# Properties

 Label 712.1.y.a Level $712$ Weight $1$ Character orbit 712.y Analytic conductor $0.355$ Analytic rank $0$ Dimension $20$ Projective image $D_{44}$ CM discriminant -8 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$712 = 2^{3} \cdot 89$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 712.y (of order $$44$$, degree $$20$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.355334288995$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q -\zeta_{44}^{20} q^{2} + ( -\zeta_{44}^{3} + \zeta_{44}^{16} ) q^{3} -\zeta_{44}^{18} q^{4} + ( -\zeta_{44} + \zeta_{44}^{14} ) q^{6} -\zeta_{44}^{16} q^{8} + ( \zeta_{44}^{6} - \zeta_{44}^{10} - \zeta_{44}^{19} ) q^{9} +O(q^{10})$$ $$q -\zeta_{44}^{20} q^{2} + ( -\zeta_{44}^{3} + \zeta_{44}^{16} ) q^{3} -\zeta_{44}^{18} q^{4} + ( -\zeta_{44} + \zeta_{44}^{14} ) q^{6} -\zeta_{44}^{16} q^{8} + ( \zeta_{44}^{6} - \zeta_{44}^{10} - \zeta_{44}^{19} ) q^{9} + ( \zeta_{44} + \zeta_{44}^{11} ) q^{11} + ( \zeta_{44}^{12} + \zeta_{44}^{21} ) q^{12} -\zeta_{44}^{14} q^{16} + ( -\zeta_{44}^{11} - \zeta_{44}^{15} ) q^{17} + ( \zeta_{44}^{4} - \zeta_{44}^{8} - \zeta_{44}^{17} ) q^{18} + ( -\zeta_{44}^{2} + \zeta_{44}^{3} ) q^{19} + ( \zeta_{44}^{9} - \zeta_{44}^{21} ) q^{22} + ( \zeta_{44}^{10} + \zeta_{44}^{19} ) q^{24} + \zeta_{44}^{10} q^{25} + ( -1 + \zeta_{44}^{4} - \zeta_{44}^{9} + \zeta_{44}^{13} ) q^{27} -\zeta_{44}^{12} q^{32} + ( -\zeta_{44}^{4} - \zeta_{44}^{5} - \zeta_{44}^{14} + \zeta_{44}^{17} ) q^{33} + ( -\zeta_{44}^{9} - \zeta_{44}^{13} ) q^{34} + ( \zeta_{44}^{2} - \zeta_{44}^{6} - \zeta_{44}^{15} ) q^{36} + ( -1 + \zeta_{44} ) q^{38} + ( \zeta_{44}^{7} + \zeta_{44}^{18} ) q^{41} + ( -\zeta_{44}^{8} + \zeta_{44}^{15} ) q^{43} + ( \zeta_{44}^{7} - \zeta_{44}^{19} ) q^{44} + ( \zeta_{44}^{8} + \zeta_{44}^{17} ) q^{48} + \zeta_{44}^{21} q^{49} + \zeta_{44}^{8} q^{50} + ( \zeta_{44}^{5} + \zeta_{44}^{9} + \zeta_{44}^{14} + \zeta_{44}^{18} ) q^{51} + ( \zeta_{44}^{2} - \zeta_{44}^{7} + \zeta_{44}^{11} + \zeta_{44}^{20} ) q^{54} + ( \zeta_{44}^{5} - \zeta_{44}^{6} - \zeta_{44}^{18} + \zeta_{44}^{19} ) q^{57} + ( -\zeta_{44}^{5} + \zeta_{44}^{20} ) q^{59} -\zeta_{44}^{10} q^{64} + ( -\zeta_{44}^{2} - \zeta_{44}^{3} - \zeta_{44}^{12} + \zeta_{44}^{15} ) q^{66} + ( -\zeta_{44} - \zeta_{44}^{7} ) q^{67} + ( -\zeta_{44}^{7} - \zeta_{44}^{11} ) q^{68} + ( 1 - \zeta_{44}^{4} - \zeta_{44}^{13} ) q^{72} + ( \zeta_{44}^{13} + \zeta_{44}^{15} ) q^{73} + ( -\zeta_{44}^{4} - \zeta_{44}^{13} ) q^{75} + ( \zeta_{44}^{20} - \zeta_{44}^{21} ) q^{76} + ( \zeta_{44}^{3} - \zeta_{44}^{7} + \zeta_{44}^{12} - \zeta_{44}^{16} + \zeta_{44}^{20} ) q^{81} + ( \zeta_{44}^{5} + \zeta_{44}^{16} ) q^{82} + ( -\zeta_{44}^{17} - \zeta_{44}^{20} ) q^{83} + ( -\zeta_{44}^{6} + \zeta_{44}^{13} ) q^{86} + ( \zeta_{44}^{5} - \zeta_{44}^{17} ) q^{88} -\zeta_{44}^{12} q^{89} + ( \zeta_{44}^{6} + \zeta_{44}^{15} ) q^{96} + ( -\zeta_{44}^{2} + \zeta_{44}^{8} ) q^{97} + \zeta_{44}^{19} q^{98} + ( \zeta_{44}^{7} + \zeta_{44}^{8} - \zeta_{44}^{11} + \zeta_{44}^{17} - \zeta_{44}^{20} - \zeta_{44}^{21} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20q + 2q^{2} - 2q^{3} - 2q^{4} + 2q^{6} + 2q^{8} + O(q^{10})$$ $$20q + 2q^{2} - 2q^{3} - 2q^{4} + 2q^{6} + 2q^{8} - 2q^{12} - 2q^{16} - 2q^{19} + 2q^{24} + 2q^{25} - 22q^{27} + 2q^{32} - 20q^{38} + 2q^{41} + 2q^{43} - 2q^{48} - 2q^{50} + 4q^{51} - 4q^{57} - 2q^{59} - 2q^{64} + 22q^{72} + 2q^{75} - 2q^{76} - 2q^{81} - 2q^{82} + 2q^{83} - 2q^{86} + 2q^{89} + 2q^{96} - 4q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/712\mathbb{Z}\right)^\times$$.

 $$n$$ $$357$$ $$535$$ $$537$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-\zeta_{44}^{19}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
99.1
 −0.540641 + 0.841254i 0.909632 − 0.415415i −0.281733 + 0.959493i −0.540641 − 0.841254i 0.989821 − 0.142315i −0.755750 + 0.654861i 0.755750 − 0.654861i −0.989821 + 0.142315i 0.540641 + 0.841254i 0.281733 − 0.959493i −0.909632 + 0.415415i 0.540641 − 0.841254i 0.909632 + 0.415415i 0.989821 + 0.142315i −0.281733 − 0.959493i 0.755750 + 0.654861i −0.755750 − 0.654861i 0.281733 + 0.959493i −0.989821 − 0.142315i −0.909632 − 0.415415i
−0.415415 + 0.909632i −1.94931 + 0.139418i −0.654861 0.755750i 0 0.682956 1.83107i 0 0.959493 0.281733i 2.79057 0.401223i 0
107.1 0.654861 + 0.755750i 0.559521 + 0.418852i −0.142315 + 0.989821i 0 0.0498610 + 0.697148i 0 −0.841254 + 0.540641i −0.144106 0.490780i 0
131.1 −0.841254 + 0.540641i −0.898064 0.334961i 0.415415 0.909632i 0 0.936593 0.203743i 0 0.142315 + 0.989821i −0.0614286 0.0532282i 0
187.1 −0.415415 0.909632i −1.94931 0.139418i −0.654861 + 0.755750i 0 0.682956 + 1.83107i 0 0.959493 + 0.281733i 2.79057 + 0.401223i 0
195.1 0.959493 + 0.281733i −1.56449 0.340335i 0.841254 + 0.540641i 0 −1.40524 0.767317i 0 0.654861 + 0.755750i 1.42218 + 0.649487i 0
227.1 0.142315 + 0.989821i −0.125226 + 0.0683785i −0.959493 + 0.281733i 0 −0.0855040 0.114220i 0 −0.415415 0.909632i −0.529635 + 0.824128i 0
307.1 0.142315 + 0.989821i 0.956056 + 1.75089i −0.959493 + 0.281733i 0 −1.59700 + 1.19550i 0 −0.415415 0.909632i −1.61092 + 2.50664i 0
339.1 0.959493 + 0.281733i 0.254771 1.17116i 0.841254 + 0.540641i 0 0.574406 1.05195i 0 0.654861 + 0.755750i −0.397086 0.181343i 0
347.1 −0.415415 0.909632i 0.0303285 0.424047i −0.654861 + 0.755750i 0 −0.398326 + 0.148568i 0 0.959493 + 0.281733i 0.810925 + 0.116593i 0
403.1 −0.841254 + 0.540641i 0.613435 1.64468i 0.415415 0.909632i 0 0.373128 + 1.71524i 0 0.142315 + 0.989821i −1.57293 1.36295i 0
427.1 0.654861 + 0.755750i 1.12299 1.50013i −0.142315 + 0.989821i 0 1.86912 0.133682i 0 −0.841254 + 0.540641i −0.707571 2.40977i 0
435.1 −0.415415 + 0.909632i 0.0303285 + 0.424047i −0.654861 0.755750i 0 −0.398326 0.148568i 0 0.959493 0.281733i 0.810925 0.116593i 0
539.1 0.654861 0.755750i 0.559521 0.418852i −0.142315 0.989821i 0 0.0498610 0.697148i 0 −0.841254 0.540641i −0.144106 + 0.490780i 0
555.1 0.959493 0.281733i −1.56449 + 0.340335i 0.841254 0.540641i 0 −1.40524 + 0.767317i 0 0.654861 0.755750i 1.42218 0.649487i 0
587.1 −0.841254 0.540641i −0.898064 + 0.334961i 0.415415 + 0.909632i 0 0.936593 + 0.203743i 0 0.142315 0.989821i −0.0614286 + 0.0532282i 0
603.1 0.142315 0.989821i 0.956056 1.75089i −0.959493 0.281733i 0 −1.59700 1.19550i 0 −0.415415 + 0.909632i −1.61092 2.50664i 0
643.1 0.142315 0.989821i −0.125226 0.0683785i −0.959493 0.281733i 0 −0.0855040 + 0.114220i 0 −0.415415 + 0.909632i −0.529635 0.824128i 0
659.1 −0.841254 0.540641i 0.613435 + 1.64468i 0.415415 + 0.909632i 0 0.373128 1.71524i 0 0.142315 0.989821i −1.57293 + 1.36295i 0
691.1 0.959493 0.281733i 0.254771 + 1.17116i 0.841254 0.540641i 0 0.574406 + 1.05195i 0 0.654861 0.755750i −0.397086 + 0.181343i 0
707.1 0.654861 0.755750i 1.12299 + 1.50013i −0.142315 0.989821i 0 1.86912 + 0.133682i 0 −0.841254 0.540641i −0.707571 + 2.40977i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 707.1 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 712.1.y.a 20
4.b odd 2 1 2848.1.cc.a 20
8.b even 2 1 2848.1.cc.a 20
8.d odd 2 1 CM 712.1.y.a 20
89.g even 44 1 inner 712.1.y.a 20
356.n odd 44 1 2848.1.cc.a 20
712.y odd 44 1 inner 712.1.y.a 20
712.z even 44 1 2848.1.cc.a 20

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

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(712, [\chi])$$.

## Hecke characteristic polynomials

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