# Properties

 Label 2848.1.cc.a Level $2848$ Weight $1$ Character orbit 2848.cc Analytic conductor $1.421$ Analytic rank $0$ Dimension $20$ Projective image $D_{44}$ CM discriminant -8 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2848 = 2^{5} \cdot 89$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2848.cc (of order $$44$$, degree $$20$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.42133715598$$ Analytic rank: $$0$$ Dimension: $$20$$ Coefficient field: $$\Q(\zeta_{44})$$ Defining polynomial: $$x^{20} - x^{18} + x^{16} - x^{14} + x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 712) Projective image: $$D_{44}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{44} - \cdots)$$

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{44} - \zeta_{44}^{20} ) q^{3} + ( \zeta_{44}^{2} - \zeta_{44}^{18} + \zeta_{44}^{21} ) q^{9} +O(q^{10})$$ $$q + ( -\zeta_{44} - \zeta_{44}^{20} ) q^{3} + ( \zeta_{44}^{2} - \zeta_{44}^{18} + \zeta_{44}^{21} ) q^{9} + ( -\zeta_{44}^{11} + \zeta_{44}^{15} ) q^{11} + ( \zeta_{44}^{5} - \zeta_{44}^{11} ) q^{17} + ( \zeta_{44} - \zeta_{44}^{8} ) q^{19} + \zeta_{44}^{18} q^{25} + ( 1 - \zeta_{44}^{3} - \zeta_{44}^{16} + \zeta_{44}^{19} ) q^{27} + ( -\zeta_{44}^{9} + \zeta_{44}^{12} + \zeta_{44}^{13} - \zeta_{44}^{16} ) q^{33} + ( \zeta_{44}^{6} - \zeta_{44}^{17} ) q^{41} + ( \zeta_{44}^{5} - \zeta_{44}^{10} ) q^{43} -\zeta_{44}^{7} q^{49} + ( \zeta_{44}^{3} - \zeta_{44}^{6} - \zeta_{44}^{9} + \zeta_{44}^{12} ) q^{51} + ( -\zeta_{44}^{2} - \zeta_{44}^{6} + \zeta_{44}^{9} - \zeta_{44}^{21} ) q^{57} + ( \zeta_{44}^{9} + \zeta_{44}^{14} ) q^{59} + ( -\zeta_{44}^{15} - \zeta_{44}^{17} ) q^{67} + ( -\zeta_{44}^{5} - \zeta_{44}^{19} ) q^{73} + ( \zeta_{44}^{16} - \zeta_{44}^{19} ) q^{75} + ( -\zeta_{44} + \zeta_{44}^{4} - \zeta_{44}^{14} + \zeta_{44}^{17} - \zeta_{44}^{20} ) q^{81} + ( \zeta_{44}^{13} - \zeta_{44}^{14} ) q^{83} -\zeta_{44}^{4} q^{89} + ( \zeta_{44}^{8} - \zeta_{44}^{10} ) q^{97} + ( -\zeta_{44}^{7} + \zeta_{44}^{10} + \zeta_{44}^{11} - \zeta_{44}^{13} - \zeta_{44}^{14} + \zeta_{44}^{17} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20q + 2q^{3} + O(q^{10})$$ $$20q + 2q^{3} + 2q^{19} + 2q^{25} + 22q^{27} + 2q^{41} - 2q^{43} - 4q^{51} - 4q^{57} + 2q^{59} - 2q^{75} - 2q^{81} - 2q^{83} + 2q^{89} - 4q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2848\mathbb{Z}\right)^\times$$.

 $$n$$ $$357$$ $$1247$$ $$1249$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$\zeta_{44}^{21}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
47.1
 0.755750 − 0.654861i 0.989821 + 0.142315i 0.755750 + 0.654861i 0.909632 + 0.415415i −0.281733 + 0.959493i 0.281733 − 0.959493i −0.909632 − 0.415415i −0.755750 − 0.654861i −0.989821 − 0.142315i −0.755750 + 0.654861i −0.909632 + 0.415415i −0.540641 − 0.841254i −0.540641 + 0.841254i 0.281733 + 0.959493i −0.989821 + 0.142315i 0.989821 − 0.142315i −0.281733 − 0.959493i 0.540641 − 0.841254i 0.540641 + 0.841254i 0.909632 − 0.415415i
0 −0.613435 + 1.64468i 0 0 0 0 0 −1.57293 1.36295i 0
79.1 0 −0.0303285 0.424047i 0 0 0 0 0 0.810925 0.116593i 0
303.1 0 −0.613435 1.64468i 0 0 0 0 0 −1.57293 + 1.36295i 0
335.1 0 −0.254771 1.17116i 0 0 0 0 0 −0.397086 + 0.181343i 0
463.1 0 −0.559521 0.418852i 0 0 0 0 0 −0.144106 0.490780i 0
783.1 0 −1.12299 + 1.50013i 0 0 0 0 0 −0.707571 2.40977i 0
911.1 0 1.56449 0.340335i 0 0 0 0 0 1.42218 0.649487i 0
943.1 0 0.898064 0.334961i 0 0 0 0 0 −0.0614286 + 0.0532282i 0
1167.1 0 1.94931 0.139418i 0 0 0 0 0 2.79057 0.401223i 0
1199.1 0 0.898064 + 0.334961i 0 0 0 0 0 −0.0614286 0.0532282i 0
1263.1 0 1.56449 + 0.340335i 0 0 0 0 0 1.42218 + 0.649487i 0
1295.1 0 0.125226 0.0683785i 0 0 0 0 0 −0.529635 + 0.824128i 0
1711.1 0 0.125226 + 0.0683785i 0 0 0 0 0 −0.529635 0.824128i 0
1775.1 0 −1.12299 1.50013i 0 0 0 0 0 −0.707571 + 2.40977i 0
1967.1 0 1.94931 + 0.139418i 0 0 0 0 0 2.79057 + 0.401223i 0
2127.1 0 −0.0303285 + 0.424047i 0 0 0 0 0 0.810925 + 0.116593i 0
2319.1 0 −0.559521 + 0.418852i 0 0 0 0 0 −0.144106 + 0.490780i 0
2383.1 0 −0.956056 + 1.75089i 0 0 0 0 0 −1.61092 2.50664i 0
2799.1 0 −0.956056 1.75089i 0 0 0 0 0 −1.61092 + 2.50664i 0
2831.1 0 −0.254771 + 1.17116i 0 0 0 0 0 −0.397086 0.181343i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2831.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$
89.g even 44 1 inner
712.y odd 44 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2848.1.cc.a 20
4.b odd 2 1 712.1.y.a 20
8.b even 2 1 712.1.y.a 20
8.d odd 2 1 CM 2848.1.cc.a 20
89.g even 44 1 inner 2848.1.cc.a 20
356.n odd 44 1 712.1.y.a 20
712.y odd 44 1 inner 2848.1.cc.a 20
712.z even 44 1 712.1.y.a 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
712.1.y.a 20 4.b odd 2 1
712.1.y.a 20 8.b even 2 1
712.1.y.a 20 356.n odd 44 1
712.1.y.a 20 712.z even 44 1
2848.1.cc.a 20 1.a even 1 1 trivial
2848.1.cc.a 20 8.d odd 2 1 CM
2848.1.cc.a 20 89.g even 44 1 inner
2848.1.cc.a 20 712.y odd 44 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(2848, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{20}$$
$3$ $$1 - 12 T + 50 T^{2} - 44 T^{3} + 250 T^{4} - 74 T^{5} - 51 T^{6} - 638 T^{7} + 713 T^{8} - 394 T^{9} + 824 T^{10} - 626 T^{11} + 214 T^{12} - 286 T^{13} + 179 T^{14} - 36 T^{15} + 40 T^{16} - 22 T^{17} + 2 T^{18} - 2 T^{19} + T^{20}$$
$5$ $$T^{20}$$
$7$ $$T^{20}$$
$11$ $$121 - 605 T^{2} + 1089 T^{4} + 484 T^{8} + 462 T^{10} + 330 T^{12} + 165 T^{14} + 55 T^{16} + 11 T^{18} + T^{20}$$
$13$ $$T^{20}$$
$17$ $$1 - 25 T^{2} + 185 T^{4} - 236 T^{6} + 224 T^{8} + 54 T^{10} + 102 T^{12} + 57 T^{14} + 27 T^{16} + 7 T^{18} + T^{20}$$
$19$ $$1 - 12 T + 105 T^{2} - 484 T^{3} + 1218 T^{4} - 1702 T^{5} + 1324 T^{6} - 484 T^{7} - 178 T^{8} + 420 T^{9} - 331 T^{10} + 122 T^{11} + 93 T^{12} - 8 T^{14} + 8 T^{15} - 4 T^{16} + 2 T^{18} - 2 T^{19} + T^{20}$$
$23$ $$T^{20}$$
$29$ $$T^{20}$$
$31$ $$T^{20}$$
$37$ $$T^{20}$$
$41$ $$1024 - 1024 T + 512 T^{2} - 256 T^{4} + 256 T^{5} - 128 T^{6} + 64 T^{8} - 64 T^{9} + 32 T^{10} - 32 T^{11} + 16 T^{12} - 8 T^{14} + 8 T^{15} - 4 T^{16} + 2 T^{18} - 2 T^{19} + T^{20}$$
$43$ $$1 - 10 T + 94 T^{2} - 462 T^{3} + 1361 T^{4} - 2412 T^{5} + 2413 T^{6} - 1100 T^{7} + 53 T^{8} - 2 T^{9} + 32 T^{10} + 32 T^{11} + 16 T^{12} - 8 T^{14} - 8 T^{15} - 4 T^{16} + 2 T^{18} + 2 T^{19} + T^{20}$$
$47$ $$T^{20}$$
$53$ $$T^{20}$$
$59$ $$1 - 12 T + 61 T^{2} - 66 T^{3} + 63 T^{4} + 454 T^{5} - 403 T^{6} + 176 T^{7} + 328 T^{8} - 658 T^{9} + 494 T^{10} - 164 T^{11} + 148 T^{12} - 66 T^{13} - 8 T^{14} + 8 T^{15} - 4 T^{16} + 2 T^{18} - 2 T^{19} + T^{20}$$
$61$ $$T^{20}$$
$67$ $$121 + 605 T^{2} + 484 T^{4} - 968 T^{6} + 484 T^{8} + 99 T^{10} + 165 T^{12} + 22 T^{16} + T^{20}$$
$71$ $$T^{20}$$
$73$ $$121 + 242 T^{2} + 1331 T^{4} + 1331 T^{6} + 121 T^{8} - 22 T^{10} + 154 T^{12} - 22 T^{14} + T^{20}$$
$79$ $$T^{20}$$
$83$ $$1 - 10 T + 17 T^{2} + 44 T^{3} + 338 T^{4} + 316 T^{5} + 400 T^{6} + 110 T^{7} - 90 T^{8} + 460 T^{9} + 505 T^{10} + 274 T^{11} + 93 T^{12} - 44 T^{13} - 52 T^{14} - 30 T^{15} - 4 T^{16} + 2 T^{18} + 2 T^{19} + T^{20}$$
$89$ $$( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2}$$
$97$ $$( 1 - 5 T + 3 T^{2} + 7 T^{3} + 20 T^{4} + 10 T^{5} + 16 T^{6} + 8 T^{7} + 4 T^{8} + 2 T^{9} + T^{10} )^{2}$$