LMFDB - Embedded newform 190.2.b.a.39.3

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			39.3
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	190.39
Dual form			190.2.b.a.39.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -0.414214i q^{3} -1.00000 q^{4} +(0.707107 - 2.12132i) q^{5} +0.414214 q^{6} -4.41421i q^{7} -1.00000i q^{8} +2.82843 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} -0.414214i q^{3} -1.00000 q^{4} +(0.707107 - 2.12132i) q^{5} +0.414214 q^{6} -4.41421i q^{7} -1.00000i q^{8} +2.82843 q^{9} +(2.12132 + 0.707107i) q^{10} -1.41421 q^{11} +0.414214i q^{12} +5.82843i q^{13} +4.41421 q^{14} +(-0.878680 - 0.292893i) q^{15} +1.00000 q^{16} -1.00000i q^{17} +2.82843i q^{18} +1.00000 q^{19} +(-0.707107 + 2.12132i) q^{20} -1.82843 q^{21} -1.41421i q^{22} +0.757359i q^{23} -0.414214 q^{24} +(-4.00000 - 3.00000i) q^{25} -5.82843 q^{26} -2.41421i q^{27} +4.41421i q^{28} +0.171573 q^{29} +(0.292893 - 0.878680i) q^{30} +6.24264 q^{31} +1.00000i q^{32} +0.585786i q^{33} +1.00000 q^{34} +(-9.36396 - 3.12132i) q^{35} -2.82843 q^{36} +8.48528i q^{37} +1.00000i q^{38} +2.41421 q^{39} +(-2.12132 - 0.707107i) q^{40} -4.24264 q^{41} -1.82843i q^{42} +1.75736i q^{43} +1.41421 q^{44} +(2.00000 - 6.00000i) q^{45} -0.757359 q^{46} -0.414214i q^{48} -12.4853 q^{49} +(3.00000 - 4.00000i) q^{50} -0.414214 q^{51} -5.82843i q^{52} +5.48528i q^{53} +2.41421 q^{54} +(-1.00000 + 3.00000i) q^{55} -4.41421 q^{56} -0.414214i q^{57} +0.171573i q^{58} -6.89949 q^{59} +(0.878680 + 0.292893i) q^{60} +14.2426 q^{61} +6.24264i q^{62} -12.4853i q^{63} -1.00000 q^{64} +(12.3640 + 4.12132i) q^{65} -0.585786 q^{66} -4.75736i q^{67} +1.00000i q^{68} +0.313708 q^{69} +(3.12132 - 9.36396i) q^{70} -13.4142 q^{71} -2.82843i q^{72} +11.4853i q^{73} -8.48528 q^{74} +(-1.24264 + 1.65685i) q^{75} -1.00000 q^{76} +6.24264i q^{77} +2.41421i q^{78} +6.48528 q^{79} +(0.707107 - 2.12132i) q^{80} +7.48528 q^{81} -4.24264i q^{82} +14.4853i q^{83} +1.82843 q^{84} +(-2.12132 - 0.707107i) q^{85} -1.75736 q^{86} -0.0710678i q^{87} +1.41421i q^{88} -7.07107 q^{89} +(6.00000 + 2.00000i) q^{90} +25.7279 q^{91} -0.757359i q^{92} -2.58579i q^{93} +(0.707107 - 2.12132i) q^{95} +0.414214 q^{96} +0.343146i q^{97} -12.4853i q^{98} -4.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$		−	0.414214i		−	0.239146i	−0.992825	−	0.119573i	$-0.961847\pi$
$3$		−	0.414214i		−	0.239146i	0.992825	−	0.119573i	$-0.0381526\pi$
$4$	−1.00000			−0.500000
$5$	0.707107	−	2.12132i	0.316228	−	0.948683i
$5$	0.707107	−	2.12132i	0.316228	−	0.948683i
$6$	0.414214			0.169102
$7$		−	4.41421i		−	1.66842i	−0.551450	−	0.834208i	$-0.685925\pi$
$7$		−	4.41421i		−	1.66842i	0.551450	−	0.834208i	$-0.314075\pi$
$8$		−	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	$-0.568389\pi$
$11$	−1.41421			−0.426401			−0.213201	+	0.977008i	$0.568389\pi$
$12$			0.414214i			0.119573i
$13$			5.82843i			1.61651i	0.588829	+	0.808257i	$0.299589\pi$
$13$			5.82843i			1.61651i	−0.588829	+	0.808257i	$0.700411\pi$
$14$	4.41421			1.17975
$15$	−0.878680	−	0.292893i	−0.226874	−	0.0756247i
$16$	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
$21$	−1.82843			−0.398996
$22$		−	1.41421i		−	0.301511i
$23$			0.757359i			0.157920i	0.996878	+	0.0789602i	$0.0251600\pi$
$23$			0.757359i			0.157920i	−0.996878	+	0.0789602i	$0.974840\pi$
$24$	−0.414214			−0.0845510
$25$	−4.00000	−	3.00000i	−0.800000	−	0.600000i
$26$	−5.82843			−1.14305
$27$		−	2.41421i		−	0.464616i
$28$			4.41421i			0.834208i
$29$	0.171573			0.0318603			0.0159301	−	0.999873i	$-0.494929\pi$
$29$	0.171573			0.0318603			0.0159301	+	0.999873i	$0.494929\pi$
$30$	0.292893	−	0.878680i	0.0534747	−	0.160424i
$31$	6.24264			1.12121			0.560606	−	0.828083i	$-0.310568\pi$
$31$	6.24264			1.12121			0.560606	+	0.828083i	$0.310568\pi$
$32$			1.00000i			0.176777i
$33$			0.585786i			0.101972i
$34$	1.00000			0.171499
$35$	−9.36396	−	3.12132i	−1.58280	−	0.527599i
$36$	−2.82843			−0.471405
$37$			8.48528i			1.39497i	0.716599	+	0.697486i	$0.245698\pi$
$37$			8.48528i			1.39497i	−0.716599	+	0.697486i	$0.754302\pi$
$38$			1.00000i			0.162221i
$39$	2.41421			0.386584
$40$	−2.12132	−	0.707107i	−0.335410	−	0.111803i
$41$	−4.24264			−0.662589			−0.331295	−	0.943527i	$-0.607485\pi$
$41$	−4.24264			−0.662589			−0.331295	+	0.943527i	$0.607485\pi$
$42$		−	1.82843i		−	0.282132i
$43$			1.75736i			0.267995i	0.990982	+	0.133997i	$0.0427814\pi$
$43$			1.75736i			0.267995i	−0.990982	+	0.133997i	$0.957219\pi$
$44$	1.41421			0.213201
$45$	2.00000	−	6.00000i	0.298142	−	0.894427i
$46$	−0.757359			−0.111667
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$		−	0.414214i		−	0.0597866i
$49$	−12.4853			−1.78361
$50$	3.00000	−	4.00000i	0.424264	−	0.565685i
$51$	−0.414214			−0.0580015
$52$		−	5.82843i		−	0.808257i
$53$			5.48528i			0.753461i	0.926323	+	0.376731i	$0.122952\pi$
$53$			5.48528i			0.753461i	−0.926323	+	0.376731i	$0.877048\pi$
$54$	2.41421			0.328533
$55$	−1.00000	+	3.00000i	−0.134840	+	0.404520i
$56$	−4.41421			−0.589874
$57$		−	0.414214i		−	0.0548639i
$58$			0.171573i			0.0225286i
$59$	−6.89949			−0.898238			−0.449119	−	0.893472i	$-0.648262\pi$
$59$	−6.89949			−0.898238			−0.449119	+	0.893472i	$0.648262\pi$
$60$	0.878680	+	0.292893i	0.113437	+	0.0378124i
$61$	14.2426			1.82358			0.911792	−	0.410653i	$-0.134699\pi$
$61$	14.2426			1.82358			0.911792	+	0.410653i	$0.134699\pi$
$62$			6.24264i			0.792816i
$63$		−	12.4853i		−	1.57300i
$64$	−1.00000			−0.125000
$65$	12.3640	+	4.12132i	1.53356	+	0.511187i
$66$	−0.585786			−0.0721053
$67$		−	4.75736i		−	0.581204i	−0.956844	−	0.290602i	$-0.906144\pi$
$67$		−	4.75736i		−	0.581204i	0.956844	−	0.290602i	$-0.0938555\pi$
$68$			1.00000i			0.121268i
$69$	0.313708			0.0377661
$70$	3.12132	−	9.36396i	0.373069	−	1.11921i
$71$	−13.4142			−1.59197			−0.795987	−	0.605314i	$-0.793048\pi$
$71$	−13.4142			−1.59197			−0.795987	+	0.605314i	$0.793048\pi$
$72$		−	2.82843i		−	0.333333i
$73$			11.4853i			1.34425i	0.740437	+	0.672125i	$0.234618\pi$
$73$			11.4853i			1.34425i	−0.740437	+	0.672125i	$0.765382\pi$
$74$	−8.48528			−0.986394
$75$	−1.24264	+	1.65685i	−0.143488	+	0.191317i
$76$	−1.00000			−0.114708
$77$			6.24264i			0.711415i
$78$			2.41421i			0.273356i
$79$	6.48528			0.729651			0.364826	−	0.931076i	$-0.381129\pi$
$79$	6.48528			0.729651			0.364826	+	0.931076i	$0.381129\pi$
$80$	0.707107	−	2.12132i	0.0790569	−	0.237171i
$81$	7.48528			0.831698
$82$		−	4.24264i		−	0.468521i
$83$			14.4853i			1.58997i	0.606632	+	0.794983i	$0.292520\pi$
$83$			14.4853i			1.58997i	−0.606632	+	0.794983i	$0.707480\pi$
$84$	1.82843			0.199498
$85$	−2.12132	−	0.707107i	−0.230089	−	0.0766965i
$86$	−1.75736			−0.189501
$87$		−	0.0710678i		−	0.00761927i
$88$			1.41421i			0.150756i
$89$	−7.07107			−0.749532			−0.374766	−	0.927119i	$-0.622277\pi$
$89$	−7.07107			−0.749532			−0.374766	+	0.927119i	$0.622277\pi$
$90$	6.00000	+	2.00000i	0.632456	+	0.210819i
$91$	25.7279			2.69702
$92$		−	0.757359i		−	0.0789602i
$93$		−	2.58579i		−	0.268134i
$94$		0			0
$95$	0.707107	−	2.12132i	0.0725476	−	0.217643i
$96$	0.414214			0.0422755
$97$			0.343146i			0.0348412i	0.999848	+	0.0174206i	$0.00554543\pi$
$97$			0.343146i			0.0348412i	−0.999848	+	0.0174206i	$0.994455\pi$
$98$		−	12.4853i		−	1.26120i
$99$	−4.00000			−0.402015
$100$	4.00000	+	3.00000i	0.400000	+	0.300000i
$101$	13.0711			1.30062			0.650310	−	0.759669i	$-0.274639\pi$
$101$	13.0711			1.30062			0.650310	+	0.759669i	$0.274639\pi$
$102$		−	0.414214i		−	0.0410133i
$103$			4.24264i			0.418040i	0.977911	+	0.209020i	$0.0670273\pi$
$103$			4.24264i			0.418040i	−0.977911	+	0.209020i	$0.932973\pi$
$104$	5.82843			0.571524
$105$	−1.29289	+	3.87868i	−0.126173	+	0.378520i
$106$	−5.48528			−0.532778
$107$		−	19.7279i		−	1.90717i	−0.301124	−	0.953585i	$-0.597362\pi$
$107$		−	19.7279i		−	1.90717i	0.301124	−	0.953585i	$-0.402638\pi$
$108$			2.41421i			0.232308i
$109$	17.9706			1.72127			0.860634	−	0.509224i	$-0.170068\pi$
$109$	17.9706			1.72127			0.860634	+	0.509224i	$0.170068\pi$
$110$	−3.00000	−	1.00000i	−0.286039	−	0.0953463i
$111$	3.51472			0.333602
$112$		−	4.41421i		−	0.417104i
$113$		−	10.2426i		−	0.963547i	−0.876296	−	0.481773i	$-0.839993\pi$
$113$		−	10.2426i		−	0.963547i	0.876296	−	0.481773i	$-0.160007\pi$
$114$	0.414214			0.0387947
$115$	1.60660	+	0.535534i	0.149816	+	0.0499388i
$116$	−0.171573			−0.0159301
$117$			16.4853i			1.52406i
$118$		−	6.89949i		−	0.635150i
$119$	−4.41421			−0.404650
$120$	−0.292893	+	0.878680i	−0.0267374	+	0.0802121i
$121$	−9.00000			−0.818182
$122$			14.2426i			1.28947i
$123$			1.75736i			0.158456i
$124$	−6.24264			−0.560606
$125$	−9.19239	+	6.36396i	−0.822192	+	0.569210i
$126$	12.4853			1.11228
$127$			2.48528i			0.220533i	0.993902	+	0.110267i	$0.0351704\pi$
$127$			2.48528i			0.220533i	−0.993902	+	0.110267i	$0.964830\pi$
$128$		−	1.00000i		−	0.0883883i
$129$	0.727922			0.0640900
$130$	−4.12132	+	12.3640i	−0.361464	+	1.08439i
$131$	−16.9706			−1.48272			−0.741362	−	0.671105i	$-0.765820\pi$
$131$	−16.9706			−1.48272			−0.741362	+	0.671105i	$0.765820\pi$
$132$		−	0.585786i		−	0.0509862i
$133$		−	4.41421i		−	0.382761i
$134$	4.75736			0.410973
$135$	−5.12132	−	1.70711i	−0.440773	−	0.146924i
$136$	−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	$-0.187409\pi$
$138$			0.313708i			0.0267046i
$139$	12.0000			1.01783			0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000			1.01783			0.508913	+	0.860818i	$0.330047\pi$
$140$	9.36396	+	3.12132i	0.791399	+	0.263800i
$141$		0			0
$142$		−	13.4142i		−	1.12570i
$143$		−	8.24264i		−	0.689284i
$144$	2.82843			0.235702
$145$	0.121320	−	0.363961i	0.0100751	−	0.0302253i
$146$	−11.4853			−0.950529
$147$			5.17157i			0.426544i
$148$		−	8.48528i		−	0.697486i
$149$	−17.6569			−1.44651			−0.723253	−	0.690583i	$-0.757354\pi$
$149$	−17.6569			−1.44651			−0.723253	+	0.690583i	$0.757354\pi$
$150$	−1.65685	−	1.24264i	−0.135282	−	0.101461i
$151$	−10.4853			−0.853280			−0.426640	−	0.904422i	$-0.640303\pi$
$151$	−10.4853			−0.853280			−0.426640	+	0.904422i	$0.640303\pi$
$152$		−	1.00000i		−	0.0811107i
$153$		−	2.82843i		−	0.228665i
$154$	−6.24264			−0.503046
$155$	4.41421	−	13.2426i	0.354558	−	1.06367i
$156$	−2.41421			−0.193292
$157$		−	11.6569i		−	0.930318i	−0.885227	−	0.465159i	$-0.845997\pi$
$157$		−	11.6569i		−	0.930318i	0.885227	−	0.465159i	$-0.154003\pi$
$158$			6.48528i			0.515941i
$159$	2.27208			0.180188
$160$	2.12132	+	0.707107i	0.167705	+	0.0559017i
$161$	3.34315			0.263477
$162$			7.48528i			0.588099i
$163$		−	10.2426i		−	0.802266i	−0.916020	−	0.401133i	$-0.868617\pi$
$163$		−	10.2426i		−	0.802266i	0.916020	−	0.401133i	$-0.131383\pi$
$164$	4.24264			0.331295
$165$	1.24264	+	0.414214i	0.0967394	+	0.0322465i
$166$	−14.4853			−1.12428
$167$			18.2426i			1.41166i	0.708382	+	0.705829i	$0.249425\pi$
$167$			18.2426i			1.41166i	−0.708382	+	0.705829i	$0.750575\pi$
$168$			1.82843i			0.141066i
$169$	−20.9706			−1.61312
$170$	0.707107	−	2.12132i	0.0542326	−	0.162698i
$171$	2.82843			0.216295
$172$		−	1.75736i		−	0.133997i
$173$			0.485281i			0.0368953i	0.999830	+	0.0184476i	$0.00587240\pi$
$173$			0.485281i			0.0368953i	−0.999830	+	0.0184476i	$0.994128\pi$
$174$	0.0710678			0.00538764
$175$	−13.2426	+	17.6569i	−1.00105	+	1.33473i
$176$	−1.41421			−0.106600
$177$			2.85786i			0.214810i
$178$		−	7.07107i		−	0.529999i
$179$	11.6569			0.871274			0.435637	−	0.900122i	$-0.356523\pi$
$179$	11.6569			0.871274			0.435637	+	0.900122i	$0.356523\pi$
$180$	−2.00000	+	6.00000i	−0.149071	+	0.447214i
$181$	−8.48528			−0.630706			−0.315353	−	0.948974i	$-0.602123\pi$
$181$	−8.48528			−0.630706			−0.315353	+	0.948974i	$0.602123\pi$
$182$			25.7279i			1.90708i
$183$		−	5.89949i		−	0.436103i
$184$	0.757359			0.0558333
$185$	18.0000	+	6.00000i	1.32339	+	0.441129i
$186$	2.58579			0.189599
$187$			1.41421i			0.103418i
$188$		0			0
$189$	−10.6569			−0.775172
$190$	2.12132	+	0.707107i	0.153897	+	0.0512989i
$191$	12.5563			0.908546			0.454273	−	0.890863i	$-0.349899\pi$
$191$	12.5563			0.908546			0.454273	+	0.890863i	$0.349899\pi$
$192$			0.414214i			0.0298933i
$193$		−	0.343146i		−	0.0247002i	−0.999924	−	0.0123501i	$-0.996069\pi$
$193$		−	0.343146i		−	0.0247002i	0.999924	−	0.0123501i	$-0.00393125\pi$
$194$	−0.343146			−0.0246364
$195$	1.70711	−	5.12132i	0.122248	−	0.366745i
$196$	12.4853			0.891806
$197$			11.7574i			0.837677i	0.908061	+	0.418839i	$0.137563\pi$
$197$			11.7574i			0.837677i	−0.908061	+	0.418839i	$0.862437\pi$
$198$		−	4.00000i		−	0.284268i
$199$	−9.24264			−0.655193			−0.327597	−	0.944818i	$-0.606239\pi$
$199$	−9.24264			−0.655193			−0.327597	+	0.944818i	$0.606239\pi$
$200$	−3.00000	+	4.00000i	−0.212132	+	0.282843i
$201$	−1.97056			−0.138993
$202$			13.0711i			0.919677i
$203$		−	0.757359i		−	0.0531562i
$204$	0.414214			0.0290008
$205$	−3.00000	+	9.00000i	−0.209529	+	0.628587i
$206$	−4.24264			−0.295599
$207$			2.14214i			0.148889i
$208$			5.82843i			0.404129i
$209$	−1.41421			−0.0978232
$210$	−3.87868	−	1.29289i	−0.267654	−	0.0892181i
$211$	−19.7279			−1.35813			−0.679063	−	0.734080i	$-0.737614\pi$
$211$	−19.7279			−1.35813			−0.679063	+	0.734080i	$0.737614\pi$
$212$		−	5.48528i		−	0.376731i
$213$			5.55635i			0.380715i
$214$	19.7279			1.34857
$215$	3.72792	+	1.24264i	0.254242	+	0.0847474i
$216$	−2.41421			−0.164266
$217$		−	27.5563i		−	1.87065i
$218$			17.9706i			1.21712i
$219$	4.75736			0.321473
$220$	1.00000	−	3.00000i	0.0674200	−	0.202260i
$221$	5.82843			0.392062
$222$			3.51472i			0.235892i
$223$		−	15.1716i		−	1.01596i	−0.861368	−	0.507982i	$-0.830392\pi$
$223$		−	15.1716i		−	1.01596i	0.861368	−	0.507982i	$-0.169608\pi$
$224$	4.41421			0.294937
$225$	−11.3137	−	8.48528i	−0.754247	−	0.565685i
$226$	10.2426			0.681330
$227$			16.7574i			1.11223i	0.831107	+	0.556113i	$0.187708\pi$
$227$			16.7574i			1.11223i	−0.831107	+	0.556113i	$0.812292\pi$
$228$			0.414214i			0.0274320i
$229$	−14.9706			−0.989283			−0.494641	−	0.869097i	$-0.664701\pi$
$229$	−14.9706			−0.989283			−0.494641	+	0.869097i	$0.664701\pi$
$230$	−0.535534	+	1.60660i	−0.0353121	+	0.105936i
$231$	2.58579			0.170132
$232$		−	0.171573i		−	0.0112643i
$233$		−	24.9706i		−	1.63588i	−0.575306	−	0.817938i	$-0.695117\pi$
$233$		−	24.9706i		−	1.63588i	0.575306	−	0.817938i	$-0.304883\pi$
$234$	−16.4853			−1.07768
$235$		0			0
$236$	6.89949			0.449119
$237$		−	2.68629i		−	0.174493i
$238$		−	4.41421i		−	0.286131i
$239$	6.89949			0.446291			0.223146	−	0.974785i	$-0.428367\pi$
$239$	6.89949			0.446291			0.223146	+	0.974785i	$0.428367\pi$
$240$	−0.878680	−	0.292893i	−0.0567185	−	0.0189062i
$241$	8.97056			0.577845			0.288922	−	0.957353i	$-0.406703\pi$
$241$	8.97056			0.577845			0.288922	+	0.957353i	$0.406703\pi$
$242$		−	9.00000i		−	0.578542i
$243$		−	10.3431i		−	0.663513i
$244$	−14.2426			−0.911792
$245$	−8.82843	+	26.4853i	−0.564028	+	1.69208i
$246$	−1.75736			−0.112045
$247$			5.82843i			0.370854i
$248$		−	6.24264i		−	0.396408i
$249$	6.00000			0.380235
$250$	−6.36396	−	9.19239i	−0.402492	−	0.581378i
$251$	3.55635			0.224475			0.112237	−	0.993681i	$-0.464198\pi$
$251$	3.55635			0.224475			0.112237	+	0.993681i	$0.464198\pi$
$252$			12.4853i			0.786499i
$253$		−	1.07107i		−	0.0673375i
$254$	−2.48528			−0.155940
$255$	−0.292893	+	0.878680i	−0.0183417	+	0.0550251i
$256$	1.00000			0.0625000
$257$		−	20.7279i		−	1.29297i	−0.762926	−	0.646486i	$-0.776238\pi$
$257$		−	20.7279i		−	1.29297i	0.762926	−	0.646486i	$-0.223762\pi$
$258$			0.727922i			0.0453184i
$259$	37.4558			2.32739
$260$	−12.3640	−	4.12132i	−0.766780	−	0.255593i
$261$	0.485281			0.0300382
$262$		−	16.9706i		−	1.04844i
$263$		−	26.9706i		−	1.66308i	−0.555468	−	0.831538i	$-0.687461\pi$
$263$		−	26.9706i		−	1.66308i	0.555468	−	0.831538i	$-0.312539\pi$
$264$	0.585786			0.0360527
$265$	11.6360	+	3.87868i	0.714796	+	0.238265i
$266$	4.41421			0.270653
$267$			2.92893i			0.179248i
$268$			4.75736i			0.290602i
$269$	16.6274			1.01379			0.506896	−	0.862007i	$-0.330793\pi$
$269$	16.6274			1.01379			0.506896	+	0.862007i	$0.330793\pi$
$270$	1.70711	−	5.12132i	0.103891	−	0.311674i
$271$	−27.2426			−1.65487			−0.827436	−	0.561560i	$-0.810202\pi$
$271$	−27.2426			−1.65487			−0.827436	+	0.561560i	$0.810202\pi$
$272$		−	1.00000i		−	0.0606339i
$273$		−	10.6569i		−	0.644982i
$274$	13.0000			0.785359
$275$	5.65685	+	4.24264i	0.341121	+	0.255841i
$276$	−0.313708			−0.0188830
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$			12.0000i			0.719712i
$279$	17.6569			1.05709
$280$	−3.12132	+	9.36396i	−0.186535	+	0.559604i
$281$	4.24264			0.253095			0.126547	−	0.991961i	$-0.459610\pi$
$281$	4.24264			0.253095			0.126547	+	0.991961i	$0.459610\pi$
$282$		0			0
$283$			32.1421i			1.91065i	0.295555	+	0.955326i	$0.404496\pi$
$283$			32.1421i			1.91065i	−0.295555	+	0.955326i	$0.595504\pi$
$284$	13.4142			0.795987
$285$	−0.878680	−	0.292893i	−0.0520485	−	0.0173495i
$286$	8.24264			0.487398
$287$			18.7279i			1.10547i
$288$			2.82843i			0.166667i
$289$	16.0000			0.941176
$290$	0.363961	+	0.121320i	0.0213725	+	0.00712418i
$291$	0.142136			0.00833214
$292$		−	11.4853i		−	0.672125i
$293$		−	5.48528i		−	0.320454i	−0.987080	−	0.160227i	$-0.948777\pi$
$293$		−	5.48528i		−	0.320454i	0.987080	−	0.160227i	$-0.0512226\pi$
$294$	−5.17157			−0.301612
$295$	−4.87868	+	14.6360i	−0.284048	+	0.852143i
$296$	8.48528			0.493197
$297$			3.41421i			0.198113i
$298$		−	17.6569i		−	1.02283i
$299$	−4.41421			−0.255281
$300$	1.24264	−	1.65685i	0.0717439	−	0.0956585i
$301$	7.75736			0.447127
$302$		−	10.4853i		−	0.603360i
$303$		−	5.41421i		−	0.311038i
$304$	1.00000			0.0573539
$305$	10.0711	−	30.2132i	0.576668	−	1.73000i
$306$	2.82843			0.161690
$307$			17.6569i			1.00773i	0.863782	+	0.503865i	$0.168089\pi$
$307$			17.6569i			1.00773i	−0.863782	+	0.503865i	$0.831911\pi$
$308$		−	6.24264i		−	0.355707i
$309$	1.75736			0.0999727
$310$	13.2426	+	4.41421i	0.752131	+	0.250710i
$311$	−4.75736			−0.269765			−0.134883	−	0.990862i	$-0.543066\pi$
$311$	−4.75736			−0.269765			−0.134883	+	0.990862i	$0.543066\pi$
$312$		−	2.41421i		−	0.136678i
$313$		−	7.97056i		−	0.450523i	−0.974298	−	0.225261i	$-0.927676\pi$
$313$		−	7.97056i		−	0.450523i	0.974298	−	0.225261i	$-0.0723236\pi$
$314$	11.6569			0.657834
$315$	−26.4853	−	8.82843i	−1.49228	−	0.497426i
$316$	−6.48528			−0.364826
$317$			9.48528i			0.532746i	0.963870	+	0.266373i	$0.0858254\pi$
$317$			9.48528i			0.532746i	−0.963870	+	0.266373i	$0.914175\pi$
$318$			2.27208i			0.127412i
$319$	−0.242641			−0.0135853
$320$	−0.707107	+	2.12132i	−0.0395285	+	0.118585i
$321$	−8.17157			−0.456093
$322$			3.34315i			0.186306i
$323$		−	1.00000i		−	0.0556415i
$324$	−7.48528			−0.415849
$325$	17.4853	−	23.3137i	0.969909	−	1.29321i
$326$	10.2426			0.567287
$327$		−	7.44365i		−	0.411635i
$328$			4.24264i			0.234261i
$329$		0			0
$330$	−0.414214	+	1.24264i	−0.0228017	+	0.0684051i
$331$	−19.2426			−1.05767			−0.528836	−	0.848724i	$-0.677371\pi$
$331$	−19.2426			−1.05767			−0.528836	+	0.848724i	$0.677371\pi$
$332$		−	14.4853i		−	0.794983i
$333$			24.0000i			1.31519i
$334$	−18.2426			−0.998193
$335$	−10.0919	−	3.36396i	−0.551378	−	0.183793i
$336$	−1.82843			−0.0997489
$337$		−	14.1005i		−	0.768103i	−0.923312	−	0.384052i	$-0.874528\pi$
$337$		−	14.1005i		−	0.768103i	0.923312	−	0.384052i	$-0.125472\pi$
$338$		−	20.9706i		−	1.14065i
$339$	−4.24264			−0.230429
$340$	2.12132	+	0.707107i	0.115045	+	0.0383482i
$341$	−8.82843			−0.478086
$342$			2.82843i			0.152944i
$343$			24.2132i			1.30739i
$344$	1.75736			0.0947505
$345$	0.221825	−	0.665476i	0.0119427	−	0.0358280i
$346$	−0.485281			−0.0260889
$347$			5.51472i			0.296046i	0.988984	+	0.148023i	$0.0472909\pi$
$347$			5.51472i			0.296046i	−0.988984	+	0.148023i	$0.952709\pi$
$348$			0.0710678i			0.00380963i
$349$	−6.00000			−0.321173			−0.160586	−	0.987022i	$-0.551338\pi$
$349$	−6.00000			−0.321173			−0.160586	+	0.987022i	$0.551338\pi$
$350$	−17.6569	−	13.2426i	−0.943799	−	0.707849i
$351$	14.0711			0.751058
$352$		−	1.41421i		−	0.0753778i
$353$		−	2.51472i		−	0.133845i	−0.997758	−	0.0669225i	$-0.978682\pi$
$353$		−	2.51472i		−	0.133845i	0.997758	−	0.0669225i	$-0.0213180\pi$
$354$	−2.85786			−0.151894
$355$	−9.48528	+	28.4558i	−0.503426	+	1.51028i
$356$	7.07107			0.374766
$357$			1.82843i			0.0967706i
$358$			11.6569i			0.616084i
$359$	22.7574			1.20109			0.600544	−	0.799592i	$-0.294951\pi$
$359$	22.7574			1.20109			0.600544	+	0.799592i	$0.294951\pi$
$360$	−6.00000	−	2.00000i	−0.316228	−	0.105409i
$361$	1.00000			0.0526316
$362$		−	8.48528i		−	0.445976i
$363$			3.72792i			0.195665i
$364$	−25.7279			−1.34851
$365$	24.3640	+	8.12132i	1.27527	+	0.425089i
$366$	5.89949			0.308372
$367$			25.4558i			1.32878i	0.747384	+	0.664392i	$0.231309\pi$
$367$			25.4558i			1.32878i	−0.747384	+	0.664392i	$0.768691\pi$
$368$			0.757359i			0.0394801i
$369$	−12.0000			−0.624695
$370$	−6.00000	+	18.0000i	−0.311925	+	0.935775i
$371$	24.2132			1.25709
$372$			2.58579i			0.134067i
$373$			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
$375$	2.63604	+	3.80761i	0.136124	+	0.196624i
$376$		0			0
$377$			1.00000i			0.0515026i
$378$		−	10.6569i		−	0.548129i
$379$	−11.2426			−0.577496			−0.288748	−	0.957405i	$-0.593239\pi$
$379$	−11.2426			−0.577496			−0.288748	+	0.957405i	$0.593239\pi$
$380$	−0.707107	+	2.12132i	−0.0362738	+	0.108821i
$381$	1.02944			0.0527397
$382$			12.5563i			0.642439i
$383$			12.2426i			0.625570i	0.949824	+	0.312785i	$0.101262\pi$
$383$			12.2426i			0.625570i	−0.949824	+	0.312785i	$0.898738\pi$
$384$	−0.414214			−0.0211377
$385$	13.2426	+	4.41421i	0.674907	+	0.224969i
$386$	0.343146			0.0174657
$387$			4.97056i			0.252668i
$388$		−	0.343146i		−	0.0174206i
$389$	−22.9289			−1.16254			−0.581272	−	0.813710i	$-0.697445\pi$
$389$	−22.9289			−1.16254			−0.581272	+	0.813710i	$0.697445\pi$
$390$	5.12132	+	1.70711i	0.259328	+	0.0864427i
$391$	0.757359			0.0383013
$392$			12.4853i			0.630602i
$393$			7.02944i			0.354588i
$394$	−11.7574			−0.592327
$395$	4.58579	−	13.7574i	0.230736	−	0.692208i
$396$	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	$-0.205734\pi$
$398$		−	9.24264i		−	0.463292i
$399$	−1.82843			−0.0915358
$400$	−4.00000	−	3.00000i	−0.200000	−	0.150000i
$401$	−25.4142			−1.26913			−0.634563	−	0.772871i	$-0.718820\pi$
$401$	−25.4142			−1.26913			−0.634563	+	0.772871i	$0.718820\pi$
$402$		−	1.97056i		−	0.0982827i
$403$			36.3848i			1.81245i
$404$	−13.0711			−0.650310
$405$	5.29289	−	15.8787i	0.263006	−	0.789018i
$406$	0.757359			0.0375871
$407$		−	12.0000i		−	0.594818i
$408$			0.414214i			0.0205066i
$409$	−25.2132			−1.24671			−0.623356	−	0.781938i	$-0.714231\pi$
$409$	−25.2132			−1.24671			−0.623356	+	0.781938i	$0.714231\pi$
$410$	−9.00000	−	3.00000i	−0.444478	−	0.148159i
$411$	−5.38478			−0.265611
$412$		−	4.24264i		−	0.209020i
$413$			30.4558i			1.49863i
$414$	−2.14214			−0.105280
$415$	30.7279	+	10.2426i	1.50837	+	0.502791i
$416$	−5.82843			−0.285762
$417$		−	4.97056i		−	0.243410i
$418$		−	1.41421i		−	0.0691714i
$419$	−19.4142			−0.948446			−0.474223	−	0.880405i	$-0.657271\pi$
$419$	−19.4142			−0.948446			−0.474223	+	0.880405i	$0.657271\pi$
$420$	1.29289	−	3.87868i	0.0630867	−	0.189260i
$421$	19.4853			0.949655			0.474827	−	0.880079i	$-0.342511\pi$
$421$	19.4853			0.949655			0.474827	+	0.880079i	$0.342511\pi$
$422$		−	19.7279i		−	0.960340i
$423$		0			0
$424$	5.48528			0.266389
$425$	−3.00000	+	4.00000i	−0.145521	+	0.194029i
$426$	−5.55635			−0.269206
$427$		−	62.8701i		−	3.04250i
$428$			19.7279i			0.953585i
$429$	−3.41421			−0.164840
$430$	−1.24264	+	3.72792i	−0.0599255	+	0.179776i
$431$	−6.38478			−0.307544			−0.153772	−	0.988106i	$-0.549142\pi$
$431$	−6.38478			−0.307544			−0.153772	+	0.988106i	$0.549142\pi$
$432$		−	2.41421i		−	0.116154i
$433$		−	9.55635i		−	0.459249i	−0.973279	−	0.229624i	$-0.926250\pi$
$433$		−	9.55635i		−	0.459249i	0.973279	−	0.229624i	$-0.0737498\pi$
$434$	27.5563			1.32275
$435$	−0.150758	−	0.0502525i	−0.00722827	−	0.00240942i
$436$	−17.9706			−0.860634
$437$			0.757359i			0.0362294i
$438$			4.75736i			0.227315i
$439$	5.75736			0.274784			0.137392	−	0.990517i	$-0.456128\pi$
$439$	5.75736			0.274784			0.137392	+	0.990517i	$0.456128\pi$
$440$	3.00000	+	1.00000i	0.143019	+	0.0476731i
$441$	−35.3137			−1.68161
$442$			5.82843i			0.277230i
$443$			4.24264i			0.201574i	0.994908	+	0.100787i	$0.0321361\pi$
$443$			4.24264i			0.201574i	−0.994908	+	0.100787i	$0.967864\pi$
$444$	−3.51472			−0.166801
$445$	−5.00000	+	15.0000i	−0.237023	+	0.711068i
$446$	15.1716			0.718395
$447$			7.31371i			0.345927i
$448$			4.41421i			0.208552i
$449$	−17.3137			−0.817084			−0.408542	−	0.912739i	$-0.633963\pi$
$449$	−17.3137			−0.817084			−0.408542	+	0.912739i	$0.633963\pi$
$450$	8.48528	−	11.3137i	0.400000	−	0.533333i
$451$	6.00000			0.282529
$452$			10.2426i			0.481773i
$453$			4.34315i			0.204059i
$454$	−16.7574			−0.786462
$455$	18.1924	−	54.5772i	0.852872	−	2.55862i
$456$	−0.414214			−0.0193973
$457$			3.00000i			0.140334i	0.997535	+	0.0701670i	$0.0223532\pi$
$457$			3.00000i			0.140334i	−0.997535	+	0.0701670i	$0.977647\pi$
$458$		−	14.9706i		−	0.699528i
$459$	−2.41421			−0.112686
$460$	−1.60660	−	0.535534i	−0.0749082	−	0.0249694i
$461$	−3.55635			−0.165636			−0.0828178	−	0.996565i	$-0.526392\pi$
$461$	−3.55635			−0.165636			−0.0828178	+	0.996565i	$0.526392\pi$
$462$			2.58579i			0.120302i
$463$			14.1421i			0.657241i	0.944462	+	0.328620i	$0.106584\pi$
$463$			14.1421i			0.657241i	−0.944462	+	0.328620i	$0.893416\pi$
$464$	0.171573			0.00796507
$465$	−5.48528	−	1.82843i	−0.254374	−	0.0847913i
$466$	24.9706			1.15674
$467$		−	0.727922i		−	0.0336842i	−0.999858	−	0.0168421i	$-0.994639\pi$
$467$		−	0.727922i		−	0.0336842i	0.999858	−	0.0168421i	$-0.00536126\pi$
$468$		−	16.4853i		−	0.762032i
$469$	−21.0000			−0.969690
$470$		0			0
$471$	−4.82843			−0.222482
$472$			6.89949i			0.317575i
$473$		−	2.48528i		−	0.114273i
$474$	2.68629			0.123385
$475$	−4.00000	−	3.00000i	−0.183533	−	0.137649i
$476$	4.41421			0.202325
$477$			15.5147i			0.710370i
$478$			6.89949i			0.315576i
$479$	31.1127			1.42158			0.710788	−	0.703407i	$-0.248339\pi$
$479$	31.1127			1.42158			0.710788	+	0.703407i	$0.248339\pi$
$480$	0.292893	−	0.878680i	0.0133687	−	0.0401061i
$481$	−49.4558			−2.25499
$482$			8.97056i			0.408598i
$483$		−	1.38478i		−	0.0630095i
$484$	9.00000			0.409091
$485$	0.727922	+	0.242641i	0.0330532	+	0.0110177i
$486$	10.3431			0.469175
$487$		−	37.7990i		−	1.71284i	−0.516283	−	0.856418i	$-0.672685\pi$
$487$		−	37.7990i		−	1.71284i	0.516283	−	0.856418i	$-0.327315\pi$
$488$		−	14.2426i		−	0.644734i
$489$	−4.24264			−0.191859
$490$	−26.4853	−	8.82843i	−1.19648	−	0.398828i
$491$	−33.5563			−1.51438			−0.757188	−	0.653197i	$-0.773428\pi$
$491$	−33.5563			−1.51438			−0.757188	+	0.653197i	$0.773428\pi$
$492$		−	1.75736i		−	0.0792279i
$493$		−	0.171573i		−	0.00772725i
$494$	−5.82843			−0.262233
$495$	−2.82843	+	8.48528i	−0.127128	+	0.381385i
$496$	6.24264			0.280303
$497$			59.2132i			2.65608i
$498$			6.00000i			0.268866i
$499$	25.7574			1.15306			0.576529	−	0.817077i	$-0.304407\pi$
$499$	25.7574			1.15306			0.576529	+	0.817077i	$0.304407\pi$
$500$	9.19239	−	6.36396i	0.411096	−	0.284605i
$501$	7.55635			0.337593
$502$			3.55635i			0.158728i
$503$			14.2721i			0.636361i	0.948030	+	0.318180i	$0.103072\pi$
$503$			14.2721i			0.636361i	−0.948030	+	0.318180i	$0.896928\pi$
$504$	−12.4853			−0.556139
$505$	9.24264	−	27.7279i	0.411292	−	1.23388i
$506$	1.07107			0.0476148
$507$			8.68629i			0.385772i
$508$		−	2.48528i		−	0.110267i
$509$	−28.9706			−1.28410			−0.642049	−	0.766664i	$-0.721915\pi$
$509$	−28.9706			−1.28410			−0.642049	+	0.766664i	$0.721915\pi$
$510$	−0.878680	−	0.292893i	−0.0389086	−	0.0129695i
$511$	50.6985			2.24277
$512$			1.00000i			0.0441942i
$513$		−	2.41421i		−	0.106590i
$514$	20.7279			0.914269
$515$	9.00000	+	3.00000i	0.396587	+	0.132196i
$516$	−0.727922			−0.0320450
$517$		0			0
$518$			37.4558i			1.64572i
$519$	0.201010			0.00882337
$520$	4.12132	−	12.3640i	0.180732	−	0.542196i
$521$	23.3137			1.02139			0.510696	−	0.859761i	$-0.329388\pi$
$521$	23.3137			1.02139			0.510696	+	0.859761i	$0.329388\pi$
$522$			0.485281i			0.0212402i
$523$		−	2.27208i		−	0.0993510i	−0.998765	−	0.0496755i	$-0.984181\pi$
$523$		−	2.27208i		−	0.0993510i	0.998765	−	0.0496755i	$-0.0158187\pi$
$524$	16.9706			0.741362
$525$	7.31371	+	5.48528i	0.319196	+	0.239397i
$526$	26.9706			1.17597
$527$		−	6.24264i		−	0.271934i
$528$			0.585786i			0.0254931i
$529$	22.4264			0.975061
$530$	−3.87868	+	11.6360i	−0.168479	+	0.505437i
$531$	−19.5147			−0.846867
$532$			4.41421i			0.191380i
$533$		−	24.7279i		−	1.07109i
$534$	−2.92893			−0.126747
$535$	−41.8492	−	13.9497i	−1.80930	−	0.603100i
$536$	−4.75736			−0.205487
$537$		−	4.82843i		−	0.208362i
$538$			16.6274i			0.716859i
$539$	17.6569			0.760535
$540$	5.12132	+	1.70711i	0.220387	+	0.0734622i
$541$	9.75736			0.419502			0.209751	−	0.977755i	$-0.432735\pi$
$541$	9.75736			0.419502			0.209751	+	0.977755i	$0.432735\pi$
$542$		−	27.2426i		−	1.17017i
$543$			3.51472i			0.150831i
$544$	1.00000			0.0428746
$545$	12.7071	−	38.1213i	0.544313	−	1.63294i
$546$	10.6569			0.456071
$547$			17.3137i			0.740281i	0.928976	+	0.370140i	$0.120690\pi$
$547$			17.3137i			0.740281i	−0.928976	+	0.370140i	$0.879310\pi$
$548$			13.0000i			0.555332i
$549$	40.2843			1.71929
$550$	−4.24264	+	5.65685i	−0.180907	+	0.241209i
$551$	0.171573			0.00730925
$552$		−	0.313708i		−	0.0133523i
$553$		−	28.6274i		−	1.21736i
$554$		0			0
$555$	2.48528	−	7.45584i	0.105494	−	0.316483i
$556$	−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	$-0.110079\pi$
$558$			17.6569i			0.747474i
$559$	−10.2426			−0.433218
$560$	−9.36396	−	3.12132i	−0.395700	−	0.131900i
$561$	0.585786			0.0247319
$562$			4.24264i			0.178965i
$563$			4.97056i			0.209484i	0.994499	+	0.104742i	$0.0334017\pi$
$563$			4.97056i			0.209484i	−0.994499	+	0.104742i	$0.966598\pi$
$564$		0			0
$565$	−21.7279	−	7.24264i	−0.914101	−	0.304700i
$566$	−32.1421			−1.35103
$567$		−	33.0416i		−	1.38762i
$568$			13.4142i			0.562848i
$569$	28.2843			1.18574			0.592869	−	0.805299i	$-0.297995\pi$
$569$	28.2843			1.18574			0.592869	+	0.805299i	$0.297995\pi$
$570$	0.292893	−	0.878680i	0.0122679	−	0.0368038i
$571$	−2.24264			−0.0938516			−0.0469258	−	0.998898i	$-0.514942\pi$
$571$	−2.24264			−0.0938516			−0.0469258	+	0.998898i	$0.514942\pi$
$572$			8.24264i			0.344642i
$573$		−	5.20101i		−	0.217275i
$574$	−18.7279			−0.781688
$575$	2.27208	−	3.02944i	0.0947522	−	0.126336i
$576$	−2.82843			−0.117851
$577$			20.3137i			0.845671i	0.906207	+	0.422835i	$0.138965\pi$
$577$			20.3137i			0.845671i	−0.906207	+	0.422835i	$0.861035\pi$
$578$			16.0000i			0.665512i
$579$	−0.142136			−0.00590695
$580$	−0.121320	+	0.363961i	−0.00503755	+	0.0151127i
$581$	63.9411			2.65272
$582$			0.142136i			0.00589171i
$583$		−	7.75736i		−	0.321277i
$584$	11.4853			0.475264
$585$	34.9706	+	11.6569i	1.44585	+	0.481952i
$586$	5.48528			0.226595
$587$		−	30.2426i		−	1.24825i	−0.781326	−	0.624124i	$-0.785456\pi$
$587$		−	30.2426i		−	1.24825i	0.781326	−	0.624124i	$-0.214544\pi$
$588$		−	5.17157i		−	0.213272i
$589$	6.24264			0.257224
$590$	−14.6360	−	4.87868i	−0.602556	−	0.200852i
$591$	4.87006			0.200327
$592$			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	$0.850812\pi$
$594$	−3.41421			−0.140087
$595$	−3.12132	+	9.36396i	−0.127962	+	0.383885i
$596$	17.6569			0.723253
$597$			3.82843i			0.156687i
$598$		−	4.41421i		−	0.180511i
$599$	15.2132			0.621595			0.310797	−	0.950476i	$-0.399404\pi$
$599$	15.2132			0.621595			0.310797	+	0.950476i	$0.399404\pi$
$600$	1.65685	+	1.24264i	0.0676408	+	0.0507306i
$601$	−20.2426			−0.825715			−0.412857	−	0.910796i	$-0.635469\pi$
$601$	−20.2426			−0.825715			−0.412857	+	0.910796i	$0.635469\pi$
$602$			7.75736i			0.316166i
$603$		−	13.4558i		−	0.547964i
$604$	10.4853			0.426640
$605$	−6.36396	+	19.0919i	−0.258732	+	0.776195i
$606$	5.41421			0.219937
$607$		−	3.17157i		−	0.128730i	−0.997926	−	0.0643651i	$-0.979498\pi$
$607$		−	3.17157i		−	0.128730i	0.997926	−	0.0643651i	$-0.0205022\pi$
$608$			1.00000i			0.0405554i
$609$	−0.313708			−0.0127121
$610$	30.2132	+	10.0711i	1.22330	+	0.407766i
$611$		0			0
$612$			2.82843i			0.114332i
$613$		−	11.6985i		−	0.472497i	−0.971693	−	0.236249i	$-0.924082\pi$
$613$		−	11.6985i		−	0.472497i	0.971693	−	0.236249i	$-0.0759180\pi$
$614$	−17.6569			−0.712573
$615$	3.72792	+	1.24264i	0.150324	+	0.0501081i
$616$	6.24264			0.251523
$617$			4.48528i			0.180571i	0.995916	+	0.0902853i	$0.0287779\pi$
$617$			4.48528i			0.180571i	−0.995916	+	0.0902853i	$0.971222\pi$
$618$			1.75736i			0.0706914i
$619$	−7.75736			−0.311795			−0.155897	−	0.987773i	$-0.549827\pi$
$619$	−7.75736			−0.311795			−0.155897	+	0.987773i	$0.549827\pi$
$620$	−4.41421	+	13.2426i	−0.177279	+	0.531837i
$621$	1.82843			0.0733723
$622$		−	4.75736i		−	0.190753i
$623$			31.2132i			1.25053i
$624$	2.41421			0.0966459
$625$	7.00000	+	24.0000i	0.280000	+	0.960000i
$626$	7.97056			0.318568
$627$			0.585786i			0.0233941i
$628$			11.6569i			0.465159i
$629$	8.48528			0.338330
$630$	8.82843	−	26.4853i	0.351733	−	1.05520i
$631$	−38.9706			−1.55139			−0.775697	−	0.631106i	$-0.782601\pi$
$631$	−38.9706			−1.55139			−0.775697	+	0.631106i	$0.782601\pi$
$632$		−	6.48528i		−	0.257971i
$633$			8.17157i			0.324791i
$634$	−9.48528			−0.376709
$635$	5.27208	+	1.75736i	0.209216	+	0.0697387i
$636$	−2.27208			−0.0900938
$637$		−	72.7696i		−	2.88323i
$638$		−	0.242641i		−	0.00960624i
$639$	−37.9411			−1.50093
$640$	−2.12132	−	0.707107i	−0.0838525	−	0.0279508i
$641$	−18.0416			−0.712602			−0.356301	−	0.934371i	$-0.615962\pi$
$641$	−18.0416			−0.712602			−0.356301	+	0.934371i	$0.615962\pi$
$642$		−	8.17157i		−	0.322506i
$643$		−	14.4853i		−	0.571244i	−0.958342	−	0.285622i	$-0.907800\pi$
$643$		−	14.4853i		−	0.571244i	0.958342	−	0.285622i	$-0.0922001\pi$
$644$	−3.34315			−0.131738
$645$	0.514719	−	1.54416i	0.0202670	−	0.0608011i
$646$	1.00000			0.0393445
$647$			18.7574i			0.737428i	0.929543	+	0.368714i	$0.120202\pi$
$647$			18.7574i			0.737428i	−0.929543	+	0.368714i	$0.879798\pi$
$648$		−	7.48528i		−	0.294050i
$649$	9.75736			0.383010
$650$	23.3137	+	17.4853i	0.914439	+	0.685829i
$651$	−11.4142			−0.447358
$652$			10.2426i			0.401133i
$653$		−	28.9706i		−	1.13371i	−0.823819	−	0.566853i	$-0.808161\pi$
$653$		−	28.9706i		−	1.13371i	0.823819	−	0.566853i	$-0.191839\pi$
$654$	7.44365			0.291070
$655$	−12.0000	+	36.0000i	−0.468879	+	1.40664i
$656$	−4.24264			−0.165647
$657$			32.4853i			1.26737i
$658$		0			0
$659$	−18.8995			−0.736220			−0.368110	−	0.929782i	$-0.619995\pi$
$659$	−18.8995			−0.736220			−0.368110	+	0.929782i	$0.619995\pi$
$660$	−1.24264	−	0.414214i	−0.0483697	−	0.0161232i
$661$	−18.4558			−0.717849			−0.358925	−	0.933367i	$-0.616856\pi$
$661$	−18.4558			−0.717849			−0.358925	+	0.933367i	$0.616856\pi$
$662$		−	19.2426i		−	0.747886i
$663$		−	2.41421i		−	0.0937603i
$664$	14.4853			0.562138
$665$	−9.36396	−	3.12132i	−0.363119	−	0.121040i
$666$	−24.0000			−0.929981
$667$			0.129942i			0.00503139i
$668$		−	18.2426i		−	0.705829i
$669$	−6.28427			−0.242964
$670$	3.36396	−	10.0919i	0.129961	−	0.389883i
$671$	−20.1421			−0.777579
$672$		−	1.82843i		−	0.0705331i
$673$		−	12.0000i		−	0.462566i	−0.972887	−	0.231283i	$-0.925708\pi$
$673$		−	12.0000i		−	0.462566i	0.972887	−	0.231283i	$-0.0742923\pi$
$674$	14.1005			0.543131
$675$	−7.24264	+	9.65685i	−0.278769	+	0.371692i
$676$	20.9706			0.806560
$677$			40.9411i			1.57350i	0.617275	+	0.786748i	$0.288237\pi$
$677$			40.9411i			1.57350i	−0.617275	+	0.786748i	$0.711763\pi$
$678$		−	4.24264i		−	0.162938i
$679$	1.51472			0.0581296
$680$	−0.707107	+	2.12132i	−0.0271163	+	0.0813489i
$681$	6.94113			0.265985
$682$		−	8.82843i		−	0.338058i
$683$			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
$686$	−24.2132			−0.924464
$687$			6.20101i			0.236583i
$688$			1.75736i			0.0669987i
$689$	−31.9706			−1.21798
$690$	0.665476	+	0.221825i	0.0253342	+	0.00844475i
$691$	22.4853			0.855380			0.427690	−	0.903925i	$-0.359327\pi$
$691$	22.4853			0.855380			0.427690	+	0.903925i	$0.359327\pi$
$692$		−	0.485281i		−	0.0184476i
$693$			17.6569i			0.670728i
$694$	−5.51472			−0.209336
$695$	8.48528	−	25.4558i	0.321865	−	0.965595i
$696$	−0.0710678			−0.00269382
$697$			4.24264i			0.160701i
$698$		−	6.00000i		−	0.227103i
$699$	−10.3431			−0.391214
$700$	13.2426	−	17.6569i	0.500525	−	0.667366i
$701$	16.9706			0.640969			0.320485	−	0.947254i	$-0.396154\pi$
$701$	16.9706			0.640969			0.320485	+	0.947254i	$0.396154\pi$
$702$			14.0711i			0.531078i
$703$			8.48528i			0.320028i
$704$	1.41421			0.0533002
$705$		0			0
$706$	2.51472			0.0946427
$707$		−	57.6985i		−	2.16997i
$708$		−	2.85786i		−	0.107405i
$709$	34.0000			1.27690			0.638448	−	0.769665i	$-0.279577\pi$
$709$	34.0000			1.27690			0.638448	+	0.769665i	$0.279577\pi$
$710$	−28.4558	−	9.48528i	−1.06793	−	0.355976i
$711$	18.3431			0.687922
$712$			7.07107i			0.264999i
$713$			4.72792i			0.177062i
$714$	−1.82843			−0.0684272
$715$	−17.4853	−	5.82843i	−0.653912	−	0.217971i
$716$	−11.6569			−0.435637
$717$		−	2.85786i		−	0.106729i
$718$			22.7574i			0.849297i
$719$	5.10051			0.190217			0.0951084	−	0.995467i	$-0.469680\pi$
$719$	5.10051			0.190217			0.0951084	+	0.995467i	$0.469680\pi$
$720$	2.00000	−	6.00000i	0.0745356	−	0.223607i
$721$	18.7279			0.697464
$722$			1.00000i			0.0372161i
$723$		−	3.71573i		−	0.138189i
$724$	8.48528			0.315353
$725$	−0.686292	−	0.514719i	−0.0254882	−	0.0191162i
$726$	−3.72792			−0.138356
$727$		−	9.72792i		−	0.360789i	−0.983594	−	0.180394i	$-0.942263\pi$
$727$		−	9.72792i		−	0.360789i	0.983594	−	0.180394i	$-0.0577374\pi$
$728$		−	25.7279i		−	0.953540i
$729$	18.1716			0.673021
$730$	−8.12132	+	24.3640i	−0.300584	+	0.901751i
$731$	1.75736			0.0649983
$732$			5.89949i			0.218052i
$733$			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
$735$	10.9706	+	3.65685i	0.404655	+	0.134885i
$736$	−0.757359			−0.0279166
$737$			6.72792i			0.247826i
$738$		−	12.0000i		−	0.441726i
$739$	−1.27208			−0.0467941			−0.0233971	−	0.999726i	$-0.507448\pi$
$739$	−1.27208			−0.0467941			−0.0233971	+	0.999726i	$0.507448\pi$
$740$	−18.0000	−	6.00000i	−0.661693	−	0.220564i
$741$	2.41421			0.0886884
$742$			24.2132i			0.888895i
$743$			6.72792i			0.246824i	0.992356	+	0.123412i	$0.0393836\pi$
$743$			6.72792i			0.246824i	−0.992356	+	0.123412i	$0.960616\pi$
$744$	−2.58579			−0.0947995
$745$	−12.4853	+	37.4558i	−0.457425	+	1.37228i
$746$	−9.00000			−0.329513
$747$			40.9706i			1.49903i
$748$		−	1.41421i		−	0.0517088i
$749$	−87.0833			−3.18195
$750$	−3.80761	+	2.63604i	−0.139034	+	0.0962545i
$751$	−26.7279			−0.975316			−0.487658	−	0.873035i	$-0.662149\pi$
$751$	−26.7279			−0.975316			−0.487658	+	0.873035i	$0.662149\pi$
$752$		0			0
$753$		−	1.47309i		−	0.0536823i
$754$	−1.00000			−0.0364179
$755$	−7.41421	+	22.2426i	−0.269831	+	0.809493i
$756$	10.6569			0.387586
$757$		−	12.3431i		−	0.448619i	−0.974518	−	0.224310i	$-0.927987\pi$
$757$		−	12.3431i		−	0.448619i	0.974518	−	0.224310i	$-0.0720127\pi$
$758$		−	11.2426i		−	0.408351i
$759$	−0.443651			−0.0161035
$760$	−2.12132	−	0.707107i	−0.0769484	−	0.0256495i
$761$	43.9706			1.59393			0.796966	−	0.604024i	$-0.206437\pi$
$761$	43.9706			1.59393			0.796966	+	0.604024i	$0.206437\pi$
$762$			1.02944i			0.0372926i
$763$		−	79.3259i		−	2.87179i
$764$	−12.5563			−0.454273
$765$	−6.00000	−	2.00000i	−0.216930	−	0.0723102i
$766$	−12.2426			−0.442345
$767$		−	40.2132i		−	1.45201i
$768$		−	0.414214i		−	0.0149466i
$769$	36.4558			1.31463			0.657316	−	0.753615i	$-0.271692\pi$
$769$	36.4558			1.31463			0.657316	+	0.753615i	$0.271692\pi$
$770$	−4.41421	+	13.2426i	−0.159077	+	0.477232i
$771$	−8.58579			−0.309210
$772$			0.343146i			0.0123501i
$773$			13.9706i			0.502486i	0.967924	+	0.251243i	$0.0808394\pi$
$773$			13.9706i			0.502486i	−0.967924	+	0.251243i	$0.919161\pi$
$774$	−4.97056			−0.178663
$775$	−24.9706	−	18.7279i	−0.896969	−	0.672727i
$776$	0.343146			0.0123182
$777$		−	15.5147i		−	0.556587i
$778$		−	22.9289i		−	0.822042i
$779$	−4.24264			−0.152008
$780$	−1.70711	+	5.12132i	−0.0611242	+	0.183373i
$781$	18.9706			0.678820
$782$			0.757359i

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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -0.414214i q^{3} -1.00000 q^{4} +(0.707107 - 2.12132i) q^{5} +0.414214 q^{6} -4.41421i q^{7} -1.00000i q^{8} +2.82843 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -0.414214i q^{3} -1.00000 q^{4} +(0.707107 - 2.12132i) q^{5} +0.414214 q^{6} -4.41421i q^{7} -1.00000i q^{8} +2.82843 q^{9} +(2.12132 + 0.707107i) q^{10} -1.41421 q^{11} +0.414214i q^{12} +5.82843i q^{13} +4.41421 q^{14} +(-0.878680 - 0.292893i) q^{15} +1.00000 q^{16} -1.00000i q^{17} +2.82843i q^{18} +1.00000 q^{19} +(-0.707107 + 2.12132i) q^{20} -1.82843 q^{21} -1.41421i q^{22} +0.757359i q^{23} -0.414214 q^{24} +(-4.00000 - 3.00000i) q^{25} -5.82843 q^{26} -2.41421i q^{27} +4.41421i q^{28} +0.171573 q^{29} +(0.292893 - 0.878680i) q^{30} +6.24264 q^{31} +1.00000i q^{32} +0.585786i q^{33} +1.00000 q^{34} +(-9.36396 - 3.12132i) q^{35} -2.82843 q^{36} +8.48528i q^{37} +1.00000i q^{38} +2.41421 q^{39} +(-2.12132 - 0.707107i) q^{40} -4.24264 q^{41} -1.82843i q^{42} +1.75736i q^{43} +1.41421 q^{44} +(2.00000 - 6.00000i) q^{45} -0.757359 q^{46} -0.414214i q^{48} -12.4853 q^{49} +(3.00000 - 4.00000i) q^{50} -0.414214 q^{51} -5.82843i q^{52} +5.48528i q^{53} +2.41421 q^{54} +(-1.00000 + 3.00000i) q^{55} -4.41421 q^{56} -0.414214i q^{57} +0.171573i q^{58} -6.89949 q^{59} +(0.878680 + 0.292893i) q^{60} +14.2426 q^{61} +6.24264i q^{62} -12.4853i q^{63} -1.00000 q^{64} +(12.3640 + 4.12132i) q^{65} -0.585786 q^{66} -4.75736i q^{67} +1.00000i q^{68} +0.313708 q^{69} +(3.12132 - 9.36396i) q^{70} -13.4142 q^{71} -2.82843i q^{72} +11.4853i q^{73} -8.48528 q^{74} +(-1.24264 + 1.65685i) q^{75} -1.00000 q^{76} +6.24264i q^{77} +2.41421i q^{78} +6.48528 q^{79} +(0.707107 - 2.12132i) q^{80} +7.48528 q^{81} -4.24264i q^{82} +14.4853i q^{83} +1.82843 q^{84} +(-2.12132 - 0.707107i) q^{85} -1.75736 q^{86} -0.0710678i q^{87} +1.41421i q^{88} -7.07107 q^{89} +(6.00000 + 2.00000i) q^{90} +25.7279 q^{91} -0.757359i q^{92} -2.58579i q^{93} +(0.707107 - 2.12132i) q^{95} +0.414214 q^{96} +0.343146i q^{97} -12.4853i q^{98} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} - 4 q^{6}+O(q^{10}) \) 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

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