LMFDB - Embedded newform 840.2.t.e.169.6

Newspace parameters

Level:	$ N $	$=$	$ 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	840.t (of order $2$, degree $1$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$6.70743376979$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.350464.1
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 $ x^6 - 2x^5 + 2x^4 + 2x^3 + 4x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			169.6
Root			$0.403032 - 0.403032i$ of defining polynomial
Character	$\chi$	$=$	840.169
Dual form			840.2.t.e.169.3

$q$-expansion

$f(q)$	$=$	$q+1.00000i q^{3} +(1.48119 - 1.67513i) q^{5} -1.00000i q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +(1.48119 - 1.67513i) q^{5} -1.00000i q^{7} -1.00000 q^{9} +0.387873 q^{11} -2.96239i q^{13} +(1.67513 + 1.48119i) q^{15} -3.35026i q^{17} +2.96239 q^{19} +1.00000 q^{21} -0.962389i q^{23} +(-0.612127 - 4.96239i) q^{25} -1.00000i q^{27} -1.22425 q^{29} +2.96239 q^{31} +0.387873i q^{33} +(-1.67513 - 1.48119i) q^{35} -5.92478i q^{37} +2.96239 q^{39} +1.03761 q^{41} +10.7005i q^{43} +(-1.48119 + 1.67513i) q^{45} -3.22425i q^{47} -1.00000 q^{49} +3.35026 q^{51} -5.66291i q^{53} +(0.574515 - 0.649738i) q^{55} +2.96239i q^{57} -3.22425 q^{59} +14.6253 q^{61} +1.00000i q^{63} +(-4.96239 - 4.38787i) q^{65} +0.962389 q^{69} +5.53690 q^{71} +6.18664i q^{73} +(4.96239 - 0.612127i) q^{75} -0.387873i q^{77} +13.9248 q^{79} +1.00000 q^{81} +3.22425i q^{83} +(-5.61213 - 4.96239i) q^{85} -1.22425i q^{87} -3.73813 q^{89} -2.96239 q^{91} +2.96239i q^{93} +(4.38787 - 4.96239i) q^{95} +7.73813i q^{97} -0.387873 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{5} - 6 q^{9}+O(q^{10}) $ 6 * q - 2 * q^5 - 6 * q^9	$ 6 q - 2 q^{5} - 6 q^{9} + 4 q^{11} - 4 q^{19} + 6 q^{21} - 2 q^{25} - 4 q^{29} - 4 q^{31} - 4 q^{39} + 28 q^{41} + 2 q^{45} - 6 q^{49} - 20 q^{55} - 16 q^{59} + 4 q^{61} - 8 q^{65} - 16 q^{69} - 12 q^{71} + 8 q^{75} + 40 q^{79} + 6 q^{81} - 32 q^{85} - 4 q^{89} + 4 q^{91} + 28 q^{95} - 4 q^{99}+O(q^{100}) $ 6 * q - 2 * q^5 - 6 * q^9 + 4 * q^11 - 4 * q^19 + 6 * q^21 - 2 * q^25 - 4 * q^29 - 4 * q^31 - 4 * q^39 + 28 * q^41 + 2 * q^45 - 6 * q^49 - 20 * q^55 - 16 * q^59 + 4 * q^61 - 8 * q^65 - 16 * q^69 - 12 * q^71 + 8 * q^75 + 40 * q^79 + 6 * q^81 - 32 * q^85 - 4 * q^89 + 4 * q^91 + 28 * q^95 - 4 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/840\mathbb{Z}\right)^\times$.

$n$	$241$	$281$	$337$	$421$	$631$
$\chi(n)$	$1$	$1$	$-1$	$1$	$1$

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
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$		0			0
$5$	1.48119	−	1.67513i	0.662410	−	0.749141i
$5$	1.48119	−	1.67513i	0.662410	−	0.749141i
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	0.387873			0.116948			0.0584741	−	0.998289i	$-0.481377\pi$
$11$	0.387873			0.116948			0.0584741	+	0.998289i	$0.481377\pi$
$12$		0			0
$13$		−	2.96239i		−	0.821619i	−0.911721	−	0.410809i	$-0.865246\pi$
$13$		−	2.96239i		−	0.821619i	0.911721	−	0.410809i	$-0.134754\pi$
$14$		0			0
$15$	1.67513	+	1.48119i	0.432517	+	0.382443i
$16$		0			0
$17$		−	3.35026i		−	0.812558i	−0.913749	−	0.406279i	$-0.866826\pi$
$17$		−	3.35026i		−	0.812558i	0.913749	−	0.406279i	$-0.133174\pi$
$18$		0			0
$19$	2.96239			0.679619			0.339809	−	0.940494i	$-0.389637\pi$
$19$	2.96239			0.679619			0.339809	+	0.940494i	$0.389637\pi$
$20$		0			0
$21$	1.00000			0.218218
$22$		0			0
$23$		−	0.962389i		−	0.200672i	−0.994954	−	0.100336i	$-0.968008\pi$
$23$		−	0.962389i		−	0.200672i	0.994954	−	0.100336i	$-0.0319918\pi$
$24$		0			0
$25$	−0.612127	−	4.96239i	−0.122425	−	0.992478i
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$	−1.22425			−0.227338			−0.113669	−	0.993519i	$-0.536260\pi$
$29$	−1.22425			−0.227338			−0.113669	+	0.993519i	$0.536260\pi$
$30$		0			0
$31$	2.96239			0.532061			0.266030	−	0.963965i	$-0.414288\pi$
$31$	2.96239			0.532061			0.266030	+	0.963965i	$0.414288\pi$
$32$		0			0
$33$			0.387873i			0.0675200i
$34$		0			0
$35$	−1.67513	−	1.48119i	−0.283149	−	0.250368i
$36$		0			0
$37$		−	5.92478i		−	0.974027i	−0.873394	−	0.487014i	$-0.838086\pi$
$37$		−	5.92478i		−	0.974027i	0.873394	−	0.487014i	$-0.161914\pi$
$38$		0			0
$39$	2.96239			0.474362
$40$		0			0
$41$	1.03761			0.162048			0.0810238	−	0.996712i	$-0.474181\pi$
$41$	1.03761			0.162048			0.0810238	+	0.996712i	$0.474181\pi$
$42$		0			0
$43$			10.7005i			1.63181i	0.578183	+	0.815907i	$0.303762\pi$
$43$			10.7005i			1.63181i	−0.578183	+	0.815907i	$0.696238\pi$
$44$		0			0
$45$	−1.48119	+	1.67513i	−0.220803	+	0.249714i
$46$		0			0
$47$		−	3.22425i		−	0.470306i	−0.971958	−	0.235153i	$-0.924441\pi$
$47$		−	3.22425i		−	0.470306i	0.971958	−	0.235153i	$-0.0755591\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	3.35026			0.469130
$52$		0			0
$53$		−	5.66291i		−	0.777861i	−0.921267	−	0.388930i	$-0.872845\pi$
$53$		−	5.66291i		−	0.777861i	0.921267	−	0.388930i	$-0.127155\pi$
$54$		0			0
$55$	0.574515	−	0.649738i	0.0774677	−	0.0876107i
$56$		0			0
$57$			2.96239i			0.392378i
$58$		0			0
$59$	−3.22425			−0.419762			−0.209881	−	0.977727i	$-0.567308\pi$
$59$	−3.22425			−0.419762			−0.209881	+	0.977727i	$0.567308\pi$
$60$		0			0
$61$	14.6253			1.87258			0.936289	−	0.351231i	$-0.114237\pi$
$61$	14.6253			1.87258			0.936289	+	0.351231i	$0.114237\pi$
$62$		0			0
$63$			1.00000i			0.125988i
$64$		0			0
$65$	−4.96239	−	4.38787i	−0.615509	−	0.544249i
$66$		0			0
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$		0			0
$69$	0.962389			0.115858
$70$		0			0
$71$	5.53690			0.657110			0.328555	−	0.944485i	$-0.393438\pi$
$71$	5.53690			0.657110			0.328555	+	0.944485i	$0.393438\pi$
$72$		0			0
$73$			6.18664i			0.724092i	0.932160	+	0.362046i	$0.117922\pi$
$73$			6.18664i			0.724092i	−0.932160	+	0.362046i	$0.882078\pi$
$74$		0			0
$75$	4.96239	−	0.612127i	0.573007	−	0.0706823i
$76$		0			0
$77$		−	0.387873i		−	0.0442022i
$78$		0			0
$79$	13.9248			1.56666			0.783330	−	0.621606i	$-0.213520\pi$
$79$	13.9248			1.56666			0.783330	+	0.621606i	$0.213520\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			3.22425i			0.353908i	0.984219	+	0.176954i	$0.0566244\pi$
$83$			3.22425i			0.353908i	−0.984219	+	0.176954i	$0.943376\pi$
$84$		0			0
$85$	−5.61213	−	4.96239i	−0.608721	−	0.538247i
$86$		0			0
$87$		−	1.22425i		−	0.131254i
$88$		0			0
$89$	−3.73813			−0.396242			−0.198121	−	0.980178i	$-0.563484\pi$
$89$	−3.73813			−0.396242			−0.198121	+	0.980178i	$0.563484\pi$
$90$		0			0
$91$	−2.96239			−0.310543
$92$		0			0
$93$			2.96239i			0.307185i
$94$		0			0
$95$	4.38787	−	4.96239i	0.450186	−	0.509130i
$96$		0			0
$97$			7.73813i			0.785689i	0.919605	+	0.392844i	$0.128509\pi$
$97$			7.73813i			0.785689i	−0.919605	+	0.392844i	$0.871491\pi$
$98$		0			0
$99$	−0.387873			−0.0389827
$100$		0			0
$101$	−7.58769			−0.755003			−0.377502	−	0.926009i	$-0.623217\pi$
$101$	−7.58769			−0.755003			−0.377502	+	0.926009i	$0.623217\pi$
$102$		0			0
$103$			14.7005i			1.44849i	0.689545	+	0.724243i	$0.257811\pi$
$103$			14.7005i			1.44849i	−0.689545	+	0.724243i	$0.742189\pi$
$104$		0			0
$105$	1.48119	−	1.67513i	0.144550	−	0.163476i
$106$		0			0
$107$		−	16.4387i		−	1.58919i	−0.607143	−	0.794593i	$-0.707685\pi$
$107$		−	16.4387i		−	1.58919i	0.607143	−	0.794593i	$-0.292315\pi$
$108$		0			0
$109$	2.77575			0.265868			0.132934	−	0.991125i	$-0.457560\pi$
$109$	2.77575			0.265868			0.132934	+	0.991125i	$0.457560\pi$
$110$		0			0
$111$	5.92478			0.562355
$112$		0			0
$113$			4.26187i			0.400923i	0.979702	+	0.200461i	$0.0642441\pi$
$113$			4.26187i			0.400923i	−0.979702	+	0.200461i	$0.935756\pi$
$114$		0			0
$115$	−1.61213	−	1.42548i	−0.150332	−	0.132927i
$116$		0			0
$117$			2.96239i			0.273873i
$118$		0			0
$119$	−3.35026			−0.307118
$120$		0			0
$121$	−10.8496			−0.986323
$122$		0			0
$123$			1.03761i			0.0935583i
$124$		0			0
$125$	−9.21933	−	6.32487i	−0.824602	−	0.565713i
$126$		0			0
$127$			14.5501i			1.29111i	0.763714	+	0.645555i	$0.223374\pi$
$127$			14.5501i			1.29111i	−0.763714	+	0.645555i	$0.776626\pi$
$128$		0			0
$129$	−10.7005			−0.942129
$130$		0			0
$131$	−4.00000			−0.349482			−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000			−0.349482			−0.174741	+	0.984614i	$0.555909\pi$
$132$		0			0
$133$		−	2.96239i		−	0.256872i
$134$		0			0
$135$	−1.67513	−	1.48119i	−0.144172	−	0.127481i
$136$		0			0
$137$		−	10.4387i		−	0.891835i	−0.895074	−	0.445917i	$-0.852878\pi$
$137$		−	10.4387i		−	0.891835i	0.895074	−	0.445917i	$-0.147122\pi$
$138$		0			0
$139$	−5.66291			−0.480322			−0.240161	−	0.970733i	$-0.577200\pi$
$139$	−5.66291			−0.480322			−0.240161	+	0.970733i	$0.577200\pi$
$140$		0			0
$141$	3.22425			0.271531
$142$		0			0
$143$		−	1.14903i		−	0.0960868i
$144$		0			0
$145$	−1.81336	+	2.05079i	−0.150591	+	0.170308i
$146$		0			0
$147$		−	1.00000i		−	0.0824786i
$148$		0			0
$149$	−20.5501			−1.68353			−0.841764	−	0.539846i	$-0.818483\pi$
$149$	−20.5501			−1.68353			−0.841764	+	0.539846i	$0.818483\pi$
$150$		0			0
$151$	−22.7005			−1.84734			−0.923671	−	0.383186i	$-0.874827\pi$
$151$	−22.7005			−1.84734			−0.923671	+	0.383186i	$0.874827\pi$
$152$		0			0
$153$			3.35026i			0.270853i
$154$		0			0
$155$	4.38787	−	4.96239i	0.352442	−	0.398589i
$156$		0			0
$157$			2.18664i			0.174513i	0.996186	+	0.0872565i	$0.0278100\pi$
$157$			2.18664i			0.174513i	−0.996186	+	0.0872565i	$0.972190\pi$
$158$		0			0
$159$	5.66291			0.449098
$160$		0			0
$161$	−0.962389			−0.0758468
$162$		0			0
$163$			7.22425i			0.565847i	0.959142	+	0.282924i	$0.0913043\pi$
$163$			7.22425i			0.565847i	−0.959142	+	0.282924i	$0.908696\pi$
$164$		0			0
$165$	0.649738	+	0.574515i	0.0505821	+	0.0447260i
$166$		0			0
$167$			15.8496i			1.22648i	0.789898	+	0.613238i	$0.210133\pi$
$167$			15.8496i			1.22648i	−0.789898	+	0.613238i	$0.789867\pi$
$168$		0			0
$169$	4.22425			0.324943
$170$		0			0
$171$	−2.96239			−0.226540
$172$		0			0
$173$			23.9756i			1.82283i	0.411490	+	0.911414i	$0.365008\pi$
$173$			23.9756i			1.82283i	−0.411490	+	0.911414i	$0.634992\pi$
$174$		0			0
$175$	−4.96239	+	0.612127i	−0.375121	+	0.0462724i
$176$		0			0
$177$		−	3.22425i		−	0.242350i
$178$		0			0
$179$	−12.2374			−0.914668			−0.457334	−	0.889295i	$-0.651196\pi$
$179$	−12.2374			−0.914668			−0.457334	+	0.889295i	$0.651196\pi$
$180$		0			0
$181$	8.70052			0.646705			0.323352	−	0.946279i	$-0.395190\pi$
$181$	8.70052			0.646705			0.323352	+	0.946279i	$0.395190\pi$
$182$		0			0
$183$			14.6253i			1.08113i
$184$		0			0
$185$	−9.92478	−	8.77575i	−0.729684	−	0.645206i
$186$		0			0
$187$		−	1.29948i		−	0.0950271i
$188$		0			0
$189$	−1.00000			−0.0727393
$190$		0			0
$191$	−13.5369			−0.979496			−0.489748	−	0.871864i	$-0.662911\pi$
$191$	−13.5369			−0.979496			−0.489748	+	0.871864i	$0.662911\pi$
$192$		0			0
$193$		−	1.92478i		−	0.138548i	−0.997598	−	0.0692742i	$-0.977932\pi$
$193$		−	1.92478i		−	0.138548i	0.997598	−	0.0692742i	$-0.0220684\pi$
$194$		0			0
$195$	4.38787	−	4.96239i	0.314222	−	0.355364i
$196$		0			0
$197$		−	20.1114i		−	1.43288i	−0.697649	−	0.716440i	$-0.745771\pi$
$197$		−	20.1114i		−	1.43288i	0.697649	−	0.716440i	$-0.254229\pi$
$198$		0			0
$199$	8.26187			0.585668			0.292834	−	0.956163i	$-0.405402\pi$
$199$	8.26187			0.585668			0.292834	+	0.956163i	$0.405402\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			1.22425i			0.0859258i
$204$		0			0
$205$	1.53690	−	1.73813i	0.107342	−	0.121397i
$206$		0			0
$207$			0.962389i			0.0668906i
$208$		0			0
$209$	1.14903			0.0794801
$210$		0			0
$211$	−18.1768			−1.25134			−0.625671	−	0.780087i	$-0.715175\pi$
$211$	−18.1768			−1.25134			−0.625671	+	0.780087i	$0.715175\pi$
$212$		0			0
$213$			5.53690i			0.379382i
$214$		0			0
$215$	17.9248	+	15.8496i	1.22246	+	1.08093i
$216$		0			0
$217$		−	2.96239i		−	0.201100i
$218$		0			0
$219$	−6.18664			−0.418055
$220$		0			0
$221$	−9.92478			−0.667613
$222$		0			0
$223$			21.4010i			1.43312i	0.697525	+	0.716560i	$0.254284\pi$
$223$			21.4010i			1.43312i	−0.697525	+	0.716560i	$0.745716\pi$
$224$		0			0
$225$	0.612127	+	4.96239i	0.0408085	+	0.330826i
$226$		0			0
$227$		−	18.5501i		−	1.23121i	−0.788054	−	0.615606i	$-0.788911\pi$
$227$		−	18.5501i		−	1.23121i	0.788054	−	0.615606i	$-0.211089\pi$
$228$		0			0
$229$	28.5501			1.88664			0.943321	−	0.331881i	$-0.107683\pi$
$229$	28.5501			1.88664			0.943321	+	0.331881i	$0.107683\pi$
$230$		0			0
$231$	0.387873			0.0255202
$232$		0			0
$233$			18.9624i			1.24227i	0.783705	+	0.621134i	$0.213328\pi$
$233$			18.9624i			1.24227i	−0.783705	+	0.621134i	$0.786672\pi$
$234$		0			0
$235$	−5.40105	−	4.77575i	−0.352325	−	0.311535i
$236$		0			0
$237$			13.9248i			0.904511i
$238$		0			0
$239$	18.1622			1.17482			0.587408	−	0.809291i	$-0.300149\pi$
$239$	18.1622			1.17482			0.587408	+	0.809291i	$0.300149\pi$
$240$		0			0
$241$	14.3733			0.925865			0.462932	−	0.886394i	$-0.346797\pi$
$241$	14.3733			0.925865			0.462932	+	0.886394i	$0.346797\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$		0			0
$245$	−1.48119	+	1.67513i	−0.0946300	+	0.107020i
$246$		0			0
$247$		−	8.77575i		−	0.558387i
$248$		0			0
$249$	−3.22425			−0.204329
$250$		0			0
$251$	−8.62530			−0.544424			−0.272212	−	0.962237i	$-0.587755\pi$
$251$	−8.62530			−0.544424			−0.272212	+	0.962237i	$0.587755\pi$
$252$		0			0
$253$		−	0.373285i		−	0.0234682i
$254$		0			0
$255$	4.96239	−	5.61213i	0.310757	−	0.351445i
$256$		0			0
$257$			27.8251i			1.73568i	0.496841	+	0.867842i	$0.334493\pi$
$257$			27.8251i			1.73568i	−0.496841	+	0.867842i	$0.665507\pi$
$258$		0			0
$259$	−5.92478			−0.368148
$260$		0			0
$261$	1.22425			0.0757794
$262$		0			0
$263$		−	0.589104i		−	0.0363257i	−0.999835	−	0.0181629i	$-0.994218\pi$
$263$		−	0.589104i		−	0.0363257i	0.999835	−	0.0181629i	$-0.00578173\pi$
$264$		0			0
$265$	−9.48612	−	8.38787i	−0.582728	−	0.515263i
$266$		0			0
$267$		−	3.73813i		−	0.228770i
$268$		0			0
$269$	−15.7381			−0.959571			−0.479786	−	0.877386i	$-0.659286\pi$
$269$	−15.7381			−0.959571			−0.479786	+	0.877386i	$0.659286\pi$
$270$		0			0
$271$	−7.11283			−0.432074			−0.216037	−	0.976385i	$-0.569313\pi$
$271$	−7.11283			−0.432074			−0.216037	+	0.976385i	$0.569313\pi$
$272$		0			0
$273$		−	2.96239i		−	0.179292i
$274$		0			0
$275$	−0.237428	−	1.92478i	−0.0143174	−	0.116068i
$276$		0			0
$277$		−	5.92478i		−	0.355985i	−0.984032	−	0.177993i	$-0.943040\pi$
$277$		−	5.92478i		−	0.355985i	0.984032	−	0.177993i	$-0.0569603\pi$
$278$		0			0
$279$	−2.96239			−0.177354
$280$		0			0
$281$	29.3258			1.74943			0.874716	−	0.484636i	$-0.161048\pi$
$281$	29.3258			1.74943			0.874716	+	0.484636i	$0.161048\pi$
$282$		0			0
$283$		−	16.1016i		−	0.957139i	−0.878050	−	0.478570i	$-0.841155\pi$
$283$		−	16.1016i		−	0.957139i	0.878050	−	0.478570i	$-0.158845\pi$
$284$		0			0
$285$	4.96239	+	4.38787i	0.293947	+	0.259915i
$286$		0			0
$287$		−	1.03761i		−	0.0612483i
$288$		0			0
$289$	5.77575			0.339750
$290$		0			0
$291$	−7.73813			−0.453617
$292$		0			0
$293$			9.27504i			0.541854i	0.962600	+	0.270927i	$0.0873301\pi$
$293$			9.27504i			0.541854i	−0.962600	+	0.270927i	$0.912670\pi$
$294$		0			0
$295$	−4.77575	+	5.40105i	−0.278055	+	0.314461i
$296$		0			0
$297$		−	0.387873i		−	0.0225067i
$298$		0			0
$299$	−2.85097			−0.164876
$300$		0			0
$301$	10.7005			0.616768
$302$		0			0
$303$		−	7.58769i		−	0.435901i
$304$		0			0
$305$	21.6629	−	24.4993i	1.24041	−	1.40283i
$306$		0			0
$307$		−	10.7005i		−	0.610711i	−0.952238	−	0.305356i	$-0.901225\pi$
$307$		−	10.7005i		−	0.610711i	0.952238	−	0.305356i	$-0.0987753\pi$
$308$		0			0
$309$	−14.7005			−0.836284
$310$		0			0
$311$	12.1016			0.686217			0.343109	−	0.939296i	$-0.388520\pi$
$311$	12.1016			0.686217			0.343109	+	0.939296i	$0.388520\pi$
$312$		0			0
$313$			28.3634i			1.60320i	0.597863	+	0.801598i	$0.296017\pi$
$313$			28.3634i			1.60320i	−0.597863	+	0.801598i	$0.703983\pi$
$314$		0			0
$315$	1.67513	+	1.48119i	0.0943829	+	0.0834558i
$316$		0			0
$317$			17.5125i			0.983598i	0.870709	+	0.491799i	$0.163661\pi$
$317$			17.5125i			0.983598i	−0.870709	+	0.491799i	$0.836339\pi$
$318$		0			0
$319$	−0.474855			−0.0265868
$320$		0			0
$321$	16.4387			0.917516
$322$		0			0
$323$		−	9.92478i		−	0.552229i
$324$		0			0
$325$	−14.7005	+	1.81336i	−0.815438	+	0.100587i
$326$		0			0
$327$			2.77575i			0.153499i
$328$		0			0
$329$	−3.22425			−0.177759
$330$		0			0
$331$	−19.2243			−1.05666			−0.528330	−	0.849039i	$-0.677182\pi$
$331$	−19.2243			−1.05666			−0.528330	+	0.849039i	$0.677182\pi$
$332$		0			0
$333$			5.92478i			0.324676i
$334$		0			0
$335$		0			0
$336$		0			0
$337$		−	16.0000i		−	0.871576i	−0.900049	−	0.435788i	$-0.856470\pi$
$337$		−	16.0000i		−	0.871576i	0.900049	−	0.435788i	$-0.143530\pi$
$338$		0			0
$339$	−4.26187			−0.231473
$340$		0			0
$341$	1.14903			0.0622235
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$	1.42548	−	1.61213i	0.0767455	−	0.0867940i
$346$		0			0
$347$			1.11283i			0.0597401i	0.999554	+	0.0298700i	$0.00950934\pi$
$347$			1.11283i			0.0597401i	−0.999554	+	0.0298700i	$0.990491\pi$
$348$		0			0
$349$	32.0263			1.71433			0.857166	−	0.515041i	$-0.172223\pi$
$349$	32.0263			1.71433			0.857166	+	0.515041i	$0.172223\pi$
$350$		0			0
$351$	−2.96239			−0.158121
$352$		0			0
$353$			6.20123i			0.330058i	0.986289	+	0.165029i	$0.0527718\pi$
$353$			6.20123i			0.330058i	−0.986289	+	0.165029i	$0.947228\pi$
$354$		0			0
$355$	8.20123	−	9.27504i	0.435276	−	0.492268i
$356$		0			0
$357$		−	3.35026i		−	0.177315i
$358$		0			0
$359$	1.68735			0.0890549			0.0445275	−	0.999008i	$-0.485822\pi$
$359$	1.68735			0.0890549			0.0445275	+	0.999008i	$0.485822\pi$
$360$		0			0
$361$	−10.2243			−0.538119
$362$		0			0
$363$		−	10.8496i		−	0.569454i
$364$		0			0
$365$	10.3634	+	9.16362i	0.542447	+	0.479646i
$366$		0			0
$367$		−	3.84955i		−	0.200945i	−0.994940	−	0.100473i	$-0.967965\pi$
$367$		−	3.84955i		−	0.200945i	0.994940	−	0.100473i	$-0.0320355\pi$
$368$		0			0
$369$	−1.03761			−0.0540159
$370$		0			0
$371$	−5.66291			−0.294004
$372$		0			0
$373$		−	24.8773i		−	1.28810i	−0.764984	−	0.644049i	$-0.777253\pi$
$373$		−	24.8773i		−	1.28810i	0.764984	−	0.644049i	$-0.222747\pi$
$374$		0			0
$375$	6.32487	−	9.21933i	0.326615	−	0.476084i
$376$		0			0
$377$			3.62672i			0.186785i
$378$		0			0
$379$	−2.70052			−0.138717			−0.0693583	−	0.997592i	$-0.522095\pi$
$379$	−2.70052			−0.138717			−0.0693583	+	0.997592i	$0.522095\pi$
$380$		0			0
$381$	−14.5501			−0.745423
$382$		0			0
$383$		−	0.523730i		−	0.0267614i	−0.999910	−	0.0133807i	$-0.995741\pi$
$383$		−	0.523730i		−	0.0267614i	0.999910	−	0.0133807i	$-0.00425933\pi$
$384$		0			0
$385$	−0.649738	−	0.574515i	−0.0331137	−	0.0292800i
$386$		0			0
$387$		−	10.7005i		−	0.543938i
$388$		0			0
$389$	−6.77575			−0.343544			−0.171772	−	0.985137i	$-0.554949\pi$
$389$	−6.77575			−0.343544			−0.171772	+	0.985137i	$0.554949\pi$
$390$		0			0
$391$	−3.22425			−0.163058
$392$		0			0
$393$		−	4.00000i		−	0.201773i
$394$		0			0
$395$	20.6253	−	23.3258i	1.03777	−	1.17365i
$396$		0			0
$397$		−	0.635150i		−	0.0318773i	−0.999873	−	0.0159386i	$-0.994926\pi$
$397$		−	0.635150i		−	0.0318773i	0.999873	−	0.0159386i	$-0.00507364\pi$
$398$		0			0
$399$	2.96239			0.148305
$400$		0			0
$401$	10.3733			0.518017			0.259009	−	0.965875i	$-0.416604\pi$
$401$	10.3733			0.518017			0.259009	+	0.965875i	$0.416604\pi$
$402$		0			0
$403$		−	8.77575i		−	0.437151i
$404$		0			0
$405$	1.48119	−	1.67513i	0.0736011	−	0.0832379i
$406$		0			0
$407$		−	2.29806i		−	0.113911i
$408$		0			0
$409$	−15.7743			−0.779991			−0.389995	−	0.920817i	$-0.627523\pi$
$409$	−15.7743			−0.779991			−0.389995	+	0.920817i	$0.627523\pi$
$410$		0			0
$411$	10.4387			0.514901
$412$		0			0
$413$			3.22425i			0.158655i
$414$		0			0
$415$	5.40105	+	4.77575i	0.265127	+	0.234432i
$416$		0			0
$417$		−	5.66291i		−	0.277314i
$418$		0			0
$419$	−25.1490			−1.22861			−0.614305	−	0.789068i	$-0.710564\pi$
$419$	−25.1490			−1.22861			−0.614305	+	0.789068i	$0.710564\pi$
$420$		0			0
$421$	20.1768			0.983357			0.491678	−	0.870777i	$-0.336384\pi$
$421$	20.1768			0.983357			0.491678	+	0.870777i	$0.336384\pi$
$422$		0			0
$423$			3.22425i			0.156769i
$424$		0			0
$425$	−16.6253	+	2.05079i	−0.806446	+	0.0994777i
$426$		0			0
$427$		−	14.6253i		−	0.707768i
$428$		0			0
$429$	1.14903			0.0554757
$430$		0			0
$431$	−38.3127			−1.84546			−0.922728	−	0.385452i	$-0.874045\pi$
$431$	−38.3127			−1.84546			−0.922728	+	0.385452i	$0.874045\pi$
$432$		0			0
$433$		−	12.8872i		−	0.619318i	−0.950848	−	0.309659i	$-0.899785\pi$
$433$		−	12.8872i		−	0.619318i	0.950848	−	0.309659i	$-0.100215\pi$
$434$		0			0
$435$	−2.05079	−	1.81336i	−0.0983276	−	0.0869438i
$436$		0			0
$437$		−	2.85097i		−	0.136380i
$438$		0			0
$439$	29.6629			1.41573			0.707867	−	0.706346i	$-0.249658\pi$
$439$	29.6629			1.41573			0.707867	+	0.706346i	$0.249658\pi$
$440$		0			0
$441$	1.00000			0.0476190
$442$		0			0
$443$		−	24.8119i		−	1.17885i	−0.807823	−	0.589425i	$-0.799354\pi$
$443$		−	24.8119i		−	1.17885i	0.807823	−	0.589425i	$-0.200646\pi$
$444$		0			0
$445$	−5.53690	+	6.26187i	−0.262474	+	0.296841i
$446$		0			0
$447$		−	20.5501i		−	0.971985i
$448$		0			0
$449$	36.6516			1.72970			0.864849	−	0.502032i	$-0.167414\pi$
$449$	36.6516			1.72970			0.864849	+	0.502032i	$0.167414\pi$
$450$		0			0
$451$	0.402462			0.0189512
$452$		0			0
$453$		−	22.7005i		−	1.06656i
$454$		0			0
$455$	−4.38787	+	4.96239i	−0.205707	+	0.232640i
$456$		0			0
$457$		−	15.4763i		−	0.723949i	−0.932188	−	0.361975i	$-0.882103\pi$
$457$		−	15.4763i		−	0.723949i	0.932188	−	0.361975i	$-0.117897\pi$
$458$		0			0
$459$	−3.35026			−0.156377
$460$		0			0
$461$	41.2605			1.92169			0.960845	−	0.277085i	$-0.0893684\pi$
$461$	41.2605			1.92169			0.960845	+	0.277085i	$0.0893684\pi$
$462$		0			0
$463$			15.5975i			0.724879i	0.932007	+	0.362440i	$0.118056\pi$
$463$			15.5975i			0.724879i	−0.932007	+	0.362440i	$0.881944\pi$
$464$		0			0
$465$	4.96239	+	4.38787i	0.230125	+	0.203483i
$466$		0			0
$467$			18.5501i			0.858395i	0.903211	+	0.429198i	$0.141204\pi$
$467$			18.5501i			0.858395i	−0.903211	+	0.429198i	$0.858796\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−2.18664			−0.100755
$472$		0			0
$473$			4.15045i			0.190838i
$474$		0			0
$475$	−1.81336	−	14.7005i	−0.0832026	−	0.674506i
$476$		0			0
$477$			5.66291i			0.259287i
$478$		0			0
$479$	41.5026			1.89630			0.948151	−	0.317819i	$-0.102950\pi$
$479$	41.5026			1.89630			0.948151	+	0.317819i	$0.102950\pi$
$480$		0			0
$481$	−17.5515			−0.800279
$482$		0			0
$483$		−	0.962389i		−	0.0437902i
$484$		0			0
$485$	12.9624	+	11.4617i	0.588592	+	0.520448i
$486$		0			0
$487$			13.6728i			0.619572i	0.950806	+	0.309786i	$0.100257\pi$
$487$			13.6728i			0.619572i	−0.950806	+	0.309786i	$0.899743\pi$
$488$		0			0
$489$	−7.22425			−0.326692
$490$		0			0
$491$	23.3404			1.05334			0.526669	−	0.850070i	$-0.323441\pi$
$491$	23.3404			1.05334			0.526669	+	0.850070i	$0.323441\pi$
$492$		0			0
$493$			4.10157i			0.184725i
$494$		0			0
$495$	−0.574515	+	0.649738i	−0.0258226	+	0.0292036i
$496$		0			0
$497$		−	5.53690i		−	0.248364i
$498$		0			0
$499$	6.85097			0.306691			0.153346	−	0.988173i	$-0.450995\pi$
$499$	6.85097			0.306691			0.153346	+	0.988173i	$0.450995\pi$
$500$		0			0
$501$	−15.8496			−0.708106
$502$		0			0
$503$			3.32582i			0.148291i	0.997247	+	0.0741456i	$0.0236230\pi$
$503$			3.32582i			0.148291i	−0.997247	+	0.0741456i	$0.976377\pi$
$504$		0			0
$505$	−11.2388	+	12.7104i	−0.500122	+	0.565604i
$506$		0			0
$507$			4.22425i			0.187606i
$508$		0			0
$509$	6.18664			0.274218			0.137109	−	0.990556i	$-0.456219\pi$
$509$	6.18664			0.274218			0.137109	+	0.990556i	$0.456219\pi$
$510$		0			0
$511$	6.18664			0.273681
$512$		0			0
$513$		−	2.96239i		−	0.130793i
$514$		0			0
$515$	24.6253	+	21.7743i	1.08512	+	0.959492i
$516$		0			0
$517$		−	1.25060i		−	0.0550014i
$518$		0			0
$519$	−23.9756			−1.05241
$520$		0			0
$521$	34.4387			1.50879			0.754393	−	0.656424i	$-0.227932\pi$
$521$	34.4387			1.50879			0.754393	+	0.656424i	$0.227932\pi$
$522$		0			0
$523$			6.59895i			0.288552i	0.989537	+	0.144276i	$0.0460853\pi$
$523$			6.59895i			0.288552i	−0.989537	+	0.144276i	$0.953915\pi$
$524$		0			0
$525$	−0.612127	−	4.96239i	−0.0267154	−	0.216576i
$526$		0			0
$527$		−	9.92478i		−	0.432330i
$528$		0			0
$529$	22.0738			0.959731
$530$		0			0
$531$	3.22425			0.139921
$532$		0			0
$533$		−	3.07381i		−	0.133141i
$534$		0			0
$535$	−27.5369	−	24.3488i	−1.19052	−	1.05269i
$536$		0			0
$537$		−	12.2374i		−	0.528084i
$538$		0			0
$539$	−0.387873			−0.0167069
$540$		0			0
$541$	−9.07381			−0.390113			−0.195057	−	0.980792i	$-0.562489\pi$
$541$	−9.07381			−0.390113			−0.195057	+	0.980792i	$0.562489\pi$
$542$		0			0
$543$			8.70052i			0.373375i
$544$		0			0
$545$	4.11142	−	4.64974i	0.176114	−	0.199173i
$546$		0			0
$547$			19.8496i			0.848706i	0.905497	+	0.424353i	$0.139498\pi$
$547$			19.8496i			0.848706i	−0.905497	+	0.424353i	$0.860502\pi$
$548$		0			0
$549$	−14.6253			−0.624193
$550$		0			0
$551$	−3.62672			−0.154503
$552$		0			0
$553$		−	13.9248i		−	0.592142i
$554$		0			0
$555$	8.77575	−	9.92478i	0.372510	−	0.421283i
$556$		0			0
$557$		−	0.785595i		−	0.0332867i	−0.999861	−	0.0166434i	$-0.994702\pi$
$557$		−	0.785595i		−	0.0332867i	0.999861	−	0.0166434i	$-0.00529799\pi$
$558$		0			0
$559$	31.6991			1.34073
$560$		0			0
$561$	1.29948			0.0548639
$562$		0			0
$563$		−	32.2228i		−	1.35803i	−0.734124	−	0.679015i	$-0.762407\pi$
$563$		−	32.2228i		−	1.35803i	0.734124	−	0.679015i	$-0.237593\pi$
$564$		0			0
$565$	7.13918	+	6.31265i	0.300348	+	0.265575i
$566$		0			0
$567$		−	1.00000i		−	0.0419961i
$568$		0			0
$569$	−16.0752			−0.673908			−0.336954	−	0.941521i	$-0.609397\pi$
$569$	−16.0752			−0.673908			−0.336954	+	0.941521i	$0.609397\pi$
$570$		0			0
$571$	−39.8496			−1.66765			−0.833826	−	0.552027i	$-0.813854\pi$
$571$	−39.8496			−1.66765			−0.833826	+	0.552027i	$0.813854\pi$
$572$		0			0
$573$		−	13.5369i		−	0.565512i
$574$		0			0
$575$	−4.77575	+	0.589104i	−0.199162	+	0.0245673i
$576$		0			0
$577$			30.1378i			1.25465i	0.778757	+	0.627326i	$0.215851\pi$
$577$			30.1378i			1.25465i	−0.778757	+	0.627326i	$0.784149\pi$
$578$		0			0
$579$	1.92478			0.0799910
$580$		0			0
$581$	3.22425			0.133765
$582$		0			0
$583$		−	2.19649i		−	0.0909694i
$584$		0			0
$585$	4.96239	+	4.38787i	0.205170	+	0.181416i
$586$		0			0
$587$			33.7743i			1.39402i	0.717063	+	0.697008i	$0.245486\pi$
$587$			33.7743i			1.39402i	−0.717063	+	0.697008i	$0.754514\pi$
$588$		0			0
$589$	8.77575			0.361598
$590$		0			0
$591$	20.1114			0.827273
$592$		0			0
$593$			7.19982i			0.295661i	0.989013	+	0.147831i	$0.0472290\pi$
$593$			7.19982i			0.295661i	−0.989013	+	0.147831i	$0.952771\pi$
$594$		0			0
$595$	−4.96239	+	5.61213i	−0.203438	+	0.230075i
$596$		0			0
$597$			8.26187i			0.338136i
$598$		0			0
$599$	29.5369			1.20685			0.603423	−	0.797422i	$-0.293803\pi$
$599$	29.5369			1.20685			0.603423	+	0.797422i	$0.293803\pi$
$600$		0			0
$601$	8.59895			0.350759			0.175379	−	0.984501i	$-0.443885\pi$
$601$	8.59895			0.350759			0.175379	+	0.984501i	$0.443885\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−16.0703	+	18.1744i	−0.653351	+	0.738895i
$606$		0			0
$607$		−	37.4010i		−	1.51806i	−0.651055	−	0.759031i	$-0.725673\pi$
$607$		−	37.4010i		−	1.51806i	0.651055	−	0.759031i	$-0.274327\pi$
$608$		0			0
$609$	−1.22425			−0.0496093
$610$		0			0
$611$	−9.55149			−0.386412
$612$		0			0
$613$		−	14.5990i		−	0.589646i	−0.955552	−	0.294823i	$-0.904739\pi$
$613$		−	14.5990i		−	0.589646i	0.955552	−	0.294823i	$-0.0952607\pi$
$614$		0			0
$615$	1.73813	+	1.53690i	0.0700884	+	0.0619740i
$616$		0			0
$617$		−	22.8119i		−	0.918374i	−0.888340	−	0.459187i	$-0.848141\pi$
$617$		−	22.8119i		−	0.918374i	0.888340	−	0.459187i	$-0.151859\pi$
$618$		0			0
$619$	−0.261865			−0.0105252			−0.00526262	−	0.999986i	$-0.501675\pi$
$619$	−0.261865			−0.0105252			−0.00526262	+	0.999986i	$0.501675\pi$
$620$		0			0
$621$	−0.962389			−0.0386193
$622$		0			0
$623$			3.73813i			0.149765i
$624$		0			0
$625$	−24.2506	+	6.07522i	−0.970024	+	0.243009i
$626$		0			0
$627$			1.14903i			0.0458879i
$628$		0			0
$629$	−19.8496			−0.791454
$630$		0			0
$631$	−5.92478			−0.235862			−0.117931	−	0.993022i	$-0.537626\pi$
$631$	−5.92478			−0.235862			−0.117931	+	0.993022i	$0.537626\pi$
$632$		0			0
$633$		−	18.1768i		−	0.722463i
$634$		0			0
$635$	24.3733	+	21.5515i	0.967224	+	0.855245i
$636$		0			0
$637$			2.96239i			0.117374i
$638$		0			0
$639$	−5.53690			−0.219037
$640$		0			0
$641$	6.87732			0.271638			0.135819	−	0.990734i	$-0.456633\pi$
$641$	6.87732			0.271638			0.135819	+	0.990734i	$0.456633\pi$
$642$		0			0
$643$		−	10.7005i		−	0.421987i	−0.977487	−	0.210994i	$-0.932330\pi$
$643$		−	10.7005i		−	0.421987i	0.977487	−	0.210994i	$-0.0676700\pi$
$644$		0			0
$645$	−15.8496	+	17.9248i	−0.624076	+	0.705787i
$646$		0			0
$647$			45.2506i			1.77898i	0.456950	+	0.889492i	$0.348942\pi$
$647$			45.2506i			1.77898i	−0.456950	+	0.889492i	$0.651058\pi$
$648$		0			0
$649$	−1.25060			−0.0490904
$650$		0			0
$651$	2.96239			0.116105
$652$		0			0
$653$			36.0625i			1.41124i	0.708592	+	0.705618i	$0.249331\pi$
$653$			36.0625i			1.41124i	−0.708592	+	0.705618i	$0.750669\pi$
$654$		0			0
$655$	−5.92478	+	6.70052i	−0.231500	+	0.261811i
$656$		0			0
$657$		−	6.18664i		−	0.241364i
$658$		0			0
$659$	−43.4617			−1.69303			−0.846513	−	0.532367i	$-0.821302\pi$
$659$	−43.4617			−1.69303			−0.846513	+	0.532367i	$0.821302\pi$
$660$		0			0
$661$	−22.4749			−0.874171			−0.437085	−	0.899420i	$-0.643989\pi$
$661$	−22.4749			−0.874171			−0.437085	+	0.899420i	$0.643989\pi$
$662$		0			0
$663$		−	9.92478i		−	0.385446i
$664$		0			0
$665$	−4.96239	−	4.38787i	−0.192433	−	0.170154i
$666$		0			0
$667$			1.17821i			0.0456204i
$668$		0			0
$669$	−21.4010			−0.827412
$670$		0			0
$671$	5.67276			0.218995
$672$		0			0
$673$			47.5487i			1.83287i	0.400188	+	0.916433i	$0.368945\pi$
$673$			47.5487i			1.83287i	−0.400188	+	0.916433i	$0.631055\pi$
$674$		0			0
$675$	−4.96239	+	0.612127i	−0.191002	+	0.0235608i
$676$		0			0
$677$		−	5.67750i		−	0.218204i	−0.994031	−	0.109102i	$-0.965202\pi$
$677$		−	5.67750i		−	0.218204i	0.994031	−	0.109102i	$-0.0347975\pi$
$678$		0			0
$679$	7.73813			0.296962
$680$		0			0
$681$	18.5501			0.710841
$682$		0			0
$683$			12.2882i			0.470195i	0.971972	+	0.235098i	$0.0755410\pi$
$683$			12.2882i			0.470195i	−0.971972	+	0.235098i	$0.924459\pi$
$684$		0			0
$685$	−17.4861	−	15.4617i	−0.668110	−	0.590760i
$686$		0			0
$687$			28.5501i			1.08925i
$688$		0			0
$689$	−16.7757			−0.639105
$690$		0			0
$691$	−1.81336			−0.0689834			−0.0344917	−	0.999405i	$-0.510981\pi$
$691$	−1.81336			−0.0689834			−0.0344917	+	0.999405i	$0.510981\pi$
$692$		0			0
$693$			0.387873i			0.0147341i
$694$		0			0
$695$	−8.38787	+	9.48612i	−0.318170	+	0.359829i
$696$		0			0
$697$		−	3.47627i		−	0.131673i
$698$		0			0
$699$	−18.9624			−0.717223
$700$		0			0
$701$	29.5778			1.11714			0.558570	−	0.829458i	$-0.311350\pi$
$701$	29.5778			1.11714			0.558570	+	0.829458i	$0.311350\pi$
$702$		0			0
$703$		−	17.5515i		−	0.661967i
$704$		0			0
$705$	4.77575	−	5.40105i	0.179865	−	0.203415i
$706$		0			0
$707$			7.58769i			0.285364i
$708$		0			0
$709$	−20.8021			−0.781239			−0.390620	−	0.920552i	$-0.627739\pi$
$709$	−20.8021			−0.781239			−0.390620	+	0.920552i	$0.627739\pi$
$710$		0			0
$711$	−13.9248			−0.522220
$712$		0			0
$713$		−	2.85097i		−	0.106770i
$714$		0			0
$715$	−1.92478	−	1.70194i	−0.0719826	−	0.0636489i
$716$		0			0
$717$			18.1622i			0.678280i
$718$		0			0
$719$	−19.3258			−0.720732			−0.360366	−	0.932811i	$-0.617348\pi$
$719$	−19.3258			−0.720732			−0.360366	+	0.932811i	$0.617348\pi$
$720$		0			0
$721$	14.7005			0.547476
$722$		0			0
$723$			14.3733i			0.534548i
$724$		0			0
$725$	0.749399	+	6.07522i	0.0278320	+	0.225628i
$726$		0			0
$727$			21.6531i			0.803068i	0.915844	+	0.401534i	$0.131523\pi$
$727$			21.6531i			0.803068i	−0.915844	+	0.401534i	$0.868477\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	35.8496			1.32594
$732$		0			0
$733$		−	48.0625i		−	1.77523i	−0.460586	−	0.887615i	$-0.652361\pi$
$733$		−	48.0625i		−	1.77523i	0.460586	−	0.887615i	$-0.347639\pi$
$734$		0			0
$735$	−1.67513	−	1.48119i	−0.0617881	−	0.0546347i
$736$		0			0
$737$		0			0
$738$		0			0
$739$	30.2981			1.11453			0.557266	−	0.830334i	$-0.311850\pi$
$739$	30.2981			1.11453			0.557266	+	0.830334i	$0.311850\pi$
$740$		0			0
$741$	8.77575			0.322385
$742$		0			0
$743$		−	40.9624i		−	1.50276i	−0.659867	−	0.751382i	$-0.729388\pi$
$743$		−	40.9624i		−	1.50276i	0.659867	−	0.751382i	$-0.270612\pi$
$744$		0			0
$745$	−30.4387	+	34.4241i	−1.11519	+	1.26120i
$746$		0			0
$747$		−	3.22425i		−	0.117969i
$748$		0			0
$749$	−16.4387			−0.600656
$750$		0			0
$751$	9.52232			0.347474			0.173737	−	0.984792i	$-0.444416\pi$
$751$	9.52232			0.347474			0.173737	+	0.984792i	$0.444416\pi$
$752$		0			0
$753$		−	8.62530i		−	0.314323i
$754$		0			0
$755$	−33.6239	+	38.0263i	−1.22370	+	1.38392i
$756$		0			0
$757$		−	31.3258i		−	1.13856i	−0.822145	−	0.569278i	$-0.807223\pi$
$757$		−	31.3258i		−	1.13856i	0.822145	−	0.569278i	$-0.192777\pi$
$758$		0			0
$759$	0.373285			0.0135494
$760$		0			0
$761$	−54.2393			−1.96617			−0.983087	−	0.183138i	$-0.941375\pi$
$761$	−54.2393			−1.96617			−0.983087	+	0.183138i	$0.941375\pi$
$762$		0			0
$763$		−	2.77575i		−	0.100489i
$764$		0			0
$765$	5.61213	+	4.96239i	0.202907	+	0.179416i
$766$		0			0
$767$			9.55149i			0.344884i
$768$		0			0
$769$	−35.1002			−1.26574			−0.632872	−	0.774256i	$-0.718124\pi$
$769$	−35.1002			−1.26574			−0.632872	+	0.774256i	$0.718124\pi$
$770$		0			0
$771$	−27.8251			−1.00210
$772$		0			0
$773$			19.8740i			0.714818i	0.933948	+	0.357409i	$0.116340\pi$
$773$			19.8740i			0.714818i	−0.933948	+	0.357409i	$0.883660\pi$
$774$		0			0
$775$	−1.81336	−	14.7005i	−0.0651377	−	0.528058i
$776$		0			0
$777$		−	5.92478i		−	0.212550i
$778$		0			0
$779$	3.07381			0.110131
$780$		0			0
$781$	2.14762			0.0768478
$782$		0			0
$783$			1.22425i			0.0437513i
$784$		0			0
$785$	3.66291	+	3.23884i	0.130735	+	0.115599i
$786$		0			0
$787$		−	13.2506i		−	0.472333i	−0.971713	−	0.236166i	$-0.924109\pi$
$787$		−	13.2506i		−	0.472333i	0.971713	−	0.236166i	$-0.0758911\pi$
$788$		0			0
$789$	0.589104			0.0209727
$790$		0			0
$791$	4.26187			0.151534
$792$		0			0
$793$		−	43.3258i		−	1.53855i
$794$		0			0
$795$	8.38787	−	9.48612i	0.297487	−	0.336438i
$796$		0			0
$797$			15.7235i			0.556957i	0.960443	+	0.278478i	$0.0898300\pi$
$797$			15.7235i			0.556957i	−0.960443	+	0.278478i	$0.910170\pi$
$798$		0			0
$799$	−10.8021			−0.382151
$800$		0			0
$801$	3.73813			0.132081

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:	\( N \)	\(=\)	\( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	840.t (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +(1.48119 - 1.67513i) q^{5} -1.00000i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +(1.48119 - 1.67513i) q^{5} -1.00000i q^{7} -1.00000 q^{9} +0.387873 q^{11} -2.96239i q^{13} +(1.67513 + 1.48119i) q^{15} -3.35026i q^{17} +2.96239 q^{19} +1.00000 q^{21} -0.962389i q^{23} +(-0.612127 - 4.96239i) q^{25} -1.00000i q^{27} -1.22425 q^{29} +2.96239 q^{31} +0.387873i q^{33} +(-1.67513 - 1.48119i) q^{35} -5.92478i q^{37} +2.96239 q^{39} +1.03761 q^{41} +10.7005i q^{43} +(-1.48119 + 1.67513i) q^{45} -3.22425i q^{47} -1.00000 q^{49} +3.35026 q^{51} -5.66291i q^{53} +(0.574515 - 0.649738i) q^{55} +2.96239i q^{57} -3.22425 q^{59} +14.6253 q^{61} +1.00000i q^{63} +(-4.96239 - 4.38787i) q^{65} +0.962389 q^{69} +5.53690 q^{71} +6.18664i q^{73} +(4.96239 - 0.612127i) q^{75} -0.387873i q^{77} +13.9248 q^{79} +1.00000 q^{81} +3.22425i q^{83} +(-5.61213 - 4.96239i) q^{85} -1.22425i q^{87} -3.73813 q^{89} -2.96239 q^{91} +2.96239i q^{93} +(4.38787 - 4.96239i) q^{95} +7.73813i q^{97} -0.387873 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{5} - 6 q^{9}+O(q^{10}) \) 6 * q - 2 * q^5 - 6 * q^9	\( 6 q - 2 q^{5} - 6 q^{9} + 4 q^{11} - 4 q^{19} + 6 q^{21} - 2 q^{25} - 4 q^{29} - 4 q^{31} - 4 q^{39} + 28 q^{41} + 2 q^{45} - 6 q^{49} - 20 q^{55} - 16 q^{59} + 4 q^{61} - 8 q^{65} - 16 q^{69} - 12 q^{71} + 8 q^{75} + 40 q^{79} + 6 q^{81} - 32 q^{85} - 4 q^{89} + 4 q^{91} + 28 q^{95} - 4 q^{99}+O(q^{100}) \) 6 * q - 2 * q^5 - 6 * q^9 + 4 * q^11 - 4 * q^19 + 6 * q^21 - 2 * q^25 - 4 * q^29 - 4 * q^31 - 4 * q^39 + 28 * q^41 + 2 * q^45 - 6 * q^49 - 20 * q^55 - 16 * q^59 + 4 * q^61 - 8 * q^65 - 16 * q^69 - 12 * q^71 + 8 * q^75 + 40 * q^79 + 6 * q^81 - 32 * q^85 - 4 * q^89 + 4 * q^91 + 28 * q^95 - 4 * q^99

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