LMFDB - Embedded newform 850.2.d.i.849.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [850,2,Mod(849,850)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("850.849"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(850, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 850 = 2 \cdot 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	850.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,-4,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$6.78728417181$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 34)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			849.4
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	850.849
Dual form			850.2.d.i.849.2

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000i q^{2} +2.82843 q^{3} -1.00000 q^{4} +2.82843i q^{6} -1.00000i q^{8} +5.00000 q^{9} +2.82843i q^{11} -2.82843 q^{12} +2.00000i q^{13} +1.00000 q^{16} +(-2.82843 + 3.00000i) q^{17} +5.00000i q^{18} +4.00000 q^{19} -2.82843 q^{22} +5.65685 q^{23} -2.82843i q^{24} -2.00000 q^{26} +5.65685 q^{27} -2.82843i q^{29} +1.00000i q^{32} +8.00000i q^{33} +(-3.00000 - 2.82843i) q^{34} -5.00000 q^{36} +8.48528 q^{37} +4.00000i q^{38} +5.65685i q^{39} -5.65685i q^{41} -4.00000i q^{43} -2.82843i q^{44} +5.65685i q^{46} +2.82843 q^{48} -7.00000 q^{49} +(-8.00000 + 8.48528i) q^{51} -2.00000i q^{52} +6.00000i q^{53} +5.65685i q^{54} +11.3137 q^{57} +2.82843 q^{58} -12.0000 q^{59} -8.48528i q^{61} -1.00000 q^{64} -8.00000 q^{66} +4.00000i q^{67} +(2.82843 - 3.00000i) q^{68} +16.0000 q^{69} -5.65685i q^{71} -5.00000i q^{72} +8.48528i q^{74} -4.00000 q^{76} -5.65685 q^{78} -16.9706i q^{79} +1.00000 q^{81} +5.65685 q^{82} -12.0000i q^{83} +4.00000 q^{86} -8.00000i q^{87} +2.82843 q^{88} -6.00000 q^{89} -5.65685 q^{92} +2.82843i q^{96} -16.9706 q^{97} -7.00000i q^{98} +14.1421i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} + 20 q^{9} + 4 q^{16} + 16 q^{19} - 8 q^{26} - 12 q^{34} - 20 q^{36} - 28 q^{49} - 32 q^{51} - 48 q^{59} - 4 q^{64} - 32 q^{66} + 64 q^{69} - 16 q^{76} + 4 q^{81} + 16 q^{86} - 24 q^{89}+O(q^{100}) $ 4 * q - 4 * q^4 + 20 * q^9 + 4 * q^16 + 16 * q^19 - 8 * q^26 - 12 * q^34 - 20 * q^36 - 28 * q^49 - 32 * q^51 - 48 * q^59 - 4 * q^64 - 32 * q^66 + 64 * q^69 - 16 * q^76 + 4 * q^81 + 16 * q^86 - 24 * q^89

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/850\mathbb{Z}\right)^\times$.

$n$	$477$	$751$
$\chi(n)$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$			1.00000i			0.707107i
$2$			1.00000i			0.707107i
$3$	2.82843			1.63299			0.816497	−	0.577350i	$-0.195913\pi$
$3$	2.82843			1.63299			0.816497	+	0.577350i	$0.195913\pi$
$4$	−1.00000			−0.500000
$5$		0			0
$5$		0			0
$6$			2.82843i			1.15470i
$7$		0			0			−	1.00000i	$-0.5\pi$
$7$		0			0				1.00000i	$0.5\pi$
$8$		−	1.00000i		−	0.353553i
$9$	5.00000			1.66667
$10$		0			0
$11$			2.82843i			0.852803i	0.904534	+	0.426401i	$0.140219\pi$
$11$			2.82843i			0.852803i	−0.904534	+	0.426401i	$0.859781\pi$
$12$	−2.82843			−0.816497
$13$			2.00000i			0.554700i	0.960769	+	0.277350i	$0.0894562\pi$
$13$			2.00000i			0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$14$		0			0
$15$		0			0
$16$	1.00000			0.250000
$17$	−2.82843	+	3.00000i	−0.685994	+	0.727607i
$17$	−2.82843	+	3.00000i	−0.685994	+	0.727607i
$18$			5.00000i			1.17851i
$19$	4.00000			0.917663			0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000			0.917663			0.458831	+	0.888523i	$0.348268\pi$
$20$		0			0
$21$		0			0
$22$	−2.82843			−0.603023
$23$	5.65685			1.17954			0.589768	−	0.807573i	$-0.299219\pi$
$23$	5.65685			1.17954			0.589768	+	0.807573i	$0.299219\pi$
$24$		−	2.82843i		−	0.577350i
$25$		0			0
$26$	−2.00000			−0.392232
$27$	5.65685			1.08866
$28$		0			0
$29$		−	2.82843i		−	0.525226i	−0.964901	−	0.262613i	$-0.915416\pi$
$29$		−	2.82843i		−	0.525226i	0.964901	−	0.262613i	$-0.0845842\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$			1.00000i			0.176777i
$33$			8.00000i			1.39262i
$34$	−3.00000	−	2.82843i	−0.514496	−	0.485071i
$35$		0			0
$36$	−5.00000			−0.833333
$37$	8.48528			1.39497			0.697486	−	0.716599i	$-0.254302\pi$
$37$	8.48528			1.39497			0.697486	+	0.716599i	$0.254302\pi$
$38$			4.00000i			0.648886i
$39$			5.65685i			0.905822i
$40$		0			0
$41$		−	5.65685i		−	0.883452i	−0.897150	−	0.441726i	$-0.854366\pi$
$41$		−	5.65685i		−	0.883452i	0.897150	−	0.441726i	$-0.145634\pi$
$42$		0			0
$43$		−	4.00000i		−	0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$		−	4.00000i		−	0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$		−	2.82843i		−	0.426401i
$45$		0			0
$46$			5.65685i			0.834058i
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$	2.82843			0.408248
$49$	−7.00000			−1.00000
$50$		0			0
$51$	−8.00000	+	8.48528i	−1.12022	+	1.18818i
$52$		−	2.00000i		−	0.277350i
$53$			6.00000i			0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$			6.00000i			0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$			5.65685i			0.769800i
$55$		0			0
$56$		0			0
$57$	11.3137			1.49854
$58$	2.82843			0.371391
$59$	−12.0000			−1.56227			−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000			−1.56227			−0.781133	+	0.624364i	$0.785358\pi$
$60$		0			0
$61$		−	8.48528i		−	1.08643i	−0.839594	−	0.543214i	$-0.817207\pi$
$61$		−	8.48528i		−	1.08643i	0.839594	−	0.543214i	$-0.182793\pi$
$62$		0			0
$63$		0			0
$64$	−1.00000			−0.125000
$65$		0			0
$66$	−8.00000			−0.984732
$67$			4.00000i			0.488678i	0.969690	+	0.244339i	$0.0785709\pi$
$67$			4.00000i			0.488678i	−0.969690	+	0.244339i	$0.921429\pi$
$68$	2.82843	−	3.00000i	0.342997	−	0.363803i
$69$	16.0000			1.92617
$70$		0			0
$71$		−	5.65685i		−	0.671345i	−0.941979	−	0.335673i	$-0.891036\pi$
$71$		−	5.65685i		−	0.671345i	0.941979	−	0.335673i	$-0.108964\pi$
$72$		−	5.00000i		−	0.589256i
$73$		0			0			−	1.00000i	$-0.5\pi$
$73$		0			0				1.00000i	$0.5\pi$
$74$			8.48528i			0.986394i
$75$		0			0
$76$	−4.00000			−0.458831
$77$		0			0
$78$	−5.65685			−0.640513
$79$		−	16.9706i		−	1.90934i	−0.297670	−	0.954669i	$-0.596210\pi$
$79$		−	16.9706i		−	1.90934i	0.297670	−	0.954669i	$-0.403790\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$	5.65685			0.624695
$83$		−	12.0000i		−	1.31717i	−0.752506	−	0.658586i	$-0.771155\pi$
$83$		−	12.0000i		−	1.31717i	0.752506	−	0.658586i	$-0.228845\pi$
$84$		0			0
$85$		0			0
$86$	4.00000			0.431331
$87$		−	8.00000i		−	0.857690i
$88$	2.82843			0.301511
$89$	−6.00000			−0.635999			−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000			−0.635999			−0.317999	+	0.948091i	$0.603011\pi$
$90$		0			0
$91$		0			0
$92$	−5.65685			−0.589768
$93$		0			0
$94$		0			0
$95$		0			0
$96$			2.82843i			0.288675i
$97$	−16.9706			−1.72310			−0.861550	−	0.507673i	$-0.830506\pi$
$97$	−16.9706			−1.72310			−0.861550	+	0.507673i	$0.830506\pi$
$98$		−	7.00000i		−	0.707107i
$99$			14.1421i			1.42134i
$100$		0			0
$101$	−6.00000			−0.597022			−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000			−0.597022			−0.298511	+	0.954406i	$0.596490\pi$
$102$	−8.48528	−	8.00000i	−0.840168	−	0.792118i
$103$			8.00000i			0.788263i	0.919054	+	0.394132i	$0.128955\pi$
$103$			8.00000i			0.788263i	−0.919054	+	0.394132i	$0.871045\pi$
$104$	2.00000			0.196116
$105$		0			0
$106$	−6.00000			−0.582772
$107$	2.82843			0.273434			0.136717	−	0.990610i	$-0.456345\pi$
$107$	2.82843			0.273434			0.136717	+	0.990610i	$0.456345\pi$
$108$	−5.65685			−0.544331
$109$		−	8.48528i		−	0.812743i	−0.913708	−	0.406371i	$-0.866794\pi$
$109$		−	8.48528i		−	0.812743i	0.913708	−	0.406371i	$-0.133206\pi$
$110$		0			0
$111$	24.0000			2.27798
$112$		0			0
$113$	−11.3137			−1.06430			−0.532152	−	0.846649i	$-0.678617\pi$
$113$	−11.3137			−1.06430			−0.532152	+	0.846649i	$0.678617\pi$
$114$			11.3137i			1.05963i
$115$		0			0
$116$			2.82843i			0.262613i
$117$			10.0000i			0.924500i
$118$		−	12.0000i		−	1.10469i
$119$		0			0
$120$		0			0
$121$	3.00000			0.272727
$122$	8.48528			0.768221
$123$		−	16.0000i		−	1.44267i
$124$		0			0
$125$		0			0
$126$		0			0
$127$			16.0000i			1.41977i	0.704317	+	0.709885i	$0.251253\pi$
$127$			16.0000i			1.41977i	−0.704317	+	0.709885i	$0.748747\pi$
$128$		−	1.00000i		−	0.0883883i
$129$		−	11.3137i		−	0.996116i
$130$		0			0
$131$			2.82843i			0.247121i	0.992337	+	0.123560i	$0.0394313\pi$
$131$			2.82843i			0.247121i	−0.992337	+	0.123560i	$0.960569\pi$
$132$		−	8.00000i		−	0.696311i
$133$		0			0
$134$	−4.00000			−0.345547
$135$		0			0
$136$	3.00000	+	2.82843i	0.257248	+	0.242536i
$137$			18.0000i			1.53784i	0.639343	+	0.768922i	$0.279207\pi$
$137$			18.0000i			1.53784i	−0.639343	+	0.768922i	$0.720793\pi$
$138$			16.0000i			1.36201i
$139$			8.48528i			0.719712i	0.933008	+	0.359856i	$0.117174\pi$
$139$			8.48528i			0.719712i	−0.933008	+	0.359856i	$0.882826\pi$
$140$		0			0
$141$		0			0
$142$	5.65685			0.474713
$143$	−5.65685			−0.473050
$144$	5.00000			0.416667
$145$		0			0
$146$		0			0
$147$	−19.7990			−1.63299
$148$	−8.48528			−0.697486
$149$	−6.00000			−0.491539			−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000			−0.491539			−0.245770	+	0.969328i	$0.579041\pi$
$150$		0			0
$151$	8.00000			0.651031			0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000			0.651031			0.325515	+	0.945537i	$0.394462\pi$
$152$		−	4.00000i		−	0.324443i
$153$	−14.1421	+	15.0000i	−1.14332	+	1.21268i
$154$		0			0
$155$		0			0
$156$		−	5.65685i		−	0.452911i
$157$		−	14.0000i		−	1.11732i	−0.829396	−	0.558661i	$-0.811315\pi$
$157$		−	14.0000i		−	1.11732i	0.829396	−	0.558661i	$-0.188685\pi$
$158$	16.9706			1.35011
$159$			16.9706i			1.34585i
$160$		0			0
$161$		0			0
$162$			1.00000i			0.0785674i
$163$	−8.48528			−0.664619			−0.332309	−	0.943170i	$-0.607828\pi$
$163$	−8.48528			−0.664619			−0.332309	+	0.943170i	$0.607828\pi$
$164$			5.65685i			0.441726i
$165$		0			0
$166$	12.0000			0.931381
$167$	11.3137			0.875481			0.437741	−	0.899101i	$-0.355779\pi$
$167$	11.3137			0.875481			0.437741	+	0.899101i	$0.355779\pi$
$168$		0			0
$169$	9.00000			0.692308
$170$		0			0
$171$	20.0000			1.52944
$172$			4.00000i			0.304997i
$173$	−2.82843			−0.215041			−0.107521	−	0.994203i	$-0.534291\pi$
$173$	−2.82843			−0.215041			−0.107521	+	0.994203i	$0.534291\pi$
$174$	8.00000			0.606478
$175$		0			0
$176$			2.82843i			0.213201i
$177$	−33.9411			−2.55117
$178$		−	6.00000i		−	0.449719i
$179$	−12.0000			−0.896922			−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000			−0.896922			−0.448461	+	0.893802i	$0.648028\pi$
$180$		0			0
$181$		−	8.48528i		−	0.630706i	−0.948974	−	0.315353i	$-0.897877\pi$
$181$		−	8.48528i		−	0.630706i	0.948974	−	0.315353i	$-0.102123\pi$
$182$		0			0
$183$		−	24.0000i		−	1.77413i
$184$		−	5.65685i		−	0.417029i
$185$		0			0
$186$		0			0
$187$	−8.48528	−	8.00000i	−0.620505	−	0.585018i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$	−2.82843			−0.204124
$193$	16.9706			1.22157			0.610784	−	0.791797i	$-0.290854\pi$
$193$	16.9706			1.22157			0.610784	+	0.791797i	$0.290854\pi$
$194$		−	16.9706i		−	1.21842i
$195$		0			0
$196$	7.00000			0.500000
$197$	2.82843			0.201517			0.100759	−	0.994911i	$-0.467873\pi$
$197$	2.82843			0.201517			0.100759	+	0.994911i	$0.467873\pi$
$198$	−14.1421			−1.00504
$199$		−	16.9706i		−	1.20301i	−0.798869	−	0.601506i	$-0.794568\pi$
$199$		−	16.9706i		−	1.20301i	0.798869	−	0.601506i	$-0.205432\pi$
$200$		0			0
$201$			11.3137i			0.798007i
$202$		−	6.00000i		−	0.422159i
$203$		0			0
$204$	8.00000	−	8.48528i	0.560112	−	0.594089i
$205$		0			0
$206$	−8.00000			−0.557386
$207$	28.2843			1.96589
$208$			2.00000i			0.138675i
$209$			11.3137i			0.782586i
$210$		0			0
$211$		−	8.48528i		−	0.584151i	−0.956395	−	0.292075i	$-0.905654\pi$
$211$		−	8.48528i		−	0.584151i	0.956395	−	0.292075i	$-0.0943458\pi$
$212$		−	6.00000i		−	0.412082i
$213$		−	16.0000i		−	1.09630i
$214$			2.82843i			0.193347i
$215$		0			0
$216$		−	5.65685i		−	0.384900i
$217$		0			0
$218$	8.48528			0.574696
$219$		0			0
$220$		0			0
$221$	−6.00000	−	5.65685i	−0.403604	−	0.380521i
$222$			24.0000i			1.61077i
$223$		−	16.0000i		−	1.07144i	−0.844396	−	0.535720i	$-0.820040\pi$
$223$		−	16.0000i		−	1.07144i	0.844396	−	0.535720i	$-0.179960\pi$
$224$		0			0
$225$		0			0
$226$		−	11.3137i		−	0.752577i
$227$	19.7990			1.31411			0.657053	−	0.753845i	$-0.271803\pi$
$227$	19.7990			1.31411			0.657053	+	0.753845i	$0.271803\pi$
$228$	−11.3137			−0.749269
$229$	22.0000			1.45380			0.726900	−	0.686743i	$-0.240960\pi$
$229$	22.0000			1.45380			0.726900	+	0.686743i	$0.240960\pi$
$230$		0			0
$231$		0			0
$232$	−2.82843			−0.185695
$233$	22.6274			1.48237			0.741186	−	0.671300i	$-0.234264\pi$
$233$	22.6274			1.48237			0.741186	+	0.671300i	$0.234264\pi$
$234$	−10.0000			−0.653720
$235$		0			0
$236$	12.0000			0.781133
$237$		−	48.0000i		−	3.11794i
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$		−	16.9706i		−	1.09317i	−0.837404	−	0.546585i	$-0.815928\pi$
$241$		−	16.9706i		−	1.09317i	0.837404	−	0.546585i	$-0.184072\pi$
$242$			3.00000i			0.192847i
$243$	−14.1421			−0.907218
$244$			8.48528i			0.543214i
$245$		0			0
$246$	16.0000			1.02012
$247$			8.00000i			0.509028i
$248$		0			0
$249$		−	33.9411i		−	2.15093i
$250$		0			0
$251$	−12.0000			−0.757433			−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000			−0.757433			−0.378717	+	0.925513i	$0.623635\pi$
$252$		0			0
$253$			16.0000i			1.00591i
$254$	−16.0000			−1.00393
$255$		0			0
$256$	1.00000			0.0625000
$257$		−	6.00000i		−	0.374270i	−0.982334	−	0.187135i	$-0.940080\pi$
$257$		−	6.00000i		−	0.374270i	0.982334	−	0.187135i	$-0.0599201\pi$
$258$	11.3137			0.704361
$259$		0			0
$260$		0			0
$261$		−	14.1421i		−	0.875376i
$262$	−2.82843			−0.174741
$263$			24.0000i			1.47990i	0.672660	+	0.739952i	$0.265152\pi$
$263$			24.0000i			1.47990i	−0.672660	+	0.739952i	$0.734848\pi$
$264$	8.00000			0.492366
$265$		0			0
$266$		0			0
$267$	−16.9706			−1.03858
$268$		−	4.00000i		−	0.244339i
$269$			31.1127i			1.89697i	0.316815	+	0.948487i	$0.397387\pi$
$269$			31.1127i			1.89697i	−0.316815	+	0.948487i	$0.602613\pi$
$270$		0			0
$271$	−16.0000			−0.971931			−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000			−0.971931			−0.485965	+	0.873978i	$0.661532\pi$
$272$	−2.82843	+	3.00000i	−0.171499	+	0.181902i
$273$		0			0
$274$	−18.0000			−1.08742
$275$		0			0
$276$	−16.0000			−0.963087
$277$	25.4558			1.52949			0.764747	−	0.644331i	$-0.222864\pi$
$277$	25.4558			1.52949			0.764747	+	0.644331i	$0.222864\pi$
$278$	−8.48528			−0.508913
$279$		0			0
$280$		0			0
$281$	18.0000			1.07379			0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000			1.07379			0.536895	+	0.843649i	$0.319597\pi$
$282$		0			0
$283$	8.48528			0.504398			0.252199	−	0.967675i	$-0.418846\pi$
$283$	8.48528			0.504398			0.252199	+	0.967675i	$0.418846\pi$
$284$			5.65685i			0.335673i
$285$		0			0
$286$		−	5.65685i		−	0.334497i
$287$		0			0
$288$			5.00000i			0.294628i
$289$	−1.00000	−	16.9706i	−0.0588235	−	0.998268i
$290$		0			0
$291$	−48.0000			−2.81381
$292$		0			0
$293$			6.00000i			0.350524i	0.984522	+	0.175262i	$0.0560772\pi$
$293$			6.00000i			0.350524i	−0.984522	+	0.175262i	$0.943923\pi$
$294$		−	19.7990i		−	1.15470i
$295$		0			0
$296$		−	8.48528i		−	0.493197i
$297$			16.0000i			0.928414i
$298$		−	6.00000i		−	0.347571i
$299$			11.3137i			0.654289i
$300$		0			0
$301$		0			0
$302$			8.00000i			0.460348i
$303$	−16.9706			−0.974933
$304$	4.00000			0.229416
$305$		0			0
$306$	−15.0000	−	14.1421i	−0.857493	−	0.808452i
$307$		−	20.0000i		−	1.14146i	−0.821138	−	0.570730i	$-0.806660\pi$
$307$		−	20.0000i		−	1.14146i	0.821138	−	0.570730i	$-0.193340\pi$
$308$		0			0
$309$			22.6274i			1.28723i
$310$		0			0
$311$			11.3137i			0.641542i	0.947157	+	0.320771i	$0.103942\pi$
$311$			11.3137i			0.641542i	−0.947157	+	0.320771i	$0.896058\pi$
$312$	5.65685			0.320256
$313$	−16.9706			−0.959233			−0.479616	−	0.877478i	$-0.659224\pi$
$313$	−16.9706			−0.959233			−0.479616	+	0.877478i	$0.659224\pi$
$314$	14.0000			0.790066
$315$		0			0
$316$			16.9706i			0.954669i
$317$	2.82843			0.158860			0.0794301	−	0.996840i	$-0.474690\pi$
$317$	2.82843			0.158860			0.0794301	+	0.996840i	$0.474690\pi$
$318$	−16.9706			−0.951662
$319$	8.00000			0.447914
$320$		0			0
$321$	8.00000			0.446516
$322$		0			0
$323$	−11.3137	+	12.0000i	−0.629512	+	0.667698i
$324$	−1.00000			−0.0555556
$325$		0			0
$326$		−	8.48528i		−	0.469956i
$327$		−	24.0000i		−	1.32720i
$328$	−5.65685			−0.312348
$329$		0			0
$330$		0			0
$331$	−4.00000			−0.219860			−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000			−0.219860			−0.109930	+	0.993939i	$0.535063\pi$
$332$			12.0000i			0.658586i
$333$	42.4264			2.32495
$334$			11.3137i			0.619059i
$335$		0			0
$336$		0			0
$337$		0			0			−	1.00000i	$-0.5\pi$
$337$		0			0				1.00000i	$0.5\pi$
$338$			9.00000i			0.489535i
$339$	−32.0000			−1.73800
$340$		0			0
$341$		0			0
$342$			20.0000i			1.08148i
$343$		0			0
$344$	−4.00000			−0.215666
$345$		0			0
$346$		−	2.82843i		−	0.152057i
$347$	−14.1421			−0.759190			−0.379595	−	0.925153i	$-0.623937\pi$
$347$	−14.1421			−0.759190			−0.379595	+	0.925153i	$0.623937\pi$
$348$			8.00000i			0.428845i
$349$	34.0000			1.81998			0.909989	−	0.414632i	$-0.136090\pi$
$349$	34.0000			1.81998			0.909989	+	0.414632i	$0.136090\pi$
$350$		0			0
$351$			11.3137i			0.603881i
$352$	−2.82843			−0.150756
$353$		−	6.00000i		−	0.319348i	−0.987170	−	0.159674i	$-0.948956\pi$
$353$		−	6.00000i		−	0.319348i	0.987170	−	0.159674i	$-0.0510443\pi$
$354$		−	33.9411i		−	1.80395i
$355$		0			0
$356$	6.00000			0.317999
$357$		0			0
$358$		−	12.0000i		−	0.634220i
$359$	−24.0000			−1.26667			−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000			−1.26667			−0.633336	+	0.773877i	$0.718315\pi$
$360$		0			0
$361$	−3.00000			−0.157895
$362$	8.48528			0.445976
$363$	8.48528			0.445362
$364$		0			0
$365$		0			0
$366$	24.0000			1.25450
$367$	−16.9706			−0.885856			−0.442928	−	0.896557i	$-0.646060\pi$
$367$	−16.9706			−0.885856			−0.442928	+	0.896557i	$0.646060\pi$
$368$	5.65685			0.294884
$369$		−	28.2843i		−	1.47242i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		−	22.0000i		−	1.13912i	−0.821951	−	0.569558i	$-0.807114\pi$
$373$		−	22.0000i		−	1.13912i	0.821951	−	0.569558i	$-0.192886\pi$
$374$	8.00000	−	8.48528i	0.413670	−	0.438763i
$375$		0			0
$376$		0			0
$377$	5.65685			0.291343
$378$		0			0
$379$			25.4558i			1.30758i	0.756677	+	0.653789i	$0.226822\pi$
$379$			25.4558i			1.30758i	−0.756677	+	0.653789i	$0.773178\pi$
$380$		0			0
$381$			45.2548i			2.31848i
$382$		0			0
$383$		0			0		1.00000			$0$
$383$		0			0		−1.00000			$\pi$
$384$		−	2.82843i		−	0.144338i
$385$		0			0
$386$			16.9706i			0.863779i
$387$		−	20.0000i		−	1.01666i
$388$	16.9706			0.861550
$389$	6.00000			0.304212			0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000			0.304212			0.152106	+	0.988364i	$0.451394\pi$
$390$		0			0
$391$	−16.0000	+	16.9706i	−0.809155	+	0.858238i
$392$			7.00000i			0.353553i
$393$			8.00000i			0.403547i
$394$			2.82843i			0.142494i
$395$		0			0
$396$		−	14.1421i		−	0.710669i
$397$	−8.48528			−0.425864			−0.212932	−	0.977067i	$-0.568301\pi$
$397$	−8.48528			−0.425864			−0.212932	+	0.977067i	$0.568301\pi$
$398$	16.9706			0.850657
$399$		0			0
$400$		0			0
$401$			28.2843i			1.41245i	0.707988	+	0.706225i	$0.249603\pi$
$401$			28.2843i			1.41245i	−0.707988	+	0.706225i	$0.750397\pi$
$402$	−11.3137			−0.564276
$403$		0			0
$404$	6.00000			0.298511
$405$		0			0
$406$		0			0
$407$			24.0000i			1.18964i
$408$	8.48528	+	8.00000i	0.420084	+	0.396059i
$409$	−2.00000			−0.0988936			−0.0494468	−	0.998777i	$-0.515746\pi$
$409$	−2.00000			−0.0988936			−0.0494468	+	0.998777i	$0.515746\pi$
$410$		0			0
$411$			50.9117i			2.51129i
$412$		−	8.00000i		−	0.394132i
$413$		0			0
$414$			28.2843i			1.39010i
$415$		0			0
$416$	−2.00000			−0.0980581
$417$			24.0000i			1.17529i
$418$	−11.3137			−0.553372
$419$			14.1421i			0.690889i	0.938439	+	0.345444i	$0.112272\pi$
$419$			14.1421i			0.690889i	−0.938439	+	0.345444i	$0.887728\pi$
$420$		0			0
$421$	26.0000			1.26716			0.633581	−	0.773676i	$-0.281584\pi$
$421$	26.0000			1.26716			0.633581	+	0.773676i	$0.281584\pi$
$422$	8.48528			0.413057
$423$		0			0
$424$	6.00000			0.291386
$425$		0			0
$426$	16.0000			0.775203
$427$		0			0
$428$	−2.82843			−0.136717
$429$	−16.0000			−0.772487
$430$		0			0
$431$		−	5.65685i		−	0.272481i	−0.990676	−	0.136241i	$-0.956498\pi$
$431$		−	5.65685i		−	0.272481i	0.990676	−	0.136241i	$-0.0435020\pi$
$432$	5.65685			0.272166
$433$			14.0000i			0.672797i	0.941720	+	0.336399i	$0.109209\pi$
$433$			14.0000i			0.672797i	−0.941720	+	0.336399i	$0.890791\pi$
$434$		0			0
$435$		0			0
$436$			8.48528i			0.406371i
$437$	22.6274			1.08242
$438$		0			0
$439$			16.9706i			0.809961i	0.914325	+	0.404980i	$0.132722\pi$
$439$			16.9706i			0.809961i	−0.914325	+	0.404980i	$0.867278\pi$
$440$		0			0
$441$	−35.0000			−1.66667
$442$	5.65685	−	6.00000i	0.269069	−	0.285391i
$443$		−	12.0000i		−	0.570137i	−0.958507	−	0.285069i	$-0.907984\pi$
$443$		−	12.0000i		−	0.570137i	0.958507	−	0.285069i	$-0.0920164\pi$
$444$	−24.0000			−1.13899
$445$		0			0
$446$	16.0000			0.757622
$447$	−16.9706			−0.802680
$448$		0			0
$449$			5.65685i			0.266963i	0.991051	+	0.133482i	$0.0426157\pi$
$449$			5.65685i			0.266963i	−0.991051	+	0.133482i	$0.957384\pi$
$450$		0			0
$451$	16.0000			0.753411
$452$	11.3137			0.532152
$453$	22.6274			1.06313
$454$			19.7990i			0.929213i
$455$		0			0
$456$		−	11.3137i		−	0.529813i
$457$			10.0000i			0.467780i	0.972263	+	0.233890i	$0.0751456\pi$
$457$			10.0000i			0.467780i	−0.972263	+	0.233890i	$0.924854\pi$
$458$			22.0000i			1.02799i
$459$	−16.0000	+	16.9706i	−0.746816	+	0.792118i
$460$		0			0
$461$	−18.0000			−0.838344			−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000			−0.838344			−0.419172	+	0.907907i	$0.637680\pi$
$462$		0			0
$463$			32.0000i			1.48717i	0.668644	+	0.743583i	$0.266875\pi$
$463$			32.0000i			1.48717i	−0.668644	+	0.743583i	$0.733125\pi$
$464$		−	2.82843i		−	0.131306i
$465$		0			0
$466$			22.6274i			1.04819i
$467$			36.0000i			1.66588i	0.553362	+	0.832941i	$0.313345\pi$
$467$			36.0000i			1.66588i	−0.553362	+	0.832941i	$0.686655\pi$
$468$		−	10.0000i		−	0.462250i
$469$		0			0
$470$		0			0
$471$		−	39.5980i		−	1.82458i
$472$			12.0000i			0.552345i
$473$	11.3137			0.520205
$474$	48.0000			2.20471
$475$		0			0
$476$		0			0
$477$			30.0000i			1.37361i
$478$		0			0
$479$			5.65685i			0.258468i	0.991614	+	0.129234i	$0.0412519\pi$
$479$			5.65685i			0.258468i	−0.991614	+	0.129234i	$0.958748\pi$
$480$		0			0
$481$			16.9706i			0.773791i
$482$	16.9706			0.772988
$483$		0			0
$484$	−3.00000			−0.136364
$485$		0			0
$486$		−	14.1421i		−	0.641500i
$487$	−16.9706			−0.769010			−0.384505	−	0.923123i	$-0.625628\pi$
$487$	−16.9706			−0.769010			−0.384505	+	0.923123i	$0.625628\pi$
$488$	−8.48528			−0.384111
$489$	−24.0000			−1.08532
$490$		0			0
$491$	−12.0000			−0.541552			−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000			−0.541552			−0.270776	+	0.962642i	$0.587280\pi$
$492$			16.0000i			0.721336i
$493$	8.48528	+	8.00000i	0.382158	+	0.360302i
$494$	−8.00000			−0.359937
$495$		0			0
$496$		0			0
$497$		0			0
$498$	33.9411			1.52094
$499$		−	25.4558i		−	1.13956i	−0.821797	−	0.569780i	$-0.807028\pi$
$499$		−	25.4558i		−	1.13956i	0.821797	−	0.569780i	$-0.192972\pi$
$500$		0			0
$501$	32.0000			1.42965
$502$		−	12.0000i		−	0.535586i
$503$	−28.2843			−1.26113			−0.630567	−	0.776135i	$-0.717177\pi$
$503$	−28.2843			−1.26113			−0.630567	+	0.776135i	$0.717177\pi$
$504$		0			0
$505$		0			0
$506$	−16.0000			−0.711287
$507$	25.4558			1.13053
$508$		−	16.0000i		−	0.709885i
$509$	18.0000			0.797836			0.398918	−	0.916987i	$-0.369386\pi$
$509$	18.0000			0.797836			0.398918	+	0.916987i	$0.369386\pi$
$510$		0			0
$511$		0			0
$512$			1.00000i			0.0441942i
$513$	22.6274			0.999025
$514$	6.00000			0.264649
$515$		0			0
$516$			11.3137i			0.498058i
$517$		0			0
$518$		0			0
$519$	−8.00000			−0.351161
$520$		0			0
$521$		−	5.65685i		−	0.247831i	−0.992293	−	0.123916i	$-0.960455\pi$
$521$		−	5.65685i		−	0.247831i	0.992293	−	0.123916i	$-0.0395452\pi$
$522$	14.1421			0.618984
$523$			20.0000i			0.874539i	0.899331	+	0.437269i	$0.144054\pi$
$523$			20.0000i			0.874539i	−0.899331	+	0.437269i	$0.855946\pi$
$524$		−	2.82843i		−	0.123560i
$525$		0			0
$526$	−24.0000			−1.04645
$527$		0			0
$528$			8.00000i			0.348155i
$529$	9.00000			0.391304
$530$		0			0
$531$	−60.0000			−2.60378
$532$		0			0
$533$	11.3137			0.490051
$534$		−	16.9706i		−	0.734388i
$535$		0			0
$536$	4.00000			0.172774
$537$	−33.9411			−1.46467
$538$	−31.1127			−1.34136
$539$		−	19.7990i		−	0.852803i
$540$		0			0
$541$			8.48528i			0.364811i	0.983223	+	0.182405i	$0.0583883\pi$
$541$			8.48528i			0.364811i	−0.983223	+	0.182405i	$0.941612\pi$
$542$		−	16.0000i		−	0.687259i
$543$		−	24.0000i		−	1.02994i
$544$	−3.00000	−	2.82843i	−0.128624	−	0.121268i
$545$		0			0
$546$		0			0
$547$	−42.4264			−1.81402			−0.907011	−	0.421107i	$-0.861642\pi$
$547$	−42.4264			−1.81402			−0.907011	+	0.421107i	$0.861642\pi$
$548$		−	18.0000i		−	0.768922i
$549$		−	42.4264i		−	1.81071i
$550$		0			0
$551$		−	11.3137i		−	0.481980i
$552$		−	16.0000i		−	0.681005i
$553$		0			0
$554$			25.4558i			1.08152i
$555$		0			0
$556$		−	8.48528i		−	0.359856i
$557$		−	18.0000i		−	0.762684i	−0.924434	−	0.381342i	$-0.875462\pi$
$557$		−	18.0000i		−	0.762684i	0.924434	−	0.381342i	$-0.124538\pi$
$558$		0			0
$559$	8.00000			0.338364
$560$		0			0
$561$	−24.0000	−	22.6274i	−1.01328	−	0.955330i
$562$			18.0000i			0.759284i
$563$			36.0000i			1.51722i	0.651546	+	0.758610i	$0.274121\pi$
$563$			36.0000i			1.51722i	−0.651546	+	0.758610i	$0.725879\pi$
$564$		0			0
$565$		0			0
$566$			8.48528i			0.356663i
$567$		0			0
$568$	−5.65685			−0.237356
$569$	6.00000			0.251533			0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000			0.251533			0.125767	+	0.992060i	$0.459861\pi$
$570$		0			0
$571$		−	25.4558i		−	1.06529i	−0.846338	−	0.532647i	$-0.821197\pi$
$571$		−	25.4558i		−	1.06529i	0.846338	−	0.532647i	$-0.178803\pi$
$572$	5.65685			0.236525
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−5.00000			−0.208333
$577$		−	38.0000i		−	1.58196i	−0.611842	−	0.790980i	$-0.709571\pi$
$577$		−	38.0000i		−	1.58196i	0.611842	−	0.790980i	$-0.290429\pi$
$578$	16.9706	−	1.00000i	0.705882	−	0.0415945i
$579$	48.0000			1.99481
$580$		0			0
$581$		0			0
$582$		−	48.0000i		−	1.98966i
$583$	−16.9706			−0.702849
$584$		0			0
$585$		0			0
$586$	−6.00000			−0.247858
$587$			12.0000i			0.495293i	0.968850	+	0.247647i	$0.0796572\pi$
$587$			12.0000i			0.495293i	−0.968850	+	0.247647i	$0.920343\pi$
$588$	19.7990			0.816497
$589$		0			0
$590$		0			0
$591$	8.00000			0.329076
$592$	8.48528			0.348743
$593$			18.0000i			0.739171i	0.929197	+	0.369586i	$0.120500\pi$
$593$			18.0000i			0.739171i	−0.929197	+	0.369586i	$0.879500\pi$
$594$	−16.0000			−0.656488
$595$		0			0
$596$	6.00000			0.245770
$597$		−	48.0000i		−	1.96451i
$598$	−11.3137			−0.462652
$599$	−24.0000			−0.980613			−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000			−0.980613			−0.490307	+	0.871550i	$0.663115\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$			20.0000i			0.814463i
$604$	−8.00000			−0.325515
$605$		0			0
$606$		−	16.9706i		−	0.689382i
$607$	33.9411			1.37763			0.688814	−	0.724938i	$-0.258132\pi$
$607$	33.9411			1.37763			0.688814	+	0.724938i	$0.258132\pi$
$608$			4.00000i			0.162221i
$609$		0			0
$610$		0			0
$611$		0			0
$612$	14.1421	−	15.0000i	0.571662	−	0.606339i
$613$		−	10.0000i		−	0.403896i	−0.979396	−	0.201948i	$-0.935273\pi$
$613$		−	10.0000i		−	0.403896i	0.979396	−	0.201948i	$-0.0647272\pi$
$614$	20.0000			0.807134
$615$		0			0
$616$		0			0
$617$	11.3137			0.455473			0.227736	−	0.973723i	$-0.426868\pi$
$617$	11.3137			0.455473			0.227736	+	0.973723i	$0.426868\pi$
$618$	−22.6274			−0.910208
$619$		−	8.48528i		−	0.341052i	−0.985353	−	0.170526i	$-0.945453\pi$
$619$		−	8.48528i		−	0.341052i	0.985353	−	0.170526i	$-0.0545467\pi$
$620$		0			0
$621$	32.0000			1.28412
$622$	−11.3137			−0.453638
$623$		0			0
$624$			5.65685i			0.226455i
$625$		0			0
$626$		−	16.9706i		−	0.678280i
$627$			32.0000i			1.27796i
$628$			14.0000i			0.558661i
$629$	−24.0000	+	25.4558i	−0.956943	+	1.01499i
$630$		0			0
$631$	8.00000			0.318475			0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000			0.318475			0.159237	+	0.987240i	$0.449096\pi$
$632$	−16.9706			−0.675053
$633$		−	24.0000i		−	0.953914i
$634$			2.82843i			0.112331i
$635$		0			0
$636$		−	16.9706i		−	0.672927i
$637$		−	14.0000i		−	0.554700i
$638$			8.00000i			0.316723i
$639$		−	28.2843i		−	1.11891i
$640$		0			0
$641$		−	5.65685i		−	0.223432i	−0.993740	−	0.111716i	$-0.964365\pi$
$641$		−	5.65685i		−	0.223432i	0.993740	−	0.111716i	$-0.0356347\pi$
$642$			8.00000i			0.315735i
$643$	−8.48528			−0.334627			−0.167313	−	0.985904i	$-0.553509\pi$
$643$	−8.48528			−0.334627			−0.167313	+	0.985904i	$0.553509\pi$
$644$		0			0
$645$		0			0
$646$	−12.0000	−	11.3137i	−0.472134	−	0.445132i
$647$			24.0000i			0.943537i	0.881722	+	0.471769i	$0.156384\pi$
$647$			24.0000i			0.943537i	−0.881722	+	0.471769i	$0.843616\pi$
$648$		−	1.00000i		−	0.0392837i
$649$		−	33.9411i		−	1.33231i
$650$		0			0
$651$		0			0
$652$	8.48528			0.332309
$653$	14.1421			0.553425			0.276712	−	0.960953i	$-0.410755\pi$
$653$	14.1421			0.553425			0.276712	+	0.960953i	$0.410755\pi$
$654$	24.0000			0.938474
$655$		0			0
$656$		−	5.65685i		−	0.220863i
$657$		0			0
$658$		0			0
$659$	12.0000			0.467454			0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000			0.467454			0.233727	+	0.972302i	$0.424908\pi$
$660$		0			0
$661$	−10.0000			−0.388955			−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000			−0.388955			−0.194477	+	0.980907i	$0.562301\pi$
$662$		−	4.00000i		−	0.155464i
$663$	−16.9706	−	16.0000i	−0.659082	−	0.621389i
$664$	−12.0000			−0.465690
$665$		0			0
$666$			42.4264i			1.64399i
$667$		−	16.0000i		−	0.619522i
$668$	−11.3137			−0.437741
$669$		−	45.2548i		−	1.74965i
$670$		0			0
$671$	24.0000			0.926510
$672$		0			0
$673$	−16.9706			−0.654167			−0.327084	−	0.944995i	$-0.606066\pi$
$673$	−16.9706			−0.654167			−0.327084	+	0.944995i	$0.606066\pi$
$674$		0			0
$675$		0			0
$676$	−9.00000			−0.346154
$677$	−14.1421			−0.543526			−0.271763	−	0.962364i	$-0.587607\pi$
$677$	−14.1421			−0.543526			−0.271763	+	0.962364i	$0.587607\pi$
$678$		−	32.0000i		−	1.22895i
$679$		0			0
$680$		0			0
$681$	56.0000			2.14592
$682$		0			0
$683$	−19.7990			−0.757587			−0.378794	−	0.925481i	$-0.623661\pi$
$683$	−19.7990			−0.757587			−0.378794	+	0.925481i	$0.623661\pi$
$684$	−20.0000			−0.764719
$685$		0			0
$686$		0			0
$687$	62.2254			2.37405
$688$		−	4.00000i		−	0.152499i
$689$	−12.0000			−0.457164
$690$		0			0
$691$		−	8.48528i		−	0.322795i	−0.986889	−	0.161398i	$-0.948400\pi$
$691$		−	8.48528i		−	0.322795i	0.986889	−	0.161398i	$-0.0516002\pi$
$692$	2.82843			0.107521
$693$		0			0
$694$		−	14.1421i		−	0.536828i
$695$		0			0
$696$	−8.00000			−0.303239
$697$	16.9706	+	16.0000i	0.642806	+	0.606043i
$698$			34.0000i			1.28692i
$699$	64.0000			2.42070
$700$		0			0
$701$	−30.0000			−1.13308			−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000			−1.13308			−0.566542	+	0.824033i	$0.691719\pi$
$702$	−11.3137			−0.427008
$703$	33.9411			1.28011
$704$		−	2.82843i		−	0.106600i
$705$		0			0
$706$	6.00000			0.225813
$707$		0			0
$708$	33.9411			1.27559
$709$			8.48528i			0.318671i	0.987224	+	0.159336i	$0.0509352\pi$
$709$			8.48528i			0.318671i	−0.987224	+	0.159336i	$0.949065\pi$
$710$		0			0
$711$		−	84.8528i		−	3.18223i
$712$			6.00000i			0.224860i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	12.0000			0.448461
$717$		0			0
$718$		−	24.0000i		−	0.895672i
$719$			39.5980i			1.47676i	0.674387	+	0.738378i	$0.264408\pi$
$719$			39.5980i			1.47676i	−0.674387	+	0.738378i	$0.735592\pi$
$720$		0			0
$721$		0			0
$722$		−	3.00000i		−	0.111648i
$723$		−	48.0000i		−	1.78514i
$724$			8.48528i			0.315353i
$725$		0			0
$726$			8.48528i			0.314918i
$727$		−	8.00000i		−	0.296704i	−0.988935	−	0.148352i	$-0.952603\pi$
$727$		−	8.00000i		−	0.296704i	0.988935	−	0.148352i	$-0.0473968\pi$
$728$		0			0
$729$	−43.0000			−1.59259
$730$		0			0
$731$	12.0000	+	11.3137i	0.443836	+	0.418453i
$732$			24.0000i			0.887066i
$733$			14.0000i			0.517102i	0.965998	+	0.258551i	$0.0832450\pi$
$733$			14.0000i			0.517102i	−0.965998	+	0.258551i	$0.916755\pi$
$734$		−	16.9706i		−	0.626395i
$735$		0			0
$736$			5.65685i			0.208514i
$737$	−11.3137			−0.416746
$738$	28.2843			1.04116
$739$	−44.0000			−1.61857			−0.809283	−	0.587419i	$-0.800144\pi$
$739$	−44.0000			−1.61857			−0.809283	+	0.587419i	$0.800144\pi$
$740$		0			0
$741$			22.6274i			0.831239i
$742$		0			0
$743$	−11.3137			−0.415060			−0.207530	−	0.978229i	$-0.566542\pi$
$743$	−11.3137			−0.415060			−0.207530	+	0.978229i	$0.566542\pi$
$744$		0			0
$745$		0			0
$746$	22.0000			0.805477
$747$		−	60.0000i		−	2.19529i
$748$	8.48528	+	8.00000i	0.310253	+	0.292509i
$749$		0			0
$750$		0			0
$751$			16.9706i			0.619265i	0.950856	+	0.309632i	$0.100206\pi$
$751$			16.9706i			0.619265i	−0.950856	+	0.309632i	$0.899794\pi$
$752$		0			0
$753$	−33.9411			−1.23688
$754$			5.65685i			0.206010i
$755$		0			0
$756$		0			0
$757$			22.0000i			0.799604i	0.916602	+	0.399802i	$0.130921\pi$
$757$			22.0000i			0.799604i	−0.916602	+	0.399802i	$0.869079\pi$
$758$	−25.4558			−0.924598
$759$			45.2548i			1.64265i
$760$		0			0
$761$	30.0000			1.08750			0.543750	−	0.839248i	$-0.317004\pi$
$761$	30.0000			1.08750			0.543750	+	0.839248i	$0.317004\pi$
$762$	−45.2548			−1.63941
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	24.0000i		−	0.866590i
$768$	2.82843			0.102062
$769$	10.0000			0.360609			0.180305	−	0.983611i	$-0.442292\pi$
$769$	10.0000			0.360609			0.180305	+	0.983611i	$0.442292\pi$
$770$		0			0
$771$		−	16.9706i		−	0.611180i
$772$	−16.9706			−0.610784
$773$		−	54.0000i		−	1.94225i	−0.238581	−	0.971123i	$-0.576682\pi$
$773$		−	54.0000i		−	1.94225i	0.238581	−	0.971123i	$-0.423318\pi$
$774$	20.0000			0.718885
$775$		0			0
$776$			16.9706i			0.609208i
$777$		0			0
$778$			6.00000i			0.215110i
$779$		−	22.6274i		−	0.810711i
$780$		0			0
$781$	16.0000			0.572525
$782$	−16.9706	−	16.0000i	−0.606866	−	0.572159i
$783$		−	16.0000i		−	0.571793i
$784$	−7.00000			−0.250000
$785$		0			0
$786$	−8.00000			−0.285351
$787$	−42.4264			−1.51234			−0.756169	−	0.654376i	$-0.772931\pi$
$787$	−42.4264			−1.51234			−0.756169	+	0.654376i	$0.772931\pi$
$788$	−2.82843			−0.100759
$789$			67.8823i			2.41667i
$790$		0			0
$791$		0			0
$792$	14.1421			0.502519
$793$	16.9706			0.602642
$794$		−	8.48528i		−	0.301131i
$795$		0			0
$796$			16.9706i			0.601506i
$797$			18.0000i			0.637593i	0.947823	+	0.318796i	$0.103279\pi$
$797$			18.0000i			0.637593i	−0.947823	+	0.318796i	$0.896721\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−30.0000			−1.06000
$802$	−28.2843			−0.998752
$803$		0			0
$804$		−	11.3137i		−	0.399004i
$805$		0			0
$806$		0			0
$807$			88.0000i			3.09775i
$808$			6.00000i			0.211079i
$809$			39.5980i			1.39219i	0.717949	+	0.696095i	$0.245081\pi$
$809$			39.5980i			1.39219i	−0.717949	+	0.696095i	$0.754919\pi$
$810$		0			0
$811$			42.4264i			1.48979i	0.667180	+	0.744896i	$0.267501\pi$
$811$			42.4264i			1.48979i	−0.667180	+	0.744896i	$0.732499\pi$
$812$		0			0
$813$	−45.2548			−1.58716
$814$	−24.0000			−0.841200
$815$		0			0
$816$	−8.00000	+	8.48528i	−0.280056	+	0.297044i
$817$		−	16.0000i		−	0.559769i
$818$		−	2.00000i		−	0.0699284i
$819$		0			0
$820$		0			0
$821$		−	48.0833i		−	1.67812i	−0.544041	−	0.839059i	$-0.683106\pi$
$821$		−	48.0833i		−	1.67812i	0.544041	−	0.839059i	$-0.316894\pi$
$822$	−50.9117			−1.77575
$823$	−16.9706			−0.591557			−0.295778	−	0.955257i	$-0.595579\pi$
$823$	−16.9706			−0.591557			−0.295778	+	0.955257i	$0.595579\pi$
$824$	8.00000			0.278693
$825$		0			0
$826$		0			0
$827$	36.7696			1.27860			0.639301	−	0.768956i	$-0.279224\pi$
$827$	36.7696			1.27860			0.639301	+	0.768956i	$0.279224\pi$
$828$	−28.2843			−0.982946
$829$	34.0000			1.18087			0.590434	−	0.807086i	$-0.298956\pi$
$829$	34.0000			1.18087			0.590434	+	0.807086i	$0.298956\pi$
$830$		0			0
$831$	72.0000			2.49765
$832$		−	2.00000i		−	0.0693375i
$833$	19.7990	−	21.0000i	0.685994	−	0.727607i
$834$	−24.0000			−0.831052
$835$		0			0
$836$		−	11.3137i		−	0.391293i
$837$		0			0
$838$	−14.1421			−0.488532
$839$		−	45.2548i		−	1.56237i	−0.624299	−	0.781185i	$-0.714615\pi$
$839$		−	45.2548i		−	1.56237i	0.624299	−	0.781185i	$-0.285385\pi$
$840$		0			0
$841$	21.0000			0.724138
$842$			26.0000i			0.896019i
$843$	50.9117			1.75349
$844$			8.48528i			0.292075i
$845$		0			0
$846$		0			0
$847$		0			0
$848$			6.00000i			0.206041i
$849$	24.0000			0.823678
$850$		0			0
$851$	48.0000			1.64542
$852$			16.0000i			0.548151i
$853$	−25.4558			−0.871592			−0.435796	−	0.900046i	$-0.643533\pi$
$853$	−25.4558			−0.871592			−0.435796	+	0.900046i	$0.643533\pi$
$854$		0			0
$855$		0			0
$856$		−	2.82843i		−	0.0966736i
$857$	28.2843			0.966172			0.483086	−	0.875573i	$-0.339516\pi$
$857$	28.2843			0.966172			0.483086	+	0.875573i	$0.339516\pi$
$858$		−	16.0000i		−	0.546231i
$859$	4.00000			0.136478			0.0682391	−	0.997669i	$-0.478262\pi$
$859$	4.00000			0.136478			0.0682391	+	0.997669i	$0.478262\pi$
$860$		0			0
$861$		0			0
$862$	5.65685			0.192673
$863$		−	48.0000i		−	1.63394i	−0.576681	−	0.816970i	$-0.695652\pi$
$863$		−	48.0000i		−	1.63394i	0.576681	−	0.816970i	$-0.304348\pi$
$864$			5.65685i			0.192450i
$865$		0			0
$866$	−14.0000			−0.475739
$867$	−2.82843	−	48.0000i	−0.0960584	−	1.63017i
$868$		0			0
$869$	48.0000			1.62829
$870$		0			0
$871$	−8.00000			−0.271070
$872$	−8.48528			−0.287348
$873$	−84.8528			−2.87183
$874$			22.6274i			0.765384i
$875$		0			0
$876$		0			0
$877$	8.48528			0.286528			0.143264	−	0.989685i	$-0.454240\pi$
$877$	8.48528			0.286528			0.143264	+	0.989685i	$0.454240\pi$
$878$	−16.9706			−0.572729
$879$			16.9706i			0.572403i
$880$		0			0
$881$			45.2548i			1.52467i	0.647180	+	0.762337i	$0.275948\pi$
$881$			45.2548i			1.52467i	−0.647180	+	0.762337i	$0.724052\pi$
$882$		−	35.0000i		−	1.17851i
$883$			44.0000i			1.48072i	0.672212	+	0.740359i	$0.265344\pi$
$883$			44.0000i			1.48072i	−0.672212	+	0.740359i	$0.734656\pi$
$884$	6.00000	+	5.65685i	0.201802	+	0.190261i
$885$		0			0
$886$	12.0000			0.403148
$887$	−39.5980			−1.32957			−0.664785	−	0.747035i	$-0.731477\pi$
$887$	−39.5980			−1.32957			−0.664785	+	0.747035i	$0.731477\pi$
$888$		−	24.0000i		−	0.805387i
$889$		0			0
$890$		0			0
$891$			2.82843i			0.0947559i
$892$			16.0000i			0.535720i
$893$		0			0
$894$		−	16.9706i		−	0.567581i
$895$		0			0
$896$		0			0
$897$			32.0000i			1.06845i
$898$	−5.65685			−0.188772
$899$		0			0
$900$		0			0
$901$	−18.0000	−	16.9706i	−0.599667	−	0.565371i
$902$			16.0000i			0.532742i
$903$		0			0
$904$			11.3137i			0.376288i
$905$		0			0
$906$			22.6274i			0.751746i
$907$	−42.4264			−1.40875			−0.704373	−	0.709830i	$-0.748772\pi$
$907$	−42.4264			−1.40875			−0.704373	+	0.709830i	$0.748772\pi$
$908$	−19.7990			−0.657053
$909$	−30.0000			−0.995037
$910$		0			0
$911$		−	56.5685i		−	1.87420i	−0.349062	−	0.937100i	$-0.613500\pi$
$911$		−	56.5685i		−	1.87420i	0.349062	−	0.937100i	$-0.386500\pi$
$912$	11.3137			0.374634
$913$	33.9411			1.12329
$914$	−10.0000			−0.330771
$915$		0			0
$916$	−22.0000			−0.726900
$917$		0			0
$918$	−16.9706	−	16.0000i	−0.560112	−	0.528079i
$919$	40.0000			1.31948			0.659739	−	0.751495i	$-0.270667\pi$
$919$	40.0000			1.31948			0.659739	+	0.751495i	$0.270667\pi$
$920$		0			0
$921$		−	56.5685i		−	1.86400i
$922$		−	18.0000i		−	0.592798i
$923$	11.3137			0.372395
$924$		0			0
$925$		0			0
$926$	−32.0000			−1.05159
$927$			40.0000i			1.31377i
$928$	2.82843			0.0928477
$929$			5.65685i			0.185595i	0.995685	+	0.0927977i	$0.0295810\pi$
$929$			5.65685i			0.185595i	−0.995685	+	0.0927977i	$0.970419\pi$
$930$		0			0
$931$	−28.0000			−0.917663
$932$	−22.6274			−0.741186
$933$			32.0000i			1.04763i
$934$	−36.0000			−1.17796
$935$		0			0
$936$	10.0000			0.326860
$937$		−	2.00000i		−	0.0653372i	−0.999466	−	0.0326686i	$-0.989599\pi$
$937$		−	2.00000i		−	0.0653372i	0.999466	−	0.0326686i	$-0.0104006\pi$
$938$		0			0
$939$	−48.0000			−1.56642
$940$		0			0
$941$			36.7696i			1.19865i	0.800505	+	0.599327i	$0.204565\pi$
$941$			36.7696i			1.19865i	−0.800505	+	0.599327i	$0.795435\pi$
$942$	39.5980			1.29017
$943$		−	32.0000i		−	1.04206i
$944$	−12.0000			−0.390567
$945$		0			0
$946$			11.3137i			0.367840i
$947$	2.82843			0.0919115			0.0459558	−	0.998943i	$-0.485367\pi$
$947$	2.82843			0.0919115			0.0459558	+	0.998943i	$0.485367\pi$
$948$			48.0000i			1.55897i
$949$		0			0
$950$		0			0
$951$	8.00000			0.259418
$952$		0			0
$953$			6.00000i			0.194359i	0.995267	+	0.0971795i	$0.0309821\pi$
$953$			6.00000i			0.194359i	−0.995267	+	0.0971795i	$0.969018\pi$
$954$	−30.0000			−0.971286
$955$		0			0
$956$		0			0
$957$	22.6274			0.731441
$958$	−5.65685			−0.182765
$959$		0			0
$960$		0			0
$961$	31.0000			1.00000
$962$	−16.9706			−0.547153
$963$	14.1421			0.455724
$964$			16.9706i			0.546585i
$965$		0			0
$966$		0			0
$967$		−	8.00000i		−	0.257263i	−0.991692	−	0.128631i	$-0.958942\pi$
$967$		−	8.00000i		−	0.257263i	0.991692	−	0.128631i	$-0.0410584\pi$
$968$		−	3.00000i		−	0.0964237i
$969$	−32.0000	+	33.9411i	−1.02799	+	1.09035i
$970$		0			0
$971$	12.0000			0.385098			0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000			0.385098			0.192549	+	0.981287i	$0.438325\pi$
$972$	14.1421			0.453609
$973$		0			0
$974$		−	16.9706i		−	0.543772i
$975$		0			0
$976$		−	8.48528i		−	0.271607i
$977$		−	18.0000i		−	0.575871i	−0.957650	−	0.287936i	$-0.907031\pi$
$977$		−	18.0000i		−	0.575871i	0.957650	−	0.287936i	$-0.0929689\pi$
$978$		−	24.0000i		−	0.767435i
$979$		−	16.9706i		−	0.542382i
$980$		0			0
$981$		−	42.4264i		−	1.35457i
$982$		−	12.0000i		−	0.382935i
$983$	5.65685			0.180426			0.0902128	−	0.995923i	$-0.471245\pi$
$983$	5.65685			0.180426			0.0902128	+	0.995923i	$0.471245\pi$
$984$	−16.0000			−0.510061
$985$		0			0
$986$	−8.00000	+	8.48528i	−0.254772	+	0.270226i
$987$		0			0
$988$		−	8.00000i		−	0.254514i
$989$		−	22.6274i		−	0.719510i
$990$		0			0
$991$			50.9117i			1.61726i	0.588315	+	0.808632i	$0.299791\pi$
$991$			50.9117i			1.61726i	−0.588315	+	0.808632i	$0.700209\pi$
$992$		0			0
$993$	−11.3137			−0.359030
$994$		0			0
$995$		0			0
$996$			33.9411i			1.07547i
$997$	59.3970			1.88112			0.940560	−	0.339626i	$-0.110301\pi$
$997$	59.3970			1.88112			0.940560	+	0.339626i	$0.110301\pi$
$998$	25.4558			0.805791
$999$	48.0000			1.51865

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	850.2.d.i.849.4		4
5.2	odd	4		34.2.b.a.33.1	✓	2
5.3	odd	4		850.2.b.f.101.2		2
5.4	even	2	inner	850.2.d.i.849.1		4
15.2	even	4		306.2.b.d.271.1		2
17.16	even	2	inner	850.2.d.i.849.3		4
20.7	even	4		272.2.b.a.33.2		2
35.27	even	4		1666.2.b.c.883.2		2
40.27	even	4		1088.2.b.a.577.1		2
40.37	odd	4		1088.2.b.b.577.2		2
60.47	odd	4		2448.2.c.n.577.1		2
85.2	odd	8		578.2.c.a.251.1		2
85.7	even	16		578.2.d.g.155.1		8
85.12	even	16		578.2.d.g.179.1		8
85.22	even	16		578.2.d.g.179.2		8
85.27	even	16		578.2.d.g.155.2		8
85.32	odd	8		578.2.c.d.251.1		2
85.33	odd	4		850.2.b.f.101.1		2
85.37	even	16		578.2.d.g.399.2		8
85.42	odd	8		578.2.c.a.327.1		2
85.47	odd	4		578.2.a.d.1.2		2
85.57	even	16		578.2.d.g.423.1		8
85.62	even	16		578.2.d.g.423.2		8
85.67	odd	4		34.2.b.a.33.2	yes	2
85.72	odd	4		578.2.a.d.1.1		2
85.77	odd	8		578.2.c.d.327.1		2
85.82	even	16		578.2.d.g.399.1		8
85.84	even	2	inner	850.2.d.i.849.2		4
255.47	even	4		5202.2.a.u.1.2		2
255.152	even	4		306.2.b.d.271.2		2
255.242	even	4		5202.2.a.u.1.1		2
340.47	even	4		4624.2.a.s.1.1		2
340.67	even	4		272.2.b.a.33.1		2
340.327	even	4		4624.2.a.s.1.2		2
595.237	even	4		1666.2.b.c.883.1		2
680.67	even	4		1088.2.b.a.577.2		2
680.237	odd	4		1088.2.b.b.577.1		2
1020.407	odd	4		2448.2.c.n.577.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
34.2.b.a.33.1	✓	2	5.2	odd	4
34.2.b.a.33.2	yes	2	85.67	odd	4
272.2.b.a.33.1		2	340.67	even	4
272.2.b.a.33.2		2	20.7	even	4
306.2.b.d.271.1		2	15.2	even	4
306.2.b.d.271.2		2	255.152	even	4
578.2.a.d.1.1		2	85.72	odd	4
578.2.a.d.1.2		2	85.47	odd	4
578.2.c.a.251.1		2	85.2	odd	8
578.2.c.a.327.1		2	85.42	odd	8
578.2.c.d.251.1		2	85.32	odd	8
578.2.c.d.327.1		2	85.77	odd	8
578.2.d.g.155.1		8	85.7	even	16
578.2.d.g.155.2		8	85.27	even	16
578.2.d.g.179.1		8	85.12	even	16
578.2.d.g.179.2		8	85.22	even	16
578.2.d.g.399.1		8	85.82	even	16
578.2.d.g.399.2		8	85.37	even	16
578.2.d.g.423.1		8	85.57	even	16
578.2.d.g.423.2		8	85.62	even	16
850.2.b.f.101.1		2	85.33	odd	4
850.2.b.f.101.2		2	5.3	odd	4
850.2.d.i.849.1		4	5.4	even	2	inner
850.2.d.i.849.2		4	85.84	even	2	inner
850.2.d.i.849.3		4	17.16	even	2	inner
850.2.d.i.849.4		4	1.1	even	1	trivial
1088.2.b.a.577.1		2	40.27	even	4
1088.2.b.a.577.2		2	680.67	even	4
1088.2.b.b.577.1		2	680.237	odd	4
1088.2.b.b.577.2		2	40.37	odd	4
1666.2.b.c.883.1		2	595.237	even	4
1666.2.b.c.883.2		2	35.27	even	4
2448.2.c.n.577.1		2	60.47	odd	4
2448.2.c.n.577.2		2	1020.407	odd	4
4624.2.a.s.1.1		2	340.47	even	4
4624.2.a.s.1.2		2	340.327	even	4
5202.2.a.u.1.1		2	255.242	even	4
5202.2.a.u.1.2		2	255.47	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 850 = 2 \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	850.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +2.82843 q^{3} -1.00000 q^{4} +2.82843i q^{6} -1.00000i q^{8} +5.00000 q^{9} +2.82843i q^{11} -2.82843 q^{12} +2.00000i q^{13} +1.00000 q^{16} +(-2.82843 + 3.00000i) q^{17} +5.00000i q^{18} +4.00000 q^{19} -2.82843 q^{22} +5.65685 q^{23} -2.82843i q^{24} -2.00000 q^{26} +5.65685 q^{27} -2.82843i q^{29} +1.00000i q^{32} +8.00000i q^{33} +(-3.00000 - 2.82843i) q^{34} -5.00000 q^{36} +8.48528 q^{37} +4.00000i q^{38} +5.65685i q^{39} -5.65685i q^{41} -4.00000i q^{43} -2.82843i q^{44} +5.65685i q^{46} +2.82843 q^{48} -7.00000 q^{49} +(-8.00000 + 8.48528i) q^{51} -2.00000i q^{52} +6.00000i q^{53} +5.65685i q^{54} +11.3137 q^{57} +2.82843 q^{58} -12.0000 q^{59} -8.48528i q^{61} -1.00000 q^{64} -8.00000 q^{66} +4.00000i q^{67} +(2.82843 - 3.00000i) q^{68} +16.0000 q^{69} -5.65685i q^{71} -5.00000i q^{72} +8.48528i q^{74} -4.00000 q^{76} -5.65685 q^{78} -16.9706i q^{79} +1.00000 q^{81} +5.65685 q^{82} -12.0000i q^{83} +4.00000 q^{86} -8.00000i q^{87} +2.82843 q^{88} -6.00000 q^{89} -5.65685 q^{92} +2.82843i q^{96} -16.9706 q^{97} -7.00000i q^{98} +14.1421i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} + 20 q^{9} + 4 q^{16} + 16 q^{19} - 8 q^{26} - 12 q^{34} - 20 q^{36} - 28 q^{49} - 32 q^{51} - 48 q^{59} - 4 q^{64} - 32 q^{66} + 64 q^{69} - 16 q^{76} + 4 q^{81} + 16 q^{86} - 24 q^{89}+O(q^{100}) \) 4 * q - 4 * q^4 + 20 * q^9 + 4 * q^16 + 16 * q^19 - 8 * q^26 - 12 * q^34 - 20 * q^36 - 28 * q^49 - 32 * q^51 - 48 * q^59 - 4 * q^64 - 32 * q^66 + 64 * q^69 - 16 * q^76 + 4 * q^81 + 16 * q^86 - 24 * q^89

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