LMFDB - Embedded newform 1120.3.c.g.209.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1120,3,Mod(209,1120)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1120, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1120.209");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1120 = 2^{5} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1120.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$30.5177896084$
Analytic rank:	$0$
Dimension:	$80$
Twist minimal:	no (minimal twist has level 280)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			209.8
Character	$\chi$	$=$	1120.209
Dual form			1120.3.c.g.209.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.23212i q^{3} +(-3.82711 + 3.21764i) q^{5} +(6.97525 - 0.588096i) q^{7} -1.44659 q^{9} +O(q^{10})$	\(q+3.23212i q^{3} +(-3.82711 + 3.21764i) q^{5} +(6.97525 - 0.588096i) q^{7} -1.44659 q^{9} -13.5631i q^{11} +21.1906i q^{13} +(-10.3998 - 12.3697i) q^{15} -0.174717 q^{17} -20.7233 q^{19} +(1.90080 + 22.5448i) q^{21} +27.0982i q^{23} +(4.29361 - 24.6285i) q^{25} +24.4135i q^{27} +18.8246i q^{29} -7.48886i q^{31} +43.8375 q^{33} +(-24.8028 + 24.6945i) q^{35} -66.6203 q^{37} -68.4904 q^{39} +40.1326i q^{41} +27.6721 q^{43} +(5.53627 - 4.65461i) q^{45} +9.05114 q^{47} +(48.3083 - 8.20423i) q^{49} -0.564706i q^{51} +60.9906 q^{53} +(43.6411 + 51.9075i) q^{55} -66.9803i q^{57} -14.5469 q^{59} -35.7308 q^{61} +(-10.0903 + 0.850734i) q^{63} +(-68.1836 - 81.0987i) q^{65} -92.9010 q^{67} -87.5845 q^{69} -66.0567 q^{71} -51.4978 q^{73} +(79.6024 + 13.8775i) q^{75} +(-7.97640 - 94.6060i) q^{77} -77.1971 q^{79} -91.9267 q^{81} -94.6032i q^{83} +(0.668662 - 0.562176i) q^{85} -60.8433 q^{87} -59.6907i q^{89} +(12.4621 + 147.810i) q^{91} +24.2049 q^{93} +(79.3106 - 66.6802i) q^{95} +28.6155 q^{97} +19.6202i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 80 q - 224 q^{9}+O(q^{10}) $ 80 * q - 224 * q^9	$ 80 q - 224 q^{9} + 72 q^{15} - 104 q^{25} + 112 q^{39} + 192 q^{49} + 472 q^{65} - 800 q^{71} - 480 q^{79} - 896 q^{81} - 1176 q^{95}+O(q^{100}) $ 80 * q - 224 * q^9 + 72 * q^15 - 104 * q^25 + 112 * q^39 + 192 * q^49 + 472 * q^65 - 800 * q^71 - 480 * q^79 - 896 * q^81 - 1176 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1120\mathbb{Z}\right)^\times$.

$n$	$351$	$421$	$801$	$897$
$\chi(n)$	$1$	$-1$	$-1$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$			3.23212i			1.07737i	0.842506	+	0.538686i	$0.181079\pi$
$3$			3.23212i			1.07737i	−0.842506	+	0.538686i	$0.818921\pi$
$4$		0			0
$5$	−3.82711	+	3.21764i	−0.765423	+	0.643528i
$5$	−3.82711	+	3.21764i	−0.765423	+	0.643528i
$6$		0			0
$7$	6.97525	−	0.588096i	0.996465	−	0.0840137i
$7$	6.97525	−	0.588096i	0.996465	−	0.0840137i
$8$		0			0
$9$	−1.44659			−0.160732
$10$		0			0
$11$		−	13.5631i		−	1.23301i	−0.787352	−	0.616504i	$-0.788548\pi$
$11$		−	13.5631i		−	1.23301i	0.787352	−	0.616504i	$-0.211452\pi$
$12$		0			0
$13$			21.1906i			1.63004i	0.579430	+	0.815022i	$0.303275\pi$
$13$			21.1906i			1.63004i	−0.579430	+	0.815022i	$0.696725\pi$
$14$		0			0
$15$	−10.3998	−	12.3697i	−0.693319	−	0.824646i
$16$		0			0
$17$	−0.174717			−0.0102775			−0.00513874	−	0.999987i	$-0.501636\pi$
$17$	−0.174717			−0.0102775			−0.00513874	+	0.999987i	$0.501636\pi$
$18$		0			0
$19$	−20.7233			−1.09070			−0.545351	−	0.838208i	$-0.683604\pi$
$19$	−20.7233			−1.09070			−0.545351	+	0.838208i	$0.683604\pi$
$20$		0			0
$21$	1.90080	+	22.5448i	0.0905141	+	1.07356i
$22$		0			0
$23$			27.0982i			1.17818i	0.808067	+	0.589091i	$0.200514\pi$
$23$			27.0982i			1.17818i	−0.808067	+	0.589091i	$0.799486\pi$
$24$		0			0
$25$	4.29361	−	24.6285i	0.171744	−	0.985142i
$26$		0			0
$27$			24.4135i			0.904204i
$28$		0			0
$29$			18.8246i			0.649124i	0.945864	+	0.324562i	$0.105217\pi$
$29$			18.8246i			0.649124i	−0.945864	+	0.324562i	$0.894783\pi$
$30$		0			0
$31$		−	7.48886i		−	0.241576i	−0.992678	−	0.120788i	$-0.961458\pi$
$31$		−	7.48886i		−	0.241576i	0.992678	−	0.120788i	$-0.0385421\pi$
$32$		0			0
$33$	43.8375			1.32841
$34$		0			0
$35$	−24.8028	+	24.6945i	−0.708652	+	0.705558i
$36$		0			0
$37$	−66.6203			−1.80055			−0.900274	−	0.435324i	$-0.856634\pi$
$37$	−66.6203			−1.80055			−0.900274	+	0.435324i	$0.856634\pi$
$38$		0			0
$39$	−68.4904			−1.75617
$40$		0			0
$41$			40.1326i			0.978844i	0.872047	+	0.489422i	$0.162792\pi$
$41$			40.1326i			0.978844i	−0.872047	+	0.489422i	$0.837208\pi$
$42$		0			0
$43$	27.6721			0.643536			0.321768	−	0.946818i	$-0.395723\pi$
$43$	27.6721			0.643536			0.321768	+	0.946818i	$0.395723\pi$
$44$		0			0
$45$	5.53627	−	4.65461i	0.123028	−	0.103436i
$46$		0			0
$47$	9.05114			0.192577			0.0962887	−	0.995353i	$-0.469303\pi$
$47$	9.05114			0.192577			0.0962887	+	0.995353i	$0.469303\pi$
$48$		0			0
$49$	48.3083	−	8.20423i	0.985883	−	0.167433i
$50$		0			0
$51$		−	0.564706i		−	0.0110727i
$52$		0			0
$53$	60.9906			1.15077			0.575383	−	0.817884i	$-0.304853\pi$
$53$	60.9906			1.15077			0.575383	+	0.817884i	$0.304853\pi$
$54$		0			0
$55$	43.6411	+	51.9075i	0.793475	+	0.943773i
$56$		0			0
$57$		−	66.9803i		−	1.17509i
$58$		0			0
$59$	−14.5469			−0.246557			−0.123279	−	0.992372i	$-0.539341\pi$
$59$	−14.5469			−0.246557			−0.123279	+	0.992372i	$0.539341\pi$
$60$		0			0
$61$	−35.7308			−0.585751			−0.292875	−	0.956151i	$-0.594612\pi$
$61$	−35.7308			−0.585751			−0.292875	+	0.956151i	$0.594612\pi$
$62$		0			0
$63$	−10.0903	+	0.850734i	−0.160164	+	0.0135037i
$64$		0			0
$65$	−68.1836	−	81.0987i	−1.04898	−	1.24767i
$66$		0			0
$67$	−92.9010			−1.38658			−0.693291	−	0.720658i	$-0.743840\pi$
$67$	−92.9010			−1.38658			−0.693291	+	0.720658i	$0.743840\pi$
$68$		0			0
$69$	−87.5845			−1.26934
$70$		0			0
$71$	−66.0567			−0.930376			−0.465188	−	0.885212i	$-0.654013\pi$
$71$	−66.0567			−0.930376			−0.465188	+	0.885212i	$0.654013\pi$
$72$		0			0
$73$	−51.4978			−0.705450			−0.352725	−	0.935727i	$-0.614745\pi$
$73$	−51.4978			−0.705450			−0.352725	+	0.935727i	$0.614745\pi$
$74$		0			0
$75$	79.6024	+	13.8775i	1.06136	+	0.185033i
$76$		0			0
$77$	−7.97640	−	94.6060i	−0.103590	−	1.22865i
$78$		0			0
$79$	−77.1971			−0.977179			−0.488589	−	0.872514i	$-0.662488\pi$
$79$	−77.1971			−0.977179			−0.488589	+	0.872514i	$0.662488\pi$
$80$		0			0
$81$	−91.9267			−1.13490
$82$		0			0
$83$		−	94.6032i		−	1.13980i	−0.821715	−	0.569899i	$-0.806982\pi$
$83$		−	94.6032i		−	1.13980i	0.821715	−	0.569899i	$-0.193018\pi$
$84$		0			0
$85$	0.668662	−	0.562176i	0.00786661	−	0.00661384i
$86$		0			0
$87$	−60.8433			−0.699348
$88$		0			0
$89$		−	59.6907i		−	0.670682i	−0.942097	−	0.335341i	$-0.891149\pi$
$89$		−	59.6907i		−	0.670682i	0.942097	−	0.335341i	$-0.108851\pi$
$90$		0			0
$91$	12.4621	+	147.810i	0.136946	+	1.62428i
$92$		0			0
$93$	24.2049			0.260267
$94$		0			0
$95$	79.3106	−	66.6802i	0.834848	−	0.701897i
$96$		0			0
$97$	28.6155			0.295006			0.147503	−	0.989062i	$-0.452876\pi$
$97$	28.6155			0.295006			0.147503	+	0.989062i	$0.452876\pi$
$98$		0			0
$99$			19.6202i			0.198184i
$100$		0			0
$101$	193.520			1.91604			0.958021	−	0.286697i	$-0.0925573\pi$
$101$	193.520			1.91604			0.958021	+	0.286697i	$0.0925573\pi$
$102$		0			0
$103$	−164.516			−1.59725			−0.798624	−	0.601831i	$-0.794438\pi$
$103$	−164.516			−1.59725			−0.798624	+	0.601831i	$0.794438\pi$
$104$		0			0
$105$	−79.8157	−	80.1656i	−0.760150	−	0.763482i
$106$		0			0
$107$	−79.2388			−0.740550			−0.370275	−	0.928922i	$-0.620737\pi$
$107$	−79.2388			−0.740550			−0.370275	+	0.928922i	$0.620737\pi$
$108$		0			0
$109$			135.174i			1.24013i	0.784551	+	0.620065i	$0.212894\pi$
$109$			135.174i			1.24013i	−0.784551	+	0.620065i	$0.787106\pi$
$110$		0			0
$111$		−	215.325i		−	1.93986i
$112$		0			0
$113$		−	40.6363i		−	0.359613i	−0.983702	−	0.179807i	$-0.942453\pi$
$113$		−	40.6363i		−	0.359613i	0.983702	−	0.179807i	$-0.0575472\pi$
$114$		0			0
$115$	−87.1921	−	103.708i	−0.758192	−	0.901807i
$116$		0			0
$117$		−	30.6541i		−	0.262001i
$118$		0			0
$119$	−1.21870	+	0.102750i	−0.0102411	+	0.000863449i
$120$		0			0
$121$	−62.9573			−0.520309
$122$		0			0
$123$	−129.713			−1.05458
$124$		0			0
$125$	62.8136	+	108.072i	0.502509	+	0.864572i
$126$		0			0
$127$		−	102.631i		−	0.808118i	−0.914733	−	0.404059i	$-0.867599\pi$
$127$		−	102.631i		−	0.808118i	0.914733	−	0.404059i	$-0.132401\pi$
$128$		0			0
$129$			89.4394i			0.693329i
$130$		0			0
$131$	−73.4099			−0.560381			−0.280191	−	0.959944i	$-0.590398\pi$
$131$	−73.4099			−0.560381			−0.280191	+	0.959944i	$0.590398\pi$
$132$		0			0
$133$	−144.550	+	12.1873i	−1.08685	+	0.0916339i
$134$		0			0
$135$	−78.5539	−	93.4333i	−0.581880	−	0.692099i
$136$		0			0
$137$			34.6820i			0.253153i	0.991957	+	0.126577i	$0.0403990\pi$
$137$			34.6820i			0.253153i	−0.991957	+	0.126577i	$0.959601\pi$
$138$		0			0
$139$	41.2578			0.296819			0.148409	−	0.988926i	$-0.452585\pi$
$139$	41.2578			0.296819			0.148409	+	0.988926i	$0.452585\pi$
$140$		0			0
$141$			29.2544i			0.207478i
$142$		0			0
$143$	287.410			2.00986
$144$		0			0
$145$	−60.5707	−	72.0438i	−0.417729	−	0.496854i
$146$		0			0
$147$	26.5171	+	156.138i	0.180388	+	1.06216i
$148$		0			0
$149$		−	197.409i		−	1.32489i	−0.749110	−	0.662446i	$-0.769518\pi$
$149$		−	197.409i		−	1.32489i	0.749110	−	0.662446i	$-0.230482\pi$
$150$		0			0
$151$	77.3543			0.512280			0.256140	−	0.966640i	$-0.417549\pi$
$151$	77.3543			0.512280			0.256140	+	0.966640i	$0.417549\pi$
$152$		0			0
$153$	0.252744			0.00165192
$154$		0			0
$155$	24.0964	+	28.6607i	0.155461	+	0.184908i
$156$		0			0
$157$		−	111.633i		−	0.711038i	−0.934669	−	0.355519i	$-0.884304\pi$
$157$		−	111.633i		−	0.711038i	0.934669	−	0.355519i	$-0.115696\pi$
$158$		0			0
$159$			197.129i			1.23980i
$160$		0			0
$161$	15.9363	+	189.017i	0.0989834	+	1.17402i
$162$		0			0
$163$	311.331			1.91001			0.955004	−	0.296592i	$-0.0958500\pi$
$163$	311.331			1.91001			0.955004	+	0.296592i	$0.0958500\pi$
$164$		0			0
$165$	−167.771	+	141.053i	−1.01679	+	0.854868i
$166$		0			0
$167$	−17.1911			−0.102941			−0.0514704	−	0.998675i	$-0.516391\pi$
$167$	−17.1911			−0.102941			−0.0514704	+	0.998675i	$0.516391\pi$
$168$		0			0
$169$	−280.040			−1.65704
$170$		0			0
$171$	29.9782			0.175311
$172$		0			0
$173$			192.100i			1.11040i	0.831716	+	0.555201i	$0.187359\pi$
$173$			192.100i			1.11040i	−0.831716	+	0.555201i	$0.812641\pi$
$174$		0			0
$175$	15.4651	−	174.315i	0.0883719	−	0.996088i
$176$		0			0
$177$		−	47.0172i		−	0.265634i
$178$		0			0
$179$			116.518i			0.650936i	0.945553	+	0.325468i	$0.105522\pi$
$179$			116.518i			0.650936i	−0.945553	+	0.325468i	$0.894478\pi$
$180$		0			0
$181$	−116.427			−0.643241			−0.321620	−	0.946869i	$-0.604227\pi$
$181$	−116.427			−0.643241			−0.321620	+	0.946869i	$0.604227\pi$
$182$		0			0
$183$		−	115.486i		−	0.631072i
$184$		0			0
$185$	254.963	−	214.360i	1.37818	−	1.15870i
$186$		0			0
$187$			2.36970i			0.0126722i
$188$		0			0
$189$	14.3575	+	170.290i	0.0759655	+	0.901008i
$190$		0			0
$191$	−126.956			−0.664692			−0.332346	−	0.943157i	$-0.607840\pi$
$191$	−126.956			−0.664692			−0.332346	+	0.943157i	$0.607840\pi$
$192$		0			0
$193$			231.635i			1.20018i	0.799932	+	0.600090i	$0.204869\pi$
$193$			231.635i			1.20018i	−0.799932	+	0.600090i	$0.795131\pi$
$194$		0			0
$195$	262.121	−	220.377i	1.34421	−	1.13014i
$196$		0			0
$197$	77.9924			0.395900			0.197950	−	0.980212i	$-0.436572\pi$
$197$	77.9924			0.395900			0.197950	+	0.980212i	$0.436572\pi$
$198$		0			0
$199$			227.874i			1.14509i	0.819872	+	0.572547i	$0.194044\pi$
$199$			227.874i			1.14509i	−0.819872	+	0.572547i	$0.805956\pi$
$200$		0			0
$201$		−	300.267i		−	1.49387i
$202$		0			0
$203$	11.0707	+	131.306i	0.0545353	+	0.646829i
$204$		0			0
$205$	−129.132	−	153.592i	−0.629913	−	0.749229i
$206$		0			0
$207$		−	39.2000i		−	0.189372i
$208$		0			0
$209$			281.072i			1.34484i
$210$		0			0
$211$		−	214.499i		−	1.01658i	−0.861185	−	0.508291i	$-0.830277\pi$
$211$		−	214.499i		−	1.01658i	0.861185	−	0.508291i	$-0.169723\pi$
$212$		0			0
$213$		−	213.503i		−	1.00236i
$214$		0			0
$215$	−105.904	+	89.0387i	−0.492578	+	0.414133i
$216$		0			0
$217$	−4.40417	−	52.2367i	−0.0202957	−	0.240722i
$218$		0			0
$219$		−	166.447i		−	0.760032i
$220$		0			0
$221$		−	3.70235i		−	0.0167527i
$222$		0			0
$223$	−10.8367			−0.0485949			−0.0242975	−	0.999705i	$-0.507735\pi$
$223$	−10.8367			−0.0485949			−0.0242975	+	0.999705i	$0.507735\pi$
$224$		0			0
$225$	−6.21110	+	35.6274i	−0.0276049	+	0.158344i
$226$		0			0
$227$			297.246i			1.30945i	0.755865	+	0.654727i	$0.227216\pi$
$227$			297.246i			1.30945i	−0.755865	+	0.654727i	$0.772784\pi$
$228$		0			0
$229$	−39.4044			−0.172072			−0.0860358	−	0.996292i	$-0.527420\pi$
$229$	−39.4044			−0.172072			−0.0860358	+	0.996292i	$0.527420\pi$
$230$		0			0
$231$	305.778	−	25.7807i	1.32371	−	0.111605i
$232$		0			0
$233$			316.557i			1.35861i	0.733855	+	0.679306i	$0.237719\pi$
$233$			316.557i			1.35861i	−0.733855	+	0.679306i	$0.762281\pi$
$234$		0			0
$235$	−34.6398	+	29.1233i	−0.147403	+	0.123929i
$236$		0			0
$237$		−	249.510i		−	1.05279i
$238$		0			0
$239$	181.500			0.759416			0.379708	−	0.925106i	$-0.376024\pi$
$239$	181.500			0.759416			0.379708	+	0.925106i	$0.376024\pi$
$240$		0			0
$241$			203.756i			0.845460i	0.906256	+	0.422730i	$0.138928\pi$
$241$			203.756i			0.845460i	−0.906256	+	0.422730i	$0.861072\pi$
$242$		0			0
$243$		−	77.3963i		−	0.318503i
$244$		0			0
$245$	−158.483	+	186.837i	−0.646870	+	0.762600i
$246$		0			0
$247$		−	439.139i		−	1.77789i
$248$		0			0
$249$	305.769			1.22799
$250$		0			0
$251$	120.326			0.479388			0.239694	−	0.970848i	$-0.422953\pi$
$251$	120.326			0.479388			0.239694	+	0.970848i	$0.422953\pi$
$252$		0			0
$253$	367.535			1.45271
$254$		0			0
$255$	1.81702	+	2.16120i	0.00712557	+	0.00847528i
$256$		0			0
$257$	214.282			0.833783			0.416892	−	0.908956i	$-0.363119\pi$
$257$	214.282			0.833783			0.416892	+	0.908956i	$0.363119\pi$
$258$		0			0
$259$	−464.693	+	39.1791i	−1.79418	+	0.151271i
$260$		0			0
$261$		−	27.2315i		−	0.104335i
$262$		0			0
$263$			388.319i			1.47650i	0.674529	+	0.738248i	$0.264347\pi$
$263$			388.319i			1.47650i	−0.674529	+	0.738248i	$0.735653\pi$
$264$		0			0
$265$	−233.418	+	196.246i	−0.880822	+	0.740549i
$266$		0			0
$267$	192.927			0.722574
$268$		0			0
$269$	−442.119			−1.64357			−0.821783	−	0.569800i	$-0.807020\pi$
$269$	−442.119			−1.64357			−0.821783	+	0.569800i	$0.807020\pi$
$270$		0			0
$271$		−	277.087i		−	1.02246i	−0.859444	−	0.511231i	$-0.829190\pi$
$271$		−	277.087i		−	1.02246i	0.859444	−	0.511231i	$-0.170810\pi$
$272$		0			0
$273$	−477.738	+	40.2789i	−1.74996	+	0.147542i
$274$		0			0
$275$	−334.039	−	58.2346i	−1.21469	−	0.211762i
$276$		0			0
$277$	−287.260			−1.03704			−0.518520	−	0.855065i	$-0.673517\pi$
$277$	−287.260			−1.03704			−0.518520	+	0.855065i	$0.673517\pi$
$278$		0			0
$279$			10.8333i			0.0388291i
$280$		0			0
$281$	234.412			0.834206			0.417103	−	0.908859i	$-0.363045\pi$
$281$	234.412			0.834206			0.417103	+	0.908859i	$0.363045\pi$
$282$		0			0
$283$		−	76.8321i		−	0.271491i	−0.990744	−	0.135746i	$-0.956657\pi$
$283$		−	76.8321i		−	0.271491i	0.990744	−	0.135746i	$-0.0433430\pi$
$284$		0			0
$285$	215.518	+	256.341i	0.756204	+	0.899443i
$286$		0			0
$287$	23.6018	+	279.935i	0.0822363	+	0.975383i
$288$		0			0
$289$	−288.969			−0.999894
$290$		0			0
$291$			92.4889i			0.317831i
$292$		0			0
$293$			248.602i			0.848472i	0.905552	+	0.424236i	$0.139457\pi$
$293$			248.602i			0.848472i	−0.905552	+	0.424236i	$0.860543\pi$
$294$		0			0
$295$	55.6725	−	46.8066i	0.188720	−	0.158666i
$296$		0			0
$297$	331.123			1.11489
$298$		0			0
$299$	−574.226			−1.92049
$300$		0			0
$301$	193.020	−	16.2738i	0.641261	−	0.0540659i
$302$		0			0
$303$			625.481i			2.06429i
$304$		0			0
$305$	136.746	−	114.969i	0.448347	−	0.376947i
$306$		0			0
$307$			244.931i			0.797822i	0.916990	+	0.398911i	$0.130612\pi$
$307$			244.931i			0.797822i	−0.916990	+	0.398911i	$0.869388\pi$
$308$		0			0
$309$		−	531.737i		−	1.72083i
$310$		0			0
$311$			556.903i			1.79069i	0.445378	+	0.895343i	$0.353069\pi$
$311$			556.903i			1.79069i	−0.445378	+	0.895343i	$0.646931\pi$
$312$		0			0
$313$	64.7781			0.206959			0.103479	−	0.994632i	$-0.467002\pi$
$313$	64.7781			0.206959			0.103479	+	0.994632i	$0.467002\pi$
$314$		0			0
$315$	35.8795	−	35.7229i	0.113903	−	0.113406i
$316$		0			0
$317$	562.104			1.77320			0.886599	−	0.462539i	$-0.153062\pi$
$317$	562.104			1.77320			0.886599	+	0.462539i	$0.153062\pi$
$318$		0			0
$319$	255.319			0.800375
$320$		0			0
$321$		−	256.109i		−	0.797848i
$322$		0			0
$323$	3.62072			0.0112097
$324$		0			0
$325$	521.893	+	90.9841i	1.60582	+	0.279951i
$326$		0			0
$327$	−436.899			−1.33608
$328$		0			0
$329$	63.1340	−	5.32294i	0.191897	−	0.0161791i
$330$		0			0
$331$		−	115.044i		−	0.347566i	−0.984784	−	0.173783i	$-0.944401\pi$
$331$		−	115.044i		−	0.347566i	0.984784	−	0.173783i	$-0.0555991\pi$
$332$		0			0
$333$	96.3722			0.289406
$334$		0			0
$335$	355.543	−	298.922i	1.06132	−	0.892304i
$336$		0			0
$337$		−	93.7824i		−	0.278286i	−0.990272	−	0.139143i	$-0.955565\pi$
$337$		−	93.7824i		−	0.278286i	0.990272	−	0.139143i	$-0.0444348\pi$
$338$		0			0
$339$	131.341			0.387437
$340$		0			0
$341$	−101.572			−0.297865
$342$		0			0
$343$	332.138	−	85.6365i	0.968331	−	0.249669i
$344$		0			0
$345$	335.196	−	281.815i	0.971582	−	0.816856i
$346$		0			0
$347$	306.423			0.883063			0.441531	−	0.897246i	$-0.354435\pi$
$347$	306.423			0.883063			0.441531	+	0.897246i	$0.354435\pi$
$348$		0			0
$349$	−386.434			−1.10726			−0.553631	−	0.832762i	$-0.686758\pi$
$349$	−386.434			−1.10726			−0.553631	+	0.832762i	$0.686758\pi$
$350$		0			0
$351$	−517.336			−1.47389
$352$		0			0
$353$	140.883			0.399103			0.199551	−	0.979887i	$-0.436052\pi$
$353$	140.883			0.399103			0.199551	+	0.979887i	$0.436052\pi$
$354$		0			0
$355$	252.807	−	212.547i	0.712131	−	0.598723i
$356$		0			0
$357$	−0.332101	−	3.93897i	−0.000930256	−	0.0110335i
$358$		0			0
$359$	331.195			0.922549			0.461274	−	0.887258i	$-0.347392\pi$
$359$	331.195			0.922549			0.461274	+	0.887258i	$0.347392\pi$
$360$		0			0
$361$	68.4565			0.189630
$362$		0			0
$363$		−	203.486i		−	0.560566i
$364$		0			0
$365$	197.088	−	165.701i	0.539967	−	0.453976i
$366$		0			0
$367$	2.40380			0.00654988			0.00327494	−	0.999995i	$-0.498958\pi$
$367$	2.40380			0.00654988			0.00327494	+	0.999995i	$0.498958\pi$
$368$		0			0
$369$		−	58.0554i		−	0.157332i
$370$		0			0
$371$	425.424	−	35.8683i	1.14670	−	0.0966800i
$372$		0			0
$373$	91.6748			0.245777			0.122889	−	0.992420i	$-0.460784\pi$
$373$	91.6748			0.245777			0.122889	+	0.992420i	$0.460784\pi$
$374$		0			0
$375$	−349.300	+	203.021i	−0.931467	+	0.541389i
$376$		0			0
$377$	−398.904			−1.05810
$378$		0			0
$379$		−	217.188i		−	0.573055i	−0.958072	−	0.286527i	$-0.907499\pi$
$379$		−	217.188i		−	0.573055i	0.958072	−	0.286527i	$-0.0925009\pi$
$380$		0			0
$381$	331.715			0.870644
$382$		0			0
$383$	−425.682			−1.11144			−0.555721	−	0.831369i	$-0.687558\pi$
$383$	−425.682			−1.11144			−0.555721	+	0.831369i	$0.687558\pi$
$384$		0			0
$385$	334.934	+	336.403i	0.869959	+	0.873773i
$386$		0			0
$387$	−40.0302			−0.103437
$388$		0			0
$389$		−	678.005i		−	1.74294i	−0.490446	−	0.871471i	$-0.663166\pi$
$389$		−	678.005i		−	1.74294i	0.490446	−	0.871471i	$-0.336834\pi$
$390$		0			0
$391$		−	4.73451i		−	0.0121087i
$392$		0			0
$393$		−	237.270i		−	0.603739i
$394$		0			0
$395$	295.442	−	248.392i	0.747955	−	0.628841i
$396$		0			0
$397$		−	8.36325i		−	0.0210661i	−0.999945	−	0.0105331i	$-0.996647\pi$
$397$		−	8.36325i		−	0.0210661i	0.999945	−	0.0105331i	$-0.00335284\pi$
$398$		0			0
$399$	−39.3908	−	467.204i	−0.0987239	−	1.17094i
$400$		0			0
$401$	579.986			1.44635			0.723174	−	0.690665i	$-0.242682\pi$
$401$	579.986			1.44635			0.723174	+	0.690665i	$0.242682\pi$
$402$		0			0
$403$	158.693			0.393780
$404$		0			0
$405$	351.814	−	295.787i	0.868677	−	0.730338i
$406$		0			0
$407$			903.576i			2.22009i
$408$		0			0
$409$			374.870i			0.916551i	0.888810	+	0.458276i	$0.151533\pi$
$409$			374.870i			0.916551i	−0.888810	+	0.458276i	$0.848467\pi$
$410$		0			0
$411$	−112.096			−0.272740
$412$		0			0
$413$	−101.468	+	8.55495i	−0.245685	+	0.0207142i
$414$		0			0
$415$	304.399	+	362.057i	0.733491	+	0.872427i
$416$		0			0
$417$			133.350i			0.319784i
$418$		0			0
$419$	−142.070			−0.339068			−0.169534	−	0.985524i	$-0.554226\pi$
$419$	−142.070			−0.339068			−0.169534	+	0.985524i	$0.554226\pi$
$420$		0			0
$421$		−	92.9811i		−	0.220858i	−0.993884	−	0.110429i	$-0.964778\pi$
$421$		−	92.9811i		−	0.220858i	0.993884	−	0.110429i	$-0.0352224\pi$
$422$		0			0
$423$	−13.0933			−0.0309534
$424$		0			0
$425$	−0.750167	+	4.30303i	−0.00176510	+	0.0101248i
$426$		0			0
$427$	−249.231	+	21.0131i	−0.583680	+	0.0492111i
$428$		0			0
$429$			928.942i			2.16537i
$430$		0			0
$431$	432.813			1.00421			0.502103	−	0.864808i	$-0.332560\pi$
$431$	432.813			1.00421			0.502103	+	0.864808i	$0.332560\pi$
$432$		0			0
$433$	681.677			1.57431			0.787156	−	0.616755i	$-0.211553\pi$
$433$	681.677			1.57431			0.787156	+	0.616755i	$0.211553\pi$
$434$		0			0
$435$	232.854	−	195.772i	0.535297	−	0.450050i
$436$		0			0
$437$		−	561.564i		−	1.28504i
$438$		0			0
$439$			676.687i			1.54143i	0.637181	+	0.770714i	$0.280101\pi$
$439$			676.687i			1.54143i	−0.637181	+	0.770714i	$0.719899\pi$
$440$		0			0
$441$	−69.8823	+	11.8682i	−0.158463	+	0.0269120i
$442$		0			0
$443$	57.8517			0.130591			0.0652954	−	0.997866i	$-0.479201\pi$
$443$	57.8517			0.130591			0.0652954	+	0.997866i	$0.479201\pi$
$444$		0			0
$445$	192.063	+	228.443i	0.431602	+	0.513355i
$446$		0			0
$447$	638.049			1.42740
$448$		0			0
$449$	509.913			1.13566			0.567832	−	0.823145i	$-0.307782\pi$
$449$	509.913			1.13566			0.567832	+	0.823145i	$0.307782\pi$
$450$		0			0
$451$	544.322			1.20692
$452$		0			0
$453$			250.018i			0.551916i
$454$		0			0
$455$	−523.292	−	525.586i	−1.15009	−	1.15513i
$456$		0			0
$457$			470.149i			1.02877i	0.857559	+	0.514386i	$0.171980\pi$
$457$			470.149i			1.02877i	−0.857559	+	0.514386i	$0.828020\pi$
$458$		0			0
$459$		−	4.26546i		−	0.00929294i
$460$		0			0
$461$	210.628			0.456893			0.228447	−	0.973556i	$-0.426635\pi$
$461$	210.628			0.456893			0.228447	+	0.973556i	$0.426635\pi$
$462$		0			0
$463$		−	708.908i		−	1.53112i	−0.643365	−	0.765559i	$-0.722462\pi$
$463$		−	708.908i		−	1.53112i	0.643365	−	0.765559i	$-0.277538\pi$
$464$		0			0
$465$	−92.6348	+	77.8825i	−0.199215	+	0.167489i
$466$		0			0
$467$			630.872i			1.35090i	0.737404	+	0.675452i	$0.236051\pi$
$467$			630.872i			1.35090i	−0.737404	+	0.675452i	$0.763949\pi$
$468$		0			0
$469$	−648.008	+	54.6347i	−1.38168	+	0.116492i
$470$		0			0
$471$	360.811			0.766053
$472$		0			0
$473$		−	375.319i		−	0.793486i
$474$		0			0
$475$	−88.9779	+	510.385i	−0.187322	+	1.07450i
$476$		0			0
$477$	−88.2284			−0.184965
$478$		0			0
$479$			235.461i			0.491569i	0.969325	+	0.245784i	$0.0790455\pi$
$479$			235.461i			0.491569i	−0.969325	+	0.245784i	$0.920954\pi$
$480$		0			0
$481$		−	1411.72i		−	2.93497i
$482$		0			0
$483$	−610.924	+	51.5081i	−1.26485	+	0.106642i
$484$		0			0
$485$	−109.515	+	92.0745i	−0.225804	+	0.189844i
$486$		0			0
$487$			11.0384i			0.0226661i	0.999936	+	0.0113331i	$0.00360750\pi$
$487$			11.0384i			0.0226661i	−0.999936	+	0.0113331i	$0.996392\pi$
$488$		0			0
$489$			1006.26i			2.05779i
$490$		0			0
$491$		−	423.608i		−	0.862745i	−0.902174	−	0.431373i	$-0.858029\pi$
$491$		−	423.608i		−	0.862745i	0.902174	−	0.431373i	$-0.141971\pi$
$492$		0			0
$493$		−	3.28898i		−	0.00667135i
$494$		0			0
$495$	−63.1308	−	75.0889i	−0.127537	−	0.151695i
$496$		0			0
$497$	−460.762	+	38.8477i	−0.927087	+	0.0781644i
$498$		0			0
$499$			16.3655i			0.0327966i	0.999866	+	0.0163983i	$0.00521997\pi$
$499$			16.3655i			0.0327966i	−0.999866	+	0.0163983i	$0.994780\pi$
$500$		0			0
$501$		−	55.5637i		−	0.110906i
$502$		0			0
$503$	−399.152			−0.793542			−0.396771	−	0.917918i	$-0.629869\pi$
$503$	−399.152			−0.793542			−0.396771	+	0.917918i	$0.629869\pi$
$504$		0			0
$505$	−740.624	+	622.678i	−1.46658	+	1.23303i
$506$		0			0
$507$		−	905.123i		−	1.78525i
$508$		0			0
$509$	−32.2063			−0.0632737			−0.0316369	−	0.999499i	$-0.510072\pi$
$509$	−32.2063			−0.0632737			−0.0316369	+	0.999499i	$0.510072\pi$
$510$		0			0
$511$	−359.210	+	30.2857i	−0.702956	+	0.0592674i
$512$		0			0
$513$		−	505.929i		−	0.986217i
$514$		0			0
$515$	629.623	−	529.354i	1.22257	−	1.02787i
$516$		0			0
$517$		−	122.761i		−	0.237450i
$518$		0			0
$519$	−620.888			−1.19632
$520$		0			0
$521$		−	180.110i		−	0.345701i	−0.984948	−	0.172850i	$-0.944702\pi$
$521$		−	180.110i		−	0.345701i	0.984948	−	0.172850i	$-0.0552977\pi$
$522$		0			0
$523$			92.5194i			0.176901i	0.996081	+	0.0884507i	$0.0281916\pi$
$523$			92.5194i			0.176901i	−0.996081	+	0.0884507i	$0.971808\pi$
$524$		0			0
$525$	563.408	+	49.9850i	1.07316	+	0.0952095i
$526$		0			0
$527$			1.30843i			0.00248279i
$528$		0			0
$529$	−205.311			−0.388111
$530$		0			0
$531$	21.0434			0.0396297
$532$		0			0
$533$	−850.432			−1.59556
$534$		0			0
$535$	303.256	−	254.962i	0.566834	−	0.476564i
$536$		0			0
$537$	−376.599			−0.701301
$538$		0			0
$539$	−111.275	−	655.210i	−0.206447	−	1.21560i
$540$		0			0
$541$			99.4342i			0.183797i	0.995768	+	0.0918986i	$0.0292936\pi$
$541$			99.4342i			0.183797i	−0.995768	+	0.0918986i	$0.970706\pi$
$542$		0			0
$543$		−	376.304i		−	0.693010i
$544$		0			0
$545$	−434.941	−	517.327i	−0.798057	−	0.949223i
$546$		0			0
$547$	103.740			0.189652			0.0948262	−	0.995494i	$-0.469770\pi$
$547$	103.740			0.189652			0.0948262	+	0.995494i	$0.469770\pi$
$548$		0			0
$549$	51.6879			0.0941491
$550$		0			0
$551$		−	390.108i		−	0.708000i
$552$		0			0
$553$	−538.469	+	45.3993i	−0.973724	+	0.0820964i
$554$		0			0
$555$	692.836	+	824.072i	1.24835	+	1.48481i
$556$		0			0
$557$	547.554			0.983041			0.491520	−	0.870866i	$-0.336441\pi$
$557$	547.554			0.983041			0.491520	+	0.870866i	$0.336441\pi$
$558$		0			0
$559$			586.387i			1.04899i
$560$		0			0
$561$	−7.65916			−0.0136527
$562$		0			0
$563$		−	801.866i		−	1.42427i	−0.702041	−	0.712137i	$-0.747728\pi$
$563$		−	801.866i		−	1.42427i	0.702041	−	0.712137i	$-0.252272\pi$
$564$		0			0
$565$	130.753	+	155.520i	0.231421	+	0.275256i
$566$		0			0
$567$	−641.212	+	54.0617i	−1.13089	+	0.0953469i
$568$		0			0
$569$	429.252			0.754398			0.377199	−	0.926132i	$-0.376887\pi$
$569$	429.252			0.754398			0.377199	+	0.926132i	$0.376887\pi$
$570$		0			0
$571$			705.248i			1.23511i	0.786527	+	0.617556i	$0.211877\pi$
$571$			705.248i			1.23511i	−0.786527	+	0.617556i	$0.788123\pi$
$572$		0			0
$573$		−	410.338i		−	0.716122i
$574$		0			0
$575$	667.388	+	116.349i	1.16068	+	0.202346i
$576$		0			0
$577$	856.994			1.48526			0.742629	−	0.669702i	$-0.233578\pi$
$577$	856.994			1.48526			0.742629	+	0.669702i	$0.233578\pi$
$578$		0			0
$579$	−748.671			−1.29304
$580$		0			0
$581$	−55.6358	−	659.881i	−0.0957586	−	1.13577i
$582$		0			0
$583$		−	827.220i		−	1.41890i
$584$		0			0
$585$	98.6338	+	117.317i	0.168605	+	0.200541i
$586$		0			0
$587$		−	87.0152i		−	0.148237i	−0.997249	−	0.0741186i	$-0.976386\pi$
$587$		−	87.0152i		−	0.148237i	0.997249	−	0.0741186i	$-0.0236143\pi$
$588$		0			0
$589$			155.194i			0.263487i
$590$		0			0
$591$			252.081i			0.426532i
$592$		0			0
$593$	−330.416			−0.557193			−0.278597	−	0.960408i	$-0.589869\pi$
$593$	−330.416			−0.557193			−0.278597	+	0.960408i	$0.589869\pi$
$594$		0			0
$595$	4.33347	−	4.31456i	0.00728315	−	0.00725136i
$596$		0			0
$597$	−736.514			−1.23369
$598$		0			0
$599$	−734.397			−1.22604			−0.613019	−	0.790068i	$-0.710045\pi$
$599$	−734.397			−1.22604			−0.613019	+	0.790068i	$0.710045\pi$
$600$		0			0
$601$			806.504i			1.34194i	0.741486	+	0.670969i	$0.234121\pi$
$601$			806.504i			1.34194i	−0.741486	+	0.670969i	$0.765879\pi$
$602$		0			0
$603$	134.390			0.222869
$604$		0			0
$605$	240.945	−	202.574i	0.398256	−	0.334833i
$606$		0			0
$607$	−771.044			−1.27025			−0.635127	−	0.772408i	$-0.719052\pi$
$607$	−771.044			−1.27025			−0.635127	+	0.772408i	$0.719052\pi$
$608$		0			0
$609$	−424.397	+	35.7817i	−0.696876	+	0.0587548i
$610$		0			0
$611$			191.799i			0.313910i
$612$		0			0
$613$	−933.654			−1.52309			−0.761545	−	0.648112i	$-0.775559\pi$
$613$	−933.654			−1.52309			−0.761545	+	0.648112i	$0.775559\pi$
$614$		0			0
$615$	496.428	−	417.370i	0.807199	−	0.678651i
$616$		0			0
$617$			256.389i			0.415542i	0.978178	+	0.207771i	$0.0666208\pi$
$617$			256.389i			0.415542i	−0.978178	+	0.207771i	$0.933379\pi$
$618$		0			0
$619$	−298.064			−0.481524			−0.240762	−	0.970584i	$-0.577397\pi$
$619$	−298.064			−0.481524			−0.240762	+	0.970584i	$0.577397\pi$
$620$		0			0
$621$	−661.562			−1.06532
$622$		0			0
$623$	−35.1038	−	416.358i	−0.0563465	−	0.668311i
$624$		0			0
$625$	−588.130	−	211.491i	−0.941008	−	0.338385i
$626$		0			0
$627$	−908.459			−1.44890
$628$		0			0
$629$	11.6397			0.0185051
$630$		0			0
$631$	526.570			0.834501			0.417250	−	0.908792i	$-0.362994\pi$
$631$	526.570			0.834501			0.417250	+	0.908792i	$0.362994\pi$
$632$		0			0
$633$	693.286			1.09524
$634$		0			0
$635$	330.229	+	392.780i	0.520046	+	0.618552i
$636$		0			0
$637$	173.852	+	1023.68i	0.272924	+	1.60703i
$638$		0			0
$639$	95.5571			0.149542
$640$		0			0
$641$	−204.664			−0.319289			−0.159644	−	0.987175i	$-0.551035\pi$
$641$	−204.664			−0.319289			−0.159644	+	0.987175i	$0.551035\pi$
$642$		0			0
$643$			580.457i			0.902733i	0.892339	+	0.451367i	$0.149063\pi$
$643$			580.457i			0.902733i	−0.892339	+	0.451367i	$0.850937\pi$
$644$		0			0
$645$	−287.784	−	342.295i	−0.446176	−	0.530690i
$646$		0			0
$647$	1027.02			1.58735			0.793676	−	0.608341i	$-0.208165\pi$
$647$	1027.02			1.58735			0.793676	+	0.608341i	$0.208165\pi$
$648$		0			0
$649$			197.300i			0.304007i
$650$		0			0
$651$	168.835	−	14.2348i	0.259347	−	0.0218660i
$652$		0			0
$653$	540.051			0.827030			0.413515	−	0.910497i	$-0.364301\pi$
$653$	540.051			0.827030			0.413515	+	0.910497i	$0.364301\pi$
$654$		0			0
$655$	280.948	−	236.207i	0.428929	−	0.360621i
$656$		0			0
$657$	74.4963			0.113389
$658$		0			0
$659$			648.051i			0.983386i	0.870769	+	0.491693i	$0.163622\pi$
$659$			648.051i			0.983386i	−0.870769	+	0.491693i	$0.836378\pi$
$660$		0			0
$661$	−63.5996			−0.0962172			−0.0481086	−	0.998842i	$-0.515319\pi$
$661$	−63.5996			−0.0962172			−0.0481086	+	0.998842i	$0.515319\pi$
$662$		0			0
$663$	11.9664			0.0180489
$664$		0			0
$665$	513.997	−	511.753i	0.772928	−	0.769554i
$666$		0			0
$667$	−510.112			−0.764785
$668$		0			0
$669$		−	35.0254i		−	0.0523549i
$670$		0			0
$671$			484.620i			0.722235i
$672$		0			0
$673$		−	499.088i		−	0.741587i	−0.928715	−	0.370793i	$-0.879086\pi$
$673$		−	499.088i		−	0.741587i	0.928715	−	0.370793i	$-0.120914\pi$
$674$		0			0
$675$	601.269	+	104.822i	0.890769	+	0.155292i
$676$		0			0
$677$			528.808i			0.781105i	0.920581	+	0.390553i	$0.127716\pi$
$677$			528.808i			0.781105i	−0.920581	+	0.390553i	$0.872284\pi$
$678$		0			0
$679$	199.601	−	16.8287i	0.293963	−	0.0247845i
$680$		0			0
$681$	−960.735			−1.41077
$682$		0			0
$683$	−300.950			−0.440629			−0.220315	−	0.975429i	$-0.570708\pi$
$683$	−300.950			−0.440629			−0.220315	+	0.975429i	$0.570708\pi$
$684$		0			0
$685$	−111.594	−	132.732i	−0.162911	−	0.193769i
$686$		0			0
$687$		−	127.360i		−	0.185385i
$688$		0			0
$689$			1292.42i			1.87580i
$690$		0			0
$691$	−761.277			−1.10170			−0.550852	−	0.834603i	$-0.685697\pi$
$691$	−761.277			−1.10170			−0.550852	+	0.834603i	$0.685697\pi$
$692$		0			0
$693$	11.5386	+	136.856i	0.0166502	+	0.197484i
$694$		0			0
$695$	−157.898	+	132.753i	−0.227192	+	0.191011i
$696$		0			0
$697$		−	7.01185i		−	0.0100600i
$698$		0			0
$699$	−1023.15			−1.46373
$700$		0			0
$701$			1086.65i			1.55015i	0.631870	+	0.775074i	$0.282288\pi$
$701$			1086.65i			1.55015i	−0.631870	+	0.775074i	$0.717712\pi$
$702$		0			0
$703$	1380.59			1.96386
$704$		0			0
$705$	−94.1299	−	111.960i	−0.133518	−	0.158808i
$706$		0			0
$707$	1349.85	−	113.808i	1.90927	−	0.160974i
$708$		0			0
$709$		−	353.798i		−	0.499010i	−0.968374	−	0.249505i	$-0.919732\pi$
$709$		−	353.798i		−	0.499010i	0.968374	−	0.249505i	$-0.0802679\pi$
$710$		0			0
$711$	111.673			0.157064
$712$		0			0
$713$	202.934			0.284620
$714$		0			0
$715$	−1099.95	+	924.780i	−1.53839	+	1.29340i
$716$		0			0
$717$			586.631i			0.818174i
$718$		0			0
$719$		−	63.4719i		−	0.0882781i	−0.999025	−	0.0441390i	$-0.985946\pi$
$719$		−	63.4719i		−	0.0882781i	0.999025	−	0.0441390i	$-0.0140545\pi$
$720$		0			0
$721$	−1147.54	+	96.7515i	−1.59160	+	0.134191i
$722$		0			0
$723$	−658.563			−0.910876
$724$		0			0
$725$	463.622	+	80.8255i	0.639479	+	0.111483i
$726$		0			0
$727$	888.016			1.22148			0.610740	−	0.791831i	$-0.290872\pi$
$727$	888.016			1.22148			0.610740	+	0.791831i	$0.290872\pi$
$728$		0			0
$729$	−577.186			−0.791750
$730$		0			0
$731$	−4.83478			−0.00661393
$732$		0			0
$733$		−	54.1893i		−	0.0739281i	−0.999317	−	0.0369640i	$-0.988231\pi$
$733$		−	54.1893i		−	0.0739281i	0.999317	−	0.0369640i	$-0.0117687\pi$
$734$		0			0
$735$	−603.880	−	512.236i	−0.821605	−	0.696920i
$736$		0			0
$737$			1260.02i			1.70967i
$738$		0			0
$739$			1414.92i			1.91464i	0.289034	+	0.957319i	$0.406666\pi$
$739$			1414.92i			1.91464i	−0.289034	+	0.957319i	$0.593334\pi$
$740$		0			0
$741$	1419.35			1.91545
$742$		0			0
$743$			51.9266i			0.0698877i	0.999389	+	0.0349439i	$0.0111252\pi$
$743$			51.9266i			0.0698877i	−0.999389	+	0.0349439i	$0.988875\pi$
$744$		0			0
$745$	635.190	+	755.506i	0.852605	+	1.01410i
$746$		0			0
$747$			136.852i			0.183202i
$748$		0			0
$749$	−552.711	+	46.6000i	−0.737932	+	0.0622163i
$750$		0			0
$751$	−926.269			−1.23338			−0.616690	−	0.787206i	$-0.711527\pi$
$751$	−926.269			−1.23338			−0.616690	+	0.787206i	$0.711527\pi$
$752$		0			0
$753$			388.909i			0.516480i
$754$		0			0
$755$	−296.044	+	248.898i	−0.392111	+	0.329666i
$756$		0			0
$757$	−899.385			−1.18809			−0.594046	−	0.804431i	$-0.702470\pi$
$757$	−899.385			−1.18809			−0.594046	+	0.804431i	$0.702470\pi$
$758$		0			0
$759$			1187.92i			1.56511i
$760$		0			0
$761$		−	673.539i		−	0.885071i	−0.896751	−	0.442536i	$-0.854079\pi$
$761$		−	673.539i		−	0.885071i	0.896751	−	0.442536i	$-0.145921\pi$
$762$		0			0
$763$	79.4953	+	942.873i	0.104188	+	1.23575i
$764$		0			0
$765$	−0.967281	+	0.813239i	−0.00126442	+	0.00106306i
$766$		0			0
$767$		−	308.256i		−	0.401899i
$768$		0			0
$769$		−	1344.86i		−	1.74884i	−0.485172	−	0.874419i	$-0.661243\pi$
$769$		−	1344.86i		−	1.74884i	0.485172	−	0.874419i	$-0.338757\pi$
$770$		0			0
$771$			692.586i			0.898295i
$772$		0			0
$773$		−	163.160i		−	0.211074i	−0.994415	−	0.105537i	$-0.966344\pi$
$773$		−	163.160i		−	0.211074i	0.994415	−	0.105537i	$-0.0336561\pi$
$774$		0			0
$775$	−184.440	−	32.1542i	−0.237987	−	0.0414894i
$776$		0			0
$777$	−126.631	−	1501.94i	−0.162975	−	1.93300i
$778$		0			0
$779$		−	831.681i		−	1.06763i
$780$		0			0
$781$			895.933i			1.14716i
$782$		0			0
$783$	−459.574			−0.586940
$784$		0			0
$785$	359.195	+	427.232i	0.457573	+	0.544245i
$786$		0			0
$787$		−	1155.66i		−	1.46844i	−0.678912	−	0.734220i	$-0.737548\pi$
$787$		−	1155.66i		−	1.46844i	0.678912	−	0.734220i	$-0.262452\pi$
$788$		0			0
$789$	−1255.09			−1.59074
$790$		0			0
$791$	−23.8980	−	283.448i	−0.0302124	−	0.358342i
$792$		0			0
$793$		−	757.156i		−	0.954800i
$794$		0			0
$795$	−634.289	−	754.434i	−0.797848	−	0.948974i
$796$		0			0
$797$		−	46.6189i		−	0.0584930i	−0.999572	−	0.0292465i	$-0.990689\pi$
$797$		−	46.6189i		−	0.0584930i	0.999572	−	0.0292465i	$-0.00931077\pi$
$798$		0			0
$799$	−1.58139			−0.00197921
$800$		0			0
$801$			86.3480i			0.107800i
$802$		0			0
$803$			698.469i			0.869825i
$804$		0			0
$805$	−669.177	−	672.111i	−0.831276	−	0.834920i
$806$		0			0
$807$		−	1428.98i		−	1.77073i
$808$		0			0
$809$	−254.107			−0.314100			−0.157050	−	0.987591i	$-0.550198\pi$
$809$	−254.107			−0.314100			−0.157050	+	0.987591i	$0.550198\pi$
$810$		0			0
$811$	1418.08			1.74855			0.874276	−	0.485429i	$-0.161337\pi$
$811$	1418.08			1.74855			0.874276	+	0.485429i	$0.161337\pi$
$812$		0			0
$813$	895.578			1.10157
$814$		0			0
$815$	−1191.50	+	1001.75i	−1.46196	+	1.22914i
$816$		0			0
$817$	−573.457			−0.701906
$818$		0			0
$819$	−18.0275	−	213.820i	−0.0220117	−	0.261074i
$820$		0			0
$821$		−	435.911i		−	0.530951i	−0.964118	−	0.265475i	$-0.914471\pi$
$821$		−	435.911i		−	0.530951i	0.964118	−	0.265475i	$-0.0855289\pi$
$822$		0			0
$823$		−	454.623i		−	0.552397i	−0.961101	−	0.276199i	$-0.910925\pi$
$823$		−	454.623i		−	0.552397i	0.961101	−	0.276199i	$-0.0890747\pi$
$824$		0			0
$825$	188.221	−	1079.65i	0.228147	−	1.30867i
$826$		0			0
$827$	−616.875			−0.745919			−0.372959	−	0.927848i	$-0.621657\pi$
$827$	−616.875			−0.745919			−0.372959	+	0.927848i	$0.621657\pi$
$828$		0			0
$829$	167.324			0.201838			0.100919	−	0.994895i	$-0.467822\pi$
$829$	167.324			0.201838			0.100919	+	0.994895i	$0.467822\pi$
$830$		0			0
$831$		−	928.459i		−	1.11728i
$832$		0			0
$833$	−8.44028	+	1.43342i	−0.0101324	+	0.00172079i
$834$		0			0
$835$	65.7923	−	55.3148i	0.0787932	−	0.0662452i
$836$		0			0
$837$	182.829			0.218434
$838$		0			0
$839$		−	713.072i		−	0.849907i	−0.905215	−	0.424954i	$-0.860290\pi$
$839$		−	713.072i		−	0.849907i	0.905215	−	0.424954i	$-0.139710\pi$
$840$		0			0
$841$	486.635			0.578639
$842$		0			0
$843$			757.647i			0.898751i
$844$		0			0
$845$	1071.75	−	901.068i	1.26834	−	1.06635i
$846$		0			0
$847$	−439.143	+	37.0249i	−0.518469	+	0.0437130i
$848$		0			0
$849$	248.330			0.292497
$850$		0			0
$851$		−	1805.29i		−	2.12137i
$852$		0			0
$853$		−	942.438i		−	1.10485i	−0.833562	−	0.552426i	$-0.813702\pi$
$853$		−	942.438i		−	1.10485i	0.833562	−	0.552426i	$-0.186298\pi$
$854$		0			0
$855$	−114.730	+	96.4589i	−0.134187	+	0.112817i
$856$		0			0
$857$	334.940			0.390828			0.195414	−	0.980721i	$-0.437395\pi$
$857$	334.940			0.390828			0.195414	+	0.980721i	$0.437395\pi$
$858$		0			0
$859$	1345.43			1.56628			0.783138	−	0.621848i	$-0.213618\pi$
$859$	1345.43			1.56628			0.783138	+	0.621848i	$0.213618\pi$
$860$		0			0
$861$	−904.783	+	76.2838i	−1.05085	+	0.0885991i
$862$		0			0
$863$			513.792i			0.595355i	0.954666	+	0.297678i	$0.0962121\pi$
$863$			513.792i			0.595355i	−0.954666	+	0.297678i	$0.903788\pi$
$864$		0			0
$865$	−618.107	−	735.187i	−0.714574	−	0.849927i
$866$		0			0
$867$		−	933.984i		−	1.07726i
$868$		0			0
$869$			1047.03i			1.20487i
$870$		0			0
$871$		−	1968.63i		−	2.26019i
$872$		0			0
$873$	−41.3950			−0.0474169
$874$		0			0
$875$	501.697	+	716.886i	0.573368	+	0.819298i
$876$		0			0
$877$	311.483			0.355169			0.177585	−	0.984106i	$-0.443172\pi$
$877$	311.483			0.355169			0.177585	+	0.984106i	$0.443172\pi$
$878$		0			0
$879$	−803.512			−0.914121
$880$		0			0
$881$			1360.45i			1.54421i	0.635495	+	0.772105i	$0.280796\pi$
$881$			1360.45i			1.54421i	−0.635495	+	0.772105i	$0.719204\pi$
$882$		0			0
$883$	−493.662			−0.559073			−0.279537	−	0.960135i	$-0.590181\pi$
$883$	−493.662			−0.559073			−0.279537	+	0.960135i	$0.590181\pi$
$884$		0			0
$885$	151.284	+	179.940i	0.170943	+	0.203322i
$886$		0			0
$887$	293.223			0.330579			0.165289	−	0.986245i	$-0.447144\pi$
$887$	293.223			0.330579			0.165289	+	0.986245i	$0.447144\pi$
$888$		0			0
$889$	−60.3569	−	715.877i	−0.0678930	−	0.805261i
$890$		0			0
$891$			1246.81i			1.39934i
$892$		0			0
$893$	−187.570			−0.210045
$894$		0			0
$895$	−374.911	−	445.926i	−0.418895	−	0.498241i
$896$		0			0
$897$		−	1855.97i		−	2.06908i
$898$		0			0
$899$	140.975			0.156813
$900$		0			0
$901$	−10.6561			−0.0118270
$902$		0			0
$903$	52.5989	+	623.862i	0.0582491	+	0.690878i
$904$		0			0
$905$	445.578	−	374.619i	0.492351	−	0.413943i
$906$		0			0
$907$	1322.64			1.45825			0.729127	−	0.684378i	$-0.239926\pi$
$907$	1322.64			1.45825			0.729127	+	0.684378i	$0.239926\pi$
$908$		0			0
$909$	−279.945			−0.307970
$910$		0			0
$911$	−49.6878			−0.0545420			−0.0272710	−	0.999628i	$-0.508682\pi$
$911$	−49.6878			−0.0545420			−0.0272710	+	0.999628i	$0.508682\pi$
$912$		0			0
$913$	−1283.11			−1.40538
$914$		0			0
$915$	371.593	+	441.979i	0.406112	+	0.483037i
$916$		0			0
$917$	−512.053	+	43.1721i	−0.558400	+	0.0470797i
$918$		0			0
$919$	−927.263			−1.00899			−0.504496	−	0.863414i	$-0.668322\pi$
$919$	−927.263			−1.00899			−0.504496	+	0.863414i	$0.668322\pi$
$920$		0			0
$921$	−791.647			−0.859552
$922$		0			0
$923$		−	1399.78i		−	1.51655i
$924$		0			0
$925$	−286.041	+	1640.76i	−0.309234	+	1.77379i
$926$		0			0
$927$	237.988			0.256729
$928$		0			0
$929$			406.847i			0.437940i	0.975732	+	0.218970i	$0.0702698\pi$
$929$			406.847i			0.437940i	−0.975732	+	0.218970i	$0.929730\pi$
$930$		0			0
$931$	−1001.11	+	170.019i	−1.07530	+	0.182620i
$932$		0			0
$933$	−1799.98			−1.92924
$934$		0			0
$935$	−7.62485	−	9.06912i	−0.00815492	−	0.00969960i
$936$		0			0
$937$	−648.308			−0.691897			−0.345949	−	0.938253i	$-0.612443\pi$
$937$	−648.308			−0.691897			−0.345949	+	0.938253i	$0.612443\pi$
$938$		0			0
$939$			209.370i			0.222972i
$940$		0			0
$941$	543.789			0.577884			0.288942	−	0.957347i	$-0.406696\pi$
$941$	543.789			0.577884			0.288942	+	0.957347i	$0.406696\pi$
$942$		0			0
$943$	−1087.52			−1.15326
$944$		0			0
$945$	−602.881	−	605.524i	−0.637969	−	0.640766i
$946$		0			0
$947$	609.736			0.643861			0.321930	−	0.946763i	$-0.395668\pi$
$947$	609.736			0.643861			0.321930	+	0.946763i	$0.395668\pi$
$948$		0			0
$949$		−	1091.27i		−	1.14991i
$950$		0			0
$951$			1816.79i			1.91039i
$952$		0			0
$953$			1454.58i			1.52632i	0.646210	+	0.763159i	$0.276353\pi$
$953$			1454.58i			1.52632i	−0.646210	+	0.763159i	$0.723647\pi$
$954$		0			0
$955$	485.876	−	408.499i	0.508771	−	0.427748i
$956$		0			0
$957$			825.223i			0.862302i
$958$		0			0
$959$	20.3963	+	241.916i	0.0212683	+	0.252258i
$960$		0			0
$961$	904.917			0.941641
$962$		0			0
$963$	114.626			0.119030
$964$		0			0
$965$	−745.317	−	886.493i	−0.772349	−	0.918646i
$966$		0			0
$967$			1095.53i			1.13292i	0.824089	+	0.566460i	$0.191687\pi$
$967$			1095.53i			1.13292i	−0.824089	+	0.566460i	$0.808313\pi$
$968$		0			0
$969$			11.7026i			0.0120770i
$970$		0			0
$971$	−1248.77			−1.28607			−0.643035	−	0.765836i	$-0.722325\pi$
$971$	−1248.77			−1.28607			−0.643035	+	0.765836i	$0.722325\pi$
$972$		0			0
$973$	287.784	−	24.2635i	0.295769	−	0.0249368i
$974$		0			0
$975$	−294.071	+	1686.82i	−0.301612	+	1.73007i
$976$		0			0
$977$			17.6376i			0.0180529i	0.999959	+	0.00902643i	$0.00287324\pi$
$977$			17.6376i			0.0180529i	−0.999959	+	0.00902643i	$0.997127\pi$
$978$		0			0
$979$	−809.590			−0.826956
$980$		0			0
$981$		−	195.542i		−	0.199329i
$982$		0			0
$983$	639.985			0.651053			0.325527	−	0.945533i	$-0.394458\pi$
$983$	639.985			0.651053			0.325527	+	0.945533i	$0.394458\pi$
$984$		0			0
$985$	−298.486	+	250.951i	−0.303031	+	0.254773i
$986$		0			0
$987$	17.2044	+	204.057i	0.0174310	+	0.206744i
$988$		0			0
$989$			749.862i			0.758203i
$990$		0			0
$991$	−978.203			−0.987086			−0.493543	−	0.869721i	$-0.664299\pi$
$991$	−978.203			−0.987086			−0.493543	+	0.869721i	$0.664299\pi$
$992$		0			0
$993$	371.837			0.374458
$994$		0			0
$995$	−733.215	−	872.098i	−0.736899	−	0.876481i
$996$		0			0
$997$		−	14.7356i		−	0.0147799i	−0.999973	−	0.00738996i	$-0.997648\pi$
$997$		−	14.7356i		−	0.0147799i	0.999973	−	0.00738996i	$-0.00235232\pi$
$998$		0			0
$999$		−	1626.43i		−	1.62806i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1120.3.c.g.209.8		80
4.3	odd	2		280.3.c.g.69.63	yes	80
5.4	even	2	inner	1120.3.c.g.209.37		80
7.6	odd	2	inner	1120.3.c.g.209.35		80
8.3	odd	2		280.3.c.g.69.20	yes	80
8.5	even	2	inner	1120.3.c.g.209.9		80
20.19	odd	2		280.3.c.g.69.18	yes	80
28.27	even	2		280.3.c.g.69.64	yes	80
35.34	odd	2	inner	1120.3.c.g.209.10		80
40.19	odd	2		280.3.c.g.69.61	yes	80
40.29	even	2	inner	1120.3.c.g.209.36		80
56.13	odd	2	inner	1120.3.c.g.209.38		80
56.27	even	2		280.3.c.g.69.19	yes	80
140.139	even	2		280.3.c.g.69.17	✓	80
280.69	odd	2	inner	1120.3.c.g.209.7		80
280.139	even	2		280.3.c.g.69.62	yes	80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.3.c.g.69.17	✓	80	140.139	even	2
280.3.c.g.69.18	yes	80	20.19	odd	2
280.3.c.g.69.19	yes	80	56.27	even	2
280.3.c.g.69.20	yes	80	8.3	odd	2
280.3.c.g.69.61	yes	80	40.19	odd	2
280.3.c.g.69.62	yes	80	280.139	even	2
280.3.c.g.69.63	yes	80	4.3	odd	2
280.3.c.g.69.64	yes	80	28.27	even	2
1120.3.c.g.209.7		80	280.69	odd	2	inner
1120.3.c.g.209.8		80	1.1	even	1	trivial
1120.3.c.g.209.9		80	8.5	even	2	inner
1120.3.c.g.209.10		80	35.34	odd	2	inner
1120.3.c.g.209.35		80	7.6	odd	2	inner
1120.3.c.g.209.36		80	40.29	even	2	inner
1120.3.c.g.209.37		80	5.4	even	2	inner
1120.3.c.g.209.38		80	56.13	odd	2	inner

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1120 = 2^{5} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1120.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.23212i q^{3} +(-3.82711 + 3.21764i) q^{5} +(6.97525 - 0.588096i) q^{7} -1.44659 q^{9} +O(q^{10})\)	\(q+3.23212i q^{3} +(-3.82711 + 3.21764i) q^{5} +(6.97525 - 0.588096i) q^{7} -1.44659 q^{9} -13.5631i q^{11} +21.1906i q^{13} +(-10.3998 - 12.3697i) q^{15} -0.174717 q^{17} -20.7233 q^{19} +(1.90080 + 22.5448i) q^{21} +27.0982i q^{23} +(4.29361 - 24.6285i) q^{25} +24.4135i q^{27} +18.8246i q^{29} -7.48886i q^{31} +43.8375 q^{33} +(-24.8028 + 24.6945i) q^{35} -66.6203 q^{37} -68.4904 q^{39} +40.1326i q^{41} +27.6721 q^{43} +(5.53627 - 4.65461i) q^{45} +9.05114 q^{47} +(48.3083 - 8.20423i) q^{49} -0.564706i q^{51} +60.9906 q^{53} +(43.6411 + 51.9075i) q^{55} -66.9803i q^{57} -14.5469 q^{59} -35.7308 q^{61} +(-10.0903 + 0.850734i) q^{63} +(-68.1836 - 81.0987i) q^{65} -92.9010 q^{67} -87.5845 q^{69} -66.0567 q^{71} -51.4978 q^{73} +(79.6024 + 13.8775i) q^{75} +(-7.97640 - 94.6060i) q^{77} -77.1971 q^{79} -91.9267 q^{81} -94.6032i q^{83} +(0.668662 - 0.562176i) q^{85} -60.8433 q^{87} -59.6907i q^{89} +(12.4621 + 147.810i) q^{91} +24.2049 q^{93} +(79.3106 - 66.6802i) q^{95} +28.6155 q^{97} +19.6202i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 224 q^{9}+O(q^{10}) \) 80 * q - 224 * q^9	\( 80 q - 224 q^{9} + 72 q^{15} - 104 q^{25} + 112 q^{39} + 192 q^{49} + 472 q^{65} - 800 q^{71} - 480 q^{79} - 896 q^{81} - 1176 q^{95}+O(q^{100}) \) 80 * q - 224 * q^9 + 72 * q^15 - 104 * q^25 + 112 * q^39 + 192 * q^49 + 472 * q^65 - 800 * q^71 - 480 * q^79 - 896 * q^81 - 1176 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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