LMFDB - Embedded newform 819.2.c.b.64.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [819,2,Mod(64,819)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(819, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("819.64"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 819 = 3^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	819.c (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	no
Analytic conductor:	$6.53974792554$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.350464.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 91)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			64.4
Root			$1.45161 - 1.45161i$ of defining polynomial
Character	$\chi$	$=$	819.64
Dual form			819.2.c.b.64.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.688892i q^{2} +1.52543 q^{4} +3.21432i q^{5} -1.00000i q^{7} +2.42864i q^{8} -2.21432 q^{10} -2.68889i q^{11} +(3.59210 - 0.311108i) q^{13} +0.688892 q^{14} +1.37778 q^{16} +3.59210 q^{17} +8.54617i q^{19} +4.90321i q^{20} +1.85236 q^{22} -3.28100 q^{23} -5.33185 q^{25} +(0.214320 + 2.47457i) q^{26} -1.52543i q^{28} -2.05086 q^{29} -5.83654i q^{31} +5.80642i q^{32} +2.47457i q^{34} +3.21432 q^{35} +3.93332i q^{37} -5.88739 q^{38} -7.80642 q^{40} +0.755569i q^{41} -8.80642 q^{43} -4.10171i q^{44} -2.26025i q^{46} -1.88247i q^{47} -1.00000 q^{49} -3.67307i q^{50} +(5.47949 - 0.474572i) q^{52} -2.52543 q^{53} +8.64296 q^{55} +2.42864 q^{56} -1.41282i q^{58} -7.33185i q^{59} +9.05086 q^{61} +4.02074 q^{62} -1.24443 q^{64} +(1.00000 + 11.5462i) q^{65} -0.428639i q^{67} +5.47949 q^{68} +2.21432i q^{70} +8.98418i q^{71} +5.79060i q^{73} -2.70964 q^{74} +13.0366i q^{76} -2.68889 q^{77} -4.47949 q^{79} +4.42864i q^{80} -0.520505 q^{82} -10.8272i q^{83} +11.5462i q^{85} -6.06668i q^{86} +6.53035 q^{88} -5.36196i q^{89} +(-0.311108 - 3.59210i) q^{91} -5.00492 q^{92} +1.29682 q^{94} -27.4701 q^{95} -9.62867i q^{97} -0.688892i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{4} + 8 q^{13} + 4 q^{14} + 8 q^{16} + 8 q^{17} + 24 q^{22} - 6 q^{23} + 8 q^{25} - 12 q^{26} + 14 q^{29} + 6 q^{35} + 4 q^{38} - 20 q^{40} - 26 q^{43} - 6 q^{49} - 20 q^{52} - 2 q^{53} + 12 q^{55}+ \cdots - 58 q^{95}+O(q^{100}) $ 6 * q - 4 * q^4 + 8 * q^13 + 4 * q^14 + 8 * q^16 + 8 * q^17 + 24 * q^22 - 6 * q^23 + 8 * q^25 - 12 * q^26 + 14 * q^29 + 6 * q^35 + 4 * q^38 - 20 * q^40 - 26 * q^43 - 6 * q^49 - 20 * q^52 - 2 * q^53 + 12 * q^55 - 12 * q^56 + 28 * q^61 - 16 * q^62 - 8 * q^64 + 6 * q^65 - 20 * q^68 + 24 * q^74 - 16 * q^77 + 26 * q^79 - 56 * q^82 - 40 * q^88 - 2 * q^91 + 36 * q^92 + 20 * q^94 - 58 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/819\mathbb{Z}\right)^\times$.

$n$	$92$	$379$	$703$
$\chi(n)$	$1$	$-1$	$1$

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 $

$2$			0.688892i			0.487120i	0.969886	+	0.243560i	$0.0783153\pi$
$2$			0.688892i			0.487120i	−0.969886	+	0.243560i	$0.921685\pi$
$3$		0			0
$3$		0			0
$4$	1.52543			0.762714
$5$			3.21432i			1.43749i	0.695275	+	0.718744i	$0.255283\pi$
$5$			3.21432i			1.43749i	−0.695275	+	0.718744i	$0.744717\pi$
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$			2.42864i			0.858654i
$9$		0			0
$10$	−2.21432			−0.700229
$11$		−	2.68889i		−	0.810731i	−0.914155	−	0.405366i	$-0.867144\pi$
$11$		−	2.68889i		−	0.810731i	0.914155	−	0.405366i	$-0.132856\pi$
$12$		0			0
$13$	3.59210	−	0.311108i	0.996270	−	0.0862858i
$13$	3.59210	−	0.311108i	0.996270	−	0.0862858i
$14$	0.688892			0.184114
$15$		0			0
$16$	1.37778			0.344446
$17$	3.59210			0.871213			0.435607	−	0.900137i	$-0.356534\pi$
$17$	3.59210			0.871213			0.435607	+	0.900137i	$0.356534\pi$
$18$		0			0
$19$			8.54617i			1.96063i	0.197449	+	0.980313i	$0.436734\pi$
$19$			8.54617i			1.96063i	−0.197449	+	0.980313i	$0.563266\pi$
$20$			4.90321i			1.09639i
$21$		0			0
$22$	1.85236			0.394924
$23$	−3.28100			−0.684135			−0.342068	−	0.939675i	$-0.611127\pi$
$23$	−3.28100			−0.684135			−0.342068	+	0.939675i	$0.611127\pi$
$24$		0			0
$25$	−5.33185			−1.06637
$26$	0.214320	+	2.47457i	0.0420316	+	0.485304i
$27$		0			0
$28$		−	1.52543i		−	0.288279i
$29$	−2.05086			−0.380834			−0.190417	−	0.981703i	$-0.560984\pi$
$29$	−2.05086			−0.380834			−0.190417	+	0.981703i	$0.560984\pi$
$30$		0			0
$31$		−	5.83654i		−	1.04827i	−0.851634	−	0.524136i	$-0.824388\pi$
$31$		−	5.83654i		−	1.04827i	0.851634	−	0.524136i	$-0.175612\pi$
$32$			5.80642i			1.02644i
$33$		0			0
$34$			2.47457i			0.424386i
$35$	3.21432			0.543319
$36$		0			0
$37$			3.93332i			0.646634i	0.946291	+	0.323317i	$0.104798\pi$
$37$			3.93332i			0.646634i	−0.946291	+	0.323317i	$0.895202\pi$
$38$	−5.88739			−0.955061
$39$		0			0
$40$	−7.80642			−1.23430
$41$			0.755569i			0.118000i	0.998258	+	0.0590000i	$0.0187912\pi$
$41$			0.755569i			0.118000i	−0.998258	+	0.0590000i	$0.981209\pi$
$42$		0			0
$43$	−8.80642			−1.34297			−0.671484	−	0.741019i	$-0.734343\pi$
$43$	−8.80642			−1.34297			−0.671484	+	0.741019i	$0.734343\pi$
$44$		−	4.10171i		−	0.618356i
$45$		0			0
$46$		−	2.26025i		−	0.333256i
$47$		−	1.88247i		−	0.274586i	−0.990530	−	0.137293i	$-0.956160\pi$
$47$		−	1.88247i		−	0.274586i	0.990530	−	0.137293i	$-0.0438402\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		−	3.67307i		−	0.519451i
$51$		0			0
$52$	5.47949	−	0.474572i	0.759869	−	0.0658114i
$53$	−2.52543			−0.346894			−0.173447	−	0.984843i	$-0.555491\pi$
$53$	−2.52543			−0.346894			−0.173447	+	0.984843i	$0.555491\pi$
$54$		0			0
$55$	8.64296			1.16542
$56$	2.42864			0.324541
$57$		0			0
$58$		−	1.41282i		−	0.185512i
$59$		−	7.33185i		−	0.954526i	−0.878761	−	0.477263i	$-0.841629\pi$
$59$		−	7.33185i		−	0.954526i	0.878761	−	0.477263i	$-0.158371\pi$
$60$		0			0
$61$	9.05086			1.15884			0.579422	−	0.815028i	$-0.303278\pi$
$61$	9.05086			1.15884			0.579422	+	0.815028i	$0.303278\pi$
$62$	4.02074			0.510635
$63$		0			0
$64$	−1.24443			−0.155554
$65$	1.00000	+	11.5462i	0.124035	+	1.43213i
$66$		0			0
$67$		−	0.428639i		−	0.0523666i	−0.999657	−	0.0261833i	$-0.991665\pi$
$67$		−	0.428639i		−	0.0523666i	0.999657	−	0.0261833i	$-0.00833536\pi$
$68$	5.47949			0.664486
$69$		0			0
$70$			2.21432i			0.264662i
$71$			8.98418i			1.06623i	0.846044	+	0.533113i	$0.178978\pi$
$71$			8.98418i			1.06623i	−0.846044	+	0.533113i	$0.821022\pi$
$72$		0			0
$73$			5.79060i			0.677739i	0.940833	+	0.338869i	$0.110044\pi$
$73$			5.79060i			0.677739i	−0.940833	+	0.338869i	$0.889956\pi$
$74$	−2.70964			−0.314989
$75$		0			0
$76$			13.0366i			1.49540i
$77$	−2.68889			−0.306428
$78$		0			0
$79$	−4.47949			−0.503983			−0.251991	−	0.967730i	$-0.581085\pi$
$79$	−4.47949			−0.503983			−0.251991	+	0.967730i	$0.581085\pi$
$80$			4.42864i			0.495137i
$81$		0			0
$82$	−0.520505			−0.0574802
$83$		−	10.8272i		−	1.18844i	−0.804304	−	0.594218i	$-0.797462\pi$
$83$		−	10.8272i		−	1.18844i	0.804304	−	0.594218i	$-0.202538\pi$
$84$		0			0
$85$			11.5462i			1.25236i
$86$		−	6.06668i		−	0.654187i
$87$		0			0
$88$	6.53035			0.696138
$89$		−	5.36196i		−	0.568367i	−0.958770	−	0.284183i	$-0.908278\pi$
$89$		−	5.36196i		−	0.568367i	0.958770	−	0.284183i	$-0.0917225\pi$
$90$		0			0
$91$	−0.311108	−	3.59210i	−0.0326130	−	0.376555i
$92$	−5.00492			−0.521799
$93$		0			0
$94$	1.29682			0.133757
$95$	−27.4701			−2.81838
$96$		0			0
$97$		−	9.62867i		−	0.977643i	−0.872384	−	0.488822i	$-0.837427\pi$
$97$		−	9.62867i		−	0.977643i	0.872384	−	0.488822i	$-0.162573\pi$
$98$		−	0.688892i		−	0.0695886i
$99$		0			0
$100$	−8.13335			−0.813335
$101$	13.6938			1.36259			0.681293	−	0.732011i	$-0.261418\pi$
$101$	13.6938			1.36259			0.681293	+	0.732011i	$0.261418\pi$
$102$		0			0
$103$	−12.2953			−1.21149			−0.605745	−	0.795659i	$-0.707125\pi$
$103$	−12.2953			−1.21149			−0.605745	+	0.795659i	$0.707125\pi$
$104$	0.755569	+	8.72393i	0.0740896	+	0.855451i
$105$		0			0
$106$		−	1.73975i		−	0.168979i
$107$	18.1891			1.75841			0.879205	−	0.476444i	$-0.158075\pi$
$107$	18.1891			1.75841			0.879205	+	0.476444i	$0.158075\pi$
$108$		0			0
$109$		−	8.36196i		−	0.800931i	−0.916312	−	0.400465i	$-0.868848\pi$
$109$		−	8.36196i		−	0.800931i	0.916312	−	0.400465i	$-0.131152\pi$
$110$			5.95407i			0.567698i
$111$		0			0
$112$		−	1.37778i		−	0.130188i
$113$	8.46520			0.796339			0.398170	−	0.917312i	$-0.369646\pi$
$113$	8.46520			0.796339			0.398170	+	0.917312i	$0.369646\pi$
$114$		0			0
$115$		−	10.5462i		−	0.983436i
$116$	−3.12843			−0.290468
$117$		0			0
$118$	5.05086			0.464969
$119$		−	3.59210i		−	0.329288i
$120$		0			0
$121$	3.76986			0.342714
$122$			6.23506i			0.564496i
$123$		0			0
$124$		−	8.90321i		−	0.799532i
$125$		−	1.06668i		−	0.0954065i
$126$		0			0
$127$	−4.08742			−0.362700			−0.181350	−	0.983419i	$-0.558047\pi$
$127$	−4.08742			−0.362700			−0.181350	+	0.983419i	$0.558047\pi$
$128$			10.7556i			0.950667i
$129$		0			0
$130$	−7.95407	+	0.688892i	−0.697618	+	0.0604198i
$131$	−5.93978			−0.518961			−0.259480	−	0.965748i	$-0.583551\pi$
$131$	−5.93978			−0.518961			−0.259480	+	0.965748i	$0.583551\pi$
$132$		0			0
$133$	8.54617			0.741047
$134$	0.295286			0.0255089
$135$		0			0
$136$			8.72393i			0.748070i
$137$		−	16.3620i		−	1.39790i	−0.715172	−	0.698948i	$-0.753652\pi$
$137$		−	16.3620i		−	1.39790i	0.715172	−	0.698948i	$-0.246348\pi$
$138$		0			0
$139$	3.03011			0.257011			0.128505	−	0.991709i	$-0.458982\pi$
$139$	3.03011			0.257011			0.128505	+	0.991709i	$0.458982\pi$
$140$	4.90321			0.414397
$141$		0			0
$142$	−6.18913			−0.519380
$143$	−0.836535	−	9.65878i	−0.0699546	−	0.807708i
$144$		0			0
$145$		−	6.59210i		−	0.547444i
$146$	−3.98910			−0.330140
$147$		0			0
$148$			6.00000i			0.493197i
$149$		−	14.5368i		−	1.19090i	−0.803392	−	0.595451i	$-0.796974\pi$
$149$		−	14.5368i		−	1.19090i	0.803392	−	0.595451i	$-0.203026\pi$
$150$		0			0
$151$		−	19.9748i		−	1.62553i	−0.582594	−	0.812764i	$-0.697962\pi$
$151$		−	19.9748i		−	1.62553i	0.582594	−	0.812764i	$-0.302038\pi$
$152$	−20.7556			−1.68350
$153$		0			0
$154$		−	1.85236i		−	0.149267i
$155$	18.7605			1.50688
$156$		0			0
$157$	7.39853			0.590467			0.295233	−	0.955425i	$-0.404603\pi$
$157$	7.39853			0.590467			0.295233	+	0.955425i	$0.404603\pi$
$158$		−	3.08589i		−	0.245500i
$159$		0			0
$160$	−18.6637			−1.47550
$161$			3.28100i			0.258579i
$162$		0			0
$163$			2.32693i			0.182259i	0.995839	+	0.0911296i	$0.0290477\pi$
$163$			2.32693i			0.182259i	−0.995839	+	0.0911296i	$0.970952\pi$
$164$			1.15257i			0.0900002i
$165$		0			0
$166$	7.45875			0.578911
$167$			3.42219i			0.264817i	0.991195	+	0.132408i	$0.0422710\pi$
$167$			3.42219i			0.264817i	−0.991195	+	0.132408i	$0.957729\pi$
$168$		0			0
$169$	12.8064	−	2.23506i	0.985110	−	0.171928i
$170$	−7.95407			−0.610049
$171$		0			0
$172$	−13.4336			−1.02430
$173$	8.27454			0.629102			0.314551	−	0.949241i	$-0.398146\pi$
$173$	8.27454			0.629102			0.314551	+	0.949241i	$0.398146\pi$
$174$		0			0
$175$			5.33185i			0.403050i
$176$		−	3.70471i		−	0.279253i
$177$		0			0
$178$	3.69381			0.276863
$179$	−16.2257			−1.21277			−0.606383	−	0.795173i	$-0.707380\pi$
$179$	−16.2257			−1.21277			−0.606383	+	0.795173i	$0.707380\pi$
$180$		0			0
$181$	9.20495			0.684199			0.342099	−	0.939664i	$-0.388862\pi$
$181$	9.20495			0.684199			0.342099	+	0.939664i	$0.388862\pi$
$182$	2.47457	−	0.214320i	0.183428	−	0.0158864i
$183$		0			0
$184$		−	7.96836i		−	0.587435i
$185$	−12.6430			−0.929529
$186$		0			0
$187$		−	9.65878i		−	0.706320i
$188$		−	2.87157i		−	0.209431i
$189$		0			0
$190$		−	18.9240i		−	1.37289i
$191$	6.66815			0.482490			0.241245	−	0.970464i	$-0.422444\pi$
$191$	6.66815			0.482490			0.241245	+	0.970464i	$0.422444\pi$
$192$		0			0
$193$			20.6035i			1.48307i	0.670914	+	0.741535i	$0.265902\pi$
$193$			20.6035i			1.48307i	−0.670914	+	0.741535i	$0.734098\pi$
$194$	6.63311			0.476230
$195$		0			0
$196$	−1.52543			−0.108959
$197$		−	8.36842i		−	0.596225i	−0.954531	−	0.298112i	$-0.903643\pi$
$197$		−	8.36842i		−	0.596225i	0.954531	−	0.298112i	$-0.0963571\pi$
$198$		0			0
$199$	0.601472			0.0426372			0.0213186	−	0.999773i	$-0.493214\pi$
$199$	0.601472			0.0426372			0.0213186	+	0.999773i	$0.493214\pi$
$200$		−	12.9491i		−	0.915643i
$201$		0			0
$202$			9.43356i			0.663743i
$203$			2.05086i			0.143942i
$204$		0			0
$205$	−2.42864			−0.169624
$206$		−	8.47013i		−	0.590142i
$207$		0			0
$208$	4.94914	−	0.428639i	0.343161	−	0.0297208i
$209$	22.9797			1.58954
$210$		0			0
$211$	−1.90321			−0.131023			−0.0655113	−	0.997852i	$-0.520868\pi$
$211$	−1.90321			−0.131023			−0.0655113	+	0.997852i	$0.520868\pi$
$212$	−3.85236			−0.264581
$213$		0			0
$214$			12.5303i			0.856557i
$215$		−	28.3067i		−	1.93050i
$216$		0			0
$217$	−5.83654			−0.396210
$218$	5.76049			0.390150
$219$		0			0
$220$	13.1842			0.888879
$221$	12.9032	−	1.11753i	0.867964	−	0.0751733i
$222$		0			0
$223$		−	10.6336i		−	0.712078i	−0.934471	−	0.356039i	$-0.884127\pi$
$223$		−	10.6336i		−	0.712078i	0.934471	−	0.356039i	$-0.115873\pi$
$224$	5.80642			0.387958
$225$		0			0
$226$			5.83161i			0.387913i
$227$			15.0509i			0.998960i	0.866325	+	0.499480i	$0.166476\pi$
$227$			15.0509i			0.998960i	−0.866325	+	0.499480i	$0.833524\pi$
$228$		0			0
$229$		−	6.53480i		−	0.431831i	−0.976412	−	0.215916i	$-0.930726\pi$
$229$		−	6.53480i		−	0.431831i	0.976412	−	0.215916i	$-0.0692736\pi$
$230$	7.26517			0.479051
$231$		0			0
$232$		−	4.98079i		−	0.327005i
$233$	−27.9590			−1.83165			−0.915827	−	0.401573i	$-0.868464\pi$
$233$	−27.9590			−1.83165			−0.915827	+	0.401573i	$0.868464\pi$
$234$		0			0
$235$	6.05086			0.394714
$236$		−	11.1842i		−	0.728030i
$237$		0			0
$238$	2.47457			0.160403
$239$		−	19.5812i		−	1.26660i	−0.773905	−	0.633301i	$-0.781699\pi$
$239$		−	19.5812i		−	1.26660i	0.773905	−	0.633301i	$-0.218301\pi$
$240$		0			0
$241$		−	13.3575i		−	0.860433i	−0.902726	−	0.430217i	$-0.858437\pi$
$241$		−	13.3575i		−	0.860433i	0.902726	−	0.430217i	$-0.141563\pi$
$242$			2.59703i			0.166943i
$243$		0			0
$244$	13.8064			0.883866
$245$		−	3.21432i		−	0.205355i
$246$		0			0
$247$	2.65878	+	30.6987i	0.169174	+	1.95331i
$248$	14.1748			0.900103
$249$		0			0
$250$	0.734825			0.0464744
$251$	3.29682			0.208093			0.104047	−	0.994572i	$-0.466821\pi$
$251$	3.29682			0.208093			0.104047	+	0.994572i	$0.466821\pi$
$252$		0			0
$253$			8.82225i			0.554650i
$254$		−	2.81579i		−	0.176678i
$255$		0			0
$256$	−9.89829			−0.618643
$257$	−23.6938			−1.47798			−0.738990	−	0.673717i	$-0.764697\pi$
$257$	−23.6938			−1.47798			−0.738990	+	0.673717i	$0.764697\pi$
$258$		0			0
$259$	3.93332			0.244405
$260$	1.52543	+	17.6128i	0.0946030	+	1.09230i
$261$		0			0
$262$		−	4.09187i		−	0.252796i
$263$	−9.99063			−0.616049			−0.308024	−	0.951378i	$-0.599668\pi$
$263$	−9.99063			−0.616049			−0.308024	+	0.951378i	$0.599668\pi$
$264$		0			0
$265$		−	8.11753i		−	0.498656i
$266$			5.88739i			0.360979i
$267$		0			0
$268$		−	0.653858i		−	0.0399408i
$269$	18.2034			1.10988			0.554941	−	0.831890i	$-0.312741\pi$
$269$	18.2034			1.10988			0.554941	+	0.831890i	$0.312741\pi$
$270$		0			0
$271$		−	24.1748i		−	1.46852i	−0.678870	−	0.734258i	$-0.737530\pi$
$271$		−	24.1748i		−	1.46852i	0.678870	−	0.734258i	$-0.262470\pi$
$272$	4.94914			0.300086
$273$		0			0
$274$	11.2716			0.680944
$275$			14.3368i			0.864540i
$276$		0			0
$277$	−1.69535			−0.101863			−0.0509317	−	0.998702i	$-0.516219\pi$
$277$	−1.69535			−0.101863			−0.0509317	+	0.998702i	$0.516219\pi$
$278$			2.08742i			0.125195i
$279$		0			0
$280$			7.80642i			0.466523i
$281$		−	11.6479i		−	0.694854i	−0.937707	−	0.347427i	$-0.887055\pi$
$281$		−	11.6479i		−	0.694854i	0.937707	−	0.347427i	$-0.112945\pi$
$282$		0			0
$283$	−12.1334			−0.721253			−0.360626	−	0.932710i	$-0.617437\pi$
$283$	−12.1334			−0.721253			−0.360626	+	0.932710i	$0.617437\pi$
$284$			13.7047i			0.813225i
$285$		0			0
$286$	6.65386	−	0.576283i	0.393451	−	0.0340763i
$287$	0.755569			0.0445998
$288$		0			0
$289$	−4.09679			−0.240988
$290$	4.54125			0.266671
$291$		0			0
$292$			8.83314i			0.516921i
$293$		−	11.4538i		−	0.669140i	−0.942371	−	0.334570i	$-0.891409\pi$
$293$		−	11.4538i		−	0.669140i	0.942371	−	0.334570i	$-0.108591\pi$
$294$		0			0
$295$	23.5669			1.37212
$296$	−9.55262			−0.555235
$297$		0			0
$298$	10.0143			0.580112
$299$	−11.7857	+	1.02074i	−0.681583	+	0.0590311i
$300$		0			0
$301$			8.80642i			0.507594i
$302$	13.7605			0.791827
$303$		0			0
$304$			11.7748i			0.675330i
$305$			29.0923i			1.66582i
$306$		0			0
$307$			3.96989i			0.226574i	0.993562	+	0.113287i	$0.0361379\pi$
$307$			3.96989i			0.226574i	−0.993562	+	0.113287i	$0.963862\pi$
$308$	−4.10171			−0.233717
$309$		0			0
$310$			12.9240i			0.734031i
$311$	27.3481			1.55077			0.775386	−	0.631488i	$-0.217556\pi$
$311$	27.3481			1.55077			0.775386	+	0.631488i	$0.217556\pi$
$312$		0			0
$313$	−19.1032			−1.07978			−0.539890	−	0.841736i	$-0.681534\pi$
$313$	−19.1032			−1.07978			−0.539890	+	0.841736i	$0.681534\pi$
$314$			5.09679i			0.287628i
$315$		0			0
$316$	−6.83314			−0.384394
$317$			9.90813i			0.556496i	0.960509	+	0.278248i	$0.0897538\pi$
$317$			9.90813i			0.556496i	−0.960509	+	0.278248i	$0.910246\pi$
$318$		0			0
$319$			5.51453i			0.308754i
$320$		−	4.00000i		−	0.223607i
$321$		0			0
$322$	−2.26025			−0.125959
$323$			30.6987i			1.70812i
$324$		0			0
$325$	−19.1526	+	1.65878i	−1.06239	+	0.0920126i
$326$	−1.60300			−0.0887821
$327$		0			0
$328$	−1.83500			−0.101321
$329$	−1.88247			−0.103784
$330$		0			0
$331$		−	23.4193i		−	1.28724i	−0.765345	−	0.643620i	$-0.777432\pi$
$331$		−	23.4193i		−	1.28724i	0.765345	−	0.643620i	$-0.222568\pi$
$332$		−	16.5161i		−	0.906437i
$333$		0			0
$334$	−2.35752			−0.128998
$335$	1.37778			0.0752764
$336$		0			0
$337$	7.51606			0.409426			0.204713	−	0.978822i	$-0.434374\pi$
$337$	7.51606			0.409426			0.204713	+	0.978822i	$0.434374\pi$
$338$	1.53972	+	8.82225i	0.0837496	+	0.479867i
$339$		0			0
$340$			17.6128i			0.955191i
$341$	−15.6938			−0.849868
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		−	21.3876i		−	1.15314i
$345$		0			0
$346$			5.70027i			0.306448i
$347$	−5.64449			−0.303012			−0.151506	−	0.988456i	$-0.548412\pi$
$347$	−5.64449			−0.303012			−0.151506	+	0.988456i	$0.548412\pi$
$348$		0			0
$349$			24.1590i			1.29320i	0.762828	+	0.646601i	$0.223810\pi$
$349$			24.1590i			1.29320i	−0.762828	+	0.646601i	$0.776190\pi$
$350$	−3.67307			−0.196334
$351$		0			0
$352$	15.6128			0.832168
$353$			36.1289i			1.92295i	0.274896	+	0.961474i	$0.411356\pi$
$353$			36.1289i			1.92295i	−0.274896	+	0.961474i	$0.588644\pi$
$354$		0			0
$355$	−28.8780			−1.53269
$356$		−	8.17929i		−	0.433501i
$357$		0			0
$358$		−	11.1778i		−	0.590763i
$359$		−	17.4128i		−	0.919013i	−0.888174	−	0.459507i	$-0.848026\pi$
$359$		−	17.4128i		−	0.919013i	0.888174	−	0.459507i	$-0.151974\pi$
$360$		0			0
$361$	−54.0370			−2.84405
$362$			6.34122i			0.333287i
$363$		0			0
$364$	−0.474572	−	5.47949i	−0.0248744	−	0.287204i
$365$	−18.6128			−0.974241
$366$		0			0
$367$	2.93825			0.153375			0.0766876	−	0.997055i	$-0.475566\pi$
$367$	2.93825			0.153375			0.0766876	+	0.997055i	$0.475566\pi$
$368$	−4.52051			−0.235648
$369$		0			0
$370$		−	8.70964i		−	0.452792i
$371$			2.52543i			0.131114i
$372$		0			0
$373$	−18.7699			−0.971866			−0.485933	−	0.873996i	$-0.661520\pi$
$373$	−18.7699			−0.971866			−0.485933	+	0.873996i	$0.661520\pi$
$374$	6.65386			0.344063
$375$		0			0
$376$	4.57184			0.235774
$377$	−7.36689	+	0.638037i	−0.379414	+	0.0328606i
$378$		0			0
$379$		−	23.6894i		−	1.21684i	−0.793615	−	0.608421i	$-0.791803\pi$
$379$		−	23.6894i		−	1.21684i	0.793615	−	0.608421i	$-0.208197\pi$
$380$	−41.9037			−2.14961
$381$		0			0
$382$			4.59364i			0.235031i
$383$			31.6128i			1.61534i	0.589634	+	0.807671i	$0.299272\pi$
$383$			31.6128i			1.61534i	−0.589634	+	0.807671i	$0.700728\pi$
$384$		0			0
$385$		−	8.64296i		−	0.440486i
$386$	−14.1936			−0.722434
$387$		0			0
$388$		−	14.6878i		−	0.745662i
$389$	−20.0558			−1.01687			−0.508434	−	0.861101i	$-0.669775\pi$
$389$	−20.0558			−1.01687			−0.508434	+	0.861101i	$0.669775\pi$
$390$		0			0
$391$	−11.7857			−0.596027
$392$		−	2.42864i		−	0.122665i
$393$		0			0
$394$	5.76494			0.290433
$395$		−	14.3985i		−	0.724469i
$396$		0			0
$397$			22.8731i			1.14797i	0.818867	+	0.573984i	$0.194603\pi$
$397$			22.8731i			1.14797i	−0.818867	+	0.573984i	$0.805397\pi$
$398$			0.414349i			0.0207695i
$399$		0			0
$400$	−7.34614			−0.367307
$401$			5.61285i			0.280292i	0.990131	+	0.140146i	$0.0447572\pi$
$401$			5.61285i			0.280292i	−0.990131	+	0.140146i	$0.955243\pi$
$402$		0			0
$403$	−1.81579	−	20.9654i	−0.0904510	−	1.04436i
$404$	20.8889			1.03926
$405$		0			0
$406$	−1.41282			−0.0701170
$407$	10.5763			0.524247
$408$		0			0
$409$			26.1175i			1.29143i	0.763579	+	0.645714i	$0.223440\pi$
$409$			26.1175i			1.29143i	−0.763579	+	0.645714i	$0.776560\pi$
$410$		−	1.67307i		−	0.0826271i
$411$		0			0
$412$	−18.7556			−0.924021
$413$	−7.33185			−0.360777
$414$		0			0
$415$	34.8020			1.70836
$416$	1.80642	+	20.8573i	0.0885672	+	1.02261i
$417$		0			0
$418$			15.8306i			0.774298i
$419$	25.2464			1.23337			0.616685	−	0.787210i	$-0.288475\pi$
$419$	25.2464			1.23337			0.616685	+	0.787210i	$0.288475\pi$
$420$		0			0
$421$			8.22861i			0.401038i	0.979690	+	0.200519i	$0.0642628\pi$
$421$			8.22861i			0.401038i	−0.979690	+	0.200519i	$0.935737\pi$
$422$		−	1.31111i		−	0.0638237i
$423$		0			0
$424$		−	6.13335i		−	0.297862i
$425$	−19.1526			−0.929036
$426$		0			0
$427$		−	9.05086i		−	0.438002i
$428$	27.7462			1.34116
$429$		0			0
$430$	19.5002			0.940385
$431$		−	1.43801i		−	0.0692664i	−0.999400	−	0.0346332i	$-0.988974\pi$
$431$		−	1.43801i		−	0.0692664i	0.999400	−	0.0346332i	$-0.0110263\pi$
$432$		0			0
$433$	16.5018			0.793024			0.396512	−	0.918029i	$-0.370220\pi$
$433$	16.5018			0.793024			0.396512	+	0.918029i	$0.370220\pi$
$434$		−	4.02074i		−	0.193002i
$435$		0			0
$436$		−	12.7556i		−	0.610881i
$437$		−	28.0400i		−	1.34133i
$438$		0			0
$439$	10.1619			0.485003			0.242501	−	0.970151i	$-0.422032\pi$
$439$	10.1619			0.485003			0.242501	+	0.970151i	$0.422032\pi$
$440$			20.9906i			1.00069i
$441$		0			0
$442$	0.769859	+	8.88892i	0.0366185	+	0.422803i
$443$	−3.20787			−0.152410			−0.0762052	−	0.997092i	$-0.524280\pi$
$443$	−3.20787			−0.152410			−0.0762052	+	0.997092i	$0.524280\pi$
$444$		0			0
$445$	17.2351			0.817020
$446$	7.32540			0.346868
$447$		0			0
$448$			1.24443i			0.0587939i
$449$			18.4099i			0.868817i	0.900716	+	0.434409i	$0.143043\pi$
$449$			18.4099i			0.868817i	−0.900716	+	0.434409i	$0.856957\pi$
$450$		0			0
$451$	2.03164			0.0956663
$452$	12.9131			0.607379
$453$		0			0
$454$	−10.3684			−0.486614
$455$	11.5462	−	1.00000i	0.541293	−	0.0468807i
$456$		0			0
$457$		−	3.40297i		−	0.159184i	−0.996828	−	0.0795922i	$-0.974638\pi$
$457$		−	3.40297i		−	0.159184i	0.996828	−	0.0795922i	$-0.0253618\pi$
$458$	4.50177			0.210354
$459$		0			0
$460$		−	16.0874i		−	0.750080i
$461$			17.5714i			0.818380i	0.912449	+	0.409190i	$0.134189\pi$
$461$			17.5714i			0.818380i	−0.912449	+	0.409190i	$0.865811\pi$
$462$		0			0
$463$			15.7714i			0.732959i	0.930426	+	0.366479i	$0.119437\pi$
$463$			15.7714i			0.732959i	−0.930426	+	0.366479i	$0.880563\pi$
$464$	−2.82564			−0.131177
$465$		0			0
$466$		−	19.2607i		−	0.892236i
$467$	−3.76694			−0.174313			−0.0871567	−	0.996195i	$-0.527778\pi$
$467$	−3.76694			−0.174313			−0.0871567	+	0.996195i	$0.527778\pi$
$468$		0			0
$469$	−0.428639			−0.0197927
$470$			4.16839i			0.192273i
$471$		0			0
$472$	17.8064			0.819607
$473$			23.6795i			1.08879i
$474$		0			0
$475$		−	45.5669i		−	2.09075i
$476$		−	5.47949i		−	0.251152i
$477$		0			0
$478$	13.4893			0.616988
$479$			12.1032i			0.553011i	0.961012	+	0.276506i	$0.0891764\pi$
$479$			12.1032i			0.553011i	−0.961012	+	0.276506i	$0.910824\pi$
$480$		0			0
$481$	1.22369	+	14.1289i	0.0557954	+	0.644223i
$482$	9.20189			0.419135
$483$		0			0
$484$	5.75065			0.261393
$485$	30.9496			1.40535
$486$		0			0
$487$			17.3778i			0.787463i	0.919226	+	0.393731i	$0.128816\pi$
$487$			17.3778i			0.787463i	−0.919226	+	0.393731i	$0.871184\pi$
$488$			21.9813i			0.995045i
$489$		0			0
$490$	2.21432			0.100033
$491$	−19.3921			−0.875152			−0.437576	−	0.899181i	$-0.644163\pi$
$491$	−19.3921			−0.875152			−0.437576	+	0.899181i	$0.644163\pi$
$492$		0			0
$493$	−7.36689			−0.331788
$494$	−21.1481	+	1.83161i	−0.951499	+	0.0824082i
$495$		0			0
$496$		−	8.04149i		−	0.361073i
$497$	8.98418			0.402995
$498$		0			0
$499$			10.6702i			0.477662i	0.971061	+	0.238831i	$0.0767642\pi$
$499$			10.6702i			0.477662i	−0.971061	+	0.238831i	$0.923236\pi$
$500$		−	1.62714i		−	0.0727678i
$501$		0			0
$502$			2.27115i			0.101366i
$503$	4.27655			0.190682			0.0953410	−	0.995445i	$-0.469606\pi$
$503$	4.27655			0.190682			0.0953410	+	0.995445i	$0.469606\pi$
$504$		0			0
$505$			44.0163i			1.95870i
$506$	−6.07758			−0.270181
$507$		0			0
$508$	−6.23506			−0.276636
$509$			33.0765i			1.46609i	0.680180	+	0.733046i	$0.261902\pi$
$509$			33.0765i			1.46609i	−0.680180	+	0.733046i	$0.738098\pi$
$510$		0			0
$511$	5.79060			0.256161
$512$			14.6923i			0.649313i
$513$		0			0
$514$		−	16.3225i		−	0.719954i
$515$		−	39.5210i		−	1.74150i
$516$		0			0
$517$	−5.06175			−0.222616
$518$			2.70964i			0.119055i
$519$		0			0
$520$	−28.0415	+	2.42864i	−1.22970	+	0.106503i
$521$	−24.3783			−1.06803			−0.534015	−	0.845475i	$-0.679318\pi$
$521$	−24.3783			−1.06803			−0.534015	+	0.845475i	$0.679318\pi$
$522$		0			0
$523$	34.9403			1.52783			0.763915	−	0.645317i	$-0.223275\pi$
$523$	34.9403			1.52783			0.763915	+	0.645317i	$0.223275\pi$
$524$	−9.06070			−0.395818
$525$		0			0
$526$		−	6.88247i		−	0.300090i
$527$		−	20.9654i		−	0.913269i
$528$		0			0
$529$	−12.2351			−0.531959
$530$	5.59210			0.242905
$531$		0			0
$532$	13.0366			0.565207
$533$	0.235063	+	2.71408i	0.0101817	+	0.117560i
$534$		0			0
$535$			58.4657i			2.52769i
$536$	1.04101			0.0449648
$537$		0			0
$538$			12.5402i			0.540646i
$539$			2.68889i			0.115819i
$540$		0			0
$541$		−	30.4953i		−	1.31110i	−0.755153	−	0.655548i	$-0.772438\pi$
$541$		−	30.4953i		−	1.31110i	0.755153	−	0.655548i	$-0.227562\pi$
$542$	16.6539			0.715344
$543$		0			0
$544$			20.8573i			0.894248i
$545$	26.8780			1.15133
$546$		0			0
$547$	−10.0049			−0.427780			−0.213890	−	0.976858i	$-0.568613\pi$
$547$	−10.0049			−0.427780			−0.213890	+	0.976858i	$0.568613\pi$
$548$		−	24.9590i		−	1.06620i
$549$		0			0
$550$	−9.87649			−0.421135
$551$		−	17.5270i		−	0.746674i
$552$		0			0
$553$			4.47949i			0.190487i
$554$		−	1.16791i		−	0.0496198i
$555$		0			0
$556$	4.62222			0.196026
$557$		−	14.1936i		−	0.601401i	−0.953719	−	0.300701i	$-0.902780\pi$
$557$		−	14.1936i		−	0.601401i	0.953719	−	0.300701i	$-0.0972205\pi$
$558$		0			0
$559$	−31.6336	+	2.73975i	−1.33796	+	0.115879i
$560$	4.42864			0.187144
$561$		0			0
$562$	8.02413			0.338478
$563$	−21.3590			−0.900177			−0.450088	−	0.892984i	$-0.648607\pi$
$563$	−21.3590			−0.900177			−0.450088	+	0.892984i	$0.648607\pi$
$564$		0			0
$565$			27.2099i			1.14473i
$566$		−	8.35857i		−	0.351337i
$567$		0			0
$568$	−21.8193			−0.915519
$569$	34.2672			1.43656			0.718278	−	0.695757i	$-0.244931\pi$
$569$	34.2672			1.43656			0.718278	+	0.695757i	$0.244931\pi$
$570$		0			0
$571$	37.6494			1.57558			0.787789	−	0.615945i	$-0.211226\pi$
$571$	37.6494			1.57558			0.787789	+	0.615945i	$0.211226\pi$
$572$	−1.27607	−	14.7338i	−0.0533553	−	0.616050i
$573$		0			0
$574$			0.520505i			0.0217255i
$575$	17.4938			0.729541
$576$		0			0
$577$		−	28.3970i		−	1.18218i	−0.806605	−	0.591091i	$-0.798697\pi$
$577$		−	28.3970i		−	1.18218i	0.806605	−	0.591091i	$-0.201303\pi$
$578$		−	2.82225i		−	0.117390i
$579$		0			0
$580$		−	10.0558i		−	0.417543i
$581$	−10.8272			−0.449187
$582$		0			0
$583$			6.79060i			0.281238i
$584$	−14.0633			−0.581943
$585$		0			0
$586$	7.89045			0.325952
$587$		−	6.23659i		−	0.257412i	−0.991683	−	0.128706i	$-0.958918\pi$
$587$		−	6.23659i		−	0.257412i	0.991683	−	0.128706i	$-0.0410823\pi$
$588$		0			0
$589$	49.8800			2.05527
$590$			16.2351i			0.668387i
$591$		0			0
$592$			5.41927i			0.222731i
$593$		−	17.5698i		−	0.721506i	−0.932661	−	0.360753i	$-0.882520\pi$
$593$		−	17.5698i		−	0.721506i	0.932661	−	0.360753i	$-0.117480\pi$
$594$		0			0
$595$	11.5462			0.473347
$596$		−	22.1748i		−	0.908317i
$597$		0			0
$598$	−0.703182	−	8.11906i	−0.0287553	−	0.332013i
$599$	−16.9813			−0.693836			−0.346918	−	0.937896i	$-0.612772\pi$
$599$	−16.9813			−0.693836			−0.346918	+	0.937896i	$0.612772\pi$
$600$		0			0
$601$	−18.7052			−0.763001			−0.381500	−	0.924369i	$-0.624592\pi$
$601$	−18.7052			−0.763001			−0.381500	+	0.924369i	$0.624592\pi$
$602$	−6.06668			−0.247259
$603$		0			0
$604$		−	30.4701i		−	1.23981i
$605$			12.1175i			0.492648i
$606$		0			0
$607$	−10.3575			−0.420399			−0.210199	−	0.977659i	$-0.567411\pi$
$607$	−10.3575			−0.420399			−0.210199	+	0.977659i	$0.567411\pi$
$608$	−49.6227			−2.01247
$609$		0			0
$610$	−20.0415			−0.811456
$611$	−0.585651	−	6.76202i	−0.0236929	−	0.273562i
$612$		0			0
$613$			7.02227i			0.283627i	0.989893	+	0.141814i	$0.0452933\pi$
$613$			7.02227i			0.283627i	−0.989893	+	0.141814i	$0.954707\pi$
$614$	−2.73483			−0.110369
$615$		0			0
$616$		−	6.53035i		−	0.263115i
$617$			29.9813i			1.20700i	0.797363	+	0.603500i	$0.206228\pi$
$617$			29.9813i			1.20700i	−0.797363	+	0.603500i	$0.793772\pi$
$618$		0			0
$619$			0.0285802i			0.00114874i	1.00000		0.000574368i	$0.000182827\pi$
$619$			0.0285802i			0.00114874i	−1.00000		0.000574368i	$0.999817\pi$
$620$	28.6178			1.14932
$621$		0			0
$622$			18.8399i			0.755412i
$623$	−5.36196			−0.214823
$624$		0			0
$625$	−23.2306			−0.929225
$626$		−	13.1601i		−	0.525982i
$627$		0			0
$628$	11.2859			0.450357
$629$			14.1289i			0.563356i
$630$		0			0
$631$		−	12.1936i		−	0.485419i	−0.970099	−	0.242709i	$-0.921964\pi$
$631$		−	12.1936i		−	0.485419i	0.970099	−	0.242709i	$-0.0780361\pi$
$632$		−	10.8791i		−	0.432746i
$633$		0			0
$634$	−6.82564			−0.271081
$635$		−	13.1383i		−	0.521377i
$636$		0			0
$637$	−3.59210	+	0.311108i	−0.142324	+	0.0123265i
$638$	−3.79892			−0.150401
$639$		0			0
$640$	−34.5718			−1.36657
$641$	−2.82516			−0.111587			−0.0557935	−	0.998442i	$-0.517769\pi$
$641$	−2.82516			−0.111587			−0.0557935	+	0.998442i	$0.517769\pi$
$642$		0			0
$643$			37.7275i			1.48783i	0.668276	+	0.743913i	$0.267032\pi$
$643$			37.7275i			1.48783i	−0.668276	+	0.743913i	$0.732968\pi$
$644$			5.00492i			0.197222i
$645$		0			0
$646$	−21.1481			−0.832062
$647$	12.3664			0.486174			0.243087	−	0.970005i	$-0.421840\pi$
$647$	12.3664			0.486174			0.243087	+	0.970005i	$0.421840\pi$
$648$		0			0
$649$	−19.7146			−0.773864
$650$	−1.14272	−	13.1941i	−0.0448212	−	0.517513i
$651$		0			0
$652$			3.54956i			0.139012i
$653$	33.0005			1.29141			0.645704	−	0.763588i	$-0.276564\pi$
$653$	33.0005			1.29141			0.645704	+	0.763588i	$0.276564\pi$
$654$		0			0
$655$		−	19.0923i		−	0.746000i
$656$			1.04101i			0.0406446i
$657$		0			0
$658$		−	1.29682i		−	0.0505552i
$659$	−32.8118			−1.27817			−0.639084	−	0.769137i	$-0.720686\pi$
$659$	−32.8118			−1.27817			−0.639084	+	0.769137i	$0.720686\pi$
$660$		0			0
$661$		−	14.7067i		−	0.572025i	−0.958226	−	0.286013i	$-0.907670\pi$
$661$		−	14.7067i		−	0.572025i	0.958226	−	0.286013i	$-0.0923299\pi$
$662$	16.1334			0.627041
$663$		0			0
$664$	26.2953			1.02046
$665$			27.4701i			1.06525i
$666$		0			0
$667$	6.72885			0.260542
$668$			5.22030i			0.201979i
$669$		0			0
$670$			0.949145i			0.0366687i
$671$		−	24.3368i		−	0.939511i
$672$		0			0
$673$	−21.2908			−0.820702			−0.410351	−	0.911928i	$-0.634594\pi$
$673$	−21.2908			−0.820702			−0.410351	+	0.911928i	$0.634594\pi$
$674$			5.17775i			0.199440i
$675$		0			0
$676$	19.5353	−	3.40943i	0.751357	−	0.131132i
$677$	3.07160			0.118051			0.0590256	−	0.998256i	$-0.481201\pi$
$677$	3.07160			0.118051			0.0590256	+	0.998256i	$0.481201\pi$
$678$		0			0
$679$	−9.62867			−0.369514
$680$	−28.0415			−1.07534
$681$		0			0
$682$		−	10.8113i		−	0.413988i
$683$			24.7971i			0.948833i	0.880301	+	0.474416i	$0.157341\pi$
$683$			24.7971i			0.948833i	−0.880301	+	0.474416i	$0.842659\pi$
$684$		0			0
$685$	52.5926			2.00946
$686$	−0.688892			−0.0263020
$687$		0			0
$688$	−12.1334			−0.462580
$689$	−9.07160	+	0.785680i	−0.345600	+	0.0299320i
$690$		0			0
$691$			27.0953i			1.03075i	0.856964	+	0.515376i	$0.172348\pi$
$691$			27.0953i			1.03075i	−0.856964	+	0.515376i	$0.827652\pi$
$692$	12.6222			0.479825
$693$		0			0
$694$		−	3.88845i		−	0.147603i
$695$			9.73975i			0.369450i
$696$		0			0
$697$			2.71408i			0.102803i
$698$	−16.6430			−0.629945
$699$		0			0
$700$			8.13335i			0.307412i
$701$	13.5205			0.510662			0.255331	−	0.966854i	$-0.417815\pi$
$701$	13.5205			0.510662			0.255331	+	0.966854i	$0.417815\pi$
$702$		0			0
$703$	−33.6149			−1.26781
$704$			3.34614i			0.126112i
$705$		0			0
$706$	−24.8889			−0.936707
$707$		−	13.6938i		−	0.515009i
$708$		0			0
$709$		−	50.9753i		−	1.91442i	−0.289399	−	0.957209i	$-0.593455\pi$
$709$		−	50.9753i		−	1.91442i	0.289399	−	0.957209i	$-0.406545\pi$
$710$		−	19.8938i		−	0.746603i
$711$		0			0
$712$	13.0223			0.488030
$713$			19.1497i			0.717160i
$714$		0			0
$715$	31.0464	−	2.68889i	1.16107	−	0.100559i
$716$	−24.7511			−0.924993
$717$		0			0
$718$	11.9956			0.447670
$719$	41.5417			1.54924			0.774622	−	0.632424i	$-0.217940\pi$
$719$	41.5417			1.54924			0.774622	+	0.632424i	$0.217940\pi$
$720$		0			0
$721$			12.2953i			0.457900i
$722$		−	37.2257i		−	1.38540i
$723$		0			0
$724$	14.0415			0.521848
$725$	10.9349			0.406110
$726$		0			0
$727$	20.8988			0.775092			0.387546	−	0.921850i	$-0.373323\pi$
$727$	20.8988			0.775092			0.387546	+	0.921850i	$0.373323\pi$
$728$	8.72393	−	0.755569i	0.323330	−	0.0280032i
$729$		0			0
$730$		−	12.8222i		−	0.474573i
$731$	−31.6336			−1.17001
$732$		0			0
$733$		−	19.3575i		−	0.714986i	−0.933916	−	0.357493i	$-0.883632\pi$
$733$		−	19.3575i		−	0.714986i	0.933916	−	0.357493i	$-0.116368\pi$
$734$			2.02413i			0.0747122i
$735$		0			0
$736$		−	19.0509i		−	0.702224i
$737$	−1.15257			−0.0424553
$738$		0			0
$739$		−	1.31111i		−	0.0482299i	−0.999709	−	0.0241149i	$-0.992323\pi$
$739$		−	1.31111i		−	0.0482299i	0.999709	−	0.0241149i	$-0.00767677\pi$
$740$	−19.2859			−0.708964
$741$		0			0
$742$	−1.73975			−0.0638681
$743$		−	21.5210i		−	0.789528i	−0.918783	−	0.394764i	$-0.870826\pi$
$743$		−	21.5210i		−	0.789528i	0.918783	−	0.394764i	$-0.129174\pi$
$744$		0			0
$745$	46.7259			1.71191
$746$		−	12.9304i		−	0.473416i
$747$		0			0
$748$		−	14.7338i		−	0.538720i
$749$		−	18.1891i		−	0.664616i
$750$		0			0
$751$	16.3176			0.595436			0.297718	−	0.954654i	$-0.403774\pi$
$751$	16.3176			0.595436			0.297718	+	0.954654i	$0.403774\pi$
$752$		−	2.59364i		−	0.0945802i
$753$		0			0
$754$	−0.439539	−	5.07499i	−0.0160071	−	0.184820i
$755$	64.2054			2.33667
$756$		0			0
$757$	−18.7462			−0.681342			−0.340671	−	0.940183i	$-0.610654\pi$
$757$	−18.7462			−0.681342			−0.340671	+	0.940183i	$0.610654\pi$
$758$	16.3194			0.592748
$759$		0			0
$760$		−	66.7150i		−	2.42001i
$761$			10.2968i			0.373259i	0.982430	+	0.186630i	$0.0597564\pi$
$761$			10.2968i			0.373259i	−0.982430	+	0.186630i	$0.940244\pi$
$762$		0			0
$763$	−8.36196			−0.302723
$764$	10.1718			0.368002
$765$		0			0
$766$	−21.7778			−0.786865
$767$	−2.28100	−	26.3368i	−0.0823620	−	0.950966i
$768$		0			0
$769$		−	9.36641i		−	0.337761i	−0.985637	−	0.168881i	$-0.945985\pi$
$769$		−	9.36641i		−	0.337761i	0.985637	−	0.168881i	$-0.0540153\pi$
$770$	5.95407			0.214570
$771$		0			0
$772$			31.4291i			1.13116i
$773$		−	3.71456i		−	0.133603i	−0.997766	−	0.0668017i	$-0.978721\pi$
$773$		−	3.71456i		−	0.133603i	0.997766	−	0.0668017i	$-0.0212795\pi$
$774$		0			0
$775$			31.1195i			1.11785i
$776$	23.3846			0.839457
$777$		0			0
$778$		−	13.8163i		−	0.495337i
$779$	−6.45722			−0.231354
$780$		0			0
$781$	24.1575			0.864423
$782$		−	8.11906i		−	0.290337i
$783$		0			0
$784$	−1.37778			−0.0492066
$785$			23.7812i			0.848789i
$786$		0			0
$787$			6.23659i			0.222311i	0.993803	+	0.111155i	$0.0354551\pi$
$787$			6.23659i			0.222311i	−0.993803	+	0.111155i	$0.964545\pi$
$788$		−	12.7654i		−	0.454749i
$789$		0			0
$790$	9.91903			0.352903
$791$		−	8.46520i		−	0.300988i
$792$		0			0
$793$	32.5116	−	2.81579i	1.15452	−	0.0999917i
$794$	−15.7571			−0.559199
$795$		0			0
$796$	0.917502			0.0325200
$797$	−40.0701			−1.41935			−0.709677	−	0.704527i	$-0.751159\pi$
$797$	−40.0701			−1.41935			−0.709677	+	0.704527i	$0.751159\pi$
$798$		0			0
$799$		−	6.76202i		−	0.239223i
$800$		−	30.9590i		−	1.09457i
$801$		0			0
$802$	−3.86665			−0.136536
$803$	15.5703			0.549464
$804$		0			0
$805$	−10.5462			−0.371704
$806$	14.4429	−	1.25088i	0.508730	−	0.0440605i
$807$		0			0
$808$			33.2573i			1.16999i
$809$	2.57136			0.0904042			0.0452021	−	0.998978i	$-0.485607\pi$
$809$	2.57136			0.0904042			0.0452021	+	0.998978i	$0.485607\pi$
$810$		0			0
$811$			28.3654i			0.996042i	0.867165	+	0.498021i	$0.165940\pi$
$811$			28.3654i			0.996042i	−0.867165	+	0.498021i	$0.834060\pi$
$812$			3.12843i			0.109786i
$813$		0			0
$814$			7.28592i			0.255371i
$815$	−7.47949			−0.261995
$816$		0			0
$817$		−	75.2612i		−	2.63306i
$818$	−17.9922			−0.629081
$819$		0			0
$820$	−3.70471			−0.129374
$821$		−	31.7846i		−	1.10929i	−0.832087	−	0.554646i	$-0.812854\pi$
$821$		−	31.7846i		−	1.10929i	0.832087	−	0.554646i	$-0.187146\pi$
$822$		0			0
$823$	−30.9131			−1.07756			−0.538781	−	0.842446i	$-0.681115\pi$
$823$	−30.9131			−1.07756			−0.538781	+	0.842446i	$0.681115\pi$
$824$		−	29.8608i		−	1.04025i
$825$		0			0
$826$		−	5.05086i		−	0.175742i
$827$		−	49.7560i		−	1.73019i	−0.501610	−	0.865094i	$-0.667259\pi$
$827$		−	49.7560i		−	1.73019i	0.501610	−	0.865094i	$-0.332741\pi$
$828$		0			0
$829$	19.9190			0.691817			0.345908	−	0.938268i	$-0.387571\pi$
$829$	19.9190			0.691817			0.345908	+	0.938268i	$0.387571\pi$
$830$			23.9748i			0.832178i
$831$		0			0
$832$	−4.47013	+	0.387152i	−0.154974	+	0.0134221i
$833$	−3.59210			−0.124459
$834$		0			0
$835$	−11.0000			−0.380671
$836$	35.0539			1.21237
$837$		0			0
$838$			17.3921i			0.600799i
$839$		−	11.5625i		−	0.399181i	−0.979879	−	0.199590i	$-0.936039\pi$
$839$		−	11.5625i		−	0.399181i	0.979879	−	0.199590i	$-0.0639611\pi$
$840$		0			0
$841$	−24.7940			−0.854965
$842$	−5.66862			−0.195354
$843$		0			0
$844$	−2.90321			−0.0999327
$845$	7.18421	+	41.1639i	0.247144	+	1.41608i
$846$		0			0
$847$		−	3.76986i		−	0.129534i
$848$	−3.47949			−0.119486
$849$		0			0
$850$		−	13.1941i		−	0.452552i
$851$		−	12.9052i		−	0.442385i
$852$		0			0
$853$			1.74467i			0.0597363i	0.999554	+	0.0298682i	$0.00950875\pi$
$853$			1.74467i			0.0597363i	−0.999554	+	0.0298682i	$0.990491\pi$
$854$	6.23506			0.213359
$855$		0			0
$856$			44.1748i			1.50986i
$857$	−42.3368			−1.44620			−0.723098	−	0.690745i	$-0.757283\pi$
$857$	−42.3368			−1.44620			−0.723098	+	0.690745i	$0.757283\pi$
$858$		0			0
$859$	−5.37778			−0.183488			−0.0917438	−	0.995783i	$-0.529244\pi$
$859$	−5.37778			−0.183488			−0.0917438	+	0.995783i	$0.529244\pi$
$860$		−	43.1798i		−	1.47242i
$861$		0			0
$862$	0.990632			0.0337411
$863$			15.4291i			0.525213i	0.964903	+	0.262607i	$0.0845821\pi$
$863$			15.4291i			0.525213i	−0.964903	+	0.262607i	$0.915418\pi$
$864$		0			0
$865$			26.5970i			0.904326i
$866$			11.3679i			0.386298i
$867$		0			0
$868$	−8.90321			−0.302195
$869$			12.0449i			0.408595i
$870$		0			0
$871$	−0.133353	−	1.53972i	−0.00451850	−	0.0521713i
$872$	20.3082			0.687722
$873$		0			0
$874$	19.3165			0.653391
$875$	−1.06668			−0.0360602
$876$		0			0
$877$		−	38.5181i		−	1.30066i	−0.759651	−	0.650331i	$-0.774630\pi$
$877$		−	38.5181i		−	1.30066i	0.759651	−	0.650331i	$-0.225370\pi$
$878$			7.00048i			0.236255i
$879$		0			0
$880$	11.9081			0.401423
$881$	−41.7373			−1.40617			−0.703083	−	0.711108i	$-0.748194\pi$
$881$	−41.7373			−1.40617			−0.703083	+	0.711108i	$0.748194\pi$
$882$		0			0
$883$	−52.8439			−1.77834			−0.889170	−	0.457577i	$-0.848717\pi$
$883$	−52.8439			−1.77834			−0.889170	+	0.457577i	$0.848717\pi$
$884$	19.6829	−	1.70471i	0.662008	−	0.0573357i
$885$		0			0
$886$		−	2.20987i		−	0.0742422i
$887$	−8.99063			−0.301876			−0.150938	−	0.988543i	$-0.548229\pi$
$887$	−8.99063			−0.301876			−0.150938	+	0.988543i	$0.548229\pi$
$888$		0			0
$889$			4.08742i			0.137088i
$890$			11.8731i			0.397987i
$891$		0			0
$892$		−	16.2208i		−	0.543112i
$893$	16.0879			0.538361
$894$		0			0
$895$		−	52.1546i		−	1.74334i
$896$	10.7556			0.359318
$897$		0			0
$898$	−12.6824			−0.423218
$899$			11.9699i			0.399218i
$900$		0			0
$901$	−9.07160			−0.302219
$902$			1.39958i			0.0466010i
$903$		0			0
$904$			20.5589i			0.683780i
$905$			29.5877i			0.983527i
$906$		0			0
$907$	20.7003			0.687341			0.343671	−	0.939090i	$-0.388330\pi$
$907$	20.7003			0.687341			0.343671	+	0.939090i	$0.388330\pi$
$908$			22.9590i			0.761921i
$909$		0			0
$910$	0.688892	+	7.95407i	0.0228366	+	0.263675i
$911$	−29.8988			−0.990590			−0.495295	−	0.868725i	$-0.664940\pi$
$911$	−29.8988			−0.990590			−0.495295	+	0.868725i	$0.664940\pi$
$912$		0			0
$913$	−29.1131			−0.963503
$914$	2.34428			0.0775420
$915$		0			0
$916$		−	9.96836i		−	0.329364i
$917$			5.93978i			0.196149i
$918$		0			0
$919$	38.5847			1.27279			0.636397	−	0.771362i	$-0.280424\pi$
$919$	38.5847			1.27279			0.636397	+	0.771362i	$0.280424\pi$
$920$	25.6128			0.844431
$921$		0			0
$922$	−12.1048			−0.398649
$923$	2.79505	+	32.2721i	0.0920001	+	1.06225i
$924$		0			0
$925$		−	20.9719i		−	0.689552i
$926$	−10.8648			−0.357039
$927$		0			0
$928$		−	11.9081i		−	0.390904i
$929$			3.25581i			0.106820i	0.998573	+	0.0534098i	$0.0170090\pi$
$929$			3.25581i			0.106820i	−0.998573	+	0.0534098i	$0.982991\pi$
$930$		0			0
$931$		−	8.54617i		−	0.280089i
$932$	−42.6494			−1.39703
$933$		0			0
$934$		−	2.59502i		−	0.0849116i
$935$	31.0464			1.01533
$936$		0			0
$937$	−11.6840			−0.381699			−0.190849	−	0.981619i	$-0.561124\pi$
$937$	−11.6840			−0.381699			−0.190849	+	0.981619i	$0.561124\pi$
$938$		−	0.295286i		−	0.00964144i
$939$		0			0
$940$	9.23014			0.301054
$941$		−	41.9699i		−	1.36818i	−0.729398	−	0.684090i	$-0.760200\pi$
$941$		−	41.9699i		−	1.36818i	0.729398	−	0.684090i	$-0.239800\pi$
$942$		0			0
$943$		−	2.47902i		−	0.0807279i
$944$		−	10.1017i		−	0.328783i
$945$		0			0
$946$	−16.3126			−0.530370
$947$			20.3555i			0.661465i	0.943725	+	0.330733i	$0.107296\pi$
$947$			20.3555i			0.661465i	−0.943725	+	0.330733i	$0.892704\pi$
$948$		0			0
$949$	1.80150	+	20.8004i	0.0584792	+	0.675211i
$950$	31.3907			1.01845
$951$		0			0
$952$	8.72393			0.282744
$953$	−30.5496			−0.989597			−0.494799	−	0.869008i	$-0.664758\pi$
$953$	−30.5496			−0.989597			−0.494799	+	0.869008i	$0.664758\pi$
$954$		0			0
$955$			21.4336i			0.693574i
$956$		−	29.8697i		−	0.966055i
$957$		0			0
$958$	−8.33783			−0.269383
$959$	−16.3620			−0.528355
$960$		0			0
$961$	−3.06515			−0.0988757
$962$	−9.73329	+	0.842989i	−0.313814	+	0.0271791i
$963$		0			0
$964$		−	20.3759i		−	0.656264i
$965$	−66.2262			−2.13190
$966$		0			0
$967$		−	54.4548i		−	1.75115i	−0.483084	−	0.875574i	$-0.660484\pi$
$967$		−	54.4548i		−	1.75115i	0.483084	−	0.875574i	$-0.339516\pi$
$968$			9.15563i			0.294273i
$969$		0			0
$970$			21.3210i			0.684575i
$971$	54.6035			1.75231			0.876155	−	0.482030i	$-0.160101\pi$
$971$	54.6035			1.75231			0.876155	+	0.482030i	$0.160101\pi$
$972$		0			0
$973$		−	3.03011i		−	0.0971409i
$974$	−11.9714			−0.383589
$975$		0			0
$976$	12.4701			0.399159
$977$			32.6474i			1.04448i	0.852798	+	0.522242i	$0.174904\pi$
$977$			32.6474i			1.04448i	−0.852798	+	0.522242i	$0.825096\pi$
$978$		0			0
$979$	−14.4177			−0.460793
$980$		−	4.90321i		−	0.156627i
$981$		0			0
$982$		−	13.3590i		−	0.426304i
$983$			27.5052i			0.877278i	0.898663	+	0.438639i	$0.144539\pi$
$983$			27.5052i			0.877278i	−0.898663	+	0.438639i	$0.855461\pi$
$984$		0			0
$985$	26.8988			0.857066
$986$		−	5.07499i		−	0.161621i
$987$		0			0
$988$	4.05578	+	46.8287i	0.129031	+	1.48982i
$989$	28.8938			0.918771
$990$		0			0
$991$	−6.29390			−0.199932			−0.0999662	−	0.994991i	$-0.531873\pi$
$991$	−6.29390			−0.199932			−0.0999662	+	0.994991i	$0.531873\pi$
$992$	33.8894			1.07599
$993$		0			0
$994$			6.18913i			0.196307i
$995$			1.93332i			0.0612905i
$996$		0			0
$997$	−20.8702			−0.660965			−0.330483	−	0.943812i	$-0.607212\pi$
$997$	−20.8702			−0.660965			−0.330483	+	0.943812i	$0.607212\pi$
$998$	−7.35059			−0.232679
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	819.2.c.b.64.4		6
3.2	odd	2		91.2.c.a.64.3	✓	6
12.11	even	2		1456.2.k.c.337.5		6
13.12	even	2	inner	819.2.c.b.64.3		6
21.2	odd	6		637.2.r.e.116.3		12
21.5	even	6		637.2.r.d.116.3		12
21.11	odd	6		637.2.r.e.324.4		12
21.17	even	6		637.2.r.d.324.4		12
21.20	even	2		637.2.c.d.246.3		6
39.5	even	4		1183.2.a.h.1.2		3
39.8	even	4		1183.2.a.j.1.2		3
39.38	odd	2		91.2.c.a.64.4	yes	6
156.155	even	2		1456.2.k.c.337.6		6
273.38	even	6		637.2.r.d.324.3		12
273.83	odd	4		8281.2.a.be.1.2		3
273.116	odd	6		637.2.r.e.324.3		12
273.125	odd	4		8281.2.a.bi.1.2		3
273.194	even	6		637.2.r.d.116.4		12
273.233	odd	6		637.2.r.e.116.4		12
273.272	even	2		637.2.c.d.246.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.c.a.64.3	✓	6	3.2	odd	2
91.2.c.a.64.4	yes	6	39.38	odd	2
637.2.c.d.246.3		6	21.20	even	2
637.2.c.d.246.4		6	273.272	even	2
637.2.r.d.116.3		12	21.5	even	6
637.2.r.d.116.4		12	273.194	even	6
637.2.r.d.324.3		12	273.38	even	6
637.2.r.d.324.4		12	21.17	even	6
637.2.r.e.116.3		12	21.2	odd	6
637.2.r.e.116.4		12	273.233	odd	6
637.2.r.e.324.3		12	273.116	odd	6
637.2.r.e.324.4		12	21.11	odd	6
819.2.c.b.64.3		6	13.12	even	2	inner
819.2.c.b.64.4		6	1.1	even	1	trivial
1183.2.a.h.1.2		3	39.5	even	4
1183.2.a.j.1.2		3	39.8	even	4
1456.2.k.c.337.5		6	12.11	even	2
1456.2.k.c.337.6		6	156.155	even	2
8281.2.a.be.1.2		3	273.83	odd	4
8281.2.a.bi.1.2		3	273.125	odd	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:	\( N \)	\(=\)	\( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	819.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.688892i q^{2} +1.52543 q^{4} +3.21432i q^{5} -1.00000i q^{7} +2.42864i q^{8} -2.21432 q^{10} -2.68889i q^{11} +(3.59210 - 0.311108i) q^{13} +0.688892 q^{14} +1.37778 q^{16} +3.59210 q^{17} +8.54617i q^{19} +4.90321i q^{20} +1.85236 q^{22} -3.28100 q^{23} -5.33185 q^{25} +(0.214320 + 2.47457i) q^{26} -1.52543i q^{28} -2.05086 q^{29} -5.83654i q^{31} +5.80642i q^{32} +2.47457i q^{34} +3.21432 q^{35} +3.93332i q^{37} -5.88739 q^{38} -7.80642 q^{40} +0.755569i q^{41} -8.80642 q^{43} -4.10171i q^{44} -2.26025i q^{46} -1.88247i q^{47} -1.00000 q^{49} -3.67307i q^{50} +(5.47949 - 0.474572i) q^{52} -2.52543 q^{53} +8.64296 q^{55} +2.42864 q^{56} -1.41282i q^{58} -7.33185i q^{59} +9.05086 q^{61} +4.02074 q^{62} -1.24443 q^{64} +(1.00000 + 11.5462i) q^{65} -0.428639i q^{67} +5.47949 q^{68} +2.21432i q^{70} +8.98418i q^{71} +5.79060i q^{73} -2.70964 q^{74} +13.0366i q^{76} -2.68889 q^{77} -4.47949 q^{79} +4.42864i q^{80} -0.520505 q^{82} -10.8272i q^{83} +11.5462i q^{85} -6.06668i q^{86} +6.53035 q^{88} -5.36196i q^{89} +(-0.311108 - 3.59210i) q^{91} -5.00492 q^{92} +1.29682 q^{94} -27.4701 q^{95} -9.62867i q^{97} -0.688892i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{4} + 8 q^{13} + 4 q^{14} + 8 q^{16} + 8 q^{17} + 24 q^{22} - 6 q^{23} + 8 q^{25} - 12 q^{26} + 14 q^{29} + 6 q^{35} + 4 q^{38} - 20 q^{40} - 26 q^{43} - 6 q^{49} - 20 q^{52} - 2 q^{53} + 12 q^{55}+ \cdots - 58 q^{95}+O(q^{100}) \) 6 * q - 4 * q^4 + 8 * q^13 + 4 * q^14 + 8 * q^16 + 8 * q^17 + 24 * q^22 - 6 * q^23 + 8 * q^25 - 12 * q^26 + 14 * q^29 + 6 * q^35 + 4 * q^38 - 20 * q^40 - 26 * q^43 - 6 * q^49 - 20 * q^52 - 2 * q^53 + 12 * q^55 - 12 * q^56 + 28 * q^61 - 16 * q^62 - 8 * q^64 + 6 * q^65 - 20 * q^68 + 24 * q^74 - 16 * q^77 + 26 * q^79 - 56 * q^82 - 40 * q^88 - 2 * q^91 + 36 * q^92 + 20 * q^94 - 58 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more