# Properties

 Label 2646.2.h.q 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: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ x^8 - x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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.

 $$f(q)$$ $$=$$ $$q + (\beta_1 - 1) q^{2} - \beta_1 q^{4} + (\beta_{6} + \beta_{5}) q^{5} + q^{8}+O(q^{10})$$ q + (b1 - 1) * q^2 - b1 * q^4 + (b6 + b5) * q^5 + q^8 $$q + (\beta_1 - 1) q^{2} - \beta_1 q^{4} + (\beta_{6} + \beta_{5}) q^{5} + q^{8} - \beta_{7} q^{10} + ( - 2 \beta_{4} - 2) q^{11} + ( - 2 \beta_{7} + \beta_{5} - \beta_{3}) q^{13} + (\beta_1 - 1) q^{16} + (4 \beta_{7} - \beta_{5} + \beta_{3}) q^{17} + ( - \beta_{7} + \beta_{6} + \beta_{5}) q^{19} + (\beta_{7} - \beta_{6} - \beta_{5}) q^{20} + (2 \beta_{4} - 2 \beta_{2} - 2 \beta_1 + 2) q^{22} + (4 \beta_{4} - 1) q^{23} + (\beta_{4} - 3) q^{25} + (2 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} + \beta_{3}) q^{26} + ( - \beta_{2} + \beta_1) q^{29} + ( - 6 \beta_{7} + 6 \beta_{6} + 6 \beta_{5} - 3 \beta_{3}) q^{31} - \beta_1 q^{32} + ( - 4 \beta_{7} + 4 \beta_{6} + 4 \beta_{5} - \beta_{3}) q^{34} + 8 \beta_1 q^{37} + ( - \beta_{6} - \beta_{5}) q^{38} + (\beta_{6} + \beta_{5}) q^{40} + (4 \beta_{5} - 4 \beta_{3}) q^{41} + ( - 3 \beta_{2} + 7 \beta_1) q^{43} + (2 \beta_{2} + 2 \beta_1) q^{44} + ( - 4 \beta_{4} + 4 \beta_{2} - \beta_1 + 1) q^{46} + ( - 2 \beta_{7} - 4 \beta_{5} + 4 \beta_{3}) q^{47} + ( - \beta_{4} + \beta_{2} - 3 \beta_1 + 3) q^{50} + (2 \beta_{6} + \beta_{5}) q^{52} + ( - \beta_{4} + \beta_{2} - 5 \beta_1 + 5) q^{53} + ( - 4 \beta_{6} - 6 \beta_{5}) q^{55} + (\beta_{4} - 1) q^{58} + (4 \beta_{7} - 4 \beta_{6} - 4 \beta_{5} + 9 \beta_{3}) q^{59} + (3 \beta_{7} - 2 \beta_{5} + 2 \beta_{3}) q^{61} + ( - 6 \beta_{6} - 3 \beta_{5}) q^{62} + q^{64} + ( - \beta_{4} + \beta_{2} + 3 \beta_1 - 3) q^{65} + (3 \beta_{2} + 9 \beta_1) q^{67} + ( - 4 \beta_{6} - 3 \beta_{5}) q^{68} + (3 \beta_{4} + 6) q^{71} + ( - 2 \beta_{7} - 4 \beta_{5} + 4 \beta_{3}) q^{73} - 8 q^{74} + \beta_{7} q^{76} + (7 \beta_{4} - 7 \beta_{2} + 2 \beta_1 - 2) q^{79} - \beta_{7} q^{80} + 4 \beta_{3} q^{82} - 7 \beta_{3} q^{83} + (3 \beta_{4} - 3 \beta_{2} - 7 \beta_1 + 7) q^{85} + (3 \beta_{4} - 7) q^{86} + ( - 2 \beta_{4} - 2) q^{88} + ( - 12 \beta_{7} + 12 \beta_{6} + 12 \beta_{5} - 7 \beta_{3}) q^{89} + ( - 4 \beta_{2} + \beta_1) q^{92} + (2 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} - 4 \beta_{3}) q^{94} + (\beta_{2} + 2 \beta_1) q^{95} + (2 \beta_{7} - 2 \beta_{6} - 2 \beta_{5}) q^{97}+O(q^{100})$$ q + (b1 - 1) * q^2 - b1 * q^4 + (b6 + b5) * q^5 + q^8 - b7 * q^10 + (-2*b4 - 2) * q^11 + (-2*b7 + b5 - b3) * q^13 + (b1 - 1) * q^16 + (4*b7 - b5 + b3) * q^17 + (-b7 + b6 + b5) * q^19 + (b7 - b6 - b5) * q^20 + (2*b4 - 2*b2 - 2*b1 + 2) * q^22 + (4*b4 - 1) * q^23 + (b4 - 3) * q^25 + (2*b7 - 2*b6 - 2*b5 + b3) * q^26 + (-b2 + b1) * q^29 + (-6*b7 + 6*b6 + 6*b5 - 3*b3) * q^31 - b1 * q^32 + (-4*b7 + 4*b6 + 4*b5 - b3) * q^34 + 8*b1 * q^37 + (-b6 - b5) * q^38 + (b6 + b5) * q^40 + (4*b5 - 4*b3) * q^41 + (-3*b2 + 7*b1) * q^43 + (2*b2 + 2*b1) * q^44 + (-4*b4 + 4*b2 - b1 + 1) * q^46 + (-2*b7 - 4*b5 + 4*b3) * q^47 + (-b4 + b2 - 3*b1 + 3) * q^50 + (2*b6 + b5) * q^52 + (-b4 + b2 - 5*b1 + 5) * q^53 + (-4*b6 - 6*b5) * q^55 + (b4 - 1) * q^58 + (4*b7 - 4*b6 - 4*b5 + 9*b3) * q^59 + (3*b7 - 2*b5 + 2*b3) * q^61 + (-6*b6 - 3*b5) * q^62 + q^64 + (-b4 + b2 + 3*b1 - 3) * q^65 + (3*b2 + 9*b1) * q^67 + (-4*b6 - 3*b5) * q^68 + (3*b4 + 6) * q^71 + (-2*b7 - 4*b5 + 4*b3) * q^73 - 8 * q^74 + b7 * q^76 + (7*b4 - 7*b2 + 2*b1 - 2) * q^79 - b7 * q^80 + 4*b3 * q^82 - 7*b3 * q^83 + (3*b4 - 3*b2 - 7*b1 + 7) * q^85 + (3*b4 - 7) * q^86 + (-2*b4 - 2) * q^88 + (-12*b7 + 12*b6 + 12*b5 - 7*b3) * q^89 + (-4*b2 + b1) * q^92 + (2*b7 - 2*b6 - 2*b5 - 4*b3) * q^94 + (b2 + 2*b1) * q^95 + (2*b7 - 2*b6 - 2*b5) * 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 $$8 q - 4 q^{2} - 4 q^{4} + 8 q^{8} - 16 q^{11} - 4 q^{16} + 8 q^{22} - 8 q^{23} - 24 q^{25} + 4 q^{29} - 4 q^{32} + 32 q^{37} + 28 q^{43} + 8 q^{44} + 4 q^{46} + 12 q^{50} + 20 q^{53} - 8 q^{58} + 8 q^{64} - 12 q^{65} + 36 q^{67} + 48 q^{71} - 64 q^{74} - 8 q^{79} + 28 q^{85} - 56 q^{86} - 16 q^{88} + 4 q^{92} + 8 q^{95}+O(q^{100})$$ 8 * q - 4 * q^2 - 4 * q^4 + 8 * q^8 - 16 * q^11 - 4 * q^16 + 8 * q^22 - 8 * q^23 - 24 * q^25 + 4 * q^29 - 4 * q^32 + 32 * q^37 + 28 * q^43 + 8 * q^44 + 4 * q^46 + 12 * q^50 + 20 * q^53 - 8 * q^58 + 8 * q^64 - 12 * q^65 + 36 * q^67 + 48 * q^71 - 64 * q^74 - 8 * q^79 + 28 * q^85 - 56 * q^86 - 16 * q^88 + 4 * q^92 + 8 * q^95

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{24}^{4}$$ v^4 $$\beta_{2}$$ $$=$$ $$\zeta_{24}^{6} + \zeta_{24}^{2}$$ v^6 + v^2 $$\beta_{3}$$ $$=$$ $$\zeta_{24}^{7} + \zeta_{24}$$ v^7 + v $$\beta_{4}$$ $$=$$ $$-\zeta_{24}^{6} + 2\zeta_{24}^{2}$$ -v^6 + 2*v^2 $$\beta_{5}$$ $$=$$ $$-\zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24}$$ -v^5 + v^3 + v $$\beta_{6}$$ $$=$$ $$-\zeta_{24}^{7} + \zeta_{24}^{5}$$ -v^7 + v^5 $$\beta_{7}$$ $$=$$ $$-\zeta_{24}^{7} - \zeta_{24}^{5} + \zeta_{24}$$ -v^7 - v^5 + v
 $$\zeta_{24}$$ $$=$$ $$( \beta_{7} + \beta_{6} + 2\beta_{3} ) / 3$$ (b7 + b6 + 2*b3) / 3 $$\zeta_{24}^{2}$$ $$=$$ $$( \beta_{4} + \beta_{2} ) / 3$$ (b4 + b2) / 3 $$\zeta_{24}^{3}$$ $$=$$ $$( -2\beta_{7} + \beta_{6} + 3\beta_{5} - \beta_{3} ) / 3$$ (-2*b7 + b6 + 3*b5 - b3) / 3 $$\zeta_{24}^{4}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{24}^{5}$$ $$=$$ $$( -\beta_{7} + 2\beta_{6} + \beta_{3} ) / 3$$ (-b7 + 2*b6 + b3) / 3 $$\zeta_{24}^{6}$$ $$=$$ $$( -\beta_{4} + 2\beta_{2} ) / 3$$ (-b4 + 2*b2) / 3 $$\zeta_{24}^{7}$$ $$=$$ $$( -\beta_{7} - \beta_{6} + \beta_{3} ) / 3$$ (-b7 - b6 + b3) / 3

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
361.1
 −0.965926 − 0.258819i −0.258819 + 0.965926i 0.258819 − 0.965926i 0.965926 + 0.258819i −0.965926 + 0.258819i −0.258819 − 0.965926i 0.258819 + 0.965926i 0.965926 − 0.258819i
−0.500000 + 0.866025i 0 −0.500000 0.866025i −1.93185 0 0 1.00000 0 0.965926 1.67303i
361.2 −0.500000 + 0.866025i 0 −0.500000 0.866025i −0.517638 0 0 1.00000 0 0.258819 0.448288i
361.3 −0.500000 + 0.866025i 0 −0.500000 0.866025i 0.517638 0 0 1.00000 0 −0.258819 + 0.448288i
361.4 −0.500000 + 0.866025i 0 −0.500000 0.866025i 1.93185 0 0 1.00000 0 −0.965926 + 1.67303i
667.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i −1.93185 0 0 1.00000 0 0.965926 + 1.67303i
667.2 −0.500000 0.866025i 0 −0.500000 + 0.866025i −0.517638 0 0 1.00000 0 0.258819 + 0.448288i
667.3 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0.517638 0 0 1.00000 0 −0.258819 0.448288i
667.4 −0.500000 0.866025i 0 −0.500000 + 0.866025i 1.93185 0 0 1.00000 0 −0.965926 1.67303i
 $$n$$: 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.q 8
3.b odd 2 1 882.2.h.t 8
7.b odd 2 1 inner 2646.2.h.q 8
7.c even 3 1 2646.2.e.t 8
7.c even 3 1 2646.2.f.q 8
7.d odd 6 1 2646.2.e.t 8
7.d odd 6 1 2646.2.f.q 8
9.c even 3 1 2646.2.e.t 8
9.d odd 6 1 882.2.e.q 8
21.c even 2 1 882.2.h.t 8
21.g even 6 1 882.2.e.q 8
21.g even 6 1 882.2.f.s 8
21.h odd 6 1 882.2.e.q 8
21.h odd 6 1 882.2.f.s 8
63.g even 3 1 inner 2646.2.h.q 8
63.g even 3 1 7938.2.a.co 4
63.h even 3 1 2646.2.f.q 8
63.i even 6 1 882.2.f.s 8
63.j odd 6 1 882.2.f.s 8
63.k odd 6 1 inner 2646.2.h.q 8
63.k odd 6 1 7938.2.a.co 4
63.l odd 6 1 2646.2.e.t 8
63.n odd 6 1 882.2.h.t 8
63.n odd 6 1 7938.2.a.cj 4
63.o even 6 1 882.2.e.q 8
63.s even 6 1 882.2.h.t 8
63.s even 6 1 7938.2.a.cj 4
63.t odd 6 1 2646.2.f.q 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
882.2.e.q 8 9.d odd 6 1
882.2.e.q 8 21.g even 6 1
882.2.e.q 8 21.h odd 6 1
882.2.e.q 8 63.o even 6 1
882.2.f.s 8 21.g even 6 1
882.2.f.s 8 21.h odd 6 1
882.2.f.s 8 63.i even 6 1
882.2.f.s 8 63.j odd 6 1
882.2.h.t 8 3.b odd 2 1
882.2.h.t 8 21.c even 2 1
882.2.h.t 8 63.n odd 6 1
882.2.h.t 8 63.s even 6 1
2646.2.e.t 8 7.c even 3 1
2646.2.e.t 8 7.d odd 6 1
2646.2.e.t 8 9.c even 3 1
2646.2.e.t 8 63.l odd 6 1
2646.2.f.q 8 7.c even 3 1
2646.2.f.q 8 7.d odd 6 1
2646.2.f.q 8 63.h even 3 1
2646.2.f.q 8 63.t odd 6 1
2646.2.h.q 8 1.a even 1 1 trivial
2646.2.h.q 8 7.b odd 2 1 inner
2646.2.h.q 8 63.g even 3 1 inner
2646.2.h.q 8 63.k odd 6 1 inner
7938.2.a.cj 4 63.n odd 6 1
7938.2.a.cj 4 63.s even 6 1
7938.2.a.co 4 63.g even 3 1
7938.2.a.co 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 $$S_{2}^{\mathrm{new}}(2646, [\chi])$$:

 $$T_{5}^{4} - 4T_{5}^{2} + 1$$ T5^4 - 4*T5^2 + 1 $$T_{11}^{2} + 4T_{11} - 8$$ T11^2 + 4*T11 - 8 $$T_{13}^{4} + 6T_{13}^{2} + 36$$ T13^4 + 6*T13^2 + 36

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + T + 1)^{4}$$
$3$ $$T^{8}$$
$5$ $$(T^{4} - 4 T^{2} + 1)^{2}$$
$7$ $$T^{8}$$
$11$ $$(T^{2} + 4 T - 8)^{4}$$
$13$ $$(T^{4} + 6 T^{2} + 36)^{2}$$
$17$ $$T^{8} + 52 T^{6} + 2220 T^{4} + \cdots + 234256$$
$19$ $$T^{8} + 4 T^{6} + 15 T^{4} + 4 T^{2} + \cdots + 1$$
$23$ $$(T^{2} + 2 T - 47)^{4}$$
$29$ $$(T^{4} - 2 T^{3} + 6 T^{2} + 4 T + 4)^{2}$$
$31$ $$(T^{4} + 54 T^{2} + 2916)^{2}$$
$37$ $$(T^{2} - 8 T + 64)^{4}$$
$41$ $$(T^{4} + 32 T^{2} + 1024)^{2}$$
$43$ $$(T^{4} - 14 T^{3} + 174 T^{2} - 308 T + 484)^{2}$$
$47$ $$T^{8} + 112 T^{6} + 10608 T^{4} + \cdots + 3748096$$
$53$ $$(T^{4} - 10 T^{3} + 78 T^{2} - 220 T + 484)^{2}$$
$59$ $$T^{8} + 244 T^{6} + \cdots + 29986576$$
$61$ $$T^{8} + 28 T^{6} + 615 T^{4} + \cdots + 28561$$
$67$ $$(T^{4} - 18 T^{3} + 270 T^{2} - 972 T + 2916)^{2}$$
$71$ $$(T^{2} - 12 T + 9)^{4}$$
$73$ $$T^{8} + 112 T^{6} + 10608 T^{4} + \cdots + 3748096$$
$79$ $$(T^{4} + 4 T^{3} + 159 T^{2} - 572 T + 20449)^{2}$$
$83$ $$(T^{4} + 98 T^{2} + 9604)^{2}$$
$89$ $$T^{8} + 436 T^{6} + \cdots + 2097273616$$
$97$ $$T^{8} + 16 T^{6} + 240 T^{4} + \cdots + 256$$