LMFDB - Embedded newform 1620.4.i.w.1081.4

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			1081.4
Root			$12.7486i$ of defining polynomial
Character	$\chi$	$=$	1620.1081
Dual form			1620.4.i.w.541.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.50000 + 4.33013i) q^{5} +(-0.292449 - 0.506536i) q^{7} +O(q^{10})$	$q+(-2.50000 + 4.33013i) q^{5} +(-0.292449 - 0.506536i) q^{7} +(-5.96034 - 10.3236i) q^{11} +(-26.8836 + 46.5638i) q^{13} -52.2388 q^{17} -144.152 q^{19} +(-27.4376 + 47.5234i) q^{23} +(-12.5000 - 21.6506i) q^{25} +(100.619 + 174.278i) q^{29} +(108.711 - 188.294i) q^{31} +2.92449 q^{35} +318.871 q^{37} +(187.806 - 325.290i) q^{41} +(19.0913 + 33.0672i) q^{43} +(-228.699 - 396.118i) q^{47} +(171.329 - 296.750i) q^{49} -356.645 q^{53} +59.6034 q^{55} +(-270.533 + 468.577i) q^{59} +(205.159 + 355.345i) q^{61} +(-134.418 - 232.819i) q^{65} +(-181.127 + 313.722i) q^{67} +175.350 q^{71} -105.897 q^{73} +(-3.48619 + 6.03826i) q^{77} +(-178.734 - 309.577i) q^{79} +(590.888 + 1023.45i) q^{83} +(130.597 - 226.201i) q^{85} -64.1496 q^{89} +31.4483 q^{91} +(360.380 - 624.197i) q^{95} +(-545.072 - 944.093i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times$.

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

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
$7$	−0.292449	−	0.506536i	−0.0157908	−	0.0273504i	0.858022	−	0.513613i	$-0.171693\pi$
$7$	−0.292449	−	0.506536i	−0.0157908	−	0.0273504i	−0.873813	+	0.486262i	$0.838360\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−5.96034	−	10.3236i	−0.163374	−	0.282972i	0.772703	−	0.634768i	$-0.218904\pi$
$11$	−5.96034	−	10.3236i	−0.163374	−	0.282972i	−0.936077	+	0.351796i	$0.885571\pi$
$12$		0			0
$13$	−26.8836	+	46.5638i	−0.573552	+	0.993421i	0.422645	+	0.906295i	$0.361102\pi$
$13$	−26.8836	+	46.5638i	−0.573552	+	0.993421i	−0.996197	+	0.0871262i	$0.972232\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−52.2388			−0.745281			−0.372641	−	0.927976i	$-0.621548\pi$
$17$	−52.2388			−0.745281			−0.372641	+	0.927976i	$0.621548\pi$
$18$		0			0
$19$	−144.152			−1.74057			−0.870283	−	0.492551i	$-0.836064\pi$
$19$	−144.152			−1.74057			−0.870283	+	0.492551i	$0.836064\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−27.4376	+	47.5234i	−0.248745	+	0.430839i	−0.963178	−	0.268865i	$-0.913352\pi$
$23$	−27.4376	+	47.5234i	−0.248745	+	0.430839i	0.714433	+	0.699704i	$0.246685\pi$
$24$		0			0
$25$	−12.5000	−	21.6506i	−0.100000	−	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	100.619	+	174.278i	0.644295	+	1.11595i	0.984464	+	0.175587i	$0.0561824\pi$
$29$	100.619	+	174.278i	0.644295	+	1.11595i	−0.340169	+	0.940364i	$0.610484\pi$
$30$		0			0
$31$	108.711	−	188.294i	0.629843	−	1.09092i	−0.357740	−	0.933821i	$-0.616453\pi$
$31$	108.711	−	188.294i	0.629843	−	1.09092i	0.987583	−	0.157099i	$-0.0502142\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.92449			0.0141237
$36$		0			0
$37$	318.871			1.41681			0.708406	−	0.705805i	$-0.249415\pi$
$37$	318.871			1.41681			0.708406	+	0.705805i	$0.249415\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	187.806	−	325.290i	0.715377	−	1.23907i	−0.247437	−	0.968904i	$-0.579588\pi$
$41$	187.806	−	325.290i	0.715377	−	1.23907i	0.962814	−	0.270165i	$-0.0870782\pi$
$42$		0			0
$43$	19.0913	+	33.0672i	0.0677070	+	0.117272i	0.897892	−	0.440217i	$-0.145098\pi$
$43$	19.0913	+	33.0672i	0.0677070	+	0.117272i	−0.830185	+	0.557489i	$0.811765\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−228.699	−	396.118i	−0.709769	−	1.22936i	−0.964943	−	0.262461i	$-0.915466\pi$
$47$	−228.699	−	396.118i	−0.709769	−	1.22936i	0.255173	−	0.966895i	$-0.417867\pi$
$48$		0			0
$49$	171.329	−	296.750i	0.499501	−	0.865162i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−356.645			−0.924320			−0.462160	−	0.886796i	$-0.652925\pi$
$53$	−356.645			−0.924320			−0.462160	+	0.886796i	$0.652925\pi$
$54$		0			0
$55$	59.6034			0.146126
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−270.533	+	468.577i	−0.596957	+	1.03396i	0.396311	+	0.918116i	$0.370290\pi$
$59$	−270.533	+	468.577i	−0.596957	+	1.03396i	−0.993268	+	0.115843i	$0.963043\pi$
$60$		0			0
$61$	205.159	+	355.345i	0.430621	+	0.745858i	0.996927	−	0.0783376i	$-0.0249612\pi$
$61$	205.159	+	355.345i	0.430621	+	0.745858i	−0.566306	+	0.824195i	$0.691628\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−134.418	−	232.819i	−0.256500	−	0.444272i
$66$		0			0
$67$	−181.127	+	313.722i	−0.330272	+	0.572048i	−0.982565	−	0.185919i	$-0.940474\pi$
$67$	−181.127	+	313.722i	−0.330272	+	0.572048i	0.652293	+	0.757967i	$0.273807\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	175.350			0.293101			0.146550	−	0.989203i	$-0.453183\pi$
$71$	175.350			0.293101			0.146550	+	0.989203i	$0.453183\pi$
$72$		0			0
$73$	−105.897			−0.169785			−0.0848924	−	0.996390i	$-0.527055\pi$
$73$	−105.897			−0.169785			−0.0848924	+	0.996390i	$0.527055\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−3.48619	+	6.03826i	−0.00515959	+	0.00893667i
$78$		0			0
$79$	−178.734	−	309.577i	−0.254547	−	0.440888i	0.710226	−	0.703974i	$-0.248593\pi$
$79$	−178.734	−	309.577i	−0.254547	−	0.440888i	−0.964772	+	0.263087i	$0.915260\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	590.888	+	1023.45i	0.781426	+	1.35347i	0.931111	+	0.364736i	$0.118841\pi$
$83$	590.888	+	1023.45i	0.781426	+	1.35347i	−0.149684	+	0.988734i	$0.547826\pi$
$84$		0			0
$85$	130.597	−	226.201i	0.166650	−	0.288646i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−64.1496			−0.0764028			−0.0382014	−	0.999270i	$-0.512163\pi$
$89$	−64.1496			−0.0764028			−0.0382014	+	0.999270i	$0.512163\pi$
$90$		0			0
$91$	31.4483			0.0362273
$92$		0			0
$93$		0			0
$94$		0			0
$95$	360.380	−	624.197i	0.389203	−	0.674119i
$96$		0			0
$97$	−545.072	−	944.093i	−0.570554	−	0.988228i	−0.996509	−	0.0834838i	$-0.973395\pi$
$97$	−545.072	−	944.093i	−0.570554	−	0.988228i	0.425955	−	0.904744i	$-0.359938\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	24.5886	+	42.5887i	0.0242243	+	0.0419577i	0.877883	−	0.478874i	$-0.158955\pi$
$101$	24.5886	+	42.5887i	0.0242243	+	0.0419577i	−0.853659	+	0.520832i	$0.825622\pi$
$102$		0			0
$103$	540.687	−	936.498i	0.517238	−	0.895883i	−0.482561	−	0.875862i	$-0.660294\pi$
$103$	540.687	−	936.498i	0.517238	−	0.895883i	0.999800	−	0.0200205i	$-0.00637316\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1477.54			−1.33495			−0.667475	−	0.744632i	$-0.732625\pi$
$107$	−1477.54			−1.33495			−0.667475	+	0.744632i	$0.732625\pi$
$108$		0			0
$109$	1079.74			0.948806			0.474403	−	0.880308i	$-0.342664\pi$
$109$	1079.74			0.948806			0.474403	+	0.880308i	$0.342664\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	453.322	−	785.177i	0.377389	−	0.653657i	−0.613293	−	0.789856i	$-0.710155\pi$
$113$	453.322	−	785.177i	0.377389	−	0.653657i	0.990681	+	0.136199i	$0.0434887\pi$
$114$		0			0
$115$	−137.188	−	237.617i	−0.111242	−	0.192677i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	15.2772	+	26.4609i	0.0117685	+	0.0203837i
$120$		0			0
$121$	594.449	−	1029.62i	0.446618	−	0.773565i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	125.000			0.0894427
$126$		0			0
$127$	1141.81			0.797790			0.398895	−	0.916997i	$-0.369394\pi$
$127$	1141.81			0.797790			0.398895	+	0.916997i	$0.369394\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	407.003	−	704.949i	0.271450	−	0.470166i	−0.697783	−	0.716309i	$-0.745830\pi$
$131$	407.003	−	704.949i	0.271450	−	0.470166i	0.969233	+	0.246143i	$0.0791634\pi$
$132$		0			0
$133$	42.1571	+	73.0183i	0.0274849	+	0.0476052i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	1409.33	+	2441.03i	0.878884	+	1.52227i	0.852567	+	0.522617i	$0.175044\pi$
$137$	1409.33	+	2441.03i	0.878884	+	1.52227i	0.0263164	+	0.999654i	$0.491622\pi$
$138$		0			0
$139$	480.823	−	832.810i	0.293402	−	0.508187i	−0.681210	−	0.732088i	$-0.738546\pi$
$139$	480.823	−	832.810i	0.293402	−	0.508187i	0.974612	+	0.223901i	$0.0718792\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	640.943			0.374813
$144$		0			0
$145$	−1006.19			−0.576275
$146$		0			0
$147$		0			0
$148$		0			0
$149$	842.961	−	1460.05i	0.463477	−	0.802766i	−0.535654	−	0.844437i	$-0.679935\pi$
$149$	842.961	−	1460.05i	0.463477	−	0.802766i	0.999131	+	0.0416715i	$0.0132683\pi$
$150$		0			0
$151$	−428.880	−	742.842i	−0.231138	−	0.400342i	0.727006	−	0.686632i	$-0.240911\pi$
$151$	−428.880	−	742.842i	−0.231138	−	0.400342i	−0.958143	+	0.286290i	$0.907578\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	543.557	+	941.468i	0.281674	+	0.487874i
$156$		0			0
$157$	1264.50	−	2190.18i	0.642790	−	1.11335i	−0.342017	−	0.939694i	$-0.611110\pi$
$157$	1264.50	−	2190.18i	0.642790	−	1.11335i	0.984807	−	0.173652i	$-0.0555567\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	32.0964			0.0157115
$162$		0			0
$163$	1616.48			0.776764			0.388382	−	0.921498i	$-0.373034\pi$
$163$	1616.48			0.776764			0.388382	+	0.921498i	$0.373034\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−122.087	+	211.461i	−0.0565712	+	0.0979841i	−0.892924	−	0.450207i	$-0.851350\pi$
$167$	−122.087	+	211.461i	−0.0565712	+	0.0979841i	0.836353	+	0.548191i	$0.184683\pi$
$168$		0			0
$169$	−346.959	−	600.951i	−0.157924	−	0.273533i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−683.784	−	1184.35i	−0.300504	−	0.520488i	0.675746	−	0.737134i	$-0.263821\pi$
$173$	−683.784	−	1184.35i	−0.300504	−	0.520488i	−0.976250	+	0.216646i	$0.930488\pi$
$174$		0			0
$175$	−7.31122	+	12.6634i	−0.00315815	+	0.00547008i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−2368.40			−0.988952			−0.494476	−	0.869191i	$-0.664640\pi$
$179$	−2368.40			−0.988952			−0.494476	+	0.869191i	$0.664640\pi$
$180$		0			0
$181$	3750.84			1.54032			0.770159	−	0.637851i	$-0.220177\pi$
$181$	3750.84			1.54032			0.770159	+	0.637851i	$0.220177\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−797.177	+	1380.75i	−0.316809	+	0.548729i
$186$		0			0
$187$	311.361	+	539.294i	0.121759	+	0.210893i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−75.5236	−	130.811i	−0.0286110	−	0.0495556i	0.851365	−	0.524573i	$-0.175775\pi$
$191$	−75.5236	−	130.811i	−0.0286110	−	0.0495556i	−0.879976	+	0.475017i	$0.842442\pi$
$192$		0			0
$193$	575.063	−	996.038i	0.214476	−	0.371484i	−0.738634	−	0.674107i	$-0.764529\pi$
$193$	575.063	−	996.038i	0.214476	−	0.371484i	0.953110	+	0.302623i	$0.0978622\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	1517.35			0.548764			0.274382	−	0.961621i	$-0.411527\pi$
$197$	1517.35			0.548764			0.274382	+	0.961621i	$0.411527\pi$
$198$		0			0
$199$	3022.98			1.07685			0.538426	−	0.842673i	$-0.319019\pi$
$199$	3022.98			1.07685			0.538426	+	0.842673i	$0.319019\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	58.8520	−	101.935i	0.0203478	−	0.0352434i
$204$		0			0
$205$	939.032	+	1626.45i	0.319926	+	0.554128i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	859.196	+	1488.17i	0.284363	+	0.492531i
$210$		0			0
$211$	912.009	−	1579.65i	0.297561	−	0.515390i	−0.678017	−	0.735047i	$-0.737160\pi$
$211$	912.009	−	1579.65i	0.297561	−	0.515390i	0.975577	+	0.219656i	$0.0704936\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−190.913			−0.0605590
$216$		0			0
$217$	−127.170			−0.0397828
$218$		0			0
$219$		0			0
$220$		0			0
$221$	1404.37	−	2432.44i	0.427458	−	0.740378i
$222$		0			0
$223$	−2772.17	−	4801.54i	−0.832459	−	1.44186i	−0.896082	−	0.443888i	$-0.853599\pi$
$223$	−2772.17	−	4801.54i	−0.832459	−	1.44186i	0.0636231	−	0.997974i	$-0.479734\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	195.822	+	339.173i	0.0572562	+	0.0991706i	0.893233	−	0.449594i	$-0.148431\pi$
$227$	195.822	+	339.173i	0.0572562	+	0.0991706i	−0.835977	+	0.548765i	$0.815098\pi$
$228$		0			0
$229$	1001.08	−	1733.92i	0.288878	−	0.500351i	−0.684664	−	0.728859i	$-0.740051\pi$
$229$	1001.08	−	1733.92i	0.288878	−	0.500351i	0.973542	+	0.228507i	$0.0733845\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−5261.60			−1.47939			−0.739697	−	0.672941i	$-0.765031\pi$
$233$	−5261.60			−1.47939			−0.739697	+	0.672941i	$0.765031\pi$
$234$		0			0
$235$	2286.99			0.634837
$236$		0			0
$237$		0			0
$238$		0			0
$239$	2396.50	−	4150.87i	0.648606	−	1.12342i	−0.334849	−	0.942272i	$-0.608685\pi$
$239$	2396.50	−	4150.87i	0.648606	−	1.12342i	0.983456	−	0.181148i	$-0.0579812\pi$
$240$		0			0
$241$	−1865.28	−	3230.76i	−0.498562	−	0.863534i	0.501437	−	0.865194i	$-0.332805\pi$
$241$	−1865.28	−	3230.76i	−0.498562	−	0.863534i	−0.999999	+	0.00165983i	$0.999472\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	856.645	+	1483.75i	0.223384	+	0.386912i
$246$		0			0
$247$	3875.33	−	6712.27i	0.998306	−	1.72912i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−6411.67			−1.61235			−0.806177	−	0.591674i	$-0.798467\pi$
$251$	−6411.67			−1.61235			−0.806177	+	0.591674i	$0.798467\pi$
$252$		0			0
$253$	654.150			0.162554
$254$		0			0
$255$		0			0
$256$		0			0
$257$	2644.89	−	4581.08i	0.641959	−	1.11191i	−0.343036	−	0.939322i	$-0.611455\pi$
$257$	2644.89	−	4581.08i	0.641959	−	1.11191i	0.984995	−	0.172584i	$-0.0552115\pi$
$258$		0			0
$259$	−93.2534	−	161.520i	−0.0223725	−	0.0387503i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	3818.75	+	6614.28i	0.895340	+	1.55077i	0.833383	+	0.552696i	$0.186401\pi$
$263$	3818.75	+	6614.28i	0.895340	+	1.55077i	0.0619573	+	0.998079i	$0.480266\pi$
$264$		0			0
$265$	891.613	−	1544.32i	0.206684	−	0.357988i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−2299.60			−0.521224			−0.260612	−	0.965444i	$-0.583924\pi$
$269$	−2299.60			−0.521224			−0.260612	+	0.965444i	$0.583924\pi$
$270$		0			0
$271$	−1962.43			−0.439886			−0.219943	−	0.975513i	$-0.570587\pi$
$271$	−1962.43			−0.439886			−0.219943	+	0.975513i	$0.570587\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−149.009	+	258.090i	−0.0326747	+	0.0565943i
$276$		0			0
$277$	1336.78	+	2315.37i	0.289961	+	0.502227i	0.973800	−	0.227406i	$-0.0730244\pi$
$277$	1336.78	+	2315.37i	0.289961	+	0.502227i	−0.683839	+	0.729633i	$0.739691\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	981.799	+	1700.53i	0.208431	+	0.361014i	0.951221	−	0.308512i	$-0.0998309\pi$
$281$	981.799	+	1700.53i	0.208431	+	0.361014i	−0.742789	+	0.669525i	$0.766498\pi$
$282$		0			0
$283$	4180.60	−	7241.02i	0.878131	−	1.52097i	0.0247409	−	0.999694i	$-0.492124\pi$
$283$	4180.60	−	7241.02i	0.878131	−	1.52097i	0.853390	−	0.521273i	$-0.174543\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−219.695			−0.0451853
$288$		0			0
$289$	−2184.10			−0.444556
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−167.953	+	290.902i	−0.0334877	+	0.0580024i	−0.882284	−	0.470718i	$-0.843995\pi$
$293$	−167.953	+	290.902i	−0.0334877	+	0.0580024i	0.848796	+	0.528721i	$0.177328\pi$
$294$		0			0
$295$	−1352.67	−	2342.89i	−0.266967	−	0.462401i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−1475.25	−	2555.20i	−0.285337	−	0.494218i
$300$		0			0
$301$	11.1665	−	19.3409i	0.00213829	−	0.00370363i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−2051.59			−0.385159
$306$		0			0
$307$	3183.28			0.591789			0.295895	−	0.955221i	$-0.404382\pi$
$307$	3183.28			0.591789			0.295895	+	0.955221i	$0.404382\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−2731.38	+	4730.89i	−0.498014	+	0.862586i	−0.999997	−	0.00229154i	$-0.999271\pi$
$311$	−2731.38	+	4730.89i	−0.498014	+	0.862586i	0.501983	+	0.864877i	$0.332604\pi$
$312$		0			0
$313$	4092.34	+	7088.14i	0.739018	+	1.28002i	0.952938	+	0.303166i	$0.0980436\pi$
$313$	4092.34	+	7088.14i	0.739018	+	1.28002i	−0.213920	+	0.976851i	$0.568623\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−2388.14	−	4136.39i	−0.423128	−	0.732879i	0.573116	−	0.819475i	$-0.305735\pi$
$317$	−2388.14	−	4136.39i	−0.423128	−	0.732879i	−0.996244	+	0.0865954i	$0.972401\pi$
$318$		0			0
$319$	1199.45	−	2077.51i	0.210522	−	0.364634i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	7530.34			1.29721
$324$		0			0
$325$	1344.18			0.229421
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−133.765	+	231.688i	−0.0224156	+	0.0388249i
$330$		0			0
$331$	−2020.51	−	3499.63i	−0.335521	−	0.581139i	0.648064	−	0.761586i	$-0.275579\pi$
$331$	−2020.51	−	3499.63i	−0.335521	−	0.581139i	−0.983585	+	0.180447i	$0.942246\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−905.637	−	1568.61i	−0.147702	−	0.255828i
$336$		0			0
$337$	−3983.28	+	6899.24i	−0.643866	+	1.11521i	0.340696	+	0.940174i	$0.389337\pi$
$337$	−3983.28	+	6899.24i	−0.643866	+	1.11521i	−0.984562	+	0.175036i	$0.943996\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−2591.83			−0.411599
$342$		0			0
$343$	−401.040			−0.0631315
$344$		0			0
$345$		0			0
$346$		0			0
$347$	2200.66	−	3811.65i	0.340454	−	0.589684i	−0.644063	−	0.764972i	$-0.722753\pi$
$347$	2200.66	−	3811.65i	0.340454	−	0.589684i	0.984517	+	0.175289i	$0.0560859\pi$
$348$		0			0
$349$	448.145	+	776.209i	0.0687353	+	0.119053i	0.898345	−	0.439291i	$-0.144770\pi$
$349$	448.145	+	776.209i	0.0687353	+	0.119053i	−0.829610	+	0.558344i	$0.811437\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	1422.25	+	2463.41i	0.214444	+	0.371428i	0.953100	−	0.302654i	$-0.0978727\pi$
$353$	1422.25	+	2463.41i	0.214444	+	0.371428i	−0.738656	+	0.674082i	$0.764539\pi$
$354$		0			0
$355$	−438.374	+	759.286i	−0.0655393	+	0.113517i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	2351.30			0.345674			0.172837	−	0.984950i	$-0.444707\pi$
$359$	2351.30			0.345674			0.172837	+	0.984950i	$0.444707\pi$
$360$		0			0
$361$	13920.8			2.02957
$362$		0			0
$363$		0			0
$364$		0			0
$365$	264.742	−	458.547i	0.0379650	−	0.0657574i
$366$		0			0
$367$	−5662.64	−	9807.98i	−0.805415	−	1.39502i	−0.916011	−	0.401154i	$-0.868609\pi$
$367$	−5662.64	−	9807.98i	−0.805415	−	1.39502i	0.110595	−	0.993866i	$-0.464724\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	104.300	+	180.654i	0.0145957	+	0.0252805i
$372$		0			0
$373$	3960.35	−	6859.53i	0.549757	−	0.952207i	−0.448534	−	0.893766i	$-0.648054\pi$
$373$	3960.35	−	6859.53i	0.549757	−	0.952207i	0.998291	−	0.0584411i	$-0.0186130\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−10820.1			−1.47815
$378$		0			0
$379$	2235.45			0.302974			0.151487	−	0.988459i	$-0.451594\pi$
$379$	2235.45			0.302974			0.151487	+	0.988459i	$0.451594\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−2215.09	+	3836.65i	−0.295524	+	0.511863i	−0.975107	−	0.221736i	$-0.928828\pi$
$383$	−2215.09	+	3836.65i	−0.295524	+	0.511863i	0.679582	+	0.733599i	$0.262161\pi$
$384$		0			0
$385$	−17.4310	−	30.1913i	−0.00230744	−	0.00399660i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	6120.61	+	10601.2i	0.797756	+	1.38175i	0.921074	+	0.389387i	$0.127313\pi$
$389$	6120.61	+	10601.2i	0.797756	+	1.38175i	−0.123318	+	0.992367i	$0.539354\pi$
$390$		0			0
$391$	1433.31	−	2482.56i	0.185385	−	0.321096i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	1787.34			0.227673
$396$		0			0
$397$	2080.41			0.263004			0.131502	−	0.991316i	$-0.458020\pi$
$397$	2080.41			0.263004			0.131502	+	0.991316i	$0.458020\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	4277.52	−	7408.88i	0.532691	−	0.922649i	−0.466580	−	0.884479i	$-0.654514\pi$
$401$	4277.52	−	7408.88i	0.532691	−	0.922649i	0.999271	−	0.0381695i	$-0.0121527\pi$
$402$		0			0
$403$	5845.11	+	10124.0i	0.722496	+	1.25140i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−1900.58	−	3291.90i	−0.231470	−	0.400917i
$408$		0			0
$409$	6048.32	−	10476.0i	0.731223	−	1.26652i	−0.225138	−	0.974327i	$-0.572283\pi$
$409$	6048.32	−	10476.0i	0.731223	−	1.26652i	0.956361	−	0.292188i	$-0.0943835\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	316.469			0.0377056
$414$		0			0
$415$	−5908.88			−0.698929
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−3251.47	+	5631.72i	−0.379105	+	0.656629i	−0.990932	−	0.134363i	$-0.957101\pi$
$419$	−3251.47	+	5631.72i	−0.379105	+	0.656629i	0.611828	+	0.790991i	$0.290435\pi$
$420$		0			0
$421$	−452.572	−	783.878i	−0.0523919	−	0.0907455i	0.838640	−	0.544686i	$-0.183351\pi$
$421$	−452.572	−	783.878i	−0.0523919	−	0.0907455i	−0.891032	+	0.453941i	$0.850018\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	652.985	+	1131.00i	0.0745281	+	0.129086i
$426$		0			0
$427$	119.997	−	207.841i	0.0135997	−	0.0235553i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−4106.21			−0.458907			−0.229454	−	0.973320i	$-0.573694\pi$
$431$	−4106.21			−0.458907			−0.229454	+	0.973320i	$0.573694\pi$
$432$		0			0
$433$	108.546			0.0120471			0.00602355	−	0.999982i	$-0.498083\pi$
$433$	108.546			0.0120471			0.00602355	+	0.999982i	$0.498083\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	3955.19	−	6850.59i	0.432958	−	0.749905i
$438$		0			0
$439$	−3195.43	−	5534.65i	−0.347403	−	0.601719i	0.638385	−	0.769717i	$-0.279603\pi$
$439$	−3195.43	−	5534.65i	−0.347403	−	0.601719i	−0.985787	+	0.167999i	$0.946270\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	2591.00	+	4487.74i	0.277882	+	0.481306i	0.970858	−	0.239654i	$-0.0770341\pi$
$443$	2591.00	+	4487.74i	0.277882	+	0.481306i	−0.692976	+	0.720961i	$0.743701\pi$
$444$		0			0
$445$	160.374	−	277.776i	0.0170842	−	0.0295907i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	6711.96			0.705472			0.352736	−	0.935723i	$-0.385251\pi$
$449$	6711.96			0.705472			0.352736	+	0.935723i	$0.385251\pi$
$450$		0			0
$451$	−4477.56			−0.467495
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−78.6209	+	136.175i	−0.00810067	+	0.0140308i
$456$		0			0
$457$	−2098.53	−	3634.76i	−0.214803	−	0.372050i	0.738408	−	0.674354i	$-0.235578\pi$
$457$	−2098.53	−	3634.76i	−0.214803	−	0.372050i	−0.953212	+	0.302304i	$0.902244\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−3621.76	−	6273.08i	−0.365905	−	0.633767i	0.623016	−	0.782209i	$-0.285907\pi$
$461$	−3621.76	−	6273.08i	−0.365905	−	0.633767i	−0.988921	+	0.148443i	$0.952574\pi$
$462$		0			0
$463$	−6275.17	+	10868.9i	−0.629875	+	1.09098i	0.357702	+	0.933836i	$0.383561\pi$
$463$	−6275.17	+	10868.9i	−0.629875	+	1.09098i	−0.987576	+	0.157139i	$0.949773\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	5357.92			0.530910			0.265455	−	0.964123i	$-0.414478\pi$
$467$	5357.92			0.530910			0.265455	+	0.964123i	$0.414478\pi$
$468$		0			0
$469$	211.882			0.0208610
$470$		0			0
$471$		0			0
$472$		0			0
$473$	227.582	−	394.183i	0.0221231	−	0.0383183i
$474$		0			0
$475$	1801.90	+	3120.99i	0.174057	+	0.301475i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−5173.10	−	8960.07i	−0.493455	−	0.854689i	0.506517	−	0.862230i	$-0.330933\pi$
$479$	−5173.10	−	8960.07i	−0.493455	−	0.854689i	−0.999972	+	0.00754133i	$0.997599\pi$
$480$		0			0
$481$	−8572.40	+	14847.8i	−0.812615	+	1.40749i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	5450.72			0.510319
$486$		0			0
$487$	1116.15			0.103855			0.0519276	−	0.998651i	$-0.483463\pi$
$487$	1116.15			0.103855			0.0519276	+	0.998651i	$0.483463\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	8404.27	−	14556.6i	0.772463	−	1.33795i	−0.163746	−	0.986503i	$-0.552358\pi$
$491$	8404.27	−	14556.6i	0.772463	−	1.33795i	0.936209	−	0.351443i	$-0.114309\pi$
$492$		0			0
$493$	−5256.24	−	9104.07i	−0.480181	−	0.831697i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−51.2808	−	88.8209i	−0.00462828	−	0.00801642i
$498$		0			0
$499$	−5192.12	+	8993.01i	−0.465794	+	0.806778i	−0.999237	−	0.0390576i	$-0.987564\pi$
$499$	−5192.12	+	8993.01i	−0.465794	+	0.806778i	0.533443	+	0.845836i	$0.320898\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	1049.28			0.0930120			0.0465060	−	0.998918i	$-0.485191\pi$
$503$	1049.28			0.0930120			0.0465060	+	0.998918i	$0.485191\pi$
$504$		0			0
$505$	−245.886			−0.0216669
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−4262.26	+	7382.45i	−0.371162	+	0.642871i	−0.989745	−	0.142848i	$-0.954374\pi$
$509$	−4262.26	+	7382.45i	−0.371162	+	0.642871i	0.618583	+	0.785720i	$0.287707\pi$
$510$		0			0
$511$	30.9694	+	53.6406i	0.00268103	+	0.00464368i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	2703.44	+	4682.49i	0.231316	+	0.400651i
$516$		0			0
$517$	−2726.25	+	4722.00i	−0.231915	+	0.401689i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	9617.07			0.808698			0.404349	−	0.914605i	$-0.367498\pi$
$521$	9617.07			0.808698			0.404349	+	0.914605i	$0.367498\pi$
$522$		0			0
$523$	−14484.2			−1.21100			−0.605498	−	0.795847i	$-0.707026\pi$
$523$	−14484.2			−1.21100			−0.605498	+	0.795847i	$0.707026\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−5678.95	+	9836.24i	−0.469410	+	0.813042i
$528$		0			0
$529$	4577.85	+	7929.08i	0.376252	+	0.651687i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	10097.8	+	17490.0i	0.820612	+	1.42134i
$534$		0			0
$535$	3693.86	−	6397.96i	0.298504	−	0.517024i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−4084.72			−0.326422
$540$		0			0
$541$	−16268.4			−1.29286			−0.646428	−	0.762975i	$-0.723738\pi$
$541$	−16268.4			−1.29286			−0.646428	+	0.762975i	$0.723738\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−2699.34	+	4675.39i	−0.212159	+	0.367471i
$546$		0			0
$547$	3771.82	+	6532.98i	0.294829	+	0.510659i	0.974945	−	0.222446i	$-0.0714040\pi$
$547$	3771.82	+	6532.98i	0.294829	+	0.510659i	−0.680116	+	0.733104i	$0.738071\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−14504.5	−	25122.5i	−1.12144	−	1.94239i
$552$		0			0
$553$	−104.541	+	181.071i	−0.00803896	+	0.0139239i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−1425.55			−0.108442			−0.0542212	−	0.998529i	$-0.517268\pi$
$557$	−1425.55			−0.108442			−0.0542212	+	0.998529i	$0.517268\pi$
$558$		0			0
$559$	−2052.98			−0.155334
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−11907.3	+	20624.0i	−0.891355	+	1.54387i	−0.0531029	+	0.998589i	$0.516911\pi$
$563$	−11907.3	+	20624.0i	−0.891355	+	1.54387i	−0.838252	+	0.545283i	$0.816422\pi$
$564$		0			0
$565$	2266.61	+	3925.88i	0.168773	+	0.292324i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−1296.10	−	2244.91i	−0.0954924	−	0.165398i	0.814322	−	0.580414i	$-0.197109\pi$
$569$	−1296.10	−	2244.91i	−0.0954924	−	0.165398i	−0.909814	+	0.415016i	$0.863776\pi$
$570$		0			0
$571$	2061.83	−	3571.20i	0.151112	−	0.261734i	−0.780525	−	0.625125i	$-0.785048\pi$
$571$	2061.83	−	3571.20i	0.151112	−	0.261734i	0.931637	+	0.363391i	$0.118381\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	1371.88			0.0994981
$576$		0			0
$577$	7216.46			0.520667			0.260334	−	0.965519i	$-0.416167\pi$
$577$	7216.46			0.520667			0.260334	+	0.965519i	$0.416167\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	345.609	−	598.613i	0.0246786	−	0.0427446i
$582$		0			0
$583$	2125.73	+	3681.87i	0.151010	+	0.261556i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	2919.58	+	5056.86i	0.205288	+	0.355569i	0.950224	−	0.311566i	$-0.100854\pi$
$587$	2919.58	+	5056.86i	0.205288	+	0.355569i	−0.744937	+	0.667135i	$0.767520\pi$
$588$		0			0
$589$	−15671.0	+	27142.9i	−1.09628	+	1.89882i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−1231.78			−0.0853007			−0.0426503	−	0.999090i	$-0.513580\pi$
$593$	−1231.78			−0.0853007			−0.0426503	+	0.999090i	$0.513580\pi$
$594$		0			0
$595$	−152.772			−0.0105261
$596$		0			0
$597$		0			0
$598$		0			0
$599$	9094.95	−	15752.9i	0.620384	−	1.07454i	−0.369031	−	0.929417i	$-0.620310\pi$
$599$	9094.95	−	15752.9i	0.620384	−	1.07454i	0.989414	−	0.145119i	$-0.0463564\pi$
$600$		0			0
$601$	8237.56	+	14267.9i	0.559097	+	0.968384i	0.997572	+	0.0696405i	$0.0221852\pi$
$601$	8237.56	+	14267.9i	0.559097	+	0.968384i	−0.438476	+	0.898743i	$0.644481\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	2972.24	+	5148.08i	0.199734	+	0.345949i
$606$		0			0
$607$	−2211.84	+	3831.01i	−0.147901	+	0.256171i	−0.930451	−	0.366415i	$-0.880585\pi$
$607$	−2211.84	+	3831.01i	−0.147901	+	0.256171i	0.782551	+	0.622587i	$0.213918\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	24593.0			1.62836
$612$		0			0
$613$	−766.478			−0.0505021			−0.0252510	−	0.999681i	$-0.508039\pi$
$613$	−766.478			−0.0505021			−0.0252510	+	0.999681i	$0.508039\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−14934.2	+	25866.9i	−0.974441	+	1.68778i	−0.292674	+	0.956212i	$0.594545\pi$
$617$	−14934.2	+	25866.9i	−0.974441	+	1.68778i	−0.681767	+	0.731569i	$0.738788\pi$
$618$		0			0
$619$	−12572.8	−	21776.8i	−0.816389	−	1.41403i	−0.908326	−	0.418263i	$-0.862639\pi$
$619$	−12572.8	−	21776.8i	−0.816389	−	1.41403i	0.0919366	−	0.995765i	$-0.470694\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	18.7605	+	32.4941i	0.00120646	+	0.00208965i
$624$		0			0
$625$	−312.500	+	541.266i	−0.0200000	+	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−16657.4			−1.05592
$630$		0			0
$631$	16777.2			1.05846			0.529232	−	0.848477i	$-0.322480\pi$
$631$	16777.2			1.05846			0.529232	+	0.848477i	$0.322480\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−2854.53	+	4944.18i	−0.178391	+	0.308983i
$636$		0			0
$637$	9211.89	+	15955.5i	0.572980	+	0.992431i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−11383.1	−	19716.2i	−0.701416	−	1.21489i	−0.967970	−	0.251067i	$-0.919218\pi$
$641$	−11383.1	−	19716.2i	−0.701416	−	1.21489i	0.266554	−	0.963820i	$-0.414115\pi$
$642$		0			0
$643$	13749.7	−	23815.2i	0.843290	−	1.46062i	−0.0438088	−	0.999040i	$-0.513949\pi$
$643$	13749.7	−	23815.2i	0.843290	−	1.46062i	0.887098	−	0.461580i	$-0.152717\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	10513.2			0.638821			0.319411	−	0.947616i	$-0.396515\pi$
$647$	10513.2			0.638821			0.319411	+	0.947616i	$0.396515\pi$
$648$		0			0
$649$	6449.88			0.390108
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−3812.30	+	6603.10i	−0.228464	+	0.395711i	−0.957353	−	0.288921i	$-0.906704\pi$
$653$	−3812.30	+	6603.10i	−0.228464	+	0.395711i	0.728889	+	0.684632i	$0.240037\pi$
$654$		0			0
$655$	2035.01	+	3524.75i	0.121396	+	0.210265i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−14744.1	−	25537.6i	−0.871547	−	1.50956i	−0.860397	−	0.509625i	$-0.829784\pi$
$659$	−14744.1	−	25537.6i	−0.871547	−	1.50956i	−0.0111499	−	0.999938i	$-0.503549\pi$
$660$		0			0
$661$	−8091.90	+	14015.6i	−0.476155	+	0.824725i	−0.999627	−	0.0273183i	$-0.991303\pi$
$661$	−8091.90	+	14015.6i	−0.476155	+	0.824725i	0.523472	+	0.852043i	$0.324637\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−421.571			−0.0245832
$666$		0			0
$667$	−11043.0			−0.641061
$668$		0			0
$669$		0			0
$670$		0			0
$671$	2445.63	−	4235.96i	0.140704	−	0.243707i
$672$		0			0
$673$	13086.5	+	22666.5i	0.749551	+	1.29826i	0.948038	+	0.318157i	$0.103064\pi$
$673$	13086.5	+	22666.5i	0.749551	+	1.29826i	−0.198487	+	0.980104i	$0.563603\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−7932.54	−	13739.6i	−0.450329	−	0.779992i	0.548078	−	0.836427i	$-0.315360\pi$
$677$	−7932.54	−	13739.6i	−0.450329	−	0.779992i	−0.998406	+	0.0564355i	$0.982026\pi$
$678$		0			0
$679$	−318.812	+	552.198i	−0.0180189	+	0.0312097i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	18101.7			1.01412			0.507059	−	0.861911i	$-0.330732\pi$
$683$	18101.7			1.01412			0.507059	+	0.861911i	$0.330732\pi$
$684$		0			0
$685$	−14093.3			−0.786097
$686$		0			0
$687$		0			0
$688$		0			0
$689$	9587.92	−	16606.8i	0.530146	−	0.918240i
$690$		0			0
$691$	−15370.5	−	26622.5i	−0.846198	−	1.46566i	−0.884577	−	0.466394i	$-0.845553\pi$
$691$	−15370.5	−	26622.5i	−0.846198	−	1.46566i	0.0383795	−	0.999263i	$-0.487780\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	2404.12	+	4164.05i	0.131213	+	0.227268i
$696$		0			0
$697$	−9810.79	+	16992.8i	−0.533157	+	0.923454i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−36996.3			−1.99334			−0.996671	−	0.0815329i	$-0.974018\pi$
$701$	−36996.3			−1.99334			−0.996671	+	0.0815329i	$0.974018\pi$
$702$		0			0
$703$	−45965.9			−2.46606
$704$		0			0
$705$		0			0
$706$		0			0
$707$	14.3818	−	24.9100i	0.000765040	−	0.00132509i
$708$		0			0
$709$	13167.8	+	22807.2i	0.697497	+	1.20810i	0.969332	+	0.245756i	$0.0790363\pi$
$709$	13167.8	+	22807.2i	0.697497	+	1.20810i	−0.271835	+	0.962344i	$0.587630\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	5965.56	+	10332.7i	0.313341	+	0.542722i
$714$		0			0
$715$	−1602.36	+	2775.36i	−0.0838108	+	0.145165i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	12816.5			0.664775			0.332388	−	0.943143i	$-0.392146\pi$
$719$	12816.5			0.664775			0.332388	+	0.943143i	$0.392146\pi$
$720$		0			0
$721$	−632.494			−0.0326703
$722$		0			0
$723$		0			0
$724$		0			0
$725$	2515.48	−	4356.95i	0.128859	−	0.223190i
$726$		0			0
$727$	−10167.1	−	17610.0i	−0.518677	−	0.898376i	−0.999764	−	0.0217028i	$-0.993091\pi$
$727$	−10167.1	−	17610.0i	−0.518677	−	0.898376i	0.481087	−	0.876673i	$-0.340242\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−997.309	−	1727.39i	−0.0504608	−	0.0874006i
$732$		0			0
$733$	−18577.1	+	32176.5i	−0.936101	+	1.62137i	−0.163441	+	0.986553i	$0.552259\pi$
$733$	−18577.1	+	32176.5i	−0.936101	+	1.62137i	−0.772659	+	0.634821i	$0.781074\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	4318.33			0.215831
$738$		0			0
$739$	−27616.3			−1.37467			−0.687335	−	0.726340i	$-0.741220\pi$
$739$	−27616.3			−1.37467			−0.687335	+	0.726340i	$0.741220\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−12428.7	+	21527.1i	−0.613678	+	1.06292i	0.376936	+	0.926239i	$0.376978\pi$
$743$	−12428.7	+	21527.1i	−0.613678	+	1.06292i	−0.990615	+	0.136683i	$0.956356\pi$
$744$		0			0
$745$	4214.81	+	7300.26i	0.207273	+	0.359008i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	432.106	+	748.430i	0.0210799	+	0.0365114i
$750$		0			0
$751$	−13339.0	+	23103.8i	−0.648132	+	1.12260i	0.335436	+	0.942063i	$0.391116\pi$
$751$	−13339.0	+	23103.8i	−0.648132	+	1.12260i	−0.983569	+	0.180535i	$0.942217\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	4288.80			0.206736
$756$		0			0
$757$	−18404.4			−0.883643			−0.441821	−	0.897103i	$-0.645667\pi$
$757$	−18404.4			−0.883643			−0.441821	+	0.897103i	$0.645667\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−321.223	+	556.375i	−0.0153014	+	0.0265027i	−0.873575	−	0.486690i	$-0.838204\pi$
$761$	−321.223	+	556.375i	−0.0153014	+	0.0265027i	0.858273	+	0.513193i	$0.171537\pi$
$762$		0			0
$763$	−315.767	−	546.925i	−0.0149824	−	0.0259502i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−14545.8	−	25194.1i	−0.684771	−	1.18606i
$768$		0			0
$769$	−13116.1	+	22717.7i	−0.615057	+	1.06531i	0.375318	+	0.926896i	$0.377534\pi$
$769$	−13116.1	+	22717.7i	−0.615057	+	1.06531i	−0.990375	+	0.138413i	$0.955800\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	42072.2			1.95761			0.978804	−	0.204799i	$-0.0656540\pi$
$773$	42072.2			1.95761			0.978804	+	0.204799i	$0.0656540\pi$
$774$		0			0
$775$	−5435.57			−0.251937
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−27072.7	+	46891.3i	−1.24516	+	2.15668i
$780$		0			0
$781$	−1045.14	−	1810.24i	−0.0478850	−	0.0829392i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	6322.50	+	10950.9i	0.287465	+	0.497903i
$786$		0			0
$787$	−10062.2	+	17428.2i	−0.455754	+	0.789389i	−0.998731	−	0.0503585i	$-0.983964\pi$
$787$	−10062.2	+	17428.2i	−0.455754	+	0.789389i	0.542977	+	0.839747i	$0.317297\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−530.294			−0.0238370
$792$		0			0
$793$	−22061.6			−0.987935
$794$		0			0
$795$		0			0
$796$		0			0
$797$	19927.7	−	34515.8i	0.885665	−	1.53402i	0.0407164	−	0.999171i	$-0.487036\pi$
$797$	19927.7	−	34515.8i	0.885665	−	1.53402i	0.844949	−	0.534847i	$-0.179631\pi$
$798$		0			0
$799$	11947.0	+	20692.7i	0.528978	+	0.916216i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	631.182	+	1093.24i	0.0277384	+	0.0480443i
$804$		0			0
$805$	−80.2410	+	138.981i	−0.00351320	+	0.00608504i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	10611.6			0.461168			0.230584	−	0.973052i	$-0.425936\pi$
$809$	10611.6			0.461168			0.230584	+	0.973052i	$0.425936\pi$
$810$		0			0
$811$	23795.8			1.03031			0.515156	−	0.857097i	$-0.327734\pi$
$811$	23795.8			1.03031			0.515156	+	0.857097i	$0.327734\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−4041.20	+	6999.57i	−0.173690	+	0.300840i
$816$		0			0
$817$	−2752.06	−	4766.70i	−0.117849	−	0.204120i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−16255.9	−	28156.1i	−0.691030	−	1.19690i	−0.971501	−	0.237037i	$-0.923824\pi$
$821$	−16255.9	−	28156.1i	−0.691030	−	1.19690i	0.280471	−	0.959863i	$-0.409509\pi$
$822$		0			0
$823$	10544.7	−	18263.9i	0.446616	−	0.773561i	−0.551548	−	0.834143i	$-0.685962\pi$
$823$	10544.7	−	18263.9i	0.446616	−	0.773561i	0.998163	+	0.0605825i	$0.0192958\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−15961.1			−0.671125			−0.335562	−	0.942018i	$-0.608926\pi$
$827$	−15961.1			−0.671125			−0.335562	+	0.942018i	$0.608926\pi$
$828$		0			0
$829$	19872.9			0.832588			0.416294	−	0.909230i	$-0.363329\pi$
$829$	19872.9			0.832588			0.416294	+	0.909230i	$0.363329\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−8950.02	+	15501.9i	−0.372269	+	0.644789i
$834$		0			0
$835$	−610.435	−	1057.31i	−0.0252994	−	0.0438198i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	11064.4	+	19164.0i	0.455285	+	0.788577i	0.998705	−	0.0508842i	$-0.0162039\pi$
$839$	11064.4	+	19164.0i	0.455285	+	0.788577i	−0.543419	+	0.839461i	$0.682871\pi$
$840$		0			0
$841$	−8054.02	+	13950.0i	−0.330232	+	0.571978i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	3469.59			0.141252
$846$		0			0
$847$	−695.383			−0.0282097
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−8749.06	+	15153.8i	−0.352425	+	0.610418i
$852$		0			0
$853$	16176.8	+	28019.0i	0.649334	+	1.12468i	0.983282	+	0.182088i	$0.0582855\pi$
$853$	16176.8	+	28019.0i	0.649334	+	1.12468i	−0.333949	+	0.942591i	$0.608381\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−8705.43	−	15078.2i	−0.346991	−	0.601007i	0.638722	−	0.769438i	$-0.279463\pi$
$857$	−8705.43	−	15078.2i	−0.346991	−	0.601007i	−0.985713	+	0.168431i	$0.946130\pi$
$858$		0			0
$859$	13602.4	−	23560.0i	0.540287	−	0.935805i	−0.458600	−	0.888643i	$-0.651649\pi$
$859$	13602.4	−	23560.0i	0.540287	−	0.935805i	0.998887	−	0.0471621i	$-0.0150177\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−14403.0			−0.568116			−0.284058	−	0.958807i	$-0.591681\pi$
$863$	−14403.0			−0.568116			−0.284058	+	0.958807i	$0.591681\pi$
$864$		0			0
$865$	6837.84			0.268779
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−2130.63	+	3690.37i	−0.0831724	+	0.144059i
$870$		0			0
$871$	−9738.73	−	16868.0i	−0.378857	−	0.656199i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−36.5561	−	63.3170i	−0.00141237	−	0.00244629i
$876$		0			0
$877$	−2267.74	+	3927.84i	−0.0873160	+	0.151236i	−0.906376	−	0.422472i	$-0.861162\pi$
$877$	−2267.74	+	3927.84i	−0.0873160	+	0.151236i	0.819060	+	0.573708i	$0.194496\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−28809.3			−1.10172			−0.550858	−	0.834599i	$-0.685699\pi$
$881$	−28809.3			−1.10172			−0.550858	+	0.834599i	$0.685699\pi$
$882$		0			0
$883$	−4304.55			−0.164054			−0.0820269	−	0.996630i	$-0.526139\pi$
$883$	−4304.55			−0.164054			−0.0820269	+	0.996630i	$0.526139\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	20085.8	−	34789.6i	0.760331	−	1.31693i	−0.182348	−	0.983234i	$-0.558370\pi$
$887$	20085.8	−	34789.6i	0.760331	−	1.31693i	0.942680	−	0.333699i	$-0.108297\pi$
$888$		0			0
$889$	−333.921	−	578.368i	−0.0125977	−	0.0218199i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	32967.4	+	57101.3i	1.23540	+	2.13978i
$894$		0			0
$895$	5921.00	−	10255.5i	0.221136	−	0.383020i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	43753.9			1.62322
$900$		0			0
$901$	18630.7			0.688878
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−9377.10	+	16241.6i	−0.344426	+	0.596563i
$906$		0			0
$907$	−10367.8	−	17957.5i	−0.379554	−	0.657407i	0.611443	−	0.791288i	$-0.290589\pi$
$907$	−10367.8	−	17957.5i	−0.379554	−	0.657407i	−0.990997	+	0.133881i	$0.957256\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−14735.6	−	25522.9i	−0.535909	−	0.928222i	−0.999119	−	0.0419735i	$-0.986636\pi$
$911$	−14735.6	−	25522.9i	−0.535909	−	0.928222i	0.463209	−	0.886249i	$-0.346698\pi$
$912$		0			0
$913$	7043.79	−	12200.2i	0.255329	−	0.442243i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−476.110			−0.0171456
$918$		0			0
$919$	−13629.2			−0.489211			−0.244605	−	0.969623i	$-0.578658\pi$
$919$	−13629.2			−0.489211			−0.244605	+	0.969623i	$0.578658\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−4714.03	+	8164.94i	−0.168109	+	0.291173i
$924$		0			0
$925$	−3985.88	−	6903.75i	−0.141681	−	0.245399i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−6877.78	−	11912.7i	−0.242898	−	0.420712i	0.718640	−	0.695382i	$-0.244765\pi$
$929$	−6877.78	−	11912.7i	−0.242898	−	0.420712i	−0.961539	+	0.274670i	$0.911431\pi$
$930$		0			0
$931$	−24697.4	+	42777.2i	−0.869415	+	1.50587i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−3113.61			−0.108905
$936$		0			0
$937$	21198.5			0.739086			0.369543	−	0.929214i	$-0.379514\pi$
$937$	21198.5			0.739086			0.369543	+	0.929214i	$0.379514\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	9919.49	−	17181.1i	0.343641	−	0.595203i	−0.641465	−	0.767152i	$-0.721673\pi$
$941$	9919.49	−	17181.1i	0.343641	−	0.595203i	0.985106	+	0.171949i	$0.0550064\pi$
$942$		0			0
$943$	10305.9	+	17850.4i	0.355893	+	0.616425i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−21682.1	−	37554.5i	−0.744006	−	1.28866i	−0.950658	−	0.310241i	$-0.899590\pi$
$947$	−21682.1	−	37554.5i	−0.744006	−	1.28866i	0.206652	−	0.978414i	$-0.433743\pi$
$948$		0			0
$949$	2846.89	−	4930.96i	0.0973805	−	0.168668i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	29215.8			0.993065			0.496533	−	0.868018i	$-0.334606\pi$
$953$	29215.8			0.993065			0.496533	+	0.868018i	$0.334606\pi$
$954$		0			0
$955$	755.236			0.0255904
$956$		0			0
$957$		0			0
$958$		0			0
$959$	824.313	−	1427.75i	0.0277565	−	0.0480756i
$960$		0			0
$961$	−8740.82	−	15139.5i	−0.293405	−	0.508192i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	2875.32	+	4980.19i	0.0959168	+	0.166133i
$966$		0			0
$967$	−17472.5	+	30263.3i	−0.581053	+	1.00641i	0.414302	+	0.910140i	$0.364026\pi$
$967$	−17472.5	+	30263.3i	−0.581053	+	1.00641i	−0.995355	+	0.0962738i	$0.969308\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	41345.4			1.36646			0.683232	−	0.730201i	$-0.260574\pi$
$971$	41345.4			1.36646			0.683232	+	0.730201i	$0.260574\pi$
$972$		0			0
$973$	−562.465			−0.0185322
$974$		0			0
$975$		0			0
$976$		0			0
$977$	3718.30	−	6440.29i	0.121760	−	0.210894i	−0.798702	−	0.601727i	$-0.794480\pi$
$977$	3718.30	−	6440.29i	0.121760	−	0.210894i	0.920462	+	0.390833i	$0.127813\pi$
$978$		0			0
$979$	382.354	+	662.256i	0.0124822	+	0.0216198i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	18609.6	+	32232.7i	0.603818	+	1.04584i	0.992237	+	0.124361i	$0.0396881\pi$
$983$	18609.6	+	32232.7i	0.603818	+	1.04584i	−0.388419	+	0.921483i	$0.626979\pi$
$984$		0			0
$985$	−3793.37	+	6570.31i	−0.122707	+	0.212535i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−2095.28			−0.0673672
$990$		0			0
$991$	−28149.2			−0.902309			−0.451154	−	0.892446i	$-0.648988\pi$
$991$	−28149.2			−0.902309			−0.451154	+	0.892446i	$0.648988\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−7557.45	+	13089.9i	−0.240791	+	0.417063i
$996$		0			0
$997$	25265.1	+	43760.5i	0.802563	+	1.39008i	0.917924	+	0.396756i	$0.129864\pi$
$997$	25265.1	+	43760.5i	0.802563	+	1.39008i	−0.115362	+	0.993324i	$0.536803\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.4.i.w.1081.4		12
3.2	odd	2		1620.4.i.x.1081.4		12
9.2	odd	6		1620.4.i.x.541.4		12
9.4	even	3		1620.4.a.j.1.3	yes	6
9.5	odd	6		1620.4.a.i.1.3	✓	6
9.7	even	3	inner	1620.4.i.w.541.4		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1620.4.a.i.1.3	✓	6	9.5	odd	6
1620.4.a.j.1.3	yes	6	9.4	even	3
1620.4.i.w.541.4		12	9.7	even	3	inner
1620.4.i.w.1081.4		12	1.1	even	1	trivial
1620.4.i.x.541.4		12	9.2	odd	6
1620.4.i.x.1081.4		12	3.2	odd	2

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)

\(f(q)\)	\(=\)	\(q+(-2.50000 + 4.33013i) q^{5} +(-0.292449 - 0.506536i) q^{7} +O(q^{10})\)	\(q+(-2.50000 + 4.33013i) q^{5} +(-0.292449 - 0.506536i) q^{7} +(-5.96034 - 10.3236i) q^{11} +(-26.8836 + 46.5638i) q^{13} -52.2388 q^{17} -144.152 q^{19} +(-27.4376 + 47.5234i) q^{23} +(-12.5000 - 21.6506i) q^{25} +(100.619 + 174.278i) q^{29} +(108.711 - 188.294i) q^{31} +2.92449 q^{35} +318.871 q^{37} +(187.806 - 325.290i) q^{41} +(19.0913 + 33.0672i) q^{43} +(-228.699 - 396.118i) q^{47} +(171.329 - 296.750i) q^{49} -356.645 q^{53} +59.6034 q^{55} +(-270.533 + 468.577i) q^{59} +(205.159 + 355.345i) q^{61} +(-134.418 - 232.819i) q^{65} +(-181.127 + 313.722i) q^{67} +175.350 q^{71} -105.897 q^{73} +(-3.48619 + 6.03826i) q^{77} +(-178.734 - 309.577i) q^{79} +(590.888 + 1023.45i) q^{83} +(130.597 - 226.201i) q^{85} -64.1496 q^{89} +31.4483 q^{91} +(360.380 - 624.197i) q^{95} +(-545.072 - 944.093i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 30 q^{5} - 12 q^{7}+O(q^{10}) \) 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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