# Properties

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

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## 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: $$\Z[a_1, \ldots, a_{17}]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 + 1) q^{2} - \beta_1 q^{4} + ( - 2 \beta_{6} - 2 \beta_{5}) q^{5} - q^{8}+O(q^{10})$$ q + (-b1 + 1) * q^2 - b1 * q^4 + (-2*b6 - 2*b5) * q^5 - q^8 $$q + ( - \beta_1 + 1) q^{2} - \beta_1 q^{4} + ( - 2 \beta_{6} - 2 \beta_{5}) q^{5} - q^{8} - 2 \beta_{7} q^{10} + (\beta_{4} + 2) q^{11} + ( - 2 \beta_{7} - 2 \beta_{5} + 2 \beta_{3}) q^{13} + (\beta_1 - 1) q^{16} + (5 \beta_{7} - 3 \beta_{5} + 3 \beta_{3}) q^{17} + ( - 3 \beta_{7} + 3 \beta_{6} + 3 \beta_{5} - 2 \beta_{3}) q^{19} + ( - 2 \beta_{7} + 2 \beta_{6} + 2 \beta_{5}) q^{20} + (\beta_{4} - \beta_{2} - 2 \beta_1 + 2) q^{22} + (2 \beta_{4} - 2) q^{23} + (4 \beta_{4} + 3) q^{25} + ( - 2 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} + 2 \beta_{3}) q^{26} - 4 \beta_1 q^{29} + ( - 2 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} - 4 \beta_{3}) q^{31} + \beta_1 q^{32} + (5 \beta_{7} - 5 \beta_{6} - 5 \beta_{5} + 3 \beta_{3}) q^{34} + (2 \beta_{2} - 4 \beta_1) q^{37} + (3 \beta_{6} + \beta_{5}) q^{38} + (2 \beta_{6} + 2 \beta_{5}) q^{40} + (3 \beta_{7} - 5 \beta_{5} + 5 \beta_{3}) q^{41} + ( - \beta_{2} - 2 \beta_1) q^{43} + ( - \beta_{2} - 2 \beta_1) q^{44} + (2 \beta_{4} - 2 \beta_{2} + 2 \beta_1 - 2) q^{46} + (4 \beta_{7} + 2 \beta_{5} - 2 \beta_{3}) q^{47} + (4 \beta_{4} - 4 \beta_{2} - 3 \beta_1 + 3) q^{50} + (2 \beta_{6} + 4 \beta_{5}) q^{52} + ( - 4 \beta_{4} + 4 \beta_{2} - 4 \beta_1 + 4) q^{53} + ( - 6 \beta_{6} - 8 \beta_{5}) q^{55} - 4 q^{58} + (3 \beta_{7} - 3 \beta_{6} - 3 \beta_{5} - 2 \beta_{3}) q^{59} + ( - 8 \beta_{7} + 6 \beta_{5} - 6 \beta_{3}) q^{61} + (2 \beta_{6} - 2 \beta_{5}) q^{62} + q^{64} + (8 \beta_{4} - 8 \beta_{2} - 12 \beta_1 + 12) q^{65} + (2 \beta_{2} - 9 \beta_1) q^{67} + ( - 5 \beta_{6} - 2 \beta_{5}) q^{68} + (2 \beta_{4} - 6) q^{71} + (5 \beta_{7} - 2 \beta_{5} + 2 \beta_{3}) q^{73} + (2 \beta_{4} - 4) q^{74} + (3 \beta_{7} - 2 \beta_{5} + 2 \beta_{3}) q^{76} + (4 \beta_{4} - 4 \beta_{2} + 2 \beta_1 - 2) q^{79} + 2 \beta_{7} q^{80} + (3 \beta_{7} - 3 \beta_{6} - 3 \beta_{5} + 5 \beta_{3}) q^{82} + (10 \beta_{7} - 10 \beta_{6} - 10 \beta_{5} + \beta_{3}) q^{83} + ( - 4 \beta_{4} + 4 \beta_{2} + 14 \beta_1 - 14) q^{85} + ( - \beta_{4} - 2) q^{86} + ( - \beta_{4} - 2) q^{88} + 5 \beta_{3} q^{89} + ( - 2 \beta_{2} + 2 \beta_1) q^{92} + (4 \beta_{7} - 4 \beta_{6} - 4 \beta_{5} - 2 \beta_{3}) q^{94} + ( - 2 \beta_{2} - 8 \beta_1) q^{95} + (5 \beta_{7} - 5 \beta_{6} - 5 \beta_{5} + 11 \beta_{3}) q^{97}+O(q^{100})$$ q + (-b1 + 1) * q^2 - b1 * q^4 + (-2*b6 - 2*b5) * q^5 - q^8 - 2*b7 * q^10 + (b4 + 2) * q^11 + (-2*b7 - 2*b5 + 2*b3) * q^13 + (b1 - 1) * q^16 + (5*b7 - 3*b5 + 3*b3) * q^17 + (-3*b7 + 3*b6 + 3*b5 - 2*b3) * q^19 + (-2*b7 + 2*b6 + 2*b5) * q^20 + (b4 - b2 - 2*b1 + 2) * q^22 + (2*b4 - 2) * q^23 + (4*b4 + 3) * q^25 + (-2*b7 + 2*b6 + 2*b5 + 2*b3) * q^26 - 4*b1 * q^29 + (-2*b7 + 2*b6 + 2*b5 - 4*b3) * q^31 + b1 * q^32 + (5*b7 - 5*b6 - 5*b5 + 3*b3) * q^34 + (2*b2 - 4*b1) * q^37 + (3*b6 + b5) * q^38 + (2*b6 + 2*b5) * q^40 + (3*b7 - 5*b5 + 5*b3) * q^41 + (-b2 - 2*b1) * q^43 + (-b2 - 2*b1) * q^44 + (2*b4 - 2*b2 + 2*b1 - 2) * q^46 + (4*b7 + 2*b5 - 2*b3) * q^47 + (4*b4 - 4*b2 - 3*b1 + 3) * q^50 + (2*b6 + 4*b5) * q^52 + (-4*b4 + 4*b2 - 4*b1 + 4) * q^53 + (-6*b6 - 8*b5) * q^55 - 4 * q^58 + (3*b7 - 3*b6 - 3*b5 - 2*b3) * q^59 + (-8*b7 + 6*b5 - 6*b3) * q^61 + (2*b6 - 2*b5) * q^62 + q^64 + (8*b4 - 8*b2 - 12*b1 + 12) * q^65 + (2*b2 - 9*b1) * q^67 + (-5*b6 - 2*b5) * q^68 + (2*b4 - 6) * q^71 + (5*b7 - 2*b5 + 2*b3) * q^73 + (2*b4 - 4) * q^74 + (3*b7 - 2*b5 + 2*b3) * q^76 + (4*b4 - 4*b2 + 2*b1 - 2) * q^79 + 2*b7 * q^80 + (3*b7 - 3*b6 - 3*b5 + 5*b3) * q^82 + (10*b7 - 10*b6 - 10*b5 + b3) * q^83 + (-4*b4 + 4*b2 + 14*b1 - 14) * q^85 + (-b4 - 2) * q^86 + (-b4 - 2) * q^88 + 5*b3 * q^89 + (-2*b2 + 2*b1) * q^92 + (4*b7 - 4*b6 - 4*b5 - 2*b3) * q^94 + (-2*b2 - 8*b1) * q^95 + (5*b7 - 5*b6 - 5*b5 + 11*b3) * 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 $$8 q + 4 q^{2} - 4 q^{4} - 8 q^{8} + 16 q^{11} - 4 q^{16} + 8 q^{22} - 16 q^{23} + 24 q^{25} - 16 q^{29} + 4 q^{32} - 16 q^{37} - 8 q^{43} - 8 q^{44} - 8 q^{46} + 12 q^{50} + 16 q^{53} - 32 q^{58} + 8 q^{64} + 48 q^{65} - 36 q^{67} - 48 q^{71} - 32 q^{74} - 8 q^{79} - 56 q^{85} - 16 q^{86} - 16 q^{88} + 8 q^{92} - 32 q^{95}+O(q^{100})$$ 8 * q + 4 * q^2 - 4 * q^4 - 8 * q^8 + 16 * q^11 - 4 * q^16 + 8 * q^22 - 16 * q^23 + 24 * q^25 - 16 * q^29 + 4 * q^32 - 16 * q^37 - 8 * q^43 - 8 * q^44 - 8 * q^46 + 12 * q^50 + 16 * q^53 - 32 * q^58 + 8 * q^64 + 48 * q^65 - 36 * q^67 - 48 * q^71 - 32 * q^74 - 8 * q^79 - 56 * q^85 - 16 * q^86 - 16 * q^88 + 8 * q^92 - 32 * 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 −3.86370 0 0 −1.00000 0 −1.93185 + 3.34607i
361.2 0.500000 0.866025i 0 −0.500000 0.866025i −1.03528 0 0 −1.00000 0 −0.517638 + 0.896575i
361.3 0.500000 0.866025i 0 −0.500000 0.866025i 1.03528 0 0 −1.00000 0 0.517638 0.896575i
361.4 0.500000 0.866025i 0 −0.500000 0.866025i 3.86370 0 0 −1.00000 0 1.93185 3.34607i
667.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −3.86370 0 0 −1.00000 0 −1.93185 3.34607i
667.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −1.03528 0 0 −1.00000 0 −0.517638 0.896575i
667.3 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 1.03528 0 0 −1.00000 0 0.517638 + 0.896575i
667.4 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 3.86370 0 0 −1.00000 0 1.93185 + 3.34607i
 $$n$$: 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.t 8
3.b odd 2 1 882.2.h.q 8
7.b odd 2 1 inner 2646.2.h.t 8
7.c even 3 1 2646.2.e.q 8
7.c even 3 1 2646.2.f.r 8
7.d odd 6 1 2646.2.e.q 8
7.d odd 6 1 2646.2.f.r 8
9.c even 3 1 2646.2.e.q 8
9.d odd 6 1 882.2.e.s 8
21.c even 2 1 882.2.h.q 8
21.g even 6 1 882.2.e.s 8
21.g even 6 1 882.2.f.q 8
21.h odd 6 1 882.2.e.s 8
21.h odd 6 1 882.2.f.q 8
63.g even 3 1 inner 2646.2.h.t 8
63.g even 3 1 7938.2.a.ci 4
63.h even 3 1 2646.2.f.r 8
63.i even 6 1 882.2.f.q 8
63.j odd 6 1 882.2.f.q 8
63.k odd 6 1 inner 2646.2.h.t 8
63.k odd 6 1 7938.2.a.ci 4
63.l odd 6 1 2646.2.e.q 8
63.n odd 6 1 882.2.h.q 8
63.n odd 6 1 7938.2.a.cp 4
63.o even 6 1 882.2.e.s 8
63.s even 6 1 882.2.h.q 8
63.s even 6 1 7938.2.a.cp 4
63.t odd 6 1 2646.2.f.r 8

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

 $$T_{5}^{4} - 16T_{5}^{2} + 16$$ T5^4 - 16*T5^2 + 16 $$T_{11}^{2} - 4T_{11} + 1$$ T11^2 - 4*T11 + 1 $$T_{13}^{8} + 48T_{13}^{6} + 2160T_{13}^{4} + 6912T_{13}^{2} + 20736$$ T13^8 + 48*T13^6 + 2160*T13^4 + 6912*T13^2 + 20736

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - T + 1)^{4}$$
$3$ $$T^{8}$$
$5$ $$(T^{4} - 16 T^{2} + 16)^{2}$$
$7$ $$T^{8}$$
$11$ $$(T^{2} - 4 T + 1)^{4}$$
$13$ $$T^{8} + 48 T^{6} + 2160 T^{4} + \cdots + 20736$$
$17$ $$T^{8} + 76 T^{6} + 4407 T^{4} + \cdots + 1874161$$
$19$ $$T^{8} + 28 T^{6} + 615 T^{4} + \cdots + 28561$$
$23$ $$(T^{2} + 4 T - 8)^{4}$$
$29$ $$(T^{2} + 4 T + 16)^{4}$$
$31$ $$T^{8} + 48 T^{6} + 2160 T^{4} + \cdots + 20736$$
$37$ $$(T^{4} + 8 T^{3} + 60 T^{2} + 32 T + 16)^{2}$$
$41$ $$T^{8} + 76 T^{6} + 5655 T^{4} + \cdots + 14641$$
$43$ $$(T^{4} + 4 T^{3} + 15 T^{2} + 4 T + 1)^{2}$$
$47$ $$T^{8} + 112 T^{6} + 12480 T^{4} + \cdots + 4096$$
$53$ $$(T^{4} - 8 T^{3} + 96 T^{2} + 256 T + 1024)^{2}$$
$59$ $$T^{8} + 76 T^{6} + 5655 T^{4} + \cdots + 14641$$
$61$ $$T^{8} + 208 T^{6} + \cdots + 59969536$$
$67$ $$(T^{4} + 18 T^{3} + 255 T^{2} + 1242 T + 4761)^{2}$$
$71$ $$(T^{2} + 12 T + 24)^{4}$$
$73$ $$T^{8} + 76 T^{6} + 4407 T^{4} + \cdots + 1874161$$
$79$ $$(T^{4} + 4 T^{3} + 60 T^{2} - 176 T + 1936)^{2}$$
$83$ $$T^{8} + 364 T^{6} + \cdots + 193877776$$
$89$ $$(T^{4} + 50 T^{2} + 2500)^{2}$$
$97$ $$T^{8} + 364 T^{6} + \cdots + 131079601$$
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