LMFDB - Embedded newform 190.2.b.a.39.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [190,2,Mod(39,190)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(190, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("190.39");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 190 = 2 \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	190.b (of order $2$, degree $1$, 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:	$1.51715763840$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			39.4
Root			$-0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	190.39
Dual form			190.2.b.a.39.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} +2.41421i q^{3} -1.00000 q^{4} +(-0.707107 + 2.12132i) q^{5} -2.41421 q^{6} -1.58579i q^{7} -1.00000i q^{8} -2.82843 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} +2.41421i q^{3} -1.00000 q^{4} +(-0.707107 + 2.12132i) q^{5} -2.41421 q^{6} -1.58579i q^{7} -1.00000i q^{8} -2.82843 q^{9} +(-2.12132 - 0.707107i) q^{10} +1.41421 q^{11} -2.41421i q^{12} +0.171573i q^{13} +1.58579 q^{14} +(-5.12132 - 1.70711i) q^{15} +1.00000 q^{16} -1.00000i q^{17} -2.82843i q^{18} +1.00000 q^{19} +(0.707107 - 2.12132i) q^{20} +3.82843 q^{21} +1.41421i q^{22} +9.24264i q^{23} +2.41421 q^{24} +(-4.00000 - 3.00000i) q^{25} -0.171573 q^{26} +0.414214i q^{27} +1.58579i q^{28} +5.82843 q^{29} +(1.70711 - 5.12132i) q^{30} -2.24264 q^{31} +1.00000i q^{32} +3.41421i q^{33} +1.00000 q^{34} +(3.36396 + 1.12132i) q^{35} +2.82843 q^{36} -8.48528i q^{37} +1.00000i q^{38} -0.414214 q^{39} +(2.12132 + 0.707107i) q^{40} +4.24264 q^{41} +3.82843i q^{42} +10.2426i q^{43} -1.41421 q^{44} +(2.00000 - 6.00000i) q^{45} -9.24264 q^{46} +2.41421i q^{48} +4.48528 q^{49} +(3.00000 - 4.00000i) q^{50} +2.41421 q^{51} -0.171573i q^{52} -11.4853i q^{53} -0.414214 q^{54} +(-1.00000 + 3.00000i) q^{55} -1.58579 q^{56} +2.41421i q^{57} +5.82843i q^{58} +12.8995 q^{59} +(5.12132 + 1.70711i) q^{60} +5.75736 q^{61} -2.24264i q^{62} +4.48528i q^{63} -1.00000 q^{64} +(-0.363961 - 0.121320i) q^{65} -3.41421 q^{66} -13.2426i q^{67} +1.00000i q^{68} -22.3137 q^{69} +(-1.12132 + 3.36396i) q^{70} -10.5858 q^{71} +2.82843i q^{72} -5.48528i q^{73} +8.48528 q^{74} +(7.24264 - 9.65685i) q^{75} -1.00000 q^{76} -2.24264i q^{77} -0.414214i q^{78} -10.4853 q^{79} +(-0.707107 + 2.12132i) q^{80} -9.48528 q^{81} +4.24264i q^{82} -2.48528i q^{83} -3.82843 q^{84} +(2.12132 + 0.707107i) q^{85} -10.2426 q^{86} +14.0711i q^{87} -1.41421i q^{88} +7.07107 q^{89} +(6.00000 + 2.00000i) q^{90} +0.272078 q^{91} -9.24264i q^{92} -5.41421i q^{93} +(-0.707107 + 2.12132i) q^{95} -2.41421 q^{96} +11.6569i q^{97} +4.48528i q^{98} -4.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} - 4 q^{6}+O(q^{10}) $ 4 * q - 4 * q^4 - 4 * q^6	$ 4 q - 4 q^{4} - 4 q^{6} + 12 q^{14} - 12 q^{15} + 4 q^{16} + 4 q^{19} + 4 q^{21} + 4 q^{24} - 16 q^{25} - 12 q^{26} + 12 q^{29} + 4 q^{30} + 8 q^{31} + 4 q^{34} - 12 q^{35} + 4 q^{39} + 8 q^{45} - 20 q^{46} - 16 q^{49} + 12 q^{50} + 4 q^{51} + 4 q^{54} - 4 q^{55} - 12 q^{56} + 12 q^{59} + 12 q^{60} + 40 q^{61} - 4 q^{64} + 24 q^{65} - 8 q^{66} - 44 q^{69} + 4 q^{70} - 48 q^{71} + 12 q^{75} - 4 q^{76} - 8 q^{79} - 4 q^{81} - 4 q^{84} - 24 q^{86} + 24 q^{90} + 52 q^{91} - 4 q^{96} - 16 q^{99}+O(q^{100}) $ 4 * q - 4 * q^4 - 4 * q^6 + 12 * q^14 - 12 * q^15 + 4 * q^16 + 4 * q^19 + 4 * q^21 + 4 * q^24 - 16 * q^25 - 12 * q^26 + 12 * q^29 + 4 * q^30 + 8 * q^31 + 4 * q^34 - 12 * q^35 + 4 * q^39 + 8 * q^45 - 20 * q^46 - 16 * q^49 + 12 * q^50 + 4 * q^51 + 4 * q^54 - 4 * q^55 - 12 * q^56 + 12 * q^59 + 12 * q^60 + 40 * q^61 - 4 * q^64 + 24 * q^65 - 8 * q^66 - 44 * q^69 + 4 * q^70 - 48 * q^71 + 12 * q^75 - 4 * q^76 - 8 * q^79 - 4 * q^81 - 4 * q^84 - 24 * q^86 + 24 * q^90 + 52 * q^91 - 4 * q^96 - 16 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/190\mathbb{Z}\right)^\times$.

$n$	$21$	$77$
$\chi(n)$	$1$	$-1$

Coefficient data

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

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

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

$2$			1.00000i			0.707107i
$2$			1.00000i			0.707107i
$3$			2.41421i			1.39385i	0.717146	+	0.696923i	$0.245448\pi$
$3$			2.41421i			1.39385i	−0.717146	+	0.696923i	$0.754552\pi$
$4$	−1.00000			−0.500000
$5$	−0.707107	+	2.12132i	−0.316228	+	0.948683i
$5$	−0.707107	+	2.12132i	−0.316228	+	0.948683i
$6$	−2.41421			−0.985599
$7$		−	1.58579i		−	0.599371i	−0.954038	−	0.299685i	$-0.903118\pi$
$7$		−	1.58579i		−	0.599371i	0.954038	−	0.299685i	$-0.0968817\pi$
$8$		−	1.00000i		−	0.353553i
$9$	−2.82843			−0.942809
$10$	−2.12132	−	0.707107i	−0.670820	−	0.223607i
$11$	1.41421			0.426401			0.213201	−	0.977008i	$-0.431611\pi$
$11$	1.41421			0.426401			0.213201	+	0.977008i	$0.431611\pi$
$12$		−	2.41421i		−	0.696923i
$13$			0.171573i			0.0475858i	0.999717	+	0.0237929i	$0.00757422\pi$
$13$			0.171573i			0.0475858i	−0.999717	+	0.0237929i	$0.992426\pi$
$14$	1.58579			0.423819
$15$	−5.12132	−	1.70711i	−1.32232	−	0.440773i
$16$	1.00000			0.250000
$17$		−	1.00000i		−	0.242536i	−0.992620	−	0.121268i	$-0.961304\pi$
$17$		−	1.00000i		−	0.242536i	0.992620	−	0.121268i	$-0.0386960\pi$
$18$		−	2.82843i		−	0.666667i
$19$	1.00000			0.229416
$19$	1.00000			0.229416
$20$	0.707107	−	2.12132i	0.158114	−	0.474342i
$21$	3.82843			0.835431
$22$			1.41421i			0.301511i
$23$			9.24264i			1.92722i	0.267305	+	0.963612i	$0.413867\pi$
$23$			9.24264i			1.92722i	−0.267305	+	0.963612i	$0.586133\pi$
$24$	2.41421			0.492799
$25$	−4.00000	−	3.00000i	−0.800000	−	0.600000i
$26$	−0.171573			−0.0336482
$27$			0.414214i			0.0797154i
$28$			1.58579i			0.299685i
$29$	5.82843			1.08231			0.541156	−	0.840922i	$-0.317987\pi$
$29$	5.82843			1.08231			0.541156	+	0.840922i	$0.317987\pi$
$30$	1.70711	−	5.12132i	0.311674	−	0.935021i
$31$	−2.24264			−0.402790			−0.201395	−	0.979510i	$-0.564548\pi$
$31$	−2.24264			−0.402790			−0.201395	+	0.979510i	$0.564548\pi$
$32$			1.00000i			0.176777i
$33$			3.41421i			0.594338i
$34$	1.00000			0.171499
$35$	3.36396	+	1.12132i	0.568613	+	0.189538i
$36$	2.82843			0.471405
$37$		−	8.48528i		−	1.39497i	−0.716599	−	0.697486i	$-0.754302\pi$
$37$		−	8.48528i		−	1.39497i	0.716599	−	0.697486i	$-0.245698\pi$
$38$			1.00000i			0.162221i
$39$	−0.414214			−0.0663273
$40$	2.12132	+	0.707107i	0.335410	+	0.111803i
$41$	4.24264			0.662589			0.331295	−	0.943527i	$-0.392515\pi$
$41$	4.24264			0.662589			0.331295	+	0.943527i	$0.392515\pi$
$42$			3.82843i			0.590739i
$43$			10.2426i			1.56199i	0.624538	+	0.780994i	$0.285287\pi$
$43$			10.2426i			1.56199i	−0.624538	+	0.780994i	$0.714713\pi$
$44$	−1.41421			−0.213201
$45$	2.00000	−	6.00000i	0.298142	−	0.894427i
$46$	−9.24264			−1.36275
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$			2.41421i			0.348462i
$49$	4.48528			0.640754
$50$	3.00000	−	4.00000i	0.424264	−	0.565685i
$51$	2.41421			0.338058
$52$		−	0.171573i		−	0.0237929i
$53$		−	11.4853i		−	1.57762i	−0.614634	−	0.788812i	$-0.710696\pi$
$53$		−	11.4853i		−	1.57762i	0.614634	−	0.788812i	$-0.289304\pi$
$54$	−0.414214			−0.0563673
$55$	−1.00000	+	3.00000i	−0.134840	+	0.404520i
$56$	−1.58579			−0.211910
$57$			2.41421i			0.319770i
$58$			5.82843i			0.765310i
$59$	12.8995			1.67937			0.839686	−	0.543073i	$-0.182739\pi$
$59$	12.8995			1.67937			0.839686	+	0.543073i	$0.182739\pi$
$60$	5.12132	+	1.70711i	0.661160	+	0.220387i
$61$	5.75736			0.737154			0.368577	−	0.929597i	$-0.379845\pi$
$61$	5.75736			0.737154			0.368577	+	0.929597i	$0.379845\pi$
$62$		−	2.24264i		−	0.284816i
$63$			4.48528i			0.565092i
$64$	−1.00000			−0.125000
$65$	−0.363961	−	0.121320i	−0.0451438	−	0.0150479i
$66$	−3.41421			−0.420261
$67$		−	13.2426i		−	1.61785i	−0.587915	−	0.808923i	$-0.700051\pi$
$67$		−	13.2426i		−	1.61785i	0.587915	−	0.808923i	$-0.299949\pi$
$68$			1.00000i			0.121268i
$69$	−22.3137			−2.68625
$70$	−1.12132	+	3.36396i	−0.134023	+	0.402070i
$71$	−10.5858			−1.25630			−0.628151	−	0.778092i	$-0.716188\pi$
$71$	−10.5858			−1.25630			−0.628151	+	0.778092i	$0.716188\pi$
$72$			2.82843i			0.333333i
$73$		−	5.48528i		−	0.642004i	−0.947079	−	0.321002i	$-0.895980\pi$
$73$		−	5.48528i		−	0.642004i	0.947079	−	0.321002i	$-0.104020\pi$
$74$	8.48528			0.986394
$75$	7.24264	−	9.65685i	0.836308	−	1.11508i
$76$	−1.00000			−0.114708
$77$		−	2.24264i		−	0.255573i
$78$		−	0.414214i		−	0.0469005i
$79$	−10.4853			−1.17969			−0.589843	−	0.807518i	$-0.700810\pi$
$79$	−10.4853			−1.17969			−0.589843	+	0.807518i	$0.700810\pi$
$80$	−0.707107	+	2.12132i	−0.0790569	+	0.237171i
$81$	−9.48528			−1.05392
$82$			4.24264i			0.468521i
$83$		−	2.48528i		−	0.272795i	−0.990654	−	0.136398i	$-0.956448\pi$
$83$		−	2.48528i		−	0.272795i	0.990654	−	0.136398i	$-0.0435524\pi$
$84$	−3.82843			−0.417716
$85$	2.12132	+	0.707107i	0.230089	+	0.0766965i
$86$	−10.2426			−1.10449
$87$			14.0711i			1.50858i
$88$		−	1.41421i		−	0.150756i
$89$	7.07107			0.749532			0.374766	−	0.927119i	$-0.377723\pi$
$89$	7.07107			0.749532			0.374766	+	0.927119i	$0.377723\pi$
$90$	6.00000	+	2.00000i	0.632456	+	0.210819i
$91$	0.272078			0.0285215
$92$		−	9.24264i		−	0.963612i
$93$		−	5.41421i		−	0.561428i
$94$		0			0
$95$	−0.707107	+	2.12132i	−0.0725476	+	0.217643i
$96$	−2.41421			−0.246400
$97$			11.6569i			1.18357i	0.806094	+	0.591787i	$0.201577\pi$
$97$			11.6569i			1.18357i	−0.806094	+	0.591787i	$0.798423\pi$
$98$			4.48528i			0.453082i
$99$	−4.00000			−0.402015
$100$	4.00000	+	3.00000i	0.400000	+	0.300000i
$101$	−1.07107			−0.106575			−0.0532876	−	0.998579i	$-0.516970\pi$
$101$	−1.07107			−0.106575			−0.0532876	+	0.998579i	$0.516970\pi$
$102$			2.41421i			0.239043i
$103$		−	4.24264i		−	0.418040i	−0.977911	−	0.209020i	$-0.932973\pi$
$103$		−	4.24264i		−	0.418040i	0.977911	−	0.209020i	$-0.0670273\pi$
$104$	0.171573			0.0168241
$105$	−2.70711	+	8.12132i	−0.264187	+	0.792560i
$106$	11.4853			1.11555
$107$			5.72792i			0.553739i	0.960908	+	0.276870i	$0.0892970\pi$
$107$			5.72792i			0.553739i	−0.960908	+	0.276870i	$0.910703\pi$
$108$		−	0.414214i		−	0.0398577i
$109$	−15.9706			−1.52970			−0.764851	−	0.644207i	$-0.777188\pi$
$109$	−15.9706			−1.52970			−0.764851	+	0.644207i	$0.777188\pi$
$110$	−3.00000	−	1.00000i	−0.286039	−	0.0953463i
$111$	20.4853			1.94438
$112$		−	1.58579i		−	0.149843i
$113$		−	1.75736i		−	0.165318i	−0.996578	−	0.0826592i	$-0.973659\pi$
$113$		−	1.75736i		−	0.165318i	0.996578	−	0.0826592i	$-0.0263413\pi$
$114$	−2.41421			−0.226112
$115$	−19.6066	−	6.53553i	−1.82833	−	0.609442i
$116$	−5.82843			−0.541156
$117$		−	0.485281i		−	0.0448643i
$118$			12.8995i			1.18749i
$119$	−1.58579			−0.145369
$120$	−1.70711	+	5.12132i	−0.155837	+	0.467510i
$121$	−9.00000			−0.818182
$122$			5.75736i			0.521247i
$123$			10.2426i			0.923548i
$124$	2.24264			0.201395
$125$	9.19239	−	6.36396i	0.822192	−	0.569210i
$126$	−4.48528			−0.399581
$127$		−	14.4853i		−	1.28536i	−0.766134	−	0.642680i	$-0.777822\pi$
$127$		−	14.4853i		−	1.28536i	0.766134	−	0.642680i	$-0.222178\pi$
$128$		−	1.00000i		−	0.0883883i
$129$	−24.7279			−2.17717
$130$	0.121320	−	0.363961i	0.0106405	−	0.0319215i
$131$	16.9706			1.48272			0.741362	−	0.671105i	$-0.234180\pi$
$131$	16.9706			1.48272			0.741362	+	0.671105i	$0.234180\pi$
$132$		−	3.41421i		−	0.297169i
$133$		−	1.58579i		−	0.137505i
$134$	13.2426			1.14399
$135$	−0.878680	−	0.292893i	−0.0756247	−	0.0252082i
$136$	−1.00000			−0.0857493
$137$		−	13.0000i		−	1.11066i	−0.831628	−	0.555332i	$-0.812591\pi$
$137$		−	13.0000i		−	1.11066i	0.831628	−	0.555332i	$-0.187409\pi$
$138$		−	22.3137i		−	1.89947i
$139$	12.0000			1.01783			0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000			1.01783			0.508913	+	0.860818i	$0.330047\pi$
$140$	−3.36396	−	1.12132i	−0.284307	−	0.0947689i
$141$		0			0
$142$		−	10.5858i		−	0.888339i
$143$			0.242641i			0.0202906i
$144$	−2.82843			−0.235702
$145$	−4.12132	+	12.3640i	−0.342257	+	1.02677i
$146$	5.48528			0.453965
$147$			10.8284i			0.893114i
$148$			8.48528i			0.697486i
$149$	−6.34315			−0.519651			−0.259825	−	0.965656i	$-0.583665\pi$
$149$	−6.34315			−0.519651			−0.259825	+	0.965656i	$0.583665\pi$
$150$	9.65685	+	7.24264i	0.788479	+	0.591359i
$151$	6.48528			0.527765			0.263882	−	0.964555i	$-0.414997\pi$
$151$	6.48528			0.527765			0.263882	+	0.964555i	$0.414997\pi$
$152$		−	1.00000i		−	0.0811107i
$153$			2.82843i			0.228665i
$154$	2.24264			0.180717
$155$	1.58579	−	4.75736i	0.127373	−	0.382120i
$156$	0.414214			0.0331636
$157$		−	0.343146i		−	0.0273860i	−0.999906	−	0.0136930i	$-0.995641\pi$
$157$		−	0.343146i		−	0.0273860i	0.999906	−	0.0136930i	$-0.00435876\pi$
$158$		−	10.4853i		−	0.834164i
$159$	27.7279			2.19897
$160$	−2.12132	−	0.707107i	−0.167705	−	0.0559017i
$161$	14.6569			1.15512
$162$		−	9.48528i		−	0.745234i
$163$		−	1.75736i		−	0.137647i	−0.997629	−	0.0688235i	$-0.978075\pi$
$163$		−	1.75736i		−	0.137647i	0.997629	−	0.0688235i	$-0.0219245\pi$
$164$	−4.24264			−0.331295
$165$	−7.24264	−	2.41421i	−0.563839	−	0.187946i
$166$	2.48528			0.192895
$167$			9.75736i			0.755047i	0.926000	+	0.377524i	$0.123224\pi$
$167$			9.75736i			0.755047i	−0.926000	+	0.377524i	$0.876776\pi$
$168$		−	3.82843i		−	0.295370i
$169$	12.9706			0.997736
$170$	−0.707107	+	2.12132i	−0.0542326	+	0.162698i
$171$	−2.82843			−0.216295
$172$		−	10.2426i		−	0.780994i
$173$		−	16.4853i		−	1.25335i	−0.779280	−	0.626676i	$-0.784415\pi$
$173$		−	16.4853i		−	1.25335i	0.779280	−	0.626676i	$-0.215585\pi$
$174$	−14.0711			−1.06672
$175$	−4.75736	+	6.34315i	−0.359623	+	0.479497i
$176$	1.41421			0.106600
$177$			31.1421i			2.34079i
$178$			7.07107i			0.529999i
$179$	0.343146			0.0256479			0.0128240	−	0.999918i	$-0.495918\pi$
$179$	0.343146			0.0256479			0.0128240	+	0.999918i	$0.495918\pi$
$180$	−2.00000	+	6.00000i	−0.149071	+	0.447214i
$181$	8.48528			0.630706			0.315353	−	0.948974i	$-0.397877\pi$
$181$	8.48528			0.630706			0.315353	+	0.948974i	$0.397877\pi$
$182$			0.272078i			0.0201678i
$183$			13.8995i			1.02748i
$184$	9.24264			0.681377
$185$	18.0000	+	6.00000i	1.32339	+	0.441129i
$186$	5.41421			0.396989
$187$		−	1.41421i		−	0.103418i
$188$		0			0
$189$	0.656854			0.0477791
$190$	−2.12132	−	0.707107i	−0.153897	−	0.0512989i
$191$	−18.5563			−1.34269			−0.671345	−	0.741145i	$-0.734283\pi$
$191$	−18.5563			−1.34269			−0.671345	+	0.741145i	$0.734283\pi$
$192$		−	2.41421i		−	0.174231i
$193$		−	11.6569i		−	0.839079i	−0.907737	−	0.419539i	$-0.862192\pi$
$193$		−	11.6569i		−	0.839079i	0.907737	−	0.419539i	$-0.137808\pi$
$194$	−11.6569			−0.836913
$195$	0.292893	−	0.878680i	0.0209745	−	0.0629236i
$196$	−4.48528			−0.320377
$197$			20.2426i			1.44223i	0.692816	+	0.721114i	$0.256370\pi$
$197$			20.2426i			1.44223i	−0.692816	+	0.721114i	$0.743630\pi$
$198$		−	4.00000i		−	0.284268i
$199$	−0.757359			−0.0536878			−0.0268439	−	0.999640i	$-0.508546\pi$
$199$	−0.757359			−0.0536878			−0.0268439	+	0.999640i	$0.508546\pi$
$200$	−3.00000	+	4.00000i	−0.212132	+	0.282843i
$201$	31.9706			2.25503
$202$		−	1.07107i		−	0.0753601i
$203$		−	9.24264i		−	0.648706i
$204$	−2.41421			−0.169029
$205$	−3.00000	+	9.00000i	−0.209529	+	0.628587i
$206$	4.24264			0.295599
$207$		−	26.1421i		−	1.81700i
$208$			0.171573i			0.0118964i
$209$	1.41421			0.0978232
$210$	−8.12132	−	2.70711i	−0.560424	−	0.186808i
$211$	5.72792			0.394326			0.197163	−	0.980371i	$-0.436827\pi$
$211$	5.72792			0.394326			0.197163	+	0.980371i	$0.436827\pi$
$212$			11.4853i			0.788812i
$213$		−	25.5563i		−	1.75109i
$214$	−5.72792			−0.391553
$215$	−21.7279	−	7.24264i	−1.48183	−	0.493944i
$216$	0.414214			0.0281837
$217$			3.55635i			0.241421i
$218$		−	15.9706i		−	1.08166i
$219$	13.2426			0.894855
$220$	1.00000	−	3.00000i	0.0674200	−	0.202260i
$221$	0.171573			0.0115412
$222$			20.4853i			1.37488i
$223$		−	20.8284i		−	1.39477i	−0.716694	−	0.697387i	$-0.754346\pi$
$223$		−	20.8284i		−	1.39477i	0.716694	−	0.697387i	$-0.245654\pi$
$224$	1.58579			0.105955
$225$	11.3137	+	8.48528i	0.754247	+	0.565685i
$226$	1.75736			0.116898
$227$			25.2426i			1.67541i	0.546121	+	0.837706i	$0.316104\pi$
$227$			25.2426i			1.67541i	−0.546121	+	0.837706i	$0.683896\pi$
$228$		−	2.41421i		−	0.159885i
$229$	18.9706			1.25361			0.626805	−	0.779176i	$-0.284362\pi$
$229$	18.9706			1.25361			0.626805	+	0.779176i	$0.284362\pi$
$230$	6.53553	−	19.6066i	0.430940	−	1.29282i
$231$	5.41421			0.356229
$232$		−	5.82843i		−	0.382655i
$233$			8.97056i			0.587681i	0.955854	+	0.293841i	$0.0949335\pi$
$233$			8.97056i			0.587681i	−0.955854	+	0.293841i	$0.905067\pi$
$234$	0.485281			0.0317238
$235$		0			0
$236$	−12.8995			−0.839686
$237$		−	25.3137i		−	1.64430i
$238$		−	1.58579i		−	0.102791i
$239$	−12.8995			−0.834399			−0.417199	−	0.908815i	$-0.636988\pi$
$239$	−12.8995			−0.834399			−0.417199	+	0.908815i	$0.636988\pi$
$240$	−5.12132	−	1.70711i	−0.330580	−	0.110193i
$241$	−24.9706			−1.60850			−0.804248	−	0.594294i	$-0.797431\pi$
$241$	−24.9706			−1.60850			−0.804248	+	0.594294i	$0.797431\pi$
$242$		−	9.00000i		−	0.578542i
$243$		−	21.6569i		−	1.38929i
$244$	−5.75736			−0.368577
$245$	−3.17157	+	9.51472i	−0.202624	+	0.607873i
$246$	−10.2426			−0.653047
$247$			0.171573i			0.0109169i
$248$			2.24264i			0.142408i
$249$	6.00000			0.380235
$250$	6.36396	+	9.19239i	0.402492	+	0.581378i
$251$	−27.5563			−1.73934			−0.869671	−	0.493632i	$-0.835669\pi$
$251$	−27.5563			−1.73934			−0.869671	+	0.493632i	$0.835669\pi$
$252$		−	4.48528i		−	0.282546i
$253$			13.0711i			0.821771i
$254$	14.4853			0.908887
$255$	−1.70711	+	5.12132i	−0.106903	+	0.320710i
$256$	1.00000			0.0625000
$257$			4.72792i			0.294920i	0.989068	+	0.147460i	$0.0471097\pi$
$257$			4.72792i			0.294920i	−0.989068	+	0.147460i	$0.952890\pi$
$258$		−	24.7279i		−	1.53949i
$259$	−13.4558			−0.836105
$260$	0.363961	+	0.121320i	0.0225719	+	0.00752397i
$261$	−16.4853			−1.02041
$262$			16.9706i			1.04844i
$263$			6.97056i			0.429823i	0.976633	+	0.214912i	$0.0689464\pi$
$263$			6.97056i			0.429823i	−0.976633	+	0.214912i	$0.931054\pi$
$264$	3.41421			0.210130
$265$	24.3640	+	8.12132i	1.49667	+	0.498889i
$266$	1.58579			0.0972308
$267$			17.0711i			1.04473i
$268$			13.2426i			0.808923i
$269$	−28.6274			−1.74544			−0.872722	−	0.488217i	$-0.837647\pi$
$269$	−28.6274			−1.74544			−0.872722	+	0.488217i	$0.837647\pi$
$270$	0.292893	−	0.878680i	0.0178249	−	0.0534747i
$271$	−18.7574			−1.13943			−0.569714	−	0.821843i	$-0.692946\pi$
$271$	−18.7574			−1.13943			−0.569714	+	0.821843i	$0.692946\pi$
$272$		−	1.00000i		−	0.0606339i
$273$			0.656854i			0.0397546i
$274$	13.0000			0.785359
$275$	−5.65685	−	4.24264i	−0.341121	−	0.255841i
$276$	22.3137			1.34313
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$			12.0000i			0.719712i
$279$	6.34315			0.379754
$280$	1.12132	−	3.36396i	0.0670117	−	0.201035i
$281$	−4.24264			−0.253095			−0.126547	−	0.991961i	$-0.540390\pi$
$281$	−4.24264			−0.253095			−0.126547	+	0.991961i	$0.540390\pi$
$282$		0			0
$283$			3.85786i			0.229326i	0.993404	+	0.114663i	$0.0365789\pi$
$283$			3.85786i			0.229326i	−0.993404	+	0.114663i	$0.963421\pi$
$284$	10.5858			0.628151
$285$	−5.12132	−	1.70711i	−0.303361	−	0.101120i
$286$	−0.242641			−0.0143476
$287$		−	6.72792i		−	0.397137i
$288$		−	2.82843i		−	0.166667i
$289$	16.0000			0.941176
$290$	−12.3640	−	4.12132i	−0.726037	−	0.242012i
$291$	−28.1421			−1.64972
$292$			5.48528i			0.321002i
$293$			11.4853i			0.670977i	0.942044	+	0.335489i	$0.108901\pi$
$293$			11.4853i			0.670977i	−0.942044	+	0.335489i	$0.891099\pi$
$294$	−10.8284			−0.631527
$295$	−9.12132	+	27.3640i	−0.531064	+	1.59319i
$296$	−8.48528			−0.493197
$297$			0.585786i			0.0339908i
$298$		−	6.34315i		−	0.367449i
$299$	−1.58579			−0.0917084
$300$	−7.24264	+	9.65685i	−0.418154	+	0.557539i
$301$	16.2426			0.936210
$302$			6.48528i			0.373186i
$303$		−	2.58579i		−	0.148550i
$304$	1.00000			0.0573539
$305$	−4.07107	+	12.2132i	−0.233109	+	0.699326i
$306$	−2.82843			−0.161690
$307$			6.34315i			0.362022i	0.983481	+	0.181011i	$0.0579370\pi$
$307$			6.34315i			0.362022i	−0.983481	+	0.181011i	$0.942063\pi$
$308$			2.24264i			0.127786i
$309$	10.2426			0.582683
$310$	4.75736	+	1.58579i	0.270200	+	0.0900666i
$311$	−13.2426			−0.750921			−0.375461	−	0.926838i	$-0.622515\pi$
$311$	−13.2426			−0.750921			−0.375461	+	0.926838i	$0.622515\pi$
$312$			0.414214i			0.0234502i
$313$			25.9706i			1.46794i	0.679180	+	0.733971i	$0.262335\pi$
$313$			25.9706i			1.46794i	−0.679180	+	0.733971i	$0.737665\pi$
$314$	0.343146			0.0193648
$315$	−9.51472	−	3.17157i	−0.536094	−	0.178698i
$316$	10.4853			0.589843
$317$		−	7.48528i		−	0.420415i	−0.977657	−	0.210208i	$-0.932586\pi$
$317$		−	7.48528i		−	0.420415i	0.977657	−	0.210208i	$-0.0674140\pi$
$318$			27.7279i			1.55490i
$319$	8.24264			0.461499
$320$	0.707107	−	2.12132i	0.0395285	−	0.118585i
$321$	−13.8284			−0.771828
$322$			14.6569i			0.816795i
$323$		−	1.00000i		−	0.0556415i
$324$	9.48528			0.526960
$325$	0.514719	−	0.686292i	0.0285515	−	0.0380686i
$326$	1.75736			0.0973311
$327$		−	38.5563i		−	2.13217i
$328$		−	4.24264i		−	0.234261i
$329$		0			0
$330$	2.41421	−	7.24264i	0.132898	−	0.398694i
$331$	−10.7574			−0.591278			−0.295639	−	0.955300i	$-0.595533\pi$
$331$	−10.7574			−0.591278			−0.295639	+	0.955300i	$0.595533\pi$
$332$			2.48528i			0.136398i
$333$			24.0000i			1.31519i
$334$	−9.75736			−0.533899
$335$	28.0919	+	9.36396i	1.53482	+	0.511608i
$336$	3.82843			0.208858
$337$		−	33.8995i		−	1.84662i	−0.384052	−	0.923312i	$-0.625472\pi$
$337$		−	33.8995i		−	1.84662i	0.384052	−	0.923312i	$-0.374528\pi$
$338$			12.9706i			0.705506i
$339$	4.24264			0.230429
$340$	−2.12132	−	0.707107i	−0.115045	−	0.0383482i
$341$	−3.17157			−0.171750
$342$		−	2.82843i		−	0.152944i
$343$		−	18.2132i		−	0.983421i
$344$	10.2426			0.552246
$345$	15.7782	−	47.3345i	0.849468	−	2.54841i
$346$	16.4853			0.886254
$347$			22.4853i			1.20707i	0.797335	+	0.603537i	$0.206242\pi$
$347$			22.4853i			1.20707i	−0.797335	+	0.603537i	$0.793758\pi$
$348$		−	14.0711i		−	0.754288i
$349$	−6.00000			−0.321173			−0.160586	−	0.987022i	$-0.551338\pi$
$349$	−6.00000			−0.321173			−0.160586	+	0.987022i	$0.551338\pi$
$350$	−6.34315	−	4.75736i	−0.339055	−	0.254292i
$351$	−0.0710678			−0.00379332
$352$			1.41421i			0.0753778i
$353$		−	19.4853i		−	1.03710i	−0.855048	−	0.518548i	$-0.826473\pi$
$353$		−	19.4853i		−	1.03710i	0.855048	−	0.518548i	$-0.173527\pi$
$354$	−31.1421			−1.65519
$355$	7.48528	−	22.4558i	0.397277	−	1.19183i
$356$	−7.07107			−0.374766
$357$		−	3.82843i		−	0.202622i
$358$			0.343146i			0.0181358i
$359$	31.2426			1.64892			0.824462	−	0.565918i	$-0.191478\pi$
$359$	31.2426			1.64892			0.824462	+	0.565918i	$0.191478\pi$
$360$	−6.00000	−	2.00000i	−0.316228	−	0.105409i
$361$	1.00000			0.0526316
$362$			8.48528i			0.445976i
$363$		−	21.7279i		−	1.14042i
$364$	−0.272078			−0.0142608
$365$	11.6360	+	3.87868i	0.609058	+	0.203019i
$366$	−13.8995			−0.726538
$367$		−	25.4558i		−	1.32878i	−0.747384	−	0.664392i	$-0.768691\pi$
$367$		−	25.4558i		−	1.32878i	0.747384	−	0.664392i	$-0.231309\pi$
$368$			9.24264i			0.481806i
$369$	−12.0000			−0.624695
$370$	−6.00000	+	18.0000i	−0.311925	+	0.935775i
$371$	−18.2132			−0.945582
$372$			5.41421i			0.280714i
$373$			9.00000i			0.466002i	0.972476	+	0.233001i	$0.0748546\pi$
$373$			9.00000i			0.466002i	−0.972476	+	0.233001i	$0.925145\pi$
$374$	1.41421			0.0731272
$375$	15.3640	+	22.1924i	0.793392	+	1.14601i
$376$		0			0
$377$			1.00000i			0.0515026i
$378$			0.656854i			0.0337849i
$379$	−2.75736			−0.141636			−0.0708180	−	0.997489i	$-0.522561\pi$
$379$	−2.75736			−0.141636			−0.0708180	+	0.997489i	$0.522561\pi$
$380$	0.707107	−	2.12132i	0.0362738	−	0.108821i
$381$	34.9706			1.79160
$382$		−	18.5563i		−	0.949425i
$383$			3.75736i			0.191992i	0.995382	+	0.0959960i	$0.0306036\pi$
$383$			3.75736i			0.191992i	−0.995382	+	0.0959960i	$0.969396\pi$
$384$	2.41421			0.123200
$385$	4.75736	+	1.58579i	0.242457	+	0.0808192i
$386$	11.6569			0.593318
$387$		−	28.9706i		−	1.47266i
$388$		−	11.6569i		−	0.591787i
$389$	−37.0711			−1.87958			−0.939789	−	0.341756i	$-0.888979\pi$
$389$	−37.0711			−1.87958			−0.939789	+	0.341756i	$0.888979\pi$
$390$	0.878680	+	0.292893i	0.0444937	+	0.0148312i
$391$	9.24264			0.467420
$392$		−	4.48528i		−	0.226541i
$393$			40.9706i			2.06669i
$394$	−20.2426			−1.01981
$395$	7.41421	−	22.2426i	0.373050	−	1.11915i
$396$	4.00000			0.201008
$397$		−	24.0000i		−	1.20453i	−0.798298	−	0.602263i	$-0.794266\pi$
$397$		−	24.0000i		−	1.20453i	0.798298	−	0.602263i	$-0.205734\pi$
$398$		−	0.757359i		−	0.0379630i
$399$	3.82843			0.191661
$400$	−4.00000	−	3.00000i	−0.200000	−	0.150000i
$401$	−22.5858			−1.12788			−0.563940	−	0.825816i	$-0.690715\pi$
$401$	−22.5858			−1.12788			−0.563940	+	0.825816i	$0.690715\pi$
$402$			31.9706i			1.59455i
$403$		−	0.384776i		−	0.0191671i
$404$	1.07107			0.0532876
$405$	6.70711	−	20.1213i	0.333279	−	0.999836i
$406$	9.24264			0.458705
$407$		−	12.0000i		−	0.594818i
$408$		−	2.41421i		−	0.119521i
$409$	17.2132			0.851138			0.425569	−	0.904926i	$-0.360074\pi$
$409$	17.2132			0.851138			0.425569	+	0.904926i	$0.360074\pi$
$410$	−9.00000	−	3.00000i	−0.444478	−	0.148159i
$411$	31.3848			1.54810
$412$			4.24264i			0.209020i
$413$		−	20.4558i		−	1.00657i
$414$	26.1421			1.28482
$415$	5.27208	+	1.75736i	0.258796	+	0.0862654i
$416$	−0.171573			−0.00841205
$417$			28.9706i			1.41869i
$418$			1.41421i			0.0691714i
$419$	−16.5858			−0.810269			−0.405134	−	0.914257i	$-0.632775\pi$
$419$	−16.5858			−0.810269			−0.405134	+	0.914257i	$0.632775\pi$
$420$	2.70711	−	8.12132i	0.132093	−	0.396280i
$421$	2.51472			0.122560			0.0612799	−	0.998121i	$-0.480482\pi$
$421$	2.51472			0.122560			0.0612799	+	0.998121i	$0.480482\pi$
$422$			5.72792i			0.278831i
$423$		0			0
$424$	−11.4853			−0.557775
$425$	−3.00000	+	4.00000i	−0.145521	+	0.194029i
$426$	25.5563			1.23821
$427$		−	9.12994i		−	0.441829i
$428$		−	5.72792i		−	0.276870i
$429$	−0.585786			−0.0282820
$430$	7.24264	−	21.7279i	0.349271	−	1.04781i
$431$	30.3848			1.46358			0.731792	−	0.681528i	$-0.238684\pi$
$431$	30.3848			1.46358			0.731792	+	0.681528i	$0.238684\pi$
$432$			0.414214i			0.0199289i
$433$			21.5563i			1.03593i	0.855401	+	0.517966i	$0.173311\pi$
$433$			21.5563i			1.03593i	−0.855401	+	0.517966i	$0.826689\pi$
$434$	−3.55635			−0.170710
$435$	−29.8492	−	9.94975i	−1.43116	−	0.477054i
$436$	15.9706			0.764851
$437$			9.24264i			0.442135i
$438$			13.2426i			0.632758i
$439$	14.2426			0.679764			0.339882	−	0.940468i	$-0.389613\pi$
$439$	14.2426			0.679764			0.339882	+	0.940468i	$0.389613\pi$
$440$	3.00000	+	1.00000i	0.143019	+	0.0476731i
$441$	−12.6863			−0.604109
$442$			0.171573i			0.00816089i
$443$		−	4.24264i		−	0.201574i	−0.994908	−	0.100787i	$-0.967864\pi$
$443$		−	4.24264i		−	0.201574i	0.994908	−	0.100787i	$-0.0321361\pi$
$444$	−20.4853			−0.972188
$445$	−5.00000	+	15.0000i	−0.237023	+	0.711068i
$446$	20.8284			0.986255
$447$		−	15.3137i		−	0.724314i
$448$			1.58579i			0.0749214i
$449$	5.31371			0.250769			0.125385	−	0.992108i	$-0.459983\pi$
$449$	5.31371			0.250769			0.125385	+	0.992108i	$0.459983\pi$
$450$	−8.48528	+	11.3137i	−0.400000	+	0.533333i
$451$	6.00000			0.282529
$452$			1.75736i			0.0826592i
$453$			15.6569i			0.735623i
$454$	−25.2426			−1.18470
$455$	−0.192388	+	0.577164i	−0.00901930	+	0.0270579i
$456$	2.41421			0.113056
$457$			3.00000i			0.140334i	0.997535	+	0.0701670i	$0.0223532\pi$
$457$			3.00000i			0.140334i	−0.997535	+	0.0701670i	$0.977647\pi$
$458$			18.9706i			0.886436i
$459$	0.414214			0.0193338
$460$	19.6066	+	6.53553i	0.914163	+	0.304721i
$461$	27.5563			1.28343			0.641714	−	0.766944i	$-0.278224\pi$
$461$	27.5563			1.28343			0.641714	+	0.766944i	$0.278224\pi$
$462$			5.41421i			0.251892i
$463$		−	14.1421i		−	0.657241i	−0.944462	−	0.328620i	$-0.893416\pi$
$463$		−	14.1421i		−	0.657241i	0.944462	−	0.328620i	$-0.106584\pi$
$464$	5.82843			0.270578
$465$	11.4853	+	3.82843i	0.532617	+	0.177539i
$466$	−8.97056			−0.415553
$467$			24.7279i			1.14427i	0.820159	+	0.572136i	$0.193885\pi$
$467$			24.7279i			1.14427i	−0.820159	+	0.572136i	$0.806115\pi$
$468$			0.485281i			0.0224321i
$469$	−21.0000			−0.969690
$470$		0			0
$471$	0.828427			0.0381719
$472$		−	12.8995i		−	0.593747i
$473$			14.4853i			0.666034i
$474$	25.3137			1.16270
$475$	−4.00000	−	3.00000i	−0.183533	−	0.137649i
$476$	1.58579			0.0726844
$477$			32.4853i			1.48740i
$478$		−	12.8995i		−	0.590009i
$479$	−31.1127			−1.42158			−0.710788	−	0.703407i	$-0.751661\pi$
$479$	−31.1127			−1.42158			−0.710788	+	0.703407i	$0.751661\pi$
$480$	1.70711	−	5.12132i	0.0779184	−	0.233755i
$481$	1.45584			0.0663808
$482$		−	24.9706i		−	1.13738i
$483$			35.3848i			1.61006i
$484$	9.00000			0.409091
$485$	−24.7279	−	8.24264i	−1.12284	−	0.374279i
$486$	21.6569			0.982375
$487$			1.79899i			0.0815200i	0.999169	+	0.0407600i	$0.0129779\pi$
$487$			1.79899i			0.0815200i	−0.999169	+	0.0407600i	$0.987022\pi$
$488$		−	5.75736i		−	0.260623i
$489$	4.24264			0.191859
$490$	−9.51472	−	3.17157i	−0.429831	−	0.143277i
$491$	−2.44365			−0.110280			−0.0551402	−	0.998479i	$-0.517561\pi$
$491$	−2.44365			−0.110280			−0.0551402	+	0.998479i	$0.517561\pi$
$492$		−	10.2426i		−	0.461774i
$493$		−	5.82843i		−	0.262499i
$494$	−0.171573			−0.00771943
$495$	2.82843	−	8.48528i	0.127128	−	0.381385i
$496$	−2.24264			−0.100698
$497$			16.7868i			0.752991i
$498$			6.00000i			0.268866i
$499$	34.2426			1.53291			0.766456	−	0.642297i	$-0.222019\pi$
$499$	34.2426			1.53291			0.766456	+	0.642297i	$0.222019\pi$
$500$	−9.19239	+	6.36396i	−0.411096	+	0.284605i
$501$	−23.5563			−1.05242
$502$		−	27.5563i		−	1.22990i
$503$			39.7279i			1.77138i	0.464277	+	0.885690i	$0.346314\pi$
$503$			39.7279i			1.77138i	−0.464277	+	0.885690i	$0.653686\pi$
$504$	4.48528			0.199790
$505$	0.757359	−	2.27208i	0.0337020	−	0.101106i
$506$	−13.0711			−0.581080
$507$			31.3137i			1.39069i
$508$			14.4853i			0.642680i
$509$	4.97056			0.220316			0.110158	−	0.993914i	$-0.464864\pi$
$509$	4.97056			0.220316			0.110158	+	0.993914i	$0.464864\pi$
$510$	−5.12132	−	1.70711i	−0.226776	−	0.0755920i
$511$	−8.69848			−0.384798
$512$			1.00000i			0.0441942i
$513$			0.414214i			0.0182880i
$514$	−4.72792			−0.208540
$515$	9.00000	+	3.00000i	0.396587	+	0.132196i
$516$	24.7279			1.08859
$517$		0			0
$518$		−	13.4558i		−	0.591216i
$519$	39.7990			1.74698
$520$	−0.121320	+	0.363961i	−0.00532025	+	0.0159607i
$521$	0.686292			0.0300670			0.0150335	−	0.999887i	$-0.495215\pi$
$521$	0.686292			0.0300670			0.0150335	+	0.999887i	$0.495215\pi$
$522$		−	16.4853i		−	0.721541i
$523$		−	27.7279i		−	1.21246i	−0.795290	−	0.606229i	$-0.792682\pi$
$523$		−	27.7279i		−	1.21246i	0.795290	−	0.606229i	$-0.207318\pi$
$524$	−16.9706			−0.741362
$525$	−15.3137	−	11.4853i	−0.668345	−	0.501259i
$526$	−6.97056			−0.303931
$527$			2.24264i			0.0976910i
$528$			3.41421i			0.148585i
$529$	−62.4264			−2.71419
$530$	−8.12132	+	24.3640i	−0.352768	+	1.05830i
$531$	−36.4853			−1.58333
$532$			1.58579i			0.0687526i
$533$			0.727922i			0.0315298i
$534$	−17.0711			−0.738737
$535$	−12.1508	−	4.05025i	−0.525323	−	0.175108i
$536$	−13.2426			−0.571995
$537$			0.828427i			0.0357493i
$538$		−	28.6274i		−	1.23422i
$539$	6.34315			0.273219
$540$	0.878680	+	0.292893i	0.0378124	+	0.0126041i
$541$	18.2426			0.784312			0.392156	−	0.919899i	$-0.371729\pi$
$541$	18.2426			0.784312			0.392156	+	0.919899i	$0.371729\pi$
$542$		−	18.7574i		−	0.805698i
$543$			20.4853i			0.879108i
$544$	1.00000			0.0428746
$545$	11.2929	−	33.8787i	0.483734	−	1.45120i
$546$	−0.656854			−0.0281108
$547$		−	5.31371i		−	0.227198i	−0.993527	−	0.113599i	$-0.963762\pi$
$547$		−	5.31371i		−	0.227198i	0.993527	−	0.113599i	$-0.0362379\pi$
$548$			13.0000i			0.555332i
$549$	−16.2843			−0.694996
$550$	4.24264	−	5.65685i	0.180907	−	0.241209i
$551$	5.82843			0.248299
$552$			22.3137i			0.949735i
$553$			16.6274i			0.707070i
$554$		0			0
$555$	−14.4853	+	43.4558i	−0.614866	+	1.84460i
$556$	−12.0000			−0.508913
$557$		−	16.0000i		−	0.677942i	−0.940797	−	0.338971i	$-0.889921\pi$
$557$		−	16.0000i		−	0.677942i	0.940797	−	0.338971i	$-0.110079\pi$
$558$			6.34315i			0.268527i
$559$	−1.75736			−0.0743284
$560$	3.36396	+	1.12132i	0.142153	+	0.0473844i
$561$	3.41421			0.144148
$562$		−	4.24264i		−	0.178965i
$563$		−	28.9706i		−	1.22096i	−0.792030	−	0.610482i	$-0.790976\pi$
$563$		−	28.9706i		−	1.22096i	0.792030	−	0.610482i	$-0.209024\pi$
$564$		0			0
$565$	3.72792	+	1.24264i	0.156835	+	0.0522783i
$566$	−3.85786			−0.162158
$567$			15.0416i			0.631689i
$568$			10.5858i			0.444170i
$569$	−28.2843			−1.18574			−0.592869	−	0.805299i	$-0.702005\pi$
$569$	−28.2843			−1.18574			−0.592869	+	0.805299i	$0.702005\pi$
$570$	1.70711	−	5.12132i	0.0715028	−	0.214509i
$571$	6.24264			0.261246			0.130623	−	0.991432i	$-0.458302\pi$
$571$	6.24264			0.261246			0.130623	+	0.991432i	$0.458302\pi$
$572$		−	0.242641i		−	0.0101453i
$573$		−	44.7990i		−	1.87150i
$574$	6.72792			0.280818
$575$	27.7279	−	36.9706i	1.15633	−	1.54178i
$576$	2.82843			0.117851
$577$		−	2.31371i		−	0.0963209i	−0.998840	−	0.0481605i	$-0.984664\pi$
$577$		−	2.31371i		−	0.0963209i	0.998840	−	0.0481605i	$-0.0153359\pi$
$578$			16.0000i			0.665512i
$579$	28.1421			1.16955
$580$	4.12132	−	12.3640i	0.171129	−	0.513386i
$581$	−3.94113			−0.163505
$582$		−	28.1421i		−	1.16653i
$583$		−	16.2426i		−	0.672701i
$584$	−5.48528			−0.226983
$585$	1.02944	+	0.343146i	0.0425620	+	0.0141873i
$586$	−11.4853			−0.474453
$587$		−	21.7574i		−	0.898022i	−0.893526	−	0.449011i	$-0.851776\pi$
$587$		−	21.7574i		−	0.898022i	0.893526	−	0.449011i	$-0.148224\pi$
$588$		−	10.8284i		−	0.446557i
$589$	−2.24264			−0.0924064
$590$	−27.3640	−	9.12132i	−1.12656	−	0.375519i
$591$	−48.8701			−2.01025
$592$		−	8.48528i		−	0.348743i
$593$			22.0000i			0.903432i	0.892162	+	0.451716i	$0.149188\pi$
$593$			22.0000i			0.903432i	−0.892162	+	0.451716i	$0.850812\pi$
$594$	−0.585786			−0.0240351
$595$	1.12132	−	3.36396i	0.0459697	−	0.137909i
$596$	6.34315			0.259825
$597$		−	1.82843i		−	0.0748325i
$598$		−	1.58579i		−	0.0648476i
$599$	−27.2132			−1.11190			−0.555951	−	0.831215i	$-0.687646\pi$
$599$	−27.2132			−1.11190			−0.555951	+	0.831215i	$0.687646\pi$
$600$	−9.65685	−	7.24264i	−0.394239	−	0.295680i
$601$	−11.7574			−0.479593			−0.239796	−	0.970823i	$-0.577081\pi$
$601$	−11.7574			−0.479593			−0.239796	+	0.970823i	$0.577081\pi$
$602$			16.2426i			0.662001i
$603$			37.4558i			1.52532i
$604$	−6.48528			−0.263882
$605$	6.36396	−	19.0919i	0.258732	−	0.776195i
$606$	2.58579			0.105040
$607$		−	8.82843i		−	0.358335i	−0.983819	−	0.179167i	$-0.942660\pi$
$607$		−	8.82843i		−	0.358335i	0.983819	−	0.179167i	$-0.0573404\pi$
$608$			1.00000i			0.0405554i
$609$	22.3137			0.904197
$610$	−12.2132	−	4.07107i	−0.494498	−	0.164833i
$611$		0			0
$612$		−	2.82843i		−	0.114332i
$613$			47.6985i			1.92652i	0.268564	+	0.963262i	$0.413451\pi$
$613$			47.6985i			1.92652i	−0.268564	+	0.963262i	$0.586549\pi$
$614$	−6.34315			−0.255989
$615$	−21.7279	−	7.24264i	−0.876154	−	0.292051i
$616$	−2.24264			−0.0903586
$617$		−	12.4853i		−	0.502639i	−0.967904	−	0.251319i	$-0.919136\pi$
$617$		−	12.4853i		−	0.502639i	0.967904	−	0.251319i	$-0.0808644\pi$
$618$			10.2426i			0.412019i
$619$	−16.2426			−0.652847			−0.326423	−	0.945224i	$-0.605844\pi$
$619$	−16.2426			−0.652847			−0.326423	+	0.945224i	$0.605844\pi$
$620$	−1.58579	+	4.75736i	−0.0636867	+	0.191060i
$621$	−3.82843			−0.153629
$622$		−	13.2426i		−	0.530982i
$623$		−	11.2132i		−	0.449248i
$624$	−0.414214			−0.0165818
$625$	7.00000	+	24.0000i	0.280000	+	0.960000i
$626$	−25.9706			−1.03799
$627$			3.41421i			0.136351i
$628$			0.343146i			0.0136930i
$629$	−8.48528			−0.338330
$630$	3.17157	−	9.51472i	0.126358	−	0.379075i
$631$	−5.02944			−0.200219			−0.100109	−	0.994976i	$-0.531919\pi$
$631$	−5.02944			−0.200219			−0.100109	+	0.994976i	$0.531919\pi$
$632$			10.4853i			0.417082i
$633$			13.8284i			0.549631i
$634$	7.48528			0.297279
$635$	30.7279	+	10.2426i	1.21940	+	0.406467i
$636$	−27.7279			−1.09948
$637$			0.769553i			0.0304908i
$638$			8.24264i			0.326329i
$639$	29.9411			1.18445
$640$	2.12132	+	0.707107i	0.0838525	+	0.0279508i
$641$	30.0416			1.18657			0.593287	−	0.804991i	$-0.297830\pi$
$641$	30.0416			1.18657			0.593287	+	0.804991i	$0.297830\pi$
$642$		−	13.8284i		−	0.545764i
$643$			2.48528i			0.0980099i	0.998799	+	0.0490050i	$0.0156050\pi$
$643$			2.48528i			0.0980099i	−0.998799	+	0.0490050i	$0.984395\pi$
$644$	−14.6569			−0.577561
$645$	17.4853	−	52.4558i	0.688482	−	2.06545i
$646$	1.00000			0.0393445
$647$			27.2426i			1.07102i	0.844529	+	0.535509i	$0.179880\pi$
$647$			27.2426i			1.07102i	−0.844529	+	0.535509i	$0.820120\pi$
$648$			9.48528i			0.372617i
$649$	18.2426			0.716086
$650$	0.686292	+	0.514719i	0.0269186	+	0.0201889i
$651$	−8.58579			−0.336504
$652$			1.75736i			0.0688235i
$653$			4.97056i			0.194513i	0.995259	+	0.0972566i	$0.0310067\pi$
$653$			4.97056i			0.194513i	−0.995259	+	0.0972566i	$0.968993\pi$
$654$	38.5563			1.50767
$655$	−12.0000	+	36.0000i	−0.468879	+	1.40664i
$656$	4.24264			0.165647
$657$			15.5147i			0.605287i
$658$		0			0
$659$	0.899495			0.0350393			0.0175197	−	0.999847i	$-0.494423\pi$
$659$	0.899495			0.0350393			0.0175197	+	0.999847i	$0.494423\pi$
$660$	7.24264	+	2.41421i	0.281919	+	0.0939731i
$661$	32.4558			1.26239			0.631193	−	0.775626i	$-0.282566\pi$
$661$	32.4558			1.26239			0.631193	+	0.775626i	$0.282566\pi$
$662$		−	10.7574i		−	0.418097i
$663$			0.414214i			0.0160867i
$664$	−2.48528			−0.0964476
$665$	3.36396	+	1.12132i	0.130449	+	0.0434829i
$666$	−24.0000			−0.929981
$667$			53.8701i			2.08586i
$668$		−	9.75736i		−	0.377524i
$669$	50.2843			1.94410
$670$	−9.36396	+	28.0919i	−0.361761	+	1.08528i
$671$	8.14214			0.314324
$672$			3.82843i			0.147685i
$673$		−	12.0000i		−	0.462566i	−0.972887	−	0.231283i	$-0.925708\pi$
$673$		−	12.0000i		−	0.462566i	0.972887	−	0.231283i	$-0.0742923\pi$
$674$	33.8995			1.30576
$675$	1.24264	−	1.65685i	0.0478293	−	0.0637723i
$676$	−12.9706			−0.498868
$677$		−	26.9411i		−	1.03543i	−0.855553	−	0.517716i	$-0.826782\pi$
$677$		−	26.9411i		−	1.03543i	0.855553	−	0.517716i	$-0.173218\pi$
$678$			4.24264i			0.162938i
$679$	18.4853			0.709400
$680$	0.707107	−	2.12132i	0.0271163	−	0.0813489i
$681$	−60.9411			−2.33527
$682$		−	3.17157i		−	0.121446i
$683$			12.0000i			0.459167i	0.973289	+	0.229584i	$0.0737364\pi$
$683$			12.0000i			0.459167i	−0.973289	+	0.229584i	$0.926264\pi$
$684$	2.82843			0.108148
$685$	27.5772	+	9.19239i	1.05367	+	0.351223i
$686$	18.2132			0.695383
$687$			45.7990i			1.74734i
$688$			10.2426i			0.390497i
$689$	1.97056			0.0750725
$690$	47.3345	+	15.7782i	1.80199	+	0.600665i
$691$	5.51472			0.209790			0.104895	−	0.994483i	$-0.466549\pi$
$691$	5.51472			0.209790			0.104895	+	0.994483i	$0.466549\pi$
$692$			16.4853i			0.626676i
$693$			6.34315i			0.240956i
$694$	−22.4853			−0.853530
$695$	−8.48528	+	25.4558i	−0.321865	+	0.965595i
$696$	14.0711			0.533362
$697$		−	4.24264i		−	0.160701i
$698$		−	6.00000i		−	0.227103i
$699$	−21.6569			−0.819137
$700$	4.75736	−	6.34315i	0.179811	−	0.239748i
$701$	−16.9706			−0.640969			−0.320485	−	0.947254i	$-0.603846\pi$
$701$	−16.9706			−0.640969			−0.320485	+	0.947254i	$0.603846\pi$
$702$		−	0.0710678i		−	0.00268228i
$703$		−	8.48528i		−	0.320028i
$704$	−1.41421			−0.0533002
$705$		0			0
$706$	19.4853			0.733338
$707$			1.69848i			0.0638781i
$708$		−	31.1421i		−	1.17039i
$709$	34.0000			1.27690			0.638448	−	0.769665i	$-0.279577\pi$
$709$	34.0000			1.27690			0.638448	+	0.769665i	$0.279577\pi$
$710$	22.4558	+	7.48528i	0.842753	+	0.280918i
$711$	29.6569			1.11222
$712$		−	7.07107i		−	0.264999i
$713$		−	20.7279i		−	0.776267i
$714$	3.82843			0.143275
$715$	−0.514719	−	0.171573i	−0.0192494	−	0.00641646i
$716$	−0.343146			−0.0128240
$717$		−	31.1421i		−	1.16302i
$718$			31.2426i			1.16596i
$719$	24.8995			0.928594			0.464297	−	0.885679i	$-0.346307\pi$
$719$	24.8995			0.928594			0.464297	+	0.885679i	$0.346307\pi$
$720$	2.00000	−	6.00000i	0.0745356	−	0.223607i
$721$	−6.72792			−0.250561
$722$			1.00000i			0.0372161i
$723$		−	60.2843i		−	2.24200i
$724$	−8.48528			−0.315353
$725$	−23.3137	−	17.4853i	−0.865849	−	0.649387i
$726$	21.7279			0.806399
$727$			15.7279i			0.583316i	0.956523	+	0.291658i	$0.0942070\pi$
$727$			15.7279i			0.583316i	−0.956523	+	0.291658i	$0.905793\pi$
$728$		−	0.272078i		−	0.0100839i
$729$	23.8284			0.882534
$730$	−3.87868	+	11.6360i	−0.143556	+	0.430669i
$731$	10.2426			0.378838
$732$		−	13.8995i		−	0.513740i
$733$			12.0000i			0.443230i	0.975134	+	0.221615i	$0.0711328\pi$
$733$			12.0000i			0.443230i	−0.975134	+	0.221615i	$0.928867\pi$
$734$	25.4558			0.939592
$735$	−22.9706	−	7.65685i	−0.847282	−	0.282427i
$736$	−9.24264			−0.340688
$737$		−	18.7279i		−	0.689852i
$738$		−	12.0000i		−	0.441726i
$739$	−26.7279			−0.983203			−0.491601	−	0.870820i	$-0.663588\pi$
$739$	−26.7279			−0.983203			−0.491601	+	0.870820i	$0.663588\pi$
$740$	−18.0000	−	6.00000i	−0.661693	−	0.220564i
$741$	−0.414214			−0.0152165
$742$		−	18.2132i		−	0.668628i
$743$		−	18.7279i		−	0.687061i	−0.939142	−	0.343530i	$-0.888377\pi$
$743$		−	18.7279i		−	0.687061i	0.939142	−	0.343530i	$-0.111623\pi$
$744$	−5.41421			−0.198495
$745$	4.48528	−	13.4558i	0.164328	−	0.492984i
$746$	−9.00000			−0.329513
$747$			7.02944i			0.257194i
$748$			1.41421i			0.0517088i
$749$	9.08326			0.331895
$750$	−22.1924	+	15.3640i	−0.810351	+	0.561013i
$751$	−1.27208			−0.0464188			−0.0232094	−	0.999731i	$-0.507388\pi$
$751$	−1.27208			−0.0464188			−0.0232094	+	0.999731i	$0.507388\pi$
$752$		0			0
$753$		−	66.5269i		−	2.42438i
$754$	−1.00000			−0.0364179
$755$	−4.58579	+	13.7574i	−0.166894	+	0.500682i
$756$	−0.656854			−0.0238896
$757$		−	23.6569i		−	0.859823i	−0.902871	−	0.429911i	$-0.858545\pi$
$757$		−	23.6569i		−	0.859823i	0.902871	−	0.429911i	$-0.141455\pi$
$758$		−	2.75736i		−	0.100152i
$759$	−31.5563			−1.14542
$760$	2.12132	+	0.707107i	0.0769484	+	0.0256495i
$761$	10.0294			0.363567			0.181783	−	0.983339i	$-0.441813\pi$
$761$	10.0294			0.363567			0.181783	+	0.983339i	$0.441813\pi$
$762$			34.9706i			1.26685i
$763$			25.3259i			0.916859i
$764$	18.5563			0.671345
$765$	−6.00000	−	2.00000i	−0.216930	−	0.0723102i
$766$	−3.75736			−0.135759
$767$			2.21320i			0.0799141i
$768$			2.41421i			0.0871154i
$769$	−14.4558			−0.521291			−0.260646	−	0.965435i	$-0.583935\pi$
$769$	−14.4558			−0.521291			−0.260646	+	0.965435i	$0.583935\pi$
$770$	−1.58579	+	4.75736i	−0.0571478	+	0.171443i
$771$	−11.4142			−0.411073
$772$			11.6569i			0.419539i
$773$		−	19.9706i		−	0.718291i	−0.933282	−	0.359146i	$-0.883068\pi$
$773$		−	19.9706i		−	0.718291i	0.933282	−	0.359146i	$-0.116932\pi$
$774$	28.9706			1.04133
$775$	8.97056	+	6.72792i	0.322232	+	0.241674i
$776$	11.6569			0.418457
$777$		−	32.4853i		−	1.16540i
$778$		−	37.0711i		−	1.32906i
$779$	4.24264			0.152008
$780$	−0.292893	+	0.878680i	−0.0104873	+	0.0314618i
$781$	−14.9706			−0.535689
$782$			9.24264i			0.330516i

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:	\( N \)	\(=\)	\( 190 = 2 \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	190.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +2.41421i q^{3} -1.00000 q^{4} +(-0.707107 + 2.12132i) q^{5} -2.41421 q^{6} -1.58579i q^{7} -1.00000i q^{8} -2.82843 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} +2.41421i q^{3} -1.00000 q^{4} +(-0.707107 + 2.12132i) q^{5} -2.41421 q^{6} -1.58579i q^{7} -1.00000i q^{8} -2.82843 q^{9} +(-2.12132 - 0.707107i) q^{10} +1.41421 q^{11} -2.41421i q^{12} +0.171573i q^{13} +1.58579 q^{14} +(-5.12132 - 1.70711i) q^{15} +1.00000 q^{16} -1.00000i q^{17} -2.82843i q^{18} +1.00000 q^{19} +(0.707107 - 2.12132i) q^{20} +3.82843 q^{21} +1.41421i q^{22} +9.24264i q^{23} +2.41421 q^{24} +(-4.00000 - 3.00000i) q^{25} -0.171573 q^{26} +0.414214i q^{27} +1.58579i q^{28} +5.82843 q^{29} +(1.70711 - 5.12132i) q^{30} -2.24264 q^{31} +1.00000i q^{32} +3.41421i q^{33} +1.00000 q^{34} +(3.36396 + 1.12132i) q^{35} +2.82843 q^{36} -8.48528i q^{37} +1.00000i q^{38} -0.414214 q^{39} +(2.12132 + 0.707107i) q^{40} +4.24264 q^{41} +3.82843i q^{42} +10.2426i q^{43} -1.41421 q^{44} +(2.00000 - 6.00000i) q^{45} -9.24264 q^{46} +2.41421i q^{48} +4.48528 q^{49} +(3.00000 - 4.00000i) q^{50} +2.41421 q^{51} -0.171573i q^{52} -11.4853i q^{53} -0.414214 q^{54} +(-1.00000 + 3.00000i) q^{55} -1.58579 q^{56} +2.41421i q^{57} +5.82843i q^{58} +12.8995 q^{59} +(5.12132 + 1.70711i) q^{60} +5.75736 q^{61} -2.24264i q^{62} +4.48528i q^{63} -1.00000 q^{64} +(-0.363961 - 0.121320i) q^{65} -3.41421 q^{66} -13.2426i q^{67} +1.00000i q^{68} -22.3137 q^{69} +(-1.12132 + 3.36396i) q^{70} -10.5858 q^{71} +2.82843i q^{72} -5.48528i q^{73} +8.48528 q^{74} +(7.24264 - 9.65685i) q^{75} -1.00000 q^{76} -2.24264i q^{77} -0.414214i q^{78} -10.4853 q^{79} +(-0.707107 + 2.12132i) q^{80} -9.48528 q^{81} +4.24264i q^{82} -2.48528i q^{83} -3.82843 q^{84} +(2.12132 + 0.707107i) q^{85} -10.2426 q^{86} +14.0711i q^{87} -1.41421i q^{88} +7.07107 q^{89} +(6.00000 + 2.00000i) q^{90} +0.272078 q^{91} -9.24264i q^{92} -5.41421i q^{93} +(-0.707107 + 2.12132i) q^{95} -2.41421 q^{96} +11.6569i q^{97} +4.48528i q^{98} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} - 4 q^{6}+O(q^{10}) \) 4 * q - 4 * q^4 - 4 * q^6	\( 4 q - 4 q^{4} - 4 q^{6} + 12 q^{14} - 12 q^{15} + 4 q^{16} + 4 q^{19} + 4 q^{21} + 4 q^{24} - 16 q^{25} - 12 q^{26} + 12 q^{29} + 4 q^{30} + 8 q^{31} + 4 q^{34} - 12 q^{35} + 4 q^{39} + 8 q^{45} - 20 q^{46} - 16 q^{49} + 12 q^{50} + 4 q^{51} + 4 q^{54} - 4 q^{55} - 12 q^{56} + 12 q^{59} + 12 q^{60} + 40 q^{61} - 4 q^{64} + 24 q^{65} - 8 q^{66} - 44 q^{69} + 4 q^{70} - 48 q^{71} + 12 q^{75} - 4 q^{76} - 8 q^{79} - 4 q^{81} - 4 q^{84} - 24 q^{86} + 24 q^{90} + 52 q^{91} - 4 q^{96} - 16 q^{99}+O(q^{100}) \) 4 * q - 4 * q^4 - 4 * q^6 + 12 * q^14 - 12 * q^15 + 4 * q^16 + 4 * q^19 + 4 * q^21 + 4 * q^24 - 16 * q^25 - 12 * q^26 + 12 * q^29 + 4 * q^30 + 8 * q^31 + 4 * q^34 - 12 * q^35 + 4 * q^39 + 8 * q^45 - 20 * q^46 - 16 * q^49 + 12 * q^50 + 4 * q^51 + 4 * q^54 - 4 * q^55 - 12 * q^56 + 12 * q^59 + 12 * q^60 + 40 * q^61 - 4 * q^64 + 24 * q^65 - 8 * q^66 - 44 * q^69 + 4 * q^70 - 48 * q^71 + 12 * q^75 - 4 * q^76 - 8 * q^79 - 4 * q^81 - 4 * q^84 - 24 * q^86 + 24 * q^90 + 52 * q^91 - 4 * q^96 - 16 * q^99

Introduction

L-functions

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