 1.a even 1 1 trivial
2646.2.h.r 8 7.b odd 2 1 inner
2646.2.h.r 8 63.g even 3 1 inner
2646.2.h.r 8 63.k odd 6 1 inner
7938.2.a.ch 4 63.n odd 6 1
7938.2.a.ch 4 63.s even 6 1
7938.2.a.cq 4 63.g even 3 1
7938.2.a.cq 4 63.k odd 6 1

## Hecke kernels

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

 $$T_{5}^{4} - 11T_{5}^{2} + 4$$ T5^4 - 11*T5^2 + 4 $$T_{11}^{2} + T_{11} - 26$$ T11^2 + T11 - 26 $$T_{13}^{8} + 44T_{13}^{6} + 1872T_{13}^{4} + 2816T_{13}^{2} + 4096$$ T13^8 + 44*T13^6 + 1872*T13^4 + 2816*T13^2 + 4096

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + T + 1)^{4}$$
$3$ $$T^{8}$$
$5$ $$(T^{4} - 11 T^{2} + 4)^{2}$$
$7$ $$T^{8}$$
$11$ $$(T^{2} + T - 26)^{4}$$
$13$ $$T^{8} + 44 T^{6} + 1872 T^{4} + \cdots + 4096$$
$17$ $$T^{8} + 39 T^{6} + 1377 T^{4} + \cdots + 20736$$
$19$ $$(T^{4} + 7 T^{2} + 49)^{2}$$
$23$ $$(T^{2} + 3 T - 24)^{4}$$
$29$ $$(T^{2} + 4 T + 16)^{4}$$
$31$ $$T^{8} + 44 T^{6} + 1872 T^{4} + \cdots + 4096$$
$37$ $$(T^{2} + 6 T + 36)^{4}$$
$41$ $$T^{8} + 39 T^{6} + 1377 T^{4} + \cdots + 20736$$
$43$ $$(T^{4} + 5 T^{3} + 45 T^{2} - 100 T + 400)^{2}$$
$47$ $$(T^{4} + 112 T^{2} + 12544)^{2}$$
$53$ $$(T^{2} + 4 T + 16)^{4}$$
$59$ $$T^{8} + 371 T^{6} + \cdots + 1097199376$$
$61$ $$T^{8} + 71 T^{6} + 4017 T^{4} + \cdots + 1048576$$
$67$ $$(T^{4} - T^{3} + 27 T^{2} + 26 T + 676)^{2}$$
$71$ $$(T^{2} - 11 T + 4)^{4}$$
$73$ $$T^{8} + 179 T^{6} + \cdots + 45212176$$
$79$ $$(T^{4} + 17 T^{3} + 243 T^{2} + 782 T + 2116)^{2}$$
$83$ $$(T^{4} + 60 T^{2} + 3600)^{2}$$
$89$ $$T^{8} + 284 T^{6} + \cdots + 268435456$$
$97$ $$T^{8} + 275 T^{6} + 73125 T^{4} + \cdots + 6250000$$