LMFDB - Embedded newform 1620.4.i.x.541.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1620,4,Mod(541,1620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1620, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1620.541");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1620 = 2^{2} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1620.i (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:	$95.5830942093$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 306x^{10} + 30777x^{8} + 1381040x^{6} + 28918584x^{4} + 243888288x^{2} + 366645904 $ x^12 + 306x^10 + 30777x^8 + 1381040x^6 + 28918584x^4 + 243888288*x^2 + 366645904
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{13} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			541.3
Root			$4.35846i$ of defining polynomial
Character	$\chi$	$=$	1620.541
Dual form			1620.4.i.x.1081.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.50000 + 4.33013i) q^{5} +(-3.10707 + 5.38160i) q^{7} +O(q^{10})$	$q+(2.50000 + 4.33013i) q^{5} +(-3.10707 + 5.38160i) q^{7} +(-14.1746 + 24.5511i) q^{11} +(4.30274 + 7.45256i) q^{13} +90.1352 q^{17} +114.170 q^{19} +(-24.2053 - 41.9248i) q^{23} +(-12.5000 + 21.6506i) q^{25} +(152.979 - 264.967i) q^{29} +(-46.7978 - 81.0562i) q^{31} -31.0707 q^{35} -282.892 q^{37} +(-14.3843 - 24.9143i) q^{41} +(177.222 - 306.957i) q^{43} +(-261.499 + 452.930i) q^{47} +(152.192 + 263.605i) q^{49} +66.9285 q^{53} -141.746 q^{55} +(-3.81611 - 6.60970i) q^{59} +(-4.70562 + 8.15037i) q^{61} +(-21.5137 + 37.2628i) q^{65} +(247.196 + 428.155i) q^{67} +560.709 q^{71} +1116.68 q^{73} +(-88.0828 - 152.564i) q^{77} +(-520.935 + 902.287i) q^{79} +(22.7554 - 39.4134i) q^{83} +(225.338 + 390.297i) q^{85} +357.159 q^{89} -53.4757 q^{91} +(285.424 + 494.369i) q^{95} +(60.0029 - 103.928i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 30 q^{5} - 12 q^{7}+O(q^{10}) $ 12 * q + 30 * q^5 - 12 * q^7	$ 12 q + 30 q^{5} - 12 q^{7} + 84 q^{13} + 24 q^{17} - 228 q^{19} - 30 q^{23} - 150 q^{25} - 168 q^{29} + 324 q^{31} - 120 q^{35} - 984 q^{37} - 312 q^{41} + 156 q^{43} - 462 q^{47} + 588 q^{49} + 2028 q^{53} - 1008 q^{59} - 36 q^{61} - 420 q^{65} - 144 q^{67} + 2424 q^{71} - 1800 q^{73} - 672 q^{77} + 936 q^{79} - 288 q^{83} + 60 q^{85} - 240 q^{89} + 4572 q^{91} - 570 q^{95} + 1188 q^{97}+O(q^{100}) $ 12 * q + 30 * q^5 - 12 * q^7 + 84 * q^13 + 24 * q^17 - 228 * q^19 - 30 * q^23 - 150 * q^25 - 168 * q^29 + 324 * q^31 - 120 * q^35 - 984 * q^37 - 312 * q^41 + 156 * q^43 - 462 * q^47 + 588 * q^49 + 2028 * q^53 - 1008 * q^59 - 36 * q^61 - 420 * q^65 - 144 * q^67 + 2424 * q^71 - 1800 * q^73 - 672 * q^77 + 936 * q^79 - 288 * q^83 + 60 * q^85 - 240 * q^89 + 4572 * q^91 - 570 * q^95 + 1188 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times$.

$n$	$811$	$1297$	$1541$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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			0
$3$		0			0
$4$		0			0
$5$	2.50000	+	4.33013i	0.223607	+	0.387298i
$5$	2.50000	+	4.33013i	0.223607	+	0.387298i
$6$		0			0
$7$	−3.10707	+	5.38160i	−0.167766	+	0.290579i	−0.937634	−	0.347624i	$-0.886989\pi$
$7$	−3.10707	+	5.38160i	−0.167766	+	0.290579i	0.769868	+	0.638203i	$0.220322\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−14.1746	+	24.5511i	−0.388527	+	0.672948i	−0.992252	−	0.124245i	$-0.960349\pi$
$11$	−14.1746	+	24.5511i	−0.388527	+	0.672948i	0.603725	+	0.797193i	$0.293683\pi$
$12$		0			0
$13$	4.30274	+	7.45256i	0.0917973	+	0.158998i	0.908267	−	0.418390i	$-0.137405\pi$
$13$	4.30274	+	7.45256i	0.0917973	+	0.158998i	−0.816470	+	0.577388i	$0.804072\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	90.1352			1.28594			0.642970	−	0.765891i	$-0.277702\pi$
$17$	90.1352			1.28594			0.642970	+	0.765891i	$0.277702\pi$
$18$		0			0
$19$	114.170			1.37854			0.689272	−	0.724503i	$-0.257931\pi$
$19$	114.170			1.37854			0.689272	+	0.724503i	$0.257931\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−24.2053	−	41.9248i	−0.219441	−	0.380083i	0.735196	−	0.677855i	$-0.237090\pi$
$23$	−24.2053	−	41.9248i	−0.219441	−	0.380083i	−0.954637	+	0.297771i	$0.903757\pi$
$24$		0			0
$25$	−12.5000	+	21.6506i	−0.100000	+	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	152.979	−	264.967i	0.979568	−	1.69666i	0.315616	−	0.948887i	$-0.397789\pi$
$29$	152.979	−	264.967i	0.979568	−	1.69666i	0.663953	−	0.747775i	$-0.268878\pi$
$30$		0			0
$31$	−46.7978	−	81.0562i	−0.271134	−	0.469617i	0.698019	−	0.716079i	$-0.254065\pi$
$31$	−46.7978	−	81.0562i	−0.271134	−	0.469617i	−0.969152	+	0.246462i	$0.920732\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−31.0707			−0.150055
$36$		0			0
$37$	−282.892			−1.25695			−0.628475	−	0.777830i	$-0.716320\pi$
$37$	−282.892			−1.25695			−0.628475	+	0.777830i	$0.716320\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−14.3843	−	24.9143i	−0.0547913	−	0.0949013i	0.837329	−	0.546700i	$-0.184116\pi$
$41$	−14.3843	−	24.9143i	−0.0547913	−	0.0949013i	−0.892120	+	0.451798i	$0.850783\pi$
$42$		0			0
$43$	177.222	−	306.957i	0.628513	−	1.08862i	−0.359338	−	0.933208i	$-0.616997\pi$
$43$	177.222	−	306.957i	0.628513	−	1.08862i	0.987850	−	0.155408i	$-0.0496694\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−261.499	+	452.930i	−0.811565	+	1.40567i	0.100203	+	0.994967i	$0.468051\pi$
$47$	−261.499	+	452.930i	−0.811565	+	1.40567i	−0.911768	+	0.410705i	$0.865282\pi$
$48$		0			0
$49$	152.192	+	263.605i	0.443709	+	0.768527i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	66.9285			0.173459			0.0867296	−	0.996232i	$-0.472358\pi$
$53$	66.9285			0.173459			0.0867296	+	0.996232i	$0.472358\pi$
$54$		0			0
$55$	−141.746			−0.347509
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−3.81611	−	6.60970i	−0.00842061	−	0.0145849i	0.861784	−	0.507275i	$-0.169347\pi$
$59$	−3.81611	−	6.60970i	−0.00842061	−	0.0145849i	−0.870205	+	0.492690i	$0.836014\pi$
$60$		0			0
$61$	−4.70562	+	8.15037i	−0.00987693	+	0.0171073i	−0.870922	−	0.491422i	$-0.836477\pi$
$61$	−4.70562	+	8.15037i	−0.00987693	+	0.0171073i	0.861045	+	0.508529i	$0.169811\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−21.5137	+	37.2628i	−0.0410530	+	0.0711059i
$66$		0			0
$67$	247.196	+	428.155i	0.450743	+	0.780709i	0.998432	−	0.0559724i	$-0.0178259\pi$
$67$	247.196	+	428.155i	0.450743	+	0.780709i	−0.547690	+	0.836682i	$0.684493\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	560.709			0.937239			0.468619	−	0.883400i	$-0.344752\pi$
$71$	560.709			0.937239			0.468619	+	0.883400i	$0.344752\pi$
$72$		0			0
$73$	1116.68			1.79038			0.895192	−	0.445680i	$-0.147038\pi$
$73$	1116.68			1.79038			0.895192	+	0.445680i	$0.147038\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−88.0828	−	152.564i	−0.130363	−	0.225796i
$78$		0			0
$79$	−520.935	+	902.287i	−0.741896	+	1.28500i	0.209734	+	0.977758i	$0.432740\pi$
$79$	−520.935	+	902.287i	−0.741896	+	1.28500i	−0.951631	+	0.307244i	$0.900593\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	22.7554	−	39.4134i	0.0300931	−	0.0521227i	−0.850587	−	0.525835i	$-0.823753\pi$
$83$	22.7554	−	39.4134i	0.0300931	−	0.0521227i	0.880680	+	0.473712i	$0.157086\pi$
$84$		0			0
$85$	225.338	+	390.297i	0.287545	+	0.498043i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	357.159			0.425379			0.212690	−	0.977120i	$-0.431778\pi$
$89$	357.159			0.425379			0.212690	+	0.977120i	$0.431778\pi$
$90$		0			0
$91$	−53.4757			−0.0616019
$92$		0			0
$93$		0			0
$94$		0			0
$95$	285.424	+	494.369i	0.308252	+	0.533908i
$96$		0			0
$97$	60.0029	−	103.928i	0.0628079	−	0.108786i	−0.832912	−	0.553406i	$-0.813328\pi$
$97$	60.0029	−	103.928i	0.0628079	−	0.108786i	0.895720	+	0.444620i	$0.146661\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−249.279	+	431.764i	−0.245586	+	0.425368i	−0.962296	−	0.272003i	$-0.912314\pi$
$101$	−249.279	+	431.764i	−0.245586	+	0.425368i	0.716710	+	0.697371i	$0.245647\pi$
$102$		0			0
$103$	127.343	+	220.565i	0.121820	+	0.210999i	0.920486	−	0.390777i	$-0.127793\pi$
$103$	127.343	+	220.565i	0.121820	+	0.210999i	−0.798665	+	0.601776i	$0.794460\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−311.077			−0.281056			−0.140528	−	0.990077i	$-0.544880\pi$
$107$	−311.077			−0.281056			−0.140528	+	0.990077i	$0.544880\pi$
$108$		0			0
$109$	−661.173			−0.580999			−0.290500	−	0.956875i	$-0.593822\pi$
$109$	−661.173			−0.580999			−0.290500	+	0.956875i	$0.593822\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−282.768	−	489.769i	−0.235404	−	0.407731i	0.723986	−	0.689814i	$-0.242308\pi$
$113$	−282.768	−	489.769i	−0.235404	−	0.407731i	−0.959390	+	0.282083i	$0.908975\pi$
$114$		0			0
$115$	121.026	−	209.624i	0.0981371	−	0.169979i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−280.056	+	485.072i	−0.215737	+	0.373668i
$120$		0			0
$121$	263.663	+	456.677i	0.198094	+	0.343109i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−125.000			−0.0894427
$126$		0			0
$127$	1809.60			1.26438			0.632190	−	0.774814i	$-0.282156\pi$
$127$	1809.60			1.26438			0.632190	+	0.774814i	$0.282156\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−1014.91	−	1757.88i	−0.676897	−	1.17242i	−0.975911	−	0.218171i	$-0.929991\pi$
$131$	−1014.91	−	1757.88i	−0.676897	−	1.17242i	0.299014	−	0.954249i	$-0.403342\pi$
$132$		0			0
$133$	−354.733	+	614.416i	−0.231273	+	0.400576i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	272.675	−	472.287i	0.170045	−	0.294527i	−0.768390	−	0.639982i	$-0.778942\pi$
$137$	272.675	−	472.287i	0.170045	−	0.294527i	0.938435	+	0.345455i	$0.112275\pi$
$138$		0			0
$139$	881.319	+	1526.49i	0.537788	+	0.931476i	0.999023	+	0.0441977i	$0.0140732\pi$
$139$	881.319	+	1526.49i	0.537788	+	0.931476i	−0.461235	+	0.887278i	$0.652594\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−243.958			−0.142663
$144$		0			0
$145$	1529.79			0.876152
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−611.765	−	1059.61i	−0.336361	−	0.582594i	0.647384	−	0.762164i	$-0.275863\pi$
$149$	−611.765	−	1059.61i	−0.336361	−	0.582594i	−0.983745	+	0.179569i	$0.942530\pi$
$150$		0			0
$151$	−1424.51	+	2467.33i	−0.767716	+	1.32972i	0.171082	+	0.985257i	$0.445274\pi$
$151$	−1424.51	+	2467.33i	−0.767716	+	1.32972i	−0.938798	+	0.344467i	$0.888060\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	233.989	−	405.281i	0.121255	−	0.210019i
$156$		0			0
$157$	471.416	+	816.516i	0.239637	+	0.415064i	0.960610	−	0.277899i	$-0.0896381\pi$
$157$	471.416	+	816.516i	0.239637	+	0.415064i	−0.720973	+	0.692963i	$0.756305\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	300.830			0.147259
$162$		0			0
$163$	417.774			0.200752			0.100376	−	0.994950i	$-0.467995\pi$
$163$	417.774			0.200752			0.100376	+	0.994950i	$0.467995\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−696.381	−	1206.17i	−0.322680	−	0.558898i	0.658360	−	0.752703i	$-0.271250\pi$
$167$	−696.381	−	1206.17i	−0.322680	−	0.558898i	−0.981040	+	0.193805i	$0.937917\pi$
$168$		0			0
$169$	1061.47	−	1838.52i	0.483146	−	0.836834i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1835.67	+	3179.48i	−0.806727	+	1.39729i	0.108392	+	0.994108i	$0.465430\pi$
$173$	−1835.67	+	3179.48i	−0.806727	+	1.39729i	−0.915119	+	0.403184i	$0.867903\pi$
$174$		0			0
$175$	−77.6768	−	134.540i	−0.0335532	−	0.0581159i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	2387.27			0.996834			0.498417	−	0.866938i	$-0.333915\pi$
$179$	2387.27			0.996834			0.498417	+	0.866938i	$0.333915\pi$
$180$		0			0
$181$	2078.63			0.853608			0.426804	−	0.904344i	$-0.359639\pi$
$181$	2078.63			0.853608			0.426804	+	0.904344i	$0.359639\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−707.229	−	1224.96i	−0.281062	−	0.486814i
$186$		0			0
$187$	−1277.63	+	2212.92i	−0.499623	+	0.865372i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−1495.51	+	2590.30i	−0.566553	+	0.981298i	0.430351	+	0.902662i	$0.358390\pi$
$191$	−1495.51	+	2590.30i	−0.566553	+	0.981298i	−0.996903	+	0.0786361i	$0.974943\pi$
$192$		0			0
$193$	2527.89	+	4378.44i	0.942807	+	1.63299i	0.760083	+	0.649826i	$0.225158\pi$
$193$	2527.89	+	4378.44i	0.942807	+	1.63299i	0.182725	+	0.983164i	$0.441508\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	3665.56			1.32569			0.662843	−	0.748758i	$-0.269349\pi$
$197$	3665.56			1.32569			0.662843	+	0.748758i	$0.269349\pi$
$198$		0			0
$199$	3217.92			1.14629			0.573147	−	0.819452i	$-0.305722\pi$
$199$	3217.92			1.14629			0.573147	+	0.819452i	$0.305722\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	950.633	+	1646.54i	0.328677	+	0.569285i
$204$		0			0
$205$	71.9213	−	124.571i	0.0245034	−	0.0424411i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−1618.31	+	2802.99i	−0.535601	+	0.927688i
$210$		0			0
$211$	1954.57	+	3385.42i	0.637718	+	1.10456i	0.985932	+	0.167145i	$0.0534548\pi$
$211$	1954.57	+	3385.42i	0.637718	+	1.10456i	−0.348214	+	0.937415i	$0.613212\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	1772.22			0.562159
$216$		0			0
$217$	581.617			0.181948
$218$		0			0
$219$		0			0
$220$		0			0
$221$	387.828	+	671.738i	0.118046	+	0.204462i
$222$		0			0
$223$	−404.361	+	700.375i	−0.121426	+	0.210316i	−0.920330	−	0.391142i	$-0.872080\pi$
$223$	−404.361	+	700.375i	−0.121426	+	0.210316i	0.798904	+	0.601458i	$0.205413\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	9.37578	−	16.2393i	0.00274138	−	0.00474821i	−0.864651	−	0.502372i	$-0.832461\pi$
$227$	9.37578	−	16.2393i	0.00274138	−	0.00474821i	0.867393	+	0.497624i	$0.165794\pi$
$228$		0			0
$229$	511.496	+	885.936i	0.147601	+	0.255652i	0.930340	−	0.366697i	$-0.119512\pi$
$229$	511.496	+	885.936i	0.147601	+	0.255652i	−0.782739	+	0.622350i	$0.786178\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−2881.43			−0.810166			−0.405083	−	0.914280i	$-0.632757\pi$
$233$	−2881.43			−0.810166			−0.405083	+	0.914280i	$0.632757\pi$
$234$		0			0
$235$	−2614.99			−0.725886
$236$		0			0
$237$		0			0
$238$		0			0
$239$	57.4142	+	99.4443i	0.0155390	+	0.0269143i	0.873690	−	0.486482i	$-0.161720\pi$
$239$	57.4142	+	99.4443i	0.0155390	+	0.0269143i	−0.858151	+	0.513397i	$0.828387\pi$
$240$		0			0
$241$	984.370	−	1704.98i	0.263107	−	0.455715i	−0.703959	−	0.710241i	$-0.748586\pi$
$241$	984.370	−	1704.98i	0.263107	−	0.455715i	0.967066	+	0.254526i	$0.0819194\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−760.961	+	1318.02i	−0.198433	+	0.343696i
$246$		0			0
$247$	491.243	+	850.857i	0.126547	+	0.219185i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	1706.47			0.429129			0.214564	−	0.976710i	$-0.431167\pi$
$251$	1706.47			0.429129			0.214564	+	0.976710i	$0.431167\pi$
$252$		0			0
$253$	1372.40			0.341035
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−3095.70	−	5361.92i	−0.751380	−	1.30143i	−0.947154	−	0.320780i	$-0.896055\pi$
$257$	−3095.70	−	5361.92i	−0.751380	−	1.30143i	0.195773	−	0.980649i	$-0.437278\pi$
$258$		0			0
$259$	878.965	−	1522.41i	0.210873	−	0.365243i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	69.3679	−	120.149i	0.0162639	−	0.0281699i	−0.857779	−	0.514019i	$-0.828156\pi$
$263$	69.3679	−	120.149i	0.0162639	−	0.0281699i	0.874043	+	0.485849i	$0.161489\pi$
$264$		0			0
$265$	167.321	+	289.809i	0.0387867	+	0.0671805i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	3362.16			0.762061			0.381030	−	0.924562i	$-0.375569\pi$
$269$	3362.16			0.762061			0.381030	+	0.924562i	$0.375569\pi$
$270$		0			0
$271$	3502.68			0.785140			0.392570	−	0.919722i	$-0.371586\pi$
$271$	3502.68			0.785140			0.392570	+	0.919722i	$0.371586\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−354.364	−	613.777i	−0.0777054	−	0.134590i
$276$		0			0
$277$	−1926.68	+	3337.11i	−0.417917	+	0.723853i	−0.995730	−	0.0923162i	$-0.970573\pi$
$277$	−1926.68	+	3337.11i	−0.417917	+	0.723853i	0.577813	+	0.816169i	$0.303906\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	4129.95	−	7153.29i	0.876770	−	1.51861i	0.0219047	−	0.999760i	$-0.493027\pi$
$281$	4129.95	−	7153.29i	0.876770	−	1.51861i	0.854865	−	0.518850i	$-0.173640\pi$
$282$		0			0
$283$	2585.53	+	4478.27i	0.543088	+	0.940656i	0.998725	+	0.0504901i	$0.0160783\pi$
$283$	2585.53	+	4478.27i	0.543088	+	0.940656i	−0.455637	+	0.890166i	$0.650588\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	178.772			0.0367685
$288$		0			0
$289$	3211.35			0.653644
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−2095.34	−	3629.24i	−0.417786	−	0.723626i	0.577931	−	0.816086i	$-0.303860\pi$
$293$	−2095.34	−	3629.24i	−0.417786	−	0.723626i	−0.995716	+	0.0924600i	$0.970527\pi$
$294$		0			0
$295$	19.0806	−	33.0485i	0.00376581	−	0.00652257i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	208.298	−	360.783i	0.0402883	−	0.0697813i
$300$		0			0
$301$	1101.28	+	1907.47i	0.210886	+	0.365266i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−47.0562			−0.00883420
$306$		0			0
$307$	3745.74			0.696355			0.348177	−	0.937429i	$-0.386801\pi$
$307$	3745.74			0.696355			0.348177	+	0.937429i	$0.386801\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−2427.39	−	4204.37i	−0.442588	−	0.766584i	0.555293	−	0.831655i	$-0.312606\pi$
$311$	−2427.39	−	4204.37i	−0.442588	−	0.766584i	−0.997881	+	0.0650705i	$0.979273\pi$
$312$		0			0
$313$	−1585.53	+	2746.22i	−0.286324	+	0.495928i	−0.972929	−	0.231102i	$-0.925767\pi$
$313$	−1585.53	+	2746.22i	−0.286324	+	0.495928i	0.686605	+	0.727030i	$0.259100\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−2918.53	+	5055.05i	−0.517101	+	0.895646i	0.482701	+	0.875785i	$0.339656\pi$
$317$	−2918.53	+	5055.05i	−0.517101	+	0.895646i	−0.999803	+	0.0198609i	$0.993678\pi$
$318$		0			0
$319$	4336.82	+	7511.60i	0.761177	+	1.31840i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	10290.7			1.77273
$324$		0			0
$325$	−215.137			−0.0367189
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−1624.99	−	2814.57i	−0.272306	−	0.471648i
$330$		0			0
$331$	−2797.65	+	4845.68i	−0.464571	+	0.804660i	−0.999182	−	0.0404378i	$-0.987125\pi$
$331$	−2797.65	+	4845.68i	−0.464571	+	0.804660i	0.534611	+	0.845098i	$0.320458\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−1235.98	+	2140.78i	−0.201578	+	0.349144i
$336$		0			0
$337$	4330.57	+	7500.78i	0.700004	+	1.21244i	0.968464	+	0.249152i	$0.0801519\pi$
$337$	4330.57	+	7500.78i	0.700004	+	1.21244i	−0.268460	+	0.963291i	$0.586515\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	2653.36			0.421371
$342$		0			0
$343$	−4022.94			−0.633289
$344$		0			0
$345$		0			0
$346$		0			0
$347$	5277.66	+	9141.18i	0.816483	+	1.41419i	0.908258	+	0.418410i	$0.137413\pi$
$347$	5277.66	+	9141.18i	0.816483	+	1.41419i	−0.0917750	+	0.995780i	$0.529254\pi$
$348$		0			0
$349$	−3199.86	+	5542.31i	−0.490786	+	0.850067i	−0.999944	−	0.0106068i	$-0.996624\pi$
$349$	−3199.86	+	5542.31i	−0.490786	+	0.850067i	0.509158	+	0.860673i	$0.329957\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−4889.06	+	8468.09i	−0.737162	+	1.27680i	0.216606	+	0.976259i	$0.430501\pi$
$353$	−4889.06	+	8468.09i	−0.737162	+	1.27680i	−0.953768	+	0.300543i	$0.902832\pi$
$354$		0			0
$355$	1401.77	+	2427.94i	0.209573	+	0.362991i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−2893.60			−0.425399			−0.212700	−	0.977118i	$-0.568226\pi$
$359$	−2893.60			−0.425399			−0.212700	+	0.977118i	$0.568226\pi$
$360$		0			0
$361$	6175.72			0.900382
$362$		0			0
$363$		0			0
$364$		0			0
$365$	2791.71	+	4835.39i	0.400342	+	0.693413i
$366$		0			0
$367$	3653.91	−	6328.76i	0.519707	−	0.900159i	−0.480031	−	0.877252i	$-0.659374\pi$
$367$	3653.91	−	6328.76i	0.519707	−	0.900159i	0.999738	−	0.0229072i	$-0.00729223\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−207.952	+	360.183i	−0.0291006	+	0.0504037i
$372$		0			0
$373$	−4136.12	−	7163.97i	−0.574156	−	0.994468i	−0.996133	−	0.0878609i	$-0.971997\pi$
$373$	−4136.12	−	7163.97i	−0.574156	−	0.994468i	0.421977	−	0.906607i	$-0.361336\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	2632.91			0.359687
$378$		0			0
$379$	−740.953			−0.100423			−0.0502113	−	0.998739i	$-0.515989\pi$
$379$	−740.953			−0.100423			−0.0502113	+	0.998739i	$0.515989\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−616.747	−	1068.24i	−0.0822828	−	0.142518i	0.821947	−	0.569564i	$-0.192888\pi$
$383$	−616.747	−	1068.24i	−0.0822828	−	0.142518i	−0.904230	+	0.427045i	$0.859554\pi$
$384$		0			0
$385$	440.414	−	762.820i	0.0583002	−	0.100979i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	3479.10	−	6025.98i	0.453464	−	0.785422i	−0.545135	−	0.838348i	$-0.683522\pi$
$389$	3479.10	−	6025.98i	0.453464	−	0.785422i	0.998598	+	0.0529264i	$0.0168549\pi$
$390$		0			0
$391$	−2181.75	−	3778.90i	−0.282189	−	0.488765i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−5209.35			−0.663572
$396$		0			0
$397$	5973.68			0.755190			0.377595	−	0.925971i	$-0.376751\pi$
$397$	5973.68			0.755190			0.377595	+	0.925971i	$0.376751\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−2358.44	−	4084.94i	−0.293703	−	0.508708i	0.680980	−	0.732302i	$-0.261554\pi$
$401$	−2358.44	−	4084.94i	−0.293703	−	0.508708i	−0.974682	+	0.223594i	$0.928221\pi$
$402$		0			0
$403$	402.718	−	697.528i	0.0497787	−	0.0862192i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	4009.87	−	6945.30i	0.488359	−	0.845862i
$408$		0			0
$409$	−3907.94	−	6768.75i	−0.472458	−	0.818321i	0.527046	−	0.849837i	$-0.323300\pi$
$409$	−3907.94	−	6768.75i	−0.472458	−	0.818321i	−0.999503	+	0.0315164i	$0.989966\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	47.4277			0.00565077
$414$		0			0
$415$	227.554			0.0269161
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−1477.58	−	2559.25i	−0.172278	−	0.298395i	0.766938	−	0.641722i	$-0.221780\pi$
$419$	−1477.58	−	2559.25i	−0.172278	−	0.298395i	−0.939216	+	0.343327i	$0.888446\pi$
$420$		0			0
$421$	3385.04	−	5863.06i	0.391869	−	0.678736i	−0.600827	−	0.799379i	$-0.705162\pi$
$421$	3385.04	−	5863.06i	0.391869	−	0.678736i	0.992696	+	0.120642i	$0.0384954\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1126.69	+	1951.48i	−0.128594	+	0.222731i
$426$		0			0
$427$	−29.2414	−	50.6476i	−0.00331403	−	0.00574006i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−6082.40			−0.679765			−0.339883	−	0.940468i	$-0.610387\pi$
$431$	−6082.40			−0.679765			−0.339883	+	0.940468i	$0.610387\pi$
$432$		0			0
$433$	−8993.04			−0.998102			−0.499051	−	0.866573i	$-0.666318\pi$
$433$	−8993.04			−0.998102			−0.499051	+	0.866573i	$0.666318\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−2763.51	−	4786.54i	−0.302509	−	0.523962i
$438$		0			0
$439$	−8590.25	+	14878.8i	−0.933919	+	1.61760i	−0.157369	+	0.987540i	$0.550301\pi$
$439$	−8590.25	+	14878.8i	−0.933919	+	1.61760i	−0.776550	+	0.630055i	$0.783032\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−6115.67	+	10592.6i	−0.655901	+	1.13605i	0.325766	+	0.945450i	$0.394378\pi$
$443$	−6115.67	+	10592.6i	−0.655901	+	1.13605i	−0.981667	+	0.190604i	$0.938955\pi$
$444$		0			0
$445$	892.897	+	1546.54i	0.0951177	+	0.164749i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	1182.08			0.124245			0.0621224	−	0.998069i	$-0.480213\pi$
$449$	1182.08			0.124245			0.0621224	+	0.998069i	$0.480213\pi$
$450$		0			0
$451$	815.563			0.0851515
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−133.689	−	231.556i	−0.0137746	−	0.0238583i
$456$		0			0
$457$	3676.52	−	6367.91i	0.376324	−	0.651813i	−0.614200	−	0.789150i	$-0.710521\pi$
$457$	3676.52	−	6367.91i	0.376324	−	0.651813i	0.990524	+	0.137338i	$0.0438545\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−1853.47	+	3210.30i	−0.187255	+	0.324335i	−0.944334	−	0.328988i	$-0.893292\pi$
$461$	−1853.47	+	3210.30i	−0.187255	+	0.324335i	0.757079	+	0.653323i	$0.226626\pi$
$462$		0			0
$463$	7007.68	+	12137.7i	0.703401	+	1.21833i	0.967266	+	0.253766i	$0.0816693\pi$
$463$	7007.68	+	12137.7i	0.703401	+	1.21833i	−0.263865	+	0.964560i	$0.584997\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−12187.1			−1.20760			−0.603801	−	0.797135i	$-0.706348\pi$
$467$	−12187.1			−1.20760			−0.603801	+	0.797135i	$0.706348\pi$
$468$		0			0
$469$	−3072.22			−0.302477
$470$		0			0
$471$		0			0
$472$		0			0
$473$	5024.08	+	8701.97i	0.488388	+	0.845913i
$474$		0			0
$475$	−1427.12	+	2471.85i	−0.137854	+	0.238771i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−32.3187	+	55.9777i	−0.00308284	+	0.00533964i	−0.867563	−	0.497328i	$-0.834315\pi$
$479$	−32.3187	+	55.9777i	−0.00308284	+	0.00533964i	0.864480	+	0.502667i	$0.167648\pi$
$480$		0			0
$481$	−1217.21	−	2108.27i	−0.115385	−	0.199852i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	600.029			0.0561771
$486$		0			0
$487$	−3763.31			−0.350168			−0.175084	−	0.984553i	$-0.556020\pi$
$487$	−3763.31			−0.350168			−0.175084	+	0.984553i	$0.556020\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−1053.47	−	1824.66i	−0.0968277	−	0.167710i	0.813542	−	0.581506i	$-0.197536\pi$
$491$	−1053.47	−	1824.66i	−0.0968277	−	0.167710i	−0.910370	+	0.413795i	$0.864203\pi$
$492$		0			0
$493$	13788.8	−	23882.9i	1.25967	−	2.18181i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−1742.16	+	3017.52i	−0.157237	+	0.272342i
$498$		0			0
$499$	−1065.49	−	1845.48i	−0.0955866	−	0.165561i	0.814267	−	0.580491i	$-0.197139\pi$
$499$	−1065.49	−	1845.48i	−0.0955866	−	0.165561i	−0.909853	+	0.414930i	$0.863806\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−12920.0			−1.14528			−0.572639	−	0.819808i	$-0.694080\pi$
$503$	−12920.0			−1.14528			−0.572639	+	0.819808i	$0.694080\pi$
$504$		0			0
$505$	−2492.79			−0.219659
$506$		0			0
$507$		0			0
$508$		0			0
$509$	342.904	+	593.927i	0.0298604	+	0.0517198i	0.880569	−	0.473917i	$-0.157160\pi$
$509$	342.904	+	593.927i	0.0298604	+	0.0517198i	−0.850709	+	0.525637i	$0.823827\pi$
$510$		0			0
$511$	−3469.62	+	6009.56i	−0.300366	+	0.520249i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−636.716	+	1102.82i	−0.0544797	+	0.0943616i
$516$		0			0
$517$	−7413.28	−	12840.2i	−0.630630	−	1.09228i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	21259.3			1.78769			0.893845	−	0.448377i	$-0.147998\pi$
$521$	21259.3			1.78769			0.893845	+	0.448377i	$0.147998\pi$
$522$		0			0
$523$	6099.14			0.509937			0.254968	−	0.966949i	$-0.417935\pi$
$523$	6099.14			0.509937			0.254968	+	0.966949i	$0.417935\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−4218.13	−	7306.02i	−0.348662	−	0.603900i
$528$		0			0
$529$	4911.71	−	8507.33i	0.403691	−	0.699213i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	123.783	−	214.399i	0.0100594	−	0.0174234i
$534$		0			0
$535$	−777.693	−	1347.00i	−0.0628460	−	0.108852i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−8629.04			−0.689572
$540$		0			0
$541$	9552.07			0.759104			0.379552	−	0.925170i	$-0.376078\pi$
$541$	9552.07			0.759104			0.379552	+	0.925170i	$0.376078\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−1652.93	−	2862.97i	−0.129915	−	0.225020i
$546$		0			0
$547$	−12439.0	+	21545.0i	−0.972309	+	1.68409i	−0.283766	+	0.958894i	$0.591584\pi$
$547$	−12439.0	+	21545.0i	−0.972309	+	1.68409i	−0.688543	+	0.725195i	$0.741749\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	17465.6	−	30251.2i	1.35038	−	2.33892i
$552$		0			0
$553$	−3237.17	−	5606.94i	−0.248930	−	0.431160i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	10625.7			0.808306			0.404153	−	0.914692i	$-0.367566\pi$
$557$	10625.7			0.808306			0.404153	+	0.914692i	$0.367566\pi$
$558$		0			0
$559$	3050.16			0.230783
$560$		0			0
$561$		0			0
$562$		0			0
$563$	8094.00	+	14019.2i	0.605900	+	1.04945i	0.991909	+	0.126954i	$0.0405200\pi$
$563$	8094.00	+	14019.2i	0.605900	+	1.04945i	−0.386009	+	0.922495i	$0.626147\pi$
$564$		0			0
$565$	1413.84	−	2448.85i	0.105276	−	0.182343i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−3959.73	+	6858.46i	−0.291741	+	0.505310i	−0.974221	−	0.225594i	$-0.927568\pi$
$569$	−3959.73	+	6858.46i	−0.291741	+	0.505310i	0.682481	+	0.730904i	$0.260901\pi$
$570$		0			0
$571$	−6752.66	−	11696.0i	−0.494904	−	0.857199i	0.505079	−	0.863073i	$-0.331463\pi$
$571$	−6752.66	−	11696.0i	−0.494904	−	0.857199i	−0.999983	+	0.00587432i	$0.998130\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	1210.26			0.0877765
$576$		0			0
$577$	−7873.07			−0.568042			−0.284021	−	0.958818i	$-0.591669\pi$
$577$	−7873.07			−0.568042			−0.284021	+	0.958818i	$0.591669\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	141.405	+	244.921i	0.0100972	+	0.0174889i
$582$		0			0
$583$	−948.684	+	1643.17i	−0.0673936	+	0.116729i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	10058.5	−	17421.8i	0.707254	−	1.22500i	−0.258617	−	0.965980i	$-0.583267\pi$
$587$	10058.5	−	17421.8i	0.707254	−	1.22500i	0.965872	−	0.259021i	$-0.0833998\pi$
$588$		0			0
$589$	−5342.90	−	9254.17i	−0.373769	−	0.647388i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−4394.15			−0.304294			−0.152147	−	0.988358i	$-0.548619\pi$
$593$	−4394.15			−0.304294			−0.152147	+	0.988358i	$0.548619\pi$
$594$		0			0
$595$	−2800.56			−0.192961
$596$		0			0
$597$		0			0
$598$		0			0
$599$	7122.37	+	12336.3i	0.485830	+	0.841482i	0.999867	−	0.0162858i	$-0.00518415\pi$
$599$	7122.37	+	12336.3i	0.485830	+	0.841482i	−0.514038	+	0.857768i	$0.671851\pi$
$600$		0			0
$601$	−8217.79	+	14233.6i	−0.557755	+	0.966059i	0.439929	+	0.898033i	$0.355004\pi$
$601$	−8217.79	+	14233.6i	−0.557755	+	0.966059i	−0.997684	+	0.0680267i	$0.978330\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−1318.31	+	2283.39i	−0.0885902	+	0.153443i
$606$		0			0
$607$	−13715.6	−	23756.1i	−0.917131	−	1.58852i	−0.803751	−	0.594965i	$-0.797166\pi$
$607$	−13715.6	−	23756.1i	−0.917131	−	1.58852i	−0.113380	−	0.993552i	$-0.536168\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−4500.65			−0.297998
$612$		0			0
$613$	−15161.9			−0.998995			−0.499497	−	0.866315i	$-0.666482\pi$
$613$	−15161.9			−0.998995			−0.499497	+	0.866315i	$0.666482\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−3080.22	−	5335.10i	−0.200981	−	0.348109i	0.747864	−	0.663852i	$-0.231079\pi$
$617$	−3080.22	−	5335.10i	−0.200981	−	0.348109i	−0.948845	+	0.315743i	$0.897746\pi$
$618$		0			0
$619$	13399.6	−	23208.8i	0.870074	−	1.50701i	0.00815380	−	0.999967i	$-0.497405\pi$
$619$	13399.6	−	23208.8i	0.870074	−	1.50701i	0.861920	−	0.507045i	$-0.169262\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−1109.72	+	1922.09i	−0.0713642	+	0.123606i
$624$		0			0
$625$	−312.500	−	541.266i	−0.0200000	−	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−25498.5			−1.61636
$630$		0			0
$631$	−26100.0			−1.64663			−0.823315	−	0.567585i	$-0.807878\pi$
$631$	−26100.0			−1.64663			−0.823315	+	0.567585i	$0.807878\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	4524.00	+	7835.80i	0.282724	+	0.489692i
$636$		0			0
$637$	−1309.69	+	2268.44i	−0.0814626	+	0.141097i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−1617.92	+	2802.32i	−0.0996944	+	0.172676i	−0.911558	−	0.411171i	$-0.865120\pi$
$641$	−1617.92	+	2802.32i	−0.0996944	+	0.172676i	0.811864	+	0.583847i	$0.198453\pi$
$642$		0			0
$643$	9224.18	+	15976.7i	0.565733	+	0.979878i	0.996981	+	0.0776445i	$0.0247399\pi$
$643$	9224.18	+	15976.7i	0.565733	+	0.979878i	−0.431248	+	0.902233i	$0.641927\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	17173.6			1.04353			0.521765	−	0.853089i	$-0.325274\pi$
$647$	17173.6			1.04353			0.521765	+	0.853089i	$0.325274\pi$
$648$		0			0
$649$	216.367			0.0130865
$650$		0			0
$651$		0			0
$652$		0			0
$653$	13019.7	+	22550.8i	0.780245	+	1.35142i	0.931799	+	0.362975i	$0.118239\pi$
$653$	13019.7	+	22550.8i	0.780245	+	1.35142i	−0.151554	+	0.988449i	$0.548428\pi$
$654$		0			0
$655$	5074.57	−	8789.42i	0.302718	−	0.524322i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	1353.06	−	2343.57i	0.0799815	−	0.138532i	−0.823260	−	0.567664i	$-0.807847\pi$
$659$	1353.06	−	2343.57i	0.0799815	−	0.138532i	0.903242	+	0.429132i	$0.141181\pi$
$660$		0			0
$661$	−13635.7	−	23617.7i	−0.802371	−	1.38975i	−0.918051	−	0.396461i	$-0.870238\pi$
$661$	−13635.7	−	23617.7i	−0.802371	−	1.38975i	0.115680	−	0.993287i	$-0.463095\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−3547.33			−0.206857
$666$		0			0
$667$	−14811.6			−0.859831
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−133.400	−	231.056i	−0.00767491	−	0.0132933i
$672$		0			0
$673$	6396.49	−	11079.0i	0.366369	−	0.634570i	−0.622626	−	0.782520i	$-0.713934\pi$
$673$	6396.49	−	11079.0i	0.366369	−	0.634570i	0.988995	+	0.147950i	$0.0472674\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	13557.0	−	23481.5i	0.769629	−	1.33304i	−0.168135	−	0.985764i	$-0.553774\pi$
$677$	13557.0	−	23481.5i	0.769629	−	1.33304i	0.937764	−	0.347273i	$-0.112892\pi$
$678$		0			0
$679$	372.866	+	645.823i	0.0210741	+	0.0365014i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−33488.7			−1.87615			−0.938075	−	0.346433i	$-0.887393\pi$
$683$	−33488.7			−1.87615			−0.938075	+	0.346433i	$0.887393\pi$
$684$		0			0
$685$	2726.75			0.152093
$686$		0			0
$687$		0			0
$688$		0			0
$689$	287.976	+	498.789i	0.0159231	+	0.0275796i
$690$		0			0
$691$	864.494	−	1497.35i	0.0475932	−	0.0824339i	−0.841247	−	0.540650i	$-0.818178\pi$
$691$	864.494	−	1497.35i	0.0475932	−	0.0824339i	0.888841	+	0.458216i	$0.151512\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−4406.59	+	7632.45i	−0.240506	+	0.416569i
$696$		0			0
$697$	−1296.53	−	2245.65i	−0.0704583	−	0.122037i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−9451.65			−0.509250			−0.254625	−	0.967040i	$-0.581952\pi$
$701$	−9451.65			−0.509250			−0.254625	+	0.967040i	$0.581952\pi$
$702$		0			0
$703$	−32297.7			−1.73276
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−1549.06	−	2683.04i	−0.0824021	−	0.142725i
$708$		0			0
$709$	11635.9	−	20153.9i	0.616353	−	1.06755i	−0.373792	−	0.927512i	$-0.621943\pi$
$709$	11635.9	−	20153.9i	0.616353	−	1.06755i	0.990145	−	0.140043i	$-0.0447239\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−2265.51	+	3923.98i	−0.118996	+	0.206107i
$714$		0			0
$715$	−609.895	−	1056.37i	−0.0319004	−	0.0552531i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	24974.3			1.29539			0.647695	−	0.761900i	$-0.275733\pi$
$719$	24974.3			1.29539			0.647695	+	0.761900i	$0.275733\pi$
$720$		0			0
$721$	−1582.66			−0.0817493
$722$		0			0
$723$		0			0
$724$		0			0
$725$	3824.47	+	6624.18i	0.195914	+	0.339332i
$726$		0			0
$727$	−833.261	+	1443.25i	−0.0425089	+	0.0736275i	−0.886497	−	0.462734i	$-0.846868\pi$
$727$	−833.261	+	1443.25i	−0.0425089	+	0.0736275i	0.843988	+	0.536362i	$0.180202\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	15973.9	−	27667.6i	0.808230	−	1.39990i
$732$		0			0
$733$	4347.46	+	7530.02i	0.219068	+	0.379437i	0.954523	−	0.298136i	$-0.0963649\pi$
$733$	4347.46	+	7530.02i	0.219068	+	0.379437i	−0.735455	+	0.677573i	$0.763032\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−14015.6			−0.700503
$738$		0			0
$739$	−19373.0			−0.964342			−0.482171	−	0.876077i	$-0.660152\pi$
$739$	−19373.0			−0.964342			−0.482171	+	0.876077i	$0.660152\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	13722.0	+	23767.2i	0.677538	+	1.17353i	0.975720	+	0.219021i	$0.0702862\pi$
$743$	13722.0	+	23767.2i	0.677538	+	1.17353i	−0.298183	+	0.954509i	$0.596380\pi$
$744$		0			0
$745$	3058.83	−	5298.04i	0.150425	−	0.260544i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	966.538	−	1674.09i	0.0471516	−	0.0816690i
$750$		0			0
$751$	1930.78	+	3344.21i	0.0938152	+	0.162493i	0.909114	−	0.416548i	$-0.136760\pi$
$751$	1930.78	+	3344.21i	0.0938152	+	0.162493i	−0.815298	+	0.579041i	$0.803427\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−14245.1			−0.686666
$756$		0			0
$757$	19846.9			0.952903			0.476452	−	0.879201i	$-0.341923\pi$
$757$	19846.9			0.952903			0.476452	+	0.879201i	$0.341923\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−15129.7	−	26205.4i	−0.720698	−	1.24828i	−0.960721	−	0.277518i	$-0.910488\pi$
$761$	−15129.7	−	26205.4i	−0.720698	−	1.24828i	0.240023	−	0.970767i	$-0.422845\pi$
$762$		0			0
$763$	2054.31	−	3558.17i	0.0974720	−	0.168826i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	32.8395	−	56.8797i	0.00154598	−	0.00267771i
$768$		0			0
$769$	−14938.6	−	25874.4i	−0.700518	−	1.21333i	−0.968285	−	0.249849i	$-0.919619\pi$
$769$	−14938.6	−	25874.4i	−0.700518	−	1.21333i	0.267766	−	0.963484i	$-0.413714\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	9509.77			0.442487			0.221244	−	0.975219i	$-0.428988\pi$
$773$	9509.77			0.442487			0.221244	+	0.975219i	$0.428988\pi$
$774$		0			0
$775$	2339.89			0.108453
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−1642.25	−	2844.45i	−0.0755322	−	0.130826i
$780$		0			0
$781$	−7947.81	+	13766.0i	−0.364142	+	0.630713i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−2357.08	+	4082.58i	−0.107169	+	0.185622i
$786$		0			0
$787$	6904.50	+	11958.9i	0.312730	+	0.541665i	0.978952	−	0.204088i	$-0.0654230\pi$
$787$	6904.50	+	11958.9i	0.312730	+	0.541665i	−0.666222	+	0.745753i	$0.732090\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	3514.33			0.157971
$792$		0			0
$793$	−80.9882			−0.00362670
$794$		0			0
$795$		0			0
$796$		0			0
$797$	8410.94	+	14568.2i	0.373815	+	0.647467i	0.990149	−	0.140018i	$-0.0447160\pi$
$797$	8410.94	+	14568.2i	0.373815	+	0.647467i	−0.616334	+	0.787485i	$0.711383\pi$
$798$		0			0
$799$	−23570.3	+	40824.9i	−1.04362	+	1.80761i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−15828.5	+	27415.8i	−0.695612	+	1.20484i
$804$		0			0
$805$	752.075	+	1302.63i	0.0329282	+	0.0570332i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	28306.2			1.23015			0.615075	−	0.788468i	$-0.289126\pi$
$809$	28306.2			1.23015			0.615075	+	0.788468i	$0.289126\pi$
$810$		0			0
$811$	13617.1			0.589594			0.294797	−	0.955560i	$-0.404748\pi$
$811$	13617.1			0.589594			0.294797	+	0.955560i	$0.404748\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	1044.43	+	1809.01i	0.0448895	+	0.0777509i
$816$		0			0
$817$	20233.3	−	35045.2i	0.866432	−	1.50070i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	15451.7	−	26763.1i	0.656841	−	1.13768i	−0.324588	−	0.945856i	$-0.605226\pi$
$821$	15451.7	−	26763.1i	0.656841	−	1.13768i	0.981429	−	0.191827i	$-0.0614411\pi$
$822$		0			0
$823$	−10136.6	−	17557.2i	−0.429333	−	0.743626i	0.567482	−	0.823386i	$-0.307918\pi$
$823$	−10136.6	−	17557.2i	−0.429333	−	0.743626i	−0.996814	+	0.0797603i	$0.974584\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−32437.6			−1.36393			−0.681963	−	0.731387i	$-0.738873\pi$
$827$	−32437.6			−1.36393			−0.681963	+	0.731387i	$0.738873\pi$
$828$		0			0
$829$	−7539.68			−0.315879			−0.157940	−	0.987449i	$-0.550485\pi$
$829$	−7539.68			−0.315879			−0.157940	+	0.987449i	$0.550485\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	13717.9	+	23760.1i	0.570584	+	0.988280i
$834$		0			0
$835$	3481.90	−	6030.83i	0.144307	−	0.249947i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	10419.7	−	18047.5i	0.428759	−	0.742632i	−0.568004	−	0.823026i	$-0.692284\pi$
$839$	10419.7	−	18047.5i	0.428759	−	0.742632i	0.996763	+	0.0803933i	$0.0256176\pi$
$840$		0			0
$841$	−34610.6	−	59947.3i	−1.41911	−	2.45797i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	10614.7			0.432139
$846$		0			0
$847$	−3276.88			−0.132934
$848$		0			0
$849$		0			0
$850$		0			0
$851$	6847.48	+	11860.2i	0.275827	+	0.477746i
$852$		0			0
$853$	18159.8	−	31453.7i	0.728934	−	1.26255i	−0.228400	−	0.973567i	$-0.573349\pi$
$853$	18159.8	−	31453.7i	0.728934	−	1.26255i	0.957334	−	0.288983i	$-0.0933172\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	5132.67	−	8890.04i	0.204584	−	0.354350i	−0.745416	−	0.666600i	$-0.767749\pi$
$857$	5132.67	−	8890.04i	0.204584	−	0.354350i	0.950000	+	0.312249i	$0.101082\pi$
$858$		0			0
$859$	−19109.3	−	33098.2i	−0.759022	−	1.31466i	−0.943350	−	0.331800i	$-0.892344\pi$
$859$	−19109.3	−	33098.2i	−0.759022	−	1.31466i	0.184328	−	0.982865i	$-0.440989\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−27385.0			−1.08018			−0.540090	−	0.841608i	$-0.681610\pi$
$863$	−27385.0			−1.08018			−0.540090	+	0.841608i	$0.681610\pi$
$864$		0			0
$865$	−18356.7			−0.721559
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−14768.1	−	25579.1i	−0.576493	−	0.998516i
$870$		0			0
$871$	−2127.24	+	3684.48i	−0.0827540	+	0.143334i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	388.384	−	672.701i	0.0150055	−	0.0259902i
$876$		0			0
$877$	−15659.8	−	27123.6i	−0.602958	−	1.04435i	−0.992371	−	0.123290i	$-0.960655\pi$
$877$	−15659.8	−	27123.6i	−0.602958	−	1.04435i	0.389413	−	0.921063i	$-0.372678\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−6309.71			−0.241293			−0.120647	−	0.992696i	$-0.538497\pi$
$881$	−6309.71			−0.241293			−0.120647	+	0.992696i	$0.538497\pi$
$882$		0			0
$883$	−24665.0			−0.940026			−0.470013	−	0.882660i	$-0.655751\pi$
$883$	−24665.0			−0.940026			−0.470013	+	0.882660i	$0.655751\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	13994.4	+	24239.0i	0.529746	+	0.917547i	0.999398	+	0.0346956i	$0.0110462\pi$
$887$	13994.4	+	24239.0i	0.529746	+	0.917547i	−0.469652	+	0.882852i	$0.655620\pi$
$888$		0			0
$889$	−5622.56	+	9738.56i	−0.212120	+	0.367402i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−29855.3	+	51710.9i	−1.11878	+	1.93778i
$894$		0			0
$895$	5968.18	+	10337.2i	0.222899	+	0.386072i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−28636.3			−1.06238
$900$		0			0
$901$	6032.62			0.223058
$902$		0			0
$903$		0			0
$904$		0			0
$905$	5196.56	+	9000.71i	0.190873	+	0.330601i
$906$		0			0
$907$	−11631.6	+	20146.5i	−0.425822	+	0.737546i	−0.996497	−	0.0836303i	$-0.973349\pi$
$907$	−11631.6	+	20146.5i	−0.425822	+	0.737546i	0.570674	+	0.821176i	$0.306682\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	19172.9	−	33208.5i	0.697286	−	1.20773i	−0.272118	−	0.962264i	$-0.587724\pi$
$911$	19172.9	−	33208.5i	0.697286	−	1.20773i	0.969404	−	0.245471i	$-0.0789425\pi$
$912$		0			0
$913$	645.095	+	1117.34i	0.0233839	+	0.0405022i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	12613.6			0.454241
$918$		0			0
$919$	−11675.3			−0.419077			−0.209539	−	0.977800i	$-0.567196\pi$
$919$	−11675.3			−0.419077			−0.209539	+	0.977800i	$0.567196\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	2412.59	+	4178.72i	0.0860360	+	0.149019i
$924$		0			0
$925$	3536.15	−	6124.79i	0.125695	−	0.217710i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	17025.2	−	29488.5i	0.601269	−	1.04143i	−0.391360	−	0.920238i	$-0.627995\pi$
$929$	17025.2	−	29488.5i	0.601269	−	1.04143i	0.992629	−	0.121191i	$-0.0386713\pi$
$930$		0			0
$931$	17375.7	+	30095.7i	0.611672	+	1.05945i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−12776.3			−0.446876
$936$		0			0
$937$	15395.5			0.536766			0.268383	−	0.963312i	$-0.413511\pi$
$937$	15395.5			0.536766			0.268383	+	0.963312i	$0.413511\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−12854.3	−	22264.4i	−0.445313	−	0.771305i	0.552761	−	0.833340i	$-0.313574\pi$
$941$	−12854.3	−	22264.4i	−0.445313	−	0.771305i	−0.998074	+	0.0620352i	$0.980241\pi$
$942$		0			0
$943$	−696.350	+	1206.11i	−0.0240469	+	0.0416505i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−3517.53	+	6092.55i	−0.120702	+	0.209061i	−0.920045	−	0.391814i	$-0.871848\pi$
$947$	−3517.53	+	6092.55i	−0.120702	+	0.209061i	0.799343	+	0.600875i	$0.205181\pi$
$948$		0			0
$949$	4804.80	+	8322.16i	0.164353	+	0.284667i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−15233.0			−0.517782			−0.258891	−	0.965907i	$-0.583357\pi$
$953$	−15233.0			−0.517782			−0.258891	+	0.965907i	$0.583357\pi$
$954$		0			0
$955$	−14955.1			−0.506740
$956$		0			0
$957$		0			0
$958$		0			0
$959$	1694.44	+	2934.86i	0.0570557	+	0.0988233i
$960$		0			0
$961$	10515.4	−	18213.2i	0.352973	−	0.611367i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−12639.5	+	21892.2i	−0.421636	+	0.730295i
$966$		0			0
$967$	−9155.83	−	15858.4i	−0.304479	−	0.527374i	0.672666	−	0.739946i	$-0.265149\pi$
$967$	−9155.83	−	15858.4i	−0.304479	−	0.527374i	−0.977145	+	0.212573i	$0.931816\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	37567.0			1.24159			0.620795	−	0.783973i	$-0.286810\pi$
$971$	37567.0			1.24159			0.620795	+	0.783973i	$0.286810\pi$
$972$		0			0
$973$	−10953.3			−0.360890
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−25020.0	−	43335.9i	−0.819305	−	1.41908i	−0.906195	−	0.422861i	$-0.861026\pi$
$977$	−25020.0	−	43335.9i	−0.819305	−	1.41908i	0.0868893	−	0.996218i	$-0.472307\pi$
$978$		0			0
$979$	−5062.58	+	8768.64i	−0.165271	+	0.286258i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	1664.03	−	2882.18i	0.0539921	−	0.0935170i	−0.837766	−	0.546029i	$-0.816139\pi$
$983$	1664.03	−	2882.18i	0.0539921	−	0.0935170i	0.891758	+	0.452512i	$0.149472\pi$
$984$		0			0
$985$	9163.90	+	15872.3i	0.296433	+	0.513436i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−17158.8			−0.551687
$990$		0			0
$991$	47485.9			1.52214			0.761069	−	0.648671i	$-0.224675\pi$
$991$	47485.9			1.52214			0.761069	+	0.648671i	$0.224675\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	8044.81	+	13934.0i	0.256319	+	0.443958i
$996$		0			0
$997$	−50.6958	+	87.8077i	−0.00161038	+	0.00278926i	−0.866829	−	0.498605i	$-0.833846\pi$
$997$	−50.6958	+	87.8077i	−0.00161038	+	0.00278926i	0.865219	+	0.501394i	$0.167179\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.4.i.x.541.3		12
3.2	odd	2		1620.4.i.w.541.3		12
9.2	odd	6		1620.4.a.j.1.4	yes	6
9.4	even	3	inner	1620.4.i.x.1081.3		12
9.5	odd	6		1620.4.i.w.1081.3		12
9.7	even	3		1620.4.a.i.1.4	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1620.4.a.i.1.4	✓	6	9.7	even	3
1620.4.a.j.1.4	yes	6	9.2	odd	6
1620.4.i.w.541.3		12	3.2	odd	2
1620.4.i.w.1081.3		12	9.5	odd	6
1620.4.i.x.541.3		12	1.1	even	1	trivial
1620.4.i.x.1081.3		12	9.4	even	3	inner

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:	\( N \)	\(=\)	\( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1620.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.50000 + 4.33013i) q^{5} +(-3.10707 + 5.38160i) q^{7} +O(q^{10})\)	\(q+(2.50000 + 4.33013i) q^{5} +(-3.10707 + 5.38160i) q^{7} +(-14.1746 + 24.5511i) q^{11} +(4.30274 + 7.45256i) q^{13} +90.1352 q^{17} +114.170 q^{19} +(-24.2053 - 41.9248i) q^{23} +(-12.5000 + 21.6506i) q^{25} +(152.979 - 264.967i) q^{29} +(-46.7978 - 81.0562i) q^{31} -31.0707 q^{35} -282.892 q^{37} +(-14.3843 - 24.9143i) q^{41} +(177.222 - 306.957i) q^{43} +(-261.499 + 452.930i) q^{47} +(152.192 + 263.605i) q^{49} +66.9285 q^{53} -141.746 q^{55} +(-3.81611 - 6.60970i) q^{59} +(-4.70562 + 8.15037i) q^{61} +(-21.5137 + 37.2628i) q^{65} +(247.196 + 428.155i) q^{67} +560.709 q^{71} +1116.68 q^{73} +(-88.0828 - 152.564i) q^{77} +(-520.935 + 902.287i) q^{79} +(22.7554 - 39.4134i) q^{83} +(225.338 + 390.297i) q^{85} +357.159 q^{89} -53.4757 q^{91} +(285.424 + 494.369i) q^{95} +(60.0029 - 103.928i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 30 q^{5} - 12 q^{7}+O(q^{10}) \) 12 * q + 30 * q^5 - 12 * q^7	\( 12 q + 30 q^{5} - 12 q^{7} + 84 q^{13} + 24 q^{17} - 228 q^{19} - 30 q^{23} - 150 q^{25} - 168 q^{29} + 324 q^{31} - 120 q^{35} - 984 q^{37} - 312 q^{41} + 156 q^{43} - 462 q^{47} + 588 q^{49} + 2028 q^{53} - 1008 q^{59} - 36 q^{61} - 420 q^{65} - 144 q^{67} + 2424 q^{71} - 1800 q^{73} - 672 q^{77} + 936 q^{79} - 288 q^{83} + 60 q^{85} - 240 q^{89} + 4572 q^{91} - 570 q^{95} + 1188 q^{97}+O(q^{100}) \) 12 * q + 30 * q^5 - 12 * q^7 + 84 * q^13 + 24 * q^17 - 228 * q^19 - 30 * q^23 - 150 * q^25 - 168 * q^29 + 324 * q^31 - 120 * q^35 - 984 * q^37 - 312 * q^41 + 156 * q^43 - 462 * q^47 + 588 * q^49 + 2028 * q^53 - 1008 * q^59 - 36 * q^61 - 420 * q^65 - 144 * q^67 + 2424 * q^71 - 1800 * q^73 - 672 * q^77 + 936 * q^79 - 288 * q^83 + 60 * q^85 - 240 * q^89 + 4572 * q^91 - 570 * q^95 + 1188 * q^97

Introduction

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