# Properties

 Label 2646.2.d.d Level $2646$ Weight $2$ Character orbit 2646.d 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.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$21.1284163748$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ x^8 - x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{6}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 378) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

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

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{24}^{6}$$ v^6 $$\beta_{2}$$ $$=$$ $$2\zeta_{24}^{4} - 1$$ 2*v^4 - 1 $$\beta_{3}$$ $$=$$ $$-\zeta_{24}^{6} + 2\zeta_{24}^{2}$$ -v^6 + 2*v^2 $$\beta_{4}$$ $$=$$ $$-2\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24}$$ -2*v^7 + v^5 + v^3 + v $$\beta_{5}$$ $$=$$ $$-2\zeta_{24}^{7} - \zeta_{24}^{5} + \zeta_{24}^{3} - \zeta_{24}$$ -2*v^7 - v^5 + v^3 - v $$\beta_{6}$$ $$=$$ $$3\zeta_{24}^{5} + 3\zeta_{24}^{3} - 3\zeta_{24}$$ 3*v^5 + 3*v^3 - 3*v $$\beta_{7}$$ $$=$$ $$-3\zeta_{24}^{5} + 3\zeta_{24}^{3} + 3\zeta_{24}$$ -3*v^5 + 3*v^3 + 3*v
 $$\zeta_{24}$$ $$=$$ $$( \beta_{7} - \beta_{6} - 3\beta_{5} + 3\beta_{4} ) / 12$$ (b7 - b6 - 3*b5 + 3*b4) / 12 $$\zeta_{24}^{2}$$ $$=$$ $$( \beta_{3} + \beta_1 ) / 2$$ (b3 + b1) / 2 $$\zeta_{24}^{3}$$ $$=$$ $$( \beta_{7} + \beta_{6} ) / 6$$ (b7 + b6) / 6 $$\zeta_{24}^{4}$$ $$=$$ $$( \beta_{2} + 1 ) / 2$$ (b2 + 1) / 2 $$\zeta_{24}^{5}$$ $$=$$ $$( -\beta_{7} + \beta_{6} - 3\beta_{5} + 3\beta_{4} ) / 12$$ (-b7 + b6 - 3*b5 + 3*b4) / 12 $$\zeta_{24}^{6}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{24}^{7}$$ $$=$$ $$( \beta_{7} + \beta_{6} - 3\beta_{5} - 3\beta_{4} ) / 12$$ (b7 + b6 - 3*b5 - 3*b4) / 12

## Character values

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

 $$n$$ $$785$$ $$1081$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2645.1
 −0.965926 + 0.258819i −0.258819 + 0.965926i 0.258819 − 0.965926i 0.965926 − 0.258819i −0.258819 − 0.965926i −0.965926 − 0.258819i 0.965926 + 0.258819i 0.258819 + 0.965926i
1.00000i 0 −1.00000 −2.44949 0 0 1.00000i 0 2.44949i
2645.2 1.00000i 0 −1.00000 −2.44949 0 0 1.00000i 0 2.44949i
2645.3 1.00000i 0 −1.00000 2.44949 0 0 1.00000i 0 2.44949i
2645.4 1.00000i 0 −1.00000 2.44949 0 0 1.00000i 0 2.44949i
2645.5 1.00000i 0 −1.00000 −2.44949 0 0 1.00000i 0 2.44949i
2645.6 1.00000i 0 −1.00000 −2.44949 0 0 1.00000i 0 2.44949i
2645.7 1.00000i 0 −1.00000 2.44949 0 0 1.00000i 0 2.44949i
2645.8 1.00000i 0 −1.00000 2.44949 0 0 1.00000i 0 2.44949i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2645.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2646.2.d.d 8
3.b odd 2 1 inner 2646.2.d.d 8
7.b odd 2 1 inner 2646.2.d.d 8
7.c even 3 1 378.2.k.d 8
7.d odd 6 1 378.2.k.d 8
21.c even 2 1 inner 2646.2.d.d 8
21.g even 6 1 378.2.k.d 8
21.h odd 6 1 378.2.k.d 8
63.g even 3 1 1134.2.t.f 8
63.h even 3 1 1134.2.l.e 8
63.i even 6 1 1134.2.l.e 8
63.j odd 6 1 1134.2.l.e 8
63.k odd 6 1 1134.2.t.f 8
63.n odd 6 1 1134.2.t.f 8
63.s even 6 1 1134.2.t.f 8
63.t odd 6 1 1134.2.l.e 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.k.d 8 7.c even 3 1
378.2.k.d 8 7.d odd 6 1
378.2.k.d 8 21.g even 6 1
378.2.k.d 8 21.h odd 6 1
1134.2.l.e 8 63.h even 3 1
1134.2.l.e 8 63.i even 6 1
1134.2.l.e 8 63.j odd 6 1
1134.2.l.e 8 63.t odd 6 1
1134.2.t.f 8 63.g even 3 1
1134.2.t.f 8 63.k odd 6 1
1134.2.t.f 8 63.n odd 6 1
1134.2.t.f 8 63.s even 6 1
2646.2.d.d 8 1.a even 1 1 trivial
2646.2.d.d 8 3.b odd 2 1 inner
2646.2.d.d 8 7.b odd 2 1 inner
2646.2.d.d 8 21.c even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 6$$ acting on $$S_{2}^{\mathrm{new}}(2646, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{4}$$
$3$ $$T^{8}$$
$5$ $$(T^{2} - 6)^{4}$$
$7$ $$T^{8}$$
$11$ $$(T^{2} + 18)^{4}$$
$13$ $$(T^{4} + 18 T^{2} + 9)^{2}$$
$17$ $$(T^{2} - 6)^{4}$$
$19$ $$(T^{2} + 24)^{4}$$
$23$ $$(T^{2} + 36)^{4}$$
$29$ $$(T^{4} + 108 T^{2} + 324)^{2}$$
$31$ $$(T^{4} + 114 T^{2} + 2601)^{2}$$
$37$ $$(T^{2} - 2 T - 17)^{4}$$
$41$ $$(T^{2} - 6)^{4}$$
$43$ $$(T - 7)^{8}$$
$47$ $$(T^{4} - 228 T^{2} + 10404)^{2}$$
$53$ $$(T^{4} + 216 T^{2} + 1296)^{2}$$
$59$ $$(T^{2} - 6)^{4}$$
$61$ $$(T^{4} + 18 T^{2} + 9)^{2}$$
$67$ $$(T^{2} - 10 T - 47)^{4}$$
$71$ $$(T^{2} + 162)^{4}$$
$73$ $$(T^{4} + 264 T^{2} + 7056)^{2}$$
$79$ $$(T^{2} - 10 T + 7)^{4}$$
$83$ $$(T^{4} - 264 T^{2} + 7056)^{2}$$
$89$ $$(T^{4} - 324 T^{2} + 2916)^{2}$$
$97$ $$(T^{4} + 54 T^{2} + 441)^{2}$$