LMFDB - Embedded newform 1520.2.q.j.961.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1520,2,Mod(881,1520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1520, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))

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

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

chi := DirichletCharacter("1520.881");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1520 = 2^{4} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1520.q (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$12.1372611072$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{3})$
Coefficient field:	6.0.3518667.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} + 7x^{4} - 8x^{3} + 43x^{2} - 42x + 49 $ x^6 - x^5 + 7x^4 - 8x^3 + 43x^2 - 42x + 49
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 95)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			961.2
Root			$0.610938 - 1.05818i$ of defining polynomial
Character	$\chi$	$=$	1520.961
Dual form			1520.2.q.j.881.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.610938 - 1.05818i) q^{3} +(0.500000 - 0.866025i) q^{5} +0.221876 q^{7} +(0.753509 + 1.30512i) q^{9} +O(q^{10})$	\(q+(0.610938 - 1.05818i) q^{3} +(0.500000 - 0.866025i) q^{5} +0.221876 q^{7} +(0.753509 + 1.30512i) q^{9} +0.778124 q^{11} +(2.50000 + 4.33013i) q^{13} +(-0.610938 - 1.05818i) q^{15} +(-3.53865 + 6.12912i) q^{17} +(-1.33281 + 4.15013i) q^{19} +(0.135553 - 0.234784i) q^{21} +(4.03865 + 6.99515i) q^{23} +(-0.500000 - 0.866025i) q^{25} +5.50702 q^{27} +(-0.110938 - 0.192150i) q^{29} -2.50702 q^{31} +(0.475385 - 0.823392i) q^{33} +(0.110938 - 0.192150i) q^{35} -1.90466 q^{37} +6.10938 q^{39} +(3.61796 - 6.26648i) q^{41} +(3.64959 - 6.32128i) q^{43} +1.50702 q^{45} +(-1.39608 - 2.41808i) q^{47} -6.95077 q^{49} +(4.32379 + 7.48903i) q^{51} +(-2.19024 - 3.79361i) q^{53} +(0.389062 - 0.673875i) q^{55} +(3.57730 + 3.94583i) q^{57} +(-1.39608 + 2.41808i) q^{59} +(6.29216 + 10.8983i) q^{61} +(0.167186 + 0.289574i) q^{63} +5.00000 q^{65} +(-5.28514 - 9.15414i) q^{67} +9.86946 q^{69} +(-4.92070 + 8.52289i) q^{71} +(7.03865 - 12.1913i) q^{73} -1.22188 q^{75} +0.172647 q^{77} +(0.792161 - 1.37206i) q^{79} +(1.10392 - 1.91204i) q^{81} +9.52106 q^{83} +(3.53865 + 6.12912i) q^{85} -0.271105 q^{87} +(-1.57028 - 2.71981i) q^{89} +(0.554690 + 0.960752i) q^{91} +(-1.53163 + 2.65287i) q^{93} +(2.92771 + 3.22932i) q^{95} +(3.18122 - 5.51004i) q^{97} +(0.586324 + 1.01554i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + q^{3} + 3 q^{5} - 4 q^{7} - 4 q^{9}+O(q^{10}) $ 6 * q + q^3 + 3 * q^5 - 4 * q^7 - 4 * q^9	$ 6 q + q^{3} + 3 q^{5} - 4 q^{7} - 4 q^{9} + 10 q^{11} + 15 q^{13} - q^{15} - q^{17} + 12 q^{21} + 4 q^{23} - 3 q^{25} + 16 q^{27} + 2 q^{29} + 2 q^{31} - 11 q^{33} - 2 q^{35} - 4 q^{37} + 10 q^{39} + 2 q^{41} - q^{43} - 8 q^{45} + 6 q^{47} - 14 q^{49} - 6 q^{51} - 11 q^{53} + 5 q^{55} - 19 q^{57} + 6 q^{59} + 9 q^{61} + 9 q^{63} + 30 q^{65} - 20 q^{67} - 10 q^{69} - 29 q^{71} + 22 q^{73} - 2 q^{75} - 32 q^{77} - 24 q^{79} + 21 q^{81} + 6 q^{83} + q^{85} - 24 q^{87} + 14 q^{89} - 10 q^{91} - 6 q^{93} - 7 q^{97} - 13 q^{99}+O(q^{100}) $ 6 * q + q^3 + 3 * q^5 - 4 * q^7 - 4 * q^9 + 10 * q^11 + 15 * q^13 - q^15 - q^17 + 12 * q^21 + 4 * q^23 - 3 * q^25 + 16 * q^27 + 2 * q^29 + 2 * q^31 - 11 * q^33 - 2 * q^35 - 4 * q^37 + 10 * q^39 + 2 * q^41 - q^43 - 8 * q^45 + 6 * q^47 - 14 * q^49 - 6 * q^51 - 11 * q^53 + 5 * q^55 - 19 * q^57 + 6 * q^59 + 9 * q^61 + 9 * q^63 + 30 * q^65 - 20 * q^67 - 10 * q^69 - 29 * q^71 + 22 * q^73 - 2 * q^75 - 32 * q^77 - 24 * q^79 + 21 * q^81 + 6 * q^83 + q^85 - 24 * q^87 + 14 * q^89 - 10 * q^91 - 6 * q^93 - 7 * q^97 - 13 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1520\mathbb{Z}\right)^\times$.

$n$	$191$	$401$	$1141$	$1217$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	0.610938	−	1.05818i	0.352725	−	0.610938i	−0.634001	−	0.773333i	$-0.718588\pi$
$3$	0.610938	−	1.05818i	0.352725	−	0.610938i	0.986726	+	0.162394i	$0.0519217\pi$
$4$		0			0
$5$	0.500000	−	0.866025i	0.223607	−	0.387298i
$5$	0.500000	−	0.866025i	0.223607	−	0.387298i
$6$		0			0
$7$	0.221876			0.0838613			0.0419307	−	0.999121i	$-0.486649\pi$
$7$	0.221876			0.0838613			0.0419307	+	0.999121i	$0.486649\pi$
$8$		0			0
$9$	0.753509	+	1.30512i	0.251170	+	0.435039i
$10$		0			0
$11$	0.778124			0.234613			0.117307	−	0.993096i	$-0.462574\pi$
$11$	0.778124			0.234613			0.117307	+	0.993096i	$0.462574\pi$
$12$		0			0
$13$	2.50000	+	4.33013i	0.693375	+	1.20096i	0.970725	+	0.240192i	$0.0772105\pi$
$13$	2.50000	+	4.33013i	0.693375	+	1.20096i	−0.277350	+	0.960769i	$0.589456\pi$
$14$		0			0
$15$	−0.610938	−	1.05818i	−0.157744	−	0.273220i
$16$		0			0
$17$	−3.53865	+	6.12912i	−0.858249	+	1.48653i	0.0153485	+	0.999882i	$0.495114\pi$
$17$	−3.53865	+	6.12912i	−0.858249	+	1.48653i	−0.873598	+	0.486649i	$0.838219\pi$
$18$		0			0
$19$	−1.33281	+	4.15013i	−0.305769	+	0.952106i
$19$	−1.33281	+	4.15013i	−0.305769	+	0.952106i
$20$		0			0
$21$	0.135553	−	0.234784i	0.0295800	−	0.0512341i
$22$		0			0
$23$	4.03865	+	6.99515i	0.842117	+	1.45859i	0.888101	+	0.459647i	$0.152024\pi$
$23$	4.03865	+	6.99515i	0.842117	+	1.45859i	−0.0459843	+	0.998942i	$0.514642\pi$
$24$		0			0
$25$	−0.500000	−	0.866025i	−0.100000	−	0.173205i
$26$		0			0
$27$	5.50702			1.05983
$28$		0			0
$29$	−0.110938	−	0.192150i	−0.0206007	−	0.0356814i	0.855541	−	0.517735i	$-0.173225\pi$
$29$	−0.110938	−	0.192150i	−0.0206007	−	0.0356814i	−0.876142	+	0.482053i	$0.839891\pi$
$30$		0			0
$31$	−2.50702			−0.450274			−0.225137	−	0.974327i	$-0.572283\pi$
$31$	−2.50702			−0.450274			−0.225137	+	0.974327i	$0.572283\pi$
$32$		0			0
$33$	0.475385	−	0.823392i	0.0827540	−	0.143334i
$34$		0			0
$35$	0.110938	−	0.192150i	0.0187520	−	0.0324793i
$36$		0			0
$37$	−1.90466			−0.313124			−0.156562	−	0.987668i	$-0.550041\pi$
$37$	−1.90466			−0.313124			−0.156562	+	0.987668i	$0.550041\pi$
$38$		0			0
$39$	6.10938			0.978284
$40$		0			0
$41$	3.61796	−	6.26648i	0.565030	−	0.978661i	−0.432017	−	0.901865i	$-0.642198\pi$
$41$	3.61796	−	6.26648i	0.565030	−	0.978661i	0.997047	−	0.0767950i	$-0.0244687\pi$
$42$		0			0
$43$	3.64959	−	6.32128i	0.556557	−	0.963985i	−0.441223	−	0.897397i	$-0.645455\pi$
$43$	3.64959	−	6.32128i	0.556557	−	0.963985i	0.997781	−	0.0665881i	$-0.0212113\pi$
$44$		0			0
$45$	1.50702			0.224653
$46$		0			0
$47$	−1.39608	−	2.41808i	−0.203639	−	0.352714i	0.746059	−	0.665880i	$-0.231944\pi$
$47$	−1.39608	−	2.41808i	−0.203639	−	0.352714i	−0.949698	+	0.313166i	$0.898610\pi$
$48$		0			0
$49$	−6.95077			−0.992967
$50$		0			0
$51$	4.32379	+	7.48903i	0.605452	+	1.04867i
$52$		0			0
$53$	−2.19024	−	3.79361i	−0.300853	−	0.521093i	0.675476	−	0.737382i	$-0.263938\pi$
$53$	−2.19024	−	3.79361i	−0.300853	−	0.521093i	−0.976329	+	0.216289i	$0.930605\pi$
$54$		0			0
$55$	0.389062	−	0.673875i	0.0524611	−	0.0908653i
$56$		0			0
$57$	3.57730	+	3.94583i	0.473825	+	0.522637i
$58$		0			0
$59$	−1.39608	+	2.41808i	−0.181754	+	0.314808i	−0.942478	−	0.334268i	$-0.891511\pi$
$59$	−1.39608	+	2.41808i	−0.181754	+	0.314808i	0.760724	+	0.649076i	$0.224844\pi$
$60$		0			0
$61$	6.29216	+	10.8983i	0.805629	+	1.39539i	0.915866	+	0.401484i	$0.131506\pi$
$61$	6.29216	+	10.8983i	0.805629	+	1.39539i	−0.110237	+	0.993905i	$0.535161\pi$
$62$		0			0
$63$	0.167186	+	0.289574i	0.0210634	+	0.0364829i
$64$		0			0
$65$	5.00000			0.620174
$66$		0			0
$67$	−5.28514	−	9.15414i	−0.645683	−	1.11836i	−0.984143	−	0.177375i	$-0.943239\pi$
$67$	−5.28514	−	9.15414i	−0.645683	−	1.11836i	0.338460	−	0.940981i	$-0.390094\pi$
$68$		0			0
$69$	9.86946			1.18814
$70$		0			0
$71$	−4.92070	+	8.52289i	−0.583979	+	1.01148i	0.411023	+	0.911625i	$0.365172\pi$
$71$	−4.92070	+	8.52289i	−0.583979	+	1.01148i	−0.995002	+	0.0998563i	$0.968162\pi$
$72$		0			0
$73$	7.03865	−	12.1913i	0.823812	−	1.42688i	−0.0790121	−	0.996874i	$-0.525177\pi$
$73$	7.03865	−	12.1913i	0.823812	−	1.42688i	0.902824	−	0.430010i	$-0.141490\pi$
$74$		0			0
$75$	−1.22188			−0.141090
$76$		0			0
$77$	0.172647			0.0196750
$78$		0			0
$79$	0.792161	−	1.37206i	0.0891251	−	0.154369i	−0.818016	−	0.575195i	$-0.804926\pi$
$79$	0.792161	−	1.37206i	0.0891251	−	0.154369i	0.907142	+	0.420826i	$0.138260\pi$
$80$		0			0
$81$	1.10392	−	1.91204i	0.122658	−	0.212449i
$82$		0			0
$83$	9.52106			1.04507			0.522536	−	0.852617i	$-0.324986\pi$
$83$	9.52106			1.04507			0.522536	+	0.852617i	$0.324986\pi$
$84$		0			0
$85$	3.53865	+	6.12912i	0.383821	+	0.664797i
$86$		0			0
$87$	−0.271105			−0.0290655
$88$		0			0
$89$	−1.57028	−	2.71981i	−0.166450	−	0.288300i	0.770719	−	0.637175i	$-0.219897\pi$
$89$	−1.57028	−	2.71981i	−0.166450	−	0.288300i	−0.937169	+	0.348875i	$0.886564\pi$
$90$		0			0
$91$	0.554690	+	0.960752i	0.0581474	+	0.100714i
$92$		0			0
$93$	−1.53163	+	2.65287i	−0.158823	+	0.275089i
$94$		0			0
$95$	2.92771	+	3.22932i	0.300377	+	0.331321i
$96$		0			0
$97$	3.18122	−	5.51004i	0.323004	−	0.559460i	−0.658102	−	0.752929i	$-0.728641\pi$
$97$	3.18122	−	5.51004i	0.323004	−	0.559460i	0.981106	+	0.193469i	$0.0619738\pi$
$98$		0			0
$99$	0.586324	+	1.01554i	0.0589277	+	0.102066i
$100$		0			0
$101$	3.69726	+	6.40385i	0.367891	+	0.637206i	0.989236	−	0.146331i	$-0.0467465\pi$
$101$	3.69726	+	6.40385i	0.367891	+	0.637206i	−0.621344	+	0.783538i	$0.713413\pi$
$102$		0			0
$103$	−12.2038			−1.20248			−0.601240	−	0.799069i	$-0.705326\pi$
$103$	−12.2038			−1.20248			−0.601240	+	0.799069i	$0.705326\pi$
$104$		0			0
$105$	−0.135553	−	0.234784i	−0.0132286	−	0.0229126i
$106$		0			0
$107$	1.63355			0.157921			0.0789607	−	0.996878i	$-0.474840\pi$
$107$	1.63355			0.157921			0.0789607	+	0.996878i	$0.474840\pi$
$108$		0			0
$109$	3.80820	−	6.59600i	0.364759	−	0.631782i	−0.623978	−	0.781442i	$-0.714485\pi$
$109$	3.80820	−	6.59600i	0.364759	−	0.631782i	0.988738	+	0.149660i	$0.0478179\pi$
$110$		0			0
$111$	−1.16363	+	2.01546i	−0.110447	+	0.191299i
$112$		0			0
$113$	12.4890			1.17486			0.587432	−	0.809273i	$-0.300139\pi$
$113$	12.4890			1.17486			0.587432	+	0.809273i	$0.300139\pi$
$114$		0			0
$115$	8.07730			0.753212
$116$		0			0
$117$	−3.76755	+	6.52558i	−0.348310	+	0.603290i
$118$		0			0
$119$	−0.785142	+	1.35991i	−0.0719739	+	0.124662i
$120$		0			0
$121$	−10.3945			−0.944957
$122$		0			0
$123$	−4.42070	−	7.65687i	−0.398601	−	0.690397i
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	−6.52106	−	11.2948i	−0.578650	−	1.00225i	−0.995635	−	0.0933378i	$-0.970246\pi$
$127$	−6.52106	−	11.2948i	−0.578650	−	1.00225i	0.416984	−	0.908914i	$-0.363087\pi$
$128$		0			0
$129$	−4.45935	−	7.72382i	−0.392624	−	0.680044i
$130$		0			0
$131$	3.55469	−	6.15690i	0.310575	−	0.537931i	−0.667912	−	0.744240i	$-0.732812\pi$
$131$	3.55469	−	6.15690i	0.310575	−	0.537931i	0.978487	+	0.206309i	$0.0661452\pi$
$132$		0			0
$133$	−0.295720	+	0.920816i	−0.0256422	+	0.0798448i
$134$		0			0
$135$	2.75351	−	4.76922i	0.236984	−	0.410469i
$136$		0			0
$137$	−2.71286	−	4.69880i	−0.231775	−	0.401446i	0.726556	−	0.687108i	$-0.241120\pi$
$137$	−2.71286	−	4.69880i	−0.231775	−	0.401446i	−0.958331	+	0.285662i	$0.907787\pi$
$138$		0			0
$139$	1.78514	+	3.09196i	0.151414	+	0.262256i	0.931747	−	0.363107i	$-0.118284\pi$
$139$	1.78514	+	3.09196i	0.151414	+	0.262256i	−0.780334	+	0.625363i	$0.784951\pi$
$140$		0			0
$141$	−3.41168			−0.287315
$142$		0			0
$143$	1.94531	+	3.36938i	0.162675	+	0.281761i
$144$		0			0
$145$	−0.221876			−0.0184258
$146$		0			0
$147$	−4.24649	+	7.35514i	−0.350245	+	0.606642i
$148$		0			0
$149$	0.468367	−	0.811235i	0.0383701	−	0.0664590i	−0.846203	−	0.532861i	$-0.821117\pi$
$149$	0.468367	−	0.811235i	0.0383701	−	0.0664590i	0.884573	+	0.466402i	$0.154450\pi$
$150$		0			0
$151$	0.971925			0.0790942			0.0395471	−	0.999218i	$-0.487408\pi$
$151$	0.971925			0.0790942			0.0395471	+	0.999218i	$0.487408\pi$
$152$		0			0
$153$	−10.6656			−0.862265
$154$		0			0
$155$	−1.25351	+	2.17114i	−0.100684	+	0.174390i
$156$		0			0
$157$	−1.35743	+	2.35114i	−0.108335	+	0.187641i	−0.915096	−	0.403237i	$-0.867885\pi$
$157$	−1.35743	+	2.35114i	−0.108335	+	0.187641i	0.806761	+	0.590878i	$0.201218\pi$
$158$		0			0
$159$	−5.35241			−0.424474
$160$		0			0
$161$	0.896081	+	1.55206i	0.0706210	+	0.122319i
$162$		0			0
$163$	3.20384			0.250944			0.125472	−	0.992097i	$-0.459956\pi$
$163$	3.20384			0.250944			0.125472	+	0.992097i	$0.459956\pi$
$164$		0			0
$165$	−0.475385	−	0.823392i	−0.0370087	−	0.0641010i
$166$		0			0
$167$	5.11094	+	8.85240i	0.395496	+	0.685020i	0.993164	−	0.116724i	$-0.0372393\pi$
$167$	5.11094	+	8.85240i	0.395496	+	0.685020i	−0.597668	+	0.801744i	$0.703906\pi$
$168$		0			0
$169$	−6.00000	+	10.3923i	−0.461538	+	0.799408i
$170$		0			0
$171$	−6.42070	+	1.38769i	−0.491003	+	0.106119i
$172$		0			0
$173$	1.33281	−	2.30850i	0.101332	−	0.175512i	−0.810902	−	0.585182i	$-0.801023\pi$
$173$	1.33281	−	2.30850i	0.101332	−	0.175512i	0.912234	+	0.409670i	$0.134356\pi$
$174$		0			0
$175$	−0.110938	−	0.192150i	−0.00838613	−	0.0145252i
$176$		0			0
$177$	1.70584	+	2.95460i	0.128219	+	0.222081i
$178$		0			0
$179$	12.9367			0.966937			0.483468	−	0.875362i	$-0.339377\pi$
$179$	12.9367			0.966937			0.483468	+	0.875362i	$0.339377\pi$
$180$		0			0
$181$	9.30620	+	16.1188i	0.691724	+	1.19810i	0.971273	+	0.237970i	$0.0764820\pi$
$181$	9.30620	+	16.1188i	0.691724	+	1.19810i	−0.279548	+	0.960132i	$0.590185\pi$
$182$		0			0
$183$	15.3765			1.13666
$184$		0			0
$185$	−0.952328	+	1.64948i	−0.0700166	+	0.121272i
$186$		0			0
$187$	−2.75351	+	4.76922i	−0.201357	+	0.348760i
$188$		0			0
$189$	1.22188			0.0888784
$190$		0			0
$191$	23.0421			1.66727			0.833634	−	0.552317i	$-0.186256\pi$
$191$	23.0421			1.66727			0.833634	+	0.552317i	$0.186256\pi$
$192$		0			0
$193$	8.53865	−	14.7894i	0.614626	−	1.06456i	−0.375824	−	0.926691i	$-0.622640\pi$
$193$	8.53865	−	14.7894i	0.614626	−	1.06456i	0.990450	−	0.137872i	$-0.0440262\pi$
$194$		0			0
$195$	3.05469	−	5.29088i	0.218751	−	0.378888i
$196$		0			0
$197$	−8.82024			−0.628416			−0.314208	−	0.949354i	$-0.601739\pi$
$197$	−8.82024			−0.628416			−0.314208	+	0.949354i	$0.601739\pi$
$198$		0			0
$199$	−1.42771	−	2.47287i	−0.101208	−	0.175297i	0.810975	−	0.585081i	$-0.198937\pi$
$199$	−1.42771	−	2.47287i	−0.101208	−	0.175297i	−0.912183	+	0.409784i	$0.865604\pi$
$200$		0			0
$201$	−12.9156			−0.910995
$202$		0			0
$203$	−0.0246145	−	0.0426336i	−0.00172760	−	0.00299229i
$204$		0			0
$205$	−3.61796	−	6.26648i	−0.252689	−	0.437670i
$206$		0			0
$207$	−6.08632	+	10.5418i	−0.423029	+	0.732707i
$208$		0			0
$209$	−1.03709	+	3.22932i	−0.0717373	+	0.223377i
$210$		0			0
$211$	−6.44375	+	11.1609i	−0.443606	+	0.768348i	−0.997954	−	0.0639367i	$-0.979634\pi$
$211$	−6.44375	+	11.1609i	−0.443606	+	0.768348i	0.554348	+	0.832285i	$0.312968\pi$
$212$		0			0
$213$	6.01248	+	10.4139i	0.411968	+	0.713550i
$214$		0			0
$215$	−3.64959	−	6.32128i	−0.248900	−	0.431107i
$216$		0			0
$217$	−0.556248			−0.0377606
$218$		0			0
$219$	−8.60036	−	14.8963i	−0.581159	−	1.00660i
$220$		0			0
$221$	−35.3865			−2.38035
$222$		0			0
$223$	10.3539	−	17.9334i	0.693346	−	1.20091i	−0.277389	−	0.960758i	$-0.589469\pi$
$223$	10.3539	−	17.9334i	0.693346	−	1.20091i	0.970735	−	0.240153i	$-0.0771978\pi$
$224$		0			0
$225$	0.753509	−	1.30512i	0.0502340	−	0.0870078i
$226$		0			0
$227$	−4.00000			−0.265489			−0.132745	−	0.991150i	$-0.542379\pi$
$227$	−4.00000			−0.265489			−0.132745	+	0.991150i	$0.542379\pi$
$228$		0			0
$229$	3.53910			0.233870			0.116935	−	0.993140i	$-0.462693\pi$
$229$	3.53910			0.233870			0.116935	+	0.993140i	$0.462693\pi$
$230$		0			0
$231$	0.105477	−	0.182691i	0.00693986	−	0.0120202i
$232$		0			0
$233$	9.89252	−	17.1344i	0.648081	−	1.12251i	−0.335500	−	0.942040i	$-0.608905\pi$
$233$	9.89252	−	17.1344i	0.648081	−	1.12251i	0.983581	−	0.180468i	$-0.0577614\pi$
$234$		0			0
$235$	−2.79216			−0.182141
$236$		0			0
$237$	−0.967923	−	1.67649i	−0.0628733	−	0.108900i
$238$		0			0
$239$	−23.7741			−1.53782			−0.768910	−	0.639357i	$-0.779201\pi$
$239$	−23.7741			−1.53782			−0.768910	+	0.639357i	$0.779201\pi$
$240$		0			0
$241$	2.27111	+	3.93367i	0.146295	+	0.253390i	0.929855	−	0.367926i	$-0.119932\pi$
$241$	2.27111	+	3.93367i	0.146295	+	0.253390i	−0.783561	+	0.621315i	$0.786599\pi$
$242$		0			0
$243$	6.91168	+	11.9714i	0.443384	+	0.767964i
$244$		0			0
$245$	−3.47539	+	6.01954i	−0.222034	+	0.384575i
$246$		0			0
$247$	−21.3026	+	4.60408i	−1.35545	+	0.292950i
$248$		0			0
$249$	5.81678	−	10.0750i	0.368623	−	0.638474i
$250$		0			0
$251$	−9.75151	−	16.8901i	−0.615510	−	1.06609i	−0.990295	−	0.138982i	$-0.955617\pi$
$251$	−9.75151	−	16.8901i	−0.615510	−	1.06609i	0.374785	−	0.927112i	$-0.377716\pi$
$252$		0			0
$253$	3.14257	+	5.44309i	0.197572	+	0.342204i
$254$		0			0
$255$	8.64759			0.541533
$256$		0			0
$257$	−0.882043	−	1.52774i	−0.0550203	−	0.0952980i	0.837203	−	0.546892i	$-0.184189\pi$
$257$	−0.882043	−	1.52774i	−0.0550203	−	0.0952980i	−0.892224	+	0.451594i	$0.850856\pi$
$258$		0			0
$259$	−0.422598			−0.0262590
$260$		0			0
$261$	0.167186	−	0.289574i	0.0103485	−	0.0179242i
$262$		0			0
$263$	−3.23591	+	5.60477i	−0.199535	+	0.345605i	−0.948378	−	0.317143i	$-0.897276\pi$
$263$	−3.23591	+	5.60477i	−0.199535	+	0.345605i	0.748843	+	0.662748i	$0.230610\pi$
$264$		0			0
$265$	−4.38049			−0.269091
$266$		0			0
$267$	−3.83739			−0.234844
$268$		0			0
$269$	−14.2429	+	24.6695i	−0.868407	+	1.50412i	−0.00478280	+	0.999989i	$0.501522\pi$
$269$	−14.2429	+	24.6695i	−0.868407	+	1.50412i	−0.863624	+	0.504136i	$0.831811\pi$
$270$		0			0
$271$	−0.246491	+	0.426934i	−0.0149732	+	0.0259344i	−0.873415	−	0.486977i	$-0.838100\pi$
$271$	−0.246491	+	0.426934i	−0.0149732	+	0.0259344i	0.858442	+	0.512911i	$0.171433\pi$
$272$		0			0
$273$	1.35553			0.0820402
$274$		0			0
$275$	−0.389062	−	0.673875i	−0.0234613	−	0.0406362i
$276$		0			0
$277$	−8.78905			−0.528083			−0.264041	−	0.964511i	$-0.585056\pi$
$277$	−8.78905			−0.528083			−0.264041	+	0.964511i	$0.585056\pi$
$278$		0			0
$279$	−1.88906	−	3.27195i	−0.113095	−	0.195887i
$280$		0			0
$281$	2.37147	+	4.10750i	0.141470	+	0.245033i	0.928050	−	0.372455i	$-0.121484\pi$
$281$	2.37147	+	4.10750i	0.141470	+	0.245033i	−0.786581	+	0.617488i	$0.788151\pi$
$282$		0			0
$283$	8.43473	−	14.6094i	0.501393	−	0.868438i	−0.498606	−	0.866829i	$-0.666155\pi$
$283$	8.43473	−	14.6094i	0.501393	−	0.868438i	0.999999	−	0.00160901i	$-0.000512164\pi$
$284$		0			0
$285$	5.20584	−	1.12512i	0.308367	−	0.0666465i
$286$		0			0
$287$	0.802738	−	1.39038i	0.0473841	−	0.0820717i
$288$		0			0
$289$	−16.5441	−	28.6552i	−0.973183	−	1.68560i
$290$		0			0
$291$	−3.88706	−	6.73259i	−0.227864	−	0.394671i
$292$		0			0
$293$	11.6336			0.679639			0.339820	−	0.940491i	$-0.389634\pi$
$293$	11.6336			0.679639			0.339820	+	0.940491i	$0.389634\pi$
$294$		0			0
$295$	1.39608	+	2.41808i	0.0812830	+	0.140786i
$296$		0			0
$297$	4.28514			0.248649
$298$		0			0
$299$	−20.1933	+	34.9758i	−1.16781	+	2.02270i
$300$		0			0
$301$	0.809757	−	1.40254i	0.0466736	−	0.0808411i
$302$		0			0
$303$	9.03519			0.519058
$304$		0			0
$305$	12.5843			0.720576
$306$		0			0
$307$	2.94531	−	5.10143i	0.168098	−	0.291154i	−0.769653	−	0.638462i	$-0.779571\pi$
$307$	2.94531	−	5.10143i	0.168098	−	0.291154i	0.937751	+	0.347308i	$0.112904\pi$
$308$		0			0
$309$	−7.45579	+	12.9138i	−0.424145	+	0.734641i
$310$		0			0
$311$	−14.8202			−0.840378			−0.420189	−	0.907437i	$-0.638036\pi$
$311$	−14.8202			−0.840378			−0.420189	+	0.907437i	$0.638036\pi$
$312$		0			0
$313$	2.94731	+	5.10489i	0.166592	+	0.288546i	0.937219	−	0.348740i	$-0.113390\pi$
$313$	2.94731	+	5.10489i	0.166592	+	0.288546i	−0.770628	+	0.637286i	$0.780057\pi$
$314$		0			0
$315$	0.334372			0.0188397
$316$		0			0
$317$	2.07930	+	3.60146i	0.116785	+	0.202278i	0.918492	−	0.395439i	$-0.129408\pi$
$317$	2.07930	+	3.60146i	0.116785	+	0.202278i	−0.801707	+	0.597718i	$0.796074\pi$
$318$		0			0
$319$	−0.0863236	−	0.149517i	−0.00483319	−	0.00837133i
$320$		0			0
$321$	0.997999	−	1.72858i	0.0557029	−	0.0964802i
$322$		0			0
$323$	−20.7203	−	22.8549i	−1.15291	−	1.27168i
$324$		0			0
$325$	2.50000	−	4.33013i	0.138675	−	0.240192i
$326$		0			0
$327$	−4.65315	−	8.05949i	−0.257320	−	0.445691i
$328$		0			0
$329$	−0.309757	−	0.536515i	−0.0170775	−	0.0295790i
$330$		0			0
$331$	−10.6797			−0.587008			−0.293504	−	0.955958i	$-0.594821\pi$
$331$	−10.6797			−0.587008			−0.293504	+	0.955958i	$0.594821\pi$
$332$		0			0
$333$	−1.43518	−	2.48580i	−0.0786472	−	0.136221i
$334$		0			0
$335$	−10.5703			−0.577516
$336$		0			0
$337$	13.1726	−	22.8157i	0.717560	−	1.24285i	−0.244404	−	0.969673i	$-0.578592\pi$
$337$	13.1726	−	22.8157i	0.717560	−	1.24285i	0.961964	−	0.273177i	$-0.0880743\pi$
$338$		0			0
$339$	7.62999	−	13.2155i	0.414404	−	0.717769i
$340$		0			0
$341$	−1.95077			−0.105640
$342$		0			0
$343$	−3.09534			−0.167133
$344$		0			0
$345$	4.93473	−	8.54721i	0.265677	−	0.460166i
$346$		0			0
$347$	−8.44175	+	14.6215i	−0.453177	+	0.784925i	−0.998581	−	0.0532476i	$-0.983043\pi$
$347$	−8.44175	+	14.6215i	−0.453177	+	0.784925i	0.545404	+	0.838173i	$0.316376\pi$
$348$		0			0
$349$	4.61640			0.247110			0.123555	−	0.992338i	$-0.460570\pi$
$349$	4.61640			0.247110			0.123555	+	0.992338i	$0.460570\pi$
$350$		0			0
$351$	13.7675	+	23.8461i	0.734857	+	1.27281i
$352$		0			0
$353$	−24.9508			−1.32800			−0.663998	−	0.747735i	$-0.731142\pi$
$353$	−24.9508			−1.32800			−0.663998	+	0.747735i	$0.731142\pi$
$354$		0			0
$355$	4.92070	+	8.52289i	0.261163	+	0.452348i
$356$		0			0
$357$	0.959347	+	1.66164i	0.0507740	+	0.0879432i
$358$		0			0
$359$	5.90110	−	10.2210i	0.311448	−	0.539444i	−0.667228	−	0.744854i	$-0.732519\pi$
$359$	5.90110	−	10.2210i	0.311448	−	0.539444i	0.978676	+	0.205410i	$0.0658527\pi$
$360$		0			0
$361$	−15.4472	−	11.0627i	−0.813011	−	0.582248i
$362$		0			0
$363$	−6.35041	+	10.9992i	−0.333310	+	0.577310i
$364$		0			0
$365$	−7.03865	−	12.1913i	−0.368420	−	0.638122i
$366$		0			0
$367$	13.1164	+	22.7183i	0.684670	+	1.18588i	0.973540	+	0.228516i	$0.0733872\pi$
$367$	13.1164	+	22.7183i	0.684670	+	1.18588i	−0.288870	+	0.957368i	$0.593279\pi$
$368$		0			0
$369$	10.9047			0.567674
$370$		0			0
$371$	−0.485963	−	0.841712i	−0.0252299	−	0.0436995i
$372$		0			0
$373$	11.9960			0.621129			0.310565	−	0.950552i	$-0.399482\pi$
$373$	11.9960			0.621129			0.310565	+	0.950552i	$0.399482\pi$
$374$		0			0
$375$	−0.610938	+	1.05818i	−0.0315487	+	0.0546440i
$376$		0			0
$377$	0.554690	−	0.960752i	0.0285680	−	0.0494812i
$378$		0			0
$379$	−0.313217			−0.0160889			−0.00804444	−	0.999968i	$-0.502561\pi$
$379$	−0.313217			−0.0160889			−0.00804444	+	0.999968i	$0.502561\pi$
$380$		0			0
$381$	−15.9358			−0.816418
$382$		0			0
$383$	15.0441	−	26.0572i	0.768718	−	1.33146i	−0.169540	−	0.985523i	$-0.554228\pi$
$383$	15.0441	−	26.0572i	0.768718	−	1.33146i	0.938258	−	0.345936i	$-0.112439\pi$
$384$		0			0
$385$	0.0863236	−	0.149517i	0.00439946	−	0.00762008i
$386$		0			0
$387$	11.0000			0.559161
$388$		0			0
$389$	17.8609	+	30.9360i	0.905583	+	1.56852i	0.820133	+	0.572173i	$0.193900\pi$
$389$	17.8609	+	30.9360i	0.905583	+	1.56852i	0.0854503	+	0.996342i	$0.472767\pi$
$390$		0			0
$391$	−57.1655			−2.89099
$392$		0			0
$393$	−4.34339	−	7.52297i	−0.219095	−	0.379484i
$394$		0			0
$395$	−0.792161	−	1.37206i	−0.0398580	−	0.0690360i
$396$		0			0
$397$	−4.77657	+	8.27326i	−0.239729	+	0.415223i	−0.960636	−	0.277809i	$-0.910392\pi$
$397$	−4.77657	+	8.27326i	−0.239729	+	0.415223i	0.720907	+	0.693031i	$0.243725\pi$
$398$		0			0
$399$	0.793718	+	0.875485i	0.0397356	+	0.0438291i
$400$		0			0
$401$	−15.4418	+	26.7459i	−0.771124	+	1.33563i	0.165823	+	0.986156i	$0.446972\pi$
$401$	−15.4418	+	26.7459i	−0.771124	+	1.33563i	−0.936947	+	0.349471i	$0.886361\pi$
$402$		0			0
$403$	−6.26755	−	10.8557i	−0.312209	−	0.540761i
$404$		0			0
$405$	−1.10392	−	1.91204i	−0.0548542	−	0.0950103i
$406$		0			0
$407$	−1.48206			−0.0734629
$408$		0			0
$409$	−15.7816	−	27.3345i	−0.780349	−	1.35160i	−0.931738	−	0.363130i	$-0.881708\pi$
$409$	−15.7816	−	27.3345i	−0.780349	−	1.35160i	0.151389	−	0.988474i	$-0.451625\pi$
$410$		0			0
$411$	−6.62955			−0.327012
$412$		0			0
$413$	−0.309757	+	0.536515i	−0.0152421	+	0.0264002i
$414$		0			0
$415$	4.76053	−	8.24548i	0.233685	−	0.404755i
$416$		0			0
$417$	4.36245			0.213630
$418$		0			0
$419$	26.5070			1.29495			0.647476	−	0.762086i	$-0.275824\pi$
$419$	26.5070			1.29495			0.647476	+	0.762086i	$0.275824\pi$
$420$		0			0
$421$	10.1180	−	17.5248i	0.493119	−	0.854107i	−0.506850	−	0.862035i	$-0.669190\pi$
$421$	10.1180	−	17.5248i	0.493119	−	0.854107i	0.999969	+	0.00792731i	$0.00252337\pi$
$422$		0			0
$423$	2.10392	−	3.64410i	0.102296	−	0.177182i
$424$		0			0
$425$	7.07730			0.343300
$426$		0			0
$427$	1.39608	+	2.41808i	0.0675611	+	0.117019i
$428$		0			0
$429$	4.75385			0.229518
$430$		0			0
$431$	−5.73045	−	9.92543i	−0.276026	−	0.478091i	0.694367	−	0.719621i	$-0.255684\pi$
$431$	−5.73045	−	9.92543i	−0.276026	−	0.478091i	−0.970394	+	0.241529i	$0.922351\pi$
$432$		0			0
$433$	−12.9734	−	22.4706i	−0.623461	−	1.07987i	−0.988836	−	0.149006i	$-0.952393\pi$
$433$	−12.9734	−	22.4706i	−0.623461	−	1.07987i	0.365375	−	0.930860i	$-0.380941\pi$
$434$		0			0
$435$	−0.135553	+	0.234784i	−0.00649925	+	0.0112570i
$436$		0			0
$437$	−34.4136	+	7.43771i	−1.64622	+	0.355794i
$438$		0			0
$439$	12.6562	−	21.9211i	0.604046	−	1.04624i	−0.388156	−	0.921594i	$-0.626888\pi$
$439$	12.6562	−	21.9211i	0.604046	−	1.04624i	0.992202	−	0.124644i	$-0.0397789\pi$
$440$		0			0
$441$	−5.23747	−	9.07157i	−0.249403	−	0.431979i
$442$		0			0
$443$	10.0125	+	17.3421i	0.475707	+	0.823949i	0.999613	−	0.0278272i	$-0.00885883\pi$
$443$	10.0125	+	17.3421i	0.475707	+	0.823949i	−0.523905	+	0.851776i	$0.675525\pi$
$444$		0			0
$445$	−3.14057			−0.148877
$446$		0			0
$447$	−0.572286	−	0.991229i	−0.0270682	−	0.0468835i
$448$		0			0
$449$	19.7601			0.932536			0.466268	−	0.884644i	$-0.345598\pi$
$449$	19.7601			0.932536			0.466268	+	0.884644i	$0.345598\pi$
$450$		0			0
$451$	2.81522	−	4.87610i	0.132563	−	0.229607i
$452$		0			0
$453$	0.593786	−	1.02847i	0.0278985	−	0.0483216i
$454$		0			0
$455$	1.10938			0.0520086
$456$		0			0
$457$	−34.4647			−1.61219			−0.806096	−	0.591785i	$-0.798423\pi$
$457$	−34.4647			−1.61219			−0.806096	+	0.591785i	$0.798423\pi$
$458$		0			0
$459$	−19.4874	+	33.7532i	−0.909595	+	1.57546i
$460$		0			0
$461$	3.96637	−	6.86995i	0.184732	−	0.319965i	−0.758754	−	0.651377i	$-0.774192\pi$
$461$	3.96637	−	6.86995i	0.184732	−	0.319965i	0.943486	+	0.331412i	$0.107525\pi$
$462$		0			0
$463$	−9.25395			−0.430068			−0.215034	−	0.976607i	$-0.568986\pi$
$463$	−9.25395			−0.430068			−0.215034	+	0.976607i	$0.568986\pi$
$464$		0			0
$465$	1.53163	+	2.65287i	0.0710278	+	0.123024i
$466$		0			0
$467$	−9.47183			−0.438304			−0.219152	−	0.975691i	$-0.570329\pi$
$467$	−9.47183			−0.438304			−0.219152	+	0.975691i	$0.570329\pi$
$468$		0			0
$469$	−1.17265	−	2.03108i	−0.0541478	−	0.0937868i
$470$		0			0
$471$	1.65861	+	2.87280i	0.0764247	+	0.132371i
$472$		0			0
$473$	2.83983	−	4.91873i	0.130576	−	0.226164i
$474$		0			0
$475$	4.26053	−	0.920816i	0.195486	−	0.0422499i
$476$		0			0
$477$	3.30074	−	5.71704i	0.151130	−	0.261765i
$478$		0			0
$479$	−17.3574	−	30.0639i	−0.793081	−	1.37366i	−0.924050	−	0.382270i	$-0.875142\pi$
$479$	−17.3574	−	30.0639i	−0.793081	−	1.37366i	0.130969	−	0.991386i	$-0.458191\pi$
$480$		0			0
$481$	−4.76164	−	8.24740i	−0.217112	−	0.376049i
$482$		0			0
$483$	2.18980			0.0996393
$484$		0			0
$485$	−3.18122	−	5.51004i	−0.144452	−	0.250198i
$486$		0			0
$487$	28.5070			1.29178			0.645888	−	0.763432i	$-0.276487\pi$
$487$	28.5070			1.29178			0.645888	+	0.763432i	$0.276487\pi$
$488$		0			0
$489$	1.95735	−	3.39022i	0.0885142	−	0.153311i
$490$		0			0
$491$	−15.5949	+	27.0112i	−0.703788	+	1.21900i	0.263339	+	0.964703i	$0.415176\pi$
$491$	−15.5949	+	27.0112i	−0.703788	+	1.21900i	−0.967127	+	0.254293i	$0.918157\pi$
$492$		0			0
$493$	1.57028			0.0707221
$494$		0			0
$495$	1.17265			0.0527066
$496$		0			0
$497$	−1.09178	+	1.89103i	−0.0489732	+	0.0848242i
$498$		0			0
$499$	8.09334	−	14.0181i	0.362308	−	0.627535i	−0.626032	−	0.779797i	$-0.715322\pi$
$499$	8.09334	−	14.0181i	0.362308	−	0.627535i	0.988340	+	0.152262i	$0.0486556\pi$
$500$		0			0
$501$	12.4899			0.558006
$502$		0			0
$503$	8.61094	+	14.9146i	0.383943	+	0.665008i	0.991622	−	0.129174i	$-0.0412327\pi$
$503$	8.61094	+	14.9146i	0.383943	+	0.665008i	−0.607679	+	0.794183i	$0.707899\pi$
$504$		0			0
$505$	7.39452			0.329052
$506$		0			0
$507$	7.33126	+	12.6981i	0.325593	+	0.563943i
$508$		0			0
$509$	18.2956	+	31.6889i	0.810939	+	1.40459i	0.912207	+	0.409729i	$0.134377\pi$
$509$	18.2956	+	31.6889i	0.810939	+	1.40459i	−0.101268	+	0.994859i	$0.532290\pi$
$510$		0			0
$511$	1.56171	−	2.70496i	0.0690859	−	0.119660i
$512$		0			0
$513$	−7.33983	+	22.8549i	−0.324062	+	1.00907i
$514$		0			0
$515$	−6.10192	+	10.5688i	−0.268883	+	0.465718i
$516$		0			0
$517$	−1.08632	−	1.88157i	−0.0477765	−	0.0827512i
$518$		0			0
$519$	−1.62853	−	2.82070i	−0.0714847	−	0.123815i
$520$		0			0
$521$	−3.63667			−0.159325			−0.0796626	−	0.996822i	$-0.525384\pi$
$521$	−3.63667			−0.159325			−0.0796626	+	0.996822i	$0.525384\pi$
$522$		0			0
$523$	−17.8117	−	30.8507i	−0.778850	−	1.34901i	−0.932605	−	0.360898i	$-0.882470\pi$
$523$	−17.8117	−	30.8507i	−0.778850	−	1.34901i	0.153756	−	0.988109i	$-0.450863\pi$
$524$		0			0
$525$	−0.271105			−0.0118320
$526$		0			0
$527$	8.87147	−	15.3658i	0.386447	−	0.669346i
$528$		0			0
$529$	−21.1214	+	36.5834i	−0.918322	+	1.59058i
$530$		0			0
$531$	−4.20784			−0.182605
$532$		0			0
$533$	36.1796			1.56711
$534$		0			0
$535$	0.816776	−	1.41470i	0.0353123	−	0.0611627i
$536$		0			0
$537$	7.90354	−	13.6893i	0.341063	−	0.590739i
$538$		0			0
$539$	−5.40856			−0.232963
$540$		0			0
$541$	−1.58632	−	2.74759i	−0.0682014	−	0.118128i	0.829908	−	0.557900i	$-0.188393\pi$
$541$	−1.58632	−	2.74759i	−0.0682014	−	0.118128i	−0.898110	+	0.439772i	$0.855059\pi$
$542$		0			0
$543$	22.7420			0.975955
$544$		0			0
$545$	−3.80820	−	6.59600i	−0.163125	−	0.282541i
$546$		0			0
$547$	−14.1902	−	24.5782i	−0.606731	−	1.05089i	−0.991775	−	0.127991i	$-0.959147\pi$
$547$	−14.1902	−	24.5782i	−0.606731	−	1.05089i	0.385044	−	0.922898i	$-0.374186\pi$
$548$		0			0
$549$	−9.48240	+	16.4240i	−0.404699	+	0.700959i
$550$		0			0
$551$	0.945310	−	0.204307i	0.0402715	−	0.00870377i
$552$		0			0
$553$	0.175762	−	0.304428i	0.00747415	−	0.0129456i
$554$		0			0
$555$	1.16363	+	2.01546i	0.0493932	+	0.0855516i
$556$		0			0
$557$	−1.06527	−	1.84510i	−0.0451368	−	0.0781793i	0.842574	−	0.538580i	$-0.181039\pi$
$557$	−1.06527	−	1.84510i	−0.0451368	−	0.0781793i	−0.887711	+	0.460401i	$0.847706\pi$
$558$		0			0
$559$	36.4959			1.54361
$560$		0			0
$561$	3.36445	+	5.82739i	0.142047	+	0.246033i
$562$		0			0
$563$	20.2609			0.853894			0.426947	−	0.904277i	$-0.359589\pi$
$563$	20.2609			0.853894			0.426947	+	0.904277i	$0.359589\pi$
$564$		0			0
$565$	6.24449	−	10.8158i	0.262708	−	0.455023i
$566$		0			0
$567$	0.244933	−	0.424237i	0.0102862	−	0.0178163i
$568$		0			0
$569$	−34.7530			−1.45692			−0.728460	−	0.685088i	$-0.759764\pi$
$569$	−34.7530			−1.45692			−0.728460	+	0.685088i	$0.759764\pi$
$570$		0			0
$571$	−32.0702			−1.34210			−0.671048	−	0.741414i	$-0.734155\pi$
$571$	−32.0702			−1.34210			−0.671048	+	0.741414i	$0.734155\pi$
$572$		0			0
$573$	14.0773	−	24.3826i	0.588088	−	1.01860i
$574$		0			0
$575$	4.03865	−	6.99515i	0.168423	−	0.291718i
$576$		0			0
$577$	30.2740			1.26032			0.630162	−	0.776464i	$-0.282988\pi$
$577$	30.2740			1.26032			0.630162	+	0.776464i	$0.282988\pi$
$578$		0			0
$579$	−10.4332	−	18.0708i	−0.433588	−	0.750996i
$580$		0			0
$581$	2.11250			0.0876411
$582$		0			0
$583$	−1.70428	−	2.95190i	−0.0705841	−	0.122255i
$584$		0			0
$585$	3.76755	+	6.52558i	0.155769	+	0.269800i
$586$		0			0
$587$	1.24805	−	2.16168i	0.0515125	−	0.0892222i	−0.839119	−	0.543947i	$-0.816929\pi$
$587$	1.24805	−	2.16168i	0.0515125	−	0.0892222i	0.890632	+	0.454725i	$0.150262\pi$
$588$		0			0
$589$	3.34139	−	10.4045i	0.137680	−	0.428708i
$590$		0			0
$591$	−5.38862	+	9.33336i	−0.221658	+	0.383923i
$592$		0			0
$593$	4.98196	+	8.62901i	0.204585	+	0.354351i	0.950000	−	0.312249i	$-0.101082\pi$
$593$	4.98196	+	8.62901i	0.204585	+	0.354351i	−0.745416	+	0.666600i	$0.767749\pi$
$594$		0			0
$595$	0.785142	+	1.35991i	0.0321877	+	0.0557507i
$596$		0			0
$597$	−3.48898			−0.142794
$598$		0			0
$599$	16.0351	+	27.7736i	0.655176	+	1.13480i	0.981850	+	0.189661i	$0.0607389\pi$
$599$	16.0351	+	27.7736i	0.655176	+	1.13480i	−0.326673	+	0.945137i	$0.605928\pi$
$600$		0			0
$601$	3.68278			0.150224			0.0751119	−	0.997175i	$-0.476069\pi$
$601$	3.68278			0.150224			0.0751119	+	0.997175i	$0.476069\pi$
$602$		0			0
$603$	7.96481	−	13.7955i	0.324352	−	0.561794i
$604$		0			0
$605$	−5.19726	+	9.00192i	−0.211299	+	0.365980i
$606$		0			0
$607$	20.4850			0.831460			0.415730	−	0.909488i	$-0.363526\pi$
$607$	20.4850			0.831460			0.415730	+	0.909488i	$0.363526\pi$
$608$		0			0
$609$	−0.0601518			−0.00243747
$610$		0			0
$611$	6.98040	−	12.0904i	0.282397	−	0.489126i
$612$		0			0
$613$	−5.35587	+	9.27664i	−0.216322	+	0.374680i	−0.953681	−	0.300821i	$-0.902739\pi$
$613$	−5.35587	+	9.27664i	−0.216322	+	0.374680i	0.737359	+	0.675501i	$0.236073\pi$
$614$		0			0
$615$	−8.84139			−0.356519
$616$		0			0
$617$	8.86600	+	15.3564i	0.356932	+	0.618224i	0.987447	−	0.157953i	$-0.0504895\pi$
$617$	8.86600	+	15.3564i	0.356932	+	0.618224i	−0.630515	+	0.776177i	$0.717156\pi$
$618$		0			0
$619$	13.2780			0.533689			0.266844	−	0.963740i	$-0.414019\pi$
$619$	13.2780			0.533689			0.266844	+	0.963740i	$0.414019\pi$
$620$		0			0
$621$	22.2409	+	38.5224i	0.892498	+	1.54585i
$622$		0			0
$623$	−0.348409	−	0.603462i	−0.0139587	−	0.0241772i
$624$		0			0
$625$	−0.500000	+	0.866025i	−0.0200000	+	0.0346410i
$626$		0			0
$627$	2.78359	+	3.07034i	0.111166	+	0.122618i
$628$		0			0
$629$	6.73992	−	11.6739i	0.268738	−	0.465468i
$630$		0			0
$631$	2.03208	+	3.51966i	0.0808957	+	0.140115i	0.903635	−	0.428304i	$-0.140889\pi$
$631$	2.03208	+	3.51966i	0.0808957	+	0.140115i	−0.822739	+	0.568419i	$0.807555\pi$
$632$		0			0
$633$	7.87347	+	13.6372i	0.312942	+	0.542032i
$634$		0			0
$635$	−13.0421			−0.517560
$636$		0			0
$637$	−17.3769	−	30.0977i	−0.688499	−	1.19252i
$638$		0			0
$639$	−14.8312			−0.586712
$640$		0			0
$641$	−2.49254	+	4.31720i	−0.0984493	+	0.170519i	−0.911043	−	0.412311i	$-0.864722\pi$
$641$	−2.49254	+	4.31720i	−0.0984493	+	0.170519i	0.812594	+	0.582831i	$0.198055\pi$
$642$		0			0
$643$	−2.93829	+	5.08927i	−0.115875	+	0.200701i	−0.918129	−	0.396281i	$-0.870300\pi$
$643$	−2.93829	+	5.08927i	−0.115875	+	0.200701i	0.802254	+	0.596983i	$0.203634\pi$
$644$		0			0
$645$	−8.91869			−0.351173
$646$		0			0
$647$	−16.9757			−0.667385			−0.333692	−	0.942682i	$-0.608295\pi$
$647$	−16.9757			−0.667385			−0.333692	+	0.942682i	$0.608295\pi$
$648$		0			0
$649$	−1.08632	+	1.88157i	−0.0426419	+	0.0738580i
$650$		0			0
$651$	−0.339833	+	0.588608i	−0.0133191	+	0.0230694i
$652$		0			0
$653$	24.6797			0.965790			0.482895	−	0.875678i	$-0.339585\pi$
$653$	24.6797			0.965790			0.482895	+	0.875678i	$0.339585\pi$
$654$		0			0
$655$	−3.55469	−	6.15690i	−0.138893	−	0.240570i
$656$		0			0
$657$	21.2148			0.827667
$658$		0			0
$659$	0.943308	+	1.63386i	0.0367461	+	0.0636461i	0.883814	−	0.467839i	$-0.154967\pi$
$659$	0.943308	+	1.63386i	0.0367461	+	0.0636461i	−0.847067	+	0.531485i	$0.821634\pi$
$660$		0			0
$661$	−22.3749	−	38.7545i	−0.870284	−	1.50738i	−0.861703	−	0.507413i	$-0.830602\pi$
$661$	−22.3749	−	38.7545i	−0.870284	−	1.50738i	−0.00858048	−	0.999963i	$-0.502731\pi$
$662$		0			0
$663$	−21.6190	+	37.4452i	−0.839611	+	1.45425i
$664$		0			0
$665$	0.649590	+	0.716509i	0.0251900	+	0.0277850i
$666$		0			0
$667$	0.896081	−	1.55206i	0.0346964	−	0.0600959i
$668$		0			0
$669$	−12.6511	−	21.9124i	−0.489122	−	0.847183i
$670$		0			0
$671$	4.89608	+	8.48026i	0.189011	+	0.327377i
$672$		0			0
$673$	25.4046			0.979274			0.489637	−	0.871926i	$-0.337129\pi$
$673$	25.4046			0.979274			0.489637	+	0.871926i	$0.337129\pi$
$674$		0			0
$675$	−2.75351	−	4.76922i	−0.105983	−	0.183567i
$676$		0			0
$677$	−3.86946			−0.148716			−0.0743578	−	0.997232i	$-0.523691\pi$
$677$	−3.86946			−0.148716			−0.0743578	+	0.997232i	$0.523691\pi$
$678$		0			0
$679$	0.705838	−	1.22255i	0.0270876	−	0.0469170i
$680$		0			0
$681$	−2.44375	+	4.23270i	−0.0936448	+	0.162198i
$682$		0			0
$683$	48.5171			1.85645			0.928227	−	0.372015i	$-0.121333\pi$
$683$	48.5171			1.85645			0.928227	+	0.372015i	$0.121333\pi$
$684$		0			0
$685$	−5.42571			−0.207306
$686$		0			0
$687$	2.16217	−	3.74499i	0.0824919	−	0.142880i
$688$		0			0
$689$	10.9512	−	18.9681i	0.417208	−	0.722626i
$690$		0			0
$691$	47.6374			1.81221			0.906105	−	0.423052i	$-0.139041\pi$
$691$	47.6374			1.81221			0.906105	+	0.423052i	$0.139041\pi$
$692$		0			0
$693$	0.130091	+	0.225325i	0.00494176	+	0.00855937i
$694$		0			0
$695$	3.57028			0.135429
$696$		0			0
$697$	25.6054	+	44.3498i	0.969873	+	1.67987i
$698$		0			0
$699$	−12.0874	−	20.9361i	−0.457189	−	0.791874i
$700$		0			0
$701$	−14.2766	+	24.7277i	−0.539218	+	0.933954i	0.459728	+	0.888060i	$0.347947\pi$
$701$	−14.2766	+	24.7277i	−0.539218	+	0.933954i	−0.998946	+	0.0458939i	$0.985386\pi$
$702$		0			0
$703$	2.53855	−	7.90458i	0.0957434	−	0.298127i
$704$		0			0
$705$	−1.70584	+	2.95460i	−0.0642456	+	0.111277i
$706$		0			0
$707$	0.820334	+	1.42086i	0.0308518	+	0.0534370i
$708$		0			0
$709$	−9.74137	−	16.8726i	−0.365845	−	0.633662i	0.623066	−	0.782169i	$-0.285887\pi$
$709$	−9.74137	−	16.8726i	−0.365845	−	0.633662i	−0.988911	+	0.148507i	$0.952553\pi$
$710$		0			0
$711$	2.38760			0.0895421
$712$		0			0
$713$	−10.1250	−	17.5370i	−0.379183	−	0.656765i
$714$		0			0
$715$	3.89062			0.145501
$716$		0			0
$717$	−14.5245	+	25.1572i	−0.542428	+	0.939513i
$718$		0			0
$719$	2.05313	−	3.55613i	0.0765689	−	0.132621i	−0.825199	−	0.564843i	$-0.808937\pi$
$719$	2.05313	−	3.55613i	0.0765689	−	0.132621i	0.901767	+	0.432221i	$0.142270\pi$
$720$		0			0
$721$	−2.70774			−0.100842
$722$		0			0
$723$	5.55002			0.206407
$724$		0			0
$725$	−0.110938	+	0.192150i	−0.00412014	+	0.00713629i
$726$		0			0
$727$	−6.20584	+	10.7488i	−0.230162	+	0.398652i	−0.957856	−	0.287250i	$-0.907259\pi$
$727$	−6.20584	+	10.7488i	−0.230162	+	0.398652i	0.727694	+	0.685902i	$0.240592\pi$
$728$		0			0
$729$	23.5139			0.870887
$730$		0			0
$731$	25.8293	+	44.7376i	0.955330	+	1.65468i
$732$		0			0
$733$	−16.3985			−0.605693			−0.302847	−	0.953039i	$-0.597937\pi$
$733$	−16.3985			−0.605693			−0.302847	+	0.953039i	$0.597937\pi$
$734$		0			0
$735$	4.24649	+	7.35514i	0.156634	+	0.271298i
$736$		0			0
$737$	−4.11250	−	7.12305i	−0.151486	−	0.262381i
$738$		0			0
$739$	−23.1018	+	40.0135i	−0.849814	+	1.47192i	0.0315597	+	0.999502i	$0.489953\pi$
$739$	−23.1018	+	40.0135i	−0.849814	+	1.47192i	−0.881374	+	0.472419i	$0.843381\pi$
$740$		0			0
$741$	−8.14267	+	25.3547i	−0.299128	+	0.931430i
$742$		0			0
$743$	−12.4066	+	21.4888i	−0.455153	+	0.788347i	−0.998697	−	0.0510331i	$-0.983749\pi$
$743$	−12.4066	+	21.4888i	−0.455153	+	0.788347i	0.543544	+	0.839380i	$0.317082\pi$
$744$		0			0
$745$	−0.468367	−	0.811235i	−0.0171596	−	0.0297214i
$746$		0			0
$747$	7.17420	+	12.4261i	0.262490	+	0.454647i
$748$		0			0
$749$	0.362446			0.0132435
$750$		0			0
$751$	−5.26955	−	9.12712i	−0.192289	−	0.333054i	0.753720	−	0.657196i	$-0.228258\pi$
$751$	−5.26955	−	9.12712i	−0.192289	−	0.333054i	−0.946008	+	0.324142i	$0.894924\pi$
$752$		0			0
$753$	−23.8303			−0.868423
$754$		0			0
$755$	0.485963	−	0.841712i	0.0176860	−	0.0306330i
$756$		0			0
$757$	3.09836	−	5.36652i	0.112612	−	0.195049i	−0.804211	−	0.594344i	$-0.797412\pi$
$757$	3.09836	−	5.36652i	0.112612	−	0.195049i	0.916823	+	0.399295i	$0.130745\pi$
$758$		0			0
$759$	7.67967			0.278754
$760$		0			0
$761$	35.6084			1.29080			0.645402	−	0.763843i	$-0.276690\pi$
$761$	35.6084			1.29080			0.645402	+	0.763843i	$0.276690\pi$
$762$		0			0
$763$	0.844949	−	1.46349i	0.0305892	−	0.0529820i
$764$		0			0
$765$	−5.33281	+	9.23671i	−0.192808	+	0.333954i
$766$		0			0
$767$	−13.9608			−0.504095
$768$		0			0
$769$	0.0777477	+	0.134663i	0.00280365	+	0.00485607i	0.867424	−	0.497570i	$-0.165774\pi$
$769$	0.0777477	+	0.134663i	0.00280365	+	0.00485607i	−0.864620	+	0.502426i	$0.832441\pi$
$770$		0			0
$771$	−2.15550			−0.0776283
$772$		0			0
$773$	2.54411	+	4.40653i	0.0915054	+	0.158492i	0.908145	−	0.418656i	$-0.137499\pi$
$773$	2.54411	+	4.40653i	0.0915054	+	0.158492i	−0.816639	+	0.577148i	$0.804165\pi$
$774$		0			0
$775$	1.25351	+	2.17114i	0.0450274	+	0.0779897i
$776$		0			0
$777$	−0.258181	+	0.447183i	−0.00926220	+	0.0160426i
$778$		0			0
$779$	21.1847	+	23.3671i	0.759020	+	0.837212i
$780$		0			0
$781$	−3.82891	+	6.63187i	−0.137009	+	0.237307i
$782$		0			0
$783$	−0.610938	−	1.05818i	−0.0218331	−	0.0378161i
$784$		0			0
$785$	1.35743	+	2.35114i	0.0484487	+	0.0839156i
$786$		0			0
$787$	22.5070			0.802289			0.401144	−	0.916015i	$-0.368613\pi$
$787$	22.5070			0.802289			0.401144	+	0.916015i	$0.368613\pi$
$788$		0			0
$789$	3.95389	+	6.84833i	0.140762	+	0.243807i
$790$		0			0
$791$	2.77101			0.0985257

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:	\( N \)	\(=\)	\( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1520.q (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.610938 - 1.05818i) q^{3} +(0.500000 - 0.866025i) q^{5} +0.221876 q^{7} +(0.753509 + 1.30512i) q^{9} +O(q^{10})\)	\(q+(0.610938 - 1.05818i) q^{3} +(0.500000 - 0.866025i) q^{5} +0.221876 q^{7} +(0.753509 + 1.30512i) q^{9} +0.778124 q^{11} +(2.50000 + 4.33013i) q^{13} +(-0.610938 - 1.05818i) q^{15} +(-3.53865 + 6.12912i) q^{17} +(-1.33281 + 4.15013i) q^{19} +(0.135553 - 0.234784i) q^{21} +(4.03865 + 6.99515i) q^{23} +(-0.500000 - 0.866025i) q^{25} +5.50702 q^{27} +(-0.110938 - 0.192150i) q^{29} -2.50702 q^{31} +(0.475385 - 0.823392i) q^{33} +(0.110938 - 0.192150i) q^{35} -1.90466 q^{37} +6.10938 q^{39} +(3.61796 - 6.26648i) q^{41} +(3.64959 - 6.32128i) q^{43} +1.50702 q^{45} +(-1.39608 - 2.41808i) q^{47} -6.95077 q^{49} +(4.32379 + 7.48903i) q^{51} +(-2.19024 - 3.79361i) q^{53} +(0.389062 - 0.673875i) q^{55} +(3.57730 + 3.94583i) q^{57} +(-1.39608 + 2.41808i) q^{59} +(6.29216 + 10.8983i) q^{61} +(0.167186 + 0.289574i) q^{63} +5.00000 q^{65} +(-5.28514 - 9.15414i) q^{67} +9.86946 q^{69} +(-4.92070 + 8.52289i) q^{71} +(7.03865 - 12.1913i) q^{73} -1.22188 q^{75} +0.172647 q^{77} +(0.792161 - 1.37206i) q^{79} +(1.10392 - 1.91204i) q^{81} +9.52106 q^{83} +(3.53865 + 6.12912i) q^{85} -0.271105 q^{87} +(-1.57028 - 2.71981i) q^{89} +(0.554690 + 0.960752i) q^{91} +(-1.53163 + 2.65287i) q^{93} +(2.92771 + 3.22932i) q^{95} +(3.18122 - 5.51004i) q^{97} +(0.586324 + 1.01554i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + q^{3} + 3 q^{5} - 4 q^{7} - 4 q^{9}+O(q^{10}) \) 6 * q + q^3 + 3 * q^5 - 4 * q^7 - 4 * q^9	\( 6 q + q^{3} + 3 q^{5} - 4 q^{7} - 4 q^{9} + 10 q^{11} + 15 q^{13} - q^{15} - q^{17} + 12 q^{21} + 4 q^{23} - 3 q^{25} + 16 q^{27} + 2 q^{29} + 2 q^{31} - 11 q^{33} - 2 q^{35} - 4 q^{37} + 10 q^{39} + 2 q^{41} - q^{43} - 8 q^{45} + 6 q^{47} - 14 q^{49} - 6 q^{51} - 11 q^{53} + 5 q^{55} - 19 q^{57} + 6 q^{59} + 9 q^{61} + 9 q^{63} + 30 q^{65} - 20 q^{67} - 10 q^{69} - 29 q^{71} + 22 q^{73} - 2 q^{75} - 32 q^{77} - 24 q^{79} + 21 q^{81} + 6 q^{83} + q^{85} - 24 q^{87} + 14 q^{89} - 10 q^{91} - 6 q^{93} - 7 q^{97} - 13 q^{99}+O(q^{100}) \) 6 * q + q^3 + 3 * q^5 - 4 * q^7 - 4 * q^9 + 10 * q^11 + 15 * q^13 - q^15 - q^17 + 12 * q^21 + 4 * q^23 - 3 * q^25 + 16 * q^27 + 2 * q^29 + 2 * q^31 - 11 * q^33 - 2 * q^35 - 4 * q^37 + 10 * q^39 + 2 * q^41 - q^43 - 8 * q^45 + 6 * q^47 - 14 * q^49 - 6 * q^51 - 11 * q^53 + 5 * q^55 - 19 * q^57 + 6 * q^59 + 9 * q^61 + 9 * q^63 + 30 * q^65 - 20 * q^67 - 10 * q^69 - 29 * q^71 + 22 * q^73 - 2 * q^75 - 32 * q^77 - 24 * q^79 + 21 * q^81 + 6 * q^83 + q^85 - 24 * q^87 + 14 * q^89 - 10 * q^91 - 6 * q^93 - 7 * q^97 - 13 * q^99

Introduction

L-functions

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