# Properties

 Label 2646.2.h.s 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: 8.0.3317760000.3 Defining polynomial: $$x^{8} - 4x^{6} + 7x^{4} - 36x^{2} + 81$$ x^8 - 4*x^6 + 7*x^4 - 36*x^2 + 81 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4x^{6} + 7x^{4} - 36x^{2} + 81$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{7} - 7\nu^{5} + 28\nu^{3} - 120\nu ) / 63$$ (v^7 - 7*v^5 + 28*v^3 - 120*v) / 63 $$\beta_{2}$$ $$=$$ $$( -4\nu^{7} + 7\nu^{5} + 35\nu^{3} + 81\nu ) / 189$$ (-4*v^7 + 7*v^5 + 35*v^3 + 81*v) / 189 $$\beta_{3}$$ $$=$$ $$( -4\nu^{6} + 7\nu^{4} - 28\nu^{2} + 144 ) / 63$$ (-4*v^6 + 7*v^4 - 28*v^2 + 144) / 63 $$\beta_{4}$$ $$=$$ $$( -5\nu^{7} - 7\nu^{5} - 35\nu^{3} + 180\nu ) / 189$$ (-5*v^7 - 7*v^5 - 35*v^3 + 180*v) / 189 $$\beta_{5}$$ $$=$$ $$( -\nu^{6} + 4\nu^{4} + 2\nu^{2} + 18 ) / 9$$ (-v^6 + 4*v^4 + 2*v^2 + 18) / 9 $$\beta_{6}$$ $$=$$ $$( -10\nu^{6} - 14\nu^{4} - 7\nu^{2} + 234 ) / 63$$ (-10*v^6 - 14*v^4 - 7*v^2 + 234) / 63 $$\beta_{7}$$ $$=$$ $$( \nu^{7} - \nu^{5} + 4\nu^{3} - 15\nu ) / 9$$ (v^7 - v^5 + 4*v^3 - 15*v) / 9
 $$\nu$$ $$=$$ $$( \beta_{7} + 2\beta_{4} + 2\beta_{2} - \beta_1 ) / 3$$ (b7 + 2*b4 + 2*b2 - b1) / 3 $$\nu^{2}$$ $$=$$ $$( \beta_{6} + 2\beta_{5} - 6\beta_{3} + 6 ) / 3$$ (b6 + 2*b5 - 6*b3 + 6) / 3 $$\nu^{3}$$ $$=$$ $$( 2\beta_{7} + \beta_{4} + 10\beta_{2} + \beta_1 ) / 3$$ (2*b7 + b4 + 10*b2 + b1) / 3 $$\nu^{4}$$ $$=$$ $$( -4\beta_{6} + 4\beta_{5} + 3\beta_{3} ) / 3$$ (-4*b6 + 4*b5 + 3*b3) / 3 $$\nu^{5}$$ $$=$$ $$( -5\beta_{7} - 31\beta_{4} + 5\beta_{2} - 10\beta_1 ) / 3$$ (-5*b7 - 31*b4 + 5*b2 - 10*b1) / 3 $$\nu^{6}$$ $$=$$ $$( -14\beta_{6} - 7\beta_{5} + 66 ) / 3$$ (-14*b6 - 7*b5 + 66) / 3 $$\nu^{7}$$ $$=$$ $$( 29\beta_{7} - 5\beta_{4} - 5\beta_{2} - 29\beta_1 ) / 3$$ (29*b7 - 5*b4 - 5*b2 - 29*b1) / 3

## Character values

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

 $$n$$ $$785$$ $$1081$$ $$\chi(n)$$ $$-\beta_{3}$$ $$-1 + \beta_{3}$$

## 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
 −1.01575 − 1.40294i −1.72286 − 0.178197i 1.72286 + 0.178197i 1.01575 + 1.40294i −1.01575 + 1.40294i −1.72286 + 0.178197i 1.72286 − 0.178197i 1.01575 − 1.40294i
0.500000 0.866025i 0 −0.500000 0.866025i −3.44572 0 0 −1.00000 0 −1.72286 + 2.98408i
361.2 0.500000 0.866025i 0 −0.500000 0.866025i −2.03151 0 0 −1.00000 0 −1.01575 + 1.75934i
361.3 0.500000 0.866025i 0 −0.500000 0.866025i 2.03151 0 0 −1.00000 0 1.01575 1.75934i
361.4 0.500000 0.866025i 0 −0.500000 0.866025i 3.44572 0 0 −1.00000 0 1.72286 2.98408i
667.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −3.44572 0 0 −1.00000 0 −1.72286 2.98408i
667.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −2.03151 0 0 −1.00000 0 −1.01575 1.75934i
667.3 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 2.03151 0 0 −1.00000 0 1.01575 + 1.75934i
667.4 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 3.44572 0 0 −1.00000 0 1.72286 + 2.98408i
 $$n$$: 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.s 8
3.b odd 2 1 882.2.h.r 8
7.b odd 2 1 inner 2646.2.h.s 8
7.c even 3 1 2646.2.e.r 8
7.c even 3 1 2646.2.f.s 8
7.d odd 6 1 2646.2.e.r 8
7.d odd 6 1 2646.2.f.s 8
9.c even 3 1 2646.2.e.r 8
9.d odd 6 1 882.2.e.t 8
21.c even 2 1 882.2.h.r 8
21.g even 6 1 882.2.e.t 8
21.g even 6 1 882.2.f.p 8
21.h odd 6 1 882.2.e.t 8
21.h odd 6 1 882.2.f.p 8
63.g even 3 1 inner 2646.2.h.s 8
63.g even 3 1 7938.2.a.cd 4
63.h even 3 1 2646.2.f.s 8
63.i even 6 1 882.2.f.p 8
63.j odd 6 1 882.2.f.p 8
63.k odd 6 1 inner 2646.2.h.s 8
63.k odd 6 1 7938.2.a.cd 4
63.l odd 6 1 2646.2.e.r 8
63.n odd 6 1 882.2.h.r 8
63.n odd 6 1 7938.2.a.cu 4
63.o even 6 1 882.2.e.t 8
63.s even 6 1 882.2.h.r 8
63.s even 6 1 7938.2.a.cu 4
63.t odd 6 1 2646.2.f.s 8

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

## Hecke kernels

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

 $$T_{5}^{4} - 16T_{5}^{2} + 49$$ T5^4 - 16*T5^2 + 49 $$T_{11} + 4$$ T11 + 4 $$T_{13}^{4} + 18T_{13}^{2} + 324$$ T13^4 + 18*T13^2 + 324

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - T + 1)^{4}$$
$3$ $$T^{8}$$
$5$ $$(T^{4} - 16 T^{2} + 49)^{2}$$
$7$ $$T^{8}$$
$11$ $$(T + 4)^{8}$$
$13$ $$(T^{4} + 18 T^{2} + 324)^{2}$$
$17$ $$(T^{4} + 2 T^{2} + 4)^{2}$$
$19$ $$T^{8} + 40 T^{6} + 1575 T^{4} + \cdots + 625$$
$23$ $$(T^{2} - 2 T - 59)^{4}$$
$29$ $$(T^{4} - 10 T^{3} + 90 T^{2} - 100 T + 100)^{2}$$
$31$ $$(T^{4} + 30 T^{2} + 900)^{2}$$
$37$ $$(T^{4} - 4 T^{3} + 72 T^{2} + 224 T + 3136)^{2}$$
$41$ $$(T^{4} + 32 T^{2} + 1024)^{2}$$
$43$ $$(T^{4} + 10 T^{3} + 90 T^{2} + 100 T + 100)^{2}$$
$47$ $$T^{8} + 64 T^{6} + 3312 T^{4} + \cdots + 614656$$
$53$ $$(T^{4} - 2 T^{3} + 138 T^{2} + 268 T + 17956)^{2}$$
$59$ $$T^{8} + 76 T^{6} + 5292 T^{4} + \cdots + 234256$$
$61$ $$T^{8} + 40 T^{6} + 1575 T^{4} + \cdots + 625$$
$67$ $$(T^{4} - 6 T^{3} + 42 T^{2} + 36 T + 36)^{2}$$
$71$ $$(T^{2} - 12 T + 21)^{4}$$
$73$ $$T^{8} + 256 T^{6} + \cdots + 21381376$$
$79$ $$(T^{4} + 4 T^{3} + 27 T^{2} - 44 T + 121)^{2}$$
$83$ $$T^{8} + 76 T^{6} + 5292 T^{4} + \cdots + 234256$$
$89$ $$(T^{4} + 50 T^{2} + 2500)^{2}$$
$97$ $$T^{8} + 544 T^{6} + \cdots + 5158686976$$